LMFDB - Embedded newform 2880.2.t.d.2161.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2880,2,Mod(721,2880)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2880, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 3, 0, 0]))

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

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

chi := DirichletCharacter("2880.721");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2880 = 2^{6} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2880.t (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:	$22.9969157821$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$10$ over $\Q(i)$
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} - 2 x^{18} - 4 x^{17} + 7 x^{16} + 16 x^{15} + 6 x^{14} - 36 x^{13} - 42 x^{12} + 40 x^{11} + \cdots + 1024 $ x^20 - 2x^18 - 4x^17 + 7x^16 + 16x^15 + 6x^14 - 36x^13 - 42x^12 + 40x^11 + 136x^10 + 80x^9 - 168x^8 - 288x^7 + 96x^6 + 512x^5 + 448x^4 - 512x^3 - 512*x^2 + 1024
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{20} $
Twist minimal:	no (minimal twist has level 240)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			2161.1
Root			$-1.13207 - 0.847599i$ of defining polynomial
Character	$\chi$	$=$	2880.2161
Dual form			2880.2.t.d.721.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.707107 - 0.707107i) q^{5} -4.27253i q^{7} +O(q^{10})$	$q+(-0.707107 - 0.707107i) q^{5} -4.27253i q^{7} +(2.94281 + 2.94281i) q^{11} +(-4.05962 + 4.05962i) q^{13} -0.160060 q^{17} +(-4.32576 + 4.32576i) q^{19} -8.40564i q^{23} +1.00000i q^{25} +(1.78072 - 1.78072i) q^{29} -7.17282 q^{31} +(-3.02114 + 3.02114i) q^{35} +(-0.669226 - 0.669226i) q^{37} +3.96632i q^{41} +(0.255733 + 0.255733i) q^{43} +0.0752658 q^{47} -11.2545 q^{49} +(2.88214 + 2.88214i) q^{53} -4.16177i q^{55} +(-5.63594 - 5.63594i) q^{59} +(-4.48857 + 4.48857i) q^{61} +5.74117 q^{65} +(0.131176 - 0.131176i) q^{67} +12.1137i q^{71} +0.382876i q^{73} +(12.5733 - 12.5733i) q^{77} -15.3239 q^{79} +(-5.54562 + 5.54562i) q^{83} +(0.113179 + 0.113179i) q^{85} +13.8991i q^{89} +(17.3449 + 17.3449i) q^{91} +6.11754 q^{95} +10.8999 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q+O(q^{10}) $ 20 * q	$ 20 q + 8 q^{11} + 24 q^{17} + 4 q^{19} - 16 q^{29} + 16 q^{37} + 8 q^{43} - 52 q^{49} + 16 q^{53} - 16 q^{59} - 4 q^{61} + 8 q^{67} + 40 q^{77} - 56 q^{79} - 48 q^{83} + 4 q^{85} + 8 q^{91} + 56 q^{97}+O(q^{100}) $ 20 * q + 8 * q^11 + 24 * q^17 + 4 * q^19 - 16 * q^29 + 16 * q^37 + 8 * q^43 - 52 * q^49 + 16 * q^53 - 16 * q^59 - 4 * q^61 + 8 * q^67 + 40 * q^77 - 56 * q^79 - 48 * q^83 + 4 * q^85 + 8 * q^91 + 56 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2880\mathbb{Z}\right)^\times$.

$n$	$577$	$641$	$901$	$2431$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{4}\right)$	$1$

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
$5$	−0.707107	−	0.707107i	−0.316228	−	0.316228i
$5$	−0.707107	−	0.707107i	−0.316228	−	0.316228i
$6$		0			0
$7$		−	4.27253i		−	1.61487i	−0.589959	−	0.807433i	$-0.700856\pi$
$7$		−	4.27253i		−	1.61487i	0.589959	−	0.807433i	$-0.299144\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	2.94281	+	2.94281i	0.887291	+	0.887291i	0.994262	−	0.106971i	$-0.0341151\pi$
$11$	2.94281	+	2.94281i	0.887291	+	0.887291i	−0.106971	+	0.994262i	$0.534115\pi$
$12$		0			0
$13$	−4.05962	+	4.05962i	−1.12594	+	1.12594i	−0.135106	+	0.990831i	$0.543137\pi$
$13$	−4.05962	+	4.05962i	−1.12594	+	1.12594i	−0.990831	+	0.135106i	$0.956863\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−0.160060			−0.0388202			−0.0194101	−	0.999812i	$-0.506179\pi$
$17$	−0.160060			−0.0388202			−0.0194101	+	0.999812i	$0.506179\pi$
$18$		0			0
$19$	−4.32576	+	4.32576i	−0.992396	+	0.992396i	−0.999971	−	0.00757497i	$-0.997589\pi$
$19$	−4.32576	+	4.32576i	−0.992396	+	0.992396i	0.00757497	+	0.999971i	$0.497589\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	8.40564i		−	1.75270i	−0.481677	−	0.876349i	$-0.659972\pi$
$23$		−	8.40564i		−	1.75270i	0.481677	−	0.876349i	$-0.340028\pi$
$24$		0			0
$25$			1.00000i			0.200000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	1.78072	−	1.78072i	0.330671	−	0.330671i	−0.522171	−	0.852841i	$-0.674878\pi$
$29$	1.78072	−	1.78072i	0.330671	−	0.330671i	0.852841	+	0.522171i	$0.174878\pi$
$30$		0			0
$31$	−7.17282			−1.28828			−0.644138	−	0.764909i	$-0.722784\pi$
$31$	−7.17282			−1.28828			−0.644138	+	0.764909i	$0.722784\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−3.02114	+	3.02114i	−0.510666	+	0.510666i
$36$		0			0
$37$	−0.669226	−	0.669226i	−0.110020	−	0.110020i	0.649954	−	0.759974i	$-0.274788\pi$
$37$	−0.669226	−	0.669226i	−0.110020	−	0.110020i	−0.759974	+	0.649954i	$0.774788\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			3.96632i			0.619435i	0.950829	+	0.309717i	$0.100234\pi$
$41$			3.96632i			0.619435i	−0.950829	+	0.309717i	$0.899766\pi$
$42$		0			0
$43$	0.255733	+	0.255733i	0.0389989	+	0.0389989i	0.726337	−	0.687338i	$-0.241221\pi$
$43$	0.255733	+	0.255733i	0.0389989	+	0.0389989i	−0.687338	+	0.726337i	$0.741221\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	0.0752658			0.0109787			0.00548933	−	0.999985i	$-0.498253\pi$
$47$	0.0752658			0.0109787			0.00548933	+	0.999985i	$0.498253\pi$
$48$		0			0
$49$	−11.2545			−1.60779
$50$		0			0
$51$		0			0
$52$		0			0
$53$	2.88214	+	2.88214i	0.395892	+	0.395892i	0.876781	−	0.480889i	$-0.159686\pi$
$53$	2.88214	+	2.88214i	0.395892	+	0.395892i	−0.480889	+	0.876781i	$0.659686\pi$
$54$		0			0
$55$		−	4.16177i		−	0.561172i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−5.63594	−	5.63594i	−0.733737	−	0.733737i	0.237621	−	0.971358i	$-0.423632\pi$
$59$	−5.63594	−	5.63594i	−0.733737	−	0.733737i	−0.971358	+	0.237621i	$0.923632\pi$
$60$		0			0
