LMFDB - Embedded newform 4008.2.a.j.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4008,2,Mod(1,4008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4008, 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("4008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4008 = 2^{3} \cdot 3 \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4008.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:	$32.0040411301$
Analytic rank:	$1$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 26x^{8} - 3x^{7} + 220x^{6} + 42x^{5} - 675x^{4} - 67x^{3} + 628x^{2} - 48x - 2 $ x^10 - 26x^8 - 3x^7 + 220x^6 + 42x^5 - 675x^4 - 67x^3 + 628x^2 - 48x - 2
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			1.10
Root			$-3.45943$ of defining polynomial
Character	$\chi$	$=$	4008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +2.45943 q^{5} -0.714036 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +2.45943 q^{5} -0.714036 q^{7} +1.00000 q^{9} -1.07869 q^{11} -6.69621 q^{13} -2.45943 q^{15} +1.59676 q^{17} +5.29870 q^{19} +0.714036 q^{21} +0.914941 q^{23} +1.04879 q^{25} -1.00000 q^{27} -8.70520 q^{29} +7.73352 q^{31} +1.07869 q^{33} -1.75612 q^{35} -5.65066 q^{37} +6.69621 q^{39} +5.51192 q^{41} -6.28625 q^{43} +2.45943 q^{45} +0.0858582 q^{47} -6.49015 q^{49} -1.59676 q^{51} -0.447201 q^{53} -2.65297 q^{55} -5.29870 q^{57} +4.05823 q^{59} -5.73166 q^{61} -0.714036 q^{63} -16.4688 q^{65} -9.03316 q^{67} -0.914941 q^{69} +0.319290 q^{71} -13.7265 q^{73} -1.04879 q^{75} +0.770226 q^{77} +10.0843 q^{79} +1.00000 q^{81} -2.88729 q^{83} +3.92712 q^{85} +8.70520 q^{87} -5.37374 q^{89} +4.78133 q^{91} -7.73352 q^{93} +13.0318 q^{95} -16.3351 q^{97} -1.07869 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 10 q^{3} - 10 q^{5} + q^{7} + 10 q^{9}+O(q^{10}) $ 10 * q - 10 * q^3 - 10 * q^5 + q^7 + 10 * q^9	$ 10 q - 10 q^{3} - 10 q^{5} + q^{7} + 10 q^{9} - q^{11} - 6 q^{13} + 10 q^{15} - 9 q^{17} + 2 q^{19} - q^{21} + 7 q^{23} + 12 q^{25} - 10 q^{27} - 13 q^{29} + 23 q^{31} + q^{33} + q^{35} - 6 q^{37} + 6 q^{39} - 12 q^{41} - 10 q^{45} + 10 q^{47} + 7 q^{49} + 9 q^{51} - 26 q^{53} + 11 q^{55} - 2 q^{57} - 10 q^{59} - 10 q^{61} + q^{63} - 22 q^{65} - 5 q^{67} - 7 q^{69} + 25 q^{71} - 8 q^{73} - 12 q^{75} - 46 q^{77} + 26 q^{79} + 10 q^{81} - 14 q^{83} + 9 q^{85} + 13 q^{87} - 31 q^{89} - 3 q^{91} - 23 q^{93} - 5 q^{95} - 32 q^{97} - q^{99}+O(q^{100}) $ 10 * q - 10 * q^3 - 10 * q^5 + q^7 + 10 * q^9 - q^11 - 6 * q^13 + 10 * q^15 - 9 * q^17 + 2 * q^19 - q^21 + 7 * q^23 + 12 * q^25 - 10 * q^27 - 13 * q^29 + 23 * q^31 + q^33 + q^35 - 6 * q^37 + 6 * q^39 - 12 * q^41 - 10 * q^45 + 10 * q^47 + 7 * q^49 + 9 * q^51 - 26 * q^53 + 11 * q^55 - 2 * q^57 - 10 * q^59 - 10 * q^61 + q^63 - 22 * q^65 - 5 * q^67 - 7 * q^69 + 25 * q^71 - 8 * q^73 - 12 * q^75 - 46 * q^77 + 26 * q^79 + 10 * q^81 - 14 * q^83 + 9 * q^85 + 13 * q^87 - 31 * q^89 - 3 * q^91 - 23 * q^93 - 5 * q^95 - 32 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	2.45943	1.09989	0.549945	−	0.835201i	$-0.314649\pi$
$5$	2.45943	1.09989	0.549945	+	0.835201i	$0.314649\pi$
$6$	0	0
$7$	−0.714036	−0.269880	−0.134940	−	0.990854i	$-0.543084\pi$
$7$	−0.714036	−0.269880	−0.134940	+	0.990854i	$0.543084\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.07869	−0.325238	−0.162619	−	0.986689i	$-0.551994\pi$
$11$	−1.07869	−0.325238	−0.162619	+	0.986689i	$0.551994\pi$
$12$	0	0
$13$	−6.69621	−1.85719	−0.928597	−	0.371091i	$-0.878984\pi$
$13$	−6.69621	−1.85719	−0.928597	+	0.371091i	$0.878984\pi$
$14$	0	0
$15$	−2.45943	−0.635022
$16$	0	0
$17$	1.59676	0.387271	0.193635	−	0.981074i	$-0.437972\pi$
$17$	1.59676	0.387271	0.193635	+	0.981074i	$0.437972\pi$
$18$	0	0
$19$	5.29870	1.21560	0.607802	−	0.794088i	$-0.292051\pi$
$19$	5.29870	1.21560	0.607802	+	0.794088i	$0.292051\pi$
$20$	0	0
$21$	0.714036	0.155815
$22$	0	0
$23$	0.914941	0.190778	0.0953892	−	0.995440i	$-0.469590\pi$
$23$	0.914941	0.190778	0.0953892	+	0.995440i	$0.469590\pi$
$24$	0	0
$25$	1.04879	0.209759
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−8.70520	−1.61651	−0.808257	−	0.588830i	$-0.799589\pi$
$29$	−8.70520	−1.61651	−0.808257	+	0.588830i	$0.799589\pi$
$30$	0	0
$31$	7.73352	1.38898	0.694491	−	0.719502i	$-0.255630\pi$
$31$	7.73352	1.38898	0.694491	+	0.719502i	$0.255630\pi$
$32$	0	0
$33$	1.07869	0.187776
$34$	0	0
$35$	−1.75612	−0.296839
$36$	0	0
$37$	−5.65066	−0.928963	−0.464482	−	0.885583i	$-0.653759\pi$
$37$	−5.65066	−0.928963	−0.464482	+	0.885583i	$0.653759\pi$
$38$	0	0
$39$	6.69621	1.07225
$40$	0	0
$41$	5.51192	0.860817	0.430409	−	0.902634i	$-0.358369\pi$
$41$	5.51192	0.860817	0.430409	+	0.902634i	$0.358369\pi$
$42$	0	0
$43$	−6.28625	−0.958644	−0.479322	−	0.877639i	$-0.659117\pi$
$43$	−6.28625	−0.958644	−0.479322	+	0.877639i	$0.659117\pi$
$44$	0	0
$45$	2.45943	0.366630
$46$	0	0
