LMFDB - Embedded newform 1152.3.m.b.991.1

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			991.1
Root			$1.40680 - 0.144584i$ of defining polynomial
Character	$\chi$	$=$	1152.991
Dual form			1152.3.m.b.415.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-4.62721 + 4.62721i) q^{5} -3.04888 q^{7} +O(q^{10})$	$q+(-4.62721 + 4.62721i) q^{5} -3.04888 q^{7} +(9.15165 + 9.15165i) q^{11} +(5.78389 + 5.78389i) q^{13} -17.6655 q^{17} +(-1.15165 + 1.15165i) q^{19} -3.45998 q^{23} -17.8222i q^{25} +(12.1950 + 12.1950i) q^{29} +38.5089i q^{31} +(14.1078 - 14.1078i) q^{35} +(0.0972356 - 0.0972356i) q^{37} -51.5266i q^{41} +(-1.70172 - 1.70172i) q^{43} -24.1533i q^{47} -39.7044 q^{49} +(27.0383 - 27.0383i) q^{53} -84.6933 q^{55} +(-19.5939 - 19.5939i) q^{59} +(-16.7250 - 16.7250i) q^{61} -53.5266 q^{65} +(-75.8560 + 75.8560i) q^{67} -134.749 q^{71} -112.210i q^{73} +(-27.9022 - 27.9022i) q^{77} -135.915i q^{79} +(-74.9250 + 74.9250i) q^{83} +(81.7422 - 81.7422i) q^{85} -31.4278i q^{89} +(-17.6344 - 17.6344i) q^{91} -10.6579i q^{95} +31.5456 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−4.62721	+	4.62721i	−0.925443	+	0.925443i	−0.997407	−	0.0719646i	$-0.977073\pi$
$5$	−4.62721	+	4.62721i	−0.925443	+	0.925443i	0.0719646	+	0.997407i	$0.477073\pi$
$6$		0			0
$7$	−3.04888			−0.435554			−0.217777	−	0.975999i	$-0.569881\pi$
$7$	−3.04888			−0.435554			−0.217777	+	0.975999i	$0.569881\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	9.15165	+	9.15165i	0.831968	+	0.831968i	0.987786	−	0.155818i	$-0.0498012\pi$
$11$	9.15165	+	9.15165i	0.831968	+	0.831968i	−0.155818	+	0.987786i	$0.549801\pi$
$12$		0			0
$13$	5.78389	+	5.78389i	0.444914	+	0.444914i	0.893660	−	0.448745i	$-0.148129\pi$
$13$	5.78389	+	5.78389i	0.444914	+	0.444914i	−0.448745	+	0.893660i	$0.648129\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−17.6655			−1.03915			−0.519574	−	0.854425i	$-0.673909\pi$
$17$	−17.6655			−1.03915			−0.519574	+	0.854425i	$0.673909\pi$
$18$		0			0
$19$	−1.15165	+	1.15165i	−0.0606132	+	0.0606132i	−0.736764	−	0.676150i	$-0.763647\pi$
$19$	−1.15165	+	1.15165i	−0.0606132	+	0.0606132i	0.676150	+	0.736764i	$0.263647\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−3.45998			−0.150434			−0.0752169	−	0.997167i	$-0.523965\pi$
$23$	−3.45998			−0.150434			−0.0752169	+	0.997167i	$0.523965\pi$
$24$		0			0
$25$		−	17.8222i		−	0.712888i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	12.1950	+	12.1950i	0.420517	+	0.420517i	0.885382	−	0.464865i	$-0.153897\pi$
$29$	12.1950	+	12.1950i	0.420517	+	0.420517i	−0.464865	+	0.885382i	$0.653897\pi$
$30$		0			0
$31$			38.5089i			1.24222i	0.783723	+	0.621111i	$0.213318\pi$
$31$			38.5089i			1.24222i	−0.783723	+	0.621111i	$0.786682\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	14.1078	−	14.1078i	0.403080	−	0.403080i
$36$		0			0
$37$	0.0972356	−	0.0972356i	0.00262799	−	0.00262799i	−0.705792	−	0.708420i	$-0.749408\pi$
$37$	0.0972356	−	0.0972356i	0.00262799	−	0.00262799i	0.708420	+	0.705792i	$0.249408\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	51.5266i		−	1.25675i	−0.777912	−	0.628373i	$-0.783721\pi$
$41$		−	51.5266i		−	1.25675i	0.777912	−	0.628373i	$-0.216279\pi$
$42$		0			0
$43$	−1.70172	−	1.70172i	−0.0395749	−	0.0395749i	0.687042	−	0.726617i	$-0.258909\pi$
$43$	−1.70172	−	1.70172i	−0.0395749	−	0.0395749i	−0.726617	+	0.687042i	$0.758909\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	24.1533i		−	0.513899i	−0.966425	−	0.256949i	$-0.917283\pi$
$47$		−	24.1533i		−	0.513899i	0.966425	−	0.256949i	$-0.0827174\pi$
$48$		0			0
$49$	−39.7044			−0.810293
$50$		0			0
$51$		0			0
$52$		0			0
$53$	27.0383	−	27.0383i	0.510157	−	0.510157i	−0.404418	−	0.914574i	$-0.632526\pi$
$53$	27.0383	−	27.0383i	0.510157	−	0.510157i	0.914574	+	0.404418i	$0.132526\pi$
$54$		0			0
$55$	−84.6933			−1.53988
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−19.5939	−	19.5939i	−0.332100	−	0.332100i	0.521283	−	0.853384i	$-0.325453\pi$
$59$	−19.5939	−	19.5939i	−0.332100	−	0.332100i	−0.853384	+	0.521283i	$0.825453\pi$
$60$		0			0
$61$	−16.7250	−	16.7250i	−0.274180	−	0.274180i	0.556601	−	0.830780i	$-0.312105\pi$
$61$	−16.7250	−	16.7250i	−0.274180	−	0.274180i	−0.830780	+	0.556601i	$0.812105\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−53.5266			−0.823485
$66$		0			0
$67$	−75.8560	+	75.8560i	−1.13218	+	1.13218i	−0.142365	+	0.989814i	$0.545471\pi$
$67$	−75.8560	+	75.8560i	−1.13218	+	1.13218i	−0.989814	+	0.142365i	$0.954529\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−134.749			−1.89787			−0.948935	−	0.315471i	$-0.897837\pi$
$71$	−134.749			−1.89787			−0.948935	+	0.315471i	$0.897837\pi$
$72$		0			0
$73$		−	112.210i		−	1.53712i	−0.639777	−	0.768560i	$-0.720974\pi$
$73$		−	112.210i		−	1.53712i	0.639777	−	0.768560i	$-0.279026\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−27.9022	−	27.9022i	−0.362367	−	0.362367i
$78$		0			0
$79$		−	135.915i		−	1.72045i	−0.509915	−	0.860225i	$-0.670323\pi$
$79$		−	135.915i		−	1.72045i	0.509915	−	0.860225i	$-0.329677\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−74.9250	+	74.9250i	−0.902711	+	0.902711i	−0.995670	−	0.0929594i	$-0.970367\pi$
$83$	−74.9250	+	74.9250i	−0.902711	+	0.902711i	0.0929594	+	0.995670i	$0.470367\pi$
$84$		0			0
$85$	81.7422	−	81.7422i	0.961672	−	0.961672i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	31.4278i		−	0.353121i	−0.984290	−	0.176561i	$-0.943503\pi$
$89$		−	31.4278i		−	0.353121i	0.984290	−	0.176561i	$-0.0564971\pi$
$90$		0			0
$91$	−17.6344	−	17.6344i	−0.193784	−	0.193784i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		−	10.6579i		−	0.112188i
