LMFDB - Embedded newform 9522.2.a.ch.1.10

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9522,2,Mod(1,9522)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("9522.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9522, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 9522 = 2 \cdot 3^{2} \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9522.a (trivial)

Newform invariants

comment:select newform

sage:traces = [10,-10,0,10,10,0,2,-10,0,-10,12,0,0,-2,0,10,24,0,-8,10,0,-12, 0,0,8,0,0,2,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(29)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$76.0335528047$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	10.10.52900342088704.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 18x^{8} + 123x^{6} - 390x^{4} + 548x^{2} - 241 $ x^10 - 18x^8 + 123x^6 - 390x^4 + 548x^2 - 241
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 414)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$2.32333$ of defining polynomial
Character	$\chi$	$=$	9522.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +4.24774 q^{5} +0.296223 q^{7} -1.00000 q^{8} -4.24774 q^{10} +2.39725 q^{11} +2.51372 q^{13} -0.296223 q^{14} +1.00000 q^{16} +4.50912 q^{17} -1.13579 q^{19} +4.24774 q^{20} -2.39725 q^{22} +13.0433 q^{25} -2.51372 q^{26} +0.296223 q^{28} -0.957087 q^{29} +8.24911 q^{31} -1.00000 q^{32} -4.50912 q^{34} +1.25827 q^{35} +8.34724 q^{37} +1.13579 q^{38} -4.24774 q^{40} -4.10669 q^{41} +3.86276 q^{43} +2.39725 q^{44} -3.35841 q^{47} -6.91225 q^{49} -13.0433 q^{50} +2.51372 q^{52} -4.84637 q^{53} +10.1829 q^{55} -0.296223 q^{56} +0.957087 q^{58} +11.3103 q^{59} -7.76531 q^{61} -8.24911 q^{62} +1.00000 q^{64} +10.6776 q^{65} +14.1030 q^{67} +4.50912 q^{68} -1.25827 q^{70} +14.7692 q^{71} -11.1912 q^{73} -8.34724 q^{74} -1.13579 q^{76} +0.710121 q^{77} -12.0724 q^{79} +4.24774 q^{80} +4.10669 q^{82} +2.12541 q^{83} +19.1536 q^{85} -3.86276 q^{86} -2.39725 q^{88} -11.4844 q^{89} +0.744621 q^{91} +3.35841 q^{94} -4.82455 q^{95} +0.795359 q^{97} +6.91225 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 10 q^{2} + 10 q^{4} + 10 q^{5} + 2 q^{7} - 10 q^{8} - 10 q^{10} + 12 q^{11} - 2 q^{14} + 10 q^{16} + 24 q^{17} - 8 q^{19} + 10 q^{20} - 12 q^{22} + 8 q^{25} + 2 q^{28} - 4 q^{29} + 18 q^{31} - 10 q^{32}+ \cdots - 36 q^{98}+O(q^{100}) $ 10 * q - 10 * q^2 + 10 * q^4 + 10 * q^5 + 2 * q^7 - 10 * q^8 - 10 * q^10 + 12 * q^11 - 2 * q^14 + 10 * q^16 + 24 * q^17 - 8 * q^19 + 10 * q^20 - 12 * q^22 + 8 * q^25 + 2 * q^28 - 4 * q^29 + 18 * q^31 - 10 * q^32 - 24 * q^34 + 24 * q^35 - 12 * q^37 + 8 * q^38 - 10 * q^40 - 28 * q^41 - 8 * q^43 + 12 * q^44 + 16 * q^47 + 36 * q^49 - 8 * q^50 + 34 * q^53 + 30 * q^55 - 2 * q^56 + 4 * q^58 + 22 * q^59 - 30 * q^61 - 18 * q^62 + 10 * q^64 + 36 * q^65 - 18 * q^67 + 24 * q^68 - 24 * q^70 + 28 * q^71 - 20 * q^73 + 12 * q^74 - 8 * q^76 + 20 * q^77 - 2 * q^79 + 10 * q^80 + 28 * q^82 + 44 * q^83 + 16 * q^85 + 8 * q^86 - 12 * q^88 + 44 * q^89 - 22 * q^91 - 16 * q^94 + 10 * q^95 - 18 * q^97 - 36 * q^98

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	4.24774	1.89964	0.949822	−	0.312790i	$-0.101263\pi$
$5$	4.24774	1.89964	0.949822	+	0.312790i	$0.101263\pi$
$6$	0	0
$7$	0.296223	0.111962	0.0559808	−	0.998432i	$-0.482171\pi$
$7$	0.296223	0.111962	0.0559808	+	0.998432i	$0.482171\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−4.24774	−1.34325
$11$	2.39725	0.722799	0.361400	−	0.932411i	$-0.382299\pi$
$11$	2.39725	0.722799	0.361400	+	0.932411i	$0.382299\pi$
$12$	0	0
$13$	2.51372	0.697181	0.348590	−	0.937275i	$-0.386660\pi$
$13$	2.51372	0.697181	0.348590	+	0.937275i	$0.386660\pi$
$14$	−0.296223	−0.0791688
$15$	0	0
$16$	1.00000	0.250000
$17$	4.50912	1.09362	0.546811	−	0.837256i	$-0.315842\pi$
$17$	4.50912	1.09362	0.546811	+	0.837256i	$0.315842\pi$
$18$	0	0
$19$	−1.13579	−0.260569	−0.130285	−	0.991477i	$-0.541589\pi$
$19$	−1.13579	−0.260569	−0.130285	+	0.991477i	$0.541589\pi$
$20$	4.24774	0.949822
$21$	0	0
$22$	−2.39725	−0.511096
$23$	0	0
$23$	0	0
$24$	0	0
$25$	13.0433	2.60865
$26$	−2.51372	−0.492981
$27$	0	0
$28$	0.296223	0.0559808
$29$	−0.957087	−0.177727	−0.0888633	−	0.996044i	$-0.528323\pi$
$29$	−0.957087	−0.177727	−0.0888633	+	0.996044i	$0.528323\pi$
$30$	0	0
$31$	8.24911	1.48158	0.740792	−	0.671735i	$-0.234451\pi$
$31$	8.24911	1.48158	0.740792	+	0.671735i	$0.234451\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−4.50912	−0.773308
$35$	1.25827	0.212687
$36$	0	0
$37$	8.34724	1.37228	0.686139	−	0.727470i	$-0.259304\pi$
$37$	8.34724	1.37228	0.686139	+	0.727470i	$0.259304\pi$
$38$	1.13579	0.184250
$39$	0	0
$40$	−4.24774	−0.671626
$41$	−4.10669	−0.641357	−0.320678	−	0.947188i	$-0.603911\pi$
$41$	−4.10669	−0.641357	−0.320678	+	0.947188i	$0.603911\pi$
$42$	0	0
$43$	3.86276	0.589066	0.294533	−	0.955641i	$-0.404836\pi$
$43$	3.86276	0.589066	0.294533	+	0.955641i	$0.404836\pi$
$44$	2.39725	0.361400
$45$	0	0
$46$	0	0
$47$	−3.35841	−0.489874	−0.244937	−	0.969539i	$-0.578767\pi$
$47$	−3.35841	−0.489874	−0.244937	+	0.969539i	$0.578767\pi$
$48$	0	0
$49$	−6.91225	−0.987465
$50$	−13.0433	−1.84459
