LMFDB - Embedded newform 6348.2.a.s.1.1

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:	$1$
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			1.1
Root			$2.35529$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} -4.08477 q^{5} +0.126013 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -4.08477 q^{5} +0.126013 q^{7} +1.00000 q^{9} -1.23408 q^{11} -1.91226 q^{13} -4.08477 q^{15} +1.89524 q^{17} +0.147654 q^{19} +0.126013 q^{21} +11.6853 q^{25} +1.00000 q^{27} +7.50497 q^{29} -2.35692 q^{31} -1.23408 q^{33} -0.514734 q^{35} +7.72570 q^{37} -1.91226 q^{39} +10.4766 q^{41} -12.0719 q^{43} -4.08477 q^{45} -5.71564 q^{47} -6.98412 q^{49} +1.89524 q^{51} +6.67854 q^{53} +5.04095 q^{55} +0.147654 q^{57} +7.61804 q^{59} +1.27140 q^{61} +0.126013 q^{63} +7.81115 q^{65} -10.6380 q^{67} +9.24001 q^{71} -12.6387 q^{73} +11.6853 q^{75} -0.155510 q^{77} -3.72925 q^{79} +1.00000 q^{81} -8.16040 q^{83} -7.74162 q^{85} +7.50497 q^{87} -7.06035 q^{89} -0.240970 q^{91} -2.35692 q^{93} -0.603133 q^{95} -13.2465 q^{97} -1.23408 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−4.08477	−1.82676	−0.913382	−	0.407103i	$-0.866539\pi$
$5$	−4.08477	−1.82676	−0.913382	+	0.407103i	$0.866539\pi$
$6$	0	0
$7$	0.126013	0.0476284	0.0238142	−	0.999716i	$-0.492419\pi$
$7$	0.126013	0.0476284	0.0238142	+	0.999716i	$0.492419\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.23408	−0.372090	−0.186045	−	0.982541i	$-0.559567\pi$
$11$	−1.23408	−0.372090	−0.186045	+	0.982541i	$0.559567\pi$
$12$	0	0
$13$	−1.91226	−0.530366	−0.265183	−	0.964198i	$-0.585432\pi$
$13$	−1.91226	−0.530366	−0.265183	+	0.964198i	$0.585432\pi$
$14$	0	0
$15$	−4.08477	−1.05468
$16$	0	0
$17$	1.89524	0.459664	0.229832	−	0.973230i	$-0.426182\pi$
$17$	1.89524	0.459664	0.229832	+	0.973230i	$0.426182\pi$
$18$	0	0
$19$	0.147654	0.0338742	0.0169371	−	0.999857i	$-0.494608\pi$
$19$	0.147654	0.0338742	0.0169371	+	0.999857i	$0.494608\pi$
$20$	0	0
$21$	0.126013	0.0274983
$22$	0	0
$23$	0	0
$23$	0	0
$24$	0	0
$25$	11.6853	2.33707
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	7.50497	1.39364	0.696819	−	0.717247i	$-0.254598\pi$
$29$	7.50497	1.39364	0.696819	+	0.717247i	$0.254598\pi$
$30$	0	0
$31$	−2.35692	−0.423316	−0.211658	−	0.977344i	$-0.567886\pi$
$31$	−2.35692	−0.423316	−0.211658	+	0.977344i	$0.567886\pi$
$32$	0	0
$33$	−1.23408	−0.214826
$34$	0	0
$35$	−0.514734	−0.0870058
$36$	0	0
$37$	7.72570	1.27010	0.635048	−	0.772472i	$-0.280980\pi$
$37$	7.72570	1.27010	0.635048	+	0.772472i	$0.280980\pi$
$38$	0	0
$39$	−1.91226	−0.306207
$40$	0	0
$41$	10.4766	1.63617	0.818086	−	0.575096i	$-0.195035\pi$
$41$	10.4766	1.63617	0.818086	+	0.575096i	$0.195035\pi$
$42$	0	0
$43$	−12.0719	−1.84095	−0.920473	−	0.390807i	$-0.872196\pi$
$43$	−12.0719	−1.84095	−0.920473	+	0.390807i	$0.872196\pi$
$44$	0	0
$45$	−4.08477	−0.608921
$46$	0	0
$47$	−5.71564	−0.833712	−0.416856	−	0.908972i	$-0.636868\pi$
$47$	−5.71564	−0.833712	−0.416856	+	0.908972i	$0.636868\pi$
$48$	0	0
$49$	−6.98412	−0.997732
$50$	0	0
$51$	1.89524	0.265387
$52$	0	0
$53$	6.67854	0.917369	0.458684	−	0.888599i	$-0.348321\pi$
$53$	6.67854	0.917369	0.458684	+	0.888599i	$0.348321\pi$
$54$	0	0
$55$	5.04095	0.679721
$56$	0	0
$57$	0.147654	0.0195573
$58$	0	0
$59$	7.61804	0.991784	0.495892	−	0.868384i	$-0.334841\pi$
$59$	7.61804	0.991784	0.495892	+	0.868384i	$0.334841\pi$
$60$	0	0
$61$	1.27140	0.162786	0.0813929	−	0.996682i	$-0.474063\pi$
$61$	1.27140	0.162786	0.0813929	+	0.996682i	$0.474063\pi$
$62$	0	0
$63$	0.126013	0.0158761
$64$	0	0
$65$	7.81115	0.968854
$66$	0	0
$67$	−10.6380	−1.29964	−0.649818	−	0.760090i	$-0.725155\pi$
$67$	−10.6380	−1.29964	−0.649818	+	0.760090i	$0.725155\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	9.24001	1.09659	0.548293	−	0.836286i	$-0.315278\pi$
$71$	9.24001	1.09659	0.548293	+	0.836286i	$0.315278\pi$
$72$	0	0
$73$	−12.6387	−1.47925	−0.739625	−	0.673020i	$-0.764997\pi$
$73$	−12.6387	−1.47925	−0.739625	+	0.673020i	$0.764997\pi$
$74$	0	0
$75$	11.6853	1.34931
$76$	0	0
$77$	−0.155510	−0.0177221
$78$	0	0
$79$	−3.72925	−0.419573	−0.209787	−	0.977747i	$-0.567277\pi$
$79$	−3.72925	−0.419573	−0.209787	+	0.977747i	$0.567277\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−8.16040	−0.895720	−0.447860	−	0.894104i	$-0.647814\pi$
$83$	−8.16040	−0.895720	−0.447860	+	0.894104i	$0.647814\pi$
$84$	0	0
$85$	−7.74162	−0.839697
$86$	0	0
$87$	7.50497	0.804617
$88$	0	0
$89$	−7.06035	−0.748395	−0.374198	−	0.927349i	$-0.622082\pi$
