LMFDB - Embedded newform 4275.2.a.br.1.1

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:	6.6.66064384.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 9x^{4} + 13x^{2} - 1 $ x^6 - 9x^4 + 13x^2 - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 95)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.30397$ 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)

$f(q)$	$=$	$q-2.41987 q^{2} +3.85577 q^{4} +3.18676 q^{7} -4.49073 q^{8} +O(q^{10})$	$q-2.41987 q^{2} +3.85577 q^{4} +3.18676 q^{7} -4.49073 q^{8} -4.15544 q^{11} -2.07086 q^{13} -7.71155 q^{14} +3.15544 q^{16} +5.79470 q^{17} -1.00000 q^{19} +10.0556 q^{22} -2.60794 q^{23} +5.01121 q^{26} +12.2874 q^{28} -6.00000 q^{29} +2.59933 q^{31} +1.34571 q^{32} -14.0224 q^{34} -4.30266 q^{37} +2.41987 q^{38} +0.599328 q^{41} -3.18676 q^{43} -16.0224 q^{44} +6.31087 q^{46} +11.7086 q^{47} +3.15544 q^{49} -7.98476 q^{52} +11.7503 q^{53} -14.3109 q^{56} +14.5192 q^{58} +1.71155 q^{59} -8.75476 q^{61} -6.29004 q^{62} -9.56732 q^{64} -4.76228 q^{67} +22.3430 q^{68} -13.7115 q^{71} +2.72714 q^{73} +10.4119 q^{74} -3.85577 q^{76} -13.2424 q^{77} -1.40067 q^{79} -1.45030 q^{82} -7.07154 q^{83} +7.71155 q^{86} +18.6609 q^{88} -16.5353 q^{89} -6.59933 q^{91} -10.0556 q^{92} -28.3333 q^{94} -2.07086 q^{97} -7.63575 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 8 q^{4}+O(q^{10}) $ 6 * q + 8 * q^4	$ 6 q + 8 q^{4} - 2 q^{11} - 16 q^{14} - 4 q^{16} - 6 q^{19} - 8 q^{26} - 36 q^{29} - 8 q^{34} - 12 q^{41} - 20 q^{44} - 8 q^{46} - 4 q^{49} - 40 q^{56} - 20 q^{59} - 14 q^{61} - 12 q^{64} - 52 q^{71} + 40 q^{74} - 8 q^{76} - 24 q^{79} + 16 q^{86} - 24 q^{89} - 24 q^{91} - 48 q^{94}+O(q^{100}) $ 6 * q + 8 * q^4 - 2 * q^11 - 16 * q^14 - 4 * q^16 - 6 * q^19 - 8 * q^26 - 36 * q^29 - 8 * q^34 - 12 * q^41 - 20 * q^44 - 8 * q^46 - 4 * q^49 - 40 * q^56 - 20 * q^59 - 14 * q^61 - 12 * q^64 - 52 * q^71 + 40 * q^74 - 8 * q^76 - 24 * q^79 + 16 * q^86 - 24 * q^89 - 24 * q^91 - 48 * q^94

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−2.41987	−1.71111	−0.855553	−	0.517715i	$-0.826783\pi$
$2$	−2.41987	−1.71111	−0.855553	+	0.517715i	$0.826783\pi$
$3$	0	0
$3$	0	0
$4$	3.85577	1.92789
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.18676	1.20448	0.602241	−	0.798314i	$-0.294275\pi$
$7$	3.18676	1.20448	0.602241	+	0.798314i	$0.294275\pi$
$8$	−4.49073	−1.58771
$9$	0	0
$10$	0	0
$11$	−4.15544	−1.25291	−0.626456	−	0.779457i	$-0.715495\pi$
$11$	−4.15544	−1.25291	−0.626456	+	0.779457i	$0.715495\pi$
$12$	0	0
$13$	−2.07086	−0.574353	−0.287176	−	0.957878i	$-0.592717\pi$
$13$	−2.07086	−0.574353	−0.287176	+	0.957878i	$0.592717\pi$
$14$	−7.71155	−2.06100
$15$	0	0
$16$	3.15544	0.788859
$17$	5.79470	1.40542	0.702710	−	0.711476i	$-0.251973\pi$
$17$	5.79470	1.40542	0.702710	+	0.711476i	$0.251973\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	10.0556	2.14386
$23$	−2.60794	−0.543793	−0.271896	−	0.962327i	$-0.587651\pi$
$23$	−2.60794	−0.543793	−0.271896	+	0.962327i	$0.587651\pi$
$24$	0	0
$25$	0	0
$26$	5.01121	0.982779
$27$	0	0
$28$	12.2874	2.32210
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	0	0
$31$	2.59933	0.466853	0.233427	−	0.972374i	$-0.425006\pi$
$31$	2.59933	0.466853	0.233427	+	0.972374i	$0.425006\pi$
$32$	1.34571	0.237890
$33$	0	0
$34$	−14.0224	−2.40482
$35$	0	0
$36$	0	0
$37$	−4.30266	−0.707353	−0.353677	−	0.935368i	$-0.615069\pi$
$37$	−4.30266	−0.707353	−0.353677	+	0.935368i	$0.615069\pi$
$38$	2.41987	0.392555
$39$	0	0
$40$	0	0
$41$	0.599328	0.0935993	0.0467997	−	0.998904i	$-0.485098\pi$
$41$	0.599328	0.0935993	0.0467997	+	0.998904i	$0.485098\pi$
$42$	0	0
$43$	−3.18676	−0.485976	−0.242988	−	0.970029i	$-0.578128\pi$
$43$	−3.18676	−0.485976	−0.242988	+	0.970029i	$0.578128\pi$
$44$	−16.0224	−2.41547
$45$	0	0
$46$	6.31087	0.930487
$47$	11.7086	1.70787	0.853937	−	0.520376i	$-0.174208\pi$
$47$	11.7086	1.70787	0.853937	+	0.520376i	$0.174208\pi$
$48$	0	0
$49$	3.15544	0.450777
$50$	0	0
$51$	0	0
$52$	−7.98476	−1.10729
$53$	11.7503	1.61403	0.807017	−	0.590529i	$-0.201081\pi$
$53$	11.7503	1.61403	0.807017	+	0.590529i	$0.201081\pi$
$54$	0	0
$55$	0	0
$56$	−14.3109	−1.91237
$57$	0	0
$58$	14.5192	1.90647
$59$	1.71155	0.222824	0.111412	−	0.993774i	$-0.464463\pi$
$59$	1.71155	0.222824	0.111412	+	0.993774i	$0.464463\pi$
$60$	0	0
$61$	−8.75476	−1.12093	−0.560466	−	0.828177i	$-0.689378\pi$
$61$	−8.75476	−1.12093	−0.560466	+	0.828177i	$0.689378\pi$
$62$	−6.29004	−0.798836
$63$	0	0
$64$	−9.56732	−1.19591
$65$	0	0
$66$	0	0
$67$	−4.76228	−0.581805	−0.290902	−	0.956753i	$-0.593956\pi$
$67$	−4.76228	−0.581805	−0.290902	+	0.956753i	$0.593956\pi$
$68$	22.3430	2.70949
$69$	0	0
$70$	0	0
$71$	−13.7115	−1.62726	−0.813631	−	0.581382i	$-0.802512\pi$
$71$	−13.7115	−1.62726	−0.813631	+	0.581382i	$0.802512\pi$
$72$	0	0
