LMFDB - Embedded newform 9464.2.a.br.1.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9464,2,Mod(1,9464)] 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("9464.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9464, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 9464 = 2^{3} \cdot 7 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9464.a (trivial)

Newform invariants

comment:select newform

sage:traces = [15,0,-3,0,-4,0,-15,0,16,0,-15,0,0,0,-8,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$75.5704204729$
Analytic rank:	$1$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 3 x^{14} - 26 x^{13} + 78 x^{12} + 253 x^{11} - 782 x^{10} - 1087 x^{9} + 3776 x^{8} + \cdots - 344 $ x^15 - 3x^14 - 26x^13 + 78x^12 + 253x^11 - 782x^10 - 1087x^9 + 3776x^8 + 1576x^7 - 8849x^6 + 1791x^5 + 8496x^4 - 5353x^3 - 1252x^2 + 1724x - 344
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$0.790243$ of defining polynomial
Character	$\chi$	$=$	9464.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.790243 q^{3} -3.98493 q^{5} -1.00000 q^{7} -2.37552 q^{9} +2.16572 q^{11} +3.14906 q^{15} -6.76042 q^{17} -5.16512 q^{19} +0.790243 q^{21} +1.26855 q^{23} +10.8797 q^{25} +4.24797 q^{27} +4.34492 q^{29} +9.78216 q^{31} -1.71144 q^{33} +3.98493 q^{35} -9.20584 q^{37} +6.34464 q^{41} +1.37775 q^{43} +9.46626 q^{45} -9.38688 q^{47} +1.00000 q^{49} +5.34238 q^{51} +9.55478 q^{53} -8.63022 q^{55} +4.08170 q^{57} +9.44610 q^{59} -4.85435 q^{61} +2.37552 q^{63} +11.7193 q^{67} -1.00246 q^{69} +1.82940 q^{71} -11.0511 q^{73} -8.59757 q^{75} -2.16572 q^{77} +5.73877 q^{79} +3.76962 q^{81} +10.4567 q^{83} +26.9398 q^{85} -3.43354 q^{87} -7.44390 q^{89} -7.73029 q^{93} +20.5826 q^{95} -11.8539 q^{97} -5.14469 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q - 3 q^{3} - 4 q^{5} - 15 q^{7} + 16 q^{9} - 15 q^{11} - 8 q^{15} + 2 q^{17} - 13 q^{19} + 3 q^{21} - 10 q^{23} + 23 q^{25} - 9 q^{27} + 25 q^{29} - 19 q^{31} + 24 q^{33} + 4 q^{35} + 2 q^{37} - 30 q^{41}+ \cdots - 70 q^{99}+O(q^{100}) $ 15 * q - 3 * q^3 - 4 * q^5 - 15 * q^7 + 16 * q^9 - 15 * q^11 - 8 * q^15 + 2 * q^17 - 13 * q^19 + 3 * q^21 - 10 * q^23 + 23 * q^25 - 9 * q^27 + 25 * q^29 - 19 * q^31 + 24 * q^33 + 4 * q^35 + 2 * q^37 - 30 * q^41 + 15 * q^43 + 2 * q^45 - 13 * q^47 + 15 * q^49 + q^51 + 62 * q^53 - 3 * q^55 + 2 * q^57 - 22 * q^59 - q^61 - 16 * q^63 - 29 * q^67 + 32 * q^69 - 20 * q^71 - 21 * q^73 - 46 * q^75 + 15 * q^77 + 29 * q^79 + 7 * q^81 - 23 * q^83 - 8 * q^85 - 22 * q^87 - 9 * q^89 - 10 * q^93 + 28 * q^95 - 11 * q^97 - 70 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	−0.790243	−0.456247	−0.228124	−	0.973632i	$-0.573259\pi$
$3$	−0.790243	−0.456247	−0.228124	+	0.973632i	$0.573259\pi$
$4$	0	0
$5$	−3.98493	−1.78211	−0.891057	−	0.453891i	$-0.850035\pi$
$5$	−3.98493	−1.78211	−0.891057	+	0.453891i	$0.850035\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−2.37552	−0.791838
$10$	0	0
$11$	2.16572	0.652988	0.326494	−	0.945199i	$-0.394133\pi$
$11$	2.16572	0.652988	0.326494	+	0.945199i	$0.394133\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	3.14906	0.813085
$16$	0	0
$17$	−6.76042	−1.63964	−0.819822	−	0.572619i	$-0.805927\pi$
$17$	−6.76042	−1.63964	−0.819822	+	0.572619i	$0.805927\pi$
$18$	0	0
$19$	−5.16512	−1.18496	−0.592480	−	0.805585i	$-0.701851\pi$
$19$	−5.16512	−1.18496	−0.592480	+	0.805585i	$0.701851\pi$
$20$	0	0
$21$	0.790243	0.172445
$22$	0	0
$23$	1.26855	0.264511	0.132255	−	0.991216i	$-0.457778\pi$
$23$	1.26855	0.264511	0.132255	+	0.991216i	$0.457778\pi$
$24$	0	0
$25$	10.8797	2.17593
$26$	0	0
$27$	4.24797	0.817521
$28$	0	0
$29$	4.34492	0.806831	0.403415	−	0.915017i	$-0.367823\pi$
$29$	4.34492	0.806831	0.403415	+	0.915017i	$0.367823\pi$
$30$	0	0
$31$	9.78216	1.75693	0.878464	−	0.477809i	$-0.158569\pi$
$31$	9.78216	1.75693	0.878464	+	0.477809i	$0.158569\pi$
$32$	0	0
$33$	−1.71144	−0.297924
$34$	0	0
$35$	3.98493	0.673576
$36$	0	0
$37$	−9.20584	−1.51343	−0.756716	−	0.653744i	$-0.773197\pi$
$37$	−9.20584	−1.51343	−0.756716	+	0.653744i	$0.773197\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	6.34464	0.990866	0.495433	−	0.868646i	$-0.335009\pi$
$41$	6.34464	0.990866	0.495433	+	0.868646i	$0.335009\pi$
$42$	0	0
$43$	1.37775	0.210104	0.105052	−	0.994467i	$-0.466499\pi$
$43$	1.37775	0.210104	0.105052	+	0.994467i	$0.466499\pi$
$44$	0	0
$45$	9.46626	1.41115
$46$	0	0
$47$	−9.38688	−1.36922	−0.684608	−	0.728911i	$-0.740027\pi$
$47$	−9.38688	−1.36922	−0.684608	+	0.728911i	$0.740027\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	5.34238	0.748083
$52$	0	0
$53$	9.55478	1.31245	0.656225	−	0.754566i	$-0.272152\pi$
$53$	9.55478	1.31245	0.656225	+	0.754566i	$0.272152\pi$
