LMFDB - Embedded newform 2925.2.a.bk.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2925,2,Mod(1,2925)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2925, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("2925.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2925 = 3^{2} \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2925.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:	$23.3562425912$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 975)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.311108$ of defining polynomial
Character	$\chi$	$=$	2925.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.21432 q^{2} -0.525428 q^{4} +2.59210 q^{7} +3.06668 q^{8} +O(q^{10})$	$q-1.21432 q^{2} -0.525428 q^{4} +2.59210 q^{7} +3.06668 q^{8} +6.11753 q^{11} +1.00000 q^{13} -3.14764 q^{14} -2.67307 q^{16} +4.37778 q^{17} +4.14764 q^{19} -7.42864 q^{22} +7.95407 q^{23} -1.21432 q^{26} -1.36196 q^{28} +3.00000 q^{29} +5.36196 q^{31} -2.88739 q^{32} -5.31603 q^{34} -6.90321 q^{37} -5.03657 q^{38} +9.19850 q^{41} -11.1383 q^{43} -3.21432 q^{44} -9.65878 q^{46} -1.21432 q^{47} -0.280996 q^{49} -0.525428 q^{52} +4.95407 q^{53} +7.94914 q^{56} -3.64296 q^{58} -5.44938 q^{59} +9.99063 q^{61} -6.51114 q^{62} +8.85236 q^{64} +4.87310 q^{67} -2.30021 q^{68} -1.39207 q^{71} -13.1383 q^{73} +8.38271 q^{74} -2.17929 q^{76} +15.8573 q^{77} -12.3827 q^{79} -11.1699 q^{82} -13.2558 q^{83} +13.5254 q^{86} +18.7605 q^{88} -8.04149 q^{89} +2.59210 q^{91} -4.17929 q^{92} +1.47457 q^{94} -15.3778 q^{97} +0.341219 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + 5 q^{4} + q^{7} + 9 q^{8}+O(q^{10}) $ 3 * q + 3 * q^2 + 5 * q^4 + q^7 + 9 * q^8	$ 3 q + 3 q^{2} + 5 q^{4} + q^{7} + 9 q^{8} + 5 q^{11} + 3 q^{13} - 3 q^{14} + 5 q^{16} + 13 q^{17} + 6 q^{19} - 9 q^{22} + 4 q^{23} + 3 q^{26} + 9 q^{28} + 9 q^{29} + 3 q^{31} + 11 q^{32} + 17 q^{34} - 14 q^{37} - 8 q^{38} + 8 q^{41} - 3 q^{44} - 22 q^{46} + 3 q^{47} + 6 q^{49} + 5 q^{52} - 5 q^{53} + 37 q^{56} + 9 q^{58} + 17 q^{59} + 3 q^{61} - 19 q^{62} + 33 q^{64} + q^{67} + 39 q^{68} + 2 q^{71} - 6 q^{73} - 8 q^{74} - 26 q^{76} + 21 q^{77} - 4 q^{79} - 26 q^{82} + 7 q^{83} + 34 q^{86} + 23 q^{88} + 16 q^{89} + q^{91} - 32 q^{92} + 11 q^{94} - 46 q^{97} + 8 q^{98}+O(q^{100}) $ 3 * q + 3 * q^2 + 5 * q^4 + q^7 + 9 * q^8 + 5 * q^11 + 3 * q^13 - 3 * q^14 + 5 * q^16 + 13 * q^17 + 6 * q^19 - 9 * q^22 + 4 * q^23 + 3 * q^26 + 9 * q^28 + 9 * q^29 + 3 * q^31 + 11 * q^32 + 17 * q^34 - 14 * q^37 - 8 * q^38 + 8 * q^41 - 3 * q^44 - 22 * q^46 + 3 * q^47 + 6 * q^49 + 5 * q^52 - 5 * q^53 + 37 * q^56 + 9 * q^58 + 17 * q^59 + 3 * q^61 - 19 * q^62 + 33 * q^64 + q^67 + 39 * q^68 + 2 * q^71 - 6 * q^73 - 8 * q^74 - 26 * q^76 + 21 * q^77 - 4 * q^79 - 26 * q^82 + 7 * q^83 + 34 * q^86 + 23 * q^88 + 16 * q^89 + q^91 - 32 * q^92 + 11 * q^94 - 46 * q^97 + 8 * q^98

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 $

$2$	−1.21432	−0.858654	−0.429327	−	0.903149i	$-0.641249\pi$
$2$	−1.21432	−0.858654	−0.429327	+	0.903149i	$0.641249\pi$
$3$	0	0
$3$	0	0
$4$	−0.525428	−0.262714
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.59210	0.979723	0.489862	−	0.871800i	$-0.337047\pi$
$7$	2.59210	0.979723	0.489862	+	0.871800i	$0.337047\pi$
$8$	3.06668	1.08423
$9$	0	0
$10$	0	0
$11$	6.11753	1.84451	0.922253	−	0.386588i	$-0.126346\pi$
$11$	6.11753	1.84451	0.922253	+	0.386588i	$0.126346\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	−3.14764	−0.841243
$15$	0	0
$16$	−2.67307	−0.668268
$17$	4.37778	1.06177	0.530884	−	0.847444i	$-0.321860\pi$
$17$	4.37778	1.06177	0.530884	+	0.847444i	$0.321860\pi$
$18$	0	0
$19$	4.14764	0.951535	0.475767	−	0.879571i	$-0.342170\pi$
$19$	4.14764	0.951535	0.475767	+	0.879571i	$0.342170\pi$
$20$	0	0
$21$	0	0
$22$	−7.42864	−1.58379
$23$	7.95407	1.65854	0.829269	−	0.558850i	$-0.188757\pi$
$23$	7.95407	1.65854	0.829269	+	0.558850i	$0.188757\pi$
$24$	0	0
$25$	0	0
$26$	−1.21432	−0.238148
$27$	0	0
$28$	−1.36196	−0.257387
$29$	3.00000	0.557086	0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000	0.557086	0.278543	+	0.960424i	$0.410149\pi$
$30$	0	0
$31$	5.36196	0.963037	0.481518	−	0.876436i	$-0.340085\pi$
$31$	5.36196	0.963037	0.481518	+	0.876436i	$0.340085\pi$
$32$	−2.88739	−0.510423
$33$	0	0
$34$	−5.31603	−0.911692
$35$	0	0
$36$	0	0
$37$	−6.90321	−1.13488	−0.567441	−	0.823414i	$-0.692066\pi$
$37$	−6.90321	−1.13488	−0.567441	+	0.823414i	$0.692066\pi$
$38$	−5.03657	−0.817039
$39$	0	0
$40$	0	0
$41$	9.19850	1.43656	0.718282	−	0.695752i	$-0.244929\pi$
$41$	9.19850	1.43656	0.718282	+	0.695752i	$0.244929\pi$
$42$	0	0
$43$	−11.1383	−1.69857	−0.849286	−	0.527934i	$-0.822967\pi$
$43$	−11.1383	−1.69857	−0.849286	+	0.527934i	$0.822967\pi$
$44$	−3.21432	−0.484577
$45$	0	0
$46$	−9.65878	−1.42411
$47$	−1.21432	−0.177127	−0.0885634	−	0.996071i	$-0.528228\pi$
$47$	−1.21432	−0.177127	−0.0885634	+	0.996071i	$0.528228\pi$
$48$	0	0
$49$	−0.280996	−0.0401423
$50$	0	0
