LMFDB - Embedded newform 1680.2.ba.d.911.15

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1680,2,Mod(911,1680)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1680, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1680.911"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1680.ba (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,0,0,0,0,-10,0,12] 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:	$13.4148675396$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 8 x^{15} + 48 x^{14} - 196 x^{13} + 640 x^{12} - 1656 x^{11} + 3522 x^{10} - 6148 x^{9} + \cdots + 13 $ x^16 - 8x^15 + 48x^14 - 196x^13 + 640x^12 - 1656x^11 + 3522x^10 - 6148x^9 + 8856x^8 - 10460x^7 + 10032x^6 - 7680x^5 + 4586x^4 - 2062x^3 + 659x^2 - 134*x + 13
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{14} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			911.15
Root			$0.500000 - 0.492896i$ of defining polynomial
Character	$\chi$	$=$	1680.911
Dual form			1680.2.ba.d.911.16

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(1.43348 - 0.972180i) q^{3} +1.00000i q^{5} -1.00000i q^{7} +(1.10973 - 2.78720i) q^{9} -0.323488 q^{11} -0.149117 q^{13} +(0.972180 + 1.43348i) q^{15} -3.33957i q^{17} -7.32126i q^{19} +(-0.972180 - 1.43348i) q^{21} -2.14146 q^{23} -1.00000 q^{25} +(-1.11888 - 5.07426i) q^{27} +6.71019i q^{29} -6.03744i q^{31} +(-0.463714 + 0.314489i) q^{33} +1.00000 q^{35} +5.36008 q^{37} +(-0.213756 + 0.144969i) q^{39} -3.14246i q^{41} +3.36679i q^{43} +(2.78720 + 1.10973i) q^{45} +8.69575 q^{47} -1.00000 q^{49} +(-3.24666 - 4.78720i) q^{51} -6.57500i q^{53} -0.323488i q^{55} +(-7.11759 - 10.4949i) q^{57} +7.92077 q^{59} -8.37615 q^{61} +(-2.78720 - 1.10973i) q^{63} -0.149117i q^{65} -4.07060i q^{67} +(-3.06975 + 2.08189i) q^{69} +9.48706 q^{71} +0.0336980 q^{73} +(-1.43348 + 0.972180i) q^{75} +0.323488i q^{77} -10.2173i q^{79} +(-6.53699 - 6.18609i) q^{81} -5.08834 q^{83} +3.33957 q^{85} +(6.52351 + 9.61893i) q^{87} +12.4661i q^{89} +0.149117i q^{91} +(-5.86948 - 8.65455i) q^{93} +7.32126 q^{95} -7.69471 q^{97} +(-0.358985 + 0.901627i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 10 q^{9} + 12 q^{11} - 2 q^{15} + 2 q^{21} + 8 q^{23} - 16 q^{25} - 12 q^{27} + 12 q^{33} + 16 q^{35} - 8 q^{37} - 14 q^{39} + 8 q^{45} + 16 q^{47} - 16 q^{49} - 34 q^{51} + 12 q^{57} + 32 q^{59}+ \cdots - 50 q^{99}+O(q^{100}) $ 16 * q - 10 * q^9 + 12 * q^11 - 2 * q^15 + 2 * q^21 + 8 * q^23 - 16 * q^25 - 12 * q^27 + 12 * q^33 + 16 * q^35 - 8 * q^37 - 14 * q^39 + 8 * q^45 + 16 * q^47 - 16 * q^49 - 34 * q^51 + 12 * q^57 + 32 * q^59 + 8 * q^61 - 8 * q^63 - 20 * q^69 - 32 * q^71 + 48 * q^73 - 22 * q^81 + 72 * q^83 - 12 * q^85 - 44 * q^87 + 28 * q^93 + 8 * q^95 - 8 * q^97 - 50 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1680\mathbb{Z}\right)^\times$.

$n$	$241$	$337$	$421$	$1121$	$1471$
$\chi(n)$	$1$	$1$	$1$	$-1$	$-1$

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 $

$2$		0			0
$2$		0			0
$3$	1.43348	−	0.972180i	0.827620	−	0.561288i
$3$	1.43348	−	0.972180i	0.827620	−	0.561288i
$4$		0			0
$5$			1.00000i			0.447214i
$5$			1.00000i			0.447214i
$6$		0			0
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$		0			0
$9$	1.10973	−	2.78720i	0.369911	−	0.929067i
$10$		0			0
$11$	−0.323488			−0.0975353			−0.0487677	−	0.998810i	$-0.515529\pi$
$11$	−0.323488			−0.0975353			−0.0487677	+	0.998810i	$0.515529\pi$
$12$		0			0
$13$	−0.149117			−0.0413576			−0.0206788	−	0.999786i	$-0.506583\pi$
$13$	−0.149117			−0.0413576			−0.0206788	+	0.999786i	$0.506583\pi$
$14$		0			0
$15$	0.972180	+	1.43348i	0.251016	+	0.370123i
$16$		0			0
$17$		−	3.33957i		−	0.809964i	−0.914325	−	0.404982i	$-0.867278\pi$
$17$		−	3.33957i		−	0.809964i	0.914325	−	0.404982i	$-0.132722\pi$
$18$		0			0
$19$		−	7.32126i		−	1.67961i	−0.542886	−	0.839806i	$-0.682668\pi$
$19$		−	7.32126i		−	1.67961i	0.542886	−	0.839806i	$-0.317332\pi$
$20$		0			0
$21$	−0.972180	−	1.43348i	−0.212147	−	0.312811i
$22$		0			0
$23$	−2.14146			−0.446526			−0.223263	−	0.974758i	$-0.571671\pi$
$23$	−2.14146			−0.446526			−0.223263	+	0.974758i	$0.571671\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$	−1.11888	−	5.07426i	−0.215329	−	0.976542i
$28$		0			0
$29$			6.71019i			1.24605i	0.782201	+	0.623026i	$0.214097\pi$
$29$			6.71019i			1.24605i	−0.782201	+	0.623026i	$0.785903\pi$
$30$		0			0
$31$		−	6.03744i		−	1.08436i	−0.840264	−	0.542178i	$-0.817600\pi$
$31$		−	6.03744i		−	1.08436i	0.840264	−	0.542178i	$-0.182400\pi$
$32$		0			0
$33$	−0.463714	+	0.314489i	−0.0807222	+	0.0547455i
$34$		0			0
$35$	1.00000			0.169031
$36$		0			0
$37$	5.36008			0.881191			0.440596	−	0.897706i	$-0.354767\pi$
$37$	5.36008			0.881191			0.440596	+	0.897706i	$0.354767\pi$
$38$		0			0
$39$	−0.213756	+	0.144969i	−0.0342284	+	0.0232136i
$40$		0			0
$41$		−	3.14246i		−	0.490770i	−0.969426	−	0.245385i	$-0.921086\pi$
$41$		−	3.14246i		−	0.490770i	0.969426	−	0.245385i	$-0.0789144\pi$
$42$		0			0
$43$			3.36679i			0.513431i	0.966487	+	0.256715i	$0.0826403\pi$
$43$			3.36679i			0.513431i	−0.966487	+	0.256715i	$0.917360\pi$
$44$		0			0
$45$	2.78720	+	1.10973i	0.415492	+	0.165429i
$46$		0			0
$47$	8.69575			1.26841			0.634203	−	0.773166i	$-0.281328\pi$
$47$	8.69575			1.26841			0.634203	+	0.773166i	$0.281328\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	−3.24666	−	4.78720i	−0.454623	−	0.670342i
$52$		0			0
$53$		−	6.57500i		−	0.903146i	−0.892234	−	0.451573i	$-0.850863\pi$
$53$		−	6.57500i		−	0.903146i	0.892234	−	0.451573i	$-0.149137\pi$
$54$		0			0
$55$		−	0.323488i		−	0.0436191i
$56$		0			0
$57$	−7.11759	−	10.4949i	−0.942747	−	1.39008i
$58$		0			0
$59$	7.92077			1.03120			0.515598	−	0.856830i	$-0.327570\pi$
$59$	7.92077			1.03120			0.515598	+	0.856830i	$0.327570\pi$
$60$		0			0
$61$	−8.37615			−1.07246			−0.536228	−	0.844073i	$-0.680151\pi$
