LMFDB - Embedded newform 2880.2.t.b.721.3

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:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	8.0.18939904.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} + 14x^{6} - 28x^{5} + 43x^{4} - 44x^{3} + 30x^{2} - 12x + 2 $ x^8 - 4x^7 + 14x^6 - 28x^5 + 43x^4 - 44x^3 + 30x^2 - 12*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 240)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			721.3
Root			$0.500000 + 2.10607i$ of defining polynomial
Character	$\chi$	$=$	2880.721
Dual form			2880.2.t.b.2161.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.707107 - 0.707107i) q^{5} -1.41421i q^{7} +O(q^{10})$	$q+(0.707107 - 0.707107i) q^{5} -1.41421i q^{7} +(-0.526602 + 0.526602i) q^{11} +(3.68554 + 3.68554i) q^{13} +1.57316 q^{17} +(0.383719 + 0.383719i) q^{19} +6.42429i q^{23} -1.00000i q^{25} +(-4.38372 - 4.38372i) q^{29} +5.75481 q^{31} +(-1.00000 - 1.00000i) q^{35} +(1.91032 - 1.91032i) q^{37} +11.9747i q^{41} +(1.12845 - 1.12845i) q^{43} -2.09311 q^{47} +5.00000 q^{49} +(-9.55274 + 9.55274i) q^{53} +0.744728i q^{55} +(4.61971 - 4.61971i) q^{59} +(4.53003 + 4.53003i) q^{61} +5.21215 q^{65} +(5.59587 + 5.59587i) q^{67} -11.8816i q^{71} -12.1995i q^{73} +(0.744728 + 0.744728i) q^{77} +3.33051 q^{79} +(6.31788 + 6.31788i) q^{83} +(1.11239 - 1.11239i) q^{85} -5.42847i q^{89} +(5.21215 - 5.21215i) q^{91} +0.542661 q^{95} +2.13853 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 8 q^{11} + 8 q^{13} - 8 q^{17} - 8 q^{19} - 24 q^{29} - 8 q^{31} - 8 q^{35} - 8 q^{37} + 40 q^{49} + 8 q^{59} - 16 q^{61} + 8 q^{65} + 8 q^{77} + 40 q^{79} + 32 q^{83} + 8 q^{85} + 8 q^{91} - 16 q^{95} - 48 q^{97}+O(q^{100}) $ 8 * q + 8 * q^11 + 8 * q^13 - 8 * q^17 - 8 * q^19 - 24 * q^29 - 8 * q^31 - 8 * q^35 - 8 * q^37 + 40 * q^49 + 8 * q^59 - 16 * q^61 + 8 * q^65 + 8 * q^77 + 40 * q^79 + 32 * q^83 + 8 * q^85 + 8 * q^91 - 16 * q^95 - 48 * q^97

Display 100 coefficients 10 coefficients

Character values

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

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

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
$7$		−	1.41421i		−	0.534522i	−0.963624	−	0.267261i	$-0.913881\pi$
$7$		−	1.41421i		−	0.534522i	0.963624	−	0.267261i	$-0.0861187\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−0.526602	+	0.526602i	−0.158777	+	0.158777i	−0.782024	−	0.623248i	$-0.785813\pi$
$11$	−0.526602	+	0.526602i	−0.158777	+	0.158777i	0.623248	+	0.782024i	$0.285813\pi$
$12$		0			0
$13$	3.68554	+	3.68554i	1.02219	+	1.02219i	0.999748	+	0.0224377i	$0.00714275\pi$
$13$	3.68554	+	3.68554i	1.02219	+	1.02219i	0.0224377	+	0.999748i	$0.492857\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	1.57316			0.381546			0.190773	−	0.981634i	$-0.438901\pi$
$17$	1.57316			0.381546			0.190773	+	0.981634i	$0.438901\pi$
$18$		0			0
$19$	0.383719	+	0.383719i	0.0880312	+	0.0880312i	0.749751	−	0.661720i	$-0.230173\pi$
$19$	0.383719	+	0.383719i	0.0880312	+	0.0880312i	−0.661720	+	0.749751i	$0.730173\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			6.42429i			1.33956i	0.742561	+	0.669779i	$0.233611\pi$
$23$			6.42429i			1.33956i	−0.742561	+	0.669779i	$0.766389\pi$
$24$		0			0
$25$		−	1.00000i		−	0.200000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−4.38372	−	4.38372i	−0.814036	−	0.814036i	0.171200	−	0.985236i	$-0.445236\pi$
$29$	−4.38372	−	4.38372i	−0.814036	−	0.814036i	−0.985236	+	0.171200i	$0.945236\pi$
$30$		0			0
$31$	5.75481			1.03359			0.516797	−	0.856108i	$-0.327124\pi$
$31$	5.75481			1.03359			0.516797	+	0.856108i	$0.327124\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−1.00000	−	1.00000i	−0.169031	−	0.169031i
$36$		0			0
$37$	1.91032	−	1.91032i	0.314055	−	0.314055i	−0.532423	−	0.846478i	$-0.678719\pi$
$37$	1.91032	−	1.91032i	0.314055	−	0.314055i	0.846478	+	0.532423i	$0.178719\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			11.9747i			1.87014i	0.354463	+	0.935070i	$0.384664\pi$
$41$			11.9747i			1.87014i	−0.354463	+	0.935070i	$0.615336\pi$
$42$		0			0
$43$	1.12845	−	1.12845i	0.172087	−	0.172087i	−0.615809	−	0.787896i	$-0.711171\pi$
$43$	1.12845	−	1.12845i	0.172087	−	0.172087i	0.787896	+	0.615809i	$0.211171\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−2.09311			−0.305311			−0.152655	−	0.988279i	$-0.548782\pi$
$47$	−2.09311			−0.305311			−0.152655	+	0.988279i	$0.548782\pi$
$48$		0			0
$49$	5.00000			0.714286
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−9.55274	+	9.55274i	−1.31217	+	1.31217i	−0.392356	+	0.919813i	$0.628340\pi$
$53$	−9.55274	+	9.55274i	−1.31217	+	1.31217i	−0.919813	+	0.392356i	$0.871660\pi$
$54$		0			0
$55$			0.744728i			0.100419i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	4.61971	−	4.61971i	0.601435	−	0.601435i	−0.339258	−	0.940693i	$-0.610176\pi$
$59$	4.61971	−	4.61971i	0.601435	−	0.601435i	0.940693	+	0.339258i	$0.110176\pi$
$60$		0			0
$61$	4.53003	+	4.53003i	0.580011	+	0.580011i	0.934906	−	0.354895i	$-0.115483\pi$
$61$	4.53003	+	4.53003i	0.580011	+	0.580011i	−0.354895	+	0.934906i	$0.615483\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	5.21215			0.646487
$66$		0			0
$67$	5.59587	+	5.59587i	0.683644	+	0.683644i	0.960819	−	0.277176i	$-0.0893984\pi$
$67$	5.59587	+	5.59587i	0.683644	+	0.683644i	−0.277176	+	0.960819i	$0.589398\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	11.8816i		−	1.41009i	−0.709162	−	0.705045i	$-0.750927\pi$
$71$		−	11.8816i		−	1.41009i	0.709162	−	0.705045i	$-0.249073\pi$
$72$		0			0
$73$		−	12.1995i		−	1.42785i	−0.700225	−	0.713923i	$-0.746917\pi$
