LMFDB - Embedded newform 75.16.a.l.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [75,16,Mod(1,75)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(75, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 16, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("75.1");

S:= CuspForms(chi, 16);

N := Newforms(S);

Level:	$ N $	$=$	$ 75 = 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 16 $
Character orbit:	$[\chi]$	$=$	75.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$107.020128825$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 206884 x^{6} + 5065964 x^{5} + 12902022496 x^{4} - 638065050800 x^{3} + \cdots + 96\!\cdots\!00 $ x^8 - x^7 - 206884x^6 + 5065964x^5 + 12902022496x^4 - 638065050800x^3 - 240460263995200x^2 + 11942226547624000x + 962566763029600000
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{14}\cdot 3^{8}\cdot 5^{14} $
Twist minimal:	no (minimal twist has level 15)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-303.266$ of defining polynomial
Character	$\chi$	$=$	75.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-269.266 q^{2} +2187.00 q^{3} +39736.0 q^{4} -588884. q^{6} -2.21965e6 q^{7} -1.87623e6 q^{8} +4.78297e6 q^{9} +O(q^{10})$	\(q-269.266 q^{2} +2187.00 q^{3} +39736.0 q^{4} -588884. q^{6} -2.21965e6 q^{7} -1.87623e6 q^{8} +4.78297e6 q^{9} -1.04447e8 q^{11} +8.69025e7 q^{12} -1.47969e8 q^{13} +5.97675e8 q^{14} -7.96863e8 q^{16} -1.95491e9 q^{17} -1.28789e9 q^{18} -6.19268e9 q^{19} -4.85437e9 q^{21} +2.81240e10 q^{22} -1.23902e10 q^{23} -4.10332e9 q^{24} +3.98429e10 q^{26} +1.04604e10 q^{27} -8.81999e10 q^{28} +1.52048e10 q^{29} -1.36970e11 q^{31} +2.76048e11 q^{32} -2.28425e11 q^{33} +5.26391e11 q^{34} +1.90056e11 q^{36} +7.49789e11 q^{37} +1.66748e12 q^{38} -3.23608e11 q^{39} +2.69443e11 q^{41} +1.30712e12 q^{42} +5.96236e11 q^{43} -4.15030e12 q^{44} +3.33626e12 q^{46} -4.90182e12 q^{47} -1.74274e12 q^{48} +1.79280e11 q^{49} -4.27539e12 q^{51} -5.87968e12 q^{52} +2.94497e12 q^{53} -2.81661e12 q^{54} +4.16458e12 q^{56} -1.35434e13 q^{57} -4.09412e12 q^{58} -9.17344e12 q^{59} -3.22985e13 q^{61} +3.68812e13 q^{62} -1.06165e13 q^{63} -4.82187e13 q^{64} +6.15071e13 q^{66} -3.17005e13 q^{67} -7.76803e13 q^{68} -2.70974e13 q^{69} -1.02794e14 q^{71} -8.97396e12 q^{72} +8.48889e13 q^{73} -2.01892e14 q^{74} -2.46072e14 q^{76} +2.31835e14 q^{77} +8.71364e13 q^{78} -8.38039e13 q^{79} +2.28768e13 q^{81} -7.25518e13 q^{82} +3.37543e14 q^{83} -1.92893e14 q^{84} -1.60546e14 q^{86} +3.32528e13 q^{87} +1.95967e14 q^{88} -7.98815e14 q^{89} +3.28439e14 q^{91} -4.92338e14 q^{92} -2.99553e14 q^{93} +1.31989e15 q^{94} +6.03717e14 q^{96} +6.39237e14 q^{97} -4.82738e13 q^{98} -4.99566e14 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 273 q^{2} + 17496 q^{3} + 160941 q^{4} + 597051 q^{6} + 1149126 q^{7} + 10053771 q^{8} + 38263752 q^{9}+O(q^{10}) $ 8 * q + 273 * q^2 + 17496 * q^3 + 160941 * q^4 + 597051 * q^6 + 1149126 * q^7 + 10053771 * q^8 + 38263752 * q^9	\( 8 q + 273 q^{2} + 17496 q^{3} + 160941 q^{4} + 597051 q^{6} + 1149126 q^{7} + 10053771 q^{8} + 38263752 q^{9} + 62424222 q^{11} + 351977967 q^{12} + 44492838 q^{13} + 306251034 q^{14} + 1871684545 q^{16} + 3593639622 q^{17} + 1305750537 q^{18} - 496736656 q^{19} + 2513138562 q^{21} + 26169930714 q^{22} + 2358873312 q^{23} + 21987597177 q^{24} + 18223186326 q^{26} + 83682825624 q^{27} - 10716504462 q^{28} + 36630923358 q^{29} + 166724197108 q^{31} + 1086177843771 q^{32} + 136521773514 q^{33} + 141055448602 q^{34} + 769775813829 q^{36} + 2497964854986 q^{37} + 785802288552 q^{38} + 97305836706 q^{39} + 4754409554748 q^{41} + 669771011358 q^{42} + 1310015295120 q^{43} + 7792818609978 q^{44} + 14727772650308 q^{46} - 3007996138620 q^{47} + 4093374099915 q^{48} + 8653589541100 q^{49} + 7859289853314 q^{51} - 22117814273898 q^{52} + 6974407032522 q^{53} + 2855676424419 q^{54} + 26685903266430 q^{56} - 1086363066672 q^{57} - 35393776551696 q^{58} + 1259020938606 q^{59} + 25651568022640 q^{61} - 43807470421128 q^{62} + 5496234035094 q^{63} + 134602047432169 q^{64} + 57233638471518 q^{66} - 27008842727148 q^{67} - 28552468844166 q^{68} + 5158855933344 q^{69} - 174179827562004 q^{71} + 48086875026099 q^{72} + 107017091040132 q^{73} - 150367200840846 q^{74} - 407245761522456 q^{76} + 192490324871652 q^{77} + 39854108494962 q^{78} + 173323732806380 q^{79} + 183014339639688 q^{81} + 969967421566242 q^{82} + 868845027534576 q^{83} - 23436995258394 q^{84} - 15\!\cdots\!48 q^{86}+ \cdots + 298573118675118 q^{99}+O(q^{100}) \) 8 * q + 273 * q^2 + 17496 * q^3 + 160941 * q^4 + 597051 * q^6 + 1149126 * q^7 + 10053771 * q^8 + 38263752 * q^9 + 62424222 * q^11 + 351977967 * q^12 + 44492838 * q^13 + 306251034 * q^14 + 1871684545 * q^16 + 3593639622 * q^17 + 1305750537 * q^18 - 496736656 * q^19 + 2513138562 * q^21 + 26169930714 * q^22 + 2358873312 * q^23 + 21987597177 * q^24 + 18223186326 * q^26 + 83682825624 * q^27 - 10716504462 * q^28 + 36630923358 * q^29 + 166724197108 * q^31 + 1086177843771 * q^32 + 136521773514 * q^33 + 141055448602 * q^34 + 769775813829 * q^36 + 2497964854986 * q^37 + 785802288552 * q^38 + 97305836706 * q^39 + 4754409554748 * q^41 + 669771011358 * q^42 + 1310015295120 * q^43 + 7792818609978 * q^44 + 14727772650308 * q^46 - 3007996138620 * q^47 + 4093374099915 * q^48 + 8653589541100 * q^49 + 7859289853314 * q^51 - 22117814273898 * q^52 + 6974407032522 * q^53 + 2855676424419 * q^54 + 26685903266430 * q^56 - 1086363066672 * q^57 - 35393776551696 * q^58 + 1259020938606 * q^59 + 25651568022640 * q^61 - 43807470421128 * q^62 + 5496234035094 * q^63 + 134602047432169 * q^64 + 57233638471518 * q^66 - 27008842727148 * q^67 - 28552468844166 * q^68 + 5158855933344 * q^69 - 174179827562004 * q^71 + 48086875026099 * q^72 + 107017091040132 * q^73 - 150367200840846 * q^74 - 407245761522456 * q^76 + 192490324871652 * q^77 + 39854108494962 * q^78 + 173323732806380 * q^79 + 183014339639688 * q^81 + 969967421566242 * q^82 + 868845027534576 * q^83 - 23436995258394 * q^84 - 1530511469933748 * q^86 + 80111829383946 * q^87 + 2334276558470406 * q^88 - 436746702974256 * q^89 + 185873985841788 * q^91 + 1698320226970356 * q^92 + 364625819075196 * q^93 + 1113775369472416 * q^94 + 2375470944327177 * q^96 - 1064926798083408 * q^97 + 3042468678281757 * q^98 + 298573118675118 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−269.266	−1.48750	−0.743748	−	0.668460i	$-0.766954\pi$
$2$	−269.266	−1.48750	−0.743748	+	0.668460i	$0.766954\pi$
$3$	2187.00	0.577350
$3$	2187.00	0.577350
$4$	39736.0	1.21265