$73$	2.72714	0.319188	0.159594	−	0.987183i	$-0.448982\pi$
$73$	2.72714	0.319188	0.159594	+	0.987183i	$0.448982\pi$
$74$	10.4119	1.21036
$75$	0	0
$76$	−3.85577	−0.442287
$77$	−13.2424	−1.50911
$78$	0	0
$79$	−1.40067	−0.157588	−0.0787939	−	0.996891i	$-0.525107\pi$
$79$	−1.40067	−0.157588	−0.0787939	+	0.996891i	$0.525107\pi$
$80$	0	0
$81$	0	0
$82$	−1.45030	−0.160158
$83$	−7.07154	−0.776203	−0.388101	−	0.921617i	$-0.626869\pi$
$83$	−7.07154	−0.776203	−0.388101	+	0.921617i	$0.626869\pi$
$84$	0	0
$85$	0	0
$86$	7.71155	0.831557
$87$	0	0
$88$	18.6609	1.98926
$89$	−16.5353	−1.75274	−0.876370	−	0.481639i	$-0.840042\pi$
$89$	−16.5353	−1.75274	−0.876370	+	0.481639i	$0.840042\pi$
$90$	0	0
$91$	−6.59933	−0.691798
$92$	−10.0556	−1.04837
$93$	0	0
$94$	−28.3333	−2.92236
$95$	0	0
$96$	0	0
$97$	−2.07086	−0.210264	−0.105132	−	0.994458i	$-0.533526\pi$
$97$	−2.07086	−0.210264	−0.105132	+	0.994458i	$0.533526\pi$
$98$	−7.63575	−0.771327
$99$	0	0
$100$	0	0
$101$	1.71155	0.170305	0.0851525	−	0.996368i	$-0.472862\pi$
$101$	1.71155	0.170305	0.0851525	+	0.996368i	$0.472862\pi$
$102$	0	0
$103$	−5.75296	−0.566856	−0.283428	−	0.958994i	$-0.591472\pi$
$103$	−5.75296	−0.566856	−0.283428	+	0.958994i	$0.591472\pi$
$104$	9.29966	0.911907
$105$	0	0
$106$	−28.4343	−2.76178
$107$	−15.4324	−1.49191	−0.745955	−	0.665996i	$-0.768007\pi$
$107$	−15.4324	−1.49191	−0.745955	+	0.665996i	$0.768007\pi$
$108$	0	0
$109$	11.7115	1.12176	0.560881	−	0.827896i	$-0.310462\pi$
$109$	11.7115	1.12176	0.560881	+	0.827896i	$0.310462\pi$
$110$	0	0
$111$	0	0
$112$	10.0556	0.950167
$113$	−10.5927	−0.996477	−0.498239	−	0.867040i	$-0.666020\pi$
$113$	−10.5927	−0.996477	−0.498239	+	0.867040i	$0.666020\pi$
$114$	0	0
$115$	0	0
$116$	−23.1346	−2.14800
$117$	0	0
$118$	−4.14172	−0.381276
$119$	18.4663	1.69280
$120$	0	0
$121$	6.26765	0.569787
$122$	21.1854	1.91804
$123$	0	0
$124$	10.0224	0.900040
$125$	0	0
$126$	0	0
$127$	6.07484	0.539055	0.269528	−	0.962993i	$-0.413132\pi$
$127$	6.07484	0.539055	0.269528	+	0.962993i	$0.413132\pi$
$128$	20.4602	1.80845
$129$	0	0
$130$	0	0
$131$	13.5785	1.18636	0.593181	−	0.805069i	$-0.297872\pi$
$131$	13.5785	1.18636	0.593181	+	0.805069i	$0.297872\pi$
$132$	0	0
$133$	−3.18676	−0.276327
$134$	11.5241	0.995530
$135$	0	0
$136$	−26.0224	−2.23140
$137$	−7.94302	−0.678618	−0.339309	−	0.940675i	$-0.610193\pi$
$137$	−7.94302	−0.678618	−0.339309	+	0.940675i	$0.610193\pi$
$138$	0	0
$139$	3.26765	0.277159	0.138579	−	0.990351i	$-0.455746\pi$
$139$	3.26765	0.277159	0.138579	+	0.990351i	$0.455746\pi$
$140$	0	0
$141$	0	0
$142$	33.1802	2.78442
$143$	8.60532	0.719613
$144$	0	0
$145$	0	0
$146$	−6.59933	−0.546164
$147$	0	0
$148$	−16.5901	−1.36370
$149$	−8.44389	−0.691751	−0.345875	−	0.938280i	$-0.612418\pi$
$149$	−8.44389	−0.691751	−0.345875	+	0.938280i	$0.612418\pi$
$150$	0	0
$151$	0.887783	0.0722468	0.0361234	−	0.999347i	$-0.488499\pi$
$151$	0.887783	0.0722468	0.0361234	+	0.999347i	$0.488499\pi$
$152$	4.49073	0.364246
$153$	0	0
$154$	32.0448	2.58225
$155$	0	0
$156$	0	0
$157$	−4.14172	−0.330545	−0.165273	−	0.986248i	$-0.552850\pi$
$157$	−4.14172	−0.330545	−0.165273	+	0.986248i	$0.552850\pi$
$158$	3.38944	0.269650
$159$	0	0
$160$	0	0
$161$	−8.31087	−0.654989
$162$	0	0
$163$	24.7126	1.93564	0.967819	−	0.251647i	$-0.0809723\pi$
$163$	24.7126	1.93564	0.967819	+	0.251647i	$0.0809723\pi$
$164$	2.31087	0.180449
$165$	0	0
$166$	17.1122	1.32817
$167$	−3.60464	−0.278935	−0.139468	−	0.990227i	$-0.544539\pi$
$167$	−3.60464	−0.278935	−0.139468	+	0.990227i	$0.544539\pi$
$168$	0	0
$169$	−8.71155	−0.670119
$170$	0	0
$171$	0	0
$172$	−12.2874	−0.936907
$173$	−22.4205	−1.70460	−0.852300	−	0.523054i	$-0.824793\pi$
$173$	−22.4205	−1.70460	−0.852300	+	0.523054i	$0.824793\pi$
$174$	0	0
$175$	0	0
$176$	−13.1122	−0.988371
$177$	0	0
$178$	40.0133	2.99912
$179$	5.13464	0.383781	0.191890	−	0.981416i	$-0.438538\pi$
$179$	5.13464	0.383781	0.191890	+	0.981416i	$0.438538\pi$
$180$	0	0
$181$	20.8462	1.54948	0.774742	−	0.632277i	$-0.217880\pi$
$181$	20.8462	1.54948	0.774742	+	0.632277i	$0.217880\pi$
$182$	15.9695	1.18374
$183$	0	0
$184$	11.7115	0.863387
$185$	0	0
$186$	0	0
$187$	−24.0795	−1.76087
$188$	45.1457	3.29259
$189$	0	0
$190$	0	0
$191$	5.26765	0.381154	0.190577	−	0.981672i	$-0.438964\pi$
$191$	5.26765	0.381154	0.190577	+	0.981672i	$0.438964\pi$
$192$	0	0
$193$	−2.07086	−0.149064	−0.0745318	−	0.997219i	$-0.523746\pi$
$193$	−2.07086	−0.149064	−0.0745318	+	0.997219i	$0.523746\pi$
$194$	5.01121	0.359784
$195$	0	0
$196$	12.1666	0.869046
$197$	10.4318	0.743232	0.371616	−	0.928387i	$-0.378804\pi$
$197$	10.4318	0.743232	0.371616	+	0.928387i	$0.378804\pi$
$198$	0	0
$199$	−2.73235	−0.193691	−0.0968455	−	0.995299i	$-0.530875\pi$
$199$	−2.73235	−0.193691	−0.0968455	+	0.995299i	$0.530875\pi$
$200$	0	0
$201$	0	0
$202$	−4.14172	−0.291410
$203$	−19.1206	−1.34200
