LMFDB - Embedded newform 4008.2.a.l.1.3

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:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4 x^{11} - 31 x^{10} + 131 x^{9} + 309 x^{8} - 1453 x^{7} - 1072 x^{6} + 6350 x^{5} + \cdots + 3008 $ x^12 - 4x^11 - 31x^10 + 131x^9 + 309x^8 - 1453x^7 - 1072x^6 + 6350x^5 + 1411x^4 - 11022x^3 - 2450x^2 + 6960*x + 3008
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.09997$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} -2.09997 q^{5} +2.97416 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -2.09997 q^{5} +2.97416 q^{7} +1.00000 q^{9} -0.138272 q^{11} +5.06303 q^{13} -2.09997 q^{15} +0.299135 q^{17} +0.432187 q^{19} +2.97416 q^{21} +5.72098 q^{23} -0.590124 q^{25} +1.00000 q^{27} -0.718106 q^{29} -4.27590 q^{31} -0.138272 q^{33} -6.24565 q^{35} -3.92212 q^{37} +5.06303 q^{39} +7.57912 q^{41} +12.3515 q^{43} -2.09997 q^{45} -9.22989 q^{47} +1.84562 q^{49} +0.299135 q^{51} +1.26409 q^{53} +0.290366 q^{55} +0.432187 q^{57} +3.01296 q^{59} +2.98537 q^{61} +2.97416 q^{63} -10.6322 q^{65} -11.3934 q^{67} +5.72098 q^{69} -7.07665 q^{71} +14.5486 q^{73} -0.590124 q^{75} -0.411242 q^{77} +13.7670 q^{79} +1.00000 q^{81} -15.7764 q^{83} -0.628175 q^{85} -0.718106 q^{87} -12.4982 q^{89} +15.0583 q^{91} -4.27590 q^{93} -0.907580 q^{95} +7.54510 q^{97} -0.138272 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 12 q^{3} + 4 q^{5} + 11 q^{7} + 12 q^{9}+O(q^{10}) $ 12 * q + 12 * q^3 + 4 * q^5 + 11 * q^7 + 12 * q^9	$ 12 q + 12 q^{3} + 4 q^{5} + 11 q^{7} + 12 q^{9} + q^{11} + 8 q^{13} + 4 q^{15} + 3 q^{17} + 12 q^{19} + 11 q^{21} + 7 q^{23} + 18 q^{25} + 12 q^{27} + 5 q^{29} + 33 q^{31} + q^{33} + 15 q^{35} + 8 q^{37} + 8 q^{39} - 6 q^{41} + 16 q^{43} + 4 q^{45} + 18 q^{47} + 25 q^{49} + 3 q^{51} + 20 q^{53} + 39 q^{55} + 12 q^{57} + 4 q^{59} + 10 q^{61} + 11 q^{63} + 9 q^{67} + 7 q^{69} + 11 q^{71} + 22 q^{73} + 18 q^{75} + 24 q^{77} + 56 q^{79} + 12 q^{81} + 26 q^{83} + 15 q^{85} + 5 q^{87} - 15 q^{89} + 11 q^{91} + 33 q^{93} + 3 q^{95} + 8 q^{97} + q^{99}+O(q^{100}) $ 12 * q + 12 * q^3 + 4 * q^5 + 11 * q^7 + 12 * q^9 + q^11 + 8 * q^13 + 4 * q^15 + 3 * q^17 + 12 * q^19 + 11 * q^21 + 7 * q^23 + 18 * q^25 + 12 * q^27 + 5 * q^29 + 33 * q^31 + q^33 + 15 * q^35 + 8 * q^37 + 8 * q^39 - 6 * q^41 + 16 * q^43 + 4 * q^45 + 18 * q^47 + 25 * q^49 + 3 * q^51 + 20 * q^53 + 39 * q^55 + 12 * q^57 + 4 * q^59 + 10 * q^61 + 11 * q^63 + 9 * q^67 + 7 * q^69 + 11 * q^71 + 22 * q^73 + 18 * q^75 + 24 * q^77 + 56 * q^79 + 12 * q^81 + 26 * q^83 + 15 * q^85 + 5 * q^87 - 15 * q^89 + 11 * q^91 + 33 * q^93 + 3 * q^95 + 8 * 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
$5$	−2.09997	−0.939135	−0.469568	−	0.882896i	$-0.655590\pi$
$5$	−2.09997	−0.939135	−0.469568	+	0.882896i	$0.655590\pi$
$6$	0	0
$7$	2.97416	1.12413	0.562063	−	0.827094i	$-0.310008\pi$
$7$	2.97416	1.12413	0.562063	+	0.827094i	$0.310008\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−0.138272	−0.0416905	−0.0208452	−	0.999783i	$-0.506636\pi$
$11$	−0.138272	−0.0416905	−0.0208452	+	0.999783i	$0.506636\pi$
$12$	0	0
$13$	5.06303	1.40423	0.702116	−	0.712063i	$-0.252239\pi$
$13$	5.06303	1.40423	0.702116	+	0.712063i	$0.252239\pi$
$14$	0	0
$15$	−2.09997	−0.542210
$16$	0	0
$17$	0.299135	0.0725509	0.0362755	−	0.999342i	$-0.488451\pi$
$17$	0.299135	0.0725509	0.0362755	+	0.999342i	$0.488451\pi$
$18$	0	0
$19$	0.432187	0.0991505	0.0495753	−	0.998770i	$-0.484213\pi$
$19$	0.432187	0.0991505	0.0495753	+	0.998770i	$0.484213\pi$
$20$	0	0
$21$	2.97416	0.649015
$22$	0	0
$23$	5.72098	1.19291	0.596454	−	0.802647i	$-0.296576\pi$
$23$	5.72098	1.19291	0.596454	+	0.802647i	$0.296576\pi$
$24$	0	0
$25$	−0.590124	−0.118025
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−0.718106	−0.133349	−0.0666745	−	0.997775i	$-0.521239\pi$
$29$	−0.718106	−0.133349	−0.0666745	+	0.997775i	$0.521239\pi$
$30$	0	0
$31$	−4.27590	−0.767974	−0.383987	−	0.923339i	$-0.625449\pi$
$31$	−4.27590	−0.767974	−0.383987	+	0.923339i	$0.625449\pi$
$32$	0	0
$33$	−0.138272	−0.0240700
$34$	0	0
$35$	−6.24565	−1.05571
$36$	0	0
$37$	−3.92212	−0.644792	−0.322396	−	0.946605i	$-0.604488\pi$
$37$	−3.92212	−0.644792	−0.322396	+	0.946605i	$0.604488\pi$
$38$	0	0
$39$	5.06303	0.810733
$40$	0	0
$41$	7.57912	1.18366	0.591830	−	0.806063i	$-0.298406\pi$
$41$	7.57912	1.18366	0.591830	+	0.806063i	$0.298406\pi$
$42$	0	0
$43$	12.3515	1.88359	0.941797	−	0.336183i	$-0.109136\pi$
$43$	12.3515	1.88359	0.941797	+	0.336183i	$0.109136\pi$
$44$	0	0
$45$	−2.09997	−0.313045
$46$	0	0
$47$	−9.22989	−1.34632	−0.673159	−	0.739498i	$-0.735063\pi$
$47$	−9.22989	−1.34632	−0.673159	+	0.739498i	$0.735063\pi$
$48$	0	0
$49$	1.84562	0.263660
$50$	0	0
$51$	0.299135	0.0418873
$52$	0	0
$53$	1.26409	0.173636	0.0868178	−	0.996224i	$-0.472330\pi$
