LMFDB - Embedded newform 350.4.c.i.99.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [350,4,Mod(99,350)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(350, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("350.99"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 350 = 2 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	350.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,-8,0,4,0,0,52,0,-70] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$20.6506685020$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			99.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	350.99
Dual form			350.4.c.i.99.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.00000i q^{2} +1.00000i q^{3} -4.00000 q^{4} +2.00000 q^{6} +7.00000i q^{7} +8.00000i q^{8} +26.0000 q^{9} -35.0000 q^{11} -4.00000i q^{12} -58.0000i q^{13} +14.0000 q^{14} +16.0000 q^{16} +107.000i q^{17} -52.0000i q^{18} -23.0000 q^{19} -7.00000 q^{21} +70.0000i q^{22} +200.000i q^{23} -8.00000 q^{24} -116.000 q^{26} +53.0000i q^{27} -28.0000i q^{28} +174.000 q^{29} +76.0000 q^{31} -32.0000i q^{32} -35.0000i q^{33} +214.000 q^{34} -104.000 q^{36} +184.000i q^{37} +46.0000i q^{38} +58.0000 q^{39} +431.000 q^{41} +14.0000i q^{42} -144.000i q^{43} +140.000 q^{44} +400.000 q^{46} +526.000i q^{47} +16.0000i q^{48} -49.0000 q^{49} -107.000 q^{51} +232.000i q^{52} -108.000i q^{53} +106.000 q^{54} -56.0000 q^{56} -23.0000i q^{57} -348.000i q^{58} -76.0000 q^{59} +118.000 q^{61} -152.000i q^{62} +182.000i q^{63} -64.0000 q^{64} -70.0000 q^{66} +687.000i q^{67} -428.000i q^{68} -200.000 q^{69} +530.000 q^{71} +208.000i q^{72} +299.000i q^{73} +368.000 q^{74} +92.0000 q^{76} -245.000i q^{77} -116.000i q^{78} -402.000 q^{79} +649.000 q^{81} -862.000i q^{82} -897.000i q^{83} +28.0000 q^{84} -288.000 q^{86} +174.000i q^{87} -280.000i q^{88} +799.000 q^{89} +406.000 q^{91} -800.000i q^{92} +76.0000i q^{93} +1052.00 q^{94} +32.0000 q^{96} +1510.00i q^{97} +98.0000i q^{98} -910.000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{4} + 4 q^{6} + 52 q^{9} - 70 q^{11} + 28 q^{14} + 32 q^{16} - 46 q^{19} - 14 q^{21} - 16 q^{24} - 232 q^{26} + 348 q^{29} + 152 q^{31} + 428 q^{34} - 208 q^{36} + 116 q^{39} + 862 q^{41} + 280 q^{44}+ \cdots - 1820 q^{99}+O(q^{100}) $ 2 * q - 8 * q^4 + 4 * q^6 + 52 * q^9 - 70 * q^11 + 28 * q^14 + 32 * q^16 - 46 * q^19 - 14 * q^21 - 16 * q^24 - 232 * q^26 + 348 * q^29 + 152 * q^31 + 428 * q^34 - 208 * q^36 + 116 * q^39 + 862 * q^41 + 280 * q^44 + 800 * q^46 - 98 * q^49 - 214 * q^51 + 212 * q^54 - 112 * q^56 - 152 * q^59 + 236 * q^61 - 128 * q^64 - 140 * q^66 - 400 * q^69 + 1060 * q^71 + 736 * q^74 + 184 * q^76 - 804 * q^79 + 1298 * q^81 + 56 * q^84 - 576 * q^86 + 1598 * q^89 + 812 * q^91 + 2104 * q^94 + 64 * q^96 - 1820 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/350\mathbb{Z}\right)^\times$.

$n$	$101$	$127$
$\chi(n)$	$1$	$-1$

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 $

$2$	− 2.00000i	− 0.707107i
$2$	− 2.00000i	− 0.707107i
$3$	1.00000i	0.192450i	0.995360	+	0.0962250i	$0.0306768\pi$
$3$	1.00000i	0.192450i	−0.995360	+	0.0962250i	$0.969323\pi$
$4$	−4.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	2.00000	0.136083
$7$	7.00000i	0.377964i
$7$	7.00000i	0.377964i
$8$	8.00000i	0.353553i
$9$	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	$0.659246\pi$
$12$	− 4.00000i	− 0.0962250i
$13$	− 58.0000i	− 1.23741i	−0.785624	−	0.618704i	$-0.787658\pi$
$13$	− 58.0000i	− 1.23741i	0.785624	−	0.618704i	$-0.212342\pi$
$14$	14.0000	0.267261
$15$	0	0
$16$	16.0000	0.250000
$17$	107.000i	1.52655i	0.646075	+	0.763274i	$0.276409\pi$
$17$	107.000i	1.52655i	−0.646075	+	0.763274i	$0.723591\pi$
$18$	− 52.0000i	− 0.680918i
$19$	−23.0000	−0.277714	−0.138857	−	0.990312i	$-0.544343\pi$
$19$	−23.0000	−0.277714	−0.138857	+	0.990312i	$0.544343\pi$
$20$	0	0
$21$	−7.00000	−0.0727393
$22$	70.0000i	0.678366i
$23$	200.000i	1.81317i	0.422025	+	0.906584i	$0.361320\pi$
$23$	200.000i	1.81317i	−0.422025	+	0.906584i	$0.638680\pi$
$24$	−8.00000	−0.0680414
$25$	0	0
$26$	−116.000	−0.874980
$27$	53.0000i	0.377772i
$28$	− 28.0000i	− 0.188982i
$29$	174.000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	174.000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	76.0000	0.440323	0.220161	−	0.975463i	$-0.429342\pi$
$31$	76.0000	0.440323	0.220161	+	0.975463i	$0.429342\pi$
$32$	− 32.0000i	− 0.176777i
$33$	− 35.0000i	− 0.184628i
$34$	214.000	1.07943
$35$	0	0
$36$	−104.000	−0.481481
$37$	184.000i	0.817552i	0.912635	+	0.408776i	$0.134044\pi$
$37$	184.000i	0.817552i	−0.912635	+	0.408776i	$0.865956\pi$
$38$	46.0000i	0.196373i
$39$	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	$0.193492\pi$
$42$	14.0000i	0.0514344i
$43$	− 144.000i	− 0.510693i	−0.966850	−	0.255346i	$-0.917810\pi$
$43$	− 144.000i	− 0.510693i	0.966850	−	0.255346i	$-0.0821895\pi$
$44$	140.000	0.479677
$45$	0	0
$46$	400.000	1.28210
$47$	526.000i	1.63245i	0.577737	+	0.816223i	$0.303936\pi$
$47$	526.000i	1.63245i	−0.577737	+	0.816223i	$0.696064\pi$
$48$	16.0000i	0.0481125i
$49$	−49.0000	−0.142857
$50$	0	0
$51$	−107.000	−0.293784
$52$	232.000i	0.618704i
$53$	− 108.000i	− 0.279905i	−0.990158	−	0.139952i	$-0.955305\pi$
$53$	− 108.000i	− 0.279905i	0.990158	−	0.139952i	$-0.0446949\pi$
