LMFDB - Embedded newform 2900.2.s.d.157.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2900,2,Mod(157,2900)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2900.157"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2900, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2900 = 2^{2} \cdot 5^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2900.s (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [30,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	no
Analytic conductor:	$23.1566165862$
Analytic rank:	$0$
Dimension:	$30$
Relative dimension:	$15$ over $\Q(i)$
Twist minimal:	no (minimal twist has level 580)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			157.1
Character	$\chi$	$=$	2900.157
Dual form			2900.2.s.d.1293.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-3.02258 q^{3} +(-0.186244 - 0.186244i) q^{7} +6.13596 q^{9} +(3.47659 + 3.47659i) q^{11} +(-1.24132 - 1.24132i) q^{13} +2.94456i q^{17} +(-2.18248 + 2.18248i) q^{19} +(0.562935 + 0.562935i) q^{21} +(-4.13617 + 4.13617i) q^{23} -9.47869 q^{27} +(4.56395 + 2.85839i) q^{29} +(-6.01854 - 6.01854i) q^{31} +(-10.5083 - 10.5083i) q^{33} +3.81738 q^{37} +(3.75197 + 3.75197i) q^{39} +(2.71137 - 2.71137i) q^{41} +7.22745 q^{43} +7.91450 q^{47} -6.93063i q^{49} -8.90015i q^{51} +(4.97626 - 4.97626i) q^{53} +(6.59672 - 6.59672i) q^{57} +8.78618i q^{59} +(-5.76599 - 5.76599i) q^{61} +(-1.14278 - 1.14278i) q^{63} +(5.62720 - 5.62720i) q^{67} +(12.5019 - 12.5019i) q^{69} +1.99299i q^{71} +2.07972i q^{73} -1.29499i q^{77} +(-6.95137 + 6.95137i) q^{79} +10.2422 q^{81} +(-12.1390 + 12.1390i) q^{83} +(-13.7949 - 8.63970i) q^{87} +(-9.41130 + 9.41130i) q^{89} +0.462375i q^{91} +(18.1915 + 18.1915i) q^{93} +0.105936 q^{97} +(21.3322 + 21.3322i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 30 q + 38 q^{9} - 4 q^{11} + 6 q^{13} - 4 q^{21} - 12 q^{27} - 4 q^{31} - 4 q^{33} + 24 q^{37} - 12 q^{39} + 10 q^{41} + 16 q^{43} + 32 q^{47} + 18 q^{53} - 24 q^{57} - 22 q^{61} - 24 q^{63} - 16 q^{67}+ \cdots + 16 q^{97}+O(q^{100}) $ 30 * q + 38 * q^9 - 4 * q^11 + 6 * q^13 - 4 * q^21 - 12 * q^27 - 4 * q^31 - 4 * q^33 + 24 * q^37 - 12 * q^39 + 10 * q^41 + 16 * q^43 + 32 * q^47 + 18 * q^53 - 24 * q^57 - 22 * q^61 - 24 * q^63 - 16 * q^67 - 8 * q^69 - 20 * q^79 + 70 * q^81 - 56 * q^83 - 48 * q^87 - 6 * q^89 - 44 * q^93 + 16 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2900\mathbb{Z}\right)^\times$.

$n$	$901$	$1277$	$1451$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$e\left(\frac{1}{4}\right)$	$1$

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
$3$	−3.02258			−1.74508			−0.872542	−	0.488538i	$-0.837530\pi$
$3$	−3.02258			−1.74508			−0.872542	+	0.488538i	$0.837530\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−0.186244	−	0.186244i	−0.0703935	−	0.0703935i	0.671034	−	0.741427i	$-0.265851\pi$
$7$	−0.186244	−	0.186244i	−0.0703935	−	0.0703935i	−0.741427	+	0.671034i	$0.765851\pi$
$8$		0			0
$9$	6.13596			2.04532
$10$		0			0
$11$	3.47659	+	3.47659i	1.04823	+	1.04823i	0.998776	+	0.0494547i	$0.0157484\pi$
$11$	3.47659	+	3.47659i	1.04823	+	1.04823i	0.0494547	+	0.998776i	$0.484252\pi$
$12$		0			0
$13$	−1.24132	−	1.24132i	−0.344279	−	0.344279i	0.513694	−	0.857973i	$-0.328277\pi$
$13$	−1.24132	−	1.24132i	−0.344279	−	0.344279i	−0.857973	+	0.513694i	$0.828277\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			2.94456i			0.714160i	0.934074	+	0.357080i	$0.116228\pi$
$17$			2.94456i			0.714160i	−0.934074	+	0.357080i	$0.883772\pi$
$18$		0			0
$19$	−2.18248	+	2.18248i	−0.500696	+	0.500696i	−0.911654	−	0.410958i	$-0.865194\pi$
$19$	−2.18248	+	2.18248i	−0.500696	+	0.500696i	0.410958	+	0.911654i	$0.365194\pi$
$20$		0			0
$21$	0.562935	+	0.562935i	0.122843	+	0.122843i
$22$		0			0
$23$	−4.13617	+	4.13617i	−0.862451	+	0.862451i	−0.991622	−	0.129172i	$-0.958768\pi$
$23$	−4.13617	+	4.13617i	−0.862451	+	0.862451i	0.129172	+	0.991622i	$0.458768\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−9.47869			−1.82417
$28$		0			0
$29$	4.56395	+	2.85839i	0.847504	+	0.530790i
$29$	4.56395	+	2.85839i	0.847504	+	0.530790i
$30$		0			0
$31$	−6.01854	−	6.01854i	−1.08096	−	1.08096i	−0.996420	−	0.0845420i	$-0.973057\pi$
$31$	−6.01854	−	6.01854i	−1.08096	−	1.08096i	−0.0845420	−	0.996420i	$-0.526943\pi$
$32$		0			0
$33$	−10.5083	−	10.5083i	−1.82925	−	1.82925i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	3.81738			0.627573			0.313786	−	0.949494i	$-0.398402\pi$
$37$	3.81738			0.627573			0.313786	+	0.949494i	$0.398402\pi$
$38$		0			0
$39$	3.75197	+	3.75197i	0.600797	+	0.600797i
$40$		0			0
$41$	2.71137	−	2.71137i	0.423445	−	0.423445i	−0.462943	−	0.886388i	$-0.653207\pi$
$41$	2.71137	−	2.71137i	0.423445	−	0.423445i	0.886388	+	0.462943i	$0.153207\pi$
$42$		0			0
$43$	7.22745			1.10218			0.551088	−	0.834447i	$-0.314213\pi$
$43$	7.22745			1.10218			0.551088	+	0.834447i	$0.314213\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	7.91450			1.15445			0.577224	−	0.816586i	$-0.304136\pi$
$47$	7.91450			1.15445			0.577224	+	0.816586i	$0.304136\pi$
$48$		0			0
$49$		−	6.93063i		−	0.990090i
$50$		0			0
$51$		−	8.90015i		−	1.24627i
$52$		0			0
$53$	4.97626	−	4.97626i	0.683541	−	0.683541i	−0.277255	−	0.960796i	$-0.589425\pi$
$53$	4.97626	−	4.97626i	0.683541	−	0.683541i	0.960796	+	0.277255i	$0.0894247\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	6.59672	−	6.59672i	0.873757	−	0.873757i
$58$		0			0
$59$			8.78618i			1.14386i	0.820301	+	0.571932i	$0.193806\pi$
$59$			8.78618i			1.14386i	−0.820301	+	0.571932i	$0.806194\pi$
$60$		0			0
$61$	−5.76599	−	5.76599i	−0.738260	−	0.738260i	0.233981	−	0.972241i	$-0.424825\pi$
$61$	−5.76599	−	5.76599i	−0.738260	−	0.738260i	−0.972241	+	0.233981i	$0.924825\pi$
$62$		0			0
$63$	−1.14278	−	1.14278i	−0.143977	−	0.143977i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	5.62720	−	5.62720i	0.687472	−	0.687472i	−0.274201	−	0.961672i	$-0.588413\pi$
