LMFDB - Embedded newform 9522.2.a.ca.1.5

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 = [5,5,0,5,11,0,1,5,0,11,11,0,10,1,0,5,11,0,1,11,0,11,0,0,30,10, 0,1,-3] 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:	$5$
Coefficient field:	$\Q(\zeta_{22})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 4x^{3} + 3x^{2} + 3x - 1 $ x^5 - x^4 - 4x^3 + 3x^2 + 3*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 138)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-0.830830$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +4.42518 q^{5} -3.31686 q^{7} +1.00000 q^{8} +4.42518 q^{10} +2.02085 q^{11} +3.02509 q^{13} -3.31686 q^{14} +1.00000 q^{16} -3.00752 q^{17} +7.15484 q^{19} +4.42518 q^{20} +2.02085 q^{22} +14.5822 q^{25} +3.02509 q^{26} -3.31686 q^{28} -1.52483 q^{29} -5.78205 q^{31} +1.00000 q^{32} -3.00752 q^{34} -14.6777 q^{35} +1.84815 q^{37} +7.15484 q^{38} +4.42518 q^{40} +1.06389 q^{41} +1.88965 q^{43} +2.02085 q^{44} +0.748608 q^{47} +4.00158 q^{49} +14.5822 q^{50} +3.02509 q^{52} +12.7722 q^{53} +8.94261 q^{55} -3.31686 q^{56} -1.52483 q^{58} -1.79797 q^{59} -3.90918 q^{61} -5.78205 q^{62} +1.00000 q^{64} +13.3866 q^{65} +3.70249 q^{67} -3.00752 q^{68} -14.6777 q^{70} +4.00069 q^{71} -5.40077 q^{73} +1.84815 q^{74} +7.15484 q^{76} -6.70287 q^{77} -7.86224 q^{79} +4.42518 q^{80} +1.06389 q^{82} -4.03434 q^{83} -13.3088 q^{85} +1.88965 q^{86} +2.02085 q^{88} -5.59734 q^{89} -10.0338 q^{91} +0.748608 q^{94} +31.6614 q^{95} -13.2254 q^{97} +4.00158 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 5 q^{2} + 5 q^{4} + 11 q^{5} + q^{7} + 5 q^{8} + 11 q^{10} + 11 q^{11} + 10 q^{13} + q^{14} + 5 q^{16} + 11 q^{17} + q^{19} + 11 q^{20} + 11 q^{22} + 30 q^{25} + 10 q^{26} + q^{28} - 3 q^{29}+ \cdots - 4 q^{98}+O(q^{100}) $ 5 * q + 5 * q^2 + 5 * q^4 + 11 * q^5 + q^7 + 5 * q^8 + 11 * q^10 + 11 * q^11 + 10 * q^13 + q^14 + 5 * q^16 + 11 * q^17 + q^19 + 11 * q^20 + 11 * q^22 + 30 * q^25 + 10 * q^26 + q^28 - 3 * q^29 + 2 * q^31 + 5 * q^32 + 11 * q^34 + 11 * q^35 - 16 * q^37 + q^38 + 11 * q^40 + 8 * q^41 + 12 * q^43 + 11 * q^44 - 4 * q^49 + 30 * q^50 + 10 * q^52 + 27 * q^53 + q^56 - 3 * q^58 + 6 * q^59 - 5 * q^61 + 2 * q^62 + 5 * q^64 + 22 * q^65 + 7 * q^67 + 11 * q^68 + 11 * q^70 + 11 * q^71 - 12 * q^73 - 16 * q^74 + q^76 - 11 * q^77 - 16 * q^79 + 11 * q^80 + 8 * q^82 + 22 * q^83 - 11 * q^85 + 12 * q^86 + 11 * q^88 + 38 * q^89 - 9 * q^91 - 11 * q^95 - 5 * q^97 - 4 * 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
$5$	4.42518	1.97900	0.989501	−	0.144529i	$-0.0461666\pi$
$5$	4.42518	1.97900	0.989501	+	0.144529i	$0.0461666\pi$
$6$	0	0
$7$	−3.31686	−1.25366	−0.626828	−	0.779157i	$-0.715647\pi$
$7$	−3.31686	−1.25366	−0.626828	+	0.779157i	$0.715647\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	4.42518	1.39937
$11$	2.02085	0.609308	0.304654	−	0.952463i	$-0.401459\pi$
$11$	2.02085	0.609308	0.304654	+	0.952463i	$0.401459\pi$
$12$	0	0
$13$	3.02509	0.839010	0.419505	−	0.907753i	$-0.362204\pi$
$13$	3.02509	0.839010	0.419505	+	0.907753i	$0.362204\pi$
$14$	−3.31686	−0.886469
$15$	0	0
$16$	1.00000	0.250000
$17$	−3.00752	−0.729430	−0.364715	−	0.931119i	$-0.618834\pi$
$17$	−3.00752	−0.729430	−0.364715	+	0.931119i	$0.618834\pi$
$18$	0	0
$19$	7.15484	1.64143	0.820716	−	0.571337i	$-0.193575\pi$
$19$	7.15484	1.64143	0.820716	+	0.571337i	$0.193575\pi$
$20$	4.42518	0.989501
$21$	0	0
$22$	2.02085	0.430846
$23$	0	0
$23$	0	0
$24$	0	0
$25$	14.5822	2.91645
$26$	3.02509	0.593269
$27$	0	0
$28$	−3.31686	−0.626828
$29$	−1.52483	−0.283154	−0.141577	−	0.989927i	$-0.545217\pi$
$29$	−1.52483	−0.283154	−0.141577	+	0.989927i	$0.545217\pi$
$30$	0	0
$31$	−5.78205	−1.03849	−0.519243	−	0.854626i	$-0.673786\pi$
$31$	−5.78205	−1.03849	−0.519243	+	0.854626i	$0.673786\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−3.00752	−0.515785
$35$	−14.6777	−2.48099
$36$	0	0
$37$	1.84815	0.303834	0.151917	−	0.988393i	$-0.451455\pi$
$37$	1.84815	0.303834	0.151917	+	0.988393i	$0.451455\pi$
$38$	7.15484	1.16067
$39$	0	0
$40$	4.42518	0.699683
$41$	1.06389	0.166152	0.0830758	−	0.996543i	$-0.473526\pi$
$41$	1.06389	0.166152	0.0830758	+	0.996543i	$0.473526\pi$
$42$	0	0
$43$	1.88965	0.288169	0.144084	−	0.989565i	$-0.453976\pi$
$43$	1.88965	0.288169	0.144084	+	0.989565i	$0.453976\pi$
$44$	2.02085	0.304654
$45$	0	0
$46$	0	0
$47$	0.748608	0.109196	0.0545978	−	0.998508i	$-0.482612\pi$
$47$	0.748608	0.109196	0.0545978	+	0.998508i	$0.482612\pi$
$48$	0	0
$49$	4.00158	0.571655
$50$	14.5822	2.06224
$51$	0	0
$52$	3.02509	0.419505
$53$	12.7722	1.75440	0.877201	−	0.480123i	$-0.159408\pi$
$53$	12.7722	1.75440	0.877201	+	0.480123i	$0.159408\pi$
$54$	0	0
$55$	8.94261	1.20582
$56$	−3.31686	−0.443235
$57$	0	0
$58$	−1.52483	−0.200220
$59$	−1.79797	−0.234075	−0.117038	−	0.993127i	$-0.537340\pi$
$59$	−1.79797	−0.234075	−0.117038	+	0.993127i	$0.537340\pi$
$60$	0	0
$61$	−3.90918	−0.500519	−0.250260	−	0.968179i	$-0.580516\pi$
$61$	−3.90918	−0.500519	−0.250260	+	0.968179i	$0.580516\pi$
$62$	−5.78205	−0.734321