$89$	−7.06035	−0.748395	−0.374198	+	0.927349i	$0.622082\pi$
$90$	0	0
$91$	−0.240970	−0.0252605
$92$	0	0
$93$	−2.35692	−0.244401
$94$	0	0
$95$	−0.603133	−0.0618802
$96$	0	0
$97$	−13.2465	−1.34498	−0.672491	−	0.740105i	$-0.734776\pi$
$97$	−13.2465	−1.34498	−0.672491	+	0.740105i	$0.734776\pi$
$98$	0	0
$99$	−1.23408	−0.124030
$100$	0	0
$101$	6.75245	0.671894	0.335947	−	0.941881i	$-0.390944\pi$
$101$	6.75245	0.671894	0.335947	+	0.941881i	$0.390944\pi$
$102$	0	0
$103$	−8.48430	−0.835983	−0.417992	−	0.908451i	$-0.637266\pi$
$103$	−8.48430	−0.835983	−0.417992	+	0.908451i	$0.637266\pi$
$104$	0	0
$105$	−0.514734	−0.0502328
$106$	0	0
$107$	−13.2333	−1.27931	−0.639655	−	0.768662i	$-0.720923\pi$
$107$	−13.2333	−1.27931	−0.639655	+	0.768662i	$0.720923\pi$
$108$	0	0
$109$	7.44781	0.713371	0.356685	−	0.934225i	$-0.383907\pi$
$109$	7.44781	0.713371	0.356685	+	0.934225i	$0.383907\pi$
$110$	0	0
$111$	7.72570	0.733291
$112$	0	0
$113$	3.18166	0.299305	0.149653	−	0.988739i	$-0.452184\pi$
$113$	3.18166	0.299305	0.149653	+	0.988739i	$0.452184\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−1.91226	−0.176789
$118$	0	0
$119$	0.238825	0.0218930
$120$	0	0
$121$	−9.47704	−0.861549
$122$	0	0
$123$	10.4766	0.944645
$124$	0	0
$125$	−27.3081	−2.44251
$126$	0	0
$127$	−20.1278	−1.78605	−0.893027	−	0.450003i	$-0.851423\pi$
$127$	−20.1278	−1.78605	−0.893027	+	0.450003i	$0.851423\pi$
$128$	0	0
$129$	−12.0719	−1.06287
$130$	0	0
$131$	−3.56351	−0.311345	−0.155673	−	0.987809i	$-0.549755\pi$
$131$	−3.56351	−0.311345	−0.155673	+	0.987809i	$0.549755\pi$
$132$	0	0
$133$	0.0186063	0.00161337
$134$	0	0
$135$	−4.08477	−0.351561
$136$	0	0
$137$	−20.4210	−1.74468	−0.872342	−	0.488897i	$-0.837399\pi$
$137$	−20.4210	−1.74468	−0.872342	+	0.488897i	$0.837399\pi$
$138$	0	0
$139$	14.3307	1.21551	0.607755	−	0.794125i	$-0.292070\pi$
$139$	14.3307	1.21551	0.607755	+	0.794125i	$0.292070\pi$
$140$	0	0
$141$	−5.71564	−0.481344
$142$	0	0
$143$	2.35989	0.197344
$144$	0	0
$145$	−30.6561	−2.54585
$146$	0	0
$147$	−6.98412	−0.576041
$148$	0	0
$149$	17.9630	1.47159	0.735795	−	0.677204i	$-0.236809\pi$
$149$	17.9630	1.47159	0.735795	+	0.677204i	$0.236809\pi$
$150$	0	0
$151$	−6.15695	−0.501045	−0.250523	−	0.968111i	$-0.580602\pi$
$151$	−6.15695	−0.501045	−0.250523	+	0.968111i	$0.580602\pi$
$152$	0	0
$153$	1.89524	0.153221
$154$	0	0
$155$	9.62748	0.773298
$156$	0	0
$157$	−7.73999	−0.617719	−0.308859	−	0.951108i	$-0.599947\pi$
$157$	−7.73999	−0.617719	−0.308859	+	0.951108i	$0.599947\pi$
$158$	0	0
$159$	6.67854	0.529643
$160$	0	0
$161$	0	0
$162$	0	0
$163$	12.8253	1.00455	0.502277	−	0.864707i	$-0.332496\pi$
$163$	12.8253	1.00455	0.502277	+	0.864707i	$0.332496\pi$
$164$	0	0
$165$	5.04095	0.392437
$166$	0	0
$167$	3.01521	0.233324	0.116662	−	0.993172i	$-0.462781\pi$
$167$	3.01521	0.233324	0.116662	+	0.993172i	$0.462781\pi$
$168$	0	0
$169$	−9.34325	−0.718712
$170$	0	0
$171$	0.147654	0.0112914
$172$	0	0
$173$	2.36422	0.179748	0.0898740	−	0.995953i	$-0.471354\pi$
$173$	2.36422	0.179748	0.0898740	+	0.995953i	$0.471354\pi$
$174$	0	0
$175$	1.47250	0.111311
$176$	0	0
$177$	7.61804	0.572607
$178$	0	0
$179$	−20.9041	−1.56245	−0.781225	−	0.624250i	$-0.785405\pi$
$179$	−20.9041	−1.56245	−0.781225	+	0.624250i	$0.785405\pi$
$180$	0	0
$181$	15.3297	1.13945	0.569726	−	0.821835i	$-0.307049\pi$
$181$	15.3297	1.13945	0.569726	+	0.821835i	$0.307049\pi$
$182$	0	0
$183$	1.27140	0.0939845
$184$	0	0
$185$	−31.5577	−2.32017
$186$	0	0
$187$	−2.33889	−0.171036
$188$	0	0
$189$	0.126013	0.00916609
$190$	0	0
$191$	−9.06388	−0.655839	−0.327920	−	0.944706i	$-0.606348\pi$
$191$	−9.06388	−0.655839	−0.327920	+	0.944706i	$0.606348\pi$
$192$	0	0
$193$	−9.53543	−0.686375	−0.343188	−	0.939267i	$-0.611507\pi$
$193$	−9.53543	−0.686375	−0.343188	+	0.939267i	$0.611507\pi$
$194$	0	0
$195$	7.81115	0.559368
$196$	0	0
$197$	2.59716	0.185040	0.0925199	−	0.995711i	$-0.470508\pi$
$197$	2.59716	0.185040	0.0925199	+	0.995711i	$0.470508\pi$
$198$	0	0
$199$	20.1487	1.42831	0.714153	−	0.699989i	$-0.246812\pi$
$199$	20.1487	1.42831	0.714153	+	0.699989i	$0.246812\pi$
$200$	0	0
$201$	−10.6380	−0.750345
$202$	0	0
$203$	0.945723	0.0663767
$204$	0	0
$205$	−42.7946	−2.98890
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−0.182218	−0.0126043
$210$	0	0
$211$	−23.8157	−1.63954	−0.819769	−	0.572695i	$-0.805898\pi$
$211$	−23.8157	−1.63954	−0.819769	+	0.572695i	$0.805898\pi$
$212$	0	0
$213$	9.24001	0.633115
$214$	0	0
$215$	49.3109	3.36297
$216$	0	0
$217$	−0.297002	−0.0201618
$218$	0	0
$219$	−12.6387	−0.854045
