LMFDB - Embedded newform 9405.2.a.bq.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9405,2,Mod(1,9405)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9405, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("9405.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9405 = 3^{2} \cdot 5 \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9405.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$75.0993031010$
Analytic rank:	$1$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - x^{13} - 19 x^{12} + 15 x^{11} + 137 x^{10} - 80 x^{9} - 467 x^{8} + 193 x^{7} + 766 x^{6} + \cdots + 1 $ x^14 - x^13 - 19x^12 + 15x^11 + 137x^10 - 80x^9 - 467x^8 + 193x^7 + 766x^6 - 240x^5 - 567x^4 + 142x^3 + 142x^2 - 25x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.40446$ of defining polynomial
Character	$\chi$	$=$	9405.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.40446 q^{2} +3.78142 q^{4} +1.00000 q^{5} -4.79843 q^{7} -4.28335 q^{8} +O(q^{10})$	\(q-2.40446 q^{2} +3.78142 q^{4} +1.00000 q^{5} -4.79843 q^{7} -4.28335 q^{8} -2.40446 q^{10} +1.00000 q^{11} +1.23940 q^{13} +11.5376 q^{14} +2.73629 q^{16} -6.60114 q^{17} -1.00000 q^{19} +3.78142 q^{20} -2.40446 q^{22} -8.39666 q^{23} +1.00000 q^{25} -2.98007 q^{26} -18.1449 q^{28} +5.32926 q^{29} +1.77631 q^{31} +1.98740 q^{32} +15.8722 q^{34} -4.79843 q^{35} -0.732022 q^{37} +2.40446 q^{38} -4.28335 q^{40} +2.92685 q^{41} -3.28752 q^{43} +3.78142 q^{44} +20.1894 q^{46} +11.5487 q^{47} +16.0249 q^{49} -2.40446 q^{50} +4.68667 q^{52} +9.80557 q^{53} +1.00000 q^{55} +20.5533 q^{56} -12.8140 q^{58} +8.35026 q^{59} +1.56507 q^{61} -4.27106 q^{62} -10.2512 q^{64} +1.23940 q^{65} -6.17773 q^{67} -24.9617 q^{68} +11.5376 q^{70} +11.4491 q^{71} -8.77977 q^{73} +1.76012 q^{74} -3.78142 q^{76} -4.79843 q^{77} -7.93451 q^{79} +2.73629 q^{80} -7.03750 q^{82} +9.71873 q^{83} -6.60114 q^{85} +7.90471 q^{86} -4.28335 q^{88} +2.98640 q^{89} -5.94715 q^{91} -31.7513 q^{92} -27.7683 q^{94} -1.00000 q^{95} +18.6360 q^{97} -38.5313 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - q^{2} + 11 q^{4} + 14 q^{5} - 12 q^{7} - 9 q^{8}+O(q^{10}) $ 14 * q - q^2 + 11 * q^4 + 14 * q^5 - 12 * q^7 - 9 * q^8	$ 14 q - q^{2} + 11 q^{4} + 14 q^{5} - 12 q^{7} - 9 q^{8} - q^{10} + 14 q^{11} - 10 q^{13} + 8 q^{14} + 13 q^{16} - 16 q^{17} - 14 q^{19} + 11 q^{20} - q^{22} - 12 q^{23} + 14 q^{25} - 12 q^{26} - 33 q^{28} - 4 q^{31} - 24 q^{32} - 2 q^{34} - 12 q^{35} - 14 q^{37} + q^{38} - 9 q^{40} - 18 q^{41} - 20 q^{43} + 11 q^{44} - 17 q^{46} - 8 q^{47} + 10 q^{49} - q^{50} - 26 q^{52} - 20 q^{53} + 14 q^{55} + 11 q^{56} - 36 q^{58} + 2 q^{59} + 4 q^{61} - 38 q^{62} + 3 q^{64} - 10 q^{65} - 22 q^{67} - 48 q^{68} + 8 q^{70} - 28 q^{73} + 19 q^{74} - 11 q^{76} - 12 q^{77} - 14 q^{79} + 13 q^{80} - 24 q^{82} - 10 q^{83} - 16 q^{85} + 23 q^{86} - 9 q^{88} + 26 q^{89} - 42 q^{91} - 12 q^{92} - 56 q^{94} - 14 q^{95} - 32 q^{97} + 4 q^{98}+O(q^{100}) $ 14 * q - q^2 + 11 * q^4 + 14 * q^5 - 12 * q^7 - 9 * q^8 - q^10 + 14 * q^11 - 10 * q^13 + 8 * q^14 + 13 * q^16 - 16 * q^17 - 14 * q^19 + 11 * q^20 - q^22 - 12 * q^23 + 14 * q^25 - 12 * q^26 - 33 * q^28 - 4 * q^31 - 24 * q^32 - 2 * q^34 - 12 * q^35 - 14 * q^37 + q^38 - 9 * q^40 - 18 * q^41 - 20 * q^43 + 11 * q^44 - 17 * q^46 - 8 * q^47 + 10 * q^49 - q^50 - 26 * q^52 - 20 * q^53 + 14 * q^55 + 11 * q^56 - 36 * q^58 + 2 * q^59 + 4 * q^61 - 38 * q^62 + 3 * q^64 - 10 * q^65 - 22 * q^67 - 48 * q^68 + 8 * q^70 - 28 * q^73 + 19 * q^74 - 11 * q^76 - 12 * q^77 - 14 * q^79 + 13 * q^80 - 24 * q^82 - 10 * q^83 - 16 * q^85 + 23 * q^86 - 9 * q^88 + 26 * q^89 - 42 * q^91 - 12 * q^92 - 56 * q^94 - 14 * q^95 - 32 * q^97 + 4 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.40446	−1.70021	−0.850104	−	0.526614i	$-0.823461\pi$
$2$	−2.40446	−1.70021	−0.850104	+	0.526614i	$0.823461\pi$
$3$	0	0
$3$	0	0
$4$	3.78142	1.89071
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−4.79843	−1.81364	−0.906818	−	0.421522i	$-0.861496\pi$
$7$	−4.79843	−1.81364	−0.906818	+	0.421522i	$0.861496\pi$
$8$	−4.28335	−1.51439
$9$	0	0
$10$	−2.40446	−0.760356
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	1.23940	0.343746	0.171873	−	0.985119i	$-0.445018\pi$
$13$	1.23940	0.343746	0.171873	+	0.985119i	$0.445018\pi$
$14$	11.5376	3.08356
$15$	0	0
$16$	2.73629	0.684072
$17$	−6.60114	−1.60101	−0.800505	−	0.599326i	$-0.795435\pi$
$17$	−6.60114	−1.60101	−0.800505	+	0.599326i	$0.795435\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	3.78142	0.845551
$21$	0	0
$22$	−2.40446	−0.512632
$23$	−8.39666	−1.75083	−0.875413	−	0.483376i	$-0.839410\pi$
$23$	−8.39666	−1.75083	−0.875413	+	0.483376i	$0.839410\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−2.98007	−0.584441
$27$	0	0
$28$	−18.1449	−3.42906
$29$	5.32926	0.989619	0.494809	−	0.869002i	$-0.335238\pi$
$29$	5.32926	0.989619	0.494809	+	0.869002i	$0.335238\pi$
$30$	0	0
$31$	1.77631	0.319034	0.159517	−	0.987195i	$-0.449006\pi$
$31$	1.77631	0.319034	0.159517	+	0.987195i	$0.449006\pi$
$32$	1.98740	0.351326
$33$	0	0
$34$	15.8722	2.72205
$35$	−4.79843	−0.811083
$36$	0	0
$37$	−0.732022	−0.120344	−0.0601719	−	0.998188i	$-0.519165\pi$
$37$	−0.732022	−0.120344	−0.0601719	+	0.998188i	$0.519165\pi$
$38$	2.40446	0.390055
$39$	0	0
$40$	−4.28335	−0.677257
