LMFDB - Embedded newform 10000.2.a.x.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:	$4$
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			1.1
Root			$-2.33275$ 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)

$f(q)$	$=$	$q-2.33275 q^{3} +3.77447 q^{7} +2.44172 q^{9} +O(q^{10})$	$q-2.33275 q^{3} +3.77447 q^{7} +2.44172 q^{9} +3.77447 q^{11} -3.17632 q^{13} +1.39250 q^{17} -3.91385 q^{19} -8.80489 q^{21} +0.891031 q^{23} +1.30233 q^{27} +0.0492169 q^{29} +5.58111 q^{31} -8.80489 q^{33} -7.04746 q^{37} +7.40955 q^{39} -1.48918 q^{41} +2.69767 q^{43} +3.77447 q^{47} +7.24660 q^{49} -3.24836 q^{51} -11.6163 q^{53} +9.13004 q^{57} +0.690074 q^{59} +10.3603 q^{61} +9.21619 q^{63} +15.3234 q^{67} -2.07855 q^{69} +6.69767 q^{71} -5.16872 q^{73} +14.2466 q^{77} +1.80664 q^{79} -10.3632 q^{81} -9.96783 q^{83} -0.114811 q^{87} +14.5103 q^{89} -11.9889 q^{91} -13.0193 q^{93} +0.0901699 q^{97} +9.21619 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−2.33275	−1.34681	−0.673407	−	0.739272i	$-0.735170\pi$
$3$	−2.33275	−1.34681	−0.673407	+	0.739272i	$0.735170\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.77447	1.42661	0.713307	−	0.700851i	$-0.247196\pi$
$7$	3.77447	1.42661	0.713307	+	0.700851i	$0.247196\pi$
$8$	0	0
$9$	2.44172	0.813906
$10$	0	0
$11$	3.77447	1.13804	0.569022	−	0.822322i	$-0.307322\pi$
$11$	3.77447	1.13804	0.569022	+	0.822322i	$0.307322\pi$
$12$	0	0
$13$	−3.17632	−0.880951	−0.440476	−	0.897765i	$-0.645190\pi$
$13$	−3.17632	−0.880951	−0.440476	+	0.897765i	$0.645190\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.39250	0.337731	0.168866	−	0.985639i	$-0.445990\pi$
$17$	1.39250	0.337731	0.168866	+	0.985639i	$0.445990\pi$
$18$	0	0
$19$	−3.91385	−0.897900	−0.448950	−	0.893557i	$-0.648202\pi$
$19$	−3.91385	−0.897900	−0.448950	+	0.893557i	$0.648202\pi$
$20$	0	0
$21$	−8.80489	−1.92138
$22$	0	0
$23$	0.891031	0.185793	0.0928964	−	0.995676i	$-0.470387\pi$
$23$	0.891031	0.185793	0.0928964	+	0.995676i	$0.470387\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.30233	0.250634
$28$	0	0
$29$	0.0492169	0.00913936	0.00456968	−	0.999990i	$-0.498545\pi$
$29$	0.0492169	0.00913936	0.00456968	+	0.999990i	$0.498545\pi$
$30$	0	0
$31$	5.58111	1.00240	0.501198	−	0.865333i	$-0.332893\pi$
$31$	5.58111	1.00240	0.501198	+	0.865333i	$0.332893\pi$
$32$	0	0
$33$	−8.80489	−1.53273
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−7.04746	−1.15860	−0.579298	−	0.815116i	$-0.696673\pi$
$37$	−7.04746	−1.15860	−0.579298	+	0.815116i	$0.696673\pi$
$38$	0	0
$39$	7.40955	1.18648
$40$	0	0
$41$	−1.48918	−0.232571	−0.116286	−	0.993216i	$-0.537099\pi$
$41$	−1.48918	−0.232571	−0.116286	+	0.993216i	$0.537099\pi$
$42$	0	0
$43$	2.69767	0.411391	0.205695	−	0.978616i	$-0.434054\pi$
$43$	2.69767	0.411391	0.205695	+	0.978616i	$0.434054\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	3.77447	0.550563	0.275281	−	0.961364i	$-0.411229\pi$
$47$	3.77447	0.550563	0.275281	+	0.961364i	$0.411229\pi$
$48$	0	0
$49$	7.24660	1.03523
$50$	0	0
$51$	−3.24836	−0.454861
$52$	0	0
$53$	−11.6163	−1.59562	−0.797809	−	0.602910i	$-0.794008\pi$
$53$	−11.6163	−1.59562	−0.797809	+	0.602910i	$0.794008\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	9.13004	1.20930
$58$	0	0
$59$	0.690074	0.0898400	0.0449200	−	0.998991i	$-0.485697\pi$
$59$	0.690074	0.0898400	0.0449200	+	0.998991i	$0.485697\pi$
$60$	0	0
$61$	10.3603	1.32650	0.663252	−	0.748396i	$-0.269176\pi$
$61$	10.3603	1.32650	0.663252	+	0.748396i	$0.269176\pi$
$62$	0	0
$63$	9.21619	1.16113
$64$	0	0
$65$	0	0
$66$	0	0
$67$	15.3234	1.87205	0.936026	−	0.351932i	$-0.114475\pi$
$67$	15.3234	1.87205	0.936026	+	0.351932i	$0.114475\pi$
$68$	0	0
$69$	−2.07855	−0.250228
$70$	0	0
$71$	6.69767	0.794867	0.397434	−	0.917631i	$-0.369901\pi$
$71$	6.69767	0.794867	0.397434	+	0.917631i	$0.369901\pi$
$72$	0	0
$73$	−5.16872	−0.604953	−0.302477	−	0.953157i	$-0.597813\pi$
$73$	−5.16872	−0.604953	−0.302477	+	0.953157i	$0.597813\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	14.2466	1.62355
$78$	0	0
$79$	1.80664	0.203263	0.101631	−	0.994822i	$-0.467594\pi$
$79$	1.80664	0.203263	0.101631	+	0.994822i	$0.467594\pi$
$80$	0	0
$81$	−10.3632	−1.15146
$82$	0	0
$83$	−9.96783	−1.09411	−0.547056	−	0.837096i	$-0.684251\pi$
$83$	−9.96783	−1.09411	−0.547056	+	0.837096i	$0.684251\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−0.114811	−0.0123090
$88$	0	0
$89$	14.5103	1.53808	0.769042	−	0.639198i	$-0.220734\pi$
$89$	14.5103	1.53808	0.769042	+	0.639198i	$0.220734\pi$
