LMFDB - Embedded newform 1080.4.f.d.649.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1080,4,Mod(649,1080)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1080, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1080.649");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1080 = 2^{3} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1080.f (of order $2$, degree $1$, 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:	$63.7220628062$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 10 x^{19} + 50 x^{18} + 352 x^{17} + 21144 x^{16} - 183248 x^{15} + 837232 x^{14} + \cdots + 2209905230625 $ x^20 - 10x^19 + 50x^18 + 352x^17 + 21144x^16 - 183248x^15 + 837232x^14 + 4991426x^13 + 123172734x^12 - 846162224x^11 + 3147448836x^10 + 8830578242x^9 + 71410987721x^8 - 374031480668x^7 + 1119018321432x^6 + 1852681857730x^5 + 5531652507775x^4 - 15867459693850x^3 + 19223534162450x^2 - 9217612347750*x + 2209905230625
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{25}\cdot 3^{12}\cdot 5^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.15
Root			$-3.24611 - 3.24611i$ of defining polynomial
Character	$\chi$	$=$	1080.649
Dual form			1080.4.f.d.649.16

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(6.82224 - 8.85760i) q^{5} +20.2612i q^{7} +O(q^{10})$	$q+(6.82224 - 8.85760i) q^{5} +20.2612i q^{7} -35.6205 q^{11} -25.8659i q^{13} +114.568i q^{17} +29.6531 q^{19} -120.823i q^{23} +(-31.9142 - 120.857i) q^{25} -270.736 q^{29} +198.633 q^{31} +(179.466 + 138.227i) q^{35} -161.239i q^{37} -201.652 q^{41} -444.035i q^{43} -18.4207i q^{47} -67.5167 q^{49} -681.836i q^{53} +(-243.011 + 315.512i) q^{55} +456.439 q^{59} -612.311 q^{61} +(-229.110 - 176.464i) q^{65} +659.266i q^{67} -455.777 q^{71} -178.511i q^{73} -721.714i q^{77} +586.281 q^{79} -778.779i q^{83} +(1014.80 + 781.609i) q^{85} +747.110 q^{89} +524.075 q^{91} +(202.301 - 262.656i) q^{95} +1172.22i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q+O(q^{10}) $ 20 * q	$ 20 q - 100 q^{19} - 24 q^{25} + 228 q^{31} - 252 q^{49} - 64 q^{55} + 748 q^{61} - 668 q^{79} - 1500 q^{85} + 1744 q^{91}+O(q^{100}) $ 20 * q - 100 * q^19 - 24 * q^25 + 228 * q^31 - 252 * q^49 - 64 * q^55 + 748 * q^61 - 668 * q^79 - 1500 * q^85 + 1744 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1080\mathbb{Z}\right)^\times$.

$n$	$217$	$271$	$541$	$1001$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$5$	6.82224	−	8.85760i	0.610199	−	0.792248i
$5$	6.82224	−	8.85760i	0.610199	−	0.792248i
$6$		0			0
$7$			20.2612i			1.09400i	0.837132	+	0.547001i	$0.184231\pi$
$7$			20.2612i			1.09400i	−0.837132	+	0.547001i	$0.815769\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−35.6205			−0.976361			−0.488180	−	0.872743i	$-0.662339\pi$
$11$	−35.6205			−0.976361			−0.488180	+	0.872743i	$0.662339\pi$
$12$		0			0
$13$		−	25.8659i		−	0.551840i	−0.961181	−	0.275920i	$-0.911017\pi$
$13$		−	25.8659i		−	0.551840i	0.961181	−	0.275920i	$-0.0889825\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			114.568i			1.63452i	0.576272	+	0.817258i	$0.304507\pi$
$17$			114.568i			1.63452i	−0.576272	+	0.817258i	$0.695493\pi$
$18$		0			0
$19$	29.6531			0.358047			0.179024	−	0.983845i	$-0.442706\pi$
$19$	29.6531			0.358047			0.179024	+	0.983845i	$0.442706\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	120.823i		−	1.09536i	−0.836687	−	0.547681i	$-0.815511\pi$
$23$		−	120.823i		−	1.09536i	0.836687	−	0.547681i	$-0.184489\pi$
$24$		0			0
$25$	−31.9142	−	120.857i	−0.255313	−	0.966858i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−270.736			−1.73360			−0.866800	−	0.498657i	$-0.833827\pi$
$29$	−270.736			−1.73360			−0.866800	+	0.498657i	$0.833827\pi$
$30$		0			0
$31$	198.633			1.15083			0.575413	−	0.817863i	$-0.304841\pi$
$31$	198.633			1.15083			0.575413	+	0.817863i	$0.304841\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	179.466	+	138.227i	0.866721	+	0.667560i
$36$		0			0
$37$		−	161.239i		−	0.716419i	−0.933641	−	0.358209i	$-0.883387\pi$
$37$		−	161.239i		−	0.716419i	0.933641	−	0.358209i	$-0.116613\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−201.652			−0.768118			−0.384059	−	0.923309i	$-0.625474\pi$
$41$	−201.652			−0.768118			−0.384059	+	0.923309i	$0.625474\pi$
$42$		0			0
$43$		−	444.035i		−	1.57476i	−0.616468	−	0.787380i	$-0.711437\pi$
$43$		−	444.035i		−	1.57476i	0.616468	−	0.787380i	$-0.288563\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	18.4207i		−	0.0571688i	−0.999591	−	0.0285844i	$-0.990900\pi$
$47$		−	18.4207i		−	0.0571688i	0.999591	−	0.0285844i	$-0.00909993\pi$
$48$		0			0
$49$	−67.5167			−0.196842
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	681.836i		−	1.76712i	−0.468318	−	0.883560i	$-0.655140\pi$
$53$		−	681.836i		−	1.76712i	0.468318	−	0.883560i	$-0.344860\pi$
$54$		0			0
$55$	−243.011	+	315.512i	−0.595775	+	0.773520i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	456.439			1.00717			0.503587	−	0.863944i	$-0.332013\pi$
$59$	456.439			1.00717			0.503587	+	0.863944i	$0.332013\pi$
$60$		0			0
$61$	−612.311			−1.28522			−0.642610	−	0.766194i	$-0.722148\pi$
