LMFDB - Embedded newform 9328.2.a.bm.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9328,2,Mod(1,9328)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9328, 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("9328.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9328 = 2^{4} \cdot 11 \cdot 53 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9328.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:	$74.4844550055$
Analytic rank:	$0$
Dimension:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - 5x^{10} - 7x^{9} + 53x^{8} + 13x^{7} - 189x^{6} - 16x^{5} + 260x^{4} + 32x^{3} - 118x^{2} - 8x + 16 $ x^11 - 5x^10 - 7x^9 + 53x^8 + 13x^7 - 189x^6 - 16x^5 + 260x^4 + 32x^3 - 118x^2 - 8x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 4664)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$0.467706$ of defining polynomial
Character	$\chi$	$=$	9328.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.532294 q^{3} -1.86524 q^{5} -3.07135 q^{7} -2.71666 q^{9} +O(q^{10})$	$q+0.532294 q^{3} -1.86524 q^{5} -3.07135 q^{7} -2.71666 q^{9} -1.00000 q^{11} -0.746823 q^{13} -0.992853 q^{15} -4.23347 q^{17} -3.33998 q^{19} -1.63486 q^{21} -2.53315 q^{23} -1.52090 q^{25} -3.04294 q^{27} +3.63994 q^{29} -4.64804 q^{31} -0.532294 q^{33} +5.72880 q^{35} -11.2073 q^{37} -0.397529 q^{39} -2.48082 q^{41} +1.67295 q^{43} +5.06722 q^{45} -6.38528 q^{47} +2.43322 q^{49} -2.25345 q^{51} +1.00000 q^{53} +1.86524 q^{55} -1.77785 q^{57} -8.90656 q^{59} -3.31219 q^{61} +8.34383 q^{63} +1.39300 q^{65} +5.81822 q^{67} -1.34838 q^{69} +14.8302 q^{71} -0.249548 q^{73} -0.809563 q^{75} +3.07135 q^{77} +9.69122 q^{79} +6.53025 q^{81} -15.2300 q^{83} +7.89642 q^{85} +1.93752 q^{87} +2.62387 q^{89} +2.29376 q^{91} -2.47413 q^{93} +6.22986 q^{95} +4.14927 q^{97} +2.71666 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q + 6 q^{3} + 3 q^{5} + 5 q^{7} + 7 q^{9}+O(q^{10}) $ 11 * q + 6 * q^3 + 3 * q^5 + 5 * q^7 + 7 * q^9	$ 11 q + 6 q^{3} + 3 q^{5} + 5 q^{7} + 7 q^{9} - 11 q^{11} - 13 q^{13} + 8 q^{15} - 7 q^{17} + 21 q^{19} + 6 q^{21} - 11 q^{23} + 4 q^{25} + 6 q^{27} + 5 q^{31} - 6 q^{33} + 25 q^{35} - 4 q^{37} + 19 q^{39} - 11 q^{41} + 16 q^{45} + 17 q^{47} - 2 q^{49} + 18 q^{51} + 11 q^{53} - 3 q^{55} - 5 q^{57} + 19 q^{59} - 2 q^{61} + 36 q^{63} - 13 q^{65} - 25 q^{67} + 3 q^{69} + 30 q^{71} + 5 q^{73} + 5 q^{75} - 5 q^{77} + 23 q^{79} - 9 q^{81} + 19 q^{83} + 2 q^{85} + 7 q^{87} + 6 q^{89} + 20 q^{91} + 43 q^{93} + 50 q^{95} - 35 q^{97} - 7 q^{99}+O(q^{100}) $ 11 * q + 6 * q^3 + 3 * q^5 + 5 * q^7 + 7 * q^9 - 11 * q^11 - 13 * q^13 + 8 * q^15 - 7 * q^17 + 21 * q^19 + 6 * q^21 - 11 * q^23 + 4 * q^25 + 6 * q^27 + 5 * q^31 - 6 * q^33 + 25 * q^35 - 4 * q^37 + 19 * q^39 - 11 * q^41 + 16 * q^45 + 17 * q^47 - 2 * q^49 + 18 * q^51 + 11 * q^53 - 3 * q^55 - 5 * q^57 + 19 * q^59 - 2 * q^61 + 36 * q^63 - 13 * q^65 - 25 * q^67 + 3 * q^69 + 30 * q^71 + 5 * q^73 + 5 * q^75 - 5 * q^77 + 23 * q^79 - 9 * q^81 + 19 * q^83 + 2 * q^85 + 7 * q^87 + 6 * q^89 + 20 * q^91 + 43 * q^93 + 50 * q^95 - 35 * q^97 - 7 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	0.532294	0.307320	0.153660	−	0.988124i	$-0.450894\pi$
$3$	0.532294	0.307320	0.153660	+	0.988124i	$0.450894\pi$
$4$	0	0
$5$	−1.86524	−0.834159	−0.417079	−	0.908870i	$-0.636946\pi$
$5$	−1.86524	−0.834159	−0.417079	+	0.908870i	$0.636946\pi$
$6$	0	0
$7$	−3.07135	−1.16086	−0.580431	−	0.814309i	$-0.697116\pi$
$7$	−3.07135	−1.16086	−0.580431	+	0.814309i	$0.697116\pi$
$8$	0	0
$9$	−2.71666	−0.905554
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−0.746823	−0.207132	−0.103566	−	0.994623i	$-0.533025\pi$
$13$	−0.746823	−0.207132	−0.103566	+	0.994623i	$0.533025\pi$
$14$	0	0
$15$	−0.992853	−0.256354
$16$	0	0
$17$	−4.23347	−1.02677	−0.513384	−	0.858159i	$-0.671608\pi$
$17$	−4.23347	−1.02677	−0.513384	+	0.858159i	$0.671608\pi$
$18$	0	0
$19$	−3.33998	−0.766245	−0.383123	−	0.923698i	$-0.625151\pi$
$19$	−3.33998	−0.766245	−0.383123	+	0.923698i	$0.625151\pi$
$20$	0	0
$21$	−1.63486	−0.356756
$22$	0	0
$23$	−2.53315	−0.528197	−0.264099	−	0.964496i	$-0.585074\pi$
$23$	−2.53315	−0.528197	−0.264099	+	0.964496i	$0.585074\pi$
$24$	0	0
$25$	−1.52090	−0.304179
$26$	0	0
$27$	−3.04294	−0.585615
$28$	0	0
$29$	3.63994	0.675920	0.337960	−	0.941160i	$-0.390263\pi$
$29$	3.63994	0.675920	0.337960	+	0.941160i	$0.390263\pi$
$30$	0	0
$31$	−4.64804	−0.834813	−0.417407	−	0.908720i	$-0.637061\pi$
$31$	−4.64804	−0.834813	−0.417407	+	0.908720i	$0.637061\pi$
$32$	0	0
$33$	−0.532294	−0.0926605
$34$	0	0
$35$	5.72880	0.968344
$36$	0	0
$37$	−11.2073	−1.84246	−0.921231	−	0.389017i	$-0.872815\pi$
$37$	−11.2073	−1.84246	−0.921231	+	0.389017i	$0.872815\pi$
$38$	0	0
$39$	−0.397529	−0.0636557
$40$	0	0
$41$	−2.48082	−0.387439	−0.193720	−	0.981057i	$-0.562055\pi$
$41$	−2.48082	−0.387439	−0.193720	+	0.981057i	$0.562055\pi$
$42$	0	0
