LMFDB - Embedded newform 6004.2.a.f.1.2

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:	$1$
Dimension:	$25$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
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)

$f(q)$	$=$	$q-2.61174 q^{3} +0.965145 q^{5} +0.0126812 q^{7} +3.82116 q^{9} +O(q^{10})$	$q-2.61174 q^{3} +0.965145 q^{5} +0.0126812 q^{7} +3.82116 q^{9} +3.64460 q^{11} +2.47728 q^{13} -2.52070 q^{15} -2.78744 q^{17} -1.00000 q^{19} -0.0331199 q^{21} +2.89278 q^{23} -4.06850 q^{25} -2.14465 q^{27} -0.637574 q^{29} -6.82405 q^{31} -9.51874 q^{33} +0.0122392 q^{35} -6.56245 q^{37} -6.47000 q^{39} +0.267111 q^{41} -11.0179 q^{43} +3.68797 q^{45} +11.9768 q^{47} -6.99984 q^{49} +7.28004 q^{51} +5.54738 q^{53} +3.51757 q^{55} +2.61174 q^{57} +8.54272 q^{59} -13.6224 q^{61} +0.0484569 q^{63} +2.39094 q^{65} -9.96214 q^{67} -7.55517 q^{69} +8.48389 q^{71} +4.92921 q^{73} +10.6258 q^{75} +0.0462179 q^{77} +1.00000 q^{79} -5.86222 q^{81} -5.48970 q^{83} -2.69028 q^{85} +1.66517 q^{87} -9.25909 q^{89} +0.0314149 q^{91} +17.8226 q^{93} -0.965145 q^{95} +2.36976 q^{97} +13.9266 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 25 q + 4 q^{3} - 8 q^{5} + 2 q^{7} + 13 q^{9}+O(q^{10}) $ 25 * q + 4 * q^3 - 8 * q^5 + 2 * q^7 + 13 * q^9	$ 25 q + 4 q^{3} - 8 q^{5} + 2 q^{7} + 13 q^{9} - 3 q^{11} + q^{13} - 5 q^{15} - 13 q^{17} - 25 q^{19} - 24 q^{21} - 31 q^{23} + 21 q^{25} + 7 q^{27} - 19 q^{29} - 7 q^{31} - 30 q^{33} - q^{35} - 29 q^{37} - 26 q^{39} - 40 q^{41} - 40 q^{45} - 8 q^{47} - 9 q^{49} + 12 q^{51} - 38 q^{53} - 29 q^{55} - 4 q^{57} + 18 q^{59} - 26 q^{61} - 40 q^{63} - 70 q^{65} - 13 q^{67} + q^{69} - 47 q^{71} - 8 q^{73} + 7 q^{75} - 19 q^{77} + 25 q^{79} - 19 q^{81} - 8 q^{83} - 33 q^{85} - 50 q^{87} - 54 q^{89} - 12 q^{91} - 24 q^{93} + 8 q^{95} - 4 q^{97} - 39 q^{99}+O(q^{100}) $ 25 * q + 4 * q^3 - 8 * q^5 + 2 * q^7 + 13 * q^9 - 3 * q^11 + q^13 - 5 * q^15 - 13 * q^17 - 25 * q^19 - 24 * q^21 - 31 * q^23 + 21 * q^25 + 7 * q^27 - 19 * q^29 - 7 * q^31 - 30 * q^33 - q^35 - 29 * q^37 - 26 * q^39 - 40 * q^41 - 40 * q^45 - 8 * q^47 - 9 * q^49 + 12 * q^51 - 38 * q^53 - 29 * q^55 - 4 * q^57 + 18 * q^59 - 26 * q^61 - 40 * q^63 - 70 * q^65 - 13 * q^67 + q^69 - 47 * q^71 - 8 * q^73 + 7 * q^75 - 19 * q^77 + 25 * q^79 - 19 * q^81 - 8 * q^83 - 33 * q^85 - 50 * q^87 - 54 * q^89 - 12 * q^91 - 24 * q^93 + 8 * q^95 - 4 * q^97 - 39 * 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
$3$	−2.61174	−1.50789	−0.753943	−	0.656940i	$-0.771851\pi$
$3$	−2.61174	−1.50789	−0.753943	+	0.656940i	$0.771851\pi$
$4$	0	0
$5$	0.965145	0.431626	0.215813	−	0.976435i	$-0.430760\pi$
$5$	0.965145	0.431626	0.215813	+	0.976435i	$0.430760\pi$
$6$	0	0
$7$	0.0126812	0.00479304	0.00239652	−	0.999997i	$-0.499237\pi$
$7$	0.0126812	0.00479304	0.00239652	+	0.999997i	$0.499237\pi$
$8$	0	0
$9$	3.82116	1.27372
$10$	0	0
$11$	3.64460	1.09889	0.549445	−	0.835530i	$-0.314839\pi$
$11$	3.64460	1.09889	0.549445	+	0.835530i	$0.314839\pi$
$12$	0	0
$13$	2.47728	0.687074	0.343537	−	0.939139i	$-0.388375\pi$
$13$	2.47728	0.687074	0.343537	+	0.939139i	$0.388375\pi$
$14$	0	0
$15$	−2.52070	−0.650843
$16$	0	0
$17$	−2.78744	−0.676053	−0.338026	−	0.941137i	$-0.609759\pi$
$17$	−2.78744	−0.676053	−0.338026	+	0.941137i	$0.609759\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−0.0331199	−0.00722736
$22$	0	0
$23$	2.89278	0.603186	0.301593	−	0.953437i	$-0.402482\pi$
$23$	2.89278	0.603186	0.301593	+	0.953437i	$0.402482\pi$
$24$	0	0
$25$	−4.06850	−0.813699
$26$	0	0
$27$	−2.14465	−0.412739
$28$	0	0
$29$	−0.637574	−0.118395	−0.0591973	−	0.998246i	$-0.518854\pi$
$29$	−0.637574	−0.118395	−0.0591973	+	0.998246i	$0.518854\pi$
$30$	0	0
$31$	−6.82405	−1.22564	−0.612818	−	0.790224i	$-0.709964\pi$
$31$	−6.82405	−1.22564	−0.612818	+	0.790224i	$0.709964\pi$
$32$	0	0
$33$	−9.51874	−1.65700
$34$	0	0
$35$	0.0122392	0.00206880
$36$	0	0
$37$	−6.56245	−1.07886	−0.539430	−	0.842031i	$-0.681360\pi$
$37$	−6.56245	−1.07886	−0.539430	+	0.842031i	$0.681360\pi$
$38$	0	0
$39$	−6.47000	−1.03603
$40$	0	0
$41$	0.267111	0.0417157	0.0208579	−	0.999782i	$-0.493360\pi$
$41$	0.267111	0.0417157	0.0208579	+	0.999782i	$0.493360\pi$
$42$	0	0
$43$	−11.0179	−1.68022	−0.840108	−	0.542419i	$-0.817508\pi$
$43$	−11.0179	−1.68022	−0.840108	+	0.542419i	$0.817508\pi$
$44$	0	0
$45$	3.68797	0.549771
$46$	0	0
$47$	11.9768	1.74700	0.873500	−	0.486824i	$-0.161845\pi$
$47$	11.9768	1.74700	0.873500	+	0.486824i	$0.161845\pi$
$48$	0	0
$49$	−6.99984	−0.999977
$50$	0	0
$51$	7.28004	1.01941
$52$	0	0
$53$	5.54738	0.761991	0.380996	−	0.924577i	$-0.375581\pi$
$53$	5.54738	0.761991	0.380996	+	0.924577i	$0.375581\pi$
$54$	0	0
$55$	3.51757	0.474309
$56$	0	0
$57$	2.61174	0.345933
$58$	0	0
$59$	8.54272	1.11217	0.556084	−	0.831126i	$-0.312303\pi$
