LMFDB - Embedded newform 4008.2.a.j.1.9

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			1.9
Root			$-2.53098$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} +1.53098 q^{5} +1.95354 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.53098 q^{5} +1.95354 q^{7} +1.00000 q^{9} -2.15240 q^{11} -0.0320505 q^{13} -1.53098 q^{15} -2.27829 q^{17} -5.52833 q^{19} -1.95354 q^{21} +6.90191 q^{23} -2.65610 q^{25} -1.00000 q^{27} -2.80212 q^{29} +1.86516 q^{31} +2.15240 q^{33} +2.99083 q^{35} -7.30674 q^{37} +0.0320505 q^{39} -11.8322 q^{41} +1.12240 q^{43} +1.53098 q^{45} -0.929399 q^{47} -3.18367 q^{49} +2.27829 q^{51} -11.1295 q^{53} -3.29529 q^{55} +5.52833 q^{57} +8.11120 q^{59} -7.44769 q^{61} +1.95354 q^{63} -0.0490687 q^{65} +10.5639 q^{67} -6.90191 q^{69} -7.98837 q^{71} +10.4948 q^{73} +2.65610 q^{75} -4.20481 q^{77} -5.86119 q^{79} +1.00000 q^{81} -14.0793 q^{83} -3.48802 q^{85} +2.80212 q^{87} +7.24231 q^{89} -0.0626121 q^{91} -1.86516 q^{93} -8.46376 q^{95} +2.96086 q^{97} -2.15240 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	1.53098	0.684675	0.342338	−	0.939577i	$-0.388781\pi$
$5$	1.53098	0.684675	0.342338	+	0.939577i	$0.388781\pi$
$6$	0	0
$7$	1.95354	0.738369	0.369185	−	0.929356i	$-0.379637\pi$
$7$	1.95354	0.738369	0.369185	+	0.929356i	$0.379637\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.15240	−0.648974	−0.324487	−	0.945890i	$-0.605192\pi$
$11$	−2.15240	−0.648974	−0.324487	+	0.945890i	$0.605192\pi$
$12$	0	0
$13$	−0.0320505	−0.00888922	−0.00444461	−	0.999990i	$-0.501415\pi$
$13$	−0.0320505	−0.00888922	−0.00444461	+	0.999990i	$0.501415\pi$
$14$	0	0
$15$	−1.53098	−0.395297
$16$	0	0
$17$	−2.27829	−0.552567	−0.276283	−	0.961076i	$-0.589103\pi$
$17$	−2.27829	−0.552567	−0.276283	+	0.961076i	$0.589103\pi$
$18$	0	0
$19$	−5.52833	−1.26829	−0.634143	−	0.773216i	$-0.718647\pi$
$19$	−5.52833	−1.26829	−0.634143	+	0.773216i	$0.718647\pi$
$20$	0	0
$21$	−1.95354	−0.426298
$22$	0	0
$23$	6.90191	1.43915	0.719574	−	0.694416i	$-0.244337\pi$
$23$	6.90191	1.43915	0.719574	+	0.694416i	$0.244337\pi$
$24$	0	0
$25$	−2.65610	−0.531220
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−2.80212	−0.520341	−0.260171	−	0.965563i	$-0.583779\pi$
$29$	−2.80212	−0.520341	−0.260171	+	0.965563i	$0.583779\pi$
$30$	0	0
$31$	1.86516	0.334992	0.167496	−	0.985873i	$-0.446432\pi$
$31$	1.86516	0.334992	0.167496	+	0.985873i	$0.446432\pi$
$32$	0	0
$33$	2.15240	0.374686
$34$	0	0
$35$	2.99083	0.505543
$36$	0	0
$37$	−7.30674	−1.20122	−0.600611	−	0.799542i	$-0.705076\pi$
$37$	−7.30674	−1.20122	−0.600611	+	0.799542i	$0.705076\pi$
$38$	0	0
$39$	0.0320505	0.00513219
$40$	0	0
$41$	−11.8322	−1.84787	−0.923936	−	0.382548i	$-0.875047\pi$
$41$	−11.8322	−1.84787	−0.923936	+	0.382548i	$0.875047\pi$
$42$	0	0
$43$	1.12240	0.171165	0.0855825	−	0.996331i	$-0.472725\pi$
$43$	1.12240	0.171165	0.0855825	+	0.996331i	$0.472725\pi$
$44$	0	0
$45$	1.53098	0.228225
$46$	0	0
$47$	−0.929399	−0.135567	−0.0677834	−	0.997700i	$-0.521593\pi$
$47$	−0.929399	−0.135567	−0.0677834	+	0.997700i	$0.521593\pi$
$48$	0	0
$49$	−3.18367	−0.454811
$50$	0	0
$51$	2.27829	0.319024
$52$	0	0
$53$	−11.1295	−1.52876	−0.764379	−	0.644768i	$-0.776954\pi$
$53$	−11.1295	−1.52876	−0.764379	+	0.644768i	$0.776954\pi$
$54$	0	0
$55$	−3.29529	−0.444337
$56$	0	0
$57$	5.52833	0.732245
$58$	0	0
$59$	8.11120	1.05599	0.527994	−	0.849248i	$-0.322944\pi$
$59$	8.11120	1.05599	0.527994	+	0.849248i	$0.322944\pi$
$60$	0	0
$61$	−7.44769	−0.953580	−0.476790	−	0.879017i	$-0.658200\pi$
$61$	−7.44769	−0.953580	−0.476790	+	0.879017i	$0.658200\pi$
$62$	0	0
$63$	1.95354	0.246123
$64$	0	0
$65$	−0.0490687	−0.00608623
$66$	0	0
$67$	10.5639	1.29058	0.645290	−	0.763938i	$-0.276737\pi$
$67$	10.5639	1.29058	0.645290	+	0.763938i	$0.276737\pi$
$68$	0	0
$69$	−6.90191	−0.830892
$70$	0	0
$71$	−7.98837	−0.948045	−0.474023	−	0.880513i	$-0.657199\pi$
$71$	−7.98837	−0.948045	−0.474023	+	0.880513i	$0.657199\pi$
$72$	0	0
$73$	10.4948	1.22833	0.614164	−	0.789179i	$-0.289493\pi$
$73$	10.4948	1.22833	0.614164	+	0.789179i	$0.289493\pi$
$74$	0	0
$75$	2.65610	0.306700
$76$	0	0
$77$	−4.20481	−0.479183
$78$	0	0
$79$	−5.86119	−0.659435	−0.329717	−	0.944080i	$-0.606953\pi$
$79$	−5.86119	−0.659435	−0.329717	+	0.944080i	$0.606953\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−14.0793	−1.54541	−0.772704	−	0.634767i	$-0.781096\pi$
$83$	−14.0793	−1.54541	−0.772704	+	0.634767i	$0.781096\pi$
$84$	0	0
$85$	−3.48802	−0.378329
$86$	0	0
