LMFDB - Embedded newform 5312.2.a.bv.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5312,2,Mod(1,5312)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5312, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("5312.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5312 = 2^{6} \cdot 83 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5312.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:	$42.4165335537$
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} - x^{10} - 25 x^{9} + 24 x^{8} + 214 x^{7} - 197 x^{6} - 721 x^{5} + 620 x^{4} + 795 x^{3} + \cdots - 16 $ x^11 - x^10 - 25x^9 + 24x^8 + 214x^7 - 197x^6 - 721x^5 + 620x^4 + 795x^3 - 514x^2 - 220*x - 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 2656)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$1.25056$ of defining polynomial
Character	$\chi$	$=$	5312.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.25056 q^{3} -3.44083 q^{5} -2.10491 q^{7} -1.43611 q^{9} +O(q^{10})$	$q+1.25056 q^{3} -3.44083 q^{5} -2.10491 q^{7} -1.43611 q^{9} -2.55473 q^{11} -5.12687 q^{13} -4.30296 q^{15} -0.576977 q^{17} -0.576818 q^{19} -2.63231 q^{21} -1.07410 q^{23} +6.83933 q^{25} -5.54761 q^{27} +1.35110 q^{29} -4.17563 q^{31} -3.19483 q^{33} +7.24264 q^{35} -1.26718 q^{37} -6.41145 q^{39} +9.26975 q^{41} +1.22309 q^{43} +4.94140 q^{45} -1.65451 q^{47} -2.56935 q^{49} -0.721543 q^{51} -12.5991 q^{53} +8.79038 q^{55} -0.721344 q^{57} -1.85829 q^{59} +5.20661 q^{61} +3.02288 q^{63} +17.6407 q^{65} -6.83947 q^{67} -1.34323 q^{69} -5.36305 q^{71} +7.83885 q^{73} +8.55297 q^{75} +5.37747 q^{77} +9.04396 q^{79} -2.62928 q^{81} +1.00000 q^{83} +1.98528 q^{85} +1.68962 q^{87} -0.496929 q^{89} +10.7916 q^{91} -5.22187 q^{93} +1.98473 q^{95} -1.45735 q^{97} +3.66886 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q + q^{3} - 5 q^{5} + 10 q^{7} + 18 q^{9}+O(q^{10}) $ 11 * q + q^3 - 5 * q^5 + 10 * q^7 + 18 * q^9	$ 11 q + q^{3} - 5 q^{5} + 10 q^{7} + 18 q^{9} - 3 q^{11} + q^{13} + 18 q^{15} + 6 q^{17} + 15 q^{19} - 16 q^{21} - 6 q^{23} + 28 q^{25} - 2 q^{27} - 31 q^{29} + 14 q^{31} - 8 q^{33} + 10 q^{35} + q^{37} + 18 q^{39} + 22 q^{41} + 37 q^{43} - 3 q^{45} - 2 q^{47} + 45 q^{49} + 24 q^{51} - 29 q^{53} + 14 q^{55} + 36 q^{57} - q^{59} - 15 q^{61} + 18 q^{63} + 14 q^{65} + 45 q^{67} - 34 q^{69} - 4 q^{73} + 17 q^{75} - 34 q^{77} + 10 q^{79} + 39 q^{81} + 11 q^{83} - 25 q^{85} - 13 q^{87} + 15 q^{91} - 2 q^{93} + 4 q^{95} - 18 q^{97} - 26 q^{99}+O(q^{100}) $ 11 * q + q^3 - 5 * q^5 + 10 * q^7 + 18 * q^9 - 3 * q^11 + q^13 + 18 * q^15 + 6 * q^17 + 15 * q^19 - 16 * q^21 - 6 * q^23 + 28 * q^25 - 2 * q^27 - 31 * q^29 + 14 * q^31 - 8 * q^33 + 10 * q^35 + q^37 + 18 * q^39 + 22 * q^41 + 37 * q^43 - 3 * q^45 - 2 * q^47 + 45 * q^49 + 24 * q^51 - 29 * q^53 + 14 * q^55 + 36 * q^57 - q^59 - 15 * q^61 + 18 * q^63 + 14 * q^65 + 45 * q^67 - 34 * q^69 - 4 * q^73 + 17 * q^75 - 34 * q^77 + 10 * q^79 + 39 * q^81 + 11 * q^83 - 25 * q^85 - 13 * q^87 + 15 * q^91 - 2 * q^93 + 4 * q^95 - 18 * q^97 - 26 * 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
$3$	1.25056	0.722010	0.361005	−	0.932564i	$-0.382434\pi$
$3$	1.25056	0.722010	0.361005	+	0.932564i	$0.382434\pi$
$4$	0	0
$5$	−3.44083	−1.53879	−0.769393	−	0.638775i	$-0.779441\pi$
$5$	−3.44083	−1.53879	−0.769393	+	0.638775i	$0.779441\pi$
$6$	0	0
$7$	−2.10491	−0.795581	−0.397791	−	0.917476i	$-0.630223\pi$
$7$	−2.10491	−0.795581	−0.397791	+	0.917476i	$0.630223\pi$
$8$	0	0
$9$	−1.43611	−0.478702
$10$	0	0
$11$	−2.55473	−0.770279	−0.385139	−	0.922858i	$-0.625847\pi$
$11$	−2.55473	−0.770279	−0.385139	+	0.922858i	$0.625847\pi$
$12$	0	0
$13$	−5.12687	−1.42194	−0.710969	−	0.703223i	$-0.751744\pi$
$13$	−5.12687	−1.42194	−0.710969	+	0.703223i	$0.751744\pi$
$14$	0	0
$15$	−4.30296	−1.11102
$16$	0	0
$17$	−0.576977	−0.139938	−0.0699688	−	0.997549i	$-0.522290\pi$
$17$	−0.576977	−0.139938	−0.0699688	+	0.997549i	$0.522290\pi$
$18$	0	0
$19$	−0.576818	−0.132331	−0.0661656	−	0.997809i	$-0.521077\pi$
$19$	−0.576818	−0.132331	−0.0661656	+	0.997809i	$0.521077\pi$
$20$	0	0
$21$	−2.63231	−0.574417
$22$	0	0
$23$	−1.07410	−0.223966	−0.111983	−	0.993710i	$-0.535720\pi$
$23$	−1.07410	−0.223966	−0.111983	+	0.993710i	$0.535720\pi$
$24$	0	0
$25$	6.83933	1.36787
$26$	0	0
$27$	−5.54761	−1.06764
$28$	0	0
$29$	1.35110	0.250892	0.125446	−	0.992100i	$-0.459964\pi$
$29$	1.35110	0.250892	0.125446	+	0.992100i	$0.459964\pi$
$30$	0	0
$31$	−4.17563	−0.749966	−0.374983	−	0.927032i	$-0.622351\pi$
$31$	−4.17563	−0.749966	−0.374983	+	0.927032i	$0.622351\pi$
$32$	0	0
$33$	−3.19483	−0.556149
$34$	0	0
$35$	7.24264	1.22423
$36$	0	0
$37$	−1.26718	−0.208323	−0.104162	−	0.994560i	$-0.533216\pi$
$37$	−1.26718	−0.208323	−0.104162	+	0.994560i	$0.533216\pi$
$38$	0	0
$39$	−6.41145	−1.02665
$40$	0	0
$41$	9.26975	1.44769	0.723845	−	0.689962i	$-0.242373\pi$
