LMFDB - Embedded newform 1773.2.a.f.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1773,2,Mod(1,1773)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1773.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1773, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1773 = 3^{2} \cdot 197 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1773.a (trivial)

Newform invariants

comment:select newform

sage:traces = [10,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	yes
Analytic conductor:	$14.1574762784$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 15x^{8} - x^{7} + 78x^{6} + 7x^{5} - 165x^{4} - 15x^{3} + 123x^{2} + 9x - 26 $ x^10 - 15x^8 - x^7 + 78x^6 + 7x^5 - 165x^4 - 15x^3 + 123x^2 + 9*x - 26
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 197)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.82964$ of defining polynomial
Character	$\chi$	$=$	1773.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.82964 q^{2} +1.34760 q^{4} -2.39480 q^{5} -2.47846 q^{7} +1.19366 q^{8} +4.38163 q^{10} -2.44912 q^{11} +0.427303 q^{13} +4.53471 q^{14} -4.87917 q^{16} +3.61425 q^{17} -0.675704 q^{19} -3.22723 q^{20} +4.48103 q^{22} -8.55393 q^{23} +0.735052 q^{25} -0.781813 q^{26} -3.33998 q^{28} -4.41033 q^{29} +0.899928 q^{31} +6.53984 q^{32} -6.61279 q^{34} +5.93542 q^{35} -9.25303 q^{37} +1.23630 q^{38} -2.85857 q^{40} +3.48904 q^{41} +4.73199 q^{43} -3.30044 q^{44} +15.6507 q^{46} -9.87379 q^{47} -0.857212 q^{49} -1.34488 q^{50} +0.575834 q^{52} -0.432240 q^{53} +5.86515 q^{55} -2.95844 q^{56} +8.06933 q^{58} +0.232893 q^{59} -8.06257 q^{61} -1.64655 q^{62} -2.20723 q^{64} -1.02330 q^{65} +12.2707 q^{67} +4.87056 q^{68} -10.8597 q^{70} +14.4805 q^{71} +10.8630 q^{73} +16.9298 q^{74} -0.910579 q^{76} +6.07007 q^{77} +8.35190 q^{79} +11.6846 q^{80} -6.38370 q^{82} +0.446491 q^{83} -8.65538 q^{85} -8.65785 q^{86} -2.92342 q^{88} -16.9275 q^{89} -1.05906 q^{91} -11.5273 q^{92} +18.0655 q^{94} +1.61817 q^{95} +6.50799 q^{97} +1.56839 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 10 q^{4} - 2 q^{5} + 11 q^{7} + 3 q^{8} - 2 q^{10} - 2 q^{11} + 8 q^{13} + 9 q^{14} - 2 q^{16} + 3 q^{17} + 17 q^{19} + 2 q^{20} + 3 q^{22} + 4 q^{23} + 6 q^{25} + 13 q^{26} + 16 q^{28} + 9 q^{29}+ \cdots - 22 q^{98}+O(q^{100}) $ 10 * q + 10 * q^4 - 2 * q^5 + 11 * q^7 + 3 * q^8 - 2 * q^10 - 2 * q^11 + 8 * q^13 + 9 * q^14 - 2 * q^16 + 3 * q^17 + 17 * q^19 + 2 * q^20 + 3 * q^22 + 4 * q^23 + 6 * q^25 + 13 * q^26 + 16 * q^28 + 9 * q^29 + 20 * q^31 + 16 * q^32 - 8 * q^34 + 12 * q^35 + 12 * q^37 + 15 * q^38 - 15 * q^40 + 18 * q^41 + 11 * q^43 + 24 * q^44 - 13 * q^46 - 5 * q^47 + q^49 + 24 * q^50 + 9 * q^52 + 6 * q^53 - 17 * q^55 + 6 * q^56 - 15 * q^58 + q^59 + 4 * q^61 + 16 * q^62 - 17 * q^64 + 17 * q^65 + 46 * q^67 - 15 * q^68 + 25 * q^70 + 5 * q^71 + 25 * q^73 - 3 * q^74 + 49 * q^76 + 14 * q^77 - q^79 - 9 * q^80 + 52 * q^82 - 28 * q^83 - 30 * q^85 - 27 * q^86 + 9 * q^88 + 6 * q^89 + 51 * q^91 - 32 * q^92 - 3 * q^94 + 13 * q^95 + 2 * q^97 - 22 * q^98

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 $

$2$	−1.82964	−1.29375	−0.646877	−	0.762594i	$-0.723925\pi$
$2$	−1.82964	−1.29375	−0.646877	+	0.762594i	$0.723925\pi$
$3$	0	0
$3$	0	0
$4$	1.34760	0.673800
$5$	−2.39480	−1.07099	−0.535493	−	0.844540i	$-0.679874\pi$
$5$	−2.39480	−1.07099	−0.535493	+	0.844540i	$0.679874\pi$
$6$	0	0
$7$	−2.47846	−0.936772	−0.468386	−	0.883524i	$-0.655164\pi$
$7$	−2.47846	−0.936772	−0.468386	+	0.883524i	$0.655164\pi$
$8$	1.19366	0.422022
$9$	0	0
$10$	4.38163	1.38559
$11$	−2.44912	−0.738438	−0.369219	−	0.929342i	$-0.620375\pi$
$11$	−2.44912	−0.738438	−0.369219	+	0.929342i	$0.620375\pi$
$12$	0	0
$13$	0.427303	0.118513	0.0592563	−	0.998243i	$-0.481127\pi$
$13$	0.427303	0.118513	0.0592563	+	0.998243i	$0.481127\pi$
$14$	4.53471	1.21195
$15$	0	0
$16$	−4.87917	−1.21979
$17$	3.61425	0.876583	0.438292	−	0.898833i	$-0.355584\pi$
$17$	3.61425	0.876583	0.438292	+	0.898833i	$0.355584\pi$
$18$	0	0
$19$	−0.675704	−0.155017	−0.0775085	−	0.996992i	$-0.524697\pi$
$19$	−0.675704	−0.155017	−0.0775085	+	0.996992i	$0.524697\pi$
$20$	−3.22723	−0.721630
$21$	0	0
$22$	4.48103	0.955358
$23$	−8.55393	−1.78362	−0.891809	−	0.452412i	$-0.850564\pi$
$23$	−8.55393	−1.78362	−0.891809	+	0.452412i	$0.850564\pi$
$24$	0	0
$25$	0.735052	0.147010
$26$	−0.781813	−0.153326
$27$	0	0
$28$	−3.33998	−0.631197
$29$	−4.41033	−0.818977	−0.409489	−	0.912315i	$-0.634293\pi$
$29$	−4.41033	−0.818977	−0.409489	+	0.912315i	$0.634293\pi$
$30$	0	0
$31$	0.899928	0.161632	0.0808159	−	0.996729i	$-0.474247\pi$
$31$	0.899928	0.161632	0.0808159	+	0.996729i	$0.474247\pi$
$32$	6.53984	1.15609
$33$	0	0
$34$	−6.61279	−1.13408
$35$	5.93542	1.00327
$36$	0	0
$37$	−9.25303	−1.52119	−0.760594	−	0.649227i	$-0.775092\pi$
$37$	−9.25303	−1.52119	−0.760594	+	0.649227i	$0.775092\pi$
$38$	1.23630	0.200554
$39$	0	0
$40$	−2.85857	−0.451980
$41$	3.48904	0.544896	0.272448	−	0.962171i	$-0.412167\pi$
$41$	3.48904	0.544896	0.272448	+	0.962171i	$0.412167\pi$
$42$	0	0
$43$	4.73199	0.721621	0.360811	−	0.932639i	$-0.382500\pi$
$43$	4.73199	0.721621	0.360811	+	0.932639i	$0.382500\pi$
$44$	−3.30044	−0.497560
$45$	0	0
