LMFDB - Embedded newform 8004.2.a.j.1.10

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:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 3 x^{15} - 49 x^{14} + 130 x^{13} + 932 x^{12} - 2028 x^{11} - 8965 x^{10} + 14400 x^{9} + \cdots + 3888 $ x^16 - 3x^15 - 49x^14 + 130x^13 + 932x^12 - 2028x^11 - 8965x^10 + 14400x^9 + 46229x^8 - 47547x^7 - 122604x^6 + 65278x^5 + 151028x^4 - 17988x^3 - 67608x^2 - 8424*x + 3888
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$0.897331$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +0.897331 q^{5} +3.81467 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +0.897331 q^{5} +3.81467 q^{7} +1.00000 q^{9} +5.23385 q^{11} -3.43366 q^{13} +0.897331 q^{15} +1.92187 q^{17} -7.80360 q^{19} +3.81467 q^{21} +1.00000 q^{23} -4.19480 q^{25} +1.00000 q^{27} -1.00000 q^{29} +3.73130 q^{31} +5.23385 q^{33} +3.42302 q^{35} +0.939833 q^{37} -3.43366 q^{39} -5.90444 q^{41} -2.82368 q^{43} +0.897331 q^{45} +13.3375 q^{47} +7.55171 q^{49} +1.92187 q^{51} +12.4108 q^{53} +4.69649 q^{55} -7.80360 q^{57} -3.06388 q^{59} +9.88695 q^{61} +3.81467 q^{63} -3.08113 q^{65} +14.3222 q^{67} +1.00000 q^{69} +10.5787 q^{71} -9.23222 q^{73} -4.19480 q^{75} +19.9654 q^{77} +2.22128 q^{79} +1.00000 q^{81} +2.48600 q^{83} +1.72455 q^{85} -1.00000 q^{87} -13.2715 q^{89} -13.0983 q^{91} +3.73130 q^{93} -7.00241 q^{95} -7.70919 q^{97} +5.23385 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 16 q^{3} + 3 q^{5} + 4 q^{7} + 16 q^{9}+O(q^{10}) $ 16 * q + 16 * q^3 + 3 * q^5 + 4 * q^7 + 16 * q^9	$ 16 q + 16 q^{3} + 3 q^{5} + 4 q^{7} + 16 q^{9} + 5 q^{11} + 6 q^{13} + 3 q^{15} + 3 q^{17} + 11 q^{19} + 4 q^{21} + 16 q^{23} + 27 q^{25} + 16 q^{27} - 16 q^{29} + 14 q^{31} + 5 q^{33} + 11 q^{35} + 4 q^{37} + 6 q^{39} + 11 q^{41} + 23 q^{43} + 3 q^{45} - 2 q^{47} + 34 q^{49} + 3 q^{51} + 19 q^{53} + 31 q^{55} + 11 q^{57} + 32 q^{59} + 19 q^{61} + 4 q^{63} + 6 q^{65} + 33 q^{67} + 16 q^{69} - 5 q^{71} + 23 q^{73} + 27 q^{75} + 42 q^{77} + 24 q^{79} + 16 q^{81} + 7 q^{83} - 16 q^{87} - 2 q^{89} + 25 q^{91} + 14 q^{93} + 7 q^{95} + 33 q^{97} + 5 q^{99}+O(q^{100}) $ 16 * q + 16 * q^3 + 3 * q^5 + 4 * q^7 + 16 * q^9 + 5 * q^11 + 6 * q^13 + 3 * q^15 + 3 * q^17 + 11 * q^19 + 4 * q^21 + 16 * q^23 + 27 * q^25 + 16 * q^27 - 16 * q^29 + 14 * q^31 + 5 * q^33 + 11 * q^35 + 4 * q^37 + 6 * q^39 + 11 * q^41 + 23 * q^43 + 3 * q^45 - 2 * q^47 + 34 * q^49 + 3 * q^51 + 19 * q^53 + 31 * q^55 + 11 * q^57 + 32 * q^59 + 19 * q^61 + 4 * q^63 + 6 * q^65 + 33 * q^67 + 16 * q^69 - 5 * q^71 + 23 * q^73 + 27 * q^75 + 42 * q^77 + 24 * q^79 + 16 * q^81 + 7 * q^83 - 16 * q^87 - 2 * q^89 + 25 * q^91 + 14 * q^93 + 7 * q^95 + 33 * 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
$5$	0.897331	0.401299	0.200649	−	0.979663i	$-0.435695\pi$
$5$	0.897331	0.401299	0.200649	+	0.979663i	$0.435695\pi$
$6$	0	0
$7$	3.81467	1.44181	0.720905	−	0.693034i	$-0.243726\pi$
$7$	3.81467	1.44181	0.720905	+	0.693034i	$0.243726\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	5.23385	1.57806	0.789032	−	0.614352i	$-0.210583\pi$
$11$	5.23385	1.57806	0.789032	+	0.614352i	$0.210583\pi$
$12$	0	0
$13$	−3.43366	−0.952327	−0.476164	−	0.879357i	$-0.657973\pi$
$13$	−3.43366	−0.952327	−0.476164	+	0.879357i	$0.657973\pi$
$14$	0	0
$15$	0.897331	0.231690
$16$	0	0
$17$	1.92187	0.466122	0.233061	−	0.972462i	$-0.425126\pi$
$17$	1.92187	0.466122	0.233061	+	0.972462i	$0.425126\pi$
$18$	0	0
$19$	−7.80360	−1.79027	−0.895134	−	0.445796i	$-0.852921\pi$
$19$	−7.80360	−1.79027	−0.895134	+	0.445796i	$0.852921\pi$
$20$	0	0
$21$	3.81467	0.832429
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−4.19480	−0.838959
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	3.73130	0.670162	0.335081	−	0.942189i	$-0.391236\pi$
$31$	3.73130	0.670162	0.335081	+	0.942189i	$0.391236\pi$
$32$	0	0
$33$	5.23385	0.911096
$34$	0	0
$35$	3.42302	0.578596
$36$	0	0
$37$	0.939833	0.154508	0.0772538	−	0.997011i	$-0.475385\pi$
$37$	0.939833	0.154508	0.0772538	+	0.997011i	$0.475385\pi$
$38$	0	0
$39$	−3.43366	−0.549826
$40$	0	0
$41$	−5.90444	−0.922119	−0.461060	−	0.887369i	$-0.652531\pi$
$41$	−5.90444	−0.922119	−0.461060	+	0.887369i	$0.652531\pi$
$42$	0	0
$43$	−2.82368	−0.430607	−0.215303	−	0.976547i	$-0.569074\pi$
$43$	−2.82368	−0.430607	−0.215303	+	0.976547i	$0.569074\pi$
$44$	0	0
$45$	0.897331	0.133766
$46$	0	0
$47$	13.3375	1.94547	0.972734	−	0.231923i	$-0.0745019\pi$
$47$	13.3375	1.94547	0.972734	+	0.231923i	$0.0745019\pi$
$48$	0	0
$49$	7.55171	1.07882
$50$	0	0
$51$	1.92187	0.269116
$52$	0	0
$53$	12.4108	1.70475	0.852377	−	0.522928i	$-0.175160\pi$
$53$	12.4108	1.70475	0.852377	+	0.522928i	$0.175160\pi$
$54$	0	0
$55$	4.69649	0.633275
$56$	0	0
$57$	−7.80360	−1.03361
$58$	0	0
$59$	−3.06388	−0.398884	−0.199442	−	0.979910i	$-0.563913\pi$
