LMFDB - Embedded newform 2880.2.t.c.721.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("2880.721");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

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

Embedding invariants

Embedding label			721.6
Root			$-0.296075 - 1.38287i$ of defining polynomial
Character	$\chi$	$=$	2880.721
Dual form			2880.2.t.c.2161.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.707107 - 0.707107i) q^{5} -2.66881i q^{7} +O(q^{10})$	$q+(0.707107 - 0.707107i) q^{5} -2.66881i q^{7} +(-3.49714 + 3.49714i) q^{11} +(2.94072 + 2.94072i) q^{13} -1.85116 q^{17} +(3.44856 + 3.44856i) q^{19} -0.707288i q^{23} -1.00000i q^{25} +(3.49909 + 3.49909i) q^{29} -6.84272 q^{31} +(-1.88714 - 1.88714i) q^{35} +(-0.0975060 + 0.0975060i) q^{37} +10.2052i q^{41} +(-4.43844 + 4.43844i) q^{43} -1.89428 q^{47} -0.122561 q^{49} +(7.43897 - 7.43897i) q^{53} +4.94571i q^{55} +(0.959574 - 0.959574i) q^{59} +(6.49825 + 6.49825i) q^{61} +4.15881 q^{65} +(-3.49691 - 3.49691i) q^{67} +7.86777i q^{71} +15.6564i q^{73} +(9.33322 + 9.33322i) q^{77} +6.70212 q^{79} +(-3.87327 - 3.87327i) q^{83} +(-1.30896 + 1.30896i) q^{85} -10.5055i q^{89} +(7.84824 - 7.84824i) q^{91} +4.87701 q^{95} +4.79937 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q - 8 q^{11} + 8 q^{19} + 16 q^{29} - 16 q^{37} - 8 q^{43} - 40 q^{47} - 16 q^{49} - 16 q^{53} - 8 q^{59} + 16 q^{61} - 40 q^{67} - 16 q^{77} - 16 q^{79} + 40 q^{83} - 16 q^{85} - 32 q^{91} + 32 q^{95}+O(q^{100}) $ 16 * q - 8 * q^11 + 8 * q^19 + 16 * q^29 - 16 * q^37 - 8 * q^43 - 40 * q^47 - 16 * q^49 - 16 * q^53 - 8 * q^59 + 16 * q^61 - 40 * q^67 - 16 * q^77 - 16 * q^79 + 40 * q^83 - 16 * q^85 - 32 * q^91 + 32 * q^95

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	0.707107	−	0.707107i	0.316228	−	0.316228i
$5$	0.707107	−	0.707107i	0.316228	−	0.316228i
$6$		0			0
$7$		−	2.66881i		−	1.00872i	−0.863495	−	0.504358i	$-0.831729\pi$
$7$		−	2.66881i		−	1.00872i	0.863495	−	0.504358i	$-0.168271\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−3.49714	+	3.49714i	−1.05443	+	1.05443i	−0.0559977	+	0.998431i	$0.517834\pi$
$11$	−3.49714	+	3.49714i	−1.05443	+	1.05443i	−0.998431	+	0.0559977i	$0.982166\pi$
$12$		0			0
$13$	2.94072	+	2.94072i	0.815610	+	0.815610i	0.985468	−	0.169858i	$-0.0543310\pi$
$13$	2.94072	+	2.94072i	0.815610	+	0.815610i	−0.169858	+	0.985468i	$0.554331\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−1.85116			−0.448971			−0.224486	−	0.974477i	$-0.572070\pi$
$17$	−1.85116			−0.448971			−0.224486	+	0.974477i	$0.572070\pi$
$18$		0			0
$19$	3.44856	+	3.44856i	0.791155	+	0.791155i	0.981682	−	0.190527i	$-0.0610197\pi$
$19$	3.44856	+	3.44856i	0.791155	+	0.791155i	−0.190527	+	0.981682i	$0.561020\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	0.707288i		−	0.147480i	−0.997278	−	0.0737399i	$-0.976507\pi$
$23$		−	0.707288i		−	0.147480i	0.997278	−	0.0737399i	$-0.0234935\pi$
$24$		0			0
$25$		−	1.00000i		−	0.200000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	3.49909	+	3.49909i	0.649766	+	0.649766i	0.952936	−	0.303171i	$-0.0980452\pi$
$29$	3.49909	+	3.49909i	0.649766	+	0.649766i	−0.303171	+	0.952936i	$0.598045\pi$
$30$		0			0
$31$	−6.84272			−1.22899			−0.614494	−	0.788921i	$-0.710640\pi$
$31$	−6.84272			−1.22899			−0.614494	+	0.788921i	$0.710640\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−1.88714	−	1.88714i	−0.318984	−	0.318984i
$36$		0			0
$37$	−0.0975060	+	0.0975060i	−0.0160299	+	0.0160299i	−0.715076	−	0.699046i	$-0.753608\pi$
$37$	−0.0975060	+	0.0975060i	−0.0160299	+	0.0160299i	0.699046	+	0.715076i	$0.253608\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			10.2052i			1.59379i	0.604117	+	0.796896i	$0.293526\pi$
$41$			10.2052i			1.59379i	−0.604117	+	0.796896i	$0.706474\pi$
$42$		0			0
$43$	−4.43844	+	4.43844i	−0.676855	+	0.676855i	−0.959287	−	0.282432i	$-0.908859\pi$
$43$	−4.43844	+	4.43844i	−0.676855	+	0.676855i	0.282432	+	0.959287i	$0.408859\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−1.89428			−0.276310			−0.138155	−	0.990411i	$-0.544117\pi$
$47$	−1.89428			−0.276310			−0.138155	+	0.990411i	$0.544117\pi$
$48$		0			0
$49$	−0.122561			−0.0175087
$50$		0			0
$51$		0			0
$52$		0			0
$53$	7.43897	−	7.43897i	1.02182	−	1.02182i	0.0220650	−	0.999757i	$-0.492976\pi$
$53$	7.43897	−	7.43897i	1.02182	−	1.02182i	0.999757	−	0.0220650i	$-0.00702407\pi$
$54$		0			0
$55$			4.94571i			0.666879i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	0.959574	−	0.959574i	0.124926	−	0.124926i	−0.641880	−	0.766805i	$-0.721845\pi$
$59$	0.959574	−	0.959574i	0.124926	−	0.124926i	0.766805	+	0.641880i	$0.221845\pi$
$60$		0			0
$61$	6.49825	+	6.49825i	0.832015	+	0.832015i	0.987792	−	0.155777i	$-0.0497881\pi$
$61$	6.49825	+	6.49825i	0.832015	+	0.832015i	−0.155777	+	0.987792i	$0.549788\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	4.15881			0.515837
$66$		0			0
$67$	−3.49691	−	3.49691i	−0.427216	−	0.427216i	0.460463	−	0.887679i	$-0.347683\pi$
$67$	−3.49691	−	3.49691i	−0.427216	−	0.427216i	−0.887679	+	0.460463i	$0.847683\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			7.86777i			0.933733i	0.884328	+	0.466866i	$0.154617\pi$
$71$			7.86777i			0.933733i	−0.884328	+	0.466866i	$0.845383\pi$
$72$		0			0
$73$			15.6564i			1.83244i	0.400675	+	0.916220i	$0.368776\pi$
$73$			15.6564i			1.83244i	−0.400675	+	0.916220i	$0.631224\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	9.33322	+	9.33322i	1.06362	+	1.06362i
$78$		0			0
$79$	6.70212			0.754047			0.377024	−	0.926204i	$-0.376948\pi$
$79$	6.70212			0.754047			0.377024	+	0.926204i	$0.376948\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−3.87327	−	3.87327i	−0.425147	−	0.425147i	0.461825	−	0.886971i	$-0.347195\pi$
