LMFDB - Embedded newform 8004.2.a.e.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:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 3x^{8} - 13x^{7} + 32x^{6} + 40x^{5} - 79x^{4} - 39x^{3} + 58x^{2} + 9x - 11 $ x^9 - 3x^8 - 13x^7 + 32x^6 + 40x^5 - 79x^4 - 39x^3 + 58x^2 + 9x - 11
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			1.9
Root			$-0.553378$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} +3.29187 q^{5} -1.02028 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +3.29187 q^{5} -1.02028 q^{7} +1.00000 q^{9} -1.88720 q^{11} -3.64262 q^{13} -3.29187 q^{15} -1.28578 q^{17} +5.72913 q^{19} +1.02028 q^{21} -1.00000 q^{23} +5.83643 q^{25} -1.00000 q^{27} +1.00000 q^{29} -7.90835 q^{31} +1.88720 q^{33} -3.35864 q^{35} -4.37633 q^{37} +3.64262 q^{39} -4.79282 q^{41} +9.68133 q^{43} +3.29187 q^{45} +6.33639 q^{47} -5.95902 q^{49} +1.28578 q^{51} -3.39752 q^{53} -6.21243 q^{55} -5.72913 q^{57} +4.11544 q^{59} -0.134655 q^{61} -1.02028 q^{63} -11.9911 q^{65} +12.1554 q^{67} +1.00000 q^{69} -1.67588 q^{71} -0.127917 q^{73} -5.83643 q^{75} +1.92548 q^{77} -0.342687 q^{79} +1.00000 q^{81} +0.0974823 q^{83} -4.23263 q^{85} -1.00000 q^{87} -14.4602 q^{89} +3.71650 q^{91} +7.90835 q^{93} +18.8596 q^{95} -18.0506 q^{97} -1.88720 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 9 q^{3} - 3 q^{5} + 7 q^{7} + 9 q^{9}+O(q^{10}) $ 9 * q - 9 * q^3 - 3 * q^5 + 7 * q^7 + 9 * q^9	$ 9 q - 9 q^{3} - 3 q^{5} + 7 q^{7} + 9 q^{9} - 2 q^{11} - 7 q^{13} + 3 q^{15} - q^{19} - 7 q^{21} - 9 q^{23} + 2 q^{25} - 9 q^{27} + 9 q^{29} + 8 q^{31} + 2 q^{33} - 5 q^{35} - 8 q^{37} + 7 q^{39} - 19 q^{41} - 3 q^{43} - 3 q^{45} - 3 q^{47} - 18 q^{49} - 17 q^{53} + 9 q^{55} + q^{57} - 10 q^{59} + q^{61} + 7 q^{63} - 16 q^{65} + 12 q^{67} + 9 q^{69} - 7 q^{71} + 13 q^{73} - 2 q^{75} - 15 q^{77} - 10 q^{79} + 9 q^{81} + 9 q^{83} - 6 q^{85} - 9 q^{87} - 5 q^{89} - 18 q^{91} - 8 q^{93} + 31 q^{95} - 7 q^{97} - 2 q^{99}+O(q^{100}) $ 9 * q - 9 * q^3 - 3 * q^5 + 7 * q^7 + 9 * q^9 - 2 * q^11 - 7 * q^13 + 3 * q^15 - q^19 - 7 * q^21 - 9 * q^23 + 2 * q^25 - 9 * q^27 + 9 * q^29 + 8 * q^31 + 2 * q^33 - 5 * q^35 - 8 * q^37 + 7 * q^39 - 19 * q^41 - 3 * q^43 - 3 * q^45 - 3 * q^47 - 18 * q^49 - 17 * q^53 + 9 * q^55 + q^57 - 10 * q^59 + q^61 + 7 * q^63 - 16 * q^65 + 12 * q^67 + 9 * q^69 - 7 * q^71 + 13 * q^73 - 2 * q^75 - 15 * q^77 - 10 * q^79 + 9 * q^81 + 9 * q^83 - 6 * q^85 - 9 * q^87 - 5 * q^89 - 18 * q^91 - 8 * q^93 + 31 * q^95 - 7 * q^97 - 2 * 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
$5$	3.29187	1.47217	0.736085	−	0.676889i	$-0.236672\pi$
$5$	3.29187	1.47217	0.736085	+	0.676889i	$0.236672\pi$
$6$	0	0
$7$	−1.02028	−0.385630	−0.192815	−	0.981235i	$-0.561762\pi$
$7$	−1.02028	−0.385630	−0.192815	+	0.981235i	$0.561762\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.88720	−0.569013	−0.284506	−	0.958674i	$-0.591830\pi$
$11$	−1.88720	−0.569013	−0.284506	+	0.958674i	$0.591830\pi$
$12$	0	0
$13$	−3.64262	−1.01028	−0.505141	−	0.863037i	$-0.668560\pi$
$13$	−3.64262	−1.01028	−0.505141	+	0.863037i	$0.668560\pi$
$14$	0	0
$15$	−3.29187	−0.849958
$16$	0	0
$17$	−1.28578	−0.311848	−0.155924	−	0.987769i	$-0.549835\pi$
$17$	−1.28578	−0.311848	−0.155924	+	0.987769i	$0.549835\pi$
$18$	0	0
$19$	5.72913	1.31435	0.657177	−	0.753737i	$-0.271751\pi$
$19$	5.72913	1.31435	0.657177	+	0.753737i	$0.271751\pi$
$20$	0	0
$21$	1.02028	0.222644
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	5.83643	1.16729
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−7.90835	−1.42038	−0.710191	−	0.704009i	$-0.751391\pi$
$31$	−7.90835	−1.42038	−0.710191	+	0.704009i	$0.751391\pi$
$32$	0	0
$33$	1.88720	0.328520
$34$	0	0
$35$	−3.35864	−0.567714
$36$	0	0
$37$	−4.37633	−0.719464	−0.359732	−	0.933056i	$-0.617132\pi$
$37$	−4.37633	−0.719464	−0.359732	+	0.933056i	$0.617132\pi$
$38$	0	0
$39$	3.64262	0.583287
$40$	0	0
$41$	−4.79282	−0.748512	−0.374256	−	0.927325i	$-0.622102\pi$
$41$	−4.79282	−0.748512	−0.374256	+	0.927325i	$0.622102\pi$
$42$	0	0
$43$	9.68133	1.47639	0.738195	−	0.674588i	$-0.235678\pi$
$43$	9.68133	1.47639	0.738195	+	0.674588i	$0.235678\pi$
$44$	0	0
$45$	3.29187	0.490723
$46$	0	0
$47$	6.33639	0.924257	0.462129	−	0.886813i	$-0.347086\pi$
$47$	6.33639	0.924257	0.462129	+	0.886813i	$0.347086\pi$
$48$	0	0
$49$	−5.95902	−0.851289
$50$	0	0
$51$	1.28578	0.180045
$52$	0	0
$53$	−3.39752	−0.466685	−0.233343	−	0.972395i	$-0.574966\pi$
$53$	−3.39752	−0.466685	−0.233343	+	0.972395i	$0.574966\pi$
$54$	0	0
$55$	−6.21243	−0.837683
$56$	0	0
$57$	−5.72913	−0.758842
$58$	0	0
$59$	4.11544	0.535784	0.267892	−	0.963449i	$-0.413673\pi$
$59$	4.11544	0.535784	0.267892	+	0.963449i	$0.413673\pi$
$60$	0	0
$61$	−0.134655	−0.0172408	−0.00862042	−	0.999963i	$-0.502744\pi$
