LMFDB - Embedded newform 6348.2.a.t.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6348,2,Mod(1,6348)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6348, 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("6348.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6348 = 2^{2} \cdot 3 \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6348.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:	$50.6890352031$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 4x^{9} - 14x^{8} + 65x^{7} + 57x^{6} - 354x^{5} - 46x^{4} + 714x^{3} - 74x^{2} - 323x - 23 $ x^10 - 4x^9 - 14x^8 + 65x^7 + 57x^6 - 354x^5 - 46x^4 + 714x^3 - 74x^2 - 323*x - 23
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 276)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$1.00266$ of defining polynomial
Character	$\chi$	$=$	6348.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +2.92409 q^{5} +1.00076 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +2.92409 q^{5} +1.00076 q^{7} +1.00000 q^{9} -2.52047 q^{11} -0.482825 q^{13} +2.92409 q^{15} +3.13735 q^{17} -0.552973 q^{19} +1.00076 q^{21} +3.55030 q^{25} +1.00000 q^{27} +0.121648 q^{29} +8.23476 q^{31} -2.52047 q^{33} +2.92630 q^{35} +9.53165 q^{37} -0.482825 q^{39} -1.39208 q^{41} -6.20411 q^{43} +2.92409 q^{45} +8.69831 q^{47} -5.99849 q^{49} +3.13735 q^{51} -7.72669 q^{53} -7.37009 q^{55} -0.552973 q^{57} -0.236931 q^{59} +1.83181 q^{61} +1.00076 q^{63} -1.41182 q^{65} +2.79746 q^{67} +11.4788 q^{71} -13.6919 q^{73} +3.55030 q^{75} -2.52238 q^{77} +17.0899 q^{79} +1.00000 q^{81} +8.06382 q^{83} +9.17389 q^{85} +0.121648 q^{87} -4.93850 q^{89} -0.483191 q^{91} +8.23476 q^{93} -1.61694 q^{95} +15.6281 q^{97} -2.52047 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 10 q^{3} + 2 q^{5} + 11 q^{7} + 10 q^{9}+O(q^{10}) $ 10 * q + 10 * q^3 + 2 * q^5 + 11 * q^7 + 10 * q^9	$ 10 q + 10 q^{3} + 2 q^{5} + 11 q^{7} + 10 q^{9} + 11 q^{11} + 2 q^{15} + 13 q^{17} + 18 q^{19} + 11 q^{21} + 10 q^{25} + 10 q^{27} + 5 q^{29} + 15 q^{31} + 11 q^{33} - 13 q^{35} + 5 q^{37} + 24 q^{41} + 40 q^{43} + 2 q^{45} - 9 q^{47} + 5 q^{49} + 13 q^{51} + 6 q^{53} - 14 q^{55} + 18 q^{57} + 28 q^{59} + 39 q^{61} + 11 q^{63} + 14 q^{65} + 32 q^{67} - 33 q^{71} - 50 q^{73} + 10 q^{75} + 19 q^{77} + 33 q^{79} + 10 q^{81} + 29 q^{83} - 21 q^{85} + 5 q^{87} + 17 q^{89} + 36 q^{91} + 15 q^{93} - 42 q^{95} + 46 q^{97} + 11 q^{99}+O(q^{100}) $ 10 * q + 10 * q^3 + 2 * q^5 + 11 * q^7 + 10 * q^9 + 11 * q^11 + 2 * q^15 + 13 * q^17 + 18 * q^19 + 11 * q^21 + 10 * q^25 + 10 * q^27 + 5 * q^29 + 15 * q^31 + 11 * q^33 - 13 * q^35 + 5 * q^37 + 24 * q^41 + 40 * q^43 + 2 * q^45 - 9 * q^47 + 5 * q^49 + 13 * q^51 + 6 * q^53 - 14 * q^55 + 18 * q^57 + 28 * q^59 + 39 * q^61 + 11 * q^63 + 14 * q^65 + 32 * q^67 - 33 * q^71 - 50 * q^73 + 10 * q^75 + 19 * q^77 + 33 * q^79 + 10 * q^81 + 29 * q^83 - 21 * q^85 + 5 * q^87 + 17 * q^89 + 36 * q^91 + 15 * q^93 - 42 * q^95 + 46 * q^97 + 11 * 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	2.92409	1.30769	0.653846	−	0.756627i	$-0.273154\pi$
$5$	2.92409	1.30769	0.653846	+	0.756627i	$0.273154\pi$
$6$	0	0
$7$	1.00076	0.378251	0.189125	−	0.981953i	$-0.439435\pi$
$7$	1.00076	0.378251	0.189125	+	0.981953i	$0.439435\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.52047	−0.759951	−0.379976	−	0.924997i	$-0.624068\pi$
$11$	−2.52047	−0.759951	−0.379976	+	0.924997i	$0.624068\pi$
$12$	0	0
$13$	−0.482825	−0.133912	−0.0669558	−	0.997756i	$-0.521329\pi$
$13$	−0.482825	−0.133912	−0.0669558	+	0.997756i	$0.521329\pi$
$14$	0	0
$15$	2.92409	0.754997
$16$	0	0
$17$	3.13735	0.760919	0.380460	−	0.924798i	$-0.375766\pi$
$17$	3.13735	0.760919	0.380460	+	0.924798i	$0.375766\pi$
$18$	0	0
$19$	−0.552973	−0.126861	−0.0634304	−	0.997986i	$-0.520204\pi$
$19$	−0.552973	−0.126861	−0.0634304	+	0.997986i	$0.520204\pi$
$20$	0	0
$21$	1.00076	0.218383
$22$	0	0
$23$	0	0
$23$	0	0
$24$	0	0
$25$	3.55030	0.710061
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	0.121648	0.0225895	0.0112948	−	0.999936i	$-0.496405\pi$
$29$	0.121648	0.0225895	0.0112948	+	0.999936i	$0.496405\pi$
$30$	0	0
$31$	8.23476	1.47901	0.739503	−	0.673153i	$-0.235061\pi$
$31$	8.23476	1.47901	0.739503	+	0.673153i	$0.235061\pi$
$32$	0	0
$33$	−2.52047	−0.438758
$34$	0	0
$35$	2.92630	0.494636
$36$	0	0
$37$	9.53165	1.56699	0.783496	−	0.621396i	$-0.213434\pi$
$37$	9.53165	1.56699	0.783496	+	0.621396i	$0.213434\pi$
$38$	0	0
$39$	−0.482825	−0.0773139
$40$	0	0
$41$	−1.39208	−0.217406	−0.108703	−	0.994074i	$-0.534670\pi$
$41$	−1.39208	−0.217406	−0.108703	+	0.994074i	$0.534670\pi$
$42$	0	0
$43$	−6.20411	−0.946117	−0.473059	−	0.881031i	$-0.656850\pi$
$43$	−6.20411	−0.946117	−0.473059	+	0.881031i	$0.656850\pi$
$44$	0	0
$45$	2.92409	0.435898
$46$	0	0
$47$	8.69831	1.26878	0.634389	−	0.773014i	$-0.281252\pi$
$47$	8.69831	1.26878	0.634389	+	0.773014i	$0.281252\pi$
