LMFDB - Embedded newform 1152.3.m.a.991.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1152,3,Mod(415,1152)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1152, base_ring=CyclotomicField(4))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1152.415");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1152 = 2^{7} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1152.m (of order $4$, degree $2$, not 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:	$31.3897264543$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(i)$
Coefficient field:	6.0.399424.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8 $ x^6 - 2x^5 + 3x^4 - 6x^3 + 6x^2 - 8*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 16)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			991.2
Root			$0.264658 + 1.38923i$ of defining polynomial
Character	$\chi$	$=$	1152.991
Dual form			1152.3.m.a.415.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.0586332 + 0.0586332i) q^{5} +4.61555 q^{7} +O(q^{10})$	$q+(-0.0586332 + 0.0586332i) q^{5} +4.61555 q^{7} +(-5.36641 - 5.36641i) q^{11} +(-11.0552 - 11.0552i) q^{13} +12.8793 q^{17} +(-2.63359 + 2.63359i) q^{19} -16.3810 q^{23} +24.9931i q^{25} +(-26.0518 - 26.0518i) q^{29} -20.2345i q^{31} +(-0.270624 + 0.270624i) q^{35} +(-41.2829 + 41.2829i) q^{37} +3.29640i q^{41} +(0.786951 + 0.786951i) q^{43} -79.7517i q^{47} -27.6967 q^{49} +(1.06207 - 1.06207i) q^{53} +0.629299 q^{55} +(32.5163 + 32.5163i) q^{59} +(-15.2897 - 15.2897i) q^{61} +1.29640 q^{65} +(60.0631 - 60.0631i) q^{67} -56.3535 q^{71} +9.70663i q^{73} +(-24.7689 - 24.7689i) q^{77} -84.4278i q^{79} +(26.7577 - 26.7577i) q^{83} +(-0.755154 + 0.755154i) q^{85} +115.555i q^{89} +(-51.0258 - 51.0258i) q^{91} -0.308832i q^{95} -146.245 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{5} - 4 q^{7}+O(q^{10}) $ 6 * q - 2 * q^5 - 4 * q^7	$ 6 q - 2 q^{5} - 4 q^{7} - 18 q^{11} + 2 q^{13} + 4 q^{17} - 30 q^{19} - 60 q^{23} - 18 q^{29} - 100 q^{35} - 46 q^{37} + 114 q^{43} - 46 q^{49} + 78 q^{53} + 252 q^{55} + 206 q^{59} - 30 q^{61} - 12 q^{65} + 226 q^{67} + 260 q^{71} - 212 q^{77} + 318 q^{83} + 212 q^{85} - 188 q^{91} - 4 q^{97}+O(q^{100}) $ 6 * q - 2 * q^5 - 4 * q^7 - 18 * q^11 + 2 * q^13 + 4 * q^17 - 30 * q^19 - 60 * q^23 - 18 * q^29 - 100 * q^35 - 46 * q^37 + 114 * q^43 - 46 * q^49 + 78 * q^53 + 252 * q^55 + 206 * q^59 - 30 * q^61 - 12 * q^65 + 226 * q^67 + 260 * q^71 - 212 * q^77 + 318 * q^83 + 212 * q^85 - 188 * q^91 - 4 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times$.

$n$	$127$	$641$	$901$
$\chi(n)$	$-1$	$1$	$e\left(\frac{3}{4}\right)$

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
$5$	−0.0586332	+	0.0586332i	−0.0117266	+	0.0117266i	−0.712946	−	0.701219i	$-0.752639\pi$
$5$	−0.0586332	+	0.0586332i	−0.0117266	+	0.0117266i	0.701219	+	0.712946i	$0.252639\pi$
$6$		0			0
$7$	4.61555			0.659364			0.329682	−	0.944092i	$-0.393058\pi$
$7$	4.61555			0.659364			0.329682	+	0.944092i	$0.393058\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−5.36641	−	5.36641i	−0.487855	−	0.487855i	0.419774	−	0.907629i	$-0.362109\pi$
$11$	−5.36641	−	5.36641i	−0.487855	−	0.487855i	−0.907629	+	0.419774i	$0.862109\pi$
$12$		0			0
$13$	−11.0552	−	11.0552i	−0.850400	−	0.850400i	0.139783	−	0.990182i	$-0.455360\pi$
$13$	−11.0552	−	11.0552i	−0.850400	−	0.850400i	−0.990182	+	0.139783i	$0.955360\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	12.8793			0.757606			0.378803	−	0.925477i	$-0.376336\pi$
$17$	12.8793			0.757606			0.378803	+	0.925477i	$0.376336\pi$
$18$		0			0
$19$	−2.63359	+	2.63359i	−0.138610	+	0.138610i	−0.773007	−	0.634397i	$-0.781248\pi$
$19$	−2.63359	+	2.63359i	−0.138610	+	0.138610i	0.634397	+	0.773007i	$0.281248\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−16.3810			−0.712218			−0.356109	−	0.934444i	$-0.615897\pi$
$23$	−16.3810			−0.712218			−0.356109	+	0.934444i	$0.615897\pi$
$24$		0			0
$25$			24.9931i			0.999725i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−26.0518	−	26.0518i	−0.898336	−	0.898336i	0.0969525	−	0.995289i	$-0.469090\pi$
$29$	−26.0518	−	26.0518i	−0.898336	−	0.898336i	−0.995289	+	0.0969525i	$0.969090\pi$
$30$		0			0
$31$		−	20.2345i		−	0.652727i	−0.945244	−	0.326363i	$-0.894177\pi$
$31$		−	20.2345i		−	0.652727i	0.945244	−	0.326363i	$-0.105823\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−0.270624	+	0.270624i	−0.00773212	+	0.00773212i
$36$		0			0
$37$	−41.2829	+	41.2829i	−1.11575	+	1.11575i	−0.123395	+	0.992358i	$0.539378\pi$
$37$	−41.2829	+	41.2829i	−1.11575	+	1.11575i	−0.992358	+	0.123395i	$0.960622\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			3.29640i			0.0804001i	0.999192	+	0.0402000i	$0.0127995\pi$
$41$			3.29640i			0.0804001i	−0.999192	+	0.0402000i	$0.987200\pi$
$42$		0			0
$43$	0.786951	+	0.786951i	0.0183012	+	0.0183012i	0.716198	−	0.697897i	$-0.245881\pi$
$43$	0.786951	+	0.786951i	0.0183012	+	0.0183012i	−0.697897	+	0.716198i	$0.745881\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	79.7517i		−	1.69685i	−0.529320	−	0.848423i	$-0.677553\pi$
$47$		−	79.7517i		−	1.69685i	0.529320	−	0.848423i	$-0.322447\pi$
$48$		0			0
$49$	−27.6967			−0.565239
$50$		0			0
$51$		0			0
$52$		0			0
$53$	1.06207	−	1.06207i	0.0200391	−	0.0200391i	−0.697016	−	0.717055i	$-0.745489\pi$
$53$	1.06207	−	1.06207i	0.0200391	−	0.0200391i	0.717055	+	0.697016i	$0.245489\pi$
$54$		0			0
$55$	0.629299			0.0114418
$56$		0			0
$57$		0			0
$58$		0			0
$59$	32.5163	+	32.5163i	0.551124	+	0.551124i	0.926765	−	0.375641i	$-0.122577\pi$
$59$	32.5163	+	32.5163i	0.551124	+	0.551124i	−0.375641	+	0.926765i	$0.622577\pi$
$60$		0			0
$61$	−15.2897	−	15.2897i	−0.250651	−	0.250651i	0.570586	−	0.821238i	$-0.306716\pi$
$61$	−15.2897	−	15.2897i	−0.250651	−	0.250651i	−0.821238	+	0.570586i	$0.806716\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	1.29640			0.0199446
$66$		0			0
$67$	60.0631	−	60.0631i	0.896465	−	0.896465i	−0.0986569	−	0.995122i	$-0.531455\pi$
