LMFDB - Embedded newform 8008.2.a.i.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8008,2,Mod(1,8008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8008, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8008.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:	$63.9442019386$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.229.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 4x - 1 $ x^3 - 4*x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.254102$ of defining polynomial
Character	$\chi$	$=$	8008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.93543 q^{3} +2.25410 q^{5} -1.00000 q^{7} +0.745898 q^{9} +O(q^{10})$	$q-1.93543 q^{3} +2.25410 q^{5} -1.00000 q^{7} +0.745898 q^{9} -1.00000 q^{11} +1.00000 q^{13} -4.36266 q^{15} +0.616763 q^{17} +2.42723 q^{19} +1.93543 q^{21} -7.69774 q^{23} +0.0809744 q^{25} +4.36266 q^{27} +8.37907 q^{29} -3.31867 q^{31} +1.93543 q^{33} -2.25410 q^{35} +4.68133 q^{37} -1.93543 q^{39} -12.2171 q^{41} +2.23769 q^{43} +1.68133 q^{45} -4.85446 q^{47} +1.00000 q^{49} -1.19370 q^{51} +6.99583 q^{53} -2.25410 q^{55} -4.69774 q^{57} +10.0440 q^{59} +2.74590 q^{61} -0.745898 q^{63} +2.25410 q^{65} -5.42723 q^{67} +14.8984 q^{69} -2.38324 q^{71} -4.98359 q^{73} -0.156721 q^{75} +1.00000 q^{77} -9.31450 q^{79} -10.6813 q^{81} -3.95184 q^{83} +1.39025 q^{85} -16.2171 q^{87} +6.91903 q^{89} -1.00000 q^{91} +6.42306 q^{93} +5.47122 q^{95} +17.2775 q^{97} -0.745898 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{3} + 6 q^{5} - 3 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q + 2 * q^3 + 6 * q^5 - 3 * q^7 + 3 * q^9	$ 3 q + 2 q^{3} + 6 q^{5} - 3 q^{7} + 3 q^{9} - 3 q^{11} + 3 q^{13} + q^{15} - 13 q^{17} + q^{19} - 2 q^{21} - 13 q^{23} + 5 q^{25} - q^{27} + 8 q^{29} - 17 q^{31} - 2 q^{33} - 6 q^{35} + 7 q^{37} + 2 q^{39} - 10 q^{41} + 9 q^{43} - 2 q^{45} - 2 q^{47} + 3 q^{49} - 27 q^{51} - 11 q^{53} - 6 q^{55} - 4 q^{57} + 9 q^{59} + 9 q^{61} - 3 q^{63} + 6 q^{65} - 10 q^{67} + 11 q^{69} - 22 q^{71} - 18 q^{73} + 2 q^{75} + 3 q^{77} - 3 q^{79} - 25 q^{81} - q^{83} - 28 q^{85} - 22 q^{87} + 16 q^{89} - 3 q^{91} - 19 q^{93} - 11 q^{95} + q^{97} - 3 q^{99}+O(q^{100}) $ 3 * q + 2 * q^3 + 6 * q^5 - 3 * q^7 + 3 * q^9 - 3 * q^11 + 3 * q^13 + q^15 - 13 * q^17 + q^19 - 2 * q^21 - 13 * q^23 + 5 * q^25 - q^27 + 8 * q^29 - 17 * q^31 - 2 * q^33 - 6 * q^35 + 7 * q^37 + 2 * q^39 - 10 * q^41 + 9 * q^43 - 2 * q^45 - 2 * q^47 + 3 * q^49 - 27 * q^51 - 11 * q^53 - 6 * q^55 - 4 * q^57 + 9 * q^59 + 9 * q^61 - 3 * q^63 + 6 * q^65 - 10 * q^67 + 11 * q^69 - 22 * q^71 - 18 * q^73 + 2 * q^75 + 3 * q^77 - 3 * q^79 - 25 * q^81 - q^83 - 28 * q^85 - 22 * q^87 + 16 * q^89 - 3 * q^91 - 19 * q^93 - 11 * q^95 + q^97 - 3 * 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
$3$	−1.93543	−1.11742	−0.558711	−	0.829362i	$-0.688704\pi$
$3$	−1.93543	−1.11742	−0.558711	+	0.829362i	$0.688704\pi$
$4$	0	0
$5$	2.25410	1.00806	0.504032	−	0.863685i	$-0.331849\pi$
$5$	2.25410	1.00806	0.504032	+	0.863685i	$0.331849\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	0.745898	0.248633
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	−4.36266	−1.12643
$16$	0	0
$17$	0.616763	0.149587	0.0747935	−	0.997199i	$-0.476170\pi$
$17$	0.616763	0.149587	0.0747935	+	0.997199i	$0.476170\pi$
$18$	0	0
$19$	2.42723	0.556845	0.278422	−	0.960459i	$-0.410189\pi$
$19$	2.42723	0.556845	0.278422	+	0.960459i	$0.410189\pi$
$20$	0	0
$21$	1.93543	0.422346
$22$	0	0
$23$	−7.69774	−1.60509	−0.802545	−	0.596592i	$-0.796521\pi$
$23$	−7.69774	−1.60509	−0.802545	+	0.596592i	$0.796521\pi$
$24$	0	0
$25$	0.0809744	0.0161949
$26$	0	0
$27$	4.36266	0.839595
$28$	0	0
$29$	8.37907	1.55595	0.777977	−	0.628293i	$-0.216246\pi$
$29$	8.37907	1.55595	0.777977	+	0.628293i	$0.216246\pi$
$30$	0	0
$31$	−3.31867	−0.596051	−0.298025	−	0.954558i	$-0.596328\pi$
$31$	−3.31867	−0.596051	−0.298025	+	0.954558i	$0.596328\pi$
$32$	0	0
$33$	1.93543	0.336916
$34$	0	0
$35$	−2.25410	−0.381013
$36$	0	0
$37$	4.68133	0.769606	0.384803	−	0.922999i	$-0.374269\pi$
$37$	4.68133	0.769606	0.384803	+	0.922999i	$0.374269\pi$
$38$	0	0
$39$	−1.93543	−0.309917
$40$	0	0
$41$	−12.2171	−1.90799	−0.953997	−	0.299817i	$-0.903074\pi$
$41$	−12.2171	−1.90799	−0.953997	+	0.299817i	$0.903074\pi$
$42$	0	0
$43$	2.23769	0.341245	0.170623	−	0.985336i	$-0.445422\pi$
$43$	2.23769	0.341245	0.170623	+	0.985336i	$0.445422\pi$
$44$	0	0
$45$	1.68133	0.250638
$46$	0	0
$47$	−4.85446	−0.708095	−0.354048	−	0.935227i	$-0.615195\pi$
$47$	−4.85446	−0.708095	−0.354048	+	0.935227i	$0.615195\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−1.19370	−0.167152
$52$	0	0
$53$	6.99583	0.960951	0.480476	−	0.877008i	$-0.340464\pi$
$53$	6.99583	0.960951	0.480476	+	0.877008i	$0.340464\pi$
$54$	0	0
$55$	−2.25410	−0.303943
$56$	0	0
$57$	−4.69774	−0.622231
