LMFDB - Embedded newform 350.4.c.a.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)

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);

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")

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,-40,0,0,-146,0,18] 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:	$\Z[a_1, \ldots, a_{7}]$
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.a.99.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.00000i q^{2} -10.0000i q^{3} -4.00000 q^{4} -20.0000 q^{6} +7.00000i q^{7} +8.00000i q^{8} -73.0000 q^{9} +9.00000 q^{11} +40.0000i q^{12} +52.0000i q^{13} +14.0000 q^{14} +16.0000 q^{16} +96.0000i q^{17} +146.000i q^{18} +10.0000 q^{19} +70.0000 q^{21} -18.0000i q^{22} -75.0000i q^{23} +80.0000 q^{24} +104.000 q^{26} +460.000i q^{27} -28.0000i q^{28} -189.000 q^{29} -232.000 q^{31} -32.0000i q^{32} -90.0000i q^{33} +192.000 q^{34} +292.000 q^{36} +305.000i q^{37} -20.0000i q^{38} +520.000 q^{39} -438.000 q^{41} -140.000i q^{42} -353.000i q^{43} -36.0000 q^{44} -150.000 q^{46} -486.000i q^{47} -160.000i q^{48} -49.0000 q^{49} +960.000 q^{51} -208.000i q^{52} +354.000i q^{53} +920.000 q^{54} -56.0000 q^{56} -100.000i q^{57} +378.000i q^{58} +672.000 q^{59} +206.000 q^{61} +464.000i q^{62} -511.000i q^{63} -64.0000 q^{64} -180.000 q^{66} +599.000i q^{67} -384.000i q^{68} -750.000 q^{69} -471.000 q^{71} -584.000i q^{72} -614.000i q^{73} +610.000 q^{74} -40.0000 q^{76} +63.0000i q^{77} -1040.00i q^{78} -743.000 q^{79} +2629.00 q^{81} +876.000i q^{82} -996.000i q^{83} -280.000 q^{84} -706.000 q^{86} +1890.00i q^{87} +72.0000i q^{88} -180.000 q^{89} -364.000 q^{91} +300.000i q^{92} +2320.00i q^{93} -972.000 q^{94} -320.000 q^{96} -184.000i q^{97} +98.0000i q^{98} -657.000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{4} - 40 q^{6} - 146 q^{9} + 18 q^{11} + 28 q^{14} + 32 q^{16} + 20 q^{19} + 140 q^{21} + 160 q^{24} + 208 q^{26} - 378 q^{29} - 464 q^{31} + 384 q^{34} + 584 q^{36} + 1040 q^{39} - 876 q^{41}+ \cdots - 1314 q^{99}+O(q^{100}) $ 2 * q - 8 * q^4 - 40 * q^6 - 146 * q^9 + 18 * q^11 + 28 * q^14 + 32 * q^16 + 20 * q^19 + 140 * q^21 + 160 * q^24 + 208 * q^26 - 378 * q^29 - 464 * q^31 + 384 * q^34 + 584 * q^36 + 1040 * q^39 - 876 * q^41 - 72 * q^44 - 300 * q^46 - 98 * q^49 + 1920 * q^51 + 1840 * q^54 - 112 * q^56 + 1344 * q^59 + 412 * q^61 - 128 * q^64 - 360 * q^66 - 1500 * q^69 - 942 * q^71 + 1220 * q^74 - 80 * q^76 - 1486 * q^79 + 5258 * q^81 - 560 * q^84 - 1412 * q^86 - 360 * q^89 - 728 * q^91 - 1944 * q^94 - 640 * q^96 - 1314 * 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
$3$	− 10.0000i	− 1.92450i	−0.272166	−	0.962250i	$-0.587740\pi$
$3$	− 10.0000i	− 1.92450i	0.272166	−	0.962250i	$-0.412260\pi$
$4$	−4.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	−20.0000	−1.36083
$7$	7.00000i	0.377964i
$7$	7.00000i	0.377964i
$8$	8.00000i	0.353553i
$9$	−73.0000	−2.70370
$10$	0	0
$11$	9.00000	0.246691	0.123346	−	0.992364i	$-0.460638\pi$
$11$	9.00000	0.246691	0.123346	+	0.992364i	$0.460638\pi$
$12$	40.0000i	0.962250i
$13$	52.0000i	1.10940i	0.832050	+	0.554700i	$0.187167\pi$
$13$	52.0000i	1.10940i	−0.832050	+	0.554700i	$0.812833\pi$
$14$	14.0000	0.267261
$15$	0	0
$16$	16.0000	0.250000
$17$	96.0000i	1.36961i	0.728725	+	0.684806i	$0.240113\pi$
$17$	96.0000i	1.36961i	−0.728725	+	0.684806i	$0.759887\pi$
$18$	146.000i	1.91181i
$19$	10.0000	0.120745	0.0603726	−	0.998176i	$-0.480771\pi$
$19$	10.0000	0.120745	0.0603726	+	0.998176i	$0.480771\pi$
$20$	0	0
$21$	70.0000	0.727393
$22$	− 18.0000i	− 0.174437i
$23$	− 75.0000i	− 0.679938i	−0.940437	−	0.339969i	$-0.889583\pi$
$23$	− 75.0000i	− 0.679938i	0.940437	−	0.339969i	$-0.110417\pi$
$24$	80.0000	0.680414
$25$	0	0
$26$	104.000	0.784465
$27$	460.000i	3.27878i
$28$	− 28.0000i	− 0.188982i
$29$	−189.000	−1.21022	−0.605111	−	0.796141i	$-0.706871\pi$
$29$	−189.000	−1.21022	−0.605111	+	0.796141i	$0.706871\pi$
$30$	0	0
$31$	−232.000	−1.34414	−0.672071	−	0.740486i	$-0.734595\pi$
$31$	−232.000	−1.34414	−0.672071	+	0.740486i	$0.734595\pi$
$32$	− 32.0000i	− 0.176777i
$33$	− 90.0000i	− 0.474757i
$34$	192.000	0.968463
$35$	0	0
$36$	292.000	1.35185
$37$	305.000i	1.35518i	0.735439	+	0.677590i	$0.236976\pi$
$37$	305.000i	1.35518i	−0.735439	+	0.677590i	$0.763024\pi$
$38$	− 20.0000i	− 0.0853797i
$39$	520.000	2.13504
$40$	0	0
$41$	−438.000	−1.66839	−0.834196	−	0.551467i	$-0.814068\pi$
$41$	−438.000	−1.66839	−0.834196	+	0.551467i	$0.814068\pi$
$42$	− 140.000i	− 0.514344i
$43$	− 353.000i	− 1.25191i	−0.779860	−	0.625953i	$-0.784710\pi$
$43$	− 353.000i	− 1.25191i	0.779860	−	0.625953i	$-0.215290\pi$
$44$	−36.0000	−0.123346
$45$	0	0
$46$	−150.000	−0.480789
$47$	− 486.000i	− 1.50831i	−0.656699	−	0.754153i	$-0.728048\pi$
$47$	− 486.000i	− 1.50831i	0.656699	−	0.754153i	$-0.271952\pi$
$48$	− 160.000i	− 0.481125i
$49$	−49.0000	−0.142857
$50$	0	0
$51$	960.000	2.63582
$52$	− 208.000i	− 0.554700i
$53$	354.000i	0.917465i	0.888574	+	0.458732i	$0.151696\pi$
$53$	354.000i	0.917465i	−0.888574	+	0.458732i	$0.848304\pi$
$54$	920.000	2.31845
$55$	0	0
$56$	−56.0000	−0.133631
$57$	− 100.000i	− 0.232374i
$58$	378.000i	0.855756i
$59$	672.000	1.48283	0.741415	−	0.671047i	$-0.234155\pi$
$59$	672.000	1.48283	0.741415	+	0.671047i	$0.234155\pi$
$60$	0	0
$61$	206.000	0.432387	0.216193	−	0.976351i	$-0.430636\pi$
$61$	206.000	0.432387	0.216193	+	0.976351i	$0.430636\pi$
$62$	464.000i	0.950453i
$63$	− 511.000i	− 1.02190i
$64$	−64.0000	−0.125000
$65$	0	0
