LMFDB - Embedded newform 405.2.e.e.271.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [405,2,Mod(136,405)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("405.136"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(405, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 405 = 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	405.e (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,2,1,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$3.23394128186$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			271.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	405.271
Dual form			405.2.e.e.136.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(1.00000 - 1.73205i) q^{4} +(0.500000 - 0.866025i) q^{5} +(-1.00000 - 1.73205i) q^{7} +(1.50000 + 2.59808i) q^{11} +(2.00000 - 3.46410i) q^{13} +(-2.00000 - 3.46410i) q^{16} -6.00000 q^{17} -1.00000 q^{19} +(-1.00000 - 1.73205i) q^{20} +(3.00000 - 5.19615i) q^{23} +(-0.500000 - 0.866025i) q^{25} -4.00000 q^{28} +(4.50000 + 7.79423i) q^{29} +(0.500000 - 0.866025i) q^{31} -2.00000 q^{35} +8.00000 q^{37} +(-1.50000 + 2.59808i) q^{41} +(2.00000 + 3.46410i) q^{43} +6.00000 q^{44} +(-6.00000 - 10.3923i) q^{47} +(1.50000 - 2.59808i) q^{49} +(-4.00000 - 6.92820i) q^{52} +6.00000 q^{53} +3.00000 q^{55} +(-1.50000 + 2.59808i) q^{59} +(5.00000 + 8.66025i) q^{61} -8.00000 q^{64} +(-2.00000 - 3.46410i) q^{65} +(-7.00000 + 12.1244i) q^{67} +(-6.00000 + 10.3923i) q^{68} -3.00000 q^{71} +2.00000 q^{73} +(-1.00000 + 1.73205i) q^{76} +(3.00000 - 5.19615i) q^{77} +(8.00000 + 13.8564i) q^{79} -4.00000 q^{80} +(6.00000 + 10.3923i) q^{83} +(-3.00000 + 5.19615i) q^{85} +15.0000 q^{89} -8.00000 q^{91} +(-6.00000 - 10.3923i) q^{92} +(-0.500000 + 0.866025i) q^{95} +(2.00000 + 3.46410i) q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{4} + q^{5} - 2 q^{7} + 3 q^{11} + 4 q^{13} - 4 q^{16} - 12 q^{17} - 2 q^{19} - 2 q^{20} + 6 q^{23} - q^{25} - 8 q^{28} + 9 q^{29} + q^{31} - 4 q^{35} + 16 q^{37} - 3 q^{41} + 4 q^{43} + 12 q^{44}+ \cdots + 4 q^{97}+O(q^{100}) $ 2 * q + 2 * q^4 + q^5 - 2 * q^7 + 3 * q^11 + 4 * q^13 - 4 * q^16 - 12 * q^17 - 2 * q^19 - 2 * q^20 + 6 * q^23 - q^25 - 8 * q^28 + 9 * q^29 + q^31 - 4 * q^35 + 16 * q^37 - 3 * q^41 + 4 * q^43 + 12 * q^44 - 12 * q^47 + 3 * q^49 - 8 * q^52 + 12 * q^53 + 6 * q^55 - 3 * q^59 + 10 * q^61 - 16 * q^64 - 4 * q^65 - 14 * q^67 - 12 * q^68 - 6 * q^71 + 4 * q^73 - 2 * q^76 + 6 * q^77 + 16 * q^79 - 8 * q^80 + 12 * q^83 - 6 * q^85 + 30 * q^89 - 16 * q^91 - 12 * q^92 - q^95 + 4 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/405\mathbb{Z}\right)^\times$.

$n$	$82$	$326$
$\chi(n)$	$1$	$e\left(\frac{1}{3}\right)$

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		0.866025	−	0.500000i	$-0.166667\pi$
$2$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$3$		0			0
$3$		0			0
$4$	1.00000	−	1.73205i	0.500000	−	0.866025i
$5$	0.500000	−	0.866025i	0.223607	−	0.387298i
$5$	0.500000	−	0.866025i	0.223607	−	0.387298i
$6$		0			0
$7$	−1.00000	−	1.73205i	−0.377964	−	0.654654i	0.612801	−	0.790237i	$-0.290043\pi$
$7$	−1.00000	−	1.73205i	−0.377964	−	0.654654i	−0.990766	+	0.135583i	$0.956709\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1.50000	+	2.59808i	0.452267	+	0.783349i	0.998526	−	0.0542666i	$-0.0172821\pi$
$11$	1.50000	+	2.59808i	0.452267	+	0.783349i	−0.546259	+	0.837616i	$0.683949\pi$
$12$		0			0
$13$	2.00000	−	3.46410i	0.554700	−	0.960769i	−0.443227	−	0.896410i	$-0.646166\pi$
$13$	2.00000	−	3.46410i	0.554700	−	0.960769i	0.997927	−	0.0643593i	$-0.0205004\pi$
$14$		0			0
$15$		0			0
$16$	−2.00000	−	3.46410i	−0.500000	−	0.866025i
$17$	−6.00000			−1.45521			−0.727607	−	0.685994i	$-0.759367\pi$
$17$	−6.00000			−1.45521			−0.727607	+	0.685994i	$0.759367\pi$
$18$		0			0
$19$	−1.00000			−0.229416			−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000			−0.229416			−0.114708	+	0.993399i	$0.536593\pi$
$20$	−1.00000	−	1.73205i	−0.223607	−	0.387298i
$21$		0			0
$22$		0			0
$23$	3.00000	−	5.19615i	0.625543	−	1.08347i	−0.362892	−	0.931831i	$-0.618211\pi$
$23$	3.00000	−	5.19615i	0.625543	−	1.08347i	0.988436	−	0.151642i	$-0.0484560\pi$
$24$		0			0
$25$	−0.500000	−	0.866025i	−0.100000	−	0.173205i
$26$		0			0
$27$		0			0
$28$	−4.00000			−0.755929
$29$	4.50000	+	7.79423i	0.835629	+	1.44735i	0.893517	+	0.449029i	$0.148230\pi$
$29$	4.50000	+	7.79423i	0.835629	+	1.44735i	−0.0578882	+	0.998323i	$0.518437\pi$
$30$		0			0
$31$	0.500000	−	0.866025i	0.0898027	−	0.155543i	−0.817625	−	0.575751i	$-0.804710\pi$
$31$	0.500000	−	0.866025i	0.0898027	−	0.155543i	0.907428	+	0.420208i	$0.138043\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−2.00000			−0.338062
$36$		0			0
$37$	8.00000			1.31519			0.657596	−	0.753371i	$-0.271573\pi$
$37$	8.00000			1.31519			0.657596	+	0.753371i	$0.271573\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−1.50000	+	2.59808i	−0.234261	+	0.405751i	−0.959058	−	0.283211i	$-0.908600\pi$
$41$	−1.50000	+	2.59808i	−0.234261	+	0.405751i	0.724797	+	0.688963i	$0.241934\pi$
$42$		0			0
$43$	2.00000	+	3.46410i	0.304997	+	0.528271i	0.977261	−	0.212041i	$-0.0680112\pi$
$43$	2.00000	+	3.46410i	0.304997	+	0.528271i	−0.672264	+	0.740312i	$0.734678\pi$
$44$	6.00000			0.904534
$45$		0			0
$46$		0			0
$47$	−6.00000	−	10.3923i	−0.875190	−	1.51587i	−0.856560	−	0.516047i	$-0.827403\pi$
$47$	−6.00000	−	10.3923i	−0.875190	−	1.51587i	−0.0186297	−	0.999826i	$-0.505930\pi$
$48$		0			0
$49$	1.50000	−	2.59808i	0.214286	−	0.371154i
$50$		0			0
$51$		0			0
$52$	−4.00000	−	6.92820i	−0.554700	−	0.960769i
$53$	6.00000			0.824163			0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000			0.824163			0.412082	+	0.911147i	$0.364802\pi$
$54$		0			0
$55$	3.00000			0.404520
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−1.50000	+	2.59808i	−0.195283	+	0.338241i	−0.946993	−	0.321253i	$-0.895896\pi$
$59$	−1.50000	+	2.59808i	−0.195283	+	0.338241i	0.751710	+	0.659494i	$0.229229\pi$
$60$		0			0
$61$	5.00000	+	8.66025i	0.640184	+	1.10883i	0.985391	+	0.170305i	$0.0544754\pi$
$61$	5.00000	+	8.66025i	0.640184	+	1.10883i	−0.345207	+	0.938527i	$0.612191\pi$
$62$		0			0
$63$		0			0
$64$	−8.00000			−1.00000
$65$	−2.00000	−	3.46410i	−0.248069	−	0.429669i
$66$		0			0
