LMFDB - Embedded newform 6004.2.a.d.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6004,2,Mod(1,6004)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6004, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6004.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6004 = 2^{2} \cdot 19 \cdot 79 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6004.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:	$47.9421813736$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 15x^{6} + 56x^{5} + 87x^{4} - 248x^{3} - 241x^{2} + 340x + 248 $ x^8 - 4x^7 - 15x^6 + 56x^5 + 87x^4 - 248x^3 - 241x^2 + 340*x + 248
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$2.82169$ of defining polynomial
Character	$\chi$	$=$	6004.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.82169 q^{5} +4.85844 q^{7} -3.00000 q^{9} +O(q^{10})$	$q+2.82169 q^{5} +4.85844 q^{7} -3.00000 q^{9} +1.69392 q^{11} +3.33980 q^{13} -2.21020 q^{17} +1.00000 q^{19} +5.35348 q^{23} +2.96195 q^{25} -6.61191 q^{29} +1.16213 q^{31} +13.7090 q^{35} +4.72858 q^{37} -3.45902 q^{41} -3.96773 q^{43} -8.46508 q^{45} +0.312939 q^{47} +16.6044 q^{49} +5.80281 q^{53} +4.77971 q^{55} +6.38954 q^{59} +3.99750 q^{61} -14.5753 q^{63} +9.42389 q^{65} -1.03190 q^{67} +15.9324 q^{71} -16.0509 q^{73} +8.22979 q^{77} +1.00000 q^{79} +9.00000 q^{81} -1.04666 q^{83} -6.23651 q^{85} -16.2736 q^{89} +16.2262 q^{91} +2.82169 q^{95} -6.30683 q^{97} -5.08175 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{5} + q^{7} - 24 q^{9}+O(q^{10}) $ 8 * q + 4 * q^5 + q^7 - 24 * q^9	$ 8 q + 4 q^{5} + q^{7} - 24 q^{9} - 7 q^{11} - 12 q^{13} - 7 q^{17} + 8 q^{19} + 10 q^{23} + 6 q^{25} + 5 q^{29} + 3 q^{31} - 11 q^{35} - 15 q^{37} - 5 q^{41} + 10 q^{43} - 12 q^{45} + 18 q^{47} + 31 q^{49} + 9 q^{53} - 17 q^{55} + 8 q^{59} + 13 q^{61} - 3 q^{63} + 4 q^{65} + 21 q^{67} + 44 q^{71} - 20 q^{73} + 15 q^{77} + 8 q^{79} + 72 q^{81} - 4 q^{83} - 9 q^{85} - 10 q^{89} + 48 q^{91} + 4 q^{95} - 22 q^{97} + 21 q^{99}+O(q^{100}) $ 8 * q + 4 * q^5 + q^7 - 24 * q^9 - 7 * q^11 - 12 * q^13 - 7 * q^17 + 8 * q^19 + 10 * q^23 + 6 * q^25 + 5 * q^29 + 3 * q^31 - 11 * q^35 - 15 * q^37 - 5 * q^41 + 10 * q^43 - 12 * q^45 + 18 * q^47 + 31 * q^49 + 9 * q^53 - 17 * q^55 + 8 * q^59 + 13 * q^61 - 3 * q^63 + 4 * q^65 + 21 * q^67 + 44 * q^71 - 20 * q^73 + 15 * q^77 + 8 * q^79 + 72 * q^81 - 4 * q^83 - 9 * q^85 - 10 * q^89 + 48 * q^91 + 4 * q^95 - 22 * q^97 + 21 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0		−	1.00000i	$-0.5\pi$
$3$	0	0			1.00000i	$0.5\pi$
$4$	0	0
$5$	2.82169	1.26190	0.630950	−	0.775824i	$-0.282665\pi$
$5$	2.82169	1.26190	0.630950	+	0.775824i	$0.282665\pi$
$6$	0	0
$7$	4.85844	1.83632	0.918158	−	0.396214i	$-0.129676\pi$
$7$	4.85844	1.83632	0.918158	+	0.396214i	$0.129676\pi$
$8$	0	0
$9$	−3.00000	−1.00000
$10$	0	0
$11$	1.69392	0.510735	0.255367	−	0.966844i	$-0.417804\pi$
$11$	1.69392	0.510735	0.255367	+	0.966844i	$0.417804\pi$
$12$	0	0
$13$	3.33980	0.926294	0.463147	−	0.886281i	$-0.346720\pi$
$13$	3.33980	0.926294	0.463147	+	0.886281i	$0.346720\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2.21020	−0.536053	−0.268026	−	0.963412i	$-0.586371\pi$
$17$	−2.21020	−0.536053	−0.268026	+	0.963412i	$0.586371\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	5.35348	1.11628	0.558139	−	0.829747i	$-0.311515\pi$
$23$	5.35348	1.11628	0.558139	+	0.829747i	$0.311515\pi$
$24$	0	0
$25$	2.96195	0.592390
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−6.61191	−1.22780	−0.613900	−	0.789383i	$-0.710400\pi$
$29$	−6.61191	−1.22780	−0.613900	+	0.789383i	$0.710400\pi$
$30$	0	0
$31$	1.16213	0.208724	0.104362	−	0.994539i	$-0.466720\pi$
$31$	1.16213	0.208724	0.104362	+	0.994539i	$0.466720\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	13.7090	2.31725
$36$	0	0
$37$	4.72858	0.777374	0.388687	−	0.921370i	$-0.372929\pi$
$37$	4.72858	0.777374	0.388687	+	0.921370i	$0.372929\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−3.45902	−0.540209	−0.270104	−	0.962831i	$-0.587058\pi$
$41$	−3.45902	−0.540209	−0.270104	+	0.962831i	$0.587058\pi$
$42$	0	0
$43$	−3.96773	−0.605073	−0.302536	−	0.953138i	$-0.597833\pi$
$43$	−3.96773	−0.605073	−0.302536	+	0.953138i	$0.597833\pi$
$44$	0	0
$45$	−8.46508	−1.26190
$46$	0	0
$47$	0.312939	0.0456468	0.0228234	−	0.999740i	$-0.492734\pi$
$47$	0.312939	0.0456468	0.0228234	+	0.999740i	$0.492734\pi$
$48$	0	0
$49$	16.6044	2.37206
$50$	0	0
$51$	0	0
$52$	0	0
$53$	5.80281	0.797077	0.398538	−	0.917152i	$-0.369518\pi$
$53$	5.80281	0.797077	0.398538	+	0.917152i	$0.369518\pi$
$54$	0	0
$55$	4.77971	0.644496
$56$	0	0
$57$	0	0
$58$	0	0
$59$	6.38954	0.831847	0.415923	−	0.909400i	$-0.363458\pi$
