LMFDB - Embedded newform 6004.2.a.g.1.25

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			1.25
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)

$f(q)$	$=$	$q+2.48243 q^{3} +1.00873 q^{5} +0.658321 q^{7} +3.16248 q^{9} +O(q^{10})$	$q+2.48243 q^{3} +1.00873 q^{5} +0.658321 q^{7} +3.16248 q^{9} -5.12660 q^{11} -2.19021 q^{13} +2.50410 q^{15} -4.93027 q^{17} +1.00000 q^{19} +1.63424 q^{21} +2.25019 q^{23} -3.98247 q^{25} +0.403349 q^{27} -3.27576 q^{29} -6.68263 q^{31} -12.7265 q^{33} +0.664065 q^{35} +4.75479 q^{37} -5.43704 q^{39} -6.29737 q^{41} -7.36027 q^{43} +3.19008 q^{45} -8.45612 q^{47} -6.56661 q^{49} -12.2391 q^{51} +6.21683 q^{53} -5.17134 q^{55} +2.48243 q^{57} +2.76767 q^{59} +1.82136 q^{61} +2.08193 q^{63} -2.20932 q^{65} +7.94750 q^{67} +5.58595 q^{69} +13.7260 q^{71} -3.62552 q^{73} -9.88622 q^{75} -3.37495 q^{77} -1.00000 q^{79} -8.48616 q^{81} -12.7858 q^{83} -4.97330 q^{85} -8.13186 q^{87} +10.3744 q^{89} -1.44186 q^{91} -16.5892 q^{93} +1.00873 q^{95} +8.58998 q^{97} -16.2128 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	2.48243	1.43323	0.716617	−	0.697467i	$-0.245690\pi$
$3$	2.48243	1.43323	0.716617	+	0.697467i	$0.245690\pi$
$4$	0	0
$5$	1.00873	0.451116	0.225558	−	0.974230i	$-0.427579\pi$
$5$	1.00873	0.451116	0.225558	+	0.974230i	$0.427579\pi$
$6$	0	0
$7$	0.658321	0.248822	0.124411	−	0.992231i	$-0.460296\pi$
$7$	0.658321	0.248822	0.124411	+	0.992231i	$0.460296\pi$
$8$	0	0
$9$	3.16248	1.05416
$10$	0	0
$11$	−5.12660	−1.54573	−0.772864	−	0.634571i	$-0.781177\pi$
$11$	−5.12660	−1.54573	−0.772864	+	0.634571i	$0.781177\pi$
$12$	0	0
$13$	−2.19021	−0.607454	−0.303727	−	0.952759i	$-0.598231\pi$
$13$	−2.19021	−0.607454	−0.303727	+	0.952759i	$0.598231\pi$
$14$	0	0
$15$	2.50410	0.646555
$16$	0	0
$17$	−4.93027	−1.19577	−0.597884	−	0.801583i	$-0.703992\pi$
$17$	−4.93027	−1.19577	−0.597884	+	0.801583i	$0.703992\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	1.63424	0.356620
$22$	0	0
$23$	2.25019	0.469197	0.234598	−	0.972092i	$-0.424622\pi$
$23$	2.25019	0.469197	0.234598	+	0.972092i	$0.424622\pi$
$24$	0	0
$25$	−3.98247	−0.796494
$26$	0	0
$27$	0.403349	0.0776246
$28$	0	0
$29$	−3.27576	−0.608294	−0.304147	−	0.952625i	$-0.598371\pi$
$29$	−3.27576	−0.608294	−0.304147	+	0.952625i	$0.598371\pi$
$30$	0	0
$31$	−6.68263	−1.20024	−0.600118	−	0.799911i	$-0.704880\pi$
$31$	−6.68263	−1.20024	−0.600118	+	0.799911i	$0.704880\pi$
$32$	0	0
$33$	−12.7265	−2.21539
$34$	0	0
$35$	0.664065	0.112248
$36$	0	0
$37$	4.75479	0.781683	0.390842	−	0.920458i	$-0.372184\pi$
$37$	4.75479	0.781683	0.390842	+	0.920458i	$0.372184\pi$
$38$	0	0
$39$	−5.43704	−0.870623
$40$	0	0
$41$	−6.29737	−0.983484	−0.491742	−	0.870741i	$-0.663640\pi$
$41$	−6.29737	−0.983484	−0.491742	+	0.870741i	$0.663640\pi$
$42$	0	0
$43$	−7.36027	−1.12243	−0.561215	−	0.827670i	$-0.689666\pi$
$43$	−7.36027	−1.12243	−0.561215	+	0.827670i	$0.689666\pi$
$44$	0	0
$45$	3.19008	0.475549
$46$	0	0
$47$	−8.45612	−1.23345	−0.616726	−	0.787178i	$-0.711541\pi$
$47$	−8.45612	−1.23345	−0.616726	+	0.787178i	$0.711541\pi$
$48$	0	0
$49$	−6.56661	−0.938088
$50$	0	0
$51$	−12.2391	−1.71381
$52$	0	0
$53$	6.21683	0.853947	0.426974	−	0.904264i	$-0.359580\pi$
$53$	6.21683	0.853947	0.426974	+	0.904264i	$0.359580\pi$
$54$	0	0
$55$	−5.17134	−0.697303
$56$	0	0
$57$	2.48243	0.328806
$58$	0	0
$59$	2.76767	0.360320	0.180160	−	0.983637i	$-0.442338\pi$
$59$	2.76767	0.360320	0.180160	+	0.983637i	$0.442338\pi$
$60$	0	0
$61$	1.82136	0.233202	0.116601	−	0.993179i	$-0.462800\pi$
$61$	1.82136	0.233202	0.116601	+	0.993179i	$0.462800\pi$
$62$	0	0
$63$	2.08193	0.262298
$64$	0	0
$65$	−2.20932	−0.274032
$66$	0	0
$67$	7.94750	0.970941	0.485471	−	0.874253i	$-0.338648\pi$
$67$	7.94750	0.970941	0.485471	+	0.874253i	$0.338648\pi$
$68$	0	0
$69$	5.58595	0.672469
$70$	0	0
$71$	13.7260	1.62898	0.814490	−	0.580177i	$-0.197017\pi$
$71$	13.7260	1.62898	0.814490	+	0.580177i	$0.197017\pi$
$72$	0	0
$73$	−3.62552	−0.424335	−0.212167	−	0.977233i	$-0.568052\pi$
$73$	−3.62552	−0.424335	−0.212167	+	0.977233i	$0.568052\pi$
$74$	0	0
$75$	−9.88622	−1.14156
$76$	0	0
$77$	−3.37495	−0.384611
$78$	0	0
$79$	−1.00000	−0.112509
$79$	−1.00000	−0.112509
$80$	0	0
$81$	−8.48616	−0.942906
$82$	0	0
$83$	−12.7858	−1.40342	−0.701712	−	0.712461i	$-0.747581\pi$
$83$	−12.7858	−1.40342	−0.701712	+	0.712461i	$0.747581\pi$
$84$	0	0
$85$	−4.97330	−0.539430
$86$	0	0
$87$	−8.13186	−0.871827
$88$	0	0
