LMFDB - Embedded newform 4000.2.a.n.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4000,2,Mod(1,4000)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4000, 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("4000.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$4000 = 2^{5} \cdot 5^{3}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	4000.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:	$31.9401608085$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.30040000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{6} - 11x^{4} - 4x^{3} + 29x^{2} + 22x + 4$ x^6 - 11x^4 - 4x^3 + 29x^2 + 22x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$1$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.29226$ of defining polynomial
Character	$\chi$	$=$	4000.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.29226 q^{3} +0.938845 q^{7} +2.25447 q^{9} +1.16354 q^{11} +2.31775 q^{13} -6.64261 q^{17} +5.70400 q^{19} -2.15208 q^{21} +1.63643 q^{23} +1.70896 q^{27} -1.93578 q^{29} +4.77507 q^{31} -2.66714 q^{33} -8.06572 q^{37} -5.31288 q^{39} +4.70589 q^{41} +3.78455 q^{43} -1.00641 q^{47} -6.11857 q^{49} +15.2266 q^{51} +4.99066 q^{53} -13.0751 q^{57} +5.79026 q^{59} -2.55900 q^{61} +2.11659 q^{63} -11.9484 q^{67} -3.75113 q^{69} +3.85157 q^{71} +0.210168 q^{73} +1.09239 q^{77} -14.6372 q^{79} -10.6808 q^{81} +5.62269 q^{83} +4.43731 q^{87} +10.8029 q^{89} +2.17600 q^{91} -10.9457 q^{93} -14.5133 q^{97} +2.62317 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$6 q + 3 q^{7} + 4 q^{9} + 13 q^{11} + 7 q^{13} - 4 q^{17} + 9 q^{19} + 7 q^{23} - 12 q^{27} + 2 q^{29} + 12 q^{31} - 6 q^{33} + 21 q^{37} + 26 q^{39} - 5 q^{41} - 8 q^{43} + 19 q^{47} - 3 q^{49} + 18 q^{51}+ \cdots + 43 q^{99}+O(q^{100})$ 6 * q + 3 * q^7 + 4 * q^9 + 13 * q^11 + 7 * q^13 - 4 * q^17 + 9 * q^19 + 7 * q^23 - 12 * q^27 + 2 * q^29 + 12 * q^31 - 6 * q^33 + 21 * q^37 + 26 * q^39 - 5 * q^41 - 8 * q^43 + 19 * q^47 - 3 * q^49 + 18 * q^51 + 11 * q^53 - 28 * q^57 + 25 * q^59 - 6 * q^61 - 3 * q^63 - 12 * q^69 + 34 * q^71 - 20 * q^73 + 14 * q^77 + 16 * q^79 - 18 * q^81 - 4 * q^83 - 26 * q^87 - 3 * q^89 + 26 * q^91 + 16 * q^93 - 20 * q^97 + 43 * q^99

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
$3$	−2.29226	−1.32344	−0.661719	−	0.749752i	$-0.730173\pi$
$3$	−2.29226	−1.32344	−0.661719	+	0.749752i	$0.730173\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0.938845	0.354850	0.177425	−	0.984134i	$-0.443223\pi$
$7$	0.938845	0.354850	0.177425	+	0.984134i	$0.443223\pi$
$8$	0	0
$9$	2.25447	0.751489
$10$	0	0
$11$	1.16354	0.350821	0.175411	−	0.984495i	$-0.443875\pi$
$11$	1.16354	0.350821	0.175411	+	0.984495i	$0.443875\pi$
$12$	0	0
$13$	2.31775	0.642827	0.321413	−	0.946939i	$-0.395842\pi$
$13$	2.31775	0.642827	0.321413	+	0.946939i	$0.395842\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−6.64261	−1.61107	−0.805535	−	0.592548i	$-0.798122\pi$
$17$	−6.64261	−1.61107	−0.805535	+	0.592548i	$0.798122\pi$
$18$	0	0
$19$	5.70400	1.30859	0.654293	−	0.756241i	$-0.272966\pi$
$19$	5.70400	1.30859	0.654293	+	0.756241i	$0.272966\pi$
$20$	0	0
$21$	−2.15208	−0.469622
$22$	0	0
$23$	1.63643	0.341220	0.170610	−	0.985339i	$-0.445426\pi$
$23$	1.63643	0.341220	0.170610	+	0.985339i	$0.445426\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.70896	0.328889
$28$	0	0
$29$	−1.93578	−0.359465	−0.179733	−	0.983716i	$-0.557523\pi$
$29$	−1.93578	−0.359465	−0.179733	+	0.983716i	$0.557523\pi$
$30$	0	0
$31$	4.77507	0.857629	0.428814	−	0.903393i	$-0.358931\pi$
$31$	4.77507	0.857629	0.428814	+	0.903393i	$0.358931\pi$
$32$	0	0
$33$	−2.66714	−0.464290
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−8.06572	−1.32600	−0.662998	−	0.748621i	$-0.730716\pi$
$37$	−8.06572	−1.32600	−0.662998	+	0.748621i	$0.730716\pi$
$38$	0	0
$39$	−5.31288	−0.850742
$40$	0	0
$41$	4.70589	0.734937	0.367468	−	0.930036i	$-0.380225\pi$
$41$	4.70589	0.734937	0.367468	+	0.930036i	$0.380225\pi$
$42$	0	0
$43$	3.78455	0.577138	0.288569	−	0.957459i	$-0.406820\pi$
$43$	3.78455	0.577138	0.288569	+	0.957459i	$0.406820\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−1.00641	−0.146800	−0.0734000	−	0.997303i	$-0.523385\pi$
$47$	−1.00641	−0.146800	−0.0734000	+	0.997303i	$0.523385\pi$
$48$	0	0
$49$	−6.11857	−0.874081
$50$	0	0
$51$	15.2266	2.13215
$52$	0	0
$53$	4.99066	0.685520	0.342760	−	0.939423i	$-0.388638\pi$
$53$	4.99066	0.685520	0.342760	+	0.939423i	$0.388638\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−13.0751	−1.73183
$58$	0	0