$73$		−	12.1995i		−	1.42785i	0.700225	−	0.713923i	$-0.253083\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	0.744728	+	0.744728i	0.0848696	+	0.0848696i
$78$		0			0
$79$	3.33051			0.374712			0.187356	−	0.982292i	$-0.440008\pi$
$79$	3.33051			0.374712			0.187356	+	0.982292i	$0.440008\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	6.31788	+	6.31788i	0.693478	+	0.693478i	0.962995	−	0.269518i	$-0.0868643\pi$
$83$	6.31788	+	6.31788i	0.693478	+	0.693478i	−0.269518	+	0.962995i	$0.586864\pi$
$84$		0			0
$85$	1.11239	−	1.11239i	0.120655	−	0.120655i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	5.42847i		−	0.575416i	−0.957718	−	0.287708i	$-0.907107\pi$
$89$		−	5.42847i		−	0.575416i	0.957718	−	0.287708i	$-0.0928933\pi$
$90$		0			0
$91$	5.21215	−	5.21215i	0.546381	−	0.546381i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	0.542661			0.0556758
$96$		0			0
$97$	2.13853			0.217134			0.108567	−	0.994089i	$-0.465374\pi$
$97$	2.13853			0.217134			0.108567	+	0.994089i	$0.465374\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−5.83851	+	5.83851i	−0.580953	+	0.580953i	−0.935165	−	0.354212i	$-0.884749\pi$
$101$	−5.83851	+	5.83851i	−0.580953	+	0.580953i	0.354212	+	0.935165i	$0.384749\pi$
$102$		0			0
$103$		−	16.4385i		−	1.61974i	−0.586611	−	0.809869i	$-0.699538\pi$
$103$		−	16.4385i		−	1.61974i	0.586611	−	0.809869i	$-0.300462\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	5.08532	−	5.08532i	0.491617	−	0.491617i	−0.417199	−	0.908815i	$-0.636988\pi$
$107$	5.08532	−	5.08532i	0.491617	−	0.491617i	0.908815	+	0.417199i	$0.136988\pi$
$108$		0			0
$109$	5.47682	+	5.47682i	0.524585	+	0.524585i	0.918953	−	0.394368i	$-0.129036\pi$
$109$	5.47682	+	5.47682i	0.524585	+	0.524585i	−0.394368	+	0.918953i	$0.629036\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	14.9575			1.40709			0.703544	−	0.710652i	$-0.251600\pi$
$113$	14.9575			1.40709			0.703544	+	0.710652i	$0.251600\pi$
$114$		0			0
$115$	4.54266	+	4.54266i	0.423605	+	0.423605i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	2.22478i		−	0.203945i
$120$		0			0
$121$			10.4454i			0.949580i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−0.707107	−	0.707107i	−0.0632456	−	0.0632456i
$126$		0			0
$127$	19.3890			1.72049			0.860246	−	0.509880i	$-0.170310\pi$
$127$	19.3890			1.72049			0.860246	+	0.509880i	$0.170310\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	4.78350	+	4.78350i	0.417936	+	0.417936i	0.884492	−	0.466556i	$-0.154505\pi$
$131$	4.78350	+	4.78350i	0.417936	+	0.417936i	−0.466556	+	0.884492i	$0.654505\pi$
$132$		0			0
$133$	0.542661	−	0.542661i	0.0470547	−	0.0470547i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			3.91630i			0.334592i	0.985907	+	0.167296i	$0.0535036\pi$
$137$			3.91630i			0.334592i	−0.985907	+	0.167296i	$0.946496\pi$
$138$		0			0
$139$	−15.1665	+	15.1665i	−1.28640	+	1.28640i	−0.349447	+	0.936956i	$0.613630\pi$
$139$	−15.1665	+	15.1665i	−1.28640	+	1.28640i	−0.936956	+	0.349447i	$0.886370\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−3.88163			−0.324598
$144$		0			0
$145$	−6.19951			−0.514842
$146$		0			0
$147$		0			0
$148$		0			0
$149$	1.53003	−	1.53003i	0.125345	−	0.125345i	−0.641651	−	0.766996i	$-0.721750\pi$
$149$	1.53003	−	1.53003i	0.125345	−	0.125345i	0.766996	+	0.641651i	$0.221750\pi$
$150$		0			0
$151$		−	11.8158i		−	0.961556i	−0.876842	−	0.480778i	$-0.840354\pi$
$151$		−	11.8158i		−	0.961556i	0.876842	−	0.480778i	$-0.159646\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	4.06926	−	4.06926i	0.326851	−	0.326851i
$156$		0			0
$157$	−10.1961	−	10.1961i	−0.813736	−	0.813736i	0.171455	−	0.985192i	$-0.445153\pi$
$157$	−10.1961	−	10.1961i	−0.813736	−	0.813736i	−0.985192	+	0.171455i	$0.945153\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	9.08532			0.716024
$162$		0			0
$163$	12.6991	+	12.6991i	0.994666	+	0.994666i	0.999986	−	0.00531949i	$-0.00169325\pi$
$163$	12.6991	+	12.6991i	0.994666	+	0.994666i	−0.00531949	+	0.999986i	$0.501693\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	19.4775i		−	1.50721i	−0.657325	−	0.753607i	$-0.728312\pi$
$167$		−	19.4775i		−	1.50721i	0.657325	−	0.753607i	$-0.271688\pi$
$168$		0			0
$169$			14.1665i			1.08973i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−10.0085	−	10.0085i	−0.760929	−	0.760929i	0.215561	−	0.976490i	$-0.430842\pi$
$173$	−10.0085	−	10.0085i	−0.760929	−	0.760929i	−0.976490	+	0.215561i	$0.930842\pi$
$174$		0			0
$175$	−1.41421			−0.106904
$176$		0			0
$177$		0			0
$178$		0			0
$179$	4.18346	+	4.18346i	0.312686	+	0.312686i	0.845949	−	0.533263i	$-0.179034\pi$
$179$	4.18346	+	4.18346i	0.312686	+	0.312686i	−0.533263	+	0.845949i	$0.679034\pi$
$180$		0			0
$181$	−10.4117	+	10.4117i	−0.773893	+	0.773893i	−0.978785	−	0.204892i	$-0.934316\pi$
$181$	−10.4117	+	10.4117i	−0.773893	+	0.773893i	0.204892	+	0.978785i	$0.434316\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	2.70160i		−	0.198626i
$186$		0			0
$187$	−0.828427	+	0.828427i	−0.0605806	+	0.0605806i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	6.47843			0.468763			0.234381	−	0.972145i	$-0.424694\pi$
$191$	6.47843			0.468763			0.234381	+	0.972145i	$0.424694\pi$
$192$		0			0
$193$	0.696756			0.0501536			0.0250768	−	0.999686i	$-0.492017\pi$
$193$	0.696756			0.0501536			0.0250768	+	0.999686i	$0.492017\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	11.6364	−	11.6364i	0.829062	−	0.829062i	−0.158325	−	0.987387i	$-0.550609\pi$
$197$	11.6364	−	11.6364i	0.829062	−	0.829062i	0.987387	+	0.158325i	$0.0506094\pi$
$198$		0			0
$199$			22.8759i			1.62163i	0.585305	+	0.810813i	$0.300975\pi$
$199$			22.8759i			1.62163i	−0.585305	+	0.810813i	$0.699025\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−6.19951	+	6.19951i	−0.435121	+	0.435121i
