LMFDB - Embedded newform 10000.2.a.i.1.2

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:	$2$
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 200)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
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)

$f(q)$	$=$	$q+2.23607 q^{3} +0.381966 q^{7} +2.00000 q^{9} +O(q^{10})$	$q+2.23607 q^{3} +0.381966 q^{7} +2.00000 q^{9} +3.23607 q^{11} -5.61803 q^{13} -0.763932 q^{17} -5.38197 q^{19} +0.854102 q^{21} -3.47214 q^{23} -2.23607 q^{27} +6.61803 q^{29} -8.70820 q^{31} +7.23607 q^{33} +3.76393 q^{37} -12.5623 q^{39} -7.70820 q^{41} +6.61803 q^{43} +6.85410 q^{47} -6.85410 q^{49} -1.70820 q^{51} +9.94427 q^{53} -12.0344 q^{57} -4.61803 q^{59} +9.18034 q^{61} +0.763932 q^{63} +0.291796 q^{67} -7.76393 q^{69} +6.38197 q^{71} -11.0000 q^{73} +1.23607 q^{77} -10.3262 q^{79} -11.0000 q^{81} -13.4721 q^{83} +14.7984 q^{87} +4.00000 q^{89} -2.14590 q^{91} -19.4721 q^{93} -18.2705 q^{97} +6.47214 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 3 q^{7} + 4 q^{9}+O(q^{10}) $ 2 * q + 3 * q^7 + 4 * q^9	$ 2 q + 3 q^{7} + 4 q^{9} + 2 q^{11} - 9 q^{13} - 6 q^{17} - 13 q^{19} - 5 q^{21} + 2 q^{23} + 11 q^{29} - 4 q^{31} + 10 q^{33} + 12 q^{37} - 5 q^{39} - 2 q^{41} + 11 q^{43} + 7 q^{47} - 7 q^{49} + 10 q^{51} + 2 q^{53} + 5 q^{57} - 7 q^{59} - 4 q^{61} + 6 q^{63} + 14 q^{67} - 20 q^{69} + 15 q^{71} - 22 q^{73} - 2 q^{77} - 5 q^{79} - 22 q^{81} - 18 q^{83} + 5 q^{87} + 8 q^{89} - 11 q^{91} - 30 q^{93} - 3 q^{97} + 4 q^{99}+O(q^{100}) $ 2 * q + 3 * q^7 + 4 * q^9 + 2 * q^11 - 9 * q^13 - 6 * q^17 - 13 * q^19 - 5 * q^21 + 2 * q^23 + 11 * q^29 - 4 * q^31 + 10 * q^33 + 12 * q^37 - 5 * q^39 - 2 * q^41 + 11 * q^43 + 7 * q^47 - 7 * q^49 + 10 * q^51 + 2 * q^53 + 5 * q^57 - 7 * q^59 - 4 * q^61 + 6 * q^63 + 14 * q^67 - 20 * q^69 + 15 * q^71 - 22 * q^73 - 2 * q^77 - 5 * q^79 - 22 * q^81 - 18 * q^83 + 5 * q^87 + 8 * q^89 - 11 * q^91 - 30 * q^93 - 3 * q^97 + 4 * 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
$3$	2.23607	1.29099	0.645497	−	0.763763i	$-0.276650\pi$
$3$	2.23607	1.29099	0.645497	+	0.763763i	$0.276650\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0.381966	0.144370	0.0721848	−	0.997391i	$-0.477003\pi$
$7$	0.381966	0.144370	0.0721848	+	0.997391i	$0.477003\pi$
$8$	0	0
$9$	2.00000	0.666667
$10$	0	0
$11$	3.23607	0.975711	0.487856	−	0.872924i	$-0.337779\pi$
$11$	3.23607	0.975711	0.487856	+	0.872924i	$0.337779\pi$
$12$	0	0
$13$	−5.61803	−1.55816	−0.779081	−	0.626923i	$-0.784314\pi$
$13$	−5.61803	−1.55816	−0.779081	+	0.626923i	$0.784314\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−0.763932	−0.185281	−0.0926404	−	0.995700i	$-0.529531\pi$
$17$	−0.763932	−0.185281	−0.0926404	+	0.995700i	$0.529531\pi$
$18$	0	0
$19$	−5.38197	−1.23471	−0.617354	−	0.786686i	$-0.711795\pi$
$19$	−5.38197	−1.23471	−0.617354	+	0.786686i	$0.711795\pi$
$20$	0	0
$21$	0.854102	0.186380
$22$	0	0
$23$	−3.47214	−0.723990	−0.361995	−	0.932180i	$-0.617904\pi$
$23$	−3.47214	−0.723990	−0.361995	+	0.932180i	$0.617904\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−2.23607	−0.430331
$28$	0	0
$29$	6.61803	1.22894	0.614469	−	0.788941i	$-0.289370\pi$
$29$	6.61803	1.22894	0.614469	+	0.788941i	$0.289370\pi$
$30$	0	0
$31$	−8.70820	−1.56404	−0.782020	−	0.623254i	$-0.785810\pi$
$31$	−8.70820	−1.56404	−0.782020	+	0.623254i	$0.785810\pi$
$32$	0	0
$33$	7.23607	1.25964
$34$	0	0
$35$	0	0
$36$	0	0
$37$	3.76393	0.618787	0.309393	−	0.950934i	$-0.399874\pi$
$37$	3.76393	0.618787	0.309393	+	0.950934i	$0.399874\pi$
$38$	0	0
$39$	−12.5623	−2.01158
$40$	0	0
$41$	−7.70820	−1.20382	−0.601910	−	0.798564i	$-0.705593\pi$
$41$	−7.70820	−1.20382	−0.601910	+	0.798564i	$0.705593\pi$
$42$	0	0
$43$	6.61803	1.00924	0.504620	−	0.863341i	$-0.331632\pi$
$43$	6.61803	1.00924	0.504620	+	0.863341i	$0.331632\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	6.85410	0.999774	0.499887	−	0.866091i	$-0.333375\pi$
$47$	6.85410	0.999774	0.499887	+	0.866091i	$0.333375\pi$
$48$	0	0
$49$	−6.85410	−0.979157
$50$	0	0
$51$	−1.70820	−0.239196
$52$	0	0
$53$	9.94427	1.36595	0.682975	−	0.730441i	$-0.260686\pi$
$53$	9.94427	1.36595	0.682975	+	0.730441i	$0.260686\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−12.0344	−1.59400
$58$	0	0
$59$	−4.61803	−0.601217	−0.300608	−	0.953748i	$-0.597190\pi$
$59$	−4.61803	−0.601217	−0.300608	+	0.953748i	$0.597190\pi$
$60$	0	0
$61$	9.18034	1.17542	0.587711	−	0.809071i	$-0.300029\pi$
$61$	9.18034	1.17542	0.587711	+	0.809071i	$0.300029\pi$
$62$	0	0
$63$	0.763932	0.0962464
$64$	0	0
$65$	0	0
$66$	0	0
$67$	0.291796	0.0356486	0.0178243	−	0.999841i	$-0.494326\pi$
$67$	0.291796	0.0356486	0.0178243	+	0.999841i	$0.494326\pi$
$68$	0	0
$69$	−7.76393	−0.934668