$59$	8.54272	1.11217	0.556084	+	0.831126i	$0.312303\pi$
$60$	0	0
$61$	−13.6224	−1.74417	−0.872086	−	0.489352i	$-0.837233\pi$
$61$	−13.6224	−1.74417	−0.872086	+	0.489352i	$0.837233\pi$
$62$	0	0
$63$	0.0484569	0.00610499
$64$	0	0
$65$	2.39094	0.296559
$66$	0	0
$67$	−9.96214	−1.21707	−0.608534	−	0.793528i	$-0.708242\pi$
$67$	−9.96214	−1.21707	−0.608534	+	0.793528i	$0.708242\pi$
$68$	0	0
$69$	−7.55517	−0.909535
$70$	0	0
$71$	8.48389	1.00685	0.503426	−	0.864038i	$-0.332073\pi$
$71$	8.48389	1.00685	0.503426	+	0.864038i	$0.332073\pi$
$72$	0	0
$73$	4.92921	0.576920	0.288460	−	0.957492i	$-0.406857\pi$
$73$	4.92921	0.576920	0.288460	+	0.957492i	$0.406857\pi$
$74$	0	0
$75$	10.6258	1.22697
$76$	0	0
$77$	0.0462179	0.00526702
$78$	0	0
$79$	1.00000	0.112509
$79$	1.00000	0.112509
$80$	0	0
$81$	−5.86222	−0.651357
$82$	0	0
$83$	−5.48970	−0.602573	−0.301286	−	0.953534i	$-0.597416\pi$
$83$	−5.48970	−0.602573	−0.301286	+	0.953534i	$0.597416\pi$
$84$	0	0
$85$	−2.69028	−0.291802
$86$	0	0
$87$	1.66517	0.178525
$88$	0	0
$89$	−9.25909	−0.981461	−0.490731	−	0.871311i	$-0.663270\pi$
$89$	−9.25909	−0.981461	−0.490731	+	0.871311i	$0.663270\pi$
$90$	0	0
$91$	0.0314149	0.00329318
$92$	0	0
$93$	17.8226	1.84812
$94$	0	0
$95$	−0.965145	−0.0990218
$96$	0	0
$97$	2.36976	0.240613	0.120306	−	0.992737i	$-0.461612\pi$
$97$	2.36976	0.240613	0.120306	+	0.992737i	$0.461612\pi$
$98$	0	0
$99$	13.9266	1.39968
$100$	0	0
$101$	6.49924	0.646699	0.323349	−	0.946280i	$-0.395191\pi$
$101$	6.49924	0.646699	0.323349	+	0.946280i	$0.395191\pi$
$102$	0	0
$103$	8.05384	0.793568	0.396784	−	0.917912i	$-0.370126\pi$
$103$	8.05384	0.793568	0.396784	+	0.917912i	$0.370126\pi$
$104$	0	0
$105$	−0.0319655	−0.00311951
$106$	0	0
$107$	4.23529	0.409441	0.204720	−	0.978820i	$-0.434371\pi$
$107$	4.23529	0.409441	0.204720	+	0.978820i	$0.434371\pi$
$108$	0	0
$109$	4.33982	0.415680	0.207840	−	0.978163i	$-0.433357\pi$
$109$	4.33982	0.415680	0.207840	+	0.978163i	$0.433357\pi$
$110$	0	0
$111$	17.1394	1.62680
$112$	0	0
$113$	0.154511	0.0145352	0.00726758	−	0.999974i	$-0.497687\pi$
$113$	0.154511	0.0145352	0.00726758	+	0.999974i	$0.497687\pi$
$114$	0	0
$115$	2.79195	0.260351
$116$	0	0
$117$	9.46609	0.875140
$118$	0	0
$119$	−0.0353480	−0.00324035
$120$	0	0
$121$	2.28313	0.207557
$122$	0	0
$123$	−0.697623	−0.0629026
$124$	0	0
$125$	−8.75241	−0.782839
$126$	0	0
$127$	19.9913	1.77394	0.886972	−	0.461823i	$-0.152804\pi$
$127$	19.9913	1.77394	0.886972	+	0.461823i	$0.152804\pi$
$128$	0	0
$129$	28.7759	2.53357
$130$	0	0
$131$	2.64369	0.230980	0.115490	−	0.993309i	$-0.463156\pi$
$131$	2.64369	0.230980	0.115490	+	0.993309i	$0.463156\pi$
$132$	0	0
$133$	−0.0126812	−0.00109960
$134$	0	0
$135$	−2.06990	−0.178149
$136$	0	0
$137$	−19.7232	−1.68507	−0.842535	−	0.538642i	$-0.818937\pi$
$137$	−19.7232	−1.68507	−0.842535	+	0.538642i	$0.818937\pi$
$138$	0	0
$139$	−4.17554	−0.354165	−0.177082	−	0.984196i	$-0.556666\pi$
$139$	−4.17554	−0.354165	−0.177082	+	0.984196i	$0.556666\pi$
$140$	0	0
$141$	−31.2803	−2.63428
$142$	0	0
$143$	9.02871	0.755019
$144$	0	0
$145$	−0.615351	−0.0511021
$146$	0	0
$147$	18.2817	1.50785
$148$	0	0
$149$	−4.54125	−0.372034	−0.186017	−	0.982547i	$-0.559558\pi$
$149$	−4.54125	−0.372034	−0.186017	+	0.982547i	$0.559558\pi$
$150$	0	0
$151$	−12.1875	−0.991806	−0.495903	−	0.868378i	$-0.665163\pi$
$151$	−12.1875	−0.991806	−0.495903	+	0.868378i	$0.665163\pi$
$152$	0	0
$153$	−10.6512	−0.861102
$154$	0	0
$155$	−6.58620	−0.529016
$156$	0	0
$157$	−2.70886	−0.216191	−0.108095	−	0.994141i	$-0.534475\pi$
$157$	−2.70886	−0.216191	−0.108095	+	0.994141i	$0.534475\pi$
$158$	0	0
$159$	−14.4883	−1.14900
$160$	0	0
$161$	0.0366839	0.00289109
$162$	0	0
$163$	−22.1982	−1.73870	−0.869350	−	0.494198i	$-0.835462\pi$
$163$	−22.1982	−1.73870	−0.869350	+	0.494198i	$0.835462\pi$
$164$	0	0
$165$	−9.18696	−0.715204
$166$	0	0
$167$	−6.32953	−0.489794	−0.244897	−	0.969549i	$-0.578754\pi$
$167$	−6.32953	−0.489794	−0.244897	+	0.969549i	$0.578754\pi$
$168$	0	0
$169$	−6.86307	−0.527929
$170$	0	0
$171$	−3.82116	−0.292211
$172$	0	0
$173$	−0.507547	−0.0385881	−0.0192940	−	0.999814i	$-0.506142\pi$
$173$	−0.507547	−0.0385881	−0.0192940	+	0.999814i	$0.506142\pi$
$174$	0	0
$175$	−0.0515934	−0.00390009
$176$	0	0
$177$	−22.3113	−1.67702
$178$	0	0
$179$	21.8489	1.63306	0.816531	−	0.577302i	$-0.195895\pi$
$179$	21.8489	1.63306	0.816531	+	0.577302i	$0.195895\pi$
$180$	0	0
$181$	−6.86668	−0.510396	−0.255198	−	0.966889i	$-0.582141\pi$
$181$	−6.86668	−0.510396	−0.255198	+	0.966889i	$0.582141\pi$
$182$	0	0
$183$	35.5782	2.63001
$184$	0	0
$185$	−6.33371	−0.465664
$186$	0	0
$187$	−10.1591	−0.742907
$188$	0	0
$189$	−0.0271968	−0.00197827
$190$	0	0
$191$	−9.78189	−0.707793	−0.353896	−	0.935285i	$-0.615143\pi$
$191$	−9.78189	−0.707793	−0.353896	+	0.935285i	$0.615143\pi$
$192$	0	0
