LMFDB - Embedded newform 6004.2.a.g.1.16

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:	$27$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.16
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)

$f(q)$	$=$	$q+0.314419 q^{3} -4.24837 q^{5} +1.07745 q^{7} -2.90114 q^{9} +O(q^{10})$	$q+0.314419 q^{3} -4.24837 q^{5} +1.07745 q^{7} -2.90114 q^{9} +3.34805 q^{11} -1.31870 q^{13} -1.33577 q^{15} -2.75429 q^{17} +1.00000 q^{19} +0.338771 q^{21} +9.53559 q^{23} +13.0487 q^{25} -1.85543 q^{27} -8.28473 q^{29} +3.02324 q^{31} +1.05269 q^{33} -4.57741 q^{35} +4.48446 q^{37} -0.414625 q^{39} -2.36401 q^{41} -10.5342 q^{43} +12.3251 q^{45} -3.65470 q^{47} -5.83910 q^{49} -0.866001 q^{51} +11.8308 q^{53} -14.2238 q^{55} +0.314419 q^{57} +10.9586 q^{59} +8.61564 q^{61} -3.12584 q^{63} +5.60233 q^{65} +3.47244 q^{67} +2.99817 q^{69} -13.6913 q^{71} +3.96374 q^{73} +4.10275 q^{75} +3.60736 q^{77} -1.00000 q^{79} +8.12004 q^{81} +7.24315 q^{83} +11.7012 q^{85} -2.60488 q^{87} +10.4935 q^{89} -1.42083 q^{91} +0.950564 q^{93} -4.24837 q^{95} -0.407329 q^{97} -9.71316 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 27 q - 4 q^{3} - 10 q^{5} - 8 q^{7} + 19 q^{9}+O(q^{10}) $ 27 * q - 4 * q^3 - 10 * q^5 - 8 * q^7 + 19 * q^9	$ 27 q - 4 q^{3} - 10 q^{5} - 8 q^{7} + 19 q^{9} + 3 q^{11} - 5 q^{13} - 11 q^{15} - 17 q^{17} + 27 q^{19} - 28 q^{21} - 11 q^{23} + 13 q^{25} - 7 q^{27} - 39 q^{29} - 27 q^{31} - 18 q^{33} - 5 q^{35} - q^{37} - 22 q^{39} - 36 q^{41} - 2 q^{43} - 18 q^{45} - 12 q^{47} + 15 q^{49} + 4 q^{51} - 28 q^{53} + 5 q^{55} - 4 q^{57} - 30 q^{59} - 6 q^{61} - 4 q^{63} - 32 q^{65} + 13 q^{67} - 27 q^{69} - 59 q^{71} - 30 q^{73} - 21 q^{75} - 39 q^{77} - 27 q^{79} - 5 q^{81} + 4 q^{83} - 3 q^{85} + 22 q^{87} - 56 q^{89} - 8 q^{91} - 38 q^{93} - 10 q^{95} - 30 q^{97} + 31 q^{99}+O(q^{100}) $ 27 * q - 4 * q^3 - 10 * q^5 - 8 * q^7 + 19 * q^9 + 3 * q^11 - 5 * q^13 - 11 * q^15 - 17 * q^17 + 27 * q^19 - 28 * q^21 - 11 * q^23 + 13 * q^25 - 7 * q^27 - 39 * q^29 - 27 * q^31 - 18 * q^33 - 5 * q^35 - q^37 - 22 * q^39 - 36 * q^41 - 2 * q^43 - 18 * q^45 - 12 * q^47 + 15 * q^49 + 4 * q^51 - 28 * q^53 + 5 * q^55 - 4 * q^57 - 30 * q^59 - 6 * q^61 - 4 * q^63 - 32 * q^65 + 13 * q^67 - 27 * q^69 - 59 * q^71 - 30 * q^73 - 21 * q^75 - 39 * q^77 - 27 * q^79 - 5 * q^81 + 4 * q^83 - 3 * q^85 + 22 * q^87 - 56 * q^89 - 8 * q^91 - 38 * q^93 - 10 * q^95 - 30 * q^97 + 31 * 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
$3$	0.314419	0.181530	0.0907650	−	0.995872i	$-0.471069\pi$
$3$	0.314419	0.181530	0.0907650	+	0.995872i	$0.471069\pi$
$4$	0	0
$5$	−4.24837	−1.89993	−0.949965	−	0.312356i	$-0.898882\pi$
$5$	−4.24837	−1.89993	−0.949965	+	0.312356i	$0.898882\pi$
$6$	0	0
$7$	1.07745	0.407238	0.203619	−	0.979050i	$-0.434730\pi$
$7$	1.07745	0.407238	0.203619	+	0.979050i	$0.434730\pi$
$8$	0	0
$9$	−2.90114	−0.967047
$10$	0	0
$11$	3.34805	1.00947	0.504737	−	0.863273i	$-0.331589\pi$
$11$	3.34805	1.00947	0.504737	+	0.863273i	$0.331589\pi$
$12$	0	0
$13$	−1.31870	−0.365742	−0.182871	−	0.983137i	$-0.558539\pi$
$13$	−1.31870	−0.365742	−0.182871	+	0.983137i	$0.558539\pi$
$14$	0	0
$15$	−1.33577	−0.344894
$16$	0	0
$17$	−2.75429	−0.668013	−0.334006	−	0.942571i	$-0.608401\pi$
$17$	−2.75429	−0.668013	−0.334006	+	0.942571i	$0.608401\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0.338771	0.0739259
$22$	0	0
$23$	9.53559	1.98831	0.994154	−	0.107973i	$-0.0344361\pi$
$23$	9.53559	1.98831	0.994154	+	0.107973i	$0.0344361\pi$
$24$	0	0
$25$	13.0487	2.60973
$26$	0	0
$27$	−1.85543	−0.357078
$28$	0	0
$29$	−8.28473	−1.53844	−0.769218	−	0.638986i	$-0.779354\pi$
$29$	−8.28473	−1.53844	−0.769218	+	0.638986i	$0.779354\pi$
$30$	0	0
$31$	3.02324	0.542990	0.271495	−	0.962440i	$-0.412482\pi$
$31$	3.02324	0.542990	0.271495	+	0.962440i	$0.412482\pi$
$32$	0	0
$33$	1.05269	0.183250
$34$	0	0
$35$	−4.57741	−0.773724
$36$	0	0
$37$	4.48446	0.737241	0.368620	−	0.929580i	$-0.379830\pi$
$37$	4.48446	0.737241	0.368620	+	0.929580i	$0.379830\pi$
$38$	0	0
$39$	−0.414625	−0.0663931
$40$	0	0
$41$	−2.36401	−0.369197	−0.184599	−	0.982814i	$-0.559098\pi$
$41$	−2.36401	−0.369197	−0.184599	+	0.982814i	$0.559098\pi$
$42$	0	0
$43$	−10.5342	−1.60645	−0.803224	−	0.595677i	$-0.796884\pi$
$43$	−10.5342	−1.60645	−0.803224	+	0.595677i	$0.796884\pi$
$44$	0	0
$45$	12.3251	1.83732
$46$	0	0
$47$	−3.65470	−0.533093	−0.266546	−	0.963822i	$-0.585883\pi$
$47$	−3.65470	−0.533093	−0.266546	+	0.963822i	$0.585883\pi$
$48$	0	0
$49$	−5.83910	−0.834157
$50$	0	0
$51$	−0.866001	−0.121264
$52$	0	0
$53$	11.8308	1.62509	0.812545	−	0.582899i	$-0.198082\pi$
$53$	11.8308	1.62509	0.812545	+	0.582899i	$0.198082\pi$
$54$	0	0
$55$	−14.2238	−1.91793
$56$	0	0
$57$	0.314419	0.0416458
$58$	0	0
$59$	10.9586	1.42669	0.713346	−	0.700812i	$-0.247179\pi$
