LMFDB - Embedded newform 4275.2.a.bq.1.2

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			1.2
Root			$3.14584$ 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)

$f(q)$	$=$	$q-2.04406 q^{2} +2.17819 q^{4} +1.43366 q^{7} -0.364240 q^{8} +5.70304 q^{11} -2.00000 q^{13} -2.93050 q^{14} -3.61186 q^{16} +3.35965 q^{17} +1.00000 q^{19} -11.6574 q^{22} -9.06269 q^{23} +4.08813 q^{26} +3.12280 q^{28} +4.45237 q^{29} -4.79005 q^{31} +8.11135 q^{32} -6.86733 q^{34} -4.35639 q^{37} -2.04406 q^{38} -8.69845 q^{41} -8.86733 q^{43} +12.4223 q^{44} +18.5247 q^{46} -3.22984 q^{47} -4.94461 q^{49} -4.35639 q^{52} -3.72389 q^{53} -0.522198 q^{56} -9.10092 q^{58} +12.7866 q^{59} -15.4474 q^{61} +9.79117 q^{62} -9.35639 q^{64} -13.6574 q^{67} +7.31796 q^{68} +0.652009 q^{71} -9.86733 q^{73} +8.90473 q^{74} +2.17819 q^{76} +8.17625 q^{77} +9.14644 q^{79} +17.7802 q^{82} -12.4223 q^{83} +18.1254 q^{86} -2.07728 q^{88} +3.72389 q^{89} -2.86733 q^{91} -19.7403 q^{92} +6.60199 q^{94} +6.09105 q^{97} +10.1071 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.04406	−1.44537	−0.722685	−	0.691177i	$-0.757092\pi$
$2$	−2.04406	−1.44537	−0.722685	+	0.691177i	$0.757092\pi$
$3$	0	0
$3$	0	0
$4$	2.17819	1.08910
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.43366	0.541874	0.270937	−	0.962597i	$-0.412666\pi$
$7$	1.43366	0.541874	0.270937	+	0.962597i	$0.412666\pi$
$8$	−0.364240	−0.128778
$9$	0	0
$10$	0	0
$11$	5.70304	1.71953	0.859766	−	0.510688i	$-0.170609\pi$
$11$	5.70304	1.71953	0.859766	+	0.510688i	$0.170609\pi$
$12$	0	0
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	−2.93050	−0.783209
$15$	0	0
$16$	−3.61186	−0.902965
$17$	3.35965	0.814834	0.407417	−	0.913242i	$-0.366430\pi$
$17$	3.35965	0.814834	0.407417	+	0.913242i	$0.366430\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	−11.6574	−2.48536
$23$	−9.06269	−1.88970	−0.944851	−	0.327501i	$-0.893793\pi$
$23$	−9.06269	−1.88970	−0.944851	+	0.327501i	$0.893793\pi$
$24$	0	0
$25$	0	0
$26$	4.08813	0.801748
$27$	0	0
$28$	3.12280	0.590154
$29$	4.45237	0.826784	0.413392	−	0.910553i	$-0.364344\pi$
$29$	4.45237	0.826784	0.413392	+	0.910553i	$0.364344\pi$
$30$	0	0
$31$	−4.79005	−0.860319	−0.430159	−	0.902753i	$-0.641543\pi$
$31$	−4.79005	−0.860319	−0.430159	+	0.902753i	$0.641543\pi$
$32$	8.11135	1.43390
$33$	0	0
$34$	−6.86733	−1.17774
$35$	0	0
$36$	0	0
$37$	−4.35639	−0.716186	−0.358093	−	0.933686i	$-0.616573\pi$
$37$	−4.35639	−0.716186	−0.358093	+	0.933686i	$0.616573\pi$
$38$	−2.04406	−0.331591
$39$	0	0
$40$	0	0
$41$	−8.69845	−1.35847	−0.679235	−	0.733921i	$-0.737688\pi$
$41$	−8.69845	−1.35847	−0.679235	+	0.733921i	$0.737688\pi$
$42$	0	0
$43$	−8.86733	−1.35226	−0.676128	−	0.736785i	$-0.736343\pi$
$43$	−8.86733	−1.35226	−0.676128	+	0.736785i	$0.736343\pi$
$44$	12.4223	1.87274
$45$	0	0
$46$	18.5247	2.73132
$47$	−3.22984	−0.471120	−0.235560	−	0.971860i	$-0.575692\pi$
$47$	−3.22984	−0.471120	−0.235560	+	0.971860i	$0.575692\pi$
$48$	0	0
$49$	−4.94461	−0.706372
$50$	0	0
$51$	0	0
$52$	−4.35639	−0.604122
$53$	−3.72389	−0.511515	−0.255758	−	0.966741i	$-0.582325\pi$
$53$	−3.72389	−0.511515	−0.255758	+	0.966741i	$0.582325\pi$
$54$	0	0
$55$	0	0
$56$	−0.522198	−0.0697816
$57$	0	0
$58$	−9.10092	−1.19501
$59$	12.7866	1.66467	0.832335	−	0.554273i	$-0.187003\pi$
$59$	12.7866	1.66467	0.832335	+	0.554273i	$0.187003\pi$
$60$	0	0
$61$	−15.4474	−1.97784	−0.988921	−	0.148445i	$-0.952573\pi$
$61$	−15.4474	−1.97784	−0.988921	+	0.148445i	$0.952573\pi$
$62$	9.79117	1.24348
$63$	0	0
$64$	−9.35639	−1.16955
$65$	0	0
$66$	0	0
$67$	−13.6574	−1.66851	−0.834257	−	0.551375i	$-0.814103\pi$
$67$	−13.6574	−1.66851	−0.834257	+	0.551375i	$0.814103\pi$
$68$	7.31796	0.887433
$69$	0	0
$70$	0	0
$71$	0.652009	0.0773792	0.0386896	−	0.999251i	$-0.487682\pi$
$71$	0.652009	0.0773792	0.0386896	+	0.999251i	$0.487682\pi$
$72$	0	0
$73$	−9.86733	−1.15488	−0.577442	−	0.816432i	$-0.695949\pi$
$73$	−9.86733	−1.15488	−0.577442	+	0.816432i	$0.695949\pi$
$74$	8.90473	1.03515
$75$	0	0
$76$	2.17819	0.249856
$77$	8.17625	0.931770
$78$	0	0
$79$	9.14644	1.02906	0.514528	−	0.857474i	$-0.327967\pi$
$79$	9.14644	1.02906	0.514528	+	0.857474i	$0.327967\pi$
$80$	0	0
$81$	0	0
$82$	17.7802	1.96349
$83$	−12.4223	−1.36353	−0.681764	−	0.731572i	$-0.738787\pi$
$83$	−12.4223	−1.36353	−0.681764	+	0.731572i	$0.738787\pi$
$84$	0	0
$85$	0	0
$86$	18.1254	1.95451
$87$	0	0
$88$	−2.07728	−0.221438
$89$	3.72389	0.394731	0.197366	−	0.980330i	$-0.436761\pi$
$89$	3.72389	0.394731	0.197366	+	0.980330i	$0.436761\pi$
$90$	0	0