$89$	10.3744	1.09969	0.549844	−	0.835267i	$-0.314687\pi$
$89$	10.3744	1.09969	0.549844	+	0.835267i	$0.314687\pi$
$90$	0	0
$91$	−1.44186	−0.151148
$92$	0	0
$93$	−16.5892	−1.72022
$94$	0	0
$95$	1.00873	0.103493
$96$	0	0
$97$	8.58998	0.872180	0.436090	−	0.899903i	$-0.356363\pi$
$97$	8.58998	0.872180	0.436090	+	0.899903i	$0.356363\pi$
$98$	0	0
$99$	−16.2128	−1.62945
$100$	0	0
$101$	−8.57318	−0.853063	−0.426531	−	0.904473i	$-0.640265\pi$
$101$	−8.57318	−0.853063	−0.426531	+	0.904473i	$0.640265\pi$
$102$	0	0
$103$	3.43742	0.338699	0.169349	−	0.985556i	$-0.445833\pi$
$103$	3.43742	0.338699	0.169349	+	0.985556i	$0.445833\pi$
$104$	0	0
$105$	1.64850	0.160877
$106$	0	0
$107$	6.52295	0.630597	0.315299	−	0.948992i	$-0.397895\pi$
$107$	6.52295	0.630597	0.315299	+	0.948992i	$0.397895\pi$
$108$	0	0
$109$	9.73287	0.932239	0.466120	−	0.884722i	$-0.345652\pi$
$109$	9.73287	0.932239	0.466120	+	0.884722i	$0.345652\pi$
$110$	0	0
$111$	11.8035	1.12034
$112$	0	0
$113$	15.0997	1.42046	0.710229	−	0.703971i	$-0.248591\pi$
$113$	15.0997	1.42046	0.710229	+	0.703971i	$0.248591\pi$
$114$	0	0
$115$	2.26983	0.211662
$116$	0	0
$117$	−6.92648	−0.640354
$118$	0	0
$119$	−3.24570	−0.297533
$120$	0	0
$121$	15.2821	1.38928
$122$	0	0
$123$	−15.6328	−1.40956
$124$	0	0
$125$	−9.06086	−0.810428
$126$	0	0
$127$	−12.4992	−1.10913	−0.554564	−	0.832141i	$-0.687115\pi$
$127$	−12.4992	−1.10913	−0.554564	+	0.832141i	$0.687115\pi$
$128$	0	0
$129$	−18.2714	−1.60871
$130$	0	0
$131$	−7.20572	−0.629567	−0.314784	−	0.949163i	$-0.601932\pi$
$131$	−7.20572	−0.629567	−0.314784	+	0.949163i	$0.601932\pi$
$132$	0	0
$133$	0.658321	0.0570836
$134$	0	0
$135$	0.406869	0.0350177
$136$	0	0
$137$	−8.01038	−0.684373	−0.342186	−	0.939632i	$-0.611167\pi$
$137$	−8.01038	−0.684373	−0.342186	+	0.939632i	$0.611167\pi$
$138$	0	0
$139$	−4.63706	−0.393311	−0.196655	−	0.980473i	$-0.563008\pi$
$139$	−4.63706	−0.393311	−0.196655	+	0.980473i	$0.563008\pi$
$140$	0	0
$141$	−20.9918	−1.76783
$142$	0	0
$143$	11.2283	0.938959
$144$	0	0
$145$	−3.30435	−0.274411
$146$	0	0
$147$	−16.3012	−1.34450
$148$	0	0
$149$	−15.7408	−1.28953	−0.644766	−	0.764380i	$-0.723045\pi$
$149$	−15.7408	−1.28953	−0.644766	+	0.764380i	$0.723045\pi$
$150$	0	0
$151$	2.04777	0.166645	0.0833225	−	0.996523i	$-0.473447\pi$
$151$	2.04777	0.166645	0.0833225	+	0.996523i	$0.473447\pi$
$152$	0	0
$153$	−15.5919	−1.26053
$154$	0	0
$155$	−6.74095	−0.541446
$156$	0	0
$157$	2.50123	0.199620	0.0998101	−	0.995007i	$-0.468176\pi$
$157$	2.50123	0.199620	0.0998101	+	0.995007i	$0.468176\pi$
$158$	0	0
$159$	15.4329	1.22391
$160$	0	0
$161$	1.48135	0.116746
$162$	0	0
$163$	−1.69880	−0.133060	−0.0665301	−	0.997784i	$-0.521193\pi$
$163$	−1.69880	−0.133060	−0.0665301	+	0.997784i	$0.521193\pi$
$164$	0	0
$165$	−12.8375	−0.999399
$166$	0	0
$167$	3.70694	0.286852	0.143426	−	0.989661i	$-0.454188\pi$
$167$	3.70694	0.286852	0.143426	+	0.989661i	$0.454188\pi$
$168$	0	0
$169$	−8.20300	−0.631000
$170$	0	0
$171$	3.16248	0.241841
$172$	0	0
$173$	−19.0688	−1.44977	−0.724887	−	0.688868i	$-0.758108\pi$
$173$	−19.0688	−1.44977	−0.724887	+	0.688868i	$0.758108\pi$
$174$	0	0
$175$	−2.62174	−0.198185
$176$	0	0
$177$	6.87056	0.516423
$178$	0	0
$179$	−7.66901	−0.573209	−0.286604	−	0.958049i	$-0.592527\pi$
$179$	−7.66901	−0.573209	−0.286604	+	0.958049i	$0.592527\pi$
$180$	0	0
$181$	−0.381147	−0.0283304	−0.0141652	−	0.999900i	$-0.504509\pi$
$181$	−0.381147	−0.0283304	−0.0141652	+	0.999900i	$0.504509\pi$
$182$	0	0
$183$	4.52142	0.334233
$184$	0	0
$185$	4.79629	0.352630
$186$	0	0
$187$	25.2756	1.84833
$188$	0	0
$189$	0.265533	0.0193147
$190$	0	0
$191$	−10.4223	−0.754134	−0.377067	−	0.926186i	$-0.623067\pi$
$191$	−10.4223	−0.754134	−0.377067	+	0.926186i	$0.623067\pi$
$192$	0	0
$193$	7.46346	0.537231	0.268616	−	0.963247i	$-0.413434\pi$
$193$	7.46346	0.537231	0.268616	+	0.963247i	$0.413434\pi$
$194$	0	0
$195$	−5.48449	−0.392752
$196$	0	0
$197$	22.3967	1.59570	0.797850	−	0.602856i	$-0.205971\pi$
$197$	22.3967	1.59570	0.797850	+	0.602856i	$0.205971\pi$
$198$	0	0
$199$	24.3863	1.72870	0.864348	−	0.502894i	$-0.167731\pi$
$199$	24.3863	1.72870	0.864348	+	0.502894i	$0.167731\pi$
$200$	0	0
$201$	19.7291	1.39159
$202$	0	0
$203$	−2.15650	−0.151357
$204$	0	0
$205$	−6.35232	−0.443666
$206$	0	0
$207$	7.11618	0.494609
$208$	0	0
$209$	−5.12660	−0.354615
$210$	0	0
$211$	9.68729	0.666901	0.333450	−	0.942768i	$-0.391787\pi$
$211$	9.68729	0.666901	0.333450	+	0.942768i	$0.391787\pi$
$212$	0	0
