LMFDB - Embedded newform 350.4.a.s.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [350,4,Mod(1,350)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(350, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("350.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 350 = 2 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	350.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:	$20.6506685020$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	350.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+2.00000 q^{2} +1.00000 q^{3} +4.00000 q^{4} +2.00000 q^{6} -7.00000 q^{7} +8.00000 q^{8} -26.0000 q^{9} +O(q^{10})$

\(q+2.00000 q^{2} +1.00000 q^{3} +4.00000 q^{4} +2.00000 q^{6} -7.00000 q^{7} +8.00000 q^{8} -26.0000 q^{9} -35.0000 q^{11} +4.00000 q^{12} -58.0000 q^{13} -14.0000 q^{14} +16.0000 q^{16} -107.000 q^{17} -52.0000 q^{18} +23.0000 q^{19} -7.00000 q^{21} -70.0000 q^{22} +200.000 q^{23} +8.00000 q^{24} -116.000 q^{26} -53.0000 q^{27} -28.0000 q^{28} -174.000 q^{29} +76.0000 q^{31} +32.0000 q^{32} -35.0000 q^{33} -214.000 q^{34} -104.000 q^{36} -184.000 q^{37} +46.0000 q^{38} -58.0000 q^{39} +431.000 q^{41} -14.0000 q^{42} -144.000 q^{43} -140.000 q^{44} +400.000 q^{46} -526.000 q^{47} +16.0000 q^{48} +49.0000 q^{49} -107.000 q^{51} -232.000 q^{52} -108.000 q^{53} -106.000 q^{54} -56.0000 q^{56} +23.0000 q^{57} -348.000 q^{58} +76.0000 q^{59} +118.000 q^{61} +152.000 q^{62} +182.000 q^{63} +64.0000 q^{64} -70.0000 q^{66} -687.000 q^{67} -428.000 q^{68} +200.000 q^{69} +530.000 q^{71} -208.000 q^{72} +299.000 q^{73} -368.000 q^{74} +92.0000 q^{76} +245.000 q^{77} -116.000 q^{78} +402.000 q^{79} +649.000 q^{81} +862.000 q^{82} -897.000 q^{83} -28.0000 q^{84} -288.000 q^{86} -174.000 q^{87} -280.000 q^{88} -799.000 q^{89} +406.000 q^{91} +800.000 q^{92} +76.0000 q^{93} -1052.00 q^{94} +32.0000 q^{96} -1510.00 q^{97} +98.0000 q^{98} +910.000 q^{99} +O(q^{100})\)

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
$3$	1.00000	0.192450	0.0962250	−	0.995360i	$-0.469323\pi$
$3$	1.00000	0.192450	0.0962250	+	0.995360i	$0.469323\pi$
$4$	4.00000	0.500000
$5$	0	0
$5$	0	0
$6$	2.00000	0.136083
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	8.00000	0.353553
$9$	−26.0000	−0.962963
$10$	0	0
$11$	−35.0000	−0.959354	−0.479677	−	0.877445i	$-0.659246\pi$
$11$	−35.0000	−0.959354	−0.479677	+	0.877445i	$0.659246\pi$
$12$	4.00000	0.0962250
$13$	−58.0000	−1.23741	−0.618704	−	0.785624i	$-0.712342\pi$
$13$	−58.0000	−1.23741	−0.618704	+	0.785624i	$0.712342\pi$
$14$	−14.0000	−0.267261
$15$	0	0
$16$	16.0000	0.250000
$17$	−107.000	−1.52655	−0.763274	−	0.646075i	$-0.776409\pi$
$17$	−107.000	−1.52655	−0.763274	+	0.646075i	$0.776409\pi$
$18$	−52.0000	−0.680918
$19$	23.0000	0.277714	0.138857	−	0.990312i	$-0.455657\pi$
$19$	23.0000	0.277714	0.138857	+	0.990312i	$0.455657\pi$
$20$	0	0
$21$	−7.00000	−0.0727393
$22$	−70.0000	−0.678366
$23$	200.000	1.81317	0.906584	−	0.422025i	$-0.138680\pi$
$23$	200.000	1.81317	0.906584	+	0.422025i	$0.138680\pi$
$24$	8.00000	0.0680414
$25$	0	0
$26$	−116.000	−0.874980
$27$	−53.0000	−0.377772
$28$	−28.0000	−0.188982
$29$	−174.000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−174.000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	0	0
$31$	76.0000	0.440323	0.220161	−	0.975463i	$-0.429342\pi$
$31$	76.0000	0.440323	0.220161	+	0.975463i	$0.429342\pi$
$32$	32.0000	0.176777
$33$	−35.0000	−0.184628
$34$	−214.000	−1.07943
$35$	0	0
$36$	−104.000	−0.481481
$37$	−184.000	−0.817552	−0.408776	−	0.912635i	$-0.634044\pi$
$37$	−184.000	−0.817552	−0.408776	+	0.912635i	$0.634044\pi$
$38$	46.0000	0.196373
$39$	−58.0000	−0.238139
$40$	0	0
$41$	431.000	1.64173	0.820865	−	0.571123i	$-0.193492\pi$
$41$	431.000	1.64173	0.820865	+	0.571123i	$0.193492\pi$
$42$	−14.0000	−0.0514344
$43$	−144.000	−0.510693	−0.255346	−	0.966850i	$-0.582190\pi$
$43$	−144.000	−0.510693	−0.255346	+	0.966850i	$0.582190\pi$
$44$	−140.000	−0.479677
$45$	0	0
$46$	400.000	1.28210
$47$	−526.000	−1.63245	−0.816223	−	0.577737i	$-0.803936\pi$
$47$	−526.000	−1.63245	−0.816223	+	0.577737i	$0.803936\pi$
$48$	16.0000	0.0481125
$49$	49.0000	0.142857
$50$	0	0
$51$	−107.000	−0.293784
$52$	−232.000	−0.618704
$53$	−108.000	−0.279905	−0.139952	−	0.990158i	$-0.544695\pi$
$53$	−108.000	−0.279905	−0.139952	+	0.990158i	$0.544695\pi$
$54$	−106.000	−0.267125
$55$	0	0
$56$	−56.0000	−0.133631
$57$	23.0000	0.0534460
$58$	−348.000	−0.787839
$59$	76.0000	0.167701	0.0838505	−	0.996478i	$-0.473278\pi$
$59$	76.0000	0.167701	0.0838505	+	0.996478i	$0.473278\pi$
$60$	0	0
$61$	118.000	0.247678	0.123839	−	0.992302i	$-0.460479\pi$
$61$	118.000	0.247678	0.123839	+	0.992302i	$0.460479\pi$
$62$	152.000	0.311355
$63$	182.000	0.363966
$64$	64.0000	0.125000
$65$	0	0
$66$	−70.0000	−0.130552
$67$	−687.000	−1.25269	−0.626346	−	0.779545i	$-0.715450\pi$
$67$	−687.000	−1.25269	−0.626346	+	0.779545i	$0.715450\pi$
$68$	−428.000	−0.763274
$69$	200.000	0.348945
$70$	0	0