$59$	10.9586	1.42669	0.713346	+	0.700812i	$0.247179\pi$
$60$	0	0
$61$	8.61564	1.10312	0.551560	−	0.834135i	$-0.314033\pi$
$61$	8.61564	1.10312	0.551560	+	0.834135i	$0.314033\pi$
$62$	0	0
$63$	−3.12584	−0.393818
$64$	0	0
$65$	5.60233	0.694884
$66$	0	0
$67$	3.47244	0.424226	0.212113	−	0.977245i	$-0.431965\pi$
$67$	3.47244	0.424226	0.212113	+	0.977245i	$0.431965\pi$
$68$	0	0
$69$	2.99817	0.360938
$70$	0	0
$71$	−13.6913	−1.62485	−0.812427	−	0.583063i	$-0.801854\pi$
$71$	−13.6913	−1.62485	−0.812427	+	0.583063i	$0.801854\pi$
$72$	0	0
$73$	3.96374	0.463920	0.231960	−	0.972725i	$-0.425486\pi$
$73$	3.96374	0.463920	0.231960	+	0.972725i	$0.425486\pi$
$74$	0	0
$75$	4.10275	0.473745
$76$	0	0
$77$	3.60736	0.411096
$78$	0	0
$79$	−1.00000	−0.112509
$79$	−1.00000	−0.112509
$80$	0	0
$81$	8.12004	0.902226
$82$	0	0
$83$	7.24315	0.795039	0.397520	−	0.917594i	$-0.369871\pi$
$83$	7.24315	0.795039	0.397520	+	0.917594i	$0.369871\pi$
$84$	0	0
$85$	11.7012	1.26918
$86$	0	0
$87$	−2.60488	−0.279272
$88$	0	0
$89$	10.4935	1.11231	0.556157	−	0.831077i	$-0.312275\pi$
$89$	10.4935	1.11231	0.556157	+	0.831077i	$0.312275\pi$
$90$	0	0
$91$	−1.42083	−0.148944
$92$	0	0
$93$	0.950564	0.0985689
$94$	0	0
$95$	−4.24837	−0.435874
$96$	0	0
$97$	−0.407329	−0.0413580	−0.0206790	−	0.999786i	$-0.506583\pi$
$97$	−0.407329	−0.0413580	−0.0206790	+	0.999786i	$0.506583\pi$
$98$	0	0
$99$	−9.71316	−0.976209
$100$	0	0
$101$	−6.78571	−0.675203	−0.337602	−	0.941289i	$-0.609616\pi$
$101$	−6.78571	−0.675203	−0.337602	+	0.941289i	$0.609616\pi$
$102$	0	0
$103$	−8.26293	−0.814170	−0.407085	−	0.913390i	$-0.633455\pi$
$103$	−8.26293	−0.814170	−0.407085	+	0.913390i	$0.633455\pi$
$104$	0	0
$105$	−1.43923	−0.140454
$106$	0	0
$107$	−8.99492	−0.869572	−0.434786	−	0.900534i	$-0.643176\pi$
$107$	−8.99492	−0.869572	−0.434786	+	0.900534i	$0.643176\pi$
$108$	0	0
$109$	−0.0110714	−0.00106044	−0.000530222	−	1.00000i	$-0.500169\pi$
$109$	−0.0110714	−0.00106044	−0.000530222		1.00000i	$0.500169\pi$
$110$	0	0
$111$	1.41000	0.133831
$112$	0	0
$113$	−18.8046	−1.76899	−0.884494	−	0.466552i	$-0.845496\pi$
$113$	−18.8046	−1.76899	−0.884494	+	0.466552i	$0.845496\pi$
$114$	0	0
$115$	−40.5107	−3.77765
$116$	0	0
$117$	3.82574	0.353689
$118$	0	0
$119$	−2.96761	−0.272040
$120$	0	0
$121$	0.209429	0.0190390
$122$	0	0
$123$	−0.743292	−0.0670204
$124$	0	0
$125$	−34.1938	−3.05838
$126$	0	0
$127$	−13.2139	−1.17255	−0.586274	−	0.810113i	$-0.699406\pi$
$127$	−13.2139	−1.17255	−0.586274	+	0.810113i	$0.699406\pi$
$128$	0	0
$129$	−3.31215	−0.291619
$130$	0	0
$131$	−7.75832	−0.677847	−0.338924	−	0.940814i	$-0.610063\pi$
$131$	−7.75832	−0.677847	−0.338924	+	0.940814i	$0.610063\pi$
$132$	0	0
$133$	1.07745	0.0934268
$134$	0	0
$135$	7.88257	0.678423
$136$	0	0
$137$	0.685371	0.0585552	0.0292776	−	0.999571i	$-0.490679\pi$
$137$	0.685371	0.0585552	0.0292776	+	0.999571i	$0.490679\pi$
$138$	0	0
$139$	14.6121	1.23938	0.619689	−	0.784847i	$-0.287259\pi$
$139$	14.6121	1.23938	0.619689	+	0.784847i	$0.287259\pi$
$140$	0	0
$141$	−1.14911	−0.0967724
$142$	0	0
$143$	−4.41507	−0.369207
$144$	0	0
$145$	35.1966	2.92292
$146$	0	0
$147$	−1.83593	−0.151425
$148$	0	0
$149$	−8.29431	−0.679496	−0.339748	−	0.940516i	$-0.610342\pi$
$149$	−8.29431	−0.679496	−0.339748	+	0.940516i	$0.610342\pi$
$150$	0	0
$151$	−2.06269	−0.167859	−0.0839296	−	0.996472i	$-0.526747\pi$
$151$	−2.06269	−0.167859	−0.0839296	+	0.996472i	$0.526747\pi$
$152$	0	0
$153$	7.99057	0.646000
$154$	0	0
$155$	−12.8438	−1.03164
$156$	0	0
$157$	6.81700	0.544056	0.272028	−	0.962289i	$-0.412306\pi$
$157$	6.81700	0.544056	0.272028	+	0.962289i	$0.412306\pi$
$158$	0	0
$159$	3.71984	0.295003
$160$	0	0
$161$	10.2741	0.809714
$162$	0	0
$163$	9.99974	0.783240	0.391620	−	0.920127i	$-0.371915\pi$
$163$	9.99974	0.783240	0.391620	+	0.920127i	$0.371915\pi$
$164$	0	0
$165$	−4.47222	−0.348162
$166$	0	0
$167$	−18.2130	−1.40937	−0.704684	−	0.709522i	$-0.748911\pi$
$167$	−18.2130	−1.40937	−0.704684	+	0.709522i	$0.748911\pi$
$168$	0	0
$169$	−11.2610	−0.866233
$170$	0	0
$171$	−2.90114	−0.221856
$172$	0	0
$173$	−13.3723	−1.01668	−0.508339	−	0.861157i	$-0.669740\pi$
$173$	−13.3723	−1.01668	−0.508339	+	0.861157i	$0.669740\pi$
$174$	0	0
$175$	14.0593	1.06278
$176$	0	0
$177$	3.44560	0.258987
$178$	0	0
$179$	4.94646	0.369716	0.184858	−	0.982765i	$-0.440818\pi$
$179$	4.94646	0.369716	0.184858	+	0.982765i	$0.440818\pi$
$180$	0	0
$181$	−5.64794	−0.419808	−0.209904	−	0.977722i	$-0.567315\pi$
$181$	−5.64794	−0.419808	−0.209904	+	0.977722i	$0.567315\pi$
$182$	0	0
$183$	2.70892	0.200249
$184$	0	0
$185$	−19.0517	−1.40071
$186$	0	0
$187$	−9.22149	−0.674342
$188$	0	0
$189$	−1.99914	−0.145416
$190$	0	0
$191$	−16.7721	−1.21358	−0.606792	−	0.794860i	$-0.707544\pi$
$191$	−16.7721	−1.21358	−0.606792	+	0.794860i	$0.707544\pi$