$47$	0.0858582	0.0125237	0.00626185	−	0.999980i	$-0.498007\pi$
$47$	0.0858582	0.0125237	0.00626185	+	0.999980i	$0.498007\pi$
$48$	0	0
$49$	−6.49015	−0.927165
$50$	0	0
$51$	−1.59676	−0.223591
$52$	0	0
$53$	−0.447201	−0.0614277	−0.0307139	−	0.999528i	$-0.509778\pi$
$53$	−0.447201	−0.0614277	−0.0307139	+	0.999528i	$0.509778\pi$
$54$	0	0
$55$	−2.65297	−0.357727
$56$	0	0
$57$	−5.29870	−0.701829
$58$	0	0
$59$	4.05823	0.528336	0.264168	−	0.964477i	$-0.414903\pi$
$59$	4.05823	0.528336	0.264168	+	0.964477i	$0.414903\pi$
$60$	0	0
$61$	−5.73166	−0.733864	−0.366932	−	0.930248i	$-0.619592\pi$
$61$	−5.73166	−0.733864	−0.366932	+	0.930248i	$0.619592\pi$
$62$	0	0
$63$	−0.714036	−0.0899601
$64$	0	0
$65$	−16.4688	−2.04271
$66$	0	0
$67$	−9.03316	−1.10358	−0.551788	−	0.833984i	$-0.686054\pi$
$67$	−9.03316	−1.10358	−0.551788	+	0.833984i	$0.686054\pi$
$68$	0	0
$69$	−0.914941	−0.110146
$70$	0	0
$71$	0.319290	0.0378928	0.0189464	−	0.999821i	$-0.493969\pi$
$71$	0.319290	0.0378928	0.0189464	+	0.999821i	$0.493969\pi$
$72$	0	0
$73$	−13.7265	−1.60657	−0.803283	−	0.595597i	$-0.796916\pi$
$73$	−13.7265	−1.60657	−0.803283	+	0.595597i	$0.796916\pi$
$74$	0	0
$75$	−1.04879	−0.121104
$76$	0	0
$77$	0.770226	0.0877754
$78$	0	0
$79$	10.0843	1.13457	0.567287	−	0.823520i	$-0.307993\pi$
$79$	10.0843	1.13457	0.567287	+	0.823520i	$0.307993\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−2.88729	−0.316922	−0.158461	−	0.987365i	$-0.550653\pi$
$83$	−2.88729	−0.316922	−0.158461	+	0.987365i	$0.550653\pi$
$84$	0	0
$85$	3.92712	0.425955
$86$	0	0
$87$	8.70520	0.933295
$88$	0	0
$89$	−5.37374	−0.569615	−0.284807	−	0.958585i	$-0.591930\pi$
$89$	−5.37374	−0.569615	−0.284807	+	0.958585i	$0.591930\pi$
$90$	0	0
$91$	4.78133	0.501220
$92$	0	0
$93$	−7.73352	−0.801929
$94$	0	0
$95$	13.0318	1.33703
$96$	0	0
$97$	−16.3351	−1.65858	−0.829288	−	0.558822i	$-0.811253\pi$
$97$	−16.3351	−1.65858	−0.829288	+	0.558822i	$0.811253\pi$
$98$	0	0
$99$	−1.07869	−0.108413
$100$	0	0
$101$	14.0849	1.40150	0.700751	−	0.713406i	$-0.252848\pi$
$101$	14.0849	1.40150	0.700751	+	0.713406i	$0.252848\pi$
$102$	0	0
$103$	15.4330	1.52066	0.760331	−	0.649536i	$-0.225037\pi$
$103$	15.4330	1.52066	0.760331	+	0.649536i	$0.225037\pi$
$104$	0	0
$105$	1.75612	0.171380
$106$	0	0
$107$	−18.2847	−1.76764	−0.883822	−	0.467822i	$-0.845039\pi$
$107$	−18.2847	−1.76764	−0.883822	+	0.467822i	$0.845039\pi$
$108$	0	0
$109$	−6.54334	−0.626738	−0.313369	−	0.949631i	$-0.601458\pi$
$109$	−6.54334	−0.626738	−0.313369	+	0.949631i	$0.601458\pi$
$110$	0	0
$111$	5.65066	0.536337
$112$	0	0
$113$	7.38633	0.694848	0.347424	−	0.937708i	$-0.387057\pi$
$113$	7.38633	0.694848	0.347424	+	0.937708i	$0.387057\pi$
$114$	0	0
$115$	2.25023	0.209835
$116$	0	0
$117$	−6.69621	−0.619064
$118$	0	0
$119$	−1.14014	−0.104517
$120$	0	0
$121$	−9.83642	−0.894220
$122$	0	0
$123$	−5.51192	−0.496993
$124$	0	0
$125$	−9.71771	−0.869179
$126$	0	0
$127$	−16.2369	−1.44079	−0.720394	−	0.693565i	$-0.756039\pi$
$127$	−16.2369	−1.44079	−0.720394	+	0.693565i	$0.756039\pi$
$128$	0	0
$129$	6.28625	0.553473
$130$	0	0
$131$	−1.49006	−0.130187	−0.0650934	−	0.997879i	$-0.520735\pi$
$131$	−1.49006	−0.130187	−0.0650934	+	0.997879i	$0.520735\pi$
$132$	0	0
$133$	−3.78346	−0.328068
$134$	0	0
$135$	−2.45943	−0.211674
$136$	0	0
$137$	11.1416	0.951888	0.475944	−	0.879476i	$-0.342106\pi$
$137$	11.1416	0.951888	0.475944	+	0.879476i	$0.342106\pi$
$138$	0	0
$139$	−1.28005	−0.108572	−0.0542861	−	0.998525i	$-0.517288\pi$
$139$	−1.28005	−0.108572	−0.0542861	+	0.998525i	$0.517288\pi$
$140$	0	0
$141$	−0.0858582	−0.00723056
$142$	0	0
$143$	7.22315	0.604030
$144$	0	0
$145$	−21.4098	−1.77799
$146$	0	0
$147$	6.49015	0.535299
$148$	0	0
$149$	6.54625	0.536290	0.268145	−	0.963379i	$-0.413589\pi$
$149$	6.54625	0.536290	0.268145	+	0.963379i	$0.413589\pi$
$150$	0	0
$151$	−10.1867	−0.828985	−0.414492	−	0.910053i	$-0.636041\pi$
$151$	−10.1867	−0.828985	−0.414492	+	0.910053i	$0.636041\pi$
$152$	0	0
$153$	1.59676	0.129090
$154$	0	0
$155$	19.0201	1.52773
$156$	0	0
$157$	6.44362	0.514257	0.257128	−	0.966377i	$-0.417224\pi$
$157$	6.44362	0.514257	0.257128	+	0.966377i	$0.417224\pi$
$158$	0	0
$159$	0.447201	0.0354653
$160$	0	0
$161$	−0.653301	−0.0514873
$162$	0	0
$163$	18.5590	1.45365	0.726827	−	0.686821i	$-0.240994\pi$
$163$	18.5590	1.45365	0.726827	+	0.686821i	$0.240994\pi$
$164$	0	0
$165$	2.65297	0.206534
$166$	0	0
$167$	−1.00000	−0.0773823
$167$	−1.00000	−0.0773823
$168$	0	0
$169$	31.8392	2.44917
$170$	0	0
$171$	5.29870	0.405201
$172$	0	0
$173$	−15.8894	−1.20805	−0.604025	−	0.796965i	$-0.706437\pi$
$173$	−15.8894	−1.20805	−0.604025	+	0.796965i	$0.706437\pi$
$174$	0	0
$175$	−0.748877	−0.0566098
$176$	0	0
$177$	−4.05823	−0.305035
$178$	0	0
$179$	−22.7303	−1.69895	−0.849473	−	0.527633i	$-0.823080\pi$
$179$	−22.7303	−1.69895	−0.849473	+	0.527633i	$0.823080\pi$
$180$	0	0