$67$	60.0631	−	60.0631i	0.896465	−	0.896465i	0.995122	+	0.0986569i	$0.0314546\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−56.3535			−0.793711			−0.396856	−	0.917881i	$-0.629899\pi$
$71$	−56.3535			−0.793711			−0.396856	+	0.917881i	$0.629899\pi$
$72$		0			0
$73$			9.70663i			0.132968i	0.997788	+	0.0664838i	$0.0211781\pi$
$73$			9.70663i			0.132968i	−0.997788	+	0.0664838i	$0.978822\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−24.7689	−	24.7689i	−0.321674	−	0.321674i
$78$		0			0
$79$		−	84.4278i		−	1.06871i	−0.845261	−	0.534353i	$-0.820555\pi$
$79$		−	84.4278i		−	1.06871i	0.845261	−	0.534353i	$-0.179445\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	26.7577	−	26.7577i	0.322382	−	0.322382i	−0.527298	−	0.849680i	$-0.676795\pi$
$83$	26.7577	−	26.7577i	0.322382	−	0.322382i	0.849680	+	0.527298i	$0.176795\pi$
$84$		0			0
$85$	−0.755154	+	0.755154i	−0.00888416	+	0.00888416i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			115.555i			1.29838i	0.760628	+	0.649188i	$0.224891\pi$
$89$			115.555i			1.29838i	−0.760628	+	0.649188i	$0.775109\pi$
$90$		0			0
$91$	−51.0258	−	51.0258i	−0.560723	−	0.560723i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		−	0.308832i		−	0.00325086i
$96$		0			0
$97$	−146.245			−1.50768			−0.753841	−	0.657056i	$-0.771801\pi$
$97$	−146.245			−1.50768			−0.753841	+	0.657056i	$0.771801\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−53.8554	+	53.8554i	−0.533222	+	0.533222i	−0.921530	−	0.388308i	$-0.873060\pi$
$101$	−53.8554	+	53.8554i	−0.533222	+	0.533222i	0.388308	+	0.921530i	$0.373060\pi$
$102$		0			0
$103$	−158.184			−1.53577			−0.767885	−	0.640588i	$-0.778691\pi$
$103$	−158.184			−1.53577			−0.767885	+	0.640588i	$0.778691\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−57.6009	−	57.6009i	−0.538327	−	0.538327i	0.384711	−	0.923037i	$-0.374301\pi$
$107$	−57.6009	−	57.6009i	−0.538327	−	0.538327i	−0.923037	+	0.384711i	$0.874301\pi$
$108$		0			0
$109$	−56.8795	−	56.8795i	−0.521830	−	0.521830i	0.396294	−	0.918124i	$-0.370296\pi$
$109$	−56.8795	−	56.8795i	−0.521830	−	0.521830i	−0.918124	+	0.396294i	$0.870296\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−135.731			−1.20116			−0.600580	−	0.799565i	$-0.705064\pi$
$113$	−135.731			−1.20116			−0.600580	+	0.799565i	$0.705064\pi$
$114$		0			0
$115$	0.960471	−	0.960471i	0.00835192	−	0.00835192i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	59.4450			0.499538
$120$		0			0
$121$		−	63.4034i		−	0.523995i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−2.93125	−	2.93125i	−0.0234500	−	0.0234500i
$126$		0			0
$127$		−	166.552i		−	1.31144i	−0.755006	−	0.655718i	$-0.772366\pi$
$127$		−	166.552i		−	1.31144i	0.755006	−	0.655718i	$-0.227634\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	22.2547	−	22.2547i	0.169883	−	0.169883i	−0.617045	−	0.786928i	$-0.711670\pi$
$131$	22.2547	−	22.2547i	0.169883	−	0.169883i	0.786928	+	0.617045i	$0.211670\pi$
$132$		0			0
$133$	−12.1555	+	12.1555i	−0.0913945	+	0.0913945i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	174.890i		−	1.27657i	−0.769800	−	0.638285i	$-0.779644\pi$
$137$		−	174.890i		−	1.27657i	0.769800	−	0.638285i	$-0.220356\pi$
$138$		0			0
$139$	−99.8891	−	99.8891i	−0.718627	−	0.718627i	0.249697	−	0.968324i	$-0.419669\pi$
$139$	−99.8891	−	99.8891i	−0.718627	−	0.718627i	−0.968324	+	0.249697i	$0.919669\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			118.653i			0.829744i
$144$		0			0
$145$	3.05499			0.0210689
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−74.8860	+	74.8860i	−0.502590	+	0.502590i	−0.912242	−	0.409652i	$-0.865650\pi$
$149$	−74.8860	+	74.8860i	−0.502590	+	0.502590i	0.409652	+	0.912242i	$0.365650\pi$
$150$		0			0
$151$	70.0357			0.463813			0.231906	−	0.972738i	$-0.425504\pi$
$151$	70.0357			0.463813			0.231906	+	0.972738i	$0.425504\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	1.18641	+	1.18641i	0.00765429	+	0.00765429i
$156$		0			0
$157$	29.5307	+	29.5307i	0.188094	+	0.188094i	0.794872	−	0.606778i	$-0.207538\pi$
$157$	29.5307	+	29.5307i	0.188094	+	0.188094i	−0.606778	+	0.794872i	$0.707538\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−75.6074			−0.469611
$162$		0			0
$163$	−47.7990	+	47.7990i	−0.293245	+	0.293245i	−0.838361	−	0.545116i	$-0.816486\pi$
$163$	−47.7990	+	47.7990i	−0.293245	+	0.293245i	0.545116	+	0.838361i	$0.316486\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	156.268			0.935734			0.467867	−	0.883799i	$-0.345023\pi$
$167$	156.268			0.935734			0.467867	+	0.883799i	$0.345023\pi$
$168$		0			0
$169$			75.4347i			0.446359i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	190.103	+	190.103i	1.09886	+	1.09886i	0.994544	+	0.104319i	$0.0332664\pi$
$173$	190.103	+	190.103i	1.09886	+	1.09886i	0.104319	+	0.994544i	$0.466734\pi$
$174$		0			0
$175$			115.357i			0.659183i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	54.2749	−	54.2749i	0.303212	−	0.303212i	−0.539057	−	0.842269i	$-0.681219\pi$
$179$	54.2749	−	54.2749i	0.303212	−	0.303212i	0.842269	+	0.539057i	$0.181219\pi$
$180$		0			0
$181$	−19.7343	+	19.7343i	−0.109029	+	0.109029i	−0.759517	−	0.650487i	$-0.774565\pi$
$181$	−19.7343	+	19.7343i	−0.109029	+	0.109029i	0.650487	+	0.759517i	$0.274565\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	4.84109i		−	0.0261680i
$186$		0			0
$187$	−69.1155	−	69.1155i	−0.369602	−	0.369602i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	166.552i		−	0.872002i	−0.899946	−	0.436001i	$-0.856394\pi$
$191$		−	166.552i		−	0.872002i	0.899946	−	0.436001i	$-0.143606\pi$
$192$		0			0
$193$	2.18257			0.0113087			0.00565434	−	0.999984i	$-0.498200\pi$
$193$	2.18257			0.0113087			0.00565434	+	0.999984i	$0.498200\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	67.4310	−	67.4310i	0.342290	−	0.342290i	−0.514938	−	0.857227i	$-0.672185\pi$
$197$	67.4310	−	67.4310i	0.342290	−	0.342290i	0.857227	+	0.514938i	$0.172185\pi$
