LMFDB - Embedded newform 4275.2.a.bq.1.4

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:	$1$
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			1.4
Root			$-1.92914$ 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)

$f(q)$	$=$	$q+0.551006 q^{2} -1.69639 q^{4} -0.574141 q^{7} -2.03673 q^{8} +3.83882 q^{11} -2.00000 q^{13} -0.316355 q^{14} +2.27053 q^{16} -5.17548 q^{17} +1.00000 q^{19} +2.11521 q^{22} +1.33666 q^{23} -1.10201 q^{26} +0.973969 q^{28} +0.934722 q^{29} +4.96693 q^{31} +5.32454 q^{32} -2.85172 q^{34} +3.39279 q^{37} +0.551006 q^{38} +3.37340 q^{41} -4.85172 q^{43} -6.51214 q^{44} +0.736508 q^{46} -9.88165 q^{47} -6.67036 q^{49} +3.39279 q^{52} +3.13874 q^{53} +1.16937 q^{56} +0.515037 q^{58} -4.47541 q^{59} +8.08214 q^{61} +2.73680 q^{62} -1.60721 q^{64} +0.115209 q^{67} +8.77964 q^{68} -16.2265 q^{71} -5.85172 q^{73} +1.86944 q^{74} -1.69639 q^{76} -2.20402 q^{77} -8.35971 q^{79} +1.85876 q^{82} +6.51214 q^{83} -2.67332 q^{86} -7.81864 q^{88} -3.13874 q^{89} +1.14828 q^{91} -2.26750 q^{92} -5.44485 q^{94} -9.68935 q^{97} -3.67541 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.551006	0.389620	0.194810	−	0.980841i	$-0.437591\pi$
$2$	0.551006	0.389620	0.194810	+	0.980841i	$0.437591\pi$
$3$	0	0
$3$	0	0
$4$	−1.69639	−0.848196
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−0.574141	−0.217005	−0.108502	−	0.994096i	$-0.534606\pi$
$7$	−0.574141	−0.217005	−0.108502	+	0.994096i	$0.534606\pi$
$8$	−2.03673	−0.720094
$9$	0	0
$10$	0	0
$11$	3.83882	1.15745	0.578723	−	0.815524i	$-0.303551\pi$
$11$	3.83882	1.15745	0.578723	+	0.815524i	$0.303551\pi$
$12$	0	0
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	−0.316355	−0.0845494
$15$	0	0
$16$	2.27053	0.567634
$17$	−5.17548	−1.25524	−0.627619	−	0.778521i	$-0.715970\pi$
$17$	−5.17548	−1.25524	−0.627619	+	0.778521i	$0.715970\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	2.11521	0.450964
$23$	1.33666	0.278713	0.139357	−	0.990242i	$-0.455497\pi$
$23$	1.33666	0.278713	0.139357	+	0.990242i	$0.455497\pi$
$24$	0	0
$25$	0	0
$26$	−1.10201	−0.216122
$27$	0	0
$28$	0.973969	0.184063
$29$	0.934722	0.173574	0.0867868	−	0.996227i	$-0.472340\pi$
$29$	0.934722	0.173574	0.0867868	+	0.996227i	$0.472340\pi$
$30$	0	0
$31$	4.96693	0.892086	0.446043	−	0.895011i	$-0.352833\pi$
$31$	4.96693	0.892086	0.446043	+	0.895011i	$0.352833\pi$
$32$	5.32454	0.941255
$33$	0	0
$34$	−2.85172	−0.489066
$35$	0	0
$36$	0	0
$37$	3.39279	0.557771	0.278885	−	0.960324i	$-0.410035\pi$
$37$	3.39279	0.557771	0.278885	+	0.960324i	$0.410035\pi$
$38$	0.551006	0.0893849
$39$	0	0
$40$	0	0
$41$	3.37340	0.526836	0.263418	−	0.964682i	$-0.415150\pi$
$41$	3.37340	0.526836	0.263418	+	0.964682i	$0.415150\pi$
$42$	0	0
$43$	−4.85172	−0.739880	−0.369940	−	0.929056i	$-0.620622\pi$
$43$	−4.85172	−0.739880	−0.369940	+	0.929056i	$0.620622\pi$
$44$	−6.51214	−0.981742
$45$	0	0
$46$	0.736508	0.108592
$47$	−9.88165	−1.44139	−0.720694	−	0.693254i	$-0.756177\pi$
$47$	−9.88165	−1.44139	−0.720694	+	0.693254i	$0.756177\pi$
$48$	0	0
$49$	−6.67036	−0.952909
$50$	0	0
$51$	0	0
$52$	3.39279	0.470495
$53$	3.13874	0.431140	0.215570	−	0.976488i	$-0.430839\pi$
$53$	3.13874	0.431140	0.215570	+	0.976488i	$0.430839\pi$
$54$	0	0
$55$	0	0
$56$	1.16937	0.156264
$57$	0	0
$58$	0.515037	0.0676277
$59$	−4.47541	−0.582648	−0.291324	−	0.956624i	$-0.594096\pi$
$59$	−4.47541	−0.582648	−0.291324	+	0.956624i	$0.594096\pi$
$60$	0	0
$61$	8.08214	1.03481	0.517406	−	0.855740i	$-0.326898\pi$
$61$	8.08214	1.03481	0.517406	+	0.855740i	$0.326898\pi$
$62$	2.73680	0.347575
$63$	0	0
$64$	−1.60721	−0.200902
$65$	0	0
$66$	0	0
$67$	0.115209	0.0140751	0.00703753	−	0.999975i	$-0.497760\pi$
$67$	0.115209	0.0140751	0.00703753	+	0.999975i	$0.497760\pi$
$68$	8.77964	1.06469
$69$	0	0
$70$	0	0
$71$	−16.2265	−1.92573	−0.962866	−	0.269980i	$-0.912983\pi$
$71$	−16.2265	−1.92573	−0.962866	+	0.269980i	$0.912983\pi$
$72$	0	0
$73$	−5.85172	−0.684892	−0.342446	−	0.939538i	$-0.611255\pi$
$73$	−5.85172	−0.684892	−0.342446	+	0.939538i	$0.611255\pi$
$74$	1.86944	0.217318
$75$	0	0
$76$	−1.69639	−0.194590
$77$	−2.20402	−0.251172
$78$	0	0
$79$	−8.35971	−0.940541	−0.470271	−	0.882522i	$-0.655844\pi$
$79$	−8.35971	−0.940541	−0.470271	+	0.882522i	$0.655844\pi$
$80$	0	0
$81$	0	0
$82$	1.85876	0.205266
$83$	6.51214	0.714800	0.357400	−	0.933951i	$-0.383663\pi$
$83$	6.51214	0.714800	0.357400	+	0.933951i	$0.383663\pi$
$84$	0	0
$85$	0	0
$86$	−2.67332	−0.288272
$87$	0	0
$88$	−7.81864	−0.833470
$89$	−3.13874	−0.332706	−0.166353	−	0.986066i	$-0.553199\pi$
$89$	−3.13874	−0.332706	−0.166353	+	0.986066i	$0.553199\pi$
