LMFDB - Embedded newform 8004.2.a.k.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8004,2,Mod(1,8004)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8004, 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("8004.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8004.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:	$63.9122617778$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 5 x^{17} - 55 x^{16} + 288 x^{15} + 1222 x^{14} - 6888 x^{13} - 13745 x^{12} + 88434 x^{11} + \cdots - 115488 $ x^18 - 5x^17 - 55x^16 + 288x^15 + 1222x^14 - 6888x^13 - 13745x^12 + 88434x^11 + 76571x^10 - 655793x^9 - 114554x^8 + 2789438x^7 - 855636x^6 - 6184176x^5 + 4260960x^4 + 5001480x^3 - 5659296x^2 + 1509552*x - 115488
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2}\cdot 3 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-1.83157$ of defining polynomial
Character	$\chi$	$=$	8004.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -1.83157 q^{5} -3.10909 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.83157 q^{5} -3.10909 q^{7} +1.00000 q^{9} +4.83870 q^{11} -4.03048 q^{13} -1.83157 q^{15} -4.45134 q^{17} -3.14088 q^{19} -3.10909 q^{21} -1.00000 q^{23} -1.64537 q^{25} +1.00000 q^{27} +1.00000 q^{29} +9.50715 q^{31} +4.83870 q^{33} +5.69451 q^{35} +4.88709 q^{37} -4.03048 q^{39} +8.35449 q^{41} -12.8921 q^{43} -1.83157 q^{45} -6.49782 q^{47} +2.66646 q^{49} -4.45134 q^{51} +3.97722 q^{53} -8.86240 q^{55} -3.14088 q^{57} +8.69780 q^{59} -12.6082 q^{61} -3.10909 q^{63} +7.38209 q^{65} +11.9816 q^{67} -1.00000 q^{69} -11.1914 q^{71} -12.9336 q^{73} -1.64537 q^{75} -15.0440 q^{77} +1.60254 q^{79} +1.00000 q^{81} +5.42371 q^{83} +8.15292 q^{85} +1.00000 q^{87} +6.27276 q^{89} +12.5311 q^{91} +9.50715 q^{93} +5.75273 q^{95} +4.00312 q^{97} +4.83870 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 18 q^{3} + 5 q^{5} + 6 q^{7} + 18 q^{9}+O(q^{10}) $ 18 * q + 18 * q^3 + 5 * q^5 + 6 * q^7 + 18 * q^9	$ 18 q + 18 q^{3} + 5 q^{5} + 6 q^{7} + 18 q^{9} + 5 q^{11} + 6 q^{13} + 5 q^{15} + 7 q^{17} + 15 q^{19} + 6 q^{21} - 18 q^{23} + 45 q^{25} + 18 q^{27} + 18 q^{29} + 10 q^{31} + 5 q^{33} - 7 q^{35} + 22 q^{37} + 6 q^{39} + 17 q^{41} + 25 q^{43} + 5 q^{45} - 6 q^{47} + 64 q^{49} + 7 q^{51} + 21 q^{53} + 3 q^{55} + 15 q^{57} + 6 q^{59} + 7 q^{61} + 6 q^{63} + 44 q^{65} + 35 q^{67} - 18 q^{69} + q^{71} + 27 q^{73} + 45 q^{75} + 4 q^{77} + 18 q^{81} + 13 q^{83} + 24 q^{85} + 18 q^{87} + 30 q^{89} + 53 q^{91} + 10 q^{93} + 17 q^{95} + 27 q^{97} + 5 q^{99}+O(q^{100}) $ 18 * q + 18 * q^3 + 5 * q^5 + 6 * q^7 + 18 * q^9 + 5 * q^11 + 6 * q^13 + 5 * q^15 + 7 * q^17 + 15 * q^19 + 6 * q^21 - 18 * q^23 + 45 * q^25 + 18 * q^27 + 18 * q^29 + 10 * q^31 + 5 * q^33 - 7 * q^35 + 22 * q^37 + 6 * q^39 + 17 * q^41 + 25 * q^43 + 5 * q^45 - 6 * q^47 + 64 * q^49 + 7 * q^51 + 21 * q^53 + 3 * q^55 + 15 * q^57 + 6 * q^59 + 7 * q^61 + 6 * q^63 + 44 * q^65 + 35 * q^67 - 18 * q^69 + q^71 + 27 * q^73 + 45 * q^75 + 4 * q^77 + 18 * q^81 + 13 * q^83 + 24 * q^85 + 18 * q^87 + 30 * q^89 + 53 * q^91 + 10 * q^93 + 17 * q^95 + 27 * q^97 + 5 * 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
$5$	−1.83157	−0.819101	−0.409550	−	0.912287i	$-0.634314\pi$
$5$	−1.83157	−0.819101	−0.409550	+	0.912287i	$0.634314\pi$
$6$	0	0
$7$	−3.10909	−1.17513	−0.587563	−	0.809178i	$-0.699913\pi$
$7$	−3.10909	−1.17513	−0.587563	+	0.809178i	$0.699913\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	4.83870	1.45892	0.729462	−	0.684022i	$-0.239771\pi$
$11$	4.83870	1.45892	0.729462	+	0.684022i	$0.239771\pi$
$12$	0	0
$13$	−4.03048	−1.11785	−0.558927	−	0.829217i	$-0.688787\pi$
$13$	−4.03048	−1.11785	−0.558927	+	0.829217i	$0.688787\pi$
$14$	0	0
$15$	−1.83157	−0.472908
$16$	0	0
$17$	−4.45134	−1.07961	−0.539804	−	0.841790i	$-0.681502\pi$
$17$	−4.45134	−1.07961	−0.539804	+	0.841790i	$0.681502\pi$
$18$	0	0
$19$	−3.14088	−0.720568	−0.360284	−	0.932843i	$-0.617320\pi$
$19$	−3.14088	−0.720568	−0.360284	+	0.932843i	$0.617320\pi$
$20$	0	0
$21$	−3.10909	−0.678460
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−1.64537	−0.329074
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	9.50715	1.70754	0.853768	−	0.520654i	$-0.174312\pi$
$31$	9.50715	1.70754	0.853768	+	0.520654i	$0.174312\pi$
$32$	0	0
$33$	4.83870	0.842310
$34$	0	0
$35$	5.69451	0.962547
$36$	0	0
$37$	4.88709	0.803433	0.401716	−	0.915764i	$-0.368414\pi$
$37$	4.88709	0.803433	0.401716	+	0.915764i	$0.368414\pi$
$38$	0	0
$39$	−4.03048	−0.645393
$40$	0	0
$41$	8.35449	1.30475	0.652376	−	0.757895i	$-0.273772\pi$
$41$	8.35449	1.30475	0.652376	+	0.757895i	$0.273772\pi$
$42$	0	0
$43$	−12.8921	−1.96603	−0.983013	−	0.183535i	$-0.941246\pi$
$43$	−12.8921	−1.96603	−0.983013	+	0.183535i	$0.941246\pi$
$44$	0	0
$45$	−1.83157	−0.273034
$46$	0	0
$47$	−6.49782	−0.947804	−0.473902	−	0.880578i	$-0.657155\pi$