$43$	1.67295	0.255122	0.127561	−	0.991831i	$-0.459285\pi$
$43$	1.67295	0.255122	0.127561	+	0.991831i	$0.459285\pi$
$44$	0	0
$45$	5.06722	0.755376
$46$	0	0
$47$	−6.38528	−0.931389	−0.465694	−	0.884946i	$-0.654195\pi$
$47$	−6.38528	−0.931389	−0.465694	+	0.884946i	$0.654195\pi$
$48$	0	0
$49$	2.43322	0.347602
$50$	0	0
$51$	−2.25345	−0.315546
$52$	0	0
$53$	1.00000	0.137361
$53$	1.00000	0.137361
$54$	0	0
$55$	1.86524	0.251508
$56$	0	0
$57$	−1.77785	−0.235482
$58$	0	0
$59$	−8.90656	−1.15954	−0.579768	−	0.814782i	$-0.696857\pi$
$59$	−8.90656	−1.15954	−0.579768	+	0.814782i	$0.696857\pi$
$60$	0	0
$61$	−3.31219	−0.424083	−0.212041	−	0.977261i	$-0.568011\pi$
$61$	−3.31219	−0.424083	−0.212041	+	0.977261i	$0.568011\pi$
$62$	0	0
$63$	8.34383	1.05122
$64$	0	0
$65$	1.39300	0.172781
$66$	0	0
$67$	5.81822	0.710808	0.355404	−	0.934713i	$-0.384343\pi$
$67$	5.81822	0.710808	0.355404	+	0.934713i	$0.384343\pi$
$68$	0	0
$69$	−1.34838	−0.162326
$70$	0	0
$71$	14.8302	1.76002	0.880009	−	0.474957i	$-0.157536\pi$
$71$	14.8302	1.76002	0.880009	+	0.474957i	$0.157536\pi$
$72$	0	0
$73$	−0.249548	−0.0292073	−0.0146037	−	0.999893i	$-0.504649\pi$
$73$	−0.249548	−0.0292073	−0.0146037	+	0.999893i	$0.504649\pi$
$74$	0	0
$75$	−0.809563	−0.0934803
$76$	0	0
$77$	3.07135	0.350013
$78$	0	0
$79$	9.69122	1.09035	0.545174	−	0.838323i	$-0.316464\pi$
$79$	9.69122	1.09035	0.545174	+	0.838323i	$0.316464\pi$
$80$	0	0
$81$	6.53025	0.725583
$82$	0	0
$83$	−15.2300	−1.67171	−0.835853	−	0.548954i	$-0.815026\pi$
$83$	−15.2300	−1.67171	−0.835853	+	0.548954i	$0.815026\pi$
$84$	0	0
$85$	7.89642	0.856487
$86$	0	0
$87$	1.93752	0.207724
$88$	0	0
$89$	2.62387	0.278129	0.139065	−	0.990283i	$-0.455590\pi$
$89$	2.62387	0.278129	0.139065	+	0.990283i	$0.455590\pi$
$90$	0	0
$91$	2.29376	0.240451
$92$	0	0
$93$	−2.47413	−0.256555
$94$	0	0
$95$	6.22986	0.639170
$96$	0	0
$97$	4.14927	0.421295	0.210648	−	0.977562i	$-0.432443\pi$
$97$	4.14927	0.421295	0.210648	+	0.977562i	$0.432443\pi$
$98$	0	0
$99$	2.71666	0.273035
$100$	0	0
$101$	−12.1578	−1.20975	−0.604875	−	0.796321i	$-0.706777\pi$
$101$	−12.1578	−1.20975	−0.604875	+	0.796321i	$0.706777\pi$
$102$	0	0
$103$	10.5709	1.04159	0.520793	−	0.853683i	$-0.325636\pi$
$103$	10.5709	1.04159	0.520793	+	0.853683i	$0.325636\pi$
$104$	0	0
$105$	3.04940	0.297591
$106$	0	0
$107$	−8.05801	−0.778997	−0.389499	−	0.921027i	$-0.627352\pi$
$107$	−8.05801	−0.778997	−0.389499	+	0.921027i	$0.627352\pi$
$108$	0	0
$109$	−19.4056	−1.85872	−0.929358	−	0.369179i	$-0.879639\pi$
$109$	−19.4056	−1.85872	−0.929358	+	0.369179i	$0.879639\pi$
$110$	0	0
$111$	−5.96555	−0.566225
$112$	0	0
$113$	−13.0506	−1.22770	−0.613849	−	0.789424i	$-0.710379\pi$
$113$	−13.0506	−1.22770	−0.613849	+	0.789424i	$0.710379\pi$
$114$	0	0
$115$	4.72491	0.440600
$116$	0	0
$117$	2.02887	0.187569
$118$	0	0
$119$	13.0025	1.19194
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−1.32053	−0.119068
$124$	0	0
$125$	12.1630	1.08789
$126$	0	0
$127$	−8.35792	−0.741646	−0.370823	−	0.928704i	$-0.620924\pi$
$127$	−8.35792	−0.741646	−0.370823	+	0.928704i	$0.620924\pi$
$128$	0	0
$129$	0.890500	0.0784042
$130$	0	0
$131$	−6.97784	−0.609657	−0.304828	−	0.952407i	$-0.598599\pi$
$131$	−6.97784	−0.609657	−0.304828	+	0.952407i	$0.598599\pi$
$132$	0	0
$133$	10.2583	0.889505
$134$	0	0
$135$	5.67581	0.488496
$136$	0	0
$137$	−3.34527	−0.285805	−0.142903	−	0.989737i	$-0.545644\pi$
$137$	−3.34527	−0.285805	−0.142903	+	0.989737i	$0.545644\pi$
$138$	0	0
$139$	6.00372	0.509229	0.254614	−	0.967043i	$-0.418051\pi$
$139$	6.00372	0.509229	0.254614	+	0.967043i	$0.418051\pi$
$140$	0	0
$141$	−3.39885	−0.286234
$142$	0	0
$143$	0.746823	0.0624525
$144$	0	0
$145$	−6.78935	−0.563825
$146$	0	0
$147$	1.29519	0.106825
$148$	0	0
$149$	−2.43636	−0.199595	−0.0997973	−	0.995008i	$-0.531819\pi$
$149$	−2.43636	−0.199595	−0.0997973	+	0.995008i	$0.531819\pi$
$150$	0	0
$151$	3.43180	0.279276	0.139638	−	0.990203i	$-0.455406\pi$
$151$	3.43180	0.279276	0.139638	+	0.990203i	$0.455406\pi$
$152$	0	0
$153$	11.5009	0.929794
$154$	0	0
$155$	8.66970	0.696367
$156$	0	0
$157$	1.00091	0.0798813	0.0399406	−	0.999202i	$-0.487283\pi$
$157$	1.00091	0.0798813	0.0399406	+	0.999202i	$0.487283\pi$
$158$	0	0
$159$	0.532294	0.0422136
$160$	0	0
$161$	7.78019	0.613165
$162$	0	0
$163$	19.8098	1.55162	0.775811	−	0.630965i	$-0.217341\pi$
$163$	19.8098	1.55162	0.775811	+	0.630965i	$0.217341\pi$
$164$	0	0
$165$	0.992853	0.0772935
$166$	0	0
$167$	12.8139	0.991573	0.495787	−	0.868444i	$-0.334880\pi$
$167$	12.8139	0.991573	0.495787	+	0.868444i	$0.334880\pi$
$168$	0	0
$169$	−12.4423	−0.957097
$170$	0	0
$171$	9.07361	0.693877
$172$	0	0
$173$	12.9471	0.984349	0.492174	−	0.870497i	$-0.336202\pi$
$173$	12.9471	0.984349	0.492174	+	0.870497i	$0.336202\pi$
$174$	0	0
$175$	4.67121	0.353110
$176$	0	0
$177$	−4.74091	−0.356348
$178$	0	0
$179$	5.58366	0.417342	0.208671	−	0.977986i	$-0.433086\pi$