$193$	26.4740	1.90564	0.952820	−	0.303537i	$-0.0981676\pi$
$193$	26.4740	1.90564	0.952820	+	0.303537i	$0.0981676\pi$
$194$	0	0
$195$	−6.24449	−0.447177
$196$	0	0
$197$	11.5885	0.825645	0.412822	−	0.910812i	$-0.364543\pi$
$197$	11.5885	0.825645	0.412822	+	0.910812i	$0.364543\pi$
$198$	0	0
$199$	4.35312	0.308584	0.154292	−	0.988025i	$-0.450690\pi$
$199$	4.35312	0.308584	0.154292	+	0.988025i	$0.450690\pi$
$200$	0	0
$201$	26.0185	1.83520
$202$	0	0
$203$	−0.00808520	−0.000567470 0
$204$	0	0
$205$	0.257801	0.0180056
$206$	0	0
$207$	11.0538	0.768290
$208$	0	0
$209$	−3.64460	−0.252102
$210$	0	0
$211$	19.2331	1.32406	0.662030	−	0.749477i	$-0.269695\pi$
$211$	19.2331	1.32406	0.662030	+	0.749477i	$0.269695\pi$
$212$	0	0
$213$	−22.1577	−1.51822
$214$	0	0
$215$	−10.6339	−0.725225
$216$	0	0
$217$	−0.0865371	−0.00587452
$218$	0	0
$219$	−12.8738	−0.869930
$220$	0	0
$221$	−6.90526	−0.464498
$222$	0	0
$223$	0.713565	0.0477839	0.0238919	−	0.999715i	$-0.492394\pi$
$223$	0.713565	0.0477839	0.0238919	+	0.999715i	$0.492394\pi$
$224$	0	0
$225$	−15.5464	−1.03642
$226$	0	0
$227$	−16.3521	−1.08532	−0.542662	−	0.839951i	$-0.682584\pi$
$227$	−16.3521	−1.08532	−0.542662	+	0.839951i	$0.682584\pi$
$228$	0	0
$229$	2.48554	0.164249	0.0821244	−	0.996622i	$-0.473830\pi$
$229$	2.48554	0.164249	0.0821244	+	0.996622i	$0.473830\pi$
$230$	0	0
$231$	−0.120709	−0.00794207
$232$	0	0
$233$	−23.1420	−1.51608	−0.758040	−	0.652208i	$-0.773843\pi$
$233$	−23.1420	−1.51608	−0.758040	+	0.652208i	$0.773843\pi$
$234$	0	0
$235$	11.5594	0.754050
$236$	0	0
$237$	−2.61174	−0.169650
$238$	0	0
$239$	−19.4687	−1.25933	−0.629663	−	0.776868i	$-0.716807\pi$
$239$	−19.4687	−1.25933	−0.629663	+	0.776868i	$0.716807\pi$
$240$	0	0
$241$	−11.9631	−0.770611	−0.385305	−	0.922789i	$-0.625904\pi$
$241$	−11.9631	−0.770611	−0.385305	+	0.922789i	$0.625904\pi$
$242$	0	0
$243$	21.7445	1.39491
$244$	0	0
$245$	−6.75586	−0.431616
$246$	0	0
$247$	−2.47728	−0.157626
$248$	0	0
$249$	14.3376	0.908611
$250$	0	0
$251$	−27.1722	−1.71510	−0.857548	−	0.514404i	$-0.828013\pi$
$251$	−27.1722	−1.71510	−0.857548	+	0.514404i	$0.828013\pi$
$252$	0	0
$253$	10.5430	0.662834
$254$	0	0
$255$	7.02630	0.440004
$256$	0	0
$257$	−3.08750	−0.192593	−0.0962964	−	0.995353i	$-0.530700\pi$
$257$	−3.08750	−0.192593	−0.0962964	+	0.995353i	$0.530700\pi$
$258$	0	0
$259$	−0.0832196	−0.00517102
$260$	0	0
$261$	−2.43627	−0.150802
$262$	0	0
$263$	21.9863	1.35573	0.677867	−	0.735185i	$-0.262905\pi$
$263$	21.9863	1.35573	0.677867	+	0.735185i	$0.262905\pi$
$264$	0	0
$265$	5.35402	0.328895
$266$	0	0
$267$	24.1823	1.47993
$268$	0	0
$269$	−1.65693	−0.101025	−0.0505125	−	0.998723i	$-0.516085\pi$
$269$	−1.65693	−0.101025	−0.0505125	+	0.998723i	$0.516085\pi$
$270$	0	0
$271$	5.79072	0.351761	0.175881	−	0.984412i	$-0.443723\pi$
$271$	5.79072	0.351761	0.175881	+	0.984412i	$0.443723\pi$
$272$	0	0
$273$	−0.0820474	−0.00496573
$274$	0	0
$275$	−14.8280	−0.894165
$276$	0	0
$277$	22.4450	1.34859	0.674295	−	0.738462i	$-0.264447\pi$
$277$	22.4450	1.34859	0.674295	+	0.738462i	$0.264447\pi$
$278$	0	0
$279$	−26.0758	−1.56112
$280$	0	0
$281$	−11.2891	−0.673449	−0.336724	−	0.941603i	$-0.609319\pi$
$281$	−11.2891	−0.673449	−0.336724	+	0.941603i	$0.609319\pi$
$282$	0	0
$283$	13.8451	0.823004	0.411502	−	0.911409i	$-0.365004\pi$
$283$	13.8451	0.823004	0.411502	+	0.911409i	$0.365004\pi$
$284$	0	0
$285$	2.52070	0.149314
$286$	0	0
$287$	0.00338729	0.000199945 0
$288$	0	0
$289$	−9.23020	−0.542953
$290$	0	0
$291$	−6.18918	−0.362816
$292$	0	0
$293$	−20.0695	−1.17248	−0.586238	−	0.810139i	$-0.699392\pi$
$293$	−20.0695	−1.17248	−0.586238	+	0.810139i	$0.699392\pi$
$294$	0	0
$295$	8.24497	0.480040
$296$	0	0
$297$	−7.81641	−0.453554
$298$	0	0
$299$	7.16623	0.414434
$300$	0	0
$301$	−0.139720	−0.00805334
$302$	0	0
$303$	−16.9743	−0.975148
$304$	0	0
$305$	−13.1476	−0.752830
$306$	0	0
$307$	−13.8393	−0.789851	−0.394926	−	0.918713i	$-0.629230\pi$
$307$	−13.8393	−0.789851	−0.394926	+	0.918713i	$0.629230\pi$
$308$	0	0
$309$	−21.0345	−1.19661
$310$	0	0
$311$	−0.561246	−0.0318254	−0.0159127	−	0.999873i	$-0.505065\pi$
$311$	−0.561246	−0.0318254	−0.0159127	+	0.999873i	$0.505065\pi$
$312$	0	0
$313$	−31.0218	−1.75345	−0.876727	−	0.480988i	$-0.840278\pi$
$313$	−31.0218	−1.75345	−0.876727	+	0.480988i	$0.840278\pi$
$314$	0	0
$315$	0.0467679	0.00263507
$316$	0	0
$317$	−22.5219	−1.26496	−0.632478	−	0.774578i	$-0.717962\pi$
$317$	−22.5219	−1.26496	−0.632478	+	0.774578i	$0.717962\pi$
$318$	0	0
$319$	−2.32370	−0.130102
$320$	0	0
$321$	−11.0615	−0.617390
$322$	0	0
$323$	2.78744	0.155097
$324$	0	0
$325$	−10.0788	−0.559072
$326$	0	0
$327$	−11.3345	−0.626797
$328$	0	0
$329$	0.151880	0.00837344
$330$	0	0
$331$	−7.98297	−0.438784	−0.219392	−	0.975637i	$-0.570407\pi$
$331$	−7.98297	−0.438784	−0.219392	+	0.975637i	$0.570407\pi$