$204$	0	0
$205$	0	0
$206$	13.9214	0.969951
$207$	0	0
$208$	−6.53446	−0.453083
$209$	4.15544	0.287438
$210$	0	0
$211$	−15.7340	−1.08317	−0.541585	−	0.840646i	$-0.682176\pi$
$211$	−15.7340	−1.08317	−0.541585	+	0.840646i	$0.682176\pi$
$212$	45.3066	3.11167
$213$	0	0
$214$	37.3445	2.55282
$215$	0	0
$216$	0	0
$217$	8.28343	0.562316
$218$	−28.3404	−1.91946
$219$	0	0
$220$	0	0
$221$	−12.0000	−0.807207
$222$	0	0
$223$	−18.8219	−1.26041	−0.630203	−	0.776430i	$-0.717028\pi$
$223$	−18.8219	−1.26041	−0.630203	+	0.776430i	$0.717028\pi$
$224$	4.28845	0.286534
$225$	0	0
$226$	25.6330	1.70508
$227$	−14.4418	−0.958533	−0.479267	−	0.877669i	$-0.659097\pi$
$227$	−14.4418	−0.958533	−0.479267	+	0.877669i	$0.659097\pi$
$228$	0	0
$229$	−4.17785	−0.276080	−0.138040	−	0.990427i	$-0.544080\pi$
$229$	−4.17785	−0.276080	−0.138040	+	0.990427i	$0.544080\pi$
$230$	0	0
$231$	0	0
$232$	26.9444	1.76898
$233$	12.0847	0.791697	0.395849	−	0.918316i	$-0.370450\pi$
$233$	12.0847	0.791697	0.395849	+	0.918316i	$0.370450\pi$
$234$	0	0
$235$	0	0
$236$	6.59933	0.429580
$237$	0	0
$238$	−44.6861	−2.89657
$239$	11.3541	0.734435	0.367218	−	0.930135i	$-0.380310\pi$
$239$	11.3541	0.734435	0.367218	+	0.930135i	$0.380310\pi$
$240$	0	0
$241$	−3.40067	−0.219057	−0.109528	−	0.993984i	$-0.534934\pi$
$241$	−3.40067	−0.219057	−0.109528	+	0.993984i	$0.534934\pi$
$242$	−15.1669	−0.974966
$243$	0	0
$244$	−33.7564	−2.16103
$245$	0	0
$246$	0	0
$247$	2.07086	0.131766
$248$	−11.6729	−0.741229
$249$	0	0
$250$	0	0
$251$	−3.04322	−0.192086	−0.0960432	−	0.995377i	$-0.530619\pi$
$251$	−3.04322	−0.192086	−0.0960432	+	0.995377i	$0.530619\pi$
$252$	0	0
$253$	10.8371	0.681324
$254$	−14.7003	−0.922381
$255$	0	0
$256$	−30.3765	−1.89853
$257$	17.2881	1.07840	0.539201	−	0.842177i	$-0.318726\pi$
$257$	17.2881	1.07840	0.539201	+	0.842177i	$0.318726\pi$
$258$	0	0
$259$	−13.7115	−0.851994
$260$	0	0
$261$	0	0
$262$	−32.8583	−2.02999
$263$	−1.19336	−0.0735859	−0.0367930	−	0.999323i	$-0.511714\pi$
$263$	−1.19336	−0.0735859	−0.0367930	+	0.999323i	$0.511714\pi$
$264$	0	0
$265$	0	0
$266$	7.71155	0.472825
$267$	0	0
$268$	−18.3623	−1.12165
$269$	−22.1089	−1.34800	−0.674000	−	0.738731i	$-0.735425\pi$
$269$	−22.1089	−1.34800	−0.674000	+	0.738731i	$0.735425\pi$
$270$	0	0
$271$	−4.08644	−0.248234	−0.124117	−	0.992268i	$-0.539610\pi$
$271$	−4.08644	−0.248234	−0.124117	+	0.992268i	$0.539610\pi$
$272$	18.2848	1.10868
$273$	0	0
$274$	19.2211	1.16119
$275$	0	0
$276$	0	0
$277$	10.0199	0.602037	0.301019	−	0.953618i	$-0.402673\pi$
$277$	10.0199	0.602037	0.301019	+	0.953618i	$0.402673\pi$
$278$	−7.90730	−0.474248
$279$	0	0
$280$	0	0
$281$	−0.599328	−0.0357529	−0.0178765	−	0.999840i	$-0.505691\pi$
$281$	−0.599328	−0.0357529	−0.0178765	+	0.999840i	$0.505691\pi$
$282$	0	0
$283$	−5.41856	−0.322100	−0.161050	−	0.986946i	$-0.551488\pi$
$283$	−5.41856	−0.322100	−0.161050	+	0.986946i	$0.551488\pi$
$284$	−52.8686	−3.13717
$285$	0	0
$286$	−20.8238	−1.23133
$287$	1.90991	0.112739
$288$	0	0
$289$	16.5785	0.975207
$290$	0	0
$291$	0	0
$292$	10.5152	0.615358
$293$	3.46691	0.202539	0.101269	−	0.994859i	$-0.467710\pi$
$293$	3.46691	0.202539	0.101269	+	0.994859i	$0.467710\pi$
$294$	0	0
$295$	0	0
$296$	19.3221	1.12307
$297$	0	0
$298$	20.4331	1.18366
$299$	5.40067	0.312329
$300$	0	0
$301$	−10.1554	−0.585350
$302$	−2.14832	−0.123622
$303$	0	0
$304$	−3.15544	−0.180977
$305$	0	0
$306$	0	0
$307$	16.5901	0.946846	0.473423	−	0.880835i	$-0.343018\pi$
$307$	16.5901	0.946846	0.473423	+	0.880835i	$0.343018\pi$
$308$	−51.0596	−2.90939
$309$	0	0
$310$	0	0
$311$	4.15544	0.235633	0.117817	−	0.993035i	$-0.462411\pi$
$311$	4.15544	0.235633	0.117817	+	0.993035i	$0.462411\pi$
$312$	0	0
$313$	0.919237	0.0519583	0.0259792	−	0.999662i	$-0.491730\pi$
$313$	0.919237	0.0519583	0.0259792	+	0.999662i	$0.491730\pi$
$314$	10.0224	0.565598
$315$	0	0
$316$	−5.40067	−0.303812
$317$	−26.7292	−1.50126	−0.750630	−	0.660723i	$-0.770250\pi$
$317$	−26.7292	−1.50126	−0.750630	+	0.660723i	$0.770250\pi$
$318$	0	0
$319$	24.9326	1.39596
$320$	0	0
$321$	0	0
$322$	20.1112	1.12076
$323$	−5.79470	−0.322426
$324$	0	0
$325$	0	0
$326$	−59.8012	−3.31208
$327$	0	0
$328$	−2.69142	−0.148609
$329$	37.3125	2.05710
$330$	0	0
$331$	8.00000	0.439720	0.219860	−	0.975531i	$-0.429440\pi$
$331$	8.00000	0.439720	0.219860	+	0.975531i	$0.429440\pi$
$332$	−27.2663	−1.49643
$333$	0	0
$334$	8.72275	0.477288
$335$	0	0
$336$	0	0
$337$	−22.5040	−1.22587	−0.612935	−	0.790133i	$-0.710011\pi$
$337$	−22.5040	−1.22587	−0.612935	+	0.790133i	$0.710011\pi$
$338$	21.0808	1.14664
$339$	0	0
$340$	0	0
$341$	−10.8013	−0.584926
$342$	0	0
$343$	−12.2517	−0.661530
$344$	14.3109	0.771591
$345$	0	0
$346$	54.2547	2.91675
$347$	−14.4543	−0.775946	−0.387973	−	0.921671i	$-0.626825\pi$
$347$	−14.4543	−0.775946	−0.387973	+	0.921671i	$0.626825\pi$
$348$	0	0