$48$	0	0
$49$	−5.99849	−0.856926
$50$	0	0
$51$	3.13735	0.439317
$52$	0	0
$53$	−7.72669	−1.06134	−0.530671	−	0.847578i	$-0.678060\pi$
$53$	−7.72669	−1.06134	−0.530671	+	0.847578i	$0.678060\pi$
$54$	0	0
$55$	−7.37009	−0.993783
$56$	0	0
$57$	−0.552973	−0.0732431
$58$	0	0
$59$	−0.236931	−0.0308458	−0.0154229	−	0.999881i	$-0.504909\pi$
$59$	−0.236931	−0.0308458	−0.0154229	+	0.999881i	$0.504909\pi$
$60$	0	0
$61$	1.83181	0.234540	0.117270	−	0.993100i	$-0.462586\pi$
$61$	1.83181	0.234540	0.117270	+	0.993100i	$0.462586\pi$
$62$	0	0
$63$	1.00076	0.126084
$64$	0	0
$65$	−1.41182	−0.175115
$66$	0	0
$67$	2.79746	0.341765	0.170882	−	0.985291i	$-0.445338\pi$
$67$	2.79746	0.341765	0.170882	+	0.985291i	$0.445338\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	11.4788	1.36228	0.681139	−	0.732154i	$-0.261485\pi$
$71$	11.4788	1.36228	0.681139	+	0.732154i	$0.261485\pi$
$72$	0	0
$73$	−13.6919	−1.60251	−0.801257	−	0.598321i	$-0.795835\pi$
$73$	−13.6919	−1.60251	−0.801257	+	0.598321i	$0.795835\pi$
$74$	0	0
$75$	3.55030	0.409954
$76$	0	0
$77$	−2.52238	−0.287452
$78$	0	0
$79$	17.0899	1.92277	0.961383	−	0.275214i	$-0.0887488\pi$
$79$	17.0899	1.92277	0.961383	+	0.275214i	$0.0887488\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	8.06382	0.885120	0.442560	−	0.896739i	$-0.354070\pi$
$83$	8.06382	0.885120	0.442560	+	0.896739i	$0.354070\pi$
$84$	0	0
$85$	9.17389	0.995049
$86$	0	0
$87$	0.121648	0.0130421
$88$	0	0
$89$	−4.93850	−0.523480	−0.261740	−	0.965138i	$-0.584296\pi$
$89$	−4.93850	−0.523480	−0.261740	+	0.965138i	$0.584296\pi$
$90$	0	0
$91$	−0.483191	−0.0506522
$92$	0	0
$93$	8.23476	0.853904
$94$	0	0
$95$	−1.61694	−0.165895
$96$	0	0
$97$	15.6281	1.58679	0.793396	−	0.608706i	$-0.208311\pi$
$97$	15.6281	1.58679	0.793396	+	0.608706i	$0.208311\pi$
$98$	0	0
$99$	−2.52047	−0.253317
$100$	0	0
$101$	−3.25218	−0.323604	−0.161802	−	0.986823i	$-0.551731\pi$
$101$	−3.25218	−0.323604	−0.161802	+	0.986823i	$0.551731\pi$
$102$	0	0
$103$	1.40182	0.138125	0.0690627	−	0.997612i	$-0.477999\pi$
$103$	1.40182	0.138125	0.0690627	+	0.997612i	$0.477999\pi$
$104$	0	0
$105$	2.92630	0.285578
$106$	0	0
$107$	10.0443	0.971016	0.485508	−	0.874232i	$-0.338635\pi$
$107$	10.0443	0.971016	0.485508	+	0.874232i	$0.338635\pi$
$108$	0	0
$109$	6.75809	0.647308	0.323654	−	0.946176i	$-0.395089\pi$
$109$	6.75809	0.647308	0.323654	+	0.946176i	$0.395089\pi$
$110$	0	0
$111$	9.53165	0.904704
$112$	0	0
$113$	10.0242	0.943000	0.471500	−	0.881866i	$-0.343713\pi$
$113$	10.0242	0.943000	0.471500	+	0.881866i	$0.343713\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−0.482825	−0.0446372
$118$	0	0
$119$	3.13973	0.287818
$120$	0	0
$121$	−4.64722	−0.422474
$122$	0	0
$123$	−1.39208	−0.125519
$124$	0	0
$125$	−4.23904	−0.379151
$126$	0	0
$127$	16.6982	1.48173	0.740865	−	0.671654i	$-0.234416\pi$
$127$	16.6982	1.48173	0.740865	+	0.671654i	$0.234416\pi$
$128$	0	0
$129$	−6.20411	−0.546241
$130$	0	0
$131$	−14.3899	−1.25725	−0.628627	−	0.777707i	$-0.716383\pi$
$131$	−14.3899	−1.25725	−0.628627	+	0.777707i	$0.716383\pi$
$132$	0	0
$133$	−0.553392	−0.0479852
$134$	0	0
$135$	2.92409	0.251666
$136$	0	0
$137$	−1.88176	−0.160769	−0.0803847	−	0.996764i	$-0.525615\pi$
$137$	−1.88176	−0.160769	−0.0803847	+	0.996764i	$0.525615\pi$
$138$	0	0
$139$	3.46292	0.293721	0.146860	−	0.989157i	$-0.453083\pi$
$139$	3.46292	0.293721	0.146860	+	0.989157i	$0.453083\pi$
$140$	0	0
$141$	8.69831	0.732530
$142$	0	0
$143$	1.21695	0.101766
$144$	0	0
$145$	0.355710	0.0295401
$146$	0	0
$147$	−5.99849	−0.494747
$148$	0	0
$149$	21.5705	1.76712	0.883560	−	0.468318i	$-0.155140\pi$
$149$	21.5705	1.76712	0.883560	+	0.468318i	$0.155140\pi$
$150$	0	0
$151$	3.27239	0.266303	0.133151	−	0.991096i	$-0.457490\pi$
$151$	3.27239	0.266303	0.133151	+	0.991096i	$0.457490\pi$
$152$	0	0
$153$	3.13735	0.253640
$154$	0	0
$155$	24.0792	1.93409
$156$	0	0
$157$	−22.7962	−1.81934	−0.909668	−	0.415335i	$-0.863664\pi$
$157$	−22.7962	−1.81934	−0.909668	+	0.415335i	$0.863664\pi$
$158$	0	0
$159$	−7.72669	−0.612766
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−12.0872	−0.946741	−0.473371	−	0.880863i	$-0.656963\pi$
$163$	−12.0872	−0.946741	−0.473371	+	0.880863i	$0.656963\pi$
$164$	0	0
$165$	−7.37009	−0.573761
$166$	0	0
$167$	8.29559	0.641932	0.320966	−	0.947091i	$-0.395992\pi$
$167$	8.29559	0.641932	0.320966	+	0.947091i	$0.395992\pi$
$168$	0	0
$169$	−12.7669	−0.982068
$170$	0	0
$171$	−0.552973	−0.0422869
$172$	0	0
$173$	−20.6611	−1.57083	−0.785417	−	0.618967i	$-0.787551\pi$
$173$	−20.6611	−1.57083	−0.785417	+	0.618967i	$0.787551\pi$
$174$	0	0
$175$	3.55299	0.268581
$176$	0	0
$177$	−0.236931	−0.0178089
$178$	0	0
$179$	3.88071	0.290058	0.145029	−	0.989427i	$-0.453672\pi$
$179$	3.88071	0.290058	0.145029	+	0.989427i	$0.453672\pi$
$180$	0	0
$181$	−7.97115	−0.592491	−0.296246	−	0.955112i	$-0.595735\pi$
