LMFDB - Embedded newform 2880.2.bl.b.1871.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2880,2,Mod(431,2880)]

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([2, 3, 2, 0]))

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

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

chi := DirichletCharacter("2880.431");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2880 = 2^{6} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2880.bl (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:	$24$
Relative dimension:	$12$ over $\Q(i)$
Twist minimal:	no (minimal twist has level 720)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1871.7
Character	$\chi$	$=$	2880.1871
Dual form			2880.2.bl.b.431.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.707107 - 0.707107i) q^{5} -4.49261 q^{7} +O(q^{10})$	$q+(0.707107 - 0.707107i) q^{5} -4.49261 q^{7} +(-4.37014 - 4.37014i) q^{11} +(1.44518 - 1.44518i) q^{13} +6.69649i q^{17} +(1.91147 + 1.91147i) q^{19} +1.18773i q^{23} -1.00000i q^{25} +(-2.16029 - 2.16029i) q^{29} +4.28465i q^{31} +(-3.17676 + 3.17676i) q^{35} +(0.00767006 + 0.00767006i) q^{37} -4.93930 q^{41} +(-2.61608 + 2.61608i) q^{43} +12.9356 q^{47} +13.1836 q^{49} +(6.04040 - 6.04040i) q^{53} -6.18031 q^{55} +(6.51088 + 6.51088i) q^{59} +(1.14755 - 1.14755i) q^{61} -2.04379i q^{65} +(5.87653 + 5.87653i) q^{67} -8.93153i q^{71} +13.7131i q^{73} +(19.6333 + 19.6333i) q^{77} +4.10759i q^{79} +(4.52750 - 4.52750i) q^{83} +(4.73514 + 4.73514i) q^{85} +12.3387 q^{89} +(-6.49261 + 6.49261i) q^{91} +2.70322 q^{95} -8.38827 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 8 q^{7}+O(q^{10}) $ 24 * q + 8 * q^7	$ 24 q + 8 q^{7} + 24 q^{19} + 8 q^{37} - 48 q^{43} + 24 q^{49} - 24 q^{55} + 40 q^{61} + 40 q^{67} + 24 q^{85} - 40 q^{91} - 16 q^{97}+O(q^{100}) $ 24 * q + 8 * q^7 + 24 * q^19 + 8 * q^37 - 48 * q^43 + 24 * q^49 - 24 * q^55 + 40 * q^61 + 40 * q^67 + 24 * q^85 - 40 * q^91 - 16 * q^97

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	0.707107	−	0.707107i	0.316228	−	0.316228i
$5$	0.707107	−	0.707107i	0.316228	−	0.316228i
$6$		0			0
$7$	−4.49261			−1.69805			−0.849024	−	0.528355i	$-0.822809\pi$
$7$	−4.49261			−1.69805			−0.849024	+	0.528355i	$0.822809\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−4.37014	−	4.37014i	−1.31765	−	1.31765i	−0.915634	−	0.402012i	$-0.868311\pi$
$11$	−4.37014	−	4.37014i	−1.31765	−	1.31765i	−0.402012	−	0.915634i	$-0.631689\pi$
$12$		0			0
$13$	1.44518	−	1.44518i	0.400820	−	0.400820i	−0.477702	−	0.878522i	$-0.658530\pi$
$13$	1.44518	−	1.44518i	0.400820	−	0.400820i	0.878522	+	0.477702i	$0.158530\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			6.69649i			1.62414i	0.583561	+	0.812069i	$0.301659\pi$
$17$			6.69649i			1.62414i	−0.583561	+	0.812069i	$0.698341\pi$
$18$		0			0
$19$	1.91147	+	1.91147i	0.438521	+	0.438521i	0.891514	−	0.452993i	$-0.149644\pi$
$19$	1.91147	+	1.91147i	0.438521	+	0.438521i	−0.452993	+	0.891514i	$0.649644\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			1.18773i			0.247659i	0.992303	+	0.123830i	$0.0395177\pi$
$23$			1.18773i			0.247659i	−0.992303	+	0.123830i	$0.960482\pi$
$24$		0			0
$25$		−	1.00000i		−	0.200000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−2.16029	−	2.16029i	−0.401155	−	0.401155i	0.477485	−	0.878640i	$-0.341549\pi$
$29$	−2.16029	−	2.16029i	−0.401155	−	0.401155i	−0.878640	+	0.477485i	$0.841549\pi$
$30$		0			0
$31$			4.28465i			0.769546i	0.923011	+	0.384773i	$0.125720\pi$
$31$			4.28465i			0.769546i	−0.923011	+	0.384773i	$0.874280\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−3.17676	+	3.17676i	−0.536970	+	0.536970i
$36$		0			0
$37$	0.00767006	+	0.00767006i	0.00126095	+	0.00126095i	0.707737	−	0.706476i	$-0.249716\pi$
$37$	0.00767006	+	0.00767006i	0.00126095	+	0.00126095i	−0.706476	+	0.707737i	$0.749716\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−4.93930			−0.771389			−0.385694	−	0.922627i	$-0.626038\pi$
$41$	−4.93930			−0.771389			−0.385694	+	0.922627i	$0.626038\pi$
$42$		0			0
$43$	−2.61608	+	2.61608i	−0.398949	+	0.398949i	−0.877862	−	0.478913i	$-0.841031\pi$
$43$	−2.61608	+	2.61608i	−0.398949	+	0.398949i	0.478913	+	0.877862i	$0.341031\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	12.9356			1.88686			0.943429	−	0.331574i	$-0.107580\pi$
$47$	12.9356			1.88686			0.943429	+	0.331574i	$0.107580\pi$
$48$		0			0
$49$	13.1836			1.88337
$50$		0			0
$51$		0			0
$52$		0			0
$53$	6.04040	−	6.04040i	0.829712	−	0.829712i	−0.157765	−	0.987477i	$-0.550429\pi$
$53$	6.04040	−	6.04040i	0.829712	−	0.829712i	0.987477	+	0.157765i	$0.0504287\pi$
$54$		0			0
$55$	−6.18031			−0.833353
$56$		0			0
$57$		0			0
$58$		0			0
$59$	6.51088	+	6.51088i	0.847644	+	0.847644i	0.989839	−	0.142195i	$-0.0454160\pi$
$59$	6.51088	+	6.51088i	0.847644	+	0.847644i	−0.142195	+	0.989839i	$0.545416\pi$
$60$		0			0
$61$	1.14755	−	1.14755i	0.146928	−	0.146928i	−0.629816	−	0.776744i	$-0.716870\pi$
$61$	1.14755	−	1.14755i	0.146928	−	0.146928i	0.776744	+	0.629816i	$0.216870\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	2.04379i		−	0.253501i
$66$		0			0
$67$	5.87653	+	5.87653i	0.717932	+	0.717932i	0.968181	−	0.250249i	$-0.0805126\pi$
$67$	5.87653	+	5.87653i	0.717932	+	0.717932i	−0.250249	+	0.968181i	$0.580513\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	8.93153i		−	1.05998i	−0.848005	−	0.529989i	$-0.822196\pi$
$71$		−	8.93153i		−	1.05998i	0.848005	−	0.529989i	$-0.177804\pi$
$72$		0			0
$73$			13.7131i			1.60500i	0.596654	+	0.802498i	$0.296496\pi$
$73$			13.7131i			1.60500i	−0.596654	+	0.802498i	$0.703504\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	19.6333	+	19.6333i	2.23743	+	2.23743i
