LMFDB - Embedded newform 2160.3.c.k.1889.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2160,3,Mod(1889,2160)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2160.1889"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2160, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 1])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 2160 = 2^{4} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2160.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,124] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	no
Analytic conductor:	$58.8557371018$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{10})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 25 $ x^4 + 25
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	no (minimal twist has level 135)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1889.2
Root			$-1.58114 - 1.58114i$ of defining polynomial
Character	$\chi$	$=$	2160.1889
Dual form			2160.3.c.k.1889.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.58114 + 4.74342i) q^{5} +3.00000i q^{7} +9.48683i q^{11} +21.0000i q^{13} -12.6491 q^{17} +31.0000 q^{19} +22.1359 q^{23} +(-20.0000 - 15.0000i) q^{25} +47.4342i q^{29} +16.0000 q^{31} +(-14.2302 - 4.74342i) q^{35} +27.0000i q^{37} -47.4342i q^{41} +48.0000i q^{43} +12.6491 q^{47} +40.0000 q^{49} -41.1096 q^{53} +(-45.0000 - 15.0000i) q^{55} -37.9473i q^{59} -1.00000 q^{61} +(-99.6117 - 33.2039i) q^{65} -21.0000i q^{67} -28.4605i q^{71} +27.0000i q^{73} -28.4605 q^{77} +1.00000 q^{79} -110.680 q^{83} +(20.0000 - 60.0000i) q^{85} +113.842i q^{89} -63.0000 q^{91} +(-49.0153 + 147.046i) q^{95} -93.0000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 124 q^{19} - 80 q^{25} + 64 q^{31} + 160 q^{49} - 180 q^{55} - 4 q^{61} + 4 q^{79} + 80 q^{85} - 252 q^{91}+O(q^{100}) $ 4 * q + 124 * q^19 - 80 * q^25 + 64 * q^31 + 160 * q^49 - 180 * q^55 - 4 * q^61 + 4 * q^79 + 80 * q^85 - 252 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2160\mathbb{Z}\right)^\times$.

$n$	$271$	$1297$	$1621$	$2081$
$\chi(n)$	$1$	$-1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−1.58114	+	4.74342i	−0.316228	+	0.948683i
$5$	−1.58114	+	4.74342i	−0.316228	+	0.948683i
$6$		0			0
$7$			3.00000i			0.428571i	0.976771	+	0.214286i	$0.0687424\pi$
$7$			3.00000i			0.428571i	−0.976771	+	0.214286i	$0.931258\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			9.48683i			0.862439i	0.902247	+	0.431220i	$0.141917\pi$
$11$			9.48683i			0.862439i	−0.902247	+	0.431220i	$0.858083\pi$
$12$		0			0
$13$			21.0000i			1.61538i	0.589604	+	0.807692i	$0.299284\pi$
$13$			21.0000i			1.61538i	−0.589604	+	0.807692i	$0.700716\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−12.6491			−0.744065			−0.372033	−	0.928220i	$-0.621339\pi$
$17$	−12.6491			−0.744065			−0.372033	+	0.928220i	$0.621339\pi$
$18$		0			0
$19$	31.0000			1.63158			0.815789	−	0.578349i	$-0.196303\pi$
$19$	31.0000			1.63158			0.815789	+	0.578349i	$0.196303\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	22.1359			0.962432			0.481216	−	0.876602i	$-0.340195\pi$
$23$	22.1359			0.962432			0.481216	+	0.876602i	$0.340195\pi$
$24$		0			0
$25$	−20.0000	−	15.0000i	−0.800000	−	0.600000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			47.4342i			1.63566i	0.575459	+	0.817830i	$0.304823\pi$
$29$			47.4342i			1.63566i	−0.575459	+	0.817830i	$0.695177\pi$
$30$		0			0
$31$	16.0000			0.516129			0.258065	−	0.966128i	$-0.416915\pi$
$31$	16.0000			0.516129			0.258065	+	0.966128i	$0.416915\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−14.2302	−	4.74342i	−0.406579	−	0.135526i
$36$		0			0
$37$			27.0000i			0.729730i	0.931060	+	0.364865i	$0.118885\pi$
$37$			27.0000i			0.729730i	−0.931060	+	0.364865i	$0.881115\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	47.4342i		−	1.15693i	−0.815707	−	0.578465i	$-0.803652\pi$
$41$		−	47.4342i		−	1.15693i	0.815707	−	0.578465i	$-0.196348\pi$
$42$		0			0
$43$			48.0000i			1.11628i	0.829747	+	0.558140i	$0.188485\pi$
$43$			48.0000i			1.11628i	−0.829747	+	0.558140i	$0.811515\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	12.6491			0.269130			0.134565	−	0.990905i	$-0.457036\pi$
$47$	12.6491			0.269130			0.134565	+	0.990905i	$0.457036\pi$
$48$		0			0
$49$	40.0000			0.816327
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−41.1096			−0.775653			−0.387827	−	0.921732i	$-0.626774\pi$
$53$	−41.1096			−0.775653			−0.387827	+	0.921732i	$0.626774\pi$
$54$		0			0
$55$	−45.0000	−	15.0000i	−0.818182	−	0.272727i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	37.9473i		−	0.643175i	−0.946880	−	0.321588i	$-0.895784\pi$
$59$		−	37.9473i		−	0.643175i	0.946880	−	0.321588i	$-0.104216\pi$
$60$		0			0
$61$	−1.00000			−0.0163934			−0.00819672	−	0.999966i	$-0.502609\pi$
$61$	−1.00000			−0.0163934			−0.00819672	+	0.999966i	$0.502609\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−99.6117	−	33.2039i	−1.53249	−	0.510829i
$66$		0			0
$67$		−	21.0000i		−	0.313433i	−0.987644	−	0.156716i	$-0.949909\pi$
$67$		−	21.0000i		−	0.313433i	0.987644	−	0.156716i	$-0.0500909\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	28.4605i		−	0.400852i	−0.979709	−	0.200426i	$-0.935767\pi$
$71$		−	28.4605i		−	0.400852i	0.979709	−	0.200426i	$-0.0642326\pi$
