LMFDB - Embedded newform 4320.2.bv.a.3601.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4320,2,Mod(721,4320)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4320, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 4, 0]))

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

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

chi := DirichletCharacter("4320.721");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4320 = 2^{5} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4320.bv (of order $6$, 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:	$34.4953736732$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 360)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			3601.1
Root			$0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	4320.3601
Dual form			4320.2.bv.a.721.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.866025 - 0.500000i) q^{5} +(-2.00000 - 3.46410i) q^{7} +O(q^{10})$	$q+(-0.866025 - 0.500000i) q^{5} +(-2.00000 - 3.46410i) q^{7} +(-4.33013 + 2.50000i) q^{11} +(-5.19615 - 3.00000i) q^{13} +3.00000 q^{17} -7.00000i q^{19} +(-1.00000 + 1.73205i) q^{23} +(0.500000 + 0.866025i) q^{25} +(2.00000 - 3.46410i) q^{31} +4.00000i q^{35} -2.00000i q^{37} +(-2.50000 + 4.33013i) q^{41} +(-4.33013 + 2.50000i) q^{43} +(-1.00000 - 1.73205i) q^{47} +(-4.50000 + 7.79423i) q^{49} -8.00000i q^{53} +5.00000 q^{55} +(6.06218 + 3.50000i) q^{59} +(3.46410 - 2.00000i) q^{61} +(3.00000 + 5.19615i) q^{65} +(-2.59808 - 1.50000i) q^{67} +2.00000 q^{71} +1.00000 q^{73} +(17.3205 + 10.0000i) q^{77} +(5.00000 + 8.66025i) q^{79} +(-10.3923 + 6.00000i) q^{83} +(-2.59808 - 1.50000i) q^{85} -2.00000 q^{89} +24.0000i q^{91} +(-3.50000 + 6.06218i) q^{95} +(3.50000 + 6.06218i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{7}+O(q^{10}) $ 4 * q - 8 * q^7	$ 4 q - 8 q^{7} + 12 q^{17} - 4 q^{23} + 2 q^{25} + 8 q^{31} - 10 q^{41} - 4 q^{47} - 18 q^{49} + 20 q^{55} + 12 q^{65} + 8 q^{71} + 4 q^{73} + 20 q^{79} - 8 q^{89} - 14 q^{95} + 14 q^{97}+O(q^{100}) $ 4 * q - 8 * q^7 + 12 * q^17 - 4 * q^23 + 2 * q^25 + 8 * q^31 - 10 * q^41 - 4 * q^47 - 18 * q^49 + 20 * q^55 + 12 * q^65 + 8 * q^71 + 4 * q^73 + 20 * q^79 - 8 * q^89 - 14 * q^95 + 14 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4320\mathbb{Z}\right)^\times$.

$n$	$2081$	$2431$	$3457$	$3781$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$1$	$1$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−0.866025	−	0.500000i	−0.387298	−	0.223607i
$5$	−0.866025	−	0.500000i	−0.387298	−	0.223607i
$6$		0			0
$7$	−2.00000	−	3.46410i	−0.755929	−	1.30931i	−0.944911	−	0.327327i	$-0.893852\pi$
$7$	−2.00000	−	3.46410i	−0.755929	−	1.30931i	0.188982	−	0.981981i	$-0.439481\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−4.33013	+	2.50000i	−1.30558	+	0.753778i	−0.981356	−	0.192201i	$-0.938437\pi$
$11$	−4.33013	+	2.50000i	−1.30558	+	0.753778i	−0.324227	+	0.945979i	$0.605104\pi$
$12$		0			0
$13$	−5.19615	−	3.00000i	−1.44115	−	0.832050i	−0.443227	−	0.896410i	$-0.646166\pi$
$13$	−5.19615	−	3.00000i	−1.44115	−	0.832050i	−0.997927	+	0.0643593i	$0.979500\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	3.00000			0.727607			0.363803	−	0.931476i	$-0.381478\pi$
$17$	3.00000			0.727607			0.363803	+	0.931476i	$0.381478\pi$
$18$		0			0
$19$		−	7.00000i		−	1.60591i	−0.596040	−	0.802955i	$-0.703260\pi$
$19$		−	7.00000i		−	1.60591i	0.596040	−	0.802955i	$-0.296740\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−1.00000	+	1.73205i	−0.208514	+	0.361158i	−0.951247	−	0.308431i	$-0.900196\pi$
$23$	−1.00000	+	1.73205i	−0.208514	+	0.361158i	0.742732	+	0.669588i	$0.233529\pi$
$24$		0			0
$25$	0.500000	+	0.866025i	0.100000	+	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$29$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$30$		0			0
$31$	2.00000	−	3.46410i	0.359211	−	0.622171i	−0.628619	−	0.777714i	$-0.716379\pi$
$31$	2.00000	−	3.46410i	0.359211	−	0.622171i	0.987829	+	0.155543i	$0.0497126\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$			4.00000i			0.676123i
$36$		0			0
$37$		−	2.00000i		−	0.328798i	−0.986394	−	0.164399i	$-0.947432\pi$
$37$		−	2.00000i		−	0.328798i	0.986394	−	0.164399i	$-0.0525685\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−2.50000	+	4.33013i	−0.390434	+	0.676252i	−0.992507	−	0.122189i	$-0.961009\pi$
$41$	−2.50000	+	4.33013i	−0.390434	+	0.676252i	0.602072	+	0.798441i	$0.294342\pi$
$42$		0			0
$43$	−4.33013	+	2.50000i	−0.660338	+	0.381246i	−0.792406	−	0.609994i	$-0.791172\pi$
$43$	−4.33013	+	2.50000i	−0.660338	+	0.381246i	0.132068	+	0.991241i	$0.457838\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−1.00000	−	1.73205i	−0.145865	−	0.252646i	0.783830	−	0.620975i	$-0.213263\pi$
$47$	−1.00000	−	1.73205i	−0.145865	−	0.252646i	−0.929695	+	0.368329i	$0.879930\pi$
$48$		0			0
$49$	−4.50000	+	7.79423i	−0.642857	+	1.11346i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	8.00000i		−	1.09888i	−0.835532	−	0.549442i	$-0.814840\pi$
$53$		−	8.00000i		−	1.09888i	0.835532	−	0.549442i	$-0.185160\pi$
$54$		0			0
$55$	5.00000			0.674200
$56$		0			0
$57$		0			0
$58$		0			0
$59$	6.06218	+	3.50000i	0.789228	+	0.455661i	0.839691	−	0.543065i	$-0.182736\pi$
$59$	6.06218	+	3.50000i	0.789228	+	0.455661i	−0.0504625	+	0.998726i	$0.516070\pi$
$60$		0			0
$61$	3.46410	−	2.00000i	0.443533	−	0.256074i	−0.261562	−	0.965187i	$-0.584238\pi$
$61$	3.46410	−	2.00000i	0.443533	−	0.256074i	0.705095	+	0.709113i	$0.250904\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	3.00000	+	5.19615i	0.372104	+	0.644503i
$66$		0			0
$67$	−2.59808	−	1.50000i	−0.317406	−	0.183254i	0.332830	−	0.942987i	$-0.391996\pi$
$67$	−2.59808	−	1.50000i	−0.317406	−	0.183254i	−0.650236	+	0.759733i	$0.725330\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	2.00000			0.237356			0.118678	−	0.992933i	$-0.462134\pi$