$53$	1.26409	0.173636	0.0868178	+	0.996224i	$0.472330\pi$
$54$	0	0
$55$	0.290366	0.0391530
$56$	0	0
$57$	0.432187	0.0572446
$58$	0	0
$59$	3.01296	0.392253	0.196127	−	0.980579i	$-0.437164\pi$
$59$	3.01296	0.392253	0.196127	+	0.980579i	$0.437164\pi$
$60$	0	0
$61$	2.98537	0.382237	0.191118	−	0.981567i	$-0.438789\pi$
$61$	2.98537	0.382237	0.191118	+	0.981567i	$0.438789\pi$
$62$	0	0
$63$	2.97416	0.374709
$64$	0	0
$65$	−10.6322	−1.31876
$66$	0	0
$67$	−11.3934	−1.39193	−0.695964	−	0.718076i	$-0.745023\pi$
$67$	−11.3934	−1.39193	−0.695964	+	0.718076i	$0.745023\pi$
$68$	0	0
$69$	5.72098	0.688726
$70$	0	0
$71$	−7.07665	−0.839844	−0.419922	−	0.907560i	$-0.637943\pi$
$71$	−7.07665	−0.839844	−0.419922	+	0.907560i	$0.637943\pi$
$72$	0	0
$73$	14.5486	1.70278	0.851392	−	0.524531i	$-0.175759\pi$
$73$	14.5486	1.70278	0.851392	+	0.524531i	$0.175759\pi$
$74$	0	0
$75$	−0.590124	−0.0681417
$76$	0	0
$77$	−0.411242	−0.0468654
$78$	0	0
$79$	13.7670	1.54891	0.774454	−	0.632630i	$-0.218025\pi$
$79$	13.7670	1.54891	0.774454	+	0.632630i	$0.218025\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−15.7764	−1.73168	−0.865840	−	0.500321i	$-0.833215\pi$
$83$	−15.7764	−1.73168	−0.865840	+	0.500321i	$0.833215\pi$
$84$	0	0
$85$	−0.628175	−0.0681351
$86$	0	0
$87$	−0.718106	−0.0769891
$88$	0	0
$89$	−12.4982	−1.32481	−0.662404	−	0.749147i	$-0.730464\pi$
$89$	−12.4982	−1.32481	−0.662404	+	0.749147i	$0.730464\pi$
$90$	0	0
$91$	15.0583	1.57853
$92$	0	0
$93$	−4.27590	−0.443390
$94$	0	0
$95$	−0.907580	−0.0931157
$96$	0	0
$97$	7.54510	0.766089	0.383044	−	0.923730i	$-0.374876\pi$
$97$	7.54510	0.766089	0.383044	+	0.923730i	$0.374876\pi$
$98$	0	0
$99$	−0.138272	−0.0138968
$100$	0	0
$101$	10.4322	1.03804	0.519021	−	0.854761i	$-0.326296\pi$
$101$	10.4322	1.03804	0.519021	+	0.854761i	$0.326296\pi$
$102$	0	0
$103$	12.4671	1.22842	0.614211	−	0.789142i	$-0.289475\pi$
$103$	12.4671	1.22842	0.614211	+	0.789142i	$0.289475\pi$
$104$	0	0
$105$	−6.24565	−0.609513
$106$	0	0
$107$	−5.84644	−0.565196	−0.282598	−	0.959238i	$-0.591196\pi$
$107$	−5.84644	−0.565196	−0.282598	+	0.959238i	$0.591196\pi$
$108$	0	0
$109$	11.4163	1.09348	0.546742	−	0.837301i	$-0.315868\pi$
$109$	11.4163	1.09348	0.546742	+	0.837301i	$0.315868\pi$
$110$	0	0
$111$	−3.92212	−0.372271
$112$	0	0
$113$	17.7140	1.66640	0.833198	−	0.552975i	$-0.186508\pi$
$113$	17.7140	1.66640	0.833198	+	0.552975i	$0.186508\pi$
$114$	0	0
$115$	−12.0139	−1.12030
$116$	0	0
$117$	5.06303	0.468077
$118$	0	0
$119$	0.889675	0.0815564
$120$	0	0
$121$	−10.9809	−0.998262
$122$	0	0
$123$	7.57912	0.683387
$124$	0	0
$125$	11.7391	1.04998
$126$	0	0
$127$	8.49463	0.753776	0.376888	−	0.926259i	$-0.376994\pi$
$127$	8.49463	0.753776	0.376888	+	0.926259i	$0.376994\pi$
$128$	0	0
$129$	12.3515	1.08749
$130$	0	0
$131$	−2.85564	−0.249499	−0.124749	−	0.992188i	$-0.539813\pi$
$131$	−2.85564	−0.249499	−0.124749	+	0.992188i	$0.539813\pi$
$132$	0	0
$133$	1.28539	0.111458
$134$	0	0
$135$	−2.09997	−0.180737
$136$	0	0
$137$	1.64533	0.140570	0.0702850	−	0.997527i	$-0.477609\pi$
$137$	1.64533	0.140570	0.0702850	+	0.997527i	$0.477609\pi$
$138$	0	0
$139$	12.1666	1.03196	0.515978	−	0.856602i	$-0.327429\pi$
$139$	12.1666	1.03196	0.515978	+	0.856602i	$0.327429\pi$
$140$	0	0
$141$	−9.22989	−0.777297
$142$	0	0
$143$	−0.700073	−0.0585431
$144$	0	0
$145$	1.50800	0.125233
$146$	0	0
$147$	1.84562	0.152224
$148$	0	0
$149$	−12.9793	−1.06331	−0.531653	−	0.846962i	$-0.678429\pi$
$149$	−12.9793	−1.06331	−0.531653	+	0.846962i	$0.678429\pi$
$150$	0	0
$151$	−20.5738	−1.67427	−0.837135	−	0.546997i	$-0.815771\pi$
$151$	−20.5738	−1.67427	−0.837135	+	0.546997i	$0.815771\pi$
$152$	0	0
$153$	0.299135	0.0241836
$154$	0	0
$155$	8.97925	0.721231
$156$	0	0
$157$	3.76196	0.300237	0.150119	−	0.988668i	$-0.452034\pi$
$157$	3.76196	0.300237	0.150119	+	0.988668i	$0.452034\pi$
$158$	0	0
$159$	1.26409	0.100249
$160$	0	0
$161$	17.0151	1.34098
$162$	0	0
$163$	18.3629	1.43829	0.719147	−	0.694858i	$-0.244533\pi$
$163$	18.3629	1.43829	0.719147	+	0.694858i	$0.244533\pi$
$164$	0	0
$165$	0.290366	0.0226050
$166$	0	0
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	12.6343	0.971866
$170$	0	0
$171$	0.432187	0.0330502
$172$	0	0
$173$	−24.3625	−1.85225	−0.926125	−	0.377217i	$-0.876881\pi$
$173$	−24.3625	−1.85225	−0.926125	+	0.377217i	$0.876881\pi$
$174$	0	0
$175$	−1.75512	−0.132675
$176$	0	0
$177$	3.01296	0.226468
$178$	0	0
$179$	−0.719729	−0.0537951	−0.0268975	−	0.999638i	$-0.508563\pi$
$179$	−0.719729	−0.0537951	−0.0268975	+	0.999638i	$0.508563\pi$
$180$	0	0
$181$	−3.81586	−0.283630	−0.141815	−	0.989893i	$-0.545294\pi$
$181$	−3.81586	−0.283630	−0.141815	+	0.989893i	$0.545294\pi$
$182$	0	0
$183$	2.98537	0.220685
$184$	0	0
$185$	8.23633	0.605547
$186$	0	0
$187$	−0.0413619	−0.00302468
$188$	0	0
$189$	2.97416	0.216338
$190$	0	0
$191$	17.9273	1.29717	0.648585	−	0.761142i	$-0.275361\pi$