$349$	−13.3541	−0.714828	−0.357414	−	0.933946i	$-0.616342\pi$
$349$	−13.3541	−0.714828	−0.357414	+	0.933946i	$0.616342\pi$
$350$	0	0
$351$	0	0
$352$	−5.59201	−0.298055
$353$	17.6410	0.938937	0.469469	−	0.882949i	$-0.344446\pi$
$353$	17.6410	0.938937	0.469469	+	0.882949i	$0.344446\pi$
$354$	0	0
$355$	0	0
$356$	−63.7564	−3.37908
$357$	0	0
$358$	−12.4252	−0.656690
$359$	−12.4663	−0.657947	−0.328973	−	0.944339i	$-0.606703\pi$
$359$	−12.4663	−0.657947	−0.328973	+	0.944339i	$0.606703\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−50.4451	−2.65133
$363$	0	0
$364$	−25.4455	−1.33371
$365$	0	0
$366$	0	0
$367$	16.4291	0.857594	0.428797	−	0.903401i	$-0.358938\pi$
$367$	16.4291	0.857594	0.428797	+	0.903401i	$0.358938\pi$
$368$	−8.22918	−0.428976
$369$	0	0
$370$	0	0
$371$	37.4455	1.94407
$372$	0	0
$373$	5.29334	0.274079	0.137039	−	0.990566i	$-0.456241\pi$
$373$	5.29334	0.274079	0.137039	+	0.990566i	$0.456241\pi$
$374$	58.2693	3.01303
$375$	0	0
$376$	−52.5801	−2.71161
$377$	12.4252	0.639928
$378$	0	0
$379$	14.5353	0.746629	0.373314	−	0.927705i	$-0.378221\pi$
$379$	14.5353	0.746629	0.373314	+	0.927705i	$0.378221\pi$
$380$	0	0
$381$	0	0
$382$	−12.7470	−0.652195
$383$	0.453598	0.0231778	0.0115889	−	0.999933i	$-0.496311\pi$
$383$	0.453598	0.0231778	0.0115889	+	0.999933i	$0.496311\pi$
$384$	0	0
$385$	0	0
$386$	5.01121	0.255064
$387$	0	0
$388$	−7.98476	−0.405365
$389$	−16.1554	−0.819113	−0.409557	−	0.912285i	$-0.634317\pi$
$389$	−16.1554	−0.819113	−0.409557	+	0.912285i	$0.634317\pi$
$390$	0	0
$391$	−15.1122	−0.764258
$392$	−14.1702	−0.715704
$393$	0	0
$394$	−25.2435	−1.27175
$395$	0	0
$396$	0	0
$397$	−32.7563	−1.64399	−0.821995	−	0.569495i	$-0.807139\pi$
$397$	−32.7563	−1.64399	−0.821995	+	0.569495i	$0.807139\pi$
$398$	6.61192	0.331426
$399$	0	0
$400$	0	0
$401$	−12.0864	−0.603568	−0.301784	−	0.953376i	$-0.597582\pi$
$401$	−12.0864	−0.603568	−0.301784	+	0.953376i	$0.597582\pi$
$402$	0	0
$403$	−5.38284	−0.268138
$404$	6.59933	0.328329
$405$	0	0
$406$	46.2693	2.29631
$407$	17.8794	0.886251
$408$	0	0
$409$	−19.1346	−0.946147	−0.473073	−	0.881023i	$-0.656855\pi$
$409$	−19.1346	−0.946147	−0.473073	+	0.881023i	$0.656855\pi$
$410$	0	0
$411$	0	0
$412$	−22.1821	−1.09283
$413$	5.45428	0.268388
$414$	0	0
$415$	0	0
$416$	−2.78678	−0.136633
$417$	0	0
$418$	−10.0556	−0.491836
$419$	−8.04484	−0.393016	−0.196508	−	0.980502i	$-0.562960\pi$
$419$	−8.04484	−0.393016	−0.196508	+	0.980502i	$0.562960\pi$
$420$	0	0
$421$	−29.3591	−1.43087	−0.715437	−	0.698678i	$-0.753772\pi$
$421$	−29.3591	−1.43087	−0.715437	+	0.698678i	$0.753772\pi$
$422$	38.0742	1.85342
$423$	0	0
$424$	−52.7676	−2.56262
$425$	0	0
$426$	0	0
$427$	−27.8993	−1.35014
$428$	−59.5040	−2.87623
$429$	0	0
$430$	0	0
$431$	32.0448	1.54355	0.771773	−	0.635898i	$-0.219370\pi$
$431$	32.0448	1.54355	0.771773	+	0.635898i	$0.219370\pi$
$432$	0	0
$433$	−0.482831	−0.0232034	−0.0116017	−	0.999933i	$-0.503693\pi$
$433$	−0.482831	−0.0232034	−0.0116017	+	0.999933i	$0.503693\pi$
$434$	−20.0448	−0.962183
$435$	0	0
$436$	45.1571	2.16263
$437$	2.60794	0.124755
$438$	0	0
$439$	−27.3591	−1.30578	−0.652889	−	0.757454i	$-0.726443\pi$
$439$	−27.3591	−1.30578	−0.652889	+	0.757454i	$0.726443\pi$
$440$	0	0
$441$	0	0
$442$	29.0384	1.38122
$443$	23.3815	1.11089	0.555444	−	0.831554i	$-0.312548\pi$
$443$	23.3815	1.11089	0.555444	+	0.831554i	$0.312548\pi$
$444$	0	0
$445$	0	0
$446$	45.5465	2.15669
$447$	0	0
$448$	−30.4887	−1.44046
$449$	−23.1346	−1.09179	−0.545895	−	0.837853i	$-0.683810\pi$
$449$	−23.1346	−1.09179	−0.545895	+	0.837853i	$0.683810\pi$
$450$	0	0
$451$	−2.49047	−0.117272
$452$	−40.8430	−1.92109
$453$	0	0
$454$	34.9472	1.64015
$455$	0	0
$456$	0	0
$457$	−21.2503	−0.994049	−0.497025	−	0.867736i	$-0.665574\pi$
$457$	−21.2503	−0.994049	−0.497025	+	0.867736i	$0.665574\pi$
$458$	10.1099	0.472403
$459$	0	0
$460$	0	0
$461$	−31.5785	−1.47076	−0.735379	−	0.677656i	$-0.762996\pi$
$461$	−31.5785	−1.47076	−0.735379	+	0.677656i	$0.762996\pi$
$462$	0	0
$463$	15.6119	0.725547	0.362774	−	0.931877i	$-0.381830\pi$
$463$	15.6119	0.725547	0.362774	+	0.931877i	$0.381830\pi$
$464$	−18.9326	−0.878925
$465$	0	0
$466$	−29.2435	−1.35468
$467$	−29.8264	−1.38020	−0.690101	−	0.723713i	$-0.742434\pi$
$467$	−29.8264	−1.38020	−0.690101	+	0.723713i	$0.742434\pi$
$468$	0	0
$469$	−15.1762	−0.700774
$470$	0	0
$471$	0	0
$472$	−7.68608	−0.353781
$473$	13.2424	0.608885
$474$	0	0
$475$	0	0
$476$	71.2019	3.26353
$477$	0	0
$478$	−27.4754	−1.25670
$479$	−4.53531	−0.207223	−0.103612	−	0.994618i	$-0.533040\pi$
$479$	−4.53531	−0.207223	−0.103612	+	0.994618i	$0.533040\pi$
$480$	0	0
$481$	8.91020	0.406270
$482$	8.22918	0.374829
$483$	0	0
$484$	24.1666	1.09848
$485$	0	0
$486$	0	0
$487$	−39.2550	−1.77881	−0.889407	−	0.457116i	$-0.848882\pi$
$487$	−39.2550	−1.77881	−0.889407	+	0.457116i	$0.848882\pi$