$220$	0	0
$221$	−3.62420	−0.243790
$222$	0	0
$223$	20.1471	1.34915	0.674576	−	0.738205i	$-0.264326\pi$
$223$	20.1471	1.34915	0.674576	+	0.738205i	$0.264326\pi$
$224$	0	0
$225$	11.6853	0.779023
$226$	0	0
$227$	9.37195	0.622038	0.311019	−	0.950404i	$-0.399330\pi$
$227$	9.37195	0.622038	0.311019	+	0.950404i	$0.399330\pi$
$228$	0	0
$229$	2.99434	0.197872	0.0989359	−	0.995094i	$-0.468456\pi$
$229$	2.99434	0.197872	0.0989359	+	0.995094i	$0.468456\pi$
$230$	0	0
$231$	−0.155510	−0.0102318
$232$	0	0
$233$	10.3570	0.678513	0.339256	−	0.940694i	$-0.389825\pi$
$233$	10.3570	0.678513	0.339256	+	0.940694i	$0.389825\pi$
$234$	0	0
$235$	23.3471	1.52300
$236$	0	0
$237$	−3.72925	−0.242241
$238$	0	0
$239$	3.39444	0.219568	0.109784	−	0.993955i	$-0.464984\pi$
$239$	3.39444	0.219568	0.109784	+	0.993955i	$0.464984\pi$
$240$	0	0
$241$	17.1828	1.10684	0.553421	−	0.832902i	$-0.313322\pi$
$241$	17.1828	1.10684	0.553421	+	0.832902i	$0.313322\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	28.5285	1.82262
$246$	0	0
$247$	−0.282354	−0.0179657
$248$	0	0
$249$	−8.16040	−0.517144
$250$	0	0
$251$	16.0561	1.01345	0.506727	−	0.862106i	$-0.330855\pi$
$251$	16.0561	1.01345	0.506727	+	0.862106i	$0.330855\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−7.74162	−0.484799
$256$	0	0
$257$	−22.0110	−1.37301	−0.686504	−	0.727126i	$-0.740856\pi$
$257$	−22.0110	−1.37301	−0.686504	+	0.727126i	$0.740856\pi$
$258$	0	0
$259$	0.973537	0.0604926
$260$	0	0
$261$	7.50497	0.464546
$262$	0	0
$263$	−22.7982	−1.40580	−0.702898	−	0.711290i	$-0.748111\pi$
$263$	−22.7982	−1.40580	−0.702898	+	0.711290i	$0.748111\pi$
$264$	0	0
$265$	−27.2803	−1.67582
$266$	0	0
$267$	−7.06035	−0.432086
$268$	0	0
$269$	5.42318	0.330657	0.165329	−	0.986239i	$-0.447132\pi$
$269$	5.42318	0.330657	0.165329	+	0.986239i	$0.447132\pi$
$270$	0	0
$271$	−25.3910	−1.54239	−0.771196	−	0.636598i	$-0.780341\pi$
$271$	−25.3910	−1.54239	−0.771196	+	0.636598i	$0.780341\pi$
$272$	0	0
$273$	−0.240970	−0.0145841
$274$	0	0
$275$	−14.4207	−0.869600
$276$	0	0
$277$	−28.3536	−1.70360	−0.851800	−	0.523867i	$-0.824489\pi$
$277$	−28.3536	−1.70360	−0.851800	+	0.523867i	$0.824489\pi$
$278$	0	0
$279$	−2.35692	−0.141105
$280$	0	0
$281$	−7.44206	−0.443956	−0.221978	−	0.975052i	$-0.571251\pi$
$281$	−7.44206	−0.443956	−0.221978	+	0.975052i	$0.571251\pi$
$282$	0	0
$283$	−24.3052	−1.44480	−0.722398	−	0.691477i	$-0.756960\pi$
$283$	−24.3052	−1.44480	−0.722398	+	0.691477i	$0.756960\pi$
$284$	0	0
$285$	−0.603133	−0.0357265
$286$	0	0
$287$	1.32019	0.0779283
$288$	0	0
$289$	−13.4081	−0.788709
$290$	0	0
$291$	−13.2465	−0.776526
$292$	0	0
$293$	16.3288	0.953939	0.476970	−	0.878920i	$-0.341735\pi$
$293$	16.3288	0.953939	0.476970	+	0.878920i	$0.341735\pi$
$294$	0	0
$295$	−31.1179	−1.81176
$296$	0	0
$297$	−1.23408	−0.0716088
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−1.52121	−0.0876812
$302$	0	0
$303$	6.75245	0.387918
$304$	0	0
$305$	−5.19337	−0.297371
$306$	0	0
$307$	24.9148	1.42196	0.710981	−	0.703211i	$-0.248251\pi$
$307$	24.9148	1.42196	0.710981	+	0.703211i	$0.248251\pi$
$308$	0	0
$309$	−8.48430	−0.482655
$310$	0	0
$311$	−28.2607	−1.60252	−0.801260	−	0.598317i	$-0.795837\pi$
$311$	−28.2607	−1.60252	−0.801260	+	0.598317i	$0.795837\pi$
$312$	0	0
$313$	−5.51343	−0.311638	−0.155819	−	0.987786i	$-0.549802\pi$
$313$	−5.51343	−0.311638	−0.155819	+	0.987786i	$0.549802\pi$
$314$	0	0
$315$	−0.514734	−0.0290019
$316$	0	0
$317$	−3.25491	−0.182814	−0.0914070	−	0.995814i	$-0.529136\pi$
$317$	−3.25491	−0.182814	−0.0914070	+	0.995814i	$0.529136\pi$
$318$	0	0
$319$	−9.26176	−0.518559
$320$	0	0
$321$	−13.2333	−0.738610
$322$	0	0
$323$	0.279840	0.0155707
$324$	0	0
$325$	−22.3454	−1.23950
$326$	0	0
$327$	7.44781	0.411865
$328$	0	0
$329$	−0.720245	−0.0397084
$330$	0	0
$331$	6.81649	0.374668	0.187334	−	0.982296i	$-0.440015\pi$
$331$	6.81649	0.374668	0.187334	+	0.982296i	$0.440015\pi$
$332$	0	0
$333$	7.72570	0.423365
$334$	0	0
$335$	43.4537	2.37413
$336$	0	0
$337$	4.60739	0.250981	0.125490	−	0.992095i	$-0.459950\pi$
$337$	4.60739	0.250981	0.125490	+	0.992095i	$0.459950\pi$
$338$	0	0
$339$	3.18166	0.172804
$340$	0	0
$341$	2.90864	0.157512
$342$	0	0
$343$	−1.76218	−0.0951487
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−22.3847	−1.20167	−0.600837	−	0.799372i	$-0.705166\pi$
$347$	−22.3847	−1.20167	−0.600837	+	0.799372i	$0.705166\pi$
$348$	0	0
$349$	21.2814	1.13917	0.569584	−	0.821933i	$-0.307104\pi$
$349$	21.2814	1.13917	0.569584	+	0.821933i	$0.307104\pi$
$350$	0	0
