LMFDB - Embedded newform 3675.2.a.k.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3675,2,Mod(1,3675)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3675, 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("3675.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3675 = 3 \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3675.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:	$29.3450227428$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 525)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	3675.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000 q^{6} -3.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

$q+1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000 q^{6} -3.00000 q^{8} +1.00000 q^{9} +1.00000 q^{12} +3.00000 q^{13} -1.00000 q^{16} -2.00000 q^{17} +1.00000 q^{18} -1.00000 q^{19} +2.00000 q^{23} +3.00000 q^{24} +3.00000 q^{26} -1.00000 q^{27} -8.00000 q^{29} +8.00000 q^{31} +5.00000 q^{32} -2.00000 q^{34} -1.00000 q^{36} +7.00000 q^{37} -1.00000 q^{38} -3.00000 q^{39} -8.00000 q^{43} +2.00000 q^{46} +10.0000 q^{47} +1.00000 q^{48} +2.00000 q^{51} -3.00000 q^{52} -14.0000 q^{53} -1.00000 q^{54} +1.00000 q^{57} -8.00000 q^{58} -10.0000 q^{59} -7.00000 q^{61} +8.00000 q^{62} +7.00000 q^{64} -5.00000 q^{67} +2.00000 q^{68} -2.00000 q^{69} -12.0000 q^{71} -3.00000 q^{72} +11.0000 q^{73} +7.00000 q^{74} +1.00000 q^{76} -3.00000 q^{78} -7.00000 q^{79} +1.00000 q^{81} -14.0000 q^{83} -8.00000 q^{86} +8.00000 q^{87} +6.00000 q^{89} -2.00000 q^{92} -8.00000 q^{93} +10.0000 q^{94} -5.00000 q^{96} -9.00000 q^{97} +O(q^{100})$

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 $

$2$	1.00000	0.707107	0.353553	−	0.935414i	$-0.384973\pi$
$2$	1.00000	0.707107	0.353553	+	0.935414i	$0.384973\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	−1.00000	−0.408248
$7$	0	0
$7$	0	0
$8$	−3.00000	−1.06066
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	1.00000	0.288675
$13$	3.00000	0.832050	0.416025	−	0.909353i	$-0.363423\pi$
$13$	3.00000	0.832050	0.416025	+	0.909353i	$0.363423\pi$
$14$	0	0
$15$	0	0
$16$	−1.00000	−0.250000
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	1.00000	0.235702
$19$	−1.00000	−0.229416	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000	−0.229416	−0.114708	+	0.993399i	$0.536593\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2.00000	0.417029	0.208514	−	0.978019i	$-0.433137\pi$
$23$	2.00000	0.417029	0.208514	+	0.978019i	$0.433137\pi$
$24$	3.00000	0.612372
$25$	0	0
$26$	3.00000	0.588348
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−8.00000	−1.48556	−0.742781	−	0.669534i	$-0.766494\pi$
$29$	−8.00000	−1.48556	−0.742781	+	0.669534i	$0.766494\pi$
$30$	0	0
$31$	8.00000	1.43684	0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000	1.43684	0.718421	+	0.695608i	$0.244865\pi$
$32$	5.00000	0.883883
$33$	0	0
$34$	−2.00000	−0.342997
$35$	0	0
$36$	−1.00000	−0.166667
$37$	7.00000	1.15079	0.575396	−	0.817875i	$-0.304848\pi$
$37$	7.00000	1.15079	0.575396	+	0.817875i	$0.304848\pi$
$38$	−1.00000	−0.162221
$39$	−3.00000	−0.480384
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	0	0
$45$	0	0
$46$	2.00000	0.294884
$47$	10.0000	1.45865	0.729325	−	0.684167i	$-0.239834\pi$
$47$	10.0000	1.45865	0.729325	+	0.684167i	$0.239834\pi$
$48$	1.00000	0.144338
$49$	0	0
$50$	0	0
$51$	2.00000	0.280056
$52$	−3.00000	−0.416025
$53$	−14.0000	−1.92305	−0.961524	−	0.274721i	$-0.911414\pi$
$53$	−14.0000	−1.92305	−0.961524	+	0.274721i	$0.911414\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	0	0
$57$	1.00000	0.132453
$58$	−8.00000	−1.05045
$59$	−10.0000	−1.30189	−0.650945	−	0.759125i	$-0.725627\pi$
$59$	−10.0000	−1.30189	−0.650945	+	0.759125i	$0.725627\pi$
$60$	0	0
$61$	−7.00000	−0.896258	−0.448129	−	0.893969i	$-0.647910\pi$
$61$	−7.00000	−0.896258	−0.448129	+	0.893969i	$0.647910\pi$
$62$	8.00000	1.01600
$63$	0	0
$64$	7.00000	0.875000
$65$	0	0
$66$	0	0
$67$	−5.00000	−0.610847	−0.305424	−	0.952217i	$-0.598798\pi$
$67$	−5.00000	−0.610847	−0.305424	+	0.952217i	$0.598798\pi$
$68$	2.00000	0.242536
$69$	−2.00000	−0.240772
$70$	0	0
$71$	−12.0000	−1.42414	−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000	−1.42414	−0.712069	+	0.702109i	$0.752242\pi$
$72$	−3.00000	−0.353553
$73$	11.0000	1.28745	0.643726	−	0.765256i	$-0.277388\pi$
$73$	11.0000	1.28745	0.643726	+	0.765256i	$0.277388\pi$
$74$	7.00000	0.813733
$75$	0	0
$76$	1.00000	0.114708
$77$	0	0
$78$	−3.00000	−0.339683
$79$	−7.00000	−0.787562	−0.393781	−	0.919204i	$-0.628833\pi$
