LMFDB - Embedded newform 10000.2.a.t.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [10000,2,Mod(1,10000)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(10000, 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("10000.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 10000 = 2^{4} \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	10000.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:	$79.8504020213$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.7625.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 9x^{2} + 4x + 16 $ x^4 - x^3 - 9x^2 + 4x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 50)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.34841$ of defining polynomial
Character	$\chi$	$=$	10000.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.34841 q^{3} +0.833366 q^{7} -1.18178 q^{9} +O(q^{10})$	$q+1.34841 q^{3} +0.833366 q^{7} -1.18178 q^{9} -0.833366 q^{11} +4.56375 q^{13} +5.45140 q^{17} +8.65392 q^{19} +1.12372 q^{21} -3.53019 q^{23} -5.63877 q^{27} +3.26962 q^{29} -6.00233 q^{31} -1.12372 q^{33} +7.31486 q^{37} +6.15382 q^{39} +1.86692 q^{41} +1.63877 q^{43} +0.833366 q^{47} -6.30550 q^{49} +7.35074 q^{51} +6.42721 q^{53} +11.6691 q^{57} -13.5325 q^{59} +5.88420 q^{61} -0.984855 q^{63} -1.49990 q^{67} -4.76016 q^{69} +2.36123 q^{71} -3.33001 q^{73} -0.694498 q^{77} -5.16896 q^{79} -4.05806 q^{81} +12.3356 q^{83} +4.40880 q^{87} -7.00579 q^{89} +3.80327 q^{91} -8.09363 q^{93} +11.0902 q^{97} +0.984855 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{3} - 2 q^{7} + 7 q^{9}+O(q^{10}) $ 4 * q - q^3 - 2 * q^7 + 7 * q^9	$ 4 q - q^{3} - 2 q^{7} + 7 q^{9} + 2 q^{11} + 11 q^{13} + 12 q^{17} + 5 q^{19} - 7 q^{21} + 4 q^{23} - 10 q^{27} + 15 q^{29} + 12 q^{31} + 7 q^{33} + 12 q^{37} + 11 q^{39} + 13 q^{41} - 6 q^{43} - 2 q^{47} - 2 q^{49} - 13 q^{51} + 11 q^{53} + 8 q^{61} - 21 q^{63} - 22 q^{67} - 31 q^{69} + 22 q^{71} + 21 q^{73} - 26 q^{77} + 10 q^{79} - 16 q^{81} + 24 q^{83} - 25 q^{87} - 5 q^{89} + 12 q^{91} - 23 q^{93} + 22 q^{97} + 21 q^{99}+O(q^{100}) $ 4 * q - q^3 - 2 * q^7 + 7 * q^9 + 2 * q^11 + 11 * q^13 + 12 * q^17 + 5 * q^19 - 7 * q^21 + 4 * q^23 - 10 * q^27 + 15 * q^29 + 12 * q^31 + 7 * q^33 + 12 * q^37 + 11 * q^39 + 13 * q^41 - 6 * q^43 - 2 * q^47 - 2 * q^49 - 13 * q^51 + 11 * q^53 + 8 * q^61 - 21 * q^63 - 22 * q^67 - 31 * q^69 + 22 * q^71 + 21 * q^73 - 26 * q^77 + 10 * q^79 - 16 * q^81 + 24 * q^83 - 25 * q^87 - 5 * q^89 + 12 * q^91 - 23 * q^93 + 22 * q^97 + 21 * 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
$3$	1.34841	0.778507	0.389254	−	0.921131i	$-0.372733\pi$
$3$	1.34841	0.778507	0.389254	+	0.921131i	$0.372733\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0.833366	0.314983	0.157491	−	0.987520i	$-0.449659\pi$
$7$	0.833366	0.314983	0.157491	+	0.987520i	$0.449659\pi$
$8$	0	0
$9$	−1.18178	−0.393927
$10$	0	0
$11$	−0.833366	−0.251269	−0.125635	−	0.992077i	$-0.540097\pi$
$11$	−0.833366	−0.251269	−0.125635	+	0.992077i	$0.540097\pi$
$12$	0	0
$13$	4.56375	1.26576	0.632878	−	0.774252i	$-0.281874\pi$
$13$	4.56375	1.26576	0.632878	+	0.774252i	$0.281874\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.45140	1.32216	0.661079	−	0.750316i	$-0.270099\pi$
$17$	5.45140	1.32216	0.661079	+	0.750316i	$0.270099\pi$
$18$	0	0
$19$	8.65392	1.98534	0.992672	−	0.120838i	$-0.0385583\pi$
$19$	8.65392	1.98534	0.992672	+	0.120838i	$0.0385583\pi$
$20$	0	0
$21$	1.12372	0.245216
$22$	0	0
$23$	−3.53019	−0.736096	−0.368048	−	0.929807i	$-0.619974\pi$
$23$	−3.53019	−0.736096	−0.368048	+	0.929807i	$0.619974\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−5.63877	−1.08518
$28$	0	0
$29$	3.26962	0.607153	0.303577	−	0.952807i	$-0.401819\pi$
$29$	3.26962	0.607153	0.303577	+	0.952807i	$0.401819\pi$
$30$	0	0
$31$	−6.00233	−1.07805	−0.539025	−	0.842290i	$-0.681207\pi$
$31$	−6.00233	−1.07805	−0.539025	+	0.842290i	$0.681207\pi$
$32$	0	0
$33$	−1.12372	−0.195615
$34$	0	0
$35$	0	0
$36$	0	0
$37$	7.31486	1.20256	0.601278	−	0.799040i	$-0.294658\pi$
$37$	7.31486	1.20256	0.601278	+	0.799040i	$0.294658\pi$
$38$	0	0
$39$	6.15382	0.985400
$40$	0	0
$41$	1.86692	0.291564	0.145782	−	0.989317i	$-0.453430\pi$
$41$	1.86692	0.291564	0.145782	+	0.989317i	$0.453430\pi$
$42$	0	0
$43$	1.63877	0.249910	0.124955	−	0.992162i	$-0.460121\pi$
$43$	1.63877	0.249910	0.124955	+	0.992162i	$0.460121\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0.833366	0.121559	0.0607794	−	0.998151i	$-0.480641\pi$
$47$	0.833366	0.121559	0.0607794	+	0.998151i	$0.480641\pi$
$48$	0	0
$49$	−6.30550	−0.900786
$50$	0	0
$51$	7.35074	1.02931
$52$	0	0
$53$	6.42721	0.882845	0.441422	−	0.897299i	$-0.354474\pi$
$53$	6.42721	0.882845	0.441422	+	0.897299i	$0.354474\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	11.6691	1.54560
$58$	0	0
$59$	−13.5325	−1.76178	−0.880892	−	0.473317i	$-0.843057\pi$
