LMFDB - Embedded newform 1170.4.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1170,4,Mod(1,1170)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1170, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1170.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1170.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:	$69.0322347067$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 390)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1170.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-2.00000 q^{2} +4.00000 q^{4} +5.00000 q^{5} -25.0000 q^{7} -8.00000 q^{8} +O(q^{10})$

\(q-2.00000 q^{2} +4.00000 q^{4} +5.00000 q^{5} -25.0000 q^{7} -8.00000 q^{8} -10.0000 q^{10} +21.0000 q^{11} +13.0000 q^{13} +50.0000 q^{14} +16.0000 q^{16} -123.000 q^{17} +146.000 q^{19} +20.0000 q^{20} -42.0000 q^{22} -99.0000 q^{23} +25.0000 q^{25} -26.0000 q^{26} -100.000 q^{28} +246.000 q^{29} +182.000 q^{31} -32.0000 q^{32} +246.000 q^{34} -125.000 q^{35} -295.000 q^{37} -292.000 q^{38} -40.0000 q^{40} -9.00000 q^{41} +452.000 q^{43} +84.0000 q^{44} +198.000 q^{46} -390.000 q^{47} +282.000 q^{49} -50.0000 q^{50} +52.0000 q^{52} -315.000 q^{53} +105.000 q^{55} +200.000 q^{56} -492.000 q^{58} +24.0000 q^{59} -727.000 q^{61} -364.000 q^{62} +64.0000 q^{64} +65.0000 q^{65} +596.000 q^{67} -492.000 q^{68} +250.000 q^{70} -771.000 q^{71} +326.000 q^{73} +590.000 q^{74} +584.000 q^{76} -525.000 q^{77} -889.000 q^{79} +80.0000 q^{80} +18.0000 q^{82} +96.0000 q^{83} -615.000 q^{85} -904.000 q^{86} -168.000 q^{88} -795.000 q^{89} -325.000 q^{91} -396.000 q^{92} +780.000 q^{94} +730.000 q^{95} +983.000 q^{97} -564.000 q^{98} +O(q^{100})\)

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 $

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	4.00000	0.500000
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	−25.0000	−1.34987	−0.674937	−	0.737876i	$-0.735829\pi$
$7$	−25.0000	−1.34987	−0.674937	+	0.737876i	$0.735829\pi$
$8$	−8.00000	−0.353553
$9$	0	0
$10$	−10.0000	−0.316228
$11$	21.0000	0.575613	0.287806	−	0.957689i	$-0.407074\pi$
$11$	21.0000	0.575613	0.287806	+	0.957689i	$0.407074\pi$
$12$	0	0
$13$	13.0000	0.277350
$13$	13.0000	0.277350
$14$	50.0000	0.954504
$15$	0	0
$16$	16.0000	0.250000
$17$	−123.000	−1.75482	−0.877408	−	0.479744i	$-0.840729\pi$
$17$	−123.000	−1.75482	−0.877408	+	0.479744i	$0.840729\pi$
$18$	0	0
$19$	146.000	1.76288	0.881439	−	0.472297i	$-0.156575\pi$
$19$	146.000	1.76288	0.881439	+	0.472297i	$0.156575\pi$
$20$	20.0000	0.223607
$21$	0	0
$22$	−42.0000	−0.407020
$23$	−99.0000	−0.897519	−0.448759	−	0.893653i	$-0.648134\pi$
$23$	−99.0000	−0.897519	−0.448759	+	0.893653i	$0.648134\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	−26.0000	−0.196116
$27$	0	0
$28$	−100.000	−0.674937
$29$	246.000	1.57521	0.787604	−	0.616181i	$-0.211321\pi$
$29$	246.000	1.57521	0.787604	+	0.616181i	$0.211321\pi$
$30$	0	0
$31$	182.000	1.05446	0.527228	−	0.849724i	$-0.323231\pi$
$31$	182.000	1.05446	0.527228	+	0.849724i	$0.323231\pi$
$32$	−32.0000	−0.176777
$33$	0	0
$34$	246.000	1.24084
$35$	−125.000	−0.603682
$36$	0	0
$37$	−295.000	−1.31075	−0.655374	−	0.755304i	$-0.727489\pi$
$37$	−295.000	−1.31075	−0.655374	+	0.755304i	$0.727489\pi$
$38$	−292.000	−1.24654
$39$	0	0
$40$	−40.0000	−0.158114
$41$	−9.00000	−0.0342820	−0.0171410	−	0.999853i	$-0.505456\pi$
$41$	−9.00000	−0.0342820	−0.0171410	+	0.999853i	$0.505456\pi$
$42$	0	0
$43$	452.000	1.60301	0.801504	−	0.597989i	$-0.204033\pi$
$43$	452.000	1.60301	0.801504	+	0.597989i	$0.204033\pi$
$44$	84.0000	0.287806
$45$	0	0
$46$	198.000	0.634641
$47$	−390.000	−1.21037	−0.605185	−	0.796085i	$-0.706901\pi$
$47$	−390.000	−1.21037	−0.605185	+	0.796085i	$0.706901\pi$
$48$	0	0
$49$	282.000	0.822157
$50$	−50.0000	−0.141421
$51$	0	0
$52$	52.0000	0.138675
$53$	−315.000	−0.816388	−0.408194	−	0.912895i	$-0.633841\pi$
$53$	−315.000	−0.816388	−0.408194	+	0.912895i	$0.633841\pi$
$54$	0	0
$55$	105.000	0.257422
$56$	200.000	0.477252
$57$	0	0
$58$	−492.000	−1.11384
$59$	24.0000	0.0529582	0.0264791	−	0.999649i	$-0.491570\pi$
$59$	24.0000	0.0529582	0.0264791	+	0.999649i	$0.491570\pi$
$60$	0	0
$61$	−727.000	−1.52595	−0.762974	−	0.646429i	$-0.776262\pi$
$61$	−727.000	−1.52595	−0.762974	+	0.646429i	$0.776262\pi$
$62$	−364.000	−0.745614
$63$	0	0
$64$	64.0000	0.125000
$65$	65.0000	0.124035
$66$	0	0
$67$	596.000	1.08676	0.543381	−	0.839487i	$-0.317144\pi$
$67$	596.000	1.08676	0.543381	+	0.839487i	$0.317144\pi$
$68$	−492.000	−0.877408
$69$	0	0
$70$	250.000	0.426867
$71$	−771.000	−1.28874	−0.644372	−	0.764712i	$-0.722881\pi$
$71$	−771.000	−1.28874	−0.644372	+	0.764712i	$0.722881\pi$
$72$	0	0
$73$	326.000	0.522677	0.261338	−	0.965247i	$-0.415836\pi$