$47$	−6.49782	−0.947804	−0.473902	+	0.880578i	$0.657155\pi$
$48$	0	0
$49$	2.66646	0.380923
$50$	0	0
$51$	−4.45134	−0.623312
$52$	0	0
$53$	3.97722	0.546313	0.273157	−	0.961970i	$-0.411932\pi$
$53$	3.97722	0.546313	0.273157	+	0.961970i	$0.411932\pi$
$54$	0	0
$55$	−8.86240	−1.19501
$56$	0	0
$57$	−3.14088	−0.416020
$58$	0	0
$59$	8.69780	1.13236	0.566178	−	0.824283i	$-0.308421\pi$
$59$	8.69780	1.13236	0.566178	+	0.824283i	$0.308421\pi$
$60$	0	0
$61$	−12.6082	−1.61432	−0.807158	−	0.590336i	$-0.798995\pi$
$61$	−12.6082	−1.61432	−0.807158	+	0.590336i	$0.798995\pi$
$62$	0	0
$63$	−3.10909	−0.391709
$64$	0	0
$65$	7.38209	0.915635
$66$	0	0
$67$	11.9816	1.46378	0.731891	−	0.681422i	$-0.238638\pi$
$67$	11.9816	1.46378	0.731891	+	0.681422i	$0.238638\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−11.1914	−1.32818	−0.664090	−	0.747653i	$-0.731181\pi$
$71$	−11.1914	−1.32818	−0.664090	+	0.747653i	$0.731181\pi$
$72$	0	0
$73$	−12.9336	−1.51376	−0.756879	−	0.653555i	$-0.773277\pi$
$73$	−12.9336	−1.51376	−0.756879	+	0.653555i	$0.773277\pi$
$74$	0	0
$75$	−1.64537	−0.189991
$76$	0	0
$77$	−15.0440	−1.71442
$78$	0	0
$79$	1.60254	0.180300	0.0901499	−	0.995928i	$-0.471265\pi$
$79$	1.60254	0.180300	0.0901499	+	0.995928i	$0.471265\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	5.42371	0.595330	0.297665	−	0.954670i	$-0.403792\pi$
$83$	5.42371	0.595330	0.297665	+	0.954670i	$0.403792\pi$
$84$	0	0
$85$	8.15292	0.884309
$86$	0	0
$87$	1.00000	0.107211
$88$	0	0
$89$	6.27276	0.664911	0.332455	−	0.943119i	$-0.392123\pi$
$89$	6.27276	0.664911	0.332455	+	0.943119i	$0.392123\pi$
$90$	0	0
$91$	12.5311	1.31362
$92$	0	0
$93$	9.50715	0.985846
$94$	0	0
$95$	5.75273	0.590218
$96$	0	0
$97$	4.00312	0.406455	0.203228	−	0.979132i	$-0.434857\pi$
$97$	4.00312	0.406455	0.203228	+	0.979132i	$0.434857\pi$
$98$	0	0
$99$	4.83870	0.486308
$100$	0	0
$101$	−17.9721	−1.78829	−0.894143	−	0.447781i	$-0.852214\pi$
$101$	−17.9721	−1.78829	−0.894143	+	0.447781i	$0.852214\pi$
$102$	0	0
$103$	16.2667	1.60280	0.801401	−	0.598127i	$-0.204088\pi$
$103$	16.2667	1.60280	0.801401	+	0.598127i	$0.204088\pi$
$104$	0	0
$105$	5.69451	0.555727
$106$	0	0
$107$	16.9817	1.64168	0.820842	−	0.571156i	$-0.193505\pi$
$107$	16.9817	1.64168	0.820842	+	0.571156i	$0.193505\pi$
$108$	0	0
$109$	4.05063	0.387980	0.193990	−	0.981004i	$-0.437857\pi$
$109$	4.05063	0.387980	0.193990	+	0.981004i	$0.437857\pi$
$110$	0	0
$111$	4.88709	0.463862
$112$	0	0
$113$	6.20069	0.583312	0.291656	−	0.956523i	$-0.405794\pi$
$113$	6.20069	0.583312	0.291656	+	0.956523i	$0.405794\pi$
$114$	0	0
$115$	1.83157	0.170794
$116$	0	0
$117$	−4.03048	−0.372618
$118$	0	0
$119$	13.8396	1.26868
$120$	0	0
$121$	12.4130	1.12846
$122$	0	0
$123$	8.35449	0.753299
$124$	0	0
$125$	12.1714	1.08865
$126$	0	0
$127$	−7.95054	−0.705496	−0.352748	−	0.935718i	$-0.614753\pi$
$127$	−7.95054	−0.705496	−0.352748	+	0.935718i	$0.614753\pi$
$128$	0	0
$129$	−12.8921	−1.13509
$130$	0	0
$131$	−11.3744	−0.993789	−0.496894	−	0.867811i	$-0.665526\pi$
$131$	−11.3744	−0.993789	−0.496894	+	0.867811i	$0.665526\pi$
$132$	0	0
$133$	9.76530	0.846759
$134$	0	0
$135$	−1.83157	−0.157636
$136$	0	0
$137$	16.6828	1.42531	0.712654	−	0.701515i	$-0.247493\pi$
$137$	16.6828	1.42531	0.712654	+	0.701515i	$0.247493\pi$
$138$	0	0
$139$	19.7097	1.67176	0.835879	−	0.548914i	$-0.184959\pi$
$139$	19.7097	1.67176	0.835879	+	0.548914i	$0.184959\pi$
$140$	0	0
$141$	−6.49782	−0.547215
$142$	0	0
$143$	−19.5023	−1.63086
$144$	0	0
$145$	−1.83157	−0.152103
$146$	0	0
$147$	2.66646	0.219926
$148$	0	0
$149$	22.6042	1.85181	0.925903	−	0.377760i	$-0.123306\pi$
$149$	22.6042	1.85181	0.925903	+	0.377760i	$0.123306\pi$
$150$	0	0
$151$	12.2717	0.998654	0.499327	−	0.866413i	$-0.333581\pi$
$151$	12.2717	0.998654	0.499327	+	0.866413i	$0.333581\pi$
$152$	0	0
$153$	−4.45134	−0.359870
$154$	0	0
$155$	−17.4130	−1.39864
$156$	0	0
$157$	−20.0414	−1.59948	−0.799739	−	0.600348i	$-0.795029\pi$
$157$	−20.0414	−1.59948	−0.799739	+	0.600348i	$0.795029\pi$
$158$	0	0
$159$	3.97722	0.315414
$160$	0	0
$161$	3.10909	0.245031
$162$	0	0
$163$	14.3286	1.12230	0.561151	−	0.827713i	$-0.310359\pi$
$163$	14.3286	1.12230	0.561151	+	0.827713i	$0.310359\pi$
$164$	0	0
$165$	−8.86240	−0.689937
$166$	0	0
$167$	10.3476	0.800719	0.400360	−	0.916358i	$-0.368885\pi$
$167$	10.3476	0.800719	0.400360	+	0.916358i	$0.368885\pi$
$168$	0	0
$169$	3.24477	0.249598
$170$	0	0
$171$	−3.14088	−0.240189
$172$	0	0
$173$	−1.38817	−0.105540	−0.0527701	−	0.998607i	$-0.516805\pi$
$173$	−1.38817	−0.105540	−0.0527701	+	0.998607i	$0.516805\pi$
$174$	0	0
$175$	5.11560	0.386703
$176$	0	0
$177$	8.69780	0.653767
$178$	0	0
$179$	5.03157	0.376077	0.188038	−	0.982162i	$-0.439787\pi$
$179$	5.03157	0.376077	0.188038	+	0.982162i	$0.439787\pi$
$180$	0	0
$181$	−15.0046	−1.11529	−0.557643	−	0.830081i	$-0.688294\pi$
