LMFDB - Embedded newform 1920.2.b.g.191.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1920,2,Mod(191,1920)] 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("1920.191"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1920, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1920 = 2^{7} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1920.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,2,0,-8,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	no
Analytic conductor:	$15.3312771881$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.619810816.2
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{5} + 14x^{4} - 8x^{3} + 2x^{2} + 2x + 1 $ x^8 - 2x^5 + 14x^4 - 8x^3 + 2x^2 + 2*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			191.1
Root			$-1.49094 + 1.49094i$ of defining polynomial
Character	$\chi$	$=$	1920.191
Dual form			1920.2.b.g.191.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-1.15558 - 1.29021i) q^{3} -1.00000 q^{5} -4.31116i q^{7} +(-0.329281 + 2.98187i) q^{9} -5.96375i q^{11} -4.89157i q^{13} +(1.15558 + 1.29021i) q^{15} -1.07217i q^{17} +4.23301 q^{19} +(-5.56229 + 4.98187i) q^{21} +5.38333 q^{23} +1.00000 q^{25} +(4.22775 - 3.02095i) q^{27} +2.80291 q^{29} -0.927826i q^{31} +(-7.69448 + 6.89157i) q^{33} +4.31116i q^{35} -8.35305i q^{37} +(-6.31116 + 5.65259i) q^{39} +6.50228i q^{41} -11.4720 q^{43} +(0.329281 - 2.98187i) q^{45} +1.92186 q^{47} -11.5861 q^{49} +(-1.38333 + 1.23898i) q^{51} +1.46147 q^{53} +5.96375i q^{55} +(-4.89157 - 5.46147i) q^{57} +5.34144i q^{59} -5.42522i q^{61} +(12.8553 + 1.41958i) q^{63} +4.89157i q^{65} +3.47199 q^{67} +(-6.22085 - 6.94562i) q^{69} +6.47796 q^{71} +14.4660 q^{73} +(-1.15558 - 1.29021i) q^{75} -25.7106 q^{77} -1.91134i q^{79} +(-8.78315 - 1.96375i) q^{81} +3.20273i q^{83} +1.07217i q^{85} +(-3.23898 - 3.61634i) q^{87} +0.538528i q^{89} -21.0883 q^{91} +(-1.19709 + 1.07217i) q^{93} -4.23301 q^{95} -7.78315 q^{97} +(17.7831 + 1.96375i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 2 q^{3} - 8 q^{5} - 4 q^{9} - 2 q^{15} - 8 q^{19} - 4 q^{21} + 12 q^{23} + 8 q^{25} + 14 q^{27} + 8 q^{29} - 8 q^{33} - 28 q^{39} - 36 q^{43} + 4 q^{45} - 4 q^{47} + 20 q^{51} + 16 q^{63} - 28 q^{67}+ \cdots + 64 q^{99}+O(q^{100}) $ 8 * q + 2 * q^3 - 8 * q^5 - 4 * q^9 - 2 * q^15 - 8 * q^19 - 4 * q^21 + 12 * q^23 + 8 * q^25 + 14 * q^27 + 8 * q^29 - 8 * q^33 - 28 * q^39 - 36 * q^43 + 4 * q^45 - 4 * q^47 + 20 * q^51 + 16 * q^63 - 28 * q^67 - 12 * q^69 + 24 * q^71 + 32 * q^73 + 2 * q^75 - 32 * q^77 + 8 * q^81 - 12 * q^87 - 40 * q^91 - 24 * q^93 + 8 * q^95 + 16 * q^97 + 64 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1920\mathbb{Z}\right)^\times$.

$n$	$511$	$641$	$901$	$1537$
$\chi(n)$	$-1$	$-1$	$-1$	$1$

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
$3$	−1.15558	−	1.29021i	−0.667173	−	0.744903i
$3$	−1.15558	−	1.29021i	−0.667173	−	0.744903i
$4$		0			0
$5$	−1.00000			−0.447214
$5$	−1.00000			−0.447214
$6$		0			0
$7$		−	4.31116i		−	1.62946i	−0.579838	−	0.814732i	$-0.696884\pi$
$7$		−	4.31116i		−	1.62946i	0.579838	−	0.814732i	$-0.303116\pi$
$8$		0			0
$9$	−0.329281	+	2.98187i	−0.109760	+	0.993958i
$10$		0			0
$11$		−	5.96375i		−	1.79814i	−0.437807	−	0.899069i	$-0.644245\pi$
$11$		−	5.96375i		−	1.79814i	0.437807	−	0.899069i	$-0.355755\pi$
$12$		0			0
$13$		−	4.89157i		−	1.35668i	−0.734749	−	0.678339i	$-0.762700\pi$
$13$		−	4.89157i		−	1.35668i	0.734749	−	0.678339i	$-0.237300\pi$
$14$		0			0
$15$	1.15558	+	1.29021i	0.298369	+	0.333131i
$16$		0			0
$17$		−	1.07217i		−	0.260040i	−0.991511	−	0.130020i	$-0.958496\pi$
$17$		−	1.07217i		−	0.260040i	0.991511	−	0.130020i	$-0.0415042\pi$
$18$		0			0
$19$	4.23301			0.971120			0.485560	−	0.874203i	$-0.338616\pi$
$19$	4.23301			0.971120			0.485560	+	0.874203i	$0.338616\pi$
$20$		0			0
$21$	−5.56229	+	4.98187i	−1.21379	+	1.08713i
$22$		0			0
$23$	5.38333			1.12250			0.561251	−	0.827646i	$-0.310320\pi$
$23$	5.38333			1.12250			0.561251	+	0.827646i	$0.310320\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$	4.22775	−	3.02095i	0.813631	−	0.581381i
$28$		0			0
$29$	2.80291			0.520487			0.260244	−	0.965543i	$-0.416197\pi$
$29$	2.80291			0.520487			0.260244	+	0.965543i	$0.416197\pi$
$30$		0			0
$31$		−	0.927826i		−	0.166642i	−0.996523	−	0.0833212i	$-0.973447\pi$
$31$		−	0.927826i		−	0.166642i	0.996523	−	0.0833212i	$-0.0265527\pi$
$32$		0			0
$33$	−7.69448	+	6.89157i	−1.33944	+	1.19967i
$34$		0			0
$35$			4.31116i			0.728718i
$36$		0			0
$37$		−	8.35305i		−	1.37323i	−0.727020	−	0.686616i	$-0.759095\pi$
$37$		−	8.35305i		−	1.37323i	0.727020	−	0.686616i	$-0.240905\pi$
$38$		0			0
$39$	−6.31116	+	5.65259i	−1.01059	+	0.905139i
$40$		0			0
$41$			6.50228i			1.01548i	0.861509	+	0.507742i	$0.169520\pi$
$41$			6.50228i			1.01548i	−0.861509	+	0.507742i	$0.830480\pi$
$42$		0			0
$43$	−11.4720			−1.74946			−0.874731	−	0.484608i	$-0.838962\pi$
$43$	−11.4720			−1.74946			−0.874731	+	0.484608i	$0.838962\pi$
$44$		0			0
$45$	0.329281	−	2.98187i	0.0490863	−	0.444512i
$46$		0			0
$47$	1.92186			0.280332			0.140166	−	0.990128i	$-0.455236\pi$
$47$	1.92186			0.280332			0.140166	+	0.990128i	$0.455236\pi$
$48$		0			0
$49$	−11.5861			−1.65515
$50$		0			0
$51$	−1.38333	+	1.23898i	−0.193705	+	0.173492i
$52$		0			0
$53$	1.46147			0.200749			0.100374	−	0.994950i	$-0.467996\pi$
$53$	1.46147			0.200749			0.100374	+	0.994950i	$0.467996\pi$
$54$		0			0
$55$			5.96375i			0.804152i
$56$		0			0
$57$	−4.89157	−	5.46147i	−0.647905	−	0.723390i
$58$		0			0
$59$			5.34144i			0.695396i	0.937607	+	0.347698i	$0.113037\pi$
$59$			5.34144i			0.695396i	−0.937607	+	0.347698i	$0.886963\pi$
$60$		0			0
$61$		−	5.42522i		−	0.694628i	−0.937749	−	0.347314i	$-0.887094\pi$
$61$		−	5.42522i		−	0.694628i	0.937749	−	0.347314i	$-0.112906\pi$
