LMFDB - Embedded newform 2880.2.t.c.721.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2880,2,Mod(721,2880)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage: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")

magma://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:traces = [16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$22.9969157821$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4 x^{15} + 4 x^{14} + 7 x^{12} - 8 x^{11} - 28 x^{10} + 28 x^{9} + 17 x^{8} + 56 x^{7} + \cdots + 256 $ x^16 - 4x^15 + 4x^14 + 7x^12 - 8x^11 - 28x^10 + 28x^9 + 17x^8 + 56x^7 - 112x^6 - 64x^5 + 112x^4 + 256x^2 - 512*x + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	no (minimal twist has level 80)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			721.4
Root			$1.26868 - 0.624862i$ of defining polynomial
Character	$\chi$	$=$	2880.721
Dual form			2880.2.t.c.2161.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.707107 + 0.707107i) q^{5} +4.02840i q^{7} +(-0.646837 + 0.646837i) q^{11} +(4.91492 + 4.91492i) q^{13} +2.70862 q^{17} +(0.438397 + 0.438397i) q^{19} +3.60080i q^{23} -1.00000i q^{25} +(-2.00921 - 2.00921i) q^{29} -4.30994 q^{31} +(-2.84851 - 2.84851i) q^{35} +(-0.743961 + 0.743961i) q^{37} +0.603979i q^{41} +(5.03010 - 5.03010i) q^{43} +10.8177 q^{47} -9.22800 q^{49} +(-4.07420 + 4.07420i) q^{53} -0.914766i q^{55} +(1.22845 - 1.22845i) q^{59} +(-6.98912 - 6.98912i) q^{61} -6.95074 q^{65} +(-5.24219 - 5.24219i) q^{67} +13.7940i q^{71} -1.30876i q^{73} +(-2.60572 - 2.60572i) q^{77} -0.611127 q^{79} +(1.29471 + 1.29471i) q^{83} +(-1.91529 + 1.91529i) q^{85} +10.9236i q^{89} +(-19.7993 + 19.7993i) q^{91} -0.619987 q^{95} -12.7571 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 8 q^{11} + 8 q^{19} + 16 q^{29} - 16 q^{37} - 8 q^{43} - 40 q^{47} - 16 q^{49} - 16 q^{53} - 8 q^{59} + 16 q^{61} - 40 q^{67} - 16 q^{77} - 16 q^{79} + 40 q^{83} - 16 q^{85} - 32 q^{91} + 32 q^{95}+O(q^{100}) $ 16 * q - 8 * q^11 + 8 * q^19 + 16 * q^29 - 16 * q^37 - 8 * q^43 - 40 * q^47 - 16 * q^49 - 16 * q^53 - 8 * q^59 + 16 * q^61 - 40 * q^67 - 16 * q^77 - 16 * q^79 + 40 * q^83 - 16 * q^85 - 32 * q^91 + 32 * q^95

Character values

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

$n$	$577$	$641$	$901$	$2431$
$\chi(n)$	$1$	$1$	$e\left(\frac{3}{4}\right)$	$1$

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
$7$			4.02840i			1.52259i	0.648405	+	0.761296i	$0.275437\pi$
$7$			4.02840i			1.52259i	−0.648405	+	0.761296i	$0.724563\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−0.646837	+	0.646837i	−0.195029	+	0.195029i	−0.797865	−	0.602836i	$-0.794037\pi$
$11$	−0.646837	+	0.646837i	−0.195029	+	0.195029i	0.602836	+	0.797865i	$0.294037\pi$
$12$		0			0
$13$	4.91492	+	4.91492i	1.36315	+	1.36315i	0.869867	+	0.493286i	$0.164204\pi$
$13$	4.91492	+	4.91492i	1.36315	+	1.36315i	0.493286	+	0.869867i	$0.335796\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	2.70862			0.656938			0.328469	−	0.944515i	$-0.393467\pi$
$17$	2.70862			0.656938			0.328469	+	0.944515i	$0.393467\pi$
$18$		0			0
$19$	0.438397	+	0.438397i	0.100575	+	0.100575i	0.755604	−	0.655029i	$-0.227344\pi$
$19$	0.438397	+	0.438397i	0.100575	+	0.100575i	−0.655029	+	0.755604i	$0.727344\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			3.60080i			0.750819i	0.926859	+	0.375410i	$0.122498\pi$
$23$			3.60080i			0.750819i	−0.926859	+	0.375410i	$0.877502\pi$
$24$		0			0
$25$		−	1.00000i		−	0.200000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−2.00921	−	2.00921i	−0.373102	−	0.373102i	0.495504	−	0.868606i	$-0.334983\pi$
$29$	−2.00921	−	2.00921i	−0.373102	−	0.373102i	−0.868606	+	0.495504i	$0.834983\pi$
$30$		0			0
$31$	−4.30994			−0.774087			−0.387044	−	0.922061i	$-0.626504\pi$
$31$	−4.30994			−0.774087			−0.387044	+	0.922061i	$0.626504\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−2.84851	−	2.84851i	−0.481486	−	0.481486i
$36$		0			0
$37$	−0.743961	+	0.743961i	−0.122306	+	0.122306i	−0.765611	−	0.643304i	$-0.777563\pi$
$37$	−0.743961	+	0.743961i	−0.122306	+	0.122306i	0.643304	+	0.765611i	$0.277563\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			0.603979i			0.0943256i	0.998887	+	0.0471628i	$0.0150180\pi$
$41$			0.603979i			0.0943256i	−0.998887	+	0.0471628i	$0.984982\pi$
$42$		0			0
$43$	5.03010	−	5.03010i	0.767083	−	0.767083i	−0.210509	−	0.977592i	$-0.567512\pi$
$43$	5.03010	−	5.03010i	0.767083	−	0.767083i	0.977592	+	0.210509i	$0.0675121\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	10.8177			1.57793			0.788963	−	0.614440i	$-0.210618\pi$
$47$	10.8177			1.57793			0.788963	+	0.614440i	$0.210618\pi$
$48$		0			0
$49$	−9.22800			−1.31829
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−4.07420	+	4.07420i	−0.559634	+	0.559634i	−0.929203	−	0.369569i	$-0.879505\pi$
$53$	−4.07420	+	4.07420i	−0.559634	+	0.559634i	0.369569	+	0.929203i	$0.379505\pi$
$54$		0			0
$55$		−	0.914766i		−	0.123347i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	1.22845	−	1.22845i	0.159931	−	0.159931i	−0.622605	−	0.782536i	$-0.713926\pi$
$59$	1.22845	−	1.22845i	0.159931	−	0.159931i	0.782536	+	0.622605i	$0.213926\pi$
$60$		0			0
$61$	−6.98912	−	6.98912i	−0.894865	−	0.894865i	0.100112	−	0.994976i	$-0.468080\pi$
$61$	−6.98912	−	6.98912i	−0.894865	−	0.894865i	−0.994976	+	0.100112i	$0.968080\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−6.95074			−0.862134
$66$		0			0
$67$	−5.24219	−	5.24219i	−0.640435	−	0.640435i	0.310227	−	0.950662i	$-0.399595\pi$
$67$	−5.24219	−	5.24219i	−0.640435	−	0.640435i	−0.950662	+	0.310227i	$0.899595\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			13.7940i			1.63704i	0.574475	+	0.818522i	$0.305206\pi$
$71$			13.7940i			1.63704i	−0.574475	+	0.818522i	$0.694794\pi$
$72$		0			0
$73$		−	1.30876i		−	0.153179i	−0.997063	−	0.0765895i	$-0.975597\pi$