$41$	9.26975	1.44769	0.723845	+	0.689962i	$0.242373\pi$
$42$	0	0
$43$	1.22309	0.186519	0.0932595	−	0.995642i	$-0.470271\pi$
$43$	1.22309	0.186519	0.0932595	+	0.995642i	$0.470271\pi$
$44$	0	0
$45$	4.94140	0.736621
$46$	0	0
$47$	−1.65451	−0.241335	−0.120667	−	0.992693i	$-0.538503\pi$
$47$	−1.65451	−0.241335	−0.120667	+	0.992693i	$0.538503\pi$
$48$	0	0
$49$	−2.56935	−0.367050
$50$	0	0
$51$	−0.721543	−0.101036
$52$	0	0
$53$	−12.5991	−1.73062	−0.865309	−	0.501239i	$-0.832878\pi$
$53$	−12.5991	−1.73062	−0.865309	+	0.501239i	$0.832878\pi$
$54$	0	0
$55$	8.79038	1.18529
$56$	0	0
$57$	−0.721344	−0.0955444
$58$	0	0
$59$	−1.85829	−0.241929	−0.120965	−	0.992657i	$-0.538599\pi$
$59$	−1.85829	−0.241929	−0.120965	+	0.992657i	$0.538599\pi$
$60$	0	0
$61$	5.20661	0.666639	0.333319	−	0.942814i	$-0.391831\pi$
$61$	5.20661	0.666639	0.333319	+	0.942814i	$0.391831\pi$
$62$	0	0
$63$	3.02288	0.380847
$64$	0	0
$65$	17.6407	2.18806
$66$	0	0
$67$	−6.83947	−0.835575	−0.417787	−	0.908545i	$-0.637194\pi$
$67$	−6.83947	−0.835575	−0.417787	+	0.908545i	$0.637194\pi$
$68$	0	0
$69$	−1.34323	−0.161706
$70$	0	0
$71$	−5.36305	−0.636477	−0.318238	−	0.948011i	$-0.603091\pi$
$71$	−5.36305	−0.636477	−0.318238	+	0.948011i	$0.603091\pi$
$72$	0	0
$73$	7.83885	0.917468	0.458734	−	0.888574i	$-0.348303\pi$
$73$	7.83885	0.917468	0.458734	+	0.888574i	$0.348303\pi$
$74$	0	0
$75$	8.55297	0.987612
$76$	0	0
$77$	5.37747	0.612819
$78$	0	0
$79$	9.04396	1.01752	0.508762	−	0.860907i	$-0.330103\pi$
$79$	9.04396	1.01752	0.508762	+	0.860907i	$0.330103\pi$
$80$	0	0
$81$	−2.62928	−0.292142
$82$	0	0
$83$	1.00000	0.109764
$83$	1.00000	0.109764
$84$	0	0
$85$	1.98528	0.215334
$86$	0	0
$87$	1.68962	0.181147
$88$	0	0
$89$	−0.496929	−0.0526743	−0.0263372	−	0.999653i	$-0.508384\pi$
$89$	−0.496929	−0.0526743	−0.0263372	+	0.999653i	$0.508384\pi$
$90$	0	0
$91$	10.7916	1.13127
$92$	0	0
$93$	−5.22187	−0.541483
$94$	0	0
$95$	1.98473	0.203630
$96$	0	0
$97$	−1.45735	−0.147971	−0.0739856	−	0.997259i	$-0.523572\pi$
$97$	−1.45735	−0.147971	−0.0739856	+	0.997259i	$0.523572\pi$
$98$	0	0
$99$	3.66886	0.368734
$100$	0	0
$101$	7.28487	0.724872	0.362436	−	0.932009i	$-0.381945\pi$
$101$	7.28487	0.724872	0.362436	+	0.932009i	$0.381945\pi$
$102$	0	0
$103$	13.4746	1.32769	0.663847	−	0.747868i	$-0.268923\pi$
$103$	13.4746	1.32769	0.663847	+	0.747868i	$0.268923\pi$
$104$	0	0
$105$	9.05734	0.883906
$106$	0	0
$107$	11.7530	1.13621	0.568105	−	0.822956i	$-0.307677\pi$
$107$	11.7530	1.13621	0.568105	+	0.822956i	$0.307677\pi$
$108$	0	0
$109$	−3.15284	−0.301987	−0.150993	−	0.988535i	$-0.548247\pi$
$109$	−3.15284	−0.301987	−0.150993	+	0.988535i	$0.548247\pi$
$110$	0	0
$111$	−1.58468	−0.150411
$112$	0	0
$113$	1.10241	0.103706	0.0518528	−	0.998655i	$-0.483487\pi$
$113$	1.10241	0.103706	0.0518528	+	0.998655i	$0.483487\pi$
$114$	0	0
$115$	3.69581	0.344636
$116$	0	0
$117$	7.36273	0.680685
$118$	0	0
$119$	1.21449	0.111332
$120$	0	0
$121$	−4.47338	−0.406671
$122$	0	0
$123$	11.5923	1.04525
$124$	0	0
$125$	−6.32882	−0.566066
$126$	0	0
$127$	−1.75770	−0.155970	−0.0779852	−	0.996955i	$-0.524849\pi$
$127$	−1.75770	−0.155970	−0.0779852	+	0.996955i	$0.524849\pi$
$128$	0	0
$129$	1.52954	0.134668
$130$	0	0
$131$	−11.4781	−1.00285	−0.501423	−	0.865202i	$-0.667190\pi$
$131$	−11.4781	−1.00285	−0.501423	+	0.865202i	$0.667190\pi$
$132$	0	0
$133$	1.21415	0.105280
$134$	0	0
$135$	19.0884	1.64287
$136$	0	0
$137$	5.06646	0.432857	0.216428	−	0.976298i	$-0.430559\pi$
$137$	5.06646	0.432857	0.216428	+	0.976298i	$0.430559\pi$
$138$	0	0
$139$	−22.2206	−1.88472	−0.942362	−	0.334595i	$-0.891401\pi$
$139$	−22.2206	−1.88472	−0.942362	+	0.334595i	$0.891401\pi$
$140$	0	0
$141$	−2.06906	−0.174246
$142$	0	0
$143$	13.0978	1.09529
$144$	0	0
$145$	−4.64889	−0.386070
$146$	0	0
$147$	−3.21312	−0.265014
$148$	0	0
$149$	0.779299	0.0638426	0.0319213	−	0.999490i	$-0.489837\pi$
$149$	0.779299	0.0638426	0.0319213	+	0.999490i	$0.489837\pi$
$150$	0	0
$151$	22.8289	1.85779	0.928895	−	0.370342i	$-0.120760\pi$
$151$	22.8289	1.85779	0.928895	+	0.370342i	$0.120760\pi$
$152$	0	0
$153$	0.828601	0.0669884
$154$	0	0
$155$	14.3677	1.15404
$156$	0	0
$157$	9.37523	0.748225	0.374113	−	0.927383i	$-0.377947\pi$
$157$	9.37523	0.748225	0.374113	+	0.927383i	$0.377947\pi$
$158$	0	0
$159$	−15.7559	−1.24952
$160$	0	0
$161$	2.26089	0.178183
$162$	0	0
$163$	9.84694	0.771272	0.385636	−	0.922651i	$-0.373982\pi$
$163$	9.84694	0.771272	0.385636	+	0.922651i	$0.373982\pi$
$164$	0	0
$165$	10.9929	0.855794
$166$	0	0
$167$	6.70742	0.519036	0.259518	−	0.965738i	$-0.416436\pi$
$167$	6.70742	0.519036	0.259518	+	0.965738i	$0.416436\pi$
$168$	0	0
$169$	13.2848	1.02191
$170$	0	0
$171$	0.828372	0.0633472
$172$	0	0
$173$	−6.45724	−0.490935	−0.245467	−	0.969405i	$-0.578941\pi$
$173$	−6.45724	−0.490935	−0.245467	+	0.969405i	$0.578941\pi$
$174$	0	0
$175$	−14.3962	−1.08825
$176$	0	0
$177$	−2.32390	−0.174675
$178$	0	0