$51$	0	0
$52$	2.51372	0.348590
$53$	−4.84637	−0.665701	−0.332850	−	0.942980i	$-0.608010\pi$
$53$	−4.84637	−0.665701	−0.332850	+	0.942980i	$0.608010\pi$
$54$	0	0
$55$	10.1829	1.37306
$56$	−0.296223	−0.0395844
$57$	0	0
$58$	0.957087	0.125672
$59$	11.3103	1.47248	0.736240	−	0.676720i	$-0.236599\pi$
$59$	11.3103	1.47248	0.736240	+	0.676720i	$0.236599\pi$
$60$	0	0
$61$	−7.76531	−0.994246	−0.497123	−	0.867680i	$-0.665610\pi$
$61$	−7.76531	−0.994246	−0.497123	+	0.867680i	$0.665610\pi$
$62$	−8.24911	−1.04764
$63$	0	0
$64$	1.00000	0.125000
$65$	10.6776	1.32440
$66$	0	0
$67$	14.1030	1.72296	0.861478	−	0.507795i	$-0.169539\pi$
$67$	14.1030	1.72296	0.861478	+	0.507795i	$0.169539\pi$
$68$	4.50912	0.546811
$69$	0	0
$70$	−1.25827	−0.150393
$71$	14.7692	1.75279	0.876393	−	0.481597i	$-0.159943\pi$
$71$	14.7692	1.75279	0.876393	+	0.481597i	$0.159943\pi$
$72$	0	0
$73$	−11.1912	−1.30983	−0.654915	−	0.755703i	$-0.727296\pi$
$73$	−11.1912	−1.30983	−0.654915	+	0.755703i	$0.727296\pi$
$74$	−8.34724	−0.970347
$75$	0	0
$76$	−1.13579	−0.130285
$77$	0.710121	0.0809258
$78$	0	0
$79$	−12.0724	−1.35826	−0.679128	−	0.734020i	$-0.737642\pi$
$79$	−12.0724	−1.35826	−0.679128	+	0.734020i	$0.737642\pi$
$80$	4.24774	0.474911
$81$	0	0
$82$	4.10669	0.453508
$83$	2.12541	0.233295	0.116647	−	0.993173i	$-0.462785\pi$
$83$	2.12541	0.233295	0.116647	+	0.993173i	$0.462785\pi$
$84$	0	0
$85$	19.1536	2.07749
$86$	−3.86276	−0.416532
$87$	0	0
$88$	−2.39725	−0.255548
$89$	−11.4844	−1.21735	−0.608673	−	0.793421i	$-0.708298\pi$
$89$	−11.4844	−1.21735	−0.608673	+	0.793421i	$0.708298\pi$
$90$	0	0
$91$	0.744621	0.0780575
$92$	0	0
$93$	0	0
$94$	3.35841	0.346393
$95$	−4.82455	−0.494989
$96$	0	0
$97$	0.795359	0.0807565	0.0403782	−	0.999184i	$-0.487144\pi$
$97$	0.795359	0.0807565	0.0403782	+	0.999184i	$0.487144\pi$
$98$	6.91225	0.698243
$99$	0	0
$100$	13.0433	1.30433
$101$	9.35035	0.930395	0.465197	−	0.885207i	$-0.345983\pi$
$101$	9.35035	0.930395	0.465197	+	0.885207i	$0.345983\pi$
$102$	0	0
$103$	−9.90643	−0.976109	−0.488055	−	0.872813i	$-0.662293\pi$
$103$	−9.90643	−0.976109	−0.488055	+	0.872813i	$0.662293\pi$
$104$	−2.51372	−0.246491
$105$	0	0
$106$	4.84637	0.470722
$107$	−18.4893	−1.78743	−0.893714	−	0.448637i	$-0.851910\pi$
$107$	−18.4893	−1.78743	−0.893714	+	0.448637i	$0.851910\pi$
$108$	0	0
$109$	−16.4814	−1.57863	−0.789314	−	0.613990i	$-0.789564\pi$
$109$	−16.4814	−1.57863	−0.789314	+	0.613990i	$0.789564\pi$
$110$	−10.1829	−0.970901
$111$	0	0
$112$	0.296223	0.0279904
$113$	−4.27218	−0.401893	−0.200946	−	0.979602i	$-0.564402\pi$
$113$	−4.27218	−0.401893	−0.200946	+	0.979602i	$0.564402\pi$
$114$	0	0
$115$	0	0
$116$	−0.957087	−0.0888633
$117$	0	0
$118$	−11.3103	−1.04120
$119$	1.33570	0.122444
$120$	0	0
$121$	−5.25317	−0.477561
$122$	7.76531	0.703038
$123$	0	0
$124$	8.24911	0.740792
$125$	34.1656	3.05586
$126$	0	0
$127$	−13.5281	−1.20042	−0.600210	−	0.799842i	$-0.704917\pi$
$127$	−13.5281	−1.20042	−0.600210	+	0.799842i	$0.704917\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−10.6776	−0.936489
$131$	13.1098	1.14541	0.572706	−	0.819761i	$-0.305894\pi$
$131$	13.1098	1.14541	0.572706	+	0.819761i	$0.305894\pi$
$132$	0	0
$133$	−0.336448	−0.0291737
$134$	−14.1030	−1.21831
$135$	0	0
$136$	−4.50912	−0.386654
$137$	1.26947	0.108458	0.0542291	−	0.998529i	$-0.482730\pi$
$137$	1.26947	0.108458	0.0542291	+	0.998529i	$0.482730\pi$
$138$	0	0
$139$	−15.5662	−1.32031	−0.660155	−	0.751129i	$-0.729509\pi$
$139$	−15.5662	−1.32031	−0.660155	+	0.751129i	$0.729509\pi$
$140$	1.25827	0.106344
$141$	0	0
$142$	−14.7692	−1.23941
$143$	6.02603	0.503922
$144$	0	0
$145$	−4.06545	−0.337617
$146$	11.1912	0.926189
$147$	0	0
$148$	8.34724	0.686139
$149$	13.6820	1.12087	0.560435	−	0.828198i	$-0.310634\pi$
$149$	13.6820	1.12087	0.560435	+	0.828198i	$0.310634\pi$
$150$	0	0
$151$	7.65087	0.622619	0.311309	−	0.950309i	$-0.399232\pi$
$151$	7.65087	0.622619	0.311309	+	0.950309i	$0.399232\pi$
$152$	1.13579	0.0921251
$153$	0	0
$154$	−0.710121	−0.0572232
$155$	35.0400	2.81448
$156$	0	0
$157$	−7.27528	−0.580631	−0.290315	−	0.956931i	$-0.593760\pi$
$157$	−7.27528	−0.580631	−0.290315	+	0.956931i	$0.593760\pi$
$158$	12.0724	0.960433
$159$	0	0
$160$	−4.24774	−0.335813
$161$	0	0
$162$	0	0
$163$	17.6142	1.37965	0.689824	−	0.723977i	$-0.257688\pi$
$163$	17.6142	1.37965	0.689824	+	0.723977i	$0.257688\pi$
$164$	−4.10669	−0.320678
$165$	0	0
$166$	−2.12541	−0.164964
$167$	−12.8331	−0.993054	−0.496527	−	0.868021i	$-0.665392\pi$
$167$	−12.8331	−0.993054	−0.496527	+	0.868021i	$0.665392\pi$
$168$	0	0
$169$	−6.68121	−0.513939
$170$	−19.1536	−1.46901
$171$	0	0
$172$	3.86276	0.294533
$173$	9.24925	0.703208	0.351604	−	0.936149i	$-0.385636\pi$
$173$	9.24925	0.703208	0.351604	+	0.936149i	$0.385636\pi$
$174$	0	0
$175$	3.86371	0.292069
$176$	2.39725	0.180700
$177$	0	0
$178$	11.4844	0.860793
$179$	18.9994	1.42008	0.710042	−	0.704160i	$-0.248676\pi$
$179$	18.9994	1.42008	0.710042	+	0.704160i	$0.248676\pi$
$180$	0	0
$181$	−2.61367	−0.194272	−0.0971362	−	0.995271i	$-0.530968\pi$
$181$	−2.61367	−0.194272	−0.0971362	+	0.995271i	$0.530968\pi$
$182$	−0.744621	−0.0551950
$183$	0	0
$184$	0	0