$59$	6.38954	0.831847	0.415923	+	0.909400i	$0.363458\pi$
$60$	0	0
$61$	3.99750	0.511828	0.255914	−	0.966700i	$-0.417624\pi$
$61$	3.99750	0.511828	0.255914	+	0.966700i	$0.417624\pi$
$62$	0	0
$63$	−14.5753	−1.83632
$64$	0	0
$65$	9.42389	1.16889
$66$	0	0
$67$	−1.03190	−0.126066	−0.0630330	−	0.998011i	$-0.520077\pi$
$67$	−1.03190	−0.126066	−0.0630330	+	0.998011i	$0.520077\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	15.9324	1.89083	0.945413	−	0.325875i	$-0.105659\pi$
$71$	15.9324	1.89083	0.945413	+	0.325875i	$0.105659\pi$
$72$	0	0
$73$	−16.0509	−1.87861	−0.939306	−	0.343080i	$-0.888530\pi$
$73$	−16.0509	−1.87861	−0.939306	+	0.343080i	$0.888530\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	8.22979	0.937871
$78$	0	0
$79$	1.00000	0.112509
$79$	1.00000	0.112509
$80$	0	0
$81$	9.00000	1.00000
$82$	0	0
$83$	−1.04666	−0.114885	−0.0574427	−	0.998349i	$-0.518295\pi$
$83$	−1.04666	−0.114885	−0.0574427	+	0.998349i	$0.518295\pi$
$84$	0	0
$85$	−6.23651	−0.676445
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−16.2736	−1.72499	−0.862497	−	0.506063i	$-0.831100\pi$
$89$	−16.2736	−1.72499	−0.862497	+	0.506063i	$0.831100\pi$
$90$	0	0
$91$	16.2262	1.70097
$92$	0	0
$93$	0	0
$94$	0	0
$95$	2.82169	0.289500
$96$	0	0
$97$	−6.30683	−0.640361	−0.320181	−	0.947357i	$-0.603744\pi$
$97$	−6.30683	−0.640361	−0.320181	+	0.947357i	$0.603744\pi$
$98$	0	0
$99$	−5.08175	−0.510735
$100$	0	0
$101$	10.7068	1.06537	0.532685	−	0.846313i	$-0.321183\pi$
$101$	10.7068	1.06537	0.532685	+	0.846313i	$0.321183\pi$
$102$	0	0
$103$	−9.74933	−0.960630	−0.480315	−	0.877096i	$-0.659478\pi$
$103$	−9.74933	−0.960630	−0.480315	+	0.877096i	$0.659478\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	17.4130	1.68338	0.841690	−	0.539961i	$-0.181561\pi$
$107$	17.4130	1.68338	0.841690	+	0.539961i	$0.181561\pi$
$108$	0	0
$109$	3.73579	0.357824	0.178912	−	0.983865i	$-0.442742\pi$
$109$	3.73579	0.357824	0.178912	+	0.983865i	$0.442742\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	16.2946	1.53287	0.766433	−	0.642324i	$-0.222030\pi$
$113$	16.2946	1.53287	0.766433	+	0.642324i	$0.222030\pi$
$114$	0	0
$115$	15.1059	1.40863
$116$	0	0
$117$	−10.0194	−0.926294
$118$	0	0
$119$	−10.7381	−0.984363
$120$	0	0
$121$	−8.13065	−0.739150
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−5.75075	−0.514362
$126$	0	0
$127$	11.8975	1.05574	0.527868	−	0.849327i	$-0.322992\pi$
$127$	11.8975	1.05574	0.527868	+	0.849327i	$0.322992\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−21.2357	−1.85537	−0.927687	−	0.373358i	$-0.878206\pi$
$131$	−21.2357	−1.85537	−0.927687	+	0.373358i	$0.878206\pi$
$132$	0	0
$133$	4.85844	0.421280
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−2.19976	−0.187938	−0.0939692	−	0.995575i	$-0.529956\pi$
$137$	−2.19976	−0.187938	−0.0939692	+	0.995575i	$0.529956\pi$
$138$	0	0
$139$	−9.83791	−0.834441	−0.417221	−	0.908805i	$-0.636996\pi$
$139$	−9.83791	−0.834441	−0.417221	+	0.908805i	$0.636996\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	5.65734	0.473091
$144$	0	0
$145$	−18.6568	−1.54936
$146$	0	0
$147$	0	0
$148$	0	0
$149$	5.64393	0.462369	0.231185	−	0.972910i	$-0.425740\pi$
$149$	5.64393	0.462369	0.231185	+	0.972910i	$0.425740\pi$
$150$	0	0
$151$	2.18465	0.177785	0.0888923	−	0.996041i	$-0.471667\pi$
$151$	2.18465	0.177785	0.0888923	+	0.996041i	$0.471667\pi$
$152$	0	0
$153$	6.63061	0.536053
$154$	0	0
$155$	3.27916	0.263389
$156$	0	0
$157$	−6.55976	−0.523526	−0.261763	−	0.965132i	$-0.584304\pi$
$157$	−6.55976	−0.523526	−0.261763	+	0.965132i	$0.584304\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	26.0096	2.04984
$162$	0	0
$163$	0.352462	0.0276069	0.0138035	−	0.999905i	$-0.495606\pi$
$163$	0.352462	0.0276069	0.0138035	+	0.999905i	$0.495606\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−5.77728	−0.447059	−0.223530	−	0.974697i	$-0.571758\pi$
$167$	−5.77728	−0.447059	−0.223530	+	0.974697i	$0.571758\pi$
$168$	0	0
$169$	−1.84573	−0.141979
$170$	0	0
$171$	−3.00000	−0.229416
$172$	0	0
$173$	11.1712	0.849330	0.424665	−	0.905350i	$-0.360392\pi$
$173$	11.1712	0.849330	0.424665	+	0.905350i	$0.360392\pi$
$174$	0	0
$175$	14.3905	1.08782
$176$	0	0
$177$	0	0
$178$	0	0
$179$	4.31850	0.322780	0.161390	−	0.986891i	$-0.448402\pi$
$179$	4.31850	0.322780	0.161390	+	0.986891i	$0.448402\pi$
$180$	0	0
$181$	−2.34931	−0.174622	−0.0873112	−	0.996181i	$-0.527827\pi$
$181$	−2.34931	−0.174622	−0.0873112	+	0.996181i	$0.527827\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	13.3426	0.980968
$186$	0	0
$187$	−3.74390	−0.273781
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−5.25296	−0.380091	−0.190045	−	0.981775i	$-0.560863\pi$