$61$	−4.48857	+	4.48857i	−0.574703	+	0.574703i	−0.933439	−	0.358736i	$-0.883208\pi$
$61$	−4.48857	+	4.48857i	−0.574703	+	0.574703i	0.358736	+	0.933439i	$0.383208\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	5.74117			0.712105
$66$		0			0
$67$	0.131176	−	0.131176i	0.0160257	−	0.0160257i	−0.699049	−	0.715074i	$-0.746393\pi$
$67$	0.131176	−	0.131176i	0.0160257	−	0.0160257i	0.715074	+	0.699049i	$0.246393\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			12.1137i			1.43764i	0.695199	+	0.718818i	$0.255316\pi$
$71$			12.1137i			1.43764i	−0.695199	+	0.718818i	$0.744684\pi$
$72$		0			0
$73$			0.382876i			0.0448123i	0.999749	+	0.0224061i	$0.00713269\pi$
$73$			0.382876i			0.0448123i	−0.999749	+	0.0224061i	$0.992867\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	12.5733	−	12.5733i	1.43286	−	1.43286i
$78$		0			0
$79$	−15.3239			−1.72408			−0.862039	−	0.506841i	$-0.830813\pi$
$79$	−15.3239			−1.72408			−0.862039	+	0.506841i	$0.830813\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−5.54562	+	5.54562i	−0.608710	+	0.608710i	−0.942609	−	0.333899i	$-0.891636\pi$
$83$	−5.54562	+	5.54562i	−0.608710	+	0.608710i	0.333899	+	0.942609i	$0.391636\pi$
$84$		0			0
$85$	0.113179	+	0.113179i	0.0122760	+	0.0122760i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			13.8991i			1.47330i	0.676275	+	0.736649i	$0.263593\pi$
$89$			13.8991i			1.47330i	−0.676275	+	0.736649i	$0.736407\pi$
$90$		0			0
$91$	17.3449	+	17.3449i	1.81824	+	1.81824i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	6.11754			0.627647
$96$		0			0
$97$	10.8999			1.10672			0.553358	−	0.832944i	$-0.313346\pi$
$97$	10.8999			1.10672			0.553358	+	0.832944i	$0.313346\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	2.98972	+	2.98972i	0.297489	+	0.297489i	0.840029	−	0.542541i	$-0.182538\pi$
$101$	2.98972	+	2.98972i	0.297489	+	0.297489i	−0.542541	+	0.840029i	$0.682538\pi$
$102$		0			0
$103$			16.8190i			1.65723i	0.559820	+	0.828615i	$0.310870\pi$
$103$			16.8190i			1.65723i	−0.559820	+	0.828615i	$0.689130\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	1.64447	+	1.64447i	0.158977	+	0.158977i	0.782113	−	0.623136i	$-0.214142\pi$
$107$	1.64447	+	1.64447i	0.158977	+	0.158977i	−0.623136	+	0.782113i	$0.714142\pi$
$108$		0			0
$109$	4.67023	−	4.67023i	0.447327	−	0.447327i	−0.447138	−	0.894465i	$-0.647557\pi$
$109$	4.67023	−	4.67023i	0.447327	−	0.447327i	0.894465	+	0.447138i	$0.147557\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−5.00533			−0.470862			−0.235431	−	0.971891i	$-0.575650\pi$
$113$	−5.00533			−0.470862			−0.235431	+	0.971891i	$0.575650\pi$
$114$		0			0
$115$	−5.94369	+	5.94369i	−0.554252	+	0.554252i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			0.683861i			0.0626894i
$120$		0			0
$121$			6.32029i			0.574572i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	0.707107	−	0.707107i	0.0632456	−	0.0632456i
$126$		0			0
$127$	−1.96679			−0.174525			−0.0872623	−	0.996185i	$-0.527812\pi$
$127$	−1.96679			−0.174525			−0.0872623	+	0.996185i	$0.527812\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−0.852904	+	0.852904i	−0.0745186	+	0.0745186i	−0.743384	−	0.668865i	$-0.766780\pi$
$131$	−0.852904	+	0.852904i	−0.0745186	+	0.0745186i	0.668865	+	0.743384i	$0.266780\pi$
$132$		0			0
$133$	18.4819	+	18.4819i	1.60259	+	1.60259i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			4.12023i			0.352015i	0.984389	+	0.176007i	$0.0563183\pi$
$137$			4.12023i			0.352015i	−0.984389	+	0.176007i	$0.943682\pi$
$138$		0			0
$139$	−5.73895	−	5.73895i	−0.486771	−	0.486771i	0.420514	−	0.907286i	$-0.361850\pi$
$139$	−5.73895	−	5.73895i	−0.486771	−	0.486771i	−0.907286	+	0.420514i	$0.861850\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−23.8934			−1.99807
$144$		0			0
$145$	−2.51831			−0.209134
$146$		0			0
$147$		0			0
$148$		0			0
$149$	9.96311	+	9.96311i	0.816210	+	0.816210i	0.985557	−	0.169346i	$-0.0541657\pi$
$149$	9.96311	+	9.96311i	0.816210	+	0.816210i	−0.169346	+	0.985557i	$0.554166\pi$
$150$		0			0
$151$			4.01882i			0.327047i	0.986539	+	0.163524i	$0.0522860\pi$
$151$			4.01882i			0.327047i	−0.986539	+	0.163524i	$0.947714\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	5.07195	+	5.07195i	0.407389	+	0.407389i
$156$		0			0
$157$	−9.14459	+	9.14459i	−0.729818	+	0.729818i	−0.970583	−	0.240766i	$-0.922601\pi$
$157$	−9.14459	+	9.14459i	−0.729818	+	0.729818i	0.240766	+	0.970583i	$0.422601\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−35.9134			−2.83037
$162$		0			0
$163$	0.346095	−	0.346095i	0.0271082	−	0.0271082i	−0.693423	−	0.720531i	$-0.743898\pi$
$163$	0.346095	−	0.346095i	0.0271082	−	0.0271082i	0.720531	+	0.693423i	$0.243898\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	10.9882i		−	0.850296i	−0.905124	−	0.425148i	$-0.860222\pi$
$167$		−	10.9882i		−	0.850296i	0.905124	−	0.425148i	$-0.139778\pi$
$168$		0			0
$169$		−	19.9611i		−	1.53547i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−2.70605	+	2.70605i	−0.205737	+	0.205737i	−0.802453	−	0.596716i	$-0.796472\pi$
$173$	−2.70605	+	2.70605i	−0.205737	+	0.205737i	0.596716	+	0.802453i	$0.296472\pi$
$174$		0			0
$175$	4.27253			0.322973
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−10.4088	+	10.4088i	−0.777990	+	0.777990i	−0.979489	−	0.201499i	$-0.935419\pi$
$179$	−10.4088	+	10.4088i	−0.777990	+	0.777990i	0.201499	+	0.979489i	$0.435419\pi$
$180$		0			0
$181$	−15.4792	−	15.4792i	−1.15056	−	1.15056i	−0.986441	−	0.164119i	$-0.947522\pi$
$181$	−15.4792	−	15.4792i	−1.15056	−	1.15056i	−0.164119	−	0.986441i	$-0.552478\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			0.946429i			0.0695828i
$186$		0			0
$187$	−0.471026	−	0.471026i	−0.0344448	−	0.0344448i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	4.75146			0.343803			0.171902	−	0.985114i	$-0.445009\pi$