$72$		0			0
$73$			27.0000i			0.369863i	0.982751	+	0.184932i	$0.0592063\pi$
$73$			27.0000i			0.369863i	−0.982751	+	0.184932i	$0.940794\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−28.4605			−0.369617
$78$		0			0
$79$	1.00000			0.0126582			0.00632911	−	0.999980i	$-0.497985\pi$
$79$	1.00000			0.0126582			0.00632911	+	0.999980i	$0.497985\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−110.680			−1.33349			−0.666745	−	0.745286i	$-0.732313\pi$
$83$	−110.680			−1.33349			−0.666745	+	0.745286i	$0.732313\pi$
$84$		0			0
$85$	20.0000	−	60.0000i	0.235294	−	0.705882i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			113.842i			1.27912i	0.768740	+	0.639562i	$0.220884\pi$
$89$			113.842i			1.27912i	−0.768740	+	0.639562i	$0.779116\pi$
$90$		0			0
$91$	−63.0000			−0.692308
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−49.0153	+	147.046i	−0.515951	+	1.54785i
$96$		0			0
$97$		−	93.0000i		−	0.958763i	−0.877607	−	0.479381i	$-0.840861\pi$
$97$		−	93.0000i		−	0.958763i	0.877607	−	0.479381i	$-0.159139\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			75.8947i			0.751432i	0.926735	+	0.375716i	$0.122603\pi$
$101$			75.8947i			0.751432i	−0.926735	+	0.375716i	$0.877397\pi$
$102$		0			0
$103$		−	147.000i		−	1.42718i	−0.700561	−	0.713592i	$-0.747067\pi$
$103$		−	147.000i		−	1.42718i	0.700561	−	0.713592i	$-0.252933\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	22.1359			0.206878			0.103439	−	0.994636i	$-0.467015\pi$
$107$	22.1359			0.206878			0.103439	+	0.994636i	$0.467015\pi$
$108$		0			0
$109$	104.000			0.954128			0.477064	−	0.878868i	$-0.341701\pi$
$109$	104.000			0.954128			0.477064	+	0.878868i	$0.341701\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−173.925			−1.53916			−0.769581	−	0.638549i	$-0.779535\pi$
$113$	−173.925			−1.53916			−0.769581	+	0.638549i	$0.779535\pi$
$114$		0			0
$115$	−35.0000	+	105.000i	−0.304348	+	0.913043i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	37.9473i		−	0.318885i
$120$		0			0
$121$	31.0000			0.256198
$122$		0			0
$123$		0			0
$124$		0			0
$125$	102.774	−	71.1512i	0.822192	−	0.569210i
$126$		0			0
$127$			144.000i			1.13386i	0.823767	+	0.566929i	$0.191869\pi$
$127$			144.000i			1.13386i	−0.823767	+	0.566929i	$0.808131\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			75.8947i			0.579349i	0.957125	+	0.289674i	$0.0935470\pi$
$131$			75.8947i			0.579349i	−0.957125	+	0.289674i	$0.906453\pi$
$132$		0			0
$133$			93.0000i			0.699248i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	53.7587			0.392399			0.196200	−	0.980564i	$-0.437140\pi$
$137$	53.7587			0.392399			0.196200	+	0.980564i	$0.437140\pi$
$138$		0			0
$139$	1.00000			0.00719424			0.00359712	−	0.999994i	$-0.498855\pi$
$139$	1.00000			0.00719424			0.00359712	+	0.999994i	$0.498855\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−199.223			−1.39317
$144$		0			0
$145$	−225.000	−	75.0000i	−1.55172	−	0.517241i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	75.8947i		−	0.509360i	−0.967025	−	0.254680i	$-0.918030\pi$
$149$		−	75.8947i		−	0.509360i	0.967025	−	0.254680i	$-0.0819702\pi$
$150$		0			0
$151$	31.0000			0.205298			0.102649	−	0.994718i	$-0.467268\pi$
$151$	31.0000			0.205298			0.102649	+	0.994718i	$0.467268\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−25.2982	+	75.8947i	−0.163214	+	0.489643i
$156$		0			0
$157$			102.000i			0.649682i	0.945769	+	0.324841i	$0.105311\pi$
$157$			102.000i			0.649682i	−0.945769	+	0.324841i	$0.894689\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			66.4078i			0.412471i
$162$		0			0
$163$		−	171.000i		−	1.04908i	−0.851386	−	0.524540i	$-0.824237\pi$
$163$		−	171.000i		−	1.04908i	0.851386	−	0.524540i	$-0.175763\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−290.930			−1.74209			−0.871047	−	0.491200i	$-0.836558\pi$
$167$	−290.930			−1.74209			−0.871047	+	0.491200i	$0.836558\pi$
$168$		0			0
$169$	−272.000			−1.60947
$170$		0			0
$171$		0			0
$172$		0			0
$173$	25.2982			0.146232			0.0731162	−	0.997323i	$-0.476706\pi$
$173$	25.2982			0.146232			0.0731162	+	0.997323i	$0.476706\pi$
$174$		0			0
$175$	45.0000	−	60.0000i	0.257143	−	0.342857i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	199.223i		−	1.11298i	−0.830854	−	0.556490i	$-0.812148\pi$
$179$		−	199.223i		−	1.11298i	0.830854	−	0.556490i	$-0.187852\pi$
$180$		0			0
$181$	−103.000			−0.569061			−0.284530	−	0.958667i	$-0.591838\pi$
$181$	−103.000			−0.569061			−0.284530	+	0.958667i	$0.591838\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−128.072	−	42.6907i	−0.692282	−	0.230761i
$186$		0			0
$187$		−	120.000i		−	0.641711i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			265.631i			1.39074i	0.718652	+	0.695370i	$0.244759\pi$
$191$			265.631i			1.39074i	−0.718652	+	0.695370i	$0.755241\pi$
$192$		0			0
$193$		−	213.000i		−	1.10363i	−0.833968	−	0.551813i	$-0.813936\pi$
$193$		−	213.000i		−	1.10363i	0.833968	−	0.551813i	$-0.186064\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−240.333			−1.21996			−0.609982	−	0.792415i	$-0.708824\pi$
$197$	−240.333			−1.21996			−0.609982	+	0.792415i	$0.708824\pi$
$198$		0			0
$199$	−47.0000			−0.236181			−0.118090	−	0.993003i	$-0.537677\pi$