$192$	0	0
$193$	−4.49497	−0.323555	−0.161777	−	0.986827i	$-0.551723\pi$
$193$	−4.49497	−0.323555	−0.161777	+	0.986827i	$0.551723\pi$
$194$	0	0
$195$	1.76148	0.126142
$196$	0	0
$197$	−9.26504	−0.660107	−0.330053	−	0.943962i	$-0.607067\pi$
$197$	−9.26504	−0.660107	−0.330053	+	0.943962i	$0.607067\pi$
$198$	0	0
$199$	8.53739	0.605199	0.302600	−	0.953118i	$-0.402145\pi$
$199$	8.53739	0.605199	0.302600	+	0.953118i	$0.402145\pi$
$200$	0	0
$201$	1.09180	0.0770098
$202$	0	0
$203$	−8.92639	−0.626510
$204$	0	0
$205$	10.0432	0.701449
$206$	0	0
$207$	−27.6641	−1.92279
$208$	0	0
$209$	3.34805	0.231589
$210$	0	0
$211$	−17.2161	−1.18520	−0.592602	−	0.805496i	$-0.701899\pi$
$211$	−17.2161	−1.18520	−0.592602	+	0.805496i	$0.701899\pi$
$212$	0	0
$213$	−4.30480	−0.294960
$214$	0	0
$215$	44.7531	3.05214
$216$	0	0
$217$	3.25739	0.221126
$218$	0	0
$219$	1.24627	0.0842154
$220$	0	0
$221$	3.63208	0.244320
$222$	0	0
$223$	4.00842	0.268424	0.134212	−	0.990953i	$-0.457150\pi$
$223$	4.00842	0.268424	0.134212	+	0.990953i	$0.457150\pi$
$224$	0	0
$225$	−37.8560	−2.52374
$226$	0	0
$227$	−24.5845	−1.63173	−0.815865	−	0.578242i	$-0.803739\pi$
$227$	−24.5845	−1.63173	−0.815865	+	0.578242i	$0.803739\pi$
$228$	0	0
$229$	20.1420	1.33102	0.665510	−	0.746389i	$-0.268214\pi$
$229$	20.1420	1.33102	0.665510	+	0.746389i	$0.268214\pi$
$230$	0	0
$231$	1.13422	0.0746264
$232$	0	0
$233$	−23.7258	−1.55433	−0.777164	−	0.629298i	$-0.783342\pi$
$233$	−23.7258	−1.55433	−0.777164	+	0.629298i	$0.783342\pi$
$234$	0	0
$235$	15.5265	1.01284
$236$	0	0
$237$	−0.314419	−0.0204237
$238$	0	0
$239$	−11.3938	−0.737001	−0.368501	−	0.929627i	$-0.620129\pi$
$239$	−11.3938	−0.737001	−0.368501	+	0.929627i	$0.620129\pi$
$240$	0	0
$241$	14.3831	0.926497	0.463248	−	0.886229i	$-0.346684\pi$
$241$	14.3831	0.926497	0.463248	+	0.886229i	$0.346684\pi$
$242$	0	0
$243$	8.11939	0.520859
$244$	0	0
$245$	24.8067	1.58484
$246$	0	0
$247$	−1.31870	−0.0839069
$248$	0	0
$249$	2.27739	0.144324
$250$	0	0
$251$	2.99892	0.189290	0.0946452	−	0.995511i	$-0.469828\pi$
$251$	2.99892	0.189290	0.0946452	+	0.995511i	$0.469828\pi$
$252$	0	0
$253$	31.9256	2.00715
$254$	0	0
$255$	3.67909	0.230394
$256$	0	0
$257$	−28.2966	−1.76509	−0.882546	−	0.470226i	$-0.844172\pi$
$257$	−28.2966	−1.76509	−0.882546	+	0.470226i	$0.844172\pi$
$258$	0	0
$259$	4.83178	0.300232
$260$	0	0
$261$	24.0352	1.48774
$262$	0	0
$263$	15.8279	0.975989	0.487994	−	0.872847i	$-0.337729\pi$
$263$	15.8279	0.975989	0.487994	+	0.872847i	$0.337729\pi$
$264$	0	0
$265$	−50.2618	−3.08756
$266$	0	0
$267$	3.29937	0.201918
$268$	0	0
$269$	−1.77829	−0.108424	−0.0542121	−	0.998529i	$-0.517265\pi$
$269$	−1.77829	−0.108424	−0.0542121	+	0.998529i	$0.517265\pi$
$270$	0	0
$271$	−24.8403	−1.50894	−0.754470	−	0.656335i	$-0.772106\pi$
$271$	−24.8403	−1.50894	−0.754470	+	0.656335i	$0.772106\pi$
$272$	0	0
$273$	−0.446738	−0.0270378
$274$	0	0
$275$	43.6876	2.63446
$276$	0	0
$277$	−16.9930	−1.02101	−0.510506	−	0.859874i	$-0.670542\pi$
$277$	−16.9930	−1.02101	−0.510506	+	0.859874i	$0.670542\pi$
$278$	0	0
$279$	−8.77084	−0.525096
$280$	0	0
$281$	12.9132	0.770335	0.385168	−	0.922847i	$-0.374144\pi$
$281$	12.9132	0.770335	0.385168	+	0.922847i	$0.374144\pi$
$282$	0	0
$283$	21.3522	1.26925	0.634627	−	0.772818i	$-0.281154\pi$
$283$	21.3522	1.26925	0.634627	+	0.772818i	$0.281154\pi$
$284$	0	0
$285$	−1.33577	−0.0791242
$286$	0	0
$287$	−2.54711	−0.150351
$288$	0	0
$289$	−9.41391	−0.553759
$290$	0	0
$291$	−0.128072	−0.00750772
$292$	0	0
$293$	9.07884	0.530392	0.265196	−	0.964195i	$-0.414563\pi$
$293$	9.07884	0.530392	0.265196	+	0.964195i	$0.414563\pi$
$294$	0	0
$295$	−46.5563	−2.71061
$296$	0	0
$297$	−6.21208	−0.360461
$298$	0	0
$299$	−12.5746	−0.727207
$300$	0	0
$301$	−11.3501	−0.654207
$302$	0	0
$303$	−2.13356	−0.122570
$304$	0	0
$305$	−36.6024	−2.09585
$306$	0	0
$307$	−5.74929	−0.328129	−0.164065	−	0.986450i	$-0.552461\pi$
$307$	−5.74929	−0.328129	−0.164065	+	0.986450i	$0.552461\pi$
$308$	0	0
$309$	−2.59802	−0.147796
$310$	0	0
$311$	−19.5978	−1.11129	−0.555645	−	0.831420i	$-0.687529\pi$
$311$	−19.5978	−1.11129	−0.555645	+	0.831420i	$0.687529\pi$
$312$	0	0
$313$	29.8598	1.68778	0.843888	−	0.536520i	$-0.180261\pi$
$313$	29.8598	1.68778	0.843888	+	0.536520i	$0.180261\pi$
$314$	0	0
$315$	13.2797	0.748227
$316$	0	0
$317$	−27.6863	−1.55502	−0.777509	−	0.628871i	$-0.783517\pi$
$317$	−27.6863	−1.55502	−0.777509	+	0.628871i	$0.783517\pi$
$318$	0	0
$319$	−27.7377	−1.55301
$320$	0	0
$321$	−2.82818	−0.157853
$322$	0	0
$323$	−2.75429	−0.153253
$324$	0	0
$325$	−17.2073	−0.954489
$326$	0	0
$327$	−0.00348105	−0.000192502 0
$328$	0	0
$329$	−3.93776	−0.217096
$330$	0	0
$331$	7.99773	0.439595	0.219797	−	0.975546i	$-0.429460\pi$
$331$	7.99773	0.439595	0.219797	+	0.975546i	$0.429460\pi$
$332$	0	0
