LMFDB - Embedded newform 2925.2.c.w.2224.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2925,2,Mod(2224,2925)]

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, 1, 0]))

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

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

chi := DirichletCharacter("2925.2224");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2925 = 3^{2} \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2925.c (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$23.3562425912$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.399424.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8 $ x^6 - 2x^5 + 3x^4 - 6x^3 + 6x^2 - 8*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 195)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2224.4
Root			$1.40680 - 0.144584i$ of defining polynomial
Character	$\chi$	$=$	2925.2224
Dual form			2925.2.c.w.2224.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.289169i q^{2} +1.91638 q^{4} +4.91638i q^{7} +1.13249i q^{8} +O(q^{10})$	$q+0.289169i q^{2} +1.91638 q^{4} +4.91638i q^{7} +1.13249i q^{8} -4.91638 q^{11} -1.00000i q^{13} -1.42166 q^{14} +3.50528 q^{16} +4.33804i q^{17} -2.57834 q^{19} -1.42166i q^{22} -6.33804i q^{23} +0.289169 q^{26} +9.42166i q^{28} +6.00000 q^{29} +1.42166 q^{31} +3.27861i q^{32} -1.25443 q^{34} +9.49472i q^{37} -0.745574i q^{38} -4.33804 q^{41} +1.15667i q^{43} -9.42166 q^{44} +1.83276 q^{46} +5.42166i q^{47} -17.1708 q^{49} -1.91638i q^{52} -0.338044i q^{53} -5.56777 q^{56} +1.73501i q^{58} -11.2544 q^{59} -10.1708 q^{61} +0.411100i q^{62} +6.06249 q^{64} -7.25443i q^{67} +8.31335i q^{68} -0.916382 q^{71} +3.15667i q^{73} -2.74557 q^{74} -4.94108 q^{76} -24.1708i q^{77} +3.49472 q^{79} -1.25443i q^{82} +11.2544i q^{83} -0.334474 q^{86} -5.56777i q^{88} -0.338044 q^{89} +4.91638 q^{91} -12.1461i q^{92} -1.56777 q^{94} -12.3380i q^{97} -4.96526i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 16 q^{4}+O(q^{10}) $ 6 * q - 16 * q^4	$ 6 q - 16 q^{4} - 2 q^{11} - 12 q^{14} + 52 q^{16} - 12 q^{19} + 36 q^{29} + 12 q^{31} + 44 q^{34} - 2 q^{41} - 60 q^{44} - 44 q^{46} - 24 q^{49} + 32 q^{56} - 16 q^{59} + 18 q^{61} - 60 q^{64} + 22 q^{71} - 68 q^{74} + 8 q^{76} - 10 q^{79} - 112 q^{86} + 22 q^{89} + 2 q^{91} + 56 q^{94}+O(q^{100}) $ 6 * q - 16 * q^4 - 2 * q^11 - 12 * q^14 + 52 * q^16 - 12 * q^19 + 36 * q^29 + 12 * q^31 + 44 * q^34 - 2 * q^41 - 60 * q^44 - 44 * q^46 - 24 * q^49 + 32 * q^56 - 16 * q^59 + 18 * q^61 - 60 * q^64 + 22 * q^71 - 68 * q^74 + 8 * q^76 - 10 * q^79 - 112 * q^86 + 22 * q^89 + 2 * q^91 + 56 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2925\mathbb{Z}\right)^\times$.

$n$	$326$	$352$	$2251$
$\chi(n)$	$1$	$-1$	$1$

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 $

$2$	0.289169i	0.204473i	0.994760	+	0.102237i	$0.0325999\pi$
$2$	0.289169i	0.204473i	−0.994760	+	0.102237i	$0.967400\pi$
$3$	0	0
$3$	0	0
$4$	1.91638	0.958191
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.91638i	1.85822i	0.369807	+	0.929109i	$0.379424\pi$
$7$	4.91638i	1.85822i	−0.369807	+	0.929109i	$0.620576\pi$
$8$	1.13249i	0.400397i
$9$	0	0
$10$	0	0
$11$	−4.91638	−1.48234	−0.741172	−	0.671315i	$-0.765730\pi$
$11$	−4.91638	−1.48234	−0.741172	+	0.671315i	$0.765730\pi$
$12$	0	0
$13$	− 1.00000i	− 0.277350i
$13$	− 1.00000i	− 0.277350i
$14$	−1.42166	−0.379955
$15$	0	0
$16$	3.50528	0.876320
$17$	4.33804i	1.05213i	0.850444	+	0.526065i	$0.176333\pi$
$17$	4.33804i	1.05213i	−0.850444	+	0.526065i	$0.823667\pi$
$18$	0	0
$19$	−2.57834	−0.591511	−0.295756	−	0.955264i	$-0.595571\pi$
$19$	−2.57834	−0.591511	−0.295756	+	0.955264i	$0.595571\pi$
$20$	0	0
$21$	0	0
$22$	− 1.42166i	− 0.303100i
$23$	− 6.33804i	− 1.32157i	−0.750574	−	0.660787i	$-0.770223\pi$
$23$	− 6.33804i	− 1.32157i	0.750574	−	0.660787i	$-0.229777\pi$
$24$	0	0
$25$	0	0
$26$	0.289169	0.0567106
$27$	0	0
$28$	9.42166i	1.78053i
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	1.42166	0.255338	0.127669	−	0.991817i	$-0.459250\pi$
$31$	1.42166	0.255338	0.127669	+	0.991817i	$0.459250\pi$
$32$	3.27861i	0.579581i
$33$	0	0
$34$	−1.25443	−0.215132
$35$	0	0
$36$	0	0
$37$	9.49472i	1.56092i	0.625204	+	0.780461i	$0.285016\pi$
$37$	9.49472i	1.56092i	−0.625204	+	0.780461i	$0.714984\pi$
$38$	− 0.745574i	− 0.120948i
$39$	0	0
$40$	0	0
$41$	−4.33804	−0.677489	−0.338744	−	0.940878i	$-0.610002\pi$
$41$	−4.33804	−0.677489	−0.338744	+	0.940878i	$0.610002\pi$
$42$	0	0
$43$	1.15667i	0.176391i	0.996103	+	0.0881956i	$0.0281100\pi$
$43$	1.15667i	0.176391i	−0.996103	+	0.0881956i	$0.971890\pi$
$44$	−9.42166	−1.42037
$45$	0	0
$46$	1.83276	0.270226
$47$	5.42166i	0.790831i	0.918502	+	0.395415i	$0.129399\pi$
$47$	5.42166i	0.790831i	−0.918502	+	0.395415i	$0.870601\pi$
$48$	0	0
$49$	−17.1708	−2.45297
$50$	0	0
$51$	0	0
$52$	− 1.91638i	− 0.265754i
$53$	− 0.338044i	− 0.0464340i	−0.999730	−	0.0232170i	$-0.992609\pi$
$53$	− 0.338044i	− 0.0464340i	0.999730	−	0.0232170i	$-0.00739086\pi$
$54$	0	0
$55$	0	0
$56$	−5.56777	−0.744025
$57$	0	0