$179$	5.58366	0.417342	0.208671	+	0.977986i	$0.433086\pi$
$180$	0	0
$181$	−14.1217	−1.04966	−0.524829	−	0.851207i	$-0.675871\pi$
$181$	−14.1217	−1.04966	−0.524829	+	0.851207i	$0.675871\pi$
$182$	0	0
$183$	−1.76306	−0.130329
$184$	0	0
$185$	20.9042	1.53691
$186$	0	0
$187$	4.23347	0.309582
$188$	0	0
$189$	9.34596	0.679818
$190$	0	0
$191$	−18.4155	−1.33250	−0.666248	−	0.745730i	$-0.732101\pi$
$191$	−18.4155	−1.33250	−0.666248	+	0.745730i	$0.732101\pi$
$192$	0	0
$193$	−9.04466	−0.651049	−0.325524	−	0.945534i	$-0.605541\pi$
$193$	−9.04466	−0.651049	−0.325524	+	0.945534i	$0.605541\pi$
$194$	0	0
$195$	0.741486	0.0530989
$196$	0	0
$197$	21.0990	1.50324	0.751621	−	0.659595i	$-0.229272\pi$
$197$	21.0990	1.50324	0.751621	+	0.659595i	$0.229272\pi$
$198$	0	0
$199$	2.73614	0.193960	0.0969800	−	0.995286i	$-0.469082\pi$
$199$	2.73614	0.193960	0.0969800	+	0.995286i	$0.469082\pi$
$200$	0	0
$201$	3.09700	0.218446
$202$	0	0
$203$	−11.1795	−0.784650
$204$	0	0
$205$	4.62732	0.323186
$206$	0	0
$207$	6.88170	0.478312
$208$	0	0
$209$	3.33998	0.231032
$210$	0	0
$211$	−3.39846	−0.233959	−0.116980	−	0.993134i	$-0.537321\pi$
$211$	−3.39846	−0.233959	−0.116980	+	0.993134i	$0.537321\pi$
$212$	0	0
$213$	7.89401	0.540889
$214$	0	0
$215$	−3.12044	−0.212813
$216$	0	0
$217$	14.2758	0.969104
$218$	0	0
$219$	−0.132833	−0.00897600
$220$	0	0
$221$	3.16165	0.212676
$222$	0	0
$223$	27.6328	1.85043	0.925215	−	0.379443i	$-0.123884\pi$
$223$	27.6328	1.85043	0.925215	+	0.379443i	$0.123884\pi$
$224$	0	0
$225$	4.13176	0.275451
$226$	0	0
$227$	1.48341	0.0984574	0.0492287	−	0.998788i	$-0.484324\pi$
$227$	1.48341	0.0984574	0.0492287	+	0.998788i	$0.484324\pi$
$228$	0	0
$229$	−6.03326	−0.398689	−0.199344	−	0.979929i	$-0.563881\pi$
$229$	−6.03326	−0.398689	−0.199344	+	0.979929i	$0.563881\pi$
$230$	0	0
$231$	1.63486	0.107566
$232$	0	0
$233$	−26.0433	−1.70615	−0.853076	−	0.521787i	$-0.825266\pi$
$233$	−26.0433	−1.70615	−0.853076	+	0.521787i	$0.825266\pi$
$234$	0	0
$235$	11.9101	0.776926
$236$	0	0
$237$	5.15857	0.335085
$238$	0	0
$239$	−19.1197	−1.23675	−0.618375	−	0.785883i	$-0.712209\pi$
$239$	−19.1197	−1.23675	−0.618375	+	0.785883i	$0.712209\pi$
$240$	0	0
$241$	−11.2355	−0.723745	−0.361872	−	0.932228i	$-0.617862\pi$
$241$	−11.2355	−0.723745	−0.361872	+	0.932228i	$0.617862\pi$
$242$	0	0
$243$	12.6048	0.808601
$244$	0	0
$245$	−4.53852	−0.289955
$246$	0	0
$247$	2.49438	0.158714
$248$	0	0
$249$	−8.10681	−0.513748
$250$	0	0
$251$	15.4457	0.974927	0.487463	−	0.873143i	$-0.337922\pi$
$251$	15.4457	0.974927	0.487463	+	0.873143i	$0.337922\pi$
$252$	0	0
$253$	2.53315	0.159258
$254$	0	0
$255$	4.20321	0.263215
$256$	0	0
$257$	−17.8001	−1.11034	−0.555169	−	0.831738i	$-0.687346\pi$
$257$	−17.8001	−1.11034	−0.555169	+	0.831738i	$0.687346\pi$
$258$	0	0
$259$	34.4214	2.13884
$260$	0	0
$261$	−9.88849	−0.612082
$262$	0	0
$263$	1.35342	0.0834557	0.0417278	−	0.999129i	$-0.486714\pi$
$263$	1.35342	0.0834557	0.0417278	+	0.999129i	$0.486714\pi$
$264$	0	0
$265$	−1.86524	−0.114581
$266$	0	0
$267$	1.39667	0.0854747
$268$	0	0
$269$	−30.1874	−1.84056	−0.920280	−	0.391261i	$-0.872039\pi$
$269$	−30.1874	−1.84056	−0.920280	+	0.391261i	$0.872039\pi$
$270$	0	0
$271$	−4.62893	−0.281187	−0.140594	−	0.990067i	$-0.544901\pi$
$271$	−4.62893	−0.281187	−0.140594	+	0.990067i	$0.544901\pi$
$272$	0	0
$273$	1.22095	0.0738955
$274$	0	0
$275$	1.52090	0.0917135
$276$	0	0
$277$	−9.12552	−0.548300	−0.274150	−	0.961687i	$-0.588396\pi$
$277$	−9.12552	−0.548300	−0.274150	+	0.961687i	$0.588396\pi$
$278$	0	0
$279$	12.6272	0.755969
$280$	0	0
$281$	−22.1878	−1.32361	−0.661806	−	0.749675i	$-0.730210\pi$
$281$	−22.1878	−1.32361	−0.661806	+	0.749675i	$0.730210\pi$
$282$	0	0
$283$	6.33820	0.376767	0.188383	−	0.982096i	$-0.439675\pi$
$283$	6.33820	0.376767	0.188383	+	0.982096i	$0.439675\pi$
$284$	0	0
$285$	3.31611	0.196430
$286$	0	0
$287$	7.61948	0.449764
$288$	0	0
$289$	0.922264	0.0542508
$290$	0	0
$291$	2.20863	0.129472
$292$	0	0
$293$	28.3394	1.65561	0.827803	−	0.561018i	$-0.189590\pi$
$293$	28.3394	1.65561	0.827803	+	0.561018i	$0.189590\pi$
$294$	0	0
$295$	16.6128	0.967237
$296$	0	0
$297$	3.04294	0.176570
$298$	0	0
$299$	1.89181	0.109406
$300$	0	0
$301$	−5.13822	−0.296162
$302$	0	0
$303$	−6.47154	−0.371780
$304$	0	0
$305$	6.17802	0.353752
$306$	0	0
$307$	10.0705	0.574752	0.287376	−	0.957818i	$-0.407217\pi$
$307$	10.0705	0.574752	0.287376	+	0.957818i	$0.407217\pi$
$308$	0	0
$309$	5.62685	0.320100
$310$	0	0
$311$	28.7817	1.63206	0.816029	−	0.578010i	$-0.196171\pi$
$311$	28.7817	1.63206	0.816029	+	0.578010i	$0.196171\pi$
$312$	0	0
$313$	28.5318	1.61271	0.806356	−	0.591430i	$-0.201436\pi$
$313$	28.5318	1.61271	0.806356	+	0.591430i	$0.201436\pi$
$314$	0	0
$315$	−15.5632	−0.876888
$316$	0	0
$317$	−10.0346	−0.563601	−0.281800	−	0.959473i	$-0.590932\pi$
$317$	−10.0346	−0.563601	−0.281800	+	0.959473i	$0.590932\pi$