$59$	−13.5325	−1.76178	−0.880892	+	0.473317i	$0.843057\pi$
$60$	0	0
$61$	5.88420	0.753394	0.376697	−	0.926336i	$-0.377060\pi$
$61$	5.88420	0.753394	0.376697	+	0.926336i	$0.377060\pi$
$62$	0	0
$63$	−0.984855	−0.124080
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−1.49990	−0.183242	−0.0916212	−	0.995794i	$-0.529205\pi$
$67$	−1.49990	−0.183242	−0.0916212	+	0.995794i	$0.529205\pi$
$68$	0	0
$69$	−4.76016	−0.573056
$70$	0	0
$71$	2.36123	0.280226	0.140113	−	0.990135i	$-0.455253\pi$
$71$	2.36123	0.280226	0.140113	+	0.990135i	$0.455253\pi$
$72$	0	0
$73$	−3.33001	−0.389748	−0.194874	−	0.980828i	$-0.562430\pi$
$73$	−3.33001	−0.389748	−0.194874	+	0.980828i	$0.562430\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−0.694498	−0.0791454
$78$	0	0
$79$	−5.16896	−0.581554	−0.290777	−	0.956791i	$-0.593914\pi$
$79$	−5.16896	−0.581554	−0.290777	+	0.956791i	$0.593914\pi$
$80$	0	0
$81$	−4.05806	−0.450895
$82$	0	0
$83$	12.3356	1.35401	0.677004	−	0.735979i	$-0.263278\pi$
$83$	12.3356	1.35401	0.677004	+	0.735979i	$0.263278\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	4.40880	0.472673
$88$	0	0
$89$	−7.00579	−0.742612	−0.371306	−	0.928511i	$-0.621090\pi$
$89$	−7.00579	−0.742612	−0.371306	+	0.928511i	$0.621090\pi$
$90$	0	0
$91$	3.80327	0.398691
$92$	0	0
$93$	−8.09363	−0.839270
$94$	0	0
$95$	0	0
$96$	0	0
$97$	11.0902	1.12604	0.563018	−	0.826445i	$-0.309640\pi$
$97$	11.0902	1.12604	0.563018	+	0.826445i	$0.309640\pi$
$98$	0	0
$99$	0.984855	0.0989816
$100$	0	0
$101$	12.7085	1.26454	0.632272	−	0.774746i	$-0.282122\pi$
$101$	12.7085	1.26454	0.632272	+	0.774746i	$0.282122\pi$
$102$	0	0
$103$	−7.01515	−0.691223	−0.345611	−	0.938378i	$-0.612328\pi$
$103$	−7.01515	−0.691223	−0.345611	+	0.938378i	$0.612328\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	10.7700	1.04118	0.520589	−	0.853807i	$-0.325712\pi$
$107$	10.7700	1.04118	0.520589	+	0.853807i	$0.325712\pi$
$108$	0	0
$109$	−10.6355	−1.01870	−0.509349	−	0.860560i	$-0.670114\pi$
$109$	−10.6355	−1.01870	−0.509349	+	0.860560i	$0.670114\pi$
$110$	0	0
$111$	9.86346	0.936198
$112$	0	0
$113$	1.74176	0.163851	0.0819253	−	0.996638i	$-0.473893\pi$
$113$	1.74176	0.163851	0.0819253	+	0.996638i	$0.473893\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−5.39334	−0.498615
$118$	0	0
$119$	4.54301	0.416457
$120$	0	0
$121$	−10.3055	−0.936864
$122$	0	0
$123$	2.51738	0.226984
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−5.66673	−0.502841	−0.251421	−	0.967878i	$-0.580898\pi$
$127$	−5.66673	−0.502841	−0.251421	+	0.967878i	$0.580898\pi$
$128$	0	0
$129$	2.20974	0.194557
$130$	0	0
$131$	16.4745	1.43938	0.719690	−	0.694295i	$-0.244284\pi$
$131$	16.4745	1.43938	0.719690	+	0.694295i	$0.244284\pi$
$132$	0	0
$133$	7.21188	0.625349
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−2.89324	−0.247187	−0.123593	−	0.992333i	$-0.539442\pi$
$137$	−2.89324	−0.247187	−0.123593	+	0.992333i	$0.539442\pi$
$138$	0	0
$139$	−4.98951	−0.423205	−0.211603	−	0.977356i	$-0.567868\pi$
$139$	−4.98951	−0.423205	−0.211603	+	0.977356i	$0.567868\pi$
$140$	0	0
$141$	1.12372	0.0946345
$142$	0	0
$143$	−3.80327	−0.318045
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−8.50243	−0.701268
$148$	0	0
$149$	2.00579	0.164320	0.0821602	−	0.996619i	$-0.473818\pi$
$149$	2.00579	0.164320	0.0821602	+	0.996619i	$0.473818\pi$
$150$	0	0
$151$	1.96971	0.160293	0.0801463	−	0.996783i	$-0.474461\pi$
$151$	1.96971	0.160293	0.0801463	+	0.996783i	$0.474461\pi$
$152$	0	0
$153$	−6.44235	−0.520833
$154$	0	0
$155$	0	0
$156$	0	0
$157$	7.78467	0.621284	0.310642	−	0.950527i	$-0.399456\pi$
$157$	7.78467	0.621284	0.310642	+	0.950527i	$0.399456\pi$
$158$	0	0
$159$	8.66654	0.687301
$160$	0	0
$161$	−2.94194	−0.231858
$162$	0	0
$163$	−14.0960	−1.10408	−0.552040	−	0.833817i	$-0.686151\pi$
$163$	−14.0960	−1.10408	−0.552040	+	0.833817i	$0.686151\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	8.15615	0.631142	0.315571	−	0.948902i	$-0.397804\pi$
$167$	8.15615	0.631142	0.315571	+	0.948902i	$0.397804\pi$
$168$	0	0
$169$	7.82777	0.602137
$170$	0	0
$171$	−10.2270	−0.782080
$172$	0	0
$173$	−2.50336	−0.190327	−0.0951634	−	0.995462i	$-0.530337\pi$
$173$	−2.50336	−0.190327	−0.0951634	+	0.995462i	$0.530337\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−18.2474	−1.37156
$178$	0	0
$179$	20.0326	1.49731	0.748654	−	0.662961i	$-0.230700\pi$
$179$	20.0326	1.49731	0.748654	+	0.662961i	$0.230700\pi$
$180$	0	0
$181$	7.87829	0.585589	0.292794	−	0.956175i	$-0.405415\pi$
$181$	7.87829	0.585589	0.292794	+	0.956175i	$0.405415\pi$
$182$	0	0
$183$	7.93434	0.586523
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−4.54301	−0.332218
$188$	0	0
$189$	−4.69916	−0.341813
$190$	0	0
$191$	5.27521	0.381701	0.190850	−	0.981619i	$-0.438875\pi$
$191$	5.27521	0.381701	0.190850	+	0.981619i	$0.438875\pi$
$192$	0	0