$199$	−47.0000			−0.236181			−0.118090	+	0.993003i	$0.537677\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−142.302			−0.700998
$204$		0			0
$205$	225.000	+	75.0000i	1.09756	+	0.365854i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			294.092i			1.40714i
$210$		0			0
$211$	223.000			1.05687			0.528436	−	0.848973i	$-0.322779\pi$
$211$	223.000			1.05687			0.528436	+	0.848973i	$0.322779\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−227.684	−	75.8947i	−1.05900	−	0.352998i
$216$		0			0
$217$			48.0000i			0.221198i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	265.631i		−	1.20195i
$222$		0			0
$223$		−	312.000i		−	1.39910i	−0.714582	−	0.699552i	$-0.753383\pi$
$223$		−	312.000i		−	1.39910i	0.714582	−	0.699552i	$-0.246617\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−338.364			−1.49059			−0.745295	−	0.666735i	$-0.767691\pi$
$227$	−338.364			−1.49059			−0.745295	+	0.666735i	$0.767691\pi$
$228$		0			0
$229$	−16.0000			−0.0698690			−0.0349345	−	0.999390i	$-0.511122\pi$
$229$	−16.0000			−0.0698690			−0.0349345	+	0.999390i	$0.511122\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−50.5964			−0.217152			−0.108576	−	0.994088i	$-0.534629\pi$
$233$	−50.5964			−0.217152			−0.108576	+	0.994088i	$0.534629\pi$
$234$		0			0
$235$	−20.0000	+	60.0000i	−0.0851064	+	0.255319i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			104.355i			0.436632i	0.975878	+	0.218316i	$0.0700564\pi$
$239$			104.355i			0.436632i	−0.975878	+	0.218316i	$0.929944\pi$
$240$		0			0
$241$	287.000			1.19087			0.595436	−	0.803403i	$-0.296979\pi$
$241$	287.000			1.19087			0.595436	+	0.803403i	$0.296979\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−63.2456	+	189.737i	−0.258145	+	0.774435i
$246$		0			0
$247$			651.000i			2.63563i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			256.144i			1.02050i	0.860027	+	0.510248i	$0.170446\pi$
$251$			256.144i			1.02050i	−0.860027	+	0.510248i	$0.829554\pi$
$252$		0			0
$253$			210.000i			0.830040i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−278.280			−1.08280			−0.541402	−	0.840764i	$-0.682106\pi$
$257$	−278.280			−1.08280			−0.541402	+	0.840764i	$0.682106\pi$
$258$		0			0
$259$	−81.0000			−0.312741
$260$		0			0
$261$		0			0
$262$		0			0
$263$	354.175			1.34667			0.673337	−	0.739336i	$-0.264860\pi$
$263$	354.175			1.34667			0.673337	+	0.739336i	$0.264860\pi$
$264$		0			0
$265$	65.0000	−	195.000i	0.245283	−	0.735849i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	113.842i		−	0.423204i	−0.977356	−	0.211602i	$-0.932132\pi$
$269$		−	113.842i		−	0.423204i	0.977356	−	0.211602i	$-0.0678681\pi$
$270$		0			0
$271$	121.000			0.446494			0.223247	−	0.974762i	$-0.428334\pi$
$271$	121.000			0.446494			0.223247	+	0.974762i	$0.428334\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	142.302	−	189.737i	0.517464	−	0.689951i
$276$		0			0
$277$		−	48.0000i		−	0.173285i	−0.996239	−	0.0866426i	$-0.972386\pi$
$277$		−	48.0000i		−	0.173285i	0.996239	−	0.0866426i	$-0.0276138\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			303.579i			1.08035i	0.841552	+	0.540176i	$0.181642\pi$
$281$			303.579i			1.08035i	−0.841552	+	0.540176i	$0.818358\pi$
$282$		0			0
$283$		−	192.000i		−	0.678445i	−0.940706	−	0.339223i	$-0.889836\pi$
$283$		−	192.000i		−	0.678445i	0.940706	−	0.339223i	$-0.110164\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	142.302			0.495828
$288$		0			0
$289$	−129.000			−0.446367
$290$		0			0
$291$		0			0
$292$		0			0
$293$	205.548			0.701529			0.350765	−	0.936464i	$-0.385922\pi$
$293$	205.548			0.701529			0.350765	+	0.936464i	$0.385922\pi$
$294$		0			0
$295$	180.000	+	60.0000i	0.610169	+	0.203390i
$296$		0			0
$297$		0			0
$298$		0			0
$299$			464.855i			1.55470i
$300$		0			0
$301$	−144.000			−0.478405
$302$		0			0
$303$		0			0
$304$		0			0
$305$	1.58114	−	4.74342i	0.00518406	−	0.0155522i
$306$		0			0
$307$			198.000i			0.644951i	0.946578	+	0.322476i	$0.104515\pi$
$307$			198.000i			0.644951i	−0.946578	+	0.322476i	$0.895485\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	493.315i		−	1.58622i	−0.609077	−	0.793111i	$-0.708460\pi$
$311$		−	493.315i		−	1.58622i	0.609077	−	0.793111i	$-0.291540\pi$
$312$		0			0
$313$			411.000i			1.31310i	0.754283	+	0.656550i	$0.227985\pi$
$313$			411.000i			1.31310i	−0.754283	+	0.656550i	$0.772015\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	252.982			0.798051			0.399026	−	0.916940i	$-0.369348\pi$
$317$	252.982			0.798051			0.399026	+	0.916940i	$0.369348\pi$
$318$		0			0
$319$	−450.000			−1.41066
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−392.122			−1.21400
$324$		0			0
$325$	315.000	−	420.000i	0.969231	−	1.29231i
$326$		0			0
$327$		0			0
$328$		0			0
$329$			37.9473i			0.115341i
$330$		0			0
$331$	−329.000			−0.993958			−0.496979	−	0.867763i	$-0.665557\pi$
$331$	−329.000			−0.993958			−0.496979	+	0.867763i	$0.665557\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	99.6117	+	33.2039i	0.297348	+	0.0991162i
$336$		0			0
$337$			381.000i			1.13056i	0.824898	+	0.565282i	$0.191233\pi$
$337$			381.000i			1.13056i	−0.824898	+	0.565282i	$0.808767\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			151.789i			0.445130i
$342$		0			0
