LMFDB - Embedded newform 6004.2.a.h.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:	$0$
Dimension:	$31$
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)

$f(q)$	$=$	$q-3.18617 q^{3} +0.345750 q^{5} -2.57023 q^{7} +7.15170 q^{9} +O(q^{10})$	$q-3.18617 q^{3} +0.345750 q^{5} -2.57023 q^{7} +7.15170 q^{9} -2.58863 q^{11} +5.15381 q^{13} -1.10162 q^{15} +7.10229 q^{17} -1.00000 q^{19} +8.18920 q^{21} +4.50456 q^{23} -4.88046 q^{25} -13.2280 q^{27} -3.13522 q^{29} +7.52954 q^{31} +8.24782 q^{33} -0.888656 q^{35} -2.97579 q^{37} -16.4209 q^{39} +11.7942 q^{41} +5.64558 q^{43} +2.47270 q^{45} +4.74211 q^{47} -0.393922 q^{49} -22.6291 q^{51} -8.86740 q^{53} -0.895017 q^{55} +3.18617 q^{57} +3.42656 q^{59} +4.45661 q^{61} -18.3815 q^{63} +1.78193 q^{65} -5.30660 q^{67} -14.3523 q^{69} -13.0343 q^{71} -1.31886 q^{73} +15.5500 q^{75} +6.65337 q^{77} -1.00000 q^{79} +20.6917 q^{81} -8.97621 q^{83} +2.45561 q^{85} +9.98935 q^{87} -10.3163 q^{89} -13.2465 q^{91} -23.9904 q^{93} -0.345750 q^{95} +5.46326 q^{97} -18.5131 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 31 q - 4 q^{3} + 10 q^{5} + 11 q^{7} + 47 q^{9}+O(q^{10}) $ 31 * q - 4 * q^3 + 10 * q^5 + 11 * q^7 + 47 * q^9	$ 31 q - 4 q^{3} + 10 q^{5} + 11 q^{7} + 47 q^{9} - 4 q^{11} + 11 q^{13} + 5 q^{15} + 14 q^{17} - 31 q^{19} + 22 q^{21} + 15 q^{23} + 59 q^{25} + 5 q^{27} + 34 q^{29} - 12 q^{31} + 10 q^{33} + 8 q^{35} + 16 q^{37} + 18 q^{39} + 27 q^{41} + 2 q^{43} + 22 q^{45} + 30 q^{47} + 62 q^{49} - 14 q^{51} + 35 q^{53} + 8 q^{55} + 4 q^{57} - 16 q^{59} + 37 q^{61} + 31 q^{63} + 80 q^{65} + 16 q^{67} + q^{69} + 19 q^{71} + 38 q^{73} + 21 q^{75} + 44 q^{77} - 31 q^{79} + 55 q^{81} - 12 q^{83} + 66 q^{85} + 58 q^{87} + 16 q^{89} - 42 q^{91} + 10 q^{93} - 10 q^{95} - 12 q^{97} + 12 q^{99}+O(q^{100}) $ 31 * q - 4 * q^3 + 10 * q^5 + 11 * q^7 + 47 * q^9 - 4 * q^11 + 11 * q^13 + 5 * q^15 + 14 * q^17 - 31 * q^19 + 22 * q^21 + 15 * q^23 + 59 * q^25 + 5 * q^27 + 34 * q^29 - 12 * q^31 + 10 * q^33 + 8 * q^35 + 16 * q^37 + 18 * q^39 + 27 * q^41 + 2 * q^43 + 22 * q^45 + 30 * q^47 + 62 * q^49 - 14 * q^51 + 35 * q^53 + 8 * q^55 + 4 * q^57 - 16 * q^59 + 37 * q^61 + 31 * q^63 + 80 * q^65 + 16 * q^67 + q^69 + 19 * q^71 + 38 * q^73 + 21 * q^75 + 44 * q^77 - 31 * q^79 + 55 * q^81 - 12 * q^83 + 66 * q^85 + 58 * q^87 + 16 * q^89 - 42 * q^91 + 10 * q^93 - 10 * q^95 - 12 * q^97 + 12 * 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
$3$	−3.18617	−1.83954	−0.919769	−	0.392460i	$-0.871624\pi$
$3$	−3.18617	−1.83954	−0.919769	+	0.392460i	$0.871624\pi$
$4$	0	0
$5$	0.345750	0.154624	0.0773119	−	0.997007i	$-0.475366\pi$
$5$	0.345750	0.154624	0.0773119	+	0.997007i	$0.475366\pi$
$6$	0	0
$7$	−2.57023	−0.971455	−0.485728	−	0.874110i	$-0.661445\pi$
$7$	−2.57023	−0.971455	−0.485728	+	0.874110i	$0.661445\pi$
$8$	0	0
$9$	7.15170	2.38390
$10$	0	0
$11$	−2.58863	−0.780501	−0.390251	−	0.920709i	$-0.627612\pi$
$11$	−2.58863	−0.780501	−0.390251	+	0.920709i	$0.627612\pi$
$12$	0	0
$13$	5.15381	1.42941	0.714705	−	0.699426i	$-0.246561\pi$
$13$	5.15381	1.42941	0.714705	+	0.699426i	$0.246561\pi$
$14$	0	0
$15$	−1.10162	−0.284437
$16$	0	0
$17$	7.10229	1.72256	0.861279	−	0.508132i	$-0.169664\pi$
$17$	7.10229	1.72256	0.861279	+	0.508132i	$0.169664\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	8.18920	1.78703
$22$	0	0
$23$	4.50456	0.939265	0.469633	−	0.882862i	$-0.344386\pi$
$23$	4.50456	0.939265	0.469633	+	0.882862i	$0.344386\pi$
$24$	0	0
$25$	−4.88046	−0.976091
$26$	0	0
$27$	−13.2280	−2.54574
$28$	0	0
$29$	−3.13522	−0.582196	−0.291098	−	0.956693i	$-0.594021\pi$
$29$	−3.13522	−0.582196	−0.291098	+	0.956693i	$0.594021\pi$
$30$	0	0
$31$	7.52954	1.35235	0.676173	−	0.736743i	$-0.263637\pi$
$31$	7.52954	1.35235	0.676173	+	0.736743i	$0.263637\pi$
$32$	0	0
$33$	8.24782	1.43576
$34$	0	0
$35$	−0.888656	−0.150210
$36$	0	0
$37$	−2.97579	−0.489216	−0.244608	−	0.969622i	$-0.578659\pi$
$37$	−2.97579	−0.489216	−0.244608	+	0.969622i	$0.578659\pi$
$38$	0	0
$39$	−16.4209	−2.62945
$40$	0	0
$41$	11.7942	1.84195	0.920973	−	0.389626i	$-0.127396\pi$
$41$	11.7942	1.84195	0.920973	+	0.389626i	$0.127396\pi$
$42$	0	0
$43$	5.64558	0.860943	0.430471	−	0.902604i	$-0.358347\pi$
$43$	5.64558	0.860943	0.430471	+	0.902604i	$0.358347\pi$
$44$	0	0
$45$	2.47270	0.368608
$46$	0	0
$47$	4.74211	0.691708	0.345854	−	0.938288i	$-0.387589\pi$
$47$	4.74211	0.691708	0.345854	+	0.938288i	$0.387589\pi$
$48$	0	0
$49$	−0.393922	−0.0562745
$50$	0	0
$51$	−22.6291	−3.16871
$52$	0	0
$53$	−8.86740	−1.21803	−0.609015	−	0.793159i	$-0.708435\pi$
$53$	−8.86740	−1.21803	−0.609015	+	0.793159i	$0.708435\pi$
$54$	0	0
$55$	−0.895017	−0.120684
$56$	0	0
$57$	3.18617	0.422019
$58$	0	0
$59$	3.42656	0.446100	0.223050	−	0.974807i	$-0.428399\pi$
