LMFDB - Embedded newform 6354.2.a.bl.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6354,2,Mod(1,6354)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6354, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6354.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6354 = 2 \cdot 3^{2} \cdot 353 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6354.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:	$50.7369454443$
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} - 2x^{9} - 26x^{8} + 44x^{7} + 239x^{6} - 340x^{5} - 946x^{4} + 1056x^{3} + 1584x^{2} - 1024x - 1024 $ x^10 - 2x^9 - 26x^8 + 44x^7 + 239x^6 - 340x^5 - 946x^4 + 1056x^3 + 1584x^2 - 1024*x - 1024
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2118)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$-0.743823$ of defining polynomial
Character	$\chi$	$=$	6354.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +4.00514 q^{5} -4.94978 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} +4.00514 q^{5} -4.94978 q^{7} +1.00000 q^{8} +4.00514 q^{10} -6.17530 q^{11} +0.916876 q^{13} -4.94978 q^{14} +1.00000 q^{16} +0.628490 q^{17} +1.49953 q^{19} +4.00514 q^{20} -6.17530 q^{22} +7.85359 q^{23} +11.0412 q^{25} +0.916876 q^{26} -4.94978 q^{28} -4.62589 q^{29} +2.73678 q^{31} +1.00000 q^{32} +0.628490 q^{34} -19.8246 q^{35} +4.78192 q^{37} +1.49953 q^{38} +4.00514 q^{40} +7.92359 q^{41} +11.2428 q^{43} -6.17530 q^{44} +7.85359 q^{46} -4.87362 q^{47} +17.5003 q^{49} +11.0412 q^{50} +0.916876 q^{52} +5.12393 q^{53} -24.7329 q^{55} -4.94978 q^{56} -4.62589 q^{58} -9.58858 q^{59} -12.7599 q^{61} +2.73678 q^{62} +1.00000 q^{64} +3.67222 q^{65} +5.81776 q^{67} +0.628490 q^{68} -19.8246 q^{70} +11.6016 q^{71} +5.32153 q^{73} +4.78192 q^{74} +1.49953 q^{76} +30.5664 q^{77} -6.48993 q^{79} +4.00514 q^{80} +7.92359 q^{82} +3.68806 q^{83} +2.51719 q^{85} +11.2428 q^{86} -6.17530 q^{88} +6.66743 q^{89} -4.53833 q^{91} +7.85359 q^{92} -4.87362 q^{94} +6.00584 q^{95} +6.38040 q^{97} +17.5003 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 10 q^{2} + 10 q^{4} + 10 q^{5} + 10 q^{8}+O(q^{10}) $ 10 * q + 10 * q^2 + 10 * q^4 + 10 * q^5 + 10 * q^8	$ 10 q + 10 q^{2} + 10 q^{4} + 10 q^{5} + 10 q^{8} + 10 q^{10} + 2 q^{11} - 4 q^{13} + 10 q^{16} + 10 q^{17} + 14 q^{19} + 10 q^{20} + 2 q^{22} + 8 q^{23} + 20 q^{25} - 4 q^{26} + 14 q^{29} + 12 q^{31} + 10 q^{32} + 10 q^{34} + 6 q^{35} - 4 q^{37} + 14 q^{38} + 10 q^{40} + 2 q^{41} + 10 q^{43} + 2 q^{44} + 8 q^{46} + 8 q^{47} + 32 q^{49} + 20 q^{50} - 4 q^{52} + 36 q^{53} + 10 q^{55} + 14 q^{58} + 10 q^{59} - 6 q^{61} + 12 q^{62} + 10 q^{64} + 26 q^{65} + 2 q^{67} + 10 q^{68} + 6 q^{70} + 12 q^{71} - 8 q^{73} - 4 q^{74} + 14 q^{76} + 56 q^{77} + 28 q^{79} + 10 q^{80} + 2 q^{82} + 22 q^{83} - 4 q^{85} + 10 q^{86} + 2 q^{88} + 28 q^{89} + 8 q^{92} + 8 q^{94} + 16 q^{95} - 8 q^{97} + 32 q^{98}+O(q^{100}) $ 10 * q + 10 * q^2 + 10 * q^4 + 10 * q^5 + 10 * q^8 + 10 * q^10 + 2 * q^11 - 4 * q^13 + 10 * q^16 + 10 * q^17 + 14 * q^19 + 10 * q^20 + 2 * q^22 + 8 * q^23 + 20 * q^25 - 4 * q^26 + 14 * q^29 + 12 * q^31 + 10 * q^32 + 10 * q^34 + 6 * q^35 - 4 * q^37 + 14 * q^38 + 10 * q^40 + 2 * q^41 + 10 * q^43 + 2 * q^44 + 8 * q^46 + 8 * q^47 + 32 * q^49 + 20 * q^50 - 4 * q^52 + 36 * q^53 + 10 * q^55 + 14 * q^58 + 10 * q^59 - 6 * q^61 + 12 * q^62 + 10 * q^64 + 26 * q^65 + 2 * q^67 + 10 * q^68 + 6 * q^70 + 12 * q^71 - 8 * q^73 - 4 * q^74 + 14 * q^76 + 56 * q^77 + 28 * q^79 + 10 * q^80 + 2 * q^82 + 22 * q^83 - 4 * q^85 + 10 * q^86 + 2 * q^88 + 28 * q^89 + 8 * q^92 + 8 * q^94 + 16 * q^95 - 8 * q^97 + 32 * q^98

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	4.00514	1.79115	0.895577	−	0.444907i	$-0.146763\pi$
$5$	4.00514	1.79115	0.895577	+	0.444907i	$0.146763\pi$
$6$	0	0
$7$	−4.94978	−1.87084	−0.935421	−	0.353537i	$-0.884979\pi$
$7$	−4.94978	−1.87084	−0.935421	+	0.353537i	$0.884979\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	4.00514	1.26654
$11$	−6.17530	−1.86192	−0.930961	−	0.365119i	$-0.881028\pi$
$11$	−6.17530	−1.86192	−0.930961	+	0.365119i	$0.881028\pi$
$12$	0	0
$13$	0.916876	0.254296	0.127148	−	0.991884i	$-0.459418\pi$
$13$	0.916876	0.254296	0.127148	+	0.991884i	$0.459418\pi$
$14$	−4.94978	−1.32288
$15$	0	0
$16$	1.00000	0.250000
$17$	0.628490	0.152431	0.0762157	−	0.997091i	$-0.475716\pi$
$17$	0.628490	0.152431	0.0762157	+	0.997091i	$0.475716\pi$
$18$	0	0
$19$	1.49953	0.344016	0.172008	−	0.985096i	$-0.444974\pi$
$19$	1.49953	0.344016	0.172008	+	0.985096i	$0.444974\pi$
$20$	4.00514	0.895577
$21$	0	0
$22$	−6.17530	−1.31658
$23$	7.85359	1.63759	0.818793	−	0.574089i	$-0.194644\pi$
$23$	7.85359	1.63759	0.818793	+	0.574089i	$0.194644\pi$
$24$	0	0
$25$	11.0412	2.20823
$26$	0.916876	0.179814
$27$	0	0
$28$	−4.94978	−0.935421
$29$	−4.62589	−0.859007	−0.429503	−	0.903065i	$-0.641311\pi$
$29$	−4.62589	−0.859007	−0.429503	+	0.903065i	$0.641311\pi$
$30$	0	0
$31$	2.73678	0.491540	0.245770	−	0.969328i	$-0.420959\pi$
$31$	2.73678	0.491540	0.245770	+	0.969328i	$0.420959\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	0.628490	0.107785
$35$	−19.8246	−3.35096
$36$	0	0
$37$	4.78192	0.786143	0.393071	−	0.919508i	$-0.371413\pi$
$37$	4.78192	0.786143	0.393071	+	0.919508i	$0.371413\pi$