$66$	−180.000	−0.335704
$67$	599.000i	1.09223i	0.837710	+	0.546116i	$0.183894\pi$
$67$	599.000i	1.09223i	−0.837710	+	0.546116i	$0.816106\pi$
$68$	− 384.000i	− 0.684806i
$69$	−750.000	−1.30854
$70$	0	0
$71$	−471.000	−0.787288	−0.393644	−	0.919263i	$-0.628786\pi$
$71$	−471.000	−0.787288	−0.393644	+	0.919263i	$0.628786\pi$
$72$	− 584.000i	− 0.955904i
$73$	− 614.000i	− 0.984428i	−0.870474	−	0.492214i	$-0.836188\pi$
$73$	− 614.000i	− 0.984428i	0.870474	−	0.492214i	$-0.163812\pi$
$74$	610.000	0.958258
$75$	0	0
$76$	−40.0000	−0.0603726
$77$	63.0000i	0.0932405i
$78$	− 1040.00i	− 1.50970i
$79$	−743.000	−1.05815	−0.529076	−	0.848574i	$-0.677461\pi$
$79$	−743.000	−1.05815	−0.529076	+	0.848574i	$0.677461\pi$
$80$	0	0
$81$	2629.00	3.60631
$82$	876.000i	1.17973i
$83$	− 996.000i	− 1.31717i	−0.752506	−	0.658586i	$-0.771155\pi$
$83$	− 996.000i	− 1.31717i	0.752506	−	0.658586i	$-0.228845\pi$
$84$	−280.000	−0.363696
$85$	0	0
$86$	−706.000	−0.885232
$87$	1890.00i	2.32907i
$88$	72.0000i	0.0872185i
$89$	−180.000	−0.214382	−0.107191	−	0.994238i	$-0.534186\pi$
$89$	−180.000	−0.214382	−0.107191	+	0.994238i	$0.534186\pi$
$90$	0	0
$91$	−364.000	−0.419314
$92$	300.000i	0.339969i
$93$	2320.00i	2.58680i
$94$	−972.000	−1.06653
$95$	0	0
$96$	−320.000	−0.340207
$97$	− 184.000i	− 0.192602i	−0.995352	−	0.0963009i	$-0.969299\pi$
$97$	− 184.000i	− 0.192602i	0.995352	−	0.0963009i	$-0.0307011\pi$
$98$	98.0000i	0.101015i
$99$	−657.000	−0.666980
$100$	0	0
$101$	−726.000	−0.715245	−0.357622	−	0.933866i	$-0.616412\pi$
$101$	−726.000	−0.715245	−0.357622	+	0.933866i	$0.616412\pi$
$102$	− 1920.00i	− 1.86381i
$103$	1798.00i	1.72002i	0.510276	+	0.860011i	$0.329543\pi$
$103$	1798.00i	1.72002i	−0.510276	+	0.860011i	$0.670457\pi$
$104$	−416.000	−0.392232
$105$	0	0
$106$	708.000	0.648746
$107$	876.000i	0.791459i	0.918367	+	0.395730i	$0.129508\pi$
$107$	876.000i	0.791459i	−0.918367	+	0.395730i	$0.870492\pi$
$108$	− 1840.00i	− 1.63939i
$109$	691.000	0.607209	0.303605	−	0.952798i	$-0.401810\pi$
$109$	691.000	0.607209	0.303605	+	0.952798i	$0.401810\pi$
$110$	0	0
$111$	3050.00	2.60805
$112$	112.000i	0.0944911i
$113$	1521.00i	1.26623i	0.774059	+	0.633113i	$0.218223\pi$
$113$	1521.00i	1.26623i	−0.774059	+	0.633113i	$0.781777\pi$
$114$	−200.000	−0.164313
$115$	0	0
$116$	756.000	0.605111
$117$	− 3796.00i	− 2.99949i
$118$	− 1344.00i	− 1.04852i
$119$	−672.000	−0.517665
$120$	0	0
$121$	−1250.00	−0.939144
$122$	− 412.000i	− 0.305744i
$123$	4380.00i	3.21082i
$124$	928.000	0.672071
$125$	0	0
$126$	−1022.00	−0.722595
$127$	1031.00i	0.720366i	0.932882	+	0.360183i	$0.117286\pi$
$127$	1031.00i	0.720366i	−0.932882	+	0.360183i	$0.882714\pi$
$128$	128.000i	0.0883883i
$129$	−3530.00	−2.40930
$130$	0	0
$131$	−1116.00	−0.744316	−0.372158	−	0.928169i	$-0.621382\pi$
$131$	−1116.00	−0.744316	−0.372158	+	0.928169i	$0.621382\pi$
$132$	360.000i	0.237379i
$133$	70.0000i	0.0456374i
$134$	1198.00	0.772324
$135$	0	0
$136$	−768.000	−0.484231
$137$	− 1398.00i	− 0.871819i	−0.899991	−	0.435909i	$-0.856427\pi$
$137$	− 1398.00i	− 0.871819i	0.899991	−	0.435909i	$-0.143573\pi$
$138$	1500.00i	0.925279i
$139$	−674.000	−0.411280	−0.205640	−	0.978628i	$-0.565928\pi$
$139$	−674.000	−0.411280	−0.205640	+	0.978628i	$0.565928\pi$
$140$	0	0
$141$	−4860.00	−2.90274
$142$	942.000i	0.556696i
$143$	468.000i	0.273679i
$144$	−1168.00	−0.675926
$145$	0	0
$146$	−1228.00	−0.696096
$147$	490.000i	0.274929i
$148$	− 1220.00i	− 0.677590i
$149$	1281.00	0.704320	0.352160	−	0.935940i	$-0.385447\pi$
$149$	1281.00	0.704320	0.352160	+	0.935940i	$0.385447\pi$
$150$	0	0
$151$	953.000	0.513603	0.256801	−	0.966464i	$-0.417331\pi$
$151$	953.000	0.513603	0.256801	+	0.966464i	$0.417331\pi$
$152$	80.0000i	0.0426898i
$153$	− 7008.00i	− 3.70303i
$154$	126.000	0.0659310
$155$	0	0
$156$	−2080.00	−1.06752
$157$	650.000i	0.330418i	0.986259	+	0.165209i	$0.0528299\pi$
$157$	650.000i	0.330418i	−0.986259	+	0.165209i	$0.947170\pi$
$158$	1486.00i	0.748227i
$159$	3540.00	1.76566
$160$	0	0
$161$	525.000	0.256993
$162$	− 5258.00i	− 2.55005i
$163$	− 932.000i	− 0.447852i	−0.974606	−	0.223926i	$-0.928113\pi$
$163$	− 932.000i	− 0.447852i	0.974606	−	0.223926i	$-0.0718874\pi$
$164$	1752.00	0.834196
$165$	0	0
$166$	−1992.00	−0.931381
$167$	− 180.000i	− 0.0834061i	−0.999130	−	0.0417030i	$-0.986722\pi$
$167$	− 180.000i	− 0.0834061i	0.999130	−	0.0417030i	$-0.0132783\pi$
$168$	560.000i	0.257172i
$169$	−507.000	−0.230769
$170$	0	0
$171$	−730.000	−0.326459
$172$	1412.00i	0.625953i
$173$	834.000i	0.366519i	0.983065	+	0.183260i	$0.0586649\pi$
$173$	834.000i	0.366519i	−0.983065	+	0.183260i	$0.941335\pi$
$174$	3780.00	1.64690
$175$	0	0
$176$	144.000	0.0616728
$177$	− 6720.00i	− 2.85371i
$178$	360.000i	0.151591i
$179$	648.000	0.270580	0.135290	−	0.990806i	$-0.456803\pi$
$179$	648.000	0.270580	0.135290	+	0.990806i	$0.456803\pi$
$180$	0	0
$181$	−2914.00	−1.19666	−0.598331	−	0.801249i	$-0.704169\pi$
$181$	−2914.00	−1.19666	−0.598331	+	0.801249i	$0.704169\pi$
$182$	728.000i	0.296500i
$183$	− 2060.00i	− 0.832129i
$184$	600.000	0.240394
$185$	0	0
$186$	4640.00	1.82915
$187$	864.000i	0.337871i
$188$	1944.00i	0.754153i
$189$	−3220.00	−1.23926
$190$	0	0
$191$	−876.000	−0.331859	−0.165930	−	0.986138i	$-0.553062\pi$
$191$	−876.000	−0.331859	−0.165930	+	0.986138i	$0.553062\pi$
$192$	640.000i	0.240563i
$193$	601.000i	0.224150i	0.993700	+	0.112075i	$0.0357497\pi$
$193$	601.000i	0.224150i	−0.993700	+	0.112075i	$0.964250\pi$
$194$	−368.000	−0.136190
$195$	0	0
$196$	196.000	0.0714286
