LMFDB - Embedded newform 9025.2.a.ck.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9025,2,Mod(1,9025)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9025, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("9025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9025 = 5^{2} \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9025.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:	$72.0649878242$
Analytic rank:	$1$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 22x^{14} + 190x^{12} - 820x^{10} + 1862x^{8} - 2154x^{6} + 1163x^{4} - 256x^{2} + 16 $ x^16 - 22x^14 + 190x^12 - 820x^10 + 1862x^8 - 2154x^6 + 1163x^4 - 256*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 1805)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-0.578047$ of defining polynomial
Character	$\chi$	$=$	9025.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.578047 q^{2} -0.551888 q^{3} -1.66586 q^{4} +0.319017 q^{6} -4.66297 q^{7} +2.11904 q^{8} -2.69542 q^{9} +O(q^{10})$	\(q-0.578047 q^{2} -0.551888 q^{3} -1.66586 q^{4} +0.319017 q^{6} -4.66297 q^{7} +2.11904 q^{8} -2.69542 q^{9} -1.22659 q^{11} +0.919368 q^{12} -5.34538 q^{13} +2.69542 q^{14} +2.10682 q^{16} +1.41400 q^{17} +1.55808 q^{18} +2.57344 q^{21} +0.709029 q^{22} +1.95642 q^{23} -1.16947 q^{24} +3.08988 q^{26} +3.14323 q^{27} +7.76787 q^{28} -7.32776 q^{29} +1.83707 q^{31} -5.45592 q^{32} +0.676942 q^{33} -0.817358 q^{34} +4.49020 q^{36} +5.59033 q^{37} +2.95005 q^{39} -8.29172 q^{41} -1.48757 q^{42} +8.30145 q^{43} +2.04333 q^{44} -1.13091 q^{46} +4.10272 q^{47} -1.16273 q^{48} +14.7433 q^{49} -0.780368 q^{51} +8.90466 q^{52} +12.9816 q^{53} -1.81694 q^{54} -9.88104 q^{56} +4.23579 q^{58} -3.76656 q^{59} -4.63515 q^{61} -1.06191 q^{62} +12.5687 q^{63} -1.05985 q^{64} -0.391304 q^{66} -4.65564 q^{67} -2.35552 q^{68} -1.07973 q^{69} +8.44003 q^{71} -5.71171 q^{72} -4.99385 q^{73} -3.23148 q^{74} +5.71957 q^{77} -1.70527 q^{78} +14.7216 q^{79} +6.35155 q^{81} +4.79301 q^{82} -1.02367 q^{83} -4.28699 q^{84} -4.79863 q^{86} +4.04410 q^{87} -2.59920 q^{88} -3.94508 q^{89} +24.9254 q^{91} -3.25913 q^{92} -1.01385 q^{93} -2.37156 q^{94} +3.01106 q^{96} -4.72554 q^{97} -8.52234 q^{98} +3.30618 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 12 q^{4} - 10 q^{6} + 6 q^{9}+O(q^{10}) $ 16 * q + 12 * q^4 - 10 * q^6 + 6 * q^9	$ 16 q + 12 q^{4} - 10 q^{6} + 6 q^{9} - 22 q^{11} - 6 q^{14} + 8 q^{16} + 20 q^{21} - 14 q^{24} - 16 q^{26} - 2 q^{29} - 16 q^{31} + 8 q^{34} + 18 q^{36} - 36 q^{39} - 26 q^{41} - 64 q^{44} - 2 q^{46} - 20 q^{49} + 38 q^{51} - 12 q^{54} - 6 q^{56} - 10 q^{59} - 30 q^{61} - 16 q^{64} + 4 q^{66} - 68 q^{69} + 20 q^{71} - 40 q^{74} - 12 q^{79} - 48 q^{81} + 2 q^{84} + 20 q^{86} + 86 q^{91} + 38 q^{94} + 22 q^{96} - 32 q^{99}+O(q^{100}) $ 16 * q + 12 * q^4 - 10 * q^6 + 6 * q^9 - 22 * q^11 - 6 * q^14 + 8 * q^16 + 20 * q^21 - 14 * q^24 - 16 * q^26 - 2 * q^29 - 16 * q^31 + 8 * q^34 + 18 * q^36 - 36 * q^39 - 26 * q^41 - 64 * q^44 - 2 * q^46 - 20 * q^49 + 38 * q^51 - 12 * q^54 - 6 * q^56 - 10 * q^59 - 30 * q^61 - 16 * q^64 + 4 * q^66 - 68 * q^69 + 20 * q^71 - 40 * q^74 - 12 * q^79 - 48 * q^81 + 2 * q^84 + 20 * q^86 + 86 * q^91 + 38 * q^94 + 22 * q^96 - 32 * q^99

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 $

$2$	−0.578047	−0.408741	−0.204371	−	0.978894i	$-0.565515\pi$
$2$	−0.578047	−0.408741	−0.204371	+	0.978894i	$0.565515\pi$
$3$	−0.551888	−0.318632	−0.159316	−	0.987228i	$-0.550929\pi$
$3$	−0.551888	−0.318632	−0.159316	+	0.987228i	$0.550929\pi$
$4$	−1.66586	−0.832931
$5$	0	0
$5$	0	0
$6$	0.319017	0.130238
$7$	−4.66297	−1.76244	−0.881219	−	0.472708i	$-0.843277\pi$
$7$	−4.66297	−1.76244	−0.881219	+	0.472708i	$0.843277\pi$
$8$	2.11904	0.749194
$9$	−2.69542	−0.898473
$10$	0	0
$11$	−1.22659	−0.369832	−0.184916	−	0.982754i	$-0.559201\pi$
$11$	−1.22659	−0.369832	−0.184916	+	0.982754i	$0.559201\pi$
$12$	0.919368	0.265399
$13$	−5.34538	−1.48254	−0.741271	−	0.671206i	$-0.765777\pi$
$13$	−5.34538	−1.48254	−0.741271	+	0.671206i	$0.765777\pi$
$14$	2.69542	0.720381
$15$	0	0
$16$	2.10682	0.526704
$17$	1.41400	0.342945	0.171472	−	0.985189i	$-0.445148\pi$
$17$	1.41400	0.342945	0.171472	+	0.985189i	$0.445148\pi$
$18$	1.55808	0.367243
$19$	0	0
$19$	0	0
$20$	0	0
$21$	2.57344	0.561570
$22$	0.709029	0.151166
$23$	1.95642	0.407942	0.203971	−	0.978977i	$-0.434615\pi$
$23$	1.95642	0.407942	0.203971	+	0.978977i	$0.434615\pi$
$24$	−1.16947	−0.238718
$25$	0	0
$26$	3.08988	0.605976
$27$	3.14323	0.604915
$28$	7.76787	1.46799
$29$	−7.32776	−1.36073	−0.680366	−	0.732873i	$-0.738179\pi$
$29$	−7.32776	−1.36073	−0.680366	+	0.732873i	$0.738179\pi$
$30$	0	0
$31$	1.83707	0.329947	0.164973	−	0.986298i	$-0.447246\pi$
$31$	1.83707	0.329947	0.164973	+	0.986298i	$0.447246\pi$
$32$	−5.45592	−0.964480
$33$	0.676942	0.117840
$34$	−0.817358	−0.140176
$35$	0	0
$36$	4.49020	0.748366
$37$	5.59033	0.919045	0.459522	−	0.888166i	$-0.348021\pi$
$37$	5.59033	0.919045	0.459522	+	0.888166i	$0.348021\pi$
$38$	0	0
$39$	2.95005	0.472386
$40$	0	0
$41$	−8.29172	−1.29495	−0.647474	−	0.762087i	$-0.724175\pi$
$41$	−8.29172	−1.29495	−0.647474	+	0.762087i	$0.724175\pi$
$42$	−1.48757	−0.229537
$43$	8.30145	1.26596	0.632980	−	0.774169i	$-0.281832\pi$
$43$	8.30145	1.26596	0.632980	+	0.774169i	$0.281832\pi$
$44$	2.04333	0.308044
$45$	0	0
$46$	−1.13091	−0.166743
$47$	4.10272	0.598443	0.299221	−	0.954184i	$-0.403273\pi$
