LMFDB - Embedded newform 75.16.a.j.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:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 140297x^{4} - 1279200x^{3} + 3920349703x^{2} - 70310137200x - 19672158033999 $ x^6 - 140297x^4 - 1279200x^3 + 3920349703x^2 - 70310137200x - 19672158033999
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{13}\cdot 3^{4}\cdot 5^{7} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$121.792$ 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)

$f(q)$	$=$	$q-82.7921 q^{2} -2187.00 q^{3} -25913.5 q^{4} +181066. q^{6} +3.04747e6 q^{7} +4.85836e6 q^{8} +4.78297e6 q^{9} +O(q^{10})$	\(q-82.7921 q^{2} -2187.00 q^{3} -25913.5 q^{4} +181066. q^{6} +3.04747e6 q^{7} +4.85836e6 q^{8} +4.78297e6 q^{9} -1.93592e7 q^{11} +5.66728e7 q^{12} +2.26145e8 q^{13} -2.52306e8 q^{14} +4.46899e8 q^{16} +2.28466e9 q^{17} -3.95992e8 q^{18} +5.59201e9 q^{19} -6.66481e9 q^{21} +1.60279e9 q^{22} +3.00417e10 q^{23} -1.06252e10 q^{24} -1.87230e10 q^{26} -1.04604e10 q^{27} -7.89704e10 q^{28} +5.45473e10 q^{29} -1.04764e11 q^{31} -1.96198e11 q^{32} +4.23386e10 q^{33} -1.89152e11 q^{34} -1.23943e11 q^{36} -7.82646e11 q^{37} -4.62974e11 q^{38} -4.94578e11 q^{39} -1.21563e11 q^{41} +5.51793e11 q^{42} +1.45074e12 q^{43} +5.01664e11 q^{44} -2.48722e12 q^{46} +2.37631e12 q^{47} -9.77368e11 q^{48} +4.53949e12 q^{49} -4.99655e12 q^{51} -5.86019e12 q^{52} -1.41969e13 q^{53} +8.66034e11 q^{54} +1.48057e13 q^{56} -1.22297e13 q^{57} -4.51608e12 q^{58} -2.10800e12 q^{59} +2.79442e13 q^{61} +8.67366e12 q^{62} +1.45759e13 q^{63} +1.59970e12 q^{64} -3.50530e12 q^{66} -3.99208e13 q^{67} -5.92035e13 q^{68} -6.57013e13 q^{69} +9.07904e13 q^{71} +2.32374e13 q^{72} +1.03132e14 q^{73} +6.47969e13 q^{74} -1.44908e14 q^{76} -5.89965e13 q^{77} +4.09472e13 q^{78} -1.55872e14 q^{79} +2.28768e13 q^{81} +1.00645e13 q^{82} +2.18588e14 q^{83} +1.72708e14 q^{84} -1.20110e14 q^{86} -1.19295e14 q^{87} -9.40540e13 q^{88} +1.18767e14 q^{89} +6.89168e14 q^{91} -7.78486e14 q^{92} +2.29120e14 q^{93} -1.96740e14 q^{94} +4.29086e14 q^{96} -1.46125e14 q^{97} -3.75834e14 q^{98} -9.25945e13 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 234 q^{2} - 13122 q^{3} + 93112 q^{4} - 511758 q^{6} - 2590222 q^{7} + 14012388 q^{8} + 28697814 q^{9}+O(q^{10}) $ 6 * q + 234 * q^2 - 13122 * q^3 + 93112 * q^4 - 511758 * q^6 - 2590222 * q^7 + 14012388 * q^8 + 28697814 * q^9	\( 6 q + 234 q^{2} - 13122 q^{3} + 93112 q^{4} - 511758 q^{6} - 2590222 q^{7} + 14012388 q^{8} + 28697814 q^{9} + 107489124 q^{11} - 203635944 q^{12} - 109881686 q^{13} - 563984442 q^{14} + 3622829560 q^{16} + 3573042876 q^{17} + 1119214746 q^{18} - 1602340942 q^{19} + 5664815514 q^{21} + 4024661012 q^{22} - 6555818844 q^{23} - 30645092556 q^{24} - 25715894778 q^{26} - 62762119218 q^{27} - 270752117896 q^{28} + 126894468996 q^{29} + 151760841646 q^{31} + 385411085208 q^{32} - 235078714188 q^{33} + 1431919606684 q^{34} + 445351809528 q^{36} + 616109002068 q^{37} - 2822785016634 q^{38} + 240311247282 q^{39} + 1091281712616 q^{41} + 1233433974654 q^{42} - 2444971199030 q^{43} + 1413344578176 q^{44} - 5480862370044 q^{46} - 8369143269660 q^{47} - 7923128247720 q^{48} + 19523846053580 q^{49} - 7814244769812 q^{51} - 10261294060344 q^{52} - 16571417665824 q^{53} - 2447722649502 q^{54} - 75252275829540 q^{56} + 3504319640154 q^{57} - 3994751501708 q^{58} + 8796604455252 q^{59} - 6959665405750 q^{61} + 52277129313066 q^{62} - 12388951529118 q^{63} + 50304241850208 q^{64} - 8801933633244 q^{66} - 53487461742094 q^{67} + 307147088145312 q^{68} + 14337575811828 q^{69} + 104634162717912 q^{71} + 67020817419972 q^{72} + 177000981923236 q^{73} - 45005277967812 q^{74} + 76188538526328 q^{76} + 117850730172876 q^{77} + 56240661879486 q^{78} + 185514024366160 q^{79} + 137260754729766 q^{81} + 654376907588896 q^{82} + 435827733256908 q^{83} + 592134881838552 q^{84} + 15\!\cdots\!14 q^{86}+ \cdots + 514117147929156 q^{99}+O(q^{100}) \) 6 * q + 234 * q^2 - 13122 * q^3 + 93112 * q^4 - 511758 * q^6 - 2590222 * q^7 + 14012388 * q^8 + 28697814 * q^9 + 107489124 * q^11 - 203635944 * q^12 - 109881686 * q^13 - 563984442 * q^14 + 3622829560 * q^16 + 3573042876 * q^17 + 1119214746 * q^18 - 1602340942 * q^19 + 5664815514 * q^21 + 4024661012 * q^22 - 6555818844 * q^23 - 30645092556 * q^24 - 25715894778 * q^26 - 62762119218 * q^27 - 270752117896 * q^28 + 126894468996 * q^29 + 151760841646 * q^31 + 385411085208 * q^32 - 235078714188 * q^33 + 1431919606684 * q^34 + 445351809528 * q^36 + 616109002068 * q^37 - 2822785016634 * q^38 + 240311247282 * q^39 + 1091281712616 * q^41 + 1233433974654 * q^42 - 2444971199030 * q^43 + 1413344578176 * q^44 - 5480862370044 * q^46 - 8369143269660 * q^47 - 7923128247720 * q^48 + 19523846053580 * q^49 - 7814244769812 * q^51 - 10261294060344 * q^52 - 16571417665824 * q^53 - 2447722649502 * q^54 - 75252275829540 * q^56 + 3504319640154 * q^57 - 3994751501708 * q^58 + 8796604455252 * q^59 - 6959665405750 * q^61 + 52277129313066 * q^62 - 12388951529118 * q^63 + 50304241850208 * q^64 - 8801933633244 * q^66 - 53487461742094 * q^67 + 307147088145312 * q^68 + 14337575811828 * q^69 + 104634162717912 * q^71 + 67020817419972 * q^72 + 177000981923236 * q^73 - 45005277967812 * q^74 + 76188538526328 * q^76 + 117850730172876 * q^77 + 56240661879486 * q^78 + 185514024366160 * q^79 + 137260754729766 * q^81 + 654376907588896 * q^82 + 435827733256908 * q^83 + 592134881838552 * q^84 + 1570057453210014 * q^86 - 277518203694252 * q^87 + 174719041441128 * q^88 + 431517021638688 * q^89 + 466827529337246 * q^91 + 50075827036128 * q^92 - 331900960679802 * q^93 + 1404749684955772 * q^94 - 842894043349896 * q^96 - 1013615616184854 * q^97 + 5117833640095236 * q^98 + 514117147929156 * 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 $

$2$	−82.7921	−0.457366	−0.228683	−	0.973501i	$-0.573442\pi$
$2$	−82.7921	−0.457366	−0.228683	+	0.973501i	$0.573442\pi$
$3$	−2187.00	−0.577350
$3$	−2187.00	−0.577350
$4$	−25913.5	−0.790816
$5$	0	0
$5$	0	0
$6$	181066.	0.264060
$7$	3.04747e6	1.39863	0.699316	−	0.714812i	$-0.253488\pi$
$7$	3.04747e6	1.39863	0.699316	+	0.714812i	$0.253488\pi$
$8$	4.85836e6	0.819058
$9$	4.78297e6	0.333333
$10$	0	0
$11$	−1.93592e7	−0.299531	−0.149766	−	0.988722i	$-0.547852\pi$
$11$	−1.93592e7	−0.299531	−0.149766	+	0.988722i	$0.547852\pi$
$12$	5.66728e7	0.456578
$13$	2.26145e8	0.999565	0.499783	−	0.866151i	$-0.333413\pi$
$13$	2.26145e8	0.999565	0.499783	+	0.866151i	$0.333413\pi$
