LMFDB - Embedded newform 6525.2.a.ca.1.6

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			1.6
Root			$-0.747618$ 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)

$f(q)$	$=$	$q+0.747618 q^{2} -1.44107 q^{4} -3.28783 q^{7} -2.57260 q^{8} +O(q^{10})$	$q+0.747618 q^{2} -1.44107 q^{4} -3.28783 q^{7} -2.57260 q^{8} +1.14171 q^{11} +2.14612 q^{13} -2.45804 q^{14} +0.958808 q^{16} +6.16470 q^{17} -7.20344 q^{19} +0.853562 q^{22} -0.227525 q^{23} +1.60448 q^{26} +4.73798 q^{28} +1.00000 q^{29} +1.59202 q^{31} +5.86203 q^{32} +4.60884 q^{34} +0.690638 q^{37} -5.38542 q^{38} +9.55584 q^{41} +0.739212 q^{43} -1.64528 q^{44} -0.170102 q^{46} -3.52745 q^{47} +3.80980 q^{49} -3.09270 q^{52} +0.756550 q^{53} +8.45827 q^{56} +0.747618 q^{58} +7.99502 q^{59} -0.836591 q^{61} +1.19023 q^{62} +2.46494 q^{64} -6.00144 q^{67} -8.88375 q^{68} -10.8715 q^{71} -6.33908 q^{73} +0.516334 q^{74} +10.3806 q^{76} -3.75374 q^{77} -7.23916 q^{79} +7.14412 q^{82} -7.46438 q^{83} +0.552648 q^{86} -2.93716 q^{88} -0.620360 q^{89} -7.05606 q^{91} +0.327878 q^{92} -2.63719 q^{94} -4.33183 q^{97} +2.84827 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.747618	0.528646	0.264323	−	0.964434i	$-0.414852\pi$
$2$	0.747618	0.528646	0.264323	+	0.964434i	$0.414852\pi$
$3$	0	0
$3$	0	0
$4$	−1.44107	−0.720534
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−3.28783	−1.24268	−0.621341	−	0.783541i	$-0.713412\pi$
$7$	−3.28783	−1.24268	−0.621341	+	0.783541i	$0.713412\pi$
$8$	−2.57260	−0.909553
$9$	0	0
$10$	0	0
$11$	1.14171	0.344238	0.172119	−	0.985076i	$-0.444939\pi$
$11$	1.14171	0.344238	0.172119	+	0.985076i	$0.444939\pi$
$12$	0	0
$13$	2.14612	0.595226	0.297613	−	0.954687i	$-0.403810\pi$
$13$	2.14612	0.595226	0.297613	+	0.954687i	$0.403810\pi$
$14$	−2.45804	−0.656938
$15$	0	0
$16$	0.958808	0.239702
$17$	6.16470	1.49516	0.747580	−	0.664172i	$-0.231216\pi$
$17$	6.16470	1.49516	0.747580	+	0.664172i	$0.231216\pi$
$18$	0	0
$19$	−7.20344	−1.65258	−0.826292	−	0.563243i	$-0.809554\pi$
$19$	−7.20344	−1.65258	−0.826292	+	0.563243i	$0.809554\pi$
$20$	0	0
$21$	0	0
$22$	0.853562	0.181980
$23$	−0.227525	−0.0474422	−0.0237211	−	0.999719i	$-0.507551\pi$
$23$	−0.227525	−0.0474422	−0.0237211	+	0.999719i	$0.507551\pi$
$24$	0	0
$25$	0	0
$26$	1.60448	0.314664
$27$	0	0
$28$	4.73798	0.895393
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	1.59202	0.285936	0.142968	−	0.989727i	$-0.454335\pi$
$31$	1.59202	0.285936	0.142968	+	0.989727i	$0.454335\pi$
$32$	5.86203	1.03627
$33$	0	0
$34$	4.60884	0.790410
$35$	0	0
$36$	0	0
$37$	0.690638	0.113540	0.0567701	−	0.998387i	$-0.481920\pi$
$37$	0.690638	0.113540	0.0567701	+	0.998387i	$0.481920\pi$
$38$	−5.38542	−0.873631
$39$	0	0
$40$	0	0
$41$	9.55584	1.49237	0.746186	−	0.665738i	$-0.231883\pi$
$41$	9.55584	1.49237	0.746186	+	0.665738i	$0.231883\pi$
$42$	0	0
$43$	0.739212	0.112729	0.0563644	−	0.998410i	$-0.482049\pi$
$43$	0.739212	0.112729	0.0563644	+	0.998410i	$0.482049\pi$
$44$	−1.64528	−0.248035
$45$	0	0
$46$	−0.170102	−0.0250801
$47$	−3.52745	−0.514531	−0.257266	−	0.966341i	$-0.582822\pi$
$47$	−3.52745	−0.514531	−0.257266	+	0.966341i	$0.582822\pi$
$48$	0	0
$49$	3.80980	0.544256
$50$	0	0
$51$	0	0
$52$	−3.09270	−0.428880
$53$	0.756550	0.103920	0.0519601	−	0.998649i	$-0.483453\pi$
$53$	0.756550	0.103920	0.0519601	+	0.998649i	$0.483453\pi$
$54$	0	0
$55$	0	0
$56$	8.45827	1.13028
$57$	0	0
$58$	0.747618	0.0981671
$59$	7.99502	1.04086	0.520431	−	0.853904i	$-0.325771\pi$
$59$	7.99502	1.04086	0.520431	+	0.853904i	$0.325771\pi$
$60$	0	0
$61$	−0.836591	−0.107115	−0.0535573	−	0.998565i	$-0.517056\pi$
$61$	−0.836591	−0.107115	−0.0535573	+	0.998565i	$0.517056\pi$
$62$	1.19023	0.151159
$63$	0	0
$64$	2.46494	0.308118
$65$	0	0
$66$	0	0
$67$	−6.00144	−0.733193	−0.366597	−	0.930380i	$-0.619477\pi$
$67$	−6.00144	−0.733193	−0.366597	+	0.930380i	$0.619477\pi$
$68$	−8.88375	−1.07731
$69$	0	0
$70$	0	0
$71$	−10.8715	−1.29021	−0.645107	−	0.764093i	$-0.723187\pi$
$71$	−10.8715	−1.29021	−0.645107	+	0.764093i	$0.723187\pi$
$72$	0	0
$73$	−6.33908	−0.741933	−0.370967	−	0.928646i	$-0.620974\pi$
$73$	−6.33908	−0.741933	−0.370967	+	0.928646i	$0.620974\pi$
$74$	0.516334	0.0600226
$75$	0	0
$76$	10.3806	1.19074
$77$	−3.75374	−0.427778
$78$	0	0
$79$	−7.23916	−0.814469	−0.407235	−	0.913324i	$-0.633507\pi$
$79$	−7.23916	−0.814469	−0.407235	+	0.913324i	$0.633507\pi$
$80$	0	0
$81$	0	0
$82$	7.14412	0.788936
$83$	−7.46438	−0.819323	−0.409661	−	0.912238i	$-0.634353\pi$
$83$	−7.46438	−0.819323	−0.409661	+	0.912238i	$0.634353\pi$
$84$	0	0
$85$	0	0
$86$	0.552648	0.0595936
