LMFDB - Embedded newform 8008.2.a.p.1.3

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:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - x^{8} - 15x^{7} + 15x^{6} + 66x^{5} - 59x^{4} - 77x^{3} + 34x^{2} + 11x - 2 $ x^9 - x^8 - 15x^7 + 15x^6 + 66x^5 - 59x^4 - 77x^3 + 34x^2 + 11*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.00974$ 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)

$f(q)$	$=$	$q-1.00974 q^{3} +2.49743 q^{5} -1.00000 q^{7} -1.98043 q^{9} +O(q^{10})$	$q-1.00974 q^{3} +2.49743 q^{5} -1.00000 q^{7} -1.98043 q^{9} +1.00000 q^{11} +1.00000 q^{13} -2.52176 q^{15} -4.53427 q^{17} -0.365921 q^{19} +1.00974 q^{21} +2.18456 q^{23} +1.23718 q^{25} +5.02893 q^{27} +7.70380 q^{29} -6.21949 q^{31} -1.00974 q^{33} -2.49743 q^{35} -6.41650 q^{37} -1.00974 q^{39} -0.881703 q^{41} -2.18214 q^{43} -4.94598 q^{45} +13.5382 q^{47} +1.00000 q^{49} +4.57843 q^{51} -5.98388 q^{53} +2.49743 q^{55} +0.369485 q^{57} +11.1921 q^{59} -4.84106 q^{61} +1.98043 q^{63} +2.49743 q^{65} -0.801261 q^{67} -2.20584 q^{69} +10.5313 q^{71} -15.0437 q^{73} -1.24923 q^{75} -1.00000 q^{77} -10.8318 q^{79} +0.863362 q^{81} -11.4116 q^{83} -11.3240 q^{85} -7.77884 q^{87} +2.15030 q^{89} -1.00000 q^{91} +6.28007 q^{93} -0.913864 q^{95} -4.92908 q^{97} -1.98043 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + q^{3} - 4 q^{5} - 9 q^{7} + 4 q^{9}+O(q^{10}) $ 9 * q + q^3 - 4 * q^5 - 9 * q^7 + 4 * q^9	$ 9 q + q^{3} - 4 q^{5} - 9 q^{7} + 4 q^{9} + 9 q^{11} + 9 q^{13} - 9 q^{15} - 11 q^{17} + 10 q^{19} - q^{21} - 14 q^{23} - q^{25} - 5 q^{27} - 10 q^{29} + 5 q^{31} + q^{33} + 4 q^{35} - 16 q^{37} + q^{39} + 2 q^{41} + 4 q^{43} - 30 q^{45} + 9 q^{49} + 3 q^{51} - 23 q^{53} - 4 q^{55} + 14 q^{57} + 9 q^{59} - 14 q^{61} - 4 q^{63} - 4 q^{65} + 8 q^{67} - 26 q^{69} - 20 q^{71} - 23 q^{73} + 32 q^{75} - 9 q^{77} + 2 q^{79} - 11 q^{81} - 9 q^{83} - 3 q^{85} - 7 q^{87} - 6 q^{89} - 9 q^{91} - 19 q^{93} - 4 q^{95} - 3 q^{97} + 4 q^{99}+O(q^{100}) $ 9 * q + q^3 - 4 * q^5 - 9 * q^7 + 4 * q^9 + 9 * q^11 + 9 * q^13 - 9 * q^15 - 11 * q^17 + 10 * q^19 - q^21 - 14 * q^23 - q^25 - 5 * q^27 - 10 * q^29 + 5 * q^31 + q^33 + 4 * q^35 - 16 * q^37 + q^39 + 2 * q^41 + 4 * q^43 - 30 * q^45 + 9 * q^49 + 3 * q^51 - 23 * q^53 - 4 * q^55 + 14 * q^57 + 9 * q^59 - 14 * q^61 - 4 * q^63 - 4 * q^65 + 8 * q^67 - 26 * q^69 - 20 * q^71 - 23 * q^73 + 32 * q^75 - 9 * q^77 + 2 * q^79 - 11 * q^81 - 9 * q^83 - 3 * q^85 - 7 * q^87 - 6 * q^89 - 9 * q^91 - 19 * q^93 - 4 * q^95 - 3 * q^97 + 4 * 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
$3$	−1.00974	−0.582974	−0.291487	−	0.956575i	$-0.594150\pi$
$3$	−1.00974	−0.582974	−0.291487	+	0.956575i	$0.594150\pi$
$4$	0	0
$5$	2.49743	1.11689	0.558443	−	0.829543i	$-0.311399\pi$
$5$	2.49743	1.11689	0.558443	+	0.829543i	$0.311399\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−1.98043	−0.660142
$10$	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
$15$	−2.52176	−0.651115
$16$	0	0
$17$	−4.53427	−1.09972	−0.549861	−	0.835256i	$-0.685319\pi$
$17$	−4.53427	−1.09972	−0.549861	+	0.835256i	$0.685319\pi$
$18$	0	0
$19$	−0.365921	−0.0839481	−0.0419740	−	0.999119i	$-0.513365\pi$
$19$	−0.365921	−0.0839481	−0.0419740	+	0.999119i	$0.513365\pi$
$20$	0	0
$21$	1.00974	0.220343
$22$	0	0
$23$	2.18456	0.455513	0.227757	−	0.973718i	$-0.426861\pi$
$23$	2.18456	0.455513	0.227757	+	0.973718i	$0.426861\pi$
$24$	0	0
$25$	1.23718	0.247436
$26$	0	0
$27$	5.02893	0.967819
$28$	0	0
$29$	7.70380	1.43056	0.715280	−	0.698838i	$-0.246299\pi$
$29$	7.70380	1.43056	0.715280	+	0.698838i	$0.246299\pi$
$30$	0	0
$31$	−6.21949	−1.11705	−0.558527	−	0.829486i	$-0.688633\pi$
$31$	−6.21949	−1.11705	−0.558527	+	0.829486i	$0.688633\pi$
$32$	0	0
$33$	−1.00974	−0.175773
$34$	0	0
$35$	−2.49743	−0.422144
$36$	0	0
$37$	−6.41650	−1.05487	−0.527433	−	0.849596i	$-0.676846\pi$
$37$	−6.41650	−1.05487	−0.527433	+	0.849596i	$0.676846\pi$
$38$	0	0
$39$	−1.00974	−0.161688
$40$	0	0
$41$	−0.881703	−0.137699	−0.0688495	−	0.997627i	$-0.521933\pi$
$41$	−0.881703	−0.137699	−0.0688495	+	0.997627i	$0.521933\pi$
$42$	0	0
$43$	−2.18214	−0.332773	−0.166386	−	0.986061i	$-0.553210\pi$
$43$	−2.18214	−0.332773	−0.166386	+	0.986061i	$0.553210\pi$
$44$	0	0
$45$	−4.94598	−0.737304
$46$	0	0
$47$	13.5382	1.97475	0.987373	−	0.158415i	$-0.0506384\pi$
$47$	13.5382	1.97475	0.987373	+	0.158415i	$0.0506384\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	4.57843	0.641109
$52$	0	0
$53$	−5.98388	−0.821949	−0.410974	−	0.911647i	$-0.634811\pi$
$53$	−5.98388	−0.821949	−0.410974	+	0.911647i	$0.634811\pi$
$54$	0	0
$55$	2.49743	0.336754
$56$	0	0
$57$	0.369485	0.0489395
$58$	0	0
$59$	11.1921	1.45709	0.728544	−	0.684999i	$-0.240197\pi$
$59$	11.1921	1.45709	0.728544	+	0.684999i	$0.240197\pi$
$60$	0	0