$47$	4.10272	0.598443	0.299221	+	0.954184i	$0.403273\pi$
$48$	−1.16273	−0.167825
$49$	14.7433	2.10619
$50$	0	0
$51$	−0.780368	−0.109273
$52$	8.90466	1.23485
$53$	12.9816	1.78317	0.891583	−	0.452858i	$-0.149596\pi$
$53$	12.9816	1.78317	0.891583	+	0.452858i	$0.149596\pi$
$54$	−1.81694	−0.247254
$55$	0	0
$56$	−9.88104	−1.32041
$57$	0	0
$58$	4.23579	0.556187
$59$	−3.76656	−0.490365	−0.245182	−	0.969477i	$-0.578848\pi$
$59$	−3.76656	−0.490365	−0.245182	+	0.969477i	$0.578848\pi$
$60$	0	0
$61$	−4.63515	−0.593470	−0.296735	−	0.954960i	$-0.595898\pi$
$61$	−4.63515	−0.593470	−0.296735	+	0.954960i	$0.595898\pi$
$62$	−1.06191	−0.134863
$63$	12.5687	1.58350
$64$	−1.05985	−0.132481
$65$	0	0
$66$	−0.391304	−0.0481662
$67$	−4.65564	−0.568778	−0.284389	−	0.958709i	$-0.591791\pi$
$67$	−4.65564	−0.568778	−0.284389	+	0.958709i	$0.591791\pi$
$68$	−2.35552	−0.285649
$69$	−1.07973	−0.129984
$70$	0	0
$71$	8.44003	1.00165	0.500824	−	0.865549i	$-0.333031\pi$
$71$	8.44003	1.00165	0.500824	+	0.865549i	$0.333031\pi$
$72$	−5.71171	−0.673131
$73$	−4.99385	−0.584486	−0.292243	−	0.956344i	$-0.594402\pi$
$73$	−4.99385	−0.584486	−0.292243	+	0.956344i	$0.594402\pi$
$74$	−3.23148	−0.375652
$75$	0	0
$76$	0	0
$77$	5.71957	0.651806
$78$	−1.70527	−0.193084
$79$	14.7216	1.65631	0.828155	−	0.560500i	$-0.189391\pi$
$79$	14.7216	1.65631	0.828155	+	0.560500i	$0.189391\pi$
$80$	0	0
$81$	6.35155	0.705728
$82$	4.79301	0.529299
$83$	−1.02367	−0.112363	−0.0561813	−	0.998421i	$-0.517892\pi$
$83$	−1.02367	−0.112363	−0.0561813	+	0.998421i	$0.517892\pi$
$84$	−4.28699	−0.467749
$85$	0	0
$86$	−4.79863	−0.517450
$87$	4.04410	0.433573
$88$	−2.59920	−0.277076
$89$	−3.94508	−0.418178	−0.209089	−	0.977897i	$-0.567050\pi$
$89$	−3.94508	−0.418178	−0.209089	+	0.977897i	$0.567050\pi$
$90$	0	0
$91$	24.9254	2.61289
$92$	−3.25913	−0.339788
$93$	−1.01385	−0.105132
$94$	−2.37156	−0.244608
$95$	0	0
$96$	3.01106	0.307315
$97$	−4.72554	−0.479806	−0.239903	−	0.970797i	$-0.577116\pi$
$97$	−4.72554	−0.479806	−0.239903	+	0.970797i	$0.577116\pi$
$98$	−8.52234	−0.860887
$99$	3.30618	0.332284
$100$	0	0
$101$	−2.69295	−0.267959	−0.133979	−	0.990984i	$-0.542776\pi$
$101$	−2.69295	−0.267959	−0.133979	+	0.990984i	$0.542776\pi$
$102$	0.451090	0.0446645
$103$	6.64948	0.655193	0.327596	−	0.944818i	$-0.393761\pi$
$103$	6.64948	0.655193	0.327596	+	0.944818i	$0.393761\pi$
$104$	−11.3271	−1.11071
$105$	0	0
$106$	−7.50400	−0.728853
$107$	−9.70596	−0.938310	−0.469155	−	0.883116i	$-0.655441\pi$
$107$	−9.70596	−0.938310	−0.469155	+	0.883116i	$0.655441\pi$
$108$	−5.23619	−0.503852
$109$	11.6767	1.11843	0.559214	−	0.829023i	$-0.311103\pi$
$109$	11.6767	1.11843	0.559214	+	0.829023i	$0.311103\pi$
$110$	0	0
$111$	−3.08523	−0.292838
$112$	−9.82403	−0.928284
$113$	−6.41337	−0.603319	−0.301659	−	0.953416i	$-0.597541\pi$
$113$	−6.41337	−0.603319	−0.301659	+	0.953416i	$0.597541\pi$
$114$	0	0
$115$	0	0
$116$	12.2070	1.13339
$117$	14.4080	1.33202
$118$	2.17725	0.200432
$119$	−6.59344	−0.604419
$120$	0	0
$121$	−9.49547	−0.863224
$122$	2.67933	0.242576
$123$	4.57610	0.412613
$124$	−3.06030	−0.274823
$125$	0	0
$126$	−7.26529	−0.647243
$127$	13.1562	1.16743	0.583713	−	0.811960i	$-0.301599\pi$
$127$	13.1562	1.16743	0.583713	+	0.811960i	$0.301599\pi$
$128$	11.5245	1.01863
$129$	−4.58147	−0.403376
$130$	0	0
$131$	17.7707	1.55263	0.776317	−	0.630342i	$-0.217086\pi$
$131$	17.7707	1.55263	0.776317	+	0.630342i	$0.217086\pi$
$132$	−1.12769	−0.0981529
$133$	0	0
$134$	2.69118	0.232483
$135$	0	0
$136$	2.99632	0.256932
$137$	14.4889	1.23787	0.618937	−	0.785441i	$-0.287564\pi$
$137$	14.4889	1.23787	0.618937	+	0.785441i	$0.287564\pi$
$138$	0.624133	0.0531297
$139$	−4.33776	−0.367924	−0.183962	−	0.982933i	$-0.558892\pi$
$139$	−4.33776	−0.367924	−0.183962	+	0.982933i	$0.558892\pi$
$140$	0	0
$141$	−2.26424	−0.190683
$142$	−4.87874	−0.409414
$143$	6.55661	0.548291
$144$	−5.67875	−0.473230
$145$	0	0
$146$	2.88668	0.238903
$147$	−8.13666	−0.671101
$148$	−9.31272	−0.765501
$149$	−11.3535	−0.930113	−0.465057	−	0.885281i	$-0.653966\pi$
$149$	−11.3535	−0.930113	−0.465057	+	0.885281i	$0.653966\pi$
$150$	0	0
$151$	−17.0316	−1.38601	−0.693004	−	0.720933i	$-0.743713\pi$
$151$	−17.0316	−1.38601	−0.693004	+	0.720933i	$0.743713\pi$
$152$	0	0
$153$	−3.81132	−0.308127
$154$	−3.30618	−0.266420
$155$	0	0
$156$	−4.91437	−0.393465
$157$	−0.137360	−0.0109625	−0.00548127	−	0.999985i	$-0.501745\pi$
$157$	−0.137360	−0.0109625	−0.00548127	+	0.999985i	$0.501745\pi$
$158$	−8.50978	−0.677002
$159$	−7.16440	−0.568174
$160$	0	0
$161$	−9.12275	−0.718974
$162$	−3.67150	−0.288460
$163$	−4.33952	−0.339897	−0.169949	−	0.985453i	$-0.554360\pi$
$163$	−4.33952	−0.339897	−0.169949	+	0.985453i	$0.554360\pi$
$164$	13.8128	1.07860
$165$	0	0
$166$	0.591731	0.0459272
$167$	15.5934	1.20665	0.603327	−	0.797494i	$-0.293841\pi$
$167$	15.5934	1.20665	0.603327	+	0.797494i	$0.293841\pi$
$168$	5.45322	0.420725
$169$	15.5731	1.19793
$170$	0	0
$171$	0	0
$172$	−13.8291	−1.05446
$173$	16.2544	1.23580	0.617900	−	0.786257i	$-0.287984\pi$
$173$	16.2544	1.23580	0.617900	+	0.786257i	$0.287984\pi$
$174$	−2.33768	−0.177219
$175$	0	0
$176$	−2.58421	−0.194792
$177$	2.07872	0.156246
$178$	2.28045	0.170927
$179$	13.8284	1.03358	0.516791	−	0.856112i	$-0.327126\pi$
$179$	13.8284	1.03358	0.516791	+	0.856112i	$0.327126\pi$
$180$	0	0
$181$	−8.95858	−0.665886	−0.332943	−	0.942947i	$-0.608042\pi$
$181$	−8.95858	−0.665886	−0.332943	+	0.942947i	$0.608042\pi$
$182$	−14.4080	−1.06800
