LMFDB - Embedded newform 9522.2.a.ch.1.2

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			1.2
Root			$0.878233$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -1.83556 q^{5} -1.24396 q^{7} -1.00000 q^{8} +1.83556 q^{10} +0.102427 q^{11} -3.87350 q^{13} +1.24396 q^{14} +1.00000 q^{16} +3.06169 q^{17} -4.97769 q^{19} -1.83556 q^{20} -0.102427 q^{22} -1.63073 q^{25} +3.87350 q^{26} -1.24396 q^{28} -6.71914 q^{29} +3.86586 q^{31} -1.00000 q^{32} -3.06169 q^{34} +2.28336 q^{35} +2.55426 q^{37} +4.97769 q^{38} +1.83556 q^{40} -1.65386 q^{41} -3.59805 q^{43} +0.102427 q^{44} +1.92036 q^{47} -5.45256 q^{49} +1.63073 q^{50} -3.87350 q^{52} -4.14290 q^{53} -0.188010 q^{55} +1.24396 q^{56} +6.71914 q^{58} -6.09859 q^{59} +5.74065 q^{61} -3.86586 q^{62} +1.00000 q^{64} +7.11003 q^{65} +1.55357 q^{67} +3.06169 q^{68} -2.28336 q^{70} +8.52403 q^{71} -12.0483 q^{73} -2.55426 q^{74} -4.97769 q^{76} -0.127415 q^{77} -13.0707 q^{79} -1.83556 q^{80} +1.65386 q^{82} -5.20439 q^{83} -5.61991 q^{85} +3.59805 q^{86} -0.102427 q^{88} +7.97976 q^{89} +4.81849 q^{91} -1.92036 q^{94} +9.13684 q^{95} -11.4726 q^{97} +5.45256 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−1.83556	−0.820886	−0.410443	−	0.911886i	$-0.634626\pi$
$5$	−1.83556	−0.820886	−0.410443	+	0.911886i	$0.634626\pi$
$6$	0	0
$7$	−1.24396	−0.470174	−0.235087	−	0.971974i	$-0.575537\pi$
$7$	−1.24396	−0.470174	−0.235087	+	0.971974i	$0.575537\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	1.83556	0.580454
$11$	0.102427	0.0308828	0.0154414	−	0.999881i	$-0.495085\pi$
$11$	0.102427	0.0308828	0.0154414	+	0.999881i	$0.495085\pi$
$12$	0	0
$13$	−3.87350	−1.07432	−0.537158	−	0.843482i	$-0.680502\pi$
$13$	−3.87350	−1.07432	−0.537158	+	0.843482i	$0.680502\pi$
$14$	1.24396	0.332463
$15$	0	0
$16$	1.00000	0.250000
$17$	3.06169	0.742570	0.371285	−	0.928519i	$-0.378917\pi$
$17$	3.06169	0.742570	0.371285	+	0.928519i	$0.378917\pi$
$18$	0	0
$19$	−4.97769	−1.14196	−0.570981	−	0.820964i	$-0.693437\pi$
$19$	−4.97769	−1.14196	−0.570981	+	0.820964i	$0.693437\pi$
$20$	−1.83556	−0.410443
$21$	0	0
$22$	−0.102427	−0.0218375
$23$	0	0
$23$	0	0
$24$	0	0
$25$	−1.63073	−0.326147
$26$	3.87350	0.759656
$27$	0	0
$28$	−1.24396	−0.235087
$29$	−6.71914	−1.24771	−0.623857	−	0.781539i	$-0.714435\pi$
$29$	−6.71914	−1.24771	−0.623857	+	0.781539i	$0.714435\pi$
$30$	0	0
$31$	3.86586	0.694329	0.347164	−	0.937804i	$-0.387145\pi$
$31$	3.86586	0.694329	0.347164	+	0.937804i	$0.387145\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−3.06169	−0.525076
$35$	2.28336	0.385959
$36$	0	0
$37$	2.55426	0.419917	0.209959	−	0.977710i	$-0.432667\pi$
$37$	2.55426	0.419917	0.209959	+	0.977710i	$0.432667\pi$
$38$	4.97769	0.807489
$39$	0	0
$40$	1.83556	0.290227
$41$	−1.65386	−0.258289	−0.129145	−	0.991626i	$-0.541223\pi$
$41$	−1.65386	−0.258289	−0.129145	+	0.991626i	$0.541223\pi$
$42$	0	0
$43$	−3.59805	−0.548697	−0.274349	−	0.961630i	$-0.588462\pi$
$43$	−3.59805	−0.548697	−0.274349	+	0.961630i	$0.588462\pi$
$44$	0.102427	0.0154414
$45$	0	0
$46$	0	0
$47$	1.92036	0.280113	0.140056	−	0.990144i	$-0.455272\pi$
$47$	1.92036	0.280113	0.140056	+	0.990144i	$0.455272\pi$
$48$	0	0
$49$	−5.45256	−0.778937
$50$	1.63073	0.230621
$51$	0	0
$52$	−3.87350	−0.537158
$53$	−4.14290	−0.569070	−0.284535	−	0.958666i	$-0.591839\pi$
$53$	−4.14290	−0.569070	−0.284535	+	0.958666i	$0.591839\pi$
$54$	0	0
$55$	−0.188010	−0.0253513
$56$	1.24396	0.166232
$57$	0	0
$58$	6.71914	0.882267
$59$	−6.09859	−0.793968	−0.396984	−	0.917825i	$-0.629943\pi$
$59$	−6.09859	−0.793968	−0.396984	+	0.917825i	$0.629943\pi$
$60$	0	0
$61$	5.74065	0.735015	0.367508	−	0.930020i	$-0.380211\pi$
$61$	5.74065	0.735015	0.367508	+	0.930020i	$0.380211\pi$
$62$	−3.86586	−0.490964
$63$	0	0
$64$	1.00000	0.125000
$65$	7.11003	0.881890
$66$	0	0
$67$	1.55357	0.189799	0.0948994	−	0.995487i	$-0.469747\pi$
$67$	1.55357	0.189799	0.0948994	+	0.995487i	$0.469747\pi$
$68$	3.06169	0.371285
$69$	0	0
$70$	−2.28336	−0.272914
$71$	8.52403	1.01162	0.505808	−	0.862646i	$-0.331194\pi$
$71$	8.52403	1.01162	0.505808	+	0.862646i	$0.331194\pi$
$72$	0	0
$73$	−12.0483	−1.41015	−0.705076	−	0.709132i	$-0.749087\pi$
$73$	−12.0483	−1.41015	−0.705076	+	0.709132i	$0.749087\pi$
$74$	−2.55426	−0.296926
$75$	0	0
$76$	−4.97769	−0.570981
$77$	−0.127415	−0.0145203
$78$	0	0
$79$	−13.0707	−1.47057	−0.735286	−	0.677757i	$-0.762952\pi$
$79$	−13.0707	−1.47057	−0.735286	+	0.677757i	$0.762952\pi$
$80$	−1.83556	−0.205221
$81$	0	0
$82$	1.65386	0.182638
$83$	−5.20439	−0.571256	−0.285628	−	0.958341i	$-0.592202\pi$
$83$	−5.20439	−0.571256	−0.285628	+	0.958341i	$0.592202\pi$
$84$	0	0
$85$	−5.61991	−0.609565
$86$	3.59805	0.387987
$87$	0	0
$88$	−0.102427	−0.0109187
$89$	7.97976	0.845853	0.422926	−	0.906164i	$-0.361003\pi$
$89$	7.97976	0.845853	0.422926	+	0.906164i	$0.361003\pi$