$71$	530.000	0.885907	0.442954	−	0.896544i	$-0.353931\pi$
$71$	530.000	0.885907	0.442954	+	0.896544i	$0.353931\pi$
$72$	−208.000	−0.340459
$73$	299.000	0.479388	0.239694	−	0.970849i	$-0.422953\pi$
$73$	299.000	0.479388	0.239694	+	0.970849i	$0.422953\pi$
$74$	−368.000	−0.578096
$75$	0	0
$76$	92.0000	0.138857
$77$	245.000	0.362602
$78$	−116.000	−0.168390
$79$	402.000	0.572513	0.286257	−	0.958153i	$-0.407589\pi$
$79$	402.000	0.572513	0.286257	+	0.958153i	$0.407589\pi$
$80$	0	0
$81$	649.000	0.890261
$82$	862.000	1.16088
$83$	−897.000	−1.18625	−0.593124	−	0.805111i	$-0.702106\pi$
$83$	−897.000	−1.18625	−0.593124	+	0.805111i	$0.702106\pi$
$84$	−28.0000	−0.0363696
$85$	0	0
$86$	−288.000	−0.361114
$87$	−174.000	−0.214423
$88$	−280.000	−0.339183
$89$	−799.000	−0.951616	−0.475808	−	0.879549i	$-0.657844\pi$
$89$	−799.000	−0.951616	−0.475808	+	0.879549i	$0.657844\pi$
$90$	0	0
$91$	406.000	0.467696
$92$	800.000	0.906584
$93$	76.0000	0.0847401
$94$	−1052.00	−1.15431
$95$	0	0
$96$	32.0000	0.0340207
$97$	−1510.00	−1.58059	−0.790295	−	0.612726i	$-0.790073\pi$
$97$	−1510.00	−1.58059	−0.790295	+	0.612726i	$0.790073\pi$
$98$	98.0000	0.101015
$99$	910.000	0.923823
$100$	0	0
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	−214.000	−0.207737
$103$	412.000	0.394132	0.197066	−	0.980390i	$-0.436859\pi$
$103$	412.000	0.394132	0.197066	+	0.980390i	$0.436859\pi$
$104$	−464.000	−0.437490
$105$	0	0
$106$	−216.000	−0.197922
$107$	851.000	0.768872	0.384436	−	0.923152i	$-0.374396\pi$
$107$	851.000	0.768872	0.384436	+	0.923152i	$0.374396\pi$
$108$	−212.000	−0.188886
$109$	2158.00	1.89632	0.948160	−	0.317793i	$-0.102942\pi$
$109$	2158.00	1.89632	0.948160	+	0.317793i	$0.102942\pi$
$110$	0	0
$111$	−184.000	−0.157338
$112$	−112.000	−0.0944911
$113$	377.000	0.313851	0.156926	−	0.987610i	$-0.449842\pi$
$113$	377.000	0.313851	0.156926	+	0.987610i	$0.449842\pi$
$114$	46.0000	0.0377921
$115$	0	0
$116$	−696.000	−0.557086
$117$	1508.00	1.19158
$118$	152.000	0.118582
$119$	749.000	0.576981
$120$	0	0
$121$	−106.000	−0.0796394
$122$	236.000	0.175135
$123$	431.000	0.315951
$124$	304.000	0.220161
$125$	0	0
$126$	364.000	0.257363
$127$	762.000	0.532414	0.266207	−	0.963916i	$-0.414230\pi$
$127$	762.000	0.532414	0.266207	+	0.963916i	$0.414230\pi$
$128$	128.000	0.0883883
$129$	−144.000	−0.0982829
$130$	0	0
$131$	−1512.00	−1.00843	−0.504214	−	0.863579i	$-0.668218\pi$
$131$	−1512.00	−1.00843	−0.504214	+	0.863579i	$0.668218\pi$
$132$	−140.000	−0.0923139
$133$	−161.000	−0.104966
$134$	−1374.00	−0.885787
$135$	0	0
$136$	−856.000	−0.539716
$137$	419.000	0.261296	0.130648	−	0.991429i	$-0.458294\pi$
$137$	419.000	0.261296	0.130648	+	0.991429i	$0.458294\pi$
$138$	400.000	0.246741
$139$	−2329.00	−1.42117	−0.710587	−	0.703609i	$-0.751571\pi$
$139$	−2329.00	−1.42117	−0.710587	+	0.703609i	$0.751571\pi$
$140$	0	0
$141$	−526.000	−0.314164
$142$	1060.00	0.626431
$143$	2030.00	1.18711
$144$	−416.000	−0.240741
$145$	0	0
$146$	598.000	0.338978
$147$	49.0000	0.0274929
$148$	−736.000	−0.408776
$149$	3108.00	1.70884	0.854420	−	0.519582i	$-0.173912\pi$
$149$	3108.00	1.70884	0.854420	+	0.519582i	$0.173912\pi$
$150$	0	0
$151$	−1192.00	−0.642408	−0.321204	−	0.947010i	$-0.604087\pi$
$151$	−1192.00	−0.642408	−0.321204	+	0.947010i	$0.604087\pi$
$152$	184.000	0.0981866
$153$	2782.00	1.47001
$154$	490.000	0.256398
$155$	0	0
$156$	−232.000	−0.119070
$157$	−1772.00	−0.900771	−0.450385	−	0.892834i	$-0.648713\pi$
$157$	−1772.00	−0.900771	−0.450385	+	0.892834i	$0.648713\pi$
$158$	804.000	0.404828
$159$	−108.000	−0.0538677
$160$	0	0
$161$	−1400.00	−0.685313
$162$	1298.00	0.629509
$163$	2973.00	1.42861	0.714305	−	0.699835i	$-0.246743\pi$
$163$	2973.00	1.42861	0.714305	+	0.699835i	$0.246743\pi$
$164$	1724.00	0.820865
$165$	0	0
$166$	−1794.00	−0.838804
$167$	4.00000	0.00185347	0.000926734	−	1.00000i	$-0.499705\pi$
$167$	4.00000	0.00185347	0.000926734		1.00000i	$0.499705\pi$
$168$	−56.0000	−0.0257172
$169$	1167.00	0.531179
$170$	0	0
$171$	−598.000	−0.267428
$172$	−576.000	−0.255346
$173$	−3434.00	−1.50915	−0.754573	−	0.656216i	$-0.772156\pi$
$173$	−3434.00	−1.50915	−0.754573	+	0.656216i	$0.772156\pi$
$174$	−348.000	−0.151620
$175$	0	0
$176$	−560.000	−0.239839
$177$	76.0000	0.0322741
$178$	−1598.00	−0.672894
$179$	397.000	0.165772	0.0828860	−	0.996559i	$-0.473586\pi$
$179$	397.000	0.165772	0.0828860	+	0.996559i	$0.473586\pi$
$180$	0	0
$181$	650.000	0.266929	0.133464	−	0.991054i	$-0.457390\pi$
$181$	650.000	0.266929	0.133464	+	0.991054i	$0.457390\pi$
$182$	812.000	0.330711
$183$	118.000	0.0476656
$184$	1600.00	0.641052
$185$	0	0
$186$	152.000	0.0599203
$187$	3745.00	1.46450
$188$	−2104.00	−0.816223
$189$	371.000	0.142785
$190$	0	0
$191$	−1426.00	−0.540219	−0.270109	−	0.962830i	$-0.587060\pi$
$191$	−1426.00	−0.540219	−0.270109	+	0.962830i	$0.587060\pi$
$192$	64.0000	0.0240563
$193$	−1357.00	−0.506109	−0.253054	−	0.967452i	$-0.581435\pi$
$193$	−1357.00	−0.506109	−0.253054	+	0.967452i	$0.581435\pi$
$194$	−3020.00	−1.11765
$195$	0	0