$71$	2.00000			0.237356			0.118678	+	0.992933i	$0.462134\pi$
$72$		0			0
$73$	1.00000			0.117041			0.0585206	−	0.998286i	$-0.481362\pi$
$73$	1.00000			0.117041			0.0585206	+	0.998286i	$0.481362\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	17.3205	+	10.0000i	1.97386	+	1.13961i
$78$		0			0
$79$	5.00000	+	8.66025i	0.562544	+	0.974355i	0.997274	+	0.0737937i	$0.0235106\pi$
$79$	5.00000	+	8.66025i	0.562544	+	0.974355i	−0.434730	+	0.900561i	$0.643156\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−10.3923	+	6.00000i	−1.14070	+	0.658586i	−0.946605	−	0.322396i	$-0.895512\pi$
$83$	−10.3923	+	6.00000i	−1.14070	+	0.658586i	−0.194099	+	0.980982i	$0.562178\pi$
$84$		0			0
$85$	−2.59808	−	1.50000i	−0.281801	−	0.162698i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−2.00000			−0.212000			−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000			−0.212000			−0.106000	+	0.994366i	$0.533804\pi$
$90$		0			0
$91$			24.0000i			2.51588i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−3.50000	+	6.06218i	−0.359092	+	0.621966i
$96$		0			0
$97$	3.50000	+	6.06218i	0.355371	+	0.615521i	0.987181	−	0.159602i	$-0.0510211\pi$
$97$	3.50000	+	6.06218i	0.355371	+	0.615521i	−0.631810	+	0.775123i	$0.717688\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−12.1244	+	7.00000i	−1.20642	+	0.696526i	−0.961975	−	0.273138i	$-0.911939\pi$
$101$	−12.1244	+	7.00000i	−1.20642	+	0.696526i	−0.244443	+	0.969664i	$0.578605\pi$
$102$		0			0
$103$	−7.00000	+	12.1244i	−0.689730	+	1.19465i	0.282194	+	0.959357i	$0.408938\pi$
$103$	−7.00000	+	12.1244i	−0.689730	+	1.19465i	−0.971925	+	0.235291i	$0.924396\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			13.0000i			1.25676i	0.777908	+	0.628379i	$0.216281\pi$
$107$			13.0000i			1.25676i	−0.777908	+	0.628379i	$0.783719\pi$
$108$		0			0
$109$			4.00000i			0.383131i	0.981480	+	0.191565i	$0.0613564\pi$
$109$			4.00000i			0.383131i	−0.981480	+	0.191565i	$0.938644\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	3.00000	−	5.19615i	0.282216	−	0.488813i	−0.689714	−	0.724082i	$-0.742264\pi$
$113$	3.00000	−	5.19615i	0.282216	−	0.488813i	0.971930	+	0.235269i	$0.0755971\pi$
$114$		0			0
$115$	1.73205	−	1.00000i	0.161515	−	0.0932505i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−6.00000	−	10.3923i	−0.550019	−	0.952661i
$120$		0			0
$121$	7.00000	−	12.1244i	0.636364	−	1.10221i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	1.00000i		−	0.0894427i
$126$		0			0
$127$	18.0000			1.59724			0.798621	−	0.601834i	$-0.205563\pi$
$127$	18.0000			1.59724			0.798621	+	0.601834i	$0.205563\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	10.3923	+	6.00000i	0.907980	+	0.524222i	0.879781	−	0.475380i	$-0.157689\pi$
$131$	10.3923	+	6.00000i	0.907980	+	0.524222i	0.0281993	+	0.999602i	$0.491023\pi$
$132$		0			0
$133$	−24.2487	+	14.0000i	−2.10263	+	1.21395i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−8.50000	−	14.7224i	−0.726204	−	1.25782i	−0.958477	−	0.285171i	$-0.907949\pi$
$137$	−8.50000	−	14.7224i	−0.726204	−	1.25782i	0.232273	−	0.972651i	$-0.425384\pi$
$138$		0			0
$139$	9.52628	+	5.50000i	0.808008	+	0.466504i	0.846264	−	0.532764i	$-0.178847\pi$
$139$	9.52628	+	5.50000i	0.808008	+	0.466504i	−0.0382553	+	0.999268i	$0.512180\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	30.0000			2.50873
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	5.19615	+	3.00000i	0.425685	+	0.245770i	0.697507	−	0.716578i	$-0.254293\pi$
$149$	5.19615	+	3.00000i	0.425685	+	0.245770i	−0.271821	+	0.962348i	$0.587626\pi$
$150$		0			0
$151$	5.00000	+	8.66025i	0.406894	+	0.704761i	0.994540	−	0.104357i	$-0.0332784\pi$
$151$	5.00000	+	8.66025i	0.406894	+	0.704761i	−0.587646	+	0.809118i	$0.699945\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−3.46410	+	2.00000i	−0.278243	+	0.160644i
$156$		0			0
$157$	−17.3205	−	10.0000i	−1.38233	−	0.798087i	−0.389892	−	0.920860i	$-0.627488\pi$
$157$	−17.3205	−	10.0000i	−1.38233	−	0.798087i	−0.992435	+	0.122774i	$0.960821\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	8.00000			0.630488
$162$		0			0
$163$		−	20.0000i		−	1.56652i	−0.621694	−	0.783260i	$-0.713555\pi$
$163$		−	20.0000i		−	1.56652i	0.621694	−	0.783260i	$-0.286445\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−3.00000	+	5.19615i	−0.232147	+	0.402090i	−0.958440	−	0.285295i	$-0.907908\pi$
$167$	−3.00000	+	5.19615i	−0.232147	+	0.402090i	0.726293	+	0.687386i	$0.241242\pi$
$168$		0			0
$169$	11.5000	+	19.9186i	0.884615	+	1.53220i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−6.92820	+	4.00000i	−0.526742	+	0.304114i	−0.739689	−	0.672949i	$-0.765027\pi$
$173$	−6.92820	+	4.00000i	−0.526742	+	0.304114i	0.212947	+	0.977064i	$0.431694\pi$
$174$		0			0
$175$	2.00000	−	3.46410i	0.151186	−	0.261861i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			12.0000i			0.896922i	0.893802	+	0.448461i	$0.148028\pi$
$179$			12.0000i			0.896922i	−0.893802	+	0.448461i	$0.851972\pi$
$180$		0			0
$181$		−	2.00000i		−	0.148659i	−0.997234	−	0.0743294i	$-0.976318\pi$
$181$		−	2.00000i		−	0.148659i	0.997234	−	0.0743294i	$-0.0236816\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−1.00000	+	1.73205i	−0.0735215	+	0.127343i
$186$		0			0
$187$	−12.9904	+	7.50000i	−0.949951	+	0.548454i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	4.00000	+	6.92820i	0.289430	+	0.501307i	0.973674	−	0.227946i	$-0.0732010\pi$
$191$	4.00000	+	6.92820i	0.289430	+	0.501307i	−0.684244	+	0.729253i	$0.739868\pi$
$192$		0			0
$193$	11.5000	−	19.9186i	0.827788	−	1.43377i	−0.0719816	−	0.997406i	$-0.522932\pi$
$193$	11.5000	−	19.9186i	0.827788	−	1.43377i	0.899770	−	0.436365i	$-0.143734\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	8.00000i		−	0.569976i	−0.958531	−	0.284988i	$-0.908010\pi$