$83$	−3.87327	−	3.87327i	−0.425147	−	0.425147i	−0.886971	+	0.461825i	$0.847195\pi$
$84$		0			0
$85$	−1.30896	+	1.30896i	−0.141977	+	0.141977i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	10.5055i		−	1.11358i	−0.830653	−	0.556790i	$-0.812033\pi$
$89$		−	10.5055i		−	1.11358i	0.830653	−	0.556790i	$-0.187967\pi$
$90$		0			0
$91$	7.84824	−	7.84824i	0.822719	−	0.822719i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	4.87701			0.500370
$96$		0			0
$97$	4.79937			0.487303			0.243651	−	0.969863i	$-0.421655\pi$
$97$	4.79937			0.487303			0.243651	+	0.969863i	$0.421655\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−0.372979	+	0.372979i	−0.0371128	+	0.0371128i	−0.725420	−	0.688307i	$-0.758354\pi$
$101$	−0.372979	+	0.372979i	−0.0371128	+	0.0371128i	0.688307	+	0.725420i	$0.258354\pi$
$102$		0			0
$103$			10.3013i			1.01502i	0.861647	+	0.507508i	$0.169433\pi$
$103$			10.3013i			1.01502i	−0.861647	+	0.507508i	$0.830567\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	14.5069	−	14.5069i	1.40244	−	1.40244i	0.610165	−	0.792274i	$-0.291103\pi$
$107$	14.5069	−	14.5069i	1.40244	−	1.40244i	0.792274	−	0.610165i	$-0.208897\pi$
$108$		0			0
$109$	0.796284	+	0.796284i	0.0762701	+	0.0762701i	0.744213	−	0.667943i	$-0.232825\pi$
$109$	0.796284	+	0.796284i	0.0762701	+	0.0762701i	−0.667943	+	0.744213i	$0.732825\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−0.842524			−0.0792580			−0.0396290	−	0.999214i	$-0.512618\pi$
$113$	−0.842524			−0.0792580			−0.0396290	+	0.999214i	$0.512618\pi$
$114$		0			0
$115$	−0.500128	−	0.500128i	−0.0466372	−	0.0466372i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			4.94039i			0.452885i
$120$		0			0
$121$		−	13.4600i		−	1.22364i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−0.707107	−	0.707107i	−0.0632456	−	0.0632456i
$126$		0			0
$127$	21.1693			1.87847			0.939234	−	0.343277i	$-0.111537\pi$
$127$	21.1693			1.87847			0.939234	+	0.343277i	$0.111537\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	4.67248	+	4.67248i	0.408237	+	0.408237i	0.881123	−	0.472887i	$-0.156788\pi$
$131$	4.67248	+	4.67248i	0.408237	+	0.408237i	−0.472887	+	0.881123i	$0.656788\pi$
$132$		0			0
$133$	9.20357	−	9.20357i	0.798051	−	0.798051i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	10.2840i		−	0.878623i	−0.898335	−	0.439312i	$-0.855222\pi$
$137$		−	10.2840i		−	0.878623i	0.898335	−	0.439312i	$-0.144778\pi$
$138$		0			0
$139$	−4.98588	+	4.98588i	−0.422897	+	0.422897i	−0.886200	−	0.463303i	$-0.846664\pi$
$139$	−4.98588	+	4.98588i	−0.422897	+	0.422897i	0.463303	+	0.886200i	$0.346664\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−20.5683			−1.72001
$144$		0			0
$145$	4.94847			0.410948
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−8.79493	+	8.79493i	−0.720509	+	0.720509i	−0.968709	−	0.248200i	$-0.920161\pi$
$149$	−8.79493	+	8.79493i	−0.720509	+	0.720509i	0.248200	+	0.968709i	$0.420161\pi$
$150$		0			0
$151$			22.1838i			1.80529i	0.430385	+	0.902645i	$0.358378\pi$
$151$			22.1838i			1.80529i	−0.430385	+	0.902645i	$0.641622\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−4.83853	+	4.83853i	−0.388640	+	0.388640i
$156$		0			0
$157$	−3.72187	−	3.72187i	−0.297038	−	0.297038i	0.542815	−	0.839852i	$-0.317359\pi$
$157$	−3.72187	−	3.72187i	−0.297038	−	0.297038i	−0.839852	+	0.542815i	$0.817359\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−1.88762			−0.148765
$162$		0			0
$163$	−2.11630	−	2.11630i	−0.165761	−	0.165761i	0.619352	−	0.785113i	$-0.287395\pi$
$163$	−2.11630	−	2.11630i	−0.165761	−	0.165761i	−0.785113	+	0.619352i	$0.787395\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			18.1604i			1.40530i	0.711538	+	0.702648i	$0.247999\pi$
$167$			18.1604i			1.40530i	−0.711538	+	0.702648i	$0.752001\pi$
$168$		0			0
$169$			4.29572i			0.330440i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	8.53542	+	8.53542i	0.648936	+	0.648936i	0.952736	−	0.303800i	$-0.0982555\pi$
$173$	8.53542	+	8.53542i	0.648936	+	0.648936i	−0.303800	+	0.952736i	$0.598255\pi$
$174$		0			0
$175$	−2.66881			−0.201743
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−2.42499	−	2.42499i	−0.181252	−	0.181252i	0.610649	−	0.791901i	$-0.290909\pi$
$179$	−2.42499	−	2.42499i	−0.181252	−	0.181252i	−0.791901	+	0.610649i	$0.790909\pi$
$180$		0			0
$181$	4.46593	−	4.46593i	0.331950	−	0.331950i	−0.521377	−	0.853327i	$-0.674581\pi$
$181$	4.46593	−	4.46593i	0.331950	−	0.331950i	0.853327	+	0.521377i	$0.174581\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			0.137894i			0.0101382i
$186$		0			0
$187$	6.47376	−	6.47376i	0.473408	−	0.473408i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	7.75030			0.560792			0.280396	−	0.959884i	$-0.409534\pi$
$191$	7.75030			0.560792			0.280396	+	0.959884i	$0.409534\pi$
$192$		0			0
$193$	−11.3388			−0.816181			−0.408091	−	0.912941i	$-0.633805\pi$
$193$	−11.3388			−0.816181			−0.408091	+	0.912941i	$0.633805\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−1.10001	+	1.10001i	−0.0783725	+	0.0783725i	−0.745206	−	0.666834i	$-0.767649\pi$
$197$	−1.10001	+	1.10001i	−0.0783725	+	0.0783725i	0.666834	+	0.745206i	$0.267649\pi$
$198$		0			0
$199$		−	14.2722i		−	1.01173i	−0.862614	−	0.505864i	$-0.831174\pi$
$199$		−	14.2722i		−	1.01173i	0.862614	−	0.505864i	$-0.168826\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	9.33843	−	9.33843i	0.655429	−	0.655429i
$204$		0			0
$205$	7.21620	+	7.21620i	0.504001	+	0.504001i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−24.1203			−1.66843
$210$		0			0
$211$	12.4716	+	12.4716i	0.858577	+	0.858577i	0.991171	−	0.132593i	$-0.0423305\pi$
$211$	12.4716	+	12.4716i	0.858577	+	0.858577i	−0.132593	+	0.991171i	$0.542330\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			6.27690i			0.428081i
$216$		0			0