$41$	2.92685	0.457098	0.228549	−	0.973532i	$-0.426602\pi$
$41$	2.92685	0.457098	0.228549	+	0.973532i	$0.426602\pi$
$42$	0	0
$43$	−3.28752	−0.501342	−0.250671	−	0.968072i	$-0.580651\pi$
$43$	−3.28752	−0.501342	−0.250671	+	0.968072i	$0.580651\pi$
$44$	3.78142	0.570070
$45$	0	0
$46$	20.1894	2.97677
$47$	11.5487	1.68455	0.842273	−	0.539052i	$-0.181217\pi$
$47$	11.5487	1.68455	0.842273	+	0.539052i	$0.181217\pi$
$48$	0	0
$49$	16.0249	2.28928
$50$	−2.40446	−0.340042
$51$	0	0
$52$	4.68667	0.649925
$53$	9.80557	1.34690	0.673449	−	0.739234i	$-0.264812\pi$
$53$	9.80557	1.34690	0.673449	+	0.739234i	$0.264812\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	20.5533	2.74656
$57$	0	0
$58$	−12.8140	−1.68256
$59$	8.35026	1.08711	0.543556	−	0.839373i	$-0.317078\pi$
$59$	8.35026	1.08711	0.543556	+	0.839373i	$0.317078\pi$
$60$	0	0
$61$	1.56507	0.200387	0.100193	−	0.994968i	$-0.468054\pi$
$61$	1.56507	0.200387	0.100193	+	0.994968i	$0.468054\pi$
$62$	−4.27106	−0.542425
$63$	0	0
$64$	−10.2512	−1.28140
$65$	1.23940	0.153728
$66$	0	0
$67$	−6.17773	−0.754730	−0.377365	−	0.926065i	$-0.623170\pi$
$67$	−6.17773	−0.754730	−0.377365	+	0.926065i	$0.623170\pi$
$68$	−24.9617	−3.02705
$69$	0	0
$70$	11.5376	1.37901
$71$	11.4491	1.35876	0.679379	−	0.733788i	$-0.262249\pi$
$71$	11.4491	1.35876	0.679379	+	0.733788i	$0.262249\pi$
$72$	0	0
$73$	−8.77977	−1.02759	−0.513797	−	0.857912i	$-0.671762\pi$
$73$	−8.77977	−1.02759	−0.513797	+	0.857912i	$0.671762\pi$
$74$	1.76012	0.204609
$75$	0	0
$76$	−3.78142	−0.433758
$77$	−4.79843	−0.546832
$78$	0	0
$79$	−7.93451	−0.892702	−0.446351	−	0.894858i	$-0.647277\pi$
$79$	−7.93451	−0.892702	−0.446351	+	0.894858i	$0.647277\pi$
$80$	2.73629	0.305926
$81$	0	0
$82$	−7.03750	−0.777161
$83$	9.71873	1.06677	0.533384	−	0.845873i	$-0.320920\pi$
$83$	9.71873	1.06677	0.533384	+	0.845873i	$0.320920\pi$
$84$	0	0
$85$	−6.60114	−0.715994
$86$	7.90471	0.852387
$87$	0	0
$88$	−4.28335	−0.456606
$89$	2.98640	0.316558	0.158279	−	0.987394i	$-0.449405\pi$
$89$	2.98640	0.316558	0.158279	+	0.987394i	$0.449405\pi$
$90$	0	0
$91$	−5.94715	−0.623431
$92$	−31.7513	−3.31030
$93$	0	0
$94$	−27.7683	−2.86408
$95$	−1.00000	−0.102598
$96$	0	0
$97$	18.6360	1.89220	0.946101	−	0.323871i	$-0.104985\pi$
$97$	18.6360	1.89220	0.946101	+	0.323871i	$0.104985\pi$
$98$	−38.5313	−3.89225
$99$	0	0
$100$	3.78142	0.378142
$101$	−15.5331	−1.54561	−0.772803	−	0.634646i	$-0.781146\pi$
$101$	−15.5331	−1.54561	−0.772803	+	0.634646i	$0.781146\pi$
$102$	0	0
$103$	−9.07442	−0.894129	−0.447065	−	0.894502i	$-0.647531\pi$
$103$	−9.07442	−0.894129	−0.447065	+	0.894502i	$0.647531\pi$
$104$	−5.30876	−0.520567
$105$	0	0
$106$	−23.5771	−2.29001
$107$	14.9297	1.44331	0.721655	−	0.692253i	$-0.243382\pi$
$107$	14.9297	1.44331	0.721655	+	0.692253i	$0.243382\pi$
$108$	0	0
$109$	−2.73194	−0.261672	−0.130836	−	0.991404i	$-0.541766\pi$
$109$	−2.73194	−0.261672	−0.130836	+	0.991404i	$0.541766\pi$
$110$	−2.40446	−0.229256
$111$	0	0
$112$	−13.1299	−1.24066
$113$	−7.95220	−0.748080	−0.374040	−	0.927413i	$-0.622028\pi$
$113$	−7.95220	−0.748080	−0.374040	+	0.927413i	$0.622028\pi$
$114$	0	0
$115$	−8.39666	−0.782993
$116$	20.1522	1.87108
$117$	0	0
$118$	−20.0778	−1.84832
$119$	31.6751	2.90365
$120$	0	0
$121$	1.00000	0.0909091
$122$	−3.76314	−0.340699
$123$	0	0
$124$	6.71696	0.603201
$125$	1.00000	0.0894427
$126$	0	0
$127$	9.78189	0.868002	0.434001	−	0.900912i	$-0.357101\pi$
$127$	9.78189	0.868002	0.434001	+	0.900912i	$0.357101\pi$
$128$	20.6738	1.82732
$129$	0	0
$130$	−2.98007	−0.261370
$131$	13.6781	1.19506	0.597530	−	0.801846i	$-0.296149\pi$
$131$	13.6781	1.19506	0.597530	+	0.801846i	$0.296149\pi$
$132$	0	0
$133$	4.79843	0.416077
$134$	14.8541	1.28320
$135$	0	0
$136$	28.2750	2.42456
$137$	15.2303	1.30121	0.650605	−	0.759417i	$-0.274516\pi$
$137$	15.2303	1.30121	0.650605	+	0.759417i	$0.274516\pi$
$138$	0	0
$139$	−6.80912	−0.577542	−0.288771	−	0.957398i	$-0.593247\pi$
$139$	−6.80912	−0.577542	−0.288771	+	0.957398i	$0.593247\pi$
$140$	−18.1449	−1.53352
$141$	0	0
$142$	−27.5289	−2.31017
$143$	1.23940	0.103643
$144$	0	0
$145$	5.32926	0.442571
$146$	21.1106	1.74712
$147$	0	0
$148$	−2.76808	−0.227535
$149$	−8.34329	−0.683509	−0.341755	−	0.939789i	$-0.611021\pi$
$149$	−8.34329	−0.683509	−0.341755	+	0.939789i	$0.611021\pi$
$150$	0	0
$151$	14.1799	1.15395	0.576974	−	0.816763i	$-0.304233\pi$
$151$	14.1799	1.15395	0.576974	+	0.816763i	$0.304233\pi$
$152$	4.28335	0.347425
$153$	0	0
$154$	11.5376	0.929729
$155$	1.77631	0.142676
$156$	0	0
$157$	−3.59527	−0.286934	−0.143467	−	0.989655i	$-0.545825\pi$
$157$	−3.59527	−0.286934	−0.143467	+	0.989655i	$0.545825\pi$
$158$	19.0782	1.51778
$159$	0	0
$160$	1.98740	0.157118
$161$	40.2908	3.17536
$162$	0	0
$163$	−0.166356	−0.0130300	−0.00651501	−	0.999979i	$-0.502074\pi$
$163$	−0.166356	−0.0130300	−0.00651501	+	0.999979i	$0.502074\pi$
$164$	11.0677	0.864239
$165$	0	0
$166$	−23.3683	−1.81373
$167$	−11.9210	−0.922477	−0.461238	−	0.887276i	$-0.652595\pi$
$167$	−11.9210	−0.922477	−0.461238	+	0.887276i	$0.652595\pi$
$168$	0	0
$169$	−11.4639	−0.881838
$170$	15.8722	1.21734
$171$	0	0
$172$	−12.4315	−0.947893
$173$	12.5436	0.953669	0.476834	−	0.878993i	$-0.341784\pi$
$173$	12.5436	0.953669	0.476834	+	0.878993i	$0.341784\pi$
$174$	0	0
$175$	−4.79843	−0.362727
$176$	2.73629	0.206256