$181$	−15.0046	−1.11529	−0.557643	+	0.830081i	$0.688294\pi$
$182$	0	0
$183$	−12.6082	−0.932026
$184$	0	0
$185$	−8.95103	−0.658093
$186$	0	0
$187$	−21.5387	−1.57507
$188$	0	0
$189$	−3.10909	−0.226153
$190$	0	0
$191$	19.4952	1.41062	0.705312	−	0.708897i	$-0.250807\pi$
$191$	19.4952	1.41062	0.705312	+	0.708897i	$0.250807\pi$
$192$	0	0
$193$	−1.47403	−0.106103	−0.0530515	−	0.998592i	$-0.516895\pi$
$193$	−1.47403	−0.106103	−0.0530515	+	0.998592i	$0.516895\pi$
$194$	0	0
$195$	7.38209	0.528642
$196$	0	0
$197$	−9.60174	−0.684096	−0.342048	−	0.939682i	$-0.611121\pi$
$197$	−9.60174	−0.684096	−0.342048	+	0.939682i	$0.611121\pi$
$198$	0	0
$199$	−1.91035	−0.135421	−0.0677105	−	0.997705i	$-0.521569\pi$
$199$	−1.91035	−0.135421	−0.0677105	+	0.997705i	$0.521569\pi$
$200$	0	0
$201$	11.9816	0.845115
$202$	0	0
$203$	−3.10909	−0.218216
$204$	0	0
$205$	−15.3018	−1.06872
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	−15.1978	−1.05125
$210$	0	0
$211$	9.08821	0.625658	0.312829	−	0.949809i	$-0.398723\pi$
$211$	9.08821	0.625658	0.312829	+	0.949809i	$0.398723\pi$
$212$	0	0
$213$	−11.1914	−0.766825
$214$	0	0
$215$	23.6127	1.61037
$216$	0	0
$217$	−29.5586	−2.00657
$218$	0	0
$219$	−12.9336	−0.873969
$220$	0	0
$221$	17.9410	1.20684
$222$	0	0
$223$	24.0320	1.60930	0.804652	−	0.593747i	$-0.202352\pi$
$223$	24.0320	1.60930	0.804652	+	0.593747i	$0.202352\pi$
$224$	0	0
$225$	−1.64537	−0.109691
$226$	0	0
$227$	8.46998	0.562172	0.281086	−	0.959683i	$-0.409305\pi$
$227$	8.46998	0.562172	0.281086	+	0.959683i	$0.409305\pi$
$228$	0	0
$229$	−9.15444	−0.604942	−0.302471	−	0.953159i	$-0.597812\pi$
$229$	−9.15444	−0.604942	−0.302471	+	0.953159i	$0.597812\pi$
$230$	0	0
$231$	−15.0440	−0.989821
$232$	0	0
$233$	4.65199	0.304762	0.152381	−	0.988322i	$-0.451306\pi$
$233$	4.65199	0.304762	0.152381	+	0.988322i	$0.451306\pi$
$234$	0	0
$235$	11.9012	0.776347
$236$	0	0
$237$	1.60254	0.104096
$238$	0	0
$239$	−21.9170	−1.41770	−0.708848	−	0.705361i	$-0.750785\pi$
$239$	−21.9170	−1.41770	−0.708848	+	0.705361i	$0.750785\pi$
$240$	0	0
$241$	17.6679	1.13809	0.569044	−	0.822307i	$-0.307313\pi$
$241$	17.6679	1.13809	0.569044	+	0.822307i	$0.307313\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−4.88380	−0.312014
$246$	0	0
$247$	12.6593	0.805490
$248$	0	0
$249$	5.42371	0.343714
$250$	0	0
$251$	20.6382	1.30267	0.651337	−	0.758789i	$-0.274208\pi$
$251$	20.6382	1.30267	0.651337	+	0.758789i	$0.274208\pi$
$252$	0	0
$253$	−4.83870	−0.304207
$254$	0	0
$255$	8.15292	0.510556
$256$	0	0
$257$	5.94398	0.370775	0.185388	−	0.982665i	$-0.440646\pi$
$257$	5.94398	0.370775	0.185388	+	0.982665i	$0.440646\pi$
$258$	0	0
$259$	−15.1944	−0.944136
$260$	0	0
$261$	1.00000	0.0618984
$262$	0	0
$263$	−13.6177	−0.839705	−0.419853	−	0.907592i	$-0.637918\pi$
$263$	−13.6177	−0.839705	−0.419853	+	0.907592i	$0.637918\pi$
$264$	0	0
$265$	−7.28454	−0.447486
$266$	0	0
$267$	6.27276	0.383886
$268$	0	0
$269$	−14.4974	−0.883923	−0.441962	−	0.897034i	$-0.645717\pi$
$269$	−14.4974	−0.883923	−0.441962	+	0.897034i	$0.645717\pi$
$270$	0	0
$271$	−5.79087	−0.351770	−0.175885	−	0.984411i	$-0.556279\pi$
$271$	−5.79087	−0.351770	−0.175885	+	0.984411i	$0.556279\pi$
$272$	0	0
$273$	12.5311	0.758419
$274$	0	0
$275$	−7.96145	−0.480093
$276$	0	0
$277$	28.7006	1.72445	0.862227	−	0.506522i	$-0.169069\pi$
$277$	28.7006	1.72445	0.862227	+	0.506522i	$0.169069\pi$
$278$	0	0
$279$	9.50715	0.569178
$280$	0	0
$281$	8.20665	0.489568	0.244784	−	0.969578i	$-0.421283\pi$
$281$	8.20665	0.489568	0.244784	+	0.969578i	$0.421283\pi$
$282$	0	0
$283$	−0.0977388	−0.00580997	−0.00290499	−	0.999996i	$-0.500925\pi$
$283$	−0.0977388	−0.00580997	−0.00290499	+	0.999996i	$0.500925\pi$
$284$	0	0
$285$	5.75273	0.340762
$286$	0	0
$287$	−25.9749	−1.53325
$288$	0	0
$289$	2.81444	0.165555
$290$	0	0
$291$	4.00312	0.234667
$292$	0	0
$293$	−22.0168	−1.28623	−0.643117	−	0.765768i	$-0.722359\pi$
$293$	−22.0168	−1.28623	−0.643117	+	0.765768i	$0.722359\pi$
$294$	0	0
$295$	−15.9306	−0.927515
$296$	0	0
$297$	4.83870	0.280770
$298$	0	0
$299$	4.03048	0.233089
$300$	0	0
$301$	40.0827	2.31033
$302$	0	0
$303$	−17.9721	−1.03247
$304$	0	0
$305$	23.0928	1.32229
$306$	0	0
$307$	−22.4854	−1.28331	−0.641655	−	0.766993i	$-0.721752\pi$
$307$	−22.4854	−1.28331	−0.641655	+	0.766993i	$0.721752\pi$
$308$	0	0
$309$	16.2667	0.925379
$310$	0	0
$311$	−16.6844	−0.946083	−0.473042	−	0.881040i	$-0.656844\pi$
$311$	−16.6844	−0.946083	−0.473042	+	0.881040i	$0.656844\pi$
$312$	0	0
$313$	−13.1953	−0.745842	−0.372921	−	0.927863i	$-0.621644\pi$
$313$	−13.1953	−0.745842	−0.372921	+	0.927863i	$0.621644\pi$
$314$	0	0
$315$	5.69451	0.320849
$316$	0	0
$317$	12.0287	0.675599	0.337800	−	0.941218i	$-0.390317\pi$
$317$	12.0287	0.675599	0.337800	+	0.941218i	$0.390317\pi$
$318$	0	0
$319$	4.83870	0.270915
$320$	0	0
$321$	16.9817	0.947826
$322$	0	0