$183$	2.55808	0.189099
$184$	4.14574	0.305628
$185$	0	0
$186$	0.586055	0.0429717
$187$	−1.73440	−0.126832
$188$	−6.83456	−0.498461
$189$	−14.6568	−1.06613
$190$	0	0
$191$	−5.51020	−0.398704	−0.199352	−	0.979928i	$-0.563884\pi$
$191$	−5.51020	−0.398704	−0.199352	+	0.979928i	$0.563884\pi$
$192$	0.584919	0.0422129
$193$	16.2322	1.16842	0.584211	−	0.811602i	$-0.301404\pi$
$193$	16.2322	1.16842	0.584211	+	0.811602i	$0.301404\pi$
$194$	2.73159	0.196117
$195$	0	0
$196$	−24.5603	−1.75431
$197$	3.87849	0.276331	0.138166	−	0.990409i	$-0.455879\pi$
$197$	3.87849	0.276331	0.138166	+	0.990409i	$0.455879\pi$
$198$	−1.91113	−0.135818
$199$	−5.13881	−0.364281	−0.182140	−	0.983273i	$-0.558303\pi$
$199$	−5.13881	−0.364281	−0.182140	+	0.983273i	$0.558303\pi$
$200$	0	0
$201$	2.56939	0.181231
$202$	1.55665	0.109526
$203$	34.1692	2.39821
$204$	1.29998	0.0910171
$205$	0	0
$206$	−3.84372	−0.267804
$207$	−5.27338	−0.366525
$208$	−11.2617	−0.780861
$209$	0	0
$210$	0	0
$211$	−11.0879	−0.763322	−0.381661	−	0.924302i	$-0.624648\pi$
$211$	−11.0879	−0.763322	−0.381661	+	0.924302i	$0.624648\pi$
$212$	−21.6256	−1.48525
$213$	−4.65795	−0.319157
$214$	5.61050	0.383526
$215$	0	0
$216$	6.66064	0.453199
$217$	−8.56619	−0.581511
$218$	−6.74970	−0.457148
$219$	2.75604	0.186236
$220$	0	0
$221$	−7.55836	−0.508430
$222$	1.78341	0.119695
$223$	−17.6708	−1.18333	−0.591664	−	0.806185i	$-0.701529\pi$
$223$	−17.6708	−1.18333	−0.591664	+	0.806185i	$0.701529\pi$
$224$	25.4408	1.69984
$225$	0	0
$226$	3.70723	0.246601
$227$	−14.1375	−0.938341	−0.469170	−	0.883108i	$-0.655447\pi$
$227$	−14.1375	−0.938341	−0.469170	+	0.883108i	$0.655447\pi$
$228$	0	0
$229$	−1.60567	−0.106106	−0.0530528	−	0.998592i	$-0.516895\pi$
$229$	−1.60567	−0.106106	−0.0530528	+	0.998592i	$0.516895\pi$
$230$	0	0
$231$	−3.15656	−0.207687
$232$	−15.5278	−1.01945
$233$	−2.15309	−0.141054	−0.0705269	−	0.997510i	$-0.522468\pi$
$233$	−2.15309	−0.141054	−0.0705269	+	0.997510i	$0.522468\pi$
$234$	−8.32853	−0.544453
$235$	0	0
$236$	6.27457	0.408440
$237$	−8.12467	−0.527754
$238$	3.81132	0.247051
$239$	−11.6863	−0.755927	−0.377963	−	0.925821i	$-0.623375\pi$
$239$	−11.6863	−0.755927	−0.377963	+	0.925821i	$0.623375\pi$
$240$	0	0
$241$	−6.95703	−0.448141	−0.224071	−	0.974573i	$-0.571935\pi$
$241$	−6.95703	−0.448141	−0.224071	+	0.974573i	$0.571935\pi$
$242$	5.48883	0.352835
$243$	−12.9350	−0.829783
$244$	7.72151	0.494319
$245$	0	0
$246$	−2.64520	−0.168652
$247$	0	0
$248$	3.89282	0.247194
$249$	0.564952	0.0358024
$250$	0	0
$251$	−17.6254	−1.11250	−0.556252	−	0.831013i	$-0.687761\pi$
$251$	−17.6254	−1.11250	−0.556252	+	0.831013i	$0.687761\pi$
$252$	−20.9377	−1.31895
$253$	−2.39974	−0.150870
$254$	−7.60491	−0.477175
$255$	0	0
$256$	−4.54200	−0.283875
$257$	16.3856	1.02210	0.511052	−	0.859550i	$-0.329256\pi$
$257$	16.3856	1.02210	0.511052	+	0.859550i	$0.329256\pi$
$258$	2.64830	0.164876
$259$	−26.0676	−1.61976
$260$	0	0
$261$	19.7514	1.22258
$262$	−10.2723	−0.634626
$263$	14.4054	0.888272	0.444136	−	0.895959i	$-0.353511\pi$
$263$	14.4054	0.888272	0.444136	+	0.895959i	$0.353511\pi$
$264$	1.43447	0.0882854
$265$	0	0
$266$	0	0
$267$	2.17724	0.133245
$268$	7.75566	0.473752
$269$	−6.13766	−0.374220	−0.187110	−	0.982339i	$-0.559912\pi$
$269$	−6.13766	−0.374220	−0.187110	+	0.982339i	$0.559912\pi$
$270$	0	0
$271$	−21.2762	−1.29244	−0.646218	−	0.763153i	$-0.723650\pi$
$271$	−21.2762	−1.29244	−0.646218	+	0.763153i	$0.723650\pi$
$272$	2.97903	0.180630
$273$	−13.7560	−0.832551
$274$	−8.37529	−0.505970
$275$	0	0
$276$	1.79867	0.108267
$277$	21.7634	1.30764	0.653819	−	0.756651i	$-0.273166\pi$
$277$	21.7634	1.30764	0.653819	+	0.756651i	$0.273166\pi$
$278$	2.50743	0.150386
$279$	−4.95166	−0.296448
$280$	0	0
$281$	17.0673	1.01815	0.509076	−	0.860722i	$-0.329987\pi$
$281$	17.0673	1.01815	0.509076	+	0.860722i	$0.329987\pi$
$282$	1.30884	0.0779401
$283$	−14.9760	−0.890229	−0.445115	−	0.895474i	$-0.646837\pi$
$283$	−14.9760	−0.890229	−0.445115	+	0.895474i	$0.646837\pi$
$284$	−14.0599	−0.834303
$285$	0	0
$286$	−3.79003	−0.224109
$287$	38.6641	2.28227
$288$	14.7060	0.866560
$289$	−15.0006	−0.882389
$290$	0	0
$291$	2.60797	0.152882
$292$	8.31906	0.486836
$293$	−14.9548	−0.873669	−0.436834	−	0.899542i	$-0.643900\pi$
$293$	−14.9548	−0.873669	−0.436834	+	0.899542i	$0.643900\pi$
$294$	4.70338	0.274306
$295$	0	0
$296$	11.8461	0.688543
$297$	−3.85547	−0.223717
$298$	6.56285	0.380176
$299$	−10.4578	−0.604792
$300$	0	0
$301$	−38.7094	−2.23118
$302$	9.84505	0.566519
$303$	1.48621	0.0853804
$304$	0	0
$305$	0	0
$306$	2.20312	0.125944
$307$	20.1616	1.15068	0.575341	−	0.817914i	$-0.304869\pi$
$307$	20.1616	1.15068	0.575341	+	0.817914i	$0.304869\pi$
$308$	−9.52802	−0.542909
$309$	−3.66977	−0.208766
$310$	0	0
$311$	−33.3688	−1.89217	−0.946084	−	0.323921i	$-0.894999\pi$
$311$	−33.3688	−1.89217	−0.946084	+	0.323921i	$0.894999\pi$
$312$	6.25128	0.353909
$313$	−24.3153	−1.37438	−0.687191	−	0.726476i	$-0.741157\pi$
$313$	−24.3153	−1.37438	−0.687191	+	0.726476i	$0.741157\pi$
$314$	0.0794007	0.00448084
$315$	0	0
$316$	−24.5241	−1.37959
$317$	−0.777616	−0.0436753	−0.0218376	−	0.999762i	$-0.506952\pi$
$317$	−0.777616	−0.0436753	−0.0218376	+	0.999762i	$0.506952\pi$
$318$	4.14137	0.232236
$319$	8.98819	0.503242
$320$	0	0
$321$	5.35660	0.298976
$322$	5.27338	0.293874
$323$	0	0
$324$	−10.5808	−0.587822
$325$	0	0
$326$	2.50845	0.138930
$327$	−6.44424	−0.356367
$328$	−17.5705	−0.970168
$329$	−19.1309	−1.05472
