LMFDB - Embedded newform 4004.2.a.d.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4004,2,Mod(1,4004)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4004, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4004.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4004.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:	$31.9721009693$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.3981.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 4x^{2} + 2x + 1 $ x^4 - x^3 - 4x^2 + 2x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.785261$ of defining polynomial
Character	$\chi$	$=$	4004.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.488200 q^{3} -3.65683 q^{5} +1.00000 q^{7} -2.76166 q^{9} +O(q^{10})$	$q+0.488200 q^{3} -3.65683 q^{5} +1.00000 q^{7} -2.76166 q^{9} +1.00000 q^{11} +1.00000 q^{13} -1.78526 q^{15} +1.65683 q^{17} -0.616636 q^{19} +0.488200 q^{21} +4.52441 q^{23} +8.37237 q^{25} -2.81284 q^{27} +3.33218 q^{29} +3.54692 q^{31} +0.488200 q^{33} -3.65683 q^{35} -9.82038 q^{37} +0.488200 q^{39} +1.93535 q^{41} +0.00506793 q^{43} +10.0989 q^{45} +4.15009 q^{47} +1.00000 q^{49} +0.808862 q^{51} +3.08123 q^{53} -3.65683 q^{55} -0.301041 q^{57} -9.09891 q^{59} -11.8069 q^{61} -2.76166 q^{63} -3.65683 q^{65} -7.88417 q^{67} +2.20882 q^{69} -7.09384 q^{71} +1.62418 q^{73} +4.08739 q^{75} +1.00000 q^{77} -2.27237 q^{79} +6.91175 q^{81} -2.98503 q^{83} -6.05872 q^{85} +1.62677 q^{87} -9.40858 q^{89} +1.00000 q^{91} +1.73161 q^{93} +2.25493 q^{95} -0.425497 q^{97} -2.76166 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 3 q^{3} - q^{5} + 4 q^{7} + q^{9}+O(q^{10}) $ 4 * q - 3 * q^3 - q^5 + 4 * q^7 + q^9	$ 4 q - 3 q^{3} - q^{5} + 4 q^{7} + q^{9} + 4 q^{11} + 4 q^{13} - 5 q^{15} - 7 q^{17} - 9 q^{19} - 3 q^{21} + 3 q^{23} - 3 q^{25} - 9 q^{27} - 3 q^{29} - 3 q^{33} - q^{35} - 18 q^{37} - 3 q^{39} - 14 q^{41} - q^{43} + 11 q^{45} - 3 q^{47} + 4 q^{49} + 11 q^{51} + 4 q^{53} - q^{55} + 6 q^{57} - 7 q^{59} - 14 q^{61} + q^{63} - q^{65} - 20 q^{69} - 6 q^{73} + 16 q^{75} + 4 q^{77} + 7 q^{79} - 4 q^{81} + 8 q^{83} - 15 q^{85} + 11 q^{87} + 9 q^{89} + 4 q^{91} + 13 q^{93} - 9 q^{95} - 16 q^{97} + q^{99}+O(q^{100}) $ 4 * q - 3 * q^3 - q^5 + 4 * q^7 + q^9 + 4 * q^11 + 4 * q^13 - 5 * q^15 - 7 * q^17 - 9 * q^19 - 3 * q^21 + 3 * q^23 - 3 * q^25 - 9 * q^27 - 3 * q^29 - 3 * q^33 - q^35 - 18 * q^37 - 3 * q^39 - 14 * q^41 - q^43 + 11 * q^45 - 3 * q^47 + 4 * q^49 + 11 * q^51 + 4 * q^53 - q^55 + 6 * q^57 - 7 * q^59 - 14 * q^61 + q^63 - q^65 - 20 * q^69 - 6 * q^73 + 16 * q^75 + 4 * q^77 + 7 * q^79 - 4 * q^81 + 8 * q^83 - 15 * q^85 + 11 * q^87 + 9 * q^89 + 4 * q^91 + 13 * q^93 - 9 * q^95 - 16 * q^97 + 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
$3$	0.488200	0.281862	0.140931	−	0.990019i	$-0.454990\pi$
$3$	0.488200	0.281862	0.140931	+	0.990019i	$0.454990\pi$
$4$	0	0
$5$	−3.65683	−1.63538	−0.817691	−	0.575657i	$-0.804746\pi$
$5$	−3.65683	−1.63538	−0.817691	+	0.575657i	$0.804746\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−2.76166	−0.920554
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	−1.78526	−0.460953
$16$	0	0
$17$	1.65683	0.401839	0.200920	−	0.979608i	$-0.435607\pi$
$17$	1.65683	0.401839	0.200920	+	0.979608i	$0.435607\pi$
$18$	0	0
$19$	−0.616636	−0.141466	−0.0707329	−	0.997495i	$-0.522534\pi$
$19$	−0.616636	−0.141466	−0.0707329	+	0.997495i	$0.522534\pi$
$20$	0	0
$21$	0.488200	0.106534
$22$	0	0
$23$	4.52441	0.943405	0.471702	−	0.881758i	$-0.343640\pi$
$23$	4.52441	0.943405	0.471702	+	0.881758i	$0.343640\pi$
$24$	0	0
$25$	8.37237	1.67447
$26$	0	0
$27$	−2.81284	−0.541332
$28$	0	0
$29$	3.33218	0.618771	0.309386	−	0.950937i	$-0.399877\pi$
$29$	3.33218	0.618771	0.309386	+	0.950937i	$0.399877\pi$
$30$	0	0
$31$	3.54692	0.637046	0.318523	−	0.947915i	$-0.396813\pi$
$31$	3.54692	0.637046	0.318523	+	0.947915i	$0.396813\pi$
$32$	0	0
$33$	0.488200	0.0849847
$34$	0	0
$35$	−3.65683	−0.618116
$36$	0	0
$37$	−9.82038	−1.61446	−0.807231	−	0.590236i	$-0.799035\pi$
$37$	−9.82038	−1.61446	−0.807231	+	0.590236i	$0.799035\pi$
$38$	0	0
$39$	0.488200	0.0781745
$40$	0	0
$41$	1.93535	0.302252	0.151126	−	0.988515i	$-0.451710\pi$
$41$	1.93535	0.302252	0.151126	+	0.988515i	$0.451710\pi$
$42$	0	0
$43$	0.00506793	0.000772852 0	0.000386426	−	1.00000i	$-0.499877\pi$
$43$	0.00506793	0.000772852 0	0.000386426		1.00000i	$0.499877\pi$
$44$	0	0
$45$	10.0989	1.50546
$46$	0	0
$47$	4.15009	0.605353	0.302677	−	0.953093i	$-0.402120\pi$
$47$	4.15009	0.605353	0.302677	+	0.953093i	$0.402120\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0.808862	0.113263
$52$	0	0
$53$	3.08123	0.423240	0.211620	−	0.977352i	$-0.432126\pi$
$53$	3.08123	0.423240	0.211620	+	0.977352i	$0.432126\pi$
$54$	0	0
$55$	−3.65683	−0.493086
$56$	0	0
$57$	−0.301041	−0.0398739