$333$	−13.0100	−0.712946
$334$	0	0
$335$	−14.7522	−0.806000
$336$	0	0
$337$	−9.04060	−0.492473	−0.246237	−	0.969210i	$-0.579194\pi$
$337$	−9.04060	−0.492473	−0.246237	+	0.969210i	$0.579194\pi$
$338$	0	0
$339$	−5.91253	−0.321124
$340$	0	0
$341$	10.1219	0.548134
$342$	0	0
$343$	−13.8335	−0.746939
$344$	0	0
$345$	−12.7374	−0.685756
$346$	0	0
$347$	2.06873	0.111055	0.0555276	−	0.998457i	$-0.482316\pi$
$347$	2.06873	0.111055	0.0555276	+	0.998457i	$0.482316\pi$
$348$	0	0
$349$	18.8973	1.01155	0.505776	−	0.862665i	$-0.331206\pi$
$349$	18.8973	1.01155	0.505776	+	0.862665i	$0.331206\pi$
$350$	0	0
$351$	2.44676	0.130598
$352$	0	0
$353$	−8.55918	−0.455559	−0.227779	−	0.973713i	$-0.573147\pi$
$353$	−8.55918	−0.455559	−0.227779	+	0.973713i	$0.573147\pi$
$354$	0	0
$355$	58.1656	3.08711
$356$	0	0
$357$	−0.933073	−0.0493835
$358$	0	0
$359$	−26.0237	−1.37348	−0.686739	−	0.726904i	$-0.740959\pi$
$359$	−26.0237	−1.37348	−0.686739	+	0.726904i	$0.740959\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0.0658485	0.00345615
$364$	0	0
$365$	−16.8394	−0.881416
$366$	0	0
$367$	−17.8267	−0.930546	−0.465273	−	0.885167i	$-0.654044\pi$
$367$	−17.8267	−0.930546	−0.465273	+	0.885167i	$0.654044\pi$
$368$	0	0
$369$	6.85834	0.357031
$370$	0	0
$371$	12.7471	0.661798
$372$	0	0
$373$	−26.4967	−1.37195	−0.685974	−	0.727626i	$-0.740624\pi$
$373$	−26.4967	−1.37195	−0.685974	+	0.727626i	$0.740624\pi$
$374$	0	0
$375$	−10.7512	−0.555188
$376$	0	0
$377$	10.9251	0.562670
$378$	0	0
$379$	−15.8384	−0.813565	−0.406782	−	0.913525i	$-0.633349\pi$
$379$	−15.8384	−0.813565	−0.406782	+	0.913525i	$0.633349\pi$
$380$	0	0
$381$	−4.15472	−0.212853
$382$	0	0
$383$	23.2204	1.18651	0.593253	−	0.805016i	$-0.297843\pi$
$383$	23.2204	1.18651	0.593253	+	0.805016i	$0.297843\pi$
$384$	0	0
$385$	−15.3254	−0.781054
$386$	0	0
$387$	30.5612	1.55351
$388$	0	0
$389$	1.64433	0.0833707	0.0416853	−	0.999131i	$-0.486727\pi$
$389$	1.64433	0.0833707	0.0416853	+	0.999131i	$0.486727\pi$
$390$	0	0
$391$	−26.2637	−1.32821
$392$	0	0
$393$	−2.43936	−0.123050
$394$	0	0
$395$	4.24837	0.213759
$396$	0	0
$397$	4.35835	0.218739	0.109370	−	0.994001i	$-0.465117\pi$
$397$	4.35835	0.218739	0.109370	+	0.994001i	$0.465117\pi$
$398$	0	0
$399$	0.338771	0.0169598
$400$	0	0
$401$	14.0493	0.701591	0.350795	−	0.936452i	$-0.385911\pi$
$401$	14.0493	0.701591	0.350795	+	0.936452i	$0.385911\pi$
$402$	0	0
$403$	−3.98675	−0.198594
$404$	0	0
$405$	−34.4969	−1.71417
$406$	0	0
$407$	15.0142	0.744226
$408$	0	0
$409$	−15.3643	−0.759716	−0.379858	−	0.925045i	$-0.624027\pi$
$409$	−15.3643	−0.759716	−0.379858	+	0.925045i	$0.624027\pi$
$410$	0	0
$411$	0.215494	0.0106295
$412$	0	0
$413$	11.8074	0.581003
$414$	0	0
$415$	−30.7716	−1.51052
$416$	0	0
$417$	4.59431	0.224984
$418$	0	0
$419$	−8.62082	−0.421155	−0.210577	−	0.977577i	$-0.567534\pi$
$419$	−8.62082	−0.421155	−0.210577	+	0.977577i	$0.567534\pi$
$420$	0	0
$421$	25.7885	1.25686	0.628428	−	0.777868i	$-0.283699\pi$
$421$	25.7885	1.25686	0.628428	+	0.777868i	$0.283699\pi$
$422$	0	0
$423$	10.6028	0.515526
$424$	0	0
$425$	−35.9398	−1.74334
$426$	0	0
$427$	9.28292	0.449232
$428$	0	0
$429$	−1.38818	−0.0670222
$430$	0	0
$431$	−27.4659	−1.32298	−0.661492	−	0.749952i	$-0.730076\pi$
$431$	−27.4659	−1.32298	−0.661492	+	0.749952i	$0.730076\pi$
$432$	0	0
$433$	38.7008	1.85984	0.929921	−	0.367759i	$-0.119875\pi$
$433$	38.7008	1.85984	0.929921	+	0.367759i	$0.119875\pi$
$434$	0	0
$435$	11.0665	0.530598
$436$	0	0
$437$	9.53559	0.456149
$438$	0	0
$439$	−4.25988	−0.203313	−0.101657	−	0.994820i	$-0.532414\pi$
$439$	−4.25988	−0.203313	−0.101657	+	0.994820i	$0.532414\pi$
$440$	0	0
$441$	16.9401	0.806669
$442$	0	0
$443$	20.6263	0.979986	0.489993	−	0.871726i	$-0.336999\pi$
$443$	20.6263	0.979986	0.489993	+	0.871726i	$0.336999\pi$
$444$	0	0
$445$	−44.5805	−2.11332
$446$	0	0
$447$	−2.60789	−0.123349
$448$	0	0
$449$	12.2174	0.576574	0.288287	−	0.957544i	$-0.406914\pi$
$449$	12.2174	0.576574	0.288287	+	0.957544i	$0.406914\pi$
$450$	0	0
$451$	−7.91484	−0.372695
$452$	0	0
$453$	−0.648549	−0.0304715
$454$	0	0
$455$	6.03624	0.282983
$456$	0	0
$457$	19.4790	0.911189	0.455594	−	0.890188i	$-0.349427\pi$
$457$	19.4790	0.911189	0.455594	+	0.890188i	$0.349427\pi$
$458$	0	0
$459$	5.11039	0.238533
$460$	0	0
$461$	7.25036	0.337683	0.168841	−	0.985643i	$-0.445997\pi$
$461$	7.25036	0.337683	0.168841	+	0.985643i	$0.445997\pi$
$462$	0	0
$463$	−21.1864	−0.984613	−0.492307	−	0.870422i	$-0.663846\pi$
$463$	−21.1864	−0.984613	−0.492307	+	0.870422i	$0.663846\pi$
$464$	0	0
$465$	−4.03835	−0.187274
$466$	0	0
$467$	−4.80986	−0.222574	−0.111287	−	0.993788i	$-0.535497\pi$
$467$	−4.80986	−0.222574	−0.111287	+	0.993788i	$0.535497\pi$
$468$	0	0
$469$	3.74138	0.172761
$470$	0	0
$471$	2.14340	0.0987624
$472$	0	0
$473$	−35.2690	−1.62167