$58$	1.73501i	0.227818i
$59$	−11.2544	−1.46520	−0.732601	−	0.680659i	$-0.761694\pi$
$59$	−11.2544	−1.46520	−0.732601	+	0.680659i	$0.761694\pi$
$60$	0	0
$61$	−10.1708	−1.30224	−0.651119	−	0.758975i	$-0.725700\pi$
$61$	−10.1708	−1.30224	−0.651119	+	0.758975i	$0.725700\pi$
$62$	0.411100i	0.0522098i
$63$	0	0
$64$	6.06249	0.757812
$65$	0	0
$66$	0	0
$67$	− 7.25443i	− 0.886269i	−0.896455	−	0.443135i	$-0.853866\pi$
$67$	− 7.25443i	− 0.886269i	0.896455	−	0.443135i	$-0.146134\pi$
$68$	8.31335i	1.00814i
$69$	0	0
$70$	0	0
$71$	−0.916382	−0.108754	−0.0543772	−	0.998520i	$-0.517317\pi$
$71$	−0.916382	−0.108754	−0.0543772	+	0.998520i	$0.517317\pi$
$72$	0	0
$73$	3.15667i	0.369461i	0.982789	+	0.184730i	$0.0591412\pi$
$73$	3.15667i	0.369461i	−0.982789	+	0.184730i	$0.940859\pi$
$74$	−2.74557	−0.319166
$75$	0	0
$76$	−4.94108	−0.566780
$77$	− 24.1708i	− 2.75452i
$78$	0	0
$79$	3.49472	0.393187	0.196593	−	0.980485i	$-0.437012\pi$
$79$	3.49472	0.393187	0.196593	+	0.980485i	$0.437012\pi$
$80$	0	0
$81$	0	0
$82$	− 1.25443i	− 0.138528i
$83$	11.2544i	1.23533i	0.786440	+	0.617667i	$0.211922\pi$
$83$	11.2544i	1.23533i	−0.786440	+	0.617667i	$0.788078\pi$
$84$	0	0
$85$	0	0
$86$	−0.334474	−0.0360672
$87$	0	0
$88$	− 5.56777i	− 0.593527i
$89$	−0.338044	−0.0358326	−0.0179163	−	0.999839i	$-0.505703\pi$
$89$	−0.338044	−0.0358326	−0.0179163	+	0.999839i	$0.505703\pi$
$90$	0	0
$91$	4.91638	0.515377
$92$	− 12.1461i	− 1.26632i
$93$	0	0
$94$	−1.56777	−0.161704
$95$	0	0
$96$	0	0
$97$	− 12.3380i	− 1.25274i	−0.779526	−	0.626369i	$-0.784540\pi$
$97$	− 12.3380i	− 1.25274i	0.779526	−	0.626369i	$-0.215460\pi$
$98$	− 4.96526i	− 0.501567i
$99$	0	0
$100$	0	0
$101$	−10.6761	−1.06231	−0.531155	−	0.847274i	$-0.678242\pi$
$101$	−10.6761	−1.06231	−0.531155	+	0.847274i	$0.678242\pi$
$102$	0	0
$103$	14.5089i	1.42960i	0.699329	+	0.714800i	$0.253482\pi$
$103$	14.5089i	1.42960i	−0.699329	+	0.714800i	$0.746518\pi$
$104$	1.13249	0.111050
$105$	0	0
$106$	0.0977518	0.00949450
$107$	− 4.17081i	− 0.403207i	−0.979467	−	0.201604i	$-0.935385\pi$
$107$	− 4.17081i	− 0.403207i	0.979467	−	0.201604i	$-0.0646153\pi$
$108$	0	0
$109$	3.83276	0.367112	0.183556	−	0.983009i	$-0.441239\pi$
$109$	3.83276	0.367112	0.183556	+	0.983009i	$0.441239\pi$
$110$	0	0
$111$	0	0
$112$	17.2333i	1.62839i
$113$	− 0.843326i	− 0.0793334i	−0.999213	−	0.0396667i	$-0.987370\pi$
$113$	− 0.843326i	− 0.0793334i	0.999213	−	0.0396667i	$-0.0126296\pi$
$114$	0	0
$115$	0	0
$116$	11.4983	1.06759
$117$	0	0
$118$	− 3.25443i	− 0.299594i
$119$	−21.3275	−1.95509
$120$	0	0
$121$	13.1708	1.19735
$122$	− 2.94108i	− 0.266273i
$123$	0	0
$124$	2.72445	0.244663
$125$	0	0
$126$	0	0
$127$	1.83276i	0.162631i	0.996688	+	0.0813157i	$0.0259122\pi$
$127$	1.83276i	0.162631i	−0.996688	+	0.0813157i	$0.974088\pi$
$128$	8.31029i	0.734533i
$129$	0	0
$130$	0	0
$131$	−5.83276	−0.509611	−0.254805	−	0.966992i	$-0.582011\pi$
$131$	−5.83276	−0.509611	−0.254805	+	0.966992i	$0.582011\pi$
$132$	0	0
$133$	− 12.6761i	− 1.09916i
$134$	2.09775	0.181218
$135$	0	0
$136$	−4.91281	−0.421270
$137$	16.5089i	1.41045i	0.708985	+	0.705223i	$0.249153\pi$
$137$	16.5089i	1.41045i	−0.708985	+	0.705223i	$0.750847\pi$
$138$	0	0
$139$	−7.49472	−0.635694	−0.317847	−	0.948142i	$-0.602960\pi$
$139$	−7.49472	−0.635694	−0.317847	+	0.948142i	$0.602960\pi$
$140$	0	0
$141$	0	0
$142$	− 0.264989i	− 0.0222374i
$143$	4.91638i	0.411128i
$144$	0	0
$145$	0	0
$146$	−0.912811	−0.0755448
$147$	0	0
$148$	18.1955i	1.49566i
$149$	20.4842	1.67813	0.839064	−	0.544033i	$-0.183103\pi$
$149$	20.4842	1.67813	0.839064	+	0.544033i	$0.183103\pi$
$150$	0	0
$151$	−16.4111	−1.33552	−0.667758	−	0.744378i	$-0.732746\pi$
$151$	−16.4111	−1.33552	−0.667758	+	0.744378i	$0.732746\pi$
$152$	− 2.91995i	− 0.236839i
$153$	0	0
$154$	6.98944	0.563225
$155$	0	0
$156$	0	0
$157$	− 21.6655i	− 1.72910i	−0.502549	−	0.864549i	$-0.667604\pi$
$157$	− 21.6655i	− 1.72910i	0.502549	−	0.864549i	$-0.332396\pi$
$158$	1.01056i	0.0803961i
$159$	0	0
$160$	0	0
$161$	31.1602	2.45577
$162$	0	0
$163$	− 6.07306i	− 0.475678i	−0.971305	−	0.237839i	$-0.923561\pi$
$163$	− 6.07306i	− 0.475678i	0.971305	−	0.237839i	$-0.0764391\pi$
$164$	−8.31335	−0.649163
$165$	0	0
$166$	−3.25443	−0.252592
$167$	0.745574i	0.0576942i	0.999584	+	0.0288471i	$0.00918360\pi$
$167$	0.745574i	0.0576942i	−0.999584	+	0.0288471i	$0.990816\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	0	0
$172$	2.21663i	0.169016i
$173$	0.843326i	0.0641169i	0.999486	+	0.0320584i	$0.0102063\pi$
$173$	0.843326i	0.0641169i	−0.999486	+	0.0320584i	$0.989794\pi$
$174$	0	0
$175$	0	0
$176$	−17.2333	−1.29901
$177$	0	0
$178$	− 0.0977518i	− 0.00732681i
$179$	−18.9894	−1.41934	−0.709669	−	0.704536i	$-0.751155\pi$
$179$	−18.9894	−1.41934	−0.709669	+	0.704536i	$0.751155\pi$
$180$	0	0
$181$	17.4947	1.30037	0.650186	−	0.759775i	$-0.274691\pi$
$181$	17.4947	1.30037	0.650186	+	0.759775i	$0.274691\pi$
$182$	1.42166i	0.105381i
$183$	0	0
$184$	7.17780	0.529154
$185$	0	0
$186$	0	0
$187$	− 21.3275i	− 1.55962i
$188$	10.3900i	0.757767i
$189$	0	0
$190$	0	0
