LMFDB - Embedded newform 75.12.a.d.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [75,12,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, 12, names="a")

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

chi := DirichletCharacter("75.1");

S:= CuspForms(chi, 12);

N := Newforms(S);

Level:	$ N $	$=$	$ 75 = 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 12 $
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:	$57.6257385420$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{1609}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 402 $ x^2 - x - 402
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 15)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-19.5562$ 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)

$f(q)$	$=$	$q+51.1123 q^{2} -243.000 q^{3} +564.472 q^{4} -12420.3 q^{6} -45751.3 q^{7} -75826.6 q^{8} +59049.0 q^{9} +O(q^{10})$	\(q+51.1123 q^{2} -243.000 q^{3} +564.472 q^{4} -12420.3 q^{6} -45751.3 q^{7} -75826.6 q^{8} +59049.0 q^{9} -597423. q^{11} -137167. q^{12} +990011. q^{13} -2.33846e6 q^{14} -5.03171e6 q^{16} +6.61148e6 q^{17} +3.01813e6 q^{18} +576856. q^{19} +1.11176e7 q^{21} -3.05357e7 q^{22} +4.64840e7 q^{23} +1.84259e7 q^{24} +5.06018e7 q^{26} -1.43489e7 q^{27} -2.58253e7 q^{28} -1.59099e8 q^{29} +1.04664e8 q^{31} -1.01890e8 q^{32} +1.45174e8 q^{33} +3.37928e8 q^{34} +3.33315e7 q^{36} -4.80096e8 q^{37} +2.94845e7 q^{38} -2.40573e8 q^{39} +6.63154e8 q^{41} +5.68245e8 q^{42} +1.76838e9 q^{43} -3.37228e8 q^{44} +2.37591e9 q^{46} +1.39342e9 q^{47} +1.22271e9 q^{48} +1.15859e8 q^{49} -1.60659e9 q^{51} +5.58833e8 q^{52} +2.28338e9 q^{53} -7.33406e8 q^{54} +3.46917e9 q^{56} -1.40176e8 q^{57} -8.13193e9 q^{58} +3.12820e9 q^{59} +6.71630e9 q^{61} +5.34962e9 q^{62} -2.70157e9 q^{63} +5.09713e9 q^{64} +7.42017e9 q^{66} +5.93071e9 q^{67} +3.73199e9 q^{68} -1.12956e10 q^{69} -1.13022e10 q^{71} -4.47749e9 q^{72} +9.39160e9 q^{73} -2.45388e10 q^{74} +3.25619e8 q^{76} +2.73329e10 q^{77} -1.22962e10 q^{78} -3.59922e10 q^{79} +3.48678e9 q^{81} +3.38953e10 q^{82} -6.01074e10 q^{83} +6.27556e9 q^{84} +9.03863e10 q^{86} +3.86611e10 q^{87} +4.53005e10 q^{88} -1.46902e9 q^{89} -4.52943e10 q^{91} +2.62389e10 q^{92} -2.54333e10 q^{93} +7.12210e10 q^{94} +2.47592e10 q^{96} +7.50721e10 q^{97} +5.92183e9 q^{98} -3.52772e10 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 22 q^{2} - 486 q^{3} - 636 q^{4} - 5346 q^{6} + 10864 q^{7} + 18744 q^{8} + 118098 q^{9}+O(q^{10}) $ 2 * q + 22 * q^2 - 486 * q^3 - 636 * q^4 - 5346 * q^6 + 10864 * q^7 + 18744 * q^8 + 118098 * q^9	\( 2 q + 22 q^{2} - 486 q^{3} - 636 q^{4} - 5346 q^{6} + 10864 q^{7} + 18744 q^{8} + 118098 q^{9} - 361792 q^{11} + 154548 q^{12} + 2133732 q^{13} - 3986664 q^{14} - 5326320 q^{16} + 7804588 q^{17} + 1299078 q^{18} - 15562224 q^{19} - 2639952 q^{21} - 37395424 q^{22} + 37450248 q^{23} - 4554792 q^{24} + 17305364 q^{26} - 28697814 q^{27} - 93790448 q^{28} - 70320668 q^{29} + 298584872 q^{31} - 286993696 q^{32} + 87915456 q^{33} + 303194188 q^{34} - 37555164 q^{36} - 236000956 q^{37} + 499330888 q^{38} - 518496876 q^{39} - 464942588 q^{41} + 968759352 q^{42} + 242208600 q^{43} - 620095808 q^{44} + 2638898832 q^{46} + 4375796920 q^{47} + 1294295760 q^{48} + 1343830130 q^{49} - 1896514884 q^{51} - 814171912 q^{52} + 2189541388 q^{53} - 315675954 q^{54} + 8823318240 q^{56} + 3781620432 q^{57} - 10716478004 q^{58} - 5480385856 q^{59} + 14557903980 q^{61} - 295874592 q^{62} + 641508336 q^{63} + 11089288256 q^{64} + 9087088032 q^{66} + 15918388888 q^{67} + 2299702856 q^{68} - 9100410264 q^{69} + 1120561024 q^{71} + 1106814456 q^{72} + 24521574348 q^{73} - 31645012364 q^{74} + 19700124976 q^{76} + 40673194368 q^{77} - 4205203452 q^{78} - 79243055560 q^{79} + 6973568802 q^{81} + 66736852348 q^{82} - 9245226696 q^{83} + 22791078864 q^{84} + 134816793608 q^{86} + 17087922324 q^{87} + 67584257664 q^{88} + 22117321236 q^{89} + 19457850112 q^{91} + 37083635424 q^{92} - 72556123896 q^{93} - 15603010256 q^{94} + 69739468128 q^{96} + 160363673468 q^{97} - 29827277722 q^{98} - 21363455808 q^{99}+O(q^{100}) \) 2 * q + 22 * q^2 - 486 * q^3 - 636 * q^4 - 5346 * q^6 + 10864 * q^7 + 18744 * q^8 + 118098 * q^9 - 361792 * q^11 + 154548 * q^12 + 2133732 * q^13 - 3986664 * q^14 - 5326320 * q^16 + 7804588 * q^17 + 1299078 * q^18 - 15562224 * q^19 - 2639952 * q^21 - 37395424 * q^22 + 37450248 * q^23 - 4554792 * q^24 + 17305364 * q^26 - 28697814 * q^27 - 93790448 * q^28 - 70320668 * q^29 + 298584872 * q^31 - 286993696 * q^32 + 87915456 * q^33 + 303194188 * q^34 - 37555164 * q^36 - 236000956 * q^37 + 499330888 * q^38 - 518496876 * q^39 - 464942588 * q^41 + 968759352 * q^42 + 242208600 * q^43 - 620095808 * q^44 + 2638898832 * q^46 + 4375796920 * q^47 + 1294295760 * q^48 + 1343830130 * q^49 - 1896514884 * q^51 - 814171912 * q^52 + 2189541388 * q^53 - 315675954 * q^54 + 8823318240 * q^56 + 3781620432 * q^57 - 10716478004 * q^58 - 5480385856 * q^59 + 14557903980 * q^61 - 295874592 * q^62 + 641508336 * q^63 + 11089288256 * q^64 + 9087088032 * q^66 + 15918388888 * q^67 + 2299702856 * q^68 - 9100410264 * q^69 + 1120561024 * q^71 + 1106814456 * q^72 + 24521574348 * q^73 - 31645012364 * q^74 + 19700124976 * q^76 + 40673194368 * q^77 - 4205203452 * q^78 - 79243055560 * q^79 + 6973568802 * q^81 + 66736852348 * q^82 - 9245226696 * q^83 + 22791078864 * q^84 + 134816793608 * q^86 + 17087922324 * q^87 + 67584257664 * q^88 + 22117321236 * q^89 + 19457850112 * q^91 + 37083635424 * q^92 - 72556123896 * q^93 - 15603010256 * q^94 + 69739468128 * q^96 + 160363673468 * q^97 - 29827277722 * q^98 - 21363455808 * 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 $

$2$	51.1123	1.12943	0.564717	−	0.825285i	$-0.308985\pi$
$2$	51.1123	1.12943	0.564717	+	0.825285i	$0.308985\pi$
$3$	−243.000	−0.577350
$3$	−243.000	−0.577350
$4$	564.472	0.275621
$5$	0	0
$5$	0	0
$6$	−12420.3	−0.652079
$7$	−45751.3	−1.02888	−0.514440	−	0.857526i	$-0.672000\pi$
$7$	−45751.3	−1.02888	−0.514440	+	0.857526i	$0.672000\pi$
$8$	−75826.6	−0.818138
$9$	59049.0	0.333333
$10$	0	0
$11$	−597423.	−1.11846	−0.559232	−	0.829011i	$-0.688904\pi$
$11$	−597423.	−1.11846	−0.559232	+	0.829011i	$0.688904\pi$
$12$	−137167.	−0.159130
$13$	990011.	0.739523	0.369761	−	0.929127i	$-0.379439\pi$
$13$	990011.	0.739523	0.369761	+	0.929127i	$0.379439\pi$
$14$	−2.33846e6	−1.16205
$15$	0	0
$16$	−5.03171e6	−1.19965
$17$	6.61148e6	1.12935	0.564677	−	0.825312i	$-0.309001\pi$