$330$	0	0
$331$	19.5153	1.07266	0.536328	−	0.844009i	$-0.319811\pi$
$331$	19.5153	1.07266	0.536328	+	0.844009i	$0.319811\pi$
$332$	1.70530	0.0935903
$333$	−15.0683	−0.825737
$334$	−9.01373	−0.493209
$335$	0	0
$336$	5.42176	0.295781
$337$	−16.3254	−0.889300	−0.444650	−	0.895704i	$-0.646672\pi$
$337$	−16.3254	−0.889300	−0.444650	+	0.895704i	$0.646672\pi$
$338$	−9.00199	−0.489643
$339$	3.53946	0.192237
$340$	0	0
$341$	−2.25333	−0.122025
$342$	0	0
$343$	−36.1070	−1.94959
$344$	17.5911	0.948449
$345$	0	0
$346$	−9.39582	−0.505122
$347$	27.2858	1.46478	0.732390	−	0.680886i	$-0.238405\pi$
$347$	27.2858	1.46478	0.732390	+	0.680886i	$0.238405\pi$
$348$	−6.73691	−0.361136
$349$	16.9079	0.905060	0.452530	−	0.891749i	$-0.350521\pi$
$349$	16.9079	0.905060	0.452530	+	0.891749i	$0.350521\pi$
$350$	0	0
$351$	−16.8018	−0.896812
$352$	6.69220	0.356695
$353$	15.5056	0.825282	0.412641	−	0.910894i	$-0.364606\pi$
$353$	15.5056	0.825282	0.412641	+	0.910894i	$0.364606\pi$
$354$	−1.20160	−0.0638642
$355$	0	0
$356$	6.57196	0.348313
$357$	3.63884	0.192588
$358$	−7.99346	−0.422468
$359$	4.25751	0.224703	0.112351	−	0.993669i	$-0.464162\pi$
$359$	4.25751	0.224703	0.112351	+	0.993669i	$0.464162\pi$
$360$	0	0
$361$	0	0
$362$	5.17848	0.272175
$363$	5.24043	0.275051
$364$	−41.5222	−2.17636
$365$	0	0
$366$	−1.47869	−0.0772924
$367$	−10.1206	−0.528294	−0.264147	−	0.964482i	$-0.585090\pi$
$367$	−10.1206	−0.528294	−0.264147	+	0.964482i	$0.585090\pi$
$368$	4.12182	0.214865
$369$	22.3497	1.16348
$370$	0	0
$371$	−60.5331	−3.14272
$372$	1.68894	0.0875674
$373$	21.1191	1.09351	0.546753	−	0.837294i	$-0.315864\pi$
$373$	21.1191	1.09351	0.546753	+	0.837294i	$0.315864\pi$
$374$	1.00257	0.0518415
$375$	0	0
$376$	8.69383	0.448350
$377$	39.1697	2.01734
$378$	8.47233	0.435770
$379$	11.6942	0.600692	0.300346	−	0.953830i	$-0.402898\pi$
$379$	11.6942	0.600692	0.300346	+	0.953830i	$0.402898\pi$
$380$	0	0
$381$	−7.26075	−0.371980
$382$	3.18516	0.162967
$383$	−1.70745	−0.0872465	−0.0436233	−	0.999048i	$-0.513890\pi$
$383$	−1.70745	−0.0872465	−0.0436233	+	0.999048i	$0.513890\pi$
$384$	−6.36022	−0.324569
$385$	0	0
$386$	−9.38300	−0.477582
$387$	−22.3759	−1.13743
$388$	7.87210	0.399645
$389$	24.3284	1.23350	0.616751	−	0.787159i	$-0.288449\pi$
$389$	24.3284	1.23350	0.616751	+	0.787159i	$0.288449\pi$
$390$	0	0
$391$	2.76638	0.139902
$392$	31.2417	1.57795
$393$	−9.80744	−0.494720
$394$	−2.24195	−0.112948
$395$	0	0
$396$	−5.50765	−0.276770
$397$	18.4591	0.926433	0.463217	−	0.886245i	$-0.346695\pi$
$397$	18.4591	0.926433	0.463217	+	0.886245i	$0.346695\pi$
$398$	2.97048	0.148896
$399$	0	0
$400$	0	0
$401$	5.16001	0.257679	0.128839	−	0.991665i	$-0.458875\pi$
$401$	5.16001	0.257679	0.128839	+	0.991665i	$0.458875\pi$
$402$	−1.48523	−0.0740766
$403$	−9.81982	−0.489160
$404$	4.48609	0.223191
$405$	0	0
$406$	−19.7514	−0.980245
$407$	−6.85707	−0.339892
$408$	−1.65363	−0.0818670
$409$	−10.1477	−0.501772	−0.250886	−	0.968017i	$-0.580722\pi$
$409$	−10.1477	−0.501772	−0.250886	+	0.968017i	$0.580722\pi$
$410$	0	0
$411$	−7.99626	−0.394427
$412$	−11.0771	−0.545730
$413$	17.5634	0.864237
$414$	3.04826	0.149814
$415$	0	0
$416$	29.1640	1.42988
$417$	2.39396	0.117233
$418$	0	0
$419$	8.33923	0.407398	0.203699	−	0.979034i	$-0.434704\pi$
$419$	8.33923	0.407398	0.203699	+	0.979034i	$0.434704\pi$
$420$	0	0
$421$	9.94235	0.484560	0.242280	−	0.970206i	$-0.422105\pi$
$421$	9.94235	0.484560	0.242280	+	0.970206i	$0.422105\pi$
$422$	6.40933	0.312001
$423$	−11.0585	−0.537685
$424$	27.5086	1.33594
$425$	0	0
$426$	2.69251	0.130453
$427$	21.6136	1.04595
$428$	16.1688	0.781547
$429$	−3.61851	−0.174703
$430$	0	0
$431$	37.2668	1.79508	0.897540	−	0.440934i	$-0.145353\pi$
$431$	37.2668	1.79508	0.897540	+	0.440934i	$0.145353\pi$
$432$	6.62221	0.318611
$433$	5.56925	0.267641	0.133821	−	0.991006i	$-0.457275\pi$
$433$	5.56925	0.267641	0.133821	+	0.991006i	$0.457275\pi$
$434$	4.95166	0.237687
$435$	0	0
$436$	−19.4518	−0.931573
$437$	0	0
$438$	−1.59312	−0.0761224
$439$	21.7559	1.03835	0.519177	−	0.854666i	$-0.326238\pi$
$439$	21.7559	1.03835	0.519177	+	0.854666i	$0.326238\pi$
$440$	0	0
$441$	−39.7395	−1.89236
$442$	4.36909	0.207816
$443$	−1.61620	−0.0767880	−0.0383940	−	0.999263i	$-0.512224\pi$
$443$	−1.61620	−0.0767880	−0.0383940	+	0.999263i	$0.512224\pi$
$444$	5.13957	0.243913
$445$	0	0
$446$	10.2146	0.483675
$447$	6.26584	0.296364
$448$	4.94206	0.233490
$449$	−3.07122	−0.144940	−0.0724700	−	0.997371i	$-0.523088\pi$
$449$	−3.07122	−0.144940	−0.0724700	+	0.997371i	$0.523088\pi$
$450$	0	0
$451$	10.1706	0.478913
$452$	10.6838	0.502523
$453$	9.39950	0.441627
$454$	8.17216	0.383539
$455$	0	0
$456$	0	0
$457$	5.23234	0.244759	0.122379	−	0.992483i	$-0.460948\pi$
$457$	5.23234	0.244759	0.122379	+	0.992483i	$0.460948\pi$
$458$	0.928152	0.0433697
$459$	4.44452	0.207453
$460$	0	0
$461$	−13.5378	−0.630519	−0.315259	−	0.949006i	$-0.602092\pi$
$461$	−13.5378	−0.630519	−0.315259	+	0.949006i	$0.602092\pi$
$462$	1.82464	0.0848900
$463$	−22.3863	−1.04038	−0.520191	−	0.854050i	$-0.674139\pi$
$463$	−22.3863	−1.04038	−0.520191	+	0.854050i	$0.674139\pi$
$464$	−15.4382	−0.716703
$465$	0	0
$466$	1.24459	0.0576545
$467$	−34.7898	−1.60988	−0.804940	−	0.593356i	$-0.797802\pi$
$467$	−34.7898	−1.60988	−0.804940	+	0.593356i	$0.797802\pi$
$468$	−24.0018	−1.10948
$469$	21.7092	1.00244
$470$	0	0
$471$	0.0758074	0.00349302
$472$	−7.98150	−0.367378
$473$	−10.1825	−0.468192
$474$	4.69644	0.215715