$87$	0	0
$88$	−2.93716	−0.313103
$89$	−0.620360	−0.0657580	−0.0328790	−	0.999459i	$-0.510468\pi$
$89$	−0.620360	−0.0657580	−0.0328790	+	0.999459i	$0.510468\pi$
$90$	0	0
$91$	−7.05606	−0.739676
$92$	0.327878	0.0341837
$93$	0	0
$94$	−2.63719	−0.272005
$95$	0	0
$96$	0	0
$97$	−4.33183	−0.439830	−0.219915	−	0.975519i	$-0.570578\pi$
$97$	−4.33183	−0.439830	−0.219915	+	0.975519i	$0.570578\pi$
$98$	2.84827	0.287719
$99$	0	0
$100$	0	0
$101$	12.4191	1.23575	0.617873	−	0.786278i	$-0.287994\pi$
$101$	12.4191	1.23575	0.617873	+	0.786278i	$0.287994\pi$
$102$	0	0
$103$	−2.24039	−0.220752	−0.110376	−	0.993890i	$-0.535205\pi$
$103$	−2.24039	−0.220752	−0.110376	+	0.993890i	$0.535205\pi$
$104$	−5.52111	−0.541389
$105$	0	0
$106$	0.565610	0.0549369
$107$	−13.7724	−1.33143	−0.665714	−	0.746207i	$-0.731873\pi$
$107$	−13.7724	−1.33143	−0.665714	+	0.746207i	$0.731873\pi$
$108$	0	0
$109$	8.13819	0.779497	0.389749	−	0.920921i	$-0.372562\pi$
$109$	8.13819	0.779497	0.389749	+	0.920921i	$0.372562\pi$
$110$	0	0
$111$	0	0
$112$	−3.15239	−0.297873
$113$	0.238009	0.0223900	0.0111950	−	0.999937i	$-0.496436\pi$
$113$	0.238009	0.0223900	0.0111950	+	0.999937i	$0.496436\pi$
$114$	0	0
$115$	0	0
$116$	−1.44107	−0.133800
$117$	0	0
$118$	5.97722	0.550248
$119$	−20.2685	−1.85801
$120$	0	0
$121$	−9.69650	−0.881500
$122$	−0.625451	−0.0566257
$123$	0	0
$124$	−2.29421	−0.206026
$125$	0	0
$126$	0	0
$127$	−0.851365	−0.0755464	−0.0377732	−	0.999286i	$-0.512026\pi$
$127$	−0.851365	−0.0755464	−0.0377732	+	0.999286i	$0.512026\pi$
$128$	−9.88123	−0.873385
$129$	0	0
$130$	0	0
$131$	18.3953	1.60720	0.803601	−	0.595168i	$-0.202914\pi$
$131$	18.3953	1.60720	0.803601	+	0.595168i	$0.202914\pi$
$132$	0	0
$133$	23.6837	2.05363
$134$	−4.48679	−0.387600
$135$	0	0
$136$	−15.8593	−1.35993
$137$	7.62484	0.651434	0.325717	−	0.945467i	$-0.394394\pi$
$137$	7.62484	0.651434	0.325717	+	0.945467i	$0.394394\pi$
$138$	0	0
$139$	−9.84900	−0.835382	−0.417691	−	0.908589i	$-0.637160\pi$
$139$	−9.84900	−0.835382	−0.417691	+	0.908589i	$0.637160\pi$
$140$	0	0
$141$	0	0
$142$	−8.12775	−0.682066
$143$	2.45024	0.204899
$144$	0	0
$145$	0	0
$146$	−4.73921	−0.392220
$147$	0	0
$148$	−0.995256	−0.0818095
$149$	−9.65138	−0.790672	−0.395336	−	0.918537i	$-0.629372\pi$
$149$	−9.65138	−0.790672	−0.395336	+	0.918537i	$0.629372\pi$
$150$	0	0
$151$	−2.60026	−0.211606	−0.105803	−	0.994387i	$-0.533741\pi$
$151$	−2.60026	−0.211606	−0.105803	+	0.994387i	$0.533741\pi$
$152$	18.5316	1.50311
$153$	0	0
$154$	−2.80636	−0.226143
$155$	0	0
$156$	0	0
$157$	5.86918	0.468411	0.234206	−	0.972187i	$-0.424751\pi$
$157$	5.86918	0.468411	0.234206	+	0.972187i	$0.424751\pi$
$158$	−5.41213	−0.430566
$159$	0	0
$160$	0	0
$161$	0.748061	0.0589555
$162$	0	0
$163$	−13.7249	−1.07501	−0.537507	−	0.843259i	$-0.680634\pi$
$163$	−13.7249	−1.07501	−0.537507	+	0.843259i	$0.680634\pi$
$164$	−13.7706	−1.07530
$165$	0	0
$166$	−5.58051	−0.433131
$167$	−23.6430	−1.82955	−0.914774	−	0.403967i	$-0.867631\pi$
$167$	−23.6430	−1.82955	−0.914774	+	0.403967i	$0.867631\pi$
$168$	0	0
$169$	−8.39418	−0.645706
$170$	0	0
$171$	0	0
$172$	−1.06525	−0.0812249
$173$	−6.38271	−0.485268	−0.242634	−	0.970118i	$-0.578011\pi$
$173$	−6.38271	−0.485268	−0.242634	+	0.970118i	$0.578011\pi$
$174$	0	0
$175$	0	0
$176$	1.09468	0.0825146
$177$	0	0
$178$	−0.463792	−0.0347627
$179$	−4.08939	−0.305656	−0.152828	−	0.988253i	$-0.548838\pi$
$179$	−4.08939	−0.305656	−0.152828	+	0.988253i	$0.548838\pi$
$180$	0	0
$181$	−2.39584	−0.178081	−0.0890407	−	0.996028i	$-0.528380\pi$
$181$	−2.39584	−0.178081	−0.0890407	+	0.996028i	$0.528380\pi$
$182$	−5.27524	−0.391026
$183$	0	0
$184$	0.585331	0.0431512
$185$	0	0
$186$	0	0
$187$	7.03829	0.514691
$188$	5.08329	0.370737
$189$	0	0
$190$	0	0
$191$	16.6078	1.20170	0.600851	−	0.799361i	$-0.294829\pi$
$191$	16.6078	1.20170	0.600851	+	0.799361i	$0.294829\pi$
$192$	0	0
$193$	−14.9574	−1.07666	−0.538328	−	0.842735i	$-0.680944\pi$
$193$	−14.9574	−1.07666	−0.538328	+	0.842735i	$0.680944\pi$
$194$	−3.23855	−0.232515
$195$	0	0
$196$	−5.49017	−0.392155
$197$	2.23776	0.159434	0.0797168	−	0.996818i	$-0.474598\pi$
$197$	2.23776	0.159434	0.0797168	+	0.996818i	$0.474598\pi$
$198$	0	0
$199$	−11.0259	−0.781608	−0.390804	−	0.920474i	$-0.627803\pi$
$199$	−11.0259	−0.781608	−0.390804	+	0.920474i	$0.627803\pi$
$200$	0	0
$201$	0	0
$202$	9.28474	0.653272
$203$	−3.28783	−0.230760
$204$	0	0
$205$	0	0
$206$	−1.67495	−0.116700
$207$	0	0
$208$	2.05771	0.142677
$209$	−8.22423	−0.568882
$210$	0	0
$211$	2.86625	0.197321	0.0986603	−	0.995121i	$-0.468544\pi$
$211$	2.86625	0.197321	0.0986603	+	0.995121i	$0.468544\pi$