$191$	17.9273	1.29717	0.648585	+	0.761142i	$0.275361\pi$
$192$	0	0
$193$	−6.31831	−0.454802	−0.227401	−	0.973801i	$-0.573023\pi$
$193$	−6.31831	−0.454802	−0.227401	+	0.973801i	$0.573023\pi$
$194$	0	0
$195$	−10.6322	−0.761388
$196$	0	0
$197$	−2.17097	−0.154675	−0.0773375	−	0.997005i	$-0.524642\pi$
$197$	−2.17097	−0.154675	−0.0773375	+	0.997005i	$0.524642\pi$
$198$	0	0
$199$	12.0022	0.850816	0.425408	−	0.905002i	$-0.360131\pi$
$199$	12.0022	0.850816	0.425408	+	0.905002i	$0.360131\pi$
$200$	0	0
$201$	−11.3934	−0.803630
$202$	0	0
$203$	−2.13576	−0.149901
$204$	0	0
$205$	−15.9159	−1.11162
$206$	0	0
$207$	5.72098	0.397636
$208$	0	0
$209$	−0.0597592	−0.00413363
$210$	0	0
$211$	12.4126	0.854521	0.427260	−	0.904129i	$-0.359479\pi$
$211$	12.4126	0.854521	0.427260	+	0.904129i	$0.359479\pi$
$212$	0	0
$213$	−7.07665	−0.484884
$214$	0	0
$215$	−25.9379	−1.76895
$216$	0	0
$217$	−12.7172	−0.863299
$218$	0	0
$219$	14.5486	0.983102
$220$	0	0
$221$	1.51453	0.101878
$222$	0	0
$223$	16.6831	1.11719	0.558593	−	0.829442i	$-0.311341\pi$
$223$	16.6831	1.11719	0.558593	+	0.829442i	$0.311341\pi$
$224$	0	0
$225$	−0.590124	−0.0393416
$226$	0	0
$227$	−7.20014	−0.477890	−0.238945	−	0.971033i	$-0.576802\pi$
$227$	−7.20014	−0.477890	−0.238945	+	0.971033i	$0.576802\pi$
$228$	0	0
$229$	16.3304	1.07914	0.539572	−	0.841939i	$-0.318586\pi$
$229$	16.3304	1.07914	0.539572	+	0.841939i	$0.318586\pi$
$230$	0	0
$231$	−0.411242	−0.0270577
$232$	0	0
$233$	16.8678	1.10504	0.552522	−	0.833498i	$-0.313666\pi$
$233$	16.8678	1.10504	0.552522	+	0.833498i	$0.313666\pi$
$234$	0	0
$235$	19.3825	1.26437
$236$	0	0
$237$	13.7670	0.894262
$238$	0	0
$239$	24.1014	1.55899	0.779494	−	0.626410i	$-0.215476\pi$
$239$	24.1014	1.55899	0.779494	+	0.626410i	$0.215476\pi$
$240$	0	0
$241$	−1.37146	−0.0883434	−0.0441717	−	0.999024i	$-0.514065\pi$
$241$	−1.37146	−0.0883434	−0.0441717	+	0.999024i	$0.514065\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−3.87575	−0.247613
$246$	0	0
$247$	2.18818	0.139230
$248$	0	0
$249$	−15.7764	−0.999786
$250$	0	0
$251$	2.57248	0.162374	0.0811868	−	0.996699i	$-0.474129\pi$
$251$	2.57248	0.162374	0.0811868	+	0.996699i	$0.474129\pi$
$252$	0	0
$253$	−0.791050	−0.0497329
$254$	0	0
$255$	−0.628175	−0.0393378
$256$	0	0
$257$	6.87895	0.429097	0.214549	−	0.976713i	$-0.431172\pi$
$257$	6.87895	0.429097	0.214549	+	0.976713i	$0.431172\pi$
$258$	0	0
$259$	−11.6650	−0.724828
$260$	0	0
$261$	−0.718106	−0.0444497
$262$	0	0
$263$	30.1737	1.86059	0.930296	−	0.366810i	$-0.119550\pi$
$263$	30.1737	1.86059	0.930296	+	0.366810i	$0.119550\pi$
$264$	0	0
$265$	−2.65454	−0.163067
$266$	0	0
$267$	−12.4982	−0.764878
$268$	0	0
$269$	8.26235	0.503764	0.251882	−	0.967758i	$-0.418950\pi$
$269$	8.26235	0.503764	0.251882	+	0.967758i	$0.418950\pi$
$270$	0	0
$271$	23.3323	1.41734	0.708669	−	0.705541i	$-0.249296\pi$
$271$	23.3323	1.41734	0.708669	+	0.705541i	$0.249296\pi$
$272$	0	0
$273$	15.0583	0.911367
$274$	0	0
$275$	0.0815975	0.00492051
$276$	0	0
$277$	−2.10722	−0.126610	−0.0633052	−	0.997994i	$-0.520164\pi$
$277$	−2.10722	−0.126610	−0.0633052	+	0.997994i	$0.520164\pi$
$278$	0	0
$279$	−4.27590	−0.255991
$280$	0	0
$281$	−7.06113	−0.421232	−0.210616	−	0.977569i	$-0.567547\pi$
$281$	−7.06113	−0.421232	−0.210616	+	0.977569i	$0.567547\pi$
$282$	0	0
$283$	−10.3953	−0.617937	−0.308969	−	0.951072i	$-0.599984\pi$
$283$	−10.3953	−0.617937	−0.308969	+	0.951072i	$0.599984\pi$
$284$	0	0
$285$	−0.907580	−0.0537604
$286$	0	0
$287$	22.5415	1.33058
$288$	0	0
$289$	−16.9105	−0.994736
$290$	0	0
$291$	7.54510	0.442302
$292$	0	0
$293$	−24.7202	−1.44417	−0.722086	−	0.691803i	$-0.756817\pi$
$293$	−24.7202	−1.44417	−0.722086	+	0.691803i	$0.756817\pi$
$294$	0	0
$295$	−6.32712	−0.368379
$296$	0	0
$297$	−0.138272	−0.00802334
$298$	0	0
$299$	28.9655	1.67512
$300$	0	0
$301$	36.7355	2.11740
$302$	0	0
$303$	10.4322	0.599314
$304$	0	0
$305$	−6.26918	−0.358972
$306$	0	0
$307$	−31.2025	−1.78082	−0.890410	−	0.455159i	$-0.849582\pi$
$307$	−31.2025	−1.78082	−0.890410	+	0.455159i	$0.849582\pi$
$308$	0	0
$309$	12.4671	0.709229
$310$	0	0
$311$	33.4235	1.89527	0.947636	−	0.319353i	$-0.103466\pi$
$311$	33.4235	1.89527	0.947636	+	0.319353i	$0.103466\pi$
$312$	0	0
$313$	−18.4281	−1.04162	−0.520808	−	0.853674i	$-0.674369\pi$
$313$	−18.4281	−1.04162	−0.520808	+	0.853674i	$0.674369\pi$
$314$	0	0
$315$	−6.24565	−0.351902
$316$	0	0
$317$	−23.6150	−1.32635	−0.663176	−	0.748463i	$-0.730792\pi$
$317$	−23.6150	−1.32635	−0.663176	+	0.748463i	$0.730792\pi$
$318$	0	0
$319$	0.0992938	0.00555938
$320$	0	0
$321$	−5.84644	−0.326316
$322$	0	0
$323$	0.129282	0.00719346
$324$	0	0
$325$	−2.98782	−0.165734
$326$	0	0
$327$	11.4163	0.631323
$328$	0	0
$329$	−27.4512	−1.51343
$330$	0	0
$331$	0.822383	0.0452023	0.0226011	−	0.999745i	$-0.492805\pi$