$177$	0	0
$178$	−7.18068	−0.538215
$179$	−9.48987	−0.709306	−0.354653	−	0.934998i	$-0.615401\pi$
$179$	−9.48987	−0.709306	−0.354653	+	0.934998i	$0.615401\pi$
$180$	0	0
$181$	1.03661	0.0770509	0.0385255	−	0.999258i	$-0.487734\pi$
$181$	1.03661	0.0770509	0.0385255	+	0.999258i	$0.487734\pi$
$182$	14.2997	1.05996
$183$	0	0
$184$	35.9658	2.65144
$185$	−0.732022	−0.0538193
$186$	0	0
$187$	−6.60114	−0.482723
$188$	43.6703	3.18499
$189$	0	0
$190$	2.40446	0.174438
$191$	−19.3372	−1.39919	−0.699595	−	0.714540i	$-0.746636\pi$
$191$	−19.3372	−1.39919	−0.699595	+	0.714540i	$0.746636\pi$
$192$	0	0
$193$	−25.7569	−1.85402	−0.927011	−	0.375035i	$-0.877631\pi$
$193$	−25.7569	−1.85402	−0.927011	+	0.375035i	$0.877631\pi$
$194$	−44.8096	−3.21714
$195$	0	0
$196$	60.5970	4.32836
$197$	−18.1459	−1.29284	−0.646422	−	0.762980i	$-0.723735\pi$
$197$	−18.1459	−1.29284	−0.646422	+	0.762980i	$0.723735\pi$
$198$	0	0
$199$	−13.0407	−0.924433	−0.462216	−	0.886767i	$-0.652946\pi$
$199$	−13.0407	−0.924433	−0.462216	+	0.886767i	$0.652946\pi$
$200$	−4.28335	−0.302878
$201$	0	0
$202$	37.3488	2.62785
$203$	−25.5721	−1.79481
$204$	0	0
$205$	2.92685	0.204420
$206$	21.8191	1.52021
$207$	0	0
$208$	3.39134	0.235147
$209$	−1.00000	−0.0691714
$210$	0	0
$211$	−22.4611	−1.54628	−0.773141	−	0.634234i	$-0.781316\pi$
$211$	−22.4611	−1.54628	−0.773141	+	0.634234i	$0.781316\pi$
$212$	37.0790	2.54659
$213$	0	0
$214$	−35.8979	−2.45393
$215$	−3.28752	−0.224207
$216$	0	0
$217$	−8.52349	−0.578612
$218$	6.56883	0.444897
$219$	0	0
$220$	3.78142	0.254943
$221$	−8.18142	−0.550342
$222$	0	0
$223$	−24.2626	−1.62475	−0.812373	−	0.583138i	$-0.801825\pi$
$223$	−24.2626	−1.62475	−0.812373	+	0.583138i	$0.801825\pi$
$224$	−9.53640	−0.637178
$225$	0	0
$226$	19.1207	1.27189
$227$	0.683640	0.0453748	0.0226874	−	0.999743i	$-0.492778\pi$
$227$	0.683640	0.0453748	0.0226874	+	0.999743i	$0.492778\pi$
$228$	0	0
$229$	12.9398	0.855088	0.427544	−	0.903995i	$-0.359379\pi$
$229$	12.9398	0.855088	0.427544	+	0.903995i	$0.359379\pi$
$230$	20.1894	1.33125
$231$	0	0
$232$	−22.8271	−1.49867
$233$	−19.0760	−1.24971	−0.624855	−	0.780741i	$-0.714842\pi$
$233$	−19.0760	−1.24971	−0.624855	+	0.780741i	$0.714842\pi$
$234$	0	0
$235$	11.5487	0.753352
$236$	31.5758	2.05541
$237$	0	0
$238$	−76.1615	−4.93681
$239$	−21.1192	−1.36609	−0.683044	−	0.730377i	$-0.739344\pi$
$239$	−21.1192	−1.36609	−0.683044	+	0.730377i	$0.739344\pi$
$240$	0	0
$241$	19.2799	1.24193	0.620964	−	0.783839i	$-0.286742\pi$
$241$	19.2799	1.24193	0.620964	+	0.783839i	$0.286742\pi$
$242$	−2.40446	−0.154564
$243$	0	0
$244$	5.91818	0.378873
$245$	16.0249	1.02380
$246$	0	0
$247$	−1.23940	−0.0788608
$248$	−7.60854	−0.483143
$249$	0	0
$250$	−2.40446	−0.152071
$251$	−10.4160	−0.657454	−0.328727	−	0.944425i	$-0.606620\pi$
$251$	−10.4160	−0.657454	−0.328727	+	0.944425i	$0.606620\pi$
$252$	0	0
$253$	−8.39666	−0.527894
$254$	−23.5201	−1.47578
$255$	0	0
$256$	−29.2068	−1.82543
$257$	11.2018	0.698749	0.349375	−	0.936983i	$-0.386394\pi$
$257$	11.2018	0.698749	0.349375	+	0.936983i	$0.386394\pi$
$258$	0	0
$259$	3.51256	0.218260
$260$	4.68667	0.290655
$261$	0	0
$262$	−32.8884	−2.03185
$263$	12.9977	0.801475	0.400737	−	0.916193i	$-0.368754\pi$
$263$	12.9977	0.801475	0.400737	+	0.916193i	$0.368754\pi$
$264$	0	0
$265$	9.80557	0.602351
$266$	−11.5376	−0.707417
$267$	0	0
$268$	−23.3606	−1.42697
$269$	9.90715	0.604050	0.302025	−	0.953300i	$-0.402337\pi$
$269$	9.90715	0.604050	0.302025	+	0.953300i	$0.402337\pi$
$270$	0	0
$271$	6.01615	0.365455	0.182728	−	0.983164i	$-0.441507\pi$
$271$	6.01615	0.365455	0.182728	+	0.983164i	$0.441507\pi$
$272$	−18.0626	−1.09521
$273$	0	0
$274$	−36.6205	−2.21233
$275$	1.00000	0.0603023
$276$	0	0
$277$	26.5244	1.59370	0.796848	−	0.604180i	$-0.206499\pi$
$277$	26.5244	1.59370	0.796848	+	0.604180i	$0.206499\pi$
$278$	16.3723	0.981943
$279$	0	0
$280$	20.5533	1.22830
$281$	−16.9232	−1.00956	−0.504778	−	0.863249i	$-0.668426\pi$
$281$	−16.9232	−1.00956	−0.504778	+	0.863249i	$0.668426\pi$
$282$	0	0
$283$	25.6564	1.52512	0.762558	−	0.646919i	$-0.223943\pi$
$283$	25.6564	1.52512	0.762558	+	0.646919i	$0.223943\pi$
$284$	43.2938	2.56901
$285$	0	0
$286$	−2.98007	−0.176215
$287$	−14.0443	−0.829009
$288$	0	0
$289$	26.5750	1.56324
$290$	−12.8140	−0.752463
$291$	0	0
$292$	−33.2000	−1.94288
$293$	21.8755	1.27798	0.638990	−	0.769215i	$-0.279353\pi$
$293$	21.8755	1.27798	0.638990	+	0.769215i	$0.279353\pi$
$294$	0	0
$295$	8.35026	0.486171
$296$	3.13550	0.182248
$297$	0	0
$298$	20.0611	1.16211
$299$	−10.4068	−0.601840
$300$	0	0
$301$	15.7750	0.909253
$302$	−34.0951	−1.96195
$303$	0	0
$304$	−2.73629	−0.156937
$305$	1.56507	0.0896156
$306$	0	0
$307$	−24.0394	−1.37200	−0.686001	−	0.727600i	$-0.740636\pi$
$307$	−24.0394	−1.37200	−0.686001	+	0.727600i	$0.740636\pi$
$308$	−18.1449	−1.03390
$309$	0	0
$310$	−4.27106	−0.242580
$311$	−7.72138	−0.437839	−0.218920	−	0.975743i	$-0.570253\pi$
$311$	−7.72138	−0.437839	−0.218920	+	0.975743i	$0.570253\pi$
$312$	0	0
$313$	−11.8314	−0.668752	−0.334376	−	0.942440i	$-0.608526\pi$
$313$	−11.8314	−0.668752	−0.334376	+	0.942440i	$0.608526\pi$
$314$	8.64467	0.487847
$315$	0	0
$316$	−30.0037	−1.68784
$317$	17.4706	0.981245	0.490622	−	0.871372i	$-0.336769\pi$
$317$	17.4706	0.981245	0.490622	+	0.871372i	$0.336769\pi$
$318$	0	0