$67$	−7.00000	+	12.1244i	−0.855186	+	1.48123i	0.0212861	+	0.999773i	$0.493224\pi$
$67$	−7.00000	+	12.1244i	−0.855186	+	1.48123i	−0.876472	+	0.481452i	$0.840109\pi$
$68$	−6.00000	+	10.3923i	−0.727607	+	1.26025i
$69$		0			0
$70$		0			0
$71$	−3.00000			−0.356034			−0.178017	−	0.984027i	$-0.556968\pi$
$71$	−3.00000			−0.356034			−0.178017	+	0.984027i	$0.556968\pi$
$72$		0			0
$73$	2.00000			0.234082			0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000			0.234082			0.117041	+	0.993127i	$0.462659\pi$
$74$		0			0
$75$		0			0
$76$	−1.00000	+	1.73205i	−0.114708	+	0.198680i
$77$	3.00000	−	5.19615i	0.341882	−	0.592157i
$78$		0			0
$79$	8.00000	+	13.8564i	0.900070	+	1.55897i	0.827401	+	0.561611i	$0.189818\pi$
$79$	8.00000	+	13.8564i	0.900070	+	1.55897i	0.0726692	+	0.997356i	$0.476848\pi$
$80$	−4.00000			−0.447214
$81$		0			0
$82$		0			0
$83$	6.00000	+	10.3923i	0.658586	+	1.14070i	0.980982	+	0.194099i	$0.0621783\pi$
$83$	6.00000	+	10.3923i	0.658586	+	1.14070i	−0.322396	+	0.946605i	$0.604488\pi$
$84$		0			0
$85$	−3.00000	+	5.19615i	−0.325396	+	0.563602i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	15.0000			1.59000			0.794998	−	0.606612i	$-0.207472\pi$
$89$	15.0000			1.59000			0.794998	+	0.606612i	$0.207472\pi$
$90$		0			0
$91$	−8.00000			−0.838628
$92$	−6.00000	−	10.3923i	−0.625543	−	1.08347i
$93$		0			0
$94$		0			0
$95$	−0.500000	+	0.866025i	−0.0512989	+	0.0888523i
$96$		0			0
$97$	2.00000	+	3.46410i	0.203069	+	0.351726i	0.949516	−	0.313719i	$-0.101575\pi$
$97$	2.00000	+	3.46410i	0.203069	+	0.351726i	−0.746447	+	0.665445i	$0.768242\pi$
$98$		0			0
$99$		0			0
$100$	−2.00000			−0.200000
$101$	−4.50000	−	7.79423i	−0.447767	−	0.775555i	0.550474	−	0.834853i	$-0.314447\pi$
$101$	−4.50000	−	7.79423i	−0.447767	−	0.775555i	−0.998240	+	0.0592978i	$0.981114\pi$
$102$		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
$107$	−18.0000			−1.74013			−0.870063	−	0.492941i	$-0.835922\pi$
$107$	−18.0000			−1.74013			−0.870063	+	0.492941i	$0.835922\pi$
$108$		0			0
$109$	−1.00000			−0.0957826			−0.0478913	−	0.998853i	$-0.515250\pi$
$109$	−1.00000			−0.0957826			−0.0478913	+	0.998853i	$0.515250\pi$
$110$		0			0
$111$		0			0
$112$	−4.00000	+	6.92820i	−0.377964	+	0.654654i
$113$	−3.00000	+	5.19615i	−0.282216	+	0.488813i	−0.971930	−	0.235269i	$-0.924403\pi$
$113$	−3.00000	+	5.19615i	−0.282216	+	0.488813i	0.689714	+	0.724082i	$0.257736\pi$
$114$		0			0
$115$	−3.00000	−	5.19615i	−0.279751	−	0.484544i
$116$	18.0000			1.67126
$117$		0			0
$118$		0			0
$119$	6.00000	+	10.3923i	0.550019	+	0.952661i
$120$		0			0
$121$	1.00000	−	1.73205i	0.0909091	−	0.157459i
$122$		0			0
$123$		0			0
$124$	−1.00000	−	1.73205i	−0.0898027	−	0.155543i
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	2.00000			0.177471			0.0887357	−	0.996055i	$-0.471717\pi$
$127$	2.00000			0.177471			0.0887357	+	0.996055i	$0.471717\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	4.50000	−	7.79423i	0.393167	−	0.680985i	−0.599699	−	0.800226i	$-0.704713\pi$
$131$	4.50000	−	7.79423i	0.393167	−	0.680985i	0.992865	+	0.119241i	$0.0380462\pi$
$132$		0			0
$133$	1.00000	+	1.73205i	0.0867110	+	0.150188i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−6.00000	−	10.3923i	−0.512615	−	0.887875i	−0.999893	−	0.0146279i	$-0.995344\pi$
$137$	−6.00000	−	10.3923i	−0.512615	−	0.887875i	0.487278	−	0.873247i	$-0.337990\pi$
$138$		0			0
$139$	3.50000	−	6.06218i	0.296866	−	0.514187i	−0.678551	−	0.734553i	$-0.737392\pi$
$139$	3.50000	−	6.06218i	0.296866	−	0.514187i	0.975417	+	0.220366i	$0.0707252\pi$
$140$	−2.00000	+	3.46410i	−0.169031	+	0.292770i
$141$		0			0
$142$		0			0
$143$	12.0000			1.00349
$144$		0			0
$145$	9.00000			0.747409
$146$		0			0
$147$		0			0
$148$	8.00000	−	13.8564i	0.657596	−	1.13899i
$149$	−3.00000	+	5.19615i	−0.245770	+	0.425685i	−0.962348	−	0.271821i	$-0.912374\pi$
$149$	−3.00000	+	5.19615i	−0.245770	+	0.425685i	0.716578	+	0.697507i	$0.245707\pi$
$150$		0			0
$151$	0.500000	+	0.866025i	0.0406894	+	0.0704761i	0.885653	−	0.464348i	$-0.153711\pi$
$151$	0.500000	+	0.866025i	0.0406894	+	0.0704761i	−0.844963	+	0.534824i	$0.820378\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−0.500000	−	0.866025i	−0.0401610	−	0.0695608i
$156$		0			0
$157$	8.00000	−	13.8564i	0.638470	−	1.10586i	−0.347299	−	0.937754i	$-0.612901\pi$
$157$	8.00000	−	13.8564i	0.638470	−	1.10586i	0.985769	−	0.168107i	$-0.0537655\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−12.0000			−0.945732
$162$		0			0
$163$	2.00000			0.156652			0.0783260	−	0.996928i	$-0.475042\pi$
$163$	2.00000			0.156652			0.0783260	+	0.996928i	$0.475042\pi$
$164$	3.00000	+	5.19615i	0.234261	+	0.405751i
$165$		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
$169$	−1.50000	−	2.59808i	−0.115385	−	0.199852i
$170$		0			0
$171$		0			0
$172$	8.00000			0.609994
$173$	3.00000	+	5.19615i	0.228086	+	0.395056i	0.957241	−	0.289292i	$-0.0934200\pi$
$173$	3.00000	+	5.19615i	0.228086	+	0.395056i	−0.729155	+	0.684349i	$0.760087\pi$
$174$		0			0
$175$	−1.00000	+	1.73205i	−0.0755929	+	0.130931i
$176$	6.00000	−	10.3923i	0.452267	−	0.783349i
$177$		0			0
$178$		0			0
$179$	−9.00000			−0.672692			−0.336346	−	0.941739i	$-0.609191\pi$
$179$	−9.00000			−0.672692			−0.336346	+	0.941739i	$0.609191\pi$
$180$		0			0
$181$	11.0000			0.817624			0.408812	−	0.912619i	$-0.365943\pi$
$181$	11.0000			0.817624			0.408812	+	0.912619i	$0.365943\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	4.00000	−	6.92820i	0.294086	−	0.509372i
$186$		0			0
$187$	−9.00000	−	15.5885i	−0.658145	−	1.13994i
$188$	−24.0000			−1.75038
$189$		0			0
$190$		0			0
$191$	7.50000	+	12.9904i	0.542681	+	0.939951i	0.998749	+	0.0500060i	$0.0159241\pi$
$191$	7.50000	+	12.9904i	0.542681	+	0.939951i	−0.456068	+	0.889945i	$0.650743\pi$
$192$		0			0
$193$	2.00000	−	3.46410i	0.143963	−	0.249351i	−0.785022	−	0.619467i	$-0.787349\pi$
$193$	2.00000	−	3.46410i	0.143963	−	0.249351i	0.928986	+	0.370116i	$0.120682\pi$
$194$		0			0
$195$		0			0
$196$	−3.00000	−	5.19615i	−0.214286	−	0.371154i
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$		0			0
$199$	−16.0000			−1.13421			−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000			−1.13421			−0.567105	+	0.823646i	$0.691937\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	9.00000	−	15.5885i	0.631676	−	1.09410i