$331$	0.822383	0.0452023	0.0226011	+	0.999745i	$0.492805\pi$
$332$	0	0
$333$	−3.92212	−0.214931
$334$	0	0
$335$	23.9259	1.30721
$336$	0	0
$337$	−14.7926	−0.805805	−0.402902	−	0.915243i	$-0.631999\pi$
$337$	−14.7926	−0.805805	−0.402902	+	0.915243i	$0.631999\pi$
$338$	0	0
$339$	17.7140	0.962094
$340$	0	0
$341$	0.591235	0.0320172
$342$	0	0
$343$	−15.3299	−0.827739
$344$	0	0
$345$	−12.0139	−0.646807
$346$	0	0
$347$	−23.3182	−1.25179	−0.625894	−	0.779908i	$-0.715266\pi$
$347$	−23.3182	−1.25179	−0.625894	+	0.779908i	$0.715266\pi$
$348$	0	0
$349$	−23.5530	−1.26076	−0.630382	−	0.776285i	$-0.717102\pi$
$349$	−23.5530	−1.26076	−0.630382	+	0.776285i	$0.717102\pi$
$350$	0	0
$351$	5.06303	0.270244
$352$	0	0
$353$	−22.7213	−1.20933	−0.604666	−	0.796479i	$-0.706693\pi$
$353$	−22.7213	−1.20933	−0.604666	+	0.796479i	$0.706693\pi$
$354$	0	0
$355$	14.8608	0.788727
$356$	0	0
$357$	0.889675	0.0470866
$358$	0	0
$359$	−4.45309	−0.235025	−0.117513	−	0.993071i	$-0.537492\pi$
$359$	−4.45309	−0.235025	−0.117513	+	0.993071i	$0.537492\pi$
$360$	0	0
$361$	−18.8132	−0.990169
$362$	0	0
$363$	−10.9809	−0.576347
$364$	0	0
$365$	−30.5516	−1.59914
$366$	0	0
$367$	31.0004	1.61821	0.809103	−	0.587667i	$-0.199953\pi$
$367$	31.0004	1.61821	0.809103	+	0.587667i	$0.199953\pi$
$368$	0	0
$369$	7.57912	0.394553
$370$	0	0
$371$	3.75960	0.195188
$372$	0	0
$373$	−14.5252	−0.752089	−0.376044	−	0.926602i	$-0.622716\pi$
$373$	−14.5252	−0.752089	−0.376044	+	0.926602i	$0.622716\pi$
$374$	0	0
$375$	11.7391	0.606204
$376$	0	0
$377$	−3.63579	−0.187253
$378$	0	0
$379$	−5.36141	−0.275397	−0.137698	−	0.990474i	$-0.543970\pi$
$379$	−5.36141	−0.275397	−0.137698	+	0.990474i	$0.543970\pi$
$380$	0	0
$381$	8.49463	0.435193
$382$	0	0
$383$	−35.4614	−1.81200	−0.905998	−	0.423283i	$-0.860878\pi$
$383$	−35.4614	−1.81200	−0.905998	+	0.423283i	$0.860878\pi$
$384$	0	0
$385$	0.863596	0.0440129
$386$	0	0
$387$	12.3515	0.627865
$388$	0	0
$389$	−11.3561	−0.575777	−0.287889	−	0.957664i	$-0.592953\pi$
$389$	−11.3561	−0.575777	−0.287889	+	0.957664i	$0.592953\pi$
$390$	0	0
$391$	1.71135	0.0865466
$392$	0	0
$393$	−2.85564	−0.144048
$394$	0	0
$395$	−28.9103	−1.45463
$396$	0	0
$397$	29.2022	1.46562	0.732808	−	0.680436i	$-0.238210\pi$
$397$	29.2022	1.46562	0.732808	+	0.680436i	$0.238210\pi$
$398$	0	0
$399$	1.28539	0.0643501
$400$	0	0
$401$	−20.5988	−1.02866	−0.514328	−	0.857594i	$-0.671959\pi$
$401$	−20.5988	−1.02866	−0.514328	+	0.857594i	$0.671959\pi$
$402$	0	0
$403$	−21.6490	−1.07841
$404$	0	0
$405$	−2.09997	−0.104348
$406$	0	0
$407$	0.542318	0.0268817
$408$	0	0
$409$	−6.71740	−0.332154	−0.166077	−	0.986113i	$-0.553110\pi$
$409$	−6.71740	−0.332154	−0.166077	+	0.986113i	$0.553110\pi$
$410$	0	0
$411$	1.64533	0.0811581
$412$	0	0
$413$	8.96101	0.440942
$414$	0	0
$415$	33.1299	1.62628
$416$	0	0
$417$	12.1666	0.595800
$418$	0	0
$419$	−2.90625	−0.141980	−0.0709898	−	0.997477i	$-0.522616\pi$
$419$	−2.90625	−0.141980	−0.0709898	+	0.997477i	$0.522616\pi$
$420$	0	0
$421$	−11.0417	−0.538140	−0.269070	−	0.963121i	$-0.586716\pi$
$421$	−11.0417	−0.538140	−0.269070	+	0.963121i	$0.586716\pi$
$422$	0	0
$423$	−9.22989	−0.448772
$424$	0	0
$425$	−0.176527	−0.00856281
$426$	0	0
$427$	8.87895	0.429683
$428$	0	0
$429$	−0.700073	−0.0337999
$430$	0	0
$431$	−17.6903	−0.852110	−0.426055	−	0.904697i	$-0.640097\pi$
$431$	−17.6903	−0.852110	−0.426055	+	0.904697i	$0.640097\pi$
$432$	0	0
$433$	−17.4174	−0.837025	−0.418513	−	0.908211i	$-0.637448\pi$
$433$	−17.4174	−0.837025	−0.418513	+	0.908211i	$0.637448\pi$
$434$	0	0
$435$	1.50800	0.0723032
$436$	0	0
$437$	2.47254	0.118277
$438$	0	0
$439$	16.5212	0.788512	0.394256	−	0.919001i	$-0.371002\pi$
$439$	16.5212	0.788512	0.394256	+	0.919001i	$0.371002\pi$
$440$	0	0
$441$	1.84562	0.0878867
$442$	0	0
$443$	−21.4189	−1.01764	−0.508822	−	0.860871i	$-0.669919\pi$
$443$	−21.4189	−1.01764	−0.508822	+	0.860871i	$0.669919\pi$
$444$	0	0
$445$	26.2459	1.24417
$446$	0	0
$447$	−12.9793	−0.613900
$448$	0	0
$449$	31.1123	1.46828	0.734141	−	0.678998i	$-0.237585\pi$
$449$	31.1123	1.46828	0.734141	+	0.678998i	$0.237585\pi$
$450$	0	0
$451$	−1.04798	−0.0493474
$452$	0	0
$453$	−20.5738	−0.966640
$454$	0	0
$455$	−31.6219	−1.48246
$456$	0	0
$457$	11.2728	0.527321	0.263660	−	0.964616i	$-0.415070\pi$
$457$	11.2728	0.527321	0.263660	+	0.964616i	$0.415070\pi$
$458$	0	0
$459$	0.299135	0.0139624
$460$	0	0
$461$	25.0888	1.16850	0.584251	−	0.811573i	$-0.301388\pi$
$461$	25.0888	1.16850	0.584251	+	0.811573i	$0.301388\pi$
$462$	0	0
$463$	29.9933	1.39391	0.696954	−	0.717116i	$-0.254538\pi$
$463$	29.9933	1.39391	0.696954	+	0.717116i	$0.254538\pi$
$464$	0	0
$465$	8.97925	0.416403
$466$	0	0
$467$	−26.6386	−1.23269	−0.616343	−	0.787478i	$-0.711386\pi$
$467$	−26.6386	−1.23269	−0.616343	+	0.787478i	$0.711386\pi$
$468$	0	0
$469$	−33.8859	−1.56470
$470$	0	0
$471$	3.76196	0.173342