$61$	−0.134655	−0.0172408	−0.00862042	+	0.999963i	$0.502744\pi$
$62$	0	0
$63$	−1.02028	−0.128543
$64$	0	0
$65$	−11.9911	−1.48731
$66$	0	0
$67$	12.1554	1.48502	0.742510	−	0.669835i	$-0.233635\pi$
$67$	12.1554	1.48502	0.742510	+	0.669835i	$0.233635\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	−1.67588	−0.198890	−0.0994450	−	0.995043i	$-0.531707\pi$
$71$	−1.67588	−0.198890	−0.0994450	+	0.995043i	$0.531707\pi$
$72$	0	0
$73$	−0.127917	−0.0149715	−0.00748577	−	0.999972i	$-0.502383\pi$
$73$	−0.127917	−0.0149715	−0.00748577	+	0.999972i	$0.502383\pi$
$74$	0	0
$75$	−5.83643	−0.673933
$76$	0	0
$77$	1.92548	0.219429
$78$	0	0
$79$	−0.342687	−0.0385553	−0.0192777	−	0.999814i	$-0.506137\pi$
$79$	−0.342687	−0.0385553	−0.0192777	+	0.999814i	$0.506137\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	0.0974823	0.0107001	0.00535004	−	0.999986i	$-0.498297\pi$
$83$	0.0974823	0.0107001	0.00535004	+	0.999986i	$0.498297\pi$
$84$	0	0
$85$	−4.23263	−0.459093
$86$	0	0
$87$	−1.00000	−0.107211
$88$	0	0
$89$	−14.4602	−1.53278	−0.766390	−	0.642375i	$-0.777949\pi$
$89$	−14.4602	−1.53278	−0.766390	+	0.642375i	$0.777949\pi$
$90$	0	0
$91$	3.71650	0.389596
$92$	0	0
$93$	7.90835	0.820058
$94$	0	0
$95$	18.8596	1.93495
$96$	0	0
$97$	−18.0506	−1.83276	−0.916382	−	0.400305i	$-0.868904\pi$
$97$	−18.0506	−1.83276	−0.916382	+	0.400305i	$0.868904\pi$
$98$	0	0
$99$	−1.88720	−0.189671
$100$	0	0
$101$	−0.307106	−0.0305582	−0.0152791	−	0.999883i	$-0.504864\pi$
$101$	−0.307106	−0.0305582	−0.0152791	+	0.999883i	$0.504864\pi$
$102$	0	0
$103$	−9.96209	−0.981594	−0.490797	−	0.871274i	$-0.663294\pi$
$103$	−9.96209	−0.981594	−0.490797	+	0.871274i	$0.663294\pi$
$104$	0	0
$105$	3.35864	0.327770
$106$	0	0
$107$	−4.95885	−0.479390	−0.239695	−	0.970848i	$-0.577047\pi$
$107$	−4.95885	−0.479390	−0.239695	+	0.970848i	$0.577047\pi$
$108$	0	0
$109$	−11.9944	−1.14886	−0.574428	−	0.818555i	$-0.694776\pi$
$109$	−11.9944	−1.14886	−0.574428	+	0.818555i	$0.694776\pi$
$110$	0	0
$111$	4.37633	0.415383
$112$	0	0
$113$	1.20091	0.112972	0.0564862	−	0.998403i	$-0.482010\pi$
$113$	1.20091	0.112972	0.0564862	+	0.998403i	$0.482010\pi$
$114$	0	0
$115$	−3.29187	−0.306969
$116$	0	0
$117$	−3.64262	−0.336761
$118$	0	0
$119$	1.31186	0.120258
$120$	0	0
$121$	−7.43847	−0.676225
$122$	0	0
$123$	4.79282	0.432154
$124$	0	0
$125$	2.75342	0.246273
$126$	0	0
$127$	−2.85660	−0.253483	−0.126741	−	0.991936i	$-0.540452\pi$
$127$	−2.85660	−0.253483	−0.126741	+	0.991936i	$0.540452\pi$
$128$	0	0
$129$	−9.68133	−0.852394
$130$	0	0
$131$	12.5796	1.09908	0.549542	−	0.835466i	$-0.314802\pi$
$131$	12.5796	1.09908	0.549542	+	0.835466i	$0.314802\pi$
$132$	0	0
$133$	−5.84533	−0.506855
$134$	0	0
$135$	−3.29187	−0.283319
$136$	0	0
$137$	−1.28529	−0.109810	−0.0549048	−	0.998492i	$-0.517486\pi$
$137$	−1.28529	−0.109810	−0.0549048	+	0.998492i	$0.517486\pi$
$138$	0	0
$139$	−5.27614	−0.447516	−0.223758	−	0.974645i	$-0.571833\pi$
$139$	−5.27614	−0.447516	−0.223758	+	0.974645i	$0.571833\pi$
$140$	0	0
$141$	−6.33639	−0.533620
$142$	0	0
$143$	6.87436	0.574863
$144$	0	0
$145$	3.29187	0.273375
$146$	0	0
$147$	5.95902	0.491492
$148$	0	0
$149$	−15.0902	−1.23624	−0.618119	−	0.786084i	$-0.712105\pi$
$149$	−15.0902	−1.23624	−0.618119	+	0.786084i	$0.712105\pi$
$150$	0	0
$151$	−0.0808502	−0.00657950	−0.00328975	−	0.999995i	$-0.501047\pi$
$151$	−0.0808502	−0.00657950	−0.00328975	+	0.999995i	$0.501047\pi$
$152$	0	0
$153$	−1.28578	−0.103949
$154$	0	0
$155$	−26.0333	−2.09104
$156$	0	0
$157$	13.2238	1.05538	0.527688	−	0.849438i	$-0.323059\pi$
$157$	13.2238	1.05538	0.527688	+	0.849438i	$0.323059\pi$
$158$	0	0
$159$	3.39752	0.269441
$160$	0	0
$161$	1.02028	0.0804095
$162$	0	0
$163$	−9.32641	−0.730501	−0.365250	−	0.930909i	$-0.619017\pi$
$163$	−9.32641	−0.730501	−0.365250	+	0.930909i	$0.619017\pi$
$164$	0	0
$165$	6.21243	0.483637
$166$	0	0
$167$	−2.65640	−0.205559	−0.102779	−	0.994704i	$-0.532774\pi$
$167$	−2.65640	−0.205559	−0.102779	+	0.994704i	$0.532774\pi$
$168$	0	0
$169$	0.268711	0.0206701
$170$	0	0
$171$	5.72913	0.438118
$172$	0	0
$173$	−20.8900	−1.58824	−0.794120	−	0.607761i	$-0.792068\pi$
$173$	−20.8900	−1.58824	−0.794120	+	0.607761i	$0.792068\pi$
$174$	0	0
$175$	−5.95480	−0.450141
$176$	0	0
$177$	−4.11544	−0.309335
$178$	0	0
$179$	7.03363	0.525718	0.262859	−	0.964834i	$-0.415335\pi$
$179$	7.03363	0.525718	0.262859	+	0.964834i	$0.415335\pi$
$180$	0	0
$181$	10.4995	0.780420	0.390210	−	0.920726i	$-0.372402\pi$
$181$	10.4995	0.780420	0.390210	+	0.920726i	$0.372402\pi$
$182$	0	0
$183$	0.134655	0.00995400
$184$	0	0
$185$	−14.4063	−1.05917
$186$	0	0
$187$	2.42653	0.177445
$188$	0	0
$189$	1.02028	0.0742146
$190$	0	0
$191$	12.9567	0.937513	0.468757	−	0.883327i	$-0.344702\pi$
$191$	12.9567	0.937513	0.468757	+	0.883327i	$0.344702\pi$
$192$	0	0