$198$		0			0
$199$	222.906			1.12013			0.560065	−	0.828449i	$-0.310776\pi$
$199$	222.906			1.12013			0.560065	+	0.828449i	$0.310776\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−120.243	−	120.243i	−0.592331	−	0.592331i
$204$		0			0
$205$	−0.193278	−	0.193278i	−0.000942822	−	0.000942822i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	28.2659			0.135243
$210$		0			0
$211$	147.118	−	147.118i	0.697240	−	0.697240i	−0.266574	−	0.963814i	$-0.585892\pi$
$211$	147.118	−	147.118i	0.697240	−	0.697240i	0.963814	+	0.266574i	$0.0858917\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−0.0922828			−0.000429223
$216$		0			0
$217$		−	93.3934i		−	0.430385i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−142.383	−	142.383i	−0.644268	−	0.644268i
$222$		0			0
$223$			60.7036i			0.272213i	0.990694	+	0.136107i	$0.0434590\pi$
$223$			60.7036i			0.272213i	−0.990694	+	0.136107i	$0.956541\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−225.526	+	225.526i	−0.993505	+	0.993505i	−0.999979	−	0.00647371i	$-0.997939\pi$
$227$	−225.526	+	225.526i	−0.993505	+	0.993505i	0.00647371	+	0.999979i	$0.497939\pi$
$228$		0			0
$229$	−227.796	+	227.796i	−0.994743	+	0.994743i	−0.999986	−	0.00524305i	$-0.998331\pi$
$229$	−227.796	+	227.796i	−0.994743	+	0.994743i	0.00524305	+	0.999986i	$0.498331\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			121.053i			0.519540i	0.965671	+	0.259770i	$0.0836467\pi$
$233$			121.053i			0.519540i	−0.965671	+	0.259770i	$0.916353\pi$
$234$		0			0
$235$	4.67610	+	4.67610i	0.0198983	+	0.0198983i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			221.393i			0.926332i	0.886271	+	0.463166i	$0.153287\pi$
$239$			221.393i			0.926332i	−0.886271	+	0.463166i	$0.846713\pi$
$240$		0			0
$241$	84.2667			0.349654			0.174827	−	0.984599i	$-0.444063\pi$
$241$	84.2667			0.349654			0.174827	+	0.984599i	$0.444063\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	1.62395	−	1.62395i	0.00662835	−	0.00662835i
$246$		0			0
$247$	58.2298			0.235748
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−176.615	−	176.615i	−0.703646	−	0.703646i	0.261545	−	0.965191i	$-0.415768\pi$
$251$	−176.615	−	176.615i	−0.703646	−	0.703646i	−0.965191	+	0.261545i	$0.915768\pi$
$252$		0			0
$253$	87.9072	+	87.9072i	0.347459	+	0.347459i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	163.001			0.634244			0.317122	−	0.948385i	$-0.397283\pi$
$257$	163.001			0.634244			0.317122	+	0.948385i	$0.397283\pi$
$258$		0			0
$259$	−190.543	+	190.543i	−0.735687	+	0.735687i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−175.001			−0.665404			−0.332702	−	0.943032i	$-0.607960\pi$
$263$	−175.001			−0.665404			−0.332702	+	0.943032i	$0.607960\pi$
$264$		0			0
$265$			0.124545i			0.000469982i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	29.7489	+	29.7489i	0.110591	+	0.110591i	0.760237	−	0.649646i	$-0.225083\pi$
$269$	29.7489	+	29.7489i	0.110591	+	0.110591i	−0.649646	+	0.760237i	$0.725083\pi$
$270$		0			0
$271$			275.891i			1.01805i	0.860753	+	0.509024i	$0.169993\pi$
$271$			275.891i			1.01805i	−0.860753	+	0.509024i	$0.830007\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	134.123	−	134.123i	0.487721	−	0.487721i
$276$		0			0
$277$	278.337	−	278.337i	1.00483	−	1.00483i	0.00484003	−	0.999988i	$-0.498459\pi$
$277$	278.337	−	278.337i	1.00483	−	1.00483i	0.999988	−	0.00484003i	$-0.00154063\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	202.356i		−	0.720128i	−0.932928	−	0.360064i	$-0.882755\pi$
$281$		−	202.356i		−	0.720128i	0.932928	−	0.360064i	$-0.117245\pi$
$282$		0			0
$283$	292.256	+	292.256i	1.03271	+	1.03271i	0.999447	+	0.0332615i	$0.0105894\pi$
$283$	292.256	+	292.256i	1.03271	+	1.03271i	0.0332615	+	0.999447i	$0.489411\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			15.2147i			0.0530129i
$288$		0			0
$289$	−123.124			−0.426034
$290$		0			0
$291$		0			0
$292$		0			0
$293$	331.170	−	331.170i	1.13027	−	1.13027i	0.140141	−	0.990132i	$-0.455244\pi$
$293$	331.170	−	331.170i	1.13027	−	1.13027i	0.990132	−	0.140141i	$-0.0447555\pi$
$294$		0			0
$295$	−3.81307			−0.0129257
$296$		0			0
$297$		0			0
$298$		0			0
$299$	181.095	+	181.095i	0.605670	+	0.605670i
$300$		0			0
$301$	3.63221	+	3.63221i	0.0120671	+	0.0120671i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	1.79297			0.00587859
$306$		0			0
$307$	−23.7513	+	23.7513i	−0.0773656	+	0.0773656i	−0.744731	−	0.667365i	$-0.767422\pi$
$307$	−23.7513	+	23.7513i	−0.0773656	+	0.0773656i	0.667365	+	0.744731i	$0.267422\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	157.757			0.507258			0.253629	−	0.967302i	$-0.418376\pi$
$311$	157.757			0.507258			0.253629	+	0.967302i	$0.418376\pi$
$312$		0			0
$313$			58.5936i			0.187200i	0.995610	+	0.0936000i	$0.0298375\pi$
$313$			58.5936i			0.187200i	−0.995610	+	0.0936000i	$0.970163\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	27.0040	+	27.0040i	0.0851863	+	0.0851863i	0.748416	−	0.663230i	$-0.230815\pi$
$317$	27.0040	+	27.0040i	0.0851863	+	0.0851863i	−0.663230	+	0.748416i	$0.730815\pi$
$318$		0			0
$319$			279.609i			0.876516i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−33.9188	+	33.9188i	−0.105012	+	0.105012i
$324$		0			0
$325$	276.304	−	276.304i	0.850166	−	0.850166i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	368.098i		−	1.11884i
$330$		0			0
$331$	182.195	+	182.195i	0.550437	+	0.550437i	0.926567	−	0.376130i	$-0.122745\pi$
$331$	182.195	+	182.195i	0.550437	+	0.550437i	−0.376130	+	0.926567i	$0.622745\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			7.04338i			0.0210250i
$336$		0			0
$337$	510.137			1.51376			0.756881	−	0.653553i	$-0.226722\pi$
$337$	510.137			1.51376			0.756881	+	0.653553i	$0.226722\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−108.587	+	108.587i	−0.318436	+	0.318436i
$342$		0			0
$343$	−353.997			−1.03206
$344$		0			0
$345$		0			0
$346$		0			0
$347$	432.614	+	432.614i	1.24673	+	1.24673i	0.957157	+	0.289570i	$0.0935122\pi$