$204$		0			0
$205$	8.46742	+	8.46742i	0.591390	+	0.591390i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−0.404135			−0.0279546
$210$		0			0
$211$	−13.2780	−	13.2780i	−0.914094	−	0.914094i	0.0824973	−	0.996591i	$-0.473710\pi$
$211$	−13.2780	−	13.2780i	−0.914094	−	0.914094i	−0.996591	+	0.0824973i	$0.973710\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	1.59587i		−	0.108837i
$216$		0			0
$217$		−	8.13853i		−	0.552479i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	5.79793	+	5.79793i	0.390011	+	0.390011i
$222$		0			0
$223$	−5.61012			−0.375681			−0.187841	−	0.982200i	$-0.560149\pi$
$223$	−5.61012			−0.375681			−0.187841	+	0.982200i	$0.560149\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	11.5349	+	11.5349i	0.765597	+	0.765597i	0.977328	−	0.211731i	$-0.0679102\pi$
$227$	11.5349	+	11.5349i	0.765597	+	0.765597i	−0.211731	+	0.977328i	$0.567910\pi$
$228$		0			0
$229$	−7.74702	+	7.74702i	−0.511938	+	0.511938i	−0.915120	−	0.403182i	$-0.867904\pi$
$229$	−7.74702	+	7.74702i	−0.511938	+	0.511938i	0.403182	+	0.915120i	$0.367904\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	6.97636i		−	0.457037i	−0.973540	−	0.228518i	$-0.926612\pi$
$233$		−	6.97636i		−	0.457037i	0.973540	−	0.228518i	$-0.0733881\pi$
$234$		0			0
$235$	−1.48005	+	1.48005i	−0.0965478	+	0.0965478i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	4.77105			0.308613			0.154307	−	0.988023i	$-0.450686\pi$
$239$	4.77105			0.308613			0.154307	+	0.988023i	$0.450686\pi$
$240$		0			0
$241$	30.5307			1.96666			0.983328	−	0.181842i	$-0.0582061\pi$
$241$	30.5307			1.96666			0.983328	+	0.181842i	$0.0582061\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	3.53553	−	3.53553i	0.225877	−	0.225877i
$246$		0			0
$247$			2.82843i			0.179969i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−12.5266	+	12.5266i	−0.790672	+	0.790672i	−0.981603	−	0.190931i	$-0.938849\pi$
$251$	−12.5266	+	12.5266i	−0.790672	+	0.790672i	0.190931	+	0.981603i	$0.438849\pi$
$252$		0			0
$253$	−3.38305	−	3.38305i	−0.212690	−	0.212690i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	23.4713			1.46410			0.732051	−	0.681250i	$-0.238563\pi$
$257$	23.4713			1.46410			0.732051	+	0.681250i	$0.238563\pi$
$258$		0			0
$259$	−2.70160	−	2.70160i	−0.167869	−	0.167869i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			24.1133i			1.48689i	0.668798	+	0.743444i	$0.266809\pi$
$263$			24.1133i			1.48689i	−0.668798	+	0.743444i	$0.733191\pi$
$264$		0			0
$265$			13.5096i			0.829889i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−8.06584	−	8.06584i	−0.491783	−	0.491783i	0.417085	−	0.908868i	$-0.363052\pi$
$269$	−8.06584	−	8.06584i	−0.491783	−	0.491783i	−0.908868	+	0.417085i	$0.863052\pi$
$270$		0			0
$271$	−15.9475			−0.968740			−0.484370	−	0.874863i	$-0.660951\pi$
$271$	−15.9475			−0.968740			−0.484370	+	0.874863i	$0.660951\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	0.526602	+	0.526602i	0.0317553	+	0.0317553i
$276$		0			0
$277$	5.23599	−	5.23599i	0.314600	−	0.314600i	−0.532089	−	0.846689i	$-0.678593\pi$
$277$	5.23599	−	5.23599i	0.314600	−	0.314600i	0.846689	+	0.532089i	$0.178593\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	4.46512i		−	0.266367i	−0.991091	−	0.133183i	$-0.957480\pi$
$281$		−	4.46512i		−	0.266367i	0.991091	−	0.133183i	$-0.0425200\pi$
$282$		0			0
$283$	−11.0211	+	11.0211i	−0.655136	+	0.655136i	−0.954225	−	0.299089i	$-0.903317\pi$
$283$	−11.0211	+	11.0211i	−0.655136	+	0.655136i	0.299089	+	0.954225i	$0.403317\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	16.9348			0.999632
$288$		0			0
$289$	−14.5252			−0.854423
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−3.78692	+	3.78692i	−0.221234	+	0.221234i	−0.809018	−	0.587784i	$-0.800001\pi$
$293$	−3.78692	+	3.78692i	−0.221234	+	0.221234i	0.587784	+	0.809018i	$0.300001\pi$
$294$		0			0
$295$		−	6.53325i		−	0.380381i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−23.6770	+	23.6770i	−1.36928	+	1.36928i
$300$		0			0
$301$	−1.59587	−	1.59587i	−0.0919841	−	0.0919841i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	6.40643			0.366831
$306$		0			0
$307$	13.3390	+	13.3390i	0.761295	+	0.761295i	0.976557	−	0.215261i	$-0.0690603\pi$
$307$	13.3390	+	13.3390i	0.761295	+	0.761295i	−0.215261	+	0.976557i	$0.569060\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	25.4977i		−	1.44584i	−0.690932	−	0.722920i	$-0.742800\pi$
$311$		−	25.4977i		−	1.44584i	0.690932	−	0.722920i	$-0.257200\pi$
$312$		0			0
$313$		−	6.04542i		−	0.341707i	−0.985296	−	0.170854i	$-0.945347\pi$
$313$		−	6.04542i		−	0.341707i	0.985296	−	0.170854i	$-0.0546525\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	8.64061	+	8.64061i	0.485305	+	0.485305i	0.906821	−	0.421516i	$-0.138502\pi$
$317$	8.64061	+	8.64061i	0.485305	+	0.485305i	−0.421516	+	0.906821i	$0.638502\pi$
$318$		0			0
$319$	4.61695			0.258500
$320$		0			0
$321$		0			0
$322$		0			0
$323$	0.603650	+	0.603650i	0.0335880	+	0.0335880i
$324$		0			0
$325$	3.68554	−	3.68554i	0.204437	−	0.204437i
$326$		0			0
$327$		0			0
$328$		0			0
$329$			2.96010i			0.163196i
$330$		0			0
$331$	16.2197	−	16.2197i	0.891514	−	0.891514i	−0.103152	−	0.994666i	$-0.532893\pi$
$331$	16.2197	−	16.2197i	0.891514	−	0.891514i	0.994666	+	0.103152i	$0.0328927\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	7.91375			0.432374
$336$		0			0
$337$	30.3702			1.65437			0.827184	−	0.561931i	$-0.189941\pi$
$337$	30.3702			1.65437			0.827184	+	0.561931i	$0.189941\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−3.03049	+	3.03049i	−0.164110	+	0.164110i
$342$		0			0
