LMFDB - Embedded newform 1380.4.a.j.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1380,4,Mod(1,1380)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1380, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1380.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1380.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:	$81.4226358079$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 1420x^{5} - 7866x^{4} + 519199x^{3} + 5329890x^{2} - 8528484x - 84125016 $ x^7 - 1420x^5 - 7866x^4 + 519199x^3 + 5329890x^2 - 8528484*x - 84125016
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$25.9626$ of defining polynomial
Character	$\chi$	$=$	1380.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000 q^{3} +5.00000 q^{5} -20.9626 q^{7} +9.00000 q^{9} +O(q^{10})$	$q+3.00000 q^{3} +5.00000 q^{5} -20.9626 q^{7} +9.00000 q^{9} -5.12906 q^{11} +22.8827 q^{13} +15.0000 q^{15} +138.573 q^{17} -93.0918 q^{19} -62.8879 q^{21} +23.0000 q^{23} +25.0000 q^{25} +27.0000 q^{27} -17.0282 q^{29} -2.74296 q^{31} -15.3872 q^{33} -104.813 q^{35} -41.4336 q^{37} +68.6480 q^{39} +455.865 q^{41} -450.537 q^{43} +45.0000 q^{45} -388.805 q^{47} +96.4321 q^{49} +415.720 q^{51} +182.360 q^{53} -25.6453 q^{55} -279.275 q^{57} +634.144 q^{59} +386.858 q^{61} -188.664 q^{63} +114.413 q^{65} +93.1921 q^{67} +69.0000 q^{69} -169.380 q^{71} +439.911 q^{73} +75.0000 q^{75} +107.519 q^{77} +402.575 q^{79} +81.0000 q^{81} +878.499 q^{83} +692.867 q^{85} -51.0846 q^{87} +1643.87 q^{89} -479.681 q^{91} -8.22889 q^{93} -465.459 q^{95} +106.397 q^{97} -46.1616 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 21 q^{3} + 35 q^{5} + 35 q^{7} + 63 q^{9}+O(q^{10}) $ 7 * q + 21 * q^3 + 35 * q^5 + 35 * q^7 + 63 * q^9	$ 7 q + 21 q^{3} + 35 q^{5} + 35 q^{7} + 63 q^{9} + 48 q^{11} + 90 q^{13} + 105 q^{15} + 163 q^{17} + 200 q^{19} + 105 q^{21} + 161 q^{23} + 175 q^{25} + 189 q^{27} + 81 q^{29} + 125 q^{31} + 144 q^{33} + 175 q^{35} + 5 q^{37} + 270 q^{39} + 369 q^{41} + 462 q^{43} + 315 q^{45} + 134 q^{47} + 614 q^{49} + 489 q^{51} + 561 q^{53} + 240 q^{55} + 600 q^{57} + 951 q^{59} + 860 q^{61} + 315 q^{63} + 450 q^{65} + 447 q^{67} + 483 q^{69} + 735 q^{71} + 1460 q^{73} + 525 q^{75} + 496 q^{77} + 18 q^{79} + 567 q^{81} + 261 q^{83} + 815 q^{85} + 243 q^{87} + 2024 q^{89} + 692 q^{91} + 375 q^{93} + 1000 q^{95} + 668 q^{97} + 432 q^{99}+O(q^{100}) $ 7 * q + 21 * q^3 + 35 * q^5 + 35 * q^7 + 63 * q^9 + 48 * q^11 + 90 * q^13 + 105 * q^15 + 163 * q^17 + 200 * q^19 + 105 * q^21 + 161 * q^23 + 175 * q^25 + 189 * q^27 + 81 * q^29 + 125 * q^31 + 144 * q^33 + 175 * q^35 + 5 * q^37 + 270 * q^39 + 369 * q^41 + 462 * q^43 + 315 * q^45 + 134 * q^47 + 614 * q^49 + 489 * q^51 + 561 * q^53 + 240 * q^55 + 600 * q^57 + 951 * q^59 + 860 * q^61 + 315 * q^63 + 450 * q^65 + 447 * q^67 + 483 * q^69 + 735 * q^71 + 1460 * q^73 + 525 * q^75 + 496 * q^77 + 18 * q^79 + 567 * q^81 + 261 * q^83 + 815 * q^85 + 243 * q^87 + 2024 * q^89 + 692 * q^91 + 375 * q^93 + 1000 * q^95 + 668 * q^97 + 432 * 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
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	0	0
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	−20.9626	−1.13188	−0.565938	−	0.824448i	$-0.691486\pi$
$7$	−20.9626	−1.13188	−0.565938	+	0.824448i	$0.691486\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−5.12906	−0.140588	−0.0702941	−	0.997526i	$-0.522394\pi$
$11$	−5.12906	−0.140588	−0.0702941	+	0.997526i	$0.522394\pi$
$12$	0	0
$13$	22.8827	0.488193	0.244096	−	0.969751i	$-0.421509\pi$
$13$	22.8827	0.488193	0.244096	+	0.969751i	$0.421509\pi$
$14$	0	0
$15$	15.0000	0.258199
$16$	0	0
$17$	138.573	1.97700	0.988500	−	0.151221i	$-0.0483206\pi$
$17$	138.573	1.97700	0.988500	+	0.151221i	$0.0483206\pi$
$18$	0	0
$19$	−93.0918	−1.12404	−0.562019	−	0.827124i	$-0.689975\pi$
$19$	−93.0918	−1.12404	−0.562019	+	0.827124i	$0.689975\pi$
$20$	0	0
$21$	−62.8879	−0.653489
$22$	0	0
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	−17.0282	−0.109037	−0.0545183	−	0.998513i	$-0.517362\pi$
$29$	−17.0282	−0.109037	−0.0545183	+	0.998513i	$0.517362\pi$
$30$	0	0
$31$	−2.74296	−0.0158920	−0.00794598	−	0.999968i	$-0.502529\pi$
$31$	−2.74296	−0.0158920	−0.00794598	+	0.999968i	$0.502529\pi$
$32$	0	0
$33$	−15.3872	−0.0811686
$34$	0	0
$35$	−104.813	−0.506190
$36$	0	0
$37$	−41.4336	−0.184099	−0.0920493	−	0.995754i	$-0.529342\pi$
$37$	−41.4336	−0.184099	−0.0920493	+	0.995754i	$0.529342\pi$
$38$	0	0
$39$	68.6480	0.281858
$40$	0	0
$41$	455.865	1.73644	0.868222	−	0.496176i	$-0.165263\pi$
$41$	455.865	1.73644	0.868222	+	0.496176i	$0.165263\pi$
$42$	0	0
$43$	−450.537	−1.59782	−0.798910	−	0.601451i	$-0.794590\pi$
$43$	−450.537	−1.59782	−0.798910	+	0.601451i	$0.794590\pi$
$44$	0	0
$45$	45.0000	0.149071
$46$	0	0
$47$	−388.805	−1.20666	−0.603330	−	0.797492i	$-0.706160\pi$
