LMFDB - Embedded newform 6525.2.a.ca.1.2

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			1.2
Root			$2.18077$ 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)

$f(q)$	$=$	$q-2.18077 q^{2} +2.75577 q^{4} -4.62113 q^{7} -1.64817 q^{8} +O(q^{10})$	$q-2.18077 q^{2} +2.75577 q^{4} -4.62113 q^{7} -1.64817 q^{8} +2.53207 q^{11} +2.08906 q^{13} +10.0776 q^{14} -1.91726 q^{16} -0.481630 q^{17} +4.25554 q^{19} -5.52188 q^{22} -5.22436 q^{23} -4.55576 q^{26} -12.7348 q^{28} +1.00000 q^{29} +0.0174694 q^{31} +7.47745 q^{32} +1.05033 q^{34} +7.52495 q^{37} -9.28036 q^{38} -5.88867 q^{41} -4.91245 q^{43} +6.97782 q^{44} +11.3932 q^{46} -7.27173 q^{47} +14.3549 q^{49} +5.75697 q^{52} -10.4262 q^{53} +7.61641 q^{56} -2.18077 q^{58} -2.12111 q^{59} -2.79999 q^{61} -0.0380968 q^{62} -12.4721 q^{64} +5.03637 q^{67} -1.32726 q^{68} -8.56725 q^{71} +1.59489 q^{73} -16.4102 q^{74} +11.7273 q^{76} -11.7010 q^{77} +11.6409 q^{79} +12.8419 q^{82} +4.69781 q^{83} +10.7129 q^{86} -4.17329 q^{88} +14.6662 q^{89} -9.65381 q^{91} -14.3972 q^{92} +15.8580 q^{94} -0.438441 q^{97} -31.3047 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 2 q^{2} + 10 q^{4} - q^{7} - 9 q^{8}+O(q^{10}) $ 9 * q - 2 * q^2 + 10 * q^4 - q^7 - 9 * q^8	$ 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}+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

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 $

$2$	−2.18077	−1.54204	−0.771020	−	0.636811i	$-0.780253\pi$
$2$	−2.18077	−1.54204	−0.771020	+	0.636811i	$0.780253\pi$
$3$	0	0
$3$	0	0
$4$	2.75577	1.37789
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−4.62113	−1.74662	−0.873312	−	0.487162i	$-0.838032\pi$
$7$	−4.62113	−1.74662	−0.873312	+	0.487162i	$0.838032\pi$
$8$	−1.64817	−0.582716
$9$	0	0
$10$	0	0
$11$	2.53207	0.763449	0.381725	−	0.924276i	$-0.375330\pi$
$11$	2.53207	0.763449	0.381725	+	0.924276i	$0.375330\pi$
$12$	0	0
$13$	2.08906	0.579400	0.289700	−	0.957117i	$-0.406444\pi$
$13$	2.08906	0.579400	0.289700	+	0.957117i	$0.406444\pi$
$14$	10.0776	2.69336
$15$	0	0
$16$	−1.91726	−0.479316
$17$	−0.481630	−0.116812	−0.0584062	−	0.998293i	$-0.518602\pi$
$17$	−0.481630	−0.116812	−0.0584062	+	0.998293i	$0.518602\pi$
$18$	0	0
$19$	4.25554	0.976287	0.488143	−	0.872763i	$-0.337674\pi$
$19$	4.25554	0.976287	0.488143	+	0.872763i	$0.337674\pi$
$20$	0	0
$21$	0	0
$22$	−5.52188	−1.17727
$23$	−5.22436	−1.08935	−0.544677	−	0.838646i	$-0.683348\pi$
$23$	−5.22436	−1.08935	−0.544677	+	0.838646i	$0.683348\pi$
$24$	0	0
$25$	0	0
$26$	−4.55576	−0.893458
$27$	0	0
$28$	−12.7348	−2.40665
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	0.0174694	0.00313760	0.00156880	−	0.999999i	$-0.499501\pi$
$31$	0.0174694	0.00313760	0.00156880	+	0.999999i	$0.499501\pi$
$32$	7.47745	1.32184
$33$	0	0
$34$	1.05033	0.180129
$35$	0	0
$36$	0	0
$37$	7.52495	1.23709	0.618547	−	0.785748i	$-0.287721\pi$
$37$	7.52495	1.23709	0.618547	+	0.785748i	$0.287721\pi$
$38$	−9.28036	−1.50547
$39$	0	0
$40$	0	0
$41$	−5.88867	−0.919655	−0.459828	−	0.888008i	$-0.652089\pi$
$41$	−5.88867	−0.919655	−0.459828	+	0.888008i	$0.652089\pi$
$42$	0	0
$43$	−4.91245	−0.749141	−0.374571	−	0.927198i	$-0.622210\pi$
$43$	−4.91245	−0.749141	−0.374571	+	0.927198i	$0.622210\pi$
$44$	6.97782	1.05195
$45$	0	0
$46$	11.3932	1.67983
$47$	−7.27173	−1.06069	−0.530346	−	0.847782i	$-0.677938\pi$
$47$	−7.27173	−1.06069	−0.530346	+	0.847782i	$0.677938\pi$
$48$	0	0
$49$	14.3549	2.05069
$50$	0	0
$51$	0	0
$52$	5.75697	0.798348
$53$	−10.4262	−1.43215	−0.716074	−	0.698024i	$-0.754063\pi$
$53$	−10.4262	−1.43215	−0.716074	+	0.698024i	$0.754063\pi$
$54$	0	0
$55$	0	0
$56$	7.61641	1.01779
$57$	0	0
$58$	−2.18077	−0.286350
$59$	−2.12111	−0.276145	−0.138073	−	0.990422i	$-0.544091\pi$
$59$	−2.12111	−0.276145	−0.138073	+	0.990422i	$0.544091\pi$
$60$	0	0
$61$	−2.79999	−0.358502	−0.179251	−	0.983803i	$-0.557367\pi$
$61$	−2.79999	−0.358502	−0.179251	+	0.983803i	$0.557367\pi$
$62$	−0.0380968	−0.00483830
$63$	0	0
$64$	−12.4721	−1.55901
$65$	0	0
$66$	0	0
$67$	5.03637	0.615290	0.307645	−	0.951501i	$-0.400459\pi$
$67$	5.03637	0.615290	0.307645	+	0.951501i	$0.400459\pi$
$68$	−1.32726	−0.160954
$69$	0	0
$70$	0	0
$71$	−8.56725	−1.01675	−0.508373	−	0.861137i	$-0.669753\pi$
$71$	−8.56725	−1.01675	−0.508373	+	0.861137i	$0.669753\pi$
$72$	0	0
$73$	1.59489	0.186668	0.0933339	−	0.995635i	$-0.470248\pi$
$73$	1.59489	0.186668	0.0933339	+	0.995635i	$0.470248\pi$
$74$	−16.4102	−1.90765
$75$	0	0
$76$	11.7273	1.34521
$77$	−11.7010	−1.33346
$78$	0	0
$79$	11.6409	1.30970	0.654852	−	0.755757i	$-0.272731\pi$
$79$	11.6409	1.30970	0.654852	+	0.755757i	$0.272731\pi$
$80$	0	0
$81$	0	0