$343$		−	16.9706i		−	0.916324i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−8.22478	+	8.22478i	−0.441529	+	0.441529i	−0.892526	−	0.450997i	$-0.851069\pi$
$347$	−8.22478	+	8.22478i	−0.441529	+	0.441529i	0.450997	+	0.892526i	$0.351069\pi$
$348$		0			0
$349$	−19.9754	−	19.9754i	−1.06926	−	1.06926i	−0.997416	−	0.0718432i	$-0.977112\pi$
$349$	−19.9754	−	19.9754i	−1.06926	−	1.06926i	−0.0718432	−	0.997416i	$-0.522888\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	8.40669			0.447443			0.223721	−	0.974653i	$-0.428179\pi$
$353$	8.40669			0.447443			0.223721	+	0.974653i	$0.428179\pi$
$354$		0			0
$355$	−8.40158	−	8.40158i	−0.445910	−	0.445910i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	13.2812i		−	0.700956i	−0.936571	−	0.350478i	$-0.886019\pi$
$359$		−	13.2812i		−	0.700956i	0.936571	−	0.350478i	$-0.113981\pi$
$360$		0			0
$361$		−	18.7055i		−	0.984501i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−8.62636	−	8.62636i	−0.451524	−	0.451524i
$366$		0			0
$367$	−27.7426			−1.44815			−0.724075	−	0.689721i	$-0.757733\pi$
$367$	−27.7426			−1.44815			−0.724075	+	0.689721i	$0.757733\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	13.5096	+	13.5096i	0.701384	+	0.701384i
$372$		0			0
$373$	14.7387	−	14.7387i	0.763143	−	0.763143i	−0.213746	−	0.976889i	$-0.568566\pi$
$373$	14.7387	−	14.7387i	0.763143	−	0.763143i	0.976889	+	0.213746i	$0.0685664\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	32.3128i		−	1.66419i
$378$		0			0
$379$	3.45249	−	3.45249i	0.177343	−	0.177343i	−0.612854	−	0.790196i	$-0.709979\pi$
$379$	3.45249	−	3.45249i	0.177343	−	0.177343i	0.790196	+	0.612854i	$0.209979\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−13.7023			−0.700154			−0.350077	−	0.936721i	$-0.613845\pi$
$383$	−13.7023			−0.700154			−0.350077	+	0.936721i	$0.613845\pi$
$384$		0			0
$385$	1.05320			0.0536763
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−8.20181	+	8.20181i	−0.415848	+	0.415848i	−0.883770	−	0.467922i	$-0.845003\pi$
$389$	−8.20181	+	8.20181i	−0.415848	+	0.415848i	0.467922	+	0.883770i	$0.345003\pi$
$390$		0			0
$391$			10.1064i			0.511103i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	2.35503	−	2.35503i	0.118494	−	0.118494i
$396$		0			0
$397$	−8.68137	−	8.68137i	−0.435705	−	0.435705i	0.454858	−	0.890564i	$-0.349690\pi$
$397$	−8.68137	−	8.68137i	−0.435705	−	0.435705i	−0.890564	+	0.454858i	$0.849690\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−23.0445			−1.15079			−0.575393	−	0.817877i	$-0.695151\pi$
$401$	−23.0445			−1.15079			−0.575393	+	0.817877i	$0.695151\pi$
$402$		0			0
$403$	21.2096	+	21.2096i	1.05653	+	1.05653i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			2.01196i			0.0997291i
$408$		0			0
$409$			15.7464i			0.778607i	0.921110	+	0.389303i	$0.127284\pi$
$409$			15.7464i			0.778607i	−0.921110	+	0.389303i	$0.872716\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−6.53325	−	6.53325i	−0.321480	−	0.321480i
$414$		0			0
$415$	8.93484			0.438594
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−24.4404	−	24.4404i	−1.19399	−	1.19399i	−0.975937	−	0.218052i	$-0.930030\pi$
$419$	−24.4404	−	24.4404i	−1.19399	−	1.19399i	−0.218052	−	0.975937i	$-0.569970\pi$
$420$		0			0
$421$	−21.7218	+	21.7218i	−1.05865	+	1.05865i	−0.0604847	+	0.998169i	$0.519265\pi$
$421$	−21.7218	+	21.7218i	−1.05865	+	1.05865i	−0.998169	+	0.0604847i	$0.980735\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	1.57316i		−	0.0763092i
$426$		0			0
$427$	6.40643	−	6.40643i	0.310029	−	0.310029i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−27.3876			−1.31922			−0.659608	−	0.751610i	$-0.729278\pi$
$431$	−27.3876			−1.31922			−0.659608	+	0.751610i	$0.729278\pi$
$432$		0			0
$433$	3.74996			0.180212			0.0901058	−	0.995932i	$-0.471279\pi$
$433$	3.74996			0.180212			0.0901058	+	0.995932i	$0.471279\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−2.46512	+	2.46512i	−0.117923	+	0.117923i
$438$		0			0
$439$			22.5580i			1.07663i	0.842743	+	0.538317i	$0.180940\pi$
$439$			22.5580i			1.07663i	−0.842743	+	0.538317i	$0.819060\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−13.0922	+	13.0922i	−0.622028	+	0.622028i	−0.946050	−	0.324022i	$-0.894965\pi$
$443$	−13.0922	+	13.0922i	−0.622028	+	0.622028i	0.324022	+	0.946050i	$0.394965\pi$
$444$		0			0
$445$	−3.83851	−	3.83851i	−0.181963	−	0.181963i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−4.13492			−0.195139			−0.0975694	−	0.995229i	$-0.531107\pi$
$449$	−4.13492			−0.195139			−0.0975694	+	0.995229i	$0.531107\pi$
$450$		0			0
$451$	−6.30592	−	6.30592i	−0.296934	−	0.296934i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	7.37109i		−	0.345562i
$456$		0			0
$457$		−	26.3008i		−	1.23030i	−0.788410	−	0.615150i	$-0.789095\pi$
$457$		−	26.3008i		−	1.23030i	0.788410	−	0.615150i	$-0.210905\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	13.3795	+	13.3795i	0.623148	+	0.623148i	0.946335	−	0.323187i	$-0.104754\pi$
$461$	13.3795	+	13.3795i	0.623148	+	0.623148i	−0.323187	+	0.946335i	$0.604754\pi$
$462$		0			0
$463$	−14.5995			−0.678496			−0.339248	−	0.940697i	$-0.610173\pi$
$463$	−14.5995			−0.678496			−0.339248	+	0.940697i	$0.610173\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	23.7344	+	23.7344i	1.09830	+	1.09830i	0.994610	+	0.103687i	$0.0330639\pi$
$467$	23.7344	+	23.7344i	1.09830	+	1.09830i	0.103687	+	0.994610i	$0.466936\pi$
$468$		0			0
$469$	7.91375	−	7.91375i	0.365423	−	0.365423i
$470$		0			0
$471$		0			0
$472$		0			0
$473$			1.18849i			0.0546466i
$474$		0			0
$475$	0.383719	−	0.383719i	0.0176062	−	0.0176062i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−18.5018			−0.845370			−0.422685	−	0.906277i	$-0.638912\pi$