$78$		0			0
$79$			4.10759i			0.462140i	0.972937	+	0.231070i	$0.0742226\pi$
$79$			4.10759i			0.462140i	−0.972937	+	0.231070i	$0.925777\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	4.52750	−	4.52750i	0.496958	−	0.496958i	−0.413532	−	0.910490i	$-0.635705\pi$
$83$	4.52750	−	4.52750i	0.496958	−	0.496958i	0.910490	+	0.413532i	$0.135705\pi$
$84$		0			0
$85$	4.73514	+	4.73514i	0.513598	+	0.513598i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	12.3387			1.30790			0.653950	−	0.756538i	$-0.273111\pi$
$89$	12.3387			1.30790			0.653950	+	0.756538i	$0.273111\pi$
$90$		0			0
$91$	−6.49261	+	6.49261i	−0.680611	+	0.680611i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	2.70322			0.277345
$96$		0			0
$97$	−8.38827			−0.851700			−0.425850	−	0.904794i	$-0.640025\pi$
$97$	−8.38827			−0.851700			−0.425850	+	0.904794i	$0.640025\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	2.91015	−	2.91015i	0.289571	−	0.289571i	−0.547339	−	0.836911i	$-0.684359\pi$
$101$	2.91015	−	2.91015i	0.289571	−	0.289571i	0.836911	+	0.547339i	$0.184359\pi$
$102$		0			0
$103$	11.5754			1.14055			0.570277	−	0.821452i	$-0.306836\pi$
$103$	11.5754			1.14055			0.570277	+	0.821452i	$0.306836\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	3.53540	+	3.53540i	0.341780	+	0.341780i	0.857036	−	0.515256i	$-0.172303\pi$
$107$	3.53540	+	3.53540i	0.341780	+	0.341780i	−0.515256	+	0.857036i	$0.672303\pi$
$108$		0			0
$109$	−6.33976	+	6.33976i	−0.607239	+	0.607239i	−0.942224	−	0.334985i	$-0.891269\pi$
$109$	−6.33976	+	6.33976i	−0.607239	+	0.607239i	0.334985	+	0.942224i	$0.391269\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	4.54008i		−	0.427095i	−0.976933	−	0.213547i	$-0.931498\pi$
$113$		−	4.54008i		−	0.427095i	0.976933	−	0.213547i	$-0.0685018\pi$
$114$		0			0
$115$	0.839854	+	0.839854i	0.0783168	+	0.0783168i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	30.0847i		−	2.75786i
$120$		0			0
$121$			27.1962i			2.47239i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−0.707107	−	0.707107i	−0.0632456	−	0.0632456i
$126$		0			0
$127$			6.51781i			0.578362i	0.957274	+	0.289181i	$0.0933830\pi$
$127$			6.51781i			0.578362i	−0.957274	+	0.289181i	$0.906617\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	2.77886	−	2.77886i	0.242790	−	0.242790i	−0.575213	−	0.818003i	$-0.695081\pi$
$131$	2.77886	−	2.77886i	0.242790	−	0.242790i	0.818003	+	0.575213i	$0.195081\pi$
$132$		0			0
$133$	−8.58748	−	8.58748i	−0.744629	−	0.744629i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−6.63676			−0.567017			−0.283508	−	0.958970i	$-0.591498\pi$
$137$	−6.63676			−0.567017			−0.283508	+	0.958970i	$0.591498\pi$
$138$		0			0
$139$	−7.05388	+	7.05388i	−0.598302	+	0.598302i	−0.939861	−	0.341558i	$-0.889045\pi$
$139$	−7.05388	+	7.05388i	−0.598302	+	0.598302i	0.341558	+	0.939861i	$0.389045\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−12.6312			−1.05628
$144$		0			0
$145$	−3.05511			−0.253713
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−9.52545	+	9.52545i	−0.780355	+	0.780355i	−0.979891	−	0.199535i	$-0.936057\pi$
$149$	−9.52545	+	9.52545i	−0.780355	+	0.780355i	0.199535	+	0.979891i	$0.436057\pi$
$150$		0			0
$151$	−23.5203			−1.91405			−0.957027	−	0.290000i	$-0.906345\pi$
$151$	−23.5203			−1.91405			−0.957027	+	0.290000i	$0.906345\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	3.02971	+	3.02971i	0.243352	+	0.243352i
$156$		0			0
$157$	10.1912	−	10.1912i	0.813348	−	0.813348i	−0.171786	−	0.985134i	$-0.554954\pi$
$157$	10.1912	−	10.1912i	0.813348	−	0.813348i	0.985134	+	0.171786i	$0.0549537\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	5.33602i		−	0.420537i
$162$		0			0
$163$	16.0686	+	16.0686i	1.25859	+	1.25859i	0.951766	+	0.306823i	$0.0992661\pi$
$163$	16.0686	+	16.0686i	1.25859	+	1.25859i	0.306823	+	0.951766i	$0.400734\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			0.149541i			0.0115718i	0.999983	+	0.00578592i	$0.00184173\pi$
$167$			0.149541i			0.0115718i	−0.999983	+	0.00578592i	$0.998158\pi$
$168$		0			0
$169$			8.82294i			0.678687i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	5.93140	+	5.93140i	0.450956	+	0.450956i	0.895672	−	0.444716i	$-0.146695\pi$
$173$	5.93140	+	5.93140i	0.450956	+	0.450956i	−0.444716	+	0.895672i	$0.646695\pi$
$174$		0			0
$175$			4.49261i			0.339609i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−13.5103	+	13.5103i	−1.00981	+	1.00981i	−0.00985834	+	0.999951i	$0.503138\pi$
$179$	−13.5103	+	13.5103i	−1.00981	+	1.00981i	−0.999951	+	0.00985834i	$0.996862\pi$
$180$		0			0
$181$	11.5235	+	11.5235i	0.856536	+	0.856536i	0.990928	−	0.134393i	$-0.0429083\pi$
$181$	11.5235	+	11.5235i	0.856536	+	0.856536i	−0.134393	+	0.990928i	$0.542908\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	0.0108471			0.000797495
$186$		0			0
$187$	29.2646	−	29.2646i	2.14004	−	2.14004i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−7.81786			−0.565680			−0.282840	−	0.959167i	$-0.591277\pi$
$191$	−7.81786			−0.565680			−0.282840	+	0.959167i	$0.591277\pi$
$192$		0			0
$193$	−23.4854			−1.69052			−0.845258	−	0.534358i	$-0.820553\pi$
$193$	−23.4854			−1.69052			−0.845258	+	0.534358i	$0.820553\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	12.3998	−	12.3998i	0.883453	−	0.883453i	−0.110431	−	0.993884i	$-0.535223\pi$
$197$	12.3998	−	12.3998i	0.883453	−	0.883453i	0.993884	+	0.110431i	$0.0352232\pi$
$198$		0			0
$199$	1.42807			0.101233			0.0506167	−	0.998718i	$-0.483881\pi$
$199$	1.42807			0.101233			0.0506167	+	0.998718i	$0.483881\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	9.70533	+	9.70533i	0.681180	+	0.681180i