$212$	−1.09024	−0.0748779
$213$	0	0
$214$	−10.2965	−0.703853
$215$	0	0
$216$	0	0
$217$	−5.23430	−0.355327
$218$	6.08426	0.412078
$219$	0	0
$220$	0	0
$221$	13.2302	0.889957
$222$	0	0
$223$	27.9990	1.87495	0.937475	−	0.348053i	$-0.113157\pi$
$223$	27.9990	1.87495	0.937475	+	0.348053i	$0.113157\pi$
$224$	−19.2733	−1.28775
$225$	0	0
$226$	0.177940	0.0118364
$227$	−5.03458	−0.334157	−0.167078	−	0.985944i	$-0.553433\pi$
$227$	−5.03458	−0.334157	−0.167078	+	0.985944i	$0.553433\pi$
$228$	0	0
$229$	−17.9628	−1.18702	−0.593508	−	0.804828i	$-0.702257\pi$
$229$	−17.9628	−1.18702	−0.593508	+	0.804828i	$0.702257\pi$
$230$	0	0
$231$	0	0
$232$	−2.57260	−0.168900
$233$	−20.1246	−1.31841	−0.659204	−	0.751964i	$-0.729107\pi$
$233$	−20.1246	−1.31841	−0.659204	+	0.751964i	$0.729107\pi$
$234$	0	0
$235$	0	0
$236$	−11.5214	−0.749976
$237$	0	0
$238$	−15.1531	−0.982227
$239$	−15.1528	−0.980151	−0.490076	−	0.871680i	$-0.663031\pi$
$239$	−15.1528	−0.980151	−0.490076	+	0.871680i	$0.663031\pi$
$240$	0	0
$241$	12.2051	0.786199	0.393100	−	0.919496i	$-0.371403\pi$
$241$	12.2051	0.786199	0.393100	+	0.919496i	$0.371403\pi$
$242$	−7.24928	−0.466001
$243$	0	0
$244$	1.20558	0.0771796
$245$	0	0
$246$	0	0
$247$	−15.4594	−0.983660
$248$	−4.09565	−0.260074
$249$	0	0
$250$	0	0
$251$	−19.9095	−1.25668	−0.628339	−	0.777940i	$-0.716265\pi$
$251$	−19.9095	−1.25668	−0.628339	+	0.777940i	$0.716265\pi$
$252$	0	0
$253$	−0.259767	−0.0163314
$254$	−0.636496	−0.0399373
$255$	0	0
$256$	−12.3173	−0.769829
$257$	−26.3424	−1.64320	−0.821598	−	0.570067i	$-0.806917\pi$
$257$	−26.3424	−1.64320	−0.821598	+	0.570067i	$0.806917\pi$
$258$	0	0
$259$	−2.27070	−0.141094
$260$	0	0
$261$	0	0
$262$	13.7526	0.849641
$263$	−12.0813	−0.744966	−0.372483	−	0.928039i	$-0.621493\pi$
$263$	−12.0813	−0.744966	−0.372483	+	0.928039i	$0.621493\pi$
$264$	0	0
$265$	0	0
$266$	17.7063	1.08565
$267$	0	0
$268$	8.64848	0.528290
$269$	−1.21176	−0.0738821	−0.0369411	−	0.999317i	$-0.511761\pi$
$269$	−1.21176	−0.0738821	−0.0369411	+	0.999317i	$0.511761\pi$
$270$	0	0
$271$	21.0190	1.27681	0.638407	−	0.769699i	$-0.279594\pi$
$271$	21.0190	1.27681	0.638407	+	0.769699i	$0.279594\pi$
$272$	5.91077	0.358393
$273$	0	0
$274$	5.70047	0.344378
$275$	0	0
$276$	0	0
$277$	5.12836	0.308133	0.154067	−	0.988060i	$-0.450763\pi$
$277$	5.12836	0.308133	0.154067	+	0.988060i	$0.450763\pi$
$278$	−7.36330	−0.441621
$279$	0	0
$280$	0	0
$281$	6.71721	0.400715	0.200357	−	0.979723i	$-0.435790\pi$
$281$	6.71721	0.400715	0.200357	+	0.979723i	$0.435790\pi$
$282$	0	0
$283$	30.4659	1.81101	0.905505	−	0.424336i	$-0.139492\pi$
$283$	30.4659	1.81101	0.905505	+	0.424336i	$0.139492\pi$
$284$	15.6666	0.929642
$285$	0	0
$286$	1.83184	0.108319
$287$	−31.4179	−1.85454
$288$	0	0
$289$	21.0035	1.23550
$290$	0	0
$291$	0	0
$292$	9.13504	0.534588
$293$	−29.3859	−1.71675	−0.858373	−	0.513027i	$-0.828524\pi$
$293$	−29.3859	−1.71675	−0.858373	+	0.513027i	$0.828524\pi$
$294$	0	0
$295$	0	0
$296$	−1.77674	−0.103271
$297$	0	0
$298$	−7.21554	−0.417985
$299$	−0.488294	−0.0282388
$300$	0	0
$301$	−2.43040	−0.140086
$302$	−1.94400	−0.111865
$303$	0	0
$304$	−6.90672	−0.396128
$305$	0	0
$306$	0	0
$307$	−17.1417	−0.978330	−0.489165	−	0.872191i	$-0.662698\pi$
$307$	−17.1417	−0.978330	−0.489165	+	0.872191i	$0.662698\pi$
$308$	5.40939	0.308229
$309$	0	0
$310$	0	0
$311$	−20.0480	−1.13682	−0.568410	−	0.822746i	$-0.692441\pi$
$311$	−20.0480	−1.13682	−0.568410	+	0.822746i	$0.692441\pi$
$312$	0	0
$313$	−14.3163	−0.809205	−0.404603	−	0.914493i	$-0.632590\pi$
$313$	−14.3163	−0.809205	−0.404603	+	0.914493i	$0.632590\pi$
$314$	4.38790	0.247624
$315$	0	0
$316$	10.4321	0.586852
$317$	−24.8223	−1.39416	−0.697080	−	0.716993i	$-0.745518\pi$
$317$	−24.8223	−1.39416	−0.697080	+	0.716993i	$0.745518\pi$
$318$	0	0
$319$	1.14171	0.0639234
$320$	0	0
$321$	0	0
$322$	0.559264	0.0311666
$323$	−44.4071	−2.47088
$324$	0	0
$325$	0	0
$326$	−10.2610	−0.568302
$327$	0	0
$328$	−24.5834	−1.35739
$329$	11.5976	0.639398
$330$	0	0
$331$	−34.8886	−1.91765	−0.958824	−	0.284001i	$-0.908338\pi$
$331$	−34.8886	−1.91765	−0.958824	+	0.284001i	$0.908338\pi$
$332$	10.7567	0.590349
$333$	0	0
$334$	−17.6759	−0.967183
$335$	0	0
$336$	0	0
$337$	−20.9359	−1.14045	−0.570227	−	0.821487i	$-0.693145\pi$
$337$	−20.9359	−1.14045	−0.570227	+	0.821487i	$0.693145\pi$
$338$	−6.27564	−0.341350
$339$	0	0
$340$	0	0
$341$	1.81763	0.0984300
$342$	0	0
$343$	10.4888	0.566344
$344$	−1.90170	−0.102533
$345$	0	0
$346$	−4.77183	−0.256535
$347$	1.68236	0.0903136	0.0451568	−	0.998980i	$-0.485621\pi$