$197$		−	8.00000i		−	0.569976i	0.958531	−	0.284988i	$-0.0919897\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	4.33013	−	2.50000i	0.302429	−	0.174608i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	17.5000	+	30.3109i	1.21050	+	2.09665i
$210$		0			0
$211$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$211$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	5.00000			0.340997
$216$		0			0
$217$	−16.0000			−1.08615
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−15.5885	−	9.00000i	−1.04859	−	0.605406i
$222$		0			0
$223$	−7.00000	−	12.1244i	−0.468755	−	0.811907i	0.530607	−	0.847618i	$-0.321964\pi$
$223$	−7.00000	−	12.1244i	−0.468755	−	0.811907i	−0.999362	+	0.0357107i	$0.988630\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	7.79423	−	4.50000i	0.517321	−	0.298675i	−0.218517	−	0.975833i	$-0.570122\pi$
$227$	7.79423	−	4.50000i	0.517321	−	0.298675i	0.735838	+	0.677158i	$0.236789\pi$
$228$		0			0
$229$	3.46410	+	2.00000i	0.228914	+	0.132164i	0.610071	−	0.792347i	$-0.291141\pi$
$229$	3.46410	+	2.00000i	0.228914	+	0.132164i	−0.381157	+	0.924510i	$0.624474\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	27.0000			1.76883			0.884414	−	0.466702i	$-0.154558\pi$
$233$	27.0000			1.76883			0.884414	+	0.466702i	$0.154558\pi$
$234$		0			0
$235$			2.00000i			0.130466i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	7.00000	−	12.1244i	0.452792	−	0.784259i	−0.545766	−	0.837938i	$-0.683761\pi$
$239$	7.00000	−	12.1244i	0.452792	−	0.784259i	0.998558	+	0.0536783i	$0.0170946\pi$
$240$		0			0
$241$	7.50000	+	12.9904i	0.483117	+	0.836784i	0.999812	−	0.0193858i	$-0.00617107\pi$
$241$	7.50000	+	12.9904i	0.483117	+	0.836784i	−0.516695	+	0.856170i	$0.672838\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	7.79423	−	4.50000i	0.497955	−	0.287494i
$246$		0			0
$247$	−21.0000	+	36.3731i	−1.33620	+	2.31436i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	9.00000i		−	0.568075i	−0.958813	−	0.284037i	$-0.908326\pi$
$251$		−	9.00000i		−	0.568075i	0.958813	−	0.284037i	$-0.0916740\pi$
$252$		0			0
$253$		−	10.0000i		−	0.628695i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	3.50000	−	6.06218i	0.218324	−	0.378148i	−0.735972	−	0.677012i	$-0.763274\pi$
$257$	3.50000	−	6.06218i	0.218324	−	0.378148i	0.954296	+	0.298864i	$0.0966077\pi$
$258$		0			0
$259$	−6.92820	+	4.00000i	−0.430498	+	0.248548i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$263$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$264$		0			0
$265$	−4.00000	+	6.92820i	−0.245718	+	0.425596i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			20.0000i			1.21942i	0.792624	+	0.609711i	$0.208714\pi$
$269$			20.0000i			1.21942i	−0.792624	+	0.609711i	$0.791286\pi$
$270$		0			0
$271$	−28.0000			−1.70088			−0.850439	−	0.526073i	$-0.823664\pi$
$271$	−28.0000			−1.70088			−0.850439	+	0.526073i	$0.823664\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−4.33013	−	2.50000i	−0.261116	−	0.150756i
$276$		0			0
$277$	1.73205	−	1.00000i	0.104069	−	0.0600842i	−0.447062	−	0.894503i	$-0.647530\pi$
$277$	1.73205	−	1.00000i	0.104069	−	0.0600842i	0.551131	+	0.834419i	$0.314196\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−11.0000	−	19.0526i	−0.656205	−	1.13658i	−0.981590	−	0.190999i	$-0.938827\pi$
$281$	−11.0000	−	19.0526i	−0.656205	−	1.13658i	0.325385	−	0.945582i	$-0.394506\pi$
$282$		0			0
$283$	−3.46410	−	2.00000i	−0.205919	−	0.118888i	0.393494	−	0.919327i	$-0.371266\pi$
$283$	−3.46410	−	2.00000i	−0.205919	−	0.118888i	−0.599414	+	0.800439i	$0.704600\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	20.0000			1.18056
$288$		0			0
$289$	−8.00000			−0.470588
$290$		0			0
$291$		0			0
$292$		0			0
$293$	3.46410	+	2.00000i	0.202375	+	0.116841i	0.597763	−	0.801673i	$-0.296056\pi$
$293$	3.46410	+	2.00000i	0.202375	+	0.116841i	−0.395388	+	0.918514i	$0.629390\pi$
$294$		0			0
$295$	−3.50000	−	6.06218i	−0.203778	−	0.352954i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	10.3923	−	6.00000i	0.601003	−	0.346989i
$300$		0			0
$301$	17.3205	+	10.0000i	0.998337	+	0.576390i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−4.00000			−0.229039
$306$		0			0
$307$			23.0000i			1.31268i	0.754466	+	0.656340i	$0.227896\pi$
$307$			23.0000i			1.31268i	−0.754466	+	0.656340i	$0.772104\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−10.0000	+	17.3205i	−0.567048	+	0.982156i	0.429808	+	0.902920i	$0.358581\pi$
$311$	−10.0000	+	17.3205i	−0.567048	+	0.982156i	−0.996856	+	0.0792356i	$0.974752\pi$
$312$		0			0
$313$	4.50000	+	7.79423i	0.254355	+	0.440556i	0.964720	−	0.263278i	$-0.0848035\pi$
$313$	4.50000	+	7.79423i	0.254355	+	0.440556i	−0.710365	+	0.703833i	$0.751470\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−15.5885	+	9.00000i	−0.875535	+	0.505490i	−0.869184	−	0.494489i	$-0.835355\pi$
$317$	−15.5885	+	9.00000i	−0.875535	+	0.505490i	−0.00635137	+	0.999980i	$0.502022\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	21.0000i		−	1.16847i
$324$		0			0
$325$		−	6.00000i		−	0.332820i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−4.00000	+	6.92820i	−0.220527	+	0.381964i
$330$		0			0
$331$	17.3205	−	10.0000i	0.952021	−	0.549650i	0.0583130	−	0.998298i	$-0.481428\pi$
$331$	17.3205	−	10.0000i	0.952021	−	0.549650i	0.893708	+	0.448649i	$0.148095\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	1.50000	+	2.59808i	0.0819538	+	0.141948i
$336$		0			0
$337$	−7.50000	+	12.9904i	−0.408551	+	0.707631i	−0.994728	−	0.102552i	$-0.967299\pi$
$337$	−7.50000	+	12.9904i	−0.408551	+	0.707631i	0.586177	+	0.810183i	$0.300632\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			20.0000i			1.08306i
$342$		0			0
$343$	8.00000			0.431959
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−16.4545	−	9.50000i	−0.883323	−	0.509987i	−0.0115703	−	0.999933i	$-0.503683\pi$
