LMFDB - Embedded newform 10000.2.a.bb.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:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{20})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} + 5 $ x^4 - 5*x^2 + 5
Coefficient ring:	$\Z[a_1, a_2, a_3]$
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			$1.17557$ 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)

$f(q)$	$=$	$q+2.17557 q^{3} +2.72654 q^{7} +1.73311 q^{9} +O(q^{10})$	$q+2.17557 q^{3} +2.72654 q^{7} +1.73311 q^{9} -5.59783 q^{11} -0.479853 q^{13} -3.90211 q^{17} +4.13818 q^{19} +5.93179 q^{21} +0.158384 q^{23} -2.75621 q^{27} -7.02967 q^{29} -0.431842 q^{31} -12.1785 q^{33} -2.65542 q^{37} -1.04395 q^{39} -8.48746 q^{41} +3.49890 q^{43} +6.76091 q^{47} +0.434034 q^{49} -8.48932 q^{51} -5.58721 q^{53} +9.00290 q^{57} +7.60845 q^{59} +7.38487 q^{61} +4.72539 q^{63} -2.06706 q^{67} +0.344577 q^{69} +7.05934 q^{71} -13.6085 q^{73} -15.2627 q^{77} -6.33801 q^{79} -11.1957 q^{81} -11.6007 q^{83} -15.2935 q^{87} +1.70820 q^{89} -1.30834 q^{91} -0.939503 q^{93} -11.8647 q^{97} -9.70164 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} + 8 q^{7} + 2 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 + 8 * q^7 + 2 * q^9	$ 4 q + 4 q^{3} + 8 q^{7} + 2 q^{9} + 2 q^{11} - 14 q^{13} - 8 q^{17} - 2 q^{21} + 4 q^{23} + 10 q^{27} - 10 q^{29} - 8 q^{31} - 8 q^{33} - 18 q^{37} - 14 q^{39} - 12 q^{41} + 14 q^{43} + 8 q^{47} + 8 q^{49} - 8 q^{51} - 4 q^{53} - 2 q^{61} - 16 q^{63} - 2 q^{67} - 6 q^{69} - 8 q^{71} - 24 q^{73} - 6 q^{77} - 20 q^{79} + 4 q^{81} + 14 q^{83} - 20 q^{87} - 20 q^{89} - 18 q^{91} - 8 q^{93} - 28 q^{97} - 14 q^{99}+O(q^{100}) $ 4 * q + 4 * q^3 + 8 * q^7 + 2 * q^9 + 2 * q^11 - 14 * q^13 - 8 * q^17 - 2 * q^21 + 4 * q^23 + 10 * q^27 - 10 * q^29 - 8 * q^31 - 8 * q^33 - 18 * q^37 - 14 * q^39 - 12 * q^41 + 14 * q^43 + 8 * q^47 + 8 * q^49 - 8 * q^51 - 4 * q^53 - 2 * q^61 - 16 * q^63 - 2 * q^67 - 6 * q^69 - 8 * q^71 - 24 * q^73 - 6 * q^77 - 20 * q^79 + 4 * q^81 + 14 * q^83 - 20 * q^87 - 20 * q^89 - 18 * q^91 - 8 * q^93 - 28 * q^97 - 14 * 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
$3$	2.17557	1.25607	0.628033	−	0.778187i	$-0.283860\pi$
$3$	2.17557	1.25607	0.628033	+	0.778187i	$0.283860\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.72654	1.03054	0.515268	−	0.857029i	$-0.327692\pi$
$7$	2.72654	1.03054	0.515268	+	0.857029i	$0.327692\pi$
$8$	0	0
$9$	1.73311	0.577702
$10$	0	0
$11$	−5.59783	−1.68781	−0.843905	−	0.536493i	$-0.819749\pi$
$11$	−5.59783	−1.68781	−0.843905	+	0.536493i	$0.819749\pi$
$12$	0	0
$13$	−0.479853	−0.133087	−0.0665436	−	0.997784i	$-0.521197\pi$
$13$	−0.479853	−0.133087	−0.0665436	+	0.997784i	$0.521197\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−3.90211	−0.946401	−0.473201	−	0.880955i	$-0.656901\pi$
$17$	−3.90211	−0.946401	−0.473201	+	0.880955i	$0.656901\pi$
$18$	0	0
$19$	4.13818	0.949364	0.474682	−	0.880157i	$-0.342563\pi$
$19$	4.13818	0.949364	0.474682	+	0.880157i	$0.342563\pi$
$20$	0	0
$21$	5.93179	1.29442
$22$	0	0
$23$	0.158384	0.0330254	0.0165127	−	0.999864i	$-0.494744\pi$
$23$	0.158384	0.0330254	0.0165127	+	0.999864i	$0.494744\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−2.75621	−0.530434
$28$	0	0
$29$	−7.02967	−1.30538	−0.652689	−	0.757626i	$-0.726359\pi$
$29$	−7.02967	−1.30538	−0.652689	+	0.757626i	$0.726359\pi$
$30$	0	0
$31$	−0.431842	−0.0775611	−0.0387805	−	0.999248i	$-0.512347\pi$
$31$	−0.431842	−0.0775611	−0.0387805	+	0.999248i	$0.512347\pi$
$32$	0	0
$33$	−12.1785	−2.12000
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−2.65542	−0.436549	−0.218274	−	0.975887i	$-0.570043\pi$
$37$	−2.65542	−0.436549	−0.218274	+	0.975887i	$0.570043\pi$
$38$	0	0
$39$	−1.04395	−0.167166
$40$	0	0
$41$	−8.48746	−1.32552	−0.662759	−	0.748833i	$-0.730615\pi$
$41$	−8.48746	−1.32552	−0.662759	+	0.748833i	$0.730615\pi$
$42$	0	0
$43$	3.49890	0.533578	0.266789	−	0.963755i	$-0.414037\pi$
$43$	3.49890	0.533578	0.266789	+	0.963755i	$0.414037\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	6.76091	0.986181	0.493090	−	0.869978i	$-0.335867\pi$
$47$	6.76091	0.986181	0.493090	+	0.869978i	$0.335867\pi$
$48$	0	0
$49$	0.434034	0.0620049
$50$	0	0
$51$	−8.48932	−1.18874
$52$	0	0
$53$	−5.58721	−0.767462	−0.383731	−	0.923445i	$-0.625361\pi$
$53$	−5.58721	−0.767462	−0.383731	+	0.923445i	$0.625361\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	9.00290	1.19246
$58$	0	0
$59$	7.60845	0.990536	0.495268	−	0.868740i	$-0.335070\pi$
$59$	7.60845	0.990536	0.495268	+	0.868740i	$0.335070\pi$
$60$	0	0
$61$	7.38487	0.945536	0.472768	−	0.881187i	$-0.343255\pi$
$61$	7.38487	0.945536	0.472768	+	0.881187i	$0.343255\pi$
$62$	0	0
$63$	4.72539	0.595343
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−2.06706	−0.252532	−0.126266	−	0.991996i	$-0.540299\pi$