$90$	0	0
$91$	4.81849	0.505115
$92$	0	0
$93$	0	0
$94$	−1.92036	−0.198070
$95$	9.13684	0.937420
$96$	0	0
$97$	−11.4726	−1.16486	−0.582431	−	0.812880i	$-0.697898\pi$
$97$	−11.4726	−1.16486	−0.582431	+	0.812880i	$0.697898\pi$
$98$	5.45256	0.550791
$99$	0	0
$100$	−1.63073	−0.163073
$101$	8.35791	0.831643	0.415822	−	0.909446i	$-0.363494\pi$
$101$	8.35791	0.831643	0.415822	+	0.909446i	$0.363494\pi$
$102$	0	0
$103$	−12.7017	−1.25154	−0.625770	−	0.780008i	$-0.715215\pi$
$103$	−12.7017	−1.25154	−0.625770	+	0.780008i	$0.715215\pi$
$104$	3.87350	0.379828
$105$	0	0
$106$	4.14290	0.402394
$107$	10.5059	1.01564	0.507822	−	0.861462i	$-0.330451\pi$
$107$	10.5059	1.01564	0.507822	+	0.861462i	$0.330451\pi$
$108$	0	0
$109$	5.49163	0.526003	0.263001	−	0.964795i	$-0.415288\pi$
$109$	5.49163	0.526003	0.263001	+	0.964795i	$0.415288\pi$
$110$	0.188010	0.0179261
$111$	0	0
$112$	−1.24396	−0.117543
$113$	18.8702	1.77516	0.887579	−	0.460655i	$-0.152386\pi$
$113$	18.8702	1.77516	0.887579	+	0.460655i	$0.152386\pi$
$114$	0	0
$115$	0	0
$116$	−6.71914	−0.623857
$117$	0	0
$118$	6.09859	0.561420
$119$	−3.80863	−0.349137
$120$	0	0
$121$	−10.9895	−0.999046
$122$	−5.74065	−0.519734
$123$	0	0
$124$	3.86586	0.347164
$125$	12.1711	1.08861
$126$	0	0
$127$	−1.17839	−0.104565	−0.0522827	−	0.998632i	$-0.516650\pi$
$127$	−1.17839	−0.104565	−0.0522827	+	0.998632i	$0.516650\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−7.11003	−0.623591
$131$	−21.8626	−1.91014	−0.955070	−	0.296380i	$-0.904220\pi$
$131$	−21.8626	−1.91014	−0.955070	+	0.296380i	$0.904220\pi$
$132$	0	0
$133$	6.19207	0.536920
$134$	−1.55357	−0.134208
$135$	0	0
$136$	−3.06169	−0.262538
$137$	17.1557	1.46571	0.732855	−	0.680385i	$-0.238188\pi$
$137$	17.1557	1.46571	0.732855	+	0.680385i	$0.238188\pi$
$138$	0	0
$139$	10.1346	0.859608	0.429804	−	0.902922i	$-0.358583\pi$
$139$	10.1346	0.859608	0.429804	+	0.902922i	$0.358583\pi$
$140$	2.28336	0.192979
$141$	0	0
$142$	−8.52403	−0.715321
$143$	−0.396750	−0.0331779
$144$	0	0
$145$	12.3334	1.02423
$146$	12.0483	0.997128
$147$	0	0
$148$	2.55426	0.209959
$149$	−0.377442	−0.0309212	−0.0154606	−	0.999880i	$-0.504921\pi$
$149$	−0.377442	−0.0309212	−0.0154606	+	0.999880i	$0.504921\pi$
$150$	0	0
$151$	−15.5480	−1.26528	−0.632638	−	0.774448i	$-0.718028\pi$
$151$	−15.5480	−1.26528	−0.632638	+	0.774448i	$0.718028\pi$
$152$	4.97769	0.403744
$153$	0	0
$154$	0.127415	0.0102674
$155$	−7.09600	−0.569964
$156$	0	0
$157$	−17.9658	−1.43383	−0.716915	−	0.697160i	$-0.754447\pi$
$157$	−17.9658	−1.43383	−0.716915	+	0.697160i	$0.754447\pi$
$158$	13.0707	1.03985
$159$	0	0
$160$	1.83556	0.145113
$161$	0	0
$162$	0	0
$163$	−4.25533	−0.333303	−0.166652	−	0.986016i	$-0.553296\pi$
$163$	−4.25533	−0.333303	−0.166652	+	0.986016i	$0.553296\pi$
$164$	−1.65386	−0.129145
$165$	0	0
$166$	5.20439	0.403939
$167$	−7.39407	−0.572170	−0.286085	−	0.958204i	$-0.592354\pi$
$167$	−7.39407	−0.572170	−0.286085	+	0.958204i	$0.592354\pi$
$168$	0	0
$169$	2.00401	0.154154
$170$	5.61991	0.431028
$171$	0	0
$172$	−3.59805	−0.274349
$173$	14.4274	1.09690	0.548448	−	0.836185i	$-0.315219\pi$
$173$	14.4274	1.09690	0.548448	+	0.836185i	$0.315219\pi$
$174$	0	0
$175$	2.02857	0.153346
$176$	0.102427	0.00772071
$177$	0	0
$178$	−7.97976	−0.598108
$179$	−25.8758	−1.93405	−0.967026	−	0.254679i	$-0.918030\pi$
$179$	−25.8758	−1.93405	−0.967026	+	0.254679i	$0.918030\pi$
$180$	0	0
$181$	13.6517	1.01472	0.507362	−	0.861733i	$-0.330621\pi$
$181$	13.6517	1.01472	0.507362	+	0.861733i	$0.330621\pi$
$182$	−4.81849	−0.357170
$183$	0	0
$184$	0	0
$185$	−4.68848	−0.344704
$186$	0	0
$187$	0.313599	0.0229327
$188$	1.92036	0.140056
$189$	0	0
$190$	−9.13684	−0.662856
$191$	22.8249	1.65155	0.825775	−	0.564000i	$-0.190738\pi$
$191$	22.8249	1.65155	0.825775	+	0.564000i	$0.190738\pi$
$192$	0	0
$193$	3.64549	0.262408	0.131204	−	0.991355i	$-0.458116\pi$
$193$	3.64549	0.262408	0.131204	+	0.991355i	$0.458116\pi$
$194$	11.4726	0.823681
$195$	0	0
$196$	−5.45256	−0.389468
$197$	−16.2366	−1.15681	−0.578405	−	0.815750i	$-0.696325\pi$
$197$	−16.2366	−1.15681	−0.578405	+	0.815750i	$0.696325\pi$
$198$	0	0
$199$	−5.87151	−0.416220	−0.208110	−	0.978105i	$-0.566731\pi$
$199$	−5.87151	−0.416220	−0.208110	+	0.978105i	$0.566731\pi$
$200$	1.63073	0.115310
$201$	0	0
$202$	−8.35791	−0.588061
$203$	8.35836	0.586642
$204$	0	0
$205$	3.03575	0.212026
$206$	12.7017	0.884972
$207$	0	0
$208$	−3.87350	−0.268579
$209$	−0.509849	−0.0352670
$210$	0	0
$211$	25.7873	1.77527	0.887637	−	0.460544i	$-0.152346\pi$
$211$	25.7873	1.77527	0.887637	+	0.460544i	$0.152346\pi$
$212$	−4.14290	−0.284535
$213$	0	0
$214$	−10.5059	−0.718169
$215$	6.60442	0.450418
$216$	0	0
$217$	−4.80898	−0.326455
$218$	−5.49163	−0.371940
$219$	0	0
$220$	−0.188010	−0.0126756
$221$	−11.8595	−0.797755
$222$	0	0