$63$	0	0
$64$	1.00000	0.125000
$65$	13.3866	1.66040
$66$	0	0
$67$	3.70249	0.452331	0.226166	−	0.974089i	$-0.427381\pi$
$67$	3.70249	0.452331	0.226166	+	0.974089i	$0.427381\pi$
$68$	−3.00752	−0.364715
$69$	0	0
$70$	−14.6777	−1.75432
$71$	4.00069	0.474794	0.237397	−	0.971413i	$-0.423706\pi$
$71$	4.00069	0.474794	0.237397	+	0.971413i	$0.423706\pi$
$72$	0	0
$73$	−5.40077	−0.632113	−0.316056	−	0.948740i	$-0.602359\pi$
$73$	−5.40077	−0.632113	−0.316056	+	0.948740i	$0.602359\pi$
$74$	1.84815	0.214843
$75$	0	0
$76$	7.15484	0.820716
$77$	−6.70287	−0.763863
$78$	0	0
$79$	−7.86224	−0.884571	−0.442286	−	0.896874i	$-0.645832\pi$
$79$	−7.86224	−0.884571	−0.442286	+	0.896874i	$0.645832\pi$
$80$	4.42518	0.494750
$81$	0	0
$82$	1.06389	0.117487
$83$	−4.03434	−0.442826	−0.221413	−	0.975180i	$-0.571067\pi$
$83$	−4.03434	−0.442826	−0.221413	+	0.975180i	$0.571067\pi$
$84$	0	0
$85$	−13.3088	−1.44354
$86$	1.88965	0.203766
$87$	0	0
$88$	2.02085	0.215423
$89$	−5.59734	−0.593317	−0.296658	−	0.954984i	$-0.595872\pi$
$89$	−5.59734	−0.593317	−0.296658	+	0.954984i	$0.595872\pi$
$90$	0	0
$91$	−10.0338	−1.05183
$92$	0	0
$93$	0	0
$94$	0.748608	0.0772130
$95$	31.6614	3.24840
$96$	0	0
$97$	−13.2254	−1.34283	−0.671417	−	0.741080i	$-0.734314\pi$
$97$	−13.2254	−1.34283	−0.671417	+	0.741080i	$0.734314\pi$
$98$	4.00158	0.404221
$99$	0	0
$100$	14.5822	1.45822
$101$	12.7657	1.27023	0.635116	−	0.772416i	$-0.280952\pi$
$101$	12.7657	1.27023	0.635116	+	0.772416i	$0.280952\pi$
$102$	0	0
$103$	1.98374	0.195464	0.0977318	−	0.995213i	$-0.468841\pi$
$103$	1.98374	0.195464	0.0977318	+	0.995213i	$0.468841\pi$
$104$	3.02509	0.296635
$105$	0	0
$106$	12.7722	1.24055
$107$	−6.19856	−0.599238	−0.299619	−	0.954059i	$-0.596859\pi$
$107$	−6.19856	−0.599238	−0.299619	+	0.954059i	$0.596859\pi$
$108$	0	0
$109$	6.05725	0.580179	0.290090	−	0.956999i	$-0.406315\pi$
$109$	6.05725	0.580179	0.290090	+	0.956999i	$0.406315\pi$
$110$	8.94261	0.852645
$111$	0	0
$112$	−3.31686	−0.313414
$113$	−3.36712	−0.316752	−0.158376	−	0.987379i	$-0.550626\pi$
$113$	−3.36712	−0.316752	−0.158376	+	0.987379i	$0.550626\pi$
$114$	0	0
$115$	0	0
$116$	−1.52483	−0.141577
$117$	0	0
$118$	−1.79797	−0.165516
$119$	9.97552	0.914454
$120$	0	0
$121$	−6.91618	−0.628743
$122$	−3.90918	−0.353921
$123$	0	0
$124$	−5.78205	−0.519243
$125$	42.4031	3.79265
$126$	0	0
$127$	10.5994	0.940545	0.470273	−	0.882521i	$-0.344156\pi$
$127$	10.5994	0.940545	0.470273	+	0.882521i	$0.344156\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	13.3866	1.17408
$131$	5.96669	0.521312	0.260656	−	0.965432i	$-0.416061\pi$
$131$	5.96669	0.521312	0.260656	+	0.965432i	$0.416061\pi$
$132$	0	0
$133$	−23.7316	−2.05779
$134$	3.70249	0.319846
$135$	0	0
$136$	−3.00752	−0.257892
$137$	10.3203	0.881721	0.440861	−	0.897576i	$-0.354673\pi$
$137$	10.3203	0.881721	0.440861	+	0.897576i	$0.354673\pi$
$138$	0	0
$139$	−13.4382	−1.13982	−0.569908	−	0.821709i	$-0.693021\pi$
$139$	−13.4382	−1.13982	−0.569908	+	0.821709i	$0.693021\pi$
$140$	−14.6777	−1.24049
$141$	0	0
$142$	4.00069	0.335730
$143$	6.11325	0.511215
$144$	0	0
$145$	−6.74765	−0.560362
$146$	−5.40077	−0.446971
$147$	0	0
$148$	1.84815	0.151917
$149$	20.8450	1.70769	0.853844	−	0.520528i	$-0.174265\pi$
$149$	20.8450	1.70769	0.853844	+	0.520528i	$0.174265\pi$
$150$	0	0
$151$	−8.70135	−0.708106	−0.354053	−	0.935225i	$-0.615197\pi$
$151$	−8.70135	−0.708106	−0.354053	+	0.935225i	$0.615197\pi$
$152$	7.15484	0.580334
$153$	0	0
$154$	−6.70287	−0.540133
$155$	−25.5866	−2.05517
$156$	0	0
$157$	22.6874	1.81065	0.905327	−	0.424716i	$-0.139626\pi$
$157$	22.6874	1.81065	0.905327	+	0.424716i	$0.139626\pi$
$158$	−7.86224	−0.625486
$159$	0	0
$160$	4.42518	0.349841
$161$	0	0
$162$	0	0
$163$	−7.06329	−0.553240	−0.276620	−	0.960979i	$-0.589214\pi$
$163$	−7.06329	−0.553240	−0.276620	+	0.960979i	$0.589214\pi$
$164$	1.06389	0.0830758
$165$	0	0
$166$	−4.03434	−0.313125
$167$	−4.71519	−0.364872	−0.182436	−	0.983218i	$-0.558398\pi$
$167$	−4.71519	−0.364872	−0.182436	+	0.983218i	$0.558398\pi$
$168$	0	0
$169$	−3.84882	−0.296063
$170$	−13.3088	−1.02074
$171$	0	0
$172$	1.88965	0.144084
$173$	13.4700	1.02411	0.512053	−	0.858954i	$-0.328885\pi$
$173$	13.4700	1.02411	0.512053	+	0.858954i	$0.328885\pi$
$174$	0	0
$175$	−48.3673	−3.65622
$176$	2.02085	0.152327
$177$	0	0
$178$	−5.59734	−0.419538
$179$	−18.3384	−1.37068	−0.685338	−	0.728225i	$-0.740346\pi$
$179$	−18.3384	−1.37068	−0.685338	+	0.728225i	$0.740346\pi$
$180$	0	0
$181$	19.7867	1.47074	0.735368	−	0.677668i	$-0.237009\pi$
$181$	19.7867	1.47074	0.735368	+	0.677668i	$0.237009\pi$
$182$	−10.0338	−0.743756
$183$	0	0
$184$	0	0
$185$	8.17840	0.601288
$186$	0	0
$187$	−6.07773	−0.444448
$188$	0.748608	0.0545978
$189$	0	0
$190$	31.6614	2.29696
$191$	−17.1492	−1.24087	−0.620437	−	0.784256i	$-0.713045\pi$
$191$	−17.1492	−1.24087	−0.620437	+	0.784256i	$0.713045\pi$
$192$	0	0
$193$	−12.6863	−0.913178	−0.456589	−	0.889678i	$-0.650929\pi$
$193$	−12.6863	−0.913178	−0.456589	+	0.889678i	$0.650929\pi$
$194$	−13.2254	−0.949527