$96$		0			0
$97$	31.5456			0.325213			0.162606	−	0.986691i	$-0.448010\pi$
$97$	31.5456			0.325213			0.162606	+	0.986691i	$0.448010\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	27.4695	−	27.4695i	0.271975	−	0.271975i	−0.557920	−	0.829895i	$-0.688400\pi$
$101$	27.4695	−	27.4695i	0.271975	−	0.271975i	0.829895	+	0.557920i	$0.188400\pi$
$102$		0			0
$103$	−102.882			−0.998854			−0.499427	−	0.866356i	$-0.666456\pi$
$103$	−102.882			−0.998854			−0.499427	+	0.866356i	$0.666456\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	79.6605	+	79.6605i	0.744491	+	0.744491i	0.973439	−	0.228948i	$-0.0735286\pi$
$107$	79.6605	+	79.6605i	0.744491	+	0.744491i	−0.228948	+	0.973439i	$0.573529\pi$
$108$		0			0
$109$	−125.408	−	125.408i	−1.15053	−	1.15053i	−0.986446	−	0.164088i	$-0.947532\pi$
$109$	−125.408	−	125.408i	−1.15053	−	1.15053i	−0.164088	−	0.986446i	$-0.552468\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	96.6199			0.855043			0.427521	−	0.904005i	$-0.359387\pi$
$113$	96.6199			0.855043			0.427521	+	0.904005i	$0.359387\pi$
$114$		0			0
$115$	16.0100	−	16.0100i	0.139218	−	0.139218i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	53.8600			0.452605
$120$		0			0
$121$			46.5054i			0.384342i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−33.2132	−	33.2132i	−0.265706	−	0.265706i
$126$		0			0
$127$			196.309i			1.54574i	0.634566	+	0.772868i	$0.281179\pi$
$127$			196.309i			1.54574i	−0.634566	+	0.772868i	$0.718821\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−17.9437	+	17.9437i	−0.136975	+	0.136975i	−0.772270	−	0.635295i	$-0.780879\pi$
$131$	−17.9437	+	17.9437i	−0.136975	+	0.136975i	0.635295	+	0.772270i	$0.280879\pi$
$132$		0			0
$133$	3.51124	−	3.51124i	0.0264003	−	0.0264003i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			51.7200i			0.377518i	0.982023	+	0.188759i	$0.0604465\pi$
$137$			51.7200i			0.377518i	−0.982023	+	0.188759i	$0.939553\pi$
$138$		0			0
$139$	−17.4640	−	17.4640i	−0.125640	−	0.125640i	0.641491	−	0.767131i	$-0.278316\pi$
$139$	−17.4640	−	17.4640i	−0.125640	−	0.125640i	−0.767131	+	0.641491i	$0.778316\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			105.864i			0.740309i
$144$		0			0
$145$	−112.858			−0.778328
$146$		0			0
$147$		0			0
$148$		0			0
$149$	11.9170	−	11.9170i	0.0799802	−	0.0799802i	−0.665985	−	0.745965i	$-0.731989\pi$
$149$	11.9170	−	11.9170i	0.0799802	−	0.0799802i	0.745965	+	0.665985i	$0.231989\pi$
$150$		0			0
$151$	132.548			0.877805			0.438902	−	0.898535i	$-0.355367\pi$
$151$	132.548			0.877805			0.438902	+	0.898535i	$0.355367\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−178.189	−	178.189i	−1.14960	−	1.14960i
$156$		0			0
$157$	−106.091	−	106.091i	−0.675742	−	0.675742i	0.283292	−	0.959034i	$-0.408573\pi$
$157$	−106.091	−	106.091i	−0.675742	−	0.675742i	−0.959034	+	0.283292i	$0.908573\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	10.5490			0.0655220
$162$		0			0
$163$	−105.577	+	105.577i	−0.647712	+	0.647712i	−0.952440	−	0.304728i	$-0.901435\pi$
$163$	−105.577	+	105.577i	−0.647712	+	0.647712i	0.304728	+	0.952440i	$0.401435\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	111.591			0.668210			0.334105	−	0.942536i	$-0.391566\pi$
$167$	111.591			0.668210			0.334105	+	0.942536i	$0.391566\pi$
$168$		0			0
$169$		−	102.093i		−	0.604102i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	14.5363	+	14.5363i	0.0840249	+	0.0840249i	0.747870	−	0.663845i	$-0.231077\pi$
$173$	14.5363	+	14.5363i	0.0840249	+	0.0840249i	−0.663845	+	0.747870i	$0.731077\pi$
$174$		0			0
$175$			54.3377i			0.310501i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	19.7371	−	19.7371i	0.110263	−	0.110263i	−0.649823	−	0.760086i	$-0.725157\pi$
$179$	19.7371	−	19.7371i	0.110263	−	0.110263i	0.760086	+	0.649823i	$0.225157\pi$
$180$		0			0
$181$	−168.153	+	168.153i	−0.929021	+	0.929021i	−0.997643	−	0.0686221i	$-0.978140\pi$
$181$	−168.153	+	168.153i	−0.929021	+	0.929021i	0.0686221	+	0.997643i	$0.478140\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			0.899859i			0.00486410i
$186$		0			0
$187$	−161.669	−	161.669i	−0.864539	−	0.864539i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			196.309i			1.02779i	0.857852	+	0.513897i	$0.171799\pi$
$191$			196.309i			1.02779i	−0.857852	+	0.513897i	$0.828201\pi$
$192$		0			0
$193$	−40.3699			−0.209170			−0.104585	−	0.994516i	$-0.533351\pi$
$193$	−40.3699			−0.209170			−0.104585	+	0.994516i	$0.533351\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−230.578	+	230.578i	−1.17045	+	1.17045i	−0.188344	+	0.982103i	$0.560312\pi$
$197$	−230.578	+	230.578i	−1.17045	+	1.17045i	−0.982103	+	0.188344i	$0.939688\pi$
$198$		0			0
$199$	−61.5598			−0.309346			−0.154673	−	0.987966i	$-0.549432\pi$
$199$	−61.5598			−0.309346			−0.154673	+	0.987966i	$0.549432\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−37.1810	−	37.1810i	−0.183158	−	0.183158i
$204$		0			0
$205$	238.424	+	238.424i	1.16305	+	1.16305i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−21.0790			−0.100857
$210$		0			0
$211$	151.149	−	151.149i	0.716346	−	0.716346i	−0.251509	−	0.967855i	$-0.580927\pi$
$211$	151.149	−	151.149i	0.716346	−	0.716346i	0.967855	+	0.251509i	$0.0809267\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	15.7485			0.0732486
$216$		0			0
$217$		−	117.409i		−	0.541054i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−102.175	−	102.175i	−0.462332	−	0.462332i
$222$		0			0
$223$		−	115.527i		−	0.518056i	−0.965870	−	0.259028i	$-0.916598\pi$
$223$		−	115.527i		−	0.518056i	0.965870	−	0.259028i	$-0.0834022\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−25.2363	+	25.2363i	−0.111173	+	0.111173i	−0.760505	−	0.649332i	$-0.775049\pi$