$351$	−1.91226	−0.102069
$352$	0	0
$353$	−6.29333	−0.334960	−0.167480	−	0.985875i	$-0.553563\pi$
$353$	−6.29333	−0.334960	−0.167480	+	0.985875i	$0.553563\pi$
$354$	0	0
$355$	−37.7433	−2.00321
$356$	0	0
$357$	0.238825	0.0126400
$358$	0	0
$359$	−15.0107	−0.792234	−0.396117	−	0.918200i	$-0.629643\pi$
$359$	−15.0107	−0.792234	−0.396117	+	0.918200i	$0.629643\pi$
$360$	0	0
$361$	−18.9782	−0.998853
$362$	0	0
$363$	−9.47704	−0.497415
$364$	0	0
$365$	51.6262	2.70224
$366$	0	0
$367$	8.18117	0.427053	0.213527	−	0.976937i	$-0.431505\pi$
$367$	8.18117	0.427053	0.213527	+	0.976937i	$0.431505\pi$
$368$	0	0
$369$	10.4766	0.545391
$370$	0	0
$371$	0.841583	0.0436928
$372$	0	0
$373$	−8.78027	−0.454625	−0.227313	−	0.973822i	$-0.572994\pi$
$373$	−8.78027	−0.454625	−0.227313	+	0.973822i	$0.572994\pi$
$374$	0	0
$375$	−27.3081	−1.41018
$376$	0	0
$377$	−14.3515	−0.739138
$378$	0	0
$379$	6.76065	0.347271	0.173636	−	0.984810i	$-0.444448\pi$
$379$	6.76065	0.347271	0.173636	+	0.984810i	$0.444448\pi$
$380$	0	0
$381$	−20.1278	−1.03118
$382$	0	0
$383$	0.531604	0.0271637	0.0135818	−	0.999908i	$-0.495677\pi$
$383$	0.531604	0.0271637	0.0135818	+	0.999908i	$0.495677\pi$
$384$	0	0
$385$	0.635224	0.0323740
$386$	0	0
$387$	−12.0719	−0.613648
$388$	0	0
$389$	−4.67978	−0.237274	−0.118637	−	0.992938i	$-0.537853\pi$
$389$	−4.67978	−0.237274	−0.118637	+	0.992938i	$0.537853\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−3.56351	−0.179755
$394$	0	0
$395$	15.2331	0.766462
$396$	0	0
$397$	3.43651	0.172473	0.0862367	−	0.996275i	$-0.472516\pi$
$397$	3.43651	0.172473	0.0862367	+	0.996275i	$0.472516\pi$
$398$	0	0
$399$	0.0186063	0.000931482 0
$400$	0	0
$401$	−13.1122	−0.654793	−0.327396	−	0.944887i	$-0.606171\pi$
$401$	−13.1122	−0.654793	−0.327396	+	0.944887i	$0.606171\pi$
$402$	0	0
$403$	4.50705	0.224512
$404$	0	0
$405$	−4.08477	−0.202974
$406$	0	0
$407$	−9.53416	−0.472591
$408$	0	0
$409$	−4.10492	−0.202975	−0.101488	−	0.994837i	$-0.532360\pi$
$409$	−4.10492	−0.202975	−0.101488	+	0.994837i	$0.532360\pi$
$410$	0	0
$411$	−20.4210	−1.00729
$412$	0	0
$413$	0.959971	0.0472371
$414$	0	0
$415$	33.3333	1.63627
$416$	0	0
$417$	14.3307	0.701775
$418$	0	0
$419$	−37.2665	−1.82059	−0.910295	−	0.413961i	$-0.864145\pi$
$419$	−37.2665	−1.82059	−0.910295	+	0.413961i	$0.864145\pi$
$420$	0	0
$421$	−9.63522	−0.469592	−0.234796	−	0.972045i	$-0.575442\pi$
$421$	−9.63522	−0.469592	−0.234796	+	0.972045i	$0.575442\pi$
$422$	0	0
$423$	−5.71564	−0.277904
$424$	0	0
$425$	22.1465	1.07427
$426$	0	0
$427$	0.160213	0.00775323
$428$	0	0
$429$	2.35989	0.113937
$430$	0	0
$431$	32.0510	1.54384	0.771920	−	0.635720i	$-0.219297\pi$
$431$	32.0510	1.54384	0.771920	+	0.635720i	$0.219297\pi$
$432$	0	0
$433$	24.7383	1.18885	0.594423	−	0.804152i	$-0.297380\pi$
$433$	24.7383	1.18885	0.594423	+	0.804152i	$0.297380\pi$
$434$	0	0
$435$	−30.6561	−1.46985
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−8.36162	−0.399078	−0.199539	−	0.979890i	$-0.563945\pi$
$439$	−8.36162	−0.399078	−0.199539	+	0.979890i	$0.563945\pi$
$440$	0	0
$441$	−6.98412	−0.332577
$442$	0	0
$443$	−8.87163	−0.421504	−0.210752	−	0.977540i	$-0.567591\pi$
$443$	−8.87163	−0.421504	−0.210752	+	0.977540i	$0.567591\pi$
$444$	0	0
$445$	28.8399	1.36714
$446$	0	0
$447$	17.9630	0.849623
$448$	0	0
$449$	−10.0137	−0.472577	−0.236288	−	0.971683i	$-0.575931\pi$
$449$	−10.0137	−0.472577	−0.236288	+	0.971683i	$0.575931\pi$
$450$	0	0
$451$	−12.9290	−0.608804
$452$	0	0
$453$	−6.15695	−0.289279
$454$	0	0
$455$	0.984306	0.0461450
$456$	0	0
$457$	−29.5735	−1.38339	−0.691695	−	0.722189i	$-0.743136\pi$
$457$	−29.5735	−1.38339	−0.691695	+	0.722189i	$0.743136\pi$
$458$	0	0
$459$	1.89524	0.0884623
$460$	0	0
$461$	−17.8301	−0.830430	−0.415215	−	0.909723i	$-0.636294\pi$
$461$	−17.8301	−0.830430	−0.415215	+	0.909723i	$0.636294\pi$
$462$	0	0
$463$	−9.85796	−0.458138	−0.229069	−	0.973410i	$-0.573568\pi$
$463$	−9.85796	−0.458138	−0.229069	+	0.973410i	$0.573568\pi$
$464$	0	0
$465$	9.62748	0.446464
$466$	0	0
$467$	38.5985	1.78613	0.893063	−	0.449931i	$-0.148551\pi$
$467$	38.5985	1.78613	0.893063	+	0.449931i	$0.148551\pi$
$468$	0	0
$469$	−1.34052	−0.0618995
$470$	0	0
$471$	−7.73999	−0.356640
$472$	0	0
$473$	14.8977	0.684998
$474$	0	0
$475$	1.72539	0.0791663
$476$	0	0
$477$	6.67854	0.305790
$478$	0	0
$479$	6.78236	0.309894	0.154947	−	0.987923i	$-0.450479\pi$
$479$	6.78236	0.309894	0.154947	+	0.987923i	$0.450479\pi$
$480$	0	0
$481$	−14.7736	−0.673616
$482$	0	0
$483$	0	0
$484$	0	0
$485$	54.1091	2.45697