$191$	−5.25296	−0.380091	−0.190045	+	0.981775i	$0.560863\pi$
$192$	0	0
$193$	14.7742	1.06347	0.531733	−	0.846912i	$-0.321541\pi$
$193$	14.7742	1.06347	0.531733	+	0.846912i	$0.321541\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0.0807586	0.00575381	0.00287690	−	0.999996i	$-0.499084\pi$
$197$	0.0807586	0.00575381	0.00287690	+	0.999996i	$0.499084\pi$
$198$	0	0
$199$	−7.68238	−0.544590	−0.272295	−	0.962214i	$-0.587783\pi$
$199$	−7.68238	−0.544590	−0.272295	+	0.962214i	$0.587783\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−32.1235	−2.25463
$204$	0	0
$205$	−9.76030	−0.681689
$206$	0	0
$207$	−16.0605	−1.11628
$208$	0	0
$209$	1.69392	0.117171
$210$	0	0
$211$	−17.6243	−1.21331	−0.606653	−	0.794967i	$-0.707488\pi$
$211$	−17.6243	−1.21331	−0.606653	+	0.794967i	$0.707488\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−11.1957	−0.763541
$216$	0	0
$217$	5.64611	0.383283
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−7.38164	−0.496543
$222$	0	0
$223$	−9.75731	−0.653398	−0.326699	−	0.945128i	$-0.605936\pi$
$223$	−9.75731	−0.653398	−0.326699	+	0.945128i	$0.605936\pi$
$224$	0	0
$225$	−8.88586	−0.592390
$226$	0	0
$227$	12.4802	0.828341	0.414170	−	0.910199i	$-0.364072\pi$
$227$	12.4802	0.828341	0.414170	+	0.910199i	$0.364072\pi$
$228$	0	0
$229$	−17.5468	−1.15953	−0.579763	−	0.814785i	$-0.696855\pi$
$229$	−17.5468	−1.15953	−0.579763	+	0.814785i	$0.696855\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−8.85189	−0.579906	−0.289953	−	0.957041i	$-0.593640\pi$
$233$	−8.85189	−0.579906	−0.289953	+	0.957041i	$0.593640\pi$
$234$	0	0
$235$	0.883017	0.0576017
$236$	0	0
$237$	0	0
$238$	0	0
$239$	25.3177	1.63767	0.818833	−	0.574031i	$-0.194621\pi$
$239$	25.3177	1.63767	0.818833	+	0.574031i	$0.194621\pi$
$240$	0	0
$241$	−7.86372	−0.506547	−0.253273	−	0.967395i	$-0.581507\pi$
$241$	−7.86372	−0.506547	−0.253273	+	0.967395i	$0.581507\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	46.8525	2.99330
$246$	0	0
$247$	3.33980	0.212506
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−18.2683	−1.15308	−0.576542	−	0.817068i	$-0.695598\pi$
$251$	−18.2683	−1.15308	−0.576542	+	0.817068i	$0.695598\pi$
$252$	0	0
$253$	9.06835	0.570123
$254$	0	0
$255$	0	0
$256$	0	0
$257$	25.3389	1.58060	0.790300	−	0.612720i	$-0.209925\pi$
$257$	25.3389	1.58060	0.790300	+	0.612720i	$0.209925\pi$
$258$	0	0
$259$	22.9735	1.42750
$260$	0	0
$261$	19.8357	1.22780
$262$	0	0
$263$	−15.9912	−0.986060	−0.493030	−	0.870012i	$-0.664111\pi$
$263$	−15.9912	−0.986060	−0.493030	+	0.870012i	$0.664111\pi$
$264$	0	0
$265$	16.3737	1.00583
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−26.3678	−1.60767	−0.803837	−	0.594850i	$-0.797211\pi$
$269$	−26.3678	−1.60767	−0.803837	+	0.594850i	$0.797211\pi$
$270$	0	0
$271$	−15.0837	−0.916267	−0.458134	−	0.888883i	$-0.651482\pi$
$271$	−15.0837	−0.916267	−0.458134	+	0.888883i	$0.651482\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	5.01730	0.302554
$276$	0	0
$277$	−32.3249	−1.94222	−0.971108	−	0.238639i	$-0.923299\pi$
$277$	−32.3249	−1.94222	−0.971108	+	0.238639i	$0.923299\pi$
$278$	0	0
$279$	−3.48638	−0.208724
$280$	0	0
$281$	27.9457	1.66710	0.833551	−	0.552443i	$-0.186304\pi$
$281$	27.9457	1.66710	0.833551	+	0.552443i	$0.186304\pi$
$282$	0	0
$283$	14.8329	0.881724	0.440862	−	0.897575i	$-0.354673\pi$
$283$	14.8329	0.881724	0.440862	+	0.897575i	$0.354673\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−16.8054	−0.991994
$288$	0	0
$289$	−12.1150	−0.712647
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−0.757761	−0.0442689	−0.0221344	−	0.999755i	$-0.507046\pi$
$293$	−0.757761	−0.0442689	−0.0221344	+	0.999755i	$0.507046\pi$
$294$	0	0
$295$	18.0293	1.04971
$296$	0	0
$297$	0	0
$298$	0	0
$299$	17.8796	1.03400
$300$	0	0
$301$	−19.2770	−1.11111
$302$	0	0
$303$	0	0
$304$	0	0
$305$	11.2797	0.645875
$306$	0	0
$307$	28.3504	1.61804	0.809022	−	0.587779i	$-0.199997\pi$
$307$	28.3504	1.61804	0.809022	+	0.587779i	$0.199997\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−18.2091	−1.03254	−0.516272	−	0.856425i	$-0.672681\pi$
$311$	−18.2091	−1.03254	−0.516272	+	0.856425i	$0.672681\pi$
$312$	0	0
$313$	−4.19041	−0.236856	−0.118428	−	0.992963i	$-0.537785\pi$
$313$	−4.19041	−0.236856	−0.118428	+	0.992963i	$0.537785\pi$
$314$	0	0
$315$	−41.1271	−2.31725
$316$	0	0
$317$	14.2686	0.801407	0.400704	−	0.916208i	$-0.368766\pi$
$317$	14.2686	0.801407	0.400704	+	0.916208i	$0.368766\pi$
$318$	0	0
$319$	−11.2000	−0.627081
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−2.21020	−0.122979
$324$	0	0
$325$	9.89233	0.548728
$326$	0	0
$327$	0	0
$328$	0	0
$329$	1.52039	0.0838220