$213$	34.0740	2.33471
$214$	0	0
$215$	−7.42449	−0.506346
$216$	0	0
$217$	−4.39931	−0.298645
$218$	0	0
$219$	−9.00011	−0.608171
$220$	0	0
$221$	10.7983	0.726373
$222$	0	0
$223$	18.7265	1.25402	0.627009	−	0.779012i	$-0.284279\pi$
$223$	18.7265	1.25402	0.627009	+	0.779012i	$0.284279\pi$
$224$	0	0
$225$	−12.5945	−0.839633
$226$	0	0
$227$	−2.30619	−0.153067	−0.0765337	−	0.997067i	$-0.524385\pi$
$227$	−2.30619	−0.153067	−0.0765337	+	0.997067i	$0.524385\pi$
$228$	0	0
$229$	20.6903	1.36725	0.683625	−	0.729833i	$-0.260402\pi$
$229$	20.6903	1.36725	0.683625	+	0.729833i	$0.260402\pi$
$230$	0	0
$231$	−8.37809	−0.551238
$232$	0	0
$233$	−24.6222	−1.61306	−0.806528	−	0.591196i	$-0.798656\pi$
$233$	−24.6222	−1.61306	−0.806528	+	0.591196i	$0.798656\pi$
$234$	0	0
$235$	−8.52991	−0.556430
$236$	0	0
$237$	−2.48243	−0.161251
$238$	0	0
$239$	23.9200	1.54726	0.773629	−	0.633638i	$-0.218439\pi$
$239$	23.9200	1.54726	0.773629	+	0.633638i	$0.218439\pi$
$240$	0	0
$241$	8.80322	0.567065	0.283533	−	0.958963i	$-0.408494\pi$
$241$	8.80322	0.567065	0.283533	+	0.958963i	$0.408494\pi$
$242$	0	0
$243$	−22.2764	−1.42903
$244$	0	0
$245$	−6.62392	−0.423187
$246$	0	0
$247$	−2.19021	−0.139359
$248$	0	0
$249$	−31.7399	−2.01144
$250$	0	0
$251$	10.6885	0.674651	0.337326	−	0.941388i	$-0.390478\pi$
$251$	10.6885	0.674651	0.337326	+	0.941388i	$0.390478\pi$
$252$	0	0
$253$	−11.5358	−0.725251
$254$	0	0
$255$	−12.3459	−0.773129
$256$	0	0
$257$	14.6042	0.910983	0.455492	−	0.890240i	$-0.349464\pi$
$257$	14.6042	0.910983	0.455492	+	0.890240i	$0.349464\pi$
$258$	0	0
$259$	3.13018	0.194500
$260$	0	0
$261$	−10.3595	−0.641239
$262$	0	0
$263$	12.2383	0.754645	0.377322	−	0.926082i	$-0.376845\pi$
$263$	12.2383	0.754645	0.377322	+	0.926082i	$0.376845\pi$
$264$	0	0
$265$	6.27108	0.385229
$266$	0	0
$267$	25.7539	1.57611
$268$	0	0
$269$	−4.76623	−0.290602	−0.145301	−	0.989387i	$-0.546415\pi$
$269$	−4.76623	−0.290602	−0.145301	+	0.989387i	$0.546415\pi$
$270$	0	0
$271$	6.12888	0.372303	0.186151	−	0.982521i	$-0.440399\pi$
$271$	6.12888	0.372303	0.186151	+	0.982521i	$0.440399\pi$
$272$	0	0
$273$	−3.57932	−0.216630
$274$	0	0
$275$	20.4165	1.23116
$276$	0	0
$277$	−12.5312	−0.752924	−0.376462	−	0.926432i	$-0.622860\pi$
$277$	−12.5312	−0.752924	−0.376462	+	0.926432i	$0.622860\pi$
$278$	0	0
$279$	−21.1337	−1.26524
$280$	0	0
$281$	−13.2783	−0.792115	−0.396058	−	0.918226i	$-0.629622\pi$
$281$	−13.2783	−0.792115	−0.396058	+	0.918226i	$0.629622\pi$
$282$	0	0
$283$	−28.2298	−1.67809	−0.839044	−	0.544064i	$-0.816885\pi$
$283$	−28.2298	−1.67809	−0.839044	+	0.544064i	$0.816885\pi$
$284$	0	0
$285$	2.50410	0.148330
$286$	0	0
$287$	−4.14569	−0.244712
$288$	0	0
$289$	7.30761	0.429859
$290$	0	0
$291$	21.3241	1.25004
$292$	0	0
$293$	−9.45824	−0.552557	−0.276278	−	0.961078i	$-0.589101\pi$
$293$	−9.45824	−0.552557	−0.276278	+	0.961078i	$0.589101\pi$
$294$	0	0
$295$	2.79182	0.162546
$296$	0	0
$297$	−2.06781	−0.119987
$298$	0	0
$299$	−4.92838	−0.285015
$300$	0	0
$301$	−4.84541	−0.279285
$302$	0	0
$303$	−21.2823	−1.22264
$304$	0	0
$305$	1.83726	0.105201
$306$	0	0
$307$	12.5716	0.717501	0.358750	−	0.933433i	$-0.383203\pi$
$307$	12.5716	0.717501	0.358750	+	0.933433i	$0.383203\pi$
$308$	0	0
$309$	8.53316	0.485435
$310$	0	0
$311$	−5.56292	−0.315444	−0.157722	−	0.987484i	$-0.550415\pi$
$311$	−5.56292	−0.315444	−0.157722	+	0.987484i	$0.550415\pi$
$312$	0	0
$313$	−7.74202	−0.437605	−0.218802	−	0.975769i	$-0.570215\pi$
$313$	−7.74202	−0.437605	−0.218802	+	0.975769i	$0.570215\pi$
$314$	0	0
$315$	2.10009	0.118327
$316$	0	0
$317$	−21.6671	−1.21695	−0.608474	−	0.793574i	$-0.708218\pi$
$317$	−21.6671	−1.21695	−0.608474	+	0.793574i	$0.708218\pi$
$318$	0	0
$319$	16.7935	0.940257
$320$	0	0
$321$	16.1928	0.903794
$322$	0	0
$323$	−4.93027	−0.274328
$324$	0	0
$325$	8.72243	0.483833
$326$	0	0
$327$	24.1612	1.33612
$328$	0	0
$329$	−5.56684	−0.306910
$330$	0	0
$331$	−19.7476	−1.08543	−0.542714	−	0.839918i	$-0.682603\pi$
$331$	−19.7476	−1.08543	−0.542714	+	0.839918i	$0.682603\pi$
$332$	0	0
$333$	15.0369	0.824020
$334$	0	0
$335$	8.01685	0.438007
$336$	0	0
$337$	−2.89769	−0.157847	−0.0789237	−	0.996881i	$-0.525148\pi$
$337$	−2.89769	−0.157847	−0.0789237	+	0.996881i	$0.525148\pi$
$338$	0	0
$339$	37.4840	2.03585
$340$	0	0
$341$	34.2592	1.85524
$342$	0	0
$343$	−8.93118	−0.482238
$344$	0	0
$345$	5.63469	0.303362
$346$	0	0
$347$	15.4000	0.826717	0.413358	−	0.910568i	$-0.364356\pi$
$347$	15.4000	0.826717	0.413358	+	0.910568i	$0.364356\pi$