$343$			267.000i			0.778426i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	117.004			0.337188			0.168594	−	0.985686i	$-0.446077\pi$
$347$	117.004			0.337188			0.168594	+	0.985686i	$0.446077\pi$
$348$		0			0
$349$	89.0000			0.255014			0.127507	−	0.991838i	$-0.459302\pi$
$349$	89.0000			0.255014			0.127507	+	0.991838i	$0.459302\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−581.859			−1.64833			−0.824163	−	0.566353i	$-0.808354\pi$
$353$	−581.859			−1.64833			−0.824163	+	0.566353i	$0.808354\pi$
$354$		0			0
$355$	135.000	+	45.0000i	0.380282	+	0.126761i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			85.3815i			0.237831i	0.992904	+	0.118916i	$0.0379418\pi$
$359$			85.3815i			0.237831i	−0.992904	+	0.118916i	$0.962058\pi$
$360$		0			0
$361$	600.000			1.66205
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−128.072	−	42.6907i	−0.350883	−	0.116961i
$366$		0			0
$367$		−	51.0000i		−	0.138965i	−0.997583	−	0.0694823i	$-0.977865\pi$
$367$		−	51.0000i		−	0.138965i	0.997583	−	0.0694823i	$-0.0221347\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	123.329i		−	0.332423i
$372$		0			0
$373$			237.000i			0.635389i	0.948193	+	0.317694i	$0.102909\pi$
$373$			237.000i			0.635389i	−0.948193	+	0.317694i	$0.897091\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−996.117			−2.64222
$378$		0			0
$379$	103.000			0.271768			0.135884	−	0.990725i	$-0.456613\pi$
$379$	103.000			0.271768			0.135884	+	0.990725i	$0.456613\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	50.5964			0.132106			0.0660528	−	0.997816i	$-0.478959\pi$
$383$	50.5964			0.132106			0.0660528	+	0.997816i	$0.478959\pi$
$384$		0			0
$385$	45.0000	−	135.000i	0.116883	−	0.350649i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	265.631i		−	0.682857i	−0.939908	−	0.341428i	$-0.889089\pi$
$389$		−	265.631i		−	0.682857i	0.939908	−	0.341428i	$-0.110911\pi$
$390$		0			0
$391$	−280.000			−0.716113
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−1.58114	+	4.74342i	−0.00400288	+	0.0120086i
$396$		0			0
$397$			72.0000i			0.181360i	0.995880	+	0.0906801i	$0.0289041\pi$
$397$			72.0000i			0.181360i	−0.995880	+	0.0906801i	$0.971096\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	104.355i		−	0.260237i	−0.991498	−	0.130119i	$-0.958464\pi$
$401$		−	104.355i		−	0.260237i	0.991498	−	0.130119i	$-0.0415358\pi$
$402$		0			0
$403$			336.000i			0.833747i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−256.144			−0.629348
$408$		0			0
$409$	137.000			0.334963			0.167482	−	0.985875i	$-0.446437\pi$
$409$	137.000			0.334963			0.167482	+	0.985875i	$0.446437\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	113.842			0.275646
$414$		0			0
$415$	175.000	−	525.000i	0.421687	−	1.26506i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	37.9473i		−	0.0905664i	−0.998974	−	0.0452832i	$-0.985581\pi$
$419$		−	37.9473i		−	0.0905664i	0.998974	−	0.0452832i	$-0.0144190\pi$
$420$		0			0
$421$	−631.000			−1.49881			−0.749406	−	0.662111i	$-0.769661\pi$
$421$	−631.000			−1.49881			−0.749406	+	0.662111i	$0.769661\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	252.982	+	189.737i	0.595252	+	0.446439i
$426$		0			0
$427$		−	3.00000i		−	0.00702576i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	540.749i		−	1.25464i	−0.778762	−	0.627320i	$-0.784152\pi$
$431$		−	540.749i		−	1.25464i	0.778762	−	0.627320i	$-0.215848\pi$
$432$		0			0
$433$		−	288.000i		−	0.665127i	−0.943081	−	0.332564i	$-0.892086\pi$
$433$		−	288.000i		−	0.665127i	0.943081	−	0.332564i	$-0.107914\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	686.214			1.57028
$438$		0			0
$439$	856.000			1.94989			0.974943	−	0.222455i	$-0.0714069\pi$
$439$	856.000			1.94989			0.974943	+	0.222455i	$0.0714069\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	430.070			0.970812			0.485406	−	0.874289i	$-0.338672\pi$
$443$	430.070			0.970812			0.485406	+	0.874289i	$0.338672\pi$
$444$		0			0
$445$	−540.000	−	180.000i	−1.21348	−	0.404494i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	796.894i		−	1.77482i	−0.460981	−	0.887410i	$-0.652503\pi$
$449$		−	796.894i		−	1.77482i	0.460981	−	0.887410i	$-0.347497\pi$
$450$		0			0
$451$	450.000			0.997783
$452$		0			0
$453$		0			0
$454$		0			0
$455$	99.6117	−	298.835i	0.218927	−	0.656781i
$456$		0			0
$457$			96.0000i			0.210066i	0.994469	+	0.105033i	$0.0334948\pi$
$457$			96.0000i			0.210066i	−0.994469	+	0.105033i	$0.966505\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			417.421i			0.905468i	0.891646	+	0.452734i	$0.149551\pi$
$461$			417.421i			0.905468i	−0.891646	+	0.452734i	$0.850449\pi$
$462$		0			0
$463$		−	237.000i		−	0.511879i	−0.966693	−	0.255940i	$-0.917615\pi$
$463$		−	237.000i		−	0.511879i	0.966693	−	0.255940i	$-0.0823848\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	468.017			1.00218			0.501089	−	0.865396i	$-0.332933\pi$
$467$	468.017			1.00218			0.501089	+	0.865396i	$0.332933\pi$
$468$		0			0
$469$	63.0000			0.134328
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−455.368			−0.962723
$474$		0			0
$475$	−620.000	−	465.000i	−1.30526	−	0.978947i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			407.934i			0.851636i	0.904809	+	0.425818i	$0.140014\pi$
$479$			407.934i			0.851636i	−0.904809	+	0.425818i	$0.859986\pi$