$58$	0	0
$59$	−9.09891	−1.18458	−0.592289	−	0.805726i	$-0.701776\pi$
$59$	−9.09891	−1.18458	−0.592289	+	0.805726i	$0.701776\pi$
$60$	0	0
$61$	−11.8069	−1.51172	−0.755861	−	0.654733i	$-0.772781\pi$
$61$	−11.8069	−1.51172	−0.755861	+	0.654733i	$0.772781\pi$
$62$	0	0
$63$	−2.76166	−0.347937
$64$	0	0
$65$	−3.65683	−0.453573
$66$	0	0
$67$	−7.88417	−0.963205	−0.481603	−	0.876390i	$-0.659945\pi$
$67$	−7.88417	−0.963205	−0.481603	+	0.876390i	$0.659945\pi$
$68$	0	0
$69$	2.20882	0.265910
$70$	0	0
$71$	−7.09384	−0.841884	−0.420942	−	0.907087i	$-0.638300\pi$
$71$	−7.09384	−0.841884	−0.420942	+	0.907087i	$0.638300\pi$
$72$	0	0
$73$	1.62418	0.190096	0.0950478	−	0.995473i	$-0.469700\pi$
$73$	1.62418	0.190096	0.0950478	+	0.995473i	$0.469700\pi$
$74$	0	0
$75$	4.08739	0.471971
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	−2.27237	−0.255662	−0.127831	−	0.991796i	$-0.540801\pi$
$79$	−2.27237	−0.255662	−0.127831	+	0.991796i	$0.540801\pi$
$80$	0	0
$81$	6.91175	0.767973
$82$	0	0
$83$	−2.98503	−0.327650	−0.163825	−	0.986489i	$-0.552383\pi$
$83$	−2.98503	−0.327650	−0.163825	+	0.986489i	$0.552383\pi$
$84$	0	0
$85$	−6.05872	−0.657161
$86$	0	0
$87$	1.62677	0.174408
$88$	0	0
$89$	−9.40858	−0.997308	−0.498654	−	0.866801i	$-0.666172\pi$
$89$	−9.40858	−0.997308	−0.498654	+	0.866801i	$0.666172\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	1.73161	0.179559
$94$	0	0
$95$	2.25493	0.231351
$96$	0	0
$97$	−0.425497	−0.0432027	−0.0216014	−	0.999767i	$-0.506876\pi$
$97$	−0.425497	−0.0432027	−0.0216014	+	0.999767i	$0.506876\pi$
$98$	0	0
$99$	−2.76166	−0.277557
$100$	0	0
$101$	0.922745	0.0918165	0.0459083	−	0.998946i	$-0.485382\pi$
$101$	0.922745	0.0918165	0.0459083	+	0.998946i	$0.485382\pi$
$102$	0	0
$103$	−16.1903	−1.59528	−0.797638	−	0.603136i	$-0.793917\pi$
$103$	−16.1903	−1.59528	−0.797638	+	0.603136i	$0.793917\pi$
$104$	0	0
$105$	−1.78526	−0.174224
$106$	0	0
$107$	−5.72568	−0.553523	−0.276761	−	0.960939i	$-0.589261\pi$
$107$	−5.72568	−0.553523	−0.276761	+	0.960939i	$0.589261\pi$
$108$	0	0
$109$	0.790329	0.0756998	0.0378499	−	0.999283i	$-0.487949\pi$
$109$	0.790329	0.0756998	0.0378499	+	0.999283i	$0.487949\pi$
$110$	0	0
$111$	−4.79431	−0.455056
$112$	0	0
$113$	7.44716	0.700569	0.350285	−	0.936643i	$-0.386085\pi$
$113$	7.44716	0.700569	0.350285	+	0.936643i	$0.386085\pi$
$114$	0	0
$115$	−16.5450	−1.54283
$116$	0	0
$117$	−2.76166	−0.255316
$118$	0	0
$119$	1.65683	0.151881
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0.944840	0.0851933
$124$	0	0
$125$	−12.3322	−1.10302
$126$	0	0
$127$	−10.0667	−0.893274	−0.446637	−	0.894715i	$-0.647378\pi$
$127$	−10.0667	−0.893274	−0.446637	+	0.894715i	$0.647378\pi$
$128$	0	0
$129$	0.00247416	0.000217838 0
$130$	0	0
$131$	−4.79431	−0.418881	−0.209440	−	0.977821i	$-0.567164\pi$
$131$	−4.79431	−0.418881	−0.209440	+	0.977821i	$0.567164\pi$
$132$	0	0
$133$	−0.616636	−0.0534691
$134$	0	0
$135$	10.2861	0.885284
$136$	0	0
$137$	2.93029	0.250351	0.125176	−	0.992135i	$-0.460051\pi$
$137$	2.93029	0.250351	0.125176	+	0.992135i	$0.460051\pi$
$138$	0	0
$139$	−14.0031	−1.18773	−0.593865	−	0.804565i	$-0.702399\pi$
$139$	−14.0031	−1.18773	−0.593865	+	0.804565i	$0.702399\pi$
$140$	0	0
$141$	2.02607	0.170626
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	−12.1852	−1.01193
$146$	0	0
$147$	0.488200	0.0402660
$148$	0	0
$149$	5.33616	0.437156	0.218578	−	0.975820i	$-0.429858\pi$
$149$	5.33616	0.437156	0.218578	+	0.975820i	$0.429858\pi$
$150$	0	0
$151$	0.753031	0.0612808	0.0306404	−	0.999530i	$-0.490245\pi$
$151$	0.753031	0.0612808	0.0306404	+	0.999530i	$0.490245\pi$
$152$	0	0
$153$	−4.57559	−0.369915
$154$	0	0
$155$	−12.9705	−1.04181
$156$	0	0
$157$	−6.24697	−0.498562	−0.249281	−	0.968431i	$-0.580194\pi$
$157$	−6.24697	−0.498562	−0.249281	+	0.968431i	$0.580194\pi$
$158$	0	0
$159$	1.50426	0.119295
$160$	0	0
$161$	4.52441	0.356573
$162$	0	0
$163$	−0.511800	−0.0400873	−0.0200436	−	0.999799i	$-0.506381\pi$
$163$	−0.511800	−0.0400873	−0.0200436	+	0.999799i	$0.506381\pi$
$164$	0	0
$165$	−1.78526	−0.138982
$166$	0	0
$167$	−13.9879	−1.08242	−0.541209	−	0.840888i	$-0.682033\pi$
$167$	−13.9879	−1.08242	−0.541209	+	0.840888i	$0.682033\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	1.70294	0.130227
$172$	0	0
$173$	−1.56805	−0.119217	−0.0596083	−	0.998222i	$-0.518985\pi$
$173$	−1.56805	−0.119217	−0.0596083	+	0.998222i	$0.518985\pi$
$174$	0	0
$175$	8.37237	0.632892
$176$	0	0
$177$	−4.44209	−0.333888
$178$	0	0
$179$	9.65338	0.721527	0.360764	−	0.932657i	$-0.382516\pi$
$179$	9.65338	0.721527	0.360764	+	0.932657i	$0.382516\pi$
$180$	0	0
$181$	11.9871	0.890992	0.445496	−	0.895284i	$-0.353027\pi$
$181$	11.9871	0.890992	0.445496	+	0.895284i	$0.353027\pi$
$182$	0	0
$183$	−5.76414	−0.426097
$184$	0	0
$185$	35.9114	2.64026
$186$	0	0
$187$	1.65683	0.121159