$204$		0			0
$205$	−3.49261	+	3.49261i	−0.243935	+	0.243935i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	16.7068i		−	1.15563i
$210$		0			0
$211$	−9.18871	−	9.18871i	−0.632577	−	0.632577i	0.316137	−	0.948714i	$-0.397614\pi$
$211$	−9.18871	−	9.18871i	−0.632577	−	0.632577i	−0.948714	+	0.316137i	$0.897614\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			3.69970i			0.252318i
$216$		0			0
$217$		−	19.2493i		−	1.30673i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	9.67761	+	9.67761i	0.650986	+	0.650986i
$222$		0			0
$223$		−	1.56615i		−	0.104877i	−0.998624	−	0.0524385i	$-0.983301\pi$
$223$		−	1.56615i		−	0.104877i	0.998624	−	0.0524385i	$-0.0166994\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	8.36008	−	8.36008i	0.554878	−	0.554878i	−0.372967	−	0.927845i	$-0.621659\pi$
$227$	8.36008	−	8.36008i	0.554878	−	0.554878i	0.927845	+	0.372967i	$0.121659\pi$
$228$		0			0
$229$	−12.1157	−	12.1157i	−0.800630	−	0.800630i	0.182564	−	0.983194i	$-0.441560\pi$
$229$	−12.1157	−	12.1157i	−0.800630	−	0.800630i	−0.983194	+	0.182564i	$0.941560\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−18.7468			−1.22815			−0.614073	−	0.789249i	$-0.710470\pi$
$233$	−18.7468			−1.22815			−0.614073	+	0.789249i	$0.710470\pi$
$234$		0			0
$235$	9.14688	−	9.14688i	0.596677	−	0.596677i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−0.624861			−0.0404189			−0.0202095	−	0.999796i	$-0.506433\pi$
$239$	−0.624861			−0.0404189			−0.0202095	+	0.999796i	$0.506433\pi$
$240$		0			0
$241$	17.9025			1.15320			0.576600	−	0.817027i	$-0.304379\pi$
$241$	17.9025			1.15320			0.576600	+	0.817027i	$0.304379\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	9.32218	−	9.32218i	0.595572	−	0.595572i
$246$		0			0
$247$	5.52481			0.351535
$248$		0			0
$249$		0			0
$250$		0			0
$251$	15.7919	+	15.7919i	0.996779	+	0.996779i	0.999995	−	0.00321589i	$-0.00102365\pi$
$251$	15.7919	+	15.7919i	0.996779	+	0.996779i	−0.00321589	+	0.999995i	$0.501024\pi$
$252$		0			0
$253$	5.19056	−	5.19056i	0.326328	−	0.326328i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			16.2199i			1.01177i	0.862601	+	0.505885i	$0.168834\pi$
$257$			16.2199i			1.01177i	−0.862601	+	0.505885i	$0.831166\pi$
$258$		0			0
$259$	−0.0344586	−	0.0344586i	−0.00214115	−	0.00214115i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	0.316457i		−	0.0195136i	−0.999952	−	0.00975679i	$-0.996894\pi$
$263$		−	0.316457i		−	0.0195136i	0.999952	−	0.00975679i	$-0.00310573\pi$
$264$		0			0
$265$		−	8.54241i		−	0.524756i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	6.71152	+	6.71152i	0.409208	+	0.409208i	0.881462	−	0.472254i	$-0.156559\pi$
$269$	6.71152	+	6.71152i	0.409208	+	0.409208i	−0.472254	+	0.881462i	$0.656559\pi$
$270$		0			0
$271$		−	11.3985i		−	0.692411i	−0.938159	−	0.346205i	$-0.887470\pi$
$271$		−	11.3985i		−	0.692411i	0.938159	−	0.346205i	$-0.112530\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−4.37014	+	4.37014i	−0.263529	+	0.263529i
$276$		0			0
$277$	17.2648	+	17.2648i	1.03734	+	1.03734i	0.999275	+	0.0380679i	$0.0121203\pi$
$277$	17.2648	+	17.2648i	1.03734	+	1.03734i	0.0380679	+	0.999275i	$0.487880\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−2.69745			−0.160916			−0.0804582	−	0.996758i	$-0.525638\pi$
$281$	−2.69745			−0.160916			−0.0804582	+	0.996758i	$0.525638\pi$
$282$		0			0
$283$	−5.38979	+	5.38979i	−0.320389	+	0.320389i	−0.848916	−	0.528527i	$-0.822745\pi$
$283$	−5.38979	+	5.38979i	−0.320389	+	0.320389i	0.528527	+	0.848916i	$0.322745\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	22.1903			1.30985
$288$		0			0
$289$	−27.8430			−1.63782
$290$		0			0
$291$		0			0
$292$		0			0
$293$	3.09213	−	3.09213i	0.180644	−	0.180644i	−0.610992	−	0.791636i	$-0.709229\pi$
$293$	3.09213	−	3.09213i	0.180644	−	0.180644i	0.791636	+	0.610992i	$0.209229\pi$
$294$		0			0
$295$	9.20777			0.536097
$296$		0			0
$297$		0			0
$298$		0			0
$299$	1.71648	+	1.71648i	0.0992667	+	0.0992667i
$300$		0			0
$301$	11.7531	−	11.7531i	0.677435	−	0.677435i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	1.62287i		−	0.0929255i
$306$		0			0
$307$	17.8156	+	17.8156i	1.01679	+	1.01679i	0.999857	+	0.0169305i	$0.00538941\pi$
$307$	17.8156	+	17.8156i	1.01679	+	1.01679i	0.0169305	+	0.999857i	$0.494611\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			16.3422i			0.926683i	0.886180	+	0.463342i	$0.153350\pi$
$311$			16.3422i			0.926683i	−0.886180	+	0.463342i	$0.846650\pi$
$312$		0			0
$313$			24.0707i			1.36056i	0.732954	+	0.680278i	$0.238141\pi$
$313$			24.0707i			1.36056i	−0.732954	+	0.680278i	$0.761859\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−13.4498	−	13.4498i	−0.755414	−	0.755414i	0.220070	−	0.975484i	$-0.429371\pi$
$317$	−13.4498	−	13.4498i	−0.755414	−	0.755414i	−0.975484	+	0.220070i	$0.929371\pi$
$318$		0			0
$319$			18.8815i			1.05716i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−12.8001	+	12.8001i	−0.712218	+	0.712218i
$324$		0			0
$325$	−1.44518	−	1.44518i	−0.0801639	−	0.0801639i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−58.1148			−3.20397
$330$		0			0
$331$	−3.82091	+	3.82091i	−0.210016	+	0.210016i	−0.804274	−	0.594258i	$-0.797446\pi$
$331$	−3.82091	+	3.82091i	−0.210016	+	0.210016i	0.594258	+	0.804274i	$0.297446\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	8.31066			0.454060
$336$		0			0
$337$	−1.44837			−0.0788978			−0.0394489	−	0.999222i	$-0.512560\pi$
$337$	−1.44837			−0.0788978			−0.0394489	+	0.999222i	$0.512560\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	18.7245	−	18.7245i	1.01399	−	1.01399i