$61$	−612.311			−1.28522			−0.642610	+	0.766194i	$0.722148\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−229.110	−	176.464i	−0.437194	−	0.336733i
$66$		0			0
$67$			659.266i			1.20212i	0.799203	+	0.601061i	$0.205255\pi$
$67$			659.266i			1.20212i	−0.799203	+	0.601061i	$0.794745\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−455.777			−0.761842			−0.380921	−	0.924608i	$-0.624393\pi$
$71$	−455.777			−0.761842			−0.380921	+	0.924608i	$0.624393\pi$
$72$		0			0
$73$		−	178.511i		−	0.286208i	−0.989708	−	0.143104i	$-0.954292\pi$
$73$		−	178.511i		−	0.286208i	0.989708	−	0.143104i	$-0.0457083\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	721.714i		−	1.06814i
$78$		0			0
$79$	586.281			0.834958			0.417479	−	0.908686i	$-0.362914\pi$
$79$	586.281			0.834958			0.417479	+	0.908686i	$0.362914\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	778.779i		−	1.02990i	−0.857219	−	0.514952i	$-0.827810\pi$
$83$		−	778.779i		−	1.02990i	0.857219	−	0.514952i	$-0.172190\pi$
$84$		0			0
$85$	1014.80	+	781.609i	1.29494	+	0.997381i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	747.110			0.889815			0.444907	−	0.895577i	$-0.353237\pi$
$89$	747.110			0.889815			0.444907	+	0.895577i	$0.353237\pi$
$90$		0			0
$91$	524.075			0.603715
$92$		0			0
$93$		0			0
$94$		0			0
$95$	202.301	−	262.656i	0.218480	−	0.283662i
$96$		0			0
$97$			1172.22i			1.22702i	0.789688	+	0.613509i	$0.210243\pi$
$97$			1172.22i			1.22702i	−0.789688	+	0.613509i	$0.789757\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−1470.54			−1.44875			−0.724376	−	0.689406i	$-0.757872\pi$
$101$	−1470.54			−1.44875			−0.724376	+	0.689406i	$0.757872\pi$
$102$		0			0
$103$		−	853.219i		−	0.816215i	−0.912934	−	0.408108i	$-0.866189\pi$
$103$		−	853.219i		−	0.816215i	0.912934	−	0.408108i	$-0.133811\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	1100.83i		−	0.994587i	−0.867582	−	0.497293i	$-0.834327\pi$
$107$		−	1100.83i		−	0.994587i	0.867582	−	0.497293i	$-0.165673\pi$
$108$		0			0
$109$	−1154.66			−1.01464			−0.507321	−	0.861757i	$-0.669364\pi$
$109$	−1154.66			−1.01464			−0.507321	+	0.861757i	$0.669364\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			722.189i			0.601220i	0.953747	+	0.300610i	$0.0971902\pi$
$113$			722.189i			0.601220i	−0.953747	+	0.300610i	$0.902810\pi$
$114$		0			0
$115$	−1070.20	−	824.283i	−0.867799	−	0.668390i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−2321.28			−1.78817
$120$		0			0
$121$	−62.1835			−0.0467194
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−1288.23	−	541.834i	−0.921784	−	0.387705i
$126$		0			0
$127$			928.596i			0.648815i	0.945917	+	0.324408i	$0.105165\pi$
$127$			928.596i			0.648815i	−0.945917	+	0.324408i	$0.894835\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−2310.64			−1.54108			−0.770540	−	0.637391i	$-0.780013\pi$
$131$	−2310.64			−1.54108			−0.770540	+	0.637391i	$0.780013\pi$
$132$		0			0
$133$			600.808i			0.391705i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	2291.21i		−	1.42884i	−0.699717	−	0.714420i	$-0.746691\pi$
$137$		−	2291.21i		−	1.42884i	0.699717	−	0.714420i	$-0.253309\pi$
$138$		0			0
$139$	−1268.99			−0.774348			−0.387174	−	0.922007i	$-0.626549\pi$
$139$	−1268.99			−0.774348			−0.387174	+	0.922007i	$0.626549\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			921.357i			0.538795i
$144$		0			0
$145$	−1847.02	+	2398.07i	−1.05784	+	1.37344i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−2756.83			−1.51576			−0.757880	−	0.652393i	$-0.773765\pi$
$149$	−2756.83			−1.51576			−0.757880	+	0.652393i	$0.773765\pi$
$150$		0			0
$151$	2196.54			1.18379			0.591894	−	0.806016i	$-0.298380\pi$
$151$	2196.54			1.18379			0.591894	+	0.806016i	$0.298380\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	1355.12	−	1759.42i	0.702234	−	0.911740i
$156$		0			0
$157$		−	2617.91i		−	1.33078i	−0.746497	−	0.665389i	$-0.768266\pi$
$157$		−	2617.91i		−	1.33078i	0.746497	−	0.665389i	$-0.231734\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	2448.02			1.19833
$162$		0			0
$163$		−	3137.27i		−	1.50755i	−0.657135	−	0.753773i	$-0.728232\pi$
$163$		−	3137.27i		−	1.50755i	0.657135	−	0.753773i	$-0.271768\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	1594.06i		−	0.738635i	−0.929303	−	0.369318i	$-0.879591\pi$
$167$		−	1594.06i		−	0.738635i	0.929303	−	0.369318i	$-0.120409\pi$
$168$		0			0
$169$	1527.95			0.695472
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	794.394i		−	0.349114i	−0.984647	−	0.174557i	$-0.944151\pi$
$173$		−	794.394i		−	0.349114i	0.984647	−	0.174557i	$-0.0558493\pi$
$174$		0			0
$175$	2448.72	−	646.620i	1.05775	−	0.279314i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−1519.77			−0.634597			−0.317298	−	0.948326i	$-0.602776\pi$
$179$	−1519.77			−0.634597			−0.317298	+	0.948326i	$0.602776\pi$
$180$		0			0
$181$	−528.342			−0.216969			−0.108484	−	0.994098i	$-0.534600\pi$
$181$	−528.342			−0.216969			−0.108484	+	0.994098i	$0.534600\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−1428.19	−	1100.01i	−0.567581	−	0.437158i
$186$		0			0
$187$		−	4080.96i		−	1.59588i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−103.092			−0.0390548			−0.0195274	−	0.999809i	$-0.506216\pi$
$191$	−103.092			−0.0390548			−0.0195274	+	0.999809i	$0.506216\pi$