$217$			18.2619i			1.23970i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−5.44374	−	5.44374i	−0.366186	−	0.366186i
$222$		0			0
$223$	−3.08673			−0.206703			−0.103351	−	0.994645i	$-0.532957\pi$
$223$	−3.08673			−0.206703			−0.103351	+	0.994645i	$0.532957\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	8.31678	+	8.31678i	0.552004	+	0.552004i	0.927019	−	0.375015i	$-0.122362\pi$
$227$	8.31678	+	8.31678i	0.552004	+	0.552004i	−0.375015	+	0.927019i	$0.622362\pi$
$228$		0			0
$229$	−9.98910	+	9.98910i	−0.660098	+	0.660098i	−0.955403	−	0.295305i	$-0.904579\pi$
$229$	−9.98910	+	9.98910i	−0.660098	+	0.660098i	0.295305	+	0.955403i	$0.404579\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			13.9015i			0.910718i	0.890308	+	0.455359i	$0.150489\pi$
$233$			13.9015i			0.910718i	−0.890308	+	0.455359i	$0.849511\pi$
$234$		0			0
$235$	−1.33946	+	1.33946i	−0.0873768	+	0.0873768i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−10.7687			−0.696569			−0.348284	−	0.937389i	$-0.613236\pi$
$239$	−10.7687			−0.696569			−0.348284	+	0.937389i	$0.613236\pi$
$240$		0			0
$241$	−12.4707			−0.803305			−0.401653	−	0.915792i	$-0.631564\pi$
$241$	−12.4707			−0.803305			−0.401653	+	0.915792i	$0.631564\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−0.0866638	+	0.0866638i	−0.00553675	+	0.00553675i
$246$		0			0
$247$			20.2826i			1.29055i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	3.69093	−	3.69093i	0.232969	−	0.232969i	−0.580962	−	0.813931i	$-0.697323\pi$
$251$	3.69093	−	3.69093i	0.232969	−	0.232969i	0.813931	+	0.580962i	$0.197323\pi$
$252$		0			0
$253$	2.47349	+	2.47349i	0.155507	+	0.155507i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−3.11011			−0.194003			−0.0970016	−	0.995284i	$-0.530925\pi$
$257$	−3.11011			−0.194003			−0.0970016	+	0.995284i	$0.530925\pi$
$258$		0			0
$259$	0.260225	+	0.260225i	0.0161696	+	0.0161696i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			17.9512i			1.10692i	0.832877	+	0.553458i	$0.186692\pi$
$263$			17.9512i			1.10692i	−0.832877	+	0.553458i	$0.813308\pi$
$264$		0			0
$265$		−	10.5203i		−	0.646257i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−1.62436	−	1.62436i	−0.0990392	−	0.0990392i	0.655851	−	0.754890i	$-0.272310\pi$
$269$	−1.62436	−	1.62436i	−0.0990392	−	0.0990392i	−0.754890	+	0.655851i	$0.772310\pi$
$270$		0			0
$271$	−18.1808			−1.10440			−0.552201	−	0.833711i	$-0.686212\pi$
$271$	−18.1808			−1.10440			−0.552201	+	0.833711i	$0.686212\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	3.49714	+	3.49714i	0.210886	+	0.210886i
$276$		0			0
$277$	−13.8675	+	13.8675i	−0.833218	+	0.833218i	−0.987956	−	0.154737i	$-0.950547\pi$
$277$	−13.8675	+	13.8675i	−0.833218	+	0.833218i	0.154737	+	0.987956i	$0.450547\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	10.7377i		−	0.640556i	−0.947324	−	0.320278i	$-0.896224\pi$
$281$		−	10.7377i		−	0.640556i	0.947324	−	0.320278i	$-0.103776\pi$
$282$		0			0
$283$	16.3679	−	16.3679i	0.972971	−	0.972971i	−0.0266735	−	0.999644i	$-0.508491\pi$
$283$	16.3679	−	16.3679i	0.972971	−	0.972971i	0.999644	+	0.0266735i	$0.00849143\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	27.2359			1.60768
$288$		0			0
$289$	−13.5732			−0.798425
$290$		0			0
$291$		0			0
$292$		0			0
$293$	4.22052	−	4.22052i	0.246566	−	0.246566i	−0.572994	−	0.819560i	$-0.694218\pi$
$293$	4.22052	−	4.22052i	0.246566	−	0.246566i	0.819560	+	0.572994i	$0.194218\pi$
$294$		0			0
$295$		−	1.35704i		−	0.0790101i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	2.07994	−	2.07994i	0.120286	−	0.120286i
$300$		0			0
$301$	11.8454	+	11.8454i	0.682755	+	0.682755i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	9.18991			0.526213
$306$		0			0
$307$	12.6363	+	12.6363i	0.721190	+	0.721190i	0.968848	−	0.247658i	$-0.0796608\pi$
$307$	12.6363	+	12.6363i	0.721190	+	0.721190i	−0.247658	+	0.968848i	$0.579661\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	8.56815i		−	0.485855i	−0.970044	−	0.242928i	$-0.921892\pi$
$311$		−	8.56815i		−	0.485855i	0.970044	−	0.242928i	$-0.0781078\pi$
$312$		0			0
$313$		−	19.1825i		−	1.08426i	−0.840295	−	0.542129i	$-0.817618\pi$
$313$		−	19.1825i		−	1.08426i	0.840295	−	0.542129i	$-0.182382\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	9.41764	+	9.41764i	0.528947	+	0.528947i	0.920258	−	0.391311i	$-0.127978\pi$
$317$	9.41764	+	9.41764i	0.528947	+	0.528947i	−0.391311	+	0.920258i	$0.627978\pi$
$318$		0			0
$319$	−24.4737			−1.37026
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−6.38383	−	6.38383i	−0.355206	−	0.355206i
$324$		0			0
$325$	2.94072	−	2.94072i	0.163122	−	0.163122i
$326$		0			0
$327$		0			0
$328$		0			0
$329$			5.05549i			0.278718i
$330$		0			0
$331$	−12.8579	+	12.8579i	−0.706733	+	0.706733i	−0.965847	−	0.259114i	$-0.916569\pi$
$331$	−12.8579	+	12.8579i	−0.706733	+	0.706733i	0.259114	+	0.965847i	$0.416569\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−4.94538			−0.270195
$336$		0			0
$337$	3.31961			0.180831			0.0904153	−	0.995904i	$-0.471181\pi$
$337$	3.31961			0.180831			0.0904153	+	0.995904i	$0.471181\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	23.9300	−	23.9300i	1.29588	−	1.29588i
$342$		0			0
$343$		−	18.3546i		−	0.991055i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	17.8860	−	17.8860i	0.960171	−	0.960171i	−0.0390656	−	0.999237i	$-0.512438\pi$
$347$	17.8860	−	17.8860i	0.960171	−	0.960171i	0.999237	+	0.0390656i	$0.0124381\pi$
$348$		0			0
$349$	−3.68796	−	3.68796i	−0.197412	−	0.197412i	0.601478	−	0.798890i	$-0.294579\pi$
$349$	−3.68796	−	3.68796i	−0.197412	−	0.197412i	−0.798890	+	0.601478i	$0.794579\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−33.0951			−1.76148			−0.880738	−	0.473604i	$-0.842953\pi$
$353$	−33.0951			−1.76148			−0.880738	+	0.473604i	$0.842953\pi$
