LMFDB - Embedded newform 8004.2.a.g.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8004, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8004.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$63.9122617778$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 3 x^{11} - 31 x^{10} + 80 x^{9} + 347 x^{8} - 697 x^{7} - 1714 x^{6} + 2146 x^{5} + 3304 x^{4} + \cdots + 16 $ x^12 - 3x^11 - 31x^10 + 80x^9 + 347x^8 - 697x^7 - 1714x^6 + 2146x^5 + 3304x^4 - 1156x^3 - 616x^2 + 136*x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$-1.42583$ of defining polynomial
Character	$\chi$	$=$	8004.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.42583 q^{5} -1.49015 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.42583 q^{5} -1.49015 q^{7} +1.00000 q^{9} +4.45964 q^{11} -0.159001 q^{13} -1.42583 q^{15} +3.47517 q^{17} +1.29574 q^{19} +1.49015 q^{21} +1.00000 q^{23} -2.96701 q^{25} -1.00000 q^{27} -1.00000 q^{29} -8.87791 q^{31} -4.45964 q^{33} -2.12470 q^{35} -9.07562 q^{37} +0.159001 q^{39} -6.40770 q^{41} -12.0404 q^{43} +1.42583 q^{45} -10.2522 q^{47} -4.77945 q^{49} -3.47517 q^{51} +8.00324 q^{53} +6.35868 q^{55} -1.29574 q^{57} +12.2266 q^{59} -10.3000 q^{61} -1.49015 q^{63} -0.226708 q^{65} -0.485962 q^{67} -1.00000 q^{69} +7.64325 q^{71} -3.71841 q^{73} +2.96701 q^{75} -6.64554 q^{77} +12.4035 q^{79} +1.00000 q^{81} +1.30451 q^{83} +4.95499 q^{85} +1.00000 q^{87} -18.2043 q^{89} +0.236935 q^{91} +8.87791 q^{93} +1.84750 q^{95} +2.78330 q^{97} +4.45964 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{3} - 3 q^{5} + 4 q^{7} + 12 q^{9}+O(q^{10}) $ 12 * q - 12 * q^3 - 3 * q^5 + 4 * q^7 + 12 * q^9	$ 12 q - 12 q^{3} - 3 q^{5} + 4 q^{7} + 12 q^{9} - 5 q^{11} - 6 q^{13} + 3 q^{15} - 7 q^{17} - 3 q^{19} - 4 q^{21} + 12 q^{23} + 11 q^{25} - 12 q^{27} - 12 q^{29} + 2 q^{31} + 5 q^{33} - 9 q^{35} - 20 q^{37} + 6 q^{39} - 3 q^{41} + 5 q^{43} - 3 q^{45} - 2 q^{49} + 7 q^{51} - 3 q^{53} + 19 q^{55} + 3 q^{57} - 20 q^{59} - 17 q^{61} + 4 q^{63} - 4 q^{65} - 9 q^{67} - 12 q^{69} + 7 q^{71} - 9 q^{73} - 11 q^{75} - 34 q^{77} + 14 q^{79} + 12 q^{81} + 5 q^{83} - 12 q^{85} + 12 q^{87} - 22 q^{89} - 3 q^{91} - 2 q^{93} - 27 q^{95} + 17 q^{97} - 5 q^{99}+O(q^{100}) $ 12 * q - 12 * q^3 - 3 * q^5 + 4 * q^7 + 12 * q^9 - 5 * q^11 - 6 * q^13 + 3 * q^15 - 7 * q^17 - 3 * q^19 - 4 * q^21 + 12 * q^23 + 11 * q^25 - 12 * q^27 - 12 * q^29 + 2 * q^31 + 5 * q^33 - 9 * q^35 - 20 * q^37 + 6 * q^39 - 3 * q^41 + 5 * q^43 - 3 * q^45 - 2 * q^49 + 7 * q^51 - 3 * q^53 + 19 * q^55 + 3 * q^57 - 20 * q^59 - 17 * q^61 + 4 * q^63 - 4 * q^65 - 9 * q^67 - 12 * q^69 + 7 * q^71 - 9 * q^73 - 11 * q^75 - 34 * q^77 + 14 * q^79 + 12 * q^81 + 5 * q^83 - 12 * q^85 + 12 * q^87 - 22 * q^89 - 3 * q^91 - 2 * q^93 - 27 * q^95 + 17 * q^97 - 5 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	1.42583	0.637650	0.318825	−	0.947814i	$-0.396712\pi$
$5$	1.42583	0.637650	0.318825	+	0.947814i	$0.396712\pi$
$6$	0	0
$7$	−1.49015	−0.563224	−0.281612	−	0.959528i	$-0.590869\pi$
$7$	−1.49015	−0.563224	−0.281612	+	0.959528i	$0.590869\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	4.45964	1.34463	0.672316	−	0.740264i	$-0.265299\pi$
$11$	4.45964	1.34463	0.672316	+	0.740264i	$0.265299\pi$
$12$	0	0
$13$	−0.159001	−0.0440988	−0.0220494	−	0.999757i	$-0.507019\pi$
$13$	−0.159001	−0.0440988	−0.0220494	+	0.999757i	$0.507019\pi$
$14$	0	0
$15$	−1.42583	−0.368147
$16$	0	0
$17$	3.47517	0.842852	0.421426	−	0.906863i	$-0.361530\pi$
$17$	3.47517	0.842852	0.421426	+	0.906863i	$0.361530\pi$
$18$	0	0
$19$	1.29574	0.297262	0.148631	−	0.988893i	$-0.452513\pi$
$19$	1.29574	0.297262	0.148631	+	0.988893i	$0.452513\pi$
$20$	0	0
$21$	1.49015	0.325177
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−2.96701	−0.593403
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−8.87791	−1.59452	−0.797260	−	0.603636i	$-0.793718\pi$
$31$	−8.87791	−1.59452	−0.797260	+	0.603636i	$0.793718\pi$
$32$	0	0
$33$	−4.45964	−0.776324
$34$	0	0
$35$	−2.12470	−0.359140
$36$	0	0
$37$	−9.07562	−1.49202	−0.746012	−	0.665933i	$-0.768034\pi$
$37$	−9.07562	−1.49202	−0.746012	+	0.665933i	$0.768034\pi$
$38$	0	0
$39$	0.159001	0.0254605
$40$	0	0
$41$	−6.40770	−1.00071	−0.500357	−	0.865819i	$-0.666798\pi$
$41$	−6.40770	−1.00071	−0.500357	+	0.865819i	$0.666798\pi$
$42$	0	0
$43$	−12.0404	−1.83614	−0.918072	−	0.396414i	$-0.870255\pi$
$43$	−12.0404	−1.83614	−0.918072	+	0.396414i	$0.870255\pi$
$44$	0	0
$45$	1.42583	0.212550
$46$	0	0
$47$	−10.2522	−1.49543	−0.747716	−	0.664019i	$-0.768850\pi$
$47$	−10.2522	−1.49543	−0.747716	+	0.664019i	$0.768850\pi$
$48$	0	0
$49$	−4.77945	−0.682779
$50$	0	0
$51$	−3.47517	−0.486621
$52$	0	0
$53$	8.00324	1.09933	0.549665	−	0.835385i	$-0.314755\pi$
$53$	8.00324	1.09933	0.549665	+	0.835385i	$0.314755\pi$
$54$	0	0
$55$	6.35868	0.857405