$332$	0	0
$333$	−25.0762	−1.37416
$334$	0	0
$335$	−9.61490	−0.525318
$336$	0	0
$337$	6.64015	0.361712	0.180856	−	0.983510i	$-0.442113\pi$
$337$	6.64015	0.361712	0.180856	+	0.983510i	$0.442113\pi$
$338$	0	0
$339$	−0.403541	−0.0219174
$340$	0	0
$341$	−24.8710	−1.34684
$342$	0	0
$343$	−0.177535	−0.00958597
$344$	0	0
$345$	−7.29183	−0.392579
$346$	0	0
$347$	16.7571	0.899567	0.449783	−	0.893138i	$-0.351501\pi$
$347$	16.7571	0.899567	0.449783	+	0.893138i	$0.351501\pi$
$348$	0	0
$349$	13.6181	0.728960	0.364480	−	0.931211i	$-0.381247\pi$
$349$	13.6181	0.728960	0.364480	+	0.931211i	$0.381247\pi$
$350$	0	0
$351$	−5.31291	−0.283582
$352$	0	0
$353$	−29.7004	−1.58079	−0.790397	−	0.612595i	$-0.790125\pi$
$353$	−29.7004	−1.58079	−0.790397	+	0.612595i	$0.790125\pi$
$354$	0	0
$355$	8.18818	0.434584
$356$	0	0
$357$	0.0923196	0.00488607
$358$	0	0
$359$	−16.2368	−0.856948	−0.428474	−	0.903554i	$-0.640949\pi$
$359$	−16.2368	−0.856948	−0.428474	+	0.903554i	$0.640949\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−5.96293	−0.312972
$364$	0	0
$365$	4.75740	0.249014
$366$	0	0
$367$	7.21872	0.376814	0.188407	−	0.982091i	$-0.439668\pi$
$367$	7.21872	0.376814	0.188407	+	0.982091i	$0.439668\pi$
$368$	0	0
$369$	1.02067	0.0531342
$370$	0	0
$371$	0.0703474	0.00365225
$372$	0	0
$373$	16.0283	0.829912	0.414956	−	0.909841i	$-0.363797\pi$
$373$	16.0283	0.829912	0.414956	+	0.909841i	$0.363797\pi$
$374$	0	0
$375$	22.8590	1.18043
$376$	0	0
$377$	−1.57945	−0.0813459
$378$	0	0
$379$	−26.3200	−1.35197	−0.675983	−	0.736917i	$-0.736281\pi$
$379$	−26.3200	−1.35197	−0.675983	+	0.736917i	$0.736281\pi$
$380$	0	0
$381$	−52.2121	−2.67491
$382$	0	0
$383$	1.76562	0.0902191	0.0451095	−	0.998982i	$-0.485636\pi$
$383$	1.76562	0.0902191	0.0451095	+	0.998982i	$0.485636\pi$
$384$	0	0
$385$	0.0446070	0.00227338
$386$	0	0
$387$	−42.1012	−2.14012
$388$	0	0
$389$	6.23371	0.316062	0.158031	−	0.987434i	$-0.449485\pi$
$389$	6.23371	0.316062	0.158031	+	0.987434i	$0.449485\pi$
$390$	0	0
$391$	−8.06343	−0.407785
$392$	0	0
$393$	−6.90462	−0.348292
$394$	0	0
$395$	0.965145	0.0485617
$396$	0	0
$397$	−27.0534	−1.35777	−0.678886	−	0.734244i	$-0.737537\pi$
$397$	−27.0534	−1.35777	−0.678886	+	0.734244i	$0.737537\pi$
$398$	0	0
$399$	0.0331199	0.00165807
$400$	0	0
$401$	6.78156	0.338655	0.169328	−	0.985560i	$-0.445840\pi$
$401$	6.78156	0.338655	0.169328	+	0.985560i	$0.445840\pi$
$402$	0	0
$403$	−16.9051	−0.842103
$404$	0	0
$405$	−5.65789	−0.281143
$406$	0	0
$407$	−23.9175	−1.18555
$408$	0	0
$409$	−21.7452	−1.07523	−0.537614	−	0.843191i	$-0.680674\pi$
$409$	−21.7452	−1.07523	−0.537614	+	0.843191i	$0.680674\pi$
$410$	0	0
$411$	51.5119	2.54089
$412$	0	0
$413$	0.108332	0.00533067
$414$	0	0
$415$	−5.29835	−0.260086
$416$	0	0
$417$	10.9054	0.534040
$418$	0	0
$419$	−2.28177	−0.111472	−0.0557359	−	0.998446i	$-0.517750\pi$
$419$	−2.28177	−0.111472	−0.0557359	+	0.998446i	$0.517750\pi$
$420$	0	0
$421$	−21.1795	−1.03222	−0.516112	−	0.856521i	$-0.672621\pi$
$421$	−21.1795	−1.03222	−0.516112	+	0.856521i	$0.672621\pi$
$422$	0	0
$423$	45.7654	2.22519
$424$	0	0
$425$	11.3407	0.550103
$426$	0	0
$427$	−0.172749	−0.00835989
$428$	0	0
$429$	−23.5806	−1.13848
$430$	0	0
$431$	−28.9419	−1.39408	−0.697040	−	0.717032i	$-0.745500\pi$
$431$	−28.9419	−1.39408	−0.697040	+	0.717032i	$0.745500\pi$
$432$	0	0
$433$	21.0202	1.01017	0.505083	−	0.863071i	$-0.331462\pi$
$433$	21.0202	1.01017	0.505083	+	0.863071i	$0.331462\pi$
$434$	0	0
$435$	1.60713	0.0770562
$436$	0	0
$437$	−2.89278	−0.138380
$438$	0	0
$439$	−26.3698	−1.25856	−0.629281	−	0.777178i	$-0.716651\pi$
$439$	−26.3698	−1.25856	−0.629281	+	0.777178i	$0.716651\pi$
$440$	0	0
$441$	−26.7475	−1.27369
$442$	0	0
$443$	41.4470	1.96921	0.984603	−	0.174808i	$-0.0559304\pi$
$443$	41.4470	1.96921	0.984603	+	0.174808i	$0.0559304\pi$
$444$	0	0
$445$	−8.93636	−0.423624
$446$	0	0
$447$	11.8606	0.560985
$448$	0	0
$449$	−7.31062	−0.345009	−0.172505	−	0.985009i	$-0.555186\pi$
$449$	−7.31062	−0.345009	−0.172505	+	0.985009i	$0.555186\pi$
$450$	0	0
$451$	0.973513	0.0458410
$452$	0	0
$453$	31.8306	1.49553
$454$	0	0
$455$	0.0303199	0.00142142
$456$	0	0
$457$	−27.7707	−1.29906	−0.649528	−	0.760337i	$-0.725034\pi$
$457$	−27.7707	−1.29906	−0.649528	+	0.760337i	$0.725034\pi$
$458$	0	0
$459$	5.97808	0.279033
$460$	0	0
$461$	34.0547	1.58608	0.793042	−	0.609167i	$-0.208496\pi$
$461$	34.0547	1.58608	0.793042	+	0.609167i	$0.208496\pi$
$462$	0	0
$463$	−41.1144	−1.91075	−0.955375	−	0.295396i	$-0.904548\pi$
$463$	−41.1144	−1.91075	−0.955375	+	0.295396i	$0.904548\pi$
$464$	0	0
$465$	17.2014	0.797696
$466$	0	0
$467$	14.3162	0.662477	0.331238	−	0.943547i	$-0.392534\pi$
$467$	14.3162	0.662477	0.331238	+	0.943547i	$0.392534\pi$
$468$	0	0
$469$	−0.126332	−0.00583346
$470$	0	0
$471$	7.07483	0.325991
$472$	0	0
$473$	−40.1559	−1.84637