$188$	0	0
$189$	−2.81284	−0.204604
$190$	0	0
$191$	8.99517	0.650867	0.325434	−	0.945565i	$-0.394490\pi$
$191$	8.99517	0.650867	0.325434	+	0.945565i	$0.394490\pi$
$192$	0	0
$193$	−5.91617	−0.425855	−0.212928	−	0.977068i	$-0.568300\pi$
$193$	−5.91617	−0.425855	−0.212928	+	0.977068i	$0.568300\pi$
$194$	0	0
$195$	−1.78526	−0.127845
$196$	0	0
$197$	−4.74905	−0.338356	−0.169178	−	0.985586i	$-0.554111\pi$
$197$	−4.74905	−0.338356	−0.169178	+	0.985586i	$0.554111\pi$
$198$	0	0
$199$	3.63237	0.257492	0.128746	−	0.991678i	$-0.458905\pi$
$199$	3.63237	0.257492	0.128746	+	0.991678i	$0.458905\pi$
$200$	0	0
$201$	−3.84905	−0.271491
$202$	0	0
$203$	3.33218	0.233873
$204$	0	0
$205$	−7.07726	−0.494297
$206$	0	0
$207$	−12.4949	−0.868455
$208$	0	0
$209$	−0.616636	−0.0426536
$210$	0	0
$211$	−25.6986	−1.76917	−0.884584	−	0.466382i	$-0.845557\pi$
$211$	−25.6986	−1.76917	−0.884584	+	0.466382i	$0.845557\pi$
$212$	0	0
$213$	−3.46321	−0.237295
$214$	0	0
$215$	−0.0185325	−0.00126391
$216$	0	0
$217$	3.54692	0.240781
$218$	0	0
$219$	0.792923	0.0535808
$220$	0	0
$221$	1.65683	0.111450
$222$	0	0
$223$	3.05763	0.204754	0.102377	−	0.994746i	$-0.467355\pi$
$223$	3.05763	0.204754	0.102377	+	0.994746i	$0.467355\pi$
$224$	0	0
$225$	−23.1217	−1.54144
$226$	0	0
$227$	−15.1301	−1.00422	−0.502109	−	0.864805i	$-0.667442\pi$
$227$	−15.1301	−1.00422	−0.502109	+	0.864805i	$0.667442\pi$
$228$	0	0
$229$	−11.3659	−0.751081	−0.375541	−	0.926806i	$-0.622543\pi$
$229$	−11.3659	−0.751081	−0.375541	+	0.926806i	$0.622543\pi$
$230$	0	0
$231$	0.488200	0.0321212
$232$	0	0
$233$	−6.13006	−0.401593	−0.200797	−	0.979633i	$-0.564353\pi$
$233$	−6.13006	−0.401593	−0.200797	+	0.979633i	$0.564353\pi$
$234$	0	0
$235$	−15.1762	−0.989984
$236$	0	0
$237$	−1.10937	−0.0720615
$238$	0	0
$239$	−19.9656	−1.29147	−0.645735	−	0.763562i	$-0.723449\pi$
$239$	−19.9656	−1.29147	−0.645735	+	0.763562i	$0.723449\pi$
$240$	0	0
$241$	−15.7116	−1.01207	−0.506036	−	0.862513i	$-0.668890\pi$
$241$	−15.7116	−1.01207	−0.506036	+	0.862513i	$0.668890\pi$
$242$	0	0
$243$	11.8128	0.757794
$244$	0	0
$245$	−3.65683	−0.233626
$246$	0	0
$247$	−0.616636	−0.0392356
$248$	0	0
$249$	−1.45729	−0.0923521
$250$	0	0
$251$	8.24285	0.520284	0.260142	−	0.965570i	$-0.416231\pi$
$251$	8.24285	0.520284	0.260142	+	0.965570i	$0.416231\pi$
$252$	0	0
$253$	4.52441	0.284447
$254$	0	0
$255$	−2.95787	−0.185229
$256$	0	0
$257$	−14.4348	−0.900421	−0.450210	−	0.892923i	$-0.648651\pi$
$257$	−14.4348	−0.900421	−0.450210	+	0.892923i	$0.648651\pi$
$258$	0	0
$259$	−9.82038	−0.610209
$260$	0	0
$261$	−9.20236	−0.569612
$262$	0	0
$263$	26.9072	1.65917	0.829585	−	0.558380i	$-0.188577\pi$
$263$	26.9072	1.65917	0.829585	+	0.558380i	$0.188577\pi$
$264$	0	0
$265$	−11.2675	−0.692159
$266$	0	0
$267$	−4.59327	−0.281103
$268$	0	0
$269$	13.7732	0.839766	0.419883	−	0.907578i	$-0.362071\pi$
$269$	13.7732	0.839766	0.419883	+	0.907578i	$0.362071\pi$
$270$	0	0
$271$	−7.43508	−0.451649	−0.225824	−	0.974168i	$-0.572508\pi$
$271$	−7.43508	−0.451649	−0.225824	+	0.974168i	$0.572508\pi$
$272$	0	0
$273$	0.488200	0.0295472
$274$	0	0
$275$	8.37237	0.504873
$276$	0	0
$277$	9.49218	0.570330	0.285165	−	0.958478i	$-0.407952\pi$
$277$	9.49218	0.570330	0.285165	+	0.958478i	$0.407952\pi$
$278$	0	0
$279$	−9.79540	−0.586435
$280$	0	0
$281$	3.72816	0.222403	0.111202	−	0.993798i	$-0.464530\pi$
$281$	3.72816	0.222403	0.111202	+	0.993798i	$0.464530\pi$
$282$	0	0
$283$	−16.7408	−0.995135	−0.497568	−	0.867425i	$-0.665773\pi$
$283$	−16.7408	−0.995135	−0.497568	+	0.867425i	$0.665773\pi$
$284$	0	0
$285$	1.10086	0.0652091
$286$	0	0
$287$	1.93535	0.114240
$288$	0	0
$289$	−14.2549	−0.838525
$290$	0	0
$291$	−0.207728	−0.0121772
$292$	0	0
$293$	−24.3410	−1.42202	−0.711010	−	0.703182i	$-0.751762\pi$
$293$	−24.3410	−1.42202	−0.711010	+	0.703182i	$0.751762\pi$
$294$	0	0
$295$	33.2731	1.93724
$296$	0	0
$297$	−2.81284	−0.163218
$298$	0	0
$299$	4.52441	0.261653
$300$	0	0
$301$	0.00506793	0.000292111 0
$302$	0	0
$303$	0.450484	0.0258796
$304$	0	0
$305$	43.1758	2.47224
$306$	0	0
$307$	−2.62483	−0.149807	−0.0749034	−	0.997191i	$-0.523865\pi$
$307$	−2.62483	−0.149807	−0.0749034	+	0.997191i	$0.523865\pi$
$308$	0	0
$309$	−7.90409	−0.449648
$310$	0	0
$311$	−2.28047	−0.129314	−0.0646569	−	0.997908i	$-0.520595\pi$
$311$	−2.28047	−0.129314	−0.0646569	+	0.997908i	$0.520595\pi$
$312$	0	0
$313$	22.0665	1.24727	0.623636	−	0.781715i	$-0.285655\pi$
$313$	22.0665	1.24727	0.623636	+	0.781715i	$0.285655\pi$
$314$	0	0
$315$	10.0989	0.569009
$316$	0	0
$317$	23.2375	1.30515	0.652574	−	0.757725i	$-0.273689\pi$
$317$	23.2375	1.30515	0.652574	+	0.757725i	$0.273689\pi$
$318$	0	0
$319$	3.33218	0.186566
$320$	0	0
$321$	−2.79528	−0.156017
$322$	0	0
$323$	−1.02166	−0.0568465
$324$	0	0
$325$	8.37237	0.464416
$326$	0	0
$327$	0.385839	0.0213369
$328$	0	0
