LMFDB - Embedded newform 8008.2.a.p.1.4

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			1.4
Root			$-0.333725$ 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)

$f(q)$	$=$	$q-0.333725 q^{3} +1.97301 q^{5} -1.00000 q^{7} -2.88863 q^{9} +O(q^{10})$	$q-0.333725 q^{3} +1.97301 q^{5} -1.00000 q^{7} -2.88863 q^{9} +1.00000 q^{11} +1.00000 q^{13} -0.658443 q^{15} +3.01194 q^{17} +7.23548 q^{19} +0.333725 q^{21} -5.73685 q^{23} -1.10722 q^{25} +1.96518 q^{27} -5.03516 q^{29} -7.65880 q^{31} -0.333725 q^{33} -1.97301 q^{35} +2.20318 q^{37} -0.333725 q^{39} -7.09291 q^{41} +8.24968 q^{43} -5.69930 q^{45} -6.28197 q^{47} +1.00000 q^{49} -1.00516 q^{51} -13.7493 q^{53} +1.97301 q^{55} -2.41466 q^{57} -11.6777 q^{59} -6.71777 q^{61} +2.88863 q^{63} +1.97301 q^{65} -0.389294 q^{67} +1.91453 q^{69} +1.03512 q^{71} +16.0040 q^{73} +0.369507 q^{75} -1.00000 q^{77} +9.98212 q^{79} +8.01005 q^{81} +4.15167 q^{83} +5.94259 q^{85} +1.68036 q^{87} +2.98890 q^{89} -1.00000 q^{91} +2.55593 q^{93} +14.2757 q^{95} +1.28773 q^{97} -2.88863 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−0.333725	−0.192676	−0.0963381	−	0.995349i	$-0.530713\pi$
$3$	−0.333725	−0.192676	−0.0963381	+	0.995349i	$0.530713\pi$
$4$	0	0
$5$	1.97301	0.882358	0.441179	−	0.897419i	$-0.354560\pi$
$5$	1.97301	0.882358	0.441179	+	0.897419i	$0.354560\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−2.88863	−0.962876
$10$	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
$15$	−0.658443	−0.170009
$16$	0	0
$17$	3.01194	0.730502	0.365251	−	0.930909i	$-0.380983\pi$
$17$	3.01194	0.730502	0.365251	+	0.930909i	$0.380983\pi$
$18$	0	0
$19$	7.23548	1.65993	0.829967	−	0.557813i	$-0.188359\pi$
$19$	7.23548	1.65993	0.829967	+	0.557813i	$0.188359\pi$
$20$	0	0
$21$	0.333725	0.0728247
$22$	0	0
$23$	−5.73685	−1.19622	−0.598108	−	0.801416i	$-0.704081\pi$
$23$	−5.73685	−1.19622	−0.598108	+	0.801416i	$0.704081\pi$
$24$	0	0
$25$	−1.10722	−0.221444
$26$	0	0
$27$	1.96518	0.378199
$28$	0	0
$29$	−5.03516	−0.935006	−0.467503	−	0.883992i	$-0.654846\pi$
$29$	−5.03516	−0.935006	−0.467503	+	0.883992i	$0.654846\pi$
$30$	0	0
$31$	−7.65880	−1.37556	−0.687781	−	0.725918i	$-0.741415\pi$
$31$	−7.65880	−1.37556	−0.687781	+	0.725918i	$0.741415\pi$
$32$	0	0
$33$	−0.333725	−0.0580940
$34$	0	0
$35$	−1.97301	−0.333500
$36$	0	0
$37$	2.20318	0.362200	0.181100	−	0.983465i	$-0.442034\pi$
$37$	2.20318	0.362200	0.181100	+	0.983465i	$0.442034\pi$
$38$	0	0
$39$	−0.333725	−0.0534387
$40$	0	0
$41$	−7.09291	−1.10773	−0.553863	−	0.832607i	$-0.686847\pi$
$41$	−7.09291	−1.10773	−0.553863	+	0.832607i	$0.686847\pi$
$42$	0	0
$43$	8.24968	1.25806	0.629032	−	0.777379i	$-0.283451\pi$
$43$	8.24968	1.25806	0.629032	+	0.777379i	$0.283451\pi$
$44$	0	0
$45$	−5.69930	−0.849601
$46$	0	0
$47$	−6.28197	−0.916320	−0.458160	−	0.888870i	$-0.651491\pi$
$47$	−6.28197	−0.916320	−0.458160	+	0.888870i	$0.651491\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−1.00516	−0.140750
$52$	0	0
$53$	−13.7493	−1.88861	−0.944303	−	0.329078i	$-0.893262\pi$
$53$	−13.7493	−1.88861	−0.944303	+	0.329078i	$0.893262\pi$
$54$	0	0
$55$	1.97301	0.266041
$56$	0	0
$57$	−2.41466	−0.319829
$58$	0	0
$59$	−11.6777	−1.52030	−0.760152	−	0.649745i	$-0.774876\pi$
$59$	−11.6777	−1.52030	−0.760152	+	0.649745i	$0.774876\pi$
$60$	0	0
$61$	−6.71777	−0.860122	−0.430061	−	0.902800i	$-0.641508\pi$
$61$	−6.71777	−0.860122	−0.430061	+	0.902800i	$0.641508\pi$
$62$	0	0
$63$	2.88863	0.363933
$64$	0	0
$65$	1.97301	0.244722
$66$	0	0
$67$	−0.389294	−0.0475598	−0.0237799	−	0.999717i	$-0.507570\pi$
$67$	−0.389294	−0.0475598	−0.0237799	+	0.999717i	$0.507570\pi$
$68$	0	0
$69$	1.91453	0.230482
$70$	0	0
$71$	1.03512	0.122846	0.0614230	−	0.998112i	$-0.480436\pi$
$71$	1.03512	0.122846	0.0614230	+	0.998112i	$0.480436\pi$
$72$	0	0
$73$	16.0040	1.87313	0.936563	−	0.350499i	$-0.113988\pi$
$73$	16.0040	1.87313	0.936563	+	0.350499i	$0.113988\pi$
$74$	0	0
$75$	0.369507	0.0426670
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	9.98212	1.12308	0.561538	−	0.827451i	$-0.310210\pi$
$79$	9.98212	1.12308	0.561538	+	0.827451i	$0.310210\pi$
$80$	0	0
$81$	8.01005	0.890006
$82$	0	0
$83$	4.15167	0.455706	0.227853	−	0.973696i	$-0.426830\pi$
$83$	4.15167	0.455706	0.227853	+	0.973696i	$0.426830\pi$
$84$	0	0
$85$	5.94259	0.644565
$86$	0	0
$87$	1.68036	0.180153
$88$	0	0
$89$	2.98890	0.316823	0.158412	−	0.987373i	$-0.449363\pi$