$90$	0	0
$91$	1.14828	0.120373
$92$	−2.26750	−0.236404
$93$	0	0
$94$	−5.44485	−0.561593
$95$	0	0
$96$	0	0
$97$	−9.68935	−0.983804	−0.491902	−	0.870650i	$-0.663698\pi$
$97$	−9.68935	−0.983804	−0.491902	+	0.870650i	$0.663698\pi$
$98$	−3.67541	−0.371272
$99$	0	0
$100$	0	0
$101$	−10.6491	−1.05962	−0.529812	−	0.848115i	$-0.677737\pi$
$101$	−10.6491	−1.05962	−0.529812	+	0.848115i	$0.677737\pi$
$102$	0	0
$103$	−12.9669	−1.27767	−0.638835	−	0.769344i	$-0.720583\pi$
$103$	−12.9669	−1.27767	−0.638835	+	0.769344i	$0.720583\pi$
$104$	4.07347	0.399436
$105$	0	0
$106$	1.72947	0.167981
$107$	9.94902	0.961808	0.480904	−	0.876773i	$-0.340309\pi$
$107$	9.94902	0.961808	0.480904	+	0.876773i	$0.340309\pi$
$108$	0	0
$109$	−2.83763	−0.271796	−0.135898	−	0.990723i	$-0.543392\pi$
$109$	−2.83763	−0.271796	−0.135898	+	0.990723i	$0.543392\pi$
$110$	0	0
$111$	0	0
$112$	−1.30361	−0.123179
$113$	1.96937	0.185263	0.0926315	−	0.995700i	$-0.470472\pi$
$113$	1.96937	0.185263	0.0926315	+	0.995700i	$0.470472\pi$
$114$	0	0
$115$	0	0
$116$	−1.58566	−0.147224
$117$	0	0
$118$	−2.46597	−0.227011
$119$	2.97146	0.272393
$120$	0	0
$121$	3.73651	0.339683
$122$	4.45330	0.403183
$123$	0	0
$124$	−8.42586	−0.756664
$125$	0	0
$126$	0	0
$127$	−5.27758	−0.468309	−0.234155	−	0.972199i	$-0.575232\pi$
$127$	−5.27758	−0.468309	−0.234155	+	0.972199i	$0.575232\pi$
$128$	−11.5347	−1.01953
$129$	0	0
$130$	0	0
$131$	19.9980	1.74723	0.873615	−	0.486618i	$-0.161770\pi$
$131$	19.9980	1.74723	0.873615	+	0.486618i	$0.161770\pi$
$132$	0	0
$133$	−0.574141	−0.0497844
$134$	0.0634810	0.00548392
$135$	0	0
$136$	10.5411	0.903889
$137$	9.24894	0.790191	0.395095	−	0.918640i	$-0.370712\pi$
$137$	9.24894	0.790191	0.395095	+	0.918640i	$0.370712\pi$
$138$	0	0
$139$	6.50799	0.552001	0.276000	−	0.961158i	$-0.410991\pi$
$139$	6.50799	0.552001	0.276000	+	0.961158i	$0.410991\pi$
$140$	0	0
$141$	0	0
$142$	−8.94090	−0.750303
$143$	−7.67763	−0.642036
$144$	0	0
$145$	0	0
$146$	−3.22433	−0.266847
$147$	0	0
$148$	−5.75550	−0.473099
$149$	8.14693	0.667423	0.333711	−	0.942675i	$-0.391699\pi$
$149$	8.14693	0.667423	0.333711	+	0.942675i	$0.391699\pi$
$150$	0	0
$151$	4.24450	0.345413	0.172706	−	0.984973i	$-0.444749\pi$
$151$	4.24450	0.345413	0.172706	+	0.984973i	$0.444749\pi$
$152$	−2.03673	−0.165201
$153$	0	0
$154$	−1.21443	−0.0978615
$155$	0	0
$156$	0	0
$157$	−13.1152	−1.04671	−0.523354	−	0.852115i	$-0.675319\pi$
$157$	−13.1152	−1.04671	−0.523354	+	0.852115i	$0.675319\pi$
$158$	−4.60625	−0.366453
$159$	0	0
$160$	0	0
$161$	−0.767433	−0.0604822
$162$	0	0
$163$	−9.65628	−0.756338	−0.378169	−	0.925737i	$-0.623446\pi$
$163$	−9.65628	−0.756338	−0.378169	+	0.925737i	$0.623446\pi$
$164$	−5.72260	−0.446860
$165$	0	0
$166$	3.58823	0.278500
$167$	−8.07957	−0.625216	−0.312608	−	0.949882i	$-0.601203\pi$
$167$	−8.07957	−0.625216	−0.312608	+	0.949882i	$0.601203\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	0	0
$172$	8.23042	0.627564
$173$	−3.60805	−0.274315	−0.137157	−	0.990549i	$-0.543797\pi$
$173$	−3.60805	−0.274315	−0.137157	+	0.990549i	$0.543797\pi$
$174$	0	0
$175$	0	0
$176$	8.71616	0.657005
$177$	0	0
$178$	−1.72947	−0.129629
$179$	−21.7001	−1.62194	−0.810971	−	0.585086i	$-0.801061\pi$
$179$	−21.7001	−1.62194	−0.810971	+	0.585086i	$0.801061\pi$
$180$	0	0
$181$	1.21443	0.0902678	0.0451339	−	0.998981i	$-0.485629\pi$
$181$	1.21443	0.0902678	0.0451339	+	0.998981i	$0.485629\pi$
$182$	0.632710	0.0468996
$183$	0	0
$184$	−2.72242	−0.200700
$185$	0	0
$186$	0	0
$187$	−19.8677	−1.45287
$188$	16.7632	1.22258
$189$	0	0
$190$	0	0
$191$	−12.3203	−0.891468	−0.445734	−	0.895166i	$-0.647057\pi$
$191$	−12.3203	−0.891468	−0.445734	+	0.895166i	$0.647057\pi$
$192$	0	0
$193$	−19.0301	−1.36981	−0.684907	−	0.728630i	$-0.740157\pi$
$193$	−19.0301	−1.36981	−0.684907	+	0.728630i	$0.740157\pi$
$194$	−5.33889	−0.383310
$195$	0	0
$196$	11.3156	0.808254
$197$	−17.3959	−1.23940	−0.619702	−	0.784837i	$-0.712747\pi$
$197$	−17.3959	−1.23940	−0.619702	+	0.784837i	$0.712747\pi$
$198$	0	0
$199$	−0.277576	−0.0196769	−0.00983843	−	0.999952i	$-0.503132\pi$
$199$	−0.277576	−0.0196769	−0.00983843	+	0.999952i	$0.503132\pi$
$200$	0	0
$201$	0	0
$202$	−5.86771	−0.412850
$203$	−0.536662	−0.0376663
$204$	0	0
$205$	0	0
$206$	−7.14485	−0.497805
$207$	0	0
$208$	−4.54107	−0.314866
$209$	3.83882	0.265536
$210$	0	0
$211$	−6.12929	−0.421958	−0.210979	−	0.977491i	$-0.567665\pi$
$211$	−6.12929	−0.421958	−0.210979	+	0.977491i	$0.567665\pi$
$212$	−5.32454	−0.365691
$213$	0	0
$214$	5.48196	0.374739
$215$	0	0
$216$	0	0
$217$	−2.85172	−0.193587
$218$	−1.56355	−0.105897