$67$	5.62720	−	5.62720i	0.687472	−	0.687472i	0.961672	+	0.274201i	$0.0884133\pi$
$68$		0			0
$69$	12.5019	−	12.5019i	1.50505	−	1.50505i
$70$		0			0
$71$			1.99299i			0.236524i	0.992982	+	0.118262i	$0.0377323\pi$
$71$			1.99299i			0.236524i	−0.992982	+	0.118262i	$0.962268\pi$
$72$		0			0
$73$			2.07972i			0.243412i	0.992566	+	0.121706i	$0.0388366\pi$
$73$			2.07972i			0.243412i	−0.992566	+	0.121706i	$0.961163\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	1.29499i		−	0.147577i
$78$		0			0
$79$	−6.95137	+	6.95137i	−0.782090	+	0.782090i	−0.980183	−	0.198093i	$-0.936525\pi$
$79$	−6.95137	+	6.95137i	−0.782090	+	0.782090i	0.198093	+	0.980183i	$0.436525\pi$
$80$		0			0
$81$	10.2422			1.13802
$82$		0			0
$83$	−12.1390	+	12.1390i	−1.33242	+	1.33242i	−0.429229	+	0.903196i	$0.641215\pi$
$83$	−12.1390	+	12.1390i	−1.33242	+	1.33242i	−0.903196	+	0.429229i	$0.858785\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−13.7949	−	8.63970i	−1.47897	−	0.926273i
$88$		0			0
$89$	−9.41130	+	9.41130i	−0.997596	+	0.997596i	−0.999997	−	0.00240145i	$-0.999236\pi$
$89$	−9.41130	+	9.41130i	−0.997596	+	0.997596i	0.00240145	+	0.999997i	$0.499236\pi$
$90$		0			0
$91$			0.462375i			0.0484700i
$92$		0			0
$93$	18.1915	+	18.1915i	1.88637	+	1.88637i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	0.105936			0.0107562			0.00537810	−	0.999986i	$-0.498288\pi$
$97$	0.105936			0.0107562			0.00537810	+	0.999986i	$0.498288\pi$
$98$		0			0
$99$	21.3322	+	21.3322i	2.14397	+	2.14397i
$100$		0			0
$101$	−2.53360	−	2.53360i	−0.252102	−	0.252102i	0.569730	−	0.821832i	$-0.307048\pi$
$101$	−2.53360	−	2.53360i	−0.252102	−	0.252102i	−0.821832	+	0.569730i	$0.807048\pi$
$102$		0			0
$103$	−6.11000	+	6.11000i	−0.602037	+	0.602037i	−0.940853	−	0.338816i	$-0.889974\pi$
$103$	−6.11000	+	6.11000i	−0.602037	+	0.602037i	0.338816	+	0.940853i	$0.389974\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−11.4600	−	11.4600i	−1.10788	−	1.10788i	−0.993429	−	0.114453i	$-0.963488\pi$
$107$	−11.4600	−	11.4600i	−1.10788	−	1.10788i	−0.114453	−	0.993429i	$-0.536512\pi$
$108$		0			0
$109$	−7.69207			−0.736766			−0.368383	−	0.929674i	$-0.620089\pi$
$109$	−7.69207			−0.736766			−0.368383	+	0.929674i	$0.620089\pi$
$110$		0			0
$111$	−11.5383			−1.09517
$112$		0			0
$113$			7.42871i			0.698834i	0.936967	+	0.349417i	$0.113620\pi$
$113$			7.42871i			0.698834i	−0.936967	+	0.349417i	$0.886380\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−7.61667	−	7.61667i	−0.704162	−	0.704162i
$118$		0			0
$119$	0.548405	−	0.548405i	0.0502722	−	0.0502722i
$120$		0			0
$121$			13.1733i			1.19758i
$122$		0			0
$123$	−8.19532	+	8.19532i	−0.738947	+	0.738947i
$124$		0			0
$125$		0			0
$126$		0			0
$127$			15.2512i			1.35332i	0.736294	+	0.676662i	$0.236574\pi$
$127$			15.2512i			1.35332i	−0.736294	+	0.676662i	$0.763426\pi$
$128$		0			0
$129$	−21.8455			−1.92339
$130$		0			0
$131$	10.5132	−	10.5132i	0.918543	−	0.918543i	−0.0783803	−	0.996924i	$-0.524975\pi$
$131$	10.5132	−	10.5132i	0.918543	−	0.918543i	0.996924	+	0.0783803i	$0.0249748\pi$
$132$		0			0
$133$	0.812947			0.0704914
$134$		0			0
$135$		0			0
$136$		0			0
$137$			3.01807i			0.257851i	0.991654	+	0.128926i	$0.0411528\pi$
$137$			3.01807i			0.257851i	−0.991654	+	0.128926i	$0.958847\pi$
$138$		0			0
$139$		−	15.3760i		−	1.30417i	−0.758144	−	0.652087i	$-0.773894\pi$
$139$		−	15.3760i		−	1.30417i	0.758144	−	0.652087i	$-0.226106\pi$
$140$		0			0
$141$	−23.9222			−2.01461
$142$		0			0
$143$		−	8.63110i		−	0.721769i
$144$		0			0
$145$		0			0
$146$		0			0
$147$			20.9483i			1.72779i
$148$		0			0
$149$	11.5948			0.949886			0.474943	−	0.880017i	$-0.342469\pi$
$149$	11.5948			0.949886			0.474943	+	0.880017i	$0.342469\pi$
$150$		0			0
$151$			17.3261i			1.40997i	0.709220	+	0.704987i	$0.249047\pi$
$151$			17.3261i			1.40997i	−0.709220	+	0.704987i	$0.750953\pi$
$152$		0			0
$153$			18.0677i			1.46069i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−5.51043			−0.439780			−0.219890	−	0.975525i	$-0.570570\pi$
$157$	−5.51043			−0.439780			−0.219890	+	0.975525i	$0.570570\pi$
$158$		0			0
$159$	−15.0411	+	15.0411i	−1.19284	+	1.19284i
$160$		0			0
$161$	1.54067			0.121422
$162$		0			0
$163$		−	11.4281i		−	0.895121i	−0.894254	−	0.447560i	$-0.852293\pi$
$163$		−	11.4281i		−	0.895121i	0.894254	−	0.447560i	$-0.147707\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−12.5024	+	12.5024i	−0.967462	+	0.967462i	−0.999487	−	0.0320255i	$-0.989804\pi$
$167$	−12.5024	+	12.5024i	−0.967462	+	0.967462i	0.0320255	+	0.999487i	$0.489804\pi$
$168$		0			0
$169$		−	9.91827i		−	0.762944i
$170$		0			0
$171$	−13.3916	+	13.3916i	−1.02408	+	1.02408i
$172$		0			0
$173$	11.7370	+	11.7370i	0.892347	+	0.892347i	0.994744	−	0.102396i	$-0.0326510\pi$
$173$	11.7370	+	11.7370i	0.892347	+	0.892347i	−0.102396	+	0.994744i	$0.532651\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		−	26.5569i		−	1.99614i
$178$		0			0
$179$	−17.4026			−1.30073			−0.650367	−	0.759620i	$-0.725385\pi$
$179$	−17.4026			−1.30073			−0.650367	+	0.759620i	$0.725385\pi$
$180$		0			0
$181$	−2.44668			−0.181860			−0.0909301	−	0.995857i	$-0.528984\pi$
$181$	−2.44668			−0.181860			−0.0909301	+	0.995857i	$0.528984\pi$
$182$		0			0
$183$	17.4281	+	17.4281i	1.28833	+	1.28833i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−10.2370	+	10.2370i	−0.748605	+	0.748605i
$188$		0			0
$189$	1.76534	+	1.76534i	0.128410	+	0.128410i
$190$		0			0
$191$	10.2554	+	10.2554i	0.742055	+	0.742055i	0.972973	−	0.230918i	$-0.0741730\pi$
$191$	10.2554	+	10.2554i	0.742055	+	0.742055i	−0.230918	+	0.972973i	$0.574173\pi$
$192$		0			0
$193$	9.64593			0.694329			0.347164	−	0.937804i	$-0.387145\pi$
$193$	9.64593			0.694329			0.347164	+	0.937804i	$0.387145\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	6.59921	+	6.59921i	0.470175	+	0.470175i	0.901971	−	0.431797i	$-0.142120\pi$
$197$	6.59921	+	6.59921i	0.470175	+	0.470175i	−0.431797	+	0.901971i	$0.642120\pi$