$90$	0	0
$91$	−11.9889	−1.25678
$92$	0	0
$93$	−13.0193	−1.35004
$94$	0	0
$95$	0	0
$96$	0	0
$97$	0.0901699	0.00915537	0.00457769	−	0.999990i	$-0.498543\pi$
$97$	0.0901699	0.00915537	0.00457769	+	0.999990i	$0.498543\pi$
$98$	0	0
$99$	9.21619	0.926261
$100$	0	0
$101$	16.3785	1.62972	0.814859	−	0.579659i	$-0.196814\pi$
$101$	16.3785	1.62972	0.814859	+	0.579659i	$0.196814\pi$
$102$	0	0
$103$	−1.21619	−0.119834	−0.0599172	−	0.998203i	$-0.519084\pi$
$103$	−1.21619	−0.119834	−0.0599172	+	0.998203i	$0.519084\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−10.8125	−1.04528	−0.522641	−	0.852553i	$-0.675053\pi$
$107$	−10.8125	−1.04528	−0.522641	+	0.852553i	$0.675053\pi$
$108$	0	0
$109$	11.4153	1.09339	0.546695	−	0.837332i	$-0.315886\pi$
$109$	11.4153	1.09339	0.546695	+	0.837332i	$0.315886\pi$
$110$	0	0
$111$	16.4400	1.56041
$112$	0	0
$113$	10.4229	0.980506	0.490253	−	0.871580i	$-0.336904\pi$
$113$	10.4229	0.980506	0.490253	+	0.871580i	$0.336904\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−7.75567	−0.717012
$118$	0	0
$119$	5.25595	0.481812
$120$	0	0
$121$	3.24660	0.295146
$122$	0	0
$123$	3.47389	0.313230
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−3.54893	−0.314917	−0.157459	−	0.987526i	$-0.550330\pi$
$127$	−3.54893	−0.314917	−0.157459	+	0.987526i	$0.550330\pi$
$128$	0	0
$129$	−6.29298	−0.554066
$130$	0	0
$131$	−4.05324	−0.354133	−0.177067	−	0.984199i	$-0.556661\pi$
$131$	−4.05324	−0.354133	−0.177067	+	0.984199i	$0.556661\pi$
$132$	0	0
$133$	−14.7727	−1.28096
$134$	0	0
$135$	0	0
$136$	0	0
$137$	3.56773	0.304812	0.152406	−	0.988318i	$-0.451298\pi$
$137$	3.56773	0.304812	0.152406	+	0.988318i	$0.451298\pi$
$138$	0	0
$139$	9.94602	0.843611	0.421805	−	0.906686i	$-0.361397\pi$
$139$	9.94602	0.843611	0.421805	+	0.906686i	$0.361397\pi$
$140$	0	0
$141$	−8.80489	−0.741505
$142$	0	0
$143$	−11.9889	−1.00256
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−16.9045	−1.39426
$148$	0	0
$149$	−19.5103	−1.59834	−0.799171	−	0.601104i	$-0.794728\pi$
$149$	−19.5103	−1.59834	−0.799171	+	0.601104i	$0.794728\pi$
$150$	0	0
$151$	18.4324	1.50001	0.750003	−	0.661435i	$-0.230052\pi$
$151$	18.4324	1.50001	0.750003	+	0.661435i	$0.230052\pi$
$152$	0	0
$153$	3.40010	0.274881
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−10.1564	−0.810572	−0.405286	−	0.914190i	$-0.632828\pi$
$157$	−10.1564	−0.810572	−0.405286	+	0.914190i	$0.632828\pi$
$158$	0	0
$159$	27.0979	2.14900
$160$	0	0
$161$	3.36317	0.265055
$162$	0	0
$163$	−18.6004	−1.45690	−0.728449	−	0.685100i	$-0.759758\pi$
$163$	−18.6004	−1.45690	−0.728449	+	0.685100i	$0.759758\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	2.17156	0.168040	0.0840201	−	0.996464i	$-0.473224\pi$
$167$	2.17156	0.168040	0.0840201	+	0.996464i	$0.473224\pi$
$168$	0	0
$169$	−2.91102	−0.223924
$170$	0	0
$171$	−9.55653	−0.730806
$172$	0	0
$173$	6.39425	0.486146	0.243073	−	0.970008i	$-0.421845\pi$
$173$	6.39425	0.486146	0.243073	+	0.970008i	$0.421845\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−1.60977	−0.120998
$178$	0	0
$179$	−8.01348	−0.598955	−0.299478	−	0.954103i	$-0.596812\pi$
$179$	−8.01348	−0.598955	−0.299478	+	0.954103i	$0.596812\pi$
$180$	0	0
$181$	−10.8629	−0.807432	−0.403716	−	0.914884i	$-0.632282\pi$
$181$	−10.8629	−0.807432	−0.403716	+	0.914884i	$0.632282\pi$
$182$	0	0
$183$	−24.1681	−1.78655
$184$	0	0
$185$	0	0
$186$	0	0
$187$	5.25595	0.384353
$188$	0	0
$189$	4.91561	0.357558
$190$	0	0
$191$	8.18577	0.592301	0.296151	−	0.955141i	$-0.404297\pi$
$191$	8.18577	0.592301	0.296151	+	0.955141i	$0.404297\pi$
$192$	0	0
$193$	17.5576	1.26382	0.631912	−	0.775040i	$-0.282270\pi$
$193$	17.5576	1.26382	0.631912	+	0.775040i	$0.282270\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−4.75742	−0.338952	−0.169476	−	0.985534i	$-0.554208\pi$
$197$	−4.75742	−0.338952	−0.169476	+	0.985534i	$0.554208\pi$
$198$	0	0
$199$	−14.5320	−1.03015	−0.515073	−	0.857146i	$-0.672235\pi$
$199$	−14.5320	−1.03015	−0.515073	+	0.857146i	$0.672235\pi$
$200$	0	0
$201$	−35.7457	−2.52130
$202$	0	0
$203$	0.185768	0.0130383
$204$	0	0
$205$	0	0
$206$	0	0
$207$	2.17565	0.151218
$208$	0	0
$209$	−14.7727	−1.02185
$210$	0	0
$211$	14.1359	0.973154	0.486577	−	0.873638i	$-0.338245\pi$
$211$	14.1359	0.973154	0.486577	+	0.873638i	$0.338245\pi$
$212$	0	0
$213$	−15.6240	−1.07054
$214$	0	0
$215$	0	0
$216$	0	0
$217$	21.0657	1.43003
$218$	0	0
$219$	12.0573	0.814759
$220$	0	0
$221$	−4.42302	−0.297525
$222$	0	0
$223$	4.61226	0.308860	0.154430	−	0.988004i	$-0.450646\pi$