$59$	3.42656	0.446100	0.223050	+	0.974807i	$0.428399\pi$
$60$	0	0
$61$	4.45661	0.570611	0.285305	−	0.958437i	$-0.407905\pi$
$61$	4.45661	0.570611	0.285305	+	0.958437i	$0.407905\pi$
$62$	0	0
$63$	−18.3815	−2.31585
$64$	0	0
$65$	1.78193	0.221021
$66$	0	0
$67$	−5.30660	−0.648304	−0.324152	−	0.946005i	$-0.605079\pi$
$67$	−5.30660	−0.648304	−0.324152	+	0.946005i	$0.605079\pi$
$68$	0	0
$69$	−14.3523	−1.72781
$70$	0	0
$71$	−13.0343	−1.54689	−0.773444	−	0.633864i	$-0.781468\pi$
$71$	−13.0343	−1.54689	−0.773444	+	0.633864i	$0.781468\pi$
$72$	0	0
$73$	−1.31886	−0.154361	−0.0771807	−	0.997017i	$-0.524592\pi$
$73$	−1.31886	−0.154361	−0.0771807	+	0.997017i	$0.524592\pi$
$74$	0	0
$75$	15.5500	1.79556
$76$	0	0
$77$	6.65337	0.758222
$78$	0	0
$79$	−1.00000	−0.112509
$79$	−1.00000	−0.112509
$80$	0	0
$81$	20.6917	2.29908
$82$	0	0
$83$	−8.97621	−0.985267	−0.492634	−	0.870237i	$-0.663966\pi$
$83$	−8.97621	−0.985267	−0.492634	+	0.870237i	$0.663966\pi$
$84$	0	0
$85$	2.45561	0.266349
$86$	0	0
$87$	9.98935	1.07097
$88$	0	0
$89$	−10.3163	−1.09352	−0.546762	−	0.837288i	$-0.684140\pi$
$89$	−10.3163	−1.09352	−0.546762	+	0.837288i	$0.684140\pi$
$90$	0	0
$91$	−13.2465	−1.38861
$92$	0	0
$93$	−23.9904	−2.48769
$94$	0	0
$95$	−0.345750	−0.0354732
$96$	0	0
$97$	5.46326	0.554710	0.277355	−	0.960768i	$-0.410542\pi$
$97$	5.46326	0.554710	0.277355	+	0.960768i	$0.410542\pi$
$98$	0	0
$99$	−18.5131	−1.86064
$100$	0	0
$101$	5.40068	0.537388	0.268694	−	0.963226i	$-0.413408\pi$
$101$	5.40068	0.537388	0.268694	+	0.963226i	$0.413408\pi$
$102$	0	0
$103$	4.94671	0.487413	0.243707	−	0.969849i	$-0.421637\pi$
$103$	4.94671	0.487413	0.243707	+	0.969849i	$0.421637\pi$
$104$	0	0
$105$	2.83141	0.276317
$106$	0	0
$107$	−16.7209	−1.61647	−0.808234	−	0.588862i	$-0.799576\pi$
$107$	−16.7209	−1.61647	−0.808234	+	0.588862i	$0.799576\pi$
$108$	0	0
$109$	16.8638	1.61526	0.807632	−	0.589687i	$-0.200749\pi$
$109$	16.8638	1.61526	0.807632	+	0.589687i	$0.200749\pi$
$110$	0	0
$111$	9.48137	0.899932
$112$	0	0
$113$	9.48841	0.892595	0.446297	−	0.894885i	$-0.352742\pi$
$113$	9.48841	0.892595	0.446297	+	0.894885i	$0.352742\pi$
$114$	0	0
$115$	1.55745	0.145233
$116$	0	0
$117$	36.8585	3.40757
$118$	0	0
$119$	−18.2545	−1.67339
$120$	0	0
$121$	−4.29900	−0.390818
$122$	0	0
$123$	−37.5784	−3.38833
$124$	0	0
$125$	−3.41616	−0.305551
$126$	0	0
$127$	4.44398	0.394340	0.197170	−	0.980369i	$-0.436825\pi$
$127$	4.44398	0.394340	0.197170	+	0.980369i	$0.436825\pi$
$128$	0	0
$129$	−17.9878	−1.58374
$130$	0	0
$131$	11.5117	1.00579	0.502893	−	0.864349i	$-0.332269\pi$
$131$	11.5117	1.00579	0.502893	+	0.864349i	$0.332269\pi$
$132$	0	0
$133$	2.57023	0.222867
$134$	0	0
$135$	−4.57359	−0.393632
$136$	0	0
$137$	−1.95519	−0.167043	−0.0835216	−	0.996506i	$-0.526617\pi$
$137$	−1.95519	−0.167043	−0.0835216	+	0.996506i	$0.526617\pi$
$138$	0	0
$139$	10.0137	0.849351	0.424676	−	0.905346i	$-0.360388\pi$
$139$	10.0137	0.849351	0.424676	+	0.905346i	$0.360388\pi$
$140$	0	0
$141$	−15.1092	−1.27242
$142$	0	0
$143$	−13.3413	−1.11566
$144$	0	0
$145$	−1.08400	−0.0900214
$146$	0	0
$147$	1.25510	0.103519
$148$	0	0
$149$	−21.2754	−1.74295	−0.871473	−	0.490443i	$-0.836835\pi$
$149$	−21.2754	−1.74295	−0.871473	+	0.490443i	$0.836835\pi$
$150$	0	0
$151$	−3.77823	−0.307468	−0.153734	−	0.988112i	$-0.549130\pi$
$151$	−3.77823	−0.307468	−0.153734	+	0.988112i	$0.549130\pi$
$152$	0	0
$153$	50.7934	4.10641
$154$	0	0
$155$	2.60334	0.209105
$156$	0	0
$157$	7.94542	0.634114	0.317057	−	0.948407i	$-0.397305\pi$
$157$	7.94542	0.634114	0.317057	+	0.948407i	$0.397305\pi$
$158$	0	0
$159$	28.2531	2.24061
$160$	0	0
$161$	−11.5777	−0.912454
$162$	0	0
$163$	−18.9184	−1.48181	−0.740903	−	0.671612i	$-0.765602\pi$
$163$	−18.9184	−1.48181	−0.740903	+	0.671612i	$0.765602\pi$
$164$	0	0
$165$	2.85168	0.222003
$166$	0	0
$167$	13.0002	1.00599	0.502994	−	0.864290i	$-0.332232\pi$
$167$	13.0002	1.00599	0.502994	+	0.864290i	$0.332232\pi$
$168$	0	0
$169$	13.5618	1.04321
$170$	0	0
$171$	−7.15170	−0.546904
$172$	0	0
$173$	−9.72605	−0.739458	−0.369729	−	0.929140i	$-0.620550\pi$
$173$	−9.72605	−0.739458	−0.369729	+	0.929140i	$0.620550\pi$
$174$	0	0
$175$	12.5439	0.948229
$176$	0	0
$177$	−10.9176	−0.820618
$178$	0	0
$179$	1.87382	0.140056	0.0700280	−	0.997545i	$-0.477691\pi$
$179$	1.87382	0.140056	0.0700280	+	0.997545i	$0.477691\pi$
$180$	0	0
$181$	−3.48868	−0.259312	−0.129656	−	0.991559i	$-0.541387\pi$
$181$	−3.48868	−0.259312	−0.129656	+	0.991559i	$0.541387\pi$
$182$	0	0
$183$	−14.1995	−1.04966
$184$	0	0
$185$	−1.02888	−0.0756445
$186$	0	0
$187$	−18.3852	−1.34446
$188$	0	0
$189$	33.9991	2.47307
$190$	0	0
$191$	16.3442	1.18262	0.591311	−	0.806444i	$-0.298611\pi$
$191$	16.3442	1.18262	0.591311	+	0.806444i	$0.298611\pi$
$192$	0	0