$488$	39.3153	1.77972
$489$	0	0
$490$	0	0
$491$	−38.8910	−1.75513	−0.877564	−	0.479460i	$-0.840832\pi$
$491$	−38.8910	−1.75513	−0.877564	+	0.479460i	$0.840832\pi$
$492$	0	0
$493$	−34.7682	−1.56588
$494$	−5.01121	−0.225465
$495$	0	0
$496$	8.20202	0.368281
$497$	−43.6954	−1.96001
$498$	0	0
$499$	4.73235	0.211849	0.105924	−	0.994374i	$-0.466220\pi$
$499$	4.73235	0.211849	0.105924	+	0.994374i	$0.466220\pi$
$500$	0	0
$501$	0	0
$502$	7.36420	0.328680
$503$	1.85567	0.0827400	0.0413700	−	0.999144i	$-0.486828\pi$
$503$	1.85567	0.0827400	0.0413700	+	0.999144i	$0.486828\pi$
$504$	0	0
$505$	0	0
$506$	−26.2244	−1.16582
$507$	0	0
$508$	23.4232	1.03924
$509$	−22.8878	−1.01448	−0.507242	−	0.861804i	$-0.669335\pi$
$509$	−22.8878	−1.01448	−0.507242	+	0.861804i	$0.669335\pi$
$510$	0	0
$511$	8.69074	0.384456
$512$	32.5867	1.44014
$513$	0	0
$514$	−41.8350	−1.84526
$515$	0	0
$516$	0	0
$517$	−48.6543	−2.13982
$518$	33.1802	1.45785
$519$	0	0
$520$	0	0
$521$	29.7340	1.30267	0.651334	−	0.758791i	$-0.274210\pi$
$521$	29.7340	1.30267	0.651334	+	0.758791i	$0.274210\pi$
$522$	0	0
$523$	−30.6497	−1.34022	−0.670109	−	0.742263i	$-0.733752\pi$
$523$	−30.6497	−1.34022	−0.670109	+	0.742263i	$0.733752\pi$
$524$	52.3557	2.28717
$525$	0	0
$526$	2.88778	0.125913
$527$	15.0623	0.656125
$528$	0	0
$529$	−16.1987	−0.704289
$530$	0	0
$531$	0	0
$532$	−12.2874	−0.532727
$533$	−1.24112	−0.0537590
$534$	0	0
$535$	0	0
$536$	21.3861	0.923739
$537$	0	0
$538$	53.5006	2.30657
$539$	−13.1122	−0.564783
$540$	0	0
$541$	2.21946	0.0954220	0.0477110	−	0.998861i	$-0.484807\pi$
$541$	2.21946	0.0954220	0.0477110	+	0.998861i	$0.484807\pi$
$542$	9.88865	0.424754
$543$	0	0
$544$	7.79798	0.334336
$545$	0	0
$546$	0	0
$547$	14.4297	0.616970	0.308485	−	0.951229i	$-0.400178\pi$
$547$	14.4297	0.616970	0.308485	+	0.951229i	$0.400178\pi$
$548$	−30.6265	−1.30830
$549$	0	0
$550$	0	0
$551$	6.00000	0.255609
$552$	0	0
$553$	−4.46360	−0.189812
$554$	−24.2469	−1.03015
$555$	0	0
$556$	12.5993	0.534331
$557$	19.4610	0.824588	0.412294	−	0.911051i	$-0.364728\pi$
$557$	19.4610	0.824588	0.412294	+	0.911051i	$0.364728\pi$
$558$	0	0
$559$	6.59933	0.279122
$560$	0	0
$561$	0	0
$562$	1.45030	0.0611771
$563$	−12.3649	−0.521118	−0.260559	−	0.965458i	$-0.583907\pi$
$563$	−12.3649	−0.521118	−0.260559	+	0.965458i	$0.583907\pi$
$564$	0	0
$565$	0	0
$566$	13.1122	0.551148
$567$	0	0
$568$	61.5748	2.58362
$569$	−15.0898	−0.632597	−0.316299	−	0.948660i	$-0.602440\pi$
$569$	−15.0898	−0.632597	−0.316299	+	0.948660i	$0.602440\pi$
$570$	0	0
$571$	17.1571	0.718000	0.359000	−	0.933337i	$-0.383118\pi$
$571$	17.1571	0.718000	0.359000	+	0.933337i	$0.383118\pi$
$572$	33.1802	1.38733
$573$	0	0
$574$	−4.62175	−0.192908
$575$	0	0
$576$	0	0
$577$	22.5165	0.937374	0.468687	−	0.883364i	$-0.344727\pi$
$577$	22.5165	0.937374	0.468687	+	0.883364i	$0.344727\pi$
$578$	−40.1179	−1.66868
$579$	0	0
$580$	0	0
$581$	−22.5353	−0.934922
$582$	0	0
$583$	−48.8278	−2.02224
$584$	−12.2469	−0.506778
$585$	0	0
$586$	−8.38946	−0.346566
$587$	7.49544	0.309370	0.154685	−	0.987964i	$-0.450564\pi$
$587$	7.49544	0.309370	0.154685	+	0.987964i	$0.450564\pi$
$588$	0	0
$589$	−2.59933	−0.107103
$590$	0	0
$591$	0	0
$592$	−13.5768	−0.558002
$593$	27.8094	1.14199	0.570997	−	0.820952i	$-0.306557\pi$
$593$	27.8094	1.14199	0.570997	+	0.820952i	$0.306557\pi$
$594$	0	0
$595$	0	0
$596$	−32.5577	−1.33362
$597$	0	0
$598$	−13.0689	−0.534428
$599$	45.4903	1.85869	0.929343	−	0.369219i	$-0.120375\pi$
$599$	45.4903	1.85869	0.929343	+	0.369219i	$0.120375\pi$
$600$	0	0
$601$	−16.5993	−0.677101	−0.338550	−	0.940948i	$-0.609937\pi$
$601$	−16.5993	−0.677101	−0.338550	+	0.940948i	$0.609937\pi$
$602$	24.5748	1.00160
$603$	0	0
$604$	3.42309	0.139284
$605$	0	0
$606$	0	0
$607$	5.08417	0.206360	0.103180	−	0.994663i	$-0.467098\pi$
$607$	5.08417	0.206360	0.103180	+	0.994663i	$0.467098\pi$
$608$	−1.34571	−0.0545758
$609$	0	0
$610$	0	0
$611$	−24.2469	−0.980923
$612$	0	0
$613$	−4.63706	−0.187289	−0.0936445	−	0.995606i	$-0.529852\pi$
$613$	−4.63706	−0.187289	−0.0936445	+	0.995606i	$0.529852\pi$
$614$	−40.1458	−1.62015
$615$	0	0
$616$	59.4679	2.39603
$617$	40.2874	1.62191	0.810955	−	0.585108i	$-0.198948\pi$
$617$	40.2874	1.62191	0.810955	+	0.585108i	$0.198948\pi$
$618$	0	0
$619$	−43.3815	−1.74365	−0.871825	−	0.489818i	$-0.837063\pi$
$619$	−43.3815	−1.74365	−0.871825	+	0.489818i	$0.837063\pi$
$620$	0	0
$621$	0	0
$622$	−10.0556	−0.403194
$623$	−52.6940	−2.11114
$624$	0	0
$625$	0	0
$626$	−2.22443	−0.0889062
$627$	0	0
$628$	−15.9695	−0.637253
$629$	−24.9326	−0.994129
$630$	0	0
$631$	7.53369	0.299911	0.149956	−	0.988693i	$-0.452087\pi$
$631$	7.53369	0.299911	0.149956	+	0.988693i	$0.452087\pi$
$632$	6.29004	0.250204
$633$	0	0
$634$	64.6812	2.56882
$635$	0	0
$636$	0	0
$637$	−6.53446	−0.258905
$638$	−60.3337	−2.38863
$639$	0	0
$640$	0	0