$223$	4.61226	0.308860	0.154430	+	0.988004i	$0.450646\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−6.28637	−0.417241	−0.208620	−	0.977997i	$-0.566897\pi$
$227$	−6.28637	−0.417241	−0.208620	+	0.977997i	$0.566897\pi$
$228$	0	0
$229$	17.7370	1.17209	0.586046	−	0.810278i	$-0.300684\pi$
$229$	17.7370	1.17209	0.586046	+	0.810278i	$0.300684\pi$
$230$	0	0
$231$	−33.2338	−2.18662
$232$	0	0
$233$	−2.56406	−0.167977	−0.0839885	−	0.996467i	$-0.526766\pi$
$233$	−2.56406	−0.167977	−0.0839885	+	0.996467i	$0.526766\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−4.21443	−0.273757
$238$	0	0
$239$	−9.44172	−0.610734	−0.305367	−	0.952235i	$-0.598779\pi$
$239$	−9.44172	−0.610734	−0.305367	+	0.952235i	$0.598779\pi$
$240$	0	0
$241$	−29.4346	−1.89605	−0.948026	−	0.318193i	$-0.896924\pi$
$241$	−29.4346	−1.89605	−0.948026	+	0.318193i	$0.896924\pi$
$242$	0	0
$243$	20.2677	1.30017
$244$	0	0
$245$	0	0
$246$	0	0
$247$	12.4316	0.791006
$248$	0	0
$249$	23.2524	1.47356
$250$	0	0
$251$	6.00759	0.379196	0.189598	−	0.981862i	$-0.439282\pi$
$251$	6.00759	0.379196	0.189598	+	0.981862i	$0.439282\pi$
$252$	0	0
$253$	3.36317	0.211440
$254$	0	0
$255$	0	0
$256$	0	0
$257$	13.6286	0.850127	0.425063	−	0.905164i	$-0.360252\pi$
$257$	13.6286	0.850127	0.425063	+	0.905164i	$0.360252\pi$
$258$	0	0
$259$	−26.6004	−1.65287
$260$	0	0
$261$	0.120174	0.00743858
$262$	0	0
$263$	−9.16294	−0.565011	−0.282506	−	0.959266i	$-0.591166\pi$
$263$	−9.16294	−0.565011	−0.282506	+	0.959266i	$0.591166\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−33.8488	−2.07151
$268$	0	0
$269$	28.1026	1.71345	0.856724	−	0.515776i	$-0.172496\pi$
$269$	28.1026	1.71345	0.856724	+	0.515776i	$0.172496\pi$
$270$	0	0
$271$	9.09860	0.552701	0.276350	−	0.961057i	$-0.410875\pi$
$271$	9.09860	0.552701	0.276350	+	0.961057i	$0.410875\pi$
$272$	0	0
$273$	27.9671	1.69265
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−18.7890	−1.12892	−0.564462	−	0.825459i	$-0.690916\pi$
$277$	−18.7890	−1.12892	−0.564462	+	0.825459i	$0.690916\pi$
$278$	0	0
$279$	13.6275	0.815856
$280$	0	0
$281$	7.88158	0.470176	0.235088	−	0.971974i	$-0.424462\pi$
$281$	7.88158	0.470176	0.235088	+	0.971974i	$0.424462\pi$
$282$	0	0
$283$	−18.2998	−1.08781	−0.543906	−	0.839146i	$-0.683055\pi$
$283$	−18.2998	−1.08781	−0.543906	+	0.839146i	$0.683055\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−5.62087	−0.331789
$288$	0	0
$289$	−15.0609	−0.885938
$290$	0	0
$291$	−0.210344	−0.0123306
$292$	0	0
$293$	29.4990	1.72335	0.861675	−	0.507461i	$-0.169416\pi$
$293$	29.4990	1.72335	0.861675	+	0.507461i	$0.169416\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	4.91561	0.285232
$298$	0	0
$299$	−2.83020	−0.163674
$300$	0	0
$301$	10.1823	0.586896
$302$	0	0
$303$	−38.2068	−2.19493
$304$	0	0
$305$	0	0
$306$	0	0
$307$	13.9131	0.794064	0.397032	−	0.917805i	$-0.370040\pi$
$307$	13.9131	0.794064	0.397032	+	0.917805i	$0.370040\pi$
$308$	0	0
$309$	2.83706	0.161394
$310$	0	0
$311$	−25.9636	−1.47226	−0.736130	−	0.676840i	$-0.763349\pi$
$311$	−25.9636	−1.47226	−0.736130	+	0.676840i	$0.763349\pi$
$312$	0	0
$313$	−9.47389	−0.535496	−0.267748	−	0.963489i	$-0.586279\pi$
$313$	−9.47389	−0.535496	−0.267748	+	0.963489i	$0.586279\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−24.5074	−1.37647	−0.688237	−	0.725486i	$-0.741615\pi$
$317$	−24.5074	−1.37647	−0.688237	+	0.725486i	$0.741615\pi$
$318$	0	0
$319$	0.185768	0.0104010
$320$	0	0
$321$	25.2228	1.40780
$322$	0	0
$323$	−5.45005	−0.303249
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−26.6291	−1.47259
$328$	0	0
$329$	14.2466	0.785441
$330$	0	0
$331$	9.91035	0.544722	0.272361	−	0.962195i	$-0.412196\pi$
$331$	9.91035	0.544722	0.272361	+	0.962195i	$0.412196\pi$
$332$	0	0
$333$	−17.2079	−0.942988
$334$	0	0
$335$	0	0
$336$	0	0
$337$	9.93566	0.541230	0.270615	−	0.962688i	$-0.412773\pi$
$337$	9.93566	0.541230	0.270615	+	0.962688i	$0.412773\pi$
$338$	0	0
$339$	−24.3141	−1.32056
$340$	0	0
$341$	21.0657	1.14077
$342$	0	0
$343$	0.930796	0.0502583
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−19.6351	−1.05407	−0.527033	−	0.849845i	$-0.676696\pi$
$347$	−19.6351	−1.05407	−0.527033	+	0.849845i	$0.676696\pi$
$348$	0	0
$349$	1.48432	0.0794538	0.0397269	−	0.999211i	$-0.487351\pi$
$349$	1.48432	0.0794538	0.0397269	+	0.999211i	$0.487351\pi$
$350$	0	0
$351$	−4.13662	−0.220796
$352$	0	0
$353$	−10.2788	−0.547084	−0.273542	−	0.961860i	$-0.588195\pi$
$353$	−10.2788	−0.547084	−0.273542	+	0.961860i	$0.588195\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−12.2608	−0.648911