$61$	−4.84106	−0.619835	−0.309917	−	0.950764i	$-0.600301\pi$
$61$	−4.84106	−0.619835	−0.309917	+	0.950764i	$0.600301\pi$
$62$	0	0
$63$	1.98043	0.249510
$64$	0	0
$65$	2.49743	0.309769
$66$	0	0
$67$	−0.801261	−0.0978897	−0.0489448	−	0.998801i	$-0.515586\pi$
$67$	−0.801261	−0.0978897	−0.0489448	+	0.998801i	$0.515586\pi$
$68$	0	0
$69$	−2.20584	−0.265552
$70$	0	0
$71$	10.5313	1.24984	0.624921	−	0.780688i	$-0.285131\pi$
$71$	10.5313	1.24984	0.624921	+	0.780688i	$0.285131\pi$
$72$	0	0
$73$	−15.0437	−1.76073	−0.880367	−	0.474294i	$-0.842703\pi$
$73$	−15.0437	−1.76073	−0.880367	+	0.474294i	$0.842703\pi$
$74$	0	0
$75$	−1.24923	−0.144249
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	−10.8318	−1.21868	−0.609339	−	0.792910i	$-0.708565\pi$
$79$	−10.8318	−1.21868	−0.609339	+	0.792910i	$0.708565\pi$
$80$	0	0
$81$	0.863362	0.0959291
$82$	0	0
$83$	−11.4116	−1.25259	−0.626294	−	0.779587i	$-0.715429\pi$
$83$	−11.4116	−1.25259	−0.626294	+	0.779587i	$0.715429\pi$
$84$	0	0
$85$	−11.3240	−1.22826
$86$	0	0
$87$	−7.77884	−0.833979
$88$	0	0
$89$	2.15030	0.227931	0.113965	−	0.993485i	$-0.463645\pi$
$89$	2.15030	0.227931	0.113965	+	0.993485i	$0.463645\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	6.28007	0.651213
$94$	0	0
$95$	−0.913864	−0.0937605
$96$	0	0
$97$	−4.92908	−0.500473	−0.250236	−	0.968185i	$-0.580508\pi$
$97$	−4.92908	−0.500473	−0.250236	+	0.968185i	$0.580508\pi$
$98$	0	0
$99$	−1.98043	−0.199040
$100$	0	0
$101$	0.302490	0.0300989	0.0150495	−	0.999887i	$-0.495209\pi$
$101$	0.302490	0.0300989	0.0150495	+	0.999887i	$0.495209\pi$
$102$	0	0
$103$	12.4841	1.23010	0.615048	−	0.788489i	$-0.289136\pi$
$103$	12.4841	1.23010	0.615048	+	0.788489i	$0.289136\pi$
$104$	0	0
$105$	2.52176	0.246099
$106$	0	0
$107$	−19.3796	−1.87350	−0.936750	−	0.349998i	$-0.886182\pi$
$107$	−19.3796	−1.87350	−0.936750	+	0.349998i	$0.886182\pi$
$108$	0	0
$109$	−10.6608	−1.02112	−0.510561	−	0.859842i	$-0.670562\pi$
$109$	−10.6608	−1.02112	−0.510561	+	0.859842i	$0.670562\pi$
$110$	0	0
$111$	6.47900	0.614959
$112$	0	0
$113$	2.85865	0.268919	0.134459	−	0.990919i	$-0.457070\pi$
$113$	2.85865	0.268919	0.134459	+	0.990919i	$0.457070\pi$
$114$	0	0
$115$	5.45581	0.508757
$116$	0	0
$117$	−1.98043	−0.183090
$118$	0	0
$119$	4.53427	0.415656
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0.890291	0.0802748
$124$	0	0
$125$	−9.39740	−0.840529
$126$	0	0
$127$	6.74502	0.598523	0.299262	−	0.954171i	$-0.403260\pi$
$127$	6.74502	0.598523	0.299262	+	0.954171i	$0.403260\pi$
$128$	0	0
$129$	2.20339	0.193998
$130$	0	0
$131$	4.19971	0.366930	0.183465	−	0.983026i	$-0.441269\pi$
$131$	4.19971	0.366930	0.183465	+	0.983026i	$0.441269\pi$
$132$	0	0
$133$	0.365921	0.0317294
$134$	0	0
$135$	12.5594	1.08094
$136$	0	0
$137$	−1.91501	−0.163610	−0.0818051	−	0.996648i	$-0.526068\pi$
$137$	−1.91501	−0.163610	−0.0818051	+	0.996648i	$0.526068\pi$
$138$	0	0
$139$	−10.7175	−0.909044	−0.454522	−	0.890735i	$-0.650190\pi$
$139$	−10.7175	−0.909044	−0.454522	+	0.890735i	$0.650190\pi$
$140$	0	0
$141$	−13.6700	−1.15122
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	19.2397	1.59777
$146$	0	0
$147$	−1.00974	−0.0832819
$148$	0	0
$149$	11.1801	0.915912	0.457956	−	0.888975i	$-0.348582\pi$
$149$	11.1801	0.915912	0.457956	+	0.888975i	$0.348582\pi$
$150$	0	0
$151$	−0.960122	−0.0781336	−0.0390668	−	0.999237i	$-0.512439\pi$
$151$	−0.960122	−0.0781336	−0.0390668	+	0.999237i	$0.512439\pi$
$152$	0	0
$153$	8.97978	0.725972
$154$	0	0
$155$	−15.5328	−1.24762
$156$	0	0
$157$	−15.2642	−1.21822	−0.609109	−	0.793086i	$-0.708473\pi$
$157$	−15.2642	−1.21822	−0.609109	+	0.793086i	$0.708473\pi$
$158$	0	0
$159$	6.04216	0.479174
$160$	0	0
$161$	−2.18456	−0.172168
$162$	0	0
$163$	−14.9718	−1.17268	−0.586341	−	0.810064i	$-0.699432\pi$
$163$	−14.9718	−1.17268	−0.586341	+	0.810064i	$0.699432\pi$
$164$	0	0
$165$	−2.52176	−0.196319
$166$	0	0
$167$	4.94970	0.383020	0.191510	−	0.981491i	$-0.438662\pi$
$167$	4.94970	0.383020	0.191510	+	0.981491i	$0.438662\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0.724680	0.0554176
$172$	0	0
$173$	−2.24942	−0.171020	−0.0855101	−	0.996337i	$-0.527252\pi$
$173$	−2.24942	−0.171020	−0.0855101	+	0.996337i	$0.527252\pi$
$174$	0	0
$175$	−1.23718	−0.0935221
$176$	0	0
$177$	−11.3011	−0.849444
$178$	0	0
$179$	−6.92785	−0.517812	−0.258906	−	0.965903i	$-0.583362\pi$
$179$	−6.92785	−0.517812	−0.258906	+	0.965903i	$0.583362\pi$
$180$	0	0
$181$	−5.74336	−0.426900	−0.213450	−	0.976954i	$-0.568470\pi$
$181$	−5.74336	−0.426900	−0.213450	+	0.976954i	$0.568470\pi$
$182$	0	0
$183$	4.88821	0.361347
$184$	0	0
$185$	−16.0248	−1.17817
$186$	0	0
$187$	−4.53427	−0.331579
$188$	0	0
$189$	−5.02893	−0.365801
$190$	0	0
$191$	−16.4151	−1.18775	−0.593877	−	0.804556i	$-0.702403\pi$