$59$	−3.06388	−0.398884	−0.199442	+	0.979910i	$0.563913\pi$
$60$	0	0
$61$	9.88695	1.26589	0.632947	−	0.774195i	$-0.281845\pi$
$61$	9.88695	1.26589	0.632947	+	0.774195i	$0.281845\pi$
$62$	0	0
$63$	3.81467	0.480603
$64$	0	0
$65$	−3.08113	−0.382168
$66$	0	0
$67$	14.3222	1.74973	0.874866	−	0.484365i	$-0.160949\pi$
$67$	14.3222	1.74973	0.874866	+	0.484365i	$0.160949\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	10.5787	1.25547	0.627733	−	0.778429i	$-0.283983\pi$
$71$	10.5787	1.25547	0.627733	+	0.778429i	$0.283983\pi$
$72$	0	0
$73$	−9.23222	−1.08055	−0.540275	−	0.841489i	$-0.681680\pi$
$73$	−9.23222	−1.08055	−0.540275	+	0.841489i	$0.681680\pi$
$74$	0	0
$75$	−4.19480	−0.484373
$76$	0	0
$77$	19.9654	2.27527
$78$	0	0
$79$	2.22128	0.249914	0.124957	−	0.992162i	$-0.460121\pi$
$79$	2.22128	0.249914	0.124957	+	0.992162i	$0.460121\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	2.48600	0.272874	0.136437	−	0.990649i	$-0.456435\pi$
$83$	2.48600	0.272874	0.136437	+	0.990649i	$0.456435\pi$
$84$	0	0
$85$	1.72455	0.187054
$86$	0	0
$87$	−1.00000	−0.107211
$88$	0	0
$89$	−13.2715	−1.40678	−0.703390	−	0.710804i	$-0.748331\pi$
$89$	−13.2715	−1.40678	−0.703390	+	0.710804i	$0.748331\pi$
$90$	0	0
$91$	−13.0983	−1.37307
$92$	0	0
$93$	3.73130	0.386918
$94$	0	0
$95$	−7.00241	−0.718432
$96$	0	0
$97$	−7.70919	−0.782750	−0.391375	−	0.920231i	$-0.628000\pi$
$97$	−7.70919	−0.782750	−0.391375	+	0.920231i	$0.628000\pi$
$98$	0	0
$99$	5.23385	0.526021
$100$	0	0
$101$	13.1086	1.30435	0.652176	−	0.758068i	$-0.273856\pi$
$101$	13.1086	1.30435	0.652176	+	0.758068i	$0.273856\pi$
$102$	0	0
$103$	18.5637	1.82913	0.914567	−	0.404435i	$-0.132532\pi$
$103$	18.5637	1.82913	0.914567	+	0.404435i	$0.132532\pi$
$104$	0	0
$105$	3.42302	0.334053
$106$	0	0
$107$	12.8057	1.23797	0.618987	−	0.785401i	$-0.287543\pi$
$107$	12.8057	1.23797	0.618987	+	0.785401i	$0.287543\pi$
$108$	0	0
$109$	−1.08922	−0.104328	−0.0521640	−	0.998639i	$-0.516612\pi$
$109$	−1.08922	−0.104328	−0.0521640	+	0.998639i	$0.516612\pi$
$110$	0	0
$111$	0.939833	0.0892050
$112$	0	0
$113$	15.8228	1.48849	0.744244	−	0.667908i	$-0.232810\pi$
$113$	15.8228	1.48849	0.744244	+	0.667908i	$0.232810\pi$
$114$	0	0
$115$	0.897331	0.0836765
$116$	0	0
$117$	−3.43366	−0.317442
$118$	0	0
$119$	7.33130	0.672059
$120$	0	0
$121$	16.3931	1.49029
$122$	0	0
$123$	−5.90444	−0.532386
$124$	0	0
$125$	−8.25078	−0.737972
$126$	0	0
$127$	20.9937	1.86289	0.931444	−	0.363885i	$-0.118550\pi$
$127$	20.9937	1.86289	0.931444	+	0.363885i	$0.118550\pi$
$128$	0	0
$129$	−2.82368	−0.248611
$130$	0	0
$131$	−11.8838	−1.03829	−0.519145	−	0.854686i	$-0.673749\pi$
$131$	−11.8838	−1.03829	−0.519145	+	0.854686i	$0.673749\pi$
$132$	0	0
$133$	−29.7682	−2.58123
$134$	0	0
$135$	0.897331	0.0772299
$136$	0	0
$137$	−9.74379	−0.832468	−0.416234	−	0.909258i	$-0.636650\pi$
$137$	−9.74379	−0.832468	−0.416234	+	0.909258i	$0.636650\pi$
$138$	0	0
$139$	−21.7119	−1.84158	−0.920790	−	0.390060i	$-0.872454\pi$
$139$	−21.7119	−1.84158	−0.920790	+	0.390060i	$0.872454\pi$
$140$	0	0
$141$	13.3375	1.12322
$142$	0	0
$143$	−17.9713	−1.50283
$144$	0	0
$145$	−0.897331	−0.0745193
$146$	0	0
$147$	7.55171	0.622854
$148$	0	0
$149$	−0.0539251	−0.00441771	−0.00220886	−	0.999998i	$-0.500703\pi$
$149$	−0.0539251	−0.00441771	−0.00220886	+	0.999998i	$0.500703\pi$
$150$	0	0
$151$	−9.95429	−0.810069	−0.405034	−	0.914301i	$-0.632740\pi$
$151$	−9.95429	−0.810069	−0.405034	+	0.914301i	$0.632740\pi$
$152$	0	0
$153$	1.92187	0.155374
$154$	0	0
$155$	3.34821	0.268935
$156$	0	0
$157$	−5.32590	−0.425053	−0.212527	−	0.977155i	$-0.568169\pi$
$157$	−5.32590	−0.425053	−0.212527	+	0.977155i	$0.568169\pi$
$158$	0	0
$159$	12.4108	0.984240
$160$	0	0
$161$	3.81467	0.300638
$162$	0	0
$163$	18.3644	1.43841	0.719206	−	0.694797i	$-0.244506\pi$
$163$	18.3644	1.43841	0.719206	+	0.694797i	$0.244506\pi$
$164$	0	0
$165$	4.69649	0.365621
$166$	0	0
$167$	−16.8360	−1.30281	−0.651405	−	0.758730i	$-0.725820\pi$
$167$	−16.8360	−1.30281	−0.651405	+	0.758730i	$0.725820\pi$
$168$	0	0
$169$	−1.20995	−0.0930729
$170$	0	0
$171$	−7.80360	−0.596756
$172$	0	0
$173$	−22.0282	−1.67478	−0.837388	−	0.546609i	$-0.815918\pi$
$173$	−22.0282	−1.67478	−0.837388	+	0.546609i	$0.815918\pi$
$174$	0	0
$175$	−16.0018	−1.20962
$176$	0	0
$177$	−3.06388	−0.230296
$178$	0	0
$179$	2.29809	0.171767	0.0858837	−	0.996305i	$-0.472629\pi$
$179$	2.29809	0.171767	0.0858837	+	0.996305i	$0.472629\pi$
$180$	0	0
$181$	1.94435	0.144522	0.0722611	−	0.997386i	$-0.476979\pi$
$181$	1.94435	0.144522	0.0722611	+	0.997386i	$0.476979\pi$
$182$	0	0
$183$	9.88695	0.730865
$184$	0	0
$185$	0.843341	0.0620037
$186$	0	0
$187$	10.0588	0.735570
$188$	0	0
$189$	3.81467	0.277476
$190$	0	0
$191$	13.7295	0.993430	0.496715	−	0.867914i	$-0.334539\pi$