$472$	0	0
$473$	−1.70787	−0.0785279
$474$	0	0
$475$	−0.255044	−0.0117022
$476$	0	0
$477$	1.26409	0.0578786
$478$	0	0
$479$	34.6774	1.58445	0.792225	−	0.610229i	$-0.208922\pi$
$479$	34.6774	1.58445	0.792225	+	0.610229i	$0.208922\pi$
$480$	0	0
$481$	−19.8578	−0.905437
$482$	0	0
$483$	17.0151	0.774215
$484$	0	0
$485$	−15.8445	−0.719461
$486$	0	0
$487$	−15.7054	−0.711678	−0.355839	−	0.934547i	$-0.615805\pi$
$487$	−15.7054	−0.711678	−0.355839	+	0.934547i	$0.615805\pi$
$488$	0	0
$489$	18.3629	0.830399
$490$	0	0
$491$	16.0372	0.723751	0.361875	−	0.932226i	$-0.382137\pi$
$491$	16.0372	0.723751	0.361875	+	0.932226i	$0.382137\pi$
$492$	0	0
$493$	−0.214811	−0.00967459
$494$	0	0
$495$	0.290366	0.0130510
$496$	0	0
$497$	−21.0471	−0.944090
$498$	0	0
$499$	−12.7331	−0.570011	−0.285005	−	0.958526i	$-0.591995\pi$
$499$	−12.7331	−0.570011	−0.285005	+	0.958526i	$0.591995\pi$
$500$	0	0
$501$	1.00000	0.0446767
$502$	0	0
$503$	3.14474	0.140217	0.0701086	−	0.997539i	$-0.477665\pi$
$503$	3.14474	0.140217	0.0701086	+	0.997539i	$0.477665\pi$
$504$	0	0
$505$	−21.9073	−0.974863
$506$	0	0
$507$	12.6343	0.561107
$508$	0	0
$509$	−4.63793	−0.205572	−0.102786	−	0.994703i	$-0.532776\pi$
$509$	−4.63793	−0.205572	−0.102786	+	0.994703i	$0.532776\pi$
$510$	0	0
$511$	43.2698	1.91414
$512$	0	0
$513$	0.432187	0.0190815
$514$	0	0
$515$	−26.1806	−1.15365
$516$	0	0
$517$	1.27623	0.0561286
$518$	0	0
$519$	−24.3625	−1.06940
$520$	0	0
$521$	37.2557	1.63220	0.816102	−	0.577908i	$-0.196131\pi$
$521$	37.2557	1.63220	0.816102	+	0.577908i	$0.196131\pi$
$522$	0	0
$523$	−27.9630	−1.22274	−0.611369	−	0.791346i	$-0.709381\pi$
$523$	−27.9630	−1.22274	−0.611369	+	0.791346i	$0.709381\pi$
$524$	0	0
$525$	−1.75512	−0.0765999
$526$	0	0
$527$	−1.27907	−0.0557172
$528$	0	0
$529$	9.72967	0.423029
$530$	0	0
$531$	3.01296	0.130751
$532$	0	0
$533$	38.3733	1.66213
$534$	0	0
$535$	12.2773	0.530796
$536$	0	0
$537$	−0.719729	−0.0310586
$538$	0	0
$539$	−0.255197	−0.0109921
$540$	0	0
$541$	−22.7551	−0.978316	−0.489158	−	0.872195i	$-0.662696\pi$
$541$	−22.7551	−0.978316	−0.489158	+	0.872195i	$0.662696\pi$
$542$	0	0
$543$	−3.81586	−0.163754
$544$	0	0
$545$	−23.9739	−1.02693
$546$	0	0
$547$	6.58788	0.281677	0.140839	−	0.990033i	$-0.455020\pi$
$547$	6.58788	0.281677	0.140839	+	0.990033i	$0.455020\pi$
$548$	0	0
$549$	2.98537	0.127412
$550$	0	0
$551$	−0.310356	−0.0132216
$552$	0	0
$553$	40.9452	1.74117
$554$	0	0
$555$	8.23633	0.349613
$556$	0	0
$557$	16.1334	0.683594	0.341797	−	0.939774i	$-0.388964\pi$
$557$	16.1334	0.683594	0.341797	+	0.939774i	$0.388964\pi$
$558$	0	0
$559$	62.5362	2.64500
$560$	0	0
$561$	−0.0413619	−0.00174630
$562$	0	0
$563$	11.2526	0.474240	0.237120	−	0.971480i	$-0.423797\pi$
$563$	11.2526	0.474240	0.237120	+	0.971480i	$0.423797\pi$
$564$	0	0
$565$	−37.1989	−1.56497
$566$	0	0
$567$	2.97416	0.124903
$568$	0	0
$569$	19.1087	0.801079	0.400539	−	0.916280i	$-0.368823\pi$
$569$	19.1087	0.801079	0.400539	+	0.916280i	$0.368823\pi$
$570$	0	0
$571$	−26.7499	−1.11945	−0.559725	−	0.828679i	$-0.689093\pi$
$571$	−26.7499	−1.11945	−0.559725	+	0.828679i	$0.689093\pi$
$572$	0	0
$573$	17.9273	0.748922
$574$	0	0
$575$	−3.37609	−0.140793
$576$	0	0
$577$	2.62594	0.109319	0.0546596	−	0.998505i	$-0.482593\pi$
$577$	2.62594	0.109319	0.0546596	+	0.998505i	$0.482593\pi$
$578$	0	0
$579$	−6.31831	−0.262580
$580$	0	0
$581$	−46.9214	−1.94663
$582$	0	0
$583$	−0.174787	−0.00723895
$584$	0	0
$585$	−10.6322	−0.439588
$586$	0	0
$587$	−24.6288	−1.01654	−0.508269	−	0.861198i	$-0.669714\pi$
$587$	−24.6288	−1.01654	−0.508269	+	0.861198i	$0.669714\pi$
$588$	0	0
$589$	−1.84799	−0.0761450
$590$	0	0
$591$	−2.17097	−0.0893016
$592$	0	0
$593$	12.2107	0.501433	0.250716	−	0.968061i	$-0.419334\pi$
$593$	12.2107	0.501433	0.250716	+	0.968061i	$0.419334\pi$
$594$	0	0
$595$	−1.86829	−0.0765925
$596$	0	0
$597$	12.0022	0.491219
$598$	0	0
$599$	−21.7233	−0.887591	−0.443796	−	0.896128i	$-0.646368\pi$
$599$	−21.7233	−0.887591	−0.443796	+	0.896128i	$0.646368\pi$
$600$	0	0
$601$	−28.7575	−1.17304	−0.586521	−	0.809934i	$-0.699503\pi$
$601$	−28.7575	−1.17304	−0.586521	+	0.809934i	$0.699503\pi$
$602$	0	0
$603$	−11.3934	−0.463976
$604$	0	0
$605$	23.0595	0.937503
$606$	0	0
$607$	−22.9370	−0.930985	−0.465493	−	0.885052i	$-0.654123\pi$
$607$	−22.9370	−0.930985	−0.465493	+	0.885052i	$0.654123\pi$
$608$	0	0
$609$	−2.13576	−0.0865455
$610$	0	0
$611$	−46.7312	−1.89054
$612$	0	0
$613$	−39.7699	−1.60629	−0.803145	−	0.595784i	$-0.796842\pi$
$613$	−39.7699	−1.60629	−0.803145	+	0.595784i	$0.796842\pi$
$614$	0	0
$615$	−15.9159	−0.641792
$616$	0	0
$617$	−16.6341	−0.669663	−0.334832	−	0.942278i	$-0.608679\pi$
$617$	−16.6341	−0.669663	−0.334832	+	0.942278i	$0.608679\pi$
$618$	0	0
$619$	−20.5361	−0.825416	−0.412708	−	0.910863i	$-0.635417\pi$
$619$	−20.5361	−0.825416	−0.412708	+	0.910863i	$0.635417\pi$