$56$	0	0
$57$	−1.29574	−0.171625
$58$	0	0
$59$	12.2266	1.59177	0.795885	−	0.605447i	$-0.207006\pi$
$59$	12.2266	1.59177	0.795885	+	0.605447i	$0.207006\pi$
$60$	0	0
$61$	−10.3000	−1.31878	−0.659388	−	0.751803i	$-0.729184\pi$
$61$	−10.3000	−1.31878	−0.659388	+	0.751803i	$0.729184\pi$
$62$	0	0
$63$	−1.49015	−0.187741
$64$	0	0
$65$	−0.226708	−0.0281196
$66$	0	0
$67$	−0.485962	−0.0593697	−0.0296848	−	0.999559i	$-0.509450\pi$
$67$	−0.485962	−0.0593697	−0.0296848	+	0.999559i	$0.509450\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	7.64325	0.907087	0.453544	−	0.891234i	$-0.350160\pi$
$71$	7.64325	0.907087	0.453544	+	0.891234i	$0.350160\pi$
$72$	0	0
$73$	−3.71841	−0.435207	−0.217603	−	0.976037i	$-0.569824\pi$
$73$	−3.71841	−0.435207	−0.217603	+	0.976037i	$0.569824\pi$
$74$	0	0
$75$	2.96701	0.342601
$76$	0	0
$77$	−6.64554	−0.757329
$78$	0	0
$79$	12.4035	1.39550	0.697751	−	0.716341i	$-0.254184\pi$
$79$	12.4035	1.39550	0.697751	+	0.716341i	$0.254184\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	1.30451	0.143188	0.0715941	−	0.997434i	$-0.477191\pi$
$83$	1.30451	0.143188	0.0715941	+	0.997434i	$0.477191\pi$
$84$	0	0
$85$	4.95499	0.537444
$86$	0	0
$87$	1.00000	0.107211
$88$	0	0
$89$	−18.2043	−1.92965	−0.964827	−	0.262884i	$-0.915326\pi$
$89$	−18.2043	−1.92965	−0.964827	+	0.262884i	$0.915326\pi$
$90$	0	0
$91$	0.236935	0.0248375
$92$	0	0
$93$	8.87791	0.920597
$94$	0	0
$95$	1.84750	0.189549
$96$	0	0
$97$	2.78330	0.282601	0.141301	−	0.989967i	$-0.454872\pi$
$97$	2.78330	0.282601	0.141301	+	0.989967i	$0.454872\pi$
$98$	0	0
$99$	4.45964	0.448211
$100$	0	0
$101$	8.42981	0.838797	0.419399	−	0.907802i	$-0.362241\pi$
$101$	8.42981	0.838797	0.419399	+	0.907802i	$0.362241\pi$
$102$	0	0
$103$	14.8072	1.45900	0.729499	−	0.683982i	$-0.239753\pi$
$103$	14.8072	1.45900	0.729499	+	0.683982i	$0.239753\pi$
$104$	0	0
$105$	2.12470	0.207349
$106$	0	0
$107$	13.5258	1.30759	0.653797	−	0.756670i	$-0.273175\pi$
$107$	13.5258	1.30759	0.653797	+	0.756670i	$0.273175\pi$
$108$	0	0
$109$	−12.8502	−1.23083	−0.615415	−	0.788203i	$-0.711012\pi$
$109$	−12.8502	−1.23083	−0.615415	+	0.788203i	$0.711012\pi$
$110$	0	0
$111$	9.07562	0.861420
$112$	0	0
$113$	0.391242	0.0368050	0.0184025	−	0.999831i	$-0.494142\pi$
$113$	0.391242	0.0368050	0.0184025	+	0.999831i	$0.494142\pi$
$114$	0	0
$115$	1.42583	0.132959
$116$	0	0
$117$	−0.159001	−0.0146996
$118$	0	0
$119$	−5.17852	−0.474714
$120$	0	0
$121$	8.88842	0.808038
$122$	0	0
$123$	6.40770	0.577763
$124$	0	0
$125$	−11.3596	−1.01603
$126$	0	0
$127$	−11.4109	−1.01256	−0.506278	−	0.862370i	$-0.668979\pi$
$127$	−11.4109	−1.01256	−0.506278	+	0.862370i	$0.668979\pi$
$128$	0	0
$129$	12.0404	1.06010
$130$	0	0
$131$	7.75502	0.677559	0.338779	−	0.940866i	$-0.389986\pi$
$131$	7.75502	0.677559	0.338779	+	0.940866i	$0.389986\pi$
$132$	0	0
$133$	−1.93084	−0.167425
$134$	0	0
$135$	−1.42583	−0.122716
$136$	0	0
$137$	8.38978	0.716787	0.358394	−	0.933571i	$-0.383325\pi$
$137$	8.38978	0.716787	0.358394	+	0.933571i	$0.383325\pi$
$138$	0	0
$139$	−13.9105	−1.17988	−0.589939	−	0.807448i	$-0.700848\pi$
$139$	−13.9105	−1.17988	−0.589939	+	0.807448i	$0.700848\pi$
$140$	0	0
$141$	10.2522	0.863388
$142$	0	0
$143$	−0.709086	−0.0592968
$144$	0	0
$145$	−1.42583	−0.118409
$146$	0	0
$147$	4.77945	0.394203
$148$	0	0
$149$	−10.7393	−0.879797	−0.439898	−	0.898048i	$-0.644986\pi$
$149$	−10.7393	−0.879797	−0.439898	+	0.898048i	$0.644986\pi$
$150$	0	0
$151$	9.86355	0.802685	0.401342	−	0.915928i	$-0.368544\pi$
$151$	9.86355	0.802685	0.401342	+	0.915928i	$0.368544\pi$
$152$	0	0
$153$	3.47517	0.280951
$154$	0	0
$155$	−12.6584	−1.01675
$156$	0	0
$157$	4.75666	0.379623	0.189812	−	0.981821i	$-0.439212\pi$
$157$	4.75666	0.379623	0.189812	+	0.981821i	$0.439212\pi$
$158$	0	0
$159$	−8.00324	−0.634698
$160$	0	0
$161$	−1.49015	−0.117440
$162$	0	0
$163$	5.52008	0.432366	0.216183	−	0.976353i	$-0.430639\pi$
$163$	5.52008	0.432366	0.216183	+	0.976353i	$0.430639\pi$
$164$	0	0
$165$	−6.35868	−0.495023
$166$	0	0
$167$	−3.79049	−0.293317	−0.146659	−	0.989187i	$-0.546852\pi$
$167$	−3.79049	−0.293317	−0.146659	+	0.989187i	$0.546852\pi$
$168$	0	0
$169$	−12.9747	−0.998055
$170$	0	0
$171$	1.29574	0.0990875
$172$	0	0
$173$	23.1910	1.76318	0.881590	−	0.472016i	$-0.156474\pi$
$173$	23.1910	1.76318	0.881590	+	0.472016i	$0.156474\pi$
$174$	0	0
$175$	4.42130	0.334219
$176$	0	0
$177$	−12.2266	−0.919009
$178$	0	0
$179$	−23.2446	−1.73738	−0.868691	−	0.495355i	$-0.835038\pi$
$179$	−23.2446	−1.73738	−0.868691	+	0.495355i	$0.835038\pi$
$180$	0	0
$181$	−4.88281	−0.362936	−0.181468	−	0.983397i	$-0.558085\pi$
$181$	−4.88281	−0.362936	−0.181468	+	0.983397i	$0.558085\pi$
$182$	0	0
$183$	10.3000	0.761395
$184$	0	0
$185$	−12.9403	−0.951388
$186$	0	0
$187$	15.4980	1.13333
$188$	0	0
$189$	1.49015	0.108392
$190$	0	0