$185$	35.4569	2.60684
$186$	0	0
$187$	10.8095	0.790470
$188$	−3.35841	−0.244937
$189$	0	0
$190$	4.82455	0.350010
$191$	5.90298	0.427125	0.213562	−	0.976929i	$-0.431493\pi$
$191$	5.90298	0.427125	0.213562	+	0.976929i	$0.431493\pi$
$192$	0	0
$193$	0.350668	0.0252416	0.0126208	−	0.999920i	$-0.495983\pi$
$193$	0.350668	0.0252416	0.0126208	+	0.999920i	$0.495983\pi$
$194$	−0.795359	−0.0571035
$195$	0	0
$196$	−6.91225	−0.493732
$197$	−18.8533	−1.34324	−0.671621	−	0.740895i	$-0.734402\pi$
$197$	−18.8533	−1.34324	−0.671621	+	0.740895i	$0.734402\pi$
$198$	0	0
$199$	16.3425	1.15849	0.579243	−	0.815155i	$-0.303348\pi$
$199$	16.3425	1.15849	0.579243	+	0.815155i	$0.303348\pi$
$200$	−13.0433	−0.922297
$201$	0	0
$202$	−9.35035	−0.657888
$203$	−0.283511	−0.0198986
$204$	0	0
$205$	−17.4441	−1.21835
$206$	9.90643	0.690213
$207$	0	0
$208$	2.51372	0.174295
$209$	−2.72279	−0.188339
$210$	0	0
$211$	13.8129	0.950918	0.475459	−	0.879738i	$-0.342282\pi$
$211$	13.8129	0.950918	0.475459	+	0.879738i	$0.342282\pi$
$212$	−4.84637	−0.332850
$213$	0	0
$214$	18.4893	1.26390
$215$	16.4080	1.11902
$216$	0	0
$217$	2.44357	0.165880
$218$	16.4814	1.11626
$219$	0	0
$220$	10.1829	0.686531
$221$	11.3347	0.762453
$222$	0	0
$223$	3.66413	0.245368	0.122684	−	0.992446i	$-0.460850\pi$
$223$	3.66413	0.245368	0.122684	+	0.992446i	$0.460850\pi$
$224$	−0.296223	−0.0197922
$225$	0	0
$226$	4.27218	0.284181
$227$	0.0570530	0.00378674	0.00189337	−	0.999998i	$-0.499397\pi$
$227$	0.0570530	0.00378674	0.00189337	+	0.999998i	$0.499397\pi$
$228$	0	0
$229$	−6.13532	−0.405433	−0.202717	−	0.979237i	$-0.564977\pi$
$229$	−6.13532	−0.405433	−0.202717	+	0.979237i	$0.564977\pi$
$230$	0	0
$231$	0	0
$232$	0.957087	0.0628358
$233$	−20.3085	−1.33045	−0.665227	−	0.746641i	$-0.731665\pi$
$233$	−20.3085	−1.33045	−0.665227	+	0.746641i	$0.731665\pi$
$234$	0	0
$235$	−14.2656	−0.930586
$236$	11.3103	0.736240
$237$	0	0
$238$	−1.33570	−0.0865808
$239$	3.65555	0.236458	0.118229	−	0.992986i	$-0.462278\pi$
$239$	3.65555	0.236458	0.118229	+	0.992986i	$0.462278\pi$
$240$	0	0
$241$	11.7513	0.756970	0.378485	−	0.925607i	$-0.376445\pi$
$241$	11.7513	0.756970	0.378485	+	0.925607i	$0.376445\pi$
$242$	5.25317	0.337687
$243$	0	0
$244$	−7.76531	−0.497123
$245$	−29.3614	−1.87583
$246$	0	0
$247$	−2.85507	−0.181664
$248$	−8.24911	−0.523819
$249$	0	0
$250$	−34.1656	−2.16082
$251$	−1.89694	−0.119734	−0.0598668	−	0.998206i	$-0.519068\pi$
$251$	−1.89694	−0.119734	−0.0598668	+	0.998206i	$0.519068\pi$
$252$	0	0
$253$	0	0
$254$	13.5281	0.848826
$255$	0	0
$256$	1.00000	0.0625000
$257$	−5.43775	−0.339198	−0.169599	−	0.985513i	$-0.554247\pi$
$257$	−5.43775	−0.339198	−0.169599	+	0.985513i	$0.554247\pi$
$258$	0	0
$259$	2.47264	0.153642
$260$	10.6776	0.662198
$261$	0	0
$262$	−13.1098	−0.809928
$263$	23.1887	1.42988	0.714940	−	0.699186i	$-0.246454\pi$
$263$	23.1887	1.42988	0.714940	+	0.699186i	$0.246454\pi$
$264$	0	0
$265$	−20.5861	−1.26460
$266$	0.336448	0.0206289
$267$	0	0
$268$	14.1030	0.861478
$269$	−5.34716	−0.326022	−0.163011	−	0.986624i	$-0.552121\pi$
$269$	−5.34716	−0.326022	−0.163011	+	0.986624i	$0.552121\pi$
$270$	0	0
$271$	−16.0856	−0.977128	−0.488564	−	0.872528i	$-0.662479\pi$
$271$	−16.0856	−0.977128	−0.488564	+	0.872528i	$0.662479\pi$
$272$	4.50912	0.273406
$273$	0	0
$274$	−1.26947	−0.0766915
$275$	31.2680	1.88553
$276$	0	0
$277$	3.44954	0.207263	0.103631	−	0.994616i	$-0.466954\pi$
$277$	3.44954	0.207263	0.103631	+	0.994616i	$0.466954\pi$
$278$	15.5662	0.933600
$279$	0	0
$280$	−1.25827	−0.0751963
$281$	−6.43983	−0.384168	−0.192084	−	0.981378i	$-0.561525\pi$
$281$	−6.43983	−0.384168	−0.192084	+	0.981378i	$0.561525\pi$
$282$	0	0
$283$	−11.6630	−0.693292	−0.346646	−	0.937996i	$-0.612679\pi$
$283$	−11.6630	−0.693292	−0.346646	+	0.937996i	$0.612679\pi$
$284$	14.7692	0.876393
$285$	0	0
$286$	−6.02603	−0.356326
$287$	−1.21649	−0.0718073
$288$	0	0
$289$	3.33217	0.196010
$290$	4.06545	0.238732
$291$	0	0
$292$	−11.1912	−0.654915
$293$	5.15331	0.301059	0.150530	−	0.988605i	$-0.451902\pi$
$293$	5.15331	0.301059	0.150530	+	0.988605i	$0.451902\pi$
$294$	0	0
$295$	48.0433	2.79719
$296$	−8.34724	−0.485174
$297$	0	0
$298$	−13.6820	−0.792575
$299$	0	0
$300$	0	0
$301$	1.14424	0.0659527
$302$	−7.65087	−0.440258
$303$	0	0
$304$	−1.13579	−0.0651423
$305$	−32.9850	−1.88871
$306$	0	0
$307$	−15.5307	−0.886382	−0.443191	−	0.896427i	$-0.646154\pi$
$307$	−15.5307	−0.886382	−0.443191	+	0.896427i	$0.646154\pi$
$308$	0.710121	0.0404629
$309$	0	0
$310$	−35.0400	−1.99014
$311$	−7.59667	−0.430767	−0.215384	−	0.976529i	$-0.569100\pi$
$311$	−7.59667	−0.430767	−0.215384	+	0.976529i	$0.569100\pi$
$312$	0	0
$313$	−26.1607	−1.47869	−0.739344	−	0.673328i	$-0.764864\pi$
$313$	−26.1607	−1.47869	−0.739344	+	0.673328i	$0.764864\pi$
$314$	7.27528	0.410568
$315$	0	0
$316$	−12.0724	−0.679128
$317$	−24.2042	−1.35944	−0.679720	−	0.733471i	$-0.737899\pi$
$317$	−24.2042	−1.35944	−0.679720	+	0.733471i	$0.737899\pi$
$318$	0	0
$319$	−2.29438	−0.128461
$320$	4.24774	0.237456
$321$	0	0
$322$	0	0
$323$	−5.12143	−0.284964
$324$	0	0
$325$	32.7871	1.81870
$326$	−17.6142	−0.975559
$327$	0	0
$328$	4.10669	0.226754