$358$	0	0
$359$	10.3725	0.547440	0.273720	−	0.961809i	$-0.411746\pi$
$359$	10.3725	0.547440	0.273720	+	0.961809i	$0.411746\pi$
$360$	0	0
$361$	−3.68175	−0.193776
$362$	0	0
$363$	−7.57351	−0.397506
$364$	0	0
$365$	0	0
$366$	0	0
$367$	26.0743	1.36107	0.680534	−	0.732717i	$-0.261748\pi$
$367$	26.0743	1.36107	0.680534	+	0.732717i	$0.261748\pi$
$368$	0	0
$369$	−3.63616	−0.189291
$370$	0	0
$371$	−43.8453	−2.27633
$372$	0	0
$373$	−9.45593	−0.489609	−0.244805	−	0.969572i	$-0.578724\pi$
$373$	−9.45593	−0.489609	−0.244805	+	0.969572i	$0.578724\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−0.156329	−0.00805133
$378$	0	0
$379$	35.3258	1.81456	0.907282	−	0.420523i	$-0.138154\pi$
$379$	35.3258	1.81456	0.907282	+	0.420523i	$0.138154\pi$
$380$	0	0
$381$	8.27877	0.424134
$382$	0	0
$383$	29.8173	1.52360	0.761798	−	0.647815i	$-0.224317\pi$
$383$	29.8173	1.52360	0.761798	+	0.647815i	$0.224317\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	6.58695	0.334833
$388$	0	0
$389$	−21.3767	−1.08384	−0.541922	−	0.840429i	$-0.682303\pi$
$389$	−21.3767	−1.08384	−0.541922	+	0.840429i	$0.682303\pi$
$390$	0	0
$391$	1.24076	0.0627480
$392$	0	0
$393$	9.45519	0.476951
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−21.0201	−1.05497	−0.527483	−	0.849566i	$-0.676864\pi$
$397$	−21.0201	−1.05497	−0.527483	+	0.849566i	$0.676864\pi$
$398$	0	0
$399$	34.4610	1.72521
$400$	0	0
$401$	16.7820	0.838053	0.419026	−	0.907974i	$-0.362371\pi$
$401$	16.7820	0.838053	0.419026	+	0.907974i	$0.362371\pi$
$402$	0	0
$403$	−17.7274	−0.883062
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−26.6004	−1.31853
$408$	0	0
$409$	36.3607	1.79792	0.898962	−	0.438028i	$-0.144323\pi$
$409$	36.3607	1.79792	0.898962	+	0.438028i	$0.144323\pi$
$410$	0	0
$411$	−8.32263	−0.410525
$412$	0	0
$413$	2.60466	0.128167
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−23.2016	−1.13619
$418$	0	0
$419$	34.5389	1.68733	0.843667	−	0.536867i	$-0.180392\pi$
$419$	34.5389	1.68733	0.843667	+	0.536867i	$0.180392\pi$
$420$	0	0
$421$	9.73403	0.474408	0.237204	−	0.971460i	$-0.423769\pi$
$421$	9.73403	0.474408	0.237204	+	0.971460i	$0.423769\pi$
$422$	0	0
$423$	9.21619	0.448106
$424$	0	0
$425$	0	0
$426$	0	0
$427$	39.1047	1.89241
$428$	0	0
$429$	27.9671	1.35026
$430$	0	0
$431$	3.09549	0.149105	0.0745523	−	0.997217i	$-0.476247\pi$
$431$	3.09549	0.149105	0.0745523	+	0.997217i	$0.476247\pi$
$432$	0	0
$433$	18.2366	0.876394	0.438197	−	0.898879i	$-0.355617\pi$
$433$	18.2366	0.876394	0.438197	+	0.898879i	$0.355617\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−3.48736	−0.166823
$438$	0	0
$439$	26.5254	1.26599	0.632994	−	0.774157i	$-0.281826\pi$
$439$	26.5254	1.26599	0.632994	+	0.774157i	$0.281826\pi$
$440$	0	0
$441$	17.6942	0.842579
$442$	0	0
$443$	35.5267	1.68793	0.843963	−	0.536401i	$-0.180217\pi$
$443$	35.5267	1.68793	0.843963	+	0.536401i	$0.180217\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	45.5125	2.15267
$448$	0	0
$449$	16.1841	0.763776	0.381888	−	0.924209i	$-0.375274\pi$
$449$	16.1841	0.763776	0.381888	+	0.924209i	$0.375274\pi$
$450$	0	0
$451$	−5.62087	−0.264676
$452$	0	0
$453$	−42.9981	−2.02023
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−23.7205	−1.10960	−0.554799	−	0.831984i	$-0.687205\pi$
$457$	−23.7205	−1.10960	−0.554799	+	0.831984i	$0.687205\pi$
$458$	0	0
$459$	1.81350	0.0846468
$460$	0	0
$461$	−7.87297	−0.366681	−0.183340	−	0.983050i	$-0.558691\pi$
$461$	−7.87297	−0.366681	−0.183340	+	0.983050i	$0.558691\pi$
$462$	0	0
$463$	15.4230	0.716769	0.358384	−	0.933574i	$-0.383328\pi$
$463$	15.4230	0.716769	0.358384	+	0.933574i	$0.383328\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−17.0373	−0.788391	−0.394196	−	0.919027i	$-0.628977\pi$
$467$	−17.0373	−0.788391	−0.394196	+	0.919027i	$0.628977\pi$
$468$	0	0
$469$	57.8377	2.67070
$470$	0	0
$471$	23.6924	1.09169
$472$	0	0
$473$	10.1823	0.468181
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−28.3637	−1.29868
$478$	0	0
$479$	25.0971	1.14672	0.573359	−	0.819304i	$-0.305640\pi$
$479$	25.0971	1.14672	0.573359	+	0.819304i	$0.305640\pi$
$480$	0	0
$481$	22.3850	1.02067
$482$	0	0
$483$	−7.84542	−0.356979
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−13.6876	−0.620244	−0.310122	−	0.950697i	$-0.600370\pi$
$487$	−13.6876	−0.620244	−0.310122	+	0.950697i	$0.600370\pi$
$488$	0	0
$489$	43.3901	1.96217
$490$	0	0
$491$	2.58009	0.116438	0.0582188	−	0.998304i	$-0.481458\pi$
$491$	2.58009	0.116438	0.0582188	+	0.998304i	$0.481458\pi$
$492$	0	0