$181$	26.2142	1.94849	0.974244	−	0.225494i	$-0.0723997\pi$
$181$	26.2142	1.94849	0.974244	+	0.225494i	$0.0723997\pi$
$182$	0	0
$183$	5.73166	0.423697
$184$	0	0
$185$	−13.8974	−1.02176
$186$	0	0
$187$	−1.72241	−0.125955
$188$	0	0
$189$	0.714036	0.0519385
$190$	0	0
$191$	−14.7644	−1.06831	−0.534157	−	0.845385i	$-0.679371\pi$
$191$	−14.7644	−1.06831	−0.534157	+	0.845385i	$0.679371\pi$
$192$	0	0
$193$	−2.51334	−0.180914	−0.0904571	−	0.995900i	$-0.528833\pi$
$193$	−2.51334	−0.180914	−0.0904571	+	0.995900i	$0.528833\pi$
$194$	0	0
$195$	16.4688	1.17936
$196$	0	0
$197$	−24.1457	−1.72031	−0.860154	−	0.510035i	$-0.829632\pi$
$197$	−24.1457	−1.72031	−0.860154	+	0.510035i	$0.829632\pi$
$198$	0	0
$199$	9.63232	0.682817	0.341409	−	0.939915i	$-0.389096\pi$
$199$	9.63232	0.682817	0.341409	+	0.939915i	$0.389096\pi$
$200$	0	0
$201$	9.03316	0.637150
$202$	0	0
$203$	6.21582	0.436265
$204$	0	0
$205$	13.5562	0.946805
$206$	0	0
$207$	0.914941	0.0635928
$208$	0	0
$209$	−5.71567	−0.395361
$210$	0	0
$211$	−10.9607	−0.754565	−0.377283	−	0.926098i	$-0.623142\pi$
$211$	−10.9607	−0.754565	−0.377283	+	0.926098i	$0.623142\pi$
$212$	0	0
$213$	−0.319290	−0.0218774
$214$	0	0
$215$	−15.4606	−1.05440
$216$	0	0
$217$	−5.52201	−0.374859
$218$	0	0
$219$	13.7265	0.927552
$220$	0	0
$221$	−10.6922	−0.719237
$222$	0	0
$223$	22.5791	1.51201	0.756004	−	0.654567i	$-0.227149\pi$
$223$	22.5791	1.51201	0.756004	+	0.654567i	$0.227149\pi$
$224$	0	0
$225$	1.04879	0.0699196
$226$	0	0
$227$	20.9478	1.39035	0.695176	−	0.718839i	$-0.255326\pi$
$227$	20.9478	1.39035	0.695176	+	0.718839i	$0.255326\pi$
$228$	0	0
$229$	−15.8591	−1.04800	−0.523999	−	0.851719i	$-0.675560\pi$
$229$	−15.8591	−1.04800	−0.523999	+	0.851719i	$0.675560\pi$
$230$	0	0
$231$	−0.770226	−0.0506772
$232$	0	0
$233$	−7.36195	−0.482298	−0.241149	−	0.970488i	$-0.577524\pi$
$233$	−7.36195	−0.482298	−0.241149	+	0.970488i	$0.577524\pi$
$234$	0	0
$235$	0.211162	0.0137747
$236$	0	0
$237$	−10.0843	−0.655046
$238$	0	0
$239$	−1.33296	−0.0862221	−0.0431110	−	0.999070i	$-0.513727\pi$
$239$	−1.33296	−0.0862221	−0.0431110	+	0.999070i	$0.513727\pi$
$240$	0	0
$241$	11.9238	0.768077	0.384039	−	0.923317i	$-0.374533\pi$
$241$	11.9238	0.768077	0.384039	+	0.923317i	$0.374533\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−15.9621	−1.01978
$246$	0	0
$247$	−35.4812	−2.25761
$248$	0	0
$249$	2.88729	0.182975
$250$	0	0
$251$	−21.4046	−1.35105	−0.675523	−	0.737339i	$-0.736082\pi$
$251$	−21.4046	−1.35105	−0.675523	+	0.737339i	$0.736082\pi$
$252$	0	0
$253$	−0.986941	−0.0620485
$254$	0	0
$255$	−3.92712	−0.245925
$256$	0	0
$257$	−7.39929	−0.461555	−0.230778	−	0.973007i	$-0.574127\pi$
$257$	−7.39929	−0.461555	−0.230778	+	0.973007i	$0.574127\pi$
$258$	0	0
$259$	4.03478	0.250709
$260$	0	0
$261$	−8.70520	−0.538838
$262$	0	0
$263$	−22.5779	−1.39221	−0.696107	−	0.717938i	$-0.745086\pi$
$263$	−22.5779	−1.39221	−0.696107	+	0.717938i	$0.745086\pi$
$264$	0	0
$265$	−1.09986	−0.0675638
$266$	0	0
$267$	5.37374	0.328867
$268$	0	0
$269$	14.9027	0.908634	0.454317	−	0.890840i	$-0.349883\pi$
$269$	14.9027	0.908634	0.454317	+	0.890840i	$0.349883\pi$
$270$	0	0
$271$	−18.6051	−1.13018	−0.565090	−	0.825029i	$-0.691159\pi$
$271$	−18.6051	−1.13018	−0.565090	+	0.825029i	$0.691159\pi$
$272$	0	0
$273$	−4.78133	−0.289379
$274$	0	0
$275$	−1.13133	−0.0682216
$276$	0	0
$277$	25.4361	1.52831	0.764153	−	0.645035i	$-0.223157\pi$
$277$	25.4361	1.52831	0.764153	+	0.645035i	$0.223157\pi$
$278$	0	0
$279$	7.73352	0.462994
$280$	0	0
$281$	−30.5282	−1.82116	−0.910581	−	0.413331i	$-0.864365\pi$
$281$	−30.5282	−1.82116	−0.910581	+	0.413331i	$0.864365\pi$
$282$	0	0
$283$	−12.1228	−0.720628	−0.360314	−	0.932831i	$-0.617331\pi$
$283$	−12.1228	−0.720628	−0.360314	+	0.932831i	$0.617331\pi$
$284$	0	0
$285$	−13.0318	−0.771936
$286$	0	0
$287$	−3.93571	−0.232318
$288$	0	0
$289$	−14.4504	−0.850021
$290$	0	0
$291$	16.3351	0.957579
$292$	0	0
$293$	0.442081	0.0258267	0.0129133	−	0.999917i	$-0.495889\pi$
$293$	0.442081	0.0258267	0.0129133	+	0.999917i	$0.495889\pi$
$294$	0	0
$295$	9.98093	0.581112
$296$	0	0
$297$	1.07869	0.0625922
$298$	0	0
$299$	−6.12663	−0.354312
$300$	0	0
$301$	4.48861	0.258719
$302$	0	0
$303$	−14.0849	−0.809158
$304$	0	0
$305$	−14.0966	−0.807170
$306$	0	0
$307$	−1.63552	−0.0933441	−0.0466720	−	0.998910i	$-0.514862\pi$
$307$	−1.63552	−0.0933441	−0.0466720	+	0.998910i	$0.514862\pi$
$308$	0	0
$309$	−15.4330	−0.877954
$310$	0	0
$311$	−12.6476	−0.717179	−0.358589	−	0.933495i	$-0.616742\pi$
$311$	−12.6476	−0.717179	−0.358589	+	0.933495i	$0.616742\pi$
$312$	0	0
$313$	−12.4801	−0.705414	−0.352707	−	0.935734i	$-0.614739\pi$
$313$	−12.4801	−0.705414	−0.352707	+	0.935734i	$0.614739\pi$
$314$	0	0
$315$	−1.75612	−0.0989463
$316$	0	0
$317$	−3.98483	−0.223810	−0.111905	−	0.993719i	$-0.535695\pi$
$317$	−3.98483	−0.223810	−0.111905	+	0.993719i	$0.535695\pi$
$318$	0	0
$319$	9.39024	0.525752