$347$	432.614	+	432.614i	1.24673	+	1.24673i	0.289570	+	0.957157i	$0.406488\pi$
$348$		0			0
$349$	148.839	+	148.839i	0.426472	+	0.426472i	0.887425	−	0.460953i	$-0.152492\pi$
$349$	148.839	+	148.839i	0.426472	+	0.426472i	−0.460953	+	0.887425i	$0.652492\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	268.587			0.760869			0.380434	−	0.924808i	$-0.375774\pi$
$353$	268.587			0.760869			0.380434	+	0.924808i	$0.375774\pi$
$354$		0			0
$355$	3.30418	−	3.30418i	0.00930756	−	0.00930756i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−628.520			−1.75075			−0.875376	−	0.483442i	$-0.839386\pi$
$359$	−628.520			−1.75075			−0.875376	+	0.483442i	$0.839386\pi$
$360$		0			0
$361$			347.128i			0.961574i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−0.569131	−	0.569131i	−0.00155926	−	0.00155926i
$366$		0			0
$367$		−	396.386i		−	1.08007i	−0.841643	−	0.540035i	$-0.818411\pi$
$367$		−	396.386i		−	1.08007i	0.841643	−	0.540035i	$-0.181589\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	4.90204	−	4.90204i	0.0132130	−	0.0132130i
$372$		0			0
$373$	−134.275	+	134.275i	−0.359987	+	0.359987i	−0.863808	−	0.503821i	$-0.831927\pi$
$373$	−134.275	+	134.275i	−0.359987	+	0.359987i	0.503821	+	0.863808i	$0.331927\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			576.015i			1.52789i
$378$		0			0
$379$	350.491	+	350.491i	0.924777	+	0.924777i	0.997362	−	0.0725851i	$-0.0231249\pi$
$379$	350.491	+	350.491i	0.924777	+	0.924777i	−0.0725851	+	0.997362i	$0.523125\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	403.778i		−	1.05425i	−0.849787	−	0.527126i	$-0.823270\pi$
$383$		−	403.778i		−	1.05425i	0.849787	−	0.527126i	$-0.176730\pi$
$384$		0			0
$385$	2.90456			0.00754431
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−125.310	+	125.310i	−0.322134	+	0.322134i	−0.849585	−	0.527452i	$-0.823148\pi$
$389$	−125.310	+	125.310i	−0.322134	+	0.322134i	0.527452	+	0.849585i	$0.323148\pi$
$390$		0			0
$391$	−210.976			−0.539580
$392$		0			0
$393$		0			0
$394$		0			0
$395$	4.95027	+	4.95027i	0.0125323	+	0.0125323i
$396$		0			0
$397$	69.8722	+	69.8722i	0.176001	+	0.176001i	0.789610	−	0.613609i	$-0.210283\pi$
$397$	69.8722	+	69.8722i	0.176001	+	0.176001i	−0.613609	+	0.789610i	$0.710283\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−11.3010			−0.0281821			−0.0140911	−	0.999901i	$-0.504485\pi$
$401$	−11.3010			−0.0281821			−0.0140911	+	0.999901i	$0.504485\pi$
$402$		0			0
$403$	−223.697	+	223.697i	−0.555079	+	0.555079i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	443.081			1.08865
$408$		0			0
$409$		−	614.595i		−	1.50268i	−0.659917	−	0.751339i	$-0.729408\pi$
$409$		−	614.595i		−	1.50268i	0.659917	−	0.751339i	$-0.270592\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	150.081	+	150.081i	0.363391	+	0.363391i
$414$		0			0
$415$			3.13778i			0.00756092i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	78.7092	−	78.7092i	0.187850	−	0.187850i	−0.606916	−	0.794766i	$-0.707594\pi$
$419$	78.7092	−	78.7092i	0.187850	−	0.187850i	0.794766	+	0.606916i	$0.207594\pi$
$420$		0			0
$421$	374.618	−	374.618i	0.889829	−	0.889829i	−0.104678	−	0.994506i	$-0.533381\pi$
$421$	374.618	−	374.618i	0.889829	−	0.889829i	0.994506	+	0.104678i	$0.0333811\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			321.894i			0.757397i
$426$		0			0
$427$	−70.5705	−	70.5705i	−0.165270	−	0.165270i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			616.593i			1.43061i	0.698813	+	0.715305i	$0.253712\pi$
$431$			616.593i			1.43061i	−0.698813	+	0.715305i	$0.746288\pi$
$432$		0			0
$433$	219.246			0.506342			0.253171	−	0.967422i	$-0.418526\pi$
$433$	219.246			0.506342			0.253171	+	0.967422i	$0.418526\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	43.1409	−	43.1409i	0.0987207	−	0.0987207i
$438$		0			0
$439$	575.292			1.31046			0.655231	−	0.755429i	$-0.272571\pi$
$439$	575.292			1.31046			0.655231	+	0.755429i	$0.272571\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	371.895	+	371.895i	0.839492	+	0.839492i	0.988792	−	0.149300i	$-0.0477021\pi$
$443$	371.895	+	371.895i	0.839492	+	0.839492i	−0.149300	+	0.988792i	$0.547702\pi$
$444$		0			0
$445$	−6.77538	−	6.77538i	−0.0152256	−	0.0152256i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	498.135			1.10943			0.554716	−	0.832040i	$-0.312827\pi$
$449$	498.135			1.10943			0.554716	+	0.832040i	$0.312827\pi$
$450$		0			0
$451$	17.6898	−	17.6898i	0.0392236	−	0.0392236i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	5.98361			0.0131508
$456$		0			0
$457$			61.1711i			0.133854i	0.997758	+	0.0669268i	$0.0213194\pi$
$457$			61.1711i			0.133854i	−0.997758	+	0.0669268i	$0.978681\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−443.183	−	443.183i	−0.961352	−	0.961352i	0.0379287	−	0.999280i	$-0.487924\pi$
$461$	−443.183	−	443.183i	−0.961352	−	0.961352i	−0.999280	+	0.0379287i	$0.987924\pi$
$462$		0			0
$463$			706.883i			1.52675i	0.645958	+	0.763373i	$0.276458\pi$
$463$			706.883i			1.52675i	−0.645958	+	0.763373i	$0.723542\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−406.857	+	406.857i	−0.871214	+	0.871214i	−0.992605	−	0.121391i	$-0.961265\pi$
$467$	−406.857	+	406.857i	−0.871214	+	0.871214i	0.121391	+	0.992605i	$0.461265\pi$
$468$		0			0
$469$	277.224	−	277.224i	0.591096	−	0.591096i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	8.44620i		−	0.0178567i
$474$		0			0
$475$	−65.8217	−	65.8217i	−0.138572	−	0.138572i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			133.063i			0.277793i	0.990307	+	0.138896i	$0.0443555\pi$
$479$			133.063i			0.277793i	−0.990307	+	0.138896i	$0.955645\pi$
$480$		0			0
$481$	912.780			1.89767
$482$		0			0
$483$		0			0
$484$		0			0
$485$	8.57482	−	8.57482i	0.0176800	−	0.0176800i
$486$		0			0
$487$	−208.075			−0.427259			−0.213629	−	0.976915i	$-0.568529\pi$
$487$	−208.075			−0.427259			−0.213629	+	0.976915i	$0.568529\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−98.9374	−	98.9374i	−0.201502	−	0.201502i	0.599141	−	0.800643i	$-0.295509\pi$