$193$	−13.4461	−0.967873	−0.483937	−	0.875103i	$-0.660793\pi$
$193$	−13.4461	−0.967873	−0.483937	+	0.875103i	$0.660793\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−5.43858	−0.387483	−0.193742	−	0.981053i	$-0.562062\pi$
$197$	−5.43858	−0.387483	−0.193742	+	0.981053i	$0.562062\pi$
$198$	0	0
$199$	17.4090	1.23409	0.617045	−	0.786927i	$-0.288329\pi$
$199$	17.4090	1.23409	0.617045	+	0.786927i	$0.288329\pi$
$200$	0	0
$201$	−2.02249	−0.142655
$202$	0	0
$203$	2.72479	0.191243
$204$	0	0
$205$	0	0
$206$	0	0
$207$	4.17191	0.289968
$208$	0	0
$209$	−7.21188	−0.498856
$210$	0	0
$211$	0.269934	0.0185830	0.00929151	−	0.999957i	$-0.497042\pi$
$211$	0.269934	0.0185830	0.00929151	+	0.999957i	$0.497042\pi$
$212$	0	0
$213$	3.18392	0.218158
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−5.00214	−0.339567
$218$	0	0
$219$	−4.49023	−0.303422
$220$	0	0
$221$	24.8788	1.67353
$222$	0	0
$223$	−23.1713	−1.55166	−0.775832	−	0.630939i	$-0.782670\pi$
$223$	−23.1713	−1.55166	−0.775832	+	0.630939i	$0.782670\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	0.252653	0.0167692	0.00838459	−	0.999965i	$-0.497331\pi$
$227$	0.252653	0.0167692	0.00838459	+	0.999965i	$0.497331\pi$
$228$	0	0
$229$	−21.0904	−1.39369	−0.696845	−	0.717222i	$-0.745414\pi$
$229$	−21.0904	−1.39369	−0.696845	+	0.717222i	$0.745414\pi$
$230$	0	0
$231$	−0.936471	−0.0616153
$232$	0	0
$233$	−14.6075	−0.956972	−0.478486	−	0.878095i	$-0.658814\pi$
$233$	−14.6075	−0.956972	−0.478486	+	0.878095i	$0.658814\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−6.96990	−0.452744
$238$	0	0
$239$	−5.81822	−0.376349	−0.188175	−	0.982136i	$-0.560257\pi$
$239$	−5.81822	−0.376349	−0.188175	+	0.982136i	$0.560257\pi$
$240$	0	0
$241$	13.7291	0.884371	0.442186	−	0.896924i	$-0.354203\pi$
$241$	13.7291	0.884371	0.442186	+	0.896924i	$0.354203\pi$
$242$	0	0
$243$	11.4444	0.734157
$244$	0	0
$245$	0	0
$246$	0	0
$247$	39.4943	2.51296
$248$	0	0
$249$	16.6335	1.05410
$250$	0	0
$251$	15.8938	1.00320	0.501602	−	0.865098i	$-0.332744\pi$
$251$	15.8938	1.00320	0.501602	+	0.865098i	$0.332744\pi$
$252$	0	0
$253$	2.94194	0.184958
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−2.31253	−0.144252	−0.0721259	−	0.997396i	$-0.522978\pi$
$257$	−2.31253	−0.144252	−0.0721259	+	0.997396i	$0.522978\pi$
$258$	0	0
$259$	6.09595	0.378784
$260$	0	0
$261$	−3.86397	−0.239174
$262$	0	0
$263$	21.4593	1.32324	0.661619	−	0.749840i	$-0.269870\pi$
$263$	21.4593	1.32324	0.661619	+	0.749840i	$0.269870\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−9.44670	−0.578129
$268$	0	0
$269$	21.3864	1.30395	0.651977	−	0.758239i	$-0.273940\pi$
$269$	21.3864	1.30395	0.651977	+	0.758239i	$0.273940\pi$
$270$	0	0
$271$	26.1305	1.58732	0.793658	−	0.608364i	$-0.208174\pi$
$271$	26.1305	1.58732	0.793658	+	0.608364i	$0.208174\pi$
$272$	0	0
$273$	5.12838	0.310384
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−18.6469	−1.12038	−0.560191	−	0.828363i	$-0.689272\pi$
$277$	−18.6469	−1.12038	−0.560191	+	0.828363i	$0.689272\pi$
$278$	0	0
$279$	7.09343	0.424673
$280$	0	0
$281$	14.8187	0.884011	0.442006	−	0.897012i	$-0.354267\pi$
$281$	14.8187	0.884011	0.442006	+	0.897012i	$0.354267\pi$
$282$	0	0
$283$	−15.7800	−0.938022	−0.469011	−	0.883192i	$-0.655389\pi$
$283$	−15.7800	−0.938022	−0.469011	+	0.883192i	$0.655389\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.55583	0.0918375
$288$	0	0
$289$	12.7178	0.748103
$290$	0	0
$291$	14.9541	0.876627
$292$	0	0
$293$	18.4003	1.07496	0.537479	−	0.843277i	$-0.319377\pi$
$293$	18.4003	1.07496	0.537479	+	0.843277i	$0.319377\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	4.69916	0.272673
$298$	0	0
$299$	−16.1109	−0.931718
$300$	0	0
$301$	1.36569	0.0787173
$302$	0	0
$303$	17.1363	0.984457
$304$	0	0
$305$	0	0
$306$	0	0
$307$	34.1179	1.94721	0.973606	−	0.228237i	$-0.0732962\pi$
$307$	34.1179	1.94721	0.973606	+	0.228237i	$0.0732962\pi$
$308$	0	0
$309$	−9.45932	−0.538122
$310$	0	0
$311$	13.0379	0.739311	0.369656	−	0.929169i	$-0.379476\pi$
$311$	13.0379	0.739311	0.369656	+	0.929169i	$0.379476\pi$
$312$	0	0
$313$	3.48262	0.196849	0.0984247	−	0.995144i	$-0.468620\pi$
$313$	3.48262	0.196849	0.0984247	+	0.995144i	$0.468620\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	4.82036	0.270738	0.135369	−	0.990795i	$-0.456778\pi$
$317$	4.82036	0.270738	0.135369	+	0.990795i	$0.456778\pi$
$318$	0	0
$319$	−2.72479	−0.152559
$320$	0	0
$321$	14.5225	0.810565
$322$	0	0
$323$	47.1760	2.62494
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−14.3411	−0.793063
$328$	0	0
$329$	0.694498	0.0382889
$330$	0	0
$331$	−8.56343	−0.470689	−0.235344	−	0.971912i	$-0.575622\pi$
$331$	−8.56343	−0.470689	−0.235344	+	0.971912i	$0.575622\pi$
$332$	0	0
$333$	−8.64456	−0.473719
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−14.6712	−0.799191	−0.399596	−	0.916692i	$-0.630849\pi$
$337$	−14.6712	−0.799191	−0.399596	+	0.916692i	$0.630849\pi$
