LMFDB - Embedded newform 6004.2.a.e.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:	$24$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
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)

$f(q)$	$=$	$q-2.24876 q^{3} -0.730072 q^{5} -4.54869 q^{7} +2.05693 q^{9} +O(q^{10})$	$q-2.24876 q^{3} -0.730072 q^{5} -4.54869 q^{7} +2.05693 q^{9} +2.49681 q^{11} +3.98439 q^{13} +1.64176 q^{15} +5.48279 q^{17} +1.00000 q^{19} +10.2289 q^{21} -7.71553 q^{23} -4.46699 q^{25} +2.12074 q^{27} +8.91307 q^{29} +1.99962 q^{31} -5.61473 q^{33} +3.32087 q^{35} -2.73646 q^{37} -8.95994 q^{39} -11.8869 q^{41} -4.19788 q^{43} -1.50171 q^{45} -2.64952 q^{47} +13.6906 q^{49} -12.3295 q^{51} -3.40137 q^{53} -1.82285 q^{55} -2.24876 q^{57} -0.194273 q^{59} +0.469019 q^{61} -9.35634 q^{63} -2.90889 q^{65} -13.8768 q^{67} +17.3504 q^{69} -9.52412 q^{71} +7.20890 q^{73} +10.0452 q^{75} -11.3572 q^{77} +1.00000 q^{79} -10.9398 q^{81} +2.01578 q^{83} -4.00283 q^{85} -20.0434 q^{87} -2.10309 q^{89} -18.1237 q^{91} -4.49667 q^{93} -0.730072 q^{95} -7.56980 q^{97} +5.13576 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + q^{3} + 9 q^{5} + 2 q^{7} + 75 q^{9}+O(q^{10}) $ 24 * q + q^3 + 9 * q^5 + 2 * q^7 + 75 * q^9	$ 24 q + q^{3} + 9 q^{5} + 2 q^{7} + 75 q^{9} + 10 q^{11} + 18 q^{13} + 16 q^{15} + 18 q^{17} + 24 q^{19} + 25 q^{21} + 9 q^{23} + 25 q^{25} + 4 q^{27} + 32 q^{29} + 20 q^{31} - 4 q^{33} + 3 q^{35} + 20 q^{37} + 13 q^{39} + 41 q^{41} - 8 q^{43} + 48 q^{45} - 5 q^{47} + 12 q^{49} + 24 q^{51} + 15 q^{53} + 14 q^{55} + q^{57} + 5 q^{59} - 13 q^{61} + 9 q^{63} + 59 q^{65} - 30 q^{67} + 51 q^{69} + 20 q^{73} - 31 q^{75} + 6 q^{77} + 24 q^{79} + 32 q^{81} + 8 q^{83} + 4 q^{85} - 32 q^{87} + 47 q^{89} - 27 q^{91} + 34 q^{93} + 9 q^{95} + 69 q^{97} - 34 q^{99}+O(q^{100}) $ 24 * q + q^3 + 9 * q^5 + 2 * q^7 + 75 * q^9 + 10 * q^11 + 18 * q^13 + 16 * q^15 + 18 * q^17 + 24 * q^19 + 25 * q^21 + 9 * q^23 + 25 * q^25 + 4 * q^27 + 32 * q^29 + 20 * q^31 - 4 * q^33 + 3 * q^35 + 20 * q^37 + 13 * q^39 + 41 * q^41 - 8 * q^43 + 48 * q^45 - 5 * q^47 + 12 * q^49 + 24 * q^51 + 15 * q^53 + 14 * q^55 + q^57 + 5 * q^59 - 13 * q^61 + 9 * q^63 + 59 * q^65 - 30 * q^67 + 51 * q^69 + 20 * q^73 - 31 * q^75 + 6 * q^77 + 24 * q^79 + 32 * q^81 + 8 * q^83 + 4 * q^85 - 32 * q^87 + 47 * q^89 - 27 * q^91 + 34 * q^93 + 9 * q^95 + 69 * q^97 - 34 * 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
$3$	−2.24876	−1.29832	−0.649162	−	0.760650i	$-0.724880\pi$
$3$	−2.24876	−1.29832	−0.649162	+	0.760650i	$0.724880\pi$
$4$	0	0
$5$	−0.730072	−0.326498	−0.163249	−	0.986585i	$-0.552197\pi$
$5$	−0.730072	−0.326498	−0.163249	+	0.986585i	$0.552197\pi$
$6$	0	0
$7$	−4.54869	−1.71924	−0.859622	−	0.510931i	$-0.829301\pi$
$7$	−4.54869	−1.71924	−0.859622	+	0.510931i	$0.829301\pi$
$8$	0	0
$9$	2.05693	0.685643
$10$	0	0
$11$	2.49681	0.752816	0.376408	−	0.926454i	$-0.377159\pi$
$11$	2.49681	0.752816	0.376408	+	0.926454i	$0.377159\pi$
$12$	0	0
$13$	3.98439	1.10507	0.552535	−	0.833490i	$-0.313661\pi$
$13$	3.98439	1.10507	0.552535	+	0.833490i	$0.313661\pi$
$14$	0	0
$15$	1.64176	0.423900
$16$	0	0
$17$	5.48279	1.32977	0.664886	−	0.746945i	$-0.268480\pi$
$17$	5.48279	1.32977	0.664886	+	0.746945i	$0.268480\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	10.2289	2.23213
$22$	0	0
$23$	−7.71553	−1.60880	−0.804400	−	0.594088i	$-0.797513\pi$
$23$	−7.71553	−1.60880	−0.804400	+	0.594088i	$0.797513\pi$
$24$	0	0
$25$	−4.46699	−0.893399
$26$	0	0
$27$	2.12074	0.408137
$28$	0	0
$29$	8.91307	1.65512	0.827558	−	0.561381i	$-0.189730\pi$
$29$	8.91307	1.65512	0.827558	+	0.561381i	$0.189730\pi$
$30$	0	0
$31$	1.99962	0.359143	0.179571	−	0.983745i	$-0.442529\pi$
$31$	1.99962	0.359143	0.179571	+	0.983745i	$0.442529\pi$
$32$	0	0
$33$	−5.61473	−0.977399
$34$	0	0
$35$	3.32087	0.561330
$36$	0	0
$37$	−2.73646	−0.449871	−0.224935	−	0.974374i	$-0.572217\pi$
$37$	−2.73646	−0.449871	−0.224935	+	0.974374i	$0.572217\pi$
$38$	0	0
$39$	−8.95994	−1.43474
$40$	0	0
$41$	−11.8869	−1.85642	−0.928211	−	0.372055i	$-0.878653\pi$
$41$	−11.8869	−1.85642	−0.928211	+	0.372055i	$0.878653\pi$
$42$	0	0
$43$	−4.19788	−0.640171	−0.320085	−	0.947389i	$-0.603712\pi$
$43$	−4.19788	−0.640171	−0.320085	+	0.947389i	$0.603712\pi$
$44$	0	0
$45$	−1.50171	−0.223861
$46$	0	0
$47$	−2.64952	−0.386472	−0.193236	−	0.981152i	$-0.561898\pi$
$47$	−2.64952	−0.386472	−0.193236	+	0.981152i	$0.561898\pi$
$48$	0	0
$49$	13.6906	1.95580
$50$	0	0
$51$	−12.3295	−1.72647
$52$	0	0
$53$	−3.40137	−0.467213	−0.233607	−	0.972331i	$-0.575053\pi$
$53$	−3.40137	−0.467213	−0.233607	+	0.972331i	$0.575053\pi$
$54$	0	0
$55$	−1.82285	−0.245793
$56$	0	0
$57$	−2.24876	−0.297856
$58$	0	0
$59$	−0.194273	−0.0252922	−0.0126461	−	0.999920i	$-0.504025\pi$