$5$	0	0
$5$	0	0
$6$	−588884.	−0.858806
$7$	−2.21965e6	−1.01871	−0.509353	−	0.860558i	$-0.670115\pi$
$7$	−2.21965e6	−1.01871	−0.509353	+	0.860558i	$0.670115\pi$
$8$	−1.87623e6	−0.316309
$9$	4.78297e6	0.333333
$10$	0	0
$11$	−1.04447e8	−1.61603	−0.808017	−	0.589159i	$-0.799459\pi$
$11$	−1.04447e8	−1.61603	−0.808017	+	0.589159i	$0.799459\pi$
$12$	8.69025e7	0.700121
$13$	−1.47969e8	−0.654026	−0.327013	−	0.945020i	$-0.606042\pi$
$13$	−1.47969e8	−0.654026	−0.327013	+	0.945020i	$0.606042\pi$
$14$	5.97675e8	1.51532
$15$	0	0
$16$	−7.96863e8	−0.742137
$17$	−1.95491e9	−1.15547	−0.577737	−	0.816223i	$-0.696064\pi$
$17$	−1.95491e9	−1.15547	−0.577737	+	0.816223i	$0.696064\pi$
$18$	−1.28789e9	−0.495832
$19$	−6.19268e9	−1.58938	−0.794689	−	0.607017i	$-0.792366\pi$
$19$	−6.19268e9	−1.58938	−0.794689	+	0.607017i	$0.792366\pi$
$20$	0	0
$21$	−4.85437e9	−0.588150
$22$	2.81240e10	2.40384
$23$	−1.23902e10	−0.758788	−0.379394	−	0.925235i	$-0.623868\pi$
$23$	−1.23902e10	−0.758788	−0.379394	+	0.925235i	$0.623868\pi$
$24$	−4.10332e9	−0.182621
$25$	0	0
$26$	3.98429e10	0.972861
$27$	1.04604e10	0.192450
$28$	−8.81999e10	−1.23533
$29$	1.52048e10	0.163680	0.0818399	−	0.996645i	$-0.473920\pi$
$29$	1.52048e10	0.163680	0.0818399	+	0.996645i	$0.473920\pi$
$30$	0	0
$31$	−1.36970e11	−0.894153	−0.447076	−	0.894496i	$-0.647535\pi$
$31$	−1.36970e11	−0.894153	−0.447076	+	0.894496i	$0.647535\pi$
$32$	2.76048e11	1.42023
$33$	−2.28425e11	−0.933017
$34$	5.26391e11	1.71876
$35$	0	0
$36$	1.90056e11	0.404215
$37$	7.49789e11	1.29845	0.649226	−	0.760595i	$-0.275093\pi$
$37$	7.49789e11	1.29845	0.649226	+	0.760595i	$0.275093\pi$
$38$	1.66748e12	2.36419
$39$	−3.23608e11	−0.377602
$40$	0	0
$41$	2.69443e11	0.216067	0.108034	−	0.994147i	$-0.465545\pi$
$41$	2.69443e11	0.216067	0.108034	+	0.994147i	$0.465545\pi$
$42$	1.30712e12	0.874871
$43$	5.96236e11	0.334507	0.167253	−	0.985914i	$-0.446510\pi$
$43$	5.96236e11	0.334507	0.167253	+	0.985914i	$0.446510\pi$
$44$	−4.15030e12	−1.95968
$45$	0	0
$46$	3.33626e12	1.12869
$47$	−4.90182e12	−1.41131	−0.705657	−	0.708554i	$-0.749348\pi$
$47$	−4.90182e12	−1.41131	−0.705657	+	0.708554i	$0.749348\pi$
$48$	−1.74274e12	−0.428473
$49$	1.79280e11	0.0377625
$50$	0	0
$51$	−4.27539e12	−0.667114
$52$	−5.87968e12	−0.793101
$53$	2.94497e12	0.344359	0.172180	−	0.985066i	$-0.444919\pi$
$53$	2.94497e12	0.344359	0.172180	+	0.985066i	$0.444919\pi$
$54$	−2.81661e12	−0.286269
$55$	0	0
$56$	4.16458e12	0.322226
$57$	−1.35434e13	−0.917627
$58$	−4.09412e12	−0.243473
$59$	−9.17344e12	−0.479891	−0.239945	−	0.970786i	$-0.577130\pi$
$59$	−9.17344e12	−0.479891	−0.239945	+	0.970786i	$0.577130\pi$
$60$	0	0
$61$	−3.22985e13	−1.31586	−0.657928	−	0.753081i	$-0.728567\pi$
$61$	−3.22985e13	−1.31586	−0.657928	+	0.753081i	$0.728567\pi$
$62$	3.68812e13	1.33005
$63$	−1.06165e13	−0.339569
$64$	−4.82187e13	−1.37046
$65$	0	0
$66$	6.15071e13	1.38786
$67$	−3.17005e13	−0.639005	−0.319503	−	0.947585i	$-0.603516\pi$
$67$	−3.17005e13	−0.639005	−0.319503	+	0.947585i	$0.603516\pi$
$68$	−7.76803e13	−1.40118
$69$	−2.70974e13	−0.438086
$70$	0	0
$71$	−1.02794e14	−1.34131	−0.670656	−	0.741769i	$-0.733987\pi$
$71$	−1.02794e14	−1.34131	−0.670656	+	0.741769i	$0.733987\pi$
$72$	−8.97396e12	−0.105436
$73$	8.48889e13	0.899352	0.449676	−	0.893192i	$-0.351539\pi$
$73$	8.48889e13	0.899352	0.449676	+	0.893192i	$0.351539\pi$
$74$	−2.01892e14	−1.93144
$75$	0	0
$76$	−2.46072e14	−1.92735
$77$	2.31835e14	1.64626
$78$	8.71364e13	0.561681
$79$	−8.38039e13	−0.490977	−0.245488	−	0.969400i	$-0.578948\pi$
$79$	−8.38039e13	−0.490977	−0.245488	+	0.969400i	$0.578948\pi$
$80$	0	0
$81$	2.28768e13	0.111111
$82$	−7.25518e13	−0.321399
$83$	3.37543e14	1.36535	0.682673	−	0.730724i	$-0.260817\pi$
$83$	3.37543e14	1.36535	0.682673	+	0.730724i	$0.260817\pi$
$84$	−1.92893e14	−0.713218
$85$	0	0
$86$	−1.60546e14	−0.497578
$87$	3.32528e13	0.0945005
$88$	1.95967e14	0.511166
$89$	−7.98815e14	−1.91435	−0.957175	−	0.289511i	$-0.906507\pi$
$89$	−7.98815e14	−1.91435	−0.957175	+	0.289511i	$0.906507\pi$
$90$	0	0
$91$	3.28439e14	0.666260
$92$	−4.92338e14	−0.920141
$93$	−2.99553e14	−0.516239
$94$	1.31989e15	2.09932
$95$	0	0
$96$	6.03717e14	0.819973
$97$	6.39237e14	0.803293	0.401646	−	0.915795i	$-0.368438\pi$
$97$	6.39237e14	0.803293	0.401646	+	0.915795i	$0.368438\pi$
$98$	−4.82738e13	−0.0561715
$99$	−4.99566e14	−0.538678
$100$	0	0
$101$	−6.53019e14	−0.606060	−0.303030	−	0.952981i	$-0.597998\pi$
$101$	−6.53019e14	−0.606060	−0.303030	+	0.952981i	$0.597998\pi$
$102$	1.15122e15	0.992329
$103$	6.11221e14	0.489688	0.244844	−	0.969563i	$-0.421263\pi$
$103$	6.11221e14	0.489688	0.244844	+	0.969563i	$0.421263\pi$
$104$	2.77624e14	0.206874
$105$	0	0
$106$	−7.92979e14	−0.512233
$107$	−2.63382e15	−1.58565	−0.792826	−	0.609448i	$-0.791391\pi$
$107$	−2.63382e15	−1.58565	−0.792826	+	0.609448i	$0.791391\pi$
$108$	4.15652e14	0.233374
$109$	2.13900e15	1.12076	0.560381	−	0.828235i	$-0.310655\pi$
$109$	2.13900e15	1.12076	0.560381	+	0.828235i	$0.310655\pi$
$110$	0	0
$111$	1.63979e15	0.749662
$112$	1.76876e15	0.756019
$113$	1.65164e15	0.660432	0.330216	−	0.943905i	$-0.392878\pi$
$113$	1.65164e15	0.660432	0.330216	+	0.943905i	$0.392878\pi$
$114$	3.64677e15	1.36497
$115$	0	0
$116$	6.04176e14	0.198485
$117$	−7.07730e14	−0.218009
$118$	2.47009e15	0.713836
$119$	4.33922e15	1.17709
$120$	0	0
$121$	6.73191e15	1.61156
$122$	8.69687e15	1.95733
$123$	5.89273e14	0.124746
$124$	−5.44262e15	−1.08429
$125$	0	0
$126$	2.85866e15	0.505107
$127$	−2.03635e15	−0.339097	−0.169548	−	0.985522i	$-0.554231\pi$
$127$	−2.03635e15	−0.339097	−0.169548	+	0.985522i	$0.554231\pi$
$128$	3.93808e15	0.618315
$129$	1.30397e15	0.193128
$130$	0	0
$131$	−5.86714e15	−0.774269	−0.387134	−	0.922023i	$-0.626535\pi$
$131$	−5.86714e15	−0.774269	−0.387134	+	0.922023i	$0.626535\pi$
$132$	−9.07670e15	−1.13142
$133$	1.37456e16	1.61911
$134$	8.53584e15	0.950518
$135$	0	0
$136$	3.66787e15	0.365487
$137$	4.12511e14	0.0389073	0.0194536	−	0.999811i	$-0.493807\pi$
$137$	4.12511e14	0.0389073	0.0194536	+	0.999811i	$0.493807\pi$
$138$	7.29640e15	0.651652
$139$	1.85574e16	1.57002	0.785010	−	0.619483i	$-0.212658\pi$
$139$	1.85574e16	1.57002	0.785010	+	0.619483i	$0.212658\pi$
$140$	0	0
$141$	−1.07203e16	−0.814822
$142$	2.76788e16	1.99520
$143$	1.54549e16	1.05693
$144$	−3.81137e15	−0.247379
$145$	0	0
$146$	−2.28577e16	−1.33778
$147$	3.92085e14	0.0218022
$148$	2.97936e16	1.57456
$149$	−2.98320e16	−1.49894	−0.749471	−	0.662037i	$-0.769692\pi$
$149$	−2.98320e16	−1.49894	−0.749471	+	0.662037i	$0.769692\pi$
$150$	0	0
$151$	2.68869e16	1.22240	0.611201	−	0.791475i	$-0.290687\pi$