$179$	−20.9370	−1.56490	−0.782451	−	0.622712i	$-0.786031\pi$
$179$	−20.9370	−1.56490	−0.782451	+	0.622712i	$0.786031\pi$
$180$	0	0
$181$	−20.5002	−1.52377	−0.761884	−	0.647714i	$-0.775725\pi$
$181$	−20.5002	−1.52377	−0.761884	+	0.647714i	$0.775725\pi$
$182$	0	0
$183$	6.51117	0.481320
$184$	0	0
$185$	4.36015	0.320565
$186$	0	0
$187$	1.47402	0.107791
$188$	0	0
$189$	11.6772	0.849392
$190$	0	0
$191$	−10.7625	−0.778747	−0.389373	−	0.921080i	$-0.627308\pi$
$191$	−10.7625	−0.778747	−0.389373	+	0.921080i	$0.627308\pi$
$192$	0	0
$193$	6.37374	0.458792	0.229396	−	0.973333i	$-0.426325\pi$
$193$	6.37374	0.458792	0.229396	+	0.973333i	$0.426325\pi$
$194$	0	0
$195$	22.0607	1.57980
$196$	0	0
$197$	13.1182	0.934630	0.467315	−	0.884091i	$-0.345221\pi$
$197$	13.1182	0.934630	0.467315	+	0.884091i	$0.345221\pi$
$198$	0	0
$199$	27.3440	1.93837	0.969183	−	0.246341i	$-0.0792283\pi$
$199$	27.3440	1.93837	0.969183	+	0.246341i	$0.0792283\pi$
$200$	0	0
$201$	−8.55315	−0.603293
$202$	0	0
$203$	−2.84393	−0.199605
$204$	0	0
$205$	−31.8956	−2.22769
$206$	0	0
$207$	1.54253	0.107213
$208$	0	0
$209$	1.47361	0.101932
$210$	0	0
$211$	11.4622	0.789092	0.394546	−	0.918876i	$-0.370902\pi$
$211$	11.4622	0.789092	0.394546	+	0.918876i	$0.370902\pi$
$212$	0	0
$213$	−6.70680	−0.459542
$214$	0	0
$215$	−4.20844	−0.287013
$216$	0	0
$217$	8.78933	0.596659
$218$	0	0
$219$	9.80293	0.662420
$220$	0	0
$221$	2.95809	0.198983
$222$	0	0
$223$	−4.59857	−0.307943	−0.153972	−	0.988075i	$-0.549206\pi$
$223$	−4.59857	−0.307943	−0.153972	+	0.988075i	$0.549206\pi$
$224$	0	0
$225$	−9.82200	−0.654800
$226$	0	0
$227$	25.2241	1.67418	0.837092	−	0.547063i	$-0.184254\pi$
$227$	25.2241	1.67418	0.837092	+	0.547063i	$0.184254\pi$
$228$	0	0
$229$	−20.3171	−1.34259	−0.671295	−	0.741190i	$-0.734262\pi$
$229$	−20.3171	−1.34259	−0.671295	+	0.741190i	$0.734262\pi$
$230$	0	0
$231$	6.72483	0.442461
$232$	0	0
$233$	−19.3727	−1.26915	−0.634575	−	0.772861i	$-0.718825\pi$
$233$	−19.3727	−1.26915	−0.634575	+	0.772861i	$0.718825\pi$
$234$	0	0
$235$	5.69289	0.371363
$236$	0	0
$237$	11.3100	0.734663
$238$	0	0
$239$	3.05789	0.197799	0.0988993	−	0.995097i	$-0.468468\pi$
$239$	3.05789	0.197799	0.0988993	+	0.995097i	$0.468468\pi$
$240$	0	0
$241$	1.25332	0.0807336	0.0403668	−	0.999185i	$-0.487147\pi$
$241$	1.25332	0.0807336	0.0403668	+	0.999185i	$0.487147\pi$
$242$	0	0
$243$	13.3548	0.856708
$244$	0	0
$245$	8.84071	0.564812
$246$	0	0
$247$	2.95727	0.188167
$248$	0	0
$249$	1.25056	0.0792508
$250$	0	0
$251$	−12.8499	−0.811077	−0.405538	−	0.914078i	$-0.632916\pi$
$251$	−12.8499	−0.811077	−0.405538	+	0.914078i	$0.632916\pi$
$252$	0	0
$253$	2.74404	0.172517
$254$	0	0
$255$	2.48271	0.155473
$256$	0	0
$257$	−12.7813	−0.797276	−0.398638	−	0.917108i	$-0.630517\pi$
$257$	−12.7813	−0.797276	−0.398638	+	0.917108i	$0.630517\pi$
$258$	0	0
$259$	2.66730	0.165738
$260$	0	0
$261$	−1.94032	−0.120103
$262$	0	0
$263$	7.84846	0.483956	0.241978	−	0.970282i	$-0.422204\pi$
$263$	7.84846	0.483956	0.241978	+	0.970282i	$0.422204\pi$
$264$	0	0
$265$	43.3513	2.66305
$266$	0	0
$267$	−0.621438	−0.0380314
$268$	0	0
$269$	−18.3040	−1.11601	−0.558006	−	0.829837i	$-0.688433\pi$
$269$	−18.3040	−1.11601	−0.558006	+	0.829837i	$0.688433\pi$
$270$	0	0
$271$	9.98662	0.606644	0.303322	−	0.952888i	$-0.401904\pi$
$271$	9.98662	0.606644	0.303322	+	0.952888i	$0.401904\pi$
$272$	0	0
$273$	13.4955	0.816786
$274$	0	0
$275$	−17.4726	−1.05364
$276$	0	0
$277$	−13.4419	−0.807646	−0.403823	−	0.914837i	$-0.632319\pi$
$277$	−13.4419	−0.807646	−0.403823	+	0.914837i	$0.632319\pi$
$278$	0	0
$279$	5.99665	0.359010
$280$	0	0
$281$	−4.55703	−0.271850	−0.135925	−	0.990719i	$-0.543401\pi$
$281$	−4.55703	−0.271850	−0.135925	+	0.990719i	$0.543401\pi$
$282$	0	0
$283$	17.7637	1.05594	0.527970	−	0.849263i	$-0.322953\pi$
$283$	17.7637	1.05594	0.527970	+	0.849263i	$0.322953\pi$
$284$	0	0
$285$	2.48202	0.147022
$286$	0	0
$287$	−19.5120	−1.15176
$288$	0	0
$289$	−16.6671	−0.980417
$290$	0	0
$291$	−1.82250	−0.106837
$292$	0	0
$293$	−30.0972	−1.75830	−0.879148	−	0.476548i	$-0.841888\pi$
$293$	−30.0972	−1.75830	−0.879148	+	0.476548i	$0.841888\pi$
$294$	0	0
$295$	6.39408	0.372278
$296$	0	0
$297$	14.1726	0.822378
$298$	0	0
$299$	5.50680	0.318466
$300$	0	0
$301$	−2.57449	−0.148391
$302$	0	0
$303$	9.11015	0.523365
$304$	0	0
$305$	−17.9151	−1.02581
$306$	0	0
$307$	6.35846	0.362897	0.181448	−	0.983400i	$-0.441921\pi$
$307$	6.35846	0.362897	0.181448	+	0.983400i	$0.441921\pi$
$308$	0	0
$309$	16.8508	0.958608
$310$	0	0
$311$	−2.36787	−0.134269	−0.0671347	−	0.997744i	$-0.521386\pi$
$311$	−2.36787	−0.134269	−0.0671347	+	0.997744i	$0.521386\pi$
$312$	0	0
$313$	1.55475	0.0878798	0.0439399	−	0.999034i	$-0.486009\pi$
$313$	1.55475	0.0878798	0.0439399	+	0.999034i	$0.486009\pi$
$314$	0	0
$315$	−10.4012	−0.586042
$316$	0	0
$317$	−24.9475	−1.40119	−0.700594	−	0.713560i	$-0.747082\pi$