$191$	22.5089	1.62868	0.814342	−	0.580386i	$-0.197098\pi$
$191$	22.5089	1.62868	0.814342	+	0.580386i	$0.197098\pi$
$192$	0	0
$193$	− 2.65139i	− 0.190851i	−0.995437	−	0.0954257i	$-0.969579\pi$
$193$	− 2.65139i	− 0.190851i	0.995437	−	0.0954257i	$-0.0304212\pi$
$194$	3.56777	0.256151
$195$	0	0
$196$	−32.9058	−2.35042
$197$	12.9894i	0.925459i	0.886500	+	0.462730i	$0.153130\pi$
$197$	12.9894i	0.925459i	−0.886500	+	0.462730i	$0.846870\pi$
$198$	0	0
$199$	2.84333	0.201558	0.100779	−	0.994909i	$-0.467866\pi$
$199$	2.84333	0.201558	0.100779	+	0.994909i	$0.467866\pi$
$200$	0	0
$201$	0	0
$202$	− 3.08719i	− 0.217214i
$203$	29.4983i	2.07037i
$204$	0	0
$205$	0	0
$206$	−4.19550	−0.292315
$207$	0	0
$208$	− 3.50528i	− 0.243048i
$209$	12.6761	0.876823
$210$	0	0
$211$	6.31335	0.434629	0.217314	−	0.976102i	$-0.430270\pi$
$211$	6.31335	0.434629	0.217314	+	0.976102i	$0.430270\pi$
$212$	− 0.647822i	− 0.0444926i
$213$	0	0
$214$	1.20607	0.0824450
$215$	0	0
$216$	0	0
$217$	6.98944i	0.474474i
$218$	1.10831i	0.0750645i
$219$	0	0
$220$	0	0
$221$	4.33804	0.291808
$222$	0	0
$223$	− 19.2544i	− 1.28937i	−0.764448	−	0.644686i	$-0.776988\pi$
$223$	− 19.2544i	− 1.28937i	0.764448	−	0.644686i	$-0.223012\pi$
$224$	−16.1189	−1.07699
$225$	0	0
$226$	0.243863	0.0162215
$227$	13.0872i	0.868627i	0.900762	+	0.434314i	$0.143009\pi$
$227$	13.0872i	0.868627i	−0.900762	+	0.434314i	$0.856991\pi$
$228$	0	0
$229$	24.5089	1.61959	0.809795	−	0.586713i	$-0.199578\pi$
$229$	24.5089	1.61959	0.809795	+	0.586713i	$0.199578\pi$
$230$	0	0
$231$	0	0
$232$	6.79497i	0.446111i
$233$	8.33804i	0.546243i	0.961979	+	0.273122i	$0.0880562\pi$
$233$	8.33804i	0.546243i	−0.961979	+	0.273122i	$0.911944\pi$
$234$	0	0
$235$	0	0
$236$	−21.5678	−1.40394
$237$	0	0
$238$	− 6.16724i	− 0.399763i
$239$	8.91638	0.576753	0.288376	−	0.957517i	$-0.406885\pi$
$239$	8.91638	0.576753	0.288376	+	0.957517i	$0.406885\pi$
$240$	0	0
$241$	−6.00000	−0.386494	−0.193247	−	0.981150i	$-0.561902\pi$
$241$	−6.00000	−0.386494	−0.193247	+	0.981150i	$0.561902\pi$
$242$	3.80858i	0.244825i
$243$	0	0
$244$	−19.4911	−1.24779
$245$	0	0
$246$	0	0
$247$	2.57834i	0.164056i
$248$	1.61003i	0.102237i
$249$	0	0
$250$	0	0
$251$	6.31335	0.398495	0.199248	−	0.979949i	$-0.436150\pi$
$251$	6.31335	0.398495	0.199248	+	0.979949i	$0.436150\pi$
$252$	0	0
$253$	31.1602i	1.95903i
$254$	−0.529977	−0.0332537
$255$	0	0
$256$	9.72191	0.607619
$257$	11.1567i	0.695934i	0.937507	+	0.347967i	$0.113128\pi$
$257$	11.1567i	0.695934i	−0.937507	+	0.347967i	$0.886872\pi$
$258$	0	0
$259$	−46.6797	−2.90053
$260$	0	0
$261$	0	0
$262$	− 1.68665i	− 0.104202i
$263$	8.00000i	0.493301i	0.969104	+	0.246651i	$0.0793300\pi$
$263$	8.00000i	0.493301i	−0.969104	+	0.246651i	$0.920670\pi$
$264$	0	0
$265$	0	0
$266$	3.66553	0.224748
$267$	0	0
$268$	− 13.9022i	− 0.849215i
$269$	18.6761	1.13870	0.569351	−	0.822095i	$-0.307195\pi$
$269$	18.6761	1.13870	0.569351	+	0.822095i	$0.307195\pi$
$270$	0	0
$271$	6.57834	0.399606	0.199803	−	0.979836i	$-0.435970\pi$
$271$	6.57834	0.399606	0.199803	+	0.979836i	$0.435970\pi$
$272$	15.2061i	0.922003i
$273$	0	0
$274$	−4.77384	−0.288398
$275$	0	0
$276$	0	0
$277$	25.6655i	1.54209i	0.636779	+	0.771046i	$0.280266\pi$
$277$	25.6655i	1.54209i	−0.636779	+	0.771046i	$0.719734\pi$
$278$	− 2.16724i	− 0.129982i
$279$	0	0
$280$	0	0
$281$	−3.15667	−0.188311	−0.0941557	−	0.995557i	$-0.530015\pi$
$281$	−3.15667	−0.188311	−0.0941557	+	0.995557i	$0.530015\pi$
$282$	0	0
$283$	3.47002i	0.206271i	0.994667	+	0.103136i	$0.0328876\pi$
$283$	3.47002i	0.206271i	−0.994667	+	0.103136i	$0.967112\pi$
$284$	−1.75614	−0.104208
$285$	0	0
$286$	−1.42166	−0.0840647
$287$	− 21.3275i	− 1.25892i
$288$	0	0
$289$	−1.81863	−0.106978
$290$	0	0
$291$	0	0
$292$	6.04939i	0.354014i
$293$	28.6550i	1.67404i	0.547172	+	0.837020i	$0.315704\pi$
$293$	28.6550i	1.67404i	−0.547172	+	0.837020i	$0.684296\pi$
$294$	0	0
$295$	0	0
$296$	−10.7527	−0.624989
$297$	0	0
$298$	5.92337i	0.343132i
$299$	−6.33804	−0.366539
$300$	0	0
$301$	−5.68665	−0.327773
$302$	− 4.74557i	− 0.273077i
$303$	0	0
$304$	−9.03780	−0.518353
$305$	0	0
$306$	0	0
$307$	− 1.92694i	− 0.109977i	−0.998487	−	0.0549883i	$-0.982488\pi$
$307$	− 1.92694i	− 0.109977i	0.998487	−	0.0549883i	$-0.0175121\pi$
$308$	− 46.3205i	− 2.63935i
$309$	0	0
$310$	0	0
$311$	29.9789	1.69995	0.849973	−	0.526826i	$-0.176618\pi$
$311$	29.9789	1.69995	0.849973	+	0.526826i	$0.176618\pi$
$312$	0	0
$313$	16.3133i	0.922085i	0.887378	+	0.461042i	$0.152524\pi$
$313$	16.3133i	0.922085i	−0.887378	+	0.461042i	$0.847476\pi$
$314$	6.26499	0.353554
$315$	0	0
$316$	6.69721	0.376748
$317$	− 30.6761i	− 1.72294i	−0.507808	−	0.861470i	$-0.669544\pi$
$317$	− 30.6761i	− 1.72294i	0.507808	−	0.861470i	$-0.330456\pi$
$318$	0	0
$319$	−29.4983	−1.65159
$320$	0	0
$321$	0	0
$322$	9.01056i	0.502139i
$323$	− 11.1849i	− 0.622347i
$324$	0	0
$325$	0	0
$326$	1.75614	0.0972634
$327$	0	0
$328$	− 4.91281i	− 0.271265i
$329$	−26.6550	−1.46954
$330$	0	0
$331$	−10.0978	−0.555023	−0.277511	−	0.960722i	$-0.589510\pi$
$331$	−10.0978	−0.555023	−0.277511	+	0.960722i	$0.589510\pi$