$354$		0			0
$355$	5.56335	+	5.56335i	0.295272	+	0.295272i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	6.52522i		−	0.344388i	−0.985063	−	0.172194i	$-0.944914\pi$
$359$		−	6.52522i		−	0.344388i	0.985063	−	0.172194i	$-0.0550856\pi$
$360$		0			0
$361$			4.78519i			0.251852i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	11.0707	+	11.0707i	0.579469	+	0.579469i
$366$		0			0
$367$	11.0338			0.575959			0.287980	−	0.957636i	$-0.407016\pi$
$367$	11.0338			0.575959			0.287980	+	0.957636i	$0.407016\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−19.8532	−	19.8532i	−1.03073	−	1.03073i
$372$		0			0
$373$	6.84468	−	6.84468i	0.354404	−	0.354404i	−0.507341	−	0.861745i	$-0.669372\pi$
$373$	6.84468	−	6.84468i	0.354404	−	0.354404i	0.861745	+	0.507341i	$0.169372\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			20.5797i			1.05991i
$378$		0			0
$379$	10.1072	−	10.1072i	0.519171	−	0.519171i	−0.398150	−	0.917321i	$-0.630347\pi$
$379$	10.1072	−	10.1072i	0.519171	−	0.519171i	0.917321	+	0.398150i	$0.130347\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	29.5283			1.50883			0.754413	−	0.656400i	$-0.227922\pi$
$383$	29.5283			1.50883			0.754413	+	0.656400i	$0.227922\pi$
$384$		0			0
$385$	13.1992			0.672692
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−0.990949	+	0.990949i	−0.0502431	+	0.0502431i	−0.731782	−	0.681539i	$-0.761311\pi$
$389$	−0.990949	+	0.990949i	−0.0502431	+	0.0502431i	0.681539	+	0.731782i	$0.261311\pi$
$390$		0			0
$391$			1.30930i			0.0662142i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	4.73911	−	4.73911i	0.238451	−	0.238451i
$396$		0			0
$397$	17.0024	+	17.0024i	0.853326	+	0.853326i	0.990541	−	0.137216i	$-0.0438153\pi$
$397$	17.0024	+	17.0024i	0.853326	+	0.853326i	−0.137216	+	0.990541i	$0.543815\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−26.7791			−1.33728			−0.668642	−	0.743585i	$-0.733124\pi$
$401$	−26.7791			−1.33728			−0.668642	+	0.743585i	$0.733124\pi$
$402$		0			0
$403$	−20.1225	−	20.1225i	−1.00238	−	1.00238i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	0.681985i		−	0.0338048i
$408$		0			0
$409$		−	13.1970i		−	0.652550i	−0.945275	−	0.326275i	$-0.894206\pi$
$409$		−	13.1970i		−	0.652550i	0.945275	−	0.326275i	$-0.105794\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−2.56092	−	2.56092i	−0.126015	−	0.126015i
$414$		0			0
$415$	−5.47763			−0.268886
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−9.92468	−	9.92468i	−0.484852	−	0.484852i	0.421825	−	0.906677i	$-0.361390\pi$
$419$	−9.92468	−	9.92468i	−0.484852	−	0.484852i	−0.906677	+	0.421825i	$0.861390\pi$
$420$		0			0
$421$	15.7930	−	15.7930i	0.769702	−	0.769702i	−0.208352	−	0.978054i	$-0.566810\pi$
$421$	15.7930	−	15.7930i	0.769702	−	0.769702i	0.978054	+	0.208352i	$0.0668100\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			1.85116i			0.0897943i
$426$		0			0
$427$	17.3426	−	17.3426i	0.839268	−	0.839268i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−0.285215			−0.0137383			−0.00686917	−	0.999976i	$-0.502187\pi$
$431$	−0.285215			−0.0137383			−0.00686917	+	0.999976i	$0.502187\pi$
$432$		0			0
$433$	−18.1101			−0.870318			−0.435159	−	0.900354i	$-0.643308\pi$
$433$	−18.1101			−0.870318			−0.435159	+	0.900354i	$0.643308\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	2.43913	−	2.43913i	0.116679	−	0.116679i
$438$		0			0
$439$		−	11.5931i		−	0.553308i	−0.960970	−	0.276654i	$-0.910774\pi$
$439$		−	11.5931i		−	0.553308i	0.960970	−	0.276654i	$-0.0892256\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−22.6855	+	22.6855i	−1.07782	+	1.07782i	−0.0811145	+	0.996705i	$0.525848\pi$
$443$	−22.6855	+	22.6855i	−1.07782	+	1.07782i	−0.996705	+	0.0811145i	$0.974152\pi$
$444$		0			0
$445$	−7.42850	−	7.42850i	−0.352145	−	0.352145i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−12.1999			−0.575747			−0.287873	−	0.957669i	$-0.592948\pi$
$449$	−12.1999			−0.575747			−0.287873	+	0.957669i	$0.592948\pi$
$450$		0			0
$451$	−35.6892	−	35.6892i	−1.68054	−	1.68054i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	11.0991i		−	0.520333i
$456$		0			0
$457$		−	1.70660i		−	0.0798314i	−0.999203	−	0.0399157i	$-0.987291\pi$
$457$		−	1.70660i		−	0.0798314i	0.999203	−	0.0399157i	$-0.0127089\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	4.74710	+	4.74710i	0.221094	+	0.221094i	0.808959	−	0.587865i	$-0.200031\pi$
$461$	4.74710	+	4.74710i	0.221094	+	0.221094i	−0.587865	+	0.808959i	$0.700031\pi$
$462$		0			0
$463$	11.1761			0.519398			0.259699	−	0.965690i	$-0.416377\pi$
$463$	11.1761			0.519398			0.259699	+	0.965690i	$0.416377\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	2.06471	+	2.06471i	0.0955435	+	0.0955435i	0.753263	−	0.657719i	$-0.228479\pi$
$467$	2.06471	+	2.06471i	0.0955435	+	0.0955435i	−0.657719	+	0.753263i	$0.728479\pi$
$468$		0			0
$469$	−9.33260	+	9.33260i	−0.430940	+	0.430940i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	31.0437i		−	1.42739i
$474$		0			0
$475$	3.44856	−	3.44856i	0.158231	−	0.158231i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−41.6214			−1.90173			−0.950864	−	0.309608i	$-0.899802\pi$
$479$	−41.6214			−1.90173			−0.950864	+	0.309608i	$0.899802\pi$
$480$		0			0
$481$	−0.573477			−0.0261483
$482$		0			0
$483$		0			0
$484$		0			0
$485$	3.39367	−	3.39367i	0.154099	−	0.154099i
$486$		0			0
$487$		−	8.25627i		−	0.374127i	−0.982348	−	0.187064i	$-0.940103\pi$
$487$		−	8.25627i		−	0.374127i	0.982348	−	0.187064i	$-0.0598970\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	4.28512	−	4.28512i	0.193385	−	0.193385i	−0.603772	−	0.797157i	$-0.706336\pi$
$491$	4.28512	−	4.28512i	0.193385	−	0.193385i	0.797157	+	0.603772i	$0.206336\pi$
$492$		0			0