$151$	2.68869e16	1.22240	0.611201	+	0.791475i	$0.290687\pi$
$152$	1.16189e16	0.502735
$153$	−9.35029e15	−0.385158
$154$	−6.24253e16	−2.44881
$155$	0	0
$156$	−1.28589e16	−0.457897
$157$	3.14336e16	1.06696	0.533478	−	0.845814i	$-0.320885\pi$
$157$	3.14336e16	1.06696	0.533478	+	0.845814i	$0.320885\pi$
$158$	2.25655e16	0.730326
$159$	6.44064e15	0.198816
$160$	0	0
$161$	2.75020e16	0.772982
$162$	−6.15993e15	−0.165277
$163$	−6.95451e16	−1.78180	−0.890901	−	0.454197i	$-0.849926\pi$
$163$	−6.95451e16	−1.78180	−0.890901	+	0.454197i	$0.849926\pi$
$164$	1.07066e16	0.262013
$165$	0	0
$166$	−9.08886e16	−2.03095
$167$	−6.81966e16	−1.45676	−0.728382	−	0.685171i	$-0.759727\pi$
$167$	−6.81966e16	−1.45676	−0.728382	+	0.685171i	$0.759727\pi$
$168$	9.10793e15	0.186037
$169$	−2.92911e16	−0.572250
$170$	0	0
$171$	−2.96194e16	−0.529792
$172$	2.36920e16	0.405638
$173$	7.66608e16	1.25669	0.628344	−	0.777936i	$-0.283733\pi$
$173$	7.66608e16	1.25669	0.628344	+	0.777936i	$0.283733\pi$
$174$	−8.95384e15	−0.140569
$175$	0	0
$176$	8.32299e16	1.19932
$177$	−2.00623e16	−0.277065
$178$	2.15093e17	2.84759
$179$	−1.44153e15	−0.0182989	−0.00914945	−	0.999958i	$-0.502912\pi$
$179$	−1.44153e15	−0.0182989	−0.00914945	+	0.999958i	$0.502912\pi$
$180$	0	0
$181$	−3.26336e16	−0.381132	−0.190566	−	0.981674i	$-0.561032\pi$
$181$	−3.26336e16	−0.381132	−0.190566	+	0.981674i	$0.561032\pi$
$182$	−8.84372e16	−0.991059
$183$	−7.06368e16	−0.759710
$184$	2.32469e16	0.240012
$185$	0	0
$186$	8.06592e16	0.767904
$187$	2.04185e17	1.86729
$188$	−1.94779e17	−1.71142
$189$	−2.32183e16	−0.196050
$190$	0	0
$191$	1.14854e17	0.896184	0.448092	−	0.893987i	$-0.352104\pi$
$191$	1.14854e17	0.896184	0.448092	+	0.893987i	$0.352104\pi$
$192$	−1.05454e17	−0.791234
$193$	6.44821e16	0.465328	0.232664	−	0.972557i	$-0.425256\pi$
$193$	6.44821e16	0.465328	0.232664	+	0.972557i	$0.425256\pi$
$194$	−1.72124e17	−1.19489
$195$	0	0
$196$	7.12385e15	0.0457925
$197$	3.53652e16	0.218816	0.109408	−	0.993997i	$-0.465104\pi$
$197$	3.53652e16	0.218816	0.109408	+	0.993997i	$0.465104\pi$
$198$	1.34516e17	0.801281
$199$	4.35340e16	0.249707	0.124853	−	0.992175i	$-0.460154\pi$
$199$	4.35340e16	0.249707	0.124853	+	0.992175i	$0.460154\pi$
$200$	0	0
$201$	−6.93289e16	−0.368930
$202$	1.75836e17	0.901512
$203$	−3.37493e16	−0.166742
$204$	−1.69887e17	−0.808972
$205$	0	0
$206$	−1.64581e17	−0.728408
$207$	−5.92621e16	−0.252929
$208$	1.17911e17	0.485376
$209$	6.46807e17	2.56849
$210$	0	0
$211$	2.08668e16	0.0771502	0.0385751	−	0.999256i	$-0.487718\pi$
$211$	2.08668e16	0.0771502	0.0385751	+	0.999256i	$0.487718\pi$
$212$	1.17021e17	0.417585
$213$	−2.24810e17	−0.774406
$214$	7.09197e17	2.35865
$215$	0	0
$216$	−1.96261e16	−0.0608737
$217$	3.04025e17	0.910879
$218$	−5.75960e17	−1.66713
$219$	1.85652e17	0.519241
$220$	0	0
$221$	2.89266e17	0.755710
$222$	−4.41539e17	−1.11512
$223$	−2.31974e17	−0.566438	−0.283219	−	0.959055i	$-0.591402\pi$
$223$	−2.31974e17	−0.566438	−0.283219	+	0.959055i	$0.591402\pi$
$224$	−6.12730e17	−1.44680
$225$	0	0
$226$	−4.44731e17	−0.982390
$227$	−7.42435e17	−1.58659	−0.793294	−	0.608839i	$-0.791636\pi$
$227$	−7.42435e17	−1.58659	−0.793294	+	0.608839i	$0.791636\pi$
$228$	−5.38160e17	−1.11276
$229$	4.36462e17	0.873334	0.436667	−	0.899623i	$-0.356159\pi$
$229$	4.36462e17	0.873334	0.436667	+	0.899623i	$0.356159\pi$
$230$	0	0
$231$	5.07024e17	0.950471
$232$	−2.85277e16	−0.0517734
$233$	−2.56591e16	−0.0450892	−0.0225446	−	0.999746i	$-0.507177\pi$
$233$	−2.56591e16	−0.0450892	−0.0225446	+	0.999746i	$0.507177\pi$
$234$	1.90567e17	0.324287
$235$	0	0
$236$	−3.64516e17	−0.581937
$237$	−1.83279e17	−0.283466
$238$	−1.16840e18	−1.75092
$239$	−3.88875e17	−0.564711	−0.282355	−	0.959310i	$-0.591116\pi$
$239$	−3.88875e17	−0.564711	−0.282355	+	0.959310i	$0.591116\pi$
$240$	0	0
$241$	1.06064e18	1.44691	0.723456	−	0.690371i	$-0.242552\pi$
$241$	1.06064e18	1.44691	0.723456	+	0.690371i	$0.242552\pi$
$242$	−1.81267e18	−2.39720
$243$	5.00315e16	0.0641500
$244$	−1.28341e18	−1.59567
$245$	0	0
$246$	−1.58671e17	−0.185560
$247$	9.16324e17	1.03949
$248$	2.56987e17	0.282829
$249$	7.38206e17	0.788283
$250$	0	0
$251$	−8.02780e17	−0.807317	−0.403658	−	0.914910i	$-0.632262\pi$
$251$	−8.02780e17	−0.807317	−0.403658	+	0.914910i	$0.632262\pi$
$252$	−4.21857e17	−0.411776
$253$	1.29412e18	1.22623
$254$	5.48318e17	0.504405
$255$	0	0
$256$	5.19639e17	0.450715
$257$	1.00876e18	0.849743	0.424872	−	0.905254i	$-0.360319\pi$
$257$	1.00876e18	0.849743	0.424872	+	0.905254i	$0.360319\pi$
$258$	−3.51114e17	−0.287277
$259$	−1.66427e18	−1.32274
$260$	0	0
$261$	7.27239e16	0.0545599
$262$	1.57982e18	1.15172
$263$	−1.98960e16	−0.0140961	−0.00704803	−	0.999975i	$-0.502243\pi$
$263$	−1.98960e16	−0.0140961	−0.00704803	+	0.999975i	$0.502243\pi$
$264$	4.28579e17	0.295122
$265$	0	0
$266$	−3.70121e18	−2.40842
$267$	−1.74701e18	−1.10525
$268$	−1.25965e18	−0.774887
$269$	−1.32630e18	−0.793411	−0.396706	−	0.917946i	$-0.629847\pi$
$269$	−1.32630e18	−0.793411	−0.396706	+	0.917946i	$0.629847\pi$
$270$	0	0
$271$	3.21961e18	1.82194	0.910970	−	0.412473i	$-0.135335\pi$
$271$	3.21961e18	1.82194	0.910970	+	0.412473i	$0.135335\pi$
$272$	1.55780e18	0.857520
$273$	7.18295e17	0.384665
$274$	−1.11075e17	−0.0578744
$275$	0	0
$276$	−1.07674e18	−0.531244
$277$	−8.93490e17	−0.429034	−0.214517	−	0.976720i	$-0.568818\pi$
$277$	−8.93490e17	−0.429034	−0.214517	+	0.976720i	$0.568818\pi$
$278$	−4.99686e18	−2.33540
$279$	−6.55122e17	−0.298051
$280$	0	0
$281$	3.34845e18	1.44393	0.721966	−	0.691928i	$-0.243239\pi$
$281$	3.34845e18	1.44393	0.721966	+	0.691928i	$0.243239\pi$
$282$	2.88660e18	1.21204
$283$	−1.23503e18	−0.504987	−0.252494	−	0.967599i	$-0.581251\pi$
$283$	−1.23503e18	−0.504987	−0.252494	+	0.967599i	$0.581251\pi$
$284$	−4.08461e18	−1.62653
$285$	0	0
$286$	−4.16147e18	−1.57218
$287$	−5.98070e17	−0.220109
$288$	1.32033e18	0.473412
$289$	9.59262e17	0.335122
$290$	0	0
$291$	1.39801e18	0.463781
$292$	3.37314e18	1.09059
$293$	−8.09279e17	−0.255030	−0.127515	−	0.991837i	$-0.540700\pi$
$293$	−8.09279e17	−0.255030	−0.127515	+	0.991837i	$0.540700\pi$
$294$	−1.05575e17	−0.0324307
$295$	0	0
$296$	−1.40678e18	−0.410712
$297$	−1.09255e18	−0.311006
$298$	8.03272e18	2.22967
$299$	1.83337e18	0.496267
$300$	0	0
$301$	−1.32344e18	−0.340764
$302$	−7.23972e18	−1.81832
$303$	−1.42815e18	−0.349909
$304$	4.93472e18	1.17954
$305$	0	0
$306$	2.51771e18	0.572922
$307$	−7.03788e18	−1.56280	−0.781401	−	0.624029i	$-0.785495\pi$
$307$	−7.03788e18	−1.56280	−0.781401	+	0.624029i	$0.785495\pi$
$308$	9.21220e18	1.99633
$309$	1.33674e18	0.282721
$310$	0	0