$191$	−21.9079	−1.58520	−0.792599	−	0.609743i	$-0.791273\pi$
$191$	−21.9079	−1.58520	−0.792599	+	0.609743i	$0.791273\pi$
$192$	0	0
$193$	−8.29118	−0.596812	−0.298406	−	0.954439i	$-0.596455\pi$
$193$	−8.29118	−0.596812	−0.298406	+	0.954439i	$0.596455\pi$
$194$	0	0
$195$	0.226708	0.0162349
$196$	0	0
$197$	4.20469	0.299572	0.149786	−	0.988718i	$-0.452142\pi$
$197$	4.20469	0.299572	0.149786	+	0.988718i	$0.452142\pi$
$198$	0	0
$199$	−12.9265	−0.916335	−0.458167	−	0.888866i	$-0.651494\pi$
$199$	−12.9265	−0.916335	−0.458167	+	0.888866i	$0.651494\pi$
$200$	0	0
$201$	0.485962	0.0342771
$202$	0	0
$203$	1.49015	0.104588
$204$	0	0
$205$	−9.13628	−0.638105
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	5.77852	0.399709
$210$	0	0
$211$	−24.3532	−1.67654	−0.838271	−	0.545254i	$-0.816433\pi$
$211$	−24.3532	−1.67654	−0.838271	+	0.545254i	$0.816433\pi$
$212$	0	0
$213$	−7.64325	−0.523707
$214$	0	0
$215$	−17.1675	−1.17082
$216$	0	0
$217$	13.2294	0.898072
$218$	0	0
$219$	3.71841	0.251267
$220$	0	0
$221$	−0.552554	−0.0371688
$222$	0	0
$223$	−8.47197	−0.567325	−0.283662	−	0.958924i	$-0.591550\pi$
$223$	−8.47197	−0.567325	−0.283662	+	0.958924i	$0.591550\pi$
$224$	0	0
$225$	−2.96701	−0.197801
$226$	0	0
$227$	17.6827	1.17364	0.586821	−	0.809717i	$-0.300379\pi$
$227$	17.6827	1.17364	0.586821	+	0.809717i	$0.300379\pi$
$228$	0	0
$229$	−4.93562	−0.326155	−0.163077	−	0.986613i	$-0.552142\pi$
$229$	−4.93562	−0.326155	−0.163077	+	0.986613i	$0.552142\pi$
$230$	0	0
$231$	6.64554	0.437244
$232$	0	0
$233$	−26.5583	−1.73989	−0.869947	−	0.493145i	$-0.835847\pi$
$233$	−26.5583	−1.73989	−0.869947	+	0.493145i	$0.835847\pi$
$234$	0	0
$235$	−14.6178	−0.953562
$236$	0	0
$237$	−12.4035	−0.805693
$238$	0	0
$239$	−0.0342729	−0.00221693	−0.00110846	−	0.999999i	$-0.500353\pi$
$239$	−0.0342729	−0.00221693	−0.00110846	+	0.999999i	$0.500353\pi$
$240$	0	0
$241$	−15.3244	−0.987134	−0.493567	−	0.869708i	$-0.664307\pi$
$241$	−15.3244	−0.987134	−0.493567	+	0.869708i	$0.664307\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−6.81468	−0.435374
$246$	0	0
$247$	−0.206023	−0.0131089
$248$	0	0
$249$	−1.30451	−0.0826698
$250$	0	0
$251$	5.44938	0.343962	0.171981	−	0.985100i	$-0.444983\pi$
$251$	5.44938	0.343962	0.171981	+	0.985100i	$0.444983\pi$
$252$	0	0
$253$	4.45964	0.280375
$254$	0	0
$255$	−4.95499	−0.310294
$256$	0	0
$257$	−1.38441	−0.0863574	−0.0431787	−	0.999067i	$-0.513748\pi$
$257$	−1.38441	−0.0863574	−0.0431787	+	0.999067i	$0.513748\pi$
$258$	0	0
$259$	13.5240	0.840343
$260$	0	0
$261$	−1.00000	−0.0618984
$262$	0	0
$263$	6.66705	0.411108	0.205554	−	0.978646i	$-0.434100\pi$
$263$	6.66705	0.411108	0.205554	+	0.978646i	$0.434100\pi$
$264$	0	0
$265$	11.4112	0.700987
$266$	0	0
$267$	18.2043	1.11409
$268$	0	0
$269$	30.4318	1.85546	0.927730	−	0.373251i	$-0.121757\pi$
$269$	30.4318	1.85546	0.927730	+	0.373251i	$0.121757\pi$
$270$	0	0
$271$	−18.2818	−1.11054	−0.555270	−	0.831670i	$-0.687385\pi$
$271$	−18.2818	−1.11054	−0.555270	+	0.831670i	$0.687385\pi$
$272$	0	0
$273$	−0.236935	−0.0143400
$274$	0	0
$275$	−13.2318	−0.797909
$276$	0	0
$277$	4.36334	0.262168	0.131084	−	0.991371i	$-0.458154\pi$
$277$	4.36334	0.262168	0.131084	+	0.991371i	$0.458154\pi$
$278$	0	0
$279$	−8.87791	−0.531507
$280$	0	0
$281$	−14.1143	−0.841990	−0.420995	−	0.907063i	$-0.638319\pi$
$281$	−14.1143	−0.841990	−0.420995	+	0.907063i	$0.638319\pi$
$282$	0	0
$283$	2.84332	0.169018	0.0845090	−	0.996423i	$-0.473068\pi$
$283$	2.84332	0.169018	0.0845090	+	0.996423i	$0.473068\pi$
$284$	0	0
$285$	−1.84750	−0.109436
$286$	0	0
$287$	9.54843	0.563626
$288$	0	0
$289$	−4.92322	−0.289601
$290$	0	0
$291$	−2.78330	−0.163160
$292$	0	0
$293$	−9.50985	−0.555571	−0.277786	−	0.960643i	$-0.589600\pi$
$293$	−9.50985	−0.555571	−0.277786	+	0.960643i	$0.589600\pi$
$294$	0	0
$295$	17.4331	1.01499
$296$	0	0
$297$	−4.45964	−0.258775
$298$	0	0
$299$	−0.159001	−0.00919524
$300$	0	0
$301$	17.9420	1.03416
$302$	0	0
$303$	−8.42981	−0.484280
$304$	0	0
$305$	−14.6860	−0.840917
$306$	0	0
$307$	−5.05037	−0.288240	−0.144120	−	0.989560i	$-0.546035\pi$
$307$	−5.05037	−0.288240	−0.144120	+	0.989560i	$0.546035\pi$
$308$	0	0
$309$	−14.8072	−0.842353
$310$	0	0
$311$	32.6238	1.84993	0.924964	−	0.380056i	$-0.124095\pi$
$311$	32.6238	1.84993	0.924964	+	0.380056i	$0.124095\pi$
$312$	0	0
$313$	−6.08094	−0.343715	−0.171858	−	0.985122i	$-0.554977\pi$
$313$	−6.08094	−0.343715	−0.171858	+	0.985122i	$0.554977\pi$
$314$	0	0
$315$	−2.12470	−0.119713
$316$	0	0
$317$	26.3432	1.47958	0.739791	−	0.672837i	$-0.234925\pi$
$317$	26.3432	1.47958	0.739791	+	0.672837i	$0.234925\pi$
$318$	0	0
$319$	−4.45964	−0.249692
$320$	0	0
$321$	−13.5258	−0.754939
$322$	0	0
$323$	4.50290	0.250548
$324$	0	0
$325$	0.471757	0.0261684
$326$	0	0
$327$	12.8502	0.710620
$328$	0	0
$329$	15.2773	0.842263
$330$	0	0
$331$	−2.49309	−0.137032	−0.0685162	−	0.997650i	$-0.521826\pi$