$323$	13.9811	0.777931
$324$	0	0
$325$	6.63163	0.367856
$326$	0	0
$327$	4.05063	0.224000
$328$	0	0
$329$	20.2023	1.11379
$330$	0	0
$331$	−0.846716	−0.0465397	−0.0232699	−	0.999729i	$-0.507408\pi$
$331$	−0.846716	−0.0465397	−0.0232699	+	0.999729i	$0.507408\pi$
$332$	0	0
$333$	4.88709	0.267811
$334$	0	0
$335$	−21.9450	−1.19898
$336$	0	0
$337$	20.3107	1.10640	0.553198	−	0.833050i	$-0.313407\pi$
$337$	20.3107	1.10640	0.553198	+	0.833050i	$0.313407\pi$
$338$	0	0
$339$	6.20069	0.336775
$340$	0	0
$341$	46.0023	2.49116
$342$	0	0
$343$	13.4734	0.727494
$344$	0	0
$345$	1.83157	0.0986082
$346$	0	0
$347$	−7.24008	−0.388668	−0.194334	−	0.980935i	$-0.562254\pi$
$347$	−7.24008	−0.388668	−0.194334	+	0.980935i	$0.562254\pi$
$348$	0	0
$349$	4.49870	0.240810	0.120405	−	0.992725i	$-0.461581\pi$
$349$	4.49870	0.240810	0.120405	+	0.992725i	$0.461581\pi$
$350$	0	0
$351$	−4.03048	−0.215131
$352$	0	0
$353$	−20.0463	−1.06696	−0.533480	−	0.845813i	$-0.679116\pi$
$353$	−20.0463	−1.06696	−0.533480	+	0.845813i	$0.679116\pi$
$354$	0	0
$355$	20.4978	1.08791
$356$	0	0
$357$	13.8396	0.732471
$358$	0	0
$359$	−1.94647	−0.102731	−0.0513653	−	0.998680i	$-0.516357\pi$
$359$	−1.94647	−0.102731	−0.0513653	+	0.998680i	$0.516357\pi$
$360$	0	0
$361$	−9.13486	−0.480782
$362$	0	0
$363$	12.4130	0.651515
$364$	0	0
$365$	23.6887	1.23992
$366$	0	0
$367$	35.8526	1.87149	0.935746	−	0.352675i	$-0.114728\pi$
$367$	35.8526	1.87149	0.935746	+	0.352675i	$0.114728\pi$
$368$	0	0
$369$	8.35449	0.434918
$370$	0	0
$371$	−12.3655	−0.641987
$372$	0	0
$373$	34.6842	1.79588	0.897940	−	0.440117i	$-0.145063\pi$
$373$	34.6842	1.79588	0.897940	+	0.440117i	$0.145063\pi$
$374$	0	0
$375$	12.1714	0.628530
$376$	0	0
$377$	−4.03048	−0.207580
$378$	0	0
$379$	23.7325	1.21906	0.609529	−	0.792764i	$-0.291359\pi$
$379$	23.7325	1.21906	0.609529	+	0.792764i	$0.291359\pi$
$380$	0	0
$381$	−7.95054	−0.407319
$382$	0	0
$383$	−22.4865	−1.14900	−0.574502	−	0.818503i	$-0.694804\pi$
$383$	−22.4865	−1.14900	−0.574502	+	0.818503i	$0.694804\pi$
$384$	0	0
$385$	27.5540	1.40428
$386$	0	0
$387$	−12.8921	−0.655342
$388$	0	0
$389$	3.64191	0.184652	0.0923261	−	0.995729i	$-0.470570\pi$
$389$	3.64191	0.184652	0.0923261	+	0.995729i	$0.470570\pi$
$390$	0	0
$391$	4.45134	0.225114
$392$	0	0
$393$	−11.3744	−0.573764
$394$	0	0
$395$	−2.93516	−0.147684
$396$	0	0
$397$	25.3873	1.27415	0.637077	−	0.770800i	$-0.280143\pi$
$397$	25.3873	1.27415	0.637077	+	0.770800i	$0.280143\pi$
$398$	0	0
$399$	9.76530	0.488876
$400$	0	0
$401$	19.9834	0.997922	0.498961	−	0.866625i	$-0.333715\pi$
$401$	19.9834	0.997922	0.498961	+	0.866625i	$0.333715\pi$
$402$	0	0
$403$	−38.3184	−1.90877
$404$	0	0
$405$	−1.83157	−0.0910112
$406$	0	0
$407$	23.6472	1.17215
$408$	0	0
$409$	17.0045	0.840821	0.420410	−	0.907334i	$-0.361886\pi$
$409$	17.0045	0.840821	0.420410	+	0.907334i	$0.361886\pi$
$410$	0	0
$411$	16.6828	0.822902
$412$	0	0
$413$	−27.0423	−1.33066
$414$	0	0
$415$	−9.93388	−0.487635
$416$	0	0
$417$	19.7097	0.965190
$418$	0	0
$419$	32.9932	1.61182	0.805912	−	0.592035i	$-0.201675\pi$
$419$	32.9932	1.61182	0.805912	+	0.592035i	$0.201675\pi$
$420$	0	0
$421$	36.1890	1.76374	0.881871	−	0.471491i	$-0.156284\pi$
$421$	36.1890	1.76374	0.881871	+	0.471491i	$0.156284\pi$
$422$	0	0
$423$	−6.49782	−0.315935
$424$	0	0
$425$	7.32410	0.355271
$426$	0	0
$427$	39.2001	1.89703
$428$	0	0
$429$	−19.5023	−0.941579
$430$	0	0
$431$	−37.5089	−1.80674	−0.903369	−	0.428863i	$-0.858914\pi$
$431$	−37.5089	−1.80674	−0.903369	+	0.428863i	$0.858914\pi$
$432$	0	0
$433$	0.895806	0.0430497	0.0215248	−	0.999768i	$-0.493148\pi$
$433$	0.895806	0.0430497	0.0215248	+	0.999768i	$0.493148\pi$
$434$	0	0
$435$	−1.83157	−0.0878168
$436$	0	0
$437$	3.14088	0.150249
$438$	0	0
$439$	24.5024	1.16944	0.584718	−	0.811237i	$-0.301205\pi$
$439$	24.5024	1.16944	0.584718	+	0.811237i	$0.301205\pi$
$440$	0	0
$441$	2.66646	0.126974
$442$	0	0
$443$	−29.2559	−1.38999	−0.694994	−	0.719015i	$-0.744593\pi$
$443$	−29.2559	−1.38999	−0.694994	+	0.719015i	$0.744593\pi$
$444$	0	0
$445$	−11.4890	−0.544629
$446$	0	0
$447$	22.6042	1.06914
$448$	0	0
$449$	0.320562	0.0151283	0.00756413	−	0.999971i	$-0.497592\pi$
$449$	0.320562	0.0151283	0.00756413	+	0.999971i	$0.497592\pi$
$450$	0	0
$451$	40.4249	1.90353
$452$	0	0
$453$	12.2717	0.576573
$454$	0	0
$455$	−22.9516	−1.07599
$456$	0	0
$457$	15.3928	0.720047	0.360024	−	0.932943i	$-0.382769\pi$
$457$	15.3928	0.720047	0.360024	+	0.932943i	$0.382769\pi$
$458$	0	0
$459$	−4.45134	−0.207771
$460$	0	0
$461$	23.6362	1.10085	0.550424	−	0.834885i	$-0.314466\pi$
$461$	23.6362	1.10085	0.550424	+	0.834885i	$0.314466\pi$
$462$	0	0
$463$	14.6247	0.679668	0.339834	−	0.940485i	$-0.389629\pi$
$463$	14.6247	0.679668	0.339834	+	0.940485i	$0.389629\pi$
$464$	0	0
$465$	−17.4130	−0.807507
$466$	0	0
$467$	−28.9948	−1.34172	−0.670859	−	0.741585i	$-0.734075\pi$