$61$	−8.37615			−1.07246			−0.536228	+	0.844073i	$0.680151\pi$
$62$		0			0
$63$	−2.78720	−	1.10973i	−0.351154	−	0.139813i
$64$		0			0
$65$		−	0.149117i		−	0.0184957i
$66$		0			0
$67$		−	4.07060i		−	0.497302i	−0.968593	−	0.248651i	$-0.920013\pi$
$67$		−	4.07060i		−	0.497302i	0.968593	−	0.248651i	$-0.0799873\pi$
$68$		0			0
$69$	−3.06975	+	2.08189i	−0.369554	+	0.250630i
$70$		0			0
$71$	9.48706			1.12591			0.562953	−	0.826489i	$-0.309665\pi$
$71$	9.48706			1.12591			0.562953	+	0.826489i	$0.309665\pi$
$72$		0			0
$73$	0.0336980			0.00394405			0.00197203	−	0.999998i	$-0.499372\pi$
$73$	0.0336980			0.00394405			0.00197203	+	0.999998i	$0.499372\pi$
$74$		0			0
$75$	−1.43348	+	0.972180i	−0.165524	+	0.112258i
$76$		0			0
$77$			0.323488i			0.0368649i
$78$		0			0
$79$		−	10.2173i		−	1.14954i	−0.818315	−	0.574771i	$-0.805091\pi$
$79$		−	10.2173i		−	1.14954i	0.818315	−	0.574771i	$-0.194909\pi$
$80$		0			0
$81$	−6.53699	−	6.18609i	−0.726332	−	0.687344i
$82$		0			0
$83$	−5.08834			−0.558518			−0.279259	−	0.960216i	$-0.590089\pi$
$83$	−5.08834			−0.558518			−0.279259	+	0.960216i	$0.590089\pi$
$84$		0			0
$85$	3.33957			0.362227
$86$		0			0
$87$	6.52351	+	9.61893i	0.699394	+	1.03126i
$88$		0			0
$89$			12.4661i			1.32140i	0.750650	+	0.660701i	$0.229741\pi$
$89$			12.4661i			1.32140i	−0.750650	+	0.660701i	$0.770259\pi$
$90$		0			0
$91$			0.149117i			0.0156317i
$92$		0			0
$93$	−5.86948	−	8.65455i	−0.608637	−	0.897435i
$94$		0			0
$95$	7.32126			0.751146
$96$		0			0
$97$	−7.69471			−0.781279			−0.390640	−	0.920544i	$-0.627746\pi$
$97$	−7.69471			−0.781279			−0.390640	+	0.920544i	$0.627746\pi$
$98$		0			0
$99$	−0.358985	+	0.901627i	−0.0360794	+	0.0906169i
$100$		0			0
$101$			11.9619i			1.19025i	0.803632	+	0.595127i	$0.202898\pi$
$101$			11.9619i			1.19025i	−0.803632	+	0.595127i	$0.797102\pi$
$102$		0			0
$103$		−	3.99886i		−	0.394019i	−0.980402	−	0.197010i	$-0.936877\pi$
$103$		−	3.99886i		−	0.394019i	0.980402	−	0.197010i	$-0.0631230\pi$
$104$		0			0
$105$	1.43348	−	0.972180i	0.139893	−	0.0948751i
$106$		0			0
$107$	12.0882			1.16861			0.584306	−	0.811534i	$-0.301367\pi$
$107$	12.0882			1.16861			0.584306	+	0.811534i	$0.301367\pi$
$108$		0			0
$109$	−0.153658			−0.0147177			−0.00735886	−	0.999973i	$-0.502342\pi$
$109$	−0.153658			−0.0147177			−0.00735886	+	0.999973i	$0.502342\pi$
$110$		0			0
$111$	7.68357	−	5.21096i	0.729292	−	0.494602i
$112$		0			0
$113$		−	9.24636i		−	0.869824i	−0.900473	−	0.434912i	$-0.856779\pi$
$113$		−	9.24636i		−	0.869824i	0.900473	−	0.434912i	$-0.143221\pi$
$114$		0			0
$115$		−	2.14146i		−	0.199693i
$116$		0			0
$117$	−0.165480	+	0.415619i	−0.0152986	+	0.0384240i
$118$		0			0
$119$	−3.33957			−0.306138
$120$		0			0
$121$	−10.8954			−0.990487
$122$		0			0
$123$	−3.05504	−	4.50466i	−0.275464	−	0.406171i
$124$		0			0
$125$		−	1.00000i		−	0.0894427i
$126$		0			0
$127$			12.8181i			1.13742i	0.822537	+	0.568712i	$0.192558\pi$
$127$			12.8181i			1.13742i	−0.822537	+	0.568712i	$0.807442\pi$
$128$		0			0
$129$	3.27313	+	4.82623i	0.288183	+	0.424926i
$130$		0			0
$131$	4.45842			0.389534			0.194767	−	0.980850i	$-0.437605\pi$
$131$	4.45842			0.389534			0.194767	+	0.980850i	$0.437605\pi$
$132$		0			0
$133$	−7.32126			−0.634834
$134$		0			0
$135$	5.07426	−	1.11888i	0.436723	−	0.0962982i
$136$		0			0
$137$			4.16349i			0.355711i	0.984057	+	0.177856i	$0.0569160\pi$
$137$			4.16349i			0.355711i	−0.984057	+	0.177856i	$0.943084\pi$
$138$		0			0
$139$		−	7.85313i		−	0.666093i	−0.942910	−	0.333047i	$-0.891923\pi$
$139$		−	7.85313i		−	0.666093i	0.942910	−	0.333047i	$-0.108077\pi$
$140$		0			0
$141$	12.4652	−	8.45384i	1.04976	−	0.711942i
$142$		0			0
$143$	0.0482376			0.00403383
$144$		0			0
$145$	−6.71019			−0.557251
$146$		0			0
$147$	−1.43348	+	0.972180i	−0.118231	+	0.0801841i
$148$		0			0
$149$			12.9773i			1.06314i	0.847013	+	0.531572i	$0.178399\pi$
$149$			12.9773i			1.06314i	−0.847013	+	0.531572i	$0.821601\pi$
$150$		0			0
$151$			21.5628i			1.75476i	0.479797	+	0.877379i	$0.340710\pi$
$151$			21.5628i			1.75476i	−0.479797	+	0.877379i	$0.659290\pi$
$152$		0			0
$153$	−9.30804	−	3.70602i	−0.752511	−	0.299614i
$154$		0			0
$155$	6.03744			0.484939
$156$		0			0
$157$	15.9701			1.27455			0.637275	−	0.770637i	$-0.280062\pi$
$157$	15.9701			1.27455			0.637275	+	0.770637i	$0.280062\pi$
$158$		0			0
$159$	−6.39209	−	9.42514i	−0.506926	−	0.747462i
$160$		0			0
$161$			2.14146i			0.168771i
$162$		0			0
$163$		−	20.5888i		−	1.61264i	−0.591481	−	0.806319i	$-0.701457\pi$
$163$		−	20.5888i		−	1.61264i	0.591481	−	0.806319i	$-0.298543\pi$
$164$		0			0
$165$	−0.314489	−	0.463714i	−0.0244829	−	0.0361001i
$166$		0			0
$167$	−0.279911			−0.0216601			−0.0108301	−	0.999941i	$-0.503447\pi$
$167$	−0.279911			−0.0216601			−0.0108301	+	0.999941i	$0.503447\pi$
$168$		0			0
$169$	−12.9778			−0.998290
$170$		0			0
$171$	−20.4058	−	8.12464i	−1.56047	−	0.621307i
$172$		0			0
$173$			13.5061i			1.02685i	0.858135	+	0.513424i	$0.171623\pi$
$173$			13.5061i			1.02685i	−0.858135	+	0.513424i	$0.828377\pi$
$174$		0			0
$175$			1.00000i			0.0755929i
$176$		0			0
$177$	11.3543	−	7.70042i	0.853439	−	0.578799i
$178$		0			0
$179$	−5.82742			−0.435562			−0.217781	−	0.975998i	$-0.569882\pi$
$179$	−5.82742			−0.435562			−0.217781	+	0.975998i	$0.569882\pi$
$180$		0			0
$181$	−7.34937			−0.546275			−0.273137	−	0.961975i	$-0.588061\pi$
$181$	−7.34937			−0.546275			−0.273137	+	0.961975i	$0.588061\pi$
$182$		0			0
$183$	−12.0071	+	8.14313i	−0.887587	+	0.601958i
$184$		0			0
$185$			5.36008i			0.394081i
$186$		0			0
$187$			1.08031i			0.0790001i
$188$		0			0
$189$	−5.07426	+	1.11888i	−0.369098	+	0.0813868i
$190$		0			0
$191$	−17.3593			−1.25608			−0.628039	−	0.778182i	$-0.716142\pi$
$191$	−17.3593			−1.25608			−0.628039	+	0.778182i	$0.716142\pi$
$192$		0			0