$191$	4.75146			0.343803			0.171902	+	0.985114i	$0.445009\pi$
$192$		0			0
$193$	−11.7915			−0.848772			−0.424386	−	0.905481i	$-0.639510\pi$
$193$	−11.7915			−0.848772			−0.424386	+	0.905481i	$0.639510\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−4.56244	−	4.56244i	−0.325060	−	0.325060i	0.525644	−	0.850705i	$-0.323824\pi$
$197$	−4.56244	−	4.56244i	−0.325060	−	0.325060i	−0.850705	+	0.525644i	$0.823824\pi$
$198$		0			0
$199$		−	3.60262i		−	0.255383i	−0.991814	−	0.127691i	$-0.959243\pi$
$199$		−	3.60262i		−	0.255383i	0.991814	−	0.127691i	$-0.0407567\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−7.60817	−	7.60817i	−0.533989	−	0.533989i
$204$		0			0
$205$	2.80461	−	2.80461i	0.195883	−	0.195883i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−25.4598			−1.76109
$210$		0			0
$211$	0.975389	−	0.975389i	0.0671486	−	0.0671486i	−0.672735	−	0.739884i	$-0.734881\pi$
$211$	0.975389	−	0.975389i	0.0671486	−	0.0671486i	0.739884	+	0.672735i	$0.234881\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	0.361661i		−	0.0246651i
$216$		0			0
$217$			30.6461i			2.08039i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	0.649782	−	0.649782i	0.0437091	−	0.0437091i
$222$		0			0
$223$	−4.66343			−0.312286			−0.156143	−	0.987734i	$-0.549906\pi$
$223$	−4.66343			−0.312286			−0.156143	+	0.987734i	$0.549906\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−1.61701	+	1.61701i	−0.107325	+	0.107325i	−0.758730	−	0.651405i	$-0.774180\pi$
$227$	−1.61701	+	1.61701i	−0.107325	+	0.107325i	0.651405	+	0.758730i	$0.274180\pi$
$228$		0			0
$229$	9.14882	+	9.14882i	0.604571	+	0.604571i	0.941522	−	0.336951i	$-0.109396\pi$
$229$	9.14882	+	9.14882i	0.604571	+	0.604571i	−0.336951	+	0.941522i	$0.609396\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	10.3002i		−	0.674789i	−0.941363	−	0.337394i	$-0.890454\pi$
$233$		−	10.3002i		−	0.674789i	0.941363	−	0.337394i	$-0.109546\pi$
$234$		0			0
$235$	−0.0532210	−	0.0532210i	−0.00347175	−	0.00347175i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	3.73825			0.241807			0.120904	−	0.992664i	$-0.461421\pi$
$239$	3.73825			0.241807			0.120904	+	0.992664i	$0.461421\pi$
$240$		0			0
$241$	−9.34283			−0.601824			−0.300912	−	0.953652i	$-0.597291\pi$
$241$	−9.34283			−0.601824			−0.300912	+	0.953652i	$0.597291\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	7.95817	+	7.95817i	0.508429	+	0.508429i
$246$		0			0
$247$		−	35.1219i		−	2.23475i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−9.31174	−	9.31174i	−0.587752	−	0.587752i	0.349270	−	0.937022i	$-0.386429\pi$
$251$	−9.31174	−	9.31174i	−0.587752	−	0.587752i	−0.937022	+	0.349270i	$0.886429\pi$
$252$		0			0
$253$	24.7362	−	24.7362i	1.55515	−	1.55515i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−8.68522			−0.541769			−0.270885	−	0.962612i	$-0.587316\pi$
$257$	−8.68522			−0.541769			−0.270885	+	0.962612i	$0.587316\pi$
$258$		0			0
$259$	−2.85929	+	2.85929i	−0.177668	+	0.177668i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	5.18971i		−	0.320011i	−0.987116	−	0.160006i	$-0.948849\pi$
$263$		−	5.18971i		−	0.320011i	0.987116	−	0.160006i	$-0.0511513\pi$
$264$		0			0
$265$		−	4.07596i		−	0.250384i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	7.99426	−	7.99426i	0.487418	−	0.487418i	−0.420072	−	0.907491i	$-0.637995\pi$
$269$	7.99426	−	7.99426i	0.487418	−	0.487418i	0.907491	+	0.420072i	$0.137995\pi$
$270$		0			0
$271$	18.8342			1.14410			0.572048	−	0.820220i	$-0.306149\pi$
$271$	18.8342			1.14410			0.572048	+	0.820220i	$0.306149\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−2.94281	+	2.94281i	−0.177458	+	0.177458i
$276$		0			0
$277$	3.71919	+	3.71919i	0.223465	+	0.223465i	0.809956	−	0.586491i	$-0.199491\pi$
$277$	3.71919	+	3.71919i	0.223465	+	0.223465i	−0.586491	+	0.809956i	$0.699491\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	11.0223i		−	0.657538i	−0.944410	−	0.328769i	$-0.893366\pi$
$281$		−	11.0223i		−	0.657538i	0.944410	−	0.328769i	$-0.106634\pi$
$282$		0			0
$283$	−10.5210	−	10.5210i	−0.625411	−	0.625411i	0.321499	−	0.946910i	$-0.395813\pi$
$283$	−10.5210	−	10.5210i	−0.625411	−	0.625411i	−0.946910	+	0.321499i	$0.895813\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	16.9462			1.00030
$288$		0			0
$289$	−16.9744			−0.998493
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−15.5643	−	15.5643i	−0.909274	−	0.909274i	0.0869399	−	0.996214i	$-0.472291\pi$
$293$	−15.5643	−	15.5643i	−0.909274	−	0.909274i	−0.996214	+	0.0869399i	$0.972291\pi$
$294$		0			0
$295$			7.97042i			0.464056i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	34.1237	+	34.1237i	1.97343	+	1.97343i
$300$		0			0
$301$	1.09263	−	1.09263i	0.0629780	−	0.0629780i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	6.34780			0.363474
$306$		0			0
$307$	−19.6795	+	19.6795i	−1.12317	+	1.12317i	−0.131908	+	0.991262i	$0.542110\pi$
$307$	−19.6795	+	19.6795i	−1.12317	+	1.12317i	−0.991262	+	0.131908i	$0.957890\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	11.8922i		−	0.674344i	−0.941443	−	0.337172i	$-0.890530\pi$
$311$		−	11.8922i		−	0.674344i	0.941443	−	0.337172i	$-0.109470\pi$
$312$		0			0
$313$			9.61193i			0.543298i	0.962396	+	0.271649i	$0.0875690\pi$
$313$			9.61193i			0.543298i	−0.962396	+	0.271649i	$0.912431\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−8.10049	+	8.10049i	−0.454969	+	0.454969i	−0.897000	−	0.442031i	$-0.854258\pi$
$317$	−8.10049	+	8.10049i	−0.454969	+	0.454969i	0.442031	+	0.897000i	$0.354258\pi$
$318$		0			0
$319$	10.4806			0.586802
$320$		0			0
$321$		0			0
$322$		0			0
$323$	0.692379	−	0.692379i	0.0385250	−	0.0385250i
$324$		0			0
$325$	−4.05962	−	4.05962i	−0.225187	−	0.225187i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	0.321576i		−	0.0177291i
$330$		0			0