$82$	12.8419	1.41815
$83$	4.69781	0.515652	0.257826	−	0.966191i	$-0.416994\pi$
$83$	4.69781	0.515652	0.257826	+	0.966191i	$0.416994\pi$
$84$	0	0
$85$	0	0
$86$	10.7129	1.15521
$87$	0	0
$88$	−4.17329	−0.444874
$89$	14.6662	1.55462	0.777309	−	0.629119i	$-0.216584\pi$
$89$	14.6662	1.55462	0.777309	+	0.629119i	$0.216584\pi$
$90$	0	0
$91$	−9.65381	−1.01199
$92$	−14.3972	−1.50101
$93$	0	0
$94$	15.8580	1.63563
$95$	0	0
$96$	0	0
$97$	−0.438441	−0.0445169	−0.0222585	−	0.999752i	$-0.507086\pi$
$97$	−0.438441	−0.0445169	−0.0222585	+	0.999752i	$0.507086\pi$
$98$	−31.3047	−3.16225
$99$	0	0
$100$	0	0
$101$	12.2631	1.22023	0.610113	−	0.792315i	$-0.291124\pi$
$101$	12.2631	1.22023	0.610113	+	0.792315i	$0.291124\pi$
$102$	0	0
$103$	13.5519	1.33531	0.667654	−	0.744471i	$-0.267298\pi$
$103$	13.5519	1.33531	0.667654	+	0.744471i	$0.267298\pi$
$104$	−3.44312	−0.337626
$105$	0	0
$106$	22.7372	2.20843
$107$	−14.0369	−1.35699	−0.678497	−	0.734603i	$-0.737368\pi$
$107$	−14.0369	−1.35699	−0.678497	+	0.734603i	$0.737368\pi$
$108$	0	0
$109$	−5.16741	−0.494948	−0.247474	−	0.968895i	$-0.579601\pi$
$109$	−5.16741	−0.494948	−0.247474	+	0.968895i	$0.579601\pi$
$110$	0	0
$111$	0	0
$112$	8.85992	0.837184
$113$	13.5837	1.27785	0.638923	−	0.769270i	$-0.279380\pi$
$113$	13.5837	1.27785	0.638923	+	0.769270i	$0.279380\pi$
$114$	0	0
$115$	0	0
$116$	2.75577	0.255867
$117$	0	0
$118$	4.62567	0.425827
$119$	2.22567	0.204027
$120$	0	0
$121$	−4.58860	−0.417145
$122$	6.10614	0.552824
$123$	0	0
$124$	0.0481417	0.00432325
$125$	0	0
$126$	0	0
$127$	21.9460	1.94739	0.973697	−	0.227847i	$-0.0731685\pi$
$127$	21.9460	1.94739	0.973697	+	0.227847i	$0.0731685\pi$
$128$	12.2439	1.08222
$129$	0	0
$130$	0	0
$131$	−0.515976	−0.0450810	−0.0225405	−	0.999746i	$-0.507175\pi$
$131$	−0.515976	−0.0450810	−0.0225405	+	0.999746i	$0.507175\pi$
$132$	0	0
$133$	−19.6654	−1.70521
$134$	−10.9832	−0.948802
$135$	0	0
$136$	0.793807	0.0680684
$137$	−4.92240	−0.420549	−0.210275	−	0.977642i	$-0.567436\pi$
$137$	−4.92240	−0.420549	−0.210275	+	0.977642i	$0.567436\pi$
$138$	0	0
$139$	−1.00838	−0.0855300	−0.0427650	−	0.999085i	$-0.513617\pi$
$139$	−1.00838	−0.0855300	−0.0427650	+	0.999085i	$0.513617\pi$
$140$	0	0
$141$	0	0
$142$	18.6832	1.56786
$143$	5.28965	0.442343
$144$	0	0
$145$	0	0
$146$	−3.47809	−0.287849
$147$	0	0
$148$	20.7371	1.70458
$149$	9.17808	0.751898	0.375949	−	0.926640i	$-0.377317\pi$
$149$	9.17808	0.751898	0.375949	+	0.926640i	$0.377317\pi$
$150$	0	0
$151$	−10.7451	−0.874421	−0.437210	−	0.899359i	$-0.644034\pi$
$151$	−10.7451	−0.874421	−0.437210	+	0.899359i	$0.644034\pi$
$152$	−7.01384	−0.568898
$153$	0	0
$154$	25.5173	2.05625
$155$	0	0
$156$	0	0
$157$	10.4159	0.831278	0.415639	−	0.909530i	$-0.363558\pi$
$157$	10.4159	0.831278	0.415639	+	0.909530i	$0.363558\pi$
$158$	−25.3862	−2.01962
$159$	0	0
$160$	0	0
$161$	24.1425	1.90269
$162$	0	0
$163$	−4.52007	−0.354039	−0.177019	−	0.984207i	$-0.556646\pi$
$163$	−4.52007	−0.354039	−0.177019	+	0.984207i	$0.556646\pi$
$164$	−16.2278	−1.26718
$165$	0	0
$166$	−10.2449	−0.795156
$167$	−7.66501	−0.593136	−0.296568	−	0.955012i	$-0.595842\pi$
$167$	−7.66501	−0.593136	−0.296568	+	0.955012i	$0.595842\pi$
$168$	0	0
$169$	−8.63584	−0.664295
$170$	0	0
$171$	0	0
$172$	−13.5376	−1.03223
$173$	−0.239810	−0.0182324	−0.00911621	−	0.999958i	$-0.502902\pi$
$173$	−0.239810	−0.0182324	−0.00911621	+	0.999958i	$0.502902\pi$
$174$	0	0
$175$	0	0
$176$	−4.85465	−0.365933
$177$	0	0
$178$	−31.9838	−2.39728
$179$	−19.3101	−1.44330	−0.721652	−	0.692256i	$-0.756617\pi$
$179$	−19.3101	−1.44330	−0.721652	+	0.692256i	$0.756617\pi$
$180$	0	0
$181$	11.1598	0.829500	0.414750	−	0.909935i	$-0.363869\pi$
$181$	11.1598	0.829500	0.414750	+	0.909935i	$0.363869\pi$
$182$	21.0528	1.56054
$183$	0	0
$184$	8.61063	0.634784
$185$	0	0
$186$	0	0
$187$	−1.21952	−0.0891803
$188$	−20.0392	−1.46151
$189$	0	0
$190$	0	0
$191$	−8.73098	−0.631751	−0.315876	−	0.948801i	$-0.602298\pi$
$191$	−8.73098	−0.631751	−0.315876	+	0.948801i	$0.602298\pi$
$192$	0	0
$193$	−13.3970	−0.964338	−0.482169	−	0.876078i	$-0.660151\pi$
$193$	−13.3970	−0.964338	−0.482169	+	0.876078i	$0.660151\pi$
$194$	0.956141	0.0686469
$195$	0	0
$196$	39.5587	2.82562
$197$	−8.46706	−0.603253	−0.301627	−	0.953426i	$-0.597530\pi$
$197$	−8.46706	−0.603253	−0.301627	+	0.953426i	$0.597530\pi$
$198$	0	0
$199$	26.8789	1.90539	0.952696	−	0.303925i	$-0.0982972\pi$
$199$	26.8789	1.90539	0.952696	+	0.303925i	$0.0982972\pi$
$200$	0	0
$201$	0	0
$202$	−26.7431	−1.88164
$203$	−4.62113	−0.324340
$204$	0	0
$205$	0	0
$206$	−29.5536	−2.05910
$207$	0	0
$208$	−4.00527	−0.277716
$209$	10.7753	0.745345
$210$	0	0