$191$	−16.4151	−1.18775	−0.593877	+	0.804556i	$0.702403\pi$
$192$	0	0
$193$	−11.7374	−0.844878	−0.422439	−	0.906391i	$-0.638826\pi$
$193$	−11.7374	−0.844878	−0.422439	+	0.906391i	$0.638826\pi$
$194$	0	0
$195$	−2.52176	−0.180587
$196$	0	0
$197$	7.58716	0.540563	0.270281	−	0.962781i	$-0.412883\pi$
$197$	7.58716	0.540563	0.270281	+	0.962781i	$0.412883\pi$
$198$	0	0
$199$	−9.79733	−0.694514	−0.347257	−	0.937770i	$-0.612887\pi$
$199$	−9.79733	−0.694514	−0.347257	+	0.937770i	$0.612887\pi$
$200$	0	0
$201$	0.809066	0.0570671
$202$	0	0
$203$	−7.70380	−0.540701
$204$	0	0
$205$	−2.20200	−0.153794
$206$	0	0
$207$	−4.32637	−0.300703
$208$	0	0
$209$	−0.365921	−0.0253113
$210$	0	0
$211$	26.3422	1.81347	0.906736	−	0.421698i	$-0.138566\pi$
$211$	26.3422	1.81347	0.906736	+	0.421698i	$0.138566\pi$
$212$	0	0
$213$	−10.6339	−0.728624
$214$	0	0
$215$	−5.44975	−0.371670
$216$	0	0
$217$	6.21949	0.422207
$218$	0	0
$219$	15.1902	1.02646
$220$	0	0
$221$	−4.53427	−0.305008
$222$	0	0
$223$	−12.3050	−0.824007	−0.412003	−	0.911182i	$-0.635171\pi$
$223$	−12.3050	−0.824007	−0.412003	+	0.911182i	$0.635171\pi$
$224$	0	0
$225$	−2.45014	−0.163343
$226$	0	0
$227$	1.54237	0.102371	0.0511855	−	0.998689i	$-0.483700\pi$
$227$	1.54237	0.102371	0.0511855	+	0.998689i	$0.483700\pi$
$228$	0	0
$229$	−17.9995	−1.18944	−0.594720	−	0.803933i	$-0.702737\pi$
$229$	−17.9995	−1.18944	−0.594720	+	0.803933i	$0.702737\pi$
$230$	0	0
$231$	1.00974	0.0664360
$232$	0	0
$233$	−15.9375	−1.04410	−0.522050	−	0.852915i	$-0.674833\pi$
$233$	−15.9375	−1.04410	−0.522050	+	0.852915i	$0.674833\pi$
$234$	0	0
$235$	33.8107	2.20557
$236$	0	0
$237$	10.9373	0.710457
$238$	0	0
$239$	18.3433	1.18653	0.593266	−	0.805006i	$-0.297838\pi$
$239$	18.3433	1.18653	0.593266	+	0.805006i	$0.297838\pi$
$240$	0	0
$241$	−8.61217	−0.554758	−0.277379	−	0.960761i	$-0.589466\pi$
$241$	−8.61217	−0.554758	−0.277379	+	0.960761i	$0.589466\pi$
$242$	0	0
$243$	−15.9586	−1.02374
$244$	0	0
$245$	2.49743	0.159555
$246$	0	0
$247$	−0.365921	−0.0232830
$248$	0	0
$249$	11.5228	0.730226
$250$	0	0
$251$	−11.4344	−0.721734	−0.360867	−	0.932617i	$-0.617519\pi$
$251$	−11.4344	−0.721734	−0.360867	+	0.932617i	$0.617519\pi$
$252$	0	0
$253$	2.18456	0.137342
$254$	0	0
$255$	11.4343	0.716046
$256$	0	0
$257$	−16.8208	−1.04925	−0.524625	−	0.851333i	$-0.675795\pi$
$257$	−16.8208	−1.04925	−0.524625	+	0.851333i	$0.675795\pi$
$258$	0	0
$259$	6.41650	0.398702
$260$	0	0
$261$	−15.2568	−0.944373
$262$	0	0
$263$	18.2677	1.12644	0.563218	−	0.826308i	$-0.309563\pi$
$263$	18.2677	1.12644	0.563218	+	0.826308i	$0.309563\pi$
$264$	0	0
$265$	−14.9443	−0.918023
$266$	0	0
$267$	−2.17124	−0.132878
$268$	0	0
$269$	−12.3169	−0.750975	−0.375487	−	0.926828i	$-0.622525\pi$
$269$	−12.3169	−0.750975	−0.375487	+	0.926828i	$0.622525\pi$
$270$	0	0
$271$	24.3006	1.47616	0.738079	−	0.674714i	$-0.235733\pi$
$271$	24.3006	1.47616	0.738079	+	0.674714i	$0.235733\pi$
$272$	0	0
$273$	1.00974	0.0611122
$274$	0	0
$275$	1.23718	0.0746048
$276$	0	0
$277$	−4.14292	−0.248924	−0.124462	−	0.992224i	$-0.539720\pi$
$277$	−4.14292	−0.248924	−0.124462	+	0.992224i	$0.539720\pi$
$278$	0	0
$279$	12.3172	0.737414
$280$	0	0
$281$	−31.2332	−1.86321	−0.931607	−	0.363467i	$-0.881593\pi$
$281$	−31.2332	−1.86321	−0.931607	+	0.363467i	$0.881593\pi$
$282$	0	0
$283$	16.6563	0.990112	0.495056	−	0.868861i	$-0.335148\pi$
$283$	16.6563	0.990112	0.495056	+	0.868861i	$0.335148\pi$
$284$	0	0
$285$	0.922765	0.0546599
$286$	0	0
$287$	0.881703	0.0520453
$288$	0	0
$289$	3.55959	0.209388
$290$	0	0
$291$	4.97709	0.291762
$292$	0	0
$293$	27.0128	1.57810	0.789052	−	0.614327i	$-0.210572\pi$
$293$	27.0128	1.57810	0.789052	+	0.614327i	$0.210572\pi$
$294$	0	0
$295$	27.9516	1.62740
$296$	0	0
$297$	5.02893	0.291808
$298$	0	0
$299$	2.18456	0.126337
$300$	0	0
$301$	2.18214	0.125776
$302$	0	0
$303$	−0.305437	−0.0175469
$304$	0	0
$305$	−12.0902	−0.692285
$306$	0	0
$307$	−15.7935	−0.901382	−0.450691	−	0.892680i	$-0.648822\pi$
$307$	−15.7935	−0.901382	−0.450691	+	0.892680i	$0.648822\pi$
$308$	0	0
$309$	−12.6057	−0.717114
$310$	0	0
$311$	17.3795	0.985501	0.492750	−	0.870171i	$-0.335992\pi$
$311$	17.3795	0.985501	0.492750	+	0.870171i	$0.335992\pi$
$312$	0	0
$313$	13.0958	0.740220	0.370110	−	0.928988i	$-0.379320\pi$
$313$	13.0958	0.740220	0.370110	+	0.928988i	$0.379320\pi$
$314$	0	0
$315$	4.94598	0.278675
$316$	0	0
$317$	33.9025	1.90416	0.952078	−	0.305857i	$-0.0989429\pi$
$317$	33.9025	1.90416	0.952078	+	0.305857i	$0.0989429\pi$
$318$	0	0
$319$	7.70380	0.431330
$320$	0	0
$321$	19.5684	1.09220
$322$	0	0
$323$	1.65918	0.0923195
$324$	0	0
$325$	1.23718	0.0686264
$326$	0	0
$327$	10.7647	0.595287
$328$	0	0
$329$	−13.5382	−0.746384
$330$	0	0
$331$	−17.4390	−0.958531	−0.479266	−	0.877670i	$-0.659097\pi$
$331$	−17.4390	−0.958531	−0.479266	+	0.877670i	$0.659097\pi$
