LMFDB - Embedded newform 9522.2.a.cg.1.8

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:	$1$
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			1.8
Root			$-1.70641$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +0.856843 q^{5} -2.82183 q^{7} -1.00000 q^{8} -0.856843 q^{10} -5.63961 q^{11} +0.572382 q^{13} +2.82183 q^{14} +1.00000 q^{16} -1.82284 q^{17} +7.67439 q^{19} +0.856843 q^{20} +5.63961 q^{22} -4.26582 q^{25} -0.572382 q^{26} -2.82183 q^{28} +4.97347 q^{29} +8.12411 q^{31} -1.00000 q^{32} +1.82284 q^{34} -2.41787 q^{35} +0.415329 q^{37} -7.67439 q^{38} -0.856843 q^{40} +2.84201 q^{41} -5.30077 q^{43} -5.63961 q^{44} -6.28555 q^{47} +0.962748 q^{49} +4.26582 q^{50} +0.572382 q^{52} -0.421889 q^{53} -4.83226 q^{55} +2.82183 q^{56} -4.97347 q^{58} +6.90614 q^{59} -5.51771 q^{61} -8.12411 q^{62} +1.00000 q^{64} +0.490442 q^{65} +12.4903 q^{67} -1.82284 q^{68} +2.41787 q^{70} +4.53693 q^{71} -2.63572 q^{73} -0.415329 q^{74} +7.67439 q^{76} +15.9140 q^{77} +4.60000 q^{79} +0.856843 q^{80} -2.84201 q^{82} -16.6586 q^{83} -1.56189 q^{85} +5.30077 q^{86} +5.63961 q^{88} +12.0581 q^{89} -1.61517 q^{91} +6.28555 q^{94} +6.57574 q^{95} -3.19471 q^{97} -0.962748 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	0.856843	0.383192	0.191596	−	0.981474i	$-0.438634\pi$
$5$	0.856843	0.383192	0.191596	+	0.981474i	$0.438634\pi$
$6$	0	0
$7$	−2.82183	−1.06655	−0.533277	−	0.845941i	$-0.679039\pi$
$7$	−2.82183	−1.06655	−0.533277	+	0.845941i	$0.679039\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−0.856843	−0.270958
$11$	−5.63961	−1.70041	−0.850203	−	0.526455i	$-0.823521\pi$
$11$	−5.63961	−1.70041	−0.850203	+	0.526455i	$0.823521\pi$
$12$	0	0
$13$	0.572382	0.158750	0.0793751	−	0.996845i	$-0.474708\pi$
$13$	0.572382	0.158750	0.0793751	+	0.996845i	$0.474708\pi$
$14$	2.82183	0.754167
$15$	0	0
$16$	1.00000	0.250000
$17$	−1.82284	−0.442104	−0.221052	−	0.975262i	$-0.570949\pi$
$17$	−1.82284	−0.442104	−0.221052	+	0.975262i	$0.570949\pi$
$18$	0	0
$19$	7.67439	1.76062	0.880312	−	0.474395i	$-0.157333\pi$
$19$	7.67439	1.76062	0.880312	+	0.474395i	$0.157333\pi$
$20$	0.856843	0.191596
$21$	0	0
$22$	5.63961	1.20237
$23$	0	0
$23$	0	0
$24$	0	0
$25$	−4.26582	−0.853164
$26$	−0.572382	−0.112253
$27$	0	0
$28$	−2.82183	−0.533277
$29$	4.97347	0.923550	0.461775	−	0.886997i	$-0.347213\pi$
$29$	4.97347	0.923550	0.461775	+	0.886997i	$0.347213\pi$
$30$	0	0
$31$	8.12411	1.45913	0.729566	−	0.683910i	$-0.239722\pi$
$31$	8.12411	1.45913	0.729566	+	0.683910i	$0.239722\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	1.82284	0.312614
$35$	−2.41787	−0.408694
$36$	0	0
$37$	0.415329	0.0682797	0.0341399	−	0.999417i	$-0.489131\pi$
$37$	0.415329	0.0682797	0.0341399	+	0.999417i	$0.489131\pi$
$38$	−7.67439	−1.24495
$39$	0	0
$40$	−0.856843	−0.135479
$41$	2.84201	0.443848	0.221924	−	0.975064i	$-0.428766\pi$
$41$	2.84201	0.443848	0.221924	+	0.975064i	$0.428766\pi$
$42$	0	0
$43$	−5.30077	−0.808359	−0.404180	−	0.914680i	$-0.632443\pi$
$43$	−5.30077	−0.808359	−0.404180	+	0.914680i	$0.632443\pi$
$44$	−5.63961	−0.850203
$45$	0	0
$46$	0	0
$47$	−6.28555	−0.916841	−0.458421	−	0.888735i	$-0.651585\pi$
$47$	−6.28555	−0.916841	−0.458421	+	0.888735i	$0.651585\pi$
$48$	0	0
$49$	0.962748	0.137535
$50$	4.26582	0.603278
$51$	0	0
$52$	0.572382	0.0793751
$53$	−0.421889	−0.0579509	−0.0289755	−	0.999580i	$-0.509224\pi$
$53$	−0.421889	−0.0579509	−0.0289755	+	0.999580i	$0.509224\pi$
$54$	0	0
$55$	−4.83226	−0.651582
$56$	2.82183	0.377083
$57$	0	0
$58$	−4.97347	−0.653049
$59$	6.90614	0.899103	0.449552	−	0.893254i	$-0.351584\pi$
$59$	6.90614	0.899103	0.449552	+	0.893254i	$0.351584\pi$
$60$	0	0
$61$	−5.51771	−0.706471	−0.353235	−	0.935535i	$-0.614918\pi$
$61$	−5.51771	−0.706471	−0.353235	+	0.935535i	$0.614918\pi$
$62$	−8.12411	−1.03176
$63$	0	0
$64$	1.00000	0.125000
$65$	0.490442	0.0608318
$66$	0	0
$67$	12.4903	1.52594	0.762968	−	0.646436i	$-0.223741\pi$
$67$	12.4903	1.52594	0.762968	+	0.646436i	$0.223741\pi$
$68$	−1.82284	−0.221052
$69$	0	0
$70$	2.41787	0.288991
$71$	4.53693	0.538434	0.269217	−	0.963080i	$-0.413235\pi$
$71$	4.53693	0.538434	0.269217	+	0.963080i	$0.413235\pi$
$72$	0	0
$73$	−2.63572	−0.308488	−0.154244	−	0.988033i	$-0.549294\pi$
$73$	−2.63572	−0.308488	−0.154244	+	0.988033i	$0.549294\pi$
$74$	−0.415329	−0.0482810
$75$	0	0
$76$	7.67439	0.880312
$77$	15.9140	1.81357
$78$	0	0
$79$	4.60000	0.517540	0.258770	−	0.965939i	$-0.416683\pi$
$79$	4.60000	0.517540	0.258770	+	0.965939i	$0.416683\pi$
$80$	0.856843	0.0957980
$81$	0	0
$82$	−2.84201	−0.313848
$83$	−16.6586	−1.82852	−0.914259	−	0.405131i	$-0.867226\pi$
$83$	−16.6586	−1.82852	−0.914259	+	0.405131i	$0.867226\pi$
$84$	0	0
$85$	−1.56189	−0.169410
$86$	5.30077	0.571596
$87$	0	0
$88$	5.63961	0.601184
$89$	12.0581	1.27816	0.639079	−	0.769141i	$-0.279316\pi$