$319$	5.32926	0.298381
$320$	−10.2512	−0.573059
$321$	0	0
$322$	−96.8776	−5.39878
$323$	6.60114	0.367297
$324$	0	0
$325$	1.23940	0.0687493
$326$	0.399996	0.0221537
$327$	0	0
$328$	−12.5367	−0.692225
$329$	−55.4155	−3.05515
$330$	0	0
$331$	−16.2117	−0.891075	−0.445538	−	0.895263i	$-0.646987\pi$
$331$	−16.2117	−0.891075	−0.445538	+	0.895263i	$0.646987\pi$
$332$	36.7506	2.01695
$333$	0	0
$334$	28.6636	1.56840
$335$	−6.17773	−0.337525
$336$	0	0
$337$	−6.76635	−0.368587	−0.184293	−	0.982871i	$-0.559000\pi$
$337$	−6.76635	−0.368587	−0.184293	+	0.982871i	$0.559000\pi$
$338$	27.5645	1.49931
$339$	0	0
$340$	−24.9617	−1.35374
$341$	1.77631	0.0961925
$342$	0	0
$343$	−43.3056	−2.33828
$344$	14.0816	0.759229
$345$	0	0
$346$	−30.1604	−1.62144
$347$	−33.2159	−1.78312	−0.891561	−	0.452901i	$-0.850389\pi$
$347$	−33.2159	−1.78312	−0.891561	+	0.452901i	$0.850389\pi$
$348$	0	0
$349$	−26.3456	−1.41025	−0.705123	−	0.709085i	$-0.749108\pi$
$349$	−26.3456	−1.41025	−0.705123	+	0.709085i	$0.749108\pi$
$350$	11.5376	0.616712
$351$	0	0
$352$	1.98740	0.105929
$353$	1.00149	0.0533038	0.0266519	−	0.999645i	$-0.491515\pi$
$353$	1.00149	0.0533038	0.0266519	+	0.999645i	$0.491515\pi$
$354$	0	0
$355$	11.4491	0.607655
$356$	11.2928	0.598519
$357$	0	0
$358$	22.8180	1.20597
$359$	−5.86337	−0.309457	−0.154728	−	0.987957i	$-0.549450\pi$
$359$	−5.86337	−0.309457	−0.154728	+	0.987957i	$0.549450\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−2.49249	−0.131003
$363$	0	0
$364$	−22.4887	−1.17873
$365$	−8.77977	−0.459554
$366$	0	0
$367$	−1.76049	−0.0918970	−0.0459485	−	0.998944i	$-0.514631\pi$
$367$	−1.76049	−0.0918970	−0.0459485	+	0.998944i	$0.514631\pi$
$368$	−22.9757	−1.19769
$369$	0	0
$370$	1.76012	0.0915041
$371$	−47.0513	−2.44278
$372$	0	0
$373$	3.13070	0.162101	0.0810506	−	0.996710i	$-0.474172\pi$
$373$	3.13070	0.162101	0.0810506	+	0.996710i	$0.474172\pi$
$374$	15.8722	0.820730
$375$	0	0
$376$	−49.4669	−2.55106
$377$	6.60506	0.340178
$378$	0	0
$379$	−28.0303	−1.43982	−0.719909	−	0.694068i	$-0.755817\pi$
$379$	−28.0303	−1.43982	−0.719909	+	0.694068i	$0.755817\pi$
$380$	−3.78142	−0.193983
$381$	0	0
$382$	46.4955	2.37892
$383$	−38.7041	−1.97769	−0.988843	−	0.148959i	$-0.952408\pi$
$383$	−38.7041	−1.97769	−0.988843	+	0.148959i	$0.952408\pi$
$384$	0	0
$385$	−4.79843	−0.244551
$386$	61.9314	3.15222
$387$	0	0
$388$	70.4706	3.57760
$389$	33.0376	1.67507	0.837536	−	0.546382i	$-0.183995\pi$
$389$	33.0376	1.67507	0.837536	+	0.546382i	$0.183995\pi$
$390$	0	0
$391$	55.4275	2.80309
$392$	−68.6404	−3.46686
$393$	0	0
$394$	43.6311	2.19810
$395$	−7.93451	−0.399228
$396$	0	0
$397$	8.03270	0.403150	0.201575	−	0.979473i	$-0.435394\pi$
$397$	8.03270	0.403150	0.201575	+	0.979473i	$0.435394\pi$
$398$	31.3559	1.57173
$399$	0	0
$400$	2.73629	0.136814
$401$	3.51084	0.175323	0.0876614	−	0.996150i	$-0.472061\pi$
$401$	3.51084	0.175323	0.0876614	+	0.996150i	$0.472061\pi$
$402$	0	0
$403$	2.20155	0.109667
$404$	−58.7373	−2.92229
$405$	0	0
$406$	61.4870	3.05155
$407$	−0.732022	−0.0362850
$408$	0	0
$409$	5.28989	0.261568	0.130784	−	0.991411i	$-0.458251\pi$
$409$	5.28989	0.261568	0.130784	+	0.991411i	$0.458251\pi$
$410$	−7.03750	−0.347557
$411$	0	0
$412$	−34.3142	−1.69054
$413$	−40.0682	−1.97162
$414$	0	0
$415$	9.71873	0.477074
$416$	2.46317	0.120767
$417$	0	0
$418$	2.40446	0.117606
$419$	−31.1475	−1.52166	−0.760828	−	0.648954i	$-0.775207\pi$
$419$	−31.1475	−1.52166	−0.760828	+	0.648954i	$0.775207\pi$
$420$	0	0
$421$	−17.9740	−0.875997	−0.437998	−	0.898976i	$-0.644312\pi$
$421$	−17.9740	−0.875997	−0.437998	+	0.898976i	$0.644312\pi$
$422$	54.0067	2.62900
$423$	0	0
$424$	−42.0006	−2.03973
$425$	−6.60114	−0.320202
$426$	0	0
$427$	−7.50988	−0.363428
$428$	56.4555	2.72888
$429$	0	0
$430$	7.90471	0.381199
$431$	35.4475	1.70745	0.853723	−	0.520728i	$-0.174340\pi$
$431$	35.4475	1.70745	0.853723	+	0.520728i	$0.174340\pi$
$432$	0	0
$433$	21.9465	1.05468	0.527341	−	0.849654i	$-0.323189\pi$
$433$	21.9465	1.05468	0.527341	+	0.849654i	$0.323189\pi$
$434$	20.4944	0.983762
$435$	0	0
$436$	−10.3306	−0.494746
$437$	8.39666	0.401667
$438$	0	0
$439$	−0.622734	−0.0297215	−0.0148607	−	0.999890i	$-0.504730\pi$
$439$	−0.622734	−0.0297215	−0.0148607	+	0.999890i	$0.504730\pi$
$440$	−4.28335	−0.204201
$441$	0	0
$442$	19.6719	0.935696
$443$	−34.2988	−1.62959	−0.814793	−	0.579753i	$-0.803149\pi$
$443$	−34.2988	−1.62959	−0.814793	+	0.579753i	$0.803149\pi$
$444$	0	0
$445$	2.98640	0.141569
$446$	58.3385	2.76241
$447$	0	0
$448$	49.1897	2.32399
$449$	25.2965	1.19381	0.596907	−	0.802310i	$-0.296396\pi$
$449$	25.2965	1.19381	0.596907	+	0.802310i	$0.296396\pi$
$450$	0	0
$451$	2.92685	0.137820
$452$	−30.0706	−1.41440
$453$	0	0
$454$	−1.64378	−0.0771466
$455$	−5.94715	−0.278807
$456$	0	0
$457$	−9.42039	−0.440667	−0.220334	−	0.975425i	$-0.570715\pi$
$457$	−9.42039	−0.440667	−0.220334	+	0.975425i	$0.570715\pi$
$458$	−31.1133	−1.45383
$459$	0	0
$460$	−31.7513	−1.48041
$461$	−21.7605	−1.01349	−0.506743	−	0.862097i	$-0.669151\pi$
$461$	−21.7605	−1.01349	−0.506743	+	0.862097i	$0.669151\pi$
$462$	0	0
$463$	16.4153	0.762885	0.381443	−	0.924392i	$-0.375427\pi$
$463$	16.4153	0.762885	0.381443	+	0.924392i	$0.375427\pi$
$464$	14.5824	0.676971
$465$	0	0
$466$	45.8674	2.12477
$467$	35.7688	1.65518	0.827592	−	0.561330i	$-0.189710\pi$