$62$		0			0
$63$	12.8553	+	1.41958i	1.61962	+	0.178850i
$64$		0			0
$65$			4.89157i			0.606725i
$66$		0			0
$67$	3.47199			0.424171			0.212086	−	0.977251i	$-0.431974\pi$
$67$	3.47199			0.424171			0.212086	+	0.977251i	$0.431974\pi$
$68$		0			0
$69$	−6.22085	−	6.94562i	−0.748903	−	0.836155i
$70$		0			0
$71$	6.47796			0.768793			0.384396	−	0.923168i	$-0.374410\pi$
$71$	6.47796			0.768793			0.384396	+	0.923168i	$0.374410\pi$
$72$		0			0
$73$	14.4660			1.69312			0.846560	−	0.532293i	$-0.178670\pi$
$73$	14.4660			1.69312			0.846560	+	0.532293i	$0.178670\pi$
$74$		0			0
$75$	−1.15558	−	1.29021i	−0.133435	−	0.148981i
$76$		0			0
$77$	−25.7106			−2.93000
$78$		0			0
$79$		−	1.91134i		−	0.215042i	−0.994203	−	0.107521i	$-0.965709\pi$
$79$		−	1.91134i		−	0.215042i	0.994203	−	0.107521i	$-0.0342913\pi$
$80$		0			0
$81$	−8.78315	−	1.96375i	−0.975905	−	0.218194i
$82$		0			0
$83$			3.20273i			0.351545i	0.984431	+	0.175773i	$0.0562423\pi$
$83$			3.20273i			0.351545i	−0.984431	+	0.175773i	$0.943758\pi$
$84$		0			0
$85$			1.07217i			0.116294i
$86$		0			0
$87$	−3.23898	−	3.61634i	−0.347255	−	0.387713i
$88$		0			0
$89$			0.538528i			0.0570838i	0.999593	+	0.0285419i	$0.00908640\pi$
$89$			0.538528i			0.0570838i	−0.999593	+	0.0285419i	$0.990914\pi$
$90$		0			0
$91$	−21.0883			−2.21066
$92$		0			0
$93$	−1.19709	+	1.07217i	−0.124132	+	0.111179i
$94$		0			0
$95$	−4.23301			−0.434298
$96$		0			0
$97$	−7.78315			−0.790259			−0.395129	−	0.918625i	$-0.629300\pi$
$97$	−7.78315			−0.790259			−0.395129	+	0.918625i	$0.629300\pi$
$98$		0			0
$99$	17.7831	+	1.96375i	1.78727	+	0.197364i
$100$		0			0
$101$	−10.4417			−1.03899			−0.519494	−	0.854474i	$-0.673880\pi$
$101$	−10.4417			−1.03899			−0.519494	+	0.854474i	$0.673880\pi$
$102$		0			0
$103$		−	15.6163i		−	1.53872i	−0.638813	−	0.769362i	$-0.720574\pi$
$103$		−	15.6163i		−	1.53872i	0.638813	−	0.769362i	$-0.279426\pi$
$104$		0			0
$105$	5.56229	−	4.98187i	0.542824	−	0.486181i
$106$		0			0
$107$		−	8.51985i		−	0.823645i	−0.911264	−	0.411823i	$-0.864892\pi$
$107$		−	8.51985i		−	0.823645i	0.911264	−	0.411823i	$-0.135108\pi$
$108$		0			0
$109$			8.64662i			0.828196i	0.910232	+	0.414098i	$0.135903\pi$
$109$			8.64662i			0.828196i	−0.910232	+	0.414098i	$0.864097\pi$
$110$		0			0
$111$	−10.7772	+	9.65259i	−1.02292	+	0.916184i
$112$		0			0
$113$			16.6385i			1.56522i	0.622515	+	0.782608i	$0.286111\pi$
$113$			16.6385i			1.56522i	−0.622515	+	0.782608i	$0.713889\pi$
$114$		0			0
$115$	−5.38333			−0.501998
$116$		0			0
$117$	14.5861	+	1.61070i	1.34848	+	0.148909i
$118$		0			0
$119$	−4.62231			−0.423726
$120$		0			0
$121$	−24.5663			−2.23330
$122$		0			0
$123$	8.38930	−	7.51388i	0.756438	−	0.677504i
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$		−	12.1549i		−	1.07857i	−0.842123	−	0.539285i	$-0.818695\pi$
$127$		−	12.1549i		−	1.07857i	0.842123	−	0.539285i	$-0.181305\pi$
$128$		0			0
$129$	13.2568	+	14.8013i	1.16719	+	1.30318i
$130$		0			0
$131$			6.74234i			0.589081i	0.955639	+	0.294541i	$0.0951667\pi$
$131$			6.74234i			0.589081i	−0.955639	+	0.294541i	$0.904833\pi$
$132$		0			0
$133$		−	18.2492i		−	1.58240i
$134$		0			0
$135$	−4.22775	+	3.02095i	−0.363867	+	0.260002i
$136$		0			0
$137$			22.8553i			1.95266i	0.216282	+	0.976331i	$0.430607\pi$
$137$			22.8553i			1.95266i	−0.216282	+	0.976331i	$0.569393\pi$
$138$		0			0
$139$	3.15596			0.267685			0.133842	−	0.991003i	$-0.457268\pi$
$139$	3.15596			0.267685			0.133842	+	0.991003i	$0.457268\pi$
$140$		0			0
$141$	−2.22085	−	2.47960i	−0.187030	−	0.208820i
$142$		0			0
$143$	−29.1721			−2.43950
$144$		0			0
$145$	−2.80291			−0.232769
$146$		0			0
$147$	13.3886	+	14.9484i	1.10427	+	1.23293i
$148$		0			0
$149$	−8.61037			−0.705389			−0.352695	−	0.935738i	$-0.614735\pi$
$149$	−8.61037			−0.705389			−0.352695	+	0.935738i	$0.614735\pi$
$150$		0			0
$151$			14.0999i			1.14744i	0.819053	+	0.573719i	$0.194500\pi$
$151$			14.0999i			1.14744i	−0.819053	+	0.573719i	$0.805500\pi$
$152$		0			0
$153$	3.19709	+	0.353047i	0.258469	+	0.0285421i
$154$		0			0
$155$			0.927826i			0.0745247i
$156$		0			0
$157$			14.2087i			1.13398i	0.823726	+	0.566989i	$0.191892\pi$
$157$			14.2087i			1.13398i	−0.823726	+	0.566989i	$0.808108\pi$
$158$		0			0
$159$	−1.68884	−	1.88561i	−0.133934	−	0.149538i
$160$		0			0
$161$		−	23.2084i		−	1.82908i
$162$		0			0
$163$	17.2551			1.35153			0.675763	−	0.737119i	$-0.263814\pi$
$163$	17.2551			1.35153			0.675763	+	0.737119i	$0.263814\pi$
$164$		0			0
$165$	7.69448	−	6.89157i	0.599015	−	0.536508i
$166$		0			0
$167$	−8.39982			−0.649997			−0.324999	−	0.945714i	$-0.605364\pi$
$167$	−8.39982			−0.649997			−0.324999	+	0.945714i	$0.605364\pi$
$168$		0			0
$169$	−10.9275			−0.840577
$170$		0			0
$171$	−1.39385	+	12.6223i	−0.106590	+	0.965252i
$172$		0			0
$173$	20.4892			1.55777			0.778884	−	0.627168i	$-0.215786\pi$
$173$	20.4892			1.55777			0.778884	+	0.627168i	$0.215786\pi$
$174$		0			0
$175$		−	4.31116i		−	0.325893i
$176$		0			0
$177$	6.89157	−	6.17245i	0.518002	−	0.463949i
$178$		0			0
$179$			14.5861i			1.09021i	0.838367	+	0.545107i	$0.183511\pi$
$179$			14.5861i			1.09021i	−0.838367	+	0.545107i	$0.816489\pi$
$180$		0			0
$181$			6.32168i			0.469886i	0.972009	+	0.234943i	$0.0754904\pi$
$181$			6.32168i			0.469886i	−0.972009	+	0.234943i	$0.924510\pi$
$182$		0			0
$183$	−6.99967	+	6.26926i	−0.517431	+	0.463437i
$184$		0			0
$185$			8.35305i			0.614128i
$186$		0			0
$187$	−6.39418			−0.467589
$188$		0			0
$189$	−13.0238	−	18.2265i	−0.947340	−	1.32578i
$190$		0			0
$191$	3.54526			0.256526			0.128263	−	0.991740i	$-0.459060\pi$
$191$	3.54526			0.256526			0.128263	+	0.991740i	$0.459060\pi$
$192$		0			0
$193$	4.86020			0.349845			0.174923	−	0.984582i	$-0.444032\pi$
$193$	4.86020			0.349845			0.174923	+	0.984582i	$0.444032\pi$
$194$		0			0