$197$	− 2013.00i	− 0.728022i	−0.931395	−	0.364011i	$-0.881407\pi$
$197$	− 2013.00i	− 0.728022i	0.931395	−	0.364011i	$-0.118593\pi$
$198$	1314.00i	0.471626i
$199$	−326.000	−0.116128	−0.0580641	−	0.998313i	$-0.518493\pi$
$199$	−326.000	−0.116128	−0.0580641	+	0.998313i	$0.518493\pi$
$200$	0	0
$201$	5990.00	2.10200
$202$	1452.00i	0.505754i
$203$	− 1323.00i	− 0.457421i
$204$	−3840.00	−1.31791
$205$	0	0
$206$	3596.00	1.21624
$207$	5475.00i	1.83835i
$208$	832.000i	0.277350i
$209$	90.0000	0.0297867
$210$	0	0
$211$	−5956.00	−1.94326	−0.971630	−	0.236505i	$-0.923998\pi$
$211$	−5956.00	−1.94326	−0.971630	+	0.236505i	$0.923998\pi$
$212$	− 1416.00i	− 0.458732i
$213$	4710.00i	1.51514i
$214$	1752.00	0.559646
$215$	0	0
$216$	−3680.00	−1.15922
$217$	− 1624.00i	− 0.508038i
$218$	− 1382.00i	− 0.429362i
$219$	−6140.00	−1.89453
$220$	0	0
$221$	−4992.00	−1.51945
$222$	− 6100.00i	− 1.84417i
$223$	3118.00i	0.936308i	0.883647	+	0.468154i	$0.155081\pi$
$223$	3118.00i	0.936308i	−0.883647	+	0.468154i	$0.844919\pi$
$224$	224.000	0.0668153
$225$	0	0
$226$	3042.00	0.895358
$227$	− 6.00000i	− 0.00175433i	−1.00000		0.000877167i	$-0.999721\pi$
$227$	− 6.00000i	− 0.00175433i	1.00000		0.000877167i	$-0.000279211\pi$
$228$	400.000i	0.116187i
$229$	586.000	0.169100	0.0845502	−	0.996419i	$-0.473055\pi$
$229$	586.000	0.169100	0.0845502	+	0.996419i	$0.473055\pi$
$230$	0	0
$231$	630.000	0.179441
$232$	− 1512.00i	− 0.427878i
$233$	− 1293.00i	− 0.363550i	−0.983340	−	0.181775i	$-0.941816\pi$
$233$	− 1293.00i	− 0.363550i	0.983340	−	0.181775i	$-0.0581843\pi$
$234$	−7592.00	−2.12096
$235$	0	0
$236$	−2688.00	−0.741415
$237$	7430.00i	2.03642i
$238$	1344.00i	0.366044i
$239$	5376.00	1.45500	0.727499	−	0.686109i	$-0.240683\pi$
$239$	5376.00	1.45500	0.727499	+	0.686109i	$0.240683\pi$
$240$	0	0
$241$	−670.000	−0.179081	−0.0895404	−	0.995983i	$-0.528540\pi$
$241$	−670.000	−0.179081	−0.0895404	+	0.995983i	$0.528540\pi$
$242$	2500.00i	0.664075i
$243$	− 13870.0i	− 3.66157i
$244$	−824.000	−0.216193
$245$	0	0
$246$	8760.00	2.27040
$247$	520.000i	0.133955i
$248$	− 1856.00i	− 0.475226i
$249$	−9960.00	−2.53490
$250$	0	0
$251$	1380.00	0.347031	0.173516	−	0.984831i	$-0.444487\pi$
$251$	1380.00	0.347031	0.173516	+	0.984831i	$0.444487\pi$
$252$	2044.00i	0.510952i
$253$	− 675.000i	− 0.167735i
$254$	2062.00	0.509376
$255$	0	0
$256$	256.000	0.0625000
$257$	− 3576.00i	− 0.867956i	−0.900923	−	0.433978i	$-0.857110\pi$
$257$	− 3576.00i	− 0.867956i	0.900923	−	0.433978i	$-0.142890\pi$
$258$	7060.00i	1.70363i
$259$	−2135.00	−0.512210
$260$	0	0
$261$	13797.0	3.27208
$262$	2232.00i	0.526311i
$263$	5919.00i	1.38776i	0.720090	+	0.693881i	$0.244100\pi$
$263$	5919.00i	1.38776i	−0.720090	+	0.693881i	$0.755900\pi$
$264$	720.000	0.167852
$265$	0	0
$266$	140.000	0.0322705
$267$	1800.00i	0.412578i
$268$	− 2396.00i	− 0.546116i
$269$	1764.00	0.399825	0.199913	−	0.979814i	$-0.435934\pi$
$269$	1764.00	0.399825	0.199913	+	0.979814i	$0.435934\pi$
$270$	0	0
$271$	−340.000	−0.0762123	−0.0381061	−	0.999274i	$-0.512132\pi$
$271$	−340.000	−0.0762123	−0.0381061	+	0.999274i	$0.512132\pi$
$272$	1536.00i	0.342403i
$273$	3640.00i	0.806970i
$274$	−2796.00	−0.616469
$275$	0	0
$276$	3000.00	0.654271
$277$	4706.00i	1.02078i	0.859943	+	0.510390i	$0.170499\pi$
$277$	4706.00i	1.02078i	−0.859943	+	0.510390i	$0.829501\pi$
$278$	1348.00i	0.290819i
$279$	16936.0	3.63416
$280$	0	0
$281$	6681.00	1.41835	0.709173	−	0.705035i	$-0.249069\pi$
$281$	6681.00	1.41835	0.709173	+	0.705035i	$0.249069\pi$
$282$	9720.00i	2.05254i
$283$	226.000i	0.0474710i	0.999718	+	0.0237355i	$0.00755596\pi$
$283$	226.000i	0.0474710i	−0.999718	+	0.0237355i	$0.992444\pi$
$284$	1884.00	0.393644
$285$	0	0
$286$	936.000	0.193520
$287$	− 3066.00i	− 0.630593i
$288$	2336.00i	0.477952i
$289$	−4303.00	−0.875840
$290$	0	0
$291$	−1840.00	−0.370662
$292$	2456.00i	0.492214i
$293$	− 4320.00i	− 0.861355i	−0.902506	−	0.430678i	$-0.858275\pi$
$293$	− 4320.00i	− 0.861355i	0.902506	−	0.430678i	$-0.141725\pi$
$294$	980.000	0.194404
$295$	0	0
$296$	−2440.00	−0.479129
$297$	4140.00i	0.808846i
$298$	− 2562.00i	− 0.498029i
$299$	3900.00	0.754324
$300$	0	0
$301$	2471.00	0.473176
$302$	− 1906.00i	− 0.363172i
$303$	7260.00i	1.37649i
$304$	160.000	0.0301863
$305$	0	0
$306$	−14016.0	−2.61844
$307$	− 6604.00i	− 1.22772i	−0.789415	−	0.613860i	$-0.789616\pi$
$307$	− 6604.00i	− 1.22772i	0.789415	−	0.613860i	$-0.210384\pi$
$308$	− 252.000i	− 0.0466202i
$309$	17980.0	3.31018
$310$	0	0
$311$	6036.00	1.10055	0.550274	−	0.834984i	$-0.314523\pi$
$311$	6036.00	1.10055	0.550274	+	0.834984i	$0.314523\pi$
$312$	4160.00i	0.754851i
$313$	− 9146.00i	− 1.65164i	−0.563936	−	0.825819i	$-0.690713\pi$
$313$	− 9146.00i	− 1.65164i	0.563936	−	0.825819i	$-0.309287\pi$
$314$	1300.00	0.233641
$315$	0	0
$316$	2972.00	0.529076
$317$	− 4449.00i	− 0.788267i	−0.919053	−	0.394134i	$-0.871045\pi$
$317$	− 4449.00i	− 0.788267i	0.919053	−	0.394134i	$-0.128955\pi$
$318$	− 7080.00i	− 1.24851i
$319$	−1701.00	−0.298551
$320$	0	0
$321$	8760.00	1.52316
$322$	− 1050.00i	− 0.181721i
$323$	960.000i	0.165374i
$324$	−10516.0	−1.80316
$325$	0	0
$326$	−1864.00	−0.316679
$327$	− 6910.00i	− 1.16857i
$328$	− 3504.00i	− 0.589866i
$329$	3402.00	0.570086
$330$	0	0
$331$	−10081.0	−1.67402	−0.837012	−	0.547185i	$-0.815700\pi$
$331$	−10081.0	−1.67402	−0.837012	+	0.547185i	$0.815700\pi$
$332$	3984.00i	0.658586i
$333$	− 22265.0i	− 3.66401i
$334$	−360.000	−0.0589770
$335$	0	0
$336$	1120.00	0.181848
$337$	7778.00i	1.25725i	0.777707	+	0.628627i	$0.216383\pi$