$317$	−24.9475	−1.40119	−0.700594	+	0.713560i	$0.747082\pi$
$318$	0	0
$319$	−3.45168	−0.193257
$320$	0	0
$321$	14.6979	0.820354
$322$	0	0
$323$	0.332811	0.0185181
$324$	0	0
$325$	−35.0644	−1.94502
$326$	0	0
$327$	−3.94280	−0.218037
$328$	0	0
$329$	3.48259	0.192001
$330$	0	0
$331$	−29.7369	−1.63449	−0.817243	−	0.576293i	$-0.804499\pi$
$331$	−29.7369	−1.63449	−0.817243	+	0.576293i	$0.804499\pi$
$332$	0	0
$333$	1.81981	0.0997247
$334$	0	0
$335$	23.5335	1.28577
$336$	0	0
$337$	5.86544	0.319511	0.159755	−	0.987157i	$-0.448929\pi$
$337$	5.86544	0.319511	0.159755	+	0.987157i	$0.448929\pi$
$338$	0	0
$339$	1.37862	0.0748765
$340$	0	0
$341$	10.6676	0.577683
$342$	0	0
$343$	20.1426	1.08760
$344$	0	0
$345$	4.62183	0.248831
$346$	0	0
$347$	23.3776	1.25498	0.627489	−	0.778626i	$-0.284083\pi$
$347$	23.3776	1.25498	0.627489	+	0.778626i	$0.284083\pi$
$348$	0	0
$349$	25.7593	1.37886	0.689432	−	0.724351i	$-0.257860\pi$
$349$	25.7593	1.37886	0.689432	+	0.724351i	$0.257860\pi$
$350$	0	0
$351$	28.4419	1.51811
$352$	0	0
$353$	7.57866	0.403371	0.201686	−	0.979450i	$-0.435358\pi$
$353$	7.57866	0.403371	0.201686	+	0.979450i	$0.435358\pi$
$354$	0	0
$355$	18.4533	0.979402
$356$	0	0
$357$	1.51878	0.0803826
$358$	0	0
$359$	19.0305	1.00439	0.502196	−	0.864754i	$-0.332526\pi$
$359$	19.0305	1.00439	0.502196	+	0.864754i	$0.332526\pi$
$360$	0	0
$361$	−18.6673	−0.982488
$362$	0	0
$363$	−5.59421	−0.293620
$364$	0	0
$365$	−26.9722	−1.41179
$366$	0	0
$367$	−2.81009	−0.146686	−0.0733428	−	0.997307i	$-0.523367\pi$
$367$	−2.81009	−0.146686	−0.0733428	+	0.997307i	$0.523367\pi$
$368$	0	0
$369$	−13.3123	−0.693013
$370$	0	0
$371$	26.5199	1.37685
$372$	0	0
$373$	15.8745	0.821951	0.410975	−	0.911646i	$-0.365188\pi$
$373$	15.8745	0.821951	0.410975	+	0.911646i	$0.365188\pi$
$374$	0	0
$375$	−7.91455	−0.408705
$376$	0	0
$377$	−6.92689	−0.356753
$378$	0	0
$379$	32.7591	1.68272	0.841360	−	0.540475i	$-0.181755\pi$
$379$	32.7591	1.68272	0.841360	+	0.540475i	$0.181755\pi$
$380$	0	0
$381$	−2.19810	−0.112612
$382$	0	0
$383$	−7.03020	−0.359227	−0.179613	−	0.983737i	$-0.557485\pi$
$383$	−7.03020	−0.359227	−0.179613	+	0.983737i	$0.557485\pi$
$384$	0	0
$385$	−18.5030	−0.942998
$386$	0	0
$387$	−1.75648	−0.0892870
$388$	0	0
$389$	8.86936	0.449695	0.224847	−	0.974394i	$-0.427812\pi$
$389$	8.86936	0.449695	0.224847	+	0.974394i	$0.427812\pi$
$390$	0	0
$391$	0.619734	0.0313413
$392$	0	0
$393$	−14.3540	−0.724065
$394$	0	0
$395$	−31.1187	−1.56575
$396$	0	0
$397$	26.5878	1.33440	0.667201	−	0.744878i	$-0.267492\pi$
$397$	26.5878	1.33440	0.667201	+	0.744878i	$0.267492\pi$
$398$	0	0
$399$	1.51837	0.0760133
$400$	0	0
$401$	3.71998	0.185767	0.0928836	−	0.995677i	$-0.470392\pi$
$401$	3.71998	0.185767	0.0928836	+	0.995677i	$0.470392\pi$
$402$	0	0
$403$	21.4079	1.06641
$404$	0	0
$405$	9.04691	0.449544
$406$	0	0
$407$	3.23730	0.160467
$408$	0	0
$409$	−10.7313	−0.530627	−0.265314	−	0.964162i	$-0.585476\pi$
$409$	−10.7313	−0.530627	−0.265314	+	0.964162i	$0.585476\pi$
$410$	0	0
$411$	6.33590	0.312527
$412$	0	0
$413$	3.91154	0.192474
$414$	0	0
$415$	−3.44083	−0.168904
$416$	0	0
$417$	−27.7881	−1.36079
$418$	0	0
$419$	−20.6546	−1.00904	−0.504521	−	0.863399i	$-0.668331\pi$
$419$	−20.6546	−1.00904	−0.504521	+	0.863399i	$0.668331\pi$
$420$	0	0
$421$	6.42250	0.313013	0.156507	−	0.987677i	$-0.449977\pi$
$421$	6.42250	0.313013	0.156507	+	0.987677i	$0.449977\pi$
$422$	0	0
$423$	2.37605	0.115528
$424$	0	0
$425$	−3.94614	−0.191416
$426$	0	0
$427$	−10.9595	−0.530365
$428$	0	0
$429$	16.3795	0.790809
$430$	0	0
$431$	−19.6752	−0.947723	−0.473861	−	0.880599i	$-0.657140\pi$
$431$	−19.6752	−0.947723	−0.473861	+	0.880599i	$0.657140\pi$
$432$	0	0
$433$	−37.8843	−1.82060	−0.910302	−	0.413946i	$-0.864150\pi$
$433$	−37.8843	−1.82060	−0.910302	+	0.413946i	$0.864150\pi$
$434$	0	0
$435$	−5.81371	−0.278746
$436$	0	0
$437$	0.619563	0.0296377
$438$	0	0
$439$	30.5289	1.45707	0.728533	−	0.685011i	$-0.240203\pi$
$439$	30.5289	1.45707	0.728533	+	0.685011i	$0.240203\pi$
$440$	0	0
$441$	3.68986	0.175708
$442$	0	0
$443$	16.2024	0.769799	0.384900	−	0.922958i	$-0.374236\pi$
$443$	16.2024	0.769799	0.384900	+	0.922958i	$0.374236\pi$
$444$	0	0
$445$	1.70985	0.0810546
$446$	0	0
$447$	0.974558	0.0460950
$448$	0	0
$449$	−35.6017	−1.68015	−0.840075	−	0.542471i	$-0.817489\pi$
$449$	−35.6017	−1.68015	−0.840075	+	0.542471i	$0.817489\pi$
$450$	0	0
$451$	−23.6817	−1.11513
$452$	0	0
$453$	28.5489	1.34134
$454$	0	0
$455$	−37.1321	−1.74078
$456$	0	0
$457$	−26.1153	−1.22162	−0.610812	−	0.791776i	$-0.709157\pi$
$457$	−26.1153	−1.22162	−0.610812	+	0.791776i	$0.709157\pi$
$458$	0	0
$459$	3.20084	0.149403
$460$	0	0
$461$	35.5310	1.65484	0.827421	−	0.561582i	$-0.189807\pi$
$461$	35.5310	1.65484	0.827421	+	0.561582i	$0.189807\pi$
$462$	0	0
$463$	−39.4855	−1.83505	−0.917524	−	0.397680i	$-0.869816\pi$
$463$	−39.4855	−1.83505	−0.917524	+	0.397680i	$0.869816\pi$
$464$	0	0