$318$	0	0
$319$	−3.63994	−0.203798
$320$	0	0
$321$	−4.28923	−0.239401
$322$	0	0
$323$	14.1397	0.786755
$324$	0	0
$325$	1.13584	0.0630051
$326$	0	0
$327$	−10.3295	−0.571221
$328$	0	0
$329$	19.6115	1.08121
$330$	0	0
$331$	3.08067	0.169329	0.0846646	−	0.996410i	$-0.473018\pi$
$331$	3.08067	0.169329	0.0846646	+	0.996410i	$0.473018\pi$
$332$	0	0
$333$	30.4463	1.66845
$334$	0	0
$335$	−10.8523	−0.592927
$336$	0	0
$337$	−12.7304	−0.693467	−0.346734	−	0.937964i	$-0.612709\pi$
$337$	−12.7304	−0.693467	−0.346734	+	0.937964i	$0.612709\pi$
$338$	0	0
$339$	−6.94675	−0.377296
$340$	0	0
$341$	4.64804	0.251706
$342$	0	0
$343$	14.0262	0.757344
$344$	0	0
$345$	2.51504	0.135405
$346$	0	0
$347$	11.9429	0.641130	0.320565	−	0.947226i	$-0.396127\pi$
$347$	11.9429	0.641130	0.320565	+	0.947226i	$0.396127\pi$
$348$	0	0
$349$	12.7427	0.682101	0.341050	−	0.940045i	$-0.389217\pi$
$349$	12.7427	0.682101	0.341050	+	0.940045i	$0.389217\pi$
$350$	0	0
$351$	2.27254	0.121299
$352$	0	0
$353$	−3.19759	−0.170190	−0.0850952	−	0.996373i	$-0.527119\pi$
$353$	−3.19759	−0.170190	−0.0850952	+	0.996373i	$0.527119\pi$
$354$	0	0
$355$	−27.6618	−1.46813
$356$	0	0
$357$	6.92114	0.366306
$358$	0	0
$359$	1.60220	0.0845610	0.0422805	−	0.999106i	$-0.486538\pi$
$359$	1.60220	0.0845610	0.0422805	+	0.999106i	$0.486538\pi$
$360$	0	0
$361$	−7.84450	−0.412869
$362$	0	0
$363$	0.532294	0.0279382
$364$	0	0
$365$	0.465465	0.0243636
$366$	0	0
$367$	−7.97142	−0.416104	−0.208052	−	0.978118i	$-0.566712\pi$
$367$	−7.97142	−0.416104	−0.208052	+	0.978118i	$0.566712\pi$
$368$	0	0
$369$	6.73956	0.350847
$370$	0	0
$371$	−3.07135	−0.159457
$372$	0	0
$373$	24.3274	1.25962	0.629812	−	0.776748i	$-0.283132\pi$
$373$	24.3274	1.25962	0.629812	+	0.776748i	$0.283132\pi$
$374$	0	0
$375$	6.47429	0.334331
$376$	0	0
$377$	−2.71839	−0.140004
$378$	0	0
$379$	−10.7797	−0.553715	−0.276858	−	0.960911i	$-0.589293\pi$
$379$	−10.7797	−0.553715	−0.276858	+	0.960911i	$0.589293\pi$
$380$	0	0
$381$	−4.44887	−0.227923
$382$	0	0
$383$	32.8899	1.68060	0.840298	−	0.542124i	$-0.182380\pi$
$383$	32.8899	1.68060	0.840298	+	0.542124i	$0.182380\pi$
$384$	0	0
$385$	−5.72880	−0.291967
$386$	0	0
$387$	−4.54484	−0.231027
$388$	0	0
$389$	25.8798	1.31216	0.656079	−	0.754692i	$-0.272214\pi$
$389$	25.8798	1.31216	0.656079	+	0.754692i	$0.272214\pi$
$390$	0	0
$391$	10.7240	0.542336
$392$	0	0
$393$	−3.71426	−0.187360
$394$	0	0
$395$	−18.0764	−0.909523
$396$	0	0
$397$	−25.2703	−1.26828	−0.634139	−	0.773219i	$-0.718645\pi$
$397$	−25.2703	−1.26828	−0.634139	+	0.773219i	$0.718645\pi$
$398$	0	0
$399$	5.46042	0.273363
$400$	0	0
$401$	−1.58579	−0.0791906	−0.0395953	−	0.999216i	$-0.512607\pi$
$401$	−1.58579	−0.0791906	−0.0395953	+	0.999216i	$0.512607\pi$
$402$	0	0
$403$	3.47127	0.172916
$404$	0	0
$405$	−12.1805	−0.605252
$406$	0	0
$407$	11.2073	0.555523
$408$	0	0
$409$	12.0701	0.596830	0.298415	−	0.954436i	$-0.403542\pi$
$409$	12.0701	0.596830	0.298415	+	0.954436i	$0.403542\pi$
$410$	0	0
$411$	−1.78066	−0.0878337
$412$	0	0
$413$	27.3552	1.34606
$414$	0	0
$415$	28.4075	1.39447
$416$	0	0
$417$	3.19574	0.156496
$418$	0	0
$419$	4.07750	0.199199	0.0995994	−	0.995028i	$-0.468244\pi$
$419$	4.07750	0.199199	0.0995994	+	0.995028i	$0.468244\pi$
$420$	0	0
$421$	−4.40436	−0.214655	−0.107328	−	0.994224i	$-0.534229\pi$
$421$	−4.40436	−0.214655	−0.107328	+	0.994224i	$0.534229\pi$
$422$	0	0
$423$	17.3467	0.843423
$424$	0	0
$425$	6.43867	0.312321
$426$	0	0
$427$	10.1729	0.492302
$428$	0	0
$429$	0.397529	0.0191929
$430$	0	0
$431$	9.89204	0.476483	0.238242	−	0.971206i	$-0.423429\pi$
$431$	9.89204	0.476483	0.238242	+	0.971206i	$0.423429\pi$
$432$	0	0
$433$	−36.5852	−1.75817	−0.879087	−	0.476661i	$-0.841847\pi$
$433$	−36.5852	−1.75817	−0.879087	+	0.476661i	$0.841847\pi$
$434$	0	0
$435$	−3.61393	−0.173275
$436$	0	0
$437$	8.46067	0.404729
$438$	0	0
$439$	6.69790	0.319673	0.159837	−	0.987143i	$-0.448903\pi$
$439$	6.69790	0.319673	0.159837	+	0.987143i	$0.448903\pi$
$440$	0	0
$441$	−6.61023	−0.314773
$442$	0	0
$443$	−28.9507	−1.37549	−0.687744	−	0.725953i	$-0.741399\pi$
$443$	−28.9507	−1.37549	−0.687744	+	0.725953i	$0.741399\pi$
$444$	0	0
$445$	−4.89413	−0.232004
$446$	0	0
$447$	−1.29686	−0.0613394
$448$	0	0
$449$	14.6576	0.691734	0.345867	−	0.938283i	$-0.387585\pi$
$449$	14.6576	0.691734	0.345867	+	0.938283i	$0.387585\pi$
$450$	0	0
$451$	2.48082	0.116817
$452$	0	0
$453$	1.82673	0.0858271
$454$	0	0
$455$	−4.27840	−0.200575
$456$	0	0
$457$	−11.5651	−0.540992	−0.270496	−	0.962721i	$-0.587188\pi$
$457$	−11.5651	−0.540992	−0.270496	+	0.962721i	$0.587188\pi$
$458$	0	0
$459$	12.8822	0.601290
$460$	0	0
$461$	32.7248	1.52415	0.762073	−	0.647492i	$-0.224182\pi$
$461$	32.7248	1.52415	0.762073	+	0.647492i	$0.224182\pi$
$462$	0	0
$463$	−39.6568	−1.84301	−0.921505	−	0.388367i	$-0.873039\pi$
$463$	−39.6568	−1.84301	−0.921505	+	0.388367i	$0.873039\pi$
$464$	0	0