$89$	12.0581	1.27816	0.639079	+	0.769141i	$0.279316\pi$
$90$	0	0
$91$	−1.61517	−0.169316
$92$	0	0
$93$	0	0
$94$	6.28555	0.648305
$95$	6.57574	0.674657
$96$	0	0
$97$	−3.19471	−0.324374	−0.162187	−	0.986760i	$-0.551855\pi$
$97$	−3.19471	−0.324374	−0.162187	+	0.986760i	$0.551855\pi$
$98$	−0.962748	−0.0972522
$99$	0	0
$100$	−4.26582	−0.426582
$101$	14.9222	1.48481	0.742407	−	0.669949i	$-0.233684\pi$
$101$	14.9222	1.48481	0.742407	+	0.669949i	$0.233684\pi$
$102$	0	0
$103$	−1.82365	−0.179689	−0.0898445	−	0.995956i	$-0.528637\pi$
$103$	−1.82365	−0.179689	−0.0898445	+	0.995956i	$0.528637\pi$
$104$	−0.572382	−0.0561267
$105$	0	0
$106$	0.421889	0.0409775
$107$	−15.3721	−1.48608	−0.743041	−	0.669246i	$-0.766617\pi$
$107$	−15.3721	−1.48608	−0.743041	+	0.669246i	$0.766617\pi$
$108$	0	0
$109$	15.5912	1.49337	0.746684	−	0.665179i	$-0.231645\pi$
$109$	15.5912	1.49337	0.746684	+	0.665179i	$0.231645\pi$
$110$	4.83226	0.460738
$111$	0	0
$112$	−2.82183	−0.266638
$113$	−12.6925	−1.19401	−0.597005	−	0.802238i	$-0.703643\pi$
$113$	−12.6925	−1.19401	−0.597005	+	0.802238i	$0.703643\pi$
$114$	0	0
$115$	0	0
$116$	4.97347	0.461775
$117$	0	0
$118$	−6.90614	−0.635762
$119$	5.14375	0.471527
$120$	0	0
$121$	20.8052	1.89138
$122$	5.51771	0.499550
$123$	0	0
$124$	8.12411	0.729566
$125$	−7.93935	−0.710117
$126$	0	0
$127$	9.02664	0.800985	0.400492	−	0.916300i	$-0.368839\pi$
$127$	9.02664	0.800985	0.400492	+	0.916300i	$0.368839\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−0.490442	−0.0430146
$131$	6.04878	0.528484	0.264242	−	0.964456i	$-0.414878\pi$
$131$	6.04878	0.528484	0.264242	+	0.964456i	$0.414878\pi$
$132$	0	0
$133$	−21.6558	−1.87780
$134$	−12.4903	−1.07900
$135$	0	0
$136$	1.82284	0.156307
$137$	−2.57420	−0.219929	−0.109964	−	0.993936i	$-0.535074\pi$
$137$	−2.57420	−0.219929	−0.109964	+	0.993936i	$0.535074\pi$
$138$	0	0
$139$	11.0030	0.933266	0.466633	−	0.884451i	$-0.345467\pi$
$139$	11.0030	0.933266	0.466633	+	0.884451i	$0.345467\pi$
$140$	−2.41787	−0.204347
$141$	0	0
$142$	−4.53693	−0.380731
$143$	−3.22801	−0.269940
$144$	0	0
$145$	4.26148	0.353897
$146$	2.63572	0.218134
$147$	0	0
$148$	0.415329	0.0341399
$149$	−8.81823	−0.722418	−0.361209	−	0.932485i	$-0.617636\pi$
$149$	−8.81823	−0.722418	−0.361209	+	0.932485i	$0.617636\pi$
$150$	0	0
$151$	−5.65385	−0.460104	−0.230052	−	0.973178i	$-0.573890\pi$
$151$	−5.65385	−0.460104	−0.230052	+	0.973178i	$0.573890\pi$
$152$	−7.67439	−0.622475
$153$	0	0
$154$	−15.9140	−1.28239
$155$	6.96108	0.559128
$156$	0	0
$157$	−20.4393	−1.63123	−0.815616	−	0.578594i	$-0.803602\pi$
$157$	−20.4393	−1.63123	−0.815616	+	0.578594i	$0.803602\pi$
$158$	−4.60000	−0.365956
$159$	0	0
$160$	−0.856843	−0.0677394
$161$	0	0
$162$	0	0
$163$	−11.3798	−0.891333	−0.445667	−	0.895199i	$-0.647033\pi$
$163$	−11.3798	−0.891333	−0.445667	+	0.895199i	$0.647033\pi$
$164$	2.84201	0.221924
$165$	0	0
$166$	16.6586	1.29296
$167$	−16.9174	−1.30911	−0.654556	−	0.756014i	$-0.727144\pi$
$167$	−16.9174	−1.30911	−0.654556	+	0.756014i	$0.727144\pi$
$168$	0	0
$169$	−12.6724	−0.974798
$170$	1.56189	0.119791
$171$	0	0
$172$	−5.30077	−0.404180
$173$	−15.4617	−1.17553	−0.587766	−	0.809031i	$-0.699992\pi$
$173$	−15.4617	−1.17553	−0.587766	+	0.809031i	$0.699992\pi$
$174$	0	0
$175$	12.0374	0.909945
$176$	−5.63961	−0.425101
$177$	0	0
$178$	−12.0581	−0.903794
$179$	−22.4108	−1.67506	−0.837532	−	0.546389i	$-0.816002\pi$
$179$	−22.4108	−1.67506	−0.837532	+	0.546389i	$0.816002\pi$
$180$	0	0
$181$	18.0215	1.33953	0.669764	−	0.742574i	$-0.266395\pi$
$181$	18.0215	1.33953	0.669764	+	0.742574i	$0.266395\pi$
$182$	1.61517	0.119724
$183$	0	0
$184$	0	0
$185$	0.355872	0.0261642
$186$	0	0
$187$	10.2801	0.751755
$188$	−6.28555	−0.458421
$189$	0	0
$190$	−6.57574	−0.477055
$191$	12.8133	0.927141	0.463570	−	0.886060i	$-0.346568\pi$
$191$	12.8133	0.927141	0.463570	+	0.886060i	$0.346568\pi$
$192$	0	0
$193$	22.9800	1.65414	0.827068	−	0.562102i	$-0.190007\pi$
$193$	22.9800	1.65414	0.827068	+	0.562102i	$0.190007\pi$
$194$	3.19471	0.229367
$195$	0	0
$196$	0.962748	0.0687677
$197$	22.5403	1.60593	0.802965	−	0.596026i	$-0.203254\pi$
$197$	22.5403	1.60593	0.802965	+	0.596026i	$0.203254\pi$
$198$	0	0
$199$	−11.8988	−0.843487	−0.421743	−	0.906715i	$-0.638582\pi$
$199$	−11.8988	−0.843487	−0.421743	+	0.906715i	$0.638582\pi$
$200$	4.26582	0.301639
$201$	0	0
$202$	−14.9222	−1.04992
$203$	−14.0343	−0.985015
$204$	0	0
$205$	2.43516	0.170079
$206$	1.82365	0.127059
$207$	0	0
$208$	0.572382	0.0396876
$209$	−43.2805	−2.99378
$210$	0	0
$211$	−16.7256	−1.15144	−0.575718	−	0.817648i	$-0.695277\pi$
$211$	−16.7256	−1.15144	−0.575718	+	0.817648i	$0.695277\pi$
$212$	−0.421889	−0.0289755
$213$	0	0
$214$	15.3721	1.05082
$215$	−4.54192	−0.309757
$216$	0	0
$217$	−22.9249	−1.55624
$218$	−15.5912	−1.05597
$219$	0	0
$220$	−4.83226	−0.325791