$329$	−0.994835	−0.0548470
$330$	0	0
$331$	−31.4228	−1.72716	−0.863578	−	0.504216i	$-0.831782\pi$
$331$	−31.4228	−1.72716	−0.863578	+	0.504216i	$0.831782\pi$
$332$	2.12541	0.116647
$333$	0	0
$334$	12.8331	0.702195
$335$	59.9058	3.27300
$336$	0	0
$337$	−5.72615	−0.311923	−0.155962	−	0.987763i	$-0.549848\pi$
$337$	−5.72615	−0.311923	−0.155962	+	0.987763i	$0.549848\pi$
$338$	6.68121	0.363410
$339$	0	0
$340$	19.1536	1.03875
$341$	19.7752	1.07089
$342$	0	0
$343$	−4.12112	−0.222520
$344$	−3.86276	−0.208266
$345$	0	0
$346$	−9.24925	−0.497243
$347$	19.6762	1.05628	0.528138	−	0.849159i	$-0.322890\pi$
$347$	19.6762	1.05628	0.528138	+	0.849159i	$0.322890\pi$
$348$	0	0
$349$	2.20578	0.118073	0.0590364	−	0.998256i	$-0.481197\pi$
$349$	2.20578	0.118073	0.0590364	+	0.998256i	$0.481197\pi$
$350$	−3.86371	−0.206524
$351$	0	0
$352$	−2.39725	−0.127774
$353$	11.8417	0.630269	0.315134	−	0.949047i	$-0.397950\pi$
$353$	11.8417	0.630269	0.315134	+	0.949047i	$0.397950\pi$
$354$	0	0
$355$	62.7358	3.32967
$356$	−11.4844	−0.608673
$357$	0	0
$358$	−18.9994	−1.00415
$359$	−22.8264	−1.20473	−0.602367	−	0.798219i	$-0.705776\pi$
$359$	−22.8264	−1.20473	−0.602367	+	0.798219i	$0.705776\pi$
$360$	0	0
$361$	−17.7100	−0.932104
$362$	2.61367	0.137371
$363$	0	0
$364$	0.744621	0.0390287
$365$	−47.5372	−2.48821
$366$	0	0
$367$	30.0634	1.56930	0.784649	−	0.619941i	$-0.212843\pi$
$367$	30.0634	1.56930	0.784649	+	0.619941i	$0.212843\pi$
$368$	0	0
$369$	0	0
$370$	−35.4569	−1.84331
$371$	−1.43561	−0.0745329
$372$	0	0
$373$	−31.5853	−1.63543	−0.817713	−	0.575625i	$-0.804759\pi$
$373$	−31.5853	−1.63543	−0.817713	+	0.575625i	$0.804759\pi$
$374$	−10.8095	−0.558946
$375$	0	0
$376$	3.35841	0.173197
$377$	−2.40585	−0.123908
$378$	0	0
$379$	25.6283	1.31644	0.658218	−	0.752828i	$-0.271311\pi$
$379$	25.6283	1.31644	0.658218	+	0.752828i	$0.271311\pi$
$380$	−4.82455	−0.247494
$381$	0	0
$382$	−5.90298	−0.302023
$383$	−22.8494	−1.16755	−0.583774	−	0.811916i	$-0.698425\pi$
$383$	−22.8494	−1.16755	−0.583774	+	0.811916i	$0.698425\pi$
$384$	0	0
$385$	3.01640	0.153730
$386$	−0.350668	−0.0178485
$387$	0	0
$388$	0.795359	0.0403782
$389$	16.2740	0.825124	0.412562	−	0.910930i	$-0.364634\pi$
$389$	16.2740	0.825124	0.412562	+	0.910930i	$0.364634\pi$
$390$	0	0
$391$	0	0
$392$	6.91225	0.349121
$393$	0	0
$394$	18.8533	0.949816
$395$	−51.2806	−2.58021
$396$	0	0
$397$	−5.21827	−0.261897	−0.130949	−	0.991389i	$-0.541802\pi$
$397$	−5.21827	−0.261897	−0.130949	+	0.991389i	$0.541802\pi$
$398$	−16.3425	−0.819173
$399$	0	0
$400$	13.0433	0.652163
$401$	8.21417	0.410196	0.205098	−	0.978741i	$-0.434249\pi$
$401$	8.21417	0.410196	0.205098	+	0.978741i	$0.434249\pi$
$402$	0	0
$403$	20.7360	1.03293
$404$	9.35035	0.465197
$405$	0	0
$406$	0.283511	0.0140704
$407$	20.0105	0.991882
$408$	0	0
$409$	−33.9872	−1.68056	−0.840280	−	0.542153i	$-0.817609\pi$
$409$	−33.9872	−1.68056	−0.840280	+	0.542153i	$0.817609\pi$
$410$	17.4441	0.861504
$411$	0	0
$412$	−9.90643	−0.488055
$413$	3.35038	0.164861
$414$	0	0
$415$	9.02820	0.443177
$416$	−2.51372	−0.123245
$417$	0	0
$418$	2.72279	0.133176
$419$	22.8282	1.11523	0.557615	−	0.830100i	$-0.311717\pi$
$419$	22.8282	1.11523	0.557615	+	0.830100i	$0.311717\pi$
$420$	0	0
$421$	−16.6661	−0.812256	−0.406128	−	0.913816i	$-0.633121\pi$
$421$	−16.6661	−0.812256	−0.406128	+	0.913816i	$0.633121\pi$
$422$	−13.8129	−0.672400
$423$	0	0
$424$	4.84637	0.235361
$425$	58.8136	2.85288
$426$	0	0
$427$	−2.30026	−0.111317
$428$	−18.4893	−0.893714
$429$	0	0
$430$	−16.4080	−0.791264
$431$	20.3455	0.980008	0.490004	−	0.871720i	$-0.336995\pi$
$431$	20.3455	0.980008	0.490004	+	0.871720i	$0.336995\pi$
$432$	0	0
$433$	29.1178	1.39931	0.699655	−	0.714481i	$-0.253337\pi$
$433$	29.1178	1.39931	0.699655	+	0.714481i	$0.253337\pi$
$434$	−2.44357	−0.117295
$435$	0	0
$436$	−16.4814	−0.789314
$437$	0	0
$438$	0	0
$439$	−19.6105	−0.935957	−0.467979	−	0.883740i	$-0.655018\pi$
$439$	−19.6105	−0.935957	−0.467979	+	0.883740i	$0.655018\pi$
$440$	−10.1829	−0.485451
$441$	0	0
$442$	−11.3347	−0.539135
$443$	8.63954	0.410477	0.205238	−	0.978712i	$-0.434203\pi$
$443$	8.63954	0.410477	0.205238	+	0.978712i	$0.434203\pi$
$444$	0	0
$445$	−48.7828	−2.31252
$446$	−3.66413	−0.173502
$447$	0	0
$448$	0.296223	0.0139952
$449$	−22.2983	−1.05232	−0.526161	−	0.850385i	$-0.676369\pi$
$449$	−22.2983	−1.05232	−0.526161	+	0.850385i	$0.676369\pi$
$450$	0	0
$451$	−9.84477	−0.463572
$452$	−4.27218	−0.200946
$453$	0	0
$454$	−0.0570530	−0.00267763
$455$	3.16295	0.148281
$456$	0	0
$457$	−3.42721	−0.160318	−0.0801591	−	0.996782i	$-0.525543\pi$
$457$	−3.42721	−0.160318	−0.0801591	+	0.996782i	$0.525543\pi$
$458$	6.13532	0.286685
$459$	0	0
$460$	0	0
$461$	28.9575	1.34869	0.674344	−	0.738418i	$-0.264427\pi$
$461$	28.9575	1.34869	0.674344	+	0.738418i	$0.264427\pi$
$462$	0	0
$463$	9.09325	0.422599	0.211299	−	0.977421i	$-0.432230\pi$
$463$	9.09325	0.422599	0.211299	+	0.977421i	$0.432230\pi$
$464$	−0.957087	−0.0444317
$465$	0	0
$466$	20.3085	0.940773
$467$	20.9454	0.969238	0.484619	−	0.874725i	$-0.338958\pi$
$467$	20.9454	0.969238	0.484619	+	0.874725i	$0.338958\pi$
$468$	0	0
$469$	4.17763	0.192905
$470$	14.2656	0.658024