$73$	326.000	0.522677	0.261338	+	0.965247i	$0.415836\pi$
$74$	590.000	0.926839
$75$	0	0
$76$	584.000	0.881439
$77$	−525.000	−0.777004
$78$	0	0
$79$	−889.000	−1.26608	−0.633040	−	0.774119i	$-0.718193\pi$
$79$	−889.000	−1.26608	−0.633040	+	0.774119i	$0.718193\pi$
$80$	80.0000	0.111803
$81$	0	0
$82$	18.0000	0.0242411
$83$	96.0000	0.126956	0.0634781	−	0.997983i	$-0.479781\pi$
$83$	96.0000	0.126956	0.0634781	+	0.997983i	$0.479781\pi$
$84$	0	0
$85$	−615.000	−0.784778
$86$	−904.000	−1.13350
$87$	0	0
$88$	−168.000	−0.203510
$89$	−795.000	−0.946852	−0.473426	−	0.880834i	$-0.656983\pi$
$89$	−795.000	−0.946852	−0.473426	+	0.880834i	$0.656983\pi$
$90$	0	0
$91$	−325.000	−0.374387
$92$	−396.000	−0.448759
$93$	0	0
$94$	780.000	0.855860
$95$	730.000	0.788383
$96$	0	0
$97$	983.000	1.02895	0.514477	−	0.857504i	$-0.327986\pi$
$97$	983.000	1.02895	0.514477	+	0.857504i	$0.327986\pi$
$98$	−564.000	−0.581353
$99$	0	0
$100$	100.000	0.100000
$101$	732.000	0.721156	0.360578	−	0.932729i	$-0.382580\pi$
$101$	732.000	0.721156	0.360578	+	0.932729i	$0.382580\pi$
$102$	0	0
$103$	−1600.00	−1.53061	−0.765304	−	0.643669i	$-0.777412\pi$
$103$	−1600.00	−1.53061	−0.765304	+	0.643669i	$0.777412\pi$
$104$	−104.000	−0.0980581
$105$	0	0
$106$	630.000	0.577274
$107$	−1011.00	−0.913430	−0.456715	−	0.889613i	$-0.650974\pi$
$107$	−1011.00	−0.913430	−0.456715	+	0.889613i	$0.650974\pi$
$108$	0	0
$109$	1100.00	0.966614	0.483307	−	0.875451i	$-0.339436\pi$
$109$	1100.00	0.966614	0.483307	+	0.875451i	$0.339436\pi$
$110$	−210.000	−0.182025
$111$	0	0
$112$	−400.000	−0.337468
$113$	1770.00	1.47352	0.736759	−	0.676155i	$-0.236355\pi$
$113$	1770.00	1.47352	0.736759	+	0.676155i	$0.236355\pi$
$114$	0	0
$115$	−495.000	−0.401383
$116$	984.000	0.787604
$117$	0	0
$118$	−48.0000	−0.0374471
$119$	3075.00	2.36878
$120$	0	0
$121$	−890.000	−0.668670
$122$	1454.00	1.07901
$123$	0	0
$124$	728.000	0.527228
$125$	125.000	0.0894427
$126$	0	0
$127$	2666.00	1.86275	0.931375	−	0.364061i	$-0.118610\pi$
$127$	2666.00	1.86275	0.931375	+	0.364061i	$0.118610\pi$
$128$	−128.000	−0.0883883
$129$	0	0
$130$	−130.000	−0.0877058
$131$	−582.000	−0.388165	−0.194082	−	0.980985i	$-0.562173\pi$
$131$	−582.000	−0.388165	−0.194082	+	0.980985i	$0.562173\pi$
$132$	0	0
$133$	−3650.00	−2.37966
$134$	−1192.00	−0.768456
$135$	0	0
$136$	984.000	0.620421
$137$	−270.000	−0.168377	−0.0841885	−	0.996450i	$-0.526830\pi$
$137$	−270.000	−0.168377	−0.0841885	+	0.996450i	$0.526830\pi$
$138$	0	0
$139$	−7.00000	−0.00427146	−0.00213573	−	0.999998i	$-0.500680\pi$
$139$	−7.00000	−0.00427146	−0.00213573	+	0.999998i	$0.500680\pi$
$140$	−500.000	−0.301841
$141$	0	0
$142$	1542.00	0.911280
$143$	273.000	0.159646
$144$	0	0
$145$	1230.00	0.704455
$146$	−652.000	−0.369588
$147$	0	0
$148$	−1180.00	−0.655374
$149$	−3099.00	−1.70389	−0.851946	−	0.523629i	$-0.824578\pi$
$149$	−3099.00	−1.70389	−0.851946	+	0.523629i	$0.824578\pi$
$150$	0	0
$151$	−2860.00	−1.54135	−0.770674	−	0.637230i	$-0.780080\pi$
$151$	−2860.00	−1.54135	−0.770674	+	0.637230i	$0.780080\pi$
$152$	−1168.00	−0.623272
$153$	0	0
$154$	1050.00	0.549425
$155$	910.000	0.471567
$156$	0	0
$157$	−2734.00	−1.38979	−0.694895	−	0.719111i	$-0.744549\pi$
$157$	−2734.00	−1.38979	−0.694895	+	0.719111i	$0.744549\pi$
$158$	1778.00	0.895254
$159$	0	0
$160$	−160.000	−0.0790569
$161$	2475.00	1.21154
$162$	0	0
$163$	−853.000	−0.409890	−0.204945	−	0.978773i	$-0.565702\pi$
$163$	−853.000	−0.409890	−0.204945	+	0.978773i	$0.565702\pi$
$164$	−36.0000	−0.0171410
$165$	0	0
$166$	−192.000	−0.0897716
$167$	−2556.00	−1.18437	−0.592183	−	0.805803i	$-0.701734\pi$
$167$	−2556.00	−1.18437	−0.592183	+	0.805803i	$0.701734\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	1230.00	0.554922
$171$	0	0
$172$	1808.00	0.801504
$173$	1134.00	0.498361	0.249180	−	0.968457i	$-0.419839\pi$
$173$	1134.00	0.498361	0.249180	+	0.968457i	$0.419839\pi$
$174$	0	0
$175$	−625.000	−0.269975
$176$	336.000	0.143903
$177$	0	0
$178$	1590.00	0.669525
$179$	−498.000	−0.207946	−0.103973	−	0.994580i	$-0.533155\pi$
$179$	−498.000	−0.207946	−0.103973	+	0.994580i	$0.533155\pi$
$180$	0	0
$181$	1901.00	0.780664	0.390332	−	0.920674i	$-0.372360\pi$
$181$	1901.00	0.780664	0.390332	+	0.920674i	$0.372360\pi$
$182$	650.000	0.264732
$183$	0	0
$184$	792.000	0.317321
$185$	−1475.00	−0.586185
$186$	0	0
$187$	−2583.00	−1.01009
$188$	−1560.00	−0.605185
$189$	0	0
$190$	−1460.00	−0.557471
$191$	−2340.00	−0.886474	−0.443237	−	0.896405i	$-0.646170\pi$
$191$	−2340.00	−0.886474	−0.443237	+	0.896405i	$0.646170\pi$
$192$	0	0
$193$	1685.00	0.628440	0.314220	−	0.949350i	$-0.398257\pi$
$193$	1685.00	0.628440	0.314220	+	0.949350i	$0.398257\pi$
$194$	−1966.00	−0.727580
$195$	0	0
$196$	1128.00	0.411079