$491$	−98.9374	−	98.9374i	−0.201502	−	0.201502i	−0.800643	+	0.599141i	$0.795509\pi$
$492$		0			0
$493$	−335.528	−	335.528i	−0.680585	−	0.680585i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−260.102			−0.523345
$498$		0			0
$499$	287.076	−	287.076i	0.575304	−	0.575304i	−0.358302	−	0.933606i	$-0.616644\pi$
$499$	287.076	−	287.076i	0.575304	−	0.575304i	0.933606	+	0.358302i	$0.116644\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	78.7359			0.156533			0.0782663	−	0.996932i	$-0.475062\pi$
$503$	78.7359			0.156533			0.0782663	+	0.996932i	$0.475062\pi$
$504$		0			0
$505$		−	6.31543i		−	0.0125058i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−242.477	−	242.477i	−0.476378	−	0.476378i	0.427593	−	0.903971i	$-0.359362\pi$
$509$	−242.477	−	242.477i	−0.476378	−	0.476378i	−0.903971	+	0.427593i	$0.859362\pi$
$510$		0			0
$511$			44.8014i			0.0876740i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	9.27484	−	9.27484i	0.0180094	−	0.0180094i
$516$		0			0
$517$	−427.980	+	427.980i	−0.827815	+	0.827815i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			561.306i			1.07736i	0.842510	+	0.538681i	$0.181077\pi$
$521$			561.306i			1.07736i	−0.842510	+	0.538681i	$0.818923\pi$
$522$		0			0
$523$	−396.152	−	396.152i	−0.757460	−	0.757460i	0.218399	−	0.975859i	$-0.429916\pi$
$523$	−396.152	−	396.152i	−0.757460	−	0.757460i	−0.975859	+	0.218399i	$0.929916\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	260.607i		−	0.494510i
$528$		0			0
$529$	−260.662			−0.492745
$530$		0			0
$531$		0			0
$532$		0			0
$533$	36.4424	−	36.4424i	0.0683722	−	0.0683722i
$534$		0			0
$535$	6.75465			0.0126255
$536$		0			0
$537$		0			0
$538$		0			0
$539$	148.632	+	148.632i	0.275755	+	0.275755i
$540$		0			0
$541$	22.5728	+	22.5728i	0.0417242	+	0.0417242i	0.727661	−	0.685937i	$-0.240607\pi$
$541$	22.5728	+	22.5728i	0.0417242	+	0.0417242i	−0.685937	+	0.727661i	$0.740607\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	6.67005			0.0122386
$546$		0			0
$547$	−601.634	+	601.634i	−1.09988	+	1.09988i	−0.105456	+	0.994424i	$0.533630\pi$
$547$	−601.634	+	601.634i	−1.09988	+	1.09988i	−0.994424	+	0.105456i	$0.966370\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	137.219			0.249037
$552$		0			0
$553$		−	389.681i		−	0.704666i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	502.883	+	502.883i	0.902841	+	0.902841i	0.995681	−	0.0928399i	$-0.0295945\pi$
$557$	502.883	+	502.883i	0.902841	+	0.902841i	−0.0928399	+	0.995681i	$0.529594\pi$
$558$		0			0
$559$		−	17.3998i		−	0.0311266i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	655.972	−	655.972i	1.16514	−	1.16514i	0.181802	−	0.983335i	$-0.441807\pi$
$563$	655.972	−	655.972i	1.16514	−	1.16514i	0.983335	−	0.181802i	$-0.0581929\pi$
$564$		0			0
$565$	7.95834	−	7.95834i	0.0140856	−	0.0140856i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	649.911i		−	1.14220i	−0.820881	−	0.571099i	$-0.806517\pi$
$569$		−	649.911i		−	1.14220i	0.820881	−	0.571099i	$-0.193483\pi$
$570$		0			0
$571$	−269.718	−	269.718i	−0.472360	−	0.472360i	0.430317	−	0.902678i	$-0.358402\pi$
$571$	−269.718	−	269.718i	−0.472360	−	0.472360i	−0.902678	+	0.430317i	$0.858402\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	409.413i		−	0.712022i
$576$		0			0
$577$	−142.675			−0.247271			−0.123635	−	0.992328i	$-0.539455\pi$
$577$	−142.675			−0.247271			−0.123635	+	0.992328i	$0.539455\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	123.502	−	123.502i	0.212567	−	0.212567i
$582$		0			0
$583$	−11.3990			−0.0195523
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−687.876	−	687.876i	−1.17185	−	1.17185i	−0.981768	−	0.190082i	$-0.939125\pi$
$587$	−687.876	−	687.876i	−1.17185	−	1.17185i	−0.190082	−	0.981768i	$-0.560875\pi$
$588$		0			0
$589$	53.2895	+	53.2895i	0.0904746	+	0.0904746i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	58.8678			0.0992711			0.0496355	−	0.998767i	$-0.484194\pi$
$593$	58.8678			0.0992711			0.0496355	+	0.998767i	$0.484194\pi$
$594$		0			0
$595$	−3.48545	+	3.48545i	−0.00585790	+	0.00585790i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	670.449			1.11928			0.559641	−	0.828735i	$-0.310939\pi$
$599$	670.449			1.11928			0.559641	+	0.828735i	$0.310939\pi$
$600$		0			0
$601$			910.721i			1.51534i	0.652636	+	0.757671i	$0.273663\pi$
$601$			910.721i			1.51534i	−0.652636	+	0.757671i	$0.726337\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	3.71754	+	3.71754i	0.00614469	+	0.00614469i
$606$		0			0
$607$		−	761.794i		−	1.25501i	−0.778611	−	0.627507i	$-0.784075\pi$
$607$		−	761.794i		−	1.25501i	0.778611	−	0.627507i	$-0.215925\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−881.671	+	881.671i	−1.44300	+	1.44300i
$612$		0			0
$613$	−273.397	+	273.397i	−0.445999	+	0.445999i	−0.894022	−	0.448023i	$-0.852128\pi$
$613$	−273.397	+	273.397i	−0.445999	+	0.445999i	0.448023	+	0.894022i	$0.352128\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	1088.68i		−	1.76448i	−0.470804	−	0.882238i	$-0.656036\pi$
$617$		−	1088.68i		−	1.76448i	0.470804	−	0.882238i	$-0.343964\pi$
$618$		0			0
$619$	129.299	+	129.299i	0.208884	+	0.208884i	0.803793	−	0.594909i	$-0.202812\pi$
$619$	129.299	+	129.299i	0.208884	+	0.208884i	−0.594909	+	0.803793i	$0.702812\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			533.351i			0.856102i
$624$		0			0
$625$	−624.484			−0.999175
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−531.694	+	531.694i	−0.845301	+	0.845301i
$630$		0			0
$631$	455.029			0.721123			0.360562	−	0.932735i	$-0.382585\pi$
$631$	455.029			0.721123			0.360562	+	0.932735i	$0.382585\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	9.76549	+	9.76549i	0.0153787	+	0.0153787i
$636$		0			0
$637$	306.193	+	306.193i	0.480679	+	0.480679i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−798.626			−1.24591			−0.622953	−	0.782259i	$-0.714067\pi$
