LMFDB - Embedded newform 9328.2.a.bm.1.4

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			1.4
Root			$1.49240$ 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)

$f(q)$	$=$	$q-0.492404 q^{3} +0.923502 q^{5} -1.84390 q^{7} -2.75754 q^{9} +O(q^{10})$	$q-0.492404 q^{3} +0.923502 q^{5} -1.84390 q^{7} -2.75754 q^{9} -1.00000 q^{11} -6.93720 q^{13} -0.454736 q^{15} -2.04682 q^{17} -3.74399 q^{19} +0.907942 q^{21} -9.37270 q^{23} -4.14714 q^{25} +2.83504 q^{27} -9.05851 q^{29} +2.98091 q^{31} +0.492404 q^{33} -1.70284 q^{35} +4.12526 q^{37} +3.41591 q^{39} -1.47532 q^{41} -4.51627 q^{43} -2.54659 q^{45} +8.14060 q^{47} -3.60005 q^{49} +1.00786 q^{51} +1.00000 q^{53} -0.923502 q^{55} +1.84356 q^{57} +2.71113 q^{59} -4.46207 q^{61} +5.08461 q^{63} -6.40652 q^{65} -1.77859 q^{67} +4.61516 q^{69} -4.40765 q^{71} +0.530085 q^{73} +2.04207 q^{75} +1.84390 q^{77} +7.81149 q^{79} +6.87663 q^{81} +14.4572 q^{83} -1.89024 q^{85} +4.46045 q^{87} +0.0829900 q^{89} +12.7915 q^{91} -1.46781 q^{93} -3.45758 q^{95} -8.81299 q^{97} +2.75754 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−0.492404	−0.284290	−0.142145	−	0.989846i	$-0.545400\pi$
$3$	−0.492404	−0.284290	−0.142145	+	0.989846i	$0.545400\pi$
$4$	0	0
$5$	0.923502	0.413003	0.206501	−	0.978446i	$-0.433792\pi$
$5$	0.923502	0.413003	0.206501	+	0.978446i	$0.433792\pi$
$6$	0	0
$7$	−1.84390	−0.696927	−0.348464	−	0.937322i	$-0.613296\pi$
$7$	−1.84390	−0.696927	−0.348464	+	0.937322i	$0.613296\pi$
$8$	0	0
$9$	−2.75754	−0.919179
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−6.93720	−1.92403	−0.962017	−	0.272991i	$-0.911987\pi$
$13$	−6.93720	−1.92403	−0.962017	+	0.272991i	$0.911987\pi$
$14$	0	0
$15$	−0.454736	−0.117412
$16$	0	0
$17$	−2.04682	−0.496426	−0.248213	−	0.968705i	$-0.579843\pi$
$17$	−2.04682	−0.496426	−0.248213	+	0.968705i	$0.579843\pi$
$18$	0	0
$19$	−3.74399	−0.858931	−0.429465	−	0.903083i	$-0.641298\pi$
$19$	−3.74399	−0.858931	−0.429465	+	0.903083i	$0.641298\pi$
$20$	0	0
$21$	0.907942	0.198129
$22$	0	0
$23$	−9.37270	−1.95434	−0.977172	−	0.212451i	$-0.931855\pi$
$23$	−9.37270	−1.95434	−0.977172	+	0.212451i	$0.931855\pi$
$24$	0	0
$25$	−4.14714	−0.829429
$26$	0	0
$27$	2.83504	0.545603
$28$	0	0
$29$	−9.05851	−1.68212	−0.841062	−	0.540939i	$-0.818069\pi$
$29$	−9.05851	−1.68212	−0.841062	+	0.540939i	$0.818069\pi$
$30$	0	0
$31$	2.98091	0.535388	0.267694	−	0.963504i	$-0.413738\pi$
$31$	2.98091	0.535388	0.267694	+	0.963504i	$0.413738\pi$
$32$	0	0
$33$	0.492404	0.0857166
$34$	0	0
$35$	−1.70284	−0.287833
$36$	0	0
$37$	4.12526	0.678189	0.339095	−	0.940752i	$-0.389879\pi$
$37$	4.12526	0.678189	0.339095	+	0.940752i	$0.389879\pi$
$38$	0	0
$39$	3.41591	0.546983
$40$	0	0
$41$	−1.47532	−0.230406	−0.115203	−	0.993342i	$-0.536752\pi$
$41$	−1.47532	−0.230406	−0.115203	+	0.993342i	$0.536752\pi$
$42$	0	0
$43$	−4.51627	−0.688724	−0.344362	−	0.938837i	$-0.611905\pi$
$43$	−4.51627	−0.688724	−0.344362	+	0.938837i	$0.611905\pi$
$44$	0	0
$45$	−2.54659	−0.379623
$46$	0	0
$47$	8.14060	1.18743	0.593714	−	0.804676i	$-0.297661\pi$
$47$	8.14060	1.18743	0.593714	+	0.804676i	$0.297661\pi$
$48$	0	0
$49$	−3.60005	−0.514293
$50$	0	0
$51$	1.00786	0.141129
$52$	0	0
$53$	1.00000	0.137361
$53$	1.00000	0.137361
$54$	0	0
$55$	−0.923502	−0.124525
$56$	0	0
$57$	1.84356	0.244185
$58$	0	0
$59$	2.71113	0.352959	0.176480	−	0.984304i	$-0.443529\pi$
$59$	2.71113	0.352959	0.176480	+	0.984304i	$0.443529\pi$
$60$	0	0
$61$	−4.46207	−0.571310	−0.285655	−	0.958333i	$-0.592211\pi$
$61$	−4.46207	−0.571310	−0.285655	+	0.958333i	$0.592211\pi$
$62$	0	0
$63$	5.08461	0.640601
$64$	0	0
$65$	−6.40652	−0.794631
$66$	0	0
$67$	−1.77859	−0.217289	−0.108644	−	0.994081i	$-0.534651\pi$
$67$	−1.77859	−0.217289	−0.108644	+	0.994081i	$0.534651\pi$
$68$	0	0
$69$	4.61516	0.555600
$70$	0	0
$71$	−4.40765	−0.523091	−0.261546	−	0.965191i	$-0.584232\pi$
$71$	−4.40765	−0.523091	−0.261546	+	0.965191i	$0.584232\pi$
$72$	0	0
$73$	0.530085	0.0620417	0.0310209	−	0.999519i	$-0.490124\pi$
$73$	0.530085	0.0620417	0.0310209	+	0.999519i	$0.490124\pi$
$74$	0	0
$75$	2.04207	0.235798
$76$	0	0
$77$	1.84390	0.210131
$78$	0	0
$79$	7.81149	0.878862	0.439431	−	0.898276i	$-0.355180\pi$
$79$	7.81149	0.878862	0.439431	+	0.898276i	$0.355180\pi$
$80$	0	0
$81$	6.87663	0.764070
$82$	0	0
$83$	14.4572	1.58688	0.793442	−	0.608646i	$-0.208287\pi$
$83$	14.4572	1.58688	0.793442	+	0.608646i	$0.208287\pi$
$84$	0	0
$85$	−1.89024	−0.205025
$86$	0	0
$87$	4.46045	0.478210