$620$	0	0
$621$	5.72098	0.229575
$622$	0	0
$623$	−37.1717	−1.48925
$624$	0	0
$625$	−21.7011	−0.868045
$626$	0	0
$627$	−0.0597592	−0.00238655
$628$	0	0
$629$	−1.17324	−0.0467802
$630$	0	0
$631$	26.0602	1.03744	0.518721	−	0.854944i	$-0.326408\pi$
$631$	26.0602	1.03744	0.518721	+	0.854944i	$0.326408\pi$
$632$	0	0
$633$	12.4126	0.493358
$634$	0	0
$635$	−17.8385	−0.707898
$636$	0	0
$637$	9.34443	0.370240
$638$	0	0
$639$	−7.07665	−0.279948
$640$	0	0
$641$	−17.1851	−0.678771	−0.339386	−	0.940647i	$-0.610219\pi$
$641$	−17.1851	−0.678771	−0.339386	+	0.940647i	$0.610219\pi$
$642$	0	0
$643$	−39.0699	−1.54077	−0.770383	−	0.637582i	$-0.779935\pi$
$643$	−39.0699	−1.54077	−0.770383	+	0.637582i	$0.779935\pi$
$644$	0	0
$645$	−25.9379	−1.02130
$646$	0	0
$647$	−1.31198	−0.0515794	−0.0257897	−	0.999667i	$-0.508210\pi$
$647$	−1.31198	−0.0515794	−0.0257897	+	0.999667i	$0.508210\pi$
$648$	0	0
$649$	−0.416606	−0.0163532
$650$	0	0
$651$	−12.7172	−0.498426
$652$	0	0
$653$	36.2903	1.42015	0.710074	−	0.704127i	$-0.248661\pi$
$653$	36.2903	1.42015	0.710074	+	0.704127i	$0.248661\pi$
$654$	0	0
$655$	5.99676	0.234313
$656$	0	0
$657$	14.5486	0.567594
$658$	0	0
$659$	7.04958	0.274613	0.137306	−	0.990529i	$-0.456156\pi$
$659$	7.04958	0.274613	0.137306	+	0.990529i	$0.456156\pi$
$660$	0	0
$661$	21.4844	0.835648	0.417824	−	0.908528i	$-0.362793\pi$
$661$	21.4844	0.835648	0.417824	+	0.908528i	$0.362793\pi$
$662$	0	0
$663$	1.51453	0.0588194
$664$	0	0
$665$	−2.69929	−0.104674
$666$	0	0
$667$	−4.10827	−0.159073
$668$	0	0
$669$	16.6831	0.645008
$670$	0	0
$671$	−0.412792	−0.0159356
$672$	0	0
$673$	−6.31641	−0.243480	−0.121740	−	0.992562i	$-0.538847\pi$
$673$	−6.31641	−0.243480	−0.121740	+	0.992562i	$0.538847\pi$
$674$	0	0
$675$	−0.590124	−0.0227139
$676$	0	0
$677$	21.2225	0.815648	0.407824	−	0.913061i	$-0.366288\pi$
$677$	21.2225	0.815648	0.407824	+	0.913061i	$0.366288\pi$
$678$	0	0
$679$	22.4403	0.861181
$680$	0	0
$681$	−7.20014	−0.275910
$682$	0	0
$683$	40.0609	1.53289	0.766443	−	0.642312i	$-0.222025\pi$
$683$	40.0609	1.53289	0.766443	+	0.642312i	$0.222025\pi$
$684$	0	0
$685$	−3.45514	−0.132014
$686$	0	0
$687$	16.3304	0.623044
$688$	0	0
$689$	6.40011	0.243825
$690$	0	0
$691$	41.3269	1.57215	0.786075	−	0.618131i	$-0.212110\pi$
$691$	41.3269	1.57215	0.786075	+	0.618131i	$0.212110\pi$
$692$	0	0
$693$	−0.411242	−0.0156218
$694$	0	0
$695$	−25.5494	−0.969146
$696$	0	0
$697$	2.26718	0.0858756
$698$	0	0
$699$	16.8678	0.637997
$700$	0	0
$701$	−26.1035	−0.985916	−0.492958	−	0.870053i	$-0.664084\pi$
$701$	−26.1035	−0.985916	−0.492958	+	0.870053i	$0.664084\pi$
$702$	0	0
$703$	−1.69509	−0.0639314
$704$	0	0
$705$	19.3825	0.729987
$706$	0	0
$707$	31.0270	1.16689
$708$	0	0
$709$	−0.259309	−0.00973854	−0.00486927	−	0.999988i	$-0.501550\pi$
$709$	−0.259309	−0.00973854	−0.00486927	+	0.999988i	$0.501550\pi$
$710$	0	0
$711$	13.7670	0.516303
$712$	0	0
$713$	−24.4623	−0.916122
$714$	0	0
$715$	1.47013	0.0549799
$716$	0	0
$717$	24.1014	0.900082
$718$	0	0
$719$	−44.6788	−1.66624	−0.833120	−	0.553093i	$-0.813447\pi$
$719$	−44.6788	−1.66624	−0.833120	+	0.553093i	$0.813447\pi$
$720$	0	0
$721$	37.0792	1.38090
$722$	0	0
$723$	−1.37146	−0.0510051
$724$	0	0
$725$	0.423772	0.0157385
$726$	0	0
$727$	11.9819	0.444384	0.222192	−	0.975003i	$-0.428679\pi$
$727$	11.9819	0.444384	0.222192	+	0.975003i	$0.428679\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	3.69478	0.136656
$732$	0	0
$733$	−18.2924	−0.675645	−0.337822	−	0.941210i	$-0.609690\pi$
$733$	−18.2924	−0.675645	−0.337822	+	0.941210i	$0.609690\pi$
$734$	0	0
$735$	−3.87575	−0.142959
$736$	0	0
$737$	1.57539	0.0580302
$738$	0	0
$739$	−11.5860	−0.426197	−0.213099	−	0.977031i	$-0.568356\pi$
$739$	−11.5860	−0.426197	−0.213099	+	0.977031i	$0.568356\pi$
$740$	0	0
$741$	2.18818	0.0803846
$742$	0	0
$743$	−14.2882	−0.524183	−0.262092	−	0.965043i	$-0.584412\pi$
$743$	−14.2882	−0.524183	−0.262092	+	0.965043i	$0.584412\pi$
$744$	0	0
$745$	27.2562	0.998589
$746$	0	0
$747$	−15.7764	−0.577226
$748$	0	0
$749$	−17.3882	−0.635352
$750$	0	0
$751$	1.28638	0.0469407	0.0234703	−	0.999725i	$-0.492528\pi$
$751$	1.28638	0.0469407	0.0234703	+	0.999725i	$0.492528\pi$
$752$	0	0
$753$	2.57248	0.0937465
$754$	0	0
$755$	43.2043	1.57237
$756$	0	0
$757$	15.5228	0.564185	0.282092	−	0.959387i	$-0.408972\pi$
$757$	15.5228	0.564185	0.282092	+	0.959387i	$0.408972\pi$
$758$	0	0
$759$	−0.791050	−0.0287133
$760$	0	0
$761$	−1.55417	−0.0563385	−0.0281693	−	0.999603i	$-0.508968\pi$
$761$	−1.55417	−0.0563385	−0.0281693	+	0.999603i	$0.508968\pi$
$762$	0	0
$763$	33.9539	1.22921
$764$	0	0
$765$	−0.628175	−0.0227117
$766$	0	0
$767$	15.2547	0.550815
$768$	0	0
$769$	−45.1817	−1.62929	−0.814647	−	0.579957i	$-0.803069\pi$
$769$	−45.1817	−1.62929	−0.814647	+	0.579957i	$0.803069\pi$
$770$	0	0
$771$	6.87895	0.247739