$223$	−7.64883	−0.512204	−0.256102	−	0.966650i	$-0.582438\pi$
$223$	−7.64883	−0.512204	−0.256102	+	0.966650i	$0.582438\pi$
$224$	1.24396	0.0831158
$225$	0	0
$226$	−18.8702	−1.25523
$227$	10.0638	0.667958	0.333979	−	0.942580i	$-0.391608\pi$
$227$	10.0638	0.667958	0.333979	+	0.942580i	$0.391608\pi$
$228$	0	0
$229$	−15.6790	−1.03609	−0.518047	−	0.855352i	$-0.673341\pi$
$229$	−15.6790	−1.03609	−0.518047	+	0.855352i	$0.673341\pi$
$230$	0	0
$231$	0	0
$232$	6.71914	0.441133
$233$	19.9526	1.30714	0.653568	−	0.756868i	$-0.273271\pi$
$233$	19.9526	1.30714	0.653568	+	0.756868i	$0.273271\pi$
$234$	0	0
$235$	−3.52493	−0.229941
$236$	−6.09859	−0.396984
$237$	0	0
$238$	3.80863	0.246877
$239$	3.63620	0.235206	0.117603	−	0.993061i	$-0.462479\pi$
$239$	3.63620	0.235206	0.117603	+	0.993061i	$0.462479\pi$
$240$	0	0
$241$	3.80034	0.244801	0.122401	−	0.992481i	$-0.460941\pi$
$241$	3.80034	0.244801	0.122401	+	0.992481i	$0.460941\pi$
$242$	10.9895	0.706432
$243$	0	0
$244$	5.74065	0.367508
$245$	10.0085	0.639418
$246$	0	0
$247$	19.2811	1.22683
$248$	−3.86586	−0.245482
$249$	0	0
$250$	−12.1711	−0.769767
$251$	14.0793	0.888675	0.444337	−	0.895859i	$-0.353439\pi$
$251$	14.0793	0.888675	0.444337	+	0.895859i	$0.353439\pi$
$252$	0	0
$253$	0	0
$254$	1.17839	0.0739390
$255$	0	0
$256$	1.00000	0.0625000
$257$	−18.4400	−1.15025	−0.575127	−	0.818064i	$-0.695047\pi$
$257$	−18.4400	−1.15025	−0.575127	+	0.818064i	$0.695047\pi$
$258$	0	0
$259$	−3.17740	−0.197434
$260$	7.11003	0.440945
$261$	0	0
$262$	21.8626	1.35067
$263$	9.32535	0.575026	0.287513	−	0.957777i	$-0.407172\pi$
$263$	9.32535	0.575026	0.287513	+	0.957777i	$0.407172\pi$
$264$	0	0
$265$	7.60452	0.467142
$266$	−6.19207	−0.379660
$267$	0	0
$268$	1.55357	0.0948994
$269$	6.53651	0.398538	0.199269	−	0.979945i	$-0.436143\pi$
$269$	6.53651	0.398538	0.199269	+	0.979945i	$0.436143\pi$
$270$	0	0
$271$	11.6098	0.705245	0.352622	−	0.935766i	$-0.385290\pi$
$271$	11.6098	0.705245	0.352622	+	0.935766i	$0.385290\pi$
$272$	3.06169	0.185642
$273$	0	0
$274$	−17.1557	−1.03641
$275$	−0.167031	−0.0100723
$276$	0	0
$277$	−12.6946	−0.762743	−0.381372	−	0.924422i	$-0.624548\pi$
$277$	−12.6946	−0.762743	−0.381372	+	0.924422i	$0.624548\pi$
$278$	−10.1346	−0.607834
$279$	0	0
$280$	−2.28336	−0.136457
$281$	−25.1876	−1.50257	−0.751284	−	0.659979i	$-0.770565\pi$
$281$	−25.1876	−1.50257	−0.751284	+	0.659979i	$0.770565\pi$
$282$	0	0
$283$	28.5549	1.69741	0.848705	−	0.528867i	$-0.177383\pi$
$283$	28.5549	1.69741	0.848705	+	0.528867i	$0.177383\pi$
$284$	8.52403	0.505808
$285$	0	0
$286$	0.396750	0.0234603
$287$	2.05734	0.121441
$288$	0	0
$289$	−7.62603	−0.448590
$290$	−12.3334	−0.724240
$291$	0	0
$292$	−12.0483	−0.705076
$293$	11.9454	0.697855	0.348928	−	0.937150i	$-0.386546\pi$
$293$	11.9454	0.697855	0.348928	+	0.937150i	$0.386546\pi$
$294$	0	0
$295$	11.1943	0.651757
$296$	−2.55426	−0.148463
$297$	0	0
$298$	0.377442	0.0218646
$299$	0	0
$300$	0	0
$301$	4.47584	0.257983
$302$	15.5480	0.894685
$303$	0	0
$304$	−4.97769	−0.285490
$305$	−10.5373	−0.603364
$306$	0	0
$307$	7.49073	0.427518	0.213759	−	0.976886i	$-0.431429\pi$
$307$	7.49073	0.427518	0.213759	+	0.976886i	$0.431429\pi$
$308$	−0.127415	−0.00726015
$309$	0	0
$310$	7.09600	0.403026
$311$	31.4287	1.78216	0.891078	−	0.453850i	$-0.149950\pi$
$311$	31.4287	1.78216	0.891078	+	0.453850i	$0.149950\pi$
$312$	0	0
$313$	27.4410	1.55106	0.775529	−	0.631312i	$-0.217483\pi$
$313$	27.4410	1.55106	0.775529	+	0.631312i	$0.217483\pi$
$314$	17.9658	1.01387
$315$	0	0
$316$	−13.0707	−0.735286
$317$	−35.1618	−1.97489	−0.987443	−	0.157976i	$-0.949503\pi$
$317$	−35.1618	−1.97489	−0.987443	+	0.157976i	$0.949503\pi$
$318$	0	0
$319$	−0.688220	−0.0385329
$320$	−1.83556	−0.102611
$321$	0	0
$322$	0	0
$323$	−15.2402	−0.847986
$324$	0	0
$325$	6.31665	0.350384
$326$	4.25533	0.235681
$327$	0	0
$328$	1.65386	0.0913191
$329$	−2.38885	−0.131702
$330$	0	0
$331$	19.6658	1.08093	0.540464	−	0.841367i	$-0.318249\pi$
$331$	19.6658	1.08093	0.540464	+	0.841367i	$0.318249\pi$
$332$	−5.20439	−0.285628
$333$	0	0
$334$	7.39407	0.404585
$335$	−2.85166	−0.155803
$336$	0	0
$337$	−4.80845	−0.261933	−0.130967	−	0.991387i	$-0.541808\pi$
$337$	−4.80845	−0.261933	−0.130967	+	0.991387i	$0.541808\pi$
$338$	−2.00401	−0.109004
$339$	0	0
$340$	−5.61991	−0.304783
$341$	0.395967	0.0214428
$342$	0	0
$343$	15.4905	0.836409
$344$	3.59805	0.193994
$345$	0	0
$346$	−14.4274	−0.775623
$347$	0.0628435	0.00337362	0.00168681	−	0.999999i	$-0.499463\pi$
$347$	0.0628435	0.00337362	0.00168681	+	0.999999i	$0.499463\pi$
$348$	0	0
$349$	−8.02078	−0.429342	−0.214671	−	0.976686i	$-0.568868\pi$
$349$	−8.02078	−0.429342	−0.214671	+	0.976686i	$0.568868\pi$
$350$	−2.02857	−0.108432
$351$	0	0
$352$	−0.102427	−0.00545937
$353$	−8.22828	−0.437947	−0.218974	−	0.975731i	$-0.570271\pi$