$67$	−2.06706	−0.252532	−0.126266	+	0.991996i	$0.540299\pi$
$68$	0	0
$69$	0.344577	0.0414821
$70$	0	0
$71$	7.05934	0.837790	0.418895	−	0.908035i	$-0.362418\pi$
$71$	7.05934	0.837790	0.418895	+	0.908035i	$0.362418\pi$
$72$	0	0
$73$	−13.6085	−1.59275	−0.796374	−	0.604804i	$-0.793251\pi$
$73$	−13.6085	−1.59275	−0.796374	+	0.604804i	$0.793251\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−15.2627	−1.73935
$78$	0	0
$79$	−6.33801	−0.713082	−0.356541	−	0.934280i	$-0.616044\pi$
$79$	−6.33801	−0.713082	−0.356541	+	0.934280i	$0.616044\pi$
$80$	0	0
$81$	−11.1957	−1.24396
$82$	0	0
$83$	−11.6007	−1.27335	−0.636673	−	0.771134i	$-0.719690\pi$
$83$	−11.6007	−1.27335	−0.636673	+	0.771134i	$0.719690\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−15.2935	−1.63964
$88$	0	0
$89$	1.70820	0.181069	0.0905346	−	0.995893i	$-0.471142\pi$
$89$	1.70820	0.181069	0.0905346	+	0.995893i	$0.471142\pi$
$90$	0	0
$91$	−1.30834	−0.137151
$92$	0	0
$93$	−0.939503	−0.0974219
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−11.8647	−1.20468	−0.602340	−	0.798240i	$-0.705765\pi$
$97$	−11.8647	−1.20468	−0.602340	+	0.798240i	$0.705765\pi$
$98$	0	0
$99$	−9.70164	−0.975051
$100$	0	0
$101$	−7.92116	−0.788185	−0.394093	−	0.919071i	$-0.628941\pi$
$101$	−7.92116	−0.788185	−0.394093	+	0.919071i	$0.628941\pi$
$102$	0	0
$103$	−6.27450	−0.618245	−0.309122	−	0.951022i	$-0.600035\pi$
$103$	−6.27450	−0.618245	−0.309122	+	0.951022i	$0.600035\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−10.9662	−1.06014	−0.530069	−	0.847954i	$-0.677834\pi$
$107$	−10.9662	−1.06014	−0.530069	+	0.847954i	$0.677834\pi$
$108$	0	0
$109$	7.81485	0.748527	0.374263	−	0.927322i	$-0.377896\pi$
$109$	7.81485	0.748527	0.374263	+	0.927322i	$0.377896\pi$
$110$	0	0
$111$	−5.77706	−0.548334
$112$	0	0
$113$	−1.38132	−0.129944	−0.0649720	−	0.997887i	$-0.520696\pi$
$113$	−1.38132	−0.129944	−0.0649720	+	0.997887i	$0.520696\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−0.831637	−0.0768848
$118$	0	0
$119$	−10.6393	−0.975301
$120$	0	0
$121$	20.3357	1.84870
$122$	0	0
$123$	−18.4651	−1.66494
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−0.208903	−0.0185371	−0.00926857	−	0.999957i	$-0.502950\pi$
$127$	−0.208903	−0.0185371	−0.00926857	+	0.999957i	$0.502950\pi$
$128$	0	0
$129$	7.61211	0.670209
$130$	0	0
$131$	5.92773	0.517908	0.258954	−	0.965890i	$-0.416622\pi$
$131$	5.92773	0.517908	0.258954	+	0.965890i	$0.416622\pi$
$132$	0	0
$133$	11.2829	0.978354
$134$	0	0
$135$	0	0
$136$	0	0
$137$	14.5054	1.23928	0.619638	−	0.784888i	$-0.287279\pi$
$137$	14.5054	1.23928	0.619638	+	0.784888i	$0.287279\pi$
$138$	0	0
$139$	6.05278	0.513390	0.256695	−	0.966492i	$-0.417366\pi$
$139$	6.05278	0.513390	0.256695	+	0.966492i	$0.417366\pi$
$140$	0	0
$141$	14.7088	1.23871
$142$	0	0
$143$	2.68614	0.224626
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0.944272	0.0778822
$148$	0	0
$149$	−13.7401	−1.12563	−0.562815	−	0.826583i	$-0.690282\pi$
$149$	−13.7401	−1.12563	−0.562815	+	0.826583i	$0.690282\pi$
$150$	0	0
$151$	−8.45089	−0.687724	−0.343862	−	0.939020i	$-0.611735\pi$
$151$	−8.45089	−0.687724	−0.343862	+	0.939020i	$0.611735\pi$
$152$	0	0
$153$	−6.76278	−0.546738
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−3.17338	−0.253263	−0.126632	−	0.991950i	$-0.540417\pi$
$157$	−3.17338	−0.253263	−0.126632	+	0.991950i	$0.540417\pi$
$158$	0	0
$159$	−12.1554	−0.963983
$160$	0	0
$161$	0.431842	0.0340339
$162$	0	0
$163$	11.0622	0.866457	0.433228	−	0.901284i	$-0.357374\pi$
$163$	11.0622	0.866457	0.433228	+	0.901284i	$0.357374\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1.63458	0.126487	0.0632437	−	0.997998i	$-0.479855\pi$
$167$	1.63458	0.126487	0.0632437	+	0.997998i	$0.479855\pi$
$168$	0	0
$169$	−12.7697	−0.982288
$170$	0	0
$171$	7.17191	0.548450
$172$	0	0
$173$	−16.9999	−1.29248	−0.646239	−	0.763135i	$-0.723659\pi$
$173$	−16.9999	−1.29248	−0.646239	+	0.763135i	$0.723659\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	16.5527	1.24418
$178$	0	0
$179$	−13.2100	−0.987363	−0.493681	−	0.869643i	$-0.664349\pi$
$179$	−13.2100	−0.987363	−0.493681	+	0.869643i	$0.664349\pi$
$180$	0	0
$181$	26.6464	1.98061	0.990307	−	0.138894i	$-0.0443549\pi$
$181$	26.6464	1.98061	0.990307	+	0.138894i	$0.0443549\pi$
$182$	0	0
$183$	16.0663	1.18766
$184$	0	0
$185$	0	0
$186$	0	0
$187$	21.8434	1.59735
$188$	0	0
$189$	−7.51494	−0.546631
$190$	0	0
$191$	−9.16850	−0.663409	−0.331705	−	0.943383i	$-0.607624\pi$
$191$	−9.16850	−0.663409	−0.331705	+	0.943383i	$0.607624\pi$
$192$	0	0
$193$	−6.00000	−0.431889	−0.215945	−	0.976406i	$-0.569283\pi$
$193$	−6.00000	−0.431889	−0.215945	+	0.976406i	$0.569283\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	13.1400	0.936190	0.468095	−	0.883678i	$-0.344941\pi$