$38$	1.49953	0.243256
$39$	0	0
$40$	4.00514	0.633268
$41$	7.92359	1.23746	0.618728	−	0.785605i	$-0.287648\pi$
$41$	7.92359	1.23746	0.618728	+	0.785605i	$0.287648\pi$
$42$	0	0
$43$	11.2428	1.71452	0.857258	−	0.514887i	$-0.172166\pi$
$43$	11.2428	1.71452	0.857258	+	0.514887i	$0.172166\pi$
$44$	−6.17530	−0.930961
$45$	0	0
$46$	7.85359	1.15795
$47$	−4.87362	−0.710891	−0.355445	−	0.934697i	$-0.615671\pi$
$47$	−4.87362	−0.710891	−0.355445	+	0.934697i	$0.615671\pi$
$48$	0	0
$49$	17.5003	2.50005
$50$	11.0412	1.56145
$51$	0	0
$52$	0.916876	0.127148
$53$	5.12393	0.703826	0.351913	−	0.936033i	$-0.385531\pi$
$53$	5.12393	0.703826	0.351913	+	0.936033i	$0.385531\pi$
$54$	0	0
$55$	−24.7329	−3.33499
$56$	−4.94978	−0.661442
$57$	0	0
$58$	−4.62589	−0.607410
$59$	−9.58858	−1.24833	−0.624163	−	0.781294i	$-0.714560\pi$
$59$	−9.58858	−1.24833	−0.624163	+	0.781294i	$0.714560\pi$
$60$	0	0
$61$	−12.7599	−1.63374	−0.816872	−	0.576819i	$-0.804294\pi$
$61$	−12.7599	−1.63374	−0.816872	+	0.576819i	$0.804294\pi$
$62$	2.73678	0.347571
$63$	0	0
$64$	1.00000	0.125000
$65$	3.67222	0.455482
$66$	0	0
$67$	5.81776	0.710752	0.355376	−	0.934723i	$-0.384353\pi$
$67$	5.81776	0.710752	0.355376	+	0.934723i	$0.384353\pi$
$68$	0.628490	0.0762157
$69$	0	0
$70$	−19.8246	−2.36949
$71$	11.6016	1.37686	0.688430	−	0.725303i	$-0.258300\pi$
$71$	11.6016	1.37686	0.688430	+	0.725303i	$0.258300\pi$
$72$	0	0
$73$	5.32153	0.622837	0.311419	−	0.950273i	$-0.399196\pi$
$73$	5.32153	0.622837	0.311419	+	0.950273i	$0.399196\pi$
$74$	4.78192	0.555887
$75$	0	0
$76$	1.49953	0.172008
$77$	30.5664	3.48336
$78$	0	0
$79$	−6.48993	−0.730175	−0.365087	−	0.930973i	$-0.618961\pi$
$79$	−6.48993	−0.730175	−0.365087	+	0.930973i	$0.618961\pi$
$80$	4.00514	0.447788
$81$	0	0
$82$	7.92359	0.875014
$83$	3.68806	0.404817	0.202409	−	0.979301i	$-0.435123\pi$
$83$	3.68806	0.404817	0.202409	+	0.979301i	$0.435123\pi$
$84$	0	0
$85$	2.51719	0.273028
$86$	11.2428	1.21235
$87$	0	0
$88$	−6.17530	−0.658289
$89$	6.66743	0.706746	0.353373	−	0.935483i	$-0.385035\pi$
$89$	6.66743	0.706746	0.353373	+	0.935483i	$0.385035\pi$
$90$	0	0
$91$	−4.53833	−0.475747
$92$	7.85359	0.818793
$93$	0	0
$94$	−4.87362	−0.502676
$95$	6.00584	0.616186
$96$	0	0
$97$	6.38040	0.647832	0.323916	−	0.946086i	$-0.395001\pi$
$97$	6.38040	0.647832	0.323916	+	0.946086i	$0.395001\pi$
$98$	17.5003	1.76780
$99$	0	0
$100$	11.0412	1.10412
$101$	−7.84955	−0.781060	−0.390530	−	0.920590i	$-0.627708\pi$
$101$	−7.84955	−0.781060	−0.390530	+	0.920590i	$0.627708\pi$
$102$	0	0
$103$	−17.8630	−1.76009	−0.880046	−	0.474888i	$-0.842489\pi$
$103$	−17.8630	−1.76009	−0.880046	+	0.474888i	$0.842489\pi$
$104$	0.916876	0.0899071
$105$	0	0
$106$	5.12393	0.497680
$107$	16.0261	1.54930	0.774652	−	0.632387i	$-0.217925\pi$
$107$	16.0261	1.54930	0.774652	+	0.632387i	$0.217925\pi$
$108$	0	0
$109$	−5.61218	−0.537549	−0.268775	−	0.963203i	$-0.586619\pi$
$109$	−5.61218	−0.537549	−0.268775	+	0.963203i	$0.586619\pi$
$110$	−24.7329	−2.35819
$111$	0	0
$112$	−4.94978	−0.467710
$113$	15.7381	1.48052	0.740258	−	0.672323i	$-0.234704\pi$
$113$	15.7381	1.48052	0.740258	+	0.672323i	$0.234704\pi$
$114$	0	0
$115$	31.4547	2.93317
$116$	−4.62589	−0.429503
$117$	0	0
$118$	−9.58858	−0.882700
$119$	−3.11089	−0.285175
$120$	0	0
$121$	27.1343	2.46675
$122$	−12.7599	−1.15523
$123$	0	0
$124$	2.73678	0.245770
$125$	24.1957	2.16413
$126$	0	0
$127$	3.71102	0.329300	0.164650	−	0.986352i	$-0.447351\pi$
$127$	3.71102	0.329300	0.164650	+	0.986352i	$0.447351\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	3.67222	0.322075
$131$	−4.66442	−0.407533	−0.203766	−	0.979020i	$-0.565318\pi$
$131$	−4.66442	−0.407533	−0.203766	+	0.979020i	$0.565318\pi$
$132$	0	0
$133$	−7.42235	−0.643600
$134$	5.81776	0.502578
$135$	0	0
$136$	0.628490	0.0538926
$137$	8.83501	0.754826	0.377413	−	0.926045i	$-0.376814\pi$
$137$	8.83501	0.754826	0.377413	+	0.926045i	$0.376814\pi$
$138$	0	0
$139$	−4.34297	−0.368366	−0.184183	−	0.982892i	$-0.558964\pi$
$139$	−4.34297	−0.368366	−0.184183	+	0.982892i	$0.558964\pi$
$140$	−19.8246	−1.67548
$141$	0	0
$142$	11.6016	0.973587
$143$	−5.66198	−0.473478
$144$	0	0
$145$	−18.5274	−1.53861
$146$	5.32153	0.440413
$147$	0	0
$148$	4.78192	0.393071
$149$	6.65476	0.545179	0.272589	−	0.962130i	$-0.412120\pi$
$149$	6.65476	0.545179	0.272589	+	0.962130i	$0.412120\pi$
$150$	0	0
$151$	16.5390	1.34592	0.672961	−	0.739678i	$-0.265022\pi$
$151$	16.5390	1.34592	0.672961	+	0.739678i	$0.265022\pi$
$152$	1.49953	0.121628
$153$	0	0
$154$	30.5664	2.46311
$155$	10.9612	0.880424
$156$	0	0
$157$	3.45448	0.275697	0.137849	−	0.990453i	$-0.455981\pi$
$157$	3.45448	0.275697	0.137849	+	0.990453i	$0.455981\pi$
$158$	−6.48993	−0.516311
$159$	0	0
$160$	4.00514	0.316634
$161$	−38.8735	−3.06366
$162$	0	0
$163$	2.69830	0.211347	0.105674	−	0.994401i	$-0.466300\pi$
$163$	2.69830	0.211347	0.105674	+	0.994401i	$0.466300\pi$
$164$	7.92359	0.618728
$165$	0	0
$166$	3.68806	0.286249
$167$	3.10427	0.240215	0.120108	−	0.992761i	$-0.461676\pi$
$167$	3.10427	0.240215	0.120108	+	0.992761i	$0.461676\pi$
$168$	0	0
$169$	−12.1593	−0.935334
$170$	2.51719	0.193060
$171$	0	0
$172$	11.2428	0.857258
$173$	0.843845	0.0641563	0.0320782	−	0.999485i	$-0.489787\pi$
$173$	0.843845	0.0641563	0.0320782	+	0.999485i	$0.489787\pi$
$174$	0	0