$353$	−8.22828	−0.437947	−0.218974	+	0.975731i	$0.570271\pi$
$354$	0	0
$355$	−15.6463	−0.830421
$356$	7.97976	0.422926
$357$	0	0
$358$	25.8758	1.36758
$359$	−29.6264	−1.56362	−0.781810	−	0.623517i	$-0.785703\pi$
$359$	−29.6264	−1.56362	−0.781810	+	0.623517i	$0.785703\pi$
$360$	0	0
$361$	5.77743	0.304075
$362$	−13.6517	−0.717519
$363$	0	0
$364$	4.81849	0.252558
$365$	22.1154	1.15757
$366$	0	0
$367$	−12.5240	−0.653750	−0.326875	−	0.945068i	$-0.605996\pi$
$367$	−12.5240	−0.653750	−0.326875	+	0.945068i	$0.605996\pi$
$368$	0	0
$369$	0	0
$370$	4.68848	0.243743
$371$	5.15361	0.267562
$372$	0	0
$373$	−20.6957	−1.07159	−0.535793	−	0.844350i	$-0.679987\pi$
$373$	−20.6957	−1.07159	−0.535793	+	0.844350i	$0.679987\pi$
$374$	−0.313599	−0.0162158
$375$	0	0
$376$	−1.92036	−0.0990349
$377$	26.0266	1.34044
$378$	0	0
$379$	−5.52787	−0.283947	−0.141974	−	0.989870i	$-0.545345\pi$
$379$	−5.52787	−0.283947	−0.141974	+	0.989870i	$0.545345\pi$
$380$	9.13684	0.468710
$381$	0	0
$382$	−22.8249	−1.16782
$383$	35.1201	1.79455	0.897276	−	0.441471i	$-0.145543\pi$
$383$	35.1201	1.79455	0.897276	+	0.441471i	$0.145543\pi$
$384$	0	0
$385$	0.233878	0.0119195
$386$	−3.64549	−0.185551
$387$	0	0
$388$	−11.4726	−0.582431
$389$	−16.5220	−0.837698	−0.418849	−	0.908056i	$-0.637566\pi$
$389$	−16.5220	−0.837698	−0.418849	+	0.908056i	$0.637566\pi$
$390$	0	0
$391$	0	0
$392$	5.45256	0.275396
$393$	0	0
$394$	16.2366	0.817988
$395$	23.9921	1.20717
$396$	0	0
$397$	31.0915	1.56044	0.780218	−	0.625507i	$-0.215108\pi$
$397$	31.0915	1.56044	0.780218	+	0.625507i	$0.215108\pi$
$398$	5.87151	0.294312
$399$	0	0
$400$	−1.63073	−0.0815367
$401$	15.5880	0.778428	0.389214	−	0.921147i	$-0.372747\pi$
$401$	15.5880	0.778428	0.389214	+	0.921147i	$0.372747\pi$
$402$	0	0
$403$	−14.9744	−0.745928
$404$	8.35791	0.415822
$405$	0	0
$406$	−8.35836	−0.414819
$407$	0.261624	0.0129682
$408$	0	0
$409$	34.0324	1.68279	0.841397	−	0.540418i	$-0.181734\pi$
$409$	34.0324	1.68279	0.841397	+	0.540418i	$0.181734\pi$
$410$	−3.03575	−0.149925
$411$	0	0
$412$	−12.7017	−0.625770
$413$	7.58642	0.373303
$414$	0	0
$415$	9.55296	0.468936
$416$	3.87350	0.189914
$417$	0	0
$418$	0.509849	0.0249375
$419$	−14.6635	−0.716357	−0.358178	−	0.933653i	$-0.616602\pi$
$419$	−14.6635	−0.716357	−0.358178	+	0.933653i	$0.616602\pi$
$420$	0	0
$421$	13.8626	0.675623	0.337811	−	0.941214i	$-0.390313\pi$
$421$	13.8626	0.675623	0.337811	+	0.941214i	$0.390313\pi$
$422$	−25.7873	−1.25531
$423$	0	0
$424$	4.14290	0.201197
$425$	−4.99281	−0.242187
$426$	0	0
$427$	−7.14116	−0.345585
$428$	10.5059	0.507822
$429$	0	0
$430$	−6.60442	−0.318493
$431$	7.04806	0.339493	0.169746	−	0.985488i	$-0.445705\pi$
$431$	7.04806	0.339493	0.169746	+	0.985488i	$0.445705\pi$
$432$	0	0
$433$	19.0914	0.917477	0.458738	−	0.888571i	$-0.348302\pi$
$433$	19.0914	0.917477	0.458738	+	0.888571i	$0.348302\pi$
$434$	4.80898	0.230839
$435$	0	0
$436$	5.49163	0.263001
$437$	0	0
$438$	0	0
$439$	−6.86262	−0.327535	−0.163768	−	0.986499i	$-0.552365\pi$
$439$	−6.86262	−0.327535	−0.163768	+	0.986499i	$0.552365\pi$
$440$	0.188010	0.00896303
$441$	0	0
$442$	11.8595	0.564098
$443$	−20.2739	−0.963243	−0.481621	−	0.876379i	$-0.659952\pi$
$443$	−20.2739	−0.963243	−0.481621	+	0.876379i	$0.659952\pi$
$444$	0	0
$445$	−14.6473	−0.694348
$446$	7.64883	0.362183
$447$	0	0
$448$	−1.24396	−0.0587717
$449$	28.2153	1.33156	0.665782	−	0.746147i	$-0.268098\pi$
$449$	28.2153	1.33156	0.665782	+	0.746147i	$0.268098\pi$
$450$	0	0
$451$	−0.169399	−0.00797671
$452$	18.8702	0.887579
$453$	0	0
$454$	−10.0638	−0.472318
$455$	−8.84461	−0.414642
$456$	0	0
$457$	29.8068	1.39430	0.697151	−	0.716925i	$-0.254451\pi$
$457$	29.8068	1.39430	0.697151	+	0.716925i	$0.254451\pi$
$458$	15.6790	0.732630
$459$	0	0
$460$	0	0
$461$	36.3247	1.69181	0.845905	−	0.533334i	$-0.179061\pi$
$461$	36.3247	1.69181	0.845905	+	0.533334i	$0.179061\pi$
$462$	0	0
$463$	25.7523	1.19681	0.598405	−	0.801193i	$-0.295801\pi$
$463$	25.7523	1.19681	0.598405	+	0.801193i	$0.295801\pi$
$464$	−6.71914	−0.311928
$465$	0	0
$466$	−19.9526	−0.924285
$467$	−6.69698	−0.309899	−0.154950	−	0.987922i	$-0.549522\pi$
$467$	−6.69698	−0.309899	−0.154950	+	0.987922i	$0.549522\pi$
$468$	0	0
$469$	−1.93258	−0.0892384
$470$	3.52493	0.162593
$471$	0	0
$472$	6.09859	0.280710
$473$	−0.368536	−0.0169453
$474$	0	0
$475$	8.11729	0.372447
$476$	−3.80863	−0.174568
$477$	0	0
$478$	−3.63620	−0.166316
$479$	−14.9165	−0.681553	−0.340777	−	0.940144i	$-0.610690\pi$
$479$	−14.9165	−0.681553	−0.340777	+	0.940144i	$0.610690\pi$
$480$	0	0
$481$	−9.89391	−0.451124
$482$	−3.80034	−0.173101
$483$	0	0
$484$	−10.9895	−0.499523
$485$	21.0585	0.956218
$486$	0	0
$487$	15.0787	0.683280	0.341640	−	0.939831i	$-0.389018\pi$
$487$	15.0787	0.683280	0.341640	+	0.939831i	$0.389018\pi$
$488$	−5.74065	−0.259867
$489$	0	0
$490$	−10.0085	−0.452137