$198$		0			0
$199$			24.6428i			1.74688i	0.486932	+	0.873440i	$0.338116\pi$
$199$			24.6428i			1.74688i	−0.486932	+	0.873440i	$0.661884\pi$
$200$		0			0
$201$	−17.0086	+	17.0086i	−1.19970	+	1.19970i
$202$		0			0
$203$	−0.317649	−	1.38236i	−0.0222946	−	0.0970229i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−25.3794	+	25.3794i	−1.76399	+	1.76399i
$208$		0			0
$209$	−15.1752			−1.04969
$210$		0			0
$211$	−6.93972	+	6.93972i	−0.477750	+	0.477750i	−0.904411	−	0.426662i	$-0.859690\pi$
$211$	−6.93972	+	6.93972i	−0.477750	+	0.477750i	0.426662	+	0.904411i	$0.359690\pi$
$212$		0			0
$213$		−	6.02395i		−	0.412754i
$214$		0			0
$215$		0			0
$216$		0			0
$217$			2.24183i			0.152185i
$218$		0			0
$219$		−	6.28610i		−	0.424775i
$220$		0			0
$221$	3.65513	−	3.65513i	0.245871	−	0.245871i
$222$		0			0
$223$	−19.5237	+	19.5237i	−1.30740	+	1.30740i	−0.384118	+	0.923284i	$0.625495\pi$
$223$	−19.5237	+	19.5237i	−1.30740	+	1.30740i	−0.923284	+	0.384118i	$0.874505\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	6.04274	+	6.04274i	0.401071	+	0.401071i	0.878610	−	0.477539i	$-0.158471\pi$
$227$	6.04274	+	6.04274i	0.401071	+	0.401071i	−0.477539	+	0.878610i	$0.658471\pi$
$228$		0			0
$229$	−0.0818182	−	0.0818182i	−0.00540670	−	0.00540670i	0.704398	−	0.709805i	$-0.251217\pi$
$229$	−0.0818182	−	0.0818182i	−0.00540670	−	0.00540670i	−0.709805	+	0.704398i	$0.751217\pi$
$230$		0			0
$231$			3.91419i			0.257535i
$232$		0			0
$233$	2.71748	−	2.71748i	0.178028	−	0.178028i	−0.612468	−	0.790496i	$-0.709823\pi$
$233$	2.71748	−	2.71748i	0.178028	−	0.178028i	0.790496	+	0.612468i	$0.209823\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	21.0110	−	21.0110i	1.36481	−	1.36481i
$238$		0			0
$239$			20.7574i			1.34268i	0.741148	+	0.671342i	$0.234282\pi$
$239$			20.7574i			1.34268i	−0.741148	+	0.671342i	$0.765718\pi$
$240$		0			0
$241$		−	9.33290i		−	0.601185i	−0.953753	−	0.300592i	$-0.902816\pi$
$241$		−	9.33290i		−	0.601185i	0.953753	−	0.300592i	$-0.0971844\pi$
$242$		0			0
$243$	−2.52163			−0.161762
$244$		0			0
$245$		0			0
$246$		0			0
$247$	5.41830			0.344758
$248$		0			0
$249$	36.6909	−	36.6909i	2.32519	−	2.32519i
$250$		0			0
$251$	−8.64957	−	8.64957i	−0.545956	−	0.545956i	0.379313	−	0.925269i	$-0.376160\pi$
$251$	−8.64957	−	8.64957i	−0.545956	−	0.545956i	−0.925269	+	0.379313i	$0.876160\pi$
$252$		0			0
$253$	−28.7595			−1.80810
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−17.7826	−	17.7826i	−1.10925	−	1.10925i	−0.993250	−	0.115995i	$-0.962994\pi$
$257$	−17.7826	−	17.7826i	−1.10925	−	1.10925i	−0.115995	−	0.993250i	$-0.537006\pi$
$258$		0			0
$259$	−0.710962	−	0.710962i	−0.0441770	−	0.0441770i
$260$		0			0
$261$	28.0042	+	17.5390i	1.73342	+	1.08564i
$262$		0			0
$263$	−7.43694			−0.458582			−0.229291	−	0.973358i	$-0.573641\pi$
$263$	−7.43694			−0.458582			−0.229291	+	0.973358i	$0.573641\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	28.4464	−	28.4464i	1.74089	−	1.74089i
$268$		0			0
$269$	9.78986	+	9.78986i	0.596898	+	0.596898i	0.939486	−	0.342588i	$-0.111303\pi$
$269$	9.78986	+	9.78986i	0.596898	+	0.596898i	−0.342588	+	0.939486i	$0.611303\pi$
$270$		0			0
$271$	7.38616	−	7.38616i	0.448677	−	0.448677i	−0.446237	−	0.894915i	$-0.647236\pi$
$271$	7.38616	−	7.38616i	0.448677	−	0.448677i	0.894915	+	0.446237i	$0.147236\pi$
$272$		0			0
$273$		−	1.39756i		−	0.0845843i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	6.37628	+	6.37628i	0.383114	+	0.383114i	0.872223	−	0.489109i	$-0.162678\pi$
$277$	6.37628	+	6.37628i	0.383114	+	0.383114i	−0.489109	+	0.872223i	$0.662678\pi$
$278$		0			0
$279$	−36.9295	−	36.9295i	−2.21091	−	2.21091i
$280$		0			0
$281$	−22.9292			−1.36784			−0.683921	−	0.729556i	$-0.739727\pi$
$281$	−22.9292			−1.36784			−0.683921	+	0.729556i	$0.739727\pi$
$282$		0			0
$283$	−13.3009	−	13.3009i	−0.790655	−	0.790655i	0.190946	−	0.981601i	$-0.438845\pi$
$283$	−13.3009	−	13.3009i	−0.790655	−	0.790655i	−0.981601	+	0.190946i	$0.938845\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−1.00995			−0.0596155
$288$		0			0
$289$	8.32958			0.489975
$290$		0			0
$291$	−0.320201			−0.0187705
$292$		0			0
$293$	−15.9198			−0.930043			−0.465022	−	0.885299i	$-0.653953\pi$
$293$	−15.9198			−0.930043			−0.465022	+	0.885299i	$0.653953\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−32.9535	−	32.9535i	−1.91216	−	1.91216i
$298$		0			0
$299$	10.2686			0.593848
$300$		0			0
$301$	−1.34607	−	1.34607i	−0.0775860	−	0.0775860i
$302$		0			0
$303$	7.65799	+	7.65799i	0.439940	+	0.439940i
$304$		0			0
$305$		0			0
$306$		0			0
$307$			14.0466i			0.801679i	0.916148	+	0.400840i	$0.131282\pi$
$307$			14.0466i			0.801679i	−0.916148	+	0.400840i	$0.868718\pi$
$308$		0			0
$309$	18.4679	−	18.4679i	1.05060	−	1.05060i
$310$		0			0
$311$	8.29913	+	8.29913i	0.470600	+	0.470600i	0.902109	−	0.431508i	$-0.142019\pi$
$311$	8.29913	+	8.29913i	0.470600	+	0.470600i	−0.431508	+	0.902109i	$0.642019\pi$
$312$		0			0
$313$	0.406578	−	0.406578i	0.0229811	−	0.0229811i	−0.695523	−	0.718504i	$-0.744827\pi$
$313$	0.406578	−	0.406578i	0.0229811	−	0.0229811i	0.718504	+	0.695523i	$0.244827\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−27.7371			−1.55787			−0.778936	−	0.627104i	$-0.784240\pi$
$317$	−27.7371			−1.55787			−0.778936	+	0.627104i	$0.784240\pi$
$318$		0			0
$319$	5.92952	+	25.8044i	0.331989	+	1.44477i
$320$		0			0
$321$	34.6388	+	34.6388i	1.93335	+	1.93335i
$322$		0			0
$323$	−6.42644	−	6.42644i	−0.357577	−	0.357577i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	23.2498			1.28572
$328$		0			0
$329$	−1.47403	−	1.47403i	−0.0812657	−	0.0812657i
$330$		0			0
$331$	−11.9539	+	11.9539i	−0.657046	+	0.657046i	−0.954680	−	0.297634i	$-0.903802\pi$
$331$	−11.9539	+	11.9539i	−0.657046	+	0.657046i	0.297634	+	0.954680i	$0.403802\pi$
$332$		0			0
$333$	23.4233			1.28359
$334$		0			0
$335$		0			0
$336$		0			0
