LMFDB - Embedded newform 8004.2.a.k.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("8004.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

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

Embedding invariants

Embedding label			1.8
Root			$0.140190$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +0.140190 q^{5} -1.18735 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +0.140190 q^{5} -1.18735 q^{7} +1.00000 q^{9} -4.10415 q^{11} +5.07840 q^{13} +0.140190 q^{15} +0.809667 q^{17} -3.72968 q^{19} -1.18735 q^{21} -1.00000 q^{23} -4.98035 q^{25} +1.00000 q^{27} +1.00000 q^{29} -1.08094 q^{31} -4.10415 q^{33} -0.166455 q^{35} +10.4285 q^{37} +5.07840 q^{39} +0.292978 q^{41} -0.309545 q^{43} +0.140190 q^{45} -3.45622 q^{47} -5.59019 q^{49} +0.809667 q^{51} +10.2648 q^{53} -0.575360 q^{55} -3.72968 q^{57} +11.3250 q^{59} +6.27295 q^{61} -1.18735 q^{63} +0.711940 q^{65} -1.24568 q^{67} -1.00000 q^{69} -4.37717 q^{71} +12.5571 q^{73} -4.98035 q^{75} +4.87307 q^{77} +16.9174 q^{79} +1.00000 q^{81} -10.7955 q^{83} +0.113507 q^{85} +1.00000 q^{87} -1.34778 q^{89} -6.02985 q^{91} -1.08094 q^{93} -0.522864 q^{95} -0.646036 q^{97} -4.10415 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 18 q^{3} + 5 q^{5} + 6 q^{7} + 18 q^{9}+O(q^{10}) $ 18 * q + 18 * q^3 + 5 * q^5 + 6 * q^7 + 18 * q^9	$ 18 q + 18 q^{3} + 5 q^{5} + 6 q^{7} + 18 q^{9} + 5 q^{11} + 6 q^{13} + 5 q^{15} + 7 q^{17} + 15 q^{19} + 6 q^{21} - 18 q^{23} + 45 q^{25} + 18 q^{27} + 18 q^{29} + 10 q^{31} + 5 q^{33} - 7 q^{35} + 22 q^{37} + 6 q^{39} + 17 q^{41} + 25 q^{43} + 5 q^{45} - 6 q^{47} + 64 q^{49} + 7 q^{51} + 21 q^{53} + 3 q^{55} + 15 q^{57} + 6 q^{59} + 7 q^{61} + 6 q^{63} + 44 q^{65} + 35 q^{67} - 18 q^{69} + q^{71} + 27 q^{73} + 45 q^{75} + 4 q^{77} + 18 q^{81} + 13 q^{83} + 24 q^{85} + 18 q^{87} + 30 q^{89} + 53 q^{91} + 10 q^{93} + 17 q^{95} + 27 q^{97} + 5 q^{99}+O(q^{100}) $ 18 * q + 18 * q^3 + 5 * q^5 + 6 * q^7 + 18 * q^9 + 5 * q^11 + 6 * q^13 + 5 * q^15 + 7 * q^17 + 15 * q^19 + 6 * q^21 - 18 * q^23 + 45 * q^25 + 18 * q^27 + 18 * q^29 + 10 * q^31 + 5 * q^33 - 7 * q^35 + 22 * q^37 + 6 * q^39 + 17 * q^41 + 25 * q^43 + 5 * q^45 - 6 * q^47 + 64 * q^49 + 7 * q^51 + 21 * q^53 + 3 * q^55 + 15 * q^57 + 6 * q^59 + 7 * q^61 + 6 * q^63 + 44 * q^65 + 35 * q^67 - 18 * q^69 + q^71 + 27 * q^73 + 45 * q^75 + 4 * q^77 + 18 * q^81 + 13 * q^83 + 24 * q^85 + 18 * q^87 + 30 * q^89 + 53 * q^91 + 10 * q^93 + 17 * q^95 + 27 * q^97 + 5 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	0.140190	0.0626949	0.0313474	−	0.999509i	$-0.490020\pi$
$5$	0.140190	0.0626949	0.0313474	+	0.999509i	$0.490020\pi$
$6$	0	0
$7$	−1.18735	−0.448777	−0.224389	−	0.974500i	$-0.572038\pi$
$7$	−1.18735	−0.448777	−0.224389	+	0.974500i	$0.572038\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−4.10415	−1.23745	−0.618723	−	0.785609i	$-0.712350\pi$
$11$	−4.10415	−1.23745	−0.618723	+	0.785609i	$0.712350\pi$
$12$	0	0
$13$	5.07840	1.40849	0.704247	−	0.709955i	$-0.251285\pi$
$13$	5.07840	1.40849	0.704247	+	0.709955i	$0.251285\pi$
$14$	0	0
$15$	0.140190	0.0361969
$16$	0	0
$17$	0.809667	0.196373	0.0981866	−	0.995168i	$-0.468696\pi$
$17$	0.809667	0.196373	0.0981866	+	0.995168i	$0.468696\pi$
$18$	0	0
$19$	−3.72968	−0.855647	−0.427823	−	0.903862i	$-0.640720\pi$
$19$	−3.72968	−0.855647	−0.427823	+	0.903862i	$0.640720\pi$
$20$	0	0
$21$	−1.18735	−0.259102
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−4.98035	−0.996069
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−1.08094	−0.194142	−0.0970712	−	0.995277i	$-0.530947\pi$
$31$	−1.08094	−0.194142	−0.0970712	+	0.995277i	$0.530947\pi$
$32$	0	0
$33$	−4.10415	−0.714440
$34$	0	0
$35$	−0.166455	−0.0281360
$36$	0	0
$37$	10.4285	1.71444	0.857219	−	0.514952i	$-0.172190\pi$
$37$	10.4285	1.71444	0.857219	+	0.514952i	$0.172190\pi$
$38$	0	0
$39$	5.07840	0.813194
$40$	0	0
$41$	0.292978	0.0457554	0.0228777	−	0.999738i	$-0.492717\pi$
$41$	0.292978	0.0457554	0.0228777	+	0.999738i	$0.492717\pi$
$42$	0	0
$43$	−0.309545	−0.0472052	−0.0236026	−	0.999721i	$-0.507514\pi$
$43$	−0.309545	−0.0472052	−0.0236026	+	0.999721i	$0.507514\pi$
$44$	0	0
$45$	0.140190	0.0208983
$46$	0	0
$47$	−3.45622	−0.504142	−0.252071	−	0.967709i	$-0.581112\pi$
$47$	−3.45622	−0.504142	−0.252071	+	0.967709i	$0.581112\pi$
$48$	0	0
$49$	−5.59019	−0.798599
$50$	0	0
$51$	0.809667	0.113376
$52$	0	0
$53$	10.2648	1.40998	0.704990	−	0.709217i	$-0.250951\pi$
$53$	10.2648	1.40998	0.704990	+	0.709217i	$0.250951\pi$
$54$	0	0
$55$	−0.575360	−0.0775815
$56$	0	0
$57$	−3.72968	−0.494008
$58$	0	0
$59$	11.3250	1.47439	0.737197	−	0.675677i	$-0.236149\pi$
$59$	11.3250	1.47439	0.737197	+	0.675677i	$0.236149\pi$
$60$	0	0
$61$	6.27295	0.803168	0.401584	−	0.915822i	$-0.368460\pi$
$61$	6.27295	0.803168	0.401584	+	0.915822i	$0.368460\pi$
