LMFDB - Embedded newform 4275.2.a.bu.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4275,2,Mod(1,4275)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4275, 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("4275.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4275 = 3^{2} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4275.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:	$34.1360468641$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.15044092.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} - 4x^{4} + 12x^{3} - x^{2} - 6x + 2 $ x^6 - 3x^5 - 4x^4 + 12x^3 - x^2 - 6x + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$0.437984$ of defining polynomial
Character	$\chi$	$=$	4275.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.34908 q^{2} +3.51820 q^{4} +4.85952 q^{7} +3.56638 q^{8} -1.93359 q^{11} +2.00000 q^{13} +11.4154 q^{14} +1.34132 q^{16} +2.43459 q^{17} +1.00000 q^{19} -4.54217 q^{22} -4.36818 q^{23} +4.69817 q^{26} +17.0968 q^{28} +8.26455 q^{29} -1.17687 q^{31} -3.98187 q^{32} +5.71905 q^{34} +7.03640 q^{37} +2.34908 q^{38} +7.93456 q^{41} -3.71905 q^{43} -6.80276 q^{44} -10.2612 q^{46} -13.2635 q^{47} +16.6150 q^{49} +7.03640 q^{52} +1.13179 q^{53} +17.3309 q^{56} +19.4141 q^{58} -3.23639 q^{59} +4.36530 q^{61} -2.76458 q^{62} -12.0364 q^{64} -2.54217 q^{67} +8.56535 q^{68} -6.50196 q^{71} -2.71905 q^{73} +16.5291 q^{74} +3.51820 q^{76} -9.39634 q^{77} +8.21327 q^{79} +18.6389 q^{82} -6.80276 q^{83} -8.73636 q^{86} -6.89592 q^{88} +1.13179 q^{89} +9.71905 q^{91} -15.3681 q^{92} -31.1571 q^{94} +16.4017 q^{97} +39.0300 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 8 q^{4} + 8 q^{7} + 12 q^{13} + 6 q^{19} + 10 q^{22} + 26 q^{28} - 2 q^{31} - 8 q^{34} + 16 q^{37} + 20 q^{43} + 18 q^{46} + 10 q^{49} + 16 q^{52} + 56 q^{58} - 6 q^{61} - 46 q^{64} + 22 q^{67} + 26 q^{73}+ \cdots + 40 q^{97}+O(q^{100}) $ 6 * q + 8 * q^4 + 8 * q^7 + 12 * q^13 + 6 * q^19 + 10 * q^22 + 26 * q^28 - 2 * q^31 - 8 * q^34 + 16 * q^37 + 20 * q^43 + 18 * q^46 + 10 * q^49 + 16 * q^52 + 56 * q^58 - 6 * q^61 - 46 * q^64 + 22 * q^67 + 26 * q^73 + 8 * q^76 + 18 * q^79 - 2 * q^82 + 6 * q^88 + 16 * q^91 - 60 * q^94 + 40 * q^97

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 $

$2$	2.34908	1.66105	0.830527	−	0.556979i	$-0.188039\pi$
$2$	2.34908	1.66105	0.830527	+	0.556979i	$0.188039\pi$
$3$	0	0
$3$	0	0
$4$	3.51820	1.75910
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.85952	1.83673	0.918364	−	0.395738i	$-0.129511\pi$
$7$	4.85952	1.83673	0.918364	+	0.395738i	$0.129511\pi$
$8$	3.56638	1.26090
$9$	0	0
$10$	0	0
$11$	−1.93359	−0.583000	−0.291500	−	0.956571i	$-0.594154\pi$
$11$	−1.93359	−0.583000	−0.291500	+	0.956571i	$0.594154\pi$
$12$	0	0
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	11.4154	3.05090
$15$	0	0
$16$	1.34132	0.335331
$17$	2.43459	0.590474	0.295237	−	0.955424i	$-0.404601\pi$
$17$	2.43459	0.590474	0.295237	+	0.955424i	$0.404601\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	−4.54217	−0.968394
$23$	−4.36818	−0.910828	−0.455414	−	0.890280i	$-0.650509\pi$
$23$	−4.36818	−0.910828	−0.455414	+	0.890280i	$0.650509\pi$
$24$	0	0
$25$	0	0
$26$	4.69817	0.921387
$27$	0	0
$28$	17.0968	3.23099
$29$	8.26455	1.53469	0.767344	−	0.641236i	$-0.221578\pi$
$29$	8.26455	1.53469	0.767344	+	0.641236i	$0.221578\pi$
$30$	0	0
$31$	−1.17687	−0.211373	−0.105686	−	0.994400i	$-0.533704\pi$
$31$	−1.17687	−0.211373	−0.105686	+	0.994400i	$0.533704\pi$
$32$	−3.98187	−0.703902
$33$	0	0
$34$	5.71905	0.980808
$35$	0	0
$36$	0	0
$37$	7.03640	1.15678	0.578388	−	0.815762i	$-0.303682\pi$
$37$	7.03640	1.15678	0.578388	+	0.815762i	$0.303682\pi$
$38$	2.34908	0.381072
$39$	0	0
$40$	0	0
$41$	7.93456	1.23917	0.619585	−	0.784930i	$-0.287301\pi$
$41$	7.93456	1.23917	0.619585	+	0.784930i	$0.287301\pi$
$42$	0	0
$43$	−3.71905	−0.567149	−0.283575	−	0.958950i	$-0.591520\pi$
$43$	−3.71905	−0.567149	−0.283575	+	0.958950i	$0.591520\pi$
$44$	−6.80276	−1.02556
$45$	0	0
$46$	−10.2612	−1.51293
$47$	−13.2635	−1.93468	−0.967342	−	0.253475i	$-0.918426\pi$
$47$	−13.2635	−1.93468	−0.967342	+	0.253475i	$0.918426\pi$
$48$	0	0
$49$	16.6150	2.37357
$50$	0	0
$51$	0	0
$52$	7.03640	0.975773
$53$	1.13179	0.155464	0.0777318	−	0.996974i	$-0.475232\pi$
$53$	1.13179	0.155464	0.0777318	+	0.996974i	$0.475232\pi$
$54$	0	0
$55$	0	0
$56$	17.3309	2.31594
$57$	0	0
$58$	19.4141	2.54920
$59$	−3.23639	−0.421342	−0.210671	−	0.977557i	$-0.567565\pi$
$59$	−3.23639	−0.421342	−0.210671	+	0.977557i	$0.567565\pi$
$60$	0	0
$61$	4.36530	0.558919	0.279460	−	0.960157i	$-0.409845\pi$
$61$	4.36530	0.558919	0.279460	+	0.960157i	$0.409845\pi$
$62$	−2.76458	−0.351102
$63$	0	0
$64$	−12.0364	−1.50455
$65$	0	0