$493$	0.0685347	0.00308665
$494$	0	0
$495$	0	0
$496$	0	0
$497$	25.2801	1.13397
$498$	0	0
$499$	39.0539	1.74829	0.874146	−	0.485664i	$-0.161422\pi$
$499$	39.0539	1.74829	0.874146	+	0.485664i	$0.161422\pi$
$500$	0	0
$501$	−5.06570	−0.226319
$502$	0	0
$503$	11.9536	0.532986	0.266493	−	0.963837i	$-0.414135\pi$
$503$	11.9536	0.532986	0.266493	+	0.963837i	$0.414135\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	6.79067	0.301584
$508$	0	0
$509$	−4.87325	−0.216003	−0.108002	−	0.994151i	$-0.534445\pi$
$509$	−4.87325	−0.216003	−0.108002	+	0.994151i	$0.534445\pi$
$510$	0	0
$511$	−19.5092	−0.863035
$512$	0	0
$513$	−5.09713	−0.225044
$514$	0	0
$515$	0	0
$516$	0	0
$517$	14.2466	0.626565
$518$	0	0
$519$	−14.9162	−0.654748
$520$	0	0
$521$	−8.74163	−0.382978	−0.191489	−	0.981495i	$-0.561332\pi$
$521$	−8.74163	−0.382978	−0.191489	+	0.981495i	$0.561332\pi$
$522$	0	0
$523$	−21.0792	−0.921728	−0.460864	−	0.887471i	$-0.652460\pi$
$523$	−21.0792	−0.921728	−0.460864	+	0.887471i	$0.652460\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	7.77170	0.338540
$528$	0	0
$529$	−22.2061	−0.965481
$530$	0	0
$531$	1.68497	0.0731213
$532$	0	0
$533$	4.73011	0.204884
$534$	0	0
$535$	0	0
$536$	0	0
$537$	18.6934	0.806681
$538$	0	0
$539$	27.3521	1.17814
$540$	0	0
$541$	32.8060	1.41044	0.705220	−	0.708988i	$-0.250848\pi$
$541$	32.8060	1.41044	0.705220	+	0.708988i	$0.250848\pi$
$542$	0	0
$543$	25.3404	1.08746
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−5.47038	−0.233897	−0.116948	−	0.993138i	$-0.537311\pi$
$547$	−5.47038	−0.233897	−0.116948	+	0.993138i	$0.537311\pi$
$548$	0	0
$549$	25.2970	1.07965
$550$	0	0
$551$	−0.192628	−0.00820623
$552$	0	0
$553$	6.81910	0.289977
$554$	0	0
$555$	0	0
$556$	0	0
$557$	5.19840	0.220263	0.110132	−	0.993917i	$-0.464873\pi$
$557$	5.19840	0.220263	0.110132	+	0.993917i	$0.464873\pi$
$558$	0	0
$559$	−8.56865	−0.362415
$560$	0	0
$561$	−12.2608	−0.517652
$562$	0	0
$563$	−5.01523	−0.211367	−0.105683	−	0.994400i	$-0.533703\pi$
$563$	−5.01523	−0.211367	−0.105683	+	0.994400i	$0.533703\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−39.1154	−1.64269
$568$	0	0
$569$	−41.8432	−1.75416	−0.877078	−	0.480347i	$-0.840511\pi$
$569$	−41.8432	−1.75416	−0.877078	+	0.480347i	$0.840511\pi$
$570$	0	0
$571$	44.5683	1.86512	0.932562	−	0.361011i	$-0.117568\pi$
$571$	44.5683	1.86512	0.932562	+	0.361011i	$0.117568\pi$
$572$	0	0
$573$	−19.0953	−0.797719
$574$	0	0
$575$	0	0
$576$	0	0
$577$	25.1417	1.04666	0.523332	−	0.852129i	$-0.324689\pi$
$577$	25.1417	1.04666	0.523332	+	0.852129i	$0.324689\pi$
$578$	0	0
$579$	−40.9575	−1.70214
$580$	0	0
$581$	−37.6232	−1.56088
$582$	0	0
$583$	−43.8453	−1.81589
$584$	0	0
$585$	0	0
$586$	0	0
$587$	21.3556	0.881439	0.440719	−	0.897645i	$-0.354723\pi$
$587$	21.3556	0.881439	0.440719	+	0.897645i	$0.354723\pi$
$588$	0	0
$589$	−21.8436	−0.900051
$590$	0	0
$591$	11.0979	0.456505
$592$	0	0
$593$	33.4430	1.37334	0.686669	−	0.726970i	$-0.259072\pi$
$593$	33.4430	1.37334	0.686669	+	0.726970i	$0.259072\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	33.8995	1.38741
$598$	0	0
$599$	−26.3174	−1.07530	−0.537650	−	0.843168i	$-0.680688\pi$
$599$	−26.3174	−1.07530	−0.537650	+	0.843168i	$0.680688\pi$
$600$	0	0
$601$	20.9964	0.856462	0.428231	−	0.903669i	$-0.359137\pi$
$601$	20.9964	0.856462	0.428231	+	0.903669i	$0.359137\pi$
$602$	0	0
$603$	37.4154	1.52367
$604$	0	0
$605$	0	0
$606$	0	0
$607$	9.32822	0.378621	0.189310	−	0.981917i	$-0.439375\pi$
$607$	9.32822	0.378621	0.189310	+	0.981917i	$0.439375\pi$
$608$	0	0
$609$	−0.433350	−0.0175602
$610$	0	0
$611$	−11.9889	−0.485019
$612$	0	0
$613$	32.1036	1.29665	0.648327	−	0.761362i	$-0.275469\pi$
$613$	32.1036	1.29665	0.648327	+	0.761362i	$0.275469\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	31.9256	1.28528	0.642639	−	0.766169i	$-0.277840\pi$
$617$	31.9256	1.28528	0.642639	+	0.766169i	$0.277840\pi$
$618$	0	0
$619$	4.54407	0.182642	0.0913208	−	0.995822i	$-0.470891\pi$
$619$	4.54407	0.182642	0.0913208	+	0.995822i	$0.470891\pi$
$620$	0	0
$621$	1.16042	0.0465659
$622$	0	0
$623$	54.7685	2.19425
$624$	0	0
$625$	0	0
$626$	0	0
$627$	34.4610	1.37624
$628$	0	0
$629$	−9.81360	−0.391294
$630$	0	0
$631$	26.9162	1.07152	0.535758	−	0.844371i	$-0.320026\pi$
$631$	26.9162	1.07152	0.535758	+	0.844371i	$0.320026\pi$
$632$	0	0
$633$	−32.9755	−1.31066
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−23.0175	−0.911987
$638$	0	0
$639$	16.3538	0.646947
$640$	0	0
$641$	35.7599	1.41243	0.706215	−	0.707998i	$-0.250401\pi$