$54$	106.000	0.267125
$55$	0	0
$56$	−56.0000	−0.133631
$57$	− 23.0000i	− 0.0534460i
$58$	− 348.000i	− 0.787839i
$59$	−76.0000	−0.167701	−0.0838505	−	0.996478i	$-0.526722\pi$
$59$	−76.0000	−0.167701	−0.0838505	+	0.996478i	$0.526722\pi$
$60$	0	0
$61$	118.000	0.247678	0.123839	−	0.992302i	$-0.460479\pi$
$61$	118.000	0.247678	0.123839	+	0.992302i	$0.460479\pi$
$62$	− 152.000i	− 0.311355i
$63$	182.000i	0.363966i
$64$	−64.0000	−0.125000
$65$	0	0
$66$	−70.0000	−0.130552
$67$	687.000i	1.25269i	0.779545	+	0.626346i	$0.215450\pi$
$67$	687.000i	1.25269i	−0.779545	+	0.626346i	$0.784550\pi$
$68$	− 428.000i	− 0.763274i
$69$	−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	$0.353931\pi$
$72$	208.000i	0.340459i
$73$	299.000i	0.479388i	0.970849	+	0.239694i	$0.0770471\pi$
$73$	299.000i	0.479388i	−0.970849	+	0.239694i	$0.922953\pi$
$74$	368.000	0.578096
$75$	0	0
$76$	92.0000	0.138857
$77$	− 245.000i	− 0.362602i
$78$	− 116.000i	− 0.168390i
$79$	−402.000	−0.572513	−0.286257	−	0.958153i	$-0.592411\pi$
$79$	−402.000	−0.572513	−0.286257	+	0.958153i	$0.592411\pi$
$80$	0	0
$81$	649.000	0.890261
$82$	− 862.000i	− 1.16088i
$83$	− 897.000i	− 1.18625i	−0.805111	−	0.593124i	$-0.797894\pi$
$83$	− 897.000i	− 1.18625i	0.805111	−	0.593124i	$-0.202106\pi$
$84$	28.0000	0.0363696
$85$	0	0
$86$	−288.000	−0.361114
$87$	174.000i	0.214423i
$88$	− 280.000i	− 0.339183i
$89$	799.000	0.951616	0.475808	−	0.879549i	$-0.342156\pi$
$89$	799.000	0.951616	0.475808	+	0.879549i	$0.342156\pi$
$90$	0	0
$91$	406.000	0.467696
$92$	− 800.000i	− 0.906584i
$93$	76.0000i	0.0847401i
$94$	1052.00	1.15431
$95$	0	0
$96$	32.0000	0.0340207
$97$	1510.00i	1.58059i	0.612726	+	0.790295i	$0.290073\pi$
$97$	1510.00i	1.58059i	−0.612726	+	0.790295i	$0.709927\pi$
$98$	98.0000i	0.101015i
$99$	−910.000	−0.923823
$100$	0	0
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	214.000i	0.207737i
$103$	412.000i	0.394132i	0.980390	+	0.197066i	$0.0631413\pi$
$103$	412.000i	0.394132i	−0.980390	+	0.197066i	$0.936859\pi$
$104$	464.000	0.437490
$105$	0	0
$106$	−216.000	−0.197922
$107$	− 851.000i	− 0.768872i	−0.923152	−	0.384436i	$-0.874396\pi$
$107$	− 851.000i	− 0.768872i	0.923152	−	0.384436i	$-0.125604\pi$
$108$	− 212.000i	− 0.188886i
$109$	−2158.00	−1.89632	−0.948160	−	0.317793i	$-0.897058\pi$
$109$	−2158.00	−1.89632	−0.948160	+	0.317793i	$0.897058\pi$
$110$	0	0
$111$	−184.000	−0.157338
$112$	112.000i	0.0944911i
$113$	377.000i	0.313851i	0.987610	+	0.156926i	$0.0501583\pi$
$113$	377.000i	0.313851i	−0.987610	+	0.156926i	$0.949842\pi$
$114$	−46.0000	−0.0377921
$115$	0	0
$116$	−696.000	−0.557086
$117$	− 1508.00i	− 1.19158i
$118$	152.000i	0.118582i
$119$	−749.000	−0.576981
$120$	0	0
$121$	−106.000	−0.0796394
$122$	− 236.000i	− 0.175135i
$123$	431.000i	0.315951i
$124$	−304.000	−0.220161
$125$	0	0
$126$	364.000	0.257363
$127$	− 762.000i	− 0.532414i	−0.963916	−	0.266207i	$-0.914230\pi$
$127$	− 762.000i	− 0.532414i	0.963916	−	0.266207i	$-0.0857705\pi$
$128$	128.000i	0.0883883i
$129$	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	$0.668218\pi$
$132$	140.000i	0.0923139i
$133$	− 161.000i	− 0.104966i
$134$	1374.00	0.885787
$135$	0	0
$136$	−856.000	−0.539716
$137$	− 419.000i	− 0.261296i	−0.991429	−	0.130648i	$-0.958294\pi$
$137$	− 419.000i	− 0.261296i	0.991429	−	0.130648i	$-0.0417058\pi$
$138$	400.000i	0.246741i
$139$	2329.00	1.42117	0.710587	−	0.703609i	$-0.248429\pi$
$139$	2329.00	1.42117	0.710587	+	0.703609i	$0.248429\pi$
$140$	0	0
$141$	−526.000	−0.314164
$142$	− 1060.00i	− 0.626431i
$143$	2030.00i	1.18711i
$144$	416.000	0.240741
$145$	0	0
$146$	598.000	0.338978
$147$	− 49.0000i	− 0.0274929i
$148$	− 736.000i	− 0.408776i
$149$	−3108.00	−1.70884	−0.854420	−	0.519582i	$-0.826088\pi$
$149$	−3108.00	−1.70884	−0.854420	+	0.519582i	$0.826088\pi$
$150$	0	0
$151$	−1192.00	−0.642408	−0.321204	−	0.947010i	$-0.604087\pi$
$151$	−1192.00	−0.642408	−0.321204	+	0.947010i	$0.604087\pi$
$152$	− 184.000i	− 0.0981866i
$153$	2782.00i	1.47001i
$154$	−490.000	−0.256398
$155$	0	0
$156$	−232.000	−0.119070
$157$	1772.00i	0.900771i	0.892834	+	0.450385i	$0.148713\pi$
$157$	1772.00i	0.900771i	−0.892834	+	0.450385i	$0.851287\pi$
$158$	804.000i	0.404828i
$159$	108.000	0.0538677
$160$	0	0
$161$	−1400.00	−0.685313
$162$	− 1298.00i	− 0.629509i
$163$	2973.00i	1.42861i	0.699835	+	0.714305i	$0.253257\pi$
$163$	2973.00i	1.42861i	−0.699835	+	0.714305i	$0.746743\pi$
$164$	−1724.00	−0.820865
$165$	0	0
$166$	−1794.00	−0.838804
$167$	− 4.00000i	− 0.00185347i	−1.00000		0.000926734i	$-0.999705\pi$
$167$	− 4.00000i	− 0.00185347i	1.00000		0.000926734i	$-0.000294989\pi$
$168$	− 56.0000i	− 0.0257172i
$169$	−1167.00	−0.531179
$170$	0	0
$171$	−598.000	−0.267428
$172$	576.000i	0.255346i
$173$	− 3434.00i	− 1.50915i	−0.656216	−	0.754573i	$-0.727844\pi$
$173$	− 3434.00i	− 1.50915i	0.656216	−	0.754573i	$-0.272156\pi$
$174$	348.000	0.151620
$175$	0	0
$176$	−560.000	−0.239839
$177$	− 76.0000i	− 0.0322741i
$178$	− 1598.00i	− 0.672894i
$179$	−397.000	−0.165772	−0.0828860	−	0.996559i	$-0.526414\pi$
$179$	−397.000	−0.165772	−0.0828860	+	0.996559i	$0.526414\pi$
$180$	0	0
$181$	650.000	0.266929	0.133464	−	0.991054i	$-0.457390\pi$
$181$	650.000	0.266929	0.133464	+	0.991054i	$0.457390\pi$
$182$	− 812.000i	− 0.330711i