$193$	−27.1033	−1.95094	−0.975470	−	0.220134i	$-0.929351\pi$
$193$	−27.1033	−1.95094	−0.975470	+	0.220134i	$0.929351\pi$
$194$	0	0
$195$	11.9911	0.858697
$196$	0	0
$197$	14.7303	1.04949	0.524744	−	0.851260i	$-0.324161\pi$
$197$	14.7303	1.04949	0.524744	+	0.851260i	$0.324161\pi$
$198$	0	0
$199$	1.88160	0.133383	0.0666915	−	0.997774i	$-0.478756\pi$
$199$	1.88160	0.133383	0.0666915	+	0.997774i	$0.478756\pi$
$200$	0	0
$201$	−12.1554	−0.857376
$202$	0	0
$203$	−1.02028	−0.0716098
$204$	0	0
$205$	−15.7773	−1.10194
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	−10.8120	−0.747883
$210$	0	0
$211$	−22.5394	−1.55167	−0.775837	−	0.630933i	$-0.782672\pi$
$211$	−22.5394	−1.55167	−0.775837	+	0.630933i	$0.782672\pi$
$212$	0	0
$213$	1.67588	0.114829
$214$	0	0
$215$	31.8697	2.17350
$216$	0	0
$217$	8.06875	0.547743
$218$	0	0
$219$	0.127917	0.00864382
$220$	0	0
$221$	4.68362	0.315054
$222$	0	0
$223$	0.0899811	0.00602558	0.00301279	−	0.999995i	$-0.499041\pi$
$223$	0.0899811	0.00602558	0.00301279	+	0.999995i	$0.499041\pi$
$224$	0	0
$225$	5.83643	0.389095
$226$	0	0
$227$	−24.3526	−1.61634	−0.808168	−	0.588951i	$-0.799541\pi$
$227$	−24.3526	−1.61634	−0.808168	+	0.588951i	$0.799541\pi$
$228$	0	0
$229$	13.3336	0.881112	0.440556	−	0.897725i	$-0.354781\pi$
$229$	13.3336	0.881112	0.440556	+	0.897725i	$0.354781\pi$
$230$	0	0
$231$	−1.92548	−0.126687
$232$	0	0
$233$	−6.28577	−0.411794	−0.205897	−	0.978574i	$-0.566011\pi$
$233$	−6.28577	−0.411794	−0.205897	+	0.978574i	$0.566011\pi$
$234$	0	0
$235$	20.8586	1.36066
$236$	0	0
$237$	0.342687	0.0222599
$238$	0	0
$239$	−13.2321	−0.855916	−0.427958	−	0.903799i	$-0.640767\pi$
$239$	−13.2321	−0.855916	−0.427958	+	0.903799i	$0.640767\pi$
$240$	0	0
$241$	19.0873	1.22952	0.614760	−	0.788714i	$-0.289253\pi$
$241$	19.0873	1.22952	0.614760	+	0.788714i	$0.289253\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−19.6164	−1.25324
$246$	0	0
$247$	−20.8691	−1.32787
$248$	0	0
$249$	−0.0974823	−0.00617769
$250$	0	0
$251$	8.09378	0.510875	0.255438	−	0.966826i	$-0.417780\pi$
$251$	8.09378	0.510875	0.255438	+	0.966826i	$0.417780\pi$
$252$	0	0
$253$	1.88720	0.118647
$254$	0	0
$255$	4.23263	0.265058
$256$	0	0
$257$	11.6298	0.725446	0.362723	−	0.931897i	$-0.381847\pi$
$257$	11.6298	0.725446	0.362723	+	0.931897i	$0.381847\pi$
$258$	0	0
$259$	4.46509	0.277447
$260$	0	0
$261$	1.00000	0.0618984
$262$	0	0
$263$	−7.51527	−0.463411	−0.231706	−	0.972786i	$-0.574431\pi$
$263$	−7.51527	−0.463411	−0.231706	+	0.972786i	$0.574431\pi$
$264$	0	0
$265$	−11.1842	−0.687040
$266$	0	0
$267$	14.4602	0.884951
$268$	0	0
$269$	1.14454	0.0697836	0.0348918	−	0.999391i	$-0.488891\pi$
$269$	1.14454	0.0697836	0.0348918	+	0.999391i	$0.488891\pi$
$270$	0	0
$271$	−12.5549	−0.762654	−0.381327	−	0.924440i	$-0.624533\pi$
$271$	−12.5549	−0.762654	−0.381327	+	0.924440i	$0.624533\pi$
$272$	0	0
$273$	−3.71650	−0.224933
$274$	0	0
$275$	−11.0145	−0.664200
$276$	0	0
$277$	−2.89067	−0.173683	−0.0868417	−	0.996222i	$-0.527677\pi$
$277$	−2.89067	−0.173683	−0.0868417	+	0.996222i	$0.527677\pi$
$278$	0	0
$279$	−7.90835	−0.473461
$280$	0	0
$281$	11.1332	0.664150	0.332075	−	0.943253i	$-0.392251\pi$
$281$	11.1332	0.664150	0.332075	+	0.943253i	$0.392251\pi$
$282$	0	0
$283$	−0.153847	−0.00914527	−0.00457264	−	0.999990i	$-0.501456\pi$
$283$	−0.153847	−0.00914527	−0.00457264	+	0.999990i	$0.501456\pi$
$284$	0	0
$285$	−18.8596	−1.11715
$286$	0	0
$287$	4.89002	0.288649
$288$	0	0
$289$	−15.3468	−0.902751
$290$	0	0
$291$	18.0506	1.05815
$292$	0	0
$293$	32.7574	1.91371	0.956855	−	0.290567i	$-0.0938440\pi$
$293$	32.7574	1.91371	0.956855	+	0.290567i	$0.0938440\pi$
$294$	0	0
$295$	13.5475	0.788766
$296$	0	0
$297$	1.88720	0.109507
$298$	0	0
$299$	3.64262	0.210658
$300$	0	0
$301$	−9.87769	−0.569341
$302$	0	0
$303$	0.307106	0.0176428
$304$	0	0
$305$	−0.443268	−0.0253815
$306$	0	0
$307$	10.3644	0.591527	0.295764	−	0.955261i	$-0.404426\pi$
$307$	10.3644	0.591527	0.295764	+	0.955261i	$0.404426\pi$
$308$	0	0
$309$	9.96209	0.566724
$310$	0	0
$311$	−0.192809	−0.0109332	−0.00546661	−	0.999985i	$-0.501740\pi$
$311$	−0.192809	−0.0109332	−0.00546661	+	0.999985i	$0.501740\pi$
$312$	0	0
$313$	−18.5661	−1.04942	−0.524709	−	0.851282i	$-0.675826\pi$
$313$	−18.5661	−1.04942	−0.524709	+	0.851282i	$0.675826\pi$
$314$	0	0
$315$	−3.35864	−0.189238
$316$	0	0
$317$	16.1487	0.907004	0.453502	−	0.891255i	$-0.350175\pi$
$317$	16.1487	0.907004	0.453502	+	0.891255i	$0.350175\pi$
$318$	0	0
$319$	−1.88720	−0.105663
$320$	0	0
$321$	4.95885	0.276776
$322$	0	0
$323$	−7.36642	−0.409878
$324$	0	0
$325$	−21.2599	−1.17929
$326$	0	0
$327$	11.9944	0.663293
$328$	0	0
$329$	−6.46491	−0.356422
$330$	0	0
$331$	−18.9714	−1.04277	−0.521383	−	0.853323i	$-0.674584\pi$
$331$	−18.9714	−1.04277	−0.521383	+	0.853323i	$0.674584\pi$
$332$	0	0
$333$	−4.37633	−0.239821