$181$	−7.97115	−0.592491	−0.296246	+	0.955112i	$0.595735\pi$
$182$	0	0
$183$	1.83181	0.135412
$184$	0	0
$185$	27.8714	2.04915
$186$	0	0
$187$	−7.90760	−0.578261
$188$	0	0
$189$	1.00076	0.0727944
$190$	0	0
$191$	−16.8859	−1.22182	−0.610910	−	0.791700i	$-0.709196\pi$
$191$	−16.8859	−1.22182	−0.610910	+	0.791700i	$0.709196\pi$
$192$	0	0
$193$	−21.3002	−1.53322	−0.766611	−	0.642112i	$-0.778058\pi$
$193$	−21.3002	−1.53322	−0.766611	+	0.642112i	$0.778058\pi$
$194$	0	0
$195$	−1.41182	−0.101103
$196$	0	0
$197$	26.9334	1.91893	0.959463	−	0.281835i	$-0.0909431\pi$
$197$	26.9334	1.91893	0.959463	+	0.281835i	$0.0909431\pi$
$198$	0	0
$199$	7.28577	0.516474	0.258237	−	0.966082i	$-0.416858\pi$
$199$	7.28577	0.516474	0.258237	+	0.966082i	$0.416858\pi$
$200$	0	0
$201$	2.79746	0.197318
$202$	0	0
$203$	0.121740	0.00854449
$204$	0	0
$205$	−4.07056	−0.284300
$206$	0	0
$207$	0	0
$208$	0	0
$209$	1.39375	0.0964080
$210$	0	0
$211$	15.6084	1.07453	0.537264	−	0.843414i	$-0.319458\pi$
$211$	15.6084	1.07453	0.537264	+	0.843414i	$0.319458\pi$
$212$	0	0
$213$	11.4788	0.786512
$214$	0	0
$215$	−18.1414	−1.23723
$216$	0	0
$217$	8.24099	0.559435
$218$	0	0
$219$	−13.6919	−0.925212
$220$	0	0
$221$	−1.51479	−0.101896
$222$	0	0
$223$	−1.40582	−0.0941407	−0.0470704	−	0.998892i	$-0.514989\pi$
$223$	−1.40582	−0.0941407	−0.0470704	+	0.998892i	$0.514989\pi$
$224$	0	0
$225$	3.55030	0.236687
$226$	0	0
$227$	4.43994	0.294689	0.147344	−	0.989085i	$-0.452927\pi$
$227$	4.43994	0.294689	0.147344	+	0.989085i	$0.452927\pi$
$228$	0	0
$229$	1.67523	0.110702	0.0553512	−	0.998467i	$-0.482372\pi$
$229$	1.67523	0.110702	0.0553512	+	0.998467i	$0.482372\pi$
$230$	0	0
$231$	−2.52238	−0.165960
$232$	0	0
$233$	−21.7578	−1.42540	−0.712700	−	0.701469i	$-0.752528\pi$
$233$	−21.7578	−1.42540	−0.712700	+	0.701469i	$0.752528\pi$
$234$	0	0
$235$	25.4346	1.65917
$236$	0	0
$237$	17.0899	1.11011
$238$	0	0
$239$	−21.0054	−1.35872	−0.679362	−	0.733803i	$-0.737743\pi$
$239$	−21.0054	−1.35872	−0.679362	+	0.733803i	$0.737743\pi$
$240$	0	0
$241$	26.3357	1.69643	0.848217	−	0.529649i	$-0.177676\pi$
$241$	26.3357	1.69643	0.848217	+	0.529649i	$0.177676\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−17.5401	−1.12060
$246$	0	0
$247$	0.266989	0.0169881
$248$	0	0
$249$	8.06382	0.511024
$250$	0	0
$251$	−26.0533	−1.64447	−0.822236	−	0.569146i	$-0.807274\pi$
$251$	−26.0533	−1.64447	−0.822236	+	0.569146i	$0.807274\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	9.17389	0.574492
$256$	0	0
$257$	7.89900	0.492726	0.246363	−	0.969178i	$-0.420764\pi$
$257$	7.89900	0.492726	0.246363	+	0.969178i	$0.420764\pi$
$258$	0	0
$259$	9.53886	0.592716
$260$	0	0
$261$	0.121648	0.00752983
$262$	0	0
$263$	12.3421	0.761044	0.380522	−	0.924772i	$-0.375744\pi$
$263$	12.3421	0.761044	0.380522	+	0.924772i	$0.375744\pi$
$264$	0	0
$265$	−22.5935	−1.38791
$266$	0	0
$267$	−4.93850	−0.302231
$268$	0	0
$269$	−11.1897	−0.682245	−0.341123	−	0.940019i	$-0.610807\pi$
$269$	−11.1897	−0.682245	−0.341123	+	0.940019i	$0.610807\pi$
$270$	0	0
$271$	23.8348	1.44786	0.723932	−	0.689871i	$-0.242333\pi$
$271$	23.8348	1.44786	0.723932	+	0.689871i	$0.242333\pi$
$272$	0	0
$273$	−0.483191	−0.0292440
$274$	0	0
$275$	−8.94844	−0.539612
$276$	0	0
$277$	−17.8542	−1.07275	−0.536376	−	0.843979i	$-0.680207\pi$
$277$	−17.8542	−1.07275	−0.536376	+	0.843979i	$0.680207\pi$
$278$	0	0
$279$	8.23476	0.493002
$280$	0	0
$281$	1.65517	0.0987390	0.0493695	−	0.998781i	$-0.484279\pi$
$281$	1.65517	0.0987390	0.0493695	+	0.998781i	$0.484279\pi$
$282$	0	0
$283$	1.36434	0.0811013	0.0405507	−	0.999177i	$-0.487089\pi$
$283$	1.36434	0.0811013	0.0405507	+	0.999177i	$0.487089\pi$
$284$	0	0
$285$	−1.61694	−0.0957795
$286$	0	0
$287$	−1.39313	−0.0822340
$288$	0	0
$289$	−7.15704	−0.421002
$290$	0	0
$291$	15.6281	0.916135
$292$	0	0
$293$	10.6055	0.619578	0.309789	−	0.950805i	$-0.399742\pi$
$293$	10.6055	0.619578	0.309789	+	0.950805i	$0.399742\pi$
$294$	0	0
$295$	−0.692809	−0.0403369
$296$	0	0
$297$	−2.52047	−0.146253
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−6.20880	−0.357870
$302$	0	0
$303$	−3.25218	−0.186833
$304$	0	0
$305$	5.35639	0.306706
$306$	0	0
$307$	−10.3224	−0.589130	−0.294565	−	0.955631i	$-0.595175\pi$
$307$	−10.3224	−0.589130	−0.294565	+	0.955631i	$0.595175\pi$
$308$	0	0
$309$	1.40182	0.0797468
$310$	0	0
$311$	−7.04897	−0.399710	−0.199855	−	0.979825i	$-0.564047\pi$
$311$	−7.04897	−0.399710	−0.199855	+	0.979825i	$0.564047\pi$
$312$	0	0
$313$	−2.26256	−0.127887	−0.0639436	−	0.997954i	$-0.520368\pi$
$313$	−2.26256	−0.127887	−0.0639436	+	0.997954i	$0.520368\pi$
$314$	0	0
$315$	2.92630	0.164879
$316$	0	0
$317$	−3.17358	−0.178246	−0.0891230	−	0.996021i	$-0.528406\pi$
$317$	−3.17358	−0.178246	−0.0891230	+	0.996021i	$0.528406\pi$
$318$	0	0
$319$	−0.306611	−0.0171669
$320$	0	0
$321$	10.0443	0.560616
$322$	0	0
$323$	−1.73487	−0.0965308
$324$	0	0
$325$	−1.71418	−0.0950854
$326$	0	0