$641$	35.7599	1.41243	0.706215	+	0.707998i	$0.250401\pi$
$642$	0	0
$643$	28.2255	1.11311	0.556553	−	0.830812i	$-0.312124\pi$
$643$	28.2255	1.11311	0.556553	+	0.830812i	$0.312124\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	1.27776	0.0502337	0.0251169	−	0.999685i	$-0.492004\pi$
$647$	1.27776	0.0502337	0.0251169	+	0.999685i	$0.492004\pi$
$648$	0	0
$649$	2.60466	0.102242
$650$	0	0
$651$	−49.1410	−1.92599
$652$	0	0
$653$	27.4108	1.07267	0.536334	−	0.844006i	$-0.319809\pi$
$653$	27.4108	1.07267	0.536334	+	0.844006i	$0.319809\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−12.6206	−0.492375
$658$	0	0
$659$	41.2271	1.60598	0.802991	−	0.595992i	$-0.203241\pi$
$659$	41.2271	1.60598	0.802991	+	0.595992i	$0.203241\pi$
$660$	0	0
$661$	−26.5326	−1.03200	−0.515999	−	0.856589i	$-0.672579\pi$
$661$	−26.5326	−1.03200	−0.515999	+	0.856589i	$0.672579\pi$
$662$	0	0
$663$	10.3178	0.400710
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0.0438538	0.00169803
$668$	0	0
$669$	−10.7592	−0.415976
$670$	0	0
$671$	39.1047	1.50962
$672$	0	0
$673$	7.93731	0.305961	0.152980	−	0.988229i	$-0.451113\pi$
$673$	7.93731	0.305961	0.152980	+	0.988229i	$0.451113\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−29.5008	−1.13381	−0.566905	−	0.823783i	$-0.691859\pi$
$677$	−29.5008	−1.13381	−0.566905	+	0.823783i	$0.691859\pi$
$678$	0	0
$679$	0.340344	0.0130612
$680$	0	0
$681$	14.6645	0.561946
$682$	0	0
$683$	37.7488	1.44442	0.722208	−	0.691676i	$-0.243127\pi$
$683$	37.7488	1.44442	0.722208	+	0.691676i	$0.243127\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−41.3759	−1.57859
$688$	0	0
$689$	36.8970	1.40566
$690$	0	0
$691$	−17.6564	−0.671681	−0.335840	−	0.941919i	$-0.609020\pi$
$691$	−17.6564	−0.671681	−0.335840	+	0.941919i	$0.609020\pi$
$692$	0	0
$693$	34.7862	1.32142
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−2.07369	−0.0785465
$698$	0	0
$699$	5.98131	0.226234
$700$	0	0
$701$	−16.8372	−0.635931	−0.317965	−	0.948102i	$-0.603000\pi$
$701$	−16.8372	−0.635931	−0.317965	+	0.948102i	$0.603000\pi$
$702$	0	0
$703$	27.5827	1.04030
$704$	0	0
$705$	0	0
$706$	0	0
$707$	61.8200	2.32498
$708$	0	0
$709$	−28.6292	−1.07519	−0.537596	−	0.843203i	$-0.680667\pi$
$709$	−28.6292	−1.07519	−0.537596	+	0.843203i	$0.680667\pi$
$710$	0	0
$711$	4.41130	0.165437
$712$	0	0
$713$	4.97294	0.186238
$714$	0	0
$715$	0	0
$716$	0	0
$717$	22.0252	0.822545
$718$	0	0
$719$	8.24734	0.307574	0.153787	−	0.988104i	$-0.450853\pi$
$719$	8.24734	0.307574	0.153787	+	0.988104i	$0.450853\pi$
$720$	0	0
$721$	−4.59045	−0.170957
$722$	0	0
$723$	68.6636	2.55363
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−2.11383	−0.0783977	−0.0391988	−	0.999231i	$-0.512481\pi$
$727$	−2.11383	−0.0783977	−0.0391988	+	0.999231i	$0.512481\pi$
$728$	0	0
$729$	−16.1899	−0.599626
$730$	0	0
$731$	3.75651	0.138939
$732$	0	0
$733$	−45.1552	−1.66785	−0.833923	−	0.551881i	$-0.813910\pi$
$733$	−45.1552	−1.66785	−0.833923	+	0.551881i	$0.813910\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	57.8377	2.13048
$738$	0	0
$739$	−29.9719	−1.10253	−0.551267	−	0.834329i	$-0.685856\pi$
$739$	−29.9719	−1.10253	−0.551267	+	0.834329i	$0.685856\pi$
$740$	0	0
$741$	−28.9999	−1.06534
$742$	0	0
$743$	17.6562	0.647741	0.323871	−	0.946101i	$-0.395016\pi$
$743$	17.6562	0.647741	0.323871	+	0.946101i	$0.395016\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−24.3386	−0.890504
$748$	0	0
$749$	−40.8113	−1.49121
$750$	0	0
$751$	−30.1342	−1.09961	−0.549806	−	0.835293i	$-0.685298\pi$
$751$	−30.1342	−1.09961	−0.549806	+	0.835293i	$0.685298\pi$
$752$	0	0
$753$	−14.0142	−0.510706
$754$	0	0
$755$	0	0
$756$	0	0
$757$	4.90706	0.178350	0.0891750	−	0.996016i	$-0.471577\pi$
$757$	4.90706	0.178350	0.0891750	+	0.996016i	$0.471577\pi$
$758$	0	0
$759$	−7.84542	−0.284771
$760$	0	0
$761$	13.2835	0.481528	0.240764	−	0.970584i	$-0.422602\pi$
$761$	13.2835	0.481528	0.240764	+	0.970584i	$0.422602\pi$
$762$	0	0
$763$	43.0868	1.55985
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−2.19189	−0.0791447
$768$	0	0
$769$	−28.5977	−1.03126	−0.515629	−	0.856812i	$-0.672442\pi$
$769$	−28.5977	−1.03126	−0.515629	+	0.856812i	$0.672442\pi$
$770$	0	0
$771$	−31.7920	−1.14496
$772$	0	0
$773$	45.0625	1.62079	0.810393	−	0.585886i	$-0.199253\pi$
$773$	45.0625	1.62079	0.810393	+	0.585886i	$0.199253\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	62.0521	2.22611
$778$	0	0
$779$	5.82844	0.208826
$780$	0	0
$781$	25.2801	0.904594
$782$	0	0
$783$	0.0640968	0.00229063
$784$	0	0
$785$	0	0
$786$	0	0