$330$	0	0
$331$	−1.22901	−0.0675523	−0.0337761	−	0.999429i	$-0.510753\pi$
$331$	−1.22901	−0.0675523	−0.0337761	+	0.999429i	$0.510753\pi$
$332$	0	0
$333$	−14.1857	−0.777374
$334$	0	0
$335$	−2.91169	−0.159083
$336$	0	0
$337$	32.8592	1.78995	0.894976	−	0.446113i	$-0.147192\pi$
$337$	32.8592	1.78995	0.894976	+	0.446113i	$0.147192\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	1.96854	0.106603
$342$	0	0
$343$	46.6624	2.51953
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−10.0851	−0.541395	−0.270698	−	0.962664i	$-0.587254\pi$
$347$	−10.0851	−0.541395	−0.270698	+	0.962664i	$0.587254\pi$
$348$	0	0
$349$	−11.7912	−0.631170	−0.315585	−	0.948897i	$-0.602201\pi$
$349$	−11.7912	−0.631170	−0.315585	+	0.948897i	$0.602201\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	25.6760	1.36660	0.683298	−	0.730139i	$-0.260545\pi$
$353$	25.6760	1.36660	0.683298	+	0.730139i	$0.260545\pi$
$354$	0	0
$355$	44.9563	2.38603
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−19.5002	−1.02918	−0.514590	−	0.857436i	$-0.672056\pi$
$359$	−19.5002	−1.02918	−0.514590	+	0.857436i	$0.672056\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−45.2906	−2.37062
$366$	0	0
$367$	−25.5547	−1.33395	−0.666974	−	0.745081i	$-0.732411\pi$
$367$	−25.5547	−1.33395	−0.666974	+	0.745081i	$0.732411\pi$
$368$	0	0
$369$	10.3771	0.540209
$370$	0	0
$371$	28.1926	1.46369
$372$	0	0
$373$	1.53066	0.0792545	0.0396272	−	0.999215i	$-0.487383\pi$
$373$	1.53066	0.0792545	0.0396272	+	0.999215i	$0.487383\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−22.0825	−1.13730
$378$	0	0
$379$	−26.6752	−1.37021	−0.685107	−	0.728442i	$-0.740245\pi$
$379$	−26.6752	−1.37021	−0.685107	+	0.728442i	$0.740245\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−4.34435	−0.221986	−0.110993	−	0.993821i	$-0.535403\pi$
$383$	−4.34435	−0.221986	−0.110993	+	0.993821i	$0.535403\pi$
$384$	0	0
$385$	23.2219	1.18350
$386$	0	0
$387$	11.9032	0.605073
$388$	0	0
$389$	−2.36016	−0.119665	−0.0598325	−	0.998208i	$-0.519057\pi$
$389$	−2.36016	−0.119665	−0.0598325	+	0.998208i	$0.519057\pi$
$390$	0	0
$391$	−11.8323	−0.598384
$392$	0	0
$393$	0	0
$394$	0	0
$395$	2.82169	0.141975
$396$	0	0
$397$	34.5939	1.73622	0.868108	−	0.496375i	$-0.165336\pi$
$397$	34.5939	1.73622	0.868108	+	0.496375i	$0.165336\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−17.0951	−0.853688	−0.426844	−	0.904325i	$-0.640375\pi$
$401$	−17.0951	−0.853688	−0.426844	+	0.904325i	$0.640375\pi$
$402$	0	0
$403$	3.88127	0.193340
$404$	0	0
$405$	25.3952	1.26190
$406$	0	0
$407$	8.00982	0.397032
$408$	0	0
$409$	−17.9216	−0.886165	−0.443082	−	0.896481i	$-0.646115\pi$
$409$	−17.9216	−0.886165	−0.443082	+	0.896481i	$0.646115\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	31.0432	1.52753
$414$	0	0
$415$	−2.95334	−0.144974
$416$	0	0
$417$	0	0
$418$	0	0
$419$	24.4187	1.19293	0.596467	−	0.802638i	$-0.296571\pi$
$419$	24.4187	1.19293	0.596467	+	0.802638i	$0.296571\pi$
$420$	0	0
$421$	24.5791	1.19791	0.598956	−	0.800782i	$-0.295583\pi$
$421$	24.5791	1.19791	0.598956	+	0.800782i	$0.295583\pi$
$422$	0	0
$423$	−0.938816	−0.0456468
$424$	0	0
$425$	−6.54651	−0.317552
$426$	0	0
$427$	19.4216	0.939877
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−31.6064	−1.52243	−0.761213	−	0.648501i	$-0.775396\pi$
$431$	−31.6064	−1.52243	−0.761213	+	0.648501i	$0.775396\pi$
$432$	0	0
$433$	−26.0874	−1.25368	−0.626839	−	0.779149i	$-0.715652\pi$
$433$	−26.0874	−1.25368	−0.626839	+	0.779149i	$0.715652\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	5.35348	0.256092
$438$	0	0
$439$	7.85137	0.374725	0.187363	−	0.982291i	$-0.440006\pi$
$439$	7.85137	0.374725	0.187363	+	0.982291i	$0.440006\pi$
$440$	0	0
$441$	−49.8132	−2.37206
$442$	0	0
$443$	19.9555	0.948115	0.474057	−	0.880494i	$-0.342789\pi$
$443$	19.9555	0.948115	0.474057	+	0.880494i	$0.342789\pi$
$444$	0	0
$445$	−45.9190	−2.17677
$446$	0	0
$447$	0	0
$448$	0	0
$449$	28.2978	1.33545	0.667727	−	0.744406i	$-0.267267\pi$
$449$	28.2978	1.33545	0.667727	+	0.744406i	$0.267267\pi$
$450$	0	0
$451$	−5.85930	−0.275904
$452$	0	0
$453$	0	0
$454$	0	0
$455$	45.7854	2.14645
$456$	0	0
$457$	10.5264	0.492405	0.246203	−	0.969218i	$-0.420817\pi$
$457$	10.5264	0.492405	0.246203	+	0.969218i	$0.420817\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	20.5589	0.957525	0.478763	−	0.877944i	$-0.341085\pi$
$461$	20.5589	0.957525	0.478763	+	0.877944i	$0.341085\pi$
$462$	0	0
$463$	17.9097	0.832334	0.416167	−	0.909288i	$-0.363373\pi$
$463$	17.9097	0.832334	0.416167	+	0.909288i	$0.363373\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0.500414	0.0231564	0.0115782	−	0.999933i	$-0.496314\pi$