$89$	2.98890	0.316823	0.158412	+	0.987373i	$0.449363\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	2.55593	0.265038
$94$	0	0
$95$	14.2757	1.46466
$96$	0	0
$97$	1.28773	0.130749	0.0653745	−	0.997861i	$-0.479176\pi$
$97$	1.28773	0.130749	0.0653745	+	0.997861i	$0.479176\pi$
$98$	0	0
$99$	−2.88863	−0.290318
$100$	0	0
$101$	−19.2561	−1.91605	−0.958024	−	0.286687i	$-0.907446\pi$
$101$	−19.2561	−1.91605	−0.958024	+	0.286687i	$0.907446\pi$
$102$	0	0
$103$	2.72365	0.268369	0.134185	−	0.990956i	$-0.457158\pi$
$103$	2.72365	0.268369	0.134185	+	0.990956i	$0.457158\pi$
$104$	0	0
$105$	0.658443	0.0642575
$106$	0	0
$107$	9.05980	0.875844	0.437922	−	0.899013i	$-0.355715\pi$
$107$	9.05980	0.875844	0.437922	+	0.899013i	$0.355715\pi$
$108$	0	0
$109$	−3.35173	−0.321037	−0.160519	−	0.987033i	$-0.551317\pi$
$109$	−3.35173	−0.321037	−0.160519	+	0.987033i	$0.551317\pi$
$110$	0	0
$111$	−0.735255	−0.0697873
$112$	0	0
$113$	−6.41177	−0.603168	−0.301584	−	0.953440i	$-0.597515\pi$
$113$	−6.41177	−0.603168	−0.301584	+	0.953440i	$0.597515\pi$
$114$	0	0
$115$	−11.3189	−1.05549
$116$	0	0
$117$	−2.88863	−0.267054
$118$	0	0
$119$	−3.01194	−0.276104
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	2.36708	0.213433
$124$	0	0
$125$	−12.0496	−1.07775
$126$	0	0
$127$	8.47402	0.751948	0.375974	−	0.926630i	$-0.377308\pi$
$127$	8.47402	0.751948	0.375974	+	0.926630i	$0.377308\pi$
$128$	0	0
$129$	−2.75312	−0.242399
$130$	0	0
$131$	−2.86075	−0.249945	−0.124972	−	0.992160i	$-0.539884\pi$
$131$	−2.86075	−0.249945	−0.124972	+	0.992160i	$0.539884\pi$
$132$	0	0
$133$	−7.23548	−0.627396
$134$	0	0
$135$	3.87733	0.333707
$136$	0	0
$137$	21.4412	1.83184	0.915922	−	0.401356i	$-0.131461\pi$
$137$	21.4412	1.83184	0.915922	+	0.401356i	$0.131461\pi$
$138$	0	0
$139$	1.37318	0.116472	0.0582359	−	0.998303i	$-0.481452\pi$
$139$	1.37318	0.116472	0.0582359	+	0.998303i	$0.481452\pi$
$140$	0	0
$141$	2.09645	0.176553
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	−9.93444	−0.825010
$146$	0	0
$147$	−0.333725	−0.0275252
$148$	0	0
$149$	−21.4379	−1.75626	−0.878131	−	0.478419i	$-0.841210\pi$
$149$	−21.4379	−1.75626	−0.878131	+	0.478419i	$0.841210\pi$
$150$	0	0
$151$	7.30293	0.594304	0.297152	−	0.954830i	$-0.403963\pi$
$151$	7.30293	0.594304	0.297152	+	0.954830i	$0.403963\pi$
$152$	0	0
$153$	−8.70037	−0.703383
$154$	0	0
$155$	−15.1109	−1.21374
$156$	0	0
$157$	−20.3412	−1.62340	−0.811701	−	0.584073i	$-0.801458\pi$
$157$	−20.3412	−1.62340	−0.811701	+	0.584073i	$0.801458\pi$
$158$	0	0
$159$	4.58847	0.363889
$160$	0	0
$161$	5.73685	0.452127
$162$	0	0
$163$	−14.9727	−1.17275	−0.586377	−	0.810038i	$-0.699446\pi$
$163$	−14.9727	−1.17275	−0.586377	+	0.810038i	$0.699446\pi$
$164$	0	0
$165$	−0.658443	−0.0512597
$166$	0	0
$167$	−11.9193	−0.922346	−0.461173	−	0.887310i	$-0.652571\pi$
$167$	−11.9193	−0.922346	−0.461173	+	0.887310i	$0.652571\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−20.9006	−1.59831
$172$	0	0
$173$	20.7129	1.57477	0.787385	−	0.616461i	$-0.211434\pi$
$173$	20.7129	1.57477	0.787385	+	0.616461i	$0.211434\pi$
$174$	0	0
$175$	1.10722	0.0836980
$176$	0	0
$177$	3.89713	0.292926
$178$	0	0
$179$	19.3446	1.44589	0.722943	−	0.690908i	$-0.242789\pi$
$179$	19.3446	1.44589	0.722943	+	0.690908i	$0.242789\pi$
$180$	0	0
$181$	−9.53877	−0.709011	−0.354506	−	0.935054i	$-0.615351\pi$
$181$	−9.53877	−0.709011	−0.354506	+	0.935054i	$0.615351\pi$
$182$	0	0
$183$	2.24189	0.165725
$184$	0	0
$185$	4.34690	0.319590
$186$	0	0
$187$	3.01194	0.220255
$188$	0	0
$189$	−1.96518	−0.142946
$190$	0	0
$191$	−26.2877	−1.90211	−0.951054	−	0.309024i	$-0.899998\pi$
$191$	−26.2877	−1.90211	−0.951054	+	0.309024i	$0.899998\pi$
$192$	0	0
$193$	−0.0136444	−0.000982144 0	−0.000491072	−	1.00000i	$-0.500156\pi$
$193$	−0.0136444	−0.000982144 0	−0.000491072		1.00000i	$0.500156\pi$
$194$	0	0
$195$	−0.658443	−0.0471521
$196$	0	0
$197$	−7.35999	−0.524377	−0.262189	−	0.965017i	$-0.584444\pi$
$197$	−7.35999	−0.524377	−0.262189	+	0.965017i	$0.584444\pi$
$198$	0	0
$199$	−18.9107	−1.34054	−0.670271	−	0.742116i	$-0.733822\pi$
$199$	−18.9107	−1.34054	−0.670271	+	0.742116i	$0.733822\pi$
$200$	0	0
$201$	0.129917	0.00916364
$202$	0	0
$203$	5.03516	0.353399
$204$	0	0
$205$	−13.9944	−0.977412
$206$	0	0
$207$	16.5716	1.15181
$208$	0	0
$209$	7.23548	0.500489
$210$	0	0
$211$	20.7642	1.42947	0.714735	−	0.699396i	$-0.246547\pi$
$211$	20.7642	1.42947	0.714735	+	0.699396i	$0.246547\pi$
$212$	0	0
$213$	−0.345445	−0.0236695
$214$	0	0