$14$	−2.52306e8	−0.639687
$15$	0	0
$16$	4.46899e8	0.416207
$17$	2.28466e9	1.35038	0.675188	−	0.737645i	$-0.264062\pi$
$17$	2.28466e9	1.35038	0.675188	+	0.737645i	$0.264062\pi$
$18$	−3.95992e8	−0.152455
$19$	5.59201e9	1.43521	0.717606	−	0.696449i	$-0.245238\pi$
$19$	5.59201e9	1.43521	0.717606	+	0.696449i	$0.245238\pi$
$20$	0	0
$21$	−6.66481e9	−0.807501
$22$	1.60279e9	0.136995
$23$	3.00417e10	1.83978	0.919891	−	0.392175i	$-0.128277\pi$
$23$	3.00417e10	1.83978	0.919891	+	0.392175i	$0.128277\pi$
$24$	−1.06252e10	−0.472884
$25$	0	0
$26$	−1.87230e10	−0.457167
$27$	−1.04604e10	−0.192450
$28$	−7.89704e10	−1.10606
$29$	5.45473e10	0.587203	0.293601	−	0.955928i	$-0.405146\pi$
$29$	5.45473e10	0.587203	0.293601	+	0.955928i	$0.405146\pi$
$30$	0	0
$31$	−1.04764e11	−0.683913	−0.341956	−	0.939716i	$-0.611090\pi$
$31$	−1.04764e11	−0.683913	−0.341956	+	0.939716i	$0.611090\pi$
$32$	−1.96198e11	−1.00942
$33$	4.23386e10	0.172935
$34$	−1.89152e11	−0.617616
$35$	0	0
$36$	−1.23943e11	−0.263605
$37$	−7.82646e11	−1.35535	−0.677677	−	0.735360i	$-0.737013\pi$
$37$	−7.82646e11	−1.35535	−0.677677	+	0.735360i	$0.737013\pi$
$38$	−4.62974e11	−0.656417
$39$	−4.94578e11	−0.577099
$40$	0	0
$41$	−1.21563e11	−0.0974816	−0.0487408	−	0.998811i	$-0.515521\pi$
$41$	−1.21563e11	−0.0974816	−0.0487408	+	0.998811i	$0.515521\pi$
$42$	5.51793e11	0.369323
$43$	1.45074e12	0.813911	0.406955	−	0.913448i	$-0.366590\pi$
$43$	1.45074e12	0.813911	0.406955	+	0.913448i	$0.366590\pi$
$44$	5.01664e11	0.236874
$45$	0	0
$46$	−2.48722e12	−0.841453
$47$	2.37631e12	0.684179	0.342090	−	0.939667i	$-0.388865\pi$
$47$	2.37631e12	0.684179	0.342090	+	0.939667i	$0.388865\pi$
$48$	−9.77368e11	−0.240297
$49$	4.53949e12	0.956174
$50$	0	0
$51$	−4.99655e12	−0.779640
$52$	−5.86019e12	−0.790473
$53$	−1.41969e13	−1.66006	−0.830032	−	0.557716i	$-0.811678\pi$
$53$	−1.41969e13	−1.66006	−0.830032	+	0.557716i	$0.811678\pi$
$54$	8.66034e11	0.0880201
$55$	0	0
$56$	1.48057e13	1.14556
$57$	−1.22297e13	−0.828620
$58$	−4.51608e12	−0.268566
$59$	−2.10800e12	−0.110276	−0.0551378	−	0.998479i	$-0.517560\pi$
$59$	−2.10800e12	−0.110276	−0.0551378	+	0.998479i	$0.517560\pi$
$60$	0	0
$61$	2.79442e13	1.13846	0.569229	−	0.822179i	$-0.307242\pi$
$61$	2.79442e13	1.13846	0.569229	+	0.822179i	$0.307242\pi$
$62$	8.67366e12	0.312798
$63$	1.45759e13	0.466211
$64$	1.59970e12	0.0454661
$65$	0	0
$66$	−3.50530e12	−0.0790944
$67$	−3.99208e13	−0.804707	−0.402353	−	0.915484i	$-0.631808\pi$
$67$	−3.99208e13	−0.804707	−0.402353	+	0.915484i	$0.631808\pi$
$68$	−5.92035e13	−1.06790
$69$	−6.57013e13	−1.06220
$70$	0	0
$71$	9.07904e13	1.18468	0.592342	−	0.805687i	$-0.298203\pi$
$71$	9.07904e13	1.18468	0.592342	+	0.805687i	$0.298203\pi$
$72$	2.32374e13	0.273019
$73$	1.03132e14	1.09263	0.546315	−	0.837580i	$-0.316030\pi$
$73$	1.03132e14	1.09263	0.546315	+	0.837580i	$0.316030\pi$
$74$	6.47969e13	0.619892
$75$	0	0
$76$	−1.44908e14	−1.13499
$77$	−5.89965e13	−0.418934
$78$	4.09472e13	0.263946
$79$	−1.55872e14	−0.913198	−0.456599	−	0.889673i	$-0.650933\pi$
$79$	−1.55872e14	−0.913198	−0.456599	+	0.889673i	$0.650933\pi$
$80$	0	0
$81$	2.28768e13	0.111111
$82$	1.00645e13	0.0445848
$83$	2.18588e14	0.884179	0.442089	−	0.896971i	$-0.354237\pi$
$83$	2.18588e14	0.884179	0.442089	+	0.896971i	$0.354237\pi$
$84$	1.72708e14	0.638585
$85$	0	0
$86$	−1.20110e14	−0.372255
$87$	−1.19295e14	−0.339022
$88$	−9.40540e13	−0.245334
$89$	1.18767e14	0.284623	0.142311	−	0.989822i	$-0.454547\pi$
$89$	1.18767e14	0.284623	0.142311	+	0.989822i	$0.454547\pi$
$90$	0	0
$91$	6.89168e14	1.39802
$92$	−7.78486e14	−1.45493
$93$	2.29120e14	0.394857
$94$	−1.96740e14	−0.312920
$95$	0	0
$96$	4.29086e14	0.582787
$97$	−1.46125e14	−0.183627	−0.0918134	−	0.995776i	$-0.529266\pi$
$97$	−1.46125e14	−0.183627	−0.0918134	+	0.995776i	$0.529266\pi$
$98$	−3.75834e14	−0.437321
$99$	−9.25945e13	−0.0998438
$100$	0	0
$101$	−1.39043e15	−1.29044	−0.645219	−	0.763997i	$-0.723234\pi$
$101$	−1.39043e15	−1.29044	−0.645219	+	0.763997i	$0.723234\pi$
$102$	4.13675e14	0.356581
$103$	9.91197e14	0.794110	0.397055	−	0.917795i	$-0.370032\pi$
$103$	9.91197e14	0.794110	0.397055	+	0.917795i	$0.370032\pi$
$104$	1.09869e15	0.818702
$105$	0	0
$106$	1.17539e15	0.759256
$107$	−1.81177e14	−0.109075	−0.0545374	−	0.998512i	$-0.517368\pi$
$107$	−1.81177e14	−0.109075	−0.0545374	+	0.998512i	$0.517368\pi$
$108$	2.71064e14	0.152193
$109$	3.54867e15	1.85938	0.929688	−	0.368348i	$-0.120077\pi$
$109$	3.54867e15	1.85938	0.929688	+	0.368348i	$0.120077\pi$
$110$	0	0
$111$	1.71165e15	0.782513
$112$	1.36191e15	0.582121
$113$	−1.28742e15	−0.514790	−0.257395	−	0.966306i	$-0.582864\pi$
$113$	−1.28742e15	−0.514790	−0.257395	+	0.966306i	$0.582864\pi$
$114$	1.01253e15	0.378983
$115$	0	0
$116$	−1.41351e15	−0.464369
$117$	1.08164e15	0.333188
$118$	1.74525e14	0.0504363
$119$	6.96243e15	1.88868
$120$	0	0
$121$	−3.80247e15	−0.910281
$122$	−2.31356e15	−0.520692
$123$	2.65858e14	0.0562810
$124$	2.71481e15	0.540849
$125$	0	0
$126$	−1.20677e15	−0.213229
$127$	−7.88543e15	−1.31310	−0.656550	−	0.754283i	$-0.727985\pi$
$127$	−7.88543e15	−1.31310	−0.656550	+	0.754283i	$0.727985\pi$
$128$	6.29659e15	0.988623
$129$	−3.17277e15	−0.469912
$130$	0	0
$131$	1.05951e16	1.39820	0.699099	−	0.715025i	$-0.253585\pi$
$131$	1.05951e16	1.39820	0.699099	+	0.715025i	$0.253585\pi$
$132$	−1.09714e15	−0.136759
$133$	1.70415e16	2.00734
$134$	3.30512e15	0.368046
$135$	0	0
$136$	1.10997e16	1.10604
$137$	−8.78465e15	−0.828553	−0.414276	−	0.910151i	$-0.635965\pi$
$137$	−8.78465e15	−0.828553	−0.414276	+	0.910151i	$0.635965\pi$
$138$	5.43955e15	0.485813
$139$	1.03867e16	0.878751	0.439376	−	0.898303i	$-0.355200\pi$
$139$	1.03867e16	0.878751	0.439376	+	0.898303i	$0.355200\pi$
$140$	0	0
$141$	−5.19700e15	−0.395011
$142$	−7.51673e15	−0.541834
$143$	−4.37798e15	−0.299401
$144$	2.13750e15	0.138736
$145$	0	0
$146$	−8.53853e15	−0.499732
$147$	−9.92787e15	−0.552047
$148$	2.02811e16	1.07184
$149$	3.09626e16	1.55575	0.777875	−	0.628419i	$-0.216298\pi$
$149$	3.09626e16	1.55575	0.777875	+	0.628419i	$0.216298\pi$
$150$	0	0
$151$	−8.22698e15	−0.374036	−0.187018	−	0.982356i	$-0.559882\pi$
$151$	−8.22698e15	−0.374036	−0.187018	+	0.982356i	$0.559882\pi$
$152$	2.71680e16	1.17552
$153$	1.09275e16	0.450126
$154$	4.88445e15	0.191606
$155$	0	0
$156$	1.28162e16	0.456380
$157$	−2.41505e16	−0.819746	−0.409873	−	0.912143i	$-0.634427\pi$
$157$	−2.41505e16	−0.819746	−0.409873	+	0.912143i	$0.634427\pi$
$158$	1.29050e16	0.417666
$159$	3.10486e16	0.958438