$329$	4.15009	0.228802
$330$	0	0
$331$	12.7925	0.703138	0.351569	−	0.936162i	$-0.385648\pi$
$331$	12.7925	0.703138	0.351569	+	0.936162i	$0.385648\pi$
$332$	0	0
$333$	27.1206	1.48620
$334$	0	0
$335$	28.8311	1.57521
$336$	0	0
$337$	−23.3785	−1.27351	−0.636755	−	0.771066i	$-0.719724\pi$
$337$	−23.3785	−1.27351	−0.636755	+	0.771066i	$0.719724\pi$
$338$	0	0
$339$	3.63570	0.197464
$340$	0	0
$341$	3.54692	0.192077
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−8.07726	−0.434865
$346$	0	0
$347$	17.9294	0.962502	0.481251	−	0.876583i	$-0.340182\pi$
$347$	17.9294	0.962502	0.481251	+	0.876583i	$0.340182\pi$
$348$	0	0
$349$	−6.29037	−0.336716	−0.168358	−	0.985726i	$-0.553846\pi$
$349$	−6.29037	−0.336716	−0.168358	+	0.985726i	$0.553846\pi$
$350$	0	0
$351$	−2.81284	−0.150138
$352$	0	0
$353$	−8.86691	−0.471938	−0.235969	−	0.971761i	$-0.575826\pi$
$353$	−8.86691	−0.471938	−0.235969	+	0.971761i	$0.575826\pi$
$354$	0	0
$355$	25.9410	1.37680
$356$	0	0
$357$	0.808862	0.0428095
$358$	0	0
$359$	8.28089	0.437049	0.218524	−	0.975832i	$-0.429876\pi$
$359$	8.28089	0.437049	0.218524	+	0.975832i	$0.429876\pi$
$360$	0	0
$361$	−18.6198	−0.979987
$362$	0	0
$363$	0.488200	0.0256238
$364$	0	0
$365$	−5.93933	−0.310879
$366$	0	0
$367$	14.6335	0.763860	0.381930	−	0.924191i	$-0.375259\pi$
$367$	14.6335	0.763860	0.381930	+	0.924191i	$0.375259\pi$
$368$	0	0
$369$	−5.34479	−0.278239
$370$	0	0
$371$	3.08123	0.159970
$372$	0	0
$373$	18.4342	0.954488	0.477244	−	0.878771i	$-0.341636\pi$
$373$	18.4342	0.954488	0.477244	+	0.878771i	$0.341636\pi$
$374$	0	0
$375$	−6.02057	−0.310901
$376$	0	0
$377$	3.33218	0.171616
$378$	0	0
$379$	5.72568	0.294109	0.147054	−	0.989128i	$-0.453021\pi$
$379$	5.72568	0.294109	0.147054	+	0.989128i	$0.453021\pi$
$380$	0	0
$381$	−4.91455	−0.251780
$382$	0	0
$383$	−11.0839	−0.566363	−0.283182	−	0.959066i	$-0.591390\pi$
$383$	−11.0839	−0.566363	−0.283182	+	0.959066i	$0.591390\pi$
$384$	0	0
$385$	−3.65683	−0.186369
$386$	0	0
$387$	−0.0139959	−0.000711452 0
$388$	0	0
$389$	3.84484	0.194941	0.0974705	−	0.995238i	$-0.468925\pi$
$389$	3.84484	0.194941	0.0974705	+	0.995238i	$0.468925\pi$
$390$	0	0
$391$	7.49616	0.379097
$392$	0	0
$393$	−2.34058	−0.118067
$394$	0	0
$395$	8.30967	0.418105
$396$	0	0
$397$	−26.2400	−1.31695	−0.658473	−	0.752604i	$-0.728797\pi$
$397$	−26.2400	−1.31695	−0.658473	+	0.752604i	$0.728797\pi$
$398$	0	0
$399$	−0.301041	−0.0150709
$400$	0	0
$401$	−37.2339	−1.85937	−0.929686	−	0.368353i	$-0.879922\pi$
$401$	−37.2339	−1.85937	−0.929686	+	0.368353i	$0.879922\pi$
$402$	0	0
$403$	3.54692	0.176685
$404$	0	0
$405$	−25.2751	−1.25593
$406$	0	0
$407$	−9.82038	−0.486778
$408$	0	0
$409$	12.2338	0.604923	0.302461	−	0.953162i	$-0.402192\pi$
$409$	12.2338	0.604923	0.302461	+	0.953162i	$0.402192\pi$
$410$	0	0
$411$	1.43057	0.0705646
$412$	0	0
$413$	−9.09891	−0.447728
$414$	0	0
$415$	10.9157	0.535832
$416$	0	0
$417$	−6.83632	−0.334776
$418$	0	0
$419$	16.0195	0.782603	0.391301	−	0.920263i	$-0.372025\pi$
$419$	16.0195	0.782603	0.391301	+	0.920263i	$0.372025\pi$
$420$	0	0
$421$	−32.4839	−1.58317	−0.791584	−	0.611060i	$-0.790743\pi$
$421$	−32.4839	−1.58317	−0.791584	+	0.611060i	$0.790743\pi$
$422$	0	0
$423$	−11.4612	−0.557260
$424$	0	0
$425$	13.8716	0.672870
$426$	0	0
$427$	−11.8069	−0.571377
$428$	0	0
$429$	0.488200	0.0235705
$430$	0	0
$431$	38.0871	1.83459	0.917295	−	0.398209i	$-0.130368\pi$
$431$	38.0871	1.83459	0.917295	+	0.398209i	$0.130368\pi$
$432$	0	0
$433$	−26.1170	−1.25510	−0.627552	−	0.778575i	$-0.715943\pi$
$433$	−26.1170	−1.25510	−0.627552	+	0.778575i	$0.715943\pi$
$434$	0	0
$435$	−5.94882	−0.285224
$436$	0	0
$437$	−2.78991	−0.133460
$438$	0	0
$439$	8.35048	0.398547	0.199273	−	0.979944i	$-0.436142\pi$
$439$	8.35048	0.398547	0.199273	+	0.979944i	$0.436142\pi$
$440$	0	0
$441$	−2.76166	−0.131508
$442$	0	0
$443$	6.23380	0.296177	0.148088	−	0.988974i	$-0.452688\pi$
$443$	6.23380	0.296177	0.148088	+	0.988974i	$0.452688\pi$
$444$	0	0
$445$	34.4056	1.63098
$446$	0	0
$447$	2.60511	0.123218
$448$	0	0
$449$	18.5150	0.873779	0.436889	−	0.899515i	$-0.356080\pi$
$449$	18.5150	0.873779	0.436889	+	0.899515i	$0.356080\pi$
$450$	0	0
$451$	1.93535	0.0911323
$452$	0	0
$453$	0.367629	0.0172727
$454$	0	0
$455$	−3.65683	−0.171435
$456$	0	0
$457$	−25.4717	−1.19152	−0.595758	−	0.803164i	$-0.703148\pi$
$457$	−25.4717	−1.19152	−0.595758	+	0.803164i	$0.703148\pi$
$458$	0	0
$459$	−4.66039	−0.217528
$460$	0	0
$461$	21.3319	0.993524	0.496762	−	0.867887i	$-0.334522\pi$
$461$	21.3319	0.993524	0.496762	+	0.867887i	$0.334522\pi$
$462$	0	0
$463$	39.4799	1.83479	0.917394	−	0.397981i	$-0.130289\pi$
$463$	39.4799	1.83479	0.917394	+	0.397981i	$0.130289\pi$
$464$	0	0
$465$	−6.33218	−0.293648
$466$	0	0
$467$	−12.8275	−0.593585	−0.296793	−	0.954942i	$-0.595917\pi$