$334$	0	0
$335$	40.0141	2.18620
$336$	0	0
$337$	19.9061	1.08436	0.542178	−	0.840264i	$-0.317600\pi$
$337$	19.9061	1.08436	0.542178	+	0.840264i	$0.317600\pi$
$338$	0	0
$339$	−1.20091	−0.0652247
$340$	0	0
$341$	14.9247	0.808215
$342$	0	0
$343$	13.2219	0.713913
$344$	0	0
$345$	3.29187	0.177228
$346$	0	0
$347$	0.225778	0.0121204	0.00606020	−	0.999982i	$-0.498071\pi$
$347$	0.225778	0.0121204	0.00606020	+	0.999982i	$0.498071\pi$
$348$	0	0
$349$	11.5780	0.619756	0.309878	−	0.950776i	$-0.399712\pi$
$349$	11.5780	0.619756	0.309878	+	0.950776i	$0.399712\pi$
$350$	0	0
$351$	3.64262	0.194429
$352$	0	0
$353$	−30.4567	−1.62105	−0.810523	−	0.585706i	$-0.800817\pi$
$353$	−30.4567	−1.62105	−0.810523	+	0.585706i	$0.800817\pi$
$354$	0	0
$355$	−5.51677	−0.292800
$356$	0	0
$357$	−1.31186	−0.0694310
$358$	0	0
$359$	12.9315	0.682499	0.341249	−	0.939973i	$-0.389150\pi$
$359$	12.9315	0.682499	0.341249	+	0.939973i	$0.389150\pi$
$360$	0	0
$361$	13.8230	0.727525
$362$	0	0
$363$	7.43847	0.390419
$364$	0	0
$365$	−0.421086	−0.0220407
$366$	0	0
$367$	20.6009	1.07536	0.537678	−	0.843150i	$-0.319302\pi$
$367$	20.6009	1.07536	0.537678	+	0.843150i	$0.319302\pi$
$368$	0	0
$369$	−4.79282	−0.249504
$370$	0	0
$371$	3.46643	0.179968
$372$	0	0
$373$	−31.6195	−1.63720	−0.818599	−	0.574365i	$-0.805249\pi$
$373$	−31.6195	−1.63720	−0.818599	+	0.574365i	$0.805249\pi$
$374$	0	0
$375$	−2.75342	−0.142186
$376$	0	0
$377$	−3.64262	−0.187605
$378$	0	0
$379$	−33.8695	−1.73976	−0.869878	−	0.493266i	$-0.835803\pi$
$379$	−33.8695	−1.73976	−0.869878	+	0.493266i	$0.835803\pi$
$380$	0	0
$381$	2.85660	0.146348
$382$	0	0
$383$	24.3113	1.24225	0.621125	−	0.783712i	$-0.286676\pi$
$383$	24.3113	1.24225	0.621125	+	0.783712i	$0.286676\pi$
$384$	0	0
$385$	6.33843	0.323036
$386$	0	0
$387$	9.68133	0.492130
$388$	0	0
$389$	30.3825	1.54046	0.770228	−	0.637769i	$-0.220143\pi$
$389$	30.3825	1.54046	0.770228	+	0.637769i	$0.220143\pi$
$390$	0	0
$391$	1.28578	0.0650248
$392$	0	0
$393$	−12.5796	−0.634556
$394$	0	0
$395$	−1.12808	−0.0567600
$396$	0	0
$397$	28.9430	1.45261	0.726304	−	0.687373i	$-0.241236\pi$
$397$	28.9430	1.45261	0.726304	+	0.687373i	$0.241236\pi$
$398$	0	0
$399$	5.84533	0.292633
$400$	0	0
$401$	−32.0873	−1.60236	−0.801182	−	0.598421i	$-0.795795\pi$
$401$	−32.0873	−1.60236	−0.801182	+	0.598421i	$0.795795\pi$
$402$	0	0
$403$	28.8072	1.43499
$404$	0	0
$405$	3.29187	0.163574
$406$	0	0
$407$	8.25902	0.409384
$408$	0	0
$409$	16.8190	0.831646	0.415823	−	0.909445i	$-0.363494\pi$
$409$	16.8190	0.831646	0.415823	+	0.909445i	$0.363494\pi$
$410$	0	0
$411$	1.28529	0.0633987
$412$	0	0
$413$	−4.19891	−0.206615
$414$	0	0
$415$	0.320899	0.0157523
$416$	0	0
$417$	5.27614	0.258374
$418$	0	0
$419$	−23.5339	−1.14971	−0.574853	−	0.818257i	$-0.694941\pi$
$419$	−23.5339	−1.14971	−0.574853	+	0.818257i	$0.694941\pi$
$420$	0	0
$421$	−4.05228	−0.197496	−0.0987481	−	0.995112i	$-0.531484\pi$
$421$	−4.05228	−0.197496	−0.0987481	+	0.995112i	$0.531484\pi$
$422$	0	0
$423$	6.33639	0.308086
$424$	0	0
$425$	−7.50437	−0.364016
$426$	0	0
$427$	0.137386	0.00664859
$428$	0	0
$429$	−6.87436	−0.331897
$430$	0	0
$431$	4.17887	0.201289	0.100644	−	0.994922i	$-0.467910\pi$
$431$	4.17887	0.201289	0.100644	+	0.994922i	$0.467910\pi$
$432$	0	0
$433$	20.8989	1.00434	0.502168	−	0.864770i	$-0.332536\pi$
$433$	20.8989	1.00434	0.502168	+	0.864770i	$0.332536\pi$
$434$	0	0
$435$	−3.29187	−0.157833
$436$	0	0
$437$	−5.72913	−0.274062
$438$	0	0
$439$	−35.9387	−1.71526	−0.857631	−	0.514266i	$-0.828064\pi$
$439$	−35.9387	−1.71526	−0.857631	+	0.514266i	$0.828064\pi$
$440$	0	0
$441$	−5.95902	−0.283763
$442$	0	0
$443$	−0.911071	−0.0432863	−0.0216432	−	0.999766i	$-0.506890\pi$
$443$	−0.911071	−0.0432863	−0.0216432	+	0.999766i	$0.506890\pi$
$444$	0	0
$445$	−47.6012	−2.25651
$446$	0	0
$447$	15.0902	0.713743
$448$	0	0
$449$	−20.5377	−0.969235	−0.484617	−	0.874726i	$-0.661041\pi$
$449$	−20.5377	−0.969235	−0.484617	+	0.874726i	$0.661041\pi$
$450$	0	0
$451$	9.04501	0.425913
$452$	0	0
$453$	0.0808502	0.00379867
$454$	0	0
$455$	12.2343	0.573551
$456$	0	0
$457$	38.8892	1.81916	0.909581	−	0.415527i	$-0.136403\pi$
$457$	38.8892	1.81916	0.909581	+	0.415527i	$0.136403\pi$
$458$	0	0
$459$	1.28578	0.0600152
$460$	0	0
$461$	−36.7804	−1.71304	−0.856518	−	0.516118i	$-0.827377\pi$
$461$	−36.7804	−1.71304	−0.856518	+	0.516118i	$0.827377\pi$
$462$	0	0
$463$	−7.42298	−0.344975	−0.172488	−	0.985012i	$-0.555180\pi$
$463$	−7.42298	−0.344975	−0.172488	+	0.985012i	$0.555180\pi$
$464$	0	0
$465$	26.0333	1.20727
$466$	0	0
$467$	−27.8689	−1.28962	−0.644808	−	0.764344i	$-0.723063\pi$
$467$	−27.8689	−1.28962	−0.644808	+	0.764344i	$0.723063\pi$
$468$	0	0
$469$	−12.4019	−0.572669
$470$	0	0
$471$	−13.2238	−0.609322
$472$	0	0
$473$	−18.2706	−0.840084
$474$	0	0
$475$	33.4377	1.53423