$215$	16.2767	1.11006
$216$	0	0
$217$	7.65880	0.519913
$218$	0	0
$219$	−5.34093	−0.360907
$220$	0	0
$221$	3.01194	0.202605
$222$	0	0
$223$	8.06011	0.539745	0.269872	−	0.962896i	$-0.413018\pi$
$223$	8.06011	0.539745	0.269872	+	0.962896i	$0.413018\pi$
$224$	0	0
$225$	3.19835	0.213223
$226$	0	0
$227$	−8.28497	−0.549893	−0.274946	−	0.961460i	$-0.588660\pi$
$227$	−8.28497	−0.549893	−0.274946	+	0.961460i	$0.588660\pi$
$228$	0	0
$229$	−12.3828	−0.818276	−0.409138	−	0.912473i	$-0.634171\pi$
$229$	−12.3828	−0.818276	−0.409138	+	0.912473i	$0.634171\pi$
$230$	0	0
$231$	0.333725	0.0219575
$232$	0	0
$233$	−16.3046	−1.06815	−0.534076	−	0.845436i	$-0.679341\pi$
$233$	−16.3046	−1.06815	−0.534076	+	0.845436i	$0.679341\pi$
$234$	0	0
$235$	−12.3944	−0.808522
$236$	0	0
$237$	−3.33128	−0.216390
$238$	0	0
$239$	−7.93137	−0.513038	−0.256519	−	0.966539i	$-0.582576\pi$
$239$	−7.93137	−0.513038	−0.256519	+	0.966539i	$0.582576\pi$
$240$	0	0
$241$	2.94072	0.189429	0.0947143	−	0.995504i	$-0.469806\pi$
$241$	2.94072	0.189429	0.0947143	+	0.995504i	$0.469806\pi$
$242$	0	0
$243$	−8.56870	−0.549682
$244$	0	0
$245$	1.97301	0.126051
$246$	0	0
$247$	7.23548	0.460383
$248$	0	0
$249$	−1.38552	−0.0878036
$250$	0	0
$251$	3.12181	0.197047	0.0985236	−	0.995135i	$-0.468588\pi$
$251$	3.12181	0.197047	0.0985236	+	0.995135i	$0.468588\pi$
$252$	0	0
$253$	−5.73685	−0.360673
$254$	0	0
$255$	−1.98319	−0.124192
$256$	0	0
$257$	−13.9680	−0.871298	−0.435649	−	0.900117i	$-0.643481\pi$
$257$	−13.9680	−0.871298	−0.435649	+	0.900117i	$0.643481\pi$
$258$	0	0
$259$	−2.20318	−0.136899
$260$	0	0
$261$	14.5447	0.900294
$262$	0	0
$263$	4.99827	0.308207	0.154103	−	0.988055i	$-0.450751\pi$
$263$	4.99827	0.308207	0.154103	+	0.988055i	$0.450751\pi$
$264$	0	0
$265$	−27.1275	−1.66643
$266$	0	0
$267$	−0.997471	−0.0610442
$268$	0	0
$269$	15.7699	0.961507	0.480753	−	0.876856i	$-0.340363\pi$
$269$	15.7699	0.961507	0.480753	+	0.876856i	$0.340363\pi$
$270$	0	0
$271$	−16.8059	−1.02088	−0.510442	−	0.859912i	$-0.670518\pi$
$271$	−16.8059	−1.02088	−0.510442	+	0.859912i	$0.670518\pi$
$272$	0	0
$273$	0.333725	0.0201979
$274$	0	0
$275$	−1.10722	−0.0667679
$276$	0	0
$277$	−23.1821	−1.39288	−0.696438	−	0.717617i	$-0.745233\pi$
$277$	−23.1821	−1.39288	−0.696438	+	0.717617i	$0.745233\pi$
$278$	0	0
$279$	22.1234	1.32450
$280$	0	0
$281$	−27.3623	−1.63230	−0.816148	−	0.577843i	$-0.803895\pi$
$281$	−27.3623	−1.63230	−0.816148	+	0.577843i	$0.803895\pi$
$282$	0	0
$283$	−9.12121	−0.542199	−0.271100	−	0.962551i	$-0.587387\pi$
$283$	−9.12121	−0.542199	−0.271100	+	0.962551i	$0.587387\pi$
$284$	0	0
$285$	−4.76415	−0.282204
$286$	0	0
$287$	7.09291	0.418681
$288$	0	0
$289$	−7.92823	−0.466366
$290$	0	0
$291$	−0.429747	−0.0251922
$292$	0	0
$293$	−23.0401	−1.34602	−0.673008	−	0.739635i	$-0.734998\pi$
$293$	−23.0401	−1.34602	−0.673008	+	0.739635i	$0.734998\pi$
$294$	0	0
$295$	−23.0402	−1.34145
$296$	0	0
$297$	1.96518	0.114031
$298$	0	0
$299$	−5.73685	−0.331771
$300$	0	0
$301$	−8.24968	−0.475504
$302$	0	0
$303$	6.42622	0.369177
$304$	0	0
$305$	−13.2542	−0.758936
$306$	0	0
$307$	−17.5530	−1.00181	−0.500903	−	0.865504i	$-0.666999\pi$
$307$	−17.5530	−1.00181	−0.500903	+	0.865504i	$0.666999\pi$
$308$	0	0
$309$	−0.908950	−0.0517083
$310$	0	0
$311$	18.9725	1.07583	0.537916	−	0.842998i	$-0.319212\pi$
$311$	18.9725	1.07583	0.537916	+	0.842998i	$0.319212\pi$
$312$	0	0
$313$	−29.3098	−1.65669	−0.828343	−	0.560221i	$-0.810716\pi$
$313$	−29.3098	−1.65669	−0.828343	+	0.560221i	$0.810716\pi$
$314$	0	0
$315$	5.69930	0.321119
$316$	0	0
$317$	−12.8597	−0.722271	−0.361136	−	0.932513i	$-0.617611\pi$
$317$	−12.8597	−0.722271	−0.361136	+	0.932513i	$0.617611\pi$
$318$	0	0
$319$	−5.03516	−0.281915
$320$	0	0
$321$	−3.02348	−0.168754
$322$	0	0
$323$	21.7928	1.21259
$324$	0	0
$325$	−1.10722	−0.0614175
$326$	0	0
$327$	1.11856	0.0618563
$328$	0	0
$329$	6.28197	0.346336
$330$	0	0
$331$	1.89233	0.104012	0.0520059	−	0.998647i	$-0.483439\pi$
$331$	1.89233	0.104012	0.0520059	+	0.998647i	$0.483439\pi$
$332$	0	0
$333$	−6.36416	−0.348754
$334$	0	0
$335$	−0.768082	−0.0419648
$336$	0	0
$337$	−6.60409	−0.359748	−0.179874	−	0.983690i	$-0.557569\pi$
$337$	−6.60409	−0.359748	−0.179874	+	0.983690i	$0.557569\pi$
$338$	0	0
$339$	2.13977	0.116216
$340$	0	0
$341$	−7.65880	−0.414747
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	3.77739	0.203368
$346$	0	0
$347$	17.2070	0.923718	0.461859	−	0.886953i	$-0.347183\pi$