$641$	23.3075	0.920591	0.460296	−	0.887766i	$-0.347743\pi$
$641$	23.3075	0.920591	0.460296	+	0.887766i	$0.347743\pi$
$642$	0	0
$643$	−1.87419	−0.0739110	−0.0369555	−	0.999317i	$-0.511766\pi$
$643$	−1.87419	−0.0739110	−0.0369555	+	0.999317i	$0.511766\pi$
$644$	−32.0448	−1.26274
$645$	0	0
$646$	14.0224	0.551705
$647$	−47.0371	−1.84922	−0.924609	−	0.380917i	$-0.875608\pi$
$647$	−47.0371	−1.84922	−0.924609	+	0.380917i	$0.875608\pi$
$648$	0	0
$649$	−7.11222	−0.279179
$650$	0	0
$651$	0	0
$652$	95.2861	3.73169
$653$	24.1630	0.945571	0.472785	−	0.881178i	$-0.343249\pi$
$653$	24.1630	0.945571	0.472785	+	0.881178i	$0.343249\pi$
$654$	0	0
$655$	0	0
$656$	1.89114	0.0738367
$657$	0	0
$658$	−90.2914	−3.51992
$659$	−10.2885	−0.400781	−0.200391	−	0.979716i	$-0.564221\pi$
$659$	−10.2885	−0.400781	−0.200391	+	0.979716i	$0.564221\pi$
$660$	0	0
$661$	5.93598	0.230883	0.115441	−	0.993314i	$-0.463172\pi$
$661$	5.93598	0.230883	0.115441	+	0.993314i	$0.463172\pi$
$662$	−19.3590	−0.752407
$663$	0	0
$664$	31.7564	1.23239
$665$	0	0
$666$	0	0
$667$	15.6476	0.605879
$668$	−13.8987	−0.537755
$669$	0	0
$670$	0	0
$671$	36.3799	1.40443
$672$	0	0
$673$	40.7053	1.56907	0.784537	−	0.620082i	$-0.212901\pi$
$673$	40.7053	1.56907	0.784537	+	0.620082i	$0.212901\pi$
$674$	54.4567	2.09759
$675$	0	0
$676$	−33.5897	−1.29191
$677$	18.8761	0.725469	0.362734	−	0.931893i	$-0.381843\pi$
$677$	18.8761	0.725469	0.362734	+	0.931893i	$0.381843\pi$
$678$	0	0
$679$	−6.59933	−0.253259
$680$	0	0
$681$	0	0
$682$	26.1379	1.00087
$683$	−19.6576	−0.752179	−0.376089	−	0.926583i	$-0.622731\pi$
$683$	−19.6576	−0.752179	−0.376089	+	0.926583i	$0.622731\pi$
$684$	0	0
$685$	0	0
$686$	29.6475	1.13195
$687$	0	0
$688$	−10.0556	−0.383367
$689$	−24.3333	−0.927025
$690$	0	0
$691$	−48.2451	−1.83533	−0.917665	−	0.397354i	$-0.869928\pi$
$691$	−48.2451	−1.83533	−0.917665	+	0.397354i	$0.869928\pi$
$692$	−86.4483	−3.28627
$693$	0	0
$694$	34.9775	1.32773
$695$	0	0
$696$	0	0
$697$	3.47293	0.131546
$698$	32.3152	1.22315
$699$	0	0
$700$	0	0
$701$	−0.512889	−0.0193715	−0.00968577	−	0.999953i	$-0.503083\pi$
$701$	−0.512889	−0.0193715	−0.00968577	+	0.999953i	$0.503083\pi$
$702$	0	0
$703$	4.30266	0.162278
$704$	39.7564	1.49838
$705$	0	0
$706$	−42.6890	−1.60662
$707$	5.45428	0.205129
$708$	0	0
$709$	−8.84618	−0.332225	−0.166113	−	0.986107i	$-0.553122\pi$
$709$	−8.84618	−0.332225	−0.166113	+	0.986107i	$0.553122\pi$
$710$	0	0
$711$	0	0
$712$	74.2556	2.78285
$713$	−6.77889	−0.253871
$714$	0	0
$715$	0	0
$716$	19.7980	0.739885
$717$	0	0
$718$	30.1669	1.12582
$719$	−7.84456	−0.292553	−0.146276	−	0.989244i	$-0.546729\pi$
$719$	−7.84456	−0.292553	−0.146276	+	0.989244i	$0.546729\pi$
$720$	0	0
$721$	−18.3333	−0.682767
$722$	−2.41987	−0.0900582
$723$	0	0
$724$	80.3781	2.98723
$725$	0	0
$726$	0	0
$727$	2.91130	0.107974	0.0539870	−	0.998542i	$-0.482807\pi$
$727$	2.91130	0.107974	0.0539870	+	0.998542i	$0.482807\pi$
$728$	29.6358	1.09838
$729$	0	0
$730$	0	0
$731$	−18.4663	−0.683001
$732$	0	0
$733$	33.7775	1.24760	0.623800	−	0.781584i	$-0.285588\pi$
$733$	33.7775	1.24760	0.623800	+	0.781584i	$0.285588\pi$
$734$	−39.7564	−1.46743
$735$	0	0
$736$	−3.50953	−0.129363
$737$	19.7893	0.728950
$738$	0	0
$739$	−35.8030	−1.31703	−0.658517	−	0.752566i	$-0.728816\pi$
$739$	−35.8030	−1.31703	−0.658517	+	0.752566i	$0.728816\pi$
$740$	0	0
$741$	0	0
$742$	−90.6133	−3.32652
$743$	−5.66948	−0.207993	−0.103996	−	0.994578i	$-0.533163\pi$
$743$	−5.66948	−0.207993	−0.103996	+	0.994578i	$0.533163\pi$
$744$	0	0
$745$	0	0
$746$	−12.8092	−0.468978
$747$	0	0
$748$	−92.8451	−3.39475
$749$	−49.1795	−1.79698
$750$	0	0
$751$	−27.4679	−1.00232	−0.501159	−	0.865355i	$-0.667093\pi$
$751$	−27.4679	−1.00232	−0.501159	+	0.865355i	$0.667093\pi$
$752$	36.9457	1.34727
$753$	0	0
$754$	−30.0673	−1.09498
$755$	0	0
$756$	0	0
$757$	41.6370	1.51332	0.756662	−	0.653806i	$-0.226829\pi$
$757$	41.6370	1.51332	0.756662	+	0.653806i	$0.226829\pi$
$758$	−35.1736	−1.27756
$759$	0	0
$760$	0	0
$761$	9.46967	0.343275	0.171638	−	0.985160i	$-0.445094\pi$
$761$	9.46967	0.343275	0.171638	+	0.985160i	$0.445094\pi$
$762$	0	0
$763$	37.3219	1.35114
$764$	20.3109	0.734822
$765$	0	0
$766$	−1.09765	−0.0396597
$767$	−3.54437	−0.127980
$768$	0	0
$769$	1.90858	0.0688253	0.0344127	−	0.999408i	$-0.489044\pi$
$769$	1.90858	0.0688253	0.0344127	+	0.999408i	$0.489044\pi$
$770$	0	0
$771$	0	0
$772$	−7.98476	−0.287378
$773$	−28.4007	−1.02150	−0.510751	−	0.859729i	$-0.670633\pi$
$773$	−28.4007	−1.02150	−0.510751	+	0.859729i	$0.670633\pi$
$774$	0	0
$775$	0	0
$776$	9.29966	0.333838
$777$	0	0
$778$	39.0941	1.40159
$779$	−0.599328	−0.0214732
$780$	0	0
$781$	56.9775	2.03881
$782$	36.5696	1.30773
$783$	0	0
$784$	9.95678	0.355599
$785$	0	0
$786$	0	0
$787$	−15.6708	−0.558605	−0.279303	−	0.960203i	$-0.590103\pi$