$66$	0	0
$67$	−2.54217	−0.310576	−0.155288	−	0.987869i	$-0.549631\pi$
$67$	−2.54217	−0.310576	−0.155288	+	0.987869i	$0.549631\pi$
$68$	8.56535	1.03870
$69$	0	0
$70$	0	0
$71$	−6.50196	−0.771640	−0.385820	−	0.922574i	$-0.626081\pi$
$71$	−6.50196	−0.771640	−0.385820	+	0.922574i	$0.626081\pi$
$72$	0	0
$73$	−2.71905	−0.318240	−0.159120	−	0.987259i	$-0.550866\pi$
$73$	−2.71905	−0.318240	−0.159120	+	0.987259i	$0.550866\pi$
$74$	16.5291	1.92147
$75$	0	0
$76$	3.51820	0.403565
$77$	−9.39634	−1.07081
$78$	0	0
$79$	8.21327	0.924065	0.462033	−	0.886863i	$-0.347120\pi$
$79$	8.21327	0.924065	0.462033	+	0.886863i	$0.347120\pi$
$80$	0	0
$81$	0	0
$82$	18.6389	2.05833
$83$	−6.80276	−0.746700	−0.373350	−	0.927691i	$-0.621791\pi$
$83$	−6.80276	−0.746700	−0.373350	+	0.927691i	$0.621791\pi$
$84$	0	0
$85$	0	0
$86$	−8.73636	−0.942065
$87$	0	0
$88$	−6.89592	−0.735108
$89$	1.13179	0.119970	0.0599849	−	0.998199i	$-0.480895\pi$
$89$	1.13179	0.119970	0.0599849	+	0.998199i	$0.480895\pi$
$90$	0	0
$91$	9.71905	1.01883
$92$	−15.3681	−1.60224
$93$	0	0
$94$	−31.1571	−3.21361
$95$	0	0
$96$	0	0
$97$	16.4017	1.66534	0.832670	−	0.553770i	$-0.186811\pi$
$97$	16.4017	1.66534	0.832670	+	0.553770i	$0.186811\pi$
$98$	39.0300	3.94262
$99$	0	0
$100$	0	0
$101$	−7.96374	−0.792422	−0.396211	−	0.918160i	$-0.629675\pi$
$101$	−7.96374	−0.792422	−0.396211	+	0.918160i	$0.629675\pi$
$102$	0	0
$103$	6.82313	0.672303	0.336151	−	0.941808i	$-0.390875\pi$
$103$	6.82313	0.672303	0.336151	+	0.941808i	$0.390875\pi$
$104$	7.13275	0.699424
$105$	0	0
$106$	2.65868	0.258233
$107$	−8.76554	−0.847397	−0.423698	−	0.905803i	$-0.639268\pi$
$107$	−8.76554	−0.847397	−0.423698	+	0.905803i	$0.639268\pi$
$108$	0	0
$109$	−18.1207	−1.73565	−0.867826	−	0.496868i	$-0.834483\pi$
$109$	−18.1207	−1.73565	−0.867826	+	0.496868i	$0.834483\pi$
$110$	0	0
$111$	0	0
$112$	6.51820	0.615912
$113$	18.4627	1.73682	0.868412	−	0.495844i	$-0.165141\pi$
$113$	18.4627	1.73682	0.868412	+	0.495844i	$0.165141\pi$
$114$	0	0
$115$	0	0
$116$	29.0763	2.69967
$117$	0	0
$118$	−7.60254	−0.699871
$119$	11.8309	1.08454
$120$	0	0
$121$	−7.26122	−0.660111
$122$	10.2545	0.928395
$123$	0	0
$124$	−4.14048	−0.371826
$125$	0	0
$126$	0	0
$127$	−7.57857	−0.672489	−0.336245	−	0.941775i	$-0.609157\pi$
$127$	−7.57857	−0.672489	−0.336245	+	0.941775i	$0.609157\pi$
$128$	−20.3108	−1.79524
$129$	0	0
$130$	0	0
$131$	−17.4607	−1.52555	−0.762774	−	0.646665i	$-0.776163\pi$
$131$	−17.4607	−1.52555	−0.762774	+	0.646665i	$0.776163\pi$
$132$	0	0
$133$	4.85952	0.421374
$134$	−5.97178	−0.515883
$135$	0	0
$136$	8.68265	0.744531
$137$	−9.56734	−0.817393	−0.408696	−	0.912670i	$-0.634017\pi$
$137$	−9.56734	−0.817393	−0.408696	+	0.912670i	$0.634017\pi$
$138$	0	0
$139$	−1.49422	−0.126738	−0.0633692	−	0.997990i	$-0.520185\pi$
$139$	−1.49422	−0.126738	−0.0633692	+	0.997990i	$0.520185\pi$
$140$	0	0
$141$	0	0
$142$	−15.2736	−1.28174
$143$	−3.86719	−0.323390
$144$	0	0
$145$	0	0
$146$	−6.38727	−0.528614
$147$	0	0
$148$	24.7554	2.03488
$149$	14.2655	1.16868	0.584338	−	0.811510i	$-0.301354\pi$
$149$	14.2655	1.16868	0.584338	+	0.811510i	$0.301354\pi$
$150$	0	0
$151$	−14.7554	−1.20078	−0.600391	−	0.799707i	$-0.704988\pi$
$151$	−14.7554	−1.20078	−0.600391	+	0.799707i	$0.704988\pi$
$152$	3.56638	0.289271
$153$	0	0
$154$	−22.0728	−1.77868
$155$	0	0
$156$	0	0
$157$	15.5422	1.24040	0.620200	−	0.784444i	$-0.287051\pi$
$157$	15.5422	1.24040	0.620200	+	0.784444i	$0.287051\pi$
$158$	19.2937	1.53492
$159$	0	0
$160$	0	0
$161$	−21.2273	−1.67294
$162$	0	0
$163$	10.2248	0.800870	0.400435	−	0.916325i	$-0.368859\pi$
$163$	10.2248	0.800870	0.400435	+	0.916325i	$0.368859\pi$
$164$	27.9153	2.17982
$165$	0	0
$166$	−15.9803	−1.24031
$167$	−7.76355	−0.600762	−0.300381	−	0.953819i	$-0.597114\pi$
$167$	−7.76355	−0.600762	−0.300381	+	0.953819i	$0.597114\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	0	0
$172$	−13.0843	−0.997672
$173$	17.0009	1.29255	0.646277	−	0.763103i	$-0.276325\pi$
$173$	17.0009	1.29255	0.646277	+	0.763103i	$0.276325\pi$
$174$	0	0
$175$	0	0
$176$	−2.59358	−0.195498
$177$	0	0
$178$	2.65868	0.199276
$179$	−12.0311	−0.899247	−0.449624	−	0.893218i	$-0.648442\pi$
$179$	−12.0311	−0.899247	−0.449624	+	0.893218i	$0.648442\pi$
$180$	0	0
$181$	22.0728	1.64066	0.820329	−	0.571892i	$-0.193790\pi$
$181$	22.0728	1.64066	0.820329	+	0.571892i	$0.193790\pi$
$182$	22.8309	1.69234
$183$	0	0
$184$	−15.5786	−1.14847
$185$	0	0
$186$	0	0
$187$	−4.70750	−0.344246
$188$	−46.6637	−3.40330
$189$	0	0
$190$	0	0
$191$	13.5935	0.983592	0.491796	−	0.870710i	$-0.336341\pi$
$191$	13.5935	0.983592	0.491796	+	0.870710i	$0.336341\pi$
$192$	0	0
$193$	−20.8282	−1.49925	−0.749625	−	0.661863i	$-0.769766\pi$
$193$	−20.8282	−1.49925	−0.749625	+	0.661863i	$0.769766\pi$