$195$	6.31116	−	5.65259i	0.451951	−	0.404791i
$196$		0			0
$197$	−1.17277			−0.0835567			−0.0417784	−	0.999127i	$-0.513302\pi$
$197$	−1.17277			−0.0835567			−0.0417784	+	0.999127i	$0.513302\pi$
$198$		0			0
$199$		−	18.0162i		−	1.27713i	−0.769567	−	0.638566i	$-0.779528\pi$
$199$		−	18.0162i		−	1.27713i	0.769567	−	0.638566i	$-0.220472\pi$
$200$		0			0
$201$	−4.01216	−	4.47960i	−0.282996	−	0.315966i
$202$		0			0
$203$		−	12.0838i		−	0.848115i
$204$		0			0
$205$		−	6.50228i		−	0.454139i
$206$		0			0
$207$	−1.77263	+	16.0524i	−0.123206	+	1.11572i
$208$		0			0
$209$		−	25.2446i		−	1.74621i
$210$		0			0
$211$	−12.6780			−0.872789			−0.436395	−	0.899755i	$-0.643745\pi$
$211$	−12.6780			−0.872789			−0.436395	+	0.899755i	$0.643745\pi$
$212$		0			0
$213$	−7.48579	−	8.35793i	−0.512918	−	0.572676i
$214$		0			0
$215$	11.4720			0.782383
$216$		0			0
$217$	−4.00000			−0.271538
$218$		0			0
$219$	−16.7166	−	18.6642i	−1.12960	−	1.26121i
$220$		0			0
$221$	−5.24462			−0.352791
$222$		0			0
$223$		−	5.54450i		−	0.371287i	−0.982617	−	0.185643i	$-0.940563\pi$
$223$		−	5.54450i		−	0.371287i	0.982617	−	0.185643i	$-0.0594369\pi$
$224$		0			0
$225$	−0.329281	+	2.98187i	−0.0219521	+	0.198792i
$226$		0			0
$227$			6.50792i			0.431946i	0.976399	+	0.215973i	$0.0692922\pi$
$227$			6.50792i			0.431946i	−0.976399	+	0.215973i	$0.930708\pi$
$228$		0			0
$229$			11.1003i			0.733527i	0.930314	+	0.366763i	$0.119534\pi$
$229$			11.1003i			0.733527i	−0.930314	+	0.366763i	$0.880466\pi$
$230$		0			0
$231$	29.7106	+	33.1721i	1.95482	+	2.18257i
$232$		0			0
$233$			3.70642i			0.242816i	0.992603	+	0.121408i	$0.0387409\pi$
$233$			3.70642i			0.242816i	−0.992603	+	0.121408i	$0.961259\pi$
$234$		0			0
$235$	−1.92186			−0.125368
$236$		0			0
$237$	−2.46602	+	2.20870i	−0.160186	+	0.143470i
$238$		0			0
$239$	24.4660			1.58258			0.791288	−	0.611444i	$-0.209411\pi$
$239$	24.4660			1.58258			0.791288	+	0.611444i	$0.209411\pi$
$240$		0			0
$241$	14.1919			0.914179			0.457090	−	0.889421i	$-0.348892\pi$
$241$	14.1919			0.914179			0.457090	+	0.889421i	$0.348892\pi$
$242$		0			0
$243$	7.61596	+	13.6014i	0.488564	+	0.872528i
$244$		0			0
$245$	11.5861			0.740206
$246$		0			0
$247$		−	20.7061i		−	1.31750i
$248$		0			0
$249$	4.13219	−	3.70100i	0.261867	−	0.234541i
$250$		0			0
$251$		−	5.96375i		−	0.376428i	−0.982128	−	0.188214i	$-0.939730\pi$
$251$		−	5.96375i		−	0.376428i	0.982128	−	0.188214i	$-0.0602699\pi$
$252$		0			0
$253$		−	32.1048i		−	2.01841i
$254$		0			0
$255$	1.38333	−	1.23898i	0.0866275	−	0.0775880i
$256$		0			0
$257$		−	0.783477i		−	0.0488719i	−0.999701	−	0.0244360i	$-0.992221\pi$
$257$		−	0.783477i		−	0.0488719i	0.999701	−	0.0244360i	$-0.00777898\pi$
$258$		0			0
$259$	−36.0113			−2.23763
$260$		0			0
$261$	−0.922945	+	8.35793i	−0.0571289	+	0.517343i
$262$		0			0
$263$	−9.45583			−0.583072			−0.291536	−	0.956560i	$-0.594166\pi$
$263$	−9.45583			−0.583072			−0.291536	+	0.956560i	$0.594166\pi$
$264$		0			0
$265$	−1.46147			−0.0897775
$266$		0			0
$267$	0.694813	−	0.622310i	0.0425219	−	0.0380848i
$268$		0			0
$269$	32.1767			1.96185			0.980923	−	0.194396i	$-0.0622747\pi$
$269$	32.1767			1.96185			0.980923	+	0.194396i	$0.0622747\pi$
$270$		0			0
$271$		−	3.31224i		−	0.201204i	−0.994927	−	0.100602i	$-0.967923\pi$
$271$		−	3.31224i		−	0.201204i	0.994927	−	0.100602i	$-0.0320769\pi$
$272$		0			0
$273$	24.3692	+	27.2084i	1.47489	+	1.64673i
$274$		0			0
$275$		−	5.96375i		−	0.359628i
$276$		0			0
$277$			27.4294i			1.64808i	0.566535	+	0.824038i	$0.308284\pi$
$277$			27.4294i			1.64808i	−0.566535	+	0.824038i	$0.691716\pi$
$278$		0			0
$279$	2.76666	+	0.305515i	0.165636	+	0.0182907i
$280$		0			0
$281$		−	11.6744i		−	0.696436i	−0.937414	−	0.348218i	$-0.886787\pi$
$281$		−	11.6744i		−	0.696436i	0.937414	−	0.348218i	$-0.113213\pi$
$282$		0			0
$283$	−11.0060			−0.654237			−0.327118	−	0.944983i	$-0.606078\pi$
$283$	−11.0060			−0.654237			−0.327118	+	0.944983i	$0.606078\pi$
$284$		0			0
$285$	4.89157	+	5.46147i	0.289752	+	0.323510i
$286$		0			0
$287$	28.0323			1.65470
$288$		0			0
$289$	15.8504			0.932379
$290$		0			0
$291$	8.99403	+	10.0419i	0.527239	+	0.588666i
$292$		0			0
$293$	13.5663			0.792551			0.396276	−	0.918132i	$-0.370302\pi$
$293$	13.5663			0.792551			0.396276	+	0.918132i	$0.370302\pi$
$294$		0			0
$295$		−	5.34144i		−	0.310991i
$296$		0			0
$297$	−18.0162	−	25.2132i	−1.04540	−	1.46302i
$298$		0			0
$299$		−	26.3330i		−	1.52287i
$300$		0			0
$301$			49.4575i			2.85069i
$302$		0			0
$303$	12.0662	+	13.4720i	0.693185	+	0.773946i
$304$		0			0
$305$			5.42522i			0.310647i
$306$		0			0
$307$	16.9215			0.965763			0.482881	−	0.875686i	$-0.339590\pi$
$307$	16.9215			0.965763			0.482881	+	0.875686i	$0.339590\pi$
$308$		0			0
$309$	−20.1484	+	18.0459i	−1.14620	+	1.02660i
$310$		0			0
$311$	−17.6896			−1.00309			−0.501543	−	0.865133i	$-0.667234\pi$
$311$	−17.6896			−1.00309			−0.501543	+	0.865133i	$0.667234\pi$
$312$		0			0
$313$	17.2776			0.976588			0.488294	−	0.872679i	$-0.337619\pi$
$313$	17.2776			0.976588			0.488294	+	0.872679i	$0.337619\pi$
$314$		0			0
$315$	−12.8553	−	1.41958i	−0.724315	−	0.0799843i
$316$		0			0
$317$	3.61558			0.203071			0.101536	−	0.994832i	$-0.467624\pi$
$317$	3.61558			0.203071			0.101536	+	0.994832i	$0.467624\pi$
$318$		0			0
$319$		−	16.7159i		−	0.935908i
$320$		0			0
$321$	−10.9924	+	9.84535i	−0.613536	+	0.549514i
$322$		0			0
$323$		−	4.53853i		−	0.252530i
$324$		0			0
$325$		−	4.89157i		−	0.271336i
$326$		0			0
$327$	11.1560	−	9.99185i	0.616926	−	0.552550i
$328$		0			0
$329$		−	8.28542i		−	0.456790i
$330$		0			0
$331$	−10.2449			−0.563113			−0.281557	−	0.959545i	$-0.590851\pi$
$331$	−10.2449			−0.563113			−0.281557	+	0.959545i	$0.590851\pi$
$332$		0			0
$333$	24.9077	+	2.75050i	1.36494	+	0.150726i
$334$		0			0
$335$	−3.47199			−0.189695
$336$		0			0