$51$	0	0
$52$	−0.525428	−0.0728637
$53$	4.95407	0.680493	0.340247	−	0.940336i	$-0.389489\pi$
$53$	4.95407	0.680493	0.340247	+	0.940336i	$0.389489\pi$
$54$	0	0
$55$	0	0
$56$	7.94914	1.06225
$57$	0	0
$58$	−3.64296	−0.478344
$59$	−5.44938	−0.709449	−0.354725	−	0.934971i	$-0.615425\pi$
$59$	−5.44938	−0.709449	−0.354725	+	0.934971i	$0.615425\pi$
$60$	0	0
$61$	9.99063	1.27917	0.639585	−	0.768721i	$-0.279106\pi$
$61$	9.99063	1.27917	0.639585	+	0.768721i	$0.279106\pi$
$62$	−6.51114	−0.826915
$63$	0	0
$64$	8.85236	1.10654
$65$	0	0
$66$	0	0
$67$	4.87310	0.595344	0.297672	−	0.954668i	$-0.403790\pi$
$67$	4.87310	0.595344	0.297672	+	0.954668i	$0.403790\pi$
$68$	−2.30021	−0.278941
$69$	0	0
$70$	0	0
$71$	−1.39207	−0.165209	−0.0826044	−	0.996582i	$-0.526324\pi$
$71$	−1.39207	−0.165209	−0.0826044	+	0.996582i	$0.526324\pi$
$72$	0	0
$73$	−13.1383	−1.53772	−0.768859	−	0.639418i	$-0.779175\pi$
$73$	−13.1383	−1.53772	−0.768859	+	0.639418i	$0.779175\pi$
$74$	8.38271	0.974470
$75$	0	0
$76$	−2.17929	−0.249981
$77$	15.8573	1.80710
$78$	0	0
$79$	−12.3827	−1.39316	−0.696582	−	0.717478i	$-0.745297\pi$
$79$	−12.3827	−1.39316	−0.696582	+	0.717478i	$0.745297\pi$
$80$	0	0
$81$	0	0
$82$	−11.1699	−1.23351
$83$	−13.2558	−1.45501	−0.727507	−	0.686100i	$-0.759321\pi$
$83$	−13.2558	−1.45501	−0.727507	+	0.686100i	$0.759321\pi$
$84$	0	0
$85$	0	0
$86$	13.5254	1.45848
$87$	0	0
$88$	18.7605	1.99988
$89$	−8.04149	−0.852396	−0.426198	−	0.904630i	$-0.640147\pi$
$89$	−8.04149	−0.852396	−0.426198	+	0.904630i	$0.640147\pi$
$90$	0	0
$91$	2.59210	0.271726
$92$	−4.17929	−0.435721
$93$	0	0
$94$	1.47457	0.152091
$95$	0	0
$96$	0	0
$97$	−15.3778	−1.56138	−0.780689	−	0.624920i	$-0.785132\pi$
$97$	−15.3778	−1.56138	−0.780689	+	0.624920i	$0.785132\pi$
$98$	0.341219	0.0344684
$99$	0	0
$100$	0	0
$101$	−6.33185	−0.630043	−0.315021	−	0.949085i	$-0.602012\pi$
$101$	−6.33185	−0.630043	−0.315021	+	0.949085i	$0.602012\pi$
$102$	0	0
$103$	−1.23014	−0.121209	−0.0606047	−	0.998162i	$-0.519303\pi$
$103$	−1.23014	−0.121209	−0.0606047	+	0.998162i	$0.519303\pi$
$104$	3.06668	0.300712
$105$	0	0
$106$	−6.01582	−0.584308
$107$	−10.4286	−1.00817	−0.504087	−	0.863653i	$-0.668171\pi$
$107$	−10.4286	−1.00817	−0.504087	+	0.863653i	$0.668171\pi$
$108$	0	0
$109$	8.94470	0.856747	0.428373	−	0.903602i	$-0.359087\pi$
$109$	8.94470	0.856747	0.428373	+	0.903602i	$0.359087\pi$
$110$	0	0
$111$	0	0
$112$	−6.92888	−0.654717
$113$	5.57136	0.524110	0.262055	−	0.965053i	$-0.415600\pi$
$113$	5.57136	0.524110	0.262055	+	0.965053i	$0.415600\pi$
$114$	0	0
$115$	0	0
$116$	−1.57628	−0.146354
$117$	0	0
$118$	6.61729	0.609171
$119$	11.3477	1.04024
$120$	0	0
$121$	26.4242	2.40220
$122$	−12.1318	−1.09836
$123$	0	0
$124$	−2.81732	−0.253003
$125$	0	0
$126$	0	0
$127$	−3.65878	−0.324664	−0.162332	−	0.986736i	$-0.551902\pi$
$127$	−3.65878	−0.324664	−0.162332	+	0.986736i	$0.551902\pi$
$128$	−4.97481	−0.439715
$129$	0	0
$130$	0	0
$131$	−2.60793	−0.227856	−0.113928	−	0.993489i	$-0.536343\pi$
$131$	−2.60793	−0.227856	−0.113928	+	0.993489i	$0.536343\pi$
$132$	0	0
$133$	10.7511	0.932241
$134$	−5.91750	−0.511194
$135$	0	0
$136$	13.4252	1.15121
$137$	−11.4795	−0.980759	−0.490380	−	0.871509i	$-0.663142\pi$
$137$	−11.4795	−0.980759	−0.490380	+	0.871509i	$0.663142\pi$
$138$	0	0
$139$	11.8064	1.00141	0.500704	−	0.865619i	$-0.333075\pi$
$139$	11.8064	1.00141	0.500704	+	0.865619i	$0.333075\pi$
$140$	0	0
$141$	0	0
$142$	1.69042	0.141857
$143$	6.11753	0.511574
$144$	0	0
$145$	0	0
$146$	15.9541	1.32037
$147$	0	0
$148$	3.62714	0.298149
$149$	9.86665	0.808307	0.404154	−	0.914691i	$-0.367566\pi$
$149$	9.86665	0.808307	0.404154	+	0.914691i	$0.367566\pi$
$150$	0	0
$151$	11.0207	0.896855	0.448428	−	0.893819i	$-0.351984\pi$
$151$	11.0207	0.896855	0.448428	+	0.893819i	$0.351984\pi$
$152$	12.7195	1.03169
$153$	0	0
$154$	−19.2558	−1.55168
$155$	0	0
$156$	0	0
$157$	12.6271	1.00776	0.503878	−	0.863775i	$-0.331906\pi$
$157$	12.6271	1.00776	0.503878	+	0.863775i	$0.331906\pi$
$158$	15.0366	1.19624
$159$	0	0
$160$	0	0
$161$	20.6178	1.62491
$162$	0	0
$163$	−3.19850	−0.250526	−0.125263	−	0.992124i	$-0.539977\pi$
$163$	−3.19850	−0.250526	−0.125263	+	0.992124i	$0.539977\pi$
$164$	−4.83314	−0.377405
$165$	0	0
$166$	16.0968	1.24935
$167$	−14.7699	−1.14293	−0.571463	−	0.820628i	$-0.693624\pi$
$167$	−14.7699	−1.14293	−0.571463	+	0.820628i	$0.693624\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	5.85236	0.446238
$173$	6.13828	0.466684	0.233342	−	0.972395i	$-0.425034\pi$
$173$	6.13828	0.466684	0.233342	+	0.972395i	$0.425034\pi$
$174$	0	0
$175$	0	0
$176$	−16.3526	−1.23262
$177$	0	0
$178$	9.76494	0.731913
$179$	−14.6494	−1.09495	−0.547474	−	0.836823i	$-0.684411\pi$
$179$	−14.6494	−1.09495	−0.547474	+	0.836823i	$0.684411\pi$
$180$	0	0
$181$	−4.16992	−0.309948	−0.154974	−	0.987919i	$-0.549529\pi$
$181$	−4.16992	−0.309948	−0.154974	+	0.987919i	$0.549529\pi$
$182$	−3.14764	−0.233319
$183$	0	0
$184$	24.3926	1.79824
$185$	0	0
$186$	0	0