$160$	0	0
$161$	9.15512e16	2.57318
$162$	−1.89402e15	−0.0508184
$163$	−3.11554e16	−0.798227	−0.399114	−	0.916901i	$-0.630682\pi$
$163$	−3.11554e16	−0.798227	−0.399114	+	0.916901i	$0.630682\pi$
$164$	3.15012e15	0.0770901
$165$	0	0
$166$	−1.80973e16	−0.404393
$167$	−6.47804e16	−1.38379	−0.691895	−	0.721998i	$-0.743224\pi$
$167$	−6.47804e16	−1.38379	−0.691895	+	0.721998i	$0.743224\pi$
$168$	−3.23801e16	−0.661391
$169$	−4.44796e13	−0.000868981 0
$170$	0	0
$171$	2.67464e16	0.478404
$172$	−3.75938e16	−0.643654
$173$	1.00247e17	1.64333	0.821665	−	0.569971i	$-0.193046\pi$
$173$	1.00247e17	1.64333	0.821665	+	0.569971i	$0.193046\pi$
$174$	9.87667e15	0.155057
$175$	0	0
$176$	−8.65160e15	−0.124667
$177$	4.61019e15	0.0636677
$178$	−9.83294e15	−0.130177
$179$	−5.90813e15	−0.0749984	−0.0374992	−	0.999297i	$-0.511939\pi$
$179$	−5.90813e15	−0.0749984	−0.0374992	+	0.999297i	$0.511939\pi$
$180$	0	0
$181$	5.07791e16	0.593056	0.296528	−	0.955024i	$-0.404171\pi$
$181$	5.07791e16	0.593056	0.296528	+	0.955024i	$0.404171\pi$
$182$	−5.70577e16	−0.639409
$183$	−6.11139e16	−0.657290
$184$	1.45954e17	1.50689
$185$	0	0
$186$	−1.89693e16	−0.180594
$187$	−4.42292e16	−0.404480
$188$	−6.15786e16	−0.541060
$189$	−3.18776e16	−0.269167
$190$	0	0
$191$	−1.48427e17	−1.15814	−0.579072	−	0.815276i	$-0.696585\pi$
$191$	−1.48427e17	−1.15814	−0.579072	+	0.815276i	$0.696585\pi$
$192$	−3.49854e15	−0.0262499
$193$	2.01057e16	0.145091	0.0725454	−	0.997365i	$-0.476888\pi$
$193$	2.01057e16	0.145091	0.0725454	+	0.997365i	$0.476888\pi$
$194$	1.20980e16	0.0839846
$195$	0	0
$196$	−1.17634e17	−0.756158
$197$	9.02210e16	0.558227	0.279114	−	0.960258i	$-0.409959\pi$
$197$	9.02210e16	0.558227	0.279114	+	0.960258i	$0.409959\pi$
$198$	7.66609e15	0.0456651
$199$	−2.12389e17	−1.21824	−0.609121	−	0.793078i	$-0.708477\pi$
$199$	−2.12389e17	−1.21824	−0.609121	+	0.793078i	$0.708477\pi$
$200$	0	0
$201$	8.73067e16	0.464598
$202$	1.15116e17	0.590203
$203$	1.66231e17	0.821281
$204$	1.29478e17	0.616552
$205$	0	0
$206$	−8.20632e16	−0.363199
$207$	1.43689e17	0.613260
$208$	1.01064e17	0.416026
$209$	−1.08257e17	−0.429891
$210$	0	0
$211$	−2.73647e17	−1.01175	−0.505873	−	0.862608i	$-0.668830\pi$
$211$	−2.73647e17	−1.01175	−0.505873	+	0.862608i	$0.668830\pi$
$212$	3.67891e17	1.31281
$213$	−1.98559e17	−0.683977
$214$	1.50000e16	0.0498871
$215$	0	0
$216$	−5.08202e16	−0.157628
$217$	−3.19266e17	−0.956543
$218$	−2.93802e17	−0.850415
$219$	−2.25550e17	−0.630830
$220$	0	0
$221$	5.16664e17	1.34979
$222$	−1.41711e17	−0.357895
$223$	2.63653e17	0.643794	0.321897	−	0.946775i	$-0.395680\pi$
$223$	2.63653e17	0.643794	0.321897	+	0.946775i	$0.395680\pi$
$224$	−5.97908e17	−1.41180
$225$	0	0
$226$	1.06588e17	0.235448
$227$	6.17091e17	1.31873	0.659364	−	0.751824i	$-0.270826\pi$
$227$	6.17091e17	1.31873	0.659364	+	0.751824i	$0.270826\pi$
$228$	3.16915e17	0.655287
$229$	−5.92458e17	−1.18547	−0.592737	−	0.805396i	$-0.701952\pi$
$229$	−5.92458e17	−1.18547	−0.592737	+	0.805396i	$0.701952\pi$
$230$	0	0
$231$	1.29025e17	0.241872
$232$	2.65010e17	0.480953
$233$	−8.06272e17	−1.41681	−0.708406	−	0.705805i	$-0.750585\pi$
$233$	−8.06272e17	−1.41681	−0.708406	+	0.705805i	$0.750585\pi$
$234$	−8.95515e16	−0.152389
$235$	0	0
$236$	5.46255e16	0.0872078
$237$	3.40892e17	0.527235
$238$	−5.76434e17	−0.863818
$239$	1.00777e18	1.46345	0.731727	−	0.681598i	$-0.238715\pi$
$239$	1.00777e18	1.46345	0.731727	+	0.681598i	$0.238715\pi$
$240$	0	0
$241$	−8.04685e17	−1.09774	−0.548868	−	0.835909i	$-0.684941\pi$
$241$	−8.04685e17	−1.09774	−0.548868	+	0.835909i	$0.684941\pi$
$242$	3.14814e17	0.416332
$243$	−5.00315e16	−0.0641500
$244$	−7.24130e17	−0.900312
$245$	0	0
$246$	−2.20110e16	−0.0257410
$247$	1.26460e18	1.43459
$248$	−5.08983e17	−0.560164
$249$	−4.78051e17	−0.510481
$250$	0	0
$251$	−1.04007e18	−1.04594	−0.522971	−	0.852350i	$-0.675177\pi$
$251$	−1.04007e18	−1.04594	−0.522971	+	0.852350i	$0.675177\pi$
$252$	−3.77713e17	−0.368687
$253$	−5.81584e17	−0.551072
$254$	6.52851e17	0.600567
$255$	0	0
$256$	−5.73727e17	−0.497628
$257$	−6.53643e17	−0.550607	−0.275304	−	0.961357i	$-0.588778\pi$
$257$	−6.53643e17	−0.550607	−0.275304	+	0.961357i	$0.588778\pi$
$258$	2.62680e17	0.214922
$259$	−2.38509e18	−1.89564
$260$	0	0
$261$	2.60898e17	0.195734
$262$	−8.77187e17	−0.639488
$263$	−3.60136e17	−0.255152	−0.127576	−	0.991829i	$-0.540720\pi$
$263$	−3.60136e17	−0.255152	−0.127576	+	0.991829i	$0.540720\pi$
$264$	2.05696e17	0.141643
$265$	0	0
$266$	−1.41090e18	−0.918087
$267$	−2.59743e17	−0.164327
$268$	1.03449e18	0.636375
$269$	−2.16614e18	−1.29582	−0.647910	−	0.761717i	$-0.724357\pi$
$269$	−2.16614e18	−1.29582	−0.647910	+	0.761717i	$0.724357\pi$
$270$	0	0
$271$	9.15235e17	0.517921	0.258960	−	0.965888i	$-0.416620\pi$
$271$	9.15235e17	0.517921	0.258960	+	0.965888i	$0.416620\pi$
$272$	1.02101e18	0.562036
$273$	−1.50721e18	−0.807150
$274$	7.27299e17	0.378952
$275$	0	0
$276$	1.70255e18	0.840004
$277$	−2.44746e18	−1.17522	−0.587609	−	0.809145i	$-0.699930\pi$
$277$	−2.44746e18	−1.17522	−0.587609	+	0.809145i	$0.699930\pi$
$278$	−8.59935e17	−0.401911
$279$	−5.01084e17	−0.227971
$280$	0	0
$281$	2.58413e17	0.111434	0.0557169	−	0.998447i	$-0.482256\pi$
$281$	2.58413e17	0.111434	0.0557169	+	0.998447i	$0.482256\pi$
$282$	4.30270e17	0.180665
$283$	−6.58483e17	−0.269244	−0.134622	−	0.990897i	$-0.542982\pi$
$283$	−6.58483e17	−0.269244	−0.134622	+	0.990897i	$0.542982\pi$
$284$	−2.35270e18	−0.936867
$285$	0	0
$286$	3.62462e17	0.136936
$287$	−3.70459e17	−0.136341
$288$	−9.38411e17	−0.336472
$289$	2.35725e18	0.823517
$290$	0	0
$291$	3.19575e17	0.106017
$292$	−2.67251e18	−0.864070
$293$	4.16212e18	1.31162	0.655810	−	0.754926i	$-0.272327\pi$
$293$	4.16212e18	1.31162	0.655810	+	0.754926i	$0.272327\pi$
$294$	8.21949e17	0.252488
$295$	0	0
$296$	−3.80238e18	−1.11011
$297$	2.02504e17	0.0576448
$298$	−2.56345e18	−0.711547
$299$	6.79378e18	1.83898
$300$	0	0
$301$	4.42109e18	1.13836
$302$	6.81129e17	0.171071
$303$	3.04086e18	0.745035
$304$	2.49906e18	0.597345
$305$	0	0
$306$	−9.04707e17	−0.205872
$307$	7.91881e18	1.75842	0.879209	−	0.476436i	$-0.158072\pi$
$307$	7.91881e18	1.75842	0.879209	+	0.476436i	$0.158072\pi$
$308$	1.52880e18	0.331300
$309$	−2.16775e18	−0.458480
$310$	0	0
$311$	3.85377e18	0.776574	0.388287	−	0.921539i	$-0.373067\pi$
$311$	3.85377e18	0.776574	0.388287	+	0.921539i	$0.373067\pi$
$312$	−2.40284e18	−0.472678
$313$	4.79993e18	0.921833	0.460917	−	0.887443i	$-0.347521\pi$
$313$	4.79993e18	0.921833	0.460917	+	0.887443i	$0.347521\pi$