$211$	−10.7357	−0.739079	−0.369540	−	0.929215i	$-0.620485\pi$
$211$	−10.7357	−0.739079	−0.369540	+	0.929215i	$0.620485\pi$
$212$	−28.7322	−1.97334
$213$	0	0
$214$	30.6112	2.09254
$215$	0	0
$216$	0	0
$217$	−0.0807284	−0.00548020
$218$	11.2690	0.763230
$219$	0	0
$220$	0	0
$221$	−1.00615	−0.0676811
$222$	0	0
$223$	−24.0249	−1.60882	−0.804412	−	0.594071i	$-0.797520\pi$
$223$	−24.0249	−1.60882	−0.804412	+	0.594071i	$0.797520\pi$
$224$	−34.5543	−2.30876
$225$	0	0
$226$	−29.6230	−1.97049
$227$	−21.3367	−1.41617	−0.708083	−	0.706130i	$-0.750440\pi$
$227$	−21.3367	−1.41617	−0.708083	+	0.706130i	$0.750440\pi$
$228$	0	0
$229$	26.2040	1.73161	0.865804	−	0.500383i	$-0.166807\pi$
$229$	26.2040	1.73161	0.865804	+	0.500383i	$0.166807\pi$
$230$	0	0
$231$	0	0
$232$	−1.64817	−0.108208
$233$	−22.7585	−1.49096	−0.745478	−	0.666530i	$-0.767779\pi$
$233$	−22.7585	−1.49096	−0.745478	+	0.666530i	$0.767779\pi$
$234$	0	0
$235$	0	0
$236$	−5.84531	−0.380497
$237$	0	0
$238$	−4.85369	−0.314618
$239$	−14.0542	−0.909089	−0.454544	−	0.890724i	$-0.650198\pi$
$239$	−14.0542	−0.909089	−0.454544	+	0.890724i	$0.650198\pi$
$240$	0	0
$241$	−19.7283	−1.27081	−0.635406	−	0.772178i	$-0.719167\pi$
$241$	−19.7283	−1.27081	−0.635406	+	0.772178i	$0.719167\pi$
$242$	10.0067	0.643255
$243$	0	0
$244$	−7.71613	−0.493975
$245$	0	0
$246$	0	0
$247$	8.89006	0.565661
$248$	−0.0287925	−0.00182833
$249$	0	0
$250$	0	0
$251$	2.03124	0.128211	0.0641055	−	0.997943i	$-0.479581\pi$
$251$	2.03124	0.128211	0.0641055	+	0.997943i	$0.479581\pi$
$252$	0	0
$253$	−13.2285	−0.831667
$254$	−47.8593	−3.00296
$255$	0	0
$256$	−1.75703	−0.109814
$257$	16.1654	1.00837	0.504184	−	0.863596i	$-0.331793\pi$
$257$	16.1654	1.00837	0.504184	+	0.863596i	$0.331793\pi$
$258$	0	0
$259$	−34.7738	−2.16074
$260$	0	0
$261$	0	0
$262$	1.12523	0.0695167
$263$	10.8268	0.667609	0.333805	−	0.942642i	$-0.391667\pi$
$263$	10.8268	0.667609	0.333805	+	0.942642i	$0.391667\pi$
$264$	0	0
$265$	0	0
$266$	42.8858	2.62949
$267$	0	0
$268$	13.8791	0.847800
$269$	−5.59873	−0.341360	−0.170680	−	0.985326i	$-0.554597\pi$
$269$	−5.59873	−0.341360	−0.170680	+	0.985326i	$0.554597\pi$
$270$	0	0
$271$	−9.13245	−0.554757	−0.277378	−	0.960761i	$-0.589466\pi$
$271$	−9.13245	−0.554757	−0.277378	+	0.960761i	$0.589466\pi$
$272$	0.923410	0.0559900
$273$	0	0
$274$	10.7346	0.648503
$275$	0	0
$276$	0	0
$277$	−11.1296	−0.668715	−0.334357	−	0.942446i	$-0.608519\pi$
$277$	−11.1296	−0.668715	−0.334357	+	0.942446i	$0.608519\pi$
$278$	2.19906	0.131891
$279$	0	0
$280$	0	0
$281$	20.9692	1.25092	0.625459	−	0.780257i	$-0.284912\pi$
$281$	20.9692	1.25092	0.625459	+	0.780257i	$0.284912\pi$
$282$	0	0
$283$	24.4468	1.45321	0.726607	−	0.687053i	$-0.241096\pi$
$283$	24.4468	1.45321	0.726607	+	0.687053i	$0.241096\pi$
$284$	−23.6094	−1.40096
$285$	0	0
$286$	−11.5355	−0.682110
$287$	27.2123	1.60629
$288$	0	0
$289$	−16.7680	−0.986355
$290$	0	0
$291$	0	0
$292$	4.39515	0.257207
$293$	32.8311	1.91801	0.959007	−	0.283382i	$-0.0914565\pi$
$293$	32.8311	1.91801	0.959007	+	0.283382i	$0.0914565\pi$
$294$	0	0
$295$	0	0
$296$	−12.4024	−0.720875
$297$	0	0
$298$	−20.0153	−1.15946
$299$	−10.9140	−0.631173
$300$	0	0
$301$	22.7011	1.30847
$302$	23.4325	1.34839
$303$	0	0
$304$	−8.15898	−0.467950
$305$	0	0
$306$	0	0
$307$	18.8856	1.07786	0.538928	−	0.842352i	$-0.318829\pi$
$307$	18.8856	1.07786	0.538928	+	0.842352i	$0.318829\pi$
$308$	−32.2454	−1.83735
$309$	0	0
$310$	0	0
$311$	5.12864	0.290819	0.145409	−	0.989372i	$-0.453550\pi$
$311$	5.12864	0.290819	0.145409	+	0.989372i	$0.453550\pi$
$312$	0	0
$313$	−7.42579	−0.419731	−0.209865	−	0.977730i	$-0.567303\pi$
$313$	−7.42579	−0.419731	−0.209865	+	0.977730i	$0.567303\pi$
$314$	−22.7147	−1.28186
$315$	0	0
$316$	32.0797	1.80462
$317$	6.50490	0.365351	0.182676	−	0.983173i	$-0.441524\pi$
$317$	6.50490	0.365351	0.182676	+	0.983173i	$0.441524\pi$
$318$	0	0
$319$	2.53207	0.141769
$320$	0	0
$321$	0	0
$322$	−52.6493	−2.93403
$323$	−2.04959	−0.114042
$324$	0	0
$325$	0	0
$326$	9.85724	0.545942
$327$	0	0
$328$	9.70552	0.535898
$329$	33.6036	1.85263
$330$	0	0
$331$	−6.95501	−0.382282	−0.191141	−	0.981563i	$-0.561219\pi$
$331$	−6.95501	−0.382282	−0.191141	+	0.981563i	$0.561219\pi$
$332$	12.9461	0.710510
$333$	0	0
$334$	16.7156	0.914639
$335$	0	0
$336$	0	0
$337$	−10.9423	−0.596067	−0.298034	−	0.954555i	$-0.596331\pi$
$337$	−10.9423	−0.596067	−0.298034	+	0.954555i	$0.596331\pi$
$338$	18.8328	1.02437
$339$	0	0
$340$	0	0
$341$	0.0442338	0.00239540
$342$	0	0
$343$	−33.9878	−1.83517
$344$	8.09655	0.436537
$345$	0	0
$346$	0.522971	0.0281151
$347$	0.705731	0.0378856	0.0189428	−	0.999821i	$-0.493970\pi$
$347$	0.705731	0.0378856	0.0189428	+	0.999821i	$0.493970\pi$