$54$	0	0
$55$	−8.63022	−1.16370
$56$	0	0
$57$	4.08170	0.540635
$58$	0	0
$59$	9.44610	1.22978	0.614889	−	0.788614i	$-0.289201\pi$
$59$	9.44610	1.22978	0.614889	+	0.788614i	$0.289201\pi$
$60$	0	0
$61$	−4.85435	−0.621536	−0.310768	−	0.950486i	$-0.600586\pi$
$61$	−4.85435	−0.621536	−0.310768	+	0.950486i	$0.600586\pi$
$62$	0	0
$63$	2.37552	0.299287
$64$	0	0
$65$	0	0
$66$	0	0
$67$	11.7193	1.43175	0.715873	−	0.698231i	$-0.246029\pi$
$67$	11.7193	1.43175	0.715873	+	0.698231i	$0.246029\pi$
$68$	0	0
$69$	−1.00246	−0.120682
$70$	0	0
$71$	1.82940	0.217110	0.108555	−	0.994090i	$-0.465378\pi$
$71$	1.82940	0.217110	0.108555	+	0.994090i	$0.465378\pi$
$72$	0	0
$73$	−11.0511	−1.29344	−0.646720	−	0.762728i	$-0.723860\pi$
$73$	−11.0511	−1.29344	−0.646720	+	0.762728i	$0.723860\pi$
$74$	0	0
$75$	−8.59757	−0.992762
$76$	0	0
$77$	−2.16572	−0.246806
$78$	0	0
$79$	5.73877	0.645662	0.322831	−	0.946457i	$-0.395365\pi$
$79$	5.73877	0.645662	0.322831	+	0.946457i	$0.395365\pi$
$80$	0	0
$81$	3.76962	0.418847
$82$	0	0
$83$	10.4567	1.14778	0.573888	−	0.818934i	$-0.305435\pi$
$83$	10.4567	1.14778	0.573888	+	0.818934i	$0.305435\pi$
$84$	0	0
$85$	26.9398	2.92203
$86$	0	0
$87$	−3.43354	−0.368114
$88$	0	0
$89$	−7.44390	−0.789052	−0.394526	−	0.918885i	$-0.629091\pi$
$89$	−7.44390	−0.789052	−0.394526	+	0.918885i	$0.629091\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−7.73029	−0.801593
$94$	0	0
$95$	20.5826	2.11173
$96$	0	0
$97$	−11.8539	−1.20358	−0.601792	−	0.798653i	$-0.705546\pi$
$97$	−11.8539	−1.20358	−0.601792	+	0.798653i	$0.705546\pi$
$98$	0	0
$99$	−5.14469	−0.517061
$100$	0	0
$101$	−12.8928	−1.28288	−0.641442	−	0.767171i	$-0.721664\pi$
$101$	−12.8928	−1.28288	−0.641442	+	0.767171i	$0.721664\pi$
$102$	0	0
$103$	19.2254	1.89434	0.947169	−	0.320736i	$-0.103930\pi$
$103$	19.2254	1.89434	0.947169	+	0.320736i	$0.103930\pi$
$104$	0	0
$105$	−3.14906	−0.307317
$106$	0	0
$107$	5.05012	0.488213	0.244107	−	0.969748i	$-0.421505\pi$
$107$	5.05012	0.488213	0.244107	+	0.969748i	$0.421505\pi$
$108$	0	0
$109$	−12.9198	−1.23749	−0.618747	−	0.785591i	$-0.712359\pi$
$109$	−12.9198	−1.23749	−0.618747	+	0.785591i	$0.712359\pi$
$110$	0	0
$111$	7.27486	0.690499
$112$	0	0
$113$	15.8160	1.48785	0.743923	−	0.668265i	$-0.232963\pi$
$113$	15.8160	1.48785	0.743923	+	0.668265i	$0.232963\pi$
$114$	0	0
$115$	−5.05507	−0.471388
$116$	0	0
$117$	0	0
$118$	0	0
$119$	6.76042	0.619727
$120$	0	0
$121$	−6.30968	−0.573607
$122$	0	0
$123$	−5.01381	−0.452080
$124$	0	0
$125$	−23.4300	−2.09564
$126$	0	0
$127$	15.8537	1.40679	0.703395	−	0.710800i	$-0.251667\pi$
$127$	15.8537	1.40679	0.703395	+	0.710800i	$0.251667\pi$
$128$	0	0
$129$	−1.08876	−0.0958596
$130$	0	0
$131$	−3.34838	−0.292549	−0.146275	−	0.989244i	$-0.546728\pi$
$131$	−3.34838	−0.292549	−0.146275	+	0.989244i	$0.546728\pi$
$132$	0	0
$133$	5.16512	0.447873
$134$	0	0
$135$	−16.9278	−1.45692
$136$	0	0
$137$	−3.33968	−0.285328	−0.142664	−	0.989771i	$-0.545567\pi$
$137$	−3.33968	−0.285328	−0.142664	+	0.989771i	$0.545567\pi$
$138$	0	0
$139$	0.920194	0.0780499	0.0390249	−	0.999238i	$-0.487575\pi$
$139$	0.920194	0.0780499	0.0390249	+	0.999238i	$0.487575\pi$
$140$	0	0
$141$	7.41792	0.624701
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−17.3142	−1.43786
$146$	0	0
$147$	−0.790243	−0.0651782
$148$	0	0
$149$	−2.65862	−0.217803	−0.108901	−	0.994053i	$-0.534733\pi$
$149$	−2.65862	−0.217803	−0.108901	+	0.994053i	$0.534733\pi$
$150$	0	0
$151$	−3.44556	−0.280395	−0.140198	−	0.990124i	$-0.544774\pi$
$151$	−3.44556	−0.280395	−0.140198	+	0.990124i	$0.544774\pi$
$152$	0	0
$153$	16.0595	1.29833
$154$	0	0
$155$	−38.9812	−3.13104
$156$	0	0
$157$	5.37075	0.428633	0.214316	−	0.976764i	$-0.431248\pi$
$157$	5.37075	0.428633	0.214316	+	0.976764i	$0.431248\pi$
$158$	0	0
$159$	−7.55060	−0.598801
$160$	0	0
$161$	−1.26855	−0.0999756
$162$	0	0
$163$	−14.6048	−1.14394	−0.571969	−	0.820275i	$-0.693820\pi$
$163$	−14.6048	−1.14394	−0.571969	+	0.820275i	$0.693820\pi$
$164$	0	0
$165$	6.81997	0.530934
$166$	0	0
$167$	−10.7468	−0.831612	−0.415806	−	0.909453i	$-0.636501\pi$
$167$	−10.7468	−0.831612	−0.415806	+	0.909453i	$0.636501\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	12.2698	0.938297
$172$	0	0
$173$	16.7477	1.27331	0.636654	−	0.771150i	$-0.280318\pi$
$173$	16.7477	1.27331	0.636654	+	0.771150i	$0.280318\pi$
$174$	0	0
$175$	−10.8797	−0.822424
$176$	0	0
$177$	−7.46472	−0.561083
$178$	0	0
$179$	−3.64304	−0.272293	−0.136147	−	0.990689i	$-0.543472\pi$
$179$	−3.64304	−0.272293	−0.136147	+	0.990689i	$0.543472\pi$
$180$	0	0
$181$	12.4673	0.926687	0.463344	−	0.886179i	$-0.346650\pi$
$181$	12.4673	0.926687	0.463344	+	0.886179i	$0.346650\pi$
$182$	0	0
$183$	3.83612	0.283574
$184$	0	0