$58$	0	0
$59$	10.0440	1.30762	0.653808	−	0.756660i	$-0.273170\pi$
$59$	10.0440	1.30762	0.653808	+	0.756660i	$0.273170\pi$
$60$	0	0
$61$	2.74590	0.351576	0.175788	−	0.984428i	$-0.443753\pi$
$61$	2.74590	0.351576	0.175788	+	0.984428i	$0.443753\pi$
$62$	0	0
$63$	−0.745898	−0.0939744
$64$	0	0
$65$	2.25410	0.279587
$66$	0	0
$67$	−5.42723	−0.663042	−0.331521	−	0.943448i	$-0.607562\pi$
$67$	−5.42723	−0.663042	−0.331521	+	0.943448i	$0.607562\pi$
$68$	0	0
$69$	14.8984	1.79356
$70$	0	0
$71$	−2.38324	−0.282838	−0.141419	−	0.989950i	$-0.545167\pi$
$71$	−2.38324	−0.282838	−0.141419	+	0.989950i	$0.545167\pi$
$72$	0	0
$73$	−4.98359	−0.583285	−0.291643	−	0.956527i	$-0.594202\pi$
$73$	−4.98359	−0.583285	−0.291643	+	0.956527i	$0.594202\pi$
$74$	0	0
$75$	−0.156721	−0.0180965
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	−9.31450	−1.04796	−0.523982	−	0.851730i	$-0.675554\pi$
$79$	−9.31450	−1.04796	−0.523982	+	0.851730i	$0.675554\pi$
$80$	0	0
$81$	−10.6813	−1.18681
$82$	0	0
$83$	−3.95184	−0.433771	−0.216885	−	0.976197i	$-0.569590\pi$
$83$	−3.95184	−0.433771	−0.216885	+	0.976197i	$0.569590\pi$
$84$	0	0
$85$	1.39025	0.150793
$86$	0	0
$87$	−16.2171	−1.73866
$88$	0	0
$89$	6.91903	0.733415	0.366708	−	0.930336i	$-0.380485\pi$
$89$	6.91903	0.733415	0.366708	+	0.930336i	$0.380485\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	6.42306	0.666040
$94$	0	0
$95$	5.47122	0.561335
$96$	0	0
$97$	17.2775	1.75427	0.877133	−	0.480247i	$-0.159453\pi$
$97$	17.2775	1.75427	0.877133	+	0.480247i	$0.159453\pi$
$98$	0	0
$99$	−0.745898	−0.0749656
$100$	0	0
$101$	−8.93543	−0.889109	−0.444554	−	0.895752i	$-0.646638\pi$
$101$	−8.93543	−0.889109	−0.444554	+	0.895752i	$0.646638\pi$
$102$	0	0
$103$	−14.7253	−1.45093	−0.725465	−	0.688260i	$-0.758375\pi$
$103$	−14.7253	−1.45093	−0.725465	+	0.688260i	$0.758375\pi$
$104$	0	0
$105$	4.36266	0.425752
$106$	0	0
$107$	10.6496	1.02953	0.514767	−	0.857330i	$-0.327879\pi$
$107$	10.6496	1.02953	0.514767	+	0.857330i	$0.327879\pi$
$108$	0	0
$109$	9.91903	0.950070	0.475035	−	0.879967i	$-0.342435\pi$
$109$	9.91903	0.950070	0.475035	+	0.879967i	$0.342435\pi$
$110$	0	0
$111$	−9.06040	−0.859975
$112$	0	0
$113$	−19.6168	−1.84539	−0.922695	−	0.385531i	$-0.874018\pi$
$113$	−19.6168	−1.84539	−0.922695	+	0.385531i	$0.874018\pi$
$114$	0	0
$115$	−17.3515	−1.61803
$116$	0	0
$117$	0.745898	0.0689583
$118$	0	0
$119$	−0.616763	−0.0565386
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	23.6454	2.13203
$124$	0	0
$125$	−11.0880	−0.991739
$126$	0	0
$127$	19.6936	1.74752	0.873761	−	0.486356i	$-0.161674\pi$
$127$	19.6936	1.74752	0.873761	+	0.486356i	$0.161674\pi$
$128$	0	0
$129$	−4.33091	−0.381315
$130$	0	0
$131$	−13.1044	−1.14494	−0.572468	−	0.819927i	$-0.694014\pi$
$131$	−13.1044	−1.14494	−0.572468	+	0.819927i	$0.694014\pi$
$132$	0	0
$133$	−2.42723	−0.210467
$134$	0	0
$135$	9.83388	0.846366
$136$	0	0
$137$	5.40665	0.461922	0.230961	−	0.972963i	$-0.425813\pi$
$137$	5.40665	0.461922	0.230961	+	0.972963i	$0.425813\pi$
$138$	0	0
$139$	−1.36266	−0.115579	−0.0577897	−	0.998329i	$-0.518405\pi$
$139$	−1.36266	−0.115579	−0.0577897	+	0.998329i	$0.518405\pi$
$140$	0	0
$141$	9.39547	0.791242
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	18.8873	1.56850
$146$	0	0
$147$	−1.93543	−0.159632
$148$	0	0
$149$	0.0481609	0.00394550	0.00197275	−	0.999998i	$-0.499372\pi$
$149$	0.0481609	0.00394550	0.00197275	+	0.999998i	$0.499372\pi$
$150$	0	0
$151$	−13.2335	−1.07693	−0.538465	−	0.842648i	$-0.680995\pi$
$151$	−13.2335	−1.07693	−0.538465	+	0.842648i	$0.680995\pi$
$152$	0	0
$153$	0.460042	0.0371922
$154$	0	0
$155$	−7.48062	−0.600858
$156$	0	0
$157$	−16.8021	−1.34096	−0.670478	−	0.741930i	$-0.733911\pi$
$157$	−16.8021	−1.34096	−0.670478	+	0.741930i	$0.733911\pi$
$158$	0	0
$159$	−13.5400	−1.07379
$160$	0	0
$161$	7.69774	0.606667
$162$	0	0
$163$	−4.93543	−0.386573	−0.193286	−	0.981142i	$-0.561915\pi$
$163$	−4.93543	−0.386573	−0.193286	+	0.981142i	$0.561915\pi$
$164$	0	0
$165$	4.36266	0.339633
$166$	0	0
$167$	−13.7623	−1.06496	−0.532480	−	0.846443i	$-0.678740\pi$
$167$	−13.7623	−1.06496	−0.532480	+	0.846443i	$0.678740\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	1.81047	0.138450
$172$	0	0
$173$	−1.53996	−0.117081	−0.0585404	−	0.998285i	$-0.518645\pi$
$173$	−1.53996	−0.117081	−0.0585404	+	0.998285i	$0.518645\pi$
$174$	0	0
$175$	−0.0809744	−0.00612109
$176$	0	0
$177$	−19.4395	−1.46116
$178$	0	0
$179$	3.43947	0.257078	0.128539	−	0.991704i	$-0.458971\pi$
$179$	3.43947	0.257078	0.128539	+	0.991704i	$0.458971\pi$
$180$	0	0
$181$	5.15672	0.383296	0.191648	−	0.981464i	$-0.438617\pi$
$181$	5.15672	0.383296	0.191648	+	0.981464i	$0.438617\pi$
$182$	0	0
$183$	−5.31450	−0.392859
$184$	0	0
$185$	10.5522	0.775813
$186$	0	0
$187$	−0.616763	−0.0451022
$188$	0	0
$189$	−4.36266	−0.317337
$190$	0	0