$79$	−7.00000	−0.787562	−0.393781	+	0.919204i	$0.628833\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−14.0000	−1.53670	−0.768350	−	0.640030i	$-0.778922\pi$
$83$	−14.0000	−1.53670	−0.768350	+	0.640030i	$0.778922\pi$
$84$	0	0
$85$	0	0
$86$	−8.00000	−0.862662
$87$	8.00000	0.857690
$88$	0	0
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	0	0
$92$	−2.00000	−0.208514
$93$	−8.00000	−0.829561
$94$	10.0000	1.03142
$95$	0	0
$96$	−5.00000	−0.510310
$97$	−9.00000	−0.913812	−0.456906	−	0.889515i	$-0.651042\pi$
$97$	−9.00000	−0.913812	−0.456906	+	0.889515i	$0.651042\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	16.0000	1.59206	0.796030	−	0.605257i	$-0.206930\pi$
$101$	16.0000	1.59206	0.796030	+	0.605257i	$0.206930\pi$
$102$	2.00000	0.198030
$103$	13.0000	1.28093	0.640464	−	0.767988i	$-0.278742\pi$
$103$	13.0000	1.28093	0.640464	+	0.767988i	$0.278742\pi$
$104$	−9.00000	−0.882523
$105$	0	0
$106$	−14.0000	−1.35980
$107$	6.00000	0.580042	0.290021	−	0.957020i	$-0.406338\pi$
$107$	6.00000	0.580042	0.290021	+	0.957020i	$0.406338\pi$
$108$	1.00000	0.0962250
$109$	−15.0000	−1.43674	−0.718370	−	0.695662i	$-0.755111\pi$
$109$	−15.0000	−1.43674	−0.718370	+	0.695662i	$0.755111\pi$
$110$	0	0
$111$	−7.00000	−0.664411
$112$	0	0
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	1.00000	0.0936586
$115$	0	0
$116$	8.00000	0.742781
$117$	3.00000	0.277350
$118$	−10.0000	−0.920575
$119$	0	0
$120$	0	0
$121$	−11.0000	−1.00000
$122$	−7.00000	−0.633750
$123$	0	0
$124$	−8.00000	−0.718421
$125$	0	0
$126$	0	0
$127$	11.0000	0.976092	0.488046	−	0.872818i	$-0.337710\pi$
$127$	11.0000	0.976092	0.488046	+	0.872818i	$0.337710\pi$
$128$	−3.00000	−0.265165
$129$	8.00000	0.704361
$130$	0	0
$131$	−14.0000	−1.22319	−0.611593	−	0.791173i	$-0.709471\pi$
$131$	−14.0000	−1.22319	−0.611593	+	0.791173i	$0.709471\pi$
$132$	0	0
$133$	0	0
$134$	−5.00000	−0.431934
$135$	0	0
$136$	6.00000	0.514496
$137$	−10.0000	−0.854358	−0.427179	−	0.904167i	$-0.640493\pi$
$137$	−10.0000	−0.854358	−0.427179	+	0.904167i	$0.640493\pi$
$138$	−2.00000	−0.170251
$139$	−3.00000	−0.254457	−0.127228	−	0.991873i	$-0.540608\pi$
$139$	−3.00000	−0.254457	−0.127228	+	0.991873i	$0.540608\pi$
$140$	0	0
$141$	−10.0000	−0.842152
$142$	−12.0000	−1.00702
$143$	0	0
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	11.0000	0.910366
$147$	0	0
$148$	−7.00000	−0.575396
$149$	−4.00000	−0.327693	−0.163846	−	0.986486i	$-0.552390\pi$
$149$	−4.00000	−0.327693	−0.163846	+	0.986486i	$0.552390\pi$
$150$	0	0
$151$	−1.00000	−0.0813788	−0.0406894	−	0.999172i	$-0.512955\pi$
$151$	−1.00000	−0.0813788	−0.0406894	+	0.999172i	$0.512955\pi$
$152$	3.00000	0.243332
$153$	−2.00000	−0.161690
$154$	0	0
$155$	0	0
$156$	3.00000	0.240192
$157$	−7.00000	−0.558661	−0.279330	−	0.960195i	$-0.590112\pi$
$157$	−7.00000	−0.558661	−0.279330	+	0.960195i	$0.590112\pi$
$158$	−7.00000	−0.556890
$159$	14.0000	1.11027
$160$	0	0
$161$	0	0
$162$	1.00000	0.0785674
$163$	−9.00000	−0.704934	−0.352467	−	0.935824i	$-0.614657\pi$
$163$	−9.00000	−0.704934	−0.352467	+	0.935824i	$0.614657\pi$
$164$	0	0
$165$	0	0
$166$	−14.0000	−1.08661
$167$	−2.00000	−0.154765	−0.0773823	−	0.997001i	$-0.524656\pi$
$167$	−2.00000	−0.154765	−0.0773823	+	0.997001i	$0.524656\pi$
$168$	0	0
$169$	−4.00000	−0.307692
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	8.00000	0.609994
$173$	−18.0000	−1.36851	−0.684257	−	0.729241i	$-0.739873\pi$
$173$	−18.0000	−1.36851	−0.684257	+	0.729241i	$0.739873\pi$
$174$	8.00000	0.606478
$175$	0	0
$176$	0	0
$177$	10.0000	0.751646
$178$	6.00000	0.449719
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	0	0
$181$	−22.0000	−1.63525	−0.817624	−	0.575753i	$-0.804709\pi$
$181$	−22.0000	−1.63525	−0.817624	+	0.575753i	$0.804709\pi$
$182$	0	0
$183$	7.00000	0.517455
$184$	−6.00000	−0.442326
$185$	0	0
$186$	−8.00000	−0.586588
$187$	0	0
$188$	−10.0000	−0.729325
$189$	0	0
$190$	0	0
$191$	−8.00000	−0.578860	−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000	−0.578860	−0.289430	+	0.957199i	$0.593466\pi$
$192$	−7.00000	−0.505181
$193$	6.00000	0.431889	0.215945	−	0.976406i	$-0.430717\pi$
$193$	6.00000	0.431889	0.215945	+	0.976406i	$0.430717\pi$
$194$	−9.00000	−0.646162
$195$	0	0
$196$	0	0
$197$	12.0000	0.854965	0.427482	−	0.904024i	$-0.359401\pi$
$197$	12.0000	0.854965	0.427482	+	0.904024i	$0.359401\pi$
$198$	0	0
$199$	−11.0000	−0.779769	−0.389885	−	0.920864i	$-0.627485\pi$
$199$	−11.0000	−0.779769	−0.389885	+	0.920864i	$0.627485\pi$
$200$	0	0
$201$	5.00000	0.352673
$202$	16.0000	1.12576
$203$	0	0
$204$	−2.00000	−0.140028
$205$	0	0
$206$	13.0000	0.905753
$207$	2.00000	0.139010
$208$	−3.00000	−0.208013