$88$	0	0
$89$	0.0829900	0.00879692	0.00439846	−	0.999990i	$-0.498600\pi$
$89$	0.0829900	0.00879692	0.00439846	+	0.999990i	$0.498600\pi$
$90$	0	0
$91$	12.7915	1.34091
$92$	0	0
$93$	−1.46781	−0.152205
$94$	0	0
$95$	−3.45758	−0.354741
$96$	0	0
$97$	−8.81299	−0.894824	−0.447412	−	0.894328i	$-0.647654\pi$
$97$	−8.81299	−0.894824	−0.447412	+	0.894328i	$0.647654\pi$
$98$	0	0
$99$	2.75754	0.277143
$100$	0	0
$101$	−1.60753	−0.159956	−0.0799778	−	0.996797i	$-0.525485\pi$
$101$	−1.60753	−0.159956	−0.0799778	+	0.996797i	$0.525485\pi$
$102$	0	0
$103$	1.26688	0.124830	0.0624148	−	0.998050i	$-0.480120\pi$
$103$	1.26688	0.124830	0.0624148	+	0.998050i	$0.480120\pi$
$104$	0	0
$105$	0.838486	0.0818278
$106$	0	0
$107$	13.7850	1.33265	0.666325	−	0.745661i	$-0.267866\pi$
$107$	13.7850	1.33265	0.666325	+	0.745661i	$0.267866\pi$
$108$	0	0
$109$	12.1995	1.16850	0.584250	−	0.811574i	$-0.301389\pi$
$109$	12.1995	1.16850	0.584250	+	0.811574i	$0.301389\pi$
$110$	0	0
$111$	−2.03130	−0.192802
$112$	0	0
$113$	−17.2981	−1.62727	−0.813636	−	0.581375i	$-0.802515\pi$
$113$	−17.2981	−1.62727	−0.813636	+	0.581375i	$0.802515\pi$
$114$	0	0
$115$	−8.65571	−0.807149
$116$	0	0
$117$	19.1296	1.76853
$118$	0	0
$119$	3.77412	0.345973
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0.726454	0.0655021
$124$	0	0
$125$	−8.44740	−0.755559
$126$	0	0
$127$	17.3494	1.53951	0.769756	−	0.638338i	$-0.220378\pi$
$127$	17.3494	1.53951	0.769756	+	0.638338i	$0.220378\pi$
$128$	0	0
$129$	2.22383	0.195797
$130$	0	0
$131$	6.09520	0.532540	0.266270	−	0.963898i	$-0.414209\pi$
$131$	6.09520	0.532540	0.266270	+	0.963898i	$0.414209\pi$
$132$	0	0
$133$	6.90353	0.598612
$134$	0	0
$135$	2.61816	0.225335
$136$	0	0
$137$	−3.17986	−0.271674	−0.135837	−	0.990731i	$-0.543372\pi$
$137$	−3.17986	−0.271674	−0.135837	+	0.990731i	$0.543372\pi$
$138$	0	0
$139$	5.79705	0.491699	0.245850	−	0.969308i	$-0.420933\pi$
$139$	5.79705	0.491699	0.245850	+	0.969308i	$0.420933\pi$
$140$	0	0
$141$	−4.00846	−0.337574
$142$	0	0
$143$	6.93720	0.580118
$144$	0	0
$145$	−8.36555	−0.694721
$146$	0	0
$147$	1.77268	0.146208
$148$	0	0
$149$	5.96445	0.488627	0.244313	−	0.969696i	$-0.421437\pi$
$149$	5.96445	0.488627	0.244313	+	0.969696i	$0.421437\pi$
$150$	0	0
$151$	3.94701	0.321203	0.160602	−	0.987019i	$-0.448657\pi$
$151$	3.94701	0.321203	0.160602	+	0.987019i	$0.448657\pi$
$152$	0	0
$153$	5.64417	0.456304
$154$	0	0
$155$	2.75288	0.221116
$156$	0	0
$157$	−7.48081	−0.597034	−0.298517	−	0.954404i	$-0.596492\pi$
$157$	−7.48081	−0.597034	−0.298517	+	0.954404i	$0.596492\pi$
$158$	0	0
$159$	−0.492404	−0.0390502
$160$	0	0
$161$	17.2823	1.36203
$162$	0	0
$163$	−24.1464	−1.89129	−0.945647	−	0.325194i	$-0.894570\pi$
$163$	−24.1464	−1.89129	−0.945647	+	0.325194i	$0.894570\pi$
$164$	0	0
$165$	0.454736	0.0354012
$166$	0	0
$167$	10.4213	0.806422	0.403211	−	0.915107i	$-0.367894\pi$
$167$	10.4213	0.806422	0.403211	+	0.915107i	$0.367894\pi$
$168$	0	0
$169$	35.1247	2.70190
$170$	0	0
$171$	10.3242	0.789512
$172$	0	0
$173$	16.3487	1.24297	0.621485	−	0.783426i	$-0.286529\pi$
$173$	16.3487	1.24297	0.621485	+	0.783426i	$0.286529\pi$
$174$	0	0
$175$	7.64690	0.578052
$176$	0	0
$177$	−1.33497	−0.100343
$178$	0	0
$179$	−22.3982	−1.67412	−0.837059	−	0.547113i	$-0.815727\pi$
$179$	−22.3982	−1.67412	−0.837059	+	0.547113i	$0.815727\pi$
$180$	0	0
$181$	−23.8231	−1.77075	−0.885377	−	0.464874i	$-0.846100\pi$
$181$	−23.8231	−1.77075	−0.885377	+	0.464874i	$0.846100\pi$
$182$	0	0
$183$	2.19714	0.162418
$184$	0	0
$185$	3.80969	0.280094
$186$	0	0
$187$	2.04682	0.149678
$188$	0	0
$189$	−5.22751	−0.380245
$190$	0	0
$191$	6.76402	0.489427	0.244714	−	0.969595i	$-0.421306\pi$
$191$	6.76402	0.489427	0.244714	+	0.969595i	$0.421306\pi$
$192$	0	0
$193$	−4.32796	−0.311534	−0.155767	−	0.987794i	$-0.549785\pi$
$193$	−4.32796	−0.311534	−0.155767	+	0.987794i	$0.549785\pi$
$194$	0	0
$195$	3.15460	0.225905
$196$	0	0
$197$	−13.1600	−0.937611	−0.468806	−	0.883301i	$-0.655316\pi$
$197$	−13.1600	−0.937611	−0.468806	+	0.883301i	$0.655316\pi$
$198$	0	0
$199$	−22.0788	−1.56512	−0.782561	−	0.622574i	$-0.786087\pi$
$199$	−22.0788	−1.56512	−0.782561	+	0.622574i	$0.786087\pi$
$200$	0	0
$201$	0.875783	0.0617730
$202$	0	0
$203$	16.7029	1.17232
$204$	0	0
$205$	−1.36246	−0.0951584
$206$	0	0
$207$	25.8456	1.79639
$208$	0	0
$209$	3.74399	0.258977
$210$	0	0
$211$	−0.0820158	−0.00564620	−0.00282310	−	0.999996i	$-0.500899\pi$
$211$	−0.0820158	−0.00564620	−0.00282310	+	0.999996i	$0.500899\pi$