$772$	0	0
$773$	−53.9167	−1.93925	−0.969625	−	0.244597i	$-0.921344\pi$
$773$	−53.9167	−1.93925	−0.969625	+	0.244597i	$0.921344\pi$
$774$	0	0
$775$	2.52331	0.0906399
$776$	0	0
$777$	−11.6650	−0.418479
$778$	0	0
$779$	3.27560	0.117361
$780$	0	0
$781$	0.978500	0.0350135
$782$	0	0
$783$	−0.718106	−0.0256630
$784$	0	0
$785$	−7.90001	−0.281963
$786$	0	0
$787$	8.43062	0.300519	0.150260	−	0.988647i	$-0.451989\pi$
$787$	8.43062	0.300519	0.150260	+	0.988647i	$0.451989\pi$
$788$	0	0
$789$	30.1737	1.07421
$790$	0	0
$791$	52.6843	1.87324
$792$	0	0
$793$	15.1150	0.536749
$794$	0	0
$795$	−2.65454	−0.0941470
$796$	0	0
$797$	24.7547	0.876856	0.438428	−	0.898766i	$-0.355535\pi$
$797$	24.7547	0.876856	0.438428	+	0.898766i	$0.355535\pi$
$798$	0	0
$799$	−2.76098	−0.0976766
$800$	0	0
$801$	−12.4982	−0.441603
$802$	0	0
$803$	−2.01166	−0.0709899
$804$	0	0
$805$	−35.7312	−1.25936
$806$	0	0
$807$	8.26235	0.290849
$808$	0	0
$809$	−27.3604	−0.961941	−0.480970	−	0.876737i	$-0.659716\pi$
$809$	−27.3604	−0.961941	−0.480970	+	0.876737i	$0.659716\pi$
$810$	0	0
$811$	−2.45887	−0.0863426	−0.0431713	−	0.999068i	$-0.513746\pi$
$811$	−2.45887	−0.0863426	−0.0431713	+	0.999068i	$0.513746\pi$
$812$	0	0
$813$	23.3323	0.818300
$814$	0	0
$815$	−38.5616	−1.35075
$816$	0	0
$817$	5.33818	0.186759
$818$	0	0
$819$	15.0583	0.526178
$820$	0	0
$821$	6.86656	0.239645	0.119822	−	0.992795i	$-0.461768\pi$
$821$	6.86656	0.239645	0.119822	+	0.992795i	$0.461768\pi$
$822$	0	0
$823$	19.4606	0.678355	0.339178	−	0.940722i	$-0.389851\pi$
$823$	19.4606	0.678355	0.339178	+	0.940722i	$0.389851\pi$
$824$	0	0
$825$	0.0815975	0.00284086
$826$	0	0
$827$	8.19257	0.284884	0.142442	−	0.989803i	$-0.454505\pi$
$827$	8.19257	0.284884	0.142442	+	0.989803i	$0.454505\pi$
$828$	0	0
$829$	−46.2894	−1.60770	−0.803849	−	0.594833i	$-0.797218\pi$
$829$	−46.2894	−1.60770	−0.803849	+	0.594833i	$0.797218\pi$
$830$	0	0
$831$	−2.10722	−0.0730986
$832$	0	0
$833$	0.552090	0.0191288
$834$	0	0
$835$	−2.09997	−0.0726725
$836$	0	0
$837$	−4.27590	−0.147797
$838$	0	0
$839$	4.64299	0.160294	0.0801468	−	0.996783i	$-0.474461\pi$
$839$	4.64299	0.160294	0.0801468	+	0.996783i	$0.474461\pi$
$840$	0	0
$841$	−28.4843	−0.982218
$842$	0	0
$843$	−7.06113	−0.243198
$844$	0	0
$845$	−26.5316	−0.912713
$846$	0	0
$847$	−32.6589	−1.12217
$848$	0	0
$849$	−10.3953	−0.356766
$850$	0	0
$851$	−22.4384	−0.769177
$852$	0	0
$853$	9.12150	0.312314	0.156157	−	0.987732i	$-0.450089\pi$
$853$	9.12150	0.312314	0.156157	+	0.987732i	$0.450089\pi$
$854$	0	0
$855$	−0.907580	−0.0310386
$856$	0	0
$857$	−7.34645	−0.250950	−0.125475	−	0.992097i	$-0.540046\pi$
$857$	−7.34645	−0.250950	−0.125475	+	0.992097i	$0.540046\pi$
$858$	0	0
$859$	−37.2638	−1.27142	−0.635712	−	0.771927i	$-0.719293\pi$
$859$	−37.2638	−1.27142	−0.635712	+	0.771927i	$0.719293\pi$
$860$	0	0
$861$	22.5415	0.768213
$862$	0	0
$863$	−17.3469	−0.590496	−0.295248	−	0.955421i	$-0.595402\pi$
$863$	−17.3469	−0.590496	−0.295248	+	0.955421i	$0.595402\pi$
$864$	0	0
$865$	51.1606	1.73951
$866$	0	0
$867$	−16.9105	−0.574311
$868$	0	0
$869$	−1.90359	−0.0645747
$870$	0	0
$871$	−57.6852	−1.95459
$872$	0	0
$873$	7.54510	0.255363
$874$	0	0
$875$	34.9139	1.18031
$876$	0	0
$877$	9.09836	0.307230	0.153615	−	0.988131i	$-0.450908\pi$
$877$	9.09836	0.307230	0.153615	+	0.988131i	$0.450908\pi$
$878$	0	0
$879$	−24.7202	−0.833793
$880$	0	0
$881$	−26.1065	−0.879550	−0.439775	−	0.898108i	$-0.644942\pi$
$881$	−26.1065	−0.879550	−0.439775	+	0.898108i	$0.644942\pi$
$882$	0	0
$883$	22.3088	0.750750	0.375375	−	0.926873i	$-0.377514\pi$
$883$	22.3088	0.750750	0.375375	+	0.926873i	$0.377514\pi$
$884$	0	0
$885$	−6.32712	−0.212684
$886$	0	0
$887$	−34.3987	−1.15500	−0.577498	−	0.816392i	$-0.695971\pi$
$887$	−34.3987	−1.15500	−0.577498	+	0.816392i	$0.695971\pi$
$888$	0	0
$889$	25.2644	0.847340
$890$	0	0
$891$	−0.138272	−0.00463228
$892$	0	0
$893$	−3.98904	−0.133488
$894$	0	0
$895$	1.51141	0.0505208
$896$	0	0
$897$	28.9655	0.967130
$898$	0	0
$899$	3.07055	0.102408
$900$	0	0
$901$	0.378133	0.0125974
$902$	0	0
$903$	36.7355	1.22248
$904$	0	0
$905$	8.01318	0.266367
$906$	0	0
$907$	18.7791	0.623551	0.311776	−	0.950156i	$-0.399076\pi$
$907$	18.7791	0.623551	0.311776	+	0.950156i	$0.399076\pi$
$908$	0	0
$909$	10.4322	0.346014
$910$	0	0
$911$	42.3954	1.40462	0.702311	−	0.711870i	$-0.252151\pi$
$911$	42.3954	1.40462	0.702311	+	0.711870i	$0.252151\pi$
$912$	0	0
$913$	2.18142	0.0721945
$914$	0	0
$915$	−6.26918	−0.207253
$916$	0	0
$917$	−8.49313	−0.280468
$918$	0	0
$919$	21.3230	0.703379	0.351690	−	0.936117i	$-0.385607\pi$
$919$	21.3230	0.703379	0.351690	+	0.936117i	$0.385607\pi$
$920$	0	0
$921$	−31.2025	−1.02816
$922$	0	0
$923$	−35.8293	−1.17933
$924$	0	0
$925$	2.31454	0.0761015
$926$	0	0
$927$	12.4671	0.409474
$928$	0	0
$929$	44.9861	1.47595	0.737973	−	0.674830i	$-0.235783\pi$