$465$	4.61483	0.214007
$466$	0	0
$467$	20.9503	0.969466	0.484733	−	0.874662i	$-0.338917\pi$
$467$	20.9503	0.969466	0.484733	+	0.874662i	$0.338917\pi$
$468$	0	0
$469$	−17.8698	−0.825151
$470$	0	0
$471$	0.532778	0.0245491
$472$	0	0
$473$	−1.67295	−0.0769223
$474$	0	0
$475$	5.07977	0.233076
$476$	0	0
$477$	−2.71666	−0.124387
$478$	0	0
$479$	24.1536	1.10361	0.551803	−	0.833975i	$-0.313940\pi$
$479$	24.1536	1.10361	0.551803	+	0.833975i	$0.313940\pi$
$480$	0	0
$481$	8.36984	0.381632
$482$	0	0
$483$	4.14135	0.188438
$484$	0	0
$485$	−7.73938	−0.351427
$486$	0	0
$487$	−9.47449	−0.429330	−0.214665	−	0.976688i	$-0.568866\pi$
$487$	−9.47449	−0.429330	−0.214665	+	0.976688i	$0.568866\pi$
$488$	0	0
$489$	10.5446	0.476845
$490$	0	0
$491$	−20.7666	−0.937184	−0.468592	−	0.883415i	$-0.655239\pi$
$491$	−20.7666	−0.937184	−0.468592	+	0.883415i	$0.655239\pi$
$492$	0	0
$493$	−15.4096	−0.694012
$494$	0	0
$495$	−5.06722	−0.227754
$496$	0	0
$497$	−45.5487	−2.04314
$498$	0	0
$499$	−13.7815	−0.616944	−0.308472	−	0.951233i	$-0.599818\pi$
$499$	−13.7815	−0.616944	−0.308472	+	0.951233i	$0.599818\pi$
$500$	0	0
$501$	6.82079	0.304730
$502$	0	0
$503$	22.3185	0.995132	0.497566	−	0.867426i	$-0.334227\pi$
$503$	22.3185	0.995132	0.497566	+	0.867426i	$0.334227\pi$
$504$	0	0
$505$	22.6772	1.00912
$506$	0	0
$507$	−6.62293	−0.294135
$508$	0	0
$509$	28.2264	1.25111	0.625556	−	0.780179i	$-0.284872\pi$
$509$	28.2264	1.25111	0.625556	+	0.780179i	$0.284872\pi$
$510$	0	0
$511$	0.766449	0.0339057
$512$	0	0
$513$	10.1634	0.448725
$514$	0	0
$515$	−19.7173	−0.868848
$516$	0	0
$517$	6.38528	0.280824
$518$	0	0
$519$	6.89165	0.302510
$520$	0	0
$521$	−30.0842	−1.31801	−0.659006	−	0.752137i	$-0.729023\pi$
$521$	−30.0842	−1.31801	−0.659006	+	0.752137i	$0.729023\pi$
$522$	0	0
$523$	−7.66001	−0.334949	−0.167474	−	0.985876i	$-0.553561\pi$
$523$	−7.66001	−0.334949	−0.167474	+	0.985876i	$0.553561\pi$
$524$	0	0
$525$	2.48646	0.108518
$526$	0	0
$527$	19.6774	0.857159
$528$	0	0
$529$	−16.5832	−0.721008
$530$	0	0
$531$	24.1961	1.05002
$532$	0	0
$533$	1.85274	0.0802509
$534$	0	0
$535$	15.0301	0.649807
$536$	0	0
$537$	2.97215	0.128258
$538$	0	0
$539$	−2.43322	−0.104806
$540$	0	0
$541$	−26.1185	−1.12292	−0.561461	−	0.827503i	$-0.689761\pi$
$541$	−26.1185	−1.12292	−0.561461	+	0.827503i	$0.689761\pi$
$542$	0	0
$543$	−7.51690	−0.322581
$544$	0	0
$545$	36.1960	1.55046
$546$	0	0
$547$	−6.81111	−0.291222	−0.145611	−	0.989342i	$-0.546515\pi$
$547$	−6.81111	−0.291222	−0.145611	+	0.989342i	$0.546515\pi$
$548$	0	0
$549$	8.99811	0.384030
$550$	0	0
$551$	−12.1573	−0.517920
$552$	0	0
$553$	−29.7652	−1.26574
$554$	0	0
$555$	11.1272	0.472322
$556$	0	0
$557$	31.7977	1.34731	0.673657	−	0.739045i	$-0.264723\pi$
$557$	31.7977	1.34731	0.673657	+	0.739045i	$0.264723\pi$
$558$	0	0
$559$	−1.24940	−0.0528439
$560$	0	0
$561$	2.25345	0.0951407
$562$	0	0
$563$	22.6512	0.954635	0.477317	−	0.878731i	$-0.341609\pi$
$563$	22.6512	0.954635	0.477317	+	0.878731i	$0.341609\pi$
$564$	0	0
$565$	24.3424	1.02409
$566$	0	0
$567$	−20.0567	−0.842303
$568$	0	0
$569$	8.53768	0.357918	0.178959	−	0.983857i	$-0.442727\pi$
$569$	8.53768	0.357918	0.178959	+	0.983857i	$0.442727\pi$
$570$	0	0
$571$	−40.6106	−1.69950	−0.849751	−	0.527184i	$-0.823248\pi$
$571$	−40.6106	−1.69950	−0.849751	+	0.527184i	$0.823248\pi$
$572$	0	0
$573$	−9.80244	−0.409503
$574$	0	0
$575$	3.85265	0.160667
$576$	0	0
$577$	−20.7576	−0.864151	−0.432076	−	0.901837i	$-0.642219\pi$
$577$	−20.7576	−0.864151	−0.432076	+	0.901837i	$0.642219\pi$
$578$	0	0
$579$	−4.81442	−0.200080
$580$	0	0
$581$	46.7766	1.94062
$582$	0	0
$583$	−1.00000	−0.0414158
$584$	0	0
$585$	−3.78432	−0.156462
$586$	0	0
$587$	10.3924	0.428942	0.214471	−	0.976730i	$-0.431197\pi$
$587$	10.3924	0.428942	0.214471	+	0.976730i	$0.431197\pi$
$588$	0	0
$589$	15.5244	0.639672
$590$	0	0
$591$	11.2309	0.461977
$592$	0	0
$593$	20.8321	0.855470	0.427735	−	0.903904i	$-0.359312\pi$
$593$	20.8321	0.855470	0.427735	+	0.903904i	$0.359312\pi$
$594$	0	0
$595$	−24.2527	−0.994264
$596$	0	0
$597$	1.45643	0.0596078
$598$	0	0
$599$	−34.5009	−1.40967	−0.704835	−	0.709371i	$-0.748979\pi$
$599$	−34.5009	−1.40967	−0.704835	+	0.709371i	$0.748979\pi$
$600$	0	0
$601$	−17.5743	−0.716870	−0.358435	−	0.933555i	$-0.616689\pi$
$601$	−17.5743	−0.716870	−0.358435	+	0.933555i	$0.616689\pi$
$602$	0	0
$603$	−15.8061	−0.643676
$604$	0	0
$605$	−1.86524	−0.0758326
$606$	0	0
$607$	5.39819	0.219106	0.109553	−	0.993981i	$-0.465058\pi$
$607$	5.39819	0.219106	0.109553	+	0.993981i	$0.465058\pi$
$608$	0	0
$609$	−5.95080	−0.241139
$610$	0	0
$611$	4.76868	0.192920
$612$	0	0
$613$	16.5734	0.669394	0.334697	−	0.942326i	$-0.391366\pi$
$613$	16.5734	0.669394	0.334697	+	0.942326i	$0.391366\pi$
$614$	0	0
$615$	2.46309	0.0993215
$616$	0	0
$617$	−7.51743	−0.302640	−0.151320	−	0.988485i	$-0.548352\pi$
$617$	−7.51743	−0.302640	−0.151320	+	0.988485i	$0.548352\pi$