$465$	17.9676	0.833226
$466$	0	0
$467$	−24.5883	−1.13781	−0.568905	−	0.822403i	$-0.692633\pi$
$467$	−24.5883	−1.13781	−0.568905	+	0.822403i	$0.692633\pi$
$468$	0	0
$469$	14.3965	0.664768
$470$	0	0
$471$	11.7243	0.540226
$472$	0	0
$473$	−3.12465	−0.143672
$474$	0	0
$475$	−3.94505	−0.181011
$476$	0	0
$477$	18.0936	0.828450
$478$	0	0
$479$	−31.0957	−1.42080	−0.710400	−	0.703798i	$-0.751486\pi$
$479$	−31.0957	−1.42080	−0.710400	+	0.703798i	$0.751486\pi$
$480$	0	0
$481$	6.49667	0.296223
$482$	0	0
$483$	2.82738	0.128650
$484$	0	0
$485$	5.01449	0.227696
$486$	0	0
$487$	0.831149	0.0376630	0.0188315	−	0.999823i	$-0.494005\pi$
$487$	0.831149	0.0376630	0.0188315	+	0.999823i	$0.494005\pi$
$488$	0	0
$489$	12.3142	0.556866
$490$	0	0
$491$	−15.7196	−0.709415	−0.354708	−	0.934977i	$-0.615420\pi$
$491$	−15.7196	−0.709415	−0.354708	+	0.934977i	$0.615420\pi$
$492$	0	0
$493$	−0.779551	−0.0351092
$494$	0	0
$495$	−12.6239	−0.567403
$496$	0	0
$497$	11.2887	0.506369
$498$	0	0
$499$	18.7317	0.838546	0.419273	−	0.907860i	$-0.362285\pi$
$499$	18.7317	0.838546	0.419273	+	0.907860i	$0.362285\pi$
$500$	0	0
$501$	8.38801	0.374749
$502$	0	0
$503$	−8.30619	−0.370355	−0.185177	−	0.982705i	$-0.559286\pi$
$503$	−8.30619	−0.370355	−0.185177	+	0.982705i	$0.559286\pi$
$504$	0	0
$505$	−25.0660	−1.11542
$506$	0	0
$507$	16.6134	0.737828
$508$	0	0
$509$	0.856595	0.0379679	0.0189839	−	0.999820i	$-0.493957\pi$
$509$	0.856595	0.0379679	0.0189839	+	0.999820i	$0.493957\pi$
$510$	0	0
$511$	−16.5001	−0.729920
$512$	0	0
$513$	3.19996	0.141282
$514$	0	0
$515$	−46.3639	−2.04304
$516$	0	0
$517$	4.22681	0.185895
$518$	0	0
$519$	−8.07514	−0.354460
$520$	0	0
$521$	5.97599	0.261813	0.130906	−	0.991395i	$-0.458211\pi$
$521$	5.97599	0.261813	0.130906	+	0.991395i	$0.458211\pi$
$522$	0	0
$523$	16.7781	0.733653	0.366827	−	0.930289i	$-0.380444\pi$
$523$	16.7781	0.733653	0.366827	+	0.930289i	$0.380444\pi$
$524$	0	0
$525$	−18.0032	−0.785726
$526$	0	0
$527$	2.40925	0.104948
$528$	0	0
$529$	−21.8463	−0.949839
$530$	0	0
$531$	2.66871	0.115812
$532$	0	0
$533$	−47.5248	−2.05853
$534$	0	0
$535$	−40.4402	−1.74838
$536$	0	0
$537$	−26.1829	−1.12987
$538$	0	0
$539$	6.56399	0.282731
$540$	0	0
$541$	4.18398	0.179883	0.0899417	−	0.995947i	$-0.471332\pi$
$541$	4.18398	0.179883	0.0899417	+	0.995947i	$0.471332\pi$
$542$	0	0
$543$	−25.6367	−1.10017
$544$	0	0
$545$	10.8484	0.464694
$546$	0	0
$547$	35.9666	1.53782	0.768910	−	0.639357i	$-0.220800\pi$
$547$	35.9666	1.53782	0.768910	+	0.639357i	$0.220800\pi$
$548$	0	0
$549$	−7.47725	−0.319121
$550$	0	0
$551$	−0.779336	−0.0332009
$552$	0	0
$553$	−19.0367	−0.809524
$554$	0	0
$555$	5.45262	0.231451
$556$	0	0
$557$	17.9697	0.761400	0.380700	−	0.924699i	$-0.375683\pi$
$557$	17.9697	0.761400	0.380700	+	0.924699i	$0.375683\pi$
$558$	0	0
$559$	−6.27061	−0.265219
$560$	0	0
$561$	1.84334	0.0778261
$562$	0	0
$563$	−26.6658	−1.12383	−0.561915	−	0.827195i	$-0.689935\pi$
$563$	−26.6658	−1.12383	−0.561915	+	0.827195i	$0.689935\pi$
$564$	0	0
$565$	−3.79320	−0.159581
$566$	0	0
$567$	5.53440	0.232423
$568$	0	0
$569$	−18.8144	−0.788740	−0.394370	−	0.918952i	$-0.629037\pi$
$569$	−18.8144	−0.788740	−0.394370	+	0.918952i	$0.629037\pi$
$570$	0	0
$571$	16.1897	0.677516	0.338758	−	0.940874i	$-0.389993\pi$
$571$	16.1897	0.677516	0.338758	+	0.940874i	$0.389993\pi$
$572$	0	0
$573$	−13.4591	−0.562263
$574$	0	0
$575$	−7.34615	−0.306356
$576$	0	0
$577$	27.5685	1.14769	0.573845	−	0.818964i	$-0.305451\pi$
$577$	27.5685	1.14769	0.573845	+	0.818964i	$0.305451\pi$
$578$	0	0
$579$	7.97072	0.331252
$580$	0	0
$581$	−2.10491	−0.0873264
$582$	0	0
$583$	32.1872	1.33306
$584$	0	0
$585$	−25.3339	−1.04743
$586$	0	0
$587$	16.9261	0.698616	0.349308	−	0.937008i	$-0.386417\pi$
$587$	16.9261	0.698616	0.349308	+	0.937008i	$0.386417\pi$
$588$	0	0
$589$	2.40858	0.0992439
$590$	0	0
$591$	16.4050	0.674812
$592$	0	0
$593$	3.39151	0.139273	0.0696364	−	0.997572i	$-0.477816\pi$
$593$	3.39151	0.139273	0.0696364	+	0.997572i	$0.477816\pi$
$594$	0	0
$595$	−4.17884	−0.171316
$596$	0	0
$597$	34.1953	1.39952
$598$	0	0
$599$	12.7915	0.522645	0.261322	−	0.965252i	$-0.415841\pi$
$599$	12.7915	0.522645	0.261322	+	0.965252i	$0.415841\pi$
$600$	0	0
$601$	−45.9371	−1.87381	−0.936907	−	0.349580i	$-0.886324\pi$
$601$	−45.9371	−1.87381	−0.936907	+	0.349580i	$0.886324\pi$
$602$	0	0
$603$	9.82221	0.399991
$604$	0	0
$605$	15.3921	0.625780
$606$	0	0
$607$	−6.12491	−0.248603	−0.124301	−	0.992245i	$-0.539669\pi$
$607$	−6.12491	−0.248603	−0.124301	+	0.992245i	$0.539669\pi$
$608$	0	0
$609$	−3.55650	−0.144117
$610$	0	0
$611$	8.48245	0.343163
$612$	0	0
$613$	−7.77951	−0.314211	−0.157106	−	0.987582i	$-0.550216\pi$
$613$	−7.77951	−0.314211	−0.157106	+	0.987582i	$0.550216\pi$
$614$	0	0
$615$	−39.8873	−1.60841
$616$	0	0
$617$	6.20863	0.249950	0.124975	−	0.992160i	$-0.460115\pi$
$617$	6.20863	0.249950	0.124975	+	0.992160i	$0.460115\pi$
$618$	0	0