$337$	7778.00i	1.25725i	−0.777707	+	0.628627i	$0.783617\pi$
$338$	1014.00i	0.163178i
$339$	15210.0	2.43685
$340$	0	0
$341$	−2088.00	−0.331588
$342$	1460.00i	0.230841i
$343$	− 343.000i	− 0.0539949i
$344$	2824.00	0.442616
$345$	0	0
$346$	1668.00	0.259168
$347$	1017.00i	0.157336i	0.996901	+	0.0786678i	$0.0250666\pi$
$347$	1017.00i	0.157336i	−0.996901	+	0.0786678i	$0.974933\pi$
$348$	− 7560.00i	− 1.16454i
$349$	−10370.0	−1.59053	−0.795263	−	0.606265i	$-0.792667\pi$
$349$	−10370.0	−1.59053	−0.795263	+	0.606265i	$0.792667\pi$
$350$	0	0
$351$	−23920.0	−3.63748
$352$	− 288.000i	− 0.0436092i
$353$	9432.00i	1.42214i	0.703122	+	0.711069i	$0.251789\pi$
$353$	9432.00i	1.42214i	−0.703122	+	0.711069i	$0.748211\pi$
$354$	−13440.0	−2.01788
$355$	0	0
$356$	720.000	0.107191
$357$	6720.00i	0.996247i
$358$	− 1296.00i	− 0.191329i
$359$	−7557.00	−1.11098	−0.555492	−	0.831522i	$-0.687470\pi$
$359$	−7557.00	−1.11098	−0.555492	+	0.831522i	$0.687470\pi$
$360$	0	0
$361$	−6759.00	−0.985421
$362$	5828.00i	0.846168i
$363$	12500.0i	1.80738i
$364$	1456.00	0.209657
$365$	0	0
$366$	−4120.00	−0.588404
$367$	− 11662.0i	− 1.65872i	−0.558712	−	0.829362i	$-0.688704\pi$
$367$	− 11662.0i	− 1.65872i	0.558712	−	0.829362i	$-0.311296\pi$
$368$	− 1200.00i	− 0.169985i
$369$	31974.0	4.51084
$370$	0	0
$371$	−2478.00	−0.346769
$372$	− 9280.00i	− 1.29340i
$373$	2377.00i	0.329964i	0.986297	+	0.164982i	$0.0527565\pi$
$373$	2377.00i	0.329964i	−0.986297	+	0.164982i	$0.947243\pi$
$374$	1728.00	0.238911
$375$	0	0
$376$	3888.00	0.533267
$377$	− 9828.00i	− 1.34262i
$378$	6440.00i	0.876291i
$379$	−4427.00	−0.599999	−0.300000	−	0.953939i	$-0.596987\pi$
$379$	−4427.00	−0.599999	−0.300000	+	0.953939i	$0.596987\pi$
$380$	0	0
$381$	10310.0	1.38634
$382$	1752.00i	0.234660i
$383$	− 4608.00i	− 0.614772i	−0.951585	−	0.307386i	$-0.900546\pi$
$383$	− 4608.00i	− 0.614772i	0.951585	−	0.307386i	$-0.0994543\pi$
$384$	1280.00	0.170103
$385$	0	0
$386$	1202.00	0.158498
$387$	25769.0i	3.38479i
$388$	736.000i	0.0963009i
$389$	699.000	0.0911072	0.0455536	−	0.998962i	$-0.485495\pi$
$389$	699.000	0.0911072	0.0455536	+	0.998962i	$0.485495\pi$
$390$	0	0
$391$	7200.00	0.931252
$392$	− 392.000i	− 0.0505076i
$393$	11160.0i	1.43244i
$394$	−4026.00	−0.514789
$395$	0	0
$396$	2628.00	0.333490
$397$	− 7630.00i	− 0.964581i	−0.876011	−	0.482291i	$-0.839805\pi$
$397$	− 7630.00i	− 0.964581i	0.876011	−	0.482291i	$-0.160195\pi$
$398$	652.000i	0.0821151i
$399$	700.000	0.0878292
$400$	0	0
$401$	5601.00	0.697508	0.348754	−	0.937214i	$-0.386605\pi$
$401$	5601.00	0.697508	0.348754	+	0.937214i	$0.386605\pi$
$402$	− 11980.0i	− 1.48634i
$403$	− 12064.0i	− 1.49119i
$404$	2904.00	0.357622
$405$	0	0
$406$	−2646.00	−0.323445
$407$	2745.00i	0.334311i
$408$	7680.00i	0.931904i
$409$	−4670.00	−0.564588	−0.282294	−	0.959328i	$-0.591095\pi$
$409$	−4670.00	−0.564588	−0.282294	+	0.959328i	$0.591095\pi$
$410$	0	0
$411$	−13980.0	−1.67782
$412$	− 7192.00i	− 0.860011i
$413$	4704.00i	0.560457i
$414$	10950.0	1.29991
$415$	0	0
$416$	1664.00	0.196116
$417$	6740.00i	0.791509i
$418$	− 180.000i	− 0.0210624i
$419$	−36.0000	−0.00419741	−0.00209871	−	0.999998i	$-0.500668\pi$
$419$	−36.0000	−0.00419741	−0.00209871	+	0.999998i	$0.500668\pi$
$420$	0	0
$421$	5495.00	0.636128	0.318064	−	0.948069i	$-0.396967\pi$
$421$	5495.00	0.636128	0.318064	+	0.948069i	$0.396967\pi$
$422$	11912.0i	1.37409i
$423$	35478.0i	4.07801i
$424$	−2832.00	−0.324373
$425$	0	0
$426$	9420.00	1.07136
$427$	1442.00i	0.163427i
$428$	− 3504.00i	− 0.395730i
$429$	4680.00	0.526696
$430$	0	0
$431$	2700.00	0.301750	0.150875	−	0.988553i	$-0.451791\pi$
$431$	2700.00	0.301750	0.150875	+	0.988553i	$0.451791\pi$
$432$	7360.00i	0.819695i
$433$	− 15104.0i	− 1.67633i	−0.545415	−	0.838166i	$-0.683628\pi$
$433$	− 15104.0i	− 1.67633i	0.545415	−	0.838166i	$-0.316372\pi$
$434$	−3248.00	−0.359237
$435$	0	0
$436$	−2764.00	−0.303605
$437$	− 750.000i	− 0.0820992i
$438$	12280.0i	1.33964i
$439$	−14948.0	−1.62512	−0.812562	−	0.582875i	$-0.801928\pi$
$439$	−14948.0	−1.62512	−0.812562	+	0.582875i	$0.801928\pi$
$440$	0	0
$441$	3577.00	0.386243
$442$	9984.00i	1.07441i
$443$	− 4980.00i	− 0.534101i	−0.963682	−	0.267051i	$-0.913951\pi$
$443$	− 4980.00i	− 0.534101i	0.963682	−	0.267051i	$-0.0860491\pi$
$444$	−12200.0	−1.30402
$445$	0	0
$446$	6236.00	0.662070
$447$	− 12810.0i	− 1.35546i
$448$	− 448.000i	− 0.0472456i
$449$	−12375.0	−1.30070	−0.650348	−	0.759637i	$-0.725377\pi$
$449$	−12375.0	−1.30070	−0.650348	+	0.759637i	$0.725377\pi$
$450$	0	0
$451$	−3942.00	−0.411578
$452$	− 6084.00i	− 0.633113i
$453$	− 9530.00i	− 0.988429i
$454$	−12.0000	−0.00124050
$455$	0	0
$456$	800.000	0.0821567
$457$	10835.0i	1.10906i	0.832164	+	0.554529i	$0.187102\pi$
$457$	10835.0i	1.10906i	−0.832164	+	0.554529i	$0.812898\pi$
$458$	− 1172.00i	− 0.119572i
$459$	−44160.0	−4.49066
$460$	0	0
$461$	5700.00	0.575869	0.287934	−	0.957650i	$-0.407032\pi$
$461$	5700.00	0.575869	0.287934	+	0.957650i	$0.407032\pi$
$462$	− 1260.00i	− 0.126884i
$463$	− 6128.00i	− 0.615102i	−0.951532	−	0.307551i	$-0.900491\pi$
$463$	− 6128.00i	− 0.615102i	0.951532	−	0.307551i	$-0.0995095\pi$
$464$	−3024.00	−0.302555
$465$	0	0
$466$	−2586.00	−0.257069
$467$	− 9810.00i	− 0.972061i	−0.873942	−	0.486031i	$-0.838444\pi$
$467$	− 9810.00i	− 0.972061i	0.873942	−	0.486031i	$-0.161556\pi$
$468$	15184.0i	1.49974i
$469$	−4193.00	−0.412825
$470$	0	0
$471$	6500.00	0.635890
$472$	5376.00i	0.524259i
$473$	− 3177.00i	− 0.308834i
$474$	14860.0	1.43996
$475$	0	0
$476$	2688.00	0.258833
$477$	− 25842.0i	− 2.48055i