$227$	−25.2363	+	25.2363i	−0.111173	+	0.111173i	0.649332	+	0.760505i	$0.275049\pi$
$228$		0			0
$229$	155.318	−	155.318i	0.678244	−	0.678244i	−0.281359	−	0.959603i	$-0.590785\pi$
$229$	155.318	−	155.318i	0.678244	−	0.678244i	0.959603	+	0.281359i	$0.0907851\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			119.738i			0.513899i	0.966425	+	0.256949i	$0.0827174\pi$
$233$			119.738i			0.513899i	−0.966425	+	0.256949i	$0.917283\pi$
$234$		0			0
$235$	111.762	+	111.762i	0.475584	+	0.475584i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	245.409i		−	1.02681i	−0.858145	−	0.513407i	$-0.828383\pi$
$239$		−	245.409i		−	1.02681i	0.858145	−	0.513407i	$-0.171617\pi$
$240$		0			0
$241$	431.216			1.78928			0.894639	−	0.446790i	$-0.147433\pi$
$241$	431.216			1.78928			0.894639	+	0.446790i	$0.147433\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	183.721	−	183.721i	0.749880	−	0.749880i
$246$		0			0
$247$	−13.3220			−0.0539354
$248$		0			0
$249$		0			0
$250$		0			0
$251$	24.0171	+	24.0171i	0.0956858	+	0.0956858i	0.753329	−	0.657643i	$-0.228447\pi$
$251$	24.0171	+	24.0171i	0.0956858	+	0.0956858i	−0.657643	+	0.753329i	$0.728447\pi$
$252$		0			0
$253$	−31.6645	−	31.6645i	−0.125156	−	0.125156i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	100.860			0.392450			0.196225	−	0.980559i	$-0.437132\pi$
$257$	100.860			0.392450			0.196225	+	0.980559i	$0.437132\pi$
$258$		0			0
$259$	−0.296459	+	0.296459i	−0.00114463	+	0.00114463i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	216.776			0.824242			0.412121	−	0.911129i	$-0.364788\pi$
$263$	216.776			0.824242			0.412121	+	0.911129i	$0.364788\pi$
$264$		0			0
$265$			250.224i			0.944242i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−256.778	−	256.778i	−0.954567	−	0.954567i	0.0444453	−	0.999012i	$-0.485848\pi$
$269$	−256.778	−	256.778i	−0.954567	−	0.954567i	−0.999012	+	0.0444453i	$0.985848\pi$
$270$		0			0
$271$			12.8603i			0.0474551i	0.999718	+	0.0237275i	$0.00755342\pi$
$271$			12.8603i			0.0474551i	−0.999718	+	0.0237275i	$0.992447\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	163.103	−	163.103i	0.593100	−	0.593100i
$276$		0			0
$277$	−77.1023	+	77.1023i	−0.278348	+	0.278348i	−0.832449	−	0.554102i	$-0.813062\pi$
$277$	−77.1023	+	77.1023i	−0.278348	+	0.278348i	0.554102	+	0.832449i	$0.313062\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	189.034i		−	0.672719i	−0.941734	−	0.336360i	$-0.890804\pi$
$281$		−	189.034i		−	0.672719i	0.941734	−	0.336360i	$-0.109196\pi$
$282$		0			0
$283$	−69.4317	−	69.4317i	−0.245342	−	0.245342i	0.573714	−	0.819056i	$-0.305502\pi$
$283$	−69.4317	−	69.4317i	−0.245342	−	0.245342i	−0.819056	+	0.573714i	$0.805502\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			157.098i			0.547380i
$288$		0			0
$289$	23.0708			0.0798298
$290$		0			0
$291$		0			0
$292$		0			0
$293$	239.919	−	239.919i	0.818837	−	0.818837i	−0.167103	−	0.985939i	$-0.553441\pi$
$293$	239.919	−	239.919i	0.818837	−	0.818837i	0.985939	+	0.167103i	$0.0534412\pi$
$294$		0			0
$295$	181.331			0.614680
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−20.0121	−	20.0121i	−0.0669301	−	0.0669301i
$300$		0			0
$301$	5.18834	+	5.18834i	0.0172370	+	0.0172370i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	154.780			0.507475
$306$		0			0
$307$	−231.185	+	231.185i	−0.753046	+	0.753046i	−0.975046	−	0.222001i	$-0.928741\pi$
$307$	−231.185	+	231.185i	−0.753046	+	0.753046i	0.222001	+	0.975046i	$0.428741\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−513.328			−1.65057			−0.825287	−	0.564714i	$-0.808987\pi$
$311$	−513.328			−1.65057			−0.825287	+	0.564714i	$0.808987\pi$
$312$		0			0
$313$			345.242i			1.10301i	0.834172	+	0.551504i	$0.185946\pi$
$313$			345.242i			1.10301i	−0.834172	+	0.551504i	$0.814054\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	345.632	+	345.632i	1.09032	+	1.09032i	0.995494	+	0.0948290i	$0.0302304\pi$
$317$	345.632	+	345.632i	1.09032	+	1.09032i	0.0948290	+	0.995494i	$0.469770\pi$
$318$		0			0
$319$			223.209i			0.699713i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	20.3445	−	20.3445i	0.0629861	−	0.0629861i
$324$		0			0
$325$	103.082	−	103.082i	0.317174	−	0.317174i
$326$		0			0
$327$		0			0
$328$		0			0
$329$			73.6403i			0.223831i
$330$		0			0
$331$	425.968	+	425.968i	1.28691	+	1.28691i	0.936652	+	0.350261i	$0.113907\pi$
$331$	425.968	+	425.968i	1.28691	+	1.28691i	0.350261	+	0.936652i	$0.386093\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	702.004i		−	2.09553i
$336$		0			0
$337$	−467.297			−1.38664			−0.693319	−	0.720631i	$-0.743852\pi$
$337$	−467.297			−1.38664			−0.693319	+	0.720631i	$0.743852\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−352.420	+	352.420i	−1.03349	+	1.03349i
$342$		0			0
$343$	270.449			0.788480
$344$		0			0
$345$		0			0
$346$		0			0
$347$	22.0463	+	22.0463i	0.0635341	+	0.0635341i	0.738160	−	0.674626i	$-0.235695\pi$
$347$	22.0463	+	22.0463i	0.0635341	+	0.0635341i	−0.674626	+	0.738160i	$0.735695\pi$
$348$		0			0
$349$	158.622	+	158.622i	0.454506	+	0.454506i	0.896847	−	0.442341i	$-0.145852\pi$
$349$	158.622	+	158.622i	0.454506	+	0.454506i	−0.442341	+	0.896847i	$0.645852\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−404.451			−1.14575			−0.572877	−	0.819642i	$-0.694173\pi$
$353$	−404.451			−1.14575			−0.572877	+	0.819642i	$0.694173\pi$
$354$		0			0
$355$	623.511	−	623.511i	1.75637	−	1.75637i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	423.833			1.18059			0.590297	−	0.807186i	$-0.299011\pi$
$359$	423.833			1.18059			0.590297	+	0.807186i	$0.299011\pi$
$360$		0			0