$87$	2.80212	0.300419
$88$	0	0
$89$	7.24231	0.767684	0.383842	−	0.923399i	$-0.374601\pi$
$89$	7.24231	0.767684	0.383842	+	0.923399i	$0.374601\pi$
$90$	0	0
$91$	−0.0626121	−0.00656353
$92$	0	0
$93$	−1.86516	−0.193408
$94$	0	0
$95$	−8.46376	−0.868364
$96$	0	0
$97$	2.96086	0.300629	0.150315	−	0.988638i	$-0.451971\pi$
$97$	2.96086	0.300629	0.150315	+	0.988638i	$0.451971\pi$
$98$	0	0
$99$	−2.15240	−0.216325
$100$	0	0
$101$	−10.7676	−1.07142	−0.535709	−	0.844403i	$-0.679955\pi$
$101$	−10.7676	−1.07142	−0.535709	+	0.844403i	$0.679955\pi$
$102$	0	0
$103$	−2.59773	−0.255962	−0.127981	−	0.991777i	$-0.540850\pi$
$103$	−2.59773	−0.255962	−0.127981	+	0.991777i	$0.540850\pi$
$104$	0	0
$105$	−2.99083	−0.291876
$106$	0	0
$107$	−2.49290	−0.240998	−0.120499	−	0.992713i	$-0.538449\pi$
$107$	−2.49290	−0.240998	−0.120499	+	0.992713i	$0.538449\pi$
$108$	0	0
$109$	8.57800	0.821624	0.410812	−	0.911720i	$-0.365245\pi$
$109$	8.57800	0.821624	0.410812	+	0.911720i	$0.365245\pi$
$110$	0	0
$111$	7.30674	0.693525
$112$	0	0
$113$	17.3592	1.63301	0.816507	−	0.577335i	$-0.195907\pi$
$113$	17.3592	1.63301	0.816507	+	0.577335i	$0.195907\pi$
$114$	0	0
$115$	10.5667	0.985349
$116$	0	0
$117$	−0.0320505	−0.00296307
$118$	0	0
$119$	−4.45074	−0.407998
$120$	0	0
$121$	−6.36715	−0.578832
$122$	0	0
$123$	11.8322	1.06687
$124$	0	0
$125$	−11.7213	−1.04839
$126$	0	0
$127$	19.8473	1.76116	0.880580	−	0.473897i	$-0.157153\pi$
$127$	19.8473	1.76116	0.880580	+	0.473897i	$0.157153\pi$
$128$	0	0
$129$	−1.12240	−0.0988221
$130$	0	0
$131$	8.00587	0.699476	0.349738	−	0.936847i	$-0.386271\pi$
$131$	8.00587	0.699476	0.349738	+	0.936847i	$0.386271\pi$
$132$	0	0
$133$	−10.7998	−0.936463
$134$	0	0
$135$	−1.53098	−0.131766
$136$	0	0
$137$	12.5851	1.07522	0.537609	−	0.843194i	$-0.319328\pi$
$137$	12.5851	1.07522	0.537609	+	0.843194i	$0.319328\pi$
$138$	0	0
$139$	−0.988330	−0.0838291	−0.0419145	−	0.999121i	$-0.513346\pi$
$139$	−0.988330	−0.0838291	−0.0419145	+	0.999121i	$0.513346\pi$
$140$	0	0
$141$	0.929399	0.0782695
$142$	0	0
$143$	0.0689857	0.00576888
$144$	0	0
$145$	−4.28999	−0.356265
$146$	0	0
$147$	3.18367	0.262585
$148$	0	0
$149$	−8.45810	−0.692915	−0.346457	−	0.938066i	$-0.612615\pi$
$149$	−8.45810	−0.692915	−0.346457	+	0.938066i	$0.612615\pi$
$150$	0	0
$151$	−0.136644	−0.0111199	−0.00555997	−	0.999985i	$-0.501770\pi$
$151$	−0.136644	−0.0111199	−0.00555997	+	0.999985i	$0.501770\pi$
$152$	0	0
$153$	−2.27829	−0.184189
$154$	0	0
$155$	2.85552	0.229361
$156$	0	0
$157$	11.3727	0.907639	0.453820	−	0.891094i	$-0.350061\pi$
$157$	11.3727	0.907639	0.453820	+	0.891094i	$0.350061\pi$
$158$	0	0
$159$	11.1295	0.882629
$160$	0	0
$161$	13.4832	1.06262
$162$	0	0
$163$	−11.0067	−0.862114	−0.431057	−	0.902325i	$-0.641859\pi$
$163$	−11.0067	−0.862114	−0.431057	+	0.902325i	$0.641859\pi$
$164$	0	0
$165$	3.29529	0.256538
$166$	0	0
$167$	−1.00000	−0.0773823
$167$	−1.00000	−0.0773823
$168$	0	0
$169$	−12.9990	−0.999921
$170$	0	0
$171$	−5.52833	−0.422762
$172$	0	0
$173$	−17.6800	−1.34418	−0.672091	−	0.740468i	$-0.734604\pi$
$173$	−17.6800	−1.34418	−0.672091	+	0.740468i	$0.734604\pi$
$174$	0	0
$175$	−5.18880	−0.392236
$176$	0	0
$177$	−8.11120	−0.609675
$178$	0	0
$179$	−23.2929	−1.74100	−0.870498	−	0.492172i	$-0.836203\pi$
$179$	−23.2929	−1.74100	−0.870498	+	0.492172i	$0.836203\pi$
$180$	0	0
$181$	12.4704	0.926918	0.463459	−	0.886118i	$-0.346608\pi$
$181$	12.4704	0.926918	0.463459	+	0.886118i	$0.346608\pi$
$182$	0	0
$183$	7.44769	0.550549
$184$	0	0
$185$	−11.1865	−0.822446
$186$	0	0
$187$	4.90380	0.358602
$188$	0	0
$189$	−1.95354	−0.142099
$190$	0	0
$191$	−10.6430	−0.770103	−0.385051	−	0.922895i	$-0.625816\pi$
$191$	−10.6430	−0.770103	−0.385051	+	0.922895i	$0.625816\pi$
$192$	0	0
$193$	−3.66898	−0.264099	−0.132050	−	0.991243i	$-0.542156\pi$
$193$	−3.66898	−0.264099	−0.132050	+	0.991243i	$0.542156\pi$
$194$	0	0
$195$	0.0490687	0.00351389
$196$	0	0
$197$	−5.11526	−0.364447	−0.182223	−	0.983257i	$-0.558329\pi$
$197$	−5.11526	−0.364447	−0.182223	+	0.983257i	$0.558329\pi$
$198$	0	0
$199$	−0.690922	−0.0489781	−0.0244891	−	0.999700i	$-0.507796\pi$
$199$	−0.690922	−0.0489781	−0.0244891	+	0.999700i	$0.507796\pi$
$200$	0	0
$201$	−10.5639	−0.745117
$202$	0	0
$203$	−5.47406	−0.384204
$204$	0	0
$205$	−18.1148	−1.26519
$206$	0	0
$207$	6.90191	0.479716
$208$	0	0
$209$	11.8992	0.823085
$210$	0	0
$211$	15.3286	1.05526	0.527631	−	0.849474i	$-0.323081\pi$
$211$	15.3286	1.05526	0.527631	+	0.849474i	$0.323081\pi$
$212$	0	0