$183$	118.000i	0.0476656i
$184$	−1600.00	−0.641052
$185$	0	0
$186$	152.000	0.0599203
$187$	− 3745.00i	− 1.46450i
$188$	− 2104.00i	− 0.816223i
$189$	−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	$0.587060\pi$
$192$	− 64.0000i	− 0.0240563i
$193$	− 1357.00i	− 0.506109i	−0.967452	−	0.253054i	$-0.918565\pi$
$193$	− 1357.00i	− 0.506109i	0.967452	−	0.253054i	$-0.0814352\pi$
$194$	3020.00	1.11765
$195$	0	0
$196$	196.000	0.0714286
$197$	− 4686.00i	− 1.69474i	−0.531003	−	0.847370i	$-0.678185\pi$
$197$	− 4686.00i	− 1.69474i	0.531003	−	0.847370i	$-0.321815\pi$
$198$	1820.00i	0.653241i
$199$	−3890.00	−1.38570	−0.692851	−	0.721081i	$-0.743646\pi$
$199$	−3890.00	−1.38570	−0.692851	+	0.721081i	$0.743646\pi$
$200$	0	0
$201$	−687.000	−0.241081
$202$	0	0
$203$	1218.00i	0.421117i
$204$	428.000	0.146892
$205$	0	0
$206$	824.000	0.278693
$207$	5200.00i	1.74601i
$208$	− 928.000i	− 0.309352i
$209$	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	$0.726913\pi$
$212$	432.000i	0.139952i
$213$	530.000i	0.170493i
$214$	−1702.00	−0.543674
$215$	0	0
$216$	−424.000	−0.133563
$217$	532.000i	0.166426i
$218$	4316.00i	1.34090i
$219$	−299.000	−0.0922582
$220$	0	0
$221$	6206.00	1.88896
$222$	368.000i	0.111255i
$223$	− 5154.00i	− 1.54770i	−0.633368	−	0.773851i	$-0.718328\pi$
$223$	− 5154.00i	− 1.54770i	0.633368	−	0.773851i	$-0.281672\pi$
$224$	224.000	0.0668153
$225$	0	0
$226$	754.000	0.221926
$227$	− 1524.00i	− 0.445601i	−0.974864	−	0.222801i	$-0.928480\pi$
$227$	− 1524.00i	− 0.445601i	0.974864	−	0.222801i	$-0.0715199\pi$
$228$	92.0000i	0.0267230i
$229$	3446.00	0.994402	0.497201	−	0.867635i	$-0.334361\pi$
$229$	3446.00	0.994402	0.497201	+	0.867635i	$0.334361\pi$
$230$	0	0
$231$	245.000	0.0697828
$232$	1392.00i	0.393919i
$233$	698.000i	0.196255i	0.995174	+	0.0981277i	$0.0312854\pi$
$233$	698.000i	0.196255i	−0.995174	+	0.0981277i	$0.968715\pi$
$234$	−3016.00	−0.842573
$235$	0	0
$236$	304.000	0.0838505
$237$	− 402.000i	− 0.110180i
$238$	1498.00i	0.407987i
$239$	6124.00	1.65744	0.828721	−	0.559662i	$-0.189069\pi$
$239$	6124.00	1.65744	0.828721	+	0.559662i	$0.189069\pi$
$240$	0	0
$241$	6975.00	1.86431	0.932156	−	0.362057i	$-0.117925\pi$
$241$	6975.00	1.86431	0.932156	+	0.362057i	$0.117925\pi$
$242$	212.000i	0.0563135i
$243$	2080.00i	0.549103i
$244$	−472.000	−0.123839
$245$	0	0
$246$	862.000	0.223411
$247$	1334.00i	0.343645i
$248$	608.000i	0.155678i
$249$	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	$0.423337\pi$
$252$	− 728.000i	− 0.181983i
$253$	− 7000.00i	− 1.73947i
$254$	−1524.00	−0.376473
$255$	0	0
$256$	256.000	0.0625000
$257$	− 4654.00i	− 1.12961i	−0.825226	−	0.564803i	$-0.808952\pi$
$257$	− 4654.00i	− 1.12961i	0.825226	−	0.564803i	$-0.191048\pi$
$258$	− 288.000i	− 0.0694965i
$259$	−1288.00	−0.309006
$260$	0	0
$261$	4524.00	1.07291
$262$	3024.00i	0.713066i
$263$	2190.00i	0.513465i	0.966483	+	0.256732i	$0.0826459\pi$
$263$	2190.00i	0.513465i	−0.966483	+	0.256732i	$0.917354\pi$
$264$	280.000	0.0652758
$265$	0	0
$266$	−322.000	−0.0742221
$267$	799.000i	0.183139i
$268$	− 2748.00i	− 0.626346i
$269$	7638.00	1.73122	0.865608	−	0.500722i	$-0.166932\pi$
$269$	7638.00	1.73122	0.865608	+	0.500722i	$0.166932\pi$
$270$	0	0
$271$	−5466.00	−1.22522	−0.612612	−	0.790384i	$-0.709881\pi$
$271$	−5466.00	−1.22522	−0.612612	+	0.790384i	$0.709881\pi$
$272$	1712.00i	0.381637i
$273$	406.000i	0.0900082i
$274$	−838.000	−0.184764
$275$	0	0
$276$	800.000	0.174472
$277$	− 310.000i	− 0.0672422i	−0.999435	−	0.0336211i	$-0.989296\pi$
$277$	− 310.000i	− 0.0672422i	0.999435	−	0.0336211i	$-0.0107039\pi$
$278$	− 4658.00i	− 1.00492i
$279$	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	$0.675937\pi$
$282$	1052.00i	0.222148i
$283$	391.000i	0.0821291i	0.999156	+	0.0410646i	$0.0130749\pi$
$283$	391.000i	0.0821291i	−0.999156	+	0.0410646i	$0.986925\pi$
$284$	−2120.00	−0.442954
$285$	0	0
$286$	4060.00	0.839415
$287$	3017.00i	0.620515i
$288$	− 832.000i	− 0.170229i
$289$	−6536.00	−1.33035
$290$	0	0
$291$	−1510.00	−0.304185
$292$	− 1196.00i	− 0.239694i
$293$	− 4122.00i	− 0.821876i	−0.911663	−	0.410938i	$-0.865201\pi$
$293$	− 4122.00i	− 0.821876i	0.911663	−	0.410938i	$-0.134799\pi$
$294$	−98.0000	−0.0194404
$295$	0	0
$296$	−1472.00	−0.289048
$297$	− 1855.00i	− 0.362418i
$298$	6216.00i	1.20833i
$299$	11600.0	2.24363
$300$	0	0
$301$	1008.00	0.193024
$302$	2384.00i	0.454251i
$303$	0	0
$304$	−368.000	−0.0694284
$305$	0	0
$306$	5564.00	1.03945
$307$	8279.00i	1.53911i	0.638579	+	0.769556i	$0.279522\pi$
$307$	8279.00i	1.53911i	−0.638579	+	0.769556i	$0.720478\pi$
$308$	980.000i	0.181301i
$309$	−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	$0.310685\pi$
$312$	464.000i	0.0841950i
$313$	− 9190.00i	− 1.65958i	−0.558073	−	0.829792i	$-0.688459\pi$
$313$	− 9190.00i	− 1.65958i	0.558073	−	0.829792i	$-0.311541\pi$
$314$	3544.00	0.636941
$315$	0	0
$316$	1608.00	0.286257
$317$	3812.00i	0.675405i	0.941253	+	0.337702i	$0.109650\pi$
$317$	3812.00i	0.675405i	−0.941253	+	0.337702i	$0.890350\pi$
$318$	− 216.000i	− 0.0380902i
$319$	−6090.00	−1.06889
$320$	0	0
$321$	851.000	0.147969
$322$	2800.00i	0.484590i
$323$	− 2461.00i	− 0.423943i
$324$	−2596.00	−0.445130
$325$	0	0
$326$	5946.00	1.01018
$327$	− 2158.00i	− 0.364947i
$328$	3448.00i	0.580439i
$329$	−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	$0.529829\pi$