$337$	6.88861			0.375247			0.187623	−	0.982241i	$-0.439922\pi$
$337$	6.88861			0.375247			0.187623	+	0.982241i	$0.439922\pi$
$338$		0			0
$339$		−	22.4538i		−	1.21952i
$340$		0			0
$341$		−	41.8480i		−	2.26620i
$342$		0			0
$343$	−2.59449	+	2.59449i	−0.140089	+	0.140089i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−1.62997	+	1.62997i	−0.0875014	+	0.0875014i	−0.749503	−	0.662001i	$-0.769707\pi$
$347$	−1.62997	+	1.62997i	−0.0875014	+	0.0875014i	0.662001	+	0.749503i	$0.269707\pi$
$348$		0			0
$349$			30.7949i			1.64841i	0.566288	+	0.824207i	$0.308379\pi$
$349$			30.7949i			1.64841i	−0.566288	+	0.824207i	$0.691621\pi$
$350$		0			0
$351$	11.7661	+	11.7661i	0.628025	+	0.628025i
$352$		0			0
$353$	26.2595	+	26.2595i	1.39765	+	1.39765i	0.806719	+	0.590936i	$0.201241\pi$
$353$	26.2595	+	26.2595i	1.39765	+	1.39765i	0.590936	+	0.806719i	$0.298759\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−1.65760	+	1.65760i	−0.0877293	+	0.0877293i
$358$		0			0
$359$	−1.78966	+	1.78966i	−0.0944545	+	0.0944545i	−0.752755	−	0.658301i	$-0.771276\pi$
$359$	−1.78966	+	1.78966i	−0.0944545	+	0.0944545i	0.658301	+	0.752755i	$0.271276\pi$
$360$		0			0
$361$			9.47354i			0.498608i
$362$		0			0
$363$		−	39.8174i		−	2.08987i
$364$		0			0
$365$		0			0
$366$		0			0
$367$			22.6566i			1.18267i	0.806428	+	0.591333i	$0.201398\pi$
$367$			22.6566i			1.18267i	−0.806428	+	0.591333i	$0.798602\pi$
$368$		0			0
$369$	16.6369	−	16.6369i	0.866081	−	0.866081i
$370$		0			0
$371$	−1.85359			−0.0962337
$372$		0			0
$373$	14.6753	−	14.6753i	0.759857	−	0.759857i	−0.216439	−	0.976296i	$-0.569444\pi$
$373$	14.6753	−	14.6753i	0.759857	−	0.759857i	0.976296	+	0.216439i	$0.0694442\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−2.11714	−	9.21347i	−0.109038	−	0.474518i
$378$		0			0
$379$	15.3792	−	15.3792i	0.789979	−	0.789979i	−0.191512	−	0.981490i	$-0.561339\pi$
$379$	15.3792	−	15.3792i	0.789979	−	0.789979i	0.981490	+	0.191512i	$0.0613390\pi$
$380$		0			0
$381$		−	46.0979i		−	2.36167i
$382$		0			0
$383$	−20.3925	−	20.3925i	−1.04201	−	1.04201i	−0.999078	−	0.0429286i	$-0.986331\pi$
$383$	−20.3925	−	20.3925i	−1.04201	−	1.04201i	−0.0429286	−	0.999078i	$-0.513669\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	44.3474			2.25430
$388$		0			0
$389$	5.44528	+	5.44528i	0.276087	+	0.276087i	0.831545	−	0.555458i	$-0.187457\pi$
$389$	5.44528	+	5.44528i	0.276087	+	0.276087i	−0.555458	+	0.831545i	$0.687457\pi$
$390$		0			0
$391$	−12.1792	−	12.1792i	−0.615928	−	0.615928i
$392$		0			0
$393$	−31.7770	+	31.7770i	−1.60294	+	1.60294i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	7.84350	+	7.84350i	0.393654	+	0.393654i	0.875988	−	0.482334i	$-0.160211\pi$
$397$	7.84350	+	7.84350i	0.393654	+	0.393654i	−0.482334	+	0.875988i	$0.660211\pi$
$398$		0			0
$399$	−2.45719			−0.123014
$400$		0			0
$401$	−18.6799			−0.932832			−0.466416	−	0.884565i	$-0.654455\pi$
$401$	−18.6799			−0.932832			−0.466416	+	0.884565i	$0.654455\pi$
$402$		0			0
$403$			14.9418i			0.744306i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	13.2714	+	13.2714i	0.657841	+	0.657841i
$408$		0			0
$409$	−19.2184	+	19.2184i	−0.950288	+	0.950288i	−0.998822	−	0.0485337i	$-0.984545\pi$
$409$	−19.2184	+	19.2184i	−0.950288	+	0.950288i	0.0485337	+	0.998822i	$0.484545\pi$
$410$		0			0
$411$		−	9.12234i		−	0.449972i
$412$		0			0
$413$	1.63637	−	1.63637i	0.0805205	−	0.0805205i
$414$		0			0
$415$		0			0
$416$		0			0
$417$			46.4750i			2.27589i
$418$		0			0
$419$	16.5340			0.807739			0.403870	−	0.914817i	$-0.367665\pi$
$419$	16.5340			0.807739			0.403870	+	0.914817i	$0.367665\pi$
$420$		0			0
$421$	−20.8265	+	20.8265i	−1.01502	+	1.01502i	−0.0151346	+	0.999885i	$0.504818\pi$
$421$	−20.8265	+	20.8265i	−1.01502	+	1.01502i	−0.999885	+	0.0151346i	$0.995182\pi$
$422$		0			0
$423$	48.5631			2.36122
$424$		0			0
$425$		0			0
$426$		0			0
$427$			2.14776i			0.103937i
$428$		0			0
$429$			26.0881i			1.25955i
$430$		0			0
$431$	−8.89803			−0.428603			−0.214301	−	0.976768i	$-0.568747\pi$
$431$	−8.89803			−0.428603			−0.214301	+	0.976768i	$0.568747\pi$
$432$		0			0
$433$		−	30.9901i		−	1.48929i	−0.667461	−	0.744644i	$-0.732619\pi$
$433$		−	30.9901i		−	1.48929i	0.667461	−	0.744644i	$-0.267381\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	18.0542i		−	0.863651i
$438$		0			0
$439$	−28.1031			−1.34129			−0.670645	−	0.741778i	$-0.733983\pi$
$439$	−28.1031			−1.34129			−0.670645	+	0.741778i	$0.733983\pi$
$440$		0			0
$441$		−	42.5261i		−	2.02505i
$442$		0			0
$443$		−	26.8865i		−	1.27742i	−0.769448	−	0.638709i	$-0.779469\pi$
$443$		−	26.8865i		−	1.27742i	0.769448	−	0.638709i	$-0.220531\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−35.0463			−1.65763
$448$		0			0
$449$	7.84155	−	7.84155i	0.370065	−	0.370065i	−0.497435	−	0.867501i	$-0.665725\pi$
$449$	7.84155	−	7.84155i	0.370065	−	0.370065i	0.867501	+	0.497435i	$0.165725\pi$
$450$		0			0
$451$	18.8526			0.887736
$452$		0			0
$453$		−	52.3693i		−	2.46053i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−19.5214	+	19.5214i	−0.913174	+	0.913174i	−0.996521	−	0.0833467i	$-0.973439\pi$
$457$	−19.5214	+	19.5214i	−0.913174	+	0.913174i	0.0833467	+	0.996521i	$0.473439\pi$
$458$		0			0
$459$		−	27.9105i		−	1.30275i
$460$		0			0
$461$	10.2212	−	10.2212i	0.476050	−	0.476050i	−0.427816	−	0.903866i	$-0.640717\pi$
$461$	10.2212	−	10.2212i	0.476050	−	0.476050i	0.903866	+	0.427816i	$0.140717\pi$
$462$		0			0
$463$	−15.9518	−	15.9518i	−0.741343	−	0.741343i	0.231493	−	0.972836i	$-0.425639\pi$
$463$	−15.9518	−	15.9518i	−0.741343	−	0.741343i	−0.972836	+	0.231493i	$0.925639\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	20.7674i		−	0.960999i	−0.876995	−	0.480500i	$-0.840455\pi$
$467$		−	20.7674i		−	0.960999i	0.876995	−	0.480500i	$-0.159545\pi$
$468$		0			0
$469$	−2.09606			−0.0967870
$470$		0			0
$471$	16.6557			0.767453
$472$		0			0
$473$	25.1269	+	25.1269i	1.15533	+	1.15533i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	30.5341	−	30.5341i	1.39806	−	1.39806i