$212$	0	0
$213$	2.17034	0.148709
$214$	0	0
$215$	−4.17078	−0.284445
$216$	0	0
$217$	−5.49649	−0.373126
$218$	0	0
$219$	−0.261016	−0.0176378
$220$	0	0
$221$	14.1992	0.955140
$222$	0	0
$223$	−9.75689	−0.653370	−0.326685	−	0.945133i	$-0.605932\pi$
$223$	−9.75689	−0.653370	−0.326685	+	0.945133i	$0.605932\pi$
$224$	0	0
$225$	11.4359	0.762394
$226$	0	0
$227$	−21.7036	−1.44052	−0.720259	−	0.693705i	$-0.755977\pi$
$227$	−21.7036	−1.44052	−0.720259	+	0.693705i	$0.755977\pi$
$228$	0	0
$229$	4.34349	0.287026	0.143513	−	0.989648i	$-0.454160\pi$
$229$	4.34349	0.287026	0.143513	+	0.989648i	$0.454160\pi$
$230$	0	0
$231$	−0.907942	−0.0597382
$232$	0	0
$233$	12.8308	0.840577	0.420288	−	0.907391i	$-0.361929\pi$
$233$	12.8308	0.840577	0.420288	+	0.907391i	$0.361929\pi$
$234$	0	0
$235$	7.51786	0.490411
$236$	0	0
$237$	−3.84641	−0.249851
$238$	0	0
$239$	−5.17355	−0.334649	−0.167325	−	0.985902i	$-0.553513\pi$
$239$	−5.17355	−0.334649	−0.167325	+	0.985902i	$0.553513\pi$
$240$	0	0
$241$	6.88187	0.443300	0.221650	−	0.975126i	$-0.428856\pi$
$241$	6.88187	0.443300	0.221650	+	0.975126i	$0.428856\pi$
$242$	0	0
$243$	−11.8912	−0.762820
$244$	0	0
$245$	−3.32465	−0.212404
$246$	0	0
$247$	25.9728	1.65261
$248$	0	0
$249$	−7.11878	−0.451134
$250$	0	0
$251$	−27.8846	−1.76006	−0.880030	−	0.474918i	$-0.842478\pi$
$251$	−27.8846	−1.76006	−0.880030	+	0.474918i	$0.842478\pi$
$252$	0	0
$253$	9.37270	0.589257
$254$	0	0
$255$	0.930761	0.0582865
$256$	0	0
$257$	−10.7730	−0.672004	−0.336002	−	0.941861i	$-0.609075\pi$
$257$	−10.7730	−0.672004	−0.336002	+	0.941861i	$0.609075\pi$
$258$	0	0
$259$	−7.60656	−0.472648
$260$	0	0
$261$	24.9792	1.54617
$262$	0	0
$263$	−22.4676	−1.38541	−0.692705	−	0.721221i	$-0.743581\pi$
$263$	−22.4676	−1.38541	−0.692705	+	0.721221i	$0.743581\pi$
$264$	0	0
$265$	0.923502	0.0567303
$266$	0	0
$267$	−0.0408646	−0.00250087
$268$	0	0
$269$	−29.5082	−1.79915	−0.899574	−	0.436769i	$-0.856123\pi$
$269$	−29.5082	−1.79915	−0.899574	+	0.436769i	$0.856123\pi$
$270$	0	0
$271$	5.28936	0.321306	0.160653	−	0.987011i	$-0.448640\pi$
$271$	5.28936	0.321306	0.160653	+	0.987011i	$0.448640\pi$
$272$	0	0
$273$	−6.29857	−0.381207
$274$	0	0
$275$	4.14714	0.250082
$276$	0	0
$277$	26.7341	1.60630	0.803148	−	0.595779i	$-0.203157\pi$
$277$	26.7341	1.60630	0.803148	+	0.595779i	$0.203157\pi$
$278$	0	0
$279$	−8.21998	−0.492117
$280$	0	0
$281$	−24.6381	−1.46978	−0.734892	−	0.678184i	$-0.762767\pi$
$281$	−24.6381	−1.46978	−0.734892	+	0.678184i	$0.762767\pi$
$282$	0	0
$283$	12.4395	0.739454	0.369727	−	0.929140i	$-0.379451\pi$
$283$	12.4395	0.739454	0.369727	+	0.929140i	$0.379451\pi$
$284$	0	0
$285$	1.70253	0.100849
$286$	0	0
$287$	2.72034	0.160576
$288$	0	0
$289$	−12.8105	−0.753561
$290$	0	0
$291$	4.33955	0.254389
$292$	0	0
$293$	−10.7925	−0.630503	−0.315251	−	0.949008i	$-0.602089\pi$
$293$	−10.7925	−0.630503	−0.315251	+	0.949008i	$0.602089\pi$
$294$	0	0
$295$	2.50374	0.145773
$296$	0	0
$297$	−2.83504	−0.164505
$298$	0	0
$299$	65.0203	3.76022
$300$	0	0
$301$	8.32752	0.479990
$302$	0	0
$303$	0.791556	0.0454737
$304$	0	0
$305$	−4.12073	−0.235952
$306$	0	0
$307$	20.7074	1.18184	0.590918	−	0.806732i	$-0.298766\pi$
$307$	20.7074	1.18184	0.590918	+	0.806732i	$0.298766\pi$
$308$	0	0
$309$	−0.623818	−0.0354878
$310$	0	0
$311$	19.6388	1.11361	0.556806	−	0.830642i	$-0.312027\pi$
$311$	19.6388	1.11361	0.556806	+	0.830642i	$0.312027\pi$
$312$	0	0
$313$	−7.35931	−0.415973	−0.207986	−	0.978132i	$-0.566691\pi$
$313$	−7.35931	−0.415973	−0.207986	+	0.978132i	$0.566691\pi$
$314$	0	0
$315$	4.69565	0.264570
$316$	0	0
$317$	22.2463	1.24948	0.624739	−	0.780834i	$-0.285205\pi$
$317$	22.2463	1.24948	0.624739	+	0.780834i	$0.285205\pi$
$318$	0	0
$319$	9.05851	0.507179
$320$	0	0
$321$	−6.78781	−0.378859
$322$	0	0
$323$	7.66326	0.426395
$324$	0	0
$325$	28.7696	1.59585
$326$	0	0
$327$	−6.00708	−0.332192
$328$	0	0
$329$	−15.0104	−0.827551
$330$	0	0
$331$	−6.92929	−0.380868	−0.190434	−	0.981700i	$-0.560990\pi$
$331$	−6.92929	−0.380868	−0.190434	+	0.981700i	$0.560990\pi$
$332$	0	0
$333$	−11.3756	−0.623378
$334$	0	0
$335$	−1.64253	−0.0897408
$336$	0	0
$337$	11.6949	0.637062	0.318531	−	0.947912i	$-0.396810\pi$
$337$	11.6949	0.637062	0.318531	+	0.947912i	$0.396810\pi$
$338$	0	0
$339$	8.51768	0.462617
$340$	0	0
$341$	−2.98091	−0.161425
$342$	0	0
$343$	19.5454	1.05535
$344$	0	0
$345$	4.26211	0.229464
$346$	0	0
$347$	7.08777	0.380491	0.190246	−	0.981737i	$-0.439072\pi$