$59$	5.79026	0.753828	0.376914	−	0.926248i	$-0.376985\pi$
$59$	5.79026	0.753828	0.376914	+	0.926248i	$0.376985\pi$
$60$	0	0
$61$	−2.55900	−0.327647	−0.163823	−	0.986490i	$-0.552383\pi$
$61$	−2.55900	−0.327647	−0.163823	+	0.986490i	$0.552383\pi$
$62$	0	0
$63$	2.11659	0.266666
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−11.9484	−1.45973	−0.729863	−	0.683594i	$-0.760416\pi$
$67$	−11.9484	−1.45973	−0.729863	+	0.683594i	$0.760416\pi$
$68$	0	0
$69$	−3.75113	−0.451583
$70$	0	0
$71$	3.85157	0.457097	0.228549	−	0.973533i	$-0.426602\pi$
$71$	3.85157	0.457097	0.228549	+	0.973533i	$0.426602\pi$
$72$	0	0
$73$	0.210168	0.0245983	0.0122992	−	0.999924i	$-0.496085\pi$
$73$	0.210168	0.0245983	0.0122992	+	0.999924i	$0.496085\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1.09239	0.124489
$78$	0	0
$79$	−14.6372	−1.64681	−0.823406	−	0.567453i	$-0.807929\pi$
$79$	−14.6372	−1.64681	−0.823406	+	0.567453i	$0.807929\pi$
$80$	0	0
$81$	−10.6808	−1.18675
$82$	0	0
$83$	5.62269	0.617171	0.308585	−	0.951197i	$-0.400144\pi$
$83$	5.62269	0.617171	0.308585	+	0.951197i	$0.400144\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	4.43731	0.475730
$88$	0	0
$89$	10.8029	1.14510	0.572550	−	0.819870i	$-0.305954\pi$
$89$	10.8029	1.14510	0.572550	+	0.819870i	$0.305954\pi$
$90$	0	0
$91$	2.17600	0.228107
$92$	0	0
$93$	−10.9457	−1.13502
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−14.5133	−1.47360	−0.736800	−	0.676111i	$-0.763664\pi$
$97$	−14.5133	−1.47360	−0.736800	+	0.676111i	$0.763664\pi$
$98$	0	0
$99$	2.62317	0.263638
$100$	0	0
$101$	14.2528	1.41820	0.709101	−	0.705107i	$-0.249101\pi$
$101$	14.2528	1.41820	0.709101	+	0.705107i	$0.249101\pi$
$102$	0	0
$103$	−14.2141	−1.40055	−0.700277	−	0.713871i	$-0.746940\pi$
$103$	−14.2141	−1.40055	−0.700277	+	0.713871i	$0.746940\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	10.6763	1.03211	0.516057	−	0.856555i	$-0.327399\pi$
$107$	10.6763	1.03211	0.516057	+	0.856555i	$0.327399\pi$
$108$	0	0
$109$	18.7091	1.79201	0.896004	−	0.444045i	$-0.146457\pi$
$109$	18.7091	1.79201	0.896004	+	0.444045i	$0.146457\pi$
$110$	0	0
$111$	18.4887	1.75487
$112$	0	0
$113$	3.44534	0.324110	0.162055	−	0.986782i	$-0.448188\pi$
$113$	3.44534	0.324110	0.162055	+	0.986782i	$0.448188\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	5.22528	0.483077
$118$	0	0
$119$	−6.23639	−0.571689
$120$	0	0
$121$	−9.64617	−0.876925
$122$	0	0
$123$	−10.7871	−0.972644
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−10.1918	−0.904373	−0.452187	−	0.891923i	$-0.649356\pi$
$127$	−10.1918	−0.904373	−0.452187	+	0.891923i	$0.649356\pi$
$128$	0	0
$129$	−8.67518	−0.763807
$130$	0	0
$131$	16.4764	1.43955	0.719776	−	0.694207i	$-0.244245\pi$
$131$	16.4764	1.43955	0.719776	+	0.694207i	$0.244245\pi$
$132$	0	0
$133$	5.35517	0.464352
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−0.370281	−0.0316353	−0.0158176	−	0.999875i	$-0.505035\pi$
$137$	−0.370281	−0.0316353	−0.0158176	+	0.999875i	$0.505035\pi$
$138$	0	0
$139$	7.37956	0.625926	0.312963	−	0.949765i	$-0.398678\pi$
$139$	7.37956	0.625926	0.312963	+	0.949765i	$0.398678\pi$
$140$	0	0
$141$	2.30696	0.194281
$142$	0	0
$143$	2.69679	0.225517
$144$	0	0
$145$	0	0
$146$	0	0
$147$	14.0254	1.15679
$148$	0	0
$149$	−13.4478	−1.10169	−0.550843	−	0.834609i	$-0.685694\pi$
$149$	−13.4478	−1.10169	−0.550843	+	0.834609i	$0.685694\pi$
$150$	0	0
$151$	8.54038	0.695006	0.347503	−	0.937679i	$-0.387030\pi$
$151$	8.54038	0.695006	0.347503	+	0.937679i	$0.387030\pi$
$152$	0	0
$153$	−14.9756	−1.21070
$154$	0	0
$155$	0	0
$156$	0	0
$157$	16.5204	1.31847	0.659236	−	0.751936i	$-0.270880\pi$
$157$	16.5204	1.31847	0.659236	+	0.751936i	$0.270880\pi$
$158$	0	0
$159$	−11.4399	−0.907243
$160$	0	0
$161$	1.53636	0.121082
$162$	0	0
$163$	0.480836	0.0376620	0.0188310	−	0.999823i	$-0.494006\pi$
$163$	0.480836	0.0376620	0.0188310	+	0.999823i	$0.494006\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−0.302600	−0.0234159	−0.0117079	−	0.999931i	$-0.503727\pi$
$167$	−0.302600	−0.0234159	−0.0117079	+	0.999931i	$0.503727\pi$
$168$	0	0
$169$	−7.62806	−0.586774
$170$	0	0
$171$	12.8595	0.983388
$172$	0	0
$173$	−7.06564	−0.537191	−0.268595	−	0.963253i	$-0.586559\pi$
$173$	−7.06564	−0.537191	−0.268595	+	0.963253i	$0.586559\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−13.2728	−0.997645
$178$	0	0
$179$	24.2142	1.80985	0.904925	−	0.425570i	$-0.139927\pi$
$179$	24.2142	1.80985	0.904925	+	0.425570i	$0.139927\pi$
$180$	0	0
$181$	−8.31698	−0.618196	−0.309098	−	0.951030i	$-0.600027\pi$