$47$	−388.805	−1.20666	−0.603330	+	0.797492i	$0.706160\pi$
$48$	0	0
$49$	96.4321	0.281143
$50$	0	0
$51$	415.720	1.14142
$52$	0	0
$53$	182.360	0.472623	0.236312	−	0.971677i	$-0.424061\pi$
$53$	182.360	0.472623	0.236312	+	0.971677i	$0.424061\pi$
$54$	0	0
$55$	−25.6453	−0.0628730
$56$	0	0
$57$	−279.275	−0.648964
$58$	0	0
$59$	634.144	1.39930	0.699648	−	0.714488i	$-0.253340\pi$
$59$	634.144	1.39930	0.699648	+	0.714488i	$0.253340\pi$
$60$	0	0
$61$	386.858	0.812002	0.406001	−	0.913873i	$-0.366923\pi$
$61$	386.858	0.812002	0.406001	+	0.913873i	$0.366923\pi$
$62$	0	0
$63$	−188.664	−0.377292
$64$	0	0
$65$	114.413	0.218326
$66$	0	0
$67$	93.1921	0.169929	0.0849644	−	0.996384i	$-0.472922\pi$
$67$	93.1921	0.169929	0.0849644	+	0.996384i	$0.472922\pi$
$68$	0	0
$69$	69.0000	0.120386
$70$	0	0
$71$	−169.380	−0.283123	−0.141561	−	0.989929i	$-0.545212\pi$
$71$	−169.380	−0.283123	−0.141561	+	0.989929i	$0.545212\pi$
$72$	0	0
$73$	439.911	0.705311	0.352655	−	0.935753i	$-0.385279\pi$
$73$	439.911	0.705311	0.352655	+	0.935753i	$0.385279\pi$
$74$	0	0
$75$	75.0000	0.115470
$76$	0	0
$77$	107.519	0.159128
$78$	0	0
$79$	402.575	0.573332	0.286666	−	0.958031i	$-0.407453\pi$
$79$	402.575	0.573332	0.286666	+	0.958031i	$0.407453\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	878.499	1.16178	0.580890	−	0.813982i	$-0.302704\pi$
$83$	878.499	1.16178	0.580890	+	0.813982i	$0.302704\pi$
$84$	0	0
$85$	692.867	0.884141
$86$	0	0
$87$	−51.0846	−0.0629523
$88$	0	0
$89$	1643.87	1.95786	0.978930	−	0.204194i	$-0.0654573\pi$
$89$	1643.87	1.95786	0.978930	+	0.204194i	$0.0654573\pi$
$90$	0	0
$91$	−479.681	−0.552574
$92$	0	0
$93$	−8.22889	−0.00917523
$94$	0	0
$95$	−465.459	−0.502685
$96$	0	0
$97$	106.397	0.111371	0.0556854	−	0.998448i	$-0.482266\pi$
$97$	106.397	0.111371	0.0556854	+	0.998448i	$0.482266\pi$
$98$	0	0
$99$	−46.1616	−0.0468627
$100$	0	0
$101$	760.278	0.749014	0.374507	−	0.927224i	$-0.377812\pi$
$101$	760.278	0.749014	0.374507	+	0.927224i	$0.377812\pi$
$102$	0	0
$103$	−421.147	−0.402882	−0.201441	−	0.979501i	$-0.564562\pi$
$103$	−421.147	−0.402882	−0.201441	+	0.979501i	$0.564562\pi$
$104$	0	0
$105$	−314.440	−0.292249
$106$	0	0
$107$	1980.69	1.78954	0.894769	−	0.446528i	$-0.147340\pi$
$107$	1980.69	1.78954	0.894769	+	0.446528i	$0.147340\pi$
$108$	0	0
$109$	2171.79	1.90844	0.954221	−	0.299103i	$-0.0966873\pi$
$109$	2171.79	1.90844	0.954221	+	0.299103i	$0.0966873\pi$
$110$	0	0
$111$	−124.301	−0.106289
$112$	0	0
$113$	−1443.64	−1.20182	−0.600912	−	0.799315i	$-0.705196\pi$
$113$	−1443.64	−1.20182	−0.600912	+	0.799315i	$0.705196\pi$
$114$	0	0
$115$	115.000	0.0932505
$116$	0	0
$117$	205.944	0.162731
$118$	0	0
$119$	−2904.86	−2.23772
$120$	0	0
$121$	−1304.69	−0.980235
$122$	0	0
$123$	1367.60	1.00254
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	1442.29	1.00774	0.503868	−	0.863780i	$-0.331910\pi$
$127$	1442.29	1.00774	0.503868	+	0.863780i	$0.331910\pi$
$128$	0	0
$129$	−1351.61	−0.922502
$130$	0	0
$131$	−1515.49	−1.01075	−0.505376	−	0.862899i	$-0.668646\pi$
$131$	−1515.49	−1.01075	−0.505376	+	0.862899i	$0.668646\pi$
$132$	0	0
$133$	1951.45	1.27227
$134$	0	0
$135$	135.000	0.0860663
$136$	0	0
$137$	1821.08	1.13566	0.567828	−	0.823147i	$-0.307784\pi$
$137$	1821.08	1.13566	0.567828	+	0.823147i	$0.307784\pi$
$138$	0	0
$139$	−1246.10	−0.760380	−0.380190	−	0.924908i	$-0.624141\pi$
$139$	−1246.10	−0.760380	−0.380190	+	0.924908i	$0.624141\pi$
$140$	0	0
$141$	−1166.41	−0.696665
$142$	0	0
$143$	−117.367	−0.0686341
$144$	0	0
$145$	−85.1410	−0.0487626
$146$	0	0
$147$	289.296	0.162318
$148$	0	0
$149$	1599.55	0.879466	0.439733	−	0.898129i	$-0.355073\pi$
$149$	1599.55	0.879466	0.439733	+	0.898129i	$0.355073\pi$
$150$	0	0
$151$	−1667.28	−0.898554	−0.449277	−	0.893392i	$-0.648318\pi$
$151$	−1667.28	−0.898554	−0.449277	+	0.893392i	$0.648318\pi$
$152$	0	0
$153$	1247.16	0.659000
$154$	0	0
$155$	−13.7148	−0.00710710
$156$	0	0
$157$	−379.604	−0.192966	−0.0964831	−	0.995335i	$-0.530759\pi$
$157$	−379.604	−0.192966	−0.0964831	+	0.995335i	$0.530759\pi$
$158$	0	0
$159$	547.079	0.272869
$160$	0	0
$161$	−482.141	−0.236012
$162$	0	0
$163$	2560.26	1.23028	0.615138	−	0.788420i	$-0.289100\pi$
$163$	2560.26	1.23028	0.615138	+	0.788420i	$0.289100\pi$
$164$	0	0
$165$	−76.9359	−0.0362997
$166$	0	0
$167$	1230.17	0.570018	0.285009	−	0.958525i	$-0.408003\pi$
$167$	1230.17	0.570018	0.285009	+	0.958525i	$0.408003\pi$
$168$	0	0
$169$	−1673.38	−0.761668
$170$	0	0
$171$	−837.826	−0.374679
$172$	0	0
$173$	−3941.33	−1.73210	−0.866051	−	0.499956i	$-0.833349\pi$
$173$	−3941.33	−1.73210	−0.866051	+	0.499956i	$0.833349\pi$
$174$	0	0
$175$	−524.066	−0.226375
$176$	0	0
$177$	1902.43	0.807884
$178$	0	0
$179$	4508.62	1.88263	0.941313	−	0.337536i	$-0.109593\pi$
$179$	4508.62	1.88263	0.941313	+	0.337536i	$0.109593\pi$
$180$	0	0
$181$	−4360.55	−1.79070	−0.895352	−	0.445359i	$-0.853076\pi$
$181$	−4360.55	−1.79070	−0.895352	+	0.445359i	$0.853076\pi$