$59$	−0.194273	−0.0252922	−0.0126461	+	0.999920i	$0.504025\pi$
$60$	0	0
$61$	0.469019	0.0600518	0.0300259	−	0.999549i	$-0.490441\pi$
$61$	0.469019	0.0600518	0.0300259	+	0.999549i	$0.490441\pi$
$62$	0	0
$63$	−9.35634	−1.17879
$64$	0	0
$65$	−2.90889	−0.360803
$66$	0	0
$67$	−13.8768	−1.69532	−0.847659	−	0.530541i	$-0.821989\pi$
$67$	−13.8768	−1.69532	−0.847659	+	0.530541i	$0.821989\pi$
$68$	0	0
$69$	17.3504	2.08874
$70$	0	0
$71$	−9.52412	−1.13030	−0.565152	−	0.824987i	$-0.691183\pi$
$71$	−9.52412	−1.13030	−0.565152	+	0.824987i	$0.691183\pi$
$72$	0	0
$73$	7.20890	0.843738	0.421869	−	0.906657i	$-0.361374\pi$
$73$	7.20890	0.843738	0.421869	+	0.906657i	$0.361374\pi$
$74$	0	0
$75$	10.0452	1.15992
$76$	0	0
$77$	−11.3572	−1.29427
$78$	0	0
$79$	1.00000	0.112509
$79$	1.00000	0.112509
$80$	0	0
$81$	−10.9398	−1.21554
$82$	0	0
$83$	2.01578	0.221261	0.110631	−	0.993862i	$-0.464713\pi$
$83$	2.01578	0.221261	0.110631	+	0.993862i	$0.464713\pi$
$84$	0	0
$85$	−4.00283	−0.434168
$86$	0	0
$87$	−20.0434	−2.14887
$88$	0	0
$89$	−2.10309	−0.222928	−0.111464	−	0.993768i	$-0.535554\pi$
$89$	−2.10309	−0.222928	−0.111464	+	0.993768i	$0.535554\pi$
$90$	0	0
$91$	−18.1237	−1.89988
$92$	0	0
$93$	−4.49667	−0.466283
$94$	0	0
$95$	−0.730072	−0.0749039
$96$	0	0
$97$	−7.56980	−0.768596	−0.384298	−	0.923209i	$-0.625557\pi$
$97$	−7.56980	−0.768596	−0.384298	+	0.923209i	$0.625557\pi$
$98$	0	0
$99$	5.13576	0.516163
$100$	0	0
$101$	10.6032	1.05506	0.527531	−	0.849536i	$-0.323118\pi$
$101$	10.6032	1.05506	0.527531	+	0.849536i	$0.323118\pi$
$102$	0	0
$103$	3.88289	0.382592	0.191296	−	0.981532i	$-0.438731\pi$
$103$	3.88289	0.382592	0.191296	+	0.981532i	$0.438731\pi$
$104$	0	0
$105$	−7.46785	−0.728788
$106$	0	0
$107$	11.5148	1.11318	0.556588	−	0.830789i	$-0.312110\pi$
$107$	11.5148	1.11318	0.556588	+	0.830789i	$0.312110\pi$
$108$	0	0
$109$	12.8178	1.22772	0.613859	−	0.789415i	$-0.289616\pi$
$109$	12.8178	1.22772	0.613859	+	0.789415i	$0.289616\pi$
$110$	0	0
$111$	6.15364	0.584078
$112$	0	0
$113$	−13.3196	−1.25301	−0.626503	−	0.779419i	$-0.715515\pi$
$113$	−13.3196	−1.25301	−0.626503	+	0.779419i	$0.715515\pi$
$114$	0	0
$115$	5.63290	0.525270
$116$	0	0
$117$	8.19560	0.757684
$118$	0	0
$119$	−24.9395	−2.28620
$120$	0	0
$121$	−4.76595	−0.433268
$122$	0	0
$123$	26.7308	2.41023
$124$	0	0
$125$	6.91159	0.618192
$126$	0	0
$127$	−9.01125	−0.799619	−0.399810	−	0.916598i	$-0.630924\pi$
$127$	−9.01125	−0.799619	−0.399810	+	0.916598i	$0.630924\pi$
$128$	0	0
$129$	9.44003	0.831148
$130$	0	0
$131$	10.9341	0.955321	0.477661	−	0.878544i	$-0.341485\pi$
$131$	10.9341	0.955321	0.477661	+	0.878544i	$0.341485\pi$
$132$	0	0
$133$	−4.54869	−0.394421
$134$	0	0
$135$	−1.54829	−0.133256
$136$	0	0
$137$	6.17260	0.527360	0.263680	−	0.964610i	$-0.415064\pi$
$137$	6.17260	0.527360	0.263680	+	0.964610i	$0.415064\pi$
$138$	0	0
$139$	9.20310	0.780596	0.390298	−	0.920688i	$-0.372372\pi$
$139$	9.20310	0.780596	0.390298	+	0.920688i	$0.372372\pi$
$140$	0	0
$141$	5.95814	0.501766
$142$	0	0
$143$	9.94825	0.831914
$144$	0	0
$145$	−6.50718	−0.540392
$146$	0	0
$147$	−30.7868	−2.53926
$148$	0	0
$149$	13.3564	1.09420	0.547100	−	0.837067i	$-0.315732\pi$
$149$	13.3564	1.09420	0.547100	+	0.837067i	$0.315732\pi$
$150$	0	0
$151$	1.03248	0.0840221	0.0420110	−	0.999117i	$-0.486624\pi$
$151$	1.03248	0.0840221	0.0420110	+	0.999117i	$0.486624\pi$
$152$	0	0
$153$	11.2777	0.911749
$154$	0	0
$155$	−1.45987	−0.117259
$156$	0	0
$157$	3.68172	0.293833	0.146917	−	0.989149i	$-0.453065\pi$
$157$	3.68172	0.293833	0.146917	+	0.989149i	$0.453065\pi$
$158$	0	0
$159$	7.64886	0.606594
$160$	0	0
$161$	35.0956	2.76592
$162$	0	0
$163$	17.8058	1.39466	0.697330	−	0.716750i	$-0.254371\pi$
$163$	17.8058	1.39466	0.697330	+	0.716750i	$0.254371\pi$
$164$	0	0
$165$	4.09916	0.319119
$166$	0	0
$167$	−12.7953	−0.990128	−0.495064	−	0.868856i	$-0.664855\pi$
$167$	−12.7953	−0.990128	−0.495064	+	0.868856i	$0.664855\pi$
$168$	0	0
$169$	2.87534	0.221180
$170$	0	0
$171$	2.05693	0.157297
$172$	0	0
$173$	15.6312	1.18842	0.594209	−	0.804311i	$-0.297465\pi$
$173$	15.6312	1.18842	0.594209	+	0.804311i	$0.297465\pi$
$174$	0	0
$175$	20.3190	1.53597
$176$	0	0
$177$	0.436874	0.0328374
$178$	0	0
$179$	−20.8897	−1.56137	−0.780686	−	0.624923i	$-0.785130\pi$
$179$	−20.8897	−1.56137	−0.780686	+	0.624923i	$0.785130\pi$
$180$	0	0
$181$	−9.82069	−0.729966	−0.364983	−	0.931014i	$-0.618925\pi$
$181$	−9.82069	−0.729966	−0.364983	+	0.931014i	$0.618925\pi$
$182$	0	0
$183$	−1.05471	−0.0779666
$184$	0	0
$185$	1.99781	0.146882
$186$	0	0
$187$	13.6895	1.00107
$188$	0	0
$189$	−9.64659	−0.701686
$190$	0	0
$191$	−2.36272	−0.170960	−0.0854801	−	0.996340i	$-0.527242\pi$
$191$	−2.36272	−0.170960	−0.0854801	+	0.996340i	$0.527242\pi$
$192$	0	0
