LMFDB - Embedded newform 572.2.f.c.441.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [572,2,Mod(441,572)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(572, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

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

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

chi := DirichletCharacter("572.441");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 572 = 2^{2} \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	572.f (of order $2$, degree $1$, 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:	$4.56744299562$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 21x^{6} + 136x^{4} + 309x^{2} + 225 $ x^8 + 21x^6 + 136x^4 + 309*x^2 + 225
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			441.2
Root			$-3.32727i$ of defining polynomial
Character	$\chi$	$=$	572.441
Dual form			572.2.f.c.441.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.32727 q^{3} +2.74343i q^{5} -4.32727i q^{7} +8.07070 q^{9} +O(q^{10})$	$q-3.32727 q^{3} +2.74343i q^{5} -4.32727i q^{7} +8.07070 q^{9} -1.00000i q^{11} +(-3.60281 - 0.140622i) q^{13} -9.12812i q^{15} +2.28124 q^{17} +5.78945i q^{19} +14.3980i q^{21} +1.58384 q^{23} -2.52641 q^{25} -16.8716 q^{27} -1.44892 q^{29} +8.74343i q^{31} +3.32727i q^{33} +11.8716 q^{35} +7.39796i q^{37} +(11.9875 + 0.467888i) q^{39} +4.33867i q^{41} -3.44892 q^{43} +22.1414i q^{45} +2.65453i q^{47} -11.7252 q^{49} -7.59031 q^{51} +11.8009 q^{53} +2.74343 q^{55} -19.2630i q^{57} +0.986729i q^{59} -5.44892 q^{61} -34.9240i q^{63} +(0.385788 - 9.88405i) q^{65} -0.180941i q^{67} -5.26984 q^{69} +15.1281i q^{71} +6.72523i q^{73} +8.40604 q^{75} -4.32727 q^{77} +12.6659 q^{79} +31.9240 q^{81} -11.5329i q^{83} +6.25844i q^{85} +4.82093 q^{87} -6.56563i q^{89} +(-0.608510 + 15.5903i) q^{91} -29.0917i q^{93} -15.8830 q^{95} +5.25657i q^{97} -8.07070i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 2 q^{3} + 18 q^{9}+O(q^{10}) $ 8 * q - 2 * q^3 + 18 * q^9	$ 8 q - 2 q^{3} + 18 q^{9} - 8 q^{13} + 16 q^{17} + 10 q^{23} - 52 q^{25} - 32 q^{27} - 4 q^{29} - 8 q^{35} + 16 q^{39} - 20 q^{43} + 2 q^{49} + 40 q^{51} + 38 q^{53} - 36 q^{61} + 36 q^{65} - 52 q^{69} + 10 q^{75} - 10 q^{77} + 40 q^{79} + 32 q^{81} + 56 q^{87} + 22 q^{91} - 12 q^{95}+O(q^{100}) $ 8 * q - 2 * q^3 + 18 * q^9 - 8 * q^13 + 16 * q^17 + 10 * q^23 - 52 * q^25 - 32 * q^27 - 4 * q^29 - 8 * q^35 + 16 * q^39 - 20 * q^43 + 2 * q^49 + 40 * q^51 + 38 * q^53 - 36 * q^61 + 36 * q^65 - 52 * q^69 + 10 * q^75 - 10 * q^77 + 40 * q^79 + 32 * q^81 + 56 * q^87 + 22 * q^91 - 12 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/572\mathbb{Z}\right)^\times$.