$182$	0	0
$183$	1160.57	0.468810
$184$	0	0
$185$	−207.168	−0.0823314
$186$	0	0
$187$	−710.752	−0.277943
$188$	0	0
$189$	−565.991	−0.217830
$190$	0	0
$191$	2007.10	0.760358	0.380179	−	0.924913i	$-0.375862\pi$
$191$	2007.10	0.760358	0.380179	+	0.924913i	$0.375862\pi$
$192$	0	0
$193$	−436.842	−0.162925	−0.0814627	−	0.996676i	$-0.525959\pi$
$193$	−436.842	−0.162925	−0.0814627	+	0.996676i	$0.525959\pi$
$194$	0	0
$195$	343.240	0.126051
$196$	0	0
$197$	4053.86	1.46612	0.733059	−	0.680165i	$-0.238092\pi$
$197$	4053.86	1.46612	0.733059	+	0.680165i	$0.238092\pi$
$198$	0	0
$199$	4868.15	1.73414	0.867070	−	0.498186i	$-0.166000\pi$
$199$	4868.15	1.73414	0.867070	+	0.498186i	$0.166000\pi$
$200$	0	0
$201$	279.576	0.0981084
$202$	0	0
$203$	356.956	0.123416
$204$	0	0
$205$	2279.33	0.776561
$206$	0	0
$207$	207.000	0.0695048
$208$	0	0
$209$	477.474	0.158027
$210$	0	0
$211$	702.776	0.229294	0.114647	−	0.993406i	$-0.463426\pi$
$211$	702.776	0.229294	0.114647	+	0.993406i	$0.463426\pi$
$212$	0	0
$213$	−508.141	−0.163461
$214$	0	0
$215$	−2252.69	−0.714567
$216$	0	0
$217$	57.4998	0.0179877
$218$	0	0
$219$	1319.73	0.407211
$220$	0	0
$221$	3170.93	0.965157
$222$	0	0
$223$	1737.77	0.521838	0.260919	−	0.965361i	$-0.415975\pi$
$223$	1737.77	0.521838	0.260919	+	0.965361i	$0.415975\pi$
$224$	0	0
$225$	225.000	0.0666667
$226$	0	0
$227$	−5086.91	−1.48736	−0.743678	−	0.668538i	$-0.766921\pi$
$227$	−5086.91	−1.48736	−0.743678	+	0.668538i	$0.766921\pi$
$228$	0	0
$229$	3454.02	0.996715	0.498358	−	0.866972i	$-0.333937\pi$
$229$	3454.02	0.996715	0.498358	+	0.866972i	$0.333937\pi$
$230$	0	0
$231$	322.556	0.0918728
$232$	0	0
$233$	−1847.38	−0.519426	−0.259713	−	0.965686i	$-0.583628\pi$
$233$	−1847.38	−0.519426	−0.259713	+	0.965686i	$0.583628\pi$
$234$	0	0
$235$	−1944.02	−0.539634
$236$	0	0
$237$	1207.73	0.331014
$238$	0	0
$239$	−7069.60	−1.91336	−0.956682	−	0.291133i	$-0.905968\pi$
$239$	−7069.60	−1.91336	−0.956682	+	0.291133i	$0.905968\pi$
$240$	0	0
$241$	−3249.49	−0.868540	−0.434270	−	0.900783i	$-0.642994\pi$
$241$	−3249.49	−0.868540	−0.434270	+	0.900783i	$0.642994\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	482.160	0.125731
$246$	0	0
$247$	−2130.19	−0.548747
$248$	0	0
$249$	2635.50	0.670754
$250$	0	0
$251$	4541.32	1.14201	0.571007	−	0.820945i	$-0.306553\pi$
$251$	4541.32	1.14201	0.571007	+	0.820945i	$0.306553\pi$
$252$	0	0
$253$	−117.968	−0.0293147
$254$	0	0
$255$	2078.60	0.510459
$256$	0	0
$257$	7331.37	1.77945	0.889725	−	0.456497i	$-0.150896\pi$
$257$	7331.37	1.77945	0.889725	+	0.456497i	$0.150896\pi$
$258$	0	0
$259$	868.558	0.208377
$260$	0	0
$261$	−153.254	−0.0363455
$262$	0	0
$263$	708.198	0.166043	0.0830216	−	0.996548i	$-0.473543\pi$
$263$	708.198	0.166043	0.0830216	+	0.996548i	$0.473543\pi$
$264$	0	0
$265$	911.798	0.211364
$266$	0	0
$267$	4931.60	1.13037
$268$	0	0
$269$	−7330.37	−1.66149	−0.830744	−	0.556654i	$-0.812085\pi$
$269$	−7330.37	−1.66149	−0.830744	+	0.556654i	$0.812085\pi$
$270$	0	0
$271$	−5258.09	−1.17862	−0.589310	−	0.807907i	$-0.700600\pi$
$271$	−5258.09	−1.17862	−0.589310	+	0.807907i	$0.700600\pi$
$272$	0	0
$273$	−1439.04	−0.319029
$274$	0	0
$275$	−128.227	−0.0281176
$276$	0	0
$277$	3496.71	0.758474	0.379237	−	0.925300i	$-0.376187\pi$
$277$	3496.71	0.758474	0.379237	+	0.925300i	$0.376187\pi$
$278$	0	0
$279$	−24.6867	−0.00529732
$280$	0	0
$281$	−5472.10	−1.16170	−0.580851	−	0.814010i	$-0.697280\pi$
$281$	−5472.10	−1.16170	−0.580851	+	0.814010i	$0.697280\pi$
$282$	0	0
$283$	2652.78	0.557213	0.278606	−	0.960405i	$-0.410128\pi$
$283$	2652.78	0.557213	0.278606	+	0.960405i	$0.410128\pi$
$284$	0	0
$285$	−1396.38	−0.290225
$286$	0	0
$287$	−9556.14	−1.96544
$288$	0	0
$289$	14289.6	2.90853
$290$	0	0
$291$	319.191	0.0643000
$292$	0	0
$293$	−2718.13	−0.541962	−0.270981	−	0.962585i	$-0.587348\pi$
$293$	−2718.13	−0.541962	−0.270981	+	0.962585i	$0.587348\pi$
$294$	0	0
$295$	3170.72	0.625784
$296$	0	0
$297$	−138.485	−0.0270562
$298$	0	0
$299$	526.301	0.101795
$300$	0	0
$301$	9444.44	1.80853
$302$	0	0
$303$	2280.83	0.432444
$304$	0	0
$305$	1934.29	0.363138
$306$	0	0
$307$	7492.52	1.39290	0.696450	−	0.717605i	$-0.254762\pi$
$307$	7492.52	1.39290	0.696450	+	0.717605i	$0.254762\pi$
$308$	0	0
$309$	−1263.44	−0.232604
$310$	0	0
$311$	7748.11	1.41272	0.706359	−	0.707854i	$-0.250337\pi$
$311$	7748.11	1.41272	0.706359	+	0.707854i	$0.250337\pi$
$312$	0	0
$313$	3808.11	0.687691	0.343845	−	0.939026i	$-0.388270\pi$
$313$	3808.11	0.687691	0.343845	+	0.939026i	$0.388270\pi$
$314$	0	0
$315$	−943.319	−0.168730
$316$	0	0
$317$	−711.554	−0.126072	−0.0630360	−	0.998011i	$-0.520078\pi$
$317$	−711.554	−0.126072	−0.0630360	+	0.998011i	$0.520078\pi$
$318$	0	0
$319$	87.3387	0.0153292
$320$	0	0
$321$	5942.07	1.03319
$322$	0	0
$323$	−12900.1	−2.22222
$324$	0	0
$325$	572.066	0.0976386
$326$	0	0
$327$	6515.38	1.10184
$328$	0	0