$193$	10.2780	0.739824	0.369912	−	0.929067i	$-0.379388\pi$
$193$	10.2780	0.739824	0.369912	+	0.929067i	$0.379388\pi$
$194$	0	0
$195$	−5.67753	−0.406576
$196$	0	0
$197$	−3.10771	−0.221415	−0.110707	−	0.993853i	$-0.535312\pi$
$197$	−3.10771	−0.221415	−0.110707	+	0.993853i	$0.535312\pi$
$198$	0	0
$199$	−19.9064	−1.41112	−0.705562	−	0.708648i	$-0.749305\pi$
$199$	−19.9064	−1.41112	−0.705562	+	0.708648i	$0.749305\pi$
$200$	0	0
$201$	16.9077	1.19258
$202$	0	0
$203$	8.05823	0.565577
$204$	0	0
$205$	4.07784	0.284809
$206$	0	0
$207$	32.2152	2.23911
$208$	0	0
$209$	2.58863	0.179059
$210$	0	0
$211$	9.39814	0.646994	0.323497	−	0.946229i	$-0.395141\pi$
$211$	9.39814	0.646994	0.323497	+	0.946229i	$0.395141\pi$
$212$	0	0
$213$	41.5296	2.84556
$214$	0	0
$215$	1.95196	0.133122
$216$	0	0
$217$	−19.3527	−1.31374
$218$	0	0
$219$	4.20213	0.283954
$220$	0	0
$221$	36.6038	2.46224
$222$	0	0
$223$	−16.6700	−1.11631	−0.558154	−	0.829737i	$-0.688490\pi$
$223$	−16.6700	−1.11631	−0.558154	+	0.829737i	$0.688490\pi$
$224$	0	0
$225$	−34.9036	−2.32690
$226$	0	0
$227$	25.7510	1.70916	0.854578	−	0.519323i	$-0.173816\pi$
$227$	25.7510	1.70916	0.854578	+	0.519323i	$0.173816\pi$
$228$	0	0
$229$	21.7871	1.43973	0.719867	−	0.694112i	$-0.244203\pi$
$229$	21.7871	1.43973	0.719867	+	0.694112i	$0.244203\pi$
$230$	0	0
$231$	−21.1988	−1.39478
$232$	0	0
$233$	27.1509	1.77871	0.889357	−	0.457213i	$-0.151152\pi$
$233$	27.1509	1.77871	0.889357	+	0.457213i	$0.151152\pi$
$234$	0	0
$235$	1.63958	0.106955
$236$	0	0
$237$	3.18617	0.206964
$238$	0	0
$239$	21.1484	1.36798	0.683988	−	0.729493i	$-0.260244\pi$
$239$	21.1484	1.36798	0.683988	+	0.729493i	$0.260244\pi$
$240$	0	0
$241$	−22.9819	−1.48039	−0.740197	−	0.672390i	$-0.765268\pi$
$241$	−22.9819	−1.48039	−0.740197	+	0.672390i	$0.765268\pi$
$242$	0	0
$243$	−26.2433	−1.68351
$244$	0	0
$245$	−0.136198	−0.00870139
$246$	0	0
$247$	−5.15381	−0.327929
$248$	0	0
$249$	28.5998	1.81244
$250$	0	0
$251$	25.7781	1.62710	0.813548	−	0.581497i	$-0.197533\pi$
$251$	25.7781	1.62710	0.813548	+	0.581497i	$0.197533\pi$
$252$	0	0
$253$	−11.6606	−0.733098
$254$	0	0
$255$	−7.82401	−0.489958
$256$	0	0
$257$	7.30912	0.455930	0.227965	−	0.973669i	$-0.426793\pi$
$257$	7.30912	0.455930	0.227965	+	0.973669i	$0.426793\pi$
$258$	0	0
$259$	7.64845	0.475252
$260$	0	0
$261$	−22.4222	−1.38790
$262$	0	0
$263$	−16.3547	−1.00848	−0.504238	−	0.863565i	$-0.668227\pi$
$263$	−16.3547	−1.00848	−0.504238	+	0.863565i	$0.668227\pi$
$264$	0	0
$265$	−3.06590	−0.188337
$266$	0	0
$267$	32.8695	2.01158
$268$	0	0
$269$	−14.1222	−0.861048	−0.430524	−	0.902579i	$-0.641671\pi$
$269$	−14.1222	−0.861048	−0.430524	+	0.902579i	$0.641671\pi$
$270$	0	0
$271$	10.2465	0.622428	0.311214	−	0.950340i	$-0.399264\pi$
$271$	10.2465	0.622428	0.311214	+	0.950340i	$0.399264\pi$
$272$	0	0
$273$	42.2056	2.55440
$274$	0	0
$275$	12.6337	0.761841
$276$	0	0
$277$	−26.1083	−1.56869	−0.784347	−	0.620323i	$-0.787002\pi$
$277$	−26.1083	−1.56869	−0.784347	+	0.620323i	$0.787002\pi$
$278$	0	0
$279$	53.8490	3.22386
$280$	0	0
$281$	−9.84338	−0.587207	−0.293604	−	0.955927i	$-0.594855\pi$
$281$	−9.84338	−0.587207	−0.293604	+	0.955927i	$0.594855\pi$
$282$	0	0
$283$	−13.6388	−0.810742	−0.405371	−	0.914152i	$-0.632858\pi$
$283$	−13.6388	−0.810742	−0.405371	+	0.914152i	$0.632858\pi$
$284$	0	0
$285$	1.10162	0.0652542
$286$	0	0
$287$	−30.3138	−1.78937
$288$	0	0
$289$	33.4425	1.96721
$290$	0	0
$291$	−17.4069	−1.02041
$292$	0	0
$293$	−11.3420	−0.662608	−0.331304	−	0.943524i	$-0.607489\pi$
$293$	−11.3420	−0.662608	−0.331304	+	0.943524i	$0.607489\pi$
$294$	0	0
$295$	1.18473	0.0689778
$296$	0	0
$297$	34.2425	1.98695
$298$	0	0
$299$	23.2156	1.34259
$300$	0	0
$301$	−14.5104	−0.836367
$302$	0	0
$303$	−17.2075	−0.988545
$304$	0	0
$305$	1.54087	0.0882300
$306$	0	0
$307$	−7.37238	−0.420764	−0.210382	−	0.977619i	$-0.567471\pi$
$307$	−7.37238	−0.420764	−0.210382	+	0.977619i	$0.567471\pi$
$308$	0	0
$309$	−15.7611	−0.896616
$310$	0	0
$311$	−12.0533	−0.683479	−0.341740	−	0.939795i	$-0.611016\pi$
$311$	−12.0533	−0.683479	−0.341740	+	0.939795i	$0.611016\pi$
$312$	0	0
$313$	19.6818	1.11248	0.556240	−	0.831022i	$-0.312244\pi$
$313$	19.6818	1.11248	0.556240	+	0.831022i	$0.312244\pi$
$314$	0	0
$315$	−6.35540	−0.358086
$316$	0	0
$317$	−17.1655	−0.964112	−0.482056	−	0.876140i	$-0.660110\pi$
$317$	−17.1655	−0.964112	−0.482056	+	0.876140i	$0.660110\pi$
$318$	0	0
$319$	8.11592	0.454404
$320$	0	0
$321$	53.2756	2.97355
$322$	0	0
$323$	−7.10229	−0.395182
$324$	0	0
$325$	−25.1530	−1.39523
$326$	0	0
$327$	−53.7311	−2.97134
$328$	0	0
$329$	−12.1883	−0.671964
$330$	0	0
$331$	12.1831	0.669646	0.334823	−	0.942281i	$-0.391324\pi$
$331$	12.1831	0.669646	0.334823	+	0.942281i	$0.391324\pi$
$332$	0	0
$333$	−21.2819	−1.16624
$334$	0	0