$195$	0	0
$196$	4.00158	0.285827
$197$	4.96665	0.353859	0.176929	−	0.984224i	$-0.443384\pi$
$197$	4.96665	0.353859	0.176929	+	0.984224i	$0.443384\pi$
$198$	0	0
$199$	4.13451	0.293088	0.146544	−	0.989204i	$-0.453185\pi$
$199$	4.13451	0.293088	0.146544	+	0.989204i	$0.453185\pi$
$200$	14.5822	1.03112
$201$	0	0
$202$	12.7657	0.898190
$203$	5.05765	0.354978
$204$	0	0
$205$	4.70790	0.328814
$206$	1.98374	0.138214
$207$	0	0
$208$	3.02509	0.209752
$209$	14.4588	1.00014
$210$	0	0
$211$	−3.98791	−0.274539	−0.137269	−	0.990534i	$-0.543833\pi$
$211$	−3.98791	−0.274539	−0.137269	+	0.990534i	$0.543833\pi$
$212$	12.7722	0.877201
$213$	0	0
$214$	−6.19856	−0.423725
$215$	8.36204	0.570286
$216$	0	0
$217$	19.1783	1.30191
$218$	6.05725	0.410249
$219$	0	0
$220$	8.94261	0.602911
$221$	−9.09801	−0.611999
$222$	0	0
$223$	−1.62025	−0.108500	−0.0542500	−	0.998527i	$-0.517277\pi$
$223$	−1.62025	−0.108500	−0.0542500	+	0.998527i	$0.517277\pi$
$224$	−3.31686	−0.221617
$225$	0	0
$226$	−3.36712	−0.223977
$227$	22.7548	1.51029	0.755143	−	0.655560i	$-0.227567\pi$
$227$	22.7548	1.51029	0.755143	+	0.655560i	$0.227567\pi$
$228$	0	0
$229$	3.35007	0.221379	0.110689	−	0.993855i	$-0.464694\pi$
$229$	3.35007	0.221379	0.110689	+	0.993855i	$0.464694\pi$
$230$	0	0
$231$	0	0
$232$	−1.52483	−0.100110
$233$	−19.5046	−1.27779	−0.638895	−	0.769294i	$-0.720608\pi$
$233$	−19.5046	−1.27779	−0.638895	+	0.769294i	$0.720608\pi$
$234$	0	0
$235$	3.31273	0.216098
$236$	−1.79797	−0.117038
$237$	0	0
$238$	9.97552	0.646617
$239$	−15.0600	−0.974153	−0.487076	−	0.873359i	$-0.661937\pi$
$239$	−15.0600	−0.974153	−0.487076	+	0.873359i	$0.661937\pi$
$240$	0	0
$241$	19.7786	1.27405	0.637024	−	0.770844i	$-0.280165\pi$
$241$	19.7786	1.27405	0.637024	+	0.770844i	$0.280165\pi$
$242$	−6.91618	−0.444589
$243$	0	0
$244$	−3.90918	−0.250260
$245$	17.7077	1.13131
$246$	0	0
$247$	21.6440	1.37718
$248$	−5.78205	−0.367161
$249$	0	0
$250$	42.4031	2.68181
$251$	28.0277	1.76909	0.884547	−	0.466451i	$-0.154468\pi$
$251$	28.0277	1.76909	0.884547	+	0.466451i	$0.154468\pi$
$252$	0	0
$253$	0	0
$254$	10.5994	0.665066
$255$	0	0
$256$	1.00000	0.0625000
$257$	−5.57881	−0.347996	−0.173998	−	0.984746i	$-0.555669\pi$
$257$	−5.57881	−0.347996	−0.173998	+	0.984746i	$0.555669\pi$
$258$	0	0
$259$	−6.13006	−0.380904
$260$	13.3866	0.830200
$261$	0	0
$262$	5.96669	0.368623
$263$	−12.4571	−0.768136	−0.384068	−	0.923305i	$-0.625477\pi$
$263$	−12.4571	−0.768136	−0.384068	+	0.923305i	$0.625477\pi$
$264$	0	0
$265$	56.5195	3.47197
$266$	−23.7316	−1.45508
$267$	0	0
$268$	3.70249	0.226166
$269$	−13.2204	−0.806064	−0.403032	−	0.915186i	$-0.632044\pi$
$269$	−13.2204	−0.806064	−0.403032	+	0.915186i	$0.632044\pi$
$270$	0	0
$271$	−25.1967	−1.53059	−0.765294	−	0.643681i	$-0.777406\pi$
$271$	−25.1967	−1.53059	−0.765294	+	0.643681i	$0.777406\pi$
$272$	−3.00752	−0.182357
$273$	0	0
$274$	10.3203	0.623471
$275$	29.4685	1.77701
$276$	0	0
$277$	23.9555	1.43934	0.719672	−	0.694314i	$-0.244292\pi$
$277$	23.9555	1.43934	0.719672	+	0.694314i	$0.244292\pi$
$278$	−13.4382	−0.805971
$279$	0	0
$280$	−14.6777	−0.877162
$281$	23.3815	1.39482	0.697412	−	0.716670i	$-0.254335\pi$
$281$	23.3815	1.39482	0.697412	+	0.716670i	$0.254335\pi$
$282$	0	0
$283$	15.2174	0.904583	0.452292	−	0.891870i	$-0.350607\pi$
$283$	15.2174	0.904583	0.452292	+	0.891870i	$0.350607\pi$
$284$	4.00069	0.237397
$285$	0	0
$286$	6.11325	0.361484
$287$	−3.52877	−0.208297
$288$	0	0
$289$	−7.95485	−0.467932
$290$	−6.74765	−0.396236
$291$	0	0
$292$	−5.40077	−0.316056
$293$	5.62259	0.328475	0.164237	−	0.986421i	$-0.447484\pi$
$293$	5.62259	0.328475	0.164237	+	0.986421i	$0.447484\pi$
$294$	0	0
$295$	−7.95633	−0.463235
$296$	1.84815	0.107422
$297$	0	0
$298$	20.8450	1.20752
$299$	0	0
$300$	0	0
$301$	−6.26771	−0.361265
$302$	−8.70135	−0.500707
$303$	0	0
$304$	7.15484	0.410358
$305$	−17.2988	−0.990529
$306$	0	0
$307$	2.53280	0.144554	0.0722772	−	0.997385i	$-0.476973\pi$
$307$	2.53280	0.144554	0.0722772	+	0.997385i	$0.476973\pi$
$308$	−6.70287	−0.381932
$309$	0	0
$310$	−25.5866	−1.45322
$311$	12.3489	0.700241	0.350120	−	0.936705i	$-0.386141\pi$
$311$	12.3489	0.700241	0.350120	+	0.936705i	$0.386141\pi$
$312$	0	0
$313$	8.90037	0.503078	0.251539	−	0.967847i	$-0.419063\pi$
$313$	8.90037	0.503078	0.251539	+	0.967847i	$0.419063\pi$
$314$	22.6874	1.28033
$315$	0	0
$316$	−7.86224	−0.442286
$317$	−7.04733	−0.395818	−0.197909	−	0.980220i	$-0.563415\pi$
$317$	−7.04733	−0.395818	−0.197909	+	0.980220i	$0.563415\pi$
$318$	0	0
$319$	−3.08145	−0.172528
$320$	4.42518	0.247375
$321$	0	0
$322$	0	0
$323$	−21.5183	−1.19731
$324$	0	0
$325$	44.1126	2.44693
$326$	−7.06329	−0.391200
$327$	0	0
$328$	1.06389	0.0587434
$329$	−2.48303	−0.136894
$330$	0	0
$331$	33.7157	1.85318	0.926592	−	0.376068i	$-0.122724\pi$
$331$	33.7157	1.85318	0.926592	+	0.376068i	$0.122724\pi$
$332$	−4.03434	−0.221413
$333$	0	0
$334$	−4.71519	−0.258004
$335$	16.3842	0.895164
$336$	0	0
$337$	−22.3614	−1.21810	−0.609052	−	0.793130i	$-0.708450\pi$
$337$	−22.3614	−1.21810	−0.609052	+	0.793130i	$0.708450\pi$