$329$	8150.37	1.36579
$330$	0	0
$331$	10258.2	1.70345	0.851725	−	0.523990i	$-0.175557\pi$
$331$	10258.2	1.70345	0.851725	+	0.523990i	$0.175557\pi$
$332$	0	0
$333$	−372.903	−0.0613662
$334$	0	0
$335$	465.960	0.0759944
$336$	0	0
$337$	−666.730	−0.107772	−0.0538859	−	0.998547i	$-0.517161\pi$
$337$	−666.730	−0.107772	−0.0538859	+	0.998547i	$0.517161\pi$
$338$	0	0
$339$	−4330.92	−0.693874
$340$	0	0
$341$	14.0688	0.00223422
$342$	0	0
$343$	5168.71	0.813657
$344$	0	0
$345$	345.000	0.0538382
$346$	0	0
$347$	−1380.75	−0.213610	−0.106805	−	0.994280i	$-0.534062\pi$
$347$	−1380.75	−0.213610	−0.106805	+	0.994280i	$0.534062\pi$
$348$	0	0
$349$	−5658.14	−0.867831	−0.433916	−	0.900954i	$-0.642868\pi$
$349$	−5658.14	−0.867831	−0.433916	+	0.900954i	$0.642868\pi$
$350$	0	0
$351$	617.832	0.0939527
$352$	0	0
$353$	−9985.62	−1.50561	−0.752806	−	0.658242i	$-0.771300\pi$
$353$	−9985.62	−1.50561	−0.752806	+	0.658242i	$0.771300\pi$
$354$	0	0
$355$	−846.901	−0.126616
$356$	0	0
$357$	−8714.59	−1.29195
$358$	0	0
$359$	−6668.11	−0.980305	−0.490153	−	0.871637i	$-0.663059\pi$
$359$	−6668.11	−0.980305	−0.490153	+	0.871637i	$0.663059\pi$
$360$	0	0
$361$	1807.09	0.263462
$362$	0	0
$363$	−3914.08	−0.565939
$364$	0	0
$365$	2199.56	0.315425
$366$	0	0
$367$	8233.69	1.17110	0.585552	−	0.810635i	$-0.300878\pi$
$367$	8233.69	1.17110	0.585552	+	0.810635i	$0.300878\pi$
$368$	0	0
$369$	4102.79	0.578815
$370$	0	0
$371$	−3822.74	−0.534951
$372$	0	0
$373$	−5040.63	−0.699715	−0.349857	−	0.936803i	$-0.613770\pi$
$373$	−5040.63	−0.699715	−0.349857	+	0.936803i	$0.613770\pi$
$374$	0	0
$375$	375.000	0.0516398
$376$	0	0
$377$	−389.651	−0.0532308
$378$	0	0
$379$	−7339.69	−0.994761	−0.497380	−	0.867533i	$-0.665705\pi$
$379$	−7339.69	−0.994761	−0.497380	+	0.867533i	$0.665705\pi$
$380$	0	0
$381$	4326.87	0.581817
$382$	0	0
$383$	11327.7	1.51128	0.755638	−	0.654990i	$-0.227327\pi$
$383$	11327.7	1.51128	0.755638	+	0.654990i	$0.227327\pi$
$384$	0	0
$385$	537.593	0.0711644
$386$	0	0
$387$	−4054.83	−0.532607
$388$	0	0
$389$	−6887.46	−0.897707	−0.448854	−	0.893605i	$-0.648168\pi$
$389$	−6887.46	−0.897707	−0.448854	+	0.893605i	$0.648168\pi$
$390$	0	0
$391$	3187.19	0.412233
$392$	0	0
$393$	−4546.46	−0.583558
$394$	0	0
$395$	2012.88	0.256402
$396$	0	0
$397$	−5068.33	−0.640736	−0.320368	−	0.947293i	$-0.603807\pi$
$397$	−5068.33	−0.640736	−0.320368	+	0.947293i	$0.603807\pi$
$398$	0	0
$399$	5854.35	0.734547
$400$	0	0
$401$	−2862.76	−0.356507	−0.178253	−	0.983985i	$-0.557045\pi$
$401$	−2862.76	−0.356507	−0.178253	+	0.983985i	$0.557045\pi$
$402$	0	0
$403$	−62.7663	−0.00775834
$404$	0	0
$405$	405.000	0.0496904
$406$	0	0
$407$	212.516	0.0258821
$408$	0	0
$409$	−14934.6	−1.80554	−0.902771	−	0.430121i	$-0.858471\pi$
$409$	−14934.6	−1.80554	−0.902771	+	0.430121i	$0.858471\pi$
$410$	0	0
$411$	5463.23	0.655672
$412$	0	0
$413$	−13293.3	−1.58383
$414$	0	0
$415$	4392.50	0.519564
$416$	0	0
$417$	−3738.30	−0.439005
$418$	0	0
$419$	−4057.49	−0.473082	−0.236541	−	0.971622i	$-0.576014\pi$
$419$	−4057.49	−0.473082	−0.236541	+	0.971622i	$0.576014\pi$
$420$	0	0
$421$	−1955.28	−0.226352	−0.113176	−	0.993575i	$-0.536102\pi$
$421$	−1955.28	−0.226352	−0.113176	+	0.993575i	$0.536102\pi$
$422$	0	0
$423$	−3499.24	−0.402220
$424$	0	0
$425$	3464.34	0.395400
$426$	0	0
$427$	−8109.57	−0.919085
$428$	0	0
$429$	−352.100	−0.0396259
$430$	0	0
$431$	−197.473	−0.0220695	−0.0110347	−	0.999939i	$-0.503513\pi$
$431$	−197.473	−0.0220695	−0.0110347	+	0.999939i	$0.503513\pi$
$432$	0	0
$433$	7918.67	0.878861	0.439431	−	0.898276i	$-0.355180\pi$
$433$	7918.67	0.878861	0.439431	+	0.898276i	$0.355180\pi$
$434$	0	0
$435$	−255.423	−0.0281531
$436$	0	0
$437$	−2141.11	−0.234378
$438$	0	0
$439$	−11479.2	−1.24800	−0.624000	−	0.781424i	$-0.714494\pi$
$439$	−11479.2	−1.24800	−0.624000	+	0.781424i	$0.714494\pi$
$440$	0	0
$441$	867.889	0.0937144
$442$	0	0
$443$	3737.80	0.400877	0.200438	−	0.979706i	$-0.435763\pi$
$443$	3737.80	0.400877	0.200438	+	0.979706i	$0.435763\pi$
$444$	0	0
$445$	8219.34	0.875582
$446$	0	0
$447$	4798.66	0.507760
$448$	0	0
$449$	12439.4	1.30747	0.653734	−	0.756724i	$-0.273202\pi$
$449$	12439.4	1.30747	0.653734	+	0.756724i	$0.273202\pi$
$450$	0	0
$451$	−2338.16	−0.244124
$452$	0	0
$453$	−5001.85	−0.518780
$454$	0	0
$455$	−2398.40	−0.247118
$456$	0	0
$457$	−2115.05	−0.216494	−0.108247	−	0.994124i	$-0.534524\pi$
$457$	−2115.05	−0.216494	−0.108247	+	0.994124i	$0.534524\pi$
$458$	0	0
$459$	3741.48	0.380474
$460$	0	0
$461$	−11441.7	−1.15595	−0.577976	−	0.816053i	$-0.696157\pi$
$461$	−11441.7	−1.15595	−0.577976	+	0.816053i	$0.696157\pi$
$462$	0	0
$463$	−15460.1	−1.55182	−0.775911	−	0.630843i	$-0.782709\pi$
$463$	−15460.1	−1.55182	−0.775911	+	0.630843i	$0.782709\pi$
$464$	0	0
$465$	−41.1445	−0.00410329
$466$	0	0
$467$	−2853.95	−0.282794	−0.141397	−	0.989953i	$-0.545159\pi$
$467$	−2853.95	−0.282794	−0.141397	+	0.989953i	$0.545159\pi$