$347$	7.08777	0.380491	0.190246	+	0.981737i	$0.439072\pi$
$348$	0	0
$349$	−27.8380	−1.49013	−0.745067	−	0.666989i	$-0.767583\pi$
$349$	−27.8380	−1.49013	−0.745067	+	0.666989i	$0.767583\pi$
$350$	0	0
$351$	−19.6672	−1.04976
$352$	0	0
$353$	−31.7162	−1.68808	−0.844041	−	0.536278i	$-0.819830\pi$
$353$	−31.7162	−1.68808	−0.844041	+	0.536278i	$0.819830\pi$
$354$	0	0
$355$	−4.07047	−0.216038
$356$	0	0
$357$	−1.85839	−0.0983564
$358$	0	0
$359$	4.19233	0.221262	0.110631	−	0.993862i	$-0.464713\pi$
$359$	4.19233	0.221262	0.110631	+	0.993862i	$0.464713\pi$
$360$	0	0
$361$	−4.98252	−0.262238
$362$	0	0
$363$	−0.492404	−0.0258445
$364$	0	0
$365$	0.489534	0.0256234
$366$	0	0
$367$	2.75611	0.143868	0.0719339	−	0.997409i	$-0.477083\pi$
$367$	2.75611	0.143868	0.0719339	+	0.997409i	$0.477083\pi$
$368$	0	0
$369$	4.06825	0.211785
$370$	0	0
$371$	−1.84390	−0.0957303
$372$	0	0
$373$	−30.6603	−1.58753	−0.793764	−	0.608226i	$-0.791882\pi$
$373$	−30.6603	−1.58753	−0.793764	+	0.608226i	$0.791882\pi$
$374$	0	0
$375$	4.15954	0.214798
$376$	0	0
$377$	62.8407	3.23646
$378$	0	0
$379$	37.4274	1.92252	0.961259	−	0.275648i	$-0.0888923\pi$
$379$	37.4274	1.92252	0.961259	+	0.275648i	$0.0888923\pi$
$380$	0	0
$381$	−8.54292	−0.437667
$382$	0	0
$383$	−3.87053	−0.197775	−0.0988875	−	0.995099i	$-0.531528\pi$
$383$	−3.87053	−0.197775	−0.0988875	+	0.995099i	$0.531528\pi$
$384$	0	0
$385$	1.70284	0.0867848
$386$	0	0
$387$	12.4538	0.633061
$388$	0	0
$389$	13.7372	0.696506	0.348253	−	0.937401i	$-0.386775\pi$
$389$	13.7372	0.696506	0.348253	+	0.937401i	$0.386775\pi$
$390$	0	0
$391$	19.1842	0.970187
$392$	0	0
$393$	−3.00130	−0.151396
$394$	0	0
$395$	7.21393	0.362972
$396$	0	0
$397$	−37.7965	−1.89695	−0.948477	−	0.316847i	$-0.897376\pi$
$397$	−37.7965	−1.89695	−0.948477	+	0.316847i	$0.897376\pi$
$398$	0	0
$399$	−3.39933	−0.170179
$400$	0	0
$401$	−8.92459	−0.445673	−0.222836	−	0.974856i	$-0.571532\pi$
$401$	−8.92459	−0.445673	−0.222836	+	0.974856i	$0.571532\pi$
$402$	0	0
$403$	−20.6792	−1.03010
$404$	0	0
$405$	6.35058	0.315563
$406$	0	0
$407$	−4.12526	−0.204482
$408$	0	0
$409$	−12.2510	−0.605773	−0.302887	−	0.953027i	$-0.597950\pi$
$409$	−12.2510	−0.605773	−0.302887	+	0.953027i	$0.597950\pi$
$410$	0	0
$411$	1.56578	0.0772342
$412$	0	0
$413$	−4.99905	−0.245987
$414$	0	0
$415$	13.3512	0.655387
$416$	0	0
$417$	−2.85449	−0.139785
$418$	0	0
$419$	7.10020	0.346868	0.173434	−	0.984846i	$-0.444514\pi$
$419$	7.10020	0.346868	0.173434	+	0.984846i	$0.444514\pi$
$420$	0	0
$421$	9.76491	0.475913	0.237956	−	0.971276i	$-0.423522\pi$
$421$	9.76491	0.475913	0.237956	+	0.971276i	$0.423522\pi$
$422$	0	0
$423$	−22.4480	−1.09146
$424$	0	0
$425$	8.48844	0.411750
$426$	0	0
$427$	8.22760	0.398161
$428$	0	0
$429$	−3.41591	−0.164921
$430$	0	0
$431$	−25.5941	−1.23282	−0.616412	−	0.787424i	$-0.711414\pi$
$431$	−25.5941	−1.23282	−0.616412	+	0.787424i	$0.711414\pi$
$432$	0	0
$433$	15.1022	0.725765	0.362883	−	0.931835i	$-0.381793\pi$
$433$	15.1022	0.725765	0.362883	+	0.931835i	$0.381793\pi$
$434$	0	0
$435$	4.11923	0.197502
$436$	0	0
$437$	35.0913	1.67865
$438$	0	0
$439$	−14.2284	−0.679083	−0.339542	−	0.940591i	$-0.610272\pi$
$439$	−14.2284	−0.679083	−0.339542	+	0.940591i	$0.610272\pi$
$440$	0	0
$441$	9.92727	0.472727
$442$	0	0
$443$	−4.71784	−0.224151	−0.112076	−	0.993700i	$-0.535750\pi$
$443$	−4.71784	−0.224151	−0.112076	+	0.993700i	$0.535750\pi$
$444$	0	0
$445$	0.0766414	0.00363315
$446$	0	0
$447$	−2.93692	−0.138911
$448$	0	0
$449$	37.7414	1.78113	0.890564	−	0.454857i	$-0.150310\pi$
$449$	37.7414	1.78113	0.890564	+	0.454857i	$0.150310\pi$
$450$	0	0
$451$	1.47532	0.0694701
$452$	0	0
$453$	−1.94352	−0.0913148
$454$	0	0
$455$	11.8129	0.553800
$456$	0	0
$457$	−25.7343	−1.20380	−0.601900	−	0.798571i	$-0.705590\pi$
$457$	−25.7343	−1.20380	−0.601900	+	0.798571i	$0.705590\pi$
$458$	0	0
$459$	−5.80280	−0.270851
$460$	0	0
$461$	−21.2430	−0.989384	−0.494692	−	0.869068i	$-0.664719\pi$
$461$	−21.2430	−0.989384	−0.494692	+	0.869068i	$0.664719\pi$
$462$	0	0
$463$	0.323287	0.0150244	0.00751221	−	0.999972i	$-0.497609\pi$
$463$	0.323287	0.0150244	0.00751221	+	0.999972i	$0.497609\pi$
$464$	0	0
$465$	−1.35553	−0.0628611
$466$	0	0
$467$	1.27864	0.0591683	0.0295842	−	0.999562i	$-0.490582\pi$
$467$	1.27864	0.0591683	0.0295842	+	0.999562i	$0.490582\pi$
$468$	0	0
$469$	3.27953	0.151434
$470$	0	0
$471$	3.68358	0.169731
$472$	0	0
$473$	4.51627	0.207658
$474$	0	0
$475$	15.5269	0.712422
$476$	0	0
$477$	−2.75754	−0.126259
$478$	0	0
