LMFDB - Embedded newform 10000.2.a.l.1.1

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:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 25)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.618034$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} -1.61803 q^{7} -2.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.61803 q^{7} -2.00000 q^{9} +0.763932 q^{11} +4.85410 q^{13} +0.763932 q^{17} +5.85410 q^{19} -1.61803 q^{21} +8.23607 q^{23} -5.00000 q^{27} -1.38197 q^{29} +3.00000 q^{31} +0.763932 q^{33} -4.23607 q^{37} +4.85410 q^{39} -5.23607 q^{41} +1.85410 q^{43} -1.61803 q^{47} -4.38197 q^{49} +0.763932 q^{51} -5.47214 q^{53} +5.85410 q^{57} +4.14590 q^{59} -4.70820 q^{61} +3.23607 q^{63} +9.23607 q^{67} +8.23607 q^{69} +4.38197 q^{71} +9.00000 q^{73} -1.23607 q^{77} -3.09017 q^{79} +1.00000 q^{81} -1.76393 q^{83} -1.38197 q^{87} +8.94427 q^{89} -7.85410 q^{91} +3.00000 q^{93} -2.85410 q^{97} -1.52786 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - q^{7} - 4 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 - q^7 - 4 * q^9	$ 2 q + 2 q^{3} - q^{7} - 4 q^{9} + 6 q^{11} + 3 q^{13} + 6 q^{17} + 5 q^{19} - q^{21} + 12 q^{23} - 10 q^{27} - 5 q^{29} + 6 q^{31} + 6 q^{33} - 4 q^{37} + 3 q^{39} - 6 q^{41} - 3 q^{43} - q^{47} - 11 q^{49} + 6 q^{51} - 2 q^{53} + 5 q^{57} + 15 q^{59} + 4 q^{61} + 2 q^{63} + 14 q^{67} + 12 q^{69} + 11 q^{71} + 18 q^{73} + 2 q^{77} + 5 q^{79} + 2 q^{81} - 8 q^{83} - 5 q^{87} - 9 q^{91} + 6 q^{93} + q^{97} - 12 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 - q^7 - 4 * q^9 + 6 * q^11 + 3 * q^13 + 6 * q^17 + 5 * q^19 - q^21 + 12 * q^23 - 10 * q^27 - 5 * q^29 + 6 * q^31 + 6 * q^33 - 4 * q^37 + 3 * q^39 - 6 * q^41 - 3 * q^43 - q^47 - 11 * q^49 + 6 * q^51 - 2 * q^53 + 5 * q^57 + 15 * q^59 + 4 * q^61 + 2 * q^63 + 14 * q^67 + 12 * q^69 + 11 * q^71 + 18 * q^73 + 2 * q^77 + 5 * q^79 + 2 * q^81 - 8 * q^83 - 5 * q^87 - 9 * q^91 + 6 * q^93 + q^97 - 12 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350	0.288675	−	0.957427i	$-0.406785\pi$
$3$	1.00000	0.577350	0.288675	+	0.957427i	$0.406785\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.61803	−0.611559	−0.305780	−	0.952102i	$-0.598917\pi$
$7$	−1.61803	−0.611559	−0.305780	+	0.952102i	$0.598917\pi$
$8$	0	0
$9$	−2.00000	−0.666667
$10$	0	0
$11$	0.763932	0.230334	0.115167	−	0.993346i	$-0.463260\pi$
$11$	0.763932	0.230334	0.115167	+	0.993346i	$0.463260\pi$
$12$	0	0
$13$	4.85410	1.34629	0.673143	−	0.739512i	$-0.264944\pi$
$13$	4.85410	1.34629	0.673143	+	0.739512i	$0.264944\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0.763932	0.185281	0.0926404	−	0.995700i	$-0.470469\pi$
$17$	0.763932	0.185281	0.0926404	+	0.995700i	$0.470469\pi$
$18$	0	0
$19$	5.85410	1.34302	0.671512	−	0.740994i	$-0.265645\pi$
$19$	5.85410	1.34302	0.671512	+	0.740994i	$0.265645\pi$
$20$	0	0
$21$	−1.61803	−0.353084
$22$	0	0
$23$	8.23607	1.71734	0.858669	−	0.512530i	$-0.171292\pi$
$23$	8.23607	1.71734	0.858669	+	0.512530i	$0.171292\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−5.00000	−0.962250
$28$	0	0
$29$	−1.38197	−0.256625	−0.128312	−	0.991734i	$-0.540956\pi$
$29$	−1.38197	−0.256625	−0.128312	+	0.991734i	$0.540956\pi$
$30$	0	0
$31$	3.00000	0.538816	0.269408	−	0.963026i	$-0.413172\pi$
$31$	3.00000	0.538816	0.269408	+	0.963026i	$0.413172\pi$
$32$	0	0
$33$	0.763932	0.132983
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−4.23607	−0.696405	−0.348203	−	0.937419i	$-0.613208\pi$
$37$	−4.23607	−0.696405	−0.348203	+	0.937419i	$0.613208\pi$
$38$	0	0
$39$	4.85410	0.777278
$40$	0	0
$41$	−5.23607	−0.817736	−0.408868	−	0.912593i	$-0.634076\pi$
$41$	−5.23607	−0.817736	−0.408868	+	0.912593i	$0.634076\pi$
$42$	0	0
$43$	1.85410	0.282748	0.141374	−	0.989956i	$-0.454848\pi$
$43$	1.85410	0.282748	0.141374	+	0.989956i	$0.454848\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−1.61803	−0.236015	−0.118007	−	0.993013i	$-0.537651\pi$
$47$	−1.61803	−0.236015	−0.118007	+	0.993013i	$0.537651\pi$
$48$	0	0
$49$	−4.38197	−0.625995
$50$	0	0
$51$	0.763932	0.106972
$52$	0	0
$53$	−5.47214	−0.751656	−0.375828	−	0.926690i	$-0.622642\pi$
$53$	−5.47214	−0.751656	−0.375828	+	0.926690i	$0.622642\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	5.85410	0.775395
$58$	0	0
$59$	4.14590	0.539750	0.269875	−	0.962895i	$-0.413018\pi$
$59$	4.14590	0.539750	0.269875	+	0.962895i	$0.413018\pi$
$60$	0	0
$61$	−4.70820	−0.602824	−0.301412	−	0.953494i	$-0.597458\pi$
$61$	−4.70820	−0.602824	−0.301412	+	0.953494i	$0.597458\pi$
$62$	0	0
$63$	3.23607	0.407706
$64$	0	0
$65$	0	0
$66$	0	0
$67$	9.23607	1.12837	0.564183	−	0.825650i	$-0.309191\pi$
$67$	9.23607	1.12837	0.564183	+	0.825650i	$0.309191\pi$
$68$	0	0
$69$	8.23607	0.991506
$70$	0	0
$71$	4.38197	0.520044	0.260022	−	0.965603i	$-0.416270\pi$