$474$	0	0
$475$	13.0487	0.598714
$476$	0	0
$477$	−34.3229	−1.57154
$478$	0	0
$479$	10.9505	0.500343	0.250171	−	0.968202i	$-0.419513\pi$
$479$	10.9505	0.500343	0.250171	+	0.968202i	$0.419513\pi$
$480$	0	0
$481$	−5.91366	−0.269640
$482$	0	0
$483$	3.23038	0.146987
$484$	0	0
$485$	1.73049	0.0785774
$486$	0	0
$487$	23.5051	1.06512	0.532559	−	0.846393i	$-0.321230\pi$
$487$	23.5051	1.06512	0.532559	+	0.846393i	$0.321230\pi$
$488$	0	0
$489$	3.14411	0.142182
$490$	0	0
$491$	−12.5450	−0.566147	−0.283073	−	0.959098i	$-0.591354\pi$
$491$	−12.5450	−0.566147	−0.283073	+	0.959098i	$0.591354\pi$
$492$	0	0
$493$	22.8185	1.02769
$494$	0	0
$495$	41.2651	1.85473
$496$	0	0
$497$	−14.7517	−0.661702
$498$	0	0
$499$	38.5083	1.72387	0.861935	−	0.507018i	$-0.169252\pi$
$499$	38.5083	1.72387	0.861935	+	0.507018i	$0.169252\pi$
$500$	0	0
$501$	−5.72653	−0.255842
$502$	0	0
$503$	−7.85492	−0.350234	−0.175117	−	0.984548i	$-0.556030\pi$
$503$	−7.85492	−0.350234	−0.175117	+	0.984548i	$0.556030\pi$
$504$	0	0
$505$	28.8282	1.28284
$506$	0	0
$507$	−3.54068	−0.157247
$508$	0	0
$509$	6.62572	0.293680	0.146840	−	0.989160i	$-0.453090\pi$
$509$	6.62572	0.293680	0.146840	+	0.989160i	$0.453090\pi$
$510$	0	0
$511$	4.27073	0.188926
$512$	0	0
$513$	−1.85543	−0.0819193
$514$	0	0
$515$	35.1040	1.54687
$516$	0	0
$517$	−12.2361	−0.538144
$518$	0	0
$519$	−4.20451	−0.184557
$520$	0	0
$521$	20.2589	0.887558	0.443779	−	0.896136i	$-0.353637\pi$
$521$	20.2589	0.887558	0.443779	+	0.896136i	$0.353637\pi$
$522$	0	0
$523$	−0.726892	−0.0317848	−0.0158924	−	0.999874i	$-0.505059\pi$
$523$	−0.726892	−0.0317848	−0.0158924	+	0.999874i	$0.505059\pi$
$524$	0	0
$525$	4.42051	0.192927
$526$	0	0
$527$	−8.32686	−0.362724
$528$	0	0
$529$	67.9274	2.95337
$530$	0	0
$531$	−31.7925	−1.37968
$532$	0	0
$533$	3.11743	0.135031
$534$	0	0
$535$	38.2138	1.65213
$536$	0	0
$537$	1.55526	0.0671145
$538$	0	0
$539$	−19.5496	−0.842061
$540$	0	0
$541$	−45.0818	−1.93822	−0.969109	−	0.246635i	$-0.920675\pi$
$541$	−45.0818	−1.93822	−0.969109	+	0.246635i	$0.920675\pi$
$542$	0	0
$543$	−1.77582	−0.0762078
$544$	0	0
$545$	0.0470353	0.00201477
$546$	0	0
$547$	−27.8385	−1.19029	−0.595144	−	0.803619i	$-0.702905\pi$
$547$	−27.8385	−1.19029	−0.595144	+	0.803619i	$0.702905\pi$
$548$	0	0
$549$	−24.9952	−1.06677
$550$	0	0
$551$	−8.28473	−0.352941
$552$	0	0
$553$	−1.07745	−0.0458179
$554$	0	0
$555$	−5.99021	−0.254270
$556$	0	0
$557$	11.4888	0.486794	0.243397	−	0.969927i	$-0.421738\pi$
$557$	11.4888	0.486794	0.243397	+	0.969927i	$0.421738\pi$
$558$	0	0
$559$	13.8914	0.587545
$560$	0	0
$561$	−2.89941	−0.122413
$562$	0	0
$563$	−17.5198	−0.738373	−0.369186	−	0.929355i	$-0.620364\pi$
$563$	−17.5198	−0.738373	−0.369186	+	0.929355i	$0.620364\pi$
$564$	0	0
$565$	79.8889	3.36095
$566$	0	0
$567$	8.74894	0.367421
$568$	0	0
$569$	−4.35993	−0.182778	−0.0913888	−	0.995815i	$-0.529131\pi$
$569$	−4.35993	−0.182778	−0.0913888	+	0.995815i	$0.529131\pi$
$570$	0	0
$571$	9.34403	0.391035	0.195518	−	0.980700i	$-0.437361\pi$
$571$	9.34403	0.391035	0.195518	+	0.980700i	$0.437361\pi$
$572$	0	0
$573$	−5.27346	−0.220302
$574$	0	0
$575$	124.427	5.18895
$576$	0	0
$577$	−11.8087	−0.491601	−0.245800	−	0.969320i	$-0.579051\pi$
$577$	−11.8087	−0.491601	−0.245800	+	0.969320i	$0.579051\pi$
$578$	0	0
$579$	−1.41330	−0.0587349
$580$	0	0
$581$	7.80414	0.323770
$582$	0	0
$583$	39.6102	1.64049
$584$	0	0
$585$	−16.2532	−0.671985
$586$	0	0
$587$	23.8366	0.983841	0.491921	−	0.870640i	$-0.336295\pi$
$587$	23.8366	0.983841	0.491921	+	0.870640i	$0.336295\pi$
$588$	0	0
$589$	3.02324	0.124570
$590$	0	0
$591$	−2.91311	−0.119829
$592$	0	0
$593$	24.2587	0.996184	0.498092	−	0.867124i	$-0.334034\pi$
$593$	24.2587	0.996184	0.498092	+	0.867124i	$0.334034\pi$
$594$	0	0
$595$	12.6075	0.516857
$596$	0	0
$597$	2.68432	0.109862
$598$	0	0
$599$	−8.03658	−0.328366	−0.164183	−	0.986430i	$-0.552499\pi$
$599$	−8.03658	−0.328366	−0.164183	+	0.986430i	$0.552499\pi$
$600$	0	0
$601$	−30.8023	−1.25645	−0.628227	−	0.778030i	$-0.716219\pi$
$601$	−30.8023	−1.25645	−0.628227	+	0.778030i	$0.716219\pi$
$602$	0	0
$603$	−10.0740	−0.410247
$604$	0	0
$605$	−0.889733	−0.0361728
$606$	0	0
$607$	−38.4188	−1.55937	−0.779686	−	0.626171i	$-0.784621\pi$
$607$	−38.4188	−1.55937	−0.779686	+	0.626171i	$0.784621\pi$
$608$	0	0
$609$	−2.80663	−0.113730
$610$	0	0
$611$	4.81946	0.194974
$612$	0	0
$613$	−26.9096	−1.08687	−0.543435	−	0.839452i	$-0.682876\pi$
$613$	−26.9096	−1.08687	−0.543435	+	0.839452i	$0.682876\pi$
$614$	0	0
$615$	3.15778	0.127334
$616$	0	0
$617$	−25.1907	−1.01414	−0.507070	−	0.861905i	$-0.669271\pi$
$617$	−25.1907	−1.01414	−0.507070	+	0.861905i	$0.669271\pi$
$618$	0	0
$619$	23.4438	0.942287	0.471144	−	0.882056i	$-0.343841\pi$
$619$	23.4438	0.942287	0.471144	+	0.882056i	$0.343841\pi$
$620$	0	0