$787$	−15.6708	−0.558605	−0.279303	+	0.960203i	$0.590103\pi$
$788$	40.2225	1.43287
$789$	0	0
$790$	0	0
$791$	−33.7564	−1.20024
$792$	0	0
$793$	18.1299	0.643811
$794$	79.2659	2.81304
$795$	0	0
$796$	−10.5353	−0.373414
$797$	−36.9225	−1.30786	−0.653932	−	0.756554i	$-0.726882\pi$
$797$	−36.9225	−1.30786	−0.653932	+	0.756554i	$0.726882\pi$
$798$	0	0
$799$	67.8478	2.40028
$800$	0	0
$801$	0	0
$802$	29.2476	1.03277
$803$	−11.3325	−0.399914
$804$	0	0
$805$	0	0
$806$	13.0258	0.458813
$807$	0	0
$808$	−7.68608	−0.270396
$809$	−43.0465	−1.51343	−0.756716	−	0.653743i	$-0.773198\pi$
$809$	−43.0465	−1.51343	−0.756716	+	0.653743i	$0.773198\pi$
$810$	0	0
$811$	32.2469	1.13234	0.566170	−	0.824288i	$-0.308425\pi$
$811$	32.2469	1.13234	0.566170	+	0.824288i	$0.308425\pi$
$812$	−73.7245	−2.58722
$813$	0	0
$814$	−43.2659	−1.51647
$815$	0	0
$816$	0	0
$817$	3.18676	0.111491
$818$	46.3033	1.61896
$819$	0	0
$820$	0	0
$821$	−5.18121	−0.180826	−0.0904128	−	0.995904i	$-0.528819\pi$
$821$	−5.18121	−0.180826	−0.0904128	+	0.995904i	$0.528819\pi$
$822$	0	0
$823$	25.8517	0.901133	0.450567	−	0.892743i	$-0.351222\pi$
$823$	25.8517	0.901133	0.450567	+	0.892743i	$0.351222\pi$
$824$	25.8350	0.900004
$825$	0	0
$826$	−13.1987	−0.459240
$827$	−9.97816	−0.346974	−0.173487	−	0.984836i	$-0.555504\pi$
$827$	−9.97816	−0.346974	−0.173487	+	0.984836i	$0.555504\pi$
$828$	0	0
$829$	21.9808	0.763425	0.381713	−	0.924281i	$-0.375334\pi$
$829$	21.9808	0.763425	0.381713	+	0.924281i	$0.375334\pi$
$830$	0	0
$831$	0	0
$832$	19.8126	0.686877
$833$	18.2848	0.633531
$834$	0	0
$835$	0	0
$836$	16.0224	0.554147
$837$	0	0
$838$	19.4675	0.672492
$839$	16.1089	0.556140	0.278070	−	0.960561i	$-0.410305\pi$
$839$	16.1089	0.556140	0.278070	+	0.960561i	$0.410305\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	71.0451	2.44838
$843$	0	0
$844$	−60.6666	−2.08823
$845$	0	0
$846$	0	0
$847$	19.9735	0.686298
$848$	37.0775	1.27324
$849$	0	0
$850$	0	0
$851$	11.2211	0.384653
$852$	0	0
$853$	−44.9615	−1.53945	−0.769727	−	0.638373i	$-0.779608\pi$
$853$	−44.9615	−1.53945	−0.769727	+	0.638373i	$0.779608\pi$
$854$	67.5128	2.31024
$855$	0	0
$856$	69.3029	2.36872
$857$	−46.2431	−1.57963	−0.789817	−	0.613343i	$-0.789824\pi$
$857$	−46.2431	−1.57963	−0.789817	+	0.613343i	$0.789824\pi$
$858$	0	0
$859$	−10.7772	−0.367713	−0.183856	−	0.982953i	$-0.558858\pi$
$859$	−10.7772	−0.367713	−0.183856	+	0.982953i	$0.558858\pi$
$860$	0	0
$861$	0	0
$862$	−77.5443	−2.64117
$863$	22.7966	0.776007	0.388003	−	0.921658i	$-0.373165\pi$
$863$	22.7966	0.776007	0.388003	+	0.921658i	$0.373165\pi$
$864$	0	0
$865$	0	0
$866$	1.16839	0.0397034
$867$	0	0
$868$	31.9390	1.08408
$869$	5.82040	0.197444
$870$	0	0
$871$	9.86201	0.334161
$872$	−52.5934	−1.78104
$873$	0	0
$874$	−6.31087	−0.213468
$875$	0	0
$876$	0	0
$877$	4.94644	0.167029	0.0835146	−	0.996507i	$-0.473385\pi$
$877$	4.94644	0.167029	0.0835146	+	0.996507i	$0.473385\pi$
$878$	66.2054	2.23432
$879$	0	0
$880$	0	0
$881$	−2.53033	−0.0852490	−0.0426245	−	0.999091i	$-0.513572\pi$
$881$	−2.53033	−0.0852490	−0.0426245	+	0.999091i	$0.513572\pi$
$882$	0	0
$883$	−29.7430	−1.00093	−0.500465	−	0.865757i	$-0.666838\pi$
$883$	−29.7430	−1.00093	−0.500465	+	0.865757i	$0.666838\pi$
$884$	−46.2693	−1.55620
$885$	0	0
$886$	−56.5801	−1.90085
$887$	45.5450	1.52925	0.764626	−	0.644474i	$-0.222924\pi$
$887$	45.5450	1.52925	0.764626	+	0.644474i	$0.222924\pi$
$888$	0	0
$889$	19.3591	0.649282
$890$	0	0
$891$	0	0
$892$	−72.5729	−2.42992
$893$	−11.7086	−0.391813
$894$	0	0
$895$	0	0
$896$	65.2019	2.17824
$897$	0	0
$898$	55.9828	1.86817
$899$	−15.5960	−0.520155
$900$	0	0
$901$	68.0897	2.26840
$902$	6.02662	0.200664
$903$	0	0
$904$	47.5689	1.58212
$905$	0	0
$906$	0	0
$907$	−33.2034	−1.10250	−0.551250	−	0.834340i	$-0.685849\pi$
$907$	−33.2034	−1.10250	−0.551250	+	0.834340i	$0.685849\pi$
$908$	−55.6841	−1.84794
$909$	0	0
$910$	0	0
$911$	−55.7788	−1.84803	−0.924017	−	0.382351i	$-0.875114\pi$
$911$	−55.7788	−1.84803	−0.924017	+	0.382351i	$0.875114\pi$
$912$	0	0
$913$	29.3853	0.972513
$914$	51.4231	1.70092
$915$	0	0
$916$	−16.1089	−0.532252
$917$	43.2715	1.42895
$918$	0	0
$919$	−28.5769	−0.942665	−0.471333	−	0.881956i	$-0.656227\pi$
$919$	−28.5769	−0.942665	−0.471333	+	0.881956i	$0.656227\pi$
$920$	0	0
$921$	0	0
$922$	76.4159	2.51662
$923$	28.3947	0.934622
$924$	0	0
$925$	0	0
$926$	−37.7788	−1.24149
$927$	0	0
$928$	−8.07426	−0.265051
$929$	54.4937	1.78788	0.893940	−	0.448186i	$-0.147930\pi$
$929$	54.4937	1.78788	0.893940	+	0.448186i	$0.147930\pi$
$930$	0	0
$931$	−3.15544	−0.103415
$932$	46.5960	1.52630
$933$	0	0
$934$	72.1761	2.36167
$935$	0	0
$936$	0	0
$937$	37.1484	1.21359	0.606793	−	0.794860i	$-0.292456\pi$
$937$	37.1484	1.21359	0.606793	+	0.794860i	$0.292456\pi$
$938$	36.7245	1.19910
$939$	0	0
$940$	0	0
$941$	−32.3973	−1.05612	−0.528061	−	0.849206i	$-0.677081\pi$