$91$	−2.86733	−0.300578
$92$	−19.7403	−2.05807
$93$	0	0
$94$	6.60199	0.680943
$95$	0	0
$96$	0	0
$97$	6.09105	0.618452	0.309226	−	0.950989i	$-0.399930\pi$
$97$	6.09105	0.618452	0.309226	+	0.950989i	$0.399930\pi$
$98$	10.1071	1.02097
$99$	0	0
$100$	0	0
$101$	−16.2227	−1.61422	−0.807109	−	0.590402i	$-0.798969\pi$
$101$	−16.2227	−1.61422	−0.807109	+	0.590402i	$0.798969\pi$
$102$	0	0
$103$	−3.20995	−0.316285	−0.158143	−	0.987416i	$-0.550551\pi$
$103$	−3.20995	−0.316285	−0.158143	+	0.987416i	$0.550551\pi$
$104$	0.728480	0.0714333
$105$	0	0
$106$	7.61186	0.739329
$107$	6.79576	0.656971	0.328486	−	0.944509i	$-0.393462\pi$
$107$	6.79576	0.656971	0.328486	+	0.944509i	$0.393462\pi$
$108$	0	0
$109$	16.9584	1.62432	0.812159	−	0.583436i	$-0.198292\pi$
$109$	16.9584	1.62432	0.812159	+	0.583436i	$0.198292\pi$
$110$	0	0
$111$	0	0
$112$	−5.17819	−0.489293
$113$	−3.20169	−0.301190	−0.150595	−	0.988596i	$-0.548119\pi$
$113$	−3.20169	−0.301190	−0.150595	+	0.988596i	$0.548119\pi$
$114$	0	0
$115$	0	0
$116$	9.69812	0.900448
$117$	0	0
$118$	−26.1366	−2.40607
$119$	4.81661	0.441538
$120$	0	0
$121$	21.5247	1.95679
$122$	31.5755	2.85871
$123$	0	0
$124$	−10.4337	−0.936971
$125$	0	0
$126$	0	0
$127$	−11.3010	−1.00280	−0.501401	−	0.865215i	$-0.667182\pi$
$127$	−11.3010	−1.00280	−0.501401	+	0.865215i	$0.667182\pi$
$128$	2.90235	0.256534
$129$	0	0
$130$	0	0
$131$	1.48511	0.129754	0.0648772	−	0.997893i	$-0.479334\pi$
$131$	1.48511	0.129754	0.0648772	+	0.997893i	$0.479334\pi$
$132$	0	0
$133$	1.43366	0.124314
$134$	27.9166	2.41162
$135$	0	0
$136$	−1.22372	−0.104933
$137$	−2.63117	−0.224796	−0.112398	−	0.993663i	$-0.535853\pi$
$137$	−2.63117	−0.224796	−0.112398	+	0.993663i	$0.535853\pi$
$138$	0	0
$139$	−15.0138	−1.27345	−0.636726	−	0.771090i	$-0.719712\pi$
$139$	−15.0138	−1.27345	−0.636726	+	0.771090i	$0.719712\pi$
$140$	0	0
$141$	0	0
$142$	−1.33275	−0.111842
$143$	−11.4061	−0.953825
$144$	0	0
$145$	0	0
$146$	20.1694	1.66924
$147$	0	0
$148$	−9.48906	−0.779996
$149$	1.45696	0.119359	0.0596794	−	0.998218i	$-0.480992\pi$
$149$	1.45696	0.119359	0.0596794	+	0.998218i	$0.480992\pi$
$150$	0	0
$151$	0.510941	0.0415798	0.0207899	−	0.999784i	$-0.493382\pi$
$151$	0.510941	0.0415798	0.0207899	+	0.999784i	$0.493382\pi$
$152$	−0.364240	−0.0295438
$153$	0	0
$154$	−16.7128	−1.34675
$155$	0	0
$156$	0	0
$157$	0.657382	0.0524648	0.0262324	−	0.999656i	$-0.491649\pi$
$157$	0.657382	0.0524648	0.0262324	+	0.999656i	$0.491649\pi$
$158$	−18.6959	−1.48737
$159$	0	0
$160$	0	0
$161$	−12.9929	−1.02398
$162$	0	0
$163$	15.8811	1.24390	0.621952	−	0.783055i	$-0.286340\pi$
$163$	15.8811	1.24390	0.621952	+	0.783055i	$0.286340\pi$
$164$	−18.9469	−1.47951
$165$	0	0
$166$	25.3920	1.97080
$167$	2.10897	0.163197	0.0815984	−	0.996665i	$-0.473997\pi$
$167$	2.10897	0.163197	0.0815984	+	0.996665i	$0.473997\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	0	0
$172$	−19.3148	−1.47274
$173$	13.6730	1.03954	0.519770	−	0.854306i	$-0.326018\pi$
$173$	13.6730	1.03954	0.519770	+	0.854306i	$0.326018\pi$
$174$	0	0
$175$	0	0
$176$	−20.5986	−1.55268
$177$	0	0
$178$	−7.61186	−0.570533
$179$	−18.9303	−1.41492	−0.707460	−	0.706754i	$-0.750159\pi$
$179$	−18.9303	−1.41492	−0.707460	+	0.706754i	$0.750159\pi$
$180$	0	0
$181$	16.7128	1.24225	0.621125	−	0.783711i	$-0.286676\pi$
$181$	16.7128	1.24225	0.621125	+	0.783711i	$0.286676\pi$
$182$	5.86100	0.434446
$183$	0	0
$184$	3.30099	0.243352
$185$	0	0
$186$	0	0
$187$	19.1602	1.40113
$188$	−7.03521	−0.513095
$189$	0	0
$190$	0	0
$191$	9.92098	0.717857	0.358928	−	0.933365i	$-0.383142\pi$
$191$	9.92098	0.717857	0.358928	+	0.933365i	$0.383142\pi$
$192$	0	0
$193$	0.201835	0.0145284	0.00726421	−	0.999974i	$-0.497688\pi$
$193$	0.201835	0.0145284	0.00726421	+	0.999974i	$0.497688\pi$
$194$	−12.4505	−0.893893
$195$	0	0
$196$	−10.7703	−0.769308
$197$	1.17421	0.0836588	0.0418294	−	0.999125i	$-0.486681\pi$
$197$	1.17421	0.0836588	0.0418294	+	0.999125i	$0.486681\pi$
$198$	0	0
$199$	−6.30099	−0.446666	−0.223333	−	0.974742i	$-0.571694\pi$
$199$	−6.30099	−0.446666	−0.223333	+	0.974742i	$0.571694\pi$
$200$	0	0
$201$	0	0
$202$	33.1602	2.33314
$203$	6.38320	0.448013
$204$	0	0
$205$	0	0
$206$	6.56133	0.457150
$207$	0	0
$208$	7.22372	0.500875
$209$	5.70304	0.394488
$210$	0	0
$211$	−16.1683	−1.11307	−0.556537	−	0.830823i	$-0.687870\pi$
$211$	−16.1683	−1.11307	−0.556537	+	0.830823i	$0.687870\pi$
$212$	−8.11135	−0.557090
$213$	0	0
$214$	−13.8910	−0.949567
$215$	0	0
$216$	0	0
$217$	−6.86733	−0.466185
$218$	−34.6640	−2.34774
$219$	0	0
$220$	0	0
$221$	−6.71929	−0.451989