$196$	196.000	0.0714286
$197$	4686.00	1.69474	0.847370	−	0.531003i	$-0.178185\pi$
$197$	4686.00	1.69474	0.847370	+	0.531003i	$0.178185\pi$
$198$	1820.00	0.653241
$199$	3890.00	1.38570	0.692851	−	0.721081i	$-0.256354\pi$
$199$	3890.00	1.38570	0.692851	+	0.721081i	$0.256354\pi$
$200$	0	0
$201$	−687.000	−0.241081
$202$	0	0
$203$	1218.00	0.421117
$204$	−428.000	−0.146892
$205$	0	0
$206$	824.000	0.278693
$207$	−5200.00	−1.74601
$208$	−928.000	−0.309352
$209$	−805.000	−0.266426
$210$	0	0
$211$	−4009.00	−1.30801	−0.654007	−	0.756489i	$-0.726913\pi$
$211$	−4009.00	−1.30801	−0.654007	+	0.756489i	$0.726913\pi$
$212$	−432.000	−0.139952
$213$	530.000	0.170493
$214$	1702.00	0.543674
$215$	0	0
$216$	−424.000	−0.133563
$217$	−532.000	−0.166426
$218$	4316.00	1.34090
$219$	299.000	0.0922582
$220$	0	0
$221$	6206.00	1.88896
$222$	−368.000	−0.111255
$223$	−5154.00	−1.54770	−0.773851	−	0.633368i	$-0.781672\pi$
$223$	−5154.00	−1.54770	−0.773851	+	0.633368i	$0.781672\pi$
$224$	−224.000	−0.0668153
$225$	0	0
$226$	754.000	0.221926
$227$	1524.00	0.445601	0.222801	−	0.974864i	$-0.428480\pi$
$227$	1524.00	0.445601	0.222801	+	0.974864i	$0.428480\pi$
$228$	92.0000	0.0267230
$229$	−3446.00	−0.994402	−0.497201	−	0.867635i	$-0.665639\pi$
$229$	−3446.00	−0.994402	−0.497201	+	0.867635i	$0.665639\pi$
$230$	0	0
$231$	245.000	0.0697828
$232$	−1392.00	−0.393919
$233$	698.000	0.196255	0.0981277	−	0.995174i	$-0.468715\pi$
$233$	698.000	0.196255	0.0981277	+	0.995174i	$0.468715\pi$
$234$	3016.00	0.842573
$235$	0	0
$236$	304.000	0.0838505
$237$	402.000	0.110180
$238$	1498.00	0.407987
$239$	−6124.00	−1.65744	−0.828721	−	0.559662i	$-0.810931\pi$
$239$	−6124.00	−1.65744	−0.828721	+	0.559662i	$0.810931\pi$
$240$	0	0
$241$	6975.00	1.86431	0.932156	−	0.362057i	$-0.117925\pi$
$241$	6975.00	1.86431	0.932156	+	0.362057i	$0.117925\pi$
$242$	−212.000	−0.0563135
$243$	2080.00	0.549103
$244$	472.000	0.123839
$245$	0	0
$246$	862.000	0.223411
$247$	−1334.00	−0.343645
$248$	608.000	0.155678
$249$	−897.000	−0.228293
$250$	0	0
$251$	1897.00	0.477042	0.238521	−	0.971137i	$-0.423337\pi$
$251$	1897.00	0.477042	0.238521	+	0.971137i	$0.423337\pi$
$252$	728.000	0.181983
$253$	−7000.00	−1.73947
$254$	1524.00	0.376473
$255$	0	0
$256$	256.000	0.0625000
$257$	4654.00	1.12961	0.564803	−	0.825226i	$-0.308952\pi$
$257$	4654.00	1.12961	0.564803	+	0.825226i	$0.308952\pi$
$258$	−288.000	−0.0694965
$259$	1288.00	0.309006
$260$	0	0
$261$	4524.00	1.07291
$262$	−3024.00	−0.713066
$263$	2190.00	0.513465	0.256732	−	0.966483i	$-0.417354\pi$
$263$	2190.00	0.513465	0.256732	+	0.966483i	$0.417354\pi$
$264$	−280.000	−0.0652758
$265$	0	0
$266$	−322.000	−0.0742221
$267$	−799.000	−0.183139
$268$	−2748.00	−0.626346
$269$	−7638.00	−1.73122	−0.865608	−	0.500722i	$-0.833068\pi$
$269$	−7638.00	−1.73122	−0.865608	+	0.500722i	$0.833068\pi$
$270$	0	0
$271$	−5466.00	−1.22522	−0.612612	−	0.790384i	$-0.709881\pi$
$271$	−5466.00	−1.22522	−0.612612	+	0.790384i	$0.709881\pi$
$272$	−1712.00	−0.381637
$273$	406.000	0.0900082
$274$	838.000	0.184764
$275$	0	0
$276$	800.000	0.174472
$277$	310.000	0.0672422	0.0336211	−	0.999435i	$-0.489296\pi$
$277$	310.000	0.0672422	0.0336211	+	0.999435i	$0.489296\pi$
$278$	−4658.00	−1.00492
$279$	−1976.00	−0.424014
$280$	0	0
$281$	−4946.00	−1.05001	−0.525006	−	0.851098i	$-0.675937\pi$
$281$	−4946.00	−1.05001	−0.525006	+	0.851098i	$0.675937\pi$
$282$	−1052.00	−0.222148
$283$	391.000	0.0821291	0.0410646	−	0.999156i	$-0.486925\pi$
$283$	391.000	0.0821291	0.0410646	+	0.999156i	$0.486925\pi$
$284$	2120.00	0.442954
$285$	0	0
$286$	4060.00	0.839415
$287$	−3017.00	−0.620515
$288$	−832.000	−0.170229
$289$	6536.00	1.33035
$290$	0	0
$291$	−1510.00	−0.304185
$292$	1196.00	0.239694
$293$	−4122.00	−0.821876	−0.410938	−	0.911663i	$-0.634799\pi$
$293$	−4122.00	−0.821876	−0.410938	+	0.911663i	$0.634799\pi$
$294$	98.0000	0.0194404
$295$	0	0
$296$	−1472.00	−0.289048
$297$	1855.00	0.362418
$298$	6216.00	1.20833
$299$	−11600.0	−2.24363
$300$	0	0
$301$	1008.00	0.193024
$302$	−2384.00	−0.454251
$303$	0	0
$304$	368.000	0.0694284
$305$	0	0
$306$	5564.00	1.03945
$307$	−8279.00	−1.53911	−0.769556	−	0.638579i	$-0.779522\pi$
$307$	−8279.00	−1.53911	−0.769556	+	0.638579i	$0.779522\pi$
$308$	980.000	0.181301
$309$	412.000	0.0758507
$310$	0	0
$311$	6146.00	1.12060	0.560302	−	0.828289i	$-0.310685\pi$
$311$	6146.00	1.12060	0.560302	+	0.828289i	$0.310685\pi$
$312$	−464.000	−0.0841950
$313$	−9190.00	−1.65958	−0.829792	−	0.558073i	$-0.811541\pi$
$313$	−9190.00	−1.65958	−0.829792	+	0.558073i	$0.811541\pi$
$314$	−3544.00	−0.636941
$315$	0	0
$316$	1608.00	0.286257
$317$	−3812.00	−0.675405	−0.337702	−	0.941253i	$-0.609650\pi$
$317$	−3812.00	−0.675405	−0.337702	+	0.941253i	$0.609650\pi$
$318$	−216.000	−0.0380902
$319$	6090.00	1.06889
$320$	0	0
$321$	851.000	0.147969
$322$	−2800.00	−0.484590
$323$	−2461.00	−0.423943
$324$	2596.00	0.445130
$325$	0	0
$326$	5946.00	1.01018
$327$	2158.00	0.364947
$328$	3448.00	0.580439
$329$	3682.00	0.617007