$192$		0			0
$193$		−	2137.32i		−	0.797139i	−0.917138	−	0.398569i	$-0.869507\pi$
$193$		−	2137.32i		−	0.797139i	0.917138	−	0.398569i	$-0.130493\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	283.711i		−	0.102607i	−0.998683	−	0.0513035i	$-0.983662\pi$
$197$		−	283.711i		−	0.102607i	0.998683	−	0.0513035i	$-0.0163376\pi$
$198$		0			0
$199$	772.154			0.275058			0.137529	−	0.990498i	$-0.456084\pi$
$199$	772.154			0.275058			0.137529	+	0.990498i	$0.456084\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	5485.44i		−	1.89656i
$204$		0			0
$205$	−1375.72	+	1786.16i	−0.468705	+	0.608539i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−1056.26			−0.349583
$210$		0			0
$211$	4059.36			1.32444			0.662222	−	0.749307i	$-0.269613\pi$
$211$	4059.36			1.32444			0.662222	+	0.749307i	$0.269613\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−3933.08	−	3029.31i	−1.24760	−	0.960918i
$216$		0			0
$217$			4024.55i			1.25901i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	2963.41			0.901992
$222$		0			0
$223$			3396.54i			1.01995i	0.860189	+	0.509976i	$0.170346\pi$
$223$			3396.54i			1.01995i	−0.860189	+	0.509976i	$0.829654\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			2458.61i			0.718870i	0.933170	+	0.359435i	$0.117031\pi$
$227$			2458.61i			0.718870i	−0.933170	+	0.359435i	$0.882969\pi$
$228$		0			0
$229$	−2397.46			−0.691829			−0.345914	−	0.938266i	$-0.612431\pi$
$229$	−2397.46			−0.691829			−0.345914	+	0.938266i	$0.612431\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	4329.19i		−	1.21723i	−0.793466	−	0.608615i	$-0.791726\pi$
$233$		−	4329.19i		−	1.21723i	0.793466	−	0.608615i	$-0.208274\pi$
$234$		0			0
$235$	−163.163	−	125.670i	−0.0452918	−	0.0348843i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	3360.71			0.909566			0.454783	−	0.890602i	$-0.349717\pi$
$239$	3360.71			0.909566			0.454783	+	0.890602i	$0.349717\pi$
$240$		0			0
$241$	−1724.33			−0.460887			−0.230444	−	0.973086i	$-0.574018\pi$
$241$	−1724.33			−0.460887			−0.230444	+	0.973086i	$0.574018\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−460.615	+	598.036i	−0.120113	+	0.155947i
$246$		0			0
$247$		−	767.006i		−	0.197585i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−4546.90			−1.14342			−0.571709	−	0.820456i	$-0.693719\pi$
$251$	−4546.90			−1.14342			−0.571709	+	0.820456i	$0.693719\pi$
$252$		0			0
$253$			4303.77i			1.06947i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			564.137i			0.136926i	0.997654	+	0.0684629i	$0.0218095\pi$
$257$			564.137i			0.136926i	−0.997654	+	0.0684629i	$0.978191\pi$
$258$		0			0
$259$	3266.89			0.783764
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	199.819i		−	0.0468493i	−0.999726	−	0.0234246i	$-0.992543\pi$
$263$		−	199.819i		−	0.0468493i	0.999726	−	0.0234246i	$-0.00745697\pi$
$264$		0			0
$265$	−6039.43	−	4651.64i	−1.40000	−	1.07830i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	336.563			0.0762848			0.0381424	−	0.999272i	$-0.487856\pi$
$269$	336.563			0.0762848			0.0381424	+	0.999272i	$0.487856\pi$
$270$		0			0
$271$	−3839.93			−0.860735			−0.430368	−	0.902654i	$-0.641616\pi$
$271$	−3839.93			−0.860735			−0.430368	+	0.902654i	$0.641616\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1136.80	+	4304.99i	0.249278	+	0.944003i
$276$		0			0
$277$		−	792.482i		−	0.171898i	−0.996300	−	0.0859488i	$-0.972608\pi$
$277$		−	792.482i		−	0.171898i	0.996300	−	0.0859488i	$-0.0273922\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	5390.87			1.14446			0.572228	−	0.820095i	$-0.306079\pi$
$281$	5390.87			1.14446			0.572228	+	0.820095i	$0.306079\pi$
$282$		0			0
$283$		−	3798.57i		−	0.797885i	−0.916976	−	0.398943i	$-0.869377\pi$
$283$		−	3798.57i		−	0.797885i	0.916976	−	0.398943i	$-0.130623\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	4085.72i		−	0.840323i
$288$		0			0
$289$	−8212.78			−1.67164
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	1031.78i		−	0.205724i	−0.994696	−	0.102862i	$-0.967200\pi$
$293$		−	1031.78i		−	0.205724i	0.994696	−	0.102862i	$-0.0328000\pi$
$294$		0			0
$295$	3113.93	−	4042.95i	0.614577	−	0.797932i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−3125.20			−0.604465
$300$		0			0
$301$	8996.69			1.72279
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−4177.33	+	5423.60i	−0.784240	+	1.01821i
$306$		0			0
$307$			4503.38i			0.837203i	0.908170	+	0.418601i	$0.137480\pi$
$307$			4503.38i			0.837203i	−0.908170	+	0.418601i	$0.862520\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−6823.68			−1.24416			−0.622082	−	0.782952i	$-0.713713\pi$
$311$	−6823.68			−1.24416			−0.622082	+	0.782952i	$0.713713\pi$
$312$		0			0
$313$			8767.48i			1.58328i	0.610987	+	0.791641i	$0.290773\pi$
$313$			8767.48i			1.58328i	−0.610987	+	0.791641i	$0.709227\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	2902.55i		−	0.514270i	−0.966376	−	0.257135i	$-0.917222\pi$
$317$		−	2902.55i		−	0.514270i	0.966376	−	0.257135i	$-0.0827784\pi$
$318$		0			0
$319$	9643.73			1.69262
$320$		0			0
$321$		0			0
$322$		0			0
$323$			3397.29i			0.585234i
$324$		0			0
$325$	−3126.09	+	825.490i	−0.533551	+	0.140892i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	373.225			0.0625428
