LMFDB - Embedded newform 5625.2.a.w.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5625,2,Mod(1,5625)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5625, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("5625.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5625 = 3^{2} \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5625.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:	$44.9158511370$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	8.8.33620000000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 10x^{6} + 30x^{4} - 25x^{2} + 5 $ x^8 - 10x^6 + 30x^4 - 25*x^2 + 5
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 225)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$0.936839$ of defining polynomial
Character	$\chi$	$=$	5625.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.936839 q^{2} -1.12233 q^{4} +2.74037 q^{7} -2.92512 q^{8} +O(q^{10})$	$q+0.936839 q^{2} -1.12233 q^{4} +2.74037 q^{7} -2.92512 q^{8} +0.291988 q^{11} -3.55634 q^{13} +2.56728 q^{14} -0.495700 q^{16} +4.37511 q^{17} +2.55634 q^{19} +0.273546 q^{22} -5.13149 q^{23} -3.33172 q^{26} -3.07561 q^{28} -9.40004 q^{29} -8.94362 q^{31} +5.38585 q^{32} +4.09877 q^{34} -0.735053 q^{37} +2.39488 q^{38} +11.2939 q^{41} +0.249982 q^{43} -0.327708 q^{44} -4.80737 q^{46} -2.88442 q^{47} +0.509614 q^{49} +3.99140 q^{52} +4.87769 q^{53} -8.01591 q^{56} -8.80632 q^{58} +5.10135 q^{59} -1.99140 q^{61} -8.37873 q^{62} +6.03708 q^{64} -13.7135 q^{67} -4.91033 q^{68} -12.4081 q^{71} +4.55038 q^{73} -0.688626 q^{74} -2.86907 q^{76} +0.800156 q^{77} +7.54669 q^{79} +10.5806 q^{82} -11.1622 q^{83} +0.234192 q^{86} -0.854102 q^{88} +7.99383 q^{89} -9.74568 q^{91} +5.75924 q^{92} -2.70224 q^{94} -16.5086 q^{97} +0.477426 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{4}+O(q^{10}) $ 8 * q + 4 * q^4	$ 8 q + 4 q^{4} + 10 q^{13} - 8 q^{16} - 18 q^{19} - 30 q^{28} - 26 q^{31} - 20 q^{34} - 40 q^{43} - 30 q^{46} - 16 q^{49} + 40 q^{52} + 10 q^{58} - 24 q^{61} - 34 q^{64} - 40 q^{67} + 40 q^{73} - 44 q^{76} - 42 q^{79} + 60 q^{82} + 20 q^{88} - 40 q^{91} - 10 q^{94} - 40 q^{97}+O(q^{100}) $ 8 * q + 4 * q^4 + 10 * q^13 - 8 * q^16 - 18 * q^19 - 30 * q^28 - 26 * q^31 - 20 * q^34 - 40 * q^43 - 30 * q^46 - 16 * q^49 + 40 * q^52 + 10 * q^58 - 24 * q^61 - 34 * q^64 - 40 * q^67 + 40 * q^73 - 44 * q^76 - 42 * q^79 + 60 * q^82 + 20 * q^88 - 40 * q^91 - 10 * q^94 - 40 * q^97

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 $

$2$	0.936839	0.662445	0.331222	−	0.943553i	$-0.392539\pi$
$2$	0.936839	0.662445	0.331222	+	0.943553i	$0.392539\pi$
$3$	0	0
$3$	0	0
$4$	−1.12233	−0.561167
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.74037	1.03576	0.517881	−	0.855453i	$-0.326721\pi$
$7$	2.74037	1.03576	0.517881	+	0.855453i	$0.326721\pi$
$8$	−2.92512	−1.03419
$9$	0	0
$10$	0	0
$11$	0.291988	0.0880378	0.0440189	−	0.999031i	$-0.485984\pi$
$11$	0.291988	0.0880378	0.0440189	+	0.999031i	$0.485984\pi$
$12$	0	0
$13$	−3.55634	−0.986352	−0.493176	−	0.869930i	$-0.664164\pi$
$13$	−3.55634	−0.986352	−0.493176	+	0.869930i	$0.664164\pi$
$14$	2.56728	0.686135
$15$	0	0
$16$	−0.495700	−0.123925
$17$	4.37511	1.06112	0.530560	−	0.847648i	$-0.321982\pi$
$17$	4.37511	1.06112	0.530560	+	0.847648i	$0.321982\pi$
$18$	0	0
$19$	2.55634	0.586465	0.293232	−	0.956041i	$-0.405269\pi$
$19$	2.55634	0.586465	0.293232	+	0.956041i	$0.405269\pi$
$20$	0	0
$21$	0	0
$22$	0.273546	0.0583202
$23$	−5.13149	−1.06999	−0.534994	−	0.844856i	$-0.679686\pi$
$23$	−5.13149	−1.06999	−0.534994	+	0.844856i	$0.679686\pi$
$24$	0	0
$25$	0	0
$26$	−3.33172	−0.653404
$27$	0	0
$28$	−3.07561	−0.581235
$29$	−9.40004	−1.74554	−0.872772	−	0.488128i	$-0.837680\pi$
$29$	−9.40004	−1.74554	−0.872772	+	0.488128i	$0.837680\pi$
$30$	0	0
$31$	−8.94362	−1.60632	−0.803161	−	0.595762i	$-0.796850\pi$
$31$	−8.94362	−1.60632	−0.803161	+	0.595762i	$0.796850\pi$
$32$	5.38585	0.952093
$33$	0	0
$34$	4.09877	0.702933
$35$	0	0
$36$	0	0
$37$	−0.735053	−0.120842	−0.0604210	−	0.998173i	$-0.519244\pi$
$37$	−0.735053	−0.120842	−0.0604210	+	0.998173i	$0.519244\pi$
$38$	2.39488	0.388501
$39$	0	0
$40$	0	0
$41$	11.2939	1.76381	0.881905	−	0.471427i	$-0.156261\pi$
$41$	11.2939	1.76381	0.881905	+	0.471427i	$0.156261\pi$
$42$	0	0
$43$	0.249982	0.0381218	0.0190609	−	0.999818i	$-0.493932\pi$
$43$	0.249982	0.0381218	0.0190609	+	0.999818i	$0.493932\pi$
$44$	−0.327708	−0.0494039
$45$	0	0
$46$	−4.80737	−0.708809
$47$	−2.88442	−0.420736	−0.210368	−	0.977622i	$-0.567466\pi$
$47$	−2.88442	−0.420736	−0.210368	+	0.977622i	$0.567466\pi$
$48$	0	0
$49$	0.509614	0.0728020
$50$	0	0
$51$	0	0
$52$	3.99140	0.553508
$53$	4.87769	0.670002	0.335001	−	0.942218i	$-0.391263\pi$
$53$	4.87769	0.670002	0.335001	+	0.942218i	$0.391263\pi$
$54$	0	0
$55$	0	0
$56$	−8.01591	−1.07117
$57$	0	0
$58$	−8.80632	−1.15633