$491$	6.87412	0.310225	0.155112	−	0.987897i	$-0.450426\pi$
$491$	6.87412	0.310225	0.155112	+	0.987897i	$0.450426\pi$
$492$	0	0
$493$	−20.5720	−0.926515
$494$	−19.2811	−0.867498
$495$	0	0
$496$	3.86586	0.173582
$497$	−10.6036	−0.475635
$498$	0	0
$499$	5.67632	0.254107	0.127054	−	0.991896i	$-0.459448\pi$
$499$	5.67632	0.254107	0.127054	+	0.991896i	$0.459448\pi$
$500$	12.1711	0.544307
$501$	0	0
$502$	−14.0793	−0.628388
$503$	42.4003	1.89053	0.945267	−	0.326297i	$-0.105801\pi$
$503$	42.4003	1.89053	0.945267	+	0.326297i	$0.105801\pi$
$504$	0	0
$505$	−15.3414	−0.682684
$506$	0	0
$507$	0	0
$508$	−1.17839	−0.0522827
$509$	3.67274	0.162791	0.0813957	−	0.996682i	$-0.474062\pi$
$509$	3.67274	0.162791	0.0813957	+	0.996682i	$0.474062\pi$
$510$	0	0
$511$	14.9877	0.663016
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	18.4400	0.813352
$515$	23.3148	1.02737
$516$	0	0
$517$	0.196696	0.00865068
$518$	3.17740	0.139607
$519$	0	0
$520$	−7.11003	−0.311795
$521$	28.9419	1.26797	0.633984	−	0.773347i	$-0.281419\pi$
$521$	28.9419	1.26797	0.633984	+	0.773347i	$0.281419\pi$
$522$	0	0
$523$	−15.9909	−0.699231	−0.349616	−	0.936893i	$-0.613688\pi$
$523$	−15.9909	−0.699231	−0.349616	+	0.936893i	$0.613688\pi$
$524$	−21.8626	−0.955070
$525$	0	0
$526$	−9.32535	−0.406605
$527$	11.8361	0.515588
$528$	0	0
$529$	0	0
$530$	−7.60452	−0.330319
$531$	0	0
$532$	6.19207	0.268460
$533$	6.40622	0.277484
$534$	0	0
$535$	−19.2842	−0.833728
$536$	−1.55357	−0.0671040
$537$	0	0
$538$	−6.53651	−0.281809
$539$	−0.558488	−0.0240558
$540$	0	0
$541$	−37.5251	−1.61333	−0.806665	−	0.591008i	$-0.798730\pi$
$541$	−37.5251	−1.61333	−0.806665	+	0.591008i	$0.798730\pi$
$542$	−11.6098	−0.498683
$543$	0	0
$544$	−3.06169	−0.131269
$545$	−10.0802	−0.431788
$546$	0	0
$547$	−12.4784	−0.533536	−0.266768	−	0.963761i	$-0.585956\pi$
$547$	−12.4784	−0.533536	−0.266768	+	0.963761i	$0.585956\pi$
$548$	17.1557	0.732855
$549$	0	0
$550$	0.167031	0.00712221
$551$	33.4458	1.42484
$552$	0	0
$553$	16.2595	0.691425
$554$	12.6946	0.539341
$555$	0	0
$556$	10.1346	0.429804
$557$	−13.6887	−0.580010	−0.290005	−	0.957025i	$-0.593657\pi$
$557$	−13.6887	−0.580010	−0.290005	+	0.957025i	$0.593657\pi$
$558$	0	0
$559$	13.9370	0.589474
$560$	2.28336	0.0964897
$561$	0	0
$562$	25.1876	1.06248
$563$	−28.6234	−1.20633	−0.603167	−	0.797615i	$-0.706095\pi$
$563$	−28.6234	−1.20633	−0.603167	+	0.797615i	$0.706095\pi$
$564$	0	0
$565$	−34.6373	−1.45720
$566$	−28.5549	−1.20025
$567$	0	0
$568$	−8.52403	−0.357660
$569$	−22.7768	−0.954851	−0.477426	−	0.878672i	$-0.658430\pi$
$569$	−22.7768	−0.954851	−0.477426	+	0.878672i	$0.658430\pi$
$570$	0	0
$571$	8.41806	0.352285	0.176142	−	0.984365i	$-0.443638\pi$
$571$	8.41806	0.352285	0.176142	+	0.984365i	$0.443638\pi$
$572$	−0.396750	−0.0165890
$573$	0	0
$574$	−2.05734	−0.0858716
$575$	0	0
$576$	0	0
$577$	17.0180	0.708467	0.354233	−	0.935157i	$-0.384742\pi$
$577$	17.0180	0.708467	0.354233	+	0.935157i	$0.384742\pi$
$578$	7.62603	0.317201
$579$	0	0
$580$	12.3334	0.512115
$581$	6.47407	0.268590
$582$	0	0
$583$	−0.424343	−0.0175745
$584$	12.0483	0.498564
$585$	0	0
$586$	−11.9454	−0.493458
$587$	−24.9116	−1.02821	−0.514106	−	0.857727i	$-0.671876\pi$
$587$	−24.9116	−1.02821	−0.514106	+	0.857727i	$0.671876\pi$
$588$	0	0
$589$	−19.2431	−0.792896
$590$	−11.1943	−0.460862
$591$	0	0
$592$	2.55426	0.104979
$593$	−11.1811	−0.459151	−0.229576	−	0.973291i	$-0.573734\pi$
$593$	−11.1811	−0.459151	−0.229576	+	0.973291i	$0.573734\pi$
$594$	0	0
$595$	6.99096	0.286601
$596$	−0.377442	−0.0154606
$597$	0	0
$598$	0	0
$599$	−40.5640	−1.65740	−0.828701	−	0.559692i	$-0.810920\pi$
$599$	−40.5640	−1.65740	−0.828701	+	0.559692i	$0.810920\pi$
$600$	0	0
$601$	−28.8610	−1.17727	−0.588633	−	0.808401i	$-0.700333\pi$
$601$	−28.8610	−1.17727	−0.588633	+	0.808401i	$0.700333\pi$
$602$	−4.47584	−0.182422
$603$	0	0
$604$	−15.5480	−0.632638
$605$	20.1719	0.820103
$606$	0	0
$607$	−28.8284	−1.17011	−0.585054	−	0.810995i	$-0.698927\pi$
$607$	−28.8284	−1.17011	−0.585054	+	0.810995i	$0.698927\pi$
$608$	4.97769	0.201872
$609$	0	0
$610$	10.5373	0.426643
$611$	−7.43851	−0.300930
$612$	0	0
$613$	−4.96802	−0.200656	−0.100328	−	0.994954i	$-0.531989\pi$
$613$	−4.96802	−0.200656	−0.100328	+	0.994954i	$0.531989\pi$
$614$	−7.49073	−0.302301
$615$	0	0
$616$	0.127415	0.00513370
$617$	24.5181	0.987061	0.493531	−	0.869728i	$-0.335706\pi$
$617$	24.5181	0.987061	0.493531	+	0.869728i	$0.335706\pi$
$618$	0	0
$619$	−6.15888	−0.247546	−0.123773	−	0.992311i	$-0.539500\pi$
$619$	−6.15888	−0.247546	−0.123773	+	0.992311i	$0.539500\pi$
$620$	−7.09600	−0.284982
$621$	0	0
$622$	−31.4287	−1.26017
$623$	−9.92652	−0.397698
$624$	0	0
$625$	−14.1870	−0.567482
$626$	−27.4410	−1.09676
$627$	0	0
$628$	−17.9658	−0.716915
$629$	7.82035	0.311818
$630$	0	0
$631$	28.0102	1.11507	0.557534	−	0.830154i	$-0.311748\pi$