$467$	0.500414	0.0231564	0.0115782	+	0.999933i	$0.496314\pi$
$468$	0	0
$469$	−5.01340	−0.231497
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−6.72100	−0.309032
$474$	0	0
$475$	2.96195	0.135904
$476$	0	0
$477$	−17.4084	−0.797077
$478$	0	0
$479$	9.94940	0.454600	0.227300	−	0.973825i	$-0.427010\pi$
$479$	9.94940	0.454600	0.227300	+	0.973825i	$0.427010\pi$
$480$	0	0
$481$	15.7925	0.720077
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−17.7959	−0.808072
$486$	0	0
$487$	−32.6003	−1.47726	−0.738631	−	0.674110i	$-0.764528\pi$
$487$	−32.6003	−1.47726	−0.738631	+	0.674110i	$0.764528\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−30.7747	−1.38884	−0.694421	−	0.719569i	$-0.744340\pi$
$491$	−30.7747	−1.38884	−0.694421	+	0.719569i	$0.744340\pi$
$492$	0	0
$493$	14.6137	0.658166
$494$	0	0
$495$	−14.3391	−0.644496
$496$	0	0
$497$	77.4065	3.47215
$498$	0	0
$499$	34.9097	1.56277	0.781386	−	0.624048i	$-0.214513\pi$
$499$	34.9097	1.56277	0.781386	+	0.624048i	$0.214513\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−9.90815	−0.441782	−0.220891	−	0.975298i	$-0.570897\pi$
$503$	−9.90815	−0.441782	−0.220891	+	0.975298i	$0.570897\pi$
$504$	0	0
$505$	30.2114	1.34439
$506$	0	0
$507$	0	0
$508$	0	0
$509$	22.7053	1.00639	0.503197	−	0.864172i	$-0.332157\pi$
$509$	22.7053	1.00639	0.503197	+	0.864172i	$0.332157\pi$
$510$	0	0
$511$	−77.9821	−3.44973
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−27.5096	−1.21222
$516$	0	0
$517$	0.530092	0.0233134
$518$	0	0
$519$	0	0
$520$	0	0
$521$	7.02911	0.307951	0.153975	−	0.988075i	$-0.450792\pi$
$521$	7.02911	0.307951	0.153975	+	0.988075i	$0.450792\pi$
$522$	0	0
$523$	24.2642	1.06100	0.530500	−	0.847685i	$-0.322004\pi$
$523$	24.2642	1.06100	0.530500	+	0.847685i	$0.322004\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−2.56853	−0.111887
$528$	0	0
$529$	5.65979	0.246078
$530$	0	0
$531$	−19.1686	−0.831847
$532$	0	0
$533$	−11.5525	−0.500392
$534$	0	0
$535$	49.1342	2.12426
$536$	0	0
$537$	0	0
$538$	0	0
$539$	28.1265	1.21149
$540$	0	0
$541$	17.8481	0.767349	0.383674	−	0.923468i	$-0.374659\pi$
$541$	17.8481	0.767349	0.383674	+	0.923468i	$0.374659\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	10.5413	0.451538
$546$	0	0
$547$	8.86098	0.378868	0.189434	−	0.981893i	$-0.439335\pi$
$547$	8.86098	0.378868	0.189434	+	0.981893i	$0.439335\pi$
$548$	0	0
$549$	−11.9925	−0.511828
$550$	0	0
$551$	−6.61191	−0.281677
$552$	0	0
$553$	4.85844	0.206602
$554$	0	0
$555$	0	0
$556$	0	0
$557$	45.3172	1.92015	0.960076	−	0.279741i	$-0.0902485\pi$
$557$	45.3172	1.92015	0.960076	+	0.279741i	$0.0902485\pi$
$558$	0	0
$559$	−13.2514	−0.560475
$560$	0	0
$561$	0	0
$562$	0	0
$563$	3.19225	0.134537	0.0672687	−	0.997735i	$-0.478572\pi$
$563$	3.19225	0.134537	0.0672687	+	0.997735i	$0.478572\pi$
$564$	0	0
$565$	45.9783	1.93432
$566$	0	0
$567$	43.7259	1.83632
$568$	0	0
$569$	−7.14434	−0.299506	−0.149753	−	0.988723i	$-0.547848\pi$
$569$	−7.14434	−0.299506	−0.149753	+	0.988723i	$0.547848\pi$
$570$	0	0
$571$	44.1456	1.84744	0.923718	−	0.383073i	$-0.125134\pi$
$571$	44.1456	1.84744	0.923718	+	0.383073i	$0.125134\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	15.8568	0.661273
$576$	0	0
$577$	12.5902	0.524139	0.262069	−	0.965049i	$-0.415595\pi$
$577$	12.5902	0.524139	0.262069	+	0.965049i	$0.415595\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−5.08511	−0.210966
$582$	0	0
$583$	9.82947	0.407095
$584$	0	0
$585$	−28.2717	−1.16889
$586$	0	0
$587$	22.7855	0.940458	0.470229	−	0.882545i	$-0.344171\pi$
$587$	22.7855	0.940458	0.470229	+	0.882545i	$0.344171\pi$
$588$	0	0
$589$	1.16213	0.0478845
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−12.8252	−0.526668	−0.263334	−	0.964705i	$-0.584822\pi$
$593$	−12.8252	−0.526668	−0.263334	+	0.964705i	$0.584822\pi$
$594$	0	0
$595$	−30.2997	−1.24217
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−41.0442	−1.67702	−0.838511	−	0.544884i	$-0.816574\pi$
$599$	−41.0442	−1.67702	−0.838511	+	0.544884i	$0.816574\pi$
$600$	0	0
$601$	−2.87696	−0.117354	−0.0586768	−	0.998277i	$-0.518688\pi$
$601$	−2.87696	−0.117354	−0.0586768	+	0.998277i	$0.518688\pi$
$602$	0	0
$603$	3.09569	0.126066
$604$	0	0
$605$	−22.9422	−0.932733
$606$	0	0
$607$	−33.3515	−1.35369	−0.676847	−	0.736124i	$-0.736654\pi$
$607$	−33.3515	−1.35369	−0.676847	+	0.736124i	$0.736654\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	1.04515	0.0422824
$612$	0	0
$613$	24.9191	1.00647	0.503237	−	0.864149i	$-0.332142\pi$
$613$	24.9191	1.00647	0.503237	+	0.864149i	$0.332142\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	46.2527	1.86207	0.931033	−	0.364936i	$-0.118909\pi$