$59$	5.10135	0.664140	0.332070	−	0.943255i	$-0.392253\pi$
$59$	5.10135	0.664140	0.332070	+	0.943255i	$0.392253\pi$
$60$	0	0
$61$	−1.99140	−0.254973	−0.127486	−	0.991840i	$-0.540691\pi$
$61$	−1.99140	−0.254973	−0.127486	+	0.991840i	$0.540691\pi$
$62$	−8.37873	−1.06410
$63$	0	0
$64$	6.03708	0.754635
$65$	0	0
$66$	0	0
$67$	−13.7135	−1.67537	−0.837686	−	0.546151i	$-0.816092\pi$
$67$	−13.7135	−1.67537	−0.837686	+	0.546151i	$0.816092\pi$
$68$	−4.91033	−0.595465
$69$	0	0
$70$	0	0
$71$	−12.4081	−1.47257	−0.736286	−	0.676670i	$-0.763422\pi$
$71$	−12.4081	−1.47257	−0.736286	+	0.676670i	$0.763422\pi$
$72$	0	0
$73$	4.55038	0.532581	0.266291	−	0.963893i	$-0.414202\pi$
$73$	4.55038	0.532581	0.266291	+	0.963893i	$0.414202\pi$
$74$	−0.688626	−0.0800511
$75$	0	0
$76$	−2.86907	−0.329105
$77$	0.800156	0.0911862
$78$	0	0
$79$	7.54669	0.849069	0.424534	−	0.905412i	$-0.360438\pi$
$79$	7.54669	0.849069	0.424534	+	0.905412i	$0.360438\pi$
$80$	0	0
$81$	0	0
$82$	10.5806	1.16843
$83$	−11.1622	−1.22521	−0.612605	−	0.790389i	$-0.709878\pi$
$83$	−11.1622	−1.22521	−0.612605	+	0.790389i	$0.709878\pi$
$84$	0	0
$85$	0	0
$86$	0.234192	0.0252536
$87$	0	0
$88$	−0.854102	−0.0910476
$89$	7.99383	0.847345	0.423672	−	0.905816i	$-0.360741\pi$
$89$	7.99383	0.847345	0.423672	+	0.905816i	$0.360741\pi$
$90$	0	0
$91$	−9.74568	−1.02163
$92$	5.75924	0.600442
$93$	0	0
$94$	−2.70224	−0.278715
$95$	0	0
$96$	0	0
$97$	−16.5086	−1.67619	−0.838095	−	0.545524i	$-0.816331\pi$
$97$	−16.5086	−1.67619	−0.838095	+	0.545524i	$0.816331\pi$
$98$	0.477426	0.0482273
$99$	0	0
$100$	0	0
$101$	9.81876	0.977003	0.488501	−	0.872563i	$-0.337544\pi$
$101$	9.81876	0.977003	0.488501	+	0.872563i	$0.337544\pi$
$102$	0	0
$103$	4.92642	0.485415	0.242707	−	0.970100i	$-0.421964\pi$
$103$	4.92642	0.485415	0.242707	+	0.970100i	$0.421964\pi$
$104$	10.4027	1.02007
$105$	0	0
$106$	4.56960	0.443839
$107$	−4.22786	−0.408722	−0.204361	−	0.978896i	$-0.565512\pi$
$107$	−4.22786	−0.408722	−0.204361	+	0.978896i	$0.565512\pi$
$108$	0	0
$109$	−10.5275	−1.00835	−0.504174	−	0.863602i	$-0.668203\pi$
$109$	−10.5275	−1.00835	−0.504174	+	0.863602i	$0.668203\pi$
$110$	0	0
$111$	0	0
$112$	−1.35840	−0.128357
$113$	−11.6955	−1.10022	−0.550111	−	0.835092i	$-0.685414\pi$
$113$	−11.6955	−1.10022	−0.550111	+	0.835092i	$0.685414\pi$
$114$	0	0
$115$	0	0
$116$	10.5500	0.979541
$117$	0	0
$118$	4.77915	0.439956
$119$	11.9894	1.09907
$120$	0	0
$121$	−10.9147	−0.992249
$122$	−1.86562	−0.168905
$123$	0	0
$124$	10.0377	0.901414
$125$	0	0
$126$	0	0
$127$	15.7072	1.39378	0.696892	−	0.717176i	$-0.254566\pi$
$127$	15.7072	1.39378	0.696892	+	0.717176i	$0.254566\pi$
$128$	−5.11594	−0.452190
$129$	0	0
$130$	0	0
$131$	8.84928	0.773165	0.386582	−	0.922255i	$-0.373655\pi$
$131$	8.84928	0.773165	0.386582	+	0.922255i	$0.373655\pi$
$132$	0	0
$133$	7.00531	0.607438
$134$	−12.8474	−1.10984
$135$	0	0
$136$	−12.7977	−1.09740
$137$	8.63561	0.737790	0.368895	−	0.929471i	$-0.379736\pi$
$137$	8.63561	0.737790	0.368895	+	0.929471i	$0.379736\pi$
$138$	0	0
$139$	−8.01720	−0.680010	−0.340005	−	0.940424i	$-0.610429\pi$
$139$	−8.01720	−0.680010	−0.340005	+	0.940424i	$0.610429\pi$
$140$	0	0
$141$	0	0
$142$	−11.6244	−0.975498
$143$	−1.03841	−0.0868363
$144$	0	0
$145$	0	0
$146$	4.26297	0.352806
$147$	0	0
$148$	0.824975	0.0678125
$149$	−12.8790	−1.05509	−0.527545	−	0.849527i	$-0.676887\pi$
$149$	−12.8790	−1.05509	−0.527545	+	0.849527i	$0.676887\pi$
$150$	0	0
$151$	−23.3541	−1.90053	−0.950263	−	0.311447i	$-0.899186\pi$
$151$	−23.3541	−1.90053	−0.950263	+	0.311447i	$0.899186\pi$
$152$	−7.47761	−0.606514
$153$	0	0
$154$	0.749617	0.0604058
$155$	0	0
$156$	0	0
$157$	9.02953	0.720635	0.360317	−	0.932830i	$-0.382668\pi$
$157$	9.02953	0.720635	0.360317	+	0.932830i	$0.382668\pi$
$158$	7.07003	0.562461
$159$	0	0
$160$	0	0
$161$	−14.0622	−1.10825
$162$	0	0
$163$	9.22970	0.722926	0.361463	−	0.932386i	$-0.382277\pi$
$163$	9.22970	0.722926	0.361463	+	0.932386i	$0.382277\pi$
$164$	−12.6755	−0.989792
$165$	0	0
$166$	−10.4572	−0.811634
$167$	−20.3209	−1.57248	−0.786239	−	0.617922i	$-0.787975\pi$
$167$	−20.3209	−1.57248	−0.786239	+	0.617922i	$0.787975\pi$
$168$	0	0
$169$	−0.352437	−0.0271105
$170$	0	0
$171$	0	0
$172$	−0.280563	−0.0213927
$173$	−15.9203	−1.21040	−0.605199	−	0.796074i	$-0.706906\pi$
$173$	−15.9203	−1.21040	−0.605199	+	0.796074i	$0.706906\pi$
$174$	0	0
$175$	0	0
$176$	−0.144739	−0.0109101
$177$	0	0
$178$	7.48893	0.561319
$179$	5.02991	0.375953	0.187977	−	0.982173i	$-0.439807\pi$
$179$	5.02991	0.375953	0.187977	+	0.982173i	$0.439807\pi$
$180$	0	0
$181$	−2.83954	−0.211061	−0.105531	−	0.994416i	$-0.533654\pi$
$181$	−2.83954	−0.211061	−0.105531	+	0.994416i	$0.533654\pi$
$182$	−9.13013	−0.676770
$183$	0	0
$184$	15.0102	1.10657
$185$	0	0
$186$	0	0
$187$	1.27748	0.0934186
$188$	3.23729	0.236103
$189$	0	0
$190$	0	0
$191$	−20.7809	−1.50365	−0.751826	−	0.659362i	$-0.770827\pi$