$70$	0	0
$71$	6.38197	0.757400	0.378700	−	0.925519i	$-0.376371\pi$
$71$	6.38197	0.757400	0.378700	+	0.925519i	$0.376371\pi$
$72$	0	0
$73$	−11.0000	−1.28745	−0.643726	−	0.765256i	$-0.722612\pi$
$73$	−11.0000	−1.28745	−0.643726	+	0.765256i	$0.722612\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1.23607	0.140863
$78$	0	0
$79$	−10.3262	−1.16179	−0.580896	−	0.813978i	$-0.697298\pi$
$79$	−10.3262	−1.16179	−0.580896	+	0.813978i	$0.697298\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	0	0
$83$	−13.4721	−1.47876	−0.739380	−	0.673289i	$-0.764881\pi$
$83$	−13.4721	−1.47876	−0.739380	+	0.673289i	$0.764881\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	14.7984	1.58655
$88$	0	0
$89$	4.00000	0.423999	0.212000	−	0.977270i	$-0.432002\pi$
$89$	4.00000	0.423999	0.212000	+	0.977270i	$0.432002\pi$
$90$	0	0
$91$	−2.14590	−0.224951
$92$	0	0
$93$	−19.4721	−2.01917
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−18.2705	−1.85509	−0.927545	−	0.373712i	$-0.878085\pi$
$97$	−18.2705	−1.85509	−0.927545	+	0.373712i	$0.878085\pi$
$98$	0	0
$99$	6.47214	0.650474
$100$	0	0
$101$	−0.0557281	−0.00554515	−0.00277258	−	0.999996i	$-0.500883\pi$
$101$	−0.0557281	−0.00554515	−0.00277258	+	0.999996i	$0.500883\pi$
$102$	0	0
$103$	−9.85410	−0.970954	−0.485477	−	0.874250i	$-0.661354\pi$
$103$	−9.85410	−0.970954	−0.485477	+	0.874250i	$0.661354\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	3.29180	0.318230	0.159115	−	0.987260i	$-0.449136\pi$
$107$	3.29180	0.318230	0.159115	+	0.987260i	$0.449136\pi$
$108$	0	0
$109$	−14.9443	−1.43140	−0.715701	−	0.698407i	$-0.753893\pi$
$109$	−14.9443	−1.43140	−0.715701	+	0.698407i	$0.753893\pi$
$110$	0	0
$111$	8.41641	0.798850
$112$	0	0
$113$	−9.56231	−0.899546	−0.449773	−	0.893143i	$-0.648495\pi$
$113$	−9.56231	−0.899546	−0.449773	+	0.893143i	$0.648495\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−11.2361	−1.03877
$118$	0	0
$119$	−0.291796	−0.0267489
$120$	0	0
$121$	−0.527864	−0.0479876
$122$	0	0
$123$	−17.2361	−1.55412
$124$	0	0
$125$	0	0
$126$	0	0
$127$	16.4721	1.46167	0.730833	−	0.682556i	$-0.239132\pi$
$127$	16.4721	1.46167	0.730833	+	0.682556i	$0.239132\pi$
$128$	0	0
$129$	14.7984	1.30292
$130$	0	0
$131$	−10.0902	−0.881582	−0.440791	−	0.897610i	$-0.645302\pi$
$131$	−10.0902	−0.881582	−0.440791	+	0.897610i	$0.645302\pi$
$132$	0	0
$133$	−2.05573	−0.178254
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−5.00000	−0.427179	−0.213589	−	0.976924i	$-0.568515\pi$
$137$	−5.00000	−0.427179	−0.213589	+	0.976924i	$0.568515\pi$
$138$	0	0
$139$	−14.2361	−1.20749	−0.603744	−	0.797178i	$-0.706325\pi$
$139$	−14.2361	−1.20749	−0.603744	+	0.797178i	$0.706325\pi$
$140$	0	0
$141$	15.3262	1.29070
$142$	0	0
$143$	−18.1803	−1.52032
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−15.3262	−1.26409
$148$	0	0
$149$	8.41641	0.689499	0.344749	−	0.938695i	$-0.387964\pi$
$149$	8.41641	0.689499	0.344749	+	0.938695i	$0.387964\pi$
$150$	0	0
$151$	−2.61803	−0.213053	−0.106526	−	0.994310i	$-0.533973\pi$
$151$	−2.61803	−0.213053	−0.106526	+	0.994310i	$0.533973\pi$
$152$	0	0
$153$	−1.52786	−0.123520
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0.236068	0.0188403	0.00942014	−	0.999956i	$-0.497001\pi$
$157$	0.236068	0.0188403	0.00942014	+	0.999956i	$0.497001\pi$
$158$	0	0
$159$	22.2361	1.76343
$160$	0	0
$161$	−1.32624	−0.104522
$162$	0	0
$163$	−7.18034	−0.562408	−0.281204	−	0.959648i	$-0.590734\pi$
$163$	−7.18034	−0.562408	−0.281204	+	0.959648i	$0.590734\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	10.0344	0.776488	0.388244	−	0.921557i	$-0.373082\pi$
$167$	10.0344	0.776488	0.388244	+	0.921557i	$0.373082\pi$
$168$	0	0
$169$	18.5623	1.42787
$170$	0	0
$171$	−10.7639	−0.823138
$172$	0	0
$173$	−15.4721	−1.17632	−0.588162	−	0.808743i	$-0.700148\pi$
$173$	−15.4721	−1.17632	−0.588162	+	0.808743i	$0.700148\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−10.3262	−0.776168
$178$	0	0
$179$	−16.7082	−1.24883	−0.624415	−	0.781093i	$-0.714663\pi$
$179$	−16.7082	−1.24883	−0.624415	+	0.781093i	$0.714663\pi$
$180$	0	0
$181$	−21.7082	−1.61356	−0.806779	−	0.590853i	$-0.798791\pi$
$181$	−21.7082	−1.61356	−0.806779	+	0.590853i	$0.798791\pi$
$182$	0	0
$183$	20.5279	1.51746
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−2.47214	−0.180780
$188$	0	0
$189$	−0.854102	−0.0621268
$190$	0	0
$191$	13.7082	0.991891	0.495945	−	0.868354i	$-0.334822\pi$
$191$	13.7082	0.991891	0.495945	+	0.868354i	$0.334822\pi$
$192$	0	0
$193$	−5.70820	−0.410886	−0.205443	−	0.978669i	$-0.565863\pi$
$193$	−5.70820	−0.410886	−0.205443	+	0.978669i	$0.565863\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	19.2361	1.37051	0.685257	−	0.728302i	$-0.259690\pi$