$327$	6.75809	0.373723
$328$	0	0
$329$	8.70489	0.479916
$330$	0	0
$331$	−20.7849	−1.14244	−0.571221	−	0.820797i	$-0.693530\pi$
$331$	−20.7849	−1.14244	−0.571221	+	0.820797i	$0.693530\pi$
$332$	0	0
$333$	9.53165	0.522331
$334$	0	0
$335$	8.18004	0.446923
$336$	0	0
$337$	26.1082	1.42221	0.711103	−	0.703088i	$-0.248196\pi$
$337$	26.1082	1.42221	0.711103	+	0.703088i	$0.248196\pi$
$338$	0	0
$339$	10.0242	0.544441
$340$	0	0
$341$	−20.7555	−1.12397
$342$	0	0
$343$	−13.0083	−0.702384
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−11.4891	−0.616769	−0.308385	−	0.951262i	$-0.599788\pi$
$347$	−11.4891	−0.616769	−0.308385	+	0.951262i	$0.599788\pi$
$348$	0	0
$349$	−29.2230	−1.56427	−0.782136	−	0.623108i	$-0.785870\pi$
$349$	−29.2230	−1.56427	−0.782136	+	0.623108i	$0.785870\pi$
$350$	0	0
$351$	−0.482825	−0.0257713
$352$	0	0
$353$	−32.5949	−1.73485	−0.867426	−	0.497565i	$-0.834227\pi$
$353$	−32.5949	−1.73485	−0.867426	+	0.497565i	$0.834227\pi$
$354$	0	0
$355$	33.5649	1.78144
$356$	0	0
$357$	3.13973	0.166172
$358$	0	0
$359$	29.1497	1.53846	0.769231	−	0.638970i	$-0.220639\pi$
$359$	29.1497	1.53846	0.769231	+	0.638970i	$0.220639\pi$
$360$	0	0
$361$	−18.6942	−0.983906
$362$	0	0
$363$	−4.64722	−0.243916
$364$	0	0
$365$	−40.0363	−2.09560
$366$	0	0
$367$	22.4367	1.17119	0.585593	−	0.810605i	$-0.300862\pi$
$367$	22.4367	1.17119	0.585593	+	0.810605i	$0.300862\pi$
$368$	0	0
$369$	−1.39208	−0.0724687
$370$	0	0
$371$	−7.73254	−0.401453
$372$	0	0
$373$	13.6209	0.705263	0.352631	−	0.935762i	$-0.385287\pi$
$373$	13.6209	0.705263	0.352631	+	0.935762i	$0.385287\pi$
$374$	0	0
$375$	−4.23904	−0.218903
$376$	0	0
$377$	−0.0587348	−0.00302500
$378$	0	0
$379$	−4.63466	−0.238066	−0.119033	−	0.992890i	$-0.537979\pi$
$379$	−4.63466	−0.238066	−0.119033	+	0.992890i	$0.537979\pi$
$380$	0	0
$381$	16.6982	0.855477
$382$	0	0
$383$	−30.4370	−1.55526	−0.777629	−	0.628724i	$-0.783578\pi$
$383$	−30.4370	−1.55526	−0.777629	+	0.628724i	$0.783578\pi$
$384$	0	0
$385$	−7.37567	−0.375899
$386$	0	0
$387$	−6.20411	−0.315372
$388$	0	0
$389$	−3.00516	−0.152368	−0.0761839	−	0.997094i	$-0.524274\pi$
$389$	−3.00516	−0.152368	−0.0761839	+	0.997094i	$0.524274\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−14.3899	−0.725876
$394$	0	0
$395$	49.9725	2.51439
$396$	0	0
$397$	−9.74084	−0.488879	−0.244439	−	0.969665i	$-0.578604\pi$
$397$	−9.74084	−0.488879	−0.244439	+	0.969665i	$0.578604\pi$
$398$	0	0
$399$	−0.553392	−0.0277042
$400$	0	0
$401$	−19.9647	−0.996988	−0.498494	−	0.866893i	$-0.666113\pi$
$401$	−19.9647	−0.996988	−0.498494	+	0.866893i	$0.666113\pi$
$402$	0	0
$403$	−3.97595	−0.198056
$404$	0	0
$405$	2.92409	0.145299
$406$	0	0
$407$	−24.0243	−1.19084
$408$	0	0
$409$	24.3311	1.20310	0.601548	−	0.798837i	$-0.294551\pi$
$409$	24.3311	1.20310	0.601548	+	0.798837i	$0.294551\pi$
$410$	0	0
$411$	−1.88176	−0.0928202
$412$	0	0
$413$	−0.237111	−0.0116675
$414$	0	0
$415$	23.5793	1.15746
$416$	0	0
$417$	3.46292	0.169580
$418$	0	0
$419$	−13.0882	−0.639399	−0.319699	−	0.947519i	$-0.603582\pi$
$419$	−13.0882	−0.639399	−0.319699	+	0.947519i	$0.603582\pi$
$420$	0	0
$421$	6.11443	0.297999	0.148999	−	0.988837i	$-0.452395\pi$
$421$	6.11443	0.297999	0.148999	+	0.988837i	$0.452395\pi$
$422$	0	0
$423$	8.69831	0.422926
$424$	0	0
$425$	11.1385	0.540299
$426$	0	0
$427$	1.83320	0.0887148
$428$	0	0
$429$	1.21695	0.0587548
$430$	0	0
$431$	24.4286	1.17668	0.588341	−	0.808613i	$-0.299781\pi$
$431$	24.4286	1.17668	0.588341	+	0.808613i	$0.299781\pi$
$432$	0	0
$433$	16.4567	0.790860	0.395430	−	0.918496i	$-0.370596\pi$
$433$	16.4567	0.790860	0.395430	+	0.918496i	$0.370596\pi$
$434$	0	0
$435$	0.355710	0.0170550
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−1.05639	−0.0504187	−0.0252093	−	0.999682i	$-0.508025\pi$
$439$	−1.05639	−0.0504187	−0.0252093	+	0.999682i	$0.508025\pi$
$440$	0	0
$441$	−5.99849	−0.285642
$442$	0	0
$443$	−19.9171	−0.946291	−0.473146	−	0.880984i	$-0.656882\pi$
$443$	−19.9171	−0.946291	−0.473146	+	0.880984i	$0.656882\pi$
$444$	0	0
$445$	−14.4406	−0.684551
$446$	0	0
$447$	21.5705	1.02025
$448$	0	0
$449$	24.4526	1.15399	0.576995	−	0.816748i	$-0.304225\pi$
$449$	24.4526	1.15399	0.576995	+	0.816748i	$0.304225\pi$
$450$	0	0
$451$	3.50869	0.165218
$452$	0	0
$453$	3.27239	0.153750
$454$	0	0
$455$	−1.41289	−0.0662375
$456$	0	0
$457$	−9.76318	−0.456703	−0.228351	−	0.973579i	$-0.573333\pi$
$457$	−9.76318	−0.456703	−0.228351	+	0.973579i	$0.573333\pi$
$458$	0	0
$459$	3.13735	0.146439
$460$	0	0
$461$	28.8585	1.34408	0.672038	−	0.740517i	$-0.265419\pi$
$461$	28.8585	1.34408	0.672038	+	0.740517i	$0.265419\pi$
$462$	0	0
$463$	20.2836	0.942658	0.471329	−	0.881958i	$-0.343774\pi$
$463$	20.2836	0.942658	0.471329	+	0.881958i	$0.343774\pi$
$464$	0	0
$465$	24.0792	1.11664
$466$	0	0
$467$	−27.6590	−1.27991	−0.639953	−	0.768414i	$-0.721046\pi$
$467$	−27.6590	−1.27991	−0.639953	+	0.768414i	$0.721046\pi$