$320$	0	0
$321$	18.2847	1.02055
$322$	0	0
$323$	8.46074	0.470768
$324$	0	0
$325$	−7.02294	−0.389563
$326$	0	0
$327$	6.54334	0.361848
$328$	0	0
$329$	−0.0613059	−0.00337990
$330$	0	0
$331$	−0.809964	−0.0445197	−0.0222598	−	0.999752i	$-0.507086\pi$
$331$	−0.809964	−0.0445197	−0.0222598	+	0.999752i	$0.507086\pi$
$332$	0	0
$333$	−5.65066	−0.309654
$334$	0	0
$335$	−22.2164	−1.21381
$336$	0	0
$337$	−26.1932	−1.42683	−0.713417	−	0.700740i	$-0.752853\pi$
$337$	−26.1932	−1.42683	−0.713417	+	0.700740i	$0.752853\pi$
$338$	0	0
$339$	−7.38633	−0.401170
$340$	0	0
$341$	−8.34210	−0.451750
$342$	0	0
$343$	9.63246	0.520104
$344$	0	0
$345$	−2.25023	−0.121148
$346$	0	0
$347$	−25.1389	−1.34952	−0.674762	−	0.738035i	$-0.735754\pi$
$347$	−25.1389	−1.34952	−0.674762	+	0.738035i	$0.735754\pi$
$348$	0	0
$349$	13.7231	0.734583	0.367291	−	0.930106i	$-0.380285\pi$
$349$	13.7231	0.734583	0.367291	+	0.930106i	$0.380285\pi$
$350$	0	0
$351$	6.69621	0.357417
$352$	0	0
$353$	−1.31939	−0.0702243	−0.0351121	−	0.999383i	$-0.511179\pi$
$353$	−1.31939	−0.0702243	−0.0351121	+	0.999383i	$0.511179\pi$
$354$	0	0
$355$	0.785272	0.0416779
$356$	0	0
$357$	1.14014	0.0603428
$358$	0	0
$359$	−34.8745	−1.84061	−0.920304	−	0.391205i	$-0.872058\pi$
$359$	−34.8745	−1.84061	−0.920304	+	0.391205i	$0.872058\pi$
$360$	0	0
$361$	9.07618	0.477694
$362$	0	0
$363$	9.83642	0.516278
$364$	0	0
$365$	−33.7594	−1.76705
$366$	0	0
$367$	−14.0984	−0.735929	−0.367964	−	0.929840i	$-0.619945\pi$
$367$	−14.0984	−0.735929	−0.367964	+	0.929840i	$0.619945\pi$
$368$	0	0
$369$	5.51192	0.286939
$370$	0	0
$371$	0.319317	0.0165781
$372$	0	0
$373$	8.87589	0.459576	0.229788	−	0.973241i	$-0.426197\pi$
$373$	8.87589	0.459576	0.229788	+	0.973241i	$0.426197\pi$
$374$	0	0
$375$	9.71771	0.501821
$376$	0	0
$377$	58.2918	3.00218
$378$	0	0
$379$	−28.1089	−1.44386	−0.721928	−	0.691968i	$-0.756744\pi$
$379$	−28.1089	−1.44386	−0.721928	+	0.691968i	$0.756744\pi$
$380$	0	0
$381$	16.2369	0.831839
$382$	0	0
$383$	19.3794	0.990240	0.495120	−	0.868825i	$-0.335124\pi$
$383$	19.3794	0.990240	0.495120	+	0.868825i	$0.335124\pi$
$384$	0	0
$385$	1.89432	0.0965434
$386$	0	0
$387$	−6.28625	−0.319548
$388$	0	0
$389$	14.2211	0.721041	0.360520	−	0.932751i	$-0.382599\pi$
$389$	14.2211	0.721041	0.360520	+	0.932751i	$0.382599\pi$
$390$	0	0
$391$	1.46094	0.0738829
$392$	0	0
$393$	1.49006	0.0751633
$394$	0	0
$395$	24.8017	1.24791
$396$	0	0
$397$	−35.3288	−1.77310	−0.886552	−	0.462629i	$-0.846906\pi$
$397$	−35.3288	−1.77310	−0.886552	+	0.462629i	$0.846906\pi$
$398$	0	0
$399$	3.78346	0.189410
$400$	0	0
$401$	−8.82943	−0.440921	−0.220460	−	0.975396i	$-0.570756\pi$
$401$	−8.82943	−0.440921	−0.220460	+	0.975396i	$0.570756\pi$
$402$	0	0
$403$	−51.7853	−2.57961
$404$	0	0
$405$	2.45943	0.122210
$406$	0	0
$407$	6.09534	0.302135
$408$	0	0
$409$	26.7117	1.32081	0.660405	−	0.750909i	$-0.270385\pi$
$409$	26.7117	1.32081	0.660405	+	0.750909i	$0.270385\pi$
$410$	0	0
$411$	−11.1416	−0.549573
$412$	0	0
$413$	−2.89772	−0.142588
$414$	0	0
$415$	−7.10110	−0.348579
$416$	0	0
$417$	1.28005	0.0626841
$418$	0	0
$419$	7.69717	0.376031	0.188016	−	0.982166i	$-0.439794\pi$
$419$	7.69717	0.376031	0.188016	+	0.982166i	$0.439794\pi$
$420$	0	0
$421$	31.1161	1.51651	0.758253	−	0.651960i	$-0.226053\pi$
$421$	31.1161	1.51651	0.758253	+	0.651960i	$0.226053\pi$
$422$	0	0
$423$	0.0858582	0.00417457
$424$	0	0
$425$	1.67467	0.0812335
$426$	0	0
$427$	4.09262	0.198056
$428$	0	0
$429$	−7.22315	−0.348737
$430$	0	0
$431$	25.0597	1.20708	0.603541	−	0.797332i	$-0.293756\pi$
$431$	25.0597	1.20708	0.603541	+	0.797332i	$0.293756\pi$
$432$	0	0
$433$	8.46748	0.406921	0.203461	−	0.979083i	$-0.434781\pi$
$433$	8.46748	0.406921	0.203461	+	0.979083i	$0.434781\pi$
$434$	0	0
$435$	21.4098	1.02652
$436$	0	0
$437$	4.84800	0.231911
$438$	0	0
$439$	29.0746	1.38766	0.693828	−	0.720141i	$-0.255923\pi$
$439$	29.0746	1.38766	0.693828	+	0.720141i	$0.255923\pi$
$440$	0	0
$441$	−6.49015	−0.309055
$442$	0	0
$443$	38.7286	1.84005	0.920026	−	0.391856i	$-0.128167\pi$
$443$	38.7286	1.84005	0.920026	+	0.391856i	$0.128167\pi$
$444$	0	0
$445$	−13.2163	−0.626514
$446$	0	0
$447$	−6.54625	−0.309627
$448$	0	0
$449$	25.4038	1.19888	0.599440	−	0.800419i	$-0.295390\pi$
$449$	25.4038	1.19888	0.599440	+	0.800419i	$0.295390\pi$
$450$	0	0
$451$	−5.94567	−0.279971
$452$	0	0
$453$	10.1867	0.478614
$454$	0	0
$455$	11.7594	0.551287
$456$	0	0
$457$	30.9832	1.44933	0.724667	−	0.689099i	$-0.241993\pi$
$457$	30.9832	1.44933	0.724667	+	0.689099i	$0.241993\pi$
$458$	0	0
$459$	−1.59676	−0.0745303
$460$	0	0
$461$	−41.3498	−1.92585	−0.962927	−	0.269762i	$-0.913055\pi$
$461$	−41.3498	−1.92585	−0.962927	+	0.269762i	$0.913055\pi$
$462$	0	0
$463$	21.6623	1.00673	0.503367	−	0.864073i	$-0.332095\pi$
$463$	21.6623	1.00673	0.503367	+	0.864073i	$0.332095\pi$
$464$	0	0
$465$	−19.0201	−0.882034
$466$	0	0
