LMFDB - Embedded newform 350.4.c.i.99.2

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

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.00000i q^{2} -1.00000i q^{3} -4.00000 q^{4} +2.00000 q^{6} -7.00000i q^{7} -8.00000i q^{8} +26.0000 q^{9} -35.0000 q^{11} +4.00000i q^{12} +58.0000i q^{13} +14.0000 q^{14} +16.0000 q^{16} -107.000i q^{17} +52.0000i q^{18} -23.0000 q^{19} -7.00000 q^{21} -70.0000i q^{22} -200.000i q^{23} -8.00000 q^{24} -116.000 q^{26} -53.0000i q^{27} +28.0000i q^{28} +174.000 q^{29} +76.0000 q^{31} +32.0000i q^{32} +35.0000i q^{33} +214.000 q^{34} -104.000 q^{36} -184.000i q^{37} -46.0000i q^{38} +58.0000 q^{39} +431.000 q^{41} -14.0000i q^{42} +144.000i q^{43} +140.000 q^{44} +400.000 q^{46} -526.000i q^{47} -16.0000i q^{48} -49.0000 q^{49} -107.000 q^{51} -232.000i q^{52} +108.000i q^{53} +106.000 q^{54} -56.0000 q^{56} +23.0000i q^{57} +348.000i q^{58} -76.0000 q^{59} +118.000 q^{61} +152.000i q^{62} -182.000i q^{63} -64.0000 q^{64} -70.0000 q^{66} -687.000i q^{67} +428.000i q^{68} -200.000 q^{69} +530.000 q^{71} -208.000i q^{72} -299.000i q^{73} +368.000 q^{74} +92.0000 q^{76} +245.000i q^{77} +116.000i q^{78} -402.000 q^{79} +649.000 q^{81} +862.000i q^{82} +897.000i q^{83} +28.0000 q^{84} -288.000 q^{86} -174.000i q^{87} +280.000i q^{88} +799.000 q^{89} +406.000 q^{91} +800.000i q^{92} -76.0000i q^{93} +1052.00 q^{94} +32.0000 q^{96} -1510.00i q^{97} -98.0000i q^{98} -910.000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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	$-0.969323\pi$
$3$	− 1.00000i	− 0.192450i	0.995360	−	0.0962250i	$-0.0306768\pi$
$4$	−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	$0.212342\pi$
$13$	58.0000i	1.23741i	−0.785624	+	0.618704i	$0.787658\pi$
$14$	14.0000	0.267261
$15$	0	0
$16$	16.0000	0.250000
$17$	− 107.000i	− 1.52655i	−0.646075	−	0.763274i	$-0.723591\pi$
$17$	− 107.000i	− 1.52655i	0.646075	−	0.763274i	$-0.276409\pi$
$18$	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	$-0.638680\pi$
$23$	− 200.000i	− 1.81317i	0.422025	−	0.906584i	$-0.361320\pi$
$24$	−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	$-0.865956\pi$
$37$	− 184.000i	− 0.817552i	0.912635	−	0.408776i	$-0.134044\pi$
$38$	− 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	$0.0821895\pi$
$43$	144.000i	0.510693i	−0.966850	+	0.255346i	$0.917810\pi$
$44$	140.000	0.479677
$45$	0	0
$46$	400.000	1.28210
$47$	− 526.000i	− 1.63245i	−0.577737	−	0.816223i	$-0.696064\pi$
$47$	− 526.000i	− 1.63245i	0.577737	−	0.816223i	$-0.303936\pi$
$48$	− 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	$0.0446949\pi$
$53$	108.000i	0.279905i	−0.990158	+	0.139952i	$0.955305\pi$
$54$	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	$-0.784550\pi$
$67$	− 687.000i	− 1.25269i	0.779545	−	0.626346i	$-0.215450\pi$
$68$	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	$-0.922953\pi$
$73$	− 299.000i	− 0.479388i	0.970849	−	0.239694i	$-0.0770471\pi$
$74$	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	$0.202106\pi$
$83$	897.000i	1.18625i	−0.805111	+	0.593124i	$0.797894\pi$
$84$	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	$-0.709927\pi$
$97$	− 1510.00i	− 1.58059i	0.612726	−	0.790295i	$-0.290073\pi$
$98$	− 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	$-0.936859\pi$
$103$	− 412.000i	− 0.394132i	0.980390	−	0.197066i	$-0.0631413\pi$
$104$	464.000	0.437490
$105$	0	0
$106$	−216.000	−0.197922
$107$	851.000i	0.768872i	0.923152	+	0.384436i	$0.125604\pi$
$107$	851.000i	0.768872i	−0.923152	+	0.384436i	$0.874396\pi$
$108$	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	$-0.949842\pi$