$361$			358.347i			0.992652i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	519.219	+	519.219i	1.42252	+	1.42252i
$366$		0			0
$367$		−	477.144i		−	1.30012i	−0.759883	−	0.650059i	$-0.774744\pi$
$367$		−	477.144i		−	1.30012i	0.759883	−	0.650059i	$-0.225256\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−82.4365	+	82.4365i	−0.222201	+	0.222201i
$372$		0			0
$373$	−112.221	+	112.221i	−0.300860	+	0.300860i	−0.841350	−	0.540490i	$-0.818239\pi$
$373$	−112.221	+	112.221i	−0.300860	+	0.300860i	0.540490	+	0.841350i	$0.318239\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			141.069i			0.374188i
$378$		0			0
$379$	52.2069	+	52.2069i	0.137749	+	0.137749i	0.772619	−	0.634870i	$-0.218946\pi$
$379$	52.2069	+	52.2069i	0.137749	+	0.137749i	−0.634870	+	0.772619i	$0.718946\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			74.8407i			0.195406i	0.995216	+	0.0977032i	$0.0311496\pi$
$383$			74.8407i			0.195406i	−0.995216	+	0.0977032i	$0.968850\pi$
$384$		0			0
$385$	258.219			0.670699
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−57.0441	+	57.0441i	−0.146643	+	0.146643i	−0.776617	−	0.629974i	$-0.783066\pi$
$389$	−57.0441	+	57.0441i	−0.146643	+	0.146643i	0.629974	+	0.776617i	$0.283066\pi$
$390$		0			0
$391$	61.1223			0.156323
$392$		0			0
$393$		0			0
$394$		0			0
$395$	628.910	+	628.910i	1.59218	+	1.59218i
$396$		0			0
$397$	355.874	+	355.874i	0.896407	+	0.896407i	0.995116	−	0.0987089i	$-0.0314713\pi$
$397$	355.874	+	355.874i	0.896407	+	0.896407i	−0.0987089	+	0.995116i	$0.531471\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−113.892			−0.284019			−0.142010	−	0.989865i	$-0.545356\pi$
$401$	−113.892			−0.284019			−0.142010	+	0.989865i	$0.545356\pi$
$402$		0			0
$403$	−222.731	+	222.731i	−0.552682	+	0.552682i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	1.77973			0.00437281
$408$		0			0
$409$			139.909i			0.342077i	0.985264	+	0.171038i	$0.0547122\pi$
$409$			139.909i			0.342077i	−0.985264	+	0.171038i	$0.945288\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	59.7394	+	59.7394i	0.144648	+	0.144648i
$414$		0			0
$415$		−	693.388i		−	1.67081i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	370.978	−	370.978i	0.885389	−	0.885389i	−0.108687	−	0.994076i	$-0.534665\pi$
$419$	370.978	−	370.978i	0.885389	−	0.885389i	0.994076	+	0.108687i	$0.0346647\pi$
$420$		0			0
$421$	−465.112	+	465.112i	−1.10478	+	1.10478i	−0.110955	+	0.993825i	$0.535391\pi$
$421$	−465.112	+	465.112i	−1.10478	+	1.10478i	−0.993825	+	0.110955i	$0.964609\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			314.839i			0.740797i
$426$		0			0
$427$	50.9923	+	50.9923i	0.119420	+	0.119420i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			409.924i			0.951099i	0.879689	+	0.475549i	$0.157751\pi$
$431$			409.924i			0.951099i	−0.879689	+	0.475549i	$0.842249\pi$
$432$		0			0
$433$	−20.6859			−0.0477735			−0.0238868	−	0.999715i	$-0.507604\pi$
$433$	−20.6859			−0.0477735			−0.0238868	+	0.999715i	$0.507604\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	3.98468	−	3.98468i	0.00911827	−	0.00911827i
$438$		0			0
$439$	63.2889			0.144166			0.0720830	−	0.997399i	$-0.477035\pi$
$439$	63.2889			0.144166			0.0720830	+	0.997399i	$0.477035\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	297.084	+	297.084i	0.670619	+	0.670619i	0.957859	−	0.287240i	$-0.0927377\pi$
$443$	297.084	+	297.084i	0.670619	+	0.670619i	−0.287240	+	0.957859i	$0.592738\pi$
$444$		0			0
$445$	145.423	+	145.423i	0.326793	+	0.326793i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−364.701			−0.812251			−0.406126	−	0.913817i	$-0.633120\pi$
$449$	−364.701			−0.812251			−0.406126	+	0.913817i	$0.633120\pi$
$450$		0			0
$451$	471.553	−	471.553i	1.04557	−	1.04557i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	163.196			0.358672
$456$		0			0
$457$		−	640.046i		−	1.40054i	−0.713879	−	0.700269i	$-0.753064\pi$
$457$		−	640.046i		−	1.40054i	0.713879	−	0.700269i	$-0.246936\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−239.416	−	239.416i	−0.519341	−	0.519341i	0.398031	−	0.917372i	$-0.369694\pi$
$461$	−239.416	−	239.416i	−0.519341	−	0.519341i	−0.917372	+	0.398031i	$0.869694\pi$
$462$		0			0
$463$			479.413i			1.03545i	0.855548	+	0.517724i	$0.173221\pi$
$463$			479.413i			1.03545i	−0.855548	+	0.517724i	$0.826779\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−403.375	+	403.375i	−0.863758	+	0.863758i	−0.991772	−	0.128015i	$-0.959140\pi$
$467$	−403.375	+	403.375i	−0.863758	+	0.863758i	0.128015	+	0.991772i	$0.459140\pi$
$468$		0			0
$469$	231.276	−	231.276i	0.493125	−	0.493125i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	31.1471i		−	0.0658501i
$474$		0			0
$475$	20.5250	+	20.5250i	0.0432104	+	0.0432104i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			460.611i			0.961609i	0.876828	+	0.480805i	$0.159655\pi$
$479$			460.611i			0.961609i	−0.876828	+	0.480805i	$0.840345\pi$
$480$		0			0
$481$	1.12480			0.00233846
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−145.968	+	145.968i	−0.300966	+	0.300966i
$486$		0			0
$487$	575.128			1.18096			0.590481	−	0.807052i	$-0.298938\pi$
$487$	575.128			1.18096			0.590481	+	0.807052i	$0.298938\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−271.375	−	271.375i	−0.552699	−	0.552699i	0.374520	−	0.927219i	$-0.377808\pi$
$491$	−271.375	−	271.375i	−0.552699	−	0.552699i	−0.927219	+	0.374520i	$0.877808\pi$
$492$		0			0
$493$	−215.431	−	215.431i	−0.436979	−	0.436979i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	410.832			0.826624
$498$		0			0
$499$	268.082	−	268.082i	0.537239	−	0.537239i	−0.385478	−	0.922717i	$-0.625963\pi$