$197$	13.1400	0.936190	0.468095	+	0.883678i	$0.344941\pi$
$198$	0	0
$199$	−7.35469	−0.521360	−0.260680	−	0.965425i	$-0.583947\pi$
$199$	−7.35469	−0.521360	−0.260680	+	0.965425i	$0.583947\pi$
$200$	0	0
$201$	−4.49704	−0.317197
$202$	0	0
$203$	−19.1667	−1.34524
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0.274497	0.0190789
$208$	0	0
$209$	−23.1648	−1.60235
$210$	0	0
$211$	20.2556	1.39446	0.697228	−	0.716849i	$-0.254416\pi$
$211$	20.2556	1.39446	0.697228	+	0.716849i	$0.254416\pi$
$212$	0	0
$213$	15.3581	1.05232
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−1.17744	−0.0799295
$218$	0	0
$219$	−29.6061	−2.00060
$220$	0	0
$221$	1.87244	0.125954
$222$	0	0
$223$	−16.3943	−1.09785	−0.548923	−	0.835873i	$-0.684962\pi$
$223$	−16.3943	−1.09785	−0.548923	+	0.835873i	$0.684962\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−7.73716	−0.513534	−0.256767	−	0.966473i	$-0.582657\pi$
$227$	−7.73716	−0.513534	−0.256767	+	0.966473i	$0.582657\pi$
$228$	0	0
$229$	−26.5674	−1.75562	−0.877812	−	0.479006i	$-0.840997\pi$
$229$	−26.5674	−1.75562	−0.877812	+	0.479006i	$0.840997\pi$
$230$	0	0
$231$	−33.2051	−2.18474
$232$	0	0
$233$	9.11507	0.597148	0.298574	−	0.954386i	$-0.403489\pi$
$233$	9.11507	0.597148	0.298574	+	0.954386i	$0.403489\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−13.7888	−0.895678
$238$	0	0
$239$	−19.9560	−1.29085	−0.645425	−	0.763824i	$-0.723320\pi$
$239$	−19.9560	−1.29085	−0.645425	+	0.763824i	$0.723320\pi$
$240$	0	0
$241$	−11.8460	−0.763067	−0.381534	−	0.924355i	$-0.624604\pi$
$241$	−11.8460	−0.763067	−0.381534	+	0.924355i	$0.624604\pi$
$242$	0	0
$243$	−16.0883	−1.03207
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−1.98572	−0.126348
$248$	0	0
$249$	−25.2382	−1.59941
$250$	0	0
$251$	−0.782668	−0.0494016	−0.0247008	−	0.999695i	$-0.507863\pi$
$251$	−0.782668	−0.0494016	−0.0247008	+	0.999695i	$0.507863\pi$
$252$	0	0
$253$	−0.886609	−0.0557406
$254$	0	0
$255$	0	0
$256$	0	0
$257$	9.43069	0.588270	0.294135	−	0.955764i	$-0.404968\pi$
$257$	9.43069	0.588270	0.294135	+	0.955764i	$0.404968\pi$
$258$	0	0
$259$	−7.24013	−0.449879
$260$	0	0
$261$	−12.1832	−0.754120
$262$	0	0
$263$	−27.0985	−1.67096	−0.835482	−	0.549518i	$-0.814812\pi$
$263$	−27.0985	−1.67096	−0.835482	+	0.549518i	$0.814812\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	3.71632	0.227435
$268$	0	0
$269$	3.51899	0.214557	0.107278	−	0.994229i	$-0.465786\pi$
$269$	3.51899	0.214557	0.107278	+	0.994229i	$0.465786\pi$
$270$	0	0
$271$	13.1186	0.796899	0.398450	−	0.917190i	$-0.369548\pi$
$271$	13.1186	0.796899	0.398450	+	0.917190i	$0.369548\pi$
$272$	0	0
$273$	−2.84638	−0.172271
$274$	0	0
$275$	0	0
$276$	0	0
$277$	3.23234	0.194212	0.0971062	−	0.995274i	$-0.469041\pi$
$277$	3.23234	0.194212	0.0971062	+	0.995274i	$0.469041\pi$
$278$	0	0
$279$	−0.748428	−0.0448072
$280$	0	0
$281$	5.50651	0.328491	0.164245	−	0.986420i	$-0.447481\pi$
$281$	5.50651	0.328491	0.164245	+	0.986420i	$0.447481\pi$
$282$	0	0
$283$	−6.23388	−0.370565	−0.185283	−	0.982685i	$-0.559320\pi$
$283$	−6.23388	−0.370565	−0.185283	+	0.982685i	$0.559320\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−23.1414	−1.36599
$288$	0	0
$289$	−1.77351	−0.104324
$290$	0	0
$291$	−25.8125	−1.51316
$292$	0	0
$293$	9.64990	0.563753	0.281877	−	0.959451i	$-0.409043\pi$
$293$	9.64990	0.563753	0.281877	+	0.959451i	$0.409043\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	15.4288	0.895271
$298$	0	0
$299$	−0.0760012	−0.00439527
$300$	0	0
$301$	9.53991	0.549871
$302$	0	0
$303$	−17.2331	−0.990013
$304$	0	0
$305$	0	0
$306$	0	0
$307$	7.69507	0.439181	0.219591	−	0.975592i	$-0.429528\pi$
$307$	7.69507	0.439181	0.219591	+	0.975592i	$0.429528\pi$
$308$	0	0
$309$	−13.6506	−0.776556
$310$	0	0
$311$	0.422260	0.0239442	0.0119721	−	0.999928i	$-0.496189\pi$
$311$	0.422260	0.0239442	0.0119721	+	0.999928i	$0.496189\pi$
$312$	0	0
$313$	23.0978	1.30556	0.652782	−	0.757546i	$-0.273602\pi$
$313$	23.0978	1.30556	0.652782	+	0.757546i	$0.273602\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	15.3735	0.863464	0.431732	−	0.902002i	$-0.357903\pi$
$317$	15.3735	0.863464	0.431732	+	0.902002i	$0.357903\pi$
$318$	0	0
$319$	39.3509	2.20323
$320$	0	0
$321$	−23.8576	−1.33160
$322$	0	0
$323$	−16.1477	−0.898479
$324$	0	0
$325$	0	0
$326$	0	0
$327$	17.0018	0.940199
$328$	0	0
$329$	18.4339	1.01629
$330$	0	0
$331$	32.0348	1.76079	0.880397	−	0.474237i	$-0.157276\pi$
$331$	32.0348	1.76079	0.880397	+	0.474237i	$0.157276\pi$
$332$	0	0
$333$	−4.60213	−0.252195
$334$	0	0
$335$	0	0
$336$	0	0
$337$	11.6870	0.636629	0.318315	−	0.947985i	$-0.396883\pi$
$337$	11.6870	0.636629	0.318315	+	0.947985i	$0.396883\pi$
$338$	0	0
$339$	−3.00517	−0.163218
$340$	0	0
$341$	2.41738	0.130908
$342$	0	0
$343$	−17.9024	−0.966638
$344$	0	0
$345$	0	0
$346$	0	0
$347$	20.7991	1.11655	0.558277	−	0.829655i	$-0.311463\pi$