$335$	−1.83475	−0.100243
$336$	0	0
$337$	23.6400	1.28775	0.643875	−	0.765131i	$-0.277326\pi$
$337$	23.6400	1.28775	0.643875	+	0.765131i	$0.277326\pi$
$338$	0	0
$339$	−30.2317	−1.64196
$340$	0	0
$341$	−19.4912	−1.05551
$342$	0	0
$343$	19.0041	1.02612
$344$	0	0
$345$	−4.96230	−0.267161
$346$	0	0
$347$	1.26865	0.0681049	0.0340525	−	0.999420i	$-0.489159\pi$
$347$	1.26865	0.0681049	0.0340525	+	0.999420i	$0.489159\pi$
$348$	0	0
$349$	5.66151	0.303054	0.151527	−	0.988453i	$-0.451581\pi$
$349$	5.66151	0.303054	0.151527	+	0.988453i	$0.451581\pi$
$350$	0	0
$351$	−68.1748	−3.63890
$352$	0	0
$353$	−10.3388	−0.550277	−0.275138	−	0.961405i	$-0.588724\pi$
$353$	−10.3388	−0.550277	−0.275138	+	0.961405i	$0.588724\pi$
$354$	0	0
$355$	−4.50661	−0.239186
$356$	0	0
$357$	58.1620	3.07826
$358$	0	0
$359$	18.1664	0.958785	0.479392	−	0.877601i	$-0.340857\pi$
$359$	18.1664	0.958785	0.479392	+	0.877601i	$0.340857\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	13.6973	0.718924
$364$	0	0
$365$	−0.455997	−0.0238680
$366$	0	0
$367$	21.9791	1.14730	0.573649	−	0.819102i	$-0.305528\pi$
$367$	21.9791	1.14730	0.573649	+	0.819102i	$0.305528\pi$
$368$	0	0
$369$	84.3487	4.39102
$370$	0	0
$371$	22.7912	1.18326
$372$	0	0
$373$	17.1488	0.887933	0.443967	−	0.896043i	$-0.353571\pi$
$373$	17.1488	0.887933	0.443967	+	0.896043i	$0.353571\pi$
$374$	0	0
$375$	10.8845	0.562073
$376$	0	0
$377$	−16.1583	−0.832196
$378$	0	0
$379$	2.76834	0.142200	0.0711000	−	0.997469i	$-0.477349\pi$
$379$	2.76834	0.142200	0.0711000	+	0.997469i	$0.477349\pi$
$380$	0	0
$381$	−14.1593	−0.725403
$382$	0	0
$383$	−8.27497	−0.422831	−0.211416	−	0.977396i	$-0.567807\pi$
$383$	−8.27497	−0.422831	−0.211416	+	0.977396i	$0.567807\pi$
$384$	0	0
$385$	2.30040	0.117239
$386$	0	0
$387$	40.3755	2.05240
$388$	0	0
$389$	29.2072	1.48086	0.740431	−	0.672133i	$-0.234622\pi$
$389$	29.2072	1.48086	0.740431	+	0.672133i	$0.234622\pi$
$390$	0	0
$391$	31.9927	1.61794
$392$	0	0
$393$	−36.6784	−1.85018
$394$	0	0
$395$	−0.345750	−0.0173965
$396$	0	0
$397$	17.9903	0.902908	0.451454	−	0.892294i	$-0.350906\pi$
$397$	17.9903	0.902908	0.451454	+	0.892294i	$0.350906\pi$
$398$	0	0
$399$	−8.18920	−0.409973
$400$	0	0
$401$	28.6795	1.43219	0.716094	−	0.698004i	$-0.245928\pi$
$401$	28.6795	1.43219	0.716094	+	0.698004i	$0.245928\pi$
$402$	0	0
$403$	38.8058	1.93306
$404$	0	0
$405$	7.15415	0.355493
$406$	0	0
$407$	7.70321	0.381834
$408$	0	0
$409$	−33.0913	−1.63626	−0.818129	−	0.575035i	$-0.804988\pi$
$409$	−33.0913	−1.63626	−0.818129	+	0.575035i	$0.804988\pi$
$410$	0	0
$411$	6.22958	0.307282
$412$	0	0
$413$	−8.80705	−0.433366
$414$	0	0
$415$	−3.10352	−0.152346
$416$	0	0
$417$	−31.9054	−1.56241
$418$	0	0
$419$	29.2636	1.42962	0.714810	−	0.699318i	$-0.246513\pi$
$419$	29.2636	1.42962	0.714810	+	0.699318i	$0.246513\pi$
$420$	0	0
$421$	11.0718	0.539605	0.269802	−	0.962916i	$-0.413042\pi$
$421$	11.0718	0.539605	0.269802	+	0.962916i	$0.413042\pi$
$422$	0	0
$423$	33.9142	1.64896
$424$	0	0
$425$	−34.6624	−1.68137
$426$	0	0
$427$	−11.4545	−0.554323
$428$	0	0
$429$	42.5077	2.05229
$430$	0	0
$431$	36.0273	1.73537	0.867686	−	0.497113i	$-0.165607\pi$
$431$	36.0273	1.73537	0.867686	+	0.497113i	$0.165607\pi$
$432$	0	0
$433$	−35.4450	−1.70338	−0.851689	−	0.524047i	$-0.824421\pi$
$433$	−35.4450	−1.70338	−0.851689	+	0.524047i	$0.824421\pi$
$434$	0	0
$435$	3.45381	0.165598
$436$	0	0
$437$	−4.50456	−0.215482
$438$	0	0
$439$	10.1849	0.486101	0.243050	−	0.970014i	$-0.421852\pi$
$439$	10.1849	0.486101	0.243050	+	0.970014i	$0.421852\pi$
$440$	0	0
$441$	−2.81721	−0.134153
$442$	0	0
$443$	13.2350	0.628815	0.314407	−	0.949288i	$-0.398194\pi$
$443$	13.2350	0.628815	0.314407	+	0.949288i	$0.398194\pi$
$444$	0	0
$445$	−3.56685	−0.169085
$446$	0	0
$447$	67.7870	3.20622
$448$	0	0
$449$	24.7773	1.16931	0.584656	−	0.811282i	$-0.301230\pi$
$449$	24.7773	1.16931	0.584656	+	0.811282i	$0.301230\pi$
$450$	0	0
$451$	−30.5308	−1.43764
$452$	0	0
$453$	12.0381	0.565599
$454$	0	0
$455$	−4.57996	−0.214712
$456$	0	0
$457$	−19.1640	−0.896456	−0.448228	−	0.893919i	$-0.647945\pi$
$457$	−19.1640	−0.896456	−0.448228	+	0.893919i	$0.647945\pi$
$458$	0	0
$459$	−93.9493	−4.38518
$460$	0	0
$461$	−9.83895	−0.458246	−0.229123	−	0.973398i	$-0.573586\pi$
$461$	−9.83895	−0.458246	−0.229123	+	0.973398i	$0.573586\pi$
$462$	0	0
$463$	9.19533	0.427343	0.213672	−	0.976906i	$-0.431458\pi$
$463$	9.19533	0.427343	0.213672	+	0.976906i	$0.431458\pi$
$464$	0	0
$465$	−8.29468	−0.384657
$466$	0	0
$467$	−18.2081	−0.842570	−0.421285	−	0.906928i	$-0.638421\pi$
$467$	−18.2081	−0.842570	−0.421285	+	0.906928i	$0.638421\pi$
$468$	0	0
$469$	13.6392	0.629799
$470$	0	0
$471$	−25.3155	−1.16648
$472$	0	0
$473$	−14.6143	−0.671967
$474$	0	0
$475$	4.88046	0.223931
$476$	0	0
$477$	−63.4170	−2.90366