$181$	−8.31698	−0.618196	−0.309098	+	0.951030i	$0.600027\pi$
$182$	0	0
$183$	5.86591	0.433620
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−7.72896	−0.565198
$188$	0	0
$189$	1.60445	0.116706
$190$	0	0
$191$	23.2852	1.68486	0.842430	−	0.538806i	$-0.181124\pi$
$191$	23.2852	1.68486	0.842430	+	0.538806i	$0.181124\pi$
$192$	0	0
$193$	20.6634	1.48738	0.743692	−	0.668523i	$-0.233073\pi$
$193$	20.6634	1.48738	0.743692	+	0.668523i	$0.233073\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	20.9457	1.49232	0.746160	−	0.665766i	$-0.231895\pi$
$197$	20.9457	1.49232	0.746160	+	0.665766i	$0.231895\pi$
$198$	0	0
$199$	−4.74212	−0.336160	−0.168080	−	0.985773i	$-0.553757\pi$
$199$	−4.74212	−0.336160	−0.168080	+	0.985773i	$0.553757\pi$
$200$	0	0
$201$	27.3888	1.93186
$202$	0	0
$203$	−1.81740	−0.127556
$204$	0	0
$205$	0	0
$206$	0	0
$207$	3.68928	0.256423
$208$	0	0
$209$	6.63684	0.459080
$210$	0	0
$211$	11.1218	0.765658	0.382829	−	0.923819i	$-0.374950\pi$
$211$	11.1218	0.765658	0.382829	+	0.923819i	$0.374950\pi$
$212$	0	0
$213$	−8.82881	−0.604940
$214$	0	0
$215$	0	0
$216$	0	0
$217$	4.48305	0.304330
$218$	0	0
$219$	−0.481761	−0.0325544
$220$	0	0
$221$	−15.3959	−1.03564
$222$	0	0
$223$	12.9371	0.866331	0.433165	−	0.901314i	$-0.357397\pi$
$223$	12.9371	0.866331	0.433165	+	0.901314i	$0.357397\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−1.04921	−0.0696385	−0.0348192	−	0.999394i	$-0.511086\pi$
$227$	−1.04921	−0.0696385	−0.0348192	+	0.999394i	$0.511086\pi$
$228$	0	0
$229$	1.86118	0.122990	0.0614950	−	0.998107i	$-0.480413\pi$
$229$	1.86118	0.122990	0.0614950	+	0.998107i	$0.480413\pi$
$230$	0	0
$231$	−2.50403	−0.164753
$232$	0	0
$233$	13.2426	0.867551	0.433775	−	0.901021i	$-0.357181\pi$
$233$	13.2426	0.867551	0.433775	+	0.901021i	$0.357181\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	33.5523	2.17945
$238$	0	0
$239$	11.1272	0.719761	0.359881	−	0.932998i	$-0.382817\pi$
$239$	11.1272	0.719761	0.359881	+	0.932998i	$0.382817\pi$
$240$	0	0
$241$	22.4775	1.44790	0.723952	−	0.689851i	$-0.242324\pi$
$241$	22.4775	1.44790	0.723952	+	0.689851i	$0.242324\pi$
$242$	0	0
$243$	19.3563	1.24171
$244$	0	0
$245$	0	0
$246$	0	0
$247$	13.2204	0.841195
$248$	0	0
$249$	−12.8887	−0.816787
$250$	0	0
$251$	−14.3075	−0.903083	−0.451541	−	0.892250i	$-0.649126\pi$
$251$	−14.3075	−0.903083	−0.451541	+	0.892250i	$0.649126\pi$
$252$	0	0
$253$	1.90406	0.119707
$254$	0	0
$255$	0	0
$256$	0	0
$257$	4.19735	0.261824	0.130912	−	0.991394i	$-0.458210\pi$
$257$	4.19735	0.261824	0.130912	+	0.991394i	$0.458210\pi$
$258$	0	0
$259$	−7.57246	−0.470530
$260$	0	0
$261$	−4.36415	−0.270134
$262$	0	0
$263$	24.1436	1.48876	0.744381	−	0.667756i	$-0.232745\pi$
$263$	24.1436	1.48876	0.744381	+	0.667756i	$0.232745\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−24.7630	−1.51547
$268$	0	0
$269$	−17.4093	−1.06146	−0.530731	−	0.847540i	$-0.678082\pi$
$269$	−17.4093	−1.06146	−0.530731	+	0.847540i	$0.678082\pi$
$270$	0	0
$271$	29.2397	1.77619	0.888093	−	0.459664i	$-0.152030\pi$
$271$	29.2397	1.77619	0.888093	+	0.459664i	$0.152030\pi$
$272$	0	0
$273$	−4.98797	−0.301886
$274$	0	0
$275$	0	0
$276$	0	0
$277$	25.2027	1.51429	0.757143	−	0.653249i	$-0.226595\pi$
$277$	25.2027	1.51429	0.757143	+	0.653249i	$0.226595\pi$
$278$	0	0
$279$	10.7652	0.644498
$280$	0	0
$281$	−31.1148	−1.85615	−0.928077	−	0.372388i	$-0.878539\pi$
$281$	−31.1148	−1.85615	−0.928077	+	0.372388i	$0.878539\pi$
$282$	0	0
$283$	−10.2160	−0.607279	−0.303640	−	0.952787i	$-0.598202\pi$
$283$	−10.2160	−0.607279	−0.303640	+	0.952787i	$0.598202\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	4.41810	0.260792
$288$	0	0
$289$	27.1243	1.59555
$290$	0	0
$291$	33.2682	1.95022
$292$	0	0
$293$	12.6454	0.738751	0.369376	−	0.929280i	$-0.379572\pi$
$293$	12.6454	0.738751	0.369376	+	0.929280i	$0.379572\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	1.98845	0.115381
$298$	0	0
$299$	3.79283	0.219345
$300$	0	0
$301$	3.55311	0.204798
$302$	0	0
$303$	−32.6711	−1.87690
$304$	0	0
$305$	0	0
$306$	0	0
$307$	9.14322	0.521831	0.260915	−	0.965362i	$-0.415976\pi$
$307$	9.14322	0.521831	0.260915	+	0.965362i	$0.415976\pi$
$308$	0	0
$309$	32.5824	1.85355
$310$	0	0
$311$	−13.2336	−0.750409	−0.375204	−	0.926942i	$-0.622427\pi$
$311$	−13.2336	−0.750409	−0.375204	+	0.926942i	$0.622427\pi$
$312$	0	0
$313$	−15.3458	−0.867398	−0.433699	−	0.901058i	$-0.642792\pi$
$313$	−15.3458	−0.867398	−0.433699	+	0.901058i	$0.642792\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	24.0301	1.34967	0.674833	−	0.737970i	$-0.264216\pi$