$474$	0	0
$475$	4.06850	0.186675
$476$	0	0
$477$	21.1974	0.970563
$478$	0	0
$479$	17.0022	0.776850	0.388425	−	0.921480i	$-0.373019\pi$
$479$	17.0022	0.776850	0.388425	+	0.921480i	$0.373019\pi$
$480$	0	0
$481$	−16.2570	−0.741257
$482$	0	0
$483$	−0.0958086	−0.00435944
$484$	0	0
$485$	2.28716	0.103855
$486$	0	0
$487$	20.4562	0.926960	0.463480	−	0.886108i	$-0.346601\pi$
$487$	20.4562	0.926960	0.463480	+	0.886108i	$0.346601\pi$
$488$	0	0
$489$	57.9759	2.62176
$490$	0	0
$491$	−18.1043	−0.817036	−0.408518	−	0.912750i	$-0.633954\pi$
$491$	−18.1043	−0.817036	−0.408518	+	0.912750i	$0.633954\pi$
$492$	0	0
$493$	1.77720	0.0800409
$494$	0	0
$495$	13.4412	0.604137
$496$	0	0
$497$	0.107586	0.00482589
$498$	0	0
$499$	−28.8607	−1.29198	−0.645990	−	0.763346i	$-0.723555\pi$
$499$	−28.8607	−1.29198	−0.645990	+	0.763346i	$0.723555\pi$
$500$	0	0
$501$	16.5311	0.738553
$502$	0	0
$503$	1.62172	0.0723090	0.0361545	−	0.999346i	$-0.488489\pi$
$503$	1.62172	0.0723090	0.0361545	+	0.999346i	$0.488489\pi$
$504$	0	0
$505$	6.27271	0.279132
$506$	0	0
$507$	17.9245	0.796056
$508$	0	0
$509$	−2.54370	−0.112748	−0.0563738	−	0.998410i	$-0.517954\pi$
$509$	−2.54370	−0.112748	−0.0563738	+	0.998410i	$0.517954\pi$
$510$	0	0
$511$	0.0625083	0.00276520
$512$	0	0
$513$	2.14465	0.0946887
$514$	0	0
$515$	7.77312	0.342524
$516$	0	0
$517$	43.6508	1.91976
$518$	0	0
$519$	1.32558	0.0581864
$520$	0	0
$521$	16.2470	0.711795	0.355897	−	0.934525i	$-0.384175\pi$
$521$	16.2470	0.711795	0.355897	+	0.934525i	$0.384175\pi$
$522$	0	0
$523$	−22.5773	−0.987235	−0.493618	−	0.869679i	$-0.664326\pi$
$523$	−22.5773	−0.987235	−0.493618	+	0.869679i	$0.664326\pi$
$524$	0	0
$525$	0.134748	0.00588090
$526$	0	0
$527$	19.0216	0.828594
$528$	0	0
$529$	−14.6318	−0.636167
$530$	0	0
$531$	32.6431	1.41659
$532$	0	0
$533$	0.661709	0.0286618
$534$	0	0
$535$	4.08767	0.176725
$536$	0	0
$537$	−57.0635	−2.46247
$538$	0	0
$539$	−25.5116	−1.09886
$540$	0	0
$541$	−33.7278	−1.45007	−0.725035	−	0.688712i	$-0.758177\pi$
$541$	−33.7278	−1.45007	−0.725035	+	0.688712i	$0.758177\pi$
$542$	0	0
$543$	17.9339	0.769619
$544$	0	0
$545$	4.18856	0.179418
$546$	0	0
$547$	−17.9848	−0.768976	−0.384488	−	0.923130i	$-0.625622\pi$
$547$	−17.9848	−0.768976	−0.384488	+	0.923130i	$0.625622\pi$
$548$	0	0
$549$	−52.0535	−2.22159
$550$	0	0
$551$	0.637574	0.0271616
$552$	0	0
$553$	0.0126812	0.000539259 0
$554$	0	0
$555$	16.5420	0.702168
$556$	0	0
$557$	17.7834	0.753505	0.376753	−	0.926314i	$-0.377041\pi$
$557$	17.7834	0.753505	0.376753	+	0.926314i	$0.377041\pi$
$558$	0	0
$559$	−27.2945	−1.15443
$560$	0	0
$561$	26.5329	1.12022
$562$	0	0
$563$	4.33675	0.182772	0.0913862	−	0.995816i	$-0.470870\pi$
$563$	4.33675	0.182772	0.0913862	+	0.995816i	$0.470870\pi$
$564$	0	0
$565$	0.149125	0.00627375
$566$	0	0
$567$	−0.0743399	−0.00312198
$568$	0	0
$569$	23.6534	0.991603	0.495802	−	0.868436i	$-0.334874\pi$
$569$	23.6534	0.991603	0.495802	+	0.868436i	$0.334874\pi$
$570$	0	0
$571$	15.1019	0.631996	0.315998	−	0.948760i	$-0.397661\pi$
$571$	15.1019	0.631996	0.315998	+	0.948760i	$0.397661\pi$
$572$	0	0
$573$	25.5477	1.06727
$574$	0	0
$575$	−11.7693	−0.490812
$576$	0	0
$577$	29.0319	1.20861	0.604307	−	0.796752i	$-0.293450\pi$
$577$	29.0319	1.20861	0.604307	+	0.796752i	$0.293450\pi$
$578$	0	0
$579$	−69.1430	−2.87349
$580$	0	0
$581$	−0.0696159	−0.00288815
$582$	0	0
$583$	20.2180	0.837344
$584$	0	0
$585$	9.13615	0.377733
$586$	0	0
$587$	29.0738	1.20000	0.600002	−	0.799998i	$-0.295166\pi$
$587$	29.0738	1.20000	0.600002	+	0.799998i	$0.295166\pi$
$588$	0	0
$589$	6.82405	0.281180
$590$	0	0
$591$	−30.2660	−1.24498
$592$	0	0
$593$	23.9146	0.982054	0.491027	−	0.871144i	$-0.336622\pi$
$593$	23.9146	0.982054	0.491027	+	0.871144i	$0.336622\pi$
$594$	0	0
$595$	−0.0341160	−0.00139862
$596$	0	0
$597$	−11.3692	−0.465310
$598$	0	0
$599$	−14.4958	−0.592282	−0.296141	−	0.955144i	$-0.595700\pi$
$599$	−14.4958	−0.592282	−0.296141	+	0.955144i	$0.595700\pi$
$600$	0	0
$601$	−23.9215	−0.975778	−0.487889	−	0.872906i	$-0.662233\pi$
$601$	−23.9215	−0.975778	−0.487889	+	0.872906i	$0.662233\pi$
$602$	0	0
$603$	−38.0669	−1.55020
$604$	0	0
$605$	2.20355	0.0895870
$606$	0	0
$607$	36.4091	1.47780	0.738899	−	0.673816i	$-0.235346\pi$
$607$	36.4091	1.47780	0.738899	+	0.673816i	$0.235346\pi$
$608$	0	0
$609$	0.0211164	0.000855680 0
$610$	0	0
$611$	29.6700	1.20032
$612$	0	0
$613$	−12.5782	−0.508030	−0.254015	−	0.967200i	$-0.581751\pi$
$613$	−12.5782	−0.508030	−0.254015	+	0.967200i	$0.581751\pi$
$614$	0	0
$615$	−0.673307	−0.0271504
$616$	0	0
$617$	−47.8597	−1.92676	−0.963379	−	0.268142i	$-0.913590\pi$
$617$	−47.8597	−1.92676	−0.963379	+	0.268142i	$0.913590\pi$
$618$	0	0
$619$	7.29187	0.293085	0.146543	−	0.989204i	$-0.453185\pi$
$619$	7.29187	0.293085	0.146543	+	0.989204i	$0.453185\pi$
$620$	0	0