$191$	13.7295	0.993430	0.496715	+	0.867914i	$0.334539\pi$
$192$	0	0
$193$	13.9644	1.00518	0.502591	−	0.864524i	$-0.332380\pi$
$193$	13.9644	1.00518	0.502591	+	0.864524i	$0.332380\pi$
$194$	0	0
$195$	−3.08113	−0.220645
$196$	0	0
$197$	6.85157	0.488154	0.244077	−	0.969756i	$-0.421515\pi$
$197$	6.85157	0.488154	0.244077	+	0.969756i	$0.421515\pi$
$198$	0	0
$199$	14.0960	0.999240	0.499620	−	0.866245i	$-0.333473\pi$
$199$	14.0960	0.999240	0.499620	+	0.866245i	$0.333473\pi$
$200$	0	0
$201$	14.3222	1.01021
$202$	0	0
$203$	−3.81467	−0.267737
$204$	0	0
$205$	−5.29824	−0.370045
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	−40.8428	−2.82516
$210$	0	0
$211$	−1.38760	−0.0955265	−0.0477632	−	0.998859i	$-0.515209\pi$
$211$	−1.38760	−0.0955265	−0.0477632	+	0.998859i	$0.515209\pi$
$212$	0	0
$213$	10.5787	0.724844
$214$	0	0
$215$	−2.53377	−0.172802
$216$	0	0
$217$	14.2337	0.966246
$218$	0	0
$219$	−9.23222	−0.623856
$220$	0	0
$221$	−6.59905	−0.443900
$222$	0	0
$223$	−8.82944	−0.591263	−0.295632	−	0.955302i	$-0.595530\pi$
$223$	−8.82944	−0.591263	−0.295632	+	0.955302i	$0.595530\pi$
$224$	0	0
$225$	−4.19480	−0.279653
$226$	0	0
$227$	27.1801	1.80401	0.902004	−	0.431727i	$-0.142096\pi$
$227$	27.1801	1.80401	0.902004	+	0.431727i	$0.142096\pi$
$228$	0	0
$229$	−3.21756	−0.212622	−0.106311	−	0.994333i	$-0.533904\pi$
$229$	−3.21756	−0.212622	−0.106311	+	0.994333i	$0.533904\pi$
$230$	0	0
$231$	19.9654	1.31363
$232$	0	0
$233$	2.82764	0.185245	0.0926225	−	0.995701i	$-0.470475\pi$
$233$	2.82764	0.185245	0.0926225	+	0.995701i	$0.470475\pi$
$234$	0	0
$235$	11.9681	0.780714
$236$	0	0
$237$	2.22128	0.144288
$238$	0	0
$239$	−25.8036	−1.66909	−0.834547	−	0.550937i	$-0.814271\pi$
$239$	−25.8036	−1.66909	−0.834547	+	0.550937i	$0.814271\pi$
$240$	0	0
$241$	−27.3656	−1.76277	−0.881387	−	0.472396i	$-0.843389\pi$
$241$	−27.3656	−1.76277	−0.881387	+	0.472396i	$0.843389\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	6.77638	0.432927
$246$	0	0
$247$	26.7950	1.70492
$248$	0	0
$249$	2.48600	0.157544
$250$	0	0
$251$	−2.14400	−0.135328	−0.0676639	−	0.997708i	$-0.521555\pi$
$251$	−2.14400	−0.135328	−0.0676639	+	0.997708i	$0.521555\pi$
$252$	0	0
$253$	5.23385	0.329049
$254$	0	0
$255$	1.72455	0.107996
$256$	0	0
$257$	−18.0033	−1.12302	−0.561508	−	0.827471i	$-0.689779\pi$
$257$	−18.0033	−1.12302	−0.561508	+	0.827471i	$0.689779\pi$
$258$	0	0
$259$	3.58515	0.222771
$260$	0	0
$261$	−1.00000	−0.0618984
$262$	0	0
$263$	−10.5249	−0.648991	−0.324495	−	0.945887i	$-0.605194\pi$
$263$	−10.5249	−0.648991	−0.324495	+	0.945887i	$0.605194\pi$
$264$	0	0
$265$	11.1366	0.684115
$266$	0	0
$267$	−13.2715	−0.812204
$268$	0	0
$269$	1.28161	0.0781413	0.0390706	−	0.999236i	$-0.487560\pi$
$269$	1.28161	0.0781413	0.0390706	+	0.999236i	$0.487560\pi$
$270$	0	0
$271$	−1.11624	−0.0678070	−0.0339035	−	0.999425i	$-0.510794\pi$
$271$	−1.11624	−0.0678070	−0.0339035	+	0.999425i	$0.510794\pi$
$272$	0	0
$273$	−13.0983	−0.792745
$274$	0	0
$275$	−21.9549	−1.32393
$276$	0	0
$277$	4.29599	0.258121	0.129061	−	0.991637i	$-0.458804\pi$
$277$	4.29599	0.258121	0.129061	+	0.991637i	$0.458804\pi$
$278$	0	0
$279$	3.73130	0.223387
$280$	0	0
$281$	−16.1931	−0.965999	−0.482999	−	0.875621i	$-0.660453\pi$
$281$	−16.1931	−0.965999	−0.482999	+	0.875621i	$0.660453\pi$
$282$	0	0
$283$	4.02545	0.239288	0.119644	−	0.992817i	$-0.461825\pi$
$283$	4.02545	0.239288	0.119644	+	0.992817i	$0.461825\pi$
$284$	0	0
$285$	−7.00241	−0.414787
$286$	0	0
$287$	−22.5235	−1.32952
$288$	0	0
$289$	−13.3064	−0.782731
$290$	0	0
$291$	−7.70919	−0.451921
$292$	0	0
$293$	−21.5224	−1.25735	−0.628676	−	0.777667i	$-0.716403\pi$
$293$	−21.5224	−1.25735	−0.628676	+	0.777667i	$0.716403\pi$
$294$	0	0
$295$	−2.74932	−0.160071
$296$	0	0
$297$	5.23385	0.303699
$298$	0	0
$299$	−3.43366	−0.198574
$300$	0	0
$301$	−10.7714	−0.620853
$302$	0	0
$303$	13.1086	0.753068
$304$	0	0
$305$	8.87187	0.508002
$306$	0	0
$307$	7.57632	0.432403	0.216202	−	0.976349i	$-0.430633\pi$
$307$	7.57632	0.432403	0.216202	+	0.976349i	$0.430633\pi$
$308$	0	0
$309$	18.5637	1.05605
$310$	0	0
$311$	−13.2050	−0.748789	−0.374394	−	0.927270i	$-0.622149\pi$
$311$	−13.2050	−0.748789	−0.374394	+	0.927270i	$0.622149\pi$
$312$	0	0
$313$	23.9026	1.35105	0.675527	−	0.737336i	$-0.263916\pi$
$313$	23.9026	1.35105	0.675527	+	0.737336i	$0.263916\pi$
$314$	0	0
$315$	3.42302	0.192865
$316$	0	0
$317$	−3.42802	−0.192537	−0.0962683	−	0.995355i	$-0.530691\pi$
$317$	−3.42802	−0.192537	−0.0962683	+	0.995355i	$0.530691\pi$
$318$	0	0
$319$	−5.23385	−0.293039
$320$	0	0
$321$	12.8057	0.714744
$322$	0	0
$323$	−14.9975	−0.834483
$324$	0	0
$325$	14.4035	0.798964
$326$	0	0
$327$	−1.08922	−0.0602338
$328$	0	0
$329$	50.8780	2.80499
$330$	0	0
$331$	19.5447	1.07428	0.537138	−	0.843494i	$-0.319505\pi$