$331$	−2.49309	−0.137032	−0.0685162	+	0.997650i	$0.521826\pi$
$332$	0	0
$333$	−9.07562	−0.497341
$334$	0	0
$335$	−0.692898	−0.0378571
$336$	0	0
$337$	−10.6144	−0.578201	−0.289101	−	0.957299i	$-0.593356\pi$
$337$	−10.6144	−0.578201	−0.289101	+	0.957299i	$0.593356\pi$
$338$	0	0
$339$	−0.391242	−0.0212494
$340$	0	0
$341$	−39.5923	−2.14404
$342$	0	0
$343$	17.5532	0.947781
$344$	0	0
$345$	−1.42583	−0.0767640
$346$	0	0
$347$	16.7762	0.900592	0.450296	−	0.892879i	$-0.351318\pi$
$347$	16.7762	0.900592	0.450296	+	0.892879i	$0.351318\pi$
$348$	0	0
$349$	9.12257	0.488320	0.244160	−	0.969735i	$-0.421488\pi$
$349$	9.12257	0.488320	0.244160	+	0.969735i	$0.421488\pi$
$350$	0	0
$351$	0.159001	0.00848683
$352$	0	0
$353$	−25.6428	−1.36483	−0.682415	−	0.730965i	$-0.739070\pi$
$353$	−25.6428	−1.36483	−0.682415	+	0.730965i	$0.739070\pi$
$354$	0	0
$355$	10.8980	0.578404
$356$	0	0
$357$	5.17852	0.274076
$358$	0	0
$359$	−21.9300	−1.15742	−0.578710	−	0.815534i	$-0.696444\pi$
$359$	−21.9300	−1.15742	−0.578710	+	0.815534i	$0.696444\pi$
$360$	0	0
$361$	−17.3211	−0.911635
$362$	0	0
$363$	−8.88842	−0.466521
$364$	0	0
$365$	−5.30181	−0.277510
$366$	0	0
$367$	35.9231	1.87517	0.937586	−	0.347755i	$-0.113056\pi$
$367$	35.9231	1.87517	0.937586	+	0.347755i	$0.113056\pi$
$368$	0	0
$369$	−6.40770	−0.333571
$370$	0	0
$371$	−11.9260	−0.619169
$372$	0	0
$373$	−14.6467	−0.758377	−0.379188	−	0.925320i	$-0.623797\pi$
$373$	−14.6467	−0.758377	−0.379188	+	0.925320i	$0.623797\pi$
$374$	0	0
$375$	11.3596	0.586607
$376$	0	0
$377$	0.159001	0.00818895
$378$	0	0
$379$	0.639273	0.0328372	0.0164186	−	0.999865i	$-0.494774\pi$
$379$	0.639273	0.0328372	0.0164186	+	0.999865i	$0.494774\pi$
$380$	0	0
$381$	11.4109	0.584600
$382$	0	0
$383$	1.78915	0.0914212	0.0457106	−	0.998955i	$-0.485445\pi$
$383$	1.78915	0.0914212	0.0457106	+	0.998955i	$0.485445\pi$
$384$	0	0
$385$	−9.47540	−0.482911
$386$	0	0
$387$	−12.0404	−0.612048
$388$	0	0
$389$	−17.9578	−0.910498	−0.455249	−	0.890364i	$-0.650450\pi$
$389$	−17.9578	−0.910498	−0.455249	+	0.890364i	$0.650450\pi$
$390$	0	0
$391$	3.47517	0.175747
$392$	0	0
$393$	−7.75502	−0.391189
$394$	0	0
$395$	17.6852	0.889841
$396$	0	0
$397$	9.23664	0.463574	0.231787	−	0.972767i	$-0.425543\pi$
$397$	9.23664	0.463574	0.231787	+	0.972767i	$0.425543\pi$
$398$	0	0
$399$	1.93084	0.0966630
$400$	0	0
$401$	−10.3628	−0.517493	−0.258746	−	0.965945i	$-0.583309\pi$
$401$	−10.3628	−0.517493	−0.258746	+	0.965945i	$0.583309\pi$
$402$	0	0
$403$	1.41159	0.0703165
$404$	0	0
$405$	1.42583	0.0708500
$406$	0	0
$407$	−40.4740	−2.00622
$408$	0	0
$409$	−14.8047	−0.732044	−0.366022	−	0.930606i	$-0.619280\pi$
$409$	−14.8047	−0.732044	−0.366022	+	0.930606i	$0.619280\pi$
$410$	0	0
$411$	−8.38978	−0.413837
$412$	0	0
$413$	−18.2195	−0.896523
$414$	0	0
$415$	1.86000	0.0913039
$416$	0	0
$417$	13.9105	0.681202
$418$	0	0
$419$	−34.8281	−1.70146	−0.850731	−	0.525601i	$-0.823841\pi$
$419$	−34.8281	−1.70146	−0.850731	+	0.525601i	$0.823841\pi$
$420$	0	0
$421$	6.17148	0.300779	0.150390	−	0.988627i	$-0.451947\pi$
$421$	6.17148	0.300779	0.150390	+	0.988627i	$0.451947\pi$
$422$	0	0
$423$	−10.2522	−0.498477
$424$	0	0
$425$	−10.3109	−0.500151
$426$	0	0
$427$	15.3485	0.742766
$428$	0	0
$429$	0.709086	0.0342350
$430$	0	0
$431$	1.31883	0.0635258	0.0317629	−	0.999495i	$-0.489888\pi$
$431$	1.31883	0.0635258	0.0317629	+	0.999495i	$0.489888\pi$
$432$	0	0
$433$	−16.3422	−0.785357	−0.392678	−	0.919676i	$-0.628451\pi$
$433$	−16.3422	−0.785357	−0.392678	+	0.919676i	$0.628451\pi$
$434$	0	0
$435$	1.42583	0.0683632
$436$	0	0
$437$	1.29574	0.0619835
$438$	0	0
$439$	29.0443	1.38621	0.693103	−	0.720838i	$-0.256243\pi$
$439$	29.0443	1.38621	0.693103	+	0.720838i	$0.256243\pi$
$440$	0	0
$441$	−4.77945	−0.227593
$442$	0	0
$443$	8.80448	0.418313	0.209157	−	0.977882i	$-0.432928\pi$
$443$	8.80448	0.418313	0.209157	+	0.977882i	$0.432928\pi$
$444$	0	0
$445$	−25.9562	−1.23044
$446$	0	0
$447$	10.7393	0.507951
$448$	0	0
$449$	6.87568	0.324483	0.162242	−	0.986751i	$-0.448128\pi$
$449$	6.87568	0.324483	0.162242	+	0.986751i	$0.448128\pi$
$450$	0	0
$451$	−28.5760	−1.34559
$452$	0	0
$453$	−9.86355	−0.463430
$454$	0	0
$455$	0.337828	0.0158376
$456$	0	0
$457$	23.2243	1.08638	0.543192	−	0.839608i	$-0.317215\pi$
$457$	23.2243	1.08638	0.543192	+	0.839608i	$0.317215\pi$
$458$	0	0
$459$	−3.47517	−0.162207
$460$	0	0
$461$	31.7004	1.47643	0.738217	−	0.674564i	$-0.235668\pi$
$461$	31.7004	1.47643	0.738217	+	0.674564i	$0.235668\pi$
$462$	0	0
$463$	29.6451	1.37772	0.688862	−	0.724892i	$-0.258110\pi$
$463$	29.6451	1.37772	0.688862	+	0.724892i	$0.258110\pi$
$464$	0	0
$465$	12.6584	0.587018
$466$	0	0
$467$	−33.5566	−1.55281	−0.776407	−	0.630232i	$-0.782960\pi$
$467$	−33.5566	−1.55281	−0.776407	+	0.630232i	$0.782960\pi$
$468$	0	0
$469$	0.724156	0.0334384
$470$	0	0