$467$	35.7688	1.65518	0.827592	+	0.561330i	$0.189710\pi$
$468$	0	0
$469$	29.6434	1.36881
$470$	−27.7683	−1.28085
$471$	0	0
$472$	−35.7671	−1.64631
$473$	−3.28752	−0.151160
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	119.777	5.48996
$477$	0	0
$478$	50.7803	2.32264
$479$	13.6114	0.621922	0.310961	−	0.950423i	$-0.399349\pi$
$479$	13.6114	0.621922	0.310961	+	0.950423i	$0.399349\pi$
$480$	0	0
$481$	−0.907265	−0.0413677
$482$	−46.3577	−2.11154
$483$	0	0
$484$	3.78142	0.171883
$485$	18.6360	0.846219
$486$	0	0
$487$	−5.45767	−0.247310	−0.123655	−	0.992325i	$-0.539462\pi$
$487$	−5.45767	−0.247310	−0.123655	+	0.992325i	$0.539462\pi$
$488$	−6.70373	−0.303464
$489$	0	0
$490$	−38.5313	−1.74067
$491$	6.56426	0.296241	0.148120	−	0.988969i	$-0.452678\pi$
$491$	6.56426	0.296241	0.148120	+	0.988969i	$0.452678\pi$
$492$	0	0
$493$	−35.1792	−1.58439
$494$	2.98007	0.134080
$495$	0	0
$496$	4.86049	0.218243
$497$	−54.9377	−2.46429
$498$	0	0
$499$	12.3668	0.553613	0.276807	−	0.960926i	$-0.410724\pi$
$499$	12.3668	0.553613	0.276807	+	0.960926i	$0.410724\pi$
$500$	3.78142	0.169110
$501$	0	0
$502$	25.0449	1.11781
$503$	−35.1042	−1.56522	−0.782611	−	0.622512i	$-0.786112\pi$
$503$	−35.1042	−1.56522	−0.782611	+	0.622512i	$0.786112\pi$
$504$	0	0
$505$	−15.5331	−0.691216
$506$	20.1894	0.897530
$507$	0	0
$508$	36.9894	1.64114
$509$	7.15309	0.317055	0.158528	−	0.987355i	$-0.449325\pi$
$509$	7.15309	0.317055	0.158528	+	0.987355i	$0.449325\pi$
$510$	0	0
$511$	42.1291	1.86368
$512$	28.8791	1.27629
$513$	0	0
$514$	−26.9343	−1.18802
$515$	−9.07442	−0.399867
$516$	0	0
$517$	11.5487	0.507910
$518$	−8.44580	−0.371087
$519$	0	0
$520$	−5.30876	−0.232805
$521$	36.0936	1.58129	0.790645	−	0.612275i	$-0.209745\pi$
$521$	36.0936	1.58129	0.790645	+	0.612275i	$0.209745\pi$
$522$	0	0
$523$	−13.4347	−0.587459	−0.293730	−	0.955889i	$-0.594896\pi$
$523$	−13.4347	−0.587459	−0.293730	+	0.955889i	$0.594896\pi$
$524$	51.7226	2.25951
$525$	0	0
$526$	−31.2525	−1.36267
$527$	−11.7257	−0.510777
$528$	0	0
$529$	47.5040	2.06539
$530$	−23.5771	−1.02412
$531$	0	0
$532$	18.1449	0.786680
$533$	3.62753	0.157126
$534$	0	0
$535$	14.9297	0.645468
$536$	26.4613	1.14296
$537$	0	0
$538$	−23.8213	−1.02701
$539$	16.0249	0.690243
$540$	0	0
$541$	15.2615	0.656144	0.328072	−	0.944653i	$-0.393601\pi$
$541$	15.2615	0.656144	0.328072	+	0.944653i	$0.393601\pi$
$542$	−14.4656	−0.621350
$543$	0	0
$544$	−13.1191	−0.562476
$545$	−2.73194	−0.117023
$546$	0	0
$547$	−21.3730	−0.913845	−0.456922	−	0.889507i	$-0.651048\pi$
$547$	−21.3730	−0.913845	−0.456922	+	0.889507i	$0.651048\pi$
$548$	57.5920	2.46021
$549$	0	0
$550$	−2.40446	−0.102526
$551$	−5.32926	−0.227034
$552$	0	0
$553$	38.0732	1.61904
$554$	−63.7768	−2.70962
$555$	0	0
$556$	−25.7481	−1.09196
$557$	10.4544	0.442967	0.221483	−	0.975164i	$-0.428910\pi$
$557$	10.4544	0.442967	0.221483	+	0.975164i	$0.428910\pi$
$558$	0	0
$559$	−4.07454	−0.172335
$560$	−13.1299	−0.554839
$561$	0	0
$562$	40.6912	1.71646
$563$	−28.7249	−1.21061	−0.605306	−	0.795993i	$-0.706949\pi$
$563$	−28.7249	−1.21061	−0.605306	+	0.795993i	$0.706949\pi$
$564$	0	0
$565$	−7.95220	−0.334552
$566$	−61.6898	−2.59302
$567$	0	0
$568$	−49.0404	−2.05769
$569$	−28.6060	−1.19923	−0.599613	−	0.800290i	$-0.704679\pi$
$569$	−28.6060	−1.19923	−0.599613	+	0.800290i	$0.704679\pi$
$570$	0	0
$571$	24.5938	1.02922	0.514608	−	0.857425i	$-0.327937\pi$
$571$	24.5938	1.02922	0.514608	+	0.857425i	$0.327937\pi$
$572$	4.68667	0.195960
$573$	0	0
$574$	33.7689	1.40949
$575$	−8.39666	−0.350165
$576$	0	0
$577$	20.3821	0.848517	0.424258	−	0.905541i	$-0.360535\pi$
$577$	20.3821	0.848517	0.424258	+	0.905541i	$0.360535\pi$
$578$	−63.8985	−2.65783
$579$	0	0
$580$	20.1522	0.836773
$581$	−46.6346	−1.93473
$582$	0	0
$583$	9.80557	0.406105
$584$	37.6068	1.55618
$585$	0	0
$586$	−52.5987	−2.17283
$587$	32.8752	1.35691	0.678453	−	0.734644i	$-0.262651\pi$
$587$	32.8752	1.35691	0.678453	+	0.734644i	$0.262651\pi$
$588$	0	0
$589$	−1.77631	−0.0731915
$590$	−20.0778	−0.826592
$591$	0	0
$592$	−2.00302	−0.0823238
$593$	−1.11824	−0.0459206	−0.0229603	−	0.999736i	$-0.507309\pi$
$593$	−1.11824	−0.0459206	−0.0229603	+	0.999736i	$0.507309\pi$
$594$	0	0
$595$	31.6751	1.29855
$596$	−31.5495	−1.29232
$597$	0	0
$598$	25.0227	1.02325
$599$	−17.4547	−0.713181	−0.356590	−	0.934261i	$-0.616061\pi$
$599$	−17.4547	−0.713181	−0.356590	+	0.934261i	$0.616061\pi$
$600$	0	0
$601$	−12.7832	−0.521436	−0.260718	−	0.965415i	$-0.583959\pi$
$601$	−12.7832	−0.521436	−0.260718	+	0.965415i	$0.583959\pi$
$602$	−37.9302	−1.54592
$603$	0	0
$604$	53.6203	2.18178
$605$	1.00000	0.0406558
$606$	0	0
$607$	26.3432	1.06924	0.534619	−	0.845093i	$-0.320455\pi$
$607$	26.3432	1.06924	0.534619	+	0.845093i	$0.320455\pi$
$608$	−1.98740	−0.0805997
$609$	0	0
$610$	−3.76314	−0.152365
$611$	14.3134	0.579057
$612$	0	0
$613$	13.9380	0.562952	0.281476	−	0.959568i	$-0.409176\pi$
$613$	13.9380	0.562952	0.281476	+	0.959568i	$0.409176\pi$
$614$	57.8018	2.33269
$615$	0	0
$616$	20.5533	0.828118
$617$	7.08082	0.285063	0.142531	−	0.989790i	$-0.454476\pi$
$617$	7.08082	0.285063	0.142531	+	0.989790i	$0.454476\pi$
$618$	0	0
$619$	−40.9716	−1.64679	−0.823394	−	0.567471i	$-0.807922\pi$
$619$	−40.9716	−1.64679	−0.823394	+	0.567471i	$0.807922\pi$
$620$	6.71696	0.269760
$621$	0	0
$622$	18.5657	0.744418