$347$	20.7991	1.11655	0.558277	+	0.829655i	$0.311463\pi$
$348$	0	0
$349$	26.1233	1.39835	0.699173	−	0.714952i	$-0.253552\pi$
$349$	26.1233	1.39835	0.699173	+	0.714952i	$0.253552\pi$
$350$	0	0
$351$	1.32258	0.0705940
$352$	0	0
$353$	−24.3378	−1.29537	−0.647685	−	0.761908i	$-0.724263\pi$
$353$	−24.3378	−1.29537	−0.647685	+	0.761908i	$0.724263\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−23.1465	−1.22504
$358$	0	0
$359$	−32.5087	−1.71574	−0.857871	−	0.513866i	$-0.828213\pi$
$359$	−32.5087	−1.71574	−0.857871	+	0.513866i	$0.828213\pi$
$360$	0	0
$361$	−1.87546	−0.0987083
$362$	0	0
$363$	44.2418	2.32209
$364$	0	0
$365$	0	0
$366$	0	0
$367$	12.0586	0.629455	0.314728	−	0.949182i	$-0.398087\pi$
$367$	12.0586	0.629455	0.314728	+	0.949182i	$0.398087\pi$
$368$	0	0
$369$	−14.7097	−0.765755
$370$	0	0
$371$	−15.2338	−0.790898
$372$	0	0
$373$	30.0597	1.55643	0.778217	−	0.627995i	$-0.216124\pi$
$373$	30.0597	1.55643	0.778217	+	0.627995i	$0.216124\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	3.37321	0.173729
$378$	0	0
$379$	25.6130	1.31565	0.657827	−	0.753169i	$-0.271476\pi$
$379$	25.6130	1.31565	0.657827	+	0.753169i	$0.271476\pi$
$380$	0	0
$381$	−0.454483	−0.0232839
$382$	0	0
$383$	−2.83556	−0.144890	−0.0724452	−	0.997372i	$-0.523080\pi$
$383$	−2.83556	−0.144890	−0.0724452	+	0.997372i	$0.523080\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	6.06397	0.308249
$388$	0	0
$389$	−36.2437	−1.83763	−0.918815	−	0.394688i	$-0.870853\pi$
$389$	−36.2437	−1.83763	−0.918815	+	0.394688i	$0.870853\pi$
$390$	0	0
$391$	−0.618034	−0.0312553
$392$	0	0
$393$	12.8962	0.650527
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−33.6851	−1.69060	−0.845302	−	0.534289i	$-0.820580\pi$
$397$	−33.6851	−1.69060	−0.845302	+	0.534289i	$0.820580\pi$
$398$	0	0
$399$	24.5468	1.22888
$400$	0	0
$401$	−21.2631	−1.06183	−0.530915	−	0.847425i	$-0.678152\pi$
$401$	−21.2631	−1.06183	−0.530915	+	0.847425i	$0.678152\pi$
$402$	0	0
$403$	0.207221	0.0103224
$404$	0	0
$405$	0	0
$406$	0	0
$407$	14.8646	0.736811
$408$	0	0
$409$	−26.0403	−1.28761	−0.643805	−	0.765190i	$-0.722645\pi$
$409$	−26.0403	−1.28761	−0.643805	+	0.765190i	$0.722645\pi$
$410$	0	0
$411$	31.5574	1.55661
$412$	0	0
$413$	20.7448	1.02078
$414$	0	0
$415$	0	0
$416$	0	0
$417$	13.1683	0.644852
$418$	0	0
$419$	−2.87681	−0.140542	−0.0702708	−	0.997528i	$-0.522386\pi$
$419$	−2.87681	−0.140542	−0.0702708	+	0.997528i	$0.522386\pi$
$420$	0	0
$421$	−15.4417	−0.752583	−0.376291	−	0.926501i	$-0.622801\pi$
$421$	−15.4417	−0.752583	−0.376291	+	0.926501i	$0.622801\pi$
$422$	0	0
$423$	11.7174	0.569719
$424$	0	0
$425$	0	0
$426$	0	0
$427$	20.1352	0.974409
$428$	0	0
$429$	5.84388	0.282145
$430$	0	0
$431$	15.9360	0.767612	0.383806	−	0.923414i	$-0.374613\pi$
$431$	15.9360	0.767612	0.383806	+	0.923414i	$0.374613\pi$
$432$	0	0
$433$	0.198566	0.00954246	0.00477123	−	0.999989i	$-0.498481\pi$
$433$	0.198566	0.00954246	0.00477123	+	0.999989i	$0.498481\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0.655423	0.0313532
$438$	0	0
$439$	−10.1400	−0.483958	−0.241979	−	0.970281i	$-0.577797\pi$
$439$	−10.1400	−0.483958	−0.241979	+	0.970281i	$0.577797\pi$
$440$	0	0
$441$	0.752228	0.0358204
$442$	0	0
$443$	30.5446	1.45122	0.725609	−	0.688107i	$-0.241558\pi$
$443$	30.5446	1.45122	0.725609	+	0.688107i	$0.241558\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−29.8925	−1.41387
$448$	0	0
$449$	20.0117	0.944412	0.472206	−	0.881488i	$-0.343458\pi$
$449$	20.0117	0.944412	0.472206	+	0.881488i	$0.343458\pi$
$450$	0	0
$451$	47.5113	2.23722
$452$	0	0
$453$	−18.3855	−0.863827
$454$	0	0
$455$	0	0
$456$	0	0
$457$	7.86472	0.367896	0.183948	−	0.982936i	$-0.441112\pi$
$457$	7.86472	0.367896	0.183948	+	0.982936i	$0.441112\pi$
$458$	0	0
$459$	10.7551	0.502003
$460$	0	0
$461$	−22.9868	−1.07060	−0.535300	−	0.844662i	$-0.679801\pi$
$461$	−22.9868	−1.07060	−0.535300	+	0.844662i	$0.679801\pi$
$462$	0	0
$463$	28.4306	1.32128	0.660640	−	0.750702i	$-0.270285\pi$
$463$	28.4306	1.32128	0.660640	+	0.750702i	$0.270285\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	1.04428	0.0483236	0.0241618	−	0.999708i	$-0.492308\pi$
$467$	1.04428	0.0483236	0.0241618	+	0.999708i	$0.492308\pi$
$468$	0	0
$469$	−5.63593	−0.260243
$470$	0	0
$471$	−6.90391	−0.318115
$472$	0	0
$473$	−19.5863	−0.900578
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−9.68323	−0.443365
$478$	0	0
$479$	0.368837	0.0168526	0.00842630	−	0.999964i	$-0.497318\pi$
$479$	0.368837	0.0168526	0.00842630	+	0.999964i	$0.497318\pi$
$480$	0	0
$481$	1.27421	0.0580991
$482$	0	0
$483$	0.939503	0.0427488
$484$	0	0
$485$	0	0
$486$	0	0
$487$	40.8702	1.85201	0.926003	−	0.377516i	$-0.123222\pi$
$487$	40.8702	1.85201	0.926003	+	0.377516i	$0.123222\pi$
$488$	0	0
$489$	24.0666	1.08833
$490$	0	0