$467$	−33.9098	−1.56916	−0.784579	−	0.620029i	$-0.787121\pi$
$467$	−33.9098	−1.56916	−0.784579	+	0.620029i	$0.787121\pi$
$468$	0	0
$469$	6.45001	0.297834
$470$	0	0
$471$	−6.44362	−0.296906
$472$	0	0
$473$	6.78093	0.311788
$474$	0	0
$475$	5.55724	0.254984
$476$	0	0
$477$	−0.447201	−0.0204759
$478$	0	0
$479$	12.1095	0.553299	0.276650	−	0.960971i	$-0.410776\pi$
$479$	12.1095	0.553299	0.276650	+	0.960971i	$0.410776\pi$
$480$	0	0
$481$	37.8380	1.72526
$482$	0	0
$483$	0.653301	0.0297262
$484$	0	0
$485$	−40.1750	−1.82425
$486$	0	0
$487$	22.2671	1.00902	0.504509	−	0.863407i	$-0.331674\pi$
$487$	22.2671	1.00902	0.504509	+	0.863407i	$0.331674\pi$
$488$	0	0
$489$	−18.5590	−0.839267
$490$	0	0
$491$	1.61654	0.0729535	0.0364767	−	0.999335i	$-0.488387\pi$
$491$	1.61654	0.0729535	0.0364767	+	0.999335i	$0.488387\pi$
$492$	0	0
$493$	−13.9001	−0.626029
$494$	0	0
$495$	−2.65297	−0.119242
$496$	0	0
$497$	−0.227985	−0.0102265
$498$	0	0
$499$	8.36274	0.374368	0.187184	−	0.982325i	$-0.440064\pi$
$499$	8.36274	0.374368	0.187184	+	0.982325i	$0.440064\pi$
$500$	0	0
$501$	1.00000	0.0446767
$502$	0	0
$503$	−3.56541	−0.158974	−0.0794869	−	0.996836i	$-0.525328\pi$
$503$	−3.56541	−0.158974	−0.0794869	+	0.996836i	$0.525328\pi$
$504$	0	0
$505$	34.6409	1.54150
$506$	0	0
$507$	−31.8392	−1.41403
$508$	0	0
$509$	−25.5053	−1.13050	−0.565252	−	0.824918i	$-0.691221\pi$
$509$	−25.5053	−1.13050	−0.565252	+	0.824918i	$0.691221\pi$
$510$	0	0
$511$	9.80123	0.433581
$512$	0	0
$513$	−5.29870	−0.233943
$514$	0	0
$515$	37.9564	1.67256
$516$	0	0
$517$	−0.0926147	−0.00407319
$518$	0	0
$519$	15.8894	0.697469
$520$	0	0
$521$	13.3805	0.586209	0.293104	−	0.956080i	$-0.405312\pi$
$521$	13.3805	0.586209	0.293104	+	0.956080i	$0.405312\pi$
$522$	0	0
$523$	17.6721	0.772748	0.386374	−	0.922342i	$-0.373727\pi$
$523$	17.6721	0.772748	0.386374	+	0.922342i	$0.373727\pi$
$524$	0	0
$525$	0.748877	0.0326837
$526$	0	0
$527$	12.3486	0.537912
$528$	0	0
$529$	−22.1629	−0.963604
$530$	0	0
$531$	4.05823	0.176112
$532$	0	0
$533$	−36.9089	−1.59870
$534$	0	0
$535$	−44.9698	−1.94422
$536$	0	0
$537$	22.7303	0.980886
$538$	0	0
$539$	7.00089	0.301550
$540$	0	0
$541$	−12.4496	−0.535248	−0.267624	−	0.963523i	$-0.586239\pi$
$541$	−12.4496	−0.535248	−0.267624	+	0.963523i	$0.586239\pi$
$542$	0	0
$543$	−26.2142	−1.12496
$544$	0	0
$545$	−16.0929	−0.689343
$546$	0	0
$547$	5.81842	0.248778	0.124389	−	0.992234i	$-0.460303\pi$
$547$	5.81842	0.248778	0.124389	+	0.992234i	$0.460303\pi$
$548$	0	0
$549$	−5.73166	−0.244621
$550$	0	0
$551$	−46.1262	−1.96504
$552$	0	0
$553$	−7.20056	−0.306199
$554$	0	0
$555$	13.8974	0.589912
$556$	0	0
$557$	−29.2821	−1.24072	−0.620361	−	0.784316i	$-0.713014\pi$
$557$	−29.2821	−1.24072	−0.620361	+	0.784316i	$0.713014\pi$
$558$	0	0
$559$	42.0940	1.78039
$560$	0	0
$561$	1.72241	0.0727203
$562$	0	0
$563$	26.4065	1.11290	0.556452	−	0.830880i	$-0.312162\pi$
$563$	26.4065	1.11290	0.556452	+	0.830880i	$0.312162\pi$
$564$	0	0
$565$	18.1662	0.764256
$566$	0	0
$567$	−0.714036	−0.0299867
$568$	0	0
$569$	3.24394	0.135993	0.0679965	−	0.997686i	$-0.478339\pi$
$569$	3.24394	0.135993	0.0679965	+	0.997686i	$0.478339\pi$
$570$	0	0
$571$	13.1836	0.551716	0.275858	−	0.961198i	$-0.411038\pi$
$571$	13.1836	0.551716	0.275858	+	0.961198i	$0.411038\pi$
$572$	0	0
$573$	14.7644	0.616791
$574$	0	0
$575$	0.959585	0.0400175
$576$	0	0
$577$	−43.7755	−1.82240	−0.911200	−	0.411965i	$-0.864842\pi$
$577$	−43.7755	−1.82240	−0.911200	+	0.411965i	$0.864842\pi$
$578$	0	0
$579$	2.51334	0.104451
$580$	0	0
$581$	2.06163	0.0855310
$582$	0	0
$583$	0.482392	0.0199787
$584$	0	0
$585$	−16.4688	−0.680903
$586$	0	0
$587$	2.22759	0.0919424	0.0459712	−	0.998943i	$-0.485362\pi$
$587$	2.22759	0.0919424	0.0459712	+	0.998943i	$0.485362\pi$
$588$	0	0
$589$	40.9776	1.68845
$590$	0	0
$591$	24.1457	0.993220
$592$	0	0
$593$	28.3810	1.16547	0.582734	−	0.812663i	$-0.301983\pi$
$593$	28.3810	1.16547	0.582734	+	0.812663i	$0.301983\pi$
$594$	0	0
$595$	−2.80410	−0.114957
$596$	0	0
$597$	−9.63232	−0.394225
$598$	0	0
$599$	29.0774	1.18807	0.594035	−	0.804439i	$-0.297534\pi$
$599$	29.0774	1.18807	0.594035	+	0.804439i	$0.297534\pi$
$600$	0	0
$601$	20.8997	0.852516	0.426258	−	0.904602i	$-0.359832\pi$
$601$	20.8997	0.852516	0.426258	+	0.904602i	$0.359832\pi$
$602$	0	0
$603$	−9.03316	−0.367859
$604$	0	0
$605$	−24.1920	−0.983544
$606$	0	0
$607$	−4.53345	−0.184007	−0.0920036	−	0.995759i	$-0.529327\pi$
$607$	−4.53345	−0.184007	−0.0920036	+	0.995759i	$0.529327\pi$
$608$	0	0
$609$	−6.21582	−0.251878
$610$	0	0
$611$	−0.574924	−0.0232589
$612$	0	0
$613$	36.3647	1.46875	0.734377	−	0.678741i	$-0.237474\pi$
$613$	36.3647	1.46875	0.734377	+	0.678741i	$0.237474\pi$
$614$	0	0
$615$	−13.5562	−0.546638
$616$	0	0
$617$	26.5735	1.06981	0.534905	−	0.844912i	$-0.320347\pi$
$617$	26.5735	1.06981	0.534905	+	0.844912i	$0.320347\pi$
$618$	0	0