$185$	36.6846	2.69711
$186$	0	0
$187$	−14.6411	−1.07067
$188$	0	0
$189$	−4.24797	−0.308994
$190$	0	0
$191$	−17.8833	−1.29399	−0.646995	−	0.762494i	$-0.723975\pi$
$191$	−17.8833	−1.29399	−0.646995	+	0.762494i	$0.723975\pi$
$192$	0	0
$193$	−0.876450	−0.0630883	−0.0315441	−	0.999502i	$-0.510042\pi$
$193$	−0.876450	−0.0630883	−0.0315441	+	0.999502i	$0.510042\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	8.57548	0.610978	0.305489	−	0.952196i	$-0.401180\pi$
$197$	8.57548	0.610978	0.305489	+	0.952196i	$0.401180\pi$
$198$	0	0
$199$	−16.5257	−1.17148	−0.585740	−	0.810499i	$-0.699196\pi$
$199$	−16.5257	−1.17148	−0.585740	+	0.810499i	$0.699196\pi$
$200$	0	0
$201$	−9.26114	−0.653230
$202$	0	0
$203$	−4.34492	−0.304953
$204$	0	0
$205$	−25.2829	−1.76584
$206$	0	0
$207$	−3.01346	−0.209450
$208$	0	0
$209$	−11.1862	−0.773764
$210$	0	0
$211$	−0.0498291	−0.00343038	−0.00171519	−	0.999999i	$-0.500546\pi$
$211$	−0.0498291	−0.00343038	−0.00171519	+	0.999999i	$0.500546\pi$
$212$	0	0
$213$	−1.44567	−0.0990559
$214$	0	0
$215$	−5.49022	−0.374430
$216$	0	0
$217$	−9.78216	−0.664056
$218$	0	0
$219$	8.73310	0.590128
$220$	0	0
$221$	0	0
$222$	0	0
$223$	24.2739	1.62550	0.812749	−	0.582614i	$-0.197970\pi$
$223$	24.2739	1.62550	0.812749	+	0.582614i	$0.197970\pi$
$224$	0	0
$225$	−25.8448	−1.72299
$226$	0	0
$227$	14.7348	0.977982	0.488991	−	0.872289i	$-0.337365\pi$
$227$	14.7348	0.977982	0.488991	+	0.872289i	$0.337365\pi$
$228$	0	0
$229$	15.1678	1.00232	0.501159	−	0.865355i	$-0.332907\pi$
$229$	15.1678	1.00232	0.501159	+	0.865355i	$0.332907\pi$
$230$	0	0
$231$	1.71144	0.112605
$232$	0	0
$233$	14.6680	0.960930	0.480465	−	0.877014i	$-0.340468\pi$
$233$	14.6680	0.960930	0.480465	+	0.877014i	$0.340468\pi$
$234$	0	0
$235$	37.4060	2.44010
$236$	0	0
$237$	−4.53502	−0.294581
$238$	0	0
$239$	−9.69605	−0.627185	−0.313593	−	0.949558i	$-0.601533\pi$
$239$	−9.69605	−0.627185	−0.313593	+	0.949558i	$0.601533\pi$
$240$	0	0
$241$	−19.2099	−1.23742	−0.618710	−	0.785620i	$-0.712344\pi$
$241$	−19.2099	−1.23742	−0.618710	+	0.785620i	$0.712344\pi$
$242$	0	0
$243$	−15.7228	−1.00862
$244$	0	0
$245$	−3.98493	−0.254588
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−8.26336	−0.523669
$250$	0	0
$251$	0.209152	0.0132016	0.00660078	−	0.999978i	$-0.497899\pi$
$251$	0.209152	0.0132016	0.00660078	+	0.999978i	$0.497899\pi$
$252$	0	0
$253$	2.74731	0.172722
$254$	0	0
$255$	−21.2890	−1.33317
$256$	0	0
$257$	10.1678	0.634250	0.317125	−	0.948384i	$-0.397283\pi$
$257$	10.1678	0.634250	0.317125	+	0.948384i	$0.397283\pi$
$258$	0	0
$259$	9.20584	0.572023
$260$	0	0
$261$	−10.3214	−0.638880
$262$	0	0
$263$	11.2382	0.692976	0.346488	−	0.938054i	$-0.387374\pi$
$263$	11.2382	0.692976	0.346488	+	0.938054i	$0.387374\pi$
$264$	0	0
$265$	−38.0751	−2.33893
$266$	0	0
$267$	5.88249	0.360003
$268$	0	0
$269$	16.2608	0.991438	0.495719	−	0.868483i	$-0.334905\pi$
$269$	16.2608	0.991438	0.495719	+	0.868483i	$0.334905\pi$
$270$	0	0
$271$	−22.2898	−1.35401	−0.677004	−	0.735979i	$-0.736722\pi$
$271$	−22.2898	−1.35401	−0.677004	+	0.735979i	$0.736722\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	23.5622	1.42086
$276$	0	0
$277$	−1.19610	−0.0718665	−0.0359333	−	0.999354i	$-0.511440\pi$
$277$	−1.19610	−0.0718665	−0.0359333	+	0.999354i	$0.511440\pi$
$278$	0	0
$279$	−23.2377	−1.39120
$280$	0	0
$281$	−3.99346	−0.238230	−0.119115	−	0.992880i	$-0.538006\pi$
$281$	−3.99346	−0.238230	−0.119115	+	0.992880i	$0.538006\pi$
$282$	0	0
$283$	−5.54700	−0.329735	−0.164867	−	0.986316i	$-0.552720\pi$
$283$	−5.54700	−0.329735	−0.164867	+	0.986316i	$0.552720\pi$
$284$	0	0
$285$	−16.2653	−0.963473
$286$	0	0
$287$	−6.34464	−0.374512
$288$	0	0
$289$	28.7033	1.68843
$290$	0	0
$291$	9.36748	0.549132
$292$	0	0
$293$	7.73762	0.452037	0.226018	−	0.974123i	$-0.427429\pi$
$293$	7.73762	0.452037	0.226018	+	0.974123i	$0.427429\pi$
$294$	0	0
$295$	−37.6420	−2.19160
$296$	0	0
$297$	9.19988	0.533831
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−1.37775	−0.0794120
$302$	0	0
$303$	10.1885	0.585313
$304$	0	0
$305$	19.3442	1.10765
$306$	0	0
$307$	1.63579	0.0933595	0.0466798	−	0.998910i	$-0.485136\pi$
$307$	1.63579	0.0933595	0.0466798	+	0.998910i	$0.485136\pi$
$308$	0	0
$309$	−15.1928	−0.864286
$310$	0	0
$311$	−27.7416	−1.57308	−0.786540	−	0.617540i	$-0.788130\pi$
$311$	−27.7416	−1.57308	−0.786540	+	0.617540i	$0.788130\pi$
$312$	0	0
$313$	9.68881	0.547644	0.273822	−	0.961780i	$-0.411712\pi$
$313$	9.68881	0.547644	0.273822	+	0.961780i	$0.411712\pi$
$314$	0	0
$315$	−9.46626	−0.533363
$316$	0	0
$317$	8.47999	0.476284	0.238142	−	0.971230i	$-0.423462\pi$
$317$	8.47999	0.476284	0.238142	+	0.971230i	$0.423462\pi$
$318$	0	0
$319$	9.40985	0.526851
$320$	0	0
