LMFDB - Embedded newform 6525.2.a.cb.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6525,2,Mod(1,6525)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6525, 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("6525.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6525 = 3^{2} \cdot 5^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6525.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:	$52.1023873189$
Analytic rank:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 2x^{8} - 12x^{7} + 21x^{6} + 48x^{5} - 68x^{4} - 73x^{3} + 66x^{2} + 40x - 10 $ x^9 - 2x^8 - 12x^7 + 21x^6 + 48x^5 - 68x^4 - 73x^3 + 66x^2 + 40x - 10
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$-2.07907$ of defining polynomial
Character	$\chi$	$=$	6525.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.07907 q^{2} +2.32253 q^{4} +0.602349 q^{7} +0.670572 q^{8} -3.75239 q^{11} +3.15004 q^{13} +1.25233 q^{14} -3.25090 q^{16} -5.57575 q^{17} -3.67364 q^{19} -7.80148 q^{22} +5.90758 q^{23} +6.54916 q^{26} +1.39898 q^{28} -1.00000 q^{29} -3.09420 q^{31} -8.10000 q^{32} -11.5924 q^{34} +9.93312 q^{37} -7.63775 q^{38} -0.0283906 q^{41} -5.05112 q^{43} -8.71505 q^{44} +12.2823 q^{46} -2.15340 q^{47} -6.63718 q^{49} +7.31608 q^{52} -11.6728 q^{53} +0.403918 q^{56} -2.07907 q^{58} -9.99216 q^{59} -5.22662 q^{61} -6.43305 q^{62} -10.3387 q^{64} +12.7239 q^{67} -12.9499 q^{68} -5.71593 q^{71} -8.39474 q^{73} +20.6517 q^{74} -8.53214 q^{76} -2.26025 q^{77} +9.33285 q^{79} -0.0590260 q^{82} -13.6956 q^{83} -10.5016 q^{86} -2.51625 q^{88} -10.8845 q^{89} +1.89742 q^{91} +13.7206 q^{92} -4.47707 q^{94} +11.3848 q^{97} -13.7992 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 2 q^{2} + 10 q^{4} + q^{7} - 9 q^{8} - 2 q^{11} + q^{13} + 3 q^{14} + 4 q^{16} - 12 q^{17} - q^{19} + 3 q^{22} - 16 q^{23} - 6 q^{26} - 4 q^{28} - 9 q^{29} + 5 q^{31} - 20 q^{32} + 3 q^{34} - 30 q^{38}+ \cdots - 51 q^{98}+O(q^{100}) $ 9 * q - 2 * q^2 + 10 * q^4 + q^7 - 9 * q^8 - 2 * q^11 + q^13 + 3 * q^14 + 4 * q^16 - 12 * q^17 - q^19 + 3 * q^22 - 16 * q^23 - 6 * q^26 - 4 * q^28 - 9 * q^29 + 5 * q^31 - 20 * q^32 + 3 * q^34 - 30 * q^38 + 10 * q^41 + 3 * q^43 + 13 * q^44 + 4 * q^46 - 26 * q^47 - 8 * q^49 - 9 * q^52 - 22 * q^53 - 22 * q^56 + 2 * q^58 - 4 * q^59 + 7 * q^61 - 28 * q^62 + 9 * q^64 + 5 * q^67 - 39 * q^68 - 10 * q^73 + 34 * q^74 - 2 * q^76 - 34 * q^77 + 10 * q^79 - 8 * q^82 - 46 * q^83 - 28 * q^86 + 2 * q^88 - 4 * q^89 - 21 * q^91 - 20 * q^92 + 5 * q^94 + 7 * q^97 - 51 * q^98

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 $

$2$	2.07907	1.47012	0.735062	−	0.677999i	$-0.237153\pi$
$2$	2.07907	1.47012	0.735062	+	0.677999i	$0.237153\pi$
$3$	0	0
$3$	0	0
$4$	2.32253	1.16127
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0.602349	0.227666	0.113833	−	0.993500i	$-0.463687\pi$
$7$	0.602349	0.227666	0.113833	+	0.993500i	$0.463687\pi$
$8$	0.670572	0.237083
$9$	0	0
$10$	0	0
$11$	−3.75239	−1.13139	−0.565694	−	0.824615i	$-0.691392\pi$
$11$	−3.75239	−1.13139	−0.565694	+	0.824615i	$0.691392\pi$
$12$	0	0
$13$	3.15004	0.873664	0.436832	−	0.899543i	$-0.356100\pi$
$13$	3.15004	0.873664	0.436832	+	0.899543i	$0.356100\pi$
$14$	1.25233	0.334698
$15$	0	0
$16$	−3.25090	−0.812726
$17$	−5.57575	−1.35232	−0.676159	−	0.736756i	$-0.736357\pi$
$17$	−5.57575	−1.35232	−0.676159	+	0.736756i	$0.736357\pi$
$18$	0	0
$19$	−3.67364	−0.842790	−0.421395	−	0.906877i	$-0.638459\pi$
$19$	−3.67364	−0.842790	−0.421395	+	0.906877i	$0.638459\pi$
$20$	0	0
$21$	0	0
$22$	−7.80148	−1.66328
$23$	5.90758	1.23182	0.615908	−	0.787818i	$-0.288789\pi$
$23$	5.90758	1.23182	0.615908	+	0.787818i	$0.288789\pi$
$24$	0	0
$25$	0	0
$26$	6.54916	1.28440
$27$	0	0
$28$	1.39898	0.264381
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−3.09420	−0.555734	−0.277867	−	0.960620i	$-0.589627\pi$
$31$	−3.09420	−0.555734	−0.277867	+	0.960620i	$0.589627\pi$
$32$	−8.10000	−1.43189
$33$	0	0
$34$	−11.5924	−1.98808
$35$	0	0
$36$	0	0
$37$	9.93312	1.63299	0.816497	−	0.577349i	$-0.195913\pi$
$37$	9.93312	1.63299	0.816497	+	0.577349i	$0.195913\pi$
$38$	−7.63775	−1.23901
$39$	0	0
$40$	0	0
$41$	−0.0283906	−0.00443386	−0.00221693	−	0.999998i	$-0.500706\pi$
$41$	−0.0283906	−0.00443386	−0.00221693	+	0.999998i	$0.500706\pi$
$42$	0	0
$43$	−5.05112	−0.770288	−0.385144	−	0.922856i	$-0.625848\pi$
$43$	−5.05112	−0.770288	−0.385144	+	0.922856i	$0.625848\pi$
$44$	−8.71505	−1.31384
$45$	0	0
$46$	12.2823	1.81092
$47$	−2.15340	−0.314105	−0.157053	−	0.987590i	$-0.550199\pi$
$47$	−2.15340	−0.314105	−0.157053	+	0.987590i	$0.550199\pi$
$48$	0	0
$49$	−6.63718	−0.948168
$50$	0	0
$51$	0	0
$52$	7.31608	1.01456
$53$	−11.6728	−1.60338	−0.801689	−	0.597742i	$-0.796065\pi$
$53$	−11.6728	−1.60338	−0.801689	+	0.597742i	$0.796065\pi$
$54$	0	0
$55$	0	0
$56$	0.403918	0.0539758
$57$	0	0