$476$	0	0
$477$	−3.39752	−0.155562
$478$	0	0
$479$	−10.8615	−0.496277	−0.248138	−	0.968725i	$-0.579819\pi$
$479$	−10.8615	−0.496277	−0.248138	+	0.968725i	$0.579819\pi$
$480$	0	0
$481$	15.9413	0.726862
$482$	0	0
$483$	−1.02028	−0.0464244
$484$	0	0
$485$	−59.4204	−2.69814
$486$	0	0
$487$	15.1469	0.686374	0.343187	−	0.939267i	$-0.388494\pi$
$487$	15.1469	0.686374	0.343187	+	0.939267i	$0.388494\pi$
$488$	0	0
$489$	9.32641	0.421755
$490$	0	0
$491$	−20.1888	−0.911107	−0.455553	−	0.890208i	$-0.650559\pi$
$491$	−20.1888	−0.911107	−0.455553	+	0.890208i	$0.650559\pi$
$492$	0	0
$493$	−1.28578	−0.0579087
$494$	0	0
$495$	−6.21243	−0.279228
$496$	0	0
$497$	1.70987	0.0766980
$498$	0	0
$499$	−2.12975	−0.0953406	−0.0476703	−	0.998863i	$-0.515180\pi$
$499$	−2.12975	−0.0953406	−0.0476703	+	0.998863i	$0.515180\pi$
$500$	0	0
$501$	2.65640	0.118679
$502$	0	0
$503$	−33.5893	−1.49767	−0.748835	−	0.662756i	$-0.769387\pi$
$503$	−33.5893	−1.49767	−0.748835	+	0.662756i	$0.769387\pi$
$504$	0	0
$505$	−1.01096	−0.0449869
$506$	0	0
$507$	−0.268711	−0.0119339
$508$	0	0
$509$	−29.0969	−1.28970	−0.644848	−	0.764311i	$-0.723079\pi$
$509$	−29.0969	−1.28970	−0.644848	+	0.764311i	$0.723079\pi$
$510$	0	0
$511$	0.130511	0.00577348
$512$	0	0
$513$	−5.72913	−0.252947
$514$	0	0
$515$	−32.7939	−1.44507
$516$	0	0
$517$	−11.9580	−0.525914
$518$	0	0
$519$	20.8900	0.916971
$520$	0	0
$521$	−39.7968	−1.74353	−0.871764	−	0.489925i	$-0.837024\pi$
$521$	−39.7968	−1.74353	−0.871764	+	0.489925i	$0.837024\pi$
$522$	0	0
$523$	10.3933	0.454469	0.227235	−	0.973840i	$-0.427032\pi$
$523$	10.3933	0.454469	0.227235	+	0.973840i	$0.427032\pi$
$524$	0	0
$525$	5.95480	0.259889
$526$	0	0
$527$	10.1684	0.442943
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	4.11544	0.178595
$532$	0	0
$533$	17.4584	0.756208
$534$	0	0
$535$	−16.3239	−0.705743
$536$	0	0
$537$	−7.03363	−0.303523
$538$	0	0
$539$	11.2459	0.484394
$540$	0	0
$541$	−0.986445	−0.0424106	−0.0212053	−	0.999775i	$-0.506750\pi$
$541$	−0.986445	−0.0424106	−0.0212053	+	0.999775i	$0.506750\pi$
$542$	0	0
$543$	−10.4995	−0.450576
$544$	0	0
$545$	−39.4841	−1.69131
$546$	0	0
$547$	−11.0767	−0.473607	−0.236804	−	0.971558i	$-0.576100\pi$
$547$	−11.0767	−0.473607	−0.236804	+	0.971558i	$0.576100\pi$
$548$	0	0
$549$	−0.134655	−0.00574695
$550$	0	0
$551$	5.72913	0.244069
$552$	0	0
$553$	0.349638	0.0148681
$554$	0	0
$555$	14.4063	0.611515
$556$	0	0
$557$	1.81205	0.0767789	0.0383895	−	0.999263i	$-0.487777\pi$
$557$	1.81205	0.0767789	0.0383895	+	0.999263i	$0.487777\pi$
$558$	0	0
$559$	−35.2655	−1.49157
$560$	0	0
$561$	−2.42653	−0.102448
$562$	0	0
$563$	2.19519	0.0925163	0.0462581	−	0.998930i	$-0.485270\pi$
$563$	2.19519	0.0925163	0.0462581	+	0.998930i	$0.485270\pi$
$564$	0	0
$565$	3.95326	0.166315
$566$	0	0
$567$	−1.02028	−0.0428478
$568$	0	0
$569$	−20.4169	−0.855921	−0.427961	−	0.903797i	$-0.640768\pi$
$569$	−20.4169	−0.855921	−0.427961	+	0.903797i	$0.640768\pi$
$570$	0	0
$571$	−23.1651	−0.969429	−0.484714	−	0.874672i	$-0.661076\pi$
$571$	−23.1651	−0.969429	−0.484714	+	0.874672i	$0.661076\pi$
$572$	0	0
$573$	−12.9567	−0.541273
$574$	0	0
$575$	−5.83643	−0.243396
$576$	0	0
$577$	−29.6341	−1.23368	−0.616842	−	0.787087i	$-0.711588\pi$
$577$	−29.6341	−1.23368	−0.616842	+	0.787087i	$0.711588\pi$
$578$	0	0
$579$	27.1033	1.12638
$580$	0	0
$581$	−0.0994595	−0.00412627
$582$	0	0
$583$	6.41181	0.265550
$584$	0	0
$585$	−11.9911	−0.495769
$586$	0	0
$587$	21.0958	0.870717	0.435359	−	0.900257i	$-0.356622\pi$
$587$	21.0958	0.870717	0.435359	+	0.900257i	$0.356622\pi$
$588$	0	0
$589$	−45.3080	−1.86688
$590$	0	0
$591$	−14.7303	−0.605922
$592$	0	0
$593$	−24.5799	−1.00938	−0.504688	−	0.863302i	$-0.668392\pi$
$593$	−24.5799	−1.00938	−0.504688	+	0.863302i	$0.668392\pi$
$594$	0	0
$595$	4.31848	0.177040
$596$	0	0
$597$	−1.88160	−0.0770087
$598$	0	0
$599$	33.2567	1.35883	0.679415	−	0.733754i	$-0.262234\pi$
$599$	33.2567	1.35883	0.679415	+	0.733754i	$0.262234\pi$
$600$	0	0
$601$	6.74958	0.275321	0.137660	−	0.990479i	$-0.456042\pi$
$601$	6.74958	0.275321	0.137660	+	0.990479i	$0.456042\pi$
$602$	0	0
$603$	12.1554	0.495006
$604$	0	0
$605$	−24.4865	−0.995518
$606$	0	0
$607$	−13.7679	−0.558821	−0.279410	−	0.960172i	$-0.590139\pi$
$607$	−13.7679	−0.558821	−0.279410	+	0.960172i	$0.590139\pi$
$608$	0	0
$609$	1.02028	0.0413439
$610$	0	0
$611$	−23.0811	−0.933761
$612$	0	0
$613$	−38.6304	−1.56027	−0.780134	−	0.625612i	$-0.784849\pi$
$613$	−38.6304	−1.56027	−0.780134	+	0.625612i	$0.784849\pi$
$614$	0	0
$615$	15.7773	0.636204
$616$	0	0
$617$	15.2750	0.614949	0.307475	−	0.951556i	$-0.400516\pi$
$617$	15.2750	0.614949	0.307475	+	0.951556i	$0.400516\pi$
$618$	0	0
$619$	−21.8995	−0.880213	−0.440107	−	0.897945i	$-0.645059\pi$