$321$	−3.99082	−0.222746
$322$	0	0
$323$	34.9184	1.94291
$324$	0	0
$325$	0	0
$326$	0	0
$327$	10.2098	0.564603
$328$	0	0
$329$	9.38688	0.517515
$330$	0	0
$331$	−16.4126	−0.902118	−0.451059	−	0.892494i	$-0.648954\pi$
$331$	−16.4126	−0.902118	−0.451059	+	0.892494i	$0.648954\pi$
$332$	0	0
$333$	21.8686	1.19839
$334$	0	0
$335$	−46.7007	−2.55153
$336$	0	0
$337$	−25.6374	−1.39656	−0.698279	−	0.715826i	$-0.746051\pi$
$337$	−25.6374	−1.39656	−0.698279	+	0.715826i	$0.746051\pi$
$338$	0	0
$339$	−12.4985	−0.678826
$340$	0	0
$341$	21.1854	1.14725
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	3.99474	0.215069
$346$	0	0
$347$	1.40073	0.0751952	0.0375976	−	0.999293i	$-0.488029\pi$
$347$	1.40073	0.0751952	0.0375976	+	0.999293i	$0.488029\pi$
$348$	0	0
$349$	−25.9125	−1.38706	−0.693532	−	0.720426i	$-0.743946\pi$
$349$	−25.9125	−1.38706	−0.693532	+	0.720426i	$0.743946\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−6.48493	−0.345158	−0.172579	−	0.984996i	$-0.555210\pi$
$353$	−6.48493	−0.345158	−0.172579	+	0.984996i	$0.555210\pi$
$354$	0	0
$355$	−7.29003	−0.386915
$356$	0	0
$357$	−5.34238	−0.282749
$358$	0	0
$359$	28.6317	1.51112	0.755562	−	0.655077i	$-0.227364\pi$
$359$	28.6317	1.51112	0.755562	+	0.655077i	$0.227364\pi$
$360$	0	0
$361$	7.67848	0.404130
$362$	0	0
$363$	4.98618	0.261707
$364$	0	0
$365$	44.0380	2.30506
$366$	0	0
$367$	−18.2503	−0.952660	−0.476330	−	0.879267i	$-0.658033\pi$
$367$	−18.2503	−0.952660	−0.476330	+	0.879267i	$0.658033\pi$
$368$	0	0
$369$	−15.0718	−0.784606
$370$	0	0
$371$	−9.55478	−0.496059
$372$	0	0
$373$	16.9749	0.878926	0.439463	−	0.898261i	$-0.355169\pi$
$373$	16.9749	0.878926	0.439463	+	0.898261i	$0.355169\pi$
$374$	0	0
$375$	18.5154	0.956131
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−28.7507	−1.47682	−0.738412	−	0.674350i	$-0.764424\pi$
$379$	−28.7507	−1.47682	−0.738412	+	0.674350i	$0.764424\pi$
$380$	0	0
$381$	−12.5283	−0.641844
$382$	0	0
$383$	−13.6961	−0.699838	−0.349919	−	0.936780i	$-0.613791\pi$
$383$	−13.6961	−0.699838	−0.349919	+	0.936780i	$0.613791\pi$
$384$	0	0
$385$	8.63022	0.439837
$386$	0	0
$387$	−3.27286	−0.166369
$388$	0	0
$389$	−17.7145	−0.898163	−0.449081	−	0.893491i	$-0.648249\pi$
$389$	−17.7145	−0.898163	−0.449081	+	0.893491i	$0.648249\pi$
$390$	0	0
$391$	−8.57592	−0.433703
$392$	0	0
$393$	2.64603	0.133475
$394$	0	0
$395$	−22.8686	−1.15064
$396$	0	0
$397$	5.82013	0.292104	0.146052	−	0.989277i	$-0.453343\pi$
$397$	5.82013	0.292104	0.146052	+	0.989277i	$0.453343\pi$
$398$	0	0
$399$	−4.08170	−0.204341
$400$	0	0
$401$	11.8537	0.591944	0.295972	−	0.955197i	$-0.404356\pi$
$401$	11.8537	0.591944	0.295972	+	0.955197i	$0.404356\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−15.0217	−0.746432
$406$	0	0
$407$	−19.9372	−0.988252
$408$	0	0
$409$	−34.4258	−1.70225	−0.851123	−	0.524967i	$-0.824078\pi$
$409$	−34.4258	−1.70225	−0.851123	+	0.524967i	$0.824078\pi$
$410$	0	0
$411$	2.63916	0.130180
$412$	0	0
$413$	−9.44610	−0.464812
$414$	0	0
$415$	−41.6693	−2.04547
$416$	0	0
$417$	−0.727178	−0.0356100
$418$	0	0
$419$	20.3825	0.995748	0.497874	−	0.867249i	$-0.334114\pi$
$419$	20.3825	0.995748	0.497874	+	0.867249i	$0.334114\pi$
$420$	0	0
$421$	16.8702	0.822203	0.411102	−	0.911589i	$-0.365144\pi$
$421$	16.8702	0.822203	0.411102	+	0.911589i	$0.365144\pi$
$422$	0	0
$423$	22.2987	1.08420
$424$	0	0
$425$	−73.5510	−3.56775
$426$	0	0
$427$	4.85435	0.234918
$428$	0	0
$429$	0	0
$430$	0	0
$431$	2.80712	0.135214	0.0676070	−	0.997712i	$-0.478464\pi$
$431$	2.80712	0.135214	0.0676070	+	0.997712i	$0.478464\pi$
$432$	0	0
$433$	7.36252	0.353820	0.176910	−	0.984227i	$-0.443390\pi$
$433$	7.36252	0.353820	0.176910	+	0.984227i	$0.443390\pi$
$434$	0	0
$435$	13.6824	0.656022
$436$	0	0
$437$	−6.55220	−0.313434
$438$	0	0
$439$	−13.7775	−0.657563	−0.328781	−	0.944406i	$-0.606638\pi$
$439$	−13.7775	−0.657563	−0.328781	+	0.944406i	$0.606638\pi$
$440$	0	0
$441$	−2.37552	−0.113120
$442$	0	0
$443$	−23.9378	−1.13732	−0.568659	−	0.822573i	$-0.692538\pi$
$443$	−23.9378	−1.13732	−0.568659	+	0.822573i	$0.692538\pi$
$444$	0	0
$445$	29.6634	1.40618
$446$	0	0
$447$	2.10096	0.0993720
$448$	0	0
$449$	−11.5378	−0.544504	−0.272252	−	0.962226i	$-0.587768\pi$
$449$	−11.5378	−0.544504	−0.272252	+	0.962226i	$0.587768\pi$
$450$	0	0
$451$	13.7407	0.647023
$452$	0	0
$453$	2.72283	0.127930
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−9.53357	−0.445961	−0.222981	−	0.974823i	$-0.571579\pi$
$457$	−9.53357	−0.445961	−0.222981	+	0.974823i	$0.571579\pi$
$458$	0	0
$459$	−28.7180	−1.34044
$460$	0	0
$461$	−39.7483	−1.85126	−0.925632	−	0.378426i	$-0.876465\pi$
$461$	−39.7483	−1.85126	−0.925632	+	0.378426i	$0.876465\pi$
$462$	0	0
$463$	−36.3393	−1.68883	−0.844416	−	0.535688i	$-0.820052\pi$