$194$	38.5290	2.76622
$195$	0	0
$196$	58.4548	4.17534
$197$	23.8328	1.69802	0.849010	−	0.528377i	$-0.177199\pi$
$197$	23.8328	1.69802	0.849010	+	0.528377i	$0.177199\pi$
$198$	0	0
$199$	12.5786	0.891671	0.445836	−	0.895115i	$-0.352907\pi$
$199$	12.5786	0.891671	0.445836	+	0.895115i	$0.352907\pi$
$200$	0	0
$201$	0	0
$202$	−18.7075	−1.31625
$203$	40.1618	2.81880
$204$	0	0
$205$	0	0
$206$	16.0281	1.11673
$207$	0	0
$208$	2.68265	0.186008
$209$	−1.93359	−0.133749
$210$	0	0
$211$	15.2976	1.05313	0.526566	−	0.850134i	$-0.323479\pi$
$211$	15.2976	1.05313	0.526566	+	0.850134i	$0.323479\pi$
$212$	3.98187	0.273476
$213$	0	0
$214$	−20.5910	−1.40757
$215$	0	0
$216$	0	0
$217$	−5.71905	−0.388234
$218$	−42.5672	−2.88301
$219$	0	0
$220$	0	0
$221$	4.86917	0.327536
$222$	0	0
$223$	−20.6398	−1.38214	−0.691072	−	0.722786i	$-0.742861\pi$
$223$	−20.6398	−1.38214	−0.691072	+	0.722786i	$0.742861\pi$
$224$	−19.3500	−1.29288
$225$	0	0
$226$	43.3704	2.88496
$227$	−2.89438	−0.192107	−0.0960535	−	0.995376i	$-0.530622\pi$
$227$	−2.89438	−0.192107	−0.0960535	+	0.995376i	$0.530622\pi$
$228$	0	0
$229$	−12.8959	−0.852186	−0.426093	−	0.904679i	$-0.640110\pi$
$229$	−12.8959	−0.852186	−0.426093	+	0.904679i	$0.640110\pi$
$230$	0	0
$231$	0	0
$232$	29.4745	1.93510
$233$	11.6599	0.763867	0.381933	−	0.924190i	$-0.375258\pi$
$233$	11.6599	0.763867	0.381933	+	0.924190i	$0.375258\pi$
$234$	0	0
$235$	0	0
$236$	−11.3862	−0.741182
$237$	0	0
$238$	27.7918	1.80148
$239$	−22.2292	−1.43789	−0.718945	−	0.695067i	$-0.755375\pi$
$239$	−22.2292	−1.43789	−0.718945	+	0.695067i	$0.755375\pi$
$240$	0	0
$241$	1.31735	0.0848580	0.0424290	−	0.999099i	$-0.486490\pi$
$241$	1.31735	0.0848580	0.0424290	+	0.999099i	$0.486490\pi$
$242$	−17.0572	−1.09648
$243$	0	0
$244$	15.3580	0.983194
$245$	0	0
$246$	0	0
$247$	2.00000	0.127257
$248$	−4.19718	−0.266521
$249$	0	0
$250$	0	0
$251$	−12.4325	−0.784735	−0.392367	−	0.919809i	$-0.628344\pi$
$251$	−12.4325	−0.784735	−0.392367	+	0.919809i	$0.628344\pi$
$252$	0	0
$253$	8.44628	0.531013
$254$	−17.8027	−1.11704
$255$	0	0
$256$	−23.6389	−1.47743
$257$	−23.5320	−1.46789	−0.733944	−	0.679210i	$-0.762323\pi$
$257$	−23.5320	−1.46789	−0.733944	+	0.679210i	$0.762323\pi$
$258$	0	0
$259$	34.1935	2.12468
$260$	0	0
$261$	0	0
$262$	−41.0167	−2.53402
$263$	−1.76259	−0.108686	−0.0543430	−	0.998522i	$-0.517306\pi$
$263$	−1.76259	−0.108686	−0.0543430	+	0.998522i	$0.517306\pi$
$264$	0	0
$265$	0	0
$266$	11.4154	0.699925
$267$	0	0
$268$	−8.94387	−0.546334
$269$	−10.6579	−0.649826	−0.324913	−	0.945744i	$-0.605335\pi$
$269$	−10.6579	−0.649826	−0.324913	+	0.945744i	$0.605335\pi$
$270$	0	0
$271$	−6.22482	−0.378131	−0.189066	−	0.981964i	$-0.560546\pi$
$271$	−6.22482	−0.378131	−0.189066	+	0.981964i	$0.560546\pi$
$272$	3.26557	0.198004
$273$	0	0
$274$	−22.4745	−1.35773
$275$	0	0
$276$	0	0
$277$	−4.79184	−0.287914	−0.143957	−	0.989584i	$-0.545983\pi$
$277$	−4.79184	−0.287914	−0.143957	+	0.989584i	$0.545983\pi$
$278$	−3.51006	−0.210519
$279$	0	0
$280$	0	0
$281$	22.3299	1.33209	0.666044	−	0.745912i	$-0.267986\pi$
$281$	22.3299	1.33209	0.666044	+	0.745912i	$0.267986\pi$
$282$	0	0
$283$	6.35375	0.377691	0.188846	−	0.982007i	$-0.439525\pi$
$283$	6.35375	0.377691	0.188846	+	0.982007i	$0.439525\pi$
$284$	−22.8752	−1.35739
$285$	0	0
$286$	−9.08435	−0.537169
$287$	38.5582	2.27602
$288$	0	0
$289$	−11.0728	−0.651341
$290$	0	0
$291$	0	0
$292$	−9.56615	−0.559816
$293$	−24.3338	−1.42160	−0.710799	−	0.703395i	$-0.751666\pi$
$293$	−24.3338	−1.42160	−0.710799	+	0.703395i	$0.751666\pi$
$294$	0	0
$295$	0	0
$296$	25.0944	1.45859
$297$	0	0
$298$	33.5109	1.94123
$299$	−8.73636	−0.505236
$300$	0	0
$301$	−18.0728	−1.04170
$302$	−34.6618	−1.99456
$303$	0	0
$304$	1.34132	0.0769302
$305$	0	0
$306$	0	0
$307$	2.18842	0.124900	0.0624500	−	0.998048i	$-0.480109\pi$
$307$	2.18842	0.124900	0.0624500	+	0.998048i	$0.480109\pi$
$308$	−33.0582	−1.88366
$309$	0	0
$310$	0	0
$311$	−11.6599	−0.661174	−0.330587	−	0.943776i	$-0.607247\pi$
$311$	−11.6599	−0.661174	−0.330587	+	0.943776i	$0.607247\pi$
$312$	0	0
$313$	−17.3225	−0.979124	−0.489562	−	0.871969i	$-0.662843\pi$
$313$	−17.3225	−0.979124	−0.489562	+	0.871969i	$0.662843\pi$
$314$	36.5099	2.06037
$315$	0	0
$316$	28.8959	1.62552
$317$	5.60059	0.314561	0.157280	−	0.987554i	$-0.449727\pi$
$317$	5.60059	0.314561	0.157280	+	0.987554i	$0.449727\pi$
$318$	0	0
$319$	−15.9803	−0.894723
$320$	0	0
$321$	0	0
$322$	−49.8646	−2.77885
$323$	2.43459	0.135464
$324$	0	0
$325$	0	0
$326$	24.0190	1.33029
$327$	0	0
$328$	28.2976	1.56247
$329$	−64.4544	−3.55349
$330$	0	0
$331$	−8.82313	−0.484963	−0.242481	−	0.970156i	$-0.577961\pi$
$331$	−8.82313	−0.484963	−0.242481	+	0.970156i	$0.577961\pi$
$332$	−23.9335	−1.31352
$333$	0	0