$209$	0	0
$210$	0	0
$211$	−5.00000	−0.344214	−0.172107	−	0.985078i	$-0.555058\pi$
$211$	−5.00000	−0.344214	−0.172107	+	0.985078i	$0.555058\pi$
$212$	14.0000	0.961524
$213$	12.0000	0.822226
$214$	6.00000	0.410152
$215$	0	0
$216$	3.00000	0.204124
$217$	0	0
$218$	−15.0000	−1.01593
$219$	−11.0000	−0.743311
$220$	0	0
$221$	−6.00000	−0.403604
$222$	−7.00000	−0.469809
$223$	15.0000	1.00447	0.502237	−	0.864730i	$-0.332510\pi$
$223$	15.0000	1.00447	0.502237	+	0.864730i	$0.332510\pi$
$224$	0	0
$225$	0	0
$226$	−6.00000	−0.399114
$227$	−20.0000	−1.32745	−0.663723	−	0.747978i	$-0.731025\pi$
$227$	−20.0000	−1.32745	−0.663723	+	0.747978i	$0.731025\pi$
$228$	−1.00000	−0.0662266
$229$	−13.0000	−0.859064	−0.429532	−	0.903052i	$-0.641321\pi$
$229$	−13.0000	−0.859064	−0.429532	+	0.903052i	$0.641321\pi$
$230$	0	0
$231$	0	0
$232$	24.0000	1.57568
$233$	24.0000	1.57229	0.786146	−	0.618041i	$-0.212073\pi$
$233$	24.0000	1.57229	0.786146	+	0.618041i	$0.212073\pi$
$234$	3.00000	0.196116
$235$	0	0
$236$	10.0000	0.650945
$237$	7.00000	0.454699
$238$	0	0
$239$	−28.0000	−1.81117	−0.905585	−	0.424165i	$-0.860568\pi$
$239$	−28.0000	−1.81117	−0.905585	+	0.424165i	$0.860568\pi$
$240$	0	0
$241$	−21.0000	−1.35273	−0.676364	−	0.736567i	$-0.736446\pi$
$241$	−21.0000	−1.35273	−0.676364	+	0.736567i	$0.736446\pi$
$242$	−11.0000	−0.707107
$243$	−1.00000	−0.0641500
$244$	7.00000	0.448129
$245$	0	0
$246$	0	0
$247$	−3.00000	−0.190885
$248$	−24.0000	−1.52400
$249$	14.0000	0.887214
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	0	0
$254$	11.0000	0.690201
$255$	0	0
$256$	−17.0000	−1.06250
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	8.00000	0.498058
$259$	0	0
$260$	0	0
$261$	−8.00000	−0.495188
$262$	−14.0000	−0.864923
$263$	16.0000	0.986602	0.493301	−	0.869859i	$-0.335790\pi$
$263$	16.0000	0.986602	0.493301	+	0.869859i	$0.335790\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−6.00000	−0.367194
$268$	5.00000	0.305424
$269$	16.0000	0.975537	0.487769	−	0.872973i	$-0.337811\pi$
$269$	16.0000	0.975537	0.487769	+	0.872973i	$0.337811\pi$
$270$	0	0
$271$	12.0000	0.728948	0.364474	−	0.931214i	$-0.381249\pi$
$271$	12.0000	0.728948	0.364474	+	0.931214i	$0.381249\pi$
$272$	2.00000	0.121268
$273$	0	0
$274$	−10.0000	−0.604122
$275$	0	0
$276$	2.00000	0.120386
$277$	5.00000	0.300421	0.150210	−	0.988654i	$-0.452005\pi$
$277$	5.00000	0.300421	0.150210	+	0.988654i	$0.452005\pi$
$278$	−3.00000	−0.179928
$279$	8.00000	0.478947
$280$	0	0
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	−10.0000	−0.595491
$283$	19.0000	1.12943	0.564716	−	0.825285i	$-0.308986\pi$
$283$	19.0000	1.12943	0.564716	+	0.825285i	$0.308986\pi$
$284$	12.0000	0.712069
$285$	0	0
$286$	0	0
$287$	0	0
$288$	5.00000	0.294628
$289$	−13.0000	−0.764706
$290$	0	0
$291$	9.00000	0.527589
$292$	−11.0000	−0.643726
$293$	16.0000	0.934730	0.467365	−	0.884064i	$-0.345203\pi$
$293$	16.0000	0.934730	0.467365	+	0.884064i	$0.345203\pi$
$294$	0	0
$295$	0	0
$296$	−21.0000	−1.22060
$297$	0	0
$298$	−4.00000	−0.231714
$299$	6.00000	0.346989
$300$	0	0
$301$	0	0
$302$	−1.00000	−0.0575435
$303$	−16.0000	−0.919176
$304$	1.00000	0.0573539
$305$	0	0
$306$	−2.00000	−0.114332
$307$	12.0000	0.684876	0.342438	−	0.939540i	$-0.388747\pi$
$307$	12.0000	0.684876	0.342438	+	0.939540i	$0.388747\pi$
$308$	0	0
$309$	−13.0000	−0.739544
$310$	0	0
$311$	30.0000	1.70114	0.850572	−	0.525859i	$-0.176256\pi$
$311$	30.0000	1.70114	0.850572	+	0.525859i	$0.176256\pi$
$312$	9.00000	0.509525
$313$	−14.0000	−0.791327	−0.395663	−	0.918396i	$-0.629485\pi$
$313$	−14.0000	−0.791327	−0.395663	+	0.918396i	$0.629485\pi$
$314$	−7.00000	−0.395033
$315$	0	0
$316$	7.00000	0.393781
$317$	14.0000	0.786318	0.393159	−	0.919470i	$-0.371382\pi$
$317$	14.0000	0.786318	0.393159	+	0.919470i	$0.371382\pi$
$318$	14.0000	0.785081
$319$	0	0
$320$	0	0
$321$	−6.00000	−0.334887
$322$	0	0
$323$	2.00000	0.111283
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	−9.00000	−0.498464
$327$	15.0000	0.829502
$328$	0	0
$329$	0	0
$330$	0	0
$331$	21.0000	1.15426	0.577132	−	0.816651i	$-0.304172\pi$
$331$	21.0000	1.15426	0.577132	+	0.816651i	$0.304172\pi$
$332$	14.0000	0.768350
$333$	7.00000	0.383598
$334$	−2.00000	−0.109435
$335$	0	0
$336$	0	0
$337$	−2.00000	−0.108947	−0.0544735	−	0.998515i	$-0.517348\pi$
$337$	−2.00000	−0.108947	−0.0544735	+	0.998515i	$0.517348\pi$
$338$	−4.00000	−0.217571
$339$	6.00000	0.325875
$340$	0	0
$341$	0	0
$342$	−1.00000	−0.0540738
$343$	0	0
$344$	24.0000	1.29399
$345$	0	0
$346$	−18.0000	−0.967686
$347$	10.0000	0.536828	0.268414	−	0.963304i	$-0.413500\pi$