$617$	46.2527	1.86207	0.931033	+	0.364936i	$0.118909\pi$
$618$	0	0
$619$	−24.7757	−0.995818	−0.497909	−	0.867229i	$-0.665899\pi$
$619$	−24.7757	−0.995818	−0.497909	+	0.867229i	$0.665899\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−79.0640	−3.16763
$624$	0	0
$625$	−31.0366	−1.24146
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−10.4511	−0.416713
$630$	0	0
$631$	15.7381	0.626526	0.313263	−	0.949666i	$-0.398578\pi$
$631$	15.7381	0.626526	0.313263	+	0.949666i	$0.398578\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	33.5712	1.33223
$636$	0	0
$637$	55.4554	2.19722
$638$	0	0
$639$	−47.7971	−1.89083
$640$	0	0
$641$	22.9675	0.907160	0.453580	−	0.891215i	$-0.350147\pi$
$641$	22.9675	0.907160	0.453580	+	0.891215i	$0.350147\pi$
$642$	0	0
$643$	−45.2197	−1.78329	−0.891646	−	0.452733i	$-0.850449\pi$
$643$	−45.2197	−1.78329	−0.891646	+	0.452733i	$0.850449\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	15.6124	0.613788	0.306894	−	0.951744i	$-0.400710\pi$
$647$	15.6124	0.613788	0.306894	+	0.951744i	$0.400710\pi$
$648$	0	0
$649$	10.8233	0.424853
$650$	0	0
$651$	0	0
$652$	0	0
$653$	13.4940	0.528060	0.264030	−	0.964514i	$-0.414948\pi$
$653$	13.4940	0.528060	0.264030	+	0.964514i	$0.414948\pi$
$654$	0	0
$655$	−59.9207	−2.34130
$656$	0	0
$657$	48.1526	1.87861
$658$	0	0
$659$	−21.0964	−0.821799	−0.410900	−	0.911681i	$-0.634785\pi$
$659$	−21.0964	−0.821799	−0.410900	+	0.911681i	$0.634785\pi$
$660$	0	0
$661$	−27.6057	−1.07374	−0.536869	−	0.843666i	$-0.680393\pi$
$661$	−27.6057	−1.07374	−0.536869	+	0.843666i	$0.680393\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	13.7090	0.531613
$666$	0	0
$667$	−35.3968	−1.37057
$668$	0	0
$669$	0	0
$670$	0	0
$671$	6.77143	0.261408
$672$	0	0
$673$	45.9966	1.77304	0.886519	−	0.462692i	$-0.153116\pi$
$673$	45.9966	1.77304	0.886519	+	0.462692i	$0.153116\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	10.1259	0.389170	0.194585	−	0.980886i	$-0.437664\pi$
$677$	10.1259	0.389170	0.194585	+	0.980886i	$0.437664\pi$
$678$	0	0
$679$	−30.6413	−1.17591
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−11.9348	−0.456672	−0.228336	−	0.973582i	$-0.573329\pi$
$683$	−11.9348	−0.456672	−0.228336	+	0.973582i	$0.573329\pi$
$684$	0	0
$685$	−6.20705	−0.237159
$686$	0	0
$687$	0	0
$688$	0	0
$689$	19.3802	0.738328
$690$	0	0
$691$	29.5500	1.12414	0.562068	−	0.827091i	$-0.310006\pi$
$691$	29.5500	1.12414	0.562068	+	0.827091i	$0.310006\pi$
$692$	0	0
$693$	−24.6894	−0.937871
$694$	0	0
$695$	−27.7596	−1.05298
$696$	0	0
$697$	7.64514	0.289580
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−34.0313	−1.28534	−0.642671	−	0.766142i	$-0.722174\pi$
$701$	−34.0313	−1.28534	−0.642671	+	0.766142i	$0.722174\pi$
$702$	0	0
$703$	4.72858	0.178342
$704$	0	0
$705$	0	0
$706$	0	0
$707$	52.0185	1.95636
$708$	0	0
$709$	−43.4999	−1.63367	−0.816837	−	0.576869i	$-0.804274\pi$
$709$	−43.4999	−1.63367	−0.816837	+	0.576869i	$0.804274\pi$
$710$	0	0
$711$	−3.00000	−0.112509
$712$	0	0
$713$	6.22142	0.232994
$714$	0	0
$715$	15.9633	0.596993
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−11.5944	−0.432398	−0.216199	−	0.976349i	$-0.569366\pi$
$719$	−11.5944	−0.432398	−0.216199	+	0.976349i	$0.569366\pi$
$720$	0	0
$721$	−47.3665	−1.76402
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−19.5842	−0.727337
$726$	0	0
$727$	3.51600	0.130401	0.0652006	−	0.997872i	$-0.479231\pi$
$727$	3.51600	0.130401	0.0652006	+	0.997872i	$0.479231\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	0	0
$731$	8.76948	0.324351
$732$	0	0
$733$	17.0432	0.629506	0.314753	−	0.949174i	$-0.398078\pi$
$733$	17.0432	0.629506	0.314753	+	0.949174i	$0.398078\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1.74794	−0.0643864
$738$	0	0
$739$	−45.0638	−1.65770	−0.828850	−	0.559471i	$-0.811004\pi$
$739$	−45.0638	−1.65770	−0.828850	+	0.559471i	$0.811004\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−19.1124	−0.701167	−0.350583	−	0.936532i	$-0.614017\pi$
$743$	−19.1124	−0.701167	−0.350583	+	0.936532i	$0.614017\pi$
$744$	0	0
$745$	15.9255	0.583463
$746$	0	0
$747$	3.13997	0.114885
$748$	0	0
$749$	84.6000	3.09122
$750$	0	0
$751$	16.9146	0.617224	0.308612	−	0.951188i	$-0.400136\pi$
$751$	16.9146	0.617224	0.308612	+	0.951188i	$0.400136\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	6.16443	0.224346
$756$	0	0
$757$	−25.6554	−0.932462	−0.466231	−	0.884663i	$-0.654388\pi$
$757$	−25.6554	−0.932462	−0.466231	+	0.884663i	$0.654388\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	29.3369	1.06346	0.531731	−	0.846913i	$-0.321542\pi$
$761$	29.3369	1.06346	0.531731	+	0.846913i	$0.321542\pi$
$762$	0	0
$763$	18.1501	0.657078
$764$	0	0