$468$	0	0
$469$	−1953.55	−0.192338
$470$	0	0
$471$	−1138.81	−0.111409
$472$	0	0
$473$	2310.83	0.224635
$474$	0	0
$475$	−2327.30	−0.224808
$476$	0	0
$477$	1641.24	0.157541
$478$	0	0
$479$	3780.30	0.360598	0.180299	−	0.983612i	$-0.442293\pi$
$479$	3780.30	0.360598	0.180299	+	0.983612i	$0.442293\pi$
$480$	0	0
$481$	−948.111	−0.0898756
$482$	0	0
$483$	−1446.42	−0.136262
$484$	0	0
$485$	531.985	0.0498066
$486$	0	0
$487$	4434.51	0.412622	0.206311	−	0.978487i	$-0.433854\pi$
$487$	4434.51	0.412622	0.206311	+	0.978487i	$0.433854\pi$
$488$	0	0
$489$	7680.77	0.710300
$490$	0	0
$491$	−15833.0	−1.45526	−0.727631	−	0.685969i	$-0.759379\pi$
$491$	−15833.0	−1.45526	−0.727631	+	0.685969i	$0.759379\pi$
$492$	0	0
$493$	−2359.66	−0.215565
$494$	0	0
$495$	−230.808	−0.0209577
$496$	0	0
$497$	3550.66	0.320460
$498$	0	0
$499$	17997.7	1.61460	0.807301	−	0.590140i	$-0.200927\pi$
$499$	17997.7	1.61460	0.807301	+	0.590140i	$0.200927\pi$
$500$	0	0
$501$	3690.50	0.329100
$502$	0	0
$503$	−7897.54	−0.700068	−0.350034	−	0.936737i	$-0.613830\pi$
$503$	−7897.54	−0.700068	−0.350034	+	0.936737i	$0.613830\pi$
$504$	0	0
$505$	3801.39	0.334969
$506$	0	0
$507$	−5020.15	−0.439749
$508$	0	0
$509$	−6231.11	−0.542611	−0.271305	−	0.962493i	$-0.587455\pi$
$509$	−6231.11	−0.542611	−0.271305	+	0.962493i	$0.587455\pi$
$510$	0	0
$511$	−9221.70	−0.798324
$512$	0	0
$513$	−2513.48	−0.216321
$514$	0	0
$515$	−2105.73	−0.180174
$516$	0	0
$517$	1994.20	0.169642
$518$	0	0
$519$	−11824.0	−1.00003
$520$	0	0
$521$	6599.95	0.554988	0.277494	−	0.960727i	$-0.410496\pi$
$521$	6599.95	0.554988	0.277494	+	0.960727i	$0.410496\pi$
$522$	0	0
$523$	12515.5	1.04640	0.523198	−	0.852211i	$-0.324739\pi$
$523$	12515.5	1.04640	0.523198	+	0.852211i	$0.324739\pi$
$524$	0	0
$525$	−1572.20	−0.130698
$526$	0	0
$527$	−380.102	−0.0314184
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	5707.29	0.466432
$532$	0	0
$533$	10431.4	0.847719
$534$	0	0
$535$	9903.46	0.800306
$536$	0	0
$537$	13525.9	1.08693
$538$	0	0
$539$	−494.606	−0.0395254
$540$	0	0
$541$	5415.46	0.430368	0.215184	−	0.976574i	$-0.430965\pi$
$541$	5415.46	0.430368	0.215184	+	0.976574i	$0.430965\pi$
$542$	0	0
$543$	−13081.7	−1.03386
$544$	0	0
$545$	10859.0	0.853481
$546$	0	0
$547$	17262.5	1.34935	0.674673	−	0.738117i	$-0.264285\pi$
$547$	17262.5	1.34935	0.674673	+	0.738117i	$0.264285\pi$
$548$	0	0
$549$	3481.72	0.270667
$550$	0	0
$551$	1585.19	0.122561
$552$	0	0
$553$	−8439.04	−0.648941
$554$	0	0
$555$	−621.505	−0.0475340
$556$	0	0
$557$	12469.0	0.948523	0.474262	−	0.880384i	$-0.342715\pi$
$557$	12469.0	0.948523	0.474262	+	0.880384i	$0.342715\pi$
$558$	0	0
$559$	−10309.5	−0.780044
$560$	0	0
$561$	−2132.26	−0.160470
$562$	0	0
$563$	4143.28	0.310157	0.155079	−	0.987902i	$-0.450437\pi$
$563$	4143.28	0.310157	0.155079	+	0.987902i	$0.450437\pi$
$564$	0	0
$565$	−7218.20	−0.537472
$566$	0	0
$567$	−1697.97	−0.125764
$568$	0	0
$569$	−6197.28	−0.456597	−0.228298	−	0.973591i	$-0.573316\pi$
$569$	−6197.28	−0.456597	−0.228298	+	0.973591i	$0.573316\pi$
$570$	0	0
$571$	−22077.5	−1.61806	−0.809030	−	0.587767i	$-0.800007\pi$
$571$	−22077.5	−1.61806	−0.809030	+	0.587767i	$0.800007\pi$
$572$	0	0
$573$	6021.29	0.438993
$574$	0	0
$575$	575.000	0.0417029
$576$	0	0
$577$	−17062.3	−1.23105	−0.615524	−	0.788118i	$-0.711056\pi$
$577$	−17062.3	−1.23105	−0.615524	+	0.788118i	$0.711056\pi$
$578$	0	0
$579$	−1310.53	−0.0940650
$580$	0	0
$581$	−18415.7	−1.31499
$582$	0	0
$583$	−935.334	−0.0664452
$584$	0	0
$585$	1029.72	0.0727755
$586$	0	0
$587$	−14215.2	−0.999528	−0.499764	−	0.866161i	$-0.666580\pi$
$587$	−14215.2	−0.999528	−0.499764	+	0.866161i	$0.666580\pi$
$588$	0	0
$589$	255.348	0.0178632
$590$	0	0
$591$	12161.6	0.846464
$592$	0	0
$593$	9911.12	0.686342	0.343171	−	0.939273i	$-0.388499\pi$
$593$	9911.12	0.686342	0.343171	+	0.939273i	$0.388499\pi$
$594$	0	0
$595$	−14524.3	−1.00074
$596$	0	0
$597$	14604.4	1.00121
$598$	0	0
$599$	−23142.3	−1.57858	−0.789288	−	0.614023i	$-0.789550\pi$
$599$	−23142.3	−1.57858	−0.789288	+	0.614023i	$0.789550\pi$
$600$	0	0
$601$	−23697.3	−1.60837	−0.804187	−	0.594376i	$-0.797399\pi$
$601$	−23697.3	−1.60837	−0.804187	+	0.594376i	$0.797399\pi$
$602$	0	0
$603$	838.729	0.0566429
$604$	0	0
$605$	−6523.46	−0.438374
$606$	0	0
$607$	15615.2	1.04415	0.522075	−	0.852899i	$-0.325158\pi$
$607$	15615.2	1.04415	0.522075	+	0.852899i	$0.325158\pi$
$608$	0	0
$609$	1070.87	0.0712542
$610$	0	0
$611$	−8896.88	−0.589082
$612$	0	0
$613$	3386.12	0.223106	0.111553	−	0.993758i	$-0.464418\pi$
$613$	3386.12	0.223106	0.111553	+	0.993758i	$0.464418\pi$
$614$	0	0
$615$	6837.98	0.448348
$616$	0	0
$617$	−8900.58	−0.580752	−0.290376	−	0.956913i	$-0.593780\pi$
$617$	−8900.58	−0.580752	−0.290376	+	0.956913i	$0.593780\pi$
$618$	0	0
$619$	−26429.1	−1.71611	−0.858056	−	0.513556i	$-0.828328\pi$
$619$	−26429.1	−1.71611	−0.858056	+	0.513556i	$0.828328\pi$
$620$	0	0