$471$	−4.75666	−0.219176
$472$	0	0
$473$	−53.6959	−2.46894
$474$	0	0
$475$	−3.84447	−0.176396
$476$	0	0
$477$	8.00324	0.366443
$478$	0	0
$479$	−15.5699	−0.711409	−0.355704	−	0.934599i	$-0.615759\pi$
$479$	−15.5699	−0.711409	−0.355704	+	0.934599i	$0.615759\pi$
$480$	0	0
$481$	1.44303	0.0657965
$482$	0	0
$483$	1.49015	0.0678042
$484$	0	0
$485$	3.96851	0.180201
$486$	0	0
$487$	−6.07824	−0.275431	−0.137716	−	0.990472i	$-0.543976\pi$
$487$	−6.07824	−0.275431	−0.137716	+	0.990472i	$0.543976\pi$
$488$	0	0
$489$	−5.52008	−0.249627
$490$	0	0
$491$	−1.40028	−0.0631935	−0.0315968	−	0.999501i	$-0.510059\pi$
$491$	−1.40028	−0.0631935	−0.0315968	+	0.999501i	$0.510059\pi$
$492$	0	0
$493$	−3.47517	−0.156514
$494$	0	0
$495$	6.35868	0.285802
$496$	0	0
$497$	−11.3896	−0.510893
$498$	0	0
$499$	29.9133	1.33910	0.669551	−	0.742766i	$-0.266487\pi$
$499$	29.9133	1.33910	0.669551	+	0.742766i	$0.266487\pi$
$500$	0	0
$501$	3.79049	0.169347
$502$	0	0
$503$	11.9620	0.533359	0.266680	−	0.963785i	$-0.414073\pi$
$503$	11.9620	0.533359	0.266680	+	0.963785i	$0.414073\pi$
$504$	0	0
$505$	12.0195	0.534859
$506$	0	0
$507$	12.9747	0.576227
$508$	0	0
$509$	9.60495	0.425732	0.212866	−	0.977081i	$-0.431720\pi$
$509$	9.60495	0.425732	0.212866	+	0.977081i	$0.431720\pi$
$510$	0	0
$511$	5.54099	0.245119
$512$	0	0
$513$	−1.29574	−0.0572082
$514$	0	0
$515$	21.1125	0.930330
$516$	0	0
$517$	−45.7210	−2.01081
$518$	0	0
$519$	−23.1910	−1.01797
$520$	0	0
$521$	24.4955	1.07317	0.536584	−	0.843847i	$-0.319715\pi$
$521$	24.4955	1.07317	0.536584	+	0.843847i	$0.319715\pi$
$522$	0	0
$523$	−16.4556	−0.719552	−0.359776	−	0.933039i	$-0.617147\pi$
$523$	−16.4556	−0.719552	−0.359776	+	0.933039i	$0.617147\pi$
$524$	0	0
$525$	−4.42130	−0.192961
$526$	0	0
$527$	−30.8522	−1.34394
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	12.2266	0.530590
$532$	0	0
$533$	1.01883	0.0441303
$534$	0	0
$535$	19.2855	0.833786
$536$	0	0
$537$	23.2446	1.00308
$538$	0	0
$539$	−21.3146	−0.918087
$540$	0	0
$541$	−25.3934	−1.09175	−0.545873	−	0.837868i	$-0.683802\pi$
$541$	−25.3934	−1.09175	−0.545873	+	0.837868i	$0.683802\pi$
$542$	0	0
$543$	4.88281	0.209541
$544$	0	0
$545$	−18.3222	−0.784839
$546$	0	0
$547$	−10.0145	−0.428190	−0.214095	−	0.976813i	$-0.568680\pi$
$547$	−10.0145	−0.428190	−0.214095	+	0.976813i	$0.568680\pi$
$548$	0	0
$549$	−10.3000	−0.439592
$550$	0	0
$551$	−1.29574	−0.0552002
$552$	0	0
$553$	−18.4831	−0.785980
$554$	0	0
$555$	12.9403	0.549284
$556$	0	0
$557$	−7.79216	−0.330164	−0.165082	−	0.986280i	$-0.552789\pi$
$557$	−7.79216	−0.330164	−0.165082	+	0.986280i	$0.552789\pi$
$558$	0	0
$559$	1.91443	0.0809718
$560$	0	0
$561$	−15.4980	−0.654326
$562$	0	0
$563$	9.81478	0.413644	0.206822	−	0.978379i	$-0.433688\pi$
$563$	9.81478	0.413644	0.206822	+	0.978379i	$0.433688\pi$
$564$	0	0
$565$	0.557844	0.0234687
$566$	0	0
$567$	−1.49015	−0.0625804
$568$	0	0
$569$	4.39753	0.184354	0.0921770	−	0.995743i	$-0.470617\pi$
$569$	4.39753	0.184354	0.0921770	+	0.995743i	$0.470617\pi$
$570$	0	0
$571$	−1.66013	−0.0694743	−0.0347372	−	0.999396i	$-0.511059\pi$
$571$	−1.66013	−0.0694743	−0.0347372	+	0.999396i	$0.511059\pi$
$572$	0	0
$573$	21.9079	0.915215
$574$	0	0
$575$	−2.96701	−0.123733
$576$	0	0
$577$	−29.3118	−1.22027	−0.610133	−	0.792299i	$-0.708884\pi$
$577$	−29.3118	−1.22027	−0.610133	+	0.792299i	$0.708884\pi$
$578$	0	0
$579$	8.29118	0.344570
$580$	0	0
$581$	−1.94391	−0.0806470
$582$	0	0
$583$	35.6916	1.47819
$584$	0	0
$585$	−0.226708	−0.00937320
$586$	0	0
$587$	−35.0457	−1.44649	−0.723246	−	0.690590i	$-0.757351\pi$
$587$	−35.0457	−1.44649	−0.723246	+	0.690590i	$0.757351\pi$
$588$	0	0
$589$	−11.5034	−0.473991
$590$	0	0
$591$	−4.20469	−0.172958
$592$	0	0
$593$	39.1421	1.60737	0.803686	−	0.595053i	$-0.202869\pi$
$593$	39.1421	1.60737	0.803686	+	0.595053i	$0.202869\pi$
$594$	0	0
$595$	−7.38368	−0.302701
$596$	0	0
$597$	12.9265	0.529046
$598$	0	0
$599$	18.8311	0.769417	0.384708	−	0.923038i	$-0.374302\pi$
$599$	18.8311	0.769417	0.384708	+	0.923038i	$0.374302\pi$
$600$	0	0
$601$	−23.4667	−0.957228	−0.478614	−	0.878025i	$-0.658861\pi$
$601$	−23.4667	−0.957228	−0.478614	+	0.878025i	$0.658861\pi$
$602$	0	0
$603$	−0.485962	−0.0197899
$604$	0	0
$605$	12.6734	0.515245
$606$	0	0
$607$	−27.8701	−1.13121	−0.565607	−	0.824675i	$-0.691358\pi$
$607$	−27.8701	−1.13121	−0.565607	+	0.824675i	$0.691358\pi$
$608$	0	0
$609$	−1.49015	−0.0603839
$610$	0	0
$611$	1.63010	0.0659468
$612$	0	0
$613$	29.5067	1.19176	0.595881	−	0.803072i	$-0.296803\pi$
$613$	29.5067	1.19176	0.595881	+	0.803072i	$0.296803\pi$
$614$	0	0
$615$	9.13628	0.368410
$616$	0	0
$617$	−31.9522	−1.28635	−0.643175	−	0.765720i	$-0.722383\pi$
$617$	−31.9522	−1.28635	−0.643175	+	0.765720i	$0.722383\pi$
$618$	0	0
$619$	−11.5996	−0.466227	−0.233114	−	0.972450i	$-0.574891\pi$