$191$	−2.33508	−0.168960	−0.0844801	−	0.996425i	$-0.526923\pi$
$191$	−2.33508	−0.168960	−0.0844801	+	0.996425i	$0.526923\pi$
$192$	0	0
$193$	−18.8339	−1.35569	−0.677846	−	0.735204i	$-0.737086\pi$
$193$	−18.8339	−1.35569	−0.677846	+	0.735204i	$0.737086\pi$
$194$	0	0
$195$	−4.36266	−0.312417
$196$	0	0
$197$	3.94661	0.281184	0.140592	−	0.990068i	$-0.455099\pi$
$197$	3.94661	0.281184	0.140592	+	0.990068i	$0.455099\pi$
$198$	0	0
$199$	20.4189	1.44746	0.723728	−	0.690085i	$-0.242427\pi$
$199$	20.4189	1.44746	0.723728	+	0.690085i	$0.242427\pi$
$200$	0	0
$201$	10.5040	0.740897
$202$	0	0
$203$	−8.37907	−0.588095
$204$	0	0
$205$	−27.5386	−1.92338
$206$	0	0
$207$	−5.74173	−0.399078
$208$	0	0
$209$	−2.42723	−0.167895
$210$	0	0
$211$	3.72949	0.256749	0.128374	−	0.991726i	$-0.459024\pi$
$211$	3.72949	0.256749	0.128374	+	0.991726i	$0.459024\pi$
$212$	0	0
$213$	4.61259	0.316050
$214$	0	0
$215$	5.04399	0.343997
$216$	0	0
$217$	3.31867	0.225286
$218$	0	0
$219$	9.64541	0.651776
$220$	0	0
$221$	0.616763	0.0414880
$222$	0	0
$223$	27.1484	1.81799	0.908995	−	0.416807i	$-0.136851\pi$
$223$	27.1484	1.81799	0.908995	+	0.416807i	$0.136851\pi$
$224$	0	0
$225$	0.0603987	0.00402658
$226$	0	0
$227$	−22.1002	−1.46684	−0.733422	−	0.679774i	$-0.762078\pi$
$227$	−22.1002	−1.46684	−0.733422	+	0.679774i	$0.762078\pi$
$228$	0	0
$229$	11.7787	0.778359	0.389180	−	0.921162i	$-0.372759\pi$
$229$	11.7787	0.778359	0.389180	+	0.921162i	$0.372759\pi$
$230$	0	0
$231$	−1.93543	−0.127342
$232$	0	0
$233$	−21.5798	−1.41374	−0.706869	−	0.707344i	$-0.749893\pi$
$233$	−21.5798	−1.41374	−0.706869	+	0.707344i	$0.749893\pi$
$234$	0	0
$235$	−10.9424	−0.713806
$236$	0	0
$237$	18.0276	1.17102
$238$	0	0
$239$	−5.61259	−0.363049	−0.181524	−	0.983386i	$-0.558103\pi$
$239$	−5.61259	−0.363049	−0.181524	+	0.983386i	$0.558103\pi$
$240$	0	0
$241$	−10.8545	−0.699197	−0.349599	−	0.936900i	$-0.613682\pi$
$241$	−10.8545	−0.699197	−0.349599	+	0.936900i	$0.613682\pi$
$242$	0	0
$243$	7.58501	0.486579
$244$	0	0
$245$	2.25410	0.144009
$246$	0	0
$247$	2.42723	0.154441
$248$	0	0
$249$	7.64852	0.484705
$250$	0	0
$251$	−18.8175	−1.18775	−0.593874	−	0.804558i	$-0.702402\pi$
$251$	−18.8175	−1.18775	−0.593874	+	0.804558i	$0.702402\pi$
$252$	0	0
$253$	7.69774	0.483953
$254$	0	0
$255$	−2.69073	−0.168500
$256$	0	0
$257$	24.6290	1.53631	0.768157	−	0.640261i	$-0.221174\pi$
$257$	24.6290	1.53631	0.768157	+	0.640261i	$0.221174\pi$
$258$	0	0
$259$	−4.68133	−0.290884
$260$	0	0
$261$	6.24993	0.386861
$262$	0	0
$263$	−8.96825	−0.553006	−0.276503	−	0.961013i	$-0.589176\pi$
$263$	−8.96825	−0.553006	−0.276503	+	0.961013i	$0.589176\pi$
$264$	0	0
$265$	15.7693	0.968701
$266$	0	0
$267$	−13.3913	−0.819535
$268$	0	0
$269$	28.7253	1.75141	0.875707	−	0.482843i	$-0.160396\pi$
$269$	28.7253	1.75141	0.875707	+	0.482843i	$0.160396\pi$
$270$	0	0
$271$	22.0674	1.34050	0.670250	−	0.742136i	$-0.266187\pi$
$271$	22.0674	1.34050	0.670250	+	0.742136i	$0.266187\pi$
$272$	0	0
$273$	1.93543	0.117138
$274$	0	0
$275$	−0.0809744	−0.00488294
$276$	0	0
$277$	−5.11273	−0.307194	−0.153597	−	0.988134i	$-0.549086\pi$
$277$	−5.11273	−0.307194	−0.153597	+	0.988134i	$0.549086\pi$
$278$	0	0
$279$	−2.47539	−0.148198
$280$	0	0
$281$	15.0286	0.896534	0.448267	−	0.893900i	$-0.352041\pi$
$281$	15.0286	0.896534	0.448267	+	0.893900i	$0.352041\pi$
$282$	0	0
$283$	−27.1372	−1.61314	−0.806570	−	0.591139i	$-0.798678\pi$
$283$	−27.1372	−1.61314	−0.806570	+	0.591139i	$0.798678\pi$
$284$	0	0
$285$	−10.5892	−0.627249
$286$	0	0
$287$	12.2171	0.721154
$288$	0	0
$289$	−16.6196	−0.977624
$290$	0	0
$291$	−33.4395	−1.96026
$292$	0	0
$293$	14.8873	0.869724	0.434862	−	0.900497i	$-0.356797\pi$
$293$	14.8873	0.869724	0.434862	+	0.900497i	$0.356797\pi$
$294$	0	0
$295$	22.6402	1.31816
$296$	0	0
$297$	−4.36266	−0.253147
$298$	0	0
$299$	−7.69774	−0.445172
$300$	0	0
$301$	−2.23769	−0.128979
$302$	0	0
$303$	17.2939	0.993510
$304$	0	0
$305$	6.18953	0.354412
$306$	0	0
$307$	−6.96302	−0.397400	−0.198700	−	0.980060i	$-0.563672\pi$
$307$	−6.96302	−0.397400	−0.198700	+	0.980060i	$0.563672\pi$
$308$	0	0
$309$	28.4999	1.62130
$310$	0	0
$311$	−29.7899	−1.68923	−0.844615	−	0.535374i	$-0.820171\pi$
$311$	−29.7899	−1.68923	−0.844615	+	0.535374i	$0.820171\pi$
$312$	0	0
$313$	15.1484	0.856237	0.428119	−	0.903723i	$-0.359177\pi$
$313$	15.1484	0.856237	0.428119	+	0.903723i	$0.359177\pi$
$314$	0	0
$315$	−1.68133	−0.0947322
$316$	0	0
$317$	−8.68133	−0.487592	−0.243796	−	0.969827i	$-0.578393\pi$
$317$	−8.68133	−0.487592	−0.243796	+	0.969827i	$0.578393\pi$
$318$	0	0
$319$	−8.37907	−0.469138
$320$	0	0
$321$	−20.6115	−1.15042
$322$	0	0
$323$	1.49702	0.0832967
$324$	0	0
$325$	0.0809744	0.00449165
$326$	0	0
$327$	−19.1976	−1.06163
$328$	0	0
$329$	4.85446	0.267635
$330$	0	0
$331$	−6.25410	−0.343757	−0.171878	−	0.985118i	$-0.554984\pi$