$621$	621.000	0.0401286
$622$	0	0
$623$	−34459.8	−2.21606
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	1432.42	0.0912367
$628$	0	0
$629$	−5741.60	−0.363963
$630$	0	0
$631$	21360.9	1.34765	0.673824	−	0.738892i	$-0.264651\pi$
$631$	21360.9	1.34765	0.673824	+	0.738892i	$0.264651\pi$
$632$	0	0
$633$	2108.33	0.132383
$634$	0	0
$635$	7211.45	0.450674
$636$	0	0
$637$	2206.62	0.137252
$638$	0	0
$639$	−1524.42	−0.0943743
$640$	0	0
$641$	−2206.40	−0.135956	−0.0679779	−	0.997687i	$-0.521655\pi$
$641$	−2206.40	−0.135956	−0.0679779	+	0.997687i	$0.521655\pi$
$642$	0	0
$643$	−8815.30	−0.540656	−0.270328	−	0.962768i	$-0.587132\pi$
$643$	−8815.30	−0.540656	−0.270328	+	0.962768i	$0.587132\pi$
$644$	0	0
$645$	−6758.06	−0.412555
$646$	0	0
$647$	−4966.00	−0.301752	−0.150876	−	0.988553i	$-0.548209\pi$
$647$	−4966.00	−0.301752	−0.150876	+	0.988553i	$0.548209\pi$
$648$	0	0
$649$	−3252.56	−0.196725
$650$	0	0
$651$	172.499	0.0103852
$652$	0	0
$653$	−16226.0	−0.972393	−0.486196	−	0.873850i	$-0.661616\pi$
$653$	−16226.0	−0.972393	−0.486196	+	0.873850i	$0.661616\pi$
$654$	0	0
$655$	−7577.43	−0.452022
$656$	0	0
$657$	3959.20	0.235104
$658$	0	0
$659$	−3565.37	−0.210755	−0.105377	−	0.994432i	$-0.533605\pi$
$659$	−3565.37	−0.210755	−0.105377	+	0.994432i	$0.533605\pi$
$660$	0	0
$661$	−2365.00	−0.139165	−0.0695824	−	0.997576i	$-0.522167\pi$
$661$	−2365.00	−0.139165	−0.0695824	+	0.997576i	$0.522167\pi$
$662$	0	0
$663$	9512.78	0.557234
$664$	0	0
$665$	9757.25	0.568977
$666$	0	0
$667$	−391.649	−0.0227357
$668$	0	0
$669$	5213.31	0.301283
$670$	0	0
$671$	−1984.22	−0.114158
$672$	0	0
$673$	29545.9	1.69229	0.846144	−	0.532955i	$-0.178918\pi$
$673$	29545.9	1.69229	0.846144	+	0.532955i	$0.178918\pi$
$674$	0	0
$675$	675.000	0.0384900
$676$	0	0
$677$	11784.9	0.669026	0.334513	−	0.942391i	$-0.391428\pi$
$677$	11784.9	0.669026	0.334513	+	0.942391i	$0.391428\pi$
$678$	0	0
$679$	−2230.36	−0.126058
$680$	0	0
$681$	−15260.7	−0.858726
$682$	0	0
$683$	−21235.0	−1.18966	−0.594828	−	0.803853i	$-0.702780\pi$
$683$	−21235.0	−1.18966	−0.594828	+	0.803853i	$0.702780\pi$
$684$	0	0
$685$	9105.38	0.507881
$686$	0	0
$687$	10362.0	0.575454
$688$	0	0
$689$	4172.87	0.230731
$690$	0	0
$691$	−17804.4	−0.980189	−0.490094	−	0.871669i	$-0.663038\pi$
$691$	−17804.4	−0.980189	−0.490094	+	0.871669i	$0.663038\pi$
$692$	0	0
$693$	967.668	0.0530428
$694$	0	0
$695$	−6230.50	−0.340052
$696$	0	0
$697$	63170.8	3.43295
$698$	0	0
$699$	−5542.15	−0.299890
$700$	0	0
$701$	18910.8	1.01890	0.509451	−	0.860499i	$-0.329848\pi$
$701$	18910.8	1.01890	0.509451	+	0.860499i	$0.329848\pi$
$702$	0	0
$703$	3857.13	0.206934
$704$	0	0
$705$	−5832.07	−0.311558
$706$	0	0
$707$	−15937.4	−0.847791
$708$	0	0
$709$	−5057.86	−0.267915	−0.133958	−	0.990987i	$-0.542769\pi$
$709$	−5057.86	−0.267915	−0.133958	+	0.990987i	$0.542769\pi$
$710$	0	0
$711$	3623.18	0.191111
$712$	0	0
$713$	−63.0882	−0.00331370
$714$	0	0
$715$	−586.833	−0.0306941
$716$	0	0
$717$	−21208.8	−1.10468
$718$	0	0
$719$	−32224.2	−1.67143	−0.835716	−	0.549162i	$-0.814947\pi$
$719$	−32224.2	−1.67143	−0.835716	+	0.549162i	$0.814947\pi$
$720$	0	0
$721$	8828.35	0.456012
$722$	0	0
$723$	−9748.48	−0.501452
$724$	0	0
$725$	−425.705	−0.0218073
$726$	0	0
$727$	−30261.9	−1.54381	−0.771907	−	0.635736i	$-0.780697\pi$
$727$	−30261.9	−1.54381	−0.771907	+	0.635736i	$0.780697\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−62432.5	−3.15889
$732$	0	0
$733$	25038.1	1.26167	0.630835	−	0.775917i	$-0.282712\pi$
$733$	25038.1	1.26167	0.630835	+	0.775917i	$0.282712\pi$
$734$	0	0
$735$	1446.48	0.0725908
$736$	0	0
$737$	−477.988	−0.0238900
$738$	0	0
$739$	34334.1	1.70906	0.854532	−	0.519398i	$-0.173844\pi$
$739$	34334.1	1.70906	0.854532	+	0.519398i	$0.173844\pi$
$740$	0	0
$741$	−6390.56	−0.316819
$742$	0	0
$743$	−7691.56	−0.379779	−0.189890	−	0.981805i	$-0.560813\pi$
$743$	−7691.56	−0.379779	−0.189890	+	0.981805i	$0.560813\pi$
$744$	0	0
$745$	7997.76	0.393309
$746$	0	0
$747$	7906.49	0.387260
$748$	0	0
$749$	−41520.5	−2.02554
$750$	0	0
$751$	−21851.4	−1.06174	−0.530871	−	0.847453i	$-0.678135\pi$
$751$	−21851.4	−1.06174	−0.530871	+	0.847453i	$0.678135\pi$
$752$	0	0
$753$	13624.0	0.659342
$754$	0	0
$755$	−8336.42	−0.401846
$756$	0	0
$757$	−18806.2	−0.902937	−0.451468	−	0.892287i	$-0.649100\pi$
$757$	−18806.2	−0.902937	−0.451468	+	0.892287i	$0.649100\pi$
$758$	0	0
$759$	−353.905	−0.0169248
$760$	0	0
$761$	−4431.61	−0.211098	−0.105549	−	0.994414i	$-0.533660\pi$
$761$	−4431.61	−0.211098	−0.105549	+	0.994414i	$0.533660\pi$
$762$	0	0
$763$	−45526.5	−2.16012
$764$	0	0
$765$	6235.81	0.294714
$766$	0	0
$767$	14510.9	0.683126
$768$	0	0
$769$	12962.4	0.607850	0.303925	−	0.952696i	$-0.401703\pi$
$769$	12962.4	0.607850	0.303925	+	0.952696i	$0.401703\pi$
$770$	0	0
$771$	21994.1	1.02737
$772$	0	0
$773$	−23003.6	−1.07035	−0.535177	−	0.844740i	$-0.679755\pi$