$73$		−	1.30876i		−	0.153179i	0.997063	−	0.0765895i	$-0.0244031\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−2.60572	−	2.60572i	−0.296949	−	0.296949i
$78$		0			0
$79$	−0.611127			−0.0687571			−0.0343786	−	0.999409i	$-0.510945\pi$
$79$	−0.611127			−0.0687571			−0.0343786	+	0.999409i	$0.510945\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	1.29471	+	1.29471i	0.142113	+	0.142113i	0.774584	−	0.632471i	$-0.217959\pi$
$83$	1.29471	+	1.29471i	0.142113	+	0.142113i	−0.632471	+	0.774584i	$0.717959\pi$
$84$		0			0
$85$	−1.91529	+	1.91529i	−0.207742	+	0.207742i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			10.9236i			1.15790i	0.815363	+	0.578950i	$0.196537\pi$
$89$			10.9236i			1.15790i	−0.815363	+	0.578950i	$0.803463\pi$
$90$		0			0
$91$	−19.7993	+	19.7993i	−2.07553	+	2.07553i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−0.619987			−0.0636093
$96$		0			0
$97$	−12.7571			−1.29528			−0.647642	−	0.761945i	$-0.724245\pi$
$97$	−12.7571			−1.29528			−0.647642	+	0.761945i	$0.724245\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	8.59804	−	8.59804i	0.855537	−	0.855537i	−0.135272	−	0.990809i	$-0.543191\pi$
$101$	8.59804	−	8.59804i	0.855537	−	0.855537i	0.990809	+	0.135272i	$0.0431908\pi$
$102$		0			0
$103$			12.0328i			1.18563i	0.805338	+	0.592815i	$0.201984\pi$
$103$			12.0328i			1.18563i	−0.805338	+	0.592815i	$0.798016\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−2.37309	+	2.37309i	−0.229415	+	0.229415i	−0.812448	−	0.583033i	$-0.801866\pi$
$107$	−2.37309	+	2.37309i	−0.229415	+	0.229415i	0.583033	+	0.812448i	$0.301866\pi$
$108$		0			0
$109$	−3.24479	−	3.24479i	−0.310794	−	0.310794i	0.534423	−	0.845217i	$-0.320529\pi$
$109$	−3.24479	−	3.24479i	−0.310794	−	0.310794i	−0.845217	+	0.534423i	$0.820529\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−17.3173			−1.62907			−0.814536	−	0.580114i	$-0.803008\pi$
$113$	−17.3173			−1.62907			−0.814536	+	0.580114i	$0.803008\pi$
$114$		0			0
$115$	−2.54615	−	2.54615i	−0.237430	−	0.237430i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			10.9114i			1.00025i
$120$		0			0
$121$			10.1632i			0.923928i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	0.707107	+	0.707107i	0.0632456	+	0.0632456i
$126$		0			0
$127$	15.5438			1.37929			0.689645	−	0.724147i	$-0.257766\pi$
$127$	15.5438			1.37929			0.689645	+	0.724147i	$0.257766\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−11.2770	−	11.2770i	−0.985280	−	0.985280i	0.0146129	−	0.999893i	$-0.495348\pi$
$131$	−11.2770	−	11.2770i	−0.985280	−	0.985280i	−0.999893	+	0.0146129i	$0.995348\pi$
$132$		0			0
$133$	−1.76604	+	1.76604i	−0.153135	+	0.153135i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	3.67273i		−	0.313782i	−0.987616	−	0.156891i	$-0.949853\pi$
$137$		−	3.67273i		−	0.313782i	0.987616	−	0.156891i	$-0.0501472\pi$
$138$		0			0
$139$	5.23552	−	5.23552i	0.444071	−	0.444071i	−0.449307	−	0.893378i	$-0.648329\pi$
$139$	5.23552	−	5.23552i	0.444071	−	0.444071i	0.893378	+	0.449307i	$0.148329\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−6.35830			−0.531708
$144$		0			0
$145$	2.84146			0.235970
$146$		0			0
$147$		0			0
$148$		0			0
$149$	3.29391	−	3.29391i	0.269848	−	0.269848i	−0.559191	−	0.829039i	$-0.688888\pi$
$149$	3.29391	−	3.29391i	0.269848	−	0.269848i	0.829039	+	0.559191i	$0.188888\pi$
$150$		0			0
$151$		−	6.93206i		−	0.564123i	−0.959396	−	0.282061i	$-0.908982\pi$
$151$		−	6.93206i		−	0.564123i	0.959396	−	0.282061i	$-0.0910182\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	3.04759	−	3.04759i	0.244788	−	0.244788i
$156$		0			0
$157$	5.65633	+	5.65633i	0.451425	+	0.451425i	0.895827	−	0.444403i	$-0.146584\pi$
$157$	5.65633	+	5.65633i	0.451425	+	0.451425i	−0.444403	+	0.895827i	$0.646584\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−14.5055			−1.14319
$162$		0			0
$163$	10.9746	+	10.9746i	0.859593	+	0.859593i	0.991290	−	0.131697i	$-0.0420425\pi$
$163$	10.9746	+	10.9746i	0.859593	+	0.859593i	−0.131697	+	0.991290i	$0.542043\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	11.7686i		−	0.910685i	−0.890316	−	0.455343i	$-0.849517\pi$
$167$		−	11.7686i		−	0.910685i	0.890316	−	0.455343i	$-0.150483\pi$
$168$		0			0
$169$			35.3128i			2.71637i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	1.40225	+	1.40225i	0.106611	+	0.106611i	0.758400	−	0.651789i	$-0.225981\pi$
$173$	1.40225	+	1.40225i	0.106611	+	0.106611i	−0.651789	+	0.758400i	$0.725981\pi$
$174$		0			0
$175$	4.02840			0.304518
$176$		0			0
$177$		0			0
$178$		0			0
$179$	9.66131	+	9.66131i	0.722120	+	0.722120i	0.969037	−	0.246917i	$-0.0794174\pi$
$179$	9.66131	+	9.66131i	0.722120	+	0.722120i	−0.246917	+	0.969037i	$0.579417\pi$
$180$		0			0
$181$	0.294844	−	0.294844i	0.0219156	−	0.0219156i	−0.696064	−	0.717980i	$-0.745067\pi$
$181$	0.294844	−	0.294844i	0.0219156	−	0.0219156i	0.717980	+	0.696064i	$0.245067\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	1.05212i		−	0.0773533i
$186$		0			0
$187$	−1.75204	+	1.75204i	−0.128122	+	0.128122i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−16.9352			−1.22539			−0.612694	−	0.790320i	$-0.709914\pi$
$191$	−16.9352			−1.22539			−0.612694	+	0.790320i	$0.709914\pi$
$192$		0			0
$193$	−16.5927			−1.19437			−0.597185	−	0.802103i	$-0.703714\pi$
$193$	−16.5927			−1.19437			−0.597185	+	0.802103i	$0.703714\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	2.38392	−	2.38392i	0.169847	−	0.169847i	−0.617065	−	0.786912i	$-0.711678\pi$
$197$	2.38392	−	2.38392i	0.169847	−	0.169847i	0.786912	+	0.617065i	$0.211678\pi$
$198$		0			0
$199$			10.1411i			0.718883i	0.933168	+	0.359442i	$0.117033\pi$
$199$			10.1411i			0.718883i	−0.933168	+	0.359442i	$0.882967\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	8.09392	−	8.09392i	0.568082	−	0.568082i
$204$		0			0