$331$	−6.25410	−0.343757	−0.171878	+	0.985118i	$0.554984\pi$
$332$	0	0
$333$	3.49180	0.191349
$334$	0	0
$335$	−12.2335	−0.668389
$336$	0	0
$337$	26.3051	1.43293	0.716465	−	0.697623i	$-0.245759\pi$
$337$	26.3051	1.43293	0.716465	+	0.697623i	$0.245759\pi$
$338$	0	0
$339$	37.9669	2.06208
$340$	0	0
$341$	3.31867	0.179716
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	33.5826	1.80803
$346$	0	0
$347$	8.08097	0.433809	0.216905	−	0.976193i	$-0.430404\pi$
$347$	8.08097	0.433809	0.216905	+	0.976193i	$0.430404\pi$
$348$	0	0
$349$	−33.6126	−1.79924	−0.899620	−	0.436673i	$-0.856157\pi$
$349$	−33.6126	−1.79924	−0.899620	+	0.436673i	$0.856157\pi$
$350$	0	0
$351$	4.36266	0.232862
$352$	0	0
$353$	−0.585009	−0.0311369	−0.0155684	−	0.999879i	$-0.504956\pi$
$353$	−0.585009	−0.0311369	−0.0155684	+	0.999879i	$0.504956\pi$
$354$	0	0
$355$	−5.37206	−0.285119
$356$	0	0
$357$	1.19370	0.0631775
$358$	0	0
$359$	−10.9201	−0.576340	−0.288170	−	0.957579i	$-0.593047\pi$
$359$	−10.9201	−0.576340	−0.288170	+	0.957579i	$0.593047\pi$
$360$	0	0
$361$	−13.1086	−0.689924
$362$	0	0
$363$	−1.93543	−0.101584
$364$	0	0
$365$	−11.2335	−0.587990
$366$	0	0
$367$	−18.5316	−0.967343	−0.483671	−	0.875250i	$-0.660697\pi$
$367$	−18.5316	−0.967343	−0.483671	+	0.875250i	$0.660697\pi$
$368$	0	0
$369$	−9.11273	−0.474390
$370$	0	0
$371$	−6.99583	−0.363205
$372$	0	0
$373$	−29.9505	−1.55078	−0.775389	−	0.631483i	$-0.782446\pi$
$373$	−29.9505	−1.55078	−0.775389	+	0.631483i	$0.782446\pi$
$374$	0	0
$375$	21.4600	1.10819
$376$	0	0
$377$	8.37907	0.431544
$378$	0	0
$379$	36.6524	1.88271	0.941354	−	0.337420i	$-0.109554\pi$
$379$	36.6524	1.88271	0.941354	+	0.337420i	$0.109554\pi$
$380$	0	0
$381$	−38.1156	−1.95272
$382$	0	0
$383$	−26.9641	−1.37780	−0.688900	−	0.724856i	$-0.741906\pi$
$383$	−26.9641	−1.37780	−0.688900	+	0.724856i	$0.741906\pi$
$384$	0	0
$385$	2.25410	0.114880
$386$	0	0
$387$	1.66909	0.0848448
$388$	0	0
$389$	−26.8175	−1.35970	−0.679850	−	0.733351i	$-0.737955\pi$
$389$	−26.8175	−1.35970	−0.679850	+	0.733351i	$0.737955\pi$
$390$	0	0
$391$	−4.74768	−0.240100
$392$	0	0
$393$	25.3627	1.27938
$394$	0	0
$395$	−20.9958	−1.05641
$396$	0	0
$397$	1.85863	0.0932818	0.0466409	−	0.998912i	$-0.485148\pi$
$397$	1.85863	0.0932818	0.0466409	+	0.998912i	$0.485148\pi$
$398$	0	0
$399$	4.69774	0.235181
$400$	0	0
$401$	26.2827	1.31250	0.656249	−	0.754545i	$-0.272142\pi$
$401$	26.2827	1.31250	0.656249	+	0.754545i	$0.272142\pi$
$402$	0	0
$403$	−3.31867	−0.165315
$404$	0	0
$405$	−24.0768	−1.19639
$406$	0	0
$407$	−4.68133	−0.232045
$408$	0	0
$409$	−32.9424	−1.62890	−0.814449	−	0.580235i	$-0.802961\pi$
$409$	−32.9424	−1.62890	−0.814449	+	0.580235i	$0.802961\pi$
$410$	0	0
$411$	−10.4642	−0.516161
$412$	0	0
$413$	−10.0440	−0.494233
$414$	0	0
$415$	−8.90785	−0.437269
$416$	0	0
$417$	2.63734	0.129151
$418$	0	0
$419$	28.5204	1.39331	0.696657	−	0.717404i	$-0.254670\pi$
$419$	28.5204	1.39331	0.696657	+	0.717404i	$0.254670\pi$
$420$	0	0
$421$	−17.4178	−0.848893	−0.424447	−	0.905453i	$-0.639531\pi$
$421$	−17.4178	−0.848893	−0.424447	+	0.905453i	$0.639531\pi$
$422$	0	0
$423$	−3.62093	−0.176056
$424$	0	0
$425$	0.0499420	0.00242254
$426$	0	0
$427$	−2.74590	−0.132883
$428$	0	0
$429$	1.93543	0.0934436
$430$	0	0
$431$	−7.45065	−0.358885	−0.179442	−	0.983768i	$-0.557429\pi$
$431$	−7.45065	−0.358885	−0.179442	+	0.983768i	$0.557429\pi$
$432$	0	0
$433$	−32.7058	−1.57174	−0.785870	−	0.618391i	$-0.787785\pi$
$433$	−32.7058	−1.57174	−0.785870	+	0.618391i	$0.787785\pi$
$434$	0	0
$435$	−36.5550	−1.75268
$436$	0	0
$437$	−18.6842	−0.893785
$438$	0	0
$439$	−17.2335	−0.822511	−0.411256	−	0.911520i	$-0.634910\pi$
$439$	−17.2335	−0.822511	−0.411256	+	0.911520i	$0.634910\pi$
$440$	0	0
$441$	0.745898	0.0355190
$442$	0	0
$443$	−30.3491	−1.44193	−0.720965	−	0.692972i	$-0.756301\pi$
$443$	−30.3491	−1.44193	−0.720965	+	0.692972i	$0.756301\pi$
$444$	0	0
$445$	15.5962	0.739330
$446$	0	0
$447$	−0.0932122	−0.00440879
$448$	0	0
$449$	−20.8656	−0.984710	−0.492355	−	0.870394i	$-0.663864\pi$
$449$	−20.8656	−0.984710	−0.492355	+	0.870394i	$0.663864\pi$
$450$	0	0
$451$	12.2171	0.575282
$452$	0	0
$453$	25.6126	1.20338
$454$	0	0
$455$	−2.25410	−0.105674
$456$	0	0
$457$	−21.3697	−0.999631	−0.499816	−	0.866132i	$-0.666599\pi$
$457$	−21.3697	−0.999631	−0.499816	+	0.866132i	$0.666599\pi$
$458$	0	0
$459$	2.69073	0.125592
$460$	0	0
$461$	27.2007	1.26686	0.633432	−	0.773799i	$-0.281646\pi$
$461$	27.2007	1.26686	0.633432	+	0.773799i	$0.281646\pi$
$462$	0	0
$463$	−33.8269	−1.57207	−0.786034	−	0.618183i	$-0.787869\pi$
$463$	−33.8269	−1.57207	−0.786034	+	0.618183i	$0.787869\pi$
$464$	0	0
$465$	14.4782	0.671412
$466$	0	0
$467$	13.6772	0.632904	0.316452	−	0.948609i	$-0.397508\pi$
$467$	13.6772	0.632904	0.316452	+	0.948609i	$0.397508\pi$
$468$	0	0
$469$	5.42723	0.250606
$470$	0	0