$347$	−16.4545	−	9.50000i	−0.883323	−	0.509987i	−0.871753	+	0.489946i	$0.837016\pi$
$348$		0			0
$349$	−13.8564	+	8.00000i	−0.741716	+	0.428230i	−0.822693	−	0.568486i	$-0.807529\pi$
$349$	−13.8564	+	8.00000i	−0.741716	+	0.428230i	0.0809766	+	0.996716i	$0.474196\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−4.50000	−	7.79423i	−0.239511	−	0.414845i	0.721063	−	0.692869i	$-0.243654\pi$
$353$	−4.50000	−	7.79423i	−0.239511	−	0.414845i	−0.960574	+	0.278024i	$0.910320\pi$
$354$		0			0
$355$	−1.73205	−	1.00000i	−0.0919277	−	0.0530745i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−12.0000			−0.633336			−0.316668	−	0.948536i	$-0.602564\pi$
$359$	−12.0000			−0.633336			−0.316668	+	0.948536i	$0.602564\pi$
$360$		0			0
$361$	−30.0000			−1.57895
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−0.866025	−	0.500000i	−0.0453298	−	0.0261712i
$366$		0			0
$367$	9.00000	+	15.5885i	0.469796	+	0.813711i	0.999404	−	0.0345320i	$-0.0109941\pi$
$367$	9.00000	+	15.5885i	0.469796	+	0.813711i	−0.529607	+	0.848243i	$0.677661\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−27.7128	+	16.0000i	−1.43878	+	0.830679i
$372$		0			0
$373$	19.0526	+	11.0000i	0.986504	+	0.569558i	0.904227	−	0.427051i	$-0.140448\pi$
$373$	19.0526	+	11.0000i	0.986504	+	0.569558i	0.0822766	+	0.996610i	$0.473781\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$			3.00000i			0.154100i	0.997027	+	0.0770498i	$0.0245501\pi$
$379$			3.00000i			0.154100i	−0.997027	+	0.0770498i	$0.975450\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−9.00000	+	15.5885i	−0.459879	+	0.796533i	−0.998954	−	0.0457244i	$-0.985440\pi$
$383$	−9.00000	+	15.5885i	−0.459879	+	0.796533i	0.539076	+	0.842257i	$0.318774\pi$
$384$		0			0
$385$	−10.0000	−	17.3205i	−0.509647	−	0.882735i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−13.8564	+	8.00000i	−0.702548	+	0.405616i	−0.808296	−	0.588777i	$-0.799610\pi$
$389$	−13.8564	+	8.00000i	−0.702548	+	0.405616i	0.105748	+	0.994393i	$0.466276\pi$
$390$		0			0
$391$	−3.00000	+	5.19615i	−0.151717	+	0.262781i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	10.0000i		−	0.503155i
$396$		0			0
$397$			26.0000i			1.30490i	0.757831	+	0.652451i	$0.226259\pi$
$397$			26.0000i			1.30490i	−0.757831	+	0.652451i	$0.773741\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−7.50000	+	12.9904i	−0.374532	+	0.648709i	−0.990257	−	0.139253i	$-0.955530\pi$
$401$	−7.50000	+	12.9904i	−0.374532	+	0.648709i	0.615725	+	0.787961i	$0.288863\pi$
$402$		0			0
$403$	−20.7846	+	12.0000i	−1.03536	+	0.597763i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	5.00000	+	8.66025i	0.247841	+	0.429273i
$408$		0			0
$409$	−7.50000	+	12.9904i	−0.370851	+	0.642333i	−0.989697	−	0.143180i	$-0.954267\pi$
$409$	−7.50000	+	12.9904i	−0.370851	+	0.642333i	0.618846	+	0.785513i	$0.287601\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	28.0000i		−	1.37779i
$414$		0			0
$415$	12.0000			0.589057
$416$		0			0
$417$		0			0
$418$		0			0
$419$	3.46410	+	2.00000i	0.169232	+	0.0977064i	0.582224	−	0.813029i	$-0.302183\pi$
$419$	3.46410	+	2.00000i	0.169232	+	0.0977064i	−0.412991	+	0.910735i	$0.635516\pi$
$420$		0			0
$421$	1.73205	−	1.00000i	0.0844150	−	0.0487370i	−0.457198	−	0.889365i	$-0.651147\pi$
$421$	1.73205	−	1.00000i	0.0844150	−	0.0487370i	0.541613	+	0.840628i	$0.317814\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	1.50000	+	2.59808i	0.0727607	+	0.126025i
$426$		0			0
$427$	−13.8564	−	8.00000i	−0.670559	−	0.387147i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−12.0000			−0.578020			−0.289010	−	0.957326i	$-0.593326\pi$
$431$	−12.0000			−0.578020			−0.289010	+	0.957326i	$0.593326\pi$
$432$		0			0
$433$	29.0000			1.39365			0.696826	−	0.717241i	$-0.254595\pi$
$433$	29.0000			1.39365			0.696826	+	0.717241i	$0.254595\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	12.1244	+	7.00000i	0.579987	+	0.334855i
$438$		0			0
$439$	−20.0000	−	34.6410i	−0.954548	−	1.65333i	−0.735399	−	0.677634i	$-0.763005\pi$
$439$	−20.0000	−	34.6410i	−0.954548	−	1.65333i	−0.219149	−	0.975691i	$-0.570328\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	21.6506	−	12.5000i	1.02865	−	0.593893i	0.112054	−	0.993702i	$-0.464257\pi$
$443$	21.6506	−	12.5000i	1.02865	−	0.593893i	0.916598	+	0.399809i	$0.130924\pi$
$444$		0			0
$445$	1.73205	+	1.00000i	0.0821071	+	0.0474045i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	27.0000			1.27421			0.637104	−	0.770778i	$-0.280132\pi$
$449$	27.0000			1.27421			0.637104	+	0.770778i	$0.280132\pi$
$450$		0			0
$451$		−	25.0000i		−	1.17720i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	12.0000	−	20.7846i	0.562569	−	0.974398i
$456$		0			0
$457$	−1.50000	−	2.59808i	−0.0701670	−	0.121533i	0.828807	−	0.559534i	$-0.189020\pi$
$457$	−1.50000	−	2.59808i	−0.0701670	−	0.121533i	−0.898974	+	0.438001i	$0.855687\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−12.1244	+	7.00000i	−0.564688	+	0.326023i	−0.755025	−	0.655696i	$-0.772375\pi$
$461$	−12.1244	+	7.00000i	−0.564688	+	0.326023i	0.190337	+	0.981719i	$0.439042\pi$
$462$		0			0
$463$	13.0000	−	22.5167i	0.604161	−	1.04644i	−0.388022	−	0.921650i	$-0.626842\pi$
$463$	13.0000	−	22.5167i	0.604161	−	1.04644i	0.992183	−	0.124788i	$-0.0398251\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			13.0000i			0.601568i	0.953692	+	0.300784i	$0.0972484\pi$
$467$			13.0000i			0.601568i	−0.953692	+	0.300784i	$0.902752\pi$
$468$		0			0
$469$			12.0000i			0.554109i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	12.5000	−	21.6506i	0.574751	−	0.995497i
$474$		0			0
$475$	6.06218	−	3.50000i	0.278152	−	0.160591i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−8.00000	−	13.8564i	−0.365529	−	0.633115i	0.623332	−	0.781958i	$-0.285779\pi$
$479$	−8.00000	−	13.8564i	−0.365529	−	0.633115i	−0.988861	+	0.148842i	$0.952445\pi$