$348$	0	0
$349$	−12.7664	−0.683369	−0.341684	−	0.939815i	$-0.610997\pi$
$349$	−12.7664	−0.683369	−0.341684	+	0.939815i	$0.610997\pi$
$350$	0	0
$351$	0	0
$352$	18.9335	1.00916
$353$	−32.7819	−1.74481	−0.872403	−	0.488788i	$-0.837439\pi$
$353$	−32.7819	−1.74481	−0.872403	+	0.488788i	$0.837439\pi$
$354$	0	0
$355$	0	0
$356$	40.4168	2.14209
$357$	0	0
$358$	42.1110	2.22563
$359$	−16.5874	−0.875448	−0.437724	−	0.899109i	$-0.644215\pi$
$359$	−16.5874	−0.875448	−0.437724	+	0.899109i	$0.644215\pi$
$360$	0	0
$361$	−0.890413	−0.0468638
$362$	−24.3370	−1.27912
$363$	0	0
$364$	−26.6037	−1.39441
$365$	0	0
$366$	0	0
$367$	−22.0945	−1.15333	−0.576663	−	0.816982i	$-0.695645\pi$
$367$	−22.0945	−1.15333	−0.576663	+	0.816982i	$0.695645\pi$
$368$	10.0165	0.522145
$369$	0	0
$370$	0	0
$371$	48.1808	2.50142
$372$	0	0
$373$	11.2414	0.582055	0.291028	−	0.956715i	$-0.406003\pi$
$373$	11.2414	0.582055	0.291028	+	0.956715i	$0.406003\pi$
$374$	2.65950	0.137520
$375$	0	0
$376$	11.9850	0.618082
$377$	2.08906	0.107592
$378$	0	0
$379$	−16.5726	−0.851276	−0.425638	−	0.904894i	$-0.639950\pi$
$379$	−16.5726	−0.851276	−0.425638	+	0.904894i	$0.639950\pi$
$380$	0	0
$381$	0	0
$382$	19.0403	0.974186
$383$	28.5900	1.46088	0.730440	−	0.682976i	$-0.239315\pi$
$383$	28.5900	1.46088	0.730440	+	0.682976i	$0.239315\pi$
$384$	0	0
$385$	0	0
$386$	29.2158	1.48705
$387$	0	0
$388$	−1.20824	−0.0613393
$389$	19.9390	1.01095	0.505473	−	0.862842i	$-0.331318\pi$
$389$	19.9390	1.01095	0.505473	+	0.862842i	$0.331318\pi$
$390$	0	0
$391$	2.51621	0.127250
$392$	−23.6592	−1.19497
$393$	0	0
$394$	18.4647	0.930240
$395$	0	0
$396$	0	0
$397$	21.5651	1.08232	0.541160	−	0.840919i	$-0.317985\pi$
$397$	21.5651	1.08232	0.541160	+	0.840919i	$0.317985\pi$
$398$	−58.6167	−2.93819
$399$	0	0
$400$	0	0
$401$	−33.4631	−1.67107	−0.835534	−	0.549439i	$-0.814842\pi$
$401$	−33.4631	−1.67107	−0.835534	+	0.549439i	$0.814842\pi$
$402$	0	0
$403$	0.0364946	0.00181792
$404$	33.7943	1.68133
$405$	0	0
$406$	10.0776	0.500145
$407$	19.0537	0.944459
$408$	0	0
$409$	−26.6886	−1.31967	−0.659833	−	0.751413i	$-0.729373\pi$
$409$	−26.6886	−1.31967	−0.659833	+	0.751413i	$0.729373\pi$
$410$	0	0
$411$	0	0
$412$	37.3460	1.83990
$413$	9.80194	0.482322
$414$	0	0
$415$	0	0
$416$	15.6208	0.765874
$417$	0	0
$418$	−23.4986	−1.14935
$419$	−16.7865	−0.820073	−0.410037	−	0.912069i	$-0.634484\pi$
$419$	−16.7865	−0.820073	−0.410037	+	0.912069i	$0.634484\pi$
$420$	0	0
$421$	10.7994	0.526332	0.263166	−	0.964751i	$-0.415233\pi$
$421$	10.7994	0.526332	0.263166	+	0.964751i	$0.415233\pi$
$422$	23.4122	1.13969
$423$	0	0
$424$	17.1841	0.834535
$425$	0	0
$426$	0	0
$427$	12.9391	0.626168
$428$	−38.6824	−1.86978
$429$	0	0
$430$	0	0
$431$	−19.9007	−0.958584	−0.479292	−	0.877655i	$-0.659107\pi$
$431$	−19.9007	−0.958584	−0.479292	+	0.877655i	$0.659107\pi$
$432$	0	0
$433$	15.1926	0.730110	0.365055	−	0.930986i	$-0.381050\pi$
$433$	15.1926	0.730110	0.365055	+	0.930986i	$0.381050\pi$
$434$	0.176050	0.00845069
$435$	0	0
$436$	−14.2402	−0.681982
$437$	−22.2325	−1.06352
$438$	0	0
$439$	−10.6767	−0.509570	−0.254785	−	0.966998i	$-0.582005\pi$
$439$	−10.6767	−0.509570	−0.254785	+	0.966998i	$0.582005\pi$
$440$	0	0
$441$	0	0
$442$	2.19419	0.104367
$443$	−39.7889	−1.89043	−0.945214	−	0.326451i	$-0.894147\pi$
$443$	−39.7889	−1.89043	−0.945214	+	0.326451i	$0.894147\pi$
$444$	0	0
$445$	0	0
$446$	52.3928	2.48087
$447$	0	0
$448$	57.6352	2.72301
$449$	−5.03661	−0.237692	−0.118846	−	0.992913i	$-0.537920\pi$
$449$	−5.03661	−0.237692	−0.118846	+	0.992913i	$0.537920\pi$
$450$	0	0
$451$	−14.9105	−0.702110
$452$	37.4336	1.76073
$453$	0	0
$454$	46.5305	2.18378
$455$	0	0
$456$	0	0
$457$	−21.9875	−1.02853	−0.514267	−	0.857630i	$-0.671936\pi$
$457$	−21.9875	−1.02853	−0.514267	+	0.857630i	$0.671936\pi$
$458$	−57.1450	−2.67021
$459$	0	0
$460$	0	0
$461$	−4.05792	−0.188996	−0.0944982	−	0.995525i	$-0.530125\pi$
$461$	−4.05792	−0.188996	−0.0944982	+	0.995525i	$0.530125\pi$
$462$	0	0
$463$	14.1164	0.656045	0.328022	−	0.944670i	$-0.393618\pi$
$463$	14.1164	0.656045	0.328022	+	0.944670i	$0.393618\pi$
$464$	−1.91726	−0.0890067
$465$	0	0
$466$	49.6310	2.29911
$467$	−17.7201	−0.819987	−0.409994	−	0.912088i	$-0.634469\pi$
$467$	−17.7201	−0.819987	−0.409994	+	0.912088i	$0.634469\pi$
$468$	0	0
$469$	−23.2737	−1.07468
$470$	0	0
$471$	0	0
$472$	3.49595	0.160914
$473$	−12.4387	−0.571931
$474$	0	0
$475$	0	0
$476$	6.13345	0.281126
$477$	0	0
$478$	30.6490	1.40185
$479$	−2.11468	−0.0966224	−0.0483112	−	0.998832i	$-0.515384\pi$
$479$	−2.11468	−0.0966224	−0.0483112	+	0.998832i	$0.515384\pi$
$480$	0	0
$481$	15.7201	0.716773
$482$	43.0230	1.95964