$621$	−17.6926	−0.709981
$622$	0	0
$623$	11.3063	0.452977
$624$	0	0
$625$	80.0245	3.20098
$626$	0	0
$627$	1.05269	0.0420404
$628$	0	0
$629$	−12.3515	−0.492486
$630$	0	0
$631$	−40.4119	−1.60877	−0.804387	−	0.594106i	$-0.797506\pi$
$631$	−40.4119	−1.60877	−0.804387	+	0.594106i	$0.797506\pi$
$632$	0	0
$633$	−5.41307	−0.215150
$634$	0	0
$635$	56.1377	2.22776
$636$	0	0
$637$	7.70003	0.305086
$638$	0	0
$639$	39.7203	1.57131
$640$	0	0
$641$	9.82300	0.387985	0.193993	−	0.981003i	$-0.437856\pi$
$641$	9.82300	0.387985	0.193993	+	0.981003i	$0.437856\pi$
$642$	0	0
$643$	24.2675	0.957019	0.478509	−	0.878082i	$-0.341177\pi$
$643$	24.2675	0.957019	0.478509	+	0.878082i	$0.341177\pi$
$644$	0	0
$645$	14.0713	0.554055
$646$	0	0
$647$	−11.9056	−0.468057	−0.234029	−	0.972230i	$-0.575191\pi$
$647$	−11.9056	−0.468057	−0.234029	+	0.972230i	$0.575191\pi$
$648$	0	0
$649$	36.6900	1.44021
$650$	0	0
$651$	1.02419	0.0401410
$652$	0	0
$653$	−14.9196	−0.583850	−0.291925	−	0.956441i	$-0.594296\pi$
$653$	−14.9196	−0.583850	−0.291925	+	0.956441i	$0.594296\pi$
$654$	0	0
$655$	32.9602	1.28786
$656$	0	0
$657$	−11.4994	−0.448633
$658$	0	0
$659$	−27.6364	−1.07656	−0.538280	−	0.842766i	$-0.680926\pi$
$659$	−27.6364	−1.07656	−0.538280	+	0.842766i	$0.680926\pi$
$660$	0	0
$661$	−1.57098	−0.0611040	−0.0305520	−	0.999533i	$-0.509727\pi$
$661$	−1.57098	−0.0611040	−0.0305520	+	0.999533i	$0.509727\pi$
$662$	0	0
$663$	1.14200	0.0443514
$664$	0	0
$665$	−4.57741	−0.177504
$666$	0	0
$667$	−78.9998	−3.05888
$668$	0	0
$669$	1.26032	0.0487269
$670$	0	0
$671$	28.8456	1.11357
$672$	0	0
$673$	−4.81563	−0.185629	−0.0928145	−	0.995683i	$-0.529586\pi$
$673$	−4.81563	−0.185629	−0.0928145	+	0.995683i	$0.529586\pi$
$674$	0	0
$675$	−24.2109	−0.931879
$676$	0	0
$677$	42.5024	1.63350	0.816751	−	0.576991i	$-0.195773\pi$
$677$	42.5024	1.63350	0.816751	+	0.576991i	$0.195773\pi$
$678$	0	0
$679$	−0.438877	−0.0168426
$680$	0	0
$681$	−7.72984	−0.296208
$682$	0	0
$683$	45.0747	1.72473	0.862367	−	0.506284i	$-0.168981\pi$
$683$	45.0747	1.72473	0.862367	+	0.506284i	$0.168981\pi$
$684$	0	0
$685$	−2.91171	−0.111251
$686$	0	0
$687$	6.33302	0.241620
$688$	0	0
$689$	−15.6013	−0.594363
$690$	0	0
$691$	31.6640	1.20456	0.602278	−	0.798287i	$-0.294260\pi$
$691$	31.6640	1.20456	0.602278	+	0.798287i	$0.294260\pi$
$692$	0	0
$693$	−10.4654	−0.397550
$694$	0	0
$695$	−62.0775	−2.35473
$696$	0	0
$697$	6.51117	0.246628
$698$	0	0
$699$	−7.45984	−0.282157
$700$	0	0
$701$	−18.6888	−0.705864	−0.352932	−	0.935649i	$-0.614815\pi$
$701$	−18.6888	−0.705864	−0.352932	+	0.935649i	$0.614815\pi$
$702$	0	0
$703$	4.48446	0.169135
$704$	0	0
$705$	4.88184	0.183861
$706$	0	0
$707$	−7.31126	−0.274968
$708$	0	0
$709$	−34.1332	−1.28190	−0.640950	−	0.767582i	$-0.721460\pi$
$709$	−34.1332	−1.28190	−0.640950	+	0.767582i	$0.721460\pi$
$710$	0	0
$711$	2.90114	0.108801
$712$	0	0
$713$	28.8284	1.07963
$714$	0	0
$715$	18.7569	0.701468
$716$	0	0
$717$	−3.58242	−0.133788
$718$	0	0
$719$	−7.44422	−0.277623	−0.138811	−	0.990319i	$-0.544328\pi$
$719$	−7.44422	−0.277623	−0.138811	+	0.990319i	$0.544328\pi$
$720$	0	0
$721$	−8.90289	−0.331561
$722$	0	0
$723$	4.52232	0.168187
$724$	0	0
$725$	−108.105	−4.01491
$726$	0	0
$727$	10.8040	0.400697	0.200348	−	0.979725i	$-0.435793\pi$
$727$	10.8040	0.400697	0.200348	+	0.979725i	$0.435793\pi$
$728$	0	0
$729$	−21.8072	−0.807675
$730$	0	0
$731$	29.0142	1.07313
$732$	0	0
$733$	−27.4847	−1.01517	−0.507584	−	0.861602i	$-0.669461\pi$
$733$	−27.4847	−1.01517	−0.507584	+	0.861602i	$0.669461\pi$
$734$	0	0
$735$	7.79970	0.287696
$736$	0	0
$737$	11.6259	0.428246
$738$	0	0
$739$	−42.9692	−1.58065	−0.790324	−	0.612689i	$-0.790088\pi$
$739$	−42.9692	−1.58065	−0.790324	+	0.612689i	$0.790088\pi$
$740$	0	0
$741$	−0.414625	−0.0152316
$742$	0	0
$743$	−28.5775	−1.04841	−0.524203	−	0.851593i	$-0.675637\pi$
$743$	−28.5775	−1.04841	−0.524203	+	0.851593i	$0.675637\pi$
$744$	0	0
$745$	35.2373	1.29099
$746$	0	0
$747$	−21.0134	−0.768840
$748$	0	0
$749$	−9.69159	−0.354123
$750$	0	0
$751$	52.7716	1.92566	0.962831	−	0.270104i	$-0.0870581\pi$
$751$	52.7716	1.92566	0.962831	+	0.270104i	$0.0870581\pi$
$752$	0	0
$753$	0.942919	0.0343619
$754$	0	0
$755$	8.76307	0.318921
$756$	0	0
$757$	12.3684	0.449539	0.224769	−	0.974412i	$-0.427837\pi$
$757$	12.3684	0.449539	0.224769	+	0.974412i	$0.427837\pi$
$758$	0	0
$759$	10.0380	0.364357
$760$	0	0
$761$	−34.9719	−1.26773	−0.633865	−	0.773443i	$-0.718533\pi$
$761$	−34.9719	−1.26773	−0.633865	+	0.773443i	$0.718533\pi$
$762$	0	0
$763$	−0.0119288	−0.000431853 0
$764$	0	0
$765$	−33.9469	−1.22735
$766$	0	0
$767$	−14.4511	−0.521801
$768$	0	0
$769$	31.5572	1.13798	0.568991	−	0.822344i	$-0.307334\pi$
$769$	31.5572	1.13798	0.568991	+	0.822344i	$0.307334\pi$
$770$	0	0
$771$	−8.89699	−0.320417
$772$	0	0