$71$	4.38197	0.520044	0.260022	+	0.965603i	$0.416270\pi$
$72$	0	0
$73$	9.00000	1.05337	0.526685	−	0.850060i	$-0.323435\pi$
$73$	9.00000	1.05337	0.526685	+	0.850060i	$0.323435\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1.23607	−0.140863
$78$	0	0
$79$	−3.09017	−0.347671	−0.173836	−	0.984775i	$-0.555616\pi$
$79$	−3.09017	−0.347671	−0.173836	+	0.984775i	$0.555616\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−1.76393	−0.193617	−0.0968083	−	0.995303i	$-0.530863\pi$
$83$	−1.76393	−0.193617	−0.0968083	+	0.995303i	$0.530863\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−1.38197	−0.148162
$88$	0	0
$89$	8.94427	0.948091	0.474045	−	0.880500i	$-0.342793\pi$
$89$	8.94427	0.948091	0.474045	+	0.880500i	$0.342793\pi$
$90$	0	0
$91$	−7.85410	−0.823334
$92$	0	0
$93$	3.00000	0.311086
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−2.85410	−0.289790	−0.144895	−	0.989447i	$-0.546284\pi$
$97$	−2.85410	−0.289790	−0.144895	+	0.989447i	$0.546284\pi$
$98$	0	0
$99$	−1.52786	−0.153556
$100$	0	0
$101$	−7.47214	−0.743505	−0.371753	−	0.928332i	$-0.621243\pi$
$101$	−7.47214	−0.743505	−0.371753	+	0.928332i	$0.621243\pi$
$102$	0	0
$103$	−11.5623	−1.13927	−0.569634	−	0.821899i	$-0.692915\pi$
$103$	−11.5623	−1.13927	−0.569634	+	0.821899i	$0.692915\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	10.4164	1.00699	0.503496	−	0.863998i	$-0.332047\pi$
$107$	10.4164	1.00699	0.503496	+	0.863998i	$0.332047\pi$
$108$	0	0
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	0	0
$111$	−4.23607	−0.402070
$112$	0	0
$113$	−10.1459	−0.954446	−0.477223	−	0.878782i	$-0.658357\pi$
$113$	−10.1459	−0.954446	−0.477223	+	0.878782i	$0.658357\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−9.70820	−0.897524
$118$	0	0
$119$	−1.23607	−0.113310
$120$	0	0
$121$	−10.4164	−0.946946
$122$	0	0
$123$	−5.23607	−0.472120
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−15.8885	−1.40988	−0.704940	−	0.709267i	$-0.749026\pi$
$127$	−15.8885	−1.40988	−0.704940	+	0.709267i	$0.749026\pi$
$128$	0	0
$129$	1.85410	0.163245
$130$	0	0
$131$	17.7984	1.55505	0.777526	−	0.628851i	$-0.216475\pi$
$131$	17.7984	1.55505	0.777526	+	0.628851i	$0.216475\pi$
$132$	0	0
$133$	−9.47214	−0.821338
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−5.94427	−0.507853	−0.253927	−	0.967223i	$-0.581722\pi$
$137$	−5.94427	−0.507853	−0.253927	+	0.967223i	$0.581722\pi$
$138$	0	0
$139$	−5.00000	−0.424094	−0.212047	−	0.977259i	$-0.568013\pi$
$139$	−5.00000	−0.424094	−0.212047	+	0.977259i	$0.568013\pi$
$140$	0	0
$141$	−1.61803	−0.136263
$142$	0	0
$143$	3.70820	0.310096
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−4.38197	−0.361418
$148$	0	0
$149$	13.9443	1.14236	0.571180	−	0.820825i	$-0.306486\pi$
$149$	13.9443	1.14236	0.571180	+	0.820825i	$0.306486\pi$
$150$	0	0
$151$	5.56231	0.452654	0.226327	−	0.974051i	$-0.427328\pi$
$151$	5.56231	0.452654	0.226327	+	0.974051i	$0.427328\pi$
$152$	0	0
$153$	−1.52786	−0.123520
$154$	0	0
$155$	0	0
$156$	0	0
$157$	9.18034	0.732671	0.366335	−	0.930483i	$-0.380612\pi$
$157$	9.18034	0.732671	0.366335	+	0.930483i	$0.380612\pi$
$158$	0	0
$159$	−5.47214	−0.433969
$160$	0	0
$161$	−13.3262	−1.05025
$162$	0	0
$163$	11.0000	0.861586	0.430793	−	0.902451i	$-0.358234\pi$
$163$	11.0000	0.861586	0.430793	+	0.902451i	$0.358234\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−5.56231	−0.430424	−0.215212	−	0.976567i	$-0.569044\pi$
$167$	−5.56231	−0.430424	−0.215212	+	0.976567i	$0.569044\pi$
$168$	0	0
$169$	10.5623	0.812485
$170$	0	0
$171$	−11.7082	−0.895349
$172$	0	0
$173$	16.8885	1.28401	0.642006	−	0.766700i	$-0.278102\pi$
$173$	16.8885	1.28401	0.642006	+	0.766700i	$0.278102\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	4.14590	0.311625
$178$	0	0
$179$	9.47214	0.707981	0.353990	−	0.935249i	$-0.384825\pi$
$179$	9.47214	0.707981	0.353990	+	0.935249i	$0.384825\pi$
$180$	0	0
$181$	13.7082	1.01892	0.509461	−	0.860494i	$-0.329845\pi$
$181$	13.7082	1.01892	0.509461	+	0.860494i	$0.329845\pi$
$182$	0	0
$183$	−4.70820	−0.348040
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0.583592	0.0426765
$188$	0	0
$189$	8.09017	0.588473
$190$	0	0
$191$	24.1803	1.74963	0.874814	−	0.484459i	$-0.160984\pi$
$191$	24.1803	1.74963	0.874814	+	0.484459i	$0.160984\pi$
$192$	0	0
$193$	5.70820	0.410886	0.205443	−	0.978669i	$-0.434137\pi$
$193$	5.70820	0.410886	0.205443	+	0.978669i	$0.434137\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	9.70820	0.691681	0.345840	−	0.938293i	$-0.387594\pi$
$197$	9.70820	0.691681	0.345840	+	0.938293i	$0.387594\pi$
$198$	0	0
$199$	−2.56231	−0.181637	−0.0908185	−	0.995867i	$-0.528948\pi$
$199$	−2.56231	−0.181637	−0.0908185	+	0.995867i	$0.528948\pi$
$200$	0	0
$201$	9.23607	0.651462
$202$	0	0
$203$	2.23607	0.156941
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−16.4721	−1.14489
$208$	0	0