$193$	18.3865	1.32349	0.661744	−	0.749730i	$-0.269817\pi$
$193$	18.3865	1.32349	0.661744	+	0.749730i	$0.269817\pi$
$194$	0	0
$195$	6.54140	0.468440
$196$	0	0
$197$	−13.5868	−0.968019	−0.484009	−	0.875063i	$-0.660820\pi$
$197$	−13.5868	−0.968019	−0.484009	+	0.875063i	$0.660820\pi$
$198$	0	0
$199$	0.0630855	0.00447201	0.00223601	−	0.999998i	$-0.499288\pi$
$199$	0.0630855	0.00447201	0.00223601	+	0.999998i	$0.499288\pi$
$200$	0	0
$201$	31.2056	2.20107
$202$	0	0
$203$	−40.5428	−2.84555
$204$	0	0
$205$	8.67829	0.606118
$206$	0	0
$207$	−15.8703	−1.10306
$208$	0	0
$209$	2.49681	0.172708
$210$	0	0
$211$	5.87051	0.404142	0.202071	−	0.979371i	$-0.435233\pi$
$211$	5.87051	0.404142	0.202071	+	0.979371i	$0.435233\pi$
$212$	0	0
$213$	21.4175	1.46750
$214$	0	0
$215$	3.06476	0.209015
$216$	0	0
$217$	−9.09565	−0.617453
$218$	0	0
$219$	−16.2111	−1.09545
$220$	0	0
$221$	21.8455	1.46949
$222$	0	0
$223$	23.7752	1.59211	0.796054	−	0.605226i	$-0.206917\pi$
$223$	23.7752	1.59211	0.796054	+	0.605226i	$0.206917\pi$
$224$	0	0
$225$	−9.18829	−0.612553
$226$	0	0
$227$	−20.8212	−1.38195	−0.690976	−	0.722878i	$-0.742819\pi$
$227$	−20.8212	−1.38195	−0.690976	+	0.722878i	$0.742819\pi$
$228$	0	0
$229$	16.5592	1.09426	0.547132	−	0.837047i	$-0.315720\pi$
$229$	16.5592	1.09426	0.547132	+	0.837047i	$0.315720\pi$
$230$	0	0
$231$	25.5397	1.68039
$232$	0	0
$233$	−12.3136	−0.806694	−0.403347	−	0.915047i	$-0.632153\pi$
$233$	−12.3136	−0.806694	−0.403347	+	0.915047i	$0.632153\pi$
$234$	0	0
$235$	1.93434	0.126183
$236$	0	0
$237$	−2.24876	−0.146073
$238$	0	0
$239$	4.70471	0.304322	0.152161	−	0.988356i	$-0.451377\pi$
$239$	4.70471	0.304322	0.152161	+	0.988356i	$0.451377\pi$
$240$	0	0
$241$	13.4517	0.866498	0.433249	−	0.901274i	$-0.357367\pi$
$241$	13.4517	0.866498	0.433249	+	0.901274i	$0.357367\pi$
$242$	0	0
$243$	18.2388	1.17002
$244$	0	0
$245$	−9.99511	−0.638564
$246$	0	0
$247$	3.98439	0.253520
$248$	0	0
$249$	−4.53302	−0.287268
$250$	0	0
$251$	18.7304	1.18225	0.591125	−	0.806580i	$-0.298684\pi$
$251$	18.7304	1.18225	0.591125	+	0.806580i	$0.298684\pi$
$252$	0	0
$253$	−19.2642	−1.21113
$254$	0	0
$255$	9.00142	0.563691
$256$	0	0
$257$	13.9006	0.867094	0.433547	−	0.901131i	$-0.357262\pi$
$257$	13.9006	0.867094	0.433547	+	0.901131i	$0.357262\pi$
$258$	0	0
$259$	12.4473	0.773437
$260$	0	0
$261$	18.3336	1.13482
$262$	0	0
$263$	1.10999	0.0684450	0.0342225	−	0.999414i	$-0.489105\pi$
$263$	1.10999	0.0684450	0.0342225	+	0.999414i	$0.489105\pi$
$264$	0	0
$265$	2.48324	0.152544
$266$	0	0
$267$	4.72936	0.289432
$268$	0	0
$269$	−2.29482	−0.139917	−0.0699587	−	0.997550i	$-0.522287\pi$
$269$	−2.29482	−0.139917	−0.0699587	+	0.997550i	$0.522287\pi$
$270$	0	0
$271$	−19.7378	−1.19899	−0.599493	−	0.800380i	$-0.704631\pi$
$271$	−19.7378	−1.19899	−0.599493	+	0.800380i	$0.704631\pi$
$272$	0	0
$273$	40.7560	2.46666
$274$	0	0
$275$	−11.1532	−0.672565
$276$	0	0
$277$	13.8599	0.832762	0.416381	−	0.909190i	$-0.363298\pi$
$277$	13.8599	0.832762	0.416381	+	0.909190i	$0.363298\pi$
$278$	0	0
$279$	4.11308	0.246244
$280$	0	0
$281$	11.5187	0.687150	0.343575	−	0.939125i	$-0.388362\pi$
$281$	11.5187	0.687150	0.343575	+	0.939125i	$0.388362\pi$
$282$	0	0
$283$	0.845788	0.0502769	0.0251384	−	0.999684i	$-0.491997\pi$
$283$	0.845788	0.0502769	0.0251384	+	0.999684i	$0.491997\pi$
$284$	0	0
$285$	1.64176	0.0972494
$286$	0	0
$287$	54.0698	3.19164
$288$	0	0
$289$	13.0610	0.768292
$290$	0	0
$291$	17.0227	0.997886
$292$	0	0
$293$	−25.7984	−1.50716	−0.753581	−	0.657356i	$-0.771675\pi$
$293$	−25.7984	−1.50716	−0.753581	+	0.657356i	$0.771675\pi$
$294$	0	0
$295$	0.141833	0.00825786
$296$	0	0
$297$	5.29508	0.307252
$298$	0	0
$299$	−30.7417	−1.77784
$300$	0	0
$301$	19.0948	1.10061
$302$	0	0
$303$	−23.8442	−1.36981
$304$	0	0
$305$	−0.342418	−0.0196068
$306$	0	0
$307$	9.81787	0.560335	0.280168	−	0.959951i	$-0.409610\pi$
$307$	9.81787	0.560335	0.280168	+	0.959951i	$0.409610\pi$
$308$	0	0
$309$	−8.73169	−0.496728
$310$	0	0
$311$	2.21175	0.125417	0.0627085	−	0.998032i	$-0.480026\pi$
$311$	2.21175	0.125417	0.0627085	+	0.998032i	$0.480026\pi$
$312$	0	0
$313$	−23.7719	−1.34367	−0.671834	−	0.740702i	$-0.734493\pi$
$313$	−23.7719	−1.34367	−0.671834	+	0.740702i	$0.734493\pi$
$314$	0	0
$315$	6.83080	0.384872
$316$	0	0
$317$	12.0667	0.677732	0.338866	−	0.940835i	$-0.389957\pi$
$317$	12.0667	0.677732	0.338866	+	0.940835i	$0.389957\pi$
$318$	0	0
$319$	22.2542	1.24600
$320$	0	0
$321$	−25.8940	−1.44526
$322$	0	0
$323$	5.48279	0.305071
$324$	0	0
$325$	−17.7982	−0.987268
$326$	0	0
$327$	−28.8241	−1.59398
$328$	0	0
$329$	12.0518	0.664440
$330$	0	0
$331$	33.7128	1.85302	0.926512	−	0.376266i	$-0.122792\pi$
$331$	33.7128	1.85302	0.926512	+	0.376266i	$0.122792\pi$
$332$	0	0
$333$	−5.62870	−0.308451
$334$	0	0