$493$	−6.47737	−	6.47737i	−0.291726	−	0.291726i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	20.9976			0.941871
$498$		0			0
$499$	15.1287	+	15.1287i	0.677253	+	0.677253i	0.959378	−	0.282125i	$-0.0910392\pi$
$499$	15.1287	+	15.1287i	0.677253	+	0.677253i	−0.282125	+	0.959378i	$0.591039\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			18.6439i			0.831291i	0.909527	+	0.415646i	$0.136444\pi$
$503$			18.6439i			0.831291i	−0.909527	+	0.415646i	$0.863556\pi$
$504$		0			0
$505$			0.527472i			0.0234722i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−11.6243	−	11.6243i	−0.515239	−	0.515239i	0.400888	−	0.916127i	$-0.368702\pi$
$509$	−11.6243	−	11.6243i	−0.515239	−	0.515239i	−0.916127	+	0.400888i	$0.868702\pi$
$510$		0			0
$511$	41.7839			1.84841
$512$		0			0
$513$		0			0
$514$		0			0
$515$	7.28411	+	7.28411i	0.320976	+	0.320976i
$516$		0			0
$517$	6.62459	−	6.62459i	0.291349	−	0.291349i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	36.9052i		−	1.61684i	−0.588603	−	0.808422i	$-0.700322\pi$
$521$		−	36.9052i		−	1.61684i	0.588603	−	0.808422i	$-0.299678\pi$
$522$		0			0
$523$	−6.04158	+	6.04158i	−0.264180	+	0.264180i	−0.826750	−	0.562570i	$-0.809813\pi$
$523$	−6.04158	+	6.04158i	−0.264180	+	0.264180i	0.562570	+	0.826750i	$0.309813\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	12.6669			0.551780
$528$		0			0
$529$	22.4997			0.978250
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−30.0108	+	30.0108i	−1.29991	+	1.29991i
$534$		0			0
$535$		−	20.5159i		−	0.886981i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	0.428614	−	0.428614i	0.0184617	−	0.0184617i
$540$		0			0
$541$	−28.4222	−	28.4222i	−1.22197	−	1.22197i	−0.966932	−	0.255035i	$-0.917913\pi$
$541$	−28.4222	−	28.4222i	−1.22197	−	1.22197i	−0.255035	−	0.966932i	$-0.582087\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	1.12612			0.0482375
$546$		0			0
$547$	−23.3562	−	23.3562i	−0.998640	−	0.998640i	0.00135902	−	0.999999i	$-0.499567\pi$
$547$	−23.3562	−	23.3562i	−0.998640	−	0.998640i	−0.999999	+	0.00135902i	$0.999567\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			24.1337i			1.02813i
$552$		0			0
$553$		−	17.8867i		−	0.760620i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	4.89520	+	4.89520i	0.207416	+	0.207416i	0.803168	−	0.595752i	$-0.203146\pi$
$557$	4.89520	+	4.89520i	0.207416	+	0.207416i	−0.595752	+	0.803168i	$0.703146\pi$
$558$		0			0
$559$	−26.1044			−1.10410
$560$		0			0
$561$		0			0
$562$		0			0
$563$	1.28613	+	1.28613i	0.0542040	+	0.0542040i	0.733689	−	0.679485i	$-0.237797\pi$
$563$	1.28613	+	1.28613i	0.0542040	+	0.0542040i	−0.679485	+	0.733689i	$0.737797\pi$
$564$		0			0
$565$	−0.595755	+	0.595755i	−0.0250636	+	0.0250636i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			11.4799i			0.481261i	0.970617	+	0.240631i	$0.0773543\pi$
$569$			11.4799i			0.481261i	−0.970617	+	0.240631i	$0.922646\pi$
$570$		0			0
$571$	−28.7069	+	28.7069i	−1.20134	+	1.20134i	−0.227587	+	0.973758i	$0.573083\pi$
$571$	−28.7069	+	28.7069i	−1.20134	+	1.20134i	−0.973758	+	0.227587i	$0.926917\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−0.707288			−0.0294960
$576$		0			0
$577$	−20.3419			−0.846842			−0.423421	−	0.905933i	$-0.639171\pi$
$577$	−20.3419			−0.846842			−0.423421	+	0.905933i	$0.639171\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−10.3370	+	10.3370i	−0.428852	+	0.428852i
$582$		0			0
$583$			52.0303i			2.15488i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−25.8136	+	25.8136i	−1.06544	+	1.06544i	−0.0677360	+	0.997703i	$0.521578\pi$
$587$	−25.8136	+	25.8136i	−1.06544	+	1.06544i	−0.997703	+	0.0677360i	$0.978422\pi$
$588$		0			0
$589$	−23.5975	−	23.5975i	−0.972320	−	0.972320i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−4.02945			−0.165470			−0.0827349	−	0.996572i	$-0.526365\pi$
$593$	−4.02945			−0.165470			−0.0827349	+	0.996572i	$0.526365\pi$
$594$		0			0
$595$	3.49338	+	3.49338i	0.143215	+	0.143215i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	31.6701i		−	1.29400i	−0.762489	−	0.647002i	$-0.776023\pi$
$599$		−	31.6701i		−	1.29400i	0.762489	−	0.647002i	$-0.223977\pi$
$600$		0			0
$601$			19.4667i			0.794065i	0.917805	+	0.397032i	$0.129960\pi$
$601$			19.4667i			0.794065i	−0.917805	+	0.397032i	$0.870040\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−9.51768	−	9.51768i	−0.386949	−	0.386949i
$606$		0			0
$607$	−13.6128			−0.552528			−0.276264	−	0.961082i	$-0.589096\pi$
$607$	−13.6128			−0.552528			−0.276264	+	0.961082i	$0.589096\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−5.57057	−	5.57057i	−0.225361	−	0.225361i
$612$		0			0
$613$	−11.1480	+	11.1480i	−0.450265	+	0.450265i	−0.895442	−	0.445177i	$-0.853141\pi$
$613$	−11.1480	+	11.1480i	−0.450265	+	0.450265i	0.445177	+	0.895442i	$0.353141\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	1.96695i		−	0.0791863i	−0.999216	−	0.0395932i	$-0.987394\pi$
$617$		−	1.96695i		−	0.0791863i	0.999216	−	0.0395932i	$-0.0126062\pi$
$618$		0			0
$619$	−7.84144	+	7.84144i	−0.315174	+	0.315174i	−0.846910	−	0.531736i	$-0.821540\pi$
$619$	−7.84144	+	7.84144i	−0.315174	+	0.315174i	0.531736	+	0.846910i	$0.321540\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−28.0372			−1.12329
$624$		0			0
$625$	−1.00000			−0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	0.180499	−	0.180499i	0.00719696	−	0.00719696i
$630$		0			0
$631$		−	0.220729i		−	0.00878708i	−0.999990	−	0.00439354i	$-0.998601\pi$
$631$		−	0.220729i		−	0.00878708i	0.999990	−	0.00439354i	$-0.00139851\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	14.9689	−	14.9689i	0.594024	−	0.594024i
$636$		0			0
$637$	−0.360418	−	0.360418i	−0.0142803	−	0.0142803i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	19.2037			0.758502			0.379251	−	0.925294i	$-0.376182\pi$