$765$	18.7095	0.676445
$766$	0	0
$767$	21.3398	0.770535
$768$	0	0
$769$	−49.3040	−1.77795	−0.888973	−	0.457960i	$-0.848580\pi$
$769$	−49.3040	−1.77795	−0.888973	+	0.457960i	$0.848580\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−37.3199	−1.34230	−0.671152	−	0.741320i	$-0.734200\pi$
$773$	−37.3199	−1.34230	−0.671152	+	0.741320i	$0.734200\pi$
$774$	0	0
$775$	3.44216	0.123646
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−3.45902	−0.123932
$780$	0	0
$781$	26.9881	0.965711
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−18.5096	−0.660637
$786$	0	0
$787$	35.4876	1.26500	0.632499	−	0.774561i	$-0.282029\pi$
$787$	35.4876	1.26500	0.632499	+	0.774561i	$0.282029\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	79.1662	2.81483
$792$	0	0
$793$	13.3509	0.474103
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−7.24576	−0.256658	−0.128329	−	0.991732i	$-0.540961\pi$
$797$	−7.24576	−0.256658	−0.128329	+	0.991732i	$0.540961\pi$
$798$	0	0
$799$	−0.691658	−0.0244691
$800$	0	0
$801$	48.8207	1.72499
$802$	0	0
$803$	−27.1888	−0.959473
$804$	0	0
$805$	73.3910	2.58669
$806$	0	0
$807$	0	0
$808$	0	0
$809$	42.8637	1.50701	0.753504	−	0.657443i	$-0.228362\pi$
$809$	42.8637	1.50701	0.753504	+	0.657443i	$0.228362\pi$
$810$	0	0
$811$	−52.3293	−1.83753	−0.918765	−	0.394805i	$-0.870812\pi$
$811$	−52.3293	−1.83753	−0.918765	+	0.394805i	$0.870812\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0.994539	0.0348372
$816$	0	0
$817$	−3.96773	−0.138813
$818$	0	0
$819$	−48.6786	−1.70097
$820$	0	0
$821$	−11.2764	−0.393548	−0.196774	−	0.980449i	$-0.563047\pi$
$821$	−11.2764	−0.393548	−0.196774	+	0.980449i	$0.563047\pi$
$822$	0	0
$823$	0.290199	0.0101157	0.00505785	−	0.999987i	$-0.498390\pi$
$823$	0.290199	0.0101157	0.00505785	+	0.999987i	$0.498390\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−1.42289	−0.0494786	−0.0247393	−	0.999694i	$-0.507876\pi$
$827$	−1.42289	−0.0494786	−0.0247393	+	0.999694i	$0.507876\pi$
$828$	0	0
$829$	−27.7482	−0.963733	−0.481867	−	0.876245i	$-0.660041\pi$
$829$	−27.7482	−0.963733	−0.481867	+	0.876245i	$0.660041\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−36.6991	−1.27155
$834$	0	0
$835$	−16.3017	−0.564144
$836$	0	0
$837$	0	0
$838$	0	0
$839$	13.7602	0.475055	0.237527	−	0.971381i	$-0.423663\pi$
$839$	13.7602	0.475055	0.237527	+	0.971381i	$0.423663\pi$
$840$	0	0
$841$	14.7174	0.507495
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−5.20808	−0.179163
$846$	0	0
$847$	−39.5022	−1.35731
$848$	0	0
$849$	0	0
$850$	0	0
$851$	25.3144	0.867766
$852$	0	0
$853$	−18.4954	−0.633269	−0.316635	−	0.948548i	$-0.602553\pi$
$853$	−18.4954	−0.633269	−0.316635	+	0.948548i	$0.602553\pi$
$854$	0	0
$855$	−8.46508	−0.289500
$856$	0	0
$857$	−19.9754	−0.682348	−0.341174	−	0.940000i	$-0.610825\pi$
$857$	−19.9754	−0.682348	−0.341174	+	0.940000i	$0.610825\pi$
$858$	0	0
$859$	−11.7152	−0.399719	−0.199859	−	0.979825i	$-0.564049\pi$
$859$	−11.7152	−0.399719	−0.199859	+	0.979825i	$0.564049\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−3.40528	−0.115917	−0.0579586	−	0.998319i	$-0.518459\pi$
$863$	−3.40528	−0.115917	−0.0579586	+	0.998319i	$0.518459\pi$
$864$	0	0
$865$	31.5217	1.07177
$866$	0	0
$867$	0	0
$868$	0	0
$869$	1.69392	0.0574622
$870$	0	0
$871$	−3.44633	−0.116774
$872$	0	0
$873$	18.9205	0.640361
$874$	0	0
$875$	−27.9396	−0.944532
$876$	0	0
$877$	−4.31093	−0.145570	−0.0727848	−	0.997348i	$-0.523189\pi$
$877$	−4.31093	−0.145570	−0.0727848	+	0.997348i	$0.523189\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	26.5401	0.894158	0.447079	−	0.894494i	$-0.352464\pi$
$881$	26.5401	0.894158	0.447079	+	0.894494i	$0.352464\pi$
$882$	0	0
$883$	9.90602	0.333364	0.166682	−	0.986011i	$-0.446695\pi$
$883$	9.90602	0.333364	0.166682	+	0.986011i	$0.446695\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−17.5043	−0.587737	−0.293869	−	0.955846i	$-0.594943\pi$
$887$	−17.5043	−0.587737	−0.293869	+	0.955846i	$0.594943\pi$
$888$	0	0
$889$	57.8034	1.93866
$890$	0	0
$891$	15.2452	0.510735
$892$	0	0
$893$	0.312939	0.0104721
$894$	0	0
$895$	12.1855	0.407315
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−7.68387	−0.256271
$900$	0	0
$901$	−12.8254	−0.427275
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−6.62902	−0.220356
$906$	0	0
$907$	−33.2867	−1.10527	−0.552634	−	0.833424i	$-0.686377\pi$
$907$	−33.2867	−1.10527	−0.552634	+	0.833424i	$0.686377\pi$
$908$	0	0
$909$	−32.1205	−1.06537
$910$	0	0
$911$	−22.2695	−0.737822	−0.368911	−	0.929465i	$-0.620269\pi$
$911$	−22.2695	−0.737822	−0.368911	+	0.929465i	$0.620269\pi$
$912$	0	0
$913$	−1.77295	−0.0586760
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−103.172	−3.40706