$483$	0	0
$484$	−12.6451	−0.574779
$485$	0	0
$486$	0	0
$487$	20.7266	0.939213	0.469607	−	0.882876i	$-0.344396\pi$
$487$	20.7266	0.939213	0.469607	+	0.882876i	$0.344396\pi$
$488$	4.61485	0.208905
$489$	0	0
$490$	0	0
$491$	−8.10793	−0.365906	−0.182953	−	0.983122i	$-0.558566\pi$
$491$	−8.10793	−0.365906	−0.182953	+	0.983122i	$0.558566\pi$
$492$	0	0
$493$	−0.481630	−0.0216915
$494$	−19.3872	−0.872272
$495$	0	0
$496$	−0.0334934	−0.00150390
$497$	39.5904	1.77587
$498$	0	0
$499$	−26.6873	−1.19469	−0.597343	−	0.801986i	$-0.703777\pi$
$499$	−26.6873	−1.19469	−0.597343	+	0.801986i	$0.703777\pi$
$500$	0	0
$501$	0	0
$502$	−4.42968	−0.197706
$503$	29.3889	1.31039	0.655193	−	0.755462i	$-0.272587\pi$
$503$	29.3889	1.31039	0.655193	+	0.755462i	$0.272587\pi$
$504$	0	0
$505$	0	0
$506$	28.8483	1.28246
$507$	0	0
$508$	60.4782	2.68329
$509$	44.1172	1.95546	0.977729	−	0.209870i	$-0.0673041\pi$
$509$	44.1172	1.95546	0.977729	+	0.209870i	$0.0673041\pi$
$510$	0	0
$511$	−7.37020	−0.326038
$512$	−20.6562	−0.912883
$513$	0	0
$514$	−35.2530	−1.55494
$515$	0	0
$516$	0	0
$517$	−18.4126	−0.809784
$518$	75.8338	3.33194
$519$	0	0
$520$	0	0
$521$	−24.4913	−1.07299	−0.536493	−	0.843905i	$-0.680251\pi$
$521$	−24.4913	−1.07299	−0.536493	+	0.843905i	$0.680251\pi$
$522$	0	0
$523$	−31.4170	−1.37377	−0.686885	−	0.726766i	$-0.741022\pi$
$523$	−31.4170	−1.37377	−0.686885	+	0.726766i	$0.741022\pi$
$524$	−1.42191	−0.0621165
$525$	0	0
$526$	−23.6108	−1.02948
$527$	−0.00841378	−0.000366510 0
$528$	0	0
$529$	4.29397	0.186694
$530$	0	0
$531$	0	0
$532$	−54.1934	−2.34958
$533$	−12.3018	−0.532849
$534$	0	0
$535$	0	0
$536$	−8.30078	−0.358539
$537$	0	0
$538$	12.2096	0.526391
$539$	36.3476	1.56560
$540$	0	0
$541$	−6.19606	−0.266389	−0.133195	−	0.991090i	$-0.542524\pi$
$541$	−6.19606	−0.266389	−0.133195	+	0.991090i	$0.542524\pi$
$542$	19.9158	0.855457
$543$	0	0
$544$	−3.60136	−0.154407
$545$	0	0
$546$	0	0
$547$	19.0729	0.815499	0.407749	−	0.913094i	$-0.366314\pi$
$547$	19.0729	0.815499	0.407749	+	0.913094i	$0.366314\pi$
$548$	−13.5650	−0.579469
$549$	0	0
$550$	0	0
$551$	4.25554	0.181292
$552$	0	0
$553$	−53.7941	−2.28756
$554$	24.2712	1.03118
$555$	0	0
$556$	−2.77888	−0.117851
$557$	−17.3672	−0.735872	−0.367936	−	0.929851i	$-0.619935\pi$
$557$	−17.3672	−0.735872	−0.367936	+	0.929851i	$0.619935\pi$
$558$	0	0
$559$	−10.2624	−0.434053
$560$	0	0
$561$	0	0
$562$	−45.7291	−1.92897
$563$	−15.0986	−0.636331	−0.318165	−	0.948035i	$-0.603067\pi$
$563$	−15.0986	−0.636331	−0.318165	+	0.948035i	$0.603067\pi$
$564$	0	0
$565$	0	0
$566$	−53.3130	−2.24091
$567$	0	0
$568$	14.1203	0.592473
$569$	−22.5431	−0.945058	−0.472529	−	0.881315i	$-0.656659\pi$
$569$	−22.5431	−0.945058	−0.472529	+	0.881315i	$0.656659\pi$
$570$	0	0
$571$	30.8973	1.29301	0.646506	−	0.762908i	$-0.276229\pi$
$571$	30.8973	1.29301	0.646506	+	0.762908i	$0.276229\pi$
$572$	14.5771	0.609498
$573$	0	0
$574$	−59.3439	−2.47697
$575$	0	0
$576$	0	0
$577$	12.4363	0.517729	0.258865	−	0.965914i	$-0.416652\pi$
$577$	12.4363	0.517729	0.258865	+	0.965914i	$0.416652\pi$
$578$	36.5673	1.52100
$579$	0	0
$580$	0	0
$581$	−21.7092	−0.900650
$582$	0	0
$583$	−26.3999	−1.09337
$584$	−2.62865	−0.108774
$585$	0	0
$586$	−71.5972	−2.95765
$587$	4.57702	0.188914	0.0944569	−	0.995529i	$-0.469889\pi$
$587$	4.57702	0.188914	0.0944569	+	0.995529i	$0.469889\pi$
$588$	0	0
$589$	0.0743417	0.00306319
$590$	0	0
$591$	0	0
$592$	−14.4273	−0.592959
$593$	−40.3499	−1.65697	−0.828486	−	0.560009i	$-0.810797\pi$
$593$	−40.3499	−1.65697	−0.828486	+	0.560009i	$0.810797\pi$
$594$	0	0
$595$	0	0
$596$	25.2927	1.03603
$597$	0	0
$598$	23.8009	0.973293
$599$	−15.5138	−0.633879	−0.316939	−	0.948446i	$-0.602655\pi$
$599$	−15.5138	−0.633879	−0.316939	+	0.948446i	$0.602655\pi$
$600$	0	0
$601$	−39.3512	−1.60517	−0.802585	−	0.596537i	$-0.796543\pi$
$601$	−39.3512	−1.60517	−0.802585	+	0.596537i	$0.796543\pi$
$602$	−49.5059	−2.01771
$603$	0	0
$604$	−29.6110	−1.20485
$605$	0	0
$606$	0	0
$607$	−27.6748	−1.12328	−0.561642	−	0.827380i	$-0.689830\pi$
$607$	−27.6748	−1.12328	−0.561642	+	0.827380i	$0.689830\pi$
$608$	31.8206	1.29049
$609$	0	0
$610$	0	0
$611$	−15.1911	−0.614565
$612$	0	0
$613$	24.9886	1.00928	0.504640	−	0.863330i	$-0.331625\pi$
$613$	24.9886	1.00928	0.504640	+	0.863330i	$0.331625\pi$
$614$	−41.1851	−1.66210
$615$	0	0
$616$	19.2853	0.777027
$617$	−39.3823	−1.58547	−0.792736	−	0.609566i	$-0.791344\pi$
$617$	−39.3823	−1.58547	−0.792736	+	0.609566i	$0.791344\pi$
$618$	0	0
$619$	22.6843	0.911760	0.455880	−	0.890041i	$-0.349325\pi$
$619$	22.6843	0.911760	0.455880	+	0.890041i	$0.349325\pi$
$620$	0	0
$621$	0	0
$622$	−11.1844	−0.448454
$623$	−67.7746	−2.71533