$332$	21.5678i	1.18369i
$333$	0	0
$334$	−0.215597	−0.0117969
$335$	0	0
$336$	0	0
$337$	− 1.32391i	− 0.0721180i	−0.999350	−	0.0360590i	$-0.988520\pi$
$337$	− 1.32391i	− 0.0721180i	0.999350	−	0.0360590i	$-0.0114804\pi$
$338$	− 0.289169i	− 0.0157287i
$339$	0	0
$340$	0	0
$341$	−6.98944	−0.378499
$342$	0	0
$343$	− 50.0036i	− 2.69994i
$344$	−1.30993	−0.0706265
$345$	0	0
$346$	−0.243863	−0.0131102
$347$	7.49472i	0.402338i	0.979557	+	0.201169i	$0.0644740\pi$
$347$	7.49472i	0.402338i	−0.979557	+	0.201169i	$0.935526\pi$
$348$	0	0
$349$	22.1461	1.18545	0.592727	−	0.805403i	$-0.298051\pi$
$349$	22.1461	1.18545	0.592727	+	0.805403i	$0.298051\pi$
$350$	0	0
$351$	0	0
$352$	− 16.1189i	− 0.859139i
$353$	4.50885i	0.239982i	0.992775	+	0.119991i	$0.0382866\pi$
$353$	4.50885i	0.239982i	−0.992775	+	0.119991i	$0.961713\pi$
$354$	0	0
$355$	0	0
$356$	−0.647822	−0.0343345
$357$	0	0
$358$	− 5.49115i	− 0.290216i
$359$	−20.4111	−1.07726	−0.538628	−	0.842543i	$-0.681057\pi$
$359$	−20.4111	−1.07726	−0.538628	+	0.842543i	$0.681057\pi$
$360$	0	0
$361$	−12.3522	−0.650115
$362$	5.05892i	0.265891i
$363$	0	0
$364$	9.42166	0.493829
$365$	0	0
$366$	0	0
$367$	− 10.3133i	− 0.538352i	−0.963091	−	0.269176i	$-0.913249\pi$
$367$	− 10.3133i	− 0.538352i	0.963091	−	0.269176i	$-0.0867514\pi$
$368$	− 22.2166i	− 1.15812i
$369$	0	0
$370$	0	0
$371$	1.66196	0.0862844
$372$	0	0
$373$	− 18.6761i	− 0.967011i	−0.875341	−	0.483506i	$-0.839363\pi$
$373$	− 18.6761i	− 0.967011i	0.875341	−	0.483506i	$-0.160637\pi$
$374$	6.16724	0.318900
$375$	0	0
$376$	−6.14000	−0.316646
$377$	− 6.00000i	− 0.309016i
$378$	0	0
$379$	28.7527	1.47693	0.738464	−	0.674293i	$-0.235552\pi$
$379$	28.7527	1.47693	0.738464	+	0.674293i	$0.235552\pi$
$380$	0	0
$381$	0	0
$382$	6.50885i	0.333022i
$383$	− 14.2439i	− 0.727827i	−0.931433	−	0.363914i	$-0.881440\pi$
$383$	− 14.2439i	− 0.727827i	0.931433	−	0.363914i	$-0.118560\pi$
$384$	0	0
$385$	0	0
$386$	0.766699	0.0390240
$387$	0	0
$388$	− 23.6444i	− 1.20036i
$389$	34.6761	1.75815	0.879074	−	0.476686i	$-0.158162\pi$
$389$	34.6761	1.75815	0.879074	+	0.476686i	$0.158162\pi$
$390$	0	0
$391$	27.4947	1.39047
$392$	− 19.4458i	− 0.982163i
$393$	0	0
$394$	−3.75614	−0.189231
$395$	0	0
$396$	0	0
$397$	7.18137i	0.360423i	0.983628	+	0.180211i	$0.0576782\pi$
$397$	7.18137i	0.360423i	−0.983628	+	0.180211i	$0.942322\pi$
$398$	0.822200i	0.0412132i
$399$	0	0
$400$	0	0
$401$	−37.8610	−1.89069	−0.945345	−	0.326072i	$-0.894275\pi$
$401$	−37.8610	−1.89069	−0.945345	+	0.326072i	$0.894275\pi$
$402$	0	0
$403$	− 1.42166i	− 0.0708181i
$404$	−20.4595	−1.01790
$405$	0	0
$406$	−8.52998	−0.423336
$407$	− 46.6797i	− 2.31382i
$408$	0	0
$409$	14.0000	0.692255	0.346128	−	0.938187i	$-0.387496\pi$
$409$	14.0000	0.692255	0.346128	+	0.938187i	$0.387496\pi$
$410$	0	0
$411$	0	0
$412$	27.8045i	1.36983i
$413$	− 55.3311i	− 2.72266i
$414$	0	0
$415$	0	0
$416$	3.27861	0.160747
$417$	0	0
$418$	3.66553i	0.179287i
$419$	−33.4983	−1.63650	−0.818249	−	0.574864i	$-0.805055\pi$
$419$	−33.4983	−1.63650	−0.818249	+	0.574864i	$0.805055\pi$
$420$	0	0
$421$	13.5194	0.658896	0.329448	−	0.944174i	$-0.393137\pi$
$421$	13.5194	0.658896	0.329448	+	0.944174i	$0.393137\pi$
$422$	1.82562i	0.0888699i
$423$	0	0
$424$	0.382833	0.0185920
$425$	0	0
$426$	0	0
$427$	− 50.0036i	− 2.41984i
$428$	− 7.99286i	− 0.386349i
$429$	0	0
$430$	0	0
$431$	−12.4111	−0.597822	−0.298911	−	0.954281i	$-0.596623\pi$
$431$	−12.4111	−0.597822	−0.298911	+	0.954281i	$0.596623\pi$
$432$	0	0
$433$	17.3239i	0.832534i	0.909242	+	0.416267i	$0.136662\pi$
$433$	17.3239i	0.832534i	−0.909242	+	0.416267i	$0.863338\pi$
$434$	−2.02113	−0.0970171
$435$	0	0
$436$	7.34504	0.351763
$437$	16.3416i	0.781725i
$438$	0	0
$439$	−0.651393	−0.0310893	−0.0155446	−	0.999879i	$-0.504948\pi$
$439$	−0.651393	−0.0310893	−0.0155446	+	0.999879i	$0.504948\pi$
$440$	0	0
$441$	0	0
$442$	1.25443i	0.0596670i
$443$	− 8.84690i	− 0.420329i	−0.977666	−	0.210164i	$-0.932600\pi$
$443$	− 8.84690i	− 0.420329i	0.977666	−	0.210164i	$-0.0673999\pi$
$444$	0	0
$445$	0	0
$446$	5.56777	0.263642
$447$	0	0
$448$	29.8055i	1.40818i
$449$	4.33804	0.204725	0.102362	−	0.994747i	$-0.467360\pi$
$449$	4.33804	0.204725	0.102362	+	0.994747i	$0.467360\pi$
$450$	0	0
$451$	21.3275	1.00427
$452$	− 1.61613i	− 0.0760166i
$453$	0	0
$454$	−3.78440	−0.177611
$455$	0	0
$456$	0	0
$457$	15.3275i	0.716989i	0.933532	+	0.358495i	$0.116710\pi$
$457$	15.3275i	0.716989i	−0.933532	+	0.358495i	$0.883290\pi$
$458$	7.08719i	0.331163i
$459$	0	0
$460$	0	0
$461$	11.8575	0.552257	0.276128	−	0.961121i	$-0.410948\pi$
$461$	11.8575	0.552257	0.276128	+	0.961121i	$0.410948\pi$
$462$	0	0
$463$	26.4147i	1.22759i	0.789464	+	0.613797i	$0.210359\pi$
$463$	26.4147i	1.22759i	−0.789464	+	0.613797i	$0.789641\pi$
$464$	21.0317	0.976372
$465$	0	0
$466$	−2.41110	−0.111692
$467$	33.6691i	1.55802i	0.627012	+	0.779010i	$0.284278\pi$
$467$	33.6691i	1.55802i	−0.627012	+	0.779010i	$0.715722\pi$
$468$	0	0
$469$	35.6655	1.64688
$470$	0	0
$471$	0	0
$472$	− 12.7456i	− 0.586663i
$473$	− 5.68665i	− 0.261473i
$474$	0	0