$311$	−4.67473e18	−0.942006	−0.471003	−	0.882131i	$-0.656108\pi$
$311$	−4.67473e18	−0.942006	−0.471003	+	0.882131i	$0.656108\pi$
$312$	6.07163e17	0.119439
$313$	−5.05896e18	−0.971581	−0.485791	−	0.874075i	$-0.661468\pi$
$313$	−5.05896e18	−0.971581	−0.485791	+	0.874075i	$0.661468\pi$
$314$	−8.46399e18	−1.58709
$315$	0	0
$316$	−3.33003e18	−0.595381
$317$	5.17964e18	0.904388	0.452194	−	0.891920i	$-0.350641\pi$
$317$	5.17964e18	0.904388	0.452194	+	0.891920i	$0.350641\pi$
$318$	−1.73424e18	−0.295738
$319$	−1.58809e18	−0.264512
$320$	0	0
$321$	−5.76017e18	−0.915477
$322$	−7.40533e18	−1.14981
$323$	1.21062e19	1.83649
$324$	9.09031e17	0.134738
$325$	0	0
$326$	1.87261e19	2.65042
$327$	4.67800e18	0.647072
$328$	−5.05539e17	−0.0683440
$329$	1.08803e19	1.43771
$330$	0	0
$331$	7.04122e18	0.889075	0.444537	−	0.895760i	$-0.353368\pi$
$331$	7.04122e18	0.889075	0.444537	+	0.895760i	$0.353368\pi$
$332$	1.34126e19	1.65568
$333$	3.58622e18	0.432817
$334$	1.83630e19	2.16693
$335$	0	0
$336$	3.86827e18	0.436488
$337$	9.93350e18	1.09617	0.548085	−	0.836423i	$-0.315357\pi$
$337$	9.93350e18	1.09617	0.548085	+	0.836423i	$0.315357\pi$
$338$	7.88710e18	0.851220
$339$	3.61215e18	0.381301
$340$	0	0
$341$	1.43061e19	1.44498
$342$	7.97549e18	0.788064
$343$	1.01400e19	0.980237
$344$	−1.11868e18	−0.105808
$345$	0	0
$346$	−2.06421e19	−1.86932
$347$	1.74750e19	1.54863	0.774313	−	0.632803i	$-0.218096\pi$
$347$	1.74750e19	1.54863	0.774313	+	0.632803i	$0.218096\pi$
$348$	1.32133e18	0.114596
$349$	3.61653e18	0.306974	0.153487	−	0.988151i	$-0.450950\pi$
$349$	3.61653e18	0.306974	0.153487	+	0.988151i	$0.450950\pi$
$350$	0	0
$351$	−1.54781e18	−0.125867
$352$	−2.88324e19	−2.29515
$353$	−1.92224e19	−1.49795	−0.748976	−	0.662597i	$-0.769454\pi$
$353$	−1.92224e19	−1.49795	−0.748976	+	0.662597i	$0.769454\pi$
$354$	5.40209e18	0.412133
$355$	0	0
$356$	−3.17417e19	−2.32143
$357$	9.48987e18	0.679593
$358$	3.88154e17	0.0272196
$359$	−7.07539e18	−0.485895	−0.242947	−	0.970039i	$-0.578114\pi$
$359$	−7.07539e18	−0.485895	−0.242947	+	0.970039i	$0.578114\pi$
$360$	0	0
$361$	2.31682e19	1.52612
$362$	8.78710e18	0.566933
$363$	1.47227e19	0.930437
$364$	1.30508e19	0.807937
$365$	0	0
$366$	1.90201e19	1.13007
$367$	4.17655e18	0.243121	0.121560	−	0.992584i	$-0.461210\pi$
$367$	4.17655e18	0.243121	0.121560	+	0.992584i	$0.461210\pi$
$368$	9.87332e18	0.563124
$369$	1.28874e18	0.0720224
$370$	0	0
$371$	−6.53679e18	−0.350801
$372$	−1.19030e19	−0.626015
$373$	−2.87435e19	−1.48157	−0.740787	−	0.671740i	$-0.765547\pi$
$373$	−2.87435e19	−1.48157	−0.740787	+	0.671740i	$0.765547\pi$
$374$	−5.49799e19	−2.77758
$375$	0	0
$376$	9.19695e18	0.446411
$377$	−2.24983e18	−0.107051
$378$	6.25189e18	0.291624
$379$	−3.34517e19	−1.52976	−0.764881	−	0.644172i	$-0.777202\pi$
$379$	−3.34517e19	−1.52976	−0.764881	+	0.644172i	$0.777202\pi$
$380$	0	0
$381$	−4.45349e18	−0.195778
$382$	−3.09263e19	−1.33307
$383$	1.11831e19	0.472687	0.236343	−	0.971670i	$-0.424051\pi$
$383$	1.11831e19	0.472687	0.236343	+	0.971670i	$0.424051\pi$
$384$	8.61259e18	0.356985
$385$	0	0
$386$	−1.73628e19	−0.692174
$387$	2.85178e18	0.111502
$388$	2.54007e19	0.974109
$389$	−1.45211e19	−0.546231	−0.273115	−	0.961981i	$-0.588054\pi$
$389$	−1.45211e19	−0.546231	−0.273115	+	0.961981i	$0.588054\pi$
$390$	0	0
$391$	2.42218e19	0.876761
$392$	−3.36370e17	−0.0119446
$393$	−1.28314e19	−0.447024
$394$	−9.52264e18	−0.325489
$395$	0	0
$396$	−1.98507e19	−0.653225
$397$	−2.84800e19	−0.919626	−0.459813	−	0.888016i	$-0.652084\pi$
$397$	−2.84800e19	−0.919626	−0.459813	+	0.888016i	$0.652084\pi$
$398$	−1.17222e19	−0.371438
$399$	3.00616e19	0.934793
$400$	0	0
$401$	−5.24890e19	−1.57212	−0.786060	−	0.618150i	$-0.787882\pi$
$401$	−5.24890e19	−1.57212	−0.786060	+	0.618150i	$0.787882\pi$
$402$	1.86679e19	0.548782
$403$	2.02672e19	0.584799
$404$	−2.59484e19	−0.734936
$405$	0	0
$406$	9.08751e18	0.248027
$407$	−7.83131e19	−2.09834
$408$	8.02163e18	0.211014
$409$	1.57729e19	0.407369	0.203684	−	0.979037i	$-0.434708\pi$
$409$	1.57729e19	0.407369	0.203684	+	0.979037i	$0.434708\pi$
$410$	0	0
$411$	9.02161e17	0.0224631
$412$	2.42875e19	0.593817
$413$	2.03618e19	0.488868
$414$	1.59572e19	0.376231
$415$	0	0
$416$	−4.08465e19	−0.928870
$417$	4.05849e19	0.906451
$418$	−1.74163e20	−3.82062
$419$	−6.29709e19	−1.35686	−0.678429	−	0.734666i	$-0.737339\pi$
$419$	−6.29709e19	−1.35686	−0.678429	+	0.734666i	$0.737339\pi$
$420$	0	0
$421$	−3.19227e19	−0.663718	−0.331859	−	0.943329i	$-0.607676\pi$
$421$	−3.19227e19	−0.663718	−0.331859	+	0.943329i	$0.607676\pi$
$422$	−5.61871e18	−0.114761
$423$	−2.34453e19	−0.470438
$424$	−5.52544e18	−0.108924
$425$	0	0
$426$	6.05336e19	1.15193
$427$	7.16913e19	1.34047
$428$	−1.04657e20	−1.92283
$429$	3.37998e19	0.610217
$430$	0	0
$431$	−4.19921e19	−0.732130	−0.366065	−	0.930589i	$-0.619295\pi$
$431$	−4.19921e19	−0.732130	−0.366065	+	0.930589i	$0.619295\pi$
$432$	−8.33547e18	−0.142824
$433$	−5.00147e18	−0.0842245	−0.0421123	−	0.999113i	$-0.513409\pi$
$433$	−5.00147e18	−0.0842245	−0.0421123	+	0.999113i	$0.513409\pi$
$434$	−8.18634e19	−1.35493
$435$	0	0
$436$	8.49954e19	1.35909
$437$	7.67288e19	1.20600
$438$	−4.99897e19	−0.772369
$439$	2.57417e19	0.390978	0.195489	−	0.980706i	$-0.437371\pi$
$439$	2.57417e19	0.390978	0.195489	+	0.980706i	$0.437371\pi$
$440$	0	0
$441$	8.57489e17	0.0125875
$442$	−7.78894e19	−1.12412
$443$	6.08443e19	0.863360	0.431680	−	0.902027i	$-0.357921\pi$
$443$	6.08443e19	0.863360	0.431680	+	0.902027i	$0.357921\pi$
$444$	6.51586e19	0.909074
$445$	0	0
$446$	6.24625e19	0.842574
$447$	−6.52425e19	−0.865415
$448$	1.07029e20	1.39609
$449$	4.01105e19	0.514530	0.257265	−	0.966341i	$-0.417179\pi$
$449$	4.01105e19	0.514530	0.257265	+	0.966341i	$0.417179\pi$
$450$	0	0
$451$	−2.81425e19	−0.349172
$452$	6.56297e19	0.800870
$453$	5.88017e19	0.705754
$454$	1.99912e20	2.36004
$455$	0	0
$456$	2.54106e19	0.290254
$457$	4.50936e19	0.506691	0.253346	−	0.967376i	$-0.418469\pi$
$457$	4.50936e19	0.506691	0.253346	+	0.967376i	$0.418469\pi$
$458$	−1.17524e20	−1.29908
$459$	−2.04491e19	−0.222371
$460$	0	0
$461$	−2.76130e19	−0.290641	−0.145321	−	0.989385i	$-0.546421\pi$
$461$	−2.76130e19	−0.290641	−0.145321	+	0.989385i	$0.546421\pi$
$462$	−1.36524e20	−1.41382
$463$	−2.81282e17	−0.00286606	−0.00143303	−	0.999999i	$-0.500456\pi$
$463$	−2.81282e17	−0.00286606	−0.00143303	+	0.999999i	$0.500456\pi$
$464$	−1.21161e19	−0.121473
$465$	0	0
$466$	6.90913e18	0.0670701
$467$	−5.75699e19	−0.549945	−0.274972	−	0.961452i	$-0.588669\pi$
$467$	−5.75699e19	−0.549945	−0.274972	+	0.961452i	$0.588669\pi$
$468$	−2.81223e19	−0.264367