$478$	− 10752.0i	− 1.02884i
$479$	204.000	0.0194593	0.00972964	−	0.999953i	$-0.496903\pi$
$479$	204.000	0.0194593	0.00972964	+	0.999953i	$0.496903\pi$
$480$	0	0
$481$	−15860.0	−1.50344
$482$	1340.00i	0.126629i
$483$	− 5250.00i	− 0.494582i
$484$	5000.00	0.469572
$485$	0	0
$486$	−27740.0	−2.58912
$487$	15401.0i	1.43303i	0.697571	+	0.716515i	$0.254264\pi$
$487$	15401.0i	1.43303i	−0.697571	+	0.716515i	$0.745736\pi$
$488$	1648.00i	0.152872i
$489$	−9320.00	−0.861892
$490$	0	0
$491$	3897.00	0.358186	0.179093	−	0.983832i	$-0.442684\pi$
$491$	3897.00	0.358186	0.179093	+	0.983832i	$0.442684\pi$
$492$	− 17520.0i	− 1.60541i
$493$	− 18144.0i	− 1.65753i
$494$	1040.00	0.0947203
$495$	0	0
$496$	−3712.00	−0.336036
$497$	− 3297.00i	− 0.297567i
$498$	19920.0i	1.79244i
$499$	−8132.00	−0.729536	−0.364768	−	0.931098i	$-0.618852\pi$
$499$	−8132.00	−0.729536	−0.364768	+	0.931098i	$0.618852\pi$
$500$	0	0
$501$	−1800.00	−0.160515
$502$	− 2760.00i	− 0.245388i
$503$	10998.0i	0.974904i	0.873150	+	0.487452i	$0.162074\pi$
$503$	10998.0i	0.974904i	−0.873150	+	0.487452i	$0.837926\pi$
$504$	4088.00	0.361298
$505$	0	0
$506$	−1350.00	−0.118606
$507$	5070.00i	0.444116i
$508$	− 4124.00i	− 0.360183i
$509$	−5940.00	−0.517261	−0.258631	−	0.965976i	$-0.583271\pi$
$509$	−5940.00	−0.517261	−0.258631	+	0.965976i	$0.583271\pi$
$510$	0	0
$511$	4298.00	0.372079
$512$	− 512.000i	− 0.0441942i
$513$	4600.00i	0.395897i
$514$	−7152.00	−0.613738
$515$	0	0
$516$	14120.0	1.20465
$517$	− 4374.00i	− 0.372086i
$518$	4270.00i	0.362187i
$519$	8340.00	0.705367
$520$	0	0
$521$	17022.0	1.43138	0.715688	−	0.698420i	$-0.246113\pi$
$521$	17022.0	1.43138	0.715688	+	0.698420i	$0.246113\pi$
$522$	− 27594.0i	− 2.31371i
$523$	15748.0i	1.31666i	0.752730	+	0.658329i	$0.228736\pi$
$523$	15748.0i	1.31666i	−0.752730	+	0.658329i	$0.771264\pi$
$524$	4464.00	0.372158
$525$	0	0
$526$	11838.0	0.981295
$527$	− 22272.0i	− 1.84096i
$528$	− 1440.00i	− 0.118689i
$529$	6542.00	0.537684
$530$	0	0
$531$	−49056.0	−4.00913
$532$	− 280.000i	− 0.0228187i
$533$	− 22776.0i	− 1.85092i
$534$	3600.00	0.291736
$535$	0	0
$536$	−4792.00	−0.386162
$537$	− 6480.00i	− 0.520731i
$538$	− 3528.00i	− 0.282719i
$539$	−441.000	−0.0352416
$540$	0	0
$541$	−3373.00	−0.268053	−0.134026	−	0.990978i	$-0.542791\pi$
$541$	−3373.00	−0.268053	−0.134026	+	0.990978i	$0.542791\pi$
$542$	680.000i	0.0538902i
$543$	29140.0i	2.30298i
$544$	3072.00	0.242116
$545$	0	0
$546$	7280.00	0.570614
$547$	− 14389.0i	− 1.12473i	−0.826888	−	0.562367i	$-0.809891\pi$
$547$	− 14389.0i	− 1.12473i	0.826888	−	0.562367i	$-0.190109\pi$
$548$	5592.00i	0.435909i
$549$	−15038.0	−1.16905
$550$	0	0
$551$	−1890.00	−0.146128
$552$	− 6000.00i	− 0.462639i
$553$	− 5201.00i	− 0.399944i
$554$	9412.00	0.721801
$555$	0	0
$556$	2696.00	0.205640
$557$	− 4929.00i	− 0.374952i	−0.982269	−	0.187476i	$-0.939969\pi$
$557$	− 4929.00i	− 0.374952i	0.982269	−	0.187476i	$-0.0600307\pi$
$558$	− 33872.0i	− 2.56974i
$559$	18356.0	1.38887
$560$	0	0
$561$	8640.00	0.650234
$562$	− 13362.0i	− 1.00292i
$563$	− 15678.0i	− 1.17362i	−0.809724	−	0.586811i	$-0.800383\pi$
$563$	− 15678.0i	− 1.17362i	0.809724	−	0.586811i	$-0.199617\pi$
$564$	19440.0	1.45137
$565$	0	0
$566$	452.000	0.0335671
$567$	18403.0i	1.36306i
$568$	− 3768.00i	− 0.278348i
$569$	14499.0	1.06824	0.534121	−	0.845408i	$-0.320643\pi$
$569$	14499.0	1.06824	0.534121	+	0.845408i	$0.320643\pi$
$570$	0	0
$571$	−9457.00	−0.693105	−0.346553	−	0.938031i	$-0.612648\pi$
$571$	−9457.00	−0.693105	−0.346553	+	0.938031i	$0.612648\pi$
$572$	− 1872.00i	− 0.136840i
$573$	8760.00i	0.638664i
$574$	−6132.00	−0.445897
$575$	0	0
$576$	4672.00	0.337963
$577$	10370.0i	0.748195i	0.927389	+	0.374098i	$0.122048\pi$
$577$	10370.0i	0.748195i	−0.927389	+	0.374098i	$0.877952\pi$
$578$	8606.00i	0.619312i
$579$	6010.00	0.431377
$580$	0	0
$581$	6972.00	0.497844
$582$	3680.00i	0.262098i
$583$	3186.00i	0.226330i
$584$	4912.00	0.348048
$585$	0	0
$586$	−8640.00	−0.609070
$587$	− 12126.0i	− 0.852630i	−0.904575	−	0.426315i	$-0.859812\pi$
$587$	− 12126.0i	− 0.852630i	0.904575	−	0.426315i	$-0.140188\pi$
$588$	− 1960.00i	− 0.137464i
$589$	−2320.00	−0.162299
$590$	0	0
$591$	−20130.0	−1.40108
$592$	4880.00i	0.338795i
$593$	− 1068.00i	− 0.0739587i	−0.999316	−	0.0369793i	$-0.988226\pi$
$593$	− 1068.00i	− 0.0739587i	0.999316	−	0.0369793i	$-0.0117736\pi$
$594$	8280.00	0.571940
$595$	0	0
$596$	−5124.00	−0.352160
$597$	3260.00i	0.223489i
$598$	− 7800.00i	− 0.533387i
$599$	−3375.00	−0.230215	−0.115107	−	0.993353i	$-0.536721\pi$
$599$	−3375.00	−0.230215	−0.115107	+	0.993353i	$0.536721\pi$
$600$	0	0
$601$	−27448.0	−1.86294	−0.931470	−	0.363817i	$-0.881473\pi$
$601$	−27448.0	−1.86294	−0.931470	+	0.363817i	$0.881473\pi$
$602$	− 4942.00i	− 0.334586i
$603$	− 43727.0i	− 2.95307i
$604$	−3812.00	−0.256801
$605$	0	0
$606$	14520.0	0.973325
$607$	3884.00i	0.259714i	0.991533	+	0.129857i	$0.0414519\pi$
$607$	3884.00i	0.259714i	−0.991533	+	0.129857i	$0.958548\pi$
$608$	− 320.000i	− 0.0213449i
$609$	−13230.0	−0.880306
$610$	0	0
$611$	25272.0	1.67332
$612$	28032.0i	1.85151i
$613$	3643.00i	0.240032i	0.992772	+	0.120016i	$0.0382945\pi$
$613$	3643.00i	0.240032i	−0.992772	+	0.120016i	$0.961705\pi$
$614$	−13208.0	−0.868129
$615$	0	0
$616$	−504.000	−0.0329655
$617$	30369.0i	1.98154i	0.135555	+	0.990770i	$0.456718\pi$
$617$	30369.0i	1.98154i	−0.135555	+	0.990770i	$0.543282\pi$
$618$	− 35960.0i	− 2.34065i
$619$	10888.0	0.706988	0.353494	−	0.935437i	$-0.384993\pi$
$619$	10888.0	0.706988	0.353494	+	0.935437i	$0.384993\pi$