$623$	−14.3301	−0.574121
$624$	0	0
$625$	1.00000	0.0400000
$626$	28.4482	1.13702
$627$	0	0
$628$	−13.5952	−0.542508
$629$	4.83218	0.192672
$630$	0	0
$631$	25.3582	1.00949	0.504747	−	0.863267i	$-0.331586\pi$
$631$	25.3582	1.00949	0.504747	+	0.863267i	$0.331586\pi$
$632$	33.9862	1.35190
$633$	0	0
$634$	−42.0073	−1.66832
$635$	9.78189	0.388182
$636$	0	0
$637$	19.8612	0.786931
$638$	−12.8140	−0.507310
$639$	0	0
$640$	20.6738	0.817203
$641$	12.3188	0.486563	0.243281	−	0.969956i	$-0.421776\pi$
$641$	12.3188	0.486563	0.243281	+	0.969956i	$0.421776\pi$
$642$	0	0
$643$	8.45730	0.333524	0.166762	−	0.985997i	$-0.446669\pi$
$643$	8.45730	0.333524	0.166762	+	0.985997i	$0.446669\pi$
$644$	152.356	6.00369
$645$	0	0
$646$	−15.8722	−0.624482
$647$	5.61144	0.220608	0.110304	−	0.993898i	$-0.464817\pi$
$647$	5.61144	0.220608	0.110304	+	0.993898i	$0.464817\pi$
$648$	0	0
$649$	8.35026	0.327776
$650$	−2.98007	−0.116888
$651$	0	0
$652$	−0.629062	−0.0246360
$653$	−44.7718	−1.75206	−0.876028	−	0.482260i	$-0.839816\pi$
$653$	−44.7718	−1.75206	−0.876028	+	0.482260i	$0.839816\pi$
$654$	0	0
$655$	13.6781	0.534447
$656$	8.00872	0.312688
$657$	0	0
$658$	133.244	5.19440
$659$	40.0450	1.55993	0.779965	−	0.625823i	$-0.215237\pi$
$659$	40.0450	1.55993	0.779965	+	0.625823i	$0.215237\pi$
$660$	0	0
$661$	−31.2038	−1.21369	−0.606843	−	0.794821i	$-0.707565\pi$
$661$	−31.2038	−1.21369	−0.606843	+	0.794821i	$0.707565\pi$
$662$	38.9803	1.51501
$663$	0	0
$664$	−41.6287	−1.61551
$665$	4.79843	0.186075
$666$	0	0
$667$	−44.7480	−1.73265
$668$	−45.0784	−1.74414
$669$	0	0
$670$	14.8541	0.573863
$671$	1.56507	0.0604188
$672$	0	0
$673$	−25.7975	−0.994421	−0.497210	−	0.867630i	$-0.665642\pi$
$673$	−25.7975	−0.994421	−0.497210	+	0.867630i	$0.665642\pi$
$674$	16.2694	0.626674
$675$	0	0
$676$	−43.3498	−1.66730
$677$	−12.9888	−0.499199	−0.249600	−	0.968349i	$-0.580299\pi$
$677$	−12.9888	−0.499199	−0.249600	+	0.968349i	$0.580299\pi$
$678$	0	0
$679$	−89.4237	−3.43177
$680$	28.2750	1.08429
$681$	0	0
$682$	−4.27106	−0.163547
$683$	6.94222	0.265637	0.132818	−	0.991140i	$-0.457597\pi$
$683$	6.94222	0.265637	0.132818	+	0.991140i	$0.457597\pi$
$684$	0	0
$685$	15.2303	0.581919
$686$	104.126	3.97557
$687$	0	0
$688$	−8.99561	−0.342955
$689$	12.1530	0.462991
$690$	0	0
$691$	−21.8893	−0.832707	−0.416354	−	0.909203i	$-0.636692\pi$
$691$	−21.8893	−0.832707	−0.416354	+	0.909203i	$0.636692\pi$
$692$	47.4324	1.80311
$693$	0	0
$694$	79.8662	3.03168
$695$	−6.80912	−0.258285
$696$	0	0
$697$	−19.3206	−0.731818
$698$	63.3468	2.39771
$699$	0	0
$700$	−18.1449	−0.685812
$701$	−21.6988	−0.819551	−0.409775	−	0.912187i	$-0.634393\pi$
$701$	−21.6988	−0.819551	−0.409775	+	0.912187i	$0.634393\pi$
$702$	0	0
$703$	0.732022	0.0276087
$704$	−10.2512	−0.386357
$705$	0	0
$706$	−2.40804	−0.0906277
$707$	74.5348	2.80317
$708$	0	0
$709$	29.1069	1.09313	0.546567	−	0.837415i	$-0.315934\pi$
$709$	29.1069	1.09313	0.546567	+	0.837415i	$0.315934\pi$
$710$	−27.5289	−1.03314
$711$	0	0
$712$	−12.7918	−0.479393
$713$	−14.9151	−0.558573
$714$	0	0
$715$	1.23940	0.0463508
$716$	−35.8852	−1.34109
$717$	0	0
$718$	14.0982	0.526141
$719$	3.14629	0.117337	0.0586683	−	0.998278i	$-0.481315\pi$
$719$	3.14629	0.117337	0.0586683	+	0.998278i	$0.481315\pi$
$720$	0	0
$721$	43.5430	1.62163
$722$	−2.40446	−0.0894847
$723$	0	0
$724$	3.91987	0.145681
$725$	5.32926	0.197924
$726$	0	0
$727$	−10.6642	−0.395513	−0.197756	−	0.980251i	$-0.563366\pi$
$727$	−10.6642	−0.395513	−0.197756	+	0.980251i	$0.563366\pi$
$728$	25.4737	0.944119
$729$	0	0
$730$	21.1106	0.781338
$731$	21.7014	0.802655
$732$	0	0
$733$	−40.9175	−1.51132	−0.755660	−	0.654964i	$-0.772684\pi$
$733$	−40.9175	−1.51132	−0.755660	+	0.654964i	$0.772684\pi$
$734$	4.23303	0.156244
$735$	0	0
$736$	−16.6875	−0.615110
$737$	−6.17773	−0.227560
$738$	0	0
$739$	−37.8430	−1.39208	−0.696038	−	0.718005i	$-0.745056\pi$
$739$	−37.8430	−1.39208	−0.696038	+	0.718005i	$0.745056\pi$
$740$	−2.76808	−0.101757
$741$	0	0
$742$	113.133	4.15324
$743$	−43.1314	−1.58234	−0.791168	−	0.611599i	$-0.790527\pi$
$743$	−43.1314	−1.58234	−0.791168	+	0.611599i	$0.790527\pi$
$744$	0	0
$745$	−8.34329	−0.305675
$746$	−7.52762	−0.275606
$747$	0	0
$748$	−24.9617	−0.912689
$749$	−71.6392	−2.61764
$750$	0	0
$751$	26.2552	0.958065	0.479032	−	0.877797i	$-0.340988\pi$
$751$	26.2552	0.958065	0.479032	+	0.877797i	$0.340988\pi$
$752$	31.6005	1.15235
$753$	0	0
$754$	−15.8816	−0.578373
$755$	14.1799	0.516061
$756$	0	0
$757$	8.70305	0.316318	0.158159	−	0.987414i	$-0.449444\pi$
$757$	8.70305	0.316318	0.158159	+	0.987414i	$0.449444\pi$
$758$	67.3976	2.44799
$759$	0	0
$760$	4.28335	0.155373
$761$	40.5667	1.47054	0.735272	−	0.677773i	$-0.237055\pi$
$761$	40.5667	1.47054	0.735272	+	0.677773i	$0.237055\pi$
$762$	0	0
$763$	13.1090	0.474578
$764$	−73.1220	−2.64546
$765$	0	0
$766$	93.0624	3.36248
$767$	10.3493	0.373691
$768$	0	0
$769$	−14.3524	−0.517559	−0.258780	−	0.965936i	$-0.583320\pi$
$769$	−14.3524	−0.517559	−0.258780	+	0.965936i	$0.583320\pi$
$770$	11.5376	0.415787
$771$	0	0
$772$	−97.3976	−3.50542
$773$	12.4276	0.446991	0.223496	−	0.974705i	$-0.428253\pi$
$773$	12.4276	0.446991	0.223496	+	0.974705i	$0.428253\pi$
$774$	0	0
$775$	1.77631	0.0638069
$776$	−79.8246	−2.86554
$777$	0	0
$778$	−79.4375	−2.84797
$779$	−2.92685	−0.104865