$918$	0	0
$919$	59.4968	1.96262	0.981309	−	0.192437i	$-0.0616391\pi$
$919$	59.4968	1.96262	0.981309	+	0.192437i	$0.0616391\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	53.2110	1.75146
$924$	0	0
$925$	14.0058	0.460509
$926$	0	0
$927$	29.2480	0.960630
$928$	0	0
$929$	−54.6282	−1.79229	−0.896146	−	0.443759i	$-0.853645\pi$
$929$	−54.6282	−1.79229	−0.896146	+	0.443759i	$0.853645\pi$
$930$	0	0
$931$	16.6044	0.544188
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−10.5641	−0.345484
$936$	0	0
$937$	6.41148	0.209454	0.104727	−	0.994501i	$-0.466603\pi$
$937$	6.41148	0.209454	0.104727	+	0.994501i	$0.466603\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−22.5191	−0.734103	−0.367051	−	0.930201i	$-0.619633\pi$
$941$	−22.5191	−0.734103	−0.367051	+	0.930201i	$0.619633\pi$
$942$	0	0
$943$	−18.5178	−0.603023
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−60.6625	−1.97127	−0.985634	−	0.168896i	$-0.945980\pi$
$947$	−60.6625	−1.97127	−0.985634	+	0.168896i	$0.945980\pi$
$948$	0	0
$949$	−53.6067	−1.74015
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−47.5880	−1.54153	−0.770763	−	0.637122i	$-0.780125\pi$
$953$	−47.5880	−1.54153	−0.770763	+	0.637122i	$0.780125\pi$
$954$	0	0
$955$	−14.8222	−0.479636
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−10.6874	−0.345114
$960$	0	0
$961$	−29.6495	−0.956434
$962$	0	0
$963$	−52.2390	−1.68338
$964$	0	0
$965$	41.6881	1.34199
$966$	0	0
$967$	−34.9569	−1.12414	−0.562069	−	0.827091i	$-0.689994\pi$
$967$	−34.9569	−1.12414	−0.562069	+	0.827091i	$0.689994\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−17.7678	−0.570196	−0.285098	−	0.958498i	$-0.592026\pi$
$971$	−17.7678	−0.570196	−0.285098	+	0.958498i	$0.592026\pi$
$972$	0	0
$973$	−47.7969	−1.53230
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−32.7343	−1.04726	−0.523632	−	0.851945i	$-0.675423\pi$
$977$	−32.7343	−1.04726	−0.523632	+	0.851945i	$0.675423\pi$
$978$	0	0
$979$	−27.5660	−0.881014
$980$	0	0
$981$	−11.2074	−0.357824
$982$	0	0
$983$	−26.5368	−0.846393	−0.423197	−	0.906038i	$-0.639092\pi$
$983$	−26.5368	−0.846393	−0.423197	+	0.906038i	$0.639092\pi$
$984$	0	0
$985$	0.227876	0.00726073
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−21.2412	−0.675430
$990$	0	0
$991$	45.3548	1.44074	0.720371	−	0.693588i	$-0.243971\pi$
$991$	45.3548	1.44074	0.720371	+	0.693588i	$0.243971\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−21.6773	−0.687218
$996$	0	0
$997$	28.1899	0.892783	0.446392	−	0.894838i	$-0.352709\pi$
$997$	28.1899	0.892783	0.446392	+	0.894838i	$0.352709\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6004.2.a.d.1.7	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6004.2.a.d.1.7	✓	8	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:	\( N \)	\(=\)	\( 6004 = 2^{2} \cdot 19 \cdot 79 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6004.a (trivial)

\(f(q)\)	\(=\)	\(q+2.82169 q^{5} +4.85844 q^{7} -3.00000 q^{9} +O(q^{10})\)	\(q+2.82169 q^{5} +4.85844 q^{7} -3.00000 q^{9} +1.69392 q^{11} +3.33980 q^{13} -2.21020 q^{17} +1.00000 q^{19} +5.35348 q^{23} +2.96195 q^{25} -6.61191 q^{29} +1.16213 q^{31} +13.7090 q^{35} +4.72858 q^{37} -3.45902 q^{41} -3.96773 q^{43} -8.46508 q^{45} +0.312939 q^{47} +16.6044 q^{49} +5.80281 q^{53} +4.77971 q^{55} +6.38954 q^{59} +3.99750 q^{61} -14.5753 q^{63} +9.42389 q^{65} -1.03190 q^{67} +15.9324 q^{71} -16.0509 q^{73} +8.22979 q^{77} +1.00000 q^{79} +9.00000 q^{81} -1.04666 q^{83} -6.23651 q^{85} -16.2736 q^{89} +16.2262 q^{91} +2.82169 q^{95} -6.30683 q^{97} -5.08175 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{5} + q^{7} - 24 q^{9}+O(q^{10}) \) 8 * q + 4 * q^5 + q^7 - 24 * q^9	\( 8 q + 4 q^{5} + q^{7} - 24 q^{9} - 7 q^{11} - 12 q^{13} - 7 q^{17} + 8 q^{19} + 10 q^{23} + 6 q^{25} + 5 q^{29} + 3 q^{31} - 11 q^{35} - 15 q^{37} - 5 q^{41} + 10 q^{43} - 12 q^{45} + 18 q^{47} + 31 q^{49} + 9 q^{53} - 17 q^{55} + 8 q^{59} + 13 q^{61} - 3 q^{63} + 4 q^{65} + 21 q^{67} + 44 q^{71} - 20 q^{73} + 15 q^{77} + 8 q^{79} + 72 q^{81} - 4 q^{83} - 9 q^{85} - 10 q^{89} + 48 q^{91} + 4 q^{95} - 22 q^{97} + 21 q^{99}+O(q^{100}) \) 8 * q + 4 * q^5 + q^7 - 24 * q^9 - 7 * q^11 - 12 * q^13 - 7 * q^17 + 8 * q^19 + 10 * q^23 + 6 * q^25 + 5 * q^29 + 3 * q^31 - 11 * q^35 - 15 * q^37 - 5 * q^41 + 10 * q^43 - 12 * q^45 + 18 * q^47 + 31 * q^49 + 9 * q^53 - 17 * q^55 + 8 * q^59 + 13 * q^61 - 3 * q^63 + 4 * q^65 + 21 * q^67 + 44 * q^71 - 20 * q^73 + 15 * q^77 + 8 * q^79 + 72 * q^81 - 4 * q^83 - 9 * q^85 - 10 * q^89 + 48 * q^91 + 4 * q^95 - 22 * q^97 + 21 * q^99

Introduction

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