$221$	−1.04336	−0.0701841
$222$	0	0
$223$	−2.37503	−0.159044	−0.0795219	−	0.996833i	$-0.525339\pi$
$223$	−2.37503	−0.159044	−0.0795219	+	0.996833i	$0.525339\pi$
$224$	2.82183	0.188542
$225$	0	0
$226$	12.6925	0.844293
$227$	−13.4894	−0.895320	−0.447660	−	0.894204i	$-0.647743\pi$
$227$	−13.4894	−0.895320	−0.447660	+	0.894204i	$0.647743\pi$
$228$	0	0
$229$	2.39089	0.157995	0.0789973	−	0.996875i	$-0.474828\pi$
$229$	2.39089	0.157995	0.0789973	+	0.996875i	$0.474828\pi$
$230$	0	0
$231$	0	0
$232$	−4.97347	−0.326524
$233$	−13.1021	−0.858349	−0.429175	−	0.903222i	$-0.641195\pi$
$233$	−13.1021	−0.858349	−0.429175	+	0.903222i	$0.641195\pi$
$234$	0	0
$235$	−5.38573	−0.351326
$236$	6.90614	0.449552
$237$	0	0
$238$	−5.14375	−0.333420
$239$	−8.65744	−0.560003	−0.280002	−	0.960000i	$-0.590335\pi$
$239$	−8.65744	−0.560003	−0.280002	+	0.960000i	$0.590335\pi$
$240$	0	0
$241$	−1.09132	−0.0702981	−0.0351491	−	0.999382i	$-0.511191\pi$
$241$	−1.09132	−0.0702981	−0.0351491	+	0.999382i	$0.511191\pi$
$242$	−20.8052	−1.33741
$243$	0	0
$244$	−5.51771	−0.353235
$245$	0.824924	0.0527024
$246$	0	0
$247$	4.39268	0.279500
$248$	−8.12411	−0.515881
$249$	0	0
$250$	7.93935	0.502129
$251$	−14.8277	−0.935917	−0.467958	−	0.883751i	$-0.655010\pi$
$251$	−14.8277	−0.935917	−0.467958	+	0.883751i	$0.655010\pi$
$252$	0	0
$253$	0	0
$254$	−9.02664	−0.566382
$255$	0	0
$256$	1.00000	0.0625000
$257$	7.38499	0.460663	0.230331	−	0.973112i	$-0.426019\pi$
$257$	7.38499	0.460663	0.230331	+	0.973112i	$0.426019\pi$
$258$	0	0
$259$	−1.17199	−0.0728239
$260$	0.490442	0.0304159
$261$	0	0
$262$	−6.04878	−0.373695
$263$	2.22728	0.137340	0.0686700	−	0.997639i	$-0.478124\pi$
$263$	2.22728	0.137340	0.0686700	+	0.997639i	$0.478124\pi$
$264$	0	0
$265$	−0.361493	−0.0222063
$266$	21.6558	1.32780
$267$	0	0
$268$	12.4903	0.762968
$269$	−21.2533	−1.29584	−0.647918	−	0.761710i	$-0.724360\pi$
$269$	−21.2533	−1.29584	−0.647918	+	0.761710i	$0.724360\pi$
$270$	0	0
$271$	−1.74483	−0.105991	−0.0529954	−	0.998595i	$-0.516877\pi$
$271$	−1.74483	−0.105991	−0.0529954	+	0.998595i	$0.516877\pi$
$272$	−1.82284	−0.110526
$273$	0	0
$274$	2.57420	0.155513
$275$	24.0575	1.45072
$276$	0	0
$277$	8.34410	0.501348	0.250674	−	0.968072i	$-0.419348\pi$
$277$	8.34410	0.501348	0.250674	+	0.968072i	$0.419348\pi$
$278$	−11.0030	−0.659919
$279$	0	0
$280$	2.41787	0.144495
$281$	−30.4405	−1.81593	−0.907965	−	0.419046i	$-0.862365\pi$
$281$	−30.4405	−1.81593	−0.907965	+	0.419046i	$0.862365\pi$
$282$	0	0
$283$	−2.84508	−0.169122	−0.0845612	−	0.996418i	$-0.526949\pi$
$283$	−2.84508	−0.169122	−0.0845612	+	0.996418i	$0.526949\pi$
$284$	4.53693	0.269217
$285$	0	0
$286$	3.22801	0.190876
$287$	−8.01969	−0.473387
$288$	0	0
$289$	−13.6773	−0.804544
$290$	−4.26148	−0.250243
$291$	0	0
$292$	−2.63572	−0.154244
$293$	17.4235	1.01789	0.508946	−	0.860799i	$-0.330035\pi$
$293$	17.4235	1.01789	0.508946	+	0.860799i	$0.330035\pi$
$294$	0	0
$295$	5.91748	0.344529
$296$	−0.415329	−0.0241405
$297$	0	0
$298$	8.81823	0.510826
$299$	0	0
$300$	0	0
$301$	14.9579	0.862158
$302$	5.65385	0.325342
$303$	0	0
$304$	7.67439	0.440156
$305$	−4.72781	−0.270714
$306$	0	0
$307$	18.2701	1.04273	0.521364	−	0.853334i	$-0.325423\pi$
$307$	18.2701	1.04273	0.521364	+	0.853334i	$0.325423\pi$
$308$	15.9140	0.906786
$309$	0	0
$310$	−6.96108	−0.395363
$311$	−29.1152	−1.65097	−0.825485	−	0.564424i	$-0.809098\pi$
$311$	−29.1152	−1.65097	−0.825485	+	0.564424i	$0.809098\pi$
$312$	0	0
$313$	17.1880	0.971525	0.485762	−	0.874091i	$-0.338542\pi$
$313$	17.1880	0.971525	0.485762	+	0.874091i	$0.338542\pi$
$314$	20.4393	1.15345
$315$	0	0
$316$	4.60000	0.258770
$317$	11.4083	0.640752	0.320376	−	0.947290i	$-0.396191\pi$
$317$	11.4083	0.640752	0.320376	+	0.947290i	$0.396191\pi$
$318$	0	0
$319$	−28.0484	−1.57041
$320$	0.856843	0.0478990
$321$	0	0
$322$	0	0
$323$	−13.9892	−0.778378
$324$	0	0
$325$	−2.44168	−0.135440
$326$	11.3798	0.630268
$327$	0	0
$328$	−2.84201	−0.156924
$329$	17.7368	0.977860
$330$	0	0
$331$	−12.6785	−0.696876	−0.348438	−	0.937332i	$-0.613288\pi$
$331$	−12.6785	−0.696876	−0.348438	+	0.937332i	$0.613288\pi$
$332$	−16.6586	−0.914259
$333$	0	0
$334$	16.9174	0.925681
$335$	10.7022	0.584726
$336$	0	0
$337$	−15.1062	−0.822886	−0.411443	−	0.911435i	$-0.634975\pi$
$337$	−15.1062	−0.822886	−0.411443	+	0.911435i	$0.634975\pi$
$338$	12.6724	0.689287
$339$	0	0
$340$	−1.56189	−0.0847052
$341$	−45.8168	−2.48112
$342$	0	0
$343$	17.0361	0.919864
$344$	5.30077	0.285798
$345$	0	0
$346$	15.4617	0.831226
$347$	−32.3996	−1.73930	−0.869650	−	0.493669i	$-0.835656\pi$
$347$	−32.3996	−1.73930	−0.869650	+	0.493669i	$0.835656\pi$
$348$	0	0
$349$	−12.8810	−0.689504	−0.344752	−	0.938694i	$-0.612037\pi$
$349$	−12.8810	−0.689504	−0.344752	+	0.938694i	$0.612037\pi$
$350$	−12.0374	−0.643428
$351$	0	0
$352$	5.63961	0.300592