$197$	−1992.00	−0.720427	−0.360214	−	0.932870i	$-0.617296\pi$
$197$	−1992.00	−0.720427	−0.360214	+	0.932870i	$0.617296\pi$
$198$	0	0
$199$	−4732.00	−1.68564	−0.842821	−	0.538195i	$-0.819107\pi$
$199$	−4732.00	−1.68564	−0.842821	+	0.538195i	$0.819107\pi$
$200$	−200.000	−0.0707107
$201$	0	0
$202$	−1464.00	−0.509934
$203$	−6150.00	−2.12633
$204$	0	0
$205$	−45.0000	−0.0153314
$206$	3200.00	1.08230
$207$	0	0
$208$	208.000	0.0693375
$209$	3066.00	1.01474
$210$	0	0
$211$	−4012.00	−1.30899	−0.654496	−	0.756065i	$-0.727119\pi$
$211$	−4012.00	−1.30899	−0.654496	+	0.756065i	$0.727119\pi$
$212$	−1260.00	−0.408194
$213$	0	0
$214$	2022.00	0.645893
$215$	2260.00	0.716887
$216$	0	0
$217$	−4550.00	−1.42338
$218$	−2200.00	−0.683499
$219$	0	0
$220$	420.000	0.128711
$221$	−1599.00	−0.486699
$222$	0	0
$223$	−1096.00	−0.329119	−0.164560	−	0.986367i	$-0.552620\pi$
$223$	−1096.00	−0.329119	−0.164560	+	0.986367i	$0.552620\pi$
$224$	800.000	0.238626
$225$	0	0
$226$	−3540.00	−1.04193
$227$	−6138.00	−1.79468	−0.897342	−	0.441335i	$-0.854505\pi$
$227$	−6138.00	−1.79468	−0.897342	+	0.441335i	$0.854505\pi$
$228$	0	0
$229$	−1294.00	−0.373406	−0.186703	−	0.982416i	$-0.559780\pi$
$229$	−1294.00	−0.373406	−0.186703	+	0.982416i	$0.559780\pi$
$230$	990.000	0.283820
$231$	0	0
$232$	−1968.00	−0.556920
$233$	−873.000	−0.245460	−0.122730	−	0.992440i	$-0.539165\pi$
$233$	−873.000	−0.245460	−0.122730	+	0.992440i	$0.539165\pi$
$234$	0	0
$235$	−1950.00	−0.541294
$236$	96.0000	0.0264791
$237$	0	0
$238$	−6150.00	−1.67498
$239$	−5163.00	−1.39735	−0.698675	−	0.715439i	$-0.746227\pi$
$239$	−5163.00	−1.39735	−0.698675	+	0.715439i	$0.746227\pi$
$240$	0	0
$241$	−2482.00	−0.663401	−0.331701	−	0.943385i	$-0.607622\pi$
$241$	−2482.00	−0.663401	−0.331701	+	0.943385i	$0.607622\pi$
$242$	1780.00	0.472821
$243$	0	0
$244$	−2908.00	−0.762974
$245$	1410.00	0.367680
$246$	0	0
$247$	1898.00	0.488935
$248$	−1456.00	−0.372807
$249$	0	0
$250$	−250.000	−0.0632456
$251$	−5172.00	−1.30061	−0.650306	−	0.759672i	$-0.725359\pi$
$251$	−5172.00	−1.30061	−0.650306	+	0.759672i	$0.725359\pi$
$252$	0	0
$253$	−2079.00	−0.516623
$254$	−5332.00	−1.31716
$255$	0	0
$256$	256.000	0.0625000
$257$	−198.000	−0.0480580	−0.0240290	−	0.999711i	$-0.507649\pi$
$257$	−198.000	−0.0480580	−0.0240290	+	0.999711i	$0.507649\pi$
$258$	0	0
$259$	7375.00	1.76934
$260$	260.000	0.0620174
$261$	0	0
$262$	1164.00	0.274474
$263$	2568.00	0.602090	0.301045	−	0.953610i	$-0.402665\pi$
$263$	2568.00	0.602090	0.301045	+	0.953610i	$0.402665\pi$
$264$	0	0
$265$	−1575.00	−0.365100
$266$	7300.00	1.68268
$267$	0	0
$268$	2384.00	0.543381
$269$	6924.00	1.56938	0.784691	−	0.619887i	$-0.212822\pi$
$269$	6924.00	1.56938	0.784691	+	0.619887i	$0.212822\pi$
$270$	0	0
$271$	254.000	0.0569351	0.0284675	−	0.999595i	$-0.490937\pi$
$271$	254.000	0.0569351	0.0284675	+	0.999595i	$0.490937\pi$
$272$	−1968.00	−0.438704
$273$	0	0
$274$	540.000	0.119061
$275$	525.000	0.115123
$276$	0	0
$277$	6590.00	1.42944	0.714720	−	0.699411i	$-0.246554\pi$
$277$	6590.00	1.42944	0.714720	+	0.699411i	$0.246554\pi$
$278$	14.0000	0.00302037
$279$	0	0
$280$	1000.00	0.213434
$281$	−2370.00	−0.503140	−0.251570	−	0.967839i	$-0.580947\pi$
$281$	−2370.00	−0.503140	−0.251570	+	0.967839i	$0.580947\pi$
$282$	0	0
$283$	−6280.00	−1.31911	−0.659553	−	0.751658i	$-0.729255\pi$
$283$	−6280.00	−1.31911	−0.659553	+	0.751658i	$0.729255\pi$
$284$	−3084.00	−0.644372
$285$	0	0
$286$	−546.000	−0.112887
$287$	225.000	0.0462764
$288$	0	0
$289$	10216.0	2.07938
$290$	−2460.00	−0.498125
$291$	0	0
$292$	1304.00	0.261338
$293$	3588.00	0.715403	0.357702	−	0.933836i	$-0.383560\pi$
$293$	3588.00	0.715403	0.357702	+	0.933836i	$0.383560\pi$
$294$	0	0
$295$	120.000	0.0236836
$296$	2360.00	0.463420
$297$	0	0
$298$	6198.00	1.20483
$299$	−1287.00	−0.248927
$300$	0	0
$301$	−11300.0	−2.16386
$302$	5720.00	1.08990
$303$	0	0
$304$	2336.00	0.440720
$305$	−3635.00	−0.682425
$306$	0	0
$307$	−6469.00	−1.20262	−0.601312	−	0.799015i	$-0.705355\pi$
$307$	−6469.00	−1.20262	−0.601312	+	0.799015i	$0.705355\pi$
$308$	−2100.00	−0.388502
$309$	0	0
$310$	−1820.00	−0.333449
$311$	−1536.00	−0.280060	−0.140030	−	0.990147i	$-0.544720\pi$
$311$	−1536.00	−0.280060	−0.140030	+	0.990147i	$0.544720\pi$
$312$	0	0
$313$	1514.00	0.273407	0.136703	−	0.990612i	$-0.456349\pi$
$313$	1514.00	0.273407	0.136703	+	0.990612i	$0.456349\pi$
$314$	5468.00	0.982730
$315$	0	0
$316$	−3556.00	−0.633040
$317$	2724.00	0.482634	0.241317	−	0.970446i	$-0.422421\pi$
$317$	2724.00	0.482634	0.241317	+	0.970446i	$0.422421\pi$
$318$	0	0
$319$	5166.00	0.906710
$320$	320.000	0.0559017
$321$	0	0
$322$	−4950.00	−0.856685
$323$	−17958.0	−3.09353
$324$	0	0
$325$	325.000	0.0554700
$326$	1706.00	0.289836
$327$	0	0
$328$	72.0000	0.0121205