$471$	32.5194	1.49841
$472$	0	0
$473$	−2.23769	−0.102889
$474$	0	0
$475$	0.196543	0.00901803
$476$	0	0
$477$	5.21818	0.238924
$478$	0	0
$479$	0.339245	0.0155005	0.00775025	−	0.999970i	$-0.497533\pi$
$479$	0.339245	0.0155005	0.00775025	+	0.999970i	$0.497533\pi$
$480$	0	0
$481$	4.68133	0.213450
$482$	0	0
$483$	−14.8984	−0.677903
$484$	0	0
$485$	38.9453	1.76841
$486$	0	0
$487$	−20.8984	−0.946999	−0.473500	−	0.880794i	$-0.657009\pi$
$487$	−20.8984	−0.946999	−0.473500	+	0.880794i	$0.657009\pi$
$488$	0	0
$489$	9.55220	0.431965
$490$	0	0
$491$	18.6373	0.841091	0.420546	−	0.907271i	$-0.361839\pi$
$491$	18.6373	0.841091	0.420546	+	0.907271i	$0.361839\pi$
$492$	0	0
$493$	5.16790	0.232750
$494$	0	0
$495$	−1.68133	−0.0755702
$496$	0	0
$497$	2.38324	0.106903
$498$	0	0
$499$	−5.85552	−0.262129	−0.131064	−	0.991374i	$-0.541839\pi$
$499$	−5.85552	−0.262129	−0.131064	+	0.991374i	$0.541839\pi$
$500$	0	0
$501$	26.6360	1.19001
$502$	0	0
$503$	29.1372	1.29916	0.649582	−	0.760292i	$-0.274944\pi$
$503$	29.1372	1.29916	0.649582	+	0.760292i	$0.274944\pi$
$504$	0	0
$505$	−20.1414	−0.896279
$506$	0	0
$507$	−1.93543	−0.0859556
$508$	0	0
$509$	−37.0192	−1.64085	−0.820425	−	0.571755i	$-0.806263\pi$
$509$	−37.0192	−1.64085	−0.820425	+	0.571755i	$0.806263\pi$
$510$	0	0
$511$	4.98359	0.220461
$512$	0	0
$513$	10.5892	0.467524
$514$	0	0
$515$	−33.1924	−1.46263
$516$	0	0
$517$	4.85446	0.213499
$518$	0	0
$519$	2.98048	0.130829
$520$	0	0
$521$	13.7201	0.601088	0.300544	−	0.953768i	$-0.402832\pi$
$521$	13.7201	0.601088	0.300544	+	0.953768i	$0.402832\pi$
$522$	0	0
$523$	−22.4754	−0.982780	−0.491390	−	0.870940i	$-0.663511\pi$
$523$	−22.4754	−0.982780	−0.491390	+	0.870940i	$0.663511\pi$
$524$	0	0
$525$	0.156721	0.00683984
$526$	0	0
$527$	−2.04683	−0.0891614
$528$	0	0
$529$	36.2552	1.57631
$530$	0	0
$531$	7.49180	0.325116
$532$	0	0
$533$	−12.2171	−0.529182
$534$	0	0
$535$	24.0052	1.03784
$536$	0	0
$537$	−6.65686	−0.287265
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	2.51654	0.108195	0.0540973	−	0.998536i	$-0.482772\pi$
$541$	2.51654	0.108195	0.0540973	+	0.998536i	$0.482772\pi$
$542$	0	0
$543$	−9.98048	−0.428304
$544$	0	0
$545$	22.3585	0.957733
$546$	0	0
$547$	−33.1924	−1.41920	−0.709602	−	0.704603i	$-0.751125\pi$
$547$	−33.1924	−1.41920	−0.709602	+	0.704603i	$0.751125\pi$
$548$	0	0
$549$	2.04816	0.0874134
$550$	0	0
$551$	20.3379	0.866424
$552$	0	0
$553$	9.31450	0.396093
$554$	0	0
$555$	−20.4231	−0.866911
$556$	0	0
$557$	−14.3309	−0.607220	−0.303610	−	0.952796i	$-0.598192\pi$
$557$	−14.3309	−0.607220	−0.303610	+	0.952796i	$0.598192\pi$
$558$	0	0
$559$	2.23769	0.0946444
$560$	0	0
$561$	1.19370	0.0503982
$562$	0	0
$563$	28.8050	1.21398	0.606992	−	0.794708i	$-0.292376\pi$
$563$	28.8050	1.21398	0.606992	+	0.794708i	$0.292376\pi$
$564$	0	0
$565$	−44.2182	−1.86027
$566$	0	0
$567$	10.6813	0.448574
$568$	0	0
$569$	−24.8873	−1.04333	−0.521664	−	0.853151i	$-0.674689\pi$
$569$	−24.8873	−1.04333	−0.521664	+	0.853151i	$0.674689\pi$
$570$	0	0
$571$	8.17730	0.342209	0.171105	−	0.985253i	$-0.445266\pi$
$571$	8.17730	0.342209	0.171105	+	0.985253i	$0.445266\pi$
$572$	0	0
$573$	4.51938	0.188800
$574$	0	0
$575$	−0.623320	−0.0259942
$576$	0	0
$577$	−37.7100	−1.56989	−0.784943	−	0.619567i	$-0.787308\pi$
$577$	−37.7100	−1.56989	−0.784943	+	0.619567i	$0.787308\pi$
$578$	0	0
$579$	36.4517	1.51488
$580$	0	0
$581$	3.95184	0.163950
$582$	0	0
$583$	−6.99583	−0.289738
$584$	0	0
$585$	1.68133	0.0695145
$586$	0	0
$587$	−18.2499	−0.753255	−0.376628	−	0.926365i	$-0.622916\pi$
$587$	−18.2499	−0.753255	−0.376628	+	0.926365i	$0.622916\pi$
$588$	0	0
$589$	−8.05517	−0.331908
$590$	0	0
$591$	−7.63840	−0.314202
$592$	0	0
$593$	6.07965	0.249661	0.124831	−	0.992178i	$-0.460161\pi$
$593$	6.07965	0.249661	0.124831	+	0.992178i	$0.460161\pi$
$594$	0	0
$595$	−1.39025	−0.0569945
$596$	0	0
$597$	−39.5194	−1.61742
$598$	0	0
$599$	−42.7498	−1.74671	−0.873355	−	0.487085i	$-0.838060\pi$
$599$	−42.7498	−1.74671	−0.873355	+	0.487085i	$0.838060\pi$
$600$	0	0
$601$	19.8831	0.811049	0.405524	−	0.914084i	$-0.367089\pi$
$601$	19.8831	0.811049	0.405524	+	0.914084i	$0.367089\pi$
$602$	0	0
$603$	−4.04816	−0.164854
$604$	0	0
$605$	2.25410	0.0916423
$606$	0	0
$607$	1.68656	0.0684553	0.0342277	−	0.999414i	$-0.489103\pi$
$607$	1.68656	0.0684553	0.0342277	+	0.999414i	$0.489103\pi$
$608$	0	0
$609$	16.2171	0.657151
$610$	0	0
$611$	−4.85446	−0.196390
$612$	0	0
$613$	4.65970	0.188203	0.0941017	−	0.995563i	$-0.470002\pi$
$613$	4.65970	0.188203	0.0941017	+	0.995563i	$0.470002\pi$
$614$	0	0
$615$	53.2992	2.14923
$616$	0	0
$617$	−2.31344	−0.0931356	−0.0465678	−	0.998915i	$-0.514828\pi$
$617$	−2.31344	−0.0931356	−0.0465678	+	0.998915i	$0.514828\pi$
$618$	0	0
$619$	−13.2859	−0.534004	−0.267002	−	0.963696i	$-0.586033\pi$
$619$	−13.2859	−0.534004	−0.267002	+	0.963696i	$0.586033\pi$