$780$	0	0
$781$	11.4491	0.409681
$782$	−133.273	−4.76584
$783$	0	0
$784$	43.8489	1.56603
$785$	−3.59527	−0.128321
$786$	0	0
$787$	20.9369	0.746321	0.373161	−	0.927767i	$-0.378274\pi$
$787$	20.9369	0.746321	0.373161	+	0.927767i	$0.378274\pi$
$788$	−68.6173	−2.44439
$789$	0	0
$790$	19.0782	0.678771
$791$	38.1581	1.35675
$792$	0	0
$793$	1.93974	0.0688822
$794$	−19.3143	−0.685438
$795$	0	0
$796$	−49.3125	−1.74783
$797$	12.0417	0.426538	0.213269	−	0.976993i	$-0.431589\pi$
$797$	12.0417	0.426538	0.213269	+	0.976993i	$0.431589\pi$
$798$	0	0
$799$	−76.2343	−2.69698
$800$	1.98740	0.0702652
$801$	0	0
$802$	−8.44166	−0.298085
$803$	−8.77977	−0.309831
$804$	0	0
$805$	40.2908	1.42006
$806$	−5.29353	−0.186457
$807$	0	0
$808$	66.5339	2.34065
$809$	−28.3618	−0.997147	−0.498574	−	0.866847i	$-0.666143\pi$
$809$	−28.3618	−0.997147	−0.498574	+	0.866847i	$0.666143\pi$
$810$	0	0
$811$	−47.9490	−1.68372	−0.841858	−	0.539699i	$-0.818538\pi$
$811$	−47.9490	−1.68372	−0.841858	+	0.539699i	$0.818538\pi$
$812$	−96.6988	−3.39346
$813$	0	0
$814$	1.76012	0.0616921
$815$	−0.166356	−0.00582720
$816$	0	0
$817$	3.28752	0.115016
$818$	−12.7193	−0.444720
$819$	0	0
$820$	11.0677	0.386499
$821$	−7.05295	−0.246150	−0.123075	−	0.992397i	$-0.539276\pi$
$821$	−7.05295	−0.246150	−0.123075	+	0.992397i	$0.539276\pi$
$822$	0	0
$823$	25.6201	0.893062	0.446531	−	0.894768i	$-0.352659\pi$
$823$	25.6201	0.893062	0.446531	+	0.894768i	$0.352659\pi$
$824$	38.8689	1.35406
$825$	0	0
$826$	96.3422	3.35217
$827$	16.8432	0.585695	0.292847	−	0.956159i	$-0.405397\pi$
$827$	16.8432	0.585695	0.292847	+	0.956159i	$0.405397\pi$
$828$	0	0
$829$	−13.4384	−0.466736	−0.233368	−	0.972388i	$-0.574975\pi$
$829$	−13.4384	−0.466736	−0.233368	+	0.972388i	$0.574975\pi$
$830$	−23.3683	−0.811124
$831$	0	0
$832$	−12.7053	−0.440477
$833$	−105.783	−3.66516
$834$	0	0
$835$	−11.9210	−0.412544
$836$	−3.78142	−0.130783
$837$	0	0
$838$	74.8929	2.58713
$839$	−13.4160	−0.463172	−0.231586	−	0.972814i	$-0.574391\pi$
$839$	−13.4160	−0.463172	−0.231586	+	0.972814i	$0.574391\pi$
$840$	0	0
$841$	−0.598999	−0.0206551
$842$	43.2176	1.48938
$843$	0	0
$844$	−84.9347	−2.92357
$845$	−11.4639	−0.394370
$846$	0	0
$847$	−4.79843	−0.164876
$848$	26.8309	0.921376
$849$	0	0
$850$	15.8722	0.544410
$851$	6.14655	0.210701
$852$	0	0
$853$	46.7681	1.60131	0.800654	−	0.599126i	$-0.204485\pi$
$853$	46.7681	1.60131	0.800654	+	0.599126i	$0.204485\pi$
$854$	18.0572	0.617904
$855$	0	0
$856$	−63.9492	−2.18574
$857$	−14.2041	−0.485203	−0.242602	−	0.970126i	$-0.578001\pi$
$857$	−14.2041	−0.485203	−0.242602	+	0.970126i	$0.578001\pi$
$858$	0	0
$859$	−8.01102	−0.273333	−0.136666	−	0.990617i	$-0.543639\pi$
$859$	−8.01102	−0.273333	−0.136666	+	0.990617i	$0.543639\pi$
$860$	−12.4315	−0.423911
$861$	0	0
$862$	−85.2320	−2.90301
$863$	−51.1105	−1.73982	−0.869910	−	0.493210i	$-0.835823\pi$
$863$	−51.1105	−1.73982	−0.869910	+	0.493210i	$0.835823\pi$
$864$	0	0
$865$	12.5436	0.426494
$866$	−52.7695	−1.79318
$867$	0	0
$868$	−32.2309	−1.09399
$869$	−7.93451	−0.269160
$870$	0	0
$871$	−7.65665	−0.259436
$872$	11.7018	0.396274
$873$	0	0
$874$	−20.1894	−0.682918
$875$	−4.79843	−0.162217
$876$	0	0
$877$	−17.6700	−0.596674	−0.298337	−	0.954461i	$-0.596432\pi$
$877$	−17.6700	−0.596674	−0.298337	+	0.954461i	$0.596432\pi$
$878$	1.49734	0.0505327
$879$	0	0
$880$	2.73629	0.0922403
$881$	−24.2078	−0.815580	−0.407790	−	0.913076i	$-0.633700\pi$
$881$	−24.2078	−0.815580	−0.407790	+	0.913076i	$0.633700\pi$
$882$	0	0
$883$	−51.3059	−1.72658	−0.863290	−	0.504708i	$-0.831600\pi$
$883$	−51.3059	−1.72658	−0.863290	+	0.504708i	$0.831600\pi$
$884$	−30.9374	−1.04054
$885$	0	0
$886$	82.4700	2.77063
$887$	24.5689	0.824942	0.412471	−	0.910971i	$-0.364666\pi$
$887$	24.5689	0.824942	0.412471	+	0.910971i	$0.364666\pi$
$888$	0	0
$889$	−46.9377	−1.57424
$890$	−7.18068	−0.240697
$891$	0	0
$892$	−91.7472	−3.07192
$893$	−11.5487	−0.386461
$894$	0	0
$895$	−9.48987	−0.317212
$896$	−99.2017	−3.31410
$897$	0	0
$898$	−60.8243	−2.02973
$899$	9.46640	0.315722
$900$	0	0
$901$	−64.7279	−2.15640
$902$	−7.03750	−0.234323
$903$	0	0
$904$	34.0620	1.13289
$905$	1.03661	0.0344582
$906$	0	0
$907$	18.8989	0.627529	0.313764	−	0.949501i	$-0.398410\pi$
$907$	18.8989	0.627529	0.313764	+	0.949501i	$0.398410\pi$
$908$	2.58513	0.0857905
$909$	0	0
$910$	14.2997	0.474030
$911$	−17.1920	−0.569596	−0.284798	−	0.958588i	$-0.591926\pi$
$911$	−17.1920	−0.569596	−0.284798	+	0.958588i	$0.591926\pi$
$912$	0	0
$913$	9.71873	0.321643
$914$	22.6509	0.749227
$915$	0	0
$916$	48.9309	1.61672
$917$	−65.6334	−2.16741
$918$	0	0
$919$	−1.25302	−0.0413332	−0.0206666	−	0.999786i	$-0.506579\pi$
$919$	−1.25302	−0.0413332	−0.0206666	+	0.999786i	$0.506579\pi$
$920$	35.9658	1.18576
$921$	0	0
$922$	52.3222	1.72314
$923$	14.1900	0.467068
$924$	0	0
$925$	−0.732022	−0.0240687
$926$	−39.4700	−1.29706
$927$	0	0
$928$	10.5914	0.347679
$929$	6.75023	0.221468	0.110734	−	0.993850i	$-0.464680\pi$
$929$	6.75023	0.221468	0.110734	+	0.993850i	$0.464680\pi$
$930$	0	0
$931$	−16.0249	−0.525196
$932$	−72.1343	−2.36284
$933$	0	0
$934$	−86.0047	−2.81416
$935$	−6.60114	−0.215880
$936$	0	0
$937$	30.0075	0.980303	0.490152	−	0.871637i	$-0.336941\pi$
$937$	30.0075	0.980303	0.490152	+	0.871637i	$0.336941\pi$
$938$	−71.2763	−2.32725
$939$	0	0
$940$	43.6703	1.42437