$941$	−32.3973	−1.05612	−0.528061	+	0.849206i	$0.677081\pi$
$942$	0	0
$943$	−1.56301	−0.0508986
$944$	5.40067	0.175777
$945$	0	0
$946$	−32.0448	−1.04187
$947$	35.4662	1.15250	0.576249	−	0.817275i	$-0.304516\pi$
$947$	35.4662	1.15250	0.576249	+	0.817275i	$0.304516\pi$
$948$	0	0
$949$	−5.64752	−0.183326
$950$	0	0
$951$	0	0
$952$	−82.9272	−2.68769
$953$	−14.3411	−0.464553	−0.232277	−	0.972650i	$-0.574617\pi$
$953$	−14.3411	−0.464553	−0.232277	+	0.972650i	$0.574617\pi$
$954$	0	0
$955$	0	0
$956$	43.7788	1.41591
$957$	0	0
$958$	10.9749	0.354581
$959$	−25.3125	−0.817383
$960$	0	0
$961$	−24.2435	−0.782048
$962$	−21.5615	−0.695172
$963$	0	0
$964$	−13.1122	−0.422316
$965$	0	0
$966$	0	0
$967$	27.9351	0.898331	0.449165	−	0.893449i	$-0.351721\pi$
$967$	27.9351	0.898331	0.449165	+	0.893449i	$0.351721\pi$
$968$	−28.1463	−0.904657
$969$	0	0
$970$	0	0
$971$	42.5993	1.36708	0.683539	−	0.729914i	$-0.260440\pi$
$971$	42.5993	1.36708	0.683539	+	0.729914i	$0.260440\pi$
$972$	0	0
$973$	10.4132	0.333833
$974$	94.9920	3.04374
$975$	0	0
$976$	−27.6251	−0.884258
$977$	39.5597	1.26563	0.632813	−	0.774304i	$-0.281900\pi$
$977$	39.5597	1.26563	0.632813	+	0.774304i	$0.281900\pi$
$978$	0	0
$979$	68.7114	2.19603
$980$	0	0
$981$	0	0
$982$	94.1112	3.00321
$983$	−39.7689	−1.26843	−0.634215	−	0.773157i	$-0.718677\pi$
$983$	−39.7689	−1.26843	−0.634215	+	0.773157i	$0.718677\pi$
$984$	0	0
$985$	0	0
$986$	84.1345	2.67939
$987$	0	0
$988$	7.98476	0.254029
$989$	8.31087	0.264270
$990$	0	0
$991$	55.2019	1.75355	0.876773	−	0.480905i	$-0.159692\pi$
$991$	55.2019	1.75355	0.876773	+	0.480905i	$0.159692\pi$
$992$	3.49794	0.111060
$993$	0	0
$994$	105.737	3.35378
$995$	0	0
$996$	0	0
$997$	11.6543	0.369097	0.184548	−	0.982823i	$-0.440918\pi$
$997$	11.6543	0.369097	0.184548	+	0.982823i	$0.440918\pi$
$998$	−11.4517	−0.362496
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4275.2.a.br.1.1		6
3.2	odd	2		475.2.a.j.1.6		6
5.2	odd	4		855.2.c.d.514.1		6
5.3	odd	4		855.2.c.d.514.6		6
5.4	even	2	inner	4275.2.a.br.1.6		6
12.11	even	2		7600.2.a.ck.1.4		6
15.2	even	4		95.2.b.b.39.6	yes	6
15.8	even	4		95.2.b.b.39.1	✓	6
15.14	odd	2		475.2.a.j.1.1		6
57.56	even	2		9025.2.a.bx.1.1		6
60.23	odd	4		1520.2.d.h.609.4		6
60.47	odd	4		1520.2.d.h.609.3		6
60.59	even	2		7600.2.a.ck.1.3		6
285.113	odd	4		1805.2.b.e.1084.6		6
285.227	odd	4		1805.2.b.e.1084.1		6
285.284	even	2		9025.2.a.bx.1.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
95.2.b.b.39.1	✓	6	15.8	even	4
95.2.b.b.39.6	yes	6	15.2	even	4
475.2.a.j.1.1		6	15.14	odd	2
475.2.a.j.1.6		6	3.2	odd	2
855.2.c.d.514.1		6	5.2	odd	4
855.2.c.d.514.6		6	5.3	odd	4
1520.2.d.h.609.3		6	60.47	odd	4
1520.2.d.h.609.4		6	60.23	odd	4
1805.2.b.e.1084.1		6	285.227	odd	4
1805.2.b.e.1084.6		6	285.113	odd	4
4275.2.a.br.1.1		6	1.1	even	1	trivial
4275.2.a.br.1.6		6	5.4	even	2	inner
7600.2.a.ck.1.3		6	60.59	even	2
7600.2.a.ck.1.4		6	12.11	even	2
9025.2.a.bx.1.1		6	57.56	even	2
9025.2.a.bx.1.6		6	285.284	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)

\(f(q)\)	\(=\)	\(q-2.41987 q^{2} +3.85577 q^{4} +3.18676 q^{7} -4.49073 q^{8} +O(q^{10})\)	\(q-2.41987 q^{2} +3.85577 q^{4} +3.18676 q^{7} -4.49073 q^{8} -4.15544 q^{11} -2.07086 q^{13} -7.71155 q^{14} +3.15544 q^{16} +5.79470 q^{17} -1.00000 q^{19} +10.0556 q^{22} -2.60794 q^{23} +5.01121 q^{26} +12.2874 q^{28} -6.00000 q^{29} +2.59933 q^{31} +1.34571 q^{32} -14.0224 q^{34} -4.30266 q^{37} +2.41987 q^{38} +0.599328 q^{41} -3.18676 q^{43} -16.0224 q^{44} +6.31087 q^{46} +11.7086 q^{47} +3.15544 q^{49} -7.98476 q^{52} +11.7503 q^{53} -14.3109 q^{56} +14.5192 q^{58} +1.71155 q^{59} -8.75476 q^{61} -6.29004 q^{62} -9.56732 q^{64} -4.76228 q^{67} +22.3430 q^{68} -13.7115 q^{71} +2.72714 q^{73} +10.4119 q^{74} -3.85577 q^{76} -13.2424 q^{77} -1.40067 q^{79} -1.45030 q^{82} -7.07154 q^{83} +7.71155 q^{86} +18.6609 q^{88} -16.5353 q^{89} -6.59933 q^{91} -10.0556 q^{92} -28.3333 q^{94} -2.07086 q^{97} -7.63575 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 8 q^{4}+O(q^{10}) \) 6 * q + 8 * q^4	\( 6 q + 8 q^{4} - 2 q^{11} - 16 q^{14} - 4 q^{16} - 6 q^{19} - 8 q^{26} - 36 q^{29} - 8 q^{34} - 12 q^{41} - 20 q^{44} - 8 q^{46} - 4 q^{49} - 40 q^{56} - 20 q^{59} - 14 q^{61} - 12 q^{64} - 52 q^{71} + 40 q^{74} - 8 q^{76} - 24 q^{79} + 16 q^{86} - 24 q^{89} - 24 q^{91} - 48 q^{94}+O(q^{100}) \) 6 * q + 8 * q^4 - 2 * q^11 - 16 * q^14 - 4 * q^16 - 6 * q^19 - 8 * q^26 - 36 * q^29 - 8 * q^34 - 12 * q^41 - 20 * q^44 - 8 * q^46 - 4 * q^49 - 40 * q^56 - 20 * q^59 - 14 * q^61 - 12 * q^64 - 52 * q^71 + 40 * q^74 - 8 * q^76 - 24 * q^79 + 16 * q^86 - 24 * q^89 - 24 * q^91 - 48 * q^94

Introduction

L-functions

Modular forms

Varieties

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Database

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