$58$	−2.07907	−0.272995
$59$	−9.99216	−1.30087	−0.650434	−	0.759562i	$-0.725413\pi$
$59$	−9.99216	−1.30087	−0.650434	+	0.759562i	$0.725413\pi$
$60$	0	0
$61$	−5.22662	−0.669200	−0.334600	−	0.942360i	$-0.608601\pi$
$61$	−5.22662	−0.669200	−0.334600	+	0.942360i	$0.608601\pi$
$62$	−6.43305	−0.816998
$63$	0	0
$64$	−10.3387	−1.29233
$65$	0	0
$66$	0	0
$67$	12.7239	1.55447	0.777237	−	0.629208i	$-0.216621\pi$
$67$	12.7239	1.55447	0.777237	+	0.629208i	$0.216621\pi$
$68$	−12.9499	−1.57040
$69$	0	0
$70$	0	0
$71$	−5.71593	−0.678356	−0.339178	−	0.940722i	$-0.610149\pi$
$71$	−5.71593	−0.678356	−0.339178	+	0.940722i	$0.610149\pi$
$72$	0	0
$73$	−8.39474	−0.982530	−0.491265	−	0.871010i	$-0.663465\pi$
$73$	−8.39474	−0.982530	−0.491265	+	0.871010i	$0.663465\pi$
$74$	20.6517	2.40071
$75$	0	0
$76$	−8.53214	−0.978704
$77$	−2.26025	−0.257579
$78$	0	0
$79$	9.33285	1.05003	0.525014	−	0.851094i	$-0.324060\pi$
$79$	9.33285	1.05003	0.525014	+	0.851094i	$0.324060\pi$
$80$	0	0
$81$	0	0
$82$	−0.0590260	−0.00651833
$83$	−13.6956	−1.50328	−0.751642	−	0.659572i	$-0.770738\pi$
$83$	−13.6956	−1.50328	−0.751642	+	0.659572i	$0.770738\pi$
$84$	0	0
$85$	0	0
$86$	−10.5016	−1.13242
$87$	0	0
$88$	−2.51625	−0.268233
$89$	−10.8845	−1.15376	−0.576878	−	0.816831i	$-0.695729\pi$
$89$	−10.8845	−1.15376	−0.576878	+	0.816831i	$0.695729\pi$
$90$	0	0
$91$	1.89742	0.198904
$92$	13.7206	1.43047
$93$	0	0
$94$	−4.47707	−0.461774
$95$	0	0
$96$	0	0
$97$	11.3848	1.15595	0.577974	−	0.816055i	$-0.303844\pi$
$97$	11.3848	1.15595	0.577974	+	0.816055i	$0.303844\pi$
$98$	−13.7992	−1.39393
$99$	0	0
$100$	0	0
$101$	16.9341	1.68501	0.842504	−	0.538691i	$-0.181081\pi$
$101$	16.9341	1.68501	0.842504	+	0.538691i	$0.181081\pi$
$102$	0	0
$103$	−13.6553	−1.34549	−0.672747	−	0.739873i	$-0.734886\pi$
$103$	−13.6553	−1.34549	−0.672747	+	0.739873i	$0.734886\pi$
$104$	2.11233	0.207131
$105$	0	0
$106$	−24.2685	−2.35716
$107$	−13.9028	−1.34403	−0.672016	−	0.740537i	$-0.734571\pi$
$107$	−13.9028	−1.34403	−0.672016	+	0.740537i	$0.734571\pi$
$108$	0	0
$109$	−16.0090	−1.53338	−0.766692	−	0.642015i	$-0.778099\pi$
$109$	−16.0090	−1.53338	−0.766692	+	0.642015i	$0.778099\pi$
$110$	0	0
$111$	0	0
$112$	−1.95818	−0.185030
$113$	16.2316	1.52694	0.763468	−	0.645845i	$-0.223495\pi$
$113$	16.2316	1.52694	0.763468	+	0.645845i	$0.223495\pi$
$114$	0	0
$115$	0	0
$116$	−2.32253	−0.215642
$117$	0	0
$118$	−20.7744	−1.91244
$119$	−3.35855	−0.307877
$120$	0	0
$121$	3.08043	0.280039
$122$	−10.8665	−0.983808
$123$	0	0
$124$	−7.18637	−0.645356
$125$	0	0
$126$	0	0
$127$	−2.82607	−0.250773	−0.125386	−	0.992108i	$-0.540017\pi$
$127$	−2.82607	−0.250773	−0.125386	+	0.992108i	$0.540017\pi$
$128$	−5.29482	−0.468000
$129$	0	0
$130$	0	0
$131$	6.34528	0.554390	0.277195	−	0.960814i	$-0.410595\pi$
$131$	6.34528	0.554390	0.277195	+	0.960814i	$0.410595\pi$
$132$	0	0
$133$	−2.21281	−0.191875
$134$	26.4539	2.28527
$135$	0	0
$136$	−3.73894	−0.320612
$137$	−10.2875	−0.878924	−0.439462	−	0.898261i	$-0.644831\pi$
$137$	−10.2875	−0.878924	−0.439462	+	0.898261i	$0.644831\pi$
$138$	0	0
$139$	0.646440	0.0548303	0.0274152	−	0.999624i	$-0.491272\pi$
$139$	0.646440	0.0548303	0.0274152	+	0.999624i	$0.491272\pi$
$140$	0	0
$141$	0	0
$142$	−11.8838	−0.997268
$143$	−11.8202	−0.988453
$144$	0	0
$145$	0	0
$146$	−17.4533	−1.44444
$147$	0	0
$148$	23.0700	1.89634
$149$	4.56864	0.374278	0.187139	−	0.982333i	$-0.440079\pi$
$149$	4.56864	0.374278	0.187139	+	0.982333i	$0.440079\pi$
$150$	0	0
$151$	16.1563	1.31478	0.657391	−	0.753550i	$-0.271660\pi$
$151$	16.1563	1.31478	0.657391	+	0.753550i	$0.271660\pi$
$152$	−2.46344	−0.199811
$153$	0	0
$154$	−4.69921	−0.378673
$155$	0	0
$156$	0	0
$157$	1.34956	0.107706	0.0538532	−	0.998549i	$-0.482850\pi$
$157$	1.34956	0.107706	0.0538532	+	0.998549i	$0.482850\pi$
$158$	19.4037	1.54367
$159$	0	0
$160$	0	0
$161$	3.55842	0.280443
$162$	0	0
$163$	−22.6410	−1.77338	−0.886691	−	0.462362i	$-0.847002\pi$
$163$	−22.6410	−1.77338	−0.886691	+	0.462362i	$0.847002\pi$
$164$	−0.0659381	−0.00514890
$165$	0	0
$166$	−28.4740	−2.21001
$167$	2.94950	0.228239	0.114120	−	0.993467i	$-0.463595\pi$
$167$	2.94950	0.228239	0.114120	+	0.993467i	$0.463595\pi$
$168$	0	0
$169$	−3.07724	−0.236711
$170$	0	0
$171$	0	0
$172$	−11.7314	−0.894510
$173$	−8.95911	−0.681149	−0.340574	−	0.940218i	$-0.610622\pi$
$173$	−8.95911	−0.681149	−0.340574	+	0.940218i	$0.610622\pi$
$174$	0	0
$175$	0	0
$176$	12.1987	0.919508
$177$	0	0
$178$	−22.6296	−1.69616
$179$	3.48533	0.260506	0.130253	−	0.991481i	$-0.458421\pi$
$179$	3.48533	0.260506	0.130253	+	0.991481i	$0.458421\pi$
$180$	0	0
$181$	−15.3531	−1.14118	−0.570592	−	0.821234i	$-0.693286\pi$
$181$	−15.3531	−1.14118	−0.570592	+	0.821234i	$0.693286\pi$
$182$	3.94488	0.292414
$183$	0	0
$184$	3.96146	0.292043
$185$	0	0
$186$	0	0
$187$	20.9224	1.53000
$188$	−5.00134	−0.364760
$189$	0	0
