LMFDB - Embedded newform 1150.4.b.f.599.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1150,4,Mod(599,1150)] 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("1150.599"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1150, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1150 = 2 \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1150.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,-8,0,4,0,0,52,0,-64] 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:	$67.8521965066$
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:	no (minimal twist has level 230)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			599.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	1150.599
Dual form			1150.4.b.f.599.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.00000i q^{2} +1.00000i q^{3} -4.00000 q^{4} +2.00000 q^{6} +18.0000i q^{7} +8.00000i q^{8} +26.0000 q^{9} -32.0000 q^{11} -4.00000i q^{12} -47.0000i q^{13} +36.0000 q^{14} +16.0000 q^{16} -20.0000i q^{17} -52.0000i q^{18} -36.0000 q^{19} -18.0000 q^{21} +64.0000i q^{22} -23.0000i q^{23} -8.00000 q^{24} -94.0000 q^{26} +53.0000i q^{27} -72.0000i q^{28} +27.0000 q^{29} -33.0000 q^{31} -32.0000i q^{32} -32.0000i q^{33} -40.0000 q^{34} -104.000 q^{36} -56.0000i q^{37} +72.0000i q^{38} +47.0000 q^{39} -157.000 q^{41} +36.0000i q^{42} +18.0000i q^{43} +128.000 q^{44} -46.0000 q^{46} -65.0000i q^{47} +16.0000i q^{48} +19.0000 q^{49} +20.0000 q^{51} +188.000i q^{52} -14.0000i q^{53} +106.000 q^{54} -144.000 q^{56} -36.0000i q^{57} -54.0000i q^{58} +744.000 q^{59} +552.000 q^{61} +66.0000i q^{62} +468.000i q^{63} -64.0000 q^{64} -64.0000 q^{66} +156.000i q^{67} +80.0000i q^{68} +23.0000 q^{69} +699.000 q^{71} +208.000i q^{72} -609.000i q^{73} -112.000 q^{74} +144.000 q^{76} -576.000i q^{77} -94.0000i q^{78} +644.000 q^{79} +649.000 q^{81} +314.000i q^{82} +512.000i q^{83} +72.0000 q^{84} +36.0000 q^{86} +27.0000i q^{87} -256.000i q^{88} +102.000 q^{89} +846.000 q^{91} +92.0000i q^{92} -33.0000i q^{93} -130.000 q^{94} +32.0000 q^{96} -578.000i q^{97} -38.0000i q^{98} -832.000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{4} + 4 q^{6} + 52 q^{9} - 64 q^{11} + 72 q^{14} + 32 q^{16} - 72 q^{19} - 36 q^{21} - 16 q^{24} - 188 q^{26} + 54 q^{29} - 66 q^{31} - 80 q^{34} - 208 q^{36} + 94 q^{39} - 314 q^{41} + 256 q^{44}+ \cdots - 1664 q^{99}+O(q^{100}) $ 2 * q - 8 * q^4 + 4 * q^6 + 52 * q^9 - 64 * q^11 + 72 * q^14 + 32 * q^16 - 72 * q^19 - 36 * q^21 - 16 * q^24 - 188 * q^26 + 54 * q^29 - 66 * q^31 - 80 * q^34 - 208 * q^36 + 94 * q^39 - 314 * q^41 + 256 * q^44 - 92 * q^46 + 38 * q^49 + 40 * q^51 + 212 * q^54 - 288 * q^56 + 1488 * q^59 + 1104 * q^61 - 128 * q^64 - 128 * q^66 + 46 * q^69 + 1398 * q^71 - 224 * q^74 + 288 * q^76 + 1288 * q^79 + 1298 * q^81 + 144 * q^84 + 72 * q^86 + 204 * q^89 + 1692 * q^91 - 260 * q^94 + 64 * q^96 - 1664 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1150\mathbb{Z}\right)^\times$.

$n$	$51$	$277$
$\chi(n)$	$1$	$-1$

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
$7$	18.0000i	0.971909i	0.873984	+	0.485954i	$0.161528\pi$
$7$	18.0000i	0.971909i	−0.873984	+	0.485954i	$0.838472\pi$
$8$	8.00000i	0.353553i
$9$	26.0000	0.962963
$10$	0	0
$11$	−32.0000	−0.877124	−0.438562	−	0.898701i	$-0.644512\pi$
$11$	−32.0000	−0.877124	−0.438562	+	0.898701i	$0.644512\pi$
$12$	− 4.00000i	− 0.0962250i
$13$	− 47.0000i	− 1.00273i	−0.865237	−	0.501364i	$-0.832832\pi$
$13$	− 47.0000i	− 1.00273i	0.865237	−	0.501364i	$-0.167168\pi$
$14$	36.0000	0.687243
$15$	0	0
$16$	16.0000	0.250000
$17$	− 20.0000i	− 0.285336i	−0.989771	−	0.142668i	$-0.954432\pi$
$17$	− 20.0000i	− 0.285336i	0.989771	−	0.142668i	$-0.0455681\pi$
$18$	− 52.0000i	− 0.680918i
$19$	−36.0000	−0.434682	−0.217341	−	0.976096i	$-0.569738\pi$
$19$	−36.0000	−0.434682	−0.217341	+	0.976096i	$0.569738\pi$
$20$	0	0
$21$	−18.0000	−0.187044
$22$	64.0000i	0.620220i
$23$	− 23.0000i	− 0.208514i
$23$	− 23.0000i	− 0.208514i
$24$	−8.00000	−0.0680414
$25$	0	0
$26$	−94.0000	−0.709035
$27$	53.0000i	0.377772i
$28$	− 72.0000i	− 0.485954i
$29$	27.0000	0.172889	0.0864444	−	0.996257i	$-0.472450\pi$
$29$	27.0000	0.172889	0.0864444	+	0.996257i	$0.472450\pi$
$30$	0	0
$31$	−33.0000	−0.191193	−0.0955964	−	0.995420i	$-0.530476\pi$
$31$	−33.0000	−0.191193	−0.0955964	+	0.995420i	$0.530476\pi$
$32$	− 32.0000i	− 0.176777i
$33$	− 32.0000i	− 0.168803i
$34$	−40.0000	−0.201763
$35$	0	0
$36$	−104.000	−0.481481
$37$	− 56.0000i	− 0.248820i	−0.992231	−	0.124410i	$-0.960296\pi$
$37$	− 56.0000i	− 0.248820i	0.992231	−	0.124410i	$-0.0397038\pi$
$38$	72.0000i	0.307367i
$39$	47.0000	0.192975
$40$	0	0
$41$	−157.000	−0.598031	−0.299016	−	0.954248i	$-0.596658\pi$
$41$	−157.000	−0.598031	−0.299016	+	0.954248i	$0.596658\pi$
$42$	36.0000i	0.132260i
$43$	18.0000i	0.0638366i	0.999490	+	0.0319183i	$0.0101616\pi$
$43$	18.0000i	0.0638366i	−0.999490	+	0.0319183i	$0.989838\pi$
$44$	128.000	0.438562
$45$	0	0
$46$	−46.0000	−0.147442
$47$	− 65.0000i	− 0.201728i	−0.994900	−	0.100864i	$-0.967839\pi$
$47$	− 65.0000i	− 0.201728i	0.994900	−	0.100864i	$-0.0321607\pi$
$48$	16.0000i	0.0481125i
$49$	19.0000	0.0553936
$50$	0	0
$51$	20.0000	0.0549129
$52$	188.000i	0.501364i
$53$	− 14.0000i	− 0.0362839i	−0.999835	−	0.0181420i	$-0.994225\pi$
$53$	− 14.0000i	− 0.0362839i	0.999835	−	0.0181420i	$-0.00577508\pi$
$54$	106.000	0.267125
$55$	0	0
$56$	−144.000	−0.343622
$57$	− 36.0000i	− 0.0836547i
$58$	− 54.0000i	− 0.122251i
$59$	744.000	1.64170	0.820852	−	0.571141i	$-0.193499\pi$
$59$	744.000	1.64170	0.820852	+	0.571141i	$0.193499\pi$
$60$	0	0
$61$	552.000	1.15863	0.579314	−	0.815104i	$-0.303320\pi$