$467$	−28.9948	−1.34172	−0.670859	+	0.741585i	$0.734075\pi$
$468$	0	0
$469$	−37.2518	−1.72013
$470$	0	0
$471$	−20.0414	−0.923459
$472$	0	0
$473$	−62.3810	−2.86828
$474$	0	0
$475$	5.16791	0.237120
$476$	0	0
$477$	3.97722	0.182104
$478$	0	0
$479$	−3.13630	−0.143301	−0.0716507	−	0.997430i	$-0.522827\pi$
$479$	−3.13630	−0.143301	−0.0716507	+	0.997430i	$0.522827\pi$
$480$	0	0
$481$	−19.6973	−0.898121
$482$	0	0
$483$	3.10909	0.141469
$484$	0	0
$485$	−7.33198	−0.332928
$486$	0	0
$487$	17.0683	0.773437	0.386719	−	0.922198i	$-0.373608\pi$
$487$	17.0683	0.773437	0.386719	+	0.922198i	$0.373608\pi$
$488$	0	0
$489$	14.3286	0.647962
$490$	0	0
$491$	8.15595	0.368073	0.184036	−	0.982919i	$-0.441084\pi$
$491$	8.15595	0.368073	0.184036	+	0.982919i	$0.441084\pi$
$492$	0	0
$493$	−4.45134	−0.200478
$494$	0	0
$495$	−8.86240	−0.398335
$496$	0	0
$497$	34.7952	1.56078
$498$	0	0
$499$	−12.9606	−0.580195	−0.290098	−	0.956997i	$-0.593688\pi$
$499$	−12.9606	−0.580195	−0.290098	+	0.956997i	$0.593688\pi$
$500$	0	0
$501$	10.3476	0.462295
$502$	0	0
$503$	−23.9798	−1.06921	−0.534604	−	0.845103i	$-0.679539\pi$
$503$	−23.9798	−1.06921	−0.534604	+	0.845103i	$0.679539\pi$
$504$	0	0
$505$	32.9170	1.46479
$506$	0	0
$507$	3.24477	0.144105
$508$	0	0
$509$	−30.9181	−1.37042	−0.685209	−	0.728346i	$-0.740289\pi$
$509$	−30.9181	−1.37042	−0.685209	+	0.728346i	$0.740289\pi$
$510$	0	0
$511$	40.2116	1.77886
$512$	0	0
$513$	−3.14088	−0.138673
$514$	0	0
$515$	−29.7935	−1.31286
$516$	0	0
$517$	−31.4410	−1.38277
$518$	0	0
$519$	−1.38817	−0.0609337
$520$	0	0
$521$	11.0628	0.484672	0.242336	−	0.970192i	$-0.422086\pi$
$521$	11.0628	0.484672	0.242336	+	0.970192i	$0.422086\pi$
$522$	0	0
$523$	−2.79862	−0.122375	−0.0611876	−	0.998126i	$-0.519489\pi$
$523$	−2.79862	−0.122375	−0.0611876	+	0.998126i	$0.519489\pi$
$524$	0	0
$525$	5.11560	0.223263
$526$	0	0
$527$	−42.3196	−1.84347
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	8.69780	0.377452
$532$	0	0
$533$	−33.6726	−1.45852
$534$	0	0
$535$	−31.1031	−1.34470
$536$	0	0
$537$	5.03157	0.217128
$538$	0	0
$539$	12.9022	0.555737
$540$	0	0
$541$	−26.1977	−1.12633	−0.563164	−	0.826345i	$-0.690416\pi$
$541$	−26.1977	−1.12633	−0.563164	+	0.826345i	$0.690416\pi$
$542$	0	0
$543$	−15.0046	−0.643910
$544$	0	0
$545$	−7.41899	−0.317795
$546$	0	0
$547$	−23.9076	−1.02221	−0.511107	−	0.859517i	$-0.670764\pi$
$547$	−23.9076	−1.02221	−0.511107	+	0.859517i	$0.670764\pi$
$548$	0	0
$549$	−12.6082	−0.538105
$550$	0	0
$551$	−3.14088	−0.133806
$552$	0	0
$553$	−4.98244	−0.211875
$554$	0	0
$555$	−8.95103	−0.379950
$556$	0	0
$557$	35.4004	1.49996	0.749981	−	0.661459i	$-0.230062\pi$
$557$	35.4004	1.49996	0.749981	+	0.661459i	$0.230062\pi$
$558$	0	0
$559$	51.9613	2.19773
$560$	0	0
$561$	−21.5387	−0.909365
$562$	0	0
$563$	−18.8707	−0.795306	−0.397653	−	0.917536i	$-0.630175\pi$
$563$	−18.8707	−0.795306	−0.397653	+	0.917536i	$0.630175\pi$
$564$	0	0
$565$	−11.3570	−0.477791
$566$	0	0
$567$	−3.10909	−0.130570
$568$	0	0
$569$	−4.91386	−0.206000	−0.103000	−	0.994681i	$-0.532844\pi$
$569$	−4.91386	−0.206000	−0.103000	+	0.994681i	$0.532844\pi$
$570$	0	0
$571$	−36.9641	−1.54690	−0.773450	−	0.633857i	$-0.781471\pi$
$571$	−36.9641	−1.54690	−0.773450	+	0.633857i	$0.781471\pi$
$572$	0	0
$573$	19.4952	0.814424
$574$	0	0
$575$	1.64537	0.0686166
$576$	0	0
$577$	−21.5606	−0.897579	−0.448790	−	0.893637i	$-0.648145\pi$
$577$	−21.5606	−0.897579	−0.448790	+	0.893637i	$0.648145\pi$
$578$	0	0
$579$	−1.47403	−0.0612586
$580$	0	0
$581$	−16.8628	−0.699588
$582$	0	0
$583$	19.2446	0.797029
$584$	0	0
$585$	7.38209	0.305212
$586$	0	0
$587$	−2.54546	−0.105062	−0.0525312	−	0.998619i	$-0.516729\pi$
$587$	−2.54546	−0.105062	−0.0525312	+	0.998619i	$0.516729\pi$
$588$	0	0
$589$	−29.8608	−1.23039
$590$	0	0
$591$	−9.60174	−0.394963
$592$	0	0
$593$	31.1345	1.27854	0.639271	−	0.768981i	$-0.279236\pi$
$593$	31.1345	1.27854	0.639271	+	0.768981i	$0.279236\pi$
$594$	0	0
$595$	−25.3482	−1.03917
$596$	0	0
$597$	−1.91035	−0.0781854
$598$	0	0
$599$	26.6912	1.09057	0.545286	−	0.838250i	$-0.316421\pi$
$599$	26.6912	1.09057	0.545286	+	0.838250i	$0.316421\pi$
$600$	0	0
$601$	17.2438	0.703390	0.351695	−	0.936115i	$-0.385605\pi$
$601$	17.2438	0.703390	0.351695	+	0.936115i	$0.385605\pi$
$602$	0	0
$603$	11.9816	0.487927
$604$	0	0
$605$	−22.7353	−0.924320
$606$	0	0
$607$	−23.7673	−0.964685	−0.482343	−	0.875983i	$-0.660214\pi$
$607$	−23.7673	−0.964685	−0.482343	+	0.875983i	$0.660214\pi$
$608$	0	0
$609$	−3.10909	−0.125987
$610$	0	0
$611$	26.1893	1.05951
$612$	0	0
$613$	−41.5084	−1.67651	−0.838255	−	0.545279i	$-0.816424\pi$
$613$	−41.5084	−1.67651	−0.838255	+	0.545279i	$0.816424\pi$
$614$	0	0
$615$	−15.3018	−0.617028
$616$	0	0
$617$	23.1306	0.931202	0.465601	−	0.884995i	$-0.345838\pi$
$617$	23.1306	0.931202	0.465601	+	0.884995i	$0.345838\pi$
$618$	0	0