$222$	0	0
$223$	23.4393	1.56961	0.784807	−	0.619741i	$-0.212762\pi$
$223$	23.4393	1.56961	0.784807	+	0.619741i	$0.212762\pi$
$224$	11.6290	0.776992
$225$	0	0
$226$	6.54445	0.435331
$227$	8.82826	0.585952	0.292976	−	0.956120i	$-0.405354\pi$
$227$	8.82826	0.585952	0.292976	+	0.956120i	$0.405354\pi$
$228$	0	0
$229$	−3.92272	−0.259221	−0.129610	−	0.991565i	$-0.541373\pi$
$229$	−3.92272	−0.259221	−0.129610	+	0.991565i	$0.541373\pi$
$230$	0	0
$231$	0	0
$232$	−1.62173	−0.106472
$233$	−15.6240	−1.02356	−0.511782	−	0.859115i	$-0.671014\pi$
$233$	−15.6240	−1.02356	−0.511782	+	0.859115i	$0.671014\pi$
$234$	0	0
$235$	0	0
$236$	27.8516	1.81299
$237$	0	0
$238$	−9.84545	−0.638186
$239$	−17.6797	−1.14360	−0.571801	−	0.820392i	$-0.693755\pi$
$239$	−17.6797	−1.14360	−0.571801	+	0.820392i	$0.693755\pi$
$240$	0	0
$241$	11.2237	0.722983	0.361492	−	0.932375i	$-0.382268\pi$
$241$	11.2237	0.722983	0.361492	+	0.932375i	$0.382268\pi$
$242$	−43.9979	−2.82829
$243$	0	0
$244$	−33.6475	−2.15406
$245$	0	0
$246$	0	0
$247$	−2.00000	−0.127257
$248$	1.74473	0.110790
$249$	0	0
$250$	0	0
$251$	18.7240	1.18185	0.590926	−	0.806726i	$-0.298763\pi$
$251$	18.7240	1.18185	0.590926	+	0.806726i	$0.298763\pi$
$252$	0	0
$253$	−51.6849	−3.24940
$254$	23.0999	1.44942
$255$	0	0
$256$	12.7802	0.798761
$257$	10.5961	0.660968	0.330484	−	0.943812i	$-0.392788\pi$
$257$	10.5961	0.660968	0.330484	+	0.943812i	$0.392788\pi$
$258$	0	0
$259$	−6.24560	−0.388083
$260$	0	0
$261$	0	0
$262$	−3.03565	−0.187543
$263$	5.10437	0.314749	0.157375	−	0.987539i	$-0.449697\pi$
$263$	5.10437	0.314749	0.157375	+	0.987539i	$0.449697\pi$
$264$	0	0
$265$	0	0
$266$	−2.93050	−0.179681
$267$	0	0
$268$	−29.7484	−1.81717
$269$	−10.9372	−0.666855	−0.333427	−	0.942776i	$-0.608205\pi$
$269$	−10.9372	−0.666855	−0.333427	+	0.942776i	$0.608205\pi$
$270$	0	0
$271$	19.8811	1.20769	0.603845	−	0.797101i	$-0.293634\pi$
$271$	19.8811	1.20769	0.603845	+	0.797101i	$0.293634\pi$
$272$	−12.1346	−0.735766
$273$	0	0
$274$	5.37827	0.324913
$275$	0	0
$276$	0	0
$277$	−13.1546	−0.790380	−0.395190	−	0.918599i	$-0.629321\pi$
$277$	−13.1546	−0.790380	−0.395190	+	0.918599i	$0.629321\pi$
$278$	30.6891	1.84061
$279$	0	0
$280$	0	0
$281$	−8.20440	−0.489433	−0.244717	−	0.969595i	$-0.578695\pi$
$281$	−8.20440	−0.489433	−0.244717	+	0.969595i	$0.578695\pi$
$282$	0	0
$283$	−13.5801	−0.807253	−0.403627	−	0.914924i	$-0.632251\pi$
$283$	−13.5801	−0.807253	−0.403627	+	0.914924i	$0.632251\pi$
$284$	1.42020	0.0842735
$285$	0	0
$286$	23.3148	1.37863
$287$	−12.4707	−0.736120
$288$	0	0
$289$	−5.71278	−0.336046
$290$	0	0
$291$	0	0
$292$	−21.4930	−1.25778
$293$	1.16919	0.0683049	0.0341524	−	0.999417i	$-0.489127\pi$
$293$	1.16919	0.0683049	0.0341524	+	0.999417i	$0.489127\pi$
$294$	0	0
$295$	0	0
$296$	1.58677	0.0922292
$297$	0	0
$298$	−2.97812	−0.172518
$299$	18.1254	1.04822
$300$	0	0
$301$	−12.7128	−0.732752
$302$	−1.04440	−0.0600982
$303$	0	0
$304$	−3.61186	−0.207154
$305$	0	0
$306$	0	0
$307$	21.2375	1.21209	0.606044	−	0.795431i	$-0.292756\pi$
$307$	21.2375	1.21209	0.606044	+	0.795431i	$0.292756\pi$
$308$	17.8095	1.01479
$309$	0	0
$310$	0	0
$311$	−15.6240	−0.885957	−0.442979	−	0.896532i	$-0.646078\pi$
$311$	−15.6240	−0.885957	−0.442979	+	0.896532i	$0.646078\pi$
$312$	0	0
$313$	10.2156	0.577420	0.288710	−	0.957417i	$-0.406774\pi$
$313$	10.2156	0.577420	0.288710	+	0.957417i	$0.406774\pi$
$314$	−1.34373	−0.0758311
$315$	0	0
$316$	19.9227	1.12074
$317$	31.2229	1.75365	0.876825	−	0.480809i	$-0.159657\pi$
$317$	31.2229	1.75365	0.876825	+	0.480809i	$0.159657\pi$
$318$	0	0
$319$	25.3920	1.42168
$320$	0	0
$321$	0	0
$322$	26.5582	1.48003
$323$	3.35965	0.186936
$324$	0	0
$325$	0	0
$326$	−32.4620	−1.79790
$327$	0	0
$328$	3.16832	0.174941
$329$	−4.63050	−0.255288
$330$	0	0
$331$	−5.20995	−0.286365	−0.143182	−	0.989696i	$-0.545734\pi$
$331$	−5.20995	−0.286365	−0.143182	+	0.989696i	$0.545734\pi$
$332$	−27.0583	−1.48501
$333$	0	0
$334$	−4.31086	−0.235880
$335$	0	0
$336$	0	0
$337$	−13.0219	−0.709347	−0.354674	−	0.934990i	$-0.615408\pi$
$337$	−13.0219	−0.709347	−0.354674	+	0.934990i	$0.615408\pi$
$338$	18.3966	1.00064
$339$	0	0
$340$	0	0
$341$	−27.3179	−1.47935
$342$	0	0
$343$	−17.1246	−0.924639
$344$	3.22984	0.174141
$345$	0	0
$346$	−27.9485	−1.50252
$347$	28.6169	1.53623	0.768117	−	0.640309i	$-0.221194\pi$
$347$	28.6169	1.53623	0.768117	+	0.640309i	$0.221194\pi$
$348$	0	0
$349$	16.3148	0.873309	0.436655	−	0.899629i	$-0.356163\pi$
$349$	16.3148	0.873309	0.436655	+	0.899629i	$0.356163\pi$
$350$	0	0
$351$	0	0
$352$	46.2594	2.46563