$337$	−4.39418			−0.239366			−0.119683	−	0.992812i	$-0.538188\pi$
$337$	−4.39418			−0.239366			−0.119683	+	0.992812i	$0.538188\pi$
$338$		0			0
$339$	21.4671	−	19.2270i	1.16593	−	1.04427i
$340$		0			0
$341$	−5.53332			−0.299646
$342$		0			0
$343$			19.7712i			1.06755i
$344$		0			0
$345$	6.22085	+	6.94562i	0.334920	+	0.373940i
$346$		0			0
$347$			7.89754i			0.423962i	0.977274	+	0.211981i	$0.0679915\pi$
$347$			7.89754i			0.423962i	−0.977274	+	0.211981i	$0.932008\pi$
$348$		0			0
$349$		−	11.4615i		−	0.613519i	−0.951787	−	0.306759i	$-0.900755\pi$
$349$		−	11.4615i		−	0.613519i	0.951787	−	0.306759i	$-0.0992447\pi$
$350$		0			0
$351$	−14.7772	−	20.6804i	−0.788748	−	1.10384i
$352$		0			0
$353$		−	31.9324i		−	1.69959i	−0.527114	−	0.849794i	$-0.676726\pi$
$353$		−	31.9324i		−	1.69959i	0.527114	−	0.849794i	$-0.323274\pi$
$354$		0			0
$355$	−6.47796			−0.343814
$356$		0			0
$357$	5.34144	+	5.96375i	0.282699	+	0.315635i
$358$		0			0
$359$	21.0673			1.11189			0.555945	−	0.831219i	$-0.312357\pi$
$359$	21.0673			1.11189			0.555945	+	0.831219i	$0.312357\pi$
$360$		0			0
$361$	−1.08161			−0.0569267
$362$		0			0
$363$	28.3883	+	31.6957i	1.49000	+	1.66359i
$364$		0			0
$365$	−14.4660			−0.757186
$366$		0			0
$367$			16.4998i			0.861281i	0.902524	+	0.430640i	$0.141712\pi$
$367$			16.4998i			0.861281i	−0.902524	+	0.430640i	$0.858288\pi$
$368$		0			0
$369$	−19.3890	−	2.14108i	−1.00935	−	0.111460i
$370$		0			0
$371$		−	6.30063i		−	0.327113i
$372$		0			0
$373$			15.2760i			0.790961i	0.918474	+	0.395480i	$0.129422\pi$
$373$			15.2760i			0.790961i	−0.918474	+	0.395480i	$0.870578\pi$
$374$		0			0
$375$	1.15558	+	1.29021i	0.0596738	+	0.0666261i
$376$		0			0
$377$		−	13.7106i		−	0.706134i
$378$		0			0
$379$	3.76699			0.193497			0.0967486	−	0.995309i	$-0.469156\pi$
$379$	3.76699			0.193497			0.0967486	+	0.995309i	$0.469156\pi$
$380$		0			0
$381$	−15.6823	+	14.0459i	−0.803430	+	0.719593i
$382$		0			0
$383$	17.7282			0.905870			0.452935	−	0.891544i	$-0.350377\pi$
$383$	17.7282			0.905870			0.452935	+	0.891544i	$0.350377\pi$
$384$		0			0
$385$	25.7106			1.31034
$386$		0			0
$387$	3.77751	−	34.2080i	0.192022	−	1.73889i
$388$		0			0
$389$	−4.92294			−0.249603			−0.124802	−	0.992182i	$-0.539829\pi$
$389$	−4.92294			−0.249603			−0.124802	+	0.992182i	$0.539829\pi$
$390$		0			0
$391$		−	5.77187i		−	0.291896i
$392$		0			0
$393$	8.69904	−	7.79130i	0.438808	−	0.393019i
$394$		0			0
$395$			1.91134i			0.0961698i
$396$		0			0
$397$		−	19.3246i		−	0.969875i	−0.874549	−	0.484937i	$-0.838842\pi$
$397$		−	19.3246i		−	0.969875i	0.874549	−	0.484937i	$-0.161158\pi$
$398$		0			0
$399$	−23.5453	+	21.0883i	−1.17874	+	1.05574i
$400$		0			0
$401$		−	24.8834i		−	1.24262i	−0.783565	−	0.621309i	$-0.786601\pi$
$401$		−	24.8834i		−	1.24262i	0.783565	−	0.621309i	$-0.213399\pi$
$402$		0			0
$403$	−4.53853			−0.226080
$404$		0			0
$405$	8.78315	+	1.96375i	0.436438	+	0.0975794i
$406$		0			0
$407$	−49.8155			−2.46926
$408$		0			0
$409$	16.5136			0.816543			0.408271	−	0.912861i	$-0.366132\pi$
$409$	16.5136			0.816543			0.408271	+	0.912861i	$0.366132\pi$
$410$		0			0
$411$	29.4882	−	26.4111i	1.45454	−	1.30276i
$412$		0			0
$413$	23.0278			1.13312
$414$		0			0
$415$		−	3.20273i		−	0.157216i
$416$		0			0
$417$	−3.64695	−	4.07185i	−0.178592	−	0.199399i
$418$		0			0
$419$			3.26959i			0.159730i	0.996806	+	0.0798650i	$0.0254489\pi$
$419$			3.26959i			0.159730i	−0.996806	+	0.0798650i	$0.974551\pi$
$420$		0			0
$421$		−	5.78642i		−	0.282013i	−0.990009	−	0.141006i	$-0.954966\pi$
$421$		−	5.78642i		−	0.282013i	0.990009	−	0.141006i	$-0.0450338\pi$
$422$		0			0
$423$	−0.632831	+	5.73074i	−0.0307693	+	0.278638i
$424$		0			0
$425$		−	1.07217i		−	0.0520081i
$426$		0			0
$427$	−23.3890			−1.13187
$428$		0			0
$429$	33.7106	+	37.6381i	1.62757	+	1.81719i
$430$		0			0
$431$	−36.3659			−1.75169			−0.875843	−	0.482597i	$-0.839694\pi$
$431$	−36.3659			−1.75169			−0.875843	+	0.482597i	$0.839694\pi$
$432$		0			0
$433$	39.3494			1.89101			0.945507	−	0.325602i	$-0.105567\pi$
$433$	39.3494			1.89101			0.945507	+	0.325602i	$0.105567\pi$
$434$		0			0
$435$	3.23898	+	3.61634i	0.155297	+	0.173390i
$436$		0			0
$437$	22.7877			1.09008
$438$		0			0
$439$		−	22.6174i		−	1.07947i	−0.841835	−	0.539736i	$-0.818524\pi$
$439$		−	22.6174i		−	1.07947i	0.841835	−	0.539736i	$-0.181476\pi$
$440$		0			0
$441$	3.81507	−	34.5482i	0.181670	−	1.64515i
$442$		0			0
$443$		−	14.1354i		−	0.671595i	−0.941934	−	0.335797i	$-0.890994\pi$
$443$		−	14.1354i		−	0.671595i	0.941934	−	0.335797i	$-0.109006\pi$
$444$		0			0
$445$		−	0.538528i		−	0.0255287i
$446$		0			0
$447$	9.94995	+	11.1092i	0.470617	+	0.525446i
$448$		0			0
$449$		−	2.03625i		−	0.0960966i	−0.998845	−	0.0480483i	$-0.984700\pi$
$449$		−	2.03625i		−	0.0960966i	0.998845	−	0.0480483i	$-0.0153001\pi$
$450$		0			0
$451$	38.7779			1.82598
$452$		0			0
$453$	18.1919	−	16.2936i	0.854729	−	0.765539i
$454$		0			0
$455$	21.0883			0.988636
$456$		0			0
$457$	6.24007			0.291898			0.145949	−	0.989292i	$-0.453376\pi$
$457$	6.24007			0.291898			0.145949	+	0.989292i	$0.453376\pi$
$458$		0			0
$459$	−3.23898	−	4.53289i	−0.151183	−	0.211577i
$460$		0			0
$461$	−33.2922			−1.55057			−0.775285	−	0.631612i	$-0.782394\pi$
$461$	−33.2922			−1.55057			−0.775285	+	0.631612i	$0.782394\pi$
$462$		0			0
$463$			7.50497i			0.348786i	0.984676	+	0.174393i	$0.0557962\pi$
$463$			7.50497i			0.348786i	−0.984676	+	0.174393i	$0.944204\pi$
$464$		0			0
$465$	1.19709	−	1.07217i	0.0555137	−	0.0497209i
$466$		0			0
$467$		−	22.9739i		−	1.06311i	−0.847025	−	0.531554i	$-0.821608\pi$
$467$		−	22.9739i		−	1.06311i	0.847025	−	0.531554i	$-0.178392\pi$
$468$		0			0
$469$		−	14.9683i		−	0.691172i
$470$		0			0
$471$	18.3322	−	16.4193i	0.844703	−	0.756559i
$472$		0			0
$473$			68.4161i			3.14577i
$474$		0			0