$335$	10.1311	0.553518
$336$	0	0
$337$	−13.6179	−0.741815	−0.370907	−	0.928670i	$-0.620953\pi$
$337$	−13.6179	−0.741815	−0.370907	+	0.928670i	$0.620953\pi$
$338$	0	0
$339$	29.9527	1.62681
$340$	0	0
$341$	4.99267	0.270368
$342$	0	0
$343$	−30.4334	−1.64325
$344$	0	0
$345$	−12.6670	−0.681971
$346$	0	0
$347$	−15.7274	−0.844292	−0.422146	−	0.906528i	$-0.638723\pi$
$347$	−15.7274	−0.844292	−0.422146	+	0.906528i	$0.638723\pi$
$348$	0	0
$349$	−9.86048	−0.527819	−0.263910	−	0.964547i	$-0.585012\pi$
$349$	−9.86048	−0.527819	−0.263910	+	0.964547i	$0.585012\pi$
$350$	0	0
$351$	8.44985	0.451020
$352$	0	0
$353$	−36.1307	−1.92304	−0.961521	−	0.274733i	$-0.911411\pi$
$353$	−36.1307	−1.92304	−0.961521	+	0.274733i	$0.911411\pi$
$354$	0	0
$355$	6.95329	0.369043
$356$	0	0
$357$	56.0830	2.96823
$358$	0	0
$359$	−5.24924	−0.277044	−0.138522	−	0.990359i	$-0.544235\pi$
$359$	−5.24924	−0.277044	−0.138522	+	0.990359i	$0.544235\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	10.7175	0.562522
$364$	0	0
$365$	−5.26302	−0.275479
$366$	0	0
$367$	10.6033	0.553488	0.276744	−	0.960944i	$-0.410745\pi$
$367$	10.6033	0.553488	0.276744	+	0.960944i	$0.410745\pi$
$368$	0	0
$369$	−24.4505	−1.27284
$370$	0	0
$371$	15.4718	0.803253
$372$	0	0
$373$	−23.9045	−1.23773	−0.618865	−	0.785498i	$-0.712407\pi$
$373$	−23.9045	−1.23773	−0.618865	+	0.785498i	$0.712407\pi$
$374$	0	0
$375$	−15.5425	−0.802612
$376$	0	0
$377$	35.5131	1.82902
$378$	0	0
$379$	−4.01957	−0.206471	−0.103236	−	0.994657i	$-0.532920\pi$
$379$	−4.01957	−0.206471	−0.103236	+	0.994657i	$0.532920\pi$
$380$	0	0
$381$	20.2642	1.03816
$382$	0	0
$383$	25.2650	1.29098	0.645490	−	0.763769i	$-0.276653\pi$
$383$	25.2650	1.29098	0.645490	+	0.763769i	$0.276653\pi$
$384$	0	0
$385$	8.29158	0.422578
$386$	0	0
$387$	−8.63474	−0.438929
$388$	0	0
$389$	32.7136	1.65865	0.829323	−	0.558770i	$-0.188727\pi$
$389$	32.7136	1.65865	0.829323	+	0.558770i	$0.188727\pi$
$390$	0	0
$391$	−42.3026	−2.13934
$392$	0	0
$393$	−24.5883	−1.24032
$394$	0	0
$395$	−0.730072	−0.0367339
$396$	0	0
$397$	20.1314	1.01036	0.505182	−	0.863013i	$-0.331425\pi$
$397$	20.1314	1.01036	0.505182	+	0.863013i	$0.331425\pi$
$398$	0	0
$399$	10.2289	0.512086
$400$	0	0
$401$	38.4867	1.92194	0.960968	−	0.276661i	$-0.0892278\pi$
$401$	38.4867	1.92194	0.960968	+	0.276661i	$0.0892278\pi$
$402$	0	0
$403$	7.96726	0.396878
$404$	0	0
$405$	7.98687	0.396871
$406$	0	0
$407$	−6.83241	−0.338670
$408$	0	0
$409$	−31.0098	−1.53334	−0.766669	−	0.642042i	$-0.778087\pi$
$409$	−31.0098	−1.53334	−0.766669	+	0.642042i	$0.778087\pi$
$410$	0	0
$411$	−13.8807	−0.684684
$412$	0	0
$413$	0.883688	0.0434834
$414$	0	0
$415$	−1.47167	−0.0722414
$416$	0	0
$417$	−20.6956	−1.01347
$418$	0	0
$419$	−16.5398	−0.808024	−0.404012	−	0.914754i	$-0.632385\pi$
$419$	−16.5398	−0.808024	−0.404012	+	0.914754i	$0.632385\pi$
$420$	0	0
$421$	22.5747	1.10022	0.550112	−	0.835091i	$-0.314585\pi$
$421$	22.5747	1.10022	0.550112	+	0.835091i	$0.314585\pi$
$422$	0	0
$423$	−5.44988	−0.264982
$424$	0	0
$425$	−24.4916	−1.18802
$426$	0	0
$427$	−2.13342	−0.103244
$428$	0	0
$429$	−22.3712	−1.08009
$430$	0	0
$431$	21.4007	1.03083	0.515417	−	0.856940i	$-0.327637\pi$
$431$	21.4007	1.03083	0.515417	+	0.856940i	$0.327637\pi$
$432$	0	0
$433$	−7.37530	−0.354434	−0.177217	−	0.984172i	$-0.556709\pi$
$433$	−7.37530	−0.354434	−0.177217	+	0.984172i	$0.556709\pi$
$434$	0	0
$435$	14.6331	0.701604
$436$	0	0
$437$	−7.71553	−0.369084
$438$	0	0
$439$	19.8077	0.945369	0.472685	−	0.881232i	$-0.343285\pi$
$439$	19.8077	0.945369	0.472685	+	0.881232i	$0.343285\pi$
$440$	0	0
$441$	28.1606	1.34098
$442$	0	0
$443$	13.9530	0.662925	0.331462	−	0.943468i	$-0.392458\pi$
$443$	13.9530	0.662925	0.331462	+	0.943468i	$0.392458\pi$
$444$	0	0
$445$	1.53541	0.0727855
$446$	0	0
$447$	−30.0354	−1.42063
$448$	0	0
$449$	−9.89242	−0.466852	−0.233426	−	0.972375i	$-0.574994\pi$
$449$	−9.89242	−0.466852	−0.233426	+	0.972375i	$0.574994\pi$
$450$	0	0
$451$	−29.6793	−1.39754
$452$	0	0
$453$	−2.32180	−0.109088
$454$	0	0
$455$	13.2316	0.620309
$456$	0	0
$457$	−29.8071	−1.39432	−0.697159	−	0.716917i	$-0.745553\pi$
$457$	−29.8071	−1.39432	−0.697159	+	0.716917i	$0.745553\pi$
$458$	0	0
$459$	11.6276	0.542728
$460$	0	0
$461$	−22.8057	−1.06217	−0.531085	−	0.847319i	$-0.678215\pi$
$461$	−22.8057	−1.06217	−0.531085	+	0.847319i	$0.678215\pi$
$462$	0	0
$463$	−21.6797	−1.00754	−0.503772	−	0.863837i	$-0.668055\pi$
$463$	−21.6797	−1.00754	−0.503772	+	0.863837i	$0.668055\pi$
$464$	0	0
$465$	3.28290	0.152241
$466$	0	0
$467$	13.1104	0.606675	0.303338	−	0.952883i	$-0.401899\pi$
$467$	13.1104	0.606675	0.303338	+	0.952883i	$0.401899\pi$
$468$	0	0
$469$	63.1211	2.91466
$470$	0	0
$471$	−8.27931	−0.381491
$472$	0	0
$473$	−10.4813	−0.481931
$474$	0	0
$475$	−4.46699	−0.204960