$619$	−21.8995	−0.880213	−0.440107	+	0.897945i	$0.645059\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	14.7535	0.591087
$624$	0	0
$625$	−20.1182	−0.804730
$626$	0	0
$627$	10.8120	0.431791
$628$	0	0
$629$	5.62701	0.224364
$630$	0	0
$631$	35.2070	1.40157	0.700784	−	0.713374i	$-0.252834\pi$
$631$	35.2070	1.40157	0.700784	+	0.713374i	$0.252834\pi$
$632$	0	0
$633$	22.5394	0.895859
$634$	0	0
$635$	−9.40358	−0.373170
$636$	0	0
$637$	21.7065	0.860042
$638$	0	0
$639$	−1.67588	−0.0662966
$640$	0	0
$641$	−38.4422	−1.51838	−0.759188	−	0.650872i	$-0.774403\pi$
$641$	−38.4422	−1.51838	−0.759188	+	0.650872i	$0.774403\pi$
$642$	0	0
$643$	11.4506	0.451569	0.225785	−	0.974177i	$-0.427505\pi$
$643$	11.4506	0.451569	0.225785	+	0.974177i	$0.427505\pi$
$644$	0	0
$645$	−31.8697	−1.25487
$646$	0	0
$647$	−29.5280	−1.16087	−0.580433	−	0.814308i	$-0.697117\pi$
$647$	−29.5280	−1.16087	−0.580433	+	0.814308i	$0.697117\pi$
$648$	0	0
$649$	−7.76666	−0.304868
$650$	0	0
$651$	−8.06875	−0.316239
$652$	0	0
$653$	−23.5186	−0.920353	−0.460176	−	0.887828i	$-0.652214\pi$
$653$	−23.5186	−0.920353	−0.460176	+	0.887828i	$0.652214\pi$
$654$	0	0
$655$	41.4104	1.61804
$656$	0	0
$657$	−0.127917	−0.00499051
$658$	0	0
$659$	−20.8705	−0.812999	−0.406499	−	0.913651i	$-0.633251\pi$
$659$	−20.8705	−0.812999	−0.406499	+	0.913651i	$0.633251\pi$
$660$	0	0
$661$	−36.2819	−1.41120	−0.705602	−	0.708608i	$-0.749323\pi$
$661$	−36.2819	−1.41120	−0.705602	+	0.708608i	$0.749323\pi$
$662$	0	0
$663$	−4.68362	−0.181897
$664$	0	0
$665$	−19.2421	−0.746176
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	0	0
$669$	−0.0899811	−0.00347887
$670$	0	0
$671$	0.254122	0.00981025
$672$	0	0
$673$	27.2490	1.05037	0.525185	−	0.850988i	$-0.323996\pi$
$673$	27.2490	1.05037	0.525185	+	0.850988i	$0.323996\pi$
$674$	0	0
$675$	−5.83643	−0.224644
$676$	0	0
$677$	35.3520	1.35869	0.679344	−	0.733820i	$-0.262264\pi$
$677$	35.3520	1.35869	0.679344	+	0.733820i	$0.262264\pi$
$678$	0	0
$679$	18.4167	0.706769
$680$	0	0
$681$	24.3526	0.933193
$682$	0	0
$683$	29.2266	1.11832	0.559162	−	0.829058i	$-0.311123\pi$
$683$	29.2266	1.11832	0.559162	+	0.829058i	$0.311123\pi$
$684$	0	0
$685$	−4.23101	−0.161659
$686$	0	0
$687$	−13.3336	−0.508710
$688$	0	0
$689$	12.3759	0.471484
$690$	0	0
$691$	50.6850	1.92815	0.964074	−	0.265635i	$-0.0855816\pi$
$691$	50.6850	1.92815	0.964074	+	0.265635i	$0.0855816\pi$
$692$	0	0
$693$	1.92548	0.0731428
$694$	0	0
$695$	−17.3684	−0.658820
$696$	0	0
$697$	6.16251	0.233422
$698$	0	0
$699$	6.28577	0.237750
$700$	0	0
$701$	−17.6090	−0.665082	−0.332541	−	0.943089i	$-0.607906\pi$
$701$	−17.6090	−0.665082	−0.332541	+	0.943089i	$0.607906\pi$
$702$	0	0
$703$	−25.0726	−0.945631
$704$	0	0
$705$	−20.8586	−0.785580
$706$	0	0
$707$	0.313335	0.0117842
$708$	0	0
$709$	16.9287	0.635771	0.317885	−	0.948129i	$-0.397027\pi$
$709$	16.9287	0.635771	0.317885	+	0.948129i	$0.397027\pi$
$710$	0	0
$711$	−0.342687	−0.0128518
$712$	0	0
$713$	7.90835	0.296170
$714$	0	0
$715$	22.6295	0.846297
$716$	0	0
$717$	13.2321	0.494163
$718$	0	0
$719$	0.578253	0.0215652	0.0107826	−	0.999942i	$-0.496568\pi$
$719$	0.578253	0.0215652	0.0107826	+	0.999942i	$0.496568\pi$
$720$	0	0
$721$	10.1641	0.378533
$722$	0	0
$723$	−19.0873	−0.709864
$724$	0	0
$725$	5.83643	0.216760
$726$	0	0
$727$	33.2629	1.23365	0.616827	−	0.787099i	$-0.288418\pi$
$727$	33.2629	1.23365	0.616827	+	0.787099i	$0.288418\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−12.4481	−0.460409
$732$	0	0
$733$	37.4181	1.38207	0.691035	−	0.722821i	$-0.257155\pi$
$733$	37.4181	1.38207	0.691035	+	0.722821i	$0.257155\pi$
$734$	0	0
$735$	19.6164	0.723560
$736$	0	0
$737$	−22.9397	−0.844995
$738$	0	0
$739$	15.7570	0.579631	0.289815	−	0.957083i	$-0.406406\pi$
$739$	15.7570	0.579631	0.289815	+	0.957083i	$0.406406\pi$
$740$	0	0
$741$	20.8691	0.766645
$742$	0	0
$743$	6.49549	0.238296	0.119148	−	0.992876i	$-0.461984\pi$
$743$	6.49549	0.238296	0.119148	+	0.992876i	$0.461984\pi$
$744$	0	0
$745$	−49.6751	−1.81995
$746$	0	0
$747$	0.0974823	0.00356669
$748$	0	0
$749$	5.05942	0.184867
$750$	0	0
$751$	16.7238	0.610260	0.305130	−	0.952311i	$-0.401300\pi$
$751$	16.7238	0.610260	0.305130	+	0.952311i	$0.401300\pi$
$752$	0	0
$753$	−8.09378	−0.294954
$754$	0	0
$755$	−0.266149	−0.00968614
$756$	0	0
$757$	13.2545	0.481742	0.240871	−	0.970557i	$-0.422567\pi$
$757$	13.2545	0.481742	0.240871	+	0.970557i	$0.422567\pi$
$758$	0	0
$759$	−1.88720	−0.0685011
$760$	0	0
$761$	−35.7001	−1.29413	−0.647064	−	0.762436i	$-0.724003\pi$
$761$	−35.7001	−1.29413	−0.647064	+	0.762436i	$0.724003\pi$
$762$	0	0
$763$	12.2377	0.443034
$764$	0	0
$765$	−4.23263	−0.153031
$766$	0	0
$767$	−14.9910	−0.541293
$768$	0	0
$769$	31.0686	1.12036	0.560182	−	0.828370i	$-0.310731\pi$
$769$	31.0686	1.12036	0.560182	+	0.828370i	$0.310731\pi$
$770$	0	0
$771$	−11.6298	−0.418837
$772$	0	0