$475$	0	0
$476$	10.9838	0.503439
$477$	−34.9910	−1.60213
$478$	6.75526	0.308978
$479$	−17.7814	−0.812454	−0.406227	−	0.913772i	$-0.633156\pi$
$479$	−17.7814	−0.812454	−0.406227	+	0.913772i	$0.633156\pi$
$480$	0	0
$481$	−29.8825	−1.36252
$482$	4.02149	0.183174
$483$	5.03473	0.229088
$484$	15.8181	0.719006
$485$	0	0
$486$	7.47706	0.339166
$487$	−2.42918	−0.110077	−0.0550383	−	0.998484i	$-0.517528\pi$
$487$	−2.42918	−0.110077	−0.0550383	+	0.998484i	$0.517528\pi$
$488$	−9.82207	−0.444624
$489$	2.39492	0.108302
$490$	0	0
$491$	−35.1354	−1.58564	−0.792819	−	0.609457i	$-0.791388\pi$
$491$	−35.1354	−1.58564	−0.792819	+	0.609457i	$0.791388\pi$
$492$	−7.62314	−0.343678
$493$	−10.3614	−0.466656
$494$	0	0
$495$	0	0
$496$	3.87036	0.173784
$497$	−39.3556	−1.76534
$498$	−0.326569	−0.0146339
$499$	26.0749	1.16727	0.583636	−	0.812015i	$-0.301629\pi$
$499$	26.0749	1.16727	0.583636	+	0.812015i	$0.301629\pi$
$500$	0	0
$501$	−8.60581	−0.384479
$502$	10.1883	0.454727
$503$	43.7728	1.95173	0.975867	−	0.218367i	$-0.0700731\pi$
$503$	43.7728	1.95173	0.975867	+	0.218367i	$0.0700731\pi$
$504$	26.6335	1.18635
$505$	0	0
$506$	1.38716	0.0616668
$507$	−8.59460	−0.381699
$508$	−21.9164	−0.972384
$509$	−4.01256	−0.177854	−0.0889268	−	0.996038i	$-0.528344\pi$
$509$	−4.01256	−0.177854	−0.0889268	+	0.996038i	$0.528344\pi$
$510$	0	0
$511$	23.2862	1.03012
$512$	−20.4235	−0.902599
$513$	0	0
$514$	−9.47164	−0.417776
$515$	0	0
$516$	7.63209	0.335984
$517$	−5.03237	−0.221323
$518$	15.0683	0.662063
$519$	−8.97060	−0.393766
$520$	0	0
$521$	−2.54253	−0.111390	−0.0556951	−	0.998448i	$-0.517737\pi$
$521$	−2.54253	−0.111390	−0.0556951	+	0.998448i	$0.517737\pi$
$522$	−11.4172	−0.499719
$523$	−25.2632	−1.10468	−0.552341	−	0.833618i	$-0.686265\pi$
$523$	−25.2632	−1.10468	−0.552341	+	0.833618i	$0.686265\pi$
$524$	−29.6035	−1.29324
$525$	0	0
$526$	−8.32698	−0.363073
$527$	2.59761	0.113154
$528$	1.42619	0.0620670
$529$	−19.1724	−0.833583
$530$	0	0
$531$	10.1525	0.440579
$532$	0	0
$533$	44.3224	1.91982
$534$	−1.25855	−0.0544628
$535$	0	0
$536$	−9.86550	−0.426125
$537$	−7.63171	−0.329333
$538$	3.54786	0.152959
$539$	−18.0841	−0.778936
$540$	0	0
$541$	−24.6342	−1.05911	−0.529553	−	0.848277i	$-0.677640\pi$
$541$	−24.6342	−1.05911	−0.529553	+	0.848277i	$0.677640\pi$
$542$	12.2986	0.528272
$543$	4.94413	0.212173
$544$	−7.71466	−0.330764
$545$	0	0
$546$	7.95162	0.340298
$547$	−17.4853	−0.747616	−0.373808	−	0.927506i	$-0.621948\pi$
$547$	−17.4853	−0.747616	−0.373808	+	0.927506i	$0.621948\pi$
$548$	−24.1366	−1.03106
$549$	12.4937	0.533217
$550$	0	0
$551$	0	0
$552$	−2.28798	−0.0973830
$553$	−68.6464	−2.91914
$554$	−12.5803	−0.534485
$555$	0	0
$556$	7.22611	0.306455
$557$	−11.4576	−0.485472	−0.242736	−	0.970092i	$-0.578045\pi$
$557$	−11.4576	−0.485472	−0.242736	+	0.970092i	$0.578045\pi$
$558$	2.86230	0.121171
$559$	−44.3744	−1.87684
$560$	0	0
$561$	0.957194	0.0404128
$562$	−9.86573	−0.416161
$563$	−24.0351	−1.01296	−0.506479	−	0.862252i	$-0.669053\pi$
$563$	−24.0351	−1.01296	−0.506479	+	0.862252i	$0.669053\pi$
$564$	3.77191	0.158826
$565$	0	0
$566$	8.65682	0.363873
$567$	−29.6171	−1.24380
$568$	17.8848	0.750428
$569$	9.61664	0.403151	0.201575	−	0.979473i	$-0.435394\pi$
$569$	9.61664	0.403151	0.201575	+	0.979473i	$0.435394\pi$
$570$	0	0
$571$	−27.2898	−1.14204	−0.571021	−	0.820936i	$-0.693452\pi$
$571$	−27.2898	−1.14204	−0.571021	+	0.820936i	$0.693452\pi$
$572$	−10.9224	−0.456689
$573$	3.04101	0.127040
$574$	−22.3497	−0.932857
$575$	0	0
$576$	2.85674	0.119031
$577$	28.7657	1.19753	0.598766	−	0.800924i	$-0.295658\pi$
$577$	28.7657	1.19753	0.598766	+	0.800924i	$0.295658\pi$
$578$	8.67106	0.360669
$579$	−8.95837	−0.372297
$580$	0	0
$581$	4.77336	0.198032
$582$	−1.50753	−0.0624891
$583$	−15.9232	−0.659471
$584$	−10.5822	−0.437893
$585$	0	0
$586$	8.64458	0.357104
$587$	−18.5265	−0.764669	−0.382335	−	0.924024i	$-0.624880\pi$
$587$	−18.5265	−0.764669	−0.382335	+	0.924024i	$0.624880\pi$
$588$	13.5545	0.558980
$589$	0	0
$590$	0	0
$591$	−2.14049	−0.0880481
$592$	11.7778	0.484065
$593$	35.2510	1.44758	0.723792	−	0.690018i	$-0.242398\pi$
$593$	35.2510	1.44758	0.723792	+	0.690018i	$0.242398\pi$
$594$	2.22864	0.0914423
$595$	0	0
$596$	18.9133	0.774720
$597$	2.83605	0.116072
$598$	6.04512	0.247203
$599$	−21.8274	−0.891845	−0.445922	−	0.895072i	$-0.647124\pi$
$599$	−21.8274	−0.891845	−0.445922	+	0.895072i	$0.647124\pi$
$600$	0	0
$601$	−36.9139	−1.50575	−0.752875	−	0.658164i	$-0.771333\pi$
$601$	−36.9139	−1.50575	−0.752875	+	0.658164i	$0.771333\pi$
$602$	22.3759	0.911973
$603$	12.5489	0.511031
$604$	28.3722	1.15445
$605$	0	0
$606$	−0.859098	−0.0348985
$607$	14.6393	0.594192	0.297096	−	0.954848i	$-0.403982\pi$
$607$	14.6393	0.594192	0.297096	+	0.954848i	$0.403982\pi$
$608$	0	0
$609$	−18.8575	−0.764146
$610$	0	0
$611$	−21.9306	−0.887216
$612$	6.34913	0.256648
$613$	32.3507	1.30663	0.653317	−	0.757085i	$-0.273377\pi$
$613$	32.3507	1.30663	0.653317	+	0.757085i	$0.273377\pi$
$614$	−11.6543	−0.470331
$615$	0	0
$616$	12.1200	0.488329
$617$	23.3753	0.941054	0.470527	−	0.882386i	$-0.344064\pi$
$617$	23.3753	0.941054	0.470527	+	0.882386i	$0.344064\pi$
$618$	2.12130	0.0853311
$619$	33.0099	1.32678	0.663390	−	0.748274i	$-0.269117\pi$
$619$	33.0099	1.32678	0.663390	+	0.748274i	$0.269117\pi$
$620$	0	0
$621$	6.14949	0.246771
$622$	19.2887	0.773407
$623$	18.3958	0.737013
$624$	6.21521	0.248808
$625$	0	0
$626$	14.0554	0.561767
$627$	0	0