$338$	−3.84882	−0.209348
$339$	0	0
$340$	−13.3088	−0.721771
$341$	−11.6846	−0.632759
$342$	0	0
$343$	9.94534	0.536998
$344$	1.88965	0.101883
$345$	0	0
$346$	13.4700	0.724152
$347$	21.1909	1.13759	0.568793	−	0.822481i	$-0.307411\pi$
$347$	21.1909	1.13759	0.568793	+	0.822481i	$0.307411\pi$
$348$	0	0
$349$	−33.2345	−1.77900	−0.889500	−	0.456936i	$-0.848947\pi$
$349$	−33.2345	−1.77900	−0.889500	+	0.456936i	$0.848947\pi$
$350$	−48.3673	−2.58534
$351$	0	0
$352$	2.02085	0.107712
$353$	−6.43611	−0.342559	−0.171280	−	0.985222i	$-0.554790\pi$
$353$	−6.43611	−0.342559	−0.171280	+	0.985222i	$0.554790\pi$
$354$	0	0
$355$	17.7038	0.939618
$356$	−5.59734	−0.296658
$357$	0	0
$358$	−18.3384	−0.969215
$359$	−5.24896	−0.277029	−0.138515	−	0.990360i	$-0.544233\pi$
$359$	−5.24896	−0.277029	−0.138515	+	0.990360i	$0.544233\pi$
$360$	0	0
$361$	32.1917	1.69430
$362$	19.7867	1.03997
$363$	0	0
$364$	−10.0338	−0.525915
$365$	−23.8994	−1.25095
$366$	0	0
$367$	−3.12177	−0.162955	−0.0814775	−	0.996675i	$-0.525964\pi$
$367$	−3.12177	−0.162955	−0.0814775	+	0.996675i	$0.525964\pi$
$368$	0	0
$369$	0	0
$370$	8.17840	0.425175
$371$	−42.3638	−2.19942
$372$	0	0
$373$	−30.8738	−1.59858	−0.799292	−	0.600943i	$-0.794792\pi$
$373$	−30.8738	−1.59858	−0.799292	+	0.600943i	$0.794792\pi$
$374$	−6.07773	−0.314272
$375$	0	0
$376$	0.748608	0.0386065
$377$	−4.61275	−0.237569
$378$	0	0
$379$	−35.5941	−1.82835	−0.914173	−	0.405324i	$-0.867159\pi$
$379$	−35.5941	−1.82835	−0.914173	+	0.405324i	$0.867159\pi$
$380$	31.6614	1.62420
$381$	0	0
$382$	−17.1492	−0.877430
$383$	22.4350	1.14638	0.573189	−	0.819424i	$-0.305706\pi$
$383$	22.4350	1.14638	0.573189	+	0.819424i	$0.305706\pi$
$384$	0	0
$385$	−29.6614	−1.51169
$386$	−12.6863	−0.645714
$387$	0	0
$388$	−13.2254	−0.671417
$389$	13.7277	0.696023	0.348012	−	0.937490i	$-0.386857\pi$
$389$	13.7277	0.696023	0.348012	+	0.937490i	$0.386857\pi$
$390$	0	0
$391$	0	0
$392$	4.00158	0.202110
$393$	0	0
$394$	4.96665	0.250216
$395$	−34.7918	−1.75057
$396$	0	0
$397$	−17.5006	−0.878332	−0.439166	−	0.898406i	$-0.644726\pi$
$397$	−17.5006	−0.878332	−0.439166	+	0.898406i	$0.644726\pi$
$398$	4.13451	0.207244
$399$	0	0
$400$	14.5822	0.729111
$401$	11.9524	0.596875	0.298438	−	0.954429i	$-0.403535\pi$
$401$	11.9524	0.596875	0.298438	+	0.954429i	$0.403535\pi$
$402$	0	0
$403$	−17.4912	−0.871300
$404$	12.7657	0.635116
$405$	0	0
$406$	5.05765	0.251007
$407$	3.73483	0.185129
$408$	0	0
$409$	−5.25400	−0.259794	−0.129897	−	0.991528i	$-0.541465\pi$
$409$	−5.25400	−0.259794	−0.129897	+	0.991528i	$0.541465\pi$
$410$	4.70790	0.232507
$411$	0	0
$412$	1.98374	0.0977318
$413$	5.96361	0.293450
$414$	0	0
$415$	−17.8527	−0.876354
$416$	3.02509	0.148317
$417$	0	0
$418$	14.4588	0.707204
$419$	4.77675	0.233359	0.116680	−	0.993170i	$-0.462775\pi$
$419$	4.77675	0.233359	0.116680	+	0.993170i	$0.462775\pi$
$420$	0	0
$421$	−27.6947	−1.34976	−0.674880	−	0.737928i	$-0.735804\pi$
$421$	−27.6947	−1.34976	−0.674880	+	0.737928i	$0.735804\pi$
$422$	−3.98791	−0.194128
$423$	0	0
$424$	12.7722	0.620275
$425$	−43.8563	−2.12734
$426$	0	0
$427$	12.9662	0.627479
$428$	−6.19856	−0.299619
$429$	0	0
$430$	8.36204	0.403253
$431$	1.52326	0.0733729	0.0366864	−	0.999327i	$-0.488320\pi$
$431$	1.52326	0.0733729	0.0366864	+	0.999327i	$0.488320\pi$
$432$	0	0
$433$	−28.3747	−1.36360	−0.681800	−	0.731539i	$-0.738803\pi$
$433$	−28.3747	−1.36360	−0.681800	+	0.731539i	$0.738803\pi$
$434$	19.1783	0.920586
$435$	0	0
$436$	6.05725	0.290090
$437$	0	0
$438$	0	0
$439$	−24.9684	−1.19168	−0.595839	−	0.803104i	$-0.703180\pi$
$439$	−24.9684	−1.19168	−0.595839	+	0.803104i	$0.703180\pi$
$440$	8.94261	0.426322
$441$	0	0
$442$	−9.09801	−0.432748
$443$	18.2215	0.865728	0.432864	−	0.901459i	$-0.357503\pi$
$443$	18.2215	0.865728	0.432864	+	0.901459i	$0.357503\pi$
$444$	0	0
$445$	−24.7692	−1.17417
$446$	−1.62025	−0.0767211
$447$	0	0
$448$	−3.31686	−0.156707
$449$	9.60417	0.453249	0.226624	−	0.973982i	$-0.427231\pi$
$449$	9.60417	0.453249	0.226624	+	0.973982i	$0.427231\pi$
$450$	0	0
$451$	2.14996	0.101237
$452$	−3.36712	−0.158376
$453$	0	0
$454$	22.7548	1.06793
$455$	−44.4015	−2.08157
$456$	0	0
$457$	−18.0768	−0.845597	−0.422798	−	0.906224i	$-0.638952\pi$
$457$	−18.0768	−0.845597	−0.422798	+	0.906224i	$0.638952\pi$
$458$	3.35007	0.156538
$459$	0	0
$460$	0	0
$461$	−4.37457	−0.203744	−0.101872	−	0.994798i	$-0.532483\pi$
$461$	−4.37457	−0.203744	−0.101872	+	0.994798i	$0.532483\pi$
$462$	0	0
$463$	−11.6481	−0.541333	−0.270667	−	0.962673i	$-0.587244\pi$
$463$	−11.6481	−0.541333	−0.270667	+	0.962673i	$0.587244\pi$
$464$	−1.52483	−0.0707885
$465$	0	0
$466$	−19.5046	−0.903534
$467$	−15.4915	−0.716860	−0.358430	−	0.933557i	$-0.616688\pi$
$467$	−15.4915	−0.716860	−0.358430	+	0.933557i	$0.616688\pi$
$468$	0	0
$469$	−12.2807	−0.567068
$470$	3.31273	0.152805
$471$	0	0
$472$	−1.79797	−0.0827581
$473$	3.81869	0.175584
$474$	0	0
$475$	104.333	4.78715
$476$	9.97552	0.457227
$477$	0	0
$478$	−15.0600	−0.688830
$479$	−9.31270	−0.425508	−0.212754	−	0.977106i	$-0.568243\pi$