$330$		0			0
$331$	7996.35			1.32785			0.663926	−	0.747798i	$-0.268889\pi$
$331$	7996.35			1.32785			0.663926	+	0.747798i	$0.268889\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	5839.51	+	4497.67i	0.952378	+	0.733534i
$336$		0			0
$337$			895.936i			0.144821i	0.997375	+	0.0724106i	$0.0230692\pi$
$337$			895.936i			0.144821i	−0.997375	+	0.0724106i	$0.976931\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−7075.41			−1.12362
$342$		0			0
$343$			5581.63i			0.878657i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			5439.32i			0.841494i	0.907178	+	0.420747i	$0.138232\pi$
$347$			5439.32i			0.841494i	−0.907178	+	0.420747i	$0.861768\pi$
$348$		0			0
$349$	−12465.8			−1.91198			−0.955988	−	0.293407i	$-0.905211\pi$
$349$	−12465.8			−1.91198			−0.955988	+	0.293407i	$0.905211\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			2354.83i			0.355056i	0.984116	+	0.177528i	$0.0568100\pi$
$353$			2354.83i			0.355056i	−0.984116	+	0.177528i	$0.943190\pi$
$354$		0			0
$355$	−3109.42	+	4037.09i	−0.464876	+	0.603568i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	10221.2			1.50266			0.751332	−	0.659925i	$-0.229412\pi$
$359$	10221.2			1.50266			0.751332	+	0.659925i	$0.229412\pi$
$360$		0			0
$361$	−5979.69			−0.871802
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−1581.18	−	1217.85i	−0.226747	−	0.174644i
$366$		0			0
$367$		−	7418.92i		−	1.05522i	−0.849488	−	0.527608i	$-0.823089\pi$
$367$		−	7418.92i		−	1.05522i	0.849488	−	0.527608i	$-0.176911\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	13814.8			1.93323
$372$		0			0
$373$			8528.87i			1.18394i	0.805962	+	0.591968i	$0.201649\pi$
$373$			8528.87i			1.18394i	−0.805962	+	0.591968i	$0.798351\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			7002.84i			0.956670i
$378$		0			0
$379$	−6762.18			−0.916490			−0.458245	−	0.888826i	$-0.651522\pi$
$379$	−6762.18			−0.916490			−0.458245	+	0.888826i	$0.651522\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			2556.23i			0.341038i	0.985354	+	0.170519i	$0.0545444\pi$
$383$			2556.23i			0.341038i	−0.985354	+	0.170519i	$0.945456\pi$
$384$		0			0
$385$	−6392.65	−	4923.70i	−0.846233	−	0.651779i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−11313.1			−1.47454			−0.737271	−	0.675597i	$-0.763886\pi$
$389$	−11313.1			−1.47454			−0.737271	+	0.675597i	$0.763886\pi$
$390$		0			0
$391$	13842.4			1.79039
$392$		0			0
$393$		0			0
$394$		0			0
$395$	3999.74	−	5193.04i	0.509491	−	0.661494i
$396$		0			0
$397$			5487.48i			0.693725i	0.937916	+	0.346862i	$0.112753\pi$
$397$			5487.48i			0.693725i	−0.937916	+	0.346862i	$0.887247\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	4524.30			0.563423			0.281712	−	0.959499i	$-0.409098\pi$
$401$	4524.30			0.563423			0.281712	+	0.959499i	$0.409098\pi$
$402$		0			0
$403$		−	5137.84i		−	0.635072i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			5743.40i			0.699483i
$408$		0			0
$409$	10690.8			1.29249			0.646243	−	0.763131i	$-0.276339\pi$
$409$	10690.8			1.29249			0.646243	+	0.763131i	$0.276339\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			9248.00i			1.10185i
$414$		0			0
$415$	−6898.11	−	5313.02i	−0.815940	−	0.628447i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−1665.69			−0.194211			−0.0971054	−	0.995274i	$-0.530958\pi$
$419$	−1665.69			−0.194211			−0.0971054	+	0.995274i	$0.530958\pi$
$420$		0			0
$421$	16244.2			1.88050			0.940252	−	0.340479i	$-0.110589\pi$
$421$	16244.2			1.88050			0.940252	+	0.340479i	$0.110589\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	13846.4	−	3656.34i	1.58035	−	0.417314i
$426$		0			0
$427$		−	12406.2i		−	1.40603i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	4138.32			0.462496			0.231248	−	0.972895i	$-0.425719\pi$
$431$	4138.32			0.462496			0.231248	+	0.972895i	$0.425719\pi$
$432$		0			0
$433$			5424.32i			0.602024i	0.953620	+	0.301012i	$0.0973244\pi$
$433$			5424.32i			0.602024i	−0.953620	+	0.301012i	$0.902676\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	3582.78i		−	0.392191i
$438$		0			0
$439$	−7513.97			−0.816907			−0.408454	−	0.912779i	$-0.633932\pi$
$439$	−7513.97			−0.816907			−0.408454	+	0.912779i	$0.633932\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			12430.0i			1.33311i	0.745455	+	0.666556i	$0.232232\pi$
$443$			12430.0i			1.33311i	−0.745455	+	0.666556i	$0.767768\pi$
$444$		0			0
$445$	5096.96	−	6617.60i	0.542964	−	0.704954i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	10565.6			1.11051			0.555257	−	0.831679i	$-0.312620\pi$
$449$	10565.6			1.11051			0.555257	+	0.831679i	$0.312620\pi$
$450$		0			0
$451$	7182.95			0.749960
$452$		0			0
$453$		0			0
$454$		0			0
$455$	3575.37	−	4642.05i	0.368386	−	0.478292i
$456$		0			0
$457$			12204.4i			1.24923i	0.780933	+	0.624615i	$0.214744\pi$
$457$			12204.4i			1.24923i	−0.780933	+	0.624615i	$0.785256\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−11031.2			−1.11448			−0.557241	−	0.830351i	$-0.688140\pi$
$461$	−11031.2			−1.11448			−0.557241	+	0.830351i	$0.688140\pi$
$462$		0			0
$463$		−	15012.5i		−	1.50689i	−0.657509	−	0.753447i	$-0.728390\pi$