$621$	−6.20400	−0.248958
$622$	0	0
$623$	−0.117416	−0.00470418
$624$	0	0
$625$	11.8951	0.475805
$626$	0	0
$627$	9.51874	0.380142
$628$	0	0
$629$	18.2924	0.729366
$630$	0	0
$631$	−34.3039	−1.36562	−0.682808	−	0.730598i	$-0.739241\pi$
$631$	−34.3039	−1.36562	−0.682808	+	0.730598i	$0.739241\pi$
$632$	0	0
$633$	−50.2317	−1.99653
$634$	0	0
$635$	19.2945	0.765680
$636$	0	0
$637$	−17.3406	−0.687059
$638$	0	0
$639$	32.4183	1.28245
$640$	0	0
$641$	−25.0871	−0.990880	−0.495440	−	0.868642i	$-0.664993\pi$
$641$	−25.0871	−0.990880	−0.495440	+	0.868642i	$0.664993\pi$
$642$	0	0
$643$	−0.0448022	−0.00176683	−0.000883413	−	1.00000i	$-0.500281\pi$
$643$	−0.0448022	−0.00176683	−0.000883413		1.00000i	$0.500281\pi$
$644$	0	0
$645$	27.7729	1.09356
$646$	0	0
$647$	−0.368668	−0.0144938	−0.00724691	−	0.999974i	$-0.502307\pi$
$647$	−0.368668	−0.0144938	−0.00724691	+	0.999974i	$0.502307\pi$
$648$	0	0
$649$	31.1348	1.22215
$650$	0	0
$651$	0.226012	0.00885811
$652$	0	0
$653$	−34.2252	−1.33934	−0.669668	−	0.742661i	$-0.733563\pi$
$653$	−34.2252	−1.33934	−0.669668	+	0.742661i	$0.733563\pi$
$654$	0	0
$655$	2.55155	0.0996971
$656$	0	0
$657$	18.8353	0.734835
$658$	0	0
$659$	−21.8883	−0.852648	−0.426324	−	0.904570i	$-0.640192\pi$
$659$	−21.8883	−0.852648	−0.426324	+	0.904570i	$0.640192\pi$
$660$	0	0
$661$	−7.13604	−0.277560	−0.138780	−	0.990323i	$-0.544318\pi$
$661$	−7.13604	−0.277560	−0.138780	+	0.990323i	$0.544318\pi$
$662$	0	0
$663$	18.0347	0.700411
$664$	0	0
$665$	−0.0122392	−0.000474615 0
$666$	0	0
$667$	−1.84436	−0.0714139
$668$	0	0
$669$	−1.86364	−0.0720526
$670$	0	0
$671$	−49.6483	−1.91665
$672$	0	0
$673$	−8.66192	−0.333893	−0.166946	−	0.985966i	$-0.553391\pi$
$673$	−8.66192	−0.333893	−0.166946	+	0.985966i	$0.553391\pi$
$674$	0	0
$675$	8.72551	0.335845
$676$	0	0
$677$	26.4183	1.01534	0.507670	−	0.861552i	$-0.330507\pi$
$677$	26.4183	1.01534	0.507670	+	0.861552i	$0.330507\pi$
$678$	0	0
$679$	0.0300514	0.00115327
$680$	0	0
$681$	42.7073	1.63655
$682$	0	0
$683$	1.98710	0.0760345	0.0380172	−	0.999277i	$-0.487896\pi$
$683$	1.98710	0.0760345	0.0380172	+	0.999277i	$0.487896\pi$
$684$	0	0
$685$	−19.0358	−0.727319
$686$	0	0
$687$	−6.49156	−0.247668
$688$	0	0
$689$	13.7424	0.523545
$690$	0	0
$691$	42.3060	1.60940	0.804698	−	0.593684i	$-0.202327\pi$
$691$	42.3060	1.60940	0.804698	+	0.593684i	$0.202327\pi$
$692$	0	0
$693$	0.176606	0.00670871
$694$	0	0
$695$	−4.03000	−0.152867
$696$	0	0
$697$	−0.744555	−0.0282020
$698$	0	0
$699$	60.4407	2.28608
$700$	0	0
$701$	−52.6138	−1.98720	−0.993599	−	0.112968i	$-0.963964\pi$
$701$	−52.6138	−1.98720	−0.993599	+	0.112968i	$0.963964\pi$
$702$	0	0
$703$	6.56245	0.247507
$704$	0	0
$705$	−30.1900	−1.13702
$706$	0	0
$707$	0.0824182	0.00309965
$708$	0	0
$709$	17.7848	0.667924	0.333962	−	0.942587i	$-0.391614\pi$
$709$	17.7848	0.667924	0.333962	+	0.942587i	$0.391614\pi$
$710$	0	0
$711$	3.82116	0.143305
$712$	0	0
$713$	−19.7405	−0.739286
$714$	0	0
$715$	8.71401	0.325886
$716$	0	0
$717$	50.8471	1.89892
$718$	0	0
$719$	11.6890	0.435927	0.217963	−	0.975957i	$-0.430059\pi$
$719$	11.6890	0.435927	0.217963	+	0.975957i	$0.430059\pi$
$720$	0	0
$721$	0.102132	0.00380360
$722$	0	0
$723$	31.2444	1.16199
$724$	0	0
$725$	2.59397	0.0963375
$726$	0	0
$727$	0.876570	0.0325102	0.0162551	−	0.999868i	$-0.494826\pi$
$727$	0.876570	0.0325102	0.0162551	+	0.999868i	$0.494826\pi$
$728$	0	0
$729$	−39.2043	−1.45201
$730$	0	0
$731$	30.7117	1.13591
$732$	0	0
$733$	17.5317	0.647548	0.323774	−	0.946134i	$-0.395048\pi$
$733$	17.5317	0.647548	0.323774	+	0.946134i	$0.395048\pi$
$734$	0	0
$735$	17.6445	0.650828
$736$	0	0
$737$	−36.3080	−1.33742
$738$	0	0
$739$	−24.0215	−0.883645	−0.441822	−	0.897103i	$-0.645668\pi$
$739$	−24.0215	−0.883645	−0.441822	+	0.897103i	$0.645668\pi$
$740$	0	0
$741$	6.47000	0.237682
$742$	0	0
$743$	−8.79370	−0.322609	−0.161305	−	0.986905i	$-0.551570\pi$
$743$	−8.79370	−0.322609	−0.161305	+	0.986905i	$0.551570\pi$
$744$	0	0
$745$	−4.38297	−0.160580
$746$	0	0
$747$	−20.9770	−0.767509
$748$	0	0
$749$	0.0537085	0.00196247
$750$	0	0
$751$	−42.3875	−1.54674	−0.773371	−	0.633954i	$-0.781431\pi$
$751$	−42.3875	−1.54674	−0.773371	+	0.633954i	$0.781431\pi$
$752$	0	0
$753$	70.9667	2.58617
$754$	0	0
$755$	−11.7627	−0.428089
$756$	0	0
$757$	39.5536	1.43760	0.718799	−	0.695217i	$-0.244692\pi$
$757$	39.5536	1.43760	0.718799	+	0.695217i	$0.244692\pi$
$758$	0	0
$759$	−27.5356	−0.999478
$760$	0	0
$761$	−14.4423	−0.523534	−0.261767	−	0.965131i	$-0.584305\pi$
$761$	−14.4423	−0.523534	−0.261767	+	0.965131i	$0.584305\pi$
$762$	0	0
$763$	0.0550341	0.00199237
$764$	0	0
$765$	−10.2800	−0.371674
$766$	0	0
$767$	21.1627	0.764142
$768$	0	0
$769$	21.7097	0.782873	0.391436	−	0.920205i	$-0.371978\pi$
$769$	21.7097	0.782873	0.391436	+	0.920205i	$0.371978\pi$
$770$	0	0
$771$	8.06372	0.290408
$772$	0	0