$347$	10.0000	0.536828	0.268414	+	0.963304i	$0.413500\pi$
$348$	−8.00000	−0.428845
$349$	22.0000	1.17763	0.588817	−	0.808267i	$-0.299594\pi$
$349$	22.0000	1.17763	0.588817	+	0.808267i	$0.299594\pi$
$350$	0	0
$351$	−3.00000	−0.160128
$352$	0	0
$353$	−24.0000	−1.27739	−0.638696	−	0.769460i	$-0.720526\pi$
$353$	−24.0000	−1.27739	−0.638696	+	0.769460i	$0.720526\pi$
$354$	10.0000	0.531494
$355$	0	0
$356$	−6.00000	−0.317999
$357$	0	0
$358$	−12.0000	−0.634220
$359$	30.0000	1.58334	0.791670	−	0.610949i	$-0.209212\pi$
$359$	30.0000	1.58334	0.791670	+	0.610949i	$0.209212\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	−22.0000	−1.15629
$363$	11.0000	0.577350
$364$	0	0
$365$	0	0
$366$	7.00000	0.365896
$367$	8.00000	0.417597	0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000	0.417597	0.208798	+	0.977959i	$0.433045\pi$
$368$	−2.00000	−0.104257
$369$	0	0
$370$	0	0
$371$	0	0
$372$	8.00000	0.414781
$373$	11.0000	0.569558	0.284779	−	0.958593i	$-0.408080\pi$
$373$	11.0000	0.569558	0.284779	+	0.958593i	$0.408080\pi$
$374$	0	0
$375$	0	0
$376$	−30.0000	−1.54713
$377$	−24.0000	−1.23606
$378$	0	0
$379$	35.0000	1.79783	0.898915	−	0.438124i	$-0.144357\pi$
$379$	35.0000	1.79783	0.898915	+	0.438124i	$0.144357\pi$
$380$	0	0
$381$	−11.0000	−0.563547
$382$	−8.00000	−0.409316
$383$	−10.0000	−0.510976	−0.255488	−	0.966812i	$-0.582236\pi$
$383$	−10.0000	−0.510976	−0.255488	+	0.966812i	$0.582236\pi$
$384$	3.00000	0.153093
$385$	0	0
$386$	6.00000	0.305392
$387$	−8.00000	−0.406663
$388$	9.00000	0.456906
$389$	−36.0000	−1.82527	−0.912636	−	0.408773i	$-0.865957\pi$
$389$	−36.0000	−1.82527	−0.912636	+	0.408773i	$0.865957\pi$
$390$	0	0
$391$	−4.00000	−0.202289
$392$	0	0
$393$	14.0000	0.706207
$394$	12.0000	0.604551
$395$	0	0
$396$	0	0
$397$	−14.0000	−0.702640	−0.351320	−	0.936255i	$-0.614267\pi$
$397$	−14.0000	−0.702640	−0.351320	+	0.936255i	$0.614267\pi$
$398$	−11.0000	−0.551380
$399$	0	0
$400$	0	0
$401$	−12.0000	−0.599251	−0.299626	−	0.954057i	$-0.596862\pi$
$401$	−12.0000	−0.599251	−0.299626	+	0.954057i	$0.596862\pi$
$402$	5.00000	0.249377
$403$	24.0000	1.19553
$404$	−16.0000	−0.796030
$405$	0	0
$406$	0	0
$407$	0	0
$408$	−6.00000	−0.297044
$409$	−5.00000	−0.247234	−0.123617	−	0.992330i	$-0.539449\pi$
$409$	−5.00000	−0.247234	−0.123617	+	0.992330i	$0.539449\pi$
$410$	0	0
$411$	10.0000	0.493264
$412$	−13.0000	−0.640464
$413$	0	0
$414$	2.00000	0.0982946
$415$	0	0
$416$	15.0000	0.735436
$417$	3.00000	0.146911
$418$	0	0
$419$	−24.0000	−1.17248	−0.586238	−	0.810139i	$-0.699392\pi$
$419$	−24.0000	−1.17248	−0.586238	+	0.810139i	$0.699392\pi$
$420$	0	0
$421$	−35.0000	−1.70580	−0.852898	−	0.522078i	$-0.825157\pi$
$421$	−35.0000	−1.70580	−0.852898	+	0.522078i	$0.825157\pi$
$422$	−5.00000	−0.243396
$423$	10.0000	0.486217
$424$	42.0000	2.03970
$425$	0	0
$426$	12.0000	0.581402
$427$	0	0
$428$	−6.00000	−0.290021
$429$	0	0
$430$	0	0
$431$	−2.00000	−0.0963366	−0.0481683	−	0.998839i	$-0.515338\pi$
$431$	−2.00000	−0.0963366	−0.0481683	+	0.998839i	$0.515338\pi$
$432$	1.00000	0.0481125
$433$	2.00000	0.0961139	0.0480569	−	0.998845i	$-0.484697\pi$
$433$	2.00000	0.0961139	0.0480569	+	0.998845i	$0.484697\pi$
$434$	0	0
$435$	0	0
$436$	15.0000	0.718370
$437$	−2.00000	−0.0956730
$438$	−11.0000	−0.525600
$439$	11.0000	0.525001	0.262501	−	0.964932i	$-0.415453\pi$
$439$	11.0000	0.525001	0.262501	+	0.964932i	$0.415453\pi$
$440$	0	0
$441$	0	0
$442$	−6.00000	−0.285391
$443$	6.00000	0.285069	0.142534	−	0.989790i	$-0.454475\pi$
$443$	6.00000	0.285069	0.142534	+	0.989790i	$0.454475\pi$
$444$	7.00000	0.332205
$445$	0	0
$446$	15.0000	0.710271
$447$	4.00000	0.189194
$448$	0	0
$449$	6.00000	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$450$	0	0
$451$	0	0
$452$	6.00000	0.282216
$453$	1.00000	0.0469841
$454$	−20.0000	−0.938647
$455$	0	0
$456$	−3.00000	−0.140488
$457$	−3.00000	−0.140334	−0.0701670	−	0.997535i	$-0.522353\pi$
$457$	−3.00000	−0.140334	−0.0701670	+	0.997535i	$0.522353\pi$
$458$	−13.0000	−0.607450
$459$	2.00000	0.0933520
$460$	0	0
$461$	2.00000	0.0931493	0.0465746	−	0.998915i	$-0.485169\pi$
$461$	2.00000	0.0931493	0.0465746	+	0.998915i	$0.485169\pi$
$462$	0	0
$463$	33.0000	1.53364	0.766820	−	0.641862i	$-0.221838\pi$
$463$	33.0000	1.53364	0.766820	+	0.641862i	$0.221838\pi$
$464$	8.00000	0.371391
$465$	0	0
$466$	24.0000	1.11178
$467$	2.00000	0.0925490	0.0462745	−	0.998929i	$-0.485265\pi$
$467$	2.00000	0.0925490	0.0462745	+	0.998929i	$0.485265\pi$
$468$	−3.00000	−0.138675
$469$	0	0
$470$	0	0
$471$	7.00000	0.322543
$472$	30.0000	1.38086
$473$	0	0
$474$	7.00000	0.321521
$475$	0	0
$476$	0	0