$619$	35.4359	1.42429	0.712146	−	0.702032i	$-0.247724\pi$
$619$	35.4359	1.42429	0.712146	+	0.702032i	$0.247724\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	−19.5026	−0.781354
$624$	0	0
$625$	−14.0659	−0.562637
$626$	0	0
$627$	−15.1978	−0.606941
$628$	0	0
$629$	−21.7541	−0.867393
$630$	0	0
$631$	37.3568	1.48715	0.743576	−	0.668651i	$-0.233128\pi$
$631$	37.3568	1.48715	0.743576	+	0.668651i	$0.233128\pi$
$632$	0	0
$633$	9.08821	0.361224
$634$	0	0
$635$	14.5619	0.577873
$636$	0	0
$637$	−10.7471	−0.425816
$638$	0	0
$639$	−11.1914	−0.442726
$640$	0	0
$641$	−4.95954	−0.195890	−0.0979451	−	0.995192i	$-0.531227\pi$
$641$	−4.95954	−0.195890	−0.0979451	+	0.995192i	$0.531227\pi$
$642$	0	0
$643$	−7.01098	−0.276486	−0.138243	−	0.990398i	$-0.544146\pi$
$643$	−7.01098	−0.276486	−0.138243	+	0.990398i	$0.544146\pi$
$644$	0	0
$645$	23.6127	0.929750
$646$	0	0
$647$	36.9377	1.45217	0.726086	−	0.687604i	$-0.241338\pi$
$647$	36.9377	1.45217	0.726086	+	0.687604i	$0.241338\pi$
$648$	0	0
$649$	42.0860	1.65202
$650$	0	0
$651$	−29.5586	−1.15849
$652$	0	0
$653$	−33.7151	−1.31937	−0.659686	−	0.751541i	$-0.729311\pi$
$653$	−33.7151	−1.31937	−0.659686	+	0.751541i	$0.729311\pi$
$654$	0	0
$655$	20.8330	0.814013
$656$	0	0
$657$	−12.9336	−0.504586
$658$	0	0
$659$	−3.80164	−0.148091	−0.0740455	−	0.997255i	$-0.523591\pi$
$659$	−3.80164	−0.148091	−0.0740455	+	0.997255i	$0.523591\pi$
$660$	0	0
$661$	33.1948	1.29113	0.645564	−	0.763706i	$-0.276622\pi$
$661$	33.1948	1.29113	0.645564	+	0.763706i	$0.276622\pi$
$662$	0	0
$663$	17.9410	0.696772
$664$	0	0
$665$	−17.8858	−0.693581
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	0	0
$669$	24.0320	0.929132
$670$	0	0
$671$	−61.0074	−2.35516
$672$	0	0
$673$	30.3365	1.16939	0.584693	−	0.811255i	$-0.301215\pi$
$673$	30.3365	1.16939	0.584693	+	0.811255i	$0.301215\pi$
$674$	0	0
$675$	−1.64537	−0.0633303
$676$	0	0
$677$	22.9494	0.882016	0.441008	−	0.897503i	$-0.354621\pi$
$677$	22.9494	0.882016	0.441008	+	0.897503i	$0.354621\pi$
$678$	0	0
$679$	−12.4461	−0.477636
$680$	0	0
$681$	8.46998	0.324570
$682$	0	0
$683$	−3.74528	−0.143309	−0.0716547	−	0.997430i	$-0.522828\pi$
$683$	−3.74528	−0.143309	−0.0716547	+	0.997430i	$0.522828\pi$
$684$	0	0
$685$	−30.5557	−1.16747
$686$	0	0
$687$	−9.15444	−0.349264
$688$	0	0
$689$	−16.0301	−0.610698
$690$	0	0
$691$	16.2703	0.618950	0.309475	−	0.950908i	$-0.399847\pi$
$691$	16.2703	0.618950	0.309475	+	0.950908i	$0.399847\pi$
$692$	0	0
$693$	−15.0440	−0.571473
$694$	0	0
$695$	−36.0997	−1.36934
$696$	0	0
$697$	−37.1887	−1.40862
$698$	0	0
$699$	4.65199	0.175954
$700$	0	0
$701$	−3.24727	−0.122648	−0.0613239	−	0.998118i	$-0.519532\pi$
$701$	−3.24727	−0.122648	−0.0613239	+	0.998118i	$0.519532\pi$
$702$	0	0
$703$	−15.3498	−0.578928
$704$	0	0
$705$	11.9012	0.448224
$706$	0	0
$707$	55.8768	2.10146
$708$	0	0
$709$	37.8398	1.42110	0.710552	−	0.703645i	$-0.248445\pi$
$709$	37.8398	1.42110	0.710552	+	0.703645i	$0.248445\pi$
$710$	0	0
$711$	1.60254	0.0600999
$712$	0	0
$713$	−9.50715	−0.356046
$714$	0	0
$715$	35.7197	1.33584
$716$	0	0
$717$	−21.9170	−0.818507
$718$	0	0
$719$	3.70119	0.138031	0.0690155	−	0.997616i	$-0.478014\pi$
$719$	3.70119	0.138031	0.0690155	+	0.997616i	$0.478014\pi$
$720$	0	0
$721$	−50.5746	−1.88350
$722$	0	0
$723$	17.6679	0.657076
$724$	0	0
$725$	−1.64537	−0.0611075
$726$	0	0
$727$	−21.7573	−0.806932	−0.403466	−	0.914995i	$-0.632195\pi$
$727$	−21.7573	−0.806932	−0.403466	+	0.914995i	$0.632195\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	57.3871	2.12254
$732$	0	0
$733$	4.28425	0.158243	0.0791213	−	0.996865i	$-0.474789\pi$
$733$	4.28425	0.158243	0.0791213	+	0.996865i	$0.474789\pi$
$734$	0	0
$735$	−4.88380	−0.180142
$736$	0	0
$737$	57.9752	2.13555
$738$	0	0
$739$	10.4991	0.386215	0.193107	−	0.981178i	$-0.438143\pi$
$739$	10.4991	0.386215	0.193107	+	0.981178i	$0.438143\pi$
$740$	0	0
$741$	12.6593	0.465050
$742$	0	0
$743$	−35.4932	−1.30212	−0.651060	−	0.759026i	$-0.725675\pi$
$743$	−35.4932	−1.30212	−0.651060	+	0.759026i	$0.725675\pi$
$744$	0	0
$745$	−41.4010	−1.51682
$746$	0	0
$747$	5.42371	0.198443
$748$	0	0
$749$	−52.7977	−1.92919
$750$	0	0
$751$	33.7618	1.23199	0.615993	−	0.787752i	$-0.288755\pi$
$751$	33.7618	1.23199	0.615993	+	0.787752i	$0.288755\pi$
$752$	0	0
$753$	20.6382	0.752099
$754$	0	0
$755$	−22.4764	−0.817999
$756$	0	0
$757$	−37.5938	−1.36637	−0.683185	−	0.730246i	$-0.739406\pi$
$757$	−37.5938	−1.36637	−0.683185	+	0.730246i	$0.739406\pi$
$758$	0	0
$759$	−4.83870	−0.175634
$760$	0	0
$761$	28.0023	1.01508	0.507542	−	0.861627i	$-0.330554\pi$
$761$	28.0023	1.01508	0.507542	+	0.861627i	$0.330554\pi$
$762$	0	0
$763$	−12.5938	−0.455925
$764$	0	0
$765$	8.15292	0.294770
$766$	0	0
$767$	−35.0563	−1.26581
$768$	0	0
$769$	−4.22693	−0.152427	−0.0762135	−	0.997092i	$-0.524283\pi$
$769$	−4.22693	−0.152427	−0.0762135	+	0.997092i	$0.524283\pi$
$770$	0	0