$17$	6.61148e6	1.12935	0.564677	+	0.825312i	$0.309001\pi$
$18$	3.01813e6	0.376478
$19$	576856.	0.0534469	0.0267235	−	0.999643i	$-0.491493\pi$
$19$	576856.	0.0534469	0.0267235	+	0.999643i	$0.491493\pi$
$20$	0	0
$21$	1.11176e7	0.594024
$22$	−3.05357e7	−1.26323
$23$	4.64840e7	1.50591	0.752957	−	0.658069i	$-0.228627\pi$
$23$	4.64840e7	1.50591	0.752957	+	0.658069i	$0.228627\pi$
$24$	1.84259e7	0.472352
$25$	0	0
$26$	5.06018e7	0.835242
$27$	−1.43489e7	−0.192450
$28$	−2.58253e7	−0.283581
$29$	−1.59099e8	−1.44039	−0.720193	−	0.693774i	$-0.755947\pi$
$29$	−1.59099e8	−1.44039	−0.720193	+	0.693774i	$0.755947\pi$
$30$	0	0
$31$	1.04664e8	0.656610	0.328305	−	0.944572i	$-0.393523\pi$
$31$	1.04664e8	0.656610	0.328305	+	0.944572i	$0.393523\pi$
$32$	−1.01890e8	−0.536792
$33$	1.45174e8	0.645745
$34$	3.37928e8	1.27553
$35$	0	0
$36$	3.33315e7	0.0918736
$37$	−4.80096e8	−1.13820	−0.569100	−	0.822268i	$-0.692708\pi$
$37$	−4.80096e8	−1.13820	−0.569100	+	0.822268i	$0.692708\pi$
$38$	2.94845e7	0.0603648
$39$	−2.40573e8	−0.426964
$40$	0	0
$41$	6.63154e8	0.893929	0.446964	−	0.894552i	$-0.352505\pi$
$41$	6.63154e8	0.893929	0.446964	+	0.894552i	$0.352505\pi$
$42$	5.68245e8	0.670911
$43$	1.76838e9	1.83443	0.917213	−	0.398398i	$-0.130434\pi$
$43$	1.76838e9	1.83443	0.917213	+	0.398398i	$0.130434\pi$
$44$	−3.37228e8	−0.308272
$45$	0	0
$46$	2.37591e9	1.70083
$47$	1.39342e9	0.886225	0.443112	−	0.896466i	$-0.353874\pi$
$47$	1.39342e9	0.886225	0.443112	+	0.896466i	$0.353874\pi$
$48$	1.22271e9	0.692621
$49$	1.15859e8	0.0585938
$50$	0	0
$51$	−1.60659e9	−0.652032
$52$	5.58833e8	0.203828
$53$	2.28338e9	0.749998	0.374999	−	0.927025i	$-0.377643\pi$
$53$	2.28338e9	0.749998	0.374999	+	0.927025i	$0.377643\pi$
$54$	−7.33406e8	−0.217360
$55$	0	0
$56$	3.46917e9	0.841766
$57$	−1.40176e8	−0.0308576
$58$	−8.13193e9	−1.62682
$59$	3.12820e9	0.569650	0.284825	−	0.958580i	$-0.408064\pi$
$59$	3.12820e9	0.569650	0.284825	+	0.958580i	$0.408064\pi$
$60$	0	0
$61$	6.71630e9	1.01816	0.509080	−	0.860719i	$-0.329986\pi$
$61$	6.71630e9	1.01816	0.509080	+	0.860719i	$0.329986\pi$
$62$	5.34962e9	0.741598
$63$	−2.70157e9	−0.342960
$64$	5.09713e9	0.593383
$65$	0	0
$66$	7.42017e9	0.729327
$67$	5.93071e9	0.536655	0.268328	−	0.963328i	$-0.413529\pi$
$67$	5.93071e9	0.536655	0.268328	+	0.963328i	$0.413529\pi$
$68$	3.73199e9	0.311273
$69$	−1.12956e10	−0.869440
$70$	0	0
$71$	−1.13022e10	−0.743430	−0.371715	−	0.928347i	$-0.621230\pi$
$71$	−1.13022e10	−0.743430	−0.371715	+	0.928347i	$0.621230\pi$
$72$	−4.47749e9	−0.272713
$73$	9.39160e9	0.530229	0.265115	−	0.964217i	$-0.414590\pi$
$73$	9.39160e9	0.530229	0.265115	+	0.964217i	$0.414590\pi$
$74$	−2.45388e10	−1.28552
$75$	0	0
$76$	3.25619e8	0.0147311
$77$	2.73329e10	1.15076
$78$	−1.22962e10	−0.482227
$79$	−3.59922e10	−1.31601	−0.658006	−	0.753013i	$-0.728600\pi$
$79$	−3.59922e10	−1.31601	−0.658006	+	0.753013i	$0.728600\pi$
$80$	0	0
$81$	3.48678e9	0.111111
$82$	3.38953e10	1.00963
$83$	−6.01074e10	−1.67494	−0.837468	−	0.546486i	$-0.815965\pi$
$83$	−6.01074e10	−1.67494	−0.837468	+	0.546486i	$0.815965\pi$
$84$	6.27556e9	0.163725
$85$	0	0
$86$	9.03863e10	2.07186
$87$	3.86611e10	0.831607
$88$	4.53005e10	0.915058
$89$	−1.46902e9	−0.0278858	−0.0139429	−	0.999903i	$-0.504438\pi$
$89$	−1.46902e9	−0.0278858	−0.0139429	+	0.999903i	$0.504438\pi$
$90$	0	0
$91$	−4.52943e10	−0.760880
$92$	2.62389e10	0.415061
$93$	−2.54333e10	−0.379094
$94$	7.12210e10	1.00093
$95$	0	0
$96$	2.47592e10	0.309917
$97$	7.50721e10	0.887634	0.443817	−	0.896117i	$-0.353624\pi$
$97$	7.50721e10	0.887634	0.443817	+	0.896117i	$0.353624\pi$
$98$	5.92183e9	0.0661779
$99$	−3.52772e10	−0.372821
$100$	0	0
$101$	1.66482e11	1.57615	0.788077	−	0.615577i	$-0.211077\pi$
$101$	1.66482e11	1.57615	0.788077	+	0.615577i	$0.211077\pi$
$102$	−8.21166e10	−0.736427
$103$	1.56404e11	1.32937	0.664683	−	0.747126i	$-0.268567\pi$
$103$	1.56404e11	1.32937	0.664683	+	0.747126i	$0.268567\pi$
$104$	−7.50692e10	−0.605032
$105$	0	0
$106$	1.16709e11	0.847073
$107$	−1.72482e11	−1.18887	−0.594435	−	0.804144i	$-0.702624\pi$
$107$	−1.72482e11	−1.18887	−0.594435	+	0.804144i	$0.702624\pi$
$108$	−8.09955e9	−0.0530433
$109$	1.98239e11	1.23408	0.617041	−	0.786931i	$-0.288331\pi$
$109$	1.98239e11	1.23408	0.617041	+	0.786931i	$0.288331\pi$
$110$	0	0
$111$	1.16663e11	0.657140
$112$	2.30208e11	1.23430
$113$	6.00761e10	0.306740	0.153370	−	0.988169i	$-0.450987\pi$
$113$	6.00761e10	0.306740	0.153370	+	0.988169i	$0.450987\pi$
$114$	−7.16473e9	−0.0348516
$115$	0	0
$116$	−8.98069e10	−0.397000
$117$	5.84591e10	0.246508
$118$	1.59890e11	0.643382
$119$	−3.02484e11	−1.16197
$120$	0	0
$121$	7.16020e10	0.250961
$122$	3.43286e11	1.14994
$123$	−1.61146e11	−0.516110
$124$	5.90798e10	0.180976
$125$	0	0
$126$	−1.38084e11	−0.387351
$127$	−6.55120e11	−1.75954	−0.879772	−	0.475395i	$-0.842305\pi$
$127$	−6.55120e11	−1.75954	−0.879772	+	0.475395i	$0.842305\pi$
$128$	4.69196e11	1.20698
$129$	−4.29717e11	−1.05911
$130$	0	0
$131$	−2.51548e11	−0.569677	−0.284839	−	0.958575i	$-0.591940\pi$
$131$	−2.51548e11	−0.569677	−0.284839	+	0.958575i	$0.591940\pi$
$132$	8.19464e10	0.177981
$133$	−2.63919e10	−0.0549905
$134$	3.03132e11	0.606116
$135$	0	0
$136$	−5.01326e11	−0.923967
$137$	−9.65207e11	−1.70867	−0.854334	−	0.519725i	$-0.826034\pi$
$137$	−9.65207e11	−1.70867	−0.854334	+	0.519725i	$0.826034\pi$
$138$	−5.77345e11	−0.981975
$139$	9.65084e11	1.57755	0.788776	−	0.614681i	$-0.210715\pi$
$139$	9.65084e11	1.57755	0.788776	+	0.614681i	$0.210715\pi$
$140$	0	0
$141$	−3.38601e11	−0.511662
$142$	−5.77679e11	−0.839655
$143$	−5.91455e11	−0.827129
$144$	−2.97118e11	−0.399885
$145$	0	0
$146$	4.80027e11	0.598859
$147$	−2.81538e10	−0.0338292
$148$	−2.71001e11	−0.313712
$149$	−2.87526e11	−0.320740	−0.160370	−	0.987057i	$-0.551269\pi$
$149$	−2.87526e11	−0.320740	−0.160370	+	0.987057i	$0.551269\pi$
$150$	0	0
$151$	−1.58923e12	−1.64745	−0.823725	−	0.566989i	$-0.808108\pi$
$151$	−1.58923e12	−1.64745	−0.823725	+	0.566989i	$0.808108\pi$
$152$	−4.37410e10	−0.0437270
$153$	3.90401e11	0.376451
$154$	1.39705e12	1.29971
$155$	0	0
$156$	−1.35796e11	−0.117680
$157$	9.15249e11	0.765757	0.382879	−	0.923799i	$-0.374933\pi$
$157$	9.15249e11	0.765757	0.382879	+	0.923799i	$0.374933\pi$
$158$	−1.83965e12	−1.48635
$159$	−5.54861e11	−0.433012
$160$	0	0
$161$	−2.12671e12	−1.54941
$162$	1.78218e11	0.125493
$163$	−5.95458e11	−0.405340	−0.202670	−	0.979247i	$-0.564962\pi$
$163$	−5.95458e11	−0.405340	−0.202670	+	0.979247i	$0.564962\pi$