$329$	9750.00	1.63384
$330$	0	0
$331$	1100.00	0.182663	0.0913315	−	0.995821i	$-0.470888\pi$
$331$	1100.00	0.182663	0.0913315	+	0.995821i	$0.470888\pi$
$332$	384.000	0.0634781
$333$	0	0
$334$	5112.00	0.837474
$335$	2980.00	0.486014
$336$	0	0
$337$	−196.000	−0.0316819	−0.0158410	−	0.999875i	$-0.505043\pi$
$337$	−196.000	−0.0316819	−0.0158410	+	0.999875i	$0.505043\pi$
$338$	−338.000	−0.0543928
$339$	0	0
$340$	−2460.00	−0.392389
$341$	3822.00	0.606959
$342$	0	0
$343$	1525.00	0.240065
$344$	−3616.00	−0.566749
$345$	0	0
$346$	−2268.00	−0.352394
$347$	−11055.0	−1.71027	−0.855135	−	0.518406i	$-0.826526\pi$
$347$	−11055.0	−1.71027	−0.855135	+	0.518406i	$0.826526\pi$
$348$	0	0
$349$	−11176.0	−1.71415	−0.857074	−	0.515194i	$-0.827720\pi$
$349$	−11176.0	−1.71415	−0.857074	+	0.515194i	$0.827720\pi$
$350$	1250.00	0.190901
$351$	0	0
$352$	−672.000	−0.101755
$353$	−2130.00	−0.321157	−0.160579	−	0.987023i	$-0.551336\pi$
$353$	−2130.00	−0.321157	−0.160579	+	0.987023i	$0.551336\pi$
$354$	0	0
$355$	−3855.00	−0.576344
$356$	−3180.00	−0.473426
$357$	0	0
$358$	996.000	0.147040
$359$	−2304.00	−0.338720	−0.169360	−	0.985554i	$-0.554170\pi$
$359$	−2304.00	−0.338720	−0.169360	+	0.985554i	$0.554170\pi$
$360$	0	0
$361$	14457.0	2.10774
$362$	−3802.00	−0.552013
$363$	0	0
$364$	−1300.00	−0.187194
$365$	1630.00	0.233748
$366$	0	0
$367$	13412.0	1.90763	0.953816	−	0.300393i	$-0.0971177\pi$
$367$	13412.0	1.90763	0.953816	+	0.300393i	$0.0971177\pi$
$368$	−1584.00	−0.224380
$369$	0	0
$370$	2950.00	0.414495
$371$	7875.00	1.10202
$372$	0	0
$373$	9056.00	1.25711	0.628555	−	0.777765i	$-0.283647\pi$
$373$	9056.00	1.25711	0.628555	+	0.777765i	$0.283647\pi$
$374$	5166.00	0.714245
$375$	0	0
$376$	3120.00	0.427930
$377$	3198.00	0.436884
$378$	0	0
$379$	−4066.00	−0.551072	−0.275536	−	0.961291i	$-0.588855\pi$
$379$	−4066.00	−0.551072	−0.275536	+	0.961291i	$0.588855\pi$
$380$	2920.00	0.394192
$381$	0	0
$382$	4680.00	0.626831
$383$	−786.000	−0.104864	−0.0524318	−	0.998625i	$-0.516697\pi$
$383$	−786.000	−0.104864	−0.0524318	+	0.998625i	$0.516697\pi$
$384$	0	0
$385$	−2625.00	−0.347487
$386$	−3370.00	−0.444374
$387$	0	0
$388$	3932.00	0.514477
$389$	−8364.00	−1.09016	−0.545079	−	0.838385i	$-0.683500\pi$
$389$	−8364.00	−1.09016	−0.545079	+	0.838385i	$0.683500\pi$
$390$	0	0
$391$	12177.0	1.57498
$392$	−2256.00	−0.290677
$393$	0	0
$394$	3984.00	0.509419
$395$	−4445.00	−0.566208
$396$	0	0
$397$	7625.00	0.963949	0.481975	−	0.876185i	$-0.339920\pi$
$397$	7625.00	0.963949	0.481975	+	0.876185i	$0.339920\pi$
$398$	9464.00	1.19193
$399$	0	0
$400$	400.000	0.0500000
$401$	−954.000	−0.118804	−0.0594021	−	0.998234i	$-0.518919\pi$
$401$	−954.000	−0.118804	−0.0594021	+	0.998234i	$0.518919\pi$
$402$	0	0
$403$	2366.00	0.292454
$404$	2928.00	0.360578
$405$	0	0
$406$	12300.0	1.50354
$407$	−6195.00	−0.754483
$408$	0	0
$409$	182.000	0.0220032	0.0110016	−	0.999939i	$-0.496498\pi$
$409$	182.000	0.0220032	0.0110016	+	0.999939i	$0.496498\pi$
$410$	90.0000	0.0108409
$411$	0	0
$412$	−6400.00	−0.765304
$413$	−600.000	−0.0714869
$414$	0	0
$415$	480.000	0.0567766
$416$	−416.000	−0.0490290
$417$	0	0
$418$	−6132.00	−0.717526
$419$	4806.00	0.560354	0.280177	−	0.959948i	$-0.409607\pi$
$419$	4806.00	0.560354	0.280177	+	0.959948i	$0.409607\pi$
$420$	0	0
$421$	−10024.0	−1.16043	−0.580214	−	0.814464i	$-0.697031\pi$
$421$	−10024.0	−1.16043	−0.580214	+	0.814464i	$0.697031\pi$
$422$	8024.00	0.925598
$423$	0	0
$424$	2520.00	0.288637
$425$	−3075.00	−0.350963
$426$	0	0
$427$	18175.0	2.05984
$428$	−4044.00	−0.456715
$429$	0	0
$430$	−4520.00	−0.506916
$431$	11544.0	1.29015	0.645075	−	0.764119i	$-0.276826\pi$
$431$	11544.0	1.29015	0.645075	+	0.764119i	$0.276826\pi$
$432$	0	0
$433$	−268.000	−0.0297442	−0.0148721	−	0.999889i	$-0.504734\pi$
$433$	−268.000	−0.0297442	−0.0148721	+	0.999889i	$0.504734\pi$
$434$	9100.00	1.00648
$435$	0	0
$436$	4400.00	0.483307
$437$	−14454.0	−1.58222
$438$	0	0
$439$	−6235.00	−0.677859	−0.338930	−	0.940812i	$-0.610065\pi$
$439$	−6235.00	−0.677859	−0.338930	+	0.940812i	$0.610065\pi$
$440$	−840.000	−0.0910123
$441$	0	0
$442$	3198.00	0.344148
$443$	−9363.00	−1.00418	−0.502088	−	0.864817i	$-0.667434\pi$
$443$	−9363.00	−1.00418	−0.502088	+	0.864817i	$0.667434\pi$
$444$	0	0
$445$	−3975.00	−0.423445
$446$	2192.00	0.232722
$447$	0	0
$448$	−1600.00	−0.168734
$449$	7023.00	0.738165	0.369082	−	0.929397i	$-0.379672\pi$
$449$	7023.00	0.738165	0.369082	+	0.929397i	$0.379672\pi$
$450$	0	0
$451$	−189.000	−0.0197332
$452$	7080.00	0.736759
$453$	0	0
$454$	12276.0	1.26903
$455$	−1625.00	−0.167431
$456$	0	0
$457$	2603.00	0.266440	0.133220	−	0.991086i	$-0.457468\pi$
$457$	2603.00	0.266440	0.133220	+	0.991086i	$0.457468\pi$
$458$	2588.00	0.264038
$459$	0	0