$480$		0			0
$481$	−567.000			−1.17879
$482$		0			0
$483$		0			0
$484$		0			0
$485$	441.138	+	147.046i	0.909562	+	0.303187i
$486$		0			0
$487$		−	261.000i		−	0.535934i	−0.963428	−	0.267967i	$-0.913648\pi$
$487$		−	261.000i		−	0.535934i	0.963428	−	0.267967i	$-0.0863519\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	749.460i		−	1.52639i	−0.646165	−	0.763197i	$-0.723628\pi$
$491$		−	749.460i		−	1.52639i	0.646165	−	0.763197i	$-0.276372\pi$
$492$		0			0
$493$		−	600.000i		−	1.21704i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	85.3815			0.171794
$498$		0			0
$499$	106.000			0.212425			0.106212	−	0.994343i	$-0.466128\pi$
$499$	106.000			0.212425			0.106212	+	0.994343i	$0.466128\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	382.636			0.760707			0.380353	−	0.924841i	$-0.375802\pi$
$503$	382.636			0.760707			0.380353	+	0.924841i	$0.375802\pi$
$504$		0			0
$505$	−360.000	−	120.000i	−0.712871	−	0.237624i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	161.276i		−	0.316849i	−0.987371	−	0.158425i	$-0.949359\pi$
$509$		−	161.276i		−	0.316849i	0.987371	−	0.158425i	$-0.0506415\pi$
$510$		0			0
$511$	−81.0000			−0.158513
$512$		0			0
$513$		0			0
$514$		0			0
$515$	697.282	+	232.427i	1.35395	+	0.451315i
$516$		0			0
$517$			120.000i			0.232108i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	369.986i		−	0.710147i	−0.934838	−	0.355073i	$-0.884456\pi$
$521$		−	369.986i		−	0.710147i	0.934838	−	0.355073i	$-0.115544\pi$
$522$		0			0
$523$		−	837.000i		−	1.60038i	−0.599745	−	0.800191i	$-0.704731\pi$
$523$		−	837.000i		−	1.60038i	0.599745	−	0.800191i	$-0.295269\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−202.386			−0.384034
$528$		0			0
$529$	−39.0000			−0.0737240
$530$		0			0
$531$		0			0
$532$		0			0
$533$	996.117			1.86889
$534$		0			0
$535$	−35.0000	+	105.000i	−0.0654206	+	0.196262i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			379.473i			0.704032i
$540$		0			0
$541$	−511.000			−0.944547			−0.472274	−	0.881452i	$-0.656567\pi$
$541$	−511.000			−0.944547			−0.472274	+	0.881452i	$0.656567\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−164.438	+	493.315i	−0.301722	+	0.905166i
$546$		0			0
$547$			1083.00i			1.97989i	0.141452	+	0.989945i	$0.454823\pi$
$547$			1083.00i			1.97989i	−0.141452	+	0.989945i	$0.545177\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			1470.46i			2.66871i
$552$		0			0
$553$			3.00000i			0.00542495i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	973.982			1.74862			0.874310	−	0.485368i	$-0.161314\pi$
$557$	973.982			1.74862			0.874310	+	0.485368i	$0.161314\pi$
$558$		0			0
$559$	−1008.00			−1.80322
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−480.666			−0.853759			−0.426879	−	0.904309i	$-0.640387\pi$
$563$	−480.666			−0.853759			−0.426879	+	0.904309i	$0.640387\pi$
$564$		0			0
$565$	275.000	−	825.000i	0.486726	−	1.46018i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			645.105i			1.13375i	0.823803	+	0.566876i	$0.191848\pi$
$569$			645.105i			1.13375i	−0.823803	+	0.566876i	$0.808152\pi$
$570$		0			0
$571$	−569.000			−0.996497			−0.498249	−	0.867034i	$-0.666023\pi$
$571$	−569.000			−0.996497			−0.498249	+	0.867034i	$0.666023\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−442.719	−	332.039i	−0.769946	−	0.577459i
$576$		0			0
$577$			837.000i			1.45061i	0.688429	+	0.725303i	$0.258300\pi$
$577$			837.000i			1.45061i	−0.688429	+	0.725303i	$0.741700\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	332.039i		−	0.571496i
$582$		0			0
$583$		−	390.000i		−	0.668954i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	629.293			1.07205			0.536025	−	0.844202i	$-0.319925\pi$
$587$	629.293			1.07205			0.536025	+	0.844202i	$0.319925\pi$
$588$		0			0
$589$	496.000			0.842105
$590$		0			0
$591$		0			0
$592$		0			0
$593$	641.942			1.08253			0.541267	−	0.840851i	$-0.317945\pi$
$593$	641.942			1.08253			0.541267	+	0.840851i	$0.317945\pi$
$594$		0			0
$595$	180.000	+	60.0000i	0.302521	+	0.100840i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	635.618i		−	1.06113i	−0.847644	−	0.530566i	$-0.821979\pi$
$599$		−	635.618i		−	1.06113i	0.847644	−	0.530566i	$-0.178021\pi$
$600$		0			0
$601$	224.000			0.372712			0.186356	−	0.982482i	$-0.440332\pi$
$601$	224.000			0.372712			0.186356	+	0.982482i	$0.440332\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−49.0153	+	147.046i	−0.0810170	+	0.243051i
$606$		0			0
$607$		−	147.000i		−	0.242175i	−0.992642	−	0.121087i	$-0.961362\pi$
$607$		−	147.000i		−	0.242175i	0.992642	−	0.121087i	$-0.0386381\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			265.631i			0.434748i
$612$		0			0
$613$		−	243.000i		−	0.396411i	−0.980160	−	0.198206i	$-0.936489\pi$
$613$		−	243.000i		−	0.396411i	0.980160	−	0.198206i	$-0.0635114\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−79.0569			−0.128131			−0.0640656	−	0.997946i	$-0.520407\pi$
$617$	−79.0569			−0.128131			−0.0640656	+	0.997946i	$0.520407\pi$
$618$		0			0
$619$	1201.00			1.94023			0.970113	−	0.242653i	$-0.0780177\pi$
$619$	1201.00			1.94023			0.970113	+	0.242653i	$0.0780177\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−341.526			−0.548196
$624$		0			0