$478$	0	0
$479$	18.2355	0.833201	0.416601	−	0.909090i	$-0.363221\pi$
$479$	18.2355	0.833201	0.416601	+	0.909090i	$0.363221\pi$
$480$	0	0
$481$	−15.3366	−0.699291
$482$	0	0
$483$	36.8887	1.67849
$484$	0	0
$485$	1.88892	0.0857714
$486$	0	0
$487$	35.0388	1.58776	0.793881	−	0.608074i	$-0.208057\pi$
$487$	35.0388	1.58776	0.793881	+	0.608074i	$0.208057\pi$
$488$	0	0
$489$	60.2774	2.72584
$490$	0	0
$491$	17.6870	0.798201	0.399101	−	0.916907i	$-0.369322\pi$
$491$	17.6870	0.798201	0.399101	+	0.916907i	$0.369322\pi$
$492$	0	0
$493$	−22.2672	−1.00287
$494$	0	0
$495$	−6.40090	−0.287699
$496$	0	0
$497$	33.5012	1.50273
$498$	0	0
$499$	−1.57022	−0.0702927	−0.0351464	−	0.999382i	$-0.511190\pi$
$499$	−1.57022	−0.0702927	−0.0351464	+	0.999382i	$0.511190\pi$
$500$	0	0
$501$	−41.4210	−1.85055
$502$	0	0
$503$	−21.4202	−0.955079	−0.477540	−	0.878610i	$-0.658471\pi$
$503$	−21.4202	−0.955079	−0.477540	+	0.878610i	$0.658471\pi$
$504$	0	0
$505$	1.86728	0.0830930
$506$	0	0
$507$	−43.2101	−1.91903
$508$	0	0
$509$	37.3290	1.65458	0.827289	−	0.561777i	$-0.189882\pi$
$509$	37.3290	1.65458	0.827289	+	0.561777i	$0.189882\pi$
$510$	0	0
$511$	3.38978	0.149955
$512$	0	0
$513$	13.2280	0.584032
$514$	0	0
$515$	1.71032	0.0753658
$516$	0	0
$517$	−12.2756	−0.539879
$518$	0	0
$519$	30.9889	1.36026
$520$	0	0
$521$	17.1668	0.752093	0.376047	−	0.926601i	$-0.377283\pi$
$521$	17.1668	0.752093	0.376047	+	0.926601i	$0.377283\pi$
$522$	0	0
$523$	39.9643	1.74752	0.873759	−	0.486359i	$-0.161675\pi$
$523$	39.9643	1.74752	0.873759	+	0.486359i	$0.161675\pi$
$524$	0	0
$525$	−39.9670	−1.74430
$526$	0	0
$527$	53.4770	2.32949
$528$	0	0
$529$	−2.70896	−0.117781
$530$	0	0
$531$	24.5057	1.06346
$532$	0	0
$533$	60.7851	2.63290
$534$	0	0
$535$	−5.78123	−0.249945
$536$	0	0
$537$	−5.97032	−0.257639
$538$	0	0
$539$	1.01972	0.0439224
$540$	0	0
$541$	10.8835	0.467920	0.233960	−	0.972246i	$-0.424832\pi$
$541$	10.8835	0.467920	0.233960	+	0.972246i	$0.424832\pi$
$542$	0	0
$543$	11.1155	0.477013
$544$	0	0
$545$	5.83067	0.249758
$546$	0	0
$547$	−23.0906	−0.987282	−0.493641	−	0.869666i	$-0.664334\pi$
$547$	−23.0906	−0.987282	−0.493641	+	0.869666i	$0.664334\pi$
$548$	0	0
$549$	31.8724	1.36028
$550$	0	0
$551$	3.13522	0.133565
$552$	0	0
$553$	2.57023	0.109297
$554$	0	0
$555$	3.27818	0.139151
$556$	0	0
$557$	−0.0572380	−0.00242525	−0.00121263	−	0.999999i	$-0.500386\pi$
$557$	−0.0572380	−0.00242525	−0.00121263	+	0.999999i	$0.500386\pi$
$558$	0	0
$559$	29.0962	1.23064
$560$	0	0
$561$	58.5784	2.47318
$562$	0	0
$563$	−19.6638	−0.828729	−0.414364	−	0.910111i	$-0.635996\pi$
$563$	−19.6638	−0.828729	−0.414364	+	0.910111i	$0.635996\pi$
$564$	0	0
$565$	3.28061	0.138016
$566$	0	0
$567$	−53.1825	−2.23345
$568$	0	0
$569$	−9.71352	−0.407212	−0.203606	−	0.979053i	$-0.565266\pi$
$569$	−9.71352	−0.407212	−0.203606	+	0.979053i	$0.565266\pi$
$570$	0	0
$571$	−24.6922	−1.03334	−0.516669	−	0.856185i	$-0.672828\pi$
$571$	−24.6922	−1.03334	−0.516669	+	0.856185i	$0.672828\pi$
$572$	0	0
$573$	−52.0753	−2.17548
$574$	0	0
$575$	−21.9843	−0.916809
$576$	0	0
$577$	13.0230	0.542157	0.271078	−	0.962557i	$-0.412620\pi$
$577$	13.0230	0.542157	0.271078	+	0.962557i	$0.412620\pi$
$578$	0	0
$579$	−32.7474	−1.36093
$580$	0	0
$581$	23.0709	0.957143
$582$	0	0
$583$	22.9544	0.950674
$584$	0	0
$585$	12.7438	0.526892
$586$	0	0
$587$	−9.95248	−0.410783	−0.205391	−	0.978680i	$-0.565847\pi$
$587$	−9.95248	−0.410783	−0.205391	+	0.978680i	$0.565847\pi$
$588$	0	0
$589$	−7.52954	−0.310249
$590$	0	0
$591$	9.90169	0.407301
$592$	0	0
$593$	−19.6140	−0.805449	−0.402724	−	0.915321i	$-0.631937\pi$
$593$	−19.6140	−0.805449	−0.402724	+	0.915321i	$0.631937\pi$
$594$	0	0
$595$	−6.31149	−0.258746
$596$	0	0
$597$	63.4251	2.59582
$598$	0	0
$599$	−10.4773	−0.428093	−0.214046	−	0.976823i	$-0.568664\pi$
$599$	−10.4773	−0.428093	−0.214046	+	0.976823i	$0.568664\pi$
$600$	0	0
$601$	11.0704	0.451571	0.225786	−	0.974177i	$-0.427505\pi$
$601$	11.0704	0.451571	0.225786	+	0.974177i	$0.427505\pi$
$602$	0	0
$603$	−37.9512	−1.54549
$604$	0	0
$605$	−1.48638	−0.0604298
$606$	0	0
$607$	−39.8787	−1.61863	−0.809314	−	0.587376i	$-0.800161\pi$
$607$	−39.8787	−1.61863	−0.809314	+	0.587376i	$0.800161\pi$
$608$	0	0
$609$	−25.6749	−1.04040
$610$	0	0
$611$	24.4400	0.988735
$612$	0	0
$613$	−2.59895	−0.104971	−0.0524854	−	0.998622i	$-0.516714\pi$
$613$	−2.59895	−0.104971	−0.0524854	+	0.998622i	$0.516714\pi$
$614$	0	0
$615$	−12.9927	−0.523917
$616$	0	0
$617$	−8.47296	−0.341109	−0.170554	−	0.985348i	$-0.554556\pi$
$617$	−8.47296	−0.341109	−0.170554	+	0.985348i	$0.554556\pi$
$618$	0	0
$619$	10.0577	0.404252	0.202126	−	0.979360i	$-0.435215\pi$
$619$	10.0577	0.404252	0.202126	+	0.979360i	$0.435215\pi$
$620$	0	0
$621$	−59.5865	−2.39112
$622$	0	0
$623$	26.5152	1.06231