$317$	24.0301	1.34967	0.674833	+	0.737970i	$0.264216\pi$
$318$	0	0
$319$	−2.25236	−0.126108
$320$	0	0
$321$	−24.4728	−1.36594
$322$	0	0
$323$	−37.8895	−2.10823
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−42.8862	−2.37161
$328$	0	0
$329$	−0.944863	−0.0520920
$330$	0	0
$331$	−14.7449	−0.810450	−0.405225	−	0.914217i	$-0.632807\pi$
$331$	−14.7449	−0.810450	−0.405225	+	0.914217i	$0.632807\pi$
$332$	0	0
$333$	−18.1839	−0.996471
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−4.52428	−0.246453	−0.123227	−	0.992379i	$-0.539324\pi$
$337$	−4.52428	−0.246453	−0.123227	+	0.992379i	$0.539324\pi$
$338$	0	0
$339$	−7.89762	−0.428940
$340$	0	0
$341$	5.55600	0.300874
$342$	0	0
$343$	−12.3163	−0.665018
$344$	0	0
$345$	0	0
$346$	0	0
$347$	13.4436	0.721690	0.360845	−	0.932626i	$-0.382488\pi$
$347$	13.4436	0.721690	0.360845	+	0.932626i	$0.382488\pi$
$348$	0	0
$349$	−27.4939	−1.47171	−0.735857	−	0.677137i	$-0.763221\pi$
$349$	−27.4939	−1.47171	−0.735857	+	0.677137i	$0.763221\pi$
$350$	0	0
$351$	3.96093	0.211419
$352$	0	0
$353$	30.0034	1.59692	0.798459	−	0.602049i	$-0.205649\pi$
$353$	30.0034	1.59692	0.798459	+	0.602049i	$0.205649\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	14.2954	0.756594
$358$	0	0
$359$	16.1270	0.851152	0.425576	−	0.904923i	$-0.360072\pi$
$359$	16.1270	0.851152	0.425576	+	0.904923i	$0.360072\pi$
$360$	0	0
$361$	13.5356	0.712400
$362$	0	0
$363$	22.1116	1.16056
$364$	0	0
$365$	0	0
$366$	0	0
$367$	20.7892	1.08519	0.542594	−	0.839995i	$-0.317442\pi$
$367$	20.7892	1.08519	0.542594	+	0.839995i	$0.317442\pi$
$368$	0	0
$369$	10.6093	0.552297
$370$	0	0
$371$	4.68546	0.243257
$372$	0	0
$373$	34.9061	1.80737	0.903684	−	0.428200i	$-0.140852\pi$
$373$	34.9061	1.80737	0.903684	+	0.428200i	$0.140852\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−4.48664	−0.231074
$378$	0	0
$379$	−26.4479	−1.35854	−0.679268	−	0.733890i	$-0.737703\pi$
$379$	−26.4479	−1.35854	−0.679268	+	0.733890i	$0.737703\pi$
$380$	0	0
$381$	23.3622	1.19688
$382$	0	0
$383$	−17.3746	−0.887802	−0.443901	−	0.896076i	$-0.646406\pi$
$383$	−17.3746	−0.887802	−0.443901	+	0.896076i	$0.646406\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	8.53214	0.433713
$388$	0	0
$389$	6.13991	0.311306	0.155653	−	0.987812i	$-0.450252\pi$
$389$	6.13991	0.311306	0.155653	+	0.987812i	$0.450252\pi$
$390$	0	0
$391$	−10.8702	−0.549729
$392$	0	0
$393$	−37.7683	−1.90516
$394$	0	0
$395$	0	0
$396$	0	0
$397$	4.21943	0.211767	0.105883	−	0.994379i	$-0.466233\pi$
$397$	4.21943	0.211767	0.105883	+	0.994379i	$0.466233\pi$
$398$	0	0
$399$	−12.2755	−0.614541
$400$	0	0
$401$	17.5061	0.874213	0.437107	−	0.899410i	$-0.356003\pi$
$401$	17.5061	0.874213	0.437107	+	0.899410i	$0.356003\pi$
$402$	0	0
$403$	11.0674	0.551307
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−9.38481	−0.465188
$408$	0	0
$409$	−5.44224	−0.269101	−0.134551	−	0.990907i	$-0.542959\pi$
$409$	−5.44224	−0.269101	−0.134551	+	0.990907i	$0.542959\pi$
$410$	0	0
$411$	0.848782	0.0418673
$412$	0	0
$413$	5.43616	0.267496
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−16.9159	−0.828375
$418$	0	0
$419$	−23.1501	−1.13095	−0.565477	−	0.824764i	$-0.691308\pi$
$419$	−23.1501	−1.13095	−0.565477	+	0.824764i	$0.691308\pi$
$420$	0	0
$421$	−10.5558	−0.514460	−0.257230	−	0.966350i	$-0.582810\pi$
$421$	−10.5558	−0.514460	−0.257230	+	0.966350i	$0.582810\pi$
$422$	0	0
$423$	−2.26892	−0.110319
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−2.40251	−0.116265
$428$	0	0
$429$	−6.18176	−0.298458
$430$	0	0
$431$	31.9502	1.53899	0.769494	−	0.638654i	$-0.220509\pi$
$431$	31.9502	1.53899	0.769494	+	0.638654i	$0.220509\pi$
$432$	0	0
$433$	−10.2586	−0.492999	−0.246499	−	0.969143i	$-0.579280\pi$
$433$	−10.2586	−0.492999	−0.246499	+	0.969143i	$0.579280\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	9.33421	0.446516
$438$	0	0
$439$	−7.16631	−0.342029	−0.171015	−	0.985268i	$-0.554705\pi$
$439$	−7.16631	−0.342029	−0.171015	+	0.985268i	$0.554705\pi$
$440$	0	0
$441$	−13.7941	−0.656862
$442$	0	0
$443$	33.0084	1.56828	0.784139	−	0.620585i	$-0.213105\pi$
$443$	33.0084	1.56828	0.784139	+	0.620585i	$0.213105\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	30.8258	1.45801
$448$	0	0
$449$	−18.6955	−0.882296	−0.441148	−	0.897434i	$-0.645429\pi$
$449$	−18.6955	−0.882296	−0.441148	+	0.897434i	$0.645429\pi$
$450$	0	0
$451$	5.47550	0.257831
$452$	0	0
$453$	−19.5768	−0.919797
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−6.12807	−0.286659	−0.143329	−	0.989675i	$-0.545781\pi$