$631$	28.0102	1.11507	0.557534	+	0.830154i	$0.311748\pi$
$632$	13.0707	0.519926
$633$	0	0
$634$	35.1618	1.39646
$635$	2.16301	0.0858363
$636$	0	0
$637$	21.1205	0.836824
$638$	0.688220	0.0272469
$639$	0	0
$640$	1.83556	0.0725567
$641$	−30.7056	−1.21280	−0.606399	−	0.795161i	$-0.707387\pi$
$641$	−30.7056	−1.21280	−0.606399	+	0.795161i	$0.707387\pi$
$642$	0	0
$643$	8.46014	0.333635	0.166818	−	0.985988i	$-0.446651\pi$
$643$	8.46014	0.333635	0.166818	+	0.985988i	$0.446651\pi$
$644$	0	0
$645$	0	0
$646$	15.2402	0.599617
$647$	2.16735	0.0852073	0.0426036	−	0.999092i	$-0.486435\pi$
$647$	2.16735	0.0852073	0.0426036	+	0.999092i	$0.486435\pi$
$648$	0	0
$649$	−0.624659	−0.0245200
$650$	−6.31665	−0.247759
$651$	0	0
$652$	−4.25533	−0.166652
$653$	8.29851	0.324746	0.162373	−	0.986729i	$-0.448085\pi$
$653$	8.29851	0.324746	0.162373	+	0.986729i	$0.448085\pi$
$654$	0	0
$655$	40.1299	1.56801
$656$	−1.65386	−0.0645723
$657$	0	0
$658$	2.38885	0.0931272
$659$	40.7076	1.58574	0.792871	−	0.609389i	$-0.208585\pi$
$659$	40.7076	1.58574	0.792871	+	0.609389i	$0.208585\pi$
$660$	0	0
$661$	38.8947	1.51283	0.756414	−	0.654093i	$-0.226949\pi$
$661$	38.8947	1.51283	0.756414	+	0.654093i	$0.226949\pi$
$662$	−19.6658	−0.764332
$663$	0	0
$664$	5.20439	0.201970
$665$	−11.3659	−0.440750
$666$	0	0
$667$	0	0
$668$	−7.39407	−0.286085
$669$	0	0
$670$	2.85166	0.110169
$671$	0.587997	0.0226994
$672$	0	0
$673$	42.5640	1.64072	0.820360	−	0.571847i	$-0.193773\pi$
$673$	42.5640	1.64072	0.820360	+	0.571847i	$0.193773\pi$
$674$	4.80845	0.185215
$675$	0	0
$676$	2.00401	0.0770772
$677$	41.5445	1.59668	0.798342	−	0.602205i	$-0.205711\pi$
$677$	41.5445	1.59668	0.798342	+	0.602205i	$0.205711\pi$
$678$	0	0
$679$	14.2714	0.547687
$680$	5.61991	0.215514
$681$	0	0
$682$	−0.395967	−0.0151624
$683$	16.7097	0.639380	0.319690	−	0.947522i	$-0.396421\pi$
$683$	16.7097	0.639380	0.319690	+	0.947522i	$0.396421\pi$
$684$	0	0
$685$	−31.4902	−1.20318
$686$	−15.4905	−0.591431
$687$	0	0
$688$	−3.59805	−0.137174
$689$	16.0475	0.611361
$690$	0	0
$691$	44.6532	1.69869	0.849344	−	0.527840i	$-0.176998\pi$
$691$	44.6532	1.69869	0.849344	+	0.527840i	$0.176998\pi$
$692$	14.4274	0.548448
$693$	0	0
$694$	−0.0628435	−0.00238551
$695$	−18.6027	−0.705640
$696$	0	0
$697$	−5.06361	−0.191798
$698$	8.02078	0.303591
$699$	0	0
$700$	2.02857	0.0766728
$701$	−22.3632	−0.844648	−0.422324	−	0.906445i	$-0.638786\pi$
$701$	−22.3632	−0.844648	−0.422324	+	0.906445i	$0.638786\pi$
$702$	0	0
$703$	−12.7143	−0.479529
$704$	0.102427	0.00386035
$705$	0	0
$706$	8.22828	0.309675
$707$	−10.3969	−0.391017
$708$	0	0
$709$	10.1024	0.379403	0.189702	−	0.981842i	$-0.439248\pi$
$709$	10.1024	0.379403	0.189702	+	0.981842i	$0.439248\pi$
$710$	15.6463	0.587197
$711$	0	0
$712$	−7.97976	−0.299054
$713$	0	0
$714$	0	0
$715$	0.728257	0.0272353
$716$	−25.8758	−0.967026
$717$	0	0
$718$	29.6264	1.10565
$719$	−8.28528	−0.308989	−0.154494	−	0.987994i	$-0.549375\pi$
$719$	−8.28528	−0.308989	−0.154494	+	0.987994i	$0.549375\pi$
$720$	0	0
$721$	15.8005	0.588441
$722$	−5.77743	−0.215014
$723$	0	0
$724$	13.6517	0.507362
$725$	10.9571	0.406938
$726$	0	0
$727$	14.8338	0.550155	0.275077	−	0.961422i	$-0.411297\pi$
$727$	14.8338	0.550155	0.275077	+	0.961422i	$0.411297\pi$
$728$	−4.81849	−0.178585
$729$	0	0
$730$	−22.1154	−0.818528
$731$	−11.0161	−0.407446
$732$	0	0
$733$	−20.4584	−0.755649	−0.377824	−	0.925877i	$-0.623328\pi$
$733$	−20.4584	−0.755649	−0.377824	+	0.925877i	$0.623328\pi$
$734$	12.5240	0.462271
$735$	0	0
$736$	0	0
$737$	0.159127	0.00586152
$738$	0	0
$739$	−38.4610	−1.41481	−0.707405	−	0.706808i	$-0.750134\pi$
$739$	−38.4610	−1.41481	−0.707405	+	0.706808i	$0.750134\pi$
$740$	−4.68848	−0.172352
$741$	0	0
$742$	−5.15361	−0.189195
$743$	16.6536	0.610961	0.305481	−	0.952198i	$-0.401183\pi$
$743$	16.6536	0.610961	0.305481	+	0.952198i	$0.401183\pi$
$744$	0	0
$745$	0.692815	0.0253828
$746$	20.6957	0.757725
$747$	0	0
$748$	0.313599	0.0114663
$749$	−13.0690	−0.477529
$750$	0	0
$751$	19.5078	0.711850	0.355925	−	0.934515i	$-0.384166\pi$
$751$	19.5078	0.711850	0.355925	+	0.934515i	$0.384166\pi$
$752$	1.92036	0.0700282
$753$	0	0
$754$	−26.0266	−0.947833
$755$	28.5392	1.03865
$756$	0	0
$757$	34.7791	1.26407	0.632034	−	0.774941i	$-0.282220\pi$
$757$	34.7791	1.26407	0.632034	+	0.774941i	$0.282220\pi$
$758$	5.52787	0.200781
$759$	0	0
$760$	−9.13684	−0.331428
$761$	−1.56785	−0.0568346	−0.0284173	−	0.999596i	$-0.509047\pi$
$761$	−1.56785	−0.0568346	−0.0284173	+	0.999596i	$0.509047\pi$
$762$	0	0
$763$	−6.83139	−0.247313
$764$	22.8249	0.825775
$765$	0	0
$766$	−35.1201	−1.26894
$767$	23.6229	0.852973
$768$	0	0
$769$	−22.8193	−0.822886	−0.411443	−	0.911435i	$-0.634975\pi$
$769$	−22.8193	−0.822886	−0.411443	+	0.911435i	$0.634975\pi$
$770$	−0.233878	−0.00842836
$771$	0	0