$624$	0	0
$625$	23.2211	0.928846
$626$	0	0
$627$	−8.24782	−0.329386
$628$	0	0
$629$	−21.1349	−0.842703
$630$	0	0
$631$	4.43464	0.176540	0.0882701	−	0.996097i	$-0.471866\pi$
$631$	4.43464	0.176540	0.0882701	+	0.996097i	$0.471866\pi$
$632$	0	0
$633$	−29.9441	−1.19017
$634$	0	0
$635$	1.53651	0.0609744
$636$	0	0
$637$	−2.03020	−0.0804394
$638$	0	0
$639$	−93.2175	−3.68763
$640$	0	0
$641$	−10.7404	−0.424220	−0.212110	−	0.977246i	$-0.568033\pi$
$641$	−10.7404	−0.424220	−0.212110	+	0.977246i	$0.568033\pi$
$642$	0	0
$643$	35.7480	1.40976	0.704882	−	0.709324i	$-0.251000\pi$
$643$	35.7480	1.40976	0.704882	+	0.709324i	$0.251000\pi$
$644$	0	0
$645$	−6.21927	−0.244884
$646$	0	0
$647$	20.7136	0.814335	0.407167	−	0.913354i	$-0.366517\pi$
$647$	20.7136	0.814335	0.407167	+	0.913354i	$0.366517\pi$
$648$	0	0
$649$	−8.87010	−0.348182
$650$	0	0
$651$	61.6609	2.41668
$652$	0	0
$653$	20.0329	0.783950	0.391975	−	0.919976i	$-0.371792\pi$
$653$	20.0329	0.783950	0.391975	+	0.919976i	$0.371792\pi$
$654$	0	0
$655$	3.98018	0.155518
$656$	0	0
$657$	−9.43212	−0.367982
$658$	0	0
$659$	−32.9415	−1.28322	−0.641610	−	0.767031i	$-0.721733\pi$
$659$	−32.9415	−1.28322	−0.641610	+	0.767031i	$0.721733\pi$
$660$	0	0
$661$	−40.9494	−1.59275	−0.796373	−	0.604806i	$-0.793251\pi$
$661$	−40.9494	−1.59275	−0.796373	+	0.604806i	$0.793251\pi$
$662$	0	0
$663$	−116.626	−4.52939
$664$	0	0
$665$	0.888656	0.0344606
$666$	0	0
$667$	−14.1228	−0.546836
$668$	0	0
$669$	53.1136	2.05349
$670$	0	0
$671$	−11.5365	−0.445362
$672$	0	0
$673$	−38.2451	−1.47424	−0.737120	−	0.675762i	$-0.763815\pi$
$673$	−38.2451	−1.47424	−0.737120	+	0.675762i	$0.763815\pi$
$674$	0	0
$675$	64.5589	2.48487
$676$	0	0
$677$	−44.9987	−1.72944	−0.864720	−	0.502254i	$-0.832504\pi$
$677$	−44.9987	−1.72944	−0.864720	+	0.502254i	$0.832504\pi$
$678$	0	0
$679$	−14.0418	−0.538876
$680$	0	0
$681$	−82.0473	−3.14406
$682$	0	0
$683$	29.0833	1.11284	0.556421	−	0.830901i	$-0.312174\pi$
$683$	29.0833	1.11284	0.556421	+	0.830901i	$0.312174\pi$
$684$	0	0
$685$	−0.676007	−0.0258289
$686$	0	0
$687$	−69.4176	−2.64845
$688$	0	0
$689$	−45.7009	−1.74106
$690$	0	0
$691$	15.6403	0.594987	0.297493	−	0.954724i	$-0.403849\pi$
$691$	15.6403	0.594987	0.297493	+	0.954724i	$0.403849\pi$
$692$	0	0
$693$	47.5829	1.80753
$694$	0	0
$695$	3.46223	0.131330
$696$	0	0
$697$	83.7659	3.17286
$698$	0	0
$699$	−86.5075	−3.27201
$700$	0	0
$701$	43.7734	1.65330	0.826648	−	0.562719i	$-0.190245\pi$
$701$	43.7734	1.65330	0.826648	+	0.562719i	$0.190245\pi$
$702$	0	0
$703$	2.97579	0.112234
$704$	0	0
$705$	−5.22400	−0.196747
$706$	0	0
$707$	−13.8810	−0.522048
$708$	0	0
$709$	8.67525	0.325806	0.162903	−	0.986642i	$-0.447914\pi$
$709$	8.67525	0.325806	0.162903	+	0.986642i	$0.447914\pi$
$710$	0	0
$711$	−7.15170	−0.268210
$712$	0	0
$713$	33.9173	1.27021
$714$	0	0
$715$	−4.61275	−0.172507
$716$	0	0
$717$	−67.3825	−2.51645
$718$	0	0
$719$	−48.1740	−1.79659	−0.898293	−	0.439398i	$-0.855192\pi$
$719$	−48.1740	−1.79659	−0.898293	+	0.439398i	$0.855192\pi$
$720$	0	0
$721$	−12.7142	−0.473500
$722$	0	0
$723$	73.2243	2.72324
$724$	0	0
$725$	15.3013	0.568276
$726$	0	0
$727$	23.0870	0.856251	0.428126	−	0.903719i	$-0.359174\pi$
$727$	23.0870	0.856251	0.428126	+	0.903719i	$0.359174\pi$
$728$	0	0
$729$	21.5405	0.797798
$730$	0	0
$731$	40.0965	1.48302
$732$	0	0
$733$	43.8014	1.61784	0.808921	−	0.587917i	$-0.200052\pi$
$733$	43.8014	1.61784	0.808921	+	0.587917i	$0.200052\pi$
$734$	0	0
$735$	0.433951	0.0160065
$736$	0	0
$737$	13.7368	0.506002
$738$	0	0
$739$	−14.6790	−0.539975	−0.269988	−	0.962864i	$-0.587020\pi$
$739$	−14.6790	−0.539975	−0.269988	+	0.962864i	$0.587020\pi$
$740$	0	0
$741$	16.4209	0.603238
$742$	0	0
$743$	3.50496	0.128585	0.0642923	−	0.997931i	$-0.479521\pi$
$743$	3.50496	0.128585	0.0642923	+	0.997931i	$0.479521\pi$
$744$	0	0
$745$	−7.35595	−0.269501
$746$	0	0
$747$	−64.1952	−2.34878
$748$	0	0
$749$	42.9765	1.57033
$750$	0	0
$751$	35.1485	1.28259	0.641293	−	0.767296i	$-0.278398\pi$
$751$	35.1485	1.28259	0.641293	+	0.767296i	$0.278398\pi$
$752$	0	0
$753$	−82.1334	−2.99311
$754$	0	0
$755$	−1.30632	−0.0475418
$756$	0	0
$757$	−2.22731	−0.0809531	−0.0404765	−	0.999180i	$-0.512888\pi$
$757$	−2.22731	−0.0809531	−0.0404765	+	0.999180i	$0.512888\pi$
$758$	0	0
$759$	37.1528	1.34856
$760$	0	0
$761$	27.1792	0.985246	0.492623	−	0.870243i	$-0.336038\pi$
$761$	27.1792	0.985246	0.492623	+	0.870243i	$0.336038\pi$
$762$	0	0
$763$	−43.3440	−1.56916
$764$	0	0
$765$	17.5618	0.634949
$766$	0	0
$767$	17.6598	0.637660
$768$	0	0
$769$	21.2037	0.764625	0.382313	−	0.924033i	$-0.375128\pi$
$769$	21.2037	0.764625	0.382313	+	0.924033i	$0.375128\pi$
$770$	0	0
$771$	−23.2881	−0.838701
$772$	0	0
$773$	−14.5254	−0.522443	−0.261221	−	0.965279i	$-0.584125\pi$