$457$	−6.12807	−0.286659	−0.143329	+	0.989675i	$0.545781\pi$
$458$	0	0
$459$	−11.3520	−0.529864
$460$	0	0
$461$	−8.95725	−0.417181	−0.208590	−	0.978003i	$-0.566888\pi$
$461$	−8.95725	−0.417181	−0.208590	+	0.978003i	$0.566888\pi$
$462$	0	0
$463$	38.1802	1.77439	0.887193	−	0.461399i	$-0.152652\pi$
$463$	38.1802	1.77439	0.887193	+	0.461399i	$0.152652\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−39.9615	−1.84920	−0.924599	−	0.380942i	$-0.875600\pi$
$467$	−39.9615	−1.84920	−0.924599	+	0.380942i	$0.875600\pi$
$468$	0	0
$469$	−11.2177	−0.517984
$470$	0	0
$471$	−37.8691	−1.74492
$472$	0	0
$473$	4.40348	0.202472
$474$	0	0
$475$	0	0
$476$	0	0
$477$	11.2513	0.515161
$478$	0	0
$479$	37.0193	1.69145	0.845727	−	0.533616i	$-0.179167\pi$
$479$	37.0193	1.69145	0.845727	+	0.533616i	$0.179167\pi$
$480$	0	0
$481$	−18.6943	−0.852386
$482$	0	0
$483$	−3.52173	−0.160244
$484$	0	0
$485$	0	0
$486$	0	0
$487$	33.1512	1.50222	0.751111	−	0.660176i	$-0.229518\pi$
$487$	33.1512	1.50222	0.751111	+	0.660176i	$0.229518\pi$
$488$	0	0
$489$	−1.10220	−0.0498434
$490$	0	0
$491$	−35.7953	−1.61542	−0.807710	−	0.589580i	$-0.799293\pi$
$491$	−35.7953	−1.61542	−0.807710	+	0.589580i	$0.799293\pi$
$492$	0	0
$493$	12.8586	0.579124
$494$	0	0
$495$	0	0
$496$	0	0
$497$	3.61603	0.162201
$498$	0	0
$499$	2.48527	0.111256	0.0556280	−	0.998452i	$-0.482284\pi$
$499$	2.48527	0.111256	0.0556280	+	0.998452i	$0.482284\pi$
$500$	0	0
$501$	0.693639	0.0309895
$502$	0	0
$503$	−24.0846	−1.07388	−0.536940	−	0.843621i	$-0.680420\pi$
$503$	−24.0846	−1.07388	−0.536940	+	0.843621i	$0.680420\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	17.4855	0.776559
$508$	0	0
$509$	10.3413	0.458371	0.229186	−	0.973383i	$-0.426394\pi$
$509$	10.3413	0.458371	0.229186	+	0.973383i	$0.426394\pi$
$510$	0	0
$511$	0.197316	0.00872872
$512$	0	0
$513$	9.74789	0.430380
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−1.17100	−0.0515006
$518$	0	0
$519$	16.1963	0.710939
$520$	0	0
$521$	−13.8975	−0.608861	−0.304430	−	0.952535i	$-0.598466\pi$
$521$	−13.8975	−0.608861	−0.304430	+	0.952535i	$0.598466\pi$
$522$	0	0
$523$	−21.7256	−0.949996	−0.474998	−	0.879987i	$-0.657551\pi$
$523$	−21.7256	−0.949996	−0.474998	+	0.879987i	$0.657551\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−31.7190	−1.38170
$528$	0	0
$529$	−20.3221	−0.883569
$530$	0	0
$531$	13.0540	0.566494
$532$	0	0
$533$	10.9071	0.472437
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−55.5052	−2.39523
$538$	0	0
$539$	−7.11921	−0.306646
$540$	0	0
$541$	27.3452	1.17566	0.587832	−	0.808983i	$-0.299982\pi$
$541$	27.3452	1.17566	0.587832	+	0.808983i	$0.299982\pi$
$542$	0	0
$543$	19.0647	0.818144
$544$	0	0
$545$	0	0
$546$	0	0
$547$	42.2665	1.80718	0.903592	−	0.428395i	$-0.140921\pi$
$547$	42.2665	1.80718	0.903592	+	0.428395i	$0.140921\pi$
$548$	0	0
$549$	−5.76919	−0.246223
$550$	0	0
$551$	−11.0417	−0.470391
$552$	0	0
$553$	−13.7420	−0.584371
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−17.2746	−0.731949	−0.365975	−	0.930625i	$-0.619264\pi$
$557$	−17.2746	−0.731949	−0.365975	+	0.930625i	$0.619264\pi$
$558$	0	0
$559$	8.77162	0.371000
$560$	0	0
$561$	17.7168	0.748004
$562$	0	0
$563$	10.8603	0.457707	0.228854	−	0.973461i	$-0.426502\pi$
$563$	10.8603	0.457707	0.228854	+	0.973461i	$0.426502\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−10.0276	−0.421120
$568$	0	0
$569$	11.6982	0.490412	0.245206	−	0.969471i	$-0.421144\pi$
$569$	11.6982	0.490412	0.245206	+	0.969471i	$0.421144\pi$
$570$	0	0
$571$	−25.0311	−1.04752	−0.523760	−	0.851866i	$-0.675471\pi$
$571$	−25.0311	−1.04752	−0.523760	+	0.851866i	$0.675471\pi$
$572$	0	0
$573$	−53.3758	−2.22981
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−29.9471	−1.24671	−0.623356	−	0.781938i	$-0.714231\pi$
$577$	−29.9471	−1.24671	−0.623356	+	0.781938i	$0.714231\pi$
$578$	0	0
$579$	−47.3659	−1.96846
$580$	0	0
$581$	5.27884	0.219003
$582$	0	0
$583$	5.80684	0.240495
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−26.1618	−1.07981	−0.539907	−	0.841725i	$-0.681541\pi$
$587$	−26.1618	−1.07981	−0.539907	+	0.841725i	$0.681541\pi$
$588$	0	0
$589$	27.2370	1.12228
$590$	0	0
$591$	−48.0131	−1.97499
$592$	0	0
$593$	−2.94588	−0.120973	−0.0604864	−	0.998169i	$-0.519265\pi$
$593$	−2.94588	−0.120973	−0.0604864	+	0.998169i	$0.519265\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	10.8702	0.444887
$598$	0	0
$599$	35.0432	1.43183	0.715914	−	0.698188i	$-0.246010\pi$
$599$	35.0432	1.43183	0.715914	+	0.698188i	$0.246010\pi$