$332$	0	0
$333$	12.7074	0.696362
$334$	0	0
$335$	−2.00110	−0.109332
$336$	0	0
$337$	−11.3030	−0.615711	−0.307856	−	0.951433i	$-0.599611\pi$
$337$	−11.3030	−0.615711	−0.307856	+	0.951433i	$0.599611\pi$
$338$	0	0
$339$	−2.88649	−0.156773
$340$	0	0
$341$	−6.21949	−0.336805
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−5.50895	−0.296592
$346$	0	0
$347$	−10.4541	−0.561205	−0.280603	−	0.959824i	$-0.590534\pi$
$347$	−10.4541	−0.561205	−0.280603	+	0.959824i	$0.590534\pi$
$348$	0	0
$349$	3.67327	0.196626	0.0983129	−	0.995156i	$-0.468655\pi$
$349$	3.67327	0.196626	0.0983129	+	0.995156i	$0.468655\pi$
$350$	0	0
$351$	5.02893	0.268425
$352$	0	0
$353$	5.58978	0.297514	0.148757	−	0.988874i	$-0.452473\pi$
$353$	5.58978	0.297514	0.148757	+	0.988874i	$0.452473\pi$
$354$	0	0
$355$	26.3014	1.39593
$356$	0	0
$357$	−4.57843	−0.242316
$358$	0	0
$359$	−18.9758	−1.00150	−0.500751	−	0.865591i	$-0.666943\pi$
$359$	−18.9758	−1.00150	−0.500751	+	0.865591i	$0.666943\pi$
$360$	0	0
$361$	−18.8661	−0.992953
$362$	0	0
$363$	−1.00974	−0.0529976
$364$	0	0
$365$	−37.5707	−1.96654
$366$	0	0
$367$	0.649035	0.0338794	0.0169397	−	0.999857i	$-0.494608\pi$
$367$	0.649035	0.0338794	0.0169397	+	0.999857i	$0.494608\pi$
$368$	0	0
$369$	1.74615	0.0909008
$370$	0	0
$371$	5.98388	0.310667
$372$	0	0
$373$	−6.93989	−0.359334	−0.179667	−	0.983727i	$-0.557502\pi$
$373$	−6.93989	−0.359334	−0.179667	+	0.983727i	$0.557502\pi$
$374$	0	0
$375$	9.48893	0.490006
$376$	0	0
$377$	7.70380	0.396766
$378$	0	0
$379$	−18.5005	−0.950309	−0.475155	−	0.879902i	$-0.657608\pi$
$379$	−18.5005	−0.950309	−0.475155	+	0.879902i	$0.657608\pi$
$380$	0	0
$381$	−6.81071	−0.348923
$382$	0	0
$383$	−11.2993	−0.577367	−0.288683	−	0.957425i	$-0.593217\pi$
$383$	−11.2993	−0.577367	−0.288683	+	0.957425i	$0.593217\pi$
$384$	0	0
$385$	−2.49743	−0.127281
$386$	0	0
$387$	4.32156	0.219677
$388$	0	0
$389$	15.0292	0.762012	0.381006	−	0.924572i	$-0.375578\pi$
$389$	15.0292	0.762012	0.381006	+	0.924572i	$0.375578\pi$
$390$	0	0
$391$	−9.90540	−0.500938
$392$	0	0
$393$	−4.24061	−0.213911
$394$	0	0
$395$	−27.0518	−1.36112
$396$	0	0
$397$	−9.95471	−0.499613	−0.249806	−	0.968296i	$-0.580367\pi$
$397$	−9.95471	−0.499613	−0.249806	+	0.968296i	$0.580367\pi$
$398$	0	0
$399$	−0.369485	−0.0184974
$400$	0	0
$401$	16.4612	0.822035	0.411017	−	0.911627i	$-0.365174\pi$
$401$	16.4612	0.822035	0.411017	+	0.911627i	$0.365174\pi$
$402$	0	0
$403$	−6.21949	−0.309815
$404$	0	0
$405$	2.15619	0.107142
$406$	0	0
$407$	−6.41650	−0.318054
$408$	0	0
$409$	−19.2681	−0.952747	−0.476373	−	0.879243i	$-0.658049\pi$
$409$	−19.2681	−0.952747	−0.476373	+	0.879243i	$0.658049\pi$
$410$	0	0
$411$	1.93366	0.0953804
$412$	0	0
$413$	−11.1921	−0.550728
$414$	0	0
$415$	−28.4998	−1.39900
$416$	0	0
$417$	10.8219	0.529949
$418$	0	0
$419$	5.68072	0.277522	0.138761	−	0.990326i	$-0.455688\pi$
$419$	5.68072	0.277522	0.138761	+	0.990326i	$0.455688\pi$
$420$	0	0
$421$	5.71031	0.278303	0.139152	−	0.990271i	$-0.455562\pi$
$421$	5.71031	0.278303	0.139152	+	0.990271i	$0.455562\pi$
$422$	0	0
$423$	−26.8113	−1.30361
$424$	0	0
$425$	−5.60971	−0.272111
$426$	0	0
$427$	4.84106	0.234275
$428$	0	0
$429$	−1.00974	−0.0487507
$430$	0	0
$431$	−20.3081	−0.978208	−0.489104	−	0.872226i	$-0.662676\pi$
$431$	−20.3081	−0.978208	−0.489104	+	0.872226i	$0.662676\pi$
$432$	0	0
$433$	9.63239	0.462903	0.231452	−	0.972846i	$-0.425653\pi$
$433$	9.63239	0.462903	0.231452	+	0.972846i	$0.425653\pi$
$434$	0	0
$435$	−19.4271	−0.931460
$436$	0	0
$437$	−0.799378	−0.0382395
$438$	0	0
$439$	39.3622	1.87865	0.939327	−	0.343024i	$-0.111451\pi$
$439$	39.3622	1.87865	0.939327	+	0.343024i	$0.111451\pi$
$440$	0	0
$441$	−1.98043	−0.0943060
$442$	0	0
$443$	−6.88885	−0.327299	−0.163650	−	0.986519i	$-0.552327\pi$
$443$	−6.88885	−0.327299	−0.163650	+	0.986519i	$0.552327\pi$
$444$	0	0
$445$	5.37022	0.254573
$446$	0	0
$447$	−11.2890	−0.533953
$448$	0	0
$449$	26.8660	1.26789	0.633943	−	0.773380i	$-0.281435\pi$
$449$	26.8660	1.26789	0.633943	+	0.773380i	$0.281435\pi$
$450$	0	0
$451$	−0.881703	−0.0415178
$452$	0	0
$453$	0.969473	0.0455498
$454$	0	0
$455$	−2.49743	−0.117082
$456$	0	0
$457$	−19.9684	−0.934082	−0.467041	−	0.884236i	$-0.654680\pi$
$457$	−19.9684	−0.934082	−0.467041	+	0.884236i	$0.654680\pi$
$458$	0	0
$459$	−22.8025	−1.06433
$460$	0	0
$461$	−35.3256	−1.64528	−0.822639	−	0.568563i	$-0.807499\pi$
$461$	−35.3256	−1.64528	−0.822639	+	0.568563i	$0.807499\pi$
$462$	0	0
$463$	−7.15838	−0.332678	−0.166339	−	0.986069i	$-0.553195\pi$
$463$	−7.15838	−0.332678	−0.166339	+	0.986069i	$0.553195\pi$
$464$	0	0
$465$	15.6841	0.727331
$466$	0	0
$467$	−40.7593	−1.88611	−0.943057	−	0.332632i	$-0.892063\pi$
$467$	−40.7593	−1.88611	−0.943057	+	0.332632i	$0.892063\pi$
$468$	0	0
$469$	0.801261	0.0369988
$470$	0	0