$348$	0	0
$349$	−8.82234	−0.472249	−0.236125	−	0.971723i	$-0.575877\pi$
$349$	−8.82234	−0.472249	−0.236125	+	0.971723i	$0.575877\pi$
$350$	0	0
$351$	−0.883417	−0.0471533
$352$	0	0
$353$	18.8313	1.00229	0.501144	−	0.865364i	$-0.332913\pi$
$353$	18.8313	1.00229	0.501144	+	0.865364i	$0.332913\pi$
$354$	0	0
$355$	13.8458	0.734860
$356$	0	0
$357$	−8.05724	−0.426434
$358$	0	0
$359$	−6.38897	−0.337197	−0.168598	−	0.985685i	$-0.553924\pi$
$359$	−6.38897	−0.337197	−0.168598	+	0.985685i	$0.553924\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	37.9367	1.99116
$364$	0	0
$365$	−3.65716	−0.191424
$366$	0	0
$367$	5.70570	0.297835	0.148918	−	0.988850i	$-0.452421\pi$
$367$	5.70570	0.297835	0.148918	+	0.988850i	$0.452421\pi$
$368$	0	0
$369$	−19.9153	−1.03675
$370$	0	0
$371$	4.09267	0.212481
$372$	0	0
$373$	17.1999	0.890579	0.445290	−	0.895387i	$-0.353101\pi$
$373$	17.1999	0.890579	0.445290	+	0.895387i	$0.353101\pi$
$374$	0	0
$375$	−22.4930	−1.16153
$376$	0	0
$377$	7.17459	0.369510
$378$	0	0
$379$	17.6285	0.905517	0.452758	−	0.891633i	$-0.350440\pi$
$379$	17.6285	0.905517	0.452758	+	0.891633i	$0.350440\pi$
$380$	0	0
$381$	−31.0285	−1.58964
$382$	0	0
$383$	−30.5078	−1.55888	−0.779438	−	0.626480i	$-0.784495\pi$
$383$	−30.5078	−1.55888	−0.779438	+	0.626480i	$0.784495\pi$
$384$	0	0
$385$	−3.40440	−0.173504
$386$	0	0
$387$	−23.2767	−1.18322
$388$	0	0
$389$	−22.4593	−1.13873	−0.569366	−	0.822084i	$-0.692811\pi$
$389$	−22.4593	−1.13873	−0.569366	+	0.822084i	$0.692811\pi$
$390$	0	0
$391$	−11.0941	−0.561050
$392$	0	0
$393$	−17.8877	−0.902317
$394$	0	0
$395$	−1.00873	−0.0507545
$396$	0	0
$397$	−5.91050	−0.296640	−0.148320	−	0.988939i	$-0.547386\pi$
$397$	−5.91050	−0.296640	−0.148320	+	0.988939i	$0.547386\pi$
$398$	0	0
$399$	1.63424	0.0818142
$400$	0	0
$401$	−9.45266	−0.472043	−0.236022	−	0.971748i	$-0.575844\pi$
$401$	−9.45266	−0.472043	−0.236022	+	0.971748i	$0.575844\pi$
$402$	0	0
$403$	14.6363	0.729088
$404$	0	0
$405$	−8.56021	−0.425360
$406$	0	0
$407$	−24.3759	−1.20827
$408$	0	0
$409$	4.79901	0.237296	0.118648	−	0.992936i	$-0.462144\pi$
$409$	4.79901	0.237296	0.118648	+	0.992936i	$0.462144\pi$
$410$	0	0
$411$	−19.8852	−0.980867
$412$	0	0
$413$	1.82201	0.0896555
$414$	0	0
$415$	−12.8974	−0.633107
$416$	0	0
$417$	−11.5112	−0.563706
$418$	0	0
$419$	−13.2479	−0.647202	−0.323601	−	0.946194i	$-0.604894\pi$
$419$	−13.2479	−0.647202	−0.323601	+	0.946194i	$0.604894\pi$
$420$	0	0
$421$	29.4889	1.43720	0.718601	−	0.695422i	$-0.244783\pi$
$421$	29.4889	1.43720	0.718601	+	0.695422i	$0.244783\pi$
$422$	0	0
$423$	−26.7423	−1.30026
$424$	0	0
$425$	19.6347	0.952422
$426$	0	0
$427$	1.19904	0.0580257
$428$	0	0
$429$	27.8736	1.34575
$430$	0	0
$431$	−30.9365	−1.49016	−0.745079	−	0.666977i	$-0.767588\pi$
$431$	−30.9365	−1.49016	−0.745079	+	0.666977i	$0.767588\pi$
$432$	0	0
$433$	21.2939	1.02332	0.511659	−	0.859188i	$-0.329031\pi$
$433$	21.2939	1.02332	0.511659	+	0.859188i	$0.329031\pi$
$434$	0	0
$435$	−8.20283	−0.393295
$436$	0	0
$437$	2.25019	0.107641
$438$	0	0
$439$	−19.7974	−0.944881	−0.472440	−	0.881363i	$-0.656627\pi$
$439$	−19.7974	−0.944881	−0.472440	+	0.881363i	$0.656627\pi$
$440$	0	0
$441$	−20.7668	−0.988895
$442$	0	0
$443$	−24.0702	−1.14361	−0.571806	−	0.820389i	$-0.693757\pi$
$443$	−24.0702	−1.14361	−0.571806	+	0.820389i	$0.693757\pi$
$444$	0	0
$445$	10.4650	0.496087
$446$	0	0
$447$	−39.0754	−1.84820
$448$	0	0
$449$	2.00537	0.0946390	0.0473195	−	0.998880i	$-0.484932\pi$
$449$	2.00537	0.0946390	0.0473195	+	0.998880i	$0.484932\pi$
$450$	0	0
$451$	32.2841	1.52020
$452$	0	0
$453$	5.08345	0.238841
$454$	0	0
$455$	−1.45444	−0.0681852
$456$	0	0
$457$	−10.8639	−0.508193	−0.254096	−	0.967179i	$-0.581778\pi$
$457$	−10.8639	−0.508193	−0.254096	+	0.967179i	$0.581778\pi$
$458$	0	0
$459$	−1.98862	−0.0928209
$460$	0	0
$461$	15.4497	0.719564	0.359782	−	0.933036i	$-0.382851\pi$
$461$	15.4497	0.719564	0.359782	+	0.933036i	$0.382851\pi$
$462$	0	0
$463$	−18.7008	−0.869100	−0.434550	−	0.900648i	$-0.643092\pi$
$463$	−18.7008	−0.869100	−0.434550	+	0.900648i	$0.643092\pi$
$464$	0	0
$465$	−16.7340	−0.776019
$466$	0	0
$467$	22.0548	1.02057	0.510287	−	0.860004i	$-0.329539\pi$
$467$	22.0548	1.02057	0.510287	+	0.860004i	$0.329539\pi$
$468$	0	0
$469$	5.23200	0.241591
$470$	0	0
$471$	6.20915	0.286103
$472$	0	0
$473$	37.7332	1.73497
$474$	0	0
$475$	−3.98247	−0.182728
$476$	0	0
$477$	19.6606	0.900197
$478$	0	0
$479$	−12.6854	−0.579611	−0.289806	−	0.957085i	$-0.593591\pi$