$342$		0			0
$343$	−27.7803			−1.50000
$344$		0			0
$345$		0			0
$346$		0			0
$347$	2.90479	+	2.90479i	0.155937	+	0.155937i	0.780764	−	0.624826i	$-0.214830\pi$
$347$	2.90479	+	2.90479i	0.155937	+	0.155937i	−0.624826	+	0.780764i	$0.714830\pi$
$348$		0			0
$349$	−22.0266	+	22.0266i	−1.17906	+	1.17906i	−0.199072	+	0.979985i	$0.563793\pi$
$349$	−22.0266	+	22.0266i	−1.17906	+	1.17906i	−0.979985	+	0.199072i	$0.936207\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	7.17990i		−	0.382147i	−0.981576	−	0.191074i	$-0.938803\pi$
$353$		−	7.17990i		−	0.382147i	0.981576	−	0.191074i	$-0.0611969\pi$
$354$		0			0
$355$	−6.31555	−	6.31555i	−0.335194	−	0.335194i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	23.4618i		−	1.23826i	−0.785287	−	0.619132i	$-0.787484\pi$
$359$		−	23.4618i		−	1.23826i	0.785287	−	0.619132i	$-0.212516\pi$
$360$		0			0
$361$		−	11.6926i		−	0.615399i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	9.69662	+	9.69662i	0.507544	+	0.507544i
$366$		0			0
$367$		−	2.96970i		−	0.155017i	−0.996992	−	0.0775086i	$-0.975303\pi$
$367$		−	2.96970i		−	0.155017i	0.996992	−	0.0775086i	$-0.0246965\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−27.1372	+	27.1372i	−1.40889	+	1.40889i
$372$		0			0
$373$	−5.38231	−	5.38231i	−0.278686	−	0.278686i	0.553899	−	0.832584i	$-0.313140\pi$
$373$	−5.38231	−	5.38231i	−0.278686	−	0.278686i	−0.832584	+	0.553899i	$0.813140\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−6.24398			−0.321582
$378$		0			0
$379$	−8.79894	+	8.79894i	−0.451971	+	0.451971i	−0.896008	−	0.444037i	$-0.853546\pi$
$379$	−8.79894	+	8.79894i	−0.451971	+	0.451971i	0.444037	+	0.896008i	$0.353546\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	15.1823			0.775780			0.387890	−	0.921706i	$-0.373204\pi$
$383$	15.1823			0.775780			0.387890	+	0.921706i	$0.373204\pi$
$384$		0			0
$385$	27.7657			1.41507
$386$		0			0
$387$		0			0
$388$		0			0
$389$	5.38360	−	5.38360i	0.272959	−	0.272959i	−0.557331	−	0.830290i	$-0.688175\pi$
$389$	5.38360	−	5.38360i	0.272959	−	0.272959i	0.830290	+	0.557331i	$0.188175\pi$
$390$		0			0
$391$	−7.95364			−0.402233
$392$		0			0
$393$		0			0
$394$		0			0
$395$	2.90450	+	2.90450i	0.146141	+	0.146141i
$396$		0			0
$397$	−4.98305	+	4.98305i	−0.250092	+	0.250092i	−0.821008	−	0.570916i	$-0.806588\pi$
$397$	−4.98305	+	4.98305i	−0.250092	+	0.250092i	0.570916	+	0.821008i	$0.306588\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	12.2573i		−	0.612099i	−0.952016	−	0.306050i	$-0.900993\pi$
$401$		−	12.2573i		−	0.612099i	0.952016	−	0.306050i	$-0.0990074\pi$
$402$		0			0
$403$	6.19207	+	6.19207i	0.308449	+	0.308449i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	0.0670385i		−	0.00332297i
$408$		0			0
$409$			28.4064i			1.40461i	0.711877	+	0.702304i	$0.247845\pi$
$409$			28.4064i			1.40461i	−0.711877	+	0.702304i	$0.752155\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−29.2508	−	29.2508i	−1.43934	−	1.43934i
$414$		0			0
$415$		−	6.40286i		−	0.314304i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−11.3131	+	11.3131i	−0.552681	+	0.552681i	−0.927214	−	0.374533i	$-0.877803\pi$
$419$	−11.3131	+	11.3131i	−0.552681	+	0.552681i	0.374533	+	0.927214i	$0.377803\pi$
$420$		0			0
$421$	−1.71988	−	1.71988i	−0.0838219	−	0.0838219i	0.663953	−	0.747775i	$-0.268878\pi$
$421$	−1.71988	−	1.71988i	−0.0838219	−	0.0838219i	−0.747775	+	0.663953i	$0.768878\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	6.69649			0.324828
$426$		0			0
$427$	−5.15548	+	5.15548i	−0.249491	+	0.249491i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	4.03049			0.194142			0.0970709	−	0.995277i	$-0.469053\pi$
$431$	4.03049			0.194142			0.0970709	+	0.995277i	$0.469053\pi$
$432$		0			0
$433$	6.01068			0.288855			0.144427	−	0.989515i	$-0.453866\pi$
$433$	6.01068			0.288855			0.144427	+	0.989515i	$0.453866\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−2.27031	+	2.27031i	−0.108604	+	0.108604i
$438$		0			0
$439$	13.1277			0.626549			0.313275	−	0.949663i	$-0.398574\pi$
$439$	13.1277			0.626549			0.313275	+	0.949663i	$0.398574\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−1.57959	−	1.57959i	−0.0750488	−	0.0750488i	0.668586	−	0.743635i	$-0.266900\pi$
$443$	−1.57959	−	1.57959i	−0.0750488	−	0.0750488i	−0.743635	+	0.668586i	$0.766900\pi$
$444$		0			0
$445$	8.72478	−	8.72478i	0.413594	−	0.413594i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	22.4970i		−	1.06170i	−0.847467	−	0.530849i	$-0.821873\pi$
$449$		−	22.4970i		−	1.06170i	0.847467	−	0.530849i	$-0.178127\pi$
$450$		0			0
$451$	21.5854	+	21.5854i	1.01642	+	1.01642i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			9.18194i			0.430456i
$456$		0			0
$457$			5.39811i			0.252513i	0.991998	+	0.126256i	$0.0402962\pi$
$457$			5.39811i			0.252513i	−0.991998	+	0.126256i	$0.959704\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−21.3939	−	21.3939i	−0.996412	−	0.996412i	0.00358156	−	0.999994i	$-0.498860\pi$
$461$	−21.3939	−	21.3939i	−0.996412	−	0.996412i	−0.999994	+	0.00358156i	$0.998860\pi$
$462$		0			0
$463$		−	24.9382i		−	1.15898i	−0.814980	−	0.579489i	$-0.803252\pi$
$463$		−	24.9382i		−	1.15898i	0.814980	−	0.579489i	$-0.196748\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	3.82512	−	3.82512i	0.177006	−	0.177006i	−0.613044	−	0.790049i	$-0.710055\pi$
$467$	3.82512	−	3.82512i	0.177006	−	0.177006i	0.790049	+	0.613044i	$0.210055\pi$
$468$		0			0
$469$	−26.4009	−	26.4009i	−1.21908	−	1.21908i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	22.8653			1.05135
$474$		0			0