$62$	0	0
$63$	−1.18735	−0.149592
$64$	0	0
$65$	0.711940	0.0883053
$66$	0	0
$67$	−1.24568	−0.152184	−0.0760921	−	0.997101i	$-0.524244\pi$
$67$	−1.24568	−0.152184	−0.0760921	+	0.997101i	$0.524244\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−4.37717	−0.519474	−0.259737	−	0.965679i	$-0.583636\pi$
$71$	−4.37717	−0.519474	−0.259737	+	0.965679i	$0.583636\pi$
$72$	0	0
$73$	12.5571	1.46969	0.734846	−	0.678234i	$-0.237254\pi$
$73$	12.5571	1.46969	0.734846	+	0.678234i	$0.237254\pi$
$74$	0	0
$75$	−4.98035	−0.575081
$76$	0	0
$77$	4.87307	0.555338
$78$	0	0
$79$	16.9174	1.90336	0.951681	−	0.307090i	$-0.0993552\pi$
$79$	16.9174	1.90336	0.951681	+	0.307090i	$0.0993552\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−10.7955	−1.18496	−0.592481	−	0.805584i	$-0.701851\pi$
$83$	−10.7955	−1.18496	−0.592481	+	0.805584i	$0.701851\pi$
$84$	0	0
$85$	0.113507	0.0123116
$86$	0	0
$87$	1.00000	0.107211
$88$	0	0
$89$	−1.34778	−0.142865	−0.0714324	−	0.997445i	$-0.522757\pi$
$89$	−1.34778	−0.142865	−0.0714324	+	0.997445i	$0.522757\pi$
$90$	0	0
$91$	−6.02985	−0.632100
$92$	0	0
$93$	−1.08094	−0.112088
$94$	0	0
$95$	−0.522864	−0.0536447
$96$	0	0
$97$	−0.646036	−0.0655951	−0.0327975	−	0.999462i	$-0.510442\pi$
$97$	−0.646036	−0.0655951	−0.0327975	+	0.999462i	$0.510442\pi$
$98$	0	0
$99$	−4.10415	−0.412482
$100$	0	0
$101$	−3.02908	−0.301404	−0.150702	−	0.988579i	$-0.548153\pi$
$101$	−3.02908	−0.301404	−0.150702	+	0.988579i	$0.548153\pi$
$102$	0	0
$103$	12.4190	1.22369	0.611843	−	0.790980i	$-0.290429\pi$
$103$	12.4190	1.22369	0.611843	+	0.790980i	$0.290429\pi$
$104$	0	0
$105$	−0.166455	−0.0162443
$106$	0	0
$107$	5.44849	0.526725	0.263363	−	0.964697i	$-0.415168\pi$
$107$	5.44849	0.526725	0.263363	+	0.964697i	$0.415168\pi$
$108$	0	0
$109$	10.4138	0.997465	0.498733	−	0.866756i	$-0.333799\pi$
$109$	10.4138	0.997465	0.498733	+	0.866756i	$0.333799\pi$
$110$	0	0
$111$	10.4285	0.989831
$112$	0	0
$113$	18.1058	1.70325	0.851624	−	0.524153i	$-0.175618\pi$
$113$	18.1058	1.70325	0.851624	+	0.524153i	$0.175618\pi$
$114$	0	0
$115$	−0.140190	−0.0130728
$116$	0	0
$117$	5.07840	0.469498
$118$	0	0
$119$	−0.961361	−0.0881278
$120$	0	0
$121$	5.84401	0.531273
$122$	0	0
$123$	0.292978	0.0264169
$124$	0	0
$125$	−1.39914	−0.125143
$126$	0	0
$127$	−2.57184	−0.228214	−0.114107	−	0.993468i	$-0.536401\pi$
$127$	−2.57184	−0.228214	−0.114107	+	0.993468i	$0.536401\pi$
$128$	0	0
$129$	−0.309545	−0.0272539
$130$	0	0
$131$	−4.33578	−0.378819	−0.189409	−	0.981898i	$-0.560657\pi$
$131$	−4.33578	−0.378819	−0.189409	+	0.981898i	$0.560657\pi$
$132$	0	0
$133$	4.42845	0.383995
$134$	0	0
$135$	0.140190	0.0120656
$136$	0	0
$137$	9.66881	0.826062	0.413031	−	0.910717i	$-0.364470\pi$
$137$	9.66881	0.826062	0.413031	+	0.910717i	$0.364470\pi$
$138$	0	0
$139$	−18.3036	−1.55249	−0.776244	−	0.630433i	$-0.782877\pi$
$139$	−18.3036	−1.55249	−0.776244	+	0.630433i	$0.782877\pi$
$140$	0	0
$141$	−3.45622	−0.291066
$142$	0	0
$143$	−20.8425	−1.74294
$144$	0	0
$145$	0.140190	0.0116421
$146$	0	0
$147$	−5.59019	−0.461071
$148$	0	0
$149$	−19.6837	−1.61255	−0.806274	−	0.591542i	$-0.798519\pi$
$149$	−19.6837	−1.61255	−0.806274	+	0.591542i	$0.798519\pi$
$150$	0	0
$151$	12.3990	1.00902	0.504509	−	0.863407i	$-0.331674\pi$
$151$	12.3990	1.00902	0.504509	+	0.863407i	$0.331674\pi$
$152$	0	0
$153$	0.809667	0.0654577
$154$	0	0
$155$	−0.151537	−0.0121717
$156$	0	0
$157$	10.4378	0.833024	0.416512	−	0.909130i	$-0.363252\pi$
$157$	10.4378	0.833024	0.416512	+	0.909130i	$0.363252\pi$
$158$	0	0
$159$	10.2648	0.814053
$160$	0	0
$161$	1.18735	0.0935765
$162$	0	0
$163$	−16.3310	−1.27914	−0.639570	−	0.768733i	$-0.720888\pi$
$163$	−16.3310	−1.27914	−0.639570	+	0.768733i	$0.720888\pi$
$164$	0	0
$165$	−0.575360	−0.0447917
$166$	0	0
$167$	2.76211	0.213738	0.106869	−	0.994273i	$-0.465917\pi$
$167$	2.76211	0.213738	0.106869	+	0.994273i	$0.465917\pi$
$168$	0	0
$169$	12.7901	0.983855
$170$	0	0
$171$	−3.72968	−0.285216
$172$	0	0
$173$	22.1926	1.68727	0.843634	−	0.536918i	$-0.180412\pi$
$173$	22.1926	1.68727	0.843634	+	0.536918i	$0.180412\pi$
$174$	0	0
$175$	5.91343	0.447013
$176$	0	0
$177$	11.3250	0.851242
$178$	0	0
$179$	−11.0849	−0.828527	−0.414264	−	0.910157i	$-0.635961\pi$
$179$	−11.0849	−0.828527	−0.414264	+	0.910157i	$0.635961\pi$
$180$	0	0
$181$	−9.80187	−0.728567	−0.364284	−	0.931288i	$-0.618686\pi$
$181$	−9.80187	−0.728567	−0.364284	+	0.931288i	$0.618686\pi$
$182$	0	0
$183$	6.27295	0.463709
$184$	0	0
$185$	1.46197	0.107486
$186$	0	0
$187$	−3.32299	−0.243001
$188$	0	0
$189$	−1.18735	−0.0863672
$190$	0	0
$191$	1.71424	0.124038	0.0620191	−	0.998075i	$-0.480246\pi$
$191$	1.71424	0.124038	0.0620191	+	0.998075i	$0.480246\pi$
$192$	0	0
$193$	8.44637	0.607983	0.303992	−	0.952675i	$-0.401681\pi$
$193$	8.44637	0.607983	0.303992	+	0.952675i	$0.401681\pi$