$46$	15.6507	2.30756
$47$	−9.87379	−1.44024	−0.720120	−	0.693850i	$-0.755913\pi$
$47$	−9.87379	−1.44024	−0.720120	+	0.693850i	$0.755913\pi$
$48$	0	0
$49$	−0.857212	−0.122459
$50$	−1.34488	−0.190195
$51$	0	0
$52$	0.575834	0.0798538
$53$	−0.432240	−0.0593728	−0.0296864	−	0.999559i	$-0.509451\pi$
$53$	−0.432240	−0.0593728	−0.0296864	+	0.999559i	$0.509451\pi$
$54$	0	0
$55$	5.86515	0.790857
$56$	−2.95844	−0.395339
$57$	0	0
$58$	8.06933	1.05956
$59$	0.232893	0.0303201	0.0151600	−	0.999885i	$-0.495174\pi$
$59$	0.232893	0.0303201	0.0151600	+	0.999885i	$0.495174\pi$
$60$	0	0
$61$	−8.06257	−1.03231	−0.516153	−	0.856496i	$-0.672637\pi$
$61$	−8.06257	−1.03231	−0.516153	+	0.856496i	$0.672637\pi$
$62$	−1.64655	−0.209112
$63$	0	0
$64$	−2.20723	−0.275904
$65$	−1.02330	−0.126925
$66$	0	0
$67$	12.2707	1.49911	0.749554	−	0.661944i	$-0.230268\pi$
$67$	12.2707	1.49911	0.749554	+	0.661944i	$0.230268\pi$
$68$	4.87056	0.590642
$69$	0	0
$70$	−10.8597	−1.29798
$71$	14.4805	1.71852	0.859259	−	0.511540i	$-0.170925\pi$
$71$	14.4805	1.71852	0.859259	+	0.511540i	$0.170925\pi$
$72$	0	0
$73$	10.8630	1.27142	0.635708	−	0.771930i	$-0.280708\pi$
$73$	10.8630	1.27142	0.635708	+	0.771930i	$0.280708\pi$
$74$	16.9298	1.96804
$75$	0	0
$76$	−0.910579	−0.104451
$77$	6.07007	0.691748
$78$	0	0
$79$	8.35190	0.939663	0.469831	−	0.882756i	$-0.344315\pi$
$79$	8.35190	0.939663	0.469831	+	0.882756i	$0.344315\pi$
$80$	11.6846	1.30638
$81$	0	0
$82$	−6.38370	−0.704961
$83$	0.446491	0.0490087	0.0245044	−	0.999700i	$-0.492199\pi$
$83$	0.446491	0.0490087	0.0245044	+	0.999700i	$0.492199\pi$
$84$	0	0
$85$	−8.65538	−0.938808
$86$	−8.65785	−0.933600
$87$	0	0
$88$	−2.92342	−0.311637
$89$	−16.9275	−1.79431	−0.897157	−	0.441711i	$-0.854372\pi$
$89$	−16.9275	−1.79431	−0.897157	+	0.441711i	$0.854372\pi$
$90$	0	0
$91$	−1.05906	−0.111019
$92$	−11.5273	−1.20180
$93$	0	0
$94$	18.0655	1.86332
$95$	1.61817	0.166021
$96$	0	0
$97$	6.50799	0.660786	0.330393	−	0.943843i	$-0.392819\pi$
$97$	6.50799	0.660786	0.330393	+	0.943843i	$0.392819\pi$
$98$	1.56839	0.158432
$99$	0	0
$100$	0.990556	0.0990556
$101$	−7.81122	−0.777245	−0.388623	−	0.921397i	$-0.627049\pi$
$101$	−7.81122	−0.777245	−0.388623	+	0.921397i	$0.627049\pi$
$102$	0	0
$103$	0.291943	0.0287660	0.0143830	−	0.999897i	$-0.495422\pi$
$103$	0.291943	0.0287660	0.0143830	+	0.999897i	$0.495422\pi$
$104$	0.510054	0.0500150
$105$	0	0
$106$	0.790846	0.0768138
$107$	15.2591	1.47515	0.737574	−	0.675266i	$-0.235971\pi$
$107$	15.2591	1.47515	0.737574	+	0.675266i	$0.235971\pi$
$108$	0	0
$109$	−4.64110	−0.444536	−0.222268	−	0.974986i	$-0.571346\pi$
$109$	−4.64110	−0.444536	−0.222268	+	0.974986i	$0.571346\pi$
$110$	−10.7311	−1.02317
$111$	0	0
$112$	12.0929	1.14267
$113$	−3.00447	−0.282637	−0.141318	−	0.989964i	$-0.545134\pi$
$113$	−3.00447	−0.282637	−0.141318	+	0.989964i	$0.545134\pi$
$114$	0	0
$115$	20.4849	1.91023
$116$	−5.94336	−0.551827
$117$	0	0
$118$	−0.426111	−0.0392267
$119$	−8.95778	−0.821158
$120$	0	0
$121$	−5.00180	−0.454709
$122$	14.7516	1.33555
$123$	0	0
$124$	1.21274	0.108908
$125$	10.2137	0.913540
$126$	0	0
$127$	−5.47730	−0.486031	−0.243016	−	0.970022i	$-0.578137\pi$
$127$	−5.47730	−0.486031	−0.243016	+	0.970022i	$0.578137\pi$
$128$	−9.04122	−0.799139
$129$	0	0
$130$	1.87228	0.164210
$131$	9.82962	0.858818	0.429409	−	0.903110i	$-0.358722\pi$
$131$	9.82962	0.858818	0.429409	+	0.903110i	$0.358722\pi$
$132$	0	0
$133$	1.67471	0.145216
$134$	−22.4511	−1.93948
$135$	0	0
$136$	4.31418	0.369938
$137$	5.13739	0.438917	0.219458	−	0.975622i	$-0.429571\pi$
$137$	5.13739	0.438917	0.219458	+	0.975622i	$0.429571\pi$
$138$	0	0
$139$	12.0692	1.02369	0.511847	−	0.859076i	$-0.328961\pi$
$139$	12.0692	1.02369	0.511847	+	0.859076i	$0.328961\pi$
$140$	7.99857	0.676003
$141$	0	0
$142$	−26.4942	−2.22334
$143$	−1.04652	−0.0875142
$144$	0	0
$145$	10.5618	0.877113
$146$	−19.8754	−1.64490
$147$	0	0
$148$	−12.4694	−1.02498
$149$	21.8887	1.79319	0.896597	−	0.442847i	$-0.146032\pi$
$149$	21.8887	1.79319	0.896597	+	0.442847i	$0.146032\pi$
$150$	0	0
$151$	4.21008	0.342611	0.171306	−	0.985218i	$-0.445201\pi$
$151$	4.21008	0.342611	0.171306	+	0.985218i	$0.445201\pi$
$152$	−0.806560	−0.0654207
$153$	0	0
$154$	−11.1061	−0.894952
$155$	−2.15514	−0.173105
$156$	0	0
$157$	−9.05645	−0.722783	−0.361392	−	0.932414i	$-0.617698\pi$
$157$	−9.05645	−0.722783	−0.361392	+	0.932414i	$0.617698\pi$
$158$	−15.2810	−1.21569
$159$	0	0
$160$	−15.6616	−1.23816
$161$	21.2006	1.67084
$162$	0	0
$163$	7.80210	0.611108	0.305554	−	0.952175i	$-0.401158\pi$
$163$	7.80210	0.611108	0.305554	+	0.952175i	$0.401158\pi$
$164$	4.70183	0.367151
$165$	0	0
$166$	−0.816920	−0.0634053
$167$	−2.46900	−0.191057	−0.0955286	−	0.995427i	$-0.530454\pi$
$167$	−2.46900	−0.191057	−0.0955286	+	0.995427i	$0.530454\pi$
$168$	0	0
$169$	−12.8174	−0.985955
$170$	15.8363	1.21459
$171$	0	0
$172$	6.37683	0.486228
$173$	14.5901	1.10926	0.554632	−	0.832096i	$-0.312859\pi$
$173$	14.5901	1.10926	0.554632	+	0.832096i	$0.312859\pi$
$174$	0	0
$175$	−1.82180	−0.137715
$176$	11.9497	0.900742
$177$	0	0
$178$	30.9714	2.32140
$179$	−1.85653	−0.138764	−0.0693820	−	0.997590i	$-0.522103\pi$
$179$	−1.85653	−0.138764	−0.0693820	+	0.997590i	$0.522103\pi$