$773$	−23003.6	−1.07035	−0.535177	+	0.844740i	$0.679755\pi$
$774$	0	0
$775$	−68.5741	−0.00317839
$776$	0	0
$777$	2605.67	0.120306
$778$	0	0
$779$	−42437.3	−1.95183
$780$	0	0
$781$	868.761	0.0398038
$782$	0	0
$783$	−459.762	−0.0209841
$784$	0	0
$785$	−1898.02	−0.0862971
$786$	0	0
$787$	−55.2903	−0.00250430	−0.00125215	−	0.999999i	$-0.500399\pi$
$787$	−55.2903	−0.00250430	−0.00125215	+	0.999999i	$0.500399\pi$
$788$	0	0
$789$	2124.60	0.0958651
$790$	0	0
$791$	30262.5	1.36032
$792$	0	0
$793$	8852.34	0.396413
$794$	0	0
$795$	2735.40	0.122031
$796$	0	0
$797$	−11413.7	−0.507269	−0.253635	−	0.967300i	$-0.581626\pi$
$797$	−11413.7	−0.507269	−0.253635	+	0.967300i	$0.581626\pi$
$798$	0	0
$799$	−53878.0	−2.38556
$800$	0	0
$801$	14794.8	0.652620
$802$	0	0
$803$	−2256.33	−0.0991584
$804$	0	0
$805$	−2410.70	−0.105548
$806$	0	0
$807$	−21991.1	−0.959261
$808$	0	0
$809$	209.406	0.00910053	0.00455026	−	0.999990i	$-0.498552\pi$
$809$	209.406	0.00910053	0.00455026	+	0.999990i	$0.498552\pi$
$810$	0	0
$811$	3756.55	0.162651	0.0813257	−	0.996688i	$-0.474085\pi$
$811$	3756.55	0.162651	0.0813257	+	0.996688i	$0.474085\pi$
$812$	0	0
$813$	−15774.3	−0.680477
$814$	0	0
$815$	12801.3	0.550196
$816$	0	0
$817$	41941.3	1.79601
$818$	0	0
$819$	−4317.13	−0.184191
$820$	0	0
$821$	6975.87	0.296540	0.148270	−	0.988947i	$-0.452630\pi$
$821$	6975.87	0.296540	0.148270	+	0.988947i	$0.452630\pi$
$822$	0	0
$823$	−23927.6	−1.01344	−0.506722	−	0.862109i	$-0.669143\pi$
$823$	−23927.6	−1.01344	−0.506722	+	0.862109i	$0.669143\pi$
$824$	0	0
$825$	−384.680	−0.0162337
$826$	0	0
$827$	−1248.67	−0.0525034	−0.0262517	−	0.999655i	$-0.508357\pi$
$827$	−1248.67	−0.0525034	−0.0262517	+	0.999655i	$0.508357\pi$
$828$	0	0
$829$	15002.4	0.628535	0.314267	−	0.949335i	$-0.398241\pi$
$829$	15002.4	0.628535	0.314267	+	0.949335i	$0.398241\pi$
$830$	0	0
$831$	10490.1	0.437905
$832$	0	0
$833$	13362.9	0.555820
$834$	0	0
$835$	6150.83	0.254920
$836$	0	0
$837$	−74.0600	−0.00305841
$838$	0	0
$839$	21653.8	0.891026	0.445513	−	0.895275i	$-0.353021\pi$
$839$	21653.8	0.891026	0.445513	+	0.895275i	$0.353021\pi$
$840$	0	0
$841$	−24099.0	−0.988111
$842$	0	0
$843$	−16416.3	−0.670709
$844$	0	0
$845$	−8366.92	−0.340628
$846$	0	0
$847$	27349.8	1.10950
$848$	0	0
$849$	7958.33	0.321707
$850$	0	0
$851$	−952.974	−0.0383872
$852$	0	0
$853$	−45309.9	−1.81873	−0.909367	−	0.415994i	$-0.863434\pi$
$853$	−45309.9	−1.81873	−0.909367	+	0.415994i	$0.863434\pi$
$854$	0	0
$855$	−4189.13	−0.167562
$856$	0	0
$857$	−10043.7	−0.400335	−0.200167	−	0.979762i	$-0.564149\pi$
$857$	−10043.7	−0.400335	−0.200167	+	0.979762i	$0.564149\pi$
$858$	0	0
$859$	−625.900	−0.0248608	−0.0124304	−	0.999923i	$-0.503957\pi$
$859$	−625.900	−0.0248608	−0.0124304	+	0.999923i	$0.503957\pi$
$860$	0	0
$861$	−28668.4	−1.13475
$862$	0	0
$863$	4993.33	0.196958	0.0984790	−	0.995139i	$-0.468602\pi$
$863$	4993.33	0.196958	0.0984790	+	0.995139i	$0.468602\pi$
$864$	0	0
$865$	−19706.6	−0.774619
$866$	0	0
$867$	42868.8	1.67924
$868$	0	0
$869$	−2064.83	−0.0806038
$870$	0	0
$871$	2132.48	0.0829580
$872$	0	0
$873$	957.572	0.0371236
$874$	0	0
$875$	−2620.33	−0.101238
$876$	0	0
$877$	9867.69	0.379941	0.189971	−	0.981790i	$-0.439161\pi$
$877$	9867.69	0.379941	0.189971	+	0.981790i	$0.439161\pi$
$878$	0	0
$879$	−8154.39	−0.312902
$880$	0	0
$881$	13326.5	0.509628	0.254814	−	0.966990i	$-0.417986\pi$
$881$	13326.5	0.509628	0.254814	+	0.966990i	$0.417986\pi$
$882$	0	0
$883$	−11867.9	−0.452307	−0.226154	−	0.974092i	$-0.572615\pi$
$883$	−11867.9	−0.452307	−0.226154	+	0.974092i	$0.572615\pi$
$884$	0	0
$885$	9512.15	0.361297
$886$	0	0
$887$	−33478.7	−1.26731	−0.633656	−	0.773615i	$-0.718446\pi$
$887$	−33478.7	−1.26731	−0.633656	+	0.773615i	$0.718446\pi$
$888$	0	0
$889$	−30234.2	−1.14063
$890$	0	0
$891$	−415.454	−0.0156209
$892$	0	0
$893$	36194.5	1.35633
$894$	0	0
$895$	22543.1	0.841936
$896$	0	0
$897$	1578.90	0.0587715
$898$	0	0
$899$	46.7078	0.00173280
$900$	0	0
$901$	25270.2	0.934376
$902$	0	0
$903$	28333.3	1.04416
$904$	0	0
$905$	−21802.8	−0.800827
$906$	0	0
$907$	16980.7	0.621647	0.310824	−	0.950468i	$-0.399395\pi$
$907$	16980.7	0.621647	0.310824	+	0.950468i	$0.399395\pi$
$908$	0	0
$909$	6842.50	0.249671
$910$	0	0
$911$	29949.8	1.08922	0.544611	−	0.838689i	$-0.316677\pi$
$911$	29949.8	1.08922	0.544611	+	0.838689i	$0.316677\pi$
$912$	0	0
$913$	−4505.88	−0.163333
$914$	0	0
$915$	5802.87	0.209658
$916$	0	0
$917$	31768.6	1.14405
$918$	0	0
$919$	−17840.8	−0.640386	−0.320193	−	0.947352i	$-0.603748\pi$
$919$	−17840.8	−0.640386	−0.320193	+	0.947352i	$0.603748\pi$
$920$	0	0
$921$	22477.5	0.804192
$922$	0	0
$923$	−3875.87	−0.138219
$924$	0	0
$925$	−1035.84	−0.0368197
$926$	0	0
$927$	−3790.32	−0.134294
$928$	0	0
$929$	−4855.49	−0.171478	−0.0857392	−	0.996318i	$-0.527325\pi$
$929$	−4855.49	−0.171478	−0.0857392	+	0.996318i	$0.527325\pi$
$930$	0	0
$931$	−8977.04	−0.316016
$932$	0	0