$61$	552.000	1.15863	0.579314	+	0.815104i	$0.303320\pi$
$62$	66.0000i	0.135194i
$63$	468.000i	0.935912i
$64$	−64.0000	−0.125000
$65$	0	0
$66$	−64.0000	−0.119361
$67$	156.000i	0.284454i	0.989834	+	0.142227i	$0.0454263\pi$
$67$	156.000i	0.284454i	−0.989834	+	0.142227i	$0.954574\pi$
$68$	80.0000i	0.142668i
$69$	23.0000	0.0401286
$70$	0	0
$71$	699.000	1.16839	0.584197	−	0.811612i	$-0.301409\pi$
$71$	699.000	1.16839	0.584197	+	0.811612i	$0.301409\pi$
$72$	208.000i	0.340459i
$73$	− 609.000i	− 0.976412i	−0.872728	−	0.488206i	$-0.837652\pi$
$73$	− 609.000i	− 0.976412i	0.872728	−	0.488206i	$-0.162348\pi$
$74$	−112.000	−0.175942
$75$	0	0
$76$	144.000	0.217341
$77$	− 576.000i	− 0.852484i
$78$	− 94.0000i	− 0.136454i
$79$	644.000	0.917160	0.458580	−	0.888653i	$-0.348358\pi$
$79$	644.000	0.917160	0.458580	+	0.888653i	$0.348358\pi$
$80$	0	0
$81$	649.000	0.890261
$82$	314.000i	0.422872i
$83$	512.000i	0.677100i	0.940948	+	0.338550i	$0.109936\pi$
$83$	512.000i	0.677100i	−0.940948	+	0.338550i	$0.890064\pi$
$84$	72.0000	0.0935220
$85$	0	0
$86$	36.0000	0.0451393
$87$	27.0000i	0.0332725i
$88$	− 256.000i	− 0.310110i
$89$	102.000	0.121483	0.0607415	−	0.998154i	$-0.480653\pi$
$89$	102.000	0.121483	0.0607415	+	0.998154i	$0.480653\pi$
$90$	0	0
$91$	846.000	0.974559
$92$	92.0000i	0.104257i
$93$	− 33.0000i	− 0.0367951i
$94$	−130.000	−0.142643
$95$	0	0
$96$	32.0000	0.0340207
$97$	− 578.000i	− 0.605021i	−0.953146	−	0.302510i	$-0.902175\pi$
$97$	− 578.000i	− 0.605021i	0.953146	−	0.302510i	$-0.0978247\pi$
$98$	− 38.0000i	− 0.0391692i
$99$	−832.000	−0.844638
$100$	0	0
$101$	−6.00000	−0.00591111	−0.00295556	−	0.999996i	$-0.500941\pi$
$101$	−6.00000	−0.00591111	−0.00295556	+	0.999996i	$0.500941\pi$
$102$	− 40.0000i	− 0.0388293i
$103$	− 160.000i	− 0.153061i	−0.997067	−	0.0765304i	$-0.975616\pi$
$103$	− 160.000i	− 0.153061i	0.997067	−	0.0765304i	$-0.0243842\pi$
$104$	376.000	0.354518
$105$	0	0
$106$	−28.0000	−0.0256566
$107$	− 380.000i	− 0.343327i	−0.985156	−	0.171663i	$-0.945086\pi$
$107$	− 380.000i	− 0.343327i	0.985156	−	0.171663i	$-0.0549142\pi$
$108$	− 212.000i	− 0.188886i
$109$	−250.000	−0.219685	−0.109842	−	0.993949i	$-0.535035\pi$
$109$	−250.000	−0.219685	−0.109842	+	0.993949i	$0.535035\pi$
$110$	0	0
$111$	56.0000	0.0478854
$112$	288.000i	0.242977i
$113$	− 390.000i	− 0.324674i	−0.986735	−	0.162337i	$-0.948097\pi$
$113$	− 390.000i	− 0.324674i	0.986735	−	0.162337i	$-0.0519031\pi$
$114$	−72.0000	−0.0591528
$115$	0	0
$116$	−108.000	−0.0864444
$117$	− 1222.00i	− 0.965589i
$118$	− 1488.00i	− 1.16086i
$119$	360.000	0.277321
$120$	0	0
$121$	−307.000	−0.230654
$122$	− 1104.00i	− 0.819274i
$123$	− 157.000i	− 0.115091i
$124$	132.000	0.0955964
$125$	0	0
$126$	936.000	0.661790
$127$	769.000i	0.537305i	0.963237	+	0.268652i	$0.0865783\pi$
$127$	769.000i	0.537305i	−0.963237	+	0.268652i	$0.913422\pi$
$128$	128.000i	0.0883883i
$129$	−18.0000	−0.0122854
$130$	0	0
$131$	−213.000	−0.142060	−0.0710301	−	0.997474i	$-0.522629\pi$
$131$	−213.000	−0.142060	−0.0710301	+	0.997474i	$0.522629\pi$
$132$	128.000i	0.0844013i
$133$	− 648.000i	− 0.422472i
$134$	312.000	0.201140
$135$	0	0
$136$	160.000	0.100882
$137$	− 2836.00i	− 1.76858i	−0.466936	−	0.884291i	$-0.654642\pi$
$137$	− 2836.00i	− 1.76858i	0.466936	−	0.884291i	$-0.345358\pi$
$138$	− 46.0000i	− 0.0283752i
$139$	1631.00	0.995249	0.497625	−	0.867393i	$-0.334206\pi$
$139$	1631.00	0.995249	0.497625	+	0.867393i	$0.334206\pi$
$140$	0	0
$141$	65.0000	0.0388226
$142$	− 1398.00i	− 0.826180i
$143$	1504.00i	0.879516i
$144$	416.000	0.240741
$145$	0	0
$146$	−1218.00	−0.690427
$147$	19.0000i	0.0106605i
$148$	224.000i	0.124410i
$149$	1966.00	1.08095	0.540473	−	0.841361i	$-0.318245\pi$
$149$	1966.00	1.08095	0.540473	+	0.841361i	$0.318245\pi$
$150$	0	0
$151$	35.0000	0.0188626	0.00943132	−	0.999956i	$-0.496998\pi$
$151$	35.0000	0.0188626	0.00943132	+	0.999956i	$0.496998\pi$
$152$	− 288.000i	− 0.153683i
$153$	− 520.000i	− 0.274768i
$154$	−1152.00	−0.602797
$155$	0	0
$156$	−188.000	−0.0964875
$157$	− 1702.00i	− 0.865187i	−0.901589	−	0.432594i	$-0.857599\pi$
$157$	− 1702.00i	− 0.865187i	0.901589	−	0.432594i	$-0.142401\pi$
$158$	− 1288.00i	− 0.648530i
$159$	14.0000	0.00698284
$160$	0	0
$161$	414.000	0.202657
$162$	− 1298.00i	− 0.629509i
$163$	− 2045.00i	− 0.982680i	−0.870968	−	0.491340i	$-0.836507\pi$
$163$	− 2045.00i	− 0.982680i	0.870968	−	0.491340i	$-0.163493\pi$
$164$	628.000	0.299016
$165$	0	0
$166$	1024.00	0.478782
$167$	− 1016.00i	− 0.470781i	−0.971901	−	0.235391i	$-0.924363\pi$
$167$	− 1016.00i	− 0.470781i	0.971901	−	0.235391i	$-0.0756369\pi$
$168$	− 144.000i	− 0.0661300i
$169$	−12.0000	−0.00546199
$170$	0	0
$171$	−936.000	−0.418583
$172$	− 72.0000i	− 0.0319183i
$173$	598.000i	0.262804i	0.991329	+	0.131402i	$0.0419479\pi$
$173$	598.000i	0.262804i	−0.991329	+	0.131402i	$0.958052\pi$
$174$	54.0000	0.0235272
$175$	0	0
$176$	−512.000	−0.219281
$177$	744.000i	0.315946i
$178$	− 204.000i	− 0.0859014i
$179$	4607.00	1.92371	0.961853	−	0.273567i	$-0.0882036\pi$
$179$	4607.00	1.92371	0.961853	+	0.273567i	$0.0882036\pi$
$180$	0	0
$181$	−1212.00	−0.497720	−0.248860	−	0.968540i	$-0.580056\pi$
$181$	−1212.00	−0.497720	−0.248860	+	0.968540i	$0.580056\pi$
$182$	− 1692.00i	− 0.689117i
$183$	552.000i	0.222978i
$184$	184.000	0.0737210
$185$	0	0
$186$	−66.0000	−0.0260180
$187$	640.000i	0.250275i
$188$	260.000i	0.100864i
$189$	−954.000	−0.367160
$190$	0	0
$191$	−1058.00	−0.400807	−0.200404	−	0.979713i	$-0.564225\pi$
$191$	−1058.00	−0.400807	−0.200404	+	0.979713i	$0.564225\pi$
$192$	− 64.0000i	− 0.0240563i
$193$	1047.00i	0.390491i	0.980754	+	0.195245i	$0.0625503\pi$
