LMFDB - Embedded newform 1440.2.x.p.127.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1440,2,Mod(127,1440)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1440, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 0, 0, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1440.127");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1440 = 2^{5} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1440.x (of order $4$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$11.4984578911$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(i, \sqrt{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9 $ x^4 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			127.2
Root			$-1.22474 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	1440.127
Dual form			1440.2.x.p.703.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.22474 - 0.224745i) q^{5} +(-1.44949 + 1.44949i) q^{7} +O(q^{10})$	$q+(2.22474 - 0.224745i) q^{5} +(-1.44949 + 1.44949i) q^{7} +2.44949i q^{11} +(2.00000 - 2.00000i) q^{13} +(0.449490 + 0.449490i) q^{17} +2.00000 q^{19} +(0.449490 + 0.449490i) q^{23} +(4.89898 - 1.00000i) q^{25} -0.449490i q^{29} +8.89898i q^{31} +(-2.89898 + 3.55051i) q^{35} +(2.89898 + 2.89898i) q^{37} -4.89898 q^{41} +(6.00000 + 6.00000i) q^{43} +(9.34847 - 9.34847i) q^{47} +2.79796i q^{49} +(-2.89898 + 2.89898i) q^{53} +(0.550510 + 5.44949i) q^{55} -11.3485 q^{59} +11.7980 q^{61} +(4.00000 - 4.89898i) q^{65} +(-6.89898 + 6.89898i) q^{67} -12.0000i q^{71} +(7.89898 - 7.89898i) q^{73} +(-3.55051 - 3.55051i) q^{77} +3.10102 q^{79} +(5.55051 + 5.55051i) q^{83} +(1.10102 + 0.898979i) q^{85} +12.0000i q^{89} +5.79796i q^{91} +(4.44949 - 0.449490i) q^{95} +(3.00000 + 3.00000i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{5} + 4 q^{7}+O(q^{10}) $ 4 * q + 4 * q^5 + 4 * q^7	$ 4 q + 4 q^{5} + 4 q^{7} + 8 q^{13} - 8 q^{17} + 8 q^{19} - 8 q^{23} + 8 q^{35} - 8 q^{37} + 24 q^{43} + 8 q^{47} + 8 q^{53} + 12 q^{55} - 16 q^{59} + 8 q^{61} + 16 q^{65} - 8 q^{67} + 12 q^{73} - 24 q^{77} + 32 q^{79} + 32 q^{83} + 24 q^{85} + 8 q^{95} + 12 q^{97}+O(q^{100}) $ 4 * q + 4 * q^5 + 4 * q^7 + 8 * q^13 - 8 * q^17 + 8 * q^19 - 8 * q^23 + 8 * q^35 - 8 * q^37 + 24 * q^43 + 8 * q^47 + 8 * q^53 + 12 * q^55 - 16 * q^59 + 8 * q^61 + 16 * q^65 - 8 * q^67 + 12 * q^73 - 24 * q^77 + 32 * q^79 + 32 * q^83 + 24 * q^85 + 8 * q^95 + 12 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1440\mathbb{Z}\right)^\times$.

$n$	$577$	$641$	$901$	$991$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$1$	$1$	$-1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	2.22474	−	0.224745i	0.994936	−	0.100509i
$5$	2.22474	−	0.224745i	0.994936	−	0.100509i
$6$		0			0
$7$	−1.44949	+	1.44949i	−0.547856	+	0.547856i	−0.925820	−	0.377964i	$-0.876624\pi$
$7$	−1.44949	+	1.44949i	−0.547856	+	0.547856i	0.377964	+	0.925820i	$0.376624\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			2.44949i			0.738549i	0.929320	+	0.369274i	$0.120394\pi$
$11$			2.44949i			0.738549i	−0.929320	+	0.369274i	$0.879606\pi$
$12$		0			0
$13$	2.00000	−	2.00000i	0.554700	−	0.554700i	−0.373094	−	0.927794i	$-0.621703\pi$
$13$	2.00000	−	2.00000i	0.554700	−	0.554700i	0.927794	+	0.373094i	$0.121703\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	0.449490	+	0.449490i	0.109017	+	0.109017i	0.759511	−	0.650494i	$-0.225438\pi$
$17$	0.449490	+	0.449490i	0.109017	+	0.109017i	−0.650494	+	0.759511i	$0.725438\pi$
$18$		0			0
$19$	2.00000			0.458831			0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000			0.458831			0.229416	+	0.973329i	$0.426318\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	0.449490	+	0.449490i	0.0937251	+	0.0937251i	0.752415	−	0.658690i	$-0.228889\pi$
$23$	0.449490	+	0.449490i	0.0937251	+	0.0937251i	−0.658690	+	0.752415i	$0.728889\pi$
$24$		0			0
$25$	4.89898	−	1.00000i	0.979796	−	0.200000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	0.449490i		−	0.0834681i	−0.999129	−	0.0417341i	$-0.986712\pi$
$29$		−	0.449490i		−	0.0834681i	0.999129	−	0.0417341i	$-0.0132882\pi$
$30$		0			0
$31$			8.89898i			1.59830i	0.601129	+	0.799152i	$0.294718\pi$
$31$			8.89898i			1.59830i	−0.601129	+	0.799152i	$0.705282\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−2.89898	+	3.55051i	−0.490017	+	0.600146i
$36$		0			0
$37$	2.89898	+	2.89898i	0.476589	+	0.476589i	0.904039	−	0.427450i	$-0.140588\pi$
$37$	2.89898	+	2.89898i	0.476589	+	0.476589i	−0.427450	+	0.904039i	$0.640588\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−4.89898			−0.765092			−0.382546	−	0.923936i	$-0.624953\pi$
$41$	−4.89898			−0.765092			−0.382546	+	0.923936i	$0.624953\pi$
$42$		0			0
$43$	6.00000	+	6.00000i	0.914991	+	0.914991i	0.996660	−	0.0816682i	$-0.0260248\pi$
$43$	6.00000	+	6.00000i	0.914991	+	0.914991i	−0.0816682	+	0.996660i	$0.526025\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	9.34847	−	9.34847i	1.36361	−	1.36361i	0.494353	−	0.869261i	$-0.335405\pi$
$47$	9.34847	−	9.34847i	1.36361	−	1.36361i	0.869261	−	0.494353i	$-0.164595\pi$
$48$		0			0
$49$			2.79796i			0.399708i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−2.89898	+	2.89898i	−0.398205	+	0.398205i	−0.877600	−	0.479394i	$-0.840856\pi$
$53$	−2.89898	+	2.89898i	−0.398205	+	0.398205i	0.479394	+	0.877600i	$0.340856\pi$
$54$		0			0
$55$	0.550510	+	5.44949i	0.0742308	+	0.734809i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−11.3485			−1.47744			−0.738722	−	0.674010i	$-0.764571\pi$
$59$	−11.3485			−1.47744			−0.738722	+	0.674010i	$0.764571\pi$
$60$		0			0
$61$	11.7980			1.51057			0.755287	−	0.655394i	$-0.227498\pi$
$61$	11.7980			1.51057			0.755287	+	0.655394i	$0.227498\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	4.00000	−	4.89898i	0.496139	−	0.607644i
$66$		0			0
$67$	−6.89898	+	6.89898i	−0.842844	+	0.842844i	−0.989228	−	0.146383i	$-0.953237\pi$