$205$	−0.427078	−	0.427078i	−0.0298284	−	0.0298284i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−0.567143			−0.0392301
$210$		0			0
$211$	2.81171	+	2.81171i	0.193566	+	0.193566i	0.797235	−	0.603669i	$-0.206295\pi$
$211$	2.81171	+	2.81171i	0.193566	+	0.193566i	−0.603669	+	0.797235i	$0.706295\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			7.11363i			0.485146i
$216$		0			0
$217$		−	17.3621i		−	1.17862i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	13.3127	+	13.3127i	0.895507	+	0.895507i
$222$		0			0
$223$	−14.0502			−0.940871			−0.470436	−	0.882434i	$-0.655903\pi$
$223$	−14.0502			−0.940871			−0.470436	+	0.882434i	$0.655903\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−13.3495	−	13.3495i	−0.886037	−	0.886037i	0.108103	−	0.994140i	$-0.465522\pi$
$227$	−13.3495	−	13.3495i	−0.886037	−	0.886037i	−0.994140	+	0.108103i	$0.965522\pi$
$228$		0			0
$229$	8.78589	−	8.78589i	0.580588	−	0.580588i	−0.354477	−	0.935065i	$-0.615341\pi$
$229$	8.78589	−	8.78589i	0.580588	−	0.580588i	0.935065	+	0.354477i	$0.115341\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			15.1472i			0.992329i	0.868229	+	0.496165i	$0.165259\pi$
$233$			15.1472i			0.992329i	−0.868229	+	0.496165i	$0.834741\pi$
$234$		0			0
$235$	−7.64928	+	7.64928i	−0.498984	+	0.498984i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	17.9151			1.15883			0.579414	−	0.815033i	$-0.303281\pi$
$239$	17.9151			1.15883			0.579414	+	0.815033i	$0.303281\pi$
$240$		0			0
$241$	25.6594			1.65287			0.826433	−	0.563035i	$-0.190366\pi$
$241$	25.6594			1.65287			0.826433	+	0.563035i	$0.190366\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	6.52518	−	6.52518i	0.416879	−	0.416879i
$246$		0			0
$247$			4.30937i			0.274199i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−5.95195	+	5.95195i	−0.375684	+	0.375684i	−0.869542	−	0.493858i	$-0.835586\pi$
$251$	−5.95195	+	5.95195i	−0.375684	+	0.375684i	0.493858	+	0.869542i	$0.335586\pi$
$252$		0			0
$253$	−2.32913	−	2.32913i	−0.146431	−	0.146431i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	4.17369			0.260348			0.130174	−	0.991491i	$-0.458446\pi$
$257$	4.17369			0.260348			0.130174	+	0.991491i	$0.458446\pi$
$258$		0			0
$259$	−2.99697	−	2.99697i	−0.186223	−	0.186223i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	9.14469i		−	0.563885i	−0.959431	−	0.281943i	$-0.909021\pi$
$263$		−	9.14469i		−	0.563885i	0.959431	−	0.281943i	$-0.0909788\pi$
$264$		0			0
$265$		−	5.76178i		−	0.353944i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	8.40029	+	8.40029i	0.512175	+	0.512175i	0.915192	−	0.403017i	$-0.132039\pi$
$269$	8.40029	+	8.40029i	0.512175	+	0.512175i	−0.403017	+	0.915192i	$0.632039\pi$
$270$		0			0
$271$	−18.7794			−1.14077			−0.570383	−	0.821379i	$-0.693205\pi$
$271$	−18.7794			−1.14077			−0.570383	+	0.821379i	$0.693205\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	0.646837	+	0.646837i	0.0390057	+	0.0390057i
$276$		0			0
$277$	−3.54167	+	3.54167i	−0.212798	+	0.212798i	−0.805455	−	0.592657i	$-0.798079\pi$
$277$	−3.54167	+	3.54167i	−0.212798	+	0.212798i	0.592657	+	0.805455i	$0.298079\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	2.31811i		−	0.138287i	−0.997607	−	0.0691433i	$-0.977973\pi$
$281$		−	2.31811i		−	0.138287i	0.997607	−	0.0691433i	$-0.0220266\pi$
$282$		0			0
$283$	1.63197	−	1.63197i	0.0970108	−	0.0970108i	−0.656936	−	0.753947i	$-0.728148\pi$
$283$	1.63197	−	1.63197i	0.0970108	−	0.0970108i	0.753947	+	0.656936i	$0.228148\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−2.43307			−0.143619
$288$		0			0
$289$	−9.66335			−0.568433
$290$		0			0
$291$		0			0
$292$		0			0
$293$	11.5789	−	11.5789i	0.676444	−	0.676444i	−0.282750	−	0.959194i	$-0.591247\pi$
$293$	11.5789	−	11.5789i	0.676444	−	0.676444i	0.959194	+	0.282750i	$0.0912466\pi$
$294$		0			0
$295$			1.73729i			0.101149i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−17.6976	+	17.6976i	−1.02348	+	1.02348i
$300$		0			0
$301$	20.2632	+	20.2632i	1.16795	+	1.16795i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	9.88410			0.565962
$306$		0			0
$307$	−11.7116	−	11.7116i	−0.668415	−	0.668415i	0.288934	−	0.957349i	$-0.406699\pi$
$307$	−11.7116	−	11.7116i	−0.668415	−	0.668415i	−0.957349	+	0.288934i	$0.906699\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			11.2068i			0.635477i	0.948178	+	0.317739i	$0.102923\pi$
$311$			11.2068i			0.635477i	−0.948178	+	0.317739i	$0.897077\pi$
$312$		0			0
$313$			7.50635i			0.424284i	0.977239	+	0.212142i	$0.0680439\pi$
$313$			7.50635i			0.424284i	−0.977239	+	0.212142i	$0.931956\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−16.2854	−	16.2854i	−0.914680	−	0.914680i	0.0819564	−	0.996636i	$-0.473883\pi$
$317$	−16.2854	−	16.2854i	−0.914680	−	0.914680i	−0.996636	+	0.0819564i	$0.973883\pi$
$318$		0			0
$319$	2.59927			0.145531
$320$		0			0
$321$		0			0
$322$		0			0
$323$	1.18745	+	1.18745i	0.0660716	+	0.0660716i
$324$		0			0
$325$	4.91492	−	4.91492i	0.272631	−	0.272631i
$326$		0			0
$327$		0			0
$328$		0			0
$329$			43.5781i			2.40254i
$330$		0			0
$331$	19.3846	−	19.3846i	1.06547	−	1.06547i	0.0677707	−	0.997701i	$-0.478411\pi$
$331$	19.3846	−	19.3846i	1.06547	−	1.06547i	0.997701	−	0.0677707i	$-0.0215886\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	7.41357			0.405047
$336$		0			0
$337$	−7.82991			−0.426522			−0.213261	−	0.976995i	$-0.568408\pi$
$337$	−7.82991			−0.426522			−0.213261	+	0.976995i	$0.568408\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	2.78783	−	2.78783i	0.150969	−	0.150969i
$342$		0			0
$343$		−	8.97529i		−	0.484620i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	8.91753	−	8.91753i	0.478718	−	0.478718i	−0.426004	−	0.904721i	$-0.640079\pi$
$347$	8.91753	−	8.91753i	0.478718	−	0.478718i	0.904721	+	0.426004i	$0.140079\pi$
$348$		0			0