$175$	−54.6513	−4.13125
$176$	−6.17530	−0.465480
$177$	0	0
$178$	6.66743	0.499745
$179$	13.5493	1.01273	0.506363	−	0.862320i	$-0.330990\pi$
$179$	13.5493	1.01273	0.506363	+	0.862320i	$0.330990\pi$
$180$	0	0
$181$	5.08269	0.377793	0.188897	−	0.981997i	$-0.439509\pi$
$181$	5.08269	0.377793	0.188897	+	0.981997i	$0.439509\pi$
$182$	−4.53833	−0.336404
$183$	0	0
$184$	7.85359	0.578974
$185$	19.1523	1.40810
$186$	0	0
$187$	−3.88111	−0.283815
$188$	−4.87362	−0.355445
$189$	0	0
$190$	6.00584	0.435709
$191$	−7.05251	−0.510302	−0.255151	−	0.966901i	$-0.582125\pi$
$191$	−7.05251	−0.510302	−0.255151	+	0.966901i	$0.582125\pi$
$192$	0	0
$193$	4.34697	0.312902	0.156451	−	0.987686i	$-0.449995\pi$
$193$	4.34697	0.312902	0.156451	+	0.987686i	$0.449995\pi$
$194$	6.38040	0.458086
$195$	0	0
$196$	17.5003	1.25002
$197$	−14.6313	−1.04243	−0.521217	−	0.853424i	$-0.674522\pi$
$197$	−14.6313	−1.04243	−0.521217	+	0.853424i	$0.674522\pi$
$198$	0	0
$199$	−3.79925	−0.269322	−0.134661	−	0.990892i	$-0.542995\pi$
$199$	−3.79925	−0.269322	−0.134661	+	0.990892i	$0.542995\pi$
$200$	11.0412	0.780727
$201$	0	0
$202$	−7.84955	−0.552293
$203$	22.8972	1.60707
$204$	0	0
$205$	31.7351	2.21647
$206$	−17.8630	−1.24457
$207$	0	0
$208$	0.916876	0.0635739
$209$	−9.26005	−0.640531
$210$	0	0
$211$	11.3329	0.780190	0.390095	−	0.920775i	$-0.372442\pi$
$211$	11.3329	0.780190	0.390095	+	0.920775i	$0.372442\pi$
$212$	5.12393	0.351913
$213$	0	0
$214$	16.0261	1.09552
$215$	45.0291	3.07096
$216$	0	0
$217$	−13.5465	−0.919593
$218$	−5.61218	−0.380105
$219$	0	0
$220$	−24.7329	−1.66749
$221$	0.576248	0.0387626
$222$	0	0
$223$	1.53924	0.103075	0.0515377	−	0.998671i	$-0.483588\pi$
$223$	1.53924	0.103075	0.0515377	+	0.998671i	$0.483588\pi$
$224$	−4.94978	−0.330721
$225$	0	0
$226$	15.7381	1.04688
$227$	25.9638	1.72328	0.861640	−	0.507520i	$-0.169438\pi$
$227$	25.9638	1.72328	0.861640	+	0.507520i	$0.169438\pi$
$228$	0	0
$229$	−0.343012	−0.0226669	−0.0113334	−	0.999936i	$-0.503608\pi$
$229$	−0.343012	−0.0226669	−0.0113334	+	0.999936i	$0.503608\pi$
$230$	31.4547	2.07406
$231$	0	0
$232$	−4.62589	−0.303705
$233$	−6.37899	−0.417901	−0.208951	−	0.977926i	$-0.567005\pi$
$233$	−6.37899	−0.417901	−0.208951	+	0.977926i	$0.567005\pi$
$234$	0	0
$235$	−19.5195	−1.27331
$236$	−9.58858	−0.624163
$237$	0	0
$238$	−3.11089	−0.201649
$239$	−25.4354	−1.64528	−0.822640	−	0.568563i	$-0.807499\pi$
$239$	−25.4354	−1.64528	−0.822640	+	0.568563i	$0.807499\pi$
$240$	0	0
$241$	−30.1312	−1.94092	−0.970461	−	0.241258i	$-0.922440\pi$
$241$	−30.1312	−1.94092	−0.970461	+	0.241258i	$0.922440\pi$
$242$	27.1343	1.74426
$243$	0	0
$244$	−12.7599	−0.816872
$245$	70.0913	4.47797
$246$	0	0
$247$	1.37488	0.0874818
$248$	2.73678	0.173786
$249$	0	0
$250$	24.1957	1.53027
$251$	−23.6296	−1.49148	−0.745742	−	0.666234i	$-0.767905\pi$
$251$	−23.6296	−1.49148	−0.745742	+	0.666234i	$0.767905\pi$
$252$	0	0
$253$	−48.4982	−3.04906
$254$	3.71102	0.232850
$255$	0	0
$256$	1.00000	0.0625000
$257$	4.65158	0.290157	0.145079	−	0.989420i	$-0.453656\pi$
$257$	4.65158	0.290157	0.145079	+	0.989420i	$0.453656\pi$
$258$	0	0
$259$	−23.6695	−1.47075
$260$	3.67222	0.227741
$261$	0	0
$262$	−4.66442	−0.288169
$263$	−23.0796	−1.42315	−0.711574	−	0.702611i	$-0.752018\pi$
$263$	−23.0796	−1.42315	−0.711574	+	0.702611i	$0.752018\pi$
$264$	0	0
$265$	20.5221	1.26066
$266$	−7.42235	−0.455094
$267$	0	0
$268$	5.81776	0.355376
$269$	22.1232	1.34888	0.674438	−	0.738331i	$-0.264386\pi$
$269$	22.1232	1.34888	0.674438	+	0.738331i	$0.264386\pi$
$270$	0	0
$271$	−10.8258	−0.657619	−0.328809	−	0.944396i	$-0.606647\pi$
$271$	−10.8258	−0.657619	−0.328809	+	0.944396i	$0.606647\pi$
$272$	0.628490	0.0381078
$273$	0	0
$274$	8.83501	0.533742
$275$	−68.1824	−4.11155
$276$	0	0
$277$	25.9833	1.56118	0.780592	−	0.625041i	$-0.214918\pi$
$277$	25.9833	1.56118	0.780592	+	0.625041i	$0.214918\pi$
$278$	−4.34297	−0.260474
$279$	0	0
$280$	−19.8246	−1.18474
$281$	12.9336	0.771552	0.385776	−	0.922592i	$-0.373934\pi$
$281$	12.9336	0.771552	0.385776	+	0.922592i	$0.373934\pi$
$282$	0	0
$283$	−8.93893	−0.531364	−0.265682	−	0.964061i	$-0.585597\pi$
$283$	−8.93893	−0.531364	−0.265682	+	0.964061i	$0.585597\pi$
$284$	11.6016	0.688430
$285$	0	0
$286$	−5.66198	−0.334800
$287$	−39.2200	−2.31508
$288$	0	0
$289$	−16.6050	−0.976765
$290$	−18.5274	−1.08796
$291$	0	0
$292$	5.32153	0.311419
$293$	12.4690	0.728445	0.364222	−	0.931312i	$-0.381335\pi$
$293$	12.4690	0.728445	0.364222	+	0.931312i	$0.381335\pi$
$294$	0	0
$295$	−38.4036	−2.23594
$296$	4.78192	0.277943
$297$	0	0
$298$	6.65476	0.385500
$299$	7.20076	0.416431
$300$	0	0
$301$	−55.6496	−3.20759
$302$	16.5390	0.951710
$303$	0	0
$304$	1.49953	0.0860040
$305$	−51.1054	−2.92628
$306$	0	0
$307$	−27.8160	−1.58754	−0.793772	−	0.608215i	$-0.791886\pi$
$307$	−27.8160	−1.58754	−0.793772	+	0.608215i	$0.791886\pi$
$308$	30.5664	1.74168
$309$	0	0
$310$	10.9612	0.622553
$311$	−24.1601	−1.37000	−0.684998	−	0.728545i	$-0.740197\pi$
$311$	−24.1601	−1.37000	−0.684998	+	0.728545i	$0.740197\pi$
$312$	0	0
$313$	−23.3715	−1.32103	−0.660517	−	0.750811i	$-0.729663\pi$
$313$	−23.3715	−1.32103	−0.660517	+	0.750811i	$0.729663\pi$
$314$	3.45448	0.194947
$315$	0	0
$316$	−6.48993	−0.365087
$317$	−21.4591	−1.20526	−0.602632	−	0.798019i	$-0.705881\pi$