$193$	−18.4993			−1.33161			−0.665804	−	0.746127i	$-0.731911\pi$
$193$	−18.4993			−1.33161			−0.665804	+	0.746127i	$0.731911\pi$
$194$		0			0
$195$	−0.144969	−	0.213756i	−0.0103814	−	0.0153074i
$196$		0			0
$197$		−	1.83126i		−	0.130472i	−0.997870	−	0.0652359i	$-0.979220\pi$
$197$		−	1.83126i		−	0.130472i	0.997870	−	0.0652359i	$-0.0207800\pi$
$198$		0			0
$199$			7.53944i			0.534457i	0.963633	+	0.267228i	$0.0861078\pi$
$199$			7.53944i			0.534457i	−0.963633	+	0.267228i	$0.913892\pi$
$200$		0			0
$201$	−3.95735	−	5.83512i	−0.279130	−	0.411578i
$202$		0			0
$203$	6.71019			0.470963
$204$		0			0
$205$	3.14246			0.219479
$206$		0			0
$207$	−2.37645	+	5.96869i	−0.165175	+	0.414853i
$208$		0			0
$209$			2.36834i			0.163822i
$210$		0			0
$211$			12.1738i			0.838081i	0.907968	+	0.419040i	$0.137633\pi$
$211$			12.1738i			0.838081i	−0.907968	+	0.419040i	$0.862367\pi$
$212$		0			0
$213$	13.5995	−	9.22313i	0.931823	−	0.631958i
$214$		0			0
$215$	−3.36679			−0.229613
$216$		0			0
$217$	−6.03744			−0.409848
$218$		0			0
$219$	0.0483054	−	0.0327605i	0.00326418	−	0.00221375i
$220$		0			0
$221$			0.497986i			0.0334982i
$222$		0			0
$223$		−	2.75770i		−	0.184669i	−0.995728	−	0.0923345i	$-0.970567\pi$
$223$		−	2.75770i		−	0.184669i	0.995728	−	0.0923345i	$-0.0294329\pi$
$224$		0			0
$225$	−1.10973	+	2.78720i	−0.0739821	+	0.185813i
$226$		0			0
$227$	28.0440			1.86134			0.930672	−	0.365854i	$-0.119223\pi$
$227$	28.0440			1.86134			0.930672	+	0.365854i	$0.119223\pi$
$228$		0			0
$229$	20.9415			1.38385			0.691927	−	0.721967i	$-0.256762\pi$
$229$	20.9415			1.38385			0.691927	+	0.721967i	$0.256762\pi$
$230$		0			0
$231$	0.314489	+	0.463714i	0.0206918	+	0.0305101i
$232$		0			0
$233$			17.7823i			1.16496i	0.812846	+	0.582479i	$0.197917\pi$
$233$			17.7823i			1.16496i	−0.812846	+	0.582479i	$0.802083\pi$
$234$		0			0
$235$			8.69575i			0.567248i
$236$		0			0
$237$	−9.93310	−	14.6464i	−0.645224	−	0.951384i
$238$		0			0
$239$	−5.66539			−0.366464			−0.183232	−	0.983070i	$-0.558656\pi$
$239$	−5.66539			−0.366464			−0.183232	+	0.983070i	$0.558656\pi$
$240$		0			0
$241$	30.3249			1.95340			0.976699	−	0.214613i	$-0.0688490\pi$
$241$	30.3249			1.95340			0.976699	+	0.214613i	$0.0688490\pi$
$242$		0			0
$243$	−15.3846	−	2.51251i	−0.986925	−	0.161178i
$244$		0			0
$245$		−	1.00000i		−	0.0638877i
$246$		0			0
$247$			1.09172i			0.0694648i
$248$		0			0
$249$	−7.29404	+	4.94678i	−0.462241	+	0.313490i
$250$		0			0
$251$	−9.18993			−0.580063			−0.290032	−	0.957017i	$-0.593666\pi$
$251$	−9.18993			−0.580063			−0.290032	+	0.957017i	$0.593666\pi$
$252$		0			0
$253$	0.692738			0.0435521
$254$		0			0
$255$	4.78720	−	3.24666i	0.299786	−	0.203314i
$256$		0			0
$257$		−	21.0025i		−	1.31010i	−0.755586	−	0.655050i	$-0.772648\pi$
$257$		−	21.0025i		−	1.31010i	0.755586	−	0.655050i	$-0.227352\pi$
$258$		0			0
$259$		−	5.36008i		−	0.333059i
$260$		0			0
$261$	18.7027	+	7.44651i	1.15767	+	0.460928i
$262$		0			0
$263$	10.1806			0.627763			0.313882	−	0.949462i	$-0.398370\pi$
$263$	10.1806			0.627763			0.313882	+	0.949462i	$0.398370\pi$
$264$		0			0
$265$	6.57500			0.403899
$266$		0			0
$267$	12.1193	+	17.8699i	0.741687	+	1.09362i
$268$		0			0
$269$			23.3443i			1.42333i	0.702519	+	0.711665i	$0.252058\pi$
$269$			23.3443i			1.42333i	−0.702519	+	0.711665i	$0.747942\pi$
$270$		0			0
$271$		−	21.5578i		−	1.30954i	−0.755827	−	0.654771i	$-0.772765\pi$
$271$		−	21.5578i		−	1.30954i	0.755827	−	0.654771i	$-0.227235\pi$
$272$		0			0
$273$	0.144969	+	0.213756i	0.00877390	+	0.0129371i
$274$		0			0
$275$	0.323488			0.0195071
$276$		0			0
$277$	32.4262			1.94830			0.974150	−	0.225901i	$-0.0725326\pi$
$277$	32.4262			1.94830			0.974150	+	0.225901i	$0.0725326\pi$
$278$		0			0
$279$	−16.8276	−	6.69994i	−1.00744	−	0.401115i
$280$		0			0
$281$		−	20.4997i		−	1.22291i	−0.791279	−	0.611455i	$-0.790584\pi$
$281$		−	20.4997i		−	1.22291i	0.791279	−	0.611455i	$-0.209416\pi$
$282$		0			0
$283$		−	0.441277i		−	0.0262312i	−0.999914	−	0.0131156i	$-0.995825\pi$
$283$		−	0.441277i		−	0.0262312i	0.999914	−	0.0131156i	$-0.00417494\pi$
$284$		0			0
$285$	10.4949	−	7.11759i	0.621663	−	0.421609i
$286$		0			0
$287$	−3.14246			−0.185494
$288$		0			0
$289$	5.84730			0.343959
$290$		0			0
$291$	−11.0302	+	7.48064i	−0.646603	+	0.438523i
$292$		0			0
$293$			13.1929i			0.770737i	0.922763	+	0.385368i	$0.125926\pi$
$293$			13.1929i			0.770737i	−0.922763	+	0.385368i	$0.874074\pi$
$294$		0			0
$295$			7.92077i			0.461165i
$296$		0			0
$297$	0.361946	+	1.64146i	0.0210022	+	0.0952473i
$298$		0			0
$299$	0.319329			0.0184673
$300$		0			0
$301$	3.36679			0.194059
$302$		0			0
$303$	11.6291	+	17.1471i	0.668075	+	0.985078i
$304$		0			0
$305$		−	8.37615i		−	0.479617i
$306$		0			0
$307$			27.2734i			1.55657i	0.627908	+	0.778287i	$0.283911\pi$
$307$			27.2734i			1.55657i	−0.627908	+	0.778287i	$0.716089\pi$
$308$		0			0
$309$	−3.88761	−	5.73229i	−0.221158	−	0.326098i
$310$		0			0
$311$	20.0451			1.13665			0.568326	−	0.822804i	$-0.307591\pi$
$311$	20.0451			1.13665			0.568326	+	0.822804i	$0.307591\pi$
$312$		0			0
$313$	−5.47548			−0.309493			−0.154746	−	0.987954i	$-0.549456\pi$
$313$	−5.47548			−0.309493			−0.154746	+	0.987954i	$0.549456\pi$
$314$		0			0
$315$	1.10973	−	2.78720i	0.0625263	−	0.157041i
$316$		0			0
$317$			15.2862i			0.858557i	0.903172	+	0.429279i	$0.141232\pi$
$317$			15.2862i			0.858557i	−0.903172	+	0.429279i	$0.858768\pi$
$318$		0			0
$319$		−	2.17067i		−	0.121534i
$320$		0			0
$321$	17.3282	−	11.7519i	0.967167	−	0.655928i
$322$		0			0
$323$	−24.4498			−1.36043
$324$		0			0
$325$	0.149117			0.00827152
$326$		0			0
$327$	−0.220265	+	0.149383i	−0.0121807	+	0.00826089i
$328$		0			0
$329$		−	8.69575i		−	0.479412i
$330$		0			0
$331$			14.2784i			0.784814i	0.919792	+	0.392407i	$0.128357\pi$