$347$	17.2070	0.923718	0.461859	+	0.886953i	$0.347183\pi$
$348$	0	0
$349$	13.5809	0.726970	0.363485	−	0.931600i	$-0.381587\pi$
$349$	13.5809	0.726970	0.363485	+	0.931600i	$0.381587\pi$
$350$	0	0
$351$	1.96518	0.104894
$352$	0	0
$353$	−35.7862	−1.90471	−0.952353	−	0.304998i	$-0.901344\pi$
$353$	−35.7862	−1.90471	−0.952353	+	0.304998i	$0.901344\pi$
$354$	0	0
$355$	2.04230	0.108394
$356$	0	0
$357$	1.00516	0.0531986
$358$	0	0
$359$	20.0878	1.06020	0.530098	−	0.847937i	$-0.322155\pi$
$359$	20.0878	1.06020	0.530098	+	0.847937i	$0.322155\pi$
$360$	0	0
$361$	33.3522	1.75538
$362$	0	0
$363$	−0.333725	−0.0175160
$364$	0	0
$365$	31.5761	1.65277
$366$	0	0
$367$	6.45117	0.336748	0.168374	−	0.985723i	$-0.446148\pi$
$367$	6.45117	0.336748	0.168374	+	0.985723i	$0.446148\pi$
$368$	0	0
$369$	20.4888	1.06660
$370$	0	0
$371$	13.7493	0.713826
$372$	0	0
$373$	24.7860	1.28337	0.641686	−	0.766967i	$-0.278235\pi$
$373$	24.7860	1.28337	0.641686	+	0.766967i	$0.278235\pi$
$374$	0	0
$375$	4.02126	0.207657
$376$	0	0
$377$	−5.03516	−0.259324
$378$	0	0
$379$	−8.93159	−0.458785	−0.229392	−	0.973334i	$-0.573674\pi$
$379$	−8.93159	−0.458785	−0.229392	+	0.973334i	$0.573674\pi$
$380$	0	0
$381$	−2.82799	−0.144882
$382$	0	0
$383$	13.9332	0.711952	0.355976	−	0.934495i	$-0.384148\pi$
$383$	13.9332	0.711952	0.355976	+	0.934495i	$0.384148\pi$
$384$	0	0
$385$	−1.97301	−0.100554
$386$	0	0
$387$	−23.8303	−1.21136
$388$	0	0
$389$	−15.9822	−0.810331	−0.405166	−	0.914243i	$-0.632786\pi$
$389$	−15.9822	−0.810331	−0.405166	+	0.914243i	$0.632786\pi$
$390$	0	0
$391$	−17.2790	−0.873839
$392$	0	0
$393$	0.954703	0.0481584
$394$	0	0
$395$	19.6949	0.990956
$396$	0	0
$397$	34.5270	1.73286	0.866431	−	0.499297i	$-0.166408\pi$
$397$	34.5270	1.73286	0.866431	+	0.499297i	$0.166408\pi$
$398$	0	0
$399$	2.41466	0.120884
$400$	0	0
$401$	13.5148	0.674896	0.337448	−	0.941344i	$-0.390436\pi$
$401$	13.5148	0.674896	0.337448	+	0.941344i	$0.390436\pi$
$402$	0	0
$403$	−7.65880	−0.381512
$404$	0	0
$405$	15.8039	0.785304
$406$	0	0
$407$	2.20318	0.109207
$408$	0	0
$409$	−23.4248	−1.15828	−0.579141	−	0.815227i	$-0.696612\pi$
$409$	−23.4248	−1.15828	−0.579141	+	0.815227i	$0.696612\pi$
$410$	0	0
$411$	−7.15546	−0.352953
$412$	0	0
$413$	11.6777	0.574621
$414$	0	0
$415$	8.19131	0.402095
$416$	0	0
$417$	−0.458265	−0.0224413
$418$	0	0
$419$	7.70241	0.376288	0.188144	−	0.982141i	$-0.439753\pi$
$419$	7.70241	0.376288	0.188144	+	0.982141i	$0.439753\pi$
$420$	0	0
$421$	−12.2520	−0.597127	−0.298563	−	0.954390i	$-0.596507\pi$
$421$	−12.2520	−0.597127	−0.298563	+	0.954390i	$0.596507\pi$
$422$	0	0
$423$	18.1463	0.882302
$424$	0	0
$425$	−3.33488	−0.161765
$426$	0	0
$427$	6.71777	0.325096
$428$	0	0
$429$	−0.333725	−0.0161124
$430$	0	0
$431$	−3.25226	−0.156656	−0.0783280	−	0.996928i	$-0.524958\pi$
$431$	−3.25226	−0.156656	−0.0783280	+	0.996928i	$0.524958\pi$
$432$	0	0
$433$	29.1913	1.40285	0.701423	−	0.712745i	$-0.252548\pi$
$433$	29.1913	1.40285	0.701423	+	0.712745i	$0.252548\pi$
$434$	0	0
$435$	3.31537	0.158960
$436$	0	0
$437$	−41.5089	−1.98564
$438$	0	0
$439$	22.5808	1.07772	0.538861	−	0.842395i	$-0.318855\pi$
$439$	22.5808	1.07772	0.538861	+	0.842395i	$0.318855\pi$
$440$	0	0
$441$	−2.88863	−0.137554
$442$	0	0
$443$	36.2552	1.72254	0.861268	−	0.508151i	$-0.169671\pi$
$443$	36.2552	1.72254	0.861268	+	0.508151i	$0.169671\pi$
$444$	0	0
$445$	5.89714	0.279551
$446$	0	0
$447$	7.15437	0.338390
$448$	0	0
$449$	−27.5209	−1.29879	−0.649397	−	0.760450i	$-0.724979\pi$
$449$	−27.5209	−1.29879	−0.649397	+	0.760450i	$0.724979\pi$
$450$	0	0
$451$	−7.09291	−0.333992
$452$	0	0
$453$	−2.43717	−0.114508
$454$	0	0
$455$	−1.97301	−0.0924963
$456$	0	0
$457$	−20.0616	−0.938442	−0.469221	−	0.883081i	$-0.655465\pi$
$457$	−20.0616	−0.938442	−0.469221	+	0.883081i	$0.655465\pi$
$458$	0	0
$459$	5.91901	0.276275
$460$	0	0
$461$	25.6492	1.19460	0.597300	−	0.802018i	$-0.296240\pi$
$461$	25.6492	1.19460	0.597300	+	0.802018i	$0.296240\pi$
$462$	0	0
$463$	−18.3062	−0.850763	−0.425381	−	0.905014i	$-0.639860\pi$
$463$	−18.3062	−0.850763	−0.425381	+	0.905014i	$0.639860\pi$
$464$	0	0
$465$	5.04289	0.233858
$466$	0	0
$467$	−14.3676	−0.664851	−0.332426	−	0.943129i	$-0.607867\pi$
$467$	−14.3676	−0.664851	−0.332426	+	0.943129i	$0.607867\pi$
$468$	0	0
$469$	0.389294	0.0179759
$470$	0	0
$471$	6.78836	0.312791
$472$	0	0
$473$	8.24968	0.379321
$474$	0	0
$475$	−8.01127	−0.367582
$476$	0	0
$477$	39.7165	1.81849
$478$	0	0
$479$	16.0193	0.731939	0.365970	−	0.930627i	$-0.380737\pi$