$317$	−21.4591	−1.20526	−0.602632	+	0.798019i	$0.705881\pi$
$318$	0	0
$319$	28.5663	1.59940
$320$	4.00514	0.223894
$321$	0	0
$322$	−38.8735	−2.16634
$323$	0.942441	0.0524388
$324$	0	0
$325$	10.1234	0.561543
$326$	2.69830	0.149445
$327$	0	0
$328$	7.92359	0.437507
$329$	24.1234	1.32996
$330$	0	0
$331$	23.3848	1.28534	0.642672	−	0.766141i	$-0.277826\pi$
$331$	23.3848	1.28534	0.642672	+	0.766141i	$0.277826\pi$
$332$	3.68806	0.202409
$333$	0	0
$334$	3.10427	0.169858
$335$	23.3009	1.27307
$336$	0	0
$337$	17.3774	0.946609	0.473305	−	0.880899i	$-0.343061\pi$
$337$	17.3774	0.946609	0.473305	+	0.880899i	$0.343061\pi$
$338$	−12.1593	−0.661381
$339$	0	0
$340$	2.51719	0.136514
$341$	−16.9004	−0.915209
$342$	0	0
$343$	−51.9743	−2.80635
$344$	11.2428	0.606173
$345$	0	0
$346$	0.843845	0.0453654
$347$	1.48461	0.0796980	0.0398490	−	0.999206i	$-0.487312\pi$
$347$	1.48461	0.0796980	0.0398490	+	0.999206i	$0.487312\pi$
$348$	0	0
$349$	−20.8966	−1.11857	−0.559285	−	0.828975i	$-0.688924\pi$
$349$	−20.8966	−1.11857	−0.559285	+	0.828975i	$0.688924\pi$
$350$	−54.6513	−2.92123
$351$	0	0
$352$	−6.17530	−0.329144
$353$	1.00000	0.0532246
$353$	1.00000	0.0532246
$354$	0	0
$355$	46.4661	2.46617
$356$	6.66743	0.353373
$357$	0	0
$358$	13.5493	0.716105
$359$	−5.97395	−0.315293	−0.157647	−	0.987496i	$-0.550391\pi$
$359$	−5.97395	−0.315293	−0.157647	+	0.987496i	$0.550391\pi$
$360$	0	0
$361$	−16.7514	−0.881653
$362$	5.08269	0.267140
$363$	0	0
$364$	−4.53833	−0.237873
$365$	21.3135	1.11560
$366$	0	0
$367$	27.1520	1.41732	0.708662	−	0.705548i	$-0.249299\pi$
$367$	27.1520	1.41732	0.708662	+	0.705548i	$0.249299\pi$
$368$	7.85359	0.409396
$369$	0	0
$370$	19.1523	0.995679
$371$	−25.3623	−1.31675
$372$	0	0
$373$	5.02933	0.260409	0.130204	−	0.991487i	$-0.458437\pi$
$373$	5.02933	0.260409	0.130204	+	0.991487i	$0.458437\pi$
$374$	−3.88111	−0.200688
$375$	0	0
$376$	−4.87362	−0.251338
$377$	−4.24137	−0.218442
$378$	0	0
$379$	−5.93013	−0.304610	−0.152305	−	0.988334i	$-0.548670\pi$
$379$	−5.93013	−0.304610	−0.152305	+	0.988334i	$0.548670\pi$
$380$	6.00584	0.308093
$381$	0	0
$382$	−7.05251	−0.360838
$383$	−19.2288	−0.982546	−0.491273	−	0.871006i	$-0.663468\pi$
$383$	−19.2288	−0.982546	−0.491273	+	0.871006i	$0.663468\pi$
$384$	0	0
$385$	122.423	6.23923
$386$	4.34697	0.221255
$387$	0	0
$388$	6.38040	0.323916
$389$	29.0093	1.47083	0.735414	−	0.677618i	$-0.236987\pi$
$389$	29.0093	1.47083	0.735414	+	0.677618i	$0.236987\pi$
$390$	0	0
$391$	4.93590	0.249619
$392$	17.5003	0.883900
$393$	0	0
$394$	−14.6313	−0.737113
$395$	−25.9931	−1.30785
$396$	0	0
$397$	−14.1358	−0.709456	−0.354728	−	0.934970i	$-0.615427\pi$
$397$	−14.1358	−0.709456	−0.354728	+	0.934970i	$0.615427\pi$
$398$	−3.79925	−0.190439
$399$	0	0
$400$	11.0412	0.552058
$401$	−11.4034	−0.569457	−0.284729	−	0.958608i	$-0.591903\pi$
$401$	−11.4034	−0.569457	−0.284729	+	0.958608i	$0.591903\pi$
$402$	0	0
$403$	2.50929	0.124996
$404$	−7.84955	−0.390530
$405$	0	0
$406$	22.8972	1.13637
$407$	−29.5298	−1.46374
$408$	0	0
$409$	10.0405	0.496471	0.248236	−	0.968700i	$-0.420149\pi$
$409$	10.0405	0.496471	0.248236	+	0.968700i	$0.420149\pi$
$410$	31.7351	1.56728
$411$	0	0
$412$	−17.8630	−0.880046
$413$	47.4614	2.33542
$414$	0	0
$415$	14.7712	0.725090
$416$	0.916876	0.0449535
$417$	0	0
$418$	−9.26005	−0.452924
$419$	30.1505	1.47295	0.736474	−	0.676466i	$-0.236489\pi$
$419$	30.1505	1.47295	0.736474	+	0.676466i	$0.236489\pi$
$420$	0	0
$421$	30.3570	1.47951	0.739754	−	0.672878i	$-0.234942\pi$
$421$	30.3570	1.47951	0.739754	+	0.672878i	$0.234942\pi$
$422$	11.3329	0.551677
$423$	0	0
$424$	5.12393	0.248840
$425$	6.93926	0.336603
$426$	0	0
$427$	63.1589	3.05647
$428$	16.0261	0.774652
$429$	0	0
$430$	45.0291	2.17150
$431$	0.554930	0.0267301	0.0133650	−	0.999911i	$-0.495746\pi$
$431$	0.554930	0.0267301	0.0133650	+	0.999911i	$0.495746\pi$
$432$	0	0
$433$	−31.8087	−1.52863	−0.764314	−	0.644844i	$-0.776922\pi$
$433$	−31.8087	−1.52863	−0.764314	+	0.644844i	$0.776922\pi$
$434$	−13.5465	−0.650251
$435$	0	0
$436$	−5.61218	−0.268775
$437$	11.7767	0.563356
$438$	0	0
$439$	5.17825	0.247145	0.123572	−	0.992336i	$-0.460565\pi$
$439$	5.17825	0.247145	0.123572	+	0.992336i	$0.460565\pi$
$440$	−24.7329	−1.17910
$441$	0	0
$442$	0.576248	0.0274093
$443$	4.70847	0.223706	0.111853	−	0.993725i	$-0.464321\pi$
$443$	4.70847	0.223706	0.111853	+	0.993725i	$0.464321\pi$
$444$	0	0
$445$	26.7040	1.26589
$446$	1.53924	0.0728853
$447$	0	0
$448$	−4.94978	−0.233855
$449$	28.5714	1.34837	0.674184	−	0.738563i	$-0.264495\pi$
$449$	28.5714	1.34837	0.674184	+	0.738563i	$0.264495\pi$
$450$	0	0
$451$	−48.9305	−2.30405
$452$	15.7381	0.740258
$453$	0	0
$454$	25.9638	1.21854
$455$	−18.1767	−0.852135
$456$	0	0
$457$	−22.0681	−1.03230	−0.516151	−	0.856498i	$-0.672636\pi$
$457$	−22.0681	−1.03230	−0.516151	+	0.856498i	$0.672636\pi$
$458$	−0.343012	−0.0160279
$459$	0	0
$460$	31.4547	1.46658
$461$	10.9166	0.508437	0.254218	−	0.967147i	$-0.418182\pi$
$461$	10.9166	0.508437	0.254218	+	0.967147i	$0.418182\pi$
$462$	0	0
$463$	−4.21298	−0.195794	−0.0978970	−	0.995197i	$-0.531212\pi$
$463$	−4.21298	−0.195794	−0.0978970	+	0.995197i	$0.531212\pi$
$464$	−4.62589	−0.214752
$465$	0	0
$466$	−6.37899	−0.295501
$467$	−10.4495	−0.483543	−0.241772	−	0.970333i	$-0.577728\pi$