$180$	0	0
$181$	21.8401	1.62336	0.811680	−	0.584103i	$-0.198553\pi$
$181$	21.8401	1.62336	0.811680	+	0.584103i	$0.198553\pi$
$182$	1.93770	0.143632
$183$	0	0
$184$	−10.2105	−0.752727
$185$	22.1591	1.62917
$186$	0	0
$187$	−8.85173	−0.647303
$188$	−13.3059	−0.970434
$189$	0	0
$190$	−2.96068	−0.214790
$191$	6.86482	0.496721	0.248361	−	0.968668i	$-0.420108\pi$
$191$	6.86482	0.496721	0.248361	+	0.968668i	$0.420108\pi$
$192$	0	0
$193$	−9.21587	−0.663373	−0.331687	−	0.943390i	$-0.607618\pi$
$193$	−9.21587	−0.663373	−0.331687	+	0.943390i	$0.607618\pi$
$194$	−11.9073	−0.854895
$195$	0	0
$196$	−1.15518	−0.0825129
$197$	−1.00000	−0.0712470
$197$	−1.00000	−0.0712470
$198$	0	0
$199$	−20.7712	−1.47243	−0.736216	−	0.676746i	$-0.763389\pi$
$199$	−20.7712	−1.47243	−0.736216	+	0.676746i	$0.763389\pi$
$200$	0.877402	0.0620417
$201$	0	0
$202$	14.2918	1.00556
$203$	10.9308	0.767195
$204$	0	0
$205$	−8.35553	−0.583576
$206$	−0.534151	−0.0372161
$207$	0	0
$208$	−2.08489	−0.144561
$209$	1.65488	0.114471
$210$	0	0
$211$	7.15252	0.492400	0.246200	−	0.969219i	$-0.420818\pi$
$211$	7.15252	0.492400	0.246200	+	0.969219i	$0.420818\pi$
$212$	−0.582487	−0.0400054
$213$	0	0
$214$	−27.9187	−1.90848
$215$	−11.3321	−0.772846
$216$	0	0
$217$	−2.23044	−0.151412
$218$	8.49156	0.575121
$219$	0	0
$220$	7.90388	0.532880
$221$	1.54438	0.103886
$222$	0	0
$223$	15.1364	1.01361	0.506803	−	0.862062i	$-0.330827\pi$
$223$	15.1364	1.01361	0.506803	+	0.862062i	$0.330827\pi$
$224$	−16.2088	−1.08299
$225$	0	0
$226$	5.49711	0.365663
$227$	23.7864	1.57876	0.789380	−	0.613905i	$-0.210402\pi$
$227$	23.7864	1.57876	0.789380	+	0.613905i	$0.210402\pi$
$228$	0	0
$229$	−19.8173	−1.30956	−0.654781	−	0.755819i	$-0.727239\pi$
$229$	−19.8173	−1.30956	−0.654781	+	0.755819i	$0.727239\pi$
$230$	−37.4802	−2.47137
$231$	0	0
$232$	−5.26443	−0.345627
$233$	29.0910	1.90581	0.952907	−	0.303261i	$-0.0980754\pi$
$233$	29.0910	1.90581	0.952907	+	0.303261i	$0.0980754\pi$
$234$	0	0
$235$	23.6457	1.54248
$236$	0.313847	0.0204297
$237$	0	0
$238$	16.3896	1.06238
$239$	17.2766	1.11753	0.558765	−	0.829326i	$-0.311275\pi$
$239$	17.2766	1.11753	0.558765	+	0.829326i	$0.311275\pi$
$240$	0	0
$241$	4.48258	0.288749	0.144374	−	0.989523i	$-0.453883\pi$
$241$	4.48258	0.288749	0.144374	+	0.989523i	$0.453883\pi$
$242$	9.15151	0.588281
$243$	0	0
$244$	−10.8651	−0.695569
$245$	2.05285	0.131152
$246$	0	0
$247$	−0.288730	−0.0183715
$248$	1.07421	0.0682122
$249$	0	0
$250$	−18.6874	−1.18190
$251$	−7.48094	−0.472193	−0.236097	−	0.971730i	$-0.575868\pi$
$251$	−7.48094	−0.472193	−0.236097	+	0.971730i	$0.575868\pi$
$252$	0	0
$253$	20.9496	1.31709
$254$	10.0215	0.628805
$255$	0	0
$256$	20.9567	1.30979
$257$	−26.1724	−1.63259	−0.816296	−	0.577634i	$-0.803976\pi$
$257$	−26.1724	−1.63259	−0.816296	+	0.577634i	$0.803976\pi$
$258$	0	0
$259$	22.9333	1.42501
$260$	−1.37901	−0.0855223
$261$	0	0
$262$	−17.9847	−1.11110
$263$	2.11510	0.130422	0.0652112	−	0.997871i	$-0.479228\pi$
$263$	2.11510	0.130422	0.0652112	+	0.997871i	$0.479228\pi$
$264$	0	0
$265$	1.03513	0.0635874
$266$	−3.06412	−0.187873
$267$	0	0
$268$	16.5360	1.01010
$269$	−5.43243	−0.331221	−0.165611	−	0.986191i	$-0.552959\pi$
$269$	−5.43243	−0.331221	−0.165611	+	0.986191i	$0.552959\pi$
$270$	0	0
$271$	8.88131	0.539501	0.269751	−	0.962930i	$-0.413059\pi$
$271$	8.88131	0.539501	0.269751	+	0.962930i	$0.413059\pi$
$272$	−17.6345	−1.06925
$273$	0	0
$274$	−9.39960	−0.567850
$275$	−1.80023	−0.108558
$276$	0	0
$277$	3.85160	0.231420	0.115710	−	0.993283i	$-0.463086\pi$
$277$	3.85160	0.231420	0.115710	+	0.993283i	$0.463086\pi$
$278$	−22.0823	−1.32441
$279$	0	0
$280$	7.08487	0.423402
$281$	16.6051	0.990575	0.495288	−	0.868729i	$-0.335063\pi$
$281$	16.6051	0.990575	0.495288	+	0.868729i	$0.335063\pi$
$282$	0	0
$283$	19.9449	1.18560	0.592801	−	0.805349i	$-0.298022\pi$
$283$	19.9449	1.18560	0.592801	+	0.805349i	$0.298022\pi$
$284$	19.5139	1.15794
$285$	0	0
$286$	1.91476	0.113222
$287$	−8.64745	−0.510443
$288$	0	0
$289$	−3.93723	−0.231602
$290$	−19.3244	−1.13477
$291$	0	0
$292$	14.6390	0.856680
$293$	−14.7094	−0.859331	−0.429665	−	0.902988i	$-0.641369\pi$
$293$	−14.7094	−0.859331	−0.429665	+	0.902988i	$0.641369\pi$
$294$	0	0
$295$	−0.557731	−0.0324724
$296$	−11.0450	−0.641976
$297$	0	0
$298$	−40.0486	−2.31995
$299$	−3.65512	−0.211381
$300$	0	0
$301$	−11.7281	−0.675994
$302$	−7.70295	−0.443255
$303$	0	0
$304$	3.29688	0.189089
$305$	19.3082	1.10559
$306$	0	0
$307$	−4.72211	−0.269505	−0.134753	−	0.990879i	$-0.543024\pi$
$307$	−4.72211	−0.269505	−0.134753	+	0.990879i	$0.543024\pi$
$308$	8.18002	0.466100
$309$	0	0
$310$	3.94315	0.223956
$311$	−17.5323	−0.994168	−0.497084	−	0.867702i	$-0.665596\pi$
$311$	−17.5323	−0.994168	−0.497084	+	0.867702i	$0.665596\pi$
$312$	0	0
$313$	12.1697	0.687873	0.343937	−	0.938993i	$-0.388239\pi$
$313$	12.1697	0.687873	0.343937	+	0.938993i	$0.388239\pi$
$314$	16.5701	0.935104
$315$	0	0
$316$	11.2550	0.633145
$317$	16.4965	0.926535	0.463268	−	0.886218i	$-0.346677\pi$
$317$	16.4965	0.926535	0.463268	+	0.886218i	$0.346677\pi$
$318$	0	0
$319$	10.8014	0.604764
$320$	5.28587	0.295489
$321$	0	0
$322$	−38.7896	−2.16166
$323$	−2.44216	−0.135885
$324$	0	0