$338$	0	0
$339$	2.34861	0.127559
$340$	0	0
$341$	5.00214	0.270881
$342$	0	0
$343$	−11.0883	−0.598715
$344$	0	0
$345$	0	0
$346$	0	0
$347$	22.9872	1.23402	0.617008	−	0.786957i	$-0.288344\pi$
$347$	22.9872	1.23402	0.617008	+	0.786957i	$0.288344\pi$
$348$	0	0
$349$	3.55023	0.190040	0.0950198	−	0.995475i	$-0.469709\pi$
$349$	3.55023	0.190040	0.0950198	+	0.995475i	$0.469709\pi$
$350$	0	0
$351$	−25.7339	−1.37357
$352$	0	0
$353$	−5.64110	−0.300245	−0.150123	−	0.988667i	$-0.547967\pi$
$353$	−5.64110	−0.300245	−0.150123	+	0.988667i	$0.547967\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	6.12586	0.324215
$358$	0	0
$359$	16.9066	0.892295	0.446147	−	0.894960i	$-0.352796\pi$
$359$	16.9066	0.892295	0.446147	+	0.894960i	$0.352796\pi$
$360$	0	0
$361$	55.8903	2.94159
$362$	0	0
$363$	−13.8961	−0.729355
$364$	0	0
$365$	0	0
$366$	0	0
$367$	12.6133	0.658411	0.329205	−	0.944258i	$-0.393219\pi$
$367$	12.6133	0.658411	0.329205	+	0.944258i	$0.393219\pi$
$368$	0	0
$369$	−2.20629	−0.114855
$370$	0	0
$371$	5.35621	0.278081
$372$	0	0
$373$	13.2496	0.686037	0.343019	−	0.939329i	$-0.388551\pi$
$373$	13.2496	0.686037	0.343019	+	0.939329i	$0.388551\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	14.9217	0.768507
$378$	0	0
$379$	−22.3360	−1.14732	−0.573661	−	0.819093i	$-0.694477\pi$
$379$	−22.3360	−1.14732	−0.573661	+	0.819093i	$0.694477\pi$
$380$	0	0
$381$	−7.64110	−0.391465
$382$	0	0
$383$	−24.3529	−1.24437	−0.622187	−	0.782869i	$-0.713756\pi$
$383$	−24.3529	−1.24437	−0.622187	+	0.782869i	$0.713756\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−1.93667	−0.0984462
$388$	0	0
$389$	30.1116	1.52672	0.763360	−	0.645974i	$-0.223548\pi$
$389$	30.1116	1.52672	0.763360	+	0.645974i	$0.223548\pi$
$390$	0	0
$391$	−19.2445	−0.973236
$392$	0	0
$393$	22.2144	1.12057
$394$	0	0
$395$	0	0
$396$	0	0
$397$	35.3704	1.77519	0.887594	−	0.460627i	$-0.152376\pi$
$397$	35.3704	1.77519	0.887594	+	0.460627i	$0.152376\pi$
$398$	0	0
$399$	9.72459	0.486839
$400$	0	0
$401$	−5.66794	−0.283043	−0.141522	−	0.989935i	$-0.545199\pi$
$401$	−5.66794	−0.283043	−0.141522	+	0.989935i	$0.545199\pi$
$402$	0	0
$403$	−27.3931	−1.36455
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−6.09595	−0.302165
$408$	0	0
$409$	19.3676	0.957668	0.478834	−	0.877905i	$-0.341060\pi$
$409$	19.3676	0.957668	0.478834	+	0.877905i	$0.341060\pi$
$410$	0	0
$411$	−3.90129	−0.192437
$412$	0	0
$413$	−11.2775	−0.554931
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−6.72793	−0.329468
$418$	0	0
$419$	−22.9792	−1.12261	−0.561304	−	0.827610i	$-0.689700\pi$
$419$	−22.9792	−1.12261	−0.561304	+	0.827610i	$0.689700\pi$
$420$	0	0
$421$	−10.1272	−0.493568	−0.246784	−	0.969070i	$-0.579374\pi$
$421$	−10.1272	−0.493568	−0.246784	+	0.969070i	$0.579374\pi$
$422$	0	0
$423$	−0.984855	−0.0478853
$424$	0	0
$425$	0	0
$426$	0	0
$427$	4.90369	0.237306
$428$	0	0
$429$	−5.12838	−0.247601
$430$	0	0
$431$	28.5024	1.37291	0.686457	−	0.727171i	$-0.259165\pi$
$431$	28.5024	1.37291	0.686457	+	0.727171i	$0.259165\pi$
$432$	0	0
$433$	15.8897	0.763609	0.381804	−	0.924243i	$-0.375303\pi$
$433$	15.8897	0.763609	0.381804	+	0.924243i	$0.375303\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−30.5500	−1.46140
$438$	0	0
$439$	−2.94660	−0.140634	−0.0703168	−	0.997525i	$-0.522401\pi$
$439$	−2.94660	−0.140634	−0.0703168	+	0.997525i	$0.522401\pi$
$440$	0	0
$441$	7.45171	0.354844
$442$	0	0
$443$	5.27327	0.250541	0.125270	−	0.992123i	$-0.460020\pi$
$443$	5.27327	0.250541	0.125270	+	0.992123i	$0.460020\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	2.70463	0.127925
$448$	0	0
$449$	−6.81659	−0.321695	−0.160847	−	0.986979i	$-0.551423\pi$
$449$	−6.81659	−0.321695	−0.160847	+	0.986979i	$0.551423\pi$
$450$	0	0
$451$	−1.55583	−0.0732609
$452$	0	0
$453$	2.65598	0.124789
$454$	0	0
$455$	0	0
$456$	0	0
$457$	4.17712	0.195397	0.0976987	−	0.995216i	$-0.468852\pi$
$457$	4.17712	0.195397	0.0976987	+	0.995216i	$0.468852\pi$
$458$	0	0
$459$	−30.7392	−1.43478
$460$	0	0
$461$	−37.4342	−1.74348	−0.871742	−	0.489965i	$-0.837010\pi$
$461$	−37.4342	−1.74348	−0.871742	+	0.489965i	$0.837010\pi$
$462$	0	0
$463$	13.8788	0.645003	0.322501	−	0.946569i	$-0.395476\pi$
$463$	13.8788	0.645003	0.322501	+	0.946569i	$0.395476\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−13.8606	−0.641391	−0.320696	−	0.947182i	$-0.603917\pi$
$467$	−13.8606	−0.641391	−0.320696	+	0.947182i	$0.603917\pi$
$468$	0	0
$469$	−1.24997	−0.0577181
$470$	0	0
$471$	10.4970	0.483674
$472$	0	0
$473$	−1.36569	−0.0627947
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−7.59554	−0.347776
$478$	0	0
$479$	−28.7974	−1.31579	−0.657894	−	0.753110i	$-0.728553\pi$
$479$	−28.7974	−1.31579	−0.657894	+	0.753110i	$0.728553\pi$
$480$	0	0
$481$	33.3832	1.52214
$482$	0	0
$483$	−3.96696	−0.180503