$463$	−36.3393	−1.68883	−0.844416	+	0.535688i	$0.820052\pi$
$464$	0	0
$465$	30.8046	1.42853
$466$	0	0
$467$	−18.0265	−0.834167	−0.417083	−	0.908868i	$-0.636948\pi$
$467$	−18.0265	−0.834167	−0.417083	+	0.908868i	$0.636948\pi$
$468$	0	0
$469$	−11.7193	−0.541149
$470$	0	0
$471$	−4.24420	−0.195563
$472$	0	0
$473$	2.98381	0.137196
$474$	0	0
$475$	−56.1947	−2.57839
$476$	0	0
$477$	−22.6975	−1.03925
$478$	0	0
$479$	8.06276	0.368397	0.184199	−	0.982889i	$-0.441031\pi$
$479$	8.06276	0.368397	0.184199	+	0.982889i	$0.441031\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	1.00246	0.0456136
$484$	0	0
$485$	47.2370	2.14492
$486$	0	0
$487$	34.2378	1.55146	0.775730	−	0.631064i	$-0.217382\pi$
$487$	34.2378	1.55146	0.775730	+	0.631064i	$0.217382\pi$
$488$	0	0
$489$	11.5414	0.521918
$490$	0	0
$491$	19.2464	0.868579	0.434290	−	0.900773i	$-0.356999\pi$
$491$	19.2464	0.868579	0.434290	+	0.900773i	$0.356999\pi$
$492$	0	0
$493$	−29.3735	−1.32291
$494$	0	0
$495$	20.5012	0.921461
$496$	0	0
$497$	−1.82940	−0.0820599
$498$	0	0
$499$	−6.09231	−0.272729	−0.136365	−	0.990659i	$-0.543542\pi$
$499$	−6.09231	−0.272729	−0.136365	+	0.990659i	$0.543542\pi$
$500$	0	0
$501$	8.49259	0.379421
$502$	0	0
$503$	−27.7308	−1.23646	−0.618228	−	0.785999i	$-0.712149\pi$
$503$	−27.7308	−1.23646	−0.618228	+	0.785999i	$0.712149\pi$
$504$	0	0
$505$	51.3770	2.28625
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−1.19215	−0.0528409	−0.0264205	−	0.999651i	$-0.508411\pi$
$509$	−1.19215	−0.0528409	−0.0264205	+	0.999651i	$0.508411\pi$
$510$	0	0
$511$	11.0511	0.488874
$512$	0	0
$513$	−21.9413	−0.968730
$514$	0	0
$515$	−76.6119	−3.37593
$516$	0	0
$517$	−20.3293	−0.894082
$518$	0	0
$519$	−13.2348	−0.580943
$520$	0	0
$521$	−1.07966	−0.0473008	−0.0236504	−	0.999720i	$-0.507529\pi$
$521$	−1.07966	−0.0473008	−0.0236504	+	0.999720i	$0.507529\pi$
$522$	0	0
$523$	−17.4831	−0.764484	−0.382242	−	0.924062i	$-0.624848\pi$
$523$	−17.4831	−0.764484	−0.382242	+	0.924062i	$0.624848\pi$
$524$	0	0
$525$	8.59757	0.375229
$526$	0	0
$527$	−66.1315	−2.88073
$528$	0	0
$529$	−21.3908	−0.930034
$530$	0	0
$531$	−22.4394	−0.973786
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−20.1244	−0.870052
$536$	0	0
$537$	2.87888	0.124233
$538$	0	0
$539$	2.16572	0.0932840
$540$	0	0
$541$	−43.7984	−1.88304	−0.941520	−	0.336958i	$-0.890602\pi$
$541$	−43.7984	−1.88304	−0.941520	+	0.336958i	$0.890602\pi$
$542$	0	0
$543$	−9.85220	−0.422799
$544$	0	0
$545$	51.4845	2.20535
$546$	0	0
$547$	2.98369	0.127573	0.0637866	−	0.997964i	$-0.479682\pi$
$547$	2.98369	0.127573	0.0637866	+	0.997964i	$0.479682\pi$
$548$	0	0
$549$	11.5316	0.492156
$550$	0	0
$551$	−22.4420	−0.956062
$552$	0	0
$553$	−5.73877	−0.244037
$554$	0	0
$555$	−28.9898	−1.23055
$556$	0	0
$557$	26.0114	1.10214	0.551069	−	0.834459i	$-0.314220\pi$
$557$	26.0114	1.10214	0.551069	+	0.834459i	$0.314220\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	11.5701	0.488489
$562$	0	0
$563$	36.1233	1.52241	0.761207	−	0.648509i	$-0.224607\pi$
$563$	36.1233	1.52241	0.761207	+	0.648509i	$0.224607\pi$
$564$	0	0
$565$	−63.0257	−2.65151
$566$	0	0
$567$	−3.76962	−0.158309
$568$	0	0
$569$	1.74691	0.0732344	0.0366172	−	0.999329i	$-0.488342\pi$
$569$	1.74691	0.0732344	0.0366172	+	0.999329i	$0.488342\pi$
$570$	0	0
$571$	−33.0744	−1.38412	−0.692061	−	0.721839i	$-0.743297\pi$
$571$	−33.0744	−1.38412	−0.692061	+	0.721839i	$0.743297\pi$
$572$	0	0
$573$	14.1322	0.590380
$574$	0	0
$575$	13.8014	0.575556
$576$	0	0
$577$	−41.2766	−1.71837	−0.859184	−	0.511667i	$-0.829028\pi$
$577$	−41.2766	−1.71837	−0.859184	+	0.511667i	$0.829028\pi$
$578$	0	0
$579$	0.692609	0.0287838
$580$	0	0
$581$	−10.4567	−0.433818
$582$	0	0
$583$	20.6929	0.857013
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−23.6087	−0.974434	−0.487217	−	0.873281i	$-0.661988\pi$
$587$	−23.6087	−0.974434	−0.487217	+	0.873281i	$0.661988\pi$
$588$	0	0
$589$	−50.5260	−2.08189
$590$	0	0
$591$	−6.77672	−0.278757
$592$	0	0
$593$	15.4602	0.634875	0.317437	−	0.948279i	$-0.397178\pi$
$593$	15.4602	0.634875	0.317437	+	0.948279i	$0.397178\pi$
$594$	0	0
$595$	−26.9398	−1.10442
$596$	0	0
$597$	13.0594	0.534484
$598$	0	0
$599$	−6.71997	−0.274570	−0.137285	−	0.990532i	$-0.543838\pi$
$599$	−6.71997	−0.274570	−0.137285	+	0.990532i	$0.543838\pi$
$600$	0	0
$601$	−26.8322	−1.09451	−0.547253	−	0.836967i	$-0.684327\pi$
$601$	−26.8322	−1.09451	−0.547253	+	0.836967i	$0.684327\pi$
$602$	0	0
$603$	−27.8395	−1.13371
$604$	0	0
$605$	25.1436	1.02223
$606$	0	0
$607$	39.1892	1.59064	0.795320	−	0.606190i	$-0.207303\pi$
$607$	39.1892	1.59064	0.795320	+	0.606190i	$0.207303\pi$
$608$	0	0
$609$	3.43354	0.139134
$610$	0	0
$611$	0	0
$612$	0	0