$187$	26.7812	1.95844
$188$	0.638037	0.0465336
$189$	0	0
$190$	0	0
$191$	−4.19358	−0.303437	−0.151718	−	0.988424i	$-0.548481\pi$
$191$	−4.19358	−0.303437	−0.151718	+	0.988424i	$0.548481\pi$
$192$	0	0
$193$	−12.4746	−0.897939	−0.448970	−	0.893547i	$-0.648209\pi$
$193$	−12.4746	−0.897939	−0.448970	+	0.893547i	$0.648209\pi$
$194$	18.6735	1.34068
$195$	0	0
$196$	0.147643	0.0105459
$197$	0.903212	0.0643512	0.0321756	−	0.999482i	$-0.489756\pi$
$197$	0.903212	0.0643512	0.0321756	+	0.999482i	$0.489756\pi$
$198$	0	0
$199$	−14.8573	−1.05320	−0.526602	−	0.850112i	$-0.676534\pi$
$199$	−14.8573	−1.05320	−0.526602	+	0.850112i	$0.676534\pi$
$200$	0	0
$201$	0	0
$202$	7.68889	0.540989
$203$	7.77631	0.545790
$204$	0	0
$205$	0	0
$206$	1.49378	0.104077
$207$	0	0
$208$	−2.67307	−0.185344
$209$	25.3733	1.75511
$210$	0	0
$211$	−20.4242	−1.40606	−0.703030	−	0.711160i	$-0.748170\pi$
$211$	−20.4242	−1.40606	−0.703030	+	0.711160i	$0.748170\pi$
$212$	−2.60300	−0.178775
$213$	0	0
$214$	12.6637	0.865673
$215$	0	0
$216$	0	0
$217$	13.8988	0.943510
$218$	−10.8617	−0.735649
$219$	0	0
$220$	0	0
$221$	4.37778	0.294482
$222$	0	0
$223$	2.90321	0.194413	0.0972067	−	0.995264i	$-0.469009\pi$
$223$	2.90321	0.194413	0.0972067	+	0.995264i	$0.469009\pi$
$224$	−7.48442	−0.500074
$225$	0	0
$226$	−6.76541	−0.450029
$227$	−3.10816	−0.206296	−0.103148	−	0.994666i	$-0.532892\pi$
$227$	−3.10816	−0.206296	−0.103148	+	0.994666i	$0.532892\pi$
$228$	0	0
$229$	−21.2257	−1.40263	−0.701317	−	0.712850i	$-0.747404\pi$
$229$	−21.2257	−1.40263	−0.701317	+	0.712850i	$0.747404\pi$
$230$	0	0
$231$	0	0
$232$	9.20003	0.604012
$233$	6.04149	0.395791	0.197895	−	0.980223i	$-0.436589\pi$
$233$	6.04149	0.395791	0.197895	+	0.980223i	$0.436589\pi$
$234$	0	0
$235$	0	0
$236$	2.86326	0.186382
$237$	0	0
$238$	−13.7797	−0.893205
$239$	20.9605	1.35582	0.677912	−	0.735143i	$-0.262885\pi$
$239$	20.9605	1.35582	0.677912	+	0.735143i	$0.262885\pi$
$240$	0	0
$241$	10.5763	0.681278	0.340639	−	0.940194i	$-0.389357\pi$
$241$	10.5763	0.681278	0.340639	+	0.940194i	$0.389357\pi$
$242$	−32.0874	−2.06266
$243$	0	0
$244$	−5.24935	−0.336055
$245$	0	0
$246$	0	0
$247$	4.14764	0.263908
$248$	16.4434	1.04416
$249$	0	0
$250$	0	0
$251$	16.2810	1.02765	0.513824	−	0.857896i	$-0.328229\pi$
$251$	16.2810	1.02765	0.513824	+	0.857896i	$0.328229\pi$
$252$	0	0
$253$	48.6593	3.05918
$254$	4.44293	0.278774
$255$	0	0
$256$	−11.6637	−0.728981
$257$	20.0049	1.24787	0.623936	−	0.781475i	$-0.285532\pi$
$257$	20.0049	1.24787	0.623936	+	0.781475i	$0.285532\pi$
$258$	0	0
$259$	−17.8938	−1.11187
$260$	0	0
$261$	0	0
$262$	3.16686	0.195649
$263$	3.15701	0.194670	0.0973348	−	0.995252i	$-0.468968\pi$
$263$	3.15701	0.194670	0.0973348	+	0.995252i	$0.468968\pi$
$264$	0	0
$265$	0	0
$266$	−13.0553	−0.800472
$267$	0	0
$268$	−2.56046	−0.156405
$269$	17.7971	1.08511	0.542553	−	0.840022i	$-0.317458\pi$
$269$	17.7971	1.08511	0.542553	+	0.840022i	$0.317458\pi$
$270$	0	0
$271$	−25.3575	−1.54036	−0.770180	−	0.637827i	$-0.779834\pi$
$271$	−25.3575	−1.54036	−0.770180	+	0.637827i	$0.779834\pi$
$272$	−11.7021	−0.709546
$273$	0	0
$274$	13.9398	0.842133
$275$	0	0
$276$	0	0
$277$	0.235063	0.0141236	0.00706179	−	0.999975i	$-0.497752\pi$
$277$	0.235063	0.0141236	0.00706179	+	0.999975i	$0.497752\pi$
$278$	−14.3368	−0.859863
$279$	0	0
$280$	0	0
$281$	−29.1338	−1.73798	−0.868989	−	0.494831i	$-0.835230\pi$
$281$	−29.1338	−1.73798	−0.868989	+	0.494831i	$0.835230\pi$
$282$	0	0
$283$	20.5718	1.22287	0.611434	−	0.791295i	$-0.290593\pi$
$283$	20.5718	1.22287	0.611434	+	0.791295i	$0.290593\pi$
$284$	0.731434	0.0434026
$285$	0	0
$286$	−7.42864	−0.439265
$287$	23.8435	1.40744
$288$	0	0
$289$	2.16500	0.127353
$290$	0	0
$291$	0	0
$292$	6.90321	0.403980
$293$	3.12399	0.182505	0.0912526	−	0.995828i	$-0.470913\pi$
$293$	3.12399	0.182505	0.0912526	+	0.995828i	$0.470913\pi$
$294$	0	0
$295$	0	0
$296$	−21.1699	−1.23048
$297$	0	0
$298$	−11.9813	−0.694056
$299$	7.95407	0.459996
$300$	0	0
$301$	−28.8716	−1.66413
$302$	−13.3827	−0.770088
$303$	0	0
$304$	−11.0869	−0.635880
$305$	0	0
$306$	0	0
$307$	14.0874	0.804012	0.402006	−	0.915637i	$-0.368313\pi$
$307$	14.0874	0.804012	0.402006	+	0.915637i	$0.368313\pi$
$308$	−8.33185	−0.474751
$309$	0	0
$310$	0	0
$311$	−17.6128	−0.998733	−0.499366	−	0.866391i	$-0.666434\pi$
$311$	−17.6128	−0.998733	−0.499366	+	0.866391i	$0.666434\pi$
$312$	0	0
$313$	−17.2810	−0.976780	−0.488390	−	0.872626i	$-0.662416\pi$
$313$	−17.2810	−0.976780	−0.488390	+	0.872626i	$0.662416\pi$
$314$	−15.3334	−0.865313
$315$	0	0
$316$	6.50622	0.366003
$317$	0.668149	0.0375270	0.0187635	−	0.999824i	$-0.494027\pi$
$317$	0.668149	0.0375270	0.0187635	+	0.999824i	$0.494027\pi$
$318$	0	0
$319$	18.3526	1.02755
$320$	0	0
$321$	0	0
$322$	−25.0366	−1.39523
$323$	18.1575	1.01031
$324$	0	0
$325$	0	0
$326$	3.88400	0.215115
$327$	0	0
$328$	28.2088	1.55757
$329$	−3.14764	−0.173535
$330$	0	0
$331$	21.3921	1.17581	0.587907	−	0.808928i	$-0.299952\pi$