$467$	−10.4495	−0.483543	−0.241772	+	0.970333i	$0.577728\pi$
$468$	0	0
$469$	−28.7966	−1.32970
$470$	−19.5195	−0.900369
$471$	0	0
$472$	−9.58858	−0.441350
$473$	−69.4278	−3.19230
$474$	0	0
$475$	16.5566	0.759667
$476$	−3.11089	−0.142587
$477$	0	0
$478$	−25.4354	−1.16339
$479$	−16.7264	−0.764251	−0.382125	−	0.924110i	$-0.624808\pi$
$479$	−16.7264	−0.764251	−0.382125	+	0.924110i	$0.624808\pi$
$480$	0	0
$481$	4.38443	0.199913
$482$	−30.1312	−1.37244
$483$	0	0
$484$	27.1343	1.23338
$485$	25.5544	1.16037
$486$	0	0
$487$	31.4739	1.42622	0.713109	−	0.701053i	$-0.247286\pi$
$487$	31.4739	1.42622	0.713109	+	0.701053i	$0.247286\pi$
$488$	−12.7599	−0.577615
$489$	0	0
$490$	70.0913	3.16640
$491$	11.4663	0.517467	0.258734	−	0.965949i	$-0.416695\pi$
$491$	11.4663	0.517467	0.258734	+	0.965949i	$0.416695\pi$
$492$	0	0
$493$	−2.90733	−0.130940
$494$	1.37488	0.0618590
$495$	0	0
$496$	2.73678	0.122885
$497$	−57.4255	−2.57589
$498$	0	0
$499$	−17.9262	−0.802485	−0.401243	−	0.915972i	$-0.631422\pi$
$499$	−17.9262	−0.802485	−0.401243	+	0.915972i	$0.631422\pi$
$500$	24.1957	1.08206
$501$	0	0
$502$	−23.6296	−1.05464
$503$	−22.3516	−0.996607	−0.498303	−	0.867003i	$-0.666043\pi$
$503$	−22.3516	−0.996607	−0.498303	+	0.867003i	$0.666043\pi$
$504$	0	0
$505$	−31.4386	−1.39900
$506$	−48.4982	−2.15601
$507$	0	0
$508$	3.71102	0.164650
$509$	21.1589	0.937851	0.468925	−	0.883238i	$-0.344641\pi$
$509$	21.1589	0.937851	0.468925	+	0.883238i	$0.344641\pi$
$510$	0	0
$511$	−26.3404	−1.16523
$512$	1.00000	0.0441942
$513$	0	0
$514$	4.65158	0.205172
$515$	−71.5438	−3.15260
$516$	0	0
$517$	30.0960	1.32362
$518$	−23.6695	−1.03998
$519$	0	0
$520$	3.67222	0.161037
$521$	22.6764	0.993469	0.496735	−	0.867902i	$-0.334532\pi$
$521$	22.6764	0.993469	0.496735	+	0.867902i	$0.334532\pi$
$522$	0	0
$523$	2.90494	0.127024	0.0635121	−	0.997981i	$-0.479770\pi$
$523$	2.90494	0.127024	0.0635121	+	0.997981i	$0.479770\pi$
$524$	−4.66442	−0.203766
$525$	0	0
$526$	−23.0796	−1.00632
$527$	1.72004	0.0749261
$528$	0	0
$529$	38.6788	1.68169
$530$	20.5221	0.891422
$531$	0	0
$532$	−7.42235	−0.321800
$533$	7.26494	0.314680
$534$	0	0
$535$	64.1869	2.77504
$536$	5.81776	0.251289
$537$	0	0
$538$	22.1232	0.953800
$539$	−108.070	−4.65489
$540$	0	0
$541$	−2.64596	−0.113759	−0.0568793	−	0.998381i	$-0.518115\pi$
$541$	−2.64596	−0.113759	−0.0568793	+	0.998381i	$0.518115\pi$
$542$	−10.8258	−0.465007
$543$	0	0
$544$	0.628490	0.0269463
$545$	−22.4776	−0.962833
$546$	0	0
$547$	−34.8476	−1.48997	−0.744987	−	0.667079i	$-0.767544\pi$
$547$	−34.8476	−1.48997	−0.744987	+	0.667079i	$0.767544\pi$
$548$	8.83501	0.377413
$549$	0	0
$550$	−68.1824	−2.90731
$551$	−6.93667	−0.295512
$552$	0	0
$553$	32.1238	1.36604
$554$	25.9833	1.10392
$555$	0	0
$556$	−4.34297	−0.184183
$557$	4.24653	0.179931	0.0899657	−	0.995945i	$-0.471324\pi$
$557$	4.24653	0.179931	0.0899657	+	0.995945i	$0.471324\pi$
$558$	0	0
$559$	10.3083	0.435994
$560$	−19.8246	−0.837741
$561$	0	0
$562$	12.9336	0.545570
$563$	−28.9747	−1.22114	−0.610569	−	0.791963i	$-0.709059\pi$
$563$	−28.9747	−1.22114	−0.610569	+	0.791963i	$0.709059\pi$
$564$	0	0
$565$	63.0333	2.65183
$566$	−8.93893	−0.375731
$567$	0	0
$568$	11.6016	0.486793
$569$	39.2167	1.64405	0.822025	−	0.569451i	$-0.192844\pi$
$569$	39.2167	1.64405	0.822025	+	0.569451i	$0.192844\pi$
$570$	0	0
$571$	−36.3006	−1.51913	−0.759566	−	0.650430i	$-0.774589\pi$
$571$	−36.3006	−1.51913	−0.759566	+	0.650430i	$0.774589\pi$
$572$	−5.66198	−0.236739
$573$	0	0
$574$	−39.2200	−1.63701
$575$	86.7126	3.61617
$576$	0	0
$577$	−25.4362	−1.05892	−0.529462	−	0.848334i	$-0.677606\pi$
$577$	−25.4362	−1.05892	−0.529462	+	0.848334i	$0.677606\pi$
$578$	−16.6050	−0.690677
$579$	0	0
$580$	−18.5274	−0.769307
$581$	−18.2551	−0.757349
$582$	0	0
$583$	−31.6418	−1.31047
$584$	5.32153	0.220206
$585$	0	0
$586$	12.4690	0.515088
$587$	23.5294	0.971162	0.485581	−	0.874192i	$-0.338608\pi$
$587$	23.5294	0.971162	0.485581	+	0.874192i	$0.338608\pi$
$588$	0	0
$589$	4.10389	0.169098
$590$	−38.4036	−1.58105
$591$	0	0
$592$	4.78192	0.196536
$593$	25.8913	1.06323	0.531613	−	0.846987i	$-0.321586\pi$
$593$	25.8913	1.06323	0.531613	+	0.846987i	$0.321586\pi$
$594$	0	0
$595$	−12.4596	−0.510792
$596$	6.65476	0.272589
$597$	0	0
$598$	7.20076	0.294461
$599$	26.7940	1.09477	0.547386	−	0.836880i	$-0.315623\pi$
$599$	26.7940	1.09477	0.547386	+	0.836880i	$0.315623\pi$
$600$	0	0
$601$	−20.5805	−0.839495	−0.419747	−	0.907641i	$-0.637881\pi$
$601$	−20.5805	−0.839495	−0.419747	+	0.907641i	$0.637881\pi$
$602$	−55.6496	−2.26811
$603$	0	0
$604$	16.5390	0.672961
$605$	108.677	4.41833
$606$	0	0
$607$	−14.7278	−0.597784	−0.298892	−	0.954287i	$-0.596617\pi$
$607$	−14.7278	−0.597784	−0.298892	+	0.954287i	$0.596617\pi$
$608$	1.49953	0.0608140
$609$	0	0
$610$	−51.1054	−2.06920
$611$	−4.46850	−0.180776
$612$	0	0
$613$	−46.2196	−1.86679	−0.933395	−	0.358850i	$-0.883169\pi$
$613$	−46.2196	−1.86679	−0.933395	+	0.358850i	$0.883169\pi$
$614$	−27.8160	−1.12256
$615$	0	0
$616$	30.5664	1.23155
$617$	2.63921	0.106251	0.0531253	−	0.998588i	$-0.483082\pi$
$617$	2.63921	0.106251	0.0531253	+	0.998588i	$0.483082\pi$
$618$	0	0
$619$	−36.3419	−1.46071	−0.730353	−	0.683070i	$-0.760644\pi$
$619$	−36.3419	−1.46071	−0.730353	+	0.683070i	$0.760644\pi$