$478$		0			0
$479$	19.8347	+	19.8347i	0.906269	+	0.906269i	0.995969	−	0.0896997i	$-0.0285907\pi$
$479$	19.8347	+	19.8347i	0.906269	+	0.906269i	−0.0896997	+	0.995969i	$0.528591\pi$
$480$		0			0
$481$	−4.73857	−	4.73857i	−0.216060	−	0.216060i
$482$		0			0
$483$	−4.65679			−0.211891
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−5.84728	−	5.84728i	−0.264966	−	0.264966i	0.562102	−	0.827068i	$-0.309993\pi$
$487$	−5.84728	−	5.84728i	−0.264966	−	0.264966i	−0.827068	+	0.562102i	$0.809993\pi$
$488$		0			0
$489$			34.5424i			1.56206i
$490$		0			0
$491$	29.6778	−	29.6778i	1.33934	−	1.33934i	0.442642	−	0.896698i	$-0.354041\pi$
$491$	29.6778	−	29.6778i	1.33934	−	1.33934i	0.896698	−	0.442642i	$-0.145959\pi$
$492$		0			0
$493$	−8.41669	+	13.4388i	−0.379069	+	0.605253i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	0.371181	−	0.371181i	0.0166497	−	0.0166497i
$498$		0			0
$499$	−18.3098			−0.819658			−0.409829	−	0.912162i	$-0.634412\pi$
$499$	−18.3098			−0.819658			−0.409829	+	0.912162i	$0.634412\pi$
$500$		0			0
$501$	37.7893	−	37.7893i	1.68830	−	1.68830i
$502$		0			0
$503$		−	2.84678i		−	0.126931i	−0.997984	−	0.0634657i	$-0.979785\pi$
$503$		−	2.84678i		−	0.126931i	0.997984	−	0.0634657i	$-0.0202154\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$			29.9787i			1.33140i
$508$		0			0
$509$			28.9616i			1.28370i	0.766831	+	0.641850i	$0.221833\pi$
$509$			28.9616i			1.28370i	−0.766831	+	0.641850i	$0.778167\pi$
$510$		0			0
$511$	0.387334	−	0.387334i	0.0171346	−	0.0171346i
$512$		0			0
$513$	20.6871	−	20.6871i	0.913356	−	0.913356i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	27.5155	+	27.5155i	1.21013	+	1.21013i
$518$		0			0
$519$	−35.4760	−	35.4760i	−1.55722	−	1.55722i
$520$		0			0
$521$			39.1235i			1.71403i	0.515291	+	0.857015i	$0.327684\pi$
$521$			39.1235i			1.71403i	−0.515291	+	0.857015i	$0.672316\pi$
$522$		0			0
$523$	−10.5906	+	10.5906i	−0.463094	+	0.463094i	−0.899668	−	0.436574i	$-0.856192\pi$
$523$	−10.5906	+	10.5906i	−0.463094	+	0.463094i	0.436574	+	0.899668i	$0.356192\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	17.7219	−	17.7219i	0.771980	−	0.771980i
$528$		0			0
$529$		−	11.2158i		−	0.487642i
$530$		0			0
$531$			53.9117i			2.33957i
$532$		0			0
$533$	−6.73134			−0.291567
$534$		0			0
$535$		0			0
$536$		0			0
$537$	52.6008			2.26989
$538$		0			0
$539$	24.0949	−	24.0949i	1.03784	−	1.03784i
$540$		0			0
$541$	18.9873	+	18.9873i	0.816330	+	0.816330i	0.985574	−	0.169244i	$-0.0541327\pi$
$541$	18.9873	+	18.9873i	0.816330	+	0.816330i	−0.169244	+	0.985574i	$0.554133\pi$
$542$		0			0
$543$	7.39527			0.317361
$544$		0			0
$545$		0			0
$546$		0			0
$547$	6.30698	+	6.30698i	0.269667	+	0.269667i	0.828966	−	0.559299i	$-0.188930\pi$
$547$	6.30698	+	6.30698i	0.269667	+	0.269667i	−0.559299	+	0.828966i	$0.688930\pi$
$548$		0			0
$549$	−35.3799	−	35.3799i	−1.50998	−	1.50998i
$550$		0			0
$551$	−16.1991	+	3.72235i	−0.690106	+	0.158577i
$552$		0			0
$553$	2.58930			0.110108
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−25.4909	+	25.4909i	−1.08008	+	1.08008i	−0.0835822	+	0.996501i	$0.526636\pi$
$557$	−25.4909	+	25.4909i	−1.08008	+	1.08008i	−0.996501	+	0.0835822i	$0.973364\pi$
$558$		0			0
$559$	−8.97155	−	8.97155i	−0.379456	−	0.379456i
$560$		0			0
$561$	30.9422	−	30.9422i	1.30638	−	1.30638i
$562$		0			0
$563$			0.898544i			0.0378691i	0.999821	+	0.0189346i	$0.00602742\pi$
$563$			0.898544i			0.0378691i	−0.999821	+	0.0189346i	$0.993973\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−1.90754	−	1.90754i	−0.0801090	−	0.0801090i
$568$		0			0
$569$	−12.6292	−	12.6292i	−0.529445	−	0.529445i	0.390962	−	0.920407i	$-0.372142\pi$
$569$	−12.6292	−	12.6292i	−0.529445	−	0.529445i	−0.920407	+	0.390962i	$0.872142\pi$
$570$		0			0
$571$	10.2347			0.428309			0.214154	−	0.976800i	$-0.431300\pi$
$571$	10.2347			0.428309			0.214154	+	0.976800i	$0.431300\pi$
$572$		0			0
$573$	−30.9977	−	30.9977i	−1.29495	−	1.29495i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	0.390421			0.0162534			0.00812671	−	0.999967i	$-0.497413\pi$
$577$	0.390421			0.0162534			0.00812671	+	0.999967i	$0.497413\pi$
$578$		0			0
$579$	−29.1555			−1.21166
$580$		0			0
$581$	4.52161			0.187588
$582$		0			0
$583$	34.6008			1.43302
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−14.0937	−	14.0937i	−0.581711	−	0.581711i	0.353662	−	0.935373i	$-0.384936\pi$
$587$	−14.0937	−	14.0937i	−0.581711	−	0.581711i	−0.935373	+	0.353662i	$0.884936\pi$
$588$		0			0
$589$	26.2707			1.08247
$590$		0			0
$591$	−19.9466	−	19.9466i	−0.820494	−	0.820494i
$592$		0			0
$593$	−23.3965	−	23.3965i	−0.960778	−	0.960778i	0.0384817	−	0.999259i	$-0.487748\pi$
$593$	−23.3965	−	23.3965i	−0.960778	−	0.960778i	−0.999259	+	0.0384817i	$0.987748\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		−	74.4846i		−	3.04845i
$598$		0			0
$599$	13.9860	−	13.9860i	0.571450	−	0.571450i	−0.361083	−	0.932534i	$-0.617593\pi$
$599$	13.9860	−	13.9860i	0.571450	−	0.571450i	0.932534	+	0.361083i	$0.117593\pi$
$600$		0			0
$601$	−23.6692	−	23.6692i	−0.965486	−	0.965486i	0.0339379	−	0.999424i	$-0.489195\pi$
$601$	−23.6692	−	23.6692i	−0.965486	−	0.965486i	−0.999424	+	0.0339379i	$0.989195\pi$
$602$		0			0
$603$	34.5283	−	34.5283i	1.40610	−	1.40610i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−10.8999			−0.442412			−0.221206	−	0.975227i	$-0.570999\pi$
$607$	−10.8999			−0.442412			−0.221206	+	0.975227i	$0.570999\pi$
$608$		0			0
$609$	0.960118	+	4.17830i	0.0389060	+	0.169313i
$610$		0			0
$611$	−9.82441	−	9.82441i	−0.397453	−	0.397453i
$612$		0			0
$613$	−12.1687	−	12.1687i	−0.491490	−	0.491490i	0.417285	−	0.908776i	$-0.362982\pi$
$613$	−12.1687	−	12.1687i	−0.491490	−	0.491490i	−0.908776	+	0.417285i	$0.862982\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	23.4321			0.943343			0.471671	−	0.881774i	$-0.343651\pi$