$468$	0	0
$469$	2.79958	0.129273
$470$	0	0
$471$	−22.7962	−1.05039
$472$	0	0
$473$	15.6373	0.719003
$474$	0	0
$475$	−1.96322	−0.0900789
$476$	0	0
$477$	−7.72669	−0.353781
$478$	0	0
$479$	−20.4525	−0.934500	−0.467250	−	0.884125i	$-0.654755\pi$
$479$	−20.4525	−0.934500	−0.467250	+	0.884125i	$0.654755\pi$
$480$	0	0
$481$	−4.60212	−0.209839
$482$	0	0
$483$	0	0
$484$	0	0
$485$	45.6980	2.07504
$486$	0	0
$487$	3.11420	0.141118	0.0705589	−	0.997508i	$-0.477522\pi$
$487$	3.11420	0.141118	0.0705589	+	0.997508i	$0.477522\pi$
$488$	0	0
$489$	−12.0872	−0.546601
$490$	0	0
$491$	−5.49988	−0.248206	−0.124103	−	0.992269i	$-0.539605\pi$
$491$	−5.49988	−0.248206	−0.124103	+	0.992269i	$0.539605\pi$
$492$	0	0
$493$	0.381653	0.0171888
$494$	0	0
$495$	−7.37009	−0.331261
$496$	0	0
$497$	11.4875	0.515283
$498$	0	0
$499$	8.38938	0.375560	0.187780	−	0.982211i	$-0.439871\pi$
$499$	8.38938	0.375560	0.187780	+	0.982211i	$0.439871\pi$
$500$	0	0
$501$	8.29559	0.370620
$502$	0	0
$503$	1.69208	0.0754461	0.0377230	−	0.999288i	$-0.487990\pi$
$503$	1.69208	0.0754461	0.0377230	+	0.999288i	$0.487990\pi$
$504$	0	0
$505$	−9.50967	−0.423175
$506$	0	0
$507$	−12.7669	−0.566997
$508$	0	0
$509$	−14.6024	−0.647241	−0.323621	−	0.946187i	$-0.604900\pi$
$509$	−14.6024	−0.647241	−0.323621	+	0.946187i	$0.604900\pi$
$510$	0	0
$511$	−13.7022	−0.606152
$512$	0	0
$513$	−0.552973	−0.0244144
$514$	0	0
$515$	4.09905	0.180626
$516$	0	0
$517$	−21.9238	−0.964210
$518$	0	0
$519$	−20.6611	−0.906921
$520$	0	0
$521$	25.7782	1.12937	0.564683	−	0.825308i	$-0.308999\pi$
$521$	25.7782	1.12937	0.564683	+	0.825308i	$0.308999\pi$
$522$	0	0
$523$	−14.0698	−0.615230	−0.307615	−	0.951511i	$-0.599531\pi$
$523$	−14.0698	−0.615230	−0.307615	+	0.951511i	$0.599531\pi$
$524$	0	0
$525$	3.55299	0.155065
$526$	0	0
$527$	25.8353	1.12540
$528$	0	0
$529$	0	0
$530$	0	0
$531$	−0.236931	−0.0102819
$532$	0	0
$533$	0.672130	0.0291132
$534$	0	0
$535$	29.3704	1.26979
$536$	0	0
$537$	3.88071	0.167465
$538$	0	0
$539$	15.1190	0.651222
$540$	0	0
$541$	−18.0495	−0.776008	−0.388004	−	0.921658i	$-0.626835\pi$
$541$	−18.0495	−0.776008	−0.388004	+	0.921658i	$0.626835\pi$
$542$	0	0
$543$	−7.97115	−0.342075
$544$	0	0
$545$	19.7613	0.846480
$546$	0	0
$547$	−16.3113	−0.697421	−0.348710	−	0.937231i	$-0.613380\pi$
$547$	−16.3113	−0.697421	−0.348710	+	0.937231i	$0.613380\pi$
$548$	0	0
$549$	1.83181	0.0781799
$550$	0	0
$551$	−0.0672682	−0.00286572
$552$	0	0
$553$	17.1029	0.727287
$554$	0	0
$555$	27.8714	1.18307
$556$	0	0
$557$	−22.3437	−0.946732	−0.473366	−	0.880866i	$-0.656961\pi$
$557$	−22.3437	−0.946732	−0.473366	+	0.880866i	$0.656961\pi$
$558$	0	0
$559$	2.99550	0.126696
$560$	0	0
$561$	−7.90760	−0.333859
$562$	0	0
$563$	14.0337	0.591452	0.295726	−	0.955273i	$-0.404439\pi$
$563$	14.0337	0.591452	0.295726	+	0.955273i	$0.404439\pi$
$564$	0	0
$565$	29.3117	1.23315
$566$	0	0
$567$	1.00076	0.0420278
$568$	0	0
$569$	−37.2949	−1.56348	−0.781741	−	0.623603i	$-0.785668\pi$
$569$	−37.2949	−1.56348	−0.781741	+	0.623603i	$0.785668\pi$
$570$	0	0
$571$	36.8538	1.54228	0.771142	−	0.636663i	$-0.219686\pi$
$571$	36.8538	1.54228	0.771142	+	0.636663i	$0.219686\pi$
$572$	0	0
$573$	−16.8859	−0.705418
$574$	0	0
$575$	0	0
$576$	0	0
$577$	21.0650	0.876946	0.438473	−	0.898744i	$-0.355519\pi$
$577$	21.0650	0.876946	0.438473	+	0.898744i	$0.355519\pi$
$578$	0	0
$579$	−21.3002	−0.885206
$580$	0	0
$581$	8.06993	0.334797
$582$	0	0
$583$	19.4749	0.806568
$584$	0	0
$585$	−1.41182	−0.0583718
$586$	0	0
$587$	0.0411633	0.00169899	0.000849496	−	1.00000i	$-0.499730\pi$
$587$	0.0411633	0.00169899	0.000849496		1.00000i	$0.499730\pi$
$588$	0	0
$589$	−4.55360	−0.187628
$590$	0	0
$591$	26.9334	1.10789
$592$	0	0
$593$	14.8649	0.610427	0.305213	−	0.952284i	$-0.401272\pi$
$593$	14.8649	0.610427	0.305213	+	0.952284i	$0.401272\pi$
$594$	0	0
$595$	9.18084	0.376378
$596$	0	0
$597$	7.28577	0.298187
$598$	0	0
$599$	44.0448	1.79962	0.899811	−	0.436280i	$-0.143704\pi$
$599$	44.0448	1.79962	0.899811	+	0.436280i	$0.143704\pi$
$600$	0	0
$601$	44.1391	1.80047	0.900237	−	0.435401i	$-0.143393\pi$
$601$	44.1391	1.80047	0.900237	+	0.435401i	$0.143393\pi$
$602$	0	0
$603$	2.79746	0.113922
$604$	0	0
$605$	−13.5889	−0.552467
$606$	0	0
$607$	−21.2355	−0.861923	−0.430961	−	0.902370i	$-0.641825\pi$
$607$	−21.2355	−0.861923	−0.430961	+	0.902370i	$0.641825\pi$
$608$	0	0
$609$	0.121740	0.00493317
$610$	0	0
$611$	−4.19976	−0.169904
$612$	0	0
$613$	34.1417	1.37897	0.689485	−	0.724300i	$-0.257837\pi$
$613$	34.1417	1.37897	0.689485	+	0.724300i	$0.257837\pi$
$614$	0	0
$615$	−4.07056	−0.164141
$616$	0	0
$617$	−41.0131	−1.65112	−0.825562	−	0.564311i	$-0.809142\pi$
$617$	−41.0131	−1.65112	−0.825562	+	0.564311i	$0.809142\pi$
$618$	0	0
$619$	−7.04481	−0.283155	−0.141577	−	0.989927i	$-0.545217\pi$
$619$	−7.04481	−0.283155	−0.141577	+	0.989927i	$0.545217\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−4.94224	−0.198007