$331$	19.5447	1.07428	0.537138	+	0.843494i	$0.319505\pi$
$332$	0	0
$333$	0.939833	0.0515025
$334$	0	0
$335$	12.8517	0.702165
$336$	0	0
$337$	−27.5463	−1.50054	−0.750272	−	0.661129i	$-0.770078\pi$
$337$	−27.5463	−1.50054	−0.750272	+	0.661129i	$0.770078\pi$
$338$	0	0
$339$	15.8228	0.859379
$340$	0	0
$341$	19.5291	1.05756
$342$	0	0
$343$	2.10458	0.113637
$344$	0	0
$345$	0.897331	0.0483107
$346$	0	0
$347$	17.9600	0.964141	0.482070	−	0.876132i	$-0.339885\pi$
$347$	17.9600	0.964141	0.482070	+	0.876132i	$0.339885\pi$
$348$	0	0
$349$	−15.2215	−0.814790	−0.407395	−	0.913252i	$-0.633563\pi$
$349$	−15.2215	−0.814790	−0.407395	+	0.913252i	$0.633563\pi$
$350$	0	0
$351$	−3.43366	−0.183275
$352$	0	0
$353$	−30.5406	−1.62551	−0.812756	−	0.582604i	$-0.802034\pi$
$353$	−30.5406	−1.62551	−0.812756	+	0.582604i	$0.802034\pi$
$354$	0	0
$355$	9.49263	0.503817
$356$	0	0
$357$	7.33130	0.388013
$358$	0	0
$359$	−29.2887	−1.54580	−0.772900	−	0.634528i	$-0.781195\pi$
$359$	−29.2887	−1.54580	−0.772900	+	0.634528i	$0.781195\pi$
$360$	0	0
$361$	41.8962	2.20506
$362$	0	0
$363$	16.3931	0.860417
$364$	0	0
$365$	−8.28436	−0.433623
$366$	0	0
$367$	5.83369	0.304516	0.152258	−	0.988341i	$-0.451346\pi$
$367$	5.83369	0.304516	0.152258	+	0.988341i	$0.451346\pi$
$368$	0	0
$369$	−5.90444	−0.307373
$370$	0	0
$371$	47.3431	2.45793
$372$	0	0
$373$	6.18034	0.320006	0.160003	−	0.987117i	$-0.448850\pi$
$373$	6.18034	0.320006	0.160003	+	0.987117i	$0.448850\pi$
$374$	0	0
$375$	−8.25078	−0.426068
$376$	0	0
$377$	3.43366	0.176843
$378$	0	0
$379$	28.9038	1.48469	0.742344	−	0.670019i	$-0.233714\pi$
$379$	28.9038	1.48469	0.742344	+	0.670019i	$0.233714\pi$
$380$	0	0
$381$	20.9937	1.07554
$382$	0	0
$383$	−3.03770	−0.155219	−0.0776096	−	0.996984i	$-0.524729\pi$
$383$	−3.03770	−0.155219	−0.0776096	+	0.996984i	$0.524729\pi$
$384$	0	0
$385$	17.9156	0.913062
$386$	0	0
$387$	−2.82368	−0.143536
$388$	0	0
$389$	−4.43947	−0.225090	−0.112545	−	0.993647i	$-0.535900\pi$
$389$	−4.43947	−0.225090	−0.112545	+	0.993647i	$0.535900\pi$
$390$	0	0
$391$	1.92187	0.0971931
$392$	0	0
$393$	−11.8838	−0.599457
$394$	0	0
$395$	1.99322	0.100290
$396$	0	0
$397$	−14.6535	−0.735436	−0.367718	−	0.929937i	$-0.619861\pi$
$397$	−14.6535	−0.735436	−0.367718	+	0.929937i	$0.619861\pi$
$398$	0	0
$399$	−29.7682	−1.49027
$400$	0	0
$401$	−5.70941	−0.285115	−0.142557	−	0.989787i	$-0.545533\pi$
$401$	−5.70941	−0.285115	−0.142557	+	0.989787i	$0.545533\pi$
$402$	0	0
$403$	−12.8120	−0.638214
$404$	0	0
$405$	0.897331	0.0445887
$406$	0	0
$407$	4.91894	0.243823
$408$	0	0
$409$	−10.1737	−0.503055	−0.251527	−	0.967850i	$-0.580933\pi$
$409$	−10.1737	−0.503055	−0.251527	+	0.967850i	$0.580933\pi$
$410$	0	0
$411$	−9.74379	−0.480626
$412$	0	0
$413$	−11.6877	−0.575115
$414$	0	0
$415$	2.23077	0.109504
$416$	0	0
$417$	−21.7119	−1.06324
$418$	0	0
$419$	26.3758	1.28854	0.644272	−	0.764796i	$-0.277160\pi$
$419$	26.3758	1.28854	0.644272	+	0.764796i	$0.277160\pi$
$420$	0	0
$421$	38.1010	1.85693	0.928464	−	0.371422i	$-0.121130\pi$
$421$	38.1010	1.85693	0.928464	+	0.371422i	$0.121130\pi$
$422$	0	0
$423$	13.3375	0.648489
$424$	0	0
$425$	−8.06185	−0.391057
$426$	0	0
$427$	37.7155	1.82518
$428$	0	0
$429$	−17.9713	−0.867661
$430$	0	0
$431$	4.20187	0.202397	0.101199	−	0.994866i	$-0.467732\pi$
$431$	4.20187	0.202397	0.101199	+	0.994866i	$0.467732\pi$
$432$	0	0
$433$	27.2424	1.30918	0.654592	−	0.755982i	$-0.272840\pi$
$433$	27.2424	1.30918	0.654592	+	0.755982i	$0.272840\pi$
$434$	0	0
$435$	−0.897331	−0.0430237
$436$	0	0
$437$	−7.80360	−0.373297
$438$	0	0
$439$	−21.4883	−1.02558	−0.512791	−	0.858513i	$-0.671389\pi$
$439$	−21.4883	−1.02558	−0.512791	+	0.858513i	$0.671389\pi$
$440$	0	0
$441$	7.55171	0.359605
$442$	0	0
$443$	12.1798	0.578681	0.289341	−	0.957226i	$-0.406564\pi$
$443$	12.1798	0.578681	0.289341	+	0.957226i	$0.406564\pi$
$444$	0	0
$445$	−11.9090	−0.564539
$446$	0	0
$447$	−0.0539251	−0.00255057
$448$	0	0
$449$	−8.96580	−0.423122	−0.211561	−	0.977365i	$-0.567855\pi$
$449$	−8.96580	−0.423122	−0.211561	+	0.977365i	$0.567855\pi$
$450$	0	0
$451$	−30.9029	−1.45516
$452$	0	0
$453$	−9.95429	−0.467693
$454$	0	0
$455$	−11.7535	−0.551013
$456$	0	0
$457$	28.6575	1.34054	0.670271	−	0.742117i	$-0.266178\pi$
$457$	28.6575	1.34054	0.670271	+	0.742117i	$0.266178\pi$
$458$	0	0
$459$	1.92187	0.0897052
$460$	0	0
$461$	−14.1620	−0.659590	−0.329795	−	0.944053i	$-0.606980\pi$
$461$	−14.1620	−0.659590	−0.329795	+	0.944053i	$0.606980\pi$
$462$	0	0
$463$	−29.2565	−1.35967	−0.679833	−	0.733367i	$-0.737948\pi$
$463$	−29.2565	−1.35967	−0.679833	+	0.733367i	$0.737948\pi$
$464$	0	0
$465$	3.34821	0.155270
$466$	0	0
$467$	31.1030	1.43927	0.719637	−	0.694351i	$-0.244308\pi$
$467$	31.1030	1.43927	0.719637	+	0.694351i	$0.244308\pi$
$468$	0	0
$469$	54.6344	2.52278
$470$	0	0
$471$	−5.32590	−0.245405