$209$	4.47214	0.309344
$210$	0	0
$211$	−13.1803	−0.907372	−0.453686	−	0.891162i	$-0.649891\pi$
$211$	−13.1803	−0.907372	−0.453686	+	0.891162i	$0.649891\pi$
$212$	0	0
$213$	4.38197	0.300247
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−4.85410	−0.329518
$218$	0	0
$219$	9.00000	0.608164
$220$	0	0
$221$	3.70820	0.249441
$222$	0	0
$223$	22.1803	1.48531	0.742653	−	0.669677i	$-0.233567\pi$
$223$	22.1803	1.48531	0.742653	+	0.669677i	$0.233567\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	19.2361	1.27674	0.638371	−	0.769729i	$-0.279608\pi$
$227$	19.2361	1.27674	0.638371	+	0.769729i	$0.279608\pi$
$228$	0	0
$229$	8.29180	0.547937	0.273969	−	0.961739i	$-0.411664\pi$
$229$	8.29180	0.547937	0.273969	+	0.961739i	$0.411664\pi$
$230$	0	0
$231$	−1.23607	−0.0813273
$232$	0	0
$233$	−14.9443	−0.979032	−0.489516	−	0.871994i	$-0.662826\pi$
$233$	−14.9443	−0.979032	−0.489516	+	0.871994i	$0.662826\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−3.09017	−0.200728
$238$	0	0
$239$	29.4721	1.90639	0.953197	−	0.302350i	$-0.0977711\pi$
$239$	29.4721	1.90639	0.953197	+	0.302350i	$0.0977711\pi$
$240$	0	0
$241$	11.4721	0.738985	0.369493	−	0.929234i	$-0.379532\pi$
$241$	11.4721	0.738985	0.369493	+	0.929234i	$0.379532\pi$
$242$	0	0
$243$	16.0000	1.02640
$244$	0	0
$245$	0	0
$246$	0	0
$247$	28.4164	1.80809
$248$	0	0
$249$	−1.76393	−0.111785
$250$	0	0
$251$	6.81966	0.430453	0.215227	−	0.976564i	$-0.430951\pi$
$251$	6.81966	0.430453	0.215227	+	0.976564i	$0.430951\pi$
$252$	0	0
$253$	6.29180	0.395562
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−16.1459	−1.00715	−0.503577	−	0.863951i	$-0.667983\pi$
$257$	−16.1459	−1.00715	−0.503577	+	0.863951i	$0.667983\pi$
$258$	0	0
$259$	6.85410	0.425893
$260$	0	0
$261$	2.76393	0.171083
$262$	0	0
$263$	−22.0902	−1.36214	−0.681069	−	0.732219i	$-0.738485\pi$
$263$	−22.0902	−1.36214	−0.681069	+	0.732219i	$0.738485\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	8.94427	0.547381
$268$	0	0
$269$	−17.2361	−1.05090	−0.525451	−	0.850824i	$-0.676103\pi$
$269$	−17.2361	−1.05090	−0.525451	+	0.850824i	$0.676103\pi$
$270$	0	0
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	0	0
$273$	−7.85410	−0.475352
$274$	0	0
$275$	0	0
$276$	0	0
$277$	11.2918	0.678458	0.339229	−	0.940704i	$-0.389834\pi$
$277$	11.2918	0.678458	0.339229	+	0.940704i	$0.389834\pi$
$278$	0	0
$279$	−6.00000	−0.359211
$280$	0	0
$281$	−1.09017	−0.0650341	−0.0325170	−	0.999471i	$-0.510352\pi$
$281$	−1.09017	−0.0650341	−0.0325170	+	0.999471i	$0.510352\pi$
$282$	0	0
$283$	−23.1459	−1.37588	−0.687940	−	0.725767i	$-0.741485\pi$
$283$	−23.1459	−1.37588	−0.687940	+	0.725767i	$0.741485\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	8.47214	0.500094
$288$	0	0
$289$	−16.4164	−0.965671
$290$	0	0
$291$	−2.85410	−0.167310
$292$	0	0
$293$	28.4721	1.66336	0.831680	−	0.555255i	$-0.187379\pi$
$293$	28.4721	1.66336	0.831680	+	0.555255i	$0.187379\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−3.81966	−0.221639
$298$	0	0
$299$	39.9787	2.31203
$300$	0	0
$301$	−3.00000	−0.172917
$302$	0	0
$303$	−7.47214	−0.429263
$304$	0	0
$305$	0	0
$306$	0	0
$307$	4.76393	0.271892	0.135946	−	0.990716i	$-0.456593\pi$
$307$	4.76393	0.271892	0.135946	+	0.990716i	$0.456593\pi$
$308$	0	0
$309$	−11.5623	−0.657757
$310$	0	0
$311$	29.5066	1.67316	0.836582	−	0.547841i	$-0.184550\pi$
$311$	29.5066	1.67316	0.836582	+	0.547841i	$0.184550\pi$
$312$	0	0
$313$	21.2361	1.20033	0.600167	−	0.799875i	$-0.295101\pi$
$313$	21.2361	1.20033	0.600167	+	0.799875i	$0.295101\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	23.6525	1.32846	0.664228	−	0.747530i	$-0.268761\pi$
$317$	23.6525	1.32846	0.664228	+	0.747530i	$0.268761\pi$
$318$	0	0
$319$	−1.05573	−0.0591094
$320$	0	0
$321$	10.4164	0.581387
$322$	0	0
$323$	4.47214	0.248836
$324$	0	0
$325$	0	0
$326$	0	0
$327$	10.0000	0.553001
$328$	0	0
$329$	2.61803	0.144337
$330$	0	0
$331$	−17.1246	−0.941254	−0.470627	−	0.882332i	$-0.655972\pi$
$331$	−17.1246	−0.941254	−0.470627	+	0.882332i	$0.655972\pi$
$332$	0	0
$333$	8.47214	0.464270
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−1.14590	−0.0624210	−0.0312105	−	0.999513i	$-0.509936\pi$
$337$	−1.14590	−0.0624210	−0.0312105	+	0.999513i	$0.509936\pi$
$338$	0	0
$339$	−10.1459	−0.551050
$340$	0	0
$341$	2.29180	0.124108
$342$	0	0
$343$	18.4164	0.994393
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−31.0902	−1.66901	−0.834504	−	0.551002i	$-0.814246\pi$
$347$	−31.0902	−1.66901	−0.834504	+	0.551002i	$0.814246\pi$
$348$	0	0
$349$	8.29180	0.443850	0.221925	−	0.975064i	$-0.428766\pi$
$349$	8.29180	0.443850	0.221925	+	0.975064i	$0.428766\pi$
$350$	0	0
$351$	−24.2705	−1.29546
$352$	0	0
$353$	−24.0902	−1.28219	−0.641095	−	0.767461i	$-0.721520\pi$
$353$	−24.0902	−1.28219	−0.641095	+	0.767461i	$0.721520\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−1.23607	−0.0654197