$314$	1.99947e18	0.374924
$315$	0	0
$316$	4.03918e18	0.722172
$317$	9.78013e18	1.70765	0.853827	−	0.520556i	$-0.174275\pi$
$317$	9.78013e18	1.70765	0.853827	+	0.520556i	$0.174275\pi$
$318$	−2.57058e18	−0.438357
$319$	−1.05599e18	−0.175886
$320$	0	0
$321$	3.96234e17	0.0629744
$322$	−7.57971e18	−1.17688
$323$	1.27759e19	1.93808
$324$	−5.92817e17	−0.0878685
$325$	0	0
$326$	2.57942e18	0.365082
$327$	−7.76095e18	−1.07351
$328$	−5.90597e17	−0.0798431
$329$	7.24174e18	0.956916
$330$	0	0
$331$	−2.85066e18	−0.359944	−0.179972	−	0.983672i	$-0.557601\pi$
$331$	−2.85066e18	−0.359944	−0.179972	+	0.983672i	$0.557601\pi$
$332$	−5.66437e18	−0.699223
$333$	−3.74337e18	−0.451784
$334$	5.36331e18	0.632899
$335$	0	0
$336$	−2.97850e18	−0.336088
$337$	−1.44591e19	−1.59557	−0.797786	−	0.602940i	$-0.793996\pi$
$337$	−1.44591e19	−1.59557	−0.797786	+	0.602940i	$0.793996\pi$
$338$	3.68256e15	0.000397442 0
$339$	2.81558e18	0.297214
$340$	0	0
$341$	2.02815e18	0.204853
$342$	−2.21439e18	−0.218806
$343$	−6.34081e17	−0.0612970
$344$	7.04823e18	0.666640
$345$	0	0
$346$	−8.29964e18	−0.751603
$347$	1.63187e19	1.44616	0.723078	−	0.690766i	$-0.242727\pi$
$347$	1.63187e19	1.44616	0.723078	+	0.690766i	$0.242727\pi$
$348$	3.09134e18	0.268104
$349$	9.81768e18	0.833332	0.416666	−	0.909060i	$-0.363198\pi$
$349$	9.81768e18	0.833332	0.416666	+	0.909060i	$0.363198\pi$
$350$	0	0
$351$	−2.36555e18	−0.192366
$352$	3.79825e18	0.302352
$353$	4.18635e18	0.326231	0.163116	−	0.986607i	$-0.447846\pi$
$353$	4.18635e18	0.326231	0.163116	+	0.986607i	$0.447846\pi$
$354$	−3.81687e17	−0.0291194
$355$	0	0
$356$	−3.07766e18	−0.225084
$357$	−1.52268e19	−1.09043
$358$	4.89146e17	0.0343017
$359$	2.38174e19	1.63563	0.817816	−	0.575480i	$-0.195185\pi$
$359$	2.38174e19	1.63563	0.817816	+	0.575480i	$0.195185\pi$
$360$	0	0
$361$	1.60895e19	1.05983
$362$	−4.20411e18	−0.271244
$363$	8.31600e18	0.525551
$364$	−1.78587e19	−1.10558
$365$	0	0
$366$	5.05975e18	0.300622
$367$	−7.36481e18	−0.428712	−0.214356	−	0.976756i	$-0.568765\pi$
$367$	−7.36481e18	−0.428712	−0.214356	+	0.976756i	$0.568765\pi$
$368$	1.34256e19	0.765730
$369$	−5.81432e17	−0.0324939
$370$	0	0
$371$	−4.32646e19	−2.32182
$372$	−5.93728e18	−0.312260
$373$	−2.20406e19	−1.13607	−0.568037	−	0.823003i	$-0.692297\pi$
$373$	−2.20406e19	−1.13607	−0.568037	+	0.823003i	$0.692297\pi$
$374$	3.66183e18	0.184995
$375$	0	0
$376$	1.15450e19	0.560383
$377$	1.23356e19	0.586947
$378$	2.63921e18	0.123108
$379$	1.90151e19	0.869571	0.434785	−	0.900534i	$-0.356824\pi$
$379$	1.90151e19	0.869571	0.434785	+	0.900534i	$0.356824\pi$
$380$	0	0
$381$	1.72454e19	0.758118
$382$	1.22886e19	0.529696
$383$	−1.10208e19	−0.465826	−0.232913	−	0.972498i	$-0.574826\pi$
$383$	−1.10208e19	−0.465826	−0.232913	+	0.972498i	$0.574826\pi$
$384$	−1.37706e19	−0.570782
$385$	0	0
$386$	−1.66459e18	−0.0663596
$387$	6.93885e18	0.271304
$388$	3.78660e18	0.145215
$389$	−2.93179e19	−1.10283	−0.551417	−	0.834230i	$-0.685913\pi$
$389$	−2.93179e19	−1.10283	−0.551417	+	0.834230i	$0.685913\pi$
$390$	0	0
$391$	6.86352e19	2.48440
$392$	2.20545e19	0.783162
$393$	−2.31714e19	−0.807250
$394$	−7.46958e18	−0.255314
$395$	0	0
$396$	2.39944e18	0.0789581
$397$	−3.45524e19	−1.11571	−0.557853	−	0.829940i	$-0.688375\pi$
$397$	−3.45524e19	−1.11571	−0.557853	+	0.829940i	$0.688375\pi$
$398$	1.75841e19	0.557182
$399$	−3.72697e19	−1.15894
$400$	0	0
$401$	−1.54511e19	−0.462783	−0.231392	−	0.972861i	$-0.574328\pi$
$401$	−1.54511e19	−0.462783	−0.231392	+	0.972861i	$0.574328\pi$
$402$	−7.22830e18	−0.212491
$403$	−2.36919e19	−0.683616
$404$	3.60307e19	1.02050
$405$	0	0
$406$	−1.37626e19	−0.375626
$407$	1.51514e19	0.405971
$408$	−2.42751e19	−0.638571
$409$	−6.33606e19	−1.63642	−0.818209	−	0.574921i	$-0.805033\pi$
$409$	−6.33606e19	−1.63642	−0.818209	+	0.574921i	$0.805033\pi$
$410$	0	0
$411$	1.92120e19	0.478365
$412$	−2.56853e19	−0.627995
$413$	−6.42405e18	−0.154235
$414$	−1.18963e19	−0.280484
$415$	0	0
$416$	−4.43692e19	−1.00898
$417$	−2.27157e19	−0.507347
$418$	8.96282e18	0.196618
$419$	−8.41310e18	−0.181280	−0.0906402	−	0.995884i	$-0.528891\pi$
$419$	−8.41310e18	−0.181280	−0.0906402	+	0.995884i	$0.528891\pi$
$420$	0	0
$421$	1.42741e19	0.296779	0.148389	−	0.988929i	$-0.452591\pi$
$421$	1.42741e19	0.296779	0.148389	+	0.988929i	$0.452591\pi$
$422$	2.26558e19	0.462738
$423$	1.13658e19	0.228060
$424$	−6.89737e19	−1.35969
$425$	0	0
$426$	1.64391e19	0.312828
$427$	8.51589e19	1.59229
$428$	4.69492e18	0.0862582
$429$	9.57464e18	0.172859
$430$	0	0
$431$	−1.01184e20	−1.76414	−0.882071	−	0.471117i	$-0.843851\pi$
$431$	−1.01184e20	−1.76414	−0.882071	+	0.471117i	$0.843851\pi$
$432$	−4.67472e18	−0.0800991
$433$	8.97859e18	0.151199	0.0755995	−	0.997138i	$-0.475913\pi$
$433$	8.97859e18	0.151199	0.0755995	+	0.997138i	$0.475913\pi$
$434$	2.64327e19	0.437490
$435$	0	0
$436$	−9.19584e19	−1.47042
$437$	1.67994e20	2.64048
$438$	1.86738e19	0.288520
$439$	1.26078e19	0.191494	0.0957470	−	0.995406i	$-0.469476\pi$
$439$	1.26078e19	0.191494	0.0957470	+	0.995406i	$0.469476\pi$
$440$	0	0
$441$	2.17123e19	0.318725
$442$	−4.27757e19	−0.617348
$443$	−8.00010e19	−1.13519	−0.567594	−	0.823309i	$-0.692126\pi$
$443$	−8.00010e19	−1.13519	−0.567594	+	0.823309i	$0.692126\pi$
$444$	−4.43547e19	−0.618824
$445$	0	0
$446$	−2.18284e19	−0.294449
$447$	−6.77151e19	−0.898213
$448$	4.87503e18	0.0635904
$449$	−1.82801e19	−0.234493	−0.117247	−	0.993103i	$-0.537407\pi$
$449$	−1.82801e19	−0.234493	−0.117247	+	0.993103i	$0.537407\pi$
$450$	0	0
$451$	2.35336e18	0.0291988
$452$	3.33614e19	0.407105
$453$	1.79924e19	0.215950
$454$	−5.10903e19	−0.603141
$455$	0	0
$456$	−5.94165e19	−0.678688
$457$	−5.86883e19	−0.659448	−0.329724	−	0.944077i	$-0.606956\pi$
$457$	−5.86883e19	−0.659448	−0.329724	+	0.944077i	$0.606956\pi$
$458$	4.90509e19	0.542195
$459$	−2.38984e19	−0.259880
$460$	0	0
$461$	1.17436e20	1.23607	0.618035	−	0.786150i	$-0.287929\pi$
$461$	1.17436e20	1.23607	0.618035	+	0.786150i	$0.287929\pi$
$462$	−1.06823e19	−0.110624
$463$	−1.68901e20	−1.72098	−0.860489	−	0.509469i	$-0.829842\pi$
$463$	−1.68901e20	−1.72098	−0.860489	+	0.509469i	$0.829842\pi$
$464$	2.43771e19	0.244398
$465$	0	0
$466$	6.67529e19	0.648001
$467$	1.71901e20	1.64210	0.821052	−	0.570853i	$-0.193388\pi$
$467$	1.71901e20	1.64210	0.821052	+	0.570853i	$0.193388\pi$
$468$	−2.80291e19	−0.263491
$469$	−1.21657e20	−1.12549
$470$	0	0
$471$	5.28172e19	0.473280
$472$	−1.02414e19	−0.0903222
$473$	−2.80852e19	−0.243792
$474$	−2.82232e19	−0.241139
$475$	0	0