$193$	1047.00i	0.390491i	−0.980754	+	0.195245i	$0.937450\pi$
$194$	−1156.00	−0.427814
$195$	0	0
$196$	−76.0000	−0.0276968
$197$	− 251.000i	− 0.0907767i	−0.998969	−	0.0453883i	$-0.985547\pi$
$197$	− 251.000i	− 0.0907767i	0.998969	−	0.0453883i	$-0.0144525\pi$
$198$	1664.00i	0.597249i
$199$	3508.00	1.24963	0.624813	−	0.780775i	$-0.285175\pi$
$199$	3508.00	1.24963	0.624813	+	0.780775i	$0.285175\pi$
$200$	0	0
$201$	−156.000	−0.0547432
$202$	12.0000i	0.00417979i
$203$	486.000i	0.168032i
$204$	−80.0000	−0.0274565
$205$	0	0
$206$	−320.000	−0.108230
$207$	− 598.000i	− 0.200792i
$208$	− 752.000i	− 0.250682i
$209$	1152.00	0.381270
$210$	0	0
$211$	−3296.00	−1.07538	−0.537692	−	0.843141i	$-0.680704\pi$
$211$	−3296.00	−1.07538	−0.537692	+	0.843141i	$0.680704\pi$
$212$	56.0000i	0.0181420i
$213$	699.000i	0.224858i
$214$	−760.000	−0.242769
$215$	0	0
$216$	−424.000	−0.133563
$217$	− 594.000i	− 0.185822i
$218$	500.000i	0.155341i
$219$	609.000	0.187911
$220$	0	0
$221$	−940.000	−0.286114
$222$	− 112.000i	− 0.0338601i
$223$	− 2720.00i	− 0.816792i	−0.912805	−	0.408396i	$-0.866088\pi$
$223$	− 2720.00i	− 0.816792i	0.912805	−	0.408396i	$-0.133912\pi$
$224$	576.000	0.171811
$225$	0	0
$226$	−780.000	−0.229579
$227$	4134.00i	1.20874i	0.796705	+	0.604368i	$0.206574\pi$
$227$	4134.00i	1.20874i	−0.796705	+	0.604368i	$0.793426\pi$
$228$	144.000i	0.0418273i
$229$	4510.00	1.30144	0.650719	−	0.759319i	$-0.274468\pi$
$229$	4510.00	1.30144	0.650719	+	0.759319i	$0.274468\pi$
$230$	0	0
$231$	576.000	0.164061
$232$	216.000i	0.0611254i
$233$	− 5003.00i	− 1.40668i	−0.710852	−	0.703342i	$-0.751690\pi$
$233$	− 5003.00i	− 1.40668i	0.710852	−	0.703342i	$-0.248310\pi$
$234$	−2444.00	−0.682775
$235$	0	0
$236$	−2976.00	−0.820852
$237$	644.000i	0.176508i
$238$	− 720.000i	− 0.196095i
$239$	6309.00	1.70751	0.853756	−	0.520674i	$-0.174319\pi$
$239$	6309.00	1.70751	0.853756	+	0.520674i	$0.174319\pi$
$240$	0	0
$241$	3038.00	0.812012	0.406006	−	0.913871i	$-0.366921\pi$
$241$	3038.00	0.812012	0.406006	+	0.913871i	$0.366921\pi$
$242$	614.000i	0.163097i
$243$	2080.00i	0.549103i
$244$	−2208.00	−0.579314
$245$	0	0
$246$	−314.000	−0.0813817
$247$	1692.00i	0.435868i
$248$	− 264.000i	− 0.0675968i
$249$	−512.000	−0.130308
$250$	0	0
$251$	−1332.00	−0.334961	−0.167480	−	0.985875i	$-0.553563\pi$
$251$	−1332.00	−0.334961	−0.167480	+	0.985875i	$0.553563\pi$
$252$	− 1872.00i	− 0.467956i
$253$	736.000i	0.182893i
$254$	1538.00	0.379932
$255$	0	0
$256$	256.000	0.0625000
$257$	− 3301.00i	− 0.801209i	−0.916251	−	0.400605i	$-0.868800\pi$
$257$	− 3301.00i	− 0.801209i	0.916251	−	0.400605i	$-0.131200\pi$
$258$	36.0000i	0.00868706i
$259$	1008.00	0.241830
$260$	0	0
$261$	702.000	0.166485
$262$	426.000i	0.100452i
$263$	2072.00i	0.485798i	0.970052	+	0.242899i	$0.0780984\pi$
$263$	2072.00i	0.485798i	−0.970052	+	0.242899i	$0.921902\pi$
$264$	256.000	0.0596807
$265$	0	0
$266$	−1296.00	−0.298733
$267$	102.000i	0.0233794i
$268$	− 624.000i	− 0.142227i
$269$	−5721.00	−1.29671	−0.648356	−	0.761337i	$-0.724543\pi$
$269$	−5721.00	−1.29671	−0.648356	+	0.761337i	$0.724543\pi$
$270$	0	0
$271$	−5900.00	−1.32251	−0.661254	−	0.750162i	$-0.729975\pi$
$271$	−5900.00	−1.32251	−0.661254	+	0.750162i	$0.729975\pi$
$272$	− 320.000i	− 0.0713340i
$273$	846.000i	0.187554i
$274$	−5672.00	−1.25058
$275$	0	0
$276$	−92.0000	−0.0200643
$277$	− 6371.00i	− 1.38194i	−0.722885	−	0.690968i	$-0.757185\pi$
$277$	− 6371.00i	− 1.38194i	0.722885	−	0.690968i	$-0.242815\pi$
$278$	− 3262.00i	− 0.703747i
$279$	−858.000	−0.184112
$280$	0	0
$281$	3190.00	0.677222	0.338611	−	0.940926i	$-0.390043\pi$
$281$	3190.00	0.677222	0.338611	+	0.940926i	$0.390043\pi$
$282$	− 130.000i	− 0.0274517i
$283$	− 4226.00i	− 0.887667i	−0.896109	−	0.443833i	$-0.853618\pi$
$283$	− 4226.00i	− 0.887667i	0.896109	−	0.443833i	$-0.146382\pi$
$284$	−2796.00	−0.584197
$285$	0	0
$286$	3008.00	0.621912
$287$	− 2826.00i	− 0.581232i
$288$	− 832.000i	− 0.170229i
$289$	4513.00	0.918583
$290$	0	0
$291$	578.000	0.116436
$292$	2436.00i	0.488206i
$293$	6048.00i	1.20590i	0.797780	+	0.602949i	$0.206008\pi$
$293$	6048.00i	1.20590i	−0.797780	+	0.602949i	$0.793992\pi$
$294$	38.0000	0.00753811
$295$	0	0
$296$	448.000	0.0879712
$297$	− 1696.00i	− 0.331353i
$298$	− 3932.00i	− 0.764344i
$299$	−1081.00	−0.209083
$300$	0	0
$301$	−324.000	−0.0620434
$302$	− 70.0000i	− 0.0133379i
$303$	− 6.00000i	− 0.00113759i
$304$	−576.000	−0.108671
$305$	0	0
$306$	−1040.00	−0.194290
$307$	− 8628.00i	− 1.60399i	−0.597328	−	0.801997i	$-0.703771\pi$
$307$	− 8628.00i	− 1.60399i	0.597328	−	0.801997i	$-0.296229\pi$
$308$	2304.00i	0.426242i
$309$	160.000	0.0294566
$310$	0	0
$311$	8247.00	1.50368	0.751840	−	0.659346i	$-0.229167\pi$
$311$	8247.00	1.50368	0.751840	+	0.659346i	$0.229167\pi$
$312$	376.000i	0.0682269i
$313$	2620.00i	0.473135i	0.971615	+	0.236567i	$0.0760224\pi$
$313$	2620.00i	0.473135i	−0.971615	+	0.236567i	$0.923978\pi$
$314$	−3404.00	−0.611780
$315$	0	0
$316$	−2576.00	−0.458580
$317$	− 9906.00i	− 1.75513i	−0.479457	−	0.877565i	$-0.659166\pi$
$317$	− 9906.00i	− 1.75513i	0.479457	−	0.877565i	$-0.340834\pi$
$318$	− 28.0000i	− 0.00493762i
$319$	−864.000	−0.151645
$320$	0	0
$321$	380.000	0.0660733
$322$	− 828.000i	− 0.143300i
$323$	720.000i	0.124031i
$324$	−2596.00	−0.445130
$325$	0	0
$326$	−4090.00	−0.694859
$327$	− 250.000i	− 0.0422784i
$328$	− 1256.00i	− 0.211436i
$329$	1170.00	0.196061
$330$	0	0
$331$	−8115.00	−1.34756	−0.673778	−	0.738934i	$-0.735329\pi$
$331$	−8115.00	−1.34756	−0.673778	+	0.738934i	$0.735329\pi$
$332$	− 2048.00i	− 0.338550i
$333$	− 1456.00i	− 0.239605i
$334$	−2032.00	−0.332892