$624$	0	0
$625$	0	0
$626$	16.1940	0.647241
$627$	0	0
$628$	28.7038	1.14541
$629$	−3.62424	−0.144508
$630$	0	0
$631$	−28.9276	−1.15159	−0.575795	−	0.817594i	$-0.695307\pi$
$631$	−28.9276	−1.15159	−0.575795	+	0.817594i	$0.695307\pi$
$632$	−19.1862	−0.763185
$633$	0	0
$634$	−14.1857	−0.563386
$635$	0	0
$636$	0	0
$637$	29.9881	1.18817
$638$	−5.52188	−0.218613
$639$	0	0
$640$	0	0
$641$	−38.4633	−1.51921	−0.759605	−	0.650384i	$-0.774608\pi$
$641$	−38.4633	−1.51921	−0.759605	+	0.650384i	$0.774608\pi$
$642$	0	0
$643$	−10.4518	−0.412179	−0.206090	−	0.978533i	$-0.566074\pi$
$643$	−10.4518	−0.412179	−0.206090	+	0.978533i	$0.566074\pi$
$644$	66.5312	2.62169
$645$	0	0
$646$	4.46970	0.175858
$647$	−31.2256	−1.22761	−0.613803	−	0.789460i	$-0.710361\pi$
$647$	−31.2256	−1.22761	−0.613803	+	0.789460i	$0.710361\pi$
$648$	0	0
$649$	−5.37082	−0.210823
$650$	0	0
$651$	0	0
$652$	−12.4563	−0.487825
$653$	19.5358	0.764494	0.382247	−	0.924060i	$-0.375150\pi$
$653$	19.5358	0.764494	0.382247	+	0.924060i	$0.375150\pi$
$654$	0	0
$655$	0	0
$656$	11.2901	0.440805
$657$	0	0
$658$	−73.2819	−2.85683
$659$	−5.49708	−0.214136	−0.107068	−	0.994252i	$-0.534146\pi$
$659$	−5.49708	−0.214136	−0.107068	+	0.994252i	$0.534146\pi$
$660$	0	0
$661$	−23.1157	−0.899097	−0.449548	−	0.893256i	$-0.648415\pi$
$661$	−23.1157	−0.899097	−0.449548	+	0.893256i	$0.648415\pi$
$662$	15.1673	0.589494
$663$	0	0
$664$	−7.74279	−0.300478
$665$	0	0
$666$	0	0
$667$	−5.22436	−0.202288
$668$	−21.1230	−0.817274
$669$	0	0
$670$	0	0
$671$	−7.08978	−0.273698
$672$	0	0
$673$	37.1375	1.43155	0.715774	−	0.698332i	$-0.246074\pi$
$673$	37.1375	1.43155	0.715774	+	0.698332i	$0.246074\pi$
$674$	23.8628	0.919159
$675$	0	0
$676$	−23.7984	−0.915324
$677$	8.34698	0.320800	0.160400	−	0.987052i	$-0.448722\pi$
$677$	8.34698	0.320800	0.160400	+	0.987052i	$0.448722\pi$
$678$	0	0
$679$	2.02609	0.0777543
$680$	0	0
$681$	0	0
$682$	−0.0964639	−0.00369379
$683$	4.60136	0.176066	0.0880330	−	0.996118i	$-0.471942\pi$
$683$	4.60136	0.176066	0.0880330	+	0.996118i	$0.471942\pi$
$684$	0	0
$685$	0	0
$686$	74.1196	2.82990
$687$	0	0
$688$	9.41845	0.359075
$689$	−21.7809	−0.829787
$690$	0	0
$691$	−37.7916	−1.43766	−0.718831	−	0.695185i	$-0.755323\pi$
$691$	−37.7916	−1.43766	−0.718831	+	0.695185i	$0.755323\pi$
$692$	−0.660862	−0.0251222
$693$	0	0
$694$	−1.53904	−0.0584211
$695$	0	0
$696$	0	0
$697$	2.83616	0.107427
$698$	27.8406	1.05378
$699$	0	0
$700$	0	0
$701$	30.4453	1.14990	0.574951	−	0.818188i	$-0.305021\pi$
$701$	30.4453	1.14990	0.574951	+	0.818188i	$0.305021\pi$
$702$	0	0
$703$	32.0227	1.20776
$704$	−31.5803	−1.19023
$705$	0	0
$706$	71.4899	2.69056
$707$	−56.6694	−2.13127
$708$	0	0
$709$	6.15061	0.230991	0.115496	−	0.993308i	$-0.463154\pi$
$709$	6.15061	0.230991	0.115496	+	0.993308i	$0.463154\pi$
$710$	0	0
$711$	0	0
$712$	−24.1724	−0.905901
$713$	−0.0912665	−0.00341796
$714$	0	0
$715$	0	0
$716$	−53.2142	−1.98871
$717$	0	0
$718$	36.1733	1.34998
$719$	−22.0551	−0.822516	−0.411258	−	0.911519i	$-0.634911\pi$
$719$	−22.0551	−0.822516	−0.411258	+	0.911519i	$0.634911\pi$
$720$	0	0
$721$	−62.6251	−2.33228
$722$	1.94179	0.0722659
$723$	0	0
$724$	30.7538	1.14296
$725$	0	0
$726$	0	0
$727$	−9.94047	−0.368672	−0.184336	−	0.982863i	$-0.559013\pi$
$727$	−9.94047	−0.368672	−0.184336	+	0.982863i	$0.559013\pi$
$728$	15.9111	0.589705
$729$	0	0
$730$	0	0
$731$	2.36598	0.0875090
$732$	0	0
$733$	−37.8281	−1.39721	−0.698606	−	0.715507i	$-0.746196\pi$
$733$	−37.8281	−1.39721	−0.698606	+	0.715507i	$0.746196\pi$
$734$	48.1832	1.77847
$735$	0	0
$736$	−39.0649	−1.43995
$737$	12.7525	0.469743
$738$	0	0
$739$	10.8388	0.398711	0.199355	−	0.979927i	$-0.436115\pi$
$739$	10.8388	0.398711	0.199355	+	0.979927i	$0.436115\pi$
$740$	0	0
$741$	0	0
$742$	−105.071	−3.85729
$743$	20.3076	0.745013	0.372506	−	0.928030i	$-0.378499\pi$
$743$	20.3076	0.745013	0.372506	+	0.928030i	$0.378499\pi$
$744$	0	0
$745$	0	0
$746$	−24.5148	−0.897552
$747$	0	0
$748$	−3.36073	−0.122880
$749$	64.8662	2.37016
$750$	0	0
$751$	−17.0745	−0.623056	−0.311528	−	0.950237i	$-0.600841\pi$
$751$	−17.0745	−0.623056	−0.311528	+	0.950237i	$0.600841\pi$
$752$	13.9418	0.508406
$753$	0	0
$754$	−4.55576	−0.165911
$755$	0	0
$756$	0	0
$757$	1.59082	0.0578194	0.0289097	−	0.999582i	$-0.490796\pi$
$757$	1.59082	0.0578194	0.0289097	+	0.999582i	$0.490796\pi$
$758$	36.1410	1.31270
$759$	0	0
$760$	0	0
$761$	−13.5039	−0.489515	−0.244758	−	0.969584i	$-0.578708\pi$
$761$	−13.5039	−0.489515	−0.244758	+	0.969584i	$0.578708\pi$
$762$	0	0
$763$	23.8793	0.864488
$764$	−24.0606	−0.870482
$765$	0	0
$766$	−62.3483	−2.25274
$767$	−4.43113	−0.159999
$768$	0	0