$331$			14.2784i			0.784814i	−0.919792	+	0.392407i	$0.871643\pi$
$332$		0			0
$333$	5.94825	−	14.9396i	0.325962	−	0.818686i
$334$		0			0
$335$	4.07060			0.222400
$336$		0			0
$337$	10.0006			0.544769			0.272385	−	0.962188i	$-0.412188\pi$
$337$	10.0006			0.544769			0.272385	+	0.962188i	$0.412188\pi$
$338$		0			0
$339$	−8.98913	−	13.2545i	−0.488222	−	0.719884i
$340$		0			0
$341$			1.95304i			0.105763i
$342$		0			0
$343$			1.00000i			0.0539949i
$344$		0			0
$345$	−2.08189	−	3.06975i	−0.112085	−	0.165270i
$346$		0			0
$347$	11.2842			0.605767			0.302884	−	0.953028i	$-0.402051\pi$
$347$	11.2842			0.605767			0.302884	+	0.953028i	$0.402051\pi$
$348$		0			0
$349$	−3.36912			−0.180345			−0.0901725	−	0.995926i	$-0.528742\pi$
$349$	−3.36912			−0.180345			−0.0901725	+	0.995926i	$0.528742\pi$
$350$		0			0
$351$	0.166845	+	0.756658i	0.00890550	+	0.0403874i
$352$		0			0
$353$			15.3206i			0.815431i	0.913109	+	0.407716i	$0.133675\pi$
$353$			15.3206i			0.815431i	−0.913109	+	0.407716i	$0.866325\pi$
$354$		0			0
$355$			9.48706i			0.503521i
$356$		0			0
$357$	−4.78720	+	3.24666i	−0.253366	+	0.171831i
$358$		0			0
$359$	−25.4322			−1.34226			−0.671131	−	0.741339i	$-0.734191\pi$
$359$	−25.4322			−1.34226			−0.671131	+	0.741339i	$0.734191\pi$
$360$		0			0
$361$	−34.6009			−1.82110
$362$		0			0
$363$	−15.6183	+	10.5922i	−0.819747	+	0.555949i
$364$		0			0
$365$			0.0336980i			0.00176383i
$366$		0			0
$367$		−	4.72097i		−	0.246432i	−0.992380	−	0.123216i	$-0.960679\pi$
$367$		−	4.72097i		−	0.246432i	0.992380	−	0.123216i	$-0.0393209\pi$
$368$		0			0
$369$	−8.75868	−	3.48729i	−0.455959	−	0.181541i
$370$		0			0
$371$	−6.57500			−0.341357
$372$		0			0
$373$	31.5109			1.63157			0.815786	−	0.578354i	$-0.196305\pi$
$373$	31.5109			1.63157			0.815786	+	0.578354i	$0.196305\pi$
$374$		0			0
$375$	−0.972180	−	1.43348i	−0.0502032	−	0.0740246i
$376$		0			0
$377$		−	1.00060i		−	0.0515337i
$378$		0			0
$379$			16.9528i			0.870806i	0.900236	+	0.435403i	$0.143394\pi$
$379$			16.9528i			0.870806i	−0.900236	+	0.435403i	$0.856606\pi$
$380$		0			0
$381$	12.4615	+	18.3745i	0.638423	+	0.941356i
$382$		0			0
$383$	−33.5173			−1.71266			−0.856328	−	0.516432i	$-0.827260\pi$
$383$	−33.5173			−1.71266			−0.856328	+	0.516432i	$0.827260\pi$
$384$		0			0
$385$	−0.323488			−0.0164865
$386$		0			0
$387$	9.38393	+	3.73624i	0.477012	+	0.189924i
$388$		0			0
$389$			18.2050i			0.923032i	0.887132	+	0.461516i	$0.152694\pi$
$389$			18.2050i			0.923032i	−0.887132	+	0.461516i	$0.847306\pi$
$390$		0			0
$391$			7.15156i			0.361670i
$392$		0			0
$393$	6.39106	−	4.33439i	0.322387	−	0.218641i
$394$		0			0
$395$	10.2173			0.514090
$396$		0			0
$397$	21.1696			1.06247			0.531237	−	0.847223i	$-0.321727\pi$
$397$	21.1696			1.06247			0.531237	+	0.847223i	$0.321727\pi$
$398$		0			0
$399$	−10.4949	+	7.11759i	−0.525401	+	0.356325i
$400$		0			0
$401$			29.6940i			1.48285i	0.671037	+	0.741424i	$0.265849\pi$
$401$			29.6940i			1.48285i	−0.671037	+	0.741424i	$0.734151\pi$
$402$		0			0
$403$			0.900285i			0.0448464i
$404$		0			0
$405$	6.18609	−	6.53699i	0.307389	−	0.324826i
$406$		0			0
$407$	−1.73392			−0.0859473
$408$		0			0
$409$	19.3765			0.958106			0.479053	−	0.877786i	$-0.340980\pi$
$409$	19.3765			0.958106			0.479053	+	0.877786i	$0.340980\pi$
$410$		0			0
$411$	4.04767	+	5.96829i	0.199657	+	0.294394i
$412$		0			0
$413$		−	7.92077i		−	0.389756i
$414$		0			0
$415$		−	5.08834i		−	0.249777i
$416$		0			0
$417$	−7.63465	−	11.2573i	−0.373871	−	0.551272i
$418$		0			0
$419$	−23.1125			−1.12912			−0.564560	−	0.825392i	$-0.690954\pi$
$419$	−23.1125			−1.12912			−0.564560	+	0.825392i	$0.690954\pi$
$420$		0			0
$421$	−30.4789			−1.48545			−0.742725	−	0.669596i	$-0.766467\pi$
$421$	−30.4789			−1.48545			−0.742725	+	0.669596i	$0.766467\pi$
$422$		0			0
$423$	9.64996	−	24.2368i	0.469197	−	1.17843i
$424$		0			0
$425$			3.33957i			0.161993i
$426$		0			0
$427$			8.37615i			0.405351i
$428$		0			0
$429$	0.0691476	−	0.0468956i	0.00333848	−	0.00226414i
$430$		0			0
$431$	−22.7846			−1.09750			−0.548748	−	0.835988i	$-0.684895\pi$
$431$	−22.7846			−1.09750			−0.548748	+	0.835988i	$0.684895\pi$
$432$		0			0
$433$	−3.39779			−0.163287			−0.0816437	−	0.996662i	$-0.526017\pi$
$433$	−3.39779			−0.163287			−0.0816437	+	0.996662i	$0.526017\pi$
$434$		0			0
$435$	−9.61893	+	6.52351i	−0.461192	+	0.312779i
$436$		0			0
$437$			15.6782i			0.749991i
$438$		0			0
$439$		−	11.2669i		−	0.537741i	−0.963176	−	0.268870i	$-0.913350\pi$
$439$		−	11.2669i		−	0.537741i	0.963176	−	0.268870i	$-0.0866503\pi$
$440$		0			0
$441$	−1.10973	+	2.78720i	−0.0528444	+	0.132724i
$442$		0			0
$443$	18.3866			0.873572			0.436786	−	0.899565i	$-0.356117\pi$
$443$	18.3866			0.873572			0.436786	+	0.899565i	$0.356117\pi$
$444$		0			0
$445$	−12.4661			−0.590949
$446$		0			0
$447$	12.6163	+	18.6028i	0.596731	+	0.879880i
$448$		0			0
$449$			27.5244i			1.29896i	0.760380	+	0.649478i	$0.225013\pi$
$449$			27.5244i			1.29896i	−0.760380	+	0.649478i	$0.774987\pi$
$450$		0			0
$451$			1.01655i			0.0478674i
$452$		0			0
$453$	20.9630	+	30.9099i	0.984926	+	1.45227i
$454$		0			0
$455$	−0.149117			−0.00699071
$456$		0			0
$457$	3.78681			0.177139			0.0885697	−	0.996070i	$-0.471770\pi$
$457$	3.78681			0.177139			0.0885697	+	0.996070i	$0.471770\pi$
$458$		0			0
$459$	−16.9458	+	3.73658i	−0.790963	+	0.174409i
$460$		0			0
$461$			33.6536i			1.56740i	0.621137	+	0.783702i	$0.286671\pi$
$461$			33.6536i			1.56740i	−0.621137	+	0.783702i	$0.713329\pi$
$462$		0			0
$463$		−	3.85486i		−	0.179151i	−0.995980	−	0.0895754i	$-0.971449\pi$
$463$		−	3.85486i		−	0.179151i	0.995980	−	0.0895754i	$-0.0285510\pi$
$464$		0			0
$465$	8.65455	−	5.86948i	0.401345	−	0.272191i
$466$		0			0
$467$	36.9726			1.71089			0.855444	−	0.517896i	$-0.173285\pi$
$467$	36.9726			1.71089			0.855444	+	0.517896i	$0.173285\pi$
$468$		0			0