$787$	25.7669	0.918490	0.459245	−	0.888310i	$-0.348120\pi$
$787$	25.7669	0.918490	0.459245	+	0.888310i	$0.348120\pi$
$788$	0	0
$789$	21.3749	0.760965
$790$	0	0
$791$	39.3410	1.39880
$792$	0	0
$793$	−32.9077	−1.16859
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−50.4446	−1.78684	−0.893420	−	0.449222i	$-0.851701\pi$
$797$	−50.4446	−1.78684	−0.893420	+	0.449222i	$0.851701\pi$
$798$	0	0
$799$	5.25595	0.185942
$800$	0	0
$801$	35.4299	1.25186
$802$	0	0
$803$	−19.5092	−0.688464
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−65.5564	−2.30769
$808$	0	0
$809$	−3.61706	−0.127169	−0.0635844	−	0.997976i	$-0.520253\pi$
$809$	−3.61706	−0.127169	−0.0635844	+	0.997976i	$0.520253\pi$
$810$	0	0
$811$	−50.4347	−1.77100	−0.885502	−	0.464636i	$-0.846185\pi$
$811$	−50.4347	−1.77100	−0.885502	+	0.464636i	$0.846185\pi$
$812$	0	0
$813$	−21.2248	−0.744385
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−10.5583	−0.369388
$818$	0	0
$819$	−29.2735	−1.02290
$820$	0	0
$821$	−11.2846	−0.393836	−0.196918	−	0.980420i	$-0.563093\pi$
$821$	−11.2846	−0.393836	−0.196918	+	0.980420i	$0.563093\pi$
$822$	0	0
$823$	1.07680	0.0375348	0.0187674	−	0.999824i	$-0.494026\pi$
$823$	1.07680	0.0375348	0.0187674	+	0.999824i	$0.494026\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−29.3800	−1.02164	−0.510821	−	0.859687i	$-0.670659\pi$
$827$	−29.3800	−1.02164	−0.510821	+	0.859687i	$0.670659\pi$
$828$	0	0
$829$	21.0798	0.732132	0.366066	−	0.930589i	$-0.380704\pi$
$829$	21.0798	0.732132	0.366066	+	0.930589i	$0.380704\pi$
$830$	0	0
$831$	43.8301	1.52045
$832$	0	0
$833$	10.0909	0.349629
$834$	0	0
$835$	0	0
$836$	0	0
$837$	7.26845	0.251234
$838$	0	0
$839$	3.41379	0.117857	0.0589285	−	0.998262i	$-0.481232\pi$
$839$	3.41379	0.117857	0.0589285	+	0.998262i	$0.481232\pi$
$840$	0	0
$841$	−28.9976	−0.999916
$842$	0	0
$843$	−18.3857	−0.633239
$844$	0	0
$845$	0	0
$846$	0	0
$847$	12.2542	0.421059
$848$	0	0
$849$	42.6889	1.46508
$850$	0	0
$851$	−6.27951	−0.215259
$852$	0	0
$853$	40.5963	1.38999	0.694995	−	0.719014i	$-0.255406\pi$
$853$	40.5963	1.38999	0.694995	+	0.719014i	$0.255406\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	43.0463	1.47043	0.735216	−	0.677833i	$-0.237081\pi$
$857$	43.0463	1.47043	0.735216	+	0.677833i	$0.237081\pi$
$858$	0	0
$859$	−7.63508	−0.260506	−0.130253	−	0.991481i	$-0.541579\pi$
$859$	−7.63508	−0.260506	−0.130253	+	0.991481i	$0.541579\pi$
$860$	0	0
$861$	13.1121	0.446858
$862$	0	0
$863$	6.47451	0.220395	0.110197	−	0.993910i	$-0.464852\pi$
$863$	6.47451	0.220395	0.110197	+	0.993910i	$0.464852\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	35.1334	1.19319
$868$	0	0
$869$	6.81910	0.231322
$870$	0	0
$871$	−48.6720	−1.64919
$872$	0	0
$873$	0.220170	0.00745161
$874$	0	0
$875$	0	0
$876$	0	0
$877$	7.87108	0.265788	0.132894	−	0.991130i	$-0.457573\pi$
$877$	7.87108	0.265788	0.132894	+	0.991130i	$0.457573\pi$
$878$	0	0
$879$	−68.8137	−2.32103
$880$	0	0
$881$	19.2652	0.649061	0.324530	−	0.945875i	$-0.394794\pi$
$881$	19.2652	0.649061	0.324530	+	0.945875i	$0.394794\pi$
$882$	0	0
$883$	6.27395	0.211135	0.105568	−	0.994412i	$-0.466334\pi$
$883$	6.27395	0.211135	0.105568	+	0.994412i	$0.466334\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	23.2728	0.781424	0.390712	−	0.920513i	$-0.372229\pi$
$887$	23.2728	0.781424	0.390712	+	0.920513i	$0.372229\pi$
$888$	0	0
$889$	−13.3953	−0.449265
$890$	0	0
$891$	−39.1154	−1.31042
$892$	0	0
$893$	−14.7727	−0.494350
$894$	0	0
$895$	0	0
$896$	0	0
$897$	6.60214	0.220439
$898$	0	0
$899$	0.274685	0.00916126
$900$	0	0
$901$	−16.1757	−0.538890
$902$	0	0
$903$	−23.7527	−0.790439
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−17.6544	−0.586206	−0.293103	−	0.956081i	$-0.594688\pi$
$907$	−17.6544	−0.586206	−0.293103	+	0.956081i	$0.594688\pi$
$908$	0	0
$909$	39.9916	1.32644
$910$	0	0
$911$	34.9519	1.15801	0.579003	−	0.815325i	$-0.303442\pi$
$911$	34.9519	1.15801	0.579003	+	0.815325i	$0.303442\pi$
$912$	0	0
$913$	−37.6232	−1.24515
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−15.2988	−0.505212
$918$	0	0
$919$	−36.3226	−1.19817	−0.599086	−	0.800684i	$-0.704469\pi$
$919$	−36.3226	−1.19817	−0.599086	+	0.800684i	$0.704469\pi$
$920$	0	0
$921$	−32.4558	−1.06946
$922$	0	0
$923$	−21.2739	−0.700239
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−2.96958	−0.0975339
$928$	0	0
$929$	19.0543	0.625152	0.312576	−	0.949893i	$-0.398808\pi$
$929$	19.0543	0.625152	0.312576	+	0.949893i	$0.398808\pi$
$930$	0	0
$931$	−28.3621	−0.929532
$932$	0	0