$476$	−1.80421e20	−1.49360
$477$	−6.79033e19	−0.553354
$478$	−8.34358e19	−0.669334
$479$	2.18113e20	1.72252	0.861260	−	0.508164i	$-0.169676\pi$
$479$	2.18113e20	1.72252	0.861260	+	0.508164i	$0.169676\pi$
$480$	0	0
$481$	−1.76991e20	−1.35476
$482$	6.66216e19	0.502067
$483$	−2.00222e20	−1.48563
$484$	9.85352e19	0.719865
$485$	0	0
$486$	4.14222e18	0.0293400
$487$	−1.98900e20	−1.38729	−0.693646	−	0.720316i	$-0.743997\pi$
$487$	−1.98900e20	−1.38729	−0.693646	+	0.720316i	$0.743997\pi$
$488$	1.35763e20	0.932464
$489$	6.81369e19	0.460857
$490$	0	0
$491$	1.96421e20	1.28848	0.644239	−	0.764824i	$-0.277174\pi$
$491$	1.96421e20	1.28848	0.644239	+	0.764824i	$0.277174\pi$
$492$	−6.88931e18	−0.0445080
$493$	1.24622e20	0.792945
$494$	−1.04699e20	−0.656132
$495$	0	0
$496$	−4.68191e19	−0.284649
$497$	2.76681e20	1.65694
$498$	3.95789e19	0.233477
$499$	−1.37993e20	−0.801867	−0.400934	−	0.916107i	$-0.631314\pi$
$499$	−1.37993e20	−0.801867	−0.400934	+	0.916107i	$0.631314\pi$
$500$	0	0
$501$	1.41675e20	0.798932
$502$	8.61092e19	0.478379
$503$	1.97196e20	1.07929	0.539645	−	0.841893i	$-0.318559\pi$
$503$	1.97196e20	1.07929	0.539645	+	0.841893i	$0.318559\pi$
$504$	7.08152e19	0.381854
$505$	0	0
$506$	4.81505e19	0.252042
$507$	9.72769e16	0.000501707 0
$508$	2.04339e20	1.03842
$509$	3.50287e20	1.75405	0.877023	−	0.480449i	$-0.159526\pi$
$509$	3.50287e20	1.75405	0.877023	+	0.480449i	$0.159526\pi$
$510$	0	0
$511$	3.14292e20	1.52819
$512$	−1.58827e20	−0.761024
$513$	−5.84944e19	−0.276207
$514$	5.41165e19	0.251829
$515$	0	0
$516$	8.22176e19	0.371614
$517$	−4.60035e19	−0.204933
$518$	1.97466e20	0.867002
$519$	−2.19240e20	−0.948777
$520$	0	0
$521$	−3.57043e20	−1.50120	−0.750598	−	0.660759i	$-0.770235\pi$
$521$	−3.57043e20	−1.50120	−0.750598	+	0.660759i	$0.770235\pi$
$522$	−2.16003e19	−0.0895222
$523$	4.32551e19	0.176716	0.0883578	−	0.996089i	$-0.471838\pi$
$523$	4.32551e19	0.176716	0.0883578	+	0.996089i	$0.471838\pi$
$524$	−2.74555e20	−1.10572
$525$	0	0
$526$	2.98164e19	0.116698
$527$	−2.39351e20	−0.923540
$528$	1.89211e19	0.0719766
$529$	6.35871e20	2.38480
$530$	0	0
$531$	−1.00825e19	−0.0367585
$532$	−4.41604e20	−1.58743
$533$	−2.74908e19	−0.0974392
$534$	2.15046e19	0.0751576
$535$	0	0
$536$	−1.93949e20	−0.659102
$537$	1.29211e19	0.0433003
$538$	1.79339e20	0.592664
$539$	−8.78810e19	−0.286404
$540$	0	0
$541$	3.39181e20	1.07511	0.537554	−	0.843229i	$-0.319348\pi$
$541$	3.39181e20	1.07511	0.537554	+	0.843229i	$0.319348\pi$
$542$	−7.57742e19	−0.236879
$543$	−1.11054e20	−0.342401
$544$	−4.48247e20	−1.36309
$545$	0	0
$546$	1.24785e20	0.369163
$547$	−4.42493e20	−1.29122	−0.645611	−	0.763666i	$-0.723397\pi$
$547$	−4.42493e20	−1.29122	−0.645611	+	0.763666i	$0.723397\pi$
$548$	2.27641e20	0.655233
$549$	1.33656e20	0.379486
$550$	0	0
$551$	3.05029e20	0.842761
$552$	−3.19201e20	−0.870002
$553$	−4.75015e20	−1.27723
$554$	2.02631e20	0.537504
$555$	0	0
$556$	−2.69155e20	−0.694931
$557$	8.40400e19	0.214078	0.107039	−	0.994255i	$-0.465863\pi$
$557$	8.40400e19	0.214078	0.107039	+	0.994255i	$0.465863\pi$
$558$	4.14858e19	0.104266
$559$	3.28078e20	0.813557
$560$	0	0
$561$	9.67293e19	0.233527
$562$	−2.13945e19	−0.0509660
$563$	−6.96374e19	−0.163693	−0.0818464	−	0.996645i	$-0.526082\pi$
$563$	−6.96374e19	−0.163693	−0.0818464	+	0.996645i	$0.526082\pi$
$564$	1.34672e20	0.312381
$565$	0	0
$566$	5.45172e19	0.123143
$567$	6.97163e19	0.155404
$568$	4.41093e20	0.970325
$569$	7.38405e20	1.60307	0.801535	−	0.597948i	$-0.204017\pi$
$569$	7.38405e20	1.60307	0.801535	+	0.597948i	$0.204017\pi$
$570$	0	0
$571$	−1.59413e20	−0.337096	−0.168548	−	0.985693i	$-0.553908\pi$
$571$	−1.59413e20	−0.337096	−0.168548	+	0.985693i	$0.553908\pi$
$572$	1.13449e20	0.236771
$573$	3.24610e20	0.668655
$574$	3.06711e19	0.0623577
$575$	0	0
$576$	7.65130e18	0.0151554
$577$	7.37398e20	1.44173	0.720864	−	0.693077i	$-0.243745\pi$
$577$	7.37398e20	1.44173	0.720864	+	0.693077i	$0.243745\pi$
$578$	−1.95162e20	−0.376649
$579$	−4.39712e19	−0.0837682
$580$	0	0
$581$	6.66139e20	1.23664
$582$	−2.64583e19	−0.0484885
$583$	2.74841e20	0.497241
$584$	5.01054e20	0.894928
$585$	0	0
$586$	−3.44591e20	−0.599890
$587$	4.64146e18	0.00797754	0.00398877	−	0.999992i	$-0.498730\pi$
$587$	4.64146e18	0.00797754	0.00398877	+	0.999992i	$0.498730\pi$
$588$	2.57266e20	0.436568
$589$	−5.85844e20	−0.981560
$590$	0	0
$591$	−1.97313e20	−0.322293
$592$	−3.49764e20	−0.564107
$593$	−4.25599e20	−0.677782	−0.338891	−	0.940826i	$-0.610052\pi$
$593$	−4.25599e20	−0.677782	−0.338891	+	0.940826i	$0.610052\pi$
$594$	−1.67657e19	−0.0263648
$595$	0	0
$596$	−8.02347e20	−1.23031
$597$	4.64494e20	0.703352
$598$	−5.62471e20	−0.841088
$599$	8.40129e20	1.24064	0.620319	−	0.784350i	$-0.287003\pi$
$599$	8.40129e20	1.24064	0.620319	+	0.784350i	$0.287003\pi$
$600$	0	0
$601$	8.65782e20	1.24695	0.623476	−	0.781842i	$-0.285720\pi$
$601$	8.65782e20	1.24695	0.623476	+	0.781842i	$0.285720\pi$
$602$	−3.66031e20	−0.520648
$603$	−1.90940e20	−0.268236
$604$	2.13190e20	0.295794
$605$	0	0
$606$	−2.51759e20	−0.340754
$607$	9.60445e20	1.28398	0.641988	−	0.766714i	$-0.278110\pi$
$607$	9.60445e20	1.28398	0.641988	+	0.766714i	$0.278110\pi$
$608$	−1.09714e21	−1.44873
$609$	−3.63547e20	−0.474167
$610$	0	0
$611$	5.37391e20	0.683882
$612$	−2.83169e20	−0.355967
$613$	4.72496e20	0.586738	0.293369	−	0.955999i	$-0.405224\pi$
$613$	4.72496e20	0.586738	0.293369	+	0.955999i	$0.405224\pi$
$614$	−6.55615e20	−0.804241
$615$	0	0
$616$	−2.86626e20	−0.343132
$617$	2.41046e20	0.285076	0.142538	−	0.989789i	$-0.454474\pi$
$617$	2.41046e20	0.285076	0.142538	+	0.989789i	$0.454474\pi$
$618$	1.79472e20	0.209693
$619$	4.71617e20	0.544389	0.272195	−	0.962242i	$-0.412251\pi$
$619$	4.71617e20	0.544389	0.272195	+	0.962242i	$0.412251\pi$
$620$	0	0
$621$	−3.14247e20	−0.354066
$622$	−3.19061e20	−0.355178
$623$	3.61938e20	0.398083
$624$	−2.21027e20	−0.240193
$625$	0	0
$626$	−3.97396e20	−0.421615
$627$	2.36758e20	0.248198
$628$	6.25824e20	0.648268
$629$	−1.78808e21	−1.83024
$630$	0	0
$631$	−6.33879e19	−0.0633558	−0.0316779	−	0.999498i	$-0.510085\pi$
$631$	−6.33879e19	−0.0633558	−0.0316779	+	0.999498i	$0.510085\pi$
$632$	−7.57282e20	−0.747962
$633$	5.98465e20	0.584132
$634$	−8.09717e20	−0.781023
$635$	0	0
$636$	−8.04578e20	−0.757948
$637$	1.02658e21	0.955758
$638$	8.74277e19	0.0804441
$639$	4.34248e20	0.394895
$640$	0	0
$641$	−2.92672e20	−0.259984	−0.129992	−	0.991515i	$-0.541495\pi$
$641$	−2.92672e20	−0.259984	−0.129992	+	0.991515i	$0.541495\pi$
$642$	−3.28050e19	−0.0288023