$190$	0	0
$191$	0.749204	0.0542105	0.0271052	−	0.999633i	$-0.491371\pi$
$191$	0.749204	0.0542105	0.0271052	+	0.999633i	$0.491371\pi$
$192$	0	0
$193$	22.5804	1.62538	0.812688	−	0.582700i	$-0.198004\pi$
$193$	22.5804	1.62538	0.812688	+	0.582700i	$0.198004\pi$
$194$	23.6697	1.69939
$195$	0	0
$196$	−15.4151	−1.10108
$197$	11.8996	0.847809	0.423905	−	0.905707i	$-0.360659\pi$
$197$	11.8996	0.847809	0.423905	+	0.905707i	$0.360659\pi$
$198$	0	0
$199$	9.59105	0.679891	0.339946	−	0.940445i	$-0.389591\pi$
$199$	9.59105	0.679891	0.339946	+	0.940445i	$0.389591\pi$
$200$	0	0
$201$	0	0
$202$	35.2072	2.47717
$203$	−0.602349	−0.0422766
$204$	0	0
$205$	0	0
$206$	−28.3903	−1.97804
$207$	0	0
$208$	−10.2405	−0.710049
$209$	13.7849	0.953522
$210$	0	0
$211$	20.1027	1.38392	0.691962	−	0.721934i	$-0.256747\pi$
$211$	20.1027	1.38392	0.691962	+	0.721934i	$0.256747\pi$
$212$	−27.1104	−1.86195
$213$	0	0
$214$	−28.9049	−1.97590
$215$	0	0
$216$	0	0
$217$	−1.86378	−0.126522
$218$	−33.2838	−2.25427
$219$	0	0
$220$	0	0
$221$	−17.5638	−1.18147
$222$	0	0
$223$	18.0107	1.20608	0.603042	−	0.797710i	$-0.293955\pi$
$223$	18.0107	1.20608	0.603042	+	0.797710i	$0.293955\pi$
$224$	−4.87902	−0.325993
$225$	0	0
$226$	33.7466	2.24479
$227$	−18.0809	−1.20007	−0.600035	−	0.799974i	$-0.704847\pi$
$227$	−18.0809	−1.20007	−0.600035	+	0.799974i	$0.704847\pi$
$228$	0	0
$229$	−8.34926	−0.551734	−0.275867	−	0.961196i	$-0.588965\pi$
$229$	−8.34926	−0.551734	−0.275867	+	0.961196i	$0.588965\pi$
$230$	0	0
$231$	0	0
$232$	−0.670572	−0.0440252
$233$	−0.184490	−0.0120864	−0.00604319	−	0.999982i	$-0.501924\pi$
$233$	−0.184490	−0.0120864	−0.00604319	+	0.999982i	$0.501924\pi$
$234$	0	0
$235$	0	0
$236$	−23.2071	−1.51066
$237$	0	0
$238$	−6.98265	−0.452618
$239$	−8.03156	−0.519519	−0.259759	−	0.965673i	$-0.583643\pi$
$239$	−8.03156	−0.519519	−0.259759	+	0.965673i	$0.583643\pi$
$240$	0	0
$241$	10.8545	0.699200	0.349600	−	0.936899i	$-0.386317\pi$
$241$	10.8545	0.699200	0.349600	+	0.936899i	$0.386317\pi$
$242$	6.40442	0.411692
$243$	0	0
$244$	−12.1390	−0.777121
$245$	0	0
$246$	0	0
$247$	−11.5721	−0.736315
$248$	−2.07488	−0.131755
$249$	0	0
$250$	0	0
$251$	8.32044	0.525182	0.262591	−	0.964907i	$-0.415423\pi$
$251$	8.32044	0.525182	0.262591	+	0.964907i	$0.415423\pi$
$252$	0	0
$253$	−22.1675	−1.39366
$254$	−5.87559	−0.368668
$255$	0	0
$256$	9.66903	0.604315
$257$	28.0860	1.75195	0.875977	−	0.482352i	$-0.160217\pi$
$257$	28.0860	1.75195	0.875977	+	0.482352i	$0.160217\pi$
$258$	0	0
$259$	5.98320	0.371778
$260$	0	0
$261$	0	0
$262$	13.1923	0.815022
$263$	−6.69144	−0.412612	−0.206306	−	0.978488i	$-0.566144\pi$
$263$	−6.69144	−0.412612	−0.206306	+	0.978488i	$0.566144\pi$
$264$	0	0
$265$	0	0
$266$	−4.60059	−0.282080
$267$	0	0
$268$	29.5517	1.80516
$269$	6.80593	0.414965	0.207482	−	0.978239i	$-0.433473\pi$
$269$	6.80593	0.414965	0.207482	+	0.978239i	$0.433473\pi$
$270$	0	0
$271$	16.0776	0.976643	0.488322	−	0.872664i	$-0.337609\pi$
$271$	16.0776	0.976643	0.488322	+	0.872664i	$0.337609\pi$
$272$	18.1262	1.09906
$273$	0	0
$274$	−21.3885	−1.29213
$275$	0	0
$276$	0	0
$277$	−7.67018	−0.460856	−0.230428	−	0.973089i	$-0.574013\pi$
$277$	−7.67018	−0.460856	−0.230428	+	0.973089i	$0.574013\pi$
$278$	1.34399	0.0806074
$279$	0	0
$280$	0	0
$281$	−0.255777	−0.0152584	−0.00762919	−	0.999971i	$-0.502428\pi$
$281$	−0.255777	−0.0152584	−0.00762919	+	0.999971i	$0.502428\pi$
$282$	0	0
$283$	−0.918149	−0.0545783	−0.0272892	−	0.999628i	$-0.508687\pi$
$283$	−0.918149	−0.0545783	−0.0272892	+	0.999628i	$0.508687\pi$
$284$	−13.2754	−0.787753
$285$	0	0
$286$	−24.5750	−1.45315
$287$	−0.0171010	−0.00100944
$288$	0	0
$289$	14.0890	0.828764
$290$	0	0
$291$	0	0
$292$	−19.4971	−1.14098
$293$	3.52768	0.206089	0.103045	−	0.994677i	$-0.467142\pi$
$293$	3.52768	0.206089	0.103045	+	0.994677i	$0.467142\pi$
$294$	0	0
$295$	0	0
$296$	6.66087	0.387155
$297$	0	0
$298$	9.49853	0.550235
$299$	18.6091	1.07619
$300$	0	0
$301$	−3.04253	−0.175369
$302$	33.5901	1.93289
$303$	0	0
$304$	11.9426	0.684957
$305$	0	0
$306$	0	0
$307$	11.2224	0.640494	0.320247	−	0.947334i	$-0.396234\pi$
$307$	11.2224	0.640494	0.320247	+	0.947334i	$0.396234\pi$
$308$	−5.24950	−0.299118
$309$	0	0
$310$	0	0
$311$	−33.3035	−1.88847	−0.944234	−	0.329274i	$-0.893196\pi$
$311$	−33.3035	−1.88847	−0.944234	+	0.329274i	$0.893196\pi$
$312$	0	0
$313$	−2.90172	−0.164015	−0.0820073	−	0.996632i	$-0.526133\pi$
$313$	−2.90172	−0.164015	−0.0820073	+	0.996632i	$0.526133\pi$
$314$	2.80583	0.158342
$315$	0	0
$316$	21.6759	1.21936
$317$	14.4137	0.809552	0.404776	−	0.914416i	$-0.367349\pi$
$317$	14.4137	0.809552	0.404776	+	0.914416i	$0.367349\pi$
$318$	0	0
$319$	3.75239	0.210093
$320$	0	0
$321$	0	0
$322$	7.39821	0.412286
$323$	20.4833	1.13972
$324$	0	0
$325$	0	0
$326$	−47.0723	−2.60709
$327$	0	0
$328$	−0.0190379	−0.00105119