$620$	10.9612	0.440212
$621$	0	0
$622$	−24.1601	−0.968734
$623$	−33.0023	−1.32221
$624$	0	0
$625$	41.7013	1.66805
$626$	−23.3715	−0.934112
$627$	0	0
$628$	3.45448	0.137849
$629$	3.00539	0.119833
$630$	0	0
$631$	17.6866	0.704092	0.352046	−	0.935983i	$-0.385486\pi$
$631$	17.6866	0.704092	0.352046	+	0.935983i	$0.385486\pi$
$632$	−6.48993	−0.258156
$633$	0	0
$634$	−21.4591	−0.852251
$635$	14.8632	0.589826
$636$	0	0
$637$	16.0456	0.635751
$638$	28.5663	1.13095
$639$	0	0
$640$	4.00514	0.158317
$641$	−9.26548	−0.365964	−0.182982	−	0.983116i	$-0.558575\pi$
$641$	−9.26548	−0.365964	−0.182982	+	0.983116i	$0.558575\pi$
$642$	0	0
$643$	25.8963	1.02125	0.510625	−	0.859804i	$-0.329414\pi$
$643$	25.8963	1.02125	0.510625	+	0.859804i	$0.329414\pi$
$644$	−38.8735	−1.53183
$645$	0	0
$646$	0.942441	0.0370799
$647$	26.3621	1.03640	0.518200	−	0.855260i	$-0.326602\pi$
$647$	26.3621	1.03640	0.518200	+	0.855260i	$0.326602\pi$
$648$	0	0
$649$	59.2123	2.32429
$650$	10.1234	0.397071
$651$	0	0
$652$	2.69830	0.105674
$653$	−26.6104	−1.04135	−0.520673	−	0.853756i	$-0.674319\pi$
$653$	−26.6104	−1.04135	−0.520673	+	0.853756i	$0.674319\pi$
$654$	0	0
$655$	−18.6817	−0.729953
$656$	7.92359	0.309364
$657$	0	0
$658$	24.1234	0.940426
$659$	39.8426	1.55205	0.776024	−	0.630703i	$-0.217233\pi$
$659$	39.8426	1.55205	0.776024	+	0.630703i	$0.217233\pi$
$660$	0	0
$661$	4.05571	0.157749	0.0788745	−	0.996885i	$-0.474867\pi$
$661$	4.05571	0.157749	0.0788745	+	0.996885i	$0.474867\pi$
$662$	23.3848	0.908876
$663$	0	0
$664$	3.68806	0.143125
$665$	−29.7276	−1.15279
$666$	0	0
$667$	−36.3299	−1.40670
$668$	3.10427	0.120108
$669$	0	0
$670$	23.3009	0.900194
$671$	78.7964	3.04190
$672$	0	0
$673$	−9.98830	−0.385021	−0.192510	−	0.981295i	$-0.561663\pi$
$673$	−9.98830	−0.385021	−0.192510	+	0.981295i	$0.561663\pi$
$674$	17.3774	0.669354
$675$	0	0
$676$	−12.1593	−0.467667
$677$	42.9675	1.65138	0.825688	−	0.564127i	$-0.190787\pi$
$677$	42.9675	1.65138	0.825688	+	0.564127i	$0.190787\pi$
$678$	0	0
$679$	−31.5816	−1.21199
$680$	2.51719	0.0965299
$681$	0	0
$682$	−16.9004	−0.647150
$683$	−17.5015	−0.669675	−0.334837	−	0.942276i	$-0.608681\pi$
$683$	−17.5015	−0.669675	−0.334837	+	0.942276i	$0.608681\pi$
$684$	0	0
$685$	35.3855	1.35201
$686$	−51.9743	−1.98439
$687$	0	0
$688$	11.2428	0.428629
$689$	4.69801	0.178980
$690$	0	0
$691$	−12.3867	−0.471212	−0.235606	−	0.971849i	$-0.575708\pi$
$691$	−12.3867	−0.471212	−0.235606	+	0.971849i	$0.575708\pi$
$692$	0.843845	0.0320782
$693$	0	0
$694$	1.48461	0.0563550
$695$	−17.3942	−0.659800
$696$	0	0
$697$	4.97990	0.188627
$698$	−20.8966	−0.790949
$699$	0	0
$700$	−54.6513	−2.06562
$701$	7.27681	0.274841	0.137421	−	0.990513i	$-0.456119\pi$
$701$	7.27681	0.274841	0.137421	+	0.990513i	$0.456119\pi$
$702$	0	0
$703$	7.17064	0.270446
$704$	−6.17530	−0.232740
$705$	0	0
$706$	1.00000	0.0376355
$707$	38.8536	1.46124
$708$	0	0
$709$	−1.28596	−0.0482952	−0.0241476	−	0.999708i	$-0.507687\pi$
$709$	−1.28596	−0.0482952	−0.0241476	+	0.999708i	$0.507687\pi$
$710$	46.4661	1.74384
$711$	0	0
$712$	6.66743	0.249872
$713$	21.4935	0.804939
$714$	0	0
$715$	−22.6770	−0.848073
$716$	13.5493	0.506363
$717$	0	0
$718$	−5.97395	−0.222946
$719$	10.7844	0.402190	0.201095	−	0.979572i	$-0.435550\pi$
$719$	10.7844	0.402190	0.201095	+	0.979572i	$0.435550\pi$
$720$	0	0
$721$	88.4179	3.29285
$722$	−16.7514	−0.623423
$723$	0	0
$724$	5.08269	0.188897
$725$	−51.0752	−1.89688
$726$	0	0
$727$	48.3571	1.79347	0.896733	−	0.442572i	$-0.145934\pi$
$727$	48.3571	1.79347	0.896733	+	0.442572i	$0.145934\pi$
$728$	−4.53833	−0.168202
$729$	0	0
$730$	21.3135	0.788847
$731$	7.06602	0.261346
$732$	0	0
$733$	8.64504	0.319312	0.159656	−	0.987173i	$-0.448962\pi$
$733$	8.64504	0.319312	0.159656	+	0.987173i	$0.448962\pi$
$734$	27.1520	1.00220
$735$	0	0
$736$	7.85359	0.289487
$737$	−35.9264	−1.32337
$738$	0	0
$739$	38.3812	1.41187	0.705937	−	0.708275i	$-0.250526\pi$
$739$	38.3812	1.41187	0.705937	+	0.708275i	$0.250526\pi$
$740$	19.1523	0.704051
$741$	0	0
$742$	−25.3623	−0.931081
$743$	−5.06300	−0.185744	−0.0928718	−	0.995678i	$-0.529605\pi$
$743$	−5.06300	−0.185744	−0.0928718	+	0.995678i	$0.529605\pi$
$744$	0	0
$745$	26.6532	0.976499
$746$	5.02933	0.184137
$747$	0	0
$748$	−3.88111	−0.141908
$749$	−79.3259	−2.89850
$750$	0	0
$751$	−41.4348	−1.51198	−0.755989	−	0.654584i	$-0.772844\pi$
$751$	−41.4348	−1.51198	−0.755989	+	0.654584i	$0.772844\pi$
$752$	−4.87362	−0.177723
$753$	0	0
$754$	−4.24137	−0.154462
$755$	66.2409	2.41075
$756$	0	0
$757$	−19.8945	−0.723079	−0.361539	−	0.932357i	$-0.617749\pi$
$757$	−19.8945	−0.723079	−0.361539	+	0.932357i	$0.617749\pi$
$758$	−5.93013	−0.215392
$759$	0	0
$760$	6.00584	0.217855
$761$	−25.0123	−0.906696	−0.453348	−	0.891333i	$-0.649771\pi$
$761$	−25.0123	−0.906696	−0.453348	+	0.891333i	$0.649771\pi$
$762$	0	0
$763$	27.7790	1.00567
$764$	−7.05251	−0.255151
$765$	0	0
$766$	−19.2288	−0.694765
$767$	−8.79153	−0.317444
$768$	0	0
$769$	12.5991	0.454336	0.227168	−	0.973856i	$-0.427053\pi$
$769$	12.5991	0.454336	0.227168	+	0.973856i	$0.427053\pi$
$770$	122.423	4.41180
$771$	0	0
$772$	4.34697	0.156451
$773$	36.7550	1.32198	0.660992	−	0.750393i	$-0.270136\pi$
$773$	36.7550	1.32198	0.660992	+	0.750393i	$0.270136\pi$
$774$	0	0
$775$	30.2172	1.08543
$776$	6.38040	0.229043