$194$	0	0
$195$	0.711940	0.0509831
$196$	0	0
$197$	−26.8409	−1.91233	−0.956167	−	0.292822i	$-0.905406\pi$
$197$	−26.8409	−1.91233	−0.956167	+	0.292822i	$0.905406\pi$
$198$	0	0
$199$	10.3515	0.733801	0.366900	−	0.930260i	$-0.380419\pi$
$199$	10.3515	0.733801	0.366900	+	0.930260i	$0.380419\pi$
$200$	0	0
$201$	−1.24568	−0.0878636
$202$	0	0
$203$	−1.18735	−0.0833358
$204$	0	0
$205$	0.0410725	0.00286863
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	15.3071	1.05882
$210$	0	0
$211$	20.5502	1.41474	0.707368	−	0.706845i	$-0.249882\pi$
$211$	20.5502	1.41474	0.707368	+	0.706845i	$0.249882\pi$
$212$	0	0
$213$	−4.37717	−0.299919
$214$	0	0
$215$	−0.0433951	−0.00295952
$216$	0	0
$217$	1.28346	0.0871267
$218$	0	0
$219$	12.5571	0.848527
$220$	0	0
$221$	4.11181	0.276590
$222$	0	0
$223$	17.0966	1.14487	0.572436	−	0.819949i	$-0.305998\pi$
$223$	17.0966	1.14487	0.572436	+	0.819949i	$0.305998\pi$
$224$	0	0
$225$	−4.98035	−0.332023
$226$	0	0
$227$	−9.44648	−0.626985	−0.313492	−	0.949591i	$-0.601499\pi$
$227$	−9.44648	−0.626985	−0.313492	+	0.949591i	$0.601499\pi$
$228$	0	0
$229$	18.6279	1.23096	0.615482	−	0.788151i	$-0.288961\pi$
$229$	18.6279	1.23096	0.615482	+	0.788151i	$0.288961\pi$
$230$	0	0
$231$	4.87307	0.320624
$232$	0	0
$233$	4.25939	0.279042	0.139521	−	0.990219i	$-0.455444\pi$
$233$	4.25939	0.279042	0.139521	+	0.990219i	$0.455444\pi$
$234$	0	0
$235$	−0.484528	−0.0316071
$236$	0	0
$237$	16.9174	1.09891
$238$	0	0
$239$	20.8179	1.34660	0.673300	−	0.739370i	$-0.264876\pi$
$239$	20.8179	1.34660	0.673300	+	0.739370i	$0.264876\pi$
$240$	0	0
$241$	−26.5631	−1.71108	−0.855539	−	0.517738i	$-0.826774\pi$
$241$	−26.5631	−1.71108	−0.855539	+	0.517738i	$0.826774\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−0.783689	−0.0500681
$246$	0	0
$247$	−18.9408	−1.20517
$248$	0	0
$249$	−10.7955	−0.684138
$250$	0	0
$251$	−11.7201	−0.739763	−0.369882	−	0.929079i	$-0.620602\pi$
$251$	−11.7201	−0.739763	−0.369882	+	0.929079i	$0.620602\pi$
$252$	0	0
$253$	4.10415	0.258025
$254$	0	0
$255$	0.113507	0.00710810
$256$	0	0
$257$	12.9362	0.806937	0.403469	−	0.914993i	$-0.367804\pi$
$257$	12.9362	0.806937	0.403469	+	0.914993i	$0.367804\pi$
$258$	0	0
$259$	−12.3823	−0.769401
$260$	0	0
$261$	1.00000	0.0618984
$262$	0	0
$263$	8.02833	0.495048	0.247524	−	0.968882i	$-0.420383\pi$
$263$	8.02833	0.495048	0.247524	+	0.968882i	$0.420383\pi$
$264$	0	0
$265$	1.43902	0.0883986
$266$	0	0
$267$	−1.34778	−0.0824830
$268$	0	0
$269$	−5.02254	−0.306230	−0.153115	−	0.988208i	$-0.548930\pi$
$269$	−5.02254	−0.306230	−0.153115	+	0.988208i	$0.548930\pi$
$270$	0	0
$271$	0.872807	0.0530192	0.0265096	−	0.999649i	$-0.491561\pi$
$271$	0.872807	0.0530192	0.0265096	+	0.999649i	$0.491561\pi$
$272$	0	0
$273$	−6.02985	−0.364943
$274$	0	0
$275$	20.4401	1.23258
$276$	0	0
$277$	8.91671	0.535753	0.267877	−	0.963453i	$-0.413678\pi$
$277$	8.91671	0.535753	0.267877	+	0.963453i	$0.413678\pi$
$278$	0	0
$279$	−1.08094	−0.0647141
$280$	0	0
$281$	24.1240	1.43912	0.719560	−	0.694430i	$-0.244344\pi$
$281$	24.1240	1.43912	0.719560	+	0.694430i	$0.244344\pi$
$282$	0	0
$283$	−22.5722	−1.34178	−0.670888	−	0.741559i	$-0.734087\pi$
$283$	−22.5722	−1.34178	−0.670888	+	0.741559i	$0.734087\pi$
$284$	0	0
$285$	−0.522864	−0.0309718
$286$	0	0
$287$	−0.347868	−0.0205340
$288$	0	0
$289$	−16.3444	−0.961438
$290$	0	0
$291$	−0.646036	−0.0378713
$292$	0	0
$293$	11.8063	0.689729	0.344864	−	0.938652i	$-0.387925\pi$
$293$	11.8063	0.689729	0.344864	+	0.938652i	$0.387925\pi$
$294$	0	0
$295$	1.58766	0.0924370
$296$	0	0
$297$	−4.10415	−0.238147
$298$	0	0
$299$	−5.07840	−0.293691
$300$	0	0
$301$	0.367539	0.0211846
$302$	0	0
$303$	−3.02908	−0.174016
$304$	0	0
$305$	0.879404	0.0503545
$306$	0	0
$307$	26.1028	1.48976	0.744882	−	0.667196i	$-0.232506\pi$
$307$	26.1028	1.48976	0.744882	+	0.667196i	$0.232506\pi$
$308$	0	0
$309$	12.4190	0.706495
$310$	0	0
$311$	−26.2697	−1.48962	−0.744808	−	0.667279i	$-0.767459\pi$
$311$	−26.2697	−1.48962	−0.744808	+	0.667279i	$0.767459\pi$
$312$	0	0
$313$	34.8344	1.96896	0.984479	−	0.175501i	$-0.0561544\pi$
$313$	34.8344	1.96896	0.984479	+	0.175501i	$0.0561544\pi$
$314$	0	0
$315$	−0.166455	−0.00937868
$316$	0	0
$317$	11.1689	0.627307	0.313654	−	0.949537i	$-0.398447\pi$
$317$	11.1689	0.627307	0.313654	+	0.949537i	$0.398447\pi$
$318$	0	0
$319$	−4.10415	−0.229788
$320$	0	0
$321$	5.44849	0.304105
$322$	0	0
$323$	−3.01980	−0.168026
$324$	0	0
$325$	−25.2922	−1.40296
$326$	0	0
$327$	10.4138	0.575887
$328$	0	0
$329$	4.10375	0.226247
$330$	0	0
$331$	12.0196	0.660658	0.330329	−	0.943866i	$-0.392840\pi$
$331$	12.0196	0.660658	0.330329	+	0.943866i	$0.392840\pi$
$332$	0	0
$333$	10.4285	0.571479
$334$	0	0
$335$	−0.174632	−0.00954117
$336$	0	0
$337$	30.3540	1.65349	0.826743	−	0.562580i	$-0.190191\pi$