$347$	1.68236	0.0903136	0.0451568	+	0.998980i	$0.485621\pi$
$348$	0	0
$349$	3.12517	0.167286	0.0836432	−	0.996496i	$-0.473344\pi$
$349$	3.12517	0.167286	0.0836432	+	0.996496i	$0.473344\pi$
$350$	0	0
$351$	0	0
$352$	6.69273	0.356724
$353$	23.4386	1.24751	0.623756	−	0.781619i	$-0.285606\pi$
$353$	23.4386	1.24751	0.623756	+	0.781619i	$0.285606\pi$
$354$	0	0
$355$	0	0
$356$	0.893980	0.0473809
$357$	0	0
$358$	−3.05730	−0.161584
$359$	1.45174	0.0766200	0.0383100	−	0.999266i	$-0.487803\pi$
$359$	1.45174	0.0766200	0.0383100	+	0.999266i	$0.487803\pi$
$360$	0	0
$361$	32.8896	1.73103
$362$	−1.79117	−0.0941420
$363$	0	0
$364$	10.1683	0.532961
$365$	0	0
$366$	0	0
$367$	−17.1372	−0.894552	−0.447276	−	0.894396i	$-0.647606\pi$
$367$	−17.1372	−0.894552	−0.447276	+	0.894396i	$0.647606\pi$
$368$	−0.218153	−0.0113720
$369$	0	0
$370$	0	0
$371$	−2.48740	−0.129140
$372$	0	0
$373$	−24.9017	−1.28936	−0.644681	−	0.764452i	$-0.723010\pi$
$373$	−24.9017	−1.28936	−0.644681	+	0.764452i	$0.723010\pi$
$374$	5.26196	0.272089
$375$	0	0
$376$	9.07473	0.467994
$377$	2.14612	0.110531
$378$	0	0
$379$	37.7156	1.93732	0.968659	−	0.248394i	$-0.0799028\pi$
$379$	37.7156	1.93732	0.968659	+	0.248394i	$0.0799028\pi$
$380$	0	0
$381$	0	0
$382$	12.4163	0.635274
$383$	−28.8331	−1.47330	−0.736650	−	0.676274i	$-0.763594\pi$
$383$	−28.8331	−1.47330	−0.736650	+	0.676274i	$0.763594\pi$
$384$	0	0
$385$	0	0
$386$	−11.1824	−0.569170
$387$	0	0
$388$	6.24245	0.316913
$389$	14.9618	0.758593	0.379296	−	0.925275i	$-0.376166\pi$
$389$	14.9618	0.758593	0.379296	+	0.925275i	$0.376166\pi$
$390$	0	0
$391$	−1.40262	−0.0709336
$392$	−9.80110	−0.495030
$393$	0	0
$394$	1.67299	0.0842839
$395$	0	0
$396$	0	0
$397$	26.4682	1.32840	0.664201	−	0.747554i	$-0.268772\pi$
$397$	26.4682	1.32840	0.664201	+	0.747554i	$0.268772\pi$
$398$	−8.24320	−0.413194
$399$	0	0
$400$	0	0
$401$	25.6298	1.27989	0.639947	−	0.768419i	$-0.278956\pi$
$401$	25.6298	1.27989	0.639947	+	0.768419i	$0.278956\pi$
$402$	0	0
$403$	3.41667	0.170196
$404$	−17.8968	−0.890397
$405$	0	0
$406$	−2.45804	−0.121990
$407$	0.788507	0.0390849
$408$	0	0
$409$	14.0081	0.692657	0.346328	−	0.938113i	$-0.387428\pi$
$409$	14.0081	0.692657	0.346328	+	0.938113i	$0.387428\pi$
$410$	0	0
$411$	0	0
$412$	3.22855	0.159059
$413$	−26.2862	−1.29346
$414$	0	0
$415$	0	0
$416$	12.5806	0.616815
$417$	0	0
$418$	−6.14859	−0.300737
$419$	−17.7803	−0.868625	−0.434313	−	0.900762i	$-0.643009\pi$
$419$	−17.7803	−0.868625	−0.434313	+	0.900762i	$0.643009\pi$
$420$	0	0
$421$	39.0762	1.90446	0.952229	−	0.305385i	$-0.0987853\pi$
$421$	39.0762	1.90446	0.952229	+	0.305385i	$0.0987853\pi$
$422$	2.14286	0.104313
$423$	0	0
$424$	−1.94630	−0.0945208
$425$	0	0
$426$	0	0
$427$	2.75057	0.133109
$428$	19.8469	0.959338
$429$	0	0
$430$	0	0
$431$	2.30354	0.110958	0.0554788	−	0.998460i	$-0.482331\pi$
$431$	2.30354	0.110958	0.0554788	+	0.998460i	$0.482331\pi$
$432$	0	0
$433$	11.7866	0.566426	0.283213	−	0.959057i	$-0.408600\pi$
$433$	11.7866	0.566426	0.283213	+	0.959057i	$0.408600\pi$
$434$	−3.91325	−0.187842
$435$	0	0
$436$	−11.7277	−0.561654
$437$	1.63896	0.0784021
$438$	0	0
$439$	20.2083	0.964492	0.482246	−	0.876036i	$-0.339821\pi$
$439$	20.2083	0.964492	0.482246	+	0.876036i	$0.339821\pi$
$440$	0	0
$441$	0	0
$442$	9.89111	0.470472
$443$	3.99092	0.189614	0.0948072	−	0.995496i	$-0.469777\pi$
$443$	3.99092	0.189614	0.0948072	+	0.995496i	$0.469777\pi$
$444$	0	0
$445$	0	0
$446$	20.9325	0.991184
$447$	0	0
$448$	−8.10431	−0.382892
$449$	35.6417	1.68204	0.841019	−	0.541006i	$-0.181956\pi$
$449$	35.6417	1.68204	0.841019	+	0.541006i	$0.181956\pi$
$450$	0	0
$451$	10.9100	0.513731
$452$	−0.342987	−0.0161328
$453$	0	0
$454$	−3.76394	−0.176650
$455$	0	0
$456$	0	0
$457$	−9.96090	−0.465951	−0.232976	−	0.972483i	$-0.574846\pi$
$457$	−9.96090	−0.465951	−0.232976	+	0.972483i	$0.574846\pi$
$458$	−13.4293	−0.627511
$459$	0	0
$460$	0	0
$461$	23.8451	1.11058	0.555288	−	0.831658i	$-0.312608\pi$
$461$	23.8451	1.11058	0.555288	+	0.831658i	$0.312608\pi$
$462$	0	0
$463$	24.0865	1.11939	0.559697	−	0.828697i	$-0.310917\pi$
$463$	24.0865	1.11939	0.559697	+	0.828697i	$0.310917\pi$
$464$	0.958808	0.0445116
$465$	0	0
$466$	−15.0455	−0.696971
$467$	−22.3707	−1.03519	−0.517596	−	0.855625i	$-0.673173\pi$
$467$	−22.3707	−1.03519	−0.517596	+	0.855625i	$0.673173\pi$
$468$	0	0
$469$	19.7317	0.911125
$470$	0	0
$471$	0	0
$472$	−20.5680	−0.946720
$473$	0.843965	0.0388055
$474$	0	0
$475$	0	0
$476$	29.2082	1.33876
$477$	0	0
$478$	−11.3285	−0.518153
$479$	18.3803	0.839816	0.419908	−	0.907567i	$-0.362062\pi$
$479$	18.3803	0.839816	0.419908	+	0.907567i	$0.362062\pi$