$197$	19.2361	1.37051	0.685257	+	0.728302i	$0.259690\pi$
$198$	0	0
$199$	−11.6180	−0.823581	−0.411790	−	0.911279i	$-0.635096\pi$
$199$	−11.6180	−0.823581	−0.411790	+	0.911279i	$0.635096\pi$
$200$	0	0
$201$	0.652476	0.0460221
$202$	0	0
$203$	2.52786	0.177421
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−6.94427	−0.482660
$208$	0	0
$209$	−17.4164	−1.20472
$210$	0	0
$211$	−2.05573	−0.141522	−0.0707611	−	0.997493i	$-0.522543\pi$
$211$	−2.05573	−0.141522	−0.0707611	+	0.997493i	$0.522543\pi$
$212$	0	0
$213$	14.2705	0.977799
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−3.32624	−0.225800
$218$	0	0
$219$	−24.5967	−1.66209
$220$	0	0
$221$	4.29180	0.288697
$222$	0	0
$223$	−7.70820	−0.516180	−0.258090	−	0.966121i	$-0.583093\pi$
$223$	−7.70820	−0.516180	−0.258090	+	0.966121i	$0.583093\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	23.5967	1.56617	0.783086	−	0.621914i	$-0.213645\pi$
$227$	23.5967	1.56617	0.783086	+	0.621914i	$0.213645\pi$
$228$	0	0
$229$	−2.18034	−0.144081	−0.0720405	−	0.997402i	$-0.522951\pi$
$229$	−2.18034	−0.144081	−0.0720405	+	0.997402i	$0.522951\pi$
$230$	0	0
$231$	2.76393	0.181853
$232$	0	0
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−23.0902	−1.49987
$238$	0	0
$239$	17.7639	1.14905	0.574527	−	0.818486i	$-0.305186\pi$
$239$	17.7639	1.14905	0.574527	+	0.818486i	$0.305186\pi$
$240$	0	0
$241$	2.52786	0.162834	0.0814170	−	0.996680i	$-0.474055\pi$
$241$	2.52786	0.162834	0.0814170	+	0.996680i	$0.474055\pi$
$242$	0	0
$243$	−17.8885	−1.14755
$244$	0	0
$245$	0	0
$246$	0	0
$247$	30.2361	1.92387
$248$	0	0
$249$	−30.1246	−1.90907
$250$	0	0
$251$	25.9443	1.63759	0.818794	−	0.574087i	$-0.194643\pi$
$251$	25.9443	1.63759	0.818794	+	0.574087i	$0.194643\pi$
$252$	0	0
$253$	−11.2361	−0.706406
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1.38197	0.0862047	0.0431023	−	0.999071i	$-0.486276\pi$
$257$	1.38197	0.0862047	0.0431023	+	0.999071i	$0.486276\pi$
$258$	0	0
$259$	1.43769	0.0893340
$260$	0	0
$261$	13.2361	0.819292
$262$	0	0
$263$	−7.14590	−0.440635	−0.220317	−	0.975428i	$-0.570709\pi$
$263$	−7.14590	−0.440635	−0.220317	+	0.975428i	$0.570709\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	8.94427	0.547381
$268$	0	0
$269$	−27.7082	−1.68940	−0.844700	−	0.535241i	$-0.820221\pi$
$269$	−27.7082	−1.68940	−0.844700	+	0.535241i	$0.820221\pi$
$270$	0	0
$271$	4.94427	0.300343	0.150172	−	0.988660i	$-0.452017\pi$
$271$	4.94427	0.300343	0.150172	+	0.988660i	$0.452017\pi$
$272$	0	0
$273$	−4.79837	−0.290411
$274$	0	0
$275$	0	0
$276$	0	0
$277$	1.18034	0.0709198	0.0354599	−	0.999371i	$-0.488710\pi$
$277$	1.18034	0.0709198	0.0354599	+	0.999371i	$0.488710\pi$
$278$	0	0
$279$	−17.4164	−1.04269
$280$	0	0
$281$	15.8541	0.945776	0.472888	−	0.881122i	$-0.343211\pi$
$281$	15.8541	0.945776	0.472888	+	0.881122i	$0.343211\pi$
$282$	0	0
$283$	26.2705	1.56162	0.780810	−	0.624769i	$-0.214807\pi$
$283$	26.2705	1.56162	0.780810	+	0.624769i	$0.214807\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−2.94427	−0.173795
$288$	0	0
$289$	−16.4164	−0.965671
$290$	0	0
$291$	−40.8541	−2.39491
$292$	0	0
$293$	8.47214	0.494947	0.247474	−	0.968895i	$-0.420400\pi$
$293$	8.47214	0.494947	0.247474	+	0.968895i	$0.420400\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−7.23607	−0.419879
$298$	0	0
$299$	19.5066	1.12809
$300$	0	0
$301$	2.52786	0.145704
$302$	0	0
$303$	−0.124612	−0.00715876
$304$	0	0
$305$	0	0
$306$	0	0
$307$	15.2361	0.869568	0.434784	−	0.900535i	$-0.356825\pi$
$307$	15.2361	0.869568	0.434784	+	0.900535i	$0.356825\pi$
$308$	0	0
$309$	−22.0344	−1.25350
$310$	0	0
$311$	16.0902	0.912390	0.456195	−	0.889880i	$-0.349212\pi$
$311$	16.0902	0.912390	0.456195	+	0.889880i	$0.349212\pi$
$312$	0	0
$313$	7.70820	0.435693	0.217847	−	0.975983i	$-0.430097\pi$
$313$	7.70820	0.435693	0.217847	+	0.975983i	$0.430097\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	9.76393	0.548397	0.274199	−	0.961673i	$-0.411587\pi$
$317$	9.76393	0.548397	0.274199	+	0.961673i	$0.411587\pi$
$318$	0	0
$319$	21.4164	1.19909
$320$	0	0
$321$	7.36068	0.410833
$322$	0	0
$323$	4.11146	0.228768
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−33.4164	−1.84793
$328$	0	0
$329$	2.61803	0.144337
$330$	0	0
$331$	26.0689	1.43288	0.716438	−	0.697651i	$-0.245771\pi$
$331$	26.0689	1.43288	0.716438	+	0.697651i	$0.245771\pi$
$332$	0	0
$333$	7.52786	0.412524
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−2.09017	−0.113859	−0.0569294	−	0.998378i	$-0.518131\pi$
$337$	−2.09017	−0.113859	−0.0569294	+	0.998378i	$0.518131\pi$
$338$	0	0
$339$	−21.3820	−1.16131
$340$	0	0
$341$	−28.1803	−1.52605
$342$	0	0
$343$	−5.29180	−0.285730
$344$	0	0
$345$	0	0
$346$	0	0
$347$	0.145898	0.00783222	0.00391611	−	0.999992i	$-0.498753\pi$