$204$		0			0
$205$	1.50000	+	2.59808i	0.104765	+	0.181458i
$206$		0			0
$207$		0			0
$208$	−16.0000			−1.10940
$209$	−1.50000	−	2.59808i	−0.103757	−	0.179713i
$210$		0			0
$211$	6.50000	−	11.2583i	0.447478	−	0.775055i	−0.550743	−	0.834675i	$-0.685655\pi$
$211$	6.50000	−	11.2583i	0.447478	−	0.775055i	0.998221	+	0.0596196i	$0.0189888\pi$
$212$	6.00000	−	10.3923i	0.412082	−	0.713746i
$213$		0			0
$214$		0			0
$215$	4.00000			0.272798
$216$		0			0
$217$	−2.00000			−0.135769
$218$		0			0
$219$		0			0
$220$	3.00000	−	5.19615i	0.202260	−	0.350325i
$221$	−12.0000	+	20.7846i	−0.807207	+	1.39812i
$222$		0			0
$223$	−10.0000	−	17.3205i	−0.669650	−	1.15987i	−0.978002	−	0.208595i	$-0.933111\pi$
$223$	−10.0000	−	17.3205i	−0.669650	−	1.15987i	0.308353	−	0.951272i	$-0.400222\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	9.00000	+	15.5885i	0.597351	+	1.03464i	0.993210	+	0.116331i	$0.0371134\pi$
$227$	9.00000	+	15.5885i	0.597351	+	1.03464i	−0.395860	+	0.918311i	$0.629553\pi$
$228$		0			0
$229$	5.00000	−	8.66025i	0.330409	−	0.572286i	−0.652183	−	0.758062i	$-0.726147\pi$
$229$	5.00000	−	8.66025i	0.330409	−	0.572286i	0.982592	+	0.185776i	$0.0594799\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	12.0000			0.786146			0.393073	−	0.919507i	$-0.371412\pi$
$233$	12.0000			0.786146			0.393073	+	0.919507i	$0.371412\pi$
$234$		0			0
$235$	−12.0000			−0.782794
$236$	3.00000	+	5.19615i	0.195283	+	0.338241i
$237$		0			0
$238$		0			0
$239$	12.0000	−	20.7846i	0.776215	−	1.34444i	−0.157893	−	0.987456i	$-0.550470\pi$
$239$	12.0000	−	20.7846i	0.776215	−	1.34444i	0.934109	−	0.356988i	$-0.116196\pi$
$240$		0			0
$241$	−8.50000	−	14.7224i	−0.547533	−	0.948355i	−0.998443	−	0.0557856i	$-0.982234\pi$
$241$	−8.50000	−	14.7224i	−0.547533	−	0.948355i	0.450910	−	0.892570i	$-0.351100\pi$
$242$		0			0
$243$		0			0
$244$	20.0000			1.28037
$245$	−1.50000	−	2.59808i	−0.0958315	−	0.165985i
$246$		0			0
$247$	−2.00000	+	3.46410i	−0.127257	+	0.220416i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−12.0000			−0.757433			−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000			−0.757433			−0.378717	+	0.925513i	$0.623635\pi$
$252$		0			0
$253$	18.0000			1.13165
$254$		0			0
$255$		0			0
$256$	−8.00000	+	13.8564i	−0.500000	+	0.866025i
$257$	−3.00000	+	5.19615i	−0.187135	+	0.324127i	−0.944294	−	0.329104i	$-0.893253\pi$
$257$	−3.00000	+	5.19615i	−0.187135	+	0.324127i	0.757159	+	0.653231i	$0.226587\pi$
$258$		0			0
$259$	−8.00000	−	13.8564i	−0.497096	−	0.860995i
$260$	−8.00000			−0.496139
$261$		0			0
$262$		0			0
$263$	−9.00000	−	15.5885i	−0.554964	−	0.961225i	−0.997906	−	0.0646755i	$-0.979399\pi$
$263$	−9.00000	−	15.5885i	−0.554964	−	0.961225i	0.442943	−	0.896550i	$-0.353935\pi$
$264$		0			0
$265$	3.00000	−	5.19615i	0.184289	−	0.319197i
$266$		0			0
$267$		0			0
$268$	14.0000	+	24.2487i	0.855186	+	1.48123i
$269$	−3.00000			−0.182913			−0.0914566	−	0.995809i	$-0.529152\pi$
$269$	−3.00000			−0.182913			−0.0914566	+	0.995809i	$0.529152\pi$
$270$		0			0
$271$	−16.0000			−0.971931			−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000			−0.971931			−0.485965	+	0.873978i	$0.661532\pi$
$272$	12.0000	+	20.7846i	0.727607	+	1.26025i
$273$		0			0
$274$		0			0
$275$	1.50000	−	2.59808i	0.0904534	−	0.156670i
$276$		0			0
$277$	8.00000	+	13.8564i	0.480673	+	0.832551i	0.999754	−	0.0221745i	$-0.00705893\pi$
$277$	8.00000	+	13.8564i	0.480673	+	0.832551i	−0.519081	+	0.854725i	$0.673726\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−3.00000	−	5.19615i	−0.178965	−	0.309976i	0.762561	−	0.646916i	$-0.223942\pi$
$281$	−3.00000	−	5.19615i	−0.178965	−	0.309976i	−0.941526	+	0.336939i	$0.890608\pi$
$282$		0			0
$283$	5.00000	−	8.66025i	0.297219	−	0.514799i	−0.678280	−	0.734804i	$-0.737274\pi$
$283$	5.00000	−	8.66025i	0.297219	−	0.514799i	0.975499	+	0.220005i	$0.0706075\pi$
$284$	−3.00000	+	5.19615i	−0.178017	+	0.308335i
$285$		0			0
$286$		0			0
$287$	6.00000			0.354169
$288$		0			0
$289$	19.0000			1.11765
$290$		0			0
$291$		0			0
$292$	2.00000	−	3.46410i	0.117041	−	0.202721i
$293$	6.00000	−	10.3923i	0.350524	−	0.607125i	−0.635818	−	0.771839i	$-0.719337\pi$
$293$	6.00000	−	10.3923i	0.350524	−	0.607125i	0.986341	+	0.164714i	$0.0526703\pi$
$294$		0			0
$295$	1.50000	+	2.59808i	0.0873334	+	0.151266i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−12.0000	−	20.7846i	−0.693978	−	1.20201i
$300$		0			0
$301$	4.00000	−	6.92820i	0.230556	−	0.399335i
$302$		0			0
$303$		0			0
$304$	2.00000	+	3.46410i	0.114708	+	0.198680i
$305$	10.0000			0.572598
$306$		0			0
$307$	−16.0000			−0.913168			−0.456584	−	0.889680i	$-0.650927\pi$
$307$	−16.0000			−0.913168			−0.456584	+	0.889680i	$0.650927\pi$
$308$	−6.00000	−	10.3923i	−0.341882	−	0.592157i
$309$		0			0
$310$		0			0
$311$	−13.5000	+	23.3827i	−0.765515	+	1.32591i	0.174459	+	0.984664i	$0.444182\pi$
$311$	−13.5000	+	23.3827i	−0.765515	+	1.32591i	−0.939974	+	0.341246i	$0.889151\pi$
$312$		0			0
$313$	−4.00000	−	6.92820i	−0.226093	−	0.391605i	0.730554	−	0.682855i	$-0.239262\pi$
$313$	−4.00000	−	6.92820i	−0.226093	−	0.391605i	−0.956647	+	0.291250i	$0.905929\pi$
$314$		0			0
$315$		0			0
$316$	32.0000			1.80014
$317$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$317$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$318$		0			0
$319$	−13.5000	+	23.3827i	−0.755855	+	1.30918i
$320$	−4.00000	+	6.92820i	−0.223607	+	0.387298i
$321$		0			0
$322$		0			0
$323$	6.00000			0.333849
$324$		0			0
$325$	−4.00000			−0.221880
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−12.0000	+	20.7846i	−0.661581	+	1.14589i
$330$		0			0
$331$	12.5000	+	21.6506i	0.687062	+	1.19003i	0.972784	+	0.231714i	$0.0744333\pi$
$331$	12.5000	+	21.6506i	0.687062	+	1.19003i	−0.285722	+	0.958313i	$0.592233\pi$
$332$	24.0000			1.31717
$333$		0			0
$334$		0			0
$335$	7.00000	+	12.1244i	0.382451	+	0.662424i
$336$		0			0
$337$	2.00000	−	3.46410i	0.108947	−	0.188702i	−0.806397	−	0.591375i	$-0.798585\pi$
$337$	2.00000	−	3.46410i	0.108947	−	0.188702i	0.915344	+	0.402673i	$0.131919\pi$
$338$		0			0
$339$		0			0
$340$	6.00000	+	10.3923i	0.325396	+	0.563602i
$341$	3.00000			0.162459
$342$		0			0