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−38.9513	−1.76505	−0.882525	−	0.470265i	$-0.844158\pi$
$487$	−38.9513	−1.76505	−0.882525	+	0.470265i	$0.844158\pi$
$488$	0	0
$489$	−19.0072	−0.859535
$490$	0	0
$491$	23.5069	1.06085	0.530426	−	0.847731i	$-0.322032\pi$
$491$	23.5069	1.06085	0.530426	+	0.847731i	$0.322032\pi$
$492$	0	0
$493$	17.8240	0.802753
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1.96777	0.0882665
$498$	0	0
$499$	−28.2651	−1.26532	−0.632660	−	0.774430i	$-0.718037\pi$
$499$	−28.2651	−1.26532	−0.632660	+	0.774430i	$0.718037\pi$
$500$	0	0
$501$	10.9979	0.491348
$502$	0	0
$503$	−6.90424	−0.307845	−0.153922	−	0.988083i	$-0.549191\pi$
$503$	−6.90424	−0.307845	−0.153922	+	0.988083i	$0.549191\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	10.5551	0.468768
$508$	0	0
$509$	33.5390	1.48659	0.743295	−	0.668964i	$-0.233262\pi$
$509$	33.5390	1.48659	0.743295	+	0.668964i	$0.233262\pi$
$510$	0	0
$511$	−2.77511	−0.122764
$512$	0	0
$513$	−48.7974	−2.15446
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−0.694498	−0.0305440
$518$	0	0
$519$	−3.37556	−0.148171
$520$	0	0
$521$	1.23342	0.0540373	0.0270186	−	0.999635i	$-0.491399\pi$
$521$	1.23342	0.0540373	0.0270186	+	0.999635i	$0.491399\pi$
$522$	0	0
$523$	−23.0305	−1.00705	−0.503526	−	0.863980i	$-0.667964\pi$
$523$	−23.0305	−1.00705	−0.503526	+	0.863980i	$0.667964\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−32.7211	−1.42535
$528$	0	0
$529$	−10.5377	−0.458162
$530$	0	0
$531$	15.9925	0.694014
$532$	0	0
$533$	8.52014	0.369048
$534$	0	0
$535$	0	0
$536$	0	0
$537$	27.0123	1.16567
$538$	0	0
$539$	5.25479	0.226340
$540$	0	0
$541$	−26.7700	−1.15093	−0.575466	−	0.817826i	$-0.695179\pi$
$541$	−26.7700	−1.15093	−0.575466	+	0.817826i	$0.695179\pi$
$542$	0	0
$543$	10.6232	0.455885
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−6.42689	−0.274794	−0.137397	−	0.990516i	$-0.543874\pi$
$547$	−6.42689	−0.274794	−0.137397	+	0.990516i	$0.543874\pi$
$548$	0	0
$549$	−6.95383	−0.296782
$550$	0	0
$551$	28.2950	1.20541
$552$	0	0
$553$	−4.30764	−0.183179
$554$	0	0
$555$	0	0
$556$	0	0
$557$	44.0843	1.86791	0.933957	−	0.357386i	$-0.116332\pi$
$557$	44.0843	1.86791	0.933957	+	0.357386i	$0.116332\pi$
$558$	0	0
$559$	7.47893	0.316325
$560$	0	0
$561$	−6.12586	−0.258634
$562$	0	0
$563$	−20.0779	−0.846181	−0.423091	−	0.906087i	$-0.639055\pi$
$563$	−20.0779	−0.846181	−0.423091	+	0.906087i	$0.639055\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−3.38185	−0.142024
$568$	0	0
$569$	−29.9339	−1.25489	−0.627447	−	0.778659i	$-0.715900\pi$
$569$	−29.9339	−1.25489	−0.627447	+	0.778659i	$0.715900\pi$
$570$	0	0
$571$	14.2396	0.595911	0.297955	−	0.954580i	$-0.403695\pi$
$571$	14.2396	0.595911	0.297955	+	0.954580i	$0.403695\pi$
$572$	0	0
$573$	7.11317	0.297157
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−18.2089	−0.758048	−0.379024	−	0.925387i	$-0.623740\pi$
$577$	−18.2089	−0.758048	−0.379024	+	0.925387i	$0.623740\pi$
$578$	0	0
$579$	−18.1309	−0.753496
$580$	0	0
$581$	10.2801	0.426489
$582$	0	0
$583$	−5.35621	−0.221832
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−5.16430	−0.213154	−0.106577	−	0.994304i	$-0.533989\pi$
$587$	−5.16430	−0.213154	−0.106577	+	0.994304i	$0.533989\pi$
$588$	0	0
$589$	−51.9437	−2.14030
$590$	0	0
$591$	−7.33346	−0.301658
$592$	0	0
$593$	−35.6286	−1.46309	−0.731546	−	0.681793i	$-0.761201\pi$
$593$	−35.6286	−1.46309	−0.731546	+	0.681793i	$0.761201\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	23.4745	0.960749
$598$	0	0
$599$	31.8284	1.30047	0.650236	−	0.759733i	$-0.274670\pi$
$599$	31.8284	1.30047	0.650236	+	0.759733i	$0.274670\pi$
$600$	0	0
$601$	−12.6378	−0.515508	−0.257754	−	0.966211i	$-0.582982\pi$
$601$	−12.6378	−0.515508	−0.257754	+	0.966211i	$0.582982\pi$
$602$	0	0
$603$	1.77255	0.0721840
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−34.9481	−1.41850	−0.709250	−	0.704957i	$-0.750966\pi$
$607$	−34.9481	−1.41850	−0.709250	+	0.704957i	$0.750966\pi$
$608$	0	0
$609$	3.67414	0.148884
$610$	0	0
$611$	3.80327	0.153864
$612$	0	0
$613$	7.12279	0.287687	0.143843	−	0.989600i	$-0.454054\pi$
$613$	7.12279	0.287687	0.143843	+	0.989600i	$0.454054\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−16.0870	−0.647639	−0.323820	−	0.946119i	$-0.604967\pi$
$617$	−16.0870	−0.647639	−0.323820	+	0.946119i	$0.604967\pi$
$618$	0	0
$619$	0.750421	0.0301619	0.0150810	−	0.999886i	$-0.495199\pi$
$619$	0.750421	0.0301619	0.0150810	+	0.999886i	$0.495199\pi$
$620$	0	0
$621$	19.9060	0.798798
$622$	0	0
$623$	−5.83838	−0.233910
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−9.72459	−0.388363
$628$	0	0
$629$	39.8762	1.58997
$630$	0	0
$631$	15.3756	0.612092	0.306046	−	0.952017i	$-0.400994\pi$
$631$	15.3756	0.612092	0.306046	+	0.952017i	$0.400994\pi$
$632$	0	0
$633$	0.363983	0.0144670
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−28.7767	−1.14017