$617$	23.4321			0.943343			0.471671	+	0.881774i	$0.343651\pi$
$618$		0			0
$619$	18.6870	+	18.6870i	0.751095	+	0.751095i	0.974684	−	0.223589i	$-0.0717772\pi$
$619$	18.6870	+	18.6870i	0.751095	+	0.751095i	−0.223589	+	0.974684i	$0.571777\pi$
$620$		0			0
$621$	39.2054	−	39.2054i	1.57326	−	1.57326i
$622$		0			0
$623$	3.50559			0.140448
$624$		0			0
$625$		0			0
$626$		0			0
$627$	45.8682			1.83180
$628$		0			0
$629$			11.2405i			0.448187i
$630$		0			0
$631$		−	15.1431i		−	0.602839i	−0.953492	−	0.301420i	$-0.902540\pi$
$631$		−	15.1431i		−	0.602839i	0.953492	−	0.301420i	$-0.0974605\pi$
$632$		0			0
$633$	20.9758	−	20.9758i	0.833714	−	0.833714i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−8.60310	+	8.60310i	−0.340867	+	0.340867i
$638$		0			0
$639$			12.2289i			0.483767i
$640$		0			0
$641$	−12.3208	−	12.3208i	−0.486644	−	0.486644i	0.420601	−	0.907246i	$-0.361819\pi$
$641$	−12.3208	−	12.3208i	−0.486644	−	0.486644i	−0.907246	+	0.420601i	$0.861819\pi$
$642$		0			0
$643$	−12.3721	−	12.3721i	−0.487907	−	0.487907i	0.419738	−	0.907645i	$-0.362122\pi$
$643$	−12.3721	−	12.3721i	−0.487907	−	0.487907i	−0.907645	+	0.419738i	$0.862122\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	13.9856	−	13.9856i	0.549832	−	0.549832i	−0.376560	−	0.926392i	$-0.622893\pi$
$647$	13.9856	−	13.9856i	0.549832	−	0.549832i	0.926392	+	0.376560i	$0.122893\pi$
$648$		0			0
$649$	−30.5460	+	30.5460i	−1.19903	+	1.19903i
$650$		0			0
$651$		−	6.77610i		−	0.265576i
$652$		0			0
$653$			8.43188i			0.329965i	0.986296	+	0.164982i	$0.0527568\pi$
$653$			8.43188i			0.329965i	−0.986296	+	0.164982i	$0.947243\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$			12.7611i			0.497856i
$658$		0			0
$659$	−10.4998	+	10.4998i	−0.409015	+	0.409015i	−0.881395	−	0.472380i	$-0.843395\pi$
$659$	−10.4998	+	10.4998i	−0.409015	+	0.409015i	0.472380	+	0.881395i	$0.343395\pi$
$660$		0			0
$661$	37.9948			1.47783			0.738913	−	0.673800i	$-0.235339\pi$
$661$	37.9948			1.47783			0.738913	+	0.673800i	$0.235339\pi$
$662$		0			0
$663$	−11.0479	+	11.0479i	−0.429065	+	0.429065i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−30.7000	+	7.05447i	−1.18871	+	0.273150i
$668$		0			0
$669$	59.0118	−	59.0118i	2.28153	−	2.28153i
$670$		0			0
$671$		−	40.0920i		−	1.54773i
$672$		0			0
$673$	20.9035	+	20.9035i	0.805771	+	0.805771i	0.983991	−	0.178220i	$-0.0570337\pi$
$673$	20.9035	+	20.9035i	0.805771	+	0.805771i	−0.178220	+	0.983991i	$0.557034\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	20.5920			0.791415			0.395708	−	0.918377i	$-0.370499\pi$
$677$	20.5920			0.791415			0.395708	+	0.918377i	$0.370499\pi$
$678$		0			0
$679$	−0.0197300	−	0.0197300i	−0.000757167	−	0.000757167i
$680$		0			0
$681$	−18.2646	−	18.2646i	−0.699903	−	0.699903i
$682$		0			0
$683$	6.89370	−	6.89370i	0.263780	−	0.263780i	−0.562808	−	0.826588i	$-0.690279\pi$
$683$	6.89370	−	6.89370i	0.263780	−	0.263780i	0.826588	+	0.562808i	$0.190279\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	0.247302	+	0.247302i	0.00943514	+	0.00943514i
$688$		0			0
$689$	−12.3542			−0.470658
$690$		0			0
$691$	8.57085			0.326051			0.163025	−	0.986622i	$-0.447875\pi$
$691$	8.57085			0.326051			0.163025	+	0.986622i	$0.447875\pi$
$692$		0			0
$693$		−	7.94598i		−	0.301843i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	7.98379	+	7.98379i	0.302407	+	0.302407i
$698$		0			0
$699$	−8.21379	+	8.21379i	−0.310674	+	0.310674i
$700$		0			0
$701$			3.49884i			0.132149i	0.997815	+	0.0660746i	$0.0210475\pi$
$701$			3.49884i			0.132149i	−0.997815	+	0.0660746i	$0.978952\pi$
$702$		0			0
$703$	−8.33135	+	8.33135i	−0.314223	+	0.314223i
$704$		0			0
$705$		0			0
$706$		0			0
$707$			0.943732i			0.0354927i
$708$		0			0
$709$	−24.1355			−0.906427			−0.453213	−	0.891402i	$-0.649722\pi$
$709$	−24.1355			−0.906427			−0.453213	+	0.891402i	$0.649722\pi$
$710$		0			0
$711$	−42.6534	+	42.6534i	−1.59963	+	1.59963i
$712$		0			0
$713$	49.7874			1.86455
$714$		0			0
$715$		0			0
$716$		0			0
$717$		−	62.7408i		−	2.34310i
$718$		0			0
$719$		−	36.9249i		−	1.37707i	−0.725205	−	0.688533i	$-0.758255\pi$
$719$		−	36.9249i		−	1.37707i	0.725205	−	0.688533i	$-0.241745\pi$
$720$		0			0
$721$	2.27590			0.0847589
$722$		0			0
$723$			28.2094i			1.04912i
$724$		0			0
$725$		0			0
$726$		0			0
$727$		−	9.19648i		−	0.341078i	−0.985351	−	0.170539i	$-0.945449\pi$
$727$		−	9.19648i		−	0.341078i	0.985351	−	0.170539i	$-0.0545509\pi$
$728$		0			0
$729$	−23.1047			−0.855728
$730$		0			0
$731$			21.2816i			0.787130i
$732$		0			0
$733$		−	32.2250i		−	1.19026i	−0.803631	−	0.595128i	$-0.797101\pi$
$733$		−	32.2250i		−	1.19026i	0.803631	−	0.595128i	$-0.202899\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	39.1269			1.44126
$738$		0			0
$739$	13.8383	−	13.8383i	0.509051	−	0.509051i	−0.405184	−	0.914235i	$-0.632793\pi$
$739$	13.8383	−	13.8383i	0.509051	−	0.509051i	0.914235	+	0.405184i	$0.132793\pi$
$740$		0			0
$741$	−16.3772			−0.601633
$742$		0			0
$743$			7.77448i			0.285218i	0.989779	+	0.142609i	$0.0455492\pi$
$743$			7.77448i			0.285218i	−0.989779	+	0.142609i	$0.954451\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−74.4842	+	74.4842i	−2.72524	+	2.72524i
$748$		0			0
$749$			4.26871i			0.155975i
$750$		0			0
$751$	13.6159	−	13.6159i	0.496853	−	0.496853i	−0.413604	−	0.910457i	$-0.635730\pi$
$751$	13.6159	−	13.6159i	0.496853	−	0.496853i	0.910457	+	0.413604i	$0.135730\pi$
$752$		0			0
$753$	26.1440	+	26.1440i	0.952739	+	0.952739i
$754$		0			0
$755$		0			0
$756$		0			0
$757$			30.5635i			1.11085i	0.831566	+	0.555426i	$0.187445\pi$
$757$			30.5635i			1.11085i	−0.831566	+	0.555426i	$0.812555\pi$
$758$		0			0
$759$	86.9278			3.15528
$760$		0			0
$761$	−49.2126			−1.78395			−0.891977	−	0.452081i	$-0.850682\pi$
$761$	−49.2126			−1.78395			−0.891977	+	0.452081i	$0.850682\pi$
$762$		0			0
$763$	1.43260	+	1.43260i	0.0518635	+	0.0518635i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	10.9064	−	10.9064i	0.393809	−	0.393809i