$773$	−14.5254	−0.522443	−0.261221	+	0.965279i	$0.584125\pi$
$774$	0	0
$775$	−36.7476	−1.32001
$776$	0	0
$777$	−24.3693	−0.874244
$778$	0	0
$779$	−11.7942	−0.422571
$780$	0	0
$781$	33.7410	1.20735
$782$	0	0
$783$	41.4728	1.48212
$784$	0	0
$785$	2.74713	0.0980491
$786$	0	0
$787$	−13.0489	−0.465141	−0.232571	−	0.972579i	$-0.574714\pi$
$787$	−13.0489	−0.465141	−0.232571	+	0.972579i	$0.574714\pi$
$788$	0	0
$789$	52.1090	1.85513
$790$	0	0
$791$	−24.3874	−0.867116
$792$	0	0
$793$	22.9685	0.815637
$794$	0	0
$795$	9.76848	0.346452
$796$	0	0
$797$	−33.3585	−1.18162	−0.590809	−	0.806811i	$-0.701191\pi$
$797$	−33.3585	−1.18162	−0.590809	+	0.806811i	$0.701191\pi$
$798$	0	0
$799$	33.6799	1.19151
$800$	0	0
$801$	−73.7790	−2.60685
$802$	0	0
$803$	3.41405	0.120479
$804$	0	0
$805$	−4.00300	−0.141087
$806$	0	0
$807$	44.9959	1.58393
$808$	0	0
$809$	24.5783	0.864128	0.432064	−	0.901843i	$-0.357785\pi$
$809$	24.5783	0.864128	0.432064	+	0.901843i	$0.357785\pi$
$810$	0	0
$811$	26.4909	0.930223	0.465111	−	0.885252i	$-0.346014\pi$
$811$	26.4909	0.930223	0.465111	+	0.885252i	$0.346014\pi$
$812$	0	0
$813$	−32.6470	−1.14498
$814$	0	0
$815$	−6.54104	−0.229123
$816$	0	0
$817$	−5.64558	−0.197514
$818$	0	0
$819$	−94.7348	−3.31030
$820$	0	0
$821$	26.7838	0.934760	0.467380	−	0.884057i	$-0.345198\pi$
$821$	26.7838	0.934760	0.467380	+	0.884057i	$0.345198\pi$
$822$	0	0
$823$	24.8412	0.865909	0.432955	−	0.901416i	$-0.357471\pi$
$823$	24.8412	0.865909	0.432955	+	0.901416i	$0.357471\pi$
$824$	0	0
$825$	−40.2531	−1.40143
$826$	0	0
$827$	43.9738	1.52912	0.764559	−	0.644554i	$-0.222957\pi$
$827$	43.9738	1.52912	0.764559	+	0.644554i	$0.222957\pi$
$828$	0	0
$829$	−20.3908	−0.708200	−0.354100	−	0.935207i	$-0.615213\pi$
$829$	−20.3908	−0.708200	−0.354100	+	0.935207i	$0.615213\pi$
$830$	0	0
$831$	83.1855	2.88567
$832$	0	0
$833$	−2.79775	−0.0969362
$834$	0	0
$835$	4.49482	0.155550
$836$	0	0
$837$	−99.6011	−3.44272
$838$	0	0
$839$	9.79242	0.338072	0.169036	−	0.985610i	$-0.445935\pi$
$839$	9.79242	0.338072	0.169036	+	0.985610i	$0.445935\pi$
$840$	0	0
$841$	−19.1704	−0.661048
$842$	0	0
$843$	31.3627	1.08019
$844$	0	0
$845$	4.68897	0.161306
$846$	0	0
$847$	11.0494	0.379662
$848$	0	0
$849$	43.4556	1.49139
$850$	0	0
$851$	−13.4046	−0.459504
$852$	0	0
$853$	−5.79772	−0.198510	−0.0992550	−	0.995062i	$-0.531646\pi$
$853$	−5.79772	−0.198510	−0.0992550	+	0.995062i	$0.531646\pi$
$854$	0	0
$855$	−2.47270	−0.0845645
$856$	0	0
$857$	−15.4044	−0.526204	−0.263102	−	0.964768i	$-0.584746\pi$
$857$	−15.4044	−0.526204	−0.263102	+	0.964768i	$0.584746\pi$
$858$	0	0
$859$	−4.16874	−0.142236	−0.0711178	−	0.997468i	$-0.522657\pi$
$859$	−4.16874	−0.142236	−0.0711178	+	0.997468i	$0.522657\pi$
$860$	0	0
$861$	96.5851	3.29161
$862$	0	0
$863$	−42.9896	−1.46338	−0.731691	−	0.681636i	$-0.761269\pi$
$863$	−42.9896	−1.46338	−0.731691	+	0.681636i	$0.761269\pi$
$864$	0	0
$865$	−3.36278	−0.114338
$866$	0	0
$867$	−106.554	−3.61875
$868$	0	0
$869$	2.58863	0.0878132
$870$	0	0
$871$	−27.3492	−0.926693
$872$	0	0
$873$	39.0716	1.32237
$874$	0	0
$875$	8.78032	0.296829
$876$	0	0
$877$	−42.9008	−1.44866	−0.724328	−	0.689456i	$-0.757850\pi$
$877$	−42.9008	−1.44866	−0.724328	+	0.689456i	$0.757850\pi$
$878$	0	0
$879$	36.1377	1.21889
$880$	0	0
$881$	44.6942	1.50579	0.752893	−	0.658143i	$-0.228658\pi$
$881$	44.6942	1.50579	0.752893	+	0.658143i	$0.228658\pi$
$882$	0	0
$883$	−46.6534	−1.57001	−0.785006	−	0.619489i	$-0.787340\pi$
$883$	−46.6534	−1.57001	−0.785006	+	0.619489i	$0.787340\pi$
$884$	0	0
$885$	−3.77476	−0.126887
$886$	0	0
$887$	−4.00717	−0.134548	−0.0672738	−	0.997735i	$-0.521430\pi$
$887$	−4.00717	−0.134548	−0.0672738	+	0.997735i	$0.521430\pi$
$888$	0	0
$889$	−11.4221	−0.383084
$890$	0	0
$891$	−53.5632	−1.79444
$892$	0	0
$893$	−4.74211	−0.158689
$894$	0	0
$895$	0.647873	0.0216560
$896$	0	0
$897$	−73.9690	−2.46975
$898$	0	0
$899$	−23.6068	−0.787330
$900$	0	0
$901$	−62.9788	−2.09813
$902$	0	0
$903$	46.2328	1.53853
$904$	0	0
$905$	−1.20621	−0.0400958
$906$	0	0
$907$	−52.9123	−1.75693	−0.878463	−	0.477811i	$-0.841430\pi$
$907$	−52.9123	−1.75693	−0.878463	+	0.477811i	$0.841430\pi$
$908$	0	0
$909$	38.6240	1.28108
$910$	0	0
$911$	−12.2999	−0.407514	−0.203757	−	0.979021i	$-0.565315\pi$
$911$	−12.2999	−0.407514	−0.203757	+	0.979021i	$0.565315\pi$
$912$	0	0
$913$	23.2361	0.769002
$914$	0	0
$915$	−4.90948	−0.162303
$916$	0	0
$917$	−29.5878	−0.977076
$918$	0	0
$919$	−6.91734	−0.228182	−0.114091	−	0.993470i	$-0.536396\pi$
$919$	−6.91734	−0.228182	−0.114091	+	0.993470i	$0.536396\pi$
$920$	0	0
$921$	23.4897	0.774011
$922$	0	0
$923$	−67.1764	−2.21114
$924$	0	0
$925$	14.5232	0.477520
$926$	0	0
$927$	35.3774	1.16194
$928$	0	0
$929$	39.9020	1.30914	0.654572	−	0.756000i	$-0.272849\pi$