$600$	0	0
$601$	20.0077	0.816133	0.408066	−	0.912952i	$-0.366203\pi$
$601$	20.0077	0.816133	0.408066	+	0.912952i	$0.366203\pi$
$602$	0	0
$603$	−26.9372	−1.09697
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−27.6672	−1.12298	−0.561488	−	0.827485i	$-0.689771\pi$
$607$	−27.6672	−1.12298	−0.561488	+	0.827485i	$0.689771\pi$
$608$	0	0
$609$	4.16595	0.168813
$610$	0	0
$611$	−2.33260	−0.0943670
$612$	0	0
$613$	42.7038	1.72479	0.862396	−	0.506235i	$-0.168963\pi$
$613$	42.7038	1.72479	0.862396	+	0.506235i	$0.168963\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−32.3048	−1.30054	−0.650271	−	0.759702i	$-0.725345\pi$
$617$	−32.3048	−1.30054	−0.650271	+	0.759702i	$0.725345\pi$
$618$	0	0
$619$	1.54036	0.0619122	0.0309561	−	0.999521i	$-0.490145\pi$
$619$	1.54036	0.0619122	0.0309561	+	0.999521i	$0.490145\pi$
$620$	0	0
$621$	2.79659	0.112223
$622$	0	0
$623$	10.1422	0.406339
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−15.2134	−0.607564
$628$	0	0
$629$	53.5775	2.13627
$630$	0	0
$631$	−25.5501	−1.01713	−0.508567	−	0.861022i	$-0.669825\pi$
$631$	−25.5501	−1.01713	−0.508567	+	0.861022i	$0.669825\pi$
$632$	0	0
$633$	−25.4941	−1.01330
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−14.1813	−0.561883
$638$	0	0
$639$	8.68323	0.343503
$640$	0	0
$641$	9.48742	0.374731	0.187365	−	0.982290i	$-0.440005\pi$
$641$	9.48742	0.374731	0.187365	+	0.982290i	$0.440005\pi$
$642$	0	0
$643$	14.6234	0.576689	0.288345	−	0.957527i	$-0.406895\pi$
$643$	14.6234	0.576689	0.288345	+	0.957527i	$0.406895\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	25.1841	0.990090	0.495045	−	0.868867i	$-0.335152\pi$
$647$	25.1841	0.990090	0.495045	+	0.868867i	$0.335152\pi$
$648$	0	0
$649$	6.73722	0.264459
$650$	0	0
$651$	−10.2763	−0.402761
$652$	0	0
$653$	−11.0960	−0.434218	−0.217109	−	0.976147i	$-0.569663\pi$
$653$	−11.0960	−0.434218	−0.217109	+	0.976147i	$0.569663\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0.473817	0.0184854
$658$	0	0
$659$	46.1162	1.79643	0.898216	−	0.439554i	$-0.144863\pi$
$659$	46.1162	1.79643	0.898216	+	0.439554i	$0.144863\pi$
$660$	0	0
$661$	−13.6876	−0.532387	−0.266194	−	0.963920i	$-0.585766\pi$
$661$	−13.6876	−0.532387	−0.266194	+	0.963920i	$0.585766\pi$
$662$	0	0
$663$	35.2914	1.37060
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−3.16777	−0.122657
$668$	0	0
$669$	−29.6552	−1.14654
$670$	0	0
$671$	−2.97751	−0.114945
$672$	0	0
$673$	23.2918	0.897831	0.448916	−	0.893574i	$-0.351810\pi$
$673$	23.2918	0.897831	0.448916	+	0.893574i	$0.351810\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−16.9928	−0.653085	−0.326543	−	0.945182i	$-0.605884\pi$
$677$	−16.9928	−0.653085	−0.326543	+	0.945182i	$0.605884\pi$
$678$	0	0
$679$	−13.6257	−0.522907
$680$	0	0
$681$	2.40506	0.0921622
$682$	0	0
$683$	−23.8721	−0.913442	−0.456721	−	0.889610i	$-0.650976\pi$
$683$	−23.8721	−0.913442	−0.456721	+	0.889610i	$0.650976\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−4.26630	−0.162770
$688$	0	0
$689$	11.5671	0.440671
$690$	0	0
$691$	−10.9306	−0.415821	−0.207910	−	0.978148i	$-0.566666\pi$
$691$	−10.9306	−0.415821	−0.207910	+	0.978148i	$0.566666\pi$
$692$	0	0
$693$	2.46275	0.0935520
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−31.2594	−1.18404
$698$	0	0
$699$	−30.3555	−1.14815
$700$	0	0
$701$	−42.6696	−1.61161	−0.805805	−	0.592181i	$-0.798267\pi$
$701$	−42.6696	−1.61161	−0.805805	+	0.592181i	$0.798267\pi$
$702$	0	0
$703$	−46.0068	−1.73518
$704$	0	0
$705$	0	0
$706$	0	0
$707$	13.3811	0.503249
$708$	0	0
$709$	−25.6653	−0.963879	−0.481940	−	0.876204i	$-0.660068\pi$
$709$	−25.6653	−0.963879	−0.481940	+	0.876204i	$0.660068\pi$
$710$	0	0
$711$	−32.9990	−1.23756
$712$	0	0
$713$	7.81409	0.292640
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−25.5066	−0.952560
$718$	0	0
$719$	49.2112	1.83527	0.917634	−	0.397427i	$-0.130097\pi$
$719$	49.2112	1.83527	0.917634	+	0.397427i	$0.130097\pi$
$720$	0	0
$721$	−13.3448	−0.496987
$722$	0	0
$723$	−51.5243	−1.91621
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−45.5639	−1.68987	−0.844935	−	0.534869i	$-0.820361\pi$
$727$	−45.5639	−1.68987	−0.844935	+	0.534869i	$0.820361\pi$
$728$	0	0
$729$	−12.3273	−0.456567
$730$	0	0
$731$	−25.1393	−0.929811
$732$	0	0
$733$	−6.36309	−0.235026	−0.117513	−	0.993071i	$-0.537492\pi$
$733$	−6.36309	−0.235026	−0.117513	+	0.993071i	$0.537492\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−13.9024	−0.512103
$738$	0	0
$739$	−35.8391	−1.31836	−0.659182	−	0.751984i	$-0.729097\pi$