$619$	28.1187	1.13018	0.565092	−	0.825028i	$-0.308841\pi$
$619$	28.1187	1.13018	0.565092	+	0.825028i	$0.308841\pi$
$620$	0	0
$621$	−0.914941	−0.0367153
$622$	0	0
$623$	3.83704	0.153728
$624$	0	0
$625$	−29.1440	−1.16576
$626$	0	0
$627$	5.71567	0.228262
$628$	0	0
$629$	−9.02275	−0.359760
$630$	0	0
$631$	−1.65222	−0.0657740	−0.0328870	−	0.999459i	$-0.510470\pi$
$631$	−1.65222	−0.0657740	−0.0328870	+	0.999459i	$0.510470\pi$
$632$	0	0
$633$	10.9607	0.435649
$634$	0	0
$635$	−39.9334	−1.58471
$636$	0	0
$637$	43.4594	1.72192
$638$	0	0
$639$	0.319290	0.0126309
$640$	0	0
$641$	−33.7903	−1.33464	−0.667318	−	0.744773i	$-0.732558\pi$
$641$	−33.7903	−1.33464	−0.667318	+	0.744773i	$0.732558\pi$
$642$	0	0
$643$	−36.9570	−1.45744	−0.728721	−	0.684811i	$-0.759885\pi$
$643$	−36.9570	−1.45744	−0.728721	+	0.684811i	$0.759885\pi$
$644$	0	0
$645$	15.4606	0.608760
$646$	0	0
$647$	19.3434	0.760468	0.380234	−	0.924890i	$-0.375843\pi$
$647$	19.3434	0.760468	0.380234	+	0.924890i	$0.375843\pi$
$648$	0	0
$649$	−4.37759	−0.171835
$650$	0	0
$651$	5.52201	0.216425
$652$	0	0
$653$	0.207939	0.00813726	0.00406863	−	0.999992i	$-0.498705\pi$
$653$	0.207939	0.00813726	0.00406863	+	0.999992i	$0.498705\pi$
$654$	0	0
$655$	−3.66469	−0.143191
$656$	0	0
$657$	−13.7265	−0.535522
$658$	0	0
$659$	9.85797	0.384012	0.192006	−	0.981394i	$-0.438501\pi$
$659$	9.85797	0.384012	0.192006	+	0.981394i	$0.438501\pi$
$660$	0	0
$661$	−13.1222	−0.510394	−0.255197	−	0.966889i	$-0.582140\pi$
$661$	−13.1222	−0.510394	−0.255197	+	0.966889i	$0.582140\pi$
$662$	0	0
$663$	10.6922	0.415252
$664$	0	0
$665$	−9.30516	−0.360838
$666$	0	0
$667$	−7.96474	−0.308396
$668$	0	0
$669$	−22.5791	−0.872958
$670$	0	0
$671$	6.18271	0.238681
$672$	0	0
$673$	17.8943	0.689776	0.344888	−	0.938644i	$-0.387917\pi$
$673$	17.8943	0.689776	0.344888	+	0.938644i	$0.387917\pi$
$674$	0	0
$675$	−1.04879	−0.0403681
$676$	0	0
$677$	−22.2692	−0.855873	−0.427937	−	0.903809i	$-0.640759\pi$
$677$	−22.2692	−0.855873	−0.427937	+	0.903809i	$0.640759\pi$
$678$	0	0
$679$	11.6638	0.447617
$680$	0	0
$681$	−20.9478	−0.802721
$682$	0	0
$683$	−45.2270	−1.73056	−0.865281	−	0.501287i	$-0.832860\pi$
$683$	−45.2270	−1.73056	−0.865281	+	0.501287i	$0.832860\pi$
$684$	0	0
$685$	27.4019	1.04697
$686$	0	0
$687$	15.8591	0.605061
$688$	0	0
$689$	2.99455	0.114083
$690$	0	0
$691$	29.9921	1.14095	0.570476	−	0.821314i	$-0.306759\pi$
$691$	29.9921	1.14095	0.570476	+	0.821314i	$0.306759\pi$
$692$	0	0
$693$	0.770226	0.0292585
$694$	0	0
$695$	−3.14818	−0.119417
$696$	0	0
$697$	8.80121	0.333369
$698$	0	0
$699$	7.36195	0.278455
$700$	0	0
$701$	29.3612	1.10896	0.554478	−	0.832198i	$-0.312918\pi$
$701$	29.3612	1.10896	0.554478	+	0.832198i	$0.312918\pi$
$702$	0	0
$703$	−29.9412	−1.12925
$704$	0	0
$705$	−0.211162	−0.00795283
$706$	0	0
$707$	−10.0571	−0.378238
$708$	0	0
$709$	13.5222	0.507837	0.253919	−	0.967226i	$-0.418280\pi$
$709$	13.5222	0.507837	0.253919	+	0.967226i	$0.418280\pi$
$710$	0	0
$711$	10.0843	0.378191
$712$	0	0
$713$	7.07572	0.264988
$714$	0	0
$715$	17.7648	0.664367
$716$	0	0
$717$	1.33296	0.0497803
$718$	0	0
$719$	−10.6152	−0.395881	−0.197940	−	0.980214i	$-0.563425\pi$
$719$	−10.6152	−0.395881	−0.197940	+	0.980214i	$0.563425\pi$
$720$	0	0
$721$	−11.0197	−0.410396
$722$	0	0
$723$	−11.9238	−0.443450
$724$	0	0
$725$	−9.12996	−0.339078
$726$	0	0
$727$	19.7131	0.731117	0.365558	−	0.930788i	$-0.380878\pi$
$727$	19.7131	0.731117	0.365558	+	0.930788i	$0.380878\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−10.0376	−0.371255
$732$	0	0
$733$	7.67318	0.283415	0.141708	−	0.989909i	$-0.454741\pi$
$733$	7.67318	0.283415	0.141708	+	0.989909i	$0.454741\pi$
$734$	0	0
$735$	15.9621	0.588770
$736$	0	0
$737$	9.74402	0.358925
$738$	0	0
$739$	52.7555	1.94064	0.970322	−	0.241817i	$-0.0777432\pi$
$739$	52.7555	1.94064	0.970322	+	0.241817i	$0.0777432\pi$
$740$	0	0
$741$	35.4812	1.30343
$742$	0	0
$743$	8.95664	0.328587	0.164294	−	0.986411i	$-0.447466\pi$
$743$	8.95664	0.328587	0.164294	+	0.986411i	$0.447466\pi$
$744$	0	0
$745$	16.1000	0.589860
$746$	0	0
$747$	−2.88729	−0.105641
$748$	0	0
$749$	13.0559	0.477053
$750$	0	0
$751$	29.0078	1.05851	0.529256	−	0.848462i	$-0.322471\pi$
$751$	29.0078	1.05851	0.529256	+	0.848462i	$0.322471\pi$
$752$	0	0
$753$	21.4046	0.780027
$754$	0	0
$755$	−25.0536	−0.911792
$756$	0	0
$757$	1.82811	0.0664438	0.0332219	−	0.999448i	$-0.489423\pi$
$757$	1.82811	0.0664438	0.0332219	+	0.999448i	$0.489423\pi$
$758$	0	0
$759$	0.986941	0.0358237
$760$	0	0
$761$	6.47749	0.234809	0.117404	−	0.993084i	$-0.462543\pi$
$761$	6.47749	0.234809	0.117404	+	0.993084i	$0.462543\pi$
$762$	0	0
$763$	4.67218	0.169144
$764$	0	0
$765$	3.92712	0.141985
$766$	0	0
$767$	−27.1747	−0.981223
$768$	0	0
$769$	−32.6384	−1.17697	−0.588486	−	0.808508i	$-0.700276\pi$
$769$	−32.6384	−1.17697	−0.588486	+	0.808508i	$0.700276\pi$
$770$	0	0