$641$	19.2037			0.758502			0.379251	+	0.925294i	$0.376182\pi$
$642$		0			0
$643$	7.17110	+	7.17110i	0.282801	+	0.282801i	0.834225	−	0.551424i	$-0.185915\pi$
$643$	7.17110	+	7.17110i	0.282801	+	0.282801i	−0.551424	+	0.834225i	$0.685915\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	26.4735i		−	1.04078i	−0.853928	−	0.520391i	$-0.825786\pi$
$647$		−	26.4735i		−	1.04078i	0.853928	−	0.520391i	$-0.174214\pi$
$648$		0			0
$649$			6.71153i			0.263451i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−10.5746	−	10.5746i	−0.413815	−	0.413815i	0.469250	−	0.883065i	$-0.344524\pi$
$653$	−10.5746	−	10.5746i	−0.413815	−	0.413815i	−0.883065	+	0.469250i	$0.844524\pi$
$654$		0			0
$655$	6.60789			0.258192
$656$		0			0
$657$		0			0
$658$		0			0
$659$	24.1291	+	24.1291i	0.939937	+	0.939937i	0.998296	−	0.0583584i	$-0.0185866\pi$
$659$	24.1291	+	24.1291i	0.939937	+	0.939937i	−0.0583584	+	0.998296i	$0.518587\pi$
$660$		0			0
$661$	23.4294	−	23.4294i	0.911299	−	0.911299i	−0.0850756	−	0.996374i	$-0.527113\pi$
$661$	23.4294	−	23.4294i	0.911299	−	0.911299i	0.996374	+	0.0850756i	$0.0271132\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	13.0158i		−	0.504732i
$666$		0			0
$667$	2.47487	−	2.47487i	0.0958273	−	0.0958273i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−45.4506			−1.75460
$672$		0			0
$673$	−41.8069			−1.61154			−0.805769	−	0.592230i	$-0.798248\pi$
$673$	−41.8069			−1.61154			−0.805769	+	0.592230i	$0.798248\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	22.7350	−	22.7350i	0.873776	−	0.873776i	−0.119106	−	0.992882i	$-0.538003\pi$
$677$	22.7350	−	22.7350i	0.873776	−	0.873776i	0.992882	+	0.119106i	$0.0380027\pi$
$678$		0			0
$679$		−	12.8086i		−	0.491550i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	34.4402	−	34.4402i	1.31782	−	1.31782i	0.402315	−	0.915501i	$-0.368206\pi$
$683$	34.4402	−	34.4402i	1.31782	−	1.31782i	0.915501	−	0.402315i	$-0.131794\pi$
$684$		0			0
$685$	−7.27190	−	7.27190i	−0.277845	−	0.277845i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	43.7519			1.66682
$690$		0			0
$691$	−16.0991	−	16.0991i	−0.612438	−	0.612438i	0.331143	−	0.943581i	$-0.392566\pi$
$691$	−16.0991	−	16.0991i	−0.612438	−	0.612438i	−0.943581	+	0.331143i	$0.892566\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			7.05110i			0.267463i
$696$		0			0
$697$		−	18.8915i		−	0.715567i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	30.0507	+	30.0507i	1.13500	+	1.13500i	0.989334	+	0.145666i	$0.0465325\pi$
$701$	30.0507	+	30.0507i	1.13500	+	1.13500i	0.145666	+	0.989334i	$0.453467\pi$
$702$		0			0
$703$	−0.672512			−0.0253643
$704$		0			0
$705$		0			0
$706$		0			0
$707$	0.995412	+	0.995412i	0.0374363	+	0.0374363i
$708$		0			0
$709$	−12.9188	+	12.9188i	−0.485176	+	0.485176i	−0.906780	−	0.421604i	$-0.861467\pi$
$709$	−12.9188	+	12.9188i	−0.485176	+	0.485176i	0.421604	+	0.906780i	$0.361467\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			4.83977i			0.181251i
$714$		0			0
$715$	−14.5440	+	14.5440i	−0.543913	+	0.543913i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	17.0356			0.635319			0.317659	−	0.948205i	$-0.397103\pi$
$719$	17.0356			0.635319			0.317659	+	0.948205i	$0.397103\pi$
$720$		0			0
$721$	27.4922			1.02386
$722$		0			0
$723$		0			0
$724$		0			0
$725$	3.49909	−	3.49909i	0.129953	−	0.129953i
$726$		0			0
$727$		−	31.7051i		−	1.17588i	−0.808905	−	0.587939i	$-0.799939\pi$
$727$		−	31.7051i		−	1.17588i	0.808905	−	0.587939i	$-0.200061\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	8.21624	−	8.21624i	0.303888	−	0.303888i
$732$		0			0
$733$	3.87657	+	3.87657i	0.143184	+	0.143184i	0.775065	−	0.631881i	$-0.217717\pi$
$733$	3.87657	+	3.87657i	0.143184	+	0.143184i	−0.631881	+	0.775065i	$0.717717\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	24.4584			0.900937
$738$		0			0
$739$	−11.3024	−	11.3024i	−0.415766	−	0.415766i	0.467975	−	0.883742i	$-0.344984\pi$
$739$	−11.3024	−	11.3024i	−0.415766	−	0.415766i	−0.883742	+	0.467975i	$0.844984\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	30.7210i		−	1.12704i	−0.826102	−	0.563521i	$-0.809446\pi$
$743$		−	30.7210i		−	1.12704i	0.826102	−	0.563521i	$-0.190554\pi$
$744$		0			0
$745$			12.4379i			0.455690i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−38.7163	−	38.7163i	−1.41466	−	1.41466i
$750$		0			0
$751$	−16.4695			−0.600981			−0.300491	−	0.953785i	$-0.597150\pi$
$751$	−16.4695			−0.600981			−0.300491	+	0.953785i	$0.597150\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	15.6863	+	15.6863i	0.570883	+	0.570883i
$756$		0			0
$757$	21.9737	−	21.9737i	0.798649	−	0.798649i	−0.184233	−	0.982883i	$-0.558980\pi$
$757$	21.9737	−	21.9737i	0.798649	−	0.798649i	0.982883	+	0.184233i	$0.0589802\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			5.91749i			0.214509i	0.994232	+	0.107254i	$0.0342060\pi$
$761$			5.91749i			0.214509i	−0.994232	+	0.107254i	$0.965794\pi$
$762$		0			0
$763$	2.12513	−	2.12513i	0.0769349	−	0.0769349i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	5.64368			0.203782
$768$		0			0
$769$	10.7206			0.386596			0.193298	−	0.981140i	$-0.438082\pi$
$769$	10.7206			0.386596			0.193298	+	0.981140i	$0.438082\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	16.8329	−	16.8329i	0.605438	−	0.605438i	−0.336312	−	0.941750i	$-0.609180\pi$
$773$	16.8329	−	16.8329i	0.605438	−	0.605438i	0.941750	+	0.336312i	$0.109180\pi$
$774$		0			0
$775$			6.84272i			0.245798i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−35.1934	+	35.1934i	−1.26094	+	1.26094i
$780$		0			0
$781$	−27.5147	−	27.5147i	−0.984554	−	0.984554i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−5.26352			−0.187863
$786$		0			0
$787$	25.4619	+	25.4619i	0.907619	+	0.907619i	0.996080	−	0.0884603i	$-0.0281946\pi$