$191$	−20.7809	−1.50365	−0.751826	+	0.659362i	$0.770827\pi$
$192$	0	0
$193$	−8.46086	−0.609026	−0.304513	−	0.952508i	$-0.598494\pi$
$193$	−8.46086	−0.609026	−0.304513	+	0.952508i	$0.598494\pi$
$194$	−15.4659	−1.11038
$195$	0	0
$196$	−0.571957	−0.0408541
$197$	6.30468	0.449190	0.224595	−	0.974452i	$-0.427894\pi$
$197$	6.30468	0.449190	0.224595	+	0.974452i	$0.427894\pi$
$198$	0	0
$199$	−19.4190	−1.37658	−0.688290	−	0.725436i	$-0.741638\pi$
$199$	−19.4190	−1.37658	−0.688290	+	0.725436i	$0.741638\pi$
$200$	0	0
$201$	0	0
$202$	9.19859	0.647211
$203$	−25.7596	−1.80797
$204$	0	0
$205$	0	0
$206$	4.61526	0.321561
$207$	0	0
$208$	1.76288	0.122234
$209$	0.746422	0.0516311
$210$	0	0
$211$	−17.2678	−1.18877	−0.594383	−	0.804182i	$-0.702604\pi$
$211$	−17.2678	−1.18877	−0.594383	+	0.804182i	$0.702604\pi$
$212$	−5.47439	−0.375983
$213$	0	0
$214$	−3.96082	−0.270756
$215$	0	0
$216$	0	0
$217$	−24.5088	−1.66377
$218$	−9.86253	−0.667975
$219$	0	0
$220$	0	0
$221$	−15.5594	−1.04664
$222$	0	0
$223$	1.42443	0.0953869	0.0476935	−	0.998862i	$-0.484813\pi$
$223$	1.42443	0.0953869	0.0476935	+	0.998862i	$0.484813\pi$
$224$	14.7592	0.986142
$225$	0	0
$226$	−10.9568	−0.728836
$227$	9.26834	0.615161	0.307581	−	0.951522i	$-0.400481\pi$
$227$	9.26834	0.615161	0.307581	+	0.951522i	$0.400481\pi$
$228$	0	0
$229$	−16.8235	−1.11173	−0.555865	−	0.831273i	$-0.687613\pi$
$229$	−16.8235	−1.11173	−0.555865	+	0.831273i	$0.687613\pi$
$230$	0	0
$231$	0	0
$232$	27.4963	1.80522
$233$	9.87404	0.646870	0.323435	−	0.946250i	$-0.395162\pi$
$233$	9.87404	0.646870	0.323435	+	0.946250i	$0.395162\pi$
$234$	0	0
$235$	0	0
$236$	−5.72542	−0.372693
$237$	0	0
$238$	11.2321	0.728071
$239$	−20.9884	−1.35763	−0.678815	−	0.734310i	$-0.737506\pi$
$239$	−20.9884	−1.35763	−0.678815	+	0.734310i	$0.737506\pi$
$240$	0	0
$241$	0.599134	0.0385936	0.0192968	−	0.999814i	$-0.493857\pi$
$241$	0.599134	0.0385936	0.0192968	+	0.999814i	$0.493857\pi$
$242$	−10.2254	−0.657311
$243$	0	0
$244$	2.23502	0.143082
$245$	0	0
$246$	0	0
$247$	−9.09122	−0.578461
$248$	26.1612	1.66124
$249$	0	0
$250$	0	0
$251$	22.8068	1.43955	0.719776	−	0.694207i	$-0.244245\pi$
$251$	22.8068	1.43955	0.719776	+	0.694207i	$0.244245\pi$
$252$	0	0
$253$	−1.49833	−0.0941995
$254$	14.7151	0.923305
$255$	0	0
$256$	−16.8670	−1.05419
$257$	−4.10576	−0.256110	−0.128055	−	0.991767i	$-0.540873\pi$
$257$	−4.10576	−0.256110	−0.128055	+	0.991767i	$0.540873\pi$
$258$	0	0
$259$	−2.01432	−0.125163
$260$	0	0
$261$	0	0
$262$	8.29034	0.512179
$263$	−12.8759	−0.793964	−0.396982	−	0.917826i	$-0.629942\pi$
$263$	−12.8759	−0.793964	−0.396982	+	0.917826i	$0.629942\pi$
$264$	0	0
$265$	0	0
$266$	6.56285	0.402394
$267$	0	0
$268$	15.3911	0.940164
$269$	−7.36300	−0.448930	−0.224465	−	0.974482i	$-0.572063\pi$
$269$	−7.36300	−0.448930	−0.224465	+	0.974482i	$0.572063\pi$
$270$	0	0
$271$	−1.31070	−0.0796191	−0.0398096	−	0.999207i	$-0.512675\pi$
$271$	−1.31070	−0.0796191	−0.0398096	+	0.999207i	$0.512675\pi$
$272$	−2.16874	−0.131499
$273$	0	0
$274$	8.09017	0.488745
$275$	0	0
$276$	0	0
$277$	11.4437	0.687583	0.343791	−	0.939046i	$-0.388289\pi$
$277$	11.4437	0.687583	0.343791	+	0.939046i	$0.388289\pi$
$278$	−7.51082	−0.450469
$279$	0	0
$280$	0	0
$281$	3.14013	0.187324	0.0936621	−	0.995604i	$-0.470143\pi$
$281$	3.14013	0.187324	0.0936621	+	0.995604i	$0.470143\pi$
$282$	0	0
$283$	18.2903	1.08725	0.543624	−	0.839329i	$-0.317052\pi$
$283$	18.2903	1.08725	0.543624	+	0.839329i	$0.317052\pi$
$284$	13.9260	0.826359
$285$	0	0
$286$	−0.972823	−0.0575242
$287$	30.9494	1.82689
$288$	0	0
$289$	2.14156	0.125974
$290$	0	0
$291$	0	0
$292$	−5.10704	−0.298867
$293$	12.2342	0.714727	0.357364	−	0.933965i	$-0.383676\pi$
$293$	12.2342	0.714727	0.357364	+	0.933965i	$0.383676\pi$
$294$	0	0
$295$	0	0
$296$	2.15012	0.124973
$297$	0	0
$298$	−12.0656	−0.698939
$299$	18.2493	1.05539
$300$	0	0
$301$	0.685041	0.0394851
$302$	−21.8790	−1.25899
$303$	0	0
$304$	−1.26718	−0.0726777
$305$	0	0
$306$	0	0
$307$	34.8868	1.99110	0.995549	−	0.0942495i	$-0.0300451\pi$
$307$	34.8868	1.99110	0.995549	+	0.0942495i	$0.0300451\pi$
$308$	−0.898042	−0.0511707
$309$	0	0
$310$	0	0
$311$	9.56937	0.542629	0.271315	−	0.962491i	$-0.412542\pi$
$311$	9.56937	0.542629	0.271315	+	0.962491i	$0.412542\pi$
$312$	0	0
$313$	−28.6193	−1.61766	−0.808828	−	0.588045i	$-0.799898\pi$
$313$	−28.6193	−1.61766	−0.808828	+	0.588045i	$0.799898\pi$
$314$	8.45921	0.477381
$315$	0	0
$316$	−8.46990	−0.476469
$317$	12.9343	0.726462	0.363231	−	0.931699i	$-0.381674\pi$
$317$	12.9343	0.726462	0.363231	+	0.931699i	$0.381674\pi$
$318$	0	0
$319$	−2.74470	−0.153674
$320$	0	0
$321$	0	0
$322$	−13.1740	−0.734157
$323$	11.1843	0.622309
$324$	0	0
$325$	0	0
$326$	8.64674	0.478899
$327$	0	0
$328$	−33.0360	−1.82411
$329$	−7.90438	−0.435783
$330$	0	0
$331$	−14.1230	−0.776269	−0.388135	−	0.921603i	$-0.626880\pi$