$334$	−18.2372	−0.997898
$335$	0	0
$336$	0	0
$337$	−17.5109	−0.953879	−0.476939	−	0.878936i	$-0.658254\pi$
$337$	−17.5109	−0.953879	−0.476939	+	0.878936i	$0.658254\pi$
$338$	−21.1418	−1.14996
$339$	0	0
$340$	0	0
$341$	2.27559	0.123230
$342$	0	0
$343$	46.7242	2.52287
$344$	−13.2635	−0.715121
$345$	0	0
$346$	39.9366	2.14700
$347$	−32.8872	−1.76548	−0.882738	−	0.469865i	$-0.844303\pi$
$347$	−32.8872	−1.76548	−0.882738	+	0.469865i	$0.844303\pi$
$348$	0	0
$349$	−16.0843	−0.860975	−0.430488	−	0.902596i	$-0.641658\pi$
$349$	−16.0843	−0.860975	−0.430488	+	0.902596i	$0.641658\pi$
$350$	0	0
$351$	0	0
$352$	7.69931	0.410375
$353$	10.3400	0.550340	0.275170	−	0.961396i	$-0.411266\pi$
$353$	10.3400	0.550340	0.275170	+	0.961396i	$0.411266\pi$
$354$	0	0
$355$	0	0
$356$	3.98187	0.211039
$357$	0	0
$358$	−28.2621	−1.49370
$359$	22.1709	1.17013	0.585067	−	0.810985i	$-0.301068\pi$
$359$	22.1709	1.17013	0.585067	+	0.810985i	$0.301068\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	51.8509	2.72522
$363$	0	0
$364$	34.1935	1.79223
$365$	0	0
$366$	0	0
$367$	−9.08435	−0.474199	−0.237099	−	0.971485i	$-0.576197\pi$
$367$	−9.08435	−0.474199	−0.237099	+	0.971485i	$0.576197\pi$
$368$	−5.85914	−0.305429
$369$	0	0
$370$	0	0
$371$	5.49997	0.285544
$372$	0	0
$373$	−6.02485	−0.311955	−0.155977	−	0.987761i	$-0.549853\pi$
$373$	−6.02485	−0.311955	−0.155977	+	0.987761i	$0.549853\pi$
$374$	−11.0583	−0.571811
$375$	0	0
$376$	−47.3027	−2.43945
$377$	16.5291	0.851292
$378$	0	0
$379$	33.6316	1.72754	0.863770	−	0.503886i	$-0.168097\pi$
$379$	33.6316	1.72754	0.863770	+	0.503886i	$0.168097\pi$
$380$	0	0
$381$	0	0
$382$	31.9323	1.63380
$383$	29.8218	1.52382	0.761911	−	0.647681i	$-0.224261\pi$
$383$	29.8218	1.52382	0.761911	+	0.647681i	$0.224261\pi$
$384$	0	0
$385$	0	0
$386$	−48.9273	−2.49033
$387$	0	0
$388$	57.7044	2.92950
$389$	2.60559	0.132108	0.0660542	−	0.997816i	$-0.478959\pi$
$389$	2.60559	0.132108	0.0660542	+	0.997816i	$0.478959\pi$
$390$	0	0
$391$	−10.6347	−0.537820
$392$	59.2552	2.99284
$393$	0	0
$394$	55.9854	2.82050
$395$	0	0
$396$	0	0
$397$	23.2497	1.16687	0.583434	−	0.812161i	$-0.301709\pi$
$397$	23.2497	1.16687	0.583434	+	0.812161i	$0.301709\pi$
$398$	29.5481	1.48111
$399$	0	0
$400$	0	0
$401$	−18.8047	−0.939061	−0.469531	−	0.882916i	$-0.655577\pi$
$401$	−18.8047	−0.939061	−0.469531	+	0.882916i	$0.655577\pi$
$402$	0	0
$403$	−2.35375	−0.117249
$404$	−28.0180	−1.39395
$405$	0	0
$406$	94.3434	4.68218
$407$	−13.6055	−0.674401
$408$	0	0
$409$	−15.1092	−0.747101	−0.373551	−	0.927610i	$-0.621860\pi$
$409$	−15.1092	−0.747101	−0.373551	+	0.927610i	$0.621860\pi$
$410$	0	0
$411$	0	0
$412$	24.0051	1.18265
$413$	−15.7273	−0.773889
$414$	0	0
$415$	0	0
$416$	−7.96374	−0.390454
$417$	0	0
$418$	−4.54217	−0.222165
$419$	12.0019	0.586332	0.293166	−	0.956061i	$-0.405291\pi$
$419$	12.0019	0.586332	0.293166	+	0.956061i	$0.405291\pi$
$420$	0	0
$421$	−24.1456	−1.17678	−0.588392	−	0.808576i	$-0.700239\pi$
$421$	−24.1456	−1.17678	−0.588392	+	0.808576i	$0.700239\pi$
$422$	35.9354	1.74931
$423$	0	0
$424$	4.03640	0.196025
$425$	0	0
$426$	0	0
$427$	21.2133	1.02658
$428$	−30.8389	−1.49065
$429$	0	0
$430$	0	0
$431$	−17.8199	−0.858353	−0.429177	−	0.903221i	$-0.641196\pi$
$431$	−17.8199	−0.858353	−0.429177	+	0.903221i	$0.641196\pi$
$432$	0	0
$433$	−9.08435	−0.436566	−0.218283	−	0.975886i	$-0.570046\pi$
$433$	−9.08435	−0.436566	−0.218283	+	0.975886i	$0.570046\pi$
$434$	−13.4345	−0.644878
$435$	0	0
$436$	−63.7524	−3.05318
$437$	−4.36818	−0.208958
$438$	0	0
$439$	15.8595	0.756934	0.378467	−	0.925615i	$-0.376451\pi$
$439$	15.8595	0.756934	0.378467	+	0.925615i	$0.376451\pi$
$440$	0	0
$441$	0	0
$442$	11.4381	0.544055
$443$	−17.2011	−0.817248	−0.408624	−	0.912703i	$-0.633991\pi$
$443$	−17.2011	−0.817248	−0.408624	+	0.912703i	$0.633991\pi$
$444$	0	0
$445$	0	0
$446$	−48.4847	−2.29582
$447$	0	0
$448$	−58.4912	−2.76345
$449$	−1.13179	−0.0534126	−0.0267063	−	0.999643i	$-0.508502\pi$
$449$	−1.13179	−0.0534126	−0.0267063	+	0.999643i	$0.508502\pi$
$450$	0	0
$451$	−15.3422	−0.722436
$452$	64.9554	3.05524
$453$	0	0
$454$	−6.79915	−0.319100
$455$	0	0
$456$	0	0
$457$	29.6150	1.38533	0.692665	−	0.721260i	$-0.256437\pi$
$457$	29.6150	1.38533	0.692665	+	0.721260i	$0.256437\pi$
$458$	−30.2936	−1.41553
$459$	0	0
$460$	0	0
$461$	−37.3560	−1.73984	−0.869921	−	0.493191i	$-0.835830\pi$
$461$	−37.3560	−1.73984	−0.869921	+	0.493191i	$0.835830\pi$
$462$	0	0
$463$	26.4993	1.23153	0.615764	−	0.787930i	$-0.288847\pi$
$463$	26.4993	1.23153	0.615764	+	0.787930i	$0.288847\pi$
$464$	11.0854	0.514629
$465$	0	0
$466$	27.3901	1.26882
$467$	34.7624	1.60861	0.804306	−	0.594215i	$-0.202537\pi$
$467$	34.7624	1.60861	0.804306	+	0.594215i	$0.202537\pi$
$468$	0	0