$619$	−11.5996	−0.466227	−0.233114	+	0.972450i	$0.574891\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	27.1272	1.08683
$624$	0	0
$625$	−1.36176	−0.0544703
$626$	0	0
$627$	−5.77852	−0.230772
$628$	0	0
$629$	−31.5393	−1.25755
$630$	0	0
$631$	−23.1670	−0.922262	−0.461131	−	0.887332i	$-0.652556\pi$
$631$	−23.1670	−0.922262	−0.461131	+	0.887332i	$0.652556\pi$
$632$	0	0
$633$	24.3532	0.967952
$634$	0	0
$635$	−16.2700	−0.645657
$636$	0	0
$637$	0.759936	0.0301098
$638$	0	0
$639$	7.64325	0.302362
$640$	0	0
$641$	−12.5953	−0.497486	−0.248743	−	0.968569i	$-0.580017\pi$
$641$	−12.5953	−0.497486	−0.248743	+	0.968569i	$0.580017\pi$
$642$	0	0
$643$	20.8087	0.820617	0.410308	−	0.911947i	$-0.365421\pi$
$643$	20.8087	0.820617	0.410308	+	0.911947i	$0.365421\pi$
$644$	0	0
$645$	17.1675	0.675971
$646$	0	0
$647$	27.5547	1.08329	0.541643	−	0.840608i	$-0.317802\pi$
$647$	27.5547	1.08329	0.541643	+	0.840608i	$0.317802\pi$
$648$	0	0
$649$	54.5264	2.14035
$650$	0	0
$651$	−13.2294	−0.518502
$652$	0	0
$653$	−35.4803	−1.38845	−0.694227	−	0.719756i	$-0.744253\pi$
$653$	−35.4803	−1.38845	−0.694227	+	0.719756i	$0.744253\pi$
$654$	0	0
$655$	11.0573	0.432045
$656$	0	0
$657$	−3.71841	−0.145069
$658$	0	0
$659$	−18.5600	−0.722995	−0.361498	−	0.932373i	$-0.617734\pi$
$659$	−18.5600	−0.722995	−0.361498	+	0.932373i	$0.617734\pi$
$660$	0	0
$661$	31.9109	1.24119	0.620595	−	0.784131i	$-0.286891\pi$
$661$	31.9109	1.24119	0.620595	+	0.784131i	$0.286891\pi$
$662$	0	0
$663$	0.552554	0.0214594
$664$	0	0
$665$	−2.75305	−0.106759
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	0	0
$669$	8.47197	0.327545
$670$	0	0
$671$	−45.9342	−1.77327
$672$	0	0
$673$	−7.56551	−0.291629	−0.145814	−	0.989312i	$-0.546580\pi$
$673$	−7.56551	−0.291629	−0.145814	+	0.989312i	$0.546580\pi$
$674$	0	0
$675$	2.96701	0.114200
$676$	0	0
$677$	−17.0944	−0.656989	−0.328495	−	0.944506i	$-0.606541\pi$
$677$	−17.0944	−0.656989	−0.328495	+	0.944506i	$0.606541\pi$
$678$	0	0
$679$	−4.14753	−0.159168
$680$	0	0
$681$	−17.6827	−0.677603
$682$	0	0
$683$	3.29592	0.126115	0.0630575	−	0.998010i	$-0.479915\pi$
$683$	3.29592	0.126115	0.0630575	+	0.998010i	$0.479915\pi$
$684$	0	0
$685$	11.9624	0.457059
$686$	0	0
$687$	4.93562	0.188306
$688$	0	0
$689$	−1.27252	−0.0484792
$690$	0	0
$691$	−4.51318	−0.171690	−0.0858448	−	0.996309i	$-0.527359\pi$
$691$	−4.51318	−0.171690	−0.0858448	+	0.996309i	$0.527359\pi$
$692$	0	0
$693$	−6.64554	−0.252443
$694$	0	0
$695$	−19.8340	−0.752348
$696$	0	0
$697$	−22.2678	−0.843454
$698$	0	0
$699$	26.5583	1.00453
$700$	0	0
$701$	−5.03168	−0.190044	−0.0950220	−	0.995475i	$-0.530292\pi$
$701$	−5.03168	−0.190044	−0.0950220	+	0.995475i	$0.530292\pi$
$702$	0	0
$703$	−11.7596	−0.443522
$704$	0	0
$705$	14.6178	0.550539
$706$	0	0
$707$	−12.5617	−0.472431
$708$	0	0
$709$	−3.07674	−0.115549	−0.0577747	−	0.998330i	$-0.518400\pi$
$709$	−3.07674	−0.115549	−0.0577747	+	0.998330i	$0.518400\pi$
$710$	0	0
$711$	12.4035	0.465167
$712$	0	0
$713$	−8.87791	−0.332480
$714$	0	0
$715$	−1.01103	−0.0378106
$716$	0	0
$717$	0.0342729	0.00127994
$718$	0	0
$719$	22.0964	0.824057	0.412028	−	0.911171i	$-0.364820\pi$
$719$	22.0964	0.824057	0.412028	+	0.911171i	$0.364820\pi$
$720$	0	0
$721$	−22.0650	−0.821743
$722$	0	0
$723$	15.3244	0.569922
$724$	0	0
$725$	2.96701	0.110192
$726$	0	0
$727$	18.9934	0.704425	0.352212	−	0.935920i	$-0.385430\pi$
$727$	18.9934	0.704425	0.352212	+	0.935920i	$0.385430\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−41.8424	−1.54760
$732$	0	0
$733$	−10.8286	−0.399964	−0.199982	−	0.979800i	$-0.564088\pi$
$733$	−10.8286	−0.399964	−0.199982	+	0.979800i	$0.564088\pi$
$734$	0	0
$735$	6.81468	0.251363
$736$	0	0
$737$	−2.16722	−0.0798304
$738$	0	0
$739$	52.1079	1.91682	0.958410	−	0.285395i	$-0.0921247\pi$
$739$	52.1079	1.91682	0.958410	+	0.285395i	$0.0921247\pi$
$740$	0	0
$741$	0.206023	0.00756844
$742$	0	0
$743$	23.1036	0.847589	0.423795	−	0.905758i	$-0.360698\pi$
$743$	23.1036	0.847589	0.423795	+	0.905758i	$0.360698\pi$
$744$	0	0
$745$	−15.3124	−0.561002
$746$	0	0
$747$	1.30451	0.0477294
$748$	0	0
$749$	−20.1555	−0.736468
$750$	0	0
$751$	−31.5778	−1.15229	−0.576146	−	0.817347i	$-0.695444\pi$
$751$	−31.5778	−1.15229	−0.576146	+	0.817347i	$0.695444\pi$
$752$	0	0
$753$	−5.44938	−0.198586
$754$	0	0
$755$	14.0637	0.511832
$756$	0	0
$757$	−4.93030	−0.179195	−0.0895974	−	0.995978i	$-0.528558\pi$
$757$	−4.93030	−0.179195	−0.0895974	+	0.995978i	$0.528558\pi$
$758$	0	0
$759$	−4.45964	−0.161875
$760$	0	0
$761$	15.0272	0.544735	0.272368	−	0.962193i	$-0.412193\pi$
$761$	15.0272	0.544735	0.272368	+	0.962193i	$0.412193\pi$
$762$	0	0
$763$	19.1488	0.693233
$764$	0	0
$765$	4.95499	0.179148
$766$	0	0
$767$	−1.94404	−0.0701952
$768$	0	0
$769$	49.6405	1.79008	0.895041	−	0.445984i	$-0.147146\pi$
$769$	49.6405	1.79008	0.895041	+	0.445984i	$0.147146\pi$
$770$	0	0
$771$	1.38441	0.0498585