$467$	−12.8275	−0.593585	−0.296793	+	0.954942i	$0.595917\pi$
$468$	0	0
$469$	−7.88417	−0.364057
$470$	0	0
$471$	−3.04977	−0.140526
$472$	0	0
$473$	0.00506793	0.000233024 0
$474$	0	0
$475$	−5.16270	−0.236881
$476$	0	0
$477$	−8.50933	−0.389615
$478$	0	0
$479$	−12.3461	−0.564106	−0.282053	−	0.959399i	$-0.591015\pi$
$479$	−12.3461	−0.564106	−0.282053	+	0.959399i	$0.591015\pi$
$480$	0	0
$481$	−9.82038	−0.447771
$482$	0	0
$483$	2.20882	0.100505
$484$	0	0
$485$	1.55597	0.0706530
$486$	0	0
$487$	24.8736	1.12713	0.563565	−	0.826072i	$-0.309429\pi$
$487$	24.8736	1.12713	0.563565	+	0.826072i	$0.309429\pi$
$488$	0	0
$489$	−0.249861	−0.0112991
$490$	0	0
$491$	−0.138572	−0.00625365	−0.00312683	−	0.999995i	$-0.500995\pi$
$491$	−0.138572	−0.00625365	−0.00312683	+	0.999995i	$0.500995\pi$
$492$	0	0
$493$	5.52085	0.248646
$494$	0	0
$495$	10.0989	0.453912
$496$	0	0
$497$	−7.09384	−0.318202
$498$	0	0
$499$	11.1553	0.499379	0.249689	−	0.968326i	$-0.419671\pi$
$499$	11.1553	0.499379	0.249689	+	0.968326i	$0.419671\pi$
$500$	0	0
$501$	−6.82890	−0.305093
$502$	0	0
$503$	−13.6019	−0.606479	−0.303239	−	0.952914i	$-0.598068\pi$
$503$	−13.6019	−0.606479	−0.303239	+	0.952914i	$0.598068\pi$
$504$	0	0
$505$	−3.37432	−0.150155
$506$	0	0
$507$	0.488200	0.0216817
$508$	0	0
$509$	28.7158	1.27280	0.636402	−	0.771357i	$-0.280422\pi$
$509$	28.7158	1.27280	0.636402	+	0.771357i	$0.280422\pi$
$510$	0	0
$511$	1.62418	0.0718494
$512$	0	0
$513$	1.73450	0.0765800
$514$	0	0
$515$	59.2050	2.60889
$516$	0	0
$517$	4.15009	0.182521
$518$	0	0
$519$	−0.765521	−0.0336026
$520$	0	0
$521$	−35.7033	−1.56419	−0.782095	−	0.623160i	$-0.785849\pi$
$521$	−35.7033	−1.56419	−0.782095	+	0.623160i	$0.785849\pi$
$522$	0	0
$523$	−32.7274	−1.43107	−0.715535	−	0.698577i	$-0.753817\pi$
$523$	−32.7274	−1.43107	−0.715535	+	0.698577i	$0.753817\pi$
$524$	0	0
$525$	4.08739	0.178388
$526$	0	0
$527$	5.87663	0.255990
$528$	0	0
$529$	−2.52971	−0.109987
$530$	0	0
$531$	25.1281	1.09047
$532$	0	0
$533$	1.93535	0.0838295
$534$	0	0
$535$	20.9378	0.905221
$536$	0	0
$537$	4.71278	0.203371
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	15.6348	0.672194	0.336097	−	0.941827i	$-0.390893\pi$
$541$	15.6348	0.672194	0.336097	+	0.941827i	$0.390893\pi$
$542$	0	0
$543$	5.85208	0.251137
$544$	0	0
$545$	−2.89010	−0.123798
$546$	0	0
$547$	−8.16645	−0.349172	−0.174586	−	0.984642i	$-0.555859\pi$
$547$	−8.16645	−0.349172	−0.174586	+	0.984642i	$0.555859\pi$
$548$	0	0
$549$	32.6067	1.39162
$550$	0	0
$551$	−2.05474	−0.0875350
$552$	0	0
$553$	−2.27237	−0.0966311
$554$	0	0
$555$	17.5320	0.744190
$556$	0	0
$557$	−3.92834	−0.166449	−0.0832246	−	0.996531i	$-0.526522\pi$
$557$	−3.92834	−0.166449	−0.0832246	+	0.996531i	$0.526522\pi$
$558$	0	0
$559$	0.00506793	0.000214351 0
$560$	0	0
$561$	0.808862	0.0341502
$562$	0	0
$563$	−13.0377	−0.549474	−0.274737	−	0.961519i	$-0.588591\pi$
$563$	−13.0377	−0.549474	−0.274737	+	0.961519i	$0.588591\pi$
$564$	0	0
$565$	−27.2329	−1.14570
$566$	0	0
$567$	6.91175	0.290266
$568$	0	0
$569$	33.8094	1.41736	0.708682	−	0.705528i	$-0.249290\pi$
$569$	33.8094	1.41736	0.708682	+	0.705528i	$0.249290\pi$
$570$	0	0
$571$	−33.0999	−1.38519	−0.692594	−	0.721328i	$-0.743532\pi$
$571$	−33.0999	−1.38519	−0.692594	+	0.721328i	$0.743532\pi$
$572$	0	0
$573$	4.39144	0.183455
$574$	0	0
$575$	37.8801	1.57971
$576$	0	0
$577$	−45.4109	−1.89048	−0.945241	−	0.326374i	$-0.894173\pi$
$577$	−45.4109	−1.89048	−0.945241	+	0.326374i	$0.894173\pi$
$578$	0	0
$579$	−2.88827	−0.120033
$580$	0	0
$581$	−2.98503	−0.123840
$582$	0	0
$583$	3.08123	0.127612
$584$	0	0
$585$	10.0989	0.417539
$586$	0	0
$587$	−31.5775	−1.30334	−0.651672	−	0.758501i	$-0.725932\pi$
$587$	−31.5775	−1.30334	−0.651672	+	0.758501i	$0.725932\pi$
$588$	0	0
$589$	−2.18716	−0.0901203
$590$	0	0
$591$	−2.31849	−0.0953698
$592$	0	0
$593$	16.4482	0.675448	0.337724	−	0.941245i	$-0.390343\pi$
$593$	16.4482	0.675448	0.337724	+	0.941245i	$0.390343\pi$
$594$	0	0
$595$	−6.05872	−0.248383
$596$	0	0
$597$	1.77332	0.0725773
$598$	0	0
$599$	−16.8120	−0.686919	−0.343460	−	0.939167i	$-0.611599\pi$
$599$	−16.8120	−0.686919	−0.343460	+	0.939167i	$0.611599\pi$
$600$	0	0
$601$	28.2319	1.15160	0.575801	−	0.817590i	$-0.304690\pi$
$601$	28.2319	1.15160	0.575801	+	0.817590i	$0.304690\pi$
$602$	0	0
$603$	21.7734	0.886682
$604$	0	0
$605$	−3.65683	−0.148671
$606$	0	0
$607$	13.8888	0.563730	0.281865	−	0.959454i	$-0.409047\pi$
$607$	13.8888	0.563730	0.281865	+	0.959454i	$0.409047\pi$
$608$	0	0
$609$	1.62677	0.0659201
$610$	0	0
$611$	4.15009	0.167895
$612$	0	0
$613$	−43.9832	−1.77646	−0.888232	−	0.459395i	$-0.848066\pi$
$613$	−43.9832	−1.77646	−0.888232	+	0.459395i	$0.848066\pi$
$614$	0	0
$615$	−3.45511	−0.139324
$616$	0	0
$617$	−13.7366	−0.553013	−0.276506	−	0.961012i	$-0.589177\pi$
$617$	−13.7366	−0.553013	−0.276506	+	0.961012i	$0.589177\pi$