$164$	3.74331e11	0.246385
$165$	0	0
$166$	−3.07223e12	−1.89173
$167$	2.27609e11	0.135597	0.0677983	−	0.997699i	$-0.478403\pi$
$167$	2.27609e11	0.135597	0.0677983	+	0.997699i	$0.478403\pi$
$168$	−8.43008e11	−0.485994
$169$	−8.12039e11	−0.453106
$170$	0	0
$171$	3.40628e10	0.0178156
$172$	9.98202e11	0.505606
$173$	3.94129e12	1.93368	0.966841	−	0.255380i	$-0.0822004\pi$
$173$	3.94129e12	1.93368	0.966841	+	0.255380i	$0.0822004\pi$
$174$	1.97606e12	0.939245
$175$	0	0
$176$	3.00606e12	1.34177
$177$	−7.60152e11	−0.328888
$178$	−7.50850e10	−0.0314951
$179$	−6.63949e11	−0.270049	−0.135025	−	0.990842i	$-0.543111\pi$
$179$	−6.63949e11	−0.270049	−0.135025	+	0.990842i	$0.543111\pi$
$180$	0	0
$181$	2.97961e12	1.14006	0.570029	−	0.821625i	$-0.306932\pi$
$181$	2.97961e12	1.14006	0.570029	+	0.821625i	$0.306932\pi$
$182$	−2.31510e12	−0.859364
$183$	−1.63206e12	−0.587835
$184$	−3.52472e12	−1.23205
$185$	0	0
$186$	−1.29996e12	−0.428162
$187$	−3.94985e12	−1.26314
$188$	7.86546e11	0.244262
$189$	6.56482e11	0.198008
$190$	0	0
$191$	2.39137e12	0.680713	0.340356	−	0.940296i	$-0.389452\pi$
$191$	2.39137e12	0.680713	0.340356	+	0.940296i	$0.389452\pi$
$192$	−1.23860e12	−0.342590
$193$	3.54987e12	0.954217	0.477108	−	0.878845i	$-0.341685\pi$
$193$	3.54987e12	0.954217	0.477108	+	0.878845i	$0.341685\pi$
$194$	3.83711e12	1.00252
$195$	0	0
$196$	6.53992e10	0.0161497
$197$	−2.10097e12	−0.504493	−0.252246	−	0.967663i	$-0.581169\pi$
$197$	−2.10097e12	−0.504493	−0.252246	+	0.967663i	$0.581169\pi$
$198$	−1.80310e12	−0.421077
$199$	1.38114e12	0.313722	0.156861	−	0.987621i	$-0.449863\pi$
$199$	1.38114e12	0.313722	0.156861	+	0.987621i	$0.449863\pi$
$200$	0	0
$201$	−1.44116e12	−0.309838
$202$	8.50926e12	1.78016
$203$	7.27900e12	1.48198
$204$	−9.06874e11	−0.179714
$205$	0	0
$206$	7.99420e12	1.50143
$207$	2.74483e12	0.501971
$208$	−4.98145e12	−0.887171
$209$	−3.44627e11	−0.0597784
$210$	0	0
$211$	5.03751e12	0.829205	0.414603	−	0.910003i	$-0.363921\pi$
$211$	5.03751e12	0.829205	0.414603	+	0.910003i	$0.363921\pi$
$212$	1.28890e12	0.206715
$213$	2.74642e12	0.429220
$214$	−8.81598e12	−1.34275
$215$	0	0
$216$	1.08803e12	0.157451
$217$	−4.78852e12	−0.675573
$218$	1.01325e13	1.39381
$219$	−2.28216e12	−0.306128
$220$	0	0
$221$	6.54544e12	0.835182
$222$	5.96294e12	0.742196
$223$	−1.54066e13	−1.87081	−0.935403	−	0.353583i	$-0.884963\pi$
$223$	−1.54066e13	−1.87081	−0.935403	+	0.353583i	$0.884963\pi$
$224$	4.66159e12	0.552294
$225$	0	0
$226$	3.07063e12	0.346442
$227$	3.09266e12	0.340557	0.170279	−	0.985396i	$-0.445533\pi$
$227$	3.09266e12	0.340557	0.170279	+	0.985396i	$0.445533\pi$
$228$	−7.91254e10	−0.00850500
$229$	−1.18942e13	−1.24808	−0.624039	−	0.781393i	$-0.714509\pi$
$229$	−1.18942e13	−1.24808	−0.624039	+	0.781393i	$0.714509\pi$
$230$	0	0
$231$	−6.64189e12	−0.664394
$232$	1.20639e13	1.17843
$233$	1.95406e13	1.86415	0.932075	−	0.362267i	$-0.117997\pi$
$233$	1.95406e13	1.86415	0.932075	+	0.362267i	$0.117997\pi$
$234$	2.98798e12	0.278414
$235$	0	0
$236$	1.76578e12	0.157007
$237$	8.74611e12	0.759800
$238$	−1.54607e13	−1.31237
$239$	4.39049e12	0.364187	0.182093	−	0.983281i	$-0.441713\pi$
$239$	4.39049e12	0.364187	0.182093	+	0.983281i	$0.441713\pi$
$240$	0	0
$241$	−9.72399e12	−0.770461	−0.385231	−	0.922820i	$-0.625878\pi$
$241$	−9.72399e12	−0.770461	−0.385231	+	0.922820i	$0.625878\pi$
$242$	3.65975e12	0.283444
$243$	−8.47289e11	−0.0641500
$244$	3.79116e12	0.280626
$245$	0	0
$246$	−8.23657e12	−0.582912
$247$	5.71094e11	0.0395252
$248$	−7.93631e12	−0.537198
$249$	1.46061e13	0.967025
$250$	0	0
$251$	1.53165e13	0.970405	0.485203	−	0.874402i	$-0.338746\pi$
$251$	1.53165e13	0.970405	0.485203	+	0.874402i	$0.338746\pi$
$252$	−1.52496e12	−0.0945269
$253$	−2.77706e13	−1.68431
$254$	−3.34847e13	−1.98729
$255$	0	0
$256$	1.35428e13	0.769819
$257$	1.18634e13	0.660050	0.330025	−	0.943972i	$-0.392943\pi$
$257$	1.18634e13	0.660050	0.330025	+	0.943972i	$0.392943\pi$
$258$	−2.19639e13	−1.19619
$259$	2.19650e13	1.17107
$260$	0	0
$261$	−9.39464e12	−0.480129
$262$	−1.28572e13	−0.643413
$263$	−5.17670e11	−0.0253686	−0.0126843	−	0.999920i	$-0.504038\pi$
$263$	−5.17670e11	−0.0253686	−0.0126843	+	0.999920i	$0.504038\pi$
$264$	−1.10080e13	−0.528309
$265$	0	0
$266$	−1.34895e12	−0.0621081
$267$	3.56972e11	0.0160999
$268$	3.34772e12	0.147913
$269$	3.74208e13	1.61985	0.809927	−	0.586531i	$-0.199507\pi$
$269$	3.74208e13	1.61985	0.809927	+	0.586531i	$0.199507\pi$
$270$	0	0
$271$	−4.10753e12	−0.170706	−0.0853532	−	0.996351i	$-0.527202\pi$
$271$	−4.10753e12	−0.170706	−0.0853532	+	0.996351i	$0.527202\pi$
$272$	−3.32671e13	−1.35483
$273$	1.10065e13	0.439294
$274$	−4.93340e13	−1.92983
$275$	0	0
$276$	−6.37605e12	−0.239636
$277$	2.08709e13	0.768959	0.384479	−	0.923134i	$-0.374381\pi$
$277$	2.08709e13	0.768959	0.384479	+	0.923134i	$0.374381\pi$
$278$	4.93277e13	1.78174
$279$	6.18030e12	0.218870
$280$	0	0
$281$	−1.03477e12	−0.0352337	−0.0176168	−	0.999845i	$-0.505608\pi$
$281$	−1.03477e12	−0.0352337	−0.0176168	+	0.999845i	$0.505608\pi$
$282$	−1.73067e13	−0.577888
$283$	4.80808e13	1.57451	0.787257	−	0.616625i	$-0.211500\pi$
$283$	4.80808e13	1.57451	0.787257	+	0.616625i	$0.211500\pi$
$284$	−6.37974e12	−0.204905
$285$	0	0
$286$	−3.02306e13	−0.934188
$287$	−3.03402e13	−0.919745
$288$	−6.01649e12	−0.178931
$289$	9.43979e12	0.275438
$290$	0	0
$291$	−1.82425e13	−0.512476
$292$	5.30129e12	0.146142
$293$	1.96764e13	0.532322	0.266161	−	0.963929i	$-0.414245\pi$
$293$	1.96764e13	0.532322	0.266161	+	0.963929i	$0.414245\pi$
$294$	−1.43901e12	−0.0382078
$295$	0	0
$296$	3.64041e13	0.931205
$297$	8.57236e12	0.215248
$298$	−1.46961e13	−0.362254
$299$	4.60196e13	1.11366
$300$	0	0
$301$	−8.09060e13	−1.88740
$302$	−8.12290e13	−1.86069
$303$	−4.04550e13	−0.909993
$304$	−2.90257e12	−0.0641178
$305$	0	0
$306$	1.99543e13	0.425177
$307$	−2.21562e13	−0.463696	−0.231848	−	0.972752i	$-0.574477\pi$
$307$	−2.21562e13	−0.463696	−0.231848	+	0.972752i	$0.574477\pi$
$308$	1.54286e13	0.317175
$309$	−3.80063e13	−0.767509
$310$	0	0
$311$	4.75136e13	0.926053	0.463026	−	0.886345i	$-0.346764\pi$
$311$	4.75136e13	0.926053	0.463026	+	0.886345i	$0.346764\pi$
$312$	1.82418e13	0.349315
$313$	2.39850e13	0.451280	0.225640	−	0.974211i	$-0.427553\pi$
$313$	2.39850e13	0.451280	0.225640	+	0.974211i	$0.427553\pi$
$314$	4.67805e13	0.864872
$315$	0	0
$316$	−2.03166e13	−0.362720
$317$	8.02864e13	1.40869	0.704346	−	0.709857i	$-0.251241\pi$