$331$	−14.1230	−0.776269	−0.388135	+	0.921603i	$0.626880\pi$
$332$	12.5277	0.687547
$333$	0	0
$334$	−19.0374	−1.04168
$335$	0	0
$336$	0	0
$337$	−26.0852	−1.42095	−0.710475	−	0.703722i	$-0.751520\pi$
$337$	−26.0852	−1.42095	−0.710475	+	0.703722i	$0.751520\pi$
$338$	−0.330176	−0.0179592
$339$	0	0
$340$	0	0
$341$	−2.61143	−0.141417
$342$	0	0
$343$	−17.7860	−0.960356
$344$	−0.731227	−0.0394251
$345$	0	0
$346$	−14.9147	−0.801822
$347$	33.3393	1.78975	0.894874	−	0.446318i	$-0.147265\pi$
$347$	33.3393	1.78975	0.894874	+	0.446318i	$0.147265\pi$
$348$	0	0
$349$	−2.42476	−0.129794	−0.0648972	−	0.997892i	$-0.520672\pi$
$349$	−2.42476	−0.129794	−0.0648972	+	0.997892i	$0.520672\pi$
$350$	0	0
$351$	0	0
$352$	1.57261	0.0838202
$353$	25.6561	1.36554	0.682768	−	0.730635i	$-0.260776\pi$
$353$	25.6561	1.36554	0.682768	+	0.730635i	$0.260776\pi$
$354$	0	0
$355$	0	0
$356$	−8.97175	−0.475502
$357$	0	0
$358$	4.71222	0.249048
$359$	0.933761	0.0492820	0.0246410	−	0.999696i	$-0.492156\pi$
$359$	0.933761	0.0492820	0.0246410	+	0.999696i	$0.492156\pi$
$360$	0	0
$361$	−12.4651	−0.656059
$362$	−2.66019	−0.139816
$363$	0	0
$364$	10.9379	0.573302
$365$	0	0
$366$	0	0
$367$	−18.1358	−0.946683	−0.473342	−	0.880879i	$-0.656952\pi$
$367$	−18.1358	−0.946683	−0.473342	+	0.880879i	$0.656952\pi$
$368$	2.54368	0.132599
$369$	0	0
$370$	0	0
$371$	13.3667	0.693962
$372$	0	0
$373$	−23.8975	−1.23736	−0.618682	−	0.785641i	$-0.712333\pi$
$373$	−23.8975	−1.23736	−0.618682	+	0.785641i	$0.712333\pi$
$374$	1.19679	0.0618847
$375$	0	0
$376$	8.43729	0.435120
$377$	33.4298	1.72172
$378$	0	0
$379$	0.812689	0.0417450	0.0208725	−	0.999782i	$-0.493356\pi$
$379$	0.812689	0.0417450	0.0208725	+	0.999782i	$0.493356\pi$
$380$	0	0
$381$	0	0
$382$	−19.4683	−0.996086
$383$	23.3827	1.19480	0.597400	−	0.801943i	$-0.296200\pi$
$383$	23.3827	1.19480	0.597400	+	0.801943i	$0.296200\pi$
$384$	0	0
$385$	0	0
$386$	−7.92646	−0.403446
$387$	0	0
$388$	18.5281	0.940622
$389$	11.5129	0.583727	0.291863	−	0.956460i	$-0.405725\pi$
$389$	11.5129	0.583727	0.291863	+	0.956460i	$0.405725\pi$
$390$	0	0
$391$	−22.4508	−1.13539
$392$	−1.49068	−0.0752909
$393$	0	0
$394$	5.90647	0.297563
$395$	0	0
$396$	0	0
$397$	4.78381	0.240093	0.120046	−	0.992768i	$-0.461696\pi$
$397$	4.78381	0.240093	0.120046	+	0.992768i	$0.461696\pi$
$398$	−18.1925	−0.911908
$399$	0	0
$400$	0	0
$401$	−29.7663	−1.48646	−0.743228	−	0.669038i	$-0.766706\pi$
$401$	−29.7663	−1.48646	−0.743228	+	0.669038i	$0.766706\pi$
$402$	0	0
$403$	31.8066	1.58440
$404$	−11.0199	−0.548261
$405$	0	0
$406$	−24.1326	−1.19768
$407$	−0.214627	−0.0106387
$408$	0	0
$409$	−5.98832	−0.296103	−0.148052	−	0.988980i	$-0.547300\pi$
$409$	−5.98832	−0.296103	−0.148052	+	0.988980i	$0.547300\pi$
$410$	0	0
$411$	0	0
$412$	−5.52909	−0.272399
$413$	13.9796	0.687890
$414$	0	0
$415$	0	0
$416$	−19.1539	−0.939099
$417$	0	0
$418$	0.699277	0.0342028
$419$	4.25686	0.207961	0.103981	−	0.994579i	$-0.466842\pi$
$419$	4.25686	0.207961	0.103981	+	0.994579i	$0.466842\pi$
$420$	0	0
$421$	−23.2688	−1.13405	−0.567026	−	0.823700i	$-0.691906\pi$
$421$	−23.2688	−1.13405	−0.567026	+	0.823700i	$0.691906\pi$
$422$	−16.1772	−0.787492
$423$	0	0
$424$	−14.2678	−0.692907
$425$	0	0
$426$	0	0
$427$	−5.45717	−0.264091
$428$	4.74507	0.229361
$429$	0	0
$430$	0	0
$431$	12.5171	0.602929	0.301465	−	0.953477i	$-0.402525\pi$
$431$	12.5171	0.602929	0.301465	+	0.953477i	$0.402525\pi$
$432$	0	0
$433$	−18.9565	−0.910990	−0.455495	−	0.890238i	$-0.650538\pi$
$433$	−18.9565	−0.910990	−0.455495	+	0.890238i	$0.650538\pi$
$434$	−22.9608	−1.10215
$435$	0	0
$436$	11.8153	0.565851
$437$	−13.1178	−0.627511
$438$	0	0
$439$	−7.29914	−0.348369	−0.174184	−	0.984713i	$-0.555729\pi$
$439$	−7.29914	−0.348369	−0.174184	+	0.984713i	$0.555729\pi$
$440$	0	0
$441$	0	0
$442$	−14.5766	−0.693339
$443$	−5.75210	−0.273291	−0.136645	−	0.990620i	$-0.543632\pi$
$443$	−5.75210	−0.273291	−0.136645	+	0.990620i	$0.543632\pi$
$444$	0	0
$445$	0	0
$446$	1.33446	0.0631886
$447$	0	0
$448$	16.5438	0.781621
$449$	−4.75061	−0.224195	−0.112098	−	0.993697i	$-0.535757\pi$
$449$	−4.75061	−0.224195	−0.112098	+	0.993697i	$0.535757\pi$
$450$	0	0
$451$	3.29769	0.155282
$452$	13.1263	0.617408
$453$	0	0
$454$	8.68294	0.407510
$455$	0	0
$456$	0	0
$457$	4.72678	0.221110	0.110555	−	0.993870i	$-0.464737\pi$
$457$	4.72678	0.221110	0.110555	+	0.993870i	$0.464737\pi$
$458$	−15.7609	−0.736460
$459$	0	0
$460$	0	0
$461$	7.71028	0.359103	0.179552	−	0.983749i	$-0.442535\pi$
$461$	7.71028	0.359103	0.179552	+	0.983749i	$0.442535\pi$
$462$	0	0
$463$	−3.82956	−0.177975	−0.0889874	−	0.996033i	$-0.528363\pi$
$463$	−3.82956	−0.177975	−0.0889874	+	0.996033i	$0.528363\pi$
$464$	4.65961	0.216317
$465$	0	0
$466$	9.25038	0.428516
$467$	19.8882	0.920318	0.460159	−	0.887837i	$-0.347792\pi$
$467$	19.8882	0.920318	0.460159	+	0.887837i	$0.347792\pi$