$618$	0	0
$619$	−2.48165	−0.0997460	−0.0498730	−	0.998756i	$-0.515882\pi$
$619$	−2.48165	−0.0997460	−0.0498730	+	0.998756i	$0.515882\pi$
$620$	0	0
$621$	7.70822	0.309320
$622$	0	0
$623$	−8.05882	−0.322870
$624$	0	0
$625$	−15.0824	−0.603296
$626$	0	0
$627$	1.77785	0.0710006
$628$	0	0
$629$	47.4456	1.89178
$630$	0	0
$631$	27.3849	1.09018	0.545088	−	0.838379i	$-0.316496\pi$
$631$	27.3849	1.09018	0.545088	+	0.838379i	$0.316496\pi$
$632$	0	0
$633$	−1.80898	−0.0719004
$634$	0	0
$635$	15.5895	0.618650
$636$	0	0
$637$	−1.81718	−0.0719994
$638$	0	0
$639$	−40.2886	−1.59379
$640$	0	0
$641$	−8.68606	−0.343079	−0.171539	−	0.985177i	$-0.554874\pi$
$641$	−8.68606	−0.343079	−0.171539	+	0.985177i	$0.554874\pi$
$642$	0	0
$643$	−28.3283	−1.11716	−0.558579	−	0.829451i	$-0.688653\pi$
$643$	−28.3283	−1.11716	−0.558579	+	0.829451i	$0.688653\pi$
$644$	0	0
$645$	−1.66099	−0.0654015
$646$	0	0
$647$	−27.7837	−1.09229	−0.546145	−	0.837691i	$-0.683905\pi$
$647$	−27.7837	−1.09229	−0.546145	+	0.837691i	$0.683905\pi$
$648$	0	0
$649$	8.90656	0.349613
$650$	0	0
$651$	7.59891	0.297825
$652$	0	0
$653$	−27.7596	−1.08632	−0.543158	−	0.839631i	$-0.682772\pi$
$653$	−27.7596	−1.08632	−0.543158	+	0.839631i	$0.682772\pi$
$654$	0	0
$655$	13.0153	0.508550
$656$	0	0
$657$	0.677937	0.0264488
$658$	0	0
$659$	−46.2580	−1.80196	−0.900978	−	0.433864i	$-0.857150\pi$
$659$	−46.2580	−1.80196	−0.900978	+	0.433864i	$0.857150\pi$
$660$	0	0
$661$	21.2435	0.826276	0.413138	−	0.910668i	$-0.364433\pi$
$661$	21.2435	0.826276	0.413138	+	0.910668i	$0.364433\pi$
$662$	0	0
$663$	1.68293	0.0653595
$664$	0	0
$665$	−19.1341	−0.741989
$666$	0	0
$667$	−9.22050	−0.357019
$668$	0	0
$669$	14.7088	0.568674
$670$	0	0
$671$	3.31219	0.127866
$672$	0	0
$673$	34.9858	1.34860	0.674301	−	0.738457i	$-0.264445\pi$
$673$	34.9858	1.34860	0.674301	+	0.738457i	$0.264445\pi$
$674$	0	0
$675$	4.62800	0.178132
$676$	0	0
$677$	−47.8700	−1.83979	−0.919897	−	0.392159i	$-0.871728\pi$
$677$	−47.8700	−1.83979	−0.919897	+	0.392159i	$0.871728\pi$
$678$	0	0
$679$	−12.7439	−0.489066
$680$	0	0
$681$	0.789610	0.0302579
$682$	0	0
$683$	5.15391	0.197209	0.0986044	−	0.995127i	$-0.468562\pi$
$683$	5.15391	0.197209	0.0986044	+	0.995127i	$0.468562\pi$
$684$	0	0
$685$	6.23971	0.238407
$686$	0	0
$687$	−3.21146	−0.122525
$688$	0	0
$689$	−0.746823	−0.0284517
$690$	0	0
$691$	−38.4158	−1.46141	−0.730703	−	0.682695i	$-0.760808\pi$
$691$	−38.4158	−1.46141	−0.730703	+	0.682695i	$0.760808\pi$
$692$	0	0
$693$	−8.34383	−0.316956
$694$	0	0
$695$	−11.1983	−0.424778
$696$	0	0
$697$	10.5025	0.397810
$698$	0	0
$699$	−13.8627	−0.524335
$700$	0	0
$701$	18.6096	0.702876	0.351438	−	0.936211i	$-0.385693\pi$
$701$	18.6096	0.702876	0.351438	+	0.936211i	$0.385693\pi$
$702$	0	0
$703$	37.4321	1.41178
$704$	0	0
$705$	6.33965	0.238765
$706$	0	0
$707$	37.3410	1.40435
$708$	0	0
$709$	1.18770	0.0446050	0.0223025	−	0.999751i	$-0.492900\pi$
$709$	1.18770	0.0446050	0.0223025	+	0.999751i	$0.492900\pi$
$710$	0	0
$711$	−26.3278	−0.987369
$712$	0	0
$713$	11.7742	0.440946
$714$	0	0
$715$	−1.39300	−0.0520953
$716$	0	0
$717$	−10.1773	−0.380078
$718$	0	0
$719$	−26.9353	−1.00452	−0.502259	−	0.864717i	$-0.667498\pi$
$719$	−26.9353	−1.00452	−0.502259	+	0.864717i	$0.667498\pi$
$720$	0	0
$721$	−32.4671	−1.20914
$722$	0	0
$723$	−5.98061	−0.222421
$724$	0	0
$725$	−5.53597	−0.205601
$726$	0	0
$727$	−1.75912	−0.0652423	−0.0326212	−	0.999468i	$-0.510385\pi$
$727$	−1.75912	−0.0652423	−0.0326212	+	0.999468i	$0.510385\pi$
$728$	0	0
$729$	−12.8813	−0.477084
$730$	0	0
$731$	−7.08238	−0.261951
$732$	0	0
$733$	17.0783	0.630803	0.315401	−	0.948958i	$-0.397861\pi$
$733$	17.0783	0.630803	0.315401	+	0.948958i	$0.397861\pi$
$734$	0	0
$735$	−2.41583	−0.0891091
$736$	0	0
$737$	−5.81822	−0.214317
$738$	0	0
$739$	−17.1054	−0.629232	−0.314616	−	0.949219i	$-0.601876\pi$
$739$	−17.1054	−0.629232	−0.314616	+	0.949219i	$0.601876\pi$
$740$	0	0
$741$	1.32774	0.0487758
$742$	0	0
$743$	−38.4915	−1.41212	−0.706059	−	0.708153i	$-0.749529\pi$
$743$	−38.4915	−1.41212	−0.706059	+	0.708153i	$0.749529\pi$
$744$	0	0
$745$	4.54439	0.166494
$746$	0	0
$747$	41.3747	1.51382
$748$	0	0
$749$	24.7490	0.904309
$750$	0	0
$751$	−12.4061	−0.452707	−0.226353	−	0.974045i	$-0.572680\pi$
$751$	−12.4061	−0.452707	−0.226353	+	0.974045i	$0.572680\pi$
$752$	0	0
$753$	8.22167	0.299614
$754$	0	0
$755$	−6.40112	−0.232961
$756$	0	0
$757$	30.7364	1.11713	0.558567	−	0.829459i	$-0.311351\pi$
$757$	30.7364	1.11713	0.558567	+	0.829459i	$0.311351\pi$
$758$	0	0
$759$	1.34838	0.0489430
$760$	0	0
$761$	−46.4417	−1.68351	−0.841755	−	0.539859i	$-0.818478\pi$
$761$	−46.4417	−1.68351	−0.841755	+	0.539859i	$0.818478\pi$
$762$	0	0
$763$	59.6014	2.15771
$764$	0	0
$765$	−21.4519	−0.775595
$766$	0	0
$767$	6.65163	0.240176
$768$	0	0
$769$	−2.56969	−0.0926653	−0.0463326	−	0.998926i	$-0.514753\pi$
$769$	−2.56969	−0.0926653	−0.0463326	+	0.998926i	$0.514753\pi$