$330$	0	0
$331$	−1127.00	−0.187147	−0.0935733	−	0.995612i	$-0.529829\pi$
$331$	−1127.00	−0.187147	−0.0935733	+	0.995612i	$0.529829\pi$
$332$	−3588.00	−0.593124
$333$	4784.00	0.787272
$334$	8.00000	0.00131060
$335$	0	0
$336$	−112.000	−0.0181848
$337$	−2003.00	−0.323770	−0.161885	−	0.986810i	$-0.551757\pi$
$337$	−2003.00	−0.323770	−0.161885	+	0.986810i	$0.551757\pi$
$338$	2334.00	0.375600
$339$	377.000	0.0604007
$340$	0	0
$341$	−2660.00	−0.422425
$342$	−1196.00	−0.189100
$343$	−343.000	−0.0539949
$344$	−1152.00	−0.180557
$345$	0	0
$346$	−6868.00	−1.06713
$347$	−10917.0	−1.68892	−0.844460	−	0.535619i	$-0.820078\pi$
$347$	−10917.0	−1.68892	−0.844460	+	0.535619i	$0.820078\pi$
$348$	−696.000	−0.107211
$349$	−7912.00	−1.21352	−0.606762	−	0.794884i	$-0.707532\pi$
$349$	−7912.00	−1.21352	−0.606762	+	0.794884i	$0.707532\pi$
$350$	0	0
$351$	3074.00	0.467459
$352$	−1120.00	−0.169591
$353$	−1854.00	−0.279542	−0.139771	−	0.990184i	$-0.544637\pi$
$353$	−1854.00	−0.279542	−0.139771	+	0.990184i	$0.544637\pi$
$354$	152.000	0.0228212
$355$	0	0
$356$	−3196.00	−0.475808
$357$	749.000	0.111040
$358$	794.000	0.117218
$359$	−11066.0	−1.62686	−0.813428	−	0.581666i	$-0.802401\pi$
$359$	−11066.0	−1.62686	−0.813428	+	0.581666i	$0.802401\pi$
$360$	0	0
$361$	−6330.00	−0.922875
$362$	1300.00	0.188747
$363$	−106.000	−0.0153266
$364$	1624.00	0.233848
$365$	0	0
$366$	236.000	0.0337047
$367$	−6136.00	−0.872743	−0.436371	−	0.899767i	$-0.643737\pi$
$367$	−6136.00	−0.872743	−0.436371	+	0.899767i	$0.643737\pi$
$368$	3200.00	0.453292
$369$	−11206.0	−1.58092
$370$	0	0
$371$	756.000	0.105794
$372$	304.000	0.0423701
$373$	4588.00	0.636884	0.318442	−	0.947942i	$-0.396840\pi$
$373$	4588.00	0.636884	0.318442	+	0.947942i	$0.396840\pi$
$374$	7490.00	1.03556
$375$	0	0
$376$	−4208.00	−0.577157
$377$	10092.0	1.37869
$378$	742.000	0.100964
$379$	−6089.00	−0.825253	−0.412627	−	0.910900i	$-0.635389\pi$
$379$	−6089.00	−0.825253	−0.412627	+	0.910900i	$0.635389\pi$
$380$	0	0
$381$	762.000	0.102463
$382$	−2852.00	−0.381992
$383$	−2958.00	−0.394639	−0.197320	−	0.980339i	$-0.563224\pi$
$383$	−2958.00	−0.394639	−0.197320	+	0.980339i	$0.563224\pi$
$384$	128.000	0.0170103
$385$	0	0
$386$	−2714.00	−0.357873
$387$	3744.00	0.491778
$388$	−6040.00	−0.790295
$389$	5472.00	0.713217	0.356609	−	0.934254i	$-0.383933\pi$
$389$	5472.00	0.713217	0.356609	+	0.934254i	$0.383933\pi$
$390$	0	0
$391$	−21400.0	−2.76789
$392$	392.000	0.0505076
$393$	−1512.00	−0.194072
$394$	9372.00	1.19836
$395$	0	0
$396$	3640.00	0.461911
$397$	3230.00	0.408335	0.204168	−	0.978936i	$-0.434551\pi$
$397$	3230.00	0.408335	0.204168	+	0.978936i	$0.434551\pi$
$398$	7780.00	0.979840
$399$	−161.000	−0.0202007
$400$	0	0
$401$	13917.0	1.73312	0.866561	−	0.499071i	$-0.166325\pi$
$401$	13917.0	1.73312	0.866561	+	0.499071i	$0.166325\pi$
$402$	−1374.00	−0.170470
$403$	−4408.00	−0.544859
$404$	0	0
$405$	0	0
$406$	2436.00	0.297775
$407$	6440.00	0.784322
$408$	−856.000	−0.103868
$409$	12271.0	1.48353	0.741763	−	0.670662i	$-0.233990\pi$
$409$	12271.0	1.48353	0.741763	+	0.670662i	$0.233990\pi$
$410$	0	0
$411$	419.000	0.0502865
$412$	1648.00	0.197066
$413$	−532.000	−0.0633850
$414$	−10400.0	−1.23462
$415$	0	0
$416$	−1856.00	−0.218745
$417$	−2329.00	−0.273505
$418$	−1610.00	−0.188392
$419$	8979.00	1.04690	0.523452	−	0.852055i	$-0.324644\pi$
$419$	8979.00	1.04690	0.523452	+	0.852055i	$0.324644\pi$
$420$	0	0
$421$	6100.00	0.706166	0.353083	−	0.935592i	$-0.385133\pi$
$421$	6100.00	0.706166	0.353083	+	0.935592i	$0.385133\pi$
$422$	−8018.00	−0.924906
$423$	13676.0	1.57199
$424$	−864.000	−0.0989612
$425$	0	0
$426$	1060.00	0.120557
$427$	−826.000	−0.0936134
$428$	3404.00	0.384436
$429$	2030.00	0.228460
$430$	0	0
$431$	324.000	0.0362100	0.0181050	−	0.999836i	$-0.494237\pi$
$431$	324.000	0.0362100	0.0181050	+	0.999836i	$0.494237\pi$
$432$	−848.000	−0.0944431
$433$	−8163.00	−0.905979	−0.452989	−	0.891516i	$-0.649642\pi$
$433$	−8163.00	−0.905979	−0.452989	+	0.891516i	$0.649642\pi$
$434$	−1064.00	−0.117681
$435$	0	0
$436$	8632.00	0.948160
$437$	4600.00	0.503542
$438$	598.000	0.0652364
$439$	−13828.0	−1.50336	−0.751679	−	0.659529i	$-0.770756\pi$
$439$	−13828.0	−1.50336	−0.751679	+	0.659529i	$0.770756\pi$
$440$	0	0
$441$	−1274.00	−0.137566
$442$	12412.0	1.33570
$443$	14941.0	1.60241	0.801206	−	0.598389i	$-0.204192\pi$
$443$	14941.0	1.60241	0.801206	+	0.598389i	$0.204192\pi$
$444$	−736.000	−0.0786690
$445$	0	0
$446$	−10308.0	−1.09439
$447$	3108.00	0.328867
$448$	−448.000	−0.0472456
$449$	1177.00	0.123711	0.0618553	−	0.998085i	$-0.480298\pi$
$449$	1177.00	0.123711	0.0618553	+	0.998085i	$0.480298\pi$
$450$	0	0
$451$	−15085.0	−1.57500
$452$	1508.00	0.156926
$453$	−1192.00	−0.123631
$454$	3048.00	0.315088
$455$	0	0
$456$	184.000	0.0188960
$457$	−4345.00	−0.444750	−0.222375	−	0.974961i	$-0.571381\pi$
$457$	−4345.00	−0.444750	−0.222375	+	0.974961i	$0.571381\pi$
$458$	−6892.00	−0.703148
$459$	5671.00	0.576688
$460$	0	0
$461$	5282.00	0.533638	0.266819	−	0.963747i	$-0.414027\pi$