$769$	9.89598	0.356858	0.178429	−	0.983953i	$-0.442899\pi$
$769$	9.89598	0.356858	0.178429	+	0.983953i	$0.442899\pi$
$770$	0	0
$771$	0	0
$772$	−36.9191	−1.32875
$773$	2.30422	0.0828771	0.0414385	−	0.999141i	$-0.486806\pi$
$773$	2.30422	0.0828771	0.0414385	+	0.999141i	$0.486806\pi$
$774$	0	0
$775$	0	0
$776$	0.722625	0.0259407
$777$	0	0
$778$	−43.4824	−1.55892
$779$	−25.0594	−0.897848
$780$	0	0
$781$	−21.6929	−0.776233
$782$	−5.48728	−0.196225
$783$	0	0
$784$	−27.5220	−0.982929
$785$	0	0
$786$	0	0
$787$	−14.6066	−0.520670	−0.260335	−	0.965518i	$-0.583833\pi$
$787$	−14.6066	−0.520670	−0.260335	+	0.965518i	$0.583833\pi$
$788$	−23.3333	−0.831214
$789$	0	0
$790$	0	0
$791$	−62.7721	−2.23192
$792$	0	0
$793$	−5.84934	−0.207716
$794$	−47.0286	−1.66898
$795$	0	0
$796$	74.0720	2.62541
$797$	0.347706	0.0123164	0.00615818	−	0.999981i	$-0.498040\pi$
$797$	0.347706	0.0123164	0.00615818	+	0.999981i	$0.498040\pi$
$798$	0	0
$799$	3.50228	0.123902
$800$	0	0
$801$	0	0
$802$	72.9755	2.57685
$803$	4.03838	0.142511
$804$	0	0
$805$	0	0
$806$	−0.0795864	−0.00280331
$807$	0	0
$808$	−20.2117	−0.711044
$809$	33.8275	1.18931	0.594656	−	0.803980i	$-0.297288\pi$
$809$	33.8275	1.18931	0.594656	+	0.803980i	$0.297288\pi$
$810$	0	0
$811$	−6.27670	−0.220405	−0.110202	−	0.993909i	$-0.535150\pi$
$811$	−6.27670	−0.220405	−0.110202	+	0.993909i	$0.535150\pi$
$812$	−12.7348	−0.446903
$813$	0	0
$814$	−41.5519	−1.45639
$815$	0	0
$816$	0	0
$817$	−20.9051	−0.731377
$818$	58.2017	2.03498
$819$	0	0
$820$	0	0
$821$	42.2966	1.47616	0.738080	−	0.674713i	$-0.235733\pi$
$821$	42.2966	1.47616	0.738080	+	0.674713i	$0.235733\pi$
$822$	0	0
$823$	6.96302	0.242716	0.121358	−	0.992609i	$-0.461275\pi$
$823$	6.96302	0.242716	0.121358	+	0.992609i	$0.461275\pi$
$824$	−22.3358	−0.778105
$825$	0	0
$826$	−21.3758	−0.743760
$827$	56.1125	1.95122	0.975612	−	0.219503i	$-0.0704436\pi$
$827$	56.1125	1.95122	0.975612	+	0.219503i	$0.0704436\pi$
$828$	0	0
$829$	−30.5026	−1.05940	−0.529699	−	0.848186i	$-0.677695\pi$
$829$	−30.5026	−1.05940	−0.529699	+	0.848186i	$0.677695\pi$
$830$	0	0
$831$	0	0
$832$	−26.0549	−0.903293
$833$	−6.91373	−0.239546
$834$	0	0
$835$	0	0
$836$	29.6944	1.02700
$837$	0	0
$838$	36.6075	1.26459
$839$	−44.8521	−1.54847	−0.774233	−	0.632901i	$-0.781864\pi$
$839$	−44.8521	−1.54847	−0.774233	+	0.632901i	$0.781864\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−23.5511	−0.811624
$843$	0	0
$844$	−29.5853	−1.01837
$845$	0	0
$846$	0	0
$847$	21.2045	0.728596
$848$	19.9898	0.686451
$849$	0	0
$850$	0	0
$851$	−39.3131	−1.34764
$852$	0	0
$853$	17.0361	0.583306	0.291653	−	0.956524i	$-0.405795\pi$
$853$	17.0361	0.583306	0.291653	+	0.956524i	$0.405795\pi$
$854$	−28.2173	−0.965575
$855$	0	0
$856$	23.1351	0.790742
$857$	−46.0976	−1.57467	−0.787333	−	0.616528i	$-0.788539\pi$
$857$	−46.0976	−1.57467	−0.787333	+	0.616528i	$0.788539\pi$
$858$	0	0
$859$	22.9553	0.783225	0.391612	−	0.920130i	$-0.371917\pi$
$859$	22.9553	0.783225	0.391612	+	0.920130i	$0.371917\pi$
$860$	0	0
$861$	0	0
$862$	43.3990	1.47817
$863$	−40.6665	−1.38430	−0.692152	−	0.721752i	$-0.743337\pi$
$863$	−40.6665	−1.38430	−0.692152	+	0.721752i	$0.743337\pi$
$864$	0	0
$865$	0	0
$866$	−33.1316	−1.12586
$867$	0	0
$868$	−0.222469	−0.00755109
$869$	29.4756	0.999892
$870$	0	0
$871$	10.5213	0.356499
$872$	8.51677	0.288414
$873$	0	0
$874$	48.4840	1.63999
$875$	0	0
$876$	0	0
$877$	47.6592	1.60934	0.804669	−	0.593724i	$-0.202343\pi$
$877$	47.6592	1.60934	0.804669	+	0.593724i	$0.202343\pi$
$878$	23.2834	0.785777
$879$	0	0
$880$	0	0
$881$	−27.9539	−0.941791	−0.470895	−	0.882189i	$-0.656069\pi$
$881$	−27.9539	−0.941791	−0.470895	+	0.882189i	$0.656069\pi$
$882$	0	0
$883$	8.66251	0.291517	0.145758	−	0.989320i	$-0.453438\pi$
$883$	8.66251	0.291517	0.145758	+	0.989320i	$0.453438\pi$
$884$	−2.77273	−0.0932569
$885$	0	0
$886$	86.7706	2.91512
$887$	−44.4792	−1.49347	−0.746733	−	0.665124i	$-0.768379\pi$
$887$	−44.4792	−1.49347	−0.746733	+	0.665124i	$0.768379\pi$
$888$	0	0
$889$	−101.415	−3.40136
$890$	0	0
$891$	0	0
$892$	−66.2071	−2.21678
$893$	−30.9451	−1.03554
$894$	0	0
$895$	0	0
$896$	−56.5808	−1.89023
$897$	0	0
$898$	10.9837	0.366531
$899$	0.0174694	0.000582637 0
$900$	0	0
$901$	5.02157	0.167293
$902$	32.5165	1.08268
$903$	0	0
$904$	−22.3882	−0.744621
$905$	0	0
$906$	0	0
$907$	−21.1492	−0.702248	−0.351124	−	0.936329i	$-0.614201\pi$
$907$	−21.1492	−0.702248	−0.351124	+	0.936329i	$0.614201\pi$
$908$	−58.7990	−1.95131
$909$	0	0
$910$	0	0
$911$	−22.2895	−0.738483	−0.369241	−	0.929334i	$-0.620382\pi$
$911$	−22.2895	−0.738483	−0.369241	+	0.929334i	$0.620382\pi$
$912$	0	0
$913$	11.8952	0.393674
$914$	47.9498	1.58604
$915$	0	0