$479$	−9.31270	−0.425508	−0.212754	+	0.977106i	$0.568243\pi$
$480$	0	0
$481$	5.59082	0.254920
$482$	19.7786	0.900889
$483$	0	0
$484$	−6.91618	−0.314372
$485$	−58.5247	−2.65747
$486$	0	0
$487$	11.0014	0.498523	0.249261	−	0.968436i	$-0.419812\pi$
$487$	11.0014	0.498523	0.249261	+	0.968436i	$0.419812\pi$
$488$	−3.90918	−0.176960
$489$	0	0
$490$	17.7077	0.799954
$491$	31.6556	1.42860	0.714299	−	0.699840i	$-0.246746\pi$
$491$	31.6556	1.42860	0.714299	+	0.699840i	$0.246746\pi$
$492$	0	0
$493$	4.58595	0.206541
$494$	21.6440	0.973811
$495$	0	0
$496$	−5.78205	−0.259622
$497$	−13.2697	−0.595229
$498$	0	0
$499$	−16.3698	−0.732813	−0.366406	−	0.930455i	$-0.619412\pi$
$499$	−16.3698	−0.732813	−0.366406	+	0.930455i	$0.619412\pi$
$500$	42.4031	1.89632
$501$	0	0
$502$	28.0277	1.25094
$503$	−19.2404	−0.857886	−0.428943	−	0.903331i	$-0.641114\pi$
$503$	−19.2404	−0.857886	−0.428943	+	0.903331i	$0.641114\pi$
$504$	0	0
$505$	56.4905	2.51379
$506$	0	0
$507$	0	0
$508$	10.5994	0.470273
$509$	12.1389	0.538046	0.269023	−	0.963134i	$-0.413299\pi$
$509$	12.1389	0.538046	0.269023	+	0.963134i	$0.413299\pi$
$510$	0	0
$511$	17.9136	0.792452
$512$	1.00000	0.0441942
$513$	0	0
$514$	−5.57881	−0.246071
$515$	8.77840	0.386823
$516$	0	0
$517$	1.51282	0.0665338
$518$	−6.13006	−0.269339
$519$	0	0
$520$	13.3866	0.587040
$521$	−6.80323	−0.298055	−0.149027	−	0.988833i	$-0.547614\pi$
$521$	−6.80323	−0.298055	−0.149027	+	0.988833i	$0.547614\pi$
$522$	0	0
$523$	−30.6583	−1.34060	−0.670298	−	0.742092i	$-0.733834\pi$
$523$	−30.6583	−1.34060	−0.670298	+	0.742092i	$0.733834\pi$
$524$	5.96669	0.260656
$525$	0	0
$526$	−12.4571	−0.543155
$527$	17.3896	0.757503
$528$	0	0
$529$	0	0
$530$	56.5195	2.45505
$531$	0	0
$532$	−23.7316	−1.02890
$533$	3.21836	0.139403
$534$	0	0
$535$	−27.4298	−1.18589
$536$	3.70249	0.159923
$537$	0	0
$538$	−13.2204	−0.569973
$539$	8.08659	0.348314
$540$	0	0
$541$	−5.12829	−0.220482	−0.110241	−	0.993905i	$-0.535162\pi$
$541$	−5.12829	−0.220482	−0.110241	+	0.993905i	$0.535162\pi$
$542$	−25.1967	−1.08229
$543$	0	0
$544$	−3.00752	−0.128946
$545$	26.8044	1.14818
$546$	0	0
$547$	−40.2374	−1.72043	−0.860214	−	0.509933i	$-0.829670\pi$
$547$	−40.2374	−1.72043	−0.860214	+	0.509933i	$0.829670\pi$
$548$	10.3203	0.440861
$549$	0	0
$550$	29.4685	1.25654
$551$	−10.9099	−0.464778
$552$	0	0
$553$	26.0780	1.10895
$554$	23.9555	1.01777
$555$	0	0
$556$	−13.4382	−0.569908
$557$	−30.7873	−1.30450	−0.652249	−	0.758004i	$-0.726175\pi$
$557$	−30.7873	−1.30450	−0.652249	+	0.758004i	$0.726175\pi$
$558$	0	0
$559$	5.71636	0.241776
$560$	−14.6777	−0.620247
$561$	0	0
$562$	23.3815	0.986290
$563$	−26.3809	−1.11182	−0.555912	−	0.831241i	$-0.687631\pi$
$563$	−26.3809	−1.11182	−0.555912	+	0.831241i	$0.687631\pi$
$564$	0	0
$565$	−14.9001	−0.626853
$566$	15.2174	0.639637
$567$	0	0
$568$	4.00069	0.167865
$569$	28.9473	1.21353	0.606766	−	0.794880i	$-0.292466\pi$
$569$	28.9473	1.21353	0.606766	+	0.794880i	$0.292466\pi$
$570$	0	0
$571$	−16.0061	−0.669833	−0.334917	−	0.942248i	$-0.608708\pi$
$571$	−16.0061	−0.669833	−0.334917	+	0.942248i	$0.608708\pi$
$572$	6.11325	0.255608
$573$	0	0
$574$	−3.52877	−0.147288
$575$	0	0
$576$	0	0
$577$	18.4542	0.768260	0.384130	−	0.923279i	$-0.374501\pi$
$577$	18.4542	0.768260	0.384130	+	0.923279i	$0.374501\pi$
$578$	−7.95485	−0.330878
$579$	0	0
$580$	−6.74765	−0.280181
$581$	13.3814	0.555152
$582$	0	0
$583$	25.8108	1.06897
$584$	−5.40077	−0.223486
$585$	0	0
$586$	5.62259	0.232267
$587$	15.6808	0.647217	0.323609	−	0.946191i	$-0.395104\pi$
$587$	15.6808	0.647217	0.323609	+	0.946191i	$0.395104\pi$
$588$	0	0
$589$	−41.3696	−1.70461
$590$	−7.95633	−0.327557
$591$	0	0
$592$	1.84815	0.0759585
$593$	−4.78133	−0.196346	−0.0981729	−	0.995169i	$-0.531300\pi$
$593$	−4.78133	−0.196346	−0.0981729	+	0.995169i	$0.531300\pi$
$594$	0	0
$595$	44.1435	1.80971
$596$	20.8450	0.853844
$597$	0	0
$598$	0	0
$599$	−25.8095	−1.05455	−0.527273	−	0.849696i	$-0.676786\pi$
$599$	−25.8095	−1.05455	−0.527273	+	0.849696i	$0.676786\pi$
$600$	0	0
$601$	9.94049	0.405481	0.202741	−	0.979232i	$-0.435015\pi$
$601$	9.94049	0.405481	0.202741	+	0.979232i	$0.435015\pi$
$602$	−6.26771	−0.255453
$603$	0	0
$604$	−8.70135	−0.354053
$605$	−30.6053	−1.24428
$606$	0	0
$607$	−39.8420	−1.61714	−0.808569	−	0.588402i	$-0.799757\pi$
$607$	−39.8420	−1.61714	−0.808569	+	0.588402i	$0.799757\pi$
$608$	7.15484	0.290167
$609$	0	0
$610$	−17.2988	−0.700409
$611$	2.26461	0.0916162
$612$	0	0
$613$	33.9706	1.37206	0.686031	−	0.727573i	$-0.259352\pi$
$613$	33.9706	1.37206	0.686031	+	0.727573i	$0.259352\pi$
$614$	2.53280	0.102215
$615$	0	0
$616$	−6.70287	−0.270066
$617$	16.6267	0.669365	0.334682	−	0.942331i	$-0.391371\pi$
$617$	16.6267	0.669365	0.334682	+	0.942331i	$0.391371\pi$
$618$	0	0
$619$	−17.1181	−0.688033	−0.344017	−	0.938964i	$-0.611788\pi$
$619$	−17.1181	−0.688033	−0.344017	+	0.938964i	$0.611788\pi$
$620$	−25.5866	−1.02758
$621$	0	0
$622$	12.3489	0.495145
$623$	18.5656	0.743815
$624$	0	0
$625$	114.730	4.58921
$626$	8.90037	0.355730