$468$	0	0
$469$	−37.5801	−1.73529
$470$	0	0
$471$	0	0
$472$	−14.9221	−0.686845
$473$	0.0729917	0.00335616
$474$	0	0
$475$	0	0
$476$	−13.4561	−0.616760
$477$	0	0
$478$	−19.6628	−0.899354
$479$	−27.8212	−1.27118	−0.635591	−	0.772026i	$-0.719244\pi$
$479$	−27.8212	−1.27118	−0.635591	+	0.772026i	$0.719244\pi$
$480$	0	0
$481$	2.61410	0.119193
$482$	0.561292	0.0255661
$483$	0	0
$484$	12.2500	0.556817
$485$	0	0
$486$	0	0
$487$	31.7141	1.43710	0.718551	−	0.695474i	$-0.244805\pi$
$487$	31.7141	1.43710	0.718551	+	0.695474i	$0.244805\pi$
$488$	5.82509	0.263690
$489$	0	0
$490$	0	0
$491$	−4.90133	−0.221194	−0.110597	−	0.993865i	$-0.535276\pi$
$491$	−4.90133	−0.221194	−0.110597	+	0.993865i	$0.535276\pi$
$492$	0	0
$493$	−41.1262	−1.85223
$494$	−8.51701	−0.383198
$495$	0	0
$496$	4.43336	0.199064
$497$	−34.0028	−1.52523
$498$	0	0
$499$	3.13604	0.140389	0.0701943	−	0.997533i	$-0.477638\pi$
$499$	3.13604	0.140389	0.0701943	+	0.997533i	$0.477638\pi$
$500$	0	0
$501$	0	0
$502$	21.3663	0.953624
$503$	19.2987	0.860485	0.430243	−	0.902713i	$-0.358428\pi$
$503$	19.2987	0.860485	0.430243	+	0.902713i	$0.358428\pi$
$504$	0	0
$505$	0	0
$506$	−1.40370	−0.0624020
$507$	0	0
$508$	−17.6287	−0.782145
$509$	41.7597	1.85097	0.925484	−	0.378787i	$-0.123659\pi$
$509$	41.7597	1.85097	0.925484	+	0.378787i	$0.123659\pi$
$510$	0	0
$511$	12.4697	0.551627
$512$	−5.56974	−0.246150
$513$	0	0
$514$	−3.84644	−0.169659
$515$	0	0
$516$	0	0
$517$	−0.842218	−0.0370407
$518$	−1.88709	−0.0829139
$519$	0	0
$520$	0	0
$521$	34.0368	1.49118	0.745589	−	0.666406i	$-0.232168\pi$
$521$	34.0368	1.49118	0.745589	+	0.666406i	$0.232168\pi$
$522$	0	0
$523$	−8.00170	−0.349890	−0.174945	−	0.984578i	$-0.555975\pi$
$523$	−8.00170	−0.349890	−0.174945	+	0.984578i	$0.555975\pi$
$524$	−9.93184	−0.433874
$525$	0	0
$526$	−12.0627	−0.525957
$527$	−39.1293	−1.70450
$528$	0	0
$529$	3.33216	0.144876
$530$	0	0
$531$	0	0
$532$	−7.86230	−0.340874
$533$	−40.1649	−1.73974
$534$	0	0
$535$	0	0
$536$	40.1137	1.73265
$537$	0	0
$538$	−6.89795	−0.297392
$539$	0.148801	0.00640933
$540$	0	0
$541$	9.69559	0.416846	0.208423	−	0.978039i	$-0.433167\pi$
$541$	9.69559	0.416846	0.208423	+	0.978039i	$0.433167\pi$
$542$	−1.22791	−0.0527433
$543$	0	0
$544$	23.5637	1.01028
$545$	0	0
$546$	0	0
$547$	−14.5504	−0.622129	−0.311065	−	0.950389i	$-0.600686\pi$
$547$	−14.5504	−0.622129	−0.311065	+	0.950389i	$0.600686\pi$
$548$	−9.69203	−0.414023
$549$	0	0
$550$	0	0
$551$	−24.0297	−1.02370
$552$	0	0
$553$	20.6807	0.879433
$554$	10.7209	0.455486
$555$	0	0
$556$	8.99797	0.381599
$557$	17.8610	0.756795	0.378397	−	0.925643i	$-0.376475\pi$
$557$	17.8610	0.756795	0.378397	+	0.925643i	$0.376475\pi$
$558$	0	0
$559$	−0.889020	−0.0376015
$560$	0	0
$561$	0	0
$562$	2.94179	0.124092
$563$	21.6519	0.912519	0.456260	−	0.889847i	$-0.349189\pi$
$563$	21.6519	0.912519	0.456260	+	0.889847i	$0.349189\pi$
$564$	0	0
$565$	0	0
$566$	17.1351	0.720242
$567$	0	0
$568$	36.2953	1.52292
$569$	32.7634	1.37351	0.686757	−	0.726887i	$-0.259034\pi$
$569$	32.7634	1.37351	0.686757	+	0.726887i	$0.259034\pi$
$570$	0	0
$571$	−3.39726	−0.142171	−0.0710854	−	0.997470i	$-0.522646\pi$
$571$	−3.39726	−0.142171	−0.0710854	+	0.997470i	$0.522646\pi$
$572$	1.16544	0.0487296
$573$	0	0
$574$	28.9946	1.21021
$575$	0	0
$576$	0	0
$577$	12.5670	0.523170	0.261585	−	0.965180i	$-0.415755\pi$
$577$	12.5670	0.523170	0.261585	+	0.965180i	$0.415755\pi$
$578$	2.00630	0.0834510
$579$	0	0
$580$	0	0
$581$	−30.5885	−1.26902
$582$	0	0
$583$	1.42423	0.0589855
$584$	−13.3104	−0.550789
$585$	0	0
$586$	11.4614	0.473467
$587$	−36.3462	−1.50017	−0.750084	−	0.661343i	$-0.769987\pi$
$587$	−36.3462	−1.50017	−0.750084	+	0.661343i	$0.769987\pi$
$588$	0	0
$589$	−22.8629	−0.942051
$590$	0	0
$591$	0	0
$592$	0.364366	0.0149754
$593$	31.7555	1.30404	0.652022	−	0.758200i	$-0.273921\pi$
$593$	31.7555	1.30404	0.652022	+	0.758200i	$0.273921\pi$
$594$	0	0
$595$	0	0
$596$	14.4545	0.592081
$597$	0	0
$598$	17.0967	0.699135
$599$	−23.6306	−0.965519	−0.482759	−	0.875753i	$-0.660365\pi$
$599$	−23.6306	−0.965519	−0.482759	+	0.875753i	$0.660365\pi$
$600$	0	0
$601$	12.9840	0.529628	0.264814	−	0.964300i	$-0.414689\pi$
$601$	12.9840	0.529628	0.264814	+	0.964300i	$0.414689\pi$
$602$	0.641773	0.0261567
$603$	0	0
$604$	26.2110	1.06651
$605$	0	0
$606$	0	0
$607$	9.23404	0.374798	0.187399	−	0.982284i	$-0.439994\pi$
$607$	9.23404	0.374798	0.187399	+	0.982284i	$0.439994\pi$
$608$	13.7681	0.558369
$609$	0	0
$610$	0	0
$611$	10.2580	0.414994
$612$	0	0
$613$	16.4276	0.663506	0.331753	−	0.943366i	$-0.392360\pi$
$613$	16.4276	0.663506	0.331753	+	0.943366i	$0.392360\pi$
$614$	32.6833	1.31899
$615$	0	0
$616$	−2.34055	−0.0943036
$617$	7.92430	0.319020	0.159510	−	0.987196i	$-0.449009\pi$
$617$	7.92430	0.319020	0.159510	+	0.987196i	$0.449009\pi$
$618$	0	0
$619$	7.20155	0.289455	0.144727	−	0.989472i	$-0.453769\pi$