$461$	5282.00	0.533638	0.266819	+	0.963747i	$0.414027\pi$
$462$	490.000	0.0493439
$463$	−7976.00	−0.800596	−0.400298	−	0.916385i	$-0.631093\pi$
$463$	−7976.00	−0.800596	−0.400298	+	0.916385i	$0.631093\pi$
$464$	−2784.00	−0.278543
$465$	0	0
$466$	1396.00	0.138774
$467$	14012.0	1.38843	0.694216	−	0.719766i	$-0.255751\pi$
$467$	14012.0	1.38843	0.694216	+	0.719766i	$0.255751\pi$
$468$	6032.00	0.595789
$469$	4809.00	0.473473
$470$	0	0
$471$	−1772.00	−0.173353
$472$	608.000	0.0592912
$473$	5040.00	0.489935
$474$	804.000	0.0779092
$475$	0	0
$476$	2996.00	0.288490
$477$	2808.00	0.269538
$478$	−12248.0	−1.17199
$479$	1446.00	0.137932	0.0689660	−	0.997619i	$-0.478030\pi$
$479$	1446.00	0.137932	0.0689660	+	0.997619i	$0.478030\pi$
$480$	0	0
$481$	10672.0	1.01165
$482$	13950.0	1.31827
$483$	−1400.00	−0.131889
$484$	−424.000	−0.0398197
$485$	0	0
$486$	4160.00	0.388275
$487$	11142.0	1.03674	0.518370	−	0.855157i	$-0.326539\pi$
$487$	11142.0	1.03674	0.518370	+	0.855157i	$0.326539\pi$
$488$	944.000	0.0875674
$489$	2973.00	0.274936
$490$	0	0
$491$	−13692.0	−1.25848	−0.629238	−	0.777213i	$-0.716633\pi$
$491$	−13692.0	−1.25848	−0.629238	+	0.777213i	$0.716633\pi$
$492$	1724.00	0.157975
$493$	18618.0	1.70084
$494$	−2668.00	−0.242994
$495$	0	0
$496$	1216.00	0.110081
$497$	−3710.00	−0.334842
$498$	−1794.00	−0.161428
$499$	19924.0	1.78742	0.893708	−	0.448649i	$-0.148095\pi$
$499$	19924.0	1.78742	0.893708	+	0.448649i	$0.148095\pi$
$500$	0	0
$501$	4.00000	0.000356700 0
$502$	3794.00	0.337320
$503$	4970.00	0.440559	0.220280	−	0.975437i	$-0.429303\pi$
$503$	4970.00	0.440559	0.220280	+	0.975437i	$0.429303\pi$
$504$	1456.00	0.128681
$505$	0	0
$506$	−14000.0	−1.22999
$507$	1167.00	0.102225
$508$	3048.00	0.266207
$509$	−2926.00	−0.254799	−0.127399	−	0.991851i	$-0.540663\pi$
$509$	−2926.00	−0.254799	−0.127399	+	0.991851i	$0.540663\pi$
$510$	0	0
$511$	−2093.00	−0.181192
$512$	512.000	0.0441942
$513$	−1219.00	−0.104913
$514$	9308.00	0.798752
$515$	0	0
$516$	−576.000	−0.0491414
$517$	18410.0	1.56609
$518$	2576.00	0.218500
$519$	−3434.00	−0.290435
$520$	0	0
$521$	18111.0	1.52295	0.761475	−	0.648194i	$-0.224475\pi$
$521$	18111.0	1.52295	0.761475	+	0.648194i	$0.224475\pi$
$522$	9048.00	0.758659
$523$	−9431.00	−0.788506	−0.394253	−	0.919002i	$-0.628997\pi$
$523$	−9431.00	−0.788506	−0.394253	+	0.919002i	$0.628997\pi$
$524$	−6048.00	−0.504214
$525$	0	0
$526$	4380.00	0.363074
$527$	−8132.00	−0.672174
$528$	−560.000	−0.0461570
$529$	27833.0	2.28758
$530$	0	0
$531$	−1976.00	−0.161490
$532$	−644.000	−0.0524830
$533$	−24998.0	−2.03149
$534$	−1598.00	−0.129499
$535$	0	0
$536$	−5496.00	−0.442894
$537$	397.000	0.0319028
$538$	−15276.0	−1.22415
$539$	−1715.00	−0.137051
$540$	0	0
$541$	−17002.0	−1.35115	−0.675576	−	0.737290i	$-0.736105\pi$
$541$	−17002.0	−1.35115	−0.675576	+	0.737290i	$0.736105\pi$
$542$	−10932.0	−0.866365
$543$	650.000	0.0513705
$544$	−3424.00	−0.269858
$545$	0	0
$546$	812.000	0.0636454
$547$	−18061.0	−1.41176	−0.705880	−	0.708332i	$-0.749448\pi$
$547$	−18061.0	−1.41176	−0.705880	+	0.708332i	$0.749448\pi$
$548$	1676.00	0.130648
$549$	−3068.00	−0.238505
$550$	0	0
$551$	−4002.00	−0.309421
$552$	1600.00	0.123371
$553$	−2814.00	−0.216390
$554$	620.000	0.0475474
$555$	0	0
$556$	−9316.00	−0.710587
$557$	17348.0	1.31967	0.659837	−	0.751409i	$-0.270625\pi$
$557$	17348.0	1.31967	0.659837	+	0.751409i	$0.270625\pi$
$558$	−3952.00	−0.299823
$559$	8352.00	0.631936
$560$	0	0
$561$	3745.00	0.281843
$562$	−9892.00	−0.742471
$563$	23460.0	1.75617	0.878083	−	0.478509i	$-0.158823\pi$
$563$	23460.0	1.75617	0.878083	+	0.478509i	$0.158823\pi$
$564$	−2104.00	−0.157082
$565$	0	0
$566$	782.000	0.0580740
$567$	−4543.00	−0.336487
$568$	4240.00	0.313216
$569$	4509.00	0.332209	0.166105	−	0.986108i	$-0.446881\pi$
$569$	4509.00	0.332209	0.166105	+	0.986108i	$0.446881\pi$
$570$	0	0
$571$	5932.00	0.434757	0.217379	−	0.976087i	$-0.430249\pi$
$571$	5932.00	0.434757	0.217379	+	0.976087i	$0.430249\pi$
$572$	8120.00	0.593556
$573$	−1426.00	−0.103965
$574$	−6034.00	−0.438771
$575$	0	0
$576$	−1664.00	−0.120370
$577$	12631.0	0.911327	0.455663	−	0.890152i	$-0.349402\pi$
$577$	12631.0	0.911327	0.455663	+	0.890152i	$0.349402\pi$
$578$	13072.0	0.940698
$579$	−1357.00	−0.0974007
$580$	0	0
$581$	6279.00	0.448359
$582$	−3020.00	−0.215091
$583$	3780.00	0.268528
$584$	2392.00	0.169489
$585$	0	0
$586$	−8244.00	−0.581154
$587$	−1041.00	−0.0731970	−0.0365985	−	0.999330i	$-0.511652\pi$
$587$	−1041.00	−0.0731970	−0.0365985	+	0.999330i	$0.511652\pi$
$588$	196.000	0.0137464
$589$	1748.00	0.122284
$590$	0	0
$591$	4686.00	0.326153
$592$	−2944.00	−0.204388
$593$	12363.0	0.856134	0.428067	−	0.903747i	$-0.359195\pi$
$593$	12363.0	0.856134	0.428067	+	0.903747i	$0.359195\pi$
$594$	3710.00	0.256268
$595$	0	0
$596$	12432.0	0.854420
$597$	3890.00	0.266679
$598$	−23200.0	−1.58649
$599$	12824.0	0.874749	0.437374	−	0.899280i	$-0.355908\pi$
$599$	12824.0	0.874749	0.437374	+	0.899280i	$0.355908\pi$
$600$	0	0