$627$	0	0
$628$	22.6874	0.905327
$629$	−5.55834	−0.221626
$630$	0	0
$631$	−24.2818	−0.966645	−0.483322	−	0.875442i	$-0.660570\pi$
$631$	−24.2818	−0.966645	−0.483322	+	0.875442i	$0.660570\pi$
$632$	−7.86224	−0.312743
$633$	0	0
$634$	−7.04733	−0.279885
$635$	46.9043	1.86134
$636$	0	0
$637$	12.1052	0.479624
$638$	−3.08145	−0.121996
$639$	0	0
$640$	4.42518	0.174921
$641$	48.1244	1.90080	0.950399	−	0.311033i	$-0.100675\pi$
$641$	48.1244	1.90080	0.950399	+	0.311033i	$0.100675\pi$
$642$	0	0
$643$	26.9065	1.06109	0.530545	−	0.847657i	$-0.321987\pi$
$643$	26.9065	1.06109	0.530545	+	0.847657i	$0.321987\pi$
$644$	0	0
$645$	0	0
$646$	−21.5183	−0.846625
$647$	−26.7555	−1.05187	−0.525934	−	0.850525i	$-0.676284\pi$
$647$	−26.7555	−1.05187	−0.525934	+	0.850525i	$0.676284\pi$
$648$	0	0
$649$	−3.63342	−0.142624
$650$	44.1126	1.73024
$651$	0	0
$652$	−7.06329	−0.276620
$653$	46.7047	1.82770	0.913848	−	0.406055i	$-0.133096\pi$
$653$	46.7047	1.82770	0.913848	+	0.406055i	$0.133096\pi$
$654$	0	0
$655$	26.4037	1.03168
$656$	1.06389	0.0415379
$657$	0	0
$658$	−2.48303	−0.0967986
$659$	3.52871	0.137459	0.0687296	−	0.997635i	$-0.478105\pi$
$659$	3.52871	0.137459	0.0687296	+	0.997635i	$0.478105\pi$
$660$	0	0
$661$	18.7448	0.729090	0.364545	−	0.931186i	$-0.381225\pi$
$661$	18.7448	0.729090	0.364545	+	0.931186i	$0.381225\pi$
$662$	33.7157	1.31040
$663$	0	0
$664$	−4.03434	−0.156563
$665$	−105.017	−4.07237
$666$	0	0
$667$	0	0
$668$	−4.71519	−0.182436
$669$	0	0
$670$	16.3842	0.632976
$671$	−7.89986	−0.304971
$672$	0	0
$673$	5.07479	0.195619	0.0978094	−	0.995205i	$-0.468816\pi$
$673$	5.07479	0.195619	0.0978094	+	0.995205i	$0.468816\pi$
$674$	−22.3614	−0.861330
$675$	0	0
$676$	−3.84882	−0.148032
$677$	32.6832	1.25612	0.628059	−	0.778166i	$-0.283850\pi$
$677$	32.6832	1.25612	0.628059	+	0.778166i	$0.283850\pi$
$678$	0	0
$679$	43.8668	1.68345
$680$	−13.3088	−0.510369
$681$	0	0
$682$	−11.6846	−0.447428
$683$	−49.2763	−1.88551	−0.942754	−	0.333490i	$-0.891774\pi$
$683$	−49.2763	−1.88551	−0.942754	+	0.333490i	$0.891774\pi$
$684$	0	0
$685$	45.6691	1.74493
$686$	9.94534	0.379715
$687$	0	0
$688$	1.88965	0.0720422
$689$	38.6372	1.47196
$690$	0	0
$691$	−36.7082	−1.39645	−0.698223	−	0.715880i	$-0.746026\pi$
$691$	−36.7082	−1.39645	−0.698223	+	0.715880i	$0.746026\pi$
$692$	13.4700	0.512053
$693$	0	0
$694$	21.1909	0.804395
$695$	−59.4666	−2.25570
$696$	0	0
$697$	−3.19966	−0.121196
$698$	−33.2345	−1.25794
$699$	0	0
$700$	−48.3673	−1.82811
$701$	−3.19304	−0.120599	−0.0602997	−	0.998180i	$-0.519206\pi$
$701$	−3.19304	−0.120599	−0.0602997	+	0.998180i	$0.519206\pi$
$702$	0	0
$703$	13.2232	0.498723
$704$	2.02085	0.0761635
$705$	0	0
$706$	−6.43611	−0.242226
$707$	−42.3420	−1.59244
$708$	0	0
$709$	6.24454	0.234519	0.117259	−	0.993101i	$-0.462589\pi$
$709$	6.24454	0.234519	0.117259	+	0.993101i	$0.462589\pi$
$710$	17.7038	0.664410
$711$	0	0
$712$	−5.59734	−0.209769
$713$	0	0
$714$	0	0
$715$	27.0522	1.01170
$716$	−18.3384	−0.685338
$717$	0	0
$718$	−5.24896	−0.195889
$719$	−36.8046	−1.37258	−0.686289	−	0.727329i	$-0.740762\pi$
$719$	−36.8046	−1.37258	−0.686289	+	0.727329i	$0.740762\pi$
$720$	0	0
$721$	−6.57979	−0.245044
$722$	32.1917	1.19805
$723$	0	0
$724$	19.7867	0.735368
$725$	−22.2354	−0.825803
$726$	0	0
$727$	37.1423	1.37753	0.688765	−	0.724985i	$-0.258153\pi$
$727$	37.1423	1.37753	0.688765	+	0.724985i	$0.258153\pi$
$728$	−10.0338	−0.371878
$729$	0	0
$730$	−23.8994	−0.884557
$731$	−5.68315	−0.210199
$732$	0	0
$733$	−21.1315	−0.780508	−0.390254	−	0.920707i	$-0.627613\pi$
$733$	−21.1315	−0.780508	−0.390254	+	0.920707i	$0.627613\pi$
$734$	−3.12177	−0.115227
$735$	0	0
$736$	0	0
$737$	7.48217	0.275609
$738$	0	0
$739$	−15.6691	−0.576396	−0.288198	−	0.957571i	$-0.593056\pi$
$739$	−15.6691	−0.576396	−0.288198	+	0.957571i	$0.593056\pi$
$740$	8.17840	0.300644
$741$	0	0
$742$	−42.3638	−1.55522
$743$	−0.814882	−0.0298951	−0.0149476	−	0.999888i	$-0.504758\pi$
$743$	−0.814882	−0.0298951	−0.0149476	+	0.999888i	$0.504758\pi$
$744$	0	0
$745$	92.2429	3.37952
$746$	−30.8738	−1.13037
$747$	0	0
$748$	−6.07773	−0.222224
$749$	20.5598	0.751238
$750$	0	0
$751$	25.2171	0.920184	0.460092	−	0.887871i	$-0.347816\pi$
$751$	25.2171	0.920184	0.460092	+	0.887871i	$0.347816\pi$
$752$	0.748608	0.0272989
$753$	0	0
$754$	−4.61275	−0.167987
$755$	−38.5051	−1.40134
$756$	0	0
$757$	−30.6339	−1.11341	−0.556704	−	0.830711i	$-0.687934\pi$
$757$	−30.6339	−1.11341	−0.556704	+	0.830711i	$0.687934\pi$
$758$	−35.5941	−1.29284
$759$	0	0
$760$	31.6614	1.14848
$761$	16.2342	0.588488	0.294244	−	0.955730i	$-0.404932\pi$
$761$	16.2342	0.588488	0.294244	+	0.955730i	$0.404932\pi$
$762$	0	0
$763$	−20.0911	−0.727345
$764$	−17.1492	−0.620437
$765$	0	0
$766$	22.4350	0.810611
$767$	−5.43901	−0.196391
$768$	0	0
$769$	35.7963	1.29085	0.645423	−	0.763825i	$-0.276681\pi$
$769$	35.7963	1.29085	0.645423	+	0.763825i	$0.276681\pi$
$770$	−29.6614	−1.06892
$771$	0	0
$772$	−12.6863	−0.456589
$773$	−46.3148	−1.66583	−0.832913	−	0.553403i	$-0.813329\pi$