$325$	0.314090	0.0174226
$326$	−14.2751	−0.790623
$327$	0	0
$328$	4.16472	0.229958
$329$	24.4718	1.34918
$330$	0	0
$331$	3.14512	0.172872	0.0864358	−	0.996257i	$-0.472452\pi$
$331$	3.14512	0.172872	0.0864358	+	0.996257i	$0.472452\pi$
$332$	0.601691	0.0330221
$333$	0	0
$334$	4.51740	0.247181
$335$	−29.3859	−1.60552
$336$	0	0
$337$	28.8759	1.57297	0.786486	−	0.617608i	$-0.211898\pi$
$337$	28.8759	1.57297	0.786486	+	0.617608i	$0.211898\pi$
$338$	23.4513	1.27558
$339$	0	0
$340$	−11.6640	−0.632569
$341$	−2.20403	−0.119355
$342$	0	0
$343$	19.4738	1.05149
$344$	5.64838	0.304540
$345$	0	0
$346$	−26.6947	−1.43511
$347$	−31.5438	−1.69336	−0.846681	−	0.532101i	$-0.821403\pi$
$347$	−31.5438	−1.69336	−0.846681	+	0.532101i	$0.821403\pi$
$348$	0	0
$349$	−6.54560	−0.350378	−0.175189	−	0.984535i	$-0.556054\pi$
$349$	−6.54560	−0.350378	−0.175189	+	0.984535i	$0.556054\pi$
$350$	3.33325	0.178170
$351$	0	0
$352$	−16.0169	−0.853702
$353$	2.38253	0.126809	0.0634047	−	0.997988i	$-0.479804\pi$
$353$	2.38253	0.126809	0.0634047	+	0.997988i	$0.479804\pi$
$354$	0	0
$355$	−34.6778	−1.84051
$356$	−22.8115	−1.20901
$357$	0	0
$358$	3.39680	0.179526
$359$	15.0342	0.793473	0.396737	−	0.917932i	$-0.370143\pi$
$359$	15.0342	0.793473	0.396737	+	0.917932i	$0.370143\pi$
$360$	0	0
$361$	−18.5434	−0.975970
$362$	−39.9596	−2.10023
$363$	0	0
$364$	−1.42718	−0.0748048
$365$	−26.0146	−1.36167
$366$	0	0
$367$	−29.0871	−1.51833	−0.759167	−	0.650896i	$-0.774393\pi$
$367$	−29.0871	−1.51833	−0.759167	+	0.650896i	$0.774393\pi$
$368$	41.7361	2.17565
$369$	0	0
$370$	−40.5433	−2.10775
$371$	1.07129	0.0556187
$372$	0	0
$373$	−3.41639	−0.176894	−0.0884469	−	0.996081i	$-0.528190\pi$
$373$	−3.41639	−0.176894	−0.0884469	+	0.996081i	$0.528190\pi$
$374$	16.1955	0.837451
$375$	0	0
$376$	−11.7859	−0.607813
$377$	−1.88455	−0.0970591
$378$	0	0
$379$	−10.8111	−0.555331	−0.277666	−	0.960678i	$-0.589561\pi$
$379$	−10.8111	−0.555331	−0.277666	+	0.960678i	$0.589561\pi$
$380$	2.18065	0.111865
$381$	0	0
$382$	−12.5602	−0.642635
$383$	20.0610	1.02507	0.512534	−	0.858667i	$-0.328707\pi$
$383$	20.0610	1.02507	0.512534	+	0.858667i	$0.328707\pi$
$384$	0	0
$385$	−14.5366	−0.740852
$386$	16.8618	0.858242
$387$	0	0
$388$	8.77017	0.445238
$389$	−15.1626	−0.768773	−0.384387	−	0.923172i	$-0.625587\pi$
$389$	−15.1626	−0.768773	−0.384387	+	0.923172i	$0.625587\pi$
$390$	0	0
$391$	−30.9160	−1.56349
$392$	−1.02322	−0.0516804
$393$	0	0
$394$	1.82964	0.0921762
$395$	−20.0011	−1.00637
$396$	0	0
$397$	33.7992	1.69633	0.848167	−	0.529729i	$-0.177706\pi$
$397$	33.7992	1.69633	0.848167	+	0.529729i	$0.177706\pi$
$398$	38.0040	1.90497
$399$	0	0
$400$	−3.58645	−0.179322
$401$	−18.8451	−0.941080	−0.470540	−	0.882379i	$-0.655941\pi$
$401$	−18.8451	−0.941080	−0.470540	+	0.882379i	$0.655941\pi$
$402$	0	0
$403$	0.384542	0.0191554
$404$	−10.5264	−0.523708
$405$	0	0
$406$	−19.9996	−0.992561
$407$	22.6618	1.12330
$408$	0	0
$409$	−27.1293	−1.34146	−0.670728	−	0.741704i	$-0.734018\pi$
$409$	−27.1293	−1.34146	−0.670728	+	0.741704i	$0.734018\pi$
$410$	15.2877	0.755004
$411$	0	0
$412$	0.393422	0.0193825
$413$	−0.577217	−0.0284030
$414$	0	0
$415$	−1.06925	−0.0524876
$416$	2.79449	0.137011
$417$	0	0
$418$	−3.02785	−0.148097
$419$	−18.7712	−0.917035	−0.458518	−	0.888685i	$-0.651619\pi$
$419$	−18.7712	−0.917035	−0.458518	+	0.888685i	$0.651619\pi$
$420$	0	0
$421$	−30.2422	−1.47392	−0.736958	−	0.675938i	$-0.763739\pi$
$421$	−30.2422	−1.47392	−0.736958	+	0.675938i	$0.763739\pi$
$422$	−13.0866	−0.637045
$423$	0	0
$424$	−0.515948	−0.0250566
$425$	2.65666	0.128867
$426$	0	0
$427$	19.9828	0.967036
$428$	20.5631	0.993956
$429$	0	0
$430$	20.7338	0.999873
$431$	−39.4571	−1.90058	−0.950292	−	0.311361i	$-0.899215\pi$
$431$	−39.4571	−1.90058	−0.950292	+	0.311361i	$0.899215\pi$
$432$	0	0
$433$	26.0606	1.25239	0.626197	−	0.779665i	$-0.284611\pi$
$433$	26.0606	1.25239	0.626197	+	0.779665i	$0.284611\pi$
$434$	4.08091	0.195890
$435$	0	0
$436$	−6.25434	−0.299529
$437$	5.77992	0.276491
$438$	0	0
$439$	−38.8850	−1.85588	−0.927940	−	0.372730i	$-0.878422\pi$
$439$	−38.8850	−1.85588	−0.927940	+	0.372730i	$0.878422\pi$
$440$	7.00099	0.333759
$441$	0	0
$442$	−2.82566	−0.134403
$443$	−30.5988	−1.45379	−0.726896	−	0.686747i	$-0.759038\pi$
$443$	−30.5988	−1.45379	−0.726896	+	0.686747i	$0.759038\pi$
$444$	0	0
$445$	40.5380	1.92169
$446$	−27.6942	−1.31136
$447$	0	0
$448$	5.47055	0.258459
$449$	−34.6750	−1.63641	−0.818207	−	0.574924i	$-0.805032\pi$
$449$	−34.6750	−1.63641	−0.818207	+	0.574924i	$0.805032\pi$
$450$	0	0
$451$	−8.54508	−0.402372
$452$	−4.04883	−0.190441
$453$	0	0
$454$	−43.5207	−2.04253
$455$	2.53622	0.118900
$456$	0	0
$457$	−8.62517	−0.403468	−0.201734	−	0.979440i	$-0.564658\pi$
$457$	−8.62517	−0.403468	−0.201734	+	0.979440i	$0.564658\pi$
$458$	36.2586	1.69425
$459$	0	0
$460$	27.6055	1.28711
$461$	−17.2908	−0.805315	−0.402657	−	0.915351i	$-0.631913\pi$
$461$	−17.2908	−0.805315	−0.402657	+	0.915351i	$0.631913\pi$
$462$	0	0
$463$	17.1023	0.794814	0.397407	−	0.917643i	$-0.369910\pi$
$463$	17.1023	0.794814	0.397407	+	0.917643i	$0.369910\pi$
$464$	21.5188	0.998983
$465$	0	0
$466$	−53.2262	−2.46566
$467$	−20.0251	−0.926650	−0.463325	−	0.886188i	$-0.653344\pi$