$337$	30.3540	1.65349	0.826743	+	0.562580i	$0.190191\pi$
$338$	0	0
$339$	18.1058	0.983371
$340$	0	0
$341$	4.43633	0.240241
$342$	0	0
$343$	14.9490	0.807170
$344$	0	0
$345$	−0.140190	−0.00754758
$346$	0	0
$347$	7.51123	0.403224	0.201612	−	0.979465i	$-0.435382\pi$
$347$	7.51123	0.403224	0.201612	+	0.979465i	$0.435382\pi$
$348$	0	0
$349$	2.57370	0.137767	0.0688834	−	0.997625i	$-0.478056\pi$
$349$	2.57370	0.137767	0.0688834	+	0.997625i	$0.478056\pi$
$350$	0	0
$351$	5.07840	0.271065
$352$	0	0
$353$	2.86078	0.152264	0.0761320	−	0.997098i	$-0.475743\pi$
$353$	2.86078	0.152264	0.0761320	+	0.997098i	$0.475743\pi$
$354$	0	0
$355$	−0.613635	−0.0325684
$356$	0	0
$357$	−0.961361	−0.0508806
$358$	0	0
$359$	−7.05142	−0.372160	−0.186080	−	0.982535i	$-0.559578\pi$
$359$	−7.05142	−0.372160	−0.186080	+	0.982535i	$0.559578\pi$
$360$	0	0
$361$	−5.08950	−0.267868
$362$	0	0
$363$	5.84401	0.306731
$364$	0	0
$365$	1.76037	0.0921421
$366$	0	0
$367$	28.8406	1.50547	0.752733	−	0.658326i	$-0.228735\pi$
$367$	28.8406	1.50547	0.752733	+	0.658326i	$0.228735\pi$
$368$	0	0
$369$	0.292978	0.0152518
$370$	0	0
$371$	−12.1880	−0.632767
$372$	0	0
$373$	29.7599	1.54091	0.770454	−	0.637496i	$-0.220030\pi$
$373$	29.7599	1.54091	0.770454	+	0.637496i	$0.220030\pi$
$374$	0	0
$375$	−1.39914	−0.0722515
$376$	0	0
$377$	5.07840	0.261551
$378$	0	0
$379$	10.3242	0.530318	0.265159	−	0.964205i	$-0.414576\pi$
$379$	10.3242	0.530318	0.265159	+	0.964205i	$0.414576\pi$
$380$	0	0
$381$	−2.57184	−0.131759
$382$	0	0
$383$	18.1963	0.929786	0.464893	−	0.885367i	$-0.346093\pi$
$383$	18.1963	0.929786	0.464893	+	0.885367i	$0.346093\pi$
$384$	0	0
$385$	0.683155	0.0348168
$386$	0	0
$387$	−0.309545	−0.0157351
$388$	0	0
$389$	22.5254	1.14209	0.571043	−	0.820920i	$-0.306539\pi$
$389$	22.5254	1.14209	0.571043	+	0.820920i	$0.306539\pi$
$390$	0	0
$391$	−0.809667	−0.0409466
$392$	0	0
$393$	−4.33578	−0.218711
$394$	0	0
$395$	2.37166	0.119331
$396$	0	0
$397$	−26.0157	−1.30569	−0.652845	−	0.757492i	$-0.726425\pi$
$397$	−26.0157	−1.30569	−0.652845	+	0.757492i	$0.726425\pi$
$398$	0	0
$399$	4.42845	0.221700
$400$	0	0
$401$	−35.6304	−1.77930	−0.889648	−	0.456648i	$-0.849050\pi$
$401$	−35.6304	−1.77930	−0.889648	+	0.456648i	$0.849050\pi$
$402$	0	0
$403$	−5.48944	−0.273448
$404$	0	0
$405$	0.140190	0.00696610
$406$	0	0
$407$	−42.8002	−2.12153
$408$	0	0
$409$	−1.82121	−0.0900532	−0.0450266	−	0.998986i	$-0.514337\pi$
$409$	−1.82121	−0.0900532	−0.0450266	+	0.998986i	$0.514337\pi$
$410$	0	0
$411$	9.66881	0.476927
$412$	0	0
$413$	−13.4468	−0.661675
$414$	0	0
$415$	−1.51342	−0.0742910
$416$	0	0
$417$	−18.3036	−0.896329
$418$	0	0
$419$	15.0551	0.735489	0.367745	−	0.929927i	$-0.380130\pi$
$419$	15.0551	0.735489	0.367745	+	0.929927i	$0.380130\pi$
$420$	0	0
$421$	−8.74661	−0.426284	−0.213142	−	0.977021i	$-0.568370\pi$
$421$	−8.74661	−0.426284	−0.213142	+	0.977021i	$0.568370\pi$
$422$	0	0
$423$	−3.45622	−0.168047
$424$	0	0
$425$	−4.03242	−0.195601
$426$	0	0
$427$	−7.44820	−0.360444
$428$	0	0
$429$	−20.8425	−1.00628
$430$	0	0
$431$	31.1437	1.50014	0.750070	−	0.661359i	$-0.230020\pi$
$431$	31.1437	1.50014	0.750070	+	0.661359i	$0.230020\pi$
$432$	0	0
$433$	−35.2685	−1.69489	−0.847447	−	0.530879i	$-0.821862\pi$
$433$	−35.2685	−1.69489	−0.847447	+	0.530879i	$0.821862\pi$
$434$	0	0
$435$	0.140190	0.00672160
$436$	0	0
$437$	3.72968	0.178415
$438$	0	0
$439$	−21.9552	−1.04786	−0.523932	−	0.851760i	$-0.675535\pi$
$439$	−21.9552	−1.04786	−0.523932	+	0.851760i	$0.675535\pi$
$440$	0	0
$441$	−5.59019	−0.266200
$442$	0	0
$443$	20.1348	0.956633	0.478316	−	0.878188i	$-0.341247\pi$
$443$	20.1348	0.956633	0.478316	+	0.878188i	$0.341247\pi$
$444$	0	0
$445$	−0.188946	−0.00895688
$446$	0	0
$447$	−19.6837	−0.931005
$448$	0	0
$449$	−25.2453	−1.19140	−0.595700	−	0.803207i	$-0.703125\pi$
$449$	−25.2453	−1.19140	−0.595700	+	0.803207i	$0.703125\pi$
$450$	0	0
$451$	−1.20242	−0.0566199
$452$	0	0
$453$	12.3990	0.582557
$454$	0	0
$455$	−0.845325	−0.0396294
$456$	0	0
$457$	−16.2740	−0.761264	−0.380632	−	0.924726i	$-0.624294\pi$
$457$	−16.2740	−0.761264	−0.380632	+	0.924726i	$0.624294\pi$
$458$	0	0
$459$	0.809667	0.0377920
$460$	0	0
$461$	−31.2274	−1.45440	−0.727202	−	0.686423i	$-0.759180\pi$
$461$	−31.2274	−1.45440	−0.727202	+	0.686423i	$0.759180\pi$
$462$	0	0
$463$	−26.3754	−1.22577	−0.612884	−	0.790173i	$-0.709991\pi$
$463$	−26.3754	−1.22577	−0.612884	+	0.790173i	$0.709991\pi$
$464$	0	0
$465$	−0.151537	−0.00702735
$466$	0	0
$467$	8.26321	0.382376	0.191188	−	0.981553i	$-0.438766\pi$
$467$	8.26321	0.382376	0.191188	+	0.981553i	$0.438766\pi$
$468$	0	0
$469$	1.47906	0.0682968
$470$	0	0
$471$	10.4378	0.480946
$472$	0	0
$473$	1.27042	0.0584139
$474$	0	0
$475$	18.5751	0.852284
$476$	0	0
$477$	10.2648	0.469994
$478$	0	0
$479$	−3.82446	−0.174744	−0.0873721	−	0.996176i	$-0.527847\pi$