$213$	7.98837	0.547354
$214$	0	0
$215$	1.71838	0.117192
$216$	0	0
$217$	3.64367	0.247348
$218$	0	0
$219$	−10.4948	−0.709175
$220$	0	0
$221$	0.0730204	0.00491188
$222$	0	0
$223$	−23.7330	−1.58928	−0.794641	−	0.607079i	$-0.792341\pi$
$223$	−23.7330	−1.58928	−0.794641	+	0.607079i	$0.792341\pi$
$224$	0	0
$225$	−2.65610	−0.177073
$226$	0	0
$227$	−20.9222	−1.38866	−0.694329	−	0.719658i	$-0.744299\pi$
$227$	−20.9222	−1.38866	−0.694329	+	0.719658i	$0.744299\pi$
$228$	0	0
$229$	−11.6369	−0.768988	−0.384494	−	0.923127i	$-0.625624\pi$
$229$	−11.6369	−0.768988	−0.384494	+	0.923127i	$0.625624\pi$
$230$	0	0
$231$	4.20481	0.276656
$232$	0	0
$233$	−7.90044	−0.517575	−0.258788	−	0.965934i	$-0.583323\pi$
$233$	−7.90044	−0.517575	−0.258788	+	0.965934i	$0.583323\pi$
$234$	0	0
$235$	−1.42289	−0.0928192
$236$	0	0
$237$	5.86119	0.380725
$238$	0	0
$239$	21.2822	1.37663	0.688316	−	0.725411i	$-0.258350\pi$
$239$	21.2822	1.37663	0.688316	+	0.725411i	$0.258350\pi$
$240$	0	0
$241$	−17.6365	−1.13607	−0.568033	−	0.823006i	$-0.692295\pi$
$241$	−17.6365	−1.13607	−0.568033	+	0.823006i	$0.692295\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−4.87414	−0.311398
$246$	0	0
$247$	0.177186	0.0112741
$248$	0	0
$249$	14.0793	0.892242
$250$	0	0
$251$	29.1497	1.83992	0.919958	−	0.392017i	$-0.128223\pi$
$251$	29.1497	1.83992	0.919958	+	0.392017i	$0.128223\pi$
$252$	0	0
$253$	−14.8557	−0.933970
$254$	0	0
$255$	3.48802	0.218428
$256$	0	0
$257$	−28.8771	−1.80131	−0.900653	−	0.434539i	$-0.856911\pi$
$257$	−28.8771	−1.80131	−0.900653	+	0.434539i	$0.856911\pi$
$258$	0	0
$259$	−14.2740	−0.886945
$260$	0	0
$261$	−2.80212	−0.173447
$262$	0	0
$263$	3.69587	0.227897	0.113948	−	0.993487i	$-0.463650\pi$
$263$	3.69587	0.227897	0.113948	+	0.993487i	$0.463650\pi$
$264$	0	0
$265$	−17.0391	−1.04670
$266$	0	0
$267$	−7.24231	−0.443222
$268$	0	0
$269$	−11.9065	−0.725952	−0.362976	−	0.931799i	$-0.618239\pi$
$269$	−11.9065	−0.725952	−0.362976	+	0.931799i	$0.618239\pi$
$270$	0	0
$271$	9.42035	0.572245	0.286123	−	0.958193i	$-0.407634\pi$
$271$	9.42035	0.572245	0.286123	+	0.958193i	$0.407634\pi$
$272$	0	0
$273$	0.0626121	0.00378945
$274$	0	0
$275$	5.71700	0.344748
$276$	0	0
$277$	−9.72432	−0.584278	−0.292139	−	0.956376i	$-0.594367\pi$
$277$	−9.72432	−0.584278	−0.292139	+	0.956376i	$0.594367\pi$
$278$	0	0
$279$	1.86516	0.111664
$280$	0	0
$281$	−9.54649	−0.569496	−0.284748	−	0.958602i	$-0.591910\pi$
$281$	−9.54649	−0.569496	−0.284748	+	0.958602i	$0.591910\pi$
$282$	0	0
$283$	0.881411	0.0523944	0.0261972	−	0.999657i	$-0.491660\pi$
$283$	0.881411	0.0523944	0.0261972	+	0.999657i	$0.491660\pi$
$284$	0	0
$285$	8.46376	0.501350
$286$	0	0
$287$	−23.1146	−1.36441
$288$	0	0
$289$	−11.8094	−0.694670
$290$	0	0
$291$	−2.96086	−0.173569
$292$	0	0
$293$	7.37269	0.430717	0.215358	−	0.976535i	$-0.430908\pi$
$293$	7.37269	0.430717	0.215358	+	0.976535i	$0.430908\pi$
$294$	0	0
$295$	12.4181	0.723009
$296$	0	0
$297$	2.15240	0.124895
$298$	0	0
$299$	−0.221210	−0.0127929
$300$	0	0
$301$	2.19266	0.126383
$302$	0	0
$303$	10.7676	0.618583
$304$	0	0
$305$	−11.4023	−0.652892
$306$	0	0
$307$	−11.3416	−0.647299	−0.323650	−	0.946177i	$-0.604910\pi$
$307$	−11.3416	−0.647299	−0.323650	+	0.946177i	$0.604910\pi$
$308$	0	0
$309$	2.59773	0.147779
$310$	0	0
$311$	−23.2572	−1.31879	−0.659397	−	0.751795i	$-0.729188\pi$
$311$	−23.2572	−1.31879	−0.659397	+	0.751795i	$0.729188\pi$
$312$	0	0
$313$	24.6287	1.39210	0.696049	−	0.717994i	$-0.254940\pi$
$313$	24.6287	1.39210	0.696049	+	0.717994i	$0.254940\pi$
$314$	0	0
$315$	2.99083	0.168514
$316$	0	0
$317$	14.6228	0.821299	0.410649	−	0.911793i	$-0.365302\pi$
$317$	14.6228	0.821299	0.410649	+	0.911793i	$0.365302\pi$
$318$	0	0
$319$	6.03130	0.337688
$320$	0	0
$321$	2.49290	0.139140
$322$	0	0
$323$	12.5951	0.700812
$324$	0	0
$325$	0.0851294	0.00472213
$326$	0	0
$327$	−8.57800	−0.474365
$328$	0	0
$329$	−1.81562	−0.100098
$330$	0	0
$331$	14.4030	0.791660	0.395830	−	0.918324i	$-0.370457\pi$
$331$	14.4030	0.791660	0.395830	+	0.918324i	$0.370457\pi$
$332$	0	0
$333$	−7.30674	−0.400407
$334$	0	0
$335$	16.1731	0.883628
$336$	0	0
$337$	−18.4014	−1.00239	−0.501193	−	0.865335i	$-0.667105\pi$
$337$	−18.4014	−1.00239	−0.501193	+	0.865335i	$0.667105\pi$
$338$	0	0
$339$	−17.3592	−0.942821
$340$	0	0
$341$	−4.01458	−0.217402
$342$	0	0
$343$	−19.8942	−1.07419
$344$	0	0
$345$	−10.5667	−0.568891
$346$	0	0
$347$	4.12491	0.221437	0.110718	−	0.993852i	$-0.464685\pi$
$347$	4.12491	0.221437	0.110718	+	0.993852i	$0.464685\pi$