$469$	−12.3537	−0.570443
$470$	0	0
$471$	0	0
$472$	−11.5422	−0.531272
$473$	7.19112	0.330648
$474$	0	0
$475$	0	0
$476$	41.6235	1.90781
$477$	0	0
$478$	−52.2184	−2.38841
$479$	−23.1609	−1.05825	−0.529123	−	0.848545i	$-0.677479\pi$
$479$	−23.1609	−1.05825	−0.529123	+	0.848545i	$0.677479\pi$
$480$	0	0
$481$	14.0728	0.641664
$482$	3.09457	0.140954
$483$	0	0
$484$	−25.5464	−1.16120
$485$	0	0
$486$	0	0
$487$	9.51089	0.430979	0.215490	−	0.976506i	$-0.430865\pi$
$487$	9.51089	0.430979	0.215490	+	0.976506i	$0.430865\pi$
$488$	15.5683	0.704744
$489$	0	0
$490$	0	0
$491$	−36.4948	−1.64699	−0.823493	−	0.567327i	$-0.807978\pi$
$491$	−36.4948	−1.64699	−0.823493	+	0.567327i	$0.807978\pi$
$492$	0	0
$493$	20.1207	0.906193
$494$	4.69817	0.211381
$495$	0	0
$496$	−1.57857	−0.0708799
$497$	−31.5964	−1.41729
$498$	0	0
$499$	40.5786	1.81655	0.908273	−	0.418378i	$-0.137401\pi$
$499$	40.5786	1.81655	0.908273	+	0.418378i	$0.137401\pi$
$500$	0	0
$501$	0	0
$502$	−29.2051	−1.30349
$503$	−3.86719	−0.172429	−0.0862146	−	0.996277i	$-0.527477\pi$
$503$	−3.86719	−0.172429	−0.0862146	+	0.996277i	$0.527477\pi$
$504$	0	0
$505$	0	0
$506$	19.8410	0.882041
$507$	0	0
$508$	−26.6629	−1.18298
$509$	−13.7937	−0.611395	−0.305697	−	0.952129i	$-0.598890\pi$
$509$	−13.7937	−0.611395	−0.305697	+	0.952129i	$0.598890\pi$
$510$	0	0
$511$	−13.2133	−0.584521
$512$	−14.9083	−0.658861
$513$	0	0
$514$	−55.2788	−2.43824
$515$	0	0
$516$	0	0
$517$	25.6463	1.12792
$518$	80.3235	3.52921
$519$	0	0
$520$	0	0
$521$	−40.3207	−1.76648	−0.883242	−	0.468917i	$-0.844644\pi$
$521$	−40.3207	−1.76648	−0.883242	+	0.468917i	$0.844644\pi$
$522$	0	0
$523$	1.74389	0.0762551	0.0381276	−	0.999273i	$-0.487861\pi$
$523$	1.74389	0.0762551	0.0381276	+	0.999273i	$0.487861\pi$
$524$	−61.4302	−2.68359
$525$	0	0
$526$	−4.14048	−0.180533
$527$	−2.86520	−0.124810
$528$	0	0
$529$	−3.91902	−0.170392
$530$	0	0
$531$	0	0
$532$	17.0968	0.741239
$533$	15.8691	0.687367
$534$	0	0
$535$	0	0
$536$	−9.06635	−0.391607
$537$	0	0
$538$	−25.0364	−1.07940
$539$	−32.1266	−1.38379
$540$	0	0
$541$	−36.3340	−1.56212	−0.781061	−	0.624455i	$-0.785321\pi$
$541$	−36.3340	−1.56212	−0.781061	+	0.624455i	$0.785321\pi$
$542$	−14.6226	−0.628096
$543$	0	0
$544$	−9.69420	−0.415635
$545$	0	0
$546$	0	0
$547$	12.5670	0.537327	0.268663	−	0.963234i	$-0.413418\pi$
$547$	12.5670	0.537327	0.268663	+	0.963234i	$0.413418\pi$
$548$	−33.6598	−1.43788
$549$	0	0
$550$	0	0
$551$	8.26455	0.352082
$552$	0	0
$553$	39.9126	1.69726
$554$	−11.2564	−0.478240
$555$	0	0
$556$	−5.25698	−0.222946
$557$	−15.1851	−0.643414	−0.321707	−	0.946839i	$-0.604257\pi$
$557$	−15.1851	−0.643414	−0.321707	+	0.946839i	$0.604257\pi$
$558$	0	0
$559$	−7.43809	−0.314598
$560$	0	0
$561$	0	0
$562$	52.4548	2.21267
$563$	−33.0290	−1.39201	−0.696003	−	0.718039i	$-0.745040\pi$
$563$	−33.0290	−1.39201	−0.696003	+	0.718039i	$0.745040\pi$
$564$	0	0
$565$	0	0
$566$	14.9255	0.627365
$567$	0	0
$568$	−23.1884	−0.972965
$569$	9.19615	0.385523	0.192761	−	0.981246i	$-0.438256\pi$
$569$	9.19615	0.385523	0.192761	+	0.981246i	$0.438256\pi$
$570$	0	0
$571$	15.9936	0.669310	0.334655	−	0.942341i	$-0.391380\pi$
$571$	15.9936	0.669310	0.334655	+	0.942341i	$0.391380\pi$
$572$	−13.6055	−0.568876
$573$	0	0
$574$	90.5764	3.78058
$575$	0	0
$576$	0	0
$577$	37.8449	1.57550	0.787752	−	0.615993i	$-0.211245\pi$
$577$	37.8449	1.57550	0.787752	+	0.615993i	$0.211245\pi$
$578$	−26.0109	−1.08191
$579$	0	0
$580$	0	0
$581$	−33.0582	−1.37148
$582$	0	0
$583$	−2.18842	−0.0906353
$584$	−9.69715	−0.401271
$585$	0	0
$586$	−57.1623	−2.36135
$587$	42.8972	1.77055	0.885277	−	0.465063i	$-0.153968\pi$
$587$	42.8972	1.77055	0.885277	+	0.465063i	$0.153968\pi$
$588$	0	0
$589$	−1.17687	−0.0484922
$590$	0	0
$591$	0	0
$592$	9.43809	0.387903
$593$	−13.9235	−0.571770	−0.285885	−	0.958264i	$-0.592288\pi$
$593$	−13.9235	−0.571770	−0.285885	+	0.958264i	$0.592288\pi$
$594$	0	0
$595$	0	0
$596$	50.1889	2.05582
$597$	0	0
$598$	−20.5224	−0.839225
$599$	25.0058	1.02171	0.510855	−	0.859667i	$-0.329329\pi$
$599$	25.0058	1.02171	0.510855	+	0.859667i	$0.329329\pi$
$600$	0	0
$601$	−38.5721	−1.57339	−0.786695	−	0.617342i	$-0.788210\pi$
$601$	−38.5721	−1.57339	−0.786695	+	0.617342i	$0.788210\pi$
$602$	−42.4545	−1.73032
$603$	0	0
$604$	−51.9126	−2.11229
$605$	0	0
$606$	0	0
$607$	−42.1010	−1.70883	−0.854414	−	0.519593i	$-0.826083\pi$
$607$	−42.1010	−1.70883	−0.854414	+	0.519593i	$0.826083\pi$
$608$	−3.98187	−0.161486
$609$	0	0
$610$	0	0
$611$	−26.5270	−1.07317
$612$	0	0
$613$	45.7606	1.84825	0.924126	−	0.382088i	$-0.124795\pi$
$613$	45.7606	1.84825	0.924126	+	0.382088i	$0.124795\pi$
$614$	5.14079	0.207466
$615$	0	0
$616$	−33.5109	−1.35019
$617$	−5.12878	−0.206477	−0.103239	−	0.994657i	$-0.532921\pi$