$219$	0	0
$220$	0	0
$221$	10.3510	0.696281
$222$	0	0
$223$	−29.0791	−1.94728	−0.973642	−	0.228084i	$-0.926754\pi$
$223$	−29.0791	−1.94728	−0.973642	+	0.228084i	$0.926754\pi$
$224$	−3.05704	−0.204257
$225$	0	0
$226$	1.08514	0.0721821
$227$	−18.4305	−1.22328	−0.611639	−	0.791137i	$-0.709489\pi$
$227$	−18.4305	−1.22328	−0.611639	+	0.791137i	$0.709489\pi$
$228$	0	0
$229$	1.81864	0.120179	0.0600897	−	0.998193i	$-0.480861\pi$
$229$	1.81864	0.120179	0.0600897	+	0.998193i	$0.480861\pi$
$230$	0	0
$231$	0	0
$232$	−1.90378	−0.124989
$233$	8.48151	0.555642	0.277821	−	0.960633i	$-0.410388\pi$
$233$	8.48151	0.555642	0.277821	+	0.960633i	$0.410388\pi$
$234$	0	0
$235$	0	0
$236$	7.59205	0.494200
$237$	0	0
$238$	1.63729	0.106130
$239$	−18.7960	−1.21581	−0.607907	−	0.794008i	$-0.707991\pi$
$239$	−18.7960	−1.21581	−0.607907	+	0.794008i	$0.707991\pi$
$240$	0	0
$241$	−0.541068	−0.0348533	−0.0174266	−	0.999848i	$-0.505547\pi$
$241$	−0.541068	−0.0348533	−0.0174266	+	0.999848i	$0.505547\pi$
$242$	2.05884	0.132347
$243$	0	0
$244$	−13.7105	−0.877723
$245$	0	0
$246$	0	0
$247$	−2.00000	−0.127257
$248$	−10.1163	−0.642386
$249$	0	0
$250$	0	0
$251$	16.4573	1.03877	0.519387	−	0.854539i	$-0.326160\pi$
$251$	16.4573	1.03877	0.519387	+	0.854539i	$0.326160\pi$
$252$	0	0
$253$	5.13120	0.322596
$254$	−2.90797	−0.182463
$255$	0	0
$256$	−3.14124	−0.196328
$257$	27.1102	1.69109	0.845545	−	0.533904i	$-0.179276\pi$
$257$	27.1102	1.69109	0.845545	+	0.533904i	$0.179276\pi$
$258$	0	0
$259$	−1.94794	−0.121039
$260$	0	0
$261$	0	0
$262$	11.0190	0.680755
$263$	−15.2918	−0.942932	−0.471466	−	0.881884i	$-0.656275\pi$
$263$	−15.2918	−0.942932	−0.471466	+	0.881884i	$0.656275\pi$
$264$	0	0
$265$	0	0
$266$	−0.316355	−0.0193970
$267$	0	0
$268$	−0.195440	−0.0119384
$269$	26.5101	1.61635	0.808175	−	0.588943i	$-0.200456\pi$
$269$	26.5101	1.61635	0.808175	+	0.588943i	$0.200456\pi$
$270$	0	0
$271$	−5.65628	−0.343594	−0.171797	−	0.985132i	$-0.554957\pi$
$271$	−5.65628	−0.343594	−0.171797	+	0.985132i	$0.554957\pi$
$272$	−11.7511	−0.712515
$273$	0	0
$274$	5.09622	0.307874
$275$	0	0
$276$	0	0
$277$	−24.6373	−1.48031	−0.740156	−	0.672436i	$-0.765248\pi$
$277$	−24.6373	−1.48031	−0.740156	+	0.672436i	$0.765248\pi$
$278$	3.58594	0.215071
$279$	0	0
$280$	0	0
$281$	−9.64700	−0.575492	−0.287746	−	0.957707i	$-0.592906\pi$
$281$	−9.64700	−0.575492	−0.287746	+	0.957707i	$0.592906\pi$
$282$	0	0
$283$	5.93385	0.352731	0.176365	−	0.984325i	$-0.443566\pi$
$283$	5.93385	0.352731	0.176365	+	0.984325i	$0.443566\pi$
$284$	27.5265	1.63340
$285$	0	0
$286$	−4.23042	−0.250150
$287$	−1.93681	−0.114326
$288$	0	0
$289$	9.78557	0.575622
$290$	0	0
$291$	0	0
$292$	9.92681	0.580923
$293$	26.4102	1.54290	0.771449	−	0.636291i	$-0.219532\pi$
$293$	26.4102	1.54290	0.771449	+	0.636291i	$0.219532\pi$
$294$	0	0
$295$	0	0
$296$	−6.91020	−0.401647
$297$	0	0
$298$	4.48901	0.260041
$299$	−2.67332	−0.154602
$300$	0	0
$301$	2.78557	0.160558
$302$	2.33875	0.134580
$303$	0	0
$304$	2.27053	0.130224
$305$	0	0
$306$	0	0
$307$	−12.0491	−0.687676	−0.343838	−	0.939029i	$-0.611727\pi$
$307$	−12.0491	−0.687676	−0.343838	+	0.939029i	$0.611727\pi$
$308$	3.73889	0.213043
$309$	0	0
$310$	0	0
$311$	8.48151	0.480942	0.240471	−	0.970656i	$-0.422698\pi$
$311$	8.48151	0.480942	0.240471	+	0.970656i	$0.422698\pi$
$312$	0	0
$313$	−30.5381	−1.72611	−0.863057	−	0.505107i	$-0.831453\pi$
$313$	−30.5381	−1.72611	−0.863057	+	0.505107i	$0.831453\pi$
$314$	−7.22655	−0.407818
$315$	0	0
$316$	14.1814	0.797764
$317$	30.2451	1.69873	0.849367	−	0.527802i	$-0.176984\pi$
$317$	30.2451	1.69873	0.849367	+	0.527802i	$0.176984\pi$
$318$	0	0
$319$	3.58823	0.200902
$320$	0	0
$321$	0	0
$322$	−0.422860	−0.0235651
$323$	−5.17548	−0.287971
$324$	0	0
$325$	0	0
$326$	−5.32066	−0.294684
$327$	0	0
$328$	−6.87071	−0.379371
$329$	5.67347	0.312788
$330$	0	0
$331$	−14.9669	−0.822657	−0.411328	−	0.911487i	$-0.634935\pi$
$331$	−14.9669	−0.822657	−0.411328	+	0.911487i	$0.634935\pi$
$332$	−11.0471	−0.606291
$333$	0	0
$334$	−4.45189	−0.243597
$335$	0	0
$336$	0	0
$337$	−20.4890	−1.11611	−0.558054	−	0.829805i	$-0.688452\pi$
$337$	−20.4890	−1.11611	−0.558054	+	0.829805i	$0.688452\pi$
$338$	−4.95905	−0.269737
$339$	0	0
$340$	0	0
$341$	19.0671	1.03254
$342$	0	0
$343$	7.84872	0.423791
$344$	9.88165	0.532783
$345$	0	0
$346$	−1.98805	−0.106878
$347$	−7.71408	−0.414113	−0.207057	−	0.978329i	$-0.566388\pi$
$347$	−7.71408	−0.414113	−0.207057	+	0.978329i	$0.566388\pi$
$348$	0	0
$349$	−11.2304	−0.601151	−0.300575	−	0.953758i	$-0.597179\pi$
$349$	−11.2304	−0.601151	−0.300575	+	0.953758i	$0.597179\pi$
$350$	0	0