$619$	−12.8999	−0.518491	−0.259246	−	0.965811i	$-0.583474\pi$
$619$	−12.8999	−0.518491	−0.259246	+	0.965811i	$0.583474\pi$
$620$	0	0
$621$	5.95871	0.239115
$622$	0	0
$623$	1.04599	0.0419067
$624$	0	0
$625$	−12.4202	−0.496810
$626$	0	0
$627$	1.84284	0.0735958
$628$	0	0
$629$	0.731134	0.0291522
$630$	0	0
$631$	12.2019	0.485750	0.242875	−	0.970058i	$-0.421910\pi$
$631$	12.2019	0.485750	0.242875	+	0.970058i	$0.421910\pi$
$632$	0	0
$633$	14.3342	0.569732
$634$	0	0
$635$	6.04794	0.240005
$636$	0	0
$637$	13.1727	0.521923
$638$	0	0
$639$	7.70191	0.304683
$640$	0	0
$641$	4.20860	0.166230	0.0831149	−	0.996540i	$-0.473513\pi$
$641$	4.20860	0.166230	0.0831149	+	0.996540i	$0.473513\pi$
$642$	0	0
$643$	−7.35145	−0.289913	−0.144957	−	0.989438i	$-0.546304\pi$
$643$	−7.35145	−0.289913	−0.144957	+	0.989438i	$0.546304\pi$
$644$	0	0
$645$	−5.26289	−0.207226
$646$	0	0
$647$	−31.6693	−1.24505	−0.622525	−	0.782600i	$-0.713893\pi$
$647$	−31.6693	−1.24505	−0.622525	+	0.782600i	$0.713893\pi$
$648$	0	0
$649$	4.74743	0.186353
$650$	0	0
$651$	10.9916	0.430793
$652$	0	0
$653$	4.40636	0.172434	0.0862170	−	0.996276i	$-0.472522\pi$
$653$	4.40636	0.172434	0.0862170	+	0.996276i	$0.472522\pi$
$654$	0	0
$655$	39.4942	1.54317
$656$	0	0
$657$	−11.2574	−0.439194
$658$	0	0
$659$	−31.6271	−1.23202	−0.616008	−	0.787740i	$-0.711251\pi$
$659$	−31.6271	−1.23202	−0.616008	+	0.787740i	$0.711251\pi$
$660$	0	0
$661$	−15.7316	−0.611889	−0.305945	−	0.952049i	$-0.598972\pi$
$661$	−15.7316	−0.611889	−0.305945	+	0.952049i	$0.598972\pi$
$662$	0	0
$663$	3.69926	0.143667
$664$	0	0
$665$	−4.17769	−0.162004
$666$	0	0
$667$	−1.45122	−0.0561914
$668$	0	0
$669$	−5.75078	−0.222338
$670$	0	0
$671$	−13.3015	−0.513498
$672$	0	0
$673$	20.7368	0.799343	0.399672	−	0.916658i	$-0.369124\pi$
$673$	20.7368	0.799343	0.399672	+	0.916658i	$0.369124\pi$
$674$	0	0
$675$	−37.9419	−1.46038
$676$	0	0
$677$	−27.6212	−1.06157	−0.530785	−	0.847506i	$-0.678103\pi$
$677$	−27.6212	−1.06157	−0.530785	+	0.847506i	$0.678103\pi$
$678$	0	0
$679$	3.06758	0.117723
$680$	0	0
$681$	31.5442	1.20878
$682$	0	0
$683$	−3.53973	−0.135444	−0.0677220	−	0.997704i	$-0.521573\pi$
$683$	−3.53973	−0.135444	−0.0677220	+	0.997704i	$0.521573\pi$
$684$	0	0
$685$	−17.4328	−0.666074
$686$	0	0
$687$	−25.4077	−0.969363
$688$	0	0
$689$	64.5939	2.46083
$690$	0	0
$691$	−0.846810	−0.0322142	−0.0161071	−	0.999870i	$-0.505127\pi$
$691$	−0.846810	−0.0322142	−0.0161071	+	0.999870i	$0.505127\pi$
$692$	0	0
$693$	−7.72262	−0.293358
$694$	0	0
$695$	76.4573	2.90019
$696$	0	0
$697$	−5.34843	−0.202586
$698$	0	0
$699$	−24.2267	−0.916338
$700$	0	0
$701$	36.4860	1.37806	0.689028	−	0.724735i	$-0.258038\pi$
$701$	36.4860	1.37806	0.689028	+	0.724735i	$0.258038\pi$
$702$	0	0
$703$	0.730933	0.0275676
$704$	0	0
$705$	7.11928	0.268128
$706$	0	0
$707$	−15.3340	−0.576695
$708$	0	0
$709$	32.4995	1.22054	0.610271	−	0.792192i	$-0.291060\pi$
$709$	32.4995	1.22054	0.610271	+	0.792192i	$0.291060\pi$
$710$	0	0
$711$	−12.9881	−0.487091
$712$	0	0
$713$	4.48507	0.167967
$714$	0	0
$715$	−45.0672	−1.68542
$716$	0	0
$717$	3.82407	0.142813
$718$	0	0
$719$	20.4516	0.762715	0.381357	−	0.924428i	$-0.375457\pi$
$719$	20.4516	0.762715	0.381357	+	0.924428i	$0.375457\pi$
$720$	0	0
$721$	−28.3629	−1.05629
$722$	0	0
$723$	1.56735	0.0582904
$724$	0	0
$725$	9.24058	0.343187
$726$	0	0
$727$	−37.4067	−1.38734	−0.693669	−	0.720294i	$-0.744007\pi$
$727$	−37.4067	−1.38734	−0.693669	+	0.720294i	$0.744007\pi$
$728$	0	0
$729$	24.5887	0.910693
$730$	0	0
$731$	−0.705693	−0.0261010
$732$	0	0
$733$	31.8719	1.17722	0.588608	−	0.808419i	$-0.299676\pi$
$733$	31.8719	1.17722	0.588608	+	0.808419i	$0.299676\pi$
$734$	0	0
$735$	11.0558	0.407800
$736$	0	0
$737$	17.4730	0.643625
$738$	0	0
$739$	−25.7977	−0.948983	−0.474491	−	0.880260i	$-0.657368\pi$
$739$	−25.7977	−0.948983	−0.474491	+	0.880260i	$0.657368\pi$
$740$	0	0
$741$	3.69824	0.135858
$742$	0	0
$743$	−38.8580	−1.42556	−0.712781	−	0.701387i	$-0.752564\pi$
$743$	−38.8580	−1.42556	−0.712781	+	0.701387i	$0.752564\pi$
$744$	0	0
$745$	−2.68144	−0.0982402
$746$	0	0
$747$	−1.43611	−0.0525444
$748$	0	0
$749$	−24.7391	−0.903947
$750$	0	0
$751$	32.3858	1.18178	0.590888	−	0.806754i	$-0.298778\pi$
$751$	32.3858	1.18178	0.590888	+	0.806754i	$0.298778\pi$
$752$	0	0
$753$	−16.0695	−0.585605
$754$	0	0
$755$	−78.5505	−2.85874
$756$	0	0
$757$	41.1768	1.49660	0.748298	−	0.663363i	$-0.230872\pi$
$757$	41.1768	1.49660	0.748298	+	0.663363i	$0.230872\pi$
$758$	0	0
$759$	3.43158	0.124559
$760$	0	0
$761$	40.6963	1.47524	0.737620	−	0.675216i	$-0.235950\pi$
$761$	40.6963	1.47524	0.737620	+	0.675216i	$0.235950\pi$
$762$	0	0
$763$	6.63644	0.240255
$764$	0	0
$765$	−2.85108	−0.103081
$766$	0	0
$767$	9.52724	0.344009
$768$	0	0
$769$	33.6235	1.21249	0.606247	−	0.795276i	$-0.292674\pi$
$769$	33.6235	1.21249	0.606247	+	0.795276i	$0.292674\pi$
$770$	0	0
$771$	−15.9838	−0.575641