$479$	16.0193	0.731939	0.365970	+	0.930627i	$0.380737\pi$
$480$	0	0
$481$	2.20318	0.100456
$482$	0	0
$483$	−1.91453	−0.0871141
$484$	0	0
$485$	2.54071	0.115367
$486$	0	0
$487$	−8.68225	−0.393430	−0.196715	−	0.980461i	$-0.563027\pi$
$487$	−8.68225	−0.393430	−0.196715	+	0.980461i	$0.563027\pi$
$488$	0	0
$489$	4.99677	0.225962
$490$	0	0
$491$	−12.1332	−0.547565	−0.273783	−	0.961792i	$-0.588275\pi$
$491$	−12.1332	−0.547565	−0.273783	+	0.961792i	$0.588275\pi$
$492$	0	0
$493$	−15.1656	−0.683024
$494$	0	0
$495$	−5.69930	−0.256164
$496$	0	0
$497$	−1.03512	−0.0464315
$498$	0	0
$499$	−33.0619	−1.48006	−0.740028	−	0.672576i	$-0.765188\pi$
$499$	−33.0619	−1.48006	−0.740028	+	0.672576i	$0.765188\pi$
$500$	0	0
$501$	3.97778	0.177714
$502$	0	0
$503$	−34.5205	−1.53920	−0.769598	−	0.638529i	$-0.779543\pi$
$503$	−34.5205	−1.53920	−0.769598	+	0.638529i	$0.779543\pi$
$504$	0	0
$505$	−37.9924	−1.69064
$506$	0	0
$507$	−0.333725	−0.0148212
$508$	0	0
$509$	26.7308	1.18482	0.592411	−	0.805636i	$-0.298176\pi$
$509$	26.7308	1.18482	0.592411	+	0.805636i	$0.298176\pi$
$510$	0	0
$511$	−16.0040	−0.707975
$512$	0	0
$513$	14.2190	0.627786
$514$	0	0
$515$	5.37380	0.236798
$516$	0	0
$517$	−6.28197	−0.276281
$518$	0	0
$519$	−6.91240	−0.303421
$520$	0	0
$521$	−11.5034	−0.503973	−0.251987	−	0.967731i	$-0.581084\pi$
$521$	−11.5034	−0.503973	−0.251987	+	0.967731i	$0.581084\pi$
$522$	0	0
$523$	35.6213	1.55761	0.778805	−	0.627267i	$-0.215826\pi$
$523$	35.6213	1.55761	0.778805	+	0.627267i	$0.215826\pi$
$524$	0	0
$525$	−0.369507	−0.0161266
$526$	0	0
$527$	−23.0678	−1.00485
$528$	0	0
$529$	9.91144	0.430932
$530$	0	0
$531$	33.7325	1.46386
$532$	0	0
$533$	−7.09291	−0.307228
$534$	0	0
$535$	17.8751	0.772808
$536$	0	0
$537$	−6.45578	−0.278588
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	30.8895	1.32804	0.664021	−	0.747714i	$-0.268849\pi$
$541$	30.8895	1.32804	0.664021	+	0.747714i	$0.268849\pi$
$542$	0	0
$543$	3.18332	0.136609
$544$	0	0
$545$	−6.61301	−0.283270
$546$	0	0
$547$	−2.17472	−0.0929841	−0.0464921	−	0.998919i	$-0.514804\pi$
$547$	−2.17472	−0.0929841	−0.0464921	+	0.998919i	$0.514804\pi$
$548$	0	0
$549$	19.4051	0.828191
$550$	0	0
$551$	−36.4318	−1.55205
$552$	0	0
$553$	−9.98212	−0.424483
$554$	0	0
$555$	−1.45067	−0.0615774
$556$	0	0
$557$	−27.2411	−1.15424	−0.577122	−	0.816658i	$-0.695824\pi$
$557$	−27.2411	−1.15424	−0.577122	+	0.816658i	$0.695824\pi$
$558$	0	0
$559$	8.24968	0.348924
$560$	0	0
$561$	−1.00516	−0.0424378
$562$	0	0
$563$	−16.2803	−0.686131	−0.343066	−	0.939311i	$-0.611465\pi$
$563$	−16.2803	−0.686131	−0.343066	+	0.939311i	$0.611465\pi$
$564$	0	0
$565$	−12.6505	−0.532210
$566$	0	0
$567$	−8.01005	−0.336391
$568$	0	0
$569$	7.80361	0.327144	0.163572	−	0.986531i	$-0.447698\pi$
$569$	7.80361	0.327144	0.163572	+	0.986531i	$0.447698\pi$
$570$	0	0
$571$	−15.4786	−0.647760	−0.323880	−	0.946098i	$-0.604987\pi$
$571$	−15.4786	−0.647760	−0.323880	+	0.946098i	$0.604987\pi$
$572$	0	0
$573$	8.77285	0.366491
$574$	0	0
$575$	6.35195	0.264895
$576$	0	0
$577$	6.80388	0.283249	0.141625	−	0.989920i	$-0.454767\pi$
$577$	6.80388	0.283249	0.141625	+	0.989920i	$0.454767\pi$
$578$	0	0
$579$	0.00455347	0.000189236 0
$580$	0	0
$581$	−4.15167	−0.172240
$582$	0	0
$583$	−13.7493	−0.569436
$584$	0	0
$585$	−5.69930	−0.235637
$586$	0	0
$587$	−5.50993	−0.227419	−0.113710	−	0.993514i	$-0.536273\pi$
$587$	−5.50993	−0.227419	−0.113710	+	0.993514i	$0.536273\pi$
$588$	0	0
$589$	−55.4151	−2.28334
$590$	0	0
$591$	2.45621	0.101035
$592$	0	0
$593$	2.55857	0.105068	0.0525339	−	0.998619i	$-0.483270\pi$
$593$	2.55857	0.105068	0.0525339	+	0.998619i	$0.483270\pi$
$594$	0	0
$595$	−5.94259	−0.243623
$596$	0	0
$597$	6.31096	0.258290
$598$	0	0
$599$	−6.57756	−0.268752	−0.134376	−	0.990930i	$-0.542903\pi$
$599$	−6.57756	−0.268752	−0.134376	+	0.990930i	$0.542903\pi$
$600$	0	0
$601$	26.7139	1.08968	0.544841	−	0.838540i	$-0.316590\pi$
$601$	26.7139	1.08968	0.544841	+	0.838540i	$0.316590\pi$
$602$	0	0
$603$	1.12453	0.0457942
$604$	0	0
$605$	1.97301	0.0802144
$606$	0	0
$607$	12.9126	0.524106	0.262053	−	0.965053i	$-0.415600\pi$
$607$	12.9126	0.524106	0.262053	+	0.965053i	$0.415600\pi$
$608$	0	0
$609$	−1.68036	−0.0680915
$610$	0	0
$611$	−6.28197	−0.254141
$612$	0	0
$613$	−41.1308	−1.66126	−0.830630	−	0.556825i	$-0.812019\pi$
$613$	−41.1308	−1.66126	−0.830630	+	0.556825i	$0.812019\pi$
$614$	0	0
$615$	4.67028	0.188324
$616$	0	0