$641$	−798.626			−1.24591			−0.622953	+	0.782259i	$0.714067\pi$
$642$		0			0
$643$	305.718	−	305.718i	0.475455	−	0.475455i	−0.428219	−	0.903675i	$-0.640859\pi$
$643$	305.718	−	305.718i	0.475455	−	0.475455i	0.903675	+	0.428219i	$0.140859\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−1161.90			−1.79583			−0.897916	−	0.440167i	$-0.854919\pi$
$647$	−1161.90			−1.79583			−0.897916	+	0.440167i	$0.854919\pi$
$648$		0			0
$649$		−	348.992i		−	0.537738i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	77.5410	+	77.5410i	0.118746	+	0.118746i	0.763983	−	0.645237i	$-0.223241\pi$
$653$	77.5410	+	77.5410i	0.118746	+	0.118746i	−0.645237	+	0.763983i	$0.723241\pi$
$654$		0			0
$655$			2.60973i			0.00398431i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	836.993	−	836.993i	1.27010	−	1.27010i	0.324059	−	0.946037i	$-0.394952\pi$
$659$	836.993	−	836.993i	1.27010	−	1.27010i	0.946037	−	0.324059i	$-0.105048\pi$
$660$		0			0
$661$	−121.071	+	121.071i	−0.183164	+	0.183164i	−0.792733	−	0.609569i	$-0.791342\pi$
$661$	−121.071	+	121.071i	−0.183164	+	0.183164i	0.609569	+	0.792733i	$0.291342\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	1.42543i		−	0.00214350i
$666$		0			0
$667$	426.754	+	426.754i	0.639811	+	0.639811i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			164.102i			0.244563i
$672$		0			0
$673$	−954.371			−1.41808			−0.709042	−	0.705166i	$-0.750872\pi$
$673$	−954.371			−1.41808			−0.709042	+	0.705166i	$0.750872\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	245.475	−	245.475i	0.362593	−	0.362593i	−0.502174	−	0.864767i	$-0.667466\pi$
$677$	245.475	−	245.475i	0.362593	−	0.362593i	0.864767	+	0.502174i	$0.167466\pi$
$678$		0			0
$679$	−675.002			−0.994112
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−911.271	−	911.271i	−1.33422	−	1.33422i	−0.901556	−	0.432663i	$-0.857574\pi$
$683$	−911.271	−	911.271i	−1.33422	−	1.33422i	−0.432663	−	0.901556i	$-0.642426\pi$
$684$		0			0
$685$	10.2544	+	10.2544i	0.0149699	+	0.0149699i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−23.4828			−0.0340824
$690$		0			0
$691$	476.155	−	476.155i	0.689081	−	0.689081i	−0.272947	−	0.962029i	$-0.587999\pi$
$691$	476.155	−	476.155i	0.689081	−	0.689081i	0.962029	+	0.272947i	$0.0879985\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	11.7136			0.0168541
$696$		0			0
$697$			42.4553i			0.0609115i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	934.966	+	934.966i	1.33376	+	1.33376i	0.901978	+	0.431782i	$0.142115\pi$
$701$	934.966	+	934.966i	1.33376	+	1.33376i	0.431782	+	0.901978i	$0.357885\pi$
$702$		0			0
$703$		−	217.444i		−	0.309309i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−248.572	+	248.572i	−0.351587	+	0.351587i
$708$		0			0
$709$	−5.89548	+	5.89548i	−0.00831520	+	0.00831520i	−0.711252	−	0.702937i	$-0.751872\pi$
$709$	−5.89548	+	5.89548i	−0.00831520	+	0.00831520i	0.702937	+	0.711252i	$0.251872\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			331.462i			0.464884i
$714$		0			0
$715$	−6.95702	−	6.95702i	−0.00973010	−	0.00973010i
$716$		0			0
$717$		0			0
$718$		0			0
$719$			19.5965i			0.0272552i	0.999907	+	0.0136276i	$0.00433793\pi$
$719$			19.5965i			0.0272552i	−0.999907	+	0.0136276i	$0.995662\pi$
$720$		0			0
$721$	−730.107			−1.01263
$722$		0			0
$723$		0			0
$724$		0			0
$725$	651.115	−	651.115i	0.898089	−	0.898089i
$726$		0			0
$727$	741.995			1.02063			0.510313	−	0.859989i	$-0.329530\pi$
$727$	741.995			1.02063			0.510313	+	0.859989i	$0.329530\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	10.1354	+	10.1354i	0.0138651	+	0.0138651i
$732$		0			0
$733$	−349.267	−	349.267i	−0.476490	−	0.476490i	0.427517	−	0.904007i	$-0.359388\pi$
$733$	−349.267	−	349.267i	−0.476490	−	0.476490i	−0.904007	+	0.427517i	$0.859388\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−644.646			−0.874690
$738$		0			0
$739$	358.932	−	358.932i	0.485700	−	0.485700i	−0.421246	−	0.906946i	$-0.638407\pi$
$739$	358.932	−	358.932i	0.485700	−	0.485700i	0.906946	+	0.421246i	$0.138407\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	856.214			1.15237			0.576187	−	0.817318i	$-0.304540\pi$
$743$	856.214			1.15237			0.576187	+	0.817318i	$0.304540\pi$
$744$		0			0
$745$		−	8.78160i		−	0.0117874i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−265.860	−	265.860i	−0.354953	−	0.354953i
$750$		0			0
$751$			442.218i			0.588839i	0.955676	+	0.294420i	$0.0951264\pi$
$751$			442.218i			0.588839i	−0.955676	+	0.294420i	$0.904874\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−4.10641	+	4.10641i	−0.00543896	+	0.00543896i
$756$		0			0
$757$	−489.198	+	489.198i	−0.646233	+	0.646233i	−0.952080	−	0.305848i	$-0.901060\pi$
$757$	−489.198	+	489.198i	−0.646233	+	0.646233i	0.305848	+	0.952080i	$0.401060\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	404.015i		−	0.530899i	−0.964125	−	0.265450i	$-0.914480\pi$
$761$		−	404.015i		−	0.530899i	0.964125	−	0.265450i	$-0.0855204\pi$
$762$		0			0
$763$	−262.530	−	262.530i	−0.344076	−	0.344076i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	718.949i		−	0.937352i
$768$		0			0
$769$	−387.336			−0.503688			−0.251844	−	0.967768i	$-0.581037\pi$
$769$	−387.336			−0.503688			−0.251844	+	0.967768i	$0.581037\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−960.396	+	960.396i	−1.24243	+	1.24243i	−0.283436	+	0.958991i	$0.591474\pi$
$773$	−960.396	+	960.396i	−1.24243	+	1.24243i	−0.958991	+	0.283436i	$0.908526\pi$
$774$		0			0
$775$	505.724			0.652547
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−8.68138	−	8.68138i	−0.0111443	−	0.0111443i
$780$		0			0
$781$	302.416	+	302.416i	0.387216	+	0.387216i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−3.46296			−0.00441141
$786$		0			0
$787$	298.374	−	298.374i	0.379129	−	0.379129i	−0.491659	−	0.870788i	$-0.663609\pi$
$787$	298.374	−	298.374i	0.379129	−	0.379129i	0.870788	+	0.491659i	$0.163609\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−626.473			−0.792002