$499$	268.082	−	268.082i	0.537239	−	0.537239i	0.922717	+	0.385478i	$0.125963\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−368.002			−0.731615			−0.365807	−	0.930691i	$-0.619207\pi$
$503$	−368.002			−0.731615			−0.365807	+	0.930691i	$0.619207\pi$
$504$		0			0
$505$			254.215i			0.503395i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	297.809	+	297.809i	0.585087	+	0.585087i	0.936297	−	0.351210i	$-0.114230\pi$
$509$	297.809	+	297.809i	0.585087	+	0.585087i	−0.351210	+	0.936297i	$0.614230\pi$
$510$		0			0
$511$			342.114i			0.669498i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	476.057	−	476.057i	0.924382	−	0.924382i
$516$		0			0
$517$	221.042	−	221.042i	0.427548	−	0.427548i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	95.5605i		−	0.183418i	−0.995786	−	0.0917088i	$-0.970767\pi$
$521$		−	95.5605i		−	0.183418i	0.995786	−	0.0917088i	$-0.0292329\pi$
$522$		0			0
$523$	−250.389	−	250.389i	−0.478756	−	0.478756i	0.425978	−	0.904734i	$-0.359930\pi$
$523$	−250.389	−	250.389i	−0.478756	−	0.478756i	−0.904734	+	0.425978i	$0.859930\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	680.279i		−	1.29085i
$528$		0			0
$529$	−517.029			−0.977370
$530$		0			0
$531$		0			0
$532$		0			0
$533$	298.024	−	298.024i	0.559144	−	0.559144i
$534$		0			0
$535$	−737.212			−1.37797
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−363.360	−	363.360i	−0.674138	−	0.674138i
$540$		0			0
$541$	81.7015	+	81.7015i	0.151019	+	0.151019i	0.778573	−	0.627554i	$-0.215944\pi$
$541$	81.7015	+	81.7015i	0.151019	+	0.151019i	−0.627554	+	0.778573i	$0.715944\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	1160.58			2.12951
$546$		0			0
$547$	−381.162	+	381.162i	−0.696823	+	0.696823i	−0.963724	−	0.266901i	$-0.914000\pi$
$547$	−381.162	+	381.162i	−0.696823	+	0.696823i	0.266901	+	0.963724i	$0.414000\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−28.0887			−0.0509777
$552$		0			0
$553$			414.389i			0.749348i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−63.7634	−	63.7634i	−0.114476	−	0.114476i	0.647548	−	0.762025i	$-0.275794\pi$
$557$	−63.7634	−	63.7634i	−0.114476	−	0.114476i	−0.762025	+	0.647548i	$0.775794\pi$
$558$		0			0
$559$		−	19.6851i		−	0.0352149i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−333.679	+	333.679i	−0.592681	+	0.592681i	−0.938355	−	0.345674i	$-0.887650\pi$
$563$	−333.679	+	333.679i	−0.592681	+	0.592681i	0.345674	+	0.938355i	$0.387650\pi$
$564$		0			0
$565$	−447.081	+	447.081i	−0.791293	+	0.791293i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	93.3114i		−	0.163992i	−0.996633	−	0.0819960i	$-0.973871\pi$
$569$		−	93.3114i		−	0.163992i	0.996633	−	0.0819960i	$-0.0261295\pi$
$570$		0			0
$571$	196.999	+	196.999i	0.345007	+	0.345007i	0.858246	−	0.513239i	$-0.171555\pi$
$571$	196.999	+	196.999i	0.345007	+	0.345007i	−0.513239	+	0.858246i	$0.671555\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			61.6644i			0.107242i
$576$		0			0
$577$	370.057			0.641347			0.320673	−	0.947190i	$-0.396091\pi$
$577$	370.057			0.641347			0.320673	+	0.947190i	$0.396091\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	228.437	−	228.437i	0.393179	−	0.393179i
$582$		0			0
$583$	494.890			0.848869
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−328.063	−	328.063i	−0.558880	−	0.558880i	0.370108	−	0.928989i	$-0.379321\pi$
$587$	−328.063	−	328.063i	−0.558880	−	0.558880i	−0.928989	+	0.370108i	$0.879321\pi$
$588$		0			0
$589$	−44.3488	−	44.3488i	−0.0752950	−	0.0752950i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−1088.78			−1.83605			−0.918024	−	0.396525i	$-0.870216\pi$
$593$	−1088.78			−1.83605			−0.918024	+	0.396525i	$0.870216\pi$
$594$		0			0
$595$	−249.222	+	249.222i	−0.418860	+	0.418860i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−350.354			−0.584899			−0.292449	−	0.956281i	$-0.594470\pi$
$599$	−350.354			−0.584899			−0.292449	+	0.956281i	$0.594470\pi$
$600$		0			0
$601$		−	1021.45i		−	1.69958i	−0.527123	−	0.849789i	$-0.676729\pi$
$601$		−	1021.45i		−	1.69958i	0.527123	−	0.849789i	$-0.323271\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−215.191	−	215.191i	−0.355687	−	0.355687i
$606$		0			0
$607$			394.204i			0.649431i	0.945812	+	0.324715i	$0.105268\pi$
$607$			394.204i			0.649431i	−0.945812	+	0.324715i	$0.894732\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	139.700	−	139.700i	0.228641	−	0.228641i
$612$		0			0
$613$	−157.606	+	157.606i	−0.257106	+	0.257106i	−0.823876	−	0.566770i	$-0.808193\pi$
$613$	−157.606	+	157.606i	−0.257106	+	0.257106i	0.566770	+	0.823876i	$0.308193\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	609.080i		−	0.987164i	−0.869699	−	0.493582i	$-0.835687\pi$
$617$		−	609.080i		−	0.987164i	0.869699	−	0.493582i	$-0.164313\pi$
$618$		0			0
$619$	−497.519	−	497.519i	−0.803747	−	0.803747i	0.179932	−	0.983679i	$-0.442412\pi$
$619$	−497.519	−	497.519i	−0.803747	−	0.803747i	−0.983679	+	0.179932i	$0.942412\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			95.8194i			0.153803i
$624$		0			0
$625$	752.924			1.20468
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−1.71772	+	1.71772i	−0.00273087	+	0.00273087i
$630$		0			0
$631$	−668.065			−1.05874			−0.529370	−	0.848391i	$-0.677572\pi$
$631$	−668.065			−1.05874			−0.529370	+	0.848391i	$0.677572\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−908.362	−	908.362i	−1.43049	−	1.43049i
$636$		0			0
$637$	−229.646	−	229.646i	−0.360511	−	0.360511i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	419.792			0.654902			0.327451	−	0.944868i	$-0.393810\pi$
$641$	419.792			0.654902			0.327451	+	0.944868i	$0.393810\pi$
$642$		0			0