$476$	0	0
$477$	−6.99637	−0.320342
$478$	0	0
$479$	−14.3169	−0.654156	−0.327078	−	0.944997i	$-0.606064\pi$
$479$	−14.3169	−0.654156	−0.327078	+	0.944997i	$0.606064\pi$
$480$	0	0
$481$	−10.9031	−0.497139
$482$	0	0
$483$	−78.9215	−3.59105
$484$	0	0
$485$	5.52650	0.250945
$486$	0	0
$487$	42.3899	1.92087	0.960434	−	0.278509i	$-0.0898401\pi$
$487$	42.3899	1.92087	0.960434	+	0.278509i	$0.0898401\pi$
$488$	0	0
$489$	−40.0411	−1.81072
$490$	0	0
$491$	−23.2327	−1.04848	−0.524238	−	0.851572i	$-0.675650\pi$
$491$	−23.2327	−1.04848	−0.524238	+	0.851572i	$0.675650\pi$
$492$	0	0
$493$	48.8685	2.20092
$494$	0	0
$495$	−3.74948	−0.168526
$496$	0	0
$497$	43.3222	1.94327
$498$	0	0
$499$	8.99347	0.402603	0.201301	−	0.979529i	$-0.435483\pi$
$499$	8.99347	0.402603	0.201301	+	0.979529i	$0.435483\pi$
$500$	0	0
$501$	28.7735	1.28551
$502$	0	0
$503$	23.2630	1.03725	0.518624	−	0.855003i	$-0.326445\pi$
$503$	23.2630	1.03725	0.518624	+	0.855003i	$0.326445\pi$
$504$	0	0
$505$	−7.74113	−0.344476
$506$	0	0
$507$	−6.46594	−0.287163
$508$	0	0
$509$	2.08351	0.0923501	0.0461750	−	0.998933i	$-0.485297\pi$
$509$	2.08351	0.0923501	0.0461750	+	0.998933i	$0.485297\pi$
$510$	0	0
$511$	−32.7911	−1.45059
$512$	0	0
$513$	2.12074	0.0936330
$514$	0	0
$515$	−2.83479	−0.124916
$516$	0	0
$517$	−6.61535	−0.290943
$518$	0	0
$519$	−35.1508	−1.54295
$520$	0	0
$521$	14.8880	0.652257	0.326129	−	0.945325i	$-0.394256\pi$
$521$	14.8880	0.652257	0.326129	+	0.945325i	$0.394256\pi$
$522$	0	0
$523$	−10.2023	−0.446117	−0.223058	−	0.974805i	$-0.571604\pi$
$523$	−10.2023	−0.446117	−0.223058	+	0.974805i	$0.571604\pi$
$524$	0	0
$525$	−45.6925	−1.99419
$526$	0	0
$527$	10.9635	0.477578
$528$	0	0
$529$	36.5294	1.58824
$530$	0	0
$531$	−0.399606	−0.0173414
$532$	0	0
$533$	−47.3620	−2.05148
$534$	0	0
$535$	−8.40662	−0.363450
$536$	0	0
$537$	46.9760	2.02717
$538$	0	0
$539$	34.1827	1.47236
$540$	0	0
$541$	26.7974	1.15211	0.576054	−	0.817411i	$-0.304592\pi$
$541$	26.7974	1.15211	0.576054	+	0.817411i	$0.304592\pi$
$542$	0	0
$543$	22.0844	0.947732
$544$	0	0
$545$	−9.35789	−0.400848
$546$	0	0
$547$	−31.3396	−1.33999	−0.669993	−	0.742368i	$-0.733703\pi$
$547$	−31.3396	−1.33999	−0.669993	+	0.742368i	$0.733703\pi$
$548$	0	0
$549$	0.964740	0.0411741
$550$	0	0
$551$	8.91307	0.379709
$552$	0	0
$553$	−4.54869	−0.193430
$554$	0	0
$555$	−4.49260	−0.190700
$556$	0	0
$557$	27.9368	1.18372	0.591861	−	0.806040i	$-0.298394\pi$
$557$	27.9368	1.18372	0.591861	+	0.806040i	$0.298394\pi$
$558$	0	0
$559$	−16.7260	−0.707433
$560$	0	0
$561$	−30.7844	−1.29972
$562$	0	0
$563$	−14.4538	−0.609154	−0.304577	−	0.952488i	$-0.598515\pi$
$563$	−14.4538	−0.609154	−0.304577	+	0.952488i	$0.598515\pi$
$564$	0	0
$565$	9.72431	0.409105
$566$	0	0
$567$	49.7619	2.08980
$568$	0	0
$569$	30.2934	1.26996	0.634982	−	0.772527i	$-0.281007\pi$
$569$	30.2934	1.26996	0.634982	+	0.772527i	$0.281007\pi$
$570$	0	0
$571$	15.7553	0.659338	0.329669	−	0.944097i	$-0.393063\pi$
$571$	15.7553	0.659338	0.329669	+	0.944097i	$0.393063\pi$
$572$	0	0
$573$	5.31319	0.221962
$574$	0	0
$575$	34.4652	1.43730
$576$	0	0
$577$	−12.0842	−0.503070	−0.251535	−	0.967848i	$-0.580935\pi$
$577$	−12.0842	−0.503070	−0.251535	+	0.967848i	$0.580935\pi$
$578$	0	0
$579$	−41.3468	−1.71831
$580$	0	0
$581$	−9.16918	−0.380402
$582$	0	0
$583$	−8.49256	−0.351726
$584$	0	0
$585$	−5.98338	−0.247382
$586$	0	0
$587$	27.5745	1.13812	0.569060	−	0.822296i	$-0.307307\pi$
$587$	27.5745	1.13812	0.569060	+	0.822296i	$0.307307\pi$
$588$	0	0
$589$	1.99962	0.0823929
$590$	0	0
$591$	30.5534	1.25680
$592$	0	0
$593$	36.6606	1.50547	0.752736	−	0.658322i	$-0.228734\pi$
$593$	36.6606	1.50547	0.752736	+	0.658322i	$0.228734\pi$
$594$	0	0
$595$	18.2076	0.746441
$596$	0	0
$597$	−0.141864	−0.00580612
$598$	0	0
$599$	−17.8540	−0.729493	−0.364747	−	0.931107i	$-0.618844\pi$
$599$	−17.8540	−0.729493	−0.364747	+	0.931107i	$0.618844\pi$
$600$	0	0
$601$	9.86207	0.402282	0.201141	−	0.979562i	$-0.435535\pi$
$601$	9.86207	0.402282	0.201141	+	0.979562i	$0.435535\pi$
$602$	0	0
$603$	−28.5436	−1.16238
$604$	0	0
$605$	3.47949	0.141461
$606$	0	0
$607$	38.5163	1.56333	0.781664	−	0.623699i	$-0.214371\pi$
$607$	38.5163	1.56333	0.781664	+	0.623699i	$0.214371\pi$
$608$	0	0
$609$	91.1710	3.69444
$610$	0	0
$611$	−10.5567	−0.427079
$612$	0	0
$613$	−7.66169	−0.309453	−0.154726	−	0.987957i	$-0.549450\pi$
$613$	−7.66169	−0.309453	−0.154726	+	0.987957i	$0.549450\pi$
$614$	0	0
$615$	−19.5154	−0.786938
$616$	0	0
$617$	18.4847	0.744167	0.372083	−	0.928199i	$-0.378644\pi$
$617$	18.4847	0.744167	0.372083	+	0.928199i	$0.378644\pi$
$618$	0	0
$619$	12.9963	0.522366	0.261183	−	0.965289i	$-0.415887\pi$
$619$	12.9963	0.522366	0.261183	+	0.965289i	$0.415887\pi$
$620$	0	0
$621$	−16.3626	−0.656610
$622$	0	0