$620$	0	0
$621$	−33.5826	−1.34762
$622$	0	0
$623$	−6.91903	−0.277205
$624$	0	0
$625$	−25.3983	−1.01593
$626$	0	0
$627$	4.69774	0.187610
$628$	0	0
$629$	2.88727	0.115123
$630$	0	0
$631$	−26.5868	−1.05840	−0.529202	−	0.848496i	$-0.677509\pi$
$631$	−26.5868	−1.05840	−0.529202	+	0.848496i	$0.677509\pi$
$632$	0	0
$633$	−7.21818	−0.286897
$634$	0	0
$635$	44.3913	1.76162
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	−1.77765	−0.0703228
$640$	0	0
$641$	−11.2346	−0.443739	−0.221870	−	0.975076i	$-0.571216\pi$
$641$	−11.2346	−0.443739	−0.221870	+	0.975076i	$0.571216\pi$
$642$	0	0
$643$	−8.96408	−0.353509	−0.176754	−	0.984255i	$-0.556560\pi$
$643$	−8.96408	−0.353509	−0.176754	+	0.984255i	$0.556560\pi$
$644$	0	0
$645$	−9.76231	−0.384390
$646$	0	0
$647$	−36.5522	−1.43702	−0.718508	−	0.695519i	$-0.755174\pi$
$647$	−36.5522	−1.43702	−0.718508	+	0.695519i	$0.755174\pi$
$648$	0	0
$649$	−10.0440	−0.394261
$650$	0	0
$651$	−6.42306	−0.251740
$652$	0	0
$653$	−42.9903	−1.68234	−0.841171	−	0.540769i	$-0.818133\pi$
$653$	−42.9903	−1.68234	−0.841171	+	0.540769i	$0.818133\pi$
$654$	0	0
$655$	−29.5386	−1.15417
$656$	0	0
$657$	−3.71725	−0.145024
$658$	0	0
$659$	22.3502	0.870638	0.435319	−	0.900276i	$-0.356636\pi$
$659$	22.3502	0.870638	0.435319	+	0.900276i	$0.356636\pi$
$660$	0	0
$661$	1.91619	0.0745310	0.0372655	−	0.999305i	$-0.488135\pi$
$661$	1.91619	0.0745310	0.0372655	+	0.999305i	$0.488135\pi$
$662$	0	0
$663$	−1.19370	−0.0463596
$664$	0	0
$665$	−5.47122	−0.212165
$666$	0	0
$667$	−64.4999	−2.49744
$668$	0	0
$669$	−52.5439	−2.03146
$670$	0	0
$671$	−2.74590	−0.106004
$672$	0	0
$673$	40.4259	1.55830	0.779152	−	0.626835i	$-0.215650\pi$
$673$	40.4259	1.55830	0.779152	+	0.626835i	$0.215650\pi$
$674$	0	0
$675$	0.353264	0.0135971
$676$	0	0
$677$	16.8422	0.647299	0.323650	−	0.946177i	$-0.395090\pi$
$677$	16.8422	0.647299	0.323650	+	0.946177i	$0.395090\pi$
$678$	0	0
$679$	−17.2775	−0.663050
$680$	0	0
$681$	42.7735	1.63908
$682$	0	0
$683$	7.92425	0.303213	0.151607	−	0.988441i	$-0.451555\pi$
$683$	7.92425	0.303213	0.151607	+	0.988441i	$0.451555\pi$
$684$	0	0
$685$	12.1871	0.465647
$686$	0	0
$687$	−22.7969	−0.869756
$688$	0	0
$689$	6.99583	0.266520
$690$	0	0
$691$	29.9477	1.13926	0.569632	−	0.821900i	$-0.307086\pi$
$691$	29.9477	1.13926	0.569632	+	0.821900i	$0.307086\pi$
$692$	0	0
$693$	0.745898	0.0283343
$694$	0	0
$695$	−3.07158	−0.116512
$696$	0	0
$697$	−7.53507	−0.285411
$698$	0	0
$699$	41.7662	1.57974
$700$	0	0
$701$	−35.1044	−1.32587	−0.662937	−	0.748675i	$-0.730690\pi$
$701$	−35.1044	−1.32587	−0.662937	+	0.748675i	$0.730690\pi$
$702$	0	0
$703$	11.3627	0.428551
$704$	0	0
$705$	21.1784	0.797623
$706$	0	0
$707$	8.93543	0.336052
$708$	0	0
$709$	−24.9969	−0.938778	−0.469389	−	0.882992i	$-0.655526\pi$
$709$	−24.9969	−0.938778	−0.469389	+	0.882992i	$0.655526\pi$
$710$	0	0
$711$	−6.94767	−0.260558
$712$	0	0
$713$	25.5462	0.956714
$714$	0	0
$715$	−2.25410	−0.0842986
$716$	0	0
$717$	10.8628	0.405679
$718$	0	0
$719$	27.0950	1.01047	0.505236	−	0.862981i	$-0.331405\pi$
$719$	27.0950	1.01047	0.505236	+	0.862981i	$0.331405\pi$
$720$	0	0
$721$	14.7253	0.548400
$722$	0	0
$723$	21.0081	0.781298
$724$	0	0
$725$	0.678490	0.0251985
$726$	0	0
$727$	−13.2070	−0.489821	−0.244910	−	0.969546i	$-0.578759\pi$
$727$	−13.2070	−0.489821	−0.244910	+	0.969546i	$0.578759\pi$
$728$	0	0
$729$	17.3637	0.643101
$730$	0	0
$731$	1.38013	0.0510459
$732$	0	0
$733$	41.6454	1.53821	0.769104	−	0.639124i	$-0.220703\pi$
$733$	41.6454	1.53821	0.769104	+	0.639124i	$0.220703\pi$
$734$	0	0
$735$	−4.36266	−0.160919
$736$	0	0
$737$	5.42723	0.199915
$738$	0	0
$739$	−10.9201	−0.401702	−0.200851	−	0.979622i	$-0.564371\pi$
$739$	−10.9201	−0.401702	−0.200851	+	0.979622i	$0.564371\pi$
$740$	0	0
$741$	−4.69774	−0.172576
$742$	0	0
$743$	−4.03281	−0.147950	−0.0739748	−	0.997260i	$-0.523568\pi$
$743$	−4.03281	−0.147950	−0.0739748	+	0.997260i	$0.523568\pi$
$744$	0	0
$745$	0.108560	0.00397732
$746$	0	0
$747$	−2.94767	−0.107850
$748$	0	0
$749$	−10.6496	−0.389127
$750$	0	0
$751$	6.38191	0.232879	0.116440	−	0.993198i	$-0.462852\pi$
$751$	6.38191	0.232879	0.116440	+	0.993198i	$0.462852\pi$
$752$	0	0
$753$	36.4200	1.32722
$754$	0	0
$755$	−29.8297	−1.08561
$756$	0	0
$757$	−39.1114	−1.42153	−0.710764	−	0.703431i	$-0.751651\pi$
$757$	−39.1114	−1.42153	−0.710764	+	0.703431i	$0.751651\pi$
$758$	0	0
$759$	−14.8984	−0.540779
$760$	0	0
$761$	−21.8625	−0.792516	−0.396258	−	0.918139i	$-0.629691\pi$
$761$	−21.8625	−0.792516	−0.396258	+	0.918139i	$0.629691\pi$
$762$	0	0
$763$	−9.91903	−0.359093
$764$	0	0
$765$	1.03698	0.0374922
$766$	0	0
$767$	10.0440	0.362668
$768$	0	0
$769$	−42.4364	−1.53029	−0.765147	−	0.643856i	$-0.777334\pi$
$769$	−42.4364	−1.53029	−0.765147	+	0.643856i	$0.777334\pi$
$770$	0	0
$771$	−47.6678	−1.71671
$772$	0	0