$625$	175.000	+	600.000i	0.280000	+	0.960000i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	341.526i		−	0.542967i
$630$		0			0
$631$	193.000			0.305864			0.152932	−	0.988237i	$-0.451128\pi$
$631$	193.000			0.305864			0.152932	+	0.988237i	$0.451128\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−683.052	−	227.684i	−1.07567	−	0.358557i
$636$		0			0
$637$			840.000i			1.31868i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			303.579i			0.473602i	0.971558	+	0.236801i	$0.0760989\pi$
$641$			303.579i			0.473602i	−0.971558	+	0.236801i	$0.923901\pi$
$642$		0			0
$643$			24.0000i			0.0373250i	0.999826	+	0.0186625i	$0.00594081\pi$
$643$			24.0000i			0.0373250i	−0.999826	+	0.0186625i	$0.994059\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	667.241			1.03128			0.515642	−	0.856804i	$-0.327554\pi$
$647$	667.241			1.03128			0.515642	+	0.856804i	$0.327554\pi$
$648$		0			0
$649$	360.000			0.554700
$650$		0			0
$651$		0			0
$652$		0			0
$653$	404.772			0.619865			0.309932	−	0.950759i	$-0.399694\pi$
$653$	404.772			0.619865			0.309932	+	0.950759i	$0.399694\pi$
$654$		0			0
$655$	−360.000	−	120.000i	−0.549618	−	0.183206i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			151.789i			0.230333i	0.993346	+	0.115166i	$0.0367401\pi$
$659$			151.789i			0.230333i	−0.993346	+	0.115166i	$0.963260\pi$
$660$		0			0
$661$	719.000			1.08775			0.543873	−	0.839168i	$-0.316957\pi$
$661$	719.000			1.08775			0.543873	+	0.839168i	$0.316957\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−441.138	−	147.046i	−0.663365	−	0.221122i
$666$		0			0
$667$			1050.00i			1.57421i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	9.48683i		−	0.0141384i
$672$		0			0
$673$		−	1173.00i		−	1.74294i	−0.490447	−	0.871471i	$-0.663166\pi$
$673$		−	1173.00i		−	1.74294i	0.490447	−	0.871471i	$-0.336834\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	1125.77			1.66288			0.831441	−	0.555613i	$-0.187517\pi$
$677$	1125.77			1.66288			0.831441	+	0.555613i	$0.187517\pi$
$678$		0			0
$679$	279.000			0.410898
$680$		0			0
$681$		0			0
$682$		0			0
$683$	894.925			1.31028			0.655142	−	0.755505i	$-0.272609\pi$
$683$	894.925			1.31028			0.655142	+	0.755505i	$0.272609\pi$
$684$		0			0
$685$	−85.0000	+	255.000i	−0.124088	+	0.372263i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	863.302i		−	1.25298i
$690$		0			0
$691$	−872.000			−1.26194			−0.630970	−	0.775808i	$-0.717343\pi$
$691$	−872.000			−1.26194			−0.630970	+	0.775808i	$0.717343\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−1.58114	+	4.74342i	−0.00227502	+	0.00682506i
$696$		0			0
$697$			600.000i			0.860832i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	1280.72i		−	1.82699i	−0.406846	−	0.913497i	$-0.633371\pi$
$701$		−	1280.72i		−	1.82699i	0.406846	−	0.913497i	$-0.366629\pi$
$702$		0			0
$703$			837.000i			1.19061i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−227.684			−0.322042
$708$		0			0
$709$	−481.000			−0.678420			−0.339210	−	0.940711i	$-0.610160\pi$
$709$	−481.000			−0.678420			−0.339210	+	0.940711i	$0.610160\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	354.175			0.496739
$714$		0			0
$715$	315.000	−	945.000i	0.440559	−	1.32168i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	939.196i		−	1.30625i	−0.757248	−	0.653127i	$-0.773457\pi$
$719$		−	939.196i		−	1.30625i	0.757248	−	0.653127i	$-0.226543\pi$
$720$		0			0
$721$	441.000			0.611650
$722$		0			0
$723$		0			0
$724$		0			0
$725$	711.512	−	948.683i	0.981397	−	1.30853i
$726$		0			0
$727$			768.000i			1.05640i	0.849121	+	0.528198i	$0.177132\pi$
$727$			768.000i			1.05640i	−0.849121	+	0.528198i	$0.822868\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	607.157i		−	0.830585i
$732$		0			0
$733$		−	1104.00i		−	1.50614i	−0.657941	−	0.753070i	$-0.728572\pi$
$733$		−	1104.00i		−	1.50614i	0.657941	−	0.753070i	$-0.271428\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	199.223			0.270317
$738$		0			0
$739$	−392.000			−0.530447			−0.265223	−	0.964187i	$-0.585446\pi$
$739$	−392.000			−0.530447			−0.265223	+	0.964187i	$0.585446\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	999.280			1.34493			0.672463	−	0.740131i	$-0.265236\pi$
$743$	999.280			1.34493			0.672463	+	0.740131i	$0.265236\pi$
$744$		0			0
$745$	360.000	+	120.000i	0.483221	+	0.161074i
$746$		0			0
$747$		0			0
$748$		0			0
$749$			66.4078i			0.0886620i
$750$		0			0
$751$	241.000			0.320905			0.160453	−	0.987044i	$-0.448705\pi$
$751$	241.000			0.320905			0.160453	+	0.987044i	$0.448705\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−49.0153	+	147.046i	−0.0649209	+	0.194763i
$756$		0			0
$757$			27.0000i			0.0356671i	0.999841	+	0.0178336i	$0.00567690\pi$
$757$			27.0000i			0.0356671i	−0.999841	+	0.0178336i	$0.994323\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			692.539i			0.910038i	0.890482	+	0.455019i	$0.150368\pi$
$761$			692.539i			0.910038i	−0.890482	+	0.455019i	$0.849632\pi$
$762$		0			0
$763$			312.000i			0.408912i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	796.894			1.03898
$768$		0			0
$769$	1127.00			1.46554			0.732770	−	0.680477i	$-0.238227\pi$
$769$	1127.00			1.46554			0.732770	+	0.680477i	$0.238227\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	243.495			0.315000			0.157500	−	0.987519i	$-0.449656\pi$