$772$	0	0
$773$	6.10858	0.219710	0.109855	−	0.993948i	$-0.464961\pi$
$773$	6.10858	0.219710	0.109855	+	0.993948i	$0.464961\pi$
$774$	0	0
$775$	−28.5585	−1.02585
$776$	0	0
$777$	3.33561	0.119664
$778$	0	0
$779$	−5.34696	−0.191575
$780$	0	0
$781$	13.7011	0.490264
$782$	0	0
$783$	−7.49534	−0.267862
$784$	0	0
$785$	−32.2586	−1.15136
$786$	0	0
$787$	−4.68133	−0.166871	−0.0834357	−	0.996513i	$-0.526589\pi$
$787$	−4.68133	−0.166871	−0.0834357	+	0.996513i	$0.526589\pi$
$788$	0	0
$789$	9.81494	0.349421
$790$	0	0
$791$	−2.32047	−0.0825063
$792$	0	0
$793$	−26.6936	−0.947919
$794$	0	0
$795$	54.2133	1.92275
$796$	0	0
$797$	25.4832	0.902661	0.451330	−	0.892357i	$-0.350950\pi$
$797$	25.4832	0.902661	0.451330	+	0.892357i	$0.350950\pi$
$798$	0	0
$799$	0.954614	0.0337718
$800$	0	0
$801$	0.713642	0.0252153
$802$	0	0
$803$	−20.0261	−0.706706
$804$	0	0
$805$	−7.77936	−0.274186
$806$	0	0
$807$	−22.8901	−0.805771
$808$	0	0
$809$	−2.64579	−0.0930209	−0.0465105	−	0.998918i	$-0.514810\pi$
$809$	−2.64579	−0.0930209	−0.0465105	+	0.998918i	$0.514810\pi$
$810$	0	0
$811$	33.9428	1.19189	0.595946	−	0.803024i	$-0.296777\pi$
$811$	33.9428	1.19189	0.595946	+	0.803024i	$0.296777\pi$
$812$	0	0
$813$	12.4888	0.438003
$814$	0	0
$815$	−33.8817	−1.18682
$816$	0	0
$817$	−0.705499	−0.0246823
$818$	0	0
$819$	−15.4979	−0.541540
$820$	0	0
$821$	−28.4095	−0.991498	−0.495749	−	0.868466i	$-0.665106\pi$
$821$	−28.4095	−0.991498	−0.495749	+	0.868466i	$0.665106\pi$
$822$	0	0
$823$	43.0706	1.50135	0.750673	−	0.660674i	$-0.229729\pi$
$823$	43.0706	1.50135	0.750673	+	0.660674i	$0.229729\pi$
$824$	0	0
$825$	−21.8505	−0.760736
$826$	0	0
$827$	−15.8528	−0.551255	−0.275628	−	0.961265i	$-0.588886\pi$
$827$	−15.8528	−0.551255	−0.275628	+	0.961265i	$0.588886\pi$
$828$	0	0
$829$	−1.14822	−0.0398793	−0.0199397	−	0.999801i	$-0.506347\pi$
$829$	−1.14822	−0.0398793	−0.0199397	+	0.999801i	$0.506347\pi$
$830$	0	0
$831$	−16.8099	−0.583128
$832$	0	0
$833$	1.48246	0.0513641
$834$	0	0
$835$	−23.0791	−0.798685
$836$	0	0
$837$	23.1648	0.800691
$838$	0	0
$839$	−40.2665	−1.39015	−0.695077	−	0.718935i	$-0.744630\pi$
$839$	−40.2665	−1.39015	−0.695077	+	0.718935i	$0.744630\pi$
$840$	0	0
$841$	−27.1745	−0.937053
$842$	0	0
$843$	−5.69883	−0.196278
$844$	0	0
$845$	−45.7108	−1.57250
$846$	0	0
$847$	9.41606	0.323540
$848$	0	0
$849$	22.2145	0.762399
$850$	0	0
$851$	1.36108	0.0466574
$852$	0	0
$853$	5.06325	0.173362	0.0866811	−	0.996236i	$-0.472374\pi$
$853$	5.06325	0.173362	0.0866811	+	0.996236i	$0.472374\pi$
$854$	0	0
$855$	−2.85029	−0.0974779
$856$	0	0
$857$	26.8280	0.916426	0.458213	−	0.888842i	$-0.348490\pi$
$857$	26.8280	0.916426	0.458213	+	0.888842i	$0.348490\pi$
$858$	0	0
$859$	32.9872	1.12551	0.562755	−	0.826624i	$-0.309742\pi$
$859$	32.9872	1.12551	0.562755	+	0.826624i	$0.309742\pi$
$860$	0	0
$861$	−24.4009	−0.831579
$862$	0	0
$863$	48.5124	1.65138	0.825690	−	0.564124i	$-0.190786\pi$
$863$	48.5124	1.65138	0.825690	+	0.564124i	$0.190786\pi$
$864$	0	0
$865$	22.2183	0.755444
$866$	0	0
$867$	−20.8432	−0.707871
$868$	0	0
$869$	−23.1048	−0.783778
$870$	0	0
$871$	35.0651	1.18814
$872$	0	0
$873$	2.09291	0.0708341
$874$	0	0
$875$	13.3216	0.450352
$876$	0	0
$877$	25.7223	0.868579	0.434290	−	0.900773i	$-0.356999\pi$
$877$	25.7223	0.868579	0.434290	+	0.900773i	$0.356999\pi$
$878$	0	0
$879$	−37.6383	−1.26951
$880$	0	0
$881$	−44.5073	−1.49949	−0.749744	−	0.661728i	$-0.769824\pi$
$881$	−44.5073	−1.49949	−0.749744	+	0.661728i	$0.769824\pi$
$882$	0	0
$883$	45.5375	1.53246	0.766229	−	0.642568i	$-0.222131\pi$
$883$	45.5375	1.53246	0.766229	+	0.642568i	$0.222131\pi$
$884$	0	0
$885$	7.99616	0.268788
$886$	0	0
$887$	22.0334	0.739809	0.369904	−	0.929070i	$-0.379390\pi$
$887$	22.0334	0.739809	0.369904	+	0.929070i	$0.379390\pi$
$888$	0	0
$889$	3.69979	0.124087
$890$	0	0
$891$	6.71709	0.225031
$892$	0	0
$893$	0.954351	0.0319361
$894$	0	0
$895$	72.0406	2.40805
$896$	0	0
$897$	6.88657	0.229936
$898$	0	0
$899$	−5.64168	−0.188161
$900$	0	0
$901$	7.26939	0.242178
$902$	0	0
$903$	−3.21954	−0.107140
$904$	0	0
$905$	70.5377	2.34475
$906$	0	0
$907$	43.7495	1.45268	0.726339	−	0.687337i	$-0.241220\pi$
$907$	43.7495	1.45268	0.726339	+	0.687337i	$0.241220\pi$
$908$	0	0
$909$	−10.4619	−0.346998
$910$	0	0
$911$	27.4848	0.910613	0.455307	−	0.890335i	$-0.349530\pi$
$911$	27.4848	0.910613	0.455307	+	0.890335i	$0.349530\pi$
$912$	0	0
$913$	−2.55473	−0.0845491
$914$	0	0
$915$	−22.4038	−0.740648
$916$	0	0
$917$	24.1604	0.797846
$918$	0	0
$919$	22.5402	0.743532	0.371766	−	0.928326i	$-0.378752\pi$
$919$	22.5402	0.743532	0.371766	+	0.928326i	$0.378752\pi$
$920$	0	0
$921$	7.95162	0.262015
$922$	0	0
$923$	27.4957	0.905031
$924$	0	0
$925$	−8.66666	−0.284958
$926$	0	0
$927$	−19.3510	−0.635570
$928$	0	0
$929$	56.4210	1.85111	0.925557	−	0.378609i	$-0.123597\pi$
$929$	56.4210	1.85111	0.925557	+	0.378609i	$0.123597\pi$
$930$	0	0
$931$	1.48205	0.0485722