$351$	0	0
$352$	20.4399	1.08945
$353$	−1.27318	−0.0677646	−0.0338823	−	0.999426i	$-0.510787\pi$
$353$	−1.27318	−0.0677646	−0.0338823	+	0.999426i	$0.510787\pi$
$354$	0	0
$355$	0	0
$356$	5.32454	0.282200
$357$	0	0
$358$	−11.9569	−0.631941
$359$	4.24464	0.224023	0.112012	−	0.993707i	$-0.464271\pi$
$359$	4.24464	0.224023	0.112012	+	0.993707i	$0.464271\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0.669157	0.0351701
$363$	0	0
$364$	−1.94794	−0.102100
$365$	0	0
$366$	0	0
$367$	4.23042	0.220826	0.110413	−	0.993886i	$-0.464783\pi$
$367$	4.23042	0.220826	0.110413	+	0.993886i	$0.464783\pi$
$368$	3.03494	0.158207
$369$	0	0
$370$	0	0
$371$	−1.80208	−0.0935595
$372$	0	0
$373$	−20.4088	−1.05673	−0.528363	−	0.849018i	$-0.677194\pi$
$373$	−20.4088	−1.05673	−0.528363	+	0.849018i	$0.677194\pi$
$374$	−10.9472	−0.566067
$375$	0	0
$376$	20.1263	1.03793
$377$	−1.86944	−0.0962813
$378$	0	0
$379$	−19.6514	−1.00942	−0.504712	−	0.863288i	$-0.668401\pi$
$379$	−19.6514	−1.00942	−0.504712	+	0.863288i	$0.668401\pi$
$380$	0	0
$381$	0	0
$382$	−6.78857	−0.347333
$383$	0.736518	0.0376343	0.0188172	−	0.999823i	$-0.494010\pi$
$383$	0.736518	0.0376343	0.0188172	+	0.999823i	$0.494010\pi$
$384$	0	0
$385$	0	0
$386$	−10.4857	−0.533707
$387$	0	0
$388$	16.4369	0.834459
$389$	16.6284	0.843096	0.421548	−	0.906806i	$-0.361487\pi$
$389$	16.6284	0.843096	0.421548	+	0.906806i	$0.361487\pi$
$390$	0	0
$391$	−6.91786	−0.349851
$392$	13.5857	0.686184
$393$	0	0
$394$	−9.58523	−0.482897
$395$	0	0
$396$	0	0
$397$	3.75250	0.188332	0.0941662	−	0.995556i	$-0.469981\pi$
$397$	3.75250	0.188332	0.0941662	+	0.995556i	$0.469981\pi$
$398$	−0.152946	−0.00766650
$399$	0	0
$400$	0	0
$401$	−20.9366	−1.04552	−0.522761	−	0.852479i	$-0.675098\pi$
$401$	−20.9366	−1.04552	−0.522761	+	0.852479i	$0.675098\pi$
$402$	0	0
$403$	−9.93385	−0.494840
$404$	18.0650	0.898769
$405$	0	0
$406$	−0.295704	−0.0146755
$407$	13.0243	0.645590
$408$	0	0
$409$	16.1784	0.799968	0.399984	−	0.916522i	$-0.369016\pi$
$409$	16.1784	0.799968	0.399984	+	0.916522i	$0.369016\pi$
$410$	0	0
$411$	0	0
$412$	21.9970	1.08371
$413$	2.56952	0.126438
$414$	0	0
$415$	0	0
$416$	−10.6491	−0.522114
$417$	0	0
$418$	2.11521	0.103458
$419$	14.4244	0.704679	0.352340	−	0.935872i	$-0.385386\pi$
$419$	14.4244	0.704679	0.352340	+	0.935872i	$0.385386\pi$
$420$	0	0
$421$	17.5711	0.856365	0.428183	−	0.903692i	$-0.359154\pi$
$421$	17.5711	0.856365	0.428183	+	0.903692i	$0.359154\pi$
$422$	−3.37728	−0.164403
$423$	0	0
$424$	−6.39279	−0.310461
$425$	0	0
$426$	0	0
$427$	−4.64029	−0.224559
$428$	−16.8774	−0.815802
$429$	0	0
$430$	0	0
$431$	15.1609	0.730277	0.365138	−	0.930953i	$-0.381022\pi$
$431$	15.1609	0.730277	0.365138	+	0.930953i	$0.381022\pi$
$432$	0	0
$433$	4.23042	0.203301	0.101650	−	0.994820i	$-0.467588\pi$
$433$	4.23042	0.203301	0.101650	+	0.994820i	$0.467588\pi$
$434$	−1.57131	−0.0754254
$435$	0	0
$436$	4.81374	0.230536
$437$	1.33666	0.0639412
$438$	0	0
$439$	11.5741	0.552404	0.276202	−	0.961100i	$-0.410924\pi$
$439$	11.5741	0.552404	0.276202	+	0.961100i	$0.410924\pi$
$440$	0	0
$441$	0	0
$442$	5.70344	0.271285
$443$	22.3367	1.06125	0.530624	−	0.847607i	$-0.321957\pi$
$443$	22.3367	1.06125	0.530624	+	0.847607i	$0.321957\pi$
$444$	0	0
$445$	0	0
$446$	−16.0228	−0.758700
$447$	0	0
$448$	0.922768	0.0435967
$449$	3.13874	0.148127	0.0740633	−	0.997254i	$-0.476403\pi$
$449$	3.13874	0.148127	0.0740633	+	0.997254i	$0.476403\pi$
$450$	0	0
$451$	12.9498	0.609784
$452$	−3.34083	−0.157139
$453$	0	0
$454$	−10.1553	−0.476613
$455$	0	0
$456$	0	0
$457$	−6.32964	−0.296088	−0.148044	−	0.988981i	$-0.547298\pi$
$457$	−6.32964	−0.296088	−0.148044	+	0.988981i	$0.547298\pi$
$458$	1.00208	0.0468243
$459$	0	0
$460$	0	0
$461$	34.8204	1.62175	0.810875	−	0.585220i	$-0.198992\pi$
$461$	34.8204	1.62175	0.810875	+	0.585220i	$0.198992\pi$
$462$	0	0
$463$	27.5050	1.27827	0.639133	−	0.769096i	$-0.279293\pi$
$463$	27.5050	1.27827	0.639133	+	0.769096i	$0.279293\pi$
$464$	2.12232	0.0985261
$465$	0	0
$466$	4.67336	0.216489
$467$	26.1043	1.20796	0.603981	−	0.796999i	$-0.293580\pi$
$467$	26.1043	1.20796	0.603981	+	0.796999i	$0.293580\pi$
$468$	0	0
$469$	−0.0661464	−0.00305436
$470$	0	0
$471$	0	0
$472$	9.11521	0.419562
$473$	−18.6249	−0.856372
$474$	0	0
$475$	0	0
$476$	−5.04076	−0.231043
$477$	0	0
$478$	−10.3567	−0.473705
$479$	3.07138	0.140335	0.0701675	−	0.997535i	$-0.477647\pi$
$479$	3.07138	0.140335	0.0701675	+	0.997535i	$0.477647\pi$
$480$	0	0
$481$	−6.78557	−0.309395
$482$	−0.298132	−0.0135795
$483$	0	0
$484$	−6.33859	−0.288118
$485$	0	0
$486$	0	0
$487$	28.4890	1.29096	0.645480	−	0.763777i	$-0.276657\pi$