$348$	0	0
$349$	−21.8747	−1.17093	−0.585463	−	0.810699i	$-0.699087\pi$
$349$	−21.8747	−1.17093	−0.585463	+	0.810699i	$0.699087\pi$
$350$	0	0
$351$	0.0320505	0.00171073
$352$	0	0
$353$	−20.7177	−1.10269	−0.551347	−	0.834276i	$-0.685886\pi$
$353$	−20.7177	−1.10269	−0.551347	+	0.834276i	$0.685886\pi$
$354$	0	0
$355$	−12.2300	−0.649103
$356$	0	0
$357$	4.45074	0.235558
$358$	0	0
$359$	−27.2856	−1.44008	−0.720040	−	0.693933i	$-0.755877\pi$
$359$	−27.2856	−1.44008	−0.720040	+	0.693933i	$0.755877\pi$
$360$	0	0
$361$	11.5624	0.608548
$362$	0	0
$363$	6.36715	0.334189
$364$	0	0
$365$	16.0674	0.841006
$366$	0	0
$367$	17.1725	0.896398	0.448199	−	0.893934i	$-0.352066\pi$
$367$	17.1725	0.896398	0.448199	+	0.893934i	$0.352066\pi$
$368$	0	0
$369$	−11.8322	−0.615957
$370$	0	0
$371$	−21.7420	−1.12879
$372$	0	0
$373$	16.5701	0.857967	0.428984	−	0.903312i	$-0.358872\pi$
$373$	16.5701	0.857967	0.428984	+	0.903312i	$0.358872\pi$
$374$	0	0
$375$	11.7213	0.605287
$376$	0	0
$377$	0.0898095	0.00462543
$378$	0	0
$379$	32.1485	1.65136	0.825680	−	0.564139i	$-0.190792\pi$
$379$	32.1485	1.65136	0.825680	+	0.564139i	$0.190792\pi$
$380$	0	0
$381$	−19.8473	−1.01681
$382$	0	0
$383$	35.1609	1.79664	0.898319	−	0.439343i	$-0.144789\pi$
$383$	35.1609	1.79664	0.898319	+	0.439343i	$0.144789\pi$
$384$	0	0
$385$	−6.43749	−0.328085
$386$	0	0
$387$	1.12240	0.0570550
$388$	0	0
$389$	−22.1942	−1.12529	−0.562646	−	0.826698i	$-0.690217\pi$
$389$	−22.1942	−1.12529	−0.562646	+	0.826698i	$0.690217\pi$
$390$	0	0
$391$	−15.7246	−0.795225
$392$	0	0
$393$	−8.00587	−0.403843
$394$	0	0
$395$	−8.97336	−0.451499
$396$	0	0
$397$	9.87019	0.495371	0.247685	−	0.968841i	$-0.420330\pi$
$397$	9.87019	0.495371	0.247685	+	0.968841i	$0.420330\pi$
$398$	0	0
$399$	10.7998	0.540667
$400$	0	0
$401$	19.5475	0.976157	0.488079	−	0.872800i	$-0.337698\pi$
$401$	19.5475	0.976157	0.488079	+	0.872800i	$0.337698\pi$
$402$	0	0
$403$	−0.0597793	−0.00297782
$404$	0	0
$405$	1.53098	0.0760750
$406$	0	0
$407$	15.7271	0.779562
$408$	0	0
$409$	6.76824	0.334668	0.167334	−	0.985900i	$-0.446484\pi$
$409$	6.76824	0.334668	0.167334	+	0.985900i	$0.446484\pi$
$410$	0	0
$411$	−12.5851	−0.620778
$412$	0	0
$413$	15.8456	0.779709
$414$	0	0
$415$	−21.5552	−1.05810
$416$	0	0
$417$	0.988330	0.0483987
$418$	0	0
$419$	1.13322	0.0553615	0.0276807	−	0.999617i	$-0.491188\pi$
$419$	1.13322	0.0553615	0.0276807	+	0.999617i	$0.491188\pi$
$420$	0	0
$421$	4.12164	0.200876	0.100438	−	0.994943i	$-0.467976\pi$
$421$	4.12164	0.200876	0.100438	+	0.994943i	$0.467976\pi$
$422$	0	0
$423$	−0.929399	−0.0451889
$424$	0	0
$425$	6.05136	0.293534
$426$	0	0
$427$	−14.5494	−0.704094
$428$	0	0
$429$	−0.0689857	−0.00333066
$430$	0	0
$431$	−6.83221	−0.329096	−0.164548	−	0.986369i	$-0.552617\pi$
$431$	−6.83221	−0.329096	−0.164548	+	0.986369i	$0.552617\pi$
$432$	0	0
$433$	31.5894	1.51809	0.759046	−	0.651037i	$-0.225666\pi$
$433$	31.5894	1.51809	0.759046	+	0.651037i	$0.225666\pi$
$434$	0	0
$435$	4.28999	0.205690
$436$	0	0
$437$	−38.1560	−1.82525
$438$	0	0
$439$	4.58677	0.218915	0.109457	−	0.993991i	$-0.465089\pi$
$439$	4.58677	0.218915	0.109457	+	0.993991i	$0.465089\pi$
$440$	0	0
$441$	−3.18367	−0.151604
$442$	0	0
$443$	−30.2286	−1.43621	−0.718103	−	0.695937i	$-0.754989\pi$
$443$	−30.2286	−1.43621	−0.718103	+	0.695937i	$0.754989\pi$
$444$	0	0
$445$	11.0878	0.525614
$446$	0	0
$447$	8.45810	0.400054
$448$	0	0
$449$	−40.1741	−1.89593	−0.947966	−	0.318370i	$-0.896865\pi$
$449$	−40.1741	−1.89593	−0.947966	+	0.318370i	$0.896865\pi$
$450$	0	0
$451$	25.4676	1.19922
$452$	0	0
$453$	0.136644	0.00642010
$454$	0	0
$455$	−0.0958578	−0.00449388
$456$	0	0
$457$	29.5905	1.38418	0.692092	−	0.721809i	$-0.256689\pi$
$457$	29.5905	1.38418	0.692092	+	0.721809i	$0.256689\pi$
$458$	0	0
$459$	2.27829	0.106341
$460$	0	0
$461$	−13.0716	−0.608807	−0.304404	−	0.952543i	$-0.598457\pi$
$461$	−13.0716	−0.608807	−0.304404	+	0.952543i	$0.598457\pi$
$462$	0	0
$463$	−12.0242	−0.558810	−0.279405	−	0.960173i	$-0.590137\pi$
$463$	−12.0242	−0.558810	−0.279405	+	0.960173i	$0.590137\pi$
$464$	0	0
$465$	−2.85552	−0.132422
$466$	0	0
$467$	7.55381	0.349549	0.174774	−	0.984609i	$-0.444080\pi$
$467$	7.55381	0.349549	0.174774	+	0.984609i	$0.444080\pi$
$468$	0	0
$469$	20.6369	0.952925
$470$	0	0
$471$	−11.3727	−0.524026
$472$	0	0
$473$	−2.41587	−0.111082
$474$	0	0
$475$	14.6838	0.673738
$476$	0	0
$477$	−11.1295	−0.509586
$478$	0	0
$479$	−26.2810	−1.20081	−0.600405	−	0.799696i	$-0.704994\pi$