$317$	8.02864e13	1.40869	0.704346	+	0.709857i	$0.251241\pi$
$318$	−2.83602e13	−0.489058
$319$	9.50494e13	1.61102
$320$	0	0
$321$	4.19132e13	0.686394
$322$	−1.08701e14	−1.74995
$323$	3.81387e12	0.0603604
$324$	1.96819e12	0.0306245
$325$	0	0
$326$	−3.04352e13	−0.457804
$327$	−4.81722e13	−0.712498
$328$	−5.02847e13	−0.731357
$329$	−6.37508e13	−0.911819
$330$	0	0
$331$	−7.35975e13	−1.01814	−0.509072	−	0.860724i	$-0.670011\pi$
$331$	−7.35975e13	−1.01814	−0.509072	+	0.860724i	$0.670011\pi$
$332$	−3.39289e13	−0.461647
$333$	−2.83492e13	−0.379400
$334$	1.16336e13	0.153147
$335$	0	0
$336$	−5.59405e13	−0.712623
$337$	1.11762e14	1.40065	0.700323	−	0.713826i	$-0.253039\pi$
$337$	1.11762e14	1.40065	0.700323	+	0.713826i	$0.253039\pi$
$338$	−4.15052e13	−0.511754
$339$	−1.45985e13	−0.177096
$340$	0	0
$341$	−6.25286e13	−0.734395
$342$	1.74103e12	0.0201216
$343$	8.51647e13	0.968594
$344$	−1.34091e14	−1.50081
$345$	0	0
$346$	2.01449e14	2.18397
$347$	2.31381e12	0.0246896	0.0123448	−	0.999924i	$-0.496070\pi$
$347$	2.31381e12	0.0246896	0.0123448	+	0.999924i	$0.496070\pi$
$348$	2.18231e13	0.229208
$349$	1.79579e14	1.85659	0.928294	−	0.371848i	$-0.121276\pi$
$349$	1.79579e14	1.85659	0.928294	+	0.371848i	$0.121276\pi$
$350$	0	0
$351$	−1.42056e13	−0.142321
$352$	6.08712e13	0.600382
$353$	1.62870e13	0.158154	0.0790768	−	0.996869i	$-0.474803\pi$
$353$	1.62870e13	0.158154	0.0790768	+	0.996869i	$0.474803\pi$
$354$	−3.88532e13	−0.371457
$355$	0	0
$356$	−8.29220e11	−0.00768590
$357$	7.35037e13	0.670863
$358$	−3.39360e13	−0.305003
$359$	1.48987e13	0.131865	0.0659325	−	0.997824i	$-0.478998\pi$
$359$	1.48987e13	0.131865	0.0659325	+	0.997824i	$0.478998\pi$
$360$	0	0
$361$	−1.16157e14	−0.997143
$362$	1.52295e14	1.28762
$363$	−1.73993e13	−0.144892
$364$	−2.55674e13	−0.209714
$365$	0	0
$366$	−8.34184e13	−0.663921
$367$	−9.81686e13	−0.769679	−0.384839	−	0.922984i	$-0.625743\pi$
$367$	−9.81686e13	−0.769679	−0.384839	+	0.922984i	$0.625743\pi$
$368$	−2.33894e14	−1.80658
$369$	3.91586e13	0.297976
$370$	0	0
$371$	−1.04468e14	−0.771658
$372$	−1.43564e13	−0.104486
$373$	−4.21169e12	−0.0302036	−0.0151018	−	0.999886i	$-0.504807\pi$
$373$	−4.21169e12	−0.0302036	−0.0151018	+	0.999886i	$0.504807\pi$
$374$	−2.01886e14	−1.42663
$375$	0	0
$376$	−1.05658e14	−0.725054
$377$	−1.57510e14	−1.06520
$378$	3.35543e13	0.223637
$379$	−8.03941e13	−0.528090	−0.264045	−	0.964510i	$-0.585057\pi$
$379$	−8.03941e13	−0.528090	−0.264045	+	0.964510i	$0.585057\pi$
$380$	0	0
$381$	1.59194e14	1.01587
$382$	1.22229e14	0.768820
$383$	4.60952e13	0.285800	0.142900	−	0.989737i	$-0.454357\pi$
$383$	4.60952e13	0.285800	0.142900	+	0.989737i	$0.454357\pi$
$384$	−1.14015e14	−0.696850
$385$	0	0
$386$	1.81442e14	1.07772
$387$	1.04421e14	0.611475
$388$	4.23761e13	0.244651
$389$	1.07321e14	0.610891	0.305445	−	0.952210i	$-0.401195\pi$
$389$	1.07321e14	0.610891	0.305445	+	0.952210i	$0.401195\pi$
$390$	0	0
$391$	3.07328e14	1.70071
$392$	−8.78521e12	−0.0479379
$393$	6.11262e13	0.328903
$394$	−1.07385e14	−0.569791
$395$	0	0
$396$	−1.99130e13	−0.102757
$397$	5.20270e13	0.264778	0.132389	−	0.991198i	$-0.457735\pi$
$397$	5.20270e13	0.264778	0.132389	+	0.991198i	$0.457735\pi$
$398$	7.05932e13	0.354328
$399$	6.41324e12	0.0317488
$400$	0	0
$401$	2.47673e14	1.19284	0.596422	−	0.802671i	$-0.296588\pi$
$401$	2.47673e14	1.19284	0.596422	+	0.802671i	$0.296588\pi$
$402$	−7.36612e13	−0.349942
$403$	1.03618e14	0.485578
$404$	9.39741e13	0.434421
$405$	0	0
$406$	3.72047e14	1.67380
$407$	2.86820e14	1.27304
$408$	1.21822e14	0.533453
$409$	7.24362e13	0.312952	0.156476	−	0.987682i	$-0.449987\pi$
$409$	7.24362e13	0.312952	0.156476	+	0.987682i	$0.449987\pi$
$410$	0	0
$411$	2.34545e14	0.986499
$412$	8.82858e13	0.366401
$413$	−1.43119e14	−0.586102
$414$	1.40295e14	0.566944
$415$	0	0
$416$	−1.00872e14	−0.396969
$417$	−2.34515e14	−0.910800
$418$	−1.76147e13	−0.0675158
$419$	1.89636e14	0.717369	0.358684	−	0.933459i	$-0.383225\pi$
$419$	1.89636e14	0.717369	0.358684	+	0.933459i	$0.383225\pi$
$420$	0	0
$421$	−1.11470e14	−0.410776	−0.205388	−	0.978681i	$-0.565846\pi$
$421$	−1.11470e14	−0.410776	−0.205388	+	0.978681i	$0.565846\pi$
$422$	2.57479e14	0.936533
$423$	8.22801e13	0.295408
$424$	−1.73141e14	−0.613602
$425$	0	0
$426$	1.40376e14	0.484775
$427$	−3.07280e14	−1.04756
$428$	−9.73614e13	−0.327677
$429$	1.43724e14	0.477543
$430$	0	0
$431$	−1.11843e14	−0.362230	−0.181115	−	0.983462i	$-0.557971\pi$
$431$	−1.11843e14	−0.362230	−0.181115	+	0.983462i	$0.557971\pi$
$432$	7.21996e13	0.230874
$433$	4.22999e14	1.33554	0.667768	−	0.744369i	$-0.267250\pi$
$433$	4.22999e14	1.33554	0.667768	+	0.744369i	$0.267250\pi$
$434$	−2.44752e14	−0.763015
$435$	0	0
$436$	1.11900e14	0.340139
$437$	2.68146e13	0.0804865
$438$	−1.16646e14	−0.345751
$439$	−5.11444e14	−1.49707	−0.748537	−	0.663093i	$-0.769243\pi$
$439$	−5.11444e14	−1.49707	−0.748537	+	0.663093i	$0.769243\pi$
$440$	0	0
$441$	6.84137e12	0.0195313
$442$	3.34553e14	0.943283
$443$	5.35642e14	1.49161	0.745803	−	0.666167i	$-0.232066\pi$
$443$	5.35642e14	1.49161	0.745803	+	0.666167i	$0.232066\pi$
$444$	6.58531e13	0.181122
$445$	0	0
$446$	−7.87465e14	−2.11295
$447$	6.98688e13	0.185179
$448$	−2.33200e14	−0.610520
$449$	1.75234e14	0.453173	0.226587	−	0.973991i	$-0.427243\pi$
$449$	1.75234e14	0.453173	0.226587	+	0.973991i	$0.427243\pi$
$450$	0	0
$451$	−3.96183e14	−0.999827
$452$	3.39112e13	0.0845439
$453$	3.86182e14	0.951156
$454$	1.58073e14	0.384637
$455$	0	0
$456$	1.06291e13	0.0252458
$457$	−2.72581e14	−0.639671	−0.319836	−	0.947473i	$-0.603628\pi$
$457$	−2.72581e14	−0.639671	−0.319836	+	0.947473i	$0.603628\pi$
$458$	−6.07942e14	−1.40962
$459$	−9.48675e13	−0.217344
$460$	0	0
$461$	−2.76659e13	−0.0618856	−0.0309428	−	0.999521i	$-0.509851\pi$
$461$	−2.76659e13	−0.0618856	−0.0309428	+	0.999521i	$0.509851\pi$
$462$	−3.39483e14	−0.750389
$463$	−7.26472e13	−0.158680	−0.0793402	−	0.996848i	$-0.525281\pi$
$463$	−7.26472e13	−0.158680	−0.0793402	+	0.996848i	$0.525281\pi$
$464$	8.00541e14	1.72796
$465$	0	0
$466$	9.98767e14	2.10543
$467$	−2.28552e14	−0.476148	−0.238074	−	0.971247i	$-0.576516\pi$
$467$	−2.28552e14	−0.476148	−0.238074	+	0.971247i	$0.576516\pi$
$468$	3.29985e13	0.0679426
$469$	−2.71338e14	−0.552154
$470$	0	0
$471$	−2.22406e14	−0.442110
$472$	−2.37201e14	−0.466053
$473$	−1.05647e15	−2.05174
$474$	4.47034e14	0.858144
$475$	0	0
$476$	−1.70744e14	−0.320263
$477$	1.34831e14	0.249999
$478$	2.24408e14	0.411325
$479$	−6.47644e14	−1.17352	−0.586761	−	0.809760i	$-0.699597\pi$