$773$	243.495			0.315000			0.157500	+	0.987519i	$0.449656\pi$
$774$		0			0
$775$	−320.000	−	240.000i	−0.412903	−	0.309677i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	1470.46i		−	1.88762i
$780$		0			0
$781$	270.000			0.345711
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−483.828	−	161.276i	−0.616342	−	0.205447i
$786$		0			0
$787$			573.000i			0.728081i	0.931383	+	0.364041i	$0.118603\pi$
$787$			573.000i			0.728081i	−0.931383	+	0.364041i	$0.881397\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	521.776i		−	0.659641i
$792$		0			0
$793$		−	21.0000i		−	0.0264817i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	632.456			0.793545			0.396773	−	0.917917i	$-0.370130\pi$
$797$	632.456			0.793545			0.396773	+	0.917917i	$0.370130\pi$
$798$		0			0
$799$	−160.000			−0.200250
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−256.144			−0.318984
$804$		0			0
$805$	−315.000	−	105.000i	−0.391304	−	0.130435i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			1138.42i			1.40719i	0.710599	+	0.703597i	$0.248424\pi$
$809$			1138.42i			1.40719i	−0.710599	+	0.703597i	$0.751576\pi$
$810$		0			0
$811$	376.000			0.463625			0.231813	−	0.972760i	$-0.425534\pi$
$811$	376.000			0.463625			0.231813	+	0.972760i	$0.425534\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	811.124	+	270.375i	0.995244	+	0.331748i
$816$		0			0
$817$			1488.00i			1.82130i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	37.9473i		−	0.0462209i	−0.999733	−	0.0231104i	$-0.992643\pi$
$821$		−	37.9473i		−	0.0462209i	0.999733	−	0.0231104i	$-0.00735693\pi$
$822$		0			0
$823$			339.000i			0.411908i	0.978562	+	0.205954i	$0.0660297\pi$
$823$			339.000i			0.411908i	−0.978562	+	0.205954i	$0.933970\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−442.719			−0.535331			−0.267666	−	0.963512i	$-0.586252\pi$
$827$	−442.719			−0.535331			−0.267666	+	0.963512i	$0.586252\pi$
$828$		0			0
$829$	−961.000			−1.15923			−0.579614	−	0.814891i	$-0.696797\pi$
$829$	−961.000			−1.15923			−0.579614	+	0.814891i	$0.696797\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−505.964			−0.607400
$834$		0			0
$835$	460.000	−	1380.00i	0.550898	−	1.65269i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			1062.53i			1.26642i	0.773981	+	0.633209i	$0.218263\pi$
$839$			1062.53i			1.26642i	−0.773981	+	0.633209i	$0.781737\pi$
$840$		0			0
$841$	−1409.00			−1.67539
$842$		0			0
$843$		0			0
$844$		0			0
$845$	430.070	−	1290.21i	0.508958	−	1.52687i
$846$		0			0
$847$			93.0000i			0.109799i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			597.670i			0.702315i
$852$		0			0
$853$			1221.00i			1.43142i	0.698398	+	0.715709i	$0.253896\pi$
$853$			1221.00i			1.43142i	−0.698398	+	0.715709i	$0.746104\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−572.372			−0.667879			−0.333939	−	0.942595i	$-0.608378\pi$
$857$	−572.372			−0.667879			−0.333939	+	0.942595i	$0.608378\pi$
$858$		0			0
$859$	−119.000			−0.138533			−0.0692666	−	0.997598i	$-0.522066\pi$
$859$	−119.000			−0.138533			−0.0692666	+	0.997598i	$0.522066\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−442.719			−0.513000			−0.256500	−	0.966544i	$-0.582569\pi$
$863$	−442.719			−0.513000			−0.256500	+	0.966544i	$0.582569\pi$
$864$		0			0
$865$	−40.0000	+	120.000i	−0.0462428	+	0.138728i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			9.48683i			0.0109170i
$870$		0			0
$871$	441.000			0.506315
$872$		0			0
$873$		0			0
$874$		0			0
$875$	213.454	+	308.322i	0.243947	+	0.352368i
$876$		0			0
$877$		−	123.000i		−	0.140251i	−0.997538	−	0.0701254i	$-0.977660\pi$
$877$		−	123.000i		−	0.140251i	0.997538	−	0.0701254i	$-0.0223400\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			996.117i			1.13067i	0.824862	+	0.565333i	$0.191252\pi$
$881$			996.117i			1.13067i	−0.824862	+	0.565333i	$0.808748\pi$
$882$		0			0
$883$		−	387.000i		−	0.438279i	−0.975694	−	0.219139i	$-0.929675\pi$
$883$		−	387.000i		−	0.438279i	0.975694	−	0.219139i	$-0.0703249\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	961.332			1.08380			0.541901	−	0.840442i	$-0.317705\pi$
$887$	961.332			1.08380			0.541901	+	0.840442i	$0.317705\pi$
$888$		0			0
$889$	−432.000			−0.485939
$890$		0			0
$891$		0			0
$892$		0			0
$893$	392.122			0.439107
$894$		0			0
$895$	945.000	+	315.000i	1.05587	+	0.351955i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			758.947i			0.844212i
$900$		0			0
$901$	520.000			0.577137
$902$		0			0
$903$		0			0
$904$		0			0
$905$	162.857	−	488.572i	0.179953	−	0.539858i
$906$		0			0
$907$			93.0000i			0.102536i	0.998685	+	0.0512679i	$0.0163262\pi$
$907$			93.0000i			0.102536i	−0.998685	+	0.0512679i	$0.983674\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	531.263i		−	0.583164i	−0.956546	−	0.291582i	$-0.905818\pi$
$911$		−	531.263i		−	0.583164i	0.956546	−	0.291582i	$-0.0941817\pi$
$912$		0			0
$913$		−	1050.00i		−	1.15005i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−227.684			−0.248292
$918$		0			0
$919$	376.000			0.409140			0.204570	−	0.978852i	$-0.434420\pi$
$919$	376.000			0.409140			0.204570	+	0.978852i	$0.434420\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	597.670			0.647530