$623$	9.56632	0.383267
$624$	0	0
$625$	17.2890	0.691560
$626$	0	0
$627$	−5.61473	−0.224231
$628$	0	0
$629$	−15.0034	−0.598225
$630$	0	0
$631$	−13.4889	−0.536983	−0.268492	−	0.963282i	$-0.586525\pi$
$631$	−13.4889	−0.536983	−0.268492	+	0.963282i	$0.586525\pi$
$632$	0	0
$633$	−13.2014	−0.524707
$634$	0	0
$635$	6.57887	0.261074
$636$	0	0
$637$	54.5485	2.16129
$638$	0	0
$639$	−19.5904	−0.774986
$640$	0	0
$641$	38.0087	1.50125	0.750626	−	0.660727i	$-0.229752\pi$
$641$	38.0087	1.50125	0.750626	+	0.660727i	$0.229752\pi$
$642$	0	0
$643$	−31.5461	−1.24406	−0.622029	−	0.782994i	$-0.713691\pi$
$643$	−31.5461	−1.24406	−0.622029	+	0.782994i	$0.713691\pi$
$644$	0	0
$645$	−6.89191	−0.271369
$646$	0	0
$647$	−40.9868	−1.61136	−0.805679	−	0.592352i	$-0.798199\pi$
$647$	−40.9868	−1.61136	−0.805679	+	0.592352i	$0.798199\pi$
$648$	0	0
$649$	−0.485063	−0.0190404
$650$	0	0
$651$	20.4540	0.801654
$652$	0	0
$653$	−0.278844	−0.0109120	−0.00545600	−	0.999985i	$-0.501737\pi$
$653$	−0.278844	−0.0109120	−0.00545600	+	0.999985i	$0.501737\pi$
$654$	0	0
$655$	−7.98272	−0.311911
$656$	0	0
$657$	14.8282	0.578503
$658$	0	0
$659$	8.02928	0.312777	0.156388	−	0.987696i	$-0.450015\pi$
$659$	8.02928	0.312777	0.156388	+	0.987696i	$0.450015\pi$
$660$	0	0
$661$	33.6624	1.30932	0.654658	−	0.755926i	$-0.272813\pi$
$661$	33.6624	1.30932	0.654658	+	0.755926i	$0.272813\pi$
$662$	0	0
$663$	−49.1254	−1.90787
$664$	0	0
$665$	3.32087	0.128778
$666$	0	0
$667$	−68.7690	−2.66275
$668$	0	0
$669$	−53.4648	−2.06707
$670$	0	0
$671$	1.17105	0.0452079
$672$	0	0
$673$	19.6323	0.756769	0.378385	−	0.925648i	$-0.376480\pi$
$673$	19.6323	0.756769	0.378385	+	0.925648i	$0.376480\pi$
$674$	0	0
$675$	−9.47333	−0.364629
$676$	0	0
$677$	−18.5363	−0.712409	−0.356204	−	0.934408i	$-0.615929\pi$
$677$	−18.5363	−0.712409	−0.356204	+	0.934408i	$0.615929\pi$
$678$	0	0
$679$	34.4326	1.32140
$680$	0	0
$681$	46.8219	1.79422
$682$	0	0
$683$	32.6614	1.24975	0.624877	−	0.780723i	$-0.285149\pi$
$683$	32.6614	1.24975	0.624877	+	0.780723i	$0.285149\pi$
$684$	0	0
$685$	−4.50644	−0.172182
$686$	0	0
$687$	−37.2377	−1.42071
$688$	0	0
$689$	−13.5524	−0.516304
$690$	0	0
$691$	20.1276	0.765690	0.382845	−	0.923813i	$-0.374944\pi$
$691$	20.1276	0.765690	0.382845	+	0.923813i	$0.374944\pi$
$692$	0	0
$693$	−23.3610	−0.887410
$694$	0	0
$695$	−6.71893	−0.254863
$696$	0	0
$697$	−65.1733	−2.46862
$698$	0	0
$699$	27.6905	1.04735
$700$	0	0
$701$	−10.9645	−0.414125	−0.207063	−	0.978328i	$-0.566390\pi$
$701$	−10.9645	−0.414125	−0.207063	+	0.978328i	$0.566390\pi$
$702$	0	0
$703$	−2.73646	−0.103207
$704$	0	0
$705$	−4.34988	−0.163826
$706$	0	0
$707$	−48.2308	−1.81391
$708$	0	0
$709$	27.7437	1.04194	0.520969	−	0.853576i	$-0.325571\pi$
$709$	27.7437	1.04194	0.520969	+	0.853576i	$0.325571\pi$
$710$	0	0
$711$	2.05693	0.0771409
$712$	0	0
$713$	−15.4281	−0.577788
$714$	0	0
$715$	−7.26294	−0.271619
$716$	0	0
$717$	−10.5798	−0.395109
$718$	0	0
$719$	0.656755	0.0244928	0.0122464	−	0.999925i	$-0.496102\pi$
$719$	0.656755	0.0244928	0.0122464	+	0.999925i	$0.496102\pi$
$720$	0	0
$721$	−17.6620	−0.657769
$722$	0	0
$723$	−30.2496	−1.12499
$724$	0	0
$725$	−39.8146	−1.47868
$726$	0	0
$727$	25.2861	0.937809	0.468904	−	0.883249i	$-0.344649\pi$
$727$	25.2861	0.937809	0.468904	+	0.883249i	$0.344649\pi$
$728$	0	0
$729$	−8.19534	−0.303531
$730$	0	0
$731$	−23.0161	−0.851281
$732$	0	0
$733$	−6.77113	−0.250097	−0.125049	−	0.992151i	$-0.539909\pi$
$733$	−6.77113	−0.250097	−0.125049	+	0.992151i	$0.539909\pi$
$734$	0	0
$735$	22.4766	0.829063
$736$	0	0
$737$	−34.6476	−1.27626
$738$	0	0
$739$	−41.0041	−1.50836	−0.754180	−	0.656668i	$-0.771965\pi$
$739$	−41.0041	−1.50836	−0.754180	+	0.656668i	$0.771965\pi$
$740$	0	0
$741$	−8.95994	−0.329151
$742$	0	0
$743$	31.2005	1.14464	0.572318	−	0.820032i	$-0.306044\pi$
$743$	31.2005	1.14464	0.572318	+	0.820032i	$0.306044\pi$
$744$	0	0
$745$	−9.75115	−0.357255
$746$	0	0
$747$	4.14633	0.151706
$748$	0	0
$749$	−52.3772	−1.91382
$750$	0	0
$751$	−2.62746	−0.0958773	−0.0479387	−	0.998850i	$-0.515265\pi$
$751$	−2.62746	−0.0958773	−0.0479387	+	0.998850i	$0.515265\pi$
$752$	0	0
$753$	−42.1201	−1.53494
$754$	0	0
$755$	−0.753786	−0.0274331
$756$	0	0
$757$	45.2153	1.64338	0.821690	−	0.569935i	$-0.193032\pi$
$757$	45.2153	1.64338	0.821690	+	0.569935i	$0.193032\pi$
$758$	0	0
$759$	43.3206	1.57244
$760$	0	0
$761$	47.6926	1.72886	0.864428	−	0.502757i	$-0.167681\pi$
$761$	47.6926	1.72886	0.864428	+	0.502757i	$0.167681\pi$
$762$	0	0
$763$	−58.3040	−2.11075
$764$	0	0
$765$	−8.23355	−0.297684
$766$	0	0
$767$	−0.774059	−0.0279496
$768$	0	0
$769$	−39.4183	−1.42146	−0.710730	−	0.703465i	$-0.751635\pi$
$769$	−39.4183	−1.42146	−0.710730	+	0.703465i	$0.751635\pi$
$770$	0	0
$771$	−31.2591	−1.12577
$772$	0	0