$349$	−6.69072	−	6.69072i	−0.358146	−	0.358146i	0.504983	−	0.863129i	$-0.331499\pi$
$349$	−6.69072	−	6.69072i	−0.358146	−	0.358146i	−0.863129	+	0.504983i	$0.831499\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	2.05215			0.109225			0.0546126	−	0.998508i	$-0.482608\pi$
$353$	2.05215			0.109225			0.0546126	+	0.998508i	$0.482608\pi$
$354$		0			0
$355$	−9.75382	−	9.75382i	−0.517679	−	0.517679i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			9.52634i			0.502781i	0.967886	+	0.251391i	$0.0808879\pi$
$359$			9.52634i			0.502781i	−0.967886	+	0.251391i	$0.919112\pi$
$360$		0			0
$361$		−	18.6156i		−	0.979769i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	0.925435	+	0.925435i	0.0484395	+	0.0484395i
$366$		0			0
$367$	3.39736			0.177341			0.0886703	−	0.996061i	$-0.471738\pi$
$367$	3.39736			0.177341			0.0886703	+	0.996061i	$0.471738\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−16.4125	−	16.4125i	−0.852094	−	0.852094i
$372$		0			0
$373$	−22.4895	+	22.4895i	−1.16446	+	1.16446i	−0.180971	+	0.983488i	$0.557924\pi$
$373$	−22.4895	+	22.4895i	−1.16446	+	1.16446i	−0.983488	+	0.180971i	$0.942076\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	19.7502i		−	1.01719i
$378$		0			0
$379$	−14.9819	+	14.9819i	−0.769567	+	0.769567i	−0.978030	−	0.208463i	$-0.933154\pi$
$379$	−14.9819	+	14.9819i	−0.769567	+	0.769567i	0.208463	+	0.978030i	$0.433154\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	26.1197			1.33466			0.667328	−	0.744764i	$-0.267438\pi$
$383$	26.1197			1.33466			0.667328	+	0.744764i	$0.267438\pi$
$384$		0			0
$385$	3.68504			0.187807
$386$		0			0
$387$		0			0
$388$		0			0
$389$	2.08395	−	2.08395i	0.105660	−	0.105660i	−0.652300	−	0.757961i	$-0.726196\pi$
$389$	2.08395	−	2.08395i	0.105660	−	0.105660i	0.757961	+	0.652300i	$0.226196\pi$
$390$		0			0
$391$			9.75322i			0.493241i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	0.432132	−	0.432132i	0.0217429	−	0.0217429i
$396$		0			0
$397$	1.43282	+	1.43282i	0.0719114	+	0.0719114i	0.742148	−	0.670236i	$-0.233807\pi$
$397$	1.43282	+	1.43282i	0.0719114	+	0.0719114i	−0.670236	+	0.742148i	$0.733807\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	29.9853			1.49739			0.748697	−	0.662912i	$-0.230680\pi$
$401$	29.9853			1.49739			0.748697	+	0.662912i	$0.230680\pi$
$402$		0			0
$403$	−21.1830	−	21.1830i	−1.05520	−	1.05520i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	0.962443i		−	0.0477065i
$408$		0			0
$409$		−	4.17833i		−	0.206605i	−0.994650	−	0.103302i	$-0.967059\pi$
$409$		−	4.17833i		−	0.206605i	0.994650	−	0.103302i	$-0.0329410\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	4.94870	+	4.94870i	0.243510	+	0.243510i
$414$		0			0
$415$	−1.83100			−0.0898801
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−24.4667	−	24.4667i	−1.19528	−	1.19528i	−0.975565	−	0.219712i	$-0.929488\pi$
$419$	−24.4667	−	24.4667i	−1.19528	−	1.19528i	−0.219712	−	0.975565i	$-0.570512\pi$
$420$		0			0
$421$	25.6017	−	25.6017i	1.24775	−	1.24775i	0.291039	−	0.956711i	$-0.405999\pi$
$421$	25.6017	−	25.6017i	1.24775	−	1.24775i	0.956711	−	0.291039i	$-0.0940008\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	2.70862i		−	0.131388i
$426$		0			0
$427$	28.1550	−	28.1550i	1.36251	−	1.36251i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	17.6126			0.848367			0.424184	−	0.905576i	$-0.360561\pi$
$431$	17.6126			0.848367			0.424184	+	0.905576i	$0.360561\pi$
$432$		0			0
$433$	27.0568			1.30027			0.650133	−	0.759820i	$-0.274713\pi$
$433$	27.0568			1.30027			0.650133	+	0.759820i	$0.274713\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−1.57858	+	1.57858i	−0.0755138	+	0.0755138i
$438$		0			0
$439$			22.9965i			1.09756i	0.835967	+	0.548780i	$0.184908\pi$
$439$			22.9965i			1.09756i	−0.835967	+	0.548780i	$0.815092\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−13.7715	+	13.7715i	−0.654303	+	0.654303i	−0.954026	−	0.299723i	$-0.903106\pi$
$443$	−13.7715	+	13.7715i	−0.654303	+	0.654303i	0.299723	+	0.954026i	$0.403106\pi$
$444$		0			0
$445$	−7.72415	−	7.72415i	−0.366160	−	0.366160i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−6.88838			−0.325083			−0.162541	−	0.986702i	$-0.551969\pi$
$449$	−6.88838			−0.325083			−0.162541	+	0.986702i	$0.551969\pi$
$450$		0			0
$451$	−0.390676	−	0.390676i	−0.0183962	−	0.0183962i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	28.0004i		−	1.31268i
$456$		0			0
$457$			2.52622i			0.118171i	0.998253	+	0.0590857i	$0.0188185\pi$
$457$			2.52622i			0.118171i	−0.998253	+	0.0590857i	$0.981181\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−9.23502	−	9.23502i	−0.430118	−	0.430118i	0.458550	−	0.888668i	$-0.348369\pi$
$461$	−9.23502	−	9.23502i	−0.430118	−	0.430118i	−0.888668	+	0.458550i	$0.848369\pi$
$462$		0			0
$463$	11.2676			0.523652			0.261826	−	0.965115i	$-0.415675\pi$
$463$	11.2676			0.523652			0.261826	+	0.965115i	$0.415675\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−25.8291	−	25.8291i	−1.19523	−	1.19523i	−0.975579	−	0.219650i	$-0.929508\pi$
$467$	−25.8291	−	25.8291i	−1.19523	−	1.19523i	−0.219650	−	0.975579i	$-0.570492\pi$
$468$		0			0
$469$	21.1176	−	21.1176i	0.975122	−	0.975122i
$470$		0			0
$471$		0			0
$472$		0			0
$473$			6.50731i			0.299206i
$474$		0			0
$475$	0.438397	−	0.438397i	0.0201150	−	0.0201150i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−15.7261			−0.718545			−0.359273	−	0.933233i	$-0.616975\pi$
$479$	−15.7261			−0.718545			−0.359273	+	0.933233i	$0.616975\pi$
$480$		0			0
$481$	−7.31301			−0.333445
$482$		0			0
$483$		0			0
$484$		0			0
$485$	9.02061	−	9.02061i	0.409605	−	0.409605i
$486$		0			0
$487$		−	35.3717i		−	1.60284i	−0.598100	−	0.801422i	$-0.704077\pi$
$487$		−	35.3717i		−	1.60284i	0.598100	−	0.801422i	$-0.295923\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	7.95703	−	7.95703i	0.359096	−	0.359096i	−0.504384	−	0.863480i	$-0.668280\pi$