$n$	$287$	$353$	$365$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	−3.32727			−1.92100			−0.960499	−	0.278284i	$-0.910234\pi$
$3$	−3.32727			−1.92100			−0.960499	+	0.278284i	$0.910234\pi$
$4$		0			0
$5$			2.74343i			1.22690i	0.789734	+	0.613450i	$0.210219\pi$
$5$			2.74343i			1.22690i	−0.789734	+	0.613450i	$0.789781\pi$
$6$		0			0
$7$		−	4.32727i		−	1.63555i	−0.575536	−	0.817776i	$-0.695207\pi$
$7$		−	4.32727i		−	1.63555i	0.575536	−	0.817776i	$-0.304793\pi$
$8$		0			0
$9$	8.07070			2.69023
$10$		0			0
$11$		−	1.00000i		−	0.301511i
$11$		−	1.00000i		−	0.301511i
$12$		0			0
$13$	−3.60281	−	0.140622i	−0.999239	−	0.0390016i
$13$	−3.60281	−	0.140622i	−0.999239	−	0.0390016i
$14$		0			0
$15$		−	9.12812i		−	2.35687i
$16$		0			0
$17$	2.28124			0.553283			0.276642	−	0.960973i	$-0.410779\pi$
$17$	2.28124			0.553283			0.276642	+	0.960973i	$0.410779\pi$
$18$		0			0
$19$			5.78945i			1.32819i	0.747648	+	0.664096i	$0.231183\pi$
$19$			5.78945i			1.32819i	−0.747648	+	0.664096i	$0.768817\pi$
$20$		0			0
$21$			14.3980i			3.14189i
$22$		0			0
$23$	1.58384			0.330252			0.165126	−	0.986272i	$-0.447197\pi$
$23$	1.58384			0.330252			0.165126	+	0.986272i	$0.447197\pi$
$24$		0			0
$25$	−2.52641			−0.505282
$26$		0			0
$27$	−16.8716			−3.24693
$28$		0			0
$29$	−1.44892			−0.269057			−0.134528	−	0.990910i	$-0.542952\pi$
$29$	−1.44892			−0.269057			−0.134528	+	0.990910i	$0.542952\pi$
$30$		0			0
$31$			8.74343i			1.57037i	0.619264	+	0.785183i	$0.287431\pi$
$31$			8.74343i			1.57037i	−0.619264	+	0.785183i	$0.712569\pi$
$32$		0			0
$33$			3.32727i			0.579203i
$34$		0			0
$35$	11.8716			2.00666
$36$		0			0
$37$			7.39796i			1.21622i	0.793854	+	0.608109i	$0.208072\pi$
$37$			7.39796i			1.21622i	−0.793854	+	0.608109i	$0.791928\pi$
$38$		0			0
$39$	11.9875	+	0.467888i	1.91954	+	0.0749220i
$40$		0			0
$41$			4.33867i			0.677586i	0.940861	+	0.338793i	$0.110019\pi$
$41$			4.33867i			0.677586i	−0.940861	+	0.338793i	$0.889981\pi$
$42$		0			0
$43$	−3.44892			−0.525955			−0.262977	−	0.964802i	$-0.584704\pi$
$43$	−3.44892			−0.525955			−0.262977	+	0.964802i	$0.584704\pi$
$44$		0			0
$45$			22.1414i			3.30064i
$46$		0			0
$47$			2.65453i			0.387203i	0.981080	+	0.193602i	$0.0620169\pi$
$47$			2.65453i			0.387203i	−0.981080	+	0.193602i	$0.937983\pi$
$48$		0			0
$49$	−11.7252			−1.67503
$50$		0			0
$51$	−7.59031			−1.06286
$52$		0			0
$53$	11.8009			1.62097			0.810486	−	0.585758i	$-0.199203\pi$
$53$	11.8009			1.62097			0.810486	+	0.585758i	$0.199203\pi$
$54$		0			0
$55$	2.74343			0.369924
$56$		0			0
$57$		−	19.2630i		−	2.55145i
$58$		0			0
$59$			0.986729i			0.128461i	0.997935	+	0.0642306i	$0.0204593\pi$
$59$			0.986729i			0.128461i	−0.997935	+	0.0642306i	$0.979541\pi$
$60$		0			0
$61$	−5.44892			−0.697662			−0.348831	−	0.937186i	$-0.613421\pi$
$61$	−5.44892			−0.697662			−0.348831	+	0.937186i	$0.613421\pi$
$62$		0			0
$63$		−	34.9240i		−	4.40002i
$64$		0			0
$65$	0.385788	−	9.88405i	0.0478511	−	1.22597i
$66$		0			0
$67$		−	0.180941i		−	0.0221055i	−0.999939	−	0.0110527i	$-0.996482\pi$
$67$		−	0.180941i		−	0.0221055i	0.999939	−	0.0110527i	$-0.00351826\pi$
$68$		0			0
$69$	−5.26984			−0.634414
$70$		0			0
$71$			15.1281i			1.79538i	0.440630	+	0.897689i	$0.354755\pi$
$71$			15.1281i			1.79538i	−0.440630	+	0.897689i	$0.645245\pi$