$613$	20.2562	0.818142	0.409071	−	0.912503i	$-0.365853\pi$
$613$	20.2562	0.818142	0.409071	+	0.912503i	$0.365853\pi$
$614$	0	0
$615$	19.9797	0.805658
$616$	0	0
$617$	23.0963	0.929823	0.464912	−	0.885357i	$-0.346086\pi$
$617$	23.0963	0.929823	0.464912	+	0.885357i	$0.346086\pi$
$618$	0	0
$619$	−4.41122	−0.177302	−0.0886510	−	0.996063i	$-0.528256\pi$
$619$	−4.41122	−0.177302	−0.0886510	+	0.996063i	$0.528256\pi$
$620$	0	0
$621$	5.38875	0.216243
$622$	0	0
$623$	7.44390	0.298234
$624$	0	0
$625$	38.9685	1.55874
$626$	0	0
$627$	8.83981	0.353028
$628$	0	0
$629$	62.2354	2.48149
$630$	0	0
$631$	23.0673	0.918295	0.459148	−	0.888360i	$-0.348155\pi$
$631$	23.0673	0.918295	0.459148	+	0.888360i	$0.348155\pi$
$632$	0	0
$633$	0.0393771	0.00156510
$634$	0	0
$635$	−63.1759	−2.50706
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−4.34577	−0.171916
$640$	0	0
$641$	−31.0018	−1.22450	−0.612249	−	0.790665i	$-0.709735\pi$
$641$	−31.0018	−1.22450	−0.612249	+	0.790665i	$0.709735\pi$
$642$	0	0
$643$	19.2476	0.759051	0.379526	−	0.925181i	$-0.376087\pi$
$643$	19.2476	0.759051	0.379526	+	0.925181i	$0.376087\pi$
$644$	0	0
$645$	4.33861	0.170833
$646$	0	0
$647$	2.23421	0.0878358	0.0439179	−	0.999035i	$-0.486016\pi$
$647$	2.23421	0.0878358	0.0439179	+	0.999035i	$0.486016\pi$
$648$	0	0
$649$	20.4576	0.803030
$650$	0	0
$651$	7.73029	0.302974
$652$	0	0
$653$	15.5215	0.607405	0.303702	−	0.952767i	$-0.401777\pi$
$653$	15.5215	0.607405	0.303702	+	0.952767i	$0.401777\pi$
$654$	0	0
$655$	13.3430	0.521356
$656$	0	0
$657$	26.2522	1.02419
$658$	0	0
$659$	48.4355	1.88678	0.943390	−	0.331685i	$-0.107617\pi$
$659$	48.4355	1.88678	0.943390	+	0.331685i	$0.107617\pi$
$660$	0	0
$661$	39.6239	1.54119	0.770596	−	0.637324i	$-0.219959\pi$
$661$	39.6239	1.54119	0.770596	+	0.637324i	$0.219959\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−20.5826	−0.798160
$666$	0	0
$667$	5.51174	0.213415
$668$	0	0
$669$	−19.1823	−0.741629
$670$	0	0
$671$	−10.5131	−0.405855
$672$	0	0
$673$	33.1518	1.27791	0.638953	−	0.769246i	$-0.279368\pi$
$673$	33.1518	1.27791	0.638953	+	0.769246i	$0.279368\pi$
$674$	0	0
$675$	46.2164	1.77887
$676$	0	0
$677$	−7.44837	−0.286264	−0.143132	−	0.989704i	$-0.545717\pi$
$677$	−7.44837	−0.286264	−0.143132	+	0.989704i	$0.545717\pi$
$678$	0	0
$679$	11.8539	0.454912
$680$	0	0
$681$	−11.6441	−0.446202
$682$	0	0
$683$	−42.2326	−1.61599	−0.807993	−	0.589193i	$-0.799446\pi$
$683$	−42.2326	−1.61599	−0.807993	+	0.589193i	$0.799446\pi$
$684$	0	0
$685$	13.3084	0.508487
$686$	0	0
$687$	−11.9863	−0.457305
$688$	0	0
$689$	0	0
$690$	0	0
$691$	14.1462	0.538148	0.269074	−	0.963119i	$-0.413282\pi$
$691$	14.1462	0.538148	0.269074	+	0.963119i	$0.413282\pi$
$692$	0	0
$693$	5.14469	0.195431
$694$	0	0
$695$	−3.66691	−0.139094
$696$	0	0
$697$	−42.8924	−1.62467
$698$	0	0
$699$	−11.5913	−0.438422
$700$	0	0
$701$	28.3144	1.06942	0.534710	−	0.845035i	$-0.320421\pi$
$701$	28.3144	1.06942	0.534710	+	0.845035i	$0.320421\pi$
$702$	0	0
$703$	47.5493	1.79336
$704$	0	0
$705$	−29.5599	−1.11329
$706$	0	0
$707$	12.8928	0.484885
$708$	0	0
$709$	30.4395	1.14318	0.571591	−	0.820539i	$-0.306327\pi$
$709$	30.4395	1.14318	0.571591	+	0.820539i	$0.306327\pi$
$710$	0	0
$711$	−13.6325	−0.511260
$712$	0	0
$713$	12.4091	0.464726
$714$	0	0
$715$	0	0
$716$	0	0
$717$	7.66224	0.286151
$718$	0	0
$719$	−25.6272	−0.955734	−0.477867	−	0.878432i	$-0.658590\pi$
$719$	−25.6272	−0.955734	−0.477867	+	0.878432i	$0.658590\pi$
$720$	0	0
$721$	−19.2254	−0.715992
$722$	0	0
$723$	15.1805	0.564569
$724$	0	0
$725$	47.2712	1.75561
$726$	0	0
$727$	−34.2130	−1.26889	−0.634444	−	0.772969i	$-0.718771\pi$
$727$	−34.2130	−1.26889	−0.634444	+	0.772969i	$0.718771\pi$
$728$	0	0
$729$	1.11599	0.0413331
$730$	0	0
$731$	−9.31415	−0.344496
$732$	0	0
$733$	42.0876	1.55454	0.777271	−	0.629166i	$-0.216603\pi$
$733$	42.0876	1.55454	0.777271	+	0.629166i	$0.216603\pi$
$734$	0	0
$735$	3.14906	0.116155
$736$	0	0
$737$	25.3808	0.934912
$738$	0	0
$739$	−33.7021	−1.23975	−0.619877	−	0.784699i	$-0.712817\pi$
$739$	−33.7021	−1.23975	−0.619877	+	0.784699i	$0.712817\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	44.1050	1.61806	0.809028	−	0.587771i	$-0.199994\pi$
$743$	44.1050	1.61806	0.809028	+	0.587771i	$0.199994\pi$
$744$	0	0
$745$	10.5944	0.388150
$746$	0	0
$747$	−24.8401	−0.908852
$748$	0	0
$749$	−5.05012	−0.184527
$750$	0	0
$751$	28.2245	1.02993	0.514963	−	0.857212i	$-0.327805\pi$
$751$	28.2245	1.02993	0.514963	+	0.857212i	$0.327805\pi$
$752$	0	0
$753$	−0.165281	−0.00602317
$754$	0	0
$755$	13.7303	0.499697
$756$	0	0
$757$	−9.15179	−0.332627	−0.166314	−	0.986073i	$-0.553186\pi$
$757$	−9.15179	−0.332627	−0.166314	+	0.986073i	$0.553186\pi$
$758$	0	0
$759$	−2.17105	−0.0788040
$760$	0	0