$113$	− 377.000i	− 0.313851i	0.987610	−	0.156926i	$-0.0501583\pi$
$114$	−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	$0.0857705\pi$
$127$	762.000i	0.532414i	−0.963916	+	0.266207i	$0.914230\pi$
$128$	− 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	$0.0417058\pi$
$137$	419.000i	0.261296i	−0.991429	+	0.130648i	$0.958294\pi$
$138$	− 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	$-0.851287\pi$
$157$	− 1772.00i	− 0.900771i	0.892834	−	0.450385i	$-0.148713\pi$
$158$	− 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	$-0.746743\pi$
$163$	− 2973.00i	− 1.42861i	0.699835	−	0.714305i	$-0.253257\pi$
$164$	−1724.00	−0.820865
$165$	0	0
$166$	−1794.00	−0.838804
$167$	4.00000i	0.00185347i	1.00000		0.000926734i	$0.000294989\pi$
$167$	4.00000i	0.00185347i	−1.00000		0.000926734i	$0.999705\pi$
$168$	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	$0.272156\pi$
$173$	3434.00i	1.50915i	−0.656216	+	0.754573i	$0.727844\pi$
$174$	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	$0.0814352\pi$
$193$	1357.00i	0.506109i	−0.967452	+	0.253054i	$0.918565\pi$
$194$	3020.00	1.11765
$195$	0	0
$196$	196.000	0.0714286
$197$	4686.00i	1.69474i	0.531003	+	0.847370i	$0.321815\pi$
$197$	4686.00i	1.69474i	−0.531003	+	0.847370i	$0.678185\pi$
$198$	− 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	$0.281672\pi$
$223$	5154.00i	1.54770i	−0.633368	+	0.773851i	$0.718328\pi$
$224$	224.000	0.0668153
$225$	0	0
$226$	754.000	0.221926
$227$	1524.00i	0.445601i	0.974864	+	0.222801i	$0.0715199\pi$
$227$	1524.00i	0.445601i	−0.974864	+	0.222801i	$0.928480\pi$
$228$	− 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	$-0.968715\pi$
$233$	− 698.000i	− 0.196255i	0.995174	−	0.0981277i	$-0.0312854\pi$
$234$	−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	$0.191048\pi$
$257$	4654.00i	1.12961i	−0.825226	+	0.564803i	$0.808952\pi$
$258$	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	$-0.917354\pi$
$263$	− 2190.00i	− 0.513465i	0.966483	−	0.256732i	$-0.0826459\pi$
$264$	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	$0.0107039\pi$
$277$	310.000i	0.0672422i	−0.999435	+	0.0336211i	$0.989296\pi$
$278$	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	$-0.986925\pi$
$283$	− 391.000i	− 0.0821291i	0.999156	−	0.0410646i	$-0.0130749\pi$
$284$	−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	$0.134799\pi$
$293$	4122.00i	0.821876i	−0.911663	+	0.410938i	$0.865201\pi$
$294$	−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	$-0.720478\pi$
$307$	− 8279.00i	− 1.53911i	0.638579	−	0.769556i	$-0.279522\pi$
$308$	− 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	$0.311541\pi$
$313$	9190.00i	1.65958i	−0.558073	+	0.829792i	$0.688459\pi$
$314$	3544.00	0.636941
$315$	0	0
$316$	1608.00	0.286257
$317$	− 3812.00i	− 0.675405i	−0.941253	−	0.337702i	$-0.890350\pi$
$317$	− 3812.00i	− 0.675405i	0.941253	−	0.337702i	$-0.109650\pi$
$318$	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	$-0.948243\pi$
$337$	− 2003.00i	− 0.323770i	0.986810	−	0.161885i	$-0.0517573\pi$
$338$	− 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	$-0.679922\pi$
$347$	− 10917.0i	− 1.68892i	0.535619	−	0.844460i	$-0.320078\pi$
$348$	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	$0.0446367\pi$
$353$	1854.00i	0.279542i	−0.990184	+	0.139771i	$0.955363\pi$
$354$	−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	$-0.856263\pi$
$367$	− 6136.00i	− 0.872743i	0.899767	−	0.436371i	$-0.143737\pi$
$368$	− 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	$-0.896840\pi$
$373$	− 4588.00i	− 0.636884i	0.947942	−	0.318442i	$-0.103160\pi$