$739$	−35.8391	−1.31836	−0.659182	+	0.751984i	$0.729097\pi$
$740$	0	0
$741$	−30.3047	−1.11327
$742$	0	0
$743$	−20.6753	−0.758504	−0.379252	−	0.925293i	$-0.623819\pi$
$743$	−20.6753	−0.758504	−0.379252	+	0.925293i	$0.623819\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	12.6762	0.463797
$748$	0	0
$749$	10.0234	0.366245
$750$	0	0
$751$	22.1073	0.806708	0.403354	−	0.915044i	$-0.367844\pi$
$751$	22.1073	0.806708	0.403354	+	0.915044i	$0.367844\pi$
$752$	0	0
$753$	32.7966	1.19517
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−15.5874	−0.566533	−0.283267	−	0.959041i	$-0.591418\pi$
$757$	−15.5874	−0.566533	−0.283267	+	0.959041i	$0.591418\pi$
$758$	0	0
$759$	−4.36460	−0.158425
$760$	0	0
$761$	18.1930	0.659496	0.329748	−	0.944069i	$-0.393036\pi$
$761$	18.1930	0.659496	0.329748	+	0.944069i	$0.393036\pi$
$762$	0	0
$763$	17.5650	0.635894
$764$	0	0
$765$	0	0
$766$	0	0
$767$	13.4204	0.484581
$768$	0	0
$769$	14.8828	0.536688	0.268344	−	0.963323i	$-0.413524\pi$
$769$	14.8828	0.536688	0.268344	+	0.963323i	$0.413524\pi$
$770$	0	0
$771$	−9.62144	−0.346508
$772$	0	0
$773$	18.8096	0.676534	0.338267	−	0.941050i	$-0.390159\pi$
$773$	18.8096	0.676534	0.338267	+	0.941050i	$0.390159\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	17.3581	0.622717
$778$	0	0
$779$	26.8424	0.961729
$780$	0	0
$781$	4.48146	0.160359
$782$	0	0
$783$	−3.30817	−0.118224
$784$	0	0
$785$	0	0
$786$	0	0
$787$	0.811893	0.0289409	0.0144704	−	0.999895i	$-0.495394\pi$
$787$	0.811893	0.0289409	0.0144704	+	0.999895i	$0.495394\pi$
$788$	0	0
$789$	−55.3436	−1.97028
$790$	0	0
$791$	3.23464	0.115011
$792$	0	0
$793$	−5.93112	−0.210620
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−42.3007	−1.49837	−0.749184	−	0.662362i	$-0.769554\pi$
$797$	−42.3007	−1.49837	−0.749184	+	0.662362i	$0.769554\pi$
$798$	0	0
$799$	6.68520	0.236505
$800$	0	0
$801$	24.3547	0.860530
$802$	0	0
$803$	0.244540	0.00862962
$804$	0	0
$805$	0	0
$806$	0	0
$807$	39.9066	1.40478
$808$	0	0
$809$	−35.8115	−1.25906	−0.629532	−	0.776975i	$-0.716753\pi$
$809$	−35.8115	−1.25906	−0.629532	+	0.776975i	$0.716753\pi$
$810$	0	0
$811$	−42.4255	−1.48976	−0.744881	−	0.667197i	$-0.767494\pi$
$811$	−42.4255	−1.48976	−0.744881	+	0.667197i	$0.767494\pi$
$812$	0	0
$813$	−67.0251	−2.35067
$814$	0	0
$815$	0	0
$816$	0	0
$817$	21.5871	0.755236
$818$	0	0
$819$	4.90573	0.171420
$820$	0	0
$821$	39.6138	1.38253	0.691265	−	0.722601i	$-0.257054\pi$
$821$	39.6138	1.38253	0.691265	+	0.722601i	$0.257054\pi$
$822$	0	0
$823$	−0.121605	−0.00423888	−0.00211944	−	0.999998i	$-0.500675\pi$
$823$	−0.121605	−0.00423888	−0.00211944	+	0.999998i	$0.500675\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	14.4621	0.502896	0.251448	−	0.967871i	$-0.419093\pi$
$827$	14.4621	0.502896	0.251448	+	0.967871i	$0.419093\pi$
$828$	0	0
$829$	14.1520	0.491518	0.245759	−	0.969331i	$-0.420963\pi$
$829$	14.1520	0.491518	0.245759	+	0.969331i	$0.420963\pi$
$830$	0	0
$831$	−57.7713	−2.00406
$832$	0	0
$833$	40.6433	1.40821
$834$	0	0
$835$	0	0
$836$	0	0
$837$	8.16040	0.282065
$838$	0	0
$839$	7.39385	0.255264	0.127632	−	0.991822i	$-0.459262\pi$
$839$	7.39385	0.255264	0.127632	+	0.991822i	$0.459262\pi$
$840$	0	0
$841$	−25.2528	−0.870785
$842$	0	0
$843$	71.3233	2.45651
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−9.05626	−0.311177
$848$	0	0
$849$	23.4178	0.803697
$850$	0	0
$851$	−13.1990	−0.452456
$852$	0	0
$853$	30.9376	1.05928	0.529641	−	0.848222i	$-0.322327\pi$
$853$	30.9376	1.05928	0.529641	+	0.848222i	$0.322327\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−29.2166	−0.998021	−0.499010	−	0.866596i	$-0.666303\pi$
$857$	−29.2166	−0.998021	−0.499010	+	0.866596i	$0.666303\pi$
$858$	0	0
$859$	−42.5991	−1.45346	−0.726731	−	0.686922i	$-0.758961\pi$
$859$	−42.5991	−1.45346	−0.726731	+	0.686922i	$0.758961\pi$
$860$	0	0
$861$	−10.1275	−0.345143
$862$	0	0
$863$	−0.0411558	−0.00140096	−0.000700480	−	1.00000i	$-0.500223\pi$
$863$	−0.0411558	−0.00140096	−0.000700480		1.00000i	$0.500223\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−62.1760	−2.11161
$868$	0	0
$869$	−17.0310	−0.577736
$870$	0	0
$871$	−27.6933	−0.938351
$872$	0	0
$873$	−32.7197	−1.10739
$874$	0	0
$875$	0	0
$876$	0	0
$877$	0.589083	0.0198919	0.00994595	−	0.999951i	$-0.496834\pi$
$877$	0.589083	0.0198919	0.00994595	+	0.999951i	$0.496834\pi$
$878$	0	0
$879$	−28.9865	−0.977692
$880$	0	0
$881$	−0.394884	−0.0133040	−0.00665200	−	0.999978i	$-0.502117\pi$
$881$	−0.394884	−0.0133040	−0.00665200	+	0.999978i	$0.502117\pi$
$882$	0	0
$883$	26.4128	0.888860	0.444430	−	0.895814i	$-0.353406\pi$