$479$	−18.5018			−0.845370			−0.422685	+	0.906277i	$0.638912\pi$
$480$		0			0
$481$	14.0811			0.642045
$482$		0			0
$483$		0			0
$484$		0			0
$485$	1.51217	−	1.51217i	0.0686639	−	0.0686639i
$486$		0			0
$487$			14.0312i			0.635813i	0.948122	+	0.317906i	$0.102980\pi$
$487$			14.0312i			0.635813i	−0.948122	+	0.317906i	$0.897020\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−8.60089	+	8.60089i	−0.388153	+	0.388153i	−0.874028	−	0.485875i	$-0.838501\pi$
$491$	−8.60089	+	8.60089i	−0.388153	+	0.388153i	0.485875	+	0.874028i	$0.338501\pi$
$492$		0			0
$493$	−6.89627	−	6.89627i	−0.310592	−	0.310592i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−16.8032			−0.753725
$498$		0			0
$499$	−10.5628	−	10.5628i	−0.472857	−	0.472857i	0.429981	−	0.902838i	$-0.358520\pi$
$499$	−10.5628	−	10.5628i	−0.472857	−	0.472857i	−0.902838	+	0.429981i	$0.858520\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	20.3857i		−	0.908955i	−0.890758	−	0.454477i	$-0.849826\pi$
$503$		−	20.3857i		−	0.908955i	0.890758	−	0.454477i	$-0.150174\pi$
$504$		0			0
$505$			8.25689i			0.367427i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	10.1972	+	10.1972i	0.451984	+	0.451984i	0.896013	−	0.444029i	$-0.146451\pi$
$509$	10.1972	+	10.1972i	0.451984	+	0.451984i	−0.444029	+	0.896013i	$0.646451\pi$
$510$		0			0
$511$	−17.2527			−0.763215
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−11.6238	−	11.6238i	−0.512206	−	0.512206i
$516$		0			0
$517$	1.10223	−	1.10223i	0.0484762	−	0.0484762i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			6.25365i			0.273977i	0.990573	+	0.136989i	$0.0437424\pi$
$521$			6.25365i			0.273977i	−0.990573	+	0.136989i	$0.956258\pi$
$522$		0			0
$523$	−17.2349	+	17.2349i	−0.753628	+	0.753628i	−0.975154	−	0.221527i	$-0.928896\pi$
$523$	−17.2349	+	17.2349i	−0.753628	+	0.753628i	0.221527	+	0.975154i	$0.428896\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	9.05320			0.394364
$528$		0			0
$529$	−18.2715			−0.794414
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−44.1334	+	44.1334i	−1.91163	+	1.91163i
$534$		0			0
$535$		−	7.19173i		−	0.310926i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−2.63301	+	2.63301i	−0.113412	+	0.113412i
$540$		0			0
$541$	−22.6231	−	22.6231i	−0.972645	−	0.972645i	0.0269911	−	0.999636i	$-0.491407\pi$
$541$	−22.6231	−	22.6231i	−0.972645	−	0.972645i	−0.999636	+	0.0269911i	$0.991407\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	7.74540			0.331776
$546$		0			0
$547$	−16.0380	−	16.0380i	−0.685736	−	0.685736i	0.275550	−	0.961287i	$-0.411140\pi$
$547$	−16.0380	−	16.0380i	−0.685736	−	0.685736i	−0.961287	+	0.275550i	$0.911140\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	3.36423i		−	0.143321i
$552$		0			0
$553$		−	4.71006i		−	0.200292i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−17.3053	−	17.3053i	−0.733247	−	0.733247i	0.238015	−	0.971262i	$-0.423503\pi$
$557$	−17.3053	−	17.3053i	−0.733247	−	0.733247i	−0.971262	+	0.238015i	$0.923503\pi$
$558$		0			0
$559$	8.31788			0.351809
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−24.4885	−	24.4885i	−1.03207	−	1.03207i	−0.999468	−	0.0325998i	$-0.989621\pi$
$563$	−24.4885	−	24.4885i	−1.03207	−	1.03207i	−0.0325998	−	0.999468i	$-0.510379\pi$
$564$		0			0
$565$	10.5766	−	10.5766i	0.444960	−	0.444960i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			21.3371i			0.894498i	0.894409	+	0.447249i	$0.147596\pi$
$569$			21.3371i			0.894498i	−0.894409	+	0.447249i	$0.852404\pi$
$570$		0			0
$571$	2.90272	−	2.90272i	0.121475	−	0.121475i	−0.643756	−	0.765231i	$-0.722625\pi$
$571$	2.90272	−	2.90272i	0.121475	−	0.121475i	0.765231	+	0.643756i	$0.222625\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	6.42429			0.267912
$576$		0			0
$577$	−0.390831			−0.0162705			−0.00813526	−	0.999967i	$-0.502590\pi$
$577$	−0.390831			−0.0162705			−0.00813526	+	0.999967i	$0.502590\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	8.93484	−	8.93484i	0.370679	−	0.370679i
$582$		0			0
$583$		−	10.0610i		−	0.416684i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	1.90689	−	1.90689i	0.0787059	−	0.0787059i	−0.666658	−	0.745364i	$-0.732276\pi$
$587$	1.90689	−	1.90689i	0.0787059	−	0.0787059i	0.745364	+	0.666658i	$0.232276\pi$
$588$		0			0
$589$	2.20823	+	2.20823i	0.0909885	+	0.0909885i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	15.9469			0.654862			0.327431	−	0.944875i	$-0.393817\pi$
$593$	15.9469			0.654862			0.327431	+	0.944875i	$0.393817\pi$
$594$		0			0
$595$	−1.57316	−	1.57316i	−0.0644931	−	0.0644931i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			19.4591i			0.795077i	0.917586	+	0.397538i	$0.130135\pi$
$599$			19.4591i			0.795077i	−0.917586	+	0.397538i	$0.869865\pi$
$600$		0			0
$601$			6.91468i			0.282056i	0.990006	+	0.141028i	$0.0450407\pi$
$601$			6.91468i			0.282056i	−0.990006	+	0.141028i	$0.954959\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	7.38600	+	7.38600i	0.300284	+	0.300284i
$606$		0			0
$607$	1.14175			0.0463422			0.0231711	−	0.999732i	$-0.492624\pi$
$607$	1.14175			0.0463422			0.0231711	+	0.999732i	$0.492624\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−7.71423	−	7.71423i	−0.312084	−	0.312084i
$612$		0			0
$613$	−24.5497	+	24.5497i	−0.991553	+	0.991553i	−0.999965	−	0.00841168i	$-0.997322\pi$
$613$	−24.5497	+	24.5497i	−0.991553	+	0.991553i	0.00841168	+	0.999965i	$0.497322\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	17.8671i		−	0.719303i	−0.933087	−	0.359652i	$-0.882896\pi$
$617$		−	17.8671i		−	0.719303i	0.933087	−	0.359652i	$-0.117104\pi$
$618$		0			0
$619$	11.3578	−	11.3578i	0.456508	−	0.456508i	−0.440999	−	0.897507i	$-0.645376\pi$