$460$	−1980.00	−0.200691
$461$	−10377.0	−1.04838	−0.524192	−	0.851600i	$-0.675633\pi$
$461$	−10377.0	−1.04838	−0.524192	+	0.851600i	$0.675633\pi$
$462$	0	0
$463$	−11401.0	−1.14438	−0.572192	−	0.820120i	$-0.693907\pi$
$463$	−11401.0	−1.14438	−0.572192	+	0.820120i	$0.693907\pi$
$464$	3936.00	0.393802
$465$	0	0
$466$	1746.00	0.173566
$467$	15381.0	1.52409	0.762043	−	0.647527i	$-0.224197\pi$
$467$	15381.0	1.52409	0.762043	+	0.647527i	$0.224197\pi$
$468$	0	0
$469$	−14900.0	−1.46699
$470$	3900.00	0.382752
$471$	0	0
$472$	−192.000	−0.0187236
$473$	9492.00	0.922712
$474$	0	0
$475$	3650.00	0.352576
$476$	12300.0	1.18439
$477$	0	0
$478$	10326.0	0.988076
$479$	1767.00	0.168552	0.0842759	−	0.996442i	$-0.473142\pi$
$479$	1767.00	0.168552	0.0842759	+	0.996442i	$0.473142\pi$
$480$	0	0
$481$	−3835.00	−0.363536
$482$	4964.00	0.469095
$483$	0	0
$484$	−3560.00	−0.334335
$485$	4915.00	0.460162
$486$	0	0
$487$	9893.00	0.920523	0.460261	−	0.887783i	$-0.347756\pi$
$487$	9893.00	0.920523	0.460261	+	0.887783i	$0.347756\pi$
$488$	5816.00	0.539504
$489$	0	0
$490$	−2820.00	−0.259989
$491$	15336.0	1.40958	0.704790	−	0.709416i	$-0.251041\pi$
$491$	15336.0	1.40958	0.704790	+	0.709416i	$0.251041\pi$
$492$	0	0
$493$	−30258.0	−2.76420
$494$	−3796.00	−0.345729
$495$	0	0
$496$	2912.00	0.263614
$497$	19275.0	1.73964
$498$	0	0
$499$	−16450.0	−1.47576	−0.737879	−	0.674933i	$-0.764172\pi$
$499$	−16450.0	−1.47576	−0.737879	+	0.674933i	$0.764172\pi$
$500$	500.000	0.0447214
$501$	0	0
$502$	10344.0	0.919672
$503$	−3444.00	−0.305289	−0.152645	−	0.988281i	$-0.548779\pi$
$503$	−3444.00	−0.305289	−0.152645	+	0.988281i	$0.548779\pi$
$504$	0	0
$505$	3660.00	0.322511
$506$	4158.00	0.365308
$507$	0	0
$508$	10664.0	0.931375
$509$	−5535.00	−0.481993	−0.240997	−	0.970526i	$-0.577474\pi$
$509$	−5535.00	−0.481993	−0.240997	+	0.970526i	$0.577474\pi$
$510$	0	0
$511$	−8150.00	−0.705548
$512$	−512.000	−0.0441942
$513$	0	0
$514$	396.000	0.0339821
$515$	−8000.00	−0.684509
$516$	0	0
$517$	−8190.00	−0.696704
$518$	−14750.0	−1.25112
$519$	0	0
$520$	−520.000	−0.0438529
$521$	6786.00	0.570634	0.285317	−	0.958433i	$-0.407901\pi$
$521$	6786.00	0.570634	0.285317	+	0.958433i	$0.407901\pi$
$522$	0	0
$523$	1928.00	0.161196	0.0805980	−	0.996747i	$-0.474317\pi$
$523$	1928.00	0.161196	0.0805980	+	0.996747i	$0.474317\pi$
$524$	−2328.00	−0.194082
$525$	0	0
$526$	−5136.00	−0.425742
$527$	−22386.0	−1.85038
$528$	0	0
$529$	−2366.00	−0.194460
$530$	3150.00	0.258165
$531$	0	0
$532$	−14600.0	−1.18983
$533$	−117.000	−0.00950813
$534$	0	0
$535$	−5055.00	−0.408499
$536$	−4768.00	−0.384228
$537$	0	0
$538$	−13848.0	−1.10972
$539$	5922.00	0.473244
$540$	0	0
$541$	−15118.0	−1.20143	−0.600715	−	0.799463i	$-0.705117\pi$
$541$	−15118.0	−1.20143	−0.600715	+	0.799463i	$0.705117\pi$
$542$	−508.000	−0.0402592
$543$	0	0
$544$	3936.00	0.310211
$545$	5500.00	0.432283
$546$	0	0
$547$	3620.00	0.282962	0.141481	−	0.989941i	$-0.454814\pi$
$547$	3620.00	0.282962	0.141481	+	0.989941i	$0.454814\pi$
$548$	−1080.00	−0.0841885
$549$	0	0
$550$	−1050.00	−0.0814039
$551$	35916.0	2.77690
$552$	0	0
$553$	22225.0	1.70905
$554$	−13180.0	−1.01077
$555$	0	0
$556$	−28.0000	−0.00213573
$557$	11886.0	0.904176	0.452088	−	0.891973i	$-0.350679\pi$
$557$	11886.0	0.904176	0.452088	+	0.891973i	$0.350679\pi$
$558$	0	0
$559$	5876.00	0.444594
$560$	−2000.00	−0.150920
$561$	0	0
$562$	4740.00	0.355774
$563$	11409.0	0.854053	0.427027	−	0.904239i	$-0.359561\pi$
$563$	11409.0	0.854053	0.427027	+	0.904239i	$0.359561\pi$
$564$	0	0
$565$	8850.00	0.658978
$566$	12560.0	0.932749
$567$	0	0
$568$	6168.00	0.455640
$569$	−11760.0	−0.866441	−0.433220	−	0.901288i	$-0.642623\pi$
$569$	−11760.0	−0.866441	−0.433220	+	0.901288i	$0.642623\pi$
$570$	0	0
$571$	−4057.00	−0.297338	−0.148669	−	0.988887i	$-0.547499\pi$
$571$	−4057.00	−0.297338	−0.148669	+	0.988887i	$0.547499\pi$
$572$	1092.00	0.0798231
$573$	0	0
$574$	−450.000	−0.0327224
$575$	−2475.00	−0.179504
$576$	0	0
$577$	−5101.00	−0.368037	−0.184019	−	0.982923i	$-0.558911\pi$
$577$	−5101.00	−0.368037	−0.184019	+	0.982923i	$0.558911\pi$
$578$	−20432.0	−1.47034
$579$	0	0
$580$	4920.00	0.352227
$581$	−2400.00	−0.171375
$582$	0	0
$583$	−6615.00	−0.469923
$584$	−2608.00	−0.184794
$585$	0	0
$586$	−7176.00	−0.505867
$587$	24186.0	1.70062	0.850309	−	0.526283i	$-0.176415\pi$
$587$	24186.0	1.70062	0.850309	+	0.526283i	$0.176415\pi$
$588$	0	0
$589$	26572.0	1.85888
$590$	−240.000	−0.0167469
$591$	0	0
$592$	−4720.00	−0.327687
$593$	−72.0000	−0.00498598	−0.00249299	−	0.999997i	$-0.500794\pi$
$593$	−72.0000	−0.00498598	−0.00249299	+	0.999997i	$0.500794\pi$
$594$	0	0
$595$	15375.0	1.05935
$596$	−12396.0	−0.851946
$597$	0	0
$598$	2574.00	0.176018
$599$	14016.0	0.956057	0.478029	−	0.878344i	$-0.341352\pi$