$471$	15.4129	0.710189
$472$	0	0
$473$	−2.18214	−0.100335
$474$	0	0
$475$	−0.452711	−0.0207718
$476$	0	0
$477$	11.8506	0.542603
$478$	0	0
$479$	−25.3387	−1.15776	−0.578878	−	0.815414i	$-0.696509\pi$
$479$	−25.3387	−1.15776	−0.578878	+	0.815414i	$0.696509\pi$
$480$	0	0
$481$	−6.41650	−0.292567
$482$	0	0
$483$	2.20584	0.100369
$484$	0	0
$485$	−12.3101	−0.558971
$486$	0	0
$487$	−4.73856	−0.214725	−0.107362	−	0.994220i	$-0.534240\pi$
$487$	−4.73856	−0.214725	−0.107362	+	0.994220i	$0.534240\pi$
$488$	0	0
$489$	15.1176	0.683643
$490$	0	0
$491$	12.7755	0.576551	0.288276	−	0.957547i	$-0.406918\pi$
$491$	12.7755	0.576551	0.288276	+	0.957547i	$0.406918\pi$
$492$	0	0
$493$	−34.9311	−1.57322
$494$	0	0
$495$	−4.94598	−0.222305
$496$	0	0
$497$	−10.5313	−0.472396
$498$	0	0
$499$	24.2772	1.08680	0.543398	−	0.839475i	$-0.317138\pi$
$499$	24.2772	1.08680	0.543398	+	0.839475i	$0.317138\pi$
$500$	0	0
$501$	−4.99791	−0.223290
$502$	0	0
$503$	34.5036	1.53844	0.769220	−	0.638984i	$-0.220645\pi$
$503$	34.5036	1.53844	0.769220	+	0.638984i	$0.220645\pi$
$504$	0	0
$505$	0.755450	0.0336171
$506$	0	0
$507$	−1.00974	−0.0448441
$508$	0	0
$509$	−29.3212	−1.29964	−0.649820	−	0.760089i	$-0.725156\pi$
$509$	−29.3212	−1.29964	−0.649820	+	0.760089i	$0.725156\pi$
$510$	0	0
$511$	15.0437	0.665495
$512$	0	0
$513$	−1.84019	−0.0812465
$514$	0	0
$515$	31.1783	1.37388
$516$	0	0
$517$	13.5382	0.595408
$518$	0	0
$519$	2.27133	0.0997002
$520$	0	0
$521$	11.7854	0.516328	0.258164	−	0.966101i	$-0.416883\pi$
$521$	11.7854	0.516328	0.258164	+	0.966101i	$0.416883\pi$
$522$	0	0
$523$	−26.6854	−1.16687	−0.583435	−	0.812160i	$-0.698292\pi$
$523$	−26.6854	−1.16687	−0.583435	+	0.812160i	$0.698292\pi$
$524$	0	0
$525$	1.24923	0.0545209
$526$	0	0
$527$	28.2009	1.22845
$528$	0	0
$529$	−18.2277	−0.792508
$530$	0	0
$531$	−22.1651	−0.961885
$532$	0	0
$533$	−0.881703	−0.0381908
$534$	0	0
$535$	−48.3994	−2.09249
$536$	0	0
$537$	6.99533	0.301871
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	−31.7426	−1.36472	−0.682360	−	0.731017i	$-0.739046\pi$
$541$	−31.7426	−1.36472	−0.682360	+	0.731017i	$0.739046\pi$
$542$	0	0
$543$	5.79930	0.248872
$544$	0	0
$545$	−26.6247	−1.14048
$546$	0	0
$547$	−2.11542	−0.0904488	−0.0452244	−	0.998977i	$-0.514400\pi$
$547$	−2.11542	−0.0904488	−0.0452244	+	0.998977i	$0.514400\pi$
$548$	0	0
$549$	9.58736	0.409179
$550$	0	0
$551$	−2.81898	−0.120093
$552$	0	0
$553$	10.8318	0.460617
$554$	0	0
$555$	16.1809	0.686840
$556$	0	0
$557$	11.5878	0.490992	0.245496	−	0.969398i	$-0.421049\pi$
$557$	11.5878	0.490992	0.245496	+	0.969398i	$0.421049\pi$
$558$	0	0
$559$	−2.18214	−0.0922946
$560$	0	0
$561$	4.57843	0.193302
$562$	0	0
$563$	23.4621	0.988811	0.494405	−	0.869231i	$-0.335386\pi$
$563$	23.4621	0.988811	0.494405	+	0.869231i	$0.335386\pi$
$564$	0	0
$565$	7.13928	0.300352
$566$	0	0
$567$	−0.863362	−0.0362578
$568$	0	0
$569$	−19.2729	−0.807963	−0.403982	−	0.914767i	$-0.632374\pi$
$569$	−19.2729	−0.807963	−0.403982	+	0.914767i	$0.632374\pi$
$570$	0	0
$571$	−4.71626	−0.197369	−0.0986847	−	0.995119i	$-0.531464\pi$
$571$	−4.71626	−0.197369	−0.0986847	+	0.995119i	$0.531464\pi$
$572$	0	0
$573$	16.5750	0.692429
$574$	0	0
$575$	2.70270	0.112710
$576$	0	0
$577$	1.74840	0.0727866	0.0363933	−	0.999338i	$-0.488413\pi$
$577$	1.74840	0.0727866	0.0363933	+	0.999338i	$0.488413\pi$
$578$	0	0
$579$	11.8517	0.492541
$580$	0	0
$581$	11.4116	0.473434
$582$	0	0
$583$	−5.98388	−0.247827
$584$	0	0
$585$	−4.94598	−0.204491
$586$	0	0
$587$	6.22227	0.256821	0.128410	−	0.991721i	$-0.459013\pi$
$587$	6.22227	0.256821	0.128410	+	0.991721i	$0.459013\pi$
$588$	0	0
$589$	2.27584	0.0937745
$590$	0	0
$591$	−7.66105	−0.315134
$592$	0	0
$593$	−39.2842	−1.61321	−0.806604	−	0.591092i	$-0.798697\pi$
$593$	−39.2842	−1.61321	−0.806604	+	0.591092i	$0.798697\pi$
$594$	0	0
$595$	11.3240	0.464240
$596$	0	0
$597$	9.89275	0.404883
$598$	0	0
$599$	15.9615	0.652169	0.326084	−	0.945341i	$-0.394271\pi$
$599$	15.9615	0.652169	0.326084	+	0.945341i	$0.394271\pi$
$600$	0	0
$601$	13.3564	0.544820	0.272410	−	0.962181i	$-0.412179\pi$
$601$	13.3564	0.544820	0.272410	+	0.962181i	$0.412179\pi$
$602$	0	0
$603$	1.58684	0.0646211
$604$	0	0
$605$	2.49743	0.101535
$606$	0	0
$607$	−44.2807	−1.79730	−0.898649	−	0.438669i	$-0.855450\pi$
$607$	−44.2807	−1.79730	−0.898649	+	0.438669i	$0.855450\pi$
$608$	0	0
$609$	7.77884	0.315214
$610$	0	0
$611$	13.5382	0.547696
$612$	0	0
$613$	10.6381	0.429670	0.214835	−	0.976650i	$-0.431079\pi$
$613$	10.6381	0.429670	0.214835	+	0.976650i	$0.431079\pi$
$614$	0	0
$615$	2.22344	0.0896579
$616$	0	0
$617$	−3.79544	−0.152799	−0.0763994	−	0.997077i	$-0.524342\pi$
$617$	−3.79544	−0.152799	−0.0763994	+	0.997077i	$0.524342\pi$
$618$	0	0
$619$	−33.1177	−1.33111	−0.665556	−	0.746348i	$-0.731806\pi$