$479$	−38.6570	−1.76628	−0.883142	−	0.469105i	$-0.844576\pi$
$479$	−38.6570	−1.76628	−0.883142	+	0.469105i	$0.844576\pi$
$480$	0	0
$481$	−28.6178	−1.30486
$482$	0	0
$483$	−8.50987	−0.387212
$484$	0	0
$485$	−8.13881	−0.369564
$486$	0	0
$487$	−16.2396	−0.735886	−0.367943	−	0.929848i	$-0.619938\pi$
$487$	−16.2396	−0.735886	−0.367943	+	0.929848i	$0.619938\pi$
$488$	0	0
$489$	11.8898	0.537675
$490$	0	0
$491$	13.1047	0.591408	0.295704	−	0.955280i	$-0.404446\pi$
$491$	13.1047	0.591408	0.295704	+	0.955280i	$0.404446\pi$
$492$	0	0
$493$	18.5411	0.835049
$494$	0	0
$495$	2.54659	0.114461
$496$	0	0
$497$	8.12724	0.364556
$498$	0	0
$499$	−22.9381	−1.02685	−0.513424	−	0.858135i	$-0.671623\pi$
$499$	−22.9381	−1.02685	−0.513424	+	0.858135i	$0.671623\pi$
$500$	0	0
$501$	−5.13147	−0.229257
$502$	0	0
$503$	−30.5639	−1.36278	−0.681388	−	0.731923i	$-0.738623\pi$
$503$	−30.5639	−1.36278	−0.681388	+	0.731923i	$0.738623\pi$
$504$	0	0
$505$	−1.48456	−0.0660620
$506$	0	0
$507$	−17.2956	−0.768123
$508$	0	0
$509$	21.1276	0.936463	0.468231	−	0.883606i	$-0.344891\pi$
$509$	21.1276	0.936463	0.468231	+	0.883606i	$0.344891\pi$
$510$	0	0
$511$	−0.977421	−0.0432386
$512$	0	0
$513$	−10.6144	−0.468635
$514$	0	0
$515$	1.16997	0.0515550
$516$	0	0
$517$	−8.14060	−0.358023
$518$	0	0
$519$	−8.05018	−0.353364
$520$	0	0
$521$	−31.7366	−1.39041	−0.695203	−	0.718814i	$-0.744685\pi$
$521$	−31.7366	−1.39041	−0.695203	+	0.718814i	$0.744685\pi$
$522$	0	0
$523$	−24.7763	−1.08339	−0.541697	−	0.840574i	$-0.682218\pi$
$523$	−24.7763	−1.08339	−0.541697	+	0.840574i	$0.682218\pi$
$524$	0	0
$525$	−3.76537	−0.164334
$526$	0	0
$527$	−6.10138	−0.265780
$528$	0	0
$529$	64.8475	2.81946
$530$	0	0
$531$	−7.47605	−0.324433
$532$	0	0
$533$	10.2346	0.443309
$534$	0	0
$535$	12.7305	0.550388
$536$	0	0
$537$	11.0289	0.475934
$538$	0	0
$539$	3.60005	0.155065
$540$	0	0
$541$	27.4283	1.17923	0.589617	−	0.807683i	$-0.299279\pi$
$541$	27.4283	1.17923	0.589617	+	0.807683i	$0.299279\pi$
$542$	0	0
$543$	11.7306	0.503407
$544$	0	0
$545$	11.2663	0.482593
$546$	0	0
$547$	26.8365	1.14745	0.573723	−	0.819049i	$-0.305498\pi$
$547$	26.8365	1.14745	0.573723	+	0.819049i	$0.305498\pi$
$548$	0	0
$549$	12.3043	0.525136
$550$	0	0
$551$	33.9150	1.44483
$552$	0	0
$553$	−14.4036	−0.612502
$554$	0	0
$555$	−1.87591	−0.0796278
$556$	0	0
$557$	25.2608	1.07033	0.535166	−	0.844747i	$-0.320249\pi$
$557$	25.2608	1.07033	0.535166	+	0.844747i	$0.320249\pi$
$558$	0	0
$559$	31.3302	1.32513
$560$	0	0
$561$	−1.00786	−0.0425519
$562$	0	0
$563$	−5.80383	−0.244602	−0.122301	−	0.992493i	$-0.539027\pi$
$563$	−5.80383	−0.244602	−0.122301	+	0.992493i	$0.539027\pi$
$564$	0	0
$565$	−15.9749	−0.672068
$566$	0	0
$567$	−12.6798	−0.532501
$568$	0	0
$569$	19.1823	0.804162	0.402081	−	0.915604i	$-0.368287\pi$
$569$	19.1823	0.804162	0.402081	+	0.915604i	$0.368287\pi$
$570$	0	0
$571$	9.04708	0.378609	0.189304	−	0.981918i	$-0.439377\pi$
$571$	9.04708	0.378609	0.189304	+	0.981918i	$0.439377\pi$
$572$	0	0
$573$	−3.33063	−0.139139
$574$	0	0
$575$	38.8699	1.62099
$576$	0	0
$577$	−13.7019	−0.570416	−0.285208	−	0.958466i	$-0.592063\pi$
$577$	−13.7019	−0.570416	−0.285208	+	0.958466i	$0.592063\pi$
$578$	0	0
$579$	2.13111	0.0885658
$580$	0	0
$581$	−26.6576	−1.10594
$582$	0	0
$583$	−1.00000	−0.0414158
$584$	0	0
$585$	17.6662	0.730408
$586$	0	0
$587$	−31.1585	−1.28605	−0.643025	−	0.765845i	$-0.722321\pi$
$587$	−31.1585	−1.28605	−0.643025	+	0.765845i	$0.722321\pi$
$588$	0	0
$589$	−11.1605	−0.459861
$590$	0	0
$591$	6.48004	0.266553
$592$	0	0
$593$	19.6272	0.805994	0.402997	−	0.915201i	$-0.367969\pi$
$593$	19.6272	0.805994	0.402997	+	0.915201i	$0.367969\pi$
$594$	0	0
$595$	3.48540	0.142888
$596$	0	0
$597$	10.8717	0.444948
$598$	0	0
$599$	−39.0137	−1.59406	−0.797029	−	0.603941i	$-0.793596\pi$
$599$	−39.0137	−1.59406	−0.797029	+	0.603941i	$0.793596\pi$
$600$	0	0
$601$	−30.6535	−1.25038	−0.625191	−	0.780472i	$-0.714979\pi$
$601$	−30.6535	−1.25038	−0.625191	+	0.780472i	$0.714979\pi$
$602$	0	0
$603$	4.90452	0.199727
$604$	0	0
$605$	0.923502	0.0375457
$606$	0	0
$607$	18.9421	0.768835	0.384417	−	0.923159i	$-0.374402\pi$
$607$	18.9421	0.768835	0.384417	+	0.923159i	$0.374402\pi$
$608$	0	0
$609$	−8.22460	−0.333278
$610$	0	0
$611$	−56.4730	−2.28465
$612$	0	0
$613$	25.2362	1.01928	0.509640	−	0.860388i	$-0.329779\pi$
$613$	25.2362	1.01928	0.509640	+	0.860388i	$0.329779\pi$
$614$	0	0
$615$	0.670881	0.0270525
$616$	0	0
$617$	−41.2187	−1.65940	−0.829702	−	0.558207i	$-0.811489\pi$