$619$	7.20155	0.289455	0.144727	+	0.989472i	$0.453769\pi$
$620$	0	0
$621$	0	0
$622$	8.96495	0.359462
$623$	21.9060	0.877647
$624$	0	0
$625$	0	0
$626$	−26.8116	−1.07161
$627$	0	0
$628$	−10.1341	−0.404396
$629$	−3.21594	−0.128228
$630$	0	0
$631$	24.2420	0.965057	0.482529	−	0.875880i	$-0.339718\pi$
$631$	24.2420	0.965057	0.482529	+	0.875880i	$0.339718\pi$
$632$	−22.0750	−0.878096
$633$	0	0
$634$	12.1173	0.481241
$635$	0	0
$636$	0	0
$637$	−1.81236	−0.0718084
$638$	−2.57134	−0.101801
$639$	0	0
$640$	0	0
$641$	12.6808	0.500863	0.250432	−	0.968134i	$-0.419427\pi$
$641$	12.6808	0.500863	0.250432	+	0.968134i	$0.419427\pi$
$642$	0	0
$643$	25.2886	0.997287	0.498643	−	0.866807i	$-0.333832\pi$
$643$	25.2886	0.997287	0.498643	+	0.866807i	$0.333832\pi$
$644$	15.7824	0.621915
$645$	0	0
$646$	10.4779	0.412246
$647$	9.57188	0.376309	0.188155	−	0.982139i	$-0.439749\pi$
$647$	9.57188	0.376309	0.188155	+	0.982139i	$0.439749\pi$
$648$	0	0
$649$	1.48954	0.0584694
$650$	0	0
$651$	0	0
$652$	−10.3588	−0.405682
$653$	12.8318	0.502147	0.251074	−	0.967968i	$-0.419216\pi$
$653$	12.8318	0.502147	0.251074	+	0.967968i	$0.419216\pi$
$654$	0	0
$655$	0	0
$656$	−5.59839	−0.218580
$657$	0	0
$658$	−7.40513	−0.288682
$659$	−29.7085	−1.15728	−0.578639	−	0.815584i	$-0.696416\pi$
$659$	−29.7085	−1.15728	−0.578639	+	0.815584i	$0.696416\pi$
$660$	0	0
$661$	−22.9708	−0.893461	−0.446730	−	0.894669i	$-0.647412\pi$
$661$	−22.9708	−0.893461	−0.446730	+	0.894669i	$0.647412\pi$
$662$	−13.2310	−0.514236
$663$	0	0
$664$	32.6508	1.26710
$665$	0	0
$666$	0	0
$667$	48.2362	1.86771
$668$	22.8068	0.882423
$669$	0	0
$670$	0	0
$671$	−0.581466	−0.0224472
$672$	0	0
$673$	15.6190	0.602069	0.301034	−	0.953613i	$-0.402668\pi$
$673$	15.6190	0.602069	0.301034	+	0.953613i	$0.402668\pi$
$674$	−24.4376	−0.941301
$675$	0	0
$676$	0.395552	0.0152135
$677$	0.209675	0.00805845	0.00402923	−	0.999992i	$-0.498717\pi$
$677$	0.209675	0.00805845	0.00402923	+	0.999992i	$0.498717\pi$
$678$	0	0
$679$	−45.2395	−1.73613
$680$	0	0
$681$	0	0
$682$	−2.44649	−0.0936810
$683$	25.7192	0.984119	0.492059	−	0.870562i	$-0.336244\pi$
$683$	25.7192	0.984119	0.492059	+	0.870562i	$0.336244\pi$
$684$	0	0
$685$	0	0
$686$	−16.6627	−0.636183
$687$	0	0
$688$	−0.123916	−0.00472425
$689$	−17.3467	−0.660857
$690$	0	0
$691$	−26.7676	−1.01829	−0.509144	−	0.860682i	$-0.670038\pi$
$691$	−26.7676	−1.01829	−0.509144	+	0.860682i	$0.670038\pi$
$692$	17.8679	0.679235
$693$	0	0
$694$	31.2336	1.18561
$695$	0	0
$696$	0	0
$697$	49.4120	1.87161
$698$	−2.27161	−0.0859816
$699$	0	0
$700$	0	0
$701$	−19.6071	−0.740551	−0.370275	−	0.928922i	$-0.620737\pi$
$701$	−19.6071	−0.740551	−0.370275	+	0.928922i	$0.620737\pi$
$702$	0	0
$703$	−1.87905	−0.0708696
$704$	1.76276	0.0664364
$705$	0	0
$706$	24.0356	0.904593
$707$	26.9070	1.01194
$708$	0	0
$709$	23.2844	0.874463	0.437232	−	0.899349i	$-0.355959\pi$
$709$	23.2844	0.874463	0.437232	+	0.899349i	$0.355959\pi$
$710$	0	0
$711$	0	0
$712$	−23.3829	−0.876313
$713$	45.8941	1.71875
$714$	0	0
$715$	0	0
$716$	−5.64524	−0.210973
$717$	0	0
$718$	0.874784	0.0326466
$719$	−16.2906	−0.607537	−0.303768	−	0.952746i	$-0.598245\pi$
$719$	−16.2906	−0.607537	−0.303768	+	0.952746i	$0.598245\pi$
$720$	0	0
$721$	13.5002	0.502774
$722$	−11.6778	−0.434603
$723$	0	0
$724$	3.18691	0.118441
$725$	0	0
$726$	0	0
$727$	−26.0407	−0.965797	−0.482899	−	0.875676i	$-0.660416\pi$
$727$	−26.0407	−0.965797	−0.482899	+	0.875676i	$0.660416\pi$
$728$	28.5073	1.05655
$729$	0	0
$730$	0	0
$731$	1.09370	0.0404518
$732$	0	0
$733$	24.1779	0.893031	0.446516	−	0.894776i	$-0.352665\pi$
$733$	24.1779	0.893031	0.446516	+	0.894776i	$0.352665\pi$
$734$	−16.9904	−0.627126
$735$	0	0
$736$	−27.6374	−1.01873
$737$	−4.00419	−0.147496
$738$	0	0
$739$	−10.7908	−0.396947	−0.198473	−	0.980106i	$-0.563598\pi$
$739$	−10.7908	−0.396947	−0.198473	+	0.980106i	$0.563598\pi$
$740$	0	0
$741$	0	0
$742$	12.5224	0.459712
$743$	−23.8051	−0.873325	−0.436662	−	0.899625i	$-0.643840\pi$
$743$	−23.8051	−0.873325	−0.436662	+	0.899625i	$0.643840\pi$
$744$	0	0
$745$	0	0
$746$	−22.3881	−0.819686
$747$	0	0
$748$	−1.43376	−0.0524234
$749$	−11.5859	−0.423339
$750$	0	0
$751$	−33.8662	−1.23580	−0.617898	−	0.786258i	$-0.712016\pi$
$751$	−33.8662	−1.23580	−0.617898	+	0.786258i	$0.712016\pi$
$752$	1.42981	0.0521398
$753$	0	0
$754$	31.3183	1.14054
$755$	0	0
$756$	0	0
$757$	−26.4066	−0.959763	−0.479881	−	0.877333i	$-0.659320\pi$
$757$	−26.4066	−0.959763	−0.479881	+	0.877333i	$0.659320\pi$
$758$	0.761359	0.0276538
$759$	0	0
$760$	0	0
$761$	20.0385	0.726396	0.363198	−	0.931712i	$-0.381685\pi$
$761$	20.0385	0.726396	0.363198	+	0.931712i	$0.381685\pi$
$762$	0	0
$763$	−28.8491	−1.04441
$764$	23.3231	0.843799
$765$	0	0
$766$	21.9058	0.791490
$767$	−18.1422	−0.655075
$768$	0	0
$769$	−16.4049	−0.591577	−0.295788	−	0.955253i	$-0.595582\pi$