$67$	−6.89898	+	6.89898i	−0.842844	+	0.842844i	0.146383	+	0.989228i	$0.453237\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	12.0000i		−	1.42414i	−0.702109	−	0.712069i	$-0.747758\pi$
$71$		−	12.0000i		−	1.42414i	0.702109	−	0.712069i	$-0.252242\pi$
$72$		0			0
$73$	7.89898	−	7.89898i	0.924506	−	0.924506i	−0.0728382	−	0.997344i	$-0.523206\pi$
$73$	7.89898	−	7.89898i	0.924506	−	0.924506i	0.997344	+	0.0728382i	$0.0232057\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−3.55051	−	3.55051i	−0.404618	−	0.404618i
$78$		0			0
$79$	3.10102			0.348892			0.174446	−	0.984667i	$-0.444187\pi$
$79$	3.10102			0.348892			0.174446	+	0.984667i	$0.444187\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	5.55051	+	5.55051i	0.609248	+	0.609248i	0.942749	−	0.333502i	$-0.108230\pi$
$83$	5.55051	+	5.55051i	0.609248	+	0.609248i	−0.333502	+	0.942749i	$0.608230\pi$
$84$		0			0
$85$	1.10102	+	0.898979i	0.119422	+	0.0975080i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			12.0000i			1.27200i	0.771690	+	0.635999i	$0.219412\pi$
$89$			12.0000i			1.27200i	−0.771690	+	0.635999i	$0.780588\pi$
$90$		0			0
$91$			5.79796i			0.607791i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	4.44949	−	0.449490i	0.456508	−	0.0461167i
$96$		0			0
$97$	3.00000	+	3.00000i	0.304604	+	0.304604i	0.842812	−	0.538208i	$-0.180899\pi$
$97$	3.00000	+	3.00000i	0.304604	+	0.304604i	−0.538208	+	0.842812i	$0.680899\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−8.44949			−0.840756			−0.420378	−	0.907349i	$-0.638102\pi$
$101$	−8.44949			−0.840756			−0.420378	+	0.907349i	$0.638102\pi$
$102$		0			0
$103$	−0.550510	−	0.550510i	−0.0542434	−	0.0542434i	0.679465	−	0.733708i	$-0.262212\pi$
$103$	−0.550510	−	0.550510i	−0.0542434	−	0.0542434i	−0.733708	+	0.679465i	$0.762212\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−7.34847	+	7.34847i	−0.710403	+	0.710403i	−0.966620	−	0.256216i	$-0.917524\pi$
$107$	−7.34847	+	7.34847i	−0.710403	+	0.710403i	0.256216	+	0.966620i	$0.417524\pi$
$108$		0			0
$109$		−	11.7980i		−	1.13004i	−0.825077	−	0.565020i	$-0.808869\pi$
$109$		−	11.7980i		−	1.13004i	0.825077	−	0.565020i	$-0.191131\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	8.44949	−	8.44949i	0.794861	−	0.794861i	−0.187419	−	0.982280i	$-0.560012\pi$
$113$	8.44949	−	8.44949i	0.794861	−	0.794861i	0.982280	+	0.187419i	$0.0600122\pi$
$114$		0			0
$115$	1.10102	+	0.898979i	0.102671	+	0.0838303i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−1.30306			−0.119451
$120$		0			0
$121$	5.00000			0.454545
$122$		0			0
$123$		0			0
$124$		0			0
$125$	10.6742	−	3.32577i	0.954733	−	0.297465i
$126$		0			0
$127$	7.44949	−	7.44949i	0.661035	−	0.661035i	−0.294589	−	0.955624i	$-0.595183\pi$
$127$	7.44949	−	7.44949i	0.661035	−	0.661035i	0.955624	+	0.294589i	$0.0951827\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			6.44949i			0.563495i	0.959489	+	0.281747i	$0.0909140\pi$
$131$			6.44949i			0.563495i	−0.959489	+	0.281747i	$0.909086\pi$
$132$		0			0
$133$	−2.89898	+	2.89898i	−0.251373	+	0.251373i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−4.44949	−	4.44949i	−0.380146	−	0.380146i	0.491009	−	0.871155i	$-0.336628\pi$
$137$	−4.44949	−	4.44949i	−0.380146	−	0.380146i	−0.871155	+	0.491009i	$0.836628\pi$
$138$		0			0
$139$	−15.7980			−1.33997			−0.669983	−	0.742377i	$-0.733698\pi$
$139$	−15.7980			−1.33997			−0.669983	+	0.742377i	$0.733698\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	4.89898	+	4.89898i	0.409673	+	0.409673i
$144$		0			0
$145$	−0.101021	−	1.00000i	−0.00838930	−	0.0830455i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			20.4495i			1.67529i	0.546217	+	0.837644i	$0.316067\pi$
$149$			20.4495i			1.67529i	−0.546217	+	0.837644i	$0.683933\pi$
$150$		0			0
$151$		−	3.79796i		−	0.309074i	−0.987987	−	0.154537i	$-0.950611\pi$
$151$		−	3.79796i		−	0.309074i	0.987987	−	0.154537i	$-0.0493885\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	2.00000	+	19.7980i	0.160644	+	1.59021i
$156$		0			0
$157$	4.89898	+	4.89898i	0.390981	+	0.390981i	0.875037	−	0.484056i	$-0.160837\pi$
$157$	4.89898	+	4.89898i	0.390981	+	0.390981i	−0.484056	+	0.875037i	$0.660837\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−1.30306			−0.102696
$162$		0			0
$163$	8.89898	+	8.89898i	0.697022	+	0.697022i	0.963767	−	0.266745i	$-0.0859482\pi$
$163$	8.89898	+	8.89898i	0.697022	+	0.697022i	−0.266745	+	0.963767i	$0.585948\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	11.5505	−	11.5505i	0.893805	−	0.893805i	−0.101074	−	0.994879i	$-0.532228\pi$
$167$	11.5505	−	11.5505i	0.893805	−	0.893805i	0.994879	+	0.101074i	$0.0322278\pi$
$168$		0			0
$169$			5.00000i			0.384615i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	16.2474	−	16.2474i	1.23527	−	1.23527i	0.273358	−	0.961912i	$-0.411866\pi$
$173$	16.2474	−	16.2474i	1.23527	−	1.23527i	0.961912	−	0.273358i	$-0.0881345\pi$
$174$		0			0
$175$	−5.65153	+	8.55051i	−0.427216	+	0.646358i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−24.2474			−1.81234			−0.906170	−	0.422914i	$-0.861007\pi$
$179$	−24.2474			−1.81234			−0.906170	+	0.422914i	$0.861007\pi$
$180$		0			0
$181$	−10.0000			−0.743294			−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000			−0.743294			−0.371647	+	0.928374i	$0.621207\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	7.10102	+	5.79796i	0.522077	+	0.426274i
$186$		0			0
$187$	−1.10102	+	1.10102i	−0.0805146	+	0.0805146i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	20.8990i		−	1.51220i	−0.654458	−	0.756099i	$-0.727103\pi$
$191$		−	20.8990i		−	1.51220i	0.654458	−	0.756099i	$-0.272897\pi$
$192$		0			0
$193$	−9.89898	+	9.89898i	−0.712544	+	0.712544i	−0.967067	−	0.254523i	$-0.918082\pi$
$193$	−9.89898	+	9.89898i	−0.712544	+	0.712544i	0.254523	+	0.967067i	$0.418082\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−7.79796	−	7.79796i	−0.555582	−	0.555582i	0.372465	−	0.928046i	$-0.378513\pi$