$477$	−14.0000	−0.641016
$478$	−28.0000	−1.28069
$479$	−2.00000	−0.0913823	−0.0456912	−	0.998956i	$-0.514549\pi$
$479$	−2.00000	−0.0913823	−0.0456912	+	0.998956i	$0.514549\pi$
$480$	0	0
$481$	21.0000	0.957518
$482$	−21.0000	−0.956524
$483$	0	0
$484$	11.0000	0.500000
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	8.00000	0.362515	0.181257	−	0.983436i	$-0.441983\pi$
$487$	8.00000	0.362515	0.181257	+	0.983436i	$0.441983\pi$
$488$	21.0000	0.950625
$489$	9.00000	0.406994
$490$	0	0
$491$	−6.00000	−0.270776	−0.135388	−	0.990793i	$-0.543228\pi$
$491$	−6.00000	−0.270776	−0.135388	+	0.990793i	$0.543228\pi$
$492$	0	0
$493$	16.0000	0.720604
$494$	−3.00000	−0.134976
$495$	0	0
$496$	−8.00000	−0.359211
$497$	0	0
$498$	14.0000	0.627355
$499$	−19.0000	−0.850557	−0.425278	−	0.905063i	$-0.639824\pi$
$499$	−19.0000	−0.850557	−0.425278	+	0.905063i	$0.639824\pi$
$500$	0	0
$501$	2.00000	0.0893534
$502$	0	0
$503$	−40.0000	−1.78351	−0.891756	−	0.452517i	$-0.850526\pi$
$503$	−40.0000	−1.78351	−0.891756	+	0.452517i	$0.850526\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	4.00000	0.177646
$508$	−11.0000	−0.488046
$509$	8.00000	0.354594	0.177297	−	0.984157i	$-0.443265\pi$
$509$	8.00000	0.354594	0.177297	+	0.984157i	$0.443265\pi$
$510$	0	0
$511$	0	0
$512$	−11.0000	−0.486136
$513$	1.00000	0.0441511
$514$	0	0
$515$	0	0
$516$	−8.00000	−0.352180
$517$	0	0
$518$	0	0
$519$	18.0000	0.790112
$520$	0	0
$521$	−34.0000	−1.48957	−0.744784	−	0.667306i	$-0.767447\pi$
$521$	−34.0000	−1.48957	−0.744784	+	0.667306i	$0.767447\pi$
$522$	−8.00000	−0.350150
$523$	16.0000	0.699631	0.349816	−	0.936819i	$-0.386244\pi$
$523$	16.0000	0.699631	0.349816	+	0.936819i	$0.386244\pi$
$524$	14.0000	0.611593
$525$	0	0
$526$	16.0000	0.697633
$527$	−16.0000	−0.696971
$528$	0	0
$529$	−19.0000	−0.826087
$530$	0	0
$531$	−10.0000	−0.433963
$532$	0	0
$533$	0	0
$534$	−6.00000	−0.259645
$535$	0	0
$536$	15.0000	0.647901
$537$	12.0000	0.517838
$538$	16.0000	0.689809
$539$	0	0
$540$	0	0
$541$	−3.00000	−0.128980	−0.0644900	−	0.997918i	$-0.520542\pi$
$541$	−3.00000	−0.128980	−0.0644900	+	0.997918i	$0.520542\pi$
$542$	12.0000	0.515444
$543$	22.0000	0.944110
$544$	−10.0000	−0.428746
$545$	0	0
$546$	0	0
$547$	−12.0000	−0.513083	−0.256541	−	0.966533i	$-0.582583\pi$
$547$	−12.0000	−0.513083	−0.256541	+	0.966533i	$0.582583\pi$
$548$	10.0000	0.427179
$549$	−7.00000	−0.298753
$550$	0	0
$551$	8.00000	0.340811
$552$	6.00000	0.255377
$553$	0	0
$554$	5.00000	0.212430
$555$	0	0
$556$	3.00000	0.127228
$557$	−4.00000	−0.169485	−0.0847427	−	0.996403i	$-0.527007\pi$
$557$	−4.00000	−0.169485	−0.0847427	+	0.996403i	$0.527007\pi$
$558$	8.00000	0.338667
$559$	−24.0000	−1.01509
$560$	0	0
$561$	0	0
$562$	6.00000	0.253095
$563$	−28.0000	−1.18006	−0.590030	−	0.807382i	$-0.700884\pi$
$563$	−28.0000	−1.18006	−0.590030	+	0.807382i	$0.700884\pi$
$564$	10.0000	0.421076
$565$	0	0
$566$	19.0000	0.798630
$567$	0	0
$568$	36.0000	1.51053
$569$	12.0000	0.503066	0.251533	−	0.967849i	$-0.419065\pi$
$569$	12.0000	0.503066	0.251533	+	0.967849i	$0.419065\pi$
$570$	0	0
$571$	−3.00000	−0.125546	−0.0627730	−	0.998028i	$-0.519994\pi$
$571$	−3.00000	−0.125546	−0.0627730	+	0.998028i	$0.519994\pi$
$572$	0	0
$573$	8.00000	0.334205
$574$	0	0
$575$	0	0
$576$	7.00000	0.291667
$577$	2.00000	0.0832611	0.0416305	−	0.999133i	$-0.486745\pi$
$577$	2.00000	0.0832611	0.0416305	+	0.999133i	$0.486745\pi$
$578$	−13.0000	−0.540729
$579$	−6.00000	−0.249351
$580$	0	0
$581$	0	0
$582$	9.00000	0.373062
$583$	0	0
$584$	−33.0000	−1.36555
$585$	0	0
$586$	16.0000	0.660954
$587$	6.00000	0.247647	0.123823	−	0.992304i	$-0.460484\pi$
$587$	6.00000	0.247647	0.123823	+	0.992304i	$0.460484\pi$
$588$	0	0
$589$	−8.00000	−0.329634
$590$	0	0
$591$	−12.0000	−0.493614
$592$	−7.00000	−0.287698
$593$	−42.0000	−1.72473	−0.862367	−	0.506284i	$-0.831019\pi$
$593$	−42.0000	−1.72473	−0.862367	+	0.506284i	$0.831019\pi$
$594$	0	0
$595$	0	0
$596$	4.00000	0.163846
$597$	11.0000	0.450200
$598$	6.00000	0.245358
$599$	26.0000	1.06233	0.531166	−	0.847268i	$-0.321754\pi$
$599$	26.0000	1.06233	0.531166	+	0.847268i	$0.321754\pi$
$600$	0	0
$601$	45.0000	1.83559	0.917794	−	0.397057i	$-0.129968\pi$
$601$	45.0000	1.83559	0.917794	+	0.397057i	$0.129968\pi$
$602$	0	0
$603$	−5.00000	−0.203616
$604$	1.00000	0.0406894
$605$	0	0
$606$	−16.0000	−0.649956
$607$	43.0000	1.74532	0.872658	−	0.488332i	$-0.162394\pi$
$607$	43.0000	1.74532	0.872658	+	0.488332i	$0.162394\pi$
$608$	−5.00000	−0.202777
$609$	0	0
$610$	0	0
$611$	30.0000	1.21367
$612$	2.00000	0.0808452
$613$	6.00000	0.242338	0.121169	−	0.992632i	$-0.461336\pi$