$773$	27.0906	0.974380	0.487190	−	0.873296i	$-0.338022\pi$
$773$	27.0906	0.974380	0.487190	+	0.873296i	$0.338022\pi$
$774$	0	0
$775$	27.7636	0.997299
$776$	0	0
$777$	0.217348	0.00779730
$778$	0	0
$779$	−0.267111	−0.00957025
$780$	0	0
$781$	30.9204	1.10642
$782$	0	0
$783$	1.36738	0.0488660
$784$	0	0
$785$	−2.61445	−0.0933136
$786$	0	0
$787$	−5.16875	−0.184246	−0.0921231	−	0.995748i	$-0.529365\pi$
$787$	−5.16875	−0.184246	−0.0921231	+	0.995748i	$0.529365\pi$
$788$	0	0
$789$	−57.4224	−2.04429
$790$	0	0
$791$	0.00195938	6.96676e−5 0
$792$	0	0
$793$	−33.7466	−1.19838
$794$	0	0
$795$	−13.9833	−0.495936
$796$	0	0
$797$	−0.853409	−0.0302293	−0.0151146	−	0.999886i	$-0.504811\pi$
$797$	−0.853409	−0.0302293	−0.0151146	+	0.999886i	$0.504811\pi$
$798$	0	0
$799$	−33.3846	−1.18106
$800$	0	0
$801$	−35.3805	−1.25011
$802$	0	0
$803$	17.9650	0.633971
$804$	0	0
$805$	0.0354052	0.00124787
$806$	0	0
$807$	4.32747	0.152334
$808$	0	0
$809$	−38.6948	−1.36044	−0.680218	−	0.733010i	$-0.738115\pi$
$809$	−38.6948	−1.36044	−0.680218	+	0.733010i	$0.738115\pi$
$810$	0	0
$811$	39.1162	1.37355	0.686777	−	0.726868i	$-0.259025\pi$
$811$	39.1162	1.37355	0.686777	+	0.726868i	$0.259025\pi$
$812$	0	0
$813$	−15.1238	−0.530416
$814$	0	0
$815$	−21.4245	−0.750468
$816$	0	0
$817$	11.0179	0.385468
$818$	0	0
$819$	0.120041	0.00419458
$820$	0	0
$821$	−14.8969	−0.519906	−0.259953	−	0.965621i	$-0.583707\pi$
$821$	−14.8969	−0.519906	−0.259953	+	0.965621i	$0.583707\pi$
$822$	0	0
$823$	−31.4552	−1.09646	−0.548229	−	0.836328i	$-0.684698\pi$
$823$	−31.4552	−1.09646	−0.548229	+	0.836328i	$0.684698\pi$
$824$	0	0
$825$	38.7269	1.34830
$826$	0	0
$827$	33.9854	1.18179	0.590893	−	0.806750i	$-0.298775\pi$
$827$	33.9854	1.18179	0.590893	+	0.806750i	$0.298775\pi$
$828$	0	0
$829$	34.4904	1.19790	0.598950	−	0.800786i	$-0.295585\pi$
$829$	34.4904	1.19790	0.598950	+	0.800786i	$0.295585\pi$
$830$	0	0
$831$	−58.6205	−2.03352
$832$	0	0
$833$	19.5116	0.676037
$834$	0	0
$835$	−6.10891	−0.211408
$836$	0	0
$837$	14.6352	0.505867
$838$	0	0
$839$	−1.25699	−0.0433961	−0.0216980	−	0.999765i	$-0.506907\pi$
$839$	−1.25699	−0.0433961	−0.0216980	+	0.999765i	$0.506907\pi$
$840$	0	0
$841$	−28.5935	−0.985983
$842$	0	0
$843$	29.4840	1.01548
$844$	0	0
$845$	−6.62386	−0.227868
$846$	0	0
$847$	0.0289528	0.000994830 0
$848$	0	0
$849$	−36.1596	−1.24100
$850$	0	0
$851$	−18.9837	−0.650753
$852$	0	0
$853$	1.17484	0.0402259	0.0201129	−	0.999798i	$-0.493597\pi$
$853$	1.17484	0.0402259	0.0201129	+	0.999798i	$0.493597\pi$
$854$	0	0
$855$	−3.68797	−0.126126
$856$	0	0
$857$	−52.7079	−1.80047	−0.900234	−	0.435406i	$-0.856605\pi$
$857$	−52.7079	−1.80047	−0.900234	+	0.435406i	$0.856605\pi$
$858$	0	0
$859$	31.1074	1.06137	0.530686	−	0.847569i	$-0.321935\pi$
$859$	31.1074	1.06137	0.530686	+	0.847569i	$0.321935\pi$
$860$	0	0
$861$	−0.00884669	−0.000301495 0
$862$	0	0
$863$	−1.66531	−0.0566877	−0.0283439	−	0.999598i	$-0.509023\pi$
$863$	−1.66531	−0.0566877	−0.0283439	+	0.999598i	$0.509023\pi$
$864$	0	0
$865$	−0.489856	−0.0166556
$866$	0	0
$867$	24.1068	0.818711
$868$	0	0
$869$	3.64460	0.123635
$870$	0	0
$871$	−24.6790	−0.836217
$872$	0	0
$873$	9.05523	0.306473
$874$	0	0
$875$	−0.110991	−0.00375218
$876$	0	0
$877$	−32.7370	−1.10545	−0.552726	−	0.833363i	$-0.686412\pi$
$877$	−32.7370	−1.10545	−0.552726	+	0.833363i	$0.686412\pi$
$878$	0	0
$879$	52.4163	1.76796
$880$	0	0
$881$	29.8735	1.00646	0.503231	−	0.864152i	$-0.332144\pi$
$881$	29.8735	1.00646	0.503231	+	0.864152i	$0.332144\pi$
$882$	0	0
$883$	0.739233	0.0248772	0.0124386	−	0.999923i	$-0.496041\pi$
$883$	0.739233	0.0248772	0.0124386	+	0.999923i	$0.496041\pi$
$884$	0	0
$885$	−21.5337	−0.723846
$886$	0	0
$887$	−49.3156	−1.65586	−0.827928	−	0.560835i	$-0.810480\pi$
$887$	−49.3156	−1.65586	−0.827928	+	0.560835i	$0.810480\pi$
$888$	0	0
$889$	0.253514	0.00850259
$890$	0	0
$891$	−21.3654	−0.715769
$892$	0	0
$893$	−11.9768	−0.400789
$894$	0	0
$895$	21.0873	0.704871
$896$	0	0
$897$	−18.7163	−0.624918
$898$	0	0
$899$	4.35084	0.145109
$900$	0	0
$901$	−15.4630	−0.515146
$902$	0	0
$903$	0.364912	0.0121435
$904$	0	0
$905$	−6.62734	−0.220300
$906$	0	0
$907$	−1.57222	−0.0522047	−0.0261023	−	0.999659i	$-0.508310\pi$
$907$	−1.57222	−0.0522047	−0.0261023	+	0.999659i	$0.508310\pi$
$908$	0	0
$909$	24.8346	0.823713
$910$	0	0
$911$	27.5014	0.911161	0.455581	−	0.890194i	$-0.349432\pi$
$911$	27.5014	0.911161	0.455581	+	0.890194i	$0.349432\pi$
$912$	0	0
$913$	−20.0078	−0.662160
$914$	0	0
$915$	34.3381	1.13518
$916$	0	0
$917$	0.0335252	0.00110710
$918$	0	0
$919$	2.87030	0.0946823	0.0473412	−	0.998879i	$-0.484925\pi$
$919$	2.87030	0.0946823	0.0473412	+	0.998879i	$0.484925\pi$
$920$	0	0
$921$	36.1446	1.19101
$922$	0	0
$923$	21.0170	0.691783
$924$	0	0
$925$	26.6993	0.877867
$926$	0	0
$927$	30.7750	1.01078
$928$	0	0