$792$		0			0
$793$			338.062i			0.426308i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−870.093	−	870.093i	−1.09171	−	1.09171i	−0.995346	−	0.0963642i	$-0.969279\pi$
$797$	−870.093	−	870.093i	−1.09171	−	1.09171i	−0.0963642	−	0.995346i	$-0.530721\pi$
$798$		0			0
$799$		−	1027.15i		−	1.28554i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	52.0897	−	52.0897i	0.0648689	−	0.0648689i
$804$		0			0
$805$	4.43310	−	4.43310i	0.00550695	−	0.00550695i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			107.642i			0.133055i	0.997785	+	0.0665277i	$0.0211921\pi$
$809$			107.642i			0.133055i	−0.997785	+	0.0665277i	$0.978808\pi$
$810$		0			0
$811$	−829.739	−	829.739i	−1.02311	−	1.02311i	−0.999727	−	0.0233795i	$-0.992557\pi$
$811$	−829.739	−	829.739i	−1.02311	−	1.02311i	−0.0233795	−	0.999727i	$-0.507443\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		−	5.60521i		−	0.00687756i
$816$		0			0
$817$	−4.14502			−0.00507346
$818$		0			0
$819$		0			0
$820$		0			0
$821$	506.899	−	506.899i	0.617416	−	0.617416i	−0.327452	−	0.944868i	$-0.606190\pi$
$821$	506.899	−	506.899i	0.617416	−	0.617416i	0.944868	+	0.327452i	$0.106190\pi$
$822$		0			0
$823$	−927.304			−1.12674			−0.563368	−	0.826206i	$-0.690495\pi$
$823$	−927.304			−1.12674			−0.563368	+	0.826206i	$0.690495\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−19.4711	−	19.4711i	−0.0235443	−	0.0235443i	0.695237	−	0.718781i	$-0.255300\pi$
$827$	−19.4711	−	19.4711i	−0.0235443	−	0.0235443i	−0.718781	+	0.695237i	$0.755300\pi$
$828$		0			0
$829$	−409.028	−	409.028i	−0.493400	−	0.493400i	0.415976	−	0.909376i	$-0.363440\pi$
$829$	−409.028	−	409.028i	−0.493400	−	0.493400i	−0.909376	+	0.415976i	$0.863440\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−356.714			−0.428228
$834$		0			0
$835$	−9.16246	+	9.16246i	−0.0109730	+	0.0109730i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−634.212			−0.755914			−0.377957	−	0.925823i	$-0.623373\pi$
$839$	−634.212			−0.755914			−0.377957	+	0.925823i	$0.623373\pi$
$840$		0			0
$841$			516.388i			0.614017i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−4.42297	−	4.42297i	−0.00523429	−	0.00523429i
$846$		0			0
$847$		−	292.641i		−	0.345503i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	676.255	−	676.255i	0.794659	−	0.794659i
$852$		0			0
$853$	−687.203	+	687.203i	−0.805630	+	0.805630i	−0.983969	−	0.178339i	$-0.942928\pi$
$853$	−687.203	+	687.203i	−0.805630	+	0.805630i	0.178339	+	0.983969i	$0.442928\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			995.675i			1.16181i	0.813970	+	0.580907i	$0.197302\pi$
$857$			995.675i			1.16181i	−0.813970	+	0.580907i	$0.802698\pi$
$858$		0			0
$859$	−430.241	−	430.241i	−0.500863	−	0.500863i	0.410843	−	0.911706i	$-0.365234\pi$
$859$	−430.241	−	430.241i	−0.500863	−	0.500863i	−0.911706	+	0.410843i	$0.865234\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	1014.03i		−	1.17501i	−0.809222	−	0.587503i	$-0.800111\pi$
$863$		−	1014.03i		−	1.17501i	0.809222	−	0.587503i	$-0.199889\pi$
$864$		0			0
$865$	−22.2927			−0.0257719
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−453.074	+	453.074i	−0.521374	+	0.521374i
$870$		0			0
$871$	−1328.02			−1.52471
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−13.5293	−	13.5293i	−0.0154621	−	0.0154621i
$876$		0			0
$877$	544.315	+	544.315i	0.620656	+	0.620656i	0.945699	−	0.325043i	$-0.105379\pi$
$877$	544.315	+	544.315i	0.620656	+	0.620656i	−0.325043	+	0.945699i	$0.605379\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	645.905			0.733150			0.366575	−	0.930388i	$-0.380530\pi$
$881$	645.905			0.733150			0.366575	+	0.930388i	$0.380530\pi$
$882$		0			0
$883$	−586.952	+	586.952i	−0.664725	+	0.664725i	−0.956490	−	0.291765i	$-0.905757\pi$
$883$	−586.952	+	586.952i	−0.664725	+	0.664725i	0.291765	+	0.956490i	$0.405757\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	1221.93			1.37759			0.688797	−	0.724955i	$-0.258139\pi$
$887$	1221.93			1.37759			0.688797	+	0.724955i	$0.258139\pi$
$888$		0			0
$889$		−	768.730i		−	0.864713i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	210.034	+	210.034i	0.235200	+	0.235200i
$894$		0			0
$895$			6.36462i			0.00711131i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−527.145	+	527.145i	−0.586368	+	0.586368i
$900$		0			0
$901$	13.6787	−	13.6787i	0.0151817	−	0.0151817i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	2.31417i		−	0.00255710i
$906$		0			0
$907$	−310.014	−	310.014i	−0.341801	−	0.341801i	0.515243	−	0.857044i	$-0.327702\pi$
$907$	−310.014	−	310.014i	−0.341801	−	0.341801i	−0.857044	+	0.515243i	$0.827702\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			1044.12i			1.14612i	0.819513	+	0.573060i	$0.194244\pi$
$911$			1044.12i			1.14612i	−0.819513	+	0.573060i	$0.805756\pi$
$912$		0			0
$913$	−287.186			−0.314552
$914$		0			0
$915$		0			0
$916$		0			0
$917$	102.718	−	102.718i	0.112015	−	0.112015i
$918$		0			0
$919$	188.522			0.205138			0.102569	−	0.994726i	$-0.467294\pi$
$919$	188.522			0.205138			0.102569	+	0.994726i	$0.467294\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	622.999	+	622.999i	0.674972	+	0.674972i
$924$		0			0
$925$	−1031.79	−	1031.79i	−1.11545	−	1.11545i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	220.366			0.237208			0.118604	−	0.992942i	$-0.462158\pi$
$929$	220.366			0.237208			0.118604	+	0.992942i	$0.462158\pi$
$930$		0			0
$931$	72.9419	−	72.9419i	0.0783479	−	0.0783479i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	8.10493			0.00866837
$936$		0			0
$937$		−	558.321i		−	0.595860i	−0.954588	−	0.297930i	$-0.903704\pi$
$937$		−	558.321i		−	0.595860i	0.954588	−	0.297930i	$-0.0962962\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−794.760	−	794.760i	−0.844591	−	0.844591i	0.144861	−	0.989452i	$-0.453726\pi$
$941$	−794.760	−	794.760i	−0.844591	−	0.844591i	−0.989452	+	0.144861i	$0.953726\pi$
$942$		0			0