$347$	0.145898	0.00783222	0.00391611	+	0.999992i	$0.498753\pi$
$348$	0	0
$349$	−15.1246	−0.809602	−0.404801	−	0.914405i	$-0.632659\pi$
$349$	−15.1246	−0.809602	−0.404801	+	0.914405i	$0.632659\pi$
$350$	0	0
$351$	12.5623	0.670526
$352$	0	0
$353$	9.43769	0.502318	0.251159	−	0.967946i	$-0.419188\pi$
$353$	9.43769	0.502318	0.251159	+	0.967946i	$0.419188\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−0.652476	−0.0345327
$358$	0	0
$359$	8.27051	0.436501	0.218250	−	0.975893i	$-0.429965\pi$
$359$	8.27051	0.436501	0.218250	+	0.975893i	$0.429965\pi$
$360$	0	0
$361$	9.96556	0.524503
$362$	0	0
$363$	−1.18034	−0.0619518
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−14.2705	−0.744915	−0.372457	−	0.928049i	$-0.621485\pi$
$367$	−14.2705	−0.744915	−0.372457	+	0.928049i	$0.621485\pi$
$368$	0	0
$369$	−15.4164	−0.802546
$370$	0	0
$371$	3.79837	0.197202
$372$	0	0
$373$	−4.90983	−0.254221	−0.127111	−	0.991889i	$-0.540570\pi$
$373$	−4.90983	−0.254221	−0.127111	+	0.991889i	$0.540570\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−37.1803	−1.91488
$378$	0	0
$379$	5.00000	0.256833	0.128416	−	0.991720i	$-0.459011\pi$
$379$	5.00000	0.256833	0.128416	+	0.991720i	$0.459011\pi$
$380$	0	0
$381$	36.8328	1.88700
$382$	0	0
$383$	18.2361	0.931820	0.465910	−	0.884832i	$-0.345727\pi$
$383$	18.2361	0.931820	0.465910	+	0.884832i	$0.345727\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	13.2361	0.672827
$388$	0	0
$389$	14.4164	0.730941	0.365470	−	0.930823i	$-0.380908\pi$
$389$	14.4164	0.730941	0.365470	+	0.930823i	$0.380908\pi$
$390$	0	0
$391$	2.65248	0.134141
$392$	0	0
$393$	−22.5623	−1.13812
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−8.90983	−0.447172	−0.223586	−	0.974684i	$-0.571776\pi$
$397$	−8.90983	−0.447172	−0.223586	+	0.974684i	$0.571776\pi$
$398$	0	0
$399$	−4.59675	−0.230125
$400$	0	0
$401$	12.5967	0.629052	0.314526	−	0.949249i	$-0.398155\pi$
$401$	12.5967	0.629052	0.314526	+	0.949249i	$0.398155\pi$
$402$	0	0
$403$	48.9230	2.43703
$404$	0	0
$405$	0	0
$406$	0	0
$407$	12.1803	0.603757
$408$	0	0
$409$	−29.4721	−1.45730	−0.728652	−	0.684884i	$-0.759853\pi$
$409$	−29.4721	−1.45730	−0.728652	+	0.684884i	$0.759853\pi$
$410$	0	0
$411$	−11.1803	−0.551485
$412$	0	0
$413$	−1.76393	−0.0867974
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−31.8328	−1.55886
$418$	0	0
$419$	14.1246	0.690032	0.345016	−	0.938597i	$-0.387873\pi$
$419$	14.1246	0.690032	0.345016	+	0.938597i	$0.387873\pi$
$420$	0	0
$421$	−12.9443	−0.630865	−0.315433	−	0.948948i	$-0.602150\pi$
$421$	−12.9443	−0.630865	−0.315433	+	0.948948i	$0.602150\pi$
$422$	0	0
$423$	13.7082	0.666516
$424$	0	0
$425$	0	0
$426$	0	0
$427$	3.50658	0.169695
$428$	0	0
$429$	−40.6525	−1.96272
$430$	0	0
$431$	−16.7082	−0.804806	−0.402403	−	0.915463i	$-0.631825\pi$
$431$	−16.7082	−0.804806	−0.402403	+	0.915463i	$0.631825\pi$
$432$	0	0
$433$	31.2705	1.50276	0.751382	−	0.659867i	$-0.229387\pi$
$433$	31.2705	1.50276	0.751382	+	0.659867i	$0.229387\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	18.6869	0.893917
$438$	0	0
$439$	7.79837	0.372196	0.186098	−	0.982531i	$-0.440416\pi$
$439$	7.79837	0.372196	0.186098	+	0.982531i	$0.440416\pi$
$440$	0	0
$441$	−13.7082	−0.652772
$442$	0	0
$443$	−39.6525	−1.88395	−0.941973	−	0.335689i	$-0.891031\pi$
$443$	−39.6525	−1.88395	−0.941973	+	0.335689i	$0.891031\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	18.8197	0.890139
$448$	0	0
$449$	−35.0902	−1.65601	−0.828004	−	0.560723i	$-0.810523\pi$
$449$	−35.0902	−1.65601	−0.828004	+	0.560723i	$0.810523\pi$
$450$	0	0
$451$	−24.9443	−1.17458
$452$	0	0
$453$	−5.85410	−0.275050
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−4.47214	−0.209198	−0.104599	−	0.994514i	$-0.533356\pi$
$457$	−4.47214	−0.209198	−0.104599	+	0.994514i	$0.533356\pi$
$458$	0	0
$459$	1.70820	0.0797321
$460$	0	0
$461$	−34.7082	−1.61652	−0.808261	−	0.588824i	$-0.799591\pi$
$461$	−34.7082	−1.61652	−0.808261	+	0.588824i	$0.799591\pi$
$462$	0	0
$463$	1.94427	0.0903580	0.0451790	−	0.998979i	$-0.485614\pi$
$463$	1.94427	0.0903580	0.0451790	+	0.998979i	$0.485614\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	26.2705	1.21565	0.607827	−	0.794069i	$-0.292041\pi$
$467$	26.2705	1.21565	0.607827	+	0.794069i	$0.292041\pi$
$468$	0	0
$469$	0.111456	0.00514657
$470$	0	0
$471$	0.527864	0.0243227
$472$	0	0
$473$	21.4164	0.984727
$474$	0	0
$475$	0	0
$476$	0	0
$477$	19.8885	0.910634
$478$	0	0
$479$	−24.6180	−1.12483	−0.562413	−	0.826856i	$-0.690127\pi$
$479$	−24.6180	−1.12483	−0.562413	+	0.826856i	$0.690127\pi$
$480$	0	0
$481$	−21.1459	−0.964170
$482$	0	0
$483$	−2.96556	−0.134938
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−13.2918	−0.602309	−0.301154	−	0.953575i	$-0.597372\pi$