$599$	14016.0	0.956057	0.478029	+	0.878344i	$0.341352\pi$
$600$	0	0
$601$	10739.0	0.728873	0.364437	−	0.931228i	$-0.381262\pi$
$601$	10739.0	0.728873	0.364437	+	0.931228i	$0.381262\pi$
$602$	22600.0	1.53008
$603$	0	0
$604$	−11440.0	−0.770674
$605$	−4450.00	−0.299038
$606$	0	0
$607$	18884.0	1.26273	0.631366	−	0.775485i	$-0.282495\pi$
$607$	18884.0	1.26273	0.631366	+	0.775485i	$0.282495\pi$
$608$	−4672.00	−0.311636
$609$	0	0
$610$	7270.00	0.482547
$611$	−5070.00	−0.335696
$612$	0	0
$613$	−3985.00	−0.262565	−0.131283	−	0.991345i	$-0.541910\pi$
$613$	−3985.00	−0.262565	−0.131283	+	0.991345i	$0.541910\pi$
$614$	12938.0	0.850383
$615$	0	0
$616$	4200.00	0.274712
$617$	−5034.00	−0.328462	−0.164231	−	0.986422i	$-0.552514\pi$
$617$	−5034.00	−0.328462	−0.164231	+	0.986422i	$0.552514\pi$
$618$	0	0
$619$	20954.0	1.36060	0.680301	−	0.732933i	$-0.261849\pi$
$619$	20954.0	1.36060	0.680301	+	0.732933i	$0.261849\pi$
$620$	3640.00	0.235784
$621$	0	0
$622$	3072.00	0.198032
$623$	19875.0	1.27813
$624$	0	0
$625$	625.000	0.0400000
$626$	−3028.00	−0.193328
$627$	0	0
$628$	−10936.0	−0.694895
$629$	36285.0	2.30012
$630$	0	0
$631$	16724.0	1.05511	0.527553	−	0.849522i	$-0.323110\pi$
$631$	16724.0	1.05511	0.527553	+	0.849522i	$0.323110\pi$
$632$	7112.00	0.447627
$633$	0	0
$634$	−5448.00	−0.341274
$635$	13330.0	0.833047
$636$	0	0
$637$	3666.00	0.228025
$638$	−10332.0	−0.641141
$639$	0	0
$640$	−640.000	−0.0395285
$641$	20520.0	1.26442	0.632208	−	0.774798i	$-0.282149\pi$
$641$	20520.0	1.26442	0.632208	+	0.774798i	$0.282149\pi$
$642$	0	0
$643$	−8953.00	−0.549101	−0.274550	−	0.961573i	$-0.588529\pi$
$643$	−8953.00	−0.549101	−0.274550	+	0.961573i	$0.588529\pi$
$644$	9900.00	0.605768
$645$	0	0
$646$	35916.0	2.18746
$647$	−17913.0	−1.08846	−0.544229	−	0.838937i	$-0.683178\pi$
$647$	−17913.0	−1.08846	−0.544229	+	0.838937i	$0.683178\pi$
$648$	0	0
$649$	504.000	0.0304834
$650$	−650.000	−0.0392232
$651$	0	0
$652$	−3412.00	−0.204945
$653$	−20718.0	−1.24159	−0.620795	−	0.783973i	$-0.713190\pi$
$653$	−20718.0	−1.24159	−0.620795	+	0.783973i	$0.713190\pi$
$654$	0	0
$655$	−2910.00	−0.173593
$656$	−144.000	−0.00857051
$657$	0	0
$658$	−19500.0	−1.15530
$659$	32040.0	1.89393	0.946966	−	0.321334i	$-0.104131\pi$
$659$	32040.0	1.89393	0.946966	+	0.321334i	$0.104131\pi$
$660$	0	0
$661$	−2176.00	−0.128043	−0.0640216	−	0.997949i	$-0.520393\pi$
$661$	−2176.00	−0.128043	−0.0640216	+	0.997949i	$0.520393\pi$
$662$	−2200.00	−0.129162
$663$	0	0
$664$	−768.000	−0.0448858
$665$	−18250.0	−1.06422
$666$	0	0
$667$	−24354.0	−1.41378
$668$	−10224.0	−0.592183
$669$	0	0
$670$	−5960.00	−0.343664
$671$	−15267.0	−0.878355
$672$	0	0
$673$	−30238.0	−1.73193	−0.865965	−	0.500104i	$-0.833295\pi$
$673$	−30238.0	−1.73193	−0.865965	+	0.500104i	$0.833295\pi$
$674$	392.000	0.0224025
$675$	0	0
$676$	676.000	0.0384615
$677$	25341.0	1.43860	0.719301	−	0.694698i	$-0.244462\pi$
$677$	25341.0	1.43860	0.719301	+	0.694698i	$0.244462\pi$
$678$	0	0
$679$	−24575.0	−1.38896
$680$	4920.00	0.277461
$681$	0	0
$682$	−7644.00	−0.429185
$683$	−6504.00	−0.364376	−0.182188	−	0.983264i	$-0.558318\pi$
$683$	−6504.00	−0.364376	−0.182188	+	0.983264i	$0.558318\pi$
$684$	0	0
$685$	−1350.00	−0.0753005
$686$	−3050.00	−0.169752
$687$	0	0
$688$	7232.00	0.400752
$689$	−4095.00	−0.226425
$690$	0	0
$691$	22358.0	1.23088	0.615440	−	0.788184i	$-0.288978\pi$
$691$	22358.0	1.23088	0.615440	+	0.788184i	$0.288978\pi$
$692$	4536.00	0.249180
$693$	0	0
$694$	22110.0	1.20934
$695$	−35.0000	−0.00191025
$696$	0	0
$697$	1107.00	0.0601587
$698$	22352.0	1.21209
$699$	0	0
$700$	−2500.00	−0.134987
$701$	2268.00	0.122199	0.0610993	−	0.998132i	$-0.480539\pi$
$701$	2268.00	0.122199	0.0610993	+	0.998132i	$0.480539\pi$
$702$	0	0
$703$	−43070.0	−2.31069
$704$	1344.00	0.0719516
$705$	0	0
$706$	4260.00	0.227092
$707$	−18300.0	−0.973469
$708$	0	0
$709$	13700.0	0.725690	0.362845	−	0.931849i	$-0.381805\pi$
$709$	13700.0	0.725690	0.362845	+	0.931849i	$0.381805\pi$
$710$	7710.00	0.407537
$711$	0	0
$712$	6360.00	0.334763
$713$	−18018.0	−0.946395
$714$	0	0
$715$	1365.00	0.0713960
$716$	−1992.00	−0.103973
$717$	0	0
$718$	4608.00	0.239511
$719$	−6540.00	−0.339222	−0.169611	−	0.985511i	$-0.554251\pi$
$719$	−6540.00	−0.339222	−0.169611	+	0.985511i	$0.554251\pi$
$720$	0	0
$721$	40000.0	2.06613
$722$	−28914.0	−1.49040
$723$	0	0
$724$	7604.00	0.390332
$725$	6150.00	0.315042
$726$	0	0
$727$	−13822.0	−0.705130	−0.352565	−	0.935787i	$-0.614690\pi$
$727$	−13822.0	−0.705130	−0.352565	+	0.935787i	$0.614690\pi$
$728$	2600.00	0.132366
$729$	0	0
$730$	−3260.00	−0.165285
$731$	−55596.0	−2.81299
$732$	0	0
$733$	−21715.0	−1.09422	−0.547109	−	0.837061i	$-0.684272\pi$
$733$	−21715.0	−1.09422	−0.547109	+	0.837061i	$0.684272\pi$