$628$	0.228823	0.00913103
$629$	7.90472	0.315182
$630$	0	0
$631$	15.2167	0.605768	0.302884	−	0.953027i	$-0.402050\pi$
$631$	15.2167	0.605768	0.302884	+	0.953027i	$0.402050\pi$
$632$	31.1957	1.24090
$633$	6.11927	0.243219
$634$	0.449499	0.0178519
$635$	0	0
$636$	11.9349	0.473250
$637$	−78.8087	−3.12252
$638$	−5.19560	−0.205696
$639$	−22.7494	−0.899953
$640$	0	0
$641$	11.0717	0.437307	0.218654	−	0.975803i	$-0.429833\pi$
$641$	11.0717	0.437307	0.218654	+	0.975803i	$0.429833\pi$
$642$	−3.09637	−0.122204
$643$	−36.4492	−1.43742	−0.718708	−	0.695312i	$-0.755266\pi$
$643$	−36.4492	−1.43742	−0.718708	+	0.695312i	$0.755266\pi$
$644$	15.1972	0.598855
$645$	0	0
$646$	0	0
$647$	−17.2616	−0.678624	−0.339312	−	0.940674i	$-0.610194\pi$
$647$	−17.2616	−0.678624	−0.339312	+	0.940674i	$0.610194\pi$
$648$	13.4592	0.528727
$649$	4.62004	0.181352
$650$	0	0
$651$	4.72757	0.185288
$652$	7.22903	0.283111
$653$	43.0613	1.68512	0.842559	−	0.538605i	$-0.181048\pi$
$653$	43.0613	1.68512	0.842559	+	0.538605i	$0.181048\pi$
$654$	3.72508	0.145662
$655$	0	0
$656$	−17.4691	−0.682055
$657$	13.4605	0.525145
$658$	11.0585	0.431107
$659$	7.86198	0.306259	0.153130	−	0.988206i	$-0.451065\pi$
$659$	7.86198	0.306259	0.153130	+	0.988206i	$0.451065\pi$
$660$	0	0
$661$	21.6266	0.841177	0.420588	−	0.907252i	$-0.361824\pi$
$661$	21.6266	0.841177	0.420588	+	0.907252i	$0.361824\pi$
$662$	−11.2808	−0.438439
$663$	4.17136	0.162002
$664$	−2.16920	−0.0841814
$665$	0	0
$666$	8.71019	0.337513
$667$	−14.3362	−0.555100
$668$	−25.9765	−1.00506
$669$	9.75232	0.377046
$670$	0	0
$671$	5.68544	0.219484
$672$	−14.0405	−0.541623
$673$	−2.49031	−0.0959943	−0.0479971	−	0.998847i	$-0.515284\pi$
$673$	−2.49031	−0.0959943	−0.0479971	+	0.998847i	$0.515284\pi$
$674$	9.43684	0.363494
$675$	0	0
$676$	−25.9426	−0.997793
$677$	−17.5426	−0.674216	−0.337108	−	0.941466i	$-0.609449\pi$
$677$	−17.5426	−0.674216	−0.337108	+	0.941466i	$0.609449\pi$
$678$	−2.04597	−0.0785752
$679$	22.0351	0.845629
$680$	0	0
$681$	7.80233	0.298986
$682$	1.30253	0.0498766
$683$	42.8977	1.64143	0.820717	−	0.571334i	$-0.193574\pi$
$683$	42.8977	1.64143	0.820717	+	0.571334i	$0.193574\pi$
$684$	0	0
$685$	0	0
$686$	20.8715	0.796879
$687$	0.886148	0.0338087
$688$	17.4896	0.666786
$689$	−69.3918	−2.64362
$690$	0	0
$691$	−21.9873	−0.836435	−0.418218	−	0.908347i	$-0.637345\pi$
$691$	−21.9873	−0.836435	−0.418218	+	0.908347i	$0.637345\pi$
$692$	−27.0776	−1.02934
$693$	−15.4167	−0.585630
$694$	−15.7725	−0.598716
$695$	0	0
$696$	8.56962	0.324831
$697$	−11.7245	−0.444096
$698$	−9.77358	−0.369935
$699$	1.18827	0.0449443
$700$	0	0
$701$	−4.73849	−0.178970	−0.0894851	−	0.995988i	$-0.528522\pi$
$701$	−4.73849	−0.178970	−0.0894851	+	0.995988i	$0.528522\pi$
$702$	9.71222	0.366564
$703$	0	0
$704$	1.30001	0.0489958
$705$	0	0
$706$	−8.96299	−0.337327
$707$	12.5572	0.472261
$708$	−3.46286	−0.130142
$709$	−4.81018	−0.180650	−0.0903251	−	0.995912i	$-0.528791\pi$
$709$	−4.81018	−0.180650	−0.0903251	+	0.995912i	$0.528791\pi$
$710$	0	0
$711$	−39.6809	−1.48815
$712$	−8.35980	−0.313297
$713$	3.59408	0.134599
$714$	−2.10342	−0.0787185
$715$	0	0
$716$	−23.0362	−0.860902
$717$	6.44955	0.240863
$718$	−2.46104	−0.0918454
$719$	−49.7230	−1.85436	−0.927178	−	0.374621i	$-0.877773\pi$
$719$	−49.7230	−1.85436	−0.927178	+	0.374621i	$0.877773\pi$
$720$	0	0
$721$	−31.0064	−1.15474
$722$	0	0
$723$	3.83950	0.142792
$724$	14.9237	0.554637
$725$	0	0
$726$	−3.02922	−0.112425
$727$	15.2548	0.565769	0.282885	−	0.959154i	$-0.408709\pi$
$727$	15.2548	0.565769	0.282885	+	0.959154i	$0.408709\pi$
$728$	52.8179	1.95756
$729$	−11.9160	−0.441332
$730$	0	0
$731$	11.7382	0.434154
$732$	−4.26141	−0.157506
$733$	−17.1375	−0.632989	−0.316495	−	0.948594i	$-0.602506\pi$
$733$	−17.1375	−0.632989	−0.316495	+	0.948594i	$0.602506\pi$
$734$	5.85021	0.215935
$735$	0	0
$736$	−10.6741	−0.393452
$737$	5.71058	0.210352
$738$	−12.9192	−0.475561
$739$	−10.0760	−0.370651	−0.185326	−	0.982677i	$-0.559334\pi$
$739$	−10.0760	−0.370651	−0.185326	+	0.982677i	$0.559334\pi$
$740$	0	0
$741$	0	0
$742$	34.9910	1.28456
$743$	2.20803	0.0810049	0.0405024	−	0.999179i	$-0.487104\pi$
$743$	2.20803	0.0810049	0.0405024	+	0.999179i	$0.487104\pi$
$744$	−2.14840	−0.0787641
$745$	0	0
$746$	−12.2079	−0.446961
$747$	2.75923	0.100955
$748$	2.88927	0.105642
$749$	45.2586	1.65371
$750$	0	0
$751$	24.9684	0.911109	0.455555	−	0.890208i	$-0.349441\pi$
$751$	24.9684	0.911109	0.455555	+	0.890208i	$0.349441\pi$
$752$	8.64367	0.315202
$753$	9.72723	0.354480
$754$	−22.6419	−0.824570
$755$	0	0
$756$	24.4162	0.888009
$757$	−26.5524	−0.965062	−0.482531	−	0.875879i	$-0.660282\pi$
$757$	−26.5524	−0.965062	−0.482531	+	0.875879i	$0.660282\pi$
$758$	−6.75981	−0.245527
$759$	1.32438	0.0480721
$760$	0	0
$761$	15.5905	0.565154	0.282577	−	0.959245i	$-0.408811\pi$
$761$	15.5905	0.565154	0.282577	+	0.959245i	$0.408811\pi$
$762$	4.19706	0.152043
$763$	−54.4483	−1.97116
$764$	9.17923	0.332093
$765$	0	0
$766$	0.986986	0.0356612
$767$	20.1337	0.726986
$768$	2.50667	0.0904517
$769$	−10.4030	−0.375142	−0.187571	−	0.982251i	$-0.560062\pi$
$769$	−10.4030	−0.375142	−0.187571	+	0.982251i	$0.560062\pi$
$770$	0	0
$771$	−9.04299	−0.325675
$772$	−27.0407	−0.973214
$773$	−54.7882	−1.97060	−0.985298	−	0.170845i	$-0.945350\pi$
$773$	−54.7882	−1.97060	−0.985298	+	0.170845i	$0.945350\pi$
$774$	12.9343	0.464915
$775$	0	0
$776$	−10.0136	−0.359468
$777$	14.3864	0.516108
$778$	−14.0630	−0.504183
$779$	0	0