$929$	44.9861	1.47595	0.737973	+	0.674830i	$0.235783\pi$
$930$	0	0
$931$	0.797654	0.0261420
$932$	0	0
$933$	33.4235	1.09424
$934$	0	0
$935$	0.0868588	0.00284059
$936$	0	0
$937$	−34.2610	−1.11926	−0.559629	−	0.828743i	$-0.689056\pi$
$937$	−34.2610	−1.11926	−0.559629	+	0.828743i	$0.689056\pi$
$938$	0	0
$939$	−18.4281	−0.601377
$940$	0	0
$941$	−18.7215	−0.610303	−0.305152	−	0.952304i	$-0.598707\pi$
$941$	−18.7215	−0.610303	−0.305152	+	0.952304i	$0.598707\pi$
$942$	0	0
$943$	43.3600	1.41200
$944$	0	0
$945$	−6.24565	−0.203171
$946$	0	0
$947$	33.2088	1.07914	0.539570	−	0.841941i	$-0.318587\pi$
$947$	33.2088	1.07914	0.539570	+	0.841941i	$0.318587\pi$
$948$	0	0
$949$	73.6599	2.39110
$950$	0	0
$951$	−23.6150	−0.765770
$952$	0	0
$953$	−29.9705	−0.970838	−0.485419	−	0.874282i	$-0.661333\pi$
$953$	−29.9705	−0.970838	−0.485419	+	0.874282i	$0.661333\pi$
$954$	0	0
$955$	−37.6467	−1.21822
$956$	0	0
$957$	0.0992938	0.00320971
$958$	0	0
$959$	4.89347	0.158018
$960$	0	0
$961$	−12.7167	−0.410217
$962$	0	0
$963$	−5.84644	−0.188399
$964$	0	0
$965$	13.2683	0.427121
$966$	0	0
$967$	−14.7207	−0.473387	−0.236693	−	0.971584i	$-0.576064\pi$
$967$	−14.7207	−0.473387	−0.236693	+	0.971584i	$0.576064\pi$
$968$	0	0
$969$	0.129282	0.00415315
$970$	0	0
$971$	11.0754	0.355428	0.177714	−	0.984082i	$-0.443130\pi$
$971$	11.0754	0.355428	0.177714	+	0.984082i	$0.443130\pi$
$972$	0	0
$973$	36.1853	1.16005
$974$	0	0
$975$	−2.98782	−0.0956867
$976$	0	0
$977$	−24.1829	−0.773681	−0.386840	−	0.922147i	$-0.626434\pi$
$977$	−24.1829	−0.773681	−0.386840	+	0.922147i	$0.626434\pi$
$978$	0	0
$979$	1.72815	0.0552319
$980$	0	0
$981$	11.4163	0.364494
$982$	0	0
$983$	−21.5383	−0.686964	−0.343482	−	0.939159i	$-0.611607\pi$
$983$	−21.5383	−0.686964	−0.343482	+	0.939159i	$0.611607\pi$
$984$	0	0
$985$	4.55897	0.145261
$986$	0	0
$987$	−27.4512	−0.873780
$988$	0	0
$989$	70.6630	2.24695
$990$	0	0
$991$	31.2346	0.992199	0.496099	−	0.868266i	$-0.334765\pi$
$991$	31.2346	0.992199	0.496099	+	0.868266i	$0.334765\pi$
$992$	0	0
$993$	0.822383	0.0260975
$994$	0	0
$995$	−25.2043	−0.799032
$996$	0	0
$997$	50.4341	1.59726	0.798632	−	0.601820i	$-0.205557\pi$
$997$	50.4341	1.59726	0.798632	+	0.601820i	$0.205557\pi$
$998$	0	0
$999$	−3.92212	−0.124090

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4008.2.a.l.1.3	✓	12
4.3	odd	2		8016.2.a.bf.1.3		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4008.2.a.l.1.3	✓	12	1.1	even	1	trivial
8016.2.a.bf.1.3		12	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -2.09997 q^{5} +2.97416 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -2.09997 q^{5} +2.97416 q^{7} +1.00000 q^{9} -0.138272 q^{11} +5.06303 q^{13} -2.09997 q^{15} +0.299135 q^{17} +0.432187 q^{19} +2.97416 q^{21} +5.72098 q^{23} -0.590124 q^{25} +1.00000 q^{27} -0.718106 q^{29} -4.27590 q^{31} -0.138272 q^{33} -6.24565 q^{35} -3.92212 q^{37} +5.06303 q^{39} +7.57912 q^{41} +12.3515 q^{43} -2.09997 q^{45} -9.22989 q^{47} +1.84562 q^{49} +0.299135 q^{51} +1.26409 q^{53} +0.290366 q^{55} +0.432187 q^{57} +3.01296 q^{59} +2.98537 q^{61} +2.97416 q^{63} -10.6322 q^{65} -11.3934 q^{67} +5.72098 q^{69} -7.07665 q^{71} +14.5486 q^{73} -0.590124 q^{75} -0.411242 q^{77} +13.7670 q^{79} +1.00000 q^{81} -15.7764 q^{83} -0.628175 q^{85} -0.718106 q^{87} -12.4982 q^{89} +15.0583 q^{91} -4.27590 q^{93} -0.907580 q^{95} +7.54510 q^{97} -0.138272 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{3} + 4 q^{5} + 11 q^{7} + 12 q^{9}+O(q^{10}) \) 12 * q + 12 * q^3 + 4 * q^5 + 11 * q^7 + 12 * q^9	\( 12 q + 12 q^{3} + 4 q^{5} + 11 q^{7} + 12 q^{9} + q^{11} + 8 q^{13} + 4 q^{15} + 3 q^{17} + 12 q^{19} + 11 q^{21} + 7 q^{23} + 18 q^{25} + 12 q^{27} + 5 q^{29} + 33 q^{31} + q^{33} + 15 q^{35} + 8 q^{37} + 8 q^{39} - 6 q^{41} + 16 q^{43} + 4 q^{45} + 18 q^{47} + 25 q^{49} + 3 q^{51} + 20 q^{53} + 39 q^{55} + 12 q^{57} + 4 q^{59} + 10 q^{61} + 11 q^{63} + 9 q^{67} + 7 q^{69} + 11 q^{71} + 22 q^{73} + 18 q^{75} + 24 q^{77} + 56 q^{79} + 12 q^{81} + 26 q^{83} + 15 q^{85} + 5 q^{87} - 15 q^{89} + 11 q^{91} + 33 q^{93} + 3 q^{95} + 8 q^{97} + q^{99}+O(q^{100}) \) 12 * q + 12 * q^3 + 4 * q^5 + 11 * q^7 + 12 * q^9 + q^11 + 8 * q^13 + 4 * q^15 + 3 * q^17 + 12 * q^19 + 11 * q^21 + 7 * q^23 + 18 * q^25 + 12 * q^27 + 5 * q^29 + 33 * q^31 + q^33 + 15 * q^35 + 8 * q^37 + 8 * q^39 - 6 * q^41 + 16 * q^43 + 4 * q^45 + 18 * q^47 + 25 * q^49 + 3 * q^51 + 20 * q^53 + 39 * q^55 + 12 * q^57 + 4 * q^59 + 10 * q^61 + 11 * q^63 + 9 * q^67 + 7 * q^69 + 11 * q^71 + 22 * q^73 + 18 * q^75 + 24 * q^77 + 56 * q^79 + 12 * q^81 + 26 * q^83 + 15 * q^85 + 5 * q^87 - 15 * q^89 + 11 * q^91 + 33 * q^93 + 3 * q^95 + 8 * q^97 + q^99

Introduction

L-functions

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Database

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