$343$	−20.0000			−1.07990
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−9.00000	+	15.5885i	−0.483145	+	0.836832i	−0.999813	−	0.0193540i	$-0.993839\pi$
$347$	−9.00000	+	15.5885i	−0.483145	+	0.836832i	0.516667	+	0.856186i	$0.327172\pi$
$348$		0			0
$349$	−2.50000	−	4.33013i	−0.133822	−	0.231786i	0.791325	−	0.611396i	$-0.209392\pi$
$349$	−2.50000	−	4.33013i	−0.133822	−	0.231786i	−0.925147	+	0.379610i	$0.876058\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−6.00000	−	10.3923i	−0.319348	−	0.553127i	0.661004	−	0.750382i	$-0.270130\pi$
$353$	−6.00000	−	10.3923i	−0.319348	−	0.553127i	−0.980352	+	0.197256i	$0.936797\pi$
$354$		0			0
$355$	−1.50000	+	2.59808i	−0.0796117	+	0.137892i
$356$	15.0000	−	25.9808i	0.794998	−	1.37698i
$357$		0			0
$358$		0			0
$359$	−15.0000			−0.791670			−0.395835	−	0.918322i	$-0.629545\pi$
$359$	−15.0000			−0.791670			−0.395835	+	0.918322i	$0.629545\pi$
$360$		0			0
$361$	−18.0000			−0.947368
$362$		0			0
$363$		0			0
$364$	−8.00000	+	13.8564i	−0.419314	+	0.726273i
$365$	1.00000	−	1.73205i	0.0523424	−	0.0906597i
$366$		0			0
$367$	5.00000	+	8.66025i	0.260998	+	0.452062i	0.966507	−	0.256639i	$-0.0826151\pi$
$367$	5.00000	+	8.66025i	0.260998	+	0.452062i	−0.705509	+	0.708700i	$0.749282\pi$
$368$	−24.0000			−1.25109
$369$		0			0
$370$		0			0
$371$	−6.00000	−	10.3923i	−0.311504	−	0.539542i
$372$		0			0
$373$	−19.0000	+	32.9090i	−0.983783	+	1.70396i	−0.336557	+	0.941663i	$0.609263\pi$
$373$	−19.0000	+	32.9090i	−0.983783	+	1.70396i	−0.647225	+	0.762299i	$0.724071\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	36.0000			1.85409
$378$		0			0
$379$	−16.0000			−0.821865			−0.410932	−	0.911666i	$-0.634797\pi$
$379$	−16.0000			−0.821865			−0.410932	+	0.911666i	$0.634797\pi$
$380$	1.00000	+	1.73205i	0.0512989	+	0.0888523i
$381$		0			0
$382$		0			0
$383$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$383$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$384$		0			0
$385$	−3.00000	−	5.19615i	−0.152894	−	0.264820i
$386$		0			0
$387$		0			0
$388$	8.00000			0.406138
$389$	−15.0000	−	25.9808i	−0.760530	−	1.31728i	−0.942578	−	0.333987i	$-0.891606\pi$
$389$	−15.0000	−	25.9808i	−0.760530	−	1.31728i	0.182047	−	0.983290i	$-0.441728\pi$
$390$		0			0
$391$	−18.0000	+	31.1769i	−0.910299	+	1.57668i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	16.0000			0.805047
$396$		0			0
$397$	−22.0000			−1.10415			−0.552074	−	0.833795i	$-0.686163\pi$
$397$	−22.0000			−1.10415			−0.552074	+	0.833795i	$0.686163\pi$
$398$		0			0
$399$		0			0
$400$	−2.00000	+	3.46410i	−0.100000	+	0.173205i
$401$	3.00000	−	5.19615i	0.149813	−	0.259483i	−0.781345	−	0.624099i	$-0.785466\pi$
$401$	3.00000	−	5.19615i	0.149813	−	0.259483i	0.931158	+	0.364615i	$0.118800\pi$
$402$		0			0
$403$	−2.00000	−	3.46410i	−0.0996271	−	0.172559i
$404$	−18.0000			−0.895533
$405$		0			0
$406$		0			0
$407$	12.0000	+	20.7846i	0.594818	+	1.03025i
$408$		0			0
$409$	−7.00000	+	12.1244i	−0.346128	+	0.599511i	−0.985558	−	0.169338i	$-0.945837\pi$
$409$	−7.00000	+	12.1244i	−0.346128	+	0.599511i	0.639430	+	0.768849i	$0.279170\pi$
$410$		0			0
$411$		0			0
$412$	14.0000	+	24.2487i	0.689730	+	1.19465i
$413$	6.00000			0.295241
$414$		0			0
$415$	12.0000			0.589057
$416$		0			0
$417$		0			0
$418$		0			0
$419$	6.00000	−	10.3923i	0.293119	−	0.507697i	−0.681426	−	0.731887i	$-0.738640\pi$
$419$	6.00000	−	10.3923i	0.293119	−	0.507697i	0.974546	+	0.224189i	$0.0719734\pi$
$420$		0			0
$421$	−2.50000	−	4.33013i	−0.121843	−	0.211037i	0.798652	−	0.601793i	$-0.205547\pi$
$421$	−2.50000	−	4.33013i	−0.121843	−	0.211037i	−0.920494	+	0.390756i	$0.872214\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	3.00000	+	5.19615i	0.145521	+	0.252050i
$426$		0			0
$427$	10.0000	−	17.3205i	0.483934	−	0.838198i
$428$	−18.0000	+	31.1769i	−0.870063	+	1.50699i
$429$		0			0
$430$		0			0
$431$	15.0000			0.722525			0.361262	−	0.932464i	$-0.382346\pi$
$431$	15.0000			0.722525			0.361262	+	0.932464i	$0.382346\pi$
$432$		0			0
$433$	26.0000			1.24948			0.624740	−	0.780833i	$-0.285205\pi$
$433$	26.0000			1.24948			0.624740	+	0.780833i	$0.285205\pi$
$434$		0			0
$435$		0			0
$436$	−1.00000	+	1.73205i	−0.0478913	+	0.0829502i
$437$	−3.00000	+	5.19615i	−0.143509	+	0.248566i
$438$		0			0
$439$	0.500000	+	0.866025i	0.0238637	+	0.0413331i	0.877711	−	0.479191i	$-0.159070\pi$
$439$	0.500000	+	0.866025i	0.0238637	+	0.0413331i	−0.853847	+	0.520524i	$0.825737\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	6.00000	+	10.3923i	0.285069	+	0.493753i	0.972626	−	0.232377i	$-0.0746503\pi$
$443$	6.00000	+	10.3923i	0.285069	+	0.493753i	−0.687557	+	0.726130i	$0.741317\pi$
$444$		0			0
$445$	7.50000	−	12.9904i	0.355534	−	0.615803i
$446$		0			0
$447$		0			0
$448$	8.00000	+	13.8564i	0.377964	+	0.654654i
$449$	−15.0000			−0.707894			−0.353947	−	0.935266i	$-0.615161\pi$
$449$	−15.0000			−0.707894			−0.353947	+	0.935266i	$0.615161\pi$
$450$		0			0
$451$	−9.00000			−0.423793
$452$	6.00000	+	10.3923i	0.282216	+	0.488813i
$453$		0			0
$454$		0			0
$455$	−4.00000	+	6.92820i	−0.187523	+	0.324799i
$456$		0			0
$457$	14.0000	+	24.2487i	0.654892	+	1.13431i	0.981921	+	0.189292i	$0.0606194\pi$
$457$	14.0000	+	24.2487i	0.654892	+	1.13431i	−0.327028	+	0.945015i	$0.606047\pi$
$458$		0			0
$459$		0			0
$460$	−12.0000			−0.559503
$461$	−10.5000	−	18.1865i	−0.489034	−	0.847031i	0.510887	−	0.859648i	$-0.329317\pi$
$461$	−10.5000	−	18.1865i	−0.489034	−	0.847031i	−0.999920	+	0.0126168i	$0.995984\pi$
$462$		0			0
$463$	−13.0000	+	22.5167i	−0.604161	+	1.04644i	0.388022	+	0.921650i	$0.373158\pi$
$463$	−13.0000	+	22.5167i	−0.604161	+	1.04644i	−0.992183	+	0.124788i	$0.960175\pi$
$464$	18.0000	−	31.1769i	0.835629	−	1.44735i
$465$		0			0
$466$		0			0
$467$	−6.00000			−0.277647			−0.138823	−	0.990317i	$-0.544332\pi$
$467$	−6.00000			−0.277647			−0.138823	+	0.990317i	$0.544332\pi$
$468$		0			0
$469$	28.0000			1.29292
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−6.00000	+	10.3923i	−0.275880	+	0.477839i
$474$		0			0
$475$	0.500000	+	0.866025i	0.0229416	+	0.0397360i
$476$	24.0000			1.10004
$477$		0			0
$478$		0			0
$479$	−16.5000	−	28.5788i	−0.753904	−	1.30580i	−0.945917	−	0.324408i	$-0.894835\pi$
$479$	−16.5000	−	28.5788i	−0.753904	−	1.30580i	0.192013	−	0.981392i	$-0.438498\pi$