$479$	−3.82446	−0.174744	−0.0873721	+	0.996176i	$0.527847\pi$
$480$	0	0
$481$	52.9602	2.41478
$482$	0	0
$483$	1.18735	0.0540264
$484$	0	0
$485$	−0.0905678	−0.00411247
$486$	0	0
$487$	14.5607	0.659806	0.329903	−	0.944015i	$-0.392984\pi$
$487$	14.5607	0.659806	0.329903	+	0.944015i	$0.392984\pi$
$488$	0	0
$489$	−16.3310	−0.738512
$490$	0	0
$491$	−31.7848	−1.43443	−0.717214	−	0.696853i	$-0.754583\pi$
$491$	−31.7848	−1.43443	−0.717214	+	0.696853i	$0.754583\pi$
$492$	0	0
$493$	0.809667	0.0364656
$494$	0	0
$495$	−0.575360	−0.0258605
$496$	0	0
$497$	5.19724	0.233128
$498$	0	0
$499$	−1.70992	−0.0765467	−0.0382733	−	0.999267i	$-0.512186\pi$
$499$	−1.70992	−0.0765467	−0.0382733	+	0.999267i	$0.512186\pi$
$500$	0	0
$501$	2.76211	0.123402
$502$	0	0
$503$	−4.76343	−0.212391	−0.106196	−	0.994345i	$-0.533867\pi$
$503$	−4.76343	−0.212391	−0.106196	+	0.994345i	$0.533867\pi$
$504$	0	0
$505$	−0.424646	−0.0188965
$506$	0	0
$507$	12.7901	0.568029
$508$	0	0
$509$	6.82904	0.302692	0.151346	−	0.988481i	$-0.451639\pi$
$509$	6.82904	0.302692	0.151346	+	0.988481i	$0.451639\pi$
$510$	0	0
$511$	−14.9097	−0.659564
$512$	0	0
$513$	−3.72968	−0.164669
$514$	0	0
$515$	1.74103	0.0767188
$516$	0	0
$517$	14.1848	0.623848
$518$	0	0
$519$	22.1926	0.974145
$520$	0	0
$521$	1.87086	0.0819638	0.0409819	−	0.999160i	$-0.486951\pi$
$521$	1.87086	0.0819638	0.0409819	+	0.999160i	$0.486951\pi$
$522$	0	0
$523$	29.6619	1.29702	0.648512	−	0.761204i	$-0.275392\pi$
$523$	29.6619	1.29702	0.648512	+	0.761204i	$0.275392\pi$
$524$	0	0
$525$	5.91343	0.258083
$526$	0	0
$527$	−0.875201	−0.0381244
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	11.3250	0.491465
$532$	0	0
$533$	1.48786	0.0644462
$534$	0	0
$535$	0.763823	0.0330230
$536$	0	0
$537$	−11.0849	−0.478350
$538$	0	0
$539$	22.9430	0.988223
$540$	0	0
$541$	−31.5959	−1.35841	−0.679206	−	0.733948i	$-0.737676\pi$
$541$	−31.5959	−1.35841	−0.679206	+	0.733948i	$0.737676\pi$
$542$	0	0
$543$	−9.80187	−0.420639
$544$	0	0
$545$	1.45992	0.0625360
$546$	0	0
$547$	−20.2518	−0.865905	−0.432953	−	0.901417i	$-0.642528\pi$
$547$	−20.2518	−0.865905	−0.432953	+	0.901417i	$0.642528\pi$
$548$	0	0
$549$	6.27295	0.267723
$550$	0	0
$551$	−3.72968	−0.158890
$552$	0	0
$553$	−20.0870	−0.854185
$554$	0	0
$555$	1.46197	0.0620573
$556$	0	0
$557$	−10.7511	−0.455540	−0.227770	−	0.973715i	$-0.573143\pi$
$557$	−10.7511	−0.455540	−0.227770	+	0.973715i	$0.573143\pi$
$558$	0	0
$559$	−1.57199	−0.0664882
$560$	0	0
$561$	−3.32299	−0.140297
$562$	0	0
$563$	−43.7398	−1.84341	−0.921707	−	0.387887i	$-0.873205\pi$
$563$	−43.7398	−1.84341	−0.921707	+	0.387887i	$0.873205\pi$
$564$	0	0
$565$	2.53825	0.106785
$566$	0	0
$567$	−1.18735	−0.0498641
$568$	0	0
$569$	−29.2901	−1.22791	−0.613953	−	0.789343i	$-0.710422\pi$
$569$	−29.2901	−1.22791	−0.613953	+	0.789343i	$0.710422\pi$
$570$	0	0
$571$	40.6028	1.69918	0.849588	−	0.527447i	$-0.176851\pi$
$571$	40.6028	1.69918	0.849588	+	0.527447i	$0.176851\pi$
$572$	0	0
$573$	1.71424	0.0716134
$574$	0	0
$575$	4.98035	0.207695
$576$	0	0
$577$	−30.3890	−1.26511	−0.632556	−	0.774515i	$-0.717994\pi$
$577$	−30.3890	−1.26511	−0.632556	+	0.774515i	$0.717994\pi$
$578$	0	0
$579$	8.44637	0.351019
$580$	0	0
$581$	12.8181	0.531784
$582$	0	0
$583$	−42.1283	−1.74478
$584$	0	0
$585$	0.711940	0.0294351
$586$	0	0
$587$	38.7617	1.59987	0.799933	−	0.600089i	$-0.204868\pi$
$587$	38.7617	1.59987	0.799933	+	0.600089i	$0.204868\pi$
$588$	0	0
$589$	4.03156	0.166117
$590$	0	0
$591$	−26.8409	−1.10409
$592$	0	0
$593$	24.7607	1.01680	0.508401	−	0.861120i	$-0.330237\pi$
$593$	24.7607	1.01680	0.508401	+	0.861120i	$0.330237\pi$
$594$	0	0
$595$	−0.134773	−0.00552516
$596$	0	0
$597$	10.3515	0.423660
$598$	0	0
$599$	−13.7986	−0.563796	−0.281898	−	0.959444i	$-0.590964\pi$
$599$	−13.7986	−0.563796	−0.281898	+	0.959444i	$0.590964\pi$
$600$	0	0
$601$	−21.4533	−0.875097	−0.437548	−	0.899195i	$-0.644153\pi$
$601$	−21.4533	−0.875097	−0.437548	+	0.899195i	$0.644153\pi$
$602$	0	0
$603$	−1.24568	−0.0507281
$604$	0	0
$605$	0.819271	0.0333081
$606$	0	0
$607$	13.8880	0.563698	0.281849	−	0.959459i	$-0.409052\pi$
$607$	13.8880	0.563698	0.281849	+	0.959459i	$0.409052\pi$
$608$	0	0
$609$	−1.18735	−0.0481140
$610$	0	0
$611$	−17.5521	−0.710080
$612$	0	0
$613$	−10.4759	−0.423119	−0.211559	−	0.977365i	$-0.567854\pi$
$613$	−10.4759	−0.423119	−0.211559	+	0.977365i	$0.567854\pi$
$614$	0	0
$615$	0.0410725	0.00165620
$616$	0	0
$617$	−2.25913	−0.0909493	−0.0454747	−	0.998965i	$-0.514480\pi$
$617$	−2.25913	−0.0909493	−0.0454747	+	0.998965i	$0.514480\pi$
$618$	0	0
$619$	−41.4122	−1.66450	−0.832248	−	0.554404i	$-0.812946\pi$
$619$	−41.4122	−1.66450	−0.832248	+	0.554404i	$0.812946\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	1.60029	0.0641144
$624$	0	0
$625$	24.7056	0.988224
$626$	0	0
$627$	15.3071	0.611308