$777$	0	0
$778$	29.0093	1.04003
$779$	11.8817	0.425705
$780$	0	0
$781$	−71.6435	−2.56361
$782$	4.93590	0.176508
$783$	0	0
$784$	17.5003	0.625012
$785$	13.8357	0.493816
$786$	0	0
$787$	9.72051	0.346499	0.173249	−	0.984878i	$-0.444573\pi$
$787$	9.72051	0.346499	0.173249	+	0.984878i	$0.444573\pi$
$788$	−14.6313	−0.521217
$789$	0	0
$790$	−25.9931	−0.924793
$791$	−77.9001	−2.76981
$792$	0	0
$793$	−11.6993	−0.415454
$794$	−14.1358	−0.501661
$795$	0	0
$796$	−3.79925	−0.134661
$797$	−3.01653	−0.106851	−0.0534256	−	0.998572i	$-0.517014\pi$
$797$	−3.01653	−0.106851	−0.0534256	+	0.998572i	$0.517014\pi$
$798$	0	0
$799$	−3.06302	−0.108362
$800$	11.0412	0.390364
$801$	0	0
$802$	−11.4034	−0.402667
$803$	−32.8620	−1.15967
$804$	0	0
$805$	−155.694	−5.48749
$806$	2.50929	0.0883858
$807$	0	0
$808$	−7.84955	−0.276146
$809$	−35.3657	−1.24339	−0.621697	−	0.783258i	$-0.713556\pi$
$809$	−35.3657	−1.24339	−0.621697	+	0.783258i	$0.713556\pi$
$810$	0	0
$811$	28.2765	0.992923	0.496461	−	0.868059i	$-0.334632\pi$
$811$	28.2765	0.992923	0.496461	+	0.868059i	$0.334632\pi$
$812$	22.8972	0.803533
$813$	0	0
$814$	−29.5298	−1.03502
$815$	10.8071	0.378555
$816$	0	0
$817$	16.8590	0.589821
$818$	10.0405	0.351058
$819$	0	0
$820$	31.7351	1.10824
$821$	−17.0691	−0.595717	−0.297858	−	0.954610i	$-0.596272\pi$
$821$	−17.0691	−0.595717	−0.297858	+	0.954610i	$0.596272\pi$
$822$	0	0
$823$	−36.4173	−1.26943	−0.634714	−	0.772747i	$-0.718882\pi$
$823$	−36.4173	−1.26943	−0.634714	+	0.772747i	$0.718882\pi$
$824$	−17.8630	−0.622287
$825$	0	0
$826$	47.4614	1.65139
$827$	−38.2603	−1.33044	−0.665220	−	0.746647i	$-0.731662\pi$
$827$	−38.2603	−1.33044	−0.665220	+	0.746647i	$0.731662\pi$
$828$	0	0
$829$	−44.6309	−1.55010	−0.775048	−	0.631902i	$-0.782274\pi$
$829$	−44.6309	−1.55010	−0.775048	+	0.631902i	$0.782274\pi$
$830$	14.7712	0.512716
$831$	0	0
$832$	0.916876	0.0317869
$833$	10.9988	0.381086
$834$	0	0
$835$	12.4330	0.430263
$836$	−9.26005	−0.320266
$837$	0	0
$838$	30.1505	1.04153
$839$	−21.2352	−0.733120	−0.366560	−	0.930394i	$-0.619465\pi$
$839$	−21.2352	−0.733120	−0.366560	+	0.930394i	$0.619465\pi$
$840$	0	0
$841$	−7.60111	−0.262107
$842$	30.3570	1.04617
$843$	0	0
$844$	11.3329	0.390095
$845$	−48.6999	−1.67533
$846$	0	0
$847$	−134.309	−4.61490
$848$	5.12393	0.175957
$849$	0	0
$850$	6.93926	0.238015
$851$	37.5552	1.28738
$852$	0	0
$853$	−8.28698	−0.283741	−0.141870	−	0.989885i	$-0.545312\pi$
$853$	−8.28698	−0.283741	−0.141870	+	0.989885i	$0.545312\pi$
$854$	63.1589	2.16125
$855$	0	0
$856$	16.0261	0.547762
$857$	8.14716	0.278302	0.139151	−	0.990271i	$-0.455563\pi$
$857$	8.14716	0.278302	0.139151	+	0.990271i	$0.455563\pi$
$858$	0	0
$859$	−24.5500	−0.837636	−0.418818	−	0.908070i	$-0.637556\pi$
$859$	−24.5500	−0.837636	−0.418818	+	0.908070i	$0.637556\pi$
$860$	45.0291	1.53548
$861$	0	0
$862$	0.554930	0.0189010
$863$	−8.18055	−0.278469	−0.139235	−	0.990259i	$-0.544464\pi$
$863$	−8.18055	−0.278469	−0.139235	+	0.990259i	$0.544464\pi$
$864$	0	0
$865$	3.37972	0.114914
$866$	−31.8087	−1.08090
$867$	0	0
$868$	−13.5465	−0.459797
$869$	40.0773	1.35953
$870$	0	0
$871$	5.33416	0.180741
$872$	−5.61218	−0.190052
$873$	0	0
$874$	11.7767	0.398353
$875$	−119.763	−4.04874
$876$	0	0
$877$	51.3052	1.73245	0.866227	−	0.499650i	$-0.166538\pi$
$877$	51.3052	1.73245	0.866227	+	0.499650i	$0.166538\pi$
$878$	5.17825	0.174758
$879$	0	0
$880$	−24.7329	−0.833747
$881$	49.5412	1.66909	0.834543	−	0.550943i	$-0.185732\pi$
$881$	49.5412	1.66909	0.834543	+	0.550943i	$0.185732\pi$
$882$	0	0
$883$	32.8476	1.10541	0.552705	−	0.833377i	$-0.313596\pi$
$883$	32.8476	1.10541	0.552705	+	0.833377i	$0.313596\pi$
$884$	0.576248	0.0193813
$885$	0	0
$886$	4.70847	0.158184
$887$	47.3175	1.58877	0.794383	−	0.607417i	$-0.207794\pi$
$887$	47.3175	1.58877	0.794383	+	0.607417i	$0.207794\pi$
$888$	0	0
$889$	−18.3687	−0.616068
$890$	26.7040	0.895120
$891$	0	0
$892$	1.53924	0.0515377
$893$	−7.30815	−0.244558
$894$	0	0
$895$	54.2670	1.81395
$896$	−4.94978	−0.165361
$897$	0	0
$898$	28.5714	0.953441
$899$	−12.6600	−0.422236
$900$	0	0
$901$	3.22034	0.107285
$902$	−48.9305	−1.62921
$903$	0	0
$904$	15.7381	0.523441
$905$	20.3569	0.676686
$906$	0	0
$907$	19.3655	0.643020	0.321510	−	0.946906i	$-0.395810\pi$
$907$	19.3655	0.643020	0.321510	+	0.946906i	$0.395810\pi$
$908$	25.9638	0.861640
$909$	0	0
$910$	−18.1767	−0.602551
$911$	29.3569	0.972637	0.486318	−	0.873782i	$-0.338340\pi$
$911$	29.3569	0.972637	0.486318	+	0.873782i	$0.338340\pi$
$912$	0	0
$913$	−22.7749	−0.753738
$914$	−22.0681	−0.729948
$915$	0	0
$916$	−0.343012	−0.0113334
$917$	23.0879	0.762429
$918$	0	0
$919$	−10.4418	−0.344443	−0.172221	−	0.985058i	$-0.555094\pi$
$919$	−10.4418	−0.344443	−0.172221	+	0.985058i	$0.555094\pi$
$920$	31.4547	1.03703
$921$	0	0
$922$	10.9166	0.359519
$923$	10.6373	0.350129
$924$	0	0
$925$	52.7979	1.73598
$926$	−4.21298	−0.138447
$927$	0	0
$928$	−4.62589	−0.151852
$929$	−30.4108	−0.997746	−0.498873	−	0.866675i	$-0.666253\pi$
$929$	−30.4108	−0.997746	−0.498873	+	0.866675i	$0.666253\pi$
$930$	0	0
$931$	26.2423	0.860057
$932$	−6.37899	−0.208951
$933$	0	0
$934$	−10.4495	−0.341917
$935$	−15.5444	−0.508357
$936$	0	0
$937$	−3.77173	−0.123217	−0.0616085	−	0.998100i	$-0.519623\pi$
$937$	−3.77173	−0.123217	−0.0616085	+	0.998100i	$0.519623\pi$