$480$		0			0
$481$	−6.00000	+	10.3923i	−0.273576	+	0.473848i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	7.00000i		−	0.317854i
$486$		0			0
$487$	12.0000			0.543772			0.271886	−	0.962329i	$-0.412353\pi$
$487$	12.0000			0.543772			0.271886	+	0.962329i	$0.412353\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−9.52628	−	5.50000i	−0.429915	−	0.248212i	0.269395	−	0.963030i	$-0.413176\pi$
$491$	−9.52628	−	5.50000i	−0.429915	−	0.248212i	−0.699310	+	0.714818i	$0.746509\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−4.00000	−	6.92820i	−0.179425	−	0.310772i
$498$		0			0
$499$	−35.5070	−	20.5000i	−1.58951	−	0.917706i	−0.993387	−	0.114816i	$-0.963372\pi$
$499$	−35.5070	−	20.5000i	−1.58951	−	0.917706i	−0.596127	−	0.802890i	$-0.703294\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	12.0000			0.535054			0.267527	−	0.963550i	$-0.413794\pi$
$503$	12.0000			0.535054			0.267527	+	0.963550i	$0.413794\pi$
$504$		0			0
$505$	14.0000			0.622992
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−10.3923	−	6.00000i	−0.460631	−	0.265945i	0.251679	−	0.967811i	$-0.419017\pi$
$509$	−10.3923	−	6.00000i	−0.460631	−	0.265945i	−0.712309	+	0.701866i	$0.752351\pi$
$510$		0			0
$511$	−2.00000	−	3.46410i	−0.0884748	−	0.153243i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	12.1244	−	7.00000i	0.534263	−	0.308457i
$516$		0			0
$517$	8.66025	+	5.00000i	0.380878	+	0.219900i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−15.0000			−0.657162			−0.328581	−	0.944476i	$-0.606570\pi$
$521$	−15.0000			−0.657162			−0.328581	+	0.944476i	$0.606570\pi$
$522$		0			0
$523$		−	4.00000i		−	0.174908i	−0.996169	−	0.0874539i	$-0.972127\pi$
$523$		−	4.00000i		−	0.174908i	0.996169	−	0.0874539i	$-0.0278730\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	6.00000	−	10.3923i	0.261364	−	0.452696i
$528$		0			0
$529$	9.50000	+	16.4545i	0.413043	+	0.715412i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	25.9808	−	15.0000i	1.12535	−	0.649722i
$534$		0			0
$535$	6.50000	−	11.2583i	0.281020	−	0.486740i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	45.0000i		−	1.93829i
$540$		0			0
$541$			22.0000i			0.945854i	0.881102	+	0.472927i	$0.156803\pi$
$541$			22.0000i			0.945854i	−0.881102	+	0.472927i	$0.843197\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	2.00000	−	3.46410i	0.0856706	−	0.148386i
$546$		0			0
$547$	11.2583	−	6.50000i	0.481371	−	0.277920i	−0.239616	−	0.970868i	$-0.577022\pi$
$547$	11.2583	−	6.50000i	0.481371	−	0.277920i	0.720988	+	0.692948i	$0.243688\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	20.0000	−	34.6410i	0.850487	−	1.47309i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		1.00000			$0$
$557$		0			0		−1.00000			$\pi$
$558$		0			0
$559$	30.0000			1.26886
$560$		0			0
$561$		0			0
$562$		0			0
$563$	33.7750	+	19.5000i	1.42345	+	0.821827i	0.996592	−	0.0824933i	$-0.0262883\pi$
$563$	33.7750	+	19.5000i	1.42345	+	0.821827i	0.426855	+	0.904320i	$0.359622\pi$
$564$		0			0
$565$	−5.19615	+	3.00000i	−0.218604	+	0.126211i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−12.5000	−	21.6506i	−0.524027	−	0.907642i	−0.999609	−	0.0279702i	$-0.991096\pi$
$569$	−12.5000	−	21.6506i	−0.524027	−	0.907642i	0.475581	−	0.879672i	$-0.342238\pi$
$570$		0			0
$571$	11.2583	+	6.50000i	0.471146	+	0.272017i	0.716720	−	0.697362i	$-0.245643\pi$
$571$	11.2583	+	6.50000i	0.471146	+	0.272017i	−0.245573	+	0.969378i	$0.578976\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−2.00000			−0.0834058
$576$		0			0
$577$	7.00000			0.291414			0.145707	−	0.989328i	$-0.453454\pi$
$577$	7.00000			0.291414			0.145707	+	0.989328i	$0.453454\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	41.5692	+	24.0000i	1.72458	+	0.995688i
$582$		0			0
$583$	20.0000	+	34.6410i	0.828315	+	1.43468i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−23.3827	+	13.5000i	−0.965107	+	0.557205i	−0.897741	−	0.440524i	$-0.854793\pi$
$587$	−23.3827	+	13.5000i	−0.965107	+	0.557205i	−0.0673658	+	0.997728i	$0.521459\pi$
$588$		0			0
$589$	−24.2487	−	14.0000i	−0.999151	−	0.576860i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−34.0000			−1.39621			−0.698106	−	0.715994i	$-0.745974\pi$
$593$	−34.0000			−1.39621			−0.698106	+	0.715994i	$0.745974\pi$
$594$		0			0
$595$			12.0000i			0.491952i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	21.0000	−	36.3731i	0.858037	−	1.48616i	−0.0157622	−	0.999876i	$-0.505017\pi$
$599$	21.0000	−	36.3731i	0.858037	−	1.48616i	0.873799	−	0.486287i	$-0.161649\pi$
$600$		0			0
$601$	−18.5000	−	32.0429i	−0.754631	−	1.30706i	−0.945558	−	0.325455i	$-0.894483\pi$
$601$	−18.5000	−	32.0429i	−0.754631	−	1.30706i	0.190927	−	0.981604i	$-0.438851\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−12.1244	+	7.00000i	−0.492925	+	0.284590i
$606$		0			0
$607$	−2.00000	+	3.46410i	−0.0811775	+	0.140604i	−0.903756	−	0.428048i	$-0.859201\pi$
$607$	−2.00000	+	3.46410i	−0.0811775	+	0.140604i	0.822578	+	0.568652i	$0.192535\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			12.0000i			0.485468i
$612$		0			0
$613$		−	26.0000i		−	1.05013i	−0.851062	−	0.525065i	$-0.824041\pi$
$613$		−	26.0000i		−	1.05013i	0.851062	−	0.525065i	$-0.175959\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−1.50000	+	2.59808i	−0.0603877	+	0.104595i	−0.894639	−	0.446790i	$-0.852567\pi$
$617$	−1.50000	+	2.59808i	−0.0603877	+	0.104595i	0.834251	+	0.551385i	$0.185900\pi$
$618$		0			0
$619$	−4.33013	+	2.50000i	−0.174042	+	0.100483i	−0.584491	−	0.811400i	$-0.698706\pi$
$619$	−4.33013	+	2.50000i	−0.174042	+	0.100483i	0.410448	+	0.911884i	$0.365372\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	4.00000	+	6.92820i	0.160257	+	0.277573i
$624$		0			0
$625$	−0.500000	+	0.866025i	−0.0200000	+	0.0346410i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	6.00000i		−	0.239236i