$620$	0	0
$621$	34500.0	2.22937
$622$	− 12072.0i	− 0.778204i
$623$	− 1260.00i	− 0.0810286i
$624$	8320.00	0.533761
$625$	0	0
$626$	−18292.0	−1.16788
$627$	− 900.000i	− 0.0573246i
$628$	− 2600.00i	− 0.165209i
$629$	−29280.0	−1.85607
$630$	0	0
$631$	28499.0	1.79798	0.898992	−	0.437966i	$-0.144301\pi$
$631$	28499.0	1.79798	0.898992	+	0.437966i	$0.144301\pi$
$632$	− 5944.00i	− 0.374113i
$633$	59560.0i	3.73981i
$634$	−8898.00	−0.557389
$635$	0	0
$636$	−14160.0	−0.882831
$637$	− 2548.00i	− 0.158486i
$638$	3402.00i	0.211107i
$639$	34383.0	2.12859
$640$	0	0
$641$	5817.00	0.358436	0.179218	−	0.983809i	$-0.442643\pi$
$641$	5817.00	0.358436	0.179218	+	0.983809i	$0.442643\pi$
$642$	− 17520.0i	− 1.07704i
$643$	− 22376.0i	− 1.37235i	−0.727435	−	0.686177i	$-0.759288\pi$
$643$	− 22376.0i	− 1.37235i	0.727435	−	0.686177i	$-0.240712\pi$
$644$	−2100.00	−0.128496
$645$	0	0
$646$	1920.00	0.116937
$647$	3018.00i	0.183385i	0.995787	+	0.0916923i	$0.0292276\pi$
$647$	3018.00i	0.183385i	−0.995787	+	0.0916923i	$0.970772\pi$
$648$	21032.0i	1.27502i
$649$	6048.00	0.365801
$650$	0	0
$651$	−16240.0	−0.977720
$652$	3728.00i	0.223926i
$653$	29682.0i	1.77878i	0.457145	+	0.889392i	$0.348872\pi$
$653$	29682.0i	1.77878i	−0.457145	+	0.889392i	$0.651128\pi$
$654$	−13820.0	−0.826307
$655$	0	0
$656$	−7008.00	−0.417098
$657$	44822.0i	2.66160i
$658$	− 6804.00i	− 0.403112i
$659$	−2052.00	−0.121297	−0.0606484	−	0.998159i	$-0.519317\pi$
$659$	−2052.00	−0.121297	−0.0606484	+	0.998159i	$0.519317\pi$
$660$	0	0
$661$	14222.0	0.836871	0.418435	−	0.908247i	$-0.362579\pi$
$661$	14222.0	0.836871	0.418435	+	0.908247i	$0.362579\pi$
$662$	20162.0i	1.18371i
$663$	49920.0i	2.92418i
$664$	7968.00	0.465690
$665$	0	0
$666$	−44530.0	−2.59084
$667$	14175.0i	0.822876i
$668$	720.000i	0.0417030i
$669$	31180.0	1.80193
$670$	0	0
$671$	1854.00	0.106666
$672$	− 2240.00i	− 0.128586i
$673$	− 20942.0i	− 1.19949i	−0.800192	−	0.599744i	$-0.795269\pi$
$673$	− 20942.0i	− 1.19949i	0.800192	−	0.599744i	$-0.204731\pi$
$674$	15556.0	0.889013
$675$	0	0
$676$	2028.00	0.115385
$677$	13074.0i	0.742208i	0.928591	+	0.371104i	$0.121021\pi$
$677$	13074.0i	0.742208i	−0.928591	+	0.371104i	$0.878979\pi$
$678$	− 30420.0i	− 1.72312i
$679$	1288.00	0.0727966
$680$	0	0
$681$	−60.0000	−0.00337622
$682$	4176.00i	0.234468i
$683$	31383.0i	1.75818i	0.476656	+	0.879090i	$0.341849\pi$
$683$	31383.0i	1.75818i	−0.476656	+	0.879090i	$0.658151\pi$
$684$	2920.00	0.163230
$685$	0	0
$686$	−686.000	−0.0381802
$687$	− 5860.00i	− 0.325434i
$688$	− 5648.00i	− 0.312977i
$689$	−18408.0	−1.01784
$690$	0	0
$691$	−622.000	−0.0342431	−0.0171216	−	0.999853i	$-0.505450\pi$
$691$	−622.000	−0.0342431	−0.0171216	+	0.999853i	$0.505450\pi$
$692$	− 3336.00i	− 0.183260i
$693$	− 4599.00i	− 0.252095i
$694$	2034.00	0.111253
$695$	0	0
$696$	−15120.0	−0.823451
$697$	− 42048.0i	− 2.28505i
$698$	20740.0i	1.12467i
$699$	−12930.0	−0.699653
$700$	0	0
$701$	7782.00	0.419290	0.209645	−	0.977778i	$-0.432769\pi$
$701$	7782.00	0.419290	0.209645	+	0.977778i	$0.432769\pi$
$702$	47840.0i	2.57209i
$703$	3050.00i	0.163631i
$704$	−576.000	−0.0308364
$705$	0	0
$706$	18864.0	1.00560
$707$	− 5082.00i	− 0.270337i
$708$	26880.0i	1.42685i
$709$	−7502.00	−0.397382	−0.198691	−	0.980062i	$-0.563669\pi$
$709$	−7502.00	−0.397382	−0.198691	+	0.980062i	$0.563669\pi$
$710$	0	0
$711$	54239.0	2.86093
$712$	− 1440.00i	− 0.0757953i
$713$	17400.0i	0.913934i
$714$	13440.0	0.704453
$715$	0	0
$716$	−2592.00	−0.135290
$717$	− 53760.0i	− 2.80015i
$718$	15114.0i	0.785584i
$719$	20814.0	1.07960	0.539799	−	0.841794i	$-0.318500\pi$
$719$	20814.0	1.07960	0.539799	+	0.841794i	$0.318500\pi$
$720$	0	0
$721$	−12586.0	−0.650107
$722$	13518.0i	0.696798i
$723$	6700.00i	0.344641i
$724$	11656.0	0.598331
$725$	0	0
$726$	25000.0	1.27801
$727$	14360.0i	0.732576i	0.930501	+	0.366288i	$0.119372\pi$
$727$	14360.0i	0.732576i	−0.930501	+	0.366288i	$0.880628\pi$
$728$	− 2912.00i	− 0.148250i
$729$	−67717.0	−3.44038
$730$	0	0
$731$	33888.0	1.71463
$732$	8240.00i	0.416064i
$733$	1588.00i	0.0800193i	0.999199	+	0.0400096i	$0.0127389\pi$
$733$	1588.00i	0.0800193i	−0.999199	+	0.0400096i	$0.987261\pi$
$734$	−23324.0	−1.17289
$735$	0	0
$736$	−2400.00	−0.120197
$737$	5391.00i	0.269444i
$738$	− 63948.0i	− 3.18965i
$739$	−2957.00	−0.147192	−0.0735961	−	0.997288i	$-0.523448\pi$
$739$	−2957.00	−0.147192	−0.0735961	+	0.997288i	$0.523448\pi$
$740$	0	0
$741$	5200.00	0.257796
$742$	4956.00i	0.245203i
$743$	12384.0i	0.611474i	0.952116	+	0.305737i	$0.0989028\pi$
$743$	12384.0i	0.611474i	−0.952116	+	0.305737i	$0.901097\pi$
$744$	−18560.0	−0.914573
$745$	0	0
$746$	4754.00	0.233319
$747$	72708.0i	3.56124i
$748$	− 3456.00i	− 0.168936i
$749$	−6132.00	−0.299143
$750$	0	0
$751$	−20236.0	−0.983252	−0.491626	−	0.870806i	$-0.663597\pi$
$751$	−20236.0	−0.983252	−0.491626	+	0.870806i	$0.663597\pi$
$752$	− 7776.00i	− 0.377077i
$753$	− 13800.0i	− 0.667862i
$754$	−19656.0	−0.949376
$755$	0	0
$756$	12880.0	0.619631
$757$	37601.0i	1.80533i	0.430348	+	0.902663i	$0.358391\pi$
$757$	37601.0i	1.80533i	−0.430348	+	0.902663i	$0.641609\pi$
$758$	8854.00i	0.424264i
$759$	−6750.00	−0.322806
$760$	0	0
$761$	−13392.0	−0.637923	−0.318962	−	0.947768i	$-0.603334\pi$
$761$	−13392.0	−0.637923	−0.318962	+	0.947768i	$0.603334\pi$
$762$	− 20620.0i	− 0.980294i
$763$	4837.00i	0.229503i
$764$	3504.00	0.165930
$765$	0	0
$766$	−9216.00	−0.434710
$767$	34944.0i	1.64505i
$768$	− 2560.00i	− 0.120281i
$769$	−22430.0	−1.05182	−0.525908	−	0.850541i	$-0.676274\pi$