$624$	0	0
$625$	−30.1469	−1.20587
$626$	0	0
$627$	1.39375	0.0556612
$628$	0	0
$629$	29.9041	1.19235
$630$	0	0
$631$	−23.3895	−0.931120	−0.465560	−	0.885016i	$-0.654147\pi$
$631$	−23.3895	−0.931120	−0.465560	+	0.885016i	$0.654147\pi$
$632$	0	0
$633$	15.6084	0.620379
$634$	0	0
$635$	48.8272	1.93765
$636$	0	0
$637$	2.89622	0.114752
$638$	0	0
$639$	11.4788	0.454093
$640$	0	0
$641$	−16.5270	−0.652778	−0.326389	−	0.945235i	$-0.605832\pi$
$641$	−16.5270	−0.652778	−0.326389	+	0.945235i	$0.605832\pi$
$642$	0	0
$643$	34.1400	1.34635	0.673176	−	0.739482i	$-0.264930\pi$
$643$	34.1400	1.34635	0.673176	+	0.739482i	$0.264930\pi$
$644$	0	0
$645$	−18.1414	−0.714316
$646$	0	0
$647$	−9.66388	−0.379926	−0.189963	−	0.981791i	$-0.560837\pi$
$647$	−9.66388	−0.379926	−0.189963	+	0.981791i	$0.560837\pi$
$648$	0	0
$649$	0.597179	0.0234413
$650$	0	0
$651$	8.24099	0.322990
$652$	0	0
$653$	5.96106	0.233274	0.116637	−	0.993175i	$-0.462789\pi$
$653$	5.96106	0.233274	0.116637	+	0.993175i	$0.462789\pi$
$654$	0	0
$655$	−42.0774	−1.64410
$656$	0	0
$657$	−13.6919	−0.534171
$658$	0	0
$659$	−37.7022	−1.46867	−0.734334	−	0.678789i	$-0.762505\pi$
$659$	−37.7022	−1.46867	−0.734334	+	0.678789i	$0.762505\pi$
$660$	0	0
$661$	32.1603	1.25089	0.625445	−	0.780268i	$-0.284917\pi$
$661$	32.1603	1.25089	0.625445	+	0.780268i	$0.284917\pi$
$662$	0	0
$663$	−1.51479	−0.0588296
$664$	0	0
$665$	−1.61817	−0.0627498
$666$	0	0
$667$	0	0
$668$	0	0
$669$	−1.40582	−0.0543522
$670$	0	0
$671$	−4.61704	−0.178239
$672$	0	0
$673$	−31.2644	−1.20516	−0.602578	−	0.798060i	$-0.705860\pi$
$673$	−31.2644	−1.20516	−0.602578	+	0.798060i	$0.705860\pi$
$674$	0	0
$675$	3.55030	0.136651
$676$	0	0
$677$	−5.46909	−0.210194	−0.105097	−	0.994462i	$-0.533515\pi$
$677$	−5.46909	−0.210194	−0.105097	+	0.994462i	$0.533515\pi$
$678$	0	0
$679$	15.6399	0.600205
$680$	0	0
$681$	4.43994	0.170139
$682$	0	0
$683$	−33.3278	−1.27525	−0.637626	−	0.770346i	$-0.720084\pi$
$683$	−33.3278	−1.27525	−0.637626	+	0.770346i	$0.720084\pi$
$684$	0	0
$685$	−5.50243	−0.210237
$686$	0	0
$687$	1.67523	0.0639141
$688$	0	0
$689$	3.73064	0.142126
$690$	0	0
$691$	23.4533	0.892206	0.446103	−	0.894982i	$-0.352811\pi$
$691$	23.4533	0.892206	0.446103	+	0.894982i	$0.352811\pi$
$692$	0	0
$693$	−2.52238	−0.0958173
$694$	0	0
$695$	10.1259	0.384097
$696$	0	0
$697$	−4.36744	−0.165428
$698$	0	0
$699$	−21.7578	−0.822955
$700$	0	0
$701$	−14.7674	−0.557757	−0.278879	−	0.960326i	$-0.589963\pi$
$701$	−14.7674	−0.557757	−0.278879	+	0.960326i	$0.589963\pi$
$702$	0	0
$703$	−5.27074	−0.198790
$704$	0	0
$705$	25.4346	0.957924
$706$	0	0
$707$	−3.25464	−0.122403
$708$	0	0
$709$	−12.9708	−0.487130	−0.243565	−	0.969885i	$-0.578317\pi$
$709$	−12.9708	−0.487130	−0.243565	+	0.969885i	$0.578317\pi$
$710$	0	0
$711$	17.0899	0.640922
$712$	0	0
$713$	0	0
$714$	0	0
$715$	3.55847	0.133079
$716$	0	0
$717$	−21.0054	−0.784459
$718$	0	0
$719$	−18.2138	−0.679259	−0.339630	−	0.940559i	$-0.610302\pi$
$719$	−18.2138	−0.679259	−0.339630	+	0.940559i	$0.610302\pi$
$720$	0	0
$721$	1.40288	0.0522460
$722$	0	0
$723$	26.3357	0.979437
$724$	0	0
$725$	0.431888	0.0160399
$726$	0	0
$727$	−8.57796	−0.318139	−0.159069	−	0.987267i	$-0.550849\pi$
$727$	−8.57796	−0.318139	−0.159069	+	0.987267i	$0.550849\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−19.4645	−0.719919
$732$	0	0
$733$	3.14291	0.116086	0.0580431	−	0.998314i	$-0.481514\pi$
$733$	3.14291	0.116086	0.0580431	+	0.998314i	$0.481514\pi$
$734$	0	0
$735$	−17.5401	−0.646977
$736$	0	0
$737$	−7.05093	−0.259724
$738$	0	0
$739$	22.5068	0.827925	0.413962	−	0.910294i	$-0.364145\pi$
$739$	22.5068	0.827925	0.413962	+	0.910294i	$0.364145\pi$
$740$	0	0
$741$	0.266989	0.00980810
$742$	0	0
$743$	−4.92937	−0.180841	−0.0904206	−	0.995904i	$-0.528821\pi$
$743$	−4.92937	−0.180841	−0.0904206	+	0.995904i	$0.528821\pi$
$744$	0	0
$745$	63.0740	2.31085
$746$	0	0
$747$	8.06382	0.295040
$748$	0	0
$749$	10.0519	0.367288
$750$	0	0
$751$	−1.04768	−0.0382306	−0.0191153	−	0.999817i	$-0.506085\pi$
$751$	−1.04768	−0.0382306	−0.0191153	+	0.999817i	$0.506085\pi$
$752$	0	0
$753$	−26.0533	−0.949437
$754$	0	0
$755$	9.56875	0.348242
$756$	0	0
$757$	2.83671	0.103102	0.0515509	−	0.998670i	$-0.483584\pi$
$757$	2.83671	0.103102	0.0515509	+	0.998670i	$0.483584\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−29.1788	−1.05773	−0.528865	−	0.848706i	$-0.677382\pi$
$761$	−29.1788	−1.05773	−0.528865	+	0.848706i	$0.677382\pi$
$762$	0	0
$763$	6.76321	0.244845
$764$	0	0
$765$	9.17389	0.331683
$766$	0	0
$767$	0.114396	0.00413062
$768$	0	0
$769$	−20.3225	−0.732849	−0.366424	−	0.930448i	$-0.619418\pi$
$769$	−20.3225	−0.732849	−0.366424	+	0.930448i	$0.619418\pi$
$770$	0	0
$771$	7.89900	0.284475
$772$	0	0
$773$	22.4407	0.807135	0.403568	−	0.914950i	$-0.367770\pi$
$773$	22.4407	0.807135	0.403568	+	0.914950i	$0.367770\pi$