$932$	0	0
$933$	−2.96115	−0.0969438
$934$	0	0
$935$	−5.07185	−0.165867
$936$	0	0
$937$	−20.7452	−0.677715	−0.338858	−	0.940838i	$-0.610040\pi$
$937$	−20.7452	−0.677715	−0.338858	+	0.940838i	$0.610040\pi$
$938$	0	0
$939$	1.94431	0.0634501
$940$	0	0
$941$	−26.9773	−0.879435	−0.439717	−	0.898136i	$-0.644921\pi$
$941$	−26.9773	−0.879435	−0.439717	+	0.898136i	$0.644921\pi$
$942$	0	0
$943$	−9.95668	−0.324234
$944$	0	0
$945$	−40.1793	−1.30703
$946$	0	0
$947$	20.9484	0.680731	0.340366	−	0.940293i	$-0.389449\pi$
$947$	20.9484	0.680731	0.340366	+	0.940293i	$0.389449\pi$
$948$	0	0
$949$	−40.1888	−1.30458
$950$	0	0
$951$	−31.1982	−1.01167
$952$	0	0
$953$	38.0330	1.23201	0.616005	−	0.787742i	$-0.288750\pi$
$953$	38.0330	1.23201	0.616005	+	0.787742i	$0.288750\pi$
$954$	0	0
$955$	37.0319	1.19833
$956$	0	0
$957$	−4.31652	−0.139533
$958$	0	0
$959$	−10.6644	−0.344373
$960$	0	0
$961$	−13.5641	−0.437551
$962$	0	0
$963$	−16.8786	−0.543906
$964$	0	0
$965$	−21.9310	−0.705982
$966$	0	0
$967$	39.6586	1.27533	0.637667	−	0.770312i	$-0.279900\pi$
$967$	39.6586	1.27533	0.637667	+	0.770312i	$0.279900\pi$
$968$	0	0
$969$	0.416199	0.0133702
$970$	0	0
$971$	15.3844	0.493710	0.246855	−	0.969052i	$-0.420603\pi$
$971$	15.3844	0.493710	0.246855	+	0.969052i	$0.420603\pi$
$972$	0	0
$973$	46.7723	1.49945
$974$	0	0
$975$	−43.8500	−1.40432
$976$	0	0
$977$	21.4691	0.686857	0.343428	−	0.939179i	$-0.388412\pi$
$977$	21.4691	0.686857	0.343428	+	0.939179i	$0.388412\pi$
$978$	0	0
$979$	1.26952	0.0405739
$980$	0	0
$981$	4.52781	0.144562
$982$	0	0
$983$	−41.0747	−1.31008	−0.655039	−	0.755595i	$-0.727348\pi$
$983$	−41.0747	−1.31008	−0.655039	+	0.755595i	$0.727348\pi$
$984$	0	0
$985$	−45.1374	−1.43820
$986$	0	0
$987$	4.35518	0.138627
$988$	0	0
$989$	−1.31372	−0.0417740
$990$	0	0
$991$	4.98997	0.158512	0.0792559	−	0.996854i	$-0.474746\pi$
$991$	4.98997	0.158512	0.0792559	+	0.996854i	$0.474746\pi$
$992$	0	0
$993$	−37.1877	−1.18011
$994$	0	0
$995$	−94.0862	−2.98273
$996$	0	0
$997$	−27.2625	−0.863411	−0.431705	−	0.902015i	$-0.642088\pi$
$997$	−27.2625	−0.863411	−0.431705	+	0.902015i	$0.642088\pi$
$998$	0	0
$999$	7.02981	0.222413

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5312.2.a.bv.1.8		11
4.3	odd	2		5312.2.a.bu.1.4		11
8.3	odd	2		2656.2.a.t.1.8	yes	11
8.5	even	2		2656.2.a.s.1.4	✓	11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2656.2.a.s.1.4	✓	11	8.5	even	2
2656.2.a.t.1.8	yes	11	8.3	odd	2
5312.2.a.bu.1.4		11	4.3	odd	2
5312.2.a.bv.1.8		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:	\( N \)	\(=\)	\( 5312 = 2^{6} \cdot 83 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5312.a (trivial)

\(f(q)\)	\(=\)	\(q+1.25056 q^{3} -3.44083 q^{5} -2.10491 q^{7} -1.43611 q^{9} +O(q^{10})\)	\(q+1.25056 q^{3} -3.44083 q^{5} -2.10491 q^{7} -1.43611 q^{9} -2.55473 q^{11} -5.12687 q^{13} -4.30296 q^{15} -0.576977 q^{17} -0.576818 q^{19} -2.63231 q^{21} -1.07410 q^{23} +6.83933 q^{25} -5.54761 q^{27} +1.35110 q^{29} -4.17563 q^{31} -3.19483 q^{33} +7.24264 q^{35} -1.26718 q^{37} -6.41145 q^{39} +9.26975 q^{41} +1.22309 q^{43} +4.94140 q^{45} -1.65451 q^{47} -2.56935 q^{49} -0.721543 q^{51} -12.5991 q^{53} +8.79038 q^{55} -0.721344 q^{57} -1.85829 q^{59} +5.20661 q^{61} +3.02288 q^{63} +17.6407 q^{65} -6.83947 q^{67} -1.34323 q^{69} -5.36305 q^{71} +7.83885 q^{73} +8.55297 q^{75} +5.37747 q^{77} +9.04396 q^{79} -2.62928 q^{81} +1.00000 q^{83} +1.98528 q^{85} +1.68962 q^{87} -0.496929 q^{89} +10.7916 q^{91} -5.22187 q^{93} +1.98473 q^{95} -1.45735 q^{97} +3.66886 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q + q^{3} - 5 q^{5} + 10 q^{7} + 18 q^{9}+O(q^{10}) \) 11 * q + q^3 - 5 * q^5 + 10 * q^7 + 18 * q^9	\( 11 q + q^{3} - 5 q^{5} + 10 q^{7} + 18 q^{9} - 3 q^{11} + q^{13} + 18 q^{15} + 6 q^{17} + 15 q^{19} - 16 q^{21} - 6 q^{23} + 28 q^{25} - 2 q^{27} - 31 q^{29} + 14 q^{31} - 8 q^{33} + 10 q^{35} + q^{37} + 18 q^{39} + 22 q^{41} + 37 q^{43} - 3 q^{45} - 2 q^{47} + 45 q^{49} + 24 q^{51} - 29 q^{53} + 14 q^{55} + 36 q^{57} - q^{59} - 15 q^{61} + 18 q^{63} + 14 q^{65} + 45 q^{67} - 34 q^{69} - 4 q^{73} + 17 q^{75} - 34 q^{77} + 10 q^{79} + 39 q^{81} + 11 q^{83} - 25 q^{85} - 13 q^{87} + 15 q^{91} - 2 q^{93} + 4 q^{95} - 18 q^{97} - 26 q^{99}+O(q^{100}) \) 11 * q + q^3 - 5 * q^5 + 10 * q^7 + 18 * q^9 - 3 * q^11 + q^13 + 18 * q^15 + 6 * q^17 + 15 * q^19 - 16 * q^21 - 6 * q^23 + 28 * q^25 - 2 * q^27 - 31 * q^29 + 14 * q^31 - 8 * q^33 + 10 * q^35 + q^37 + 18 * q^39 + 22 * q^41 + 37 * q^43 - 3 * q^45 - 2 * q^47 + 45 * q^49 + 24 * q^51 - 29 * q^53 + 14 * q^55 + 36 * q^57 - q^59 - 15 * q^61 + 18 * q^63 + 14 * q^65 + 45 * q^67 - 34 * q^69 - 4 * q^73 + 17 * q^75 - 34 * q^77 + 10 * q^79 + 39 * q^81 + 11 * q^83 - 25 * q^85 - 13 * q^87 + 15 * q^91 - 2 * q^93 + 4 * q^95 - 18 * q^97 - 26 * q^99

Introduction

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