$475$	4.23301			0.194224
$476$		0			0
$477$	−0.481235	+	4.35793i	−0.0220342	+	0.199536i
$478$		0			0
$479$	29.4219			1.34432			0.672162	−	0.740405i	$-0.265366\pi$
$479$	29.4219			1.34432			0.672162	+	0.740405i	$0.265366\pi$
$480$		0			0
$481$	−40.8595			−1.86303
$482$		0			0
$483$	−29.9437	+	26.8191i	−1.36248	+	1.22031i
$484$		0			0
$485$	7.78315			0.353415
$486$		0			0
$487$		−	5.39949i		−	0.244674i	−0.992489	−	0.122337i	$-0.960961\pi$
$487$		−	5.39949i		−	0.244674i	0.992489	−	0.122337i	$-0.0390389\pi$
$488$		0			0
$489$	−19.9397	−	22.2627i	−0.901702	−	1.00676i
$490$		0			0
$491$			6.42977i			0.290172i	0.989419	+	0.145086i	$0.0463458\pi$
$491$			6.42977i			0.290172i	−0.989419	+	0.145086i	$0.953654\pi$
$492$		0			0
$493$		−	3.00521i		−	0.135348i
$494$		0			0
$495$	−17.7831	−	1.96375i	−0.799293	−	0.0882639i
$496$		0			0
$497$		−	27.9275i		−	1.25272i
$498$		0			0
$499$	−12.2330			−0.547625			−0.273812	−	0.961783i	$-0.588285\pi$
$499$	−12.2330			−0.547625			−0.273812	+	0.961783i	$0.588285\pi$
$500$		0			0
$501$	9.70664	+	10.8375i	0.433661	+	0.484185i
$502$		0			0
$503$	4.68852			0.209051			0.104525	−	0.994522i	$-0.466668\pi$
$503$	4.68852			0.209051			0.104525	+	0.994522i	$0.466668\pi$
$504$		0			0
$505$	10.4417			0.464650
$506$		0			0
$507$	12.6276	+	14.0988i	0.560810	+	0.626148i
$508$		0			0
$509$	2.87541			0.127450			0.0637252	−	0.997967i	$-0.479702\pi$
$509$	2.87541			0.127450			0.0637252	+	0.997967i	$0.479702\pi$
$510$		0			0
$511$		−	62.3653i		−	2.75888i
$512$		0			0
$513$	17.8961	−	12.7877i	0.790133	−	0.564591i
$514$		0			0
$515$			15.6163i			0.688138i
$516$		0			0
$517$		−	11.4615i		−	0.504075i
$518$		0			0
$519$	−23.6769	−	26.4354i	−1.03930	−	1.16039i
$520$		0			0
$521$		−	7.27760i		−	0.318837i	−0.987211	−	0.159419i	$-0.949038\pi$
$521$		−	7.27760i		−	0.318837i	0.987211	−	0.159419i	$-0.0509620\pi$
$522$		0			0
$523$	−33.5551			−1.46726			−0.733631	−	0.679548i	$-0.762176\pi$
$523$	−33.5551			−1.46726			−0.733631	+	0.679548i	$0.762176\pi$
$524$		0			0
$525$	−5.56229	+	4.98187i	−0.242758	+	0.217427i
$526$		0			0
$527$	−0.994791			−0.0433338
$528$		0			0
$529$	5.98024			0.260010
$530$		0			0
$531$	−15.9275	−	1.75883i	−0.691194	−	0.0763269i
$532$		0			0
$533$	31.8064			1.37769
$534$		0			0
$535$			8.51985i			0.368345i
$536$		0			0
$537$	18.8191	−	16.8553i	0.812103	−	0.727361i
$538$		0			0
$539$			69.0963i			2.97619i
$540$		0			0
$541$			26.2492i			1.12854i	0.825590	+	0.564270i	$0.190842\pi$
$541$			26.2492i			1.12854i	−0.825590	+	0.564270i	$0.809158\pi$
$542$		0			0
$543$	8.15629	−	7.30519i	0.350020	−	0.313496i
$544$		0			0
$545$		−	8.64662i		−	0.370381i
$546$		0			0
$547$	−36.8490			−1.57555			−0.787775	−	0.615963i	$-0.788767\pi$
$547$	−36.8490			−1.57555			−0.787775	+	0.615963i	$0.788767\pi$
$548$		0			0
$549$	16.1773	+	1.78642i	0.690431	+	0.0762426i
$550$		0			0
$551$	11.8648			0.505456
$552$		0			0
$553$	−8.24007			−0.350403
$554$		0			0
$555$	10.7772	−	9.65259i	0.457466	−	0.409730i
$556$		0			0
$557$	−32.5379			−1.37867			−0.689337	−	0.724441i	$-0.742098\pi$
$557$	−32.5379			−1.37867			−0.689337	+	0.724441i	$0.742098\pi$
$558$		0			0
$559$			56.1161i			2.37346i
$560$		0			0
$561$	7.38897	+	8.24983i	0.311963	+	0.348308i
$562$		0			0
$563$		−	16.5199i		−	0.696229i	−0.937452	−	0.348114i	$-0.886822\pi$
$563$		−	16.5199i		−	0.696229i	0.937452	−	0.348114i	$-0.113178\pi$
$564$		0			0
$565$		−	16.6385i		−	0.699986i
$566$		0			0
$567$	−8.46602	+	37.8655i	−0.355540	+	1.59020i
$568$		0			0
$569$			31.5696i			1.32347i	0.749740	+	0.661733i	$0.230179\pi$
$569$			31.5696i			1.32347i	−0.749740	+	0.661733i	$0.769821\pi$
$570$		0			0
$571$	19.9556			0.835116			0.417558	−	0.908650i	$-0.362886\pi$
$571$	19.9556			0.835116			0.417558	+	0.908650i	$0.362886\pi$
$572$		0			0
$573$	−4.09682	−	4.57412i	−0.171147	−	0.191087i
$574$		0			0
$575$	5.38333			0.224500
$576$		0			0
$577$	4.63425			0.192926			0.0964631	−	0.995337i	$-0.469247\pi$
$577$	4.63425			0.192926			0.0964631	+	0.995337i	$0.469247\pi$
$578$		0			0
$579$	−5.61634	−	6.27068i	−0.233407	−	0.260601i
$580$		0			0
$581$	13.8075			0.572830
$582$		0			0
$583$		−	8.71585i		−	0.360974i
$584$		0			0
$585$	−14.5861	−	1.61070i	−0.603059	−	0.0665943i
$586$		0			0
$587$			11.4638i			0.473163i	0.971612	+	0.236582i	$0.0760271\pi$
$587$			11.4638i			0.473163i	−0.971612	+	0.236582i	$0.923973\pi$
$588$		0			0
$589$		−	3.92750i		−	0.161830i
$590$		0			0
$591$	1.35523	+	1.51313i	0.0557468	+	0.0622416i
$592$		0			0
$593$			10.5569i			0.433518i	0.976225	+	0.216759i	$0.0695487\pi$
$593$			10.5569i			0.433518i	−0.976225	+	0.216759i	$0.930451\pi$
$594$		0			0
$595$	4.62231			0.189496
$596$		0			0
$597$	−23.2446	+	20.8191i	−0.951339	+	0.852068i
$598$		0			0
$599$	−16.3210			−0.666859			−0.333429	−	0.942775i	$-0.608206\pi$
$599$	−16.3210			−0.666859			−0.333429	+	0.942775i	$0.608206\pi$
$600$		0			0
$601$	17.8793			0.729312			0.364656	−	0.931142i	$-0.381187\pi$
$601$	17.8793			0.729312			0.364656	+	0.931142i	$0.381187\pi$
$602$		0			0
$603$	−1.14326	+	10.3530i	−0.0465572	+	0.421609i
$604$		0			0
$605$	24.5663			0.998762
$606$		0			0
$607$			24.2499i			0.984274i	0.870518	+	0.492137i	$0.163784\pi$
$607$			24.2499i			0.984274i	−0.870518	+	0.492137i	$0.836216\pi$
$608$		0			0
$609$	−15.5906	+	13.9637i	−0.631763	+	0.565840i
$610$		0			0
$611$		−	9.40091i		−	0.380320i
$612$		0			0
$613$			22.3201i			0.901499i	0.892650	+	0.450750i	$0.148843\pi$
$613$			22.3201i			0.901499i	−0.892650	+	0.450750i	$0.851157\pi$
$614$		0			0
$615$	−8.38930	+	7.51388i	−0.338289	+	0.302989i
$616$		0			0
$617$		−	27.5614i		−	1.10958i	−0.831990	−	0.554790i	$-0.812798\pi$
$617$		−	27.5614i		−	1.10958i	0.831990	−	0.554790i	$-0.187202\pi$
$618$		0			0
$619$	28.7552			1.15577			0.577885	−	0.816118i	$-0.303878\pi$