$628$	0	0
$629$	8.44363	0.336670
$630$	0	0
$631$	−7.02851	−0.279801	−0.139900	−	0.990166i	$-0.544678\pi$
$631$	−7.02851	−0.279801	−0.139900	+	0.990166i	$0.544678\pi$
$632$	0	0
$633$	20.5502	0.816798
$634$	0	0
$635$	−0.360546	−0.0143078
$636$	0	0
$637$	−28.3892	−1.12482
$638$	0	0
$639$	−4.37717	−0.173158
$640$	0	0
$641$	23.5616	0.930626	0.465313	−	0.885146i	$-0.345942\pi$
$641$	23.5616	0.930626	0.465313	+	0.885146i	$0.345942\pi$
$642$	0	0
$643$	34.0953	1.34459	0.672294	−	0.740285i	$-0.265309\pi$
$643$	34.0953	1.34459	0.672294	+	0.740285i	$0.265309\pi$
$644$	0	0
$645$	−0.0433951	−0.00170868
$646$	0	0
$647$	35.0500	1.37796	0.688978	−	0.724782i	$-0.258059\pi$
$647$	35.0500	1.37796	0.688978	+	0.724782i	$0.258059\pi$
$648$	0	0
$649$	−46.4796	−1.82448
$650$	0	0
$651$	1.28346	0.0503026
$652$	0	0
$653$	−16.0552	−0.628289	−0.314144	−	0.949375i	$-0.601718\pi$
$653$	−16.0552	−0.628289	−0.314144	+	0.949375i	$0.601718\pi$
$654$	0	0
$655$	−0.607832	−0.0237500
$656$	0	0
$657$	12.5571	0.489897
$658$	0	0
$659$	25.4954	0.993161	0.496581	−	0.867991i	$-0.334589\pi$
$659$	25.4954	0.993161	0.496581	+	0.867991i	$0.334589\pi$
$660$	0	0
$661$	17.1380	0.666590	0.333295	−	0.942823i	$-0.391839\pi$
$661$	17.1380	0.666590	0.333295	+	0.942823i	$0.391839\pi$
$662$	0	0
$663$	4.11181	0.159690
$664$	0	0
$665$	0.620824	0.0240745
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	0	0
$669$	17.0966	0.660993
$670$	0	0
$671$	−25.7451	−0.993878
$672$	0	0
$673$	34.9137	1.34582	0.672912	−	0.739722i	$-0.265043\pi$
$673$	34.9137	1.34582	0.672912	+	0.739722i	$0.265043\pi$
$674$	0	0
$675$	−4.98035	−0.191694
$676$	0	0
$677$	−29.1425	−1.12004	−0.560018	−	0.828480i	$-0.689206\pi$
$677$	−29.1425	−1.12004	−0.560018	+	0.828480i	$0.689206\pi$
$678$	0	0
$679$	0.767073	0.0294376
$680$	0	0
$681$	−9.44648	−0.361990
$682$	0	0
$683$	2.01735	0.0771919	0.0385960	−	0.999255i	$-0.487711\pi$
$683$	2.01735	0.0771919	0.0385960	+	0.999255i	$0.487711\pi$
$684$	0	0
$685$	1.35547	0.0517899
$686$	0	0
$687$	18.6279	0.710698
$688$	0	0
$689$	52.1288	1.98595
$690$	0	0
$691$	15.4812	0.588931	0.294466	−	0.955662i	$-0.404858\pi$
$691$	15.4812	0.588931	0.294466	+	0.955662i	$0.404858\pi$
$692$	0	0
$693$	4.87307	0.185113
$694$	0	0
$695$	−2.56598	−0.0973330
$696$	0	0
$697$	0.237214	0.00898514
$698$	0	0
$699$	4.25939	0.161105
$700$	0	0
$701$	10.8282	0.408976	0.204488	−	0.978869i	$-0.434447\pi$
$701$	10.8282	0.408976	0.204488	+	0.978869i	$0.434447\pi$
$702$	0	0
$703$	−38.8950	−1.46695
$704$	0	0
$705$	−0.484528	−0.0182484
$706$	0	0
$707$	3.59658	0.135263
$708$	0	0
$709$	6.57645	0.246984	0.123492	−	0.992346i	$-0.460591\pi$
$709$	6.57645	0.246984	0.123492	+	0.992346i	$0.460591\pi$
$710$	0	0
$711$	16.9174	0.634454
$712$	0	0
$713$	1.08094	0.0404815
$714$	0	0
$715$	−2.92191	−0.109273
$716$	0	0
$717$	20.8179	0.777460
$718$	0	0
$719$	−18.3035	−0.682604	−0.341302	−	0.939954i	$-0.610868\pi$
$719$	−18.3035	−0.682604	−0.341302	+	0.939954i	$0.610868\pi$
$720$	0	0
$721$	−14.7458	−0.549162
$722$	0	0
$723$	−26.5631	−0.987892
$724$	0	0
$725$	−4.98035	−0.184965
$726$	0	0
$727$	14.9858	0.555793	0.277897	−	0.960611i	$-0.410363\pi$
$727$	14.9858	0.555793	0.277897	+	0.960611i	$0.410363\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−0.250629	−0.00926984
$732$	0	0
$733$	1.44188	0.0532571	0.0266286	−	0.999645i	$-0.491523\pi$
$733$	1.44188	0.0532571	0.0266286	+	0.999645i	$0.491523\pi$
$734$	0	0
$735$	−0.783689	−0.0289068
$736$	0	0
$737$	5.11246	0.188320
$738$	0	0
$739$	41.6379	1.53168	0.765838	−	0.643034i	$-0.222324\pi$
$739$	41.6379	1.53168	0.765838	+	0.643034i	$0.222324\pi$
$740$	0	0
$741$	−18.9408	−0.695807
$742$	0	0
$743$	18.0218	0.661157	0.330578	−	0.943779i	$-0.392756\pi$
$743$	18.0218	0.661157	0.330578	+	0.943779i	$0.392756\pi$
$744$	0	0
$745$	−2.75945	−0.101098
$746$	0	0
$747$	−10.7955	−0.394987
$748$	0	0
$749$	−6.46928	−0.236382
$750$	0	0
$751$	−15.1052	−0.551196	−0.275598	−	0.961273i	$-0.588876\pi$
$751$	−15.1052	−0.551196	−0.275598	+	0.961273i	$0.588876\pi$
$752$	0	0
$753$	−11.7201	−0.427102
$754$	0	0
$755$	1.73822	0.0632602
$756$	0	0
$757$	20.8631	0.758283	0.379141	−	0.925339i	$-0.376219\pi$
$757$	20.8631	0.758283	0.379141	+	0.925339i	$0.376219\pi$
$758$	0	0
$759$	4.10415	0.148971
$760$	0	0
$761$	−54.7908	−1.98617	−0.993083	−	0.117417i	$-0.962539\pi$
$761$	−54.7908	−1.98617	−0.993083	+	0.117417i	$0.962539\pi$
$762$	0	0
$763$	−12.3649	−0.447640
$764$	0	0
$765$	0.113507	0.00410386
$766$	0	0
$767$	57.5131	2.07668
$768$	0	0
$769$	−6.62115	−0.238765	−0.119382	−	0.992848i	$-0.538091\pi$
$769$	−6.62115	−0.238765	−0.119382	+	0.992848i	$0.538091\pi$
$770$	0	0
$771$	12.9362	0.465885
$772$	0	0
$773$	−20.7193	−0.745223	−0.372612	−	0.927987i	$-0.621538\pi$
$773$	−20.7193	−0.745223	−0.372612	+	0.927987i	$0.621538\pi$
$774$	0	0