$619$	−33.1177	−1.33111	−0.665556	+	0.746348i	$0.731806\pi$
$620$	0	0
$621$	10.9860	0.440854
$622$	0	0
$623$	−2.15030	−0.0861498
$624$	0	0
$625$	−29.6553	−1.18621
$626$	0	0
$627$	0.369485	0.0147558
$628$	0	0
$629$	29.0942	1.16006
$630$	0	0
$631$	−15.6084	−0.621360	−0.310680	−	0.950515i	$-0.600557\pi$
$631$	−15.6084	−0.621360	−0.310680	+	0.950515i	$0.600557\pi$
$632$	0	0
$633$	−26.5988	−1.05721
$634$	0	0
$635$	16.8452	0.668483
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	−20.8566	−0.825072
$640$	0	0
$641$	−11.1102	−0.438825	−0.219413	−	0.975632i	$-0.570414\pi$
$641$	−11.1102	−0.438825	−0.219413	+	0.975632i	$0.570414\pi$
$642$	0	0
$643$	−1.12970	−0.0445510	−0.0222755	−	0.999752i	$-0.507091\pi$
$643$	−1.12970	−0.0445510	−0.0222755	+	0.999752i	$0.507091\pi$
$644$	0	0
$645$	5.50283	0.216674
$646$	0	0
$647$	1.67659	0.0659134	0.0329567	−	0.999457i	$-0.489508\pi$
$647$	1.67659	0.0659134	0.0329567	+	0.999457i	$0.489508\pi$
$648$	0	0
$649$	11.1921	0.439329
$650$	0	0
$651$	−6.28007	−0.246135
$652$	0	0
$653$	40.4887	1.58445	0.792224	−	0.610231i	$-0.208923\pi$
$653$	40.4887	1.58445	0.792224	+	0.610231i	$0.208923\pi$
$654$	0	0
$655$	10.4885	0.409820
$656$	0	0
$657$	29.7929	1.16233
$658$	0	0
$659$	15.6442	0.609411	0.304706	−	0.952447i	$-0.401442\pi$
$659$	15.6442	0.609411	0.304706	+	0.952447i	$0.401442\pi$
$660$	0	0
$661$	8.38907	0.326297	0.163149	−	0.986602i	$-0.447835\pi$
$661$	8.38907	0.326297	0.163149	+	0.986602i	$0.447835\pi$
$662$	0	0
$663$	4.57843	0.177812
$664$	0	0
$665$	0.913864	0.0354381
$666$	0	0
$667$	16.8295	0.651639
$668$	0	0
$669$	12.4249	0.480374
$670$	0	0
$671$	−4.84106	−0.186887
$672$	0	0
$673$	−37.1583	−1.43235	−0.716173	−	0.697923i	$-0.754108\pi$
$673$	−37.1583	−1.43235	−0.716173	+	0.697923i	$0.754108\pi$
$674$	0	0
$675$	6.22170	0.239473
$676$	0	0
$677$	21.5564	0.828479	0.414239	−	0.910168i	$-0.364048\pi$
$677$	21.5564	0.828479	0.414239	+	0.910168i	$0.364048\pi$
$678$	0	0
$679$	4.92908	0.189161
$680$	0	0
$681$	−1.55740	−0.0596795
$682$	0	0
$683$	−10.5299	−0.402917	−0.201458	−	0.979497i	$-0.564568\pi$
$683$	−10.5299	−0.402917	−0.201458	+	0.979497i	$0.564568\pi$
$684$	0	0
$685$	−4.78261	−0.182734
$686$	0	0
$687$	18.1748	0.693412
$688$	0	0
$689$	−5.98388	−0.227968
$690$	0	0
$691$	52.3175	1.99025	0.995125	−	0.0986202i	$-0.0314429\pi$
$691$	52.3175	1.99025	0.995125	+	0.0986202i	$0.0314429\pi$
$692$	0	0
$693$	1.98043	0.0752301
$694$	0	0
$695$	−26.7662	−1.01530
$696$	0	0
$697$	3.99788	0.151431
$698$	0	0
$699$	16.0927	0.608683
$700$	0	0
$701$	−5.79701	−0.218950	−0.109475	−	0.993990i	$-0.534917\pi$
$701$	−5.79701	−0.218950	−0.109475	+	0.993990i	$0.534917\pi$
$702$	0	0
$703$	2.34793	0.0885540
$704$	0	0
$705$	−34.1400	−1.28579
$706$	0	0
$707$	−0.302490	−0.0113763
$708$	0	0
$709$	−27.5275	−1.03382	−0.516908	−	0.856041i	$-0.672917\pi$
$709$	−27.5275	−1.03382	−0.516908	+	0.856041i	$0.672917\pi$
$710$	0	0
$711$	21.4517	0.804500
$712$	0	0
$713$	−13.5869	−0.508833
$714$	0	0
$715$	2.49743	0.0933988
$716$	0	0
$717$	−18.5220	−0.691717
$718$	0	0
$719$	−23.2588	−0.867408	−0.433704	−	0.901055i	$-0.642794\pi$
$719$	−23.2588	−0.867408	−0.433704	+	0.901055i	$0.642794\pi$
$720$	0	0
$721$	−12.4841	−0.464933
$722$	0	0
$723$	8.69605	0.323410
$724$	0	0
$725$	9.53099	0.353972
$726$	0	0
$727$	−34.9978	−1.29800	−0.648999	−	0.760790i	$-0.724812\pi$
$727$	−34.9978	−1.29800	−0.648999	+	0.760790i	$0.724812\pi$
$728$	0	0
$729$	13.5239	0.500886
$730$	0	0
$731$	9.89440	0.365958
$732$	0	0
$733$	22.1423	0.817845	0.408922	−	0.912569i	$-0.365905\pi$
$733$	22.1423	0.817845	0.408922	+	0.912569i	$0.365905\pi$
$734$	0	0
$735$	−2.52176	−0.0930165
$736$	0	0
$737$	−0.801261	−0.0295148
$738$	0	0
$739$	−30.3036	−1.11474	−0.557368	−	0.830266i	$-0.688189\pi$
$739$	−30.3036	−1.11474	−0.557368	+	0.830266i	$0.688189\pi$
$740$	0	0
$741$	0.369485	0.0135734
$742$	0	0
$743$	−34.5396	−1.26714	−0.633568	−	0.773687i	$-0.718410\pi$
$743$	−34.5396	−1.26714	−0.633568	+	0.773687i	$0.718410\pi$
$744$	0	0
$745$	27.9217	1.02297
$746$	0	0
$747$	22.5999	0.826886
$748$	0	0
$749$	19.3796	0.708117
$750$	0	0
$751$	47.4273	1.73065	0.865323	−	0.501214i	$-0.167113\pi$
$751$	47.4273	1.73065	0.865323	+	0.501214i	$0.167113\pi$
$752$	0	0
$753$	11.5458	0.420752
$754$	0	0
$755$	−2.39784	−0.0872664
$756$	0	0
$757$	−6.64799	−0.241625	−0.120813	−	0.992675i	$-0.538550\pi$
$757$	−6.64799	−0.241625	−0.120813	+	0.992675i	$0.538550\pi$
$758$	0	0
$759$	−2.20584	−0.0800670
$760$	0	0
$761$	35.9591	1.30352	0.651758	−	0.758427i	$-0.274032\pi$
$761$	35.9591	1.30352	0.651758	+	0.758427i	$0.274032\pi$
$762$	0	0
$763$	10.6608	0.385948
$764$	0	0
$765$	22.4264	0.810829
$766$	0	0
$767$	11.1921	0.404124
$768$	0	0
$769$	−12.7818	−0.460924	−0.230462	−	0.973081i	$-0.574024\pi$
$769$	−12.7818	−0.460924	−0.230462	+	0.973081i	$0.574024\pi$