$780$	0	0
$781$	−10.3525	−0.370441
$782$	−1.59910	−0.0571836
$783$	−23.0329	−0.823127
$784$	31.0615	1.10934
$785$	0	0
$786$	5.66916	0.202212
$787$	−3.32906	−0.118668	−0.0593341	−	0.998238i	$-0.518898\pi$
$787$	−3.32906	−0.118668	−0.0593341	+	0.998238i	$0.518898\pi$
$788$	−6.46103	−0.230165
$789$	−7.95014	−0.283032
$790$	0	0
$791$	29.9054	1.06331
$792$	7.00594	0.248945
$793$	24.7766	0.879844
$794$	−10.6702	−0.378672
$795$	0	0
$796$	8.56055	0.303420
$797$	−28.1743	−0.997985	−0.498993	−	0.866606i	$-0.666297\pi$
$797$	−28.1743	−0.997985	−0.498993	+	0.866606i	$0.666297\pi$
$798$	0	0
$799$	5.80123	0.205233
$800$	0	0
$801$	10.6337	0.375722
$802$	−2.98273	−0.105324
$803$	6.12542	0.216161
$804$	−4.28025	−0.150953
$805$	0	0
$806$	5.67632	0.199940
$807$	3.38730	0.119239
$808$	−5.70648	−0.200753
$809$	13.5668	0.476983	0.238492	−	0.971145i	$-0.423347\pi$
$809$	13.5668	0.476983	0.238492	+	0.971145i	$0.423347\pi$
$810$	0	0
$811$	4.52088	0.158750	0.0793748	−	0.996845i	$-0.474708\pi$
$811$	4.52088	0.158750	0.0793748	+	0.996845i	$0.474708\pi$
$812$	−56.9211	−1.99754
$813$	11.7421	0.411812
$814$	3.96371	0.138928
$815$	0	0
$816$	−1.64409	−0.0575547
$817$	0	0
$818$	5.86586	0.205095
$819$	−67.1844	−2.34761
$820$	0	0
$821$	−20.1939	−0.704771	−0.352386	−	0.935855i	$-0.614629\pi$
$821$	−20.1939	−0.704771	−0.352386	+	0.935855i	$0.614629\pi$
$822$	4.62222	0.161218
$823$	−42.5280	−1.48243	−0.741216	−	0.671267i	$-0.765751\pi$
$823$	−42.5280	−1.48243	−0.741216	+	0.671267i	$0.765751\pi$
$824$	14.0905	0.490867
$825$	0	0
$826$	−10.1525	−0.353249
$827$	42.2171	1.46803	0.734017	−	0.679131i	$-0.237643\pi$
$827$	42.2171	1.46803	0.734017	+	0.679131i	$0.237643\pi$
$828$	8.78472	0.305290
$829$	3.78122	0.131327	0.0656637	−	0.997842i	$-0.479084\pi$
$829$	3.78122	0.131327	0.0656637	+	0.997842i	$0.479084\pi$
$830$	0	0
$831$	−12.0110	−0.416656
$832$	5.66531	0.196409
$833$	20.8470	0.722307
$834$	−1.38382	−0.0479178
$835$	0	0
$836$	0	0
$837$	5.77432	0.199590
$838$	−4.82047	−0.166520
$839$	49.7370	1.71711	0.858557	−	0.512719i	$-0.171362\pi$
$839$	49.7370	1.71711	0.858557	+	0.512719i	$0.171362\pi$
$840$	0	0
$841$	24.6961	0.851590
$842$	−5.74715	−0.198060
$843$	−9.41925	−0.324416
$844$	18.4709	0.635794
$845$	0	0
$846$	6.39236	0.219774
$847$	44.2771	1.52138
$848$	27.3499	0.939200
$849$	8.26505	0.283656
$850$	0	0
$851$	10.9371	0.374917
$852$	7.75949	0.265836
$853$	21.4908	0.735831	0.367916	−	0.929859i	$-0.380072\pi$
$853$	21.4908	0.735831	0.367916	+	0.929859i	$0.380072\pi$
$854$	−12.4937	−0.427525
$855$	0	0
$856$	−20.5673	−0.702977
$857$	−2.96612	−0.101321	−0.0506604	−	0.998716i	$-0.516133\pi$
$857$	−2.96612	−0.101321	−0.0506604	+	0.998716i	$0.516133\pi$
$858$	2.09167	0.0714085
$859$	−21.3696	−0.729120	−0.364560	−	0.931180i	$-0.618781\pi$
$859$	−21.3696	−0.729120	−0.364560	+	0.931180i	$0.618781\pi$
$860$	0	0
$861$	−21.3382	−0.727204
$862$	−21.5420	−0.733723
$863$	8.17747	0.278364	0.139182	−	0.990267i	$-0.455553\pi$
$863$	8.17747	0.278364	0.139182	+	0.990267i	$0.455553\pi$
$864$	−17.1492	−0.583429
$865$	0	0
$866$	−3.21929	−0.109396
$867$	8.27865	0.281158
$868$	14.2701	0.484358
$869$	−18.0574	−0.612556
$870$	0	0
$871$	24.8862	0.843236
$872$	24.7435	0.837920
$873$	12.7373	0.431093
$874$	0	0
$875$	0	0
$876$	−4.59118	−0.155122
$877$	7.84379	0.264866	0.132433	−	0.991192i	$-0.457721\pi$
$877$	7.84379	0.264866	0.132433	+	0.991192i	$0.457721\pi$
$878$	−12.5760	−0.424418
$879$	8.25337	0.278379
$880$	0	0
$881$	27.9372	0.941229	0.470614	−	0.882339i	$-0.344032\pi$
$881$	27.9372	0.941229	0.470614	+	0.882339i	$0.344032\pi$
$882$	22.9713	0.773484
$883$	8.68312	0.292210	0.146105	−	0.989269i	$-0.453326\pi$
$883$	8.68312	0.292210	0.146105	+	0.989269i	$0.453326\pi$
$884$	12.5912	0.423487
$885$	0	0
$886$	0.934240	0.0313864
$887$	−52.7474	−1.77108	−0.885542	−	0.464559i	$-0.846213\pi$
$887$	−52.7474	−1.77108	−0.885542	+	0.464559i	$0.846213\pi$
$888$	−6.53774	−0.219392
$889$	−61.3471	−2.05752
$890$	0	0
$891$	−7.79077	−0.261001
$892$	29.4372	0.985629
$893$	0	0
$894$	−3.62195	−0.121136
$895$	0	0
$896$	−53.7384	−1.79527
$897$	5.77154	0.192706
$898$	1.77531	0.0592429
$899$	−13.4616	−0.448969
$900$	0	0
$901$	18.3560	0.611528
$902$	−5.87907	−0.195752
$903$	21.3633	0.710925
$904$	−13.5902	−0.452003
$905$	0	0
$906$	−5.43336	−0.180511
$907$	11.6721	0.387567	0.193783	−	0.981044i	$-0.437924\pi$
$907$	11.6721	0.387567	0.193783	+	0.981044i	$0.437924\pi$
$908$	23.5512	0.781573
$909$	7.25864	0.240754
$910$	0	0
$911$	−39.3098	−1.30239	−0.651196	−	0.758909i	$-0.725733\pi$
$911$	−39.3098	−1.30239	−0.651196	+	0.758909i	$0.725733\pi$
$912$	0	0
$913$	1.25563	0.0415553
$914$	−3.02454	−0.100043
$915$	0	0
$916$	2.67482	0.0883785
$917$	−82.8644	−2.73642
$918$	−2.56915	−0.0847944
$919$	−48.2908	−1.59297	−0.796483	−	0.604660i	$-0.793309\pi$
$919$	−48.2908	−1.59297	−0.796483	+	0.604660i	$0.793309\pi$
$920$	0	0
$921$	−11.1269	−0.366644
$922$	7.82550	0.257719
$923$	−45.1152	−1.48498
$924$	5.25839	0.172988
$925$	0	0
$926$	12.9404	0.425247
$927$	−17.9231	−0.588673
$928$	39.9797	1.31240
$929$	−50.1999	−1.64700	−0.823502	−	0.567314i	$-0.807983\pi$
$929$	−50.1999	−1.64700	−0.823502	+	0.567314i	$0.807983\pi$
$930$	0	0
$931$	0	0
$932$	3.58676	0.117488
$933$	18.4158	0.602906
$934$	20.1101	0.658024
$935$	0	0
$936$	30.5312	0.997945
$937$	−12.7749	−0.417339	−0.208669	−	0.977986i	$-0.566913\pi$
$937$	−12.7749	−0.417339	−0.208669	+	0.977986i	$0.566913\pi$
$938$	−12.5489	−0.409737