$772$	0	0
$773$	−4.88562	−0.175724	−0.0878618	−	0.996133i	$-0.528003\pi$
$773$	−4.88562	−0.175724	−0.0878618	+	0.996133i	$0.528003\pi$
$774$	0	0
$775$	26.3409	0.946193
$776$	0	0
$777$	−13.5240	−0.485172
$778$	0	0
$779$	−8.30269	−0.297475
$780$	0	0
$781$	34.0862	1.21970
$782$	0	0
$783$	1.00000	0.0357371
$784$	0	0
$785$	6.78219	0.242067
$786$	0	0
$787$	−3.31990	−0.118342	−0.0591708	−	0.998248i	$-0.518846\pi$
$787$	−3.31990	−0.118342	−0.0591708	+	0.998248i	$0.518846\pi$
$788$	0	0
$789$	−6.66705	−0.237353
$790$	0	0
$791$	−0.583010	−0.0207294
$792$	0	0
$793$	1.63770	0.0581565
$794$	0	0
$795$	−11.4112	−0.404715
$796$	0	0
$797$	−28.8324	−1.02129	−0.510647	−	0.859790i	$-0.670594\pi$
$797$	−28.8324	−1.02129	−0.510647	+	0.859790i	$0.670594\pi$
$798$	0	0
$799$	−35.6280	−1.26043
$800$	0	0
$801$	−18.2043	−0.643218
$802$	0	0
$803$	−16.5828	−0.585194
$804$	0	0
$805$	−2.12470	−0.0748858
$806$	0	0
$807$	−30.4318	−1.07125
$808$	0	0
$809$	−44.7935	−1.57486	−0.787428	−	0.616407i	$-0.788588\pi$
$809$	−44.7935	−1.57486	−0.787428	+	0.616407i	$0.788588\pi$
$810$	0	0
$811$	−17.1891	−0.603591	−0.301795	−	0.953373i	$-0.597586\pi$
$811$	−17.1891	−0.603591	−0.301795	+	0.953373i	$0.597586\pi$
$812$	0	0
$813$	18.2818	0.641170
$814$	0	0
$815$	7.87069	0.275698
$816$	0	0
$817$	−15.6012	−0.545816
$818$	0	0
$819$	0.236935	0.00827917
$820$	0	0
$821$	−5.07521	−0.177126	−0.0885630	−	0.996071i	$-0.528227\pi$
$821$	−5.07521	−0.177126	−0.0885630	+	0.996071i	$0.528227\pi$
$822$	0	0
$823$	34.8400	1.21444	0.607222	−	0.794532i	$-0.292284\pi$
$823$	34.8400	1.21444	0.607222	+	0.794532i	$0.292284\pi$
$824$	0	0
$825$	13.2318	0.460673
$826$	0	0
$827$	−6.00246	−0.208726	−0.104363	−	0.994539i	$-0.533280\pi$
$827$	−6.00246	−0.208726	−0.104363	+	0.994539i	$0.533280\pi$
$828$	0	0
$829$	−6.25611	−0.217284	−0.108642	−	0.994081i	$-0.534650\pi$
$829$	−6.25611	−0.217284	−0.108642	+	0.994081i	$0.534650\pi$
$830$	0	0
$831$	−4.36334	−0.151363
$832$	0	0
$833$	−16.6094	−0.575481
$834$	0	0
$835$	−5.40459	−0.187034
$836$	0	0
$837$	8.87791	0.306866
$838$	0	0
$839$	−0.807807	−0.0278886	−0.0139443	−	0.999903i	$-0.504439\pi$
$839$	−0.807807	−0.0278886	−0.0139443	+	0.999903i	$0.504439\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	14.1143	0.486123
$844$	0	0
$845$	−18.4997	−0.636410
$846$	0	0
$847$	−13.2451	−0.455106
$848$	0	0
$849$	−2.84332	−0.0975826
$850$	0	0
$851$	−9.07562	−0.311108
$852$	0	0
$853$	17.6367	0.603868	0.301934	−	0.953329i	$-0.402368\pi$
$853$	17.6367	0.603868	0.301934	+	0.953329i	$0.402368\pi$
$854$	0	0
$855$	1.84750	0.0631831
$856$	0	0
$857$	−26.1242	−0.892387	−0.446193	−	0.894937i	$-0.647221\pi$
$857$	−26.1242	−0.892387	−0.446193	+	0.894937i	$0.647221\pi$
$858$	0	0
$859$	51.9607	1.77288	0.886438	−	0.462848i	$-0.153172\pi$
$859$	51.9607	1.77288	0.886438	+	0.462848i	$0.153172\pi$
$860$	0	0
$861$	−9.54843	−0.325410
$862$	0	0
$863$	37.1896	1.26595	0.632975	−	0.774172i	$-0.281834\pi$
$863$	37.1896	1.26595	0.632975	+	0.774172i	$0.281834\pi$
$864$	0	0
$865$	33.0664	1.12429
$866$	0	0
$867$	4.92322	0.167201
$868$	0	0
$869$	55.3151	1.87644
$870$	0	0
$871$	0.0772682	0.00261813
$872$	0	0
$873$	2.78330	0.0942004
$874$	0	0
$875$	16.9275	0.572254
$876$	0	0
$877$	2.89445	0.0977385	0.0488692	−	0.998805i	$-0.484438\pi$
$877$	2.89445	0.0977385	0.0488692	+	0.998805i	$0.484438\pi$
$878$	0	0
$879$	9.50985	0.320759
$880$	0	0
$881$	−7.22220	−0.243322	−0.121661	−	0.992572i	$-0.538822\pi$
$881$	−7.22220	−0.243322	−0.121661	+	0.992572i	$0.538822\pi$
$882$	0	0
$883$	19.5749	0.658747	0.329374	−	0.944200i	$-0.393162\pi$
$883$	19.5749	0.658747	0.329374	+	0.944200i	$0.393162\pi$
$884$	0	0
$885$	−17.4331	−0.586006
$886$	0	0
$887$	14.0585	0.472038	0.236019	−	0.971748i	$-0.424157\pi$
$887$	14.0585	0.472038	0.236019	+	0.971748i	$0.424157\pi$
$888$	0	0
$889$	17.0040	0.570296
$890$	0	0
$891$	4.45964	0.149404
$892$	0	0
$893$	−13.2841	−0.444536
$894$	0	0
$895$	−33.1428	−1.10784
$896$	0	0
$897$	0.159001	0.00530888
$898$	0	0
$899$	8.87791	0.296095
$900$	0	0
$901$	27.8126	0.926572
$902$	0	0
$903$	−17.9420	−0.597073
$904$	0	0
$905$	−6.96205	−0.231426
$906$	0	0
$907$	53.7323	1.78415	0.892076	−	0.451886i	$-0.149249\pi$
$907$	53.7323	1.78415	0.892076	+	0.451886i	$0.149249\pi$
$908$	0	0
$909$	8.42981	0.279599
$910$	0	0
$911$	20.9457	0.693962	0.346981	−	0.937872i	$-0.387207\pi$
$911$	20.9457	0.693962	0.346981	+	0.937872i	$0.387207\pi$
$912$	0	0
$913$	5.81763	0.192536
$914$	0	0
$915$	14.6860	0.485504
$916$	0	0
$917$	−11.5561	−0.381617
$918$	0	0
$919$	27.6524	0.912169	0.456085	−	0.889936i	$-0.349251\pi$
$919$	27.6524	0.912169	0.456085	+	0.889936i	$0.349251\pi$
$920$	0	0
$921$	5.05037	0.166415
$922$	0	0
$923$	−1.21528	−0.0400015
$924$	0	0
$925$	26.9275	0.885371
$926$	0	0
$927$	14.8072	0.486333
$928$	0	0
$929$	16.7689	0.550168	0.275084	−	0.961420i	$-0.411294\pi$