$467$	−20.0251	−0.926650	−0.463325	+	0.886188i	$0.653344\pi$
$468$	0	0
$469$	−30.4125	−1.40432
$470$	−43.2633	−1.99559
$471$	0	0
$472$	0.277995	0.0127958
$473$	−11.5892	−0.532873
$474$	0	0
$475$	−0.496677	−0.0227891
$476$	−12.0715	−0.553297
$477$	0	0
$478$	−31.6101	−1.44581
$479$	27.6009	1.26112	0.630559	−	0.776141i	$-0.282825\pi$
$479$	27.6009	1.26112	0.630559	+	0.776141i	$0.282825\pi$
$480$	0	0
$481$	−3.95385	−0.180280
$482$	−8.20154	−0.373570
$483$	0	0
$484$	−6.74042	−0.306383
$485$	−15.5853	−0.707693
$486$	0	0
$487$	10.3523	0.469109	0.234554	−	0.972103i	$-0.424637\pi$
$487$	10.3523	0.469109	0.234554	+	0.972103i	$0.424637\pi$
$488$	−9.62397	−0.435657
$489$	0	0
$490$	−3.75599	−0.169678
$491$	36.6785	1.65528	0.827638	−	0.561262i	$-0.189684\pi$
$491$	36.6785	1.65528	0.827638	+	0.561262i	$0.189684\pi$
$492$	0	0
$493$	−15.9400	−0.717902
$494$	0.528274	0.0237682
$495$	0	0
$496$	−4.39091	−0.197157
$497$	−35.8894	−1.60986
$498$	0	0
$499$	14.0562	0.629243	0.314622	−	0.949217i	$-0.398122\pi$
$499$	14.0562	0.629243	0.314622	+	0.949217i	$0.398122\pi$
$500$	13.7640	0.615543
$501$	0	0
$502$	13.6875	0.610902
$503$	−25.4260	−1.13369	−0.566844	−	0.823825i	$-0.691836\pi$
$503$	−25.4260	−1.13369	−0.566844	+	0.823825i	$0.691836\pi$
$504$	0	0
$505$	18.7063	0.832418
$506$	−38.3304	−1.70399
$507$	0	0
$508$	−7.38121	−0.327488
$509$	15.1509	0.671551	0.335775	−	0.941942i	$-0.391002\pi$
$509$	15.1509	0.671551	0.335775	+	0.941942i	$0.391002\pi$
$510$	0	0
$511$	−26.9235	−1.19103
$512$	−20.2609	−0.895412
$513$	0	0
$514$	47.8863	2.11217
$515$	−0.699143	−0.0308079
$516$	0	0
$517$	24.1821	1.06353
$518$	−41.9598	−1.84361
$519$	0	0
$520$	−1.22148	−0.0535653
$521$	−23.8145	−1.04333	−0.521666	−	0.853150i	$-0.674689\pi$
$521$	−23.8145	−1.04333	−0.521666	+	0.853150i	$0.674689\pi$
$522$	0	0
$523$	38.0421	1.66346	0.831732	−	0.555177i	$-0.187350\pi$
$523$	38.0421	1.66346	0.831732	+	0.555177i	$0.187350\pi$
$524$	13.2464	0.578672
$525$	0	0
$526$	−3.86988	−0.168735
$527$	3.25256	0.141684
$528$	0	0
$529$	50.1698	2.18129
$530$	−1.89392	−0.0822664
$531$	0	0
$532$	2.25684	0.0978463
$533$	1.49088	0.0645770
$534$	0	0
$535$	−36.5423	−1.57986
$536$	14.6471	0.632657
$537$	0	0
$538$	9.93942	0.428519
$539$	2.09942	0.0904284
$540$	0	0
$541$	−23.8834	−1.02683	−0.513413	−	0.858141i	$-0.671619\pi$
$541$	−23.8834	−1.02683	−0.513413	+	0.858141i	$0.671619\pi$
$542$	−16.2496	−0.697982
$543$	0	0
$544$	23.6366	1.01341
$545$	11.1145	0.476092
$546$	0	0
$547$	−10.8990	−0.466006	−0.233003	−	0.972476i	$-0.574855\pi$
$547$	−10.8990	−0.466006	−0.233003	+	0.972476i	$0.574855\pi$
$548$	6.92315	0.295742
$549$	0	0
$550$	3.29379	0.140448
$551$	2.98007	0.126955
$552$	0	0
$553$	−20.6999	−0.880249
$554$	−7.04707	−0.299401
$555$	0	0
$556$	16.2644	0.689766
$557$	13.5365	0.573559	0.286779	−	0.957997i	$-0.407415\pi$
$557$	13.5365	0.573559	0.286779	+	0.957997i	$0.407415\pi$
$558$	0	0
$559$	2.02199	0.0855212
$560$	−28.9599	−1.22378
$561$	0	0
$562$	−30.3814	−1.28156
$563$	43.2131	1.82121	0.910607	−	0.413274i	$-0.135615\pi$
$563$	43.2131	1.82121	0.910607	+	0.413274i	$0.135615\pi$
$564$	0	0
$565$	7.19510	0.302700
$566$	−36.4921	−1.53388
$567$	0	0
$568$	17.2848	0.725253
$569$	19.9320	0.835594	0.417797	−	0.908540i	$-0.362802\pi$
$569$	19.9320	0.835594	0.417797	+	0.908540i	$0.362802\pi$
$570$	0	0
$571$	−24.7452	−1.03556	−0.517778	−	0.855515i	$-0.673241\pi$
$571$	−24.7452	−1.03556	−0.517778	+	0.855515i	$0.673241\pi$
$572$	−1.41029	−0.0589671
$573$	0	0
$574$	15.8218	0.660388
$575$	−6.28758	−0.262210
$576$	0	0
$577$	33.3646	1.38899	0.694493	−	0.719499i	$-0.255629\pi$
$577$	33.3646	1.38899	0.694493	+	0.719499i	$0.255629\pi$
$578$	7.20373	0.299636
$579$	0	0
$580$	14.2331	0.590999
$581$	−1.10661	−0.0459100
$582$	0	0
$583$	1.05861	0.0438431
$584$	12.9667	0.536566
$585$	0	0
$586$	26.9129	1.11176
$587$	−39.7834	−1.64204	−0.821019	−	0.570901i	$-0.806594\pi$
$587$	−39.7834	−1.64204	−0.821019	+	0.570901i	$0.806594\pi$
$588$	0	0
$589$	−0.608085	−0.0250557
$590$	1.02045	0.0420113
$591$	0	0
$592$	45.1471	1.85554
$593$	3.83642	0.157543	0.0787715	−	0.996893i	$-0.474900\pi$
$593$	3.83642	0.157543	0.0787715	+	0.996893i	$0.474900\pi$
$594$	0	0
$595$	21.4521	0.879449
$596$	29.4973	1.20825
$597$	0	0
$598$	6.68758	0.273475
$599$	41.3503	1.68953	0.844763	−	0.535140i	$-0.179741\pi$
$599$	41.3503	1.68953	0.844763	+	0.535140i	$0.179741\pi$
$600$	0	0
$601$	−39.5659	−1.61393	−0.806963	−	0.590602i	$-0.798890\pi$
$601$	−39.5659	−1.61393	−0.806963	+	0.590602i	$0.798890\pi$
$602$	21.4582	0.874570
$603$	0	0
$604$	5.67351	0.230852
$605$	11.9783	0.486987
$606$	0	0
$607$	−20.6997	−0.840174	−0.420087	−	0.907484i	$-0.638000\pi$
$607$	−20.6997	−0.840174	−0.420087	+	0.907484i	$0.638000\pi$
$608$	−4.41899	−0.179214
$609$	0	0
$610$	−35.3272	−1.43036
$611$	−4.21910	−0.170687
$612$	0	0
$613$	−1.30007	−0.0525092	−0.0262546	−	0.999655i	$-0.508358\pi$
$613$	−1.30007	−0.0525092	−0.0262546	+	0.999655i	$0.508358\pi$
$614$	8.63979	0.348674
$615$	0	0
$616$	7.24559	0.291933
$617$	−3.57289	−0.143839	−0.0719196	−	0.997410i	$-0.522913\pi$
$617$	−3.57289	−0.143839	−0.0719196	+	0.997410i	$0.522913\pi$
$618$	0	0