$353$	36.9792	1.96821	0.984103	−	0.177597i	$-0.0568325\pi$
$353$	36.9792	1.96821	0.984103	+	0.177597i	$0.0568325\pi$
$354$	0	0
$355$	0	0
$356$	8.11135	0.429901
$357$	0	0
$358$	38.6948	2.04508
$359$	−32.1626	−1.69748	−0.848739	−	0.528811i	$-0.822638\pi$
$359$	−32.1626	−1.69748	−0.848739	+	0.528811i	$0.822638\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−34.1620	−1.79551
$363$	0	0
$364$	−6.24560	−0.327358
$365$	0	0
$366$	0	0
$367$	−23.3148	−1.21702	−0.608510	−	0.793546i	$-0.708233\pi$
$367$	−23.3148	−1.21702	−0.608510	+	0.793546i	$0.708233\pi$
$368$	32.7332	1.70633
$369$	0	0
$370$	0	0
$371$	−5.33880	−0.277177
$372$	0	0
$373$	30.3839	1.57322	0.786610	−	0.617450i	$-0.211834\pi$
$373$	30.3839	1.57322	0.786610	+	0.617450i	$0.211834\pi$
$374$	−39.1647	−2.02516
$375$	0	0
$376$	1.17644	0.0606700
$377$	−8.90473	−0.458617
$378$	0	0
$379$	−31.9803	−1.64272	−0.821358	−	0.570413i	$-0.806783\pi$
$379$	−31.9803	−1.64272	−0.821358	+	0.570413i	$0.806783\pi$
$380$	0	0
$381$	0	0
$382$	−20.2791	−1.03757
$383$	−30.5960	−1.56338	−0.781692	−	0.623664i	$-0.785643\pi$
$383$	−30.5960	−1.56338	−0.781692	+	0.623664i	$0.785643\pi$
$384$	0	0
$385$	0	0
$386$	−0.412564	−0.0209990
$387$	0	0
$388$	13.2675	0.673554
$389$	−14.1671	−0.718299	−0.359149	−	0.933280i	$-0.616933\pi$
$389$	−14.1671	−0.718299	−0.359149	+	0.933280i	$0.616933\pi$
$390$	0	0
$391$	−30.4474	−1.53979
$392$	1.80102	0.0909654
$393$	0	0
$394$	−2.40015	−0.120918
$395$	0	0
$396$	0	0
$397$	−21.5028	−1.07920	−0.539598	−	0.841923i	$-0.681424\pi$
$397$	−21.5028	−1.07920	−0.539598	+	0.841923i	$0.681424\pi$
$398$	12.8796	0.645597
$399$	0	0
$400$	0	0
$401$	18.4131	0.919509	0.459754	−	0.888046i	$-0.347937\pi$
$401$	18.4131	0.919509	0.459754	+	0.888046i	$0.347937\pi$
$402$	0	0
$403$	9.58011	0.477219
$404$	−35.3362	−1.75804
$405$	0	0
$406$	−13.0477	−0.647545
$407$	−24.8447	−1.23150
$408$	0	0
$409$	−7.06916	−0.349548	−0.174774	−	0.984609i	$-0.555919\pi$
$409$	−7.06916	−0.349548	−0.174774	+	0.984609i	$0.555919\pi$
$410$	0	0
$411$	0	0
$412$	−6.99189	−0.344466
$413$	18.3317	0.902042
$414$	0	0
$415$	0	0
$416$	−16.2227	−0.795383
$417$	0	0
$418$	−11.6574	−0.570181
$419$	−5.99081	−0.292670	−0.146335	−	0.989235i	$-0.546748\pi$
$419$	−5.99081	−0.292670	−0.146335	+	0.989235i	$0.546748\pi$
$420$	0	0
$421$	−13.4256	−0.654321	−0.327161	−	0.944969i	$-0.606092\pi$
$421$	−13.4256	−0.654321	−0.327161	+	0.944969i	$0.606092\pi$
$422$	33.0491	1.60880
$423$	0	0
$424$	1.35639	0.0658720
$425$	0	0
$426$	0	0
$427$	−22.1464	−1.07174
$428$	14.8025	0.715505
$429$	0	0
$430$	0	0
$431$	−36.5869	−1.76233	−0.881163	−	0.472812i	$-0.843239\pi$
$431$	−36.5869	−1.76233	−0.881163	+	0.472812i	$0.843239\pi$
$432$	0	0
$433$	−23.3148	−1.12044	−0.560218	−	0.828345i	$-0.689283\pi$
$433$	−23.3148	−1.12044	−0.560218	+	0.828345i	$0.689283\pi$
$434$	14.0373	0.673810
$435$	0	0
$436$	36.9386	1.76904
$437$	−9.06269	−0.433527
$438$	0	0
$439$	9.56634	0.456576	0.228288	−	0.973594i	$-0.426687\pi$
$439$	9.56634	0.456576	0.228288	+	0.973594i	$0.426687\pi$
$440$	0	0
$441$	0	0
$442$	13.7347	0.653291
$443$	0.440711	0.0209388	0.0104694	−	0.999945i	$-0.496667\pi$
$443$	0.440711	0.0209388	0.0104694	+	0.999945i	$0.496667\pi$
$444$	0	0
$445$	0	0
$446$	−47.9115	−2.26867
$447$	0	0
$448$	−13.4139	−0.633748
$449$	−3.72389	−0.175741	−0.0878705	−	0.996132i	$-0.528006\pi$
$449$	−3.72389	−0.175741	−0.0878705	+	0.996132i	$0.528006\pi$
$450$	0	0
$451$	−49.6076	−2.33593
$452$	−6.97390	−0.328025
$453$	0	0
$454$	−18.0455	−0.846918
$455$	0	0
$456$	0	0
$457$	−8.05539	−0.376815	−0.188408	−	0.982091i	$-0.560333\pi$
$457$	−8.05539	−0.376815	−0.188408	+	0.982091i	$0.560333\pi$
$458$	8.01829	0.374670
$459$	0	0
$460$	0	0
$461$	6.32986	0.294811	0.147405	−	0.989076i	$-0.452908\pi$
$461$	6.32986	0.294811	0.147405	+	0.989076i	$0.452908\pi$
$462$	0	0
$463$	−23.0057	−1.06916	−0.534582	−	0.845117i	$-0.679531\pi$
$463$	−23.0057	−1.06916	−0.534582	+	0.845117i	$0.679531\pi$
$464$	−16.0813	−0.746556
$465$	0	0
$466$	31.9365	1.47943
$467$	26.9284	1.24610	0.623050	−	0.782182i	$-0.285893\pi$
$467$	26.9284	1.24610	0.623050	+	0.782182i	$0.285893\pi$
$468$	0	0
$469$	−19.5801	−0.904125
$470$	0	0
$471$	0	0
$472$	−4.65738	−0.214373
$473$	−50.5708	−2.32525
$474$	0	0
$475$	0	0
$476$	10.4915	0.480877
$477$	0	0
$478$	36.1383	1.65293
$479$	−7.28981	−0.333080	−0.166540	−	0.986035i	$-0.553260\pi$
$479$	−7.28981	−0.333080	−0.166540	+	0.986035i	$0.553260\pi$
$480$	0	0
$481$	8.71278	0.397268
$482$	−22.9420	−1.04498
$483$	0	0
$484$	46.8850	2.13114
$485$	0	0
$486$	0	0