$358$	0	0
$359$	28.7426	1.51698	0.758489	−	0.651685i	$-0.225938\pi$
$359$	28.7426	1.51698	0.758489	+	0.651685i	$0.225938\pi$
$360$	0	0
$361$	15.2705	0.803711
$362$	0	0
$363$	−10.4164	−0.546720
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−5.43769	−0.283845	−0.141923	−	0.989878i	$-0.545328\pi$
$367$	−5.43769	−0.283845	−0.141923	+	0.989878i	$0.545328\pi$
$368$	0	0
$369$	10.4721	0.545158
$370$	0	0
$371$	8.85410	0.459682
$372$	0	0
$373$	−5.27051	−0.272897	−0.136448	−	0.990647i	$-0.543569\pi$
$373$	−5.27051	−0.272897	−0.136448	+	0.990647i	$0.543569\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−6.70820	−0.345490
$378$	0	0
$379$	34.5967	1.77712	0.888558	−	0.458765i	$-0.151708\pi$
$379$	34.5967	1.77712	0.888558	+	0.458765i	$0.151708\pi$
$380$	0	0
$381$	−15.8885	−0.813995
$382$	0	0
$383$	−11.3607	−0.580504	−0.290252	−	0.956950i	$-0.593739\pi$
$383$	−11.3607	−0.580504	−0.290252	+	0.956950i	$0.593739\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−3.70820	−0.188499
$388$	0	0
$389$	15.0000	0.760530	0.380265	−	0.924878i	$-0.375833\pi$
$389$	15.0000	0.760530	0.380265	+	0.924878i	$0.375833\pi$
$390$	0	0
$391$	6.29180	0.318190
$392$	0	0
$393$	17.7984	0.897809
$394$	0	0
$395$	0	0
$396$	0	0
$397$	0.0344419	0.00172859	0.000864294	−	1.00000i	$-0.499725\pi$
$397$	0.0344419	0.00172859	0.000864294		1.00000i	$0.499725\pi$
$398$	0	0
$399$	−9.47214	−0.474200
$400$	0	0
$401$	−22.5967	−1.12843	−0.564214	−	0.825629i	$-0.690821\pi$
$401$	−22.5967	−1.12843	−0.564214	+	0.825629i	$0.690821\pi$
$402$	0	0
$403$	14.5623	0.725400
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−3.23607	−0.160406
$408$	0	0
$409$	28.4164	1.40510	0.702550	−	0.711634i	$-0.252045\pi$
$409$	28.4164	1.40510	0.702550	+	0.711634i	$0.252045\pi$
$410$	0	0
$411$	−5.94427	−0.293209
$412$	0	0
$413$	−6.70820	−0.330089
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−5.00000	−0.244851
$418$	0	0
$419$	0.527864	0.0257878	0.0128939	−	0.999917i	$-0.495896\pi$
$419$	0.527864	0.0257878	0.0128939	+	0.999917i	$0.495896\pi$
$420$	0	0
$421$	32.0000	1.55958	0.779792	−	0.626038i	$-0.215325\pi$
$421$	32.0000	1.55958	0.779792	+	0.626038i	$0.215325\pi$
$422$	0	0
$423$	3.23607	0.157343
$424$	0	0
$425$	0	0
$426$	0	0
$427$	7.61803	0.368663
$428$	0	0
$429$	3.70820	0.179034
$430$	0	0
$431$	−23.8328	−1.14799	−0.573993	−	0.818860i	$-0.694606\pi$
$431$	−23.8328	−1.14799	−0.573993	+	0.818860i	$0.694606\pi$
$432$	0	0
$433$	−20.1459	−0.968150	−0.484075	−	0.875026i	$-0.660844\pi$
$433$	−20.1459	−0.968150	−0.484075	+	0.875026i	$0.660844\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	48.2148	2.30643
$438$	0	0
$439$	−5.97871	−0.285348	−0.142674	−	0.989770i	$-0.545570\pi$
$439$	−5.97871	−0.285348	−0.142674	+	0.989770i	$0.545570\pi$
$440$	0	0
$441$	8.76393	0.417330
$442$	0	0
$443$	12.0557	0.572785	0.286392	−	0.958112i	$-0.407544\pi$
$443$	12.0557	0.572785	0.286392	+	0.958112i	$0.407544\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	13.9443	0.659541
$448$	0	0
$449$	20.3262	0.959254	0.479627	−	0.877472i	$-0.340772\pi$
$449$	20.3262	0.959254	0.479627	+	0.877472i	$0.340772\pi$
$450$	0	0
$451$	−4.00000	−0.188353
$452$	0	0
$453$	5.56231	0.261340
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−5.41641	−0.253369	−0.126684	−	0.991943i	$-0.540434\pi$
$457$	−5.41641	−0.253369	−0.126684	+	0.991943i	$0.540434\pi$
$458$	0	0
$459$	−3.81966	−0.178286
$460$	0	0
$461$	23.1803	1.07962	0.539808	−	0.841788i	$-0.318497\pi$
$461$	23.1803	1.07962	0.539808	+	0.841788i	$0.318497\pi$
$462$	0	0
$463$	16.1246	0.749374	0.374687	−	0.927151i	$-0.377750\pi$
$463$	16.1246	0.749374	0.374687	+	0.927151i	$0.377750\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−28.4508	−1.31655	−0.658274	−	0.752778i	$-0.728713\pi$
$467$	−28.4508	−1.31655	−0.658274	+	0.752778i	$0.728713\pi$
$468$	0	0
$469$	−14.9443	−0.690062
$470$	0	0
$471$	9.18034	0.423008
$472$	0	0
$473$	1.41641	0.0651265
$474$	0	0
$475$	0	0
$476$	0	0
$477$	10.9443	0.501104
$478$	0	0
$479$	4.14590	0.189431	0.0947155	−	0.995504i	$-0.469806\pi$
$479$	4.14590	0.189431	0.0947155	+	0.995504i	$0.469806\pi$
$480$	0	0
$481$	−20.5623	−0.937560
$482$	0	0
$483$	−13.3262	−0.606365
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−9.58359	−0.434274	−0.217137	−	0.976141i	$-0.569672\pi$
$487$	−9.58359	−0.434274	−0.217137	+	0.976141i	$0.569672\pi$
$488$	0	0
$489$	11.0000	0.497437
$490$	0	0
$491$	−37.2492	−1.68103	−0.840517	−	0.541785i	$-0.817749\pi$
$491$	−37.2492	−1.68103	−0.840517	+	0.541785i	$0.817749\pi$
$492$	0	0
$493$	−1.05573	−0.0475476
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−7.09017	−0.318038
$498$	0	0
$499$	12.5623	0.562366	0.281183	−	0.959654i	$-0.409273\pi$
$499$	12.5623	0.562366	0.281183	+	0.959654i	$0.409273\pi$
$500$	0	0