$479$	−6.47644e14	−1.17352	−0.586761	+	0.809760i	$0.699597\pi$
$480$	0	0
$481$	−4.75300e14	−0.841725
$482$	−4.97016e14	−0.870185
$483$	5.16789e14	0.894549
$484$	4.04173e13	0.0691700
$485$	0	0
$486$	−4.33069e13	−0.0724532
$487$	−9.59498e14	−1.58721	−0.793606	−	0.608433i	$-0.791799\pi$
$487$	−9.59498e14	−1.58721	−0.793606	+	0.608433i	$0.791799\pi$
$488$	−5.09274e14	−0.832995
$489$	1.44696e14	0.234023
$490$	0	0
$491$	−8.15780e14	−1.29010	−0.645052	−	0.764138i	$-0.723164\pi$
$491$	−8.15780e14	−1.29010	−0.645052	+	0.764138i	$0.723164\pi$
$492$	−9.09625e13	−0.142251
$493$	−1.05188e15	−1.62670
$494$	2.91899e13	0.0446411
$495$	0	0
$496$	−5.26639e14	−0.787705
$497$	5.17089e14	0.764900
$498$	7.46552e14	1.09219
$499$	−2.28979e14	−0.331316	−0.165658	−	0.986183i	$-0.552975\pi$
$499$	−2.28979e14	−0.331316	−0.165658	+	0.986183i	$0.552975\pi$
$500$	0	0
$501$	−5.53090e13	−0.0782867
$502$	7.82861e14	1.09601
$503$	−5.81915e14	−0.805815	−0.402908	−	0.915241i	$-0.632000\pi$
$503$	−5.81915e14	−0.805815	−0.402908	+	0.915241i	$0.632000\pi$
$504$	2.04851e14	0.280589
$505$	0	0
$506$	−1.41942e15	−1.90232
$507$	1.97326e14	0.261601
$508$	−3.69797e14	−0.484967
$509$	−2.09435e14	−0.271707	−0.135854	−	0.990729i	$-0.543378\pi$
$509$	−2.09435e14	−0.271707	−0.135854	+	0.990729i	$0.543378\pi$
$510$	0	0
$511$	−4.29678e14	−0.545542
$512$	−2.68709e14	−0.337519
$513$	−8.27725e12	−0.0102859
$514$	6.06366e14	0.745483
$515$	0	0
$516$	−2.42563e14	−0.291912
$517$	−8.32461e14	−0.991210
$518$	1.12268e15	1.32265
$519$	−9.57734e14	−1.11641
$520$	0	0
$521$	4.47986e14	0.511278	0.255639	−	0.966772i	$-0.417714\pi$
$521$	4.47986e14	0.511278	0.255639	+	0.966772i	$0.417714\pi$
$522$	−4.80182e14	−0.542273
$523$	−6.17551e14	−0.690102	−0.345051	−	0.938584i	$-0.612138\pi$
$523$	−6.17551e14	−0.690102	−0.345051	+	0.938584i	$0.612138\pi$
$524$	−1.41992e14	−0.157015
$525$	0	0
$526$	−2.64593e13	−0.0286522
$527$	6.91984e14	0.741545
$528$	−7.30472e14	−0.774671
$529$	1.20795e15	1.26778
$530$	0	0
$531$	1.84717e14	0.189883
$532$	−1.48975e13	−0.0151565
$533$	6.56529e14	0.661081
$534$	1.82457e13	0.0181837
$535$	0	0
$536$	−4.49706e14	−0.439058
$537$	1.61340e14	0.155913
$538$	1.91267e15	1.82952
$539$	−6.92169e13	−0.0655351
$540$	0	0
$541$	1.60818e15	1.49194	0.745969	−	0.665981i	$-0.231987\pi$
$541$	1.60818e15	1.49194	0.745969	+	0.665981i	$0.231987\pi$
$542$	−2.09946e14	−0.192802
$543$	−7.24045e14	−0.658213
$544$	−6.73642e14	−0.606227
$545$	0	0
$546$	5.62569e14	0.496154
$547$	2.14344e13	0.0187146	0.00935732	−	0.999956i	$-0.497021\pi$
$547$	2.14344e13	0.0187146	0.00935732	+	0.999956i	$0.497021\pi$
$548$	−5.44832e14	−0.470944
$549$	3.96591e14	0.339387
$550$	0	0
$551$	−9.17773e13	−0.0769842
$552$	8.56508e14	0.711322
$553$	1.64669e15	1.35402
$554$	1.06676e15	0.868488
$555$	0	0
$556$	5.44762e14	0.434806
$557$	3.80343e14	0.300588	0.150294	−	0.988641i	$-0.451978\pi$
$557$	3.80343e14	0.300588	0.150294	+	0.988641i	$0.451978\pi$
$558$	3.15890e14	0.247199
$559$	1.75072e15	1.35660
$560$	0	0
$561$	9.59813e14	0.729274
$562$	−5.28894e13	−0.0397941
$563$	4.56848e14	0.340389	0.170194	−	0.985411i	$-0.445560\pi$
$563$	4.56848e14	0.340389	0.170194	+	0.985411i	$0.445560\pi$
$564$	−1.91131e14	−0.141025
$565$	0	0
$566$	2.45752e15	1.77831
$567$	−1.59525e14	−0.114320
$568$	8.57004e14	0.608229
$569$	−1.02008e15	−0.716993	−0.358496	−	0.933531i	$-0.616710\pi$
$569$	−1.02008e15	−0.716993	−0.358496	+	0.933531i	$0.616710\pi$
$570$	0	0
$571$	−1.16801e15	−0.805285	−0.402642	−	0.915357i	$-0.631908\pi$
$571$	−1.16801e15	−0.805285	−0.402642	+	0.915357i	$0.631908\pi$
$572$	−3.33859e14	−0.227974
$573$	−5.81104e14	−0.393010
$574$	−1.55076e15	−1.03879
$575$	0	0
$576$	3.00980e14	0.197794
$577$	4.17243e14	0.271595	0.135798	−	0.990737i	$-0.456640\pi$
$577$	4.17243e14	0.271595	0.135798	+	0.990737i	$0.456640\pi$
$578$	4.82490e14	0.311089
$579$	−8.62618e14	−0.550917
$580$	0	0
$581$	2.74999e15	1.72331
$582$	−9.32418e14	−0.578808
$583$	−1.36414e15	−0.838846
$584$	−7.12133e14	−0.433801
$585$	0	0
$586$	1.00571e15	0.601223
$587$	3.15560e14	0.186884	0.0934421	−	0.995625i	$-0.470213\pi$
$587$	3.15560e14	0.186884	0.0934421	+	0.995625i	$0.470213\pi$
$588$	−1.58920e13	−0.00932402
$589$	6.03760e13	0.0350938
$590$	0	0
$591$	5.10535e14	0.291269
$592$	2.41571e15	1.36545
$593$	−1.66344e15	−0.931551	−0.465775	−	0.884903i	$-0.654225\pi$
$593$	−1.66344e15	−0.931551	−0.465775	+	0.884903i	$0.654225\pi$
$594$	4.38153e14	0.243109
$595$	0	0
$596$	−1.62300e14	−0.0884026
$597$	−3.35616e14	−0.181128
$598$	2.35217e15	1.25780
$599$	6.19005e14	0.327979	0.163990	−	0.986462i	$-0.447564\pi$
$599$	6.19005e14	0.327979	0.163990	+	0.986462i	$0.447564\pi$
$600$	0	0
$601$	−1.90367e15	−0.990333	−0.495167	−	0.868798i	$-0.664893\pi$
$601$	−1.90367e15	−0.990333	−0.495167	+	0.868798i	$0.664893\pi$
$602$	−4.13529e15	−2.13170
$603$	3.50202e14	0.178885
$604$	−8.97072e14	−0.454072
$605$	0	0
$606$	−2.06775e15	−1.02778
$607$	1.45859e15	0.718450	0.359225	−	0.933251i	$-0.383041\pi$
$607$	1.45859e15	0.718450	0.359225	+	0.933251i	$0.383041\pi$
$608$	−5.87757e13	−0.0286899
$609$	−1.76880e15	−0.855624
$610$	0	0
$611$	1.37950e15	0.655383
$612$	2.20370e14	0.103758
$613$	−2.48221e14	−0.115826	−0.0579130	−	0.998322i	$-0.518445\pi$
$613$	−2.48221e14	−0.115826	−0.0579130	+	0.998322i	$0.518445\pi$
$614$	−1.13245e15	−0.523714
$615$	0	0
$616$	−2.07256e15	−0.941485
$617$	2.70824e15	1.21932	0.609662	−	0.792662i	$-0.291305\pi$
$617$	2.70824e15	1.21932	0.609662	+	0.792662i	$0.291305\pi$
$618$	−1.94259e15	−0.866851
$619$	−1.22881e14	−0.0543483	−0.0271741	−	0.999631i	$-0.508651\pi$
$619$	−1.22881e14	−0.0543483	−0.0271741	+	0.999631i	$0.508651\pi$
$620$	0	0
$621$	−6.66994e14	−0.289813
$622$	2.42853e15	1.04592
$623$	6.72096e13	0.0286911
$624$	1.21049e15	0.512209
$625$	0	0
$626$	1.22593e15	0.509691
$627$	8.37443e13	0.0345131
$628$	5.16632e14	0.211059
$629$	−3.17415e15	−1.28543
$630$	0	0
$631$	2.44958e15	0.974832	0.487416	−	0.873170i	$-0.337940\pi$
$631$	2.44958e15	0.974832	0.487416	+	0.873170i	$0.337940\pi$
$632$	2.72917e15	1.07668
$633$	−1.22411e15	−0.478742
$634$	4.10362e15	1.59102
$635$	0	0
$636$	−3.13203e14	−0.119347
$637$	1.14702e14	0.0433315
$638$	4.85820e15	1.81954
$639$	−6.67381e14	−0.247810
$640$	0	0
$641$	2.91178e15	1.06277	0.531386	−	0.847130i	$-0.321671\pi$
$641$	2.91178e15	1.06277	0.531386	+	0.847130i	$0.321671\pi$
$642$	2.14228e15	0.775237
$643$	−2.55085e15	−0.915219	−0.457610	−	0.889153i	$-0.651294\pi$