$770$	0	0
$771$	16.9846	0.611685
$772$	0	0
$773$	−30.0166	−1.07962	−0.539812	−	0.841786i	$-0.681505\pi$
$773$	−30.0166	−1.07962	−0.539812	+	0.841786i	$0.681505\pi$
$774$	0	0
$775$	−7.69464	−0.276400
$776$	0	0
$777$	−6.47900	−0.232433
$778$	0	0
$779$	0.322634	0.0115596
$780$	0	0
$781$	10.5313	0.376841
$782$	0	0
$783$	38.7419	1.38452
$784$	0	0
$785$	−38.1214	−1.36061
$786$	0	0
$787$	−35.4408	−1.26333	−0.631664	−	0.775242i	$-0.717628\pi$
$787$	−35.4408	−1.26333	−0.631664	+	0.775242i	$0.717628\pi$
$788$	0	0
$789$	−18.4457	−0.656683
$790$	0	0
$791$	−2.85865	−0.101642
$792$	0	0
$793$	−4.84106	−0.171911
$794$	0	0
$795$	15.0899	0.535183
$796$	0	0
$797$	43.5538	1.54275	0.771377	−	0.636379i	$-0.219568\pi$
$797$	43.5538	1.54275	0.771377	+	0.636379i	$0.219568\pi$
$798$	0	0
$799$	−61.3857	−2.17167
$800$	0	0
$801$	−4.25850	−0.150467
$802$	0	0
$803$	−15.0437	−0.530881
$804$	0	0
$805$	−5.45581	−0.192292
$806$	0	0
$807$	12.4369	0.437798
$808$	0	0
$809$	−28.7702	−1.01151	−0.505753	−	0.862678i	$-0.668785\pi$
$809$	−28.7702	−1.01151	−0.505753	+	0.862678i	$0.668785\pi$
$810$	0	0
$811$	6.53736	0.229558	0.114779	−	0.993391i	$-0.463384\pi$
$811$	6.53736	0.229558	0.114779	+	0.993391i	$0.463384\pi$
$812$	0	0
$813$	−24.5373	−0.860561
$814$	0	0
$815$	−37.3911	−1.30975
$816$	0	0
$817$	0.798491	0.0279356
$818$	0	0
$819$	1.98043	0.0692017
$820$	0	0
$821$	−54.3259	−1.89599	−0.947994	−	0.318287i	$-0.896892\pi$
$821$	−54.3259	−1.89599	−0.947994	+	0.318287i	$0.896892\pi$
$822$	0	0
$823$	−17.7051	−0.617161	−0.308580	−	0.951198i	$-0.599854\pi$
$823$	−17.7051	−0.617161	−0.308580	+	0.951198i	$0.599854\pi$
$824$	0	0
$825$	−1.24923	−0.0434926
$826$	0	0
$827$	5.06631	0.176173	0.0880865	−	0.996113i	$-0.471925\pi$
$827$	5.06631	0.176173	0.0880865	+	0.996113i	$0.471925\pi$
$828$	0	0
$829$	−39.1760	−1.36064	−0.680319	−	0.732916i	$-0.738159\pi$
$829$	−39.1760	−1.36064	−0.680319	+	0.732916i	$0.738159\pi$
$830$	0	0
$831$	4.18327	0.145116
$832$	0	0
$833$	−4.53427	−0.157103
$834$	0	0
$835$	12.3616	0.427790
$836$	0	0
$837$	−31.2774	−1.08111
$838$	0	0
$839$	14.3925	0.496884	0.248442	−	0.968647i	$-0.420082\pi$
$839$	14.3925	0.496884	0.248442	+	0.968647i	$0.420082\pi$
$840$	0	0
$841$	30.3486	1.04650
$842$	0	0
$843$	31.5374	1.08620
$844$	0	0
$845$	2.49743	0.0859144
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	−16.8185	−0.577209
$850$	0	0
$851$	−14.0173	−0.480506
$852$	0	0
$853$	−22.5823	−0.773202	−0.386601	−	0.922247i	$-0.626351\pi$
$853$	−22.5823	−0.773202	−0.386601	+	0.922247i	$0.626351\pi$
$854$	0	0
$855$	1.80984	0.0618952
$856$	0	0
$857$	−25.3719	−0.866688	−0.433344	−	0.901229i	$-0.642666\pi$
$857$	−25.3719	−0.866688	−0.433344	+	0.901229i	$0.642666\pi$
$858$	0	0
$859$	11.2873	0.385117	0.192558	−	0.981286i	$-0.438322\pi$
$859$	11.2873	0.385117	0.192558	+	0.981286i	$0.438322\pi$
$860$	0	0
$861$	−0.890291	−0.0303410
$862$	0	0
$863$	−48.2809	−1.64350	−0.821750	−	0.569848i	$-0.807002\pi$
$863$	−48.2809	−1.64350	−0.821750	+	0.569848i	$0.807002\pi$
$864$	0	0
$865$	−5.61778	−0.191010
$866$	0	0
$867$	−3.59426	−0.122068
$868$	0	0
$869$	−10.8318	−0.367445
$870$	0	0
$871$	−0.801261	−0.0271497
$872$	0	0
$873$	9.76168	0.330383
$874$	0	0
$875$	9.39740	0.317690
$876$	0	0
$877$	54.8479	1.85208	0.926040	−	0.377425i	$-0.123190\pi$
$877$	54.8479	1.85208	0.926040	+	0.377425i	$0.123190\pi$
$878$	0	0
$879$	−27.2759	−0.919993
$880$	0	0
$881$	17.6520	0.594710	0.297355	−	0.954767i	$-0.403896\pi$
$881$	17.6520	0.594710	0.297355	+	0.954767i	$0.403896\pi$
$882$	0	0
$883$	−47.7267	−1.60613	−0.803065	−	0.595891i	$-0.796799\pi$
$883$	−47.7267	−1.60613	−0.803065	+	0.595891i	$0.796799\pi$
$884$	0	0
$885$	−28.2238	−0.948733
$886$	0	0
$887$	13.0401	0.437843	0.218922	−	0.975742i	$-0.429746\pi$
$887$	13.0401	0.437843	0.218922	+	0.975742i	$0.429746\pi$
$888$	0	0
$889$	−6.74502	−0.226221
$890$	0	0
$891$	0.863362	0.0289237
$892$	0	0
$893$	−4.95390	−0.165776
$894$	0	0
$895$	−17.3019	−0.578337
$896$	0	0
$897$	−2.20584	−0.0736509
$898$	0	0
$899$	−47.9138	−1.59801
$900$	0	0
$901$	27.1325	0.903915
$902$	0	0
$903$	−2.20339	−0.0733243
$904$	0	0
$905$	−14.3437	−0.476799
$906$	0	0
$907$	−10.4071	−0.345563	−0.172782	−	0.984960i	$-0.555275\pi$
$907$	−10.4071	−0.345563	−0.172782	+	0.984960i	$0.555275\pi$
$908$	0	0
$909$	−0.599060	−0.0198696
$910$	0	0
$911$	8.38463	0.277795	0.138898	−	0.990307i	$-0.455644\pi$
$911$	8.38463	0.277795	0.138898	+	0.990307i	$0.455644\pi$
$912$	0	0
$913$	−11.4116	−0.377670
$914$	0	0
$915$	12.2080	0.403584
$916$	0	0
$917$	−4.19971	−0.138687
$918$	0	0
$919$	40.9995	1.35245	0.676224	−	0.736696i	$-0.263615\pi$
$919$	40.9995	1.35245	0.676224	+	0.736696i	$0.263615\pi$
$920$	0	0
$921$	15.9473	0.525482
$922$	0	0
$923$	10.5313	0.346644
$924$	0	0
$925$	−7.93837	−0.261012