$771$	5.94398	0.214067
$772$	0	0
$773$	−38.2902	−1.37720	−0.688602	−	0.725139i	$-0.741775\pi$
$773$	−38.2902	−1.37720	−0.688602	+	0.725139i	$0.741775\pi$
$774$	0	0
$775$	−15.6428	−0.561905
$776$	0	0
$777$	−15.1944	−0.545097
$778$	0	0
$779$	−26.2405	−0.940163
$780$	0	0
$781$	−54.1520	−1.93771
$782$	0	0
$783$	1.00000	0.0357371
$784$	0	0
$785$	36.7071	1.31013
$786$	0	0
$787$	−36.7226	−1.30902	−0.654509	−	0.756054i	$-0.727125\pi$
$787$	−36.7226	−1.30902	−0.654509	+	0.756054i	$0.727125\pi$
$788$	0	0
$789$	−13.6177	−0.484804
$790$	0	0
$791$	−19.2785	−0.685465
$792$	0	0
$793$	50.8171	1.80457
$794$	0	0
$795$	−7.28454	−0.258356
$796$	0	0
$797$	−7.04095	−0.249403	−0.124702	−	0.992194i	$-0.539797\pi$
$797$	−7.04095	−0.249403	−0.124702	+	0.992194i	$0.539797\pi$
$798$	0	0
$799$	28.9240	1.02326
$800$	0	0
$801$	6.27276	0.221637
$802$	0	0
$803$	−62.5816	−2.20846
$804$	0	0
$805$	−5.69451	−0.200705
$806$	0	0
$807$	−14.4974	−0.510333
$808$	0	0
$809$	−29.8928	−1.05098	−0.525488	−	0.850801i	$-0.676117\pi$
$809$	−29.8928	−1.05098	−0.525488	+	0.850801i	$0.676117\pi$
$810$	0	0
$811$	27.1299	0.952659	0.476330	−	0.879267i	$-0.341967\pi$
$811$	27.1299	0.952659	0.476330	+	0.879267i	$0.341967\pi$
$812$	0	0
$813$	−5.79087	−0.203095
$814$	0	0
$815$	−26.2438	−0.919279
$816$	0	0
$817$	40.4926	1.41666
$818$	0	0
$819$	12.5311	0.437873
$820$	0	0
$821$	−2.81157	−0.0981246	−0.0490623	−	0.998796i	$-0.515623\pi$
$821$	−2.81157	−0.0981246	−0.0490623	+	0.998796i	$0.515623\pi$
$822$	0	0
$823$	−21.4840	−0.748883	−0.374442	−	0.927250i	$-0.622166\pi$
$823$	−21.4840	−0.748883	−0.374442	+	0.927250i	$0.622166\pi$
$824$	0	0
$825$	−7.96145	−0.277182
$826$	0	0
$827$	35.2327	1.22516	0.612580	−	0.790409i	$-0.290132\pi$
$827$	35.2327	1.22516	0.612580	+	0.790409i	$0.290132\pi$
$828$	0	0
$829$	34.8597	1.21073	0.605363	−	0.795949i	$-0.293028\pi$
$829$	34.8597	1.21073	0.605363	+	0.795949i	$0.293028\pi$
$830$	0	0
$831$	28.7006	0.995614
$832$	0	0
$833$	−11.8693	−0.411248
$834$	0	0
$835$	−18.9523	−0.655870
$836$	0	0
$837$	9.50715	0.328615
$838$	0	0
$839$	25.7081	0.887543	0.443772	−	0.896140i	$-0.353640\pi$
$839$	25.7081	0.887543	0.443772	+	0.896140i	$0.353640\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	8.20665	0.282652
$844$	0	0
$845$	−5.94301	−0.204446
$846$	0	0
$847$	−38.5933	−1.32608
$848$	0	0
$849$	−0.0977388	−0.00335439
$850$	0	0
$851$	−4.88709	−0.167527
$852$	0	0
$853$	13.0372	0.446387	0.223193	−	0.974774i	$-0.428352\pi$
$853$	13.0372	0.446387	0.223193	+	0.974774i	$0.428352\pi$
$854$	0	0
$855$	5.75273	0.196739
$856$	0	0
$857$	−30.3866	−1.03799	−0.518994	−	0.854778i	$-0.673693\pi$
$857$	−30.3866	−1.03799	−0.518994	+	0.854778i	$0.673693\pi$
$858$	0	0
$859$	3.07450	0.104901	0.0524503	−	0.998624i	$-0.483297\pi$
$859$	3.07450	0.104901	0.0524503	+	0.998624i	$0.483297\pi$
$860$	0	0
$861$	−25.9749	−0.885222
$862$	0	0
$863$	53.7487	1.82963	0.914813	−	0.403877i	$-0.132338\pi$
$863$	53.7487	1.82963	0.914813	+	0.403877i	$0.132338\pi$
$864$	0	0
$865$	2.54252	0.0864481
$866$	0	0
$867$	2.81444	0.0955833
$868$	0	0
$869$	7.75421	0.263044
$870$	0	0
$871$	−48.2915	−1.63629
$872$	0	0
$873$	4.00312	0.135485
$874$	0	0
$875$	−37.8421	−1.27930
$876$	0	0
$877$	−45.9284	−1.55089	−0.775446	−	0.631414i	$-0.782475\pi$
$877$	−45.9284	−1.55089	−0.775446	+	0.631414i	$0.782475\pi$
$878$	0	0
$879$	−22.0168	−0.742608
$880$	0	0
$881$	−37.2828	−1.25609	−0.628045	−	0.778177i	$-0.716144\pi$
$881$	−37.2828	−1.25609	−0.628045	+	0.778177i	$0.716144\pi$
$882$	0	0
$883$	10.4023	0.350066	0.175033	−	0.984563i	$-0.443997\pi$
$883$	10.4023	0.350066	0.175033	+	0.984563i	$0.443997\pi$
$884$	0	0
$885$	−15.9306	−0.535501
$886$	0	0
$887$	−27.9270	−0.937696	−0.468848	−	0.883279i	$-0.655331\pi$
$887$	−27.9270	−0.937696	−0.468848	+	0.883279i	$0.655331\pi$
$888$	0	0
$889$	24.7190	0.829048
$890$	0	0
$891$	4.83870	0.162103
$892$	0	0
$893$	20.4089	0.682957
$894$	0	0
$895$	−9.21564	−0.308045
$896$	0	0
$897$	4.03048	0.134574
$898$	0	0
$899$	9.50715	0.317081
$900$	0	0
$901$	−17.7040	−0.589804
$902$	0	0
$903$	40.0827	1.33387
$904$	0	0
$905$	27.4820	0.913531
$906$	0	0
$907$	−17.1754	−0.570301	−0.285150	−	0.958483i	$-0.592044\pi$
$907$	−17.1754	−0.570301	−0.285150	+	0.958483i	$0.592044\pi$
$908$	0	0
$909$	−17.9721	−0.596096
$910$	0	0
$911$	−47.3579	−1.56904	−0.784519	−	0.620105i	$-0.787090\pi$
$911$	−47.3579	−1.56904	−0.784519	+	0.620105i	$0.787090\pi$
$912$	0	0
$913$	26.2437	0.868541
$914$	0	0
$915$	23.0928	0.763423
$916$	0	0
$917$	35.3642	1.16783
$918$	0	0
$919$	19.8387	0.654418	0.327209	−	0.944952i	$-0.393892\pi$
$919$	19.8387	0.654418	0.327209	+	0.944952i	$0.393892\pi$
$920$	0	0
$921$	−22.4854	−0.740920
$922$	0	0
$923$	45.1069	1.48471
$924$	0	0
$925$	−8.04107	−0.264389
$926$	0	0
$927$	16.2667	0.534268
$928$	0	0
$929$	16.1487	0.529820	0.264910	−	0.964273i	$-0.414658\pi$