$643$	−2.55085e15	−0.915219	−0.457610	+	0.889153i	$0.651294\pi$
$644$	−1.20046e15	−0.427048
$645$	0	0
$646$	1.94936e14	0.0681731
$647$	2.35946e15	0.818163	0.409082	−	0.912498i	$-0.365849\pi$
$647$	2.35946e15	0.818163	0.409082	+	0.912498i	$0.365849\pi$
$648$	−2.64391e14	−0.0909043
$649$	−1.86886e15	−0.637133
$650$	0	0
$651$	1.16361e15	0.390042
$652$	−3.36119e14	−0.111720
$653$	−3.85495e15	−1.27056	−0.635282	−	0.772280i	$-0.719116\pi$
$653$	−3.85495e15	−1.27056	−0.635282	+	0.772280i	$0.719116\pi$
$654$	−2.46219e15	−0.804719
$655$	0	0
$656$	−3.33680e15	−1.07241
$657$	5.54565e14	0.176743
$658$	−3.25846e15	−1.02984
$659$	2.43185e15	0.762198	0.381099	−	0.924534i	$-0.375546\pi$
$659$	2.43185e15	0.762198	0.381099	+	0.924534i	$0.375546\pi$
$660$	0	0
$661$	−2.91546e15	−0.898668	−0.449334	−	0.893364i	$-0.648339\pi$
$661$	−2.91546e15	−0.898668	−0.449334	+	0.893364i	$0.648339\pi$
$662$	−3.76174e15	−1.14993
$663$	−1.59054e15	−0.482193
$664$	4.55774e15	1.37033
$665$	0	0
$666$	−1.44899e15	−0.428507
$667$	−7.39556e15	−2.16910
$668$	1.28479e14	0.0373732
$669$	3.74379e15	1.08011
$670$	0	0
$671$	−4.01247e15	−1.13877
$672$	−1.13277e15	−0.318867
$673$	2.73025e14	0.0762289	0.0381145	−	0.999273i	$-0.487865\pi$
$673$	2.73025e14	0.0762289	0.0381145	+	0.999273i	$0.487865\pi$
$674$	5.71240e15	1.58194
$675$	0	0
$676$	−4.58373e14	−0.124886
$677$	−5.49970e14	−0.148628	−0.0743142	−	0.997235i	$-0.523677\pi$
$677$	−5.49970e14	−0.148628	−0.0743142	+	0.997235i	$0.523677\pi$
$678$	−7.46163e14	−0.200019
$679$	−3.43465e15	−0.913269
$680$	0	0
$681$	−7.51517e14	−0.196621
$682$	−3.19598e15	−0.829450
$683$	5.37455e15	1.38366	0.691828	−	0.722062i	$-0.256806\pi$
$683$	5.37455e15	1.38366	0.691828	+	0.722062i	$0.256806\pi$
$684$	1.92275e13	0.00491036
$685$	0	0
$686$	4.35296e15	1.09396
$687$	2.89030e15	0.720578
$688$	−8.89800e15	−2.20068
$689$	2.26057e15	0.554641
$690$	0	0
$691$	1.45481e15	0.351298	0.175649	−	0.984453i	$-0.443798\pi$
$691$	1.45481e15	0.351298	0.175649	+	0.984453i	$0.443798\pi$
$692$	2.22475e15	0.532963
$693$	1.61398e15	0.383588
$694$	1.18264e14	0.0278853
$695$	0	0
$696$	−2.93154e15	−0.680370
$697$	4.38443e15	1.00956
$698$	9.17870e15	2.09689
$699$	−4.74837e15	−1.07627
$700$	0	0
$701$	−8.94470e14	−0.199580	−0.0997899	−	0.995009i	$-0.531817\pi$
$701$	−8.94470e14	−0.199580	−0.0997899	+	0.995009i	$0.531817\pi$
$702$	−7.26080e14	−0.160742
$703$	−2.76946e14	−0.0608333
$704$	−3.04514e15	−0.663678
$705$	0	0
$706$	8.32465e14	0.178624
$707$	−7.61676e15	−1.62167
$708$	−4.29084e14	−0.0906483
$709$	−6.98755e15	−1.46477	−0.732387	−	0.680889i	$-0.761594\pi$
$709$	−6.98755e15	−1.46477	−0.732387	+	0.680889i	$0.761594\pi$
$710$	0	0
$711$	−2.12531e15	−0.438671
$712$	1.11391e14	0.0228144
$713$	4.86520e15	0.988799
$714$	3.75694e15	0.757695
$715$	0	0
$716$	−3.74780e14	−0.0744312
$717$	−1.06689e15	−0.210263
$718$	7.61508e14	0.148933
$719$	4.29436e15	0.833469	0.416735	−	0.909028i	$-0.363174\pi$
$719$	4.29436e15	0.833469	0.416735	+	0.909028i	$0.363174\pi$
$720$	0	0
$721$	−7.15571e15	−1.36776
$722$	−5.93708e15	−1.12621
$723$	2.36293e15	0.444826
$724$	1.68190e15	0.314224
$725$	0	0
$726$	−8.89319e14	−0.163646
$727$	1.43942e15	0.262874	0.131437	−	0.991325i	$-0.458041\pi$
$727$	1.43942e15	0.262874	0.131437	+	0.991325i	$0.458041\pi$
$728$	3.43452e15	0.622505
$729$	2.05891e14	0.0370370
$730$	0	0
$731$	1.16916e16	2.07171
$732$	−9.21252e14	−0.162020
$733$	5.59957e15	0.977424	0.488712	−	0.872445i	$-0.337467\pi$
$733$	5.59957e15	0.977424	0.488712	+	0.872445i	$0.337467\pi$
$734$	−5.01763e15	−0.869301
$735$	0	0
$736$	−4.73624e15	−0.808362
$737$	−3.54314e15	−0.600229
$738$	2.00149e15	0.336544
$739$	2.27349e15	0.379446	0.189723	−	0.981838i	$-0.439241\pi$
$739$	2.27349e15	0.379446	0.189723	+	0.981838i	$0.439241\pi$
$740$	0	0
$741$	−1.38776e14	−0.0228199
$742$	−5.33958e15	−0.871537
$743$	−8.38494e14	−0.135851	−0.0679253	−	0.997690i	$-0.521638\pi$
$743$	−8.38494e14	−0.135851	−0.0679253	+	0.997690i	$0.521638\pi$
$744$	1.92852e15	0.310151
$745$	0	0
$746$	−2.15270e14	−0.0341129
$747$	−3.54928e15	−0.558312
$748$	−2.22958e15	−0.348148
$749$	7.89130e15	1.22320
$750$	0	0
$751$	4.57452e15	0.698756	0.349378	−	0.936982i	$-0.386393\pi$
$751$	4.57452e15	0.698756	0.349378	+	0.936982i	$0.386393\pi$
$752$	−7.01129e15	−1.06316
$753$	−3.72190e15	−0.560264
$754$	−8.05070e15	−1.20307
$755$	0	0
$756$	3.70565e14	0.0545751
$757$	2.03687e15	0.297808	0.148904	−	0.988852i	$-0.452425\pi$
$757$	2.03687e15	0.297808	0.148904	+	0.988852i	$0.452425\pi$
$758$	−4.10913e15	−0.596443
$759$	6.74825e15	0.972437
$760$	0	0
$761$	−1.18948e16	−1.68943	−0.844716	−	0.535215i	$-0.820230\pi$
$761$	−1.18948e16	−1.68943	−0.844716	+	0.535215i	$0.820230\pi$
$762$	8.13678e15	1.14736
$763$	−9.06972e15	−1.26972
$764$	1.34986e15	0.187619
$765$	0	0
$766$	2.35603e15	0.322792
$767$	3.09695e15	0.421269
$768$	−3.29090e15	−0.444455
$769$	−1.03701e16	−1.39056	−0.695280	−	0.718739i	$-0.744720\pi$
$769$	−1.03701e16	−1.39056	−0.695280	+	0.718739i	$0.744720\pi$
$770$	0	0
$771$	−2.88280e15	−0.381080
$772$	2.00380e15	0.263002
$773$	−5.86680e15	−0.764564	−0.382282	−	0.924046i	$-0.624862\pi$
$773$	−5.86680e15	−0.764564	−0.382282	+	0.924046i	$0.624862\pi$
$774$	5.33722e15	0.690621
$775$	0	0
$776$	−5.69246e15	−0.726208
$777$	−5.33750e15	−0.676118
$778$	5.48545e15	0.689961
$779$	3.82544e14	0.0477777
$780$	0	0
$781$	6.75216e15	0.831500
$782$	1.57083e16	1.92084
$783$	2.28290e15	0.277202
$784$	−5.82970e14	−0.0702923
$785$	0	0
$786$	3.12430e15	0.371475
$787$	−4.35191e15	−0.513829	−0.256915	−	0.966434i	$-0.582706\pi$
$787$	−4.35191e15	−0.513829	−0.256915	+	0.966434i	$0.582706\pi$
$788$	−1.18594e15	−0.139049
$789$	1.25794e14	0.0146466
$790$	0	0
$791$	−2.74856e15	−0.315598
$792$	2.67495e15	0.305019
$793$	6.64921e15	0.752952
$794$	2.65922e15	0.299049
$795$	0	0
$796$	7.79613e14	0.0864683
$797$	−3.97211e15	−0.437522	−0.218761	−	0.975778i	$-0.570202\pi$
$797$	−3.97211e15	−0.437522	−0.218761	+	0.975778i	$0.570202\pi$
$798$	3.27796e14	0.0358581
$799$	9.21257e15	1.00086
$800$	0	0
$801$	−8.67441e13	−0.00929526
$802$	1.26591e16	1.34724
$803$	−5.61075e15	−0.593042
$804$	−8.13495e14	−0.0853978
$805$	0	0
$806$	5.29618e15	0.548428
$807$	−9.09326e15	−0.935223
$808$	−1.26237e16	−1.28951
$809$	−9.34419e15	−0.948036	−0.474018	−	0.880515i	$-0.657197\pi$
$809$	−9.34419e15	−0.948036	−0.474018	+	0.880515i	$0.657197\pi$
$810$	0	0
$811$	−1.25244e16	−1.25355	−0.626777	−	0.779199i	$-0.715626\pi$
$811$	−1.25244e16	−1.25355	−0.626777	+	0.779199i	$0.715626\pi$