$335$	0	0
$336$	−288.000	−0.0467610
$337$	7586.00i	1.22622i	0.789998	+	0.613109i	$0.210082\pi$
$337$	7586.00i	1.22622i	−0.789998	+	0.613109i	$0.789918\pi$
$338$	24.0000i	0.00386221i
$339$	390.000	0.0624835
$340$	0	0
$341$	1056.00	0.167700
$342$	1872.00i	0.295983i
$343$	6516.00i	1.02575i
$344$	−144.000	−0.0225697
$345$	0	0
$346$	1196.00	0.185831
$347$	− 1356.00i	− 0.209781i	−0.994484	−	0.104890i	$-0.966551\pi$
$347$	− 1356.00i	− 0.209781i	0.994484	−	0.104890i	$-0.0334492\pi$
$348$	− 108.000i	− 0.0166362i
$349$	−6649.00	−1.01981	−0.509904	−	0.860231i	$-0.670319\pi$
$349$	−6649.00	−1.01981	−0.509904	+	0.860231i	$0.670319\pi$
$350$	0	0
$351$	2491.00	0.378803
$352$	1024.00i	0.155055i
$353$	10691.0i	1.61197i	0.591938	+	0.805984i	$0.298363\pi$
$353$	10691.0i	1.61197i	−0.591938	+	0.805984i	$0.701637\pi$
$354$	1488.00	0.223408
$355$	0	0
$356$	−408.000	−0.0607415
$357$	360.000i	0.0533704i
$358$	− 9214.00i	− 1.36027i
$359$	6420.00	0.943829	0.471915	−	0.881644i	$-0.343563\pi$
$359$	6420.00	0.943829	0.471915	+	0.881644i	$0.343563\pi$
$360$	0	0
$361$	−5563.00	−0.811051
$362$	2424.00i	0.351941i
$363$	− 307.000i	− 0.0443893i
$364$	−3384.00	−0.487280
$365$	0	0
$366$	1104.00	0.157669
$367$	524.000i	0.0745302i	0.999305	+	0.0372651i	$0.0118646\pi$
$367$	524.000i	0.0745302i	−0.999305	+	0.0372651i	$0.988135\pi$
$368$	− 368.000i	− 0.0521286i
$369$	−4082.00	−0.575882
$370$	0	0
$371$	252.000	0.0352647
$372$	132.000i	0.0183975i
$373$	5566.00i	0.772645i	0.922364	+	0.386322i	$0.126255\pi$
$373$	5566.00i	0.772645i	−0.922364	+	0.386322i	$0.873745\pi$
$374$	1280.00	0.176971
$375$	0	0
$376$	520.000	0.0713217
$377$	− 1269.00i	− 0.173360i
$378$	1908.00i	0.259622i
$379$	−2240.00	−0.303591	−0.151796	−	0.988412i	$-0.548506\pi$
$379$	−2240.00	−0.303591	−0.151796	+	0.988412i	$0.548506\pi$
$380$	0	0
$381$	−769.000	−0.103404
$382$	2116.00i	0.283414i
$383$	8778.00i	1.17111i	0.810633	+	0.585555i	$0.199123\pi$
$383$	8778.00i	1.17111i	−0.810633	+	0.585555i	$0.800877\pi$
$384$	−128.000	−0.0170103
$385$	0	0
$386$	2094.00	0.276119
$387$	468.000i	0.0614723i
$388$	2312.00i	0.302510i
$389$	−4056.00	−0.528656	−0.264328	−	0.964433i	$-0.585150\pi$
$389$	−4056.00	−0.528656	−0.264328	+	0.964433i	$0.585150\pi$
$390$	0	0
$391$	−460.000	−0.0594967
$392$	152.000i	0.0195846i
$393$	− 213.000i	− 0.0273395i
$394$	−502.000	−0.0641888
$395$	0	0
$396$	3328.00	0.422319
$397$	9151.00i	1.15687i	0.815730	+	0.578433i	$0.196335\pi$
$397$	9151.00i	1.15687i	−0.815730	+	0.578433i	$0.803665\pi$
$398$	− 7016.00i	− 0.883619i
$399$	648.000	0.0813047
$400$	0	0
$401$	15930.0	1.98381	0.991903	−	0.126997i	$-0.0405340\pi$
$401$	15930.0	1.98381	0.991903	+	0.126997i	$0.0405340\pi$
$402$	312.000i	0.0387093i
$403$	1551.00i	0.191714i
$404$	24.0000	0.00295556
$405$	0	0
$406$	972.000	0.118817
$407$	1792.00i	0.218246i
$408$	160.000i	0.0194147i
$409$	5891.00	0.712203	0.356102	−	0.934447i	$-0.384106\pi$
$409$	5891.00	0.712203	0.356102	+	0.934447i	$0.384106\pi$
$410$	0	0
$411$	2836.00	0.340364
$412$	640.000i	0.0765304i
$413$	13392.0i	1.59559i
$414$	−1196.00	−0.141981
$415$	0	0
$416$	−1504.00	−0.177259
$417$	1631.00i	0.191536i
$418$	− 2304.00i	− 0.269599i
$419$	−15282.0	−1.78180	−0.890900	−	0.454199i	$-0.849926\pi$
$419$	−15282.0	−1.78180	−0.890900	+	0.454199i	$0.849926\pi$
$420$	0	0
$421$	−10934.0	−1.26577	−0.632887	−	0.774244i	$-0.718130\pi$
$421$	−10934.0	−1.26577	−0.632887	+	0.774244i	$0.718130\pi$
$422$	6592.00i	0.760411i
$423$	− 1690.00i	− 0.194257i
$424$	112.000	0.0128283
$425$	0	0
$426$	1398.00	0.158998
$427$	9936.00i	1.12608i
$428$	1520.00i	0.171663i
$429$	−1504.00	−0.169263
$430$	0	0
$431$	−2794.00	−0.312256	−0.156128	−	0.987737i	$-0.549901\pi$
$431$	−2794.00	−0.312256	−0.156128	+	0.987737i	$0.549901\pi$
$432$	848.000i	0.0944431i
$433$	− 15062.0i	− 1.67167i	−0.548980	−	0.835835i	$-0.684984\pi$
$433$	− 15062.0i	− 1.67167i	0.548980	−	0.835835i	$-0.315016\pi$
$434$	−1188.00	−0.131396
$435$	0	0
$436$	1000.00	0.109842
$437$	828.000i	0.0906376i
$438$	− 1218.00i	− 0.132873i
$439$	−261.000	−0.0283755	−0.0141878	−	0.999899i	$-0.504516\pi$
$439$	−261.000	−0.0283755	−0.0141878	+	0.999899i	$0.504516\pi$
$440$	0	0
$441$	494.000	0.0533420
$442$	1880.00i	0.202313i
$443$	− 7083.00i	− 0.759647i	−0.925059	−	0.379823i	$-0.875985\pi$
$443$	− 7083.00i	− 0.759647i	0.925059	−	0.379823i	$-0.124015\pi$
$444$	−224.000	−0.0239427
$445$	0	0
$446$	−5440.00	−0.577559
$447$	1966.00i	0.208028i
$448$	− 1152.00i	− 0.121489i
$449$	10370.0	1.08996	0.544978	−	0.838450i	$-0.316538\pi$
$449$	10370.0	1.08996	0.544978	+	0.838450i	$0.316538\pi$
$450$	0	0
$451$	5024.00	0.524547
$452$	1560.00i	0.162337i
$453$	35.0000i	0.00363012i
$454$	8268.00	0.854706
$455$	0	0
$456$	288.000	0.0295764
$457$	10496.0i	1.07436i	0.843468	+	0.537180i	$0.180510\pi$
$457$	10496.0i	1.07436i	−0.843468	+	0.537180i	$0.819490\pi$
$458$	− 9020.00i	− 0.920255i
$459$	1060.00	0.107792
$460$	0	0
$461$	18021.0	1.82065	0.910327	−	0.413889i	$-0.135830\pi$
$461$	18021.0	1.82065	0.910327	+	0.413889i	$0.135830\pi$
$462$	− 1152.00i	− 0.116008i
$463$	− 17188.0i	− 1.72526i	−0.505838	−	0.862629i	$-0.668817\pi$
$463$	− 17188.0i	− 1.72526i	0.505838	−	0.862629i	$-0.331183\pi$
$464$	432.000	0.0432222
$465$	0	0
$466$	−10006.0	−0.994676
$467$	15246.0i	1.51071i	0.655317	+	0.755354i	$0.272535\pi$
$467$	15246.0i	1.51071i	−0.655317	+	0.755354i	$0.727465\pi$
$468$	4888.00i	0.482795i
$469$	−2808.00	−0.276464
$470$	0	0
$471$	1702.00	0.166505
$472$	5952.00i	0.580430i
$473$	− 576.000i	− 0.0559926i
$474$	1288.00	0.124810
$475$	0	0
$476$	−1440.00	−0.138660
$477$	− 364.000i	− 0.0349401i
$478$	− 12618.0i	− 1.20739i