$479$	−12.6854	−0.579611	−0.289806	+	0.957085i	$0.593591\pi$
$480$	0	0
$481$	−10.4140	−0.474837
$482$	0	0
$483$	3.67734	0.167325
$484$	0	0
$485$	8.66494	0.393455
$486$	0	0
$487$	−25.4984	−1.15544	−0.577722	−	0.816234i	$-0.696058\pi$
$487$	−25.4984	−1.15544	−0.577722	+	0.816234i	$0.696058\pi$
$488$	0	0
$489$	−4.21715	−0.190706
$490$	0	0
$491$	24.9883	1.12771	0.563853	−	0.825875i	$-0.309318\pi$
$491$	24.9883	1.12771	0.563853	+	0.825875i	$0.309318\pi$
$492$	0	0
$493$	16.1504	0.727378
$494$	0	0
$495$	−16.3543	−0.735069
$496$	0	0
$497$	9.03613	0.405326
$498$	0	0
$499$	−23.3219	−1.04403	−0.522016	−	0.852935i	$-0.674820\pi$
$499$	−23.3219	−1.04403	−0.522016	+	0.852935i	$0.674820\pi$
$500$	0	0
$501$	9.20223	0.411126
$502$	0	0
$503$	−11.6957	−0.521487	−0.260744	−	0.965408i	$-0.583968\pi$
$503$	−11.6957	−0.521487	−0.260744	+	0.965408i	$0.583968\pi$
$504$	0	0
$505$	−8.64799	−0.384830
$506$	0	0
$507$	−20.3634	−0.904371
$508$	0	0
$509$	−14.9289	−0.661710	−0.330855	−	0.943682i	$-0.607337\pi$
$509$	−14.9289	−0.661710	−0.330855	+	0.943682i	$0.607337\pi$
$510$	0	0
$511$	−2.38675	−0.105584
$512$	0	0
$513$	0.403349	0.0178083
$514$	0	0
$515$	3.46741	0.152792
$516$	0	0
$517$	43.3512	1.90658
$518$	0	0
$519$	−47.3370	−2.07786
$520$	0	0
$521$	28.4538	1.24658	0.623291	−	0.781990i	$-0.285795\pi$
$521$	28.4538	1.24658	0.623291	+	0.781990i	$0.285795\pi$
$522$	0	0
$523$	6.27044	0.274187	0.137094	−	0.990558i	$-0.456224\pi$
$523$	6.27044	0.274187	0.137094	+	0.990558i	$0.456224\pi$
$524$	0	0
$525$	−6.50830	−0.284046
$526$	0	0
$527$	32.9472	1.43520
$528$	0	0
$529$	−17.9366	−0.779854
$530$	0	0
$531$	8.75271	0.379835
$532$	0	0
$533$	13.7925	0.597421
$534$	0	0
$535$	6.57987	0.284473
$536$	0	0
$537$	−19.0378	−0.821542
$538$	0	0
$539$	33.6644	1.45003
$540$	0	0
$541$	−4.76648	−0.204927	−0.102464	−	0.994737i	$-0.532672\pi$
$541$	−4.76648	−0.204927	−0.102464	+	0.994737i	$0.532672\pi$
$542$	0	0
$543$	−0.946173	−0.0406042
$544$	0	0
$545$	9.81780	0.420548
$546$	0	0
$547$	−3.37305	−0.144221	−0.0721106	−	0.997397i	$-0.522973\pi$
$547$	−3.37305	−0.144221	−0.0721106	+	0.997397i	$0.522973\pi$
$548$	0	0
$549$	5.76003	0.245832
$550$	0	0
$551$	−3.27576	−0.139552
$552$	0	0
$553$	−0.658321	−0.0279946
$554$	0	0
$555$	11.9065	0.505401
$556$	0	0
$557$	−45.3138	−1.92001	−0.960003	−	0.279990i	$-0.909669\pi$
$557$	−45.3138	−1.92001	−0.960003	+	0.279990i	$0.909669\pi$
$558$	0	0
$559$	16.1205	0.681824
$560$	0	0
$561$	62.7449	2.64909
$562$	0	0
$563$	−5.07491	−0.213882	−0.106941	−	0.994265i	$-0.534106\pi$
$563$	−5.07491	−0.213882	−0.106941	+	0.994265i	$0.534106\pi$
$564$	0	0
$565$	15.2314	0.640791
$566$	0	0
$567$	−5.58661	−0.234616
$568$	0	0
$569$	−21.8671	−0.916718	−0.458359	−	0.888767i	$-0.651563\pi$
$569$	−21.8671	−0.916718	−0.458359	+	0.888767i	$0.651563\pi$
$570$	0	0
$571$	22.5429	0.943389	0.471695	−	0.881762i	$-0.343642\pi$
$571$	22.5429	0.943389	0.471695	+	0.881762i	$0.343642\pi$
$572$	0	0
$573$	−25.8728	−1.08085
$574$	0	0
$575$	−8.96131	−0.373713
$576$	0	0
$577$	−39.0368	−1.62512	−0.812562	−	0.582874i	$-0.801928\pi$
$577$	−39.0368	−1.62512	−0.812562	+	0.582874i	$0.801928\pi$
$578$	0	0
$579$	18.5275	0.769978
$580$	0	0
$581$	−8.41716	−0.349203
$582$	0	0
$583$	−31.8712	−1.31997
$584$	0	0
$585$	−6.98693	−0.288874
$586$	0	0
$587$	−12.5986	−0.520002	−0.260001	−	0.965608i	$-0.583723\pi$
$587$	−12.5986	−0.520002	−0.260001	+	0.965608i	$0.583723\pi$
$588$	0	0
$589$	−6.68263	−0.275353
$590$	0	0
$591$	55.5984	2.28701
$592$	0	0
$593$	27.3483	1.12306	0.561530	−	0.827457i	$-0.310213\pi$
$593$	27.3483	1.12306	0.561530	+	0.827457i	$0.310213\pi$
$594$	0	0
$595$	−3.27402	−0.134222
$596$	0	0
$597$	60.5373	2.47763
$598$	0	0
$599$	−17.0298	−0.695818	−0.347909	−	0.937528i	$-0.613108\pi$
$599$	−17.0298	−0.695818	−0.347909	+	0.937528i	$0.613108\pi$
$600$	0	0
$601$	8.57587	0.349817	0.174909	−	0.984585i	$-0.444037\pi$
$601$	8.57587	0.349817	0.174909	+	0.984585i	$0.444037\pi$
$602$	0	0
$603$	25.1338	1.02353
$604$	0	0
$605$	15.4154	0.626726
$606$	0	0
$607$	41.0258	1.66519	0.832594	−	0.553884i	$-0.186855\pi$
$607$	41.0258	1.66519	0.832594	+	0.553884i	$0.186855\pi$
$608$	0	0
$609$	−5.35337	−0.216930
$610$	0	0
$611$	18.5206	0.749265
$612$	0	0
$613$	−33.2855	−1.34439	−0.672193	−	0.740376i	$-0.734648\pi$
$613$	−33.2855	−1.34439	−0.672193	+	0.740376i	$0.734648\pi$
$614$	0	0
$615$	−15.7692	−0.635877
$616$	0	0
$617$	37.8994	1.52577	0.762887	−	0.646532i	$-0.223781\pi$