$480$		0			0
$481$	16.0000	−	27.7128i	0.729537	−	1.26360i
$482$		0			0
$483$		0			0
$484$	−2.00000	−	3.46410i	−0.0909091	−	0.157459i
$485$	4.00000			0.181631
$486$		0			0
$487$	−16.0000			−0.725029			−0.362515	−	0.931978i	$-0.618082\pi$
$487$	−16.0000			−0.725029			−0.362515	+	0.931978i	$0.618082\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	4.50000	−	7.79423i	0.203082	−	0.351749i	−0.746438	−	0.665455i	$-0.768237\pi$
$491$	4.50000	−	7.79423i	0.203082	−	0.351749i	0.949520	+	0.313707i	$0.101571\pi$
$492$		0			0
$493$	−27.0000	−	46.7654i	−1.21602	−	2.10621i
$494$		0			0
$495$		0			0
$496$	−4.00000			−0.179605
$497$	3.00000	+	5.19615i	0.134568	+	0.233079i
$498$		0			0
$499$	6.50000	−	11.2583i	0.290980	−	0.503992i	−0.683062	−	0.730361i	$-0.739352\pi$
$499$	6.50000	−	11.2583i	0.290980	−	0.503992i	0.974042	+	0.226369i	$0.0726854\pi$
$500$	−1.00000	+	1.73205i	−0.0447214	+	0.0774597i
$501$		0			0
$502$		0			0
$503$	−12.0000			−0.535054			−0.267527	−	0.963550i	$-0.586206\pi$
$503$	−12.0000			−0.535054			−0.267527	+	0.963550i	$0.586206\pi$
$504$		0			0
$505$	−9.00000			−0.400495
$506$		0			0
$507$		0			0
$508$	2.00000	−	3.46410i	0.0887357	−	0.153695i
$509$	−15.0000	+	25.9808i	−0.664863	+	1.15158i	0.314459	+	0.949271i	$0.398177\pi$
$509$	−15.0000	+	25.9808i	−0.664863	+	1.15158i	−0.979322	+	0.202306i	$0.935156\pi$
$510$		0			0
$511$	−2.00000	−	3.46410i	−0.0884748	−	0.153243i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	7.00000	+	12.1244i	0.308457	+	0.534263i
$516$		0			0
$517$	18.0000	−	31.1769i	0.791639	−	1.37116i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−30.0000			−1.31432			−0.657162	−	0.753749i	$-0.728243\pi$
$521$	−30.0000			−1.31432			−0.657162	+	0.753749i	$0.728243\pi$
$522$		0			0
$523$	20.0000			0.874539			0.437269	−	0.899331i	$-0.355946\pi$
$523$	20.0000			0.874539			0.437269	+	0.899331i	$0.355946\pi$
$524$	−9.00000	−	15.5885i	−0.393167	−	0.680985i
$525$		0			0
$526$		0			0
$527$	−3.00000	+	5.19615i	−0.130682	+	0.226348i
$528$		0			0
$529$	−6.50000	−	11.2583i	−0.282609	−	0.489493i
$530$		0			0
$531$		0			0
$532$	4.00000			0.173422
$533$	6.00000	+	10.3923i	0.259889	+	0.450141i
$534$		0			0
$535$	−9.00000	+	15.5885i	−0.389104	+	0.673948i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	9.00000			0.387657
$540$		0			0
$541$	23.0000			0.988847			0.494424	−	0.869221i	$-0.335379\pi$
$541$	23.0000			0.988847			0.494424	+	0.869221i	$0.335379\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−0.500000	+	0.866025i	−0.0214176	+	0.0370965i
$546$		0			0
$547$	−4.00000	−	6.92820i	−0.171028	−	0.296229i	0.767752	−	0.640747i	$-0.221375\pi$
$547$	−4.00000	−	6.92820i	−0.171028	−	0.296229i	−0.938779	+	0.344519i	$0.888042\pi$
$548$	−24.0000			−1.02523
$549$		0			0
$550$		0			0
$551$	−4.50000	−	7.79423i	−0.191706	−	0.332045i
$552$		0			0
$553$	16.0000	−	27.7128i	0.680389	−	1.17847i
$554$		0			0
$555$		0			0
$556$	−7.00000	−	12.1244i	−0.296866	−	0.514187i
$557$	24.0000			1.01691			0.508456	−	0.861088i	$-0.330216\pi$
$557$	24.0000			1.01691			0.508456	+	0.861088i	$0.330216\pi$
$558$		0			0
$559$	16.0000			0.676728
$560$	4.00000	+	6.92820i	0.169031	+	0.292770i
$561$		0			0
$562$		0			0
$563$	12.0000	−	20.7846i	0.505740	−	0.875967i	−0.494238	−	0.869326i	$-0.664553\pi$
$563$	12.0000	−	20.7846i	0.505740	−	0.875967i	0.999978	−	0.00664037i	$-0.00211371\pi$
$564$		0			0
$565$	3.00000	+	5.19615i	0.126211	+	0.218604i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	13.5000	+	23.3827i	0.565949	+	0.980253i	0.996961	+	0.0779066i	$0.0248236\pi$
$569$	13.5000	+	23.3827i	0.565949	+	0.980253i	−0.431011	+	0.902347i	$0.641843\pi$
$570$		0			0
$571$	9.50000	−	16.4545i	0.397563	−	0.688599i	−0.595862	−	0.803087i	$-0.703189\pi$
$571$	9.50000	−	16.4545i	0.397563	−	0.688599i	0.993425	+	0.114488i	$0.0365228\pi$
$572$	12.0000	−	20.7846i	0.501745	−	0.869048i
$573$		0			0
$574$		0			0
$575$	−6.00000			−0.250217
$576$		0			0
$577$	−34.0000			−1.41544			−0.707719	−	0.706494i	$-0.750276\pi$
$577$	−34.0000			−1.41544			−0.707719	+	0.706494i	$0.750276\pi$
$578$		0			0
$579$		0			0
$580$	9.00000	−	15.5885i	0.373705	−	0.647275i
$581$	12.0000	−	20.7846i	0.497844	−	0.862291i
$582$		0			0
$583$	9.00000	+	15.5885i	0.372742	+	0.645608i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	24.0000	+	41.5692i	0.990586	+	1.71575i	0.613845	+	0.789427i	$0.289622\pi$
$587$	24.0000	+	41.5692i	0.990586	+	1.71575i	0.376741	+	0.926319i	$0.377045\pi$
$588$		0			0
$589$	−0.500000	+	0.866025i	−0.0206021	+	0.0356840i
$590$		0			0
$591$		0			0
$592$	−16.0000	−	27.7128i	−0.657596	−	1.13899i
$593$	18.0000			0.739171			0.369586	−	0.929197i	$-0.379500\pi$
$593$	18.0000			0.739171			0.369586	+	0.929197i	$0.379500\pi$
$594$		0			0
$595$	12.0000			0.491952
$596$	6.00000	+	10.3923i	0.245770	+	0.425685i
$597$		0			0
$598$		0			0
$599$	10.5000	−	18.1865i	0.429018	−	0.743082i	−0.567768	−	0.823189i	$-0.692193\pi$
$599$	10.5000	−	18.1865i	0.429018	−	0.743082i	0.996786	+	0.0801071i	$0.0255262\pi$
$600$		0			0
$601$	15.5000	+	26.8468i	0.632258	+	1.09510i	0.987089	+	0.160173i	$0.0512051\pi$
$601$	15.5000	+	26.8468i	0.632258	+	1.09510i	−0.354831	+	0.934931i	$0.615462\pi$
$602$		0			0
$603$		0			0
$604$	2.00000			0.0813788
$605$	−1.00000	−	1.73205i	−0.0406558	−	0.0704179i
$606$		0			0
$607$	−10.0000	+	17.3205i	−0.405887	+	0.703018i	−0.994424	−	0.105453i	$-0.966371\pi$
$607$	−10.0000	+	17.3205i	−0.405887	+	0.703018i	0.588537	+	0.808470i	$0.299704\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−48.0000			−1.94187
$612$		0			0
$613$	−16.0000			−0.646234			−0.323117	−	0.946359i	$-0.604731\pi$
$613$	−16.0000			−0.646234			−0.323117	+	0.946359i	$0.604731\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	12.0000	−	20.7846i	0.483102	−	0.836757i	−0.516710	−	0.856161i	$-0.672843\pi$
$617$	12.0000	−	20.7846i	0.483102	−	0.836757i	0.999812	+	0.0194037i	$0.00617676\pi$
$618$		0			0
$619$	−22.0000	−	38.1051i	−0.884255	−	1.53157i	−0.846566	−	0.532284i	$-0.821334\pi$
$619$	−22.0000	−	38.1051i	−0.884255	−	1.53157i	−0.0376891	−	0.999290i	$-0.512000\pi$
$620$	−2.00000			−0.0803219
$621$		0			0
$622$		0			0
$623$	−15.0000	−	25.9808i	−0.600962	−	1.04090i