$929$	34.5862	1.13474	0.567368	−	0.823464i	$-0.307962\pi$
$929$	34.5862	1.13474	0.567368	+	0.823464i	$0.307962\pi$
$930$	0	0
$931$	6.99984	0.229410
$932$	0	0
$933$	1.46583	0.0479890
$934$	0	0
$935$	−9.80500	−0.320658
$936$	0	0
$937$	−38.7454	−1.26576	−0.632879	−	0.774251i	$-0.718127\pi$
$937$	−38.7454	−1.26576	−0.632879	+	0.774251i	$0.718127\pi$
$938$	0	0
$939$	81.0206	2.64401
$940$	0	0
$941$	−6.73269	−0.219479	−0.109740	−	0.993960i	$-0.535002\pi$
$941$	−6.73269	−0.219479	−0.109740	+	0.993960i	$0.535002\pi$
$942$	0	0
$943$	0.772693	0.0251623
$944$	0	0
$945$	−0.0262488	−0.000853874 0
$946$	0	0
$947$	32.2599	1.04831	0.524154	−	0.851624i	$-0.324382\pi$
$947$	32.2599	1.04831	0.524154	+	0.851624i	$0.324382\pi$
$948$	0	0
$949$	12.2110	0.396387
$950$	0	0
$951$	58.8212	1.90741
$952$	0	0
$953$	39.6989	1.28597	0.642987	−	0.765877i	$-0.277695\pi$
$953$	39.6989	1.28597	0.642987	+	0.765877i	$0.277695\pi$
$954$	0	0
$955$	−9.44094	−0.305502
$956$	0	0
$957$	6.06890	0.196180
$958$	0	0
$959$	−0.250114	−0.00807661
$960$	0	0
$961$	15.5677	0.502183
$962$	0	0
$963$	16.1837	0.521513
$964$	0	0
$965$	25.5512	0.822523
$966$	0	0
$967$	−41.4916	−1.33428	−0.667140	−	0.744932i	$-0.732482\pi$
$967$	−41.4916	−1.33428	−0.667140	+	0.744932i	$0.732482\pi$
$968$	0	0
$969$	−7.28004	−0.233869
$970$	0	0
$971$	−23.0110	−0.738458	−0.369229	−	0.929338i	$-0.620378\pi$
$971$	−23.0110	−0.738458	−0.369229	+	0.929338i	$0.620378\pi$
$972$	0	0
$973$	−0.0529508	−0.00169753
$974$	0	0
$975$	26.3232	0.843017
$976$	0	0
$977$	−30.6073	−0.979213	−0.489606	−	0.871944i	$-0.662860\pi$
$977$	−30.6073	−0.979213	−0.489606	+	0.871944i	$0.662860\pi$
$978$	0	0
$979$	−33.7457	−1.07852
$980$	0	0
$981$	16.5832	0.529460
$982$	0	0
$983$	1.27415	0.0406391	0.0203196	−	0.999794i	$-0.493532\pi$
$983$	1.27415	0.0406391	0.0203196	+	0.999794i	$0.493532\pi$
$984$	0	0
$985$	11.1846	0.356370
$986$	0	0
$987$	−0.396672	−0.0126262
$988$	0	0
$989$	−31.8724	−1.01348
$990$	0	0
$991$	56.0104	1.77923	0.889614	−	0.456713i	$-0.150973\pi$
$991$	56.0104	1.77923	0.889614	+	0.456713i	$0.150973\pi$
$992$	0	0
$993$	20.8494	0.661636
$994$	0	0
$995$	4.20139	0.133193
$996$	0	0
$997$	31.1122	0.985332	0.492666	−	0.870218i	$-0.336022\pi$
$997$	31.1122	0.985332	0.492666	+	0.870218i	$0.336022\pi$
$998$	0	0
$999$	14.0742	0.445287

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6004.2.a.f.1.2	✓	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6004.2.a.f.1.2	✓	25	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)

\(f(q)\)	\(=\)	\(q-2.61174 q^{3} +0.965145 q^{5} +0.0126812 q^{7} +3.82116 q^{9} +O(q^{10})\)	\(q-2.61174 q^{3} +0.965145 q^{5} +0.0126812 q^{7} +3.82116 q^{9} +3.64460 q^{11} +2.47728 q^{13} -2.52070 q^{15} -2.78744 q^{17} -1.00000 q^{19} -0.0331199 q^{21} +2.89278 q^{23} -4.06850 q^{25} -2.14465 q^{27} -0.637574 q^{29} -6.82405 q^{31} -9.51874 q^{33} +0.0122392 q^{35} -6.56245 q^{37} -6.47000 q^{39} +0.267111 q^{41} -11.0179 q^{43} +3.68797 q^{45} +11.9768 q^{47} -6.99984 q^{49} +7.28004 q^{51} +5.54738 q^{53} +3.51757 q^{55} +2.61174 q^{57} +8.54272 q^{59} -13.6224 q^{61} +0.0484569 q^{63} +2.39094 q^{65} -9.96214 q^{67} -7.55517 q^{69} +8.48389 q^{71} +4.92921 q^{73} +10.6258 q^{75} +0.0462179 q^{77} +1.00000 q^{79} -5.86222 q^{81} -5.48970 q^{83} -2.69028 q^{85} +1.66517 q^{87} -9.25909 q^{89} +0.0314149 q^{91} +17.8226 q^{93} -0.965145 q^{95} +2.36976 q^{97} +13.9266 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q + 4 q^{3} - 8 q^{5} + 2 q^{7} + 13 q^{9}+O(q^{10}) \) 25 * q + 4 * q^3 - 8 * q^5 + 2 * q^7 + 13 * q^9	\( 25 q + 4 q^{3} - 8 q^{5} + 2 q^{7} + 13 q^{9} - 3 q^{11} + q^{13} - 5 q^{15} - 13 q^{17} - 25 q^{19} - 24 q^{21} - 31 q^{23} + 21 q^{25} + 7 q^{27} - 19 q^{29} - 7 q^{31} - 30 q^{33} - q^{35} - 29 q^{37} - 26 q^{39} - 40 q^{41} - 40 q^{45} - 8 q^{47} - 9 q^{49} + 12 q^{51} - 38 q^{53} - 29 q^{55} - 4 q^{57} + 18 q^{59} - 26 q^{61} - 40 q^{63} - 70 q^{65} - 13 q^{67} + q^{69} - 47 q^{71} - 8 q^{73} + 7 q^{75} - 19 q^{77} + 25 q^{79} - 19 q^{81} - 8 q^{83} - 33 q^{85} - 50 q^{87} - 54 q^{89} - 12 q^{91} - 24 q^{93} + 8 q^{95} - 4 q^{97} - 39 q^{99}+O(q^{100}) \) 25 * q + 4 * q^3 - 8 * q^5 + 2 * q^7 + 13 * q^9 - 3 * q^11 + q^13 - 5 * q^15 - 13 * q^17 - 25 * q^19 - 24 * q^21 - 31 * q^23 + 21 * q^25 + 7 * q^27 - 19 * q^29 - 7 * q^31 - 30 * q^33 - q^35 - 29 * q^37 - 26 * q^39 - 40 * q^41 - 40 * q^45 - 8 * q^47 - 9 * q^49 + 12 * q^51 - 38 * q^53 - 29 * q^55 - 4 * q^57 + 18 * q^59 - 26 * q^61 - 40 * q^63 - 70 * q^65 - 13 * q^67 + q^69 - 47 * q^71 - 8 * q^73 + 7 * q^75 - 19 * q^77 + 25 * q^79 - 19 * q^81 - 8 * q^83 - 33 * q^85 - 50 * q^87 - 54 * q^89 - 12 * q^91 - 24 * q^93 + 8 * q^95 - 4 * q^97 - 39 * q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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