$472$	0	0
$473$	−14.7787	−0.679525
$474$	0	0
$475$	32.7345	1.50196
$476$	0	0
$477$	12.4108	0.568251
$478$	0	0
$479$	3.86249	0.176482	0.0882409	−	0.996099i	$-0.471875\pi$
$479$	3.86249	0.176482	0.0882409	+	0.996099i	$0.471875\pi$
$480$	0	0
$481$	−3.22707	−0.147142
$482$	0	0
$483$	3.81467	0.173573
$484$	0	0
$485$	−6.91770	−0.314116
$486$	0	0
$487$	−16.7922	−0.760929	−0.380465	−	0.924795i	$-0.624236\pi$
$487$	−16.7922	−0.760929	−0.380465	+	0.924795i	$0.624236\pi$
$488$	0	0
$489$	18.3644	0.830468
$490$	0	0
$491$	−15.8110	−0.713540	−0.356770	−	0.934192i	$-0.616122\pi$
$491$	−15.8110	−0.713540	−0.356770	+	0.934192i	$0.616122\pi$
$492$	0	0
$493$	−1.92187	−0.0865566
$494$	0	0
$495$	4.69649	0.211092
$496$	0	0
$497$	40.3544	1.81014
$498$	0	0
$499$	−34.0511	−1.52433	−0.762167	−	0.647380i	$-0.775865\pi$
$499$	−34.0511	−1.52433	−0.762167	+	0.647380i	$0.775865\pi$
$500$	0	0
$501$	−16.8360	−0.752178
$502$	0	0
$503$	2.51408	0.112097	0.0560486	−	0.998428i	$-0.482150\pi$
$503$	2.51408	0.112097	0.0560486	+	0.998428i	$0.482150\pi$
$504$	0	0
$505$	11.7627	0.523434
$506$	0	0
$507$	−1.20995	−0.0537356
$508$	0	0
$509$	−17.5905	−0.779686	−0.389843	−	0.920881i	$-0.627471\pi$
$509$	−17.5905	−0.779686	−0.389843	+	0.920881i	$0.627471\pi$
$510$	0	0
$511$	−35.2179	−1.55795
$512$	0	0
$513$	−7.80360	−0.344537
$514$	0	0
$515$	16.6578	0.734029
$516$	0	0
$517$	69.8062	3.07007
$518$	0	0
$519$	−22.0282	−0.966932
$520$	0	0
$521$	39.9368	1.74966	0.874832	−	0.484426i	$-0.160972\pi$
$521$	39.9368	1.74966	0.874832	+	0.484426i	$0.160972\pi$
$522$	0	0
$523$	−6.76095	−0.295636	−0.147818	−	0.989015i	$-0.547225\pi$
$523$	−6.76095	−0.295636	−0.147818	+	0.989015i	$0.547225\pi$
$524$	0	0
$525$	−16.0018	−0.698374
$526$	0	0
$527$	7.17108	0.312377
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−3.06388	−0.132961
$532$	0	0
$533$	20.2739	0.878159
$534$	0	0
$535$	11.4909	0.496797
$536$	0	0
$537$	2.29809	0.0991700
$538$	0	0
$539$	39.5245	1.70244
$540$	0	0
$541$	−13.6532	−0.586997	−0.293498	−	0.955960i	$-0.594820\pi$
$541$	−13.6532	−0.586997	−0.293498	+	0.955960i	$0.594820\pi$
$542$	0	0
$543$	1.94435	0.0834399
$544$	0	0
$545$	−0.977388	−0.0418667
$546$	0	0
$547$	27.3579	1.16974	0.584870	−	0.811127i	$-0.301146\pi$
$547$	27.3579	1.16974	0.584870	+	0.811127i	$0.301146\pi$
$548$	0	0
$549$	9.88695	0.421965
$550$	0	0
$551$	7.80360	0.332445
$552$	0	0
$553$	8.47345	0.360328
$554$	0	0
$555$	0.843341	0.0357978
$556$	0	0
$557$	−21.0863	−0.893456	−0.446728	−	0.894670i	$-0.647411\pi$
$557$	−21.0863	−0.893456	−0.446728	+	0.894670i	$0.647411\pi$
$558$	0	0
$559$	9.69556	0.410078
$560$	0	0
$561$	10.0588	0.424681
$562$	0	0
$563$	−17.2843	−0.728446	−0.364223	−	0.931312i	$-0.618665\pi$
$563$	−17.2843	−0.728446	−0.364223	+	0.931312i	$0.618665\pi$
$564$	0	0
$565$	14.1983	0.597328
$566$	0	0
$567$	3.81467	0.160201
$568$	0	0
$569$	−14.4233	−0.604658	−0.302329	−	0.953204i	$-0.597764\pi$
$569$	−14.4233	−0.604658	−0.302329	+	0.953204i	$0.597764\pi$
$570$	0	0
$571$	−8.58461	−0.359255	−0.179627	−	0.983735i	$-0.557489\pi$
$571$	−8.58461	−0.359255	−0.179627	+	0.983735i	$0.557489\pi$
$572$	0	0
$573$	13.7295	0.573557
$574$	0	0
$575$	−4.19480	−0.174935
$576$	0	0
$577$	−11.1434	−0.463907	−0.231953	−	0.972727i	$-0.574512\pi$
$577$	−11.1434	−0.463907	−0.231953	+	0.972727i	$0.574512\pi$
$578$	0	0
$579$	13.9644	0.580342
$580$	0	0
$581$	9.48328	0.393433
$582$	0	0
$583$	64.9562	2.69021
$584$	0	0
$585$	−3.08113	−0.127389
$586$	0	0
$587$	−28.2597	−1.16640	−0.583200	−	0.812328i	$-0.698200\pi$
$587$	−28.2597	−1.16640	−0.583200	+	0.812328i	$0.698200\pi$
$588$	0	0
$589$	−29.1176	−1.19977
$590$	0	0
$591$	6.85157	0.281836
$592$	0	0
$593$	25.4794	1.04631	0.523157	−	0.852236i	$-0.324754\pi$
$593$	25.4794	1.04631	0.523157	+	0.852236i	$0.324754\pi$
$594$	0	0
$595$	6.57860	0.269696
$596$	0	0
$597$	14.0960	0.576911
$598$	0	0
$599$	−39.7316	−1.62339	−0.811694	−	0.584082i	$-0.801454\pi$
$599$	−39.7316	−1.62339	−0.811694	+	0.584082i	$0.801454\pi$
$600$	0	0
$601$	−12.5941	−0.513725	−0.256863	−	0.966448i	$-0.582689\pi$
$601$	−12.5941	−0.513725	−0.256863	+	0.966448i	$0.582689\pi$
$602$	0	0
$603$	14.3222	0.583244
$604$	0	0
$605$	14.7101	0.598049
$606$	0	0
$607$	−35.7473	−1.45094	−0.725470	−	0.688254i	$-0.758378\pi$
$607$	−35.7473	−1.45094	−0.725470	+	0.688254i	$0.758378\pi$
$608$	0	0
$609$	−3.81467	−0.154578
$610$	0	0
$611$	−45.7964	−1.85272
$612$	0	0
$613$	−24.5319	−0.990835	−0.495418	−	0.868655i	$-0.664985\pi$
$613$	−24.5319	−0.990835	−0.495418	+	0.868655i	$0.664985\pi$
$614$	0	0
$615$	−5.29824	−0.213646
$616$	0	0
$617$	−31.9825	−1.28757	−0.643784	−	0.765207i	$-0.722637\pi$
$617$	−31.9825	−1.28757	−0.643784	+	0.765207i	$0.722637\pi$
$618$	0	0
$619$	−21.6271	−0.869266	−0.434633	−	0.900608i	$-0.643122\pi$