$469$	7.03639e19	0.650959
$470$	0	0
$471$	6.87453e19	0.616008
$472$	1.72115e19	0.151794
$473$	−6.22750e19	−0.540574
$474$	4.93508e19	0.421654
$475$	0	0
$476$	1.72423e20	1.42739
$477$	1.40857e19	0.114786
$478$	1.04711e20	0.840005
$479$	4.10568e19	0.324241	0.162121	−	0.986771i	$-0.448167\pi$
$479$	4.10568e19	0.324241	0.162121	+	0.986771i	$0.448167\pi$
$480$	0	0
$481$	−1.10945e20	−0.849221
$482$	−2.85595e20	−2.15228
$483$	6.01468e19	0.446281
$484$	2.67499e20	1.95426
$485$	0	0
$486$	−1.34718e19	−0.0954229
$487$	−9.97767e19	−0.695924	−0.347962	−	0.937509i	$-0.613126\pi$
$487$	−9.97767e19	−0.695924	−0.347962	+	0.937509i	$0.613126\pi$
$488$	6.05994e19	0.416217
$489$	−1.52095e20	−1.02872
$490$	0	0
$491$	1.96756e20	1.29068	0.645338	−	0.763897i	$-0.276717\pi$
$491$	1.96756e20	1.29068	0.645338	+	0.763897i	$0.276717\pi$
$492$	2.34153e19	0.151273
$493$	−2.97240e19	−0.189128
$494$	−2.46734e20	−1.54624
$495$	0	0
$496$	1.09146e20	0.663584
$497$	2.28166e20	1.36640
$498$	−1.98773e20	−1.17257
$499$	−5.57267e19	−0.323824	−0.161912	−	0.986805i	$-0.551766\pi$
$499$	−5.57267e19	−0.323824	−0.161912	+	0.986805i	$0.551766\pi$
$500$	0	0
$501$	−1.49146e20	−0.841063
$502$	2.16161e20	1.20088
$503$	−1.58990e20	−0.870182	−0.435091	−	0.900386i	$-0.643284\pi$
$503$	−1.58990e20	−0.870182	−0.435091	+	0.900386i	$0.643284\pi$
$504$	1.99190e19	0.107409
$505$	0	0
$506$	−3.48462e20	−1.82401
$507$	−6.40597e19	−0.330389
$508$	−8.09162e19	−0.411204
$509$	−1.93989e20	−0.971390	−0.485695	−	0.874128i	$-0.661433\pi$
$509$	−1.93989e20	−0.971390	−0.485695	+	0.874128i	$0.661433\pi$
$510$	0	0
$511$	−1.88424e20	−0.916175
$512$	−2.68964e20	−1.28875
$513$	−6.47777e19	−0.305876
$514$	−2.71623e20	−1.26399
$515$	0	0
$516$	5.18145e19	0.234195
$517$	5.11980e20	2.28073
$518$	4.48130e20	1.96757
$519$	1.67657e20	0.725549
$520$	0	0
$521$	−1.77348e20	−0.745664	−0.372832	−	0.927899i	$-0.621613\pi$
$521$	−1.77348e20	−0.745664	−0.372832	+	0.927899i	$0.621613\pi$
$522$	−1.95821e19	−0.0811577
$523$	3.25540e20	1.32997	0.664986	−	0.746856i	$-0.268438\pi$
$523$	3.25540e20	1.32997	0.664986	+	0.746856i	$0.268438\pi$
$524$	−2.33137e20	−0.938914
$525$	0	0
$526$	5.35731e18	0.0209678
$527$	2.67764e20	1.03317
$528$	1.82024e20	0.692426
$529$	−1.13118e20	−0.424241
$530$	0	0
$531$	−4.38763e19	−0.159964
$532$	5.46194e20	1.96340
$533$	−3.98692e19	−0.141313
$534$	4.70409e20	1.64406
$535$	0	0
$536$	5.94774e19	0.202123
$537$	−3.15262e18	−0.0105649
$538$	3.57126e20	1.18020
$539$	−1.87252e19	−0.0610254
$540$	0	0
$541$	−2.48259e20	−0.786912	−0.393456	−	0.919343i	$-0.628721\pi$
$541$	−2.48259e20	−0.786912	−0.393456	+	0.919343i	$0.628721\pi$
$542$	−8.66931e20	−2.71013
$543$	−7.13697e19	−0.220047
$544$	−5.39650e20	−1.64105
$545$	0	0
$546$	−1.93412e20	−0.572188
$547$	3.93762e20	1.14902	0.574511	−	0.818497i	$-0.305192\pi$
$547$	3.93762e20	1.14902	0.574511	+	0.818497i	$0.305192\pi$
$548$	1.63915e19	0.0471807
$549$	−1.54483e20	−0.438619
$550$	0	0
$551$	−9.41584e19	−0.260149
$552$	5.08411e19	0.138571
$553$	1.86015e20	0.500161
$554$	2.40586e20	0.638186
$555$	0	0
$556$	7.37394e20	1.90388
$557$	2.64334e20	0.673348	0.336674	−	0.941621i	$-0.390698\pi$
$557$	2.64334e20	0.673348	0.336674	+	0.941621i	$0.390698\pi$
$558$	1.76402e20	0.443350
$559$	−8.82243e19	−0.218776
$560$	0	0
$561$	4.46552e20	1.07808
$562$	−9.01623e20	−2.14784
$563$	6.50268e20	1.52855	0.764274	−	0.644891i	$-0.223097\pi$
$563$	6.50268e20	1.52855	0.764274	+	0.644891i	$0.223097\pi$
$564$	−4.25981e20	−0.988090
$565$	0	0
$566$	3.32552e20	0.751167
$567$	−5.07784e19	−0.113190
$568$	1.92865e20	0.424269
$569$	−3.00610e20	−0.652622	−0.326311	−	0.945262i	$-0.605806\pi$
$569$	−3.00610e20	−0.652622	−0.326311	+	0.945262i	$0.605806\pi$
$570$	0	0
$571$	−1.43433e19	−0.0303304	−0.0151652	−	0.999885i	$-0.504827\pi$
$571$	−1.43433e19	−0.0303304	−0.0151652	+	0.999885i	$0.504827\pi$
$572$	6.14114e20	1.28168
$573$	2.51187e20	0.517412
$574$	1.61040e20	0.327411
$575$	0	0
$576$	−2.30628e20	−0.456819
$577$	−1.29132e20	−0.252472	−0.126236	−	0.992000i	$-0.540290\pi$
$577$	−1.29132e20	−0.252472	−0.126236	+	0.992000i	$0.540290\pi$
$578$	−2.58296e20	−0.498493
$579$	1.41022e20	0.268657
$580$	0	0
$581$	−7.49226e20	−1.39089
$582$	−3.76436e20	−0.689873
$583$	−3.07593e20	−0.556496
$584$	−1.59271e20	−0.284473
$585$	0	0
$586$	2.17911e20	0.379356
$587$	6.12184e20	1.05219	0.526097	−	0.850424i	$-0.323655\pi$
$587$	6.12184e20	1.05219	0.526097	+	0.850424i	$0.323655\pi$
$588$	1.55799e19	0.0264383
$589$	8.48210e20	1.42115
$590$	0	0
$591$	7.73438e19	0.126334
$592$	−5.97479e20	−0.963629
$593$	3.70587e18	0.00590175	0.00295087	−	0.999996i	$-0.499061\pi$
$593$	3.70587e18	0.00590175	0.00295087	+	0.999996i	$0.499061\pi$
$594$	2.94187e20	0.462620
$595$	0	0
$596$	−1.18540e21	−1.81769
$597$	9.52088e19	0.144168
$598$	−4.93662e20	−0.738195
$599$	4.04448e20	0.597257	0.298628	−	0.954369i	$-0.403471\pi$
$599$	4.04448e20	0.597257	0.298628	+	0.954369i	$0.403471\pi$
$600$	0	0
$601$	1.23096e21	1.77290	0.886450	−	0.462824i	$-0.153164\pi$
$601$	1.23096e21	1.77290	0.886450	+	0.462824i	$0.153164\pi$
$602$	3.56356e20	0.506886
$603$	−1.51622e20	−0.213002
$604$	1.06838e21	1.48234
$605$	0	0
$606$	3.84553e20	0.520488
$607$	−1.16825e21	−1.56179	−0.780894	−	0.624663i	$-0.785236\pi$
$607$	−1.16825e21	−1.56179	−0.780894	+	0.624663i	$0.785236\pi$
$608$	−1.70948e21	−2.25729
$609$	−7.38096e19	−0.0962683
$610$	0	0
$611$	7.25316e20	0.923035
$612$	−3.71543e20	−0.467060
$613$	−1.39705e21	−1.73484	−0.867420	−	0.497576i	$-0.834224\pi$
$613$	−1.39705e21	−1.73484	−0.867420	+	0.497576i	$0.834224\pi$
$614$	1.89506e21	2.32466
$615$	0	0
$616$	−4.34977e20	−0.520728
$617$	−1.59207e20	−0.188288	−0.0941440	−	0.995559i	$-0.530011\pi$
$617$	−1.59207e20	−0.188288	−0.0941440	+	0.995559i	$0.530011\pi$
$618$	−3.59938e20	−0.420547
$619$	−1.56513e19	−0.0180664	−0.00903318	−	0.999959i	$-0.502875\pi$
$619$	−1.56513e19	−0.0180664	−0.00903318	+	0.999959i	$0.502875\pi$
$620$	0	0
$621$	−1.29606e20	−0.146029
$622$	1.25874e21	1.40123
$623$	1.77309e21	1.95016
$624$	2.57871e20	0.280232
$625$	0	0
$626$	1.36221e21	1.44522
$627$	1.41457e21	1.48292
$628$	1.24904e21	1.29384
$629$	−1.46577e21	−1.50033
$630$	0	0
$631$	−3.14481e20	−0.314321	−0.157161	−	0.987573i	$-0.550234\pi$
$631$	−3.14481e20	−0.314321	−0.157161	+	0.987573i	$0.550234\pi$
$632$	1.57236e20	0.155300
$633$	4.56357e19	0.0445427
$634$	−1.39470e21	−1.34527
$635$	0	0
$636$	2.55925e20	0.241093
$637$	−2.65278e19	−0.0246976
$638$	4.27618e20	0.393461
$639$	−4.91660e20	−0.447104
$640$	0	0
$641$	6.36192e20	0.565136	0.282568	−	0.959247i	$-0.408814\pi$