$773$	−21.0149	−0.755855	−0.377927	−	0.925835i	$-0.623363\pi$
$773$	−21.0149	−0.755855	−0.377927	+	0.925835i	$0.623363\pi$
$774$	0	0
$775$	39.4492	1.41706
$776$	0	0
$777$	1.51921	0.0545012
$778$	0	0
$779$	−2.36401	−0.0846996
$780$	0	0
$781$	−45.8390	−1.64025
$782$	0	0
$783$	15.3718	0.549342
$784$	0	0
$785$	−28.9611	−1.03367
$786$	0	0
$787$	9.73925	0.347167	0.173583	−	0.984819i	$-0.444465\pi$
$787$	9.73925	0.347167	0.173583	+	0.984819i	$0.444465\pi$
$788$	0	0
$789$	4.97659	0.177171
$790$	0	0
$791$	−20.2610	−0.720399
$792$	0	0
$793$	−11.3614	−0.403457
$794$	0	0
$795$	−15.8033	−0.560484
$796$	0	0
$797$	−48.5600	−1.72008	−0.860041	−	0.510225i	$-0.829562\pi$
$797$	−48.5600	−1.72008	−0.860041	+	0.510225i	$0.829562\pi$
$798$	0	0
$799$	10.0661	0.356113
$800$	0	0
$801$	−30.4433	−1.07566
$802$	0	0
$803$	13.2708	0.468316
$804$	0	0
$805$	−43.6483	−1.53840
$806$	0	0
$807$	−0.559128	−0.0196822
$808$	0	0
$809$	−42.3081	−1.48747	−0.743736	−	0.668473i	$-0.766948\pi$
$809$	−42.3081	−1.48747	−0.743736	+	0.668473i	$0.766948\pi$
$810$	0	0
$811$	10.2832	0.361093	0.180546	−	0.983566i	$-0.442213\pi$
$811$	10.2832	0.361093	0.180546	+	0.983566i	$0.442213\pi$
$812$	0	0
$813$	−7.81026	−0.273918
$814$	0	0
$815$	−42.4826	−1.48810
$816$	0	0
$817$	−10.5342	−0.368544
$818$	0	0
$819$	4.12204	0.144036
$820$	0	0
$821$	−14.5056	−0.506250	−0.253125	−	0.967434i	$-0.581458\pi$
$821$	−14.5056	−0.506250	−0.253125	+	0.967434i	$0.581458\pi$
$822$	0	0
$823$	42.6451	1.48651	0.743257	−	0.669006i	$-0.233280\pi$
$823$	42.6451	1.48651	0.743257	+	0.669006i	$0.233280\pi$
$824$	0	0
$825$	13.7362	0.478234
$826$	0	0
$827$	27.6660	0.962040	0.481020	−	0.876710i	$-0.340266\pi$
$827$	27.6660	0.962040	0.481020	+	0.876710i	$0.340266\pi$
$828$	0	0
$829$	40.1502	1.39447	0.697236	−	0.716841i	$-0.254413\pi$
$829$	40.1502	1.39447	0.697236	+	0.716841i	$0.254413\pi$
$830$	0	0
$831$	−5.34293	−0.185344
$832$	0	0
$833$	16.0826	0.557228
$834$	0	0
$835$	77.3758	2.67770
$836$	0	0
$837$	−5.60941	−0.193890
$838$	0	0
$839$	6.56916	0.226792	0.113396	−	0.993550i	$-0.463827\pi$
$839$	6.56916	0.226792	0.113396	+	0.993550i	$0.463827\pi$
$840$	0	0
$841$	39.6368	1.36679
$842$	0	0
$843$	4.06015	0.139839
$844$	0	0
$845$	47.8410	1.64578
$846$	0	0
$847$	0.225649	0.00775341
$848$	0	0
$849$	6.71353	0.230408
$850$	0	0
$851$	42.7620	1.46586
$852$	0	0
$853$	−17.4858	−0.598703	−0.299351	−	0.954143i	$-0.596770\pi$
$853$	−17.4858	−0.598703	−0.299351	+	0.954143i	$0.596770\pi$
$854$	0	0
$855$	12.3251	0.421510
$856$	0	0
$857$	−2.76241	−0.0943620	−0.0471810	−	0.998886i	$-0.515024\pi$
$857$	−2.76241	−0.0943620	−0.0471810	+	0.998886i	$0.515024\pi$
$858$	0	0
$859$	5.81499	0.198405	0.0992024	−	0.995067i	$-0.468371\pi$
$859$	5.81499	0.198405	0.0992024	+	0.995067i	$0.468371\pi$
$860$	0	0
$861$	−0.800860	−0.0272932
$862$	0	0
$863$	5.42653	0.184721	0.0923607	−	0.995726i	$-0.470559\pi$
$863$	5.42653	0.184721	0.0923607	+	0.995726i	$0.470559\pi$
$864$	0	0
$865$	56.8105	1.93162
$866$	0	0
$867$	−2.95991	−0.100524
$868$	0	0
$869$	−3.34805	−0.113575
$870$	0	0
$871$	−4.57911	−0.155157
$872$	0	0
$873$	1.18172	0.0399951
$874$	0	0
$875$	−36.8421	−1.24549
$876$	0	0
$877$	29.9751	1.01219	0.506093	−	0.862479i	$-0.331089\pi$
$877$	29.9751	1.01219	0.506093	+	0.862479i	$0.331089\pi$
$878$	0	0
$879$	2.85456	0.0962820
$880$	0	0
$881$	23.1094	0.778574	0.389287	−	0.921116i	$-0.372721\pi$
$881$	23.1094	0.778574	0.389287	+	0.921116i	$0.372721\pi$
$882$	0	0
$883$	−42.2997	−1.42350	−0.711749	−	0.702434i	$-0.752097\pi$
$883$	−42.2997	−1.42350	−0.711749	+	0.702434i	$0.752097\pi$
$884$	0	0
$885$	−14.6382	−0.492058
$886$	0	0
$887$	31.1128	1.04467	0.522333	−	0.852742i	$-0.325062\pi$
$887$	31.1128	1.04467	0.522333	+	0.852742i	$0.325062\pi$
$888$	0	0
$889$	−14.2374	−0.477506
$890$	0	0
$891$	27.1863	0.910775
$892$	0	0
$893$	−3.65470	−0.122300
$894$	0	0
$895$	−21.0144	−0.702434
$896$	0	0
$897$	−3.95369	−0.132010
$898$	0	0
$899$	−25.0467	−0.835355
$900$	0	0
$901$	−32.5855	−1.08558
$902$	0	0
$903$	−3.56868	−0.118758
$904$	0	0
$905$	23.9946	0.797606
$906$	0	0
$907$	21.1778	0.703196	0.351598	−	0.936151i	$-0.385638\pi$
$907$	21.1778	0.703196	0.351598	+	0.936151i	$0.385638\pi$
$908$	0	0
$909$	19.6863	0.652953
$910$	0	0
$911$	−23.3822	−0.774686	−0.387343	−	0.921936i	$-0.626607\pi$
$911$	−23.3822	−0.774686	−0.387343	+	0.921936i	$0.626607\pi$
$912$	0	0
$913$	24.2504	0.802572
$914$	0	0
$915$	−11.5085	−0.380460
$916$	0	0
$917$	−8.35920	−0.276045
$918$	0	0
$919$	−57.9593	−1.91190	−0.955951	−	0.293526i	$-0.905171\pi$
$919$	−57.9593	−1.91190	−0.955951	+	0.293526i	$0.905171\pi$
$920$	0	0
$921$	−1.80769	−0.0595654
$922$	0	0
$923$	18.0547	0.594277
$924$	0	0
$925$	58.5162	1.92400
$926$	0	0
$927$	23.9719	0.787341
$928$	0	0
$929$	−23.2594	−0.763116	−0.381558	−	0.924345i	$-0.624612\pi$