$933$	60.5665	1.98286
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−59.5279	−1.94469	−0.972345	−	0.233548i	$-0.924966\pi$
$937$	−59.5279	−1.94469	−0.972345	+	0.233548i	$0.924966\pi$
$938$	0	0
$939$	22.1002	0.721213
$940$	0	0
$941$	15.1539	0.494005	0.247002	−	0.969015i	$-0.420554\pi$
$941$	15.1539	0.494005	0.247002	+	0.969015i	$0.420554\pi$
$942$	0	0
$943$	−1.32691	−0.0432101
$944$	0	0
$945$	0	0
$946$	0	0
$947$	2.52089	0.0819180	0.0409590	−	0.999161i	$-0.486959\pi$
$947$	2.52089	0.0819180	0.0409590	+	0.999161i	$0.486959\pi$
$948$	0	0
$949$	16.4175	0.532934
$950$	0	0
$951$	57.1697	1.85385
$952$	0	0
$953$	−31.3684	−1.01612	−0.508061	−	0.861321i	$-0.669637\pi$
$953$	−31.3684	−1.01612	−0.508061	+	0.861321i	$0.669637\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−0.433350	−0.0140082
$958$	0	0
$959$	13.4663	0.434849
$960$	0	0
$961$	0.148734	0.00479787
$962$	0	0
$963$	−26.4010	−0.850761
$964$	0	0
$965$	0	0
$966$	0	0
$967$	49.1061	1.57915	0.789573	−	0.613656i	$-0.210302\pi$
$967$	49.1061	1.57915	0.789573	+	0.613656i	$0.210302\pi$
$968$	0	0
$969$	12.7136	0.408419
$970$	0	0
$971$	−24.9840	−0.801776	−0.400888	−	0.916127i	$-0.631298\pi$
$971$	−24.9840	−0.801776	−0.400888	+	0.916127i	$0.631298\pi$
$972$	0	0
$973$	37.5409	1.20351
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−15.5861	−0.498645	−0.249323	−	0.968421i	$-0.580208\pi$
$977$	−15.5861	−0.498645	−0.249323	+	0.968421i	$0.580208\pi$
$978$	0	0
$979$	54.7685	1.75041
$980$	0	0
$981$	27.8730	0.889917
$982$	0	0
$983$	−13.5589	−0.432462	−0.216231	−	0.976342i	$-0.569376\pi$
$983$	−13.5589	−0.432462	−0.216231	+	0.976342i	$0.569376\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−33.2338	−1.05784
$988$	0	0
$989$	2.40371	0.0764334
$990$	0	0
$991$	−31.0401	−0.986023	−0.493011	−	0.870023i	$-0.664104\pi$
$991$	−31.0401	−0.986023	−0.493011	+	0.870023i	$0.664104\pi$
$992$	0	0
$993$	−23.1184	−0.733639
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−1.54918	−0.0490630	−0.0245315	−	0.999699i	$-0.507809\pi$
$997$	−1.54918	−0.0490630	−0.0245315	+	0.999699i	$0.507809\pi$
$998$	0	0
$999$	−9.17813	−0.290383

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	10000.2.a.x.1.1		4
4.3	odd	2		1250.2.a.f.1.4		4
5.4	even	2		10000.2.a.t.1.4		4
20.3	even	4		1250.2.b.e.1249.8		8
20.7	even	4		1250.2.b.e.1249.1		8
20.19	odd	2		1250.2.a.l.1.1		4
25.4	even	10		400.2.u.d.241.2		8
25.19	even	10		400.2.u.d.161.2		8
100.3	even	20		250.2.e.c.49.3		16
100.19	odd	10		50.2.d.b.11.1	✓	8
100.31	odd	10		250.2.d.d.51.2		8
100.47	even	20		250.2.e.c.49.2		16
100.67	even	20		250.2.e.c.199.3		16
100.71	odd	10		250.2.d.d.201.2		8
100.79	odd	10		50.2.d.b.41.1	yes	8
100.83	even	20		250.2.e.c.199.2		16
300.119	even	10		450.2.h.e.361.2		8
300.179	even	10		450.2.h.e.91.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
50.2.d.b.11.1	✓	8	100.19	odd	10
50.2.d.b.41.1	yes	8	100.79	odd	10
250.2.d.d.51.2		8	100.31	odd	10
250.2.d.d.201.2		8	100.71	odd	10
250.2.e.c.49.2		16	100.47	even	20
250.2.e.c.49.3		16	100.3	even	20
250.2.e.c.199.2		16	100.83	even	20
250.2.e.c.199.3		16	100.67	even	20
400.2.u.d.161.2		8	25.19	even	10
400.2.u.d.241.2		8	25.4	even	10
450.2.h.e.91.2		8	300.179	even	10
450.2.h.e.361.2		8	300.119	even	10
1250.2.a.f.1.4		4	4.3	odd	2
1250.2.a.l.1.1		4	20.19	odd	2
1250.2.b.e.1249.1		8	20.7	even	4
1250.2.b.e.1249.8		8	20.3	even	4
10000.2.a.t.1.4		4	5.4	even	2
10000.2.a.x.1.1		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)

\(f(q)\)	\(=\)	\(q-2.33275 q^{3} +3.77447 q^{7} +2.44172 q^{9} +O(q^{10})\)	\(q-2.33275 q^{3} +3.77447 q^{7} +2.44172 q^{9} +3.77447 q^{11} -3.17632 q^{13} +1.39250 q^{17} -3.91385 q^{19} -8.80489 q^{21} +0.891031 q^{23} +1.30233 q^{27} +0.0492169 q^{29} +5.58111 q^{31} -8.80489 q^{33} -7.04746 q^{37} +7.40955 q^{39} -1.48918 q^{41} +2.69767 q^{43} +3.77447 q^{47} +7.24660 q^{49} -3.24836 q^{51} -11.6163 q^{53} +9.13004 q^{57} +0.690074 q^{59} +10.3603 q^{61} +9.21619 q^{63} +15.3234 q^{67} -2.07855 q^{69} +6.69767 q^{71} -5.16872 q^{73} +14.2466 q^{77} +1.80664 q^{79} -10.3632 q^{81} -9.96783 q^{83} -0.114811 q^{87} +14.5103 q^{89} -11.9889 q^{91} -13.0193 q^{93} +0.0901699 q^{97} +9.21619 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{3} + 2 q^{7} + 7 q^{9}+O(q^{10}) \) 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

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Representations

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Database

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