$331$	13.3267	+	13.3267i	0.732501	+	0.732501i	0.971115	−	0.238613i	$-0.0766928\pi$
$331$	13.3267	+	13.3267i	0.732501	+	0.732501i	−0.238613	+	0.971115i	$0.576693\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−0.185511			−0.0101356
$336$		0			0
$337$	32.6763			1.77999			0.889996	−	0.455967i	$-0.150707\pi$
$337$	32.6763			1.77999			0.889996	+	0.455967i	$0.150707\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−21.1083	−	21.1083i	−1.14308	−	1.14308i
$342$		0			0
$343$			18.1777i			0.981504i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	22.4898	+	22.4898i	1.20732	+	1.20732i	0.971893	+	0.235423i	$0.0756477\pi$
$347$	22.4898	+	22.4898i	1.20732	+	1.20732i	0.235423	+	0.971893i	$0.424352\pi$
$348$		0			0
$349$	−22.6617	+	22.6617i	−1.21305	+	1.21305i	−0.243032	+	0.970018i	$0.578142\pi$
$349$	−22.6617	+	22.6617i	−1.21305	+	1.21305i	−0.970018	+	0.243032i	$0.921858\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	10.6347			0.566026			0.283013	−	0.959116i	$-0.408666\pi$
$353$	10.6347			0.566026			0.283013	+	0.959116i	$0.408666\pi$
$354$		0			0
$355$	8.56570	−	8.56570i	0.454620	−	0.454620i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	17.0191i		−	0.898235i	−0.893473	−	0.449118i	$-0.851738\pi$
$359$		−	17.0191i		−	0.898235i	0.893473	−	0.449118i	$-0.148262\pi$
$360$		0			0
$361$		−	18.4243i		−	0.969701i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	0.270734	−	0.270734i	0.0141709	−	0.0141709i
$366$		0			0
$367$	25.7040			1.34174			0.670868	−	0.741577i	$-0.265922\pi$
$367$	25.7040			1.34174			0.670868	+	0.741577i	$0.265922\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	12.3140	−	12.3140i	0.639313	−	0.639313i
$372$		0			0
$373$	−14.7290	−	14.7290i	−0.762641	−	0.762641i	0.214158	−	0.976799i	$-0.431299\pi$
$373$	−14.7290	−	14.7290i	−0.762641	−	0.762641i	−0.976799	+	0.214158i	$0.931299\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			14.4581i			0.744628i
$378$		0			0
$379$	−10.0142	−	10.0142i	−0.514395	−	0.514395i	0.401475	−	0.915870i	$-0.368498\pi$
$379$	−10.0142	−	10.0142i	−0.514395	−	0.514395i	−0.915870	+	0.401475i	$0.868498\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−28.1834			−1.44010			−0.720052	−	0.693920i	$-0.755882\pi$
$383$	−28.1834			−1.44010			−0.720052	+	0.693920i	$0.755882\pi$
$384$		0			0
$385$	−17.7813			−0.906218
$386$		0			0
$387$		0			0
$388$		0			0
$389$	11.1551	+	11.1551i	0.565586	+	0.565586i	0.930889	−	0.365303i	$-0.119035\pi$
$389$	11.1551	+	11.1551i	0.565586	+	0.565586i	−0.365303	+	0.930889i	$0.619035\pi$
$390$		0			0
$391$			1.34540i			0.0680400i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	10.8357	+	10.8357i	0.545202	+	0.545202i
$396$		0			0
$397$	2.98216	−	2.98216i	0.149670	−	0.149670i	−0.628300	−	0.777971i	$-0.716249\pi$
$397$	2.98216	−	2.98216i	0.149670	−	0.149670i	0.777971	+	0.628300i	$0.216249\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	15.9896			0.798485			0.399242	−	0.916845i	$-0.369273\pi$
$401$	15.9896			0.798485			0.399242	+	0.916845i	$0.369273\pi$
$402$		0			0
$403$	29.1189	−	29.1189i	1.45052	−	1.45052i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	3.93881i		−	0.195240i
$408$		0			0
$409$			27.5110i			1.36033i	0.733059	+	0.680165i	$0.238092\pi$
$409$			27.5110i			1.36033i	−0.733059	+	0.680165i	$0.761908\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−24.0797	+	24.0797i	−1.18489	+	1.18489i
$414$		0			0
$415$	7.84269			0.384982
$416$		0			0
$417$		0			0
$418$		0			0
$419$	1.95916	−	1.95916i	0.0957114	−	0.0957114i	−0.657630	−	0.753341i	$-0.728441\pi$
$419$	1.95916	−	1.95916i	0.0957114	−	0.0957114i	0.753341	+	0.657630i	$0.228441\pi$
$420$		0			0
$421$	6.23743	+	6.23743i	0.303994	+	0.303994i	0.842574	−	0.538580i	$-0.181039\pi$
$421$	6.23743	+	6.23743i	0.303994	+	0.303994i	−0.538580	+	0.842574i	$0.681039\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	0.160060i		−	0.00776404i
$426$		0			0
$427$	19.1776	+	19.1776i	0.928068	+	0.928068i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	5.88180			0.283317			0.141658	−	0.989916i	$-0.454757\pi$
$431$	5.88180			0.283317			0.141658	+	0.989916i	$0.454757\pi$
$432$		0			0
$433$	−6.37830			−0.306522			−0.153261	−	0.988186i	$-0.548977\pi$
$433$	−6.37830			−0.306522			−0.153261	+	0.988186i	$0.548977\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	36.3608	+	36.3608i	1.73937	+	1.73937i
$438$		0			0
$439$		−	37.2899i		−	1.77975i	−0.456206	−	0.889874i	$-0.650792\pi$
$439$		−	37.2899i		−	1.77975i	0.456206	−	0.889874i	$-0.349208\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	28.4238	+	28.4238i	1.35046	+	1.35046i	0.885149	+	0.465308i	$0.154056\pi$
$443$	28.4238	+	28.4238i	1.35046	+	1.35046i	0.465308	+	0.885149i	$0.345944\pi$
$444$		0			0
$445$	9.82813	−	9.82813i	0.465898	−	0.465898i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−11.9658			−0.564701			−0.282350	−	0.959311i	$-0.591114\pi$
$449$	−11.9658			−0.564701			−0.282350	+	0.959311i	$0.591114\pi$
$450$		0			0
$451$	−11.6721	+	11.6721i	−0.549619	+	0.549619i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	24.5294i		−	1.14995i
$456$		0			0
$457$		−	33.0504i		−	1.54603i	−0.634385	−	0.773017i	$-0.718747\pi$
$457$		−	33.0504i		−	1.54603i	0.634385	−	0.773017i	$-0.281253\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	20.7242	−	20.7242i	0.965221	−	0.965221i	−0.0341939	−	0.999415i	$-0.510886\pi$
$461$	20.7242	−	20.7242i	0.965221	−	0.965221i	0.999415	+	0.0341939i	$0.0108864\pi$
$462$		0			0
$463$	−14.2359			−0.661599			−0.330800	−	0.943701i	$-0.607318\pi$
$463$	−14.2359			−0.661599			−0.330800	+	0.943701i	$0.607318\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	18.7991	−	18.7991i	0.869918	−	0.869918i	−0.122545	−	0.992463i	$-0.539106\pi$
$467$	18.7991	−	18.7991i	0.869918	−	0.869918i	0.992463	+	0.122545i	$0.0391056\pi$