$929$	−23.2594	−0.763116	−0.381558	+	0.924345i	$0.624612\pi$
$930$	0	0
$931$	−5.83910	−0.191369
$932$	0	0
$933$	−6.16193	−0.201732
$934$	0	0
$935$	39.1763	1.28120
$936$	0	0
$937$	−11.6400	−0.380263	−0.190132	−	0.981759i	$-0.560891\pi$
$937$	−11.6400	−0.380263	−0.190132	+	0.981759i	$0.560891\pi$
$938$	0	0
$939$	9.38850	0.306382
$940$	0	0
$941$	−31.2134	−1.01753	−0.508764	−	0.860906i	$-0.669898\pi$
$941$	−31.2134	−1.01753	−0.508764	+	0.860906i	$0.669898\pi$
$942$	0	0
$943$	−22.5423	−0.734077
$944$	0	0
$945$	8.49308	0.276280
$946$	0	0
$947$	18.5160	0.601688	0.300844	−	0.953673i	$-0.402732\pi$
$947$	18.5160	0.601688	0.300844	+	0.953673i	$0.402732\pi$
$948$	0	0
$949$	−5.22698	−0.169675
$950$	0	0
$951$	−8.70511	−0.282283
$952$	0	0
$953$	−13.1901	−0.427270	−0.213635	−	0.976913i	$-0.568530\pi$
$953$	−13.1901	−0.427270	−0.213635	+	0.976913i	$0.568530\pi$
$954$	0	0
$955$	71.2540	2.30573
$956$	0	0
$957$	−8.72126	−0.281918
$958$	0	0
$959$	0.738453	0.0238459
$960$	0	0
$961$	−21.8600	−0.705162
$962$	0	0
$963$	26.0955	0.840917
$964$	0	0
$965$	19.0963	0.614732
$966$	0	0
$967$	−20.3247	−0.653599	−0.326800	−	0.945094i	$-0.605970\pi$
$967$	−20.3247	−0.653599	−0.326800	+	0.945094i	$0.605970\pi$
$968$	0	0
$969$	−0.866001	−0.0278200
$970$	0	0
$971$	−6.42856	−0.206302	−0.103151	−	0.994666i	$-0.532893\pi$
$971$	−6.42856	−0.206302	−0.103151	+	0.994666i	$0.532893\pi$
$972$	0	0
$973$	15.7438	0.504722
$974$	0	0
$975$	−5.41030	−0.173268
$976$	0	0
$977$	3.52455	0.112760	0.0563802	−	0.998409i	$-0.482044\pi$
$977$	3.52455	0.112760	0.0563802	+	0.998409i	$0.482044\pi$
$978$	0	0
$979$	35.1329	1.12285
$980$	0	0
$981$	0.0321196	0.00102550
$982$	0	0
$983$	−52.7012	−1.68091	−0.840453	−	0.541885i	$-0.817711\pi$
$983$	−52.7012	−1.68091	−0.840453	+	0.541885i	$0.817711\pi$
$984$	0	0
$985$	39.3613	1.25416
$986$	0	0
$987$	−1.23811	−0.0394094
$988$	0	0
$989$	−100.450	−3.19411
$990$	0	0
$991$	3.68444	0.117040	0.0585200	−	0.998286i	$-0.481362\pi$
$991$	3.68444	0.117040	0.0585200	+	0.998286i	$0.481362\pi$
$992$	0	0
$993$	2.51464	0.0797997
$994$	0	0
$995$	−36.2700	−1.14984
$996$	0	0
$997$	20.5510	0.650855	0.325428	−	0.945567i	$-0.394492\pi$
$997$	20.5510	0.650855	0.325428	+	0.945567i	$0.394492\pi$
$998$	0	0
$999$	−8.32061	−0.263253

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6004.2.a.g.1.16	✓	27

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6004.2.a.g.1.16	✓	27	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)

\(f(q)\)	\(=\)	\(q+0.314419 q^{3} -4.24837 q^{5} +1.07745 q^{7} -2.90114 q^{9} +O(q^{10})\)	\(q+0.314419 q^{3} -4.24837 q^{5} +1.07745 q^{7} -2.90114 q^{9} +3.34805 q^{11} -1.31870 q^{13} -1.33577 q^{15} -2.75429 q^{17} +1.00000 q^{19} +0.338771 q^{21} +9.53559 q^{23} +13.0487 q^{25} -1.85543 q^{27} -8.28473 q^{29} +3.02324 q^{31} +1.05269 q^{33} -4.57741 q^{35} +4.48446 q^{37} -0.414625 q^{39} -2.36401 q^{41} -10.5342 q^{43} +12.3251 q^{45} -3.65470 q^{47} -5.83910 q^{49} -0.866001 q^{51} +11.8308 q^{53} -14.2238 q^{55} +0.314419 q^{57} +10.9586 q^{59} +8.61564 q^{61} -3.12584 q^{63} +5.60233 q^{65} +3.47244 q^{67} +2.99817 q^{69} -13.6913 q^{71} +3.96374 q^{73} +4.10275 q^{75} +3.60736 q^{77} -1.00000 q^{79} +8.12004 q^{81} +7.24315 q^{83} +11.7012 q^{85} -2.60488 q^{87} +10.4935 q^{89} -1.42083 q^{91} +0.950564 q^{93} -4.24837 q^{95} -0.407329 q^{97} -9.71316 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 27 q - 4 q^{3} - 10 q^{5} - 8 q^{7} + 19 q^{9}+O(q^{10}) \) 27 * q - 4 * q^3 - 10 * q^5 - 8 * q^7 + 19 * q^9	\( 27 q - 4 q^{3} - 10 q^{5} - 8 q^{7} + 19 q^{9} + 3 q^{11} - 5 q^{13} - 11 q^{15} - 17 q^{17} + 27 q^{19} - 28 q^{21} - 11 q^{23} + 13 q^{25} - 7 q^{27} - 39 q^{29} - 27 q^{31} - 18 q^{33} - 5 q^{35} - q^{37} - 22 q^{39} - 36 q^{41} - 2 q^{43} - 18 q^{45} - 12 q^{47} + 15 q^{49} + 4 q^{51} - 28 q^{53} + 5 q^{55} - 4 q^{57} - 30 q^{59} - 6 q^{61} - 4 q^{63} - 32 q^{65} + 13 q^{67} - 27 q^{69} - 59 q^{71} - 30 q^{73} - 21 q^{75} - 39 q^{77} - 27 q^{79} - 5 q^{81} + 4 q^{83} - 3 q^{85} + 22 q^{87} - 56 q^{89} - 8 q^{91} - 38 q^{93} - 10 q^{95} - 30 q^{97} + 31 q^{99}+O(q^{100}) \) 27 * q - 4 * q^3 - 10 * q^5 - 8 * q^7 + 19 * q^9 + 3 * q^11 - 5 * q^13 - 11 * q^15 - 17 * q^17 + 27 * q^19 - 28 * q^21 - 11 * q^23 + 13 * q^25 - 7 * q^27 - 39 * q^29 - 27 * q^31 - 18 * q^33 - 5 * q^35 - q^37 - 22 * q^39 - 36 * q^41 - 2 * q^43 - 18 * q^45 - 12 * q^47 + 15 * q^49 + 4 * q^51 - 28 * q^53 + 5 * q^55 - 4 * q^57 - 30 * q^59 - 6 * q^61 - 4 * q^63 - 32 * q^65 + 13 * q^67 - 27 * q^69 - 59 * q^71 - 30 * q^73 - 21 * q^75 - 39 * q^77 - 27 * q^79 - 5 * q^81 + 4 * q^83 - 3 * q^85 + 22 * q^87 - 56 * q^89 - 8 * q^91 - 38 * q^93 - 10 * q^95 - 30 * q^97 + 31 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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