$619$	11.3578	−	11.3578i	0.456508	−	0.456508i	0.897507	+	0.440999i	$0.145376\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−7.67701			−0.307573
$624$		0			0
$625$	−1.00000			−0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	3.00523	−	3.00523i	0.119826	−	0.119826i
$630$		0			0
$631$		−	20.7001i		−	0.824058i	−0.911171	−	0.412029i	$-0.864820\pi$
$631$		−	20.7001i		−	0.824058i	0.911171	−	0.412029i	$-0.135180\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	13.7101	−	13.7101i	0.544067	−	0.544067i
$636$		0			0
$637$	18.4277	+	18.4277i	0.730133	+	0.730133i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−23.4435			−0.925963			−0.462982	−	0.886368i	$-0.653220\pi$
$641$	−23.4435			−0.925963			−0.462982	+	0.886368i	$0.653220\pi$
$642$		0			0
$643$	−14.5738	−	14.5738i	−0.574736	−	0.574736i	0.358712	−	0.933448i	$-0.383216\pi$
$643$	−14.5738	−	14.5738i	−0.574736	−	0.574736i	−0.933448	+	0.358712i	$0.883216\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	14.9417i		−	0.587418i	−0.955895	−	0.293709i	$-0.905110\pi$
$647$		−	14.9417i		−	0.587418i	0.955895	−	0.293709i	$-0.0948897\pi$
$648$		0			0
$649$			4.86550i			0.190987i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−1.78262	−	1.78262i	−0.0697594	−	0.0697594i	0.671366	−	0.741126i	$-0.265708\pi$
$653$	−1.78262	−	1.78262i	−0.0697594	−	0.0697594i	−0.741126	+	0.671366i	$0.765708\pi$
$654$		0			0
$655$	6.76489			0.264326
$656$		0			0
$657$		0			0
$658$		0			0
$659$	11.7844	+	11.7844i	0.459056	+	0.459056i	0.898346	−	0.439289i	$-0.144770\pi$
$659$	11.7844	+	11.7844i	0.459056	+	0.459056i	−0.439289	+	0.898346i	$0.644770\pi$
$660$		0			0
$661$	−3.89751	+	3.89751i	−0.151595	+	0.151595i	−0.778830	−	0.627235i	$-0.784187\pi$
$661$	−3.89751	+	3.89751i	−0.151595	+	0.151595i	0.627235	+	0.778830i	$0.284187\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	0.767438i		−	0.0297600i
$666$		0			0
$667$	28.1623	−	28.1623i	1.09045	−	1.09045i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−4.77105			−0.184184
$672$		0			0
$673$	−42.2243			−1.62763			−0.813813	−	0.581127i	$-0.802612\pi$
$673$	−42.2243			−1.62763			−0.813813	+	0.581127i	$0.802612\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	24.7733	−	24.7733i	0.952117	−	0.952117i	−0.0467880	−	0.998905i	$-0.514899\pi$
$677$	24.7733	−	24.7733i	0.952117	−	0.952117i	0.998905	+	0.0467880i	$0.0148985\pi$
$678$		0			0
$679$		−	3.02433i		−	0.116063i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	13.3912	−	13.3912i	0.512402	−	0.512402i	−0.402860	−	0.915262i	$-0.631984\pi$
$683$	13.3912	−	13.3912i	0.512402	−	0.512402i	0.915262	+	0.402860i	$0.131984\pi$
$684$		0			0
$685$	2.76924	+	2.76924i	0.105807	+	0.105807i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−70.4141			−2.68256
$690$		0			0
$691$	19.8648	+	19.8648i	0.755694	+	0.755694i	0.975536	−	0.219842i	$-0.0705542\pi$
$691$	19.8648	+	19.8648i	0.755694	+	0.755694i	−0.219842	+	0.975536i	$0.570554\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			21.4486i			0.813593i
$696$		0			0
$697$			18.8381i			0.713545i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−0.771998	−	0.771998i	−0.0291580	−	0.0291580i	0.692377	−	0.721535i	$-0.256563\pi$
$701$	−0.771998	−	0.771998i	−0.0291580	−	0.0291580i	−0.721535	+	0.692377i	$0.756563\pi$
$702$		0			0
$703$	1.46605			0.0552933
$704$		0			0
$705$		0			0
$706$		0			0
$707$	8.25689	+	8.25689i	0.310532	+	0.310532i
$708$		0			0
$709$	7.16580	−	7.16580i	0.269117	−	0.269117i	−0.559627	−	0.828744i	$-0.689056\pi$
$709$	7.16580	−	7.16580i	0.269117	−	0.269117i	0.828744	+	0.559627i	$0.189056\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			36.9706i			1.38456i
$714$		0			0
$715$	−2.74473	+	2.74473i	−0.102647	+	0.102647i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−4.88942			−0.182344			−0.0911722	−	0.995835i	$-0.529061\pi$
$719$	−4.88942			−0.182344			−0.0911722	+	0.995835i	$0.529061\pi$
$720$		0			0
$721$	−23.2476			−0.865786
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−4.38372	+	4.38372i	−0.162807	+	0.162807i
$726$		0			0
$727$		−	14.1531i		−	0.524911i	−0.964944	−	0.262456i	$-0.915468\pi$
$727$		−	14.1531i		−	0.524911i	0.964944	−	0.262456i	$-0.0845323\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	1.77522	−	1.77522i	0.0656590	−	0.0656590i
$732$		0			0
$733$	11.2850	+	11.2850i	0.416822	+	0.416822i	0.884107	−	0.467285i	$-0.154768\pi$
$733$	11.2850	+	11.2850i	0.416822	+	0.416822i	−0.467285	+	0.884107i	$0.654768\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−5.89359			−0.217093
$738$		0			0
$739$	−30.9164	−	30.9164i	−1.13728	−	1.13728i	−0.988936	−	0.148343i	$-0.952606\pi$
$739$	−30.9164	−	30.9164i	−1.13728	−	1.13728i	−0.148343	−	0.988936i	$-0.547394\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			17.5161i			0.642602i	0.946977	+	0.321301i	$0.104120\pi$
$743$			17.5161i			0.642602i	−0.946977	+	0.321301i	$0.895880\pi$
$744$		0			0
$745$		−	2.16379i		−	0.0792751i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−7.19173	−	7.19173i	−0.262780	−	0.262780i
$750$		0			0
$751$	1.99154			0.0726725			0.0363362	−	0.999340i	$-0.488431\pi$
$751$	1.99154			0.0726725			0.0363362	+	0.999340i	$0.488431\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−8.35503	−	8.35503i	−0.304071	−	0.304071i
$756$		0			0
$757$	−12.3823	+	12.3823i	−0.450042	+	0.450042i	−0.895368	−	0.445326i	$-0.853088\pi$
$757$	−12.3823	+	12.3823i	−0.450042	+	0.450042i	0.445326	+	0.895368i	$0.353088\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	34.4159i		−	1.24758i	−0.781593	−	0.623788i	$-0.785593\pi$
$761$		−	34.4159i		−	1.24758i	0.781593	−	0.623788i	$-0.214407\pi$
$762$		0			0