$941$	49.7943	1.62325	0.811624	−	0.584180i	$-0.198584\pi$
$941$	49.7943	1.62325	0.811624	+	0.584180i	$0.198584\pi$
$942$	0	0
$943$	−24.5758	−0.800298
$944$	22.8487	0.743663
$945$	0	0
$946$	7.90471	0.257004
$947$	27.7203	0.900789	0.450394	−	0.892830i	$-0.351283\pi$
$947$	27.7203	0.900789	0.450394	+	0.892830i	$0.351283\pi$
$948$	0	0
$949$	−10.8816	−0.353232
$950$	2.40446	0.0780109
$951$	0	0
$952$	−135.675	−4.39727
$953$	18.0599	0.585016	0.292508	−	0.956263i	$-0.405510\pi$
$953$	18.0599	0.585016	0.292508	+	0.956263i	$0.405510\pi$
$954$	0	0
$955$	−19.3372	−0.625737
$956$	−79.8606	−2.58288
$957$	0	0
$958$	−32.7281	−1.05740
$959$	−73.0814	−2.35992
$960$	0	0
$961$	−27.8447	−0.898217
$962$	2.18148	0.0703338
$963$	0	0
$964$	72.9053	2.34812
$965$	−25.7569	−0.829144
$966$	0	0
$967$	−2.42624	−0.0780225	−0.0390113	−	0.999239i	$-0.512421\pi$
$967$	−2.42624	−0.0780225	−0.0390113	+	0.999239i	$0.512421\pi$
$968$	−4.28335	−0.137672
$969$	0	0
$970$	−44.8096	−1.43875
$971$	35.9498	1.15368	0.576842	−	0.816856i	$-0.304285\pi$
$971$	35.9498	1.15368	0.576842	+	0.816856i	$0.304285\pi$
$972$	0	0
$973$	32.6731	1.04745
$974$	13.1227	0.420479
$975$	0	0
$976$	4.28248	0.137079
$977$	−45.3025	−1.44935	−0.724677	−	0.689088i	$-0.758011\pi$
$977$	−45.3025	−1.44935	−0.724677	+	0.689088i	$0.758011\pi$
$978$	0	0
$979$	2.98640	0.0954459
$980$	60.5970	1.93570
$981$	0	0
$982$	−15.7835	−0.503671
$983$	7.85485	0.250531	0.125265	−	0.992123i	$-0.460022\pi$
$983$	7.85485	0.250531	0.125265	+	0.992123i	$0.460022\pi$
$984$	0	0
$985$	−18.1459	−0.578177
$986$	84.5868	2.69379
$987$	0	0
$988$	−4.68667	−0.149103
$989$	27.6042	0.877763
$990$	0	0
$991$	−29.9656	−0.951888	−0.475944	−	0.879476i	$-0.657893\pi$
$991$	−29.9656	−0.951888	−0.475944	+	0.879476i	$0.657893\pi$
$992$	3.53023	0.112085
$993$	0	0
$994$	132.095	4.18981
$995$	−13.0407	−0.413419
$996$	0	0
$997$	38.2300	1.21076	0.605379	−	0.795938i	$-0.293022\pi$
$997$	38.2300	1.21076	0.605379	+	0.795938i	$0.293022\pi$
$998$	−29.7354	−0.941258
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9405.2.a.bq.1.2	✓	14
3.2	odd	2		9405.2.a.br.1.13	yes	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9405.2.a.bq.1.2	✓	14	1.1	even	1	trivial
9405.2.a.br.1.13	yes	14	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 9405 = 3^{2} \cdot 5 \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9405.a (trivial)

\(f(q)\)	\(=\)	\(q-2.40446 q^{2} +3.78142 q^{4} +1.00000 q^{5} -4.79843 q^{7} -4.28335 q^{8} +O(q^{10})\)	\(q-2.40446 q^{2} +3.78142 q^{4} +1.00000 q^{5} -4.79843 q^{7} -4.28335 q^{8} -2.40446 q^{10} +1.00000 q^{11} +1.23940 q^{13} +11.5376 q^{14} +2.73629 q^{16} -6.60114 q^{17} -1.00000 q^{19} +3.78142 q^{20} -2.40446 q^{22} -8.39666 q^{23} +1.00000 q^{25} -2.98007 q^{26} -18.1449 q^{28} +5.32926 q^{29} +1.77631 q^{31} +1.98740 q^{32} +15.8722 q^{34} -4.79843 q^{35} -0.732022 q^{37} +2.40446 q^{38} -4.28335 q^{40} +2.92685 q^{41} -3.28752 q^{43} +3.78142 q^{44} +20.1894 q^{46} +11.5487 q^{47} +16.0249 q^{49} -2.40446 q^{50} +4.68667 q^{52} +9.80557 q^{53} +1.00000 q^{55} +20.5533 q^{56} -12.8140 q^{58} +8.35026 q^{59} +1.56507 q^{61} -4.27106 q^{62} -10.2512 q^{64} +1.23940 q^{65} -6.17773 q^{67} -24.9617 q^{68} +11.5376 q^{70} +11.4491 q^{71} -8.77977 q^{73} +1.76012 q^{74} -3.78142 q^{76} -4.79843 q^{77} -7.93451 q^{79} +2.73629 q^{80} -7.03750 q^{82} +9.71873 q^{83} -6.60114 q^{85} +7.90471 q^{86} -4.28335 q^{88} +2.98640 q^{89} -5.94715 q^{91} -31.7513 q^{92} -27.7683 q^{94} -1.00000 q^{95} +18.6360 q^{97} -38.5313 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - q^{2} + 11 q^{4} + 14 q^{5} - 12 q^{7} - 9 q^{8}+O(q^{10}) \) 14 * q - q^2 + 11 * q^4 + 14 * q^5 - 12 * q^7 - 9 * q^8	\( 14 q - q^{2} + 11 q^{4} + 14 q^{5} - 12 q^{7} - 9 q^{8} - q^{10} + 14 q^{11} - 10 q^{13} + 8 q^{14} + 13 q^{16} - 16 q^{17} - 14 q^{19} + 11 q^{20} - q^{22} - 12 q^{23} + 14 q^{25} - 12 q^{26} - 33 q^{28} - 4 q^{31} - 24 q^{32} - 2 q^{34} - 12 q^{35} - 14 q^{37} + q^{38} - 9 q^{40} - 18 q^{41} - 20 q^{43} + 11 q^{44} - 17 q^{46} - 8 q^{47} + 10 q^{49} - q^{50} - 26 q^{52} - 20 q^{53} + 14 q^{55} + 11 q^{56} - 36 q^{58} + 2 q^{59} + 4 q^{61} - 38 q^{62} + 3 q^{64} - 10 q^{65} - 22 q^{67} - 48 q^{68} + 8 q^{70} - 28 q^{73} + 19 q^{74} - 11 q^{76} - 12 q^{77} - 14 q^{79} + 13 q^{80} - 24 q^{82} - 10 q^{83} - 16 q^{85} + 23 q^{86} - 9 q^{88} + 26 q^{89} - 42 q^{91} - 12 q^{92} - 56 q^{94} - 14 q^{95} - 32 q^{97} + 4 q^{98}+O(q^{100}) \) 14 * q - q^2 + 11 * q^4 + 14 * q^5 - 12 * q^7 - 9 * q^8 - q^10 + 14 * q^11 - 10 * q^13 + 8 * q^14 + 13 * q^16 - 16 * q^17 - 14 * q^19 + 11 * q^20 - q^22 - 12 * q^23 + 14 * q^25 - 12 * q^26 - 33 * q^28 - 4 * q^31 - 24 * q^32 - 2 * q^34 - 12 * q^35 - 14 * q^37 + q^38 - 9 * q^40 - 18 * q^41 - 20 * q^43 + 11 * q^44 - 17 * q^46 - 8 * q^47 + 10 * q^49 - q^50 - 26 * q^52 - 20 * q^53 + 14 * q^55 + 11 * q^56 - 36 * q^58 + 2 * q^59 + 4 * q^61 - 38 * q^62 + 3 * q^64 - 10 * q^65 - 22 * q^67 - 48 * q^68 + 8 * q^70 - 28 * q^73 + 19 * q^74 - 11 * q^76 - 12 * q^77 - 14 * q^79 + 13 * q^80 - 24 * q^82 - 10 * q^83 - 16 * q^85 + 23 * q^86 - 9 * q^88 + 26 * q^89 - 42 * q^91 - 12 * q^92 - 56 * q^94 - 14 * q^95 - 32 * q^97 + 4 * q^98

Introduction

L-functions

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