$491$	7.95703	−	7.95703i	0.359096	−	0.359096i	0.863480	+	0.504384i	$0.168280\pi$
$492$		0			0
$493$	−5.44221	−	5.44221i	−0.245105	−	0.245105i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−55.5677			−2.49255
$498$		0			0
$499$	−11.5864	−	11.5864i	−0.518677	−	0.518677i	0.398494	−	0.917171i	$-0.369533\pi$
$499$	−11.5864	−	11.5864i	−0.518677	−	0.518677i	−0.917171	+	0.398494i	$0.869533\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			23.5051i			1.04804i	0.851706	+	0.524020i	$0.175568\pi$
$503$			23.5051i			1.04804i	−0.851706	+	0.524020i	$0.824432\pi$
$504$		0			0
$505$			12.1595i			0.541089i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	3.08381	+	3.08381i	0.136687	+	0.136687i	0.772140	−	0.635452i	$-0.219186\pi$
$509$	3.08381	+	3.08381i	0.136687	+	0.136687i	−0.635452	+	0.772140i	$0.719186\pi$
$510$		0			0
$511$	5.27222			0.233229
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−8.50850	−	8.50850i	−0.374929	−	0.374929i
$516$		0			0
$517$	−6.99730	+	6.99730i	−0.307741	+	0.307741i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	11.5762i		−	0.507161i	−0.967314	−	0.253580i	$-0.918392\pi$
$521$		−	11.5762i		−	0.507161i	0.967314	−	0.253580i	$-0.0816083\pi$
$522$		0			0
$523$	−3.97900	+	3.97900i	−0.173990	+	0.173990i	−0.788730	−	0.614740i	$-0.789261\pi$
$523$	−3.97900	+	3.97900i	−0.173990	+	0.173990i	0.614740	+	0.788730i	$0.289261\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−11.6740			−0.508527
$528$		0			0
$529$	10.0342			0.436271
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−2.96851	+	2.96851i	−0.128580	+	0.128580i
$534$		0			0
$535$		−	3.35605i		−	0.145095i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	5.96902	−	5.96902i	0.257104	−	0.257104i
$540$		0			0
$541$	17.2148	+	17.2148i	0.740123	+	0.740123i	0.972602	−	0.232478i	$-0.0746835\pi$
$541$	17.2148	+	17.2148i	0.740123	+	0.740123i	−0.232478	+	0.972602i	$0.574684\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	4.58882			0.196564
$546$		0			0
$547$	20.3610	+	20.3610i	0.870573	+	0.870573i	0.992535	−	0.121962i	$-0.0389187\pi$
$547$	20.3610	+	20.3610i	0.870573	+	0.870573i	−0.121962	+	0.992535i	$0.538919\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	1.76167i		−	0.0750495i
$552$		0			0
$553$		−	2.46186i		−	0.104689i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	22.7029	+	22.7029i	0.961954	+	0.961954i	0.999302	−	0.0373478i	$-0.0118910\pi$
$557$	22.7029	+	22.7029i	0.961954	+	0.961954i	−0.0373478	+	0.999302i	$0.511891\pi$
$558$		0			0
$559$	49.4451			2.09130
$560$		0			0
$561$		0			0
$562$		0			0
$563$	15.4153	+	15.4153i	0.649676	+	0.649676i	0.952915	−	0.303238i	$-0.0980678\pi$
$563$	15.4153	+	15.4153i	0.649676	+	0.649676i	−0.303238	+	0.952915i	$0.598068\pi$
$564$		0			0
$565$	12.2452	−	12.2452i	0.515158	−	0.515158i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	22.6529i		−	0.949660i	−0.880078	−	0.474830i	$-0.842510\pi$
$569$		−	22.6529i		−	0.949660i	0.880078	−	0.474830i	$-0.157490\pi$
$570$		0			0
$571$	13.4941	−	13.4941i	0.564710	−	0.564710i	−0.365931	−	0.930642i	$-0.619250\pi$
$571$	13.4941	−	13.4941i	0.564710	−	0.564710i	0.930642	+	0.365931i	$0.119250\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	3.60080			0.150164
$576$		0			0
$577$	6.08684			0.253398			0.126699	−	0.991941i	$-0.459562\pi$
$577$	6.08684			0.253398			0.126699	+	0.991941i	$0.459562\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−5.21561	+	5.21561i	−0.216380	+	0.216380i
$582$		0			0
$583$		−	5.27068i		−	0.218289i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	21.4418	−	21.4418i	0.884999	−	0.884999i	−0.109039	−	0.994038i	$-0.534777\pi$
$587$	21.4418	−	21.4418i	0.884999	−	0.884999i	0.994038	+	0.109039i	$0.0347772\pi$
$588$		0			0
$589$	−1.88946	−	1.88946i	−0.0778540	−	0.0778540i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	28.2005			1.15806			0.579028	−	0.815308i	$-0.303432\pi$
$593$	28.2005			1.15806			0.579028	+	0.815308i	$0.303432\pi$
$594$		0			0
$595$	−7.71554	−	7.71554i	−0.316306	−	0.316306i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			38.0516i			1.55475i	0.629039	+	0.777374i	$0.283449\pi$
$599$			38.0516i			1.55475i	−0.629039	+	0.777374i	$0.716551\pi$
$600$		0			0
$601$		−	19.0716i		−	0.777947i	−0.921249	−	0.388974i	$-0.872830\pi$
$601$		−	19.0716i		−	0.777947i	0.921249	−	0.388974i	$-0.127170\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−7.18647	−	7.18647i	−0.292172	−	0.292172i
$606$		0			0
$607$	−5.73433			−0.232749			−0.116375	−	0.993205i	$-0.537127\pi$
$607$	−5.73433			−0.232749			−0.116375	+	0.993205i	$0.537127\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	53.1682	+	53.1682i	2.15096	+	2.15096i
$612$		0			0
$613$	5.36917	−	5.36917i	0.216859	−	0.216859i	−0.590315	−	0.807173i	$-0.700996\pi$
$613$	5.36917	−	5.36917i	0.216859	−	0.216859i	0.807173	+	0.590315i	$0.200996\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	28.2915i		−	1.13897i	−0.822000	−	0.569487i	$-0.807142\pi$
$617$		−	28.2915i		−	1.13897i	0.822000	−	0.569487i	$-0.192858\pi$
$618$		0			0
$619$	−18.9669	+	18.9669i	−0.762345	+	0.762345i	−0.976746	−	0.214401i	$-0.931220\pi$
$619$	−18.9669	+	18.9669i	−0.762345	+	0.762345i	0.214401	+	0.976746i	$0.431220\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−44.0046			−1.76301
$624$		0			0
$625$	−1.00000			−0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−2.01511	+	2.01511i	−0.0803477	+	0.0803477i
$630$		0			0
$631$			41.7662i			1.66269i	0.555758	+	0.831344i	$0.312428\pi$
$631$			41.7662i			1.66269i	−0.555758	+	0.831344i	$0.687572\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−10.9911	+	10.9911i	−0.436170	+	0.436170i
$636$		0			0
$637$	−45.3549	−	45.3549i	−1.79703	−	1.79703i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	2.85195			0.112645			0.0563227	−	0.998413i	$-0.482062\pi$