$619$	−9.76477	−0.392479	−0.196240	−	0.980556i	$-0.562873\pi$
$619$	−9.76477	−0.392479	−0.196240	+	0.980556i	$0.562873\pi$
$620$	−2.90427	−0.116638
$621$	0	0
$622$	32.0780	1.28621
$623$	41.9543	1.68086
$624$	0	0
$625$	−28.1350	−1.12540
$626$	−22.2663	−0.889939
$627$	0	0
$628$	−12.2045	−0.487012
$629$	−33.4427	−1.33345
$630$	0	0
$631$	13.9942	0.557102	0.278551	−	0.960421i	$-0.410146\pi$
$631$	13.9942	0.557102	0.278551	+	0.960421i	$0.410146\pi$
$632$	9.96933	0.396559
$633$	0	0
$634$	−30.1827	−1.19871
$635$	13.1170	0.520533
$636$	0	0
$637$	−0.366290	−0.0145129
$638$	−19.7628	−0.782416
$639$	0	0
$640$	21.6519	0.855866
$641$	−9.77562	−0.386114	−0.193057	−	0.981188i	$-0.561840\pi$
$641$	−9.77562	−0.386114	−0.193057	+	0.981188i	$0.561840\pi$
$642$	0	0
$643$	−7.28129	−0.287146	−0.143573	−	0.989640i	$-0.545859\pi$
$643$	−7.28129	−0.287146	−0.143573	+	0.989640i	$0.545859\pi$
$644$	28.5700	1.12581
$645$	0	0
$646$	4.46828	0.175802
$647$	23.9804	0.942767	0.471384	−	0.881928i	$-0.343755\pi$
$647$	23.9804	0.942767	0.471384	+	0.881928i	$0.343755\pi$
$648$	0	0
$649$	−0.570384	−0.0223895
$650$	−0.574673	−0.0225405
$651$	0	0
$652$	10.5141	0.411765
$653$	38.0906	1.49060	0.745299	−	0.666730i	$-0.232307\pi$
$653$	38.0906	1.49060	0.745299	+	0.666730i	$0.232307\pi$
$654$	0	0
$655$	−23.5399	−0.919782
$656$	−17.0236	−0.664660
$657$	0	0
$658$	−44.7748	−1.74550
$659$	−14.7028	−0.572739	−0.286370	−	0.958119i	$-0.592449\pi$
$659$	−14.7028	−0.572739	−0.286370	+	0.958119i	$0.592449\pi$
$660$	0	0
$661$	25.4067	0.988208	0.494104	−	0.869403i	$-0.335496\pi$
$661$	25.4067	0.988208	0.494104	+	0.869403i	$0.335496\pi$
$662$	−5.75446	−0.223653
$663$	0	0
$664$	0.532958	0.0206828
$665$	−4.01059	−0.155524
$666$	0	0
$667$	37.7256	1.46074
$668$	−3.32723	−0.128734
$669$	0	0
$670$	53.7657	2.07715
$671$	19.7462	0.762295
$672$	0	0
$673$	30.5437	1.17737	0.588687	−	0.808361i	$-0.299645\pi$
$673$	30.5437	1.17737	0.588687	+	0.808361i	$0.299645\pi$
$674$	−52.8327	−2.03504
$675$	0	0
$676$	−17.2728	−0.664337
$677$	−14.2822	−0.548908	−0.274454	−	0.961600i	$-0.588497\pi$
$677$	−14.2822	−0.548908	−0.274454	+	0.961600i	$0.588497\pi$
$678$	0	0
$679$	−16.1298	−0.619006
$680$	−10.3316	−0.396198
$681$	0	0
$682$	4.03260	0.154416
$683$	25.9111	0.991461	0.495731	−	0.868476i	$-0.334900\pi$
$683$	25.9111	0.991461	0.495731	+	0.868476i	$0.334900\pi$
$684$	0	0
$685$	−12.3030	−0.470074
$686$	−35.6302	−1.36037
$687$	0	0
$688$	−23.0882	−0.880229
$689$	−0.184698	−0.00703642
$690$	0	0
$691$	24.8832	0.946603	0.473302	−	0.880900i	$-0.343062\pi$
$691$	24.8832	0.946603	0.473302	+	0.880900i	$0.343062\pi$
$692$	19.6616	0.747422
$693$	0	0
$694$	57.7140	2.19079
$695$	−28.9032	−1.09636
$696$	0	0
$697$	12.6102	0.477647
$698$	11.9761	0.453303
$699$	0	0
$700$	−2.45506	−0.0927925
$701$	−21.1982	−0.800644	−0.400322	−	0.916375i	$-0.631102\pi$
$701$	−21.1982	−0.800644	−0.400322	+	0.916375i	$0.631102\pi$
$702$	0	0
$703$	6.25231	0.235810
$704$	5.40578	0.203738
$705$	0	0
$706$	−4.35919	−0.164060
$707$	19.3598	0.728101
$708$	0	0
$709$	25.8806	0.971964	0.485982	−	0.873969i	$-0.338462\pi$
$709$	25.8806	0.971964	0.485982	+	0.873969i	$0.338462\pi$
$710$	63.4481	2.38117
$711$	0	0
$712$	−20.2057	−0.757241
$713$	−7.69792	−0.288290
$714$	0	0
$715$	2.50620	0.0937265
$716$	−2.50187	−0.0934992
$717$	0	0
$718$	−27.5072	−1.02656
$719$	33.5697	1.25194	0.625968	−	0.779848i	$-0.284704\pi$
$719$	33.5697	1.25194	0.625968	+	0.779848i	$0.284704\pi$
$720$	0	0
$721$	−0.723570	−0.0269471
$722$	33.9279	1.26267
$723$	0	0
$724$	29.4317	1.09382
$725$	−3.24182	−0.120398
$726$	0	0
$727$	48.7995	1.80987	0.904937	−	0.425545i	$-0.139918\pi$
$727$	48.7995	1.80987	0.904937	+	0.425545i	$0.139918\pi$
$728$	−1.26415	−0.0468526
$729$	0	0
$730$	47.5976	1.76166
$731$	17.1026	0.632561
$732$	0	0
$733$	26.7007	0.986214	0.493107	−	0.869969i	$-0.335861\pi$
$733$	26.7007	0.986214	0.493107	+	0.869969i	$0.335861\pi$
$734$	53.2191	1.96435
$735$	0	0
$736$	−55.9413	−2.06202
$737$	−30.0525	−1.10700
$738$	0	0
$739$	17.3969	0.639954	0.319977	−	0.947425i	$-0.396325\pi$
$739$	17.3969	0.639954	0.319977	+	0.947425i	$0.396325\pi$
$740$	29.8617	1.09774
$741$	0	0
$742$	−1.96008	−0.0719569
$743$	−50.6579	−1.85846	−0.929229	−	0.369505i	$-0.879527\pi$
$743$	−50.6579	−1.85846	−0.929229	+	0.369505i	$0.879527\pi$
$744$	0	0
$745$	−52.4190	−1.92049
$746$	6.25077	0.228857
$747$	0	0
$748$	−11.9286	−0.436153
$749$	−37.8190	−1.38188
$750$	0	0
$751$	6.22588	0.227186	0.113593	−	0.993527i	$-0.463764\pi$
$751$	6.22588	0.227186	0.113593	+	0.993527i	$0.463764\pi$
$752$	48.1759	1.75679
$753$	0	0
$754$	3.44805	0.125571
$755$	−10.0823	−0.366932
$756$	0	0
$757$	−1.43773	−0.0522550	−0.0261275	−	0.999659i	$-0.508318\pi$
$757$	−1.43773	−0.0522550	−0.0261275	+	0.999659i	$0.508318\pi$
$758$	19.7806	0.718462
$759$	0	0
$760$	1.93155	0.0700646
$761$	−9.52056	−0.345120	−0.172560	−	0.984999i	$-0.555204\pi$
$761$	−9.52056	−0.345120	−0.172560	+	0.984999i	$0.555204\pi$
$762$	0	0
$763$	11.5028	0.416429
$764$	9.25104	0.334691
$765$	0	0
$766$	−36.7044	−1.32619
$767$	0.0995159	0.00359331
$768$	0	0
$769$	−0.107823	−0.00388818	−0.00194409	−	0.999998i	$-0.500619\pi$
$769$	−0.107823	−0.00388818	−0.00194409	+	0.999998i	$0.500619\pi$