$638$	0	0
$639$	−2.79045	−0.110389
$640$	0	0
$641$	9.45385	0.373405	0.186702	−	0.982417i	$-0.440220\pi$
$641$	9.45385	0.373405	0.186702	+	0.982417i	$0.440220\pi$
$642$	0	0
$643$	−32.8334	−1.29482	−0.647411	−	0.762141i	$-0.724148\pi$
$643$	−32.8334	−1.29482	−0.647411	+	0.762141i	$0.724148\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−17.8681	−0.702468	−0.351234	−	0.936288i	$-0.614238\pi$
$647$	−17.8681	−0.702468	−0.351234	+	0.936288i	$0.614238\pi$
$648$	0	0
$649$	11.2775	0.442682
$650$	0	0
$651$	−6.74495	−0.264355
$652$	0	0
$653$	−31.9642	−1.25086	−0.625428	−	0.780282i	$-0.715076\pi$
$653$	−31.9642	−1.25086	−0.625428	+	0.780282i	$0.715076\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	3.93534	0.153532
$658$	0	0
$659$	11.3989	0.444037	0.222018	−	0.975042i	$-0.428736\pi$
$659$	11.3989	0.444037	0.222018	+	0.975042i	$0.428736\pi$
$660$	0	0
$661$	7.33427	0.285270	0.142635	−	0.989775i	$-0.454442\pi$
$661$	7.33427	0.285270	0.142635	+	0.989775i	$0.454442\pi$
$662$	0	0
$663$	33.5469	1.30285
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−11.5424	−0.446923
$668$	0	0
$669$	−31.2445	−1.20798
$670$	0	0
$671$	−4.90369	−0.189305
$672$	0	0
$673$	−32.7630	−1.26292	−0.631460	−	0.775408i	$-0.717544\pi$
$673$	−32.7630	−1.26292	−0.631460	+	0.775408i	$0.717544\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−32.5826	−1.25225	−0.626126	−	0.779722i	$-0.715360\pi$
$677$	−32.5826	−1.25225	−0.626126	+	0.779722i	$0.715360\pi$
$678$	0	0
$679$	9.24217	0.354682
$680$	0	0
$681$	0.340681	0.0130549
$682$	0	0
$683$	4.34942	0.166426	0.0832130	−	0.996532i	$-0.473482\pi$
$683$	4.34942	0.166426	0.0832130	+	0.996532i	$0.473482\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−28.4385	−1.08500
$688$	0	0
$689$	29.3321	1.11747
$690$	0	0
$691$	20.2595	0.770708	0.385354	−	0.922769i	$-0.374079\pi$
$691$	20.2595	0.770708	0.385354	+	0.922769i	$0.374079\pi$
$692$	0	0
$693$	0.820744	0.0311775
$694$	0	0
$695$	0	0
$696$	0	0
$697$	10.1773	0.385493
$698$	0	0
$699$	−19.6970	−0.745010
$700$	0	0
$701$	12.5964	0.475758	0.237879	−	0.971295i	$-0.423548\pi$
$701$	12.5964	0.475758	0.237879	+	0.971295i	$0.423548\pi$
$702$	0	0
$703$	63.3022	2.38749
$704$	0	0
$705$	0	0
$706$	0	0
$707$	10.5908	0.398310
$708$	0	0
$709$	29.4781	1.10707	0.553537	−	0.832825i	$-0.313278\pi$
$709$	29.4781	1.10707	0.553537	+	0.832825i	$0.313278\pi$
$710$	0	0
$711$	6.10858	0.229090
$712$	0	0
$713$	21.1894	0.793549
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−7.84537	−0.292991
$718$	0	0
$719$	30.1585	1.12472	0.562361	−	0.826892i	$-0.309893\pi$
$719$	30.1585	1.12472	0.562361	+	0.826892i	$0.309893\pi$
$720$	0	0
$721$	−5.84618	−0.217723
$722$	0	0
$723$	18.5126	0.688489
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−5.94735	−0.220575	−0.110287	−	0.993900i	$-0.535177\pi$
$727$	−5.94735	−0.220575	−0.110287	+	0.993900i	$0.535177\pi$
$728$	0	0
$729$	27.6059	1.02244
$730$	0	0
$731$	8.93359	0.330421
$732$	0	0
$733$	10.1763	0.375871	0.187935	−	0.982181i	$-0.439820\pi$
$733$	10.1763	0.375871	0.187935	+	0.982181i	$0.439820\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.24997	0.0460432
$738$	0	0
$739$	−36.3198	−1.33605	−0.668023	−	0.744140i	$-0.732859\pi$
$739$	−36.3198	−1.33605	−0.668023	+	0.744140i	$0.732859\pi$
$740$	0	0
$741$	53.2546	1.95636
$742$	0	0
$743$	−2.84832	−0.104495	−0.0522473	−	0.998634i	$-0.516638\pi$
$743$	−2.84832	−0.104495	−0.0522473	+	0.998634i	$0.516638\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−14.5780	−0.533380
$748$	0	0
$749$	8.97537	0.327953
$750$	0	0
$751$	31.5502	1.15128	0.575641	−	0.817702i	$-0.304752\pi$
$751$	31.5502	1.15128	0.575641	+	0.817702i	$0.304752\pi$
$752$	0	0
$753$	21.4314	0.781002
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−45.6446	−1.65898	−0.829491	−	0.558520i	$-0.811369\pi$
$757$	−45.6446	−1.65898	−0.829491	+	0.558520i	$0.811369\pi$
$758$	0	0
$759$	3.96696	0.143991
$760$	0	0
$761$	9.07879	0.329106	0.164553	−	0.986368i	$-0.447382\pi$
$761$	9.07879	0.329106	0.164553	+	0.986368i	$0.447382\pi$
$762$	0	0
$763$	−8.86327	−0.320872
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−61.7590	−2.22999
$768$	0	0
$769$	−25.4585	−0.918057	−0.459029	−	0.888421i	$-0.651802\pi$
$769$	−25.4585	−0.918057	−0.459029	+	0.888421i	$0.651802\pi$
$770$	0	0
$771$	−3.11825	−0.112301
$772$	0	0
$773$	17.1523	0.616924	0.308462	−	0.951237i	$-0.400186\pi$
$773$	17.1523	0.616924	0.308462	+	0.951237i	$0.400186\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	8.21987	0.294886
$778$	0	0
$779$	16.1561	0.578854
$780$	0	0
$781$	−1.96777	−0.0704123
$782$	0	0
$783$	−18.4366	−0.658872
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−11.2729	−0.401835	−0.200917	−	0.979608i	$-0.564392\pi$
$787$	−11.2729	−0.401835	−0.200917	+	0.979608i	$0.564392\pi$
$788$	0	0
$789$	28.9360	1.03015
$790$	0	0
$791$	1.45152	0.0516101