$617$	−41.2187	−1.65940	−0.829702	+	0.558207i	$0.811489\pi$
$618$	0	0
$619$	−5.55508	−0.223277	−0.111639	−	0.993749i	$-0.535610\pi$
$619$	−5.55508	−0.223277	−0.111639	+	0.993749i	$0.535610\pi$
$620$	0	0
$621$	−26.5719	−1.06630
$622$	0	0
$623$	−0.153025	−0.00613081
$624$	0	0
$625$	12.9345	0.517381
$626$	0	0
$627$	−1.84356	−0.0736246
$628$	0	0
$629$	−8.44366	−0.336671
$630$	0	0
$631$	40.1408	1.59798	0.798991	−	0.601343i	$-0.205367\pi$
$631$	40.1408	1.59798	0.798991	+	0.601343i	$0.205367\pi$
$632$	0	0
$633$	0.0403849	0.00160516
$634$	0	0
$635$	16.0222	0.635822
$636$	0	0
$637$	24.9743	0.989516
$638$	0	0
$639$	12.1542	0.480815
$640$	0	0
$641$	−47.8022	−1.88807	−0.944036	−	0.329842i	$-0.893004\pi$
$641$	−47.8022	−1.88807	−0.944036	+	0.329842i	$0.893004\pi$
$642$	0	0
$643$	32.8118	1.29397	0.646985	−	0.762503i	$-0.276030\pi$
$643$	32.8118	1.29397	0.646985	+	0.762503i	$0.276030\pi$
$644$	0	0
$645$	2.05371	0.0808647
$646$	0	0
$647$	−13.6408	−0.536273	−0.268137	−	0.963381i	$-0.586408\pi$
$647$	−13.6408	−0.536273	−0.268137	+	0.963381i	$0.586408\pi$
$648$	0	0
$649$	−2.71113	−0.106421
$650$	0	0
$651$	2.70650	0.106076
$652$	0	0
$653$	3.22191	0.126083	0.0630416	−	0.998011i	$-0.479920\pi$
$653$	3.22191	0.126083	0.0630416	+	0.998011i	$0.479920\pi$
$654$	0	0
$655$	5.62893	0.219940
$656$	0	0
$657$	−1.46173	−0.0570275
$658$	0	0
$659$	−31.3324	−1.22054	−0.610268	−	0.792195i	$-0.708938\pi$
$659$	−31.3324	−1.22054	−0.610268	+	0.792195i	$0.708938\pi$
$660$	0	0
$661$	−10.8137	−0.420604	−0.210302	−	0.977636i	$-0.567445\pi$
$661$	−10.8137	−0.420604	−0.210302	+	0.977636i	$0.567445\pi$
$662$	0	0
$663$	−6.99173	−0.271536
$664$	0	0
$665$	6.37542	0.247228
$666$	0	0
$667$	84.9027	3.28745
$668$	0	0
$669$	4.80433	0.185746
$670$	0	0
$671$	4.46207	0.172256
$672$	0	0
$673$	−11.8821	−0.458020	−0.229010	−	0.973424i	$-0.573549\pi$
$673$	−11.8821	−0.458020	−0.229010	+	0.973424i	$0.573549\pi$
$674$	0	0
$675$	−11.7573	−0.452539
$676$	0	0
$677$	20.6861	0.795030	0.397515	−	0.917596i	$-0.369873\pi$
$677$	20.6861	0.795030	0.397515	+	0.917596i	$0.369873\pi$
$678$	0	0
$679$	16.2502	0.623627
$680$	0	0
$681$	10.6869	0.409525
$682$	0	0
$683$	−10.6221	−0.406442	−0.203221	−	0.979133i	$-0.565141\pi$
$683$	−10.6221	−0.406442	−0.203221	+	0.979133i	$0.565141\pi$
$684$	0	0
$685$	−2.93661	−0.112202
$686$	0	0
$687$	−2.13875	−0.0815985
$688$	0	0
$689$	−6.93720	−0.264286
$690$	0	0
$691$	−52.1110	−1.98239	−0.991197	−	0.132393i	$-0.957734\pi$
$691$	−52.1110	−1.98239	−0.991197	+	0.132393i	$0.957734\pi$
$692$	0	0
$693$	−5.08461	−0.193148
$694$	0	0
$695$	5.35358	0.203073
$696$	0	0
$697$	3.01971	0.114380
$698$	0	0
$699$	−6.31796	−0.238967
$700$	0	0
$701$	18.5198	0.699482	0.349741	−	0.936846i	$-0.386270\pi$
$701$	18.5198	0.699482	0.349741	+	0.936846i	$0.386270\pi$
$702$	0	0
$703$	−15.4450	−0.582518
$704$	0	0
$705$	−3.70182	−0.139419
$706$	0	0
$707$	2.96412	0.111477
$708$	0	0
$709$	−1.24655	−0.0468153	−0.0234077	−	0.999726i	$-0.507452\pi$
$709$	−1.24655	−0.0468153	−0.0234077	+	0.999726i	$0.507452\pi$
$710$	0	0
$711$	−21.5405	−0.807831
$712$	0	0
$713$	−27.9392	−1.04633
$714$	0	0
$715$	6.40652	0.239590
$716$	0	0
$717$	2.54748	0.0951373
$718$	0	0
$719$	4.64926	0.173388	0.0866941	−	0.996235i	$-0.472370\pi$
$719$	4.64926	0.173388	0.0866941	+	0.996235i	$0.472370\pi$
$720$	0	0
$721$	−2.33600	−0.0869972
$722$	0	0
$723$	−3.38866	−0.126026
$724$	0	0
$725$	37.5670	1.39520
$726$	0	0
$727$	−32.9600	−1.22242	−0.611210	−	0.791469i	$-0.709317\pi$
$727$	−32.9600	−1.22242	−0.611210	+	0.791469i	$0.709317\pi$
$728$	0	0
$729$	−14.7746	−0.547208
$730$	0	0
$731$	9.24397	0.341900
$732$	0	0
$733$	38.0787	1.40647	0.703234	−	0.710958i	$-0.251738\pi$
$733$	38.0787	1.40647	0.703234	+	0.710958i	$0.251738\pi$
$734$	0	0
$735$	1.63707	0.0603843
$736$	0	0
$737$	1.77859	0.0655150
$738$	0	0
$739$	26.5619	0.977095	0.488548	−	0.872537i	$-0.337527\pi$
$739$	26.5619	0.977095	0.488548	+	0.872537i	$0.337527\pi$
$740$	0	0
$741$	−12.7891	−0.469820
$742$	0	0
$743$	21.9417	0.804963	0.402482	−	0.915428i	$-0.368148\pi$
$743$	21.9417	0.804963	0.402482	+	0.915428i	$0.368148\pi$
$744$	0	0
$745$	5.50818	0.201804
$746$	0	0
$747$	−39.8663	−1.45863
$748$	0	0
$749$	−25.4182	−0.928760
$750$	0	0
$751$	19.2751	0.703360	0.351680	−	0.936120i	$-0.385611\pi$
$751$	19.2751	0.703360	0.351680	+	0.936120i	$0.385611\pi$
$752$	0	0
$753$	13.7305	0.500367
$754$	0	0
$755$	3.64507	0.132658
$756$	0	0
$757$	49.2523	1.79011	0.895053	−	0.445959i	$-0.147137\pi$