$772$	3.64549	0.131204
$773$	44.5857	1.60364	0.801819	−	0.597567i	$-0.203866\pi$
$773$	44.5857	1.60364	0.801819	+	0.597567i	$0.203866\pi$
$774$	0	0
$775$	−6.30418	−0.226453
$776$	11.4726	0.411841
$777$	0	0
$778$	16.5220	0.592342
$779$	8.23240	0.294956
$780$	0	0
$781$	0.873089	0.0312416
$782$	0	0
$783$	0	0
$784$	−5.45256	−0.194734
$785$	32.9773	1.17701
$786$	0	0
$787$	−37.7715	−1.34641	−0.673204	−	0.739457i	$-0.735082\pi$
$787$	−37.7715	−1.34641	−0.673204	+	0.739457i	$0.735082\pi$
$788$	−16.2366	−0.578405
$789$	0	0
$790$	−23.9921	−0.853600
$791$	−23.4738	−0.834633
$792$	0	0
$793$	−22.2364	−0.789639
$794$	−31.0915	−1.10340
$795$	0	0
$796$	−5.87151	−0.208110
$797$	−11.8638	−0.420238	−0.210119	−	0.977676i	$-0.567385\pi$
$797$	−11.8638	−0.420238	−0.210119	+	0.977676i	$0.567385\pi$
$798$	0	0
$799$	5.87955	0.208003
$800$	1.63073	0.0576551
$801$	0	0
$802$	−15.5880	−0.550432
$803$	−1.23407	−0.0435495
$804$	0	0
$805$	0	0
$806$	14.9744	0.527451
$807$	0	0
$808$	−8.35791	−0.294030
$809$	7.51083	0.264067	0.132033	−	0.991245i	$-0.457849\pi$
$809$	7.51083	0.264067	0.132033	+	0.991245i	$0.457849\pi$
$810$	0	0
$811$	26.9552	0.946526	0.473263	−	0.880921i	$-0.343076\pi$
$811$	26.9552	0.946526	0.473263	+	0.880921i	$0.343076\pi$
$812$	8.35836	0.293321
$813$	0	0
$814$	−0.261624	−0.00916992
$815$	7.81090	0.273604
$816$	0	0
$817$	17.9100	0.626591
$818$	−34.0324	−1.18991
$819$	0	0
$820$	3.03575	0.106013
$821$	−39.4005	−1.37509	−0.687543	−	0.726143i	$-0.741311\pi$
$821$	−39.4005	−1.37509	−0.687543	+	0.726143i	$0.741311\pi$
$822$	0	0
$823$	48.8334	1.70222	0.851112	−	0.524984i	$-0.175929\pi$
$823$	48.8334	1.70222	0.851112	+	0.524984i	$0.175929\pi$
$824$	12.7017	0.442486
$825$	0	0
$826$	−7.58642	−0.263965
$827$	20.7802	0.722599	0.361299	−	0.932450i	$-0.382333\pi$
$827$	20.7802	0.722599	0.361299	+	0.932450i	$0.382333\pi$
$828$	0	0
$829$	11.8167	0.410410	0.205205	−	0.978719i	$-0.434214\pi$
$829$	11.8167	0.410410	0.205205	+	0.978719i	$0.434214\pi$
$830$	−9.55296	−0.331588
$831$	0	0
$832$	−3.87350	−0.134289
$833$	−16.6941	−0.578415
$834$	0	0
$835$	13.5722	0.469686
$836$	−0.509849	−0.0176335
$837$	0	0
$838$	14.6635	0.506541
$839$	27.0142	0.932633	0.466317	−	0.884618i	$-0.345581\pi$
$839$	27.0142	0.932633	0.466317	+	0.884618i	$0.345581\pi$
$840$	0	0
$841$	16.1469	0.556789
$842$	−13.8626	−0.477737
$843$	0	0
$844$	25.7873	0.887637
$845$	−3.67847	−0.126543
$846$	0	0
$847$	13.6705	0.469725
$848$	−4.14290	−0.142268
$849$	0	0
$850$	4.99281	0.171252
$851$	0	0
$852$	0	0
$853$	−21.5059	−0.736349	−0.368174	−	0.929757i	$-0.620017\pi$
$853$	−21.5059	−0.736349	−0.368174	+	0.929757i	$0.620017\pi$
$854$	7.14116	0.244365
$855$	0	0
$856$	−10.5059	−0.359085
$857$	38.4955	1.31498	0.657490	−	0.753463i	$-0.271618\pi$
$857$	38.4955	1.31498	0.657490	+	0.753463i	$0.271618\pi$
$858$	0	0
$859$	−29.3919	−1.00284	−0.501419	−	0.865204i	$-0.667189\pi$
$859$	−29.3919	−1.00284	−0.501419	+	0.865204i	$0.667189\pi$
$860$	6.60442	0.225209
$861$	0	0
$862$	−7.04806	−0.240058
$863$	27.9880	0.952722	0.476361	−	0.879250i	$-0.341956\pi$
$863$	27.9880	0.952722	0.476361	+	0.879250i	$0.341956\pi$
$864$	0	0
$865$	−26.4823	−0.900426
$866$	−19.0914	−0.648754
$867$	0	0
$868$	−4.80898	−0.163228
$869$	−1.33879	−0.0454154
$870$	0	0
$871$	−6.01775	−0.203904
$872$	−5.49163	−0.185970
$873$	0	0
$874$	0	0
$875$	−15.1404	−0.511838
$876$	0	0
$877$	−6.72130	−0.226962	−0.113481	−	0.993540i	$-0.536200\pi$
$877$	−6.72130	−0.226962	−0.113481	+	0.993540i	$0.536200\pi$
$878$	6.86262	0.231602
$879$	0	0
$880$	−0.188010	−0.00633782
$881$	2.65811	0.0895539	0.0447770	−	0.998997i	$-0.485742\pi$
$881$	2.65811	0.0895539	0.0447770	+	0.998997i	$0.485742\pi$
$882$	0	0
$883$	−46.3081	−1.55839	−0.779195	−	0.626781i	$-0.784372\pi$
$883$	−46.3081	−1.55839	−0.779195	+	0.626781i	$0.784372\pi$
$884$	−11.8595	−0.398877
$885$	0	0
$886$	20.2739	0.681115
$887$	12.8359	0.430988	0.215494	−	0.976505i	$-0.430864\pi$
$887$	12.8359	0.430988	0.215494	+	0.976505i	$0.430864\pi$
$888$	0	0
$889$	1.46588	0.0491639
$890$	14.6473	0.490978
$891$	0	0
$892$	−7.64883	−0.256102
$893$	−9.55895	−0.319878
$894$	0	0
$895$	47.4966	1.58764
$896$	1.24396	0.0415579
$897$	0	0
$898$	−28.2153	−0.941557
$899$	−25.9753	−0.866323
$900$	0	0
$901$	−12.6843	−0.422575
$902$	0.169399	0.00564038
$903$	0	0
$904$	−18.8702	−0.627613
$905$	−25.0585	−0.832973
$906$	0	0
$907$	21.9963	0.730374	0.365187	−	0.930934i	$-0.381005\pi$
$907$	21.9963	0.730374	0.365187	+	0.930934i	$0.381005\pi$
$908$	10.0638	0.333979
$909$	0	0
$910$	8.84461	0.293196
$911$	−26.1783	−0.867325	−0.433663	−	0.901075i	$-0.642779\pi$
$911$	−26.1783	−0.867325	−0.433663	+	0.901075i	$0.642779\pi$
$912$	0	0
$913$	−0.533069	−0.0176420
$914$	−29.8068	−0.985920
$915$	0	0
$916$	−15.6790	−0.518047
$917$	27.1962	0.898098
$918$	0	0