$480$	0	0
$481$	1.48219	0.0675820
$482$	9.12475	0.415621
$483$	0	0
$484$	13.9733	0.635150
$485$	0	0
$486$	0	0
$487$	7.49492	0.339627	0.169814	−	0.985476i	$-0.445683\pi$
$487$	7.49492	0.339627	0.169814	+	0.985476i	$0.445683\pi$
$488$	2.15222	0.0974264
$489$	0	0
$490$	0	0
$491$	2.85995	0.129068	0.0645338	−	0.997916i	$-0.479444\pi$
$491$	2.85995	0.129068	0.0645338	+	0.997916i	$0.479444\pi$
$492$	0	0
$493$	6.16470	0.277644
$494$	−11.5577	−0.520008
$495$	0	0
$496$	1.52645	0.0685394
$497$	35.7437	1.60332
$498$	0	0
$499$	−23.5907	−1.05606	−0.528032	−	0.849224i	$-0.677070\pi$
$499$	−23.5907	−1.05606	−0.528032	+	0.849224i	$0.677070\pi$
$500$	0	0
$501$	0	0
$502$	−14.8847	−0.664337
$503$	−18.3177	−0.816748	−0.408374	−	0.912815i	$-0.633904\pi$
$503$	−18.3177	−0.816748	−0.408374	+	0.912815i	$0.633904\pi$
$504$	0	0
$505$	0	0
$506$	−0.194206	−0.00863353
$507$	0	0
$508$	1.22687	0.0544337
$509$	22.5521	0.999605	0.499803	−	0.866139i	$-0.333406\pi$
$509$	22.5521	0.999605	0.499803	+	0.866139i	$0.333406\pi$
$510$	0	0
$511$	20.8418	0.921986
$512$	10.5538	0.466418
$513$	0	0
$514$	−19.6941	−0.868669
$515$	0	0
$516$	0	0
$517$	−4.02732	−0.177121
$518$	−1.69761	−0.0745889
$519$	0	0
$520$	0	0
$521$	−32.8127	−1.43755	−0.718776	−	0.695242i	$-0.755297\pi$
$521$	−32.8127	−1.43755	−0.718776	+	0.695242i	$0.755297\pi$
$522$	0	0
$523$	−23.7123	−1.03687	−0.518433	−	0.855118i	$-0.673484\pi$
$523$	−23.7123	−1.03687	−0.518433	+	0.855118i	$0.673484\pi$
$524$	−26.5088	−1.15804
$525$	0	0
$526$	−9.03221	−0.393823
$527$	9.81435	0.427520
$528$	0	0
$529$	−22.9482	−0.997749
$530$	0	0
$531$	0	0
$532$	−34.1297	−1.47971
$533$	20.5080	0.888298
$534$	0	0
$535$	0	0
$536$	15.4393	0.666878
$537$	0	0
$538$	−0.905932	−0.0390575
$539$	4.34968	0.187354
$540$	0	0
$541$	12.8337	0.551766	0.275883	−	0.961191i	$-0.411030\pi$
$541$	12.8337	0.551766	0.275883	+	0.961191i	$0.411030\pi$
$542$	15.7142	0.674982
$543$	0	0
$544$	36.1377	1.54939
$545$	0	0
$546$	0	0
$547$	−0.0964595	−0.00412431	−0.00206216	−	0.999998i	$-0.500656\pi$
$547$	−0.0964595	−0.00412431	−0.00206216	+	0.999998i	$0.500656\pi$
$548$	−10.9879	−0.469380
$549$	0	0
$550$	0	0
$551$	−7.20344	−0.306877
$552$	0	0
$553$	23.8011	1.01213
$554$	3.83405	0.162893
$555$	0	0
$556$	14.1931	0.601921
$557$	−13.1379	−0.556670	−0.278335	−	0.960484i	$-0.589783\pi$
$557$	−13.1379	−0.556670	−0.278335	+	0.960484i	$0.589783\pi$
$558$	0	0
$559$	1.58643	0.0670991
$560$	0	0
$561$	0	0
$562$	5.02191	0.211836
$563$	13.5992	0.573138	0.286569	−	0.958060i	$-0.407485\pi$
$563$	13.5992	0.573138	0.286569	+	0.958060i	$0.407485\pi$
$564$	0	0
$565$	0	0
$566$	22.7769	0.957383
$567$	0	0
$568$	27.9681	1.17352
$569$	−9.13357	−0.382899	−0.191450	−	0.981502i	$-0.561319\pi$
$569$	−9.13357	−0.382899	−0.191450	+	0.981502i	$0.561319\pi$
$570$	0	0
$571$	−31.1678	−1.30433	−0.652166	−	0.758076i	$-0.726140\pi$
$571$	−31.1678	−1.30433	−0.652166	+	0.758076i	$0.726140\pi$
$572$	−3.53096	−0.147637
$573$	0	0
$574$	−23.4886	−0.980396
$575$	0	0
$576$	0	0
$577$	11.3819	0.473833	0.236916	−	0.971530i	$-0.423863\pi$
$577$	11.3819	0.473833	0.236916	+	0.971530i	$0.423863\pi$
$578$	15.7026	0.653143
$579$	0	0
$580$	0	0
$581$	24.5416	1.01816
$582$	0	0
$583$	0.863760	0.0357733
$584$	16.3079	0.674828
$585$	0	0
$586$	−21.9695	−0.907550
$587$	−25.1969	−1.03999	−0.519994	−	0.854170i	$-0.674066\pi$
$587$	−25.1969	−1.03999	−0.519994	+	0.854170i	$0.674066\pi$
$588$	0	0
$589$	−11.4681	−0.472533
$590$	0	0
$591$	0	0
$592$	0.662190	0.0272158
$593$	−11.8015	−0.484628	−0.242314	−	0.970198i	$-0.577906\pi$
$593$	−11.8015	−0.484628	−0.242314	+	0.970198i	$0.577906\pi$
$594$	0	0
$595$	0	0
$596$	13.9083	0.569705
$597$	0	0
$598$	−0.365058	−0.0149283
$599$	−19.6438	−0.802626	−0.401313	−	0.915941i	$-0.631446\pi$
$599$	−19.6438	−0.802626	−0.401313	+	0.915941i	$0.631446\pi$
$600$	0	0
$601$	−37.2951	−1.52130	−0.760649	−	0.649163i	$-0.775119\pi$
$601$	−37.2951	−1.52130	−0.760649	+	0.649163i	$0.775119\pi$
$602$	−1.81701	−0.0740558
$603$	0	0
$604$	3.74715	0.152469
$605$	0	0
$606$	0	0
$607$	33.4688	1.35846	0.679228	−	0.733927i	$-0.262315\pi$
$607$	33.4688	1.35846	0.679228	+	0.733927i	$0.262315\pi$
$608$	−42.2268	−1.71252
$609$	0	0
$610$	0	0
$611$	−7.57032	−0.306262
$612$	0	0
$613$	9.99980	0.403888	0.201944	−	0.979397i	$-0.435274\pi$
$613$	9.99980	0.403888	0.201944	+	0.979397i	$0.435274\pi$
$614$	−12.8155	−0.517190
$615$	0	0
$616$	9.65689	0.389087
$617$	−5.23113	−0.210597	−0.105299	−	0.994441i	$-0.533580\pi$
$617$	−5.23113	−0.210597	−0.105299	+	0.994441i	$0.533580\pi$
$618$	0	0
$619$	4.44318	0.178586	0.0892932	−	0.996005i	$-0.471539\pi$