$773$	31.5280	1.13398	0.566992	−	0.823723i	$-0.308107\pi$
$773$	31.5280	1.13398	0.566992	+	0.823723i	$0.308107\pi$
$774$	0	0
$775$	−8.93230	−0.320858
$776$	0	0
$777$	−27.9910	−1.00417
$778$	0	0
$779$	−11.8869	−0.425892
$780$	0	0
$781$	−23.7799	−0.850911
$782$	0	0
$783$	18.9023	0.675513
$784$	0	0
$785$	−2.68792	−0.0959361
$786$	0	0
$787$	9.91581	0.353461	0.176730	−	0.984259i	$-0.443448\pi$
$787$	9.91581	0.353461	0.176730	+	0.984259i	$0.443448\pi$
$788$	0	0
$789$	−2.49611	−0.0888638
$790$	0	0
$791$	60.5869	2.15422
$792$	0	0
$793$	1.86875	0.0663614
$794$	0	0
$795$	−5.58422	−0.198052
$796$	0	0
$797$	10.9124	0.386539	0.193269	−	0.981146i	$-0.438091\pi$
$797$	10.9124	0.386539	0.193269	+	0.981146i	$0.438091\pi$
$798$	0	0
$799$	−14.5268	−0.513920
$800$	0	0
$801$	−4.32592	−0.152849
$802$	0	0
$803$	17.9992	0.635180
$804$	0	0
$805$	−25.6223	−0.903067
$806$	0	0
$807$	5.16050	0.181658
$808$	0	0
$809$	49.9387	1.75575	0.877876	−	0.478889i	$-0.158960\pi$
$809$	49.9387	1.75575	0.877876	+	0.478889i	$0.158960\pi$
$810$	0	0
$811$	−7.68830	−0.269973	−0.134986	−	0.990847i	$-0.543099\pi$
$811$	−7.68830	−0.269973	−0.134986	+	0.990847i	$0.543099\pi$
$812$	0	0
$813$	44.3856	1.55667
$814$	0	0
$815$	−12.9995	−0.455354
$816$	0	0
$817$	−4.19788	−0.146865
$818$	0	0
$819$	−37.2793	−1.30264
$820$	0	0
$821$	43.4470	1.51631	0.758156	−	0.652073i	$-0.226100\pi$
$821$	43.4470	1.51631	0.758156	+	0.652073i	$0.226100\pi$
$822$	0	0
$823$	49.0948	1.71134	0.855668	−	0.517526i	$-0.173147\pi$
$823$	49.0948	1.71134	0.855668	+	0.517526i	$0.173147\pi$
$824$	0	0
$825$	25.0810	0.873207
$826$	0	0
$827$	11.6871	0.406402	0.203201	−	0.979137i	$-0.434866\pi$
$827$	11.6871	0.406402	0.203201	+	0.979137i	$0.434866\pi$
$828$	0	0
$829$	−11.1849	−0.388469	−0.194234	−	0.980955i	$-0.562222\pi$
$829$	−11.1849	−0.388469	−0.194234	+	0.980955i	$0.562222\pi$
$830$	0	0
$831$	−31.1677	−1.08119
$832$	0	0
$833$	75.0625	2.60076
$834$	0	0
$835$	9.34148	0.323275
$836$	0	0
$837$	4.24068	0.146579
$838$	0	0
$839$	37.6959	1.30141	0.650704	−	0.759332i	$-0.274474\pi$
$839$	37.6959	1.30141	0.650704	+	0.759332i	$0.274474\pi$
$840$	0	0
$841$	50.4428	1.73941
$842$	0	0
$843$	−25.9029	−0.892142
$844$	0	0
$845$	−2.09920	−0.0722148
$846$	0	0
$847$	21.6788	0.744893
$848$	0	0
$849$	−1.90198	−0.0652756
$850$	0	0
$851$	21.1132	0.723752
$852$	0	0
$853$	49.2608	1.68666	0.843329	−	0.537398i	$-0.180593\pi$
$853$	49.2608	1.68666	0.843329	+	0.537398i	$0.180593\pi$
$854$	0	0
$855$	−1.50171	−0.0513573
$856$	0	0
$857$	25.7877	0.880891	0.440445	−	0.897779i	$-0.354821\pi$
$857$	25.7877	0.880891	0.440445	+	0.897779i	$0.354821\pi$
$858$	0	0
$859$	−32.4792	−1.10818	−0.554088	−	0.832458i	$-0.686933\pi$
$859$	−32.4792	−1.10818	−0.554088	+	0.832458i	$0.686933\pi$
$860$	0	0
$861$	−121.590	−4.14378
$862$	0	0
$863$	−1.30842	−0.0445390	−0.0222695	−	0.999752i	$-0.507089\pi$
$863$	−1.30842	−0.0445390	−0.0222695	+	0.999752i	$0.507089\pi$
$864$	0	0
$865$	−11.4119	−0.388017
$866$	0	0
$867$	−29.3710	−0.997492
$868$	0	0
$869$	2.49681	0.0846984
$870$	0	0
$871$	−55.2904	−1.87344
$872$	0	0
$873$	−15.5705	−0.526983
$874$	0	0
$875$	−31.4387	−1.06282
$876$	0	0
$877$	−13.0264	−0.439869	−0.219934	−	0.975515i	$-0.570584\pi$
$877$	−13.0264	−0.439869	−0.219934	+	0.975515i	$0.570584\pi$
$878$	0	0
$879$	58.0146	1.95678
$880$	0	0
$881$	−21.2111	−0.714621	−0.357310	−	0.933986i	$-0.616306\pi$
$881$	−21.2111	−0.714621	−0.357310	+	0.933986i	$0.616306\pi$
$882$	0	0
$883$	−32.4901	−1.09338	−0.546689	−	0.837335i	$-0.684112\pi$
$883$	−32.4901	−1.09338	−0.546689	+	0.837335i	$0.684112\pi$
$884$	0	0
$885$	−0.318950	−0.0107214
$886$	0	0
$887$	19.6312	0.659152	0.329576	−	0.944129i	$-0.393094\pi$
$887$	19.6312	0.659152	0.329576	+	0.944129i	$0.393094\pi$
$888$	0	0
$889$	40.9894	1.37474
$890$	0	0
$891$	−27.3147	−0.915075
$892$	0	0
$893$	−2.64952	−0.0886628
$894$	0	0
$895$	15.2510	0.509785
$896$	0	0
$897$	69.1307	2.30821
$898$	0	0
$899$	17.8228	0.594422
$900$	0	0
$901$	−18.6490	−0.621287
$902$	0	0
$903$	−42.9398	−1.42895
$904$	0	0
$905$	7.16982	0.238333
$906$	0	0
$907$	10.0644	0.334183	0.167092	−	0.985941i	$-0.446562\pi$
$907$	10.0644	0.334183	0.167092	+	0.985941i	$0.446562\pi$
$908$	0	0
$909$	21.8101	0.723396
$910$	0	0
$911$	46.1372	1.52859	0.764297	−	0.644865i	$-0.223086\pi$
$911$	46.1372	1.52859	0.764297	+	0.644865i	$0.223086\pi$
$912$	0	0
$913$	5.03303	0.166569
$914$	0	0
$915$	0.770017	0.0254560
$916$	0	0
$917$	−49.7361	−1.64243
$918$	0	0
$919$	−16.4400	−0.542307	−0.271154	−	0.962536i	$-0.587405\pi$
$919$	−16.4400	−0.542307	−0.271154	+	0.962536i	$0.587405\pi$
$920$	0	0
$921$	−22.0781	−0.727497
$922$	0	0
$923$	−37.9478	−1.24907
$924$	0	0
$925$	12.2237	0.401914
$926$	0	0
$927$	7.98682	0.262322
$928$	0	0
$929$	−53.9627	−1.77046	−0.885229	−	0.465156i	$-0.845998\pi$