$486$	0	0
$487$	−6.67523	−0.302484	−0.151242	−	0.988497i	$-0.548327\pi$
$487$	−6.67523	−0.302484	−0.151242	+	0.988497i	$0.548327\pi$
$488$	0	0
$489$	12.8253	0.579980
$490$	0	0
$491$	−17.7889	−0.802803	−0.401401	−	0.915902i	$-0.631477\pi$
$491$	−17.7889	−0.802803	−0.401401	+	0.915902i	$0.631477\pi$
$492$	0	0
$493$	14.2237	0.640605
$494$	0	0
$495$	5.04095	0.226574
$496$	0	0
$497$	1.16436	0.0522287
$498$	0	0
$499$	−1.51520	−0.0678295	−0.0339147	−	0.999425i	$-0.510797\pi$
$499$	−1.51520	−0.0678295	−0.0339147	+	0.999425i	$0.510797\pi$
$500$	0	0
$501$	3.01521	0.134710
$502$	0	0
$503$	−17.9057	−0.798377	−0.399188	−	0.916869i	$-0.630708\pi$
$503$	−17.9057	−0.798377	−0.399188	+	0.916869i	$0.630708\pi$
$504$	0	0
$505$	−27.5822	−1.22739
$506$	0	0
$507$	−9.34325	−0.414948
$508$	0	0
$509$	−18.9719	−0.840914	−0.420457	−	0.907312i	$-0.638130\pi$
$509$	−18.9719	−0.840914	−0.420457	+	0.907312i	$0.638130\pi$
$510$	0	0
$511$	−1.59264	−0.0704543
$512$	0	0
$513$	0.147654	0.00651909
$514$	0	0
$515$	34.6564	1.52714
$516$	0	0
$517$	7.05358	0.310216
$518$	0	0
$519$	2.36422	0.103778
$520$	0	0
$521$	31.5083	1.38040	0.690201	−	0.723617i	$-0.257522\pi$
$521$	31.5083	1.38040	0.690201	+	0.723617i	$0.257522\pi$
$522$	0	0
$523$	−17.2164	−0.752821	−0.376410	−	0.926453i	$-0.622842\pi$
$523$	−17.2164	−0.752821	−0.376410	+	0.926453i	$0.622842\pi$
$524$	0	0
$525$	1.47250	0.0642653
$526$	0	0
$527$	−4.46694	−0.194583
$528$	0	0
$529$	0	0
$530$	0	0
$531$	7.61804	0.330595
$532$	0	0
$533$	−20.0340	−0.867770
$534$	0	0
$535$	54.0549	2.33700
$536$	0	0
$537$	−20.9041	−0.902081
$538$	0	0
$539$	8.61899	0.371246
$540$	0	0
$541$	0.273187	0.0117452	0.00587261	−	0.999983i	$-0.498131\pi$
$541$	0.273187	0.0117452	0.00587261	+	0.999983i	$0.498131\pi$
$542$	0	0
$543$	15.3297	0.657863
$544$	0	0
$545$	−30.4226	−1.30316
$546$	0	0
$547$	26.4427	1.13061	0.565305	−	0.824882i	$-0.308758\pi$
$547$	26.4427	1.13061	0.565305	+	0.824882i	$0.308758\pi$
$548$	0	0
$549$	1.27140	0.0542620
$550$	0	0
$551$	1.10814	0.0472084
$552$	0	0
$553$	−0.469934	−0.0199836
$554$	0	0
$555$	−31.5577	−1.33955
$556$	0	0
$557$	24.7095	1.04698	0.523488	−	0.852033i	$-0.324631\pi$
$557$	24.7095	1.04698	0.523488	+	0.852033i	$0.324631\pi$
$558$	0	0
$559$	23.0846	0.976375
$560$	0	0
$561$	−2.33889	−0.0987479
$562$	0	0
$563$	16.8824	0.711507	0.355754	−	0.934580i	$-0.384224\pi$
$563$	16.8824	0.711507	0.355754	+	0.934580i	$0.384224\pi$
$564$	0	0
$565$	−12.9963	−0.546760
$566$	0	0
$567$	0.126013	0.00529204
$568$	0	0
$569$	9.76100	0.409202	0.204601	−	0.978845i	$-0.434410\pi$
$569$	9.76100	0.409202	0.204601	+	0.978845i	$0.434410\pi$
$570$	0	0
$571$	−14.6385	−0.612603	−0.306302	−	0.951935i	$-0.599092\pi$
$571$	−14.6385	−0.612603	−0.306302	+	0.951935i	$0.599092\pi$
$572$	0	0
$573$	−9.06388	−0.378649
$574$	0	0
$575$	0	0
$576$	0	0
$577$	12.3100	0.512472	0.256236	−	0.966614i	$-0.417518\pi$
$577$	12.3100	0.512472	0.256236	+	0.966614i	$0.417518\pi$
$578$	0	0
$579$	−9.53543	−0.396279
$580$	0	0
$581$	−1.02831	−0.0426617
$582$	0	0
$583$	−8.24188	−0.341344
$584$	0	0
$585$	7.81115	0.322951
$586$	0	0
$587$	12.0812	0.498644	0.249322	−	0.968421i	$-0.419792\pi$
$587$	12.0812	0.498644	0.249322	+	0.968421i	$0.419792\pi$
$588$	0	0
$589$	−0.348009	−0.0143395
$590$	0	0
$591$	2.59716	0.106833
$592$	0	0
$593$	−43.4007	−1.78225	−0.891127	−	0.453754i	$-0.850085\pi$
$593$	−43.4007	−1.78225	−0.891127	+	0.453754i	$0.850085\pi$
$594$	0	0
$595$	−0.975544	−0.0399934
$596$	0	0
$597$	20.1487	0.824633
$598$	0	0
$599$	−36.5272	−1.49246	−0.746230	−	0.665688i	$-0.768138\pi$
$599$	−36.5272	−1.49246	−0.746230	+	0.665688i	$0.768138\pi$
$600$	0	0
$601$	−0.669700	−0.0273176	−0.0136588	−	0.999907i	$-0.504348\pi$
$601$	−0.669700	−0.0273176	−0.0136588	+	0.999907i	$0.504348\pi$
$602$	0	0
$603$	−10.6380	−0.433212
$604$	0	0
$605$	38.7115	1.57385
$606$	0	0
$607$	36.1392	1.46685	0.733423	−	0.679772i	$-0.237921\pi$
$607$	36.1392	1.46685	0.733423	+	0.679772i	$0.237921\pi$
$608$	0	0
$609$	0.945723	0.0383226
$610$	0	0
$611$	10.9298	0.442173
$612$	0	0
$613$	−39.5553	−1.59762	−0.798811	−	0.601582i	$-0.794537\pi$
$613$	−39.5553	−1.59762	−0.798811	+	0.601582i	$0.794537\pi$
$614$	0	0
$615$	−42.7946	−1.72564
$616$	0	0
$617$	−5.73065	−0.230707	−0.115354	−	0.993324i	$-0.536800\pi$
$617$	−5.73065	−0.230707	−0.115354	+	0.993324i	$0.536800\pi$
$618$	0	0
$619$	26.3197	1.05788	0.528938	−	0.848660i	$-0.322590\pi$
$619$	26.3197	1.05788	0.528938	+	0.848660i	$0.322590\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−0.889695	−0.0356449