$774$	0	0
$775$	29.2359	1.05018
$776$	0	0
$777$	9.53886	0.342205
$778$	0	0
$779$	0.769782	0.0275803
$780$	0	0
$781$	−28.9319	−1.03526
$782$	0	0
$783$	0.121648	0.00434735
$784$	0	0
$785$	−66.6582	−2.37913
$786$	0	0
$787$	14.0329	0.500219	0.250109	−	0.968218i	$-0.419533\pi$
$787$	14.0329	0.500219	0.250109	+	0.968218i	$0.419533\pi$
$788$	0	0
$789$	12.3421	0.439389
$790$	0	0
$791$	10.0318	0.356690
$792$	0	0
$793$	−0.884446	−0.0314076
$794$	0	0
$795$	−22.5935	−0.801310
$796$	0	0
$797$	−52.6768	−1.86591	−0.932954	−	0.359994i	$-0.882779\pi$
$797$	−52.6768	−1.86591	−0.932954	+	0.359994i	$0.882779\pi$
$798$	0	0
$799$	27.2896	0.965438
$800$	0	0
$801$	−4.93850	−0.174493
$802$	0	0
$803$	34.5100	1.21783
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−11.1897	−0.393895
$808$	0	0
$809$	52.8291	1.85737	0.928686	−	0.370868i	$-0.120940\pi$
$809$	52.8291	1.85737	0.928686	+	0.370868i	$0.120940\pi$
$810$	0	0
$811$	42.5497	1.49412	0.747061	−	0.664756i	$-0.231464\pi$
$811$	42.5497	1.49412	0.747061	+	0.664756i	$0.231464\pi$
$812$	0	0
$813$	23.8348	0.835925
$814$	0	0
$815$	−35.3440	−1.23805
$816$	0	0
$817$	3.43070	0.120025
$818$	0	0
$819$	−0.483191	−0.0168841
$820$	0	0
$821$	10.3213	0.360214	0.180107	−	0.983647i	$-0.442356\pi$
$821$	10.3213	0.360214	0.180107	+	0.983647i	$0.442356\pi$
$822$	0	0
$823$	−54.3933	−1.89603	−0.948016	−	0.318223i	$-0.896914\pi$
$823$	−54.3933	−1.89603	−0.948016	+	0.318223i	$0.896914\pi$
$824$	0	0
$825$	−8.94844	−0.311545
$826$	0	0
$827$	−27.7576	−0.965227	−0.482613	−	0.875834i	$-0.660312\pi$
$827$	−27.7576	−0.965227	−0.482613	+	0.875834i	$0.660312\pi$
$828$	0	0
$829$	−31.3925	−1.09031	−0.545153	−	0.838336i	$-0.683529\pi$
$829$	−31.3925	−1.09031	−0.545153	+	0.838336i	$0.683529\pi$
$830$	0	0
$831$	−17.8542	−0.619354
$832$	0	0
$833$	−18.8193	−0.652052
$834$	0	0
$835$	24.2571	0.839450
$836$	0	0
$837$	8.23476	0.284635
$838$	0	0
$839$	−3.42802	−0.118348	−0.0591742	−	0.998248i	$-0.518847\pi$
$839$	−3.42802	−0.118348	−0.0591742	+	0.998248i	$0.518847\pi$
$840$	0	0
$841$	−28.9852	−0.999490
$842$	0	0
$843$	1.65517	0.0570070
$844$	0	0
$845$	−37.3315	−1.28424
$846$	0	0
$847$	−4.65074	−0.159801
$848$	0	0
$849$	1.36434	0.0468239
$850$	0	0
$851$	0	0
$852$	0	0
$853$	32.5301	1.11381	0.556904	−	0.830577i	$-0.311989\pi$
$853$	32.5301	1.11381	0.556904	+	0.830577i	$0.311989\pi$
$854$	0	0
$855$	−1.61694	−0.0552983
$856$	0	0
$857$	−35.8564	−1.22483	−0.612416	−	0.790536i	$-0.709802\pi$
$857$	−35.8564	−1.22483	−0.612416	+	0.790536i	$0.709802\pi$
$858$	0	0
$859$	9.07094	0.309497	0.154748	−	0.987954i	$-0.450543\pi$
$859$	9.07094	0.309497	0.154748	+	0.987954i	$0.450543\pi$
$860$	0	0
$861$	−1.39313	−0.0474778
$862$	0	0
$863$	42.4039	1.44344	0.721722	−	0.692183i	$-0.243351\pi$
$863$	42.4039	1.44344	0.721722	+	0.692183i	$0.243351\pi$
$864$	0	0
$865$	−60.4149	−2.05417
$866$	0	0
$867$	−7.15704	−0.243066
$868$	0	0
$869$	−43.0747	−1.46121
$870$	0	0
$871$	−1.35069	−0.0457662
$872$	0	0
$873$	15.6281	0.528931
$874$	0	0
$875$	−4.24225	−0.143414
$876$	0	0
$877$	41.3885	1.39759	0.698795	−	0.715322i	$-0.253720\pi$
$877$	41.3885	1.39759	0.698795	+	0.715322i	$0.253720\pi$
$878$	0	0
$879$	10.6055	0.357714
$880$	0	0
$881$	−17.7850	−0.599192	−0.299596	−	0.954066i	$-0.596852\pi$
$881$	−17.7850	−0.599192	−0.299596	+	0.954066i	$0.596852\pi$
$882$	0	0
$883$	8.22699	0.276860	0.138430	−	0.990372i	$-0.455794\pi$
$883$	8.22699	0.276860	0.138430	+	0.990372i	$0.455794\pi$
$884$	0	0
$885$	−0.692809	−0.0232885
$886$	0	0
$887$	−44.6339	−1.49866	−0.749330	−	0.662197i	$-0.769624\pi$
$887$	−44.6339	−1.49866	−0.749330	+	0.662197i	$0.769624\pi$
$888$	0	0
$889$	16.7109	0.560465
$890$	0	0
$891$	−2.52047	−0.0844390
$892$	0	0
$893$	−4.80993	−0.160958
$894$	0	0
$895$	11.3476	0.379307
$896$	0	0
$897$	0	0
$898$	0	0
$899$	1.00174	0.0334100
$900$	0	0
$901$	−24.2413	−0.807596
$902$	0	0
$903$	−6.20880	−0.206616
$904$	0	0
$905$	−23.3084	−0.774797
$906$	0	0
$907$	−40.8967	−1.35795	−0.678977	−	0.734160i	$-0.737576\pi$
$907$	−40.8967	−1.35795	−0.678977	+	0.734160i	$0.737576\pi$
$908$	0	0
$909$	−3.25218	−0.107868
$910$	0	0
$911$	−0.868077	−0.0287607	−0.0143803	−	0.999897i	$-0.504578\pi$
$911$	−0.868077	−0.0287607	−0.0143803	+	0.999897i	$0.504578\pi$
$912$	0	0
$913$	−20.3246	−0.672648
$914$	0	0
$915$	5.35639	0.177077
$916$	0	0
$917$	−14.4008	−0.475557
$918$	0	0
$919$	53.6469	1.76965	0.884824	−	0.465926i	$-0.154279\pi$
$919$	53.6469	1.76965	0.884824	+	0.465926i	$0.154279\pi$
$920$	0	0
$921$	−10.3224	−0.340135
$922$	0	0
$923$	−5.54224	−0.182425
$924$	0	0
$925$	33.8402	1.11266
$926$	0	0
$927$	1.40182	0.0460418
$928$	0	0
$929$	−4.37588	−0.143568	−0.0717840	−	0.997420i	$-0.522869\pi$
$929$	−4.37588	−0.143568	−0.0717840	+	0.997420i	$0.522869\pi$
$930$	0	0
$931$	3.31700	0.108710
$932$	0	0
$933$	−7.04897	−0.230773
$934$	0	0
$935$	−23.1225	−0.756188
$936$	0	0