$630$		0			0
$631$	6.00000			0.238856			0.119428	−	0.992843i	$-0.461894\pi$
$631$	6.00000			0.238856			0.119428	+	0.992843i	$0.461894\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−15.5885	−	9.00000i	−0.618609	−	0.357154i
$636$		0			0
$637$	46.7654	−	27.0000i	1.85291	−	1.06978i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	8.50000	+	14.7224i	0.335730	+	0.581501i	0.983625	−	0.180229i	$-0.0576838\pi$
$641$	8.50000	+	14.7224i	0.335730	+	0.581501i	−0.647895	+	0.761730i	$0.724350\pi$
$642$		0			0
$643$	−18.1865	−	10.5000i	−0.717207	−	0.414080i	0.0965169	−	0.995331i	$-0.469230\pi$
$643$	−18.1865	−	10.5000i	−0.717207	−	0.414080i	−0.813724	+	0.581252i	$0.802563\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−24.0000			−0.943537			−0.471769	−	0.881722i	$-0.656384\pi$
$647$	−24.0000			−0.943537			−0.471769	+	0.881722i	$0.656384\pi$
$648$		0			0
$649$	−35.0000			−1.37387
$650$		0			0
$651$		0			0
$652$		0			0
$653$	10.3923	+	6.00000i	0.406682	+	0.234798i	0.689363	−	0.724416i	$-0.257890\pi$
$653$	10.3923	+	6.00000i	0.406682	+	0.234798i	−0.282681	+	0.959214i	$0.591224\pi$
$654$		0			0
$655$	−6.00000	−	10.3923i	−0.234439	−	0.406061i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−31.1769	+	18.0000i	−1.21448	+	0.701180i	−0.963732	−	0.266872i	$-0.914010\pi$
$659$	−31.1769	+	18.0000i	−1.21448	+	0.701180i	−0.250748	+	0.968052i	$0.580677\pi$
$660$		0			0
$661$	39.8372	+	23.0000i	1.54949	+	0.894596i	0.998181	+	0.0602929i	$0.0192035\pi$
$661$	39.8372	+	23.0000i	1.54949	+	0.894596i	0.551306	+	0.834303i	$0.314130\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	28.0000			1.08579
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−10.0000	+	17.3205i	−0.386046	+	0.668651i
$672$		0			0
$673$	−5.00000	−	8.66025i	−0.192736	−	0.333828i	0.753420	−	0.657539i	$-0.228403\pi$
$673$	−5.00000	−	8.66025i	−0.192736	−	0.333828i	−0.946156	+	0.323711i	$0.895069\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−13.8564	+	8.00000i	−0.532545	+	0.307465i	−0.742052	−	0.670342i	$-0.766147\pi$
$677$	−13.8564	+	8.00000i	−0.532545	+	0.307465i	0.209507	+	0.977807i	$0.432814\pi$
$678$		0			0
$679$	14.0000	−	24.2487i	0.537271	−	0.930580i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	51.0000i		−	1.95146i	−0.218975	−	0.975730i	$-0.570271\pi$
$683$		−	51.0000i		−	1.95146i	0.218975	−	0.975730i	$-0.429729\pi$
$684$		0			0
$685$			17.0000i			0.649537i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−24.0000	+	41.5692i	−0.914327	+	1.58366i
$690$		0			0
$691$	10.3923	−	6.00000i	0.395342	−	0.228251i	−0.289130	−	0.957290i	$-0.593366\pi$
$691$	10.3923	−	6.00000i	0.395342	−	0.228251i	0.684472	+	0.729039i	$0.260033\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−5.50000	−	9.52628i	−0.208627	−	0.361352i
$696$		0			0
$697$	−7.50000	+	12.9904i	−0.284083	+	0.492046i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			10.0000i			0.377695i	0.982006	+	0.188847i	$0.0604752\pi$
$701$			10.0000i			0.377695i	−0.982006	+	0.188847i	$0.939525\pi$
$702$		0			0
$703$	−14.0000			−0.528020
$704$		0			0
$705$		0			0
$706$		0			0
$707$	48.4974	+	28.0000i	1.82393	+	1.05305i
$708$		0			0
$709$	27.7128	−	16.0000i	1.04078	−	0.600893i	0.120723	−	0.992686i	$-0.461479\pi$
$709$	27.7128	−	16.0000i	1.04078	−	0.600893i	0.920053	+	0.391794i	$0.128145\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	4.00000	+	6.92820i	0.149801	+	0.259463i
$714$		0			0
$715$	−25.9808	−	15.0000i	−0.971625	−	0.560968i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−18.0000			−0.671287			−0.335643	−	0.941989i	$-0.608954\pi$
$719$	−18.0000			−0.671287			−0.335643	+	0.941989i	$0.608954\pi$
$720$		0			0
$721$	56.0000			2.08555
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−25.0000	−	43.3013i	−0.927199	−	1.60596i	−0.787986	−	0.615693i	$-0.788876\pi$
$727$	−25.0000	−	43.3013i	−0.927199	−	1.60596i	−0.139212	−	0.990263i	$-0.544457\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−12.9904	+	7.50000i	−0.480467	+	0.277398i
$732$		0			0
$733$	−31.1769	−	18.0000i	−1.15155	−	0.664845i	−0.202282	−	0.979327i	$-0.564836\pi$
$733$	−31.1769	−	18.0000i	−1.15155	−	0.664845i	−0.949263	+	0.314482i	$0.898169\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	15.0000			0.552532
$738$		0			0
$739$		−	15.0000i		−	0.551784i	−0.961189	−	0.275892i	$-0.911027\pi$
$739$		−	15.0000i		−	0.551784i	0.961189	−	0.275892i	$-0.0889732\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−18.0000	+	31.1769i	−0.660356	+	1.14377i	0.320166	+	0.947361i	$0.396261\pi$
$743$	−18.0000	+	31.1769i	−0.660356	+	1.14377i	−0.980522	+	0.196409i	$0.937072\pi$
$744$		0			0
$745$	−3.00000	−	5.19615i	−0.109911	−	0.190372i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	45.0333	−	26.0000i	1.64548	−	0.950019i
$750$		0			0
$751$	−23.0000	+	39.8372i	−0.839282	+	1.45368i	0.0512140	+	0.998688i	$0.483691\pi$
$751$	−23.0000	+	39.8372i	−0.839282	+	1.45368i	−0.890496	+	0.454991i	$0.849642\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	10.0000i		−	0.363937i
$756$		0			0
$757$			4.00000i			0.145382i	0.997354	+	0.0726912i	$0.0231588\pi$
$757$			4.00000i			0.145382i	−0.997354	+	0.0726912i	$0.976841\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−25.0000	+	43.3013i	−0.906249	+	1.56967i	−0.0870179	+	0.996207i	$0.527734\pi$
$761$	−25.0000	+	43.3013i	−0.906249	+	1.56967i	−0.819231	+	0.573463i	$0.805600\pi$
$762$		0			0
$763$	13.8564	−	8.00000i	0.501636	−	0.289619i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−21.0000	−	36.3731i	−0.758266	−	1.31336i
$768$		0			0
$769$	15.0000	−	25.9808i	0.540914	−	0.936890i	−0.457938	−	0.888984i	$-0.651412\pi$
$769$	15.0000	−	25.9808i	0.540914	−	0.936890i	0.998852	−	0.0479061i	$-0.0152548\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			42.0000i			1.51064i	0.655359	+	0.755318i	$0.272517\pi$