$943$		−	53.9984i		−	0.0572624i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−44.9362	+	44.9362i	−0.0474511	+	0.0474511i	−0.730434	−	0.682983i	$-0.760682\pi$
$947$	−44.9362	+	44.9362i	−0.0474511	+	0.0474511i	0.682983	+	0.730434i	$0.260682\pi$
$948$		0			0
$949$	107.309	−	107.309i	0.113076	−	0.113076i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			304.232i			0.319236i	0.987179	+	0.159618i	$0.0510262\pi$
$953$			304.232i			0.319236i	−0.987179	+	0.159618i	$0.948974\pi$
$954$		0			0
$955$	9.76549	+	9.76549i	0.0102256	+	0.0102256i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	807.213i		−	0.841724i
$960$		0			0
$961$	551.564			0.573948
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−0.127971	+	0.127971i	−0.000132613	+	0.000132613i
$966$		0			0
$967$	834.409			0.862884			0.431442	−	0.902141i	$-0.358005\pi$
$967$	834.409			0.862884			0.431442	+	0.902141i	$0.358005\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	211.499	+	211.499i	0.217816	+	0.217816i	0.807577	−	0.589761i	$-0.200778\pi$
$971$	211.499	+	211.499i	0.217816	+	0.217816i	−0.589761	+	0.807577i	$0.700778\pi$
$972$		0			0
$973$	−461.043	−	461.043i	−0.473837	−	0.473837i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−891.561			−0.912549			−0.456275	−	0.889839i	$-0.650817\pi$
$977$	−891.561			−0.912549			−0.456275	+	0.889839i	$0.650817\pi$
$978$		0			0
$979$	620.117	−	620.117i	0.633419	−	0.633419i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	181.589			0.184730			0.0923648	−	0.995725i	$-0.470557\pi$
$983$	181.589			0.184730			0.0923648	+	0.995725i	$0.470557\pi$
$984$		0			0
$985$			7.90739i			0.00802781i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−12.8911	−	12.8911i	−0.0130344	−	0.0130344i
$990$		0			0
$991$		−	1140.89i		−	1.15125i	−0.817715	−	0.575624i	$-0.804759\pi$
$991$		−	1140.89i		−	1.15125i	0.817715	−	0.575624i	$-0.195241\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−13.0697	+	13.0697i	−0.0131354	+	0.0131354i
$996$		0			0
$997$	742.946	−	742.946i	0.745182	−	0.745182i	−0.228388	−	0.973570i	$-0.573346\pi$
$997$	742.946	−	742.946i	0.745182	−	0.745182i	0.973570	+	0.228388i	$0.0733455\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1152.3.m.a.991.2		6
3.2	odd	2		128.3.f.b.95.3		6
4.3	odd	2		1152.3.m.b.991.2		6
8.3	odd	2		576.3.m.a.559.2		6
8.5	even	2		144.3.m.a.19.1		6
12.11	even	2		128.3.f.a.95.1		6
16.3	odd	4		144.3.m.a.91.1		6
16.5	even	4		1152.3.m.b.415.2		6
16.11	odd	4	inner	1152.3.m.a.415.2		6
16.13	even	4		576.3.m.a.271.2		6
24.5	odd	2		16.3.f.a.3.3	✓	6
24.11	even	2		64.3.f.a.47.3		6
48.5	odd	4		128.3.f.a.31.1		6
48.11	even	4		128.3.f.b.31.3		6
48.29	odd	4		64.3.f.a.15.3		6
48.35	even	4		16.3.f.a.11.3	yes	6
96.5	odd	8		1024.3.c.j.1023.12		12
96.11	even	8		1024.3.c.j.1023.11		12
96.29	odd	8		1024.3.d.k.511.12		12
96.35	even	8		1024.3.d.k.511.2		12
96.53	odd	8		1024.3.c.j.1023.1		12
96.59	even	8		1024.3.c.j.1023.2		12
96.77	odd	8		1024.3.d.k.511.1		12
96.83	even	8		1024.3.d.k.511.11		12
120.29	odd	2		400.3.r.c.51.1		6
120.53	even	4		400.3.k.c.99.1		6
120.77	even	4		400.3.k.d.99.3		6
240.83	odd	4		400.3.k.d.299.3		6
240.179	even	4		400.3.r.c.251.1		6
240.227	odd	4		400.3.k.c.299.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.3.f.a.3.3	✓	6	24.5	odd	2
16.3.f.a.11.3	yes	6	48.35	even	4
64.3.f.a.15.3		6	48.29	odd	4
64.3.f.a.47.3		6	24.11	even	2
128.3.f.a.31.1		6	48.5	odd	4
128.3.f.a.95.1		6	12.11	even	2
128.3.f.b.31.3		6	48.11	even	4
128.3.f.b.95.3		6	3.2	odd	2
144.3.m.a.19.1		6	8.5	even	2
144.3.m.a.91.1		6	16.3	odd	4
400.3.k.c.99.1		6	120.53	even	4
400.3.k.c.299.1		6	240.227	odd	4
400.3.k.d.99.3		6	120.77	even	4
400.3.k.d.299.3		6	240.83	odd	4
400.3.r.c.51.1		6	120.29	odd	2
400.3.r.c.251.1		6	240.179	even	4
576.3.m.a.271.2		6	16.13	even	4
576.3.m.a.559.2		6	8.3	odd	2
1024.3.c.j.1023.1		12	96.53	odd	8
1024.3.c.j.1023.2		12	96.59	even	8
1024.3.c.j.1023.11		12	96.11	even	8
1024.3.c.j.1023.12		12	96.5	odd	8
1024.3.d.k.511.1		12	96.77	odd	8
1024.3.d.k.511.2		12	96.35	even	8
1024.3.d.k.511.11		12	96.83	even	8
1024.3.d.k.511.12		12	96.29	odd	8
1152.3.m.a.415.2		6	16.11	odd	4	inner
1152.3.m.a.991.2		6	1.1	even	1	trivial
1152.3.m.b.415.2		6	16.5	even	4
1152.3.m.b.991.2		6	4.3	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1152 = 2^{7} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1152.m (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.0586332 + 0.0586332i) q^{5} +4.61555 q^{7} +O(q^{10})\)	\(q+(-0.0586332 + 0.0586332i) q^{5} +4.61555 q^{7} +(-5.36641 - 5.36641i) q^{11} +(-11.0552 - 11.0552i) q^{13} +12.8793 q^{17} +(-2.63359 + 2.63359i) q^{19} -16.3810 q^{23} +24.9931i q^{25} +(-26.0518 - 26.0518i) q^{29} -20.2345i q^{31} +(-0.270624 + 0.270624i) q^{35} +(-41.2829 + 41.2829i) q^{37} +3.29640i q^{41} +(0.786951 + 0.786951i) q^{43} -79.7517i q^{47} -27.6967 q^{49} +(1.06207 - 1.06207i) q^{53} +0.629299 q^{55} +(32.5163 + 32.5163i) q^{59} +(-15.2897 - 15.2897i) q^{61} +1.29640 q^{65} +(60.0631 - 60.0631i) q^{67} -56.3535 q^{71} +9.70663i q^{73} +(-24.7689 - 24.7689i) q^{77} -84.4278i q^{79} +(26.7577 - 26.7577i) q^{83} +(-0.755154 + 0.755154i) q^{85} +115.555i q^{89} +(-51.0258 - 51.0258i) q^{91} -0.308832i q^{95} -146.245 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{5} - 4 q^{7}+O(q^{10}) \) 6 * q - 2 * q^5 - 4 * q^7	\( 6 q - 2 q^{5} - 4 q^{7} - 18 q^{11} + 2 q^{13} + 4 q^{17} - 30 q^{19} - 60 q^{23} - 18 q^{29} - 100 q^{35} - 46 q^{37} + 114 q^{43} - 46 q^{49} + 78 q^{53} + 252 q^{55} + 206 q^{59} - 30 q^{61} - 12 q^{65} + 226 q^{67} + 260 q^{71} - 212 q^{77} + 318 q^{83} + 212 q^{85} - 188 q^{91} - 4 q^{97}+O(q^{100}) \) 6 * q - 2 * q^5 - 4 * q^7 - 18 * q^11 + 2 * q^13 + 4 * q^17 - 30 * q^19 - 60 * q^23 - 18 * q^29 - 100 * q^35 - 46 * q^37 + 114 * q^43 - 46 * q^49 + 78 * q^53 + 252 * q^55 + 206 * q^59 - 30 * q^61 - 12 * q^65 + 226 * q^67 + 260 * q^71 - 212 * q^77 + 318 * q^83 + 212 * q^85 - 188 * q^91 - 4 * q^97

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