$491$	39.1992	1.76903	0.884517	−	0.466508i	$-0.154488\pi$
$491$	39.1992	1.76903	0.884517	+	0.466508i	$0.154488\pi$
$492$	0	0
$493$	27.4306	1.23541
$494$	0	0
$495$	0	0
$496$	0	0
$497$	19.2476	0.863373
$498$	0	0
$499$	−42.9767	−1.92390	−0.961950	−	0.273226i	$-0.911909\pi$
$499$	−42.9767	−1.92390	−0.961950	+	0.273226i	$0.911909\pi$
$500$	0	0
$501$	3.55614	0.158877
$502$	0	0
$503$	−29.2420	−1.30384	−0.651918	−	0.758289i	$-0.726036\pi$
$503$	−29.2420	−1.30384	−0.651918	+	0.758289i	$0.726036\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−27.7815	−1.23382
$508$	0	0
$509$	43.2330	1.91627	0.958135	−	0.286318i	$-0.0924313\pi$
$509$	43.2330	1.91627	0.958135	+	0.286318i	$0.0924313\pi$
$510$	0	0
$511$	−37.1040	−1.64139
$512$	0	0
$513$	−11.4057	−0.503575
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−37.8465	−1.66449
$518$	0	0
$519$	−36.9845	−1.62344
$520$	0	0
$521$	−8.33948	−0.365359	−0.182680	−	0.983172i	$-0.558477\pi$
$521$	−8.33948	−0.365359	−0.182680	+	0.983172i	$0.558477\pi$
$522$	0	0
$523$	38.0652	1.66447	0.832236	−	0.554421i	$-0.187060\pi$
$523$	38.0652	1.66447	0.832236	+	0.554421i	$0.187060\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.68510	0.0734039
$528$	0	0
$529$	−22.9749	−0.998909
$530$	0	0
$531$	13.1863	0.572235
$532$	0	0
$533$	4.07273	0.176410
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−28.7393	−1.24019
$538$	0	0
$539$	−2.42965	−0.104652
$540$	0	0
$541$	−20.1518	−0.866395	−0.433198	−	0.901299i	$-0.642615\pi$
$541$	−20.1518	−0.866395	−0.433198	+	0.901299i	$0.642615\pi$
$542$	0	0
$543$	57.9712	2.48778
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−40.4648	−1.73015	−0.865076	−	0.501641i	$-0.832730\pi$
$547$	−40.4648	−1.73015	−0.865076	+	0.501641i	$0.832730\pi$
$548$	0	0
$549$	12.7988	0.546238
$550$	0	0
$551$	−29.0901	−1.23928
$552$	0	0
$553$	−17.2809	−0.734857
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−21.4012	−0.906799	−0.453400	−	0.891307i	$-0.649789\pi$
$557$	−21.4012	−0.906799	−0.453400	+	0.891307i	$0.649789\pi$
$558$	0	0
$559$	−1.67896	−0.0710124
$560$	0	0
$561$	47.5218	2.00637
$562$	0	0
$563$	−7.60200	−0.320386	−0.160193	−	0.987086i	$-0.551212\pi$
$563$	−7.60200	−0.320386	−0.160193	+	0.987086i	$0.551212\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−30.5254	−1.28195
$568$	0	0
$569$	15.8422	0.664140	0.332070	−	0.943255i	$-0.392253\pi$
$569$	15.8422	0.664140	0.332070	+	0.943255i	$0.392253\pi$
$570$	0	0
$571$	−23.3620	−0.977668	−0.488834	−	0.872377i	$-0.662578\pi$
$571$	−23.3620	−0.977668	−0.488834	+	0.872377i	$0.662578\pi$
$572$	0	0
$573$	−19.9467	−0.833286
$574$	0	0
$575$	0	0
$576$	0	0
$577$	8.06573	0.335781	0.167890	−	0.985806i	$-0.446305\pi$
$577$	8.06573	0.335781	0.167890	+	0.985806i	$0.446305\pi$
$578$	0	0
$579$	−13.0534	−0.542482
$580$	0	0
$581$	−31.6299	−1.31223
$582$	0	0
$583$	31.2762	1.29533
$584$	0	0
$585$	0	0
$586$	0	0
$587$	36.4139	1.50296	0.751480	−	0.659756i	$-0.229340\pi$
$587$	36.4139	1.50296	0.751480	+	0.659756i	$0.229340\pi$
$588$	0	0
$589$	−1.78704	−0.0736337
$590$	0	0
$591$	28.5871	1.17592
$592$	0	0
$593$	−15.2692	−0.627029	−0.313515	−	0.949583i	$-0.601506\pi$
$593$	−15.2692	−0.627029	−0.313515	+	0.949583i	$0.601506\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−16.0006	−0.654863
$598$	0	0
$599$	8.19555	0.334861	0.167431	−	0.985884i	$-0.446453\pi$
$599$	8.19555	0.334861	0.167431	+	0.985884i	$0.446453\pi$
$600$	0	0
$601$	42.6346	1.73910	0.869551	−	0.493844i	$-0.164408\pi$
$601$	42.6346	1.73910	0.869551	+	0.493844i	$0.164408\pi$
$602$	0	0
$603$	−3.58244	−0.145888
$604$	0	0
$605$	0	0
$606$	0	0
$607$	30.2136	1.22633	0.613165	−	0.789955i	$-0.289896\pi$
$607$	30.2136	1.22633	0.613165	+	0.789955i	$0.289896\pi$
$608$	0	0
$609$	−41.6985	−1.68971
$610$	0	0
$611$	−3.24425	−0.131248
$612$	0	0
$613$	−7.48872	−0.302467	−0.151233	−	0.988498i	$-0.548324\pi$
$613$	−7.48872	−0.302467	−0.151233	+	0.988498i	$0.548324\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	39.6524	1.59634	0.798172	−	0.602430i	$-0.205801\pi$
$617$	39.6524	1.59634	0.798172	+	0.602430i	$0.205801\pi$
$618$	0	0
$619$	9.10072	0.365789	0.182894	−	0.983133i	$-0.441453\pi$
$619$	9.10072	0.365789	0.182894	+	0.983133i	$0.441453\pi$
$620$	0	0
$621$	−0.436542	−0.0175178
$622$	0	0
$623$	4.65749	0.186598
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−50.3967	−2.01265
$628$	0	0
$629$	10.3618	0.413151
$630$	0	0
$631$	−14.2434	−0.567021	−0.283511	−	0.958969i	$-0.591499\pi$
$631$	−14.2434	−0.567021	−0.283511	+	0.958969i	$0.591499\pi$
$632$	0	0
$633$	44.0676	1.75153
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−0.208273	−0.00825206
$638$	0	0
$639$	12.2346	0.483993
$640$	0	0
$641$	9.28357	0.366679	0.183339	−	0.983050i	$-0.441309\pi$
$641$	9.28357	0.366679	0.183339	+	0.983050i	$0.441309\pi$
$642$	0	0