$618$	0	0
$619$	6.37980	0.256426	0.128213	−	0.991747i	$-0.459076\pi$
$619$	6.37980	0.256426	0.128213	+	0.991747i	$0.459076\pi$
$620$	0	0
$621$	−12.7264	−0.510695
$622$	0	0
$623$	−9.40858	−0.376947
$624$	0	0
$625$	3.23478	0.129391
$626$	0	0
$627$	−0.301041	−0.0120224
$628$	0	0
$629$	−16.2707	−0.648754
$630$	0	0
$631$	2.30622	0.0918093	0.0459047	−	0.998946i	$-0.485383\pi$
$631$	2.30622	0.0918093	0.0459047	+	0.998946i	$0.485383\pi$
$632$	0	0
$633$	−12.5461	−0.498661
$634$	0	0
$635$	36.8121	1.46084
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	19.5908	0.775000
$640$	0	0
$641$	31.3870	1.23971	0.619856	−	0.784715i	$-0.287191\pi$
$641$	31.3870	1.23971	0.619856	+	0.784715i	$0.287191\pi$
$642$	0	0
$643$	10.4208	0.410955	0.205477	−	0.978662i	$-0.434125\pi$
$643$	10.4208	0.410955	0.205477	+	0.978662i	$0.434125\pi$
$644$	0	0
$645$	−0.00904758	−0.000356248 0
$646$	0	0
$647$	−28.4908	−1.12009	−0.560044	−	0.828463i	$-0.689216\pi$
$647$	−28.4908	−1.12009	−0.560044	+	0.828463i	$0.689216\pi$
$648$	0	0
$649$	−9.09891	−0.357164
$650$	0	0
$651$	1.73161	0.0678670
$652$	0	0
$653$	13.7711	0.538907	0.269453	−	0.963013i	$-0.413157\pi$
$653$	13.7711	0.538907	0.269453	+	0.963013i	$0.413157\pi$
$654$	0	0
$655$	17.5320	0.685030
$656$	0	0
$657$	−4.48543	−0.174993
$658$	0	0
$659$	34.9761	1.36247	0.681237	−	0.732063i	$-0.261442\pi$
$659$	34.9761	1.36247	0.681237	+	0.732063i	$0.261442\pi$
$660$	0	0
$661$	−9.20213	−0.357921	−0.178961	−	0.983856i	$-0.557273\pi$
$661$	−9.20213	−0.357921	−0.178961	+	0.983856i	$0.557273\pi$
$662$	0	0
$663$	0.808862	0.0314136
$664$	0	0
$665$	2.25493	0.0874424
$666$	0	0
$667$	15.0762	0.583752
$668$	0	0
$669$	1.49274	0.0577125
$670$	0	0
$671$	−11.8069	−0.455801
$672$	0	0
$673$	2.26606	0.0873502	0.0436751	−	0.999046i	$-0.486093\pi$
$673$	2.26606	0.0873502	0.0436751	+	0.999046i	$0.486093\pi$
$674$	0	0
$675$	−23.5502	−0.906446
$676$	0	0
$677$	21.3559	0.820773	0.410387	−	0.911912i	$-0.365394\pi$
$677$	21.3559	0.820773	0.410387	+	0.911912i	$0.365394\pi$
$678$	0	0
$679$	−0.425497	−0.0163291
$680$	0	0
$681$	−7.38649	−0.283051
$682$	0	0
$683$	−13.7168	−0.524857	−0.262428	−	0.964951i	$-0.584523\pi$
$683$	−13.7168	−0.524857	−0.262428	+	0.964951i	$0.584523\pi$
$684$	0	0
$685$	−10.7155	−0.409420
$686$	0	0
$687$	−5.54884	−0.211701
$688$	0	0
$689$	3.08123	0.117386
$690$	0	0
$691$	−29.4678	−1.12101	−0.560505	−	0.828151i	$-0.689393\pi$
$691$	−29.4678	−1.12101	−0.560505	+	0.828151i	$0.689393\pi$
$692$	0	0
$693$	−2.76166	−0.104907
$694$	0	0
$695$	51.2070	1.94239
$696$	0	0
$697$	3.20655	0.121457
$698$	0	0
$699$	−2.99269	−0.113194
$700$	0	0
$701$	−4.66890	−0.176342	−0.0881710	−	0.996105i	$-0.528102\pi$
$701$	−4.66890	−0.176342	−0.0881710	+	0.996105i	$0.528102\pi$
$702$	0	0
$703$	6.05560	0.228391
$704$	0	0
$705$	−7.40900	−0.279039
$706$	0	0
$707$	0.922745	0.0347034
$708$	0	0
$709$	−29.2119	−1.09708	−0.548539	−	0.836125i	$-0.684816\pi$
$709$	−29.2119	−1.09708	−0.548539	+	0.836125i	$0.684816\pi$
$710$	0	0
$711$	6.27552	0.235351
$712$	0	0
$713$	16.0477	0.600992
$714$	0	0
$715$	−3.65683	−0.136758
$716$	0	0
$717$	−9.74722	−0.364017
$718$	0	0
$719$	−11.1464	−0.415691	−0.207845	−	0.978162i	$-0.566645\pi$
$719$	−11.1464	−0.415691	−0.207845	+	0.978162i	$0.566645\pi$
$720$	0	0
$721$	−16.1903	−0.602958
$722$	0	0
$723$	−7.67038	−0.285265
$724$	0	0
$725$	27.8983	1.03612
$726$	0	0
$727$	26.5305	0.983963	0.491982	−	0.870606i	$-0.336273\pi$
$727$	26.5305	0.983963	0.491982	+	0.870606i	$0.336273\pi$
$728$	0	0
$729$	−14.9682	−0.554379
$730$	0	0
$731$	0.00839668	0.000310562 0
$732$	0	0
$733$	−26.6618	−0.984775	−0.492388	−	0.870376i	$-0.663876\pi$
$733$	−26.6618	−0.984775	−0.492388	+	0.870376i	$0.663876\pi$
$734$	0	0
$735$	−1.78526	−0.0658504
$736$	0	0
$737$	−7.88417	−0.290417
$738$	0	0
$739$	35.6109	1.30997	0.654985	−	0.755642i	$-0.272675\pi$
$739$	35.6109	1.30997	0.654985	+	0.755642i	$0.272675\pi$
$740$	0	0
$741$	−0.301041	−0.0110590
$742$	0	0
$743$	8.92793	0.327534	0.163767	−	0.986499i	$-0.447635\pi$
$743$	8.92793	0.327534	0.163767	+	0.986499i	$0.447635\pi$
$744$	0	0
$745$	−19.5134	−0.714916
$746$	0	0
$747$	8.24364	0.301619
$748$	0	0
$749$	−5.72568	−0.209212
$750$	0	0
$751$	27.5755	1.00624	0.503121	−	0.864216i	$-0.332185\pi$
$751$	27.5755	1.00624	0.503121	+	0.864216i	$0.332185\pi$
$752$	0	0
$753$	4.02416	0.146648
$754$	0	0
$755$	−2.75370	−0.100217
$756$	0	0
$757$	−18.5693	−0.674914	−0.337457	−	0.941341i	$-0.609567\pi$
$757$	−18.5693	−0.674914	−0.337457	+	0.941341i	$0.609567\pi$
$758$	0	0
$759$	2.20882	0.0801749
$760$	0	0
$761$	21.0495	0.763045	0.381523	−	0.924360i	$-0.375400\pi$
$761$	21.0495	0.763045	0.381523	+	0.924360i	$0.375400\pi$
$762$	0	0
$763$	0.790329	0.0286118
$764$	0	0
$765$	16.7321	0.604952
$766$	0	0
$767$	−9.09891	−0.328543
$768$	0	0