$619$	−21.6271	−0.869266	−0.434633	+	0.900608i	$0.643122\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	−50.6265	−2.02831
$624$	0	0
$625$	13.5703	0.542812
$626$	0	0
$627$	−40.8428	−1.63111
$628$	0	0
$629$	1.80624	0.0720194
$630$	0	0
$631$	−2.28377	−0.0909156	−0.0454578	−	0.998966i	$-0.514475\pi$
$631$	−2.28377	−0.0909156	−0.0454578	+	0.998966i	$0.514475\pi$
$632$	0	0
$633$	−1.38760	−0.0551522
$634$	0	0
$635$	18.8383	0.747574
$636$	0	0
$637$	−25.9300	−1.02739
$638$	0	0
$639$	10.5787	0.418489
$640$	0	0
$641$	13.9618	0.551459	0.275730	−	0.961235i	$-0.411081\pi$
$641$	13.9618	0.551459	0.275730	+	0.961235i	$0.411081\pi$
$642$	0	0
$643$	−15.5694	−0.613997	−0.306999	−	0.951710i	$-0.599325\pi$
$643$	−15.5694	−0.613997	−0.306999	+	0.951710i	$0.599325\pi$
$644$	0	0
$645$	−2.53377	−0.0997672
$646$	0	0
$647$	−49.3101	−1.93858	−0.969289	−	0.245924i	$-0.920909\pi$
$647$	−49.3101	−1.93858	−0.969289	+	0.245924i	$0.920909\pi$
$648$	0	0
$649$	−16.0359	−0.629464
$650$	0	0
$651$	14.2337	0.557863
$652$	0	0
$653$	7.26719	0.284387	0.142194	−	0.989839i	$-0.454584\pi$
$653$	7.26719	0.284387	0.142194	+	0.989839i	$0.454584\pi$
$654$	0	0
$655$	−10.6637	−0.416664
$656$	0	0
$657$	−9.23222	−0.360183
$658$	0	0
$659$	8.90275	0.346802	0.173401	−	0.984851i	$-0.444524\pi$
$659$	8.90275	0.346802	0.173401	+	0.984851i	$0.444524\pi$
$660$	0	0
$661$	4.56464	0.177544	0.0887720	−	0.996052i	$-0.471706\pi$
$661$	4.56464	0.177544	0.0887720	+	0.996052i	$0.471706\pi$
$662$	0	0
$663$	−6.59905	−0.256286
$664$	0	0
$665$	−26.7119	−1.03584
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	0	0
$669$	−8.82944	−0.341366
$670$	0	0
$671$	51.7468	1.99766
$672$	0	0
$673$	−5.28579	−0.203752	−0.101876	−	0.994797i	$-0.532485\pi$
$673$	−5.28579	−0.203752	−0.101876	+	0.994797i	$0.532485\pi$
$674$	0	0
$675$	−4.19480	−0.161458
$676$	0	0
$677$	26.6869	1.02566	0.512831	−	0.858490i	$-0.328597\pi$
$677$	26.6869	1.02566	0.512831	+	0.858490i	$0.328597\pi$
$678$	0	0
$679$	−29.4080	−1.12858
$680$	0	0
$681$	27.1801	1.04154
$682$	0	0
$683$	−7.20362	−0.275639	−0.137819	−	0.990457i	$-0.544009\pi$
$683$	−7.20362	−0.275639	−0.137819	+	0.990457i	$0.544009\pi$
$684$	0	0
$685$	−8.74340	−0.334068
$686$	0	0
$687$	−3.21756	−0.122758
$688$	0	0
$689$	−42.6145	−1.62348
$690$	0	0
$691$	30.4661	1.15898	0.579492	−	0.814978i	$-0.303251\pi$
$691$	30.4661	1.15898	0.579492	+	0.814978i	$0.303251\pi$
$692$	0	0
$693$	19.9654	0.758423
$694$	0	0
$695$	−19.4828	−0.739023
$696$	0	0
$697$	−11.3476	−0.429820
$698$	0	0
$699$	2.82764	0.106951
$700$	0	0
$701$	20.7783	0.784784	0.392392	−	0.919798i	$-0.371648\pi$
$701$	20.7783	0.784784	0.392392	+	0.919798i	$0.371648\pi$
$702$	0	0
$703$	−7.33408	−0.276610
$704$	0	0
$705$	11.9681	0.450745
$706$	0	0
$707$	50.0049	1.88063
$708$	0	0
$709$	19.4473	0.730359	0.365179	−	0.930937i	$-0.381008\pi$
$709$	19.4473	0.730359	0.365179	+	0.930937i	$0.381008\pi$
$710$	0	0
$711$	2.22128	0.0833046
$712$	0	0
$713$	3.73130	0.139738
$714$	0	0
$715$	−16.1262	−0.603085
$716$	0	0
$717$	−25.8036	−0.963652
$718$	0	0
$719$	26.2733	0.979827	0.489914	−	0.871771i	$-0.337028\pi$
$719$	26.2733	0.979827	0.489914	+	0.871771i	$0.337028\pi$
$720$	0	0
$721$	70.8143	2.63726
$722$	0	0
$723$	−27.3656	−1.01774
$724$	0	0
$725$	4.19480	0.155791
$726$	0	0
$727$	28.3921	1.05300	0.526502	−	0.850174i	$-0.323503\pi$
$727$	28.3921	1.05300	0.526502	+	0.850174i	$0.323503\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−5.42674	−0.200715
$732$	0	0
$733$	46.1551	1.70478	0.852389	−	0.522909i	$-0.175153\pi$
$733$	46.1551	1.70478	0.852389	+	0.522909i	$0.175153\pi$
$734$	0	0
$735$	6.77638	0.249951
$736$	0	0
$737$	74.9600	2.76119
$738$	0	0
$739$	−4.13404	−0.152073	−0.0760365	−	0.997105i	$-0.524227\pi$
$739$	−4.13404	−0.152073	−0.0760365	+	0.997105i	$0.524227\pi$
$740$	0	0
$741$	26.7950	0.984337
$742$	0	0
$743$	−31.3373	−1.14965	−0.574826	−	0.818276i	$-0.694930\pi$
$743$	−31.3373	−1.14965	−0.574826	+	0.818276i	$0.694930\pi$
$744$	0	0
$745$	−0.0483886	−0.00177282
$746$	0	0
$747$	2.48600	0.0909581
$748$	0	0
$749$	48.8495	1.78492
$750$	0	0
$751$	15.2978	0.558226	0.279113	−	0.960258i	$-0.409960\pi$
$751$	15.2978	0.558226	0.279113	+	0.960258i	$0.409960\pi$
$752$	0	0
$753$	−2.14400	−0.0781315
$754$	0	0
$755$	−8.93229	−0.325079
$756$	0	0
$757$	−38.9382	−1.41523	−0.707617	−	0.706596i	$-0.750230\pi$
$757$	−38.9382	−1.41523	−0.707617	+	0.706596i	$0.750230\pi$
$758$	0	0
$759$	5.23385	0.189977
$760$	0	0
$761$	−16.3325	−0.592054	−0.296027	−	0.955180i	$-0.595662\pi$
$761$	−16.3325	−0.592054	−0.296027	+	0.955180i	$0.595662\pi$
$762$	0	0
$763$	−4.15500	−0.150421
$764$	0	0
$765$	1.72455	0.0623513
$766$	0	0
$767$	10.5204	0.379868
$768$	0	0
$769$	−52.3619	−1.88822	−0.944109	−	0.329634i	$-0.893075\pi$
$769$	−52.3619	−1.88822	−0.944109	+	0.329634i	$0.893075\pi$