$487$	−13.2918	−0.602309	−0.301154	+	0.953575i	$0.597372\pi$
$488$	0	0
$489$	−16.0557	−0.726065
$490$	0	0
$491$	−8.23607	−0.371689	−0.185844	−	0.982579i	$-0.559502\pi$
$491$	−8.23607	−0.371689	−0.185844	+	0.982579i	$0.559502\pi$
$492$	0	0
$493$	−5.05573	−0.227699
$494$	0	0
$495$	0	0
$496$	0	0
$497$	2.43769	0.109346
$498$	0	0
$499$	−9.43769	−0.422489	−0.211245	−	0.977433i	$-0.567752\pi$
$499$	−9.43769	−0.422489	−0.211245	+	0.977433i	$0.567752\pi$
$500$	0	0
$501$	22.4377	1.00244
$502$	0	0
$503$	19.8885	0.886786	0.443393	−	0.896327i	$-0.353775\pi$
$503$	19.8885	0.886786	0.443393	+	0.896327i	$0.353775\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	41.5066	1.84337
$508$	0	0
$509$	19.9098	0.882488	0.441244	−	0.897387i	$-0.354537\pi$
$509$	19.9098	0.882488	0.441244	+	0.897387i	$0.354537\pi$
$510$	0	0
$511$	−4.20163	−0.185869
$512$	0	0
$513$	12.0344	0.531334
$514$	0	0
$515$	0	0
$516$	0	0
$517$	22.1803	0.975490
$518$	0	0
$519$	−34.5967	−1.51863
$520$	0	0
$521$	−18.4164	−0.806837	−0.403419	−	0.915015i	$-0.632178\pi$
$521$	−18.4164	−0.806837	−0.403419	+	0.915015i	$0.632178\pi$
$522$	0	0
$523$	−20.6180	−0.901564	−0.450782	−	0.892634i	$-0.648855\pi$
$523$	−20.6180	−0.901564	−0.450782	+	0.892634i	$0.648855\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	6.65248	0.289786
$528$	0	0
$529$	−10.9443	−0.475838
$530$	0	0
$531$	−9.23607	−0.400811
$532$	0	0
$533$	43.3050	1.87575
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−37.3607	−1.61223
$538$	0	0
$539$	−22.1803	−0.955375
$540$	0	0
$541$	−22.6525	−0.973906	−0.486953	−	0.873428i	$-0.661892\pi$
$541$	−22.6525	−0.973906	−0.486953	+	0.873428i	$0.661892\pi$
$542$	0	0
$543$	−48.5410	−2.08309
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−35.3607	−1.51191	−0.755957	−	0.654622i	$-0.772828\pi$
$547$	−35.3607	−1.51191	−0.755957	+	0.654622i	$0.772828\pi$
$548$	0	0
$549$	18.3607	0.783615
$550$	0	0
$551$	−35.6180	−1.51738
$552$	0	0
$553$	−3.94427	−0.167728
$554$	0	0
$555$	0	0
$556$	0	0
$557$	30.1803	1.27878	0.639391	−	0.768882i	$-0.279187\pi$
$557$	30.1803	1.27878	0.639391	+	0.768882i	$0.279187\pi$
$558$	0	0
$559$	−37.1803	−1.57256
$560$	0	0
$561$	−5.52786	−0.233387
$562$	0	0
$563$	31.0344	1.30795	0.653973	−	0.756518i	$-0.273101\pi$
$563$	31.0344	1.30795	0.653973	+	0.756518i	$0.273101\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−4.20163	−0.176452
$568$	0	0
$569$	8.52786	0.357507	0.178753	−	0.983894i	$-0.442794\pi$
$569$	8.52786	0.357507	0.178753	+	0.983894i	$0.442794\pi$
$570$	0	0
$571$	−30.3050	−1.26822	−0.634111	−	0.773242i	$-0.718634\pi$
$571$	−30.3050	−1.26822	−0.634111	+	0.773242i	$0.718634\pi$
$572$	0	0
$573$	30.6525	1.28053
$574$	0	0
$575$	0	0
$576$	0	0
$577$	19.8885	0.827971	0.413985	−	0.910283i	$-0.364136\pi$
$577$	19.8885	0.827971	0.413985	+	0.910283i	$0.364136\pi$
$578$	0	0
$579$	−12.7639	−0.530451
$580$	0	0
$581$	−5.14590	−0.213488
$582$	0	0
$583$	32.1803	1.33277
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−22.5279	−0.929824	−0.464912	−	0.885357i	$-0.653914\pi$
$587$	−22.5279	−0.929824	−0.464912	+	0.885357i	$0.653914\pi$
$588$	0	0
$589$	46.8673	1.93113
$590$	0	0
$591$	43.0132	1.76932
$592$	0	0
$593$	0.562306	0.0230911	0.0115456	−	0.999933i	$-0.496325\pi$
$593$	0.562306	0.0230911	0.0115456	+	0.999933i	$0.496325\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−25.9787	−1.06324
$598$	0	0
$599$	−7.18034	−0.293381	−0.146690	−	0.989182i	$-0.546862\pi$
$599$	−7.18034	−0.293381	−0.146690	+	0.989182i	$0.546862\pi$
$600$	0	0
$601$	−42.5623	−1.73615	−0.868076	−	0.496431i	$-0.834644\pi$
$601$	−42.5623	−1.73615	−0.868076	+	0.496431i	$0.834644\pi$
$602$	0	0
$603$	0.583592	0.0237657
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−32.2705	−1.30982	−0.654910	−	0.755707i	$-0.727293\pi$
$607$	−32.2705	−1.30982	−0.654910	+	0.755707i	$0.727293\pi$
$608$	0	0
$609$	5.65248	0.229050
$610$	0	0
$611$	−38.5066	−1.55781
$612$	0	0
$613$	−19.0344	−0.768794	−0.384397	−	0.923168i	$-0.625591\pi$
$613$	−19.0344	−0.768794	−0.384397	+	0.923168i	$0.625591\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	18.5967	0.748677	0.374338	−	0.927292i	$-0.377870\pi$
$617$	18.5967	0.748677	0.374338	+	0.927292i	$0.377870\pi$
$618$	0	0
$619$	27.2918	1.09695	0.548475	−	0.836167i	$-0.315209\pi$
$619$	27.2918	1.09695	0.548475	+	0.836167i	$0.315209\pi$
$620$	0	0
$621$	7.76393	0.311556
$622$	0	0
$623$	1.52786	0.0612126
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−38.9443	−1.55528
$628$	0	0
$629$	−2.87539	−0.114649
$630$	0	0
$631$	45.8328	1.82458	0.912288	−	0.409550i	$-0.134314\pi$
$631$	45.8328	1.82458	0.912288	+	0.409550i	$0.134314\pi$
$632$	0	0
$633$	−4.59675	−0.182704