$475$	1.91147	−	1.91147i	0.0877042	−	0.0877042i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	10.0760			0.460383			0.230192	−	0.973145i	$-0.426065\pi$
$479$	10.0760			0.460383			0.230192	+	0.973145i	$0.426065\pi$
$480$		0			0
$481$	0.0221692			0.00101083
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−5.93140	+	5.93140i	−0.269331	+	0.269331i
$486$		0			0
$487$	11.1510			0.505302			0.252651	−	0.967557i	$-0.418698\pi$
$487$	11.1510			0.505302			0.252651	+	0.967557i	$0.418698\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	10.7851	+	10.7851i	0.486725	+	0.486725i	0.907271	−	0.420546i	$-0.138161\pi$
$491$	10.7851	+	10.7851i	0.486725	+	0.486725i	−0.420546	+	0.907271i	$0.638161\pi$
$492$		0			0
$493$	14.4663	−	14.4663i	0.651531	−	0.651531i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			40.1259i			1.79989i
$498$		0			0
$499$	−10.6160	−	10.6160i	−0.475238	−	0.475238i	0.428367	−	0.903605i	$-0.359089\pi$
$499$	−10.6160	−	10.6160i	−0.475238	−	0.475238i	−0.903605	+	0.428367i	$0.859089\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	18.3508i		−	0.818224i	−0.912484	−	0.409112i	$-0.865839\pi$
$503$		−	18.3508i		−	0.818224i	0.912484	−	0.409112i	$-0.134161\pi$
$504$		0			0
$505$		−	4.11558i		−	0.183141i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	15.9467	+	15.9467i	0.706824	+	0.706824i	0.965866	−	0.259042i	$-0.0834069\pi$
$509$	15.9467	+	15.9467i	0.706824	+	0.706824i	−0.259042	+	0.965866i	$0.583407\pi$
$510$		0			0
$511$		−	61.6076i		−	2.72536i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	8.18502	−	8.18502i	0.360675	−	0.360675i
$516$		0			0
$517$	−56.5306	−	56.5306i	−2.48621	−	2.48621i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	22.6636			0.992908			0.496454	−	0.868063i	$-0.334635\pi$
$521$	22.6636			0.992908			0.496454	+	0.868063i	$0.334635\pi$
$522$		0			0
$523$	1.12095	−	1.12095i	0.0490158	−	0.0490158i	−0.682174	−	0.731190i	$-0.738965\pi$
$523$	1.12095	−	1.12095i	0.0490158	−	0.0490158i	0.731190	+	0.682174i	$0.238965\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−28.6921			−1.24985
$528$		0			0
$529$	21.5893			0.938665
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−7.13815	+	7.13815i	−0.309188	+	0.309188i
$534$		0			0
$535$	4.99981			0.216161
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−57.6140	−	57.6140i	−2.48161	−	2.48161i
$540$		0			0
$541$	32.3559	−	32.3559i	1.39109	−	1.39109i	0.568195	−	0.822894i	$-0.307642\pi$
$541$	32.3559	−	32.3559i	1.39109	−	1.39109i	0.822894	−	0.568195i	$-0.192358\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			8.96577i			0.384051i
$546$		0			0
$547$	−19.5398	−	19.5398i	−0.835463	−	0.835463i	0.152795	−	0.988258i	$-0.451172\pi$
$547$	−19.5398	−	19.5398i	−0.835463	−	0.835463i	−0.988258	+	0.152795i	$0.951172\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	8.25864i		−	0.351830i
$552$		0			0
$553$		−	18.4538i		−	0.784735i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	25.8255	+	25.8255i	1.09426	+	1.09426i	0.995068	+	0.0991915i	$0.0316256\pi$
$557$	25.8255	+	25.8255i	1.09426	+	1.09426i	0.0991915	+	0.995068i	$0.468374\pi$
$558$		0			0
$559$			7.56140i			0.319813i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−10.3693	+	10.3693i	−0.437013	+	0.437013i	−0.891006	−	0.453992i	$-0.849999\pi$
$563$	−10.3693	+	10.3693i	−0.437013	+	0.437013i	0.453992	+	0.891006i	$0.349999\pi$
$564$		0			0
$565$	−3.21032	−	3.21032i	−0.135059	−	0.135059i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−44.4509			−1.86348			−0.931738	−	0.363130i	$-0.881708\pi$
$569$	−44.4509			−1.86348			−0.931738	+	0.363130i	$0.881708\pi$
$570$		0			0
$571$	23.7391	−	23.7391i	0.993450	−	0.993450i	−0.00652855	−	0.999979i	$-0.502078\pi$
$571$	23.7391	−	23.7391i	0.993450	−	0.993450i	0.999979	+	0.00652855i	$0.00207812\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	1.18773			0.0495319
$576$		0			0
$577$	−2.99852			−0.124830			−0.0624150	−	0.998050i	$-0.519880\pi$
$577$	−2.99852			−0.124830			−0.0624150	+	0.998050i	$0.519880\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−20.3403	+	20.3403i	−0.843858	+	0.843858i
$582$		0			0
$583$	−52.7947			−2.18653
$584$		0			0
$585$		0			0
$586$		0			0
$587$	20.4168	+	20.4168i	0.842691	+	0.842691i	0.989208	−	0.146517i	$-0.0468064\pi$
$587$	20.4168	+	20.4168i	0.842691	+	0.842691i	−0.146517	+	0.989208i	$0.546806\pi$
$588$		0			0
$589$	−8.18998	+	8.18998i	−0.337462	+	0.337462i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			2.16808i			0.0890322i	0.999009	+	0.0445161i	$0.0141746\pi$
$593$			2.16808i			0.0890322i	−0.999009	+	0.0445161i	$0.985825\pi$
$594$		0			0
$595$	−21.2731	−	21.2731i	−0.872113	−	0.872113i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			7.37935i			0.301512i	0.988571	+	0.150756i	$0.0481708\pi$
$599$			7.37935i			0.301512i	−0.988571	+	0.150756i	$0.951829\pi$
$600$		0			0
$601$			2.20199i			0.0898209i	0.998991	+	0.0449105i	$0.0143003\pi$
$601$			2.20199i			0.0898209i	−0.998991	+	0.0449105i	$0.985700\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	19.2306	+	19.2306i	0.781837	+	0.781837i
$606$		0			0
$607$		−	1.10100i		−	0.0446880i	−0.999750	−	0.0223440i	$-0.992887\pi$
$607$		−	1.10100i		−	0.0446880i	0.999750	−	0.0223440i	$-0.00711291\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	18.6943	−	18.6943i	0.756290	−	0.756290i
$612$		0			0
$613$	25.9748	+	25.9748i	1.04911	+	1.04911i	0.998730	+	0.0503809i	$0.0160435\pi$
$613$	25.9748	+	25.9748i	1.04911	+	1.04911i	0.0503809	+	0.998730i	$0.483956\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	31.4011			1.26416			0.632081	−	0.774903i	$-0.282201\pi$
$617$	31.4011			1.26416			0.632081	+	0.774903i	$0.282201\pi$