$197$	−7.79796	−	7.79796i	−0.555582	−	0.555582i	−0.928046	+	0.372465i	$0.878513\pi$
$198$		0			0
$199$	15.7980			1.11989			0.559944	−	0.828531i	$-0.310823\pi$
$199$	15.7980			1.11989			0.559944	+	0.828531i	$0.310823\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	0.651531	+	0.651531i	0.0457285	+	0.0457285i
$204$		0			0
$205$	−10.8990	+	1.10102i	−0.761218	+	0.0768986i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			4.89898i			0.338869i
$210$		0			0
$211$		−	17.5959i		−	1.21135i	−0.795711	−	0.605676i	$-0.792903\pi$
$211$		−	17.5959i		−	1.21135i	0.795711	−	0.605676i	$-0.207097\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	14.6969	+	12.0000i	1.00232	+	0.818393i
$216$		0			0
$217$	−12.8990	−	12.8990i	−0.875640	−	0.875640i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	1.79796			0.120944
$222$		0			0
$223$	−18.3485	−	18.3485i	−1.22870	−	1.22870i	−0.964454	−	0.264250i	$-0.914876\pi$
$223$	−18.3485	−	18.3485i	−1.22870	−	1.22870i	−0.264250	−	0.964454i	$-0.585124\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	3.10102	−	3.10102i	0.205822	−	0.205822i	−0.596667	−	0.802489i	$-0.703509\pi$
$227$	3.10102	−	3.10102i	0.205822	−	0.205822i	0.802489	+	0.596667i	$0.203509\pi$
$228$		0			0
$229$			10.0000i			0.660819i	0.943838	+	0.330409i	$0.107187\pi$
$229$			10.0000i			0.660819i	−0.943838	+	0.330409i	$0.892813\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−9.34847	+	9.34847i	−0.612439	+	0.612439i	−0.943581	−	0.331142i	$-0.892566\pi$
$233$	−9.34847	+	9.34847i	−0.612439	+	0.612439i	0.331142	+	0.943581i	$0.392566\pi$
$234$		0			0
$235$	18.6969	−	22.8990i	1.21965	−	1.49376i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−10.2020			−0.659915			−0.329958	−	0.943996i	$-0.607034\pi$
$239$	−10.2020			−0.659915			−0.329958	+	0.943996i	$0.607034\pi$
$240$		0			0
$241$	−13.7980			−0.888805			−0.444402	−	0.895827i	$-0.646584\pi$
$241$	−13.7980			−0.888805			−0.444402	+	0.895827i	$0.646584\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	0.628827	+	6.22474i	0.0401743	+	0.397684i
$246$		0			0
$247$	4.00000	−	4.00000i	0.254514	−	0.254514i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	17.1464i		−	1.08227i	−0.840935	−	0.541136i	$-0.817994\pi$
$251$		−	17.1464i		−	1.08227i	0.840935	−	0.541136i	$-0.182006\pi$
$252$		0			0
$253$	−1.10102	+	1.10102i	−0.0692206	+	0.0692206i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−21.3485	−	21.3485i	−1.33168	−	1.33168i	−0.903867	−	0.427814i	$-0.859284\pi$
$257$	−21.3485	−	21.3485i	−1.33168	−	1.33168i	−0.427814	−	0.903867i	$-0.640716\pi$
$258$		0			0
$259$	−8.40408			−0.522204
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−6.24745	−	6.24745i	−0.385234	−	0.385234i	0.487749	−	0.872984i	$-0.337818\pi$
$263$	−6.24745	−	6.24745i	−0.385234	−	0.385234i	−0.872984	+	0.487749i	$0.837818\pi$
$264$		0			0
$265$	−5.79796	+	7.10102i	−0.356166	+	0.436212i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			4.44949i			0.271290i	0.990757	+	0.135645i	$0.0433107\pi$
$269$			4.44949i			0.271290i	−0.990757	+	0.135645i	$0.956689\pi$
$270$		0			0
$271$		−	4.20204i		−	0.255256i	−0.991822	−	0.127628i	$-0.959264\pi$
$271$		−	4.20204i		−	0.255256i	0.991822	−	0.127628i	$-0.0407363\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	2.44949	+	12.0000i	0.147710	+	0.723627i
$276$		0			0
$277$	−8.69694	−	8.69694i	−0.522548	−	0.522548i	0.395792	−	0.918340i	$-0.370470\pi$
$277$	−8.69694	−	8.69694i	−0.522548	−	0.522548i	−0.918340	+	0.395792i	$0.870470\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−18.6969			−1.11537			−0.557683	−	0.830054i	$-0.688309\pi$
$281$	−18.6969			−1.11537			−0.557683	+	0.830054i	$0.688309\pi$
$282$		0			0
$283$	−8.69694	−	8.69694i	−0.516979	−	0.516979i	0.399677	−	0.916656i	$-0.369122\pi$
$283$	−8.69694	−	8.69694i	−0.516979	−	0.516979i	−0.916656	+	0.399677i	$0.869122\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	7.10102	−	7.10102i	0.419160	−	0.419160i
$288$		0			0
$289$		−	16.5959i		−	0.976230i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−6.89898	+	6.89898i	−0.403043	+	0.403043i	−0.879304	−	0.476261i	$-0.841992\pi$
$293$	−6.89898	+	6.89898i	−0.403043	+	0.403043i	0.476261	+	0.879304i	$0.341992\pi$
$294$		0			0
$295$	−25.2474	+	2.55051i	−1.46996	+	0.148496i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	1.79796			0.103979
$300$		0			0
$301$	−17.3939			−1.00257
$302$		0			0
$303$		0			0
$304$		0			0
$305$	26.2474	−	2.65153i	1.50292	−	0.151826i
$306$		0			0
$307$	7.79796	−	7.79796i	0.445053	−	0.445053i	−0.448653	−	0.893706i	$-0.648096\pi$
$307$	7.79796	−	7.79796i	0.445053	−	0.445053i	0.893706	+	0.448653i	$0.148096\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	5.79796i		−	0.328772i	−0.986396	−	0.164386i	$-0.947436\pi$
$311$		−	5.79796i		−	0.328772i	0.986396	−	0.164386i	$-0.0525643\pi$
$312$		0			0
$313$	−7.89898	+	7.89898i	−0.446477	+	0.446477i	−0.894181	−	0.447705i	$-0.852242\pi$
$313$	−7.89898	+	7.89898i	−0.446477	+	0.446477i	0.447705	+	0.894181i	$0.352242\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−8.65153	−	8.65153i	−0.485918	−	0.485918i	0.421097	−	0.907016i	$-0.361645\pi$
$317$	−8.65153	−	8.65153i	−0.485918	−	0.485918i	−0.907016	+	0.421097i	$0.861645\pi$
$318$		0			0
$319$	1.10102			0.0616453
$320$		0			0
$321$		0			0
$322$		0			0
$323$	0.898979	+	0.898979i	0.0500206	+	0.0500206i
$324$		0			0
$325$	7.79796	−	11.7980i	0.432553	−	0.654433i
$326$		0			0
$327$		0			0
$328$		0			0
$329$			27.1010i			1.49413i
$330$		0			0
$331$		−	18.0000i		−	0.989369i	−0.869072	−	0.494685i	$-0.835284\pi$
$331$		−	18.0000i		−	0.989369i	0.869072	−	0.494685i	$-0.164716\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−13.7980	+	16.8990i	−0.753863	+	0.923290i