$641$	6.36192e20	0.565136	0.282568	+	0.959247i	$0.408814\pi$
$642$	1.55101e21	1.36177
$643$	8.27139e19	0.0717788	0.0358894	−	0.999356i	$-0.488574\pi$
$643$	8.27139e19	0.0717788	0.0358894	+	0.999356i	$0.488574\pi$
$644$	1.09282e21	0.937353
$645$	0	0
$646$	−3.25977e21	−2.73177
$647$	−5.97156e20	−0.494659	−0.247330	−	0.968931i	$-0.579553\pi$
$647$	−5.97156e20	−0.494659	−0.247330	+	0.968931i	$0.579553\pi$
$648$	−4.29222e19	−0.0351455
$649$	9.58138e20	0.775520
$650$	0	0
$651$	6.64902e20	0.525896
$652$	−2.76344e21	−2.16069
$653$	2.55343e20	0.197367	0.0986837	−	0.995119i	$-0.468537\pi$
$653$	2.55343e20	0.197367	0.0986837	+	0.995119i	$0.468537\pi$
$654$	−1.25963e21	−0.962517
$655$	0	0
$656$	−2.14710e20	−0.160351
$657$	4.06021e20	0.299784
$658$	−2.92970e21	−2.13859
$659$	2.14276e21	1.54644	0.773219	−	0.634139i	$-0.218645\pi$
$659$	2.14276e21	1.54644	0.773219	+	0.634139i	$0.218645\pi$
$660$	0	0
$661$	−1.01666e21	−0.717244	−0.358622	−	0.933483i	$-0.616753\pi$
$661$	−1.01666e21	−0.717244	−0.358622	+	0.933483i	$0.616753\pi$
$662$	−1.89596e21	−1.32250
$663$	6.32625e20	0.436310
$664$	−6.33308e20	−0.431872
$665$	0	0
$666$	−9.65645e20	−0.643814
$667$	−1.88391e20	−0.124198
$668$	−2.70986e21	−1.76654
$669$	−5.07326e20	−0.327033
$670$	0	0
$671$	3.37348e21	2.12647
$672$	−1.34004e21	−0.835312
$673$	9.99280e20	0.615991	0.307995	−	0.951388i	$-0.400342\pi$
$673$	9.99280e20	0.615991	0.307995	+	0.951388i	$0.400342\pi$
$674$	−2.67475e21	−1.63055
$675$	0	0
$676$	−1.16391e21	−0.693937
$677$	1.16089e21	0.684504	0.342252	−	0.939608i	$-0.388810\pi$
$677$	1.16089e21	0.684504	0.342252	+	0.939608i	$0.388810\pi$
$678$	−9.72627e20	−0.567183
$679$	−1.41888e21	−0.818319
$680$	0	0
$681$	−1.62370e21	−0.916017
$682$	−3.85213e21	−2.14940
$683$	−1.39260e21	−0.768546	−0.384273	−	0.923220i	$-0.625548\pi$
$683$	−1.39260e21	−0.768546	−0.384273	+	0.923220i	$0.625548\pi$
$684$	−1.17696e21	−0.642450
$685$	0	0
$686$	−2.73035e21	−1.45810
$687$	9.54543e20	0.504220
$688$	−4.75119e20	−0.248250
$689$	−4.35763e20	−0.225220
$690$	0	0
$691$	1.89176e21	0.956713	0.478356	−	0.878166i	$-0.341233\pi$
$691$	1.89176e21	0.956713	0.478356	+	0.878166i	$0.341233\pi$
$692$	3.04619e21	1.52392
$693$	1.10886e21	0.548755
$694$	−4.70542e21	−2.30357
$695$	0	0
$696$	−6.23900e19	−0.0298914
$697$	−5.26738e20	−0.249660
$698$	−9.73807e20	−0.456622
$699$	−5.61166e19	−0.0260323
$700$	0	0
$701$	9.64903e20	0.438125	0.219063	−	0.975711i	$-0.429700\pi$
$701$	9.64903e20	0.438125	0.219063	+	0.975711i	$0.429700\pi$
$702$	4.16771e20	0.187227
$703$	−4.64321e21	−2.06373
$704$	5.03629e21	2.21471
$705$	0	0
$706$	5.17594e21	2.22820
$707$	1.44947e21	0.617397
$708$	−7.97196e20	−0.335982
$709$	2.68680e21	1.12044	0.560220	−	0.828344i	$-0.310716\pi$
$709$	2.68680e21	1.12044	0.560220	+	0.828344i	$0.310716\pi$
$710$	0	0
$711$	−4.00831e20	−0.163659
$712$	1.49876e21	0.605526
$713$	1.69709e21	0.678472
$714$	−2.55530e21	−1.01089
$715$	0	0
$716$	−5.72805e19	−0.0221901
$717$	−8.50470e20	−0.326036
$718$	1.90516e21	0.722766
$719$	2.38998e21	0.897279	0.448640	−	0.893713i	$-0.351909\pi$
$719$	2.38998e21	0.897279	0.448640	+	0.893713i	$0.351909\pi$
$720$	0	0
$721$	−1.35670e21	−0.498848
$722$	−6.23840e21	−2.27010
$723$	2.31963e21	0.835375
$724$	−1.29673e21	−0.462178
$725$	0	0
$726$	−3.96431e21	−1.38402
$727$	3.01018e21	1.04012	0.520060	−	0.854130i	$-0.325909\pi$
$727$	3.01018e21	1.04012	0.520060	+	0.854130i	$0.325909\pi$
$728$	−6.16227e20	−0.210744
$729$	1.09419e20	0.0370370
$730$	0	0
$731$	−1.16559e21	−0.386514
$732$	−2.80682e21	−0.921258
$733$	−5.81164e17	−0.000188807 0	−9.44037e−5	−	1.00000i	$-0.500030\pi$
$733$	−5.81164e17	−0.000188807 0	−9.44037e−5		1.00000i	$0.500030\pi$
$734$	−1.12460e21	−0.361641
$735$	0	0
$736$	−3.42030e21	−1.07766
$737$	3.31102e21	1.03265
$738$	−3.47013e20	−0.107133
$739$	−4.24170e21	−1.29630	−0.648151	−	0.761512i	$-0.724458\pi$
$739$	−4.24170e21	−1.29630	−0.648151	+	0.761512i	$0.724458\pi$
$740$	0	0
$741$	2.00400e21	0.600152
$742$	1.76013e21	0.521815
$743$	−5.77221e20	−0.169405	−0.0847025	−	0.996406i	$-0.526994\pi$
$743$	−5.77221e20	−0.169405	−0.0847025	+	0.996406i	$0.526994\pi$
$744$	5.62030e20	0.163291
$745$	0	0
$746$	7.73963e21	2.20383
$747$	1.61446e21	0.455116
$748$	8.11347e21	2.26436
$749$	5.84616e21	1.61531
$750$	0	0
$751$	−1.64783e21	−0.446287	−0.223143	−	0.974786i	$-0.571632\pi$
$751$	−1.64783e21	−0.446287	−0.223143	+	0.974786i	$0.571632\pi$
$752$	3.90608e21	1.04739
$753$	−1.75568e21	−0.466104
$754$	6.05802e20	0.159238
$755$	0	0
$756$	−9.22602e20	−0.237739
$757$	−2.60997e21	−0.665911	−0.332955	−	0.942943i	$-0.608046\pi$
$757$	−2.60997e21	−0.665911	−0.332955	+	0.942943i	$0.608046\pi$
$758$	9.00738e21	2.27551
$759$	2.83024e21	0.707963
$760$	0	0
$761$	−7.91755e20	−0.194181	−0.0970903	−	0.995276i	$-0.530954\pi$
$761$	−7.91755e20	−0.194181	−0.0970903	+	0.995276i	$0.530954\pi$
$762$	1.19917e21	0.291218
$763$	−4.74784e21	−1.14173
$764$	4.56385e21	1.08675
$765$	0	0
$766$	−3.01124e21	−0.703120
$767$	1.35738e21	0.313861
$768$	1.13645e21	0.260221
$769$	−5.96173e21	−1.35184	−0.675919	−	0.736976i	$-0.736253\pi$
$769$	−5.96173e21	−1.35184	−0.675919	+	0.736976i	$0.736253\pi$
$770$	0	0
$771$	2.20615e21	0.490599
$772$	2.56226e21	0.564278
$773$	5.48630e21	1.19656	0.598279	−	0.801288i	$-0.295851\pi$
$773$	5.48630e21	1.19656	0.598279	+	0.801288i	$0.295851\pi$
$774$	−7.67886e20	−0.165859
$775$	0	0
$776$	−1.19936e21	−0.254089
$777$	−3.63975e21	−0.763685
$778$	3.91002e21	0.812516
$779$	−1.66858e21	−0.343412
$780$	0	0
$781$	1.07365e22	2.16760
$782$	−6.52210e21	−1.30418
$783$	1.59047e20	0.0315002
$784$	−1.42861e20	−0.0280249
$785$	0	0
$786$	3.45506e21	0.664947
$787$	7.61931e21	1.45246	0.726232	−	0.687450i	$-0.241270\pi$
$787$	7.61931e21	1.45246	0.726232	+	0.687450i	$0.241270\pi$
$788$	1.40527e21	0.265347
$789$	−4.35126e19	−0.00813837
$790$	0	0
$791$	−3.66607e21	−0.672786
$792$	9.37302e20	0.170389
$793$	4.77916e21	0.860603
$794$	7.66869e21	1.36794
$795$	0	0
$796$	1.72986e21	0.302806
$797$	−8.62089e21	−1.49491	−0.747454	−	0.664313i	$-0.768724\pi$
$797$	−8.62089e21	−1.49491	−0.747454	+	0.664313i	$0.768724\pi$
$798$	−8.09455e21	−1.39050
$799$	9.58263e21	1.63074
$800$	0	0
$801$	−3.82071e21	−0.638116
$802$	1.41335e22	2.33852
$803$	−8.86639e21	−1.45338
$804$	−2.75485e21	−0.447381
$805$	0	0
$806$	−5.45727e21	−0.869886
$807$	−2.90061e21	−0.458076
$808$	1.22522e21	0.191702
$809$	−1.43024e21	−0.221714	−0.110857	−	0.993836i	$-0.535360\pi$
$809$	−1.43024e21	−0.221714	−0.110857	+	0.993836i	$0.535360\pi$
$810$	0	0
$811$	−1.87574e21	−0.285440	−0.142720	−	0.989763i	$-0.545585\pi$