$929$	16.1487	0.529820	0.264910	+	0.964273i	$0.414658\pi$
$930$	0	0
$931$	−8.37504	−0.274481
$932$	0	0
$933$	−16.6844	−0.546221
$934$	0	0
$935$	39.4496	1.29014
$936$	0	0
$937$	48.6162	1.58822	0.794110	−	0.607774i	$-0.207937\pi$
$937$	48.6162	1.58822	0.794110	+	0.607774i	$0.207937\pi$
$938$	0	0
$939$	−13.1953	−0.430612
$940$	0	0
$941$	−10.0398	−0.327289	−0.163644	−	0.986519i	$-0.552325\pi$
$941$	−10.0398	−0.327289	−0.163644	+	0.986519i	$0.552325\pi$
$942$	0	0
$943$	−8.35449	−0.272060
$944$	0	0
$945$	5.69451	0.185242
$946$	0	0
$947$	−8.37566	−0.272172	−0.136086	−	0.990697i	$-0.543452\pi$
$947$	−8.37566	−0.272172	−0.136086	+	0.990697i	$0.543452\pi$
$948$	0	0
$949$	52.1284	1.69216
$950$	0	0
$951$	12.0287	0.390057
$952$	0	0
$953$	−36.8507	−1.19371	−0.596856	−	0.802349i	$-0.703584\pi$
$953$	−36.8507	−1.19371	−0.596856	+	0.802349i	$0.703584\pi$
$954$	0	0
$955$	−35.7067	−1.15544
$956$	0	0
$957$	4.83870	0.156413
$958$	0	0
$959$	−51.8684	−1.67492
$960$	0	0
$961$	59.3860	1.91568
$962$	0	0
$963$	16.9817	0.547228
$964$	0	0
$965$	2.69978	0.0869091
$966$	0	0
$967$	30.7974	0.990377	0.495189	−	0.868786i	$-0.335099\pi$
$967$	30.7974	0.990377	0.495189	+	0.868786i	$0.335099\pi$
$968$	0	0
$969$	13.9811	0.449139
$970$	0	0
$971$	32.1169	1.03068	0.515340	−	0.856986i	$-0.327666\pi$
$971$	32.1169	1.03068	0.515340	+	0.856986i	$0.327666\pi$
$972$	0	0
$973$	−61.2794	−1.96453
$974$	0	0
$975$	6.63163	0.212382
$976$	0	0
$977$	−51.0375	−1.63283	−0.816416	−	0.577464i	$-0.804043\pi$
$977$	−51.0375	−1.63283	−0.816416	+	0.577464i	$0.804043\pi$
$978$	0	0
$979$	30.3520	0.970054
$980$	0	0
$981$	4.05063	0.129327
$982$	0	0
$983$	13.9284	0.444246	0.222123	−	0.975019i	$-0.428701\pi$
$983$	13.9284	0.444246	0.222123	+	0.975019i	$0.428701\pi$
$984$	0	0
$985$	17.5862	0.560344
$986$	0	0
$987$	20.2023	0.643047
$988$	0	0
$989$	12.8921	0.409945
$990$	0	0
$991$	−53.2564	−1.69174	−0.845872	−	0.533386i	$-0.820919\pi$
$991$	−53.2564	−1.69174	−0.845872	+	0.533386i	$0.820919\pi$
$992$	0	0
$993$	−0.846716	−0.0268697
$994$	0	0
$995$	3.49893	0.110924
$996$	0	0
$997$	−22.0378	−0.697943	−0.348971	−	0.937133i	$-0.613469\pi$
$997$	−22.0378	−0.697943	−0.348971	+	0.937133i	$0.613469\pi$
$998$	0	0
$999$	4.88709	0.154621

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8004.2.a.k.1.7	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8004.2.a.k.1.7	✓	18	1.1	even	1	trivial

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Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8004.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.83157 q^{5} -3.10909 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.83157 q^{5} -3.10909 q^{7} +1.00000 q^{9} +4.83870 q^{11} -4.03048 q^{13} -1.83157 q^{15} -4.45134 q^{17} -3.14088 q^{19} -3.10909 q^{21} -1.00000 q^{23} -1.64537 q^{25} +1.00000 q^{27} +1.00000 q^{29} +9.50715 q^{31} +4.83870 q^{33} +5.69451 q^{35} +4.88709 q^{37} -4.03048 q^{39} +8.35449 q^{41} -12.8921 q^{43} -1.83157 q^{45} -6.49782 q^{47} +2.66646 q^{49} -4.45134 q^{51} +3.97722 q^{53} -8.86240 q^{55} -3.14088 q^{57} +8.69780 q^{59} -12.6082 q^{61} -3.10909 q^{63} +7.38209 q^{65} +11.9816 q^{67} -1.00000 q^{69} -11.1914 q^{71} -12.9336 q^{73} -1.64537 q^{75} -15.0440 q^{77} +1.60254 q^{79} +1.00000 q^{81} +5.42371 q^{83} +8.15292 q^{85} +1.00000 q^{87} +6.27276 q^{89} +12.5311 q^{91} +9.50715 q^{93} +5.75273 q^{95} +4.00312 q^{97} +4.83870 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 18 q^{3} + 5 q^{5} + 6 q^{7} + 18 q^{9}+O(q^{10}) \) 18 * q + 18 * q^3 + 5 * q^5 + 6 * q^7 + 18 * q^9	\( 18 q + 18 q^{3} + 5 q^{5} + 6 q^{7} + 18 q^{9} + 5 q^{11} + 6 q^{13} + 5 q^{15} + 7 q^{17} + 15 q^{19} + 6 q^{21} - 18 q^{23} + 45 q^{25} + 18 q^{27} + 18 q^{29} + 10 q^{31} + 5 q^{33} - 7 q^{35} + 22 q^{37} + 6 q^{39} + 17 q^{41} + 25 q^{43} + 5 q^{45} - 6 q^{47} + 64 q^{49} + 7 q^{51} + 21 q^{53} + 3 q^{55} + 15 q^{57} + 6 q^{59} + 7 q^{61} + 6 q^{63} + 44 q^{65} + 35 q^{67} - 18 q^{69} + q^{71} + 27 q^{73} + 45 q^{75} + 4 q^{77} + 18 q^{81} + 13 q^{83} + 24 q^{85} + 18 q^{87} + 30 q^{89} + 53 q^{91} + 10 q^{93} + 17 q^{95} + 27 q^{97} + 5 q^{99}+O(q^{100}) \) 18 * q + 18 * q^3 + 5 * q^5 + 6 * q^7 + 18 * q^9 + 5 * q^11 + 6 * q^13 + 5 * q^15 + 7 * q^17 + 15 * q^19 + 6 * q^21 - 18 * q^23 + 45 * q^25 + 18 * q^27 + 18 * q^29 + 10 * q^31 + 5 * q^33 - 7 * q^35 + 22 * q^37 + 6 * q^39 + 17 * q^41 + 25 * q^43 + 5 * q^45 - 6 * q^47 + 64 * q^49 + 7 * q^51 + 21 * q^53 + 3 * q^55 + 15 * q^57 + 6 * q^59 + 7 * q^61 + 6 * q^63 + 44 * q^65 + 35 * q^67 - 18 * q^69 + q^71 + 27 * q^73 + 45 * q^75 + 4 * q^77 + 18 * q^81 + 13 * q^83 + 24 * q^85 + 18 * q^87 + 30 * q^89 + 53 * q^91 + 10 * q^93 + 17 * q^95 + 27 * q^97 + 5 * q^99

Introduction

L-functions

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Database

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