$329$	−1.29710	−0.0715112
$330$	0	0
$331$	−4.56720	−0.251036	−0.125518	−	0.992091i	$-0.540059\pi$
$331$	−4.56720	−0.251036	−0.125518	+	0.992091i	$0.540059\pi$
$332$	−31.8084	−1.74571
$333$	0	0
$334$	6.13222	0.335540
$335$	0	0
$336$	0	0
$337$	9.05419	0.493213	0.246607	−	0.969116i	$-0.420684\pi$
$337$	9.05419	0.493213	0.246607	+	0.969116i	$0.420684\pi$
$338$	−6.39781	−0.347995
$339$	0	0
$340$	0	0
$341$	11.6106	0.628751
$342$	0	0
$343$	−8.21433	−0.443532
$344$	−3.38714	−0.182622
$345$	0	0
$346$	−18.6266	−1.00137
$347$	4.08659	0.219380	0.109690	−	0.993966i	$-0.465014\pi$
$347$	4.08659	0.219380	0.109690	+	0.993966i	$0.465014\pi$
$348$	0	0
$349$	−34.4132	−1.84209	−0.921047	−	0.389451i	$-0.872665\pi$
$349$	−34.4132	−1.84209	−0.921047	+	0.389451i	$0.872665\pi$
$350$	0	0
$351$	0	0
$352$	30.3944	1.62002
$353$	−15.4658	−0.823162	−0.411581	−	0.911373i	$-0.635023\pi$
$353$	−15.4658	−0.823162	−0.411581	+	0.911373i	$0.635023\pi$
$354$	0	0
$355$	0	0
$356$	−25.2796	−1.33982
$357$	0	0
$358$	7.24625	0.382976
$359$	−12.3787	−0.653323	−0.326662	−	0.945141i	$-0.605924\pi$
$359$	−12.3787	−0.653323	−0.326662	+	0.945141i	$0.605924\pi$
$360$	0	0
$361$	−5.50440	−0.289705
$362$	−31.9201	−1.67768
$363$	0	0
$364$	4.40683	0.230981
$365$	0	0
$366$	0	0
$367$	−30.9404	−1.61508	−0.807539	−	0.589815i	$-0.799201\pi$
$367$	−30.9404	−1.61508	−0.807539	+	0.589815i	$0.799201\pi$
$368$	−19.2050	−1.00113
$369$	0	0
$370$	0	0
$371$	−7.03107	−0.365035
$372$	0	0
$373$	−34.6136	−1.79222	−0.896112	−	0.443828i	$-0.853620\pi$
$373$	−34.6136	−1.79222	−0.896112	+	0.443828i	$0.853620\pi$
$374$	43.4991	2.24929
$375$	0	0
$376$	−1.44401	−0.0744691
$377$	−3.15004	−0.162235
$378$	0	0
$379$	−7.78953	−0.400122	−0.200061	−	0.979783i	$-0.564114\pi$
$379$	−7.78953	−0.400122	−0.200061	+	0.979783i	$0.564114\pi$
$380$	0	0
$381$	0	0
$382$	1.55765	0.0796962
$383$	−26.3868	−1.34830	−0.674152	−	0.738592i	$-0.735491\pi$
$383$	−26.3868	−1.34830	−0.674152	+	0.738592i	$0.735491\pi$
$384$	0	0
$385$	0	0
$386$	46.9463	2.38950
$387$	0	0
$388$	26.4415	1.34236
$389$	31.7173	1.60813	0.804065	−	0.594541i	$-0.202666\pi$
$389$	31.7173	1.60813	0.804065	+	0.594541i	$0.202666\pi$
$390$	0	0
$391$	−32.9392	−1.66581
$392$	−4.45070	−0.224795
$393$	0	0
$394$	24.7400	1.24639
$395$	0	0
$396$	0	0
$397$	26.2946	1.31969	0.659845	−	0.751402i	$-0.270622\pi$
$397$	26.2946	1.31969	0.659845	+	0.751402i	$0.270622\pi$
$398$	19.9405	0.999525
$399$	0	0
$400$	0	0
$401$	23.3208	1.16459	0.582294	−	0.812979i	$-0.302155\pi$
$401$	23.3208	1.16459	0.582294	+	0.812979i	$0.302155\pi$
$402$	0	0
$403$	−9.74684	−0.485525
$404$	39.3301	1.95674
$405$	0	0
$406$	−1.25233	−0.0621519
$407$	−37.2729	−1.84755
$408$	0	0
$409$	−14.8687	−0.735210	−0.367605	−	0.929982i	$-0.619822\pi$
$409$	−14.8687	−0.735210	−0.367605	+	0.929982i	$0.619822\pi$
$410$	0	0
$411$	0	0
$412$	−31.7148	−1.56248
$413$	−6.01877	−0.296164
$414$	0	0
$415$	0	0
$416$	−25.5153	−1.25099
$417$	0	0
$418$	28.6598	1.40180
$419$	10.6506	0.520314	0.260157	−	0.965566i	$-0.416226\pi$
$419$	10.6506	0.520314	0.260157	+	0.965566i	$0.416226\pi$
$420$	0	0
$421$	21.0394	1.02540	0.512698	−	0.858569i	$-0.328646\pi$
$421$	21.0394	1.02540	0.512698	+	0.858569i	$0.328646\pi$
$422$	41.7948	2.03454
$423$	0	0
$424$	−7.82743	−0.380133
$425$	0	0
$426$	0	0
$427$	−3.14825	−0.152354
$428$	−32.2897	−1.56078
$429$	0	0
$430$	0	0
$431$	−32.3891	−1.56013	−0.780064	−	0.625699i	$-0.784814\pi$
$431$	−32.3891	−1.56013	−0.780064	+	0.625699i	$0.784814\pi$
$432$	0	0
$433$	31.7159	1.52417	0.762084	−	0.647479i	$-0.224176\pi$
$433$	31.7159	1.52417	0.762084	+	0.647479i	$0.224176\pi$
$434$	−3.87494	−0.186003
$435$	0	0
$436$	−37.1815	−1.78067
$437$	−21.7023	−1.03816
$438$	0	0
$439$	34.4377	1.64362	0.821810	−	0.569761i	$-0.192964\pi$
$439$	34.4377	1.64362	0.821810	+	0.569761i	$0.192964\pi$
$440$	0	0
$441$	0	0
$442$	−36.5165	−1.73691
$443$	−11.7927	−0.560287	−0.280144	−	0.959958i	$-0.590382\pi$
$443$	−11.7927	−0.560287	−0.280144	+	0.959958i	$0.590382\pi$
$444$	0	0
$445$	0	0
$446$	37.4454	1.77309
$447$	0	0
$448$	−6.22748	−0.294221
$449$	1.55151	0.0732205	0.0366102	−	0.999330i	$-0.488344\pi$
$449$	1.55151	0.0732205	0.0366102	+	0.999330i	$0.488344\pi$
$450$	0	0
$451$	0.106532	0.00501642
$452$	37.6984	1.77318
$453$	0	0
$454$	−37.5914	−1.76425
$455$	0	0
$456$	0	0
$457$	−12.5511	−0.587114	−0.293557	−	0.955942i	$-0.594839\pi$
$457$	−12.5511	−0.587114	−0.293557	+	0.955942i	$0.594839\pi$
$458$	−17.3587	−0.811118
$459$	0	0
$460$	0	0
$461$	−21.2069	−0.987705	−0.493853	−	0.869546i	$-0.664412\pi$
$461$	−21.2069	−0.987705	−0.493853	+	0.869546i	$0.664412\pi$
$462$	0	0
$463$	−20.0687	−0.932674	−0.466337	−	0.884607i	$-0.654427\pi$
$463$	−20.0687	−0.932674	−0.466337	+	0.884607i	$0.654427\pi$
$464$	3.25090	0.150919
$465$	0	0
$466$	−0.383569	−0.0177685
$467$	−25.5687	−1.18318	−0.591590	−	0.806239i	$-0.701499\pi$