$353$	16.2767	0.866323	0.433161	−	0.901316i	$-0.357398\pi$
$353$	16.2767	0.866323	0.433161	+	0.901316i	$0.357398\pi$
$354$	0	0
$355$	3.88743	0.206324
$356$	12.0581	0.639079
$357$	0	0
$358$	22.4108	1.18445
$359$	−30.3447	−1.60153	−0.800765	−	0.598978i	$-0.795574\pi$
$359$	−30.3447	−1.60153	−0.800765	+	0.598978i	$0.795574\pi$
$360$	0	0
$361$	39.8962	2.09980
$362$	−18.0215	−0.947189
$363$	0	0
$364$	−1.61517	−0.0846578
$365$	−2.25840	−0.118210
$366$	0	0
$367$	−3.34854	−0.174793	−0.0873963	−	0.996174i	$-0.527855\pi$
$367$	−3.34854	−0.174793	−0.0873963	+	0.996174i	$0.527855\pi$
$368$	0	0
$369$	0	0
$370$	−0.355872	−0.0185009
$371$	1.19050	0.0618077
$372$	0	0
$373$	−3.28590	−0.170137	−0.0850686	−	0.996375i	$-0.527111\pi$
$373$	−3.28590	−0.170137	−0.0850686	+	0.996375i	$0.527111\pi$
$374$	−10.2801	−0.531571
$375$	0	0
$376$	6.28555	0.324152
$377$	2.84673	0.146614
$378$	0	0
$379$	−8.01850	−0.411883	−0.205941	−	0.978564i	$-0.566026\pi$
$379$	−8.01850	−0.411883	−0.205941	+	0.978564i	$0.566026\pi$
$380$	6.57574	0.337329
$381$	0	0
$382$	−12.8133	−0.655587
$383$	−27.8632	−1.42374	−0.711871	−	0.702310i	$-0.752152\pi$
$383$	−27.8632	−1.42374	−0.711871	+	0.702310i	$0.752152\pi$
$384$	0	0
$385$	13.6358	0.694946
$386$	−22.9800	−1.16965
$387$	0	0
$388$	−3.19471	−0.162187
$389$	−5.11899	−0.259543	−0.129772	−	0.991544i	$-0.541424\pi$
$389$	−5.11899	−0.259543	−0.129772	+	0.991544i	$0.541424\pi$
$390$	0	0
$391$	0	0
$392$	−0.962748	−0.0486261
$393$	0	0
$394$	−22.5403	−1.13556
$395$	3.94148	0.198317
$396$	0	0
$397$	−6.71598	−0.337065	−0.168533	−	0.985696i	$-0.553903\pi$
$397$	−6.71598	−0.337065	−0.168533	+	0.985696i	$0.553903\pi$
$398$	11.8988	0.596435
$399$	0	0
$400$	−4.26582	−0.213291
$401$	−24.8487	−1.24089	−0.620443	−	0.784251i	$-0.713047\pi$
$401$	−24.8487	−1.24089	−0.620443	+	0.784251i	$0.713047\pi$
$402$	0	0
$403$	4.65009	0.231638
$404$	14.9222	0.742407
$405$	0	0
$406$	14.0343	0.696511
$407$	−2.34229	−0.116103
$408$	0	0
$409$	26.0081	1.28602	0.643008	−	0.765860i	$-0.277686\pi$
$409$	26.0081	1.28602	0.643008	+	0.765860i	$0.277686\pi$
$410$	−2.43516	−0.120264
$411$	0	0
$412$	−1.82365	−0.0898445
$413$	−19.4880	−0.958941
$414$	0	0
$415$	−14.2738	−0.700673
$416$	−0.572382	−0.0280633
$417$	0	0
$418$	43.2805	2.11692
$419$	−6.95184	−0.339620	−0.169810	−	0.985477i	$-0.554315\pi$
$419$	−6.95184	−0.339620	−0.169810	+	0.985477i	$0.554315\pi$
$420$	0	0
$421$	−3.97308	−0.193636	−0.0968180	−	0.995302i	$-0.530866\pi$
$421$	−3.97308	−0.193636	−0.0968180	+	0.995302i	$0.530866\pi$
$422$	16.7256	0.814188
$423$	0	0
$424$	0.421889	0.0204887
$425$	7.77591	0.377187
$426$	0	0
$427$	15.5701	0.753488
$428$	−15.3721	−0.743041
$429$	0	0
$430$	4.54192	0.219031
$431$	24.9334	1.20100	0.600500	−	0.799625i	$-0.294968\pi$
$431$	24.9334	1.20100	0.600500	+	0.799625i	$0.294968\pi$
$432$	0	0
$433$	34.2887	1.64781	0.823905	−	0.566727i	$-0.191791\pi$
$433$	34.2887	1.64781	0.823905	+	0.566727i	$0.191791\pi$
$434$	22.9249	1.10043
$435$	0	0
$436$	15.5912	0.746684
$437$	0	0
$438$	0	0
$439$	−29.8744	−1.42583	−0.712914	−	0.701251i	$-0.752625\pi$
$439$	−29.8744	−1.42583	−0.712914	+	0.701251i	$0.752625\pi$
$440$	4.83226	0.230369
$441$	0	0
$442$	1.04336	0.0496276
$443$	−18.0904	−0.859499	−0.429749	−	0.902948i	$-0.641398\pi$
$443$	−18.0904	−0.859499	−0.429749	+	0.902948i	$0.641398\pi$
$444$	0	0
$445$	10.3319	0.489779
$446$	2.37503	0.112461
$447$	0	0
$448$	−2.82183	−0.133319
$449$	31.8510	1.50314	0.751571	−	0.659652i	$-0.229296\pi$
$449$	31.8510	1.50314	0.751571	+	0.659652i	$0.229296\pi$
$450$	0	0
$451$	−16.0278	−0.754721
$452$	−12.6925	−0.597005
$453$	0	0
$454$	13.4894	0.633087
$455$	−1.38395	−0.0648804
$456$	0	0
$457$	−30.5977	−1.43130	−0.715650	−	0.698459i	$-0.753869\pi$
$457$	−30.5977	−1.43130	−0.715650	+	0.698459i	$0.753869\pi$
$458$	−2.39089	−0.111719
$459$	0	0
$460$	0	0
$461$	19.9053	0.927080	0.463540	−	0.886076i	$-0.346579\pi$
$461$	19.9053	0.927080	0.463540	+	0.886076i	$0.346579\pi$
$462$	0	0
$463$	40.2973	1.87278	0.936388	−	0.350967i	$-0.114147\pi$
$463$	40.2973	1.87278	0.936388	+	0.350967i	$0.114147\pi$
$464$	4.97347	0.230888
$465$	0	0
$466$	13.1021	0.606944
$467$	31.8816	1.47531	0.737653	−	0.675181i	$-0.235934\pi$
$467$	31.8816	1.47531	0.737653	+	0.675181i	$0.235934\pi$
$468$	0	0
$469$	−35.2456	−1.62749
$470$	5.38573	0.248425
$471$	0	0
$472$	−6.90614	−0.317881
$473$	29.8942	1.37454
$474$	0	0
$475$	−32.7375	−1.50210
$476$	5.14375	0.235763
$477$	0	0
$478$	8.65744	0.395982
$479$	−35.5148	−1.62271	−0.811356	−	0.584553i	$-0.801270\pi$
$479$	−35.5148	−1.62271	−0.811356	+	0.584553i	$0.801270\pi$
$480$	0	0
$481$	0.237727	0.0108394
$482$	1.09132	0.0497083
$483$	0	0
$484$	20.8052	0.945689
$485$	−2.73737	−0.124297
$486$	0	0
$487$	−37.3370	−1.69190	−0.845951	−	0.533260i	$-0.820967\pi$
$487$	−37.3370	−1.69190	−0.845951	+	0.533260i	$0.820967\pi$