$479$	−26.2810	−1.20081	−0.600405	+	0.799696i	$0.704994\pi$
$480$	0	0
$481$	0.234185	0.0106779
$482$	0	0
$483$	−13.4832	−0.613506
$484$	0	0
$485$	4.53301	0.205834
$486$	0	0
$487$	6.63742	0.300770	0.150385	−	0.988628i	$-0.451949\pi$
$487$	6.63742	0.300770	0.150385	+	0.988628i	$0.451949\pi$
$488$	0	0
$489$	11.0067	0.497742
$490$	0	0
$491$	13.0991	0.591156	0.295578	−	0.955319i	$-0.404488\pi$
$491$	13.0991	0.591156	0.295578	+	0.955319i	$0.404488\pi$
$492$	0	0
$493$	6.38405	0.287523
$494$	0	0
$495$	−3.29529	−0.148112
$496$	0	0
$497$	−15.6056	−0.700008
$498$	0	0
$499$	22.1353	0.990912	0.495456	−	0.868633i	$-0.335001\pi$
$499$	22.1353	0.990912	0.495456	+	0.868633i	$0.335001\pi$
$500$	0	0
$501$	1.00000	0.0446767
$502$	0	0
$503$	22.7703	1.01528	0.507639	−	0.861570i	$-0.330518\pi$
$503$	22.7703	1.01528	0.507639	+	0.861570i	$0.330518\pi$
$504$	0	0
$505$	−16.4850	−0.733573
$506$	0	0
$507$	12.9990	0.577305
$508$	0	0
$509$	1.95493	0.0866506	0.0433253	−	0.999061i	$-0.486205\pi$
$509$	1.95493	0.0866506	0.0433253	+	0.999061i	$0.486205\pi$
$510$	0	0
$511$	20.5021	0.906960
$512$	0	0
$513$	5.52833	0.244082
$514$	0	0
$515$	−3.97707	−0.175251
$516$	0	0
$517$	2.00044	0.0879794
$518$	0	0
$519$	17.6800	0.776064
$520$	0	0
$521$	−7.26095	−0.318108	−0.159054	−	0.987270i	$-0.550844\pi$
$521$	−7.26095	−0.318108	−0.159054	+	0.987270i	$0.550844\pi$
$522$	0	0
$523$	−25.6991	−1.12374	−0.561871	−	0.827225i	$-0.689918\pi$
$523$	−25.6991	−1.12374	−0.561871	+	0.827225i	$0.689918\pi$
$524$	0	0
$525$	5.18880	0.226458
$526$	0	0
$527$	−4.24937	−0.185106
$528$	0	0
$529$	24.6364	1.07115
$530$	0	0
$531$	8.11120	0.351996
$532$	0	0
$533$	0.379227	0.0164261
$534$	0	0
$535$	−3.81659	−0.165006
$536$	0	0
$537$	23.2929	1.00516
$538$	0	0
$539$	6.85256	0.295160
$540$	0	0
$541$	21.5546	0.926703	0.463352	−	0.886174i	$-0.346647\pi$
$541$	21.5546	0.926703	0.463352	+	0.886174i	$0.346647\pi$
$542$	0	0
$543$	−12.4704	−0.535156
$544$	0	0
$545$	13.1328	0.562545
$546$	0	0
$547$	−11.8562	−0.506936	−0.253468	−	0.967344i	$-0.581571\pi$
$547$	−11.8562	−0.506936	−0.253468	+	0.967344i	$0.581571\pi$
$548$	0	0
$549$	−7.44769	−0.317860
$550$	0	0
$551$	15.4911	0.659941
$552$	0	0
$553$	−11.4501	−0.486907
$554$	0	0
$555$	11.1865	0.474840
$556$	0	0
$557$	−44.9125	−1.90300	−0.951502	−	0.307642i	$-0.900460\pi$
$557$	−44.9125	−1.90300	−0.951502	+	0.307642i	$0.900460\pi$
$558$	0	0
$559$	−0.0359736	−0.00152152
$560$	0	0
$561$	−4.90380	−0.207039
$562$	0	0
$563$	−8.14999	−0.343481	−0.171741	−	0.985142i	$-0.554939\pi$
$563$	−8.14999	−0.343481	−0.171741	+	0.985142i	$0.554939\pi$
$564$	0	0
$565$	26.5766	1.11808
$566$	0	0
$567$	1.95354	0.0820410
$568$	0	0
$569$	39.1874	1.64282	0.821410	−	0.570338i	$-0.193188\pi$
$569$	39.1874	1.64282	0.821410	+	0.570338i	$0.193188\pi$
$570$	0	0
$571$	6.40716	0.268131	0.134066	−	0.990972i	$-0.457197\pi$
$571$	6.40716	0.268131	0.134066	+	0.990972i	$0.457197\pi$
$572$	0	0
$573$	10.6430	0.444619
$574$	0	0
$575$	−18.3322	−0.764504
$576$	0	0
$577$	45.5385	1.89579	0.947895	−	0.318582i	$-0.103207\pi$
$577$	45.5385	1.89579	0.947895	+	0.318582i	$0.103207\pi$
$578$	0	0
$579$	3.66898	0.152478
$580$	0	0
$581$	−27.5046	−1.14108
$582$	0	0
$583$	23.9552	0.992125
$584$	0	0
$585$	−0.0490687	−0.00202874
$586$	0	0
$587$	−36.1537	−1.49222	−0.746111	−	0.665821i	$-0.768081\pi$
$587$	−36.1537	−1.49222	−0.746111	+	0.665821i	$0.768081\pi$
$588$	0	0
$589$	−10.3112	−0.424866
$590$	0	0
$591$	5.11526	0.210414
$592$	0	0
$593$	20.9600	0.860724	0.430362	−	0.902656i	$-0.358386\pi$
$593$	20.9600	0.860724	0.430362	+	0.902656i	$0.358386\pi$
$594$	0	0
$595$	−6.81399	−0.279346
$596$	0	0
$597$	0.690922	0.0282775
$598$	0	0
$599$	22.3654	0.913826	0.456913	−	0.889512i	$-0.348955\pi$
$599$	22.3654	0.913826	0.456913	+	0.889512i	$0.348955\pi$
$600$	0	0
$601$	8.29446	0.338338	0.169169	−	0.985587i	$-0.445892\pi$
$601$	8.29446	0.338338	0.169169	+	0.985587i	$0.445892\pi$
$602$	0	0
$603$	10.5639	0.430193
$604$	0	0
$605$	−9.74799	−0.396312
$606$	0	0
$607$	−4.58092	−0.185934	−0.0929670	−	0.995669i	$-0.529635\pi$
$607$	−4.58092	−0.185934	−0.0929670	+	0.995669i	$0.529635\pi$
$608$	0	0
$609$	5.47406	0.221820
$610$	0	0
$611$	0.0297877	0.00120508
$612$	0	0
$613$	8.46801	0.342020	0.171010	−	0.985269i	$-0.445297\pi$
$613$	8.46801	0.342020	0.171010	+	0.985269i	$0.445297\pi$
$614$	0	0
$615$	18.1148	0.730459
$616$	0	0
$617$	23.3651	0.940642	0.470321	−	0.882495i	$-0.344138\pi$
$617$	23.3651	0.940642	0.470321	+	0.882495i	$0.344138\pi$