$643$	−138.767	+	138.767i	−0.215813	+	0.215813i	−0.806731	−	0.590919i	$-0.798765\pi$
$643$	−138.767	+	138.767i	−0.215813	+	0.215813i	0.590919	+	0.806731i	$0.298765\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	647.036			1.00006			0.500028	−	0.866009i	$-0.333323\pi$
$647$	647.036			1.00006			0.500028	+	0.866009i	$0.333323\pi$
$648$		0			0
$649$		−	358.633i		−	0.552594i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−452.293	−	452.293i	−0.692639	−	0.692639i	0.270173	−	0.962812i	$-0.412919\pi$
$653$	−452.293	−	452.293i	−0.692639	−	0.692639i	−0.962812	+	0.270173i	$0.912919\pi$
$654$		0			0
$655$		−	166.059i		−	0.253525i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−382.858	+	382.858i	−0.580969	+	0.580969i	−0.935169	−	0.354201i	$-0.884753\pi$
$659$	−382.858	+	382.858i	−0.580969	+	0.580969i	0.354201	+	0.935169i	$0.384753\pi$
$660$		0			0
$661$	841.606	−	841.606i	1.27323	−	1.27323i	0.328849	−	0.944383i	$-0.393339\pi$
$661$	841.606	−	841.606i	1.27323	−	1.27323i	0.944383	−	0.328849i	$-0.106661\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			32.4945i			0.0488639i
$666$		0			0
$667$	−42.1944	−	42.1944i	−0.0632599	−	0.0632599i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	306.122i		−	0.456218i
$672$		0			0
$673$	−506.103			−0.752010			−0.376005	−	0.926618i	$-0.622703\pi$
$673$	−506.103			−0.752010			−0.376005	+	0.926618i	$0.622703\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−430.816	+	430.816i	−0.636361	+	0.636361i	−0.949656	−	0.313295i	$-0.898567\pi$
$677$	−430.816	+	430.816i	−0.636361	+	0.636361i	0.313295	+	0.949656i	$0.398567\pi$
$678$		0			0
$679$	−96.1787			−0.141648
$680$		0			0
$681$		0			0
$682$		0			0
$683$	910.083	+	910.083i	1.33248	+	1.33248i	0.903146	+	0.429333i	$0.141251\pi$
$683$	910.083	+	910.083i	1.33248	+	1.33248i	0.429333	+	0.903146i	$0.358749\pi$
$684$		0			0
$685$	−239.319	−	239.319i	−0.349371	−	0.349371i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	312.773			0.453952
$690$		0			0
$691$	−601.836	+	601.836i	−0.870964	+	0.870964i	−0.992577	−	0.121614i	$-0.961193\pi$
$691$	−601.836	+	601.836i	−0.870964	+	0.870964i	0.121614	+	0.992577i	$0.461193\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	161.619			0.232545
$696$		0			0
$697$			910.244i			1.30595i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−555.343	−	555.343i	−0.792215	−	0.792215i	0.189639	−	0.981854i	$-0.439268\pi$
$701$	−555.343	−	555.343i	−0.792215	−	0.792215i	−0.981854	+	0.189639i	$0.939268\pi$
$702$		0			0
$703$			0.223963i			0.000318582i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−83.7511	+	83.7511i	−0.118460	+	0.118460i
$708$		0			0
$709$	412.979	−	412.979i	0.582480	−	0.582480i	−0.353104	−	0.935584i	$-0.614874\pi$
$709$	412.979	−	412.979i	0.582480	−	0.582480i	0.935584	+	0.353104i	$0.114874\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	133.240i		−	0.186872i
$714$		0			0
$715$	−489.856	−	489.856i	−0.685114	−	0.685114i
$716$		0			0
$717$		0			0
$718$		0			0
$719$			1173.98i			1.63279i	0.577495	+	0.816394i	$0.304030\pi$
$719$			1173.98i			1.63279i	−0.577495	+	0.816394i	$0.695970\pi$
$720$		0			0
$721$	313.674			0.435054
$722$		0			0
$723$		0			0
$724$		0			0
$725$	217.342	−	217.342i	0.299781	−	0.299781i
$726$		0			0
$727$	−678.813			−0.933718			−0.466859	−	0.884332i	$-0.654614\pi$
$727$	−678.813			−0.933718			−0.466859	+	0.884332i	$0.654614\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	30.0618	+	30.0618i	0.0411242	+	0.0411242i
$732$		0			0
$733$	−336.854	−	336.854i	−0.459556	−	0.459556i	0.438954	−	0.898510i	$-0.355349\pi$
$733$	−336.854	−	336.854i	−0.459556	−	0.459556i	−0.898510	+	0.438954i	$0.855349\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−1388.42			−1.88387
$738$		0			0
$739$	178.478	−	178.478i	0.241513	−	0.241513i	−0.575963	−	0.817476i	$-0.695373\pi$
$739$	178.478	−	178.478i	0.241513	−	0.241513i	0.817476	+	0.575963i	$0.195373\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	795.320			1.07042			0.535208	−	0.844720i	$-0.320233\pi$
$743$	795.320			1.07042			0.535208	+	0.844720i	$0.320233\pi$
$744$		0			0
$745$			110.285i			0.148034i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−242.875	−	242.875i	−0.324266	−	0.324266i
$750$		0			0
$751$		−	102.850i		−	0.136951i	−0.997653	−	0.0684755i	$-0.978186\pi$
$751$		−	102.850i		−	0.136951i	0.997653	−	0.0684755i	$-0.0218135\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−613.330	+	613.330i	−0.812358	+	0.812358i
$756$		0			0
$757$	−48.6324	+	48.6324i	−0.0642436	+	0.0642436i	−0.738499	−	0.674255i	$-0.764465\pi$
$757$	−48.6324	+	48.6324i	−0.0642436	+	0.0642436i	0.674255	+	0.738499i	$0.264465\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			947.802i			1.24547i	0.782433	+	0.622734i	$0.213978\pi$
$761$			947.802i			1.24547i	−0.782433	+	0.622734i	$0.786022\pi$
$762$		0			0
$763$	382.354	+	382.354i	0.501119	+	0.501119i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	226.658i		−	0.295512i
$768$		0			0
$769$	−183.427			−0.238527			−0.119263	−	0.992863i	$-0.538053\pi$
$769$	−183.427			−0.238527			−0.119263	+	0.992863i	$0.538053\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−178.338	+	178.338i	−0.230710	+	0.230710i	−0.812989	−	0.582279i	$-0.802161\pi$
$773$	−178.338	+	178.338i	−0.230710	+	0.230710i	0.582279	+	0.812989i	$0.302161\pi$
$774$		0			0
$775$	686.312			0.885564
$776$		0			0
$777$		0			0
$778$		0			0
$779$	59.3406	+	59.3406i	0.0761754	+	0.0761754i
$780$		0			0
$781$	−1233.17	−	1233.17i	−1.57897	−	1.57897i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	981.815			1.25072
$786$		0			0
$787$	−480.981	+	480.981i	−0.611158	+	0.611158i	−0.943248	−	0.332090i	$-0.892246\pi$