$463$		−	15012.5i		−	1.50689i	0.657509	−	0.753447i	$-0.271610\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			4232.05i			0.419349i	0.977771	+	0.209674i	$0.0672404\pi$
$467$			4232.05i			0.419349i	−0.977771	+	0.209674i	$0.932760\pi$
$468$		0			0
$469$	−13357.5			−1.31512
$470$		0			0
$471$		0			0
$472$		0			0
$473$			15816.7i			1.53753i
$474$		0			0
$475$	−946.355	−	3583.80i	−0.0914142	−	0.346181i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	16192.1			1.54455			0.772273	−	0.635291i	$-0.219120\pi$
$479$	16192.1			1.54455			0.772273	+	0.635291i	$0.219120\pi$
$480$		0			0
$481$	−4170.59			−0.395349
$482$		0			0
$483$		0			0
$484$		0			0
$485$	10383.0	+	7997.15i	0.972103	+	0.748726i
$486$		0			0
$487$		−	10417.3i		−	0.969312i	−0.874705	−	0.484656i	$-0.838945\pi$
$487$		−	10417.3i		−	0.969312i	0.874705	−	0.484656i	$-0.161055\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	949.021			0.0872275			0.0436138	−	0.999048i	$-0.486113\pi$
$491$	949.021			0.0872275			0.0436138	+	0.999048i	$0.486113\pi$
$492$		0			0
$493$		−	31017.6i		−	2.83360i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	9234.60i		−	0.833458i
$498$		0			0
$499$	11844.7			1.06261			0.531306	−	0.847180i	$-0.321701\pi$
$499$	11844.7			1.06261			0.531306	+	0.847180i	$0.321701\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	8481.56i		−	0.751837i	−0.926653	−	0.375918i	$-0.877327\pi$
$503$		−	8481.56i		−	0.751837i	0.926653	−	0.375918i	$-0.122673\pi$
$504$		0			0
$505$	−10032.3	+	13025.4i	−0.884027	+	1.14777i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	22153.8			1.92918			0.964588	−	0.263760i	$-0.0849628\pi$
$509$	22153.8			1.92918			0.964588	+	0.263760i	$0.0849628\pi$
$510$		0			0
$511$	3616.85			0.313112
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−7557.48	−	5820.86i	−0.646645	−	0.498054i
$516$		0			0
$517$			656.153i			0.0558173i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	7768.66			0.653265			0.326632	−	0.945151i	$-0.394086\pi$
$521$	7768.66			0.653265			0.326632	+	0.945151i	$0.394086\pi$
$522$		0			0
$523$			18785.3i			1.57060i	0.619116	+	0.785299i	$0.287491\pi$
$523$			18785.3i			1.57060i	−0.619116	+	0.785299i	$0.712509\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			22757.0i			1.88104i
$528$		0			0
$529$	−2431.20			−0.199819
$530$		0			0
$531$		0			0
$532$		0			0
$533$			5215.93i			0.423878i
$534$		0			0
$535$	−9750.67	−	7510.09i	−0.787959	−	0.606896i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	2404.98			0.192189
$540$		0			0
$541$	−1952.50			−0.155165			−0.0775826	−	0.996986i	$-0.524720\pi$
$541$	−1952.50			−0.155165			−0.0775826	+	0.996986i	$0.524720\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−7877.34	+	10227.5i	−0.619134	+	0.803848i
$546$		0			0
$547$			3586.09i			0.280311i	0.990129	+	0.140156i	$0.0447603\pi$
$547$			3586.09i			0.280311i	−0.990129	+	0.140156i	$0.955240\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−8028.16			−0.620710
$552$		0			0
$553$			11878.8i			0.913447i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			6722.16i			0.511359i	0.966762	+	0.255680i	$0.0822992\pi$
$557$			6722.16i			0.511359i	−0.966762	+	0.255680i	$0.917701\pi$
$558$		0			0
$559$	−11485.4			−0.869016
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	4620.43i		−	0.345876i	−0.984933	−	0.172938i	$-0.944674\pi$
$563$		−	4620.43i		−	0.345876i	0.984933	−	0.172938i	$-0.0553260\pi$
$564$		0			0
$565$	6396.86	+	4926.94i	0.476315	+	0.366864i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−11133.2			−0.820258			−0.410129	−	0.912027i	$-0.634516\pi$
$569$	−11133.2			−0.820258			−0.410129	+	0.912027i	$0.634516\pi$
$570$		0			0
$571$	24399.4			1.78824			0.894119	−	0.447829i	$-0.147803\pi$
$571$	24399.4			1.78824			0.894119	+	0.447829i	$0.147803\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−14602.3	+	3855.97i	−1.05906	+	0.279661i
$576$		0			0
$577$			726.720i			0.0524328i	0.999656	+	0.0262164i	$0.00834590\pi$
$577$			726.720i			0.0524328i	−0.999656	+	0.0262164i	$0.991654\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	15779.0			1.12672
$582$		0			0
$583$			24287.3i			1.72535i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			17538.4i			1.23320i	0.787277	+	0.616599i	$0.211490\pi$
$587$			17538.4i			1.23320i	−0.787277	+	0.616599i	$0.788510\pi$
$588$		0			0
$589$	5890.10			0.412050
$590$		0			0
$591$		0			0
$592$		0			0
$593$			14480.6i			1.00278i	0.865221	+	0.501390i	$0.167178\pi$
$593$			14480.6i			1.00278i	−0.865221	+	0.501390i	$0.832822\pi$
$594$		0			0
$595$	−15836.3	+	20561.0i	−1.09114	+	1.41667i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−8691.65			−0.592874			−0.296437	−	0.955052i	$-0.595798\pi$
$599$	−8691.65			−0.592874			−0.296437	+	0.955052i	$0.595798\pi$
$600$		0			0
$601$	6135.04			0.416395			0.208198	−	0.978087i	$-0.433240\pi$
$601$	6135.04			0.416395			0.208198	+	0.978087i	$0.433240\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−424.231	+	550.797i	−0.0285082	+	0.0370133i
$606$		0			0
$607$		−	6700.57i		−	0.448052i	−0.974583	−	0.224026i	$-0.928080\pi$
$607$		−	6700.57i		−	0.448052i	0.974583	−	0.224026i	$-0.0719201\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−476.468			−0.0315480