$734$	−26824.0	−1.34890
$735$	0	0
$736$	3168.00	0.158660
$737$	12516.0	0.625553
$738$	0	0
$739$	−3310.00	−0.164764	−0.0823818	−	0.996601i	$-0.526253\pi$
$739$	−3310.00	−0.164764	−0.0823818	+	0.996601i	$0.526253\pi$
$740$	−5900.00	−0.293092
$741$	0	0
$742$	−15750.0	−0.779246
$743$	−11178.0	−0.551926	−0.275963	−	0.961168i	$-0.588997\pi$
$743$	−11178.0	−0.551926	−0.275963	+	0.961168i	$0.588997\pi$
$744$	0	0
$745$	−15495.0	−0.762004
$746$	−18112.0	−0.888911
$747$	0	0
$748$	−10332.0	−0.505047
$749$	25275.0	1.23302
$750$	0	0
$751$	16337.0	0.793802	0.396901	−	0.917861i	$-0.370086\pi$
$751$	16337.0	0.793802	0.396901	+	0.917861i	$0.370086\pi$
$752$	−6240.00	−0.302592
$753$	0	0
$754$	−6396.00	−0.308924
$755$	−14300.0	−0.689312
$756$	0	0
$757$	23096.0	1.10890	0.554451	−	0.832217i	$-0.312928\pi$
$757$	23096.0	1.10890	0.554451	+	0.832217i	$0.312928\pi$
$758$	8132.00	0.389667
$759$	0	0
$760$	−5840.00	−0.278736
$761$	4578.00	0.218071	0.109036	−	0.994038i	$-0.465224\pi$
$761$	4578.00	0.218071	0.109036	+	0.994038i	$0.465224\pi$
$762$	0	0
$763$	−27500.0	−1.30481
$764$	−9360.00	−0.443237
$765$	0	0
$766$	1572.00	0.0741497
$767$	312.000	0.0146880
$768$	0	0
$769$	29036.0	1.36159	0.680796	−	0.732473i	$-0.261634\pi$
$769$	29036.0	1.36159	0.680796	+	0.732473i	$0.261634\pi$
$770$	5250.00	0.245710
$771$	0	0
$772$	6740.00	0.314220
$773$	37056.0	1.72421	0.862103	−	0.506733i	$-0.169147\pi$
$773$	37056.0	1.72421	0.862103	+	0.506733i	$0.169147\pi$
$774$	0	0
$775$	4550.00	0.210891
$776$	−7864.00	−0.363790
$777$	0	0
$778$	16728.0	0.770858
$779$	−1314.00	−0.0604351
$780$	0	0
$781$	−16191.0	−0.741818
$782$	−24354.0	−1.11368
$783$	0	0
$784$	4512.00	0.205539
$785$	−13670.0	−0.621533
$786$	0	0
$787$	−17332.0	−0.785031	−0.392515	−	0.919745i	$-0.628395\pi$
$787$	−17332.0	−0.785031	−0.392515	+	0.919745i	$0.628395\pi$
$788$	−7968.00	−0.360214
$789$	0	0
$790$	8890.00	0.400370
$791$	−44250.0	−1.98906
$792$	0	0
$793$	−9451.00	−0.423222
$794$	−15250.0	−0.681615
$795$	0	0
$796$	−18928.0	−0.842821
$797$	30759.0	1.36705	0.683526	−	0.729927i	$-0.260446\pi$
$797$	30759.0	1.36705	0.683526	+	0.729927i	$0.260446\pi$
$798$	0	0
$799$	47970.0	2.12398
$800$	−800.000	−0.0353553
$801$	0	0
$802$	1908.00	0.0840073
$803$	6846.00	0.300859
$804$	0	0
$805$	12375.0	0.541815
$806$	−4732.00	−0.206796
$807$	0	0
$808$	−5856.00	−0.254967
$809$	30102.0	1.30820	0.654098	−	0.756410i	$-0.273049\pi$
$809$	30102.0	1.30820	0.654098	+	0.756410i	$0.273049\pi$
$810$	0	0
$811$	−14308.0	−0.619509	−0.309755	−	0.950817i	$-0.600247\pi$
$811$	−14308.0	−0.619509	−0.309755	+	0.950817i	$0.600247\pi$
$812$	−24600.0	−1.06317
$813$	0	0
$814$	12390.0	0.533500
$815$	−4265.00	−0.183309
$816$	0	0
$817$	65992.0	2.82591
$818$	−364.000	−0.0155586
$819$	0	0
$820$	−180.000	−0.00766570
$821$	−27501.0	−1.16905	−0.584526	−	0.811375i	$-0.698719\pi$
$821$	−27501.0	−1.16905	−0.584526	+	0.811375i	$0.698719\pi$
$822$	0	0
$823$	2288.00	0.0969072	0.0484536	−	0.998825i	$-0.484571\pi$
$823$	2288.00	0.0969072	0.0484536	+	0.998825i	$0.484571\pi$
$824$	12800.0	0.541152
$825$	0	0
$826$	1200.00	0.0505488
$827$	1422.00	0.0597918	0.0298959	−	0.999553i	$-0.490482\pi$
$827$	1422.00	0.0597918	0.0298959	+	0.999553i	$0.490482\pi$
$828$	0	0
$829$	−42298.0	−1.77210	−0.886050	−	0.463590i	$-0.846561\pi$
$829$	−42298.0	−1.77210	−0.886050	+	0.463590i	$0.846561\pi$
$830$	−960.000	−0.0401471
$831$	0	0
$832$	832.000	0.0346688
$833$	−34686.0	−1.44274
$834$	0	0
$835$	−12780.0	−0.529665
$836$	12264.0	0.507368
$837$	0	0
$838$	−9612.00	−0.396230
$839$	−17595.0	−0.724013	−0.362006	−	0.932176i	$-0.617908\pi$
$839$	−17595.0	−0.724013	−0.362006	+	0.932176i	$0.617908\pi$
$840$	0	0
$841$	36127.0	1.48128
$842$	20048.0	0.820546
$843$	0	0
$844$	−16048.0	−0.654496
$845$	845.000	0.0344010
$846$	0	0
$847$	22250.0	0.902620
$848$	−5040.00	−0.204097
$849$	0	0
$850$	6150.00	0.248169
$851$	29205.0	1.17642
$852$	0	0
$853$	−13633.0	−0.547227	−0.273614	−	0.961840i	$-0.588219\pi$
$853$	−13633.0	−0.547227	−0.273614	+	0.961840i	$0.588219\pi$
$854$	−36350.0	−1.45652
$855$	0	0
$856$	8088.00	0.322946
$857$	29319.0	1.16863	0.584316	−	0.811526i	$-0.301363\pi$
$857$	29319.0	1.16863	0.584316	+	0.811526i	$0.301363\pi$
$858$	0	0
$859$	−24235.0	−0.962616	−0.481308	−	0.876551i	$-0.659838\pi$
$859$	−24235.0	−0.962616	−0.481308	+	0.876551i	$0.659838\pi$
$860$	9040.00	0.358444
$861$	0	0
$862$	−23088.0	−0.912274
$863$	7470.00	0.294649	0.147324	−	0.989088i	$-0.452934\pi$
$863$	7470.00	0.294649	0.147324	+	0.989088i	$0.452934\pi$
$864$	0	0
$865$	5670.00	0.222874
$866$	536.000	0.0210324
$867$	0	0
$868$	−18200.0	−0.711692
$869$	−18669.0	−0.728772
$870$	0	0
$871$	7748.00	0.301413
$872$	−8800.00	−0.341750
$873$	0	0
$874$	28908.0	1.11880