$331$	21.3921	1.17581	0.587907	+	0.808928i	$0.299952\pi$
$332$	6.96497	0.382252
$333$	0	0
$334$	17.9353	0.981378
$335$	0	0
$336$	0	0
$337$	−27.5259	−1.49943	−0.749716	−	0.661760i	$-0.769810\pi$
$337$	−27.5259	−1.49943	−0.749716	+	0.661760i	$0.769810\pi$
$338$	−1.21432	−0.0660503
$339$	0	0
$340$	0	0
$341$	32.8020	1.77633
$342$	0	0
$343$	−18.8731	−1.01905
$344$	−34.1575	−1.84165
$345$	0	0
$346$	−7.45383	−0.400720
$347$	3.53972	0.190022	0.0950110	−	0.995476i	$-0.469711\pi$
$347$	3.53972	0.190022	0.0950110	+	0.995476i	$0.469711\pi$
$348$	0	0
$349$	−3.46520	−0.185488	−0.0927441	−	0.995690i	$-0.529564\pi$
$349$	−3.46520	−0.185488	−0.0927441	+	0.995690i	$0.529564\pi$
$350$	0	0
$351$	0	0
$352$	−17.6637	−0.941479
$353$	16.6365	0.885472	0.442736	−	0.896652i	$-0.354008\pi$
$353$	16.6365	0.885472	0.442736	+	0.896652i	$0.354008\pi$
$354$	0	0
$355$	0	0
$356$	4.22522	0.223936
$357$	0	0
$358$	17.7891	0.940182
$359$	30.2652	1.59733	0.798667	−	0.601773i	$-0.205539\pi$
$359$	30.2652	1.59733	0.798667	+	0.601773i	$0.205539\pi$
$360$	0	0
$361$	−1.79706	−0.0945819
$362$	5.06361	0.266138
$363$	0	0
$364$	−1.36196	−0.0713863
$365$	0	0
$366$	0	0
$367$	−21.1240	−1.10266	−0.551332	−	0.834286i	$-0.685880\pi$
$367$	−21.1240	−1.10266	−0.551332	+	0.834286i	$0.685880\pi$
$368$	−21.2618	−1.10835
$369$	0	0
$370$	0	0
$371$	12.8415	0.666695
$372$	0	0
$373$	−16.9541	−0.877848	−0.438924	−	0.898524i	$-0.644640\pi$
$373$	−16.9541	−0.877848	−0.438924	+	0.898524i	$0.644640\pi$
$374$	−32.5210	−1.68162
$375$	0	0
$376$	−3.72393	−0.192047
$377$	3.00000	0.154508
$378$	0	0
$379$	−8.27946	−0.425288	−0.212644	−	0.977130i	$-0.568207\pi$
$379$	−8.27946	−0.425288	−0.212644	+	0.977130i	$0.568207\pi$
$380$	0	0
$381$	0	0
$382$	5.09234	0.260547
$383$	−21.7418	−1.11095	−0.555476	−	0.831533i	$-0.687464\pi$
$383$	−21.7418	−1.11095	−0.555476	+	0.831533i	$0.687464\pi$
$384$	0	0
$385$	0	0
$386$	15.1481	0.771019
$387$	0	0
$388$	8.07991	0.410195
$389$	−21.7748	−1.10403	−0.552013	−	0.833836i	$-0.686140\pi$
$389$	−21.7748	−1.10403	−0.552013	+	0.833836i	$0.686140\pi$
$390$	0	0
$391$	34.8212	1.76098
$392$	−0.861725	−0.0435237
$393$	0	0
$394$	−1.09679	−0.0552554
$395$	0	0
$396$	0	0
$397$	−4.91750	−0.246802	−0.123401	−	0.992357i	$-0.539380\pi$
$397$	−4.91750	−0.246802	−0.123401	+	0.992357i	$0.539380\pi$
$398$	18.0415	0.904338
$399$	0	0
$400$	0	0
$401$	−2.60793	−0.130234	−0.0651168	−	0.997878i	$-0.520742\pi$
$401$	−2.60793	−0.130234	−0.0651168	+	0.997878i	$0.520742\pi$
$402$	0	0
$403$	5.36196	0.267098
$404$	3.32693	0.165521
$405$	0	0
$406$	−9.44293	−0.468645
$407$	−42.2306	−2.09329
$408$	0	0
$409$	18.7096	0.925132	0.462566	−	0.886585i	$-0.346929\pi$
$409$	18.7096	0.925132	0.462566	+	0.886585i	$0.346929\pi$
$410$	0	0
$411$	0	0
$412$	0.646350	0.0318434
$413$	−14.1254	−0.695064
$414$	0	0
$415$	0	0
$416$	−2.88739	−0.141566
$417$	0	0
$418$	−30.8113	−1.50703
$419$	−14.2623	−0.696757	−0.348379	−	0.937354i	$-0.613268\pi$
$419$	−14.2623	−0.696757	−0.348379	+	0.937354i	$0.613268\pi$
$420$	0	0
$421$	−21.8064	−1.06278	−0.531390	−	0.847127i	$-0.678330\pi$
$421$	−21.8064	−1.06278	−0.531390	+	0.847127i	$0.678330\pi$
$422$	24.8015	1.20732
$423$	0	0
$424$	15.1925	0.737814
$425$	0	0
$426$	0	0
$427$	25.8968	1.25323
$428$	5.47949	0.264861
$429$	0	0
$430$	0	0
$431$	4.53480	0.218433	0.109217	−	0.994018i	$-0.465166\pi$
$431$	4.53480	0.218433	0.109217	+	0.994018i	$0.465166\pi$
$432$	0	0
$433$	19.4608	0.935224	0.467612	−	0.883934i	$-0.345114\pi$
$433$	19.4608	0.935224	0.467612	+	0.883934i	$0.345114\pi$
$434$	−16.8775	−0.810148
$435$	0	0
$436$	−4.69979	−0.225079
$437$	32.9906	1.57816
$438$	0	0
$439$	−3.17976	−0.151762	−0.0758809	−	0.997117i	$-0.524177\pi$
$439$	−3.17976	−0.151762	−0.0758809	+	0.997117i	$0.524177\pi$
$440$	0	0
$441$	0	0
$442$	−5.31603	−0.252858
$443$	−32.6780	−1.55258	−0.776289	−	0.630377i	$-0.782900\pi$
$443$	−32.6780	−1.55258	−0.776289	+	0.630377i	$0.782900\pi$
$444$	0	0
$445$	0	0
$446$	−3.52543	−0.166934
$447$	0	0
$448$	22.9462	1.08411
$449$	20.1748	0.952110	0.476055	−	0.879416i	$-0.342066\pi$
$449$	20.1748	0.952110	0.476055	+	0.879416i	$0.342066\pi$
$450$	0	0
$451$	56.2721	2.64975
$452$	−2.92735	−0.137691
$453$	0	0
$454$	3.77430	0.177137
$455$	0	0
$456$	0	0
$457$	19.5669	0.915302	0.457651	−	0.889132i	$-0.348691\pi$
$457$	19.5669	0.915302	0.457651	+	0.889132i	$0.348691\pi$
$458$	25.7748	1.20438
$459$	0	0
$460$	0	0
$461$	−33.1338	−1.54320	−0.771598	−	0.636110i	$-0.780542\pi$
$461$	−33.1338	−1.54320	−0.771598	+	0.636110i	$0.780542\pi$
$462$	0	0
$463$	−11.1684	−0.519039	−0.259519	−	0.965738i	$-0.583564\pi$
$463$	−11.1684	−0.519039	−0.259519	+	0.965738i	$0.583564\pi$
$464$	−8.01921	−0.372283
$465$	0	0
$466$	−7.33630	−0.339847
$467$	39.2672	1.81707	0.908534	−	0.417810i	$-0.137202\pi$
$467$	39.2672	1.81707	0.908534	+	0.417810i	$0.137202\pi$
$468$	0	0
$469$	12.6316	0.583272
$470$	0	0
$471$	0	0
$472$	−16.7115	−0.769209
$473$	−68.1388	−3.13302
$474$	0	0
$475$	0	0
$476$	−5.96238	−0.273285