$643$	−1.98146	−0.0781411	−0.0390706	−	0.999236i	$-0.512440\pi$
$643$	−1.98146	−0.0781411	−0.0390706	+	0.999236i	$0.512440\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−3.07187	−0.120768	−0.0603839	−	0.998175i	$-0.519232\pi$
$647$	−3.07187	−0.120768	−0.0603839	+	0.998175i	$0.519232\pi$
$648$	0	0
$649$	−42.5908	−1.67184
$650$	0	0
$651$	−2.56159	−0.100397
$652$	0	0
$653$	1.09050	0.0426745	0.0213372	−	0.999772i	$-0.493208\pi$
$653$	1.09050	0.0426745	0.0213372	+	0.999772i	$0.493208\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−23.5849	−0.920135
$658$	0	0
$659$	1.72924	0.0673618	0.0336809	−	0.999433i	$-0.489277\pi$
$659$	1.72924	0.0673618	0.0336809	+	0.999433i	$0.489277\pi$
$660$	0	0
$661$	23.9209	0.930417	0.465208	−	0.885201i	$-0.345979\pi$
$661$	23.9209	0.930417	0.465208	+	0.885201i	$0.345979\pi$
$662$	0	0
$663$	4.07363	0.158207
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−1.11339	−0.0431107
$668$	0	0
$669$	−35.6670	−1.37897
$670$	0	0
$671$	−41.3393	−1.59588
$672$	0	0
$673$	28.8112	1.11059	0.555295	−	0.831653i	$-0.312605\pi$
$673$	28.8112	1.11059	0.555295	+	0.831653i	$0.312605\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	43.3441	1.66585	0.832924	−	0.553387i	$-0.186665\pi$
$677$	43.3441	1.66585	0.832924	+	0.553387i	$0.186665\pi$
$678$	0	0
$679$	−32.3497	−1.24147
$680$	0	0
$681$	−16.8327	−0.645032
$682$	0	0
$683$	25.4622	0.974285	0.487143	−	0.873322i	$-0.338039\pi$
$683$	25.4622	0.974285	0.487143	+	0.873322i	$0.338039\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−57.7993	−2.20518
$688$	0	0
$689$	2.68104	0.102139
$690$	0	0
$691$	4.95084	0.188339	0.0941693	−	0.995556i	$-0.469980\pi$
$691$	4.95084	0.188339	0.0941693	+	0.995556i	$0.469980\pi$
$692$	0	0
$693$	−26.4519	−1.00483
$694$	0	0
$695$	0	0
$696$	0	0
$697$	33.1190	1.25447
$698$	0	0
$699$	19.8305	0.750058
$700$	0	0
$701$	−17.5718	−0.663677	−0.331838	−	0.943336i	$-0.607669\pi$
$701$	−17.5718	−0.663677	−0.331838	+	0.943336i	$0.607669\pi$
$702$	0	0
$703$	−10.9886	−0.414444
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−21.5974	−0.812253
$708$	0	0
$709$	28.1943	1.05886	0.529430	−	0.848354i	$-0.322406\pi$
$709$	28.1943	1.05886	0.529430	+	0.848354i	$0.322406\pi$
$710$	0	0
$711$	−10.9845	−0.411949
$712$	0	0
$713$	−0.0683970	−0.00256149
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−43.4158	−1.62139
$718$	0	0
$719$	−18.1328	−0.676238	−0.338119	−	0.941103i	$-0.609791\pi$
$719$	−18.1328	−0.676238	−0.338119	+	0.941103i	$0.609791\pi$
$720$	0	0
$721$	−17.1077	−0.637123
$722$	0	0
$723$	−25.7718	−0.958463
$724$	0	0
$725$	0	0
$726$	0	0
$727$	45.6984	1.69486	0.847431	−	0.530906i	$-0.178148\pi$
$727$	45.6984	1.69486	0.847431	+	0.530906i	$0.178148\pi$
$728$	0	0
$729$	−1.41426	−0.0523800
$730$	0	0
$731$	−13.6531	−0.504979
$732$	0	0
$733$	−13.4008	−0.494971	−0.247486	−	0.968892i	$-0.579604\pi$
$733$	−13.4008	−0.494971	−0.247486	+	0.968892i	$0.579604\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	11.5711	0.426226
$738$	0	0
$739$	−44.9630	−1.65399	−0.826995	−	0.562209i	$-0.809952\pi$
$739$	−44.9630	−1.65399	−0.826995	+	0.562209i	$0.809952\pi$
$740$	0	0
$741$	−4.32007	−0.158702
$742$	0	0
$743$	−19.5304	−0.716502	−0.358251	−	0.933625i	$-0.616627\pi$
$743$	−19.5304	−0.716502	−0.358251	+	0.933625i	$0.616627\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−20.1053	−0.735615
$748$	0	0
$749$	−29.8997	−1.09251
$750$	0	0
$751$	19.8988	0.726117	0.363058	−	0.931766i	$-0.381733\pi$
$751$	19.8988	0.726117	0.363058	+	0.931766i	$0.381733\pi$
$752$	0	0
$753$	−1.70275	−0.0620516
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−13.8807	−0.504502	−0.252251	−	0.967662i	$-0.581171\pi$
$757$	−13.8807	−0.504502	−0.252251	+	0.967662i	$0.581171\pi$
$758$	0	0
$759$	−1.92888	−0.0700139
$760$	0	0
$761$	−12.9113	−0.468033	−0.234016	−	0.972233i	$-0.575187\pi$
$761$	−12.9113	−0.468033	−0.234016	+	0.972233i	$0.575187\pi$
$762$	0	0
$763$	21.3075	0.771384
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−3.65094	−0.131828
$768$	0	0
$769$	4.55295	0.164184	0.0820918	−	0.996625i	$-0.473840\pi$
$769$	4.55295	0.164184	0.0820918	+	0.996625i	$0.473840\pi$
$770$	0	0
$771$	20.5171	0.738906
$772$	0	0
$773$	−42.3671	−1.52384	−0.761920	−	0.647671i	$-0.775743\pi$
$773$	−42.3671	−1.52384	−0.761920	+	0.647671i	$0.775743\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−15.7514	−0.565078
$778$	0	0
$779$	−35.1226	−1.25840
$780$	0	0
$781$	−39.5170	−1.41403
$782$	0	0
$783$	19.3753	0.692416
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−4.55862	−0.162497	−0.0812486	−	0.996694i	$-0.525891\pi$
$787$	−4.55862	−0.162497	−0.0812486	+	0.996694i	$0.525891\pi$
$788$	0	0
$789$	−58.9547	−2.09884
$790$	0	0
$791$	−3.76624	−0.133912
$792$	0	0
$793$	−3.54365	−0.125839