$488$	5.51771	0.249775
$489$	0	0
$490$	−0.824924	−0.0372662
$491$	17.3451	0.782773	0.391386	−	0.920226i	$-0.371996\pi$
$491$	17.3451	0.782773	0.391386	+	0.920226i	$0.371996\pi$
$492$	0	0
$493$	−9.06584	−0.408305
$494$	−4.39268	−0.197636
$495$	0	0
$496$	8.12411	0.364783
$497$	−12.8025	−0.574269
$498$	0	0
$499$	−14.7162	−0.658789	−0.329394	−	0.944192i	$-0.606845\pi$
$499$	−14.7162	−0.658789	−0.329394	+	0.944192i	$0.606845\pi$
$500$	−7.93935	−0.355059
$501$	0	0
$502$	14.8277	0.661793
$503$	30.2265	1.34773	0.673866	−	0.738853i	$-0.264632\pi$
$503$	30.2265	1.34773	0.673866	+	0.738853i	$0.264632\pi$
$504$	0	0
$505$	12.7860	0.568969
$506$	0	0
$507$	0	0
$508$	9.02664	0.400492
$509$	15.0259	0.666013	0.333007	−	0.942924i	$-0.391937\pi$
$509$	15.0259	0.666013	0.333007	+	0.942924i	$0.391937\pi$
$510$	0	0
$511$	7.43757	0.329019
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−7.38499	−0.325738
$515$	−1.56258	−0.0688554
$516$	0	0
$517$	35.4480	1.55900
$518$	1.17199	0.0514943
$519$	0	0
$520$	−0.490442	−0.0215073
$521$	−15.1022	−0.661638	−0.330819	−	0.943694i	$-0.607325\pi$
$521$	−15.1022	−0.661638	−0.330819	+	0.943694i	$0.607325\pi$
$522$	0	0
$523$	−40.0817	−1.75265	−0.876325	−	0.481721i	$-0.840012\pi$
$523$	−40.0817	−1.75265	−0.876325	+	0.481721i	$0.840012\pi$
$524$	6.04878	0.264242
$525$	0	0
$526$	−2.22728	−0.0971141
$527$	−14.8089	−0.645088
$528$	0	0
$529$	0	0
$530$	0.361493	0.0157022
$531$	0	0
$532$	−21.6558	−0.938900
$533$	1.62672	0.0704610
$534$	0	0
$535$	−13.1715	−0.569454
$536$	−12.4903	−0.539500
$537$	0	0
$538$	21.2533	0.916294
$539$	−5.42952	−0.233866
$540$	0	0
$541$	−26.4260	−1.13614	−0.568071	−	0.822979i	$-0.692310\pi$
$541$	−26.4260	−1.13614	−0.568071	+	0.822979i	$0.692310\pi$
$542$	1.74483	0.0749468
$543$	0	0
$544$	1.82284	0.0781536
$545$	13.3592	0.572246
$546$	0	0
$547$	16.5623	0.708152	0.354076	−	0.935217i	$-0.384795\pi$
$547$	16.5623	0.708152	0.354076	+	0.935217i	$0.384795\pi$
$548$	−2.57420	−0.109964
$549$	0	0
$550$	−24.0575	−1.02582
$551$	38.1683	1.62603
$552$	0	0
$553$	−12.9804	−0.551984
$554$	−8.34410	−0.354507
$555$	0	0
$556$	11.0030	0.466633
$557$	−2.01500	−0.0853784	−0.0426892	−	0.999088i	$-0.513593\pi$
$557$	−2.01500	−0.0853784	−0.0426892	+	0.999088i	$0.513593\pi$
$558$	0	0
$559$	−3.03406	−0.128327
$560$	−2.41787	−0.102174
$561$	0	0
$562$	30.4405	1.28406
$563$	−12.1107	−0.510405	−0.255202	−	0.966888i	$-0.582142\pi$
$563$	−12.1107	−0.510405	−0.255202	+	0.966888i	$0.582142\pi$
$564$	0	0
$565$	−10.8755	−0.457535
$566$	2.84508	0.119588
$567$	0	0
$568$	−4.53693	−0.190365
$569$	3.99574	0.167510	0.0837550	−	0.996486i	$-0.473309\pi$
$569$	3.99574	0.167510	0.0837550	+	0.996486i	$0.473309\pi$
$570$	0	0
$571$	−25.5965	−1.07118	−0.535590	−	0.844478i	$-0.679911\pi$
$571$	−25.5965	−1.07118	−0.535590	+	0.844478i	$0.679911\pi$
$572$	−3.22801	−0.134970
$573$	0	0
$574$	8.01969	0.334735
$575$	0	0
$576$	0	0
$577$	6.65310	0.276972	0.138486	−	0.990364i	$-0.455776\pi$
$577$	6.65310	0.276972	0.138486	+	0.990364i	$0.455776\pi$
$578$	13.6773	0.568899
$579$	0	0
$580$	4.26148	0.176948
$581$	47.0078	1.95021
$582$	0	0
$583$	2.37929	0.0985401
$584$	2.63572	0.109067
$585$	0	0
$586$	−17.4235	−0.719758
$587$	−20.1965	−0.833597	−0.416799	−	0.908999i	$-0.636848\pi$
$587$	−20.1965	−0.833597	−0.416799	+	0.908999i	$0.636848\pi$
$588$	0	0
$589$	62.3475	2.56898
$590$	−5.91748	−0.243619
$591$	0	0
$592$	0.415329	0.0170699
$593$	−17.2338	−0.707708	−0.353854	−	0.935301i	$-0.615129\pi$
$593$	−17.2338	−0.707708	−0.353854	+	0.935301i	$0.615129\pi$
$594$	0	0
$595$	4.40739	0.180685
$596$	−8.81823	−0.361209
$597$	0	0
$598$	0	0
$599$	−14.1802	−0.579389	−0.289694	−	0.957119i	$-0.593554\pi$
$599$	−14.1802	−0.579389	−0.289694	+	0.957119i	$0.593554\pi$
$600$	0	0
$601$	37.7713	1.54072	0.770362	−	0.637607i	$-0.220075\pi$
$601$	37.7713	1.54072	0.770362	+	0.637607i	$0.220075\pi$
$602$	−14.9579	−0.609638
$603$	0	0
$604$	−5.65385	−0.230052
$605$	17.8268	0.724761
$606$	0	0
$607$	28.9169	1.17370	0.586850	−	0.809696i	$-0.300368\pi$
$607$	28.9169	1.17370	0.586850	+	0.809696i	$0.300368\pi$
$608$	−7.67439	−0.311237
$609$	0	0
$610$	4.72781	0.191424
$611$	−3.59774	−0.145549
$612$	0	0
$613$	−0.0168695	−0.000681354 0	−0.000340677	−	1.00000i	$-0.500108\pi$
$613$	−0.0168695	−0.000681354 0	−0.000340677		1.00000i	$0.500108\pi$
$614$	−18.2701	−0.737320
$615$	0	0
$616$	−15.9140	−0.641195
$617$	22.9765	0.924999	0.462499	−	0.886620i	$-0.346953\pi$
$617$	22.9765	0.924999	0.462499	+	0.886620i	$0.346953\pi$
$618$	0	0
$619$	−14.8725	−0.597778	−0.298889	−	0.954288i	$-0.596616\pi$
$619$	−14.8725	−0.597778	−0.298889	+	0.954288i	$0.596616\pi$
$620$	6.96108	0.279564
$621$	0	0
$622$	29.1152	1.16741
$623$	−34.0260	−1.36322
$624$	0	0
$625$	14.5263	0.581053
$626$	−17.1880	−0.686972
$627$	0	0
$628$	−20.4393	−0.815616
$629$	−0.757079	−0.0301867