$619$	4.44318	0.178586	0.0892932	+	0.996005i	$0.471539\pi$
$620$	0	0
$621$	0	0
$622$	−14.9883	−0.600975
$623$	2.03964	0.0817163
$624$	0	0
$625$	0	0
$626$	−10.7031	−0.427783
$627$	0	0
$628$	−8.45788	−0.337506
$629$	4.25758	0.169761
$630$	0	0
$631$	28.0756	1.11767	0.558836	−	0.829278i	$-0.311248\pi$
$631$	28.0756	1.11767	0.558836	+	0.829278i	$0.311248\pi$
$632$	18.6235	0.740803
$633$	0	0
$634$	−18.5576	−0.737017
$635$	0	0
$636$	0	0
$637$	8.17626	0.323955
$638$	0.853562	0.0337929
$639$	0	0
$640$	0	0
$641$	−28.9980	−1.14535	−0.572676	−	0.819782i	$-0.694095\pi$
$641$	−28.9980	−1.14535	−0.572676	+	0.819782i	$0.694095\pi$
$642$	0	0
$643$	−8.78346	−0.346386	−0.173193	−	0.984888i	$-0.555408\pi$
$643$	−8.78346	−0.346386	−0.173193	+	0.984888i	$0.555408\pi$
$644$	−1.07801	−0.0424794
$645$	0	0
$646$	−33.1995	−1.30622
$647$	14.6873	0.577416	0.288708	−	0.957417i	$-0.406774\pi$
$647$	14.6873	0.577416	0.288708	+	0.957417i	$0.406774\pi$
$648$	0	0
$649$	9.12798	0.358305
$650$	0	0
$651$	0	0
$652$	19.7785	0.774584
$653$	−33.8232	−1.32360	−0.661802	−	0.749678i	$-0.730208\pi$
$653$	−33.8232	−1.32360	−0.661802	+	0.749678i	$0.730208\pi$
$654$	0	0
$655$	0	0
$656$	9.16222	0.357725
$657$	0	0
$658$	8.67060	0.338015
$659$	8.05417	0.313746	0.156873	−	0.987619i	$-0.449859\pi$
$659$	8.05417	0.313746	0.156873	+	0.987619i	$0.449859\pi$
$660$	0	0
$661$	−2.01657	−0.0784357	−0.0392178	−	0.999231i	$-0.512487\pi$
$661$	−2.01657	−0.0784357	−0.0392178	+	0.999231i	$0.512487\pi$
$662$	−26.0833	−1.01376
$663$	0	0
$664$	19.2029	0.745217
$665$	0	0
$666$	0	0
$667$	−0.227525	−0.00880979
$668$	34.0711	1.31825
$669$	0	0
$670$	0	0
$671$	−0.955144	−0.0368729
$672$	0	0
$673$	−35.0573	−1.35136	−0.675679	−	0.737196i	$-0.736149\pi$
$673$	−35.0573	−1.35136	−0.675679	+	0.737196i	$0.736149\pi$
$674$	−15.6521	−0.602896
$675$	0	0
$676$	12.0966	0.465253
$677$	43.5224	1.67270	0.836351	−	0.548194i	$-0.184684\pi$
$677$	43.5224	1.67270	0.836351	+	0.548194i	$0.184684\pi$
$678$	0	0
$679$	14.2423	0.546569
$680$	0	0
$681$	0	0
$682$	1.35889	0.0520346
$683$	−33.6868	−1.28899	−0.644494	−	0.764609i	$-0.722932\pi$
$683$	−33.6868	−1.28899	−0.644494	+	0.764609i	$0.722932\pi$
$684$	0	0
$685$	0	0
$686$	7.84164	0.299395
$687$	0	0
$688$	0.708763	0.0270213
$689$	1.62364	0.0618559
$690$	0	0
$691$	−22.7655	−0.866039	−0.433020	−	0.901384i	$-0.642552\pi$
$691$	−22.7655	−0.866039	−0.433020	+	0.901384i	$0.642552\pi$
$692$	9.19791	0.349652
$693$	0	0
$694$	1.25776	0.0477439
$695$	0	0
$696$	0	0
$697$	58.9089	2.23133
$698$	2.33643	0.0884352
$699$	0	0
$700$	0	0
$701$	−39.1196	−1.47753	−0.738763	−	0.673966i	$-0.764589\pi$
$701$	−39.1196	−1.47753	−0.738763	+	0.673966i	$0.764589\pi$
$702$	0	0
$703$	−4.97497	−0.187635
$704$	2.81425	0.106066
$705$	0	0
$706$	17.5232	0.659493
$707$	−40.8318	−1.53564
$708$	0	0
$709$	−8.56919	−0.321823	−0.160911	−	0.986969i	$-0.551443\pi$
$709$	−8.56919	−0.321823	−0.160911	+	0.986969i	$0.551443\pi$
$710$	0	0
$711$	0	0
$712$	1.59594	0.0598104
$713$	−0.362225	−0.0135654
$714$	0	0
$715$	0	0
$716$	5.89309	0.220235
$717$	0	0
$718$	1.08535	0.0405049
$719$	9.04221	0.337218	0.168609	−	0.985683i	$-0.446073\pi$
$719$	9.04221	0.337218	0.168609	+	0.985683i	$0.446073\pi$
$720$	0	0
$721$	7.36600	0.274324
$722$	24.5889	0.915102
$723$	0	0
$724$	3.45257	0.128314
$725$	0	0
$726$	0	0
$727$	25.6760	0.952270	0.476135	−	0.879372i	$-0.342037\pi$
$727$	25.6760	0.952270	0.476135	+	0.879372i	$0.342037\pi$
$728$	18.1524	0.672774
$729$	0	0
$730$	0	0
$731$	4.55702	0.168547
$732$	0	0
$733$	1.30522	0.0482093	0.0241047	−	0.999709i	$-0.492327\pi$
$733$	1.30522	0.0482093	0.0241047	+	0.999709i	$0.492327\pi$
$734$	−12.8121	−0.472901
$735$	0	0
$736$	−1.33376	−0.0491629
$737$	−6.85190	−0.252393
$738$	0	0
$739$	5.31974	0.195690	0.0978449	−	0.995202i	$-0.468805\pi$
$739$	5.31974	0.195690	0.0978449	+	0.995202i	$0.468805\pi$
$740$	0	0
$741$	0	0
$742$	−1.85963	−0.0682691
$743$	38.7894	1.42304	0.711522	−	0.702664i	$-0.248006\pi$
$743$	38.7894	1.42304	0.711522	+	0.702664i	$0.248006\pi$
$744$	0	0
$745$	0	0
$746$	−18.6170	−0.681616
$747$	0	0
$748$	−10.1427	−0.370852
$749$	45.2812	1.65454
$750$	0	0
$751$	41.9496	1.53076	0.765382	−	0.643576i	$-0.222550\pi$
$751$	41.9496	1.53076	0.765382	+	0.643576i	$0.222550\pi$
$752$	−3.38215	−0.123334
$753$	0	0
$754$	1.60448	0.0584316
$755$	0	0
$756$	0	0
$757$	−44.0799	−1.60211	−0.801057	−	0.598589i	$-0.795728\pi$
$757$	−44.0799	−1.60211	−0.801057	+	0.598589i	$0.795728\pi$
$758$	28.1968	1.02416
$759$	0	0
$760$	0	0
$761$	−48.8024	−1.76909	−0.884543	−	0.466459i	$-0.845529\pi$