$773$	−21.8747	−0.786779	−0.393390	−	0.919372i	$-0.628698\pi$
$773$	−21.8747	−0.786779	−0.393390	+	0.919372i	$0.628698\pi$
$774$	0	0
$775$	−46.1565	−1.65799
$776$	0	0
$777$	−4.46509	−0.160184
$778$	0	0
$779$	−27.4587	−0.983809
$780$	0	0
$781$	3.16272	0.113171
$782$	0	0
$783$	−1.00000	−0.0357371
$784$	0	0
$785$	43.5312	1.55369
$786$	0	0
$787$	−6.97490	−0.248628	−0.124314	−	0.992243i	$-0.539673\pi$
$787$	−6.97490	−0.248628	−0.124314	+	0.992243i	$0.539673\pi$
$788$	0	0
$789$	7.51527	0.267551
$790$	0	0
$791$	−1.22527	−0.0435656
$792$	0	0
$793$	0.490498	0.0174181
$794$	0	0
$795$	11.1842	0.396663
$796$	0	0
$797$	12.1684	0.431025	0.215513	−	0.976501i	$-0.430858\pi$
$797$	12.1684	0.431025	0.215513	+	0.976501i	$0.430858\pi$
$798$	0	0
$799$	−8.14722	−0.288228
$800$	0	0
$801$	−14.4602	−0.510927
$802$	0	0
$803$	0.241405	0.00851900
$804$	0	0
$805$	3.35864	0.118376
$806$	0	0
$807$	−1.14454	−0.0402896
$808$	0	0
$809$	29.8416	1.04918	0.524588	−	0.851356i	$-0.324220\pi$
$809$	29.8416	1.04918	0.524588	+	0.851356i	$0.324220\pi$
$810$	0	0
$811$	−13.5969	−0.477451	−0.238725	−	0.971087i	$-0.576730\pi$
$811$	−13.5969	−0.477451	−0.238725	+	0.971087i	$0.576730\pi$
$812$	0	0
$813$	12.5549	0.440319
$814$	0	0
$815$	−30.7014	−1.07542
$816$	0	0
$817$	55.4657	1.94050
$818$	0	0
$819$	3.71650	0.129865
$820$	0	0
$821$	−4.64965	−0.162274	−0.0811369	−	0.996703i	$-0.525855\pi$
$821$	−4.64965	−0.162274	−0.0811369	+	0.996703i	$0.525855\pi$
$822$	0	0
$823$	20.5466	0.716210	0.358105	−	0.933681i	$-0.383423\pi$
$823$	20.5466	0.716210	0.358105	+	0.933681i	$0.383423\pi$
$824$	0	0
$825$	11.0145	0.383476
$826$	0	0
$827$	−3.31525	−0.115282	−0.0576412	−	0.998337i	$-0.518358\pi$
$827$	−3.31525	−0.115282	−0.0576412	+	0.998337i	$0.518358\pi$
$828$	0	0
$829$	9.08377	0.315492	0.157746	−	0.987480i	$-0.449577\pi$
$829$	9.08377	0.315492	0.157746	+	0.987480i	$0.449577\pi$
$830$	0	0
$831$	2.89067	0.100276
$832$	0	0
$833$	7.66201	0.265473
$834$	0	0
$835$	−8.74455	−0.302617
$836$	0	0
$837$	7.90835	0.273353
$838$	0	0
$839$	−36.9667	−1.27623	−0.638116	−	0.769940i	$-0.720286\pi$
$839$	−36.9667	−1.27623	−0.638116	+	0.769940i	$0.720286\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	−11.1332	−0.383447
$844$	0	0
$845$	0.884563	0.0304299
$846$	0	0
$847$	7.58934	0.260773
$848$	0	0
$849$	0.153847	0.00528002
$850$	0	0
$851$	4.37633	0.150019
$852$	0	0
$853$	4.56076	0.156158	0.0780788	−	0.996947i	$-0.475121\pi$
$853$	4.56076	0.156158	0.0780788	+	0.996947i	$0.475121\pi$
$854$	0	0
$855$	18.8596	0.644984
$856$	0	0
$857$	−30.9348	−1.05671	−0.528356	−	0.849023i	$-0.677191\pi$
$857$	−30.9348	−1.05671	−0.528356	+	0.849023i	$0.677191\pi$
$858$	0	0
$859$	39.4985	1.34767	0.673835	−	0.738882i	$-0.264646\pi$
$859$	39.4985	1.34767	0.673835	+	0.738882i	$0.264646\pi$
$860$	0	0
$861$	−4.89002	−0.166652
$862$	0	0
$863$	−5.03237	−0.171304	−0.0856520	−	0.996325i	$-0.527297\pi$
$863$	−5.03237	−0.171304	−0.0856520	+	0.996325i	$0.527297\pi$
$864$	0	0
$865$	−68.7674	−2.33816
$866$	0	0
$867$	15.3468	0.521203
$868$	0	0
$869$	0.646719	0.0219385
$870$	0	0
$871$	−44.2776	−1.50029
$872$	0	0
$873$	−18.0506	−0.610921
$874$	0	0
$875$	−2.80926	−0.0949704
$876$	0	0
$877$	−1.80048	−0.0607979	−0.0303989	−	0.999538i	$-0.509678\pi$
$877$	−1.80048	−0.0607979	−0.0303989	+	0.999538i	$0.509678\pi$
$878$	0	0
$879$	−32.7574	−1.10488
$880$	0	0
$881$	−17.4943	−0.589397	−0.294698	−	0.955590i	$-0.595219\pi$
$881$	−17.4943	−0.589397	−0.294698	+	0.955590i	$0.595219\pi$
$882$	0	0
$883$	12.9385	0.435415	0.217708	−	0.976014i	$-0.430142\pi$
$883$	12.9385	0.435415	0.217708	+	0.976014i	$0.430142\pi$
$884$	0	0
$885$	−13.5475	−0.455394
$886$	0	0
$887$	−16.9871	−0.570372	−0.285186	−	0.958472i	$-0.592055\pi$
$887$	−16.9871	−0.570372	−0.285186	+	0.958472i	$0.592055\pi$
$888$	0	0
$889$	2.91454	0.0977506
$890$	0	0
$891$	−1.88720	−0.0632236
$892$	0	0
$893$	36.3020	1.21480
$894$	0	0
$895$	23.1538	0.773947
$896$	0	0
$897$	−3.64262	−0.121624
$898$	0	0
$899$	−7.90835	−0.263758
$900$	0	0
$901$	4.36847	0.145535
$902$	0	0
$903$	9.87769	0.328709
$904$	0	0
$905$	34.5629	1.14891
$906$	0	0
$907$	−53.7530	−1.78484	−0.892419	−	0.451207i	$-0.850994\pi$
$907$	−53.7530	−1.78484	−0.892419	+	0.451207i	$0.850994\pi$
$908$	0	0
$909$	−0.307106	−0.0101861
$910$	0	0
$911$	−0.961864	−0.0318680	−0.0159340	−	0.999873i	$-0.505072\pi$
$911$	−0.961864	−0.0318680	−0.0159340	+	0.999873i	$0.505072\pi$
$912$	0	0
$913$	−0.183969	−0.00608848
$914$	0	0
$915$	0.443268	0.0146540
$916$	0	0
$917$	−12.8347	−0.423840
$918$	0	0
$919$	−25.5693	−0.843454	−0.421727	−	0.906723i	$-0.638576\pi$
$919$	−25.5693	−0.843454	−0.421727	+	0.906723i	$0.638576\pi$
$920$	0	0
$921$	−10.3644	−0.341518
$922$	0	0
$923$	6.10459	0.200935
$924$	0	0
$925$	−25.5421	−0.839821
$926$	0	0
$927$	−9.96209	−0.327198
$928$	0	0