$469$	−4.07060			−0.187963
$470$		0			0
$471$	22.8928	−	15.5258i	1.05484	−	0.715390i
$472$		0			0
$473$		−	1.08912i		−	0.0500777i
$474$		0			0
$475$			7.32126i			0.335923i
$476$		0			0
$477$	−18.3259	−	7.29649i	−0.839084	−	0.334083i
$478$		0			0
$479$	−15.6283			−0.714076			−0.357038	−	0.934090i	$-0.616213\pi$
$479$	−15.6283			−0.714076			−0.357038	+	0.934090i	$0.616213\pi$
$480$		0			0
$481$	−0.799279			−0.0364440
$482$		0			0
$483$	2.08189	+	3.06975i	0.0947292	+	0.139678i
$484$		0			0
$485$		−	7.69471i		−	0.349399i
$486$		0			0
$487$			25.2408i			1.14377i	0.820333	+	0.571886i	$0.193788\pi$
$487$			25.2408i			1.14377i	−0.820333	+	0.571886i	$0.806212\pi$
$488$		0			0
$489$	−20.0160	−	29.5136i	−0.905155	−	1.33465i
$490$		0			0
$491$	−19.0619			−0.860250			−0.430125	−	0.902769i	$-0.641531\pi$
$491$	−19.0619			−0.860250			−0.430125	+	0.902769i	$0.641531\pi$
$492$		0			0
$493$	22.4091			1.00926
$494$		0			0
$495$	−0.901627	−	0.358985i	−0.0405251	−	0.0161352i
$496$		0			0
$497$		−	9.48706i		−	0.425553i
$498$		0			0
$499$		−	4.22462i		−	0.189120i	−0.995519	−	0.0945600i	$-0.969856\pi$
$499$		−	4.22462i		−	0.189120i	0.995519	−	0.0945600i	$-0.0301444\pi$
$500$		0			0
$501$	−0.401246	+	0.272123i	−0.0179264	+	0.0121576i
$502$		0			0
$503$	−7.35455			−0.327923			−0.163962	−	0.986467i	$-0.552427\pi$
$503$	−7.35455			−0.327923			−0.163962	+	0.986467i	$0.552427\pi$
$504$		0			0
$505$	−11.9619			−0.532297
$506$		0			0
$507$	−18.6034	+	12.6167i	−0.826205	+	0.560328i
$508$		0			0
$509$		−	43.0336i		−	1.90743i	−0.300712	−	0.953715i	$-0.597224\pi$
$509$		−	43.0336i		−	1.90743i	0.300712	−	0.953715i	$-0.402776\pi$
$510$		0			0
$511$		−	0.0336980i		−	0.00149071i
$512$		0			0
$513$	−37.1500	+	8.19164i	−1.64021	+	0.361670i
$514$		0			0
$515$	3.99886			0.176211
$516$		0			0
$517$	−2.81297			−0.123714
$518$		0			0
$519$	13.1303	+	19.3607i	0.576357	+	0.849839i
$520$		0			0
$521$		−	16.0937i		−	0.705076i	−0.935797	−	0.352538i	$-0.885319\pi$
$521$		−	16.0937i		−	0.705076i	0.935797	−	0.352538i	$-0.114681\pi$
$522$		0			0
$523$			34.4109i			1.50468i	0.658773	+	0.752342i	$0.271076\pi$
$523$			34.4109i			1.50468i	−0.658773	+	0.752342i	$0.728924\pi$
$524$		0			0
$525$	0.972180	+	1.43348i	0.0424294	+	0.0625622i
$526$		0			0
$527$	−20.1624			−0.878289
$528$		0			0
$529$	−18.4141			−0.800614
$530$		0			0
$531$	8.78993	−	22.0768i	0.381451	−	0.958051i
$532$		0			0
$533$			0.468595i			0.0202971i
$534$		0			0
$535$			12.0882i			0.522619i
$536$		0			0
$537$	−8.35349	+	5.66530i	−0.360480	+	0.244476i
$538$		0			0
$539$	0.323488			0.0139336
$540$		0			0
$541$	−13.9111			−0.598085			−0.299042	−	0.954240i	$-0.596667\pi$
$541$	−13.9111			−0.598085			−0.299042	+	0.954240i	$0.596667\pi$
$542$		0			0
$543$	−10.5352	+	7.14491i	−0.452108	+	0.306618i
$544$		0			0
$545$		−	0.153658i		−	0.00658197i
$546$		0			0
$547$		−	6.84919i		−	0.292850i	−0.989222	−	0.146425i	$-0.953223\pi$
$547$		−	6.84919i		−	0.292850i	0.989222	−	0.146425i	$-0.0467768\pi$
$548$		0			0
$549$	−9.29529	+	23.3460i	−0.396713	+	0.996385i
$550$		0			0
$551$	49.1271			2.09288
$552$		0			0
$553$	−10.2173			−0.434486
$554$		0			0
$555$	5.21096	+	7.68357i	0.221193	+	0.326149i
$556$		0			0
$557$		−	30.0492i		−	1.27322i	−0.771184	−	0.636612i	$-0.780335\pi$
$557$		−	30.0492i		−	1.27322i	0.771184	−	0.636612i	$-0.219665\pi$
$558$		0			0
$559$		−	0.502046i		−	0.0212343i
$560$		0			0
$561$	1.05026	+	1.54860i	0.0443418	+	0.0653821i
$562$		0			0
$563$	16.4071			0.691477			0.345739	−	0.938331i	$-0.387628\pi$
$563$	16.4071			0.691477			0.345739	+	0.938331i	$0.387628\pi$
$564$		0			0
$565$	9.24636			0.388997
$566$		0			0
$567$	−6.18609	+	6.53699i	−0.259792	+	0.274528i
$568$		0			0
$569$		−	8.48964i		−	0.355904i	−0.984039	−	0.177952i	$-0.943053\pi$
$569$		−	8.48964i		−	0.355904i	0.984039	−	0.177952i	$-0.0569472\pi$
$570$		0			0
$571$			0.443759i			0.0185707i	0.999957	+	0.00928536i	$0.00295567\pi$
$571$			0.443759i			0.0185707i	−0.999957	+	0.00928536i	$0.997044\pi$
$572$		0			0
$573$	−24.8843	+	16.8764i	−1.03955	+	0.705022i
$574$		0			0
$575$	2.14146			0.0893052
$576$		0			0
$577$	40.3709			1.68066			0.840331	−	0.542074i	$-0.182361\pi$
$577$	40.3709			1.68066			0.840331	+	0.542074i	$0.182361\pi$
$578$		0			0
$579$	−26.5184	+	17.9846i	−1.10207	+	0.747416i
$580$		0			0
$581$			5.08834i			0.211100i
$582$		0			0
$583$			2.12694i			0.0880887i
$584$		0			0
$585$	−0.415619	−	0.165480i	−0.0171837	−	0.00684175i
$586$		0			0
$587$	39.6680			1.63727			0.818636	−	0.574313i	$-0.194731\pi$
$587$	39.6680			1.63727			0.818636	+	0.574313i	$0.194731\pi$
$588$		0			0
$589$	−44.2017			−1.82130
$590$		0			0
$591$	−1.78031	−	2.62507i	−0.0732323	−	0.107981i
$592$		0			0
$593$			44.1724i			1.81394i	0.421193	+	0.906971i	$0.361611\pi$
$593$			44.1724i			1.81394i	−0.421193	+	0.906971i	$0.638389\pi$
$594$		0			0
$595$		−	3.33957i		−	0.136909i
$596$		0			0
$597$	7.32969	+	10.8076i	0.299984	+	0.442327i
$598$		0			0
$599$	−23.2508			−0.950004			−0.475002	−	0.879985i	$-0.657553\pi$
$599$	−23.2508			−0.950004			−0.475002	+	0.879985i	$0.657553\pi$
$600$		0			0
$601$	−18.4518			−0.752664			−0.376332	−	0.926485i	$-0.622815\pi$
$601$	−18.4518			−0.752664			−0.376332	+	0.926485i	$0.622815\pi$
$602$		0			0
$603$	−11.3456	−	4.51727i	−0.462027	−	0.183957i
$604$		0			0
$605$		−	10.8954i		−	0.442959i
$606$		0			0
$607$		−	26.1636i		−	1.06195i	−0.847389	−	0.530973i	$-0.821827\pi$
$607$		−	26.1636i		−	1.06195i	0.847389	−	0.530973i	$-0.178173\pi$
$608$		0			0
$609$	9.61893	−	6.52351i	0.389779	−	0.264346i
$610$		0			0
$611$	−1.29668			−0.0524583
$612$		0			0
$613$	17.3607			0.701193			0.350596	−	0.936527i	$-0.385979\pi$
$613$	17.3607			0.701193			0.350596	+	0.936527i	$0.385979\pi$
$614$		0			0
$615$	4.50466	−	3.05504i	0.181645	−	0.123191i
$616$		0			0