$336$		0			0
$337$	8.10102	+	8.10102i	0.441291	+	0.441291i	0.892446	−	0.451155i	$-0.148988\pi$
$337$	8.10102	+	8.10102i	0.441291	+	0.441291i	−0.451155	+	0.892446i	$0.648988\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−21.7980			−1.18043
$342$		0			0
$343$	−14.2020	−	14.2020i	−0.766838	−	0.766838i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	12.0000	−	12.0000i	0.644194	−	0.644194i	−0.307390	−	0.951584i	$-0.599456\pi$
$347$	12.0000	−	12.0000i	0.644194	−	0.644194i	0.951584	+	0.307390i	$0.0994556\pi$
$348$		0			0
$349$			21.5959i			1.15600i	0.816036	+	0.578001i	$0.196167\pi$
$349$			21.5959i			1.15600i	−0.816036	+	0.578001i	$0.803833\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	18.2474	−	18.2474i	0.971214	−	0.971214i	−0.0283834	−	0.999597i	$-0.509036\pi$
$353$	18.2474	−	18.2474i	0.971214	−	0.971214i	0.999597	+	0.0283834i	$0.00903593\pi$
$354$		0			0
$355$	−2.69694	−	26.6969i	−0.143139	−	1.41693i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−10.6969			−0.564563			−0.282281	−	0.959332i	$-0.591091\pi$
$359$	−10.6969			−0.564563			−0.282281	+	0.959332i	$0.591091\pi$
$360$		0			0
$361$	−15.0000			−0.789474
$362$		0			0
$363$		0			0
$364$		0			0
$365$	15.7980	−	19.3485i	0.826903	−	1.01275i
$366$		0			0
$367$	8.34847	−	8.34847i	0.435787	−	0.435787i	−0.454805	−	0.890591i	$-0.650291\pi$
$367$	8.34847	−	8.34847i	0.435787	−	0.435787i	0.890591	+	0.454805i	$0.150291\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	8.40408i		−	0.436318i
$372$		0			0
$373$	−7.10102	+	7.10102i	−0.367677	+	0.367677i	−0.866629	−	0.498952i	$-0.833718\pi$
$373$	−7.10102	+	7.10102i	−0.367677	+	0.367677i	0.498952	+	0.866629i	$0.333718\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−0.898979	−	0.898979i	−0.0462998	−	0.0462998i
$378$		0			0
$379$	−0.202041			−0.0103782			−0.00518908	−	0.999987i	$-0.501652\pi$
$379$	−0.202041			−0.0103782			−0.00518908	+	0.999987i	$0.501652\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	9.34847	+	9.34847i	0.477684	+	0.477684i	0.904390	−	0.426706i	$-0.140326\pi$
$383$	9.34847	+	9.34847i	0.477684	+	0.477684i	−0.426706	+	0.904390i	$0.640326\pi$
$384$		0			0
$385$	−8.69694	−	7.10102i	−0.443237	−	0.361902i
$386$		0			0
$387$		0			0
$388$		0			0
$389$			35.1464i			1.78199i	0.454009	+	0.890997i	$0.349994\pi$
$389$			35.1464i			1.78199i	−0.454009	+	0.890997i	$0.650006\pi$
$390$		0			0
$391$			0.404082i			0.0204353i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	6.89898	−	0.696938i	0.347125	−	0.0350668i
$396$		0			0
$397$	−27.5959	−	27.5959i	−1.38500	−	1.38500i	−0.835483	−	0.549517i	$-0.814812\pi$
$397$	−27.5959	−	27.5959i	−1.38500	−	1.38500i	−0.549517	−	0.835483i	$-0.685188\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	23.5959			1.17832			0.589162	−	0.808015i	$-0.299458\pi$
$401$	23.5959			1.17832			0.589162	+	0.808015i	$0.299458\pi$
$402$		0			0
$403$	17.7980	+	17.7980i	0.886579	+	0.886579i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−7.10102	+	7.10102i	−0.351985	+	0.351985i
$408$		0			0
$409$			20.0000i			0.988936i	0.869196	+	0.494468i	$0.164637\pi$
$409$			20.0000i			0.988936i	−0.869196	+	0.494468i	$0.835363\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	16.4495	−	16.4495i	0.809426	−	0.809426i
$414$		0			0
$415$	13.5959	+	11.1010i	0.667397	+	0.544928i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−11.7526			−0.574150			−0.287075	−	0.957908i	$-0.592683\pi$
$419$	−11.7526			−0.574150			−0.287075	+	0.957908i	$0.592683\pi$
$420$		0			0
$421$	−10.0000			−0.487370			−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000			−0.487370			−0.243685	+	0.969854i	$0.578356\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	2.65153	+	1.75255i	0.128618	+	0.0850112i
$426$		0			0
$427$	−17.1010	+	17.1010i	−0.827576	+	0.827576i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			20.4949i			0.987204i	0.869688	+	0.493602i	$0.164320\pi$
$431$			20.4949i			0.987204i	−0.869688	+	0.493602i	$0.835680\pi$
$432$		0			0
$433$	2.79796	−	2.79796i	0.134461	−	0.134461i	−0.636673	−	0.771134i	$-0.719690\pi$
$433$	2.79796	−	2.79796i	0.134461	−	0.134461i	0.771134	+	0.636673i	$0.219690\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	0.898979	+	0.898979i	0.0430040	+	0.0430040i
$438$		0			0
$439$	19.7980			0.944905			0.472453	−	0.881356i	$-0.343369\pi$
$439$	19.7980			0.944905			0.472453	+	0.881356i	$0.343369\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	12.8990	+	12.8990i	0.612849	+	0.612849i	0.943687	−	0.330838i	$-0.107332\pi$
$443$	12.8990	+	12.8990i	0.612849	+	0.612849i	−0.330838	+	0.943687i	$0.607332\pi$
$444$		0			0
$445$	2.69694	+	26.6969i	0.127847	+	1.26556i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	20.4949i		−	0.967214i	−0.875285	−	0.483607i	$-0.839326\pi$
$449$		−	20.4949i		−	0.967214i	0.875285	−	0.483607i	$-0.160674\pi$
$450$		0			0
$451$		−	12.0000i		−	0.565058i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	1.30306	+	12.8990i	0.0610885	+	0.604713i
$456$		0			0
$457$	0.797959	+	0.797959i	0.0373269	+	0.0373269i	0.725524	−	0.688197i	$-0.241598\pi$
$457$	0.797959	+	0.797959i	0.0373269	+	0.0373269i	−0.688197	+	0.725524i	$0.741598\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−1.75255			−0.0816245			−0.0408122	−	0.999167i	$-0.512995\pi$
$461$	−1.75255			−0.0816245			−0.0408122	+	0.999167i	$0.512995\pi$
$462$		0			0
$463$	−23.2474	−	23.2474i	−1.08040	−	1.08040i	−0.996472	−	0.0839288i	$-0.973253\pi$
$463$	−23.2474	−	23.2474i	−1.08040	−	1.08040i	−0.0839288	−	0.996472i	$-0.526747\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	21.1464	−	21.1464i	0.978540	−	0.978540i	−0.0212345	−	0.999775i	$-0.506760\pi$
$467$	21.1464	−	21.1464i	0.978540	−	0.978540i	0.999775	+	0.0212345i	$0.00675967\pi$
$468$		0			0