$769$	−16.4049	−0.591577	−0.295788	+	0.955253i	$0.595582\pi$
$770$	0	0
$771$	0	0
$772$	9.49590	0.341765
$773$	−11.4461	−0.411688	−0.205844	−	0.978585i	$-0.565994\pi$
$773$	−11.4461	−0.411688	−0.205844	+	0.978585i	$0.565994\pi$
$774$	0	0
$775$	0	0
$776$	48.2896	1.73349
$777$	0	0
$778$	10.7857	0.386687
$779$	28.8710	1.03441
$780$	0	0
$781$	−3.62303	−0.129642
$782$	−21.0328	−0.752131
$783$	0	0
$784$	−0.252616	−0.00902200
$785$	0	0
$786$	0	0
$787$	6.50658	0.231934	0.115967	−	0.993253i	$-0.463003\pi$
$787$	6.50658	0.231934	0.115967	+	0.993253i	$0.463003\pi$
$788$	−7.07595	−0.252070
$789$	0	0
$790$	0	0
$791$	−32.0500	−1.13957
$792$	0	0
$793$	7.08210	0.251493
$794$	4.48166	0.159048
$795$	0	0
$796$	21.7946	0.772491
$797$	−35.6753	−1.26368	−0.631841	−	0.775098i	$-0.717701\pi$
$797$	−35.6753	−1.26368	−0.631841	+	0.775098i	$0.717701\pi$
$798$	0	0
$799$	−12.6197	−0.446452
$800$	0	0
$801$	0	0
$802$	−27.8862	−0.984696
$803$	1.32866	0.0468873
$804$	0	0
$805$	0	0
$806$	29.7976	1.04958
$807$	0	0
$808$	−28.7211	−1.01040
$809$	15.8037	0.555630	0.277815	−	0.960635i	$-0.410390\pi$
$809$	15.8037	0.555630	0.277815	+	0.960635i	$0.410390\pi$
$810$	0	0
$811$	30.3394	1.06536	0.532681	−	0.846316i	$-0.321185\pi$
$811$	30.3394	1.06536	0.532681	+	0.846316i	$0.321185\pi$
$812$	28.9108	1.01457
$813$	0	0
$814$	−0.201071	−0.00704753
$815$	0	0
$816$	0	0
$817$	0.639038	0.0223571
$818$	−5.61009	−0.196152
$819$	0	0
$820$	0	0
$821$	31.4259	1.09677	0.548386	−	0.836225i	$-0.315242\pi$
$821$	31.4259	1.09677	0.548386	+	0.836225i	$0.315242\pi$
$822$	0	0
$823$	45.0336	1.56977	0.784886	−	0.619641i	$-0.212722\pi$
$823$	45.0336	1.56977	0.784886	+	0.619641i	$0.212722\pi$
$824$	−14.4104	−0.502010
$825$	0	0
$826$	13.0966	0.455690
$827$	11.9965	0.417161	0.208580	−	0.978005i	$-0.433116\pi$
$827$	11.9965	0.417161	0.208580	+	0.978005i	$0.433116\pi$
$828$	0	0
$829$	−30.9847	−1.07614	−0.538071	−	0.842899i	$-0.680847\pi$
$829$	−30.9847	−1.07614	−0.538071	+	0.842899i	$0.680847\pi$
$830$	0	0
$831$	0	0
$832$	−21.4699	−0.744335
$833$	2.22962	0.0772516
$834$	0	0
$835$	0	0
$836$	−0.837735	−0.0289737
$837$	0	0
$838$	3.98799	0.137763
$839$	−36.2024	−1.24985	−0.624923	−	0.780687i	$-0.714870\pi$
$839$	−36.2024	−1.24985	−0.624923	+	0.780687i	$0.714870\pi$
$840$	0	0
$841$	59.3608	2.04692
$842$	−21.7991	−0.751247
$843$	0	0
$844$	19.3803	0.667096
$845$	0	0
$846$	0	0
$847$	−29.9104	−1.02773
$848$	−2.41787	−0.0830300
$849$	0	0
$850$	0	0
$851$	3.77191	0.129300
$852$	0	0
$853$	32.6123	1.11662	0.558312	−	0.829631i	$-0.311449\pi$
$853$	32.6123	1.11662	0.558312	+	0.829631i	$0.311449\pi$
$854$	−5.11249	−0.174946
$855$	0	0
$856$	12.3670	0.422695
$857$	−4.85223	−0.165749	−0.0828744	−	0.996560i	$-0.526410\pi$
$857$	−4.85223	−0.165749	−0.0828744	+	0.996560i	$0.526410\pi$
$858$	0	0
$859$	−32.3815	−1.10484	−0.552421	−	0.833565i	$-0.686296\pi$
$859$	−32.3815	−1.10484	−0.552421	+	0.833565i	$0.686296\pi$
$860$	0	0
$861$	0	0
$862$	11.7265	0.399407
$863$	−21.9498	−0.747180	−0.373590	−	0.927594i	$-0.621873\pi$
$863$	−21.9498	−0.747180	−0.373590	+	0.927594i	$0.621873\pi$
$864$	0	0
$865$	0	0
$866$	−17.7592	−0.603481
$867$	0	0
$868$	27.5071	0.933650
$869$	2.20355	0.0747502
$870$	0	0
$871$	48.7700	1.65251
$872$	30.7941	1.04282
$873$	0	0
$874$	−12.2893	−0.415691
$875$	0	0
$876$	0	0
$877$	4.27196	0.144254	0.0721270	−	0.997395i	$-0.477021\pi$
$877$	4.27196	0.144254	0.0721270	+	0.997395i	$0.477021\pi$
$878$	−6.83812	−0.230775
$879$	0	0
$880$	0	0
$881$	−29.1356	−0.981604	−0.490802	−	0.871271i	$-0.663296\pi$
$881$	−29.1356	−0.981604	−0.490802	+	0.871271i	$0.663296\pi$
$882$	0	0
$883$	49.1547	1.65419	0.827093	−	0.562065i	$-0.189993\pi$
$883$	49.1547	1.65419	0.827093	+	0.562065i	$0.189993\pi$
$884$	17.4628	0.587338
$885$	0	0
$886$	−5.38879	−0.181040
$887$	30.4384	1.02202	0.511010	−	0.859575i	$-0.329271\pi$
$887$	30.4384	1.02202	0.511010	+	0.859575i	$0.329271\pi$
$888$	0	0
$889$	43.0434	1.44363
$890$	0	0
$891$	0	0
$892$	−1.59869	−0.0535280
$893$	−7.37357	−0.246747
$894$	0	0
$895$	0	0
$896$	−14.0196	−0.468361
$897$	0	0
$898$	−4.45056	−0.148517
$899$	84.0704	2.80391
$900$	0	0
$901$	21.3404	0.710952
$902$	3.08940	0.102866
$903$	0	0
$904$	34.2108	1.13783
$905$	0	0
$906$	0	0
$907$	−14.9403	−0.496084	−0.248042	−	0.968749i	$-0.579787\pi$
$907$	−14.9403	−0.496084	−0.248042	+	0.968749i	$0.579787\pi$
$908$	−10.4022	−0.345208
$909$	0	0
$910$	0	0
$911$	−11.8295	−0.391928	−0.195964	−	0.980611i	$-0.562784\pi$
$911$	−11.8295	−0.391928	−0.195964	+	0.980611i	$0.562784\pi$
$912$	0	0
$913$	−3.25923	−0.107865
$914$	4.42823	0.146473
$915$	0	0
$916$	18.8816	0.623866
$917$	24.2503	0.800814
$918$	0	0
$919$	−30.6610	−1.01141	−0.505706	−	0.862706i	$-0.668768\pi$
$919$	−30.6610	−1.01141	−0.505706	+	0.862706i	$0.668768\pi$
$920$	0	0
$921$	0	0
$922$	7.22329	0.237886
$923$	44.1275	1.45247
$924$	0	0
$925$	0	0
$926$	−3.58768	−0.117898