$471$	0	0
$472$	−11.3103	−0.520601
$473$	9.26002	0.425776
$474$	0	0
$475$	−14.8145	−0.679734
$476$	1.33570	0.0612219
$477$	0	0
$478$	−3.65555	−0.167201
$479$	−9.92872	−0.453655	−0.226827	−	0.973935i	$-0.572835\pi$
$479$	−9.92872	−0.453655	−0.226827	+	0.973935i	$0.572835\pi$
$480$	0	0
$481$	20.9826	0.956726
$482$	−11.7513	−0.535259
$483$	0	0
$484$	−5.25317	−0.238781
$485$	3.37848	0.153409
$486$	0	0
$487$	−15.0194	−0.680595	−0.340298	−	0.940318i	$-0.610528\pi$
$487$	−15.0194	−0.680595	−0.340298	+	0.940318i	$0.610528\pi$
$488$	7.76531	0.351519
$489$	0	0
$490$	29.3614	1.32641
$491$	−4.08834	−0.184504	−0.0922522	−	0.995736i	$-0.529407\pi$
$491$	−4.08834	−0.184504	−0.0922522	+	0.995736i	$0.529407\pi$
$492$	0	0
$493$	−4.31562	−0.194366
$494$	2.85507	0.128456
$495$	0	0
$496$	8.24911	0.370396
$497$	4.37498	0.196245
$498$	0	0
$499$	−15.0954	−0.675763	−0.337881	−	0.941189i	$-0.609710\pi$
$499$	−15.0954	−0.675763	−0.337881	+	0.941189i	$0.609710\pi$
$500$	34.1656	1.52793
$501$	0	0
$502$	1.89694	0.0846644
$503$	33.3581	1.48736	0.743681	−	0.668534i	$-0.233078\pi$
$503$	33.3581	1.48736	0.743681	+	0.668534i	$0.233078\pi$
$504$	0	0
$505$	39.7178	1.76742
$506$	0	0
$507$	0	0
$508$	−13.5281	−0.600210
$509$	−33.1267	−1.46832	−0.734158	−	0.678979i	$-0.762423\pi$
$509$	−33.1267	−1.46832	−0.734158	+	0.678979i	$0.762423\pi$
$510$	0	0
$511$	−3.31508	−0.146651
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	5.43775	0.239849
$515$	−42.0799	−1.85426
$516$	0	0
$517$	−8.05095	−0.354080
$518$	−2.47264	−0.108642
$519$	0	0
$520$	−10.6776	−0.468245
$521$	13.8143	0.605214	0.302607	−	0.953115i	$-0.402143\pi$
$521$	13.8143	0.605214	0.302607	+	0.953115i	$0.402143\pi$
$522$	0	0
$523$	−16.0204	−0.700524	−0.350262	−	0.936652i	$-0.613907\pi$
$523$	−16.0204	−0.700524	−0.350262	+	0.936652i	$0.613907\pi$
$524$	13.1098	0.572706
$525$	0	0
$526$	−23.1887	−1.01108
$527$	37.1962	1.62029
$528$	0	0
$529$	0	0
$530$	20.5861	0.894204
$531$	0	0
$532$	−0.336448	−0.0145869
$533$	−10.3231	−0.447142
$534$	0	0
$535$	−78.5377	−3.39548
$536$	−14.1030	−0.609157
$537$	0	0
$538$	5.34716	0.230532
$539$	−16.5704	−0.713739
$540$	0	0
$541$	−12.0153	−0.516579	−0.258289	−	0.966068i	$-0.583159\pi$
$541$	−12.0153	−0.516579	−0.258289	+	0.966068i	$0.583159\pi$
$542$	16.0856	0.690934
$543$	0	0
$544$	−4.50912	−0.193327
$545$	−70.0084	−2.99883
$546$	0	0
$547$	29.6869	1.26932	0.634661	−	0.772791i	$-0.281140\pi$
$547$	29.6869	1.26932	0.634661	+	0.772791i	$0.281140\pi$
$548$	1.26947	0.0542291
$549$	0	0
$550$	−31.2680	−1.33327
$551$	1.08705	0.0463101
$552$	0	0
$553$	−3.57613	−0.152073
$554$	−3.44954	−0.146557
$555$	0	0
$556$	−15.5662	−0.660155
$557$	30.1816	1.27883	0.639417	−	0.768860i	$-0.279176\pi$
$557$	30.1816	1.27883	0.639417	+	0.768860i	$0.279176\pi$
$558$	0	0
$559$	9.70991	0.410685
$560$	1.25827	0.0531718
$561$	0	0
$562$	6.43983	0.271648
$563$	2.17257	0.0915630	0.0457815	−	0.998951i	$-0.485422\pi$
$563$	2.17257	0.0915630	0.0457815	+	0.998951i	$0.485422\pi$
$564$	0	0
$565$	−18.1471	−0.763454
$566$	11.6630	0.490231
$567$	0	0
$568$	−14.7692	−0.619703
$569$	2.10059	0.0880614	0.0440307	−	0.999030i	$-0.485980\pi$
$569$	2.10059	0.0880614	0.0440307	+	0.999030i	$0.485980\pi$
$570$	0	0
$571$	−33.2711	−1.39235	−0.696177	−	0.717871i	$-0.745117\pi$
$571$	−33.2711	−1.39235	−0.696177	+	0.717871i	$0.745117\pi$
$572$	6.02603	0.251961
$573$	0	0
$574$	1.21649	0.0507754
$575$	0	0
$576$	0	0
$577$	−3.58875	−0.149402	−0.0747008	−	0.997206i	$-0.523800\pi$
$577$	−3.58875	−0.149402	−0.0747008	+	0.997206i	$0.523800\pi$
$578$	−3.33217	−0.138600
$579$	0	0
$580$	−4.06545	−0.168809
$581$	0.629596	0.0261200
$582$	0	0
$583$	−11.6180	−0.481168
$584$	11.1912	0.463095
$585$	0	0
$586$	−5.15331	−0.212881
$587$	37.5313	1.54908	0.774541	−	0.632524i	$-0.217981\pi$
$587$	37.5313	1.54908	0.774541	+	0.632524i	$0.217981\pi$
$588$	0	0
$589$	−9.36929	−0.386055
$590$	−48.0433	−1.97791
$591$	0	0
$592$	8.34724	0.343069
$593$	−20.9062	−0.858515	−0.429258	−	0.903182i	$-0.641225\pi$
$593$	−20.9062	−0.858515	−0.429258	+	0.903182i	$0.641225\pi$
$594$	0	0
$595$	5.67371	0.232600
$596$	13.6820	0.560435
$597$	0	0
$598$	0	0
$599$	−2.81949	−0.115201	−0.0576007	−	0.998340i	$-0.518345\pi$
$599$	−2.81949	−0.115201	−0.0576007	+	0.998340i	$0.518345\pi$
$600$	0	0
$601$	29.7258	1.21254	0.606269	−	0.795259i	$-0.292665\pi$
$601$	29.7258	1.21254	0.606269	+	0.795259i	$0.292665\pi$
$602$	−1.14424	−0.0466356
$603$	0	0
$604$	7.65087	0.311309
$605$	−22.3141	−0.907197
$606$	0	0
$607$	−22.9379	−0.931021	−0.465511	−	0.885042i	$-0.654129\pi$
$607$	−22.9379	−0.931021	−0.465511	+	0.885042i	$0.654129\pi$
$608$	1.13579	0.0460625
$609$	0	0
$610$	32.9850	1.33552
$611$	−8.44209	−0.341531
$612$	0	0
$613$	26.7781	1.08156	0.540779	−	0.841164i	$-0.318129\pi$
$613$	26.7781	1.08156	0.540779	+	0.841164i	$0.318129\pi$
$614$	15.5307	0.626766
$615$	0	0
$616$	−0.710121	−0.0286116
$617$	−34.1884	−1.37637	−0.688187	−	0.725533i	$-0.741593\pi$
$617$	−34.1884	−1.37637	−0.688187	+	0.725533i	$0.741593\pi$
$618$	0	0
$619$	−8.18439	−0.328958	−0.164479	−	0.986381i	$-0.552594\pi$
$619$	−8.18439	−0.328958	−0.164479	+	0.986381i	$0.552594\pi$