$332$	3588.00i	0.593124i
$333$	4784.00i	0.787272i
$334$	−8.00000	−0.00131060
$335$	0	0
$336$	−112.000	−0.0181848
$337$	2003.00i	0.323770i	0.986810	+	0.161885i	$0.0517573\pi$
$337$	2003.00i	0.323770i	−0.986810	+	0.161885i	$0.948243\pi$
$338$	2334.00i	0.375600i
$339$	−377.000	−0.0604007
$340$	0	0
$341$	−2660.00	−0.422425
$342$	1196.00i	0.189100i
$343$	− 343.000i	− 0.0539949i
$344$	1152.00	0.180557
$345$	0	0
$346$	−6868.00	−1.06713
$347$	10917.0i	1.68892i	0.535619	+	0.844460i	$0.320078\pi$
$347$	10917.0i	1.68892i	−0.535619	+	0.844460i	$0.679922\pi$
$348$	− 696.000i	− 0.107211i
$349$	7912.00	1.21352	0.606762	−	0.794884i	$-0.292468\pi$
$349$	7912.00	1.21352	0.606762	+	0.794884i	$0.292468\pi$
$350$	0	0
$351$	3074.00	0.467459
$352$	1120.00i	0.169591i
$353$	− 1854.00i	− 0.279542i	−0.990184	−	0.139771i	$-0.955363\pi$
$353$	− 1854.00i	− 0.279542i	0.990184	−	0.139771i	$-0.0446367\pi$
$354$	−152.000	−0.0228212
$355$	0	0
$356$	−3196.00	−0.475808
$357$	− 749.000i	− 0.111040i
$358$	794.000i	0.117218i
$359$	11066.0	1.62686	0.813428	−	0.581666i	$-0.197599\pi$
$359$	11066.0	1.62686	0.813428	+	0.581666i	$0.197599\pi$
$360$	0	0
$361$	−6330.00	−0.922875
$362$	− 1300.00i	− 0.188747i
$363$	− 106.000i	− 0.0153266i
$364$	−1624.00	−0.233848
$365$	0	0
$366$	236.000	0.0337047
$367$	6136.00i	0.872743i	0.899767	+	0.436371i	$0.143737\pi$
$367$	6136.00i	0.872743i	−0.899767	+	0.436371i	$0.856263\pi$
$368$	3200.00i	0.453292i
$369$	11206.0	1.58092
$370$	0	0
$371$	756.000	0.105794
$372$	− 304.000i	− 0.0423701i
$373$	4588.00i	0.636884i	0.947942	+	0.318442i	$0.103160\pi$
$373$	4588.00i	0.636884i	−0.947942	+	0.318442i	$0.896840\pi$
$374$	−7490.00	−1.03556
$375$	0	0
$376$	−4208.00	−0.577157
$377$	− 10092.0i	− 1.37869i
$378$	742.000i	0.100964i
$379$	6089.00	0.825253	0.412627	−	0.910900i	$-0.364611\pi$
$379$	6089.00	0.825253	0.412627	+	0.910900i	$0.364611\pi$
$380$	0	0
$381$	762.000	0.102463
$382$	2852.00i	0.381992i
$383$	− 2958.00i	− 0.394639i	−0.980339	−	0.197320i	$-0.936776\pi$
$383$	− 2958.00i	− 0.394639i	0.980339	−	0.197320i	$-0.0632236\pi$
$384$	−128.000	−0.0170103
$385$	0	0
$386$	−2714.00	−0.357873
$387$	− 3744.00i	− 0.491778i
$388$	− 6040.00i	− 0.790295i
$389$	−5472.00	−0.713217	−0.356609	−	0.934254i	$-0.616067\pi$
$389$	−5472.00	−0.713217	−0.356609	+	0.934254i	$0.616067\pi$
$390$	0	0
$391$	−21400.0	−2.76789
$392$	− 392.000i	− 0.0505076i
$393$	− 1512.00i	− 0.194072i
$394$	−9372.00	−1.19836
$395$	0	0
$396$	3640.00	0.461911
$397$	− 3230.00i	− 0.408335i	−0.978936	−	0.204168i	$-0.934551\pi$
$397$	− 3230.00i	− 0.408335i	0.978936	−	0.204168i	$-0.0654487\pi$
$398$	7780.00i	0.979840i
$399$	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	$0.166325\pi$
$402$	1374.00i	0.170470i
$403$	− 4408.00i	− 0.544859i
$404$	0	0
$405$	0	0
$406$	2436.00	0.297775
$407$	− 6440.00i	− 0.784322i
$408$	− 856.000i	− 0.103868i
$409$	−12271.0	−1.48353	−0.741763	−	0.670662i	$-0.766010\pi$
$409$	−12271.0	−1.48353	−0.741763	+	0.670662i	$0.766010\pi$
$410$	0	0
$411$	419.000	0.0502865
$412$	− 1648.00i	− 0.197066i
$413$	− 532.000i	− 0.0633850i
$414$	10400.0	1.23462
$415$	0	0
$416$	−1856.00	−0.218745
$417$	2329.00i	0.273505i
$418$	− 1610.00i	− 0.188392i
$419$	−8979.00	−1.04690	−0.523452	−	0.852055i	$-0.675356\pi$
$419$	−8979.00	−1.04690	−0.523452	+	0.852055i	$0.675356\pi$
$420$	0	0
$421$	6100.00	0.706166	0.353083	−	0.935592i	$-0.385133\pi$
$421$	6100.00	0.706166	0.353083	+	0.935592i	$0.385133\pi$
$422$	8018.00i	0.924906i
$423$	13676.0i	1.57199i
$424$	864.000	0.0989612
$425$	0	0
$426$	1060.00	0.120557
$427$	826.000i	0.0936134i
$428$	3404.00i	0.384436i
$429$	−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	$0.494237\pi$
$432$	848.000i	0.0944431i
$433$	− 8163.00i	− 0.905979i	−0.891516	−	0.452989i	$-0.850358\pi$
$433$	− 8163.00i	− 0.905979i	0.891516	−	0.452989i	$-0.149642\pi$
$434$	1064.00	0.117681
$435$	0	0
$436$	8632.00	0.948160
$437$	− 4600.00i	− 0.503542i
$438$	598.000i	0.0652364i
$439$	13828.0	1.50336	0.751679	−	0.659529i	$-0.229244\pi$
$439$	13828.0	1.50336	0.751679	+	0.659529i	$0.229244\pi$
$440$	0	0
$441$	−1274.00	−0.137566
$442$	− 12412.0i	− 1.33570i
$443$	14941.0i	1.60241i	0.598389	+	0.801206i	$0.295808\pi$
$443$	14941.0i	1.60241i	−0.598389	+	0.801206i	$0.704192\pi$
$444$	736.000	0.0786690
$445$	0	0
$446$	−10308.0	−1.09439
$447$	− 3108.00i	− 0.328867i
$448$	− 448.000i	− 0.0472456i
$449$	−1177.00	−0.123711	−0.0618553	−	0.998085i	$-0.519702\pi$
$449$	−1177.00	−0.123711	−0.0618553	+	0.998085i	$0.519702\pi$
$450$	0	0
$451$	−15085.0	−1.57500
$452$	− 1508.00i	− 0.156926i
$453$	− 1192.00i	− 0.123631i
$454$	−3048.00	−0.315088
$455$	0	0
$456$	184.000	0.0188960
$457$	4345.00i	0.444750i	0.974961	+	0.222375i	$0.0713808\pi$
$457$	4345.00i	0.444750i	−0.974961	+	0.222375i	$0.928619\pi$
$458$	− 6892.00i	− 0.703148i
$459$	−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	$0.414027\pi$
$462$	− 490.000i	− 0.0493439i
$463$	− 7976.00i	− 0.800596i	−0.916385	−	0.400298i	$-0.868907\pi$
$463$	− 7976.00i	− 0.800596i	0.916385	−	0.400298i	$-0.131093\pi$
$464$	2784.00	0.278543
$465$	0	0
$466$	1396.00	0.138774
$467$	− 14012.0i	− 1.38843i	−0.719766	−	0.694216i	$-0.755751\pi$
$467$	− 14012.0i	− 1.38843i	0.719766	−	0.694216i	$-0.244249\pi$