$487$	21.0219	0.952592	0.476296	−	0.879285i	$-0.341979\pi$
$487$	21.0219	0.952592	0.476296	+	0.879285i	$0.341979\pi$
$488$	5.62657	0.254703
$489$	0	0
$490$	0	0
$491$	−19.1366	−0.863623	−0.431812	−	0.901964i	$-0.642125\pi$
$491$	−19.1366	−0.863623	−0.431812	+	0.901964i	$0.642125\pi$
$492$	0	0
$493$	14.9584	0.673691
$494$	4.08813	0.183933
$495$	0	0
$496$	17.3010	0.776838
$497$	0.934762	0.0419298
$498$	0	0
$499$	21.6990	0.971381	0.485690	−	0.874131i	$-0.338568\pi$
$499$	21.6990	0.971381	0.485690	+	0.874131i	$0.338568\pi$
$500$	0	0
$501$	0	0
$502$	−38.2731	−1.70821
$503$	−11.4061	−0.508572	−0.254286	−	0.967129i	$-0.581841\pi$
$503$	−11.4061	−0.508572	−0.254286	+	0.967129i	$0.581841\pi$
$504$	0	0
$505$	0	0
$506$	105.647	4.69659
$507$	0	0
$508$	−24.6158	−1.09215
$509$	−24.0347	−1.06532	−0.532660	−	0.846329i	$-0.678807\pi$
$509$	−24.0347	−1.06532	−0.532660	+	0.846329i	$0.678807\pi$
$510$	0	0
$511$	−14.1464	−0.625802
$512$	−31.9282	−1.41104
$513$	0	0
$514$	−21.6591	−0.955344
$515$	0	0
$516$	0	0
$517$	−18.4199	−0.810106
$518$	12.7664	0.560923
$519$	0	0
$520$	0	0
$521$	−17.5750	−0.769976	−0.384988	−	0.922922i	$-0.625794\pi$
$521$	−17.5750	−0.769976	−0.384988	+	0.922922i	$0.625794\pi$
$522$	0	0
$523$	−13.5166	−0.591040	−0.295520	−	0.955337i	$-0.595493\pi$
$523$	−13.5166	−0.591040	−0.295520	+	0.955337i	$0.595493\pi$
$524$	3.23485	0.141315
$525$	0	0
$526$	−10.4337	−0.454929
$527$	−16.0929	−0.701017
$528$	0	0
$529$	59.1324	2.57097
$530$	0	0
$531$	0	0
$532$	3.12280	0.135391
$533$	17.3969	0.753543
$534$	0	0
$535$	0	0
$536$	4.97456	0.214868
$537$	0	0
$538$	22.3564	0.963852
$539$	−28.1993	−1.21463
$540$	0	0
$541$	−2.18807	−0.0940723	−0.0470361	−	0.998893i	$-0.514978\pi$
$541$	−2.18807	−0.0940723	−0.0470361	+	0.998893i	$0.514978\pi$
$542$	−40.6382	−1.74556
$543$	0	0
$544$	27.2513	1.16839
$545$	0	0
$546$	0	0
$547$	−20.7265	−0.886203	−0.443102	−	0.896471i	$-0.646122\pi$
$547$	−20.7265	−0.886203	−0.443102	+	0.896471i	$0.646122\pi$
$548$	−5.73119	−0.244824
$549$	0	0
$550$	0	0
$551$	4.45237	0.189677
$552$	0	0
$553$	13.1129	0.557618
$554$	26.8887	1.14239
$555$	0	0
$556$	−32.7029	−1.38691
$557$	25.8328	1.09457	0.547285	−	0.836946i	$-0.315661\pi$
$557$	25.8328	1.09457	0.547285	+	0.836946i	$0.315661\pi$
$558$	0	0
$559$	17.7347	0.750096
$560$	0	0
$561$	0	0
$562$	16.7703	0.707413
$563$	−7.11168	−0.299722	−0.149861	−	0.988707i	$-0.547883\pi$
$563$	−7.11168	−0.299722	−0.149861	+	0.988707i	$0.547883\pi$
$564$	0	0
$565$	0	0
$566$	27.7586	1.16678
$567$	0	0
$568$	−0.237488	−0.00996476
$569$	−5.93747	−0.248912	−0.124456	−	0.992225i	$-0.539719\pi$
$569$	−5.93747	−0.248912	−0.124456	+	0.992225i	$0.539719\pi$
$570$	0	0
$571$	26.0194	1.08888	0.544440	−	0.838800i	$-0.316742\pi$
$571$	26.0194	1.08888	0.544440	+	0.838800i	$0.316742\pi$
$572$	−24.8447	−1.03881
$573$	0	0
$574$	25.4908	1.06397
$575$	0	0
$576$	0	0
$577$	26.8338	1.11711	0.558553	−	0.829469i	$-0.311357\pi$
$577$	26.8338	1.11711	0.558553	+	0.829469i	$0.311357\pi$
$578$	11.6773	0.485711
$579$	0	0
$580$	0	0
$581$	−17.8095	−0.738861
$582$	0	0
$583$	−21.2375	−0.879567
$584$	3.59408	0.148724
$585$	0	0
$586$	−2.38990	−0.0987259
$587$	21.5132	0.887944	0.443972	−	0.896041i	$-0.353569\pi$
$587$	21.5132	0.887944	0.443972	+	0.896041i	$0.353569\pi$
$588$	0	0
$589$	−4.79005	−0.197371
$590$	0	0
$591$	0	0
$592$	15.7347	0.646690
$593$	23.0718	0.947445	0.473723	−	0.880674i	$-0.342910\pi$
$593$	23.0718	0.947445	0.473723	+	0.880674i	$0.342910\pi$
$594$	0	0
$595$	0	0
$596$	3.17354	0.129993
$597$	0	0
$598$	−37.0494	−1.51506
$599$	−7.29483	−0.298059	−0.149029	−	0.988833i	$-0.547615\pi$
$599$	−7.29483	−0.298059	−0.149029	+	0.988833i	$0.547615\pi$
$600$	0	0
$601$	−29.7184	−1.21224	−0.606120	−	0.795373i	$-0.707275\pi$
$601$	−29.7184	−1.21224	−0.606120	+	0.795373i	$0.707275\pi$
$602$	25.9857	1.05910
$603$	0	0
$604$	1.11293	0.0452844
$605$	0	0
$606$	0	0
$607$	−34.3504	−1.39424	−0.697120	−	0.716954i	$-0.745536\pi$
$607$	−34.3504	−1.39424	−0.697120	+	0.716954i	$0.745536\pi$
$608$	8.11135	0.328959
$609$	0	0
$610$	0	0
$611$	6.45967	0.261330
$612$	0	0
$613$	−13.4809	−0.544490	−0.272245	−	0.962228i	$-0.587766\pi$
$613$	−13.4809	−0.544490	−0.272245	+	0.962228i	$0.587766\pi$
$614$	−43.4108	−1.75192
$615$	0	0
$616$	−2.97812	−0.119992
$617$	−8.64511	−0.348039	−0.174020	−	0.984742i	$-0.555676\pi$
$617$	−8.64511	−0.348039	−0.174020	+	0.984742i	$0.555676\pi$
$618$	0	0
$619$	47.6158	1.91384	0.956919	−	0.290354i	$-0.0937729\pi$
$619$	47.6158	1.91384	0.956919	+	0.290354i	$0.0937729\pi$
$620$	0	0
$621$	0	0