$643$	−1.73682e21	−1.50720	−0.753601	−	0.657332i	$-0.771685\pi$
$643$	−1.73682e21	−1.50720	−0.753601	+	0.657332i	$0.771685\pi$
$644$	−2.37241e21	−2.03491
$645$	0	0
$646$	−1.05774e21	−0.886411
$647$	−3.12346e20	−0.258734	−0.129367	−	0.991597i	$-0.541295\pi$
$647$	−3.12346e20	−0.258734	−0.129367	+	0.991597i	$0.541295\pi$
$648$	1.11144e20	0.0910065
$649$	4.08091e19	0.0330310
$650$	0	0
$651$	6.98234e20	0.552260
$652$	8.07345e20	0.631251
$653$	2.34057e20	0.180914	0.0904571	−	0.995900i	$-0.471167\pi$
$653$	2.34057e20	0.180914	0.0904571	+	0.995900i	$0.471167\pi$
$654$	6.42545e20	0.490987
$655$	0	0
$656$	−5.43264e19	−0.0405725
$657$	4.93278e20	0.364210
$658$	−5.99559e20	−0.437661
$659$	−1.83728e21	−1.32597	−0.662985	−	0.748633i	$-0.730711\pi$
$659$	−1.83728e21	−1.32597	−0.662985	+	0.748633i	$0.730711\pi$
$660$	0	0
$661$	1.01470e21	0.715858	0.357929	−	0.933749i	$-0.383483\pi$
$661$	1.01470e21	0.715858	0.357929	+	0.933749i	$0.383483\pi$
$662$	2.36012e20	0.164626
$663$	−1.12994e21	−0.779301
$664$	1.06198e21	0.724194
$665$	0	0
$666$	3.09922e20	0.206631
$667$	1.63869e21	1.08032
$668$	1.67869e21	1.09432
$669$	−5.76610e20	−0.371695
$670$	0	0
$671$	−5.40977e20	−0.341004
$672$	1.30763e21	0.815106
$673$	8.72384e20	0.537768	0.268884	−	0.963173i	$-0.413345\pi$
$673$	8.72384e20	0.537768	0.268884	+	0.963173i	$0.413345\pi$
$674$	1.19710e21	0.729761
$675$	0	0
$676$	1.15262e18	0.000687205 0
$677$	6.97330e19	0.0411172	0.0205586	−	0.999789i	$-0.493456\pi$
$677$	6.97330e19	0.0411172	0.0205586	+	0.999789i	$0.493456\pi$
$678$	−2.33107e20	−0.135936
$679$	−4.45311e20	−0.256826
$680$	0	0
$681$	−1.34958e21	−0.761368
$682$	−1.67915e20	−0.0936929
$683$	−2.50322e21	−1.38148	−0.690739	−	0.723104i	$-0.742715\pi$
$683$	−2.50322e21	−1.38148	−0.690739	+	0.723104i	$0.742715\pi$
$684$	−6.93093e20	−0.378330
$685$	0	0
$686$	5.24969e19	0.0280351
$687$	1.29571e21	0.684433
$688$	6.48335e20	0.338755
$689$	−3.21055e21	−1.65934
$690$	0	0
$691$	−4.07072e20	−0.205867	−0.102933	−	0.994688i	$-0.532823\pi$
$691$	−4.07072e20	−0.205867	−0.102933	+	0.994688i	$0.532823\pi$
$692$	−2.59774e21	−1.29957
$693$	−2.82179e20	−0.139645
$694$	−1.35106e21	−0.661422
$695$	0	0
$696$	−5.79577e20	−0.277678
$697$	−2.77730e20	−0.131637
$698$	−8.12826e20	−0.381138
$699$	1.76332e21	0.817996
$700$	0	0
$701$	3.75615e21	1.70552	0.852762	−	0.522299i	$-0.174926\pi$
$701$	3.75615e21	1.70552	0.852762	+	0.522299i	$0.174926\pi$
$702$	1.95849e20	0.0879819
$703$	−4.37657e21	−1.94522
$704$	−3.09689e19	−0.0136185
$705$	0	0
$706$	−3.46597e20	−0.149207
$707$	−4.23728e21	−1.80485
$708$	−1.19466e20	−0.0503494
$709$	1.54239e21	0.643201	0.321601	−	0.946875i	$-0.395779\pi$
$709$	1.54239e21	0.643201	0.321601	+	0.946875i	$0.395779\pi$
$710$	0	0
$711$	−7.45531e20	−0.304399
$712$	5.77011e20	0.233123
$713$	−3.14730e21	−1.25825
$714$	1.26066e21	0.498726
$715$	0	0
$716$	1.53100e20	0.0593100
$717$	−2.20400e21	−0.844926
$718$	−1.97189e21	−0.748082
$719$	−1.25201e21	−0.470048	−0.235024	−	0.971990i	$-0.575517\pi$
$719$	−1.25201e21	−0.470048	−0.235024	+	0.971990i	$0.575517\pi$
$720$	0	0
$721$	3.02064e21	1.11067
$722$	−1.33208e21	−0.484732
$723$	1.75985e21	0.633779
$724$	−1.31586e21	−0.468999
$725$	0	0
$726$	−6.88499e20	−0.240369
$727$	2.62650e21	0.907546	0.453773	−	0.891117i	$-0.350078\pi$
$727$	2.62650e21	0.907546	0.453773	+	0.891117i	$0.350078\pi$
$728$	3.34823e21	1.14506
$729$	1.09419e20	0.0370370
$730$	0	0
$731$	3.31445e21	1.09909
$732$	1.58367e21	0.519795
$733$	2.26879e21	0.737079	0.368540	−	0.929612i	$-0.379858\pi$
$733$	2.26879e21	0.737079	0.368540	+	0.929612i	$0.379858\pi$
$734$	6.09748e20	0.196078
$735$	0	0
$736$	−5.89414e21	−1.85711
$737$	7.72834e20	0.241035
$738$	4.81380e19	0.0148616
$739$	2.59587e21	0.793323	0.396661	−	0.917965i	$-0.370169\pi$
$739$	2.59587e21	0.793323	0.396661	+	0.917965i	$0.370169\pi$
$740$	0	0
$741$	−2.76569e21	−0.828260
$742$	3.58197e21	1.06192
$743$	1.61834e21	0.474957	0.237478	−	0.971393i	$-0.423679\pi$
$743$	1.61834e21	0.474957	0.237478	+	0.971393i	$0.423679\pi$
$744$	1.11315e21	0.323411
$745$	0	0
$746$	1.82478e21	0.519601
$747$	1.04550e21	0.294726
$748$	1.14613e21	0.319870
$749$	−5.52131e20	−0.152556
$750$	0	0
$751$	−4.84564e21	−1.31236	−0.656178	−	0.754606i	$-0.727828\pi$
$751$	−4.84564e21	−1.31236	−0.656178	+	0.754606i	$0.727828\pi$
$752$	1.06197e21	0.284760
$753$	2.27462e21	0.603875
$754$	−1.02129e21	−0.268450
$755$	0	0
$756$	8.26059e20	0.212862
$757$	−7.01199e21	−1.78905	−0.894524	−	0.447019i	$-0.852486\pi$
$757$	−7.01199e21	−1.78905	−0.894524	+	0.447019i	$0.852486\pi$
$758$	−1.57430e21	−0.397712
$759$	1.27192e21	0.318162
$760$	0	0
$761$	3.72850e21	0.914428	0.457214	−	0.889357i	$-0.348847\pi$
$761$	3.72850e21	0.914428	0.457214	+	0.889357i	$0.348847\pi$
$762$	−1.42779e21	−0.346737
$763$	1.08145e22	2.60058
$764$	3.84626e21	0.915880
$765$	0	0
$766$	9.12438e20	0.213053
$767$	−4.76712e20	−0.110228
$768$	1.25474e21	0.287306
$769$	6.72004e21	1.52379	0.761894	−	0.647702i	$-0.224270\pi$
$769$	6.72004e21	1.52379	0.761894	+	0.647702i	$0.224270\pi$
$770$	0	0
$771$	1.42952e21	0.317893
$772$	−5.21009e20	−0.114740
$773$	−3.44422e21	−0.751181	−0.375590	−	0.926786i	$-0.622560\pi$
$773$	−3.44422e21	−0.751181	−0.375590	+	0.926786i	$0.622560\pi$
$774$	−5.74482e20	−0.124085
$775$	0	0
$776$	−7.09927e20	−0.150401
$777$	5.21619e21	1.09445
$778$	2.42729e21	0.504399
$779$	−6.79782e20	−0.139907
$780$	0	0
$781$	−1.75763e21	−0.354850
$782$	−5.68245e21	−1.13628
$783$	−5.70584e20	−0.113007
$784$	2.02869e21	0.397966
$785$	0	0
$786$	1.91841e21	0.369209
$787$	−2.13375e21	−0.406755	−0.203377	−	0.979100i	$-0.565192\pi$
$787$	−2.13375e21	−0.406755	−0.203377	+	0.979100i	$0.565192\pi$
$788$	−2.33794e21	−0.441455
$789$	7.87617e20	0.147312
$790$	0	0
$791$	−3.92335e21	−0.720002
$792$	−4.49857e20	−0.0817779
$793$	6.31942e21	1.13796
$794$	2.86067e21	0.510286
$795$	0	0
$796$	5.50373e21	0.963405
$797$	−7.26600e21	−1.25996	−0.629981	−	0.776610i	$-0.716938\pi$
$797$	−7.26600e21	−1.25996	−0.629981	+	0.776610i	$0.716938\pi$
$798$	3.08564e21	0.530058
$799$	5.42907e21	0.923900
$800$	0	0
$801$	5.68057e20	0.0948742
$802$	1.27923e21	0.211661
$803$	−1.99656e21	−0.327277
$804$	−2.26242e21	−0.367412
$805$	0	0
$806$	1.96150e21	0.312662
$807$	4.73735e21	0.748142
$808$	−6.75519e21	−1.05694
$809$	4.22149e20	0.0654413	0.0327207	−	0.999465i	$-0.489583\pi$
$809$	4.22149e20	0.0654413	0.0327207	+	0.999465i	$0.489583\pi$
$810$	0	0
$811$	6.44753e21	0.981153	0.490576	−	0.871398i	$-0.336786\pi$