$487$	28.4890	1.29096	0.645480	+	0.763777i	$0.276657\pi$
$488$	−16.4612	−0.745162
$489$	0	0
$490$	0	0
$491$	−26.9430	−1.21592	−0.607959	−	0.793968i	$-0.708012\pi$
$491$	−26.9430	−1.21592	−0.607959	+	0.793968i	$0.708012\pi$
$492$	0	0
$493$	−4.83763	−0.217876
$494$	−1.10201	−0.0495818
$495$	0	0
$496$	11.2776	0.506378
$497$	9.31631	0.417893
$498$	0	0
$499$	27.7224	1.24103	0.620513	−	0.784196i	$-0.286924\pi$
$499$	27.7224	1.24103	0.620513	+	0.784196i	$0.286924\pi$
$500$	0	0
$501$	0	0
$502$	9.06805	0.404727
$503$	−7.67763	−0.342329	−0.171164	−	0.985242i	$-0.554753\pi$
$503$	−7.67763	−0.342329	−0.171164	+	0.985242i	$0.554753\pi$
$504$	0	0
$505$	0	0
$506$	2.82732	0.125690
$507$	0	0
$508$	8.95284	0.397218
$509$	−6.40833	−0.284044	−0.142022	−	0.989863i	$-0.545360\pi$
$509$	−6.40833	−0.284044	−0.142022	+	0.989863i	$0.545360\pi$
$510$	0	0
$511$	3.35971	0.148625
$512$	21.3385	0.943037
$513$	0	0
$514$	14.9379	0.658882
$515$	0	0
$516$	0	0
$517$	−37.9339	−1.66833
$518$	−1.07332	−0.0471592
$519$	0	0
$520$	0	0
$521$	13.3550	0.585092	0.292546	−	0.956251i	$-0.405498\pi$
$521$	13.3550	0.585092	0.292546	+	0.956251i	$0.405498\pi$
$522$	0	0
$523$	33.2605	1.45438	0.727190	−	0.686436i	$-0.240826\pi$
$523$	33.2605	1.45438	0.727190	+	0.686436i	$0.240826\pi$
$524$	−33.9244	−1.48199
$525$	0	0
$526$	−8.42586	−0.367385
$527$	−25.7062	−1.11978
$528$	0	0
$529$	−21.2133	−0.922319
$530$	0	0
$531$	0	0
$532$	0.973969	0.0422269
$533$	−6.74679	−0.292236
$534$	0	0
$535$	0	0
$536$	−0.234651	−0.0101354
$537$	0	0
$538$	14.6072	0.629762
$539$	−25.6063	−1.10294
$540$	0	0
$541$	−4.47792	−0.192521	−0.0962604	−	0.995356i	$-0.530688\pi$
$541$	−4.47792	−0.192521	−0.0962604	+	0.995356i	$0.530688\pi$
$542$	−3.11664	−0.133871
$543$	0	0
$544$	−27.5571	−1.18150
$545$	0	0
$546$	0	0
$547$	16.2936	0.696663	0.348331	−	0.937371i	$-0.386748\pi$
$547$	16.2936	0.696663	0.348331	+	0.937371i	$0.386748\pi$
$548$	−15.6898	−0.670237
$549$	0	0
$550$	0	0
$551$	0.934722	0.0398205
$552$	0	0
$553$	4.79966	0.204102
$554$	−13.5753	−0.576759
$555$	0	0
$556$	−11.0401	−0.468205
$557$	−39.0651	−1.65524	−0.827620	−	0.561288i	$-0.810306\pi$
$557$	−39.0651	−1.65524	−0.827620	+	0.561288i	$0.810306\pi$
$558$	0	0
$559$	9.70344	0.410412
$560$	0	0
$561$	0	0
$562$	−5.31555	−0.224223
$563$	−3.53680	−0.149059	−0.0745293	−	0.997219i	$-0.523745\pi$
$563$	−3.53680	−0.149059	−0.0745293	+	0.997219i	$0.523745\pi$
$564$	0	0
$565$	0	0
$566$	3.26959	0.137431
$567$	0	0
$568$	33.0491	1.38671
$569$	−20.9327	−0.877544	−0.438772	−	0.898599i	$-0.644586\pi$
$569$	−20.9327	−0.877544	−0.438772	+	0.898599i	$0.644586\pi$
$570$	0	0
$571$	−46.0130	−1.92558	−0.962792	−	0.270244i	$-0.912896\pi$
$571$	−46.0130	−1.92558	−0.962792	+	0.270244i	$0.912896\pi$
$572$	13.0243	0.544572
$573$	0	0
$574$	−1.06719	−0.0445437
$575$	0	0
$576$	0	0
$577$	32.0111	1.33264	0.666319	−	0.745666i	$-0.267869\pi$
$577$	32.0111	1.33264	0.666319	+	0.745666i	$0.267869\pi$
$578$	5.39190	0.224274
$579$	0	0
$580$	0	0
$581$	−3.73889	−0.155115
$582$	0	0
$583$	12.0491	0.499021
$584$	11.9184	0.493186
$585$	0	0
$586$	14.5522	0.601144
$587$	4.00222	0.165190	0.0825948	−	0.996583i	$-0.473679\pi$
$587$	4.00222	0.165190	0.0825948	+	0.996583i	$0.473679\pi$
$588$	0	0
$589$	4.96693	0.204659
$590$	0	0
$591$	0	0
$592$	7.70344	0.316609
$593$	−14.7590	−0.606080	−0.303040	−	0.952978i	$-0.598002\pi$
$593$	−14.7590	−0.606080	−0.303040	+	0.952978i	$0.598002\pi$
$594$	0	0
$595$	0	0
$596$	−13.8204	−0.566106
$597$	0	0
$598$	−1.47302	−0.0602361
$599$	46.8774	1.91536	0.957680	−	0.287835i	$-0.0929355\pi$
$599$	46.8774	1.91536	0.957680	+	0.287835i	$0.0929355\pi$
$600$	0	0
$601$	36.2906	1.48032	0.740162	−	0.672429i	$-0.234749\pi$
$601$	36.2906	1.48032	0.740162	+	0.672429i	$0.234749\pi$
$602$	1.53487	0.0625564
$603$	0	0
$604$	−7.20034	−0.292978
$605$	0	0
$606$	0	0
$607$	7.24941	0.294244	0.147122	−	0.989118i	$-0.452999\pi$
$607$	7.24941	0.294244	0.147122	+	0.989118i	$0.452999\pi$
$608$	5.32454	0.215939
$609$	0	0
$610$	0	0
$611$	19.7633	0.799538
$612$	0	0
$613$	19.2415	0.777157	0.388579	−	0.921416i	$-0.372966\pi$
$613$	19.2415	0.777157	0.388579	+	0.921416i	$0.372966\pi$
$614$	−6.63910	−0.267932
$615$	0	0
$616$	4.48901	0.180867
$617$	−31.9837	−1.28762	−0.643808	−	0.765187i	$-0.722646\pi$
$617$	−31.9837	−1.28762	−0.643808	+	0.765187i	$0.722646\pi$
$618$	0	0
$619$	14.0472	0.564603	0.282301	−	0.959326i	$-0.408902\pi$
$619$	14.0472	0.564603	0.282301	+	0.959326i	$0.408902\pi$
$620$	0	0
$621$	0	0
$622$	4.67336	0.187385
$623$	1.80208	0.0721989
$624$	0	0
$625$	0	0
$626$	−16.8266	−0.672528