$468$		0			0
$469$	−0.560456	−	0.560456i	−0.0258794	−	0.0258794i
$470$		0			0
$471$		0			0
$472$		0			0
$473$			1.50515i			0.0692068i
$474$		0			0
$475$	−4.32576	−	4.32576i	−0.198479	−	0.198479i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−26.7178			−1.22077			−0.610384	−	0.792106i	$-0.708985\pi$
$479$	−26.7178			−1.22077			−0.610384	+	0.792106i	$0.708985\pi$
$480$		0			0
$481$	5.43361			0.247751
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−7.70738	−	7.70738i	−0.349974	−	0.349974i
$486$		0			0
$487$			27.0801i			1.22712i	0.789650	+	0.613558i	$0.210262\pi$
$487$			27.0801i			1.22712i	−0.789650	+	0.613558i	$0.789738\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	1.51536	+	1.51536i	0.0683872	+	0.0683872i	0.740473	−	0.672086i	$-0.234601\pi$
$491$	1.51536	+	1.51536i	0.0683872	+	0.0683872i	−0.672086	+	0.740473i	$0.734601\pi$
$492$		0			0
$493$	−0.285021	+	0.285021i	−0.0128367	+	0.0128367i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	51.7563			2.32159
$498$		0			0
$499$	−7.31012	+	7.31012i	−0.327246	+	0.327246i	−0.851538	−	0.524292i	$-0.824330\pi$
$499$	−7.31012	+	7.31012i	−0.327246	+	0.327246i	0.524292	+	0.851538i	$0.324330\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	37.4924i		−	1.67170i	−0.548954	−	0.835852i	$-0.684974\pi$
$503$		−	37.4924i		−	1.67170i	0.548954	−	0.835852i	$-0.315026\pi$
$504$		0			0
$505$		−	4.22811i		−	0.188148i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−27.7592	+	27.7592i	−1.23040	+	1.23040i	−0.266595	+	0.963809i	$0.585899\pi$
$509$	−27.7592	+	27.7592i	−1.23040	+	1.23040i	−0.963809	+	0.266595i	$0.914101\pi$
$510$		0			0
$511$	1.63585			0.0723658
$512$		0			0
$513$		0			0
$514$		0			0
$515$	11.8929	−	11.8929i	0.524062	−	0.524062i
$516$		0			0
$517$	0.221493	+	0.221493i	0.00974126	+	0.00974126i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	10.0782i		−	0.441533i	−0.975327	−	0.220767i	$-0.929144\pi$
$521$		−	10.0782i		−	0.441533i	0.975327	−	0.220767i	$-0.0708559\pi$
$522$		0			0
$523$	−8.16934	−	8.16934i	−0.357220	−	0.357220i	0.505567	−	0.862787i	$-0.331283\pi$
$523$	−8.16934	−	8.16934i	−0.357220	−	0.357220i	−0.862787	+	0.505567i	$0.831283\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	1.14808			0.0500111
$528$		0			0
$529$	−47.6548			−2.07195
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−16.1018	−	16.1018i	−0.697445	−	0.697445i
$534$		0			0
$535$		−	2.32563i		−	0.100546i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−33.1200	−	33.1200i	−1.42658	−	1.42658i
$540$		0			0
$541$	1.98301	−	1.98301i	0.0852564	−	0.0852564i	−0.663192	−	0.748449i	$-0.730799\pi$
$541$	1.98301	−	1.98301i	0.0852564	−	0.0852564i	0.748449	+	0.663192i	$0.230799\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−6.60470			−0.282914
$546$		0			0
$547$	14.5464	−	14.5464i	0.621959	−	0.621959i	−0.324073	−	0.946032i	$-0.605052\pi$
$547$	14.5464	−	14.5464i	0.621959	−	0.621959i	0.946032	+	0.324073i	$0.105052\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			15.4059i			0.656313i
$552$		0			0
$553$			65.4721i			2.78416i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	25.6935	−	25.6935i	1.08867	−	1.08867i	0.0930007	−	0.995666i	$-0.470354\pi$
$557$	25.6935	−	25.6935i	1.08867	−	1.08867i	0.995666	−	0.0930007i	$-0.0296459\pi$
$558$		0			0
$559$	−2.07636			−0.0878206
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−15.8673	+	15.8673i	−0.668729	+	0.668729i	−0.957422	−	0.288693i	$-0.906779\pi$
$563$	−15.8673	+	15.8673i	−0.668729	+	0.668729i	0.288693	+	0.957422i	$0.406779\pi$
$564$		0			0
$565$	3.53930	+	3.53930i	0.148900	+	0.148900i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	45.0379i		−	1.88809i	−0.329819	−	0.944044i	$-0.606988\pi$
$569$		−	45.0379i		−	1.88809i	0.329819	−	0.944044i	$-0.393012\pi$
$570$		0			0
$571$	−23.0995	−	23.0995i	−0.966684	−	0.966684i	0.0327783	−	0.999463i	$-0.489564\pi$
$571$	−23.0995	−	23.0995i	−0.966684	−	0.966684i	−0.999463	+	0.0327783i	$0.989564\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	8.40564			0.350540
$576$		0			0
$577$	−10.3572			−0.431178			−0.215589	−	0.976484i	$-0.569167\pi$
$577$	−10.3572			−0.431178			−0.215589	+	0.976484i	$0.569167\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	23.6938	+	23.6938i	0.982986	+	0.982986i
$582$		0			0
$583$			16.9632i			0.702543i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−23.6336	−	23.6336i	−0.975462	−	0.975462i	0.0242436	−	0.999706i	$-0.492282\pi$
$587$	−23.6336	−	23.6336i	−0.975462	−	0.975462i	−0.999706	+	0.0242436i	$0.992282\pi$
$588$		0			0
$589$	31.0279	−	31.0279i	1.27848	−	1.27848i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	30.5649			1.25515			0.627575	−	0.778556i	$-0.284048\pi$
$593$	30.5649			1.25515			0.627575	+	0.778556i	$0.284048\pi$
$594$		0			0
$595$	0.483562	−	0.483562i	0.0198241	−	0.0198241i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			10.2816i			0.420096i	0.977691	+	0.210048i	$0.0673620\pi$
$599$			10.2816i			0.420096i	−0.977691	+	0.210048i	$0.932638\pi$
$600$		0			0
$601$		−	22.9223i		−	0.935021i	−0.883987	−	0.467511i	$-0.845151\pi$
$601$		−	22.9223i		−	0.935021i	0.883987	−	0.467511i	$-0.154849\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	4.46912	−	4.46912i	0.181696	−	0.181696i
$606$		0			0
$607$	−39.7832			−1.61475			−0.807374	−	0.590040i	$-0.799112\pi$
$607$	−39.7832			−1.61475			−0.807374	+	0.590040i	$0.799112\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−0.305551	+	0.305551i	−0.0123613	+	0.0123613i
$612$		0			0
$613$	−26.9534	−	26.9534i	−1.08864	−	1.08864i	−0.995669	−	0.0929665i	$-0.970365\pi$
$613$	−26.9534	−	26.9534i	−1.08864	−	1.08864i	−0.0929665	−	0.995669i	$-0.529635\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	12.2306i		−	0.492385i	−0.969221	−	0.246193i	$-0.920820\pi$