$475$	0	0
$476$	−40.8716	−1.87335
$477$	0	0
$478$	2.57834i	0.117930i
$479$	10.7491	0.491141	0.245570	−	0.969379i	$-0.421025\pi$
$479$	10.7491	0.491141	0.245570	+	0.969379i	$0.421025\pi$
$480$	0	0
$481$	9.49472	0.432922
$482$	− 1.73501i	− 0.0790276i
$483$	0	0
$484$	25.2403	1.14729
$485$	0	0
$486$	0	0
$487$	22.7491i	1.03086i	0.856931	+	0.515431i	$0.172368\pi$
$487$	22.7491i	1.03086i	−0.856931	+	0.515431i	$0.827632\pi$
$488$	− 11.5184i	− 0.521413i
$489$	0	0
$490$	0	0
$491$	17.6867	0.798187	0.399094	−	0.916910i	$-0.369325\pi$
$491$	17.6867	0.798187	0.399094	+	0.916910i	$0.369325\pi$
$492$	0	0
$493$	26.0283i	1.17225i
$494$	−0.745574	−0.0335450
$495$	0	0
$496$	4.98333	0.223758
$497$	− 4.50528i	− 0.202089i
$498$	0	0
$499$	−19.9305	−0.892212	−0.446106	−	0.894980i	$-0.647190\pi$
$499$	−19.9305	−0.892212	−0.446106	+	0.894980i	$0.647190\pi$
$500$	0	0
$501$	0	0
$502$	1.82562i	0.0814815i
$503$	33.3522i	1.48710i	0.668680	+	0.743550i	$0.266860\pi$
$503$	33.3522i	1.48710i	−0.668680	+	0.743550i	$0.733140\pi$
$504$	0	0
$505$	0	0
$506$	−9.01056	−0.400568
$507$	0	0
$508$	3.51227i	0.155832i
$509$	13.8363	0.613285	0.306642	−	0.951825i	$-0.400794\pi$
$509$	13.8363	0.613285	0.306642	+	0.951825i	$0.400794\pi$
$510$	0	0
$511$	−15.5194	−0.686538
$512$	19.4319i	0.858775i
$513$	0	0
$514$	−3.22616	−0.142300
$515$	0	0
$516$	0	0
$517$	− 26.6550i	− 1.17228i
$518$	− 13.4983i	− 0.593081i
$519$	0	0
$520$	0	0
$521$	23.3522	1.02308	0.511539	−	0.859260i	$-0.329076\pi$
$521$	23.3522	1.02308	0.511539	+	0.859260i	$0.329076\pi$
$522$	0	0
$523$	4.00000i	0.174908i	0.996169	+	0.0874539i	$0.0278730\pi$
$523$	4.00000i	0.174908i	−0.996169	+	0.0874539i	$0.972127\pi$
$524$	−11.1778	−0.488304
$525$	0	0
$526$	−2.31335	−0.100867
$527$	6.16724i	0.268649i
$528$	0	0
$529$	−17.1708	−0.746557
$530$	0	0
$531$	0	0
$532$	− 24.2922i	− 1.05320i
$533$	4.33804i	0.187902i
$534$	0	0
$535$	0	0
$536$	8.21560	0.354860
$537$	0	0
$538$	5.40054i	0.232834i
$539$	84.4182	3.63615
$540$	0	0
$541$	16.1744	0.695391	0.347695	−	0.937608i	$-0.386964\pi$
$541$	16.1744	0.695391	0.347695	+	0.937608i	$0.386964\pi$
$542$	1.90225i	0.0817086i
$543$	0	0
$544$	−14.2227	−0.609795
$545$	0	0
$546$	0	0
$547$	9.68665i	0.414171i	0.978323	+	0.207086i	$0.0663979\pi$
$547$	9.68665i	0.414171i	−0.978323	+	0.207086i	$0.933602\pi$
$548$	31.6373i	1.35148i
$549$	0	0
$550$	0	0
$551$	−15.4700	−0.659045
$552$	0	0
$553$	17.1814i	0.730626i
$554$	−7.42166	−0.315316
$555$	0	0
$556$	−14.3627	−0.609116
$557$	− 0.647822i	− 0.0274491i	−0.999906	−	0.0137246i	$-0.995631\pi$
$557$	− 0.647822i	− 0.0274491i	0.999906	−	0.0137246i	$-0.00436880\pi$
$558$	0	0
$559$	1.15667	0.0489221
$560$	0	0
$561$	0	0
$562$	− 0.912811i	− 0.0385046i
$563$	− 16.3169i	− 0.687676i	−0.939029	−	0.343838i	$-0.888273\pi$
$563$	− 16.3169i	− 0.687676i	0.939029	−	0.343838i	$-0.111727\pi$
$564$	0	0
$565$	0	0
$566$	−1.00342	−0.0421769
$567$	0	0
$568$	− 1.03780i	− 0.0435450i
$569$	44.6550	1.87203	0.936017	−	0.351956i	$-0.114483\pi$
$569$	44.6550	1.87203	0.936017	+	0.351956i	$0.114483\pi$
$570$	0	0
$571$	−6.67252	−0.279236	−0.139618	−	0.990205i	$-0.544587\pi$
$571$	−6.67252	−0.279236	−0.139618	+	0.990205i	$0.544587\pi$
$572$	9.42166i	0.393940i
$573$	0	0
$574$	6.16724	0.257415
$575$	0	0
$576$	0	0
$577$	15.3275i	0.638091i	0.947739	+	0.319046i	$0.103362\pi$
$577$	15.3275i	0.638091i	−0.947739	+	0.319046i	$0.896638\pi$
$578$	− 0.525891i	− 0.0218742i
$579$	0	0
$580$	0	0
$581$	−55.3311	−2.29552
$582$	0	0
$583$	1.66196i	0.0688312i
$584$	−3.57492	−0.147931
$585$	0	0
$586$	−8.28611	−0.342296
$587$	− 12.2650i	− 0.506230i	−0.967436	−	0.253115i	$-0.918545\pi$
$587$	− 12.2650i	− 0.506230i	0.967436	−	0.253115i	$-0.0814551\pi$
$588$	0	0
$589$	−3.66553	−0.151035
$590$	0	0
$591$	0	0
$592$	33.2817i	1.36787i
$593$	− 5.85389i	− 0.240390i	−0.992750	−	0.120195i	$-0.961648\pi$
$593$	− 5.85389i	− 0.240390i	0.992750	−	0.120195i	$-0.0383520\pi$
$594$	0	0
$595$	0	0
$596$	39.2555	1.60797
$597$	0	0
$598$	− 1.83276i	− 0.0749473i
$599$	27.1355	1.10873	0.554364	−	0.832274i	$-0.312961\pi$
$599$	27.1355	1.10873	0.554364	+	0.832274i	$0.312961\pi$
$600$	0	0
$601$	34.1708	1.39386	0.696928	−	0.717141i	$-0.254550\pi$
$601$	34.1708	1.39386	0.696928	+	0.717141i	$0.254550\pi$
$602$	− 1.64440i	− 0.0670208i
$603$	0	0
$604$	−31.4499	−1.27968
$605$	0	0
$606$	0	0
$607$	10.3133i	0.418606i	0.977851	+	0.209303i	$0.0671195\pi$
$607$	10.3133i	0.418606i	−0.977851	+	0.209303i	$0.932881\pi$
$608$	− 8.45335i	− 0.342829i
$609$	0	0
$610$	0	0
$611$	5.42166	0.219337
$612$	0	0
$613$	− 0.484156i	− 0.0195549i	−0.999952	−	0.00977744i	$-0.996888\pi$
$613$	− 0.484156i	− 0.0195549i	0.999952	−	0.00977744i	$-0.00311230\pi$
$614$	0.557212	0.0224872
$615$	0	0
$616$	27.3733	1.10290
$617$	7.15667i	0.288117i	0.989569	+	0.144058i	$0.0460153\pi$
$617$	7.15667i	0.288117i	−0.989569	+	0.144058i	$0.953985\pi$
$618$	0	0
$619$	−5.42166	−0.217915	−0.108958	−	0.994046i	$-0.534751\pi$
$619$	−5.42166	−0.217915	−0.108958	+	0.994046i	$0.534751\pi$
$620$	0	0
$621$	0	0
$622$	8.66895i	0.347593i
$623$	− 1.66196i	− 0.0665848i