$624$		0			0
$625$	−0.500000	+	0.866025i	−0.0200000	+	0.0346410i
$626$		0			0
$627$		0			0
$628$	−16.0000	−	27.7128i	−0.638470	−	1.10586i
$629$	−48.0000			−1.91389
$630$		0			0
$631$	−25.0000			−0.995234			−0.497617	−	0.867397i	$-0.665792\pi$
$631$	−25.0000			−0.995234			−0.497617	+	0.867397i	$0.665792\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	1.00000	−	1.73205i	0.0396838	−	0.0687343i
$636$		0			0
$637$	−6.00000	−	10.3923i	−0.237729	−	0.411758i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	16.5000	+	28.5788i	0.651711	+	1.12880i	0.982708	+	0.185164i	$0.0592817\pi$
$641$	16.5000	+	28.5788i	0.651711	+	1.12880i	−0.330997	+	0.943632i	$0.607385\pi$
$642$		0			0
$643$	2.00000	−	3.46410i	0.0788723	−	0.136611i	−0.823891	−	0.566748i	$-0.808201\pi$
$643$	2.00000	−	3.46410i	0.0788723	−	0.136611i	0.902764	+	0.430137i	$0.141535\pi$
$644$	−12.0000	+	20.7846i	−0.472866	+	0.819028i
$645$		0			0
$646$		0			0
$647$	6.00000			0.235884			0.117942	−	0.993020i	$-0.462370\pi$
$647$	6.00000			0.235884			0.117942	+	0.993020i	$0.462370\pi$
$648$		0			0
$649$	−9.00000			−0.353281
$650$		0			0
$651$		0			0
$652$	2.00000	−	3.46410i	0.0783260	−	0.135665i
$653$	12.0000	−	20.7846i	0.469596	−	0.813365i	−0.529799	−	0.848123i	$-0.677733\pi$
$653$	12.0000	−	20.7846i	0.469596	−	0.813365i	0.999396	+	0.0347583i	$0.0110661\pi$
$654$		0			0
$655$	−4.50000	−	7.79423i	−0.175830	−	0.304546i
$656$	12.0000			0.468521
$657$		0			0
$658$		0			0
$659$	12.0000	+	20.7846i	0.467454	+	0.809653i	0.999309	−	0.0371821i	$-0.0118382\pi$
$659$	12.0000	+	20.7846i	0.467454	+	0.809653i	−0.531855	+	0.846836i	$0.678505\pi$
$660$		0			0
$661$	−2.50000	+	4.33013i	−0.0972387	+	0.168422i	−0.910541	−	0.413419i	$-0.864334\pi$
$661$	−2.50000	+	4.33013i	−0.0972387	+	0.168422i	0.813302	+	0.581842i	$0.197668\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	2.00000			0.0775567
$666$		0			0
$667$	54.0000			2.09089
$668$	6.00000	+	10.3923i	0.232147	+	0.402090i
$669$		0			0
$670$		0			0
$671$	−15.0000	+	25.9808i	−0.579069	+	1.00298i
$672$		0			0
$673$	17.0000	+	29.4449i	0.655302	+	1.13502i	0.981818	+	0.189824i	$0.0607919\pi$
$673$	17.0000	+	29.4449i	0.655302	+	1.13502i	−0.326516	+	0.945192i	$0.605875\pi$
$674$		0			0
$675$		0			0
$676$	−6.00000			−0.230769
$677$	−15.0000	−	25.9808i	−0.576497	−	0.998522i	−0.995877	−	0.0907112i	$-0.971086\pi$
$677$	−15.0000	−	25.9808i	−0.576497	−	0.998522i	0.419380	−	0.907811i	$-0.362247\pi$
$678$		0			0
$679$	4.00000	−	6.92820i	0.153506	−	0.265880i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−12.0000			−0.459167			−0.229584	−	0.973289i	$-0.573736\pi$
$683$	−12.0000			−0.459167			−0.229584	+	0.973289i	$0.573736\pi$
$684$		0			0
$685$	−12.0000			−0.458496
$686$		0			0
$687$		0			0
$688$	8.00000	−	13.8564i	0.304997	−	0.528271i
$689$	12.0000	−	20.7846i	0.457164	−	0.791831i
$690$		0			0
$691$	−22.0000	−	38.1051i	−0.836919	−	1.44959i	−0.892458	−	0.451130i	$-0.851021\pi$
$691$	−22.0000	−	38.1051i	−0.836919	−	1.44959i	0.0555386	−	0.998457i	$-0.482312\pi$
$692$	12.0000			0.456172
$693$		0			0
$694$		0			0
$695$	−3.50000	−	6.06218i	−0.132763	−	0.229952i
$696$		0			0
$697$	9.00000	−	15.5885i	0.340899	−	0.590455i
$698$		0			0
$699$		0			0
$700$	2.00000	+	3.46410i	0.0755929	+	0.130931i
$701$	15.0000			0.566542			0.283271	−	0.959040i	$-0.408580\pi$
$701$	15.0000			0.566542			0.283271	+	0.959040i	$0.408580\pi$
$702$		0			0
$703$	−8.00000			−0.301726
$704$	−12.0000	−	20.7846i	−0.452267	−	0.783349i
$705$		0			0
$706$		0			0
$707$	−9.00000	+	15.5885i	−0.338480	+	0.586264i
$708$		0			0
$709$	11.0000	+	19.0526i	0.413114	+	0.715534i	0.995228	−	0.0975728i	$-0.0311079\pi$
$709$	11.0000	+	19.0526i	0.413114	+	0.715534i	−0.582115	+	0.813107i	$0.697775\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−3.00000	−	5.19615i	−0.112351	−	0.194597i
$714$		0			0
$715$	6.00000	−	10.3923i	0.224387	−	0.388650i
$716$	−9.00000	+	15.5885i	−0.336346	+	0.582568i
$717$		0			0
$718$		0			0
$719$	27.0000			1.00693			0.503465	−	0.864016i	$-0.332058\pi$
$719$	27.0000			1.00693			0.503465	+	0.864016i	$0.332058\pi$
$720$		0			0
$721$	28.0000			1.04277
$722$		0			0
$723$		0			0
$724$	11.0000	−	19.0526i	0.408812	−	0.708083i
$725$	4.50000	−	7.79423i	0.167126	−	0.289470i
$726$		0			0
$727$	14.0000	+	24.2487i	0.519231	+	0.899335i	0.999750	+	0.0223506i	$0.00711500\pi$
$727$	14.0000	+	24.2487i	0.519231	+	0.899335i	−0.480519	+	0.876984i	$0.659552\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−12.0000	−	20.7846i	−0.443836	−	0.768747i
$732$		0			0
$733$	−7.00000	+	12.1244i	−0.258551	+	0.447823i	−0.965854	−	0.259087i	$-0.916578\pi$
$733$	−7.00000	+	12.1244i	−0.258551	+	0.447823i	0.707303	+	0.706910i	$0.249912\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−42.0000			−1.54709
$738$		0			0
$739$	−7.00000			−0.257499			−0.128750	−	0.991677i	$-0.541096\pi$
$739$	−7.00000			−0.257499			−0.128750	+	0.991677i	$0.541096\pi$
$740$	−8.00000	−	13.8564i	−0.294086	−	0.509372i
$741$		0			0
$742$		0			0
$743$	9.00000	−	15.5885i	0.330178	−	0.571885i	−0.652369	−	0.757902i	$-0.726225\pi$
$743$	9.00000	−	15.5885i	0.330178	−	0.571885i	0.982547	+	0.186017i	$0.0595579\pi$
$744$		0			0
$745$	3.00000	+	5.19615i	0.109911	+	0.190372i
$746$		0			0
$747$		0			0
$748$	−36.0000			−1.31629
$749$	18.0000	+	31.1769i	0.657706	+	1.13918i
$750$		0			0
$751$	8.00000	−	13.8564i	0.291924	−	0.505627i	−0.682341	−	0.731034i	$-0.739038\pi$
$751$	8.00000	−	13.8564i	0.291924	−	0.505627i	0.974265	+	0.225407i	$0.0723712\pi$
$752$	−24.0000	+	41.5692i	−0.875190	+	1.51587i
$753$		0			0
$754$		0			0
$755$	1.00000			0.0363937
$756$		0			0
$757$	20.0000			0.726912			0.363456	−	0.931611i	$-0.381597\pi$
$757$	20.0000			0.726912			0.363456	+	0.931611i	$0.381597\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−16.5000	+	28.5788i	−0.598125	+	1.03598i	0.394973	+	0.918693i	$0.370754\pi$
$761$	−16.5000	+	28.5788i	−0.598125	+	1.03598i	−0.993098	+	0.117289i	$0.962579\pi$
$762$		0			0
$763$	1.00000	+	1.73205i	0.0362024	+	0.0627044i
$764$	30.0000			1.08536
$765$		0			0
$766$		0			0
$767$	6.00000	+	10.3923i	0.216647	+	0.375244i
$768$		0			0
$769$	−20.5000	+	35.5070i	−0.739249	+	1.28042i	0.213585	+	0.976924i	$0.431486\pi$