$773$	−46.3148	−1.66583	−0.832913	+	0.553403i	$0.813329\pi$
$774$	0	0
$775$	−84.3152	−3.02869
$776$	−13.2254	−0.474763
$777$	0	0
$778$	13.7277	0.492163
$779$	7.61195	0.272726
$780$	0	0
$781$	8.08477	0.289296
$782$	0	0
$783$	0	0
$784$	4.00158	0.142914
$785$	100.396	3.58328
$786$	0	0
$787$	−0.264656	−0.00943397	−0.00471698	−	0.999989i	$-0.501501\pi$
$787$	−0.264656	−0.00943397	−0.00471698	+	0.999989i	$0.501501\pi$
$788$	4.96665	0.176929
$789$	0	0
$790$	−34.7918	−1.23784
$791$	11.1683	0.397098
$792$	0	0
$793$	−11.8256	−0.419941
$794$	−17.5006	−0.621074
$795$	0	0
$796$	4.13451	0.146544
$797$	−18.3016	−0.648276	−0.324138	−	0.946010i	$-0.605074\pi$
$797$	−18.3016	−0.648276	−0.324138	+	0.946010i	$0.605074\pi$
$798$	0	0
$799$	−2.25145	−0.0796506
$800$	14.5822	0.515560
$801$	0	0
$802$	11.9524	0.422055
$803$	−10.9141	−0.385152
$804$	0	0
$805$	0	0
$806$	−17.4912	−0.616102
$807$	0	0
$808$	12.7657	0.449095
$809$	−9.92873	−0.349075	−0.174538	−	0.984650i	$-0.555843\pi$
$809$	−9.92873	−0.349075	−0.174538	+	0.984650i	$0.555843\pi$
$810$	0	0
$811$	−19.6936	−0.691535	−0.345767	−	0.938320i	$-0.612381\pi$
$811$	−19.6936	−0.691535	−0.345767	+	0.938320i	$0.612381\pi$
$812$	5.05765	0.177489
$813$	0	0
$814$	3.73483	0.130906
$815$	−31.2563	−1.09486
$816$	0	0
$817$	13.5201	0.473009
$818$	−5.25400	−0.183702
$819$	0	0
$820$	4.70790	0.164407
$821$	−41.2626	−1.44008	−0.720038	−	0.693935i	$-0.755875\pi$
$821$	−41.2626	−1.44008	−0.720038	+	0.693935i	$0.755875\pi$
$822$	0	0
$823$	29.7378	1.03659	0.518297	−	0.855201i	$-0.326566\pi$
$823$	29.7378	1.03659	0.518297	+	0.855201i	$0.326566\pi$
$824$	1.98374	0.0691068
$825$	0	0
$826$	5.96361	0.207501
$827$	−39.0877	−1.35921	−0.679606	−	0.733577i	$-0.737849\pi$
$827$	−39.0877	−1.35921	−0.679606	+	0.733577i	$0.737849\pi$
$828$	0	0
$829$	−49.3414	−1.71370	−0.856849	−	0.515568i	$-0.827581\pi$
$829$	−49.3414	−1.71370	−0.856849	+	0.515568i	$0.827581\pi$
$830$	−17.8527	−0.619676
$831$	0	0
$832$	3.02509	0.104876
$833$	−12.0348	−0.416982
$834$	0	0
$835$	−20.8656	−0.722083
$836$	14.4588	0.500069
$837$	0	0
$838$	4.77675	0.165010
$839$	0.0380549	0.00131380	0.000656901	−	1.00000i	$-0.499791\pi$
$839$	0.0380549	0.00131380	0.000656901		1.00000i	$0.499791\pi$
$840$	0	0
$841$	−26.6749	−0.919824
$842$	−27.6947	−0.954424
$843$	0	0
$844$	−3.98791	−0.137269
$845$	−17.0317	−0.585909
$846$	0	0
$847$	22.9400	0.788228
$848$	12.7722	0.438601
$849$	0	0
$850$	−43.8563	−1.50426
$851$	0	0
$852$	0	0
$853$	2.35009	0.0804655	0.0402328	−	0.999190i	$-0.487190\pi$
$853$	2.35009	0.0804655	0.0402328	+	0.999190i	$0.487190\pi$
$854$	12.9662	0.443695
$855$	0	0
$856$	−6.19856	−0.211862
$857$	−8.89113	−0.303715	−0.151858	−	0.988402i	$-0.548525\pi$
$857$	−8.89113	−0.303715	−0.151858	+	0.988402i	$0.548525\pi$
$858$	0	0
$859$	26.2380	0.895228	0.447614	−	0.894227i	$-0.352274\pi$
$859$	26.2380	0.895228	0.447614	+	0.894227i	$0.352274\pi$
$860$	8.36204	0.285143
$861$	0	0
$862$	1.52326	0.0518824
$863$	55.3473	1.88404	0.942021	−	0.335553i	$-0.108923\pi$
$863$	55.3473	1.88404	0.942021	+	0.335553i	$0.108923\pi$
$864$	0	0
$865$	59.6072	2.02671
$866$	−28.3747	−0.964211
$867$	0	0
$868$	19.1783	0.650953
$869$	−15.8884	−0.538976
$870$	0	0
$871$	11.2004	0.379510
$872$	6.05725	0.205124
$873$	0	0
$874$	0	0
$875$	−140.645	−4.75468
$876$	0	0
$877$	50.9910	1.72184	0.860922	−	0.508737i	$-0.169887\pi$
$877$	50.9910	1.72184	0.860922	+	0.508737i	$0.169887\pi$
$878$	−24.9684	−0.842643
$879$	0	0
$880$	8.94261	0.301455
$881$	26.4450	0.890953	0.445477	−	0.895294i	$-0.353034\pi$
$881$	26.4450	0.890953	0.445477	+	0.895294i	$0.353034\pi$
$882$	0	0
$883$	20.6260	0.694120	0.347060	−	0.937843i	$-0.387180\pi$
$883$	20.6260	0.694120	0.347060	+	0.937843i	$0.387180\pi$
$884$	−9.09801	−0.305999
$885$	0	0
$886$	18.2215	0.612162
$887$	12.7487	0.428061	0.214030	−	0.976827i	$-0.431341\pi$
$887$	12.7487	0.428061	0.214030	+	0.976827i	$0.431341\pi$
$888$	0	0
$889$	−35.1568	−1.17912
$890$	−24.7692	−0.830266
$891$	0	0
$892$	−1.62025	−0.0542500
$893$	5.35617	0.179237
$894$	0	0
$895$	−81.1508	−2.71257
$896$	−3.31686	−0.110809
$897$	0	0
$898$	9.60417	0.320495
$899$	8.81664	0.294052
$900$	0	0
$901$	−38.4127	−1.27971
$902$	2.14996	0.0715857
$903$	0	0
$904$	−3.36712	−0.111989
$905$	87.5598	2.91059
$906$	0	0
$907$	−15.3266	−0.508912	−0.254456	−	0.967084i	$-0.581896\pi$
$907$	−15.3266	−0.508912	−0.254456	+	0.967084i	$0.581896\pi$
$908$	22.7548	0.755143
$909$	0	0
$910$	−44.4015	−1.47189
$911$	−27.0655	−0.896720	−0.448360	−	0.893853i	$-0.647992\pi$
$911$	−27.0655	−0.896720	−0.448360	+	0.893853i	$0.647992\pi$
$912$	0	0
$913$	−8.15278	−0.269818
$914$	−18.0768	−0.597927
$915$	0	0
$916$	3.35007	0.110689
$917$	−19.7907	−0.653546
$918$	0	0
$919$	−4.93575	−0.162815	−0.0814076	−	0.996681i	$-0.525942\pi$
$919$	−4.93575	−0.162815	−0.0814076	+	0.996681i	$0.525942\pi$
$920$	0	0
$921$	0	0
$922$	−4.37457	−0.144069
$923$	12.1024	0.398357
$924$	0	0
$925$	26.9502	0.886115
$926$	−11.6481	−0.382780
$927$	0	0
$928$	−1.52483	−0.0500550
$929$	−30.2582	−0.992741	−0.496370	−	0.868111i	$-0.665334\pi$