$477$	0	0
$478$	−25.4528	−1.16418
$479$	12.8000	0.584846	0.292423	−	0.956289i	$-0.405539\pi$
$479$	12.8000	0.584846	0.292423	+	0.956289i	$0.405539\pi$
$480$	0	0
$481$	−6.90321	−0.314759
$482$	−12.8430	−0.584982
$483$	0	0
$484$	−13.8840	−0.631091
$485$	0	0
$486$	0	0
$487$	−3.06668	−0.138964	−0.0694822	−	0.997583i	$-0.522135\pi$
$487$	−3.06668	−0.138964	−0.0694822	+	0.997583i	$0.522135\pi$
$488$	30.6380	1.38692
$489$	0	0
$490$	0	0
$491$	−1.17130	−0.0528601	−0.0264300	−	0.999651i	$-0.508414\pi$
$491$	−1.17130	−0.0528601	−0.0264300	+	0.999651i	$0.508414\pi$
$492$	0	0
$493$	13.1334	0.591496
$494$	−5.03657	−0.226606
$495$	0	0
$496$	−14.3329	−0.643566
$497$	−3.60840	−0.161859
$498$	0	0
$499$	38.4499	1.72125	0.860626	−	0.509237i	$-0.170073\pi$
$499$	38.4499	1.72125	0.860626	+	0.509237i	$0.170073\pi$
$500$	0	0
$501$	0	0
$502$	−19.7703	−0.882393
$503$	11.1985	0.499316	0.249658	−	0.968334i	$-0.419682\pi$
$503$	11.1985	0.499316	0.249658	+	0.968334i	$0.419682\pi$
$504$	0	0
$505$	0	0
$506$	−59.0879	−2.62678
$507$	0	0
$508$	1.92242	0.0852938
$509$	6.48442	0.287417	0.143708	−	0.989620i	$-0.454097\pi$
$509$	6.48442	0.287417	0.143708	+	0.989620i	$0.454097\pi$
$510$	0	0
$511$	−34.0558	−1.50654
$512$	24.1131	1.06566
$513$	0	0
$514$	−24.2924	−1.07149
$515$	0	0
$516$	0	0
$517$	−7.42864	−0.326711
$518$	21.7288	0.954711
$519$	0	0
$520$	0	0
$521$	−8.75557	−0.383588	−0.191794	−	0.981435i	$-0.561431\pi$
$521$	−8.75557	−0.383588	−0.191794	+	0.981435i	$0.561431\pi$
$522$	0	0
$523$	−0.447375	−0.0195624	−0.00978118	−	0.999952i	$-0.503113\pi$
$523$	−0.447375	−0.0195624	−0.00978118	+	0.999952i	$0.503113\pi$
$524$	1.37028	0.0598608
$525$	0	0
$526$	−3.83362	−0.167154
$527$	23.4735	1.02252
$528$	0	0
$529$	40.2672	1.75075
$530$	0	0
$531$	0	0
$532$	−5.64894	−0.244912
$533$	9.19850	0.398431
$534$	0	0
$535$	0	0
$536$	14.9442	0.645492
$537$	0	0
$538$	−21.6113	−0.931730
$539$	−1.71900	−0.0740427
$540$	0	0
$541$	−38.9733	−1.67559	−0.837796	−	0.545983i	$-0.816156\pi$
$541$	−38.9733	−1.67559	−0.837796	+	0.545983i	$0.816156\pi$
$542$	30.7921	1.32264
$543$	0	0
$544$	−12.6404	−0.541952
$545$	0	0
$546$	0	0
$547$	−13.1669	−0.562974	−0.281487	−	0.959565i	$-0.590828\pi$
$547$	−13.1669	−0.562974	−0.281487	+	0.959565i	$0.590828\pi$
$548$	6.03164	0.257659
$549$	0	0
$550$	0	0
$551$	12.4429	0.530087
$552$	0	0
$553$	−32.0973	−1.36491
$554$	−0.285442	−0.0121273
$555$	0	0
$556$	−6.20342	−0.263084
$557$	34.3926	1.45726	0.728630	−	0.684908i	$-0.240158\pi$
$557$	34.3926	1.45726	0.728630	+	0.684908i	$0.240158\pi$
$558$	0	0
$559$	−11.1383	−0.471099
$560$	0	0
$561$	0	0
$562$	35.3778	1.49232
$563$	−15.3921	−0.648699	−0.324349	−	0.945937i	$-0.605145\pi$
$563$	−15.3921	−0.648699	−0.324349	+	0.945937i	$0.605145\pi$
$564$	0	0
$565$	0	0
$566$	−24.9808	−1.05002
$567$	0	0
$568$	−4.26904	−0.179125
$569$	−17.6049	−0.738034	−0.369017	−	0.929423i	$-0.620306\pi$
$569$	−17.6049	−0.738034	−0.369017	+	0.929423i	$0.620306\pi$
$570$	0	0
$571$	35.8336	1.49959	0.749795	−	0.661670i	$-0.230152\pi$
$571$	35.8336	1.49959	0.749795	+	0.661670i	$0.230152\pi$
$572$	−3.21432	−0.134397
$573$	0	0
$574$	−28.9536	−1.20850
$575$	0	0
$576$	0	0
$577$	−27.3145	−1.13712	−0.568559	−	0.822643i	$-0.692499\pi$
$577$	−27.3145	−1.13712	−0.568559	+	0.822643i	$0.692499\pi$
$578$	−2.62900	−0.109352
$579$	0	0
$580$	0	0
$581$	−34.3604	−1.42551
$582$	0	0
$583$	30.3067	1.25517
$584$	−40.2908	−1.66725
$585$	0	0
$586$	−3.79352	−0.156709
$587$	6.21924	0.256696	0.128348	−	0.991729i	$-0.459033\pi$
$587$	6.21924	0.256696	0.128348	+	0.991729i	$0.459033\pi$
$588$	0	0
$589$	22.2395	0.916363
$590$	0	0
$591$	0	0
$592$	18.4528	0.758404
$593$	12.3970	0.509084	0.254542	−	0.967062i	$-0.418075\pi$
$593$	12.3970	0.509084	0.254542	+	0.967062i	$0.418075\pi$
$594$	0	0
$595$	0	0
$596$	−5.18421	−0.212353
$597$	0	0
$598$	−9.65878	−0.394977
$599$	−44.2306	−1.80721	−0.903607	−	0.428362i	$-0.859091\pi$
$599$	−44.2306	−1.80721	−0.903607	+	0.428362i	$0.859091\pi$
$600$	0	0
$601$	−42.7003	−1.74178	−0.870890	−	0.491478i	$-0.836457\pi$
$601$	−42.7003	−1.74178	−0.870890	+	0.491478i	$0.836457\pi$
$602$	35.0593	1.42891
$603$	0	0
$604$	−5.79060	−0.235616
$605$	0	0
$606$	0	0
$607$	37.5625	1.52461	0.762307	−	0.647216i	$-0.224067\pi$
$607$	37.5625	1.52461	0.762307	+	0.647216i	$0.224067\pi$
$608$	−11.9759	−0.485685
$609$	0	0
$610$	0	0
$611$	−1.21432	−0.0491261
$612$	0	0
$613$	−29.1052	−1.17555	−0.587775	−	0.809024i	$-0.699996\pi$
$613$	−29.1052	−1.17555	−0.587775	+	0.809024i	$0.699996\pi$
$614$	−17.1066	−0.690367
$615$	0	0
$616$	48.6291	1.95932
$617$	38.4286	1.54708	0.773539	−	0.633748i	$-0.218484\pi$
$617$	38.4286	1.54708	0.773539	+	0.633748i	$0.218484\pi$
$618$	0	0
$619$	−28.0143	−1.12599	−0.562995	−	0.826461i	$-0.690351\pi$
$619$	−28.0143	−1.12599	−0.562995	+	0.826461i	$0.690351\pi$
$620$	0	0
$621$	0	0
$622$	21.3876	0.857566
$623$	−20.8444	−0.835112
$624$	0	0
$625$	0	0
$626$	20.9847	0.838715
$627$	0	0
$628$	−6.63465	−0.264751