$467$	−25.5687	−1.18318	−0.591590	+	0.806239i	$0.701499\pi$
$468$	0	0
$469$	7.66423	0.353901
$470$	0	0
$471$	0	0
$472$	−6.70047	−0.308414
$473$	18.9538	0.871495
$474$	0	0
$475$	0	0
$476$	−7.80034	−0.357528
$477$	0	0
$478$	−16.6982	−0.763757
$479$	11.3166	0.517069	0.258534	−	0.966002i	$-0.416761\pi$
$479$	11.3166	0.517069	0.258534	+	0.966002i	$0.416761\pi$
$480$	0	0
$481$	31.2897	1.42669
$482$	22.5673	1.02791
$483$	0	0
$484$	7.15439	0.325200
$485$	0	0
$486$	0	0
$487$	3.19292	0.144685	0.0723426	−	0.997380i	$-0.476953\pi$
$487$	3.19292	0.144685	0.0723426	+	0.997380i	$0.476953\pi$
$488$	−3.50483	−0.158656
$489$	0	0
$490$	0	0
$491$	39.9490	1.80287	0.901436	−	0.432912i	$-0.142514\pi$
$491$	39.9490	1.80287	0.901436	+	0.432912i	$0.142514\pi$
$492$	0	0
$493$	5.57575	0.251119
$494$	−24.0592	−1.08248
$495$	0	0
$496$	10.0589	0.451659
$497$	−3.44298	−0.154439
$498$	0	0
$499$	32.5068	1.45520	0.727601	−	0.686000i	$-0.240635\pi$
$499$	32.5068	1.45520	0.727601	+	0.686000i	$0.240635\pi$
$500$	0	0
$501$	0	0
$502$	17.2988	0.772083
$503$	6.24593	0.278492	0.139246	−	0.990258i	$-0.455532\pi$
$503$	6.24593	0.278492	0.139246	+	0.990258i	$0.455532\pi$
$504$	0	0
$505$	0	0
$506$	−46.0879	−2.04886
$507$	0	0
$508$	−6.56364	−0.291214
$509$	5.13547	0.227626	0.113813	−	0.993502i	$-0.463694\pi$
$509$	5.13547	0.227626	0.113813	+	0.993502i	$0.463694\pi$
$510$	0	0
$511$	−5.05656	−0.223689
$512$	30.6922	1.35642
$513$	0	0
$514$	58.3927	2.57559
$515$	0	0
$516$	0	0
$517$	8.08039	0.355375
$518$	12.4395	0.546560
$519$	0	0
$520$	0	0
$521$	−6.01802	−0.263654	−0.131827	−	0.991273i	$-0.542084\pi$
$521$	−6.01802	−0.263654	−0.131827	+	0.991273i	$0.542084\pi$
$522$	0	0
$523$	23.4215	1.02415	0.512075	−	0.858940i	$-0.328877\pi$
$523$	23.4215	1.02415	0.512075	+	0.858940i	$0.328877\pi$
$524$	14.7371	0.643795
$525$	0	0
$526$	−13.9120	−0.606591
$527$	17.2525	0.751529
$528$	0	0
$529$	11.8995	0.517369
$530$	0	0
$531$	0	0
$532$	−5.13933	−0.222818
$533$	−0.0894315	−0.00387371
$534$	0	0
$535$	0	0
$536$	8.53230	0.368539
$537$	0	0
$538$	14.1500	0.610050
$539$	24.9053	1.07275
$540$	0	0
$541$	42.0863	1.80943	0.904715	−	0.426017i	$-0.140084\pi$
$541$	42.0863	1.80943	0.904715	+	0.426017i	$0.140084\pi$
$542$	33.4264	1.43579
$543$	0	0
$544$	45.1636	1.93637
$545$	0	0
$546$	0	0
$547$	−15.6945	−0.671046	−0.335523	−	0.942032i	$-0.608913\pi$
$547$	−15.6945	−0.671046	−0.335523	+	0.942032i	$0.608913\pi$
$548$	−23.8932	−1.02067
$549$	0	0
$550$	0	0
$551$	3.67364	0.156502
$552$	0	0
$553$	5.62163	0.239056
$554$	−15.9468	−0.677517
$555$	0	0
$556$	1.50138	0.0636726
$557$	6.74754	0.285902	0.142951	−	0.989730i	$-0.454341\pi$
$557$	6.74754	0.285902	0.142951	+	0.989730i	$0.454341\pi$
$558$	0	0
$559$	−15.9112	−0.672973
$560$	0	0
$561$	0	0
$562$	−0.531779	−0.0224317
$563$	−12.4598	−0.525117	−0.262558	−	0.964916i	$-0.584566\pi$
$563$	−12.4598	−0.525117	−0.262558	+	0.964916i	$0.584566\pi$
$564$	0	0
$565$	0	0
$566$	−1.90890	−0.0802369
$567$	0	0
$568$	−3.83294	−0.160827
$569$	−40.8410	−1.71214	−0.856072	−	0.516856i	$-0.827102\pi$
$569$	−40.8410	−1.71214	−0.856072	+	0.516856i	$0.827102\pi$
$570$	0	0
$571$	−11.7126	−0.490157	−0.245078	−	0.969503i	$-0.578814\pi$
$571$	−11.7126	−0.490157	−0.245078	+	0.969503i	$0.578814\pi$
$572$	−27.4528	−1.14786
$573$	0	0
$574$	−0.0355542	−0.00148401
$575$	0	0
$576$	0	0
$577$	11.6090	0.483288	0.241644	−	0.970365i	$-0.422313\pi$
$577$	11.6090	0.483288	0.241644	+	0.970365i	$0.422313\pi$
$578$	29.2920	1.21839
$579$	0	0
$580$	0	0
$581$	−8.24950	−0.342247
$582$	0	0
$583$	43.8007	1.81404
$584$	−5.62928	−0.232941
$585$	0	0
$586$	7.33430	0.302977
$587$	9.69989	0.400357	0.200179	−	0.979759i	$-0.435848\pi$
$587$	9.69989	0.400357	0.200179	+	0.979759i	$0.435848\pi$
$588$	0	0
$589$	11.3669	0.468367
$590$	0	0
$591$	0	0
$592$	−32.2916	−1.32718
$593$	1.61655	0.0663837	0.0331919	−	0.999449i	$-0.489433\pi$
$593$	1.61655	0.0663837	0.0331919	+	0.999449i	$0.489433\pi$
$594$	0	0
$595$	0	0
$596$	10.6108	0.434636
$597$	0	0
$598$	38.6897	1.58214
$599$	−16.8939	−0.690265	−0.345132	−	0.938554i	$-0.612166\pi$
$599$	−16.8939	−0.690265	−0.345132	+	0.938554i	$0.612166\pi$
$600$	0	0
$601$	29.6603	1.20987	0.604934	−	0.796275i	$-0.293199\pi$
$601$	29.6603	1.20987	0.604934	+	0.796275i	$0.293199\pi$
$602$	−6.32564	−0.257814
$603$	0	0
$604$	37.5236	1.52681
$605$	0	0
$606$	0	0
$607$	14.7374	0.598172	0.299086	−	0.954226i	$-0.403318\pi$
$607$	14.7374	0.598172	0.299086	+	0.954226i	$0.403318\pi$
$608$	29.7564	1.20678
$609$	0	0
$610$	0	0
$611$	−6.78329	−0.274423
$612$	0	0
$613$	−15.9688	−0.644975	−0.322488	−	0.946574i	$-0.604519\pi$
$613$	−15.9688	−0.644975	−0.322488	+	0.946574i	$0.604519\pi$
$614$	23.3321	0.941607
$615$	0	0
$616$	−1.51566	−0.0610676
$617$	−8.94217	−0.359998	−0.179999	−	0.983667i	$-0.557609\pi$
$617$	−8.94217	−0.359998	−0.179999	+	0.983667i	$0.557609\pi$