$770$	0	0
$771$	−18.0033	−0.648373
$772$	0	0
$773$	−20.5849	−0.740388	−0.370194	−	0.928955i	$-0.620709\pi$
$773$	−20.5849	−0.740388	−0.370194	+	0.928955i	$0.620709\pi$
$774$	0	0
$775$	−15.6521	−0.562239
$776$	0	0
$777$	3.58515	0.128617
$778$	0	0
$779$	46.0759	1.65084
$780$	0	0
$781$	55.3675	1.98121
$782$	0	0
$783$	−1.00000	−0.0357371
$784$	0	0
$785$	−4.77910	−0.170573
$786$	0	0
$787$	−54.1435	−1.93001	−0.965003	−	0.262239i	$-0.915539\pi$
$787$	−54.1435	−1.93001	−0.965003	+	0.262239i	$0.915539\pi$
$788$	0	0
$789$	−10.5249	−0.374695
$790$	0	0
$791$	60.3589	2.14612
$792$	0	0
$793$	−33.9485	−1.20555
$794$	0	0
$795$	11.1366	0.394974
$796$	0	0
$797$	50.2674	1.78056	0.890281	−	0.455412i	$-0.150508\pi$
$797$	50.2674	1.78056	0.890281	+	0.455412i	$0.150508\pi$
$798$	0	0
$799$	25.6328	0.906825
$800$	0	0
$801$	−13.2715	−0.468926
$802$	0	0
$803$	−48.3200	−1.70518
$804$	0	0
$805$	3.42302	0.120646
$806$	0	0
$807$	1.28161	0.0451149
$808$	0	0
$809$	7.91149	0.278153	0.139077	−	0.990282i	$-0.455587\pi$
$809$	7.91149	0.278153	0.139077	+	0.990282i	$0.455587\pi$
$810$	0	0
$811$	−8.86464	−0.311280	−0.155640	−	0.987814i	$-0.549744\pi$
$811$	−8.86464	−0.311280	−0.155640	+	0.987814i	$0.549744\pi$
$812$	0	0
$813$	−1.11624	−0.0391484
$814$	0	0
$815$	16.4790	0.577233
$816$	0	0
$817$	22.0349	0.770902
$818$	0	0
$819$	−13.0983	−0.457692
$820$	0	0
$821$	−1.24346	−0.0433970	−0.0216985	−	0.999765i	$-0.506907\pi$
$821$	−1.24346	−0.0433970	−0.0216985	+	0.999765i	$0.506907\pi$
$822$	0	0
$823$	−17.8486	−0.622163	−0.311081	−	0.950383i	$-0.600691\pi$
$823$	−17.8486	−0.622163	−0.311081	+	0.950383i	$0.600691\pi$
$824$	0	0
$825$	−21.9549	−0.764372
$826$	0	0
$827$	−14.7403	−0.512570	−0.256285	−	0.966601i	$-0.582499\pi$
$827$	−14.7403	−0.512570	−0.256285	+	0.966601i	$0.582499\pi$
$828$	0	0
$829$	−2.83338	−0.0984075	−0.0492038	−	0.998789i	$-0.515668\pi$
$829$	−2.83338	−0.0984075	−0.0492038	+	0.998789i	$0.515668\pi$
$830$	0	0
$831$	4.29599	0.149026
$832$	0	0
$833$	14.5134	0.502859
$834$	0	0
$835$	−15.1075	−0.522816
$836$	0	0
$837$	3.73130	0.128973
$838$	0	0
$839$	−17.7053	−0.611255	−0.305627	−	0.952151i	$-0.598866\pi$
$839$	−17.7053	−0.611255	−0.305627	+	0.952151i	$0.598866\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	−16.1931	−0.557720
$844$	0	0
$845$	−1.08572	−0.0373500
$846$	0	0
$847$	62.5344	2.14871
$848$	0	0
$849$	4.02545	0.138153
$850$	0	0
$851$	0.939833	0.0322171
$852$	0	0
$853$	−9.48557	−0.324780	−0.162390	−	0.986727i	$-0.551920\pi$
$853$	−9.48557	−0.324780	−0.162390	+	0.986727i	$0.551920\pi$
$854$	0	0
$855$	−7.00241	−0.239477
$856$	0	0
$857$	13.4094	0.458058	0.229029	−	0.973420i	$-0.426445\pi$
$857$	13.4094	0.458058	0.229029	+	0.973420i	$0.426445\pi$
$858$	0	0
$859$	−34.9045	−1.19093	−0.595463	−	0.803382i	$-0.703031\pi$
$859$	−34.9045	−1.19093	−0.595463	+	0.803382i	$0.703031\pi$
$860$	0	0
$861$	−22.5235	−0.767599
$862$	0	0
$863$	38.8001	1.32077	0.660386	−	0.750926i	$-0.270393\pi$
$863$	38.8001	1.32077	0.660386	+	0.750926i	$0.270393\pi$
$864$	0	0
$865$	−19.7666	−0.672085
$866$	0	0
$867$	−13.3064	−0.451910
$868$	0	0
$869$	11.6258	0.394380
$870$	0	0
$871$	−49.1775	−1.66632
$872$	0	0
$873$	−7.70919	−0.260917
$874$	0	0
$875$	−31.4740	−1.06401
$876$	0	0
$877$	−22.4583	−0.758365	−0.379182	−	0.925322i	$-0.623795\pi$
$877$	−22.4583	−0.758365	−0.379182	+	0.925322i	$0.623795\pi$
$878$	0	0
$879$	−21.5224	−0.725932
$880$	0	0
$881$	24.8079	0.835799	0.417900	−	0.908493i	$-0.362766\pi$
$881$	24.8079	0.835799	0.417900	+	0.908493i	$0.362766\pi$
$882$	0	0
$883$	1.03936	0.0349773	0.0174886	−	0.999847i	$-0.494433\pi$
$883$	1.03936	0.0349773	0.0174886	+	0.999847i	$0.494433\pi$
$884$	0	0
$885$	−2.74932	−0.0924173
$886$	0	0
$887$	−5.32456	−0.178781	−0.0893906	−	0.995997i	$-0.528492\pi$
$887$	−5.32456	−0.178781	−0.0893906	+	0.995997i	$0.528492\pi$
$888$	0	0
$889$	80.0840	2.68593
$890$	0	0
$891$	5.23385	0.175340
$892$	0	0
$893$	−104.080	−3.48291
$894$	0	0
$895$	2.06215	0.0689300
$896$	0	0
$897$	−3.43366	−0.114647
$898$	0	0
$899$	−3.73130	−0.124446
$900$	0	0
$901$	23.8519	0.794623
$902$	0	0
$903$	−10.7714	−0.358450
$904$	0	0
$905$	1.74472	0.0579965
$906$	0	0
$907$	−25.7645	−0.855497	−0.427749	−	0.903898i	$-0.640693\pi$
$907$	−25.7645	−0.855497	−0.427749	+	0.903898i	$0.640693\pi$
$908$	0	0
$909$	13.1086	0.434784
$910$	0	0
$911$	−36.3906	−1.20567	−0.602837	−	0.797864i	$-0.705963\pi$
$911$	−36.3906	−1.20567	−0.602837	+	0.797864i	$0.705963\pi$
$912$	0	0
$913$	13.0114	0.430613
$914$	0	0
$915$	8.87187	0.293295
$916$	0	0
$917$	−45.3326	−1.49702
$918$	0	0
$919$	30.3593	1.00146	0.500731	−	0.865603i	$-0.333065\pi$
$919$	30.3593	1.00146	0.500731	+	0.865603i	$0.333065\pi$
$920$	0	0
$921$	7.57632	0.249648
$922$	0	0
$923$	−36.3239	−1.19561
$924$	0	0
$925$	−3.94241	−0.129626
$926$	0	0