$617$		−	12.2306i		−	0.492385i	0.969221	−	0.246193i	$-0.0791795\pi$
$618$		0			0
$619$	4.13984	+	4.13984i	0.166394	+	0.166394i	0.785392	−	0.618998i	$-0.212461\pi$
$619$	4.13984	+	4.13984i	0.166394	+	0.166394i	−0.618998	+	0.785392i	$0.712461\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	59.3843			2.37918
$624$		0			0
$625$	−1.00000			−0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	0.107116	+	0.107116i	0.00427100	+	0.00427100i
$630$		0			0
$631$		−	25.3923i		−	1.01085i	−0.862870	−	0.505426i	$-0.831336\pi$
$631$		−	25.3923i		−	1.01085i	0.862870	−	0.505426i	$-0.168664\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	1.39073	+	1.39073i	0.0551895	+	0.0551895i
$636$		0			0
$637$	45.6892	−	45.6892i	1.81027	−	1.81027i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−42.0683			−1.66160			−0.830799	−	0.556572i	$-0.812116\pi$
$641$	−42.0683			−1.66160			−0.830799	+	0.556572i	$0.812116\pi$
$642$		0			0
$643$	8.73724	−	8.73724i	0.344563	−	0.344563i	−0.513517	−	0.858080i	$-0.671657\pi$
$643$	8.73724	−	8.73724i	0.344563	−	0.344563i	0.858080	+	0.513517i	$0.171657\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			4.04843i			0.159160i	0.996828	+	0.0795802i	$0.0253580\pi$
$647$			4.04843i			0.159160i	−0.996828	+	0.0795802i	$0.974642\pi$
$648$		0			0
$649$		−	33.1710i		−	1.30208i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	12.8095	−	12.8095i	0.501275	−	0.501275i	−0.410559	−	0.911834i	$-0.634666\pi$
$653$	12.8095	−	12.8095i	0.501275	−	0.501275i	0.911834	+	0.410559i	$0.134666\pi$
$654$		0			0
$655$	1.20619			0.0471297
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−29.8196	+	29.8196i	−1.16161	+	1.16161i	−0.177481	+	0.984124i	$0.556795\pi$
$659$	−29.8196	+	29.8196i	−1.16161	+	1.16161i	−0.984124	+	0.177481i	$0.943205\pi$
$660$		0			0
$661$	−30.2458	−	30.2458i	−1.17642	−	1.17642i	−0.980649	−	0.195775i	$-0.937278\pi$
$661$	−30.2458	−	30.2458i	−1.17642	−	1.17642i	−0.195775	−	0.980649i	$-0.562722\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	26.1374i		−	1.01357i
$666$		0			0
$667$	−14.9681	−	14.9681i	−0.579566	−	0.579566i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−26.4181			−1.01986
$672$		0			0
$673$	38.3587			1.47862			0.739309	−	0.673366i	$-0.235152\pi$
$673$	38.3587			1.47862			0.739309	+	0.673366i	$0.235152\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	19.3263	+	19.3263i	0.742772	+	0.742772i	0.973111	−	0.230339i	$-0.0739834\pi$
$677$	19.3263	+	19.3263i	0.742772	+	0.742772i	−0.230339	+	0.973111i	$0.573983\pi$
$678$		0			0
$679$		−	46.5701i		−	1.78720i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	21.3741	+	21.3741i	0.817856	+	0.817856i	0.985797	−	0.167941i	$-0.0537119\pi$
$683$	21.3741	+	21.3741i	0.817856	+	0.817856i	−0.167941	+	0.985797i	$0.553712\pi$
$684$		0			0
$685$	2.91344	−	2.91344i	0.111317	−	0.111317i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−23.4008			−0.891499
$690$		0			0
$691$	−3.96331	+	3.96331i	−0.150771	+	0.150771i	−0.778462	−	0.627691i	$-0.784000\pi$
$691$	−3.96331	+	3.96331i	−0.150771	+	0.150771i	0.627691	+	0.778462i	$0.284000\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			8.11610i			0.307861i
$696$		0			0
$697$		−	0.634848i		−	0.0240466i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	27.4177	−	27.4177i	1.03555	−	1.03555i	0.0362086	−	0.999344i	$-0.488472\pi$
$701$	27.4177	−	27.4177i	1.03555	−	1.03555i	0.999344	−	0.0362086i	$-0.0115281\pi$
$702$		0			0
$703$	5.78982			0.218367
$704$		0			0
$705$		0			0
$706$		0			0
$707$	12.7737	−	12.7737i	0.480404	−	0.480404i
$708$		0			0
$709$	36.8453	+	36.8453i	1.38376	+	1.38376i	0.837840	+	0.545915i	$0.183818\pi$
$709$	36.8453	+	36.8453i	1.38376	+	1.38376i	0.545915	+	0.837840i	$0.316182\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			60.2922i			2.25796i
$714$		0			0
$715$	16.8952	+	16.8952i	0.631845	+	0.631845i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−26.8739			−1.00223			−0.501114	−	0.865382i	$-0.667076\pi$
$719$	−26.8739			−1.00223			−0.501114	+	0.865382i	$0.667076\pi$
$720$		0			0
$721$	71.8599			2.67620
$722$		0			0
$723$		0			0
$724$		0			0
$725$	1.78072	+	1.78072i	0.0661341	+	0.0661341i
$726$		0			0
$727$			9.81023i			0.363841i	0.983313	+	0.181921i	$0.0582314\pi$
$727$			9.81023i			0.363841i	−0.983313	+	0.181921i	$0.941769\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−0.0409325	−	0.0409325i	−0.00151394	−	0.00151394i
$732$		0			0
$733$	−1.66941	+	1.66941i	−0.0616610	+	0.0616610i	−0.737265	−	0.675604i	$-0.763883\pi$
$733$	−1.66941	+	1.66941i	−0.0616610	+	0.0616610i	0.675604	+	0.737265i	$0.263883\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	0.772055			0.0284390
$738$		0			0
$739$	−36.2208	+	36.2208i	−1.33240	+	1.33240i	−0.429191	+	0.903214i	$0.641201\pi$
$739$	−36.2208	+	36.2208i	−1.33240	+	1.33240i	−0.903214	+	0.429191i	$0.858799\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			32.7601i			1.20185i	0.799305	+	0.600926i	$0.205201\pi$
$743$			32.7601i			1.20185i	−0.799305	+	0.600926i	$0.794799\pi$
$744$		0			0
$745$		−	14.0900i		−	0.516217i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	7.02606	−	7.02606i	0.256727	−	0.256727i
$750$		0			0
$751$	−8.98910			−0.328017			−0.164008	−	0.986459i	$-0.552442\pi$
$751$	−8.98910			−0.328017			−0.164008	+	0.986459i	$0.552442\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	2.84174	−	2.84174i	0.103421	−	0.103421i
$756$		0			0
$757$	27.4805	+	27.4805i	0.998796	+	0.998796i	0.999999	−	0.00120342i	$-0.000383062\pi$
$757$	27.4805	+	27.4805i	0.998796	+	0.998796i	−0.00120342	+	0.999999i	$0.500383\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			1.80961i			0.0655984i	0.999462	+	0.0327992i	$0.0104422\pi$
$761$			1.80961i			0.0655984i	−0.999462	+	0.0327992i	$0.989558\pi$