$72$		0			0
$73$			6.72523i			0.787128i	0.919297	+	0.393564i	$0.128758\pi$
$73$			6.72523i			0.787128i	−0.919297	+	0.393564i	$0.871242\pi$
$74$		0			0
$75$	8.40604			0.970646
$76$		0			0
$77$	−4.32727			−0.493138
$78$		0			0
$79$	12.6659			1.42503			0.712515	−	0.701657i	$-0.247556\pi$
$79$	12.6659			1.42503			0.712515	+	0.701657i	$0.247556\pi$
$80$		0			0
$81$	31.9240			3.54712
$82$		0			0
$83$		−	11.5329i		−	1.26590i	−0.774193	−	0.632949i	$-0.781844\pi$
$83$		−	11.5329i		−	1.26590i	0.774193	−	0.632949i	$-0.218156\pi$
$84$		0			0
$85$			6.25844i			0.678823i
$86$		0			0
$87$	4.82093			0.516857
$88$		0			0
$89$		−	6.56563i		−	0.695956i	−0.937503	−	0.347978i	$-0.886868\pi$
$89$		−	6.56563i		−	0.695956i	0.937503	−	0.347978i	$-0.113132\pi$
$90$		0			0
$91$	−0.608510	+	15.5903i	−0.0637892	+	1.63431i
$92$		0			0
$93$		−	29.0917i		−	3.01667i
$94$		0			0
$95$	−15.8830			−1.62956
$96$		0			0
$97$			5.25657i			0.533724i	0.963735	+	0.266862i	$0.0859868\pi$
$97$			5.25657i			0.533724i	−0.963735	+	0.266862i	$0.914013\pi$
$98$		0			0
$99$		−	8.07070i		−	0.811135i
$100$		0			0
$101$	−6.00000			−0.597022			−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000			−0.597022			−0.298511	+	0.954406i	$0.596490\pi$
$102$		0			0
$103$	−2.61991			−0.258148			−0.129074	−	0.991635i	$-0.541200\pi$
$103$	−2.61991			−0.258148			−0.129074	+	0.991635i	$0.541200\pi$
$104$		0			0
$105$	−39.4998			−3.85479
$106$		0			0
$107$	2.93578			0.283812			0.141906	−	0.989880i	$-0.454677\pi$
$107$	2.93578			0.283812			0.141906	+	0.989880i	$0.454677\pi$
$108$		0			0
$109$		−	0.865080i		−	0.0828596i	−0.999141	−	0.0414298i	$-0.986809\pi$
$109$		−	0.865080i		−	0.0828596i	0.999141	−	0.0414298i	$-0.0131913\pi$
$110$		0			0
$111$		−	24.6150i		−	2.33635i
$112$		0			0
$113$	−7.46866			−0.702592			−0.351296	−	0.936264i	$-0.614259\pi$
$113$	−7.46866			−0.702592			−0.351296	+	0.936264i	$0.614259\pi$
$114$		0			0
$115$			4.34514i			0.405187i
$116$		0			0
$117$	−29.0772	−	1.13492i	−2.68819	−	0.104923i
$118$		0			0
$119$		−	9.87155i		−	0.904924i
$120$		0			0
$121$	−1.00000			−0.0909091
$122$		0			0
$123$		−	14.4359i		−	1.30164i
$124$		0			0
$125$			6.78612i			0.606969i
$126$		0			0
$127$	−14.4112			−1.27879			−0.639395	−	0.768878i	$-0.720815\pi$
$127$	−14.4112			−1.27879			−0.639395	+	0.768878i	$0.720815\pi$
$128$		0			0
$129$	11.4755			1.01036
$130$		0			0
$131$	10.7959			0.943244			0.471622	−	0.881801i	$-0.343669\pi$
$131$	10.7959			0.943244			0.471622	+	0.881801i	$0.343669\pi$
$132$		0			0
$133$	25.0525			2.17233
$134$		0			0
$135$		−	46.2859i		−	3.98366i
$136$		0			0
$137$			16.0753i			1.37341i	0.726938	+	0.686703i	$0.240943\pi$
$137$			16.0753i			1.37341i	−0.726938	+	0.686703i	$0.759057\pi$
$138$		0			0
$139$	3.87155			0.328381			0.164190	−	0.986429i	$-0.447499\pi$
$139$	3.87155			0.328381			0.164190	+	0.986429i	$0.447499\pi$
$140$		0			0
$141$		−	8.83233i		−	0.743816i
$142$		0			0
$143$	−0.140622	+	3.60281i	−0.0117594	+	0.301282i
$144$		0			0
$145$		−	3.97500i		−	0.330106i
$146$		0			0
$147$	39.0129			3.21773
$148$		0			0
$149$		−	9.49494i		−	0.777855i	−0.921268	−	0.388928i	$-0.872846\pi$
$149$		−	9.49494i		−	0.777855i	0.921268	−	0.388928i	$-0.127154\pi$
$150$		0			0
$151$			15.0506i			1.22480i	0.790547	+	0.612401i	$0.209796\pi$