$622$	31.9365	1.28054
$623$	5.33880	0.213895
$624$	0	0
$625$	0	0
$626$	−20.8813	−0.834586
$627$	0	0
$628$	1.43191	0.0571393
$629$	−14.6359	−0.583572
$630$	0	0
$631$	41.0770	1.63525	0.817624	−	0.575752i	$-0.195291\pi$
$631$	41.0770	1.63525	0.817624	+	0.575752i	$0.195291\pi$
$632$	−3.33150	−0.132520
$633$	0	0
$634$	−63.8215	−2.53468
$635$	0	0
$636$	0	0
$637$	9.88921	0.391825
$638$	−51.9029	−2.05486
$639$	0	0
$640$	0	0
$641$	20.5704	0.812484	0.406242	−	0.913766i	$-0.366839\pi$
$641$	20.5704	0.812484	0.406242	+	0.913766i	$0.366839\pi$
$642$	0	0
$643$	38.5939	1.52199	0.760997	−	0.648755i	$-0.224710\pi$
$643$	38.5939	1.52199	0.760997	+	0.648755i	$0.224710\pi$
$644$	−28.3010	−1.11521
$645$	0	0
$646$	−6.86733	−0.270191
$647$	−39.0399	−1.53482	−0.767408	−	0.641159i	$-0.778454\pi$
$647$	−39.0399	−1.53482	−0.767408	+	0.641159i	$0.778454\pi$
$648$	0	0
$649$	72.9224	2.86245
$650$	0	0
$651$	0	0
$652$	34.5921	1.35473
$653$	−13.1790	−0.515733	−0.257866	−	0.966181i	$-0.583019\pi$
$653$	−13.1790	−0.515733	−0.257866	+	0.966181i	$0.583019\pi$
$654$	0	0
$655$	0	0
$656$	31.4176	1.22665
$657$	0	0
$658$	9.46504	0.368986
$659$	17.5498	0.683645	0.341822	−	0.939765i	$-0.388956\pi$
$659$	17.5498	0.683645	0.341822	+	0.939765i	$0.388956\pi$
$660$	0	0
$661$	20.2731	0.788534	0.394267	−	0.918996i	$-0.370999\pi$
$661$	20.2731	0.788534	0.394267	+	0.918996i	$0.370999\pi$
$662$	10.6495	0.413903
$663$	0	0
$664$	4.52471	0.175593
$665$	0	0
$666$	0	0
$667$	−40.3504	−1.56237
$668$	4.59374	0.177737
$669$	0	0
$670$	0	0
$671$	−88.0974	−3.40096
$672$	0	0
$673$	−39.1602	−1.50952	−0.754758	−	0.656004i	$-0.772246\pi$
$673$	−39.1602	−1.50952	−0.754758	+	0.656004i	$0.772246\pi$
$674$	26.6175	1.02527
$675$	0	0
$676$	−19.6037	−0.753990
$677$	−5.49676	−0.211258	−0.105629	−	0.994406i	$-0.533686\pi$
$677$	−5.49676	−0.211258	−0.105629	+	0.994406i	$0.533686\pi$
$678$	0	0
$679$	8.73252	0.335123
$680$	0	0
$681$	0	0
$682$	55.8395	2.13820
$683$	−35.5988	−1.36215	−0.681074	−	0.732214i	$-0.738487\pi$
$683$	−35.5988	−1.36215	−0.681074	+	0.732214i	$0.738487\pi$
$684$	0	0
$685$	0	0
$686$	35.0037	1.33645
$687$	0	0
$688$	32.0275	1.22104
$689$	7.44777	0.283738
$690$	0	0
$691$	−12.8673	−0.489496	−0.244748	−	0.969587i	$-0.578705\pi$
$691$	−12.8673	−0.489496	−0.244748	+	0.969587i	$0.578705\pi$
$692$	29.7825	1.13216
$693$	0	0
$694$	−58.4947	−2.22043
$695$	0	0
$696$	0	0
$697$	−29.2237	−1.10693
$698$	−33.3484	−1.26226
$699$	0	0
$700$	0	0
$701$	1.17421	0.0443492	0.0221746	−	0.999754i	$-0.492941\pi$
$701$	1.17421	0.0443492	0.0221746	+	0.999754i	$0.492941\pi$
$702$	0	0
$703$	−4.35639	−0.164304
$704$	−53.3599	−2.01108
$705$	0	0
$706$	−75.5879	−2.84479
$707$	−23.2579	−0.874703
$708$	0	0
$709$	−38.6076	−1.44994	−0.724970	−	0.688780i	$-0.758147\pi$
$709$	−38.6076	−1.44994	−0.724970	+	0.688780i	$0.758147\pi$
$710$	0	0
$711$	0	0
$712$	−1.35639	−0.0508328
$713$	43.4108	1.62575
$714$	0	0
$715$	0	0
$716$	−41.2339	−1.54098
$717$	0	0
$718$	65.7425	2.45349
$719$	−1.33216	−0.0496814	−0.0248407	−	0.999691i	$-0.507908\pi$
$719$	−1.33216	−0.0496814	−0.0248407	+	0.999691i	$0.507908\pi$
$720$	0	0
$721$	−4.60199	−0.171387
$722$	−2.04406	−0.0760722
$723$	0	0
$724$	36.4037	1.35293
$725$	0	0
$726$	0	0
$727$	−39.7703	−1.47500	−0.737500	−	0.675348i	$-0.763994\pi$
$727$	−39.7703	−1.47500	−0.737500	+	0.675348i	$0.763994\pi$
$728$	1.04440	0.0387079
$729$	0	0
$730$	0	0
$731$	−29.7911	−1.10186
$732$	0	0
$733$	9.40367	0.347332	0.173666	−	0.984805i	$-0.444439\pi$
$733$	9.40367	0.347332	0.173666	+	0.984805i	$0.444439\pi$
$734$	47.6568	1.75905
$735$	0	0
$736$	−73.5106	−2.70964
$737$	−77.8887	−2.86907
$738$	0	0
$739$	−27.3067	−1.00449	−0.502246	−	0.864725i	$-0.667493\pi$
$739$	−27.3067	−1.00449	−0.502246	+	0.864725i	$0.667493\pi$
$740$	0	0
$741$	0	0
$742$	10.9129	0.400623
$743$	38.7723	1.42242	0.711209	−	0.702981i	$-0.248148\pi$
$743$	38.7723	1.42242	0.711209	+	0.702981i	$0.248148\pi$
$744$	0	0
$745$	0	0
$746$	−62.1067	−2.27389
$747$	0	0
$748$	41.7347	1.52597
$749$	9.74285	0.355996
$750$	0	0
$751$	−26.4474	−0.965081	−0.482540	−	0.875874i	$-0.660286\pi$
$751$	−26.4474	−0.965081	−0.482540	+	0.875874i	$0.660286\pi$
$752$	11.6657	0.425405
$753$	0	0
$754$	18.2018	0.662872
$755$	0	0
$756$	0	0
$757$	−41.4750	−1.50743	−0.753717	−	0.657199i	$-0.771741\pi$
$757$	−41.4750	−1.50743	−0.753717	+	0.657199i	$0.771741\pi$
$758$	65.3697	2.37433
$759$	0	0
$760$	0	0
$761$	14.4498	0.523806	0.261903	−	0.965094i	$-0.415650\pi$
$761$	14.4498	0.523806	0.261903	+	0.965094i	$0.415650\pi$
$762$	0	0
$763$	24.3126	0.880176