$617$	−5.12878	−0.206477	−0.103239	+	0.994657i	$0.532921\pi$
$618$	0	0
$619$	−3.66292	−0.147225	−0.0736125	−	0.997287i	$-0.523453\pi$
$619$	−3.66292	−0.147225	−0.0736125	+	0.997287i	$0.523453\pi$
$620$	0	0
$621$	0	0
$622$	−27.3901	−1.09824
$623$	5.49997	0.220352
$624$	0	0
$625$	0	0
$626$	−40.6919	−1.62638
$627$	0	0
$628$	54.6805	2.18199
$629$	17.1307	0.683046
$630$	0	0
$631$	−43.5340	−1.73306	−0.866530	−	0.499124i	$-0.833655\pi$
$631$	−43.5340	−1.73306	−0.866530	+	0.499124i	$0.833655\pi$
$632$	29.2916	1.16516
$633$	0	0
$634$	13.1563	0.522502
$635$	0	0
$636$	0	0
$637$	33.2299	1.31662
$638$	−37.5390	−1.48618
$639$	0	0
$640$	0	0
$641$	34.3198	1.35555	0.677775	−	0.735269i	$-0.262944\pi$
$641$	34.3198	1.35555	0.677775	+	0.735269i	$0.262944\pi$
$642$	0	0
$643$	−17.8480	−0.703855	−0.351928	−	0.936027i	$-0.614474\pi$
$643$	−17.8480	−0.703855	−0.351928	+	0.936027i	$0.614474\pi$
$644$	−74.6817	−2.94287
$645$	0	0
$646$	5.71905	0.225013
$647$	−47.9373	−1.88461	−0.942305	−	0.334755i	$-0.891347\pi$
$647$	−47.9373	−1.88461	−0.942305	+	0.334755i	$0.891347\pi$
$648$	0	0
$649$	6.25785	0.245642
$650$	0	0
$651$	0	0
$652$	35.9730	1.40881
$653$	−31.3962	−1.22863	−0.614314	−	0.789061i	$-0.710567\pi$
$653$	−31.3962	−1.22863	−0.614314	+	0.789061i	$0.710567\pi$
$654$	0	0
$655$	0	0
$656$	10.6428	0.415532
$657$	0	0
$658$	−151.409	−5.90253
$659$	11.4003	0.444093	0.222046	−	0.975036i	$-0.428726\pi$
$659$	11.4003	0.444093	0.222046	+	0.975036i	$0.428726\pi$
$660$	0	0
$661$	−47.2051	−1.83607	−0.918033	−	0.396505i	$-0.870223\pi$
$661$	−47.2051	−1.83607	−0.918033	+	0.396505i	$0.870223\pi$
$662$	−20.7263	−0.805549
$663$	0	0
$664$	−24.2612	−0.941518
$665$	0	0
$666$	0	0
$667$	−36.1010	−1.39784
$668$	−27.3137	−1.05680
$669$	0	0
$670$	0	0
$671$	−8.44071	−0.325850
$672$	0	0
$673$	24.7075	0.952404	0.476202	−	0.879336i	$-0.342013\pi$
$673$	24.7075	0.952404	0.476202	+	0.879336i	$0.342013\pi$
$674$	−41.1346	−1.58444
$675$	0	0
$676$	−31.6638	−1.21784
$677$	−26.3972	−1.01453	−0.507264	−	0.861791i	$-0.669343\pi$
$677$	−26.3972	−1.01453	−0.507264	+	0.861791i	$0.669343\pi$
$678$	0	0
$679$	79.7044	3.05878
$680$	0	0
$681$	0	0
$682$	5.34557	0.204692
$683$	−10.9708	−0.419784	−0.209892	−	0.977725i	$-0.567311\pi$
$683$	−10.9708	−0.419784	−0.209892	+	0.977725i	$0.567311\pi$
$684$	0	0
$685$	0	0
$686$	109.759	4.19062
$687$	0	0
$688$	−4.98845	−0.190183
$689$	2.26358	0.0862357
$690$	0	0
$691$	−0.280953	−0.0106880	−0.00534398	−	0.999986i	$-0.501701\pi$
$691$	−0.280953	−0.0106880	−0.00534398	+	0.999986i	$0.501701\pi$
$692$	59.8126	2.27373
$693$	0	0
$694$	−77.2548	−2.93255
$695$	0	0
$696$	0	0
$697$	19.3174	0.731697
$698$	−37.7835	−1.43013
$699$	0	0
$700$	0	0
$701$	−23.8328	−0.900154	−0.450077	−	0.892990i	$-0.648604\pi$
$701$	−23.8328	−0.900154	−0.450077	+	0.892990i	$0.648604\pi$
$702$	0	0
$703$	7.03640	0.265383
$704$	23.2735	0.877153
$705$	0	0
$706$	24.2894	0.914145
$707$	−38.7000	−1.45546
$708$	0	0
$709$	−4.34220	−0.163075	−0.0815373	−	0.996670i	$-0.525983\pi$
$709$	−4.34220	−0.163075	−0.0815373	+	0.996670i	$0.525983\pi$
$710$	0	0
$711$	0	0
$712$	4.03640	0.151270
$713$	5.14079	0.192524
$714$	0	0
$715$	0	0
$716$	−42.3278	−1.58187
$717$	0	0
$718$	52.0813	1.94366
$719$	44.7301	1.66815	0.834076	−	0.551650i	$-0.186001\pi$
$719$	44.7301	1.66815	0.834076	+	0.551650i	$0.186001\pi$
$720$	0	0
$721$	33.1571	1.23484
$722$	2.34908	0.0874239
$723$	0	0
$724$	77.6565	2.88608
$725$	0	0
$726$	0	0
$727$	−29.4548	−1.09242	−0.546208	−	0.837649i	$-0.683929\pi$
$727$	−29.4548	−1.09242	−0.546208	+	0.837649i	$0.683929\pi$
$728$	34.6618	1.28465
$729$	0	0
$730$	0	0
$731$	−9.05434	−0.334887
$732$	0	0
$733$	−50.6565	−1.87104	−0.935520	−	0.353274i	$-0.885068\pi$
$733$	−50.6565	−1.87104	−0.935520	+	0.353274i	$0.885068\pi$
$734$	−21.3399	−0.787670
$735$	0	0
$736$	17.3935	0.641133
$737$	4.91553	0.181066
$738$	0	0
$739$	−11.9208	−0.438513	−0.219256	−	0.975667i	$-0.570363\pi$
$739$	−11.9208	−0.438513	−0.219256	+	0.975667i	$0.570363\pi$
$740$	0	0
$741$	0	0
$742$	12.9199	0.474304
$743$	−39.2181	−1.43877	−0.719387	−	0.694610i	$-0.755577\pi$
$743$	−39.2181	−1.43877	−0.719387	+	0.694610i	$0.755577\pi$
$744$	0	0
$745$	0	0
$746$	−14.1529	−0.518174
$747$	0	0
$748$	−16.5619	−0.605563
$749$	−42.5963	−1.55644
$750$	0	0
$751$	−6.63470	−0.242104	−0.121052	−	0.992646i	$-0.538627\pi$
$751$	−6.63470	−0.242104	−0.121052	+	0.992646i	$0.538627\pi$
$752$	−17.7907	−0.648760
$753$	0	0
$754$	38.8282	1.41404
$755$	0	0
$756$	0	0
$757$	−5.37685	−0.195425	−0.0977125	−	0.995215i	$-0.531153\pi$
$757$	−5.37685	−0.195425	−0.0977125	+	0.995215i	$0.531153\pi$
$758$	79.0035	2.86954
$759$	0	0
$760$	0	0
$761$	35.4928	1.28661	0.643306	−	0.765609i	$-0.277562\pi$
$761$	35.4928	1.28661	0.643306	+	0.765609i	$0.277562\pi$