$620$	35.0400	1.40724
$621$	0	0
$622$	7.59667	0.304599
$623$	−3.40194	−0.136296
$624$	0	0
$625$	79.9102	3.19641
$626$	26.1607	1.04559
$627$	0	0
$628$	−7.27528	−0.290315
$629$	37.6387	1.50075
$630$	0	0
$631$	−37.1546	−1.47910	−0.739550	−	0.673102i	$-0.764962\pi$
$631$	−37.1546	−1.47910	−0.739550	+	0.673102i	$0.764962\pi$
$632$	12.0724	0.480216
$633$	0	0
$634$	24.2042	0.961270
$635$	−57.4636	−2.28037
$636$	0	0
$637$	−17.3755	−0.688441
$638$	2.29438	0.0908354
$639$	0	0
$640$	−4.24774	−0.167906
$641$	2.08457	0.0823357	0.0411679	−	0.999152i	$-0.486892\pi$
$641$	2.08457	0.0823357	0.0411679	+	0.999152i	$0.486892\pi$
$642$	0	0
$643$	−38.7677	−1.52885	−0.764424	−	0.644713i	$-0.776977\pi$
$643$	−38.7677	−1.52885	−0.764424	+	0.644713i	$0.776977\pi$
$644$	0	0
$645$	0	0
$646$	5.12143	0.201500
$647$	−7.60261	−0.298889	−0.149445	−	0.988770i	$-0.547749\pi$
$647$	−7.60261	−0.298889	−0.149445	+	0.988770i	$0.547749\pi$
$648$	0	0
$649$	27.1138	1.06431
$650$	−32.7871	−1.28602
$651$	0	0
$652$	17.6142	0.689824
$653$	−9.86403	−0.386009	−0.193005	−	0.981198i	$-0.561823\pi$
$653$	−9.86403	−0.386009	−0.193005	+	0.981198i	$0.561823\pi$
$654$	0	0
$655$	55.6871	2.17587
$656$	−4.10669	−0.160339
$657$	0	0
$658$	0.994835	0.0387827
$659$	−18.2003	−0.708981	−0.354491	−	0.935060i	$-0.615346\pi$
$659$	−18.2003	−0.708981	−0.354491	+	0.935060i	$0.615346\pi$
$660$	0	0
$661$	11.1697	0.434451	0.217226	−	0.976121i	$-0.430299\pi$
$661$	11.1697	0.434451	0.217226	+	0.976121i	$0.430299\pi$
$662$	31.4228	1.22128
$663$	0	0
$664$	−2.12541	−0.0824821
$665$	−1.42914	−0.0554197
$666$	0	0
$667$	0	0
$668$	−12.8331	−0.496527
$669$	0	0
$670$	−59.9058	−2.31436
$671$	−18.6154	−0.718640
$672$	0	0
$673$	39.7021	1.53040	0.765201	−	0.643792i	$-0.222640\pi$
$673$	39.7021	1.53040	0.765201	+	0.643792i	$0.222640\pi$
$674$	5.72615	0.220563
$675$	0	0
$676$	−6.68121	−0.256970
$677$	−7.95806	−0.305853	−0.152926	−	0.988238i	$-0.548870\pi$
$677$	−7.95806	−0.305853	−0.152926	+	0.988238i	$0.548870\pi$
$678$	0	0
$679$	0.235603	0.00904162
$680$	−19.1536	−0.734505
$681$	0	0
$682$	−19.7752	−0.757232
$683$	17.4650	0.668280	0.334140	−	0.942523i	$-0.391554\pi$
$683$	17.4650	0.668280	0.334140	+	0.942523i	$0.391554\pi$
$684$	0	0
$685$	5.39237	0.206032
$686$	4.12112	0.157345
$687$	0	0
$688$	3.86276	0.147266
$689$	−12.1824	−0.464114
$690$	0	0
$691$	−14.4115	−0.548240	−0.274120	−	0.961696i	$-0.588386\pi$
$691$	−14.4115	−0.548240	−0.274120	+	0.961696i	$0.588386\pi$
$692$	9.24925	0.351604
$693$	0	0
$694$	−19.6762	−0.746900
$695$	−66.1212	−2.50812
$696$	0	0
$697$	−18.5175	−0.701402
$698$	−2.20578	−0.0834901
$699$	0	0
$700$	3.86371	0.146034
$701$	27.1514	1.02550	0.512748	−	0.858539i	$-0.328628\pi$
$701$	27.1514	1.02550	0.512748	+	0.858539i	$0.328628\pi$
$702$	0	0
$703$	−9.48075	−0.357573
$704$	2.39725	0.0903499
$705$	0	0
$706$	−11.8417	−0.445667
$707$	2.76978	0.104168
$708$	0	0
$709$	−14.8838	−0.558972	−0.279486	−	0.960150i	$-0.590164\pi$
$709$	−14.8838	−0.558972	−0.279486	+	0.960150i	$0.590164\pi$
$710$	−62.7358	−2.35443
$711$	0	0
$712$	11.4844	0.430397
$713$	0	0
$714$	0	0
$715$	25.5970	0.957272
$716$	18.9994	0.710042
$717$	0	0
$718$	22.8264	0.851876
$719$	−3.29160	−0.122756	−0.0613780	−	0.998115i	$-0.519550\pi$
$719$	−3.29160	−0.122756	−0.0613780	+	0.998115i	$0.519550\pi$
$720$	0	0
$721$	−2.93451	−0.109287
$722$	17.7100	0.659097
$723$	0	0
$724$	−2.61367	−0.0971362
$725$	−12.4835	−0.463627
$726$	0	0
$727$	32.1610	1.19279	0.596393	−	0.802693i	$-0.296600\pi$
$727$	32.1610	1.19279	0.596393	+	0.802693i	$0.296600\pi$
$728$	−0.744621	−0.0275975
$729$	0	0
$730$	47.5372	1.75943
$731$	17.4177	0.644216
$732$	0	0
$733$	52.0402	1.92215	0.961074	−	0.276291i	$-0.0891054\pi$
$733$	52.0402	1.92215	0.961074	+	0.276291i	$0.0891054\pi$
$734$	−30.0634	−1.10966
$735$	0	0
$736$	0	0
$737$	33.8085	1.24535
$738$	0	0
$739$	−16.5749	−0.609717	−0.304859	−	0.952398i	$-0.598609\pi$
$739$	−16.5749	−0.609717	−0.304859	+	0.952398i	$0.598609\pi$
$740$	35.4569	1.30342
$741$	0	0
$742$	1.43561	0.0527027
$743$	9.13697	0.335203	0.167601	−	0.985855i	$-0.446398\pi$
$743$	9.13697	0.335203	0.167601	+	0.985855i	$0.446398\pi$
$744$	0	0
$745$	58.1173	2.12925
$746$	31.5853	1.15642
$747$	0	0
$748$	10.8095	0.395235
$749$	−5.47695	−0.200123
$750$	0	0
$751$	42.0794	1.53550	0.767750	−	0.640750i	$-0.221376\pi$
$751$	42.0794	1.53550	0.767750	+	0.640750i	$0.221376\pi$
$752$	−3.35841	−0.122468
$753$	0	0
$754$	2.40585	0.0876159
$755$	32.4989	1.18275
$756$	0	0
$757$	7.58553	0.275701	0.137850	−	0.990453i	$-0.455981\pi$
$757$	7.58553	0.275701	0.137850	+	0.990453i	$0.455981\pi$
$758$	−25.6283	−0.930860
$759$	0	0
$760$	4.82455	0.175005
$761$	46.0739	1.67018	0.835089	−	0.550115i	$-0.185416\pi$
$761$	46.0739	1.67018	0.835089	+	0.550115i	$0.185416\pi$
$762$	0	0
$763$	−4.88215	−0.176746
$764$	5.90298	0.213562
$765$	0	0
$766$	22.8494	0.825581
$767$	28.4310	1.02659
$768$	0	0
$769$	−12.3780	−0.446364	−0.223182	−	0.974777i	$-0.571644\pi$
$769$	−12.3780	−0.446364	−0.223182	+	0.974777i	$0.571644\pi$
$770$	−3.01640	−0.108704
$771$	0	0
$772$	0.350668	0.0126208
$773$	46.1635	1.66038	0.830192	−	0.557477i	$-0.188231\pi$