$624$	0	0
$625$	53.1205	2.12482
$626$	0	0
$627$	−0.182218	−0.00727707
$628$	0	0
$629$	14.6421	0.583817
$630$	0	0
$631$	28.7295	1.14370	0.571852	−	0.820357i	$-0.306225\pi$
$631$	28.7295	1.14370	0.571852	+	0.820357i	$0.306225\pi$
$632$	0	0
$633$	−23.8157	−0.946588
$634$	0	0
$635$	82.2174	3.26270
$636$	0	0
$637$	13.3555	0.529163
$638$	0	0
$639$	9.24001	0.365529
$640$	0	0
$641$	−22.1048	−0.873089	−0.436544	−	0.899683i	$-0.643798\pi$
$641$	−22.1048	−0.873089	−0.436544	+	0.899683i	$0.643798\pi$
$642$	0	0
$643$	−11.1875	−0.441193	−0.220596	−	0.975365i	$-0.570800\pi$
$643$	−11.1875	−0.441193	−0.220596	+	0.975365i	$0.570800\pi$
$644$	0	0
$645$	49.3109	1.94161
$646$	0	0
$647$	26.0083	1.02249	0.511245	−	0.859435i	$-0.329184\pi$
$647$	26.0083	1.02249	0.511245	+	0.859435i	$0.329184\pi$
$648$	0	0
$649$	−9.40129	−0.369033
$650$	0	0
$651$	−0.297002	−0.0116404
$652$	0	0
$653$	45.6222	1.78533	0.892667	−	0.450718i	$-0.148832\pi$
$653$	45.6222	1.78533	0.892667	+	0.450718i	$0.148832\pi$
$654$	0	0
$655$	14.5561	0.568755
$656$	0	0
$657$	−12.6387	−0.493083
$658$	0	0
$659$	−30.6399	−1.19356	−0.596781	−	0.802404i	$-0.703554\pi$
$659$	−30.6399	−1.19356	−0.596781	+	0.802404i	$0.703554\pi$
$660$	0	0
$661$	15.7545	0.612777	0.306389	−	0.951906i	$-0.400879\pi$
$661$	15.7545	0.612777	0.306389	+	0.951906i	$0.400879\pi$
$662$	0	0
$663$	−3.62420	−0.140752
$664$	0	0
$665$	−0.0760026	−0.00294725
$666$	0	0
$667$	0	0
$668$	0	0
$669$	20.1471	0.778933
$670$	0	0
$671$	−1.56901	−0.0605710
$672$	0	0
$673$	10.1298	0.390475	0.195237	−	0.980756i	$-0.437452\pi$
$673$	10.1298	0.390475	0.195237	+	0.980756i	$0.437452\pi$
$674$	0	0
$675$	11.6853	0.449769
$676$	0	0
$677$	−11.8067	−0.453769	−0.226884	−	0.973922i	$-0.572854\pi$
$677$	−11.8067	−0.453769	−0.226884	+	0.973922i	$0.572854\pi$
$678$	0	0
$679$	−1.66924	−0.0640594
$680$	0	0
$681$	9.37195	0.359134
$682$	0	0
$683$	−11.9913	−0.458835	−0.229418	−	0.973328i	$-0.573682\pi$
$683$	−11.9913	−0.458835	−0.229418	+	0.973328i	$0.573682\pi$
$684$	0	0
$685$	83.4150	3.18712
$686$	0	0
$687$	2.99434	0.114241
$688$	0	0
$689$	−12.7711	−0.486541
$690$	0	0
$691$	20.6233	0.784548	0.392274	−	0.919848i	$-0.371689\pi$
$691$	20.6233	0.784548	0.392274	+	0.919848i	$0.371689\pi$
$692$	0	0
$693$	−0.155510	−0.00590735
$694$	0	0
$695$	−58.5374	−2.22045
$696$	0	0
$697$	19.8557	0.752089
$698$	0	0
$699$	10.3570	0.391740
$700$	0	0
$701$	−44.8202	−1.69284	−0.846418	−	0.532519i	$-0.821245\pi$
$701$	−44.8202	−1.69284	−0.846418	+	0.532519i	$0.821245\pi$
$702$	0	0
$703$	1.14073	0.0430235
$704$	0	0
$705$	23.3471	0.879302
$706$	0	0
$707$	0.850895	0.0320012
$708$	0	0
$709$	37.9378	1.42479	0.712393	−	0.701781i	$-0.247612\pi$
$709$	37.9378	1.42479	0.712393	+	0.701781i	$0.247612\pi$
$710$	0	0
$711$	−3.72925	−0.139858
$712$	0	0
$713$	0	0
$714$	0	0
$715$	−9.63961	−0.360501
$716$	0	0
$717$	3.39444	0.126768
$718$	0	0
$719$	−36.9829	−1.37923	−0.689614	−	0.724177i	$-0.742220\pi$
$719$	−36.9829	−1.37923	−0.689614	+	0.724177i	$0.742220\pi$
$720$	0	0
$721$	−1.06913	−0.0398165
$722$	0	0
$723$	17.1828	0.639035
$724$	0	0
$725$	87.6981	3.25703
$726$	0	0
$727$	−46.5240	−1.72548	−0.862740	−	0.505648i	$-0.831254\pi$
$727$	−46.5240	−1.72548	−0.862740	+	0.505648i	$0.831254\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−22.8791	−0.846215
$732$	0	0
$733$	−3.07625	−0.113624	−0.0568119	−	0.998385i	$-0.518094\pi$
$733$	−3.07625	−0.113624	−0.0568119	+	0.998385i	$0.518094\pi$
$734$	0	0
$735$	28.5285	1.05229
$736$	0	0
$737$	13.1282	0.483582
$738$	0	0
$739$	23.4363	0.862117	0.431058	−	0.902324i	$-0.358140\pi$
$739$	23.4363	0.862117	0.431058	+	0.902324i	$0.358140\pi$
$740$	0	0
$741$	−0.282354	−0.0103725
$742$	0	0
$743$	1.63755	0.0600758	0.0300379	−	0.999549i	$-0.490437\pi$
$743$	1.63755	0.0600758	0.0300379	+	0.999549i	$0.490437\pi$
$744$	0	0
$745$	−73.3749	−2.68825
$746$	0	0
$747$	−8.16040	−0.298573
$748$	0	0
$749$	−1.66756	−0.0609315
$750$	0	0
$751$	−42.5273	−1.55184	−0.775921	−	0.630830i	$-0.782715\pi$
$751$	−42.5273	−1.55184	−0.775921	+	0.630830i	$0.782715\pi$
$752$	0	0
$753$	16.0561	0.585118
$754$	0	0
$755$	25.1497	0.915292
$756$	0	0
$757$	−8.90630	−0.323705	−0.161853	−	0.986815i	$-0.551747\pi$
$757$	−8.90630	−0.323705	−0.161853	+	0.986815i	$0.551747\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	38.5544	1.39759	0.698797	−	0.715320i	$-0.253719\pi$
$761$	38.5544	1.39759	0.698797	+	0.715320i	$0.253719\pi$
$762$	0	0
$763$	0.938520	0.0339767
$764$	0	0
$765$	−7.74162	−0.279899
$766$	0	0