$919$	56.1530	1.85232	0.926158	−	0.377135i	$-0.123091\pi$
$919$	56.1530	1.85232	0.926158	+	0.377135i	$0.123091\pi$
$920$	0	0
$921$	0	0
$922$	−36.3247	−1.19629
$923$	−33.0178	−1.08680
$924$	0	0
$925$	−4.16531	−0.136955
$926$	−25.7523	−0.846273
$927$	0	0
$928$	6.71914	0.220567
$929$	−1.40355	−0.0460489	−0.0230244	−	0.999735i	$-0.507330\pi$
$929$	−1.40355	−0.0460489	−0.0230244	+	0.999735i	$0.507330\pi$
$930$	0	0
$931$	27.1412	0.889515
$932$	19.9526	0.653568
$933$	0	0
$934$	6.69698	0.219132
$935$	−0.575630	−0.0188251
$936$	0	0
$937$	7.27065	0.237522	0.118761	−	0.992923i	$-0.462108\pi$
$937$	7.27065	0.237522	0.118761	+	0.992923i	$0.462108\pi$
$938$	1.93258	0.0631011
$939$	0	0
$940$	−3.52493	−0.114970
$941$	−24.2465	−0.790414	−0.395207	−	0.918592i	$-0.629327\pi$
$941$	−24.2465	−0.790414	−0.395207	+	0.918592i	$0.629327\pi$
$942$	0	0
$943$	0	0
$944$	−6.09859	−0.198492
$945$	0	0
$946$	0.368536	0.0119822
$947$	−23.9123	−0.777046	−0.388523	−	0.921439i	$-0.627015\pi$
$947$	−23.9123	−0.777046	−0.388523	+	0.921439i	$0.627015\pi$
$948$	0	0
$949$	46.6693	1.51495
$950$	−8.11729	−0.263360
$951$	0	0
$952$	3.80863	0.123439
$953$	−18.1561	−0.588134	−0.294067	−	0.955785i	$-0.595009\pi$
$953$	−18.1561	−0.588134	−0.294067	+	0.955785i	$0.595009\pi$
$954$	0	0
$955$	−41.8963	−1.35573
$956$	3.63620	0.117603
$957$	0	0
$958$	14.9165	0.481931
$959$	−21.3410	−0.689138
$960$	0	0
$961$	−16.0551	−0.517908
$962$	9.89391	0.318993
$963$	0	0
$964$	3.80034	0.122401
$965$	−6.69150	−0.215407
$966$	0	0
$967$	40.5149	1.30287	0.651436	−	0.758704i	$-0.274167\pi$
$967$	40.5149	1.30287	0.651436	+	0.758704i	$0.274167\pi$
$968$	10.9895	0.353216
$969$	0	0
$970$	−21.0585	−0.676148
$971$	2.84248	0.0912195	0.0456098	−	0.998959i	$-0.485477\pi$
$971$	2.84248	0.0912195	0.0456098	+	0.998959i	$0.485477\pi$
$972$	0	0
$973$	−12.6071	−0.404165
$974$	−15.0787	−0.483152
$975$	0	0
$976$	5.74065	0.183754
$977$	5.39050	0.172457	0.0862287	−	0.996275i	$-0.472518\pi$
$977$	5.39050	0.172457	0.0862287	+	0.996275i	$0.472518\pi$
$978$	0	0
$979$	0.817341	0.0261223
$980$	10.0085	0.319709
$981$	0	0
$982$	−6.87412	−0.219362
$983$	35.8444	1.14326	0.571629	−	0.820512i	$-0.306312\pi$
$983$	35.8444	1.14326	0.571629	+	0.820512i	$0.306312\pi$
$984$	0	0
$985$	29.8032	0.949609
$986$	20.5720	0.655145
$987$	0	0
$988$	19.2811	0.613413
$989$	0	0
$990$	0	0
$991$	−25.1975	−0.800427	−0.400213	−	0.916422i	$-0.631064\pi$
$991$	−25.1975	−0.800427	−0.400213	+	0.916422i	$0.631064\pi$
$992$	−3.86586	−0.122741
$993$	0	0
$994$	10.6036	0.336325
$995$	10.7775	0.341669
$996$	0	0
$997$	6.91886	0.219122	0.109561	−	0.993980i	$-0.465055\pi$
$997$	6.91886	0.219122	0.109561	+	0.993980i	$0.465055\pi$
$998$	−5.67632	−0.179681
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9522.2.a.ch.1.2		10
3.2	odd	2		9522.2.a.ci.1.9		10
23.5	odd	22		414.2.i.h.163.1	yes	20
23.14	odd	22		414.2.i.h.127.1	yes	20
23.22	odd	2		9522.2.a.cg.1.9		10
69.5	even	22		414.2.i.g.163.2	yes	20
69.14	even	22		414.2.i.g.127.2	✓	20
69.68	even	2		9522.2.a.cj.1.2		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
414.2.i.g.127.2	✓	20	69.14	even	22
414.2.i.g.163.2	yes	20	69.5	even	22
414.2.i.h.127.1	yes	20	23.14	odd	22
414.2.i.h.163.1	yes	20	23.5	odd	22
9522.2.a.cg.1.9		10	23.22	odd	2
9522.2.a.ch.1.2		10	1.1	even	1	trivial
9522.2.a.ci.1.9		10	3.2	odd	2
9522.2.a.cj.1.2		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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.83556 q^{5} -1.24396 q^{7} -1.00000 q^{8} +1.83556 q^{10} +0.102427 q^{11} -3.87350 q^{13} +1.24396 q^{14} +1.00000 q^{16} +3.06169 q^{17} -4.97769 q^{19} -1.83556 q^{20} -0.102427 q^{22} -1.63073 q^{25} +3.87350 q^{26} -1.24396 q^{28} -6.71914 q^{29} +3.86586 q^{31} -1.00000 q^{32} -3.06169 q^{34} +2.28336 q^{35} +2.55426 q^{37} +4.97769 q^{38} +1.83556 q^{40} -1.65386 q^{41} -3.59805 q^{43} +0.102427 q^{44} +1.92036 q^{47} -5.45256 q^{49} +1.63073 q^{50} -3.87350 q^{52} -4.14290 q^{53} -0.188010 q^{55} +1.24396 q^{56} +6.71914 q^{58} -6.09859 q^{59} +5.74065 q^{61} -3.86586 q^{62} +1.00000 q^{64} +7.11003 q^{65} +1.55357 q^{67} +3.06169 q^{68} -2.28336 q^{70} +8.52403 q^{71} -12.0483 q^{73} -2.55426 q^{74} -4.97769 q^{76} -0.127415 q^{77} -13.0707 q^{79} -1.83556 q^{80} +1.65386 q^{82} -5.20439 q^{83} -5.61991 q^{85} +3.59805 q^{86} -0.102427 q^{88} +7.97976 q^{89} +4.81849 q^{91} -1.92036 q^{94} +9.13684 q^{95} -11.4726 q^{97} +5.45256 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 10 q^{2} + 10 q^{4} + 10 q^{5} + 2 q^{7} - 10 q^{8} - 10 q^{10} + 12 q^{11} - 2 q^{14} + 10 q^{16} + 24 q^{17} - 8 q^{19} + 10 q^{20} - 12 q^{22} + 8 q^{25} + 2 q^{28} - 4 q^{29} + 18 q^{31} - 10 q^{32}+ \cdots - 36 q^{98}+O(q^{100}) \) 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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