$613$	6.00000	0.242338	0.121169	+	0.992632i	$0.461336\pi$
$614$	12.0000	0.484281
$615$	0	0
$616$	0	0
$617$	−24.0000	−0.966204	−0.483102	−	0.875564i	$-0.660490\pi$
$617$	−24.0000	−0.966204	−0.483102	+	0.875564i	$0.660490\pi$
$618$	−13.0000	−0.522937
$619$	12.0000	0.482321	0.241160	−	0.970485i	$-0.422472\pi$
$619$	12.0000	0.482321	0.241160	+	0.970485i	$0.422472\pi$
$620$	0	0
$621$	−2.00000	−0.0802572
$622$	30.0000	1.20289
$623$	0	0
$624$	3.00000	0.120096
$625$	0	0
$626$	−14.0000	−0.559553
$627$	0	0
$628$	7.00000	0.279330
$629$	−14.0000	−0.558217
$630$	0	0
$631$	45.0000	1.79142	0.895711	−	0.444637i	$-0.146667\pi$
$631$	45.0000	1.79142	0.895711	+	0.444637i	$0.146667\pi$
$632$	21.0000	0.835335
$633$	5.00000	0.198732
$634$	14.0000	0.556011
$635$	0	0
$636$	−14.0000	−0.555136
$637$	0	0
$638$	0	0
$639$	−12.0000	−0.474713
$640$	0	0
$641$	24.0000	0.947943	0.473972	−	0.880540i	$-0.342820\pi$
$641$	24.0000	0.947943	0.473972	+	0.880540i	$0.342820\pi$
$642$	−6.00000	−0.236801
$643$	29.0000	1.14365	0.571824	−	0.820376i	$-0.306236\pi$
$643$	29.0000	1.14365	0.571824	+	0.820376i	$0.306236\pi$
$644$	0	0
$645$	0	0
$646$	2.00000	0.0786889
$647$	−12.0000	−0.471769	−0.235884	−	0.971781i	$-0.575799\pi$
$647$	−12.0000	−0.471769	−0.235884	+	0.971781i	$0.575799\pi$
$648$	−3.00000	−0.117851
$649$	0	0
$650$	0	0
$651$	0	0
$652$	9.00000	0.352467
$653$	44.0000	1.72185	0.860927	−	0.508729i	$-0.169885\pi$
$653$	44.0000	1.72185	0.860927	+	0.508729i	$0.169885\pi$
$654$	15.0000	0.586546
$655$	0	0
$656$	0	0
$657$	11.0000	0.429151
$658$	0	0
$659$	−24.0000	−0.934907	−0.467454	−	0.884018i	$-0.654829\pi$
$659$	−24.0000	−0.934907	−0.467454	+	0.884018i	$0.654829\pi$
$660$	0	0
$661$	−11.0000	−0.427850	−0.213925	−	0.976850i	$-0.568625\pi$
$661$	−11.0000	−0.427850	−0.213925	+	0.976850i	$0.568625\pi$
$662$	21.0000	0.816188
$663$	6.00000	0.233021
$664$	42.0000	1.62992
$665$	0	0
$666$	7.00000	0.271244
$667$	−16.0000	−0.619522
$668$	2.00000	0.0773823
$669$	−15.0000	−0.579934
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−1.00000	−0.0385472	−0.0192736	−	0.999814i	$-0.506135\pi$
$673$	−1.00000	−0.0385472	−0.0192736	+	0.999814i	$0.506135\pi$
$674$	−2.00000	−0.0770371
$675$	0	0
$676$	4.00000	0.153846
$677$	14.0000	0.538064	0.269032	−	0.963131i	$-0.413296\pi$
$677$	14.0000	0.538064	0.269032	+	0.963131i	$0.413296\pi$
$678$	6.00000	0.230429
$679$	0	0
$680$	0	0
$681$	20.0000	0.766402
$682$	0	0
$683$	0	0		−	1.00000i	$-0.5\pi$
$683$	0	0			1.00000i	$0.5\pi$
$684$	1.00000	0.0382360
$685$	0	0
$686$	0	0
$687$	13.0000	0.495981
$688$	8.00000	0.304997
$689$	−42.0000	−1.60007
$690$	0	0
$691$	15.0000	0.570627	0.285313	−	0.958434i	$-0.407902\pi$
$691$	15.0000	0.570627	0.285313	+	0.958434i	$0.407902\pi$
$692$	18.0000	0.684257
$693$	0	0
$694$	10.0000	0.379595
$695$	0	0
$696$	−24.0000	−0.909718
$697$	0	0
$698$	22.0000	0.832712
$699$	−24.0000	−0.907763
$700$	0	0
$701$	34.0000	1.28416	0.642081	−	0.766637i	$-0.278071\pi$
$701$	34.0000	1.28416	0.642081	+	0.766637i	$0.278071\pi$
$702$	−3.00000	−0.113228
$703$	−7.00000	−0.264010
$704$	0	0
$705$	0	0
$706$	−24.0000	−0.903252
$707$	0	0
$708$	−10.0000	−0.375823
$709$	1.00000	0.0375558	0.0187779	−	0.999824i	$-0.494022\pi$
$709$	1.00000	0.0375558	0.0187779	+	0.999824i	$0.494022\pi$
$710$	0	0
$711$	−7.00000	−0.262521
$712$	−18.0000	−0.674579
$713$	16.0000	0.599205
$714$	0	0
$715$	0	0
$716$	12.0000	0.448461
$717$	28.0000	1.04568
$718$	30.0000	1.11959
$719$	4.00000	0.149175	0.0745874	−	0.997214i	$-0.476236\pi$
$719$	4.00000	0.149175	0.0745874	+	0.997214i	$0.476236\pi$
$720$	0	0
$721$	0	0
$722$	−18.0000	−0.669891
$723$	21.0000	0.780998
$724$	22.0000	0.817624
$725$	0	0
$726$	11.0000	0.408248
$727$	−43.0000	−1.59478	−0.797391	−	0.603463i	$-0.793787\pi$
$727$	−43.0000	−1.59478	−0.797391	+	0.603463i	$0.793787\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	16.0000	0.591781
$732$	−7.00000	−0.258727
$733$	31.0000	1.14501	0.572506	−	0.819901i	$-0.305971\pi$
$733$	31.0000	1.14501	0.572506	+	0.819901i	$0.305971\pi$
$734$	8.00000	0.295285
$735$	0	0
$736$	10.0000	0.368605
$737$	0	0
$738$	0	0
$739$	−49.0000	−1.80249	−0.901247	−	0.433306i	$-0.857347\pi$
$739$	−49.0000	−1.80249	−0.901247	+	0.433306i	$0.857347\pi$
$740$	0	0
$741$	3.00000	0.110208
$742$	0	0
$743$	6.00000	0.220119	0.110059	−	0.993925i	$-0.464896\pi$
$743$	6.00000	0.220119	0.110059	+	0.993925i	$0.464896\pi$
$744$	24.0000	0.879883
$745$	0	0
$746$	11.0000	0.402739
$747$	−14.0000	−0.512233
$748$	0	0
$749$	0	0
$750$	0	0
$751$	23.0000	0.839282	0.419641	−	0.907690i	$-0.362156\pi$
$751$	23.0000	0.839282	0.419641	+	0.907690i	$0.362156\pi$