$468$	6032.00i	0.595789i
$469$	−4809.00	−0.473473
$470$	0	0
$471$	−1772.00	−0.173353
$472$	− 608.000i	− 0.0592912i
$473$	5040.00i	0.489935i
$474$	−804.000	−0.0779092
$475$	0	0
$476$	2996.00	0.288490
$477$	− 2808.00i	− 0.269538i
$478$	− 12248.0i	− 1.17199i
$479$	−1446.00	−0.137932	−0.0689660	−	0.997619i	$-0.521970\pi$
$479$	−1446.00	−0.137932	−0.0689660	+	0.997619i	$0.521970\pi$
$480$	0	0
$481$	10672.0	1.01165
$482$	− 13950.0i	− 1.31827i
$483$	− 1400.00i	− 0.131889i
$484$	424.000	0.0398197
$485$	0	0
$486$	4160.00	0.388275
$487$	− 11142.0i	− 1.03674i	−0.855157	−	0.518370i	$-0.826539\pi$
$487$	− 11142.0i	− 1.03674i	0.855157	−	0.518370i	$-0.173461\pi$
$488$	944.000i	0.0875674i
$489$	−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	$0.716633\pi$
$492$	− 1724.00i	− 0.157975i
$493$	18618.0i	1.70084i
$494$	2668.00	0.242994
$495$	0	0
$496$	1216.00	0.110081
$497$	3710.00i	0.334842i
$498$	− 1794.00i	− 0.161428i
$499$	−19924.0	−1.78742	−0.893708	−	0.448649i	$-0.851905\pi$
$499$	−19924.0	−1.78742	−0.893708	+	0.448649i	$0.851905\pi$
$500$	0	0
$501$	4.00000	0.000356700 0
$502$	− 3794.00i	− 0.337320i
$503$	4970.00i	0.440559i	0.975437	+	0.220280i	$0.0706970\pi$
$503$	4970.00i	0.440559i	−0.975437	+	0.220280i	$0.929303\pi$
$504$	−1456.00	−0.128681
$505$	0	0
$506$	−14000.0	−1.22999
$507$	− 1167.00i	− 0.102225i
$508$	3048.00i	0.266207i
$509$	2926.00	0.254799	0.127399	−	0.991851i	$-0.459337\pi$
$509$	2926.00	0.254799	0.127399	+	0.991851i	$0.459337\pi$
$510$	0	0
$511$	−2093.00	−0.181192
$512$	− 512.000i	− 0.0441942i
$513$	− 1219.00i	− 0.104913i
$514$	−9308.00	−0.798752
$515$	0	0
$516$	−576.000	−0.0491414
$517$	− 18410.0i	− 1.56609i
$518$	2576.00i	0.218500i
$519$	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	$0.224475\pi$
$522$	− 9048.00i	− 0.758659i
$523$	− 9431.00i	− 0.788506i	−0.919002	−	0.394253i	$-0.871003\pi$
$523$	− 9431.00i	− 0.788506i	0.919002	−	0.394253i	$-0.128997\pi$
$524$	6048.00	0.504214
$525$	0	0
$526$	4380.00	0.363074
$527$	8132.00i	0.672174i
$528$	− 560.000i	− 0.0461570i
$529$	−27833.0	−2.28758
$530$	0	0
$531$	−1976.00	−0.161490
$532$	644.000i	0.0524830i
$533$	− 24998.0i	− 2.03149i
$534$	1598.00	0.129499
$535$	0	0
$536$	−5496.00	−0.442894
$537$	− 397.000i	− 0.0319028i
$538$	− 15276.0i	− 1.22415i
$539$	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	$0.736105\pi$
$542$	10932.0i	0.866365i
$543$	650.000i	0.0513705i
$544$	3424.00	0.269858
$545$	0	0
$546$	812.000	0.0636454
$547$	18061.0i	1.41176i	0.708332	+	0.705880i	$0.249448\pi$
$547$	18061.0i	1.41176i	−0.708332	+	0.705880i	$0.750552\pi$
$548$	1676.00i	0.130648i
$549$	3068.00	0.238505
$550$	0	0
$551$	−4002.00	−0.309421
$552$	− 1600.00i	− 0.123371i
$553$	− 2814.00i	− 0.216390i
$554$	−620.000	−0.0475474
$555$	0	0
$556$	−9316.00	−0.710587
$557$	− 17348.0i	− 1.31967i	−0.751409	−	0.659837i	$-0.770625\pi$
$557$	− 17348.0i	− 1.31967i	0.751409	−	0.659837i	$-0.229375\pi$
$558$	− 3952.00i	− 0.299823i
$559$	−8352.00	−0.631936
$560$	0	0
$561$	3745.00	0.281843
$562$	9892.00i	0.742471i
$563$	23460.0i	1.75617i	0.478509	+	0.878083i	$0.341177\pi$
$563$	23460.0i	1.75617i	−0.478509	+	0.878083i	$0.658823\pi$
$564$	2104.00	0.157082
$565$	0	0
$566$	782.000	0.0580740
$567$	4543.00i	0.336487i
$568$	4240.00i	0.313216i
$569$	−4509.00	−0.332209	−0.166105	−	0.986108i	$-0.553119\pi$
$569$	−4509.00	−0.332209	−0.166105	+	0.986108i	$0.553119\pi$
$570$	0	0
$571$	5932.00	0.434757	0.217379	−	0.976087i	$-0.430249\pi$
$571$	5932.00	0.434757	0.217379	+	0.976087i	$0.430249\pi$
$572$	− 8120.00i	− 0.593556i
$573$	− 1426.00i	− 0.103965i
$574$	6034.00	0.438771
$575$	0	0
$576$	−1664.00	−0.120370
$577$	− 12631.0i	− 0.911327i	−0.890152	−	0.455663i	$-0.849402\pi$
$577$	− 12631.0i	− 0.911327i	0.890152	−	0.455663i	$-0.150598\pi$
$578$	13072.0i	0.940698i
$579$	1357.00	0.0974007
$580$	0	0
$581$	6279.00	0.448359
$582$	3020.00i	0.215091i
$583$	3780.00i	0.268528i
$584$	−2392.00	−0.169489
$585$	0	0
$586$	−8244.00	−0.581154
$587$	1041.00i	0.0731970i	0.999330	+	0.0365985i	$0.0116523\pi$
$587$	1041.00i	0.0731970i	−0.999330	+	0.0365985i	$0.988348\pi$
$588$	196.000i	0.0137464i
$589$	−1748.00	−0.122284
$590$	0	0
$591$	4686.00	0.326153
$592$	2944.00i	0.204388i
$593$	12363.0i	0.856134i	0.903747	+	0.428067i	$0.140805\pi$
$593$	12363.0i	0.856134i	−0.903747	+	0.428067i	$0.859195\pi$
$594$	−3710.00	−0.256268
$595$	0	0
$596$	12432.0	0.854420
$597$	− 3890.00i	− 0.266679i
$598$	− 23200.0i	− 1.58649i
$599$	−12824.0	−0.874749	−0.437374	−	0.899280i	$-0.644092\pi$
$599$	−12824.0	−0.874749	−0.437374	+	0.899280i	$0.644092\pi$
$600$	0	0
$601$	41.0000	0.00278274	0.00139137	−	0.999999i	$-0.499557\pi$
$601$	41.0000	0.00278274	0.00139137	+	0.999999i	$0.499557\pi$
$602$	− 2016.00i	− 0.136488i
$603$	17862.0i	1.20630i
$604$	4768.00	0.321204
$605$	0	0
$606$	0	0
$607$	19284.0i	1.28948i	0.764403	+	0.644739i	$0.223034\pi$
$607$	19284.0i	1.28948i	−0.764403	+	0.644739i	$0.776966\pi$
$608$	736.000i	0.0490933i
$609$	−1218.00	−0.0810441
$610$	0	0
$611$	30508.0	2.02000
$612$	− 11128.0i	− 0.735004i
$613$	− 11306.0i	− 0.744935i	−0.928045	−	0.372467i	$-0.878512\pi$
$613$	− 11306.0i	− 0.744935i	0.928045	−	0.372467i	$-0.121488\pi$
$614$	16558.0	1.08832
$615$	0	0
$616$	1960.00	0.128199
$617$	4222.00i	0.275480i	0.990468	+	0.137740i	$0.0439839\pi$