$617$	37.8994	1.52577	0.762887	+	0.646532i	$0.223781\pi$
$618$	0	0
$619$	−1.58989	−0.0639029	−0.0319515	−	0.999489i	$-0.510172\pi$
$619$	−1.58989	−0.0639029	−0.0319515	+	0.999489i	$0.510172\pi$
$620$	0	0
$621$	0.907612	0.0364212
$622$	0	0
$623$	6.82971	0.273627
$624$	0	0
$625$	10.7724	0.430897
$626$	0	0
$627$	−12.7265	−0.508246
$628$	0	0
$629$	−23.4424	−0.934711
$630$	0	0
$631$	−21.3500	−0.849929	−0.424965	−	0.905210i	$-0.639713\pi$
$631$	−21.3500	−0.849929	−0.424965	+	0.905210i	$0.639713\pi$
$632$	0	0
$633$	24.0481	0.955825
$634$	0	0
$635$	−12.6083	−0.500345
$636$	0	0
$637$	14.3822	0.569845
$638$	0	0
$639$	43.4083	1.71721
$640$	0	0
$641$	−9.25486	−0.365545	−0.182773	−	0.983155i	$-0.558507\pi$
$641$	−9.25486	−0.365545	−0.182773	+	0.983155i	$0.558507\pi$
$642$	0	0
$643$	36.9308	1.45641	0.728204	−	0.685360i	$-0.240355\pi$
$643$	36.9308	1.45641	0.728204	+	0.685360i	$0.240355\pi$
$644$	0	0
$645$	−18.4308	−0.725713
$646$	0	0
$647$	4.43717	0.174443	0.0872215	−	0.996189i	$-0.472201\pi$
$647$	4.43717	0.174443	0.0872215	+	0.996189i	$0.472201\pi$
$648$	0	0
$649$	−14.1887	−0.556957
$650$	0	0
$651$	−10.9210	−0.428028
$652$	0	0
$653$	−31.7716	−1.24332	−0.621660	−	0.783287i	$-0.713541\pi$
$653$	−31.7716	−1.24332	−0.621660	+	0.783287i	$0.713541\pi$
$654$	0	0
$655$	−7.26860	−0.284008
$656$	0	0
$657$	−11.4656	−0.447317
$658$	0	0
$659$	−18.8368	−0.733778	−0.366889	−	0.930265i	$-0.619577\pi$
$659$	−18.8368	−0.733778	−0.366889	+	0.930265i	$0.619577\pi$
$660$	0	0
$661$	2.00428	0.0779576	0.0389788	−	0.999240i	$-0.487590\pi$
$661$	2.00428	0.0779576	0.0389788	+	0.999240i	$0.487590\pi$
$662$	0	0
$663$	26.8061	1.04106
$664$	0	0
$665$	0.664065	0.0257513
$666$	0	0
$667$	−7.37108	−0.285410
$668$	0	0
$669$	46.4873	1.79730
$670$	0	0
$671$	−9.33741	−0.360467
$672$	0	0
$673$	17.5254	0.675555	0.337778	−	0.941226i	$-0.390325\pi$
$673$	17.5254	0.675555	0.337778	+	0.941226i	$0.390325\pi$
$674$	0	0
$675$	−1.60633	−0.0618275
$676$	0	0
$677$	−10.0753	−0.387227	−0.193614	−	0.981078i	$-0.562021\pi$
$677$	−10.0753	−0.387227	−0.193614	+	0.981078i	$0.562021\pi$
$678$	0	0
$679$	5.65496	0.217017
$680$	0	0
$681$	−5.72497	−0.219382
$682$	0	0
$683$	−6.96926	−0.266671	−0.133336	−	0.991071i	$-0.542569\pi$
$683$	−6.96926	−0.266671	−0.133336	+	0.991071i	$0.542569\pi$
$684$	0	0
$685$	−8.08028	−0.308732
$686$	0	0
$687$	51.3622	1.95959
$688$	0	0
$689$	−13.6161	−0.518733
$690$	0	0
$691$	−36.3762	−1.38382	−0.691908	−	0.721985i	$-0.743230\pi$
$691$	−36.3762	−1.38382	−0.691908	+	0.721985i	$0.743230\pi$
$692$	0	0
$693$	−10.6732	−0.405442
$694$	0	0
$695$	−4.67753	−0.177429
$696$	0	0
$697$	31.0478	1.17602
$698$	0	0
$699$	−61.1230	−2.31189
$700$	0	0
$701$	20.7487	0.783668	0.391834	−	0.920036i	$-0.371841\pi$
$701$	20.7487	0.783668	0.391834	+	0.920036i	$0.371841\pi$
$702$	0	0
$703$	4.75479	0.179330
$704$	0	0
$705$	−21.1749	−0.797495
$706$	0	0
$707$	−5.64390	−0.212261
$708$	0	0
$709$	16.7842	0.630345	0.315172	−	0.949034i	$-0.397938\pi$
$709$	16.7842	0.630345	0.315172	+	0.949034i	$0.397938\pi$
$710$	0	0
$711$	−3.16248	−0.118602
$712$	0	0
$713$	−15.0372	−0.563147
$714$	0	0
$715$	11.3263	0.423579
$716$	0	0
$717$	59.3799	2.21758
$718$	0	0
$719$	−12.4046	−0.462615	−0.231307	−	0.972881i	$-0.574300\pi$
$719$	−12.4046	−0.462615	−0.231307	+	0.972881i	$0.574300\pi$
$720$	0	0
$721$	2.26292	0.0842756
$722$	0	0
$723$	21.8534	0.812738
$724$	0	0
$725$	13.0456	0.484502
$726$	0	0
$727$	−44.4741	−1.64945	−0.824727	−	0.565530i	$-0.808671\pi$
$727$	−44.4741	−1.64945	−0.824727	+	0.565530i	$0.808671\pi$
$728$	0	0
$729$	−29.8412	−1.10523
$730$	0	0
$731$	36.2881	1.34217
$732$	0	0
$733$	−20.2150	−0.746656	−0.373328	−	0.927699i	$-0.621783\pi$
$733$	−20.2150	−0.746656	−0.373328	+	0.927699i	$0.621783\pi$
$734$	0	0
$735$	−16.4434	−0.606525
$736$	0	0
$737$	−40.7437	−1.50081
$738$	0	0
$739$	40.0180	1.47209	0.736043	−	0.676935i	$-0.236692\pi$
$739$	40.0180	1.47209	0.736043	+	0.676935i	$0.236692\pi$
$740$	0	0
$741$	−5.43704	−0.199735
$742$	0	0
$743$	13.4838	0.494673	0.247337	−	0.968930i	$-0.420445\pi$
$743$	13.4838	0.494673	0.247337	+	0.968930i	$0.420445\pi$
$744$	0	0
$745$	−15.8781	−0.581729
$746$	0	0
$747$	−40.4349	−1.47943
$748$	0	0
$749$	4.29419	0.156906
$750$	0	0
$751$	−25.4034	−0.926984	−0.463492	−	0.886101i	$-0.653404\pi$
$751$	−25.4034	−0.926984	−0.463492	+	0.886101i	$0.653404\pi$
$752$	0	0
$753$	26.5335	0.966933
$754$	0	0
$755$	2.06564	0.0751762
$756$	0	0