$469$		−	20.0000i		−	0.923514i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−14.6969	+	14.6969i	−0.675766	+	0.675766i
$474$		0			0
$475$	9.79796	−	2.00000i	0.449561	−	0.0917663i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−41.3939			−1.89133			−0.945667	−	0.325136i	$-0.894590\pi$
$479$	−41.3939			−1.89133			−0.945667	+	0.325136i	$0.894590\pi$
$480$		0			0
$481$	11.5959			0.528728
$482$		0			0
$483$		0			0
$484$		0			0
$485$	7.34847	+	6.00000i	0.333677	+	0.272446i
$486$		0			0
$487$	23.4495	−	23.4495i	1.06260	−	1.06260i	0.0646926	−	0.997905i	$-0.479393\pi$
$487$	23.4495	−	23.4495i	1.06260	−	1.06260i	0.997905	−	0.0646926i	$-0.0206067\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	12.6515i		−	0.570956i	−0.958385	−	0.285478i	$-0.907848\pi$
$491$		−	12.6515i		−	0.570956i	0.958385	−	0.285478i	$-0.0921523\pi$
$492$		0			0
$493$	0.202041	−	0.202041i	0.00909947	−	0.00909947i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	17.3939	+	17.3939i	0.780222	+	0.780222i
$498$		0			0
$499$	−29.5959			−1.32490			−0.662448	−	0.749108i	$-0.730482\pi$
$499$	−29.5959			−1.32490			−0.662448	+	0.749108i	$0.730482\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	26.2474	+	26.2474i	1.17032	+	1.17032i	0.982134	+	0.188182i	$0.0602594\pi$
$503$	26.2474	+	26.2474i	1.17032	+	1.17032i	0.188182	+	0.982134i	$0.439741\pi$
$504$		0			0
$505$	−18.7980	+	1.89898i	−0.836498	+	0.0845035i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	16.0454i		−	0.711200i	−0.934638	−	0.355600i	$-0.884277\pi$
$509$		−	16.0454i		−	0.711200i	0.934638	−	0.355600i	$-0.115723\pi$
$510$		0			0
$511$			22.8990i			1.01299i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−1.34847	−	1.10102i	−0.0594207	−	0.0485168i
$516$		0			0
$517$	22.8990	+	22.8990i	1.00710	+	1.00710i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−36.4949			−1.59887			−0.799435	−	0.600752i	$-0.794868\pi$
$521$	−36.4949			−1.59887			−0.799435	+	0.600752i	$0.794868\pi$
$522$		0			0
$523$	26.8990	+	26.8990i	1.17621	+	1.17621i	0.980702	+	0.195508i	$0.0626356\pi$
$523$	26.8990	+	26.8990i	1.17621	+	1.17621i	0.195508	+	0.980702i	$0.437364\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−4.00000	+	4.00000i	−0.174243	+	0.174243i
$528$		0			0
$529$		−	22.5959i		−	0.982431i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−9.79796	+	9.79796i	−0.424397	+	0.424397i
$534$		0			0
$535$	−14.6969	+	18.0000i	−0.635404	+	0.778208i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−6.85357			−0.295204
$540$		0			0
$541$	15.7980			0.679207			0.339604	−	0.940569i	$-0.389707\pi$
$541$	15.7980			0.679207			0.339604	+	0.940569i	$0.389707\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−2.65153	−	26.2474i	−0.113579	−	1.12432i
$546$		0			0
$547$	−25.5959	+	25.5959i	−1.09440	+	1.09440i	−0.0993499	+	0.995053i	$0.531676\pi$
$547$	−25.5959	+	25.5959i	−1.09440	+	1.09440i	−0.995053	+	0.0993499i	$0.968324\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	0.898979i		−	0.0382978i
$552$		0			0
$553$	−4.49490	+	4.49490i	−0.191142	+	0.191142i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−1.10102	−	1.10102i	−0.0466517	−	0.0466517i	0.683396	−	0.730048i	$-0.260502\pi$
$557$	−1.10102	−	1.10102i	−0.0466517	−	0.0466517i	−0.730048	+	0.683396i	$0.760502\pi$
$558$		0			0
$559$	24.0000			1.01509
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−2.20204	−	2.20204i	−0.0928050	−	0.0928050i	0.659180	−	0.751985i	$-0.270903\pi$
$563$	−2.20204	−	2.20204i	−0.0928050	−	0.0928050i	−0.751985	+	0.659180i	$0.770903\pi$
$564$		0			0
$565$	16.8990	−	20.6969i	0.710945	−	0.870727i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			33.3939i			1.39994i	0.714170	+	0.699972i	$0.246804\pi$
$569$			33.3939i			1.39994i	−0.714170	+	0.699972i	$0.753196\pi$
$570$		0			0
$571$			5.59592i			0.234182i	0.993121	+	0.117091i	$0.0373569\pi$
$571$			5.59592i			0.234182i	−0.993121	+	0.117091i	$0.962643\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	2.65153	+	1.75255i	0.110576	+	0.0730864i
$576$		0			0
$577$	11.6969	+	11.6969i	0.486950	+	0.486950i	0.907342	−	0.420392i	$-0.138108\pi$
$577$	11.6969	+	11.6969i	0.486950	+	0.486950i	−0.420392	+	0.907342i	$0.638108\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−16.0908			−0.667560
$582$		0			0
$583$	−7.10102	−	7.10102i	−0.294094	−	0.294094i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	25.1464	−	25.1464i	1.03790	−	1.03790i	0.0386514	−	0.999253i	$-0.487694\pi$
$587$	25.1464	−	25.1464i	1.03790	−	1.03790i	0.999253	−	0.0386514i	$-0.0123062\pi$
$588$		0			0
$589$			17.7980i			0.733352i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	19.5505	−	19.5505i	0.802843	−	0.802843i	−0.180696	−	0.983539i	$-0.557835\pi$
$593$	19.5505	−	19.5505i	0.802843	−	0.802843i	0.983539	+	0.180696i	$0.0578349\pi$
$594$		0			0
$595$	−2.89898	+	0.292856i	−0.118847	+	0.0120059i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−28.4949			−1.16427			−0.582135	−	0.813092i	$-0.697782\pi$
$599$	−28.4949			−1.16427			−0.582135	+	0.813092i	$0.697782\pi$
$600$		0			0
$601$	34.0000			1.38689			0.693444	−	0.720510i	$-0.256092\pi$
$601$	34.0000			1.38689			0.693444	+	0.720510i	$0.256092\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	11.1237	−	1.12372i	0.452244	−	0.0456859i
$606$		0			0
$607$	6.55051	−	6.55051i	0.265877	−	0.265877i	−0.561559	−	0.827436i	$-0.689798\pi$
$607$	6.55051	−	6.55051i	0.265877	−	0.265877i	0.827436	+	0.561559i	$0.189798\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	37.3939i		−	1.51279i
$612$		0			0
$613$	19.5959	−	19.5959i	0.791472	−	0.791472i	−0.190262	−	0.981733i	$-0.560934\pi$
$613$	19.5959	−	19.5959i	0.791472	−	0.791472i	0.981733	+	0.190262i	$0.0609337\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−9.34847	−	9.34847i	−0.376355	−	0.376355i	0.493430	−	0.869785i	$-0.335743\pi$