$752$	−10.0000	−0.364662
$753$	0	0
$754$	−24.0000	−0.874028
$755$	0	0
$756$	0	0
$757$	−43.0000	−1.56286	−0.781431	−	0.623992i	$-0.785510\pi$
$757$	−43.0000	−1.56286	−0.781431	+	0.623992i	$0.785510\pi$
$758$	35.0000	1.27126
$759$	0	0
$760$	0	0
$761$	6.00000	0.217500	0.108750	−	0.994069i	$-0.465315\pi$
$761$	6.00000	0.217500	0.108750	+	0.994069i	$0.465315\pi$
$762$	−11.0000	−0.398488
$763$	0	0
$764$	8.00000	0.289430
$765$	0	0
$766$	−10.0000	−0.361315
$767$	−30.0000	−1.08324
$768$	17.0000	0.613435
$769$	10.0000	0.360609	0.180305	−	0.983611i	$-0.442292\pi$
$769$	10.0000	0.360609	0.180305	+	0.983611i	$0.442292\pi$
$770$	0	0
$771$	0	0
$772$	−6.00000	−0.215945
$773$	−48.0000	−1.72644	−0.863220	−	0.504828i	$-0.831556\pi$
$773$	−48.0000	−1.72644	−0.863220	+	0.504828i	$0.831556\pi$
$774$	−8.00000	−0.287554
$775$	0	0
$776$	27.0000	0.969244
$777$	0	0
$778$	−36.0000	−1.29066
$779$	0	0
$780$	0	0
$781$	0	0
$782$	−4.00000	−0.143040
$783$	8.00000	0.285897
$784$	0	0
$785$	0	0
$786$	14.0000	0.499363
$787$	17.0000	0.605985	0.302992	−	0.952993i	$-0.402014\pi$
$787$	17.0000	0.605985	0.302992	+	0.952993i	$0.402014\pi$
$788$	−12.0000	−0.427482
$789$	−16.0000	−0.569615
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−21.0000	−0.745732
$794$	−14.0000	−0.496841
$795$	0	0
$796$	11.0000	0.389885
$797$	−30.0000	−1.06265	−0.531327	−	0.847167i	$-0.678307\pi$
$797$	−30.0000	−1.06265	−0.531327	+	0.847167i	$0.678307\pi$
$798$	0	0
$799$	−20.0000	−0.707549
$800$	0	0
$801$	6.00000	0.212000
$802$	−12.0000	−0.423735
$803$	0	0
$804$	−5.00000	−0.176336
$805$	0	0
$806$	24.0000	0.845364
$807$	−16.0000	−0.563227
$808$	−48.0000	−1.68863
$809$	−6.00000	−0.210949	−0.105474	−	0.994422i	$-0.533636\pi$
$809$	−6.00000	−0.210949	−0.105474	+	0.994422i	$0.533636\pi$
$810$	0	0
$811$	−3.00000	−0.105344	−0.0526721	−	0.998612i	$-0.516774\pi$
$811$	−3.00000	−0.105344	−0.0526721	+	0.998612i	$0.516774\pi$
$812$	0	0
$813$	−12.0000	−0.420858
$814$	0	0
$815$	0	0
$816$	−2.00000	−0.0700140
$817$	8.00000	0.279885
$818$	−5.00000	−0.174821
$819$	0	0
$820$	0	0
$821$	−48.0000	−1.67521	−0.837606	−	0.546275i	$-0.816045\pi$
$821$	−48.0000	−1.67521	−0.837606	+	0.546275i	$0.816045\pi$
$822$	10.0000	0.348790
$823$	5.00000	0.174289	0.0871445	−	0.996196i	$-0.472226\pi$
$823$	5.00000	0.174289	0.0871445	+	0.996196i	$0.472226\pi$
$824$	−39.0000	−1.35863
$825$	0	0
$826$	0	0
$827$	−24.0000	−0.834562	−0.417281	−	0.908778i	$-0.637017\pi$
$827$	−24.0000	−0.834562	−0.417281	+	0.908778i	$0.637017\pi$
$828$	−2.00000	−0.0695048
$829$	15.0000	0.520972	0.260486	−	0.965478i	$-0.416117\pi$
$829$	15.0000	0.520972	0.260486	+	0.965478i	$0.416117\pi$
$830$	0	0
$831$	−5.00000	−0.173448
$832$	21.0000	0.728044
$833$	0	0
$834$	3.00000	0.103882
$835$	0	0
$836$	0	0
$837$	−8.00000	−0.276520
$838$	−24.0000	−0.829066
$839$	42.0000	1.45000	0.725001	−	0.688748i	$-0.241839\pi$
$839$	42.0000	1.45000	0.725001	+	0.688748i	$0.241839\pi$
$840$	0	0
$841$	35.0000	1.20690
$842$	−35.0000	−1.20618
$843$	−6.00000	−0.206651
$844$	5.00000	0.172107
$845$	0	0
$846$	10.0000	0.343807
$847$	0	0
$848$	14.0000	0.480762
$849$	−19.0000	−0.652078
$850$	0	0
$851$	14.0000	0.479914
$852$	−12.0000	−0.411113
$853$	6.00000	0.205436	0.102718	−	0.994711i	$-0.467246\pi$
$853$	6.00000	0.205436	0.102718	+	0.994711i	$0.467246\pi$
$854$	0	0
$855$	0	0
$856$	−18.0000	−0.615227
$857$	−18.0000	−0.614868	−0.307434	−	0.951569i	$-0.599470\pi$
$857$	−18.0000	−0.614868	−0.307434	+	0.951569i	$0.599470\pi$
$858$	0	0
$859$	52.0000	1.77422	0.887109	−	0.461561i	$-0.152710\pi$
$859$	52.0000	1.77422	0.887109	+	0.461561i	$0.152710\pi$
$860$	0	0
$861$	0	0
$862$	−2.00000	−0.0681203
$863$	6.00000	0.204242	0.102121	−	0.994772i	$-0.467437\pi$
$863$	6.00000	0.204242	0.102121	+	0.994772i	$0.467437\pi$
$864$	−5.00000	−0.170103
$865$	0	0
$866$	2.00000	0.0679628
$867$	13.0000	0.441503
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−15.0000	−0.508256
$872$	45.0000	1.52389
$873$	−9.00000	−0.304604
$874$	−2.00000	−0.0676510
$875$	0	0
$876$	11.0000	0.371656
$877$	23.0000	0.776655	0.388327	−	0.921521i	$-0.373053\pi$
$877$	23.0000	0.776655	0.388327	+	0.921521i	$0.373053\pi$
$878$	11.0000	0.371232
$879$	−16.0000	−0.539667
$880$	0	0
$881$	−12.0000	−0.404290	−0.202145	−	0.979356i	$-0.564791\pi$
$881$	−12.0000	−0.404290	−0.202145	+	0.979356i	$0.564791\pi$
$882$	0	0
$883$	31.0000	1.04323	0.521617	−	0.853180i	$-0.325329\pi$
$883$	31.0000	1.04323	0.521617	+	0.853180i	$0.325329\pi$
$884$	6.00000	0.201802
$885$	0	0
$886$	6.00000	0.201574
$887$	−58.0000	−1.94745	−0.973725	−	0.227728i	$-0.926870\pi$