$641$	2.85195			0.112645			0.0563227	+	0.998413i	$0.482062\pi$
$642$		0			0
$643$	31.8921	+	31.8921i	1.25770	+	1.25770i	0.952187	+	0.305516i	$0.0988288\pi$
$643$	31.8921	+	31.8921i	1.25770	+	1.25770i	0.305516	+	0.952187i	$0.401171\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	7.83402i		−	0.307987i	−0.988072	−	0.153994i	$-0.950786\pi$
$647$		−	7.83402i		−	0.307987i	0.988072	−	0.153994i	$-0.0492135\pi$
$648$		0			0
$649$			1.58922i			0.0623823i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	12.6822	+	12.6822i	0.496292	+	0.496292i	0.910282	−	0.413989i	$-0.135865\pi$
$653$	12.6822	+	12.6822i	0.496292	+	0.496292i	−0.413989	+	0.910282i	$0.635865\pi$
$654$		0			0
$655$	15.9482			0.623146
$656$		0			0
$657$		0			0
$658$		0			0
$659$	12.9694	+	12.9694i	0.505217	+	0.505217i	0.913055	−	0.407837i	$-0.133717\pi$
$659$	12.9694	+	12.9694i	0.505217	+	0.505217i	−0.407837	+	0.913055i	$0.633717\pi$
$660$		0			0
$661$	−6.85796	+	6.85796i	−0.266744	+	0.266744i	−0.827787	−	0.561043i	$-0.810400\pi$
$661$	−6.85796	+	6.85796i	−0.266744	+	0.266744i	0.561043	+	0.827787i	$0.310400\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	2.49756i		−	0.0968511i
$666$		0			0
$667$	7.23478	−	7.23478i	0.280132	−	0.280132i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	9.04164			0.349049
$672$		0			0
$673$	23.1277			0.891508			0.445754	−	0.895155i	$-0.352936\pi$
$673$	23.1277			0.891508			0.445754	+	0.895155i	$0.352936\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	20.2521	−	20.2521i	0.778352	−	0.778352i	−0.201199	−	0.979550i	$-0.564484\pi$
$677$	20.2521	−	20.2521i	0.778352	−	0.778352i	0.979550	+	0.201199i	$0.0644837\pi$
$678$		0			0
$679$		−	51.3905i		−	1.97219i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	26.2957	−	26.2957i	1.00618	−	1.00618i	0.00619708	−	0.999981i	$-0.498027\pi$
$683$	26.2957	−	26.2957i	1.00618	−	1.00618i	0.999981	−	0.00619708i	$-0.00197260\pi$
$684$		0			0
$685$	2.59701	+	2.59701i	0.0992267	+	0.0992267i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−40.0487			−1.52573
$690$		0			0
$691$	4.91230	+	4.91230i	0.186873	+	0.186873i	0.794343	−	0.607470i	$-0.207815\pi$
$691$	4.91230	+	4.91230i	0.186873	+	0.186873i	−0.607470	+	0.794343i	$0.707815\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			7.40414i			0.280855i
$696$		0			0
$697$			1.63595i			0.0619661i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	12.3598	+	12.3598i	0.466824	+	0.466824i	0.900884	−	0.434060i	$-0.142919\pi$
$701$	12.3598	+	12.3598i	0.466824	+	0.466824i	−0.434060	+	0.900884i	$0.642919\pi$
$702$		0			0
$703$	−0.652300			−0.0246020
$704$		0			0
$705$		0			0
$706$		0			0
$707$	34.6363	+	34.6363i	1.30263	+	1.30263i
$708$		0			0
$709$	−26.6076	+	26.6076i	−0.999270	+	0.999270i	−1.00000		0.000729493i	$-0.999768\pi$
$709$	−26.6076	+	26.6076i	−0.999270	+	0.999270i	0.000729493		1.00000i	$0.499768\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	15.5192i		−	0.581200i
$714$		0			0
$715$	4.49600	−	4.49600i	0.168141	−	0.168141i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	50.9765			1.90110			0.950551	−	0.310570i	$-0.100520\pi$
$719$	50.9765			1.90110			0.950551	+	0.310570i	$0.100520\pi$
$720$		0			0
$721$	−48.4731			−1.80523
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−2.00921	+	2.00921i	−0.0746203	+	0.0746203i
$726$		0			0
$727$		−	13.2824i		−	0.492616i	−0.969192	−	0.246308i	$-0.920782\pi$
$727$		−	13.2824i		−	0.492616i	0.969192	−	0.246308i	$-0.0792175\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	13.6246	−	13.6246i	0.503926	−	0.503926i
$732$		0			0
$733$	21.3075	+	21.3075i	0.787012	+	0.787012i	0.981003	−	0.193991i	$-0.0621434\pi$
$733$	21.3075	+	21.3075i	0.787012	+	0.787012i	−0.193991	+	0.981003i	$0.562143\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	6.78168			0.249807
$738$		0			0
$739$	−5.83841	−	5.83841i	−0.214769	−	0.214769i	0.591521	−	0.806290i	$-0.298528\pi$
$739$	−5.83841	−	5.83841i	−0.214769	−	0.214769i	−0.806290	+	0.591521i	$0.798528\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	3.25778i		−	0.119516i	−0.998213	−	0.0597582i	$-0.980967\pi$
$743$		−	3.25778i		−	0.119516i	0.998213	−	0.0597582i	$-0.0190330\pi$
$744$		0			0
$745$			4.65829i			0.170667i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−9.55975	−	9.55975i	−0.349306	−	0.349306i
$750$		0			0
$751$	−20.7322			−0.756530			−0.378265	−	0.925697i	$-0.623479\pi$
$751$	−20.7322			−0.756530			−0.378265	+	0.925697i	$0.623479\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	4.90171	+	4.90171i	0.178391	+	0.178391i
$756$		0			0
$757$	23.3278	−	23.3278i	0.847862	−	0.847862i	−0.142004	−	0.989866i	$-0.545355\pi$
$757$	23.3278	−	23.3278i	0.847862	−	0.847862i	0.989866	+	0.142004i	$0.0453547\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			24.0242i			0.870878i	0.900218	+	0.435439i	$0.143407\pi$
$761$			24.0242i			0.870878i	−0.900218	+	0.435439i	$0.856593\pi$
$762$		0			0
$763$	13.0713	−	13.0713i	0.473213	−	0.473213i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	12.0755			0.436021
$768$		0			0
$769$	−2.70862			−0.0976754			−0.0488377	−	0.998807i	$-0.515552\pi$
$769$	−2.70862			−0.0976754			−0.0488377	+	0.998807i	$0.515552\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−22.9473	+	22.9473i	−0.825358	+	0.825358i	−0.986871	−	0.161513i	$-0.948363\pi$
$773$	−22.9473	+	22.9473i	−0.825358	+	0.825358i	0.161513	+	0.986871i	$0.448363\pi$
$774$		0			0
$775$			4.30994i			0.154817i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−0.264783	+	0.264783i	−0.00948682	+	0.00948682i
$780$		0			0
$781$	−8.92246	−	8.92246i	−0.319271	−	0.319271i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−7.99926			−0.285506
$786$		0			0
$787$	14.0592	+	14.0592i	0.501157	+	0.501157i	0.911797	−	0.410641i	$-0.134695\pi$