$811$	6.44753e21	0.981153	0.490576	+	0.871398i	$0.336786\pi$
$812$	−4.30762e21	−0.649482
$813$	−2.00162e21	−0.299022
$814$	−1.25442e21	−0.185677
$815$	0	0
$816$	−2.23295e21	−0.324492
$817$	8.11257e21	1.16813
$818$	5.24575e21	0.748442
$819$	3.29627e21	0.466008
$820$	0	0
$821$	8.36694e21	1.16143	0.580715	−	0.814107i	$-0.302773\pi$
$821$	8.36694e21	1.16143	0.580715	+	0.814107i	$0.302773\pi$
$822$	−1.59060e21	−0.218788
$823$	2.84179e21	0.387341	0.193670	−	0.981067i	$-0.437961\pi$
$823$	2.84179e21	0.387341	0.193670	+	0.981067i	$0.437961\pi$
$824$	4.81559e21	0.650422
$825$	0	0
$826$	5.31860e20	0.0705419
$827$	2.75693e21	0.362355	0.181177	−	0.983450i	$-0.442009\pi$
$827$	2.75693e21	0.362355	0.181177	+	0.983450i	$0.442009\pi$
$828$	−3.72347e21	−0.484976
$829$	−1.04802e22	−1.35273	−0.676367	−	0.736565i	$-0.736446\pi$
$829$	−1.04802e22	−1.35273	−0.676367	+	0.736565i	$0.736446\pi$
$830$	0	0
$831$	5.35260e21	0.678512
$832$	3.61763e20	0.0454464
$833$	1.03712e22	1.29119
$834$	1.88068e21	0.232043
$835$	0	0
$836$	2.80531e21	0.339965
$837$	1.09587e21	0.131619
$838$	6.96538e20	0.0829115
$839$	1.94410e21	0.229353	0.114677	−	0.993403i	$-0.463417\pi$
$839$	1.94410e21	0.229353	0.114677	+	0.993403i	$0.463417\pi$
$840$	0	0
$841$	−5.65379e21	−0.655193
$842$	−1.18178e21	−0.135736
$843$	−5.65149e20	−0.0643363
$844$	7.09114e21	0.800106
$845$	0	0
$846$	−9.41001e20	−0.104307
$847$	−1.15879e22	−1.27315
$848$	−6.34458e21	−0.690930
$849$	1.44010e21	0.155448
$850$	0	0
$851$	−2.35120e22	−2.49355
$852$	5.14534e21	0.540901
$853$	−8.65398e21	−0.901775	−0.450887	−	0.892581i	$-0.648892\pi$
$853$	−8.65398e21	−0.901775	−0.450887	+	0.892581i	$0.648892\pi$
$854$	−7.05048e21	−0.728257
$855$	0	0
$856$	−8.80223e20	−0.0893387
$857$	−6.49635e21	−0.653602	−0.326801	−	0.945093i	$-0.605971\pi$
$857$	−6.49635e21	−0.653602	−0.326801	+	0.945093i	$0.605971\pi$
$858$	−7.92705e20	−0.0790600
$859$	1.48724e22	1.47039	0.735196	−	0.677855i	$-0.237090\pi$
$859$	1.48724e22	1.47039	0.735196	+	0.677855i	$0.237090\pi$
$860$	0	0
$861$	8.10195e20	0.0787165
$862$	8.37726e21	0.806858
$863$	−1.46940e22	−1.40300	−0.701502	−	0.712668i	$-0.747487\pi$
$863$	−1.46940e22	−1.40300	−0.701502	+	0.712668i	$0.747487\pi$
$864$	2.05231e21	0.194262
$865$	0	0
$866$	−7.43356e20	−0.0691533
$867$	−5.15531e21	−0.475458
$868$	8.27329e21	0.756450
$869$	3.01756e21	0.273531
$870$	0	0
$871$	−9.02787e21	−0.804357
$872$	1.72407e22	1.52294
$873$	−6.98910e20	−0.0612089
$874$	−1.39086e22	−1.20766
$875$	0	0
$876$	5.84479e21	0.498871
$877$	−1.90581e22	−1.61281	−0.806403	−	0.591367i	$-0.798589\pi$
$877$	−1.90581e22	−1.61281	−0.806403	+	0.591367i	$0.798589\pi$
$878$	−1.04383e21	−0.0875829
$879$	−9.10256e21	−0.757264
$880$	0	0
$881$	7.91825e21	0.647604	0.323802	−	0.946125i	$-0.395039\pi$
$881$	7.91825e21	0.647604	0.323802	+	0.946125i	$0.395039\pi$
$882$	−1.79760e21	−0.145774
$883$	−1.50259e22	−1.20819	−0.604095	−	0.796913i	$-0.706465\pi$
$883$	−1.50259e22	−1.20819	−0.604095	+	0.796913i	$0.706465\pi$
$884$	−1.33886e22	−1.06744
$885$	0	0
$886$	6.62345e21	0.519196
$887$	−3.13787e21	−0.243898	−0.121949	−	0.992536i	$-0.538914\pi$
$887$	−3.13787e21	−0.243898	−0.121949	+	0.992536i	$0.538914\pi$
$888$	8.31580e21	0.640924
$889$	−2.40306e22	−1.83654
$890$	0	0
$891$	−4.42876e20	−0.0332813
$892$	−6.83217e21	−0.509123
$893$	1.32884e22	0.981943
$894$	5.60627e21	0.410812
$895$	0	0
$896$	1.91886e22	1.38272
$897$	−1.48580e22	−1.06174
$898$	1.51345e21	0.107249
$899$	−5.71461e21	−0.401595
$900$	0	0
$901$	−3.24351e22	−2.24171
$902$	−1.94840e20	−0.0133545
$903$	−9.66892e21	−0.657234
$904$	−6.25473e21	−0.421643
$905$	0	0
$906$	−1.48963e21	−0.0987681
$907$	2.45168e22	1.61217	0.806083	−	0.591802i	$-0.201583\pi$
$907$	2.45168e22	1.61217	0.806083	+	0.591802i	$0.201583\pi$
$908$	−1.59910e22	−1.04287
$909$	−6.65036e21	−0.430146
$910$	0	0
$911$	−1.75648e22	−1.11752	−0.558760	−	0.829329i	$-0.688723\pi$
$911$	−1.75648e22	−1.11752	−0.558760	+	0.829329i	$0.688723\pi$
$912$	−5.46545e21	−0.344878
$913$	−4.23168e21	−0.264839
$914$	4.85893e21	0.301609
$915$	0	0
$916$	1.53527e22	0.937491
$917$	3.22881e22	1.95557
$918$	1.97860e21	0.118860
$919$	−1.34689e22	−0.802539	−0.401269	−	0.915960i	$-0.631431\pi$
$919$	−1.34689e22	−0.802539	−0.401269	+	0.915960i	$0.631431\pi$
$920$	0	0
$921$	−1.73184e22	−1.01522
$922$	−9.72275e21	−0.565337
$923$	2.05318e22	1.18417
$924$	−3.34350e21	−0.191276
$925$	0	0
$926$	1.39837e22	0.787116
$927$	4.74086e21	0.264703
$928$	−1.07021e22	−0.592732
$929$	2.85420e21	0.156808	0.0784038	−	0.996922i	$-0.475018\pi$
$929$	2.85420e21	0.156808	0.0784038	+	0.996922i	$0.475018\pi$
$930$	0	0
$931$	2.53849e22	1.37231
$932$	2.08933e22	1.12044
$933$	−8.42819e21	−0.448355
$934$	−1.42320e22	−0.751042
$935$	0	0
$936$	5.25501e21	0.272901
$937$	1.33439e22	0.687440	0.343720	−	0.939072i	$-0.388313\pi$
$937$	1.33439e22	0.687440	0.343720	+	0.939072i	$0.388313\pi$
$938$	1.00723e22	0.514761
$939$	−1.04974e22	−0.532221
$940$	0	0
$941$	7.36149e20	0.0367319	0.0183660	−	0.999831i	$-0.494154\pi$
$941$	7.36149e20	0.0367319	0.0183660	+	0.999831i	$0.494154\pi$
$942$	−4.37285e21	−0.216462
$943$	−3.65196e21	−0.179345
$944$	−9.42061e20	−0.0458975
$945$	0	0
$946$	2.32523e21	0.111502
$947$	1.62457e22	0.772881	0.386441	−	0.922314i	$-0.373704\pi$
$947$	1.62457e22	0.772881	0.386441	+	0.922314i	$0.373704\pi$
$948$	−8.83370e21	−0.416946
$949$	2.33228e22	1.09215
$950$	0	0
$951$	−2.13891e22	−0.985915
$952$	3.38260e22	1.54694
$953$	8.56439e21	0.388597	0.194299	−	0.980942i	$-0.437757\pi$
$953$	8.56439e21	0.388597	0.194299	+	0.980942i	$0.437757\pi$
$954$	5.62186e21	0.253085
$955$	0	0
$956$	−2.61149e22	−1.15732
$957$	2.30945e21	0.101548
$958$	−1.80580e22	−0.787822
$959$	−2.67709e22	−1.15884
$960$	0	0
$961$	−1.24897e22	−0.532263
$962$	1.46535e22	0.619623
$963$	−8.66564e20	−0.0363583
$964$	2.08522e22	0.868108
$965$	0	0
$966$	1.65768e22	0.679474
$967$	−4.32414e22	−1.75874	−0.879370	−	0.476139i	$-0.842036\pi$
$967$	−4.32414e22	−1.75874	−0.879370	+	0.476139i	$0.842036\pi$
$968$	−1.84738e22	−0.745573
$969$	−2.79408e22	−1.11895
$970$	0	0
$971$	−3.83444e22	−1.51202	−0.756011	−	0.654559i	$-0.772854\pi$
$971$	−3.83444e22	−1.51202	−0.756011	+	0.654559i	$0.772854\pi$
$972$	1.29649e21	0.0507309
$973$	3.16531e22	1.22905
$974$	1.64674e22	0.634500
$975$	0	0
$976$	1.24882e22	0.473835
$977$	3.49369e22	1.31545	0.657727	−	0.753257i	$-0.271518\pi$
$977$	3.49369e22	1.31545	0.657727	+	0.753257i	$0.271518\pi$
$978$	−5.64120e21	−0.210780
$979$	−2.29923e21	−0.0852534
$980$	0	0