$624$	0	0
$625$	0	0
$626$	−4.71731	−0.188542
$627$	0	0
$628$	− 41.5194i	− 1.65681i
$629$	−41.1885	−1.64229
$630$	0	0
$631$	−10.7244	−0.426934	−0.213467	−	0.976950i	$-0.568476\pi$
$631$	−10.7244	−0.426934	−0.213467	+	0.976950i	$0.568476\pi$
$632$	3.95775i	0.157431i
$633$	0	0
$634$	8.87056	0.352295
$635$	0	0
$636$	0	0
$637$	17.1708i	0.680332i
$638$	− 8.52998i	− 0.337705i
$639$	0	0
$640$	0	0
$641$	0.362741	0.0143274	0.00716370	−	0.999974i	$-0.497720\pi$
$641$	0.362741	0.0143274	0.00716370	+	0.999974i	$0.497720\pi$
$642$	0	0
$643$	9.39697i	0.370580i	0.982684	+	0.185290i	$0.0593225\pi$
$643$	9.39697i	0.370580i	−0.982684	+	0.185290i	$0.940678\pi$
$644$	59.7149	2.35310
$645$	0	0
$646$	3.23433	0.127253
$647$	− 18.0036i	− 0.707793i	−0.935285	−	0.353897i	$-0.884856\pi$
$647$	− 18.0036i	− 0.707793i	0.935285	−	0.353897i	$-0.115144\pi$
$648$	0	0
$649$	55.3311	2.17193
$650$	0	0
$651$	0	0
$652$	− 11.6383i	− 0.455791i
$653$	− 34.8222i	− 1.36270i	−0.731959	−	0.681349i	$-0.761394\pi$
$653$	− 34.8222i	− 1.36270i	0.731959	−	0.681349i	$-0.238606\pi$
$654$	0	0
$655$	0	0
$656$	−15.2061	−0.593697
$657$	0	0
$658$	− 7.70778i	− 0.300480i
$659$	11.4700	0.446809	0.223404	−	0.974726i	$-0.428283\pi$
$659$	11.4700	0.446809	0.223404	+	0.974726i	$0.428283\pi$
$660$	0	0
$661$	12.1672	0.473251	0.236625	−	0.971601i	$-0.423959\pi$
$661$	12.1672	0.473251	0.236625	+	0.971601i	$0.423959\pi$
$662$	− 2.91995i	− 0.113487i
$663$	0	0
$664$	−12.7456	−0.494624
$665$	0	0
$666$	0	0
$667$	− 38.0283i	− 1.47246i
$668$	1.42880i	0.0552821i
$669$	0	0
$670$	0	0
$671$	50.0036	1.93037
$672$	0	0
$673$	27.9789i	1.07851i	0.842144	+	0.539253i	$0.181293\pi$
$673$	27.9789i	1.07851i	−0.842144	+	0.539253i	$0.818707\pi$
$674$	0.382833	0.0147462
$675$	0	0
$676$	−1.91638	−0.0737070
$677$	− 22.9930i	− 0.883693i	−0.897091	−	0.441847i	$-0.854324\pi$
$677$	− 22.9930i	− 0.883693i	0.897091	−	0.441847i	$-0.145676\pi$
$678$	0	0
$679$	60.6585	2.32786
$680$	0	0
$681$	0	0
$682$	− 2.02113i	− 0.0773929i
$683$	− 28.6066i	− 1.09460i	−0.836936	−	0.547301i	$-0.815655\pi$
$683$	− 28.6066i	− 1.09460i	0.836936	−	0.547301i	$-0.184345\pi$
$684$	0	0
$685$	0	0
$686$	14.4595	0.552065
$687$	0	0
$688$	4.05447i	0.154575i
$689$	−0.338044	−0.0128785
$690$	0	0
$691$	−19.4005	−0.738031	−0.369016	−	0.929423i	$-0.620305\pi$
$691$	−19.4005	−0.738031	−0.369016	+	0.929423i	$0.620305\pi$
$692$	1.61613i	0.0614362i
$693$	0	0
$694$	−2.16724	−0.0822672
$695$	0	0
$696$	0	0
$697$	− 18.8186i	− 0.712806i
$698$	6.40396i	0.242393i
$699$	0	0
$700$	0	0
$701$	38.9683	1.47181	0.735906	−	0.677083i	$-0.236756\pi$
$701$	38.9683	1.47181	0.735906	+	0.677083i	$0.236756\pi$
$702$	0	0
$703$	− 24.4806i	− 0.923303i
$704$	−29.8055	−1.12334
$705$	0	0
$706$	−1.30382	−0.0490698
$707$	− 52.4877i	− 1.97400i
$708$	0	0
$709$	17.5194	0.657955	0.328978	−	0.944338i	$-0.393296\pi$
$709$	17.5194	0.657955	0.328978	+	0.944338i	$0.393296\pi$
$710$	0	0
$711$	0	0
$712$	− 0.382833i	− 0.0143473i
$713$	− 9.01056i	− 0.337448i
$714$	0	0
$715$	0	0
$716$	−36.3910	−1.36000
$717$	0	0
$718$	− 5.90225i	− 0.220270i
$719$	−4.33447	−0.161649	−0.0808243	−	0.996728i	$-0.525755\pi$
$719$	−4.33447	−0.161649	−0.0808243	+	0.996728i	$0.525755\pi$
$720$	0	0
$721$	−71.3311	−2.65651
$722$	− 3.57186i	− 0.132931i
$723$	0	0
$724$	33.5266	1.24600
$725$	0	0
$726$	0	0
$727$	22.1672i	0.822137i	0.911604	+	0.411069i	$0.134844\pi$
$727$	22.1672i	0.822137i	−0.911604	+	0.411069i	$0.865156\pi$
$728$	5.56777i	0.206355i
$729$	0	0
$730$	0	0
$731$	−5.01770	−0.185586
$732$	0	0
$733$	2.83976i	0.104889i	0.998624	+	0.0524444i	$0.0167012\pi$
$733$	2.83976i	0.104889i	−0.998624	+	0.0524444i	$0.983299\pi$
$734$	2.98230	0.110079
$735$	0	0
$736$	20.7799	0.765959
$737$	35.6655i	1.31376i
$738$	0	0
$739$	43.9305	1.61601	0.808005	−	0.589176i	$-0.200547\pi$
$739$	43.9305	1.61601	0.808005	+	0.589176i	$0.200547\pi$
$740$	0	0
$741$	0	0
$742$	0.480585i	0.0176428i
$743$	41.0872i	1.50734i	0.657251	+	0.753671i	$0.271719\pi$
$743$	41.0872i	1.50734i	−0.657251	+	0.753671i	$0.728281\pi$
$744$	0	0
$745$	0	0
$746$	5.40054	0.197728
$747$	0	0
$748$	− 40.8716i	− 1.49441i
$749$	20.5053	0.749247
$750$	0	0
$751$	23.6902	0.864468	0.432234	−	0.901761i	$-0.357725\pi$
$751$	23.6902	0.864468	0.432234	+	0.901761i	$0.357725\pi$
$752$	19.0045i	0.693021i
$753$	0	0
$754$	1.73501	0.0631854
$755$	0	0
$756$	0	0
$757$	9.32391i	0.338883i	0.985540	+	0.169442i	$0.0541964\pi$
$757$	9.32391i	0.338883i	−0.985540	+	0.169442i	$0.945804\pi$
$758$	8.31438i	0.301992i
$759$	0	0
$760$	0	0
$761$	42.8222	1.55230	0.776152	−	0.630546i	$-0.217169\pi$
$761$	42.8222	1.55230	0.776152	+	0.630546i	$0.217169\pi$
$762$	0	0
$763$	18.8433i	0.682174i
$764$	43.1355	1.56059
$765$	0	0
$766$	4.11888	0.148821
$767$	11.2544i	0.406374i
$768$	0	0
$769$	17.3239	0.624716	0.312358	−	0.949964i	$-0.398881\pi$
$769$	17.3239	0.624716	0.312358	+	0.949964i	$0.398881\pi$
$770$	0	0
$771$	0	0
$772$	− 5.08108i	− 0.182872i
$773$	− 11.6373i	− 0.418563i	−0.977855	−	0.209282i	$-0.932887\pi$
$773$	− 11.6373i	− 0.418563i	0.977855	−	0.209282i	$-0.0671125\pi$