$629$	−30.2208	−1.20498
$630$	0	0
$631$	−30.7610	−1.22457	−0.612287	−	0.790635i	$-0.709750\pi$
$631$	−30.7610	−1.22457	−0.612287	+	0.790635i	$0.709750\pi$
$632$	−37.9738	−1.51051
$633$	0	0
$634$	−0.811346	−0.0322227
$635$	0	0
$636$	0	0
$637$	−0.280996	−0.0111335
$638$	−22.2859	−0.882308
$639$	0	0
$640$	0	0
$641$	−15.2351	−0.601749	−0.300874	−	0.953664i	$-0.597278\pi$
$641$	−15.2351	−0.601749	−0.300874	+	0.953664i	$0.597278\pi$
$642$	0	0
$643$	−4.88448	−0.192625	−0.0963125	−	0.995351i	$-0.530705\pi$
$643$	−4.88448	−0.192625	−0.0963125	+	0.995351i	$0.530705\pi$
$644$	−10.8331	−0.426886
$645$	0	0
$646$	−22.0490	−0.867506
$647$	−41.1655	−1.61838	−0.809191	−	0.587546i	$-0.800094\pi$
$647$	−41.1655	−1.61838	−0.809191	+	0.587546i	$0.800094\pi$
$648$	0	0
$649$	−33.3368	−1.30858
$650$	0	0
$651$	0	0
$652$	1.68058	0.0658166
$653$	−20.7748	−0.812980	−0.406490	−	0.913655i	$-0.633247\pi$
$653$	−20.7748	−0.812980	−0.406490	+	0.913655i	$0.633247\pi$
$654$	0	0
$655$	0	0
$656$	−24.5882	−0.960009
$657$	0	0
$658$	3.82225	0.149007
$659$	37.4148	1.45747	0.728737	−	0.684793i	$-0.240108\pi$
$659$	37.4148	1.45747	0.728737	+	0.684793i	$0.240108\pi$
$660$	0	0
$661$	−17.7605	−0.690803	−0.345402	−	0.938455i	$-0.612257\pi$
$661$	−17.7605	−0.690803	−0.345402	+	0.938455i	$0.612257\pi$
$662$	−25.9768	−1.00962
$663$	0	0
$664$	−40.6513	−1.57758
$665$	0	0
$666$	0	0
$667$	23.8622	0.923948
$668$	7.76049	0.300262
$669$	0	0
$670$	0	0
$671$	61.1180	2.35943
$672$	0	0
$673$	−26.9353	−1.03828	−0.519140	−	0.854689i	$-0.673748\pi$
$673$	−26.9353	−1.03828	−0.519140	+	0.854689i	$0.673748\pi$
$674$	33.4252	1.28749
$675$	0	0
$676$	−0.525428	−0.0202088
$677$	4.46028	0.171423	0.0857113	−	0.996320i	$-0.472684\pi$
$677$	4.46028	0.171423	0.0857113	+	0.996320i	$0.472684\pi$
$678$	0	0
$679$	−39.8608	−1.52972
$680$	0	0
$681$	0	0
$682$	−39.8321	−1.52525
$683$	−9.62867	−0.368431	−0.184215	−	0.982886i	$-0.558974\pi$
$683$	−9.62867	−0.368431	−0.184215	+	0.982886i	$0.558974\pi$
$684$	0	0
$685$	0	0
$686$	22.9180	0.875012
$687$	0	0
$688$	29.7734	1.13510
$689$	4.95407	0.188735
$690$	0	0
$691$	32.9605	1.25388	0.626939	−	0.779069i	$-0.284308\pi$
$691$	32.9605	1.25388	0.626939	+	0.779069i	$0.284308\pi$
$692$	−3.22522	−0.122604
$693$	0	0
$694$	−4.29835	−0.163163
$695$	0	0
$696$	0	0
$697$	40.2690	1.52530
$698$	4.20787	0.159270
$699$	0	0
$700$	0	0
$701$	19.8385	0.749291	0.374646	−	0.927168i	$-0.377764\pi$
$701$	19.8385	0.749291	0.374646	+	0.927168i	$0.377764\pi$
$702$	0	0
$703$	−28.6321	−1.07988
$704$	54.1546	2.04103
$705$	0	0
$706$	−20.2020	−0.760314
$707$	−16.4128	−0.617268
$708$	0	0
$709$	16.7828	0.630290	0.315145	−	0.949044i	$-0.397947\pi$
$709$	16.7828	0.630290	0.315145	+	0.949044i	$0.397947\pi$
$710$	0	0
$711$	0	0
$712$	−24.6606	−0.924197
$713$	42.6494	1.59723
$714$	0	0
$715$	0	0
$716$	7.69721	0.287658
$717$	0	0
$718$	−36.7516	−1.37156
$719$	7.89384	0.294391	0.147195	−	0.989107i	$-0.452975\pi$
$719$	7.89384	0.294391	0.147195	+	0.989107i	$0.452975\pi$
$720$	0	0
$721$	−3.18865	−0.118752
$722$	2.18220	0.0812131
$723$	0	0
$724$	2.19099	0.0814275
$725$	0	0
$726$	0	0
$727$	−47.6182	−1.76606	−0.883031	−	0.469314i	$-0.844501\pi$
$727$	−47.6182	−1.76606	−0.883031	+	0.469314i	$0.844501\pi$
$728$	7.94914	0.294615
$729$	0	0
$730$	0	0
$731$	−48.7610	−1.80349
$732$	0	0
$733$	−10.3970	−0.384022	−0.192011	−	0.981393i	$-0.561501\pi$
$733$	−10.3970	−0.384022	−0.192011	+	0.981393i	$0.561501\pi$
$734$	25.6513	0.946806
$735$	0	0
$736$	−22.9665	−0.846556
$737$	29.8113	1.09812
$738$	0	0
$739$	40.1175	1.47575	0.737874	−	0.674939i	$-0.235830\pi$
$739$	40.1175	1.47575	0.737874	+	0.674939i	$0.235830\pi$
$740$	0	0
$741$	0	0
$742$	−15.5936	−0.572460
$743$	−41.1497	−1.50963	−0.754817	−	0.655935i	$-0.772274\pi$
$743$	−41.1497	−1.50963	−0.754817	+	0.655935i	$0.772274\pi$
$744$	0	0
$745$	0	0
$746$	20.5877	0.753768
$747$	0	0
$748$	−14.0716	−0.514509
$749$	−27.0321	−0.987732
$750$	0	0
$751$	46.6178	1.70111	0.850553	−	0.525889i	$-0.176267\pi$
$751$	46.6178	1.70111	0.850553	+	0.525889i	$0.176267\pi$
$752$	3.24596	0.118368
$753$	0	0
$754$	−3.64296	−0.132669
$755$	0	0
$756$	0	0
$757$	−11.1748	−0.406156	−0.203078	−	0.979163i	$-0.565095\pi$
$757$	−11.1748	−0.406156	−0.203078	+	0.979163i	$0.565095\pi$
$758$	10.0539	0.365175
$759$	0	0
$760$	0	0
$761$	36.9260	1.33857	0.669283	−	0.743008i	$-0.266602\pi$
$761$	36.9260	1.33857	0.669283	+	0.743008i	$0.266602\pi$
$762$	0	0
$763$	23.1856	0.839375
$764$	2.20342	0.0797170
$765$	0	0
$766$	26.4014	0.953923
$767$	−5.44938	−0.196766
$768$	0	0
$769$	32.9273	1.18739	0.593695	−	0.804690i	$-0.297668\pi$
$769$	32.9273	1.18739	0.593695	+	0.804690i	$0.297668\pi$
$770$	0	0
$771$	0	0
$772$	6.55448	0.235901
$773$	−11.7649	−0.423155	−0.211578	−	0.977361i	$-0.567860\pi$
$773$	−11.7649	−0.423155	−0.211578	+	0.977361i	$0.567860\pi$
$774$	0	0
$775$	0	0
$776$	−47.1587	−1.69290
$777$	0	0
$778$	26.4415	0.947975
$779$	38.1521	1.36694
$780$	0	0
$781$	−8.51606	−0.304729