$618$	0	0
$619$	−11.4277	−0.459318	−0.229659	−	0.973271i	$-0.573761\pi$
$619$	−11.4277	−0.459318	−0.229659	+	0.973271i	$0.573761\pi$
$620$	0	0
$621$	−6.90191	−0.276964
$622$	0	0
$623$	14.1482	0.566834
$624$	0	0
$625$	−4.66464	−0.186586
$626$	0	0
$627$	−11.8992	−0.475208
$628$	0	0
$629$	16.6469	0.663755
$630$	0	0
$631$	−3.40993	−0.135747	−0.0678736	−	0.997694i	$-0.521621\pi$
$631$	−3.40993	−0.135747	−0.0678736	+	0.997694i	$0.521621\pi$
$632$	0	0
$633$	−15.3286	−0.609255
$634$	0	0
$635$	30.3858	1.20582
$636$	0	0
$637$	0.102038	0.00404291
$638$	0	0
$639$	−7.98837	−0.316015
$640$	0	0
$641$	−19.8009	−0.782088	−0.391044	−	0.920372i	$-0.627886\pi$
$641$	−19.8009	−0.782088	−0.391044	+	0.920372i	$0.627886\pi$
$642$	0	0
$643$	−39.9862	−1.57690	−0.788451	−	0.615097i	$-0.789117\pi$
$643$	−39.9862	−1.57690	−0.788451	+	0.615097i	$0.789117\pi$
$644$	0	0
$645$	−1.71838	−0.0676611
$646$	0	0
$647$	−23.7997	−0.935664	−0.467832	−	0.883817i	$-0.654965\pi$
$647$	−23.7997	−0.935664	−0.467832	+	0.883817i	$0.654965\pi$
$648$	0	0
$649$	−17.4586	−0.685309
$650$	0	0
$651$	−3.64367	−0.142807
$652$	0	0
$653$	−3.24958	−0.127166	−0.0635829	−	0.997977i	$-0.520253\pi$
$653$	−3.24958	−0.127166	−0.0635829	+	0.997977i	$0.520253\pi$
$654$	0	0
$655$	12.2568	0.478914
$656$	0	0
$657$	10.4948	0.409443
$658$	0	0
$659$	47.1312	1.83597	0.917985	−	0.396616i	$-0.129816\pi$
$659$	47.1312	1.83597	0.917985	+	0.396616i	$0.129816\pi$
$660$	0	0
$661$	49.8545	1.93912	0.969558	−	0.244861i	$-0.0787424\pi$
$661$	49.8545	1.93912	0.969558	+	0.244861i	$0.0787424\pi$
$662$	0	0
$663$	−0.0730204	−0.00283588
$664$	0	0
$665$	−16.5343	−0.641173
$666$	0	0
$667$	−19.3400	−0.748848
$668$	0	0
$669$	23.7330	0.917573
$670$	0	0
$671$	16.0305	0.618849
$672$	0	0
$673$	29.5519	1.13914	0.569571	−	0.821942i	$-0.307109\pi$
$673$	29.5519	1.13914	0.569571	+	0.821942i	$0.307109\pi$
$674$	0	0
$675$	2.65610	0.102233
$676$	0	0
$677$	−1.97155	−0.0757729	−0.0378864	−	0.999282i	$-0.512063\pi$
$677$	−1.97155	−0.0757729	−0.0378864	+	0.999282i	$0.512063\pi$
$678$	0	0
$679$	5.78416	0.221976
$680$	0	0
$681$	20.9222	0.801742
$682$	0	0
$683$	14.7594	0.564751	0.282375	−	0.959304i	$-0.408878\pi$
$683$	14.7594	0.564751	0.282375	+	0.959304i	$0.408878\pi$
$684$	0	0
$685$	19.2676	0.736175
$686$	0	0
$687$	11.6369	0.443976
$688$	0	0
$689$	0.356707	0.0135895
$690$	0	0
$691$	−8.62826	−0.328234	−0.164117	−	0.986441i	$-0.552478\pi$
$691$	−8.62826	−0.328234	−0.164117	+	0.986441i	$0.552478\pi$
$692$	0	0
$693$	−4.20481	−0.159728
$694$	0	0
$695$	−1.51311	−0.0573957
$696$	0	0
$697$	26.9571	1.02107
$698$	0	0
$699$	7.90044	0.298822
$700$	0	0
$701$	−33.1406	−1.25170	−0.625851	−	0.779943i	$-0.715248\pi$
$701$	−33.1406	−1.25170	−0.625851	+	0.779943i	$0.715248\pi$
$702$	0	0
$703$	40.3941	1.52349
$704$	0	0
$705$	1.42289	0.0535892
$706$	0	0
$707$	−21.0350	−0.791102
$708$	0	0
$709$	35.0277	1.31549	0.657746	−	0.753240i	$-0.271510\pi$
$709$	35.0277	1.31549	0.657746	+	0.753240i	$0.271510\pi$
$710$	0	0
$711$	−5.86119	−0.219812
$712$	0	0
$713$	12.8732	0.482104
$714$	0	0
$715$	0.105616	0.00394981
$716$	0	0
$717$	−21.2822	−0.794798
$718$	0	0
$719$	45.7903	1.70769	0.853846	−	0.520526i	$-0.174264\pi$
$719$	45.7903	1.70769	0.853846	+	0.520526i	$0.174264\pi$
$720$	0	0
$721$	−5.07477	−0.188994
$722$	0	0
$723$	17.6365	0.655908
$724$	0	0
$725$	7.44271	0.276415
$726$	0	0
$727$	12.5668	0.466076	0.233038	−	0.972468i	$-0.425133\pi$
$727$	12.5668	0.466076	0.233038	+	0.972468i	$0.425133\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−2.55716	−0.0945800
$732$	0	0
$733$	35.3051	1.30402	0.652012	−	0.758209i	$-0.273925\pi$
$733$	35.3051	1.30402	0.652012	+	0.758209i	$0.273925\pi$
$734$	0	0
$735$	4.87414	0.179785
$736$	0	0
$737$	−22.7377	−0.837553
$738$	0	0
$739$	−30.2423	−1.11248	−0.556241	−	0.831021i	$-0.687757\pi$
$739$	−30.2423	−1.11248	−0.556241	+	0.831021i	$0.687757\pi$
$740$	0	0
$741$	−0.177186	−0.00650908
$742$	0	0
$743$	16.5002	0.605332	0.302666	−	0.953097i	$-0.402123\pi$
$743$	16.5002	0.605332	0.302666	+	0.953097i	$0.402123\pi$
$744$	0	0
$745$	−12.9492	−0.474422
$746$	0	0
$747$	−14.0793	−0.515136
$748$	0	0
$749$	−4.86999	−0.177946
$750$	0	0
$751$	43.4211	1.58446	0.792229	−	0.610224i	$-0.208920\pi$
$751$	43.4211	1.58446	0.792229	+	0.610224i	$0.208920\pi$
$752$	0	0
$753$	−29.1497	−1.06228
$754$	0	0
$755$	−0.209199	−0.00761354
$756$	0	0
$757$	−10.8584	−0.394655	−0.197327	−	0.980338i	$-0.563226\pi$
$757$	−10.8584	−0.394655	−0.197327	+	0.980338i	$0.563226\pi$