$774$	0	0
$775$	0	0
$776$	13.9728	0.501593
$777$	0	0
$778$	10.0272i	0.359494i
$779$	11.1849	0.400742
$780$	0	0
$781$	4.50528	0.161212
$782$	7.95061i	0.284313i
$783$	0	0
$784$	−60.1885	−2.14959
$785$	0	0
$786$	0	0
$787$	20.9411i	0.746469i	0.927737	+	0.373234i	$0.121751\pi$
$787$	20.9411i	0.746469i	−0.927737	+	0.373234i	$0.878249\pi$
$788$	24.8927i	0.886766i
$789$	0	0
$790$	0	0
$791$	4.14611	0.147419
$792$	0	0
$793$	10.1708i	0.361176i
$794$	−2.07663	−0.0736967
$795$	0	0
$796$	5.44890	0.193131
$797$	− 20.3380i	− 0.720410i	−0.932873	−	0.360205i	$-0.882707\pi$
$797$	− 20.3380i	− 0.720410i	0.932873	−	0.360205i	$-0.117293\pi$
$798$	0	0
$799$	−23.5194	−0.832057
$800$	0	0
$801$	0	0
$802$	− 10.9482i	− 0.386595i
$803$	− 15.5194i	− 0.547668i
$804$	0	0
$805$	0	0
$806$	0.411100	0.0144804
$807$	0	0
$808$	− 12.0906i	− 0.425346i
$809$	7.68665	0.270248	0.135124	−	0.990829i	$-0.456857\pi$
$809$	7.68665	0.270248	0.135124	+	0.990829i	$0.456857\pi$
$810$	0	0
$811$	−44.4111	−1.55948	−0.779742	−	0.626101i	$-0.784650\pi$
$811$	−44.4111	−1.55948	−0.779742	+	0.626101i	$0.784650\pi$
$812$	56.5300i	1.98381i
$813$	0	0
$814$	13.4983	0.473115
$815$	0	0
$816$	0	0
$817$	− 2.98230i	− 0.104337i
$818$	4.04836i	0.141548i
$819$	0	0
$820$	0	0
$821$	46.4630	1.62157	0.810785	−	0.585343i	$-0.199040\pi$
$821$	46.4630	1.62157	0.810785	+	0.585343i	$0.199040\pi$
$822$	0	0
$823$	− 46.5089i	− 1.62120i	−0.585603	−	0.810598i	$-0.699142\pi$
$823$	− 46.5089i	− 1.62120i	0.585603	−	0.810598i	$-0.300858\pi$
$824$	−16.4312	−0.572408
$825$	0	0
$826$	16.0000	0.556711
$827$	− 39.4005i	− 1.37009i	−0.728500	−	0.685045i	$-0.759782\pi$
$827$	− 39.4005i	− 1.37009i	0.728500	−	0.685045i	$-0.240218\pi$
$828$	0	0
$829$	−47.6444	−1.65476	−0.827379	−	0.561644i	$-0.810169\pi$
$829$	−47.6444	−1.65476	−0.827379	+	0.561644i	$0.810169\pi$
$830$	0	0
$831$	0	0
$832$	− 6.06249i	− 0.210179i
$833$	− 74.4877i	− 2.58085i
$834$	0	0
$835$	0	0
$836$	24.2922	0.840164
$837$	0	0
$838$	− 9.68665i	− 0.334620i
$839$	−39.9058	−1.37770	−0.688851	−	0.724903i	$-0.741885\pi$
$839$	−39.9058	−1.37770	−0.688851	+	0.724903i	$0.741885\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	3.90939i	0.134726i
$843$	0	0
$844$	12.0988	0.416457
$845$	0	0
$846$	0	0
$847$	64.7527i	2.22493i
$848$	− 1.18494i	− 0.0406910i
$849$	0	0
$850$	0	0
$851$	60.1779	2.06287
$852$	0	0
$853$	− 29.5019i	− 1.01012i	−0.863083	−	0.505062i	$-0.831470\pi$
$853$	− 29.5019i	− 1.01012i	0.863083	−	0.505062i	$-0.168530\pi$
$854$	14.4595	0.494793
$855$	0	0
$856$	4.72342	0.161443
$857$	− 8.33804i	− 0.284822i	−0.989808	−	0.142411i	$-0.954515\pi$
$857$	− 8.33804i	− 0.284822i	0.989808	−	0.142411i	$-0.0454855\pi$
$858$	0	0
$859$	4.17081	0.142306	0.0711531	−	0.997465i	$-0.477332\pi$
$859$	4.17081	0.142306	0.0711531	+	0.997465i	$0.477332\pi$
$860$	0	0
$861$	0	0
$862$	− 3.58890i	− 0.122238i
$863$	− 3.93051i	− 0.133796i	−0.997760	−	0.0668981i	$-0.978690\pi$
$863$	− 3.93051i	− 0.133796i	0.997760	−	0.0668981i	$-0.0213103\pi$
$864$	0	0
$865$	0	0
$866$	−5.00953	−0.170231
$867$	0	0
$868$	13.3944i	0.454637i
$869$	−17.1814	−0.582838
$870$	0	0
$871$	−7.25443	−0.245807
$872$	4.34058i	0.146991i
$873$	0	0
$874$	−4.72548	−0.159842
$875$	0	0
$876$	0	0
$877$	− 23.0177i	− 0.777253i	−0.921395	−	0.388626i	$-0.872950\pi$
$877$	− 23.0177i	− 0.777253i	0.921395	−	0.388626i	$-0.127050\pi$
$878$	− 0.188362i	− 0.00635692i
$879$	0	0
$880$	0	0
$881$	15.3522	0.517228	0.258614	−	0.965981i	$-0.416734\pi$
$881$	15.3522	0.517228	0.258614	+	0.965981i	$0.416734\pi$
$882$	0	0
$883$	− 42.8011i	− 1.44037i	−0.693782	−	0.720185i	$-0.744057\pi$
$883$	− 42.8011i	− 1.44037i	0.693782	−	0.720185i	$-0.255943\pi$
$884$	8.31335	0.279608
$885$	0	0
$886$	2.55824	0.0859459
$887$	53.1885i	1.78590i	0.450160	+	0.892948i	$0.351367\pi$
$887$	53.1885i	1.78590i	−0.450160	+	0.892948i	$0.648633\pi$
$888$	0	0
$889$	−9.01056	−0.302205
$890$	0	0
$891$	0	0
$892$	− 36.8988i	− 1.23546i
$893$	− 13.9789i	− 0.467785i
$894$	0	0
$895$	0	0
$896$	−40.8566	−1.36492
$897$	0	0
$898$	1.25443i	0.0418607i
$899$	8.52998	0.284491
$900$	0	0
$901$	1.46645	0.0488546
$902$	6.16724i	0.205347i
$903$	0	0
$904$	0.955062	0.0317649
$905$	0	0
$906$	0	0
$907$	11.8116i	0.392199i	0.980584	+	0.196099i	$0.0628276\pi$
$907$	11.8116i	0.392199i	−0.980584	+	0.196099i	$0.937172\pi$
$908$	25.0800i	0.832311i
$909$	0	0
$910$	0	0
$911$	−44.1955	−1.46426	−0.732131	−	0.681164i	$-0.761474\pi$
$911$	−44.1955	−1.46426	−0.732131	+	0.681164i	$0.761474\pi$
$912$	0	0
$913$	− 55.3311i	− 1.83119i
$914$	−4.43223	−0.146605
$915$	0	0
$916$	46.9683	1.55188
$917$	− 28.6761i	− 0.946968i
$918$	0	0
$919$	−55.2096	−1.82120	−0.910599	−	0.413291i	$-0.864379\pi$
$919$	−55.2096	−1.82120	−0.910599	+	0.413291i	$0.864379\pi$
$920$	0	0
$921$	0	0
$922$	3.42880i	0.112922i
$923$	0.916382i	0.0301631i
$924$	0	0
$925$	0	0
$926$	−7.63829	−0.251010
$927$	0	0
$928$	19.6716i	0.645753i
$929$	22.9930	0.754376	0.377188	−	0.926137i	$-0.376891\pi$
$929$	22.9930	0.754376	0.377188	+	0.926137i	$0.376891\pi$
$930$	0	0
$931$	44.2721	1.45096