$501$	−5.56231	−0.248506
$502$	0	0
$503$	−10.5836	−0.471899	−0.235950	−	0.971765i	$-0.575820\pi$
$503$	−10.5836	−0.471899	−0.235950	+	0.971765i	$0.575820\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	10.5623	0.469088
$508$	0	0
$509$	−4.67376	−0.207161	−0.103580	−	0.994621i	$-0.533030\pi$
$509$	−4.67376	−0.207161	−0.103580	+	0.994621i	$0.533030\pi$
$510$	0	0
$511$	−14.5623	−0.644198
$512$	0	0
$513$	−29.2705	−1.29232
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−1.23607	−0.0543622
$518$	0	0
$519$	16.8885	0.741325
$520$	0	0
$521$	−15.3607	−0.672964	−0.336482	−	0.941690i	$-0.609237\pi$
$521$	−15.3607	−0.672964	−0.336482	+	0.941690i	$0.609237\pi$
$522$	0	0
$523$	−19.8541	−0.868159	−0.434080	−	0.900875i	$-0.642926\pi$
$523$	−19.8541	−0.868159	−0.434080	+	0.900875i	$0.642926\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	2.29180	0.0998322
$528$	0	0
$529$	44.8328	1.94925
$530$	0	0
$531$	−8.29180	−0.359833
$532$	0	0
$533$	−25.4164	−1.10091
$534$	0	0
$535$	0	0
$536$	0	0
$537$	9.47214	0.408753
$538$	0	0
$539$	−3.34752	−0.144188
$540$	0	0
$541$	−13.1246	−0.564271	−0.282136	−	0.959375i	$-0.591043\pi$
$541$	−13.1246	−0.564271	−0.282136	+	0.959375i	$0.591043\pi$
$542$	0	0
$543$	13.7082	0.588275
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−34.7082	−1.48402	−0.742008	−	0.670391i	$-0.766126\pi$
$547$	−34.7082	−1.48402	−0.742008	+	0.670391i	$0.766126\pi$
$548$	0	0
$549$	9.41641	0.401882
$550$	0	0
$551$	−8.09017	−0.344653
$552$	0	0
$553$	5.00000	0.212622
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−9.23607	−0.391345	−0.195672	−	0.980669i	$-0.562689\pi$
$557$	−9.23607	−0.391345	−0.195672	+	0.980669i	$0.562689\pi$
$558$	0	0
$559$	9.00000	0.380659
$560$	0	0
$561$	0.583592	0.0246393
$562$	0	0
$563$	9.61803	0.405352	0.202676	−	0.979246i	$-0.435036\pi$
$563$	9.61803	0.405352	0.202676	+	0.979246i	$0.435036\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−1.61803	−0.0679510
$568$	0	0
$569$	29.4721	1.23554	0.617768	−	0.786360i	$-0.288037\pi$
$569$	29.4721	1.23554	0.617768	+	0.786360i	$0.288037\pi$
$570$	0	0
$571$	−32.1246	−1.34437	−0.672187	−	0.740382i	$-0.734645\pi$
$571$	−32.1246	−1.34437	−0.672187	+	0.740382i	$0.734645\pi$
$572$	0	0
$573$	24.1803	1.01015
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−37.7771	−1.57268	−0.786340	−	0.617794i	$-0.788027\pi$
$577$	−37.7771	−1.57268	−0.786340	+	0.617794i	$0.788027\pi$
$578$	0	0
$579$	5.70820	0.237225
$580$	0	0
$581$	2.85410	0.118408
$582$	0	0
$583$	−4.18034	−0.173132
$584$	0	0
$585$	0	0
$586$	0	0
$587$	18.7082	0.772170	0.386085	−	0.922463i	$-0.373827\pi$
$587$	18.7082	0.772170	0.386085	+	0.922463i	$0.373827\pi$
$588$	0	0
$589$	17.5623	0.723642
$590$	0	0
$591$	9.70820	0.399342
$592$	0	0
$593$	22.0902	0.907135	0.453567	−	0.891222i	$-0.350151\pi$
$593$	22.0902	0.907135	0.453567	+	0.891222i	$0.350151\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−2.56231	−0.104868
$598$	0	0
$599$	0.527864	0.0215679	0.0107840	−	0.999942i	$-0.496567\pi$
$599$	0.527864	0.0215679	0.0107840	+	0.999942i	$0.496567\pi$
$600$	0	0
$601$	36.2705	1.47950	0.739752	−	0.672879i	$-0.234943\pi$
$601$	36.2705	1.47950	0.739752	+	0.672879i	$0.234943\pi$
$602$	0	0
$603$	−18.4721	−0.752244
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−15.4377	−0.626597	−0.313298	−	0.949655i	$-0.601434\pi$
$607$	−15.4377	−0.626597	−0.313298	+	0.949655i	$0.601434\pi$
$608$	0	0
$609$	2.23607	0.0906100
$610$	0	0
$611$	−7.85410	−0.317743
$612$	0	0
$613$	−31.9787	−1.29161	−0.645804	−	0.763503i	$-0.723478\pi$
$613$	−31.9787	−1.29161	−0.645804	+	0.763503i	$0.723478\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−9.76393	−0.393081	−0.196541	−	0.980496i	$-0.562971\pi$
$617$	−9.76393	−0.393081	−0.196541	+	0.980496i	$0.562971\pi$
$618$	0	0
$619$	−39.4721	−1.58652	−0.793260	−	0.608884i	$-0.791618\pi$
$619$	−39.4721	−1.58652	−0.793260	+	0.608884i	$0.791618\pi$
$620$	0	0
$621$	−41.1803	−1.65251
$622$	0	0
$623$	−14.4721	−0.579814
$624$	0	0
$625$	0	0
$626$	0	0
$627$	4.47214	0.178600
$628$	0	0
$629$	−3.23607	−0.129030
$630$	0	0
$631$	5.76393	0.229459	0.114729	−	0.993397i	$-0.463400\pi$
$631$	5.76393	0.229459	0.114729	+	0.993397i	$0.463400\pi$
$632$	0	0
$633$	−13.1803	−0.523871
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−21.2705	−0.842768
$638$	0	0
$639$	−8.76393	−0.346696
$640$	0	0
$641$	10.0902	0.398538	0.199269	−	0.979945i	$-0.436143\pi$
$641$	10.0902	0.398538	0.199269	+	0.979945i	$0.436143\pi$
$642$	0	0
$643$	22.8328	0.900438	0.450219	−	0.892918i	$-0.351346\pi$
$643$	22.8328	0.900438	0.450219	+	0.892918i	$0.351346\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	30.5410	1.20069	0.600346	−	0.799741i	$-0.295030\pi$
$647$	30.5410	1.20069	0.600346	+	0.799741i	$0.295030\pi$
$648$	0	0
$649$	3.16718	0.124323
$650$	0	0
$651$	−4.85410	−0.190247