$916$	72.2122	2.38596
$917$	2.38439	0.0787396
$918$	0	0
$919$	−52.4155	−1.72903	−0.864513	−	0.502610i	$-0.832373\pi$
$919$	−52.4155	−1.72903	−0.864513	+	0.502610i	$0.832373\pi$
$920$	0	0
$921$	0	0
$922$	8.84941	0.291440
$923$	−17.8975	−0.589102
$924$	0	0
$925$	0	0
$926$	−30.7847	−1.01165
$927$	0	0
$928$	7.47745	0.245459
$929$	4.32654	0.141949	0.0709746	−	0.997478i	$-0.477389\pi$
$929$	4.32654	0.141949	0.0709746	+	0.997478i	$0.477389\pi$
$930$	0	0
$931$	61.0876	2.00207
$932$	−62.7171	−2.05437
$933$	0	0
$934$	38.6435	1.26445
$935$	0	0
$936$	0	0
$937$	5.59734	0.182857	0.0914286	−	0.995812i	$-0.470857\pi$
$937$	5.59734	0.182857	0.0914286	+	0.995812i	$0.470857\pi$
$938$	50.7547	1.65720
$939$	0	0
$940$	0	0
$941$	−48.3823	−1.57722	−0.788608	−	0.614896i	$-0.789198\pi$
$941$	−48.3823	−1.57722	−0.788608	+	0.614896i	$0.789198\pi$
$942$	0	0
$943$	30.7645	1.00183
$944$	4.06673	0.132361
$945$	0	0
$946$	27.1260	0.881941
$947$	−54.4744	−1.77018	−0.885091	−	0.465419i	$-0.845904\pi$
$947$	−54.4744	−1.77018	−0.885091	+	0.465419i	$0.845904\pi$
$948$	0	0
$949$	3.33182	0.108155
$950$	0	0
$951$	0	0
$952$	−3.66829	−0.118890
$953$	−0.143303	−0.00464203	−0.00232101	−	0.999997i	$-0.500739\pi$
$953$	−0.143303	−0.00464203	−0.00232101	+	0.999997i	$0.500739\pi$
$954$	0	0
$955$	0	0
$956$	−38.7301	−1.25262
$957$	0	0
$958$	4.61165	0.148996
$959$	22.7471	0.734541
$960$	0	0
$961$	−30.9997	−0.999990
$962$	−34.2819	−1.10529
$963$	0	0
$964$	−54.3667	−1.75103
$965$	0	0
$966$	0	0
$967$	8.05985	0.259187	0.129594	−	0.991567i	$-0.458633\pi$
$967$	8.05985	0.259187	0.129594	+	0.991567i	$0.458633\pi$
$968$	7.56279	0.243077
$969$	0	0
$970$	0	0
$971$	−9.00391	−0.288949	−0.144475	−	0.989509i	$-0.546149\pi$
$971$	−9.00391	−0.288949	−0.144475	+	0.989509i	$0.546149\pi$
$972$	0	0
$973$	4.65987	0.149389
$974$	−45.2001	−1.44830
$975$	0	0
$976$	5.36831	0.171835
$977$	58.2212	1.86266	0.931330	−	0.364176i	$-0.118649\pi$
$977$	58.2212	1.86266	0.931330	+	0.364176i	$0.118649\pi$
$978$	0	0
$979$	37.1360	1.18687
$980$	0	0
$981$	0	0
$982$	17.6816	0.564241
$983$	−9.19101	−0.293148	−0.146574	−	0.989200i	$-0.546825\pi$
$983$	−9.19101	−0.293148	−0.146574	+	0.989200i	$0.546825\pi$
$984$	0	0
$985$	0	0
$986$	1.05033	0.0334492
$987$	0	0
$988$	24.4990	0.779416
$989$	25.6644	0.816081
$990$	0	0
$991$	−18.4929	−0.587447	−0.293724	−	0.955890i	$-0.594895\pi$
$991$	−18.4929	−0.587447	−0.293724	+	0.955890i	$0.594895\pi$
$992$	0.130627	0.00414740
$993$	0	0
$994$	−86.3376	−2.73846
$995$	0	0
$996$	0	0
$997$	24.7326	0.783288	0.391644	−	0.920117i	$-0.371906\pi$
$997$	24.7326	0.783288	0.391644	+	0.920117i	$0.371906\pi$
$998$	58.1989	1.84225
$999$	0	0

Twists

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

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6525 = 3^{2} \cdot 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6525.a (trivial)

\(f(q)\)	\(=\)	\(q-2.18077 q^{2} +2.75577 q^{4} -4.62113 q^{7} -1.64817 q^{8} +O(q^{10})\)	\(q-2.18077 q^{2} +2.75577 q^{4} -4.62113 q^{7} -1.64817 q^{8} +2.53207 q^{11} +2.08906 q^{13} +10.0776 q^{14} -1.91726 q^{16} -0.481630 q^{17} +4.25554 q^{19} -5.52188 q^{22} -5.22436 q^{23} -4.55576 q^{26} -12.7348 q^{28} +1.00000 q^{29} +0.0174694 q^{31} +7.47745 q^{32} +1.05033 q^{34} +7.52495 q^{37} -9.28036 q^{38} -5.88867 q^{41} -4.91245 q^{43} +6.97782 q^{44} +11.3932 q^{46} -7.27173 q^{47} +14.3549 q^{49} +5.75697 q^{52} -10.4262 q^{53} +7.61641 q^{56} -2.18077 q^{58} -2.12111 q^{59} -2.79999 q^{61} -0.0380968 q^{62} -12.4721 q^{64} +5.03637 q^{67} -1.32726 q^{68} -8.56725 q^{71} +1.59489 q^{73} -16.4102 q^{74} +11.7273 q^{76} -11.7010 q^{77} +11.6409 q^{79} +12.8419 q^{82} +4.69781 q^{83} +10.7129 q^{86} -4.17329 q^{88} +14.6662 q^{89} -9.65381 q^{91} -14.3972 q^{92} +15.8580 q^{94} -0.438441 q^{97} -31.3047 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 2 q^{2} + 10 q^{4} - q^{7} - 9 q^{8}+O(q^{10}) \) 9 * q - 2 * q^2 + 10 * q^4 - q^7 - 9 * q^8	\( 9 q - 2 q^{2} + 10 q^{4} - q^{7} - 9 q^{8} + 2 q^{11} - q^{13} - 3 q^{14} + 4 q^{16} - 12 q^{17} - q^{19} - 3 q^{22} - 16 q^{23} + 6 q^{26} + 4 q^{28} + 9 q^{29} + 5 q^{31} - 20 q^{32} + 3 q^{34} - 30 q^{38} - 10 q^{41} - 3 q^{43} - 13 q^{44} + 4 q^{46} - 26 q^{47} - 8 q^{49} + 9 q^{52} - 22 q^{53} + 22 q^{56} - 2 q^{58} + 4 q^{59} + 7 q^{61} - 28 q^{62} + 9 q^{64} - 5 q^{67} - 39 q^{68} + 10 q^{73} - 34 q^{74} - 2 q^{76} - 34 q^{77} + 10 q^{79} + 8 q^{82} - 46 q^{83} + 28 q^{86} - 2 q^{88} + 4 q^{89} - 21 q^{91} - 20 q^{92} + 5 q^{94} - 7 q^{97} - 51 q^{98}+O(q^{100}) \) 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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