$612$		0			0
$613$			1647.58i			0.108556i	0.998526	+	0.0542781i	$0.0172858\pi$
$613$			1647.58i			0.108556i	−0.998526	+	0.0542781i	$0.982714\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			1466.68i			0.0956991i	0.998855	+	0.0478495i	$0.0152368\pi$
$617$			1466.68i			0.0956991i	−0.998855	+	0.0478495i	$0.984763\pi$
$618$		0			0
$619$	−250.842			−0.0162879			−0.00814394	−	0.999967i	$-0.502592\pi$
$619$	−250.842			−0.0162879			−0.00814394	+	0.999967i	$0.502592\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			15137.4i			0.973460i
$624$		0			0
$625$	−13588.0	+	7714.12i	−0.869630	+	0.493704i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	18472.8			1.17100
$630$		0			0
$631$	24074.6			1.51885			0.759425	−	0.650595i	$-0.225480\pi$
$631$	24074.6			1.51885			0.759425	+	0.650595i	$0.225480\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	8225.13	+	6335.10i	0.514023	+	0.395907i
$636$		0			0
$637$			1746.38i			0.108625i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−12057.7			−0.742983			−0.371492	−	0.928436i	$-0.621154\pi$
$641$	−12057.7			−0.742983			−0.371492	+	0.928436i	$0.621154\pi$
$642$		0			0
$643$		−	29896.3i		−	1.83358i	−0.399365	−	0.916792i	$-0.630769\pi$
$643$		−	29896.3i		−	1.83358i	0.399365	−	0.916792i	$-0.369231\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			4109.14i			0.249686i	0.992177	+	0.124843i	$0.0398427\pi$
$647$			4109.14i			0.249686i	−0.992177	+	0.124843i	$0.960157\pi$
$648$		0			0
$649$	−16258.6			−0.983365
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	26942.2i		−	1.61459i	−0.590146	−	0.807297i	$-0.700930\pi$
$653$		−	26942.2i		−	1.61459i	0.590146	−	0.807297i	$-0.299070\pi$
$654$		0			0
$655$	−15763.7	+	20466.7i	−0.940366	+	1.22092i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−23600.5			−1.39506			−0.697530	−	0.716555i	$-0.745718\pi$
$659$	−23600.5			−1.39506			−0.697530	+	0.716555i	$0.745718\pi$
$660$		0			0
$661$	20942.3			1.23232			0.616159	−	0.787622i	$-0.288688\pi$
$661$	20942.3			1.23232			0.616159	+	0.787622i	$0.288688\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	5321.72	+	4098.86i	0.310327	+	0.239018i
$666$		0			0
$667$			32711.1i			1.89892i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	21810.8			1.25484
$672$		0			0
$673$		−	19417.2i		−	1.11215i	−0.831132	−	0.556076i	$-0.812307\pi$
$673$		−	19417.2i		−	1.11215i	0.831132	−	0.556076i	$-0.187693\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	2573.36i		−	0.146089i	−0.997329	−	0.0730444i	$-0.976729\pi$
$677$		−	2573.36i		−	0.146089i	0.997329	−	0.0730444i	$-0.0232715\pi$
$678$		0			0
$679$	−23750.6			−1.34236
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	13059.4i		−	0.731631i	−0.930687	−	0.365816i	$-0.880790\pi$
$683$		−	13059.4i		−	0.731631i	0.930687	−	0.365816i	$-0.119210\pi$
$684$		0			0
$685$	−20294.6	−	15631.2i	−1.13200	−	0.871877i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−17636.3			−0.975168
$690$		0			0
$691$	21599.0			1.18910			0.594548	−	0.804060i	$-0.297331\pi$
$691$	21599.0			1.18910			0.594548	+	0.804060i	$0.297331\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−8657.35	+	11240.2i	−0.472506	+	0.613475i
$696$		0			0
$697$		−	23102.9i		−	1.25550i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	11688.0			0.629744			0.314872	−	0.949134i	$-0.398038\pi$
$701$	11688.0			0.629744			0.314872	+	0.949134i	$0.398038\pi$
$702$		0			0
$703$		−	4781.24i		−	0.256512i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	29794.9i		−	1.58494i
$708$		0			0
$709$	24799.9			1.31365			0.656826	−	0.754042i	$-0.271898\pi$
$709$	24799.9			1.31365			0.656826	+	0.754042i	$0.271898\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	23999.5i		−	1.26057i
$714$		0			0
$715$	8161.01	+	6285.71i	0.426859	+	0.328773i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−23713.3			−1.22998			−0.614991	−	0.788534i	$-0.710840\pi$
$719$	−23713.3			−1.22998			−0.614991	+	0.788534i	$0.710840\pi$
$720$		0			0
$721$	17287.3			0.892942
$722$		0			0
$723$		0			0
$724$		0			0
$725$	8640.31	+	32720.4i	0.442611	+	1.67614i
$726$		0			0
$727$			9476.00i			0.483419i	0.970349	+	0.241709i	$0.0777081\pi$
$727$			9476.00i			0.483419i	−0.970349	+	0.241709i	$0.922292\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	50872.1			2.57397
$732$		0			0
$733$		−	17191.6i		−	0.866285i	−0.901325	−	0.433143i	$-0.857405\pi$
$733$		−	17191.6i		−	0.866285i	0.901325	−	0.433143i	$-0.142595\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	23483.3i		−	1.17370i
$738$		0			0
$739$	−28205.7			−1.40401			−0.702004	−	0.712173i	$-0.747711\pi$
$739$	−28205.7			−1.40401			−0.702004	+	0.712173i	$0.747711\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	28305.4i		−	1.39761i	−0.715311	−	0.698806i	$-0.753715\pi$
$743$		−	28305.4i		−	1.39761i	0.715311	−	0.698806i	$-0.246285\pi$
$744$		0			0
$745$	−18807.8	+	24418.9i	−0.924916	+	1.20086i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	22304.0			1.08808
$750$		0			0
$751$	4888.09			0.237509			0.118754	−	0.992924i	$-0.462110\pi$
$751$	4888.09			0.237509			0.118754	+	0.992924i	$0.462110\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	14985.3	−	19456.1i	0.722347	−	0.937854i