$619$	28.7552			1.15577			0.577885	+	0.816118i	$0.303878\pi$
$620$		0			0
$621$	22.7594	−	16.2627i	0.913303	−	0.652602i
$622$		0			0
$623$	2.32168			0.0930160
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	−32.5708	+	29.1721i	−1.30075	+	1.16502i
$628$		0			0
$629$	−8.95592			−0.357096
$630$		0			0
$631$			39.9070i			1.58867i	0.607479	+	0.794336i	$0.292181\pi$
$631$			39.9070i			1.58867i	−0.607479	+	0.794336i	$0.707819\pi$
$632$		0			0
$633$	14.6504	+	16.3573i	0.582301	+	0.650143i
$634$		0			0
$635$			12.1549i			0.482351i
$636$		0			0
$637$			56.6741i			2.24551i
$638$		0			0
$639$	−2.13307	+	19.3165i	−0.0843829	+	0.764148i
$640$		0			0
$641$			15.3857i			0.607698i	0.952720	+	0.303849i	$0.0982719\pi$
$641$			15.3857i			0.607698i	−0.952720	+	0.303849i	$0.901728\pi$
$642$		0			0
$643$	2.75614			0.108691			0.0543457	−	0.998522i	$-0.482693\pi$
$643$	2.75614			0.108691			0.0543457	+	0.998522i	$0.482693\pi$
$644$		0			0
$645$	−13.2568	−	14.8013i	−0.521985	−	0.582800i
$646$		0			0
$647$	4.25765			0.167385			0.0836927	−	0.996492i	$-0.473329\pi$
$647$	4.25765			0.167385			0.0836927	+	0.996492i	$0.473329\pi$
$648$		0			0
$649$	31.8550			1.25042
$650$		0			0
$651$	4.62231	+	5.16084i	0.181163	+	0.202269i
$652$		0			0
$653$	5.07640			0.198655			0.0993274	−	0.995055i	$-0.468331\pi$
$653$	5.07640			0.198655			0.0993274	+	0.995055i	$0.468331\pi$
$654$		0			0
$655$		−	6.74234i		−	0.263445i
$656$		0			0
$657$	−4.76339	+	43.1359i	−0.185837	+	1.68289i
$658$		0			0
$659$			32.5531i			1.26809i	0.773297	+	0.634044i	$0.218606\pi$
$659$			32.5531i			1.26809i	−0.773297	+	0.634044i	$0.781394\pi$
$660$		0			0
$661$			21.9961i			0.855548i	0.903886	+	0.427774i	$0.140702\pi$
$661$			21.9961i			0.855548i	−0.903886	+	0.427774i	$0.859298\pi$
$662$		0			0
$663$	6.06057	+	6.76666i	0.235373	+	0.262795i
$664$		0			0
$665$			18.2492i			0.707673i
$666$		0			0
$667$	15.0890			0.584248
$668$		0			0
$669$	−7.15356	+	6.40710i	−0.276573	+	0.247713i
$670$		0			0
$671$	−32.3547			−1.24904
$672$		0			0
$673$	−6.70609			−0.258501			−0.129250	−	0.991612i	$-0.541257\pi$
$673$	−6.70609			−0.258501			−0.129250	+	0.991612i	$0.541257\pi$
$674$		0			0
$675$	4.22775	−	3.02095i	0.162726	−	0.116276i
$676$		0			0
$677$	−25.0764			−0.963764			−0.481882	−	0.876236i	$-0.660047\pi$
$677$	−25.0764			−0.963764			−0.481882	+	0.876236i	$0.660047\pi$
$678$		0			0
$679$			33.5544i			1.28770i
$680$		0			0
$681$	8.39657	−	7.52040i	0.321757	−	0.288182i
$682$		0			0
$683$		−	3.30282i		−	0.126379i	−0.998002	−	0.0631895i	$-0.979873\pi$
$683$		−	3.30282i		−	0.126379i	0.998002	−	0.0631895i	$-0.0201272\pi$
$684$		0			0
$685$		−	22.8553i		−	0.873257i
$686$		0			0
$687$	14.3217	−	12.8272i	0.546406	−	0.489389i
$688$		0			0
$689$		−	7.14890i		−	0.272351i
$690$		0			0
$691$	−3.32200			−0.126375			−0.0631875	−	0.998002i	$-0.520127\pi$
$691$	−3.32200			−0.126375			−0.0631875	+	0.998002i	$0.520127\pi$
$692$		0			0
$693$	8.46602	−	76.6659i	0.321598	−	2.91230i
$694$		0			0
$695$	−3.15596			−0.119712
$696$		0			0
$697$	6.97157			0.264067
$698$		0			0
$699$	4.78206	−	4.28306i	0.180874	−	0.162000i
$700$		0			0
$701$	22.2887			0.841832			0.420916	−	0.907100i	$-0.361709\pi$
$701$	22.2887			0.841832			0.420916	+	0.907100i	$0.361709\pi$
$702$		0			0
$703$		−	35.3585i		−	1.33357i
$704$		0			0
$705$	2.22085	+	2.47960i	0.0836422	+	0.0933871i
$706$		0			0
$707$			45.0158i			1.69299i
$708$		0			0
$709$			24.1048i			0.905276i	0.891694	+	0.452638i	$0.149517\pi$
$709$			24.1048i			0.905276i	−0.891694	+	0.452638i	$0.850483\pi$
$710$		0			0
$711$	5.69937	+	0.629367i	0.213743	+	0.0236031i
$712$		0			0
$713$		−	4.99479i		−	0.187056i
$714$		0			0
$715$	29.1721			1.09098
$716$		0			0
$717$	−28.2724	−	31.5663i	−1.05585	−	1.17886i
$718$		0			0
$719$	−43.5889			−1.62559			−0.812795	−	0.582550i	$-0.802055\pi$
$719$	−43.5889			−1.62559			−0.812795	+	0.582550i	$0.802055\pi$
$720$		0			0
$721$	−67.3245			−2.50729
$722$		0			0
$723$	−16.3998	−	18.3105i	−0.609916	−	0.680975i
$724$		0			0
$725$	2.80291			0.104097
$726$		0			0
$727$		−	33.2784i		−	1.23423i	−0.786875	−	0.617113i	$-0.788302\pi$
$727$		−	33.2784i		−	1.23423i	0.786875	−	0.617113i	$-0.211698\pi$
$728$		0			0
$729$	8.74777	−	25.5436i	0.323992	−	0.946060i
$730$		0			0
$731$			12.3000i			0.454931i
$732$		0			0
$733$		−	27.2132i		−	1.00514i	−0.864535	−	0.502572i	$-0.832387\pi$
$733$		−	27.2132i		−	1.00514i	0.864535	−	0.502572i	$-0.167613\pi$
$734$		0			0
$735$	−13.3886	−	14.9484i	−0.493846	−	0.551382i
$736$		0			0
$737$		−	20.7061i		−	0.762719i
$738$		0			0
$739$	6.81106			0.250549			0.125275	−	0.992122i	$-0.460019\pi$
$739$	6.81106			0.250549			0.125275	+	0.992122i	$0.460019\pi$
$740$		0			0
$741$	−26.7152	+	23.9275i	−0.981407	+	0.878999i
$742$		0			0
$743$	−32.6160			−1.19657			−0.598283	−	0.801285i	$-0.704150\pi$
$743$	−32.6160			−1.19657			−0.598283	+	0.801285i	$0.704150\pi$
$744$		0			0
$745$	8.61037			0.315460
$746$		0			0
$747$	−9.55014	−	1.05460i	−0.349421	−	0.0385857i
$748$		0			0
$749$	−36.7304			−1.34210
$750$		0			0
$751$		−	42.1950i		−	1.53972i	−0.638214	−	0.769859i	$-0.720327\pi$
$751$		−	42.1950i		−	1.53972i	0.638214	−	0.769859i	$-0.279673\pi$
$752$		0			0
$753$	−7.69448	+	6.89157i	−0.280403	+	0.251143i
$754$		0			0
$755$		−	14.0999i		−	0.513149i
$756$		0			0
$757$		−	16.5692i		−	0.602219i	−0.953590	−	0.301110i	$-0.902643\pi$
$757$		−	16.5692i		−	0.602219i	0.953590	−	0.301110i	$-0.0973570\pi$
$758$		0			0
$759$	−41.4219	+	37.0996i	−1.50352	+	1.34663i
$760$		0			0
$761$		−	49.6284i		−	1.79903i	−0.436893	−	0.899514i	$-0.643921\pi$
$761$		−	49.6284i		−	1.79903i	0.436893	−	0.899514i	$-0.356079\pi$
$762$		0			0
$763$	37.2769			1.34952
$764$		0			0
$765$	−3.19709	−	0.353047i	−0.115591	−	0.0127644i
$766$		0			0
$767$	26.1280			0.943429
$768$		0			0