$787$	−480.981	+	480.981i	−0.611158	+	0.611158i	0.332090	+	0.943248i	$0.392246\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−294.582			−0.372417
$792$		0			0
$793$		−	193.471i		−	0.243973i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−558.478	−	558.478i	−0.700725	−	0.700725i	0.263841	−	0.964566i	$-0.415011\pi$
$797$	−558.478	−	558.478i	−0.700725	−	0.700725i	−0.964566	+	0.263841i	$0.915011\pi$
$798$		0			0
$799$			426.680i			0.534017i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	1026.90	−	1026.90i	1.27884	−	1.27884i
$804$		0			0
$805$	−48.8126	+	48.8126i	−0.0606368	+	0.0606368i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			1152.43i			1.42451i	0.701918	+	0.712257i	$0.252327\pi$
$809$			1152.43i			1.42451i	−0.701918	+	0.712257i	$0.747673\pi$
$810$		0			0
$811$	364.890	+	364.890i	0.449926	+	0.449926i	0.895330	−	0.445404i	$-0.146940\pi$
$811$	364.890	+	364.890i	0.449926	+	0.449926i	−0.445404	+	0.895330i	$0.646940\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		−	977.055i		−	1.19884i
$816$		0			0
$817$	3.91958			0.00479753
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−618.975	+	618.975i	−0.753928	+	0.753928i	−0.975210	−	0.221282i	$-0.928976\pi$
$821$	−618.975	+	618.975i	−0.753928	+	0.753928i	0.221282	+	0.975210i	$0.428976\pi$
$822$		0			0
$823$	−626.066			−0.760712			−0.380356	−	0.924840i	$-0.624199\pi$
$823$	−626.066			−0.760712			−0.380356	+	0.924840i	$0.624199\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−375.666	−	375.666i	−0.454252	−	0.454252i	0.442511	−	0.896763i	$-0.354088\pi$
$827$	−375.666	−	375.666i	−0.454252	−	0.454252i	−0.896763	+	0.442511i	$0.854088\pi$
$828$		0			0
$829$	299.648	+	299.648i	0.361457	+	0.361457i	0.864349	−	0.502892i	$-0.167731\pi$
$829$	299.648	+	299.648i	0.361457	+	0.361457i	−0.502892	+	0.864349i	$0.667731\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	701.398			0.842015
$834$		0			0
$835$	−516.356	+	516.356i	−0.618390	+	0.618390i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−1477.80			−1.76138			−0.880689	−	0.473694i	$-0.842920\pi$
$839$	−1477.80			−1.76138			−0.880689	+	0.473694i	$0.842920\pi$
$840$		0			0
$841$		−	543.565i		−	0.646331i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	472.407	+	472.407i	0.559062	+	0.559062i
$846$		0			0
$847$		−	141.789i		−	0.167402i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−0.336433	+	0.336433i	−0.000395338	+	0.000395338i
$852$		0			0
$853$	−404.051	+	404.051i	−0.473682	+	0.473682i	−0.903104	−	0.429422i	$-0.858717\pi$
$853$	−404.051	+	404.051i	−0.473682	+	0.473682i	0.429422	+	0.903104i	$0.358717\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	892.363i		−	1.04126i	−0.853781	−	0.520632i	$-0.825696\pi$
$857$		−	892.363i		−	1.04126i	0.853781	−	0.520632i	$-0.174304\pi$
$858$		0			0
$859$	−378.424	−	378.424i	−0.440540	−	0.440540i	0.451654	−	0.892193i	$-0.350834\pi$
$859$	−378.424	−	378.424i	−0.440540	−	0.440540i	−0.892193	+	0.451654i	$0.850834\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			1457.30i			1.68865i	0.535833	+	0.844324i	$0.319998\pi$
$863$			1457.30i			1.68865i	−0.535833	+	0.844324i	$0.680002\pi$
$864$		0			0
$865$	−134.525			−0.155520
$866$		0			0
$867$		0			0
$868$		0			0
$869$	1243.85	−	1243.85i	1.43136	−	1.43136i
$870$		0			0
$871$	−877.485			−1.00745
$872$		0			0
$873$		0			0
$874$		0			0
$875$	101.263	+	101.263i	0.115729	+	0.115729i
$876$		0			0
$877$	571.322	+	571.322i	0.651450	+	0.651450i	0.953342	−	0.301892i	$-0.0976181\pi$
$877$	571.322	+	571.322i	0.651450	+	0.651450i	−0.301892	+	0.953342i	$0.597618\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−994.662			−1.12901			−0.564507	−	0.825428i	$-0.690934\pi$
$881$	−994.662			−1.12901			−0.564507	+	0.825428i	$0.690934\pi$
$882$		0			0
$883$	74.0725	−	74.0725i	0.0838873	−	0.0838873i	−0.663918	−	0.747805i	$-0.731108\pi$
$883$	74.0725	−	74.0725i	0.0838873	−	0.0838873i	0.747805	+	0.663918i	$0.231108\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−522.759			−0.589356			−0.294678	−	0.955597i	$-0.595212\pi$
$887$	−522.759			−0.589356			−0.294678	+	0.955597i	$0.595212\pi$
$888$		0			0
$889$		−	598.520i		−	0.673251i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	27.8161	+	27.8161i	0.0311491	+	0.0311491i
$894$		0			0
$895$			182.656i			0.204085i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−469.615	+	469.615i	−0.522375	+	0.522375i
$900$		0			0
$901$	−477.646	+	477.646i	−0.530129	+	0.530129i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	1556.16i		−	1.71951i
$906$		0			0
$907$	442.760	+	442.760i	0.488159	+	0.488159i	0.907725	−	0.419566i	$-0.137818\pi$
$907$	442.760	+	442.760i	0.488159	+	0.488159i	−0.419566	+	0.907725i	$0.637818\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	835.738i		−	0.917385i	−0.888595	−	0.458692i	$-0.848318\pi$
$911$		−	835.738i		−	0.917385i	0.888595	−	0.458692i	$-0.151682\pi$
$912$		0			0
$913$	−1371.37			−1.50205
$914$		0			0
$915$		0			0
$916$		0			0
$917$	54.7082	−	54.7082i	0.0596599	−	0.0596599i
$918$		0			0
$919$	−776.423			−0.844856			−0.422428	−	0.906396i	$-0.638822\pi$
$919$	−776.423			−0.844856			−0.422428	+	0.906396i	$0.638822\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−779.372	−	779.372i	−0.844390	−	0.844390i
$924$		0			0
$925$	−1.73295	−	1.73295i	−0.00187346	−	0.00187346i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	144.945			0.156022			0.0780112	−	0.996952i	$-0.475143\pi$
$929$	144.945			0.156022			0.0780112	+	0.996952i	$0.475143\pi$
$930$		0			0
$931$	45.7256	−	45.7256i	0.0491145	−	0.0491145i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	1496.15			1.60016