$630$	0	0
$631$	−18.1776	−0.723638	−0.361819	−	0.932248i	$-0.617844\pi$
$631$	−18.1776	−0.723638	−0.361819	+	0.932248i	$0.617844\pi$
$632$	−4.60000	−0.182978
$633$	0	0
$634$	−11.4083	−0.453080
$635$	7.73441	0.306931
$636$	0	0
$637$	0.551060	0.0218338
$638$	28.0484	1.11045
$639$	0	0
$640$	−0.856843	−0.0338697
$641$	14.8258	0.585583	0.292792	−	0.956176i	$-0.405416\pi$
$641$	14.8258	0.585583	0.292792	+	0.956176i	$0.405416\pi$
$642$	0	0
$643$	−21.6321	−0.853087	−0.426544	−	0.904467i	$-0.640269\pi$
$643$	−21.6321	−0.853087	−0.426544	+	0.904467i	$0.640269\pi$
$644$	0	0
$645$	0	0
$646$	13.9892	0.550397
$647$	48.5849	1.91007	0.955035	−	0.296494i	$-0.0958174\pi$
$647$	48.5849	1.91007	0.955035	+	0.296494i	$0.0958174\pi$
$648$	0	0
$649$	−38.9479	−1.52884
$650$	2.44168	0.0957706
$651$	0	0
$652$	−11.3798	−0.445667
$653$	−14.7377	−0.576731	−0.288366	−	0.957520i	$-0.593112\pi$
$653$	−14.7377	−0.576731	−0.288366	+	0.957520i	$0.593112\pi$
$654$	0	0
$655$	5.18285	0.202511
$656$	2.84201	0.110962
$657$	0	0
$658$	−17.7368	−0.691451
$659$	−1.06806	−0.0416057	−0.0208028	−	0.999784i	$-0.506622\pi$
$659$	−1.06806	−0.0416057	−0.0208028	+	0.999784i	$0.506622\pi$
$660$	0	0
$661$	4.34646	0.169058	0.0845289	−	0.996421i	$-0.473061\pi$
$661$	4.34646	0.169058	0.0845289	+	0.996421i	$0.473061\pi$
$662$	12.6785	0.492765
$663$	0	0
$664$	16.6586	0.646479
$665$	−18.5557	−0.719557
$666$	0	0
$667$	0	0
$668$	−16.9174	−0.654556
$669$	0	0
$670$	−10.7022	−0.413464
$671$	31.1177	1.20129
$672$	0	0
$673$	9.86439	0.380244	0.190122	−	0.981760i	$-0.439112\pi$
$673$	9.86439	0.380244	0.190122	+	0.981760i	$0.439112\pi$
$674$	15.1062	0.581868
$675$	0	0
$676$	−12.6724	−0.487399
$677$	34.8588	1.33973	0.669866	−	0.742482i	$-0.266351\pi$
$677$	34.8588	1.33973	0.669866	+	0.742482i	$0.266351\pi$
$678$	0	0
$679$	9.01495	0.345962
$680$	1.56189	0.0598956
$681$	0	0
$682$	45.8168	1.75441
$683$	−24.5739	−0.940293	−0.470146	−	0.882588i	$-0.655799\pi$
$683$	−24.5739	−0.940293	−0.470146	+	0.882588i	$0.655799\pi$
$684$	0	0
$685$	−2.20569	−0.0842749
$686$	−17.0361	−0.650442
$687$	0	0
$688$	−5.30077	−0.202090
$689$	−0.241482	−0.00919973
$690$	0	0
$691$	−47.0341	−1.78926	−0.894631	−	0.446806i	$-0.852561\pi$
$691$	−47.0341	−1.78926	−0.894631	+	0.446806i	$0.852561\pi$
$692$	−15.4617	−0.587766
$693$	0	0
$694$	32.3996	1.22987
$695$	9.42788	0.357620
$696$	0	0
$697$	−5.18053	−0.196227
$698$	12.8810	0.487553
$699$	0	0
$700$	12.0374	0.454972
$701$	−43.9728	−1.66083	−0.830414	−	0.557146i	$-0.811896\pi$
$701$	−43.9728	−1.66083	−0.830414	+	0.557146i	$0.811896\pi$
$702$	0	0
$703$	3.18740	0.120215
$704$	−5.63961	−0.212551
$705$	0	0
$706$	−16.2767	−0.612583
$707$	−42.1080	−1.58363
$708$	0	0
$709$	−44.6012	−1.67503	−0.837517	−	0.546411i	$-0.815994\pi$
$709$	−44.6012	−1.67503	−0.837517	+	0.546411i	$0.815994\pi$
$710$	−3.88743	−0.145893
$711$	0	0
$712$	−12.0581	−0.451897
$713$	0	0
$714$	0	0
$715$	−2.76590	−0.103439
$716$	−22.4108	−0.837532
$717$	0	0
$718$	30.3447	1.13245
$719$	25.1192	0.936789	0.468395	−	0.883519i	$-0.344833\pi$
$719$	25.1192	0.936789	0.468395	+	0.883519i	$0.344833\pi$
$720$	0	0
$721$	5.14602	0.191648
$722$	−39.8962	−1.48478
$723$	0	0
$724$	18.0215	0.669764
$725$	−21.2159	−0.787940
$726$	0	0
$727$	−32.6652	−1.21149	−0.605743	−	0.795660i	$-0.707124\pi$
$727$	−32.6652	−1.21149	−0.605743	+	0.795660i	$0.707124\pi$
$728$	1.61517	0.0598621
$729$	0	0
$730$	2.25840	0.0835871
$731$	9.66245	0.357379
$732$	0	0
$733$	36.9762	1.36575	0.682874	−	0.730536i	$-0.260730\pi$
$733$	36.9762	1.36575	0.682874	+	0.730536i	$0.260730\pi$
$734$	3.34854	0.123597
$735$	0	0
$736$	0	0
$737$	−70.4405	−2.59471
$738$	0	0
$739$	2.26037	0.0831490	0.0415745	−	0.999135i	$-0.486763\pi$
$739$	2.26037	0.0831490	0.0415745	+	0.999135i	$0.486763\pi$
$740$	0.355872	0.0130821
$741$	0	0
$742$	−1.19050	−0.0437047
$743$	−23.6574	−0.867907	−0.433953	−	0.900935i	$-0.642882\pi$
$743$	−23.6574	−0.867907	−0.433953	+	0.900935i	$0.642882\pi$
$744$	0	0
$745$	−7.55584	−0.276825
$746$	3.28590	0.120305
$747$	0	0
$748$	10.2801	0.375878
$749$	43.3776	1.58498
$750$	0	0
$751$	18.9668	0.692110	0.346055	−	0.938214i	$-0.387521\pi$
$751$	18.9668	0.692110	0.346055	+	0.938214i	$0.387521\pi$
$752$	−6.28555	−0.229210
$753$	0	0
$754$	−2.84673	−0.103672
$755$	−4.84446	−0.176308
$756$	0	0
$757$	−8.87010	−0.322389	−0.161195	−	0.986923i	$-0.551535\pi$
$757$	−8.87010	−0.322389	−0.161195	+	0.986923i	$0.551535\pi$
$758$	8.01850	0.291245
$759$	0	0
$760$	−6.57574	−0.238527
$761$	3.22026	0.116734	0.0583671	−	0.998295i	$-0.481411\pi$
$761$	3.22026	0.116734	0.0583671	+	0.998295i	$0.481411\pi$
$762$	0	0
$763$	−43.9958	−1.59276
$764$	12.8133	0.463570
$765$	0	0
$766$	27.8632	1.00674
$767$	3.95295	0.142733
$768$	0	0
$769$	8.47283	0.305538	0.152769	−	0.988262i	$-0.451181\pi$
$769$	8.47283	0.305538	0.152769	+	0.988262i	$0.451181\pi$