$762$		0			0
$763$	−19.9537	−	19.9537i	−0.722373	−	0.722373i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	45.7596			1.65228
$768$		0			0
$769$	41.3952			1.49275			0.746374	−	0.665527i	$-0.231793\pi$
$769$	41.3952			1.49275			0.746374	+	0.665527i	$0.231793\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−20.2439	−	20.2439i	−0.728122	−	0.728122i	0.242123	−	0.970246i	$-0.422156\pi$
$773$	−20.2439	−	20.2439i	−0.728122	−	0.728122i	−0.970246	+	0.242123i	$0.922156\pi$
$774$		0			0
$775$		−	7.17282i		−	0.257655i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−17.1573	−	17.1573i	−0.614725	−	0.614725i
$780$		0			0
$781$	−35.6484	+	35.6484i	−1.27560	+	1.27560i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	12.9324			0.461577
$786$		0			0
$787$	36.5446	−	36.5446i	1.30267	−	1.30267i	0.376091	−	0.926583i	$-0.377268\pi$
$787$	36.5446	−	36.5446i	1.30267	−	1.30267i	0.926583	−	0.376091i	$-0.122732\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			21.3854i			0.760379i
$792$		0			0
$793$		−	36.4438i		−	1.29416i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	12.7215	−	12.7215i	0.450619	−	0.450619i	−0.444941	−	0.895560i	$-0.646775\pi$
$797$	12.7215	−	12.7215i	0.450619	−	0.450619i	0.895560	+	0.444941i	$0.146775\pi$
$798$		0			0
$799$	−0.0120470			−0.000426193
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−1.12673	+	1.12673i	−0.0397615	+	0.0397615i
$804$		0			0
$805$	25.3946	+	25.3946i	0.895042	+	0.895042i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			5.24547i			0.184421i	0.995740	+	0.0922105i	$0.0293933\pi$
$809$			5.24547i			0.184421i	−0.995740	+	0.0922105i	$0.970607\pi$
$810$		0			0
$811$	2.66407	+	2.66407i	0.0935481	+	0.0935481i	0.752332	−	0.658784i	$-0.228929\pi$
$811$	2.66407	+	2.66407i	0.0935481	+	0.0935481i	−0.658784	+	0.752332i	$0.728929\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−0.489452			−0.0171447
$816$		0			0
$817$	−2.21248			−0.0774047
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−1.11276	−	1.11276i	−0.0388357	−	0.0388357i	0.687422	−	0.726258i	$-0.258742\pi$
$821$	−1.11276	−	1.11276i	−0.0388357	−	0.0388357i	−0.726258	+	0.687422i	$0.758742\pi$
$822$		0			0
$823$			11.7558i			0.409781i	0.978785	+	0.204891i	$0.0656838\pi$
$823$			11.7558i			0.409781i	−0.978785	+	0.204891i	$0.934316\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−30.1964	−	30.1964i	−1.05003	−	1.05003i	−0.998681	−	0.0513511i	$-0.983647\pi$
$827$	−30.1964	−	30.1964i	−1.05003	−	1.05003i	−0.0513511	−	0.998681i	$-0.516353\pi$
$828$		0			0
$829$	0.0724462	−	0.0724462i	0.00251616	−	0.00251616i	−0.705848	−	0.708364i	$-0.749434\pi$
$829$	0.0724462	−	0.0724462i	0.00251616	−	0.00251616i	0.708364	+	0.705848i	$0.249434\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	1.80140			0.0624148
$834$		0			0
$835$	−7.76986	+	7.76986i	−0.268887	+	0.268887i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			7.91971i			0.273419i	0.990611	+	0.136709i	$0.0436527\pi$
$839$			7.91971i			0.273419i	−0.990611	+	0.136709i	$0.956347\pi$
$840$		0			0
$841$			22.6581i			0.781314i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−14.1146	+	14.1146i	−0.485558	+	0.485558i
$846$		0			0
$847$	27.0037			0.927857
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−5.62528	+	5.62528i	−0.192832	+	0.192832i
$852$		0			0
$853$	−14.1305	−	14.1305i	−0.483817	−	0.483817i	0.422531	−	0.906348i	$-0.361142\pi$
$853$	−14.1305	−	14.1305i	−0.483817	−	0.483817i	−0.906348	+	0.422531i	$0.861142\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			4.80762i			0.164225i	0.996623	+	0.0821126i	$0.0261667\pi$
$857$			4.80762i			0.164225i	−0.996623	+	0.0821126i	$0.973833\pi$
$858$		0			0
$859$	34.8123	+	34.8123i	1.18778	+	1.18778i	0.977680	+	0.210101i	$0.0673794\pi$
$859$	34.8123	+	34.8123i	1.18778	+	1.18778i	0.210101	+	0.977680i	$0.432621\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−51.5605			−1.75514			−0.877570	−	0.479449i	$-0.840837\pi$
$863$	−51.5605			−1.75514			−0.877570	+	0.479449i	$0.840837\pi$
$864$		0			0
$865$	3.82694			0.130120
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−45.0955	−	45.0955i	−1.52976	−	1.52976i
$870$		0			0
$871$			1.06505i			0.0360880i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−3.02114	−	3.02114i	−0.102133	−	0.102133i
$876$		0			0
$877$	−12.4082	+	12.4082i	−0.418994	+	0.418994i	−0.884857	−	0.465863i	$-0.845744\pi$
$877$	−12.4082	+	12.4082i	−0.418994	+	0.418994i	0.465863	+	0.884857i	$0.345744\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	45.7147			1.54017			0.770084	−	0.637942i	$-0.220214\pi$
$881$	45.7147			1.54017			0.770084	+	0.637942i	$0.220214\pi$
$882$		0			0
$883$	8.68322	−	8.68322i	0.292213	−	0.292213i	−0.545741	−	0.837954i	$-0.683752\pi$
$883$	8.68322	−	8.68322i	0.292213	−	0.292213i	0.837954	+	0.545741i	$0.183752\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			5.87579i			0.197290i	0.995123	+	0.0986449i	$0.0314508\pi$
$887$			5.87579i			0.197290i	−0.995123	+	0.0986449i	$0.968549\pi$
$888$		0			0
$889$			8.40319i			0.281834i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−0.325582	+	0.325582i	−0.0108952	+	0.0108952i
$894$		0			0
$895$	14.7203			0.492044
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−12.7727	+	12.7727i	−0.425995	+	0.425995i
$900$		0			0
$901$	−0.461314	−	0.461314i	−0.0153686	−	0.0153686i
$902$		0			0
$903$		0			0
$904$		0			0
$905$			21.8909i			0.727678i
$906$		0			0
$907$	−0.656960	−	0.656960i	−0.0218140	−	0.0218140i	0.696116	−	0.717930i	$-0.254910\pi$
$907$	−0.656960	−	0.656960i	−0.0218140	−	0.0218140i	−0.717930	+	0.696116i	$0.754910\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−3.71828			−0.123192			−0.0615960	−	0.998101i	$-0.519619\pi$
$911$	−3.71828			−0.123192			−0.0615960	+	0.998101i	$0.519619\pi$