$929$	16.7689	0.550168	0.275084	+	0.961420i	$0.411294\pi$
$930$	0	0
$931$	−6.19291	−0.202964
$932$	0	0
$933$	−32.6238	−1.06806
$934$	0	0
$935$	22.0975	0.722665
$936$	0	0
$937$	−38.0173	−1.24197	−0.620985	−	0.783822i	$-0.713267\pi$
$937$	−38.0173	−1.24197	−0.620985	+	0.783822i	$0.713267\pi$
$938$	0	0
$939$	6.08094	0.198444
$940$	0	0
$941$	−0.763245	−0.0248811	−0.0124405	−	0.999923i	$-0.503960\pi$
$941$	−0.763245	−0.0248811	−0.0124405	+	0.999923i	$0.503960\pi$
$942$	0	0
$943$	−6.40770	−0.208663
$944$	0	0
$945$	2.12470	0.0691164
$946$	0	0
$947$	−27.1708	−0.882932	−0.441466	−	0.897278i	$-0.645541\pi$
$947$	−27.1708	−0.882932	−0.441466	+	0.897278i	$0.645541\pi$
$948$	0	0
$949$	0.591229	0.0191921
$950$	0	0
$951$	−26.3432	−0.854237
$952$	0	0
$953$	24.2213	0.784605	0.392302	−	0.919836i	$-0.371679\pi$
$953$	24.2213	0.784605	0.392302	+	0.919836i	$0.371679\pi$
$954$	0	0
$955$	−31.2369	−1.01080
$956$	0	0
$957$	4.45964	0.144160
$958$	0	0
$959$	−12.5020	−0.403712
$960$	0	0
$961$	47.8173	1.54249
$962$	0	0
$963$	13.5258	0.435864
$964$	0	0
$965$	−11.8218	−0.380557
$966$	0	0
$967$	29.2987	0.942184	0.471092	−	0.882084i	$-0.343860\pi$
$967$	29.2987	0.942184	0.471092	+	0.882084i	$0.343860\pi$
$968$	0	0
$969$	−4.50290	−0.144654
$970$	0	0
$971$	−41.1612	−1.32092	−0.660462	−	0.750859i	$-0.729640\pi$
$971$	−41.1612	−1.32092	−0.660462	+	0.750859i	$0.729640\pi$
$972$	0	0
$973$	20.7288	0.664535
$974$	0	0
$975$	−0.471757	−0.0151083
$976$	0	0
$977$	12.6314	0.404115	0.202057	−	0.979374i	$-0.435237\pi$
$977$	12.6314	0.404115	0.202057	+	0.979374i	$0.435237\pi$
$978$	0	0
$979$	−81.1848	−2.59468
$980$	0	0
$981$	−12.8502	−0.410277
$982$	0	0
$983$	−9.56242	−0.304994	−0.152497	−	0.988304i	$-0.548731\pi$
$983$	−9.56242	−0.304994	−0.152497	+	0.988304i	$0.548731\pi$
$984$	0	0
$985$	5.99516	0.191022
$986$	0	0
$987$	−15.2773	−0.486281
$988$	0	0
$989$	−12.0404	−0.382862
$990$	0	0
$991$	18.8307	0.598177	0.299089	−	0.954225i	$-0.403317\pi$
$991$	18.8307	0.598177	0.299089	+	0.954225i	$0.403317\pi$
$992$	0	0
$993$	2.49309	0.0791157
$994$	0	0
$995$	−18.4310	−0.584300
$996$	0	0
$997$	−39.5586	−1.25283	−0.626416	−	0.779489i	$-0.715479\pi$
$997$	−39.5586	−1.25283	−0.626416	+	0.779489i	$0.715479\pi$
$998$	0	0
$999$	9.07562	0.287140

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8004.2.a.g.1.9	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8004.2.a.g.1.9	✓	12	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:	\( N \)	\(=\)	\( 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8004.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.42583 q^{5} -1.49015 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.42583 q^{5} -1.49015 q^{7} +1.00000 q^{9} +4.45964 q^{11} -0.159001 q^{13} -1.42583 q^{15} +3.47517 q^{17} +1.29574 q^{19} +1.49015 q^{21} +1.00000 q^{23} -2.96701 q^{25} -1.00000 q^{27} -1.00000 q^{29} -8.87791 q^{31} -4.45964 q^{33} -2.12470 q^{35} -9.07562 q^{37} +0.159001 q^{39} -6.40770 q^{41} -12.0404 q^{43} +1.42583 q^{45} -10.2522 q^{47} -4.77945 q^{49} -3.47517 q^{51} +8.00324 q^{53} +6.35868 q^{55} -1.29574 q^{57} +12.2266 q^{59} -10.3000 q^{61} -1.49015 q^{63} -0.226708 q^{65} -0.485962 q^{67} -1.00000 q^{69} +7.64325 q^{71} -3.71841 q^{73} +2.96701 q^{75} -6.64554 q^{77} +12.4035 q^{79} +1.00000 q^{81} +1.30451 q^{83} +4.95499 q^{85} +1.00000 q^{87} -18.2043 q^{89} +0.236935 q^{91} +8.87791 q^{93} +1.84750 q^{95} +2.78330 q^{97} +4.45964 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{3} - 3 q^{5} + 4 q^{7} + 12 q^{9}+O(q^{10}) \) 12 * q - 12 * q^3 - 3 * q^5 + 4 * q^7 + 12 * q^9	\( 12 q - 12 q^{3} - 3 q^{5} + 4 q^{7} + 12 q^{9} - 5 q^{11} - 6 q^{13} + 3 q^{15} - 7 q^{17} - 3 q^{19} - 4 q^{21} + 12 q^{23} + 11 q^{25} - 12 q^{27} - 12 q^{29} + 2 q^{31} + 5 q^{33} - 9 q^{35} - 20 q^{37} + 6 q^{39} - 3 q^{41} + 5 q^{43} - 3 q^{45} - 2 q^{49} + 7 q^{51} - 3 q^{53} + 19 q^{55} + 3 q^{57} - 20 q^{59} - 17 q^{61} + 4 q^{63} - 4 q^{65} - 9 q^{67} - 12 q^{69} + 7 q^{71} - 9 q^{73} - 11 q^{75} - 34 q^{77} + 14 q^{79} + 12 q^{81} + 5 q^{83} - 12 q^{85} + 12 q^{87} - 22 q^{89} - 3 q^{91} - 2 q^{93} - 27 q^{95} + 17 q^{97} - 5 q^{99}+O(q^{100}) \) 12 * q - 12 * q^3 - 3 * q^5 + 4 * q^7 + 12 * q^9 - 5 * q^11 - 6 * q^13 + 3 * q^15 - 7 * q^17 - 3 * q^19 - 4 * q^21 + 12 * q^23 + 11 * q^25 - 12 * q^27 - 12 * q^29 + 2 * q^31 + 5 * q^33 - 9 * q^35 - 20 * q^37 + 6 * q^39 - 3 * q^41 + 5 * q^43 - 3 * q^45 - 2 * q^49 + 7 * q^51 - 3 * q^53 + 19 * q^55 + 3 * q^57 - 20 * q^59 - 17 * q^61 + 4 * q^63 - 4 * q^65 - 9 * q^67 - 12 * q^69 + 7 * q^71 - 9 * q^73 - 11 * q^75 - 34 * q^77 + 14 * q^79 + 12 * q^81 + 5 * q^83 - 12 * q^85 + 12 * q^87 - 22 * q^89 - 3 * q^91 - 2 * q^93 - 27 * q^95 + 17 * q^97 - 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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