$811$	−1.87574e21	−0.285440	−0.142720	+	0.989763i	$0.545585\pi$
$812$	−1.34106e21	−0.202198
$813$	7.04129e21	1.05190
$814$	2.10870e22	3.12128
$815$	0	0
$816$	3.40690e21	0.495090
$817$	−3.69230e21	−0.531658
$818$	−4.24711e21	−0.605959
$819$	1.57091e21	0.222087
$820$	0	0
$821$	−1.01899e22	−1.41447	−0.707236	−	0.706977i	$-0.750058\pi$
$821$	−1.01899e22	−1.41447	−0.707236	+	0.706977i	$0.750058\pi$
$822$	−2.42921e20	−0.0334138
$823$	3.95704e21	0.539352	0.269676	−	0.962951i	$-0.413083\pi$
$823$	3.95704e21	0.539352	0.269676	+	0.962951i	$0.413083\pi$
$824$	−1.14679e21	−0.154893
$825$	0	0
$826$	−5.48274e21	−0.727189
$827$	2.77322e21	0.364496	0.182248	−	0.983253i	$-0.441663\pi$
$827$	2.77322e21	0.364496	0.182248	+	0.983253i	$0.441663\pi$
$828$	−2.35484e21	−0.306714
$829$	−1.03785e22	−1.33960	−0.669802	−	0.742540i	$-0.733621\pi$
$829$	−1.03785e22	−1.33960	−0.669802	+	0.742540i	$0.733621\pi$
$830$	0	0
$831$	−1.95406e21	−0.247703
$832$	7.13486e21	0.896314
$833$	−3.50476e20	−0.0436336
$834$	−1.09281e22	−1.34834
$835$	0	0
$836$	2.57015e22	3.11466
$837$	−1.43275e21	−0.172080
$838$	1.69559e22	2.01832
$839$	−1.02185e22	−1.20551	−0.602756	−	0.797926i	$-0.705931\pi$
$839$	−1.02185e22	−1.20551	−0.602756	+	0.797926i	$0.705931\pi$
$840$	0	0
$841$	−8.39800e21	−0.973209
$842$	8.59568e21	0.987279
$843$	7.32307e21	0.833655
$844$	8.29162e20	0.0935559
$845$	0	0
$846$	6.31300e21	0.699774
$847$	−1.49425e22	−1.64171
$848$	−2.34674e21	−0.255561
$849$	−2.70102e21	−0.291555
$850$	0	0
$851$	−9.29005e21	−0.985250
$852$	−8.93305e21	−0.939080
$853$	−1.31509e22	−1.37037	−0.685183	−	0.728371i	$-0.740278\pi$
$853$	−1.31509e22	−1.37037	−0.685183	+	0.728371i	$0.740278\pi$
$854$	−1.93040e22	−1.99394
$855$	0	0
$856$	4.94166e21	0.501556
$857$	−1.45347e21	−0.146235	−0.0731173	−	0.997323i	$-0.523295\pi$
$857$	−1.45347e21	−0.146235	−0.0731173	+	0.997323i	$0.523295\pi$
$858$	−9.10113e21	−0.907696
$859$	−8.04641e21	−0.795524	−0.397762	−	0.917489i	$-0.630213\pi$
$859$	−8.04641e21	−0.795524	−0.397762	+	0.917489i	$0.630213\pi$
$860$	0	0
$861$	−1.30798e21	−0.127080
$862$	1.13070e22	1.08904
$863$	−3.87179e21	−0.369685	−0.184842	−	0.982768i	$-0.559177\pi$
$863$	−3.87179e21	−0.369685	−0.184842	+	0.982768i	$0.559177\pi$
$864$	2.88756e21	0.273324
$865$	0	0
$866$	1.34672e21	0.125284
$867$	2.09790e21	0.193483
$868$	1.20807e22	1.10457
$869$	8.75306e21	0.793435
$870$	0	0
$871$	4.69068e21	0.417926
$872$	−4.01327e21	−0.354507
$873$	3.05745e21	0.267764
$874$	−2.06604e22	−1.79392
$875$	0	0
$876$	7.37707e21	0.629655
$877$	4.15578e21	0.351686	0.175843	−	0.984418i	$-0.443735\pi$
$877$	4.15578e21	0.351686	0.175843	+	0.984418i	$0.443735\pi$
$878$	−6.93135e21	−0.581579
$879$	−1.76989e21	−0.147242
$880$	0	0
$881$	9.19831e20	0.0752296	0.0376148	−	0.999292i	$-0.488024\pi$
$881$	9.19831e20	0.0752296	0.0376148	+	0.999292i	$0.488024\pi$
$882$	−2.30892e20	−0.0187238
$883$	−1.12103e22	−0.901388	−0.450694	−	0.892679i	$-0.648823\pi$
$883$	−1.12103e22	−0.901388	−0.450694	+	0.892679i	$0.648823\pi$
$884$	1.14943e22	0.916408
$885$	0	0
$886$	−1.63833e22	−1.28425
$887$	−2.15361e22	−1.67394	−0.836970	−	0.547249i	$-0.815675\pi$
$887$	−2.15361e22	−1.67394	−0.836970	+	0.547249i	$0.815675\pi$
$888$	−3.07662e21	−0.237125
$889$	4.51997e21	0.345440
$890$	0	0
$891$	−2.38941e21	−0.179559
$892$	−9.21770e21	−0.686888
$893$	3.03554e22	2.24311
$894$	1.75676e22	1.28730
$895$	0	0
$896$	−8.74116e21	−0.629882
$897$	4.00957e21	0.286520
$898$	−1.08004e22	−0.765361
$899$	−2.08259e21	−0.146355
$900$	0	0
$901$	−5.75716e21	−0.397898
$902$	7.57782e21	0.519392
$903$	−2.89435e21	−0.196740
$904$	−3.09887e21	−0.208901
$905$	0	0
$906$	−1.58333e22	−1.04981
$907$	5.75170e21	0.378217	0.189109	−	0.981956i	$-0.439440\pi$
$907$	5.75170e21	0.378217	0.189109	+	0.981956i	$0.439440\pi$
$908$	−2.95013e22	−1.92397
$909$	−3.12337e21	−0.202020
$910$	0	0
$911$	1.14422e22	0.727985	0.363993	−	0.931402i	$-0.381413\pi$
$911$	1.14422e22	0.727985	0.363993	+	0.931402i	$0.381413\pi$
$912$	1.07922e22	0.681005
$913$	−3.52553e22	−2.20645
$914$	−1.21422e22	−0.753702
$915$	0	0
$916$	1.73432e22	1.05904
$917$	1.30230e22	0.788753
$918$	5.50623e21	0.330776
$919$	−5.20338e21	−0.310041	−0.155021	−	0.987911i	$-0.549544\pi$
$919$	−5.20338e21	−0.310041	−0.155021	+	0.987911i	$0.549544\pi$
$920$	0	0
$921$	−1.53918e22	−0.902284
$922$	7.43524e21	0.432328
$923$	1.52103e22	0.877252
$924$	2.01471e22	1.15258
$925$	0	0
$926$	7.57396e19	0.00426325
$927$	2.92345e21	0.163229
$928$	4.19725e21	0.232464
$929$	−1.95888e22	−1.07619	−0.538095	−	0.842884i	$-0.680856\pi$
$929$	−1.95888e22	−1.07619	−0.538095	+	0.842884i	$0.680856\pi$
$930$	0	0
$931$	−1.11022e21	−0.0600188
$932$	−1.01959e21	−0.0546772
$933$	−1.02236e22	−0.543868
$934$	1.55016e22	0.818041
$935$	0	0
$936$	1.32787e21	0.0689581
$937$	−3.08090e22	−1.58720	−0.793598	−	0.608443i	$-0.791795\pi$
$937$	−3.08090e22	−1.58720	−0.793598	+	0.608443i	$0.791795\pi$
$938$	−1.89466e22	−0.968299
$939$	−1.10640e22	−0.560943
$940$	0	0
$941$	9.00651e21	0.449401	0.224701	−	0.974428i	$-0.427860\pi$
$941$	9.00651e21	0.449401	0.224701	+	0.974428i	$0.427860\pi$
$942$	−1.85107e22	−0.916309
$943$	−3.33847e21	−0.163949
$944$	7.30998e21	0.356144
$945$	0	0
$946$	1.67685e22	0.804102
$947$	−1.46518e22	−0.697055	−0.348528	−	0.937299i	$-0.613318\pi$
$947$	−1.46518e22	−0.697055	−0.348528	+	0.937299i	$0.613318\pi$
$948$	−7.28277e21	−0.343743
$949$	−1.25609e22	−0.588199
$950$	0	0
$951$	1.13279e22	0.522149
$952$	−8.14139e21	−0.372324
$953$	4.57863e21	0.207749	0.103874	−	0.994590i	$-0.466876\pi$
$953$	4.57863e21	0.207749	0.103874	+	0.994590i	$0.466876\pi$
$954$	−3.79279e21	−0.170744
$955$	0	0
$956$	−1.54523e22	−0.684794
$957$	−3.47316e21	−0.152716
$958$	−1.10552e22	−0.482308
$959$	−9.15629e20	−0.0396351
$960$	0	0
$961$	−4.70457e21	−0.200491
$962$	2.98738e22	1.26321
$963$	−1.25975e22	−0.528551
$964$	4.21457e22	1.75459
$965$	0	0
$966$	−1.61955e22	−0.663842
$967$	−1.65508e22	−0.673165	−0.336583	−	0.941654i	$-0.609271\pi$
$967$	−1.65508e22	−0.673165	−0.336583	+	0.941654i	$0.609271\pi$
$968$	−1.26306e22	−0.509753
$969$	2.64762e22	1.06030
$970$	0	0
$971$	−2.02921e22	−0.800172	−0.400086	−	0.916478i	$-0.631020\pi$
$971$	−2.02921e22	−0.800172	−0.400086	+	0.916478i	$0.631020\pi$
$972$	1.98805e21	0.0777912
$973$	−4.11908e22	−1.59939
$974$	2.68664e22	1.03518
$975$	0	0
$976$	2.57375e22	0.976545
$977$	2.62183e22	0.987177	0.493589	−	0.869695i	$-0.335685\pi$
$977$	2.62183e22	0.987177	0.493589	+	0.869695i	$0.335685\pi$
$978$	4.09540e22	1.53022
$979$	8.34338e22	3.09365
$980$	0	0
$981$	1.02308e22	0.373587
$982$	−5.29797e22	−1.91988