$927$	18.5637	0.609711
$928$	0	0
$929$	8.91679	0.292550	0.146275	−	0.989244i	$-0.453271\pi$
$929$	8.91679	0.292550	0.146275	+	0.989244i	$0.453271\pi$
$930$	0	0
$931$	−58.9305	−1.93137
$932$	0	0
$933$	−13.2050	−0.432313
$934$	0	0
$935$	9.02604	0.295183
$936$	0	0
$937$	30.9839	1.01220	0.506099	−	0.862475i	$-0.331087\pi$
$937$	30.9839	1.01220	0.506099	+	0.862475i	$0.331087\pi$
$938$	0	0
$939$	23.9026	0.780031
$940$	0	0
$941$	53.3625	1.73957	0.869783	−	0.493434i	$-0.164259\pi$
$941$	53.3625	1.73957	0.869783	+	0.493434i	$0.164259\pi$
$942$	0	0
$943$	−5.90444	−0.192275
$944$	0	0
$945$	3.42302	0.111351
$946$	0	0
$947$	−45.6156	−1.48231	−0.741155	−	0.671334i	$-0.765722\pi$
$947$	−45.6156	−1.48231	−0.741155	+	0.671334i	$0.765722\pi$
$948$	0	0
$949$	31.7003	1.02904
$950$	0	0
$951$	−3.42802	−0.111161
$952$	0	0
$953$	22.5093	0.729147	0.364574	−	0.931175i	$-0.381215\pi$
$953$	22.5093	0.729147	0.364574	+	0.931175i	$0.381215\pi$
$954$	0	0
$955$	12.3199	0.398662
$956$	0	0
$957$	−5.23385	−0.169186
$958$	0	0
$959$	−37.1693	−1.20026
$960$	0	0
$961$	−17.0774	−0.550883
$962$	0	0
$963$	12.8057	0.412658
$964$	0	0
$965$	12.5307	0.403378
$966$	0	0
$967$	−2.31451	−0.0744295	−0.0372147	−	0.999307i	$-0.511849\pi$
$967$	−2.31451	−0.0744295	−0.0372147	+	0.999307i	$0.511849\pi$
$968$	0	0
$969$	−14.9975	−0.481789
$970$	0	0
$971$	29.4829	0.946152	0.473076	−	0.881022i	$-0.343144\pi$
$971$	29.4829	0.946152	0.473076	+	0.881022i	$0.343144\pi$
$972$	0	0
$973$	−82.8237	−2.65521
$974$	0	0
$975$	14.4035	0.461282
$976$	0	0
$977$	13.0559	0.417697	0.208848	−	0.977948i	$-0.433029\pi$
$977$	13.0559	0.417697	0.208848	+	0.977948i	$0.433029\pi$
$978$	0	0
$979$	−69.4611	−2.21999
$980$	0	0
$981$	−1.08922	−0.0347760
$982$	0	0
$983$	−7.30342	−0.232943	−0.116471	−	0.993194i	$-0.537158\pi$
$983$	−7.30342	−0.232943	−0.116471	+	0.993194i	$0.537158\pi$
$984$	0	0
$985$	6.14812	0.195895
$986$	0	0
$987$	50.8780	1.61946
$988$	0	0
$989$	−2.82368	−0.0897877
$990$	0	0
$991$	4.49027	0.142638	0.0713190	−	0.997454i	$-0.477279\pi$
$991$	4.49027	0.142638	0.0713190	+	0.997454i	$0.477279\pi$
$992$	0	0
$993$	19.5447	0.620234
$994$	0	0
$995$	12.6488	0.400993
$996$	0	0
$997$	14.3467	0.454365	0.227183	−	0.973852i	$-0.427049\pi$
$997$	14.3467	0.454365	0.227183	+	0.973852i	$0.427049\pi$
$998$	0	0
$999$	0.939833	0.0297350

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8004.2.a.j.1.10	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8004.2.a.j.1.10	✓	16	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +0.897331 q^{5} +3.81467 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +0.897331 q^{5} +3.81467 q^{7} +1.00000 q^{9} +5.23385 q^{11} -3.43366 q^{13} +0.897331 q^{15} +1.92187 q^{17} -7.80360 q^{19} +3.81467 q^{21} +1.00000 q^{23} -4.19480 q^{25} +1.00000 q^{27} -1.00000 q^{29} +3.73130 q^{31} +5.23385 q^{33} +3.42302 q^{35} +0.939833 q^{37} -3.43366 q^{39} -5.90444 q^{41} -2.82368 q^{43} +0.897331 q^{45} +13.3375 q^{47} +7.55171 q^{49} +1.92187 q^{51} +12.4108 q^{53} +4.69649 q^{55} -7.80360 q^{57} -3.06388 q^{59} +9.88695 q^{61} +3.81467 q^{63} -3.08113 q^{65} +14.3222 q^{67} +1.00000 q^{69} +10.5787 q^{71} -9.23222 q^{73} -4.19480 q^{75} +19.9654 q^{77} +2.22128 q^{79} +1.00000 q^{81} +2.48600 q^{83} +1.72455 q^{85} -1.00000 q^{87} -13.2715 q^{89} -13.0983 q^{91} +3.73130 q^{93} -7.00241 q^{95} -7.70919 q^{97} +5.23385 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 16 q^{3} + 3 q^{5} + 4 q^{7} + 16 q^{9}+O(q^{10}) \) 16 * q + 16 * q^3 + 3 * q^5 + 4 * q^7 + 16 * q^9	\( 16 q + 16 q^{3} + 3 q^{5} + 4 q^{7} + 16 q^{9} + 5 q^{11} + 6 q^{13} + 3 q^{15} + 3 q^{17} + 11 q^{19} + 4 q^{21} + 16 q^{23} + 27 q^{25} + 16 q^{27} - 16 q^{29} + 14 q^{31} + 5 q^{33} + 11 q^{35} + 4 q^{37} + 6 q^{39} + 11 q^{41} + 23 q^{43} + 3 q^{45} - 2 q^{47} + 34 q^{49} + 3 q^{51} + 19 q^{53} + 31 q^{55} + 11 q^{57} + 32 q^{59} + 19 q^{61} + 4 q^{63} + 6 q^{65} + 33 q^{67} + 16 q^{69} - 5 q^{71} + 23 q^{73} + 27 q^{75} + 42 q^{77} + 24 q^{79} + 16 q^{81} + 7 q^{83} - 16 q^{87} - 2 q^{89} + 25 q^{91} + 14 q^{93} + 7 q^{95} + 33 q^{97} + 5 q^{99}+O(q^{100}) \) 16 * q + 16 * q^3 + 3 * q^5 + 4 * q^7 + 16 * q^9 + 5 * q^11 + 6 * q^13 + 3 * q^15 + 3 * q^17 + 11 * q^19 + 4 * q^21 + 16 * q^23 + 27 * q^25 + 16 * q^27 - 16 * q^29 + 14 * q^31 + 5 * q^33 + 11 * q^35 + 4 * q^37 + 6 * q^39 + 11 * q^41 + 23 * q^43 + 3 * q^45 - 2 * q^47 + 34 * q^49 + 3 * q^51 + 19 * q^53 + 31 * q^55 + 11 * q^57 + 32 * q^59 + 19 * q^61 + 4 * q^63 + 6 * q^65 + 33 * q^67 + 16 * q^69 - 5 * q^71 + 23 * q^73 + 27 * q^75 + 42 * q^77 + 24 * q^79 + 16 * q^81 + 7 * q^83 - 16 * q^87 - 2 * q^89 + 25 * q^91 + 14 * q^93 + 7 * q^95 + 33 * q^97 + 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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