$601$	41.0000	0.00278274	0.00139137	−	0.999999i	$-0.499557\pi$
$601$	41.0000	0.00278274	0.00139137	+	0.999999i	$0.499557\pi$
$602$	2016.00	0.136488
$603$	17862.0	1.20630
$604$	−4768.00	−0.321204
$605$	0	0
$606$	0	0
$607$	−19284.0	−1.28948	−0.644739	−	0.764403i	$-0.723034\pi$
$607$	−19284.0	−1.28948	−0.644739	+	0.764403i	$0.723034\pi$
$608$	736.000	0.0490933
$609$	1218.00	0.0810441
$610$	0	0
$611$	30508.0	2.02000
$612$	11128.0	0.735004
$613$	−11306.0	−0.744935	−0.372467	−	0.928045i	$-0.621488\pi$
$613$	−11306.0	−0.744935	−0.372467	+	0.928045i	$0.621488\pi$
$614$	−16558.0	−1.08832
$615$	0	0
$616$	1960.00	0.128199
$617$	−4222.00	−0.275480	−0.137740	−	0.990468i	$-0.543984\pi$
$617$	−4222.00	−0.275480	−0.137740	+	0.990468i	$0.543984\pi$
$618$	824.000	0.0536345
$619$	−28796.0	−1.86980	−0.934902	−	0.354905i	$-0.884513\pi$
$619$	−28796.0	−1.86980	−0.934902	+	0.354905i	$0.884513\pi$
$620$	0	0
$621$	−10600.0	−0.684965
$622$	12292.0	0.792386
$623$	5593.00	0.359677
$624$	−928.000	−0.0595348
$625$	0	0
$626$	−18380.0	−1.17350
$627$	−805.000	−0.0512737
$628$	−7088.00	−0.450385
$629$	19688.0	1.24803
$630$	0	0
$631$	−12696.0	−0.800982	−0.400491	−	0.916301i	$-0.631160\pi$
$631$	−12696.0	−0.800982	−0.400491	+	0.916301i	$0.631160\pi$
$632$	3216.00	0.202414
$633$	−4009.00	−0.251727
$634$	−7624.00	−0.477583
$635$	0	0
$636$	−432.000	−0.0269338
$637$	−2842.00	−0.176773
$638$	12180.0	0.755816
$639$	−13780.0	−0.853096
$640$	0	0
$641$	4002.00	0.246598	0.123299	−	0.992370i	$-0.460653\pi$
$641$	4002.00	0.246598	0.123299	+	0.992370i	$0.460653\pi$
$642$	1702.00	0.104630
$643$	2528.00	0.155046	0.0775230	−	0.996991i	$-0.475299\pi$
$643$	2528.00	0.155046	0.0775230	+	0.996991i	$0.475299\pi$
$644$	−5600.00	−0.342657
$645$	0	0
$646$	−4922.00	−0.299773
$647$	−5416.00	−0.329096	−0.164548	−	0.986369i	$-0.552616\pi$
$647$	−5416.00	−0.329096	−0.164548	+	0.986369i	$0.552616\pi$
$648$	5192.00	0.314755
$649$	−2660.00	−0.160885
$650$	0	0
$651$	−532.000	−0.0320288
$652$	11892.0	0.714305
$653$	3436.00	0.205913	0.102956	−	0.994686i	$-0.467170\pi$
$653$	3436.00	0.205913	0.102956	+	0.994686i	$0.467170\pi$
$654$	4316.00	0.258057
$655$	0	0
$656$	6896.00	0.410432
$657$	−7774.00	−0.461633
$658$	7364.00	0.436290
$659$	10159.0	0.600514	0.300257	−	0.953858i	$-0.402928\pi$
$659$	10159.0	0.600514	0.300257	+	0.953858i	$0.402928\pi$
$660$	0	0
$661$	−9010.00	−0.530179	−0.265090	−	0.964224i	$-0.585402\pi$
$661$	−9010.00	−0.530179	−0.265090	+	0.964224i	$0.585402\pi$
$662$	−2254.00	−0.132333
$663$	6206.00	0.363531
$664$	−7176.00	−0.419402
$665$	0	0
$666$	9568.00	0.556685
$667$	−34800.0	−2.02018
$668$	16.0000	0.000926734 0
$669$	−5154.00	−0.297855
$670$	0	0
$671$	−4130.00	−0.237611
$672$	−224.000	−0.0128586
$673$	−1538.00	−0.0880914	−0.0440457	−	0.999030i	$-0.514025\pi$
$673$	−1538.00	−0.0880914	−0.0440457	+	0.999030i	$0.514025\pi$
$674$	−4006.00	−0.228940
$675$	0	0
$676$	4668.00	0.265589
$677$	−18816.0	−1.06818	−0.534090	−	0.845428i	$-0.679346\pi$
$677$	−18816.0	−1.06818	−0.534090	+	0.845428i	$0.679346\pi$
$678$	754.000	0.0427097
$679$	10570.0	0.597407
$680$	0	0
$681$	1524.00	0.0857560
$682$	−5320.00	−0.298700
$683$	−4829.00	−0.270537	−0.135268	−	0.990809i	$-0.543190\pi$
$683$	−4829.00	−0.270537	−0.135268	+	0.990809i	$0.543190\pi$
$684$	−2392.00	−0.133714
$685$	0	0
$686$	−686.000	−0.0381802
$687$	−3446.00	−0.191373
$688$	−2304.00	−0.127673
$689$	6264.00	0.346356
$690$	0	0
$691$	−6221.00	−0.342486	−0.171243	−	0.985229i	$-0.554778\pi$
$691$	−6221.00	−0.342486	−0.171243	+	0.985229i	$0.554778\pi$
$692$	−13736.0	−0.754573
$693$	−6370.00	−0.349172
$694$	−21834.0	−1.19425
$695$	0	0
$696$	−1392.00	−0.0758098
$697$	−46117.0	−2.50618
$698$	−15824.0	−0.858091
$699$	698.000	0.0377694
$700$	0	0
$701$	2568.00	0.138362	0.0691812	−	0.997604i	$-0.477961\pi$
$701$	2568.00	0.138362	0.0691812	+	0.997604i	$0.477961\pi$
$702$	6148.00	0.330543
$703$	−4232.00	−0.227045
$704$	−2240.00	−0.119919
$705$	0	0
$706$	−3708.00	−0.197666
$707$	0	0
$708$	304.000	0.0161370
$709$	−34364.0	−1.82026	−0.910132	−	0.414319i	$-0.864020\pi$
$709$	−34364.0	−1.82026	−0.910132	+	0.414319i	$0.864020\pi$
$710$	0	0
$711$	−10452.0	−0.551309
$712$	−6392.00	−0.336447
$713$	15200.0	0.798379
$714$	1498.00	0.0785171
$715$	0	0
$716$	1588.00	0.0828860
$717$	−6124.00	−0.318975
$718$	−22132.0	−1.15036
$719$	19248.0	0.998372	0.499186	−	0.866495i	$-0.333632\pi$
$719$	19248.0	0.998372	0.499186	+	0.866495i	$0.333632\pi$
$720$	0	0
$721$	−2884.00	−0.148968
$722$	−12660.0	−0.652571
$723$	6975.00	0.358787
$724$	2600.00	0.133464
$725$	0	0
$726$	−212.000	−0.0108375
$727$	14262.0	0.727577	0.363788	−	0.931482i	$-0.381483\pi$
$727$	14262.0	0.727577	0.363788	+	0.931482i	$0.381483\pi$
$728$	3248.00	0.165356
$729$	−15443.0	−0.784586
$730$	0	0
$731$	15408.0	0.779597
$732$	472.000	0.0238328
$733$	−13680.0	−0.689335	−0.344667	−	0.938725i	$-0.612008\pi$
$733$	−13680.0	−0.689335	−0.344667	+	0.938725i	$0.612008\pi$
$734$	−12272.0	−0.617122
$735$	0	0