$933$	23244.3	0.815633
$934$	0	0
$935$	−3553.76	−0.124300
$936$	0	0
$937$	45218.6	1.57655	0.788275	−	0.615323i	$-0.210975\pi$
$937$	45218.6	1.57655	0.788275	+	0.615323i	$0.210975\pi$
$938$	0	0
$939$	11424.3	0.397038
$940$	0	0
$941$	−5355.04	−0.185515	−0.0927574	−	0.995689i	$-0.529568\pi$
$941$	−5355.04	−0.185515	−0.0927574	+	0.995689i	$0.529568\pi$
$942$	0	0
$943$	10484.9	0.362074
$944$	0	0
$945$	−2829.96	−0.0974164
$946$	0	0
$947$	53576.1	1.83842	0.919212	−	0.393762i	$-0.128827\pi$
$947$	53576.1	1.83842	0.919212	+	0.393762i	$0.128827\pi$
$948$	0	0
$949$	10066.3	0.344328
$950$	0	0
$951$	−2134.66	−0.0727877
$952$	0	0
$953$	31346.1	1.06548	0.532738	−	0.846280i	$-0.321163\pi$
$953$	31346.1	1.06548	0.532738	+	0.846280i	$0.321163\pi$
$954$	0	0
$955$	10035.5	0.340042
$956$	0	0
$957$	262.016	0.00885035
$958$	0	0
$959$	−38174.5	−1.28542
$960$	0	0
$961$	−29783.5	−0.999747
$962$	0	0
$963$	17826.2	0.596513
$964$	0	0
$965$	−2184.21	−0.0728625
$966$	0	0
$967$	−28993.7	−0.964191	−0.482096	−	0.876119i	$-0.660124\pi$
$967$	−28993.7	−0.964191	−0.482096	+	0.876119i	$0.660124\pi$
$968$	0	0
$969$	−38700.2	−1.28300
$970$	0	0
$971$	12687.0	0.419305	0.209653	−	0.977776i	$-0.432767\pi$
$971$	12687.0	0.419305	0.209653	+	0.977776i	$0.432767\pi$
$972$	0	0
$973$	26121.5	0.860655
$974$	0	0
$975$	1716.20	0.0563716
$976$	0	0
$977$	50214.0	1.64431	0.822154	−	0.569265i	$-0.192772\pi$
$977$	50214.0	1.64431	0.822154	+	0.569265i	$0.192772\pi$
$978$	0	0
$979$	−8431.50	−0.275252
$980$	0	0
$981$	19546.1	0.636147
$982$	0	0
$983$	19596.0	0.635825	0.317913	−	0.948120i	$-0.397018\pi$
$983$	19596.0	0.635825	0.317913	+	0.948120i	$0.397018\pi$
$984$	0	0
$985$	20269.3	0.655668
$986$	0	0
$987$	24451.1	0.788538
$988$	0	0
$989$	−10362.4	−0.333168
$990$	0	0
$991$	20774.7	0.665923	0.332962	−	0.942940i	$-0.391952\pi$
$991$	20774.7	0.665923	0.332962	+	0.942940i	$0.391952\pi$
$992$	0	0
$993$	30774.6	0.983487
$994$	0	0
$995$	24340.7	0.775531
$996$	0	0
$997$	53443.5	1.69767	0.848833	−	0.528661i	$-0.177306\pi$
$997$	53443.5	1.69767	0.848833	+	0.528661i	$0.177306\pi$
$998$	0	0
$999$	−1118.71	−0.0354298

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1380.4.a.j.1.2	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1380.4.a.j.1.2	✓	7	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:	\( N \)	\(=\)	\( 1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1380.a (trivial)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} +5.00000 q^{5} -20.9626 q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q+3.00000 q^{3} +5.00000 q^{5} -20.9626 q^{7} +9.00000 q^{9} -5.12906 q^{11} +22.8827 q^{13} +15.0000 q^{15} +138.573 q^{17} -93.0918 q^{19} -62.8879 q^{21} +23.0000 q^{23} +25.0000 q^{25} +27.0000 q^{27} -17.0282 q^{29} -2.74296 q^{31} -15.3872 q^{33} -104.813 q^{35} -41.4336 q^{37} +68.6480 q^{39} +455.865 q^{41} -450.537 q^{43} +45.0000 q^{45} -388.805 q^{47} +96.4321 q^{49} +415.720 q^{51} +182.360 q^{53} -25.6453 q^{55} -279.275 q^{57} +634.144 q^{59} +386.858 q^{61} -188.664 q^{63} +114.413 q^{65} +93.1921 q^{67} +69.0000 q^{69} -169.380 q^{71} +439.911 q^{73} +75.0000 q^{75} +107.519 q^{77} +402.575 q^{79} +81.0000 q^{81} +878.499 q^{83} +692.867 q^{85} -51.0846 q^{87} +1643.87 q^{89} -479.681 q^{91} -8.22889 q^{93} -465.459 q^{95} +106.397 q^{97} -46.1616 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 21 q^{3} + 35 q^{5} + 35 q^{7} + 63 q^{9}+O(q^{10}) \) 7 * q + 21 * q^3 + 35 * q^5 + 35 * q^7 + 63 * q^9	\( 7 q + 21 q^{3} + 35 q^{5} + 35 q^{7} + 63 q^{9} + 48 q^{11} + 90 q^{13} + 105 q^{15} + 163 q^{17} + 200 q^{19} + 105 q^{21} + 161 q^{23} + 175 q^{25} + 189 q^{27} + 81 q^{29} + 125 q^{31} + 144 q^{33} + 175 q^{35} + 5 q^{37} + 270 q^{39} + 369 q^{41} + 462 q^{43} + 315 q^{45} + 134 q^{47} + 614 q^{49} + 489 q^{51} + 561 q^{53} + 240 q^{55} + 600 q^{57} + 951 q^{59} + 860 q^{61} + 315 q^{63} + 450 q^{65} + 447 q^{67} + 483 q^{69} + 735 q^{71} + 1460 q^{73} + 525 q^{75} + 496 q^{77} + 18 q^{79} + 567 q^{81} + 261 q^{83} + 815 q^{85} + 243 q^{87} + 2024 q^{89} + 692 q^{91} + 375 q^{93} + 1000 q^{95} + 668 q^{97} + 432 q^{99}+O(q^{100}) \) 7 * q + 21 * q^3 + 35 * q^5 + 35 * q^7 + 63 * q^9 + 48 * q^11 + 90 * q^13 + 105 * q^15 + 163 * q^17 + 200 * q^19 + 105 * q^21 + 161 * q^23 + 175 * q^25 + 189 * q^27 + 81 * q^29 + 125 * q^31 + 144 * q^33 + 175 * q^35 + 5 * q^37 + 270 * q^39 + 369 * q^41 + 462 * q^43 + 315 * q^45 + 134 * q^47 + 614 * q^49 + 489 * q^51 + 561 * q^53 + 240 * q^55 + 600 * q^57 + 951 * q^59 + 860 * q^61 + 315 * q^63 + 450 * q^65 + 447 * q^67 + 483 * q^69 + 735 * q^71 + 1460 * q^73 + 525 * q^75 + 496 * q^77 + 18 * q^79 + 567 * q^81 + 261 * q^83 + 815 * q^85 + 243 * q^87 + 2024 * q^89 + 692 * q^91 + 375 * q^93 + 1000 * q^95 + 668 * q^97 + 432 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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