$769$	−33.8497	−1.22065	−0.610326	−	0.792151i	$-0.708961\pi$
$769$	−33.8497	−1.22065	−0.610326	+	0.792151i	$0.708961\pi$
$770$	0	0
$771$	−7.04709	−0.253795
$772$	0	0
$773$	−21.4092	−0.770037	−0.385018	−	0.922909i	$-0.625805\pi$
$773$	−21.4092	−0.770037	−0.385018	+	0.922909i	$0.625805\pi$
$774$	0	0
$775$	29.6962	1.06672
$776$	0	0
$777$	−4.79431	−0.171995
$778$	0	0
$779$	−1.19341	−0.0427583
$780$	0	0
$781$	−7.09384	−0.253838
$782$	0	0
$783$	−9.37291	−0.334960
$784$	0	0
$785$	22.8441	0.815340
$786$	0	0
$787$	−15.3196	−0.546084	−0.273042	−	0.962002i	$-0.588030\pi$
$787$	−15.3196	−0.546084	−0.273042	+	0.962002i	$0.588030\pi$
$788$	0	0
$789$	13.1361	0.467657
$790$	0	0
$791$	7.44716	0.264790
$792$	0	0
$793$	−11.8069	−0.419276
$794$	0	0
$795$	−5.50081	−0.195094
$796$	0	0
$797$	21.7200	0.769362	0.384681	−	0.923050i	$-0.374312\pi$
$797$	21.7200	0.769362	0.384681	+	0.923050i	$0.374312\pi$
$798$	0	0
$799$	6.87598	0.243255
$800$	0	0
$801$	25.9833	0.918075
$802$	0	0
$803$	1.62418	0.0573160
$804$	0	0
$805$	−16.5450	−0.583134
$806$	0	0
$807$	6.72406	0.236698
$808$	0	0
$809$	42.9612	1.51044	0.755218	−	0.655474i	$-0.227531\pi$
$809$	42.9612	1.51044	0.755218	+	0.655474i	$0.227531\pi$
$810$	0	0
$811$	−36.7875	−1.29178	−0.645892	−	0.763428i	$-0.723515\pi$
$811$	−36.7875	−1.29178	−0.645892	+	0.763428i	$0.723515\pi$
$812$	0	0
$813$	−3.62980	−0.127303
$814$	0	0
$815$	1.87156	0.0655580
$816$	0	0
$817$	−0.00312507	−0.000109332 0
$818$	0	0
$819$	−2.76166	−0.0965002
$820$	0	0
$821$	−34.7917	−1.21424	−0.607120	−	0.794610i	$-0.707675\pi$
$821$	−34.7917	−1.21424	−0.607120	+	0.794610i	$0.707675\pi$
$822$	0	0
$823$	11.8136	0.411795	0.205897	−	0.978574i	$-0.433989\pi$
$823$	11.8136	0.411795	0.205897	+	0.978574i	$0.433989\pi$
$824$	0	0
$825$	4.08739	0.142305
$826$	0	0
$827$	−22.6996	−0.789342	−0.394671	−	0.918823i	$-0.629141\pi$
$827$	−22.6996	−0.789342	−0.394671	+	0.918823i	$0.629141\pi$
$828$	0	0
$829$	−2.77091	−0.0962378	−0.0481189	−	0.998842i	$-0.515323\pi$
$829$	−2.77091	−0.0962378	−0.0481189	+	0.998842i	$0.515323\pi$
$830$	0	0
$831$	4.63408	0.160754
$832$	0	0
$833$	1.65683	0.0574056
$834$	0	0
$835$	51.1514	1.77017
$836$	0	0
$837$	−9.97693	−0.344853
$838$	0	0
$839$	9.55995	0.330046	0.165023	−	0.986290i	$-0.447230\pi$
$839$	9.55995	0.330046	0.165023	+	0.986290i	$0.447230\pi$
$840$	0	0
$841$	−17.8966	−0.617122
$842$	0	0
$843$	1.82009	0.0626871
$844$	0	0
$845$	−3.65683	−0.125799
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	−8.17284	−0.280491
$850$	0	0
$851$	−44.4314	−1.52309
$852$	0	0
$853$	33.7731	1.15637	0.578184	−	0.815906i	$-0.303761\pi$
$853$	33.7731	1.15637	0.578184	+	0.815906i	$0.303761\pi$
$854$	0	0
$855$	−6.22735	−0.212971
$856$	0	0
$857$	−12.4095	−0.423902	−0.211951	−	0.977280i	$-0.567982\pi$
$857$	−12.4095	−0.423902	−0.211951	+	0.977280i	$0.567982\pi$
$858$	0	0
$859$	21.0757	0.719092	0.359546	−	0.933127i	$-0.382932\pi$
$859$	21.0757	0.719092	0.359546	+	0.933127i	$0.382932\pi$
$860$	0	0
$861$	0.944840	0.0322001
$862$	0	0
$863$	−25.8044	−0.878393	−0.439196	−	0.898391i	$-0.644737\pi$
$863$	−25.8044	−0.878393	−0.439196	+	0.898391i	$0.644737\pi$
$864$	0	0
$865$	5.73408	0.194965
$866$	0	0
$867$	−6.95925	−0.236349
$868$	0	0
$869$	−2.27237	−0.0770850
$870$	0	0
$871$	−7.88417	−0.267145
$872$	0	0
$873$	1.17508	0.0397704
$874$	0	0
$875$	−12.3322	−0.416904
$876$	0	0
$877$	50.5624	1.70737	0.853686	−	0.520788i	$-0.174362\pi$
$877$	50.5624	1.70737	0.853686	+	0.520788i	$0.174362\pi$
$878$	0	0
$879$	−11.8833	−0.400814
$880$	0	0
$881$	24.5120	0.825831	0.412915	−	0.910769i	$-0.364511\pi$
$881$	24.5120	0.825831	0.412915	+	0.910769i	$0.364511\pi$
$882$	0	0
$883$	29.8752	1.00538	0.502691	−	0.864466i	$-0.332344\pi$
$883$	29.8752	1.00538	0.502691	+	0.864466i	$0.332344\pi$
$884$	0	0
$885$	16.2439	0.546034
$886$	0	0
$887$	10.4200	0.349869	0.174935	−	0.984580i	$-0.444029\pi$
$887$	10.4200	0.349869	0.174935	+	0.984580i	$0.444029\pi$
$888$	0	0
$889$	−10.0667	−0.337626
$890$	0	0
$891$	6.91175	0.231552
$892$	0	0
$893$	−2.55910	−0.0856369
$894$	0	0
$895$	−35.3007	−1.17997
$896$	0	0
$897$	2.20882	0.0737502
$898$	0	0
$899$	11.8190	0.394186
$900$	0	0
$901$	5.10507	0.170075
$902$	0	0
$903$	0.00247416	8.23350e−5 0
$904$	0	0
$905$	−43.8346	−1.45711
$906$	0	0
$907$	−35.4994	−1.17874	−0.589369	−	0.807864i	$-0.700624\pi$
$907$	−35.4994	−1.17874	−0.589369	+	0.807864i	$0.700624\pi$
$908$	0	0
$909$	−2.54831	−0.0845221
$910$	0	0
$911$	44.9900	1.49058	0.745292	−	0.666738i	$-0.232310\pi$
$911$	44.9900	1.49058	0.745292	+	0.666738i	$0.232310\pi$
$912$	0	0
$913$	−2.98503	−0.0987901
$914$	0	0
$915$	21.0784	0.696832
$916$	0	0
$917$	−4.79431	−0.158322
$918$	0	0
$919$	2.54065	0.0838084	0.0419042	−	0.999122i	$-0.486658\pi$
$919$	2.54065	0.0838084	0.0419042	+	0.999122i	$0.486658\pi$
$920$	0	0
$921$	−1.28144	−0.0422249
$922$	0	0