$479$	8556.00	0.816145	0.408073	−	0.912949i	$-0.366201\pi$
$479$	8556.00	0.816145	0.408073	+	0.912949i	$0.366201\pi$
$480$	0	0
$481$	−2632.00	−0.249499
$482$	− 6076.00i	− 0.574179i
$483$	414.000i	0.0390014i
$484$	1228.00	0.115327
$485$	0	0
$486$	4160.00	0.388275
$487$	1805.00i	0.167951i	0.996468	+	0.0839757i	$0.0267618\pi$
$487$	1805.00i	0.167951i	−0.996468	+	0.0839757i	$0.973238\pi$
$488$	4416.00i	0.409637i
$489$	2045.00	0.189117
$490$	0	0
$491$	5245.00	0.482085	0.241042	−	0.970515i	$-0.422511\pi$
$491$	5245.00	0.482085	0.241042	+	0.970515i	$0.422511\pi$
$492$	628.000i	0.0575456i
$493$	− 540.000i	− 0.0493314i
$494$	3384.00	0.308205
$495$	0	0
$496$	−528.000	−0.0477982
$497$	12582.0i	1.13557i
$498$	1024.00i	0.0921416i
$499$	−9027.00	−0.809828	−0.404914	−	0.914355i	$-0.632698\pi$
$499$	−9027.00	−0.809828	−0.404914	+	0.914355i	$0.632698\pi$
$500$	0	0
$501$	1016.00	0.0906019
$502$	2664.00i	0.236853i
$503$	3522.00i	0.312203i	0.987741	+	0.156102i	$0.0498927\pi$
$503$	3522.00i	0.312203i	−0.987741	+	0.156102i	$0.950107\pi$
$504$	−3744.00	−0.330895
$505$	0	0
$506$	1472.00	0.129325
$507$	− 12.0000i	− 0.00105116i
$508$	− 3076.00i	− 0.268652i
$509$	−3949.00	−0.343883	−0.171941	−	0.985107i	$-0.555004\pi$
$509$	−3949.00	−0.343883	−0.171941	+	0.985107i	$0.555004\pi$
$510$	0	0
$511$	10962.0	0.948983
$512$	− 512.000i	− 0.0441942i
$513$	− 1908.00i	− 0.164211i
$514$	−6602.00	−0.566540
$515$	0	0
$516$	72.0000	0.00614268
$517$	2080.00i	0.176941i
$518$	− 2016.00i	− 0.171000i
$519$	−598.000	−0.0505767
$520$	0	0
$521$	3236.00	0.272115	0.136057	−	0.990701i	$-0.456557\pi$
$521$	3236.00	0.272115	0.136057	+	0.990701i	$0.456557\pi$
$522$	− 1404.00i	− 0.117723i
$523$	12394.0i	1.03624i	0.855309	+	0.518118i	$0.173367\pi$
$523$	12394.0i	1.03624i	−0.855309	+	0.518118i	$0.826633\pi$
$524$	852.000	0.0710301
$525$	0	0
$526$	4144.00	0.343511
$527$	660.000i	0.0545542i
$528$	− 512.000i	− 0.0422006i
$529$	−529.000	−0.0434783
$530$	0	0
$531$	19344.0	1.58090
$532$	2592.00i	0.211236i
$533$	7379.00i	0.599662i
$534$	204.000	0.0165317
$535$	0	0
$536$	−1248.00	−0.100570
$537$	4607.00i	0.370217i
$538$	11442.0i	0.916914i
$539$	−608.000	−0.0485870
$540$	0	0
$541$	−7159.00	−0.568927	−0.284463	−	0.958687i	$-0.591815\pi$
$541$	−7159.00	−0.568927	−0.284463	+	0.958687i	$0.591815\pi$
$542$	11800.0i	0.935154i
$543$	− 1212.00i	− 0.0957862i
$544$	−640.000	−0.0504408
$545$	0	0
$546$	1692.00	0.132621
$547$	19761.0i	1.54464i	0.635232	+	0.772321i	$0.280904\pi$
$547$	19761.0i	1.54464i	−0.635232	+	0.772321i	$0.719096\pi$
$548$	11344.0i	0.884291i
$549$	14352.0	1.11572
$550$	0	0
$551$	−972.000	−0.0751517
$552$	184.000i	0.0141876i
$553$	11592.0i	0.891396i
$554$	−12742.0	−0.977176
$555$	0	0
$556$	−6524.00	−0.497625
$557$	− 18010.0i	− 1.37003i	−0.728528	−	0.685016i	$-0.759795\pi$
$557$	− 18010.0i	− 1.37003i	0.728528	−	0.685016i	$-0.240205\pi$
$558$	1716.00i	0.130187i
$559$	846.000	0.0640107
$560$	0	0
$561$	−640.000	−0.0481655
$562$	− 6380.00i	− 0.478868i
$563$	− 2648.00i	− 0.198224i	−0.995076	−	0.0991118i	$-0.968400\pi$
$563$	− 2648.00i	− 0.198224i	0.995076	−	0.0991118i	$-0.0316001\pi$
$564$	−260.000	−0.0194113
$565$	0	0
$566$	−8452.00	−0.627675
$567$	11682.0i	0.865252i
$568$	5592.00i	0.413090i
$569$	1566.00	0.115378	0.0576890	−	0.998335i	$-0.481627\pi$
$569$	1566.00	0.115378	0.0576890	+	0.998335i	$0.481627\pi$
$570$	0	0
$571$	2864.00	0.209903	0.104952	−	0.994477i	$-0.466531\pi$
$571$	2864.00	0.209903	0.104952	+	0.994477i	$0.466531\pi$
$572$	− 6016.00i	− 0.439758i
$573$	− 1058.00i	− 0.0771354i
$574$	−5652.00	−0.410993
$575$	0	0
$576$	−1664.00	−0.120370
$577$	929.000i	0.0670273i	0.999438	+	0.0335137i	$0.0106697\pi$
$577$	929.000i	0.0670273i	−0.999438	+	0.0335137i	$0.989330\pi$
$578$	− 9026.00i	− 0.649537i
$579$	−1047.00	−0.0751500
$580$	0	0
$581$	−9216.00	−0.658079
$582$	− 1156.00i	− 0.0823329i
$583$	448.000i	0.0318255i
$584$	4872.00	0.345214
$585$	0	0
$586$	12096.0	0.852698
$587$	19499.0i	1.37106i	0.728046	+	0.685528i	$0.240429\pi$
$587$	19499.0i	1.37106i	−0.728046	+	0.685528i	$0.759571\pi$
$588$	− 76.0000i	− 0.00533025i
$589$	1188.00	0.0831081
$590$	0	0
$591$	251.000	0.0174700
$592$	− 896.000i	− 0.0622050i
$593$	6570.00i	0.454971i	0.973782	+	0.227485i	$0.0730504\pi$
$593$	6570.00i	0.454971i	−0.973782	+	0.227485i	$0.926950\pi$
$594$	−3392.00	−0.234302
$595$	0	0
$596$	−7864.00	−0.540473
$597$	3508.00i	0.240491i
$598$	2162.00i	0.147844i
$599$	−1880.00	−0.128238	−0.0641191	−	0.997942i	$-0.520424\pi$
$599$	−1880.00	−0.128238	−0.0641191	+	0.997942i	$0.520424\pi$
$600$	0	0
$601$	3701.00	0.251193	0.125596	−	0.992081i	$-0.459916\pi$
$601$	3701.00	0.251193	0.125596	+	0.992081i	$0.459916\pi$
$602$	648.000i	0.0438713i
$603$	4056.00i	0.273919i
$604$	−140.000	−0.00943132
$605$	0	0
$606$	−12.0000	−0.000804400 0
$607$	3080.00i	0.205953i	0.994684	+	0.102976i	$0.0328366\pi$
$607$	3080.00i	0.205953i	−0.994684	+	0.102976i	$0.967163\pi$
$608$	1152.00i	0.0768417i
$609$	−486.000	−0.0323378
$610$	0	0
$611$	−3055.00	−0.202278
$612$	2080.00i	0.137384i
$613$	24004.0i	1.58159i	0.612083	+	0.790793i	$0.290332\pi$
$613$	24004.0i	1.58159i	−0.612083	+	0.790793i	$0.709668\pi$
$614$	−17256.0	−1.13419
$615$	0	0
$616$	4608.00	0.301399
$617$	− 780.000i	− 0.0508940i	−0.999676	−	0.0254470i	$-0.991899\pi$
$617$	− 780.000i	− 0.0508940i	0.999676	−	0.0254470i	$-0.00810091\pi$
$618$	− 320.000i	− 0.0208289i
$619$	−21892.0	−1.42151	−0.710754	−	0.703440i	$-0.751646\pi$
$619$	−21892.0	−1.42151	−0.710754	+	0.703440i	$0.751646\pi$
$620$	0	0
$621$	1219.00	0.0787710
$622$	− 16494.0i	− 1.06326i
$623$	1836.00i	0.118070i
$624$	752.000	0.0482437
$625$	0	0