$770$	0	0
$771$	−9.47486	−0.341229
$772$	0	0
$773$	−26.7459	−0.961982	−0.480991	−	0.876726i	$-0.659723\pi$
$773$	−26.7459	−0.961982	−0.480991	+	0.876726i	$0.659723\pi$
$774$	0	0
$775$	7.06919	0.253933
$776$	0	0
$777$	18.3223	0.657310
$778$	0	0
$779$	8.28591	0.296874
$780$	0	0
$781$	−14.8302	−0.530666
$782$	0	0
$783$	−11.0761	−0.395829
$784$	0	0
$785$	−1.86693	−0.0666337
$786$	0	0
$787$	−33.2583	−1.18553	−0.592766	−	0.805375i	$-0.701964\pi$
$787$	−33.2583	−1.18553	−0.592766	+	0.805375i	$0.701964\pi$
$788$	0	0
$789$	0.720419	0.0256476
$790$	0	0
$791$	40.0830	1.42519
$792$	0	0
$793$	2.47362	0.0878409
$794$	0	0
$795$	−0.992853	−0.0352129
$796$	0	0
$797$	4.85450	0.171955	0.0859776	−	0.996297i	$-0.472599\pi$
$797$	4.85450	0.171955	0.0859776	+	0.996297i	$0.472599\pi$
$798$	0	0
$799$	27.0319	0.956320
$800$	0	0
$801$	−7.12816	−0.251861
$802$	0	0
$803$	0.249548	0.00880635
$804$	0	0
$805$	−14.5119	−0.511477
$806$	0	0
$807$	−16.0686	−0.565641
$808$	0	0
$809$	18.2829	0.642793	0.321396	−	0.946945i	$-0.395848\pi$
$809$	18.2829	0.642793	0.321396	+	0.946945i	$0.395848\pi$
$810$	0	0
$811$	34.7252	1.21937	0.609683	−	0.792645i	$-0.291297\pi$
$811$	34.7252	1.21937	0.609683	+	0.792645i	$0.291297\pi$
$812$	0	0
$813$	−2.46395	−0.0864145
$814$	0	0
$815$	−36.9499	−1.29430
$816$	0	0
$817$	−5.58762	−0.195486
$818$	0	0
$819$	−6.23137	−0.217742
$820$	0	0
$821$	−4.40704	−0.153807	−0.0769034	−	0.997039i	$-0.524503\pi$
$821$	−4.40704	−0.153807	−0.0769034	+	0.997039i	$0.524503\pi$
$822$	0	0
$823$	−12.9467	−0.451292	−0.225646	−	0.974209i	$-0.572449\pi$
$823$	−12.9467	−0.451292	−0.225646	+	0.974209i	$0.572449\pi$
$824$	0	0
$825$	0.809563	0.0281854
$826$	0	0
$827$	7.08039	0.246209	0.123105	−	0.992394i	$-0.460715\pi$
$827$	7.08039	0.246209	0.123105	+	0.992394i	$0.460715\pi$
$828$	0	0
$829$	−39.1956	−1.36132	−0.680661	−	0.732599i	$-0.738307\pi$
$829$	−39.1956	−1.36132	−0.680661	+	0.732599i	$0.738307\pi$
$830$	0	0
$831$	−4.85746	−0.168503
$832$	0	0
$833$	−10.3009	−0.356906
$834$	0	0
$835$	−23.9010	−0.827129
$836$	0	0
$837$	14.1437	0.488879
$838$	0	0
$839$	20.2490	0.699075	0.349537	−	0.936922i	$-0.386339\pi$
$839$	20.2490	0.699075	0.349537	+	0.936922i	$0.386339\pi$
$840$	0	0
$841$	−15.7508	−0.543132
$842$	0	0
$843$	−11.8104	−0.406773
$844$	0	0
$845$	23.2077	0.798370
$846$	0	0
$847$	−3.07135	−0.105533
$848$	0	0
$849$	3.37378	0.115788
$850$	0	0
$851$	28.3896	0.973183
$852$	0	0
$853$	−4.23680	−0.145065	−0.0725326	−	0.997366i	$-0.523108\pi$
$853$	−4.23680	−0.145065	−0.0725326	+	0.997366i	$0.523108\pi$
$854$	0	0
$855$	−16.9244	−0.578803
$856$	0	0
$857$	45.9251	1.56877	0.784385	−	0.620274i	$-0.212979\pi$
$857$	45.9251	1.56877	0.784385	+	0.620274i	$0.212979\pi$
$858$	0	0
$859$	3.91493	0.133576	0.0667879	−	0.997767i	$-0.478725\pi$
$859$	3.91493	0.133576	0.0667879	+	0.997767i	$0.478725\pi$
$860$	0	0
$861$	4.05580	0.138221
$862$	0	0
$863$	−53.0342	−1.80530	−0.902652	−	0.430371i	$-0.858383\pi$
$863$	−53.0342	−1.80530	−0.902652	+	0.430371i	$0.858383\pi$
$864$	0	0
$865$	−24.1494	−0.821103
$866$	0	0
$867$	0.490915	0.0166724
$868$	0	0
$869$	−9.69122	−0.328752
$870$	0	0
$871$	−4.34518	−0.147231
$872$	0	0
$873$	−11.2722	−0.381506
$874$	0	0
$875$	−37.3569	−1.26289
$876$	0	0
$877$	−31.8326	−1.07491	−0.537455	−	0.843292i	$-0.680614\pi$
$877$	−31.8326	−1.07491	−0.537455	+	0.843292i	$0.680614\pi$
$878$	0	0
$879$	15.0849	0.508801
$880$	0	0
$881$	16.1927	0.545547	0.272773	−	0.962078i	$-0.412059\pi$
$881$	16.1927	0.545547	0.272773	+	0.962078i	$0.412059\pi$
$882$	0	0
$883$	31.0139	1.04370	0.521850	−	0.853037i	$-0.325242\pi$
$883$	31.0139	1.04370	0.521850	+	0.853037i	$0.325242\pi$
$884$	0	0
$885$	8.84291	0.297251
$886$	0	0
$887$	−49.3740	−1.65782	−0.828909	−	0.559384i	$-0.811038\pi$
$887$	−49.3740	−1.65782	−0.828909	+	0.559384i	$0.811038\pi$
$888$	0	0
$889$	25.6701	0.860949
$890$	0	0
$891$	−6.53025	−0.218772
$892$	0	0
$893$	21.3267	0.713672
$894$	0	0
$895$	−10.4148	−0.348130
$896$	0	0
$897$	1.00700	0.0336228
$898$	0	0
$899$	−16.9186	−0.564267
$900$	0	0
$901$	−4.23347	−0.141037
$902$	0	0
$903$	−2.73504	−0.0910165
$904$	0	0
$905$	26.3403	0.875582
$906$	0	0
$907$	−27.0043	−0.896664	−0.448332	−	0.893867i	$-0.647982\pi$
$907$	−27.0043	−0.896664	−0.448332	+	0.893867i	$0.647982\pi$
$908$	0	0
$909$	33.0287	1.09549
$910$	0	0
$911$	−3.83442	−0.127040	−0.0635199	−	0.997981i	$-0.520233\pi$
$911$	−3.83442	−0.127040	−0.0635199	+	0.997981i	$0.520233\pi$
$912$	0	0
$913$	15.2300	0.504038
$914$	0	0
$915$	3.28852	0.108715
$916$	0	0
$917$	21.4314	0.707728
$918$	0	0
$919$	−36.2307	−1.19514	−0.597571	−	0.801816i	$-0.703867\pi$
$919$	−36.2307	−1.19514	−0.597571	+	0.801816i	$0.703867\pi$
$920$	0	0
$921$	5.36044	0.176633
$922$	0	0
$923$	−11.0755	−0.364555
$924$	0	0
$925$	17.0451	0.560438
$926$	0	0
$927$	−28.7177	−0.943213
$928$	0	0
$929$	−30.0785	−0.986844	−0.493422	−	0.869790i	$-0.664254\pi$