$787$	25.4619	+	25.4619i	0.907619	+	0.907619i	−0.0884603	+	0.996080i	$0.528195\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			2.24854i			0.0799489i
$792$		0			0
$793$			38.2191i			1.35720i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−7.14518	−	7.14518i	−0.253095	−	0.253095i	0.569143	−	0.822238i	$-0.307275\pi$
$797$	−7.14518	−	7.14518i	−0.253095	−	0.253095i	−0.822238	+	0.569143i	$0.807275\pi$
$798$		0			0
$799$	3.50662			0.124055
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−54.7526	−	54.7526i	−1.93218	−	1.93218i
$804$		0			0
$805$	−1.33475	+	1.33475i	−0.0470437	+	0.0470437i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	12.4413i		−	0.437412i	−0.975791	−	0.218706i	$-0.929817\pi$
$809$		−	12.4413i		−	0.437412i	0.975791	−	0.218706i	$-0.0701835\pi$
$810$		0			0
$811$	30.6494	−	30.6494i	1.07624	−	1.07624i	0.0794022	−	0.996843i	$-0.474699\pi$
$811$	30.6494	−	30.6494i	1.07624	−	1.07624i	0.996843	−	0.0794022i	$-0.0253011\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−2.99290			−0.104837
$816$		0			0
$817$	−30.6125			−1.07099
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−8.84907	+	8.84907i	−0.308835	+	0.308835i	−0.844457	−	0.535623i	$-0.820077\pi$
$821$	−8.84907	+	8.84907i	−0.308835	+	0.308835i	0.535623	+	0.844457i	$0.320077\pi$
$822$		0			0
$823$		−	11.7501i		−	0.409583i	−0.978806	−	0.204792i	$-0.934348\pi$
$823$		−	11.7501i		−	0.409583i	0.978806	−	0.204792i	$-0.0656516\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−3.40407	+	3.40407i	−0.118371	+	0.118371i	−0.763811	−	0.645440i	$-0.776674\pi$
$827$	−3.40407	+	3.40407i	−0.118371	+	0.118371i	0.645440	+	0.763811i	$0.276674\pi$
$828$		0			0
$829$	24.8718	+	24.8718i	0.863834	+	0.863834i	0.991781	−	0.127947i	$-0.0408389\pi$
$829$	24.8718	+	24.8718i	0.863834	+	0.863834i	−0.127947	+	0.991781i	$0.540839\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	0.226880			0.00786092
$834$		0			0
$835$	12.8414	+	12.8414i	0.444393	+	0.444393i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			3.26196i			0.112615i	0.998413	+	0.0563076i	$0.0179328\pi$
$839$			3.26196i			0.112615i	−0.998413	+	0.0563076i	$0.982067\pi$
$840$		0			0
$841$		−	4.51268i		−	0.155610i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	3.03753	+	3.03753i	0.104494	+	0.104494i
$846$		0			0
$847$	−35.9223			−1.23430
$848$		0			0
$849$		0			0
$850$		0			0
$851$	0.0689649	+	0.0689649i	0.00236409	+	0.00236409i
$852$		0			0
$853$	4.02276	−	4.02276i	0.137737	−	0.137737i	−0.634877	−	0.772613i	$-0.718949\pi$
$853$	4.02276	−	4.02276i	0.137737	−	0.137737i	0.772613	+	0.634877i	$0.218949\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			44.6563i			1.52543i	0.646736	+	0.762714i	$0.276134\pi$
$857$			44.6563i			1.52543i	−0.646736	+	0.762714i	$0.723866\pi$
$858$		0			0
$859$	−5.22864	+	5.22864i	−0.178399	+	0.178399i	−0.790658	−	0.612259i	$-0.790261\pi$
$859$	−5.22864	+	5.22864i	−0.178399	+	0.178399i	0.612259	+	0.790658i	$0.290261\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	36.9653			1.25831			0.629157	−	0.777278i	$-0.283400\pi$
$863$	36.9653			1.25831			0.629157	+	0.777278i	$0.283400\pi$
$864$		0			0
$865$	12.0709			0.410423
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−23.4383	+	23.4383i	−0.795089	+	0.795089i
$870$		0			0
$871$		−	20.5669i		−	0.696883i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−1.88714	+	1.88714i	−0.0637968	+	0.0637968i
$876$		0			0
$877$	9.40192	+	9.40192i	0.317480	+	0.317480i	0.847799	−	0.530318i	$-0.177928\pi$
$877$	9.40192	+	9.40192i	0.317480	+	0.317480i	−0.530318	+	0.847799i	$0.677928\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	10.3069			0.347248			0.173624	−	0.984812i	$-0.444452\pi$
$881$	10.3069			0.347248			0.173624	+	0.984812i	$0.444452\pi$
$882$		0			0
$883$	−34.3375	−	34.3375i	−1.15555	−	1.15555i	−0.985422	−	0.170128i	$-0.945582\pi$
$883$	−34.3375	−	34.3375i	−1.15555	−	1.15555i	−0.170128	−	0.985422i	$-0.554418\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			6.79523i			0.228161i	0.993471	+	0.114081i	$0.0363922\pi$
$887$			6.79523i			0.228161i	−0.993471	+	0.114081i	$0.963608\pi$
$888$		0			0
$889$		−	56.4968i		−	1.89484i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−6.53256	−	6.53256i	−0.218604	−	0.218604i
$894$		0			0
$895$	−3.42945			−0.114634
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−23.9433	−	23.9433i	−0.798554	−	0.798554i
$900$		0			0
$901$	−13.7707	+	13.7707i	−0.458769	+	0.458769i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	6.31578i		−	0.209944i
$906$		0			0
$907$	3.82391	−	3.82391i	0.126971	−	0.126971i	−0.640766	−	0.767737i	$-0.721383\pi$
$907$	3.82391	−	3.82391i	0.126971	−	0.126971i	0.767737	+	0.640766i	$0.221383\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−18.9169			−0.626743			−0.313372	−	0.949631i	$-0.601459\pi$
$911$	−18.9169			−0.626743			−0.313372	+	0.949631i	$0.601459\pi$
$912$		0			0
$913$	27.0908			0.896574
$914$		0			0
$915$		0			0
$916$		0			0
$917$	12.4700	−	12.4700i	0.411795	−	0.411795i
$918$		0			0
$919$		−	48.9075i		−	1.61331i	−0.591022	−	0.806655i	$-0.701276\pi$
$919$		−	48.9075i		−	1.61331i	0.591022	−	0.806655i	$-0.298724\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−23.1369	+	23.1369i	−0.761562	+	0.761562i
$924$		0			0
$925$	0.0975060	+	0.0975060i	0.00320598	+	0.00320598i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−35.4660			−1.16360			−0.581801	−	0.813331i	$-0.697652\pi$
$929$	−35.4660			−1.16360			−0.581801	+	0.813331i	$0.697652\pi$
$930$		0			0
$931$	−0.422660	−	0.422660i	−0.0138521	−	0.0138521i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	9.15528i		−	0.299410i
$936$		0			0
$937$		−	56.4991i		−	1.84575i	−0.385105	−	0.922873i	$-0.625835\pi$