$926$	0	0
$927$	−24.7239	−0.812038
$928$	0	0
$929$	−7.83579	−0.257084	−0.128542	−	0.991704i	$-0.541030\pi$
$929$	−7.83579	−0.257084	−0.128542	+	0.991704i	$0.541030\pi$
$930$	0	0
$931$	−0.365921	−0.0119926
$932$	0	0
$933$	−17.5488	−0.574521
$934$	0	0
$935$	−11.3240	−0.370336
$936$	0	0
$937$	−18.1617	−0.593317	−0.296659	−	0.954984i	$-0.595872\pi$
$937$	−18.1617	−0.593317	−0.296659	+	0.954984i	$0.595872\pi$
$938$	0	0
$939$	−13.2234	−0.431529
$940$	0	0
$941$	−31.5504	−1.02851	−0.514257	−	0.857636i	$-0.671932\pi$
$941$	−31.5504	−1.02851	−0.514257	+	0.857636i	$0.671932\pi$
$942$	0	0
$943$	−1.92614	−0.0627237
$944$	0	0
$945$	−12.5594	−0.408558
$946$	0	0
$947$	27.3514	0.888802	0.444401	−	0.895828i	$-0.353417\pi$
$947$	27.3514	0.888802	0.444401	+	0.895828i	$0.353417\pi$
$948$	0	0
$949$	−15.0437	−0.488339
$950$	0	0
$951$	−34.2327	−1.11007
$952$	0	0
$953$	12.2584	0.397088	0.198544	−	0.980092i	$-0.436379\pi$
$953$	12.2584	0.397088	0.198544	+	0.980092i	$0.436379\pi$
$954$	0	0
$955$	−40.9956	−1.32659
$956$	0	0
$957$	−7.77884	−0.251454
$958$	0	0
$959$	1.91501	0.0618388
$960$	0	0
$961$	7.68211	0.247810
$962$	0	0
$963$	38.3799	1.23678
$964$	0	0
$965$	−29.3134	−0.943633
$966$	0	0
$967$	11.2786	0.362697	0.181348	−	0.983419i	$-0.441954\pi$
$967$	11.2786	0.362697	0.181348	+	0.983419i	$0.441954\pi$
$968$	0	0
$969$	−1.67535	−0.0538198
$970$	0	0
$971$	40.8741	1.31171	0.655856	−	0.754886i	$-0.272308\pi$
$971$	40.8741	1.31171	0.655856	+	0.754886i	$0.272308\pi$
$972$	0	0
$973$	10.7175	0.343586
$974$	0	0
$975$	−1.24923	−0.0400074
$976$	0	0
$977$	38.3903	1.22821	0.614107	−	0.789223i	$-0.289516\pi$
$977$	38.3903	1.22821	0.614107	+	0.789223i	$0.289516\pi$
$978$	0	0
$979$	2.15030	0.0687238
$980$	0	0
$981$	21.1130	0.674085
$982$	0	0
$983$	15.4016	0.491233	0.245617	−	0.969367i	$-0.421010\pi$
$983$	15.4016	0.491233	0.245617	+	0.969367i	$0.421010\pi$
$984$	0	0
$985$	18.9484	0.603747
$986$	0	0
$987$	13.6700	0.435122
$988$	0	0
$989$	−4.76702	−0.151582
$990$	0	0
$991$	−19.0694	−0.605759	−0.302879	−	0.953029i	$-0.597948\pi$
$991$	−19.0694	−0.605759	−0.302879	+	0.953029i	$0.597948\pi$
$992$	0	0
$993$	17.6088	0.558798
$994$	0	0
$995$	−24.4682	−0.775694
$996$	0	0
$997$	62.7971	1.98881	0.994403	−	0.105656i	$-0.0336944\pi$
$997$	62.7971	1.98881	0.994403	+	0.105656i	$0.0336944\pi$
$998$	0	0
$999$	−32.2682	−1.02092

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.p.1.3	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.p.1.3	✓	9	1.1	even	1	trivial

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

\(f(q)\)	\(=\)	\(q-1.00974 q^{3} +2.49743 q^{5} -1.00000 q^{7} -1.98043 q^{9} +O(q^{10})\)	\(q-1.00974 q^{3} +2.49743 q^{5} -1.00000 q^{7} -1.98043 q^{9} +1.00000 q^{11} +1.00000 q^{13} -2.52176 q^{15} -4.53427 q^{17} -0.365921 q^{19} +1.00974 q^{21} +2.18456 q^{23} +1.23718 q^{25} +5.02893 q^{27} +7.70380 q^{29} -6.21949 q^{31} -1.00974 q^{33} -2.49743 q^{35} -6.41650 q^{37} -1.00974 q^{39} -0.881703 q^{41} -2.18214 q^{43} -4.94598 q^{45} +13.5382 q^{47} +1.00000 q^{49} +4.57843 q^{51} -5.98388 q^{53} +2.49743 q^{55} +0.369485 q^{57} +11.1921 q^{59} -4.84106 q^{61} +1.98043 q^{63} +2.49743 q^{65} -0.801261 q^{67} -2.20584 q^{69} +10.5313 q^{71} -15.0437 q^{73} -1.24923 q^{75} -1.00000 q^{77} -10.8318 q^{79} +0.863362 q^{81} -11.4116 q^{83} -11.3240 q^{85} -7.77884 q^{87} +2.15030 q^{89} -1.00000 q^{91} +6.28007 q^{93} -0.913864 q^{95} -4.92908 q^{97} -1.98043 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + q^{3} - 4 q^{5} - 9 q^{7} + 4 q^{9}+O(q^{10}) \) 9 * q + q^3 - 4 * q^5 - 9 * q^7 + 4 * q^9	\( 9 q + q^{3} - 4 q^{5} - 9 q^{7} + 4 q^{9} + 9 q^{11} + 9 q^{13} - 9 q^{15} - 11 q^{17} + 10 q^{19} - q^{21} - 14 q^{23} - q^{25} - 5 q^{27} - 10 q^{29} + 5 q^{31} + q^{33} + 4 q^{35} - 16 q^{37} + q^{39} + 2 q^{41} + 4 q^{43} - 30 q^{45} + 9 q^{49} + 3 q^{51} - 23 q^{53} - 4 q^{55} + 14 q^{57} + 9 q^{59} - 14 q^{61} - 4 q^{63} - 4 q^{65} + 8 q^{67} - 26 q^{69} - 20 q^{71} - 23 q^{73} + 32 q^{75} - 9 q^{77} + 2 q^{79} - 11 q^{81} - 9 q^{83} - 3 q^{85} - 7 q^{87} - 6 q^{89} - 9 q^{91} - 19 q^{93} - 4 q^{95} - 3 q^{97} + 4 q^{99}+O(q^{100}) \) 9 * q + q^3 - 4 * q^5 - 9 * q^7 + 4 * q^9 + 9 * q^11 + 9 * q^13 - 9 * q^15 - 11 * q^17 + 10 * q^19 - q^21 - 14 * q^23 - q^25 - 5 * q^27 - 10 * q^29 + 5 * q^31 + q^33 + 4 * q^35 - 16 * q^37 + q^39 + 2 * q^41 + 4 * q^43 - 30 * q^45 + 9 * q^49 + 3 * q^51 - 23 * q^53 - 4 * q^55 + 14 * q^57 + 9 * q^59 - 14 * q^61 - 4 * q^63 - 4 * q^65 + 8 * q^67 - 26 * q^69 - 20 * q^71 - 23 * q^73 + 32 * q^75 - 9 * q^77 + 2 * q^79 - 11 * q^81 - 9 * q^83 - 3 * q^85 - 7 * q^87 - 6 * q^89 - 9 * q^91 - 19 * q^93 - 4 * q^95 - 3 * q^97 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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