$758$	0	0
$759$	14.8557	0.539228
$760$	0	0
$761$	−42.4505	−1.53883	−0.769415	−	0.638750i	$-0.779452\pi$
$761$	−42.4505	−1.53883	−0.769415	+	0.638750i	$0.779452\pi$
$762$	0	0
$763$	16.7575	0.606662
$764$	0	0
$765$	−3.48802	−0.126110
$766$	0	0
$767$	−0.259968	−0.00938691
$768$	0	0
$769$	24.3994	0.879865	0.439933	−	0.898031i	$-0.355002\pi$
$769$	24.3994	0.879865	0.439933	+	0.898031i	$0.355002\pi$
$770$	0	0
$771$	28.8771	1.03998
$772$	0	0
$773$	−6.70612	−0.241203	−0.120601	−	0.992701i	$-0.538482\pi$
$773$	−6.70612	−0.241203	−0.120601	+	0.992701i	$0.538482\pi$
$774$	0	0
$775$	−4.95405	−0.177955
$776$	0	0
$777$	14.2740	0.512078
$778$	0	0
$779$	65.4120	2.34363
$780$	0	0
$781$	17.1942	0.615257
$782$	0	0
$783$	2.80212	0.100140
$784$	0	0
$785$	17.4114	0.621438
$786$	0	0
$787$	−44.6064	−1.59005	−0.795023	−	0.606579i	$-0.792541\pi$
$787$	−44.6064	−1.59005	−0.795023	+	0.606579i	$0.792541\pi$
$788$	0	0
$789$	−3.69587	−0.131576
$790$	0	0
$791$	33.9119	1.20577
$792$	0	0
$793$	0.238703	0.00847658
$794$	0	0
$795$	17.0391	0.604314
$796$	0	0
$797$	23.7264	0.840433	0.420217	−	0.907424i	$-0.361954\pi$
$797$	23.7264	0.840433	0.420217	+	0.907424i	$0.361954\pi$
$798$	0	0
$799$	2.11744	0.0749097
$800$	0	0
$801$	7.24231	0.255895
$802$	0	0
$803$	−22.5891	−0.797153
$804$	0	0
$805$	20.6425	0.727552
$806$	0	0
$807$	11.9065	0.419128
$808$	0	0
$809$	−39.3941	−1.38502	−0.692512	−	0.721407i	$-0.743496\pi$
$809$	−39.3941	−1.38502	−0.692512	+	0.721407i	$0.743496\pi$
$810$	0	0
$811$	−40.3386	−1.41648	−0.708240	−	0.705971i	$-0.750511\pi$
$811$	−40.3386	−1.41648	−0.708240	+	0.705971i	$0.750511\pi$
$812$	0	0
$813$	−9.42035	−0.330386
$814$	0	0
$815$	−16.8511	−0.590268
$816$	0	0
$817$	−6.20502	−0.217086
$818$	0	0
$819$	−0.0626121	−0.00218784
$820$	0	0
$821$	−42.3862	−1.47929	−0.739645	−	0.672997i	$-0.765007\pi$
$821$	−42.3862	−1.47929	−0.739645	+	0.672997i	$0.765007\pi$
$822$	0	0
$823$	−6.66255	−0.232242	−0.116121	−	0.993235i	$-0.537046\pi$
$823$	−6.66255	−0.232242	−0.116121	+	0.993235i	$0.537046\pi$
$824$	0	0
$825$	−5.71700	−0.199040
$826$	0	0
$827$	−19.1082	−0.664456	−0.332228	−	0.943199i	$-0.607800\pi$
$827$	−19.1082	−0.664456	−0.332228	+	0.943199i	$0.607800\pi$
$828$	0	0
$829$	−50.4467	−1.75209	−0.876044	−	0.482231i	$-0.839827\pi$
$829$	−50.4467	−1.75209	−0.876044	+	0.482231i	$0.839827\pi$
$830$	0	0
$831$	9.72432	0.337333
$832$	0	0
$833$	7.25333	0.251313
$834$	0	0
$835$	−1.53098	−0.0529818
$836$	0	0
$837$	−1.86516	−0.0644693
$838$	0	0
$839$	−27.7449	−0.957862	−0.478931	−	0.877853i	$-0.658976\pi$
$839$	−27.7449	−0.957862	−0.478931	+	0.877853i	$0.658976\pi$
$840$	0	0
$841$	−21.1481	−0.729245
$842$	0	0
$843$	9.54649	0.328799
$844$	0	0
$845$	−19.9012	−0.684621
$846$	0	0
$847$	−12.4385	−0.427392
$848$	0	0
$849$	−0.881411	−0.0302499
$850$	0	0
$851$	−50.4305	−1.72873
$852$	0	0
$853$	49.0988	1.68111	0.840556	−	0.541725i	$-0.182229\pi$
$853$	49.0988	1.68111	0.840556	+	0.541725i	$0.182229\pi$
$854$	0	0
$855$	−8.46376	−0.289455
$856$	0	0
$857$	47.7973	1.63272	0.816362	−	0.577541i	$-0.195988\pi$
$857$	47.7973	1.63272	0.816362	+	0.577541i	$0.195988\pi$
$858$	0	0
$859$	34.6308	1.18159	0.590794	−	0.806822i	$-0.298814\pi$
$859$	34.6308	1.18159	0.590794	+	0.806822i	$0.298814\pi$
$860$	0	0
$861$	23.1146	0.787743
$862$	0	0
$863$	53.2646	1.81315	0.906574	−	0.422047i	$-0.138688\pi$
$863$	53.2646	1.81315	0.906574	+	0.422047i	$0.138688\pi$
$864$	0	0
$865$	−27.0677	−0.920329
$866$	0	0
$867$	11.8094	0.401068
$868$	0	0
$869$	12.6156	0.427956
$870$	0	0
$871$	−0.338577	−0.0114722
$872$	0	0
$873$	2.96086	0.100210
$874$	0	0
$875$	−22.8981	−0.774098
$876$	0	0
$877$	−49.6892	−1.67789	−0.838943	−	0.544219i	$-0.816826\pi$
$877$	−49.6892	−1.67789	−0.838943	+	0.544219i	$0.816826\pi$
$878$	0	0
$879$	−7.37269	−0.248675
$880$	0	0
$881$	0.172379	0.00580759	0.00290379	−	0.999996i	$-0.499076\pi$
$881$	0.172379	0.00580759	0.00290379	+	0.999996i	$0.499076\pi$
$882$	0	0
$883$	55.2229	1.85840	0.929200	−	0.369578i	$-0.120498\pi$
$883$	55.2229	1.85840	0.929200	+	0.369578i	$0.120498\pi$
$884$	0	0
$885$	−12.4181	−0.417429
$886$	0	0
$887$	−39.2457	−1.31774	−0.658871	−	0.752256i	$-0.728966\pi$
$887$	−39.2457	−1.31774	−0.658871	+	0.752256i	$0.728966\pi$
$888$	0	0
$889$	38.7725	1.30039
$890$	0	0
$891$	−2.15240	−0.0721083
$892$	0	0
$893$	5.13802	0.171937
$894$	0	0
$895$	−35.6610	−1.19202
$896$	0	0
$897$	0.221210	0.00738598
$898$	0	0
$899$	−5.22640	−0.174310
$900$	0	0
$901$	25.3563	0.844740