$626$	5240.00	0.334557
$627$	1152.00i	0.0733755i
$628$	6808.00i	0.432594i
$629$	−1120.00	−0.0709973
$630$	0	0
$631$	8050.00	0.507869	0.253935	−	0.967221i	$-0.418275\pi$
$631$	8050.00	0.507869	0.253935	+	0.967221i	$0.418275\pi$
$632$	5152.00i	0.324265i
$633$	− 3296.00i	− 0.206958i
$634$	−19812.0	−1.24106
$635$	0	0
$636$	−56.0000	−0.00349142
$637$	− 893.000i	− 0.0555447i
$638$	1728.00i	0.107229i
$639$	18174.0	1.12512
$640$	0	0
$641$	−25890.0	−1.59531	−0.797655	−	0.603114i	$-0.793926\pi$
$641$	−25890.0	−1.59531	−0.797655	+	0.603114i	$0.793926\pi$
$642$	− 760.000i	− 0.0467209i
$643$	4774.00i	0.292797i	0.989226	+	0.146398i	$0.0467681\pi$
$643$	4774.00i	0.292797i	−0.989226	+	0.146398i	$0.953232\pi$
$644$	−1656.00	−0.101328
$645$	0	0
$646$	1440.00	0.0877029
$647$	− 3349.00i	− 0.203497i	−0.994810	−	0.101749i	$-0.967556\pi$
$647$	− 3349.00i	− 0.203497i	0.994810	−	0.101749i	$-0.0324438\pi$
$648$	5192.00i	0.314755i
$649$	−23808.0	−1.43998
$650$	0	0
$651$	594.000	0.0357614
$652$	8180.00i	0.491340i
$653$	24813.0i	1.48699i	0.668739	+	0.743497i	$0.266834\pi$
$653$	24813.0i	1.48699i	−0.668739	+	0.743497i	$0.733166\pi$
$654$	−500.000	−0.0298953
$655$	0	0
$656$	−2512.00	−0.149508
$657$	− 15834.0i	− 0.940248i
$658$	− 2340.00i	− 0.138636i
$659$	−18180.0	−1.07465	−0.537323	−	0.843376i	$-0.680565\pi$
$659$	−18180.0	−1.07465	−0.537323	+	0.843376i	$0.680565\pi$
$660$	0	0
$661$	−29250.0	−1.72117	−0.860585	−	0.509307i	$-0.829902\pi$
$661$	−29250.0	−1.72117	−0.860585	+	0.509307i	$0.829902\pi$
$662$	16230.0i	0.952865i
$663$	− 940.000i	− 0.0550627i
$664$	−4096.00	−0.239391
$665$	0	0
$666$	−2912.00	−0.169426
$667$	− 621.000i	− 0.0360498i
$668$	4064.00i	0.235391i
$669$	2720.00	0.157192
$670$	0	0
$671$	−17664.0	−1.01626
$672$	576.000i	0.0330650i
$673$	− 23027.0i	− 1.31891i	−0.751745	−	0.659454i	$-0.770787\pi$
$673$	− 23027.0i	− 1.31891i	0.751745	−	0.659454i	$-0.229213\pi$
$674$	15172.0	0.867068
$675$	0	0
$676$	48.0000	0.00273100
$677$	− 20106.0i	− 1.14141i	−0.821154	−	0.570706i	$-0.806669\pi$
$677$	− 20106.0i	− 1.14141i	0.821154	−	0.570706i	$-0.193331\pi$
$678$	− 780.000i	− 0.0441825i
$679$	10404.0	0.588025
$680$	0	0
$681$	−4134.00	−0.232621
$682$	− 2112.00i	− 0.118582i
$683$	− 18745.0i	− 1.05016i	−0.851054	−	0.525079i	$-0.824036\pi$
$683$	− 18745.0i	− 1.05016i	0.851054	−	0.525079i	$-0.175964\pi$
$684$	3744.00	0.209292
$685$	0	0
$686$	13032.0	0.725312
$687$	4510.00i	0.250462i
$688$	288.000i	0.0159592i
$689$	−658.000	−0.0363829
$690$	0	0
$691$	24424.0	1.34462	0.672310	−	0.740270i	$-0.265302\pi$
$691$	24424.0	1.34462	0.672310	+	0.740270i	$0.265302\pi$
$692$	− 2392.00i	− 0.131402i
$693$	− 14976.0i	− 0.820911i
$694$	−2712.00	−0.148337
$695$	0	0
$696$	−216.000	−0.0117636
$697$	3140.00i	0.170640i
$698$	13298.0i	0.721113i
$699$	5003.00	0.270717
$700$	0	0
$701$	−27278.0	−1.46972	−0.734862	−	0.678217i	$-0.762753\pi$
$701$	−27278.0	−1.46972	−0.734862	+	0.678217i	$0.762753\pi$
$702$	− 4982.00i	− 0.267854i
$703$	2016.00i	0.108158i
$704$	2048.00	0.109640
$705$	0	0
$706$	21382.0	1.13983
$707$	− 108.000i	− 0.00574506i
$708$	− 2976.00i	− 0.157973i
$709$	−12214.0	−0.646977	−0.323488	−	0.946232i	$-0.604856\pi$
$709$	−12214.0	−0.646977	−0.323488	+	0.946232i	$0.604856\pi$
$710$	0	0
$711$	16744.0	0.883191
$712$	816.000i	0.0429507i
$713$	759.000i	0.0398664i
$714$	720.000	0.0377385
$715$	0	0
$716$	−18428.0	−0.961853
$717$	6309.00i	0.328611i
$718$	− 12840.0i	− 0.667388i
$719$	−12932.0	−0.670768	−0.335384	−	0.942082i	$-0.608866\pi$
$719$	−12932.0	−0.670768	−0.335384	+	0.942082i	$0.608866\pi$
$720$	0	0
$721$	2880.00	0.148761
$722$	11126.0i	0.573500i
$723$	3038.00i	0.156272i
$724$	4848.00	0.248860
$725$	0	0
$726$	−614.000	−0.0313880
$727$	− 10046.0i	− 0.512497i	−0.966611	−	0.256249i	$-0.917513\pi$
$727$	− 10046.0i	− 0.512497i	0.966611	−	0.256249i	$-0.0824866\pi$
$728$	6768.00i	0.344559i
$729$	15443.0	0.784586
$730$	0	0
$731$	360.000	0.0182149
$732$	− 2208.00i	− 0.111489i
$733$	− 5924.00i	− 0.298510i	−0.988799	−	0.149255i	$-0.952312\pi$
$733$	− 5924.00i	− 0.298510i	0.988799	−	0.149255i	$-0.0476876\pi$
$734$	1048.00	0.0527008
$735$	0	0
$736$	−736.000	−0.0368605
$737$	− 4992.00i	− 0.249502i
$738$	8164.00i	0.407210i
$739$	−829.000	−0.0412656	−0.0206328	−	0.999787i	$-0.506568\pi$
$739$	−829.000	−0.0412656	−0.0206328	+	0.999787i	$0.506568\pi$
$740$	0	0
$741$	−1692.00	−0.0838828
$742$	− 504.000i	− 0.0249359i
$743$	7072.00i	0.349188i	0.984641	+	0.174594i	$0.0558613\pi$
$743$	7072.00i	0.349188i	−0.984641	+	0.174594i	$0.944139\pi$
$744$	264.000	0.0130090
$745$	0	0
$746$	11132.0	0.546342
$747$	13312.0i	0.652022i
$748$	− 2560.00i	− 0.125138i
$749$	6840.00	0.333682
$750$	0	0
$751$	16234.0	0.788798	0.394399	−	0.918939i	$-0.370953\pi$
$751$	16234.0	0.788798	0.394399	+	0.918939i	$0.370953\pi$
$752$	− 1040.00i	− 0.0504320i
$753$	− 1332.00i	− 0.0644632i
$754$	−2538.00	−0.122584
$755$	0	0
$756$	3816.00	0.183580
$757$	− 9128.00i	− 0.438260i	−0.975696	−	0.219130i	$-0.929678\pi$
$757$	− 9128.00i	− 0.438260i	0.975696	−	0.219130i	$-0.0703219\pi$
$758$	4480.00i	0.214671i
$759$	−736.000	−0.0351978
$760$	0	0
$761$	165.000	0.00785972	0.00392986	−	0.999992i	$-0.498749\pi$
$761$	165.000	0.00785972	0.00392986	+	0.999992i	$0.498749\pi$
$762$	1538.00i	0.0731179i
$763$	− 4500.00i	− 0.213514i
$764$	4232.00	0.200404
$765$	0	0
$766$	17556.0	0.828099
$767$	− 34968.0i	− 1.64618i
$768$	256.000i	0.0120281i
$769$	20834.0	0.976974	0.488487	−	0.872571i	$-0.337549\pi$
$769$	20834.0	0.976974	0.488487	+	0.872571i	$0.337549\pi$
$770$	0	0
$771$	3301.00	0.154193
$772$	− 4188.00i	− 0.195245i
$773$	− 31782.0i	− 1.47881i	−0.673262	−	0.739404i	$-0.735107\pi$