$757$	49.2523	1.79011	0.895053	+	0.445959i	$0.147137\pi$
$758$	0	0
$759$	−4.61516	−0.167520
$760$	0	0
$761$	−1.24521	−0.0451387	−0.0225693	−	0.999745i	$-0.507185\pi$
$761$	−1.24521	−0.0451387	−0.0225693	+	0.999745i	$0.507185\pi$
$762$	0	0
$763$	−22.4946	−0.814359
$764$	0	0
$765$	5.21240	0.188455
$766$	0	0
$767$	−18.8077	−0.679106
$768$	0	0
$769$	32.6021	1.17566	0.587832	−	0.808983i	$-0.299982\pi$
$769$	32.6021	1.17566	0.587832	+	0.808983i	$0.299982\pi$
$770$	0	0
$771$	5.30469	0.191044
$772$	0	0
$773$	40.9615	1.47328	0.736642	−	0.676283i	$-0.236410\pi$
$773$	40.9615	1.47328	0.736642	+	0.676283i	$0.236410\pi$
$774$	0	0
$775$	−12.3623	−0.444066
$776$	0	0
$777$	3.74550	0.134369
$778$	0	0
$779$	5.52359	0.197903
$780$	0	0
$781$	4.40765	0.157718
$782$	0	0
$783$	−25.6812	−0.917771
$784$	0	0
$785$	−6.90854	−0.246576
$786$	0	0
$787$	51.6284	1.84035	0.920177	−	0.391502i	$-0.128044\pi$
$787$	51.6284	1.84035	0.920177	+	0.391502i	$0.128044\pi$
$788$	0	0
$789$	11.0631	0.393858
$790$	0	0
$791$	31.8960	1.13409
$792$	0	0
$793$	30.9543	1.09922
$794$	0	0
$795$	−0.454736	−0.0161278
$796$	0	0
$797$	53.7023	1.90223	0.951117	−	0.308830i	$-0.0999373\pi$
$797$	53.7023	1.90223	0.951117	+	0.308830i	$0.0999373\pi$
$798$	0	0
$799$	−16.6623	−0.589470
$800$	0	0
$801$	−0.228848	−0.00808595
$802$	0	0
$803$	−0.530085	−0.0187063
$804$	0	0
$805$	15.9602	0.562524
$806$	0	0
$807$	14.5300	0.511479
$808$	0	0
$809$	15.7090	0.552299	0.276150	−	0.961115i	$-0.410942\pi$
$809$	15.7090	0.552299	0.276150	+	0.961115i	$0.410942\pi$
$810$	0	0
$811$	30.0739	1.05604	0.528020	−	0.849232i	$-0.322935\pi$
$811$	30.0739	1.05604	0.528020	+	0.849232i	$0.322935\pi$
$812$	0	0
$813$	−2.60450	−0.0913439
$814$	0	0
$815$	−22.2993	−0.781109
$816$	0	0
$817$	16.9089	0.591566
$818$	0	0
$819$	−35.2730	−1.23254
$820$	0	0
$821$	−42.4655	−1.48206	−0.741028	−	0.671475i	$-0.765661\pi$
$821$	−42.4655	−1.48206	−0.741028	+	0.671475i	$0.765661\pi$
$822$	0	0
$823$	6.70042	0.233562	0.116781	−	0.993158i	$-0.462742\pi$
$823$	6.70042	0.233562	0.116781	+	0.993158i	$0.462742\pi$
$824$	0	0
$825$	−2.04207	−0.0710958
$826$	0	0
$827$	−9.53613	−0.331604	−0.165802	−	0.986159i	$-0.553021\pi$
$827$	−9.53613	−0.331604	−0.165802	+	0.986159i	$0.553021\pi$
$828$	0	0
$829$	1.03518	0.0359533	0.0179766	−	0.999838i	$-0.494278\pi$
$829$	1.03518	0.0359533	0.0179766	+	0.999838i	$0.494278\pi$
$830$	0	0
$831$	−13.1640	−0.456653
$832$	0	0
$833$	7.36864	0.255308
$834$	0	0
$835$	9.62405	0.333054
$836$	0	0
$837$	8.45099	0.292109
$838$	0	0
$839$	46.9174	1.61977	0.809885	−	0.586589i	$-0.199530\pi$
$839$	46.9174	1.61977	0.809885	+	0.586589i	$0.199530\pi$
$840$	0	0
$841$	53.0566	1.82954
$842$	0	0
$843$	12.1319	0.417845
$844$	0	0
$845$	32.4378	1.11589
$846$	0	0
$847$	−1.84390	−0.0633570
$848$	0	0
$849$	−6.12528	−0.210219
$850$	0	0
$851$	−38.6649	−1.32541
$852$	0	0
$853$	−26.6742	−0.913307	−0.456654	−	0.889645i	$-0.650952\pi$
$853$	−26.6742	−0.913307	−0.456654	+	0.889645i	$0.650952\pi$
$854$	0	0
$855$	9.53442	0.326070
$856$	0	0
$857$	10.6566	0.364022	0.182011	−	0.983297i	$-0.441739\pi$
$857$	10.6566	0.364022	0.182011	+	0.983297i	$0.441739\pi$
$858$	0	0
$859$	−42.1018	−1.43650	−0.718248	−	0.695787i	$-0.755056\pi$
$859$	−42.1018	−1.43650	−0.718248	+	0.695787i	$0.755056\pi$
$860$	0	0
$861$	−1.33951	−0.0456502
$862$	0	0
$863$	−4.82480	−0.164238	−0.0821190	−	0.996623i	$-0.526169\pi$
$863$	−4.82480	−0.164238	−0.0821190	+	0.996623i	$0.526169\pi$
$864$	0	0
$865$	15.0981	0.513350
$866$	0	0
$867$	6.30796	0.214230
$868$	0	0
$869$	−7.81149	−0.264987
$870$	0	0
$871$	12.3384	0.418071
$872$	0	0
$873$	24.3022	0.822503
$874$	0	0
$875$	15.5761	0.526569
$876$	0	0
$877$	24.0983	0.813740	0.406870	−	0.913486i	$-0.366620\pi$
$877$	24.0983	0.813740	0.406870	+	0.913486i	$0.366620\pi$
$878$	0	0
$879$	5.31426	0.179245
$880$	0	0
$881$	−8.80511	−0.296652	−0.148326	−	0.988939i	$-0.547388\pi$
$881$	−8.80511	−0.296652	−0.148326	+	0.988939i	$0.547388\pi$
$882$	0	0
$883$	−19.5528	−0.658005	−0.329003	−	0.944329i	$-0.606713\pi$
$883$	−19.5528	−0.658005	−0.329003	+	0.944329i	$0.606713\pi$
$884$	0	0
$885$	−1.23285	−0.0414418
$886$	0	0
$887$	28.0943	0.943313	0.471657	−	0.881782i	$-0.343656\pi$
$887$	28.0943	0.943313	0.471657	+	0.881782i	$0.343656\pi$
$888$	0	0
$889$	−31.9905	−1.07293
$890$	0	0
$891$	−6.87663	−0.230376
$892$	0	0
$893$	−30.4783	−1.01992
$894$	0	0
$895$	−20.6847	−0.691415
$896$	0	0
$897$	−32.0163	−1.06899
$898$	0	0
$899$	−27.0026	−0.900588