$757$	−28.1858	−1.02443	−0.512215	−	0.858857i	$-0.671175\pi$
$757$	−28.1858	−1.02443	−0.512215	+	0.858857i	$0.671175\pi$
$758$	0	0
$759$	−28.6369	−1.03945
$760$	0	0
$761$	−28.9203	−1.04836	−0.524179	−	0.851608i	$-0.675628\pi$
$761$	−28.9203	−1.04836	−0.524179	+	0.851608i	$0.675628\pi$
$762$	0	0
$763$	6.40734	0.231961
$764$	0	0
$765$	−15.7280	−0.568646
$766$	0	0
$767$	−6.06177	−0.218878
$768$	0	0
$769$	−24.2439	−0.874256	−0.437128	−	0.899399i	$-0.644004\pi$
$769$	−24.2439	−0.874256	−0.437128	+	0.899399i	$0.644004\pi$
$770$	0	0
$771$	36.2539	1.30565
$772$	0	0
$773$	13.2842	0.477801	0.238901	−	0.971044i	$-0.423213\pi$
$773$	13.2842	0.477801	0.238901	+	0.971044i	$0.423213\pi$
$774$	0	0
$775$	26.6134	0.955981
$776$	0	0
$777$	7.77046	0.278764
$778$	0	0
$779$	−6.29737	−0.225627
$780$	0	0
$781$	−70.3679	−2.51796
$782$	0	0
$783$	−1.32128	−0.0472185
$784$	0	0
$785$	2.52306	0.0900519
$786$	0	0
$787$	51.9098	1.85038	0.925192	−	0.379498i	$-0.123903\pi$
$787$	51.9098	1.85038	0.925192	+	0.379498i	$0.123903\pi$
$788$	0	0
$789$	30.3807	1.08158
$790$	0	0
$791$	9.94042	0.353441
$792$	0	0
$793$	−3.98916	−0.141659
$794$	0	0
$795$	15.5675	0.552124
$796$	0	0
$797$	−19.9085	−0.705197	−0.352598	−	0.935775i	$-0.614702\pi$
$797$	−19.9085	−0.705197	−0.352598	+	0.935775i	$0.614702\pi$
$798$	0	0
$799$	41.6910	1.47492
$800$	0	0
$801$	32.8090	1.15925
$802$	0	0
$803$	18.5866	0.655906
$804$	0	0
$805$	1.49427	0.0526662
$806$	0	0
$807$	−11.8319	−0.416501
$808$	0	0
$809$	7.23883	0.254504	0.127252	−	0.991870i	$-0.459384\pi$
$809$	7.23883	0.254504	0.127252	+	0.991870i	$0.459384\pi$
$810$	0	0
$811$	9.70986	0.340959	0.170480	−	0.985361i	$-0.445468\pi$
$811$	9.70986	0.340959	0.170480	+	0.985361i	$0.445468\pi$
$812$	0	0
$813$	15.2145	0.533597
$814$	0	0
$815$	−1.71362	−0.0600256
$816$	0	0
$817$	−7.36027	−0.257503
$818$	0	0
$819$	−4.55985	−0.159334
$820$	0	0
$821$	−30.3133	−1.05794	−0.528970	−	0.848640i	$-0.677422\pi$
$821$	−30.3133	−1.05794	−0.528970	+	0.848640i	$0.677422\pi$
$822$	0	0
$823$	22.4937	0.784083	0.392041	−	0.919948i	$-0.371769\pi$
$823$	22.4937	0.784083	0.392041	+	0.919948i	$0.371769\pi$
$824$	0	0
$825$	50.6827	1.76455
$826$	0	0
$827$	−31.3105	−1.08877	−0.544386	−	0.838835i	$-0.683237\pi$
$827$	−31.3105	−1.08877	−0.544386	+	0.838835i	$0.683237\pi$
$828$	0	0
$829$	−0.349214	−0.0121287	−0.00606435	−	0.999982i	$-0.501930\pi$
$829$	−0.349214	−0.0121287	−0.00606435	+	0.999982i	$0.501930\pi$
$830$	0	0
$831$	−31.1078	−1.07912
$832$	0	0
$833$	32.3752	1.12173
$834$	0	0
$835$	3.73929	0.129403
$836$	0	0
$837$	−2.69543	−0.0931678
$838$	0	0
$839$	27.1615	0.937720	0.468860	−	0.883273i	$-0.344665\pi$
$839$	27.1615	0.937720	0.468860	+	0.883273i	$0.344665\pi$
$840$	0	0
$841$	−18.2694	−0.629979
$842$	0	0
$843$	−32.9624	−1.13529
$844$	0	0
$845$	−8.27458	−0.284654
$846$	0	0
$847$	10.0605	0.345683
$848$	0	0
$849$	−70.0786	−2.40509
$850$	0	0
$851$	10.6992	0.366763
$852$	0	0
$853$	−7.70825	−0.263926	−0.131963	−	0.991255i	$-0.542128\pi$
$853$	−7.70825	−0.263926	−0.131963	+	0.991255i	$0.542128\pi$
$854$	0	0
$855$	3.19008	0.109098
$856$	0	0
$857$	44.6750	1.52607	0.763034	−	0.646358i	$-0.223709\pi$
$857$	44.6750	1.52607	0.763034	+	0.646358i	$0.223709\pi$
$858$	0	0
$859$	29.2965	0.999583	0.499791	−	0.866146i	$-0.333410\pi$
$859$	29.2965	0.999583	0.499791	+	0.866146i	$0.333410\pi$
$860$	0	0
$861$	−10.2914	−0.350730
$862$	0	0
$863$	−32.3740	−1.10202	−0.551011	−	0.834498i	$-0.685758\pi$
$863$	−32.3740	−1.10202	−0.551011	+	0.834498i	$0.685758\pi$
$864$	0	0
$865$	−19.2352	−0.654016
$866$	0	0
$867$	18.1407	0.616089
$868$	0	0
$869$	5.12660	0.173908
$870$	0	0
$871$	−17.4067	−0.589802
$872$	0	0
$873$	27.1656	0.919418
$874$	0	0
$875$	−5.96495	−0.201652
$876$	0	0
$877$	2.26761	0.0765717	0.0382859	−	0.999267i	$-0.487810\pi$
$877$	2.26761	0.0765717	0.0382859	+	0.999267i	$0.487810\pi$
$878$	0	0
$879$	−23.4795	−0.791943
$880$	0	0
$881$	3.18348	0.107254	0.0536271	−	0.998561i	$-0.482922\pi$
$881$	3.18348	0.107254	0.0536271	+	0.998561i	$0.482922\pi$
$882$	0	0
$883$	−16.1955	−0.545021	−0.272510	−	0.962153i	$-0.587854\pi$
$883$	−16.1955	−0.545021	−0.272510	+	0.962153i	$0.587854\pi$
$884$	0	0
$885$	6.93052	0.232967
$886$	0	0
$887$	−12.2848	−0.412483	−0.206241	−	0.978501i	$-0.566123\pi$
$887$	−12.2848	−0.412483	−0.206241	+	0.978501i	$0.566123\pi$
$888$	0	0
$889$	−8.22850	−0.275975
$890$	0	0
$891$	43.5051	1.45748
$892$	0	0
$893$	−8.45612	−0.282973
$894$	0	0
$895$	−7.73593	−0.258584
$896$	0	0