$929$	39.9020	1.30914	0.654572	+	0.756000i	$0.272849\pi$
$930$	0	0
$931$	0.393922	0.0129103
$932$	0	0
$933$	38.4039	1.25729
$934$	0	0
$935$	−6.35667	−0.207885
$936$	0	0
$937$	1.55272	0.0507253	0.0253626	−	0.999678i	$-0.491926\pi$
$937$	1.55272	0.0507253	0.0253626	+	0.999678i	$0.491926\pi$
$938$	0	0
$939$	−62.7095	−2.04645
$940$	0	0
$941$	11.4329	0.372703	0.186351	−	0.982483i	$-0.440334\pi$
$941$	11.4329	0.372703	0.186351	+	0.982483i	$0.440334\pi$
$942$	0	0
$943$	53.1277	1.73008
$944$	0	0
$945$	11.7552	0.382396
$946$	0	0
$947$	11.5530	0.375423	0.187712	−	0.982224i	$-0.439893\pi$
$947$	11.5530	0.375423	0.187712	+	0.982224i	$0.439893\pi$
$948$	0	0
$949$	−6.79717	−0.220646
$950$	0	0
$951$	54.6923	1.77352
$952$	0	0
$953$	13.1935	0.427380	0.213690	−	0.976902i	$-0.431452\pi$
$953$	13.1935	0.427380	0.213690	+	0.976902i	$0.431452\pi$
$954$	0	0
$955$	5.65098	0.182862
$956$	0	0
$957$	−25.8587	−0.835894
$958$	0	0
$959$	5.02529	0.162275
$960$	0	0
$961$	25.6940	0.828839
$962$	0	0
$963$	−119.583	−3.85350
$964$	0	0
$965$	3.55360	0.114394
$966$	0	0
$967$	21.5397	0.692671	0.346336	−	0.938111i	$-0.387426\pi$
$967$	21.5397	0.692671	0.346336	+	0.938111i	$0.387426\pi$
$968$	0	0
$969$	22.6291	0.726952
$970$	0	0
$971$	39.4613	1.26637	0.633186	−	0.774000i	$-0.281747\pi$
$971$	39.4613	1.26637	0.633186	+	0.774000i	$0.281747\pi$
$972$	0	0
$973$	−25.7375	−0.825107
$974$	0	0
$975$	80.1417	2.56659
$976$	0	0
$977$	−52.7862	−1.68878	−0.844390	−	0.535729i	$-0.820037\pi$
$977$	−52.7862	−1.68878	−0.844390	+	0.535729i	$0.820037\pi$
$978$	0	0
$979$	26.7051	0.853497
$980$	0	0
$981$	120.605	3.85063
$982$	0	0
$983$	−27.8920	−0.889616	−0.444808	−	0.895626i	$-0.646728\pi$
$983$	−27.8920	−0.889616	−0.444808	+	0.895626i	$0.646728\pi$
$984$	0	0
$985$	−1.07449	−0.0342360
$986$	0	0
$987$	38.8341	1.23610
$988$	0	0
$989$	25.4308	0.808653
$990$	0	0
$991$	−58.5431	−1.85968	−0.929841	−	0.367961i	$-0.880056\pi$
$991$	−58.5431	−1.85968	−0.929841	+	0.367961i	$0.880056\pi$
$992$	0	0
$993$	−38.8176	−1.23184
$994$	0	0
$995$	−6.88261	−0.218194
$996$	0	0
$997$	−17.7666	−0.562673	−0.281336	−	0.959609i	$-0.590778\pi$
$997$	−17.7666	−0.562673	−0.281336	+	0.959609i	$0.590778\pi$
$998$	0	0
$999$	39.3638	1.24542

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6004.2.a.h.1.2	✓	31

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6004.2.a.h.1.2	✓	31	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)

\(f(q)\)	\(=\)	\(q-3.18617 q^{3} +0.345750 q^{5} -2.57023 q^{7} +7.15170 q^{9} +O(q^{10})\)	\(q-3.18617 q^{3} +0.345750 q^{5} -2.57023 q^{7} +7.15170 q^{9} -2.58863 q^{11} +5.15381 q^{13} -1.10162 q^{15} +7.10229 q^{17} -1.00000 q^{19} +8.18920 q^{21} +4.50456 q^{23} -4.88046 q^{25} -13.2280 q^{27} -3.13522 q^{29} +7.52954 q^{31} +8.24782 q^{33} -0.888656 q^{35} -2.97579 q^{37} -16.4209 q^{39} +11.7942 q^{41} +5.64558 q^{43} +2.47270 q^{45} +4.74211 q^{47} -0.393922 q^{49} -22.6291 q^{51} -8.86740 q^{53} -0.895017 q^{55} +3.18617 q^{57} +3.42656 q^{59} +4.45661 q^{61} -18.3815 q^{63} +1.78193 q^{65} -5.30660 q^{67} -14.3523 q^{69} -13.0343 q^{71} -1.31886 q^{73} +15.5500 q^{75} +6.65337 q^{77} -1.00000 q^{79} +20.6917 q^{81} -8.97621 q^{83} +2.45561 q^{85} +9.98935 q^{87} -10.3163 q^{89} -13.2465 q^{91} -23.9904 q^{93} -0.345750 q^{95} +5.46326 q^{97} -18.5131 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 31 q - 4 q^{3} + 10 q^{5} + 11 q^{7} + 47 q^{9}+O(q^{10}) \) 31 * q - 4 * q^3 + 10 * q^5 + 11 * q^7 + 47 * q^9	\( 31 q - 4 q^{3} + 10 q^{5} + 11 q^{7} + 47 q^{9} - 4 q^{11} + 11 q^{13} + 5 q^{15} + 14 q^{17} - 31 q^{19} + 22 q^{21} + 15 q^{23} + 59 q^{25} + 5 q^{27} + 34 q^{29} - 12 q^{31} + 10 q^{33} + 8 q^{35} + 16 q^{37} + 18 q^{39} + 27 q^{41} + 2 q^{43} + 22 q^{45} + 30 q^{47} + 62 q^{49} - 14 q^{51} + 35 q^{53} + 8 q^{55} + 4 q^{57} - 16 q^{59} + 37 q^{61} + 31 q^{63} + 80 q^{65} + 16 q^{67} + q^{69} + 19 q^{71} + 38 q^{73} + 21 q^{75} + 44 q^{77} - 31 q^{79} + 55 q^{81} - 12 q^{83} + 66 q^{85} + 58 q^{87} + 16 q^{89} - 42 q^{91} + 10 q^{93} - 10 q^{95} - 12 q^{97} + 12 q^{99}+O(q^{100}) \) 31 * q - 4 * q^3 + 10 * q^5 + 11 * q^7 + 47 * q^9 - 4 * q^11 + 11 * q^13 + 5 * q^15 + 14 * q^17 - 31 * q^19 + 22 * q^21 + 15 * q^23 + 59 * q^25 + 5 * q^27 + 34 * q^29 - 12 * q^31 + 10 * q^33 + 8 * q^35 + 16 * q^37 + 18 * q^39 + 27 * q^41 + 2 * q^43 + 22 * q^45 + 30 * q^47 + 62 * q^49 - 14 * q^51 + 35 * q^53 + 8 * q^55 + 4 * q^57 - 16 * q^59 + 37 * q^61 + 31 * q^63 + 80 * q^65 + 16 * q^67 + q^69 + 19 * q^71 + 38 * q^73 + 21 * q^75 + 44 * q^77 - 31 * q^79 + 55 * q^81 - 12 * q^83 + 66 * q^85 + 58 * q^87 + 16 * q^89 - 42 * q^91 + 10 * q^93 - 10 * q^95 - 12 * q^97 + 12 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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