$618$		0			0
$619$	10.6007	−	10.6007i	0.426078	−	0.426078i	−0.461212	−	0.887290i	$-0.652585\pi$
$619$	10.6007	−	10.6007i	0.426078	−	0.426078i	0.887290	+	0.461212i	$0.152585\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−55.4330			−2.22088
$624$		0			0
$625$	−1.00000			−0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−0.0513625	+	0.0513625i	−0.00204796	+	0.00204796i
$630$		0			0
$631$	−13.5474			−0.539313			−0.269657	−	0.962957i	$-0.586910\pi$
$631$	−13.5474			−0.539313			−0.269657	+	0.962957i	$0.586910\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	4.60879	+	4.60879i	0.182894	+	0.182894i
$636$		0			0
$637$	19.0526	−	19.0526i	0.754890	−	0.754890i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			40.4032i			1.59583i	0.602770	+	0.797915i	$0.294063\pi$
$641$			40.4032i			1.59583i	−0.602770	+	0.797915i	$0.705937\pi$
$642$		0			0
$643$	2.22995	+	2.22995i	0.0879407	+	0.0879407i	0.749709	−	0.661768i	$-0.230194\pi$
$643$	2.22995	+	2.22995i	0.0879407	+	0.0879407i	−0.661768	+	0.749709i	$0.730194\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	2.10482i		−	0.0827490i	−0.999144	−	0.0413745i	$-0.986826\pi$
$647$		−	2.10482i		−	0.0827490i	0.999144	−	0.0413745i	$-0.0131737\pi$
$648$		0			0
$649$		−	56.9069i		−	2.23379i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−24.4398	−	24.4398i	−0.956402	−	0.956402i	0.0426867	−	0.999089i	$-0.486408\pi$
$653$	−24.4398	−	24.4398i	−0.956402	−	0.956402i	−0.999089	+	0.0426867i	$0.986408\pi$
$654$		0			0
$655$		−	3.92990i		−	0.153554i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	5.67541	−	5.67541i	0.221082	−	0.221082i	−0.587872	−	0.808954i	$-0.700034\pi$
$659$	5.67541	−	5.67541i	0.221082	−	0.221082i	0.808954	+	0.587872i	$0.200034\pi$
$660$		0			0
$661$	−9.18787	−	9.18787i	−0.357367	−	0.357367i	0.505475	−	0.862841i	$-0.331317\pi$
$661$	−9.18787	−	9.18787i	−0.357367	−	0.357367i	−0.862841	+	0.505475i	$0.831317\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−12.1445			−0.470945
$666$		0			0
$667$	2.56584	−	2.56584i	0.0993498	−	0.0993498i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−10.0299			−0.387199
$672$		0			0
$673$	−5.85854			−0.225830			−0.112915	−	0.993605i	$-0.536019\pi$
$673$	−5.85854			−0.225830			−0.112915	+	0.993605i	$0.536019\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−29.6489	+	29.6489i	−1.13950	+	1.13950i	−0.150958	+	0.988540i	$0.548236\pi$
$677$	−29.6489	+	29.6489i	−1.13950	+	1.13950i	−0.988540	+	0.150958i	$0.951764\pi$
$678$		0			0
$679$	37.6852			1.44623
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−29.4194	−	29.4194i	−1.12570	−	1.12570i	−0.990868	−	0.134834i	$-0.956950\pi$
$683$	−29.4194	−	29.4194i	−1.12570	−	1.12570i	−0.134834	−	0.990868i	$-0.543050\pi$
$684$		0			0
$685$	−4.69290	+	4.69290i	−0.179307	+	0.179307i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	17.4589i		−	0.665130i
$690$		0			0
$691$	−20.6070	−	20.6070i	−0.783926	−	0.783926i	0.196565	−	0.980491i	$-0.437021\pi$
$691$	−20.6070	−	20.6070i	−0.783926	−	0.783926i	−0.980491	+	0.196565i	$0.937021\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			9.97570i			0.378400i
$696$		0			0
$697$		−	33.0760i		−	1.25284i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	3.37293	+	3.37293i	0.127394	+	0.127394i	0.767929	−	0.640535i	$-0.221287\pi$
$701$	3.37293	+	3.37293i	0.127394	+	0.127394i	−0.640535	+	0.767929i	$0.721287\pi$
$702$		0			0
$703$			0.0293221i			0.00110591i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−13.0742	+	13.0742i	−0.491706	+	0.491706i
$708$		0			0
$709$	34.1260	+	34.1260i	1.28163	+	1.28163i	0.939741	+	0.341886i	$0.111066\pi$
$709$	34.1260	+	34.1260i	1.28163	+	1.28163i	0.341886	+	0.939741i	$0.388934\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−5.08902			−0.190585
$714$		0			0
$715$	−8.93163	+	8.93163i	−0.334024	+	0.334024i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−41.4927			−1.54742			−0.773708	−	0.633542i	$-0.781600\pi$
$719$	−41.4927			−1.54742			−0.773708	+	0.633542i	$0.781600\pi$
$720$		0			0
$721$	−52.0036			−1.93672
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−2.16029	+	2.16029i	−0.0802310	+	0.0802310i
$726$		0			0
$727$	15.1442			0.561669			0.280835	−	0.959756i	$-0.409389\pi$
$727$	15.1442			0.561669			0.280835	+	0.959756i	$0.409389\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−17.5186	−	17.5186i	−0.647949	−	0.647949i
$732$		0			0
$733$	−1.53796	+	1.53796i	−0.0568060	+	0.0568060i	−0.734939	−	0.678133i	$-0.762789\pi$
$733$	−1.53796	+	1.53796i	−0.0568060	+	0.0568060i	0.678133	+	0.734939i	$0.262789\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	51.3625i		−	1.89196i
$738$		0			0
$739$	−35.3121	−	35.3121i	−1.29898	−	1.29898i	−0.929068	−	0.369910i	$-0.879389\pi$
$739$	−35.3121	−	35.3121i	−1.29898	−	1.29898i	−0.369910	−	0.929068i	$-0.620611\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	38.6663i		−	1.41853i	−0.704943	−	0.709264i	$-0.749028\pi$
$743$		−	38.6663i		−	1.41853i	0.704943	−	0.709264i	$-0.250972\pi$
$744$		0			0
$745$			13.4710i			0.493540i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−15.8832	−	15.8832i	−0.580359	−	0.580359i
$750$		0			0
$751$			53.8013i			1.96324i	0.190853	+	0.981619i	$0.438875\pi$
$751$			53.8013i			1.96324i	−0.190853	+	0.981619i	$0.561125\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−16.6314	+	16.6314i	−0.605277	+	0.605277i
$756$		0			0
$757$	10.7306	+	10.7306i	0.390010	+	0.390010i	0.874691	−	0.484681i	$-0.161064\pi$
$757$	10.7306	+	10.7306i	0.390010	+	0.390010i	−0.484681	+	0.874691i	$0.661064\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−19.3367			−0.700954			−0.350477	−	0.936571i	$-0.613981\pi$