$771$	7.39929	0.266479
$772$	0	0
$773$	20.6326	0.742103	0.371051	−	0.928612i	$-0.378997\pi$
$773$	20.6326	0.742103	0.371051	+	0.928612i	$0.378997\pi$
$774$	0	0
$775$	8.11087	0.291351
$776$	0	0
$777$	−4.03478	−0.144747
$778$	0	0
$779$	29.2060	1.04641
$780$	0	0
$781$	−0.344417	−0.0123242
$782$	0	0
$783$	8.70520	0.311098
$784$	0	0
$785$	15.8476	0.565626
$786$	0	0
$787$	2.74461	0.0978347	0.0489174	−	0.998803i	$-0.484423\pi$
$787$	2.74461	0.0978347	0.0489174	+	0.998803i	$0.484423\pi$
$788$	0	0
$789$	22.5779	0.803795
$790$	0	0
$791$	−5.27411	−0.187526
$792$	0	0
$793$	38.3804	1.36293
$794$	0	0
$795$	1.09986	0.0390080
$796$	0	0
$797$	−19.3372	−0.684960	−0.342480	−	0.939525i	$-0.611267\pi$
$797$	−19.3372	−0.684960	−0.342480	+	0.939525i	$0.611267\pi$
$798$	0	0
$799$	0.137095	0.00485007
$800$	0	0
$801$	−5.37374	−0.189872
$802$	0	0
$803$	14.8067	0.522517
$804$	0	0
$805$	−1.60675	−0.0566304
$806$	0	0
$807$	−14.9027	−0.524600
$808$	0	0
$809$	−21.3507	−0.750650	−0.375325	−	0.926893i	$-0.622469\pi$
$809$	−21.3507	−0.750650	−0.375325	+	0.926893i	$0.622469\pi$
$810$	0	0
$811$	−9.58052	−0.336417	−0.168209	−	0.985751i	$-0.553798\pi$
$811$	−9.58052	−0.336417	−0.168209	+	0.985751i	$0.553798\pi$
$812$	0	0
$813$	18.6051	0.652510
$814$	0	0
$815$	45.6446	1.59886
$816$	0	0
$817$	−33.3089	−1.16533
$818$	0	0
$819$	4.78133	0.167073
$820$	0	0
$821$	−5.12041	−0.178704	−0.0893518	−	0.996000i	$-0.528480\pi$
$821$	−5.12041	−0.178704	−0.0893518	+	0.996000i	$0.528480\pi$
$822$	0	0
$823$	−3.19493	−0.111368	−0.0556840	−	0.998448i	$-0.517734\pi$
$823$	−3.19493	−0.111368	−0.0556840	+	0.998448i	$0.517734\pi$
$824$	0	0
$825$	1.13133	0.0393878
$826$	0	0
$827$	41.7978	1.45345	0.726726	−	0.686927i	$-0.241041\pi$
$827$	41.7978	1.45345	0.726726	+	0.686927i	$0.241041\pi$
$828$	0	0
$829$	30.9371	1.07449	0.537246	−	0.843426i	$-0.319465\pi$
$829$	30.9371	1.07449	0.537246	+	0.843426i	$0.319465\pi$
$830$	0	0
$831$	−25.4361	−0.882368
$832$	0	0
$833$	−10.3632	−0.359064
$834$	0	0
$835$	−2.45943	−0.0851121
$836$	0	0
$837$	−7.73352	−0.267310
$838$	0	0
$839$	57.1213	1.97205	0.986024	−	0.166605i	$-0.0532805\pi$
$839$	57.1213	1.97205	0.986024	+	0.166605i	$0.0532805\pi$
$840$	0	0
$841$	46.7804	1.61312
$842$	0	0
$843$	30.5282	1.05145
$844$	0	0
$845$	78.3062	2.69381
$846$	0	0
$847$	7.02356	0.241332
$848$	0	0
$849$	12.1228	0.416055
$850$	0	0
$851$	−5.17003	−0.177226
$852$	0	0
$853$	5.38670	0.184437	0.0922186	−	0.995739i	$-0.470604\pi$
$853$	5.38670	0.184437	0.0922186	+	0.995739i	$0.470604\pi$
$854$	0	0
$855$	13.0318	0.445677
$856$	0	0
$857$	26.6989	0.912016	0.456008	−	0.889976i	$-0.349279\pi$
$857$	26.6989	0.912016	0.456008	+	0.889976i	$0.349279\pi$
$858$	0	0
$859$	−14.5880	−0.497737	−0.248868	−	0.968537i	$-0.580059\pi$
$859$	−14.5880	−0.497737	−0.248868	+	0.968537i	$0.580059\pi$
$860$	0	0
$861$	3.93571	0.134129
$862$	0	0
$863$	−1.93356	−0.0658192	−0.0329096	−	0.999458i	$-0.510477\pi$
$863$	−1.93356	−0.0658192	−0.0329096	+	0.999458i	$0.510477\pi$
$864$	0	0
$865$	−39.0789	−1.32872
$866$	0	0
$867$	14.4504	0.490760
$868$	0	0
$869$	−10.8779	−0.369007
$870$	0	0
$871$	60.4879	2.04955
$872$	0	0
$873$	−16.3351	−0.552858
$874$	0	0
$875$	6.93880	0.234574
$876$	0	0
$877$	−36.2567	−1.22430	−0.612150	−	0.790742i	$-0.709695\pi$
$877$	−36.2567	−1.22430	−0.612150	+	0.790742i	$0.709695\pi$
$878$	0	0
$879$	−0.442081	−0.0149110
$880$	0	0
$881$	6.95686	0.234382	0.117191	−	0.993109i	$-0.462611\pi$
$881$	6.95686	0.234382	0.117191	+	0.993109i	$0.462611\pi$
$882$	0	0
$883$	9.89795	0.333093	0.166546	−	0.986034i	$-0.446738\pi$
$883$	9.89795	0.333093	0.166546	+	0.986034i	$0.446738\pi$
$884$	0	0
$885$	−9.98093	−0.335505
$886$	0	0
$887$	36.8798	1.23830	0.619151	−	0.785272i	$-0.287477\pi$
$887$	36.8798	1.23830	0.619151	+	0.785272i	$0.287477\pi$
$888$	0	0
$889$	11.5937	0.388840
$890$	0	0
$891$	−1.07869	−0.0361376
$892$	0	0
$893$	0.454937	0.0152239
$894$	0	0
$895$	−55.9037	−1.86865
$896$	0	0
$897$	6.12663	0.204562
$898$	0	0
$899$	−67.3218	−2.24531
$900$	0	0
$901$	−0.714071	−0.0237892
$902$	0	0
$903$	−4.48861	−0.149372
$904$	0	0
$905$	64.4721	2.14312
$906$	0	0
$907$	38.3270	1.27263	0.636313	−	0.771431i	$-0.280459\pi$
$907$	38.3270	1.27263	0.636313	+	0.771431i	$0.280459\pi$
$908$	0	0
$909$	14.0849	0.467167
$910$	0	0
$911$	−9.86649	−0.326891	−0.163446	−	0.986552i	$-0.552261\pi$
$911$	−9.86649	−0.326891	−0.163446	+	0.986552i	$0.552261\pi$
$912$	0	0
$913$	3.11451	0.103075
$914$	0	0
$915$	14.0966	0.466020
$916$	0	0
$917$	1.06395	0.0351348
$918$	0	0
$919$	−22.0614	−0.727739	−0.363869	−	0.931450i	$-0.618545\pi$
$919$	−22.0614	−0.727739	−0.363869	+	0.931450i	$0.618545\pi$
$920$	0	0
$921$	1.63552	0.0538922
$922$	0	0
$923$	−2.13803	−0.0703743
$924$	0	0
$925$	−5.92638	−0.194858
$926$	0	0
$927$	15.4330	0.506887
$928$	0	0
$929$	0.0300225	0.000985006 0	0.000492503	−	1.00000i	$-0.499843\pi$