$617$	−9.34847	−	9.34847i	−0.376355	−	0.376355i	−0.869785	+	0.493430i	$0.835743\pi$
$618$		0			0
$619$	−31.3939			−1.26183			−0.630913	−	0.775853i	$-0.717320\pi$
$619$	−31.3939			−1.26183			−0.630913	+	0.775853i	$0.717320\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−17.3939	−	17.3939i	−0.696871	−	0.696871i
$624$		0			0
$625$	23.0000	−	9.79796i	0.920000	−	0.391918i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			2.60612i			0.103913i
$630$		0			0
$631$			12.8990i			0.513500i	0.966478	+	0.256750i	$0.0826517\pi$
$631$			12.8990i			0.513500i	−0.966478	+	0.256750i	$0.917348\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	14.8990	−	18.2474i	0.591248	−	0.724128i
$636$		0			0
$637$	5.59592	+	5.59592i	0.221718	+	0.221718i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−33.7980			−1.33494			−0.667470	−	0.744637i	$-0.732623\pi$
$641$	−33.7980			−1.33494			−0.667470	+	0.744637i	$0.732623\pi$
$642$		0			0
$643$	23.1010	+	23.1010i	0.911015	+	0.911015i	0.996352	−	0.0853368i	$-0.0271966\pi$
$643$	23.1010	+	23.1010i	0.911015	+	0.911015i	−0.0853368	+	0.996352i	$0.527197\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−10.2474	+	10.2474i	−0.402869	+	0.402869i	−0.879243	−	0.476374i	$-0.841951\pi$
$647$	−10.2474	+	10.2474i	−0.402869	+	0.402869i	0.476374	+	0.879243i	$0.341951\pi$
$648$		0			0
$649$		−	27.7980i		−	1.09117i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	1.55051	−	1.55051i	0.0606762	−	0.0606762i	−0.676118	−	0.736794i	$-0.736339\pi$
$653$	1.55051	−	1.55051i	0.0606762	−	0.0606762i	0.736794	+	0.676118i	$0.236339\pi$
$654$		0			0
$655$	1.44949	+	14.3485i	0.0566363	+	0.560641i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	40.2474			1.56782			0.783909	−	0.620876i	$-0.213223\pi$
$659$	40.2474			1.56782			0.783909	+	0.620876i	$0.213223\pi$
$660$		0			0
$661$	8.20204			0.319022			0.159511	−	0.987196i	$-0.449008\pi$
$661$	8.20204			0.319022			0.159511	+	0.987196i	$0.449008\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−5.79796	+	7.10102i	−0.224835	+	0.275366i
$666$		0			0
$667$	0.202041	−	0.202041i	0.00782306	−	0.00782306i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			28.8990i			1.11563i
$672$		0			0
$673$	8.79796	−	8.79796i	0.339136	−	0.339136i	−0.516906	−	0.856042i	$-0.672916\pi$
$673$	8.79796	−	8.79796i	0.339136	−	0.339136i	0.856042	+	0.516906i	$0.172916\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	4.65153	+	4.65153i	0.178773	+	0.178773i	0.790821	−	0.612048i	$-0.209654\pi$
$677$	4.65153	+	4.65153i	0.178773	+	0.178773i	−0.612048	+	0.790821i	$0.709654\pi$
$678$		0			0
$679$	−8.69694			−0.333758
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−12.6515	−	12.6515i	−0.484097	−	0.484097i	0.422340	−	0.906437i	$-0.361209\pi$
$683$	−12.6515	−	12.6515i	−0.484097	−	0.484097i	−0.906437	+	0.422340i	$0.861209\pi$
$684$		0			0
$685$	−10.8990	−	8.89898i	−0.416429	−	0.340013i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			11.5959i			0.441769i
$690$		0			0
$691$		−	7.79796i		−	0.296648i	−0.988939	−	0.148324i	$-0.952612\pi$
$691$		−	7.79796i		−	0.296648i	0.988939	−	0.148324i	$-0.0473879\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−35.1464	+	3.55051i	−1.33318	+	0.134679i
$696$		0			0
$697$	−2.20204	−	2.20204i	−0.0834083	−	0.0834083i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	26.7423			1.01004			0.505022	−	0.863106i	$-0.331484\pi$
$701$	26.7423			1.01004			0.505022	+	0.863106i	$0.331484\pi$
$702$		0			0
$703$	5.79796	+	5.79796i	0.218674	+	0.218674i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	12.2474	−	12.2474i	0.460613	−	0.460613i
$708$		0			0
$709$		−	8.20204i		−	0.308034i	−0.988068	−	0.154017i	$-0.950779\pi$
$709$		−	8.20204i		−	0.308034i	0.988068	−	0.154017i	$-0.0492211\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−4.00000	+	4.00000i	−0.149801	+	0.149801i
$714$		0			0
$715$	12.0000	+	9.79796i	0.448775	+	0.366423i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	25.7980			0.962102			0.481051	−	0.876693i	$-0.340255\pi$
$719$	25.7980			0.962102			0.481051	+	0.876693i	$0.340255\pi$
$720$		0			0
$721$	1.59592			0.0594351
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−0.449490	−	2.20204i	−0.0166936	−	0.0817818i
$726$		0			0
$727$	−8.14643	+	8.14643i	−0.302134	+	0.302134i	−0.841848	−	0.539714i	$-0.818532\pi$
$727$	−8.14643	+	8.14643i	−0.302134	+	0.302134i	0.539714	+	0.841848i	$0.318532\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			5.39388i			0.199500i
$732$		0			0
$733$	13.1010	−	13.1010i	0.483897	−	0.483897i	−0.422477	−	0.906374i	$-0.638839\pi$
$733$	13.1010	−	13.1010i	0.483897	−	0.483897i	0.906374	+	0.422477i	$0.138839\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−16.8990	−	16.8990i	−0.622482	−	0.622482i
$738$		0			0
$739$	27.3939			1.00770			0.503850	−	0.863791i	$-0.331916\pi$
$739$	27.3939			1.00770			0.503850	+	0.863791i	$0.331916\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−8.44949	−	8.44949i	−0.309982	−	0.309982i	0.534921	−	0.844902i	$-0.320342\pi$
$743$	−8.44949	−	8.44949i	−0.309982	−	0.309982i	−0.844902	+	0.534921i	$0.820342\pi$
$744$		0			0
$745$	4.59592	+	45.4949i	0.168381	+	1.66680i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	21.3031i		−	0.778397i
$750$		0			0
$751$			42.6969i			1.55803i	0.627003	+	0.779017i	$0.284281\pi$
$751$			42.6969i			1.55803i	−0.627003	+	0.779017i	$0.715719\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−0.853572	−	8.44949i	−0.0310647	−	0.307508i
$756$		0			0
$757$	16.6969	+	16.6969i	0.606861	+	0.606861i	0.942124	−	0.335264i	$-0.108825\pi$
$757$	16.6969	+	16.6969i	0.606861	+	0.606861i	−0.335264	+	0.942124i	$0.608825\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	34.6969			1.25776			0.628882	−	0.777501i	$-0.283513\pi$
$761$	34.6969			1.25776			0.628882	+	0.777501i	$0.283513\pi$
$762$		0			0