$761$	−19.3367			−0.700954			−0.350477	+	0.936571i	$0.613981\pi$
$762$		0			0
$763$	28.4821	−	28.4821i	1.03112	−	1.03112i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	18.8187			0.679504
$768$		0			0
$769$	16.6094			0.598949			0.299474	−	0.954104i	$-0.403189\pi$
$769$	16.6094			0.598949			0.299474	+	0.954104i	$0.403189\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−27.9353	+	27.9353i	−1.00476	+	1.00476i	−0.00477424	+	0.999989i	$0.501520\pi$
$773$	−27.9353	+	27.9353i	−1.00476	+	1.00476i	−0.999989	+	0.00477424i	$0.998480\pi$
$774$		0			0
$775$	4.28465			0.153909
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−9.44131	−	9.44131i	−0.338270	−	0.338270i
$780$		0			0
$781$	−39.0320	+	39.0320i	−1.39668	+	1.39668i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	14.4126i		−	0.514407i
$786$		0			0
$787$	9.37684	+	9.37684i	0.334248	+	0.334248i	0.854197	−	0.519949i	$-0.174049\pi$
$787$	9.37684	+	9.37684i	0.334248	+	0.334248i	−0.519949	+	0.854197i	$0.674049\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			20.3968i			0.725227i
$792$		0			0
$793$		−	3.31681i		−	0.117783i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	12.3229	+	12.3229i	0.436498	+	0.436498i	0.890832	−	0.454334i	$-0.150123\pi$
$797$	12.3229	+	12.3229i	0.436498	+	0.436498i	−0.454334	+	0.890832i	$0.650123\pi$
$798$		0			0
$799$			86.6235i			3.06452i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	59.9281	−	59.9281i	2.11482	−	2.11482i
$804$		0			0
$805$	−3.77314	−	3.77314i	−0.132986	−	0.132986i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−19.4755			−0.684721			−0.342360	−	0.939569i	$-0.611226\pi$
$809$	−19.4755			−0.684721			−0.342360	+	0.939569i	$0.611226\pi$
$810$		0			0
$811$	−5.32021	+	5.32021i	−0.186818	+	0.186818i	−0.794319	−	0.607501i	$-0.792172\pi$
$811$	−5.32021	+	5.32021i	−0.186818	+	0.186818i	0.607501	+	0.794319i	$0.292172\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	22.7244			0.796002
$816$		0			0
$817$	−10.0011			−0.349895
$818$		0			0
$819$		0			0
$820$		0			0
$821$	21.3478	−	21.3478i	0.745042	−	0.745042i	−0.228502	−	0.973544i	$-0.573383\pi$
$821$	21.3478	−	21.3478i	0.745042	−	0.745042i	0.973544	+	0.228502i	$0.0733826\pi$
$822$		0			0
$823$	−17.9419			−0.625414			−0.312707	−	0.949850i	$-0.601236\pi$
$823$	−17.9419			−0.625414			−0.312707	+	0.949850i	$0.601236\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	25.7459	+	25.7459i	0.895272	+	0.895272i	0.995013	−	0.0997415i	$-0.0318016\pi$
$827$	25.7459	+	25.7459i	0.895272	+	0.895272i	−0.0997415	+	0.995013i	$0.531802\pi$
$828$		0			0
$829$	−17.3731	+	17.3731i	−0.603394	+	0.603394i	−0.941211	−	0.337818i	$-0.890311\pi$
$829$	−17.3731	+	17.3731i	−0.603394	+	0.603394i	0.337818	+	0.941211i	$0.390311\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			88.2836i			3.05885i
$834$		0			0
$835$	0.105742	+	0.105742i	0.00365934	+	0.00365934i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			19.5757i			0.675827i	0.941177	+	0.337913i	$0.109721\pi$
$839$			19.5757i			0.675827i	−0.941177	+	0.337913i	$0.890279\pi$
$840$		0			0
$841$		−	19.6663i		−	0.678149i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	6.23876	+	6.23876i	0.214620	+	0.214620i
$846$		0			0
$847$		−	122.182i		−	4.19823i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−0.00910998	+	0.00910998i	−0.000312286	+	0.000312286i
$852$		0			0
$853$	6.26218	+	6.26218i	0.214413	+	0.214413i	0.806139	−	0.591726i	$-0.201553\pi$
$853$	6.26218	+	6.26218i	0.214413	+	0.214413i	−0.591726	+	0.806139i	$0.701553\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	40.9392			1.39846			0.699229	−	0.714898i	$-0.253527\pi$
$857$	40.9392			1.39846			0.699229	+	0.714898i	$0.253527\pi$
$858$		0			0
$859$	24.9076	−	24.9076i	0.849837	−	0.849837i	−0.140275	−	0.990113i	$-0.544799\pi$
$859$	24.9076	−	24.9076i	0.849837	−	0.849837i	0.990113	+	0.140275i	$0.0447987\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−9.89680			−0.336891			−0.168445	−	0.985711i	$-0.553875\pi$
$863$	−9.89680			−0.336891			−0.168445	+	0.985711i	$0.553875\pi$
$864$		0			0
$865$	8.38827			0.285210
$866$		0			0
$867$		0			0
$868$		0			0
$869$	17.9507	−	17.9507i	0.608937	−	0.608937i
$870$		0			0
$871$	16.9852			0.575522
$872$		0			0
$873$		0			0
$874$		0			0
$875$	3.17676	+	3.17676i	0.107394	+	0.107394i
$876$		0			0
$877$	22.2824	−	22.2824i	0.752424	−	0.752424i	−0.222507	−	0.974931i	$-0.571424\pi$
$877$	22.2824	−	22.2824i	0.752424	−	0.752424i	0.974931	+	0.222507i	$0.0714240\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	47.9458i		−	1.61533i	−0.589639	−	0.807667i	$-0.700730\pi$
$881$		−	47.9458i		−	1.61533i	0.589639	−	0.807667i	$-0.299270\pi$
$882$		0			0
$883$	23.2635	+	23.2635i	0.782880	+	0.782880i	0.980316	−	0.197436i	$-0.0632615\pi$
$883$	23.2635	+	23.2635i	0.782880	+	0.782880i	−0.197436	+	0.980316i	$0.563261\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			32.4020i			1.08795i	0.839100	+	0.543977i	$0.183082\pi$
$887$			32.4020i			1.08795i	−0.839100	+	0.543977i	$0.816918\pi$
$888$		0			0
$889$		−	29.2820i		−	0.982087i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	24.7261	+	24.7261i	0.827427	+	0.827427i
$894$		0			0
$895$			19.1065i			0.638660i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	9.25608	−	9.25608i	0.308707	−	0.308707i
$900$		0			0
$901$	40.4495	+	40.4495i	1.34757	+	1.34757i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	16.2967			0.541721
$906$		0			0
$907$	−15.2567	+	15.2567i	−0.506589	+	0.506589i	−0.913478	−	0.406888i	$-0.866614\pi$