$770$	−13.6358	−0.491401
$771$	0	0
$772$	22.9800	0.827068
$773$	6.09325	0.219159	0.109579	−	0.993978i	$-0.465050\pi$
$773$	6.09325	0.219159	0.109579	+	0.993978i	$0.465050\pi$
$774$	0	0
$775$	−34.6560	−1.24488
$776$	3.19471	0.114683
$777$	0	0
$778$	5.11899	0.183525
$779$	21.8107	0.781449
$780$	0	0
$781$	−25.5865	−0.915557
$782$	0	0
$783$	0	0
$784$	0.962748	0.0343838
$785$	−17.5132	−0.625075
$786$	0	0
$787$	−20.4654	−0.729514	−0.364757	−	0.931103i	$-0.618848\pi$
$787$	−20.4654	−0.729514	−0.364757	+	0.931103i	$0.618848\pi$
$788$	22.5403	0.802965
$789$	0	0
$790$	−3.94148	−0.140231
$791$	35.8161	1.27347
$792$	0	0
$793$	−3.15824	−0.112152
$794$	6.71598	0.238341
$795$	0	0
$796$	−11.8988	−0.421743
$797$	−18.1217	−0.641905	−0.320953	−	0.947095i	$-0.604003\pi$
$797$	−18.1217	−0.641905	−0.320953	+	0.947095i	$0.604003\pi$
$798$	0	0
$799$	11.4575	0.405339
$800$	4.26582	0.150820
$801$	0	0
$802$	24.8487	0.877439
$803$	14.8644	0.524554
$804$	0	0
$805$	0	0
$806$	−4.65009	−0.163793
$807$	0	0
$808$	−14.9222	−0.524961
$809$	37.1140	1.30486	0.652429	−	0.757849i	$-0.273750\pi$
$809$	37.1140	1.30486	0.652429	+	0.757849i	$0.273750\pi$
$810$	0	0
$811$	9.24026	0.324470	0.162235	−	0.986752i	$-0.448130\pi$
$811$	9.24026	0.324470	0.162235	+	0.986752i	$0.448130\pi$
$812$	−14.0343	−0.492508
$813$	0	0
$814$	2.34229	0.0820974
$815$	−9.75069	−0.341552
$816$	0	0
$817$	−40.6801	−1.42322
$818$	−26.0081	−0.909351
$819$	0	0
$820$	2.43516	0.0850394
$821$	20.6232	0.719756	0.359878	−	0.932999i	$-0.382818\pi$
$821$	20.6232	0.719756	0.359878	+	0.932999i	$0.382818\pi$
$822$	0	0
$823$	−14.7434	−0.513923	−0.256961	−	0.966422i	$-0.582721\pi$
$823$	−14.7434	−0.513923	−0.256961	+	0.966422i	$0.582721\pi$
$824$	1.82365	0.0635297
$825$	0	0
$826$	19.4880	0.678074
$827$	−7.95122	−0.276491	−0.138245	−	0.990398i	$-0.544146\pi$
$827$	−7.95122	−0.276491	−0.138245	+	0.990398i	$0.544146\pi$
$828$	0	0
$829$	−21.0393	−0.730725	−0.365363	−	0.930865i	$-0.619055\pi$
$829$	−21.0393	−0.730725	−0.365363	+	0.930865i	$0.619055\pi$
$830$	14.2738	0.495451
$831$	0	0
$832$	0.572382	0.0198438
$833$	−1.75493	−0.0608049
$834$	0	0
$835$	−14.4956	−0.501641
$836$	−43.2805	−1.49689
$837$	0	0
$838$	6.95184	0.240147
$839$	7.87061	0.271724	0.135862	−	0.990728i	$-0.456620\pi$
$839$	7.87061	0.271724	0.135862	+	0.990728i	$0.456620\pi$
$840$	0	0
$841$	−4.26459	−0.147055
$842$	3.97308	0.136921
$843$	0	0
$844$	−16.7256	−0.575718
$845$	−10.8582	−0.373535
$846$	0	0
$847$	−58.7087	−2.01726
$848$	−0.421889	−0.0144877
$849$	0	0
$850$	−7.77591	−0.266711
$851$	0	0
$852$	0	0
$853$	36.9189	1.26408	0.632040	−	0.774936i	$-0.282218\pi$
$853$	36.9189	1.26408	0.632040	+	0.774936i	$0.282218\pi$
$854$	−15.5701	−0.532797
$855$	0	0
$856$	15.3721	0.525409
$857$	−41.3713	−1.41322	−0.706608	−	0.707605i	$-0.749776\pi$
$857$	−41.3713	−1.41322	−0.706608	+	0.707605i	$0.749776\pi$
$858$	0	0
$859$	−48.2101	−1.64491	−0.822453	−	0.568833i	$-0.807395\pi$
$859$	−48.2101	−1.64491	−0.822453	+	0.568833i	$0.807395\pi$
$860$	−4.54192	−0.154878
$861$	0	0
$862$	−24.9334	−0.849235
$863$	15.9016	0.541296	0.270648	−	0.962678i	$-0.412762\pi$
$863$	15.9016	0.541296	0.270648	+	0.962678i	$0.412762\pi$
$864$	0	0
$865$	−13.2483	−0.450454
$866$	−34.2887	−1.16518
$867$	0	0
$868$	−22.9249	−0.778121
$869$	−25.9422	−0.880028
$870$	0	0
$871$	7.14924	0.242243
$872$	−15.5912	−0.527985
$873$	0	0
$874$	0	0
$875$	22.4035	0.757378
$876$	0	0
$877$	−28.0234	−0.946283	−0.473142	−	0.880986i	$-0.656880\pi$
$877$	−28.0234	−0.946283	−0.473142	+	0.880986i	$0.656880\pi$
$878$	29.8744	1.00821
$879$	0	0
$880$	−4.83226	−0.162895
$881$	−13.5794	−0.457500	−0.228750	−	0.973485i	$-0.573464\pi$
$881$	−13.5794	−0.457500	−0.228750	+	0.973485i	$0.573464\pi$
$882$	0	0
$883$	19.1253	0.643617	0.321808	−	0.946805i	$-0.395709\pi$
$883$	19.1253	0.643617	0.321808	+	0.946805i	$0.395709\pi$
$884$	−1.04336	−0.0350920
$885$	0	0
$886$	18.0904	0.607758
$887$	2.37914	0.0798839	0.0399419	−	0.999202i	$-0.487283\pi$
$887$	2.37914	0.0798839	0.0399419	+	0.999202i	$0.487283\pi$
$888$	0	0
$889$	−25.4717	−0.854293
$890$	−10.3319	−0.346326
$891$	0	0
$892$	−2.37503	−0.0795219
$893$	−48.2377	−1.61421
$894$	0	0
$895$	−19.2026	−0.641871
$896$	2.82183	0.0942709
$897$	0	0
$898$	−31.8510	−1.06288
$899$	40.4050	1.34758
$900$	0	0
$901$	0.769036	0.0256203
$902$	16.0278	0.533668
$903$	0	0
$904$	12.6925	0.422146
$905$	15.4416	0.513296
$906$	0	0
$907$	−5.80904	−0.192886	−0.0964430	−	0.995339i	$-0.530747\pi$
$907$	−5.80904	−0.192886	−0.0964430	+	0.995339i	$0.530747\pi$
$908$	−13.4894	−0.447660
$909$	0	0
$910$	1.38395	0.0458773
$911$	−10.5674	−0.350115	−0.175057	−	0.984558i	$-0.556011\pi$
$911$	−10.5674	−0.350115	−0.175057	+	0.984558i	$0.556011\pi$
$912$	0	0
$913$	93.9479	3.10922
$914$	30.5977	1.01208
$915$	0	0
$916$	2.39089	0.0789973
$917$	−17.0686	−0.563656