$736$	6400.00	0.320526
$737$	24045.0	1.20178
$738$	−22412.0	−1.11788
$739$	−32364.0	−1.61100	−0.805500	−	0.592596i	$-0.798103\pi$
$739$	−32364.0	−1.61100	−0.805500	+	0.592596i	$0.798103\pi$
$740$	0	0
$741$	−1334.00	−0.0661346
$742$	1512.00	0.0748076
$743$	−25632.0	−1.26561	−0.632804	−	0.774312i	$-0.718096\pi$
$743$	−25632.0	−1.26561	−0.632804	+	0.774312i	$0.718096\pi$
$744$	608.000	0.0299602
$745$	0	0
$746$	9176.00	0.450345
$747$	23322.0	1.14231
$748$	14980.0	0.732250
$749$	−5957.00	−0.290606
$750$	0	0
$751$	7990.00	0.388228	0.194114	−	0.980979i	$-0.437817\pi$
$751$	7990.00	0.388228	0.194114	+	0.980979i	$0.437817\pi$
$752$	−8416.00	−0.408112
$753$	1897.00	0.0918068
$754$	20184.0	0.974878
$755$	0	0
$756$	1484.00	0.0713923
$757$	−13214.0	−0.634440	−0.317220	−	0.948352i	$-0.602749\pi$
$757$	−13214.0	−0.634440	−0.317220	+	0.948352i	$0.602749\pi$
$758$	−12178.0	−0.583542
$759$	−7000.00	−0.334761
$760$	0	0
$761$	37967.0	1.80854	0.904272	−	0.426956i	$-0.140414\pi$
$761$	37967.0	1.80854	0.904272	+	0.426956i	$0.140414\pi$
$762$	1524.00	0.0724524
$763$	−15106.0	−0.716742
$764$	−5704.00	−0.270109
$765$	0	0
$766$	−5916.00	−0.279052
$767$	−4408.00	−0.207515
$768$	256.000	0.0120281
$769$	−9569.00	−0.448722	−0.224361	−	0.974506i	$-0.572029\pi$
$769$	−9569.00	−0.448722	−0.224361	+	0.974506i	$0.572029\pi$
$770$	0	0
$771$	4654.00	0.217393
$772$	−5428.00	−0.253054
$773$	27908.0	1.29855	0.649276	−	0.760553i	$-0.275072\pi$
$773$	27908.0	1.29855	0.649276	+	0.760553i	$0.275072\pi$
$774$	7488.00	0.347740
$775$	0	0
$776$	−12080.0	−0.558823
$777$	1288.00	0.0594681
$778$	10944.0	0.504321
$779$	9913.00	0.455931
$780$	0	0
$781$	−18550.0	−0.849899
$782$	−42800.0	−1.95719
$783$	9222.00	0.420903
$784$	784.000	0.0357143
$785$	0	0
$786$	−3024.00	−0.137230
$787$	8884.00	0.402389	0.201195	−	0.979551i	$-0.435518\pi$
$787$	8884.00	0.402389	0.201195	+	0.979551i	$0.435518\pi$
$788$	18744.0	0.847370
$789$	2190.00	0.0988163
$790$	0	0
$791$	−2639.00	−0.118625
$792$	7280.00	0.326621
$793$	−6844.00	−0.306479
$794$	6460.00	0.288737
$795$	0	0
$796$	15560.0	0.692851
$797$	−31866.0	−1.41625	−0.708125	−	0.706087i	$-0.750459\pi$
$797$	−31866.0	−1.41625	−0.708125	+	0.706087i	$0.750459\pi$
$798$	−322.000	−0.0142841
$799$	56282.0	2.49201
$800$	0	0
$801$	20774.0	0.916371
$802$	27834.0	1.22550
$803$	−10465.0	−0.459903
$804$	−2748.00	−0.120540
$805$	0	0
$806$	−8816.00	−0.385273
$807$	−7638.00	−0.333173
$808$	0	0
$809$	8854.00	0.384784	0.192392	−	0.981318i	$-0.438376\pi$
$809$	8854.00	0.384784	0.192392	+	0.981318i	$0.438376\pi$
$810$	0	0
$811$	16812.0	0.727927	0.363964	−	0.931413i	$-0.381423\pi$
$811$	16812.0	0.727927	0.363964	+	0.931413i	$0.381423\pi$
$812$	4872.00	0.210559
$813$	−5466.00	−0.235795
$814$	12880.0	0.554599
$815$	0	0
$816$	−1712.00	−0.0734461
$817$	−3312.00	−0.141826
$818$	24542.0	1.04901
$819$	−10556.0	−0.450374
$820$	0	0
$821$	20416.0	0.867872	0.433936	−	0.900944i	$-0.357124\pi$
$821$	20416.0	0.867872	0.433936	+	0.900944i	$0.357124\pi$
$822$	838.000	0.0355579
$823$	−23110.0	−0.978814	−0.489407	−	0.872055i	$-0.662787\pi$
$823$	−23110.0	−0.978814	−0.489407	+	0.872055i	$0.662787\pi$
$824$	3296.00	0.139347
$825$	0	0
$826$	−1064.00	−0.0448200
$827$	−7983.00	−0.335666	−0.167833	−	0.985815i	$-0.553677\pi$
$827$	−7983.00	−0.335666	−0.167833	+	0.985815i	$0.553677\pi$
$828$	−20800.0	−0.873007
$829$	−10156.0	−0.425492	−0.212746	−	0.977108i	$-0.568241\pi$
$829$	−10156.0	−0.425492	−0.212746	+	0.977108i	$0.568241\pi$
$830$	0	0
$831$	310.000	0.0129408
$832$	−3712.00	−0.154676
$833$	−5243.00	−0.218078
$834$	−4658.00	−0.193397
$835$	0	0
$836$	−3220.00	−0.133213
$837$	−4028.00	−0.166342
$838$	17958.0	0.740273
$839$	−19962.0	−0.821412	−0.410706	−	0.911768i	$-0.634718\pi$
$839$	−19962.0	−0.821412	−0.410706	+	0.911768i	$0.634718\pi$
$840$	0	0
$841$	5887.00	0.241379
$842$	12200.0	0.499335
$843$	−4946.00	−0.202075
$844$	−16036.0	−0.654007
$845$	0	0
$846$	27352.0	1.11156
$847$	742.000	0.0301009
$848$	−1728.00	−0.0699761
$849$	391.000	0.0158058
$850$	0	0
$851$	−36800.0	−1.48236
$852$	2120.00	0.0852465
$853$	−47422.0	−1.90352	−0.951758	−	0.306851i	$-0.900725\pi$
$853$	−47422.0	−1.90352	−0.951758	+	0.306851i	$0.900725\pi$
$854$	−1652.00	−0.0661947
$855$	0	0
$856$	6808.00	0.271837
$857$	17641.0	0.703156	0.351578	−	0.936159i	$-0.385645\pi$
$857$	17641.0	0.703156	0.351578	+	0.936159i	$0.385645\pi$
$858$	4060.00	0.161546
$859$	8831.00	0.350768	0.175384	−	0.984500i	$-0.443883\pi$
$859$	8831.00	0.350768	0.175384	+	0.984500i	$0.443883\pi$
$860$	0	0
$861$	−3017.00	−0.119418
$862$	648.000	0.0256044
$863$	−19908.0	−0.785256	−0.392628	−	0.919697i	$-0.628434\pi$
$863$	−19908.0	−0.785256	−0.392628	+	0.919697i	$0.628434\pi$
$864$	−1696.00	−0.0667814
$865$	0	0
$866$	−16326.0	−0.640624
$867$	6536.00	0.256026
$868$	−2128.00	−0.0832132
$869$	−14070.0	−0.549243
$870$	0	0
$871$	39846.0	1.55009
$872$	17264.0	0.670450
$873$	39260.0	1.52205
$874$	9200.00	0.356058
$875$	0	0
$876$	1196.00	0.0461291