$773$			42.0000i			1.51064i	−0.655359	+	0.755318i	$0.727483\pi$
$774$		0			0
$775$	4.00000			0.143684
$776$		0			0
$777$		0			0
$778$		0			0
$779$	30.3109	+	17.5000i	1.08600	+	0.627003i
$780$		0			0
$781$	−8.66025	+	5.00000i	−0.309888	+	0.178914i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	10.0000	+	17.3205i	0.356915	+	0.618195i
$786$		0			0
$787$	45.0333	+	26.0000i	1.60526	+	0.926800i	0.990410	+	0.138159i	$0.0441186\pi$
$787$	45.0333	+	26.0000i	1.60526	+	0.926800i	0.614855	+	0.788641i	$0.289215\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−24.0000			−0.853342
$792$		0			0
$793$	−24.0000			−0.852265
$794$		0			0
$795$		0			0
$796$		0			0
$797$	5.19615	+	3.00000i	0.184057	+	0.106265i	0.589197	−	0.807989i	$-0.299444\pi$
$797$	5.19615	+	3.00000i	0.184057	+	0.106265i	−0.405140	+	0.914255i	$0.632777\pi$
$798$		0			0
$799$	−3.00000	−	5.19615i	−0.106132	−	0.183827i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−4.33013	+	2.50000i	−0.152807	+	0.0882231i
$804$		0			0
$805$	−6.92820	−	4.00000i	−0.244187	−	0.140981i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−7.00000			−0.246107			−0.123053	−	0.992400i	$-0.539269\pi$
$809$	−7.00000			−0.246107			−0.123053	+	0.992400i	$0.539269\pi$
$810$		0			0
$811$			1.00000i			0.0351147i	0.999846	+	0.0175574i	$0.00558897\pi$
$811$			1.00000i			0.0351147i	−0.999846	+	0.0175574i	$0.994411\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−10.0000	+	17.3205i	−0.350285	+	0.606711i
$816$		0			0
$817$	17.5000	+	30.3109i	0.612247	+	1.06044i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	3.46410	−	2.00000i	0.120898	−	0.0698005i	−0.438331	−	0.898813i	$-0.644430\pi$
$821$	3.46410	−	2.00000i	0.120898	−	0.0698005i	0.559229	+	0.829013i	$0.311097\pi$
$822$		0			0
$823$	−3.00000	+	5.19615i	−0.104573	+	0.181126i	−0.913564	−	0.406695i	$-0.866681\pi$
$823$	−3.00000	+	5.19615i	−0.104573	+	0.181126i	0.808990	+	0.587822i	$0.200014\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	16.0000i		−	0.556375i	−0.960527	−	0.278187i	$-0.910266\pi$
$827$		−	16.0000i		−	0.556375i	0.960527	−	0.278187i	$-0.0897336\pi$
$828$		0			0
$829$		−	8.00000i		−	0.277851i	−0.990303	−	0.138926i	$-0.955635\pi$
$829$		−	8.00000i		−	0.277851i	0.990303	−	0.138926i	$-0.0443649\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−13.5000	+	23.3827i	−0.467747	+	0.810162i
$834$		0			0
$835$	5.19615	−	3.00000i	0.179820	−	0.103819i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−14.0000	−	24.2487i	−0.483334	−	0.837158i	0.516483	−	0.856297i	$-0.327241\pi$
$839$	−14.0000	−	24.2487i	−0.483334	−	0.837158i	−0.999817	+	0.0191389i	$0.993908\pi$
$840$		0			0
$841$	−14.5000	+	25.1147i	−0.500000	+	0.866025i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		−	23.0000i		−	0.791224i
$846$		0			0
$847$	−56.0000			−1.92418
$848$		0			0
$849$		0			0
$850$		0			0
$851$	3.46410	+	2.00000i	0.118748	+	0.0685591i
$852$		0			0
$853$	39.8372	−	23.0000i	1.36400	−	0.787505i	0.373845	−	0.927491i	$-0.378039\pi$
$853$	39.8372	−	23.0000i	1.36400	−	0.787505i	0.990153	+	0.139986i	$0.0447058\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−21.0000	−	36.3731i	−0.717346	−	1.24248i	−0.962048	−	0.272882i	$-0.912023\pi$
$857$	−21.0000	−	36.3731i	−0.717346	−	1.24248i	0.244701	−	0.969599i	$-0.421310\pi$
$858$		0			0
$859$	32.0429	+	18.5000i	1.09329	+	0.631212i	0.934451	−	0.356092i	$-0.115891\pi$
$859$	32.0429	+	18.5000i	1.09329	+	0.631212i	0.158840	+	0.987304i	$0.449225\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	30.0000			1.02121			0.510606	−	0.859815i	$-0.329421\pi$
$863$	30.0000			1.02121			0.510606	+	0.859815i	$0.329421\pi$
$864$		0			0
$865$	8.00000			0.272008
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−43.3013	−	25.0000i	−1.46889	−	0.848067i
$870$		0			0
$871$	9.00000	+	15.5885i	0.304953	+	0.528195i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−3.46410	+	2.00000i	−0.117108	+	0.0676123i
$876$		0			0
$877$	−27.7128	−	16.0000i	−0.935795	−	0.540282i	−0.0471555	−	0.998888i	$-0.515016\pi$
$877$	−27.7128	−	16.0000i	−0.935795	−	0.540282i	−0.888640	+	0.458606i	$0.848349\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−30.0000			−1.01073			−0.505363	−	0.862907i	$-0.668641\pi$
$881$	−30.0000			−1.01073			−0.505363	+	0.862907i	$0.668641\pi$
$882$		0			0
$883$		−	41.0000i		−	1.37976i	−0.723924	−	0.689880i	$-0.757663\pi$
$883$		−	41.0000i		−	1.37976i	0.723924	−	0.689880i	$-0.242337\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−11.0000	+	19.0526i	−0.369344	+	0.639722i	−0.989463	−	0.144785i	$-0.953751\pi$
$887$	−11.0000	+	19.0526i	−0.369344	+	0.639722i	0.620119	+	0.784508i	$0.287084\pi$
$888$		0			0
$889$	−36.0000	−	62.3538i	−1.20740	−	2.09128i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−12.1244	+	7.00000i	−0.405726	+	0.234246i
$894$		0			0
$895$	6.00000	−	10.3923i	0.200558	−	0.347376i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		−	24.0000i		−	0.799556i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−1.00000	+	1.73205i	−0.0332411	+	0.0575753i
$906$		0			0
$907$	21.6506	−	12.5000i	0.718898	−	0.415056i	−0.0954492	−	0.995434i	$-0.530429\pi$
$907$	21.6506	−	12.5000i	0.718898	−	0.415056i	0.814347	+	0.580379i	$0.197095\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−18.0000	−	31.1769i	−0.596367	−	1.03294i	−0.993352	−	0.115113i	$-0.963277\pi$
$911$	−18.0000	−	31.1769i	−0.596367	−	1.03294i	0.396986	−	0.917825i	$-0.370056\pi$
$912$		0			0
$913$	30.0000	−	51.9615i	0.992855	−	1.71968i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	48.0000i		−	1.58510i
$918$		0			0
$919$	26.0000			0.857661			0.428830	−	0.903385i	$-0.358926\pi$
$919$	26.0000			0.857661			0.428830	+	0.903385i	$0.358926\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−10.3923	−	6.00000i	−0.342067	−	0.197492i