$763$	17.1010	+	17.1010i	0.619099	+	0.619099i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−22.6969	+	22.6969i	−0.819539	+	0.819539i
$768$		0			0
$769$			25.1918i			0.908441i	0.890889	+	0.454220i	$0.150082\pi$
$769$			25.1918i			0.908441i	−0.890889	+	0.454220i	$0.849918\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−23.3485	+	23.3485i	−0.839786	+	0.839786i	−0.988830	−	0.149044i	$-0.952380\pi$
$773$	−23.3485	+	23.3485i	−0.839786	+	0.839786i	0.149044	+	0.988830i	$0.452380\pi$
$774$		0			0
$775$	8.89898	+	43.5959i	0.319661	+	1.56601i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−9.79796			−0.351048
$780$		0			0
$781$	29.3939			1.05180
$782$		0			0
$783$		0			0
$784$		0			0
$785$	12.0000	+	9.79796i	0.428298	+	0.349704i
$786$		0			0
$787$	−25.1010	+	25.1010i	−0.894755	+	0.894755i	−0.994966	−	0.100211i	$-0.968048\pi$
$787$	−25.1010	+	25.1010i	−0.894755	+	0.894755i	0.100211	+	0.994966i	$0.468048\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			24.4949i			0.870938i
$792$		0			0
$793$	23.5959	−	23.5959i	0.837916	−	0.837916i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−10.4495	−	10.4495i	−0.370140	−	0.370140i	0.497388	−	0.867528i	$-0.334292\pi$
$797$	−10.4495	−	10.4495i	−0.370140	−	0.370140i	−0.867528	+	0.497388i	$0.834292\pi$
$798$		0			0
$799$	8.40408			0.297315
$800$		0			0
$801$		0			0
$802$		0			0
$803$	19.3485	+	19.3485i	0.682793	+	0.682793i
$804$		0			0
$805$	−2.89898	+	0.292856i	−0.102176	+	0.0103218i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	9.79796i		−	0.344478i	−0.985055	−	0.172239i	$-0.944900\pi$
$809$		−	9.79796i		−	0.344478i	0.985055	−	0.172239i	$-0.0551001\pi$
$810$		0			0
$811$			15.7980i			0.554741i	0.960763	+	0.277371i	$0.0894630\pi$
$811$			15.7980i			0.554741i	−0.960763	+	0.277371i	$0.910537\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	21.7980	+	17.7980i	0.763549	+	0.623435i
$816$		0			0
$817$	12.0000	+	12.0000i	0.419827	+	0.419827i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−7.55051			−0.263515			−0.131757	−	0.991282i	$-0.542062\pi$
$821$	−7.55051			−0.263515			−0.131757	+	0.991282i	$0.542062\pi$
$822$		0			0
$823$	−10.3485	−	10.3485i	−0.360725	−	0.360725i	0.503355	−	0.864080i	$-0.332099\pi$
$823$	−10.3485	−	10.3485i	−0.360725	−	0.360725i	−0.864080	+	0.503355i	$0.832099\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−8.00000	+	8.00000i	−0.278187	+	0.278187i	−0.832385	−	0.554198i	$-0.813025\pi$
$827$	−8.00000	+	8.00000i	−0.278187	+	0.278187i	0.554198	+	0.832385i	$0.313025\pi$
$828$		0			0
$829$		−	3.79796i		−	0.131909i	−0.997823	−	0.0659543i	$-0.978991\pi$
$829$		−	3.79796i		−	0.131909i	0.997823	−	0.0659543i	$-0.0210092\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−1.25765	+	1.25765i	−0.0435751	+	0.0435751i
$834$		0			0
$835$	23.1010	−	28.2929i	0.799444	−	0.979115i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	2.69694			0.0931087			0.0465543	−	0.998916i	$-0.485176\pi$
$839$	2.69694			0.0931087			0.0465543	+	0.998916i	$0.485176\pi$
$840$		0			0
$841$	28.7980			0.993033
$842$		0			0
$843$		0			0
$844$		0			0
$845$	1.12372	+	11.1237i	0.0386573	+	0.382668i
$846$		0			0
$847$	−7.24745	+	7.24745i	−0.249025	+	0.249025i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			2.60612i			0.0893367i
$852$		0			0
$853$	−21.7980	+	21.7980i	−0.746348	+	0.746348i	−0.973791	−	0.227443i	$-0.926963\pi$
$853$	−21.7980	+	21.7980i	−0.746348	+	0.746348i	0.227443	+	0.973791i	$0.426963\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	23.5505	+	23.5505i	0.804470	+	0.804470i	0.983791	−	0.179321i	$-0.0573899\pi$
$857$	23.5505	+	23.5505i	0.804470	+	0.804470i	−0.179321	+	0.983791i	$0.557390\pi$
$858$		0			0
$859$	−4.20204			−0.143372			−0.0716859	−	0.997427i	$-0.522838\pi$
$859$	−4.20204			−0.143372			−0.0716859	+	0.997427i	$0.522838\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	21.3485	+	21.3485i	0.726710	+	0.726710i	0.969963	−	0.243253i	$-0.0782144\pi$
$863$	21.3485	+	21.3485i	0.726710	+	0.726710i	−0.243253	+	0.969963i	$0.578214\pi$
$864$		0			0
$865$	32.4949	−	39.7980i	1.10486	−	1.35317i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			7.59592i			0.257674i
$870$		0			0
$871$			27.5959i			0.935052i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−10.6515	+	20.2929i	−0.360087	+	0.686024i
$876$		0			0
$877$	−7.10102	−	7.10102i	−0.239784	−	0.239784i	0.576976	−	0.816761i	$-0.304232\pi$
$877$	−7.10102	−	7.10102i	−0.239784	−	0.239784i	−0.816761	+	0.576976i	$0.804232\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	53.7980			1.81250			0.906250	−	0.422743i	$-0.138933\pi$
$881$	53.7980			1.81250			0.906250	+	0.422743i	$0.138933\pi$
$882$		0			0
$883$	22.6969	+	22.6969i	0.763813	+	0.763813i	0.977009	−	0.213197i	$-0.0683875\pi$
$883$	22.6969	+	22.6969i	0.763813	+	0.763813i	−0.213197	+	0.977009i	$0.568387\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−3.14643	+	3.14643i	−0.105647	+	0.105647i	−0.757954	−	0.652308i	$-0.773801\pi$
$887$	−3.14643	+	3.14643i	−0.105647	+	0.105647i	0.652308	+	0.757954i	$0.273801\pi$
$888$		0			0
$889$			21.5959i			0.724304i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	18.6969	−	18.6969i	0.625669	−	0.625669i
$894$		0			0
$895$	−53.9444	+	5.44949i	−1.80316	+	0.182156i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	4.00000			0.133407
$900$		0			0
$901$	−2.60612			−0.0868225
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−22.2474	+	2.24745i	−0.739530	+	0.0747077i
$906$		0			0
$907$	9.79796	−	9.79796i	0.325336	−	0.325336i	−0.525474	−	0.850810i	$-0.676112\pi$
$907$	9.79796	−	9.79796i	0.325336	−	0.325336i	0.850810	+	0.525474i	$0.176112\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	40.0908i		−	1.32827i	−0.747614	−	0.664134i	$-0.768800\pi$
$911$		−	40.0908i		−	1.32827i	0.747614	−	0.664134i	$-0.231200\pi$