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−0.431709	−0.0152919	−0.00764596	−	0.999971i	$-0.502434\pi$
$797$	−0.431709	−0.0152919	−0.00764596	+	0.999971i	$0.502434\pi$
$798$	0	0
$799$	−26.3819	−0.933323
$800$	0	0
$801$	2.96050	0.104604
$802$	0	0
$803$	76.1778	2.68826
$804$	0	0
$805$	0	0
$806$	0	0
$807$	7.65582	0.269498
$808$	0	0
$809$	14.5872	0.512859	0.256429	−	0.966563i	$-0.417454\pi$
$809$	14.5872	0.512859	0.256429	+	0.966563i	$0.417454\pi$
$810$	0	0
$811$	−12.8283	−0.450462	−0.225231	−	0.974305i	$-0.572314\pi$
$811$	−12.8283	−0.450462	−0.225231	+	0.974305i	$0.572314\pi$
$812$	0	0
$813$	28.5405	1.00096
$814$	0	0
$815$	0	0
$816$	0	0
$817$	14.4791	0.506560
$818$	0	0
$819$	−2.26749	−0.0792326
$820$	0	0
$821$	−38.9754	−1.36025	−0.680125	−	0.733096i	$-0.738075\pi$
$821$	−38.9754	−1.36025	−0.680125	+	0.733096i	$0.738075\pi$
$822$	0	0
$823$	−27.1304	−0.945707	−0.472854	−	0.881141i	$-0.656776\pi$
$823$	−27.1304	−0.945707	−0.472854	+	0.881141i	$0.656776\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−35.2817	−1.22686	−0.613432	−	0.789748i	$-0.710212\pi$
$827$	−35.2817	−1.22686	−0.613432	+	0.789748i	$0.710212\pi$
$828$	0	0
$829$	−54.7593	−1.90187	−0.950935	−	0.309391i	$-0.899875\pi$
$829$	−54.7593	−1.90187	−0.950935	+	0.309391i	$0.899875\pi$
$830$	0	0
$831$	7.03218	0.243944
$832$	0	0
$833$	−1.69365	−0.0586815
$834$	0	0
$835$	0	0
$836$	0	0
$837$	1.19025	0.0411410
$838$	0	0
$839$	−36.5395	−1.26148	−0.630742	−	0.775993i	$-0.717249\pi$
$839$	−36.5395	−1.26148	−0.630742	+	0.775993i	$0.717249\pi$
$840$	0	0
$841$	20.4163	0.704010
$842$	0	0
$843$	11.9798	0.412606
$844$	0	0
$845$	0	0
$846$	0	0
$847$	55.4462	1.90515
$848$	0	0
$849$	−13.5622	−0.465455
$850$	0	0
$851$	−0.420578	−0.0144172
$852$	0	0
$853$	25.0484	0.857641	0.428820	−	0.903390i	$-0.358929\pi$
$853$	25.0484	0.857641	0.428820	+	0.903390i	$0.358929\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	26.2947	0.898211	0.449105	−	0.893479i	$-0.351743\pi$
$857$	26.2947	0.898211	0.449105	+	0.893479i	$0.351743\pi$
$858$	0	0
$859$	7.85601	0.268044	0.134022	−	0.990978i	$-0.457211\pi$
$859$	7.85601	0.268044	0.134022	+	0.990978i	$0.457211\pi$
$860$	0	0
$861$	−50.3458	−1.71578
$862$	0	0
$863$	52.7226	1.79470	0.897350	−	0.441320i	$-0.145490\pi$
$863$	52.7226	1.79470	0.897350	+	0.441320i	$0.145490\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−3.85840	−0.131038
$868$	0	0
$869$	35.4791	1.20355
$870$	0	0
$871$	0.991886	0.0336088
$872$	0	0
$873$	−20.5628	−0.695947
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−16.9295	−0.571668	−0.285834	−	0.958279i	$-0.592271\pi$
$877$	−16.9295	−0.571668	−0.285834	+	0.958279i	$0.592271\pi$
$878$	0	0
$879$	20.9940	0.708111
$880$	0	0
$881$	26.2491	0.884354	0.442177	−	0.896928i	$-0.354206\pi$
$881$	26.2491	0.884354	0.442177	+	0.896928i	$0.354206\pi$
$882$	0	0
$883$	2.10581	0.0708660	0.0354330	−	0.999372i	$-0.488719\pi$
$883$	2.10581	0.0708660	0.0354330	+	0.999372i	$0.488719\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−46.3196	−1.55526	−0.777629	−	0.628723i	$-0.783578\pi$
$887$	−46.3196	−1.55526	−0.777629	+	0.628723i	$0.783578\pi$
$888$	0	0
$889$	−0.569583	−0.0191032
$890$	0	0
$891$	62.6714	2.09957
$892$	0	0
$893$	27.9779	0.936244
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−0.165346	−0.00552074
$898$	0	0
$899$	3.03571	0.101247
$900$	0	0
$901$	21.8019	0.726327
$902$	0	0
$903$	20.7547	0.690675
$904$	0	0
$905$	0	0
$906$	0	0
$907$	19.1755	0.636711	0.318355	−	0.947971i	$-0.396869\pi$
$907$	19.1755	0.636711	0.318355	+	0.947971i	$0.396869\pi$
$908$	0	0
$909$	−13.7282	−0.455336
$910$	0	0
$911$	−20.6395	−0.683816	−0.341908	−	0.939733i	$-0.611073\pi$
$911$	−20.6395	−0.683816	−0.341908	+	0.939733i	$0.611073\pi$
$912$	0	0
$913$	64.9389	2.14917
$914$	0	0
$915$	0	0
$916$	0	0
$917$	16.1622	0.533723
$918$	0	0
$919$	−6.25939	−0.206478	−0.103239	−	0.994657i	$-0.532921\pi$
$919$	−6.25939	−0.206478	−0.103239	+	0.994657i	$0.532921\pi$
$920$	0	0
$921$	16.7412	0.551641
$922$	0	0
$923$	−3.38745	−0.111499
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−10.8744	−0.357161
$928$	0	0
$929$	26.5755	0.871914	0.435957	−	0.899967i	$-0.356410\pi$
$929$	26.5755	0.871914	0.435957	+	0.899967i	$0.356410\pi$
$930$	0	0
$931$	1.79611	0.0588652
$932$	0	0
$933$	0.918657	0.0300755
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−8.07155	−0.263686	−0.131843	−	0.991271i	$-0.542089\pi$
$937$	−8.07155	−0.263686	−0.131843	+	0.991271i	$0.542089\pi$
$938$	0	0
$939$	50.2508	1.63987
$940$	0	0
$941$	−11.4342	−0.372746	−0.186373	−	0.982479i	$-0.559673\pi$
$941$	−11.4342	−0.372746	−0.186373	+	0.982479i	$0.559673\pi$
$942$	0	0
$943$	−1.34428	−0.0437758
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−3.59030	−0.116669	−0.0583345	−	0.998297i	$-0.518579\pi$
$947$	−3.59030	−0.116669	−0.0583345	+	0.998297i	$0.518579\pi$