$773$	46.1635	1.66038	0.830192	+	0.557477i	$0.188231\pi$
$774$	0	0
$775$	107.595	3.86493
$776$	−0.795359	−0.0285517
$777$	0	0
$778$	−16.2740	−0.583451
$779$	4.66435	0.167118
$780$	0	0
$781$	35.4056	1.26691
$782$	0	0
$783$	0	0
$784$	−6.91225	−0.246866
$785$	−30.9035	−1.10299
$786$	0	0
$787$	−12.7894	−0.455893	−0.227946	−	0.973674i	$-0.573201\pi$
$787$	−12.7894	−0.455893	−0.227946	+	0.973674i	$0.573201\pi$
$788$	−18.8533	−0.671621
$789$	0	0
$790$	51.2806	1.82448
$791$	−1.26552	−0.0449966
$792$	0	0
$793$	−19.5198	−0.693169
$794$	5.21827	0.185189
$795$	0	0
$796$	16.3425	0.579243
$797$	17.8798	0.633334	0.316667	−	0.948537i	$-0.397436\pi$
$797$	17.8798	0.633334	0.316667	+	0.948537i	$0.397436\pi$
$798$	0	0
$799$	−15.1435	−0.535737
$800$	−13.0433	−0.461149
$801$	0	0
$802$	−8.21417	−0.290052
$803$	−26.8281	−0.946744
$804$	0	0
$805$	0	0
$806$	−20.7360	−0.730393
$807$	0	0
$808$	−9.35035	−0.328944
$809$	−37.7595	−1.32755	−0.663776	−	0.747931i	$-0.731047\pi$
$809$	−37.7595	−1.32755	−0.663776	+	0.747931i	$0.731047\pi$
$810$	0	0
$811$	21.3002	0.747950	0.373975	−	0.927439i	$-0.377995\pi$
$811$	21.3002	0.747950	0.373975	+	0.927439i	$0.377995\pi$
$812$	−0.283511	−0.00994928
$813$	0	0
$814$	−20.0105	−0.701366
$815$	74.8204	2.62084
$816$	0	0
$817$	−4.38730	−0.153492
$818$	33.9872	1.18834
$819$	0	0
$820$	−17.4441	−0.609175
$821$	−7.82220	−0.272997	−0.136498	−	0.990640i	$-0.543585\pi$
$821$	−7.82220	−0.272997	−0.136498	+	0.990640i	$0.543585\pi$
$822$	0	0
$823$	0.0399456	0.00139242	0.000696208	−	1.00000i	$-0.499778\pi$
$823$	0.0399456	0.00139242	0.000696208		1.00000i	$0.499778\pi$
$824$	9.90643	0.345107
$825$	0	0
$826$	−3.35038	−0.116575
$827$	35.9303	1.24942	0.624710	−	0.780857i	$-0.285217\pi$
$827$	35.9303	1.24942	0.624710	+	0.780857i	$0.285217\pi$
$828$	0	0
$829$	11.7357	0.407597	0.203799	−	0.979013i	$-0.434671\pi$
$829$	11.7357	0.407597	0.203799	+	0.979013i	$0.434671\pi$
$830$	−9.02820	−0.313373
$831$	0	0
$832$	2.51372	0.0871476
$833$	−31.1682	−1.07991
$834$	0	0
$835$	−54.5115	−1.88645
$836$	−2.72279	−0.0941696
$837$	0	0
$838$	−22.8282	−0.788586
$839$	−54.3029	−1.87475	−0.937373	−	0.348328i	$-0.886749\pi$
$839$	−54.3029	−1.87475	−0.937373	+	0.348328i	$0.886749\pi$
$840$	0	0
$841$	−28.0840	−0.968413
$842$	16.6661	0.574352
$843$	0	0
$844$	13.8129	0.475459
$845$	−28.3800	−0.976302
$846$	0	0
$847$	−1.55611	−0.0534685
$848$	−4.84637	−0.166425
$849$	0	0
$850$	−58.8136	−2.01729
$851$	0	0
$852$	0	0
$853$	44.1406	1.51135	0.755673	−	0.654949i	$-0.227310\pi$
$853$	44.1406	1.51135	0.755673	+	0.654949i	$0.227310\pi$
$854$	2.30026	0.0787132
$855$	0	0
$856$	18.4893	0.631951
$857$	25.1720	0.859860	0.429930	−	0.902862i	$-0.358538\pi$
$857$	25.1720	0.859860	0.429930	+	0.902862i	$0.358538\pi$
$858$	0	0
$859$	17.2254	0.587722	0.293861	−	0.955848i	$-0.405060\pi$
$859$	17.2254	0.587722	0.293861	+	0.955848i	$0.405060\pi$
$860$	16.4080	0.559508
$861$	0	0
$862$	−20.3455	−0.692971
$863$	−37.7817	−1.28610	−0.643052	−	0.765823i	$-0.722332\pi$
$863$	−37.7817	−1.28610	−0.643052	+	0.765823i	$0.722332\pi$
$864$	0	0
$865$	39.2884	1.33585
$866$	−29.1178	−0.989462
$867$	0	0
$868$	2.44357	0.0829402
$869$	−28.9407	−0.981747
$870$	0	0
$871$	35.4510	1.20121
$872$	16.4814	0.558129
$873$	0	0
$874$	0	0
$875$	10.1206	0.342139
$876$	0	0
$877$	−57.5016	−1.94169	−0.970845	−	0.239708i	$-0.922948\pi$
$877$	−57.5016	−1.94169	−0.970845	+	0.239708i	$0.922948\pi$
$878$	19.6105	0.661822
$879$	0	0
$880$	10.1829	0.343265
$881$	33.9249	1.14296	0.571479	−	0.820616i	$-0.306370\pi$
$881$	33.9249	1.14296	0.571479	+	0.820616i	$0.306370\pi$
$882$	0	0
$883$	−15.3770	−0.517477	−0.258738	−	0.965947i	$-0.583307\pi$
$883$	−15.3770	−0.517477	−0.258738	+	0.965947i	$0.583307\pi$
$884$	11.3347	0.381226
$885$	0	0
$886$	−8.63954	−0.290251
$887$	7.10730	0.238640	0.119320	−	0.992856i	$-0.461929\pi$
$887$	7.10730	0.238640	0.119320	+	0.992856i	$0.461929\pi$
$888$	0	0
$889$	−4.00732	−0.134401
$890$	48.7828	1.63520
$891$	0	0
$892$	3.66413	0.122684
$893$	3.81446	0.127646
$894$	0	0
$895$	80.7045	2.69765
$896$	−0.296223	−0.00989610
$897$	0	0
$898$	22.2983	0.744104
$899$	−7.89511	−0.263317
$900$	0	0
$901$	−21.8529	−0.728025
$902$	9.84477	0.327795
$903$	0	0
$904$	4.27218	0.142091
$905$	−11.1022	−0.369048
$906$	0	0
$907$	−28.9535	−0.961386	−0.480693	−	0.876889i	$-0.659615\pi$
$907$	−28.9535	−0.961386	−0.480693	+	0.876889i	$0.659615\pi$
$908$	0.0570530	0.00189337
$909$	0	0
$910$	−3.16295	−0.104851
$911$	−3.10995	−0.103037	−0.0515186	−	0.998672i	$-0.516406\pi$
$911$	−3.10995	−0.103037	−0.0515186	+	0.998672i	$0.516406\pi$
$912$	0	0
$913$	5.09516	0.168625
$914$	3.42721	0.113362
$915$	0	0
$916$	−6.13532	−0.202717
$917$	3.88343	0.128242
$918$	0	0
$919$	14.9768	0.494038	0.247019	−	0.969011i	$-0.420549\pi$
$919$	14.9768	0.494038	0.247019	+	0.969011i	$0.420549\pi$
$920$	0	0
$921$	0	0
$922$	−28.9575	−0.953666
$923$	37.1257	1.22201
$924$	0	0
$925$	108.875	3.57979
$926$	−9.09325	−0.298823
$927$	0	0
$928$	0.957087	0.0314179
$929$	−31.9251	−1.04743	−0.523715	−	0.851893i	$-0.675454\pi$
$929$	−31.9251	−1.04743	−0.523715	+	0.851893i	$0.675454\pi$
$930$	0	0
$931$	7.85090	0.257303
$932$	−20.3085	−0.665227