$912$		0			0
$913$	−13.5959	+	13.5959i	−0.449959	+	0.449959i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−9.34847	−	9.34847i	−0.308714	−	0.308714i
$918$		0			0
$919$	40.8990			1.34913			0.674566	−	0.738214i	$-0.264331\pi$
$919$	40.8990			1.34913			0.674566	+	0.738214i	$0.264331\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−24.0000	−	24.0000i	−0.789970	−	0.789970i
$924$		0			0
$925$	17.1010	+	11.3031i	0.562278	+	0.371642i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	56.0908i		−	1.84028i	−0.391590	−	0.920140i	$-0.628075\pi$
$929$		−	56.0908i		−	1.84028i	0.391590	−	0.920140i	$-0.371925\pi$
$930$		0			0
$931$			5.59592i			0.183399i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−2.20204	+	2.69694i	−0.0720144	+	0.0881993i
$936$		0			0
$937$	−22.5959	−	22.5959i	−0.738176	−	0.738176i	0.234048	−	0.972225i	$-0.424803\pi$
$937$	−22.5959	−	22.5959i	−0.738176	−	0.738176i	−0.972225	+	0.234048i	$0.924803\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	27.1464			0.884948			0.442474	−	0.896781i	$-0.354101\pi$
$941$	27.1464			0.884948			0.442474	+	0.896781i	$0.354101\pi$
$942$		0			0
$943$	−2.20204	−	2.20204i	−0.0717083	−	0.0717083i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−20.4949	+	20.4949i	−0.665995	+	0.665995i	−0.956786	−	0.290792i	$-0.906081\pi$
$947$	−20.4949	+	20.4949i	−0.665995	+	0.665995i	0.290792	+	0.956786i	$0.406081\pi$
$948$		0			0
$949$		−	31.5959i		−	1.02565i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−1.34847	+	1.34847i	−0.0436812	+	0.0436812i	−0.728610	−	0.684929i	$-0.759833\pi$
$953$	−1.34847	+	1.34847i	−0.0436812	+	0.0436812i	0.684929	+	0.728610i	$0.259833\pi$
$954$		0			0
$955$	−4.69694	−	46.4949i	−0.151989	−	1.50454i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	12.8990			0.416530
$960$		0			0
$961$	−48.1918			−1.55458
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−19.7980	+	24.2474i	−0.637319	+	0.780553i
$966$		0			0
$967$	−6.34847	+	6.34847i	−0.204153	+	0.204153i	−0.801777	−	0.597624i	$-0.796112\pi$
$967$	−6.34847	+	6.34847i	−0.204153	+	0.204153i	0.597624	+	0.801777i	$0.296112\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	6.04541i		−	0.194006i	−0.995284	−	0.0970032i	$-0.969074\pi$
$971$		−	6.04541i		−	0.194006i	0.995284	−	0.0970032i	$-0.0309257\pi$
$972$		0			0
$973$	22.8990	−	22.8990i	0.734108	−	0.734108i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	20.4495	+	20.4495i	0.654237	+	0.654237i	0.954010	−	0.299773i	$-0.0969110\pi$
$977$	20.4495	+	20.4495i	0.654237	+	0.654237i	−0.299773	+	0.954010i	$0.596911\pi$
$978$		0			0
$979$	−29.3939			−0.939432
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−26.2474	−	26.2474i	−0.837164	−	0.837164i	0.151321	−	0.988485i	$-0.451647\pi$
$983$	−26.2474	−	26.2474i	−0.837164	−	0.837164i	−0.988485	+	0.151321i	$0.951647\pi$
$984$		0			0
$985$	−19.1010	−	15.5959i	−0.608609	−	0.496927i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			5.39388i			0.171515i
$990$		0			0
$991$		−	24.2020i		−	0.768803i	−0.923166	−	0.384402i	$-0.874408\pi$
$991$		−	24.2020i		−	0.768803i	0.923166	−	0.384402i	$-0.125592\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	35.1464	−	3.55051i	1.11422	−	0.112559i
$996$		0			0
$997$	20.2020	+	20.2020i	0.639805	+	0.639805i	0.950507	−	0.310702i	$-0.100564\pi$
$997$	20.2020	+	20.2020i	0.639805	+	0.639805i	−0.310702	+	0.950507i	$0.600564\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1440.2.x.p.127.2	yes	4
3.2	odd	2		1440.2.x.l.127.1	yes	4
4.3	odd	2		1440.2.x.o.127.2	yes	4
5.3	odd	4		1440.2.x.o.703.2	yes	4
12.11	even	2		1440.2.x.k.127.1	✓	4
15.8	even	4		1440.2.x.k.703.1	yes	4
20.3	even	4	inner	1440.2.x.p.703.2	yes	4
60.23	odd	4		1440.2.x.l.703.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1440.2.x.k.127.1	✓	4	12.11	even	2
1440.2.x.k.703.1	yes	4	15.8	even	4
1440.2.x.l.127.1	yes	4	3.2	odd	2
1440.2.x.l.703.1	yes	4	60.23	odd	4
1440.2.x.o.127.2	yes	4	4.3	odd	2
1440.2.x.o.703.2	yes	4	5.3	odd	4
1440.2.x.p.127.2	yes	4	1.1	even	1	trivial
1440.2.x.p.703.2	yes	4	20.3	even	4	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1440 = 2^{5} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1440.x (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(2.22474 - 0.224745i) q^{5} +(-1.44949 + 1.44949i) q^{7} +O(q^{10})\)	\(q+(2.22474 - 0.224745i) q^{5} +(-1.44949 + 1.44949i) q^{7} +2.44949i q^{11} +(2.00000 - 2.00000i) q^{13} +(0.449490 + 0.449490i) q^{17} +2.00000 q^{19} +(0.449490 + 0.449490i) q^{23} +(4.89898 - 1.00000i) q^{25} -0.449490i q^{29} +8.89898i q^{31} +(-2.89898 + 3.55051i) q^{35} +(2.89898 + 2.89898i) q^{37} -4.89898 q^{41} +(6.00000 + 6.00000i) q^{43} +(9.34847 - 9.34847i) q^{47} +2.79796i q^{49} +(-2.89898 + 2.89898i) q^{53} +(0.550510 + 5.44949i) q^{55} -11.3485 q^{59} +11.7980 q^{61} +(4.00000 - 4.89898i) q^{65} +(-6.89898 + 6.89898i) q^{67} -12.0000i q^{71} +(7.89898 - 7.89898i) q^{73} +(-3.55051 - 3.55051i) q^{77} +3.10102 q^{79} +(5.55051 + 5.55051i) q^{83} +(1.10102 + 0.898979i) q^{85} +12.0000i q^{89} +5.79796i q^{91} +(4.44949 - 0.449490i) q^{95} +(3.00000 + 3.00000i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{5} + 4 q^{7}+O(q^{10}) \) 4 * q + 4 * q^5 + 4 * q^7	\( 4 q + 4 q^{5} + 4 q^{7} + 8 q^{13} - 8 q^{17} + 8 q^{19} - 8 q^{23} + 8 q^{35} - 8 q^{37} + 24 q^{43} + 8 q^{47} + 8 q^{53} + 12 q^{55} - 16 q^{59} + 8 q^{61} + 16 q^{65} - 8 q^{67} + 12 q^{73} - 24 q^{77} + 32 q^{79} + 32 q^{83} + 24 q^{85} + 8 q^{95} + 12 q^{97}+O(q^{100}) \) 4 * q + 4 * q^5 + 4 * q^7 + 8 * q^13 - 8 * q^17 + 8 * q^19 - 8 * q^23 + 8 * q^35 - 8 * q^37 + 24 * q^43 + 8 * q^47 + 8 * q^53 + 12 * q^55 - 16 * q^59 + 8 * q^61 + 16 * q^65 - 8 * q^67 + 12 * q^73 - 24 * q^77 + 32 * q^79 + 32 * q^83 + 24 * q^85 + 8 * q^95 + 12 * q^97

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