$634$	0	0
$635$	0	0
$636$	0	0
$637$	38.5066	1.52569
$638$	0	0
$639$	12.7639	0.504933
$640$	0	0
$641$	38.4508	1.51872	0.759359	−	0.650672i	$-0.225513\pi$
$641$	38.4508	1.51872	0.759359	+	0.650672i	$0.225513\pi$
$642$	0	0
$643$	36.9443	1.45694	0.728470	−	0.685078i	$-0.240232\pi$
$643$	36.9443	1.45694	0.728470	+	0.685078i	$0.240232\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	8.65248	0.340164	0.170082	−	0.985430i	$-0.445597\pi$
$647$	8.65248	0.340164	0.170082	+	0.985430i	$0.445597\pi$
$648$	0	0
$649$	−14.9443	−0.586614
$650$	0	0
$651$	−7.43769	−0.291506
$652$	0	0
$653$	7.50658	0.293755	0.146878	−	0.989155i	$-0.453078\pi$
$653$	7.50658	0.293755	0.146878	+	0.989155i	$0.453078\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−22.0000	−0.858302
$658$	0	0
$659$	−35.8885	−1.39802	−0.699010	−	0.715112i	$-0.746376\pi$
$659$	−35.8885	−1.39802	−0.699010	+	0.715112i	$0.746376\pi$
$660$	0	0
$661$	−14.4377	−0.561561	−0.280781	−	0.959772i	$-0.590593\pi$
$661$	−14.4377	−0.561561	−0.280781	+	0.959772i	$0.590593\pi$
$662$	0	0
$663$	9.59675	0.372707
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−22.9787	−0.889739
$668$	0	0
$669$	−17.2361	−0.666385
$670$	0	0
$671$	29.7082	1.14687
$672$	0	0
$673$	−7.12461	−0.274634	−0.137317	−	0.990527i	$-0.543848\pi$
$673$	−7.12461	−0.274634	−0.137317	+	0.990527i	$0.543848\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1.14590	0.0440404	0.0220202	−	0.999758i	$-0.492990\pi$
$677$	1.14590	0.0440404	0.0220202	+	0.999758i	$0.492990\pi$
$678$	0	0
$679$	−6.97871	−0.267818
$680$	0	0
$681$	52.7639	2.02192
$682$	0	0
$683$	−7.29180	−0.279013	−0.139506	−	0.990221i	$-0.544552\pi$
$683$	−7.29180	−0.279013	−0.139506	+	0.990221i	$0.544552\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−4.87539	−0.186008
$688$	0	0
$689$	−55.8673	−2.12837
$690$	0	0
$691$	40.1459	1.52722	0.763611	−	0.645677i	$-0.223425\pi$
$691$	40.1459	1.52722	0.763611	+	0.645677i	$0.223425\pi$
$692$	0	0
$693$	2.47214	0.0939087
$694$	0	0
$695$	0	0
$696$	0	0
$697$	5.88854	0.223045
$698$	0	0
$699$	−13.4164	−0.507455
$700$	0	0
$701$	−20.1803	−0.762201	−0.381100	−	0.924534i	$-0.624455\pi$
$701$	−20.1803	−0.762201	−0.381100	+	0.924534i	$0.624455\pi$
$702$	0	0
$703$	−20.2574	−0.764021
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−0.0212862	−0.000800551 0
$708$	0	0
$709$	12.5967	0.473081	0.236540	−	0.971622i	$-0.423986\pi$
$709$	12.5967	0.473081	0.236540	+	0.971622i	$0.423986\pi$
$710$	0	0
$711$	−20.6525	−0.774528
$712$	0	0
$713$	30.2361	1.13235
$714$	0	0
$715$	0	0
$716$	0	0
$717$	39.7214	1.48342
$718$	0	0
$719$	−24.3050	−0.906422	−0.453211	−	0.891403i	$-0.649722\pi$
$719$	−24.3050	−0.906422	−0.453211	+	0.891403i	$0.649722\pi$
$720$	0	0
$721$	−3.76393	−0.140176
$722$	0	0
$723$	5.65248	0.210218
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−4.32624	−0.160451	−0.0802256	−	0.996777i	$-0.525564\pi$
$727$	−4.32624	−0.160451	−0.0802256	+	0.996777i	$0.525564\pi$
$728$	0	0
$729$	−7.00000	−0.259259
$730$	0	0
$731$	−5.05573	−0.186993
$732$	0	0
$733$	−4.14590	−0.153132	−0.0765661	−	0.997065i	$-0.524396\pi$
$733$	−4.14590	−0.153132	−0.0765661	+	0.997065i	$0.524396\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0.944272	0.0347827
$738$	0	0
$739$	4.32624	0.159143	0.0795716	−	0.996829i	$-0.474645\pi$
$739$	4.32624	0.159143	0.0795716	+	0.996829i	$0.474645\pi$
$740$	0	0
$741$	67.6099	2.48371
$742$	0	0
$743$	34.2492	1.25648	0.628241	−	0.778019i	$-0.283775\pi$
$743$	34.2492	1.25648	0.628241	+	0.778019i	$0.283775\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−26.9443	−0.985839
$748$	0	0
$749$	1.25735	0.0459427
$750$	0	0
$751$	36.2361	1.32227	0.661136	−	0.750266i	$-0.270074\pi$
$751$	36.2361	1.32227	0.661136	+	0.750266i	$0.270074\pi$
$752$	0	0
$753$	58.0132	2.11412
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−15.0000	−0.545184	−0.272592	−	0.962130i	$-0.587881\pi$
$757$	−15.0000	−0.545184	−0.272592	+	0.962130i	$0.587881\pi$
$758$	0	0
$759$	−25.1246	−0.911966
$760$	0	0
$761$	−31.8541	−1.15471	−0.577355	−	0.816493i	$-0.695915\pi$
$761$	−31.8541	−1.15471	−0.577355	+	0.816493i	$0.695915\pi$
$762$	0	0
$763$	−5.70820	−0.206651
$764$	0	0
$765$	0	0
$766$	0	0
$767$	25.9443	0.936793
$768$	0	0
$769$	48.4721	1.74795	0.873975	−	0.485971i	$-0.161534\pi$
$769$	48.4721	1.74795	0.873975	+	0.485971i	$0.161534\pi$
$770$	0	0
$771$	3.09017	0.111290
$772$	0	0
$773$	−42.3262	−1.52237	−0.761184	−	0.648535i	$-0.775382\pi$
$773$	−42.3262	−1.52237	−0.761184	+	0.648535i	$0.775382\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	3.21478	0.115330
$778$	0	0
$779$	41.4853	1.48636
$780$	0	0
$781$	20.6525	0.739004
$782$	0	0
$783$	−14.7984	−0.528851
$784$	0	0
$785$	0	0
$786$	0	0