$617$	4222.00i	0.275480i	−0.990468	+	0.137740i	$0.956016\pi$
$618$	824.000i	0.0536345i
$619$	28796.0	1.86980	0.934902	−	0.354905i	$-0.115487\pi$
$619$	28796.0	1.86980	0.934902	+	0.354905i	$0.115487\pi$
$620$	0	0
$621$	−10600.0	−0.684965
$622$	− 12292.0i	− 0.792386i
$623$	5593.00i	0.359677i
$624$	928.000	0.0595348
$625$	0	0
$626$	−18380.0	−1.17350
$627$	805.000i	0.0512737i
$628$	− 7088.00i	− 0.450385i
$629$	−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	$0.631160\pi$
$632$	− 3216.00i	− 0.202414i
$633$	− 4009.00i	− 0.251727i
$634$	7624.00	0.477583
$635$	0	0
$636$	−432.000	−0.0269338
$637$	2842.00i	0.176773i
$638$	12180.0i	0.755816i
$639$	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	$0.460653\pi$
$642$	− 1702.00i	− 0.104630i
$643$	2528.00i	0.155046i	0.996991	+	0.0775230i	$0.0247011\pi$
$643$	2528.00i	0.155046i	−0.996991	+	0.0775230i	$0.975299\pi$
$644$	5600.00	0.342657
$645$	0	0
$646$	−4922.00	−0.299773
$647$	5416.00i	0.329096i	0.986369	+	0.164548i	$0.0526165\pi$
$647$	5416.00i	0.329096i	−0.986369	+	0.164548i	$0.947384\pi$
$648$	5192.00i	0.314755i
$649$	2660.00	0.160885
$650$	0	0
$651$	−532.000	−0.0320288
$652$	− 11892.0i	− 0.714305i
$653$	3436.00i	0.205913i	0.994686	+	0.102956i	$0.0328302\pi$
$653$	3436.00i	0.205913i	−0.994686	+	0.102956i	$0.967170\pi$
$654$	−4316.00	−0.258057
$655$	0	0
$656$	6896.00	0.410432
$657$	7774.00i	0.461633i
$658$	7364.00i	0.436290i
$659$	−10159.0	−0.600514	−0.300257	−	0.953858i	$-0.597072\pi$
$659$	−10159.0	−0.600514	−0.300257	+	0.953858i	$0.597072\pi$
$660$	0	0
$661$	−9010.00	−0.530179	−0.265090	−	0.964224i	$-0.585402\pi$
$661$	−9010.00	−0.530179	−0.265090	+	0.964224i	$0.585402\pi$
$662$	2254.00i	0.132333i
$663$	6206.00i	0.363531i
$664$	7176.00	0.419402
$665$	0	0
$666$	9568.00	0.556685
$667$	34800.0i	2.02018i
$668$	16.0000i	0 0.000926734i
$669$	5154.00	0.297855
$670$	0	0
$671$	−4130.00	−0.237611
$672$	224.000i	0.0128586i
$673$	− 1538.00i	− 0.0880914i	−0.999030	−	0.0440457i	$-0.985975\pi$
$673$	− 1538.00i	− 0.0880914i	0.999030	−	0.0440457i	$-0.0140247\pi$
$674$	4006.00	0.228940
$675$	0	0
$676$	4668.00	0.265589
$677$	18816.0i	1.06818i	0.845428	+	0.534090i	$0.179346\pi$
$677$	18816.0i	1.06818i	−0.845428	+	0.534090i	$0.820654\pi$
$678$	754.000i	0.0427097i
$679$	−10570.0	−0.597407
$680$	0	0
$681$	1524.00	0.0857560
$682$	5320.00i	0.298700i
$683$	− 4829.00i	− 0.270537i	−0.990809	−	0.135268i	$-0.956810\pi$
$683$	− 4829.00i	− 0.270537i	0.990809	−	0.135268i	$-0.0431896\pi$
$684$	2392.00	0.133714
$685$	0	0
$686$	−686.000	−0.0381802
$687$	3446.00i	0.191373i
$688$	− 2304.00i	− 0.127673i
$689$	−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	$0.554778\pi$
$692$	13736.0i	0.754573i
$693$	− 6370.00i	− 0.349172i
$694$	21834.0	1.19425
$695$	0	0
$696$	−1392.00	−0.0758098
$697$	46117.0i	2.50618i
$698$	− 15824.0i	− 0.858091i
$699$	−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	$0.477961\pi$
$702$	− 6148.00i	− 0.330543i
$703$	− 4232.00i	− 0.227045i
$704$	2240.00	0.119919
$705$	0	0
$706$	−3708.00	−0.197666
$707$	0	0
$708$	304.000i	0.0161370i
$709$	34364.0	1.82026	0.910132	−	0.414319i	$-0.135980\pi$
$709$	34364.0	1.82026	0.910132	+	0.414319i	$0.135980\pi$
$710$	0	0
$711$	−10452.0	−0.551309
$712$	6392.00i	0.336447i
$713$	15200.0i	0.798379i
$714$	−1498.00	−0.0785171
$715$	0	0
$716$	1588.00	0.0828860
$717$	6124.00i	0.318975i
$718$	− 22132.0i	− 1.15036i
$719$	−19248.0	−0.998372	−0.499186	−	0.866495i	$-0.666368\pi$
$719$	−19248.0	−0.998372	−0.499186	+	0.866495i	$0.666368\pi$
$720$	0	0
$721$	−2884.00	−0.148968
$722$	12660.0i	0.652571i
$723$	6975.00i	0.358787i
$724$	−2600.00	−0.133464
$725$	0	0
$726$	−212.000	−0.0108375
$727$	− 14262.0i	− 0.727577i	−0.931482	−	0.363788i	$-0.881483\pi$
$727$	− 14262.0i	− 0.727577i	0.931482	−	0.363788i	$-0.118517\pi$
$728$	3248.00i	0.165356i
$729$	15443.0	0.784586
$730$	0	0
$731$	15408.0	0.779597
$732$	− 472.000i	− 0.0238328i
$733$	− 13680.0i	− 0.689335i	−0.938725	−	0.344667i	$-0.887992\pi$
$733$	− 13680.0i	− 0.689335i	0.938725	−	0.344667i	$-0.112008\pi$
$734$	12272.0	0.617122
$735$	0	0
$736$	6400.00	0.320526
$737$	− 24045.0i	− 1.20178i
$738$	− 22412.0i	− 1.11788i
$739$	32364.0	1.61100	0.805500	−	0.592596i	$-0.201897\pi$
$739$	32364.0	1.61100	0.805500	+	0.592596i	$0.201897\pi$
$740$	0	0
$741$	−1334.00	−0.0661346
$742$	− 1512.00i	− 0.0748076i
$743$	− 25632.0i	− 1.26561i	−0.774312	−	0.632804i	$-0.781904\pi$
$743$	− 25632.0i	− 1.26561i	0.774312	−	0.632804i	$-0.218096\pi$
$744$	−608.000	−0.0299602
$745$	0	0
$746$	9176.00	0.450345
$747$	− 23322.0i	− 1.14231i
$748$	14980.0i	0.732250i
$749$	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	$0.437817\pi$
$752$	8416.00i	0.408112i
$753$	1897.00i	0.0918068i
$754$	−20184.0	−0.974878
$755$	0	0
$756$	1484.00	0.0713923
$757$	13214.0i	0.634440i	0.948352	+	0.317220i	$0.102749\pi$
$757$	13214.0i	0.634440i	−0.948352	+	0.317220i	$0.897251\pi$
$758$	− 12178.0i	− 0.583542i
$759$	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	$0.140414\pi$
$762$	− 1524.00i	− 0.0724524i
$763$	− 15106.0i	− 0.716742i
$764$	5704.00	0.270109
$765$	0	0
$766$	−5916.00	−0.279052
$767$	4408.00i	0.207515i
$768$	256.000i	0.0120281i
$769$	9569.00	0.448722	0.224361	−	0.974506i	$-0.427971\pi$
$769$	9569.00	0.448722	0.224361	+	0.974506i	$0.427971\pi$