$929$	−30.2582	−0.992741	−0.496370	+	0.868111i	$0.665334\pi$
$930$	0	0
$931$	28.6307	0.938332
$932$	−19.5046	−0.638895
$933$	0	0
$934$	−15.4915	−0.506897
$935$	−26.8951	−0.879562
$936$	0	0
$937$	−12.4306	−0.406091	−0.203046	−	0.979169i	$-0.565084\pi$
$937$	−12.4306	−0.406091	−0.203046	+	0.979169i	$0.565084\pi$
$938$	−12.2807	−0.400978
$939$	0	0
$940$	3.31273	0.108049
$941$	17.2277	0.561606	0.280803	−	0.959765i	$-0.409399\pi$
$941$	17.2277	0.561606	0.280803	+	0.959765i	$0.409399\pi$
$942$	0	0
$943$	0	0
$944$	−1.79797	−0.0585188
$945$	0	0
$946$	3.81869	0.124156
$947$	22.3831	0.727352	0.363676	−	0.931526i	$-0.381522\pi$
$947$	22.3831	0.727352	0.363676	+	0.931526i	$0.381522\pi$
$948$	0	0
$949$	−16.3378	−0.530349
$950$	104.333	3.38502
$951$	0	0
$952$	9.97552	0.323308
$953$	−32.0271	−1.03746	−0.518731	−	0.854938i	$-0.673595\pi$
$953$	−32.0271	−1.03746	−0.518731	+	0.854938i	$0.673595\pi$
$954$	0	0
$955$	−75.8884	−2.45569
$956$	−15.0600	−0.487076
$957$	0	0
$958$	−9.31270	−0.300880
$959$	−34.2310	−1.10538
$960$	0	0
$961$	2.43210	0.0784548
$962$	5.59082	0.180255
$963$	0	0
$964$	19.7786	0.637024
$965$	−56.1390	−1.80718
$966$	0	0
$967$	34.6187	1.11326	0.556631	−	0.830760i	$-0.312094\pi$
$967$	34.6187	1.11326	0.556631	+	0.830760i	$0.312094\pi$
$968$	−6.91618	−0.222294
$969$	0	0
$970$	−58.5247	−1.87911
$971$	9.41932	0.302280	0.151140	−	0.988512i	$-0.451706\pi$
$971$	9.41932	0.302280	0.151140	+	0.988512i	$0.451706\pi$
$972$	0	0
$973$	44.5728	1.42894
$974$	11.0014	0.352509
$975$	0	0
$976$	−3.90918	−0.125130
$977$	30.0128	0.960196	0.480098	−	0.877215i	$-0.340601\pi$
$977$	30.0128	0.960196	0.480098	+	0.877215i	$0.340601\pi$
$978$	0	0
$979$	−11.3114	−0.361513
$980$	17.7077	0.565653
$981$	0	0
$982$	31.6556	1.01017
$983$	31.1346	0.993041	0.496521	−	0.868025i	$-0.334611\pi$
$983$	31.1346	0.993041	0.496521	+	0.868025i	$0.334611\pi$
$984$	0	0
$985$	21.9783	0.700287
$986$	4.58595	0.146046
$987$	0	0
$988$	21.6440	0.688588
$989$	0	0
$990$	0	0
$991$	−33.6697	−1.06955	−0.534776	−	0.844994i	$-0.679604\pi$
$991$	−33.6697	−1.06955	−0.534776	+	0.844994i	$0.679604\pi$
$992$	−5.78205	−0.183580
$993$	0	0
$994$	−13.2697	−0.420890
$995$	18.2960	0.580021
$996$	0	0
$997$	−28.0726	−0.889069	−0.444534	−	0.895762i	$-0.646631\pi$
$997$	−28.0726	−0.889069	−0.444534	+	0.895762i	$0.646631\pi$
$998$	−16.3698	−0.518177
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9522.2.a.ca.1.5		5
3.2	odd	2		3174.2.a.y.1.1		5
23.5	odd	22		414.2.i.b.163.1		10
23.14	odd	22		414.2.i.b.127.1		10
23.22	odd	2		9522.2.a.bv.1.1		5
69.5	even	22		138.2.e.c.25.1	✓	10
69.14	even	22		138.2.e.c.127.1	yes	10
69.68	even	2		3174.2.a.z.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.2.e.c.25.1	✓	10	69.5	even	22
138.2.e.c.127.1	yes	10	69.14	even	22
414.2.i.b.127.1		10	23.14	odd	22
414.2.i.b.163.1		10	23.5	odd	22
3174.2.a.y.1.1		5	3.2	odd	2
3174.2.a.z.1.5		5	69.68	even	2
9522.2.a.bv.1.1		5	23.22	odd	2
9522.2.a.ca.1.5		5	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +4.42518 q^{5} -3.31686 q^{7} +1.00000 q^{8} +4.42518 q^{10} +2.02085 q^{11} +3.02509 q^{13} -3.31686 q^{14} +1.00000 q^{16} -3.00752 q^{17} +7.15484 q^{19} +4.42518 q^{20} +2.02085 q^{22} +14.5822 q^{25} +3.02509 q^{26} -3.31686 q^{28} -1.52483 q^{29} -5.78205 q^{31} +1.00000 q^{32} -3.00752 q^{34} -14.6777 q^{35} +1.84815 q^{37} +7.15484 q^{38} +4.42518 q^{40} +1.06389 q^{41} +1.88965 q^{43} +2.02085 q^{44} +0.748608 q^{47} +4.00158 q^{49} +14.5822 q^{50} +3.02509 q^{52} +12.7722 q^{53} +8.94261 q^{55} -3.31686 q^{56} -1.52483 q^{58} -1.79797 q^{59} -3.90918 q^{61} -5.78205 q^{62} +1.00000 q^{64} +13.3866 q^{65} +3.70249 q^{67} -3.00752 q^{68} -14.6777 q^{70} +4.00069 q^{71} -5.40077 q^{73} +1.84815 q^{74} +7.15484 q^{76} -6.70287 q^{77} -7.86224 q^{79} +4.42518 q^{80} +1.06389 q^{82} -4.03434 q^{83} -13.3088 q^{85} +1.88965 q^{86} +2.02085 q^{88} -5.59734 q^{89} -10.0338 q^{91} +0.748608 q^{94} +31.6614 q^{95} -13.2254 q^{97} +4.00158 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 5 q^{2} + 5 q^{4} + 11 q^{5} + q^{7} + 5 q^{8} + 11 q^{10} + 11 q^{11} + 10 q^{13} + q^{14} + 5 q^{16} + 11 q^{17} + q^{19} + 11 q^{20} + 11 q^{22} + 30 q^{25} + 10 q^{26} + q^{28} - 3 q^{29}+ \cdots - 4 q^{98}+O(q^{100}) \) 5 * q + 5 * q^2 + 5 * q^4 + 11 * q^5 + q^7 + 5 * q^8 + 11 * q^10 + 11 * q^11 + 10 * q^13 + q^14 + 5 * q^16 + 11 * q^17 + q^19 + 11 * q^20 + 11 * q^22 + 30 * q^25 + 10 * q^26 + q^28 - 3 * q^29 + 2 * q^31 + 5 * q^32 + 11 * q^34 + 11 * q^35 - 16 * q^37 + q^38 + 11 * q^40 + 8 * q^41 + 12 * q^43 + 11 * q^44 - 4 * q^49 + 30 * q^50 + 10 * q^52 + 27 * q^53 + q^56 - 3 * q^58 + 6 * q^59 - 5 * q^61 + 2 * q^62 + 5 * q^64 + 22 * q^65 + 7 * q^67 + 11 * q^68 + 11 * q^70 + 11 * q^71 - 12 * q^73 - 16 * q^74 + q^76 - 11 * q^77 - 16 * q^79 + 11 * q^80 + 8 * q^82 + 22 * q^83 - 11 * q^85 + 12 * q^86 + 11 * q^88 + 38 * q^89 - 9 * q^91 - 11 * q^95 - 5 * q^97 - 4 * q^98

Introduction

L-functions

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