$769$	−20.5000	+	35.5070i	−0.739249	+	1.28042i	−0.952834	+	0.303492i	$0.901847\pi$
$770$		0			0
$771$		0			0
$772$	−4.00000	−	6.92820i	−0.143963	−	0.249351i
$773$	24.0000			0.863220			0.431610	−	0.902060i	$-0.357946\pi$
$773$	24.0000			0.863220			0.431610	+	0.902060i	$0.357946\pi$
$774$		0			0
$775$	−1.00000			−0.0359211
$776$		0			0
$777$		0			0
$778$		0			0
$779$	1.50000	−	2.59808i	0.0537431	−	0.0930857i
$780$		0			0
$781$	−4.50000	−	7.79423i	−0.161023	−	0.278899i
$782$		0			0
$783$		0			0
$784$	−12.0000			−0.428571
$785$	−8.00000	−	13.8564i	−0.285532	−	0.494556i
$786$		0			0
$787$	−7.00000	+	12.1244i	−0.249523	+	0.432187i	−0.963394	−	0.268091i	$-0.913607\pi$
$787$	−7.00000	+	12.1244i	−0.249523	+	0.432187i	0.713871	+	0.700278i	$0.246941\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	12.0000			0.426671
$792$		0			0
$793$	40.0000			1.42044
$794$		0			0
$795$		0			0
$796$	−16.0000	+	27.7128i	−0.567105	+	0.982255i
$797$	12.0000	−	20.7846i	0.425062	−	0.736229i	−0.571364	−	0.820696i	$-0.693586\pi$
$797$	12.0000	−	20.7846i	0.425062	−	0.736229i	0.996426	+	0.0844678i	$0.0269190\pi$
$798$		0			0
$799$	36.0000	+	62.3538i	1.27359	+	2.20592i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	3.00000	+	5.19615i	0.105868	+	0.183368i
$804$		0			0
$805$	−6.00000	+	10.3923i	−0.211472	+	0.366281i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	15.0000			0.527372			0.263686	−	0.964609i	$-0.415062\pi$
$809$	15.0000			0.527372			0.263686	+	0.964609i	$0.415062\pi$
$810$		0			0
$811$	23.0000			0.807639			0.403820	−	0.914839i	$-0.367682\pi$
$811$	23.0000			0.807639			0.403820	+	0.914839i	$0.367682\pi$
$812$	−18.0000	−	31.1769i	−0.631676	−	1.09410i
$813$		0			0
$814$		0			0
$815$	1.00000	−	1.73205i	0.0350285	−	0.0606711i
$816$		0			0
$817$	−2.00000	−	3.46410i	−0.0699711	−	0.121194i
$818$		0			0
$819$		0			0
$820$	6.00000			0.209529
$821$	22.5000	+	38.9711i	0.785255	+	1.36010i	0.928846	+	0.370465i	$0.120802\pi$
$821$	22.5000	+	38.9711i	0.785255	+	1.36010i	−0.143591	+	0.989637i	$0.545865\pi$
$822$		0			0
$823$	11.0000	−	19.0526i	0.383436	−	0.664130i	−0.608115	−	0.793849i	$-0.708074\pi$
$823$	11.0000	−	19.0526i	0.383436	−	0.664130i	0.991551	+	0.129719i	$0.0414074\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−36.0000			−1.25184			−0.625921	−	0.779886i	$-0.715277\pi$
$827$	−36.0000			−1.25184			−0.625921	+	0.779886i	$0.715277\pi$
$828$		0			0
$829$	−13.0000			−0.451509			−0.225754	−	0.974184i	$-0.572485\pi$
$829$	−13.0000			−0.451509			−0.225754	+	0.974184i	$0.572485\pi$
$830$		0			0
$831$		0			0
$832$	−16.0000	+	27.7128i	−0.554700	+	0.960769i
$833$	−9.00000	+	15.5885i	−0.311832	+	0.540108i
$834$		0			0
$835$	3.00000	+	5.19615i	0.103819	+	0.179820i
$836$	−6.00000			−0.207514
$837$		0			0
$838$		0			0
$839$	−22.5000	−	38.9711i	−0.776786	−	1.34543i	−0.933785	−	0.357834i	$-0.883515\pi$
$839$	−22.5000	−	38.9711i	−0.776786	−	1.34543i	0.156999	−	0.987599i	$-0.449818\pi$
$840$		0			0
$841$	−26.0000	+	45.0333i	−0.896552	+	1.55287i
$842$		0			0
$843$		0			0
$844$	−13.0000	−	22.5167i	−0.447478	−	0.775055i
$845$	−3.00000			−0.103203
$846$		0			0
$847$	−4.00000			−0.137442
$848$	−12.0000	−	20.7846i	−0.412082	−	0.713746i
$849$		0			0
$850$		0			0
$851$	24.0000	−	41.5692i	0.822709	−	1.42497i
$852$		0			0
$853$	23.0000	+	39.8372i	0.787505	+	1.36400i	0.927491	+	0.373845i	$0.121961\pi$
$853$	23.0000	+	39.8372i	0.787505	+	1.36400i	−0.139986	+	0.990153i	$0.544706\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−24.0000	−	41.5692i	−0.819824	−	1.41998i	−0.905811	−	0.423681i	$-0.860738\pi$
$857$	−24.0000	−	41.5692i	−0.819824	−	1.41998i	0.0859870	−	0.996296i	$-0.472596\pi$
$858$		0			0
$859$	12.5000	−	21.6506i	0.426494	−	0.738710i	−0.570064	−	0.821600i	$-0.693082\pi$
$859$	12.5000	−	21.6506i	0.426494	−	0.738710i	0.996559	+	0.0828900i	$0.0264150\pi$
$860$	4.00000	−	6.92820i	0.136399	−	0.236250i
$861$		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
$865$	6.00000			0.204006
$866$		0			0
$867$		0			0
$868$	−2.00000	+	3.46410i	−0.0678844	+	0.117579i
$869$	−24.0000	+	41.5692i	−0.814144	+	1.41014i
$870$		0			0
$871$	28.0000	+	48.4974i	0.948744	+	1.64327i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	1.00000	+	1.73205i	0.0338062	+	0.0585540i
$876$		0			0
$877$	−22.0000	+	38.1051i	−0.742887	+	1.28672i	0.208288	+	0.978068i	$0.433211\pi$
$877$	−22.0000	+	38.1051i	−0.742887	+	1.28672i	−0.951175	+	0.308651i	$0.900123\pi$
$878$		0			0
$879$		0			0
$880$	−6.00000	−	10.3923i	−0.202260	−	0.350325i
$881$	39.0000			1.31394			0.656972	−	0.753915i	$-0.271837\pi$
$881$	39.0000			1.31394			0.656972	+	0.753915i	$0.271837\pi$
$882$		0			0
$883$	−46.0000			−1.54802			−0.774012	−	0.633171i	$-0.781753\pi$
$883$	−46.0000			−1.54802			−0.774012	+	0.633171i	$0.781753\pi$
$884$	24.0000	+	41.5692i	0.807207	+	1.39812i
$885$		0			0
$886$		0			0
$887$	−18.0000	+	31.1769i	−0.604381	+	1.04682i	0.387768	+	0.921757i	$0.373246\pi$
$887$	−18.0000	+	31.1769i	−0.604381	+	1.04682i	−0.992149	+	0.125061i	$0.960087\pi$
$888$		0			0
$889$	−2.00000	−	3.46410i	−0.0670778	−	0.116182i
$890$		0			0
$891$		0			0
$892$	−40.0000			−1.33930
$893$	6.00000	+	10.3923i	0.200782	+	0.347765i
$894$		0			0
$895$	−4.50000	+	7.79423i	−0.150418	+	0.260532i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	9.00000			0.300167
$900$		0			0
$901$	−36.0000			−1.19933
$902$		0			0
$903$		0			0
$904$		0			0
$905$	5.50000	−	9.52628i	0.182826	−	0.316664i
$906$		0			0
$907$	−19.0000	−	32.9090i	−0.630885	−	1.09272i	−0.987371	−	0.158424i	$-0.949359\pi$
$907$	−19.0000	−	32.9090i	−0.630885	−	1.09272i	0.356487	−	0.934300i	$-0.383975\pi$
$908$	36.0000			1.19470
$909$		0			0
$910$		0			0
$911$	4.50000	+	7.79423i	0.149092	+	0.258234i	0.930892	−	0.365295i	$-0.119032\pi$
$911$	4.50000	+	7.79423i	0.149092	+	0.258234i	−0.781800	+	0.623529i	$0.785698\pi$
$912$		0			0
$913$	−18.0000	+	31.1769i	−0.595713	+	1.03181i
$914$		0			0
$915$		0			0
$916$	−10.0000	−	17.3205i	−0.330409	−	0.572286i
$917$	−18.0000			−0.594412
$918$		0			0
$919$	−7.00000			−0.230909			−0.115454	−	0.993313i	$-0.536832\pi$
$919$	−7.00000			−0.230909			−0.115454	+	0.993313i	$0.536832\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−6.00000	+	10.3923i	−0.197492	+	0.342067i