$770$	0	0
$771$	4654.00	0.217393
$772$	5428.00i	0.253054i
$773$	27908.0i	1.29855i	0.760553	+	0.649276i	$0.224928\pi$
$773$	27908.0i	1.29855i	−0.760553	+	0.649276i	$0.775072\pi$
$774$	−7488.00	−0.347740
$775$	0	0
$776$	−12080.0	−0.558823
$777$	− 1288.00i	− 0.0594681i
$778$	10944.0i	0.504321i
$779$	−9913.00	−0.455931
$780$	0	0
$781$	−18550.0	−0.849899
$782$	42800.0i	1.95719i
$783$	9222.00i	0.420903i
$784$	−784.000	−0.0357143
$785$	0	0
$786$	−3024.00	−0.137230
$787$	− 8884.00i	− 0.402389i	−0.979551	−	0.201195i	$-0.935518\pi$
$787$	− 8884.00i	− 0.402389i	0.979551	−	0.201195i	$-0.0644824\pi$
$788$	18744.0i	0.847370i
$789$	−2190.00	−0.0988163
$790$	0	0
$791$	−2639.00	−0.118625
$792$	− 7280.00i	− 0.326621i
$793$	− 6844.00i	− 0.306479i
$794$	−6460.00	−0.288737
$795$	0	0
$796$	15560.0	0.692851
$797$	31866.0i	1.41625i	0.706087	+	0.708125i	$0.250459\pi$
$797$	31866.0i	1.41625i	−0.706087	+	0.708125i	$0.749541\pi$
$798$	− 322.000i	− 0.0142841i
$799$	−56282.0	−2.49201
$800$	0	0
$801$	20774.0	0.916371
$802$	− 27834.0i	− 1.22550i
$803$	− 10465.0i	− 0.459903i
$804$	2748.00	0.120540
$805$	0	0
$806$	−8816.00	−0.385273
$807$	7638.00i	0.333173i
$808$	0	0
$809$	−8854.00	−0.384784	−0.192392	−	0.981318i	$-0.561624\pi$
$809$	−8854.00	−0.384784	−0.192392	+	0.981318i	$0.561624\pi$
$810$	0	0
$811$	16812.0	0.727927	0.363964	−	0.931413i	$-0.381423\pi$
$811$	16812.0	0.727927	0.363964	+	0.931413i	$0.381423\pi$
$812$	− 4872.00i	− 0.210559i
$813$	− 5466.00i	− 0.235795i
$814$	−12880.0	−0.554599
$815$	0	0
$816$	−1712.00	−0.0734461
$817$	3312.00i	0.141826i
$818$	24542.0i	1.04901i
$819$	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	$0.357124\pi$
$822$	− 838.000i	− 0.0355579i
$823$	− 23110.0i	− 0.978814i	−0.872055	−	0.489407i	$-0.837213\pi$
$823$	− 23110.0i	− 0.978814i	0.872055	−	0.489407i	$-0.162787\pi$
$824$	−3296.00	−0.139347
$825$	0	0
$826$	−1064.00	−0.0448200
$827$	7983.00i	0.335666i	0.985815	+	0.167833i	$0.0536770\pi$
$827$	7983.00i	0.335666i	−0.985815	+	0.167833i	$0.946323\pi$
$828$	− 20800.0i	− 0.873007i
$829$	10156.0	0.425492	0.212746	−	0.977108i	$-0.431759\pi$
$829$	10156.0	0.425492	0.212746	+	0.977108i	$0.431759\pi$
$830$	0	0
$831$	310.000	0.0129408
$832$	3712.00i	0.154676i
$833$	− 5243.00i	− 0.218078i
$834$	4658.00	0.193397
$835$	0	0
$836$	−3220.00	−0.133213
$837$	4028.00i	0.166342i
$838$	17958.0i	0.740273i
$839$	19962.0	0.821412	0.410706	−	0.911768i	$-0.365282\pi$
$839$	19962.0	0.821412	0.410706	+	0.911768i	$0.365282\pi$
$840$	0	0
$841$	5887.00	0.241379
$842$	− 12200.0i	− 0.499335i
$843$	− 4946.00i	− 0.202075i
$844$	16036.0	0.654007
$845$	0	0
$846$	27352.0	1.11156
$847$	− 742.000i	− 0.0301009i
$848$	− 1728.00i	− 0.0699761i
$849$	−391.000	−0.0158058
$850$	0	0
$851$	−36800.0	−1.48236
$852$	− 2120.00i	− 0.0852465i
$853$	− 47422.0i	− 1.90352i	−0.306851	−	0.951758i	$-0.599275\pi$
$853$	− 47422.0i	− 1.90352i	0.306851	−	0.951758i	$-0.400725\pi$
$854$	1652.00	0.0661947
$855$	0	0
$856$	6808.00	0.271837
$857$	− 17641.0i	− 0.703156i	−0.936159	−	0.351578i	$-0.885645\pi$
$857$	− 17641.0i	− 0.703156i	0.936159	−	0.351578i	$-0.114355\pi$
$858$	4060.00i	0.161546i
$859$	−8831.00	−0.350768	−0.175384	−	0.984500i	$-0.556117\pi$
$859$	−8831.00	−0.350768	−0.175384	+	0.984500i	$0.556117\pi$
$860$	0	0
$861$	−3017.00	−0.119418
$862$	− 648.000i	− 0.0256044i
$863$	− 19908.0i	− 0.785256i	−0.919697	−	0.392628i	$-0.871566\pi$
$863$	− 19908.0i	− 0.785256i	0.919697	−	0.392628i	$-0.128434\pi$
$864$	1696.00	0.0667814
$865$	0	0
$866$	−16326.0	−0.640624
$867$	− 6536.00i	− 0.256026i
$868$	− 2128.00i	− 0.0832132i
$869$	14070.0	0.549243
$870$	0	0
$871$	39846.0	1.55009
$872$	− 17264.0i	− 0.670450i
$873$	39260.0i	1.52205i
$874$	−9200.00	−0.356058
$875$	0	0
$876$	1196.00	0.0461291
$877$	− 41584.0i	− 1.60113i	−0.599245	−	0.800566i	$-0.704533\pi$
$877$	− 41584.0i	− 1.60113i	0.599245	−	0.800566i	$-0.295467\pi$
$878$	− 27656.0i	− 1.06304i
$879$	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	$0.634280\pi$
$882$	2548.00i	0.0972739i
$883$	14975.0i	0.570724i	0.958420	+	0.285362i	$0.0921137\pi$
$883$	14975.0i	0.570724i	−0.958420	+	0.285362i	$0.907886\pi$
$884$	−24824.0	−0.944481
$885$	0	0
$886$	29882.0	1.13308
$887$	− 11702.0i	− 0.442970i	−0.975164	−	0.221485i	$-0.928910\pi$
$887$	− 11702.0i	− 0.442970i	0.975164	−	0.221485i	$-0.0710904\pi$
$888$	− 1472.00i	− 0.0556273i
$889$	5334.00	0.201234
$890$	0	0
$891$	−22715.0	−0.854075
$892$	20616.0i	0.773851i
$893$	− 12098.0i	− 0.453353i
$894$	−6216.00	−0.232544
$895$	0	0
$896$	−896.000	−0.0334077
$897$	11600.0i	0.431787i
$898$	2354.00i	0.0874766i
$899$	13224.0	0.490595
$900$	0	0
$901$	11556.0	0.427288
$902$	30170.0i	1.11369i
$903$	1008.00i	0.0371474i
$904$	−3016.00	−0.110963
$905$	0	0
$906$	−2384.00	−0.0874206
$907$	− 15252.0i	− 0.558362i	−0.960238	−	0.279181i	$-0.909937\pi$
$907$	− 15252.0i	− 0.558362i	0.960238	−	0.279181i	$-0.0900629\pi$
$908$	6096.00i	0.222801i
$909$	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	$0.450556\pi$
$912$	− 368.000i	− 0.0133615i
$913$	31395.0i	1.13803i
$914$	8690.00	0.314485
$915$	0	0
$916$	−13784.0	−0.497201
$917$	− 10584.0i	− 0.381150i
$918$	11342.0i	0.407780i
$919$	22406.0	0.804250	0.402125	−	0.915585i	$-0.368272\pi$
$919$	22406.0	0.804250	0.402125	+	0.915585i	$0.368272\pi$