$929$	−20.9124	−0.686113	−0.343057	−	0.939315i	$-0.611462\pi$
$929$	−20.9124	−0.686113	−0.343057	+	0.939315i	$0.611462\pi$
$930$	0	0
$931$	−34.1400	−1.11889
$932$	0	0
$933$	0.192809	0.00631229
$934$	0	0
$935$	7.98783	0.261230
$936$	0	0
$937$	49.7499	1.62526	0.812629	−	0.582782i	$-0.198036\pi$
$937$	49.7499	1.62526	0.812629	+	0.582782i	$0.198036\pi$
$938$	0	0
$939$	18.5661	0.605882
$940$	0	0
$941$	22.8626	0.745300	0.372650	−	0.927972i	$-0.378449\pi$
$941$	22.8626	0.745300	0.372650	+	0.927972i	$0.378449\pi$
$942$	0	0
$943$	4.79282	0.156076
$944$	0	0
$945$	3.35864	0.109257
$946$	0	0
$947$	−44.1781	−1.43559	−0.717797	−	0.696253i	$-0.754849\pi$
$947$	−44.1781	−1.43559	−0.717797	+	0.696253i	$0.754849\pi$
$948$	0	0
$949$	0.465953	0.0151255
$950$	0	0
$951$	−16.1487	−0.523659
$952$	0	0
$953$	43.7614	1.41757	0.708786	−	0.705424i	$-0.249243\pi$
$953$	43.7614	1.41757	0.708786	+	0.705424i	$0.249243\pi$
$954$	0	0
$955$	42.6518	1.38018
$956$	0	0
$957$	1.88720	0.0610045
$958$	0	0
$959$	1.31136	0.0423460
$960$	0	0
$961$	31.5421	1.01749
$962$	0	0
$963$	−4.95885	−0.159797
$964$	0	0
$965$	−89.2207	−2.87212
$966$	0	0
$967$	−12.6578	−0.407048	−0.203524	−	0.979070i	$-0.565240\pi$
$967$	−12.6578	−0.407048	−0.203524	+	0.979070i	$0.565240\pi$
$968$	0	0
$969$	7.36642	0.236643
$970$	0	0
$971$	25.3998	0.815118	0.407559	−	0.913179i	$-0.366380\pi$
$971$	25.3998	0.815118	0.407559	+	0.913179i	$0.366380\pi$
$972$	0	0
$973$	5.38315	0.172576
$974$	0	0
$975$	21.2599	0.680862
$976$	0	0
$977$	30.7723	0.984494	0.492247	−	0.870455i	$-0.336176\pi$
$977$	30.7723	0.984494	0.492247	+	0.870455i	$0.336176\pi$
$978$	0	0
$979$	27.2893	0.872171
$980$	0	0
$981$	−11.9944	−0.382952
$982$	0	0
$983$	14.3927	0.459057	0.229528	−	0.973302i	$-0.426282\pi$
$983$	14.3927	0.459057	0.229528	+	0.973302i	$0.426282\pi$
$984$	0	0
$985$	48.4902	1.54502
$986$	0	0
$987$	6.46491	0.205780
$988$	0	0
$989$	−9.68133	−0.307849
$990$	0	0
$991$	−19.9935	−0.635116	−0.317558	−	0.948239i	$-0.602863\pi$
$991$	−19.9935	−0.635116	−0.317558	+	0.948239i	$0.602863\pi$
$992$	0	0
$993$	18.9714	0.602041
$994$	0	0
$995$	6.19398	0.196362
$996$	0	0
$997$	8.84129	0.280006	0.140003	−	0.990151i	$-0.455289\pi$
$997$	8.84129	0.280006	0.140003	+	0.990151i	$0.455289\pi$
$998$	0	0
$999$	4.37633	0.138461

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8004.2.a.e.1.9	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8004.2.a.e.1.9	✓	9	1.1	even	1	trivial

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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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +3.29187 q^{5} -1.02028 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +3.29187 q^{5} -1.02028 q^{7} +1.00000 q^{9} -1.88720 q^{11} -3.64262 q^{13} -3.29187 q^{15} -1.28578 q^{17} +5.72913 q^{19} +1.02028 q^{21} -1.00000 q^{23} +5.83643 q^{25} -1.00000 q^{27} +1.00000 q^{29} -7.90835 q^{31} +1.88720 q^{33} -3.35864 q^{35} -4.37633 q^{37} +3.64262 q^{39} -4.79282 q^{41} +9.68133 q^{43} +3.29187 q^{45} +6.33639 q^{47} -5.95902 q^{49} +1.28578 q^{51} -3.39752 q^{53} -6.21243 q^{55} -5.72913 q^{57} +4.11544 q^{59} -0.134655 q^{61} -1.02028 q^{63} -11.9911 q^{65} +12.1554 q^{67} +1.00000 q^{69} -1.67588 q^{71} -0.127917 q^{73} -5.83643 q^{75} +1.92548 q^{77} -0.342687 q^{79} +1.00000 q^{81} +0.0974823 q^{83} -4.23263 q^{85} -1.00000 q^{87} -14.4602 q^{89} +3.71650 q^{91} +7.90835 q^{93} +18.8596 q^{95} -18.0506 q^{97} -1.88720 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 9 q^{3} - 3 q^{5} + 7 q^{7} + 9 q^{9}+O(q^{10}) \) 9 * q - 9 * q^3 - 3 * q^5 + 7 * q^7 + 9 * q^9	\( 9 q - 9 q^{3} - 3 q^{5} + 7 q^{7} + 9 q^{9} - 2 q^{11} - 7 q^{13} + 3 q^{15} - q^{19} - 7 q^{21} - 9 q^{23} + 2 q^{25} - 9 q^{27} + 9 q^{29} + 8 q^{31} + 2 q^{33} - 5 q^{35} - 8 q^{37} + 7 q^{39} - 19 q^{41} - 3 q^{43} - 3 q^{45} - 3 q^{47} - 18 q^{49} - 17 q^{53} + 9 q^{55} + q^{57} - 10 q^{59} + q^{61} + 7 q^{63} - 16 q^{65} + 12 q^{67} + 9 q^{69} - 7 q^{71} + 13 q^{73} - 2 q^{75} - 15 q^{77} - 10 q^{79} + 9 q^{81} + 9 q^{83} - 6 q^{85} - 9 q^{87} - 5 q^{89} - 18 q^{91} - 8 q^{93} + 31 q^{95} - 7 q^{97} - 2 q^{99}+O(q^{100}) \) 9 * q - 9 * q^3 - 3 * q^5 + 7 * q^7 + 9 * q^9 - 2 * q^11 - 7 * q^13 + 3 * q^15 - q^19 - 7 * q^21 - 9 * q^23 + 2 * q^25 - 9 * q^27 + 9 * q^29 + 8 * q^31 + 2 * q^33 - 5 * q^35 - 8 * q^37 + 7 * q^39 - 19 * q^41 - 3 * q^43 - 3 * q^45 - 3 * q^47 - 18 * q^49 - 17 * q^53 + 9 * q^55 + q^57 - 10 * q^59 + q^61 + 7 * q^63 - 16 * q^65 + 12 * q^67 + 9 * q^69 - 7 * q^71 + 13 * q^73 - 2 * q^75 - 15 * q^77 - 10 * q^79 + 9 * q^81 + 9 * q^83 - 6 * q^85 - 9 * q^87 - 5 * q^89 - 18 * q^91 - 8 * q^93 + 31 * q^95 - 7 * q^97 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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