$773$	− 31782.0i	− 1.47881i	0.673262	−	0.739404i	$-0.264893\pi$
$774$	936.000	0.0434675
$775$	0	0
$776$	4624.00	0.213907
$777$	1008.00i	0.0465403i
$778$	8112.00i	0.373817i
$779$	5652.00	0.259954
$780$	0	0
$781$	−22368.0	−1.02483
$782$	920.000i	0.0420705i
$783$	1431.00i	0.0653126i
$784$	304.000	0.0138484
$785$	0	0
$786$	−426.000	−0.0193320
$787$	− 33104.0i	− 1.49940i	−0.661776	−	0.749701i	$-0.730197\pi$
$787$	− 33104.0i	− 1.49940i	0.661776	−	0.749701i	$-0.269803\pi$
$788$	1004.00i	0.0453883i
$789$	−2072.00	−0.0934920
$790$	0	0
$791$	7020.00	0.315553
$792$	− 6656.00i	− 0.298625i
$793$	− 25944.0i	− 1.16179i
$794$	18302.0	0.818027
$795$	0	0
$796$	−14032.0	−0.624813
$797$	− 4736.00i	− 0.210486i	−0.994447	−	0.105243i	$-0.966438\pi$
$797$	− 4736.00i	− 0.210486i	0.994447	−	0.105243i	$-0.0335621\pi$
$798$	− 1296.00i	− 0.0574911i
$799$	−1300.00	−0.0575603
$800$	0	0
$801$	2652.00	0.116984
$802$	− 31860.0i	− 1.40276i
$803$	19488.0i	0.856434i
$804$	624.000	0.0273716
$805$	0	0
$806$	3102.00	0.135562
$807$	− 5721.00i	− 0.249552i
$808$	− 48.0000i	− 0.00208989i
$809$	7470.00	0.324637	0.162318	−	0.986738i	$-0.448103\pi$
$809$	7470.00	0.324637	0.162318	+	0.986738i	$0.448103\pi$
$810$	0	0
$811$	19919.0	0.862455	0.431227	−	0.902243i	$-0.358081\pi$
$811$	19919.0	0.862455	0.431227	+	0.902243i	$0.358081\pi$
$812$	− 1944.00i	− 0.0840160i
$813$	− 5900.00i	− 0.254517i
$814$	3584.00	0.154323
$815$	0	0
$816$	320.000	0.0137282
$817$	− 648.000i	− 0.0277487i
$818$	− 11782.0i	− 0.503604i
$819$	21996.0	0.938465
$820$	0	0
$821$	−22694.0	−0.964709	−0.482354	−	0.875976i	$-0.660218\pi$
$821$	−22694.0	−0.964709	−0.482354	+	0.875976i	$0.660218\pi$
$822$	− 5672.00i	− 0.240674i
$823$	− 31907.0i	− 1.35141i	−0.737173	−	0.675704i	$-0.763840\pi$
$823$	− 31907.0i	− 1.35141i	0.737173	−	0.675704i	$-0.236160\pi$
$824$	1280.00	0.0541152
$825$	0	0
$826$	26784.0	1.12825
$827$	15236.0i	0.640638i	0.947310	+	0.320319i	$0.103790\pi$
$827$	15236.0i	0.640638i	−0.947310	+	0.320319i	$0.896210\pi$
$828$	2392.00i	0.100396i
$829$	−27286.0	−1.14316	−0.571581	−	0.820545i	$-0.693670\pi$
$829$	−27286.0	−1.14316	−0.571581	+	0.820545i	$0.693670\pi$
$830$	0	0
$831$	6371.00	0.265954
$832$	3008.00i	0.125341i
$833$	− 380.000i	− 0.0158058i
$834$	3262.00	0.135436
$835$	0	0
$836$	−4608.00	−0.190635
$837$	− 1749.00i	− 0.0722273i
$838$	30564.0i	1.25992i
$839$	−23054.0	−0.948644	−0.474322	−	0.880351i	$-0.657307\pi$
$839$	−23054.0	−0.948644	−0.474322	+	0.880351i	$0.657307\pi$
$840$	0	0
$841$	−23660.0	−0.970109
$842$	21868.0i	0.895037i
$843$	3190.00i	0.130331i
$844$	13184.0	0.537692
$845$	0	0
$846$	−3380.00	−0.137360
$847$	− 5526.00i	− 0.224174i
$848$	− 224.000i	− 0.00907098i
$849$	4226.00	0.170832
$850$	0	0
$851$	−1288.00	−0.0518826
$852$	− 2796.00i	− 0.112429i
$853$	− 34506.0i	− 1.38507i	−0.721385	−	0.692534i	$-0.756494\pi$
$853$	− 34506.0i	− 1.38507i	0.721385	−	0.692534i	$-0.243506\pi$
$854$	19872.0	0.796260
$855$	0	0
$856$	3040.00	0.121384
$857$	22263.0i	0.887386i	0.896179	+	0.443693i	$0.146332\pi$
$857$	22263.0i	0.887386i	−0.896179	+	0.443693i	$0.853668\pi$
$858$	3008.00i	0.119687i
$859$	−12851.0	−0.510443	−0.255221	−	0.966883i	$-0.582148\pi$
$859$	−12851.0	−0.510443	−0.255221	+	0.966883i	$0.582148\pi$
$860$	0	0
$861$	2826.00	0.111858
$862$	5588.00i	0.220798i
$863$	15723.0i	0.620182i	0.950707	+	0.310091i	$0.100360\pi$
$863$	15723.0i	0.620182i	−0.950707	+	0.310091i	$0.899640\pi$
$864$	1696.00	0.0667814
$865$	0	0
$866$	−30124.0	−1.18205
$867$	4513.00i	0.176781i
$868$	2376.00i	0.0929109i
$869$	−20608.0	−0.804463
$870$	0	0
$871$	7332.00	0.285230
$872$	− 2000.00i	− 0.0776704i
$873$	− 15028.0i	− 0.582613i
$874$	1656.00	0.0640904
$875$	0	0
$876$	−2436.00	−0.0939553
$877$	886.000i	0.0341141i	0.999855	+	0.0170571i	$0.00542970\pi$
$877$	886.000i	0.0341141i	−0.999855	+	0.0170571i	$0.994570\pi$
$878$	522.000i	0.0200645i
$879$	−6048.00	−0.232075
$880$	0	0
$881$	−37120.0	−1.41953	−0.709764	−	0.704439i	$-0.751199\pi$
$881$	−37120.0	−1.41953	−0.709764	+	0.704439i	$0.751199\pi$
$882$	− 988.000i	− 0.0377185i
$883$	− 7524.00i	− 0.286753i	−0.989668	−	0.143376i	$-0.954204\pi$
$883$	− 7524.00i	− 0.286753i	0.989668	−	0.143376i	$-0.0457960\pi$
$884$	3760.00	0.143057
$885$	0	0
$886$	−14166.0	−0.537151
$887$	− 9221.00i	− 0.349054i	−0.984652	−	0.174527i	$-0.944160\pi$
$887$	− 9221.00i	− 0.349054i	0.984652	−	0.174527i	$-0.0558396\pi$
$888$	448.000i	0.0169301i
$889$	−13842.0	−0.522211
$890$	0	0
$891$	−20768.0	−0.780869
$892$	10880.0i	0.408396i
$893$	2340.00i	0.0876877i
$894$	3932.00	0.147098
$895$	0	0
$896$	−2304.00	−0.0859054
$897$	− 1081.00i	− 0.0402381i
$898$	− 20740.0i	− 0.770716i
$899$	−891.000	−0.0330551
$900$	0	0
$901$	−280.000	−0.0103531
$902$	− 10048.0i	− 0.370911i
$903$	− 324.000i	− 0.0119402i
$904$	3120.00	0.114789
$905$	0	0
$906$	70.0000	0.00256688
$907$	− 29116.0i	− 1.06591i	−0.846143	−	0.532955i	$-0.821081\pi$
$907$	− 29116.0i	− 1.06591i	0.846143	−	0.532955i	$-0.178919\pi$
$908$	− 16536.0i	− 0.604368i
$909$	−156.000	−0.00569218
$910$	0	0
$911$	11440.0	0.416053	0.208026	−	0.978123i	$-0.433296\pi$
$911$	11440.0	0.416053	0.208026	+	0.978123i	$0.433296\pi$
$912$	− 576.000i	− 0.0209137i
$913$	− 16384.0i	− 0.593901i
$914$	20992.0	0.759687
$915$	0	0
$916$	−18040.0	−0.650719
$917$	− 3834.00i	− 0.138070i
$918$	− 2120.00i	− 0.0762205i
$919$	−2958.00	−0.106176	−0.0530878	−	0.998590i	$-0.516906\pi$
$919$	−2958.00	−0.106176	−0.0530878	+	0.998590i	$0.516906\pi$
$920$	0	0
$921$	8628.00	0.308689
$922$	− 36042.0i	− 1.28740i
$923$	− 32853.0i	− 1.17158i
$924$	−2304.00	−0.0820303
$925$	0	0
$926$	−34376.0	−1.21994
$927$	− 4160.00i	− 0.147392i