$617$	−12.0996	−0.487112	−0.243556	−	0.969887i	$-0.578314\pi$
$617$	−12.0996	−0.487112	−0.243556	+	0.969887i	$0.578314\pi$
$618$	0	0
$619$	−1.42334	−0.0572087	−0.0286043	−	0.999591i	$-0.509106\pi$
$619$	−1.42334	−0.0572087	−0.0286043	+	0.999591i	$0.509106\pi$
$620$	0	0
$621$	−11.2739	−0.452408
$622$	0	0
$623$	−2.98890	−0.119748
$624$	0	0
$625$	−18.2380	−0.729519
$626$	0	0
$627$	−2.41466	−0.0964322
$628$	0	0
$629$	6.63584	0.264588
$630$	0	0
$631$	31.9088	1.27027	0.635135	−	0.772401i	$-0.280945\pi$
$631$	31.9088	1.27027	0.635135	+	0.772401i	$0.280945\pi$
$632$	0	0
$633$	−6.92954	−0.275425
$634$	0	0
$635$	16.7194	0.663488
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	−2.99008	−0.118286
$640$	0	0
$641$	−46.8794	−1.85162	−0.925812	−	0.377984i	$-0.876617\pi$
$641$	−46.8794	−1.85162	−0.925812	+	0.377984i	$0.876617\pi$
$642$	0	0
$643$	4.36336	0.172074	0.0860370	−	0.996292i	$-0.472580\pi$
$643$	4.36336	0.172074	0.0860370	+	0.996292i	$0.472580\pi$
$644$	0	0
$645$	−5.43195	−0.213883
$646$	0	0
$647$	−6.75159	−0.265432	−0.132716	−	0.991154i	$-0.542370\pi$
$647$	−6.75159	−0.265432	−0.132716	+	0.991154i	$0.542370\pi$
$648$	0	0
$649$	−11.6777	−0.458389
$650$	0	0
$651$	−2.55593	−0.100175
$652$	0	0
$653$	28.8122	1.12751	0.563754	−	0.825943i	$-0.309357\pi$
$653$	28.8122	1.12751	0.563754	+	0.825943i	$0.309357\pi$
$654$	0	0
$655$	−5.64430	−0.220541
$656$	0	0
$657$	−46.2296	−1.80359
$658$	0	0
$659$	−36.0871	−1.40575	−0.702876	−	0.711312i	$-0.748101\pi$
$659$	−36.0871	−1.40575	−0.702876	+	0.711312i	$0.748101\pi$
$660$	0	0
$661$	−5.98285	−0.232706	−0.116353	−	0.993208i	$-0.537120\pi$
$661$	−5.98285	−0.232706	−0.116353	+	0.993208i	$0.537120\pi$
$662$	0	0
$663$	−1.00516	−0.0390371
$664$	0	0
$665$	−14.2757	−0.553588
$666$	0	0
$667$	28.8860	1.11847
$668$	0	0
$669$	−2.68986	−0.103996
$670$	0	0
$671$	−6.71777	−0.259337
$672$	0	0
$673$	−5.18162	−0.199737	−0.0998683	−	0.995001i	$-0.531842\pi$
$673$	−5.18162	−0.199737	−0.0998683	+	0.995001i	$0.531842\pi$
$674$	0	0
$675$	−2.17589	−0.0837500
$676$	0	0
$677$	19.6368	0.754704	0.377352	−	0.926070i	$-0.376835\pi$
$677$	19.6368	0.754704	0.377352	+	0.926070i	$0.376835\pi$
$678$	0	0
$679$	−1.28773	−0.0494185
$680$	0	0
$681$	2.76490	0.105951
$682$	0	0
$683$	−31.1124	−1.19048	−0.595241	−	0.803547i	$-0.702944\pi$
$683$	−31.1124	−1.19048	−0.595241	+	0.803547i	$0.702944\pi$
$684$	0	0
$685$	42.3037	1.61634
$686$	0	0
$687$	4.13244	0.157662
$688$	0	0
$689$	−13.7493	−0.523805
$690$	0	0
$691$	−19.6363	−0.747001	−0.373500	−	0.927630i	$-0.621843\pi$
$691$	−19.6363	−0.747001	−0.373500	+	0.927630i	$0.621843\pi$
$692$	0	0
$693$	2.88863	0.109730
$694$	0	0
$695$	2.70931	0.102770
$696$	0	0
$697$	−21.3634	−0.809197
$698$	0	0
$699$	5.44126	0.205807
$700$	0	0
$701$	13.4620	0.508452	0.254226	−	0.967145i	$-0.418179\pi$
$701$	13.4620	0.508452	0.254226	+	0.967145i	$0.418179\pi$
$702$	0	0
$703$	15.9411	0.601228
$704$	0	0
$705$	4.13632	0.155783
$706$	0	0
$707$	19.2561	0.724198
$708$	0	0
$709$	−30.9345	−1.16177	−0.580884	−	0.813986i	$-0.697293\pi$
$709$	−30.9345	−1.16177	−0.580884	+	0.813986i	$0.697293\pi$
$710$	0	0
$711$	−28.8346	−1.08138
$712$	0	0
$713$	43.9374	1.64547
$714$	0	0
$715$	1.97301	0.0737865
$716$	0	0
$717$	2.64690	0.0988502
$718$	0	0
$719$	39.0701	1.45707	0.728534	−	0.685009i	$-0.240202\pi$
$719$	39.0701	1.45707	0.728534	+	0.685009i	$0.240202\pi$
$720$	0	0
$721$	−2.72365	−0.101434
$722$	0	0
$723$	−0.981392	−0.0364984
$724$	0	0
$725$	5.57503	0.207051
$726$	0	0
$727$	−4.29250	−0.159200	−0.0795999	−	0.996827i	$-0.525364\pi$
$727$	−4.29250	−0.159200	−0.0795999	+	0.996827i	$0.525364\pi$
$728$	0	0
$729$	−21.1706	−0.784095
$730$	0	0
$731$	24.8475	0.919019
$732$	0	0
$733$	−0.420890	−0.0155459	−0.00777297	−	0.999970i	$-0.502474\pi$
$733$	−0.420890	−0.0155459	−0.00777297	+	0.999970i	$0.502474\pi$
$734$	0	0
$735$	−0.658443	−0.0242870
$736$	0	0
$737$	−0.389294	−0.0143398
$738$	0	0
$739$	41.3074	1.51952	0.759759	−	0.650204i	$-0.225317\pi$
$739$	41.3074	1.51952	0.759759	+	0.650204i	$0.225317\pi$
$740$	0	0
$741$	−2.41466	−0.0887047
$742$	0	0
$743$	39.3027	1.44188	0.720938	−	0.692999i	$-0.243711\pi$
$743$	39.3027	1.44188	0.720938	+	0.692999i	$0.243711\pi$
$744$	0	0
$745$	−42.2973	−1.54965
$746$	0	0
$747$	−11.9926	−0.438788
$748$	0	0
$749$	−9.05980	−0.331038
$750$	0	0
$751$	−13.9043	−0.507375	−0.253688	−	0.967286i	$-0.581643\pi$
$751$	−13.9043	−0.507375	−0.253688	+	0.967286i	$0.581643\pi$
$752$	0	0
$753$	−1.04183	−0.0379663