$627$	0	0
$628$	22.2485	0.887814
$629$	−17.5593	−0.700135
$630$	0	0
$631$	−37.5430	−1.49456	−0.747281	−	0.664508i	$-0.768641\pi$
$631$	−37.5430	−1.49456	−0.747281	+	0.664508i	$0.768641\pi$
$632$	17.0265	0.677278
$633$	0	0
$634$	16.6652	0.661861
$635$	0	0
$636$	0	0
$637$	13.3407	0.528579
$638$	1.97713	0.0782754
$639$	0	0
$640$	0	0
$641$	−20.5672	−0.812355	−0.406178	−	0.913794i	$-0.633139\pi$
$641$	−20.5672	−0.812355	−0.406178	+	0.913794i	$0.633139\pi$
$642$	0	0
$643$	−2.44185	−0.0962971	−0.0481486	−	0.998840i	$-0.515332\pi$
$643$	−2.44185	−0.0962971	−0.0481486	+	0.998840i	$0.515332\pi$
$644$	1.30187	0.0513008
$645$	0	0
$646$	−2.85172	−0.112199
$647$	17.8017	0.699857	0.349928	−	0.936776i	$-0.386206\pi$
$647$	17.8017	0.699857	0.349928	+	0.936776i	$0.386206\pi$
$648$	0	0
$649$	−17.1803	−0.674384
$650$	0	0
$651$	0	0
$652$	16.3808	0.641523
$653$	−9.41235	−0.368334	−0.184167	−	0.982895i	$-0.558959\pi$
$653$	−9.41235	−0.368334	−0.184167	+	0.982895i	$0.558959\pi$
$654$	0	0
$655$	0	0
$656$	7.65941	0.299050
$657$	0	0
$658$	3.12611	0.121869
$659$	33.8532	1.31873	0.659366	−	0.751822i	$-0.270825\pi$
$659$	33.8532	1.31873	0.659366	+	0.751822i	$0.270825\pi$
$660$	0	0
$661$	−27.0681	−1.05283	−0.526413	−	0.850229i	$-0.676463\pi$
$661$	−27.0681	−1.05283	−0.526413	+	0.850229i	$0.676463\pi$
$662$	−8.24686	−0.320523
$663$	0	0
$664$	−13.2635	−0.514723
$665$	0	0
$666$	0	0
$667$	1.24941	0.0483772
$668$	13.7061	0.530306
$669$	0	0
$670$	0	0
$671$	31.0258	1.19774
$672$	0	0
$673$	−0.132293	−0.00509951	−0.00254976	−	0.999997i	$-0.500812\pi$
$673$	−0.132293	−0.00509951	−0.00254976	+	0.999997i	$0.500812\pi$
$674$	−11.2896	−0.434858
$675$	0	0
$676$	15.2675	0.587213
$677$	1.40402	0.0539610	0.0269805	−	0.999636i	$-0.491411\pi$
$677$	1.40402	0.0539610	0.0269805	+	0.999636i	$0.491411\pi$
$678$	0	0
$679$	5.56306	0.213490
$680$	0	0
$681$	0	0
$682$	10.5061	0.402299
$683$	−10.8799	−0.416306	−0.208153	−	0.978096i	$-0.566745\pi$
$683$	−10.8799	−0.416306	−0.208153	+	0.978096i	$0.566745\pi$
$684$	0	0
$685$	0	0
$686$	4.32469	0.165117
$687$	0	0
$688$	−11.0160	−0.419981
$689$	−6.27749	−0.239153
$690$	0	0
$691$	−8.85172	−0.336735	−0.168368	−	0.985724i	$-0.553850\pi$
$691$	−8.85172	−0.336735	−0.168368	+	0.985724i	$0.553850\pi$
$692$	6.12066	0.232673
$693$	0	0
$694$	−4.25050	−0.161347
$695$	0	0
$696$	0	0
$697$	−17.4589	−0.661304
$698$	−6.18802	−0.234220
$699$	0	0
$700$	0	0
$701$	−17.3959	−0.657033	−0.328517	−	0.944498i	$-0.606549\pi$
$701$	−17.3959	−0.657033	−0.328517	+	0.944498i	$0.606549\pi$
$702$	0	0
$703$	3.39279	0.127961
$704$	−6.16980	−0.232533
$705$	0	0
$706$	−0.701530	−0.0264024
$707$	6.11408	0.229944
$708$	0	0
$709$	23.9498	0.899455	0.449728	−	0.893166i	$-0.351521\pi$
$709$	23.9498	0.899455	0.449728	+	0.893166i	$0.351521\pi$
$710$	0	0
$711$	0	0
$712$	6.39279	0.239580
$713$	6.63910	0.248636
$714$	0	0
$715$	0	0
$716$	36.8119	1.37573
$717$	0	0
$718$	2.33882	0.0872839
$719$	20.6020	0.768324	0.384162	−	0.923266i	$-0.374490\pi$
$719$	20.6020	0.768324	0.384162	+	0.923266i	$0.374490\pi$
$720$	0	0
$721$	7.44485	0.277261
$722$	0.551006	0.0205063
$723$	0	0
$724$	−2.06015	−0.0765648
$725$	0	0
$726$	0	0
$727$	−17.6844	−0.655880	−0.327940	−	0.944699i	$-0.606354\pi$
$727$	−17.6844	−0.655880	−0.327940	+	0.944699i	$0.606354\pi$
$728$	−2.33875	−0.0866797
$729$	0	0
$730$	0	0
$731$	25.1100	0.928725
$732$	0	0
$733$	−29.0601	−1.07336	−0.536681	−	0.843786i	$-0.680322\pi$
$733$	−29.0601	−1.07336	−0.536681	+	0.843786i	$0.680322\pi$
$734$	2.33098	0.0860382
$735$	0	0
$736$	7.11711	0.262340
$737$	0.442267	0.0162911
$738$	0	0
$739$	29.2274	1.07515	0.537574	−	0.843217i	$-0.319341\pi$
$739$	29.2274	1.07515	0.537574	+	0.843217i	$0.319341\pi$
$740$	0	0
$741$	0	0
$742$	−0.992958	−0.0364526
$743$	−2.94054	−0.107878	−0.0539390	−	0.998544i	$-0.517178\pi$
$743$	−2.94054	−0.107878	−0.0539390	+	0.998544i	$0.517178\pi$
$744$	0	0
$745$	0	0
$746$	−11.2454	−0.411722
$747$	0	0
$748$	33.7034	1.23232
$749$	−5.71214	−0.208717
$750$	0	0
$751$	−2.91786	−0.106474	−0.0532372	−	0.998582i	$-0.516954\pi$
$751$	−2.91786	−0.106474	−0.0532372	+	0.998582i	$0.516954\pi$
$752$	−22.4366	−0.818180
$753$	0	0
$754$	−1.03007	−0.0375131
$755$	0	0
$756$	0	0
$757$	25.0981	0.912207	0.456103	−	0.889927i	$-0.349245\pi$
$757$	25.0981	0.912207	0.456103	+	0.889927i	$0.349245\pi$
$758$	−10.8280	−0.393291
$759$	0	0
$760$	0	0
$761$	8.91437	0.323146	0.161573	−	0.986861i	$-0.448343\pi$
$761$	8.91437	0.323146	0.161573	+	0.986861i	$0.448343\pi$
$762$	0	0
$763$	1.62920	0.0589811
$764$	20.9001	0.756140
$765$	0	0
$766$	0.405826	0.0146631
$767$	8.95081	0.323195