$761$	−25.4695	−0.923268	−0.461634	−	0.887071i	$-0.652737\pi$
$761$	−25.4695	−0.923268	−0.461634	+	0.887071i	$0.652737\pi$
$762$	0	0
$763$	12.9198	0.467728
$764$	0	0
$765$	−63.9959	−2.31378
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−6.02093	−0.217120	−0.108560	−	0.994090i	$-0.534624\pi$
$769$	−6.02093	−0.217120	−0.108560	+	0.994090i	$0.534624\pi$
$770$	0	0
$771$	−8.03504	−0.289375
$772$	0	0
$773$	−3.65274	−0.131380	−0.0656900	−	0.997840i	$-0.520925\pi$
$773$	−3.65274	−0.131380	−0.0656900	+	0.997840i	$0.520925\pi$
$774$	0	0
$775$	106.426	3.82295
$776$	0	0
$777$	−7.27486	−0.260984
$778$	0	0
$779$	−32.7708	−1.17414
$780$	0	0
$781$	3.96196	0.141770
$782$	0	0
$783$	18.4571	0.659602
$784$	0	0
$785$	−21.4021	−0.763873
$786$	0	0
$787$	34.3751	1.22534	0.612670	−	0.790339i	$-0.290096\pi$
$787$	34.3751	1.22534	0.612670	+	0.790339i	$0.290096\pi$
$788$	0	0
$789$	−8.88089	−0.316168
$790$	0	0
$791$	−15.8160	−0.562353
$792$	0	0
$793$	0	0
$794$	0	0
$795$	30.0886	1.06713
$796$	0	0
$797$	32.4388	1.14904	0.574520	−	0.818490i	$-0.305189\pi$
$797$	32.4388	1.14904	0.574520	+	0.818490i	$0.305189\pi$
$798$	0	0
$799$	63.4593	2.24503
$800$	0	0
$801$	17.6831	0.624802
$802$	0	0
$803$	−23.9336	−0.844600
$804$	0	0
$805$	5.05507	0.178168
$806$	0	0
$807$	−12.8500	−0.452341
$808$	0	0
$809$	37.9210	1.33323	0.666615	−	0.745402i	$-0.267742\pi$
$809$	37.9210	1.33323	0.666615	+	0.745402i	$0.267742\pi$
$810$	0	0
$811$	−31.2365	−1.09686	−0.548431	−	0.836196i	$-0.684775\pi$
$811$	−31.2365	−1.09686	−0.548431	+	0.836196i	$0.684775\pi$
$812$	0	0
$813$	17.6144	0.617763
$814$	0	0
$815$	58.1991	2.03863
$816$	0	0
$817$	−7.11623	−0.248965
$818$	0	0
$819$	0	0
$820$	0	0
$821$	47.0884	1.64340	0.821699	−	0.569922i	$-0.193026\pi$
$821$	47.0884	1.64340	0.821699	+	0.569922i	$0.193026\pi$
$822$	0	0
$823$	2.33731	0.0814733	0.0407367	−	0.999170i	$-0.487030\pi$
$823$	2.33731	0.0814733	0.0407367	+	0.999170i	$0.487030\pi$
$824$	0	0
$825$	−18.6199	−0.648261
$826$	0	0
$827$	−13.6433	−0.474425	−0.237213	−	0.971458i	$-0.576234\pi$
$827$	−13.6433	−0.474425	−0.237213	+	0.971458i	$0.576234\pi$
$828$	0	0
$829$	−28.9610	−1.00586	−0.502929	−	0.864327i	$-0.667744\pi$
$829$	−28.9610	−1.00586	−0.502929	+	0.864327i	$0.667744\pi$
$830$	0	0
$831$	0.945208	0.0327889
$832$	0	0
$833$	−6.76042	−0.234235
$834$	0	0
$835$	42.8252	1.48203
$836$	0	0
$837$	41.5543	1.43633
$838$	0	0
$839$	22.5924	0.779978	0.389989	−	0.920820i	$-0.372479\pi$
$839$	22.5924	0.779978	0.389989	+	0.920820i	$0.372479\pi$
$840$	0	0
$841$	−10.1217	−0.349024
$842$	0	0
$843$	3.15581	0.108692
$844$	0	0
$845$	0	0
$846$	0	0
$847$	6.30968	0.216803
$848$	0	0
$849$	4.38348	0.150441
$850$	0	0
$851$	−11.6781	−0.400319
$852$	0	0
$853$	10.8220	0.370539	0.185269	−	0.982688i	$-0.440684\pi$
$853$	10.8220	0.370539	0.185269	+	0.982688i	$0.440684\pi$
$854$	0	0
$855$	−48.8944	−1.67215
$856$	0	0
$857$	−35.4224	−1.21001	−0.605004	−	0.796223i	$-0.706828\pi$
$857$	−35.4224	−1.21001	−0.605004	+	0.796223i	$0.706828\pi$
$858$	0	0
$859$	15.1785	0.517884	0.258942	−	0.965893i	$-0.416626\pi$
$859$	15.1785	0.517884	0.258942	+	0.965893i	$0.416626\pi$
$860$	0	0
$861$	5.01381	0.170870
$862$	0	0
$863$	12.9661	0.441372	0.220686	−	0.975345i	$-0.429170\pi$
$863$	12.9661	0.441372	0.220686	+	0.975345i	$0.429170\pi$
$864$	0	0
$865$	−66.7385	−2.26918
$866$	0	0
$867$	−22.6826	−0.770342
$868$	0	0
$869$	12.4285	0.421609
$870$	0	0
$871$	0	0
$872$	0	0
$873$	28.1592	0.953044
$874$	0	0
$875$	23.4300	0.792078
$876$	0	0
$877$	−34.3085	−1.15852	−0.579258	−	0.815144i	$-0.696658\pi$
$877$	−34.3085	−1.15852	−0.579258	+	0.815144i	$0.696658\pi$
$878$	0	0
$879$	−6.11460	−0.206240
$880$	0	0
$881$	−29.1949	−0.983601	−0.491800	−	0.870708i	$-0.663661\pi$
$881$	−29.1949	−0.983601	−0.491800	+	0.870708i	$0.663661\pi$
$882$	0	0
$883$	−35.0623	−1.17994	−0.589970	−	0.807425i	$-0.700860\pi$
$883$	−35.0623	−1.17994	−0.589970	+	0.807425i	$0.700860\pi$
$884$	0	0
$885$	29.7464	0.999914
$886$	0	0
$887$	−15.4554	−0.518943	−0.259471	−	0.965751i	$-0.583548\pi$
$887$	−15.4554	−0.518943	−0.259471	+	0.965751i	$0.583548\pi$
$888$	0	0
$889$	−15.8537	−0.531716
$890$	0	0
$891$	8.16392	0.273502
$892$	0	0
$893$	48.4844	1.62247
$894$	0	0
$895$	14.5172	0.485258
$896$	0	0
$897$	0	0
$898$	0	0
$899$	42.5027	1.41754
$900$	0	0
$901$	−64.5943	−2.15195
$902$	0	0
$903$	1.08876	0.0362315
$904$	0	0
$905$	−49.6813	−1.65146
$906$	0	0
$907$	17.8916	0.594081	0.297041	−	0.954865i	$-0.404000\pi$
$907$	17.8916	0.594081	0.297041	+	0.954865i	$0.404000\pi$
$908$	0	0
$909$	30.6271	1.01584
$910$	0	0
$911$	0.535509	0.0177422	0.00887110	−	0.999961i	$-0.497176\pi$
$911$	0.535509	0.0177422	0.00887110	+	0.999961i	$0.497176\pi$
$912$	0	0
$913$	22.6463	0.749483