$782$	−42.2841	−1.51207
$783$	0	0
$784$	0.751123	0.0268258
$785$	0	0
$786$	0	0
$787$	11.2888	0.402403	0.201202	−	0.979550i	$-0.435515\pi$
$787$	11.2888	0.402403	0.201202	+	0.979550i	$0.435515\pi$
$788$	−0.474572	−0.0169059
$789$	0	0
$790$	0	0
$791$	14.4415	0.513482
$792$	0	0
$793$	9.99063	0.354778
$794$	5.97142	0.211918
$795$	0	0
$796$	7.80642	0.276691
$797$	10.6030	0.375578	0.187789	−	0.982209i	$-0.439868\pi$
$797$	10.6030	0.375578	0.187789	+	0.982209i	$0.439868\pi$
$798$	0	0
$799$	−5.31603	−0.188068
$800$	0	0
$801$	0	0
$802$	3.16686	0.111826
$803$	−80.3738	−2.83633
$804$	0	0
$805$	0	0
$806$	−6.51114	−0.229345
$807$	0	0
$808$	−19.4177	−0.683114
$809$	22.9906	0.808308	0.404154	−	0.914691i	$-0.367566\pi$
$809$	22.9906	0.808308	0.404154	+	0.914691i	$0.367566\pi$
$810$	0	0
$811$	−8.71255	−0.305939	−0.152970	−	0.988231i	$-0.548884\pi$
$811$	−8.71255	−0.305939	−0.152970	+	0.988231i	$0.548884\pi$
$812$	−4.08589	−0.143387
$813$	0	0
$814$	51.2815	1.79741
$815$	0	0
$816$	0	0
$817$	−46.1976	−1.61625
$818$	−22.7195	−0.794368
$819$	0	0
$820$	0	0
$821$	22.7828	0.795124	0.397562	−	0.917575i	$-0.369856\pi$
$821$	22.7828	0.795124	0.397562	+	0.917575i	$0.369856\pi$
$822$	0	0
$823$	−28.8988	−1.00735	−0.503674	−	0.863894i	$-0.668019\pi$
$823$	−28.8988	−1.00735	−0.503674	+	0.863894i	$0.668019\pi$
$824$	−3.77245	−0.131419
$825$	0	0
$826$	17.1527	0.596819
$827$	1.09832	0.0381923	0.0190962	−	0.999818i	$-0.493921\pi$
$827$	1.09832	0.0381923	0.0190962	+	0.999818i	$0.493921\pi$
$828$	0	0
$829$	−54.3689	−1.88831	−0.944155	−	0.329502i	$-0.893119\pi$
$829$	−54.3689	−1.88831	−0.944155	+	0.329502i	$0.893119\pi$
$830$	0	0
$831$	0	0
$832$	8.85236	0.306900
$833$	−1.23014	−0.0426219
$834$	0	0
$835$	0	0
$836$	−13.3319	−0.461092
$837$	0	0
$838$	17.3189	0.598273
$839$	21.2716	0.734378	0.367189	−	0.930146i	$-0.380320\pi$
$839$	21.2716	0.734378	0.367189	+	0.930146i	$0.380320\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	26.4800	0.912560
$843$	0	0
$844$	10.7314	0.369391
$845$	0	0
$846$	0	0
$847$	68.4943	2.35349
$848$	−13.2426	−0.454752
$849$	0	0
$850$	0	0
$851$	−54.9086	−1.88224
$852$	0	0
$853$	5.04101	0.172601	0.0863005	−	0.996269i	$-0.472495\pi$
$853$	5.04101	0.172601	0.0863005	+	0.996269i	$0.472495\pi$
$854$	−31.4469	−1.07609
$855$	0	0
$856$	−31.9813	−1.09310
$857$	−26.3497	−0.900088	−0.450044	−	0.893006i	$-0.648592\pi$
$857$	−26.3497	−0.900088	−0.450044	+	0.893006i	$0.648592\pi$
$858$	0	0
$859$	−12.5575	−0.428458	−0.214229	−	0.976783i	$-0.568724\pi$
$859$	−12.5575	−0.428458	−0.214229	+	0.976783i	$0.568724\pi$
$860$	0	0
$861$	0	0
$862$	−5.50669	−0.187559
$863$	42.5788	1.44940	0.724699	−	0.689066i	$-0.241979\pi$
$863$	42.5788	1.44940	0.724699	+	0.689066i	$0.241979\pi$
$864$	0	0
$865$	0	0
$866$	−23.6316	−0.803034
$867$	0	0
$868$	−7.30279	−0.247873
$869$	−75.7516	−2.56970
$870$	0	0
$871$	4.87310	0.165119
$872$	27.4305	0.928914
$873$	0	0
$874$	−40.0612	−1.35509
$875$	0	0
$876$	0	0
$877$	39.3590	1.32906	0.664530	−	0.747261i	$-0.268632\pi$
$877$	39.3590	1.32906	0.664530	+	0.747261i	$0.268632\pi$
$878$	3.86125	0.130311
$879$	0	0
$880$	0	0
$881$	−33.8845	−1.14160	−0.570799	−	0.821090i	$-0.693366\pi$
$881$	−33.8845	−1.14160	−0.570799	+	0.821090i	$0.693366\pi$
$882$	0	0
$883$	40.6035	1.36642	0.683208	−	0.730224i	$-0.260584\pi$
$883$	40.6035	1.36642	0.683208	+	0.730224i	$0.260584\pi$
$884$	−2.30021	−0.0773644
$885$	0	0
$886$	39.6815	1.33313
$887$	−56.9733	−1.91298	−0.956488	−	0.291772i	$-0.905755\pi$
$887$	−56.9733	−1.91298	−0.956488	+	0.291772i	$0.905755\pi$
$888$	0	0
$889$	−9.48394	−0.318081
$890$	0	0
$891$	0	0
$892$	−1.52543	−0.0510751
$893$	−5.03657	−0.168542
$894$	0	0
$895$	0	0
$896$	−12.8952	−0.430799
$897$	0	0
$898$	−24.4987	−0.817532
$899$	16.0859	0.536494
$900$	0	0
$901$	21.6878	0.722527
$902$	−68.3323	−2.27522
$903$	0	0
$904$	17.0856	0.568257
$905$	0	0
$906$	0	0
$907$	−21.4479	−0.712164	−0.356082	−	0.934455i	$-0.615888\pi$
$907$	−21.4479	−0.712164	−0.356082	+	0.934455i	$0.615888\pi$
$908$	1.63311	0.0541968
$909$	0	0
$910$	0	0
$911$	−17.0781	−0.565821	−0.282911	−	0.959146i	$-0.591300\pi$
$911$	−17.0781	−0.565821	−0.282911	+	0.959146i	$0.591300\pi$
$912$	0	0
$913$	−81.0928	−2.68378
$914$	−23.7605	−0.785927
$915$	0	0
$916$	11.1526	0.368491
$917$	−6.76001	−0.223235
$918$	0	0
$919$	22.8069	0.752330	0.376165	−	0.926553i	$-0.377243\pi$
$919$	22.8069	0.752330	0.376165	+	0.926553i	$0.377243\pi$
$920$	0	0
$921$	0	0
$922$	40.2351	1.32507
$923$	−1.39207	−0.0458207
$924$	0	0
$925$	0	0
$926$	13.5620	0.445675
$927$	0	0
$928$	−8.66217	−0.284350
$929$	22.7654	0.746909	0.373454	−	0.927649i	$-0.378173\pi$
$929$	22.7654	0.746909	0.373454	+	0.927649i	$0.378173\pi$
$930$	0	0
$931$	−1.16547	−0.0381968
$932$	−3.17436	−0.103980
$933$	0	0
$934$	−47.6829	−1.56023
$935$	0	0
$936$	0	0
$937$	29.2065	0.954134	0.477067	−	0.878867i	$-0.341700\pi$
$937$	29.2065	0.954134	0.477067	+	0.878867i	$0.341700\pi$
$938$	−15.3388	−0.500829
$939$	0	0
$940$	0	0