$652$	0	0
$653$	−7.90983	−0.309536	−0.154768	−	0.987951i	$-0.549463\pi$
$653$	−7.90983	−0.309536	−0.154768	+	0.987951i	$0.549463\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−18.0000	−0.702247
$658$	0	0
$659$	−24.4721	−0.953299	−0.476650	−	0.879093i	$-0.658149\pi$
$659$	−24.4721	−0.953299	−0.476650	+	0.879093i	$0.658149\pi$
$660$	0	0
$661$	−40.6869	−1.58254	−0.791269	−	0.611468i	$-0.790579\pi$
$661$	−40.6869	−1.58254	−0.791269	+	0.611468i	$0.790579\pi$
$662$	0	0
$663$	3.70820	0.144015
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−11.3820	−0.440711
$668$	0	0
$669$	22.1803	0.857541
$670$	0	0
$671$	−3.59675	−0.138851
$672$	0	0
$673$	10.1803	0.392423	0.196212	−	0.980562i	$-0.437136\pi$
$673$	10.1803	0.392423	0.196212	+	0.980562i	$0.437136\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−8.38197	−0.322145	−0.161073	−	0.986943i	$-0.551495\pi$
$677$	−8.38197	−0.322145	−0.161073	+	0.986943i	$0.551495\pi$
$678$	0	0
$679$	4.61803	0.177224
$680$	0	0
$681$	19.2361	0.737128
$682$	0	0
$683$	−4.52786	−0.173254	−0.0866270	−	0.996241i	$-0.527609\pi$
$683$	−4.52786	−0.173254	−0.0866270	+	0.996241i	$0.527609\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	8.29180	0.316352
$688$	0	0
$689$	−26.5623	−1.01194
$690$	0	0
$691$	−2.72949	−0.103835	−0.0519173	−	0.998651i	$-0.516533\pi$
$691$	−2.72949	−0.103835	−0.0519173	+	0.998651i	$0.516533\pi$
$692$	0	0
$693$	2.47214	0.0939087
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−4.00000	−0.151511
$698$	0	0
$699$	−14.9443	−0.565244
$700$	0	0
$701$	35.0132	1.32243	0.661214	−	0.750197i	$-0.270041\pi$
$701$	35.0132	1.32243	0.661214	+	0.750197i	$0.270041\pi$
$702$	0	0
$703$	−24.7984	−0.935288
$704$	0	0
$705$	0	0
$706$	0	0
$707$	12.0902	0.454698
$708$	0	0
$709$	33.5410	1.25966	0.629830	−	0.776733i	$-0.283125\pi$
$709$	33.5410	1.25966	0.629830	+	0.776733i	$0.283125\pi$
$710$	0	0
$711$	6.18034	0.231781
$712$	0	0
$713$	24.7082	0.925330
$714$	0	0
$715$	0	0
$716$	0	0
$717$	29.4721	1.10066
$718$	0	0
$719$	36.7082	1.36899	0.684493	−	0.729020i	$-0.260024\pi$
$719$	36.7082	1.36899	0.684493	+	0.729020i	$0.260024\pi$
$720$	0	0
$721$	18.7082	0.696730
$722$	0	0
$723$	11.4721	0.426653
$724$	0	0
$725$	0	0
$726$	0	0
$727$	4.43769	0.164585	0.0822925	−	0.996608i	$-0.473776\pi$
$727$	4.43769	0.164585	0.0822925	+	0.996608i	$0.473776\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	1.41641	0.0523877
$732$	0	0
$733$	−26.9787	−0.996482	−0.498241	−	0.867039i	$-0.666020\pi$
$733$	−26.9787	−0.996482	−0.498241	+	0.867039i	$0.666020\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	7.05573	0.259901
$738$	0	0
$739$	30.9787	1.13957	0.569785	−	0.821794i	$-0.307026\pi$
$739$	30.9787	1.13957	0.569785	+	0.821794i	$0.307026\pi$
$740$	0	0
$741$	28.4164	1.04390
$742$	0	0
$743$	−16.3607	−0.600215	−0.300108	−	0.953905i	$-0.597023\pi$
$743$	−16.3607	−0.600215	−0.300108	+	0.953905i	$0.597023\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	3.52786	0.129078
$748$	0	0
$749$	−16.8541	−0.615835
$750$	0	0
$751$	40.8885	1.49204	0.746022	−	0.665921i	$-0.231961\pi$
$751$	40.8885	1.49204	0.746022	+	0.665921i	$0.231961\pi$
$752$	0	0
$753$	6.81966	0.248522
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−3.58359	−0.130248	−0.0651239	−	0.997877i	$-0.520744\pi$
$757$	−3.58359	−0.130248	−0.0651239	+	0.997877i	$0.520744\pi$
$758$	0	0
$759$	6.29180	0.228378
$760$	0	0
$761$	37.4508	1.35759	0.678796	−	0.734327i	$-0.262502\pi$
$761$	37.4508	1.35759	0.678796	+	0.734327i	$0.262502\pi$
$762$	0	0
$763$	−16.1803	−0.585768
$764$	0	0
$765$	0	0
$766$	0	0
$767$	20.1246	0.726658
$768$	0	0
$769$	−13.4164	−0.483808	−0.241904	−	0.970300i	$-0.577772\pi$
$769$	−13.4164	−0.483808	−0.241904	+	0.970300i	$0.577772\pi$
$770$	0	0
$771$	−16.1459	−0.581480
$772$	0	0
$773$	−33.1591	−1.19265	−0.596324	−	0.802744i	$-0.703373\pi$
$773$	−33.1591	−1.19265	−0.596324	+	0.802744i	$0.703373\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	6.85410	0.245890
$778$	0	0
$779$	−30.6525	−1.09824
$780$	0	0
$781$	3.34752	0.119784
$782$	0	0
$783$	6.90983	0.246937
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−34.1803	−1.21840	−0.609199	−	0.793018i	$-0.708509\pi$
$787$	−34.1803	−1.21840	−0.609199	+	0.793018i	$0.708509\pi$
$788$	0	0
$789$	−22.0902	−0.786431
$790$	0	0
$791$	16.4164	0.583700
$792$	0	0
$793$	−22.8541	−0.811573
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−14.2361	−0.504267	−0.252134	−	0.967692i	$-0.581132\pi$
$797$	−14.2361	−0.504267	−0.252134	+	0.967692i	$0.581132\pi$
$798$	0	0
$799$	−1.23607	−0.0437289
$800$	0	0
$801$	−17.8885	−0.632061
$802$	0	0
$803$	6.87539	0.242627
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−17.2361	−0.606738
$808$	0	0
$809$	−15.9787	−0.561782	−0.280891	−	0.959740i	$-0.590630\pi$
$809$	−15.9787	−0.561782	−0.280891	+	0.959740i	$0.590630\pi$