$914$	0	0
$915$	−15.2866	−0.505361
$916$	0	0
$917$	3.34838	0.110573
$918$	0	0
$919$	38.5396	1.27130	0.635652	−	0.771976i	$-0.280731\pi$
$919$	38.5396	1.27130	0.635652	+	0.771976i	$0.280731\pi$
$920$	0	0
$921$	−1.29267	−0.0425950
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−100.156	−3.29312
$926$	0	0
$927$	−45.6703	−1.50001
$928$	0	0
$929$	−11.6669	−0.382778	−0.191389	−	0.981514i	$-0.561299\pi$
$929$	−11.6669	−0.382778	−0.191389	+	0.981514i	$0.561299\pi$
$930$	0	0
$931$	−5.16512	−0.169280
$932$	0	0
$933$	21.9226	0.717713
$934$	0	0
$935$	58.3439	1.90805
$936$	0	0
$937$	41.2645	1.34805	0.674027	−	0.738707i	$-0.264563\pi$
$937$	41.2645	1.34805	0.674027	+	0.738707i	$0.264563\pi$
$938$	0	0
$939$	−7.65652	−0.249861
$940$	0	0
$941$	14.1891	0.462550	0.231275	−	0.972888i	$-0.425710\pi$
$941$	14.1891	0.462550	0.231275	+	0.972888i	$0.425710\pi$
$942$	0	0
$943$	8.04848	0.262095
$944$	0	0
$945$	16.9278	0.550663
$946$	0	0
$947$	−14.0264	−0.455797	−0.227899	−	0.973685i	$-0.573186\pi$
$947$	−14.0264	−0.455797	−0.227899	+	0.973685i	$0.573186\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−6.70126	−0.217303
$952$	0	0
$953$	−39.7628	−1.28804	−0.644021	−	0.765007i	$-0.722735\pi$
$953$	−39.7628	−1.28804	−0.644021	+	0.765007i	$0.722735\pi$
$954$	0	0
$955$	71.2637	2.30604
$956$	0	0
$957$	−7.43608	−0.240374
$958$	0	0
$959$	3.33968	0.107844
$960$	0	0
$961$	64.6906	2.08679
$962$	0	0
$963$	−11.9966	−0.386586
$964$	0	0
$965$	3.49259	0.112430
$966$	0	0
$967$	−47.2674	−1.52002	−0.760009	−	0.649912i	$-0.774806\pi$
$967$	−47.2674	−1.52002	−0.760009	+	0.649912i	$0.774806\pi$
$968$	0	0
$969$	−27.5940	−0.886448
$970$	0	0
$971$	18.0869	0.580437	0.290219	−	0.956960i	$-0.406272\pi$
$971$	18.0869	0.580437	0.290219	+	0.956960i	$0.406272\pi$
$972$	0	0
$973$	−0.920194	−0.0295001
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−32.6963	−1.04605	−0.523024	−	0.852318i	$-0.675196\pi$
$977$	−32.6963	−1.04605	−0.523024	+	0.852318i	$0.675196\pi$
$978$	0	0
$979$	−16.1214	−0.515241
$980$	0	0
$981$	30.6912	0.979895
$982$	0	0
$983$	−59.5124	−1.89815	−0.949076	−	0.315047i	$-0.897980\pi$
$983$	−59.5124	−1.89815	−0.949076	+	0.315047i	$0.897980\pi$
$984$	0	0
$985$	−34.1727	−1.08883
$986$	0	0
$987$	−7.41792	−0.236115
$988$	0	0
$989$	1.74774	0.0555748
$990$	0	0
$991$	−52.3822	−1.66397	−0.831987	−	0.554795i	$-0.812797\pi$
$991$	−52.3822	−1.66397	−0.831987	+	0.554795i	$0.812797\pi$
$992$	0	0
$993$	12.9699	0.411589
$994$	0	0
$995$	65.8539	2.08771
$996$	0	0
$997$	−36.3479	−1.15115	−0.575575	−	0.817749i	$-0.695222\pi$
$997$	−36.3479	−1.15115	−0.575575	+	0.817749i	$0.695222\pi$
$998$	0	0
$999$	−39.1061	−1.23726

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9464.2.a.br.1.7	✓	15
13.12	even	2		9464.2.a.bs.1.7	yes	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9464.2.a.br.1.7	✓	15	1.1	even	1	trivial
9464.2.a.bs.1.7	yes	15	13.12	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:	\( N \)	\(=\)	\( 9464 = 2^{3} \cdot 7 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9464.a (trivial)

\(f(q)\)	\(=\)	\(q-0.790243 q^{3} -3.98493 q^{5} -1.00000 q^{7} -2.37552 q^{9} +2.16572 q^{11} +3.14906 q^{15} -6.76042 q^{17} -5.16512 q^{19} +0.790243 q^{21} +1.26855 q^{23} +10.8797 q^{25} +4.24797 q^{27} +4.34492 q^{29} +9.78216 q^{31} -1.71144 q^{33} +3.98493 q^{35} -9.20584 q^{37} +6.34464 q^{41} +1.37775 q^{43} +9.46626 q^{45} -9.38688 q^{47} +1.00000 q^{49} +5.34238 q^{51} +9.55478 q^{53} -8.63022 q^{55} +4.08170 q^{57} +9.44610 q^{59} -4.85435 q^{61} +2.37552 q^{63} +11.7193 q^{67} -1.00246 q^{69} +1.82940 q^{71} -11.0511 q^{73} -8.59757 q^{75} -2.16572 q^{77} +5.73877 q^{79} +3.76962 q^{81} +10.4567 q^{83} +26.9398 q^{85} -3.43354 q^{87} -7.44390 q^{89} -7.73029 q^{93} +20.5826 q^{95} -11.8539 q^{97} -5.14469 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - 3 q^{3} - 4 q^{5} - 15 q^{7} + 16 q^{9} - 15 q^{11} - 8 q^{15} + 2 q^{17} - 13 q^{19} + 3 q^{21} - 10 q^{23} + 23 q^{25} - 9 q^{27} + 25 q^{29} - 19 q^{31} + 24 q^{33} + 4 q^{35} + 2 q^{37} - 30 q^{41}+ \cdots - 70 q^{99}+O(q^{100}) \) 15 * q - 3 * q^3 - 4 * q^5 - 15 * q^7 + 16 * q^9 - 15 * q^11 - 8 * q^15 + 2 * q^17 - 13 * q^19 + 3 * q^21 - 10 * q^23 + 23 * q^25 - 9 * q^27 + 25 * q^29 - 19 * q^31 + 24 * q^33 + 4 * q^35 + 2 * q^37 - 30 * q^41 + 15 * q^43 + 2 * q^45 - 13 * q^47 + 15 * q^49 + q^51 + 62 * q^53 - 3 * q^55 + 2 * q^57 - 22 * q^59 - q^61 - 16 * q^63 - 29 * q^67 + 32 * q^69 - 20 * q^71 - 21 * q^73 - 46 * q^75 + 15 * q^77 + 29 * q^79 + 7 * q^81 - 23 * q^83 - 8 * q^85 - 22 * q^87 - 9 * q^89 - 10 * q^93 + 28 * q^95 - 11 * q^97 - 70 * q^99

Introduction

L-functions

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