$937$		−	56.4991i		−	1.84575i	0.385105	−	0.922873i	$-0.374165\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	2.86034	+	2.86034i	0.0932445	+	0.0932445i	0.752190	−	0.658946i	$-0.228997\pi$
$941$	2.86034	+	2.86034i	0.0932445	+	0.0932445i	−0.658946	+	0.752190i	$0.728997\pi$
$942$		0			0
$943$	7.21805			0.235052
$944$		0			0
$945$		0			0
$946$		0			0
$947$	5.86681	+	5.86681i	0.190646	+	0.190646i	0.795975	−	0.605329i	$-0.206959\pi$
$947$	5.86681	+	5.86681i	0.190646	+	0.190646i	−0.605329	+	0.795975i	$0.706959\pi$
$948$		0			0
$949$	−46.0411	+	46.0411i	−1.49456	+	1.49456i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			25.0238i			0.810599i	0.914184	+	0.405299i	$0.132833\pi$
$953$			25.0238i			0.810599i	−0.914184	+	0.405299i	$0.867167\pi$
$954$		0			0
$955$	5.48029	−	5.48029i	0.177338	−	0.177338i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−27.4461			−0.886281
$960$		0			0
$961$	15.8228			0.510412
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−8.01771	+	8.01771i	−0.258099	+	0.258099i
$966$		0			0
$967$			16.6523i			0.535502i	0.963488	+	0.267751i	$0.0862804\pi$
$967$			16.6523i			0.535502i	−0.963488	+	0.267751i	$0.913720\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−30.6552	+	30.6552i	−0.983771	+	0.983771i	−0.999870	−	0.0160991i	$-0.994875\pi$
$971$	−30.6552	+	30.6552i	−0.983771	+	0.983771i	0.0160991	+	0.999870i	$0.494875\pi$
$972$		0			0
$973$	13.3064	+	13.3064i	0.426583	+	0.426583i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	13.4307			0.429687			0.214844	−	0.976648i	$-0.431076\pi$
$977$	13.4307			0.429687			0.214844	+	0.976648i	$0.431076\pi$
$978$		0			0
$979$	36.7392	+	36.7392i	1.17419	+	1.17419i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	7.94549i		−	0.253422i	−0.991940	−	0.126711i	$-0.959558\pi$
$983$		−	7.94549i		−	0.253422i	0.991940	−	0.126711i	$-0.0404420\pi$
$984$		0			0
$985$			1.55565i			0.0495672i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	3.13925	+	3.13925i	0.0998225	+	0.0998225i
$990$		0			0
$991$	25.0787			0.796652			0.398326	−	0.917244i	$-0.369591\pi$
$991$	25.0787			0.796652			0.398326	+	0.917244i	$0.369591\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−10.0919	−	10.0919i	−0.319936	−	0.319936i
$996$		0			0
$997$	36.1819	−	36.1819i	1.14589	−	1.14589i	0.158539	−	0.987353i	$-0.449322\pi$
$997$	36.1819	−	36.1819i	1.14589	−	1.14589i	0.987353	−	0.158539i	$-0.0506783\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2880.2.t.c.721.6		16
3.2	odd	2		320.2.l.a.81.5		16
4.3	odd	2		720.2.t.c.541.2		16
12.11	even	2		80.2.l.a.61.7	yes	16
15.2	even	4		1600.2.q.g.849.5		16
15.8	even	4		1600.2.q.h.849.4		16
15.14	odd	2		1600.2.l.i.401.4		16
16.5	even	4	inner	2880.2.t.c.2161.7		16
16.11	odd	4		720.2.t.c.181.2		16
24.5	odd	2		640.2.l.a.161.4		16
24.11	even	2		640.2.l.b.161.5		16
48.5	odd	4		320.2.l.a.241.5		16
48.11	even	4		80.2.l.a.21.7	✓	16
48.29	odd	4		640.2.l.a.481.4		16
48.35	even	4		640.2.l.b.481.5		16
60.23	odd	4		400.2.q.g.349.3		16
60.47	odd	4		400.2.q.h.349.6		16
60.59	even	2		400.2.l.h.301.2		16
96.5	odd	8		5120.2.a.u.1.4		8
96.11	even	8		5120.2.a.v.1.4		8
96.53	odd	8		5120.2.a.t.1.5		8
96.59	even	8		5120.2.a.s.1.5		8
240.53	even	4		1600.2.q.g.49.5		16
240.59	even	4		400.2.l.h.101.2		16
240.107	odd	4		400.2.q.g.149.3		16
240.149	odd	4		1600.2.l.i.1201.4		16
240.197	even	4		1600.2.q.h.49.4		16
240.203	odd	4		400.2.q.h.149.6		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
80.2.l.a.21.7	✓	16	48.11	even	4
80.2.l.a.61.7	yes	16	12.11	even	2
320.2.l.a.81.5		16	3.2	odd	2
320.2.l.a.241.5		16	48.5	odd	4
400.2.l.h.101.2		16	240.59	even	4
400.2.l.h.301.2		16	60.59	even	2
400.2.q.g.149.3		16	240.107	odd	4
400.2.q.g.349.3		16	60.23	odd	4
400.2.q.h.149.6		16	240.203	odd	4
400.2.q.h.349.6		16	60.47	odd	4
640.2.l.a.161.4		16	24.5	odd	2
640.2.l.a.481.4		16	48.29	odd	4
640.2.l.b.161.5		16	24.11	even	2
640.2.l.b.481.5		16	48.35	even	4
720.2.t.c.181.2		16	16.11	odd	4
720.2.t.c.541.2		16	4.3	odd	2
1600.2.l.i.401.4		16	15.14	odd	2
1600.2.l.i.1201.4		16	240.149	odd	4
1600.2.q.g.49.5		16	240.53	even	4
1600.2.q.g.849.5		16	15.2	even	4
1600.2.q.h.49.4		16	240.197	even	4
1600.2.q.h.849.4		16	15.8	even	4
2880.2.t.c.721.6		16	1.1	even	1	trivial
2880.2.t.c.2161.7		16	16.5	even	4	inner
5120.2.a.s.1.5		8	96.59	even	8
5120.2.a.t.1.5		8	96.53	odd	8
5120.2.a.u.1.4		8	96.5	odd	8
5120.2.a.v.1.4		8	96.11	even	8

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(0.707107 - 0.707107i) q^{5} -2.66881i q^{7} +O(q^{10})\)	\(q+(0.707107 - 0.707107i) q^{5} -2.66881i q^{7} +(-3.49714 + 3.49714i) q^{11} +(2.94072 + 2.94072i) q^{13} -1.85116 q^{17} +(3.44856 + 3.44856i) q^{19} -0.707288i q^{23} -1.00000i q^{25} +(3.49909 + 3.49909i) q^{29} -6.84272 q^{31} +(-1.88714 - 1.88714i) q^{35} +(-0.0975060 + 0.0975060i) q^{37} +10.2052i q^{41} +(-4.43844 + 4.43844i) q^{43} -1.89428 q^{47} -0.122561 q^{49} +(7.43897 - 7.43897i) q^{53} +4.94571i q^{55} +(0.959574 - 0.959574i) q^{59} +(6.49825 + 6.49825i) q^{61} +4.15881 q^{65} +(-3.49691 - 3.49691i) q^{67} +7.86777i q^{71} +15.6564i q^{73} +(9.33322 + 9.33322i) q^{77} +6.70212 q^{79} +(-3.87327 - 3.87327i) q^{83} +(-1.30896 + 1.30896i) q^{85} -10.5055i q^{89} +(7.84824 - 7.84824i) q^{91} +4.87701 q^{95} +4.79937 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q - 8 q^{11} + 8 q^{19} + 16 q^{29} - 16 q^{37} - 8 q^{43} - 40 q^{47} - 16 q^{49} - 16 q^{53} - 8 q^{59} + 16 q^{61} - 40 q^{67} - 16 q^{77} - 16 q^{79} + 40 q^{83} - 16 q^{85} - 32 q^{91} + 32 q^{95}+O(q^{100}) \) 16 * q - 8 * q^11 + 8 * q^19 + 16 * q^29 - 16 * q^37 - 8 * q^43 - 40 * q^47 - 16 * q^49 - 16 * q^53 - 8 * q^59 + 16 * q^61 - 40 * q^67 - 16 * q^77 - 16 * q^79 + 40 * q^83 - 16 * q^85 - 32 * q^91 + 32 * q^95

Introduction

L-functions

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