$767$	−14.5677	−0.526008
$768$	0	0
$769$	51.4184	1.85420	0.927098	−	0.374820i	$-0.122295\pi$
$769$	51.4184	1.85420	0.927098	+	0.374820i	$0.122295\pi$
$770$	0	0
$771$	−22.0110	−0.792706
$772$	0	0
$773$	−25.5148	−0.917703	−0.458852	−	0.888513i	$-0.651739\pi$
$773$	−25.5148	−0.917703	−0.458852	+	0.888513i	$0.651739\pi$
$774$	0	0
$775$	−27.5414	−0.989317
$776$	0	0
$777$	0.973537	0.0349254
$778$	0	0
$779$	1.54692	0.0554240
$780$	0	0
$781$	−11.4029	−0.408029
$782$	0	0
$783$	7.50497	0.268206
$784$	0	0
$785$	31.6161	1.12843
$786$	0	0
$787$	−10.2359	−0.364869	−0.182434	−	0.983218i	$-0.558398\pi$
$787$	−10.2359	−0.364869	−0.182434	+	0.983218i	$0.558398\pi$
$788$	0	0
$789$	−22.7982	−0.811637
$790$	0	0
$791$	0.400930	0.0142554
$792$	0	0
$793$	−2.43125	−0.0863361
$794$	0	0
$795$	−27.2803	−0.967533
$796$	0	0
$797$	1.66514	0.0589821	0.0294911	−	0.999565i	$-0.490611\pi$
$797$	1.66514	0.0589821	0.0294911	+	0.999565i	$0.490611\pi$
$798$	0	0
$799$	−10.8325	−0.383227
$800$	0	0
$801$	−7.06035	−0.249465
$802$	0	0
$803$	15.5972	0.550414
$804$	0	0
$805$	0	0
$806$	0	0
$807$	5.42318	0.190905
$808$	0	0
$809$	8.02810	0.282253	0.141126	−	0.989992i	$-0.454928\pi$
$809$	8.02810	0.282253	0.141126	+	0.989992i	$0.454928\pi$
$810$	0	0
$811$	54.9737	1.93039	0.965194	−	0.261534i	$-0.0842283\pi$
$811$	54.9737	1.93039	0.965194	+	0.261534i	$0.0842283\pi$
$812$	0	0
$813$	−25.3910	−0.890500
$814$	0	0
$815$	−52.3884	−1.83508
$816$	0	0
$817$	−1.78246	−0.0623605
$818$	0	0
$819$	−0.240970	−0.00842016
$820$	0	0
$821$	19.1373	0.667897	0.333948	−	0.942591i	$-0.391619\pi$
$821$	19.1373	0.667897	0.333948	+	0.942591i	$0.391619\pi$
$822$	0	0
$823$	21.1905	0.738655	0.369327	−	0.929299i	$-0.379588\pi$
$823$	21.1905	0.738655	0.369327	+	0.929299i	$0.379588\pi$
$824$	0	0
$825$	−14.4207	−0.502064
$826$	0	0
$827$	29.2748	1.01798	0.508992	−	0.860771i	$-0.330018\pi$
$827$	29.2748	1.01798	0.508992	+	0.860771i	$0.330018\pi$
$828$	0	0
$829$	−31.5164	−1.09461	−0.547304	−	0.836934i	$-0.684346\pi$
$829$	−31.5164	−1.09461	−0.547304	+	0.836934i	$0.684346\pi$
$830$	0	0
$831$	−28.3536	−0.983574
$832$	0	0
$833$	−13.2366	−0.458621
$834$	0	0
$835$	−12.3164	−0.426228
$836$	0	0
$837$	−2.35692	−0.0814671
$838$	0	0
$839$	−10.4437	−0.360556	−0.180278	−	0.983616i	$-0.557700\pi$
$839$	−10.4437	−0.360556	−0.180278	+	0.983616i	$0.557700\pi$
$840$	0	0
$841$	27.3246	0.942227
$842$	0	0
$843$	−7.44206	−0.256318
$844$	0	0
$845$	38.1650	1.31292
$846$	0	0
$847$	−1.19423	−0.0410342
$848$	0	0
$849$	−24.3052	−0.834154
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−18.2810	−0.625930	−0.312965	−	0.949765i	$-0.601322\pi$
$853$	−18.2810	−0.625930	−0.312965	+	0.949765i	$0.601322\pi$
$854$	0	0
$855$	−0.603133	−0.0206267
$856$	0	0
$857$	−31.2900	−1.06884	−0.534422	−	0.845218i	$-0.679471\pi$
$857$	−31.2900	−1.06884	−0.534422	+	0.845218i	$0.679471\pi$
$858$	0	0
$859$	−9.64053	−0.328930	−0.164465	−	0.986383i	$-0.552590\pi$
$859$	−9.64053	−0.328930	−0.164465	+	0.986383i	$0.552590\pi$
$860$	0	0
$861$	1.32019	0.0449919
$862$	0	0
$863$	−10.5991	−0.360798	−0.180399	−	0.983593i	$-0.557739\pi$
$863$	−10.5991	−0.360798	−0.180399	+	0.983593i	$0.557739\pi$
$864$	0	0
$865$	−9.65728	−0.328357
$866$	0	0
$867$	−13.4081	−0.455362
$868$	0	0
$869$	4.60221	0.156119
$870$	0	0
$871$	20.3426	0.689283
$872$	0	0
$873$	−13.2465	−0.448328
$874$	0	0
$875$	−3.44117	−0.116333
$876$	0	0
$877$	5.65848	0.191073	0.0955367	−	0.995426i	$-0.469543\pi$
$877$	5.65848	0.191073	0.0955367	+	0.995426i	$0.469543\pi$
$878$	0	0
$879$	16.3288	0.550757
$880$	0	0
$881$	50.3047	1.69481	0.847404	−	0.530949i	$-0.178164\pi$
$881$	50.3047	1.69481	0.847404	+	0.530949i	$0.178164\pi$
$882$	0	0
$883$	17.5390	0.590234	0.295117	−	0.955461i	$-0.404641\pi$
$883$	17.5390	0.590234	0.295117	+	0.955461i	$0.404641\pi$
$884$	0	0
$885$	−31.1179	−1.04602
$886$	0	0
$887$	−27.1689	−0.912242	−0.456121	−	0.889918i	$-0.650762\pi$
$887$	−27.1689	−0.912242	−0.456121	+	0.889918i	$0.650762\pi$
$888$	0	0
$889$	−2.53636	−0.0850669
$890$	0	0
$891$	−1.23408	−0.0413434
$892$	0	0
$893$	−0.843939	−0.0282413
$894$	0	0
$895$	85.3886	2.85423
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−17.6886	−0.589949
$900$	0	0
$901$	12.6575	0.421681
$902$	0	0
$903$	−1.52121	−0.0506228
$904$	0	0
$905$	−62.6185	−2.08151
$906$	0	0
$907$	−33.2686	−1.10466	−0.552332	−	0.833624i	$-0.686262\pi$
$907$	−33.2686	−1.10466	−0.552332	+	0.833624i	$0.686262\pi$
$908$	0	0
$909$	6.75245	0.223965
$910$	0	0
$911$	38.0645	1.26113	0.630567	−	0.776135i	$-0.282822\pi$