$938$	−28.7966	−0.940243
$939$	0	0
$940$	−19.5195	−0.636657
$941$	0.632202	0.0206092	0.0103046	−	0.999947i	$-0.496720\pi$
$941$	0.632202	0.0206092	0.0103046	+	0.999947i	$0.496720\pi$
$942$	0	0
$943$	62.2286	2.02644
$944$	−9.58858	−0.312082
$945$	0	0
$946$	−69.4278	−2.25729
$947$	13.0616	0.424444	0.212222	−	0.977221i	$-0.431930\pi$
$947$	13.0616	0.424444	0.212222	+	0.977221i	$0.431930\pi$
$948$	0	0
$949$	4.87918	0.158385
$950$	16.5566	0.537166
$951$	0	0
$952$	−3.11089	−0.100825
$953$	24.5130	0.794053	0.397026	−	0.917807i	$-0.370042\pi$
$953$	24.5130	0.794053	0.397026	+	0.917807i	$0.370042\pi$
$954$	0	0
$955$	−28.2463	−0.914028
$956$	−25.4354	−0.822640
$957$	0	0
$958$	−16.7264	−0.540407
$959$	−43.7314	−1.41216
$960$	0	0
$961$	−23.5100	−0.758388
$962$	4.38443	0.141360
$963$	0	0
$964$	−30.1312	−0.970461
$965$	17.4102	0.560455
$966$	0	0
$967$	−2.77588	−0.0892664	−0.0446332	−	0.999003i	$-0.514212\pi$
$967$	−2.77588	−0.0892664	−0.0446332	+	0.999003i	$0.514212\pi$
$968$	27.1343	0.872129
$969$	0	0
$970$	25.5544	0.820503
$971$	13.3573	0.428656	0.214328	−	0.976762i	$-0.431244\pi$
$971$	13.3573	0.428656	0.214328	+	0.976762i	$0.431244\pi$
$972$	0	0
$973$	21.4968	0.689154
$974$	31.4739	1.00849
$975$	0	0
$976$	−12.7599	−0.408436
$977$	−18.5166	−0.592397	−0.296199	−	0.955126i	$-0.595719\pi$
$977$	−18.5166	−0.592397	−0.296199	+	0.955126i	$0.595719\pi$
$978$	0	0
$979$	−41.1733	−1.31591
$980$	70.0913	2.23898
$981$	0	0
$982$	11.4663	0.365904
$983$	−62.3378	−1.98827	−0.994133	−	0.108163i	$-0.965503\pi$
$983$	−62.3378	−1.98827	−0.994133	+	0.108163i	$0.965503\pi$
$984$	0	0
$985$	−58.6003	−1.86716
$986$	−2.90733	−0.0925882
$987$	0	0
$988$	1.37488	0.0437409
$989$	88.2966	2.80767
$990$	0	0
$991$	16.4715	0.523234	0.261617	−	0.965172i	$-0.415744\pi$
$991$	16.4715	0.523234	0.261617	+	0.965172i	$0.415744\pi$
$992$	2.73678	0.0868928
$993$	0	0
$994$	−57.4255	−1.82143
$995$	−15.2165	−0.482397
$996$	0	0
$997$	24.6726	0.781388	0.390694	−	0.920521i	$-0.372235\pi$
$997$	24.6726	0.781388	0.390694	+	0.920521i	$0.372235\pi$
$998$	−17.9262	−0.567443
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6354.2.a.bl.1.8		10
3.2	odd	2		2118.2.a.t.1.3	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2118.2.a.t.1.3	✓	10	3.2	odd	2
6354.2.a.bl.1.8		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:	\( N \)	\(=\)	\( 6354 = 2 \cdot 3^{2} \cdot 353 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6354.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +4.00514 q^{5} -4.94978 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +4.00514 q^{5} -4.94978 q^{7} +1.00000 q^{8} +4.00514 q^{10} -6.17530 q^{11} +0.916876 q^{13} -4.94978 q^{14} +1.00000 q^{16} +0.628490 q^{17} +1.49953 q^{19} +4.00514 q^{20} -6.17530 q^{22} +7.85359 q^{23} +11.0412 q^{25} +0.916876 q^{26} -4.94978 q^{28} -4.62589 q^{29} +2.73678 q^{31} +1.00000 q^{32} +0.628490 q^{34} -19.8246 q^{35} +4.78192 q^{37} +1.49953 q^{38} +4.00514 q^{40} +7.92359 q^{41} +11.2428 q^{43} -6.17530 q^{44} +7.85359 q^{46} -4.87362 q^{47} +17.5003 q^{49} +11.0412 q^{50} +0.916876 q^{52} +5.12393 q^{53} -24.7329 q^{55} -4.94978 q^{56} -4.62589 q^{58} -9.58858 q^{59} -12.7599 q^{61} +2.73678 q^{62} +1.00000 q^{64} +3.67222 q^{65} +5.81776 q^{67} +0.628490 q^{68} -19.8246 q^{70} +11.6016 q^{71} +5.32153 q^{73} +4.78192 q^{74} +1.49953 q^{76} +30.5664 q^{77} -6.48993 q^{79} +4.00514 q^{80} +7.92359 q^{82} +3.68806 q^{83} +2.51719 q^{85} +11.2428 q^{86} -6.17530 q^{88} +6.66743 q^{89} -4.53833 q^{91} +7.85359 q^{92} -4.87362 q^{94} +6.00584 q^{95} +6.38040 q^{97} +17.5003 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 10 q^{2} + 10 q^{4} + 10 q^{5} + 10 q^{8}+O(q^{10}) \) 10 * q + 10 * q^2 + 10 * q^4 + 10 * q^5 + 10 * q^8	\( 10 q + 10 q^{2} + 10 q^{4} + 10 q^{5} + 10 q^{8} + 10 q^{10} + 2 q^{11} - 4 q^{13} + 10 q^{16} + 10 q^{17} + 14 q^{19} + 10 q^{20} + 2 q^{22} + 8 q^{23} + 20 q^{25} - 4 q^{26} + 14 q^{29} + 12 q^{31} + 10 q^{32} + 10 q^{34} + 6 q^{35} - 4 q^{37} + 14 q^{38} + 10 q^{40} + 2 q^{41} + 10 q^{43} + 2 q^{44} + 8 q^{46} + 8 q^{47} + 32 q^{49} + 20 q^{50} - 4 q^{52} + 36 q^{53} + 10 q^{55} + 14 q^{58} + 10 q^{59} - 6 q^{61} + 12 q^{62} + 10 q^{64} + 26 q^{65} + 2 q^{67} + 10 q^{68} + 6 q^{70} + 12 q^{71} - 8 q^{73} - 4 q^{74} + 14 q^{76} + 56 q^{77} + 28 q^{79} + 10 q^{80} + 2 q^{82} + 22 q^{83} - 4 q^{85} + 10 q^{86} + 2 q^{88} + 28 q^{89} + 8 q^{92} + 8 q^{94} + 16 q^{95} - 8 q^{97} + 32 q^{98}+O(q^{100}) \) 10 * q + 10 * q^2 + 10 * q^4 + 10 * q^5 + 10 * q^8 + 10 * q^10 + 2 * q^11 - 4 * q^13 + 10 * q^16 + 10 * q^17 + 14 * q^19 + 10 * q^20 + 2 * q^22 + 8 * q^23 + 20 * q^25 - 4 * q^26 + 14 * q^29 + 12 * q^31 + 10 * q^32 + 10 * q^34 + 6 * q^35 - 4 * q^37 + 14 * q^38 + 10 * q^40 + 2 * q^41 + 10 * q^43 + 2 * q^44 + 8 * q^46 + 8 * q^47 + 32 * q^49 + 20 * q^50 - 4 * q^52 + 36 * q^53 + 10 * q^55 + 14 * q^58 + 10 * q^59 - 6 * q^61 + 12 * q^62 + 10 * q^64 + 26 * q^65 + 2 * q^67 + 10 * q^68 + 6 * q^70 + 12 * q^71 - 8 * q^73 - 4 * q^74 + 14 * q^76 + 56 * q^77 + 28 * q^79 + 10 * q^80 + 2 * q^82 + 22 * q^83 - 4 * q^85 + 10 * q^86 + 2 * q^88 + 28 * q^89 + 8 * q^92 + 8 * q^94 + 16 * q^95 - 8 * q^97 + 32 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more