$773$	−6.20700	−0.223250	−0.111625	−	0.993750i	$-0.535606\pi$
$773$	−6.20700	−0.223250	−0.111625	+	0.993750i	$0.535606\pi$
$774$	0	0
$775$	−0.268727	−0.00965297
$776$	0	0
$777$	9.06040	0.325040
$778$	0	0
$779$	−29.6537	−1.06246
$780$	0	0
$781$	2.38324	0.0852789
$782$	0	0
$783$	36.5550	1.30637
$784$	0	0
$785$	−37.8737	−1.35177
$786$	0	0
$787$	24.3913	0.869456	0.434728	−	0.900562i	$-0.356844\pi$
$787$	24.3913	0.869456	0.434728	+	0.900562i	$0.356844\pi$
$788$	0	0
$789$	17.3574	0.617941
$790$	0	0
$791$	19.6168	0.697492
$792$	0	0
$793$	2.74590	0.0975097
$794$	0	0
$795$	−30.5204	−1.08245
$796$	0	0
$797$	32.6178	1.15538	0.577691	−	0.816255i	$-0.303954\pi$
$797$	32.6178	1.15538	0.577691	+	0.816255i	$0.303954\pi$
$798$	0	0
$799$	−2.99405	−0.105922
$800$	0	0
$801$	5.16089	0.182351
$802$	0	0
$803$	4.98359	0.175867
$804$	0	0
$805$	17.3515	0.611559
$806$	0	0
$807$	−55.5959	−1.95707
$808$	0	0
$809$	5.77242	0.202948	0.101474	−	0.994838i	$-0.467644\pi$
$809$	5.77242	0.202948	0.101474	+	0.994838i	$0.467644\pi$
$810$	0	0
$811$	18.2653	0.641381	0.320690	−	0.947184i	$-0.396085\pi$
$811$	18.2653	0.641381	0.320690	+	0.947184i	$0.396085\pi$
$812$	0	0
$813$	−42.7100	−1.49790
$814$	0	0
$815$	−11.1250	−0.389691
$816$	0	0
$817$	5.43140	0.190021
$818$	0	0
$819$	−0.745898	−0.0260638
$820$	0	0
$821$	−40.7927	−1.42368	−0.711838	−	0.702344i	$-0.752137\pi$
$821$	−40.7927	−1.42368	−0.711838	+	0.702344i	$0.752137\pi$
$822$	0	0
$823$	−41.0521	−1.43098	−0.715492	−	0.698620i	$-0.753798\pi$
$823$	−41.0521	−1.43098	−0.715492	+	0.698620i	$0.753798\pi$
$824$	0	0
$825$	0.156721	0.00545631
$826$	0	0
$827$	−1.52461	−0.0530159	−0.0265079	−	0.999649i	$-0.508439\pi$
$827$	−1.52461	−0.0530159	−0.0265079	+	0.999649i	$0.508439\pi$
$828$	0	0
$829$	25.4067	0.882410	0.441205	−	0.897406i	$-0.354551\pi$
$829$	25.4067	0.882410	0.441205	+	0.897406i	$0.354551\pi$
$830$	0	0
$831$	9.89534	0.343266
$832$	0	0
$833$	0.616763	0.0213696
$834$	0	0
$835$	−31.0216	−1.07355
$836$	0	0
$837$	−14.4782	−0.500441
$838$	0	0
$839$	28.1648	0.972357	0.486178	−	0.873860i	$-0.338391\pi$
$839$	28.1648	0.972357	0.486178	+	0.873860i	$0.338391\pi$
$840$	0	0
$841$	41.2088	1.42099
$842$	0	0
$843$	−29.0869	−1.00181
$844$	0	0
$845$	2.25410	0.0775435
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	52.5222	1.80256
$850$	0	0
$851$	−36.0357	−1.23529
$852$	0	0
$853$	8.78049	0.300638	0.150319	−	0.988638i	$-0.451970\pi$
$853$	8.78049	0.300638	0.150319	+	0.988638i	$0.451970\pi$
$854$	0	0
$855$	4.08097	0.139566
$856$	0	0
$857$	−41.5561	−1.41953	−0.709765	−	0.704439i	$-0.751199\pi$
$857$	−41.5561	−1.41953	−0.709765	+	0.704439i	$0.751199\pi$
$858$	0	0
$859$	−23.7941	−0.811843	−0.405921	−	0.913908i	$-0.633049\pi$
$859$	−23.7941	−0.811843	−0.405921	+	0.913908i	$0.633049\pi$
$860$	0	0
$861$	−23.6454	−0.805833
$862$	0	0
$863$	23.4835	0.799386	0.399693	−	0.916649i	$-0.369117\pi$
$863$	23.4835	0.799386	0.399693	+	0.916649i	$0.369117\pi$
$864$	0	0
$865$	−3.47122	−0.118025
$866$	0	0
$867$	32.1661	1.09242
$868$	0	0
$869$	9.31450	0.315973
$870$	0	0
$871$	−5.42723	−0.183895
$872$	0	0
$873$	12.8873	0.436168
$874$	0	0
$875$	11.0880	0.374842
$876$	0	0
$877$	−4.18252	−0.141234	−0.0706169	−	0.997504i	$-0.522497\pi$
$877$	−4.18252	−0.141234	−0.0706169	+	0.997504i	$0.522497\pi$
$878$	0	0
$879$	−28.8133	−0.971849
$880$	0	0
$881$	−35.1484	−1.18418	−0.592090	−	0.805872i	$-0.701697\pi$
$881$	−35.1484	−1.18418	−0.592090	+	0.805872i	$0.701697\pi$
$882$	0	0
$883$	5.58812	0.188055	0.0940276	−	0.995570i	$-0.470026\pi$
$883$	5.58812	0.188055	0.0940276	+	0.995570i	$0.470026\pi$
$884$	0	0
$885$	−43.8185	−1.47294
$886$	0	0
$887$	−14.2911	−0.479848	−0.239924	−	0.970792i	$-0.577122\pi$
$887$	−14.2911	−0.479848	−0.239924	+	0.970792i	$0.577122\pi$
$888$	0	0
$889$	−19.6936	−0.660501
$890$	0	0
$891$	10.6813	0.357838
$892$	0	0
$893$	−11.7829	−0.394299
$894$	0	0
$895$	7.75291	0.259151
$896$	0	0
$897$	14.8984	0.497445
$898$	0	0
$899$	−27.8074	−0.927427
$900$	0	0
$901$	4.31477	0.143746
$902$	0	0
$903$	4.33091	0.144124
$904$	0	0
$905$	11.6238	0.386387
$906$	0	0
$907$	54.2884	1.80262	0.901309	−	0.433177i	$-0.142608\pi$
$907$	54.2884	1.80262	0.901309	+	0.433177i	$0.142608\pi$
$908$	0	0
$909$	−6.66492	−0.221062
$910$	0	0
$911$	−47.0081	−1.55745	−0.778723	−	0.627367i	$-0.784132\pi$
$911$	−47.0081	−1.55745	−0.778723	+	0.627367i	$0.784132\pi$
$912$	0	0
$913$	3.95184	0.130787
$914$	0	0
$915$	−11.9794	−0.396028
$916$	0	0
$917$	13.1044	0.432745
$918$	0	0
$919$	53.8831	1.77744	0.888720	−	0.458451i	$-0.151595\pi$
$919$	53.8831	1.77744	0.888720	+	0.458451i	$0.151595\pi$
$920$	0	0
$921$	13.4764	0.444064
$922$	0	0
$923$	−2.38324	−0.0784452
$924$	0	0
$925$	0.379068	0.0124637
$926$	0	0
$927$	−10.9836	−0.360749
$928$	0	0
$929$	34.8891	1.14467	0.572336	−	0.820019i	$-0.306037\pi$
$929$	34.8891	1.14467	0.572336	+	0.820019i	$0.306037\pi$