$754$	0	0
$755$	14.4088	0.524389
$756$	0	0
$757$	1.48474	0.0539638	0.0269819	−	0.999636i	$-0.491410\pi$
$757$	1.48474	0.0539638	0.0269819	+	0.999636i	$0.491410\pi$
$758$	0	0
$759$	1.91453	0.0694930
$760$	0	0
$761$	42.4869	1.54015	0.770075	−	0.637953i	$-0.220219\pi$
$761$	42.4869	1.54015	0.770075	+	0.637953i	$0.220219\pi$
$762$	0	0
$763$	3.35173	0.121341
$764$	0	0
$765$	−17.1659	−0.620636
$766$	0	0
$767$	−11.6777	−0.421657
$768$	0	0
$769$	42.3380	1.52675	0.763374	−	0.645956i	$-0.223541\pi$
$769$	42.3380	1.52675	0.763374	+	0.645956i	$0.223541\pi$
$770$	0	0
$771$	4.66146	0.167878
$772$	0	0
$773$	−15.2131	−0.547177	−0.273589	−	0.961847i	$-0.588211\pi$
$773$	−15.2131	−0.547177	−0.273589	+	0.961847i	$0.588211\pi$
$774$	0	0
$775$	8.47998	0.304610
$776$	0	0
$777$	0.735255	0.0263771
$778$	0	0
$779$	−51.3206	−1.83875
$780$	0	0
$781$	1.03512	0.0370395
$782$	0	0
$783$	−9.89500	−0.353618
$784$	0	0
$785$	−40.1334	−1.43242
$786$	0	0
$787$	−52.9559	−1.88767	−0.943837	−	0.330412i	$-0.892812\pi$
$787$	−52.9559	−1.88767	−0.943837	+	0.330412i	$0.892812\pi$
$788$	0	0
$789$	−1.66805	−0.0593840
$790$	0	0
$791$	6.41177	0.227976
$792$	0	0
$793$	−6.71777	−0.238555
$794$	0	0
$795$	9.05311	0.321081
$796$	0	0
$797$	12.8354	0.454652	0.227326	−	0.973819i	$-0.427002\pi$
$797$	12.8354	0.454652	0.227326	+	0.973819i	$0.427002\pi$
$798$	0	0
$799$	−18.9209	−0.669374
$800$	0	0
$801$	−8.63383	−0.305061
$802$	0	0
$803$	16.0040	0.564769
$804$	0	0
$805$	11.3189	0.398938
$806$	0	0
$807$	−5.26280	−0.185259
$808$	0	0
$809$	−46.9610	−1.65106	−0.825531	−	0.564356i	$-0.809124\pi$
$809$	−46.9610	−1.65106	−0.825531	+	0.564356i	$0.809124\pi$
$810$	0	0
$811$	−23.4840	−0.824635	−0.412317	−	0.911040i	$-0.635280\pi$
$811$	−23.4840	−0.824635	−0.412317	+	0.911040i	$0.635280\pi$
$812$	0	0
$813$	5.60854	0.196700
$814$	0	0
$815$	−29.5414	−1.03479
$816$	0	0
$817$	59.6904	2.08830
$818$	0	0
$819$	2.88863	0.100937
$820$	0	0
$821$	25.5084	0.890250	0.445125	−	0.895468i	$-0.353159\pi$
$821$	25.5084	0.890250	0.445125	+	0.895468i	$0.353159\pi$
$822$	0	0
$823$	−14.1797	−0.494272	−0.247136	−	0.968981i	$-0.579490\pi$
$823$	−14.1797	−0.494272	−0.247136	+	0.968981i	$0.579490\pi$
$824$	0	0
$825$	0.369507	0.0128646
$826$	0	0
$827$	−49.7107	−1.72861	−0.864305	−	0.502969i	$-0.832241\pi$
$827$	−49.7107	−1.72861	−0.864305	+	0.502969i	$0.832241\pi$
$828$	0	0
$829$	30.0671	1.04427	0.522137	−	0.852861i	$-0.325135\pi$
$829$	30.0671	1.04427	0.522137	+	0.852861i	$0.325135\pi$
$830$	0	0
$831$	7.73643	0.268374
$832$	0	0
$833$	3.01194	0.104357
$834$	0	0
$835$	−23.5170	−0.813839
$836$	0	0
$837$	−15.0509	−0.520236
$838$	0	0
$839$	−37.9182	−1.30908	−0.654541	−	0.756027i	$-0.727138\pi$
$839$	−37.9182	−1.30908	−0.654541	+	0.756027i	$0.727138\pi$
$840$	0	0
$841$	−3.64717	−0.125764
$842$	0	0
$843$	9.13147	0.314504
$844$	0	0
$845$	1.97301	0.0678737
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	3.04397	0.104469
$850$	0	0
$851$	−12.6393	−0.433270
$852$	0	0
$853$	−44.3174	−1.51740	−0.758700	−	0.651440i	$-0.774165\pi$
$853$	−44.3174	−1.51740	−0.758700	+	0.651440i	$0.774165\pi$
$854$	0	0
$855$	−41.2372	−1.41028
$856$	0	0
$857$	−28.8189	−0.984435	−0.492218	−	0.870472i	$-0.663814\pi$
$857$	−28.8189	−0.984435	−0.492218	+	0.870472i	$0.663814\pi$
$858$	0	0
$859$	11.9287	0.407003	0.203501	−	0.979075i	$-0.434768\pi$
$859$	11.9287	0.407003	0.203501	+	0.979075i	$0.434768\pi$
$860$	0	0
$861$	−2.36708	−0.0806699
$862$	0	0
$863$	14.9535	0.509022	0.254511	−	0.967070i	$-0.418085\pi$
$863$	14.9535	0.509022	0.254511	+	0.967070i	$0.418085\pi$
$864$	0	0
$865$	40.8668	1.38951
$866$	0	0
$867$	2.64585	0.0898576
$868$	0	0
$869$	9.98212	0.338620
$870$	0	0
$871$	−0.389294	−0.0131907
$872$	0	0
$873$	−3.71977	−0.125895
$874$	0	0
$875$	12.0496	0.407352
$876$	0	0
$877$	−39.4530	−1.33223	−0.666116	−	0.745848i	$-0.732044\pi$
$877$	−39.4530	−1.33223	−0.666116	+	0.745848i	$0.732044\pi$
$878$	0	0
$879$	7.68905	0.259345
$880$	0	0
$881$	36.6337	1.23422	0.617110	−	0.786877i	$-0.288303\pi$
$881$	36.6337	1.23422	0.617110	+	0.786877i	$0.288303\pi$
$882$	0	0
$883$	47.3345	1.59293	0.796467	−	0.604682i	$-0.206700\pi$
$883$	47.3345	1.59293	0.796467	+	0.604682i	$0.206700\pi$
$884$	0	0
$885$	7.68909	0.258466
$886$	0	0
$887$	−33.9341	−1.13940	−0.569698	−	0.821854i	$-0.692940\pi$
$887$	−33.9341	−1.13940	−0.569698	+	0.821854i	$0.692940\pi$
$888$	0	0
$889$	−8.47402	−0.284210
$890$	0	0
$891$	8.01005	0.268347
$892$	0	0
$893$	−45.4531	−1.52103