$929$	−30.0785	−0.986844	−0.493422	+	0.869790i	$0.664254\pi$
$930$	0	0
$931$	−8.12690	−0.266348
$932$	0	0
$933$	15.3203	0.501564
$934$	0	0
$935$	−7.89642	−0.258240
$936$	0	0
$937$	−6.42371	−0.209854	−0.104927	−	0.994480i	$-0.533461\pi$
$937$	−6.42371	−0.209854	−0.104927	+	0.994480i	$0.533461\pi$
$938$	0	0
$939$	15.1873	0.495619
$940$	0	0
$941$	−18.8138	−0.613312	−0.306656	−	0.951820i	$-0.599210\pi$
$941$	−18.8138	−0.613312	−0.306656	+	0.951820i	$0.599210\pi$
$942$	0	0
$943$	6.28429	0.204644
$944$	0	0
$945$	−17.4324	−0.567077
$946$	0	0
$947$	33.9318	1.10264	0.551318	−	0.834295i	$-0.314125\pi$
$947$	33.9318	1.10264	0.551318	+	0.834295i	$0.314125\pi$
$948$	0	0
$949$	0.186368	0.00604976
$950$	0	0
$951$	−5.34137	−0.173206
$952$	0	0
$953$	26.0766	0.844703	0.422352	−	0.906432i	$-0.361205\pi$
$953$	26.0766	0.844703	0.422352	+	0.906432i	$0.361205\pi$
$954$	0	0
$955$	34.3492	1.11151
$956$	0	0
$957$	−1.93752	−0.0626311
$958$	0	0
$959$	10.2745	0.331781
$960$	0	0
$961$	−9.39568	−0.303086
$962$	0	0
$963$	21.8909	0.705425
$964$	0	0
$965$	16.8704	0.543078
$966$	0	0
$967$	−10.6606	−0.342823	−0.171412	−	0.985200i	$-0.554833\pi$
$967$	−10.6606	−0.342823	−0.171412	+	0.985200i	$0.554833\pi$
$968$	0	0
$969$	7.52649	0.241786
$970$	0	0
$971$	13.7555	0.441436	0.220718	−	0.975338i	$-0.429160\pi$
$971$	13.7555	0.441436	0.220718	+	0.975338i	$0.429160\pi$
$972$	0	0
$973$	−18.4395	−0.591145
$974$	0	0
$975$	0.604601	0.0193627
$976$	0	0
$977$	−36.3303	−1.16231	−0.581154	−	0.813794i	$-0.697399\pi$
$977$	−36.3303	−1.16231	−0.581154	+	0.813794i	$0.697399\pi$
$978$	0	0
$979$	−2.62387	−0.0838591
$980$	0	0
$981$	52.7184	1.68317
$982$	0	0
$983$	47.1102	1.50258	0.751292	−	0.659970i	$-0.229431\pi$
$983$	47.1102	1.50258	0.751292	+	0.659970i	$0.229431\pi$
$984$	0	0
$985$	−39.3546	−1.25394
$986$	0	0
$987$	10.4391	0.332279
$988$	0	0
$989$	−4.23782	−0.134755
$990$	0	0
$991$	−24.6098	−0.781755	−0.390878	−	0.920443i	$-0.627828\pi$
$991$	−24.6098	−0.781755	−0.390878	+	0.920443i	$0.627828\pi$
$992$	0	0
$993$	1.63982	0.0520382
$994$	0	0
$995$	−5.10355	−0.161794
$996$	0	0
$997$	12.9141	0.408994	0.204497	−	0.978867i	$-0.434444\pi$
$997$	12.9141	0.408994	0.204497	+	0.978867i	$0.434444\pi$
$998$	0	0
$999$	34.1031	1.07897

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9328.2.a.bm.1.6		11
4.3	odd	2		4664.2.a.k.1.6	✓	11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4664.2.a.k.1.6	✓	11	4.3	odd	2
9328.2.a.bm.1.6		11	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 9328 = 2^{4} \cdot 11 \cdot 53 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9328.a (trivial)

\(f(q)\)	\(=\)	\(q+0.532294 q^{3} -1.86524 q^{5} -3.07135 q^{7} -2.71666 q^{9} +O(q^{10})\)	\(q+0.532294 q^{3} -1.86524 q^{5} -3.07135 q^{7} -2.71666 q^{9} -1.00000 q^{11} -0.746823 q^{13} -0.992853 q^{15} -4.23347 q^{17} -3.33998 q^{19} -1.63486 q^{21} -2.53315 q^{23} -1.52090 q^{25} -3.04294 q^{27} +3.63994 q^{29} -4.64804 q^{31} -0.532294 q^{33} +5.72880 q^{35} -11.2073 q^{37} -0.397529 q^{39} -2.48082 q^{41} +1.67295 q^{43} +5.06722 q^{45} -6.38528 q^{47} +2.43322 q^{49} -2.25345 q^{51} +1.00000 q^{53} +1.86524 q^{55} -1.77785 q^{57} -8.90656 q^{59} -3.31219 q^{61} +8.34383 q^{63} +1.39300 q^{65} +5.81822 q^{67} -1.34838 q^{69} +14.8302 q^{71} -0.249548 q^{73} -0.809563 q^{75} +3.07135 q^{77} +9.69122 q^{79} +6.53025 q^{81} -15.2300 q^{83} +7.89642 q^{85} +1.93752 q^{87} +2.62387 q^{89} +2.29376 q^{91} -2.47413 q^{93} +6.22986 q^{95} +4.14927 q^{97} +2.71666 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q + 6 q^{3} + 3 q^{5} + 5 q^{7} + 7 q^{9}+O(q^{10}) \) 11 * q + 6 * q^3 + 3 * q^5 + 5 * q^7 + 7 * q^9	\( 11 q + 6 q^{3} + 3 q^{5} + 5 q^{7} + 7 q^{9} - 11 q^{11} - 13 q^{13} + 8 q^{15} - 7 q^{17} + 21 q^{19} + 6 q^{21} - 11 q^{23} + 4 q^{25} + 6 q^{27} + 5 q^{31} - 6 q^{33} + 25 q^{35} - 4 q^{37} + 19 q^{39} - 11 q^{41} + 16 q^{45} + 17 q^{47} - 2 q^{49} + 18 q^{51} + 11 q^{53} - 3 q^{55} - 5 q^{57} + 19 q^{59} - 2 q^{61} + 36 q^{63} - 13 q^{65} - 25 q^{67} + 3 q^{69} + 30 q^{71} + 5 q^{73} + 5 q^{75} - 5 q^{77} + 23 q^{79} - 9 q^{81} + 19 q^{83} + 2 q^{85} + 7 q^{87} + 6 q^{89} + 20 q^{91} + 43 q^{93} + 50 q^{95} - 35 q^{97} - 7 q^{99}+O(q^{100}) \) 11 * q + 6 * q^3 + 3 * q^5 + 5 * q^7 + 7 * q^9 - 11 * q^11 - 13 * q^13 + 8 * q^15 - 7 * q^17 + 21 * q^19 + 6 * q^21 - 11 * q^23 + 4 * q^25 + 6 * q^27 + 5 * q^31 - 6 * q^33 + 25 * q^35 - 4 * q^37 + 19 * q^39 - 11 * q^41 + 16 * q^45 + 17 * q^47 - 2 * q^49 + 18 * q^51 + 11 * q^53 - 3 * q^55 - 5 * q^57 + 19 * q^59 - 2 * q^61 + 36 * q^63 - 13 * q^65 - 25 * q^67 + 3 * q^69 + 30 * q^71 + 5 * q^73 + 5 * q^75 - 5 * q^77 + 23 * q^79 - 9 * q^81 + 19 * q^83 + 2 * q^85 + 7 * q^87 + 6 * q^89 + 20 * q^91 + 43 * q^93 + 50 * q^95 - 35 * q^97 - 7 * q^99

Introduction

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