$948$	0	0
$949$	6.53006	0.211975
$950$	0	0
$951$	33.4462	1.08457
$952$	0	0
$953$	39.6616	1.28477	0.642383	−	0.766384i	$-0.277946\pi$
$953$	39.6616	1.28477	0.642383	+	0.766384i	$0.277946\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	85.6107	2.76740
$958$	0	0
$959$	39.5495	1.27712
$960$	0	0
$961$	−30.8135	−0.993984
$962$	0	0
$963$	−19.0055	−0.612444
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−8.12436	−0.261262	−0.130631	−	0.991431i	$-0.541700\pi$
$967$	−8.12436	−0.261262	−0.130631	+	0.991431i	$0.541700\pi$
$968$	0	0
$969$	−35.1304	−1.12855
$970$	0	0
$971$	27.8883	0.894979	0.447489	−	0.894289i	$-0.352318\pi$
$971$	27.8883	0.894979	0.447489	+	0.894289i	$0.352318\pi$
$972$	0	0
$973$	16.5032	0.529067
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−42.8561	−1.37109	−0.685543	−	0.728032i	$-0.740435\pi$
$977$	−42.8561	−1.37109	−0.685543	+	0.728032i	$0.740435\pi$
$978$	0	0
$979$	−9.56224	−0.305610
$980$	0	0
$981$	13.5440	0.432426
$982$	0	0
$983$	−32.8773	−1.04862	−0.524311	−	0.851527i	$-0.675677\pi$
$983$	−32.8773	−1.04862	−0.524311	+	0.851527i	$0.675677\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	40.1043	1.27653
$988$	0	0
$989$	0.554172	0.0176216
$990$	0	0
$991$	−22.0131	−0.699269	−0.349634	−	0.936886i	$-0.613694\pi$
$991$	−22.0131	−0.699269	−0.349634	+	0.936886i	$0.613694\pi$
$992$	0	0
$993$	69.6941	2.21167
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−31.6453	−1.00222	−0.501108	−	0.865385i	$-0.667074\pi$
$997$	−31.6453	−1.00222	−0.501108	+	0.865385i	$0.667074\pi$
$998$	0	0
$999$	7.31892	0.231560

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	10000.2.a.bb.1.3		4
4.3	odd	2		1250.2.a.i.1.2		4
5.4	even	2		10000.2.a.o.1.2		4
20.3	even	4		1250.2.b.c.1249.2		8
20.7	even	4		1250.2.b.c.1249.7		8
20.19	odd	2		1250.2.a.h.1.3		4
25.3	odd	20		400.2.y.a.209.1		8
25.17	odd	20		400.2.y.a.289.1		8
100.3	even	20		50.2.e.a.9.1	✓	8
100.19	odd	10		250.2.d.c.51.2		8
100.31	odd	10		250.2.d.b.51.1		8
100.47	even	20		250.2.e.a.49.2		8
100.67	even	20		50.2.e.a.39.1	yes	8
100.71	odd	10		250.2.d.b.201.1		8
100.79	odd	10		250.2.d.c.201.2		8
100.83	even	20		250.2.e.a.199.2		8
300.167	odd	20		450.2.l.b.289.2		8
300.203	odd	20		450.2.l.b.109.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
50.2.e.a.9.1	✓	8	100.3	even	20
50.2.e.a.39.1	yes	8	100.67	even	20
250.2.d.b.51.1		8	100.31	odd	10
250.2.d.b.201.1		8	100.71	odd	10
250.2.d.c.51.2		8	100.19	odd	10
250.2.d.c.201.2		8	100.79	odd	10
250.2.e.a.49.2		8	100.47	even	20
250.2.e.a.199.2		8	100.83	even	20
400.2.y.a.209.1		8	25.3	odd	20
400.2.y.a.289.1		8	25.17	odd	20
450.2.l.b.109.2		8	300.203	odd	20
450.2.l.b.289.2		8	300.167	odd	20
1250.2.a.h.1.3		4	20.19	odd	2
1250.2.a.i.1.2		4	4.3	odd	2
1250.2.b.c.1249.2		8	20.3	even	4
1250.2.b.c.1249.7		8	20.7	even	4
10000.2.a.o.1.2		4	5.4	even	2
10000.2.a.bb.1.3		4	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:	\( N \)	\(=\)	\( 10000 = 2^{4} \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	10000.a (trivial)

\(f(q)\)	\(=\)	\(q+2.17557 q^{3} +2.72654 q^{7} +1.73311 q^{9} +O(q^{10})\)	\(q+2.17557 q^{3} +2.72654 q^{7} +1.73311 q^{9} -5.59783 q^{11} -0.479853 q^{13} -3.90211 q^{17} +4.13818 q^{19} +5.93179 q^{21} +0.158384 q^{23} -2.75621 q^{27} -7.02967 q^{29} -0.431842 q^{31} -12.1785 q^{33} -2.65542 q^{37} -1.04395 q^{39} -8.48746 q^{41} +3.49890 q^{43} +6.76091 q^{47} +0.434034 q^{49} -8.48932 q^{51} -5.58721 q^{53} +9.00290 q^{57} +7.60845 q^{59} +7.38487 q^{61} +4.72539 q^{63} -2.06706 q^{67} +0.344577 q^{69} +7.05934 q^{71} -13.6085 q^{73} -15.2627 q^{77} -6.33801 q^{79} -11.1957 q^{81} -11.6007 q^{83} -15.2935 q^{87} +1.70820 q^{89} -1.30834 q^{91} -0.939503 q^{93} -11.8647 q^{97} -9.70164 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 8 q^{7} + 2 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 + 8 * q^7 + 2 * q^9	\( 4 q + 4 q^{3} + 8 q^{7} + 2 q^{9} + 2 q^{11} - 14 q^{13} - 8 q^{17} - 2 q^{21} + 4 q^{23} + 10 q^{27} - 10 q^{29} - 8 q^{31} - 8 q^{33} - 18 q^{37} - 14 q^{39} - 12 q^{41} + 14 q^{43} + 8 q^{47} + 8 q^{49} - 8 q^{51} - 4 q^{53} - 2 q^{61} - 16 q^{63} - 2 q^{67} - 6 q^{69} - 8 q^{71} - 24 q^{73} - 6 q^{77} - 20 q^{79} + 4 q^{81} + 14 q^{83} - 20 q^{87} - 20 q^{89} - 18 q^{91} - 8 q^{93} - 28 q^{97} - 14 q^{99}+O(q^{100}) \) 4 * q + 4 * q^3 + 8 * q^7 + 2 * q^9 + 2 * q^11 - 14 * q^13 - 8 * q^17 - 2 * q^21 + 4 * q^23 + 10 * q^27 - 10 * q^29 - 8 * q^31 - 8 * q^33 - 18 * q^37 - 14 * q^39 - 12 * q^41 + 14 * q^43 + 8 * q^47 + 8 * q^49 - 8 * q^51 - 4 * q^53 - 2 * q^61 - 16 * q^63 - 2 * q^67 - 6 * q^69 - 8 * q^71 - 24 * q^73 - 6 * q^77 - 20 * q^79 + 4 * q^81 + 14 * q^83 - 20 * q^87 - 20 * q^89 - 18 * q^91 - 8 * q^93 - 28 * q^97 - 14 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more