$907$	−15.2567	+	15.2567i	−0.506589	+	0.506589i	0.406888	+	0.913478i	$0.366614\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−45.6786			−1.51340			−0.756700	−	0.653762i	$-0.773190\pi$
$911$	−45.6786			−1.51340			−0.756700	+	0.653762i	$0.773190\pi$
$912$		0			0
$913$	−39.5716			−1.30963
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−12.4843	+	12.4843i	−0.412269	+	0.412269i
$918$		0			0
$919$	−48.1001			−1.58668			−0.793338	−	0.608782i	$-0.791658\pi$
$919$	−48.1001			−1.58668			−0.793338	+	0.608782i	$0.791658\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−12.9076	−	12.9076i	−0.424860	−	0.424860i
$924$		0			0
$925$	0.00767006	−	0.00767006i	0.000252190	−	0.000252190i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	10.8562i		−	0.356180i	−0.984014	−	0.178090i	$-0.943008\pi$
$929$		−	10.8562i		−	0.356180i	0.984014	−	0.178090i	$-0.0569918\pi$
$930$		0			0
$931$	25.1999	+	25.1999i	0.825895	+	0.825895i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	41.3864i		−	1.35348i
$936$		0			0
$937$			52.5798i			1.71771i	0.512220	+	0.858854i	$0.328823\pi$
$937$			52.5798i			1.71771i	−0.512220	+	0.858854i	$0.671177\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−34.9046	−	34.9046i	−1.13786	−	1.13786i	−0.988834	−	0.149023i	$-0.952387\pi$
$941$	−34.9046	−	34.9046i	−1.13786	−	1.13786i	−0.149023	−	0.988834i	$-0.547613\pi$
$942$		0			0
$943$		−	5.86657i		−	0.191042i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−15.6778	+	15.6778i	−0.509459	+	0.509459i	−0.914360	−	0.404902i	$-0.867306\pi$
$947$	−15.6778	+	15.6778i	−0.509459	+	0.509459i	0.404902	+	0.914360i	$0.367306\pi$
$948$		0			0
$949$	19.8178	+	19.8178i	0.643314	+	0.643314i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	4.85833			0.157377			0.0786884	−	0.996899i	$-0.474927\pi$
$953$	4.85833			0.157377			0.0786884	+	0.996899i	$0.474927\pi$
$954$		0			0
$955$	−5.52806	+	5.52806i	−0.178884	+	0.178884i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	29.8164			0.962822
$960$		0			0
$961$	12.6418			0.407799
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−16.6067	+	16.6067i	−0.534588	+	0.534588i
$966$		0			0
$967$	24.0916			0.774733			0.387367	−	0.921926i	$-0.373385\pi$
$967$	24.0916			0.774733			0.387367	+	0.921926i	$0.373385\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−13.3267	−	13.3267i	−0.427673	−	0.427673i	0.460162	−	0.887835i	$-0.347791\pi$
$971$	−13.3267	−	13.3267i	−0.427673	−	0.427673i	−0.887835	+	0.460162i	$0.847791\pi$
$972$		0			0
$973$	31.6904	−	31.6904i	1.01595	−	1.01595i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			34.7266i			1.11100i	0.831516	+	0.555501i	$0.187474\pi$
$977$			34.7266i			1.11100i	−0.831516	+	0.555501i	$0.812526\pi$
$978$		0			0
$979$	−53.9219	−	53.9219i	−1.72335	−	1.72335i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			52.7711i			1.68314i	0.540152	+	0.841568i	$0.318367\pi$
$983$			52.7711i			1.68314i	−0.540152	+	0.841568i	$0.681633\pi$
$984$		0			0
$985$		−	17.5360i		−	0.558744i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−3.10721	−	3.10721i	−0.0988035	−	0.0988035i
$990$		0			0
$991$			29.3280i			0.931634i	0.884881	+	0.465817i	$0.154239\pi$
$991$			29.3280i			0.931634i	−0.884881	+	0.465817i	$0.845761\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	1.00980	−	1.00980i	0.0320128	−	0.0320128i
$996$		0			0
$997$	−17.1758	−	17.1758i	−0.543964	−	0.543964i	0.380725	−	0.924688i	$-0.375675\pi$
$997$	−17.1758	−	17.1758i	−0.543964	−	0.543964i	−0.924688	+	0.380725i	$0.875675\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2880.2.bl.b.1871.7		24
3.2	odd	2	inner	2880.2.bl.b.1871.1		24
4.3	odd	2		720.2.bl.b.251.8	yes	24
12.11	even	2		720.2.bl.b.251.5	✓	24
16.3	odd	4	inner	2880.2.bl.b.431.1		24
16.13	even	4		720.2.bl.b.611.5	yes	24
48.29	odd	4		720.2.bl.b.611.8	yes	24
48.35	even	4	inner	2880.2.bl.b.431.7		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
720.2.bl.b.251.5	✓	24	12.11	even	2
720.2.bl.b.251.8	yes	24	4.3	odd	2
720.2.bl.b.611.5	yes	24	16.13	even	4
720.2.bl.b.611.8	yes	24	48.29	odd	4
2880.2.bl.b.431.1		24	16.3	odd	4	inner
2880.2.bl.b.431.7		24	48.35	even	4	inner
2880.2.bl.b.1871.1		24	3.2	odd	2	inner
2880.2.bl.b.1871.7		24	1.1	even	1	trivial

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.bl (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.707107 - 0.707107i) q^{5} -4.49261 q^{7} +O(q^{10})\)	\(q+(0.707107 - 0.707107i) q^{5} -4.49261 q^{7} +(-4.37014 - 4.37014i) q^{11} +(1.44518 - 1.44518i) q^{13} +6.69649i q^{17} +(1.91147 + 1.91147i) q^{19} +1.18773i q^{23} -1.00000i q^{25} +(-2.16029 - 2.16029i) q^{29} +4.28465i q^{31} +(-3.17676 + 3.17676i) q^{35} +(0.00767006 + 0.00767006i) q^{37} -4.93930 q^{41} +(-2.61608 + 2.61608i) q^{43} +12.9356 q^{47} +13.1836 q^{49} +(6.04040 - 6.04040i) q^{53} -6.18031 q^{55} +(6.51088 + 6.51088i) q^{59} +(1.14755 - 1.14755i) q^{61} -2.04379i q^{65} +(5.87653 + 5.87653i) q^{67} -8.93153i q^{71} +13.7131i q^{73} +(19.6333 + 19.6333i) q^{77} +4.10759i q^{79} +(4.52750 - 4.52750i) q^{83} +(4.73514 + 4.73514i) q^{85} +12.3387 q^{89} +(-6.49261 + 6.49261i) q^{91} +2.70322 q^{95} -8.38827 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 8 q^{7}+O(q^{10}) \) 24 * q + 8 * q^7	\( 24 q + 8 q^{7} + 24 q^{19} + 8 q^{37} - 48 q^{43} + 24 q^{49} - 24 q^{55} + 40 q^{61} + 40 q^{67} + 24 q^{85} - 40 q^{91} - 16 q^{97}+O(q^{100}) \) 24 * q + 8 * q^7 + 24 * q^19 + 8 * q^37 - 48 * q^43 + 24 * q^49 - 24 * q^55 + 40 * q^61 + 40 * q^67 + 24 * q^85 - 40 * q^91 - 16 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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