$770$	26.5968	0.958481
$771$	0	0
$772$	−12.4193	−0.446981
$773$	18.3261	0.659145	0.329573	−	0.944130i	$-0.393095\pi$
$773$	18.3261	0.659145	0.329573	+	0.944130i	$0.393095\pi$
$774$	0	0
$775$	0.661494	0.0237616
$776$	7.76832	0.278867
$777$	0	0
$778$	27.7421	0.994604
$779$	−2.35755	−0.0844682
$780$	0	0
$781$	−35.4645	−1.26902
$782$	56.5653	2.02277
$783$	0	0
$784$	4.18249	0.149375
$785$	21.6884	0.774091
$786$	0	0
$787$	−12.3630	−0.440694	−0.220347	−	0.975422i	$-0.570719\pi$
$787$	−12.3630	−0.440694	−0.220347	+	0.975422i	$0.570719\pi$
$788$	−1.34760	−0.0480063
$789$	0	0
$790$	36.5949	1.30199
$791$	7.44648	0.264766
$792$	0	0
$793$	−3.44516	−0.122341
$794$	−61.8406	−2.19464
$795$	0	0
$796$	−27.9913	−0.992125
$797$	−4.56337	−0.161643	−0.0808213	−	0.996729i	$-0.525754\pi$
$797$	−4.56337	−0.161643	−0.0808213	+	0.996729i	$0.525754\pi$
$798$	0	0
$799$	−35.6863	−1.26249
$800$	4.80712	0.169957
$801$	0	0
$802$	34.4799	1.21753
$803$	−26.6048	−0.938862
$804$	0	0
$805$	−50.7712	−1.78945
$806$	−0.703575	−0.0247824
$807$	0	0
$808$	−9.32393	−0.328015
$809$	27.7914	0.977095	0.488547	−	0.872537i	$-0.337527\pi$
$809$	27.7914	0.977095	0.488547	+	0.872537i	$0.337527\pi$
$810$	0	0
$811$	25.2664	0.887225	0.443612	−	0.896219i	$-0.353697\pi$
$811$	25.2664	0.887225	0.443612	+	0.896219i	$0.353697\pi$
$812$	14.7304	0.516936
$813$	0	0
$814$	−41.4631	−1.45328
$815$	−18.6844	−0.654488
$816$	0	0
$817$	−3.19742	−0.111864
$818$	49.6369	1.73551
$819$	0	0
$820$	−11.2599	−0.393213
$821$	41.2375	1.43920	0.719599	−	0.694390i	$-0.244326\pi$
$821$	41.2375	1.43920	0.719599	+	0.694390i	$0.244326\pi$
$822$	0	0
$823$	−17.9367	−0.625234	−0.312617	−	0.949879i	$-0.601206\pi$
$823$	−17.9367	−0.625234	−0.312617	+	0.949879i	$0.601206\pi$
$824$	0.348480	0.0121399
$825$	0	0
$826$	1.05610	0.0367465
$827$	−44.4745	−1.54653	−0.773265	−	0.634083i	$-0.781378\pi$
$827$	−44.4745	−1.54653	−0.773265	+	0.634083i	$0.781378\pi$
$828$	0	0
$829$	1.60966	0.0559059	0.0279529	−	0.999609i	$-0.491101\pi$
$829$	1.60966	0.0559059	0.0279529	+	0.999609i	$0.491101\pi$
$830$	1.95636	0.0679061
$831$	0	0
$832$	−0.943157	−0.0326981
$833$	−3.09818	−0.107345
$834$	0	0
$835$	5.91276	0.204620
$836$	2.23012	0.0771303
$837$	0	0
$838$	34.3447	1.18642
$839$	47.6291	1.64434	0.822169	−	0.569243i	$-0.192764\pi$
$839$	47.6291	1.64434	0.822169	+	0.569243i	$0.192764\pi$
$840$	0	0
$841$	−9.54901	−0.329276
$842$	55.3326	1.90689
$843$	0	0
$844$	9.63874	0.331779
$845$	30.6951	1.05594
$846$	0	0
$847$	12.3968	0.425958
$848$	2.10897	0.0724225
$849$	0	0
$850$	−4.86074	−0.166722
$851$	79.1498	2.71322
$852$	0	0
$853$	41.7275	1.42872	0.714361	−	0.699778i	$-0.246718\pi$
$853$	41.7275	1.42872	0.714361	+	0.699778i	$0.246718\pi$
$854$	−36.5614	−1.25111
$855$	0	0
$856$	18.2141	0.622546
$857$	−55.8573	−1.90805	−0.954024	−	0.299729i	$-0.903104\pi$
$857$	−55.8573	−1.90805	−0.954024	+	0.299729i	$0.903104\pi$
$858$	0	0
$859$	−13.5803	−0.463355	−0.231677	−	0.972793i	$-0.574421\pi$
$859$	−13.5803	−0.463355	−0.231677	+	0.972793i	$0.574421\pi$
$860$	−15.2712	−0.520744
$861$	0	0
$862$	72.1926	2.45889
$863$	47.5067	1.61715	0.808574	−	0.588395i	$-0.200240\pi$
$863$	47.5067	1.61715	0.808574	+	0.588395i	$0.200240\pi$
$864$	0	0
$865$	−34.9403	−1.18800
$866$	−47.6817	−1.62029
$867$	0	0
$868$	−3.00574	−0.102022
$869$	−20.4548	−0.693883
$870$	0	0
$871$	5.24332	0.177663
$872$	−5.53989	−0.187604
$873$	0	0
$874$	−10.5752	−0.357712
$875$	−25.3143	−0.855778
$876$	0	0
$877$	−8.06528	−0.272345	−0.136173	−	0.990685i	$-0.543480\pi$
$877$	−8.06528	−0.272345	−0.136173	+	0.990685i	$0.543480\pi$
$878$	71.1457	2.40105
$879$	0	0
$880$	−28.6171	−0.964682
$881$	−38.7817	−1.30659	−0.653294	−	0.757104i	$-0.726613\pi$
$881$	−38.7817	−1.30659	−0.653294	+	0.757104i	$0.726613\pi$
$882$	0	0
$883$	13.6849	0.460535	0.230268	−	0.973127i	$-0.426040\pi$
$883$	13.6849	0.460535	0.230268	+	0.973127i	$0.426040\pi$
$884$	2.08121	0.0699985
$885$	0	0
$886$	55.9849	1.88085
$887$	4.89122	0.164231	0.0821156	−	0.996623i	$-0.473832\pi$
$887$	4.89122	0.164231	0.0821156	+	0.996623i	$0.473832\pi$
$888$	0	0
$889$	13.5753	0.455300
$890$	−74.1701	−2.48619
$891$	0	0
$892$	20.3978	0.682969
$893$	6.67175	0.223262
$894$	0	0
$895$	4.44602	0.148614
$896$	22.4084	0.748611
$897$	0	0
$898$	63.4429	2.11712
$899$	−3.96898	−0.132373
$900$	0	0
$901$	−1.56222	−0.0520452
$902$	15.6345	0.520571
$903$	0	0
$904$	−3.58631	−0.119279
$905$	−52.3025	−1.73860
$906$	0	0
$907$	−4.79442	−0.159196	−0.0795981	−	0.996827i	$-0.525364\pi$
$907$	−4.79442	−0.159196	−0.0795981	+	0.996827i	$0.525364\pi$
$908$	32.0546	1.06377
$909$	0	0
$910$	−4.64039	−0.153827
$911$	−27.3364	−0.905694	−0.452847	−	0.891588i	$-0.649592\pi$
$911$	−27.3364	−0.905694	−0.452847	+	0.891588i	$0.649592\pi$
$912$	0	0
$913$	−1.09351	−0.0361899
$914$	15.7810	0.521989
$915$	0	0
$916$	−26.7058	−0.882383
$917$	−24.3624	−0.804516
$918$	0	0
$919$	54.3152	1.79169	0.895846	−	0.444365i	$-0.146571\pi$
$919$	54.3152	1.79169	0.895846	+	0.444365i	$0.146571\pi$
$920$	24.4520	0.806160
$921$	0	0
$922$	31.6361	1.04188
$923$	6.18756	0.203666
$924$	0	0
$925$	−6.80146	−0.223631
$926$	−31.2912	−1.02829
$927$	0	0
$928$	−28.8428	−0.946812