$924$		0			0
$925$	−4.00000	−	6.92820i	−0.131519	−	0.227798i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	19.5000	+	33.7750i	0.639774	+	1.10812i	0.985482	+	0.169779i	$0.0543055\pi$
$929$	19.5000	+	33.7750i	0.639774	+	1.10812i	−0.345708	+	0.938342i	$0.612361\pi$
$930$		0			0
$931$	−1.50000	+	2.59808i	−0.0491605	+	0.0851485i
$932$	12.0000	−	20.7846i	0.393073	−	0.680823i
$933$		0			0
$934$		0			0
$935$	−18.0000			−0.588663
$936$		0			0
$937$	14.0000			0.457360			0.228680	−	0.973502i	$-0.426559\pi$
$937$	14.0000			0.457360			0.228680	+	0.973502i	$0.426559\pi$
$938$		0			0
$939$		0			0
$940$	−12.0000	+	20.7846i	−0.391397	+	0.677919i
$941$	−15.0000	+	25.9808i	−0.488986	+	0.846949i	−0.999920	−	0.0126715i	$-0.995966\pi$
$941$	−15.0000	+	25.9808i	−0.488986	+	0.846949i	0.510934	+	0.859620i	$0.329300\pi$
$942$		0			0
$943$	9.00000	+	15.5885i	0.293080	+	0.507630i
$944$	12.0000			0.390567
$945$		0			0
$946$		0			0
$947$	−15.0000	−	25.9808i	−0.487435	−	0.844261i	0.512461	−	0.858710i	$-0.328734\pi$
$947$	−15.0000	−	25.9808i	−0.487435	−	0.844261i	−0.999896	+	0.0144491i	$0.995401\pi$
$948$		0			0
$949$	4.00000	−	6.92820i	0.129845	−	0.224899i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−48.0000			−1.55487			−0.777436	−	0.628962i	$-0.783480\pi$
$953$	−48.0000			−1.55487			−0.777436	+	0.628962i	$0.783480\pi$
$954$		0			0
$955$	15.0000			0.485389
$956$	−24.0000	−	41.5692i	−0.776215	−	1.34444i
$957$		0			0
$958$		0			0
$959$	−12.0000	+	20.7846i	−0.387500	+	0.671170i
$960$		0			0
$961$	15.0000	+	25.9808i	0.483871	+	0.838089i
$962$		0			0
$963$		0			0
$964$	−34.0000			−1.09507
$965$	−2.00000	−	3.46410i	−0.0643823	−	0.111513i
$966$		0			0
$967$	−25.0000	+	43.3013i	−0.803946	+	1.39247i	0.113055	+	0.993589i	$0.463936\pi$
$967$	−25.0000	+	43.3013i	−0.803946	+	1.39247i	−0.917000	+	0.398886i	$0.869397\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−3.00000			−0.0962746			−0.0481373	−	0.998841i	$-0.515328\pi$
$971$	−3.00000			−0.0962746			−0.0481373	+	0.998841i	$0.515328\pi$
$972$		0			0
$973$	−14.0000			−0.448819
$974$		0			0
$975$		0			0
$976$	20.0000	−	34.6410i	0.640184	−	1.10883i
$977$	24.0000	−	41.5692i	0.767828	−	1.32992i	−0.170910	−	0.985287i	$-0.554671\pi$
$977$	24.0000	−	41.5692i	0.767828	−	1.32992i	0.938738	−	0.344631i	$-0.111996\pi$
$978$		0			0
$979$	22.5000	+	38.9711i	0.719103	+	1.24552i
$980$	−6.00000			−0.191663
$981$		0			0
$982$		0			0
$983$	3.00000	+	5.19615i	0.0956851	+	0.165732i	0.909894	−	0.414840i	$-0.136162\pi$
$983$	3.00000	+	5.19615i	0.0956851	+	0.165732i	−0.814209	+	0.580572i	$0.802829\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	4.00000	+	6.92820i	0.127257	+	0.220416i
$989$	24.0000			0.763156
$990$		0			0
$991$	47.0000			1.49300			0.746502	−	0.665383i	$-0.231732\pi$
$991$	47.0000			1.49300			0.746502	+	0.665383i	$0.231732\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−8.00000	+	13.8564i	−0.253617	+	0.439278i
$996$		0			0
$997$	5.00000	+	8.66025i	0.158352	+	0.274273i	0.934274	−	0.356555i	$-0.116049\pi$
$997$	5.00000	+	8.66025i	0.158352	+	0.274273i	−0.775923	+	0.630828i	$0.782715\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	405.2.e.e.271.1		2
3.2	odd	2		405.2.e.d.271.1		2
9.2	odd	6		405.2.e.d.136.1		2
9.4	even	3		405.2.a.c.1.1	✓	1
9.5	odd	6		405.2.a.d.1.1	yes	1
9.7	even	3	inner	405.2.e.e.136.1		2
36.23	even	6		6480.2.a.o.1.1		1
36.31	odd	6		6480.2.a.c.1.1		1
45.4	even	6		2025.2.a.c.1.1		1
45.13	odd	12		2025.2.b.e.649.1		2
45.14	odd	6		2025.2.a.d.1.1		1
45.22	odd	12		2025.2.b.e.649.2		2
45.23	even	12		2025.2.b.f.649.1		2
45.32	even	12		2025.2.b.f.649.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
405.2.a.c.1.1	✓	1	9.4	even	3
405.2.a.d.1.1	yes	1	9.5	odd	6
405.2.e.d.136.1		2	9.2	odd	6
405.2.e.d.271.1		2	3.2	odd	2
405.2.e.e.136.1		2	9.7	even	3	inner
405.2.e.e.271.1		2	1.1	even	1	trivial
2025.2.a.c.1.1		1	45.4	even	6
2025.2.a.d.1.1		1	45.14	odd	6
2025.2.b.e.649.1		2	45.13	odd	12
2025.2.b.e.649.2		2	45.22	odd	12
2025.2.b.f.649.1		2	45.23	even	12
2025.2.b.f.649.2		2	45.32	even	12
6480.2.a.c.1.1		1	36.31	odd	6
6480.2.a.o.1.1		1	36.23	even	6

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:	\( N \)	\(=\)	\( 405 = 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	405.e (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.00000 - 1.73205i) q^{4} +(0.500000 - 0.866025i) q^{5} +(-1.00000 - 1.73205i) q^{7} +(1.50000 + 2.59808i) q^{11} +(2.00000 - 3.46410i) q^{13} +(-2.00000 - 3.46410i) q^{16} -6.00000 q^{17} -1.00000 q^{19} +(-1.00000 - 1.73205i) q^{20} +(3.00000 - 5.19615i) q^{23} +(-0.500000 - 0.866025i) q^{25} -4.00000 q^{28} +(4.50000 + 7.79423i) q^{29} +(0.500000 - 0.866025i) q^{31} -2.00000 q^{35} +8.00000 q^{37} +(-1.50000 + 2.59808i) q^{41} +(2.00000 + 3.46410i) q^{43} +6.00000 q^{44} +(-6.00000 - 10.3923i) q^{47} +(1.50000 - 2.59808i) q^{49} +(-4.00000 - 6.92820i) q^{52} +6.00000 q^{53} +3.00000 q^{55} +(-1.50000 + 2.59808i) q^{59} +(5.00000 + 8.66025i) q^{61} -8.00000 q^{64} +(-2.00000 - 3.46410i) q^{65} +(-7.00000 + 12.1244i) q^{67} +(-6.00000 + 10.3923i) q^{68} -3.00000 q^{71} +2.00000 q^{73} +(-1.00000 + 1.73205i) q^{76} +(3.00000 - 5.19615i) q^{77} +(8.00000 + 13.8564i) q^{79} -4.00000 q^{80} +(6.00000 + 10.3923i) q^{83} +(-3.00000 + 5.19615i) q^{85} +15.0000 q^{89} -8.00000 q^{91} +(-6.00000 - 10.3923i) q^{92} +(-0.500000 + 0.866025i) q^{95} +(2.00000 + 3.46410i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{4} + q^{5} - 2 q^{7} + 3 q^{11} + 4 q^{13} - 4 q^{16} - 12 q^{17} - 2 q^{19} - 2 q^{20} + 6 q^{23} - q^{25} - 8 q^{28} + 9 q^{29} + q^{31} - 4 q^{35} + 16 q^{37} - 3 q^{41} + 4 q^{43} + 12 q^{44}+ \cdots + 4 q^{97}+O(q^{100}) \) 2 * q + 2 * q^4 + q^5 - 2 * q^7 + 3 * q^11 + 4 * q^13 - 4 * q^16 - 12 * q^17 - 2 * q^19 - 2 * q^20 + 6 * q^23 - q^25 - 8 * q^28 + 9 * q^29 + q^31 - 4 * q^35 + 16 * q^37 - 3 * q^41 + 4 * q^43 + 12 * q^44 - 12 * q^47 + 3 * q^49 - 8 * q^52 + 12 * q^53 + 6 * q^55 - 3 * q^59 + 10 * q^61 - 16 * q^64 - 4 * q^65 - 14 * q^67 - 12 * q^68 - 6 * q^71 + 4 * q^73 - 2 * q^76 + 6 * q^77 + 16 * q^79 - 8 * q^80 + 12 * q^83 - 6 * q^85 + 30 * q^89 - 16 * q^91 - 12 * q^92 - q^95 + 4 * q^97

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