$769$	−22430.0	−1.05182	−0.525908	+	0.850541i	$0.676274\pi$
$770$	0	0
$771$	−35760.0	−1.67038
$772$	− 2404.00i	− 0.112075i
$773$	− 34704.0i	− 1.61477i	−0.590026	−	0.807384i	$-0.700882\pi$
$773$	− 34704.0i	− 1.61477i	0.590026	−	0.807384i	$-0.299118\pi$
$774$	51538.0	2.39340
$775$	0	0
$776$	1472.00	0.0680950
$777$	21350.0i	0.985749i
$778$	− 1398.00i	− 0.0644225i
$779$	−4380.00	−0.201450
$780$	0	0
$781$	−4239.00	−0.194217
$782$	− 14400.0i	− 0.658495i
$783$	− 86940.0i	− 3.96805i
$784$	−784.000	−0.0357143
$785$	0	0
$786$	22320.0	1.01289
$787$	37646.0i	1.70513i	0.522624	+	0.852564i	$0.324953\pi$
$787$	37646.0i	1.70513i	−0.522624	+	0.852564i	$0.675047\pi$
$788$	8052.00i	0.364011i
$789$	59190.0	2.67075
$790$	0	0
$791$	−10647.0	−0.478589
$792$	− 5256.00i	− 0.235813i
$793$	10712.0i	0.479690i
$794$	−15260.0	−0.682062
$795$	0	0
$796$	1304.00	0.0580641
$797$	− 8988.00i	− 0.399462i	−0.979851	−	0.199731i	$-0.935993\pi$
$797$	− 8988.00i	− 0.399462i	0.979851	−	0.199731i	$-0.0640068\pi$
$798$	− 1400.00i	− 0.0621046i
$799$	46656.0	2.06580
$800$	0	0
$801$	13140.0	0.579624
$802$	− 11202.0i	− 0.493212i
$803$	− 5526.00i	− 0.242850i
$804$	−23960.0	−1.05100
$805$	0	0
$806$	−24128.0	−1.05443
$807$	− 17640.0i	− 0.769464i
$808$	− 5808.00i	− 0.252877i
$809$	28029.0	1.21811	0.609053	−	0.793130i	$-0.291550\pi$
$809$	28029.0	1.21811	0.609053	+	0.793130i	$0.291550\pi$
$810$	0	0
$811$	8078.00	0.349762	0.174881	−	0.984590i	$-0.444046\pi$
$811$	8078.00	0.349762	0.174881	+	0.984590i	$0.444046\pi$
$812$	5292.00i	0.228710i
$813$	3400.00i	0.146671i
$814$	5490.00	0.236394
$815$	0	0
$816$	15360.0	0.658955
$817$	− 3530.00i	− 0.151162i
$818$	9340.00i	0.399224i
$819$	26572.0	1.13370
$820$	0	0
$821$	−35574.0	−1.51223	−0.756115	−	0.654439i	$-0.772905\pi$
$821$	−35574.0	−1.51223	−0.756115	+	0.654439i	$0.772905\pi$
$822$	27960.0i	1.18640i
$823$	− 6599.00i	− 0.279498i	−0.990187	−	0.139749i	$-0.955370\pi$
$823$	− 6599.00i	− 0.279498i	0.990187	−	0.139749i	$-0.0446295\pi$
$824$	−14384.0	−0.608119
$825$	0	0
$826$	9408.00	0.396303
$827$	− 663.000i	− 0.0278776i	−0.999903	−	0.0139388i	$-0.995563\pi$
$827$	− 663.000i	− 0.0278776i	0.999903	−	0.0139388i	$-0.00443700\pi$
$828$	− 21900.0i	− 0.919176i
$829$	22564.0	0.945332	0.472666	−	0.881242i	$-0.343292\pi$
$829$	22564.0	0.945332	0.472666	+	0.881242i	$0.343292\pi$
$830$	0	0
$831$	47060.0	1.96449
$832$	− 3328.00i	− 0.138675i
$833$	− 4704.00i	− 0.195659i
$834$	13480.0	0.559681
$835$	0	0
$836$	−360.000	−0.0148934
$837$	− 106720.i	− 4.40715i
$838$	72.0000i	0.00296802i
$839$	294.000	0.0120977	0.00604887	−	0.999982i	$-0.498075\pi$
$839$	294.000	0.0120977	0.00604887	+	0.999982i	$0.498075\pi$
$840$	0	0
$841$	11332.0	0.464636
$842$	− 10990.0i	− 0.449810i
$843$	− 66810.0i	− 2.72961i
$844$	23824.0	0.971630
$845$	0	0
$846$	70956.0	2.88359
$847$	− 8750.00i	− 0.354963i
$848$	5664.00i	0.229366i
$849$	2260.00	0.0913581
$850$	0	0
$851$	22875.0	0.921439
$852$	− 18840.0i	− 0.757568i
$853$	28852.0i	1.15812i	0.815286	+	0.579058i	$0.196580\pi$
$853$	28852.0i	1.15812i	−0.815286	+	0.579058i	$0.803420\pi$
$854$	2884.00	0.115560
$855$	0	0
$856$	−7008.00	−0.279823
$857$	− 7422.00i	− 0.295835i	−0.989000	−	0.147918i	$-0.952743\pi$
$857$	− 7422.00i	− 0.295835i	0.989000	−	0.147918i	$-0.0472570\pi$
$858$	− 9360.00i	− 0.372430i
$859$	−8138.00	−0.323242	−0.161621	−	0.986853i	$-0.551672\pi$
$859$	−8138.00	−0.323242	−0.161621	+	0.986853i	$0.551672\pi$
$860$	0	0
$861$	−30660.0	−1.21358
$862$	− 5400.00i	− 0.213370i
$863$	32199.0i	1.27007i	0.772485	+	0.635033i	$0.219013\pi$
$863$	32199.0i	1.27007i	−0.772485	+	0.635033i	$0.780987\pi$
$864$	14720.0	0.579612
$865$	0	0
$866$	−30208.0	−1.18535
$867$	43030.0i	1.68555i
$868$	6496.00i	0.254019i
$869$	−6687.00	−0.261037
$870$	0	0
$871$	−31148.0	−1.21172
$872$	5528.00i	0.214681i
$873$	13432.0i	0.520738i
$874$	−1500.00	−0.0580529
$875$	0	0
$876$	24560.0	0.947267
$877$	− 11158.0i	− 0.429622i	−0.976656	−	0.214811i	$-0.931086\pi$
$877$	− 11158.0i	− 0.429622i	0.976656	−	0.214811i	$-0.0689136\pi$
$878$	29896.0i	1.14914i
$879$	−43200.0	−1.65768
$880$	0	0
$881$	16272.0	0.622267	0.311134	−	0.950366i	$-0.399291\pi$
$881$	16272.0	0.622267	0.311134	+	0.950366i	$0.399291\pi$
$882$	− 7154.00i	− 0.273115i
$883$	− 9071.00i	− 0.345712i	−0.984947	−	0.172856i	$-0.944701\pi$
$883$	− 9071.00i	− 0.345712i	0.984947	−	0.172856i	$-0.0552995\pi$
$884$	19968.0	0.759725
$885$	0	0
$886$	−9960.00	−0.377667
$887$	− 30138.0i	− 1.14085i	−0.821349	−	0.570426i	$-0.806778\pi$
$887$	− 30138.0i	− 1.14085i	0.821349	−	0.570426i	$-0.193222\pi$
$888$	24400.0i	0.922084i
$889$	−7217.00	−0.272273
$890$	0	0
$891$	23661.0	0.889645
$892$	− 12472.0i	− 0.468154i
$893$	− 4860.00i	− 0.182121i
$894$	−25620.0	−0.958457
$895$	0	0
$896$	−896.000	−0.0334077
$897$	− 39000.0i	− 1.45170i
$898$	24750.0i	0.919731i
$899$	43848.0	1.62671
$900$	0	0
$901$	−33984.0	−1.25657
$902$	7884.00i	0.291029i
$903$	− 24710.0i	− 0.910628i
$904$	−12168.0	−0.447679
$905$	0	0
$906$	−19060.0	−0.698925
$907$	25844.0i	0.946126i	0.881029	+	0.473063i	$0.156852\pi$
$907$	25844.0i	0.946126i	−0.881029	+	0.473063i	$0.843148\pi$
$908$	24.0000i	0 0.000877167i
$909$	52998.0	1.93381
$910$	0	0
$911$	40815.0	1.48437	0.742185	−	0.670195i	$-0.233790\pi$
$911$	40815.0	1.48437	0.742185	+	0.670195i	$0.233790\pi$
$912$	− 1600.00i	− 0.0580935i
$913$	− 8964.00i	− 0.324934i
$914$	21670.0	0.784223
$915$	0	0
$916$	−2344.00	−0.0845502
$917$	− 7812.00i	− 0.281325i
$918$	88320.0i	3.17538i
$919$	5389.00	0.193435	0.0967175	−	0.995312i	$-0.469166\pi$