$900$	0	0
$901$	−2.04682	−0.0681893
$902$	0	0
$903$	−4.10051	−0.136456
$904$	0	0
$905$	−22.0006	−0.731326
$906$	0	0
$907$	−17.5441	−0.582541	−0.291271	−	0.956641i	$-0.594078\pi$
$907$	−17.5441	−0.582541	−0.291271	+	0.956641i	$0.594078\pi$
$908$	0	0
$909$	4.43283	0.147028
$910$	0	0
$911$	21.3695	0.708005	0.354002	−	0.935245i	$-0.384820\pi$
$911$	21.3695	0.708005	0.354002	+	0.935245i	$0.384820\pi$
$912$	0	0
$913$	−14.4572	−0.478463
$914$	0	0
$915$	2.02907	0.0670788
$916$	0	0
$917$	−11.2389	−0.371142
$918$	0	0
$919$	−0.368526	−0.0121566	−0.00607828	−	0.999982i	$-0.501935\pi$
$919$	−0.368526	−0.0121566	−0.00607828	+	0.999982i	$0.501935\pi$
$920$	0	0
$921$	−10.1964	−0.335984
$922$	0	0
$923$	30.5767	1.00644
$924$	0	0
$925$	−17.1081	−0.562510
$926$	0	0
$927$	−3.49348	−0.114741
$928$	0	0
$929$	36.8795	1.20998	0.604988	−	0.796234i	$-0.293178\pi$
$929$	36.8795	1.20998	0.604988	+	0.796234i	$0.293178\pi$
$930$	0	0
$931$	13.4786	0.441742
$932$	0	0
$933$	−9.67021	−0.316588
$934$	0	0
$935$	1.89024	0.0618174
$936$	0	0
$937$	16.4329	0.536839	0.268420	−	0.963302i	$-0.413499\pi$
$937$	16.4329	0.536839	0.268420	+	0.963302i	$0.413499\pi$
$938$	0	0
$939$	3.62375	0.118257
$940$	0	0
$941$	37.4051	1.21937	0.609686	−	0.792643i	$-0.291296\pi$
$941$	37.4051	1.21937	0.609686	+	0.792643i	$0.291296\pi$
$942$	0	0
$943$	13.8277	0.450293
$944$	0	0
$945$	−4.82761	−0.157042
$946$	0	0
$947$	48.6867	1.58210	0.791052	−	0.611749i	$-0.209534\pi$
$947$	48.6867	1.58210	0.791052	+	0.611749i	$0.209534\pi$
$948$	0	0
$949$	−3.67730	−0.119370
$950$	0	0
$951$	−10.9542	−0.355214
$952$	0	0
$953$	38.6032	1.25048	0.625240	−	0.780433i	$-0.285001\pi$
$953$	38.6032	1.25048	0.625240	+	0.780433i	$0.285001\pi$
$954$	0	0
$955$	6.24658	0.202135
$956$	0	0
$957$	−4.46045	−0.144186
$958$	0	0
$959$	5.86334	0.189337
$960$	0	0
$961$	−22.1142	−0.713360
$962$	0	0
$963$	−38.0128	−1.22494
$964$	0	0
$965$	−3.99688	−0.128664
$966$	0	0
$967$	−13.8135	−0.444213	−0.222107	−	0.975022i	$-0.571293\pi$
$967$	−13.8135	−0.444213	−0.222107	+	0.975022i	$0.571293\pi$
$968$	0	0
$969$	−3.77342	−0.121220
$970$	0	0
$971$	17.5525	0.563285	0.281643	−	0.959519i	$-0.409121\pi$
$971$	17.5525	0.563285	0.281643	+	0.959519i	$0.409121\pi$
$972$	0	0
$973$	−10.6892	−0.342678
$974$	0	0
$975$	−14.1663	−0.453683
$976$	0	0
$977$	36.8851	1.18006	0.590030	−	0.807381i	$-0.299116\pi$
$977$	36.8851	1.18006	0.590030	+	0.807381i	$0.299116\pi$
$978$	0	0
$979$	−0.0829900	−0.00265237
$980$	0	0
$981$	−33.6406	−1.07406
$982$	0	0
$983$	−31.9091	−1.01774	−0.508872	−	0.860842i	$-0.669937\pi$
$983$	−31.9091	−1.01774	−0.508872	+	0.860842i	$0.669937\pi$
$984$	0	0
$985$	−12.1533	−0.387236
$986$	0	0
$987$	7.39119	0.235264
$988$	0	0
$989$	42.3296	1.34600
$990$	0	0
$991$	10.1417	0.322163	0.161081	−	0.986941i	$-0.448502\pi$
$991$	10.1417	0.322163	0.161081	+	0.986941i	$0.448502\pi$
$992$	0	0
$993$	3.41201	0.108277
$994$	0	0
$995$	−20.3898	−0.646399
$996$	0	0
$997$	2.26539	0.0717457	0.0358728	−	0.999356i	$-0.488579\pi$
$997$	2.26539	0.0717457	0.0358728	+	0.999356i	$0.488579\pi$
$998$	0	0
$999$	11.6953	0.370022

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9328.2.a.bm.1.4		11
4.3	odd	2		4664.2.a.k.1.8	✓	11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4664.2.a.k.1.8	✓	11	4.3	odd	2
9328.2.a.bm.1.4		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)

\(f(q)\)	\(=\)	\(q-0.492404 q^{3} +0.923502 q^{5} -1.84390 q^{7} -2.75754 q^{9} +O(q^{10})\)	\(q-0.492404 q^{3} +0.923502 q^{5} -1.84390 q^{7} -2.75754 q^{9} -1.00000 q^{11} -6.93720 q^{13} -0.454736 q^{15} -2.04682 q^{17} -3.74399 q^{19} +0.907942 q^{21} -9.37270 q^{23} -4.14714 q^{25} +2.83504 q^{27} -9.05851 q^{29} +2.98091 q^{31} +0.492404 q^{33} -1.70284 q^{35} +4.12526 q^{37} +3.41591 q^{39} -1.47532 q^{41} -4.51627 q^{43} -2.54659 q^{45} +8.14060 q^{47} -3.60005 q^{49} +1.00786 q^{51} +1.00000 q^{53} -0.923502 q^{55} +1.84356 q^{57} +2.71113 q^{59} -4.46207 q^{61} +5.08461 q^{63} -6.40652 q^{65} -1.77859 q^{67} +4.61516 q^{69} -4.40765 q^{71} +0.530085 q^{73} +2.04207 q^{75} +1.84390 q^{77} +7.81149 q^{79} +6.87663 q^{81} +14.4572 q^{83} -1.89024 q^{85} +4.46045 q^{87} +0.0829900 q^{89} +12.7915 q^{91} -1.46781 q^{93} -3.45758 q^{95} -8.81299 q^{97} +2.75754 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q + 6 q^{3} + 3 q^{5} + 5 q^{7} + 7 q^{9}+O(q^{10}) \) 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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