$761$	−48.8024	−1.76909	−0.884543	+	0.466459i	$0.845529\pi$
$762$	0	0
$763$	−26.7569	−0.968666
$764$	−23.9330	−0.865866
$765$	0	0
$766$	−21.5561	−0.778854
$767$	17.1582	0.619548
$768$	0	0
$769$	10.5748	0.381336	0.190668	−	0.981655i	$-0.438935\pi$
$769$	10.5748	0.381336	0.190668	+	0.981655i	$0.438935\pi$
$770$	0	0
$771$	0	0
$772$	21.5546	0.775767
$773$	1.99543	0.0717706	0.0358853	−	0.999356i	$-0.488575\pi$
$773$	1.99543	0.0717706	0.0358853	+	0.999356i	$0.488575\pi$
$774$	0	0
$775$	0	0
$776$	11.1441	0.400049
$777$	0	0
$778$	11.1857	0.401027
$779$	−68.8350	−2.46627
$780$	0	0
$781$	−12.4121	−0.444141
$782$	−1.04862	−0.0374987
$783$	0	0
$784$	3.65286	0.130459
$785$	0	0
$786$	0	0
$787$	6.26378	0.223280	0.111640	−	0.993749i	$-0.464390\pi$
$787$	6.26378	0.223280	0.111640	+	0.993749i	$0.464390\pi$
$788$	−3.22476	−0.114877
$789$	0	0
$790$	0	0
$791$	−0.782533	−0.0278237
$792$	0	0
$793$	−1.79542	−0.0637573
$794$	19.7881	0.702254
$795$	0	0
$796$	15.8891	0.563175
$797$	−23.6779	−0.838715	−0.419358	−	0.907821i	$-0.637745\pi$
$797$	−23.6779	−0.838715	−0.419358	+	0.907821i	$0.637745\pi$
$798$	0	0
$799$	−21.7457	−0.769306
$800$	0	0
$801$	0	0
$802$	19.1613	0.676610
$803$	−7.23738	−0.255402
$804$	0	0
$805$	0	0
$806$	2.55436	0.0899736
$807$	0	0
$808$	−31.9494	−1.12398
$809$	3.13385	0.110180	0.0550901	−	0.998481i	$-0.482455\pi$
$809$	3.13385	0.110180	0.0550901	+	0.998481i	$0.482455\pi$
$810$	0	0
$811$	22.4677	0.788949	0.394474	−	0.918907i	$-0.370927\pi$
$811$	22.4677	0.788949	0.394474	+	0.918907i	$0.370927\pi$
$812$	4.73798	0.166270
$813$	0	0
$814$	0.589503	0.0206621
$815$	0	0
$816$	0	0
$817$	−5.32487	−0.186294
$818$	10.4727	0.366170
$819$	0	0
$820$	0	0
$821$	−34.1236	−1.19092	−0.595462	−	0.803384i	$-0.703031\pi$
$821$	−34.1236	−1.19092	−0.595462	+	0.803384i	$0.703031\pi$
$822$	0	0
$823$	−49.5543	−1.72735	−0.863677	−	0.504046i	$-0.831844\pi$
$823$	−49.5543	−1.72735	−0.863677	+	0.504046i	$0.831844\pi$
$824$	5.76363	0.200786
$825$	0	0
$826$	−19.6521	−0.683782
$827$	−10.1137	−0.351687	−0.175843	−	0.984418i	$-0.556265\pi$
$827$	−10.1137	−0.351687	−0.175843	+	0.984418i	$0.556265\pi$
$828$	0	0
$829$	40.3973	1.40306	0.701528	−	0.712642i	$-0.252502\pi$
$829$	40.3973	1.40306	0.701528	+	0.712642i	$0.252502\pi$
$830$	0	0
$831$	0	0
$832$	5.29006	0.183400
$833$	23.4862	0.813750
$834$	0	0
$835$	0	0
$836$	11.8517	0.409899
$837$	0	0
$838$	−13.2929	−0.459195
$839$	−42.2304	−1.45795	−0.728977	−	0.684538i	$-0.760004\pi$
$839$	−42.2304	−1.45795	−0.728977	+	0.684538i	$0.760004\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	29.2141	1.00678
$843$	0	0
$844$	−4.13046	−0.142176
$845$	0	0
$846$	0	0
$847$	31.8804	1.09542
$848$	0.725386	0.0249099
$849$	0	0
$850$	0	0
$851$	−0.157137	−0.00538659
$852$	0	0
$853$	−53.7233	−1.83945	−0.919726	−	0.392562i	$-0.871589\pi$
$853$	−53.7233	−1.83945	−0.919726	+	0.392562i	$0.871589\pi$
$854$	2.05637	0.0703676
$855$	0	0
$856$	35.4309	1.21100
$857$	37.9764	1.29725	0.648625	−	0.761109i	$-0.275345\pi$
$857$	37.9764	1.29725	0.648625	+	0.761109i	$0.275345\pi$
$858$	0	0
$859$	52.2536	1.78287	0.891435	−	0.453148i	$-0.149699\pi$
$859$	52.2536	1.78287	0.891435	+	0.453148i	$0.149699\pi$
$860$	0	0
$861$	0	0
$862$	1.72217	0.0586573
$863$	46.9411	1.59789	0.798946	−	0.601402i	$-0.205391\pi$
$863$	46.9411	1.59789	0.798946	+	0.601402i	$0.205391\pi$
$864$	0	0
$865$	0	0
$866$	8.81186	0.299439
$867$	0	0
$868$	7.54297	0.256025
$869$	−8.26501	−0.280371
$870$	0	0
$871$	−12.8798	−0.436415
$872$	−20.9363	−0.708994
$873$	0	0
$874$	1.22532	0.0414469
$875$	0	0
$876$	0	0
$877$	−37.5859	−1.26919	−0.634593	−	0.772846i	$-0.718832\pi$
$877$	−37.5859	−1.26919	−0.634593	+	0.772846i	$0.718832\pi$
$878$	15.1081	0.509875
$879$	0	0
$880$	0	0
$881$	−34.1047	−1.14902	−0.574509	−	0.818499i	$-0.694807\pi$
$881$	−34.1047	−1.14902	−0.574509	+	0.818499i	$0.694807\pi$
$882$	0	0
$883$	−5.27332	−0.177461	−0.0887307	−	0.996056i	$-0.528281\pi$
$883$	−5.27332	−0.177461	−0.0887307	+	0.996056i	$0.528281\pi$
$884$	−19.0656	−0.641244
$885$	0	0
$886$	2.98368	0.100239
$887$	−15.9851	−0.536728	−0.268364	−	0.963318i	$-0.586483\pi$
$887$	−15.9851	−0.536728	−0.268364	+	0.963318i	$0.586483\pi$
$888$	0	0
$889$	2.79914	0.0938801
$890$	0	0
$891$	0	0
$892$	−40.3484	−1.35096
$893$	25.4098	0.850306
$894$	0	0
$895$	0	0
$896$	32.4877	1.08534
$897$	0	0
$898$	26.6464	0.889202
$899$	1.59202	0.0530970
$900$	0	0
$901$	4.66390	0.155377
$902$	8.15651	0.271582
$903$	0	0
$904$	−0.612303	−0.0203649
$905$	0	0
$906$	0	0
$907$	33.0649	1.09790	0.548950	−	0.835855i	$-0.315028\pi$
$907$	33.0649	1.09790	0.548950	+	0.835855i	$0.315028\pi$