$617$		−	33.7808i		−	1.35996i	−0.733229	−	0.679981i	$-0.761988\pi$
$617$		−	33.7808i		−	1.35996i	0.733229	−	0.679981i	$-0.238012\pi$
$618$		0			0
$619$		−	42.1957i		−	1.69599i	−0.530004	−	0.847995i	$-0.677810\pi$
$619$		−	42.1957i		−	1.69599i	0.530004	−	0.847995i	$-0.322190\pi$
$620$		0			0
$621$	2.39605	+	10.8663i	0.0961501	+	0.436051i
$622$		0			0
$623$	12.4661			0.499443
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	2.30245	+	3.39497i	0.0919512	+	0.135582i
$628$		0			0
$629$		−	17.9003i		−	0.713733i
$630$		0			0
$631$		−	28.1746i		−	1.12161i	−0.827947	−	0.560807i	$-0.810491\pi$
$631$		−	28.1746i		−	1.12161i	0.827947	−	0.560807i	$-0.189509\pi$
$632$		0			0
$633$	11.8352	+	17.4509i	0.470405	+	0.693613i
$634$		0			0
$635$	−12.8181			−0.508672
$636$		0			0
$637$	0.149117			0.00590823
$638$		0			0
$639$	10.5281	−	26.4423i	0.416485	−	1.04604i
$640$		0			0
$641$		−	4.45528i		−	0.175973i	−0.996122	−	0.0879864i	$-0.971957\pi$
$641$		−	4.45528i		−	0.175973i	0.996122	−	0.0879864i	$-0.0280432\pi$
$642$		0			0
$643$			11.4662i			0.452185i	0.974106	+	0.226092i	$0.0725951\pi$
$643$			11.4662i			0.452185i	−0.974106	+	0.226092i	$0.927405\pi$
$644$		0			0
$645$	−4.82623	+	3.27313i	−0.190033	+	0.128879i
$646$		0			0
$647$	−19.1859			−0.754274			−0.377137	−	0.926157i	$-0.623091\pi$
$647$	−19.1859			−0.754274			−0.377137	+	0.926157i	$0.623091\pi$
$648$		0			0
$649$	−2.56228			−0.100578
$650$		0			0
$651$	−8.65455	+	5.86948i	−0.339199	+	0.230043i
$652$		0			0
$653$			1.84828i			0.0723286i	0.999346	+	0.0361643i	$0.0115140\pi$
$653$			1.84828i			0.0723286i	−0.999346	+	0.0361643i	$0.988486\pi$
$654$		0			0
$655$			4.45842i			0.174205i
$656$		0			0
$657$	0.0373957	−	0.0939231i	0.00145895	−	0.00366429i
$658$		0			0
$659$	8.45557			0.329382			0.164691	−	0.986345i	$-0.447337\pi$
$659$	8.45557			0.329382			0.164691	+	0.986345i	$0.447337\pi$
$660$		0			0
$661$	10.9536			0.426047			0.213024	−	0.977047i	$-0.431669\pi$
$661$	10.9536			0.426047			0.213024	+	0.977047i	$0.431669\pi$
$662$		0			0
$663$	0.484132	+	0.713853i	0.0188021	+	0.0277238i
$664$		0			0
$665$		−	7.32126i		−	0.283906i
$666$		0			0
$667$		−	14.3696i		−	0.556394i
$668$		0			0
$669$	−2.68098	−	3.95310i	−0.103653	−	0.152836i
$670$		0			0
$671$	2.70959			0.104602
$672$		0			0
$673$	5.49441			0.211794			0.105897	−	0.994377i	$-0.466229\pi$
$673$	5.49441			0.211794			0.105897	+	0.994377i	$0.466229\pi$
$674$		0			0
$675$	1.11888	+	5.07426i	0.0430658	+	0.195308i
$676$		0			0
$677$			23.7957i			0.914543i	0.889327	+	0.457271i	$0.151173\pi$
$677$			23.7957i			0.914543i	−0.889327	+	0.457271i	$0.848827\pi$
$678$		0			0
$679$			7.69471i			0.295296i
$680$		0			0
$681$	40.2005	−	27.2638i	1.54049	−	1.04475i
$682$		0			0
$683$	31.8861			1.22009			0.610044	−	0.792368i	$-0.291152\pi$
$683$	31.8861			1.22009			0.610044	+	0.792368i	$0.291152\pi$
$684$		0			0
$685$	−4.16349			−0.159079
$686$		0			0
$687$	30.0192	−	20.3589i	1.14531	−	0.776741i
$688$		0			0
$689$			0.980445i			0.0373520i
$690$		0			0
$691$			25.8822i			0.984606i	0.870424	+	0.492303i	$0.163845\pi$
$691$			25.8822i			0.984606i	−0.870424	+	0.492303i	$0.836155\pi$
$692$		0			0
$693$	0.901627	+	0.358985i	0.0342500	+	0.0136367i
$694$		0			0
$695$	7.85313			0.297886
$696$		0			0
$697$	−10.4945			−0.397506
$698$		0			0
$699$	17.2876	+	25.4906i	0.653877	+	0.964142i
$700$		0			0
$701$		−	3.89184i		−	0.146993i	−0.997295	−	0.0734964i	$-0.976584\pi$
$701$		−	3.89184i		−	0.146993i	0.997295	−	0.0734964i	$-0.0234157\pi$
$702$		0			0
$703$		−	39.2425i		−	1.48006i
$704$		0			0
$705$	8.45384	+	12.4652i	0.318390	+	0.469466i
$706$		0			0
$707$	11.9619			0.449873
$708$		0			0
$709$	27.3378			1.02669			0.513347	−	0.858181i	$-0.328405\pi$
$709$	27.3378			1.02669			0.513347	+	0.858181i	$0.328405\pi$
$710$		0			0
$711$	−28.4778	−	11.3385i	−1.06800	−	0.425227i
$712$		0			0
$713$			12.9290i			0.484193i
$714$		0			0
$715$			0.0482376i			0.00180398i
$716$		0			0
$717$	−8.12122	+	5.50778i	−0.303293	+	0.205692i
$718$		0			0
$719$	3.28236			0.122411			0.0612057	−	0.998125i	$-0.480505\pi$
$719$	3.28236			0.122411			0.0612057	+	0.998125i	$0.480505\pi$
$720$		0			0
$721$	−3.99886			−0.148925
$722$		0			0
$723$	43.4701	−	29.4813i	1.61667	−	1.09642i
$724$		0			0
$725$		−	6.71019i		−	0.249210i
$726$		0			0
$727$		−	1.66432i		−	0.0617263i	−0.999524	−	0.0308631i	$-0.990174\pi$
$727$		−	1.66432i		−	0.0617263i	0.999524	−	0.0308631i	$-0.00982560\pi$
$728$		0			0
$729$	−24.4962	+	11.3550i	−0.907267	+	0.420556i
$730$		0			0
$731$	11.2436			0.415860
$732$		0			0
$733$	−8.16593			−0.301615			−0.150808	−	0.988563i	$-0.548187\pi$
$733$	−8.16593			−0.301615			−0.150808	+	0.988563i	$0.548187\pi$
$734$		0			0
$735$	−0.972180	−	1.43348i	−0.0358594	−	0.0528747i
$736$		0			0
$737$			1.31679i			0.0485046i
$738$		0			0
$739$		−	25.1763i		−	0.926125i	−0.886326	−	0.463063i	$-0.846751\pi$
$739$		−	25.1763i		−	0.926125i	0.886326	−	0.463063i	$-0.153249\pi$
$740$		0			0
$741$	1.06135	+	1.56497i	0.0389898	+	0.0574905i
$742$		0			0
$743$	−36.0524			−1.32264			−0.661318	−	0.750106i	$-0.730002\pi$
$743$	−36.0524			−1.32264			−0.661318	+	0.750106i	$0.730002\pi$
$744$		0			0
$745$	−12.9773			−0.475453
$746$		0			0
$747$	−5.64669	+	14.1822i	−0.206602	+	0.518901i
$748$		0			0
$749$		−	12.0882i		−	0.441694i
$750$		0			0
$751$			34.1513i			1.24620i	0.782143	+	0.623099i	$0.214127\pi$
$751$			34.1513i			1.24620i	−0.782143	+	0.623099i	$0.785873\pi$
$752$		0			0
$753$	−13.1736	+	8.93427i	−0.480072	+	0.325583i
$754$		0			0
$755$	−21.5628			−0.784752
$756$		0			0
$757$	−3.59204			−0.130555			−0.0652774	−	0.997867i	$-0.520793\pi$
$757$	−3.59204			−0.130555			−0.0652774	+	0.997867i	$0.520793\pi$
$758$		0			0
$759$	0.993026	−	0.673466i	0.0360446	−	0.0244453i
$760$		0			0
$761$		−	27.9356i		−	1.01267i	−0.862338	−	0.506333i	$-0.831001\pi$