$929$	0.0300225	0.000985006 0	0.000492503		1.00000i	$0.499843\pi$
$930$	0	0
$931$	−34.3893	−1.12707
$932$	0	0
$933$	12.6476	0.414063
$934$	0	0
$935$	−4.23615	−0.138537
$936$	0	0
$937$	4.95051	0.161726	0.0808631	−	0.996725i	$-0.474232\pi$
$937$	4.95051	0.161726	0.0808631	+	0.996725i	$0.474232\pi$
$938$	0	0
$939$	12.4801	0.407271
$940$	0	0
$941$	−22.3268	−0.727833	−0.363917	−	0.931432i	$-0.618561\pi$
$941$	−22.3268	−0.727833	−0.363917	+	0.931432i	$0.618561\pi$
$942$	0	0
$943$	5.04308	0.164225
$944$	0	0
$945$	1.75612	0.0571266
$946$	0	0
$947$	−32.0422	−1.04123	−0.520615	−	0.853791i	$-0.674297\pi$
$947$	−32.0422	−1.04123	−0.520615	+	0.853791i	$0.674297\pi$
$948$	0	0
$949$	91.9156	2.98371
$950$	0	0
$951$	3.98483	0.129217
$952$	0	0
$953$	−24.0457	−0.778917	−0.389458	−	0.921044i	$-0.627338\pi$
$953$	−24.0457	−0.778917	−0.389458	+	0.921044i	$0.627338\pi$
$954$	0	0
$955$	−36.3120	−1.17503
$956$	0	0
$957$	−9.39024	−0.303543
$958$	0	0
$959$	−7.95548	−0.256896
$960$	0	0
$961$	28.8074	0.929270
$962$	0	0
$963$	−18.2847	−0.589215
$964$	0	0
$965$	−6.18138	−0.198986
$966$	0	0
$967$	24.3610	0.783396	0.391698	−	0.920094i	$-0.371888\pi$
$967$	24.3610	0.783396	0.391698	+	0.920094i	$0.371888\pi$
$968$	0	0
$969$	−8.46074	−0.271798
$970$	0	0
$971$	−9.75015	−0.312897	−0.156449	−	0.987686i	$-0.550005\pi$
$971$	−9.75015	−0.312897	−0.156449	+	0.987686i	$0.550005\pi$
$972$	0	0
$973$	0.914000	0.0293015
$974$	0	0
$975$	7.02294	0.224914
$976$	0	0
$977$	−25.6461	−0.820490	−0.410245	−	0.911975i	$-0.634557\pi$
$977$	−25.6461	−0.820490	−0.410245	+	0.911975i	$0.634557\pi$
$978$	0	0
$979$	5.79661	0.185261
$980$	0	0
$981$	−6.54334	−0.208913
$982$	0	0
$983$	29.1839	0.930821	0.465410	−	0.885095i	$-0.345907\pi$
$983$	29.1839	0.930821	0.465410	+	0.885095i	$0.345907\pi$
$984$	0	0
$985$	−59.3846	−1.89215
$986$	0	0
$987$	0.0613059	0.00195139
$988$	0	0
$989$	−5.75155	−0.182889
$990$	0	0
$991$	7.13100	0.226524	0.113262	−	0.993565i	$-0.463870\pi$
$991$	7.13100	0.226524	0.113262	+	0.993565i	$0.463870\pi$
$992$	0	0
$993$	0.809964	0.0257034
$994$	0	0
$995$	23.6900	0.751024
$996$	0	0
$997$	7.78283	0.246485	0.123242	−	0.992377i	$-0.460671\pi$
$997$	7.78283	0.246485	0.123242	+	0.992377i	$0.460671\pi$
$998$	0	0
$999$	5.65066	0.178779

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4008.2.a.j.1.10	✓	10
4.3	odd	2		8016.2.a.bd.1.10		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4008.2.a.j.1.10	✓	10	1.1	even	1	trivial
8016.2.a.bd.1.10		10	4.3	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 4008 = 2^{3} \cdot 3 \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4008.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +2.45943 q^{5} -0.714036 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +2.45943 q^{5} -0.714036 q^{7} +1.00000 q^{9} -1.07869 q^{11} -6.69621 q^{13} -2.45943 q^{15} +1.59676 q^{17} +5.29870 q^{19} +0.714036 q^{21} +0.914941 q^{23} +1.04879 q^{25} -1.00000 q^{27} -8.70520 q^{29} +7.73352 q^{31} +1.07869 q^{33} -1.75612 q^{35} -5.65066 q^{37} +6.69621 q^{39} +5.51192 q^{41} -6.28625 q^{43} +2.45943 q^{45} +0.0858582 q^{47} -6.49015 q^{49} -1.59676 q^{51} -0.447201 q^{53} -2.65297 q^{55} -5.29870 q^{57} +4.05823 q^{59} -5.73166 q^{61} -0.714036 q^{63} -16.4688 q^{65} -9.03316 q^{67} -0.914941 q^{69} +0.319290 q^{71} -13.7265 q^{73} -1.04879 q^{75} +0.770226 q^{77} +10.0843 q^{79} +1.00000 q^{81} -2.88729 q^{83} +3.92712 q^{85} +8.70520 q^{87} -5.37374 q^{89} +4.78133 q^{91} -7.73352 q^{93} +13.0318 q^{95} -16.3351 q^{97} -1.07869 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 10 q^{3} - 10 q^{5} + q^{7} + 10 q^{9}+O(q^{10}) \) 10 * q - 10 * q^3 - 10 * q^5 + q^7 + 10 * q^9	\( 10 q - 10 q^{3} - 10 q^{5} + q^{7} + 10 q^{9} - q^{11} - 6 q^{13} + 10 q^{15} - 9 q^{17} + 2 q^{19} - q^{21} + 7 q^{23} + 12 q^{25} - 10 q^{27} - 13 q^{29} + 23 q^{31} + q^{33} + q^{35} - 6 q^{37} + 6 q^{39} - 12 q^{41} - 10 q^{45} + 10 q^{47} + 7 q^{49} + 9 q^{51} - 26 q^{53} + 11 q^{55} - 2 q^{57} - 10 q^{59} - 10 q^{61} + q^{63} - 22 q^{65} - 5 q^{67} - 7 q^{69} + 25 q^{71} - 8 q^{73} - 12 q^{75} - 46 q^{77} + 26 q^{79} + 10 q^{81} - 14 q^{83} + 9 q^{85} + 13 q^{87} - 31 q^{89} - 3 q^{91} - 23 q^{93} - 5 q^{95} - 32 q^{97} - q^{99}+O(q^{100}) \) 10 * q - 10 * q^3 - 10 * q^5 + q^7 + 10 * q^9 - q^11 - 6 * q^13 + 10 * q^15 - 9 * q^17 + 2 * q^19 - q^21 + 7 * q^23 + 12 * q^25 - 10 * q^27 - 13 * q^29 + 23 * q^31 + q^33 + q^35 - 6 * q^37 + 6 * q^39 - 12 * q^41 - 10 * q^45 + 10 * q^47 + 7 * q^49 + 9 * q^51 - 26 * q^53 + 11 * q^55 - 2 * q^57 - 10 * q^59 - 10 * q^61 + q^63 - 22 * q^65 - 5 * q^67 - 7 * q^69 + 25 * q^71 - 8 * q^73 - 12 * q^75 - 46 * q^77 + 26 * q^79 + 10 * q^81 - 14 * q^83 + 9 * q^85 + 13 * q^87 - 31 * q^89 - 3 * q^91 - 23 * q^93 - 5 * q^95 - 32 * q^97 - q^99

Introduction

L-functions

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