$923$	−7.09384	−0.233497
$924$	0	0
$925$	−82.2199	−2.70337
$926$	0	0
$927$	44.7121	1.46854
$928$	0	0
$929$	−42.1208	−1.38194	−0.690969	−	0.722884i	$-0.742816\pi$
$929$	−42.1208	−1.38194	−0.690969	+	0.722884i	$0.742816\pi$
$930$	0	0
$931$	−0.616636	−0.0202094
$932$	0	0
$933$	−1.11333	−0.0364487
$934$	0	0
$935$	−6.05872	−0.198141
$936$	0	0
$937$	27.9796	0.914055	0.457027	−	0.889453i	$-0.348914\pi$
$937$	27.9796	0.914055	0.457027	+	0.889453i	$0.348914\pi$
$938$	0	0
$939$	10.7729	0.351559
$940$	0	0
$941$	−20.3019	−0.661825	−0.330912	−	0.943662i	$-0.607356\pi$
$941$	−20.3019	−0.661825	−0.330912	+	0.943662i	$0.607356\pi$
$942$	0	0
$943$	8.75634	0.285146
$944$	0	0
$945$	10.2861	0.334606
$946$	0	0
$947$	−43.9288	−1.42749	−0.713747	−	0.700404i	$-0.753003\pi$
$947$	−43.9288	−1.42749	−0.713747	+	0.700404i	$0.753003\pi$
$948$	0	0
$949$	1.62418	0.0527230
$950$	0	0
$951$	11.3445	0.367872
$952$	0	0
$953$	27.6282	0.894965	0.447483	−	0.894293i	$-0.352321\pi$
$953$	27.6282	0.894965	0.447483	+	0.894293i	$0.352321\pi$
$954$	0	0
$955$	−32.8938	−1.06442
$956$	0	0
$957$	1.62677	0.0525861
$958$	0	0
$959$	2.93029	0.0946239
$960$	0	0
$961$	−18.4193	−0.594172
$962$	0	0
$963$	15.8124	0.509547
$964$	0	0
$965$	21.6344	0.696436
$966$	0	0
$967$	34.5045	1.10959	0.554794	−	0.831987i	$-0.312797\pi$
$967$	34.5045	1.10959	0.554794	+	0.831987i	$0.312797\pi$
$968$	0	0
$969$	−0.498773	−0.0160229
$970$	0	0
$971$	11.9996	0.385085	0.192542	−	0.981289i	$-0.438327\pi$
$971$	11.9996	0.385085	0.192542	+	0.981289i	$0.438327\pi$
$972$	0	0
$973$	−14.0031	−0.448920
$974$	0	0
$975$	4.08739	0.130901
$976$	0	0
$977$	−31.0686	−0.993973	−0.496986	−	0.867758i	$-0.665560\pi$
$977$	−31.0686	−0.993973	−0.496986	+	0.867758i	$0.665560\pi$
$978$	0	0
$979$	−9.40858	−0.300700
$980$	0	0
$981$	−2.18262	−0.0696858
$982$	0	0
$983$	−60.4819	−1.92907	−0.964536	−	0.263952i	$-0.914974\pi$
$983$	−60.4819	−1.92907	−0.964536	+	0.263952i	$0.914974\pi$
$984$	0	0
$985$	17.3665	0.553341
$986$	0	0
$987$	2.02607	0.0644907
$988$	0	0
$989$	0.0229294	0.000729112 0
$990$	0	0
$991$	−5.41819	−0.172114	−0.0860572	−	0.996290i	$-0.527427\pi$
$991$	−5.41819	−0.172114	−0.0860572	+	0.996290i	$0.527427\pi$
$992$	0	0
$993$	6.24529	0.198188
$994$	0	0
$995$	−13.2829	−0.421098
$996$	0	0
$997$	15.4974	0.490806	0.245403	−	0.969421i	$-0.421080\pi$
$997$	15.4974	0.490806	0.245403	+	0.969421i	$0.421080\pi$
$998$	0	0
$999$	27.6232	0.873959

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4004.2.a.d.1.3	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4004.2.a.d.1.3	✓	4	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4004.a (trivial)

\(f(q)\)	\(=\)	\(q+0.488200 q^{3} -3.65683 q^{5} +1.00000 q^{7} -2.76166 q^{9} +O(q^{10})\)	\(q+0.488200 q^{3} -3.65683 q^{5} +1.00000 q^{7} -2.76166 q^{9} +1.00000 q^{11} +1.00000 q^{13} -1.78526 q^{15} +1.65683 q^{17} -0.616636 q^{19} +0.488200 q^{21} +4.52441 q^{23} +8.37237 q^{25} -2.81284 q^{27} +3.33218 q^{29} +3.54692 q^{31} +0.488200 q^{33} -3.65683 q^{35} -9.82038 q^{37} +0.488200 q^{39} +1.93535 q^{41} +0.00506793 q^{43} +10.0989 q^{45} +4.15009 q^{47} +1.00000 q^{49} +0.808862 q^{51} +3.08123 q^{53} -3.65683 q^{55} -0.301041 q^{57} -9.09891 q^{59} -11.8069 q^{61} -2.76166 q^{63} -3.65683 q^{65} -7.88417 q^{67} +2.20882 q^{69} -7.09384 q^{71} +1.62418 q^{73} +4.08739 q^{75} +1.00000 q^{77} -2.27237 q^{79} +6.91175 q^{81} -2.98503 q^{83} -6.05872 q^{85} +1.62677 q^{87} -9.40858 q^{89} +1.00000 q^{91} +1.73161 q^{93} +2.25493 q^{95} -0.425497 q^{97} -2.76166 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 3 q^{3} - q^{5} + 4 q^{7} + q^{9}+O(q^{10}) \) 4 * q - 3 * q^3 - q^5 + 4 * q^7 + q^9	\( 4 q - 3 q^{3} - q^{5} + 4 q^{7} + q^{9} + 4 q^{11} + 4 q^{13} - 5 q^{15} - 7 q^{17} - 9 q^{19} - 3 q^{21} + 3 q^{23} - 3 q^{25} - 9 q^{27} - 3 q^{29} - 3 q^{33} - q^{35} - 18 q^{37} - 3 q^{39} - 14 q^{41} - q^{43} + 11 q^{45} - 3 q^{47} + 4 q^{49} + 11 q^{51} + 4 q^{53} - q^{55} + 6 q^{57} - 7 q^{59} - 14 q^{61} + q^{63} - q^{65} - 20 q^{69} - 6 q^{73} + 16 q^{75} + 4 q^{77} + 7 q^{79} - 4 q^{81} + 8 q^{83} - 15 q^{85} + 11 q^{87} + 9 q^{89} + 4 q^{91} + 13 q^{93} - 9 q^{95} - 16 q^{97} + q^{99}+O(q^{100}) \) 4 * q - 3 * q^3 - q^5 + 4 * q^7 + q^9 + 4 * q^11 + 4 * q^13 - 5 * q^15 - 7 * q^17 - 9 * q^19 - 3 * q^21 + 3 * q^23 - 3 * q^25 - 9 * q^27 - 3 * q^29 - 3 * q^33 - q^35 - 18 * q^37 - 3 * q^39 - 14 * q^41 - q^43 + 11 * q^45 - 3 * q^47 + 4 * q^49 + 11 * q^51 + 4 * q^53 - q^55 + 6 * q^57 - 7 * q^59 - 14 * q^61 + q^63 - q^65 - 20 * q^69 - 6 * q^73 + 16 * q^75 + 4 * q^77 + 7 * q^79 - 4 * q^81 + 8 * q^83 - 15 * q^85 + 11 * q^87 + 9 * q^89 + 4 * q^91 + 13 * q^93 - 9 * q^95 - 16 * q^97 + q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

Properties

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