$768$	0	0
$769$	32.3126	1.16522	0.582610	−	0.812752i	$-0.302031\pi$
$769$	32.3126	1.16522	0.582610	+	0.812752i	$0.302031\pi$
$770$	0	0
$771$	0	0
$772$	32.2825	1.16187
$773$	46.0426	1.65604	0.828019	−	0.560700i	$-0.189468\pi$
$773$	46.0426	1.65604	0.828019	+	0.560700i	$0.189468\pi$
$774$	0	0
$775$	0	0
$776$	19.7346	0.708432
$777$	0	0
$778$	9.16237	0.328487
$779$	3.37340	0.120864
$780$	0	0
$781$	−62.2906	−2.22893
$782$	−3.81178	−0.136309
$783$	0	0
$784$	−15.1453	−0.540903
$785$	0	0
$786$	0	0
$787$	−18.0350	−0.642878	−0.321439	−	0.946930i	$-0.604166\pi$
$787$	−18.0350	−0.642878	−0.321439	+	0.946930i	$0.604166\pi$
$788$	29.5102	1.05126
$789$	0	0
$790$	0	0
$791$	−1.13070	−0.0402030
$792$	0	0
$793$	−16.1643	−0.574010
$794$	2.06765	0.0733781
$795$	0	0
$796$	0.470878	0.0166898
$797$	5.23508	0.185436	0.0927180	−	0.995692i	$-0.470444\pi$
$797$	5.23508	0.185436	0.0927180	+	0.995692i	$0.470444\pi$
$798$	0	0
$799$	51.1423	1.80928
$800$	0	0
$801$	0	0
$802$	−11.5362	−0.407356
$803$	−22.4637	−0.792726
$804$	0	0
$805$	0	0
$806$	−5.47361	−0.192800
$807$	0	0
$808$	21.6894	0.763029
$809$	26.6448	0.936782	0.468391	−	0.883521i	$-0.344834\pi$
$809$	26.6448	0.936782	0.468391	+	0.883521i	$0.344834\pi$
$810$	0	0
$811$	−6.96693	−0.244642	−0.122321	−	0.992491i	$-0.539034\pi$
$811$	−6.96693	−0.244642	−0.122321	+	0.992491i	$0.539034\pi$
$812$	0.910390	0.0319484
$813$	0	0
$814$	7.17645	0.251534
$815$	0	0
$816$	0	0
$817$	−4.85172	−0.169740
$818$	8.91437	0.311683
$819$	0	0
$820$	0	0
$821$	−36.4282	−1.27135	−0.635676	−	0.771956i	$-0.719279\pi$
$821$	−36.4282	−1.27135	−0.635676	+	0.771956i	$0.719279\pi$
$822$	0	0
$823$	39.2556	1.36836	0.684182	−	0.729311i	$-0.260159\pi$
$823$	39.2556	1.36836	0.684182	+	0.729311i	$0.260159\pi$
$824$	26.4102	0.920042
$825$	0	0
$826$	1.41582	0.0492626
$827$	−3.61193	−0.125599	−0.0627995	−	0.998026i	$-0.520003\pi$
$827$	−3.61193	−0.125599	−0.0627995	+	0.998026i	$0.520003\pi$
$828$	0	0
$829$	−42.3567	−1.47111	−0.735555	−	0.677465i	$-0.763078\pi$
$829$	−42.3567	−1.47111	−0.735555	+	0.677465i	$0.763078\pi$
$830$	0	0
$831$	0	0
$832$	3.21443	0.111440
$833$	34.5223	1.19613
$834$	0	0
$835$	0	0
$836$	−6.51214	−0.225227
$837$	0	0
$838$	7.94794	0.274557
$839$	11.6837	0.403367	0.201684	−	0.979451i	$-0.435359\pi$
$839$	11.6837	0.403367	0.201684	+	0.979451i	$0.435359\pi$
$840$	0	0
$841$	−28.1263	−0.969872
$842$	9.68180	0.333657
$843$	0	0
$844$	10.3977	0.357903
$845$	0	0
$846$	0	0
$847$	−2.14528	−0.0737128
$848$	7.12663	0.244729
$849$	0	0
$850$	0	0
$851$	4.53501	0.155458
$852$	0	0
$853$	2.19735	0.0752357	0.0376178	−	0.999292i	$-0.488023\pi$
$853$	2.19735	0.0752357	0.0376178	+	0.999292i	$0.488023\pi$
$854$	−2.55682	−0.0874927
$855$	0	0
$856$	−20.2635	−0.692592
$857$	17.1938	0.587329	0.293664	−	0.955909i	$-0.405125\pi$
$857$	17.1938	0.587329	0.293664	+	0.955909i	$0.405125\pi$
$858$	0	0
$859$	50.7666	1.73213	0.866067	−	0.499928i	$-0.166640\pi$
$859$	50.7666	1.73213	0.866067	+	0.499928i	$0.166640\pi$
$860$	0	0
$861$	0	0
$862$	8.35376	0.284530
$863$	−21.1039	−0.718383	−0.359192	−	0.933264i	$-0.616948\pi$
$863$	−21.1039	−0.718383	−0.359192	+	0.933264i	$0.616948\pi$
$864$	0	0
$865$	0	0
$866$	2.33098	0.0792101
$867$	0	0
$868$	4.83763	0.164200
$869$	−32.0914	−1.08863
$870$	0	0
$871$	−0.230419	−0.00780744
$872$	5.77950	0.195719
$873$	0	0
$874$	0.736508	0.0249128
$875$	0	0
$876$	0	0
$877$	−44.9498	−1.51785	−0.758924	−	0.651179i	$-0.774275\pi$
$877$	−44.9498	−1.51785	−0.758924	+	0.651179i	$0.774275\pi$
$878$	6.37742	0.215227
$879$	0	0
$880$	0	0
$881$	−50.3546	−1.69649	−0.848245	−	0.529603i	$-0.822341\pi$
$881$	−50.3546	−1.69649	−0.848245	+	0.529603i	$0.822341\pi$
$882$	0	0
$883$	36.4700	1.22731	0.613657	−	0.789573i	$-0.289698\pi$
$883$	36.4700	1.22731	0.613657	+	0.789573i	$0.289698\pi$
$884$	−17.5593	−0.590583
$885$	0	0
$886$	12.3077	0.413484
$887$	−38.2612	−1.28469	−0.642343	−	0.766418i	$-0.722037\pi$
$887$	−38.2612	−1.28469	−0.642343	+	0.766418i	$0.722037\pi$
$888$	0	0
$889$	3.03007	0.101625
$890$	0	0
$891$	0	0
$892$	49.3296	1.65168
$893$	−9.88165	−0.330677
$894$	0	0
$895$	0	0
$896$	6.62253	0.221243
$897$	0	0
$898$	1.72947	0.0577130
$899$	4.64270	0.154843
$900$	0	0
$901$	−16.2445	−0.541183
$902$	7.13544	0.237584
$903$	0	0
$904$	−4.01109	−0.133407
$905$	0	0
$906$	0	0
$907$	−38.3426	−1.27315	−0.636573	−	0.771216i	$-0.719649\pi$
$907$	−38.3426	−1.27315	−0.636573	+	0.771216i	$0.719649\pi$
$908$	31.2654	1.03758
$909$	0	0
$910$	0	0
$911$	−30.6509	−1.01551	−0.507755	−	0.861501i	$-0.669525\pi$
$911$	−30.6509	−1.01551	−0.507755	+	0.861501i	$0.669525\pi$
$912$	0	0