$933$	0	0
$934$	−20.9454	−0.685355
$935$	45.9159	1.50161
$936$	0	0
$937$	21.6312	0.706661	0.353330	−	0.935499i	$-0.385049\pi$
$937$	21.6312	0.706661	0.353330	+	0.935499i	$0.385049\pi$
$938$	−4.17763	−0.136404
$939$	0	0
$940$	−14.2656	−0.465293
$941$	2.95773	0.0964193	0.0482097	−	0.998837i	$-0.484648\pi$
$941$	2.95773	0.0964193	0.0482097	+	0.998837i	$0.484648\pi$
$942$	0	0
$943$	0	0
$944$	11.3103	0.368120
$945$	0	0
$946$	−9.26002	−0.301069
$947$	1.61057	0.0523364	0.0261682	−	0.999658i	$-0.491669\pi$
$947$	1.61057	0.0523364	0.0261682	+	0.999658i	$0.491669\pi$
$948$	0	0
$949$	−28.1315	−0.913188
$950$	14.8145	0.480644
$951$	0	0
$952$	−1.33570	−0.0432904
$953$	−31.4202	−1.01780	−0.508899	−	0.860826i	$-0.669947\pi$
$953$	−31.4202	−1.01780	−0.508899	+	0.860826i	$0.669947\pi$
$954$	0	0
$955$	25.0743	0.811385
$956$	3.65555	0.118229
$957$	0	0
$958$	9.92872	0.320782
$959$	0.376046	0.0121431
$960$	0	0
$961$	37.0478	1.19509
$962$	−20.9826	−0.676507
$963$	0	0
$964$	11.7513	0.378485
$965$	1.48955	0.0479502
$966$	0	0
$967$	15.2718	0.491108	0.245554	−	0.969383i	$-0.421030\pi$
$967$	15.2718	0.491108	0.245554	+	0.969383i	$0.421030\pi$
$968$	5.25317	0.168843
$969$	0	0
$970$	−3.37848	−0.108476
$971$	−25.7653	−0.826848	−0.413424	−	0.910539i	$-0.635667\pi$
$971$	−25.7653	−0.826848	−0.413424	+	0.910539i	$0.635667\pi$
$972$	0	0
$973$	−4.61107	−0.147824
$974$	15.0194	0.481253
$975$	0	0
$976$	−7.76531	−0.248561
$977$	8.88263	0.284180	0.142090	−	0.989854i	$-0.454618\pi$
$977$	8.88263	0.284180	0.142090	+	0.989854i	$0.454618\pi$
$978$	0	0
$979$	−27.5311	−0.879897
$980$	−29.3614	−0.937916
$981$	0	0
$982$	4.08834	0.130464
$983$	3.56769	0.113792	0.0568959	−	0.998380i	$-0.481880\pi$
$983$	3.56769	0.113792	0.0568959	+	0.998380i	$0.481880\pi$
$984$	0	0
$985$	−80.0838	−2.55168
$986$	4.31562	0.137437
$987$	0	0
$988$	−2.85507	−0.0908319
$989$	0	0
$990$	0	0
$991$	0.467875	0.0148625	0.00743127	−	0.999972i	$-0.497635\pi$
$991$	0.467875	0.0148625	0.00743127	+	0.999972i	$0.497635\pi$
$992$	−8.24911	−0.261909
$993$	0	0
$994$	−4.37498	−0.138766
$995$	69.4184	2.20071
$996$	0	0
$997$	16.2383	0.514273	0.257137	−	0.966375i	$-0.417221\pi$
$997$	16.2383	0.514273	0.257137	+	0.966375i	$0.417221\pi$
$998$	15.0954	0.477836
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9522.2.a.ch.1.10		10
3.2	odd	2		9522.2.a.ci.1.1		10
23.11	odd	22		414.2.i.h.397.1	yes	20
23.21	odd	22		414.2.i.h.73.1	yes	20
23.22	odd	2		9522.2.a.cg.1.1		10
69.11	even	22		414.2.i.g.397.2	yes	20
69.44	even	22		414.2.i.g.73.2	✓	20
69.68	even	2		9522.2.a.cj.1.10		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
414.2.i.g.73.2	✓	20	69.44	even	22
414.2.i.g.397.2	yes	20	69.11	even	22
414.2.i.h.73.1	yes	20	23.21	odd	22
414.2.i.h.397.1	yes	20	23.11	odd	22
9522.2.a.cg.1.1		10	23.22	odd	2
9522.2.a.ch.1.10		10	1.1	even	1	trivial
9522.2.a.ci.1.1		10	3.2	odd	2
9522.2.a.cj.1.10		10	69.68	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 9522 = 2 \cdot 3^{2} \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9522.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +4.24774 q^{5} +0.296223 q^{7} -1.00000 q^{8} -4.24774 q^{10} +2.39725 q^{11} +2.51372 q^{13} -0.296223 q^{14} +1.00000 q^{16} +4.50912 q^{17} -1.13579 q^{19} +4.24774 q^{20} -2.39725 q^{22} +13.0433 q^{25} -2.51372 q^{26} +0.296223 q^{28} -0.957087 q^{29} +8.24911 q^{31} -1.00000 q^{32} -4.50912 q^{34} +1.25827 q^{35} +8.34724 q^{37} +1.13579 q^{38} -4.24774 q^{40} -4.10669 q^{41} +3.86276 q^{43} +2.39725 q^{44} -3.35841 q^{47} -6.91225 q^{49} -13.0433 q^{50} +2.51372 q^{52} -4.84637 q^{53} +10.1829 q^{55} -0.296223 q^{56} +0.957087 q^{58} +11.3103 q^{59} -7.76531 q^{61} -8.24911 q^{62} +1.00000 q^{64} +10.6776 q^{65} +14.1030 q^{67} +4.50912 q^{68} -1.25827 q^{70} +14.7692 q^{71} -11.1912 q^{73} -8.34724 q^{74} -1.13579 q^{76} +0.710121 q^{77} -12.0724 q^{79} +4.24774 q^{80} +4.10669 q^{82} +2.12541 q^{83} +19.1536 q^{85} -3.86276 q^{86} -2.39725 q^{88} -11.4844 q^{89} +0.744621 q^{91} +3.35841 q^{94} -4.82455 q^{95} +0.795359 q^{97} +6.91225 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 10 q^{2} + 10 q^{4} + 10 q^{5} + 2 q^{7} - 10 q^{8} - 10 q^{10} + 12 q^{11} - 2 q^{14} + 10 q^{16} + 24 q^{17} - 8 q^{19} + 10 q^{20} - 12 q^{22} + 8 q^{25} + 2 q^{28} - 4 q^{29} + 18 q^{31} - 10 q^{32}+ \cdots - 36 q^{98}+O(q^{100}) \) 10 * q - 10 * q^2 + 10 * q^4 + 10 * q^5 + 2 * q^7 - 10 * q^8 - 10 * q^10 + 12 * q^11 - 2 * q^14 + 10 * q^16 + 24 * q^17 - 8 * q^19 + 10 * q^20 - 12 * q^22 + 8 * q^25 + 2 * q^28 - 4 * q^29 + 18 * q^31 - 10 * q^32 - 24 * q^34 + 24 * q^35 - 12 * q^37 + 8 * q^38 - 10 * q^40 - 28 * q^41 - 8 * q^43 + 12 * q^44 + 16 * q^47 + 36 * q^49 - 8 * q^50 + 34 * q^53 + 30 * q^55 - 2 * q^56 + 4 * q^58 + 22 * q^59 - 30 * q^61 - 18 * q^62 + 10 * q^64 + 36 * q^65 - 18 * q^67 + 24 * q^68 - 24 * q^70 + 28 * q^71 - 20 * q^73 + 12 * q^74 - 8 * q^76 + 20 * q^77 - 2 * q^79 + 10 * q^80 + 28 * q^82 + 44 * q^83 + 16 * q^85 + 8 * q^86 - 12 * q^88 + 44 * q^89 - 22 * q^91 - 16 * q^94 + 10 * q^95 - 18 * q^97 - 36 * q^98

Introduction

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