$883$	26.4128	0.888860	0.444430	+	0.895814i	$0.353406\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	29.4493	0.988809	0.494405	−	0.869232i	$-0.335386\pi$
$887$	29.4493	0.988809	0.494405	+	0.869232i	$0.335386\pi$
$888$	0	0
$889$	−9.56849	−0.320917
$890$	0	0
$891$	−12.4275	−0.416338
$892$	0	0
$893$	−5.74056	−0.192101
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−8.69417	−0.290290
$898$	0	0
$899$	−9.24349	−0.308288
$900$	0	0
$901$	−33.1510	−1.10442
$902$	0	0
$903$	−8.14465	−0.271037
$904$	0	0
$905$	0	0
$906$	0	0
$907$	34.3520	1.14064	0.570319	−	0.821423i	$-0.306819\pi$
$907$	34.3520	1.14064	0.570319	+	0.821423i	$0.306819\pi$
$908$	0	0
$909$	32.1324	1.06576
$910$	0	0
$911$	−2.10204	−0.0696437	−0.0348218	−	0.999394i	$-0.511086\pi$
$911$	−2.10204	−0.0696437	−0.0348218	+	0.999394i	$0.511086\pi$
$912$	0	0
$913$	6.54224	0.216516
$914$	0	0
$915$	0	0
$916$	0	0
$917$	15.4688	0.510825
$918$	0	0
$919$	−31.2470	−1.03074	−0.515372	−	0.856967i	$-0.672346\pi$
$919$	−31.2470	−1.03074	−0.515372	+	0.856967i	$0.672346\pi$
$920$	0	0
$921$	−20.9586	−0.690611
$922$	0	0
$923$	8.92696	0.293834
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−32.0451	−1.05250
$928$	0	0
$929$	−39.1073	−1.28307	−0.641535	−	0.767094i	$-0.721702\pi$
$929$	−39.1073	−1.28307	−0.641535	+	0.767094i	$0.721702\pi$
$930$	0	0
$931$	−34.9003	−1.14381
$932$	0	0
$933$	30.3349	0.993119
$934$	0	0
$935$	0	0
$936$	0	0
$937$	43.9556	1.43597	0.717983	−	0.696060i	$-0.245065\pi$
$937$	43.9556	1.43597	0.717983	+	0.696060i	$0.245065\pi$
$938$	0	0
$939$	35.1767	1.14795
$940$	0	0
$941$	16.8894	0.550579	0.275290	−	0.961361i	$-0.411226\pi$
$941$	16.8894	0.550579	0.275290	+	0.961361i	$0.411226\pi$
$942$	0	0
$943$	7.70088	0.250775
$944$	0	0
$945$	0	0
$946$	0	0
$947$	45.2730	1.47118	0.735588	−	0.677429i	$-0.236906\pi$
$947$	45.2730	1.47118	0.735588	+	0.677429i	$0.236906\pi$
$948$	0	0
$949$	0.487117	0.0158125
$950$	0	0
$951$	−55.0833	−1.78620
$952$	0	0
$953$	−23.2174	−0.752084	−0.376042	−	0.926603i	$-0.622715\pi$
$953$	−23.2174	−0.752084	−0.376042	+	0.926603i	$0.622715\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	5.16300	0.166896
$958$	0	0
$959$	−0.347637	−0.0112258
$960$	0	0
$961$	−8.19867	−0.264473
$962$	0	0
$963$	24.0693	0.775621
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−43.0803	−1.38537	−0.692685	−	0.721240i	$-0.743572\pi$
$967$	−43.0803	−1.38537	−0.692685	+	0.721240i	$0.743572\pi$
$968$	0	0
$969$	86.8526	2.79011
$970$	0	0
$971$	−19.7402	−0.633495	−0.316747	−	0.948510i	$-0.602591\pi$
$971$	−19.7402	−0.633495	−0.316747	+	0.948510i	$0.602591\pi$
$972$	0	0
$973$	6.92827	0.222110
$974$	0	0
$975$	0	0
$976$	0	0
$977$	11.2292	0.359254	0.179627	−	0.983735i	$-0.442511\pi$
$977$	11.2292	0.359254	0.179627	+	0.983735i	$0.442511\pi$
$978$	0	0
$979$	12.5696	0.401725
$980$	0	0
$981$	42.1791	1.34667
$982$	0	0
$983$	38.7873	1.23712	0.618561	−	0.785737i	$-0.287716\pi$
$983$	38.7873	1.23712	0.618561	+	0.785737i	$0.287716\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	2.16587	0.0689406
$988$	0	0
$989$	6.19316	0.196931
$990$	0	0
$991$	37.1021	1.17859	0.589294	−	0.807919i	$-0.299406\pi$
$991$	37.1021	1.17859	0.589294	+	0.807919i	$0.299406\pi$
$992$	0	0
$993$	33.7991	1.07258
$994$	0	0
$995$	0	0
$996$	0	0
$997$	5.14469	0.162934	0.0814670	−	0.996676i	$-0.474039\pi$
$997$	5.14469	0.162934	0.0814670	+	0.996676i	$0.474039\pi$
$998$	0	0
$999$	−13.7840	−0.436106

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4000.2.a.n.1.2	yes	6
4.3	odd	2		4000.2.a.k.1.5	✓	6
5.2	odd	4		4000.2.c.g.1249.10		12
5.3	odd	4		4000.2.c.g.1249.3		12
5.4	even	2		4000.2.a.l.1.5	yes	6
8.3	odd	2		8000.2.a.bv.1.2		6
8.5	even	2		8000.2.a.bw.1.5		6
20.3	even	4		4000.2.c.f.1249.10		12
20.7	even	4		4000.2.c.f.1249.3		12
20.19	odd	2		4000.2.a.m.1.2	yes	6
40.19	odd	2		8000.2.a.bx.1.5		6
40.29	even	2		8000.2.a.bu.1.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4000.2.a.k.1.5	✓	6	4.3	odd	2
4000.2.a.l.1.5	yes	6	5.4	even	2
4000.2.a.m.1.2	yes	6	20.19	odd	2
4000.2.a.n.1.2	yes	6	1.1	even	1	trivial
4000.2.c.f.1249.3		12	20.7	even	4
4000.2.c.f.1249.10		12	20.3	even	4
4000.2.c.g.1249.3		12	5.3	odd	4
4000.2.c.g.1249.10		12	5.2	odd	4
8000.2.a.bu.1.2		6	40.29	even	2
8000.2.a.bv.1.2		6	8.3	odd	2
8000.2.a.bw.1.5		6	8.5	even	2
8000.2.a.bx.1.5		6	40.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	4000.2.a.n.1.2

Level	$4000$
Weight	$2$
Character	4000.1
Self dual	yes
Analytic conductor	$31.940$
Analytic rank	$0$
Dimension	$6$
CM	no
Inner twists	$1$