$929$	−53.9627	−1.77046	−0.885229	+	0.465156i	$0.845998\pi$
$930$	0	0
$931$	13.6906	0.448691
$932$	0	0
$933$	−4.97371	−0.162832
$934$	0	0
$935$	−9.99431	−0.326849
$936$	0	0
$937$	14.7221	0.480949	0.240475	−	0.970655i	$-0.422697\pi$
$937$	14.7221	0.480949	0.240475	+	0.970655i	$0.422697\pi$
$938$	0	0
$939$	53.4574	1.74452
$940$	0	0
$941$	−25.1243	−0.819030	−0.409515	−	0.912303i	$-0.634302\pi$
$941$	−25.1243	−0.819030	−0.409515	+	0.912303i	$0.634302\pi$
$942$	0	0
$943$	91.7137	2.98661
$944$	0	0
$945$	7.04271	0.229099
$946$	0	0
$947$	−21.5577	−0.700530	−0.350265	−	0.936651i	$-0.613908\pi$
$947$	−21.5577	−0.700530	−0.350265	+	0.936651i	$0.613908\pi$
$948$	0	0
$949$	28.7231	0.932390
$950$	0	0
$951$	−27.1351	−0.879915
$952$	0	0
$953$	−54.0916	−1.75220	−0.876099	−	0.482132i	$-0.839863\pi$
$953$	−54.0916	−1.75220	−0.876099	+	0.482132i	$0.839863\pi$
$954$	0	0
$955$	1.72496	0.0558182
$956$	0	0
$957$	−50.0444	−1.61771
$958$	0	0
$959$	−28.0772	−0.906661
$960$	0	0
$961$	−27.0015	−0.871017
$962$	0	0
$963$	23.6851	0.763242
$964$	0	0
$965$	−13.4235	−0.432116
$966$	0	0
$967$	52.5627	1.69030	0.845151	−	0.534528i	$-0.179511\pi$
$967$	52.5627	1.69030	0.845151	+	0.534528i	$0.179511\pi$
$968$	0	0
$969$	−12.3295	−0.396080
$970$	0	0
$971$	21.5700	0.692215	0.346107	−	0.938195i	$-0.387503\pi$
$971$	21.5700	0.692215	0.346107	+	0.938195i	$0.387503\pi$
$972$	0	0
$973$	−41.8620	−1.34203
$974$	0	0
$975$	40.0240	1.28179
$976$	0	0
$977$	35.4295	1.13349	0.566746	−	0.823893i	$-0.308202\pi$
$977$	35.4295	1.13349	0.566746	+	0.823893i	$0.308202\pi$
$978$	0	0
$979$	−5.25102	−0.167823
$980$	0	0
$981$	26.3652	0.841777
$982$	0	0
$983$	12.8218	0.408952	0.204476	−	0.978872i	$-0.434451\pi$
$983$	12.8218	0.408952	0.204476	+	0.978872i	$0.434451\pi$
$984$	0	0
$985$	9.91934	0.316056
$986$	0	0
$987$	−27.1017	−0.862658
$988$	0	0
$989$	32.3889	1.02991
$990$	0	0
$991$	28.5689	0.907522	0.453761	−	0.891123i	$-0.350082\pi$
$991$	28.5689	0.907522	0.453761	+	0.891123i	$0.350082\pi$
$992$	0	0
$993$	−75.8121	−2.40582
$994$	0	0
$995$	−0.0460570	−0.00146010
$996$	0	0
$997$	−35.2593	−1.11667	−0.558336	−	0.829615i	$-0.688560\pi$
$997$	−35.2593	−1.11667	−0.558336	+	0.829615i	$0.688560\pi$
$998$	0	0
$999$	−5.80331	−0.183609

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6004.2.a.e.1.7	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6004.2.a.e.1.7	✓	24	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)

\(f(q)\)	\(=\)	\(q-2.24876 q^{3} -0.730072 q^{5} -4.54869 q^{7} +2.05693 q^{9} +O(q^{10})\)	\(q-2.24876 q^{3} -0.730072 q^{5} -4.54869 q^{7} +2.05693 q^{9} +2.49681 q^{11} +3.98439 q^{13} +1.64176 q^{15} +5.48279 q^{17} +1.00000 q^{19} +10.2289 q^{21} -7.71553 q^{23} -4.46699 q^{25} +2.12074 q^{27} +8.91307 q^{29} +1.99962 q^{31} -5.61473 q^{33} +3.32087 q^{35} -2.73646 q^{37} -8.95994 q^{39} -11.8869 q^{41} -4.19788 q^{43} -1.50171 q^{45} -2.64952 q^{47} +13.6906 q^{49} -12.3295 q^{51} -3.40137 q^{53} -1.82285 q^{55} -2.24876 q^{57} -0.194273 q^{59} +0.469019 q^{61} -9.35634 q^{63} -2.90889 q^{65} -13.8768 q^{67} +17.3504 q^{69} -9.52412 q^{71} +7.20890 q^{73} +10.0452 q^{75} -11.3572 q^{77} +1.00000 q^{79} -10.9398 q^{81} +2.01578 q^{83} -4.00283 q^{85} -20.0434 q^{87} -2.10309 q^{89} -18.1237 q^{91} -4.49667 q^{93} -0.730072 q^{95} -7.56980 q^{97} +5.13576 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + q^{3} + 9 q^{5} + 2 q^{7} + 75 q^{9}+O(q^{10}) \) 24 * q + q^3 + 9 * q^5 + 2 * q^7 + 75 * q^9	\( 24 q + q^{3} + 9 q^{5} + 2 q^{7} + 75 q^{9} + 10 q^{11} + 18 q^{13} + 16 q^{15} + 18 q^{17} + 24 q^{19} + 25 q^{21} + 9 q^{23} + 25 q^{25} + 4 q^{27} + 32 q^{29} + 20 q^{31} - 4 q^{33} + 3 q^{35} + 20 q^{37} + 13 q^{39} + 41 q^{41} - 8 q^{43} + 48 q^{45} - 5 q^{47} + 12 q^{49} + 24 q^{51} + 15 q^{53} + 14 q^{55} + q^{57} + 5 q^{59} - 13 q^{61} + 9 q^{63} + 59 q^{65} - 30 q^{67} + 51 q^{69} + 20 q^{73} - 31 q^{75} + 6 q^{77} + 24 q^{79} + 32 q^{81} + 8 q^{83} + 4 q^{85} - 32 q^{87} + 47 q^{89} - 27 q^{91} + 34 q^{93} + 9 q^{95} + 69 q^{97} - 34 q^{99}+O(q^{100}) \) 24 * q + q^3 + 9 * q^5 + 2 * q^7 + 75 * q^9 + 10 * q^11 + 18 * q^13 + 16 * q^15 + 18 * q^17 + 24 * q^19 + 25 * q^21 + 9 * q^23 + 25 * q^25 + 4 * q^27 + 32 * q^29 + 20 * q^31 - 4 * q^33 + 3 * q^35 + 20 * q^37 + 13 * q^39 + 41 * q^41 - 8 * q^43 + 48 * q^45 - 5 * q^47 + 12 * q^49 + 24 * q^51 + 15 * q^53 + 14 * q^55 + q^57 + 5 * q^59 - 13 * q^61 + 9 * q^63 + 59 * q^65 - 30 * q^67 + 51 * q^69 + 20 * q^73 - 31 * q^75 + 6 * q^77 + 24 * q^79 + 32 * q^81 + 8 * q^83 + 4 * q^85 - 32 * q^87 + 47 * q^89 - 27 * q^91 + 34 * q^93 + 9 * q^95 + 69 * q^97 - 34 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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