$810$	0	0
$811$	1.29180	0.0453611	0.0226805	−	0.999743i	$-0.492780\pi$
$811$	1.29180	0.0453611	0.0226805	+	0.999743i	$0.492780\pi$
$812$	0	0
$813$	8.00000	0.280572
$814$	0	0
$815$	0	0
$816$	0	0
$817$	10.8541	0.379737
$818$	0	0
$819$	15.7082	0.548889
$820$	0	0
$821$	19.6869	0.687078	0.343539	−	0.939138i	$-0.388374\pi$
$821$	19.6869	0.687078	0.343539	+	0.939138i	$0.388374\pi$
$822$	0	0
$823$	34.2918	1.19534	0.597668	−	0.801743i	$-0.296094\pi$
$823$	34.2918	1.19534	0.597668	+	0.801743i	$0.296094\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−30.0344	−1.04440	−0.522200	−	0.852823i	$-0.674889\pi$
$827$	−30.0344	−1.04440	−0.522200	+	0.852823i	$0.674889\pi$
$828$	0	0
$829$	−29.1459	−1.01228	−0.506139	−	0.862452i	$-0.668928\pi$
$829$	−29.1459	−1.01228	−0.506139	+	0.862452i	$0.668928\pi$
$830$	0	0
$831$	11.2918	0.391708
$832$	0	0
$833$	−3.34752	−0.115985
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−15.0000	−0.518476
$838$	0	0
$839$	−4.14590	−0.143132	−0.0715661	−	0.997436i	$-0.522800\pi$
$839$	−4.14590	−0.143132	−0.0715661	+	0.997436i	$0.522800\pi$
$840$	0	0
$841$	−27.0902	−0.934144
$842$	0	0
$843$	−1.09017	−0.0375474
$844$	0	0
$845$	0	0
$846$	0	0
$847$	16.8541	0.579114
$848$	0	0
$849$	−23.1459	−0.794365
$850$	0	0
$851$	−34.8885	−1.19596
$852$	0	0
$853$	−47.3050	−1.61969	−0.809845	−	0.586643i	$-0.800449\pi$
$853$	−47.3050	−1.61969	−0.809845	+	0.586643i	$0.800449\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	40.6869	1.38984	0.694919	−	0.719088i	$-0.255440\pi$
$857$	40.6869	1.38984	0.694919	+	0.719088i	$0.255440\pi$
$858$	0	0
$859$	28.4164	0.969555	0.484778	−	0.874637i	$-0.338901\pi$
$859$	28.4164	0.969555	0.484778	+	0.874637i	$0.338901\pi$
$860$	0	0
$861$	8.47214	0.288730
$862$	0	0
$863$	−41.5623	−1.41480	−0.707399	−	0.706815i	$-0.750131\pi$
$863$	−41.5623	−1.41480	−0.707399	+	0.706815i	$0.750131\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−16.4164	−0.557530
$868$	0	0
$869$	−2.36068	−0.0800806
$870$	0	0
$871$	44.8328	1.51910
$872$	0	0
$873$	5.70820	0.193193
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−30.5410	−1.03130	−0.515648	−	0.856800i	$-0.672449\pi$
$877$	−30.5410	−1.03130	−0.515648	+	0.856800i	$0.672449\pi$
$878$	0	0
$879$	28.4721	0.960341
$880$	0	0
$881$	4.36068	0.146915	0.0734575	−	0.997298i	$-0.476597\pi$
$881$	4.36068	0.146915	0.0734575	+	0.997298i	$0.476597\pi$
$882$	0	0
$883$	−47.4164	−1.59569	−0.797845	−	0.602863i	$-0.794026\pi$
$883$	−47.4164	−1.59569	−0.797845	+	0.602863i	$0.794026\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−5.88854	−0.197718	−0.0988590	−	0.995101i	$-0.531519\pi$
$887$	−5.88854	−0.197718	−0.0988590	+	0.995101i	$0.531519\pi$
$888$	0	0
$889$	25.7082	0.862225
$890$	0	0
$891$	0.763932	0.0255927
$892$	0	0
$893$	−9.47214	−0.316973
$894$	0	0
$895$	0	0
$896$	0	0
$897$	39.9787	1.33485
$898$	0	0
$899$	−4.14590	−0.138273
$900$	0	0
$901$	−4.18034	−0.139267
$902$	0	0
$903$	−3.00000	−0.0998337
$904$	0	0
$905$	0	0
$906$	0	0
$907$	47.2492	1.56888	0.784442	−	0.620202i	$-0.212949\pi$
$907$	47.2492	1.56888	0.784442	+	0.620202i	$0.212949\pi$
$908$	0	0
$909$	14.9443	0.495670
$910$	0	0
$911$	35.7639	1.18491	0.592456	−	0.805603i	$-0.298158\pi$
$911$	35.7639	1.18491	0.592456	+	0.805603i	$0.298158\pi$
$912$	0	0
$913$	−1.34752	−0.0445965
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−28.7984	−0.951006
$918$	0	0
$919$	1.78522	0.0588889	0.0294445	−	0.999566i	$-0.490626\pi$
$919$	1.78522	0.0588889	0.0294445	+	0.999566i	$0.490626\pi$
$920$	0	0
$921$	4.76393	0.156977
$922$	0	0
$923$	21.2705	0.700127
$924$	0	0
$925$	0	0
$926$	0	0
$927$	23.1246	0.759512
$928$	0	0
$929$	−36.6312	−1.20183	−0.600915	−	0.799313i	$-0.705197\pi$
$929$	−36.6312	−1.20183	−0.600915	+	0.799313i	$0.705197\pi$
$930$	0	0
$931$	−25.6525	−0.840726
$932$	0	0
$933$	29.5066	0.966002
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−51.2705	−1.67493	−0.837467	−	0.546487i	$-0.815965\pi$
$937$	−51.2705	−1.67493	−0.837467	+	0.546487i	$0.815965\pi$
$938$	0	0
$939$	21.2361	0.693013
$940$	0	0
$941$	−19.5836	−0.638407	−0.319203	−	0.947686i	$-0.603415\pi$
$941$	−19.5836	−0.638407	−0.319203	+	0.947686i	$0.603415\pi$
$942$	0	0
$943$	−43.1246	−1.40433
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−28.6525	−0.931080	−0.465540	−	0.885027i	$-0.654140\pi$
$947$	−28.6525	−0.931080	−0.465540	+	0.885027i	$0.654140\pi$
$948$	0	0
$949$	43.6869	1.41814
$950$	0	0
$951$	23.6525	0.766984
$952$	0	0
$953$	−34.7426	−1.12542	−0.562712	−	0.826653i	$-0.690242\pi$
$953$	−34.7426	−1.12542	−0.562712	+	0.826653i	$0.690242\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−1.05573	−0.0341268
$958$	0	0
$959$	9.61803	0.310583
$960$	0	0
$961$	−22.0000	−0.709677
$962$	0	0
$963$	−20.8328	−0.671328
$964$	0	0
$965$	0	0
$966$	0	0
$967$	39.8885	1.28273	0.641365	−	0.767236i	$-0.278369\pi$