$920$	0	0
$921$	−8279.00	−0.296202
$922$	− 10564.0i	− 0.377339i
$923$	− 30740.0i	− 1.09623i
$924$	−980.000	−0.0348914
$925$	0	0
$926$	−15952.0	−0.566107
$927$	10712.0i	0.379534i
$928$	− 5568.00i	− 0.196960i
$929$	−49294.0	−1.74089	−0.870443	−	0.492269i	$-0.836168\pi$
$929$	−49294.0	−1.74089	−0.870443	+	0.492269i	$0.836168\pi$
$930$	0	0
$931$	1127.00	0.0396734
$932$	− 2792.00i	− 0.0981277i
$933$	6146.00i	0.215660i
$934$	−28024.0	−0.981770
$935$	0	0
$936$	12064.0	0.421287
$937$	24669.0i	0.860087i	0.902808	+	0.430043i	$0.141502\pi$
$937$	24669.0i	0.860087i	−0.902808	+	0.430043i	$0.858498\pi$
$938$	9618.00i	0.334796i
$939$	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	$0.523310\pi$
$942$	3544.00i	0.122579i
$943$	86200.0i	2.97673i
$944$	−1216.00	−0.0419252
$945$	0	0
$946$	10080.0	0.346437
$947$	− 39252.0i	− 1.34690i	−0.739231	−	0.673452i	$-0.764811\pi$
$947$	− 39252.0i	− 1.34690i	0.739231	−	0.673452i	$-0.235189\pi$
$948$	1608.00i	0.0550901i
$949$	17342.0	0.593198
$950$	0	0
$951$	−3812.00	−0.129982
$952$	− 5992.00i	− 0.203994i
$953$	33567.0i	1.14097i	0.821309	+	0.570484i	$0.193244\pi$
$953$	33567.0i	1.14097i	−0.821309	+	0.570484i	$0.806756\pi$
$954$	−5616.00	−0.190592
$955$	0	0
$956$	−24496.0	−0.828721
$957$	− 6090.00i	− 0.205707i
$958$	2892.00i	0.0975327i
$959$	2933.00	0.0987607
$960$	0	0
$961$	−24015.0	−0.806116
$962$	− 21344.0i	− 0.715341i
$963$	− 22126.0i	− 0.740395i
$964$	−27900.0	−0.932156
$965$	0	0
$966$	−2800.00	−0.0932593
$967$	− 56126.0i	− 1.86648i	−0.359248	−	0.933242i	$-0.616967\pi$
$967$	− 56126.0i	− 1.86648i	0.359248	−	0.933242i	$-0.383033\pi$
$968$	− 848.000i	− 0.0281568i
$969$	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	$0.464373\pi$
$972$	− 8320.00i	− 0.274552i
$973$	16303.0i	0.537153i
$974$	−22284.0	−0.733086
$975$	0	0
$976$	1888.00	0.0619195
$977$	− 35547.0i	− 1.16402i	−0.813181	−	0.582011i	$-0.802266\pi$
$977$	− 35547.0i	− 1.16402i	0.813181	−	0.582011i	$-0.197734\pi$
$978$	5946.00i	0.194409i
$979$	−27965.0	−0.912937
$980$	0	0
$981$	−56108.0	−1.82609
$982$	27384.0i	0.889876i
$983$	− 35252.0i	− 1.14381i	−0.820320	−	0.571904i	$-0.806205\pi$
$983$	− 35252.0i	− 1.14381i	0.820320	−	0.571904i	$-0.193795\pi$
$984$	−3448.00	−0.111706
$985$	0	0
$986$	37236.0	1.20267
$987$	− 3682.00i	− 0.118743i
$988$	− 5336.00i	− 0.171823i
$989$	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	$0.469937\pi$
$992$	− 2432.00i	− 0.0778388i
$993$	− 1127.00i	− 0.0360164i
$994$	7420.00	0.236769
$995$	0	0
$996$	−3588.00	−0.114147
$997$	− 39874.0i	− 1.26662i	−0.773897	−	0.633311i	$-0.781695\pi$
$997$	− 39874.0i	− 1.26662i	0.773897	−	0.633311i	$-0.218305\pi$
$998$	39848.0i	1.26389i
$999$	−9752.00	−0.308848

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
350.4.a.d.1.1	✓	1	5.3	odd	4
350.4.a.s.1.1	yes	1	5.2	odd	4
350.4.c.i.99.1		2	1.1	even	1	trivial
350.4.c.i.99.2		2	5.4	even	2	inner
2450.4.a.l.1.1		1	35.13	even	4
2450.4.a.bd.1.1		1	35.27	even	4

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	350.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.00000i q^{2} +1.00000i q^{3} -4.00000 q^{4} +2.00000 q^{6} +7.00000i q^{7} +8.00000i q^{8} +26.0000 q^{9} -35.0000 q^{11} -4.00000i q^{12} -58.0000i q^{13} +14.0000 q^{14} +16.0000 q^{16} +107.000i q^{17} -52.0000i q^{18} -23.0000 q^{19} -7.00000 q^{21} +70.0000i q^{22} +200.000i q^{23} -8.00000 q^{24} -116.000 q^{26} +53.0000i q^{27} -28.0000i q^{28} +174.000 q^{29} +76.0000 q^{31} -32.0000i q^{32} -35.0000i q^{33} +214.000 q^{34} -104.000 q^{36} +184.000i q^{37} +46.0000i q^{38} +58.0000 q^{39} +431.000 q^{41} +14.0000i q^{42} -144.000i q^{43} +140.000 q^{44} +400.000 q^{46} +526.000i q^{47} +16.0000i q^{48} -49.0000 q^{49} -107.000 q^{51} +232.000i q^{52} -108.000i q^{53} +106.000 q^{54} -56.0000 q^{56} -23.0000i q^{57} -348.000i q^{58} -76.0000 q^{59} +118.000 q^{61} -152.000i q^{62} +182.000i q^{63} -64.0000 q^{64} -70.0000 q^{66} +687.000i q^{67} -428.000i q^{68} -200.000 q^{69} +530.000 q^{71} +208.000i q^{72} +299.000i q^{73} +368.000 q^{74} +92.0000 q^{76} -245.000i q^{77} -116.000i q^{78} -402.000 q^{79} +649.000 q^{81} -862.000i q^{82} -897.000i q^{83} +28.0000 q^{84} -288.000 q^{86} +174.000i q^{87} -280.000i q^{88} +799.000 q^{89} +406.000 q^{91} -800.000i q^{92} +76.0000i q^{93} +1052.00 q^{94} +32.0000 q^{96} +1510.00i q^{97} +98.0000i q^{98} -910.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{4} + 4 q^{6} + 52 q^{9} - 70 q^{11} + 28 q^{14} + 32 q^{16} - 46 q^{19} - 14 q^{21} - 16 q^{24} - 232 q^{26} + 348 q^{29} + 152 q^{31} + 428 q^{34} - 208 q^{36} + 116 q^{39} + 862 q^{41} + 280 q^{44}+ \cdots - 1820 q^{99}+O(q^{100}) \) 2 * q - 8 * q^4 + 4 * q^6 + 52 * q^9 - 70 * q^11 + 28 * q^14 + 32 * q^16 - 46 * q^19 - 14 * q^21 - 16 * q^24 - 232 * q^26 + 348 * q^29 + 152 * q^31 + 428 * q^34 - 208 * q^36 + 116 * q^39 + 862 * q^41 + 280 * q^44 + 800 * q^46 - 98 * q^49 - 214 * q^51 + 212 * q^54 - 112 * q^56 - 152 * q^59 + 236 * q^61 - 128 * q^64 - 140 * q^66 - 400 * q^69 + 1060 * q^71 + 736 * q^74 + 184 * q^76 - 804 * q^79 + 1298 * q^81 + 56 * q^84 - 576 * q^86 + 1598 * q^89 + 812 * q^91 + 2104 * q^94 + 64 * q^96 - 1820 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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