$877$	41584.0	1.60113	0.800566	−	0.599245i	$-0.204533\pi$
$877$	41584.0	1.60113	0.800566	+	0.599245i	$0.204533\pi$
$878$	−27656.0	−1.06304
$879$	−4122.00	−0.158170
$880$	0	0
$881$	−21414.0	−0.818906	−0.409453	−	0.912331i	$-0.634280\pi$
$881$	−21414.0	−0.818906	−0.409453	+	0.912331i	$0.634280\pi$
$882$	−2548.00	−0.0972739
$883$	14975.0	0.570724	0.285362	−	0.958420i	$-0.407886\pi$
$883$	14975.0	0.570724	0.285362	+	0.958420i	$0.407886\pi$
$884$	24824.0	0.944481
$885$	0	0
$886$	29882.0	1.13308
$887$	11702.0	0.442970	0.221485	−	0.975164i	$-0.428910\pi$
$887$	11702.0	0.442970	0.221485	+	0.975164i	$0.428910\pi$
$888$	−1472.00	−0.0556273
$889$	−5334.00	−0.201234
$890$	0	0
$891$	−22715.0	−0.854075
$892$	−20616.0	−0.773851
$893$	−12098.0	−0.453353
$894$	6216.00	0.232544
$895$	0	0
$896$	−896.000	−0.0334077
$897$	−11600.0	−0.431787
$898$	2354.00	0.0874766
$899$	−13224.0	−0.490595
$900$	0	0
$901$	11556.0	0.427288
$902$	−30170.0	−1.11369
$903$	1008.00	0.0371474
$904$	3016.00	0.110963
$905$	0	0
$906$	−2384.00	−0.0874206
$907$	15252.0	0.558362	0.279181	−	0.960238i	$-0.409937\pi$
$907$	15252.0	0.558362	0.279181	+	0.960238i	$0.409937\pi$
$908$	6096.00	0.222801
$909$	0	0
$910$	0	0
$911$	8508.00	0.309421	0.154711	−	0.987960i	$-0.450556\pi$
$911$	8508.00	0.309421	0.154711	+	0.987960i	$0.450556\pi$
$912$	368.000	0.0133615
$913$	31395.0	1.13803
$914$	−8690.00	−0.314485
$915$	0	0
$916$	−13784.0	−0.497201
$917$	10584.0	0.381150
$918$	11342.0	0.407780
$919$	−22406.0	−0.804250	−0.402125	−	0.915585i	$-0.631728\pi$
$919$	−22406.0	−0.804250	−0.402125	+	0.915585i	$0.631728\pi$
$920$	0	0
$921$	−8279.00	−0.296202
$922$	10564.0	0.377339
$923$	−30740.0	−1.09623
$924$	980.000	0.0348914
$925$	0	0
$926$	−15952.0	−0.566107
$927$	−10712.0	−0.379534
$928$	−5568.00	−0.196960
$929$	49294.0	1.74089	0.870443	−	0.492269i	$-0.163832\pi$
$929$	49294.0	1.74089	0.870443	+	0.492269i	$0.163832\pi$
$930$	0	0
$931$	1127.00	0.0396734
$932$	2792.00	0.0981277
$933$	6146.00	0.215660
$934$	28024.0	0.981770
$935$	0	0
$936$	12064.0	0.421287
$937$	−24669.0	−0.860087	−0.430043	−	0.902808i	$-0.641502\pi$
$937$	−24669.0	−0.860087	−0.430043	+	0.902808i	$0.641502\pi$
$938$	9618.00	0.334796
$939$	−9190.00	−0.319387
$940$	0	0
$941$	−4224.00	−0.146332	−0.0731660	−	0.997320i	$-0.523310\pi$
$941$	−4224.00	−0.146332	−0.0731660	+	0.997320i	$0.523310\pi$
$942$	−3544.00	−0.122579
$943$	86200.0	2.97673
$944$	1216.00	0.0419252
$945$	0	0
$946$	10080.0	0.346437
$947$	39252.0	1.34690	0.673452	−	0.739231i	$-0.264811\pi$
$947$	39252.0	1.34690	0.673452	+	0.739231i	$0.264811\pi$
$948$	1608.00	0.0550901
$949$	−17342.0	−0.593198
$950$	0	0
$951$	−3812.00	−0.129982
$952$	5992.00	0.203994
$953$	33567.0	1.14097	0.570484	−	0.821309i	$-0.306756\pi$
$953$	33567.0	1.14097	0.570484	+	0.821309i	$0.306756\pi$
$954$	5616.00	0.190592
$955$	0	0
$956$	−24496.0	−0.828721
$957$	6090.00	0.205707
$958$	2892.00	0.0975327
$959$	−2933.00	−0.0987607
$960$	0	0
$961$	−24015.0	−0.806116
$962$	21344.0	0.715341
$963$	−22126.0	−0.740395
$964$	27900.0	0.932156
$965$	0	0
$966$	−2800.00	−0.0932593
$967$	56126.0	1.86648	0.933242	−	0.359248i	$-0.116967\pi$
$967$	56126.0	1.86648	0.933242	+	0.359248i	$0.116967\pi$
$968$	−848.000	−0.0281568
$969$	−2461.00	−0.0815879
$970$	0	0
$971$	6759.00	0.223385	0.111692	−	0.993743i	$-0.464373\pi$
$971$	6759.00	0.223385	0.111692	+	0.993743i	$0.464373\pi$
$972$	8320.00	0.274552
$973$	16303.0	0.537153
$974$	22284.0	0.733086
$975$	0	0
$976$	1888.00	0.0619195
$977$	35547.0	1.16402	0.582011	−	0.813181i	$-0.302266\pi$
$977$	35547.0	1.16402	0.582011	+	0.813181i	$0.302266\pi$
$978$	5946.00	0.194409
$979$	27965.0	0.912937
$980$	0	0
$981$	−56108.0	−1.82609
$982$	−27384.0	−0.889876
$983$	−35252.0	−1.14381	−0.571904	−	0.820320i	$-0.693795\pi$
$983$	−35252.0	−1.14381	−0.571904	+	0.820320i	$0.693795\pi$
$984$	3448.00	0.111706
$985$	0	0
$986$	37236.0	1.20267
$987$	3682.00	0.118743
$988$	−5336.00	−0.171823
$989$	−28800.0	−0.925972
$990$	0	0
$991$	5884.00	0.188609	0.0943045	−	0.995543i	$-0.469937\pi$
$991$	5884.00	0.188609	0.0943045	+	0.995543i	$0.469937\pi$
$992$	2432.00	0.0778388
$993$	−1127.00	−0.0360164
$994$	−7420.00	−0.236769
$995$	0	0
$996$	−3588.00	−0.114147
$997$	39874.0	1.26662	0.633311	−	0.773897i	$-0.281695\pi$
$997$	39874.0	1.26662	0.633311	+	0.773897i	$0.281695\pi$
$998$	39848.0	1.26389
$999$	9752.00	0.308848

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	350.4.a.s.1.1	yes	1
5.2	odd	4		350.4.c.i.99.2		2
5.3	odd	4		350.4.c.i.99.1		2
5.4	even	2		350.4.a.d.1.1	✓	1
7.6	odd	2		2450.4.a.bd.1.1		1
35.34	odd	2		2450.4.a.l.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
350.4.a.d.1.1	✓	1	5.4	even	2
350.4.a.s.1.1	yes	1	1.1	even	1	trivial
350.4.c.i.99.1		2	5.3	odd	4
350.4.c.i.99.2		2	5.2	odd	4
2450.4.a.l.1.1		1	35.34	odd	2
2450.4.a.bd.1.1		1	7.6	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:	\( N \)	\(=\)	\( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	350.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more