$918$	0	0
$919$	−44.4966	−1.46781	−0.733904	−	0.679253i	$-0.762304\pi$
$919$	−44.4966	−1.46781	−0.733904	+	0.679253i	$0.762304\pi$
$920$	0	0
$921$	0	0
$922$	−19.9053	−0.655545
$923$	2.59686	0.0854766
$924$	0	0
$925$	−1.77172	−0.0582538
$926$	−40.2973	−1.32425
$927$	0	0
$928$	−4.97347	−0.163262
$929$	8.96397	0.294098	0.147049	−	0.989129i	$-0.453022\pi$
$929$	8.96397	0.294098	0.147049	+	0.989129i	$0.453022\pi$
$930$	0	0
$931$	7.38850	0.242148
$932$	−13.1021	−0.429175
$933$	0	0
$934$	−31.8816	−1.04320
$935$	8.80843	0.288066
$936$	0	0
$937$	−1.76897	−0.0577898	−0.0288949	−	0.999582i	$-0.509199\pi$
$937$	−1.76897	−0.0577898	−0.0288949	+	0.999582i	$0.509199\pi$
$938$	35.2456	1.15081
$939$	0	0
$940$	−5.38573	−0.175663
$941$	22.3560	0.728786	0.364393	−	0.931245i	$-0.381277\pi$
$941$	22.3560	0.728786	0.364393	+	0.931245i	$0.381277\pi$
$942$	0	0
$943$	0	0
$944$	6.90614	0.224776
$945$	0	0
$946$	−29.8942	−0.971946
$947$	−47.0751	−1.52973	−0.764867	−	0.644188i	$-0.777195\pi$
$947$	−47.0751	−1.52973	−0.764867	+	0.644188i	$0.777195\pi$
$948$	0	0
$949$	−1.50864	−0.0489725
$950$	32.7375	1.06215
$951$	0	0
$952$	−5.14375	−0.166710
$953$	−23.2393	−0.752793	−0.376397	−	0.926459i	$-0.622837\pi$
$953$	−23.2393	−0.752793	−0.376397	+	0.926459i	$0.622837\pi$
$954$	0	0
$955$	10.9790	0.355273
$956$	−8.65744	−0.280002
$957$	0	0
$958$	35.5148	1.14743
$959$	7.26397	0.234566
$960$	0	0
$961$	35.0011	1.12907
$962$	−0.237727	−0.00766463
$963$	0	0
$964$	−1.09132	−0.0351491
$965$	19.6902	0.633851
$966$	0	0
$967$	−55.2673	−1.77728	−0.888638	−	0.458610i	$-0.848347\pi$
$967$	−55.2673	−1.77728	−0.888638	+	0.458610i	$0.848347\pi$
$968$	−20.8052	−0.668703
$969$	0	0
$970$	2.73737	0.0878915
$971$	−16.6667	−0.534861	−0.267431	−	0.963577i	$-0.586175\pi$
$971$	−16.6667	−0.534861	−0.267431	+	0.963577i	$0.586175\pi$
$972$	0	0
$973$	−31.0488	−0.995378
$974$	37.3370	1.19636
$975$	0	0
$976$	−5.51771	−0.176618
$977$	9.70281	0.310420	0.155210	−	0.987881i	$-0.450395\pi$
$977$	9.70281	0.310420	0.155210	+	0.987881i	$0.450395\pi$
$978$	0	0
$979$	−68.0030	−2.17339
$980$	0.824924	0.0263512
$981$	0	0
$982$	−17.3451	−0.553504
$983$	−35.5109	−1.13262	−0.566311	−	0.824191i	$-0.691630\pi$
$983$	−35.5109	−1.13262	−0.566311	+	0.824191i	$0.691630\pi$
$984$	0	0
$985$	19.3135	0.615380
$986$	9.06584	0.288715
$987$	0	0
$988$	4.39268	0.139750
$989$	0	0
$990$	0	0
$991$	29.2889	0.930391	0.465195	−	0.885208i	$-0.345984\pi$
$991$	29.2889	0.930391	0.465195	+	0.885208i	$0.345984\pi$
$992$	−8.12411	−0.257941
$993$	0	0
$994$	12.8025	0.406069
$995$	−10.1954	−0.323217
$996$	0	0
$997$	20.7364	0.656729	0.328365	−	0.944551i	$-0.393503\pi$
$997$	20.7364	0.656729	0.328365	+	0.944551i	$0.393503\pi$
$998$	14.7162	0.465834
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9522.2.a.cg.1.8		10
3.2	odd	2		9522.2.a.cj.1.3		10
23.4	even	11		414.2.i.h.361.2	yes	20
23.6	even	11		414.2.i.h.289.2	yes	20
23.22	odd	2		9522.2.a.ch.1.3		10
69.29	odd	22		414.2.i.g.289.1	✓	20
69.50	odd	22		414.2.i.g.361.1	yes	20
69.68	even	2		9522.2.a.ci.1.8		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
414.2.i.g.289.1	✓	20	69.29	odd	22
414.2.i.g.361.1	yes	20	69.50	odd	22
414.2.i.h.289.2	yes	20	23.6	even	11
414.2.i.h.361.2	yes	20	23.4	even	11
9522.2.a.cg.1.8		10	1.1	even	1	trivial
9522.2.a.ch.1.3		10	23.22	odd	2
9522.2.a.ci.1.8		10	69.68	even	2
9522.2.a.cj.1.3		10	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +0.856843 q^{5} -2.82183 q^{7} -1.00000 q^{8} -0.856843 q^{10} -5.63961 q^{11} +0.572382 q^{13} +2.82183 q^{14} +1.00000 q^{16} -1.82284 q^{17} +7.67439 q^{19} +0.856843 q^{20} +5.63961 q^{22} -4.26582 q^{25} -0.572382 q^{26} -2.82183 q^{28} +4.97347 q^{29} +8.12411 q^{31} -1.00000 q^{32} +1.82284 q^{34} -2.41787 q^{35} +0.415329 q^{37} -7.67439 q^{38} -0.856843 q^{40} +2.84201 q^{41} -5.30077 q^{43} -5.63961 q^{44} -6.28555 q^{47} +0.962748 q^{49} +4.26582 q^{50} +0.572382 q^{52} -0.421889 q^{53} -4.83226 q^{55} +2.82183 q^{56} -4.97347 q^{58} +6.90614 q^{59} -5.51771 q^{61} -8.12411 q^{62} +1.00000 q^{64} +0.490442 q^{65} +12.4903 q^{67} -1.82284 q^{68} +2.41787 q^{70} +4.53693 q^{71} -2.63572 q^{73} -0.415329 q^{74} +7.67439 q^{76} +15.9140 q^{77} +4.60000 q^{79} +0.856843 q^{80} -2.84201 q^{82} -16.6586 q^{83} -1.56189 q^{85} +5.30077 q^{86} +5.63961 q^{88} +12.0581 q^{89} -1.61517 q^{91} +6.28555 q^{94} +6.57574 q^{95} -3.19471 q^{97} -0.962748 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 10 q^{2} + 10 q^{4} - 10 q^{5} - 2 q^{7} - 10 q^{8} + 10 q^{10} - 12 q^{11} + 2 q^{14} + 10 q^{16} - 24 q^{17} + 8 q^{19} - 10 q^{20} + 12 q^{22} + 8 q^{25} - 2 q^{28} - 4 q^{29} + 18 q^{31} - 10 q^{32}+ \cdots - 36 q^{98}+O(q^{100}) \) 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

L-functions

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