$941$	−0.189130	−0.00616547	−0.00308274	−	0.999995i	$-0.500981\pi$
$941$	−0.189130	−0.00616547	−0.00308274	+	0.999995i	$0.500981\pi$
$942$	0	0
$943$	73.1655	2.38260
$944$	14.5666	0.474102
$945$	0	0
$946$	82.7422	2.69018
$947$	−30.7812	−1.00026	−0.500128	−	0.865952i	$-0.666714\pi$
$947$	−30.7812	−1.00026	−0.500128	+	0.865952i	$0.666714\pi$
$948$	0	0
$949$	−13.1383	−0.426486
$950$	0	0
$951$	0	0
$952$	34.7996	1.12786
$953$	−52.2400	−1.69222	−0.846110	−	0.533009i	$-0.821061\pi$
$953$	−52.2400	−1.69222	−0.846110	+	0.533009i	$0.821061\pi$
$954$	0	0
$955$	0	0
$956$	−11.0132	−0.356193
$957$	0	0
$958$	−15.5433	−0.502180
$959$	−29.7560	−0.960873
$960$	0	0
$961$	−2.24935	−0.0725598
$962$	8.38271	0.270269
$963$	0	0
$964$	−5.55707	−0.178981
$965$	0	0
$966$	0	0
$967$	−47.5279	−1.52839	−0.764197	−	0.644983i	$-0.776865\pi$
$967$	−47.5279	−1.52839	−0.764197	+	0.644983i	$0.776865\pi$
$968$	81.0345	2.60455
$969$	0	0
$970$	0	0
$971$	15.6445	0.502056	0.251028	−	0.967980i	$-0.419231\pi$
$971$	15.6445	0.502056	0.251028	+	0.967980i	$0.419231\pi$
$972$	0	0
$973$	30.6035	0.981103
$974$	3.72393	0.119322
$975$	0	0
$976$	−26.7057	−0.854828
$977$	14.3398	0.458772	0.229386	−	0.973336i	$-0.426328\pi$
$977$	14.3398	0.458772	0.229386	+	0.973336i	$0.426328\pi$
$978$	0	0
$979$	−49.1941	−1.57225
$980$	0	0
$981$	0	0
$982$	1.42233	0.0453885
$983$	6.95206	0.221736	0.110868	−	0.993835i	$-0.464637\pi$
$983$	6.95206	0.221736	0.110868	+	0.993835i	$0.464637\pi$
$984$	0	0
$985$	0	0
$986$	−15.9481	−0.507891
$987$	0	0
$988$	−2.17929	−0.0693323
$989$	−88.5946	−2.81714
$990$	0	0
$991$	13.5768	0.431280	0.215640	−	0.976473i	$-0.430816\pi$
$991$	13.5768	0.431280	0.215640	+	0.976473i	$0.430816\pi$
$992$	−15.4821	−0.491557
$993$	0	0
$994$	4.38175	0.138981
$995$	0	0
$996$	0	0
$997$	13.6953	0.433736	0.216868	−	0.976201i	$-0.430416\pi$
$997$	13.6953	0.433736	0.216868	+	0.976201i	$0.430416\pi$
$998$	−46.6904	−1.47796
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2925.2.a.bk.1.1		3
3.2	odd	2		975.2.a.m.1.3	✓	3
5.2	odd	4		2925.2.c.y.2224.3		6
5.3	odd	4		2925.2.c.y.2224.4		6
5.4	even	2		2925.2.a.be.1.3		3
15.2	even	4		975.2.c.k.274.4		6
15.8	even	4		975.2.c.k.274.3		6
15.14	odd	2		975.2.a.q.1.1	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
975.2.a.m.1.3	✓	3	3.2	odd	2
975.2.a.q.1.1	yes	3	15.14	odd	2
975.2.c.k.274.3		6	15.8	even	4
975.2.c.k.274.4		6	15.2	even	4
2925.2.a.be.1.3		3	5.4	even	2
2925.2.a.bk.1.1		3	1.1	even	1	trivial
2925.2.c.y.2224.3		6	5.2	odd	4
2925.2.c.y.2224.4		6	5.3	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2925 = 3^{2} \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2925.a (trivial)

\(f(q)\)	\(=\)	\(q-1.21432 q^{2} -0.525428 q^{4} +2.59210 q^{7} +3.06668 q^{8} +O(q^{10})\)	\(q-1.21432 q^{2} -0.525428 q^{4} +2.59210 q^{7} +3.06668 q^{8} +6.11753 q^{11} +1.00000 q^{13} -3.14764 q^{14} -2.67307 q^{16} +4.37778 q^{17} +4.14764 q^{19} -7.42864 q^{22} +7.95407 q^{23} -1.21432 q^{26} -1.36196 q^{28} +3.00000 q^{29} +5.36196 q^{31} -2.88739 q^{32} -5.31603 q^{34} -6.90321 q^{37} -5.03657 q^{38} +9.19850 q^{41} -11.1383 q^{43} -3.21432 q^{44} -9.65878 q^{46} -1.21432 q^{47} -0.280996 q^{49} -0.525428 q^{52} +4.95407 q^{53} +7.94914 q^{56} -3.64296 q^{58} -5.44938 q^{59} +9.99063 q^{61} -6.51114 q^{62} +8.85236 q^{64} +4.87310 q^{67} -2.30021 q^{68} -1.39207 q^{71} -13.1383 q^{73} +8.38271 q^{74} -2.17929 q^{76} +15.8573 q^{77} -12.3827 q^{79} -11.1699 q^{82} -13.2558 q^{83} +13.5254 q^{86} +18.7605 q^{88} -8.04149 q^{89} +2.59210 q^{91} -4.17929 q^{92} +1.47457 q^{94} -15.3778 q^{97} +0.341219 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 5 q^{4} + q^{7} + 9 q^{8}+O(q^{10}) \) 3 * q + 3 * q^2 + 5 * q^4 + q^7 + 9 * q^8	\( 3 q + 3 q^{2} + 5 q^{4} + q^{7} + 9 q^{8} + 5 q^{11} + 3 q^{13} - 3 q^{14} + 5 q^{16} + 13 q^{17} + 6 q^{19} - 9 q^{22} + 4 q^{23} + 3 q^{26} + 9 q^{28} + 9 q^{29} + 3 q^{31} + 11 q^{32} + 17 q^{34} - 14 q^{37} - 8 q^{38} + 8 q^{41} - 3 q^{44} - 22 q^{46} + 3 q^{47} + 6 q^{49} + 5 q^{52} - 5 q^{53} + 37 q^{56} + 9 q^{58} + 17 q^{59} + 3 q^{61} - 19 q^{62} + 33 q^{64} + q^{67} + 39 q^{68} + 2 q^{71} - 6 q^{73} - 8 q^{74} - 26 q^{76} + 21 q^{77} - 4 q^{79} - 26 q^{82} + 7 q^{83} + 34 q^{86} + 23 q^{88} + 16 q^{89} + q^{91} - 32 q^{92} + 11 q^{94} - 46 q^{97} + 8 q^{98}+O(q^{100}) \) 3 * q + 3 * q^2 + 5 * q^4 + q^7 + 9 * q^8 + 5 * q^11 + 3 * q^13 - 3 * q^14 + 5 * q^16 + 13 * q^17 + 6 * q^19 - 9 * q^22 + 4 * q^23 + 3 * q^26 + 9 * q^28 + 9 * q^29 + 3 * q^31 + 11 * q^32 + 17 * q^34 - 14 * q^37 - 8 * q^38 + 8 * q^41 - 3 * q^44 - 22 * q^46 + 3 * q^47 + 6 * q^49 + 5 * q^52 - 5 * q^53 + 37 * q^56 + 9 * q^58 + 17 * q^59 + 3 * q^61 - 19 * q^62 + 33 * q^64 + q^67 + 39 * q^68 + 2 * q^71 - 6 * q^73 - 8 * q^74 - 26 * q^76 + 21 * q^77 - 4 * q^79 - 26 * q^82 + 7 * q^83 + 34 * q^86 + 23 * q^88 + 16 * q^89 + q^91 - 32 * q^92 + 11 * q^94 - 46 * q^97 + 8 * q^98

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