$787$	14.0592	+	14.0592i	0.501157	+	0.501157i	−0.410641	+	0.911797i	$0.634695\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	69.7609i		−	2.48041i
$792$		0			0
$793$		−	68.7019i		−	2.43967i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−35.4258	−	35.4258i	−1.25485	−	1.25485i	−0.953521	−	0.301326i	$-0.902571\pi$
$797$	−35.4258	−	35.4258i	−1.25485	−	1.25485i	−0.301326	−	0.953521i	$-0.597429\pi$
$798$		0			0
$799$	29.3011			1.03660
$800$		0			0
$801$		0			0
$802$		0			0
$803$	0.846556	+	0.846556i	0.0298743	+	0.0298743i
$804$		0			0
$805$	10.2569	−	10.2569i	0.361509	−	0.361509i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			16.9217i			0.594935i	0.954732	+	0.297467i	$0.0961420\pi$
$809$			16.9217i			0.594935i	−0.954732	+	0.297467i	$0.903858\pi$
$810$		0			0
$811$	20.4270	−	20.4270i	0.717288	−	0.717288i	−0.250761	−	0.968049i	$-0.580681\pi$
$811$	20.4270	−	20.4270i	0.717288	−	0.717288i	0.968049	+	0.250761i	$0.0806807\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−15.5204			−0.543655
$816$		0			0
$817$	4.41036			0.154299
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−32.4563	+	32.4563i	−1.13273	+	1.13273i	−0.143013	+	0.989721i	$0.545679\pi$
$821$	−32.4563	+	32.4563i	−1.13273	+	1.13273i	−0.989721	+	0.143013i	$0.954321\pi$
$822$		0			0
$823$			6.50705i			0.226821i	0.993548	+	0.113411i	$0.0361776\pi$
$823$			6.50705i			0.226821i	−0.993548	+	0.113411i	$0.963822\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	27.0528	−	27.0528i	0.940718	−	0.940718i	−0.0576204	−	0.998339i	$-0.518351\pi$
$827$	27.0528	−	27.0528i	0.940718	−	0.940718i	0.998339	+	0.0576204i	$0.0183513\pi$
$828$		0			0
$829$	−8.54216	−	8.54216i	−0.296682	−	0.296682i	0.543031	−	0.839713i	$-0.317277\pi$
$829$	−8.54216	−	8.54216i	−0.296682	−	0.296682i	−0.839713	+	0.543031i	$0.817277\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−24.9952			−0.866032
$834$		0			0
$835$	8.32169	+	8.32169i	0.287984	+	0.287984i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	24.4138i		−	0.842860i	−0.906861	−	0.421430i	$-0.861528\pi$
$839$		−	24.4138i		−	0.842860i	0.906861	−	0.421430i	$-0.138472\pi$
$840$		0			0
$841$		−	20.9261i		−	0.721590i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−24.9700	−	24.9700i	−0.858993	−	0.858993i
$846$		0			0
$847$	−40.9414			−1.40676
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−2.67885	−	2.67885i	−0.0918299	−	0.0918299i
$852$		0			0
$853$	8.23270	−	8.23270i	0.281882	−	0.281882i	−0.551977	−	0.833859i	$-0.686126\pi$
$853$	8.23270	−	8.23270i	0.281882	−	0.281882i	0.833859	+	0.551977i	$0.186126\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	4.73909i		−	0.161884i	−0.996719	−	0.0809421i	$-0.974207\pi$
$857$		−	4.73909i		−	0.161884i	0.996719	−	0.0809421i	$-0.0257929\pi$
$858$		0			0
$859$	9.65120	−	9.65120i	0.329295	−	0.329295i	−0.523024	−	0.852318i	$-0.675196\pi$
$859$	9.65120	−	9.65120i	0.329295	−	0.329295i	0.852318	+	0.523024i	$0.175196\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−3.80368			−0.129479			−0.0647393	−	0.997902i	$-0.520622\pi$
$863$	−3.80368			−0.129479			−0.0647393	+	0.997902i	$0.520622\pi$
$864$		0			0
$865$	−1.98309			−0.0674269
$866$		0			0
$867$		0			0
$868$		0			0
$869$	0.395300	−	0.395300i	0.0134096	−	0.0134096i
$870$		0			0
$871$		−	51.5299i		−	1.74602i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−2.84851	+	2.84851i	−0.0962972	+	0.0962972i
$876$		0			0
$877$	2.38917	+	2.38917i	0.0806765	+	0.0806765i	0.746294	−	0.665617i	$-0.231832\pi$
$877$	2.38917	+	2.38917i	0.0806765	+	0.0806765i	−0.665617	+	0.746294i	$0.731832\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	23.6195			0.795762			0.397881	−	0.917437i	$-0.369746\pi$
$881$	23.6195			0.795762			0.397881	+	0.917437i	$0.369746\pi$
$882$		0			0
$883$	9.64752	+	9.64752i	0.324665	+	0.324665i	0.850553	−	0.525889i	$-0.176267\pi$
$883$	9.64752	+	9.64752i	0.324665	+	0.324665i	−0.525889	+	0.850553i	$0.676267\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	8.91140i		−	0.299215i	−0.988745	−	0.149608i	$-0.952199\pi$
$887$		−	8.91140i		−	0.299215i	0.988745	−	0.149608i	$-0.0478011\pi$
$888$		0			0
$889$			62.6167i			2.10010i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	4.74246	+	4.74246i	0.158700	+	0.158700i
$894$		0			0
$895$	−13.6632			−0.456709
$896$		0			0
$897$		0			0
$898$		0			0
$899$	8.65959	+	8.65959i	0.288813	+	0.288813i
$900$		0			0
$901$	−11.0355	+	11.0355i	−0.367645	+	0.367645i
$902$		0			0
$903$		0			0
$904$		0			0
$905$			0.416973i			0.0138606i
$906$		0			0
$907$	−23.4874	+	23.4874i	−0.779886	+	0.779886i	−0.979811	−	0.199925i	$-0.935930\pi$
$907$	−23.4874	+	23.4874i	−0.779886	+	0.779886i	0.199925	+	0.979811i	$0.435930\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−1.77171			−0.0586993			−0.0293497	−	0.999569i	$-0.509344\pi$
$911$	−1.77171			−0.0586993			−0.0293497	+	0.999569i	$0.509344\pi$
$912$		0			0
$913$	−1.67493			−0.0554322
$914$		0			0
$915$		0			0
$916$		0			0
$917$	45.4285	−	45.4285i	1.50018	−	1.50018i
$918$		0			0
$919$			46.2001i			1.52400i	0.647576	+	0.762001i	$0.275783\pi$
$919$			46.2001i			1.52400i	−0.647576	+	0.762001i	$0.724217\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−67.7963	+	67.7963i	−2.23154	+	2.23154i
$924$		0			0
$925$	0.743961	+	0.743961i	0.0244613	+	0.0244613i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	35.5011			1.16475			0.582376	−	0.812920i	$-0.302123\pi$
$929$	35.5011			1.16475			0.582376	+	0.812920i	$0.302123\pi$
$930$		0			0
$931$	−4.04553	−	4.04553i	−0.132587	−	0.132587i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	2.47776i		−	0.0810313i
$936$		0			0
$937$			10.0385i			0.327945i	0.986465	+	0.163972i	$0.0524308\pi$
$937$			10.0385i			0.327945i	−0.986465	+	0.163972i	$0.947569\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−42.3367	−	42.3367i	−1.38014	−	1.38014i	−0.844359	−	0.535777i	$-0.820019\pi$