$912$		0			0
$913$	−32.6394			−1.08021
$914$		0			0
$915$		0			0
$916$		0			0
$917$	3.64406	+	3.64406i	0.120338	+	0.120338i
$918$		0			0
$919$			33.0548i			1.09038i	0.838313	+	0.545189i	$0.183542\pi$
$919$			33.0548i			1.09038i	−0.838313	+	0.545189i	$0.816458\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−49.1772	−	49.1772i	−1.61869	−	1.61869i
$924$		0			0
$925$	0.669226	−	0.669226i	0.0220040	−	0.0220040i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	17.6732			0.579839			0.289919	−	0.957051i	$-0.406371\pi$
$929$	17.6732			0.579839			0.289919	+	0.957051i	$0.406371\pi$
$930$		0			0
$931$	48.6844	−	48.6844i	1.59557	−	1.59557i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			0.666131i			0.0217848i
$936$		0			0
$937$		−	25.9902i		−	0.849061i	−0.905414	−	0.424531i	$-0.860439\pi$
$937$		−	25.9902i		−	0.849061i	0.905414	−	0.424531i	$-0.139561\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−29.0860	+	29.0860i	−0.948176	+	0.948176i	−0.998722	−	0.0505455i	$-0.983904\pi$
$941$	−29.0860	+	29.0860i	−0.948176	+	0.948176i	0.0505455	+	0.998722i	$0.483904\pi$
$942$		0			0
$943$	33.3395			1.08568
$944$		0			0
$945$		0			0
$946$		0			0
$947$	30.8896	−	30.8896i	1.00378	−	1.00378i	0.00378509	−	0.999993i	$-0.498795\pi$
$947$	30.8896	−	30.8896i	1.00378	−	1.00378i	0.999993	−	0.00378509i	$-0.00120483\pi$
$948$		0			0
$949$	−1.55433	−	1.55433i	−0.0504558	−	0.0504558i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	51.5794i		−	1.67082i	−0.549626	−	0.835411i	$-0.685230\pi$
$953$		−	51.5794i		−	1.67082i	0.549626	−	0.835411i	$-0.314770\pi$
$954$		0			0
$955$	−3.35979	−	3.35979i	−0.108720	−	0.108720i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	17.6038			0.568457
$960$		0			0
$961$	20.4493			0.659656
$962$		0			0
$963$		0			0
$964$		0			0
$965$	8.33786	+	8.33786i	0.268405	+	0.268405i
$966$		0			0
$967$		−	5.55245i		−	0.178555i	−0.996007	−	0.0892774i	$-0.971544\pi$
$967$		−	5.55245i		−	0.178555i	0.996007	−	0.0892774i	$-0.0284558\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−30.5095	−	30.5095i	−0.979098	−	0.979098i	0.0206884	−	0.999786i	$-0.493414\pi$
$971$	−30.5095	−	30.5095i	−0.979098	−	0.979098i	−0.999786	+	0.0206884i	$0.993414\pi$
$972$		0			0
$973$	−24.5199	+	24.5199i	−0.786071	+	0.786071i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−31.8649			−1.01945			−0.509723	−	0.860338i	$-0.670252\pi$
$977$	−31.8649			−1.01945			−0.509723	+	0.860338i	$0.670252\pi$
$978$		0			0
$979$	−40.9024	+	40.9024i	−1.30725	+	1.30725i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			24.2289i			0.772782i	0.922335	+	0.386391i	$0.126279\pi$
$983$			24.2289i			0.772782i	−0.922335	+	0.386391i	$0.873721\pi$
$984$		0			0
$985$			6.45226i			0.205586i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	2.14960	−	2.14960i	0.0683533	−	0.0683533i
$990$		0			0
$991$	−38.9268			−1.23655			−0.618276	−	0.785961i	$-0.712168\pi$
$991$	−38.9268			−1.23655			−0.618276	+	0.785961i	$0.712168\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−2.54744	+	2.54744i	−0.0807592	+	0.0807592i
$996$		0			0
$997$	−8.00046	−	8.00046i	−0.253377	−	0.253377i	0.568977	−	0.822354i	$-0.307340\pi$
$997$	−8.00046	−	8.00046i	−0.253377	−	0.253377i	−0.822354	+	0.568977i	$0.807340\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2880.2.t.d.2161.1		20
3.2	odd	2		960.2.s.c.241.1		20
4.3	odd	2		720.2.t.d.181.9		20
12.11	even	2		240.2.s.c.181.2	yes	20
16.3	odd	4		720.2.t.d.541.9		20
16.13	even	4	inner	2880.2.t.d.721.5		20
24.5	odd	2		1920.2.s.f.481.6		20
24.11	even	2		1920.2.s.e.481.5		20
48.5	odd	4		1920.2.s.f.1441.10		20
48.11	even	4		1920.2.s.e.1441.1		20
48.29	odd	4		960.2.s.c.721.5		20
48.35	even	4		240.2.s.c.61.2	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
240.2.s.c.61.2	✓	20	48.35	even	4
240.2.s.c.181.2	yes	20	12.11	even	2
720.2.t.d.181.9		20	4.3	odd	2
720.2.t.d.541.9		20	16.3	odd	4
960.2.s.c.241.1		20	3.2	odd	2
960.2.s.c.721.5		20	48.29	odd	4
1920.2.s.e.481.5		20	24.11	even	2
1920.2.s.e.1441.1		20	48.11	even	4
1920.2.s.f.481.6		20	24.5	odd	2
1920.2.s.f.1441.10		20	48.5	odd	4
2880.2.t.d.721.5		20	16.13	even	4	inner
2880.2.t.d.2161.1		20	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:	\( N \)	\(=\)	\( 2880 = 2^{6} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2880.t (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.707107 - 0.707107i) q^{5} -4.27253i q^{7} +O(q^{10})\)	\(q+(-0.707107 - 0.707107i) q^{5} -4.27253i q^{7} +(2.94281 + 2.94281i) q^{11} +(-4.05962 + 4.05962i) q^{13} -0.160060 q^{17} +(-4.32576 + 4.32576i) q^{19} -8.40564i q^{23} +1.00000i q^{25} +(1.78072 - 1.78072i) q^{29} -7.17282 q^{31} +(-3.02114 + 3.02114i) q^{35} +(-0.669226 - 0.669226i) q^{37} +3.96632i q^{41} +(0.255733 + 0.255733i) q^{43} +0.0752658 q^{47} -11.2545 q^{49} +(2.88214 + 2.88214i) q^{53} -4.16177i q^{55} +(-5.63594 - 5.63594i) q^{59} +(-4.48857 + 4.48857i) q^{61} +5.74117 q^{65} +(0.131176 - 0.131176i) q^{67} +12.1137i q^{71} +0.382876i q^{73} +(12.5733 - 12.5733i) q^{77} -15.3239 q^{79} +(-5.54562 + 5.54562i) q^{83} +(0.113179 + 0.113179i) q^{85} +13.8991i q^{89} +(17.3449 + 17.3449i) q^{91} +6.11754 q^{95} +10.8999 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q+O(q^{10}) \) 20 * q	\( 20 q + 8 q^{11} + 24 q^{17} + 4 q^{19} - 16 q^{29} + 16 q^{37} + 8 q^{43} - 52 q^{49} + 16 q^{53} - 16 q^{59} - 4 q^{61} + 8 q^{67} + 40 q^{77} - 56 q^{79} - 48 q^{83} + 4 q^{85} + 8 q^{91} + 56 q^{97}+O(q^{100}) \) 20 * q + 8 * q^11 + 24 * q^17 + 4 * q^19 - 16 * q^29 + 16 * q^37 + 8 * q^43 - 52 * q^49 + 16 * q^53 - 16 * q^59 - 4 * q^61 + 8 * q^67 + 40 * q^77 - 56 * q^79 - 48 * q^83 + 4 * q^85 + 8 * q^91 + 56 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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