$762$	0	0
$763$	−88.0582	−3.18792
$764$	47.8247	1.73024
$765$	0	0
$766$	70.0539	2.53115
$767$	−6.47277	−0.233718
$768$	0	0
$769$	33.4496	1.20623	0.603113	−	0.797656i	$-0.293927\pi$
$769$	33.4496	1.20623	0.603113	+	0.797656i	$0.293927\pi$
$770$	0	0
$771$	0	0
$772$	−73.2779	−2.63733
$773$	18.2505	0.656425	0.328212	−	0.944604i	$-0.393554\pi$
$773$	18.2505	0.656425	0.328212	+	0.944604i	$0.393554\pi$
$774$	0	0
$775$	0	0
$776$	58.4946	2.09984
$777$	0	0
$778$	6.12074	0.219439
$779$	7.93456	0.284285
$780$	0	0
$781$	12.5721	0.449866
$782$	−24.9818	−0.893348
$783$	0	0
$784$	22.2861	0.795931
$785$	0	0
$786$	0	0
$787$	32.0282	1.14168	0.570841	−	0.821060i	$-0.306617\pi$
$787$	32.0282	1.14168	0.570841	+	0.821060i	$0.306617\pi$
$788$	83.8487	2.98699
$789$	0	0
$790$	0	0
$791$	89.7199	3.19007
$792$	0	0
$793$	8.73060	0.310033
$794$	54.6154	1.93823
$795$	0	0
$796$	44.2539	1.56854
$797$	31.5380	1.11713	0.558567	−	0.829459i	$-0.311351\pi$
$797$	31.5380	1.11713	0.558567	+	0.829459i	$0.311351\pi$
$798$	0	0
$799$	−32.2912	−1.14238
$800$	0	0
$801$	0	0
$802$	−44.1738	−1.55983
$803$	5.25753	0.185534
$804$	0	0
$805$	0	0
$806$	−5.52915	−0.194756
$807$	0	0
$808$	−28.4017	−0.999168
$809$	33.4002	1.17429	0.587144	−	0.809482i	$-0.300252\pi$
$809$	33.4002	1.17429	0.587144	+	0.809482i	$0.300252\pi$
$810$	0	0
$811$	−0.823126	−0.0289039	−0.0144519	−	0.999896i	$-0.504600\pi$
$811$	−0.823126	−0.0289039	−0.0144519	+	0.999896i	$0.504600\pi$
$812$	141.297	4.95855
$813$	0	0
$814$	−31.9605	−1.12022
$815$	0	0
$816$	0	0
$817$	−3.71905	−0.130113
$818$	−35.4928	−1.24098
$819$	0	0
$820$	0	0
$821$	52.9415	1.84767	0.923835	−	0.382791i	$-0.125037\pi$
$821$	52.9415	1.84767	0.923835	+	0.382791i	$0.125037\pi$
$822$	0	0
$823$	49.6003	1.72896	0.864480	−	0.502668i	$-0.167648\pi$
$823$	49.6003	1.72896	0.864480	+	0.502668i	$0.167648\pi$
$824$	24.3338	0.847710
$825$	0	0
$826$	−36.9447	−1.28547
$827$	45.0017	1.56486	0.782432	−	0.622736i	$-0.213979\pi$
$827$	45.0017	1.56486	0.782432	+	0.622736i	$0.213979\pi$
$828$	0	0
$829$	20.2184	0.702214	0.351107	−	0.936335i	$-0.385805\pi$
$829$	20.2184	0.702214	0.351107	+	0.936335i	$0.385805\pi$
$830$	0	0
$831$	0	0
$832$	−24.0728	−0.834574
$833$	40.4506	1.40153
$834$	0	0
$835$	0	0
$836$	−6.80276	−0.235278
$837$	0	0
$838$	28.1935	0.973930
$839$	−18.7635	−0.647788	−0.323894	−	0.946093i	$-0.604992\pi$
$839$	−18.7635	−0.647788	−0.323894	+	0.946093i	$0.604992\pi$
$840$	0	0
$841$	39.3027	1.35527
$842$	−56.7200	−1.95470
$843$	0	0
$844$	53.8201	1.85256
$845$	0	0
$846$	0	0
$847$	−35.2861	−1.21244
$848$	1.51810	0.0521318
$849$	0	0
$850$	0	0
$851$	−30.7362	−1.05362
$852$	0	0
$853$	−0.907472	−0.0310712	−0.0155356	−	0.999879i	$-0.504945\pi$
$853$	−0.907472	−0.0310712	−0.0155356	+	0.999879i	$0.504945\pi$
$854$	49.8318	1.70521
$855$	0	0
$856$	−31.2612	−1.06849
$857$	42.2543	1.44338	0.721690	−	0.692216i	$-0.243365\pi$
$857$	42.2543	1.44338	0.721690	+	0.692216i	$0.243365\pi$
$858$	0	0
$859$	−0.0894585	−0.00305229	−0.00152614	−	0.999999i	$-0.500486\pi$
$859$	−0.0894585	−0.00305229	−0.00152614	+	0.999999i	$0.500486\pi$
$860$	0	0
$861$	0	0
$862$	−41.8604	−1.42577
$863$	5.84197	0.198863	0.0994315	−	0.995044i	$-0.468298\pi$
$863$	5.84197	0.198863	0.0994315	+	0.995044i	$0.468298\pi$
$864$	0	0
$865$	0	0
$866$	−21.3399	−0.725159
$867$	0	0
$868$	−20.1207	−0.682943
$869$	−15.8811	−0.538730
$870$	0	0
$871$	−5.08435	−0.172276
$872$	−64.6254	−2.18849
$873$	0	0
$874$	−10.2612	−0.347091
$875$	0	0
$876$	0	0
$877$	16.6578	0.562494	0.281247	−	0.959635i	$-0.409252\pi$
$877$	16.6578	0.562494	0.281247	+	0.959635i	$0.409252\pi$
$878$	37.2554	1.25731
$879$	0	0
$880$	0	0
$881$	−25.9495	−0.874259	−0.437130	−	0.899399i	$-0.644005\pi$
$881$	−25.9495	−0.874259	−0.437130	+	0.899399i	$0.644005\pi$
$882$	0	0
$883$	31.5276	1.06099	0.530493	−	0.847689i	$-0.322007\pi$
$883$	31.5276	1.06099	0.530493	+	0.847689i	$0.322007\pi$
$884$	17.1307	0.576168
$885$	0	0
$886$	−40.4068	−1.35749
$887$	−7.39237	−0.248211	−0.124106	−	0.992269i	$-0.539606\pi$
$887$	−7.39237	−0.248211	−0.124106	+	0.992269i	$0.539606\pi$
$888$	0	0
$889$	−36.8282	−1.23518
$890$	0	0
$891$	0	0
$892$	−72.6150	−2.43133
$893$	−13.2635	−0.443847
$894$	0	0
$895$	0	0
$896$	−98.7007	−3.29736
$897$	0	0
$898$	−2.65868	−0.0887211
$899$	−9.72633	−0.324391
$900$	0	0
$901$	2.75544	0.0917972
$902$	−36.0401	−1.20000
$903$	0	0
$904$	65.8449	2.18997
$905$	0	0
$906$	0	0
$907$	−0.378595	−0.0125710	−0.00628552	−	0.999980i	$-0.502001\pi$
$907$	−0.378595	−0.0125710	−0.00628552	+	0.999980i	$0.502001\pi$
$908$	−10.1830	−0.337935
$909$	0	0
$910$	0	0
$911$	−18.5039	−0.613061	−0.306530	−	0.951861i	$-0.599168\pi$
$911$	−18.5039	−0.613061	−0.306530	+	0.951861i	$0.599168\pi$
$912$	0	0
$913$	13.1538	0.435326