$775$	5.38345	0.193379
$776$	0	0
$777$	−12.3823	−0.444214
$778$	0	0
$779$	−1.09271	−0.0391505
$780$	0	0
$781$	17.9645	0.642821
$782$	0	0
$783$	1.00000	0.0357371
$784$	0	0
$785$	1.46327	0.0522263
$786$	0	0
$787$	21.1928	0.755442	0.377721	−	0.925919i	$-0.376708\pi$
$787$	21.1928	0.755442	0.377721	+	0.925919i	$0.376708\pi$
$788$	0	0
$789$	8.02833	0.285816
$790$	0	0
$791$	−21.4979	−0.764379
$792$	0	0
$793$	31.8565	1.13126
$794$	0	0
$795$	1.43902	0.0510369
$796$	0	0
$797$	26.4412	0.936595	0.468297	−	0.883571i	$-0.344868\pi$
$797$	26.4412	0.936595	0.468297	+	0.883571i	$0.344868\pi$
$798$	0	0
$799$	−2.79839	−0.0989999
$800$	0	0
$801$	−1.34778	−0.0476216
$802$	0	0
$803$	−51.5360	−1.81866
$804$	0	0
$805$	0.166455	0.00586677
$806$	0	0
$807$	−5.02254	−0.176802
$808$	0	0
$809$	−21.1670	−0.744191	−0.372095	−	0.928194i	$-0.621361\pi$
$809$	−21.1670	−0.744191	−0.372095	+	0.928194i	$0.621361\pi$
$810$	0	0
$811$	47.6537	1.67335	0.836674	−	0.547701i	$-0.184497\pi$
$811$	47.6537	1.67335	0.836674	+	0.547701i	$0.184497\pi$
$812$	0	0
$813$	0.872807	0.0306107
$814$	0	0
$815$	−2.28944	−0.0801956
$816$	0	0
$817$	1.15450	0.0403910
$818$	0	0
$819$	−6.02985	−0.210700
$820$	0	0
$821$	−26.6885	−0.931437	−0.465718	−	0.884933i	$-0.654204\pi$
$821$	−26.6885	−0.931437	−0.465718	+	0.884933i	$0.654204\pi$
$822$	0	0
$823$	45.6800	1.59231	0.796153	−	0.605096i	$-0.206865\pi$
$823$	45.6800	1.59231	0.796153	+	0.605096i	$0.206865\pi$
$824$	0	0
$825$	20.4401	0.711632
$826$	0	0
$827$	−28.1277	−0.978097	−0.489049	−	0.872256i	$-0.662656\pi$
$827$	−28.1277	−0.978097	−0.489049	+	0.872256i	$0.662656\pi$
$828$	0	0
$829$	34.7778	1.20788	0.603941	−	0.797029i	$-0.293596\pi$
$829$	34.7778	1.20788	0.603941	+	0.797029i	$0.293596\pi$
$830$	0	0
$831$	8.91671	0.309317
$832$	0	0
$833$	−4.52620	−0.156823
$834$	0	0
$835$	0.387220	0.0134003
$836$	0	0
$837$	−1.08094	−0.0373627
$838$	0	0
$839$	−31.3802	−1.08337	−0.541683	−	0.840583i	$-0.682213\pi$
$839$	−31.3802	−1.08337	−0.541683	+	0.840583i	$0.682213\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	24.1240	0.830876
$844$	0	0
$845$	1.79305	0.0616826
$846$	0	0
$847$	−6.93890	−0.238423
$848$	0	0
$849$	−22.5722	−0.774675
$850$	0	0
$851$	−10.4285	−0.357485
$852$	0	0
$853$	30.2686	1.03638	0.518188	−	0.855267i	$-0.326607\pi$
$853$	30.2686	1.03638	0.518188	+	0.855267i	$0.326607\pi$
$854$	0	0
$855$	−0.522864	−0.0178816
$856$	0	0
$857$	−1.68752	−0.0576447	−0.0288223	−	0.999585i	$-0.509176\pi$
$857$	−1.68752	−0.0576447	−0.0288223	+	0.999585i	$0.509176\pi$
$858$	0	0
$859$	−42.8880	−1.46332	−0.731661	−	0.681669i	$-0.761254\pi$
$859$	−42.8880	−1.46332	−0.731661	+	0.681669i	$0.761254\pi$
$860$	0	0
$861$	−0.347868	−0.0118553
$862$	0	0
$863$	−31.8179	−1.08309	−0.541547	−	0.840671i	$-0.682161\pi$
$863$	−31.8179	−1.08309	−0.541547	+	0.840671i	$0.682161\pi$
$864$	0	0
$865$	3.11117	0.105783
$866$	0	0
$867$	−16.3444	−0.555086
$868$	0	0
$869$	−69.4316	−2.35531
$870$	0	0
$871$	−6.32607	−0.214351
$872$	0	0
$873$	−0.646036	−0.0218650
$874$	0	0
$875$	1.66128	0.0561615
$876$	0	0
$877$	−17.7353	−0.598879	−0.299440	−	0.954115i	$-0.596800\pi$
$877$	−17.7353	−0.598879	−0.299440	+	0.954115i	$0.596800\pi$
$878$	0	0
$879$	11.8063	0.398215
$880$	0	0
$881$	−20.1727	−0.679637	−0.339818	−	0.940491i	$-0.610366\pi$
$881$	−20.1727	−0.679637	−0.339818	+	0.940491i	$0.610366\pi$
$882$	0	0
$883$	30.5769	1.02899	0.514497	−	0.857492i	$-0.327979\pi$
$883$	30.5769	1.02899	0.514497	+	0.857492i	$0.327979\pi$
$884$	0	0
$885$	1.58766	0.0533685
$886$	0	0
$887$	−8.67491	−0.291275	−0.145637	−	0.989338i	$-0.546523\pi$
$887$	−8.67491	−0.291275	−0.145637	+	0.989338i	$0.546523\pi$
$888$	0	0
$889$	3.05368	0.102417
$890$	0	0
$891$	−4.10415	−0.137494
$892$	0	0
$893$	12.8906	0.431367
$894$	0	0
$895$	−1.55400	−0.0519444
$896$	0	0
$897$	−5.07840	−0.169563
$898$	0	0
$899$	−1.08094	−0.0360513
$900$	0	0
$901$	8.31109	0.276882
$902$	0	0
$903$	0.367539	0.0122309
$904$	0	0
$905$	−1.37412	−0.0456774
$906$	0	0
$907$	6.69431	0.222281	0.111140	−	0.993805i	$-0.464550\pi$
$907$	6.69431	0.222281	0.111140	+	0.993805i	$0.464550\pi$
$908$	0	0
$909$	−3.02908	−0.100468
$910$	0	0
$911$	−50.2397	−1.66452	−0.832258	−	0.554389i	$-0.812952\pi$
$911$	−50.2397	−1.66452	−0.832258	+	0.554389i	$0.812952\pi$
$912$	0	0
$913$	44.3064	1.46633
$914$	0	0
$915$	0.879404	0.0290722
$916$	0	0
$917$	5.14810	0.170005
$918$	0	0
$919$	−13.7002	−0.451927	−0.225963	−	0.974136i	$-0.572553\pi$
$919$	−13.7002	−0.451927	−0.225963	+	0.974136i	$0.572553\pi$
$920$	0	0
$921$	26.1028	0.860115
$922$	0	0
$923$	−22.2290	−0.731676
$924$	0	0
$925$	−51.9376	−1.70770
$926$	0	0
$927$	12.4190	0.407895
$928$	0	0
$929$	−33.6596	−1.10433	−0.552167	−	0.833733i	$-0.686199\pi$
$929$	−33.6596	−1.10433	−0.552167	+	0.833733i	$0.686199\pi$
$930$	0	0
$931$	20.8496	0.683319
$932$	0	0