$981$	1.69732e22	0.619792
$982$	−1.62621e22	−0.589306
$983$	5.59409e21	0.201177	0.100588	−	0.994928i	$-0.467927\pi$
$983$	5.59409e21	0.201177	0.100588	+	0.994928i	$0.467927\pi$
$984$	1.29164e21	0.0460975
$985$	0	0
$986$	−1.03177e22	−0.362666
$987$	−1.58377e22	−0.552475
$988$	−3.27703e22	−1.13450
$989$	4.35828e22	1.49742
$990$	0	0
$991$	3.93649e22	1.33216	0.666080	−	0.745880i	$-0.267971\pi$
$991$	3.93649e22	1.33216	0.666080	+	0.745880i	$0.267971\pi$
$992$	2.05546e22	0.690353
$993$	6.23439e21	0.207814
$994$	−2.29070e22	−0.757827
$995$	0	0
$996$	1.23880e22	0.403697
$997$	−7.34569e21	−0.237585	−0.118792	−	0.992919i	$-0.537902\pi$
$997$	−7.34569e21	−0.237585	−0.118792	+	0.992919i	$0.537902\pi$
$998$	1.14247e22	0.366747
$999$	8.18675e21	0.260838

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.16.a.j.1.2	yes	6
5.2	odd	4		75.16.b.h.49.5		12
5.3	odd	4		75.16.b.h.49.8		12
5.4	even	2		75.16.a.i.1.5	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.16.a.i.1.5	✓	6	5.4	even	2
75.16.a.j.1.2	yes	6	1.1	even	1	trivial
75.16.b.h.49.5		12	5.2	odd	4
75.16.b.h.49.8		12	5.3	odd	4

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)

\(f(q)\)	\(=\)	\(q-82.7921 q^{2} -2187.00 q^{3} -25913.5 q^{4} +181066. q^{6} +3.04747e6 q^{7} +4.85836e6 q^{8} +4.78297e6 q^{9} +O(q^{10})\)	\(q-82.7921 q^{2} -2187.00 q^{3} -25913.5 q^{4} +181066. q^{6} +3.04747e6 q^{7} +4.85836e6 q^{8} +4.78297e6 q^{9} -1.93592e7 q^{11} +5.66728e7 q^{12} +2.26145e8 q^{13} -2.52306e8 q^{14} +4.46899e8 q^{16} +2.28466e9 q^{17} -3.95992e8 q^{18} +5.59201e9 q^{19} -6.66481e9 q^{21} +1.60279e9 q^{22} +3.00417e10 q^{23} -1.06252e10 q^{24} -1.87230e10 q^{26} -1.04604e10 q^{27} -7.89704e10 q^{28} +5.45473e10 q^{29} -1.04764e11 q^{31} -1.96198e11 q^{32} +4.23386e10 q^{33} -1.89152e11 q^{34} -1.23943e11 q^{36} -7.82646e11 q^{37} -4.62974e11 q^{38} -4.94578e11 q^{39} -1.21563e11 q^{41} +5.51793e11 q^{42} +1.45074e12 q^{43} +5.01664e11 q^{44} -2.48722e12 q^{46} +2.37631e12 q^{47} -9.77368e11 q^{48} +4.53949e12 q^{49} -4.99655e12 q^{51} -5.86019e12 q^{52} -1.41969e13 q^{53} +8.66034e11 q^{54} +1.48057e13 q^{56} -1.22297e13 q^{57} -4.51608e12 q^{58} -2.10800e12 q^{59} +2.79442e13 q^{61} +8.67366e12 q^{62} +1.45759e13 q^{63} +1.59970e12 q^{64} -3.50530e12 q^{66} -3.99208e13 q^{67} -5.92035e13 q^{68} -6.57013e13 q^{69} +9.07904e13 q^{71} +2.32374e13 q^{72} +1.03132e14 q^{73} +6.47969e13 q^{74} -1.44908e14 q^{76} -5.89965e13 q^{77} +4.09472e13 q^{78} -1.55872e14 q^{79} +2.28768e13 q^{81} +1.00645e13 q^{82} +2.18588e14 q^{83} +1.72708e14 q^{84} -1.20110e14 q^{86} -1.19295e14 q^{87} -9.40540e13 q^{88} +1.18767e14 q^{89} +6.89168e14 q^{91} -7.78486e14 q^{92} +2.29120e14 q^{93} -1.96740e14 q^{94} +4.29086e14 q^{96} -1.46125e14 q^{97} -3.75834e14 q^{98} -9.25945e13 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 234 q^{2} - 13122 q^{3} + 93112 q^{4} - 511758 q^{6} - 2590222 q^{7} + 14012388 q^{8} + 28697814 q^{9}+O(q^{10}) \) 6 * q + 234 * q^2 - 13122 * q^3 + 93112 * q^4 - 511758 * q^6 - 2590222 * q^7 + 14012388 * q^8 + 28697814 * q^9	\( 6 q + 234 q^{2} - 13122 q^{3} + 93112 q^{4} - 511758 q^{6} - 2590222 q^{7} + 14012388 q^{8} + 28697814 q^{9} + 107489124 q^{11} - 203635944 q^{12} - 109881686 q^{13} - 563984442 q^{14} + 3622829560 q^{16} + 3573042876 q^{17} + 1119214746 q^{18} - 1602340942 q^{19} + 5664815514 q^{21} + 4024661012 q^{22} - 6555818844 q^{23} - 30645092556 q^{24} - 25715894778 q^{26} - 62762119218 q^{27} - 270752117896 q^{28} + 126894468996 q^{29} + 151760841646 q^{31} + 385411085208 q^{32} - 235078714188 q^{33} + 1431919606684 q^{34} + 445351809528 q^{36} + 616109002068 q^{37} - 2822785016634 q^{38} + 240311247282 q^{39} + 1091281712616 q^{41} + 1233433974654 q^{42} - 2444971199030 q^{43} + 1413344578176 q^{44} - 5480862370044 q^{46} - 8369143269660 q^{47} - 7923128247720 q^{48} + 19523846053580 q^{49} - 7814244769812 q^{51} - 10261294060344 q^{52} - 16571417665824 q^{53} - 2447722649502 q^{54} - 75252275829540 q^{56} + 3504319640154 q^{57} - 3994751501708 q^{58} + 8796604455252 q^{59} - 6959665405750 q^{61} + 52277129313066 q^{62} - 12388951529118 q^{63} + 50304241850208 q^{64} - 8801933633244 q^{66} - 53487461742094 q^{67} + 307147088145312 q^{68} + 14337575811828 q^{69} + 104634162717912 q^{71} + 67020817419972 q^{72} + 177000981923236 q^{73} - 45005277967812 q^{74} + 76188538526328 q^{76} + 117850730172876 q^{77} + 56240661879486 q^{78} + 185514024366160 q^{79} + 137260754729766 q^{81} + 654376907588896 q^{82} + 435827733256908 q^{83} + 592134881838552 q^{84} + 15\!\cdots\!14 q^{86}+ \cdots + 514117147929156 q^{99}+O(q^{100}) \) 6 * q + 234 * q^2 - 13122 * q^3 + 93112 * q^4 - 511758 * q^6 - 2590222 * q^7 + 14012388 * q^8 + 28697814 * q^9 + 107489124 * q^11 - 203635944 * q^12 - 109881686 * q^13 - 563984442 * q^14 + 3622829560 * q^16 + 3573042876 * q^17 + 1119214746 * q^18 - 1602340942 * q^19 + 5664815514 * q^21 + 4024661012 * q^22 - 6555818844 * q^23 - 30645092556 * q^24 - 25715894778 * q^26 - 62762119218 * q^27 - 270752117896 * q^28 + 126894468996 * q^29 + 151760841646 * q^31 + 385411085208 * q^32 - 235078714188 * q^33 + 1431919606684 * q^34 + 445351809528 * q^36 + 616109002068 * q^37 - 2822785016634 * q^38 + 240311247282 * q^39 + 1091281712616 * q^41 + 1233433974654 * q^42 - 2444971199030 * q^43 + 1413344578176 * q^44 - 5480862370044 * q^46 - 8369143269660 * q^47 - 7923128247720 * q^48 + 19523846053580 * q^49 - 7814244769812 * q^51 - 10261294060344 * q^52 - 16571417665824 * q^53 - 2447722649502 * q^54 - 75252275829540 * q^56 + 3504319640154 * q^57 - 3994751501708 * q^58 + 8796604455252 * q^59 - 6959665405750 * q^61 + 52277129313066 * q^62 - 12388951529118 * q^63 + 50304241850208 * q^64 - 8801933633244 * q^66 - 53487461742094 * q^67 + 307147088145312 * q^68 + 14337575811828 * q^69 + 104634162717912 * q^71 + 67020817419972 * q^72 + 177000981923236 * q^73 - 45005277967812 * q^74 + 76188538526328 * q^76 + 117850730172876 * q^77 + 56240661879486 * q^78 + 185514024366160 * q^79 + 137260754729766 * q^81 + 654376907588896 * q^82 + 435827733256908 * q^83 + 592134881838552 * q^84 + 1570057453210014 * q^86 - 277518203694252 * q^87 + 174719041441128 * q^88 + 431517021638688 * q^89 + 466827529337246 * q^91 + 50075827036128 * q^92 - 331900960679802 * q^93 + 1404749684955772 * q^94 - 842894043349896 * q^96 - 1013615616184854 * q^97 + 5117833640095236 * q^98 + 514117147929156 * q^99

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