$927$	0	0
$928$	−50.6273	−1.66192
$929$	−6.46400	−0.212077	−0.106039	−	0.994362i	$-0.533817\pi$
$929$	−6.46400	−0.212077	−0.106039	+	0.994362i	$0.533817\pi$
$930$	0	0
$931$	1.30275	0.0426958
$932$	−11.0820	−0.363002
$933$	0	0
$934$	18.6321	0.609660
$935$	0	0
$936$	0	0
$937$	−12.2409	−0.399894	−0.199947	−	0.979807i	$-0.564077\pi$
$937$	−12.2409	−0.399894	−0.199947	+	0.979807i	$0.564077\pi$
$938$	−35.2065	−1.14953
$939$	0	0
$940$	0	0
$941$	7.39466	0.241059	0.120530	−	0.992710i	$-0.461541\pi$
$941$	7.39466	0.241059	0.120530	+	0.992710i	$0.461541\pi$
$942$	0	0
$943$	−57.9545	−1.88726
$944$	−2.52874	−0.0823036
$945$	0	0
$946$	0.0683815	0.00222327
$947$	28.9928	0.942139	0.471070	−	0.882096i	$-0.343868\pi$
$947$	28.9928	0.942139	0.471070	+	0.882096i	$0.343868\pi$
$948$	0	0
$949$	−16.1827	−0.525312
$950$	0	0
$951$	0	0
$952$	−35.0705	−1.13664
$953$	−43.0634	−1.39496	−0.697481	−	0.716604i	$-0.745696\pi$
$953$	−43.0634	−1.39496	−0.697481	+	0.716604i	$0.745696\pi$
$954$	0	0
$955$	0	0
$956$	23.5560	0.761856
$957$	0	0
$958$	−26.0640	−0.842089
$959$	23.6647	0.764174
$960$	0	0
$961$	48.9884	1.58027
$962$	2.44899	0.0789586
$963$	0	0
$964$	−0.672428	−0.0216574
$965$	0	0
$966$	0	0
$967$	31.9201	1.02648	0.513240	−	0.858245i	$-0.328445\pi$
$967$	31.9201	1.02648	0.513240	+	0.858245i	$0.328445\pi$
$968$	31.9270	1.02617
$969$	0	0
$970$	0	0
$971$	11.8218	0.379380	0.189690	−	0.981844i	$-0.439252\pi$
$971$	11.8218	0.379380	0.189690	+	0.981844i	$0.439252\pi$
$972$	0	0
$973$	−21.9701	−0.704328
$974$	29.7110	0.952001
$975$	0	0
$976$	0.987138	0.0315975
$977$	−15.7868	−0.505066	−0.252533	−	0.967588i	$-0.581264\pi$
$977$	−15.7868	−0.505066	−0.252533	+	0.967588i	$0.581264\pi$
$978$	0	0
$979$	2.33411	0.0745984
$980$	0	0
$981$	0	0
$982$	−4.59176	−0.146529
$983$	25.5427	0.814685	0.407342	−	0.913276i	$-0.366456\pi$
$983$	25.5427	0.814685	0.407342	+	0.913276i	$0.366456\pi$
$984$	0	0
$985$	0	0
$986$	−38.5286	−1.22700
$987$	0	0
$988$	10.2034	0.324613
$989$	−1.28278	−0.0407899
$990$	0	0
$991$	−1.33158	−0.0422991	−0.0211495	−	0.999776i	$-0.506733\pi$
$991$	−1.33158	−0.0422991	−0.0211495	+	0.999776i	$0.506733\pi$
$992$	−48.1690	−1.52937
$993$	0	0
$994$	−31.8551	−1.01038
$995$	0	0
$996$	0	0
$997$	51.6975	1.63728	0.818639	−	0.574309i	$-0.194729\pi$
$997$	51.6975	1.63728	0.818639	+	0.574309i	$0.194729\pi$
$998$	2.93797	0.0929997
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5625.2.a.w.1.6		8
3.2	odd	2	inner	5625.2.a.w.1.3		8
5.4	even	2		5625.2.a.v.1.3		8
15.14	odd	2		5625.2.a.v.1.6		8
25.6	even	5		225.2.h.e.136.2	yes	16
25.21	even	5		225.2.h.e.91.2	✓	16
75.56	odd	10		225.2.h.e.136.3	yes	16
75.71	odd	10		225.2.h.e.91.3	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
225.2.h.e.91.2	✓	16	25.21	even	5
225.2.h.e.91.3	yes	16	75.71	odd	10
225.2.h.e.136.2	yes	16	25.6	even	5
225.2.h.e.136.3	yes	16	75.56	odd	10
5625.2.a.v.1.3		8	5.4	even	2
5625.2.a.v.1.6		8	15.14	odd	2
5625.2.a.w.1.3		8	3.2	odd	2	inner
5625.2.a.w.1.6		8	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 5625 = 3^{2} \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5625.a (trivial)

\(f(q)\)	\(=\)	\(q+0.936839 q^{2} -1.12233 q^{4} +2.74037 q^{7} -2.92512 q^{8} +O(q^{10})\)	\(q+0.936839 q^{2} -1.12233 q^{4} +2.74037 q^{7} -2.92512 q^{8} +0.291988 q^{11} -3.55634 q^{13} +2.56728 q^{14} -0.495700 q^{16} +4.37511 q^{17} +2.55634 q^{19} +0.273546 q^{22} -5.13149 q^{23} -3.33172 q^{26} -3.07561 q^{28} -9.40004 q^{29} -8.94362 q^{31} +5.38585 q^{32} +4.09877 q^{34} -0.735053 q^{37} +2.39488 q^{38} +11.2939 q^{41} +0.249982 q^{43} -0.327708 q^{44} -4.80737 q^{46} -2.88442 q^{47} +0.509614 q^{49} +3.99140 q^{52} +4.87769 q^{53} -8.01591 q^{56} -8.80632 q^{58} +5.10135 q^{59} -1.99140 q^{61} -8.37873 q^{62} +6.03708 q^{64} -13.7135 q^{67} -4.91033 q^{68} -12.4081 q^{71} +4.55038 q^{73} -0.688626 q^{74} -2.86907 q^{76} +0.800156 q^{77} +7.54669 q^{79} +10.5806 q^{82} -11.1622 q^{83} +0.234192 q^{86} -0.854102 q^{88} +7.99383 q^{89} -9.74568 q^{91} +5.75924 q^{92} -2.70224 q^{94} -16.5086 q^{97} +0.477426 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{4}+O(q^{10}) \) 8 * q + 4 * q^4	\( 8 q + 4 q^{4} + 10 q^{13} - 8 q^{16} - 18 q^{19} - 30 q^{28} - 26 q^{31} - 20 q^{34} - 40 q^{43} - 30 q^{46} - 16 q^{49} + 40 q^{52} + 10 q^{58} - 24 q^{61} - 34 q^{64} - 40 q^{67} + 40 q^{73} - 44 q^{76} - 42 q^{79} + 60 q^{82} + 20 q^{88} - 40 q^{91} - 10 q^{94} - 40 q^{97}+O(q^{100}) \) 8 * q + 4 * q^4 + 10 * q^13 - 8 * q^16 - 18 * q^19 - 30 * q^28 - 26 * q^31 - 20 * q^34 - 40 * q^43 - 30 * q^46 - 16 * q^49 + 40 * q^52 + 10 * q^58 - 24 * q^61 - 34 * q^64 - 40 * q^67 + 40 * q^73 - 44 * q^76 - 42 * q^79 + 60 * q^82 + 20 * q^88 - 40 * q^91 - 10 * q^94 - 40 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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