$928$	− 864.000i	− 0.0305627i
$929$	−20907.0	−0.738360	−0.369180	−	0.929358i	$-0.620361\pi$
$929$	−20907.0	−0.738360	−0.369180	+	0.929358i	$0.620361\pi$
$930$	0	0
$931$	−684.000	−0.0240786
$932$	20012.0i	0.703342i
$933$	8247.00i	0.289383i
$934$	30492.0	1.06823
$935$	0	0
$936$	9776.00	0.341387
$937$	− 9748.00i	− 0.339865i	−0.985456	−	0.169932i	$-0.945645\pi$
$937$	− 9748.00i	− 0.339865i	0.985456	−	0.169932i	$-0.0543549\pi$
$938$	5616.00i	0.195489i
$939$	−2620.00	−0.0910548
$940$	0	0
$941$	19624.0	0.679834	0.339917	−	0.940455i	$-0.389601\pi$
$941$	19624.0	0.679834	0.339917	+	0.940455i	$0.389601\pi$
$942$	− 3404.00i	− 0.117737i
$943$	3611.00i	0.124698i
$944$	11904.0	0.410426
$945$	0	0
$946$	−1152.00	−0.0395928
$947$	41859.0i	1.43636i	0.695856	+	0.718181i	$0.255025\pi$
$947$	41859.0i	1.43636i	−0.695856	+	0.718181i	$0.744975\pi$
$948$	− 2576.00i	− 0.0882538i
$949$	−28623.0	−0.979075
$950$	0	0
$951$	9906.00	0.337775
$952$	2880.00i	0.0980476i
$953$	29226.0i	0.993413i	0.867918	+	0.496707i	$0.165458\pi$
$953$	29226.0i	0.993413i	−0.867918	+	0.496707i	$0.834542\pi$
$954$	−728.000	−0.0247064
$955$	0	0
$956$	−25236.0	−0.853756
$957$	− 864.000i	− 0.0291841i
$958$	− 17112.0i	− 0.577102i
$959$	51048.0	1.71890
$960$	0	0
$961$	−28702.0	−0.963445
$962$	5264.00i	0.176422i
$963$	− 9880.00i	− 0.330611i
$964$	−12152.0	−0.406006
$965$	0	0
$966$	828.000	0.0275781
$967$	29849.0i	0.992636i	0.868141	+	0.496318i	$0.165315\pi$
$967$	29849.0i	0.992636i	−0.868141	+	0.496318i	$0.834685\pi$
$968$	− 2456.00i	− 0.0815484i
$969$	−720.000	−0.0238697
$970$	0	0
$971$	−9390.00	−0.310339	−0.155170	−	0.987888i	$-0.549592\pi$
$971$	−9390.00	−0.310339	−0.155170	+	0.987888i	$0.549592\pi$
$972$	− 8320.00i	− 0.274552i
$973$	29358.0i	0.967291i
$974$	3610.00	0.118760
$975$	0	0
$976$	8832.00	0.289657
$977$	− 33536.0i	− 1.09817i	−0.835767	−	0.549085i	$-0.814976\pi$
$977$	− 33536.0i	− 1.09817i	0.835767	−	0.549085i	$-0.185024\pi$
$978$	− 4090.00i	− 0.133726i
$979$	−3264.00	−0.106556
$980$	0	0
$981$	−6500.00	−0.211548
$982$	− 10490.0i	− 0.340885i
$983$	28994.0i	0.940758i	0.882465	+	0.470379i	$0.155883\pi$
$983$	28994.0i	0.940758i	−0.882465	+	0.470379i	$0.844117\pi$
$984$	1256.00	0.0406909
$985$	0	0
$986$	−1080.00	−0.0348826
$987$	1170.00i	0.0377320i
$988$	− 6768.00i	− 0.217934i
$989$	414.000	0.0133109
$990$	0	0
$991$	11272.0	0.361319	0.180659	−	0.983546i	$-0.442177\pi$
$991$	11272.0	0.361319	0.180659	+	0.983546i	$0.442177\pi$
$992$	1056.00i	0.0337984i
$993$	− 8115.00i	− 0.259337i
$994$	25164.0	0.802971
$995$	0	0
$996$	2048.00	0.0651540
$997$	− 61186.0i	− 1.94361i	−0.235784	−	0.971805i	$-0.575766\pi$
$997$	− 61186.0i	− 1.94361i	0.235784	−	0.971805i	$-0.424234\pi$
$998$	18054.0i	0.572635i
$999$	2968.00	0.0939974

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1150.4.b.f.599.1		2
5.2	odd	4		230.4.a.e.1.1	✓	1
5.3	odd	4		1150.4.a.b.1.1		1
5.4	even	2	inner	1150.4.b.f.599.2		2
15.2	even	4		2070.4.a.e.1.1		1
20.7	even	4		1840.4.a.d.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.4.a.e.1.1	✓	1	5.2	odd	4
1150.4.a.b.1.1		1	5.3	odd	4
1150.4.b.f.599.1		2	1.1	even	1	trivial
1150.4.b.f.599.2		2	5.4	even	2	inner
1840.4.a.d.1.1		1	20.7	even	4
2070.4.a.e.1.1		1	15.2	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:	\( N \)	\(=\)	\( 1150 = 2 \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1150.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.00000i q^{2} +1.00000i q^{3} -4.00000 q^{4} +2.00000 q^{6} +18.0000i q^{7} +8.00000i q^{8} +26.0000 q^{9} -32.0000 q^{11} -4.00000i q^{12} -47.0000i q^{13} +36.0000 q^{14} +16.0000 q^{16} -20.0000i q^{17} -52.0000i q^{18} -36.0000 q^{19} -18.0000 q^{21} +64.0000i q^{22} -23.0000i q^{23} -8.00000 q^{24} -94.0000 q^{26} +53.0000i q^{27} -72.0000i q^{28} +27.0000 q^{29} -33.0000 q^{31} -32.0000i q^{32} -32.0000i q^{33} -40.0000 q^{34} -104.000 q^{36} -56.0000i q^{37} +72.0000i q^{38} +47.0000 q^{39} -157.000 q^{41} +36.0000i q^{42} +18.0000i q^{43} +128.000 q^{44} -46.0000 q^{46} -65.0000i q^{47} +16.0000i q^{48} +19.0000 q^{49} +20.0000 q^{51} +188.000i q^{52} -14.0000i q^{53} +106.000 q^{54} -144.000 q^{56} -36.0000i q^{57} -54.0000i q^{58} +744.000 q^{59} +552.000 q^{61} +66.0000i q^{62} +468.000i q^{63} -64.0000 q^{64} -64.0000 q^{66} +156.000i q^{67} +80.0000i q^{68} +23.0000 q^{69} +699.000 q^{71} +208.000i q^{72} -609.000i q^{73} -112.000 q^{74} +144.000 q^{76} -576.000i q^{77} -94.0000i q^{78} +644.000 q^{79} +649.000 q^{81} +314.000i q^{82} +512.000i q^{83} +72.0000 q^{84} +36.0000 q^{86} +27.0000i q^{87} -256.000i q^{88} +102.000 q^{89} +846.000 q^{91} +92.0000i q^{92} -33.0000i q^{93} -130.000 q^{94} +32.0000 q^{96} -578.000i q^{97} -38.0000i q^{98} -832.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{4} + 4 q^{6} + 52 q^{9} - 64 q^{11} + 72 q^{14} + 32 q^{16} - 72 q^{19} - 36 q^{21} - 16 q^{24} - 188 q^{26} + 54 q^{29} - 66 q^{31} - 80 q^{34} - 208 q^{36} + 94 q^{39} - 314 q^{41} + 256 q^{44}+ \cdots - 1664 q^{99}+O(q^{100}) \) 2 * q - 8 * q^4 + 4 * q^6 + 52 * q^9 - 64 * q^11 + 72 * q^14 + 32 * q^16 - 72 * q^19 - 36 * q^21 - 16 * q^24 - 188 * q^26 + 54 * q^29 - 66 * q^31 - 80 * q^34 - 208 * q^36 + 94 * q^39 - 314 * q^41 + 256 * q^44 - 92 * q^46 + 38 * q^49 + 40 * q^51 + 212 * q^54 - 288 * q^56 + 1488 * q^59 + 1104 * q^61 - 128 * q^64 - 128 * q^66 + 46 * q^69 + 1398 * q^71 - 224 * q^74 + 288 * q^76 + 1288 * q^79 + 1298 * q^81 + 144 * q^84 + 72 * q^86 + 204 * q^89 + 1692 * q^91 - 260 * q^94 + 64 * q^96 - 1664 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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