$768$		0			0
$769$	−37.7124	−	37.7124i	−1.35994	−	1.35994i	−0.873979	−	0.485964i	$-0.838468\pi$
$769$	−37.7124	−	37.7124i	−1.35994	−	1.35994i	−0.485964	−	0.873979i	$-0.661532\pi$
$770$		0			0
$771$	53.7491	+	53.7491i	1.93573	+	1.93573i
$772$		0			0
$773$	39.0189			1.40341			0.701705	−	0.712467i	$-0.252422\pi$
$773$	39.0189			1.40341			0.701705	+	0.712467i	$0.252422\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	2.14894	+	2.14894i	0.0770927	+	0.0770927i
$778$		0			0
$779$			11.8350i			0.424034i
$780$		0			0
$781$	−6.92880	+	6.92880i	−0.247932	+	0.247932i
$782$		0			0
$783$	−43.2602	−	27.0938i	−1.54599	−	0.968253i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	2.53126	−	2.53126i	0.0902298	−	0.0902298i	−0.660551	−	0.750781i	$-0.729677\pi$
$787$	2.53126	−	2.53126i	0.0902298	−	0.0902298i	0.750781	+	0.660551i	$0.229677\pi$
$788$		0			0
$789$	22.4787			0.800264
$790$		0			0
$791$	1.38355	−	1.38355i	0.0491933	−	0.0491933i
$792$		0			0
$793$			14.3148i			0.508335i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			18.9107i			0.669853i	0.942244	+	0.334926i	$0.108711\pi$
$797$			18.9107i			0.669853i	−0.942244	+	0.334926i	$0.891289\pi$
$798$		0			0
$799$			23.3047i			0.824461i
$800$		0			0
$801$	−57.7474	+	57.7474i	−2.04040	+	2.04040i
$802$		0			0
$803$	−7.23032	+	7.23032i	−0.255152	+	0.255152i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−29.5906	−	29.5906i	−1.04164	−	1.04164i
$808$		0			0
$809$	−26.1951	−	26.1951i	−0.920969	−	0.920969i	0.0761287	−	0.997098i	$-0.475744\pi$
$809$	−26.1951	−	26.1951i	−0.920969	−	0.920969i	−0.997098	+	0.0761287i	$0.975744\pi$
$810$		0			0
$811$			20.7298i			0.727923i	0.931414	+	0.363962i	$0.118576\pi$
$811$			20.7298i			0.727923i	−0.931414	+	0.363962i	$0.881424\pi$
$812$		0			0
$813$	−22.3252	+	22.3252i	−0.782980	+	0.782980i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−15.7738	+	15.7738i	−0.551855	+	0.551855i
$818$		0			0
$819$			2.83711i			0.0991368i
$820$		0			0
$821$			2.75383i			0.0961093i	0.998845	+	0.0480547i	$0.0153022\pi$
$821$			2.75383i			0.0961093i	−0.998845	+	0.0480547i	$0.984698\pi$
$822$		0			0
$823$	−10.7412			−0.374415			−0.187208	−	0.982320i	$-0.559944\pi$
$823$	−10.7412			−0.374415			−0.187208	+	0.982320i	$0.559944\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	33.3096			1.15829			0.579144	−	0.815225i	$-0.303387\pi$
$827$	33.3096			1.15829			0.579144	+	0.815225i	$0.303387\pi$
$828$		0			0
$829$	9.46860	−	9.46860i	0.328858	−	0.328858i	−0.523294	−	0.852152i	$-0.675297\pi$
$829$	9.46860	−	9.46860i	0.328858	−	0.328858i	0.852152	+	0.523294i	$0.175297\pi$
$830$		0			0
$831$	−19.2728	−	19.2728i	−0.668566	−	0.668566i
$832$		0			0
$833$	20.4076			0.707082
$834$		0			0
$835$		0			0
$836$		0			0
$837$	57.0479	+	57.0479i	1.97186	+	1.97186i
$838$		0			0
$839$	−29.1671	−	29.1671i	−1.00696	−	1.00696i	−0.999976	−	0.00698587i	$-0.997776\pi$
$839$	−29.1671	−	29.1671i	−1.00696	−	1.00696i	−0.00698587	−	0.999976i	$-0.502224\pi$
$840$		0			0
$841$	12.6592	+	26.0911i	0.436525	+	0.899692i
$842$		0			0
$843$	69.3053			2.38700
$844$		0			0
$845$		0			0
$846$		0			0
$847$	2.45345	−	2.45345i	0.0843016	−	0.0843016i
$848$		0			0
$849$	40.2029	+	40.2029i	1.37976	+	1.37976i
$850$		0			0
$851$	−15.7893	+	15.7893i	−0.541250	+	0.541250i
$852$		0			0
$853$			25.2009i			0.862861i	0.902146	+	0.431430i	$0.141991\pi$
$853$			25.2009i			0.862861i	−0.902146	+	0.431430i	$0.858009\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−17.4654	−	17.4654i	−0.596606	−	0.596606i	0.342801	−	0.939408i	$-0.388624\pi$
$857$	−17.4654	−	17.4654i	−0.596606	−	0.596606i	−0.939408	+	0.342801i	$0.888624\pi$
$858$		0			0
$859$	3.61444	+	3.61444i	0.123323	+	0.123323i	0.766075	−	0.642752i	$-0.222207\pi$
$859$	3.61444	+	3.61444i	0.123323	+	0.123323i	−0.642752	+	0.766075i	$0.722207\pi$
$860$		0			0
$861$	3.05265			0.104034
$862$		0			0
$863$	0.261748	+	0.261748i	0.00890999	+	0.00890999i	0.711548	−	0.702638i	$-0.247995\pi$
$863$	0.261748	+	0.261748i	0.00890999	+	0.00890999i	−0.702638	+	0.711548i	$0.747995\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−25.1768			−0.855048
$868$		0			0
$869$	−48.3341			−1.63962
$870$		0			0
$871$	−13.9703			−0.473364
$872$		0			0
$873$	0.650021			0.0219999
$874$		0			0
$875$		0			0
$876$		0			0
$877$	2.67832	+	2.67832i	0.0904404	+	0.0904404i	0.750880	−	0.660439i	$-0.229630\pi$
$877$	2.67832	+	2.67832i	0.0904404	+	0.0904404i	−0.660439	+	0.750880i	$0.729630\pi$
$878$		0			0
$879$	48.1187			1.62300
$880$		0			0
$881$	26.6157	+	26.6157i	0.896707	+	0.896707i	0.995143	−	0.0984364i	$-0.0313841\pi$
$881$	26.6157	+	26.6157i	0.896707	+	0.896707i	−0.0984364	+	0.995143i	$0.531384\pi$
$882$		0			0
$883$	−33.3666	−	33.3666i	−1.12288	−	1.12288i	−0.991307	−	0.131569i	$-0.957998\pi$
$883$	−33.3666	−	33.3666i	−1.12288	−	1.12288i	−0.131569	−	0.991307i	$-0.542002\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	34.5625i		−	1.16050i	−0.814440	−	0.580248i	$-0.802956\pi$
$887$		−	34.5625i		−	1.16050i	0.814440	−	0.580248i	$-0.197044\pi$
$888$		0			0
$889$	2.84044	−	2.84044i	0.0952652	−	0.0952652i
$890$		0			0
$891$	35.6078	+	35.6078i	1.19290	+	1.19290i
$892$		0			0
$893$	−17.2733	+	17.2733i	−0.578028	+	0.578028i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−31.0376			−1.03631
$898$		0			0
$899$	−10.2650	−	44.6716i	−0.342356	−	1.48988i
$900$		0			0
$901$	14.6529	+	14.6529i	0.488158	+	0.488158i
$902$		0			0
$903$	4.06859	+	4.06859i	0.135394	+	0.135394i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−0.682191			−0.0226518			−0.0113259	−	0.999936i	$-0.503605\pi$
$907$	−0.682191			−0.0226518			−0.0113259	+	0.999936i	$0.503605\pi$
$908$		0			0
$909$	−15.5461	−	15.5461i	−0.515630	−	0.515630i
$910$		0			0
$911$	−11.7012	+	11.7012i	−0.387680	+	0.387680i	−0.873859	−	0.486179i	$-0.838390\pi$
$911$	−11.7012	+	11.7012i	−0.387680	+	0.387680i	0.486179	+	0.873859i	$0.338390\pi$