$887$	−58.0000	−1.94745	−0.973725	+	0.227728i	$0.926870\pi$
$888$	21.0000	0.704714
$889$	0	0
$890$	0	0
$891$	0	0
$892$	−15.0000	−0.502237
$893$	−10.0000	−0.334637
$894$	4.00000	0.133780
$895$	0	0
$896$	0	0
$897$	−6.00000	−0.200334
$898$	6.00000	0.200223
$899$	−64.0000	−2.13452
$900$	0	0
$901$	28.0000	0.932815
$902$	0	0
$903$	0	0
$904$	18.0000	0.598671
$905$	0	0
$906$	1.00000	0.0332228
$907$	1.00000	0.0332045	0.0166022	−	0.999862i	$-0.494715\pi$
$907$	1.00000	0.0332045	0.0166022	+	0.999862i	$0.494715\pi$
$908$	20.0000	0.663723
$909$	16.0000	0.530687
$910$	0	0
$911$	54.0000	1.78910	0.894550	−	0.446968i	$-0.147496\pi$
$911$	54.0000	1.78910	0.894550	+	0.446968i	$0.147496\pi$
$912$	−1.00000	−0.0331133
$913$	0	0
$914$	−3.00000	−0.0992312
$915$	0	0
$916$	13.0000	0.429532
$917$	0	0
$918$	2.00000	0.0660098
$919$	24.0000	0.791687	0.395843	−	0.918318i	$-0.370452\pi$
$919$	24.0000	0.791687	0.395843	+	0.918318i	$0.370452\pi$
$920$	0	0
$921$	−12.0000	−0.395413
$922$	2.00000	0.0658665
$923$	−36.0000	−1.18495
$924$	0	0
$925$	0	0
$926$	33.0000	1.08445
$927$	13.0000	0.426976
$928$	−40.0000	−1.31306
$929$	44.0000	1.44359	0.721797	−	0.692105i	$-0.243317\pi$
$929$	44.0000	1.44359	0.721797	+	0.692105i	$0.243317\pi$
$930$	0	0
$931$	0	0
$932$	−24.0000	−0.786146
$933$	−30.0000	−0.982156
$934$	2.00000	0.0654420
$935$	0	0
$936$	−9.00000	−0.294174
$937$	−22.0000	−0.718709	−0.359354	−	0.933201i	$-0.617003\pi$
$937$	−22.0000	−0.718709	−0.359354	+	0.933201i	$0.617003\pi$
$938$	0	0
$939$	14.0000	0.456873
$940$	0	0
$941$	−42.0000	−1.36916	−0.684580	−	0.728937i	$-0.740015\pi$
$941$	−42.0000	−1.36916	−0.684580	+	0.728937i	$0.740015\pi$
$942$	7.00000	0.228072
$943$	0	0
$944$	10.0000	0.325472
$945$	0	0
$946$	0	0
$947$	−22.0000	−0.714904	−0.357452	−	0.933932i	$-0.616354\pi$
$947$	−22.0000	−0.714904	−0.357452	+	0.933932i	$0.616354\pi$
$948$	−7.00000	−0.227349
$949$	33.0000	1.07123
$950$	0	0
$951$	−14.0000	−0.453981
$952$	0	0
$953$	−20.0000	−0.647864	−0.323932	−	0.946080i	$-0.605005\pi$
$953$	−20.0000	−0.647864	−0.323932	+	0.946080i	$0.605005\pi$
$954$	−14.0000	−0.453267
$955$	0	0
$956$	28.0000	0.905585
$957$	0	0
$958$	−2.00000	−0.0646171
$959$	0	0
$960$	0	0
$961$	33.0000	1.06452
$962$	21.0000	0.677067
$963$	6.00000	0.193347
$964$	21.0000	0.676364
$965$	0	0
$966$	0	0
$967$	−23.0000	−0.739630	−0.369815	−	0.929105i	$-0.620579\pi$
$967$	−23.0000	−0.739630	−0.369815	+	0.929105i	$0.620579\pi$
$968$	33.0000	1.06066
$969$	−2.00000	−0.0642493
$970$	0	0
$971$	−8.00000	−0.256732	−0.128366	−	0.991727i	$-0.540973\pi$
$971$	−8.00000	−0.256732	−0.128366	+	0.991727i	$0.540973\pi$
$972$	1.00000	0.0320750
$973$	0	0
$974$	8.00000	0.256337
$975$	0	0
$976$	7.00000	0.224065
$977$	−40.0000	−1.27971	−0.639857	−	0.768494i	$-0.721006\pi$
$977$	−40.0000	−1.27971	−0.639857	+	0.768494i	$0.721006\pi$
$978$	9.00000	0.287788
$979$	0	0
$980$	0	0
$981$	−15.0000	−0.478913
$982$	−6.00000	−0.191468
$983$	22.0000	0.701691	0.350846	−	0.936433i	$-0.385894\pi$
$983$	22.0000	0.701691	0.350846	+	0.936433i	$0.385894\pi$
$984$	0	0
$985$	0	0
$986$	16.0000	0.509544
$987$	0	0
$988$	3.00000	0.0954427
$989$	−16.0000	−0.508770
$990$	0	0
$991$	0	0		−	1.00000i	$-0.5\pi$
$991$	0	0			1.00000i	$0.5\pi$
$992$	40.0000	1.27000
$993$	−21.0000	−0.666415
$994$	0	0
$995$	0	0
$996$	−14.0000	−0.443607
$997$	37.0000	1.17180	0.585901	−	0.810383i	$-0.300741\pi$
$997$	37.0000	1.17180	0.585901	+	0.810383i	$0.300741\pi$
$998$	−19.0000	−0.601434
$999$	−7.00000	−0.221470

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3675.2.a.k.1.1		1
5.4	even	2		3675.2.a.g.1.1		1
7.3	odd	6		525.2.i.b.226.1	yes	2
7.5	odd	6		525.2.i.b.151.1	✓	2
7.6	odd	2		3675.2.a.m.1.1		1
35.3	even	12		525.2.r.c.499.2		4
35.12	even	12		525.2.r.c.424.2		4
35.17	even	12		525.2.r.c.499.1		4
35.19	odd	6		525.2.i.d.151.1	yes	2
35.24	odd	6		525.2.i.d.226.1	yes	2
35.33	even	12		525.2.r.c.424.1		4
35.34	odd	2		3675.2.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
525.2.i.b.151.1	✓	2	7.5	odd	6
525.2.i.b.226.1	yes	2	7.3	odd	6
525.2.i.d.151.1	yes	2	35.19	odd	6
525.2.i.d.226.1	yes	2	35.24	odd	6
525.2.r.c.424.1		4	35.33	even	12
525.2.r.c.424.2		4	35.12	even	12
525.2.r.c.499.1		4	35.17	even	12
525.2.r.c.499.2		4	35.3	even	12
3675.2.a.e.1.1		1	35.34	odd	2
3675.2.a.g.1.1		1	5.4	even	2
3675.2.a.k.1.1		1	1.1	even	1	trivial
3675.2.a.m.1.1		1	7.6	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:	\( N \)	\(=\)	\( 3675 = 3 \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3675.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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