$908$	7.25516	0.240771
$909$	0	0
$910$	0	0
$911$	14.5224	0.481148	0.240574	−	0.970631i	$-0.422664\pi$
$911$	14.5224	0.481148	0.240574	+	0.970631i	$0.422664\pi$
$912$	0	0
$913$	−8.52215	−0.282042
$914$	−7.44695	−0.246323
$915$	0	0
$916$	25.8856	0.855284
$917$	−60.4805	−1.99724
$918$	0	0
$919$	−1.83140	−0.0604122	−0.0302061	−	0.999544i	$-0.509616\pi$
$919$	−1.83140	−0.0604122	−0.0302061	+	0.999544i	$0.509616\pi$
$920$	0	0
$921$	0	0
$922$	17.8270	0.587101
$923$	−23.3316	−0.767968
$924$	0	0
$925$	0	0
$926$	18.0075	0.591763
$927$	0	0
$928$	5.86203	0.192431
$929$	17.5449	0.575631	0.287815	−	0.957686i	$-0.407071\pi$
$929$	17.5449	0.575631	0.287815	+	0.957686i	$0.407071\pi$
$930$	0	0
$931$	−27.4436	−0.899429
$932$	29.0010	0.949958
$933$	0	0
$934$	−16.7247	−0.547250
$935$	0	0
$936$	0	0
$937$	−32.4416	−1.05982	−0.529911	−	0.848053i	$-0.677775\pi$
$937$	−32.4416	−1.05982	−0.529911	+	0.848053i	$0.677775\pi$
$938$	14.7518	0.481663
$939$	0	0
$940$	0	0
$941$	17.5838	0.573216	0.286608	−	0.958048i	$-0.407472\pi$
$941$	17.5838	0.573216	0.286608	+	0.958048i	$0.407472\pi$
$942$	0	0
$943$	−2.17419	−0.0708013
$944$	7.66569	0.249497
$945$	0	0
$946$	0.630963	0.0205144
$947$	27.3823	0.889805	0.444902	−	0.895579i	$-0.353238\pi$
$947$	27.3823	0.889805	0.444902	+	0.895579i	$0.353238\pi$
$948$	0	0
$949$	−13.6044	−0.441618
$950$	0	0
$951$	0	0
$952$	52.1427	1.68996
$953$	41.5899	1.34723	0.673614	−	0.739083i	$-0.264741\pi$
$953$	41.5899	1.34723	0.673614	+	0.739083i	$0.264741\pi$
$954$	0	0
$955$	0	0
$956$	21.8362	0.706232
$957$	0	0
$958$	13.7414	0.443965
$959$	−25.0691	−0.809525
$960$	0	0
$961$	−28.4655	−0.918241
$962$	1.10811	0.0357270
$963$	0	0
$964$	−17.5884	−0.566483
$965$	0	0
$966$	0	0
$967$	−24.3025	−0.781517	−0.390759	−	0.920493i	$-0.627787\pi$
$967$	−24.3025	−0.781517	−0.390759	+	0.920493i	$0.627787\pi$
$968$	24.9453	0.801771
$969$	0	0
$970$	0	0
$971$	−56.8121	−1.82319	−0.911593	−	0.411093i	$-0.865147\pi$
$971$	−56.8121	−1.82319	−0.911593	+	0.411093i	$0.865147\pi$
$972$	0	0
$973$	32.3818	1.03811
$974$	5.60334	0.179543
$975$	0	0
$976$	−0.802131	−0.0256756
$977$	25.2789	0.808743	0.404371	−	0.914595i	$-0.367490\pi$
$977$	25.2789	0.808743	0.404371	+	0.914595i	$0.367490\pi$
$978$	0	0
$979$	−0.708271	−0.0226364
$980$	0	0
$981$	0	0
$982$	2.13815	0.0682311
$983$	16.8975	0.538945	0.269473	−	0.963008i	$-0.413151\pi$
$983$	16.8975	0.538945	0.269473	+	0.963008i	$0.413151\pi$
$984$	0	0
$985$	0	0
$986$	4.60884	0.146775
$987$	0	0
$988$	22.2781	0.708760
$989$	−0.168189	−0.00534810
$990$	0	0
$991$	−41.6666	−1.32358	−0.661792	−	0.749688i	$-0.730204\pi$
$991$	−41.6666	−1.32358	−0.661792	+	0.749688i	$0.730204\pi$
$992$	9.33249	0.296307
$993$	0	0
$994$	26.7226	0.847590
$995$	0	0
$996$	0	0
$997$	15.6779	0.496523	0.248262	−	0.968693i	$-0.420141\pi$
$997$	15.6779	0.496523	0.248262	+	0.968693i	$0.420141\pi$
$998$	−17.6368	−0.558284
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6525.2.a.ca.1.6	✓	9
3.2	odd	2		6525.2.a.cc.1.4	yes	9
5.4	even	2		6525.2.a.cd.1.4	yes	9
15.14	odd	2		6525.2.a.cb.1.6	yes	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6525.2.a.ca.1.6	✓	9	1.1	even	1	trivial
6525.2.a.cb.1.6	yes	9	15.14	odd	2
6525.2.a.cc.1.4	yes	9	3.2	odd	2
6525.2.a.cd.1.4	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)

\(f(q)\)	\(=\)	\(q+0.747618 q^{2} -1.44107 q^{4} -3.28783 q^{7} -2.57260 q^{8} +O(q^{10})\)	\(q+0.747618 q^{2} -1.44107 q^{4} -3.28783 q^{7} -2.57260 q^{8} +1.14171 q^{11} +2.14612 q^{13} -2.45804 q^{14} +0.958808 q^{16} +6.16470 q^{17} -7.20344 q^{19} +0.853562 q^{22} -0.227525 q^{23} +1.60448 q^{26} +4.73798 q^{28} +1.00000 q^{29} +1.59202 q^{31} +5.86203 q^{32} +4.60884 q^{34} +0.690638 q^{37} -5.38542 q^{38} +9.55584 q^{41} +0.739212 q^{43} -1.64528 q^{44} -0.170102 q^{46} -3.52745 q^{47} +3.80980 q^{49} -3.09270 q^{52} +0.756550 q^{53} +8.45827 q^{56} +0.747618 q^{58} +7.99502 q^{59} -0.836591 q^{61} +1.19023 q^{62} +2.46494 q^{64} -6.00144 q^{67} -8.88375 q^{68} -10.8715 q^{71} -6.33908 q^{73} +0.516334 q^{74} +10.3806 q^{76} -3.75374 q^{77} -7.23916 q^{79} +7.14412 q^{82} -7.46438 q^{83} +0.552648 q^{86} -2.93716 q^{88} -0.620360 q^{89} -7.05606 q^{91} +0.327878 q^{92} -2.63719 q^{94} -4.33183 q^{97} +2.84827 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 2 q^{2} + 10 q^{4} - q^{7} - 9 q^{8}+O(q^{10}) \) 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

Modular forms

Varieties

Fields

Representations

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Database

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