$939$	13.4193	0.437923
$940$	0	0
$941$	−0.911554	−0.0297158	−0.0148579	−	0.999890i	$-0.504730\pi$
$941$	−0.911554	−0.0297158	−0.0148579	+	0.999890i	$0.504730\pi$
$942$	−0.0438203	−0.00142774
$943$	−16.2221	−0.528264
$944$	−7.93545	−0.258277
$945$	0	0
$946$	5.88597	0.191369
$947$	14.6832	0.477139	0.238569	−	0.971125i	$-0.423322\pi$
$947$	14.6832	0.477139	0.238569	+	0.971125i	$0.423322\pi$
$948$	13.5346	0.439582
$949$	26.6940	0.866524
$950$	0	0
$951$	0.429157	0.0139164
$952$	−13.9718	−0.452828
$953$	17.0223	0.551406	0.275703	−	0.961243i	$-0.411089\pi$
$953$	17.0223	0.551406	0.275703	+	0.961243i	$0.411089\pi$
$954$	20.2264	0.654855
$955$	0	0
$956$	19.4678	0.629635
$957$	−4.96047	−0.160349
$958$	10.2785	0.332083
$959$	−67.5615	−2.18168
$960$	0	0
$961$	−27.6252	−0.891135
$962$	17.2735	0.556919
$963$	26.1616	0.843047
$964$	11.5894	0.373271
$965$	0	0
$966$	−2.91031	−0.0936378
$967$	−42.8810	−1.37896	−0.689480	−	0.724305i	$-0.742161\pi$
$967$	−42.8810	−1.37896	−0.689480	+	0.724305i	$0.742161\pi$
$968$	−20.1213	−0.646723
$969$	0	0
$970$	0	0
$971$	43.9092	1.40911	0.704556	−	0.709648i	$-0.251146\pi$
$971$	43.9092	1.40911	0.704556	+	0.709648i	$0.251146\pi$
$972$	21.5480	0.691152
$973$	20.2269	0.648444
$974$	1.40418	0.0449928
$975$	0	0
$976$	−9.76540	−0.312583
$977$	18.8399	0.602740	0.301370	−	0.953507i	$-0.402556\pi$
$977$	18.8399	0.602740	0.301370	+	0.953507i	$0.402556\pi$
$978$	−1.38438	−0.0442676
$979$	4.83901	0.154656
$980$	0	0
$981$	−31.4737	−1.00488
$982$	20.3099	0.648116
$983$	−48.3532	−1.54223	−0.771114	−	0.636697i	$-0.780300\pi$
$983$	−48.3532	−1.54223	−0.771114	+	0.636697i	$0.780300\pi$
$984$	9.69694	0.309127
$985$	0	0
$986$	5.98941	0.190742
$987$	10.5581	0.336068
$988$	0	0
$989$	16.2411	0.516438
$990$	0	0
$991$	−7.19452	−0.228542	−0.114271	−	0.993450i	$-0.536453\pi$
$991$	−7.19452	−0.228542	−0.114271	+	0.993450i	$0.536453\pi$
$992$	−10.0229	−0.318227
$993$	−10.7702	−0.341783
$994$	22.7494	0.721568
$995$	0	0
$996$	−0.941132	−0.0298209
$997$	−50.6587	−1.60438	−0.802189	−	0.597071i	$-0.796331\pi$
$997$	−50.6587	−1.60438	−0.802189	+	0.597071i	$0.796331\pi$
$998$	−15.0725	−0.477112
$999$	17.5717	0.555944

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9025.2.a.ck.1.7		16
5.2	odd	4		1805.2.b.j.1084.7	yes	16
5.3	odd	4		1805.2.b.j.1084.10	yes	16
5.4	even	2	inner	9025.2.a.ck.1.10		16
19.18	odd	2		9025.2.a.cl.1.10		16
95.18	even	4		1805.2.b.i.1084.7	✓	16
95.37	even	4		1805.2.b.i.1084.10	yes	16
95.94	odd	2		9025.2.a.cl.1.7		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1805.2.b.i.1084.7	✓	16	95.18	even	4
1805.2.b.i.1084.10	yes	16	95.37	even	4
1805.2.b.j.1084.7	yes	16	5.2	odd	4
1805.2.b.j.1084.10	yes	16	5.3	odd	4
9025.2.a.ck.1.7		16	1.1	even	1	trivial
9025.2.a.ck.1.10		16	5.4	even	2	inner
9025.2.a.cl.1.7		16	95.94	odd	2
9025.2.a.cl.1.10		16	19.18	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:	\( N \)	\(=\)	\( 9025 = 5^{2} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9025.a (trivial)

\(f(q)\)	\(=\)	\(q-0.578047 q^{2} -0.551888 q^{3} -1.66586 q^{4} +0.319017 q^{6} -4.66297 q^{7} +2.11904 q^{8} -2.69542 q^{9} +O(q^{10})\)	\(q-0.578047 q^{2} -0.551888 q^{3} -1.66586 q^{4} +0.319017 q^{6} -4.66297 q^{7} +2.11904 q^{8} -2.69542 q^{9} -1.22659 q^{11} +0.919368 q^{12} -5.34538 q^{13} +2.69542 q^{14} +2.10682 q^{16} +1.41400 q^{17} +1.55808 q^{18} +2.57344 q^{21} +0.709029 q^{22} +1.95642 q^{23} -1.16947 q^{24} +3.08988 q^{26} +3.14323 q^{27} +7.76787 q^{28} -7.32776 q^{29} +1.83707 q^{31} -5.45592 q^{32} +0.676942 q^{33} -0.817358 q^{34} +4.49020 q^{36} +5.59033 q^{37} +2.95005 q^{39} -8.29172 q^{41} -1.48757 q^{42} +8.30145 q^{43} +2.04333 q^{44} -1.13091 q^{46} +4.10272 q^{47} -1.16273 q^{48} +14.7433 q^{49} -0.780368 q^{51} +8.90466 q^{52} +12.9816 q^{53} -1.81694 q^{54} -9.88104 q^{56} +4.23579 q^{58} -3.76656 q^{59} -4.63515 q^{61} -1.06191 q^{62} +12.5687 q^{63} -1.05985 q^{64} -0.391304 q^{66} -4.65564 q^{67} -2.35552 q^{68} -1.07973 q^{69} +8.44003 q^{71} -5.71171 q^{72} -4.99385 q^{73} -3.23148 q^{74} +5.71957 q^{77} -1.70527 q^{78} +14.7216 q^{79} +6.35155 q^{81} +4.79301 q^{82} -1.02367 q^{83} -4.28699 q^{84} -4.79863 q^{86} +4.04410 q^{87} -2.59920 q^{88} -3.94508 q^{89} +24.9254 q^{91} -3.25913 q^{92} -1.01385 q^{93} -2.37156 q^{94} +3.01106 q^{96} -4.72554 q^{97} -8.52234 q^{98} +3.30618 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 12 q^{4} - 10 q^{6} + 6 q^{9}+O(q^{10}) \) 16 * q + 12 * q^4 - 10 * q^6 + 6 * q^9	\( 16 q + 12 q^{4} - 10 q^{6} + 6 q^{9} - 22 q^{11} - 6 q^{14} + 8 q^{16} + 20 q^{21} - 14 q^{24} - 16 q^{26} - 2 q^{29} - 16 q^{31} + 8 q^{34} + 18 q^{36} - 36 q^{39} - 26 q^{41} - 64 q^{44} - 2 q^{46} - 20 q^{49} + 38 q^{51} - 12 q^{54} - 6 q^{56} - 10 q^{59} - 30 q^{61} - 16 q^{64} + 4 q^{66} - 68 q^{69} + 20 q^{71} - 40 q^{74} - 12 q^{79} - 48 q^{81} + 2 q^{84} + 20 q^{86} + 86 q^{91} + 38 q^{94} + 22 q^{96} - 32 q^{99}+O(q^{100}) \) 16 * q + 12 * q^4 - 10 * q^6 + 6 * q^9 - 22 * q^11 - 6 * q^14 + 8 * q^16 + 20 * q^21 - 14 * q^24 - 16 * q^26 - 2 * q^29 - 16 * q^31 + 8 * q^34 + 18 * q^36 - 36 * q^39 - 26 * q^41 - 64 * q^44 - 2 * q^46 - 20 * q^49 + 38 * q^51 - 12 * q^54 - 6 * q^56 - 10 * q^59 - 30 * q^61 - 16 * q^64 + 4 * q^66 - 68 * q^69 + 20 * q^71 - 40 * q^74 - 12 * q^79 - 48 * q^81 + 2 * q^84 + 20 * q^86 + 86 * q^91 + 38 * q^94 + 22 * q^96 - 32 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more