$769$	−20.5617			−0.741475			−0.370738	−	0.928738i	$-0.620895\pi$
$769$	−20.5617			−0.741475			−0.370738	+	0.928738i	$0.620895\pi$
$770$		0			0
$771$	−1.01085	+	0.905368i	−0.0364048	+	0.0326060i
$772$		0			0
$773$	−38.3935			−1.38092			−0.690459	−	0.723371i	$-0.742592\pi$
$773$	−38.3935			−1.38092			−0.690459	+	0.723371i	$0.742592\pi$
$774$		0			0
$775$		−	0.927826i		−	0.0333285i
$776$		0			0
$777$	41.6138	+	46.4621i	1.49289	+	1.66682i
$778$		0			0
$779$			27.5242i			0.986157i
$780$		0			0
$781$		−	38.6329i		−	1.38240i
$782$		0			0
$783$	11.8500	−	8.46744i	0.423485	−	0.302602i
$784$		0			0
$785$		−	14.2087i		−	0.507130i
$786$		0			0
$787$	−24.9426			−0.889107			−0.444553	−	0.895752i	$-0.646638\pi$
$787$	−24.9426			−0.889107			−0.444553	+	0.895752i	$0.646638\pi$
$788$		0			0
$789$	10.9269	+	12.2000i	0.389010	+	0.434332i
$790$		0			0
$791$	71.7310			2.55046
$792$		0			0
$793$	−26.5379			−0.942387
$794$		0			0
$795$	1.68884	+	1.88561i	0.0598972	+	0.0668755i
$796$		0			0
$797$	48.2394			1.70873			0.854364	−	0.519675i	$-0.173947\pi$
$797$	48.2394			1.70873			0.854364	+	0.519675i	$0.173947\pi$
$798$		0			0
$799$		−	2.06057i		−	0.0728976i
$800$		0			0
$801$	−1.60582	−	0.177327i	−0.0567389	−	0.00626554i
$802$		0			0
$803$		−	86.2717i		−	3.04446i
$804$		0			0
$805$			23.2084i			0.817988i
$806$		0			0
$807$	−37.1826	−	41.5146i	−1.30889	−	1.46138i
$808$		0			0
$809$			22.7877i			0.801173i	0.916259	+	0.400586i	$0.131194\pi$
$809$			22.7877i			0.801173i	−0.916259	+	0.400586i	$0.868806\pi$
$810$		0			0
$811$	31.2981			1.09903			0.549513	−	0.835485i	$-0.314813\pi$
$811$	31.2981			1.09903			0.549513	+	0.835485i	$0.314813\pi$
$812$		0			0
$813$	−4.27349	+	3.82755i	−0.149878	+	0.134238i
$814$		0			0
$815$	−17.2551			−0.604421
$816$		0			0
$817$	−48.5611			−1.69894
$818$		0			0
$819$	6.94399	−	62.8828i	0.242643	−	2.19730i
$820$		0			0
$821$	−41.4452			−1.44645			−0.723223	−	0.690614i	$-0.757340\pi$
$821$	−41.4452			−1.44645			−0.723223	+	0.690614i	$0.757340\pi$
$822$		0			0
$823$		−	29.1497i		−	1.01609i	−0.861330	−	0.508047i	$-0.830368\pi$
$823$		−	29.1497i		−	1.01609i	0.861330	−	0.508047i	$-0.169632\pi$
$824$		0			0
$825$	−7.69448	+	6.89157i	−0.267888	+	0.239934i
$826$		0			0
$827$		−	49.2956i		−	1.71418i	−0.515170	−	0.857088i	$-0.672271\pi$
$827$		−	49.2956i		−	1.71418i	0.515170	−	0.857088i	$-0.327729\pi$
$828$		0			0
$829$		−	0.646625i		−	0.0224582i	−0.999937	−	0.0112291i	$-0.996426\pi$
$829$		−	0.646625i		−	0.0224582i	0.999937	−	0.0112291i	$-0.00357441\pi$
$830$		0			0
$831$	35.3897	−	31.6969i	1.22766	−	1.09955i
$832$		0			0
$833$			12.4223i			0.430406i
$834$		0			0
$835$	8.39982			0.290688
$836$		0			0
$837$	−2.80291	−	3.92262i	−0.0968828	−	0.135585i
$838$		0			0
$839$	32.6336			1.12664			0.563318	−	0.826240i	$-0.309524\pi$
$839$	32.6336			1.12664			0.563318	+	0.826240i	$0.309524\pi$
$840$		0			0
$841$	−21.1437			−0.729093
$842$		0			0
$843$	−15.0624	+	13.4907i	−0.518777	+	0.464643i
$844$		0			0
$845$	10.9275			0.375917
$846$		0			0
$847$			105.909i			3.63908i
$848$		0			0
$849$	12.7183	+	14.2000i	0.436489	+	0.487343i
$850$		0			0
$851$		−	44.9672i		−	1.54146i
$852$		0			0
$853$			12.1855i			0.417223i	0.977999	+	0.208611i	$0.0668944\pi$
$853$			12.1855i			0.417223i	−0.977999	+	0.208611i	$0.933106\pi$
$854$		0			0
$855$	1.39385	−	12.6223i	0.0476687	−	0.431674i
$856$		0			0
$857$			11.2890i			0.385626i	0.981236	+	0.192813i	$0.0617610\pi$
$857$			11.2890i			0.385626i	−0.981236	+	0.192813i	$0.938239\pi$
$858$		0			0
$859$	−15.0348			−0.512982			−0.256491	−	0.966547i	$-0.582566\pi$
$859$	−15.0348			−0.512982			−0.256491	+	0.966547i	$0.582566\pi$
$860$		0			0
$861$	−32.3935	−	36.1676i	−1.10397	−	1.23259i
$862$		0			0
$863$	−28.5561			−0.972061			−0.486031	−	0.873942i	$-0.661556\pi$
$863$	−28.5561			−0.972061			−0.486031	+	0.873942i	$0.661556\pi$
$864$		0			0
$865$	−20.4892			−0.696655
$866$		0			0
$867$	−18.3164	−	20.4504i	−0.622058	−	0.694532i
$868$		0			0
$869$	−11.3987			−0.386675
$870$		0			0
$871$		−	16.9835i		−	0.575464i
$872$		0			0
$873$	2.56284	−	23.2084i	0.0867391	−	0.785484i
$874$		0			0
$875$			4.31116i			0.145744i
$876$		0			0
$877$			0.00815412i			0.000275345i	1.00000		0.000137673i	$4.38225e-5\pi$
$877$			0.00815412i			0.000275345i	−1.00000		0.000137673i	$0.999956\pi$
$878$		0			0
$879$	−15.6769	−	17.5034i	−0.528769	−	0.590374i
$880$		0			0
$881$		−	0.420669i		−	0.0141727i	−0.999975	−	0.00708635i	$-0.997744\pi$
$881$		−	0.420669i		−	0.0141727i	0.999975	−	0.00708635i	$-0.00225567\pi$
$882$		0			0
$883$	−4.48782			−0.151027			−0.0755137	−	0.997145i	$-0.524060\pi$
$883$	−4.48782			−0.151027			−0.0755137	+	0.997145i	$0.524060\pi$
$884$		0			0
$885$	−6.89157	+	6.17245i	−0.231658	+	0.207485i
$886$		0			0
$887$	36.5225			1.22630			0.613152	−	0.789965i	$-0.289901\pi$
$887$	36.5225			1.22630			0.613152	+	0.789965i	$0.289901\pi$
$888$		0			0
$889$	−52.4015			−1.75749
$890$		0			0
$891$	−11.7113	+	52.3805i	−0.392343	+	1.75481i
$892$		0			0
$893$	8.13524			0.272236
$894$		0			0
$895$		−	14.5861i		−	0.487558i
$896$		0			0
$897$	−33.9750	+	30.4298i	−1.13439	+	1.01602i
$898$		0			0
$899$		−	2.60061i		−	0.0867353i
$900$		0			0
$901$		−	1.56695i		−	0.0522028i
$902$		0			0
$903$	63.8106	−	57.1520i	2.12348	−	1.90190i
$904$		0			0
$905$		−	6.32168i		−	0.210140i
$906$		0			0
$907$	28.4040			0.943141			0.471570	−	0.881828i	$-0.343687\pi$
$907$	28.4040			0.943141			0.471570	+	0.881828i	$0.343687\pi$
$908$		0			0
$909$	3.43826	−	31.1359i	0.114040	−	1.03271i
$910$		0			0
$911$	−57.8993			−1.91829			−0.959144	−	0.282919i	$-0.908697\pi$
$911$	−57.8993			−1.91829			−0.959144	+	0.282919i	$0.908697\pi$
$912$		0			0
$913$	19.1003			0.632127
$914$		0			0
$915$	6.99967	−	6.26926i	0.231402	−	0.207255i
$916$		0			0
$917$	29.0673			0.959887
$918$		0			0