$756$		0			0
$757$		−	498.409i		−	0.0239300i	−0.999928	−	0.0119650i	$-0.996191\pi$
$757$		−	498.409i		−	0.0239300i	0.999928	−	0.0119650i	$-0.00380866\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−19859.0			−0.945975			−0.472988	−	0.881069i	$-0.656824\pi$
$761$	−19859.0			−0.945975			−0.472988	+	0.881069i	$0.656824\pi$
$762$		0			0
$763$		−	23394.7i		−	1.11002i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	11806.2i		−	0.555799i
$768$		0			0
$769$	−29037.2			−1.36165			−0.680825	−	0.732447i	$-0.738378\pi$
$769$	−29037.2			−1.36165			−0.680825	+	0.732447i	$0.738378\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			36859.0i			1.71504i	0.514449	+	0.857521i	$0.327996\pi$
$773$			36859.0i			1.71504i	−0.514449	+	0.857521i	$0.672004\pi$
$774$		0			0
$775$	−6339.22	−	24006.3i	−0.293821	−	1.11269i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−5979.63			−0.275022
$780$		0			0
$781$	16235.0			0.743833
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−23188.4	−	17860.0i	−1.05431	−	0.812040i
$786$		0			0
$787$			30387.1i			1.37634i	0.725548	+	0.688171i	$0.241586\pi$
$787$			30387.1i			1.37634i	−0.725548	+	0.688171i	$0.758414\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−14632.4			−0.657736
$792$		0			0
$793$			15838.0i			0.709236i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			7178.90i			0.319059i	0.987193	+	0.159529i	$0.0509976\pi$
$797$			7178.90i			0.319059i	−0.987193	+	0.159529i	$0.949002\pi$
$798$		0			0
$799$	2110.42			0.0934433
$800$		0			0
$801$		0			0
$802$		0			0
$803$			6358.65i			0.279442i
$804$		0			0
$805$	16701.0	−	21683.6i	0.731220	−	0.949374i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−129.015			−0.00560683			−0.00280342	−	0.999996i	$-0.500892\pi$
$809$	−129.015			−0.00560683			−0.00280342	+	0.999996i	$0.500892\pi$
$810$		0			0
$811$	−40638.1			−1.75955			−0.879776	−	0.475388i	$-0.842308\pi$
$811$	−40638.1			−1.75955			−0.879776	+	0.475388i	$0.842308\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−27788.7	−	21403.2i	−1.19435	−	0.919904i
$816$		0			0
$817$		−	13167.0i		−	0.563838i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−28389.4			−1.20682			−0.603410	−	0.797431i	$-0.706192\pi$
$821$	−28389.4			−1.20682			−0.603410	+	0.797431i	$0.706192\pi$
$822$		0			0
$823$		−	5037.70i		−	0.213370i	−0.994293	−	0.106685i	$-0.965976\pi$
$823$		−	5037.70i		−	0.213370i	0.994293	−	0.106685i	$-0.0340236\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			16989.2i			0.714358i	0.934036	+	0.357179i	$0.116261\pi$
$827$			16989.2i			0.714358i	−0.934036	+	0.357179i	$0.883739\pi$
$828$		0			0
$829$	43599.9			1.82664			0.913321	−	0.407241i	$-0.133509\pi$
$829$	43599.9			1.82664			0.913321	+	0.407241i	$0.133509\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	7735.24i		−	0.321741i
$834$		0			0
$835$	−14119.5	−	10875.1i	−0.585182	−	0.450715i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	33929.9			1.39618			0.698088	−	0.716012i	$-0.254035\pi$
$839$	33929.9			1.39618			0.698088	+	0.716012i	$0.254035\pi$
$840$		0			0
$841$	48908.9			2.00537
$842$		0			0
$843$		0			0
$844$		0			0
$845$	10424.1	−	13534.0i	0.424377	−	0.550986i
$846$		0			0
$847$		−	1259.91i		−	0.0511111i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−19481.4			−0.784738
$852$		0			0
$853$		−	4176.52i		−	0.167645i	−0.996481	−	0.0838226i	$-0.973287\pi$
$853$		−	4176.52i		−	0.167645i	0.996481	−	0.0838226i	$-0.0267129\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	40750.3i		−	1.62428i	−0.583465	−	0.812138i	$-0.698303\pi$
$857$		−	40750.3i		−	1.62428i	0.583465	−	0.812138i	$-0.301697\pi$
$858$		0			0
$859$	−8930.54			−0.354722			−0.177361	−	0.984146i	$-0.556756\pi$
$859$	−8930.54			−0.354722			−0.177361	+	0.984146i	$0.556756\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	30496.1i		−	1.20290i	−0.798911	−	0.601449i	$-0.794590\pi$
$863$		−	30496.1i		−	1.20290i	0.798911	−	0.601449i	$-0.205410\pi$
$864$		0			0
$865$	−7036.43	−	5419.55i	−0.276585	−	0.213029i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−20883.6			−0.815221
$870$		0			0
$871$	17052.5			0.663379
$872$		0			0
$873$		0			0
$874$		0			0
$875$	10978.2	−	26101.1i	0.424150	−	1.00843i
$876$		0			0
$877$		−	22779.9i		−	0.877107i	−0.898705	−	0.438553i	$-0.855491\pi$
$877$		−	22779.9i		−	0.877107i	0.898705	−	0.438553i	$-0.144509\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−25260.7			−0.966011			−0.483005	−	0.875617i	$-0.660455\pi$
$881$	−25260.7			−0.966011			−0.483005	+	0.875617i	$0.660455\pi$
$882$		0			0
$883$		−	7722.97i		−	0.294336i	−0.989112	−	0.147168i	$-0.952984\pi$
$883$		−	7722.97i		−	0.294336i	0.989112	−	0.147168i	$-0.0470158\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			36874.9i			1.39587i	0.716161	+	0.697935i	$0.245898\pi$
$887$			36874.9i			1.39587i	−0.716161	+	0.697935i	$0.754102\pi$
$888$		0			0
$889$	−18814.5			−0.709806
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	546.231i		−	0.0204691i
$894$		0			0
$895$	−10368.2	+	13461.5i	−0.387231	+	0.502758i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−53777.2			−1.99507