$897$	−12.2344	−0.408494
$898$	0	0
$899$	21.8907	0.730096
$900$	0	0
$901$	−30.6507	−1.02112
$902$	0	0
$903$	−12.0284	−0.400281
$904$	0	0
$905$	−0.384473	−0.0127803
$906$	0	0
$907$	38.1373	1.26633	0.633164	−	0.774018i	$-0.281756\pi$
$907$	38.1373	1.26633	0.633164	+	0.774018i	$0.281756\pi$
$908$	0	0
$909$	−27.1125	−0.899265
$910$	0	0
$911$	59.0466	1.95630	0.978150	−	0.207899i	$-0.0666624\pi$
$911$	59.0466	1.95630	0.978150	+	0.207899i	$0.0666624\pi$
$912$	0	0
$913$	65.5477	2.16931
$914$	0	0
$915$	4.56087	0.150778
$916$	0	0
$917$	−4.74368	−0.156650
$918$	0	0
$919$	48.6032	1.60327	0.801635	−	0.597814i	$-0.203964\pi$
$919$	48.6032	1.60327	0.801635	+	0.597814i	$0.203964\pi$
$920$	0	0
$921$	31.2083	1.02835
$922$	0	0
$923$	−30.0628	−0.989530
$924$	0	0
$925$	−18.9358	−0.622606
$926$	0	0
$927$	10.8708	0.357043
$928$	0	0
$929$	−36.1498	−1.18604	−0.593018	−	0.805189i	$-0.702064\pi$
$929$	−36.1498	−1.18604	−0.593018	+	0.805189i	$0.702064\pi$
$930$	0	0
$931$	−6.56661	−0.215212
$932$	0	0
$933$	−13.8096	−0.452105
$934$	0	0
$935$	25.4961	0.833812
$936$	0	0
$937$	1.54777	0.0505633	0.0252817	−	0.999680i	$-0.491952\pi$
$937$	1.54777	0.0505633	0.0252817	+	0.999680i	$0.491952\pi$
$938$	0	0
$939$	−19.2191	−0.627190
$940$	0	0
$941$	−46.3173	−1.50990	−0.754950	−	0.655782i	$-0.772339\pi$
$941$	−46.3173	−1.50990	−0.754950	+	0.655782i	$0.772339\pi$
$942$	0	0
$943$	−14.1703	−0.461448
$944$	0	0
$945$	0.267850	0.00871316
$946$	0	0
$947$	−47.0761	−1.52977	−0.764884	−	0.644169i	$-0.777204\pi$
$947$	−47.0761	−1.52977	−0.764884	+	0.644169i	$0.777204\pi$
$948$	0	0
$949$	7.94063	0.257764
$950$	0	0
$951$	−53.7872	−1.74417
$952$	0	0
$953$	12.7537	0.413134	0.206567	−	0.978432i	$-0.433771\pi$
$953$	12.7537	0.413134	0.206567	+	0.978432i	$0.433771\pi$
$954$	0	0
$955$	−10.5133	−0.340202
$956$	0	0
$957$	41.6888	1.34761
$958$	0	0
$959$	−5.27340	−0.170287
$960$	0	0
$961$	13.6576	0.440567
$962$	0	0
$963$	20.6287	0.664751
$964$	0	0
$965$	7.52859	0.242354
$966$	0	0
$967$	−9.92906	−0.319297	−0.159649	−	0.987174i	$-0.551036\pi$
$967$	−9.92906	−0.319297	−0.159649	+	0.987174i	$0.551036\pi$
$968$	0	0
$969$	−12.2391	−0.393176
$970$	0	0
$971$	36.8091	1.18126	0.590630	−	0.806943i	$-0.298879\pi$
$971$	36.8091	1.18126	0.590630	+	0.806943i	$0.298879\pi$
$972$	0	0
$973$	−3.05267	−0.0978642
$974$	0	0
$975$	21.6529	0.693447
$976$	0	0
$977$	−31.2825	−1.00082	−0.500408	−	0.865790i	$-0.666817\pi$
$977$	−31.2825	−1.00082	−0.500408	+	0.865790i	$0.666817\pi$
$978$	0	0
$979$	−53.1856	−1.69982
$980$	0	0
$981$	30.7800	0.982730
$982$	0	0
$983$	28.0864	0.895816	0.447908	−	0.894080i	$-0.352169\pi$
$983$	28.0864	0.895816	0.447908	+	0.894080i	$0.352169\pi$
$984$	0	0
$985$	22.5922	0.719846
$986$	0	0
$987$	−13.8193	−0.439873
$988$	0	0
$989$	−16.5620	−0.526641
$990$	0	0
$991$	−13.2215	−0.419995	−0.209997	−	0.977702i	$-0.567346\pi$
$991$	−13.2215	−0.419995	−0.209997	+	0.977702i	$0.567346\pi$
$992$	0	0
$993$	−49.0222	−1.55567
$994$	0	0
$995$	24.5991	0.779843
$996$	0	0
$997$	40.5073	1.28288	0.641440	−	0.767173i	$-0.278337\pi$
$997$	40.5073	1.28288	0.641440	+	0.767173i	$0.278337\pi$
$998$	0	0
$999$	1.91784	0.0606778

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6004.2.a.g.1.25	✓	27

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6004.2.a.g.1.25	✓	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)

\(f(q)\)	\(=\)	\(q+2.48243 q^{3} +1.00873 q^{5} +0.658321 q^{7} +3.16248 q^{9} +O(q^{10})\)	\(q+2.48243 q^{3} +1.00873 q^{5} +0.658321 q^{7} +3.16248 q^{9} -5.12660 q^{11} -2.19021 q^{13} +2.50410 q^{15} -4.93027 q^{17} +1.00000 q^{19} +1.63424 q^{21} +2.25019 q^{23} -3.98247 q^{25} +0.403349 q^{27} -3.27576 q^{29} -6.68263 q^{31} -12.7265 q^{33} +0.664065 q^{35} +4.75479 q^{37} -5.43704 q^{39} -6.29737 q^{41} -7.36027 q^{43} +3.19008 q^{45} -8.45612 q^{47} -6.56661 q^{49} -12.2391 q^{51} +6.21683 q^{53} -5.17134 q^{55} +2.48243 q^{57} +2.76767 q^{59} +1.82136 q^{61} +2.08193 q^{63} -2.20932 q^{65} +7.94750 q^{67} +5.58595 q^{69} +13.7260 q^{71} -3.62552 q^{73} -9.88622 q^{75} -3.37495 q^{77} -1.00000 q^{79} -8.48616 q^{81} -12.7858 q^{83} -4.97330 q^{85} -8.13186 q^{87} +10.3744 q^{89} -1.44186 q^{91} -16.5892 q^{93} +1.00873 q^{95} +8.58998 q^{97} -16.2128 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 27 q - 4 q^{3} - 10 q^{5} - 8 q^{7} + 19 q^{9}+O(q^{10}) \) 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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