$967$	39.8885	1.28273	0.641365	+	0.767236i	$0.278369\pi$
$968$	0	0
$969$	4.47214	0.143666
$970$	0	0
$971$	−3.38197	−0.108532	−0.0542662	−	0.998527i	$-0.517282\pi$
$971$	−3.38197	−0.108532	−0.0542662	+	0.998527i	$0.517282\pi$
$972$	0	0
$973$	8.09017	0.259359
$974$	0	0
$975$	0	0
$976$	0	0
$977$	33.6525	1.07664	0.538319	−	0.842741i	$-0.319060\pi$
$977$	33.6525	1.07664	0.538319	+	0.842741i	$0.319060\pi$
$978$	0	0
$979$	6.83282	0.218378
$980$	0	0
$981$	−20.0000	−0.638551
$982$	0	0
$983$	7.38197	0.235448	0.117724	−	0.993046i	$-0.462440\pi$
$983$	7.38197	0.235448	0.117724	+	0.993046i	$0.462440\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	2.61803	0.0833329
$988$	0	0
$989$	15.2705	0.485574
$990$	0	0
$991$	−29.3607	−0.932673	−0.466336	−	0.884607i	$-0.654426\pi$
$991$	−29.3607	−0.932673	−0.466336	+	0.884607i	$0.654426\pi$
$992$	0	0
$993$	−17.1246	−0.543433
$994$	0	0
$995$	0	0
$996$	0	0
$997$	10.8885	0.344844	0.172422	−	0.985023i	$-0.444841\pi$
$997$	10.8885	0.344844	0.172422	+	0.985023i	$0.444841\pi$
$998$	0	0
$999$	21.1803	0.670116

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	10000.2.a.l.1.1		2
4.3	odd	2		625.2.a.c.1.1		2
5.4	even	2		10000.2.a.c.1.2		2
12.11	even	2		5625.2.a.d.1.2		2
20.3	even	4		625.2.b.a.624.3		4
20.7	even	4		625.2.b.a.624.2		4
20.19	odd	2		625.2.a.b.1.2		2
25.4	even	10		400.2.u.b.241.1		4
25.19	even	10		400.2.u.b.161.1		4
60.59	even	2		5625.2.a.f.1.1		2
100.3	even	20		125.2.e.a.49.2		8
100.11	odd	10		625.2.d.b.501.1		4
100.19	odd	10		25.2.d.a.11.1	✓	4
100.23	even	20		625.2.e.c.124.2		8
100.27	even	20		625.2.e.c.124.1		8
100.31	odd	10		125.2.d.a.51.1		4
100.39	odd	10		625.2.d.h.501.1		4
100.47	even	20		125.2.e.a.49.1		8
100.59	odd	10		625.2.d.h.126.1		4
100.63	even	20		625.2.e.c.499.1		8
100.67	even	20		125.2.e.a.74.2		8
100.71	odd	10		125.2.d.a.76.1		4
100.79	odd	10		25.2.d.a.16.1	yes	4
100.83	even	20		125.2.e.a.74.1		8
100.87	even	20		625.2.e.c.499.2		8
100.91	odd	10		625.2.d.b.126.1		4
300.119	even	10		225.2.h.b.136.1		4
300.179	even	10		225.2.h.b.91.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.2.d.a.11.1	✓	4	100.19	odd	10
25.2.d.a.16.1	yes	4	100.79	odd	10
125.2.d.a.51.1		4	100.31	odd	10
125.2.d.a.76.1		4	100.71	odd	10
125.2.e.a.49.1		8	100.47	even	20
125.2.e.a.49.2		8	100.3	even	20
125.2.e.a.74.1		8	100.83	even	20
125.2.e.a.74.2		8	100.67	even	20
225.2.h.b.91.1		4	300.179	even	10
225.2.h.b.136.1		4	300.119	even	10
400.2.u.b.161.1		4	25.19	even	10
400.2.u.b.241.1		4	25.4	even	10
625.2.a.b.1.2		2	20.19	odd	2
625.2.a.c.1.1		2	4.3	odd	2
625.2.b.a.624.2		4	20.7	even	4
625.2.b.a.624.3		4	20.3	even	4
625.2.d.b.126.1		4	100.91	odd	10
625.2.d.b.501.1		4	100.11	odd	10
625.2.d.h.126.1		4	100.59	odd	10
625.2.d.h.501.1		4	100.39	odd	10
625.2.e.c.124.1		8	100.27	even	20
625.2.e.c.124.2		8	100.23	even	20
625.2.e.c.499.1		8	100.63	even	20
625.2.e.c.499.2		8	100.87	even	20
5625.2.a.d.1.2		2	12.11	even	2
5625.2.a.f.1.1		2	60.59	even	2
10000.2.a.c.1.2		2	5.4	even	2
10000.2.a.l.1.1		2	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.61803 q^{7} -2.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.61803 q^{7} -2.00000 q^{9} +0.763932 q^{11} +4.85410 q^{13} +0.763932 q^{17} +5.85410 q^{19} -1.61803 q^{21} +8.23607 q^{23} -5.00000 q^{27} -1.38197 q^{29} +3.00000 q^{31} +0.763932 q^{33} -4.23607 q^{37} +4.85410 q^{39} -5.23607 q^{41} +1.85410 q^{43} -1.61803 q^{47} -4.38197 q^{49} +0.763932 q^{51} -5.47214 q^{53} +5.85410 q^{57} +4.14590 q^{59} -4.70820 q^{61} +3.23607 q^{63} +9.23607 q^{67} +8.23607 q^{69} +4.38197 q^{71} +9.00000 q^{73} -1.23607 q^{77} -3.09017 q^{79} +1.00000 q^{81} -1.76393 q^{83} -1.38197 q^{87} +8.94427 q^{89} -7.85410 q^{91} +3.00000 q^{93} -2.85410 q^{97} -1.52786 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - q^{7} - 4 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - q^7 - 4 * q^9	\( 2 q + 2 q^{3} - q^{7} - 4 q^{9} + 6 q^{11} + 3 q^{13} + 6 q^{17} + 5 q^{19} - q^{21} + 12 q^{23} - 10 q^{27} - 5 q^{29} + 6 q^{31} + 6 q^{33} - 4 q^{37} + 3 q^{39} - 6 q^{41} - 3 q^{43} - q^{47} - 11 q^{49} + 6 q^{51} - 2 q^{53} + 5 q^{57} + 15 q^{59} + 4 q^{61} + 2 q^{63} + 14 q^{67} + 12 q^{69} + 11 q^{71} + 18 q^{73} + 2 q^{77} + 5 q^{79} + 2 q^{81} - 8 q^{83} - 5 q^{87} - 9 q^{91} + 6 q^{93} + q^{97} - 12 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 - q^7 - 4 * q^9 + 6 * q^11 + 3 * q^13 + 6 * q^17 + 5 * q^19 - q^21 + 12 * q^23 - 10 * q^27 - 5 * q^29 + 6 * q^31 + 6 * q^33 - 4 * q^37 + 3 * q^39 - 6 * q^41 - 3 * q^43 - q^47 - 11 * q^49 + 6 * q^51 - 2 * q^53 + 5 * q^57 + 15 * q^59 + 4 * q^61 + 2 * q^63 + 14 * q^67 + 12 * q^69 + 11 * q^71 + 18 * q^73 + 2 * q^77 + 5 * q^79 + 2 * q^81 - 8 * q^83 - 5 * q^87 - 9 * q^91 + 6 * q^93 + q^97 - 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more