$911$	38.0645	1.26113	0.630567	+	0.776135i	$0.282822\pi$
$912$	0	0
$913$	10.0706	0.333289
$914$	0	0
$915$	−5.19337	−0.171688
$916$	0	0
$917$	−0.449048	−0.0148289
$918$	0	0
$919$	24.8732	0.820491	0.410246	−	0.911975i	$-0.365443\pi$
$919$	24.8732	0.820491	0.410246	+	0.911975i	$0.365443\pi$
$920$	0	0
$921$	24.9148	0.820970
$922$	0	0
$923$	−17.6693	−0.581593
$924$	0	0
$925$	90.2774	2.96830
$926$	0	0
$927$	−8.48430	−0.278661
$928$	0	0
$929$	−16.4725	−0.540446	−0.270223	−	0.962798i	$-0.587097\pi$
$929$	−16.4725	−0.540446	−0.270223	+	0.962798i	$0.587097\pi$
$930$	0	0
$931$	−1.03123	−0.0337974
$932$	0	0
$933$	−28.2607	−0.925215
$934$	0	0
$935$	9.55381	0.312443
$936$	0	0
$937$	−8.26555	−0.270024	−0.135012	−	0.990844i	$-0.543107\pi$
$937$	−8.26555	−0.270024	−0.135012	+	0.990844i	$0.543107\pi$
$938$	0	0
$939$	−5.51343	−0.179924
$940$	0	0
$941$	5.63931	0.183836	0.0919181	−	0.995767i	$-0.470700\pi$
$941$	5.63931	0.183836	0.0919181	+	0.995767i	$0.470700\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	−0.514734	−0.0167443
$946$	0	0
$947$	−24.8510	−0.807549	−0.403774	−	0.914859i	$-0.632302\pi$
$947$	−24.8510	−0.807549	−0.403774	+	0.914859i	$0.632302\pi$
$948$	0	0
$949$	24.1685	0.784544
$950$	0	0
$951$	−3.25491	−0.105548
$952$	0	0
$953$	50.4672	1.63479	0.817396	−	0.576076i	$-0.195417\pi$
$953$	50.4672	1.63479	0.817396	+	0.576076i	$0.195417\pi$
$954$	0	0
$955$	37.0238	1.19806
$956$	0	0
$957$	−9.26176	−0.299390
$958$	0	0
$959$	−2.57331	−0.0830964
$960$	0	0
$961$	−25.4449	−0.820804
$962$	0	0
$963$	−13.2333	−0.426437
$964$	0	0
$965$	38.9500	1.25385
$966$	0	0
$967$	1.31499	0.0422872	0.0211436	−	0.999776i	$-0.493269\pi$
$967$	1.31499	0.0422872	0.0211436	+	0.999776i	$0.493269\pi$
$968$	0	0
$969$	0.279840	0.00898977
$970$	0	0
$971$	21.9465	0.704297	0.352148	−	0.935944i	$-0.385451\pi$
$971$	21.9465	0.704297	0.352148	+	0.935944i	$0.385451\pi$
$972$	0	0
$973$	1.80585	0.0578928
$974$	0	0
$975$	−22.3454	−0.715627
$976$	0	0
$977$	22.6258	0.723864	0.361932	−	0.932204i	$-0.382117\pi$
$977$	22.6258	0.723864	0.361932	+	0.932204i	$0.382117\pi$
$978$	0	0
$979$	8.71306	0.278471
$980$	0	0
$981$	7.44781	0.237790
$982$	0	0
$983$	−20.0305	−0.638873	−0.319437	−	0.947608i	$-0.603494\pi$
$983$	−20.0305	−0.638873	−0.319437	+	0.947608i	$0.603494\pi$
$984$	0	0
$985$	−10.6088	−0.338024
$986$	0	0
$987$	−0.720245	−0.0229256
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−55.0019	−1.74719	−0.873596	−	0.486651i	$-0.838218\pi$
$991$	−55.0019	−1.74719	−0.873596	+	0.486651i	$0.838218\pi$
$992$	0	0
$993$	6.81649	0.216315
$994$	0	0
$995$	−82.3030	−2.60918
$996$	0	0
$997$	33.9561	1.07540	0.537700	−	0.843136i	$-0.319293\pi$
$997$	33.9561	1.07540	0.537700	+	0.843136i	$0.319293\pi$
$998$	0	0
$999$	7.72570	0.244430

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6348.2.a.s.1.1		10
23.9	even	11		276.2.i.a.265.2	yes	20
23.18	even	11		276.2.i.a.25.2	✓	20
23.22	odd	2		6348.2.a.t.1.10		10
69.32	odd	22		828.2.q.c.541.1		20
69.41	odd	22		828.2.q.c.577.1		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
276.2.i.a.25.2	✓	20	23.18	even	11
276.2.i.a.265.2	yes	20	23.9	even	11
828.2.q.c.541.1		20	69.32	odd	22
828.2.q.c.577.1		20	69.41	odd	22
6348.2.a.s.1.1		10	1.1	even	1	trivial
6348.2.a.t.1.10		10	23.22	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6348 = 2^{2} \cdot 3 \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6348.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -4.08477 q^{5} +0.126013 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -4.08477 q^{5} +0.126013 q^{7} +1.00000 q^{9} -1.23408 q^{11} -1.91226 q^{13} -4.08477 q^{15} +1.89524 q^{17} +0.147654 q^{19} +0.126013 q^{21} +11.6853 q^{25} +1.00000 q^{27} +7.50497 q^{29} -2.35692 q^{31} -1.23408 q^{33} -0.514734 q^{35} +7.72570 q^{37} -1.91226 q^{39} +10.4766 q^{41} -12.0719 q^{43} -4.08477 q^{45} -5.71564 q^{47} -6.98412 q^{49} +1.89524 q^{51} +6.67854 q^{53} +5.04095 q^{55} +0.147654 q^{57} +7.61804 q^{59} +1.27140 q^{61} +0.126013 q^{63} +7.81115 q^{65} -10.6380 q^{67} +9.24001 q^{71} -12.6387 q^{73} +11.6853 q^{75} -0.155510 q^{77} -3.72925 q^{79} +1.00000 q^{81} -8.16040 q^{83} -7.74162 q^{85} +7.50497 q^{87} -7.06035 q^{89} -0.240970 q^{91} -2.35692 q^{93} -0.603133 q^{95} -13.2465 q^{97} -1.23408 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 10 q^{3} - 2 q^{5} - 11 q^{7} + 10 q^{9}+O(q^{10}) \) 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

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