$792$	0	0
$793$	26.8540	0.953613
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−26.6713	−0.944745	−0.472372	−	0.881399i	$-0.656602\pi$
$797$	−26.6713	−0.944745	−0.472372	+	0.881399i	$0.656602\pi$
$798$	0	0
$799$	4.54301	0.160720
$800$	0	0
$801$	8.27929	0.292534
$802$	0	0
$803$	2.77511	0.0979316
$804$	0	0
$805$	0	0
$806$	0	0
$807$	28.8378	1.01514
$808$	0	0
$809$	10.9742	0.385832	0.192916	−	0.981215i	$-0.438206\pi$
$809$	10.9742	0.385832	0.192916	+	0.981215i	$0.438206\pi$
$810$	0	0
$811$	9.86618	0.346448	0.173224	−	0.984882i	$-0.444581\pi$
$811$	9.86618	0.346448	0.173224	+	0.984882i	$0.444581\pi$
$812$	0	0
$813$	35.2348	1.23574
$814$	0	0
$815$	0	0
$816$	0	0
$817$	14.1818	0.496158
$818$	0	0
$819$	−4.49463	−0.157055
$820$	0	0
$821$	−2.29790	−0.0801971	−0.0400985	−	0.999196i	$-0.512767\pi$
$821$	−2.29790	−0.0801971	−0.0400985	+	0.999196i	$0.512767\pi$
$822$	0	0
$823$	−0.805405	−0.0280746	−0.0140373	−	0.999901i	$-0.504468\pi$
$823$	−0.805405	−0.0280746	−0.0140373	+	0.999901i	$0.504468\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−36.4543	−1.26764	−0.633820	−	0.773480i	$-0.718514\pi$
$827$	−36.4543	−1.26764	−0.633820	+	0.773480i	$0.718514\pi$
$828$	0	0
$829$	29.5705	1.02703	0.513514	−	0.858081i	$-0.328344\pi$
$829$	29.5705	1.02703	0.513514	+	0.858081i	$0.328344\pi$
$830$	0	0
$831$	−25.1437	−0.872226
$832$	0	0
$833$	−34.3738	−1.19098
$834$	0	0
$835$	0	0
$836$	0	0
$837$	33.8458	1.16988
$838$	0	0
$839$	43.5273	1.50273	0.751365	−	0.659887i	$-0.229396\pi$
$839$	43.5273	1.50273	0.751365	+	0.659887i	$0.229396\pi$
$840$	0	0
$841$	−18.3096	−0.631365
$842$	0	0
$843$	19.9818	0.688209
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−8.58825	−0.295096
$848$	0	0
$849$	−21.2779	−0.730257
$850$	0	0
$851$	−25.8229	−0.885197
$852$	0	0
$853$	50.6065	1.73273	0.866367	−	0.499408i	$-0.166449\pi$
$853$	50.6065	1.73273	0.866367	+	0.499408i	$0.166449\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	34.1589	1.16684	0.583422	−	0.812169i	$-0.301713\pi$
$857$	34.1589	1.16684	0.583422	+	0.812169i	$0.301713\pi$
$858$	0	0
$859$	−10.9872	−0.374878	−0.187439	−	0.982276i	$-0.560019\pi$
$859$	−10.9872	−0.374878	−0.187439	+	0.982276i	$0.560019\pi$
$860$	0	0
$861$	2.09790	0.0714961
$862$	0	0
$863$	46.3080	1.57634	0.788172	−	0.615455i	$-0.211028\pi$
$863$	46.3080	1.57634	0.788172	+	0.615455i	$0.211028\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	17.1488	0.582404
$868$	0	0
$869$	4.30764	0.146127
$870$	0	0
$871$	−6.84518	−0.231940
$872$	0	0
$873$	−13.1061	−0.443576
$874$	0	0
$875$	0	0
$876$	0	0
$877$	20.9772	0.708350	0.354175	−	0.935179i	$-0.384762\pi$
$877$	20.9772	0.708350	0.354175	+	0.935179i	$0.384762\pi$
$878$	0	0
$879$	24.8113	0.836863
$880$	0	0
$881$	−50.8631	−1.71362	−0.856811	−	0.515631i	$-0.827558\pi$
$881$	−50.8631	−1.71362	−0.856811	+	0.515631i	$0.827558\pi$
$882$	0	0
$883$	49.0893	1.65199	0.825994	−	0.563680i	$-0.190615\pi$
$883$	49.0893	1.65199	0.825994	+	0.563680i	$0.190615\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	36.9694	1.24131	0.620655	−	0.784084i	$-0.286867\pi$
$887$	36.9694	1.24131	0.620655	+	0.784084i	$0.286867\pi$
$888$	0	0
$889$	−4.72246	−0.158386
$890$	0	0
$891$	3.38185	0.113296
$892$	0	0
$893$	7.21188	0.241336
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−21.7242	−0.725349
$898$	0	0
$899$	−19.6253	−0.654542
$900$	0	0
$901$	35.0373	1.16726
$902$	0	0
$903$	1.84152	0.0612820
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−44.9718	−1.49326	−0.746632	−	0.665237i	$-0.768330\pi$
$907$	−44.9718	−1.49326	−0.746632	+	0.665237i	$0.768330\pi$
$908$	0	0
$909$	−15.0187	−0.498138
$910$	0	0
$911$	26.9495	0.892876	0.446438	−	0.894815i	$-0.352692\pi$
$911$	26.9495	0.892876	0.446438	+	0.894815i	$0.352692\pi$
$912$	0	0
$913$	−10.2801	−0.340220
$914$	0	0
$915$	0	0
$916$	0	0
$917$	13.7293	0.453380
$918$	0	0
$919$	−31.9013	−1.05233	−0.526163	−	0.850384i	$-0.676370\pi$
$919$	−31.9013	−1.05233	−0.526163	+	0.850384i	$0.676370\pi$
$920$	0	0
$921$	46.0050	1.51592
$922$	0	0
$923$	10.7761	0.354698
$924$	0	0
$925$	0	0
$926$	0	0
$927$	8.29036	0.272291
$928$	0	0
$929$	−6.25536	−0.205232	−0.102616	−	0.994721i	$-0.532721\pi$
$929$	−6.25536	−0.205232	−0.102616	+	0.994721i	$0.532721\pi$
$930$	0	0
$931$	−54.5673	−1.78837
$932$	0	0
$933$	17.5805	0.575559
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−30.9195	−1.01010	−0.505049	−	0.863091i	$-0.668525\pi$
$937$	−30.9195	−1.01010	−0.505049	+	0.863091i	$0.668525\pi$
$938$	0	0
$939$	4.69602	0.153249
$940$	0	0
$941$	−25.6341	−0.835647	−0.417823	−	0.908528i	$-0.637207\pi$
$941$	−25.6341	−0.835647	−0.417823	+	0.908528i	$0.637207\pi$
$942$	0	0
$943$	−6.59058	−0.214619
$944$	0	0
$945$	0	0
$946$	0	0
$947$	45.2123	1.46920	0.734601	−	0.678500i	$-0.237370\pi$
$947$	45.2123	1.46920	0.734601	+	0.678500i	$0.237370\pi$
$948$	0	0
$949$	−15.1973	−0.493325