$914$	69.5681	2.30111
$915$	0	0
$916$	−45.3704	−1.49908
$917$	−84.8507	−2.80202
$918$	0	0
$919$	−7.49422	−0.247212	−0.123606	−	0.992331i	$-0.539446\pi$
$919$	−7.49422	−0.247212	−0.123606	+	0.992331i	$0.539446\pi$
$920$	0	0
$921$	0	0
$922$	−87.7524	−2.88997
$923$	−13.0039	−0.428029
$924$	0	0
$925$	0	0
$926$	62.2492	2.04564
$927$	0	0
$928$	−32.9083	−1.08027
$929$	55.5470	1.82244	0.911220	−	0.411921i	$-0.135142\pi$
$929$	55.5470	1.82244	0.911220	+	0.411921i	$0.135142\pi$
$930$	0	0
$931$	16.6150	0.544534
$932$	41.0219	1.34372
$933$	0	0
$934$	81.6598	2.67199
$935$	0	0
$936$	0	0
$937$	21.7606	0.710886	0.355443	−	0.934698i	$-0.384330\pi$
$937$	21.7606	0.710886	0.355443	+	0.934698i	$0.384330\pi$
$938$	−29.0200	−0.947537
$939$	0	0
$940$	0	0
$941$	36.2774	1.18261	0.591305	−	0.806448i	$-0.298613\pi$
$941$	36.2774	1.18261	0.591305	+	0.806448i	$0.298613\pi$
$942$	0	0
$943$	−34.6595	−1.12867
$944$	−4.34104	−0.141289
$945$	0	0
$946$	16.8926	0.549224
$947$	44.6597	1.45125	0.725623	−	0.688092i	$-0.241552\pi$
$947$	44.6597	1.45125	0.725623	+	0.688092i	$0.241552\pi$
$948$	0	0
$949$	−5.43809	−0.176528
$950$	0	0
$951$	0	0
$952$	42.1935	1.36750
$953$	−6.26058	−0.202800	−0.101400	−	0.994846i	$-0.532332\pi$
$953$	−6.26058	−0.202800	−0.101400	+	0.994846i	$0.532332\pi$
$954$	0	0
$955$	0	0
$956$	−78.2069	−2.52939
$957$	0	0
$958$	−54.4068	−1.75780
$959$	−46.4927	−1.50133
$960$	0	0
$961$	−29.6150	−0.955322
$962$	33.0582	1.06584
$963$	0	0
$964$	4.63470	0.149274
$965$	0	0
$966$	0	0
$967$	22.2976	0.717043	0.358522	−	0.933521i	$-0.383281\pi$
$967$	22.2976	0.717043	0.358522	+	0.933521i	$0.383281\pi$
$968$	−25.8962	−0.832337
$969$	0	0
$970$	0	0
$971$	24.6106	0.789793	0.394896	−	0.918726i	$-0.370781\pi$
$971$	24.6106	0.789793	0.394896	+	0.918726i	$0.370781\pi$
$972$	0	0
$973$	−7.26122	−0.232784
$974$	22.3419	0.715880
$975$	0	0
$976$	5.85528	0.187423
$977$	−5.78877	−0.185199	−0.0925995	−	0.995703i	$-0.529518\pi$
$977$	−5.78877	−0.185199	−0.0925995	+	0.995703i	$0.529518\pi$
$978$	0	0
$979$	−2.18842	−0.0699424
$980$	0	0
$981$	0	0
$982$	−85.7293	−2.73573
$983$	34.9214	1.11382	0.556910	−	0.830573i	$-0.311987\pi$
$983$	34.9214	1.11382	0.556910	+	0.830573i	$0.311987\pi$
$984$	0	0
$985$	0	0
$986$	47.2653	1.50523
$987$	0	0
$988$	7.03640	0.223858
$989$	16.2455	0.516576
$990$	0	0
$991$	−23.8844	−0.758712	−0.379356	−	0.925251i	$-0.623854\pi$
$991$	−23.8844	−0.758712	−0.379356	+	0.925251i	$0.623854\pi$
$992$	4.68616	0.148786
$993$	0	0
$994$	−74.2226	−2.35420
$995$	0	0
$996$	0	0
$997$	−37.4681	−1.18663	−0.593313	−	0.804972i	$-0.702180\pi$
$997$	−37.4681	−1.18663	−0.593313	+	0.804972i	$0.702180\pi$
$998$	95.3225	3.01738
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4275.2.a.bu.1.6	yes	6
3.2	odd	2	inner	4275.2.a.bu.1.1	yes	6
5.4	even	2		4275.2.a.bq.1.1	✓	6
15.14	odd	2		4275.2.a.bq.1.6	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4275.2.a.bq.1.1	✓	6	5.4	even	2
4275.2.a.bq.1.6	yes	6	15.14	odd	2
4275.2.a.bu.1.1	yes	6	3.2	odd	2	inner
4275.2.a.bu.1.6	yes	6	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 4275 = 3^{2} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4275.a (trivial)

\(f(q)\)	\(=\)	\(q+2.34908 q^{2} +3.51820 q^{4} +4.85952 q^{7} +3.56638 q^{8} -1.93359 q^{11} +2.00000 q^{13} +11.4154 q^{14} +1.34132 q^{16} +2.43459 q^{17} +1.00000 q^{19} -4.54217 q^{22} -4.36818 q^{23} +4.69817 q^{26} +17.0968 q^{28} +8.26455 q^{29} -1.17687 q^{31} -3.98187 q^{32} +5.71905 q^{34} +7.03640 q^{37} +2.34908 q^{38} +7.93456 q^{41} -3.71905 q^{43} -6.80276 q^{44} -10.2612 q^{46} -13.2635 q^{47} +16.6150 q^{49} +7.03640 q^{52} +1.13179 q^{53} +17.3309 q^{56} +19.4141 q^{58} -3.23639 q^{59} +4.36530 q^{61} -2.76458 q^{62} -12.0364 q^{64} -2.54217 q^{67} +8.56535 q^{68} -6.50196 q^{71} -2.71905 q^{73} +16.5291 q^{74} +3.51820 q^{76} -9.39634 q^{77} +8.21327 q^{79} +18.6389 q^{82} -6.80276 q^{83} -8.73636 q^{86} -6.89592 q^{88} +1.13179 q^{89} +9.71905 q^{91} -15.3681 q^{92} -31.1571 q^{94} +16.4017 q^{97} +39.0300 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 8 q^{4} + 8 q^{7} + 12 q^{13} + 6 q^{19} + 10 q^{22} + 26 q^{28} - 2 q^{31} - 8 q^{34} + 16 q^{37} + 20 q^{43} + 18 q^{46} + 10 q^{49} + 16 q^{52} + 56 q^{58} - 6 q^{61} - 46 q^{64} + 22 q^{67} + 26 q^{73}+ \cdots + 40 q^{97}+O(q^{100}) \) 6 * q + 8 * q^4 + 8 * q^7 + 12 * q^13 + 6 * q^19 + 10 * q^22 + 26 * q^28 - 2 * q^31 - 8 * q^34 + 16 * q^37 + 20 * q^43 + 18 * q^46 + 10 * q^49 + 16 * q^52 + 56 * q^58 - 6 * q^61 - 46 * q^64 + 22 * q^67 + 26 * q^73 + 8 * q^76 + 18 * q^79 - 2 * q^82 + 6 * q^88 + 16 * q^91 - 60 * q^94 + 40 * q^97

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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