$763$	7.74540	−	7.74540i	0.280402	−	0.280402i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	34.0523			1.22956
$768$		0			0
$769$	−48.5054			−1.74915			−0.874575	−	0.484889i	$-0.838860\pi$
$769$	−48.5054			−1.74915			−0.874575	+	0.484889i	$0.838860\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−0.980514	+	0.980514i	−0.0352666	+	0.0352666i	−0.724520	−	0.689254i	$-0.757939\pi$
$773$	−0.980514	+	0.980514i	−0.0352666	+	0.0352666i	0.689254	+	0.724520i	$0.257939\pi$
$774$		0			0
$775$		−	5.75481i		−	0.206719i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−4.59494	+	4.59494i	−0.164631	+	0.164631i
$780$		0			0
$781$	6.25689	+	6.25689i	0.223889	+	0.223889i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−14.4194			−0.514652
$786$		0			0
$787$	−17.6390	−	17.6390i	−0.628762	−	0.628762i	0.318994	−	0.947757i	$-0.396655\pi$
$787$	−17.6390	−	17.6390i	−0.628762	−	0.628762i	−0.947757	+	0.318994i	$0.896655\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	21.1532i		−	0.752120i
$792$		0			0
$793$			33.3912i			1.18576i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−0.451563	−	0.451563i	−0.0159952	−	0.0159952i	0.699064	−	0.715059i	$-0.253600\pi$
$797$	−0.451563	−	0.451563i	−0.0159952	−	0.0159952i	−0.715059	+	0.699064i	$0.753600\pi$
$798$		0			0
$799$	−3.29278			−0.116490
$800$		0			0
$801$		0			0
$802$		0			0
$803$	6.42429	+	6.42429i	0.226708	+	0.226708i
$804$		0			0
$805$	6.42429	−	6.42429i	0.226427	−	0.226427i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			20.0844i			0.706130i	0.935599	+	0.353065i	$0.114860\pi$
$809$			20.0844i			0.706130i	−0.935599	+	0.353065i	$0.885140\pi$
$810$		0			0
$811$	−21.4675	+	21.4675i	−0.753827	+	0.753827i	−0.975191	−	0.221364i	$-0.928949\pi$
$811$	−21.4675	+	21.4675i	−0.753827	+	0.753827i	0.221364	+	0.975191i	$0.428949\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	17.9592			0.629082
$816$		0			0
$817$	0.866013			0.0302980
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−5.82776	+	5.82776i	−0.203390	+	0.203390i	−0.801451	−	0.598061i	$-0.795938\pi$
$821$	−5.82776	+	5.82776i	−0.203390	+	0.203390i	0.598061	+	0.801451i	$0.295938\pi$
$822$		0			0
$823$		−	39.1302i		−	1.36399i	−0.731355	−	0.681996i	$-0.761112\pi$
$823$		−	39.1302i		−	1.36399i	0.731355	−	0.681996i	$-0.238888\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	37.3252	−	37.3252i	1.29792	−	1.29792i	0.368160	−	0.929762i	$-0.379988\pi$
$827$	37.3252	−	37.3252i	1.29792	−	1.29792i	0.929762	−	0.368160i	$-0.120012\pi$
$828$		0			0
$829$	−17.7182	−	17.7182i	−0.615377	−	0.615377i	0.328965	−	0.944342i	$-0.393300\pi$
$829$	−17.7182	−	17.7182i	−0.615377	−	0.615377i	−0.944342	+	0.328965i	$0.893300\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	7.86578			0.272533
$834$		0			0
$835$	−13.7727	−	13.7727i	−0.476623	−	0.476623i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			35.3371i			1.21997i	0.792412	+	0.609986i	$0.208825\pi$
$839$			35.3371i			1.21997i	−0.792412	+	0.609986i	$0.791175\pi$
$840$		0			0
$841$			9.43399i			0.325310i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	10.0172	+	10.0172i	0.344602	+	0.344602i
$846$		0			0
$847$	14.7720			0.507572
$848$		0			0
$849$		0			0
$850$		0			0
$851$	12.2725	+	12.2725i	0.420695	+	0.420695i
$852$		0			0
$853$	−35.7126	+	35.7126i	−1.22277	+	1.22277i	−0.256133	+	0.966642i	$0.582449\pi$
$853$	−35.7126	+	35.7126i	−1.22277	+	1.22277i	−0.966642	+	0.256133i	$0.917551\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			28.0974i			0.959789i	0.877326	+	0.479895i	$0.159325\pi$
$857$			28.0974i			0.959789i	−0.877326	+	0.479895i	$0.840675\pi$
$858$		0			0
$859$	−26.5677	+	26.5677i	−0.906477	+	0.906477i	−0.995986	−	0.0895090i	$-0.971470\pi$
$859$	−26.5677	+	26.5677i	−0.906477	+	0.906477i	0.0895090	+	0.995986i	$0.471470\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	12.4481			0.423737			0.211868	−	0.977298i	$-0.432045\pi$
$863$	12.4481			0.423737			0.211868	+	0.977298i	$0.432045\pi$
$864$		0			0
$865$	−14.1541			−0.481254
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−1.75386	+	1.75386i	−0.0594955	+	0.0594955i
$870$		0			0
$871$			41.2476i			1.39762i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−1.00000	+	1.00000i	−0.0338062	+	0.0338062i
$876$		0			0
$877$	6.42087	+	6.42087i	0.216817	+	0.216817i	0.807156	−	0.590339i	$-0.201006\pi$
$877$	6.42087	+	6.42087i	0.216817	+	0.216817i	−0.590339	+	0.807156i	$0.701006\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	14.0591			0.473664			0.236832	−	0.971551i	$-0.423891\pi$
$881$	14.0591			0.473664			0.236832	+	0.971551i	$0.423891\pi$
$882$		0			0
$883$	37.6339	+	37.6339i	1.26648	+	1.26648i	0.947893	+	0.318588i	$0.103209\pi$
$883$	37.6339	+	37.6339i	1.26648	+	1.26648i	0.318588	+	0.947893i	$0.396791\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	22.1691i		−	0.744367i	−0.928159	−	0.372184i	$-0.878609\pi$
$887$		−	22.1691i		−	0.744367i	0.928159	−	0.372184i	$-0.121391\pi$
$888$		0			0
$889$		−	27.4201i		−	0.919641i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−0.803165	−	0.803165i	−0.0268769	−	0.0268769i
$894$		0			0
$895$	5.91630			0.197760
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−25.2275	−	25.2275i	−0.841383	−	0.841383i
$900$		0			0
$901$	−15.0279	+	15.0279i	−0.500653	+	0.500653i
$902$		0			0
$903$		0			0
$904$		0			0
$905$			14.7243i			0.489453i
$906$		0			0
$907$	−31.1389	+	31.1389i	−1.03395	+	1.03395i	−0.0345475	+	0.999403i	$0.510999\pi$
$907$	−31.1389	+	31.1389i	−1.03395	+	1.03395i	−0.999403	+	0.0345475i	$0.989001\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−25.7283			−0.852417			−0.426208	−	0.904625i	$-0.640151\pi$