$924$		0			0
$925$	1.73205	−	1.00000i	0.0569495	−	0.0328798i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−25.0000	−	43.3013i	−0.820223	−	1.42067i	−0.905516	−	0.424313i	$-0.860516\pi$
$929$	−25.0000	−	43.3013i	−0.820223	−	1.42067i	0.0852924	−	0.996356i	$-0.472818\pi$
$930$		0			0
$931$	54.5596	+	31.5000i	1.78812	+	1.03237i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	15.0000			0.490552
$936$		0			0
$937$	−14.0000			−0.457360			−0.228680	−	0.973502i	$-0.573441\pi$
$937$	−14.0000			−0.457360			−0.228680	+	0.973502i	$0.573441\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−6.92820	−	4.00000i	−0.225853	−	0.130396i	0.382804	−	0.923829i	$-0.374958\pi$
$941$	−6.92820	−	4.00000i	−0.225853	−	0.130396i	−0.608657	+	0.793433i	$0.708292\pi$
$942$		0			0
$943$	−5.00000	−	8.66025i	−0.162822	−	0.282017i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	44.1673	−	25.5000i	1.43524	−	0.828639i	0.437730	−	0.899106i	$-0.355783\pi$
$947$	44.1673	−	25.5000i	1.43524	−	0.828639i	0.997514	+	0.0704677i	$0.0224492\pi$
$948$		0			0
$949$	−5.19615	−	3.00000i	−0.168674	−	0.0973841i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	29.0000			0.939402			0.469701	−	0.882826i	$-0.344362\pi$
$953$	29.0000			0.939402			0.469701	+	0.882826i	$0.344362\pi$
$954$		0			0
$955$		−	8.00000i		−	0.258874i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−34.0000	+	58.8897i	−1.09792	+	1.90165i
$960$		0			0
$961$	7.50000	+	12.9904i	0.241935	+	0.419045i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−19.9186	+	11.5000i	−0.641202	+	0.370198i
$966$		0			0
$967$	14.0000	−	24.2487i	0.450210	−	0.779786i	−0.548189	−	0.836354i	$-0.684683\pi$
$967$	14.0000	−	24.2487i	0.450210	−	0.779786i	0.998399	+	0.0565684i	$0.0180159\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			36.0000i			1.15529i	0.816286	+	0.577647i	$0.196029\pi$
$971$			36.0000i			1.15529i	−0.816286	+	0.577647i	$0.803971\pi$
$972$		0			0
$973$		−	44.0000i		−	1.41058i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−27.5000	+	47.6314i	−0.879803	+	1.52386i	−0.0282462	+	0.999601i	$0.508992\pi$
$977$	−27.5000	+	47.6314i	−0.879803	+	1.52386i	−0.851557	+	0.524262i	$0.824341\pi$
$978$		0			0
$979$	8.66025	−	5.00000i	0.276783	−	0.159801i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	25.0000	+	43.3013i	0.797376	+	1.38110i	0.921319	+	0.388807i	$0.127113\pi$
$983$	25.0000	+	43.3013i	0.797376	+	1.38110i	−0.123943	+	0.992289i	$0.539554\pi$
$984$		0			0
$985$	−4.00000	+	6.92820i	−0.127451	+	0.220751i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	10.0000i		−	0.317982i
$990$		0			0
$991$	−8.00000			−0.254128			−0.127064	−	0.991894i	$-0.540555\pi$
$991$	−8.00000			−0.254128			−0.127064	+	0.991894i	$0.540555\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−24.2487	+	14.0000i	−0.767964	+	0.443384i	−0.832148	−	0.554554i	$-0.812889\pi$
$997$	−24.2487	+	14.0000i	−0.767964	+	0.443384i	0.0641836	+	0.997938i	$0.479556\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4320.2.bv.a.3601.1		4
3.2	odd	2		1440.2.bv.a.1201.2		4
4.3	odd	2		1080.2.bf.a.901.2		4
8.3	odd	2		1080.2.bf.a.901.1		4
8.5	even	2	inner	4320.2.bv.a.3601.2		4
9.2	odd	6		1440.2.bv.a.241.1		4
9.7	even	3	inner	4320.2.bv.a.721.2		4
12.11	even	2		360.2.bf.a.301.1	yes	4
24.5	odd	2		1440.2.bv.a.1201.1		4
24.11	even	2		360.2.bf.a.301.2	yes	4
36.7	odd	6		1080.2.bf.a.181.1		4
36.11	even	6		360.2.bf.a.61.2	yes	4
72.11	even	6		360.2.bf.a.61.1	✓	4
72.29	odd	6		1440.2.bv.a.241.2		4
72.43	odd	6		1080.2.bf.a.181.2		4
72.61	even	6	inner	4320.2.bv.a.721.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
360.2.bf.a.61.1	✓	4	72.11	even	6
360.2.bf.a.61.2	yes	4	36.11	even	6
360.2.bf.a.301.1	yes	4	12.11	even	2
360.2.bf.a.301.2	yes	4	24.11	even	2
1080.2.bf.a.181.1		4	36.7	odd	6
1080.2.bf.a.181.2		4	72.43	odd	6
1080.2.bf.a.901.1		4	8.3	odd	2
1080.2.bf.a.901.2		4	4.3	odd	2
1440.2.bv.a.241.1		4	9.2	odd	6
1440.2.bv.a.241.2		4	72.29	odd	6
1440.2.bv.a.1201.1		4	24.5	odd	2
1440.2.bv.a.1201.2		4	3.2	odd	2
4320.2.bv.a.721.1		4	72.61	even	6	inner
4320.2.bv.a.721.2		4	9.7	even	3	inner
4320.2.bv.a.3601.1		4	1.1	even	1	trivial
4320.2.bv.a.3601.2		4	8.5	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4320 = 2^{5} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4320.bv (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.866025 - 0.500000i) q^{5} +(-2.00000 - 3.46410i) q^{7} +O(q^{10})\)	\(q+(-0.866025 - 0.500000i) q^{5} +(-2.00000 - 3.46410i) q^{7} +(-4.33013 + 2.50000i) q^{11} +(-5.19615 - 3.00000i) q^{13} +3.00000 q^{17} -7.00000i q^{19} +(-1.00000 + 1.73205i) q^{23} +(0.500000 + 0.866025i) q^{25} +(2.00000 - 3.46410i) q^{31} +4.00000i q^{35} -2.00000i q^{37} +(-2.50000 + 4.33013i) q^{41} +(-4.33013 + 2.50000i) q^{43} +(-1.00000 - 1.73205i) q^{47} +(-4.50000 + 7.79423i) q^{49} -8.00000i q^{53} +5.00000 q^{55} +(6.06218 + 3.50000i) q^{59} +(3.46410 - 2.00000i) q^{61} +(3.00000 + 5.19615i) q^{65} +(-2.59808 - 1.50000i) q^{67} +2.00000 q^{71} +1.00000 q^{73} +(17.3205 + 10.0000i) q^{77} +(5.00000 + 8.66025i) q^{79} +(-10.3923 + 6.00000i) q^{83} +(-2.59808 - 1.50000i) q^{85} -2.00000 q^{89} +24.0000i q^{91} +(-3.50000 + 6.06218i) q^{95} +(3.50000 + 6.06218i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{7}+O(q^{10}) \) 4 * q - 8 * q^7	\( 4 q - 8 q^{7} + 12 q^{17} - 4 q^{23} + 2 q^{25} + 8 q^{31} - 10 q^{41} - 4 q^{47} - 18 q^{49} + 20 q^{55} + 12 q^{65} + 8 q^{71} + 4 q^{73} + 20 q^{79} - 8 q^{89} - 14 q^{95} + 14 q^{97}+O(q^{100}) \) 4 * q - 8 * q^7 + 12 * q^17 - 4 * q^23 + 2 * q^25 + 8 * q^31 - 10 * q^41 - 4 * q^47 - 18 * q^49 + 20 * q^55 + 12 * q^65 + 8 * q^71 + 4 * q^73 + 20 * q^79 - 8 * q^89 - 14 * q^95 + 14 * q^97

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