$937$	−5.70488	−0.186370	−0.0931851	−	0.995649i	$-0.529705\pi$
$937$	−5.70488	−0.186370	−0.0931851	+	0.995649i	$0.529705\pi$
$938$	0	0
$939$	−2.26256	−0.0738357
$940$	0	0
$941$	11.1076	0.362098	0.181049	−	0.983474i	$-0.442051\pi$
$941$	11.1076	0.362098	0.181049	+	0.983474i	$0.442051\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	2.92630	0.0951927
$946$	0	0
$947$	−14.9499	−0.485807	−0.242904	−	0.970050i	$-0.578100\pi$
$947$	−14.9499	−0.485807	−0.242904	+	0.970050i	$0.578100\pi$
$948$	0	0
$949$	6.61079	0.214595
$950$	0	0
$951$	−3.17358	−0.102910
$952$	0	0
$953$	29.5217	0.956301	0.478150	−	0.878278i	$-0.341307\pi$
$953$	29.5217	0.956301	0.478150	+	0.878278i	$0.341307\pi$
$954$	0	0
$955$	−49.3758	−1.59776
$956$	0	0
$957$	−0.306611	−0.00991132
$958$	0	0
$959$	−1.88318	−0.0608111
$960$	0	0
$961$	36.8112	1.18746
$962$	0	0
$963$	10.0443	0.323672
$964$	0	0
$965$	−62.2837	−2.00498
$966$	0	0
$967$	−44.5676	−1.43320	−0.716598	−	0.697486i	$-0.754302\pi$
$967$	−44.5676	−1.43320	−0.716598	+	0.697486i	$0.754302\pi$
$968$	0	0
$969$	−1.73487	−0.0557321
$970$	0	0
$971$	−53.2047	−1.70742	−0.853710	−	0.520749i	$-0.825653\pi$
$971$	−53.2047	−1.70742	−0.853710	+	0.520749i	$0.825653\pi$
$972$	0	0
$973$	3.46554	0.111100
$974$	0	0
$975$	−1.71418	−0.0548976
$976$	0	0
$977$	36.8076	1.17758	0.588790	−	0.808286i	$-0.299604\pi$
$977$	36.8076	1.17758	0.588790	+	0.808286i	$0.299604\pi$
$978$	0	0
$979$	12.4474	0.397819
$980$	0	0
$981$	6.75809	0.215769
$982$	0	0
$983$	41.2307	1.31506	0.657528	−	0.753430i	$-0.271602\pi$
$983$	41.2307	1.31506	0.657528	+	0.753430i	$0.271602\pi$
$984$	0	0
$985$	78.7557	2.50937
$986$	0	0
$987$	8.70489	0.277080
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−40.3806	−1.28273	−0.641366	−	0.767235i	$-0.721632\pi$
$991$	−40.3806	−1.28273	−0.641366	+	0.767235i	$0.721632\pi$
$992$	0	0
$993$	−20.7849	−0.659589
$994$	0	0
$995$	21.3042	0.675390
$996$	0	0
$997$	26.4577	0.837923	0.418961	−	0.908004i	$-0.362394\pi$
$997$	26.4577	0.837923	0.418961	+	0.908004i	$0.362394\pi$
$998$	0	0
$999$	9.53165	0.301568

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6348.2.a.t.1.9		10
23.17	odd	22		276.2.i.a.13.1	✓	20
23.19	odd	22		276.2.i.a.85.1	yes	20
23.22	odd	2		6348.2.a.s.1.2		10
69.17	even	22		828.2.q.c.289.2		20
69.65	even	22		828.2.q.c.361.2		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
276.2.i.a.13.1	✓	20	23.17	odd	22
276.2.i.a.85.1	yes	20	23.19	odd	22
828.2.q.c.289.2		20	69.17	even	22
828.2.q.c.361.2		20	69.65	even	22
6348.2.a.s.1.2		10	23.22	odd	2
6348.2.a.t.1.9		10	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:	\( N \)	\(=\)	\( 6348 = 2^{2} \cdot 3 \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6348.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +2.92409 q^{5} +1.00076 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +2.92409 q^{5} +1.00076 q^{7} +1.00000 q^{9} -2.52047 q^{11} -0.482825 q^{13} +2.92409 q^{15} +3.13735 q^{17} -0.552973 q^{19} +1.00076 q^{21} +3.55030 q^{25} +1.00000 q^{27} +0.121648 q^{29} +8.23476 q^{31} -2.52047 q^{33} +2.92630 q^{35} +9.53165 q^{37} -0.482825 q^{39} -1.39208 q^{41} -6.20411 q^{43} +2.92409 q^{45} +8.69831 q^{47} -5.99849 q^{49} +3.13735 q^{51} -7.72669 q^{53} -7.37009 q^{55} -0.552973 q^{57} -0.236931 q^{59} +1.83181 q^{61} +1.00076 q^{63} -1.41182 q^{65} +2.79746 q^{67} +11.4788 q^{71} -13.6919 q^{73} +3.55030 q^{75} -2.52238 q^{77} +17.0899 q^{79} +1.00000 q^{81} +8.06382 q^{83} +9.17389 q^{85} +0.121648 q^{87} -4.93850 q^{89} -0.483191 q^{91} +8.23476 q^{93} -1.61694 q^{95} +15.6281 q^{97} -2.52047 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 10 q^{3} + 2 q^{5} + 11 q^{7} + 10 q^{9}+O(q^{10}) \) 10 * q + 10 * q^3 + 2 * q^5 + 11 * q^7 + 10 * q^9	\( 10 q + 10 q^{3} + 2 q^{5} + 11 q^{7} + 10 q^{9} + 11 q^{11} + 2 q^{15} + 13 q^{17} + 18 q^{19} + 11 q^{21} + 10 q^{25} + 10 q^{27} + 5 q^{29} + 15 q^{31} + 11 q^{33} - 13 q^{35} + 5 q^{37} + 24 q^{41} + 40 q^{43} + 2 q^{45} - 9 q^{47} + 5 q^{49} + 13 q^{51} + 6 q^{53} - 14 q^{55} + 18 q^{57} + 28 q^{59} + 39 q^{61} + 11 q^{63} + 14 q^{65} + 32 q^{67} - 33 q^{71} - 50 q^{73} + 10 q^{75} + 19 q^{77} + 33 q^{79} + 10 q^{81} + 29 q^{83} - 21 q^{85} + 5 q^{87} + 17 q^{89} + 36 q^{91} + 15 q^{93} - 42 q^{95} + 46 q^{97} + 11 q^{99}+O(q^{100}) \) 10 * q + 10 * q^3 + 2 * q^5 + 11 * q^7 + 10 * q^9 + 11 * q^11 + 2 * q^15 + 13 * q^17 + 18 * q^19 + 11 * q^21 + 10 * q^25 + 10 * q^27 + 5 * q^29 + 15 * q^31 + 11 * q^33 - 13 * q^35 + 5 * q^37 + 24 * q^41 + 40 * q^43 + 2 * q^45 - 9 * q^47 + 5 * q^49 + 13 * q^51 + 6 * q^53 - 14 * q^55 + 18 * q^57 + 28 * q^59 + 39 * q^61 + 11 * q^63 + 14 * q^65 + 32 * q^67 - 33 * q^71 - 50 * q^73 + 10 * q^75 + 19 * q^77 + 33 * q^79 + 10 * q^81 + 29 * q^83 - 21 * q^85 + 5 * q^87 + 17 * q^89 + 36 * q^91 + 15 * q^93 - 42 * q^95 + 46 * q^97 + 11 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more