$618$	0	0
$619$	10.3149	0.414590	0.207295	−	0.978279i	$-0.433534\pi$
$619$	10.3149	0.414590	0.207295	+	0.978279i	$0.433534\pi$
$620$	0	0
$621$	0	0
$622$	−69.2404	−2.77629
$623$	−6.55627	−0.262671
$624$	0	0
$625$	0	0
$626$	−6.03287	−0.241122
$627$	0	0
$628$	3.13440	0.125076
$629$	−55.3846	−2.20833
$630$	0	0
$631$	14.0815	0.560577	0.280288	−	0.959916i	$-0.409570\pi$
$631$	14.0815	0.560577	0.280288	+	0.959916i	$0.409570\pi$
$632$	6.25835	0.248944
$633$	0	0
$634$	29.9670	1.19014
$635$	0	0
$636$	0	0
$637$	−20.9074	−0.828380
$638$	7.80148	0.308864
$639$	0	0
$640$	0	0
$641$	10.8916	0.430193	0.215097	−	0.976593i	$-0.430993\pi$
$641$	10.8916	0.430193	0.215097	+	0.976593i	$0.430993\pi$
$642$	0	0
$643$	−35.4722	−1.39889	−0.699443	−	0.714689i	$-0.746568\pi$
$643$	−35.4722	−1.39889	−0.699443	+	0.714689i	$0.746568\pi$
$644$	8.26456	0.325669
$645$	0	0
$646$	42.5862	1.67553
$647$	−32.7006	−1.28559	−0.642796	−	0.766037i	$-0.722226\pi$
$647$	−32.7006	−1.28559	−0.642796	+	0.766037i	$0.722226\pi$
$648$	0	0
$649$	37.4945	1.47179
$650$	0	0
$651$	0	0
$652$	−52.5846	−2.05937
$653$	12.2635	0.479908	0.239954	−	0.970784i	$-0.422868\pi$
$653$	12.2635	0.479908	0.239954	+	0.970784i	$0.422868\pi$
$654$	0	0
$655$	0	0
$656$	0.0922950	0.00360351
$657$	0	0
$658$	−2.69676	−0.105130
$659$	−23.6010	−0.919364	−0.459682	−	0.888084i	$-0.652037\pi$
$659$	−23.6010	−0.919364	−0.459682	+	0.888084i	$0.652037\pi$
$660$	0	0
$661$	20.9083	0.813237	0.406618	−	0.913598i	$-0.366708\pi$
$661$	20.9083	0.813237	0.406618	+	0.913598i	$0.366708\pi$
$662$	−9.49553	−0.369054
$663$	0	0
$664$	−9.18386	−0.356403
$665$	0	0
$666$	0	0
$667$	−5.90758	−0.228742
$668$	6.85031	0.265047
$669$	0	0
$670$	0	0
$671$	19.6123	0.757125
$672$	0	0
$673$	−10.0198	−0.386234	−0.193117	−	0.981176i	$-0.561860\pi$
$673$	−10.0198	−0.386234	−0.193117	+	0.981176i	$0.561860\pi$
$674$	18.8243	0.725085
$675$	0	0
$676$	−7.14700	−0.274885
$677$	9.48674	0.364605	0.182303	−	0.983242i	$-0.441645\pi$
$677$	9.48674	0.364605	0.182303	+	0.983242i	$0.441645\pi$
$678$	0	0
$679$	6.85760	0.263170
$680$	0	0
$681$	0	0
$682$	24.1393	0.924342
$683$	−2.04161	−0.0781201	−0.0390600	−	0.999237i	$-0.512436\pi$
$683$	−2.04161	−0.0781201	−0.0390600	+	0.999237i	$0.512436\pi$
$684$	0	0
$685$	0	0
$686$	−17.0782	−0.652048
$687$	0	0
$688$	16.4207	0.626033
$689$	−36.7697	−1.40081
$690$	0	0
$691$	33.7822	1.28513	0.642567	−	0.766229i	$-0.277869\pi$
$691$	33.7822	1.28513	0.642567	+	0.766229i	$0.277869\pi$
$692$	−20.8078	−0.790996
$693$	0	0
$694$	8.49631	0.322515
$695$	0	0
$696$	0	0
$697$	0.158299	0.00599599
$698$	−71.5474	−2.70811
$699$	0	0
$700$	0	0
$701$	34.0512	1.28610	0.643048	−	0.765826i	$-0.277670\pi$
$701$	34.0512	1.28610	0.643048	+	0.765826i	$0.277670\pi$
$702$	0	0
$703$	−36.4907	−1.37627
$704$	38.7947	1.46213
$705$	0	0
$706$	−32.1545	−1.21015
$707$	10.2002	0.383619
$708$	0	0
$709$	21.4032	0.803814	0.401907	−	0.915680i	$-0.368347\pi$
$709$	21.4032	0.803814	0.401907	+	0.915680i	$0.368347\pi$
$710$	0	0
$711$	0	0
$712$	−7.29884	−0.273536
$713$	−18.2792	−0.684562
$714$	0	0
$715$	0	0
$716$	8.09480	0.302517
$717$	0	0
$718$	−25.7362	−0.960467
$719$	6.12025	0.228247	0.114123	−	0.993467i	$-0.463594\pi$
$719$	6.12025	0.228247	0.114123	+	0.993467i	$0.463594\pi$
$720$	0	0
$721$	−8.22523	−0.306324
$722$	−11.4440	−0.425903
$723$	0	0
$724$	−35.6580	−1.32522
$725$	0	0
$726$	0	0
$727$	3.62879	0.134584	0.0672922	−	0.997733i	$-0.478564\pi$
$727$	3.62879	0.134584	0.0672922	+	0.997733i	$0.478564\pi$
$728$	1.27236	0.0471567
$729$	0	0
$730$	0	0
$731$	28.1638	1.04167
$732$	0	0
$733$	37.4985	1.38504	0.692519	−	0.721400i	$-0.256501\pi$
$733$	37.4985	1.38504	0.692519	+	0.721400i	$0.256501\pi$
$734$	−64.3273	−2.37437
$735$	0	0
$736$	−47.8514	−1.76383
$737$	−47.7451	−1.75871
$738$	0	0
$739$	2.03701	0.0749325	0.0374663	−	0.999298i	$-0.488071\pi$
$739$	2.03701	0.0749325	0.0374663	+	0.999298i	$0.488071\pi$
$740$	0	0
$741$	0	0
$742$	−14.6181	−0.536647
$743$	37.5670	1.37820	0.689099	−	0.724667i	$-0.258006\pi$
$743$	37.5670	1.37820	0.689099	+	0.724667i	$0.258006\pi$
$744$	0	0
$745$	0	0
$746$	−71.9641	−2.63479
$747$	0	0
$748$	48.5930	1.77673
$749$	−8.37432	−0.305991
$750$	0	0
$751$	−18.3132	−0.668260	−0.334130	−	0.942527i	$-0.608442\pi$
$751$	−18.3132	−0.668260	−0.334130	+	0.942527i	$0.608442\pi$
$752$	7.00049	0.255282
$753$	0	0
$754$	−6.54916	−0.238506
$755$	0	0
$756$	0	0
$757$	32.6004	1.18488	0.592441	−	0.805614i	$-0.298164\pi$
$757$	32.6004	1.18488	0.592441	+	0.805614i	$0.298164\pi$
$758$	−16.1950	−0.588229
$759$	0	0
$760$	0	0
$761$	17.1784	0.622715	0.311358	−	0.950293i	$-0.399216\pi$
$761$	17.1784	0.622715	0.311358	+	0.950293i	$0.399216\pi$
$762$	0	0
$763$	−9.64300	−0.349100
$764$	1.74005	0.0629528
$765$	0	0
$766$	−54.8601	−1.98218
$767$	−31.4757	−1.13652
$768$	0	0