$787$	43.7082	1.55803	0.779015	−	0.627006i	$-0.215720\pi$
$787$	43.7082	1.55803	0.779015	+	0.627006i	$0.215720\pi$
$788$	0	0
$789$	−15.9787	−0.568857
$790$	0	0
$791$	−3.65248	−0.129867
$792$	0	0
$793$	−51.5755	−1.83150
$794$	0	0
$795$	0	0
$796$	0	0
$797$	18.7082	0.662679	0.331339	−	0.943512i	$-0.392500\pi$
$797$	18.7082	0.662679	0.331339	+	0.943512i	$0.392500\pi$
$798$	0	0
$799$	−5.23607	−0.185239
$800$	0	0
$801$	8.00000	0.282666
$802$	0	0
$803$	−35.5967	−1.25618
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−61.9574	−2.18100
$808$	0	0
$809$	31.7984	1.11797	0.558986	−	0.829177i	$-0.311191\pi$
$809$	31.7984	1.11797	0.558986	+	0.829177i	$0.311191\pi$
$810$	0	0
$811$	0.639320	0.0224496	0.0112248	−	0.999937i	$-0.496427\pi$
$811$	0.639320	0.0224496	0.0112248	+	0.999937i	$0.496427\pi$
$812$	0	0
$813$	11.0557	0.387741
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−35.6180	−1.24612
$818$	0	0
$819$	−4.29180	−0.149967
$820$	0	0
$821$	−7.14590	−0.249394	−0.124697	−	0.992195i	$-0.539796\pi$
$821$	−7.14590	−0.249394	−0.124697	+	0.992195i	$0.539796\pi$
$822$	0	0
$823$	35.5967	1.24082	0.620412	−	0.784276i	$-0.286965\pi$
$823$	35.5967	1.24082	0.620412	+	0.784276i	$0.286965\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	26.1459	0.909182	0.454591	−	0.890700i	$-0.349785\pi$
$827$	26.1459	0.909182	0.454591	+	0.890700i	$0.349785\pi$
$828$	0	0
$829$	−24.5623	−0.853084	−0.426542	−	0.904468i	$-0.640268\pi$
$829$	−24.5623	−0.853084	−0.426542	+	0.904468i	$0.640268\pi$
$830$	0	0
$831$	2.63932	0.0915570
$832$	0	0
$833$	5.23607	0.181419
$834$	0	0
$835$	0	0
$836$	0	0
$837$	19.4721	0.673055
$838$	0	0
$839$	−14.7984	−0.510897	−0.255448	−	0.966823i	$-0.582223\pi$
$839$	−14.7984	−0.510897	−0.255448	+	0.966823i	$0.582223\pi$
$840$	0	0
$841$	14.7984	0.510289
$842$	0	0
$843$	35.4508	1.22099
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−0.201626	−0.00692796
$848$	0	0
$849$	58.7426	2.01604
$850$	0	0
$851$	−13.0689	−0.447996
$852$	0	0
$853$	21.4164	0.733284	0.366642	−	0.930362i	$-0.380507\pi$
$853$	21.4164	0.733284	0.366642	+	0.930362i	$0.380507\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	9.85410	0.336610	0.168305	−	0.985735i	$-0.446171\pi$
$857$	9.85410	0.336610	0.168305	+	0.985735i	$0.446171\pi$
$858$	0	0
$859$	18.5967	0.634513	0.317256	−	0.948340i	$-0.397238\pi$
$859$	18.5967	0.634513	0.317256	+	0.948340i	$0.397238\pi$
$860$	0	0
$861$	−6.58359	−0.224368
$862$	0	0
$863$	24.5066	0.834214	0.417107	−	0.908857i	$-0.363044\pi$
$863$	24.5066	0.834214	0.417107	+	0.908857i	$0.363044\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−36.7082	−1.24668
$868$	0	0
$869$	−33.4164	−1.13357
$870$	0	0
$871$	−1.63932	−0.0555462
$872$	0	0
$873$	−36.5410	−1.23673
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−3.34752	−0.113038	−0.0565189	−	0.998402i	$-0.518000\pi$
$877$	−3.34752	−0.113038	−0.0565189	+	0.998402i	$0.518000\pi$
$878$	0	0
$879$	18.9443	0.638974
$880$	0	0
$881$	−16.5836	−0.558715	−0.279358	−	0.960187i	$-0.590122\pi$
$881$	−16.5836	−0.558715	−0.279358	+	0.960187i	$0.590122\pi$
$882$	0	0
$883$	21.5279	0.724470	0.362235	−	0.932087i	$-0.382014\pi$
$883$	21.5279	0.724470	0.362235	+	0.932087i	$0.382014\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−37.8885	−1.27217	−0.636086	−	0.771618i	$-0.719448\pi$
$887$	−37.8885	−1.27217	−0.636086	+	0.771618i	$0.719448\pi$
$888$	0	0
$889$	6.29180	0.211020
$890$	0	0
$891$	−35.5967	−1.19254
$892$	0	0
$893$	−36.8885	−1.23443
$894$	0	0
$895$	0	0
$896$	0	0
$897$	43.6180	1.45636
$898$	0	0
$899$	−57.6312	−1.92211
$900$	0	0
$901$	−7.59675	−0.253084
$902$	0	0
$903$	5.65248	0.188103
$904$	0	0
$905$	0	0
$906$	0	0
$907$	10.8197	0.359261	0.179630	−	0.983734i	$-0.442510\pi$
$907$	10.8197	0.359261	0.179630	+	0.983734i	$0.442510\pi$
$908$	0	0
$909$	−0.111456	−0.00369677
$910$	0	0
$911$	17.3607	0.575185	0.287592	−	0.957753i	$-0.407145\pi$
$911$	17.3607	0.575185	0.287592	+	0.957753i	$0.407145\pi$
$912$	0	0
$913$	−43.5967	−1.44284
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−3.85410	−0.127274
$918$	0	0
$919$	−15.5623	−0.513353	−0.256677	−	0.966497i	$-0.582628\pi$
$919$	−15.5623	−0.513353	−0.256677	+	0.966497i	$0.582628\pi$
$920$	0	0
$921$	34.0689	1.12261
$922$	0	0
$923$	−35.8541	−1.18015
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−19.7082	−0.647302
$928$	0	0
$929$	58.5623	1.92137	0.960683	−	0.277646i	$-0.0895542\pi$
$929$	58.5623	1.92137	0.960683	+	0.277646i	$0.0895542\pi$
$930$	0	0
$931$	36.8885	1.20897
$932$	0	0
$933$	35.9787	1.17789
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−0.437694	−0.0142988	−0.00714942	−	0.999974i	$-0.502276\pi$
$937$	−0.437694	−0.0142988	−0.00714942	+	0.999974i	$0.502276\pi$
$938$	0	0