$932$	15.9789i	0.523405i
$933$	0	0
$934$	−9.73604	−0.318573
$935$	0	0
$936$	0	0
$937$	7.97887i	0.260658i	0.991471	+	0.130329i	$0.0416034\pi$
$937$	7.97887i	0.260658i	−0.991471	+	0.130329i	$0.958397\pi$
$938$	10.3133i	0.336743i
$939$	0	0
$940$	0	0
$941$	−41.5019	−1.35292	−0.676461	−	0.736478i	$-0.736487\pi$
$941$	−41.5019	−1.35292	−0.676461	+	0.736478i	$0.736487\pi$
$942$	0	0
$943$	27.4947i	0.895351i
$944$	−39.4499	−1.28399
$945$	0	0
$946$	1.64440	0.0534641
$947$	− 47.4499i	− 1.54192i	−0.636886	−	0.770958i	$-0.719778\pi$
$947$	− 47.4499i	− 1.54192i	0.636886	−	0.770958i	$-0.280222\pi$
$948$	0	0
$949$	3.15667	0.102470
$950$	0	0
$951$	0	0
$952$	− 24.1533i	− 0.782811i
$953$	30.3663i	0.983661i	0.870691	+	0.491831i	$0.163672\pi$
$953$	30.3663i	0.983661i	−0.870691	+	0.491831i	$0.836328\pi$
$954$	0	0
$955$	0	0
$956$	17.0872	0.552639
$957$	0	0
$958$	3.10831i	0.100425i
$959$	−81.1638	−2.62092
$960$	0	0
$961$	−28.9789	−0.934802
$962$	2.74557i	0.0885209i
$963$	0	0
$964$	−11.4983	−0.370335
$965$	0	0
$966$	0	0
$967$	− 41.0943i	− 1.32150i	−0.750604	−	0.660752i	$-0.770237\pi$
$967$	− 41.0943i	− 1.32150i	0.750604	−	0.660752i	$-0.229763\pi$
$968$	14.9159i	0.479414i
$969$	0	0
$970$	0	0
$971$	−38.1744	−1.22507	−0.612537	−	0.790442i	$-0.709851\pi$
$971$	−38.1744	−1.22507	−0.612537	+	0.790442i	$0.709851\pi$
$972$	0	0
$973$	− 36.8469i	− 1.18126i
$974$	−6.57834	−0.210784
$975$	0	0
$976$	−35.6515	−1.14118
$977$	10.4806i	0.335304i	0.985846	+	0.167652i	$0.0536184\pi$
$977$	10.4806i	0.335304i	−0.985846	+	0.167652i	$0.946382\pi$
$978$	0	0
$979$	1.66196	0.0531163
$980$	0	0
$981$	0	0
$982$	5.11442i	0.163208i
$983$	− 0.0766264i	− 0.00244400i	−0.999999	−	0.00122200i	$-0.999611\pi$
$983$	− 0.0766264i	− 0.00244400i	0.999999	−	0.00122200i	$-0.000388975\pi$
$984$	0	0
$985$	0	0
$986$	−7.52656	−0.239694
$987$	0	0
$988$	4.94108i	0.157197i
$989$	7.33105	0.233114
$990$	0	0
$991$	−13.8575	−0.440197	−0.220098	−	0.975478i	$-0.570638\pi$
$991$	−13.8575	−0.440197	−0.220098	+	0.975478i	$0.570638\pi$
$992$	4.66107i	0.147989i
$993$	0	0
$994$	1.30279	0.0413219
$995$	0	0
$996$	0	0
$997$	− 10.3416i	− 0.327522i	−0.986500	−	0.163761i	$-0.947637\pi$
$997$	− 10.3416i	− 0.327522i	0.986500	−	0.163761i	$-0.0523626\pi$
$998$	− 5.76328i	− 0.182433i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2925.2.c.w.2224.4		6
3.2	odd	2		975.2.c.i.274.3		6
5.2	odd	4		2925.2.a.bh.1.2		3
5.3	odd	4		585.2.a.n.1.2		3
5.4	even	2	inner	2925.2.c.w.2224.3		6
15.2	even	4		975.2.a.o.1.2		3
15.8	even	4		195.2.a.e.1.2	✓	3
15.14	odd	2		975.2.c.i.274.4		6
20.3	even	4		9360.2.a.dd.1.1		3
60.23	odd	4		3120.2.a.bj.1.1		3
65.38	odd	4		7605.2.a.bx.1.2		3
105.83	odd	4		9555.2.a.bq.1.2		3
195.38	even	4		2535.2.a.bc.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
195.2.a.e.1.2	✓	3	15.8	even	4
585.2.a.n.1.2		3	5.3	odd	4
975.2.a.o.1.2		3	15.2	even	4
975.2.c.i.274.3		6	3.2	odd	2
975.2.c.i.274.4		6	15.14	odd	2
2535.2.a.bc.1.2		3	195.38	even	4
2925.2.a.bh.1.2		3	5.2	odd	4
2925.2.c.w.2224.3		6	5.4	even	2	inner
2925.2.c.w.2224.4		6	1.1	even	1	trivial
3120.2.a.bj.1.1		3	60.23	odd	4
7605.2.a.bx.1.2		3	65.38	odd	4
9360.2.a.dd.1.1		3	20.3	even	4
9555.2.a.bq.1.2		3	105.83	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.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.289169i q^{2} +1.91638 q^{4} +4.91638i q^{7} +1.13249i q^{8} +O(q^{10})\)	\(q+0.289169i q^{2} +1.91638 q^{4} +4.91638i q^{7} +1.13249i q^{8} -4.91638 q^{11} -1.00000i q^{13} -1.42166 q^{14} +3.50528 q^{16} +4.33804i q^{17} -2.57834 q^{19} -1.42166i q^{22} -6.33804i q^{23} +0.289169 q^{26} +9.42166i q^{28} +6.00000 q^{29} +1.42166 q^{31} +3.27861i q^{32} -1.25443 q^{34} +9.49472i q^{37} -0.745574i q^{38} -4.33804 q^{41} +1.15667i q^{43} -9.42166 q^{44} +1.83276 q^{46} +5.42166i q^{47} -17.1708 q^{49} -1.91638i q^{52} -0.338044i q^{53} -5.56777 q^{56} +1.73501i q^{58} -11.2544 q^{59} -10.1708 q^{61} +0.411100i q^{62} +6.06249 q^{64} -7.25443i q^{67} +8.31335i q^{68} -0.916382 q^{71} +3.15667i q^{73} -2.74557 q^{74} -4.94108 q^{76} -24.1708i q^{77} +3.49472 q^{79} -1.25443i q^{82} +11.2544i q^{83} -0.334474 q^{86} -5.56777i q^{88} -0.338044 q^{89} +4.91638 q^{91} -12.1461i q^{92} -1.56777 q^{94} -12.3380i q^{97} -4.96526i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 16 q^{4}+O(q^{10}) \) 6 * q - 16 * q^4	\( 6 q - 16 q^{4} - 2 q^{11} - 12 q^{14} + 52 q^{16} - 12 q^{19} + 36 q^{29} + 12 q^{31} + 44 q^{34} - 2 q^{41} - 60 q^{44} - 44 q^{46} - 24 q^{49} + 32 q^{56} - 16 q^{59} + 18 q^{61} - 60 q^{64} + 22 q^{71} - 68 q^{74} + 8 q^{76} - 10 q^{79} - 112 q^{86} + 22 q^{89} + 2 q^{91} + 56 q^{94}+O(q^{100}) \) 6 * q - 16 * q^4 - 2 * q^11 - 12 * q^14 + 52 * q^16 - 12 * q^19 + 36 * q^29 + 12 * q^31 + 44 * q^34 - 2 * q^41 - 60 * q^44 - 44 * q^46 - 24 * q^49 + 32 * q^56 - 16 * q^59 + 18 * q^61 - 60 * q^64 + 22 * q^71 - 68 * q^74 + 8 * q^76 - 10 * q^79 - 112 * q^86 + 22 * q^89 + 2 * q^91 + 56 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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