$941$	−42.3367	−	42.3367i	−1.38014	−	1.38014i	−0.535777	−	0.844359i	$-0.679981\pi$
$942$		0			0
$943$	−2.17481			−0.0708215
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−36.3384	−	36.3384i	−1.18084	−	1.18084i	−0.979527	−	0.201313i	$-0.935479\pi$
$947$	−36.3384	−	36.3384i	−1.18084	−	1.18084i	−0.201313	−	0.979527i	$-0.564521\pi$
$948$		0			0
$949$	6.43246	−	6.43246i	0.208807	−	0.208807i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	32.7338i		−	1.06035i	−0.847888	−	0.530176i	$-0.822126\pi$
$953$		−	32.7338i		−	1.06035i	0.847888	−	0.530176i	$-0.177874\pi$
$954$		0			0
$955$	11.9750	−	11.9750i	0.387502	−	0.387502i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	14.7952			0.477762
$960$		0			0
$961$	−12.4244			−0.400789
$962$		0			0
$963$		0			0
$964$		0			0
$965$	11.7328	−	11.7328i	0.377693	−	0.377693i
$966$		0			0
$967$		−	49.7169i		−	1.59879i	−0.600807	−	0.799394i	$-0.705154\pi$
$967$		−	49.7169i		−	1.59879i	0.600807	−	0.799394i	$-0.294846\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−24.7937	+	24.7937i	−0.795667	+	0.795667i	−0.982409	−	0.186742i	$-0.940207\pi$
$971$	−24.7937	+	24.7937i	−0.795667	+	0.795667i	0.186742	+	0.982409i	$0.440207\pi$
$972$		0			0
$973$	21.0908	+	21.0908i	0.676139	+	0.676139i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	49.4546			1.58219			0.791096	−	0.611692i	$-0.209511\pi$
$977$	49.4546			1.58219			0.791096	+	0.611692i	$0.209511\pi$
$978$		0			0
$979$	−7.06579	−	7.06579i	−0.225824	−	0.225824i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	23.9656i		−	0.764383i	−0.924083	−	0.382191i	$-0.875170\pi$
$983$		−	23.9656i		−	0.764383i	0.924083	−	0.382191i	$-0.124830\pi$
$984$		0			0
$985$			3.37137i			0.107421i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	18.1124	+	18.1124i	0.575940	+	0.575940i
$990$		0			0
$991$	−28.8345			−0.915957			−0.457978	−	0.888963i	$-0.651426\pi$
$991$	−28.8345			−0.915957			−0.457978	+	0.888963i	$0.651426\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−7.17084	−	7.17084i	−0.227331	−	0.227331i
$996$		0			0
$997$	6.03212	−	6.03212i	0.191039	−	0.191039i	−0.605106	−	0.796145i	$-0.706869\pi$
$997$	6.03212	−	6.03212i	0.191039	−	0.191039i	0.796145	+	0.605106i	$0.206869\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2880.2.t.c.721.4		16
3.2	odd	2		320.2.l.a.81.6		16
4.3	odd	2		720.2.t.c.541.8		16
12.11	even	2		80.2.l.a.61.1	yes	16
15.2	even	4		1600.2.q.g.849.6		16
15.8	even	4		1600.2.q.h.849.3		16
15.14	odd	2		1600.2.l.i.401.3		16
16.5	even	4	inner	2880.2.t.c.2161.1		16
16.11	odd	4		720.2.t.c.181.8		16
24.5	odd	2		640.2.l.a.161.3		16
24.11	even	2		640.2.l.b.161.6		16
48.5	odd	4		320.2.l.a.241.6		16
48.11	even	4		80.2.l.a.21.1	✓	16
48.29	odd	4		640.2.l.a.481.3		16
48.35	even	4		640.2.l.b.481.6		16
60.23	odd	4		400.2.q.g.349.5		16
60.47	odd	4		400.2.q.h.349.4		16
60.59	even	2		400.2.l.h.301.8		16
96.5	odd	8		5120.2.a.t.1.4		8
96.11	even	8		5120.2.a.s.1.4		8
96.53	odd	8		5120.2.a.u.1.5		8
96.59	even	8		5120.2.a.v.1.5		8
240.53	even	4		1600.2.q.g.49.6		16
240.59	even	4		400.2.l.h.101.8		16
240.107	odd	4		400.2.q.g.149.5		16
240.149	odd	4		1600.2.l.i.1201.3		16
240.197	even	4		1600.2.q.h.49.3		16
240.203	odd	4		400.2.q.h.149.4		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
80.2.l.a.21.1	✓	16	48.11	even	4
80.2.l.a.61.1	yes	16	12.11	even	2
320.2.l.a.81.6		16	3.2	odd	2
320.2.l.a.241.6		16	48.5	odd	4
400.2.l.h.101.8		16	240.59	even	4
400.2.l.h.301.8		16	60.59	even	2
400.2.q.g.149.5		16	240.107	odd	4
400.2.q.g.349.5		16	60.23	odd	4
400.2.q.h.149.4		16	240.203	odd	4
400.2.q.h.349.4		16	60.47	odd	4
640.2.l.a.161.3		16	24.5	odd	2
640.2.l.a.481.3		16	48.29	odd	4
640.2.l.b.161.6		16	24.11	even	2
640.2.l.b.481.6		16	48.35	even	4
720.2.t.c.181.8		16	16.11	odd	4
720.2.t.c.541.8		16	4.3	odd	2
1600.2.l.i.401.3		16	15.14	odd	2
1600.2.l.i.1201.3		16	240.149	odd	4
1600.2.q.g.49.6		16	240.53	even	4
1600.2.q.g.849.6		16	15.2	even	4
1600.2.q.h.49.3		16	240.197	even	4
1600.2.q.h.849.3		16	15.8	even	4
2880.2.t.c.721.4		16	1.1	even	1	trivial
2880.2.t.c.2161.1		16	16.5	even	4	inner
5120.2.a.s.1.4		8	96.11	even	8
5120.2.a.t.1.4		8	96.5	odd	8
5120.2.a.u.1.5		8	96.53	odd	8
5120.2.a.v.1.5		8	96.59	even	8

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:	\( N \)	\(=\)	\( 2880 = 2^{6} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2880.t (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.707107 + 0.707107i) q^{5} +4.02840i q^{7} +(-0.646837 + 0.646837i) q^{11} +(4.91492 + 4.91492i) q^{13} +2.70862 q^{17} +(0.438397 + 0.438397i) q^{19} +3.60080i q^{23} -1.00000i q^{25} +(-2.00921 - 2.00921i) q^{29} -4.30994 q^{31} +(-2.84851 - 2.84851i) q^{35} +(-0.743961 + 0.743961i) q^{37} +0.603979i q^{41} +(5.03010 - 5.03010i) q^{43} +10.8177 q^{47} -9.22800 q^{49} +(-4.07420 + 4.07420i) q^{53} -0.914766i q^{55} +(1.22845 - 1.22845i) q^{59} +(-6.98912 - 6.98912i) q^{61} -6.95074 q^{65} +(-5.24219 - 5.24219i) q^{67} +13.7940i q^{71} -1.30876i q^{73} +(-2.60572 - 2.60572i) q^{77} -0.611127 q^{79} +(1.29471 + 1.29471i) q^{83} +(-1.91529 + 1.91529i) q^{85} +10.9236i q^{89} +(-19.7993 + 19.7993i) q^{91} -0.619987 q^{95} -12.7571 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 8 q^{11} + 8 q^{19} + 16 q^{29} - 16 q^{37} - 8 q^{43} - 40 q^{47} - 16 q^{49} - 16 q^{53} - 8 q^{59} + 16 q^{61} - 40 q^{67} - 16 q^{77} - 16 q^{79} + 40 q^{83} - 16 q^{85} - 32 q^{91} + 32 q^{95}+O(q^{100}) \) 16 * q - 8 * q^11 + 8 * q^19 + 16 * q^29 - 16 * q^37 - 8 * q^43 - 40 * q^47 - 16 * q^49 - 16 * q^53 - 8 * q^59 + 16 * q^61 - 40 * q^67 - 16 * q^77 - 16 * q^79 + 40 * q^83 - 16 * q^85 - 32 * q^91 + 32 * q^95

Introduction

L-functions

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