$936$		0			0
$937$		−	851.499i		−	0.908750i	−0.890811	−	0.454375i	$-0.849863\pi$
$937$		−	851.499i		−	0.908750i	0.890811	−	0.454375i	$-0.150137\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	1251.60	+	1251.60i	1.33008	+	1.33008i	0.905297	+	0.424778i	$0.139648\pi$
$941$	1251.60	+	1251.60i	1.33008	+	1.33008i	0.424778	+	0.905297i	$0.360352\pi$
$942$		0			0
$943$			178.281i			0.189057i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−919.818	+	919.818i	−0.971296	+	0.971296i	−0.999599	−	0.0283032i	$-0.990990\pi$
$947$	−919.818	+	919.818i	−0.971296	+	0.971296i	0.0283032	+	0.999599i	$0.490990\pi$
$948$		0			0
$949$	649.009	−	649.009i	0.683887	−	0.683887i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	489.450i		−	0.513589i	−0.966466	−	0.256794i	$-0.917334\pi$
$953$		−	489.450i		−	0.513589i	0.966466	−	0.256794i	$-0.0826663\pi$
$954$		0			0
$955$	−908.362	−	908.362i	−0.951164	−	0.951164i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	157.688i		−	0.164429i
$960$		0			0
$961$	−521.932			−0.543113
$962$		0			0
$963$		0			0
$964$		0			0
$965$	186.800	−	186.800i	0.193575	−	0.193575i
$966$		0			0
$967$	1368.49			1.41519			0.707594	−	0.706619i	$-0.249780\pi$
$967$	1368.49			1.41519			0.707594	+	0.706619i	$0.249780\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	1013.79	+	1013.79i	1.04407	+	1.04407i	0.998983	+	0.0450900i	$0.0143575\pi$
$971$	1013.79	+	1013.79i	1.04407	+	1.04407i	0.0450900	+	0.998983i	$0.485643\pi$
$972$		0			0
$973$	53.2455	+	53.2455i	0.0547230	+	0.0547230i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	5.19534			0.00531765			0.00265882	−	0.999996i	$-0.499154\pi$
$977$	5.19534			0.00531765			0.00265882	+	0.999996i	$0.499154\pi$
$978$		0			0
$979$	287.616	−	287.616i	0.293785	−	0.293785i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−1591.90			−1.61943			−0.809714	−	0.586825i	$-0.800378\pi$
$983$	−1591.90			−1.61943			−0.809714	+	0.586825i	$0.800378\pi$
$984$		0			0
$985$		−	2133.87i		−	2.16636i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	5.88792	+	5.88792i	0.00595340	+	0.00595340i
$990$		0			0
$991$			622.896i			0.628553i	0.949331	+	0.314277i	$0.101762\pi$
$991$			622.896i			0.628553i	−0.949331	+	0.314277i	$0.898238\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	284.850	−	284.850i	0.286282	−	0.286282i
$996$		0			0
$997$	635.503	−	635.503i	0.637415	−	0.637415i	−0.312502	−	0.949917i	$-0.601167\pi$
$997$	635.503	−	635.503i	0.637415	−	0.637415i	0.949917	+	0.312502i	$0.101167\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1152.3.m.b.991.1		6
3.2	odd	2		128.3.f.a.95.3		6
4.3	odd	2		1152.3.m.a.991.1		6
8.3	odd	2		144.3.m.a.19.3		6
8.5	even	2		576.3.m.a.559.3		6
12.11	even	2		128.3.f.b.95.1		6
16.3	odd	4		576.3.m.a.271.3		6
16.5	even	4		1152.3.m.a.415.1		6
16.11	odd	4	inner	1152.3.m.b.415.1		6
16.13	even	4		144.3.m.a.91.3		6
24.5	odd	2		64.3.f.a.47.1		6
24.11	even	2		16.3.f.a.3.1	✓	6
48.5	odd	4		128.3.f.b.31.1		6
48.11	even	4		128.3.f.a.31.3		6
48.29	odd	4		16.3.f.a.11.1	yes	6
48.35	even	4		64.3.f.a.15.1		6
96.5	odd	8		1024.3.c.j.1023.10		12
96.11	even	8		1024.3.c.j.1023.9		12
96.29	odd	8		1024.3.d.k.511.10		12
96.35	even	8		1024.3.d.k.511.4		12
96.53	odd	8		1024.3.c.j.1023.3		12
96.59	even	8		1024.3.c.j.1023.4		12
96.77	odd	8		1024.3.d.k.511.3		12
96.83	even	8		1024.3.d.k.511.9		12
120.59	even	2		400.3.r.c.51.3		6
120.83	odd	4		400.3.k.c.99.2		6
120.107	odd	4		400.3.k.d.99.2		6
240.29	odd	4		400.3.r.c.251.3		6
240.77	even	4		400.3.k.c.299.2		6
240.173	even	4		400.3.k.d.299.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.3.f.a.3.1	✓	6	24.11	even	2
16.3.f.a.11.1	yes	6	48.29	odd	4
64.3.f.a.15.1		6	48.35	even	4
64.3.f.a.47.1		6	24.5	odd	2
128.3.f.a.31.3		6	48.11	even	4
128.3.f.a.95.3		6	3.2	odd	2
128.3.f.b.31.1		6	48.5	odd	4
128.3.f.b.95.1		6	12.11	even	2
144.3.m.a.19.3		6	8.3	odd	2
144.3.m.a.91.3		6	16.13	even	4
400.3.k.c.99.2		6	120.83	odd	4
400.3.k.c.299.2		6	240.77	even	4
400.3.k.d.99.2		6	120.107	odd	4
400.3.k.d.299.2		6	240.173	even	4
400.3.r.c.51.3		6	120.59	even	2
400.3.r.c.251.3		6	240.29	odd	4
576.3.m.a.271.3		6	16.3	odd	4
576.3.m.a.559.3		6	8.5	even	2
1024.3.c.j.1023.3		12	96.53	odd	8
1024.3.c.j.1023.4		12	96.59	even	8
1024.3.c.j.1023.9		12	96.11	even	8
1024.3.c.j.1023.10		12	96.5	odd	8
1024.3.d.k.511.3		12	96.77	odd	8
1024.3.d.k.511.4		12	96.35	even	8
1024.3.d.k.511.9		12	96.83	even	8
1024.3.d.k.511.10		12	96.29	odd	8
1152.3.m.a.415.1		6	16.5	even	4
1152.3.m.a.991.1		6	4.3	odd	2
1152.3.m.b.415.1		6	16.11	odd	4	inner
1152.3.m.b.991.1		6	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+(-4.62721 + 4.62721i) q^{5} -3.04888 q^{7} +O(q^{10})\)	\(q+(-4.62721 + 4.62721i) q^{5} -3.04888 q^{7} +(9.15165 + 9.15165i) q^{11} +(5.78389 + 5.78389i) q^{13} -17.6655 q^{17} +(-1.15165 + 1.15165i) q^{19} -3.45998 q^{23} -17.8222i q^{25} +(12.1950 + 12.1950i) q^{29} +38.5089i q^{31} +(14.1078 - 14.1078i) q^{35} +(0.0972356 - 0.0972356i) q^{37} -51.5266i q^{41} +(-1.70172 - 1.70172i) q^{43} -24.1533i q^{47} -39.7044 q^{49} +(27.0383 - 27.0383i) q^{53} -84.6933 q^{55} +(-19.5939 - 19.5939i) q^{59} +(-16.7250 - 16.7250i) q^{61} -53.5266 q^{65} +(-75.8560 + 75.8560i) q^{67} -134.749 q^{71} -112.210i q^{73} +(-27.9022 - 27.9022i) q^{77} -135.915i q^{79} +(-74.9250 + 74.9250i) q^{83} +(81.7422 - 81.7422i) q^{85} -31.4278i q^{89} +(-17.6344 - 17.6344i) q^{91} -10.6579i q^{95} +31.5456 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{5} + 4 q^{7}+O(q^{10}) \) 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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more