$983$	−1.36375e22	−0.490436	−0.245218	−	0.969468i	$-0.578859\pi$
$983$	−1.36375e22	−0.490436	−0.245218	+	0.969468i	$0.578859\pi$
$984$	−1.10561e21	−0.0394584
$985$	0	0
$986$	8.00365e21	0.281327
$987$	2.37953e22	0.830064
$988$	3.64110e22	1.26054
$989$	−7.38750e21	−0.253820
$990$	0	0
$991$	1.07904e22	0.365160	0.182580	−	0.983191i	$-0.441555\pi$
$991$	1.07904e22	0.365160	0.182580	+	0.983191i	$0.441555\pi$
$992$	−3.78102e22	−1.26991
$993$	1.53992e22	0.513307
$994$	−6.14373e22	−2.03252
$995$	0	0
$996$	2.93333e22	0.955908
$997$	−4.68858e22	−1.51645	−0.758225	−	0.651994i	$-0.773933\pi$
$997$	−4.68858e22	−1.51645	−0.758225	+	0.651994i	$0.773933\pi$
$998$	1.50053e22	0.481687
$999$	7.84306e21	0.249887

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.16.a.l.1.2		8
5.2	odd	4		15.16.b.a.4.3	✓	16
5.3	odd	4		15.16.b.a.4.14	yes	16
5.4	even	2		75.16.a.k.1.7		8
15.2	even	4		45.16.b.d.19.14		16
15.8	even	4		45.16.b.d.19.3		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.16.b.a.4.3	✓	16	5.2	odd	4
15.16.b.a.4.14	yes	16	5.3	odd	4
45.16.b.d.19.3		16	15.8	even	4
45.16.b.d.19.14		16	15.2	even	4
75.16.a.k.1.7		8	5.4	even	2
75.16.a.l.1.2		8	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 16 \)
Character orbit:	\([\chi]\)	\(=\)	75.a (trivial)

\(f(q)\)	\(=\)	\(q-269.266 q^{2} +2187.00 q^{3} +39736.0 q^{4} -588884. q^{6} -2.21965e6 q^{7} -1.87623e6 q^{8} +4.78297e6 q^{9} +O(q^{10})\)	\(q-269.266 q^{2} +2187.00 q^{3} +39736.0 q^{4} -588884. q^{6} -2.21965e6 q^{7} -1.87623e6 q^{8} +4.78297e6 q^{9} -1.04447e8 q^{11} +8.69025e7 q^{12} -1.47969e8 q^{13} +5.97675e8 q^{14} -7.96863e8 q^{16} -1.95491e9 q^{17} -1.28789e9 q^{18} -6.19268e9 q^{19} -4.85437e9 q^{21} +2.81240e10 q^{22} -1.23902e10 q^{23} -4.10332e9 q^{24} +3.98429e10 q^{26} +1.04604e10 q^{27} -8.81999e10 q^{28} +1.52048e10 q^{29} -1.36970e11 q^{31} +2.76048e11 q^{32} -2.28425e11 q^{33} +5.26391e11 q^{34} +1.90056e11 q^{36} +7.49789e11 q^{37} +1.66748e12 q^{38} -3.23608e11 q^{39} +2.69443e11 q^{41} +1.30712e12 q^{42} +5.96236e11 q^{43} -4.15030e12 q^{44} +3.33626e12 q^{46} -4.90182e12 q^{47} -1.74274e12 q^{48} +1.79280e11 q^{49} -4.27539e12 q^{51} -5.87968e12 q^{52} +2.94497e12 q^{53} -2.81661e12 q^{54} +4.16458e12 q^{56} -1.35434e13 q^{57} -4.09412e12 q^{58} -9.17344e12 q^{59} -3.22985e13 q^{61} +3.68812e13 q^{62} -1.06165e13 q^{63} -4.82187e13 q^{64} +6.15071e13 q^{66} -3.17005e13 q^{67} -7.76803e13 q^{68} -2.70974e13 q^{69} -1.02794e14 q^{71} -8.97396e12 q^{72} +8.48889e13 q^{73} -2.01892e14 q^{74} -2.46072e14 q^{76} +2.31835e14 q^{77} +8.71364e13 q^{78} -8.38039e13 q^{79} +2.28768e13 q^{81} -7.25518e13 q^{82} +3.37543e14 q^{83} -1.92893e14 q^{84} -1.60546e14 q^{86} +3.32528e13 q^{87} +1.95967e14 q^{88} -7.98815e14 q^{89} +3.28439e14 q^{91} -4.92338e14 q^{92} -2.99553e14 q^{93} +1.31989e15 q^{94} +6.03717e14 q^{96} +6.39237e14 q^{97} -4.82738e13 q^{98} -4.99566e14 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 273 q^{2} + 17496 q^{3} + 160941 q^{4} + 597051 q^{6} + 1149126 q^{7} + 10053771 q^{8} + 38263752 q^{9}+O(q^{10}) \) 8 * q + 273 * q^2 + 17496 * q^3 + 160941 * q^4 + 597051 * q^6 + 1149126 * q^7 + 10053771 * q^8 + 38263752 * q^9	\( 8 q + 273 q^{2} + 17496 q^{3} + 160941 q^{4} + 597051 q^{6} + 1149126 q^{7} + 10053771 q^{8} + 38263752 q^{9} + 62424222 q^{11} + 351977967 q^{12} + 44492838 q^{13} + 306251034 q^{14} + 1871684545 q^{16} + 3593639622 q^{17} + 1305750537 q^{18} - 496736656 q^{19} + 2513138562 q^{21} + 26169930714 q^{22} + 2358873312 q^{23} + 21987597177 q^{24} + 18223186326 q^{26} + 83682825624 q^{27} - 10716504462 q^{28} + 36630923358 q^{29} + 166724197108 q^{31} + 1086177843771 q^{32} + 136521773514 q^{33} + 141055448602 q^{34} + 769775813829 q^{36} + 2497964854986 q^{37} + 785802288552 q^{38} + 97305836706 q^{39} + 4754409554748 q^{41} + 669771011358 q^{42} + 1310015295120 q^{43} + 7792818609978 q^{44} + 14727772650308 q^{46} - 3007996138620 q^{47} + 4093374099915 q^{48} + 8653589541100 q^{49} + 7859289853314 q^{51} - 22117814273898 q^{52} + 6974407032522 q^{53} + 2855676424419 q^{54} + 26685903266430 q^{56} - 1086363066672 q^{57} - 35393776551696 q^{58} + 1259020938606 q^{59} + 25651568022640 q^{61} - 43807470421128 q^{62} + 5496234035094 q^{63} + 134602047432169 q^{64} + 57233638471518 q^{66} - 27008842727148 q^{67} - 28552468844166 q^{68} + 5158855933344 q^{69} - 174179827562004 q^{71} + 48086875026099 q^{72} + 107017091040132 q^{73} - 150367200840846 q^{74} - 407245761522456 q^{76} + 192490324871652 q^{77} + 39854108494962 q^{78} + 173323732806380 q^{79} + 183014339639688 q^{81} + 969967421566242 q^{82} + 868845027534576 q^{83} - 23436995258394 q^{84} - 15\!\cdots\!48 q^{86}+ \cdots + 298573118675118 q^{99}+O(q^{100}) \) 8 * q + 273 * q^2 + 17496 * q^3 + 160941 * q^4 + 597051 * q^6 + 1149126 * q^7 + 10053771 * q^8 + 38263752 * q^9 + 62424222 * q^11 + 351977967 * q^12 + 44492838 * q^13 + 306251034 * q^14 + 1871684545 * q^16 + 3593639622 * q^17 + 1305750537 * q^18 - 496736656 * q^19 + 2513138562 * q^21 + 26169930714 * q^22 + 2358873312 * q^23 + 21987597177 * q^24 + 18223186326 * q^26 + 83682825624 * q^27 - 10716504462 * q^28 + 36630923358 * q^29 + 166724197108 * q^31 + 1086177843771 * q^32 + 136521773514 * q^33 + 141055448602 * q^34 + 769775813829 * q^36 + 2497964854986 * q^37 + 785802288552 * q^38 + 97305836706 * q^39 + 4754409554748 * q^41 + 669771011358 * q^42 + 1310015295120 * q^43 + 7792818609978 * q^44 + 14727772650308 * q^46 - 3007996138620 * q^47 + 4093374099915 * q^48 + 8653589541100 * q^49 + 7859289853314 * q^51 - 22117814273898 * q^52 + 6974407032522 * q^53 + 2855676424419 * q^54 + 26685903266430 * q^56 - 1086363066672 * q^57 - 35393776551696 * q^58 + 1259020938606 * q^59 + 25651568022640 * q^61 - 43807470421128 * q^62 + 5496234035094 * q^63 + 134602047432169 * q^64 + 57233638471518 * q^66 - 27008842727148 * q^67 - 28552468844166 * q^68 + 5158855933344 * q^69 - 174179827562004 * q^71 + 48086875026099 * q^72 + 107017091040132 * q^73 - 150367200840846 * q^74 - 407245761522456 * q^76 + 192490324871652 * q^77 + 39854108494962 * q^78 + 173323732806380 * q^79 + 183014339639688 * q^81 + 969967421566242 * q^82 + 868845027534576 * q^83 - 23436995258394 * q^84 - 1530511469933748 * q^86 + 80111829383946 * q^87 + 2334276558470406 * q^88 - 436746702974256 * q^89 + 185873985841788 * q^91 + 1698320226970356 * q^92 + 364625819075196 * q^93 + 1113775369472416 * q^94 + 2375470944327177 * q^96 - 1064926798083408 * q^97 + 3042468678281757 * q^98 + 298573118675118 * q^99

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