$919$			25.1327i			0.829053i	0.910037	+	0.414526i	$0.136053\pi$
$919$			25.1327i			0.829053i	−0.910037	+	0.414526i	$0.863947\pi$
$920$		0			0
$921$	−19.5541	−	21.8323i	−0.644331	−	0.719399i
$922$		0			0
$923$		−	31.6874i		−	1.04300i
$924$		0			0
$925$		−	8.35305i		−	0.274646i
$926$		0			0
$927$	46.5660	+	5.14216i	1.52943	+	0.168891i
$928$		0			0
$929$			29.4575i			0.966471i	0.875491	+	0.483235i	$0.160538\pi$
$929$			29.4575i			0.966471i	−0.875491	+	0.483235i	$0.839462\pi$
$930$		0			0
$931$	−49.0439			−1.60735
$932$		0			0
$933$	20.4417	+	22.8233i	0.669232	+	0.747201i
$934$		0			0
$935$	6.39418			0.209112
$936$		0			0
$937$	−43.8782			−1.43344			−0.716719	−	0.697362i	$-0.754357\pi$
$937$	−43.8782			−1.43344			−0.716719	+	0.697362i	$0.754357\pi$
$938$		0			0
$939$	−19.9656	−	22.2917i	−0.651553	−	0.727463i
$940$		0			0
$941$	19.6138			0.639393			0.319696	−	0.947520i	$-0.396419\pi$
$941$	19.6138			0.639393			0.319696	+	0.947520i	$0.396419\pi$
$942$		0			0
$943$			35.0039i			1.13988i
$944$		0			0
$945$	13.0238	+	18.2265i	0.423663	+	0.592908i
$946$		0			0
$947$		−	30.7241i		−	0.998399i	−0.866487	−	0.499200i	$-0.833627\pi$
$947$		−	30.7241i		−	0.998399i	0.866487	−	0.499200i	$-0.166373\pi$
$948$		0			0
$949$		−	70.7616i		−	2.29702i
$950$		0			0
$951$	−4.17809	−	4.66486i	−0.135484	−	0.151268i
$952$		0			0
$953$			44.2048i			1.43193i	0.698135	+	0.715966i	$0.254014\pi$
$953$			44.2048i			1.43193i	−0.698135	+	0.715966i	$0.745986\pi$
$954$		0			0
$955$	−3.54526			−0.114722
$956$		0			0
$957$	−21.5670	+	19.3165i	−0.697161	+	0.624413i
$958$		0			0
$959$	98.5328			3.18179
$960$		0			0
$961$	30.1391			0.972230
$962$		0			0
$963$	25.4051	+	2.80542i	0.818669	+	0.0904036i
$964$		0			0
$965$	−4.86020			−0.156455
$966$		0			0
$967$			26.2057i			0.842718i	0.906894	+	0.421359i	$0.138447\pi$
$967$			26.2057i			0.842718i	−0.906894	+	0.421359i	$0.861553\pi$
$968$		0			0
$969$	−5.85565	+	5.24462i	−0.188111	+	0.168481i
$970$		0			0
$971$			13.6025i			0.436527i	0.975890	+	0.218263i	$0.0700391\pi$
$971$			13.6025i			0.436527i	−0.975890	+	0.218263i	$0.929961\pi$
$972$		0			0
$973$		−	13.6058i		−	0.436183i
$974$		0			0
$975$	−6.31116	+	5.65259i	−0.202119	+	0.181028i
$976$		0			0
$977$		−	11.9226i		−	0.381438i	−0.981645	−	0.190719i	$-0.938918\pi$
$977$		−	11.9226i		−	0.381438i	0.981645	−	0.190719i	$-0.0610820\pi$
$978$		0			0
$979$	3.21164			0.102645
$980$		0			0
$981$	−25.7831	−	2.84717i	−0.823193	−	0.0909031i
$982$		0			0
$983$	−50.9377			−1.62466			−0.812330	−	0.583198i	$-0.801801\pi$
$983$	−50.9377			−1.62466			−0.812330	+	0.583198i	$0.801801\pi$
$984$		0			0
$985$	1.17277			0.0373677
$986$		0			0
$987$	−10.6899	+	9.57445i	−0.340264	+	0.304758i
$988$		0			0
$989$	−61.7575			−1.96377
$990$		0			0
$991$			26.3661i			0.837546i	0.908091	+	0.418773i	$0.137540\pi$
$991$			26.3661i			0.837546i	−0.908091	+	0.418773i	$0.862460\pi$
$992$		0			0
$993$	11.8388	+	13.2181i	0.375694	+	0.419465i
$994$		0			0
$995$			18.0162i			0.571151i
$996$		0			0
$997$		−	4.18548i		−	0.132556i	−0.997801	−	0.0662778i	$-0.978888\pi$
$997$		−	4.18548i		−	0.132556i	0.997801	−	0.0662778i	$-0.0211123\pi$
$998$		0			0
$999$	−25.2341	−	35.3146i	−0.798372	−	1.11730i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1920.2.b.g.191.1	yes	8
3.2	odd	2		1920.2.b.h.191.2	yes	8
4.3	odd	2		1920.2.b.a.191.8	yes	8
8.3	odd	2		1920.2.b.h.191.1	yes	8
8.5	even	2		1920.2.b.b.191.8	yes	8
12.11	even	2		1920.2.b.b.191.7	yes	8
24.5	odd	2		1920.2.b.a.191.7	✓	8
24.11	even	2	inner	1920.2.b.g.191.2	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1920.2.b.a.191.7	✓	8	24.5	odd	2
1920.2.b.a.191.8	yes	8	4.3	odd	2
1920.2.b.b.191.7	yes	8	12.11	even	2
1920.2.b.b.191.8	yes	8	8.5	even	2
1920.2.b.g.191.1	yes	8	1.1	even	1	trivial
1920.2.b.g.191.2	yes	8	24.11	even	2	inner
1920.2.b.h.191.1	yes	8	8.3	odd	2
1920.2.b.h.191.2	yes	8	3.2	odd	2

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:	\( N \)	\(=\)	\( 1920 = 2^{7} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1920.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.15558 - 1.29021i) q^{3} -1.00000 q^{5} -4.31116i q^{7} +(-0.329281 + 2.98187i) q^{9} -5.96375i q^{11} -4.89157i q^{13} +(1.15558 + 1.29021i) q^{15} -1.07217i q^{17} +4.23301 q^{19} +(-5.56229 + 4.98187i) q^{21} +5.38333 q^{23} +1.00000 q^{25} +(4.22775 - 3.02095i) q^{27} +2.80291 q^{29} -0.927826i q^{31} +(-7.69448 + 6.89157i) q^{33} +4.31116i q^{35} -8.35305i q^{37} +(-6.31116 + 5.65259i) q^{39} +6.50228i q^{41} -11.4720 q^{43} +(0.329281 - 2.98187i) q^{45} +1.92186 q^{47} -11.5861 q^{49} +(-1.38333 + 1.23898i) q^{51} +1.46147 q^{53} +5.96375i q^{55} +(-4.89157 - 5.46147i) q^{57} +5.34144i q^{59} -5.42522i q^{61} +(12.8553 + 1.41958i) q^{63} +4.89157i q^{65} +3.47199 q^{67} +(-6.22085 - 6.94562i) q^{69} +6.47796 q^{71} +14.4660 q^{73} +(-1.15558 - 1.29021i) q^{75} -25.7106 q^{77} -1.91134i q^{79} +(-8.78315 - 1.96375i) q^{81} +3.20273i q^{83} +1.07217i q^{85} +(-3.23898 - 3.61634i) q^{87} +0.538528i q^{89} -21.0883 q^{91} +(-1.19709 + 1.07217i) q^{93} -4.23301 q^{95} -7.78315 q^{97} +(17.7831 + 1.96375i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 2 q^{3} - 8 q^{5} - 4 q^{9} - 2 q^{15} - 8 q^{19} - 4 q^{21} + 12 q^{23} + 8 q^{25} + 14 q^{27} + 8 q^{29} - 8 q^{33} - 28 q^{39} - 36 q^{43} + 4 q^{45} - 4 q^{47} + 20 q^{51} + 16 q^{63} - 28 q^{67}+ \cdots + 64 q^{99}+O(q^{100}) \) 8 * q + 2 * q^3 - 8 * q^5 - 4 * q^9 - 2 * q^15 - 8 * q^19 - 4 * q^21 + 12 * q^23 + 8 * q^25 + 14 * q^27 + 8 * q^29 - 8 * q^33 - 28 * q^39 - 36 * q^43 + 4 * q^45 - 4 * q^47 + 20 * q^51 + 16 * q^63 - 28 * q^67 - 12 * q^69 + 24 * q^71 + 32 * q^73 + 2 * q^75 - 32 * q^77 + 8 * q^81 - 12 * q^87 - 40 * q^91 - 24 * q^93 + 8 * q^95 + 16 * q^97 + 64 * q^99

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