$769$	21.6261	0.779858	0.389929	−	0.920845i	$-0.372500\pi$
$769$	21.6261	0.779858	0.389929	+	0.920845i	$0.372500\pi$
$770$	0	0
$771$	0	0
$772$	52.4438	1.88749
$773$	−43.4266	−1.56195	−0.780973	−	0.624565i	$-0.785276\pi$
$773$	−43.4266	−1.56195	−0.780973	+	0.624565i	$0.785276\pi$
$774$	0	0
$775$	0	0
$776$	7.63430	0.274056
$777$	0	0
$778$	65.9425	2.36415
$779$	0.104297	0.00373681
$780$	0	0
$781$	21.4484	0.767484
$782$	−68.4829	−2.44894
$783$	0	0
$784$	21.5768	0.770600
$785$	0	0
$786$	0	0
$787$	−28.4274	−1.01333	−0.506663	−	0.862144i	$-0.669121\pi$
$787$	−28.4274	−1.01333	−0.506663	+	0.862144i	$0.669121\pi$
$788$	27.6372	0.984533
$789$	0	0
$790$	0	0
$791$	9.77706	0.347632
$792$	0	0
$793$	−16.4641	−0.584656
$794$	54.6684	1.94011
$795$	0	0
$796$	22.2755	0.789535
$797$	−29.4872	−1.04449	−0.522246	−	0.852795i	$-0.674906\pi$
$797$	−29.4872	−1.04449	−0.522246	+	0.852795i	$0.674906\pi$
$798$	0	0
$799$	12.0068	0.424770
$800$	0	0
$801$	0	0
$802$	48.4857	1.71209
$803$	31.5003	1.11162
$804$	0	0
$805$	0	0
$806$	−20.2644	−0.713782
$807$	0	0
$808$	11.3555	0.399487
$809$	−26.1311	−0.918721	−0.459361	−	0.888250i	$-0.651921\pi$
$809$	−26.1311	−0.918721	−0.459361	+	0.888250i	$0.651921\pi$
$810$	0	0
$811$	40.1295	1.40914	0.704568	−	0.709637i	$-0.251141\pi$
$811$	40.1295	1.40914	0.704568	+	0.709637i	$0.251141\pi$
$812$	−1.39898	−0.0490944
$813$	0	0
$814$	−77.4930	−2.71613
$815$	0	0
$816$	0	0
$817$	18.5560	0.649191
$818$	−30.9131	−1.08085
$819$	0	0
$820$	0	0
$821$	3.29593	0.115029	0.0575143	−	0.998345i	$-0.481683\pi$
$821$	3.29593	0.115029	0.0575143	+	0.998345i	$0.481683\pi$
$822$	0	0
$823$	11.4922	0.400593	0.200297	−	0.979735i	$-0.435809\pi$
$823$	11.4922	0.400593	0.200297	+	0.979735i	$0.435809\pi$
$824$	−9.15684	−0.318994
$825$	0	0
$826$	−12.5134	−0.435398
$827$	−39.3099	−1.36694	−0.683470	−	0.729979i	$-0.739530\pi$
$827$	−39.3099	−1.36694	−0.683470	+	0.729979i	$0.739530\pi$
$828$	0	0
$829$	5.76236	0.200135	0.100068	−	0.994981i	$-0.468094\pi$
$829$	5.76236	0.200135	0.100068	+	0.994981i	$0.468094\pi$
$830$	0	0
$831$	0	0
$832$	−32.5672	−1.12907
$833$	37.0072	1.28222
$834$	0	0
$835$	0	0
$836$	32.0159	1.10729
$837$	0	0
$838$	22.1433	0.764927
$839$	12.0263	0.415193	0.207596	−	0.978215i	$-0.433436\pi$
$839$	12.0263	0.415193	0.207596	+	0.978215i	$0.433436\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	43.7423	1.50746
$843$	0	0
$844$	46.6891	1.60711
$845$	0	0
$846$	0	0
$847$	1.85549	0.0637554
$848$	37.9470	1.30311
$849$	0	0
$850$	0	0
$851$	58.6807	2.01155
$852$	0	0
$853$	−38.8644	−1.33069	−0.665346	−	0.746535i	$-0.731716\pi$
$853$	−38.8644	−1.33069	−0.665346	+	0.746535i	$0.731716\pi$
$854$	−6.54543	−0.223980
$855$	0	0
$856$	−9.32281	−0.318647
$857$	18.9047	0.645772	0.322886	−	0.946438i	$-0.395347\pi$
$857$	18.9047	0.645772	0.322886	+	0.946438i	$0.395347\pi$
$858$	0	0
$859$	26.1799	0.893245	0.446623	−	0.894722i	$-0.352627\pi$
$859$	26.1799	0.893245	0.446623	+	0.894722i	$0.352627\pi$
$860$	0	0
$861$	0	0
$862$	−67.3392	−2.29358
$863$	−9.88255	−0.336406	−0.168203	−	0.985752i	$-0.553796\pi$
$863$	−9.88255	−0.336406	−0.168203	+	0.985752i	$0.553796\pi$
$864$	0	0
$865$	0	0
$866$	65.9395	2.24072
$867$	0	0
$868$	−4.32870	−0.146926
$869$	−35.0205	−1.18799
$870$	0	0
$871$	40.0808	1.35809
$872$	−10.7352	−0.363539
$873$	0	0
$874$	−45.1206	−1.52623
$875$	0	0
$876$	0	0
$877$	−44.4465	−1.50085	−0.750426	−	0.660955i	$-0.770151\pi$
$877$	−44.4465	−1.50085	−0.750426	+	0.660955i	$0.770151\pi$
$878$	71.5984	2.41633
$879$	0	0
$880$	0	0
$881$	48.1307	1.62156	0.810782	−	0.585348i	$-0.199042\pi$
$881$	48.1307	1.62156	0.810782	+	0.585348i	$0.199042\pi$
$882$	0	0
$883$	−53.0155	−1.78411	−0.892056	−	0.451924i	$-0.850738\pi$
$883$	−53.0155	−1.78411	−0.892056	+	0.451924i	$0.850738\pi$
$884$	−40.7926	−1.37200
$885$	0	0
$886$	−24.5178	−0.823692
$887$	3.01914	0.101373	0.0506863	−	0.998715i	$-0.483859\pi$
$887$	3.01914	0.101373	0.0506863	+	0.998715i	$0.483859\pi$
$888$	0	0
$889$	−1.70228	−0.0570926
$890$	0	0
$891$	0	0
$892$	41.8304	1.40059
$893$	7.91080	0.264725
$894$	0	0
$895$	0	0
$896$	−3.18933	−0.106548
$897$	0	0
$898$	3.22571	0.107643
$899$	3.09420	0.103197
$900$	0	0
$901$	65.0844	2.16828
$902$	0.221489	0.00737476
$903$	0	0
$904$	10.8844	0.362011
$905$	0	0
$906$	0	0
$907$	4.46070	0.148115	0.0740575	−	0.997254i	$-0.476405\pi$
$907$	4.46070	0.148115	0.0740575	+	0.997254i	$0.476405\pi$
$908$	−41.9934	−1.39360
$909$	0	0
$910$	0	0
$911$	−33.9155	−1.12367	−0.561835	−	0.827249i	$-0.689905\pi$
$911$	−33.9155	−1.12367	−0.561835	+	0.827249i	$0.689905\pi$
$912$	0	0
$913$	51.3911	1.70080
$914$	−26.0945	−0.863131
$915$	0	0
$916$	−19.3914	−0.640711
$917$	3.82207	0.126216
$918$	0	0
$919$	−17.9083	−0.590739	−0.295370	−	0.955383i	$-0.595443\pi$
$919$	−17.9083	−0.590739	−0.295370	+	0.955383i	$0.595443\pi$
$920$	0	0
$921$	0	0
$922$	−44.0907	−1.45205