$900$		0			0
$901$	78116.4			2.88839
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−3604.48	+	4679.84i	−0.132394	+	0.171893i
$906$		0			0
$907$		−	24260.5i		−	0.888156i	−0.895988	−	0.444078i	$-0.853531\pi$
$907$		−	24260.5i		−	0.888156i	0.895988	−	0.444078i	$-0.146469\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	2992.94			0.108848			0.0544241	−	0.998518i	$-0.482668\pi$
$911$	2992.94			0.108848			0.0544241	+	0.998518i	$0.482668\pi$
$912$		0			0
$913$			27740.5i			1.00556i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	46816.4i		−	1.68595i
$918$		0			0
$919$	−1473.94			−0.0529061			−0.0264531	−	0.999650i	$-0.508421\pi$
$919$	−1473.94			−0.0529061			−0.0264531	+	0.999650i	$0.508421\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			11789.1i			0.420415i
$924$		0			0
$925$	−19486.9	+	5145.80i	−0.692675	+	0.182911i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−2894.52			−0.102224			−0.0511120	−	0.998693i	$-0.516277\pi$
$929$	−2894.52			−0.102224			−0.0511120	+	0.998693i	$0.516277\pi$
$930$		0			0
$931$	−2002.08			−0.0704786
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−36147.5	−	27841.3i	−1.26433	−	0.973804i
$936$		0			0
$937$			11564.1i			0.403183i	0.979470	+	0.201591i	$0.0646113\pi$
$937$			11564.1i			0.403183i	−0.979470	+	0.201591i	$0.935389\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−31624.5			−1.09557			−0.547784	−	0.836620i	$-0.684528\pi$
$941$	−31624.5			−1.09557			−0.547784	+	0.836620i	$0.684528\pi$
$942$		0			0
$943$			24364.3i			0.841367i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			50297.6i			1.72593i	0.505268	+	0.862963i	$0.331394\pi$
$947$			50297.6i			1.72593i	−0.505268	+	0.862963i	$0.668606\pi$
$948$		0			0
$949$	−4617.36			−0.157941
$950$		0			0
$951$		0			0
$952$		0			0
$953$			12.0683i			0.000410211i	1.00000		0.000205105i	$6.52871e-5\pi$
$953$			12.0683i			0.000410211i	−1.00000		0.000205105i	$0.999935\pi$
$954$		0			0
$955$	−703.317	+	913.147i	−0.0238312	+	0.0309411i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	46422.6			1.56315
$960$		0			0
$961$	9664.24			0.324401
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−18931.5	−	14581.3i	−0.631531	−	0.486413i
$966$		0			0
$967$			33887.2i			1.12693i	0.826141	+	0.563464i	$0.190532\pi$
$967$			33887.2i			1.12693i	−0.826141	+	0.563464i	$0.809468\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−24364.0			−0.805231			−0.402615	−	0.915369i	$-0.631899\pi$
$971$	−24364.0			−0.805231			−0.402615	+	0.915369i	$0.631899\pi$
$972$		0			0
$973$		−	25711.3i		−	0.847138i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	19185.8i		−	0.628259i	−0.949380	−	0.314130i	$-0.898287\pi$
$977$		−	19185.8i		−	0.628259i	0.949380	−	0.314130i	$-0.101713\pi$
$978$		0			0
$979$	−26612.4			−0.868780
$980$		0			0
$981$		0			0
$982$		0			0
$983$			49118.4i			1.59373i	0.604158	+	0.796864i	$0.293509\pi$
$983$			49118.4i			1.59373i	−0.604158	+	0.796864i	$0.706491\pi$
$984$		0			0
$985$	−2513.00	−	1935.54i	−0.0812901	−	0.0626107i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−53649.6			−1.72493
$990$		0			0
$991$	−9741.07			−0.312246			−0.156123	−	0.987738i	$-0.549900\pi$
$991$	−9741.07			−0.312246			−0.156123	+	0.987738i	$0.549900\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	5267.82	−	6839.43i	0.167840	−	0.217914i
$996$		0			0
$997$		−	36968.6i		−	1.17433i	−0.809467	−	0.587166i	$-0.800244\pi$
$997$		−	36968.6i		−	1.17433i	0.809467	−	0.587166i	$-0.199756\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1080.4.f.d.649.15	yes	20
3.2	odd	2	inner	1080.4.f.d.649.6	yes	20
5.4	even	2	inner	1080.4.f.d.649.16	yes	20
15.14	odd	2	inner	1080.4.f.d.649.5	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1080.4.f.d.649.5	✓	20	15.14	odd	2	inner
1080.4.f.d.649.6	yes	20	3.2	odd	2	inner
1080.4.f.d.649.15	yes	20	1.1	even	1	trivial
1080.4.f.d.649.16	yes	20	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1080 = 2^{3} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1080.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(6.82224 - 8.85760i) q^{5} +20.2612i q^{7} +O(q^{10})\)	\(q+(6.82224 - 8.85760i) q^{5} +20.2612i q^{7} -35.6205 q^{11} -25.8659i q^{13} +114.568i q^{17} +29.6531 q^{19} -120.823i q^{23} +(-31.9142 - 120.857i) q^{25} -270.736 q^{29} +198.633 q^{31} +(179.466 + 138.227i) q^{35} -161.239i q^{37} -201.652 q^{41} -444.035i q^{43} -18.4207i q^{47} -67.5167 q^{49} -681.836i q^{53} +(-243.011 + 315.512i) q^{55} +456.439 q^{59} -612.311 q^{61} +(-229.110 - 176.464i) q^{65} +659.266i q^{67} -455.777 q^{71} -178.511i q^{73} -721.714i q^{77} +586.281 q^{79} -778.779i q^{83} +(1014.80 + 781.609i) q^{85} +747.110 q^{89} +524.075 q^{91} +(202.301 - 262.656i) q^{95} +1172.22i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q+O(q^{10}) \) 20 * q	\( 20 q - 100 q^{19} - 24 q^{25} + 228 q^{31} - 252 q^{49} - 64 q^{55} + 748 q^{61} - 668 q^{79} - 1500 q^{85} + 1744 q^{91}+O(q^{100}) \) 20 * q - 100 * q^19 - 24 * q^25 + 228 * q^31 - 252 * q^49 - 64 * q^55 + 748 * q^61 - 668 * q^79 - 1500 * q^85 + 1744 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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