$950$	0	0
$951$	6.49984	0.210772
$952$	0	0
$953$	−20.6040	−0.667428	−0.333714	−	0.942674i	$-0.608302\pi$
$953$	−20.6040	−0.667428	−0.333714	+	0.942674i	$0.608302\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−3.67414	−0.118768
$958$	0	0
$959$	−2.41113	−0.0778595
$960$	0	0
$961$	5.02796	0.162192
$962$	0	0
$963$	−12.7278	−0.410148
$964$	0	0
$965$	0	0
$966$	0	0
$967$	51.0957	1.64313	0.821564	−	0.570116i	$-0.193102\pi$
$967$	51.0957	1.64313	0.821564	+	0.570116i	$0.193102\pi$
$968$	0	0
$969$	63.6127	2.04353
$970$	0	0
$971$	−14.6139	−0.468982	−0.234491	−	0.972118i	$-0.575342\pi$
$971$	−14.6139	−0.468982	−0.234491	+	0.972118i	$0.575342\pi$
$972$	0	0
$973$	−4.15809	−0.133302
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−52.2556	−1.67181	−0.835903	−	0.548877i	$-0.815055\pi$
$977$	−52.2556	−1.67181	−0.835903	+	0.548877i	$0.815055\pi$
$978$	0	0
$979$	5.83838	0.186595
$980$	0	0
$981$	12.5688	0.401292
$982$	0	0
$983$	−29.6089	−0.944376	−0.472188	−	0.881498i	$-0.656536\pi$
$983$	−29.6089	−0.944376	−0.472188	+	0.881498i	$0.656536\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0.936471	0.0298082
$988$	0	0
$989$	−5.78518	−0.183958
$990$	0	0
$991$	−14.8753	−0.472531	−0.236265	−	0.971689i	$-0.575923\pi$
$991$	−14.8753	−0.472531	−0.236265	+	0.971689i	$0.575923\pi$
$992$	0	0
$993$	−11.5471	−0.366435
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−30.7746	−0.974640	−0.487320	−	0.873223i	$-0.662026\pi$
$997$	−30.7746	−0.974640	−0.487320	+	0.873223i	$0.662026\pi$
$998$	0	0
$999$	−41.2468	−1.30499

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	10000.2.a.t.1.3		4
4.3	odd	2		1250.2.a.l.1.2		4
5.4	even	2		10000.2.a.x.1.2		4
20.3	even	4		1250.2.b.e.1249.2		8
20.7	even	4		1250.2.b.e.1249.7		8
20.19	odd	2		1250.2.a.f.1.3		4
25.11	even	5		400.2.u.d.321.1		8
25.16	even	5		400.2.u.d.81.1		8
100.11	odd	10		50.2.d.b.21.2	✓	8
100.23	even	20		250.2.e.c.149.1		16
100.27	even	20		250.2.e.c.149.4		16
100.39	odd	10		250.2.d.d.101.1		8
100.59	odd	10		250.2.d.d.151.1		8
100.63	even	20		250.2.e.c.99.4		16
100.87	even	20		250.2.e.c.99.1		16
100.91	odd	10		50.2.d.b.31.2	yes	8
300.11	even	10		450.2.h.e.271.2		8
300.191	even	10		450.2.h.e.181.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
50.2.d.b.21.2	✓	8	100.11	odd	10
50.2.d.b.31.2	yes	8	100.91	odd	10
250.2.d.d.101.1		8	100.39	odd	10
250.2.d.d.151.1		8	100.59	odd	10
250.2.e.c.99.1		16	100.87	even	20
250.2.e.c.99.4		16	100.63	even	20
250.2.e.c.149.1		16	100.23	even	20
250.2.e.c.149.4		16	100.27	even	20
400.2.u.d.81.1		8	25.16	even	5
400.2.u.d.321.1		8	25.11	even	5
450.2.h.e.181.2		8	300.191	even	10
450.2.h.e.271.2		8	300.11	even	10
1250.2.a.f.1.3		4	20.19	odd	2
1250.2.a.l.1.2		4	4.3	odd	2
1250.2.b.e.1249.2		8	20.3	even	4
1250.2.b.e.1249.7		8	20.7	even	4
10000.2.a.t.1.3		4	1.1	even	1	trivial
10000.2.a.x.1.2		4	5.4	even	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:	\( N \)	\(=\)	\( 10000 = 2^{4} \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	10000.a (trivial)

\(f(q)\)	\(=\)	\(q+1.34841 q^{3} +0.833366 q^{7} -1.18178 q^{9} +O(q^{10})\)	\(q+1.34841 q^{3} +0.833366 q^{7} -1.18178 q^{9} -0.833366 q^{11} +4.56375 q^{13} +5.45140 q^{17} +8.65392 q^{19} +1.12372 q^{21} -3.53019 q^{23} -5.63877 q^{27} +3.26962 q^{29} -6.00233 q^{31} -1.12372 q^{33} +7.31486 q^{37} +6.15382 q^{39} +1.86692 q^{41} +1.63877 q^{43} +0.833366 q^{47} -6.30550 q^{49} +7.35074 q^{51} +6.42721 q^{53} +11.6691 q^{57} -13.5325 q^{59} +5.88420 q^{61} -0.984855 q^{63} -1.49990 q^{67} -4.76016 q^{69} +2.36123 q^{71} -3.33001 q^{73} -0.694498 q^{77} -5.16896 q^{79} -4.05806 q^{81} +12.3356 q^{83} +4.40880 q^{87} -7.00579 q^{89} +3.80327 q^{91} -8.09363 q^{93} +11.0902 q^{97} +0.984855 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{3} - 2 q^{7} + 7 q^{9}+O(q^{10}) \) 4 * q - q^3 - 2 * q^7 + 7 * q^9	\( 4 q - q^{3} - 2 q^{7} + 7 q^{9} + 2 q^{11} + 11 q^{13} + 12 q^{17} + 5 q^{19} - 7 q^{21} + 4 q^{23} - 10 q^{27} + 15 q^{29} + 12 q^{31} + 7 q^{33} + 12 q^{37} + 11 q^{39} + 13 q^{41} - 6 q^{43} - 2 q^{47} - 2 q^{49} - 13 q^{51} + 11 q^{53} + 8 q^{61} - 21 q^{63} - 22 q^{67} - 31 q^{69} + 22 q^{71} + 21 q^{73} - 26 q^{77} + 10 q^{79} - 16 q^{81} + 24 q^{83} - 25 q^{87} - 5 q^{89} + 12 q^{91} - 23 q^{93} + 22 q^{97} + 21 q^{99}+O(q^{100}) \) 4 * q - q^3 - 2 * q^7 + 7 * q^9 + 2 * q^11 + 11 * q^13 + 12 * q^17 + 5 * q^19 - 7 * q^21 + 4 * q^23 - 10 * q^27 + 15 * q^29 + 12 * q^31 + 7 * q^33 + 12 * q^37 + 11 * q^39 + 13 * q^41 - 6 * q^43 - 2 * q^47 - 2 * q^49 - 13 * q^51 + 11 * q^53 + 8 * q^61 - 21 * q^63 - 22 * q^67 - 31 * q^69 + 22 * q^71 + 21 * q^73 - 26 * q^77 + 10 * q^79 - 16 * q^81 + 24 * q^83 - 25 * q^87 - 5 * q^89 + 12 * q^91 - 23 * q^93 + 22 * q^97 + 21 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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