$902$	0	0
$903$	−2.19266	−0.0729673
$904$	0	0
$905$	19.0919	0.634638
$906$	0	0
$907$	17.0928	0.567559	0.283779	−	0.958890i	$-0.408412\pi$
$907$	17.0928	0.567559	0.283779	+	0.958890i	$0.408412\pi$
$908$	0	0
$909$	−10.7676	−0.357139
$910$	0	0
$911$	−18.5880	−0.615850	−0.307925	−	0.951411i	$-0.599635\pi$
$911$	−18.5880	−0.615850	−0.307925	+	0.951411i	$0.599635\pi$
$912$	0	0
$913$	30.3044	1.00293
$914$	0	0
$915$	11.4023	0.376948
$916$	0	0
$917$	15.6398	0.516472
$918$	0	0
$919$	25.4303	0.838868	0.419434	−	0.907786i	$-0.362229\pi$
$919$	25.4303	0.838868	0.419434	+	0.907786i	$0.362229\pi$
$920$	0	0
$921$	11.3416	0.373718
$922$	0	0
$923$	0.256032	0.00842738
$924$	0	0
$925$	19.4074	0.638112
$926$	0	0
$927$	−2.59773	−0.0853205
$928$	0	0
$929$	6.00391	0.196982	0.0984909	−	0.995138i	$-0.468598\pi$
$929$	6.00391	0.196982	0.0984909	+	0.995138i	$0.468598\pi$
$930$	0	0
$931$	17.6004	0.576830
$932$	0	0
$933$	23.2572	0.761406
$934$	0	0
$935$	7.50763	0.245526
$936$	0	0
$937$	−47.8575	−1.56344	−0.781719	−	0.623631i	$-0.785657\pi$
$937$	−47.8575	−1.56344	−0.781719	+	0.623631i	$0.785657\pi$
$938$	0	0
$939$	−24.6287	−0.803728
$940$	0	0
$941$	−1.11153	−0.0362348	−0.0181174	−	0.999836i	$-0.505767\pi$
$941$	−1.11153	−0.0362348	−0.0181174	+	0.999836i	$0.505767\pi$
$942$	0	0
$943$	−81.6644	−2.65936
$944$	0	0
$945$	−2.99083	−0.0972919
$946$	0	0
$947$	−15.8001	−0.513434	−0.256717	−	0.966487i	$-0.582641\pi$
$947$	−15.8001	−0.513434	−0.256717	+	0.966487i	$0.582641\pi$
$948$	0	0
$949$	−0.336365	−0.0109189
$950$	0	0
$951$	−14.6228	−0.474177
$952$	0	0
$953$	12.6651	0.410263	0.205131	−	0.978734i	$-0.434238\pi$
$953$	12.6651	0.410263	0.205131	+	0.978734i	$0.434238\pi$
$954$	0	0
$955$	−16.2943	−0.527270
$956$	0	0
$957$	−6.03130	−0.194964
$958$	0	0
$959$	24.5855	0.793908
$960$	0	0
$961$	−27.5212	−0.887780
$962$	0	0
$963$	−2.49290	−0.0803327
$964$	0	0
$965$	−5.61714	−0.180822
$966$	0	0
$967$	−31.0282	−0.997801	−0.498901	−	0.866659i	$-0.666263\pi$
$967$	−31.0282	−0.997801	−0.498901	+	0.866659i	$0.666263\pi$
$968$	0	0
$969$	−12.5951	−0.404614
$970$	0	0
$971$	12.5965	0.404242	0.202121	−	0.979361i	$-0.435217\pi$
$971$	12.5965	0.404242	0.202121	+	0.979361i	$0.435217\pi$
$972$	0	0
$973$	−1.93074	−0.0618968
$974$	0	0
$975$	−0.0851294	−0.00272632
$976$	0	0
$977$	−21.7953	−0.697295	−0.348647	−	0.937254i	$-0.613359\pi$
$977$	−21.7953	−0.697295	−0.348647	+	0.937254i	$0.613359\pi$
$978$	0	0
$979$	−15.5884	−0.498207
$980$	0	0
$981$	8.57800	0.273875
$982$	0	0
$983$	32.0842	1.02333	0.511664	−	0.859186i	$-0.329029\pi$
$983$	32.0842	1.02333	0.511664	+	0.859186i	$0.329029\pi$
$984$	0	0
$985$	−7.83136	−0.249528
$986$	0	0
$987$	1.81562	0.0577918
$988$	0	0
$989$	7.74673	0.246332
$990$	0	0
$991$	33.9196	1.07749	0.538745	−	0.842469i	$-0.318899\pi$
$991$	33.9196	1.07749	0.538745	+	0.842469i	$0.318899\pi$
$992$	0	0
$993$	−14.4030	−0.457065
$994$	0	0
$995$	−1.05779	−0.0335341
$996$	0	0
$997$	−55.7384	−1.76525	−0.882626	−	0.470076i	$-0.844226\pi$
$997$	−55.7384	−1.76525	−0.882626	+	0.470076i	$0.844226\pi$
$998$	0	0
$999$	7.30674	0.231175

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4008.2.a.j.1.9	✓	10
4.3	odd	2		8016.2.a.bd.1.9		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4008.2.a.j.1.9	✓	10	1.1	even	1	trivial
8016.2.a.bd.1.9		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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.53098 q^{5} +1.95354 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.53098 q^{5} +1.95354 q^{7} +1.00000 q^{9} -2.15240 q^{11} -0.0320505 q^{13} -1.53098 q^{15} -2.27829 q^{17} -5.52833 q^{19} -1.95354 q^{21} +6.90191 q^{23} -2.65610 q^{25} -1.00000 q^{27} -2.80212 q^{29} +1.86516 q^{31} +2.15240 q^{33} +2.99083 q^{35} -7.30674 q^{37} +0.0320505 q^{39} -11.8322 q^{41} +1.12240 q^{43} +1.53098 q^{45} -0.929399 q^{47} -3.18367 q^{49} +2.27829 q^{51} -11.1295 q^{53} -3.29529 q^{55} +5.52833 q^{57} +8.11120 q^{59} -7.44769 q^{61} +1.95354 q^{63} -0.0490687 q^{65} +10.5639 q^{67} -6.90191 q^{69} -7.98837 q^{71} +10.4948 q^{73} +2.65610 q^{75} -4.20481 q^{77} -5.86119 q^{79} +1.00000 q^{81} -14.0793 q^{83} -3.48802 q^{85} +2.80212 q^{87} +7.24231 q^{89} -0.0626121 q^{91} -1.86516 q^{93} -8.46376 q^{95} +2.96086 q^{97} -2.15240 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 10 q^{3} - 10 q^{5} + q^{7} + 10 q^{9}+O(q^{10}) \) 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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