$924$		0			0
$925$	405.000	−	540.000i	0.437838	−	0.583784i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			1413.54i			1.52157i	0.649004	+	0.760785i	$0.275186\pi$
$929$			1413.54i			1.52157i	−0.649004	+	0.760785i	$0.724814\pi$
$930$		0			0
$931$	1240.00			1.33190
$932$		0			0
$933$		0			0
$934$		0			0
$935$	569.210	+	189.737i	0.608781	+	0.202927i
$936$		0			0
$937$		−	243.000i		−	0.259338i	−0.991557	−	0.129669i	$-0.958608\pi$
$937$		−	243.000i		−	0.259338i	0.991557	−	0.129669i	$-0.0413915\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	388.960i		−	0.413348i	−0.978410	−	0.206674i	$-0.933736\pi$
$941$		−	388.960i		−	0.413348i	0.978410	−	0.206674i	$-0.0662639\pi$
$942$		0			0
$943$		−	1050.00i		−	1.11347i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	316.228			0.333926			0.166963	−	0.985963i	$-0.446604\pi$
$947$	316.228			0.333926			0.166963	+	0.985963i	$0.446604\pi$
$948$		0			0
$949$	−567.000			−0.597471
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−1530.54			−1.60603			−0.803013	−	0.595962i	$-0.796771\pi$
$953$	−1530.54			−1.60603			−0.803013	+	0.595962i	$0.796771\pi$
$954$		0			0
$955$	−1260.00	−	420.000i	−1.31937	−	0.439791i
$956$		0			0
$957$		0			0
$958$		0			0
$959$			161.276i			0.168171i
$960$		0			0
$961$	−705.000			−0.733611
$962$		0			0
$963$		0			0
$964$		0			0
$965$	1010.35	+	336.783i	1.04699	+	0.348997i
$966$		0			0
$967$		−	237.000i		−	0.245088i	−0.992463	−	0.122544i	$-0.960895\pi$
$967$		−	237.000i		−	0.245088i	0.992463	−	0.122544i	$-0.0391052\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	569.210i		−	0.586210i	−0.956080	−	0.293105i	$-0.905311\pi$
$971$		−	569.210i		−	0.586210i	0.956080	−	0.293105i	$-0.0946886\pi$
$972$		0			0
$973$			3.00000i			0.00308325i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−515.451			−0.527586			−0.263793	−	0.964579i	$-0.584974\pi$
$977$	−515.451			−0.527586			−0.263793	+	0.964579i	$0.584974\pi$
$978$		0			0
$979$	−1080.00			−1.10317
$980$		0			0
$981$		0			0
$982$		0			0
$983$	629.293			0.640176			0.320088	−	0.947388i	$-0.396287\pi$
$983$	629.293			0.640176			0.320088	+	0.947388i	$0.396287\pi$
$984$		0			0
$985$	380.000	−	1140.00i	0.385787	−	1.15736i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			1062.53i			1.07434i
$990$		0			0
$991$	751.000			0.757820			0.378910	−	0.925433i	$-0.376299\pi$
$991$	751.000			0.757820			0.378910	+	0.925433i	$0.376299\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	74.3135	−	222.941i	0.0746870	−	0.224061i
$996$		0			0
$997$			1536.00i			1.54062i	0.637668	+	0.770311i	$0.279899\pi$
$997$			1536.00i			1.54062i	−0.637668	+	0.770311i	$0.720101\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2160.3.c.k.1889.2		4
3.2	odd	2	inner	2160.3.c.k.1889.3		4
4.3	odd	2		135.3.d.g.134.2	yes	4
5.4	even	2	inner	2160.3.c.k.1889.4		4
12.11	even	2		135.3.d.g.134.3	yes	4
15.14	odd	2	inner	2160.3.c.k.1889.1		4
20.3	even	4		675.3.c.f.26.2		2
20.7	even	4		675.3.c.g.26.1		2
20.19	odd	2		135.3.d.g.134.4	yes	4
36.7	odd	6		405.3.h.i.134.3		8
36.11	even	6		405.3.h.i.134.2		8
36.23	even	6		405.3.h.i.269.1		8
36.31	odd	6		405.3.h.i.269.4		8
60.23	odd	4		675.3.c.f.26.1		2
60.47	odd	4		675.3.c.g.26.2		2
60.59	even	2		135.3.d.g.134.1	✓	4
180.59	even	6		405.3.h.i.269.3		8
180.79	odd	6		405.3.h.i.134.1		8
180.119	even	6		405.3.h.i.134.4		8
180.139	odd	6		405.3.h.i.269.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
135.3.d.g.134.1	✓	4	60.59	even	2
135.3.d.g.134.2	yes	4	4.3	odd	2
135.3.d.g.134.3	yes	4	12.11	even	2
135.3.d.g.134.4	yes	4	20.19	odd	2
405.3.h.i.134.1		8	180.79	odd	6
405.3.h.i.134.2		8	36.11	even	6
405.3.h.i.134.3		8	36.7	odd	6
405.3.h.i.134.4		8	180.119	even	6
405.3.h.i.269.1		8	36.23	even	6
405.3.h.i.269.2		8	180.139	odd	6
405.3.h.i.269.3		8	180.59	even	6
405.3.h.i.269.4		8	36.31	odd	6
675.3.c.f.26.1		2	60.23	odd	4
675.3.c.f.26.2		2	20.3	even	4
675.3.c.g.26.1		2	20.7	even	4
675.3.c.g.26.2		2	60.47	odd	4
2160.3.c.k.1889.1		4	15.14	odd	2	inner
2160.3.c.k.1889.2		4	1.1	even	1	trivial
2160.3.c.k.1889.3		4	3.2	odd	2	inner
2160.3.c.k.1889.4		4	5.4	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2160 = 2^{4} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2160.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.58114 + 4.74342i) q^{5} +3.00000i q^{7} +9.48683i q^{11} +21.0000i q^{13} -12.6491 q^{17} +31.0000 q^{19} +22.1359 q^{23} +(-20.0000 - 15.0000i) q^{25} +47.4342i q^{29} +16.0000 q^{31} +(-14.2302 - 4.74342i) q^{35} +27.0000i q^{37} -47.4342i q^{41} +48.0000i q^{43} +12.6491 q^{47} +40.0000 q^{49} -41.1096 q^{53} +(-45.0000 - 15.0000i) q^{55} -37.9473i q^{59} -1.00000 q^{61} +(-99.6117 - 33.2039i) q^{65} -21.0000i q^{67} -28.4605i q^{71} +27.0000i q^{73} -28.4605 q^{77} +1.00000 q^{79} -110.680 q^{83} +(20.0000 - 60.0000i) q^{85} +113.842i q^{89} -63.0000 q^{91} +(-49.0153 + 147.046i) q^{95} -93.0000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 124 q^{19} - 80 q^{25} + 64 q^{31} + 160 q^{49} - 180 q^{55} - 4 q^{61} + 4 q^{79} + 80 q^{85} - 252 q^{91}+O(q^{100}) \) 4 * q + 124 * q^19 - 80 * q^25 + 64 * q^31 + 160 * q^49 - 180 * q^55 - 4 * q^61 + 4 * q^79 + 80 * q^85 - 252 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more