$894$	0	0
$895$	38.1672	1.27579
$896$	0	0
$897$	1.91453	0.0639243
$898$	0	0
$899$	38.5633	1.28616
$900$	0	0
$901$	−41.4119	−1.37963
$902$	0	0
$903$	2.75312	0.0916182
$904$	0	0
$905$	−18.8201	−0.625602
$906$	0	0
$907$	55.4136	1.83998	0.919989	−	0.391943i	$-0.128197\pi$
$907$	55.4136	1.83998	0.919989	+	0.391943i	$0.128197\pi$
$908$	0	0
$909$	55.6236	1.84492
$910$	0	0
$911$	31.7810	1.05295	0.526476	−	0.850190i	$-0.323513\pi$
$911$	31.7810	1.05295	0.526476	+	0.850190i	$0.323513\pi$
$912$	0	0
$913$	4.15167	0.137400
$914$	0	0
$915$	4.42327	0.146229
$916$	0	0
$917$	2.86075	0.0944703
$918$	0	0
$919$	11.4409	0.377399	0.188700	−	0.982035i	$-0.439573\pi$
$919$	11.4409	0.377399	0.188700	+	0.982035i	$0.439573\pi$
$920$	0	0
$921$	5.85789	0.193024
$922$	0	0
$923$	1.03512	0.0340714
$924$	0	0
$925$	−2.43940	−0.0802071
$926$	0	0
$927$	−7.86761	−0.258406
$928$	0	0
$929$	12.9478	0.424804	0.212402	−	0.977182i	$-0.431871\pi$
$929$	12.9478	0.424804	0.212402	+	0.977182i	$0.431871\pi$
$930$	0	0
$931$	7.23548	0.237133
$932$	0	0
$933$	−6.33160	−0.207287
$934$	0	0
$935$	5.94259	0.194344
$936$	0	0
$937$	2.59120	0.0846507	0.0423253	−	0.999104i	$-0.486523\pi$
$937$	2.59120	0.0846507	0.0423253	+	0.999104i	$0.486523\pi$
$938$	0	0
$939$	9.78140	0.319204
$940$	0	0
$941$	5.51308	0.179721	0.0898607	−	0.995954i	$-0.471358\pi$
$941$	5.51308	0.179721	0.0898607	+	0.995954i	$0.471358\pi$
$942$	0	0
$943$	40.6910	1.32508
$944$	0	0
$945$	−3.87733	−0.126129
$946$	0	0
$947$	5.91068	0.192071	0.0960356	−	0.995378i	$-0.469384\pi$
$947$	5.91068	0.192071	0.0960356	+	0.995378i	$0.469384\pi$
$948$	0	0
$949$	16.0040	0.519512
$950$	0	0
$951$	4.29159	0.139164
$952$	0	0
$953$	39.7042	1.28615	0.643073	−	0.765805i	$-0.277659\pi$
$953$	39.7042	1.28615	0.643073	+	0.765805i	$0.277659\pi$
$954$	0	0
$955$	−51.8659	−1.67834
$956$	0	0
$957$	1.68036	0.0543182
$958$	0	0
$959$	−21.4412	−0.692372
$960$	0	0
$961$	27.6573	0.892170
$962$	0	0
$963$	−26.1704	−0.843329
$964$	0	0
$965$	−0.0269205	−0.000866603 0
$966$	0	0
$967$	38.3582	1.23352	0.616759	−	0.787152i	$-0.288445\pi$
$967$	38.3582	1.23352	0.616759	+	0.787152i	$0.288445\pi$
$968$	0	0
$969$	−7.27281	−0.233636
$970$	0	0
$971$	−23.9671	−0.769142	−0.384571	−	0.923095i	$-0.625651\pi$
$971$	−23.9671	−0.769142	−0.384571	+	0.923095i	$0.625651\pi$
$972$	0	0
$973$	−1.37318	−0.0440222
$974$	0	0
$975$	0.369507	0.0118337
$976$	0	0
$977$	9.19563	0.294194	0.147097	−	0.989122i	$-0.453007\pi$
$977$	9.19563	0.294194	0.147097	+	0.989122i	$0.453007\pi$
$978$	0	0
$979$	2.98890	0.0955257
$980$	0	0
$981$	9.68190	0.309119
$982$	0	0
$983$	33.4086	1.06557	0.532784	−	0.846251i	$-0.321146\pi$
$983$	33.4086	1.06557	0.532784	+	0.846251i	$0.321146\pi$
$984$	0	0
$985$	−14.5214	−0.462689
$986$	0	0
$987$	−2.09645	−0.0667307
$988$	0	0
$989$	−47.3272	−1.50492
$990$	0	0
$991$	7.27070	0.230961	0.115481	−	0.993310i	$-0.463159\pi$
$991$	7.27070	0.230961	0.115481	+	0.993310i	$0.463159\pi$
$992$	0	0
$993$	−0.631517	−0.0200406
$994$	0	0
$995$	−37.3110	−1.18284
$996$	0	0
$997$	0.815713	0.0258339	0.0129170	−	0.999917i	$-0.495888\pi$
$997$	0.815713	0.0258339	0.0129170	+	0.999917i	$0.495888\pi$
$998$	0	0
$999$	4.32964	0.136984

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.p.1.4	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.p.1.4	✓	9	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-0.333725 q^{3} +1.97301 q^{5} -1.00000 q^{7} -2.88863 q^{9} +O(q^{10})\)	\(q-0.333725 q^{3} +1.97301 q^{5} -1.00000 q^{7} -2.88863 q^{9} +1.00000 q^{11} +1.00000 q^{13} -0.658443 q^{15} +3.01194 q^{17} +7.23548 q^{19} +0.333725 q^{21} -5.73685 q^{23} -1.10722 q^{25} +1.96518 q^{27} -5.03516 q^{29} -7.65880 q^{31} -0.333725 q^{33} -1.97301 q^{35} +2.20318 q^{37} -0.333725 q^{39} -7.09291 q^{41} +8.24968 q^{43} -5.69930 q^{45} -6.28197 q^{47} +1.00000 q^{49} -1.00516 q^{51} -13.7493 q^{53} +1.97301 q^{55} -2.41466 q^{57} -11.6777 q^{59} -6.71777 q^{61} +2.88863 q^{63} +1.97301 q^{65} -0.389294 q^{67} +1.91453 q^{69} +1.03512 q^{71} +16.0040 q^{73} +0.369507 q^{75} -1.00000 q^{77} +9.98212 q^{79} +8.01005 q^{81} +4.15167 q^{83} +5.94259 q^{85} +1.68036 q^{87} +2.98890 q^{89} -1.00000 q^{91} +2.55593 q^{93} +14.2757 q^{95} +1.28773 q^{97} -2.88863 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + q^{3} - 4 q^{5} - 9 q^{7} + 4 q^{9}+O(q^{10}) \) 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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