$911$	−25.7283			−0.852417			−0.426208	+	0.904625i	$0.640151\pi$
$912$		0			0
$913$	−6.65402			−0.220216
$914$		0			0
$915$		0			0
$916$		0			0
$917$	6.76489	−	6.76489i	0.223396	−	0.223396i
$918$		0			0
$919$			2.17776i			0.0718375i	0.999355	+	0.0359188i	$0.0114358\pi$
$919$			2.17776i			0.0718375i	−0.999355	+	0.0359188i	$0.988564\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	43.7903	−	43.7903i	1.44137	−	1.44137i
$924$		0			0
$925$	−1.91032	−	1.91032i	−0.0628110	−	0.0628110i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	25.3234			0.830834			0.415417	−	0.909631i	$-0.363636\pi$
$929$	25.3234			0.830834			0.415417	+	0.909631i	$0.363636\pi$
$930$		0			0
$931$	1.91860	+	1.91860i	0.0628794	+	0.0628794i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			1.17157i			0.0383145i
$936$		0			0
$937$		−	36.5560i		−	1.19423i	−0.802155	−	0.597116i	$-0.796313\pi$
$937$		−	36.5560i		−	1.19423i	0.802155	−	0.597116i	$-0.203687\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−10.0143	−	10.0143i	−0.326455	−	0.326455i	0.524782	−	0.851237i	$-0.324147\pi$
$941$	−10.0143	−	10.0143i	−0.326455	−	0.326455i	−0.851237	+	0.524782i	$0.824147\pi$
$942$		0			0
$943$	−76.9292			−2.50516
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−39.9949	−	39.9949i	−1.29966	−	1.29966i	−0.928615	−	0.371044i	$-0.879000\pi$
$947$	−39.9949	−	39.9949i	−1.29966	−	1.29966i	−0.371044	−	0.928615i	$-0.621000\pi$
$948$		0			0
$949$	44.9618	−	44.9618i	1.45952	−	1.45952i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	49.8263i		−	1.61403i	−0.590530	−	0.807016i	$-0.701081\pi$
$953$		−	49.8263i		−	1.61403i	0.590530	−	0.807016i	$-0.298919\pi$
$954$		0			0
$955$	4.58094	−	4.58094i	0.148236	−	0.148236i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	5.53849			0.178847
$960$		0			0
$961$	2.11780			0.0683162
$962$		0			0
$963$		0			0
$964$		0			0
$965$	0.492681	−	0.492681i	0.0158599	−	0.0158599i
$966$		0			0
$967$			21.0321i			0.676347i	0.941084	+	0.338173i	$0.109809\pi$
$967$			21.0321i			0.676347i	−0.941084	+	0.338173i	$0.890191\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	37.7385	−	37.7385i	1.21109	−	1.21109i	0.240416	−	0.970670i	$-0.422716\pi$
$971$	37.7385	−	37.7385i	1.21109	−	1.21109i	0.970670	−	0.240416i	$-0.0772838\pi$
$972$		0			0
$973$	21.4486	+	21.4486i	0.687611	+	0.687611i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	53.5629			1.71363			0.856815	−	0.515624i	$-0.172440\pi$
$977$	53.5629			1.71363			0.856815	+	0.515624i	$0.172440\pi$
$978$		0			0
$979$	2.85864	+	2.85864i	0.0913626	+	0.0913626i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	44.1169i		−	1.40711i	−0.710641	−	0.703555i	$-0.751595\pi$
$983$		−	44.1169i		−	1.40711i	0.710641	−	0.703555i	$-0.248405\pi$
$984$		0			0
$985$		−	16.4564i		−	0.524345i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	7.24947	+	7.24947i	0.230520	+	0.230520i
$990$		0			0
$991$	1.51348			0.0480773			0.0240387	−	0.999711i	$-0.492348\pi$
$991$	1.51348			0.0480773			0.0240387	+	0.999711i	$0.492348\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	16.1757	+	16.1757i	0.512803	+	0.512803i
$996$		0			0
$997$	−3.84609	+	3.84609i	−0.121807	+	0.121807i	−0.765382	−	0.643576i	$-0.777450\pi$
$997$	−3.84609	+	3.84609i	−0.121807	+	0.121807i	0.643576	+	0.765382i	$0.277450\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2880.2.t.b.721.3		8
3.2	odd	2		960.2.s.b.721.2		8
4.3	odd	2		720.2.t.b.541.4		8
12.11	even	2		240.2.s.b.61.1	✓	8
16.5	even	4	inner	2880.2.t.b.2161.3		8
16.11	odd	4		720.2.t.b.181.4		8
24.5	odd	2		1920.2.s.d.1441.3		8
24.11	even	2		1920.2.s.c.1441.2		8
48.5	odd	4		960.2.s.b.241.2		8
48.11	even	4		240.2.s.b.181.1	yes	8
48.29	odd	4		1920.2.s.d.481.3		8
48.35	even	4		1920.2.s.c.481.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
240.2.s.b.61.1	✓	8	12.11	even	2
240.2.s.b.181.1	yes	8	48.11	even	4
720.2.t.b.181.4		8	16.11	odd	4
720.2.t.b.541.4		8	4.3	odd	2
960.2.s.b.241.2		8	48.5	odd	4
960.2.s.b.721.2		8	3.2	odd	2
1920.2.s.c.481.2		8	48.35	even	4
1920.2.s.c.1441.2		8	24.11	even	2
1920.2.s.d.481.3		8	48.29	odd	4
1920.2.s.d.1441.3		8	24.5	odd	2
2880.2.t.b.721.3		8	1.1	even	1	trivial
2880.2.t.b.2161.3		8	16.5	even	4	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(0.707107 - 0.707107i) q^{5} -1.41421i q^{7} +O(q^{10})\)	\(q+(0.707107 - 0.707107i) q^{5} -1.41421i q^{7} +(-0.526602 + 0.526602i) q^{11} +(3.68554 + 3.68554i) q^{13} +1.57316 q^{17} +(0.383719 + 0.383719i) q^{19} +6.42429i q^{23} -1.00000i q^{25} +(-4.38372 - 4.38372i) q^{29} +5.75481 q^{31} +(-1.00000 - 1.00000i) q^{35} +(1.91032 - 1.91032i) q^{37} +11.9747i q^{41} +(1.12845 - 1.12845i) q^{43} -2.09311 q^{47} +5.00000 q^{49} +(-9.55274 + 9.55274i) q^{53} +0.744728i q^{55} +(4.61971 - 4.61971i) q^{59} +(4.53003 + 4.53003i) q^{61} +5.21215 q^{65} +(5.59587 + 5.59587i) q^{67} -11.8816i q^{71} -12.1995i q^{73} +(0.744728 + 0.744728i) q^{77} +3.33051 q^{79} +(6.31788 + 6.31788i) q^{83} +(1.11239 - 1.11239i) q^{85} -5.42847i q^{89} +(5.21215 - 5.21215i) q^{91} +0.542661 q^{95} +2.13853 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 8 q^{11} + 8 q^{13} - 8 q^{17} - 8 q^{19} - 24 q^{29} - 8 q^{31} - 8 q^{35} - 8 q^{37} + 40 q^{49} + 8 q^{59} - 16 q^{61} + 8 q^{65} + 8 q^{77} + 40 q^{79} + 32 q^{83} + 8 q^{85} + 8 q^{91} - 16 q^{95} - 48 q^{97}+O(q^{100}) \) 8 * q + 8 * q^11 + 8 * q^13 - 8 * q^17 - 8 * q^19 - 24 * q^29 - 8 * q^31 - 8 * q^35 - 8 * q^37 + 40 * q^49 + 8 * q^59 - 16 * q^61 + 8 * q^65 + 8 * q^77 + 40 * q^79 + 32 * q^83 + 8 * q^85 + 8 * q^91 - 16 * q^95 - 48 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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