$913$	24.9989	0.827343
$914$	−3.48767	−0.115362
$915$	0	0
$916$	−3.08514	−0.101936
$917$	−11.4817	−0.379158
$918$	0	0
$919$	0.507995	0.0167572	0.00837860	−	0.999965i	$-0.497333\pi$
$919$	0.507995	0.0167572	0.00837860	+	0.999965i	$0.497333\pi$
$920$	0	0
$921$	0	0
$922$	19.1863	0.631866
$923$	32.4530	1.06820
$924$	0	0
$925$	0	0
$926$	15.1554	0.498038
$927$	0	0
$928$	4.97697	0.163377
$929$	−19.7998	−0.649609	−0.324804	−	0.945781i	$-0.605298\pi$
$929$	−19.7998	−0.649609	−0.324804	+	0.945781i	$0.605298\pi$
$930$	0	0
$931$	−6.67036	−0.218612
$932$	−14.3880	−0.471294
$933$	0	0
$934$	14.3836	0.470646
$935$	0	0
$936$	0	0
$937$	43.2415	1.41264	0.706319	−	0.707894i	$-0.250354\pi$
$937$	43.2415	1.41264	0.706319	+	0.707894i	$0.250354\pi$
$938$	−0.0364470	−0.00119004
$939$	0	0
$940$	0	0
$941$	26.2832	0.856808	0.428404	−	0.903587i	$-0.359076\pi$
$941$	26.2832	0.856808	0.428404	+	0.903587i	$0.359076\pi$
$942$	0	0
$943$	4.50909	0.146836
$944$	−10.1616	−0.330731
$945$	0	0
$946$	−10.2624	−0.333659
$947$	19.2940	0.626971	0.313486	−	0.949593i	$-0.398503\pi$
$947$	19.2940	0.626971	0.313486	+	0.949593i	$0.398503\pi$
$948$	0	0
$949$	11.7034	0.379910
$950$	0	0
$951$	0	0
$952$	−6.05206	−0.196148
$953$	−35.1225	−1.13773	−0.568864	−	0.822432i	$-0.692617\pi$
$953$	−35.1225	−1.13773	−0.568864	+	0.822432i	$0.692617\pi$
$954$	0	0
$955$	0	0
$956$	31.8854	1.03125
$957$	0	0
$958$	1.69235	0.0546773
$959$	−5.31020	−0.171475
$960$	0	0
$961$	−6.32964	−0.204182
$962$	−3.73889	−0.120547
$963$	0	0
$964$	0.917864	0.0295624
$965$	0	0
$966$	0	0
$967$	−0.870706	−0.0280000	−0.0140000	−	0.999902i	$-0.504456\pi$
$967$	−0.870706	−0.0280000	−0.0140000	+	0.999902i	$0.504456\pi$
$968$	−7.61027	−0.244603
$969$	0	0
$970$	0	0
$971$	−33.9934	−1.09090	−0.545450	−	0.838143i	$-0.683641\pi$
$971$	−33.9934	−1.09090	−0.545450	+	0.838143i	$0.683641\pi$
$972$	0	0
$973$	−3.73651	−0.119787
$974$	15.6976	0.502984
$975$	0	0
$976$	18.3508	0.587394
$977$	−36.8611	−1.17929	−0.589645	−	0.807663i	$-0.700732\pi$
$977$	−36.8611	−1.17929	−0.589645	+	0.807663i	$0.700732\pi$
$978$	0	0
$979$	−12.0491	−0.385090
$980$	0	0
$981$	0	0
$982$	−14.8457	−0.473746
$983$	39.9959	1.27567	0.637836	−	0.770172i	$-0.279830\pi$
$983$	39.9959	1.27567	0.637836	+	0.770172i	$0.279830\pi$
$984$	0	0
$985$	0	0
$986$	−2.66556	−0.0848888
$987$	0	0
$988$	3.39279	0.107939
$989$	−6.48511	−0.206214
$990$	0	0
$991$	6.83463	0.217109	0.108555	−	0.994090i	$-0.465378\pi$
$991$	6.83463	0.217109	0.108555	+	0.994090i	$0.465378\pi$
$992$	26.4466	0.839681
$993$	0	0
$994$	5.13334	0.162820
$995$	0	0
$996$	0	0
$997$	−52.1092	−1.65032	−0.825158	−	0.564903i	$-0.808914\pi$
$997$	−52.1092	−1.65032	−0.825158	+	0.564903i	$0.808914\pi$
$998$	15.2752	0.483528
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4275.2.a.bq.1.4	yes	6
3.2	odd	2	inner	4275.2.a.bq.1.3	✓	6
5.4	even	2		4275.2.a.bu.1.3	yes	6
15.14	odd	2		4275.2.a.bu.1.4	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4275.2.a.bq.1.3	✓	6	3.2	odd	2	inner
4275.2.a.bq.1.4	yes	6	1.1	even	1	trivial
4275.2.a.bu.1.3	yes	6	5.4	even	2
4275.2.a.bu.1.4	yes	6	15.14	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:	\( N \)	\(=\)	\( 4275 = 3^{2} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4275.a (trivial)

\(f(q)\)	\(=\)	\(q+0.551006 q^{2} -1.69639 q^{4} -0.574141 q^{7} -2.03673 q^{8} +3.83882 q^{11} -2.00000 q^{13} -0.316355 q^{14} +2.27053 q^{16} -5.17548 q^{17} +1.00000 q^{19} +2.11521 q^{22} +1.33666 q^{23} -1.10201 q^{26} +0.973969 q^{28} +0.934722 q^{29} +4.96693 q^{31} +5.32454 q^{32} -2.85172 q^{34} +3.39279 q^{37} +0.551006 q^{38} +3.37340 q^{41} -4.85172 q^{43} -6.51214 q^{44} +0.736508 q^{46} -9.88165 q^{47} -6.67036 q^{49} +3.39279 q^{52} +3.13874 q^{53} +1.16937 q^{56} +0.515037 q^{58} -4.47541 q^{59} +8.08214 q^{61} +2.73680 q^{62} -1.60721 q^{64} +0.115209 q^{67} +8.77964 q^{68} -16.2265 q^{71} -5.85172 q^{73} +1.86944 q^{74} -1.69639 q^{76} -2.20402 q^{77} -8.35971 q^{79} +1.85876 q^{82} +6.51214 q^{83} -2.67332 q^{86} -7.81864 q^{88} -3.13874 q^{89} +1.14828 q^{91} -2.26750 q^{92} -5.44485 q^{94} -9.68935 q^{97} -3.67541 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 8 q^{4} - 8 q^{7} - 12 q^{13} + 6 q^{19} - 10 q^{22} - 26 q^{28} - 2 q^{31} - 8 q^{34} - 16 q^{37} - 20 q^{43} + 18 q^{46} + 10 q^{49} - 16 q^{52} - 56 q^{58} - 6 q^{61} - 46 q^{64} - 22 q^{67} - 26 q^{73}+ \cdots - 40 q^{97}+O(q^{100}) \) 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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Groups

Database

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