$933$	−26.2697	−0.860030
$934$	0	0
$935$	−0.465850	−0.0152349
$936$	0	0
$937$	32.6231	1.06575	0.532874	−	0.846194i	$-0.321112\pi$
$937$	32.6231	1.06575	0.532874	+	0.846194i	$0.321112\pi$
$938$	0	0
$939$	34.8344	1.13678
$940$	0	0
$941$	−1.94881	−0.0635295	−0.0317647	−	0.999495i	$-0.510113\pi$
$941$	−1.94881	−0.0635295	−0.0317647	+	0.999495i	$0.510113\pi$
$942$	0	0
$943$	−0.292978	−0.00954067
$944$	0	0
$945$	−0.166455	−0.00541478
$946$	0	0
$947$	−34.1407	−1.10942	−0.554712	−	0.832042i	$-0.687172\pi$
$947$	−34.1407	−1.10942	−0.554712	+	0.832042i	$0.687172\pi$
$948$	0	0
$949$	63.7697	2.07005
$950$	0	0
$951$	11.1689	0.362176
$952$	0	0
$953$	34.4738	1.11672	0.558359	−	0.829600i	$-0.311431\pi$
$953$	34.4738	1.11672	0.558359	+	0.829600i	$0.311431\pi$
$954$	0	0
$955$	0.240319	0.00777655
$956$	0	0
$957$	−4.10415	−0.132668
$958$	0	0
$959$	−11.4803	−0.370718
$960$	0	0
$961$	−29.8316	−0.962309
$962$	0	0
$963$	5.44849	0.175575
$964$	0	0
$965$	1.18410	0.0381174
$966$	0	0
$967$	21.1281	0.679434	0.339717	−	0.940528i	$-0.389669\pi$
$967$	21.1281	0.679434	0.339717	+	0.940528i	$0.389669\pi$
$968$	0	0
$969$	−3.01980	−0.0970099
$970$	0	0
$971$	−30.9048	−0.991783	−0.495892	−	0.868384i	$-0.665159\pi$
$971$	−30.9048	−0.991783	−0.495892	+	0.868384i	$0.665159\pi$
$972$	0	0
$973$	21.7328	0.696721
$974$	0	0
$975$	−25.2922	−0.809998
$976$	0	0
$977$	45.3234	1.45002	0.725012	−	0.688736i	$-0.241834\pi$
$977$	45.3234	1.45002	0.725012	+	0.688736i	$0.241834\pi$
$978$	0	0
$979$	5.53150	0.176787
$980$	0	0
$981$	10.4138	0.332488
$982$	0	0
$983$	56.0286	1.78703	0.893517	−	0.449030i	$-0.148230\pi$
$983$	56.0286	1.78703	0.893517	+	0.449030i	$0.148230\pi$
$984$	0	0
$985$	−3.76282	−0.119894
$986$	0	0
$987$	4.10375	0.130624
$988$	0	0
$989$	0.309545	0.00984297
$990$	0	0
$991$	−13.9641	−0.443585	−0.221793	−	0.975094i	$-0.571191\pi$
$991$	−13.9641	−0.443585	−0.221793	+	0.975094i	$0.571191\pi$
$992$	0	0
$993$	12.0196	0.381431
$994$	0	0
$995$	1.45118	0.0460056
$996$	0	0
$997$	−0.562112	−0.0178023	−0.00890114	−	0.999960i	$-0.502833\pi$
$997$	−0.562112	−0.0178023	−0.00890114	+	0.999960i	$0.502833\pi$
$998$	0	0
$999$	10.4285	0.329944

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8004.2.a.k.1.8	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8004.2.a.k.1.8	✓	18	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:	\( N \)	\(=\)	\( 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8004.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +0.140190 q^{5} -1.18735 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +0.140190 q^{5} -1.18735 q^{7} +1.00000 q^{9} -4.10415 q^{11} +5.07840 q^{13} +0.140190 q^{15} +0.809667 q^{17} -3.72968 q^{19} -1.18735 q^{21} -1.00000 q^{23} -4.98035 q^{25} +1.00000 q^{27} +1.00000 q^{29} -1.08094 q^{31} -4.10415 q^{33} -0.166455 q^{35} +10.4285 q^{37} +5.07840 q^{39} +0.292978 q^{41} -0.309545 q^{43} +0.140190 q^{45} -3.45622 q^{47} -5.59019 q^{49} +0.809667 q^{51} +10.2648 q^{53} -0.575360 q^{55} -3.72968 q^{57} +11.3250 q^{59} +6.27295 q^{61} -1.18735 q^{63} +0.711940 q^{65} -1.24568 q^{67} -1.00000 q^{69} -4.37717 q^{71} +12.5571 q^{73} -4.98035 q^{75} +4.87307 q^{77} +16.9174 q^{79} +1.00000 q^{81} -10.7955 q^{83} +0.113507 q^{85} +1.00000 q^{87} -1.34778 q^{89} -6.02985 q^{91} -1.08094 q^{93} -0.522864 q^{95} -0.646036 q^{97} -4.10415 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 18 q^{3} + 5 q^{5} + 6 q^{7} + 18 q^{9}+O(q^{10}) \) 18 * q + 18 * q^3 + 5 * q^5 + 6 * q^7 + 18 * q^9	\( 18 q + 18 q^{3} + 5 q^{5} + 6 q^{7} + 18 q^{9} + 5 q^{11} + 6 q^{13} + 5 q^{15} + 7 q^{17} + 15 q^{19} + 6 q^{21} - 18 q^{23} + 45 q^{25} + 18 q^{27} + 18 q^{29} + 10 q^{31} + 5 q^{33} - 7 q^{35} + 22 q^{37} + 6 q^{39} + 17 q^{41} + 25 q^{43} + 5 q^{45} - 6 q^{47} + 64 q^{49} + 7 q^{51} + 21 q^{53} + 3 q^{55} + 15 q^{57} + 6 q^{59} + 7 q^{61} + 6 q^{63} + 44 q^{65} + 35 q^{67} - 18 q^{69} + q^{71} + 27 q^{73} + 45 q^{75} + 4 q^{77} + 18 q^{81} + 13 q^{83} + 24 q^{85} + 18 q^{87} + 30 q^{89} + 53 q^{91} + 10 q^{93} + 17 q^{95} + 27 q^{97} + 5 q^{99}+O(q^{100}) \) 18 * q + 18 * q^3 + 5 * q^5 + 6 * q^7 + 18 * q^9 + 5 * q^11 + 6 * q^13 + 5 * q^15 + 7 * q^17 + 15 * q^19 + 6 * q^21 - 18 * q^23 + 45 * q^25 + 18 * q^27 + 18 * q^29 + 10 * q^31 + 5 * q^33 - 7 * q^35 + 22 * q^37 + 6 * q^39 + 17 * q^41 + 25 * q^43 + 5 * q^45 - 6 * q^47 + 64 * q^49 + 7 * q^51 + 21 * q^53 + 3 * q^55 + 15 * q^57 + 6 * q^59 + 7 * q^61 + 6 * q^63 + 44 * q^65 + 35 * q^67 - 18 * q^69 + q^71 + 27 * q^73 + 45 * q^75 + 4 * q^77 + 18 * q^81 + 13 * q^83 + 24 * q^85 + 18 * q^87 + 30 * q^89 + 53 * q^91 + 10 * q^93 + 17 * q^95 + 27 * q^97 + 5 * q^99

Introduction

L-functions

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Database

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