$761$		−	27.9356i		−	1.01267i	0.862338	−	0.506333i	$-0.168999\pi$
$762$		0			0
$763$			0.153658i			0.00556278i
$764$		0			0
$765$	3.70602	−	9.30804i	0.133992	−	0.336533i
$766$		0			0
$767$	−1.18112			−0.0426478
$768$		0			0
$769$	3.96273			0.142900			0.0714499	−	0.997444i	$-0.477237\pi$
$769$	3.96273			0.142900			0.0714499	+	0.997444i	$0.477237\pi$
$770$		0			0
$771$	−20.4182	−	30.1067i	−0.735344	−	1.08426i
$772$		0			0
$773$		−	26.3807i		−	0.948847i	−0.880297	−	0.474424i	$-0.842657\pi$
$773$		−	26.3807i		−	0.948847i	0.880297	−	0.474424i	$-0.157343\pi$
$774$		0			0
$775$			6.03744i			0.216871i
$776$		0			0
$777$	−5.21096	−	7.68357i	−0.186942	−	0.275646i
$778$		0			0
$779$	−23.0068			−0.824304
$780$		0			0
$781$	−3.06895			−0.109816
$782$		0			0
$783$	34.0492	−	7.50792i	1.21682	−	0.268311i
$784$		0			0
$785$			15.9701i			0.569996i
$786$		0			0
$787$		−	54.0742i		−	1.92754i	−0.266739	−	0.963769i	$-0.585946\pi$
$787$		−	54.0742i		−	1.92754i	0.266739	−	0.963769i	$-0.414054\pi$
$788$		0			0
$789$	14.5937	−	9.89738i	0.519550	−	0.352356i
$790$		0			0
$791$	−9.24636			−0.328763
$792$		0			0
$793$	1.24903			0.0443543
$794$		0			0
$795$	9.42514	−	6.39209i	0.334275	−	0.226704i
$796$		0			0
$797$		−	38.7047i		−	1.37099i	−0.728077	−	0.685495i	$-0.759586\pi$
$797$		−	38.7047i		−	1.37099i	0.728077	−	0.685495i	$-0.240414\pi$
$798$		0			0
$799$		−	29.0400i		−	1.02736i
$800$		0			0
$801$	34.7455	+	13.8340i	1.22767	+	0.488800i
$802$		0			0
$803$	−0.0109009			−0.000384684
$804$		0			0
$805$	−2.14146			−0.0754767
$806$		0			0
$807$	22.6949	+	33.4636i	0.798898	+	1.17798i
$808$		0			0
$809$		−	51.8638i		−	1.82343i	−0.410821	−	0.911716i	$-0.634758\pi$
$809$		−	51.8638i		−	1.82343i	0.410821	−	0.911716i	$-0.365242\pi$
$810$		0			0
$811$			20.7071i			0.727123i	0.931570	+	0.363562i	$0.118439\pi$
$811$			20.7071i			0.727123i	−0.931570	+	0.363562i	$0.881561\pi$
$812$		0			0
$813$	−20.9580	−	30.9027i	−0.735031	−	1.08380i
$814$		0			0
$815$	20.5888			0.721193
$816$		0			0
$817$	24.6492			0.862365
$818$		0			0
$819$	0.415619	+	0.165480i	0.0145229	+	0.00578234i
$820$		0			0
$821$			40.1707i			1.40197i	0.713178	+	0.700983i	$0.247255\pi$
$821$			40.1707i			1.40197i	−0.713178	+	0.700983i	$0.752745\pi$
$822$		0			0
$823$		−	27.4546i		−	0.957006i	−0.878086	−	0.478503i	$-0.841180\pi$
$823$		−	27.4546i		−	0.957006i	0.878086	−	0.478503i	$-0.158820\pi$
$824$		0			0
$825$	0.463714	−	0.314489i	0.0161444	−	0.0109491i
$826$		0			0
$827$	−1.32264			−0.0459927			−0.0229964	−	0.999736i	$-0.507321\pi$
$827$	−1.32264			−0.0459927			−0.0229964	+	0.999736i	$0.507321\pi$
$828$		0			0
$829$	−5.62094			−0.195223			−0.0976116	−	0.995225i	$-0.531120\pi$
$829$	−5.62094			−0.195223			−0.0976116	+	0.995225i	$0.531120\pi$
$830$		0			0
$831$	46.4823	−	31.5241i	1.61245	−	1.09356i
$832$		0			0
$833$			3.33957i			0.115709i
$834$		0			0
$835$		−	0.279911i		−	0.00968670i
$836$		0			0
$837$	−30.6355	+	6.75519i	−1.05892	+	0.233494i
$838$		0			0
$839$	−41.2351			−1.42360			−0.711798	−	0.702385i	$-0.752119\pi$
$839$	−41.2351			−1.42360			−0.711798	+	0.702385i	$0.752119\pi$
$840$		0			0
$841$	−16.0267			−0.552643
$842$		0			0
$843$	−19.9294	−	29.3859i	−0.686406	−	1.01211i
$844$		0			0
$845$		−	12.9778i		−	0.446449i
$846$		0			0
$847$			10.8954i			0.374369i
$848$		0			0
$849$	−0.429000	−	0.632562i	−0.0147233	−	0.0217095i
$850$		0			0
$851$	−11.4784			−0.393475
$852$		0			0
$853$	1.63724			0.0560580			0.0280290	−	0.999607i	$-0.491077\pi$
$853$	1.63724			0.0560580			0.0280290	+	0.999607i	$0.491077\pi$
$854$		0			0
$855$	8.12464	−	20.4058i	0.277857	−	0.697865i
$856$		0			0
$857$			6.88726i			0.235264i	0.993057	+	0.117632i	$0.0375304\pi$
$857$			6.88726i			0.235264i	−0.993057	+	0.117632i	$0.962470\pi$
$858$		0			0
$859$		−	22.0906i		−	0.753723i	−0.926270	−	0.376861i	$-0.877003\pi$
$859$		−	22.0906i		−	0.753723i	0.926270	−	0.376861i	$-0.122997\pi$
$860$		0			0
$861$	−4.50466	+	3.05504i	−0.153518	+	0.104115i
$862$		0			0
$863$	−20.2484			−0.689263			−0.344632	−	0.938738i	$-0.611996\pi$
$863$	−20.2484			−0.689263			−0.344632	+	0.938738i	$0.611996\pi$
$864$		0			0
$865$	−13.5061			−0.459220
$866$		0			0
$867$	8.38199	−	5.68463i	0.284667	−	0.193060i
$868$		0			0
$869$			3.30519i			0.112121i
$870$		0			0
$871$			0.606995i			0.0205672i
$872$		0			0
$873$	−8.53906	+	21.4467i	−0.289004	+	0.725861i
$874$		0			0
$875$	−1.00000			−0.0338062
$876$		0			0
$877$	2.99329			0.101076			0.0505380	−	0.998722i	$-0.483906\pi$
$877$	2.99329			0.101076			0.0505380	+	0.998722i	$0.483906\pi$
$878$		0			0
$879$	12.8259	+	18.9117i	0.432606	+	0.637877i
$880$		0			0
$881$		−	24.3411i		−	0.820073i	−0.912069	−	0.410036i	$-0.865516\pi$
$881$		−	24.3411i		−	0.820073i	0.912069	−	0.410036i	$-0.134484\pi$
$882$		0			0
$883$		−	22.5699i		−	0.759538i	−0.925081	−	0.379769i	$-0.876004\pi$
$883$		−	22.5699i		−	0.759538i	0.925081	−	0.379769i	$-0.123996\pi$
$884$		0			0
$885$	7.70042	+	11.3543i	0.258847	+	0.381670i
$886$		0			0
$887$	46.2301			1.55225			0.776127	−	0.630577i	$-0.217182\pi$
$887$	46.2301			1.55225			0.776127	+	0.630577i	$0.217182\pi$
$888$		0			0
$889$	12.8181			0.429906
$890$		0			0
$891$	2.11464	+	2.00113i	0.0708431	+	0.0670403i
$892$		0			0
$893$		−	63.6639i		−	2.13043i
$894$		0			0
$895$		−	5.82742i		−	0.194789i
$896$		0			0
$897$	0.457751	−	0.310445i	0.0152839	−	0.0103655i
$898$		0			0
$899$	40.5124			1.35116
$900$		0			0
$901$	−21.9577			−0.731516
$902$		0			0
$903$	4.82623	−	3.27313i	0.160607	−	0.108923i
$904$		0			0
$905$		−	7.34937i		−	0.244301i
$906$		0			0
$907$		−	8.40278i		−	0.279010i	−0.990221	−	0.139505i	$-0.955449\pi$
$907$		−	8.40278i		−	0.279010i	0.990221	−	0.139505i	$-0.0445511\pi$
$908$		0			0
$909$	33.3402	+	13.2745i	1.10583	+	0.440287i
$910$		0			0
$911$	12.1429			0.402312			0.201156	−	0.979559i	$-0.435530\pi$