$939$	17.2361	0.562478
$940$	0	0
$941$	−30.0557	−0.979789	−0.489894	−	0.871782i	$-0.662965\pi$
$941$	−30.0557	−0.979789	−0.489894	+	0.871782i	$0.662965\pi$
$942$	0	0
$943$	26.7639	0.871554
$944$	0	0
$945$	0	0
$946$	0	0
$947$	21.0132	0.682836	0.341418	−	0.939912i	$-0.389093\pi$
$947$	21.0132	0.682836	0.341418	+	0.939912i	$0.389093\pi$
$948$	0	0
$949$	61.7984	2.00606
$950$	0	0
$951$	21.8328	0.707978
$952$	0	0
$953$	−9.21478	−0.298496	−0.149248	−	0.988800i	$-0.547685\pi$
$953$	−9.21478	−0.298496	−0.149248	+	0.988800i	$0.547685\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	47.8885	1.54802
$958$	0	0
$959$	−1.90983	−0.0616716
$960$	0	0
$961$	44.8328	1.44622
$962$	0	0
$963$	6.58359	0.212153
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−8.47214	−0.272446	−0.136223	−	0.990678i	$-0.543496\pi$
$967$	−8.47214	−0.272446	−0.136223	+	0.990678i	$0.543496\pi$
$968$	0	0
$969$	9.19350	0.295338
$970$	0	0
$971$	−47.5623	−1.52635	−0.763174	−	0.646194i	$-0.776360\pi$
$971$	−47.5623	−1.52635	−0.763174	+	0.646194i	$0.776360\pi$
$972$	0	0
$973$	−5.43769	−0.174324
$974$	0	0
$975$	0	0
$976$	0	0
$977$	5.29180	0.169300	0.0846498	−	0.996411i	$-0.473023\pi$
$977$	5.29180	0.169300	0.0846498	+	0.996411i	$0.473023\pi$
$978$	0	0
$979$	12.9443	0.413701
$980$	0	0
$981$	−29.8885	−0.954268
$982$	0	0
$983$	−20.7984	−0.663365	−0.331683	−	0.943391i	$-0.607616\pi$
$983$	−20.7984	−0.663365	−0.331683	+	0.943391i	$0.607616\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	5.85410	0.186338
$988$	0	0
$989$	−22.9787	−0.730681
$990$	0	0
$991$	28.8197	0.915487	0.457743	−	0.889084i	$-0.348658\pi$
$991$	28.8197	0.915487	0.457743	+	0.889084i	$0.348658\pi$
$992$	0	0
$993$	58.2918	1.84983
$994$	0	0
$995$	0	0
$996$	0	0
$997$	47.4721	1.50346	0.751729	−	0.659472i	$-0.229220\pi$
$997$	47.4721	1.50346	0.751729	+	0.659472i	$0.229220\pi$
$998$	0	0
$999$	−8.41641	−0.266283

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	10000.2.a.i.1.2		2
4.3	odd	2		5000.2.a.a.1.1		2
5.4	even	2		10000.2.a.g.1.1		2
20.19	odd	2		5000.2.a.c.1.2		2
25.4	even	10		400.2.u.a.241.1		4
25.19	even	10		400.2.u.a.161.1		4
100.3	even	20		1000.2.q.a.49.2		8
100.19	odd	10		200.2.m.a.161.1	yes	4
100.31	odd	10		1000.2.m.a.801.1		4
100.47	even	20		1000.2.q.a.49.1		8
100.67	even	20		1000.2.q.a.449.2		8
100.71	odd	10		1000.2.m.a.201.1		4
100.79	odd	10		200.2.m.a.41.1	✓	4
100.83	even	20		1000.2.q.a.449.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
200.2.m.a.41.1	✓	4	100.79	odd	10
200.2.m.a.161.1	yes	4	100.19	odd	10
400.2.u.a.161.1		4	25.19	even	10
400.2.u.a.241.1		4	25.4	even	10
1000.2.m.a.201.1		4	100.71	odd	10
1000.2.m.a.801.1		4	100.31	odd	10
1000.2.q.a.49.1		8	100.47	even	20
1000.2.q.a.49.2		8	100.3	even	20
1000.2.q.a.449.1		8	100.83	even	20
1000.2.q.a.449.2		8	100.67	even	20
5000.2.a.a.1.1		2	4.3	odd	2
5000.2.a.c.1.2		2	20.19	odd	2
10000.2.a.g.1.1		2	5.4	even	2
10000.2.a.i.1.2		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)

\(f(q)\)	\(=\)	\(q+2.23607 q^{3} +0.381966 q^{7} +2.00000 q^{9} +O(q^{10})\)	\(q+2.23607 q^{3} +0.381966 q^{7} +2.00000 q^{9} +3.23607 q^{11} -5.61803 q^{13} -0.763932 q^{17} -5.38197 q^{19} +0.854102 q^{21} -3.47214 q^{23} -2.23607 q^{27} +6.61803 q^{29} -8.70820 q^{31} +7.23607 q^{33} +3.76393 q^{37} -12.5623 q^{39} -7.70820 q^{41} +6.61803 q^{43} +6.85410 q^{47} -6.85410 q^{49} -1.70820 q^{51} +9.94427 q^{53} -12.0344 q^{57} -4.61803 q^{59} +9.18034 q^{61} +0.763932 q^{63} +0.291796 q^{67} -7.76393 q^{69} +6.38197 q^{71} -11.0000 q^{73} +1.23607 q^{77} -10.3262 q^{79} -11.0000 q^{81} -13.4721 q^{83} +14.7984 q^{87} +4.00000 q^{89} -2.14590 q^{91} -19.4721 q^{93} -18.2705 q^{97} +6.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 3 q^{7} + 4 q^{9}+O(q^{10}) \) 2 * q + 3 * q^7 + 4 * q^9	\( 2 q + 3 q^{7} + 4 q^{9} + 2 q^{11} - 9 q^{13} - 6 q^{17} - 13 q^{19} - 5 q^{21} + 2 q^{23} + 11 q^{29} - 4 q^{31} + 10 q^{33} + 12 q^{37} - 5 q^{39} - 2 q^{41} + 11 q^{43} + 7 q^{47} - 7 q^{49} + 10 q^{51} + 2 q^{53} + 5 q^{57} - 7 q^{59} - 4 q^{61} + 6 q^{63} + 14 q^{67} - 20 q^{69} + 15 q^{71} - 22 q^{73} - 2 q^{77} - 5 q^{79} - 22 q^{81} - 18 q^{83} + 5 q^{87} + 8 q^{89} - 11 q^{91} - 30 q^{93} - 3 q^{97} + 4 q^{99}+O(q^{100}) \) 2 * q + 3 * q^7 + 4 * q^9 + 2 * q^11 - 9 * q^13 - 6 * q^17 - 13 * q^19 - 5 * q^21 + 2 * q^23 + 11 * q^29 - 4 * q^31 + 10 * q^33 + 12 * q^37 - 5 * q^39 - 2 * q^41 + 11 * q^43 + 7 * q^47 - 7 * q^49 + 10 * q^51 + 2 * q^53 + 5 * q^57 - 7 * q^59 - 4 * q^61 + 6 * q^63 + 14 * q^67 - 20 * q^69 + 15 * q^71 - 22 * q^73 - 2 * q^77 - 5 * q^79 - 22 * q^81 - 18 * q^83 + 5 * q^87 + 8 * q^89 - 11 * q^91 - 30 * q^93 - 3 * q^97 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more