$764$	21.6098	0.781816
$765$	0	0
$766$	62.5402	2.25967
$767$	−25.5732	−0.923393
$768$	0	0
$769$	−18.7622	−0.676582	−0.338291	−	0.941042i	$-0.609849\pi$
$769$	−18.7622	−0.676582	−0.338291	+	0.941042i	$0.609849\pi$
$770$	0	0
$771$	0	0
$772$	0.439636	0.0158229
$773$	−23.8536	−0.857955	−0.428977	−	0.903315i	$-0.641126\pi$
$773$	−23.8536	−0.857955	−0.428977	+	0.903315i	$0.641126\pi$
$774$	0	0
$775$	0	0
$776$	−2.21860	−0.0796432
$777$	0	0
$778$	28.9584	1.03821
$779$	−8.69845	−0.311654
$780$	0	0
$781$	3.71843	0.133056
$782$	62.2365	2.22557
$783$	0	0
$784$	17.8592	0.637829
$785$	0	0
$786$	0	0
$787$	39.0632	1.39245	0.696226	−	0.717822i	$-0.254861\pi$
$787$	39.0632	1.39245	0.696226	+	0.717822i	$0.254861\pi$
$788$	2.55765	0.0911125
$789$	0	0
$790$	0	0
$791$	−4.59015	−0.163207
$792$	0	0
$793$	30.8949	1.09711
$794$	43.9531	1.55984
$795$	0	0
$796$	−13.7248	−0.486462
$797$	48.9075	1.73239	0.866197	−	0.499703i	$-0.166558\pi$
$797$	48.9075	1.73239	0.866197	+	0.499703i	$0.166558\pi$
$798$	0	0
$799$	−10.8511	−0.383885
$800$	0	0
$801$	0	0
$802$	−37.6376	−1.32903
$803$	−56.2738	−1.98586
$804$	0	0
$805$	0	0
$806$	−19.5823	−0.689759
$807$	0	0
$808$	5.90895	0.207876
$809$	−3.80537	−0.133790	−0.0668949	−	0.997760i	$-0.521309\pi$
$809$	−3.80537	−0.133790	−0.0668949	+	0.997760i	$0.521309\pi$
$810$	0	0
$811$	2.79005	0.0979720	0.0489860	−	0.998799i	$-0.484401\pi$
$811$	2.79005	0.0979720	0.0489860	+	0.998799i	$0.484401\pi$
$812$	13.9038	0.487929
$813$	0	0
$814$	50.7841	1.77998
$815$	0	0
$816$	0	0
$817$	−8.86733	−0.310229
$818$	14.4498	0.505226
$819$	0	0
$820$	0	0
$821$	47.7304	1.66580	0.832901	−	0.553423i	$-0.186679\pi$
$821$	47.7304	1.66580	0.832901	+	0.553423i	$0.186679\pi$
$822$	0	0
$823$	30.3448	1.05775	0.528876	−	0.848699i	$-0.322614\pi$
$823$	30.3448	1.05775	0.528876	+	0.848699i	$0.322614\pi$
$824$	1.16919	0.0407307
$825$	0	0
$826$	−37.4711	−1.30379
$827$	38.0236	1.32221	0.661106	−	0.750293i	$-0.270087\pi$
$827$	38.0236	1.32221	0.661106	+	0.750293i	$0.270087\pi$
$828$	0	0
$829$	4.13833	0.143730	0.0718651	−	0.997414i	$-0.477105\pi$
$829$	4.13833	0.143730	0.0718651	+	0.997414i	$0.477105\pi$
$830$	0	0
$831$	0	0
$832$	18.7128	0.648749
$833$	−16.6121	−0.575576
$834$	0	0
$835$	0	0
$836$	12.4223	0.429636
$837$	0	0
$838$	12.2456	0.423017
$839$	8.56864	0.295822	0.147911	−	0.989001i	$-0.452745\pi$
$839$	8.56864	0.295822	0.147911	+	0.989001i	$0.452745\pi$
$840$	0	0
$841$	−9.17644	−0.316429
$842$	27.4427	0.945737
$843$	0	0
$844$	−35.2177	−1.21224
$845$	0	0
$846$	0	0
$847$	30.8592	1.06034
$848$	13.4502	0.461880
$849$	0	0
$850$	0	0
$851$	39.4806	1.35338
$852$	0	0
$853$	−35.1048	−1.20197	−0.600983	−	0.799262i	$-0.705224\pi$
$853$	−35.1048	−1.20197	−0.600983	+	0.799262i	$0.705224\pi$
$854$	45.2687	1.54906
$855$	0	0
$856$	−2.47529	−0.0846036
$857$	−11.8720	−0.405540	−0.202770	−	0.979226i	$-0.564994\pi$
$857$	−11.8720	−0.405540	−0.202770	+	0.979226i	$0.564994\pi$
$858$	0	0
$859$	49.3229	1.68287	0.841437	−	0.540355i	$-0.181710\pi$
$859$	49.3229	1.68287	0.841437	+	0.540355i	$0.181710\pi$
$860$	0	0
$861$	0	0
$862$	74.7858	2.54722
$863$	26.9536	0.917513	0.458756	−	0.888562i	$-0.348295\pi$
$863$	26.9536	0.917513	0.458756	+	0.888562i	$0.348295\pi$
$864$	0	0
$865$	0	0
$866$	47.6568	1.61945
$867$	0	0
$868$	−14.9584	−0.507720
$869$	52.1626	1.76949
$870$	0	0
$871$	27.3148	0.925525
$872$	−6.17692	−0.209177
$873$	0	0
$874$	18.5247	0.626608
$875$	0	0
$876$	0	0
$877$	17.6076	0.594568	0.297284	−	0.954789i	$-0.403919\pi$
$877$	17.6076	0.594568	0.297284	+	0.954789i	$0.403919\pi$
$878$	−19.5542	−0.659922
$879$	0	0
$880$	0	0
$881$	52.4503	1.76710	0.883548	−	0.468340i	$-0.155148\pi$
$881$	52.4503	1.76710	0.883548	+	0.468340i	$0.155148\pi$
$882$	0	0
$883$	43.0575	1.44900	0.724501	−	0.689274i	$-0.242070\pi$
$883$	43.0575	1.44900	0.724501	+	0.689274i	$0.242070\pi$
$884$	−14.6359	−0.492259
$885$	0	0
$886$	−0.900841	−0.0302644
$887$	−1.19734	−0.0402027	−0.0201013	−	0.999798i	$-0.506399\pi$
$887$	−1.19734	−0.0402027	−0.0201013	+	0.999798i	$0.506399\pi$
$888$	0	0
$889$	−16.2018	−0.543392
$890$	0	0
$891$	0	0
$892$	51.0554	1.70946
$893$	−3.22984	−0.108082
$894$	0	0
$895$	0	0
$896$	4.16100	0.139009
$897$	0	0
$898$	7.61186	0.254011
$899$	−21.3271	−0.711298
$900$	0	0
$901$	−12.5109	−0.416800
$902$	101.401	3.37629
$903$	0	0
$904$	1.16618	0.0387867
$905$	0	0
$906$	0	0
$907$	31.9640	1.06135	0.530674	−	0.847576i	$-0.321939\pi$
$907$	31.9640	1.06135	0.530674	+	0.847576i	$0.321939\pi$
$908$	19.2297	0.638159
$909$	0	0
$910$	0	0
$911$	6.64282	0.220086	0.110043	−	0.993927i	$-0.464901\pi$