$374$	−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	$0.0632236\pi$
$383$	2958.00i	0.394639i	−0.980339	+	0.197320i	$0.936776\pi$
$384$	−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	$0.0654487\pi$
$397$	3230.00i	0.408335i	−0.978936	+	0.204168i	$0.934551\pi$
$398$	− 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	$0.149642\pi$
$433$	8163.00i	0.905979i	−0.891516	+	0.452989i	$0.850358\pi$
$434$	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	$-0.704192\pi$
$443$	− 14941.0i	− 1.60241i	0.598389	−	0.801206i	$-0.295808\pi$
$444$	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	$-0.928619\pi$
$457$	− 4345.00i	− 0.444750i	0.974961	−	0.222375i	$-0.0713808\pi$
$458$	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	$0.131093\pi$
$463$	7976.00i	0.800596i	−0.916385	+	0.400298i	$0.868907\pi$
$464$	2784.00	0.278543
$465$	0	0
$466$	1396.00	0.138774
$467$	14012.0i	1.38843i	0.719766	+	0.694216i	$0.244249\pi$
$467$	14012.0i	1.38843i	−0.719766	+	0.694216i	$0.755751\pi$
$468$	− 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	$0.173461\pi$
$487$	11142.0i	1.03674i	−0.855157	+	0.518370i	$0.826539\pi$
$488$	− 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	$-0.929303\pi$
$503$	− 4970.00i	− 0.440559i	0.975437	−	0.220280i	$-0.0706970\pi$
$504$	−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	$0.128997\pi$
$523$	9431.00i	0.788506i	−0.919002	+	0.394253i	$0.871003\pi$
$524$	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	$-0.750552\pi$
$547$	− 18061.0i	− 1.41176i	0.708332	−	0.705880i	$-0.249448\pi$
$548$	− 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	$0.229375\pi$
$557$	17348.0i	1.31967i	−0.751409	+	0.659837i	$0.770625\pi$
$558$	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	$-0.658823\pi$
$563$	− 23460.0i	− 1.75617i	0.478509	−	0.878083i	$-0.341177\pi$
$564$	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	$0.150598\pi$
$577$	12631.0i	0.911327i	−0.890152	+	0.455663i	$0.849402\pi$
$578$	− 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	$-0.988348\pi$
$587$	− 1041.00i	− 0.0731970i	0.999330	−	0.0365985i	$-0.0116523\pi$
$588$	− 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	$-0.859195\pi$
$593$	− 12363.0i	− 0.856134i	0.903747	−	0.428067i	$-0.140805\pi$
$594$	−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	$-0.776966\pi$
$607$	− 19284.0i	− 1.28948i	0.764403	−	0.644739i	$-0.223034\pi$
$608$	− 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	$0.121488\pi$
$613$	11306.0i	0.744935i	−0.928045	+	0.372467i	$0.878512\pi$
$614$	16558.0	1.08832
$615$	0	0
$616$	1960.00	0.128199
$617$	− 4222.00i	− 0.275480i	−0.990468	−	0.137740i	$-0.956016\pi$
$617$	− 4222.00i	− 0.275480i	0.990468	−	0.137740i	$-0.0439839\pi$
$618$	− 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	$-0.975299\pi$
$643$	− 2528.00i	− 0.155046i	0.996991	−	0.0775230i	$-0.0247011\pi$
$644$	5600.00	0.342657
$645$	0	0
$646$	−4922.00	−0.299773
$647$	− 5416.00i	− 0.329096i	−0.986369	−	0.164548i	$-0.947384\pi$
$647$	− 5416.00i	− 0.329096i	0.986369	−	0.164548i	$-0.0526165\pi$
$648$	− 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	$-0.967170\pi$
$653$	− 3436.00i	− 0.205913i	0.994686	−	0.102956i	$-0.0328302\pi$
$654$	−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	$0.0140247\pi$
$673$	1538.00i	0.0880914i	−0.999030	+	0.0440457i	$0.985975\pi$
$674$	4006.00	0.228940
$675$	0	0
$676$	4668.00	0.265589
$677$	− 18816.0i	− 1.06818i	−0.845428	−	0.534090i	$-0.820654\pi$
$677$	− 18816.0i	− 1.06818i	0.845428	−	0.534090i	$-0.179346\pi$
$678$	− 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	$0.0431896\pi$
$683$	4829.00i	0.270537i	−0.990809	+	0.135268i	$0.956810\pi$