$875$	−3125.00	−0.120736
$876$	0	0
$877$	−28762.0	−1.10744	−0.553719	−	0.832703i	$-0.686792\pi$
$877$	−28762.0	−1.10744	−0.553719	+	0.832703i	$0.686792\pi$
$878$	12470.0	0.479319
$879$	0	0
$880$	1680.00	0.0643554
$881$	−11442.0	−0.437560	−0.218780	−	0.975774i	$-0.570208\pi$
$881$	−11442.0	−0.437560	−0.218780	+	0.975774i	$0.570208\pi$
$882$	0	0
$883$	−15208.0	−0.579604	−0.289802	−	0.957087i	$-0.593589\pi$
$883$	−15208.0	−0.579604	−0.289802	+	0.957087i	$0.593589\pi$
$884$	−6396.00	−0.243349
$885$	0	0
$886$	18726.0	0.710059
$887$	−25461.0	−0.963807	−0.481903	−	0.876224i	$-0.660054\pi$
$887$	−25461.0	−0.963807	−0.481903	+	0.876224i	$0.660054\pi$
$888$	0	0
$889$	−66650.0	−2.51448
$890$	7950.00	0.299421
$891$	0	0
$892$	−4384.00	−0.164560
$893$	−56940.0	−2.13373
$894$	0	0
$895$	−2490.00	−0.0929961
$896$	3200.00	0.119313
$897$	0	0
$898$	−14046.0	−0.521961
$899$	44772.0	1.66099
$900$	0	0
$901$	38745.0	1.43261
$902$	378.000	0.0139535
$903$	0	0
$904$	−14160.0	−0.520967
$905$	9505.00	0.349124
$906$	0	0
$907$	27326.0	1.00038	0.500190	−	0.865916i	$-0.333263\pi$
$907$	27326.0	1.00038	0.500190	+	0.865916i	$0.333263\pi$
$908$	−24552.0	−0.897342
$909$	0	0
$910$	3250.00	0.118392
$911$	−24420.0	−0.888113	−0.444056	−	0.895999i	$-0.646461\pi$
$911$	−24420.0	−0.888113	−0.444056	+	0.895999i	$0.646461\pi$
$912$	0	0
$913$	2016.00	0.0730776
$914$	−5206.00	−0.188402
$915$	0	0
$916$	−5176.00	−0.186703
$917$	14550.0	0.523973
$918$	0	0
$919$	32951.0	1.18276	0.591378	−	0.806394i	$-0.298584\pi$
$919$	32951.0	1.18276	0.591378	+	0.806394i	$0.298584\pi$
$920$	3960.00	0.141910
$921$	0	0
$922$	20754.0	0.741320
$923$	−10023.0	−0.357433
$924$	0	0
$925$	−7375.00	−0.262150
$926$	22802.0	0.809201
$927$	0	0
$928$	−7872.00	−0.278460
$929$	−49779.0	−1.75802	−0.879008	−	0.476808i	$-0.841794\pi$
$929$	−49779.0	−1.75802	−0.879008	+	0.476808i	$0.841794\pi$
$930$	0	0
$931$	41172.0	1.44936
$932$	−3492.00	−0.122730
$933$	0	0
$934$	−30762.0	−1.07769
$935$	−12915.0	−0.451728
$936$	0	0
$937$	−44494.0	−1.55129	−0.775643	−	0.631171i	$-0.782574\pi$
$937$	−44494.0	−1.55129	−0.775643	+	0.631171i	$0.782574\pi$
$938$	29800.0	1.03732
$939$	0	0
$940$	−7800.00	−0.270647
$941$	−41235.0	−1.42850	−0.714252	−	0.699888i	$-0.753233\pi$
$941$	−41235.0	−1.42850	−0.714252	+	0.699888i	$0.753233\pi$
$942$	0	0
$943$	891.000	0.0307688
$944$	384.000	0.0132396
$945$	0	0
$946$	−18984.0	−0.652456
$947$	34536.0	1.18508	0.592539	−	0.805542i	$-0.298126\pi$
$947$	34536.0	1.18508	0.592539	+	0.805542i	$0.298126\pi$
$948$	0	0
$949$	4238.00	0.144964
$950$	−7300.00	−0.249309
$951$	0	0
$952$	−24600.0	−0.837490
$953$	48693.0	1.65511	0.827556	−	0.561384i	$-0.189731\pi$
$953$	48693.0	1.65511	0.827556	+	0.561384i	$0.189731\pi$
$954$	0	0
$955$	−11700.0	−0.396443
$956$	−20652.0	−0.698675
$957$	0	0
$958$	−3534.00	−0.119184
$959$	6750.00	0.227288
$960$	0	0
$961$	3333.00	0.111879
$962$	7670.00	0.257059
$963$	0	0
$964$	−9928.00	−0.331701
$965$	8425.00	0.281047
$966$	0	0
$967$	−18088.0	−0.601521	−0.300761	−	0.953700i	$-0.597240\pi$
$967$	−18088.0	−0.601521	−0.300761	+	0.953700i	$0.597240\pi$
$968$	7120.00	0.236411
$969$	0	0
$970$	−9830.00	−0.325384
$971$	−15396.0	−0.508837	−0.254419	−	0.967094i	$-0.581884\pi$
$971$	−15396.0	−0.508837	−0.254419	+	0.967094i	$0.581884\pi$
$972$	0	0
$973$	175.000	0.00576592
$974$	−19786.0	−0.650908
$975$	0	0
$976$	−11632.0	−0.381487
$977$	−29700.0	−0.972556	−0.486278	−	0.873804i	$-0.661646\pi$
$977$	−29700.0	−0.972556	−0.486278	+	0.873804i	$0.661646\pi$
$978$	0	0
$979$	−16695.0	−0.545020
$980$	5640.00	0.183840
$981$	0	0
$982$	−30672.0	−0.996724
$983$	−54444.0	−1.76652	−0.883262	−	0.468879i	$-0.844658\pi$
$983$	−54444.0	−1.76652	−0.883262	+	0.468879i	$0.844658\pi$
$984$	0	0
$985$	−9960.00	−0.322185
$986$	60516.0	1.95459
$987$	0	0
$988$	7592.00	0.244467
$989$	−44748.0	−1.43873
$990$	0	0
$991$	−12571.0	−0.402958	−0.201479	−	0.979493i	$-0.564575\pi$
$991$	−12571.0	−0.402958	−0.201479	+	0.979493i	$0.564575\pi$
$992$	−5824.00	−0.186403
$993$	0	0
$994$	−38550.0	−1.23011
$995$	−23660.0	−0.753842
$996$	0	0
$997$	−3796.00	−0.120582	−0.0602911	−	0.998181i	$-0.519203\pi$
$997$	−3796.00	−0.120582	−0.0602911	+	0.998181i	$0.519203\pi$
$998$	32900.0	1.04352
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1170.4.a.e.1.1		1
3.2	odd	2		390.4.a.g.1.1	✓	1
15.14	odd	2		1950.4.a.g.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.4.a.g.1.1	✓	1	3.2	odd	2
1170.4.a.e.1.1		1	1.1	even	1	trivial
1950.4.a.g.1.1		1	15.14	odd	2

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:	\( N \)	\(=\)	\( 1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1170.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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