$911$	12.1429			0.402312			0.201156	+	0.979559i	$0.435530\pi$
$912$		0			0
$913$	1.64602			0.0544752
$914$		0			0
$915$	−8.14313	−	12.0071i	−0.269204	−	0.396941i
$916$		0			0
$917$		−	4.45842i		−	0.147230i
$918$		0			0
$919$			1.86348i			0.0614706i	0.999528	+	0.0307353i	$0.00978489\pi$
$919$			1.86348i			0.0614706i	−0.999528	+	0.0307353i	$0.990215\pi$
$920$		0			0
$921$	26.5146	+	39.0959i	0.873687	+	1.28825i
$922$		0			0
$923$	−1.41468			−0.0465648
$924$		0			0
$925$	−5.36008			−0.176238
$926$		0			0
$927$	−11.1456	−	4.43766i	−0.366070	−	0.145752i
$928$		0			0
$929$		−	1.74263i		−	0.0571737i	−0.999591	−	0.0285868i	$-0.990899\pi$
$929$		−	1.74263i		−	0.0571737i	0.999591	−	0.0285868i	$-0.00910072\pi$
$930$		0			0
$931$			7.32126i			0.239945i
$932$		0			0
$933$	28.7342	−	19.4874i	0.940716	−	0.637989i
$934$		0			0
$935$	−1.08031			−0.0353299
$936$		0			0
$937$	56.8886			1.85847			0.929234	−	0.369491i	$-0.120468\pi$
$937$	56.8886			1.85847			0.929234	+	0.369491i	$0.120468\pi$
$938$		0			0
$939$	−7.84900	+	5.32315i	−0.256142	+	0.173715i
$940$		0			0
$941$			1.42589i			0.0464826i	0.999730	+	0.0232413i	$0.00739861\pi$
$941$			1.42589i			0.0464826i	−0.999730	+	0.0232413i	$0.992601\pi$
$942$		0			0
$943$			6.72947i			0.219142i
$944$		0			0
$945$	−1.11888	−	5.07426i	−0.0363973	−	0.165066i
$946$		0			0
$947$	−31.0660			−1.00951			−0.504754	−	0.863263i	$-0.668417\pi$
$947$	−31.0660			−1.00951			−0.504754	+	0.863263i	$0.668417\pi$
$948$		0			0
$949$	−0.00502494			−0.000163117
$950$		0			0
$951$	14.8609	+	21.9124i	0.481898	+	0.710560i
$952$		0			0
$953$		−	20.4058i		−	0.661010i	−0.943804	−	0.330505i	$-0.892781\pi$
$953$		−	20.4058i		−	0.661010i	0.943804	−	0.330505i	$-0.107219\pi$
$954$		0			0
$955$		−	17.3593i		−	0.561735i
$956$		0			0
$957$	−2.11028	−	3.11161i	−0.0682156	−	0.100584i
$958$		0			0
$959$	4.16349			0.134446
$960$		0			0
$961$	−5.45067			−0.175828
$962$		0			0
$963$	13.4147	−	33.6923i	0.432282	−	1.08572i
$964$		0			0
$965$		−	18.4993i		−	0.595513i
$966$		0			0
$967$		−	0.962974i		−	0.0309671i	−0.999880	−	0.0154836i	$-0.995071\pi$
$967$		−	0.962974i		−	0.0309671i	0.999880	−	0.0154836i	$-0.00492877\pi$
$968$		0			0
$969$	−35.0484	+	23.7696i	−1.12592	+	0.763591i
$970$		0			0
$971$	30.9386			0.992866			0.496433	−	0.868075i	$-0.334643\pi$
$971$	30.9386			0.992866			0.496433	+	0.868075i	$0.334643\pi$
$972$		0			0
$973$	−7.85313			−0.251760
$974$		0			0
$975$	0.213756	−	0.144969i	0.00684568	−	0.00464271i
$976$		0			0
$977$			15.8558i			0.507272i	0.967300	+	0.253636i	$0.0816265\pi$
$977$			15.8558i			0.507272i	−0.967300	+	0.253636i	$0.918373\pi$
$978$		0			0
$979$		−	4.03263i		−	0.128883i
$980$		0			0
$981$	−0.170519	+	0.428275i	−0.00544424	+	0.0136738i
$982$		0			0
$983$	34.6971			1.10667			0.553333	−	0.832960i	$-0.313356\pi$
$983$	34.6971			1.10667			0.553333	+	0.832960i	$0.313356\pi$
$984$		0			0
$985$	1.83126			0.0583488
$986$		0			0
$987$	−8.45384	−	12.4652i	−0.269089	−	0.396771i
$988$		0			0
$989$		−	7.20986i		−	0.229260i
$990$		0			0
$991$		−	19.1032i		−	0.606834i	−0.952858	−	0.303417i	$-0.901872\pi$
$991$		−	19.1032i		−	0.606834i	0.952858	−	0.303417i	$-0.0981276\pi$
$992$		0			0
$993$	13.8812	+	20.4679i	0.440507	+	0.649528i
$994$		0			0
$995$	−7.53944			−0.239016
$996$		0			0
$997$	−35.8525			−1.13546			−0.567730	−	0.823215i	$-0.692178\pi$
$997$	−35.8525			−1.13546			−0.567730	+	0.823215i	$0.692178\pi$
$998$		0			0
$999$	−5.99730	−	27.1984i	−0.189746	−	0.860520i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1680.2.ba.d.911.15	yes	16
3.2	odd	2		1680.2.ba.c.911.1	✓	16
4.3	odd	2		1680.2.ba.c.911.2	yes	16
12.11	even	2	inner	1680.2.ba.d.911.16	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1680.2.ba.c.911.1	✓	16	3.2	odd	2
1680.2.ba.c.911.2	yes	16	4.3	odd	2
1680.2.ba.d.911.15	yes	16	1.1	even	1	trivial
1680.2.ba.d.911.16	yes	16	12.11	even	2	inner

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:	\( N \)	\(=\)	\( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1680.ba (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(1.43348 - 0.972180i) q^{3} +1.00000i q^{5} -1.00000i q^{7} +(1.10973 - 2.78720i) q^{9} -0.323488 q^{11} -0.149117 q^{13} +(0.972180 + 1.43348i) q^{15} -3.33957i q^{17} -7.32126i q^{19} +(-0.972180 - 1.43348i) q^{21} -2.14146 q^{23} -1.00000 q^{25} +(-1.11888 - 5.07426i) q^{27} +6.71019i q^{29} -6.03744i q^{31} +(-0.463714 + 0.314489i) q^{33} +1.00000 q^{35} +5.36008 q^{37} +(-0.213756 + 0.144969i) q^{39} -3.14246i q^{41} +3.36679i q^{43} +(2.78720 + 1.10973i) q^{45} +8.69575 q^{47} -1.00000 q^{49} +(-3.24666 - 4.78720i) q^{51} -6.57500i q^{53} -0.323488i q^{55} +(-7.11759 - 10.4949i) q^{57} +7.92077 q^{59} -8.37615 q^{61} +(-2.78720 - 1.10973i) q^{63} -0.149117i q^{65} -4.07060i q^{67} +(-3.06975 + 2.08189i) q^{69} +9.48706 q^{71} +0.0336980 q^{73} +(-1.43348 + 0.972180i) q^{75} +0.323488i q^{77} -10.2173i q^{79} +(-6.53699 - 6.18609i) q^{81} -5.08834 q^{83} +3.33957 q^{85} +(6.52351 + 9.61893i) q^{87} +12.4661i q^{89} +0.149117i q^{91} +(-5.86948 - 8.65455i) q^{93} +7.32126 q^{95} -7.69471 q^{97} +(-0.358985 + 0.901627i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 10 q^{9} + 12 q^{11} - 2 q^{15} + 2 q^{21} + 8 q^{23} - 16 q^{25} - 12 q^{27} + 12 q^{33} + 16 q^{35} - 8 q^{37} - 14 q^{39} + 8 q^{45} + 16 q^{47} - 16 q^{49} - 34 q^{51} + 12 q^{57} + 32 q^{59}+ \cdots - 50 q^{99}+O(q^{100}) \) 16 * q - 10 * q^9 + 12 * q^11 - 2 * q^15 + 2 * q^21 + 8 * q^23 - 16 * q^25 - 12 * q^27 + 12 * q^33 + 16 * q^35 - 8 * q^37 - 14 * q^39 + 8 * q^45 + 16 * q^47 - 16 * q^49 - 34 * q^51 + 12 * q^57 + 32 * q^59 + 8 * q^61 - 8 * q^63 - 20 * q^69 - 32 * q^71 + 48 * q^73 - 22 * q^81 + 72 * q^83 - 12 * q^85 - 44 * q^87 + 28 * q^93 + 8 * q^95 - 8 * q^97 - 50 * q^99

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