$929$	−21.7090	−0.712249	−0.356124	−	0.934439i	$-0.615902\pi$
$929$	−21.7090	−0.712249	−0.356124	+	0.934439i	$0.615902\pi$
$930$	0	0
$931$	0.579222	0.0189832
$932$	39.2031	1.28414
$933$	0	0
$934$	36.6388	1.19886
$935$	21.1981	0.693252
$936$	0	0
$937$	−14.5142	−0.474158	−0.237079	−	0.971490i	$-0.576190\pi$
$937$	−14.5142	−0.474158	−0.237079	+	0.971490i	$0.576190\pi$
$938$	55.6442	1.81685
$939$	0	0
$940$	31.8650	1.03932
$941$	44.5490	1.45226	0.726128	−	0.687560i	$-0.241318\pi$
$941$	44.5490	1.45226	0.726128	+	0.687560i	$0.241318\pi$
$942$	0	0
$943$	−29.8450	−0.971886
$944$	−1.13633	−0.0369842
$945$	0	0
$946$	21.2042	0.689406
$947$	40.2068	1.30655	0.653274	−	0.757122i	$-0.273395\pi$
$947$	40.2068	1.30655	0.653274	+	0.757122i	$0.273395\pi$
$948$	0	0
$949$	4.64179	0.150679
$950$	0.908743	0.0294835
$951$	0	0
$952$	−10.6925	−0.346547
$953$	−25.6261	−0.830109	−0.415055	−	0.909797i	$-0.636238\pi$
$953$	−25.6261	−0.830109	−0.415055	+	0.909797i	$0.636238\pi$
$954$	0	0
$955$	−16.4399	−0.531981
$956$	23.2820	0.752993
$957$	0	0
$958$	−50.4999	−1.63158
$959$	−12.7328	−0.411165
$960$	0	0
$961$	−30.1901	−0.973875
$962$	7.23414	0.233238
$963$	0	0
$964$	6.04073	0.194559
$965$	22.0701	0.710463
$966$	0	0
$967$	30.8216	0.991156	0.495578	−	0.868563i	$-0.334956\pi$
$967$	30.8216	0.991156	0.495578	+	0.868563i	$0.334956\pi$
$968$	−5.97044	−0.191897
$969$	0	0
$970$	28.5156	0.915580
$971$	−5.70753	−0.183163	−0.0915817	−	0.995798i	$-0.529192\pi$
$971$	−5.70753	−0.183163	−0.0915817	+	0.995798i	$0.529192\pi$
$972$	0	0
$973$	−29.9130	−0.958968
$974$	−18.9411	−0.606912
$975$	0	0
$976$	39.3387	1.25920
$977$	4.69258	0.150129	0.0750644	−	0.997179i	$-0.476084\pi$
$977$	4.69258	0.150129	0.0750644	+	0.997179i	$0.476084\pi$
$978$	0	0
$979$	41.4576	1.32499
$980$	2.76642	0.0883701
$981$	0	0
$982$	−67.1086	−2.14152
$983$	−10.5465	−0.336381	−0.168190	−	0.985755i	$-0.553792\pi$
$983$	−10.5465	−0.336381	−0.168190	+	0.985755i	$0.553792\pi$
$984$	0	0
$985$	2.39480	0.0763046
$986$	29.1646	0.928789
$987$	0	0
$988$	−0.389093	−0.0123787
$989$	−40.4771	−1.28710
$990$	0	0
$991$	−30.6070	−0.972265	−0.486132	−	0.873885i	$-0.661593\pi$
$991$	−30.6070	−0.972265	−0.486132	+	0.873885i	$0.661593\pi$
$992$	5.88538	0.186861
$993$	0	0
$994$	65.6648	2.08276
$995$	49.7428	1.57695
$996$	0	0
$997$	56.0308	1.77451	0.887257	−	0.461275i	$-0.152608\pi$
$997$	56.0308	1.77451	0.887257	+	0.461275i	$0.152608\pi$
$998$	−25.7179	−0.814086
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1773.2.a.f.1.3		10
3.2	odd	2		197.2.a.c.1.8	✓	10
12.11	even	2		3152.2.a.m.1.7		10
15.14	odd	2		4925.2.a.i.1.3		10
21.20	even	2		9653.2.a.j.1.8		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
197.2.a.c.1.8	✓	10	3.2	odd	2
1773.2.a.f.1.3		10	1.1	even	1	trivial
3152.2.a.m.1.7		10	12.11	even	2
4925.2.a.i.1.3		10	15.14	odd	2
9653.2.a.j.1.8		10	21.20	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1773 = 3^{2} \cdot 197 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1773.a (trivial)

\(f(q)\)	\(=\)	\(q-1.82964 q^{2} +1.34760 q^{4} -2.39480 q^{5} -2.47846 q^{7} +1.19366 q^{8} +4.38163 q^{10} -2.44912 q^{11} +0.427303 q^{13} +4.53471 q^{14} -4.87917 q^{16} +3.61425 q^{17} -0.675704 q^{19} -3.22723 q^{20} +4.48103 q^{22} -8.55393 q^{23} +0.735052 q^{25} -0.781813 q^{26} -3.33998 q^{28} -4.41033 q^{29} +0.899928 q^{31} +6.53984 q^{32} -6.61279 q^{34} +5.93542 q^{35} -9.25303 q^{37} +1.23630 q^{38} -2.85857 q^{40} +3.48904 q^{41} +4.73199 q^{43} -3.30044 q^{44} +15.6507 q^{46} -9.87379 q^{47} -0.857212 q^{49} -1.34488 q^{50} +0.575834 q^{52} -0.432240 q^{53} +5.86515 q^{55} -2.95844 q^{56} +8.06933 q^{58} +0.232893 q^{59} -8.06257 q^{61} -1.64655 q^{62} -2.20723 q^{64} -1.02330 q^{65} +12.2707 q^{67} +4.87056 q^{68} -10.8597 q^{70} +14.4805 q^{71} +10.8630 q^{73} +16.9298 q^{74} -0.910579 q^{76} +6.07007 q^{77} +8.35190 q^{79} +11.6846 q^{80} -6.38370 q^{82} +0.446491 q^{83} -8.65538 q^{85} -8.65785 q^{86} -2.92342 q^{88} -16.9275 q^{89} -1.05906 q^{91} -11.5273 q^{92} +18.0655 q^{94} +1.61817 q^{95} +6.50799 q^{97} +1.56839 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 10 q^{4} - 2 q^{5} + 11 q^{7} + 3 q^{8} - 2 q^{10} - 2 q^{11} + 8 q^{13} + 9 q^{14} - 2 q^{16} + 3 q^{17} + 17 q^{19} + 2 q^{20} + 3 q^{22} + 4 q^{23} + 6 q^{25} + 13 q^{26} + 16 q^{28} + 9 q^{29}+ \cdots - 22 q^{98}+O(q^{100}) \) 10 * q + 10 * q^4 - 2 * q^5 + 11 * q^7 + 3 * q^8 - 2 * q^10 - 2 * q^11 + 8 * q^13 + 9 * q^14 - 2 * q^16 + 3 * q^17 + 17 * q^19 + 2 * q^20 + 3 * q^22 + 4 * q^23 + 6 * q^25 + 13 * q^26 + 16 * q^28 + 9 * q^29 + 20 * q^31 + 16 * q^32 - 8 * q^34 + 12 * q^35 + 12 * q^37 + 15 * q^38 - 15 * q^40 + 18 * q^41 + 11 * q^43 + 24 * q^44 - 13 * q^46 - 5 * q^47 + q^49 + 24 * q^50 + 9 * q^52 + 6 * q^53 - 17 * q^55 + 6 * q^56 - 15 * q^58 + q^59 + 4 * q^61 + 16 * q^62 - 17 * q^64 + 17 * q^65 + 46 * q^67 - 15 * q^68 + 25 * q^70 + 5 * q^71 + 25 * q^73 - 3 * q^74 + 49 * q^76 + 14 * q^77 - q^79 - 9 * q^80 + 52 * q^82 - 28 * q^83 - 30 * q^85 - 27 * q^86 + 9 * q^88 + 6 * q^89 + 51 * q^91 - 32 * q^92 - 3 * q^94 + 13 * q^95 + 2 * q^97 - 22 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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