$684$	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	$0.118517\pi$
$727$	14262.0i	0.727577i	−0.931482	+	0.363788i	$0.881483\pi$
$728$	− 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	$0.112008\pi$
$733$	13680.0i	0.689335i	−0.938725	+	0.344667i	$0.887992\pi$
$734$	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	$0.218096\pi$
$743$	25632.0i	1.26561i	−0.774312	+	0.632804i	$0.781904\pi$
$744$	−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	$-0.897251\pi$
$757$	− 13214.0i	− 0.634440i	0.948352	−	0.317220i	$-0.102749\pi$
$758$	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	$-0.775072\pi$
$773$	− 27908.0i	− 1.29855i	0.760553	−	0.649276i	$-0.224928\pi$
$774$	−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	$0.0644824\pi$
$787$	8884.00i	0.402389i	−0.979551	+	0.201195i	$0.935518\pi$
$788$	− 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	$-0.749541\pi$
$797$	− 31866.0i	− 1.41625i	0.706087	−	0.708125i	$-0.250459\pi$
$798$	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	$0.162787\pi$
$823$	23110.0i	0.978814i	−0.872055	+	0.489407i	$0.837213\pi$
$824$	−3296.00	−0.139347
$825$	0	0
$826$	−1064.00	−0.0448200
$827$	− 7983.00i	− 0.335666i	−0.985815	−	0.167833i	$-0.946323\pi$
$827$	− 7983.00i	− 0.335666i	0.985815	−	0.167833i	$-0.0536770\pi$
$828$	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	$0.400725\pi$
$853$	47422.0i	1.90352i	−0.306851	+	0.951758i	$0.599275\pi$
$854$	1652.00	0.0661947
$855$	0	0
$856$	6808.00	0.271837
$857$	17641.0i	0.703156i	0.936159	+	0.351578i	$0.114355\pi$
$857$	17641.0i	0.703156i	−0.936159	+	0.351578i	$0.885645\pi$
$858$	− 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	$0.128434\pi$
$863$	19908.0i	0.785256i	−0.919697	+	0.392628i	$0.871566\pi$
$864$	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	$0.295467\pi$
$877$	41584.0i	1.60113i	−0.599245	+	0.800566i	$0.704533\pi$
$878$	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	$-0.907886\pi$
$883$	− 14975.0i	− 0.570724i	0.958420	−	0.285362i	$-0.0921137\pi$
$884$	−24824.0	−0.944481
$885$	0	0
$886$	29882.0	1.13308
$887$	11702.0i	0.442970i	0.975164	+	0.221485i	$0.0710904\pi$
$887$	11702.0i	0.442970i	−0.975164	+	0.221485i	$0.928910\pi$
$888$	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	$0.0900629\pi$
$907$	15252.0i	0.558362i	−0.960238	+	0.279181i	$0.909937\pi$
$908$	− 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	$-0.858498\pi$
$937$	− 24669.0i	− 0.860087i	0.902808	−	0.430043i	$-0.141502\pi$
$938$	− 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	$0.235189\pi$
$947$	39252.0i	1.34690i	−0.739231	+	0.673452i	$0.764811\pi$
$948$	− 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	$-0.806756\pi$
$953$	− 33567.0i	− 1.14097i	0.821309	−	0.570484i	$-0.193244\pi$
$954$	−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	$0.383033\pi$
$967$	56126.0i	1.86648i	−0.359248	+	0.933242i	$0.616967\pi$
$968$	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	$0.197734\pi$
$977$	35547.0i	1.16402i	−0.813181	+	0.582011i	$0.802266\pi$
$978$	− 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	$0.193795\pi$
$983$	35252.0i	1.14381i	−0.820320	+	0.571904i	$0.806205\pi$
$984$	−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	$0.218305\pi$
$997$	39874.0i	1.26662i	−0.773897	+	0.633311i	$0.781695\pi$
$998$	− 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.99.2		2
5.2	odd	4		350.4.a.d.1.1	✓	1
5.3	odd	4		350.4.a.s.1.1	yes	1
5.4	even	2	inner	350.4.c.i.99.1		2
35.13	even	4		2450.4.a.bd.1.1		1
35.27	even	4		2450.4.a.l.1.1		1

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