$930$	0	0
$931$	2.42723	0.0795492
$932$	0	0
$933$	57.6563	1.88758
$934$	0	0
$935$	−1.39025	−0.0454659
$936$	0	0
$937$	18.0726	0.590407	0.295204	−	0.955434i	$-0.404613\pi$
$937$	18.0726	0.590407	0.295204	+	0.955434i	$0.404613\pi$
$938$	0	0
$939$	−29.3187	−0.956779
$940$	0	0
$941$	10.8873	0.354915	0.177457	−	0.984128i	$-0.443213\pi$
$941$	10.8873	0.354915	0.177457	+	0.984128i	$0.443213\pi$
$942$	0	0
$943$	94.0442	3.06250
$944$	0	0
$945$	−9.83388	−0.319896
$946$	0	0
$947$	45.2158	1.46932	0.734658	−	0.678438i	$-0.237343\pi$
$947$	45.2158	1.46932	0.734658	+	0.678438i	$0.237343\pi$
$948$	0	0
$949$	−4.98359	−0.161774
$950$	0	0
$951$	16.8021	0.544846
$952$	0	0
$953$	−47.2007	−1.52898	−0.764491	−	0.644635i	$-0.777009\pi$
$953$	−47.2007	−1.52898	−0.764491	+	0.644635i	$0.777009\pi$
$954$	0	0
$955$	−5.26350	−0.170323
$956$	0	0
$957$	16.2171	0.524225
$958$	0	0
$959$	−5.40665	−0.174590
$960$	0	0
$961$	−19.9864	−0.644724
$962$	0	0
$963$	7.94350	0.255976
$964$	0	0
$965$	−42.4535	−1.36663
$966$	0	0
$967$	43.5163	1.39939	0.699694	−	0.714442i	$-0.253319\pi$
$967$	43.5163	1.39939	0.699694	+	0.714442i	$0.253319\pi$
$968$	0	0
$969$	−2.89739	−0.0930776
$970$	0	0
$971$	43.1247	1.38394	0.691969	−	0.721928i	$-0.256744\pi$
$971$	43.1247	1.38394	0.691969	+	0.721928i	$0.256744\pi$
$972$	0	0
$973$	1.36266	0.0436849
$974$	0	0
$975$	−0.156721	−0.00501907
$976$	0	0
$977$	−35.4311	−1.13354	−0.566771	−	0.823875i	$-0.691808\pi$
$977$	−35.4311	−1.13354	−0.566771	+	0.823875i	$0.691808\pi$
$978$	0	0
$979$	−6.91903	−0.221133
$980$	0	0
$981$	7.39858	0.236219
$982$	0	0
$983$	−51.0192	−1.62726	−0.813631	−	0.581382i	$-0.802512\pi$
$983$	−51.0192	−1.62726	−0.813631	+	0.581382i	$0.802512\pi$
$984$	0	0
$985$	8.89606	0.283452
$986$	0	0
$987$	−9.39547	−0.299061
$988$	0	0
$989$	−17.2252	−0.547729
$990$	0	0
$991$	16.0000	0.508257	0.254128	−	0.967170i	$-0.418211\pi$
$991$	16.0000	0.508257	0.254128	+	0.967170i	$0.418211\pi$
$992$	0	0
$993$	12.1044	0.384121
$994$	0	0
$995$	46.0263	1.45913
$996$	0	0
$997$	−26.7240	−0.846357	−0.423179	−	0.906046i	$-0.639086\pi$
$997$	−26.7240	−0.846357	−0.423179	+	0.906046i	$0.639086\pi$
$998$	0	0
$999$	20.4231	0.646157

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.i.1.1	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.i.1.1	✓	3	1.1	even	1	trivial

Overview	Random
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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:	\( N \)	\(=\)	\( 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8008.a (trivial)

\(f(q)\)	\(=\)	\(q-1.93543 q^{3} +2.25410 q^{5} -1.00000 q^{7} +0.745898 q^{9} +O(q^{10})\)	\(q-1.93543 q^{3} +2.25410 q^{5} -1.00000 q^{7} +0.745898 q^{9} -1.00000 q^{11} +1.00000 q^{13} -4.36266 q^{15} +0.616763 q^{17} +2.42723 q^{19} +1.93543 q^{21} -7.69774 q^{23} +0.0809744 q^{25} +4.36266 q^{27} +8.37907 q^{29} -3.31867 q^{31} +1.93543 q^{33} -2.25410 q^{35} +4.68133 q^{37} -1.93543 q^{39} -12.2171 q^{41} +2.23769 q^{43} +1.68133 q^{45} -4.85446 q^{47} +1.00000 q^{49} -1.19370 q^{51} +6.99583 q^{53} -2.25410 q^{55} -4.69774 q^{57} +10.0440 q^{59} +2.74590 q^{61} -0.745898 q^{63} +2.25410 q^{65} -5.42723 q^{67} +14.8984 q^{69} -2.38324 q^{71} -4.98359 q^{73} -0.156721 q^{75} +1.00000 q^{77} -9.31450 q^{79} -10.6813 q^{81} -3.95184 q^{83} +1.39025 q^{85} -16.2171 q^{87} +6.91903 q^{89} -1.00000 q^{91} +6.42306 q^{93} +5.47122 q^{95} +17.2775 q^{97} -0.745898 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{3} + 6 q^{5} - 3 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q + 2 * q^3 + 6 * q^5 - 3 * q^7 + 3 * q^9	\( 3 q + 2 q^{3} + 6 q^{5} - 3 q^{7} + 3 q^{9} - 3 q^{11} + 3 q^{13} + q^{15} - 13 q^{17} + q^{19} - 2 q^{21} - 13 q^{23} + 5 q^{25} - q^{27} + 8 q^{29} - 17 q^{31} - 2 q^{33} - 6 q^{35} + 7 q^{37} + 2 q^{39} - 10 q^{41} + 9 q^{43} - 2 q^{45} - 2 q^{47} + 3 q^{49} - 27 q^{51} - 11 q^{53} - 6 q^{55} - 4 q^{57} + 9 q^{59} + 9 q^{61} - 3 q^{63} + 6 q^{65} - 10 q^{67} + 11 q^{69} - 22 q^{71} - 18 q^{73} + 2 q^{75} + 3 q^{77} - 3 q^{79} - 25 q^{81} - q^{83} - 28 q^{85} - 22 q^{87} + 16 q^{89} - 3 q^{91} - 19 q^{93} - 11 q^{95} + q^{97} - 3 q^{99}+O(q^{100}) \) 3 * q + 2 * q^3 + 6 * q^5 - 3 * q^7 + 3 * q^9 - 3 * q^11 + 3 * q^13 + q^15 - 13 * q^17 + q^19 - 2 * q^21 - 13 * q^23 + 5 * q^25 - q^27 + 8 * q^29 - 17 * q^31 - 2 * q^33 - 6 * q^35 + 7 * q^37 + 2 * q^39 - 10 * q^41 + 9 * q^43 - 2 * q^45 - 2 * q^47 + 3 * q^49 - 27 * q^51 - 11 * q^53 - 6 * q^55 - 4 * q^57 + 9 * q^59 + 9 * q^61 - 3 * q^63 + 6 * q^65 - 10 * q^67 + 11 * q^69 - 22 * q^71 - 18 * q^73 + 2 * q^75 + 3 * q^77 - 3 * q^79 - 25 * q^81 - q^83 - 28 * q^85 - 22 * q^87 + 16 * q^89 - 3 * q^91 - 19 * q^93 - 11 * q^95 + q^97 - 3 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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