$919$	5389.00	0.193435	0.0967175	+	0.995312i	$0.469166\pi$
$920$	0	0
$921$	−66040.0	−2.36275
$922$	− 11400.0i	− 0.407201i
$923$	− 24492.0i	− 0.873417i
$924$	−2520.00	−0.0897207
$925$	0	0
$926$	−12256.0	−0.434943
$927$	− 131254.i	− 4.65043i
$928$	6048.00i	0.213939i
$929$	−43662.0	−1.54198	−0.770992	−	0.636844i	$-0.780239\pi$
$929$	−43662.0	−1.54198	−0.770992	+	0.636844i	$0.780239\pi$
$930$	0	0
$931$	−490.000	−0.0172493
$932$	5172.00i	0.181775i
$933$	− 60360.0i	− 2.11800i
$934$	−19620.0	−0.687351
$935$	0	0
$936$	30368.0	1.06048
$937$	− 11950.0i	− 0.416638i	−0.978061	−	0.208319i	$-0.933201\pi$
$937$	− 11950.0i	− 0.416638i	0.978061	−	0.208319i	$-0.0667992\pi$
$938$	8386.00i	0.291911i
$939$	−91460.0	−3.17858
$940$	0	0
$941$	−8448.00	−0.292664	−0.146332	−	0.989236i	$-0.546747\pi$
$941$	−8448.00	−0.292664	−0.146332	+	0.989236i	$0.546747\pi$
$942$	− 13000.0i	− 0.449642i
$943$	32850.0i	1.13440i
$944$	10752.0	0.370707
$945$	0	0
$946$	−6354.00	−0.218379
$947$	25692.0i	0.881603i	0.897605	+	0.440801i	$0.145306\pi$
$947$	25692.0i	0.881603i	−0.897605	+	0.440801i	$0.854694\pi$
$948$	− 29720.0i	− 1.01821i
$949$	31928.0	1.09213
$950$	0	0
$951$	−44490.0	−1.51702
$952$	− 5376.00i	− 0.183022i
$953$	− 47547.0i	− 1.61616i	−0.589074	−	0.808079i	$-0.700507\pi$
$953$	− 47547.0i	− 1.61616i	0.589074	−	0.808079i	$-0.299493\pi$
$954$	−51684.0	−1.75402
$955$	0	0
$956$	−21504.0	−0.727499
$957$	17010.0i	0.574561i
$958$	− 408.000i	− 0.0137598i
$959$	9786.00	0.329517
$960$	0	0
$961$	24033.0	0.806720
$962$	31720.0i	1.06309i
$963$	− 63948.0i	− 2.13987i
$964$	2680.00	0.0895404
$965$	0	0
$966$	−10500.0	−0.349723
$967$	51608.0i	1.71624i	0.513452	+	0.858119i	$0.328367\pi$
$967$	51608.0i	1.71624i	−0.513452	+	0.858119i	$0.671633\pi$
$968$	− 10000.0i	− 0.332037i
$969$	9600.00	0.318263
$970$	0	0
$971$	−11754.0	−0.388469	−0.194235	−	0.980955i	$-0.562222\pi$
$971$	−11754.0	−0.388469	−0.194235	+	0.980955i	$0.562222\pi$
$972$	55480.0i	1.83078i
$973$	− 4718.00i	− 0.155449i
$974$	30802.0	1.01331
$975$	0	0
$976$	3296.00	0.108097
$977$	12765.0i	0.418003i	0.977915	+	0.209001i	$0.0670213\pi$
$977$	12765.0i	0.418003i	−0.977915	+	0.209001i	$0.932979\pi$
$978$	18640.0i	0.609449i
$979$	−1620.00	−0.0528860
$980$	0	0
$981$	−50443.0	−1.64171
$982$	− 7794.00i	− 0.253275i
$983$	32112.0i	1.04193i	0.853579	+	0.520963i	$0.174427\pi$
$983$	32112.0i	1.04193i	−0.853579	+	0.520963i	$0.825573\pi$
$984$	−35040.0	−1.13520
$985$	0	0
$986$	−36288.0	−1.17205
$987$	− 34020.0i	− 1.09713i
$988$	− 2080.00i	− 0.0669773i
$989$	−26475.0	−0.851219
$990$	0	0
$991$	−42505.0	−1.36248	−0.681239	−	0.732061i	$-0.738559\pi$
$991$	−42505.0	−1.36248	−0.681239	+	0.732061i	$0.738559\pi$
$992$	7424.00i	0.237613i
$993$	100810.i	3.22166i
$994$	−6594.00	−0.210411
$995$	0	0
$996$	39840.0	1.26745
$997$	59654.0i	1.89495i	0.319836	+	0.947473i	$0.396372\pi$
$997$	59654.0i	1.89495i	−0.319836	+	0.947473i	$0.603628\pi$
$998$	16264.0i	0.515860i
$999$	−140300.	−4.44334

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
350.4.a.j.1.1	✓	1	5.3	odd	4
350.4.a.k.1.1	yes	1	5.2	odd	4
350.4.c.a.99.1		2	1.1	even	1	trivial
350.4.c.a.99.2		2	5.4	even	2	inner
2450.4.a.a.1.1		1	35.13	even	4
2450.4.a.bp.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)

\(f(q)\)	\(=\)	\(q-2.00000i q^{2} -10.0000i q^{3} -4.00000 q^{4} -20.0000 q^{6} +7.00000i q^{7} +8.00000i q^{8} -73.0000 q^{9} +9.00000 q^{11} +40.0000i q^{12} +52.0000i q^{13} +14.0000 q^{14} +16.0000 q^{16} +96.0000i q^{17} +146.000i q^{18} +10.0000 q^{19} +70.0000 q^{21} -18.0000i q^{22} -75.0000i q^{23} +80.0000 q^{24} +104.000 q^{26} +460.000i q^{27} -28.0000i q^{28} -189.000 q^{29} -232.000 q^{31} -32.0000i q^{32} -90.0000i q^{33} +192.000 q^{34} +292.000 q^{36} +305.000i q^{37} -20.0000i q^{38} +520.000 q^{39} -438.000 q^{41} -140.000i q^{42} -353.000i q^{43} -36.0000 q^{44} -150.000 q^{46} -486.000i q^{47} -160.000i q^{48} -49.0000 q^{49} +960.000 q^{51} -208.000i q^{52} +354.000i q^{53} +920.000 q^{54} -56.0000 q^{56} -100.000i q^{57} +378.000i q^{58} +672.000 q^{59} +206.000 q^{61} +464.000i q^{62} -511.000i q^{63} -64.0000 q^{64} -180.000 q^{66} +599.000i q^{67} -384.000i q^{68} -750.000 q^{69} -471.000 q^{71} -584.000i q^{72} -614.000i q^{73} +610.000 q^{74} -40.0000 q^{76} +63.0000i q^{77} -1040.00i q^{78} -743.000 q^{79} +2629.00 q^{81} +876.000i q^{82} -996.000i q^{83} -280.000 q^{84} -706.000 q^{86} +1890.00i q^{87} +72.0000i q^{88} -180.000 q^{89} -364.000 q^{91} +300.000i q^{92} +2320.00i q^{93} -972.000 q^{94} -320.000 q^{96} -184.000i q^{97} +98.0000i q^{98} -657.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{4} - 40 q^{6} - 146 q^{9} + 18 q^{11} + 28 q^{14} + 32 q^{16} + 20 q^{19} + 140 q^{21} + 160 q^{24} + 208 q^{26} - 378 q^{29} - 464 q^{31} + 384 q^{34} + 584 q^{36} + 1040 q^{39} - 876 q^{41}+ \cdots - 1314 q^{99}+O(q^{100}) \) 2 * q - 8 * q^4 - 40 * q^6 - 146 * q^9 + 18 * q^11 + 28 * q^14 + 32 * q^16 + 20 * q^19 + 140 * q^21 + 160 * q^24 + 208 * q^26 - 378 * q^29 - 464 * q^31 + 384 * q^34 + 584 * q^36 + 1040 * q^39 - 876 * q^41 - 72 * q^44 - 300 * q^46 - 98 * q^49 + 1920 * q^51 + 1840 * q^54 - 112 * q^56 + 1344 * q^59 + 412 * q^61 - 128 * q^64 - 360 * q^66 - 1500 * q^69 - 942 * q^71 + 1220 * q^74 - 80 * q^76 - 1486 * q^79 + 5258 * q^81 - 560 * q^84 - 1412 * q^86 - 360 * q^89 - 728 * q^91 - 1944 * q^94 - 640 * q^96 - 1314 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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