$923$	−18.0054	−0.592655
$924$	0	0
$925$	0	0
$926$	−41.7243	−1.37115
$927$	0	0
$928$	8.10000	0.265896
$929$	31.3508	1.02859	0.514293	−	0.857615i	$-0.328054\pi$
$929$	31.3508	1.02859	0.514293	+	0.857615i	$0.328054\pi$
$930$	0	0
$931$	24.3826	0.799106
$932$	−0.428485	−0.0140355
$933$	0	0
$934$	−53.1592	−1.73942
$935$	0	0
$936$	0	0
$937$	7.95458	0.259865	0.129932	−	0.991523i	$-0.458524\pi$
$937$	7.95458	0.259865	0.129932	+	0.991523i	$0.458524\pi$
$938$	15.9345	0.520279
$939$	0	0
$940$	0	0
$941$	−40.7463	−1.32829	−0.664146	−	0.747603i	$-0.731204\pi$
$941$	−40.7463	−1.32829	−0.664146	+	0.747603i	$0.731204\pi$
$942$	0	0
$943$	−0.167720	−0.00546170
$944$	32.4836	1.05725
$945$	0	0
$946$	39.4062	1.28121
$947$	−9.40892	−0.305749	−0.152874	−	0.988246i	$-0.548853\pi$
$947$	−9.40892	−0.305749	−0.152874	+	0.988246i	$0.548853\pi$
$948$	0	0
$949$	−26.4438	−0.858401
$950$	0	0
$951$	0	0
$952$	−2.25215	−0.0729925
$953$	−32.6522	−1.05771	−0.528854	−	0.848713i	$-0.677378\pi$
$953$	−32.6522	−1.05771	−0.528854	+	0.848713i	$0.677378\pi$
$954$	0	0
$955$	0	0
$956$	−18.6536	−0.603300
$957$	0	0
$958$	23.5280	0.760155
$959$	−6.19669	−0.200101
$960$	0	0
$961$	−21.4260	−0.691160
$962$	65.0535	2.09741
$963$	0	0
$964$	25.2100	0.811958
$965$	0	0
$966$	0	0
$967$	−9.12653	−0.293489	−0.146745	−	0.989174i	$-0.546880\pi$
$967$	−9.12653	−0.293489	−0.146745	+	0.989174i	$0.546880\pi$
$968$	2.06565	0.0663924
$969$	0	0
$970$	0	0
$971$	−5.64921	−0.181292	−0.0906459	−	0.995883i	$-0.528893\pi$
$971$	−5.64921	−0.181292	−0.0906459	+	0.995883i	$0.528893\pi$
$972$	0	0
$973$	0.389382	0.0124830
$974$	6.63831	0.212705
$975$	0	0
$976$	16.9912	0.543876
$977$	−32.7475	−1.04768	−0.523842	−	0.851815i	$-0.675502\pi$
$977$	−32.7475	−1.04768	−0.523842	+	0.851815i	$0.675502\pi$
$978$	0	0
$979$	40.8429	1.30534
$980$	0	0
$981$	0	0
$982$	83.0567	2.65045
$983$	−33.5033	−1.06859	−0.534295	−	0.845298i	$-0.679423\pi$
$983$	−33.5033	−1.06859	−0.534295	+	0.845298i	$0.679423\pi$
$984$	0	0
$985$	0	0
$986$	11.5924	0.369177
$987$	0	0
$988$	−26.8766	−0.855059
$989$	−29.8399	−0.948853
$990$	0	0
$991$	22.5805	0.717295	0.358647	−	0.933473i	$-0.383238\pi$
$991$	22.5805	0.717295	0.358647	+	0.933473i	$0.383238\pi$
$992$	25.0630	0.795750
$993$	0	0
$994$	−7.15820	−0.227044
$995$	0	0
$996$	0	0
$997$	22.5164	0.713102	0.356551	−	0.934276i	$-0.383953\pi$
$997$	22.5164	0.713102	0.356551	+	0.934276i	$0.383953\pi$
$998$	67.5839	2.13933
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6525.2.a.cb.1.8	yes	9
3.2	odd	2		6525.2.a.cd.1.2	yes	9
5.4	even	2		6525.2.a.cc.1.2	yes	9
15.14	odd	2		6525.2.a.ca.1.8	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6525.2.a.ca.1.8	✓	9	15.14	odd	2
6525.2.a.cb.1.8	yes	9	1.1	even	1	trivial
6525.2.a.cc.1.2	yes	9	5.4	even	2
6525.2.a.cd.1.2	yes	9	3.2	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:	\( N \)	\(=\)	\( 6525 = 3^{2} \cdot 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6525.a (trivial)

\(f(q)\)	\(=\)	\(q+2.07907 q^{2} +2.32253 q^{4} +0.602349 q^{7} +0.670572 q^{8} -3.75239 q^{11} +3.15004 q^{13} +1.25233 q^{14} -3.25090 q^{16} -5.57575 q^{17} -3.67364 q^{19} -7.80148 q^{22} +5.90758 q^{23} +6.54916 q^{26} +1.39898 q^{28} -1.00000 q^{29} -3.09420 q^{31} -8.10000 q^{32} -11.5924 q^{34} +9.93312 q^{37} -7.63775 q^{38} -0.0283906 q^{41} -5.05112 q^{43} -8.71505 q^{44} +12.2823 q^{46} -2.15340 q^{47} -6.63718 q^{49} +7.31608 q^{52} -11.6728 q^{53} +0.403918 q^{56} -2.07907 q^{58} -9.99216 q^{59} -5.22662 q^{61} -6.43305 q^{62} -10.3387 q^{64} +12.7239 q^{67} -12.9499 q^{68} -5.71593 q^{71} -8.39474 q^{73} +20.6517 q^{74} -8.53214 q^{76} -2.26025 q^{77} +9.33285 q^{79} -0.0590260 q^{82} -13.6956 q^{83} -10.5016 q^{86} -2.51625 q^{88} -10.8845 q^{89} +1.89742 q^{91} +13.7206 q^{92} -4.47707 q^{94} +11.3848 q^{97} -13.7992 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 2 q^{2} + 10 q^{4} + q^{7} - 9 q^{8} - 2 q^{11} + q^{13} + 3 q^{14} + 4 q^{16} - 12 q^{17} - q^{19} + 3 q^{22} - 16 q^{23} - 6 q^{26} - 4 q^{28} - 9 q^{29} + 5 q^{31} - 20 q^{32} + 3 q^{34} - 30 q^{38}+ \cdots - 51 q^{98}+O(q^{100}) \) 9 * q - 2 * q^2 + 10 * q^4 + q^7 - 9 * q^8 - 2 * q^11 + q^13 + 3 * q^14 + 4 * q^16 - 12 * q^17 - q^19 + 3 * q^22 - 16 * q^23 - 6 * q^26 - 4 * q^28 - 9 * q^29 + 5 * q^31 - 20 * q^32 + 3 * q^34 - 30 * q^38 + 10 * q^41 + 3 * q^43 + 13 * q^44 + 4 * q^46 - 26 * q^47 - 8 * q^49 - 9 * q^52 - 22 * q^53 - 22 * q^56 + 2 * q^58 - 4 * q^59 + 7 * q^61 - 28 * q^62 + 9 * q^64 + 5 * q^67 - 39 * q^68 - 10 * q^73 + 34 * q^74 - 2 * q^76 - 34 * q^77 + 10 * q^79 - 8 * q^82 - 46 * q^83 - 28 * q^86 + 2 * q^88 - 4 * q^89 - 21 * q^91 - 20 * q^92 + 5 * q^94 + 7 * q^97 - 51 * q^98

Introduction

L-functions

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Database

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