$812$	4.10879e15	0.408466
$813$	9.98130e14	0.0985574
$814$	1.46601e16	1.43781
$815$	0	0
$816$	8.08390e15	0.782213
$817$	1.02010e15	0.0980444
$818$	3.70238e15	0.353458
$819$	−2.67458e15	−0.253627
$820$	0	0
$821$	1.78922e16	1.67408	0.837042	−	0.547138i	$-0.184283\pi$
$821$	1.78922e16	1.67408	0.837042	+	0.547138i	$0.184283\pi$
$822$	1.19882e16	1.11419
$823$	−1.07441e16	−0.991910	−0.495955	−	0.868348i	$-0.665182\pi$
$823$	−1.07441e16	−0.991910	−0.495955	+	0.868348i	$0.665182\pi$
$824$	−1.18596e16	−1.08760
$825$	0	0
$826$	−7.31516e15	−0.661963
$827$	1.91483e16	1.72128	0.860638	−	0.509217i	$-0.170065\pi$
$827$	1.91483e16	1.72128	0.860638	+	0.509217i	$0.170065\pi$
$828$	1.54938e15	0.138354
$829$	−1.76350e16	−1.56432	−0.782158	−	0.623080i	$-0.785881\pi$
$829$	−1.76350e16	−1.56432	−0.782158	+	0.623080i	$0.785881\pi$
$830$	0	0
$831$	−5.07164e15	−0.443959
$832$	5.04621e15	0.438821
$833$	7.66001e14	0.0661731
$834$	−1.19866e16	−1.02869
$835$	0	0
$836$	−1.94532e14	−0.0164762
$837$	−1.50181e15	−0.126365
$838$	9.69271e15	0.810221
$839$	9.56616e15	0.794414	0.397207	−	0.917729i	$-0.369979\pi$
$839$	9.56616e15	0.794414	0.397207	+	0.917729i	$0.369979\pi$
$840$	0	0
$841$	1.31120e16	1.07471
$842$	−5.69748e15	−0.463945
$843$	2.51449e14	0.0203422
$844$	2.84353e15	0.228546
$845$	0	0
$846$	4.20553e15	0.333644
$847$	−3.27589e15	−0.258209
$848$	−1.14893e16	−0.899738
$849$	−1.16836e16	−0.909046
$850$	0	0
$851$	−2.23168e16	−1.71403
$852$	1.55028e15	0.118302
$853$	1.51540e16	1.14896	0.574482	−	0.818517i	$-0.305203\pi$
$853$	1.51540e16	1.14896	0.574482	+	0.818517i	$0.305203\pi$
$854$	−1.57058e16	−1.18315
$855$	0	0
$856$	1.30788e16	0.972659
$857$	1.68905e16	1.24810	0.624048	−	0.781386i	$-0.285487\pi$
$857$	1.68905e16	1.24810	0.624048	+	0.781386i	$0.285487\pi$
$858$	7.34605e15	0.539354
$859$	−3.39202e15	−0.247455	−0.123727	−	0.992316i	$-0.539485\pi$
$859$	−3.39202e15	−0.247455	−0.123727	+	0.992316i	$0.539485\pi$
$860$	0	0
$861$	7.37266e15	0.531015
$862$	−5.71656e15	−0.409115
$863$	8.29475e15	0.589853	0.294927	−	0.955520i	$-0.404705\pi$
$863$	8.29475e15	0.589853	0.294927	+	0.955520i	$0.404705\pi$
$864$	1.46201e15	0.103306
$865$	0	0
$866$	2.16205e16	1.50840
$867$	−2.29387e15	−0.159024
$868$	−2.70298e15	−0.186202
$869$	2.15026e16	1.47191
$870$	0	0
$871$	5.87147e15	0.396869
$872$	−1.50318e16	−1.00965
$873$	4.43293e15	0.295878
$874$	1.37056e15	0.0909042
$875$	0	0
$876$	−1.28821e15	−0.0843753
$877$	4.24748e15	0.276461	0.138230	−	0.990400i	$-0.455859\pi$
$877$	4.24748e15	0.276461	0.138230	+	0.990400i	$0.455859\pi$
$878$	−2.61411e16	−1.69085
$879$	−4.78138e15	−0.307336
$880$	0	0
$881$	−4.11863e15	−0.261448	−0.130724	−	0.991419i	$-0.541730\pi$
$881$	−4.11863e15	−0.261448	−0.130724	+	0.991419i	$0.541730\pi$
$882$	3.49678e14	0.0220593
$883$	1.48075e16	0.928319	0.464160	−	0.885752i	$-0.346356\pi$
$883$	1.48075e16	0.928319	0.464160	+	0.885752i	$0.346356\pi$
$884$	3.69471e15	0.230194
$885$	0	0
$886$	2.73779e16	1.68467
$887$	8.21370e15	0.502295	0.251147	−	0.967949i	$-0.419192\pi$
$887$	8.21370e15	0.502295	0.251147	+	0.967949i	$0.419192\pi$
$888$	−8.84619e15	−0.537632
$889$	2.99726e16	1.81036
$890$	0	0
$891$	−2.08308e15	−0.124274
$892$	−8.69656e15	−0.515633
$893$	8.03803e14	0.0473660
$894$	3.57116e15	0.209148
$895$	0	0
$896$	−2.14664e16	−1.24184
$897$	−1.11828e16	−0.642971
$898$	8.95664e15	0.511829
$899$	−1.66519e16	−0.945772
$900$	0	0
$901$	1.50965e16	0.847013
$902$	−2.02498e16	−1.12924
$903$	1.96601e16	1.08969
$904$	−4.55537e15	−0.250956
$905$	0	0
$906$	1.97387e16	1.07427
$907$	−2.79307e16	−1.51092	−0.755462	−	0.655193i	$-0.772587\pi$
$907$	−2.79307e16	−1.51092	−0.755462	+	0.655193i	$0.772587\pi$
$908$	1.74572e15	0.0938647
$909$	9.83057e15	0.525385
$910$	0	0
$911$	−9.25991e15	−0.488940	−0.244470	−	0.969657i	$-0.578614\pi$
$911$	−9.25991e15	−0.488940	−0.244470	+	0.969657i	$0.578614\pi$
$912$	7.05326e14	0.0370184
$913$	3.59095e16	1.87335
$914$	−1.39323e16	−0.722466
$915$	0	0
$916$	−6.71396e15	−0.343996
$917$	1.15087e16	0.586129
$918$	−4.84890e15	−0.245476
$919$	3.66664e16	1.84516	0.922578	−	0.385810i	$-0.126078\pi$
$919$	3.66664e16	1.84516	0.922578	+	0.385810i	$0.126078\pi$
$920$	0	0
$921$	5.38395e15	0.267715
$922$	−1.41407e15	−0.0698956
$923$	−1.11893e16	−0.549784
$924$	−3.74916e15	−0.183121
$925$	0	0
$926$	−3.71317e15	−0.179219
$927$	9.23553e15	0.443122
$928$	1.62106e16	0.773187
$929$	−2.64404e16	−1.25367	−0.626834	−	0.779153i	$-0.715649\pi$
$929$	−2.64404e16	−1.25367	−0.626834	+	0.779153i	$0.715649\pi$
$930$	0	0
$931$	6.68341e13	0.00313166
$932$	1.10301e16	0.513798
$933$	−1.15458e16	−0.534657
$934$	−1.16818e16	−0.537778
$935$	0	0
$936$	−4.43276e15	−0.201677
$937$	2.56311e16	1.15931	0.579655	−	0.814862i	$-0.303187\pi$
$937$	2.56311e16	1.15931	0.579655	+	0.814862i	$0.303187\pi$
$938$	−1.38687e16	−0.623621
$939$	−5.82836e15	−0.260547
$940$	0	0
$941$	7.37922e15	0.326038	0.163019	−	0.986623i	$-0.447877\pi$
$941$	7.37922e15	0.326038	0.163019	+	0.986623i	$0.447877\pi$
$942$	−1.13677e16	−0.499334
$943$	3.08260e16	1.34618
$944$	−1.57402e16	−0.683383
$945$	0	0
$946$	−5.39988e16	−2.31730
$947$	1.91401e16	0.816619	0.408309	−	0.912844i	$-0.366118\pi$
$947$	1.91401e16	0.816619	0.408309	+	0.912844i	$0.366118\pi$
$948$	4.93693e15	0.209417
$949$	9.29778e15	0.392117
$950$	0	0
$951$	−1.95096e16	−0.813308
$952$	2.29364e16	0.950651
$953$	−2.15693e16	−0.888845	−0.444422	−	0.895817i	$-0.646591\pi$
$953$	−2.15693e16	−0.888845	−0.444422	+	0.895817i	$0.646591\pi$
$954$	6.89153e15	0.282358
$955$	0	0
$956$	2.47831e15	0.100378
$957$	−2.30970e16	−0.930122
$958$	−3.31026e16	−1.32542
$959$	4.41595e16	1.75801
$960$	0	0
$961$	−1.44539e16	−0.568863
$962$	−2.42937e16	−0.950672
$963$	−1.01849e16	−0.396290
$964$	−5.48891e15	−0.212355
$965$	0	0
$966$	2.64143e16	1.01033
$967$	2.47540e16	0.941456	0.470728	−	0.882278i	$-0.343991\pi$
$967$	2.47540e16	0.941456	0.470728	+	0.882278i	$0.343991\pi$
$968$	−5.42934e15	−0.205321
$969$	−9.26771e14	−0.0348491
$970$	0	0
$971$	−4.43638e16	−1.64939	−0.824695	−	0.565578i	$-0.808653\pi$
$971$	−4.43638e16	−1.64939	−0.824695	+	0.565578i	$0.808653\pi$
$972$	−4.78270e14	−0.0176811
$973$	−4.41539e16	−1.62311
$974$	−4.90422e16	−1.79265
$975$	0	0
$976$	−3.37945e16	−1.22144
$977$	−2.48061e16	−0.891536	−0.445768	−	0.895149i	$-0.647069\pi$
$977$	−2.48061e16	−0.891536	−0.445768	+	0.895149i	$0.647069\pi$
$978$	7.39576e15	0.264313
$979$	8.77626e14	0.0311892
$980$	0	0