$912$		0			0
$913$	−84.4044			−2.79338
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−3.91604			−0.129319
$918$		0			0
$919$			7.87894i			0.259902i	0.991520	+	0.129951i	$0.0414821\pi$
$919$			7.87894i			0.259902i	−0.991520	+	0.129951i	$0.958518\pi$
$920$		0			0
$921$		−	42.4568i		−	1.39900i
$922$		0			0
$923$	2.47393	−	2.47393i	0.0814303	−	0.0814303i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−37.4908	+	37.4908i	−1.23136	+	1.23136i
$928$		0			0
$929$			3.81328i			0.125110i	0.998042	+	0.0625549i	$0.0199248\pi$
$929$			3.81328i			0.125110i	−0.998042	+	0.0625549i	$0.980075\pi$
$930$		0			0
$931$	15.1260	+	15.1260i	0.495734	+	0.495734i
$932$		0			0
$933$	−25.0847	−	25.0847i	−0.821238	−	0.821238i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−26.5659	+	26.5659i	−0.867870	+	0.867870i	−0.992236	−	0.124367i	$-0.960310\pi$
$937$	−26.5659	+	26.5659i	−0.867870	+	0.867870i	0.124367	+	0.992236i	$0.460310\pi$
$938$		0			0
$939$	−1.22891	+	1.22891i	−0.0401040	+	0.0401040i
$940$		0			0
$941$			21.2195i			0.691735i	0.938283	+	0.345868i	$0.112415\pi$
$941$			21.2195i			0.691735i	−0.938283	+	0.345868i	$0.887585\pi$
$942$		0			0
$943$			22.4294i			0.730401i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			55.8929i			1.81627i	0.418673	+	0.908137i	$0.362495\pi$
$947$			55.8929i			1.81627i	−0.418673	+	0.908137i	$0.637505\pi$
$948$		0			0
$949$	2.58159	−	2.58159i	0.0838018	−	0.0838018i
$950$		0			0
$951$	83.8375			2.71862
$952$		0			0
$953$	−36.1253	+	36.1253i	−1.17021	+	1.17021i	−0.188053	+	0.982159i	$0.560218\pi$
$953$	−36.1253	+	36.1253i	−1.17021	+	1.17021i	−0.982159	+	0.188053i	$0.939782\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−17.9224	−	77.9958i	−0.579350	−	2.52125i
$958$		0			0
$959$	0.562096	−	0.562096i	0.0181510	−	0.0181510i
$960$		0			0
$961$			41.4457i			1.33696i
$962$		0			0
$963$	−70.3183	−	70.3183i	−2.26597	−	2.26597i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	7.16476			0.230403			0.115202	−	0.993342i	$-0.463249\pi$
$967$	7.16476			0.230403			0.115202	+	0.993342i	$0.463249\pi$
$968$		0			0
$969$	19.4244	+	19.4244i	0.624002	+	0.624002i
$970$		0			0
$971$	0.232530	+	0.232530i	0.00746225	+	0.00746225i	0.710828	−	0.703366i	$-0.248320\pi$
$971$	0.232530	+	0.232530i	0.00746225	+	0.00746225i	−0.703366	+	0.710828i	$0.748320\pi$
$972$		0			0
$973$	−2.86368	+	2.86368i	−0.0918053	+	0.0918053i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−4.97909	−	4.97909i	−0.159295	−	0.159295i	0.622959	−	0.782254i	$-0.285930\pi$
$977$	−4.97909	−	4.97909i	−0.159295	−	0.159295i	−0.782254	+	0.622959i	$0.785930\pi$
$978$		0			0
$979$	−65.4384			−2.09142
$980$		0			0
$981$	−47.1982			−1.50692
$982$		0			0
$983$		−	29.0039i		−	0.925080i	−0.886599	−	0.462540i	$-0.846938\pi$
$983$		−	29.0039i		−	0.925080i	0.886599	−	0.462540i	$-0.153062\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	4.45535	+	4.45535i	0.141815	+	0.141815i
$988$		0			0
$989$	−29.8939	+	29.8939i	−0.950572	+	0.950572i
$990$		0			0
$991$			55.1215i			1.75099i	0.483224	+	0.875497i	$0.339466\pi$
$991$			55.1215i			1.75099i	−0.483224	+	0.875497i	$0.660534\pi$
$992$		0			0
$993$	36.1316	−	36.1316i	1.14660	−	1.14660i
$994$		0			0
$995$		0			0
$996$		0			0
$997$			26.2683i			0.831925i	0.909382	+	0.415963i	$0.136555\pi$
$997$			26.2683i			0.831925i	−0.909382	+	0.415963i	$0.863445\pi$
$998$		0			0
$999$	−36.1837			−1.14480

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2900.2.s.d.157.1		30
5.2	odd	4		580.2.j.a.273.15	yes	30
5.3	odd	4		2900.2.j.d.2593.1		30
5.4	even	2		580.2.s.a.157.15	yes	30
29.17	odd	4		2900.2.j.d.1757.15		30
145.17	even	4		580.2.s.a.133.15	yes	30
145.104	odd	4		580.2.j.a.17.1	✓	30
145.133	even	4	inner	2900.2.s.d.1293.1		30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
580.2.j.a.17.1	✓	30	145.104	odd	4
580.2.j.a.273.15	yes	30	5.2	odd	4
580.2.s.a.133.15	yes	30	145.17	even	4
580.2.s.a.157.15	yes	30	5.4	even	2
2900.2.j.d.1757.15		30	29.17	odd	4
2900.2.j.d.2593.1		30	5.3	odd	4
2900.2.s.d.157.1		30	1.1	even	1	trivial
2900.2.s.d.1293.1		30	145.133	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2900 = 2^{2} \cdot 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2900.s (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q-3.02258 q^{3} +(-0.186244 - 0.186244i) q^{7} +6.13596 q^{9} +(3.47659 + 3.47659i) q^{11} +(-1.24132 - 1.24132i) q^{13} +2.94456i q^{17} +(-2.18248 + 2.18248i) q^{19} +(0.562935 + 0.562935i) q^{21} +(-4.13617 + 4.13617i) q^{23} -9.47869 q^{27} +(4.56395 + 2.85839i) q^{29} +(-6.01854 - 6.01854i) q^{31} +(-10.5083 - 10.5083i) q^{33} +3.81738 q^{37} +(3.75197 + 3.75197i) q^{39} +(2.71137 - 2.71137i) q^{41} +7.22745 q^{43} +7.91450 q^{47} -6.93063i q^{49} -8.90015i q^{51} +(4.97626 - 4.97626i) q^{53} +(6.59672 - 6.59672i) q^{57} +8.78618i q^{59} +(-5.76599 - 5.76599i) q^{61} +(-1.14278 - 1.14278i) q^{63} +(5.62720 - 5.62720i) q^{67} +(12.5019 - 12.5019i) q^{69} +1.99299i q^{71} +2.07972i q^{73} -1.29499i q^{77} +(-6.95137 + 6.95137i) q^{79} +10.2422 q^{81} +(-12.1390 + 12.1390i) q^{83} +(-13.7949 - 8.63970i) q^{87} +(-9.41130 + 9.41130i) q^{89} +0.462375i q^{91} +(18.1915 + 18.1915i) q^{93} +0.105936 q^{97} +(21.3322 + 21.3322i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 38 q^{9} - 4 q^{11} + 6 q^{13} - 4 q^{21} - 12 q^{27} - 4 q^{31} - 4 q^{33} + 24 q^{37} - 12 q^{39} + 10 q^{41} + 16 q^{43} + 32 q^{47} + 18 q^{53} - 24 q^{57} - 22 q^{61} - 24 q^{63} - 16 q^{67}+ \cdots + 16 q^{97}+O(q^{100}) \) 30 * q + 38 * q^9 - 4 * q^11 + 6 * q^13 - 4 * q^21 - 12 * q^27 - 4 * q^31 - 4 * q^33 + 24 * q^37 - 12 * q^39 + 10 * q^41 + 16 * q^43 + 32 * q^47 + 18 * q^53 - 24 * q^57 - 22 * q^61 - 24 * q^63 - 16 * q^67 - 8 * q^69 - 20 * q^79 + 70 * q^81 - 56 * q^83 - 48 * q^87 - 6 * q^89 - 44 * q^93 + 16 * q^97

Introduction

L-functions

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