$911$	6.64282	0.220086	0.110043	+	0.993927i	$0.464901\pi$
$912$	0	0
$913$	−70.8451	−2.34463
$914$	16.4657	0.544638
$915$	0	0
$916$	−8.54445	−0.282317
$917$	2.12915	0.0703106
$918$	0	0
$919$	−21.0138	−0.693180	−0.346590	−	0.938017i	$-0.612661\pi$
$919$	−21.0138	−0.693180	−0.346590	+	0.938017i	$0.612661\pi$
$920$	0	0
$921$	0	0
$922$	−12.9386	−0.426111
$923$	−1.30402	−0.0429223
$924$	0	0
$925$	0	0
$926$	47.0250	1.54534
$927$	0	0
$928$	36.1147	1.18552
$929$	33.5633	1.10118	0.550588	−	0.834777i	$-0.314404\pi$
$929$	33.5633	1.10118	0.550588	+	0.834777i	$0.314404\pi$
$930$	0	0
$931$	−4.94461	−0.162053
$932$	−34.0322	−1.11476
$933$	0	0
$934$	−55.0435	−1.80108
$935$	0	0
$936$	0	0
$937$	10.5191	0.343642	0.171821	−	0.985128i	$-0.445035\pi$
$937$	10.5191	0.343642	0.171821	+	0.985128i	$0.445035\pi$
$938$	40.0230	1.30680
$939$	0	0
$940$	0	0
$941$	−54.6639	−1.78199	−0.890996	−	0.454011i	$-0.849993\pi$
$941$	−54.6639	−1.78199	−0.890996	+	0.454011i	$0.849993\pi$
$942$	0	0
$943$	78.8314	2.56710
$944$	−46.1833	−1.50314
$945$	0	0
$946$	103.370	3.36084
$947$	16.4088	0.533214	0.266607	−	0.963805i	$-0.414097\pi$
$947$	16.4088	0.533214	0.266607	+	0.963805i	$0.414097\pi$
$948$	0	0
$949$	19.7347	0.640614
$950$	0	0
$951$	0	0
$952$	−1.75440	−0.0568604
$953$	−4.92122	−0.159414	−0.0797070	−	0.996818i	$-0.525398\pi$
$953$	−4.92122	−0.159414	−0.0797070	+	0.996818i	$0.525398\pi$
$954$	0	0
$955$	0	0
$956$	−38.5097	−1.24549
$957$	0	0
$958$	14.9008	0.481424
$959$	−3.77221	−0.121811
$960$	0	0
$961$	−8.05539	−0.259851
$962$	−17.8095	−0.574200
$963$	0	0
$964$	24.4474	0.787399
$965$	0	0
$966$	0	0
$967$	9.16832	0.294833	0.147417	−	0.989074i	$-0.452904\pi$
$967$	9.16832	0.294833	0.147417	+	0.989074i	$0.452904\pi$
$968$	−7.84016	−0.251992
$969$	0	0
$970$	0	0
$971$	58.9302	1.89116	0.945580	−	0.325391i	$-0.105496\pi$
$971$	58.9302	1.89116	0.945580	+	0.325391i	$0.105496\pi$
$972$	0	0
$973$	−21.5247	−0.690051
$974$	−42.9701	−1.37685
$975$	0	0
$976$	55.7940	1.78592
$977$	17.6565	0.564882	0.282441	−	0.959285i	$-0.408856\pi$
$977$	17.6565	0.564882	0.282441	+	0.959285i	$0.408856\pi$
$978$	0	0
$979$	21.2375	0.678753
$980$	0	0
$981$	0	0
$982$	39.1164	1.24826
$983$	2.97021	0.0947351	0.0473676	−	0.998878i	$-0.484917\pi$
$983$	2.97021	0.0947351	0.0473676	+	0.998878i	$0.484917\pi$
$984$	0	0
$985$	0	0
$986$	−30.5759	−0.973734
$987$	0	0
$988$	−4.35639	−0.138595
$989$	80.3619	2.55536
$990$	0	0
$991$	−41.9503	−1.33259	−0.666297	−	0.745686i	$-0.732122\pi$
$991$	−41.9503	−1.33259	−0.666297	+	0.745686i	$0.732122\pi$
$992$	−38.8538	−1.23361
$993$	0	0
$994$	−1.91071	−0.0606041
$995$	0	0
$996$	0	0
$997$	19.6412	0.622042	0.311021	−	0.950403i	$-0.399329\pi$
$997$	19.6412	0.622042	0.311021	+	0.950403i	$0.399329\pi$
$998$	−44.3541	−1.40401
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4275.2.a.bq.1.2	✓	6
3.2	odd	2	inner	4275.2.a.bq.1.5	yes	6
5.4	even	2		4275.2.a.bu.1.5	yes	6
15.14	odd	2		4275.2.a.bu.1.2	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4275.2.a.bq.1.2	✓	6	1.1	even	1	trivial
4275.2.a.bq.1.5	yes	6	3.2	odd	2	inner
4275.2.a.bu.1.2	yes	6	15.14	odd	2
4275.2.a.bu.1.5	yes	6	5.4	even	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)

\(f(q)\)	\(=\)	\(q-2.04406 q^{2} +2.17819 q^{4} +1.43366 q^{7} -0.364240 q^{8} +5.70304 q^{11} -2.00000 q^{13} -2.93050 q^{14} -3.61186 q^{16} +3.35965 q^{17} +1.00000 q^{19} -11.6574 q^{22} -9.06269 q^{23} +4.08813 q^{26} +3.12280 q^{28} +4.45237 q^{29} -4.79005 q^{31} +8.11135 q^{32} -6.86733 q^{34} -4.35639 q^{37} -2.04406 q^{38} -8.69845 q^{41} -8.86733 q^{43} +12.4223 q^{44} +18.5247 q^{46} -3.22984 q^{47} -4.94461 q^{49} -4.35639 q^{52} -3.72389 q^{53} -0.522198 q^{56} -9.10092 q^{58} +12.7866 q^{59} -15.4474 q^{61} +9.79117 q^{62} -9.35639 q^{64} -13.6574 q^{67} +7.31796 q^{68} +0.652009 q^{71} -9.86733 q^{73} +8.90473 q^{74} +2.17819 q^{76} +8.17625 q^{77} +9.14644 q^{79} +17.7802 q^{82} -12.4223 q^{83} +18.1254 q^{86} -2.07728 q^{88} +3.72389 q^{89} -2.86733 q^{91} -19.7403 q^{92} +6.60199 q^{94} +6.09105 q^{97} +10.1071 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 8 q^{4} - 8 q^{7} - 12 q^{13} + 6 q^{19} - 10 q^{22} - 26 q^{28} - 2 q^{31} - 8 q^{34} - 16 q^{37} - 20 q^{43} + 18 q^{46} + 10 q^{49} - 16 q^{52} - 56 q^{58} - 6 q^{61} - 46 q^{64} - 22 q^{67} - 26 q^{73}+ \cdots - 40 q^{97}+O(q^{100}) \) 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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Database

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