$981$	1.17058e16	0.411361
$982$	−4.16964e16	−1.45709
$983$	−4.74780e16	−1.64986	−0.824932	−	0.565232i	$-0.808787\pi$
$983$	−4.74780e16	−1.64986	−0.824932	+	0.565232i	$0.808787\pi$
$984$	1.22192e16	0.422249
$985$	0	0
$986$	−5.37641e16	−1.83725
$987$	1.54915e16	0.526439
$988$	3.22366e14	0.0108940
$989$	8.22015e16	2.76249
$990$	0	0
$991$	2.01588e16	0.669976	0.334988	−	0.942222i	$-0.391268\pi$
$991$	2.01588e16	0.669976	0.334988	+	0.942222i	$0.391268\pi$
$992$	−1.06642e16	−0.352463
$993$	1.78842e16	0.587825
$994$	2.64296e16	0.863904
$995$	0	0
$996$	8.24472e15	0.266532
$997$	−3.44516e16	−1.10761	−0.553805	−	0.832647i	$-0.686825\pi$
$997$	−3.44516e16	−1.10761	−0.553805	+	0.832647i	$0.686825\pi$
$998$	−1.17036e16	−0.374199
$999$	6.88885e15	0.219047

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.12.a.d.1.2		2
3.2	odd	2		225.12.a.g.1.1		2
5.2	odd	4		75.12.b.c.49.4		4
5.3	odd	4		75.12.b.c.49.1		4
5.4	even	2		15.12.a.b.1.1	✓	2
15.2	even	4		225.12.b.h.199.1		4
15.8	even	4		225.12.b.h.199.4		4
15.14	odd	2		45.12.a.e.1.2		2
20.19	odd	2		240.12.a.j.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.12.a.b.1.1	✓	2	5.4	even	2
45.12.a.e.1.2		2	15.14	odd	2
75.12.a.d.1.2		2	1.1	even	1	trivial
75.12.b.c.49.1		4	5.3	odd	4
75.12.b.c.49.4		4	5.2	odd	4
225.12.a.g.1.1		2	3.2	odd	2
225.12.b.h.199.1		4	15.2	even	4
225.12.b.h.199.4		4	15.8	even	4
240.12.a.j.1.1		2	20.19	odd	2

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:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	75.a (trivial)

\(f(q)\)	\(=\)	\(q+51.1123 q^{2} -243.000 q^{3} +564.472 q^{4} -12420.3 q^{6} -45751.3 q^{7} -75826.6 q^{8} +59049.0 q^{9} +O(q^{10})\)	\(q+51.1123 q^{2} -243.000 q^{3} +564.472 q^{4} -12420.3 q^{6} -45751.3 q^{7} -75826.6 q^{8} +59049.0 q^{9} -597423. q^{11} -137167. q^{12} +990011. q^{13} -2.33846e6 q^{14} -5.03171e6 q^{16} +6.61148e6 q^{17} +3.01813e6 q^{18} +576856. q^{19} +1.11176e7 q^{21} -3.05357e7 q^{22} +4.64840e7 q^{23} +1.84259e7 q^{24} +5.06018e7 q^{26} -1.43489e7 q^{27} -2.58253e7 q^{28} -1.59099e8 q^{29} +1.04664e8 q^{31} -1.01890e8 q^{32} +1.45174e8 q^{33} +3.37928e8 q^{34} +3.33315e7 q^{36} -4.80096e8 q^{37} +2.94845e7 q^{38} -2.40573e8 q^{39} +6.63154e8 q^{41} +5.68245e8 q^{42} +1.76838e9 q^{43} -3.37228e8 q^{44} +2.37591e9 q^{46} +1.39342e9 q^{47} +1.22271e9 q^{48} +1.15859e8 q^{49} -1.60659e9 q^{51} +5.58833e8 q^{52} +2.28338e9 q^{53} -7.33406e8 q^{54} +3.46917e9 q^{56} -1.40176e8 q^{57} -8.13193e9 q^{58} +3.12820e9 q^{59} +6.71630e9 q^{61} +5.34962e9 q^{62} -2.70157e9 q^{63} +5.09713e9 q^{64} +7.42017e9 q^{66} +5.93071e9 q^{67} +3.73199e9 q^{68} -1.12956e10 q^{69} -1.13022e10 q^{71} -4.47749e9 q^{72} +9.39160e9 q^{73} -2.45388e10 q^{74} +3.25619e8 q^{76} +2.73329e10 q^{77} -1.22962e10 q^{78} -3.59922e10 q^{79} +3.48678e9 q^{81} +3.38953e10 q^{82} -6.01074e10 q^{83} +6.27556e9 q^{84} +9.03863e10 q^{86} +3.86611e10 q^{87} +4.53005e10 q^{88} -1.46902e9 q^{89} -4.52943e10 q^{91} +2.62389e10 q^{92} -2.54333e10 q^{93} +7.12210e10 q^{94} +2.47592e10 q^{96} +7.50721e10 q^{97} +5.92183e9 q^{98} -3.52772e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 22 q^{2} - 486 q^{3} - 636 q^{4} - 5346 q^{6} + 10864 q^{7} + 18744 q^{8} + 118098 q^{9}+O(q^{10}) \) 2 * q + 22 * q^2 - 486 * q^3 - 636 * q^4 - 5346 * q^6 + 10864 * q^7 + 18744 * q^8 + 118098 * q^9	\( 2 q + 22 q^{2} - 486 q^{3} - 636 q^{4} - 5346 q^{6} + 10864 q^{7} + 18744 q^{8} + 118098 q^{9} - 361792 q^{11} + 154548 q^{12} + 2133732 q^{13} - 3986664 q^{14} - 5326320 q^{16} + 7804588 q^{17} + 1299078 q^{18} - 15562224 q^{19} - 2639952 q^{21} - 37395424 q^{22} + 37450248 q^{23} - 4554792 q^{24} + 17305364 q^{26} - 28697814 q^{27} - 93790448 q^{28} - 70320668 q^{29} + 298584872 q^{31} - 286993696 q^{32} + 87915456 q^{33} + 303194188 q^{34} - 37555164 q^{36} - 236000956 q^{37} + 499330888 q^{38} - 518496876 q^{39} - 464942588 q^{41} + 968759352 q^{42} + 242208600 q^{43} - 620095808 q^{44} + 2638898832 q^{46} + 4375796920 q^{47} + 1294295760 q^{48} + 1343830130 q^{49} - 1896514884 q^{51} - 814171912 q^{52} + 2189541388 q^{53} - 315675954 q^{54} + 8823318240 q^{56} + 3781620432 q^{57} - 10716478004 q^{58} - 5480385856 q^{59} + 14557903980 q^{61} - 295874592 q^{62} + 641508336 q^{63} + 11089288256 q^{64} + 9087088032 q^{66} + 15918388888 q^{67} + 2299702856 q^{68} - 9100410264 q^{69} + 1120561024 q^{71} + 1106814456 q^{72} + 24521574348 q^{73} - 31645012364 q^{74} + 19700124976 q^{76} + 40673194368 q^{77} - 4205203452 q^{78} - 79243055560 q^{79} + 6973568802 q^{81} + 66736852348 q^{82} - 9245226696 q^{83} + 22791078864 q^{84} + 134816793608 q^{86} + 17087922324 q^{87} + 67584257664 q^{88} + 22117321236 q^{89} + 19457850112 q^{91} + 37083635424 q^{92} - 72556123896 q^{93} - 15603010256 q^{94} + 69739468128 q^{96} + 160363673468 q^{97} - 29827277722 q^{98} - 21363455808 q^{99}+O(q^{100}) \) 2 * q + 22 * q^2 - 486 * q^3 - 636 * q^4 - 5346 * q^6 + 10864 * q^7 + 18744 * q^8 + 118098 * q^9 - 361792 * q^11 + 154548 * q^12 + 2133732 * q^13 - 3986664 * q^14 - 5326320 * q^16 + 7804588 * q^17 + 1299078 * q^18 - 15562224 * q^19 - 2639952 * q^21 - 37395424 * q^22 + 37450248 * q^23 - 4554792 * q^24 + 17305364 * q^26 - 28697814 * q^27 - 93790448 * q^28 - 70320668 * q^29 + 298584872 * q^31 - 286993696 * q^32 + 87915456 * q^33 + 303194188 * q^34 - 37555164 * q^36 - 236000956 * q^37 + 499330888 * q^38 - 518496876 * q^39 - 464942588 * q^41 + 968759352 * q^42 + 242208600 * q^43 - 620095808 * q^44 + 2638898832 * q^46 + 4375796920 * q^47 + 1294295760 * q^48 + 1343830130 * q^49 - 1896514884 * q^51 - 814171912 * q^52 + 2189541388 * q^53 - 315675954 * q^54 + 8823318240 * q^56 + 3781620432 * q^57 - 10716478004 * q^58 - 5480385856 * q^59 + 14557903980 * q^61 - 295874592 * q^62 + 641508336 * q^63 + 11089288256 * q^64 + 9087088032 * q^66 + 15918388888 * q^67 + 2299702856 * q^68 - 9100410264 * q^69 + 1120561024 * q^71 + 1106814456 * q^72 + 24521574348 * q^73 - 31645012364 * q^74 + 19700124976 * q^76 + 40673194368 * q^77 - 4205203452 * q^78 - 79243055560 * q^79 + 6973568802 * q^81 + 66736852348 * q^82 - 9245226696 * q^83 + 22791078864 * q^84 + 134816793608 * q^86 + 17087922324 * q^87 + 67584257664 * q^88 + 22117321236 * q^89 + 19457850112 * q^91 + 37083635424 * q^92 - 72556123896 * q^93 - 15603010256 * q^94 + 69739468128 * q^96 + 160363673468 * q^97 - 29827277722 * q^98 - 21363455808 * q^99

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