$151$			15.0506i			1.22480i	−0.790547	+	0.612401i	$0.790204\pi$
$152$		0			0
$153$	18.4112			1.48846
$154$		0			0
$155$	−23.9870			−1.92668
$156$		0			0
$157$	13.4554			1.07386			0.536928	−	0.843628i	$-0.319585\pi$
$157$	13.4554			1.07386			0.536928	+	0.843628i	$0.319585\pi$
$158$		0			0
$159$	−39.2646			−3.11388
$160$		0			0
$161$		−	6.85368i		−	0.540145i
$162$		0			0
$163$			2.56249i			0.200710i	0.994952	+	0.100355i	$0.0319978\pi$
$163$			2.56249i			0.200710i	−0.994952	+	0.100355i	$0.968002\pi$
$164$		0			0
$165$	−9.12812			−0.710623
$166$		0			0
$167$			10.1855i			0.788181i	0.919072	+	0.394091i	$0.128940\pi$
$167$			10.1855i			0.788181i	−0.919072	+	0.394091i	$0.871060\pi$
$168$		0			0
$169$	12.9605	+	1.01327i	0.996958	+	0.0779439i
$170$		0			0
$171$			46.7249i			3.57314i
$172$		0			0
$173$	19.0278			1.44666			0.723329	−	0.690503i	$-0.242611\pi$
$173$	19.0278			1.44666			0.723329	+	0.690503i	$0.242611\pi$
$174$		0			0
$175$			10.9324i			0.826415i
$176$		0			0
$177$		−	3.28311i		−	0.246774i
$178$		0			0
$179$	8.45078			0.631641			0.315821	−	0.948819i	$-0.397720\pi$
$179$	8.45078			0.631641			0.315821	+	0.948819i	$0.397720\pi$
$180$		0			0
$181$	15.1068			1.12288			0.561439	−	0.827518i	$-0.310248\pi$
$181$	15.1068			1.12288			0.561439	+	0.827518i	$0.310248\pi$
$182$		0			0
$183$	18.1300			1.34021
$184$		0			0
$185$	−20.2958			−1.49218
$186$		0			0
$187$		−	2.28124i		−	0.166821i
$188$		0			0
$189$			73.0077i			5.31053i
$190$		0			0
$191$	−11.8898			−0.860312			−0.430156	−	0.902754i	$-0.641542\pi$
$191$	−11.8898			−0.860312			−0.430156	+	0.902754i	$0.641542\pi$
$192$		0			0
$193$		−	9.22510i		−	0.664037i	−0.943273	−	0.332018i	$-0.892270\pi$
$193$		−	9.22510i		−	0.664037i	0.943273	−	0.332018i	$-0.107730\pi$
$194$		0			0
$195$	−1.28362	+	32.8869i	−0.0919218	+	2.35508i
$196$		0			0
$197$		−	17.9422i		−	1.27833i	−0.769069	−	0.639166i	$-0.779280\pi$
$197$		−	17.9422i		−	1.27833i	0.769069	−	0.639166i	$-0.220720\pi$
$198$		0			0
$199$	1.70881			0.121135			0.0605673	−	0.998164i	$-0.480709\pi$
$199$	1.70881			0.121135			0.0605673	+	0.998164i	$0.480709\pi$
$200$		0			0
$201$			0.602039i			0.0424645i
$202$		0			0
$203$			6.26984i			0.440057i
$204$		0			0
$205$	−11.9028			−0.831330
$206$		0			0
$207$	12.7827			0.888456
$208$		0			0
$209$	5.78945			0.400465
$210$		0			0
$211$	−8.04935			−0.554140			−0.277070	−	0.960850i	$-0.589363\pi$
$211$	−8.04935			−0.554140			−0.277070	+	0.960850i	$0.589363\pi$
$212$		0			0
$213$		−	50.3353i		−	3.44892i
$214$		0			0
$215$		−	9.46186i		−	0.645293i
$216$		0			0
$217$	37.8351			2.56842
$218$		0			0
$219$		−	22.3766i		−	1.51207i
$220$		0			0
$221$	−8.21889	−	0.320794i	−0.552862	−	0.0215789i
$222$		0			0
$223$			21.1775i			1.41815i	0.705134	+	0.709074i	$0.250887\pi$
$223$			21.1775i			1.41815i	−0.705134	+	0.709074i	$0.749113\pi$
$224$		0			0
$225$	−20.3899			−1.35933
$226$		0			0
$227$			16.3026i			1.08204i	0.841009	+	0.541020i	$0.181962\pi$
$227$			16.3026i			1.08204i	−0.841009	+	0.541020i	$0.818038\pi$
$228$		0			0
$229$			2.76997i			0.183045i	0.995803	+	0.0915224i	$0.0291733\pi$
$229$			2.76997i			0.183045i	−0.995803	+	0.0915224i	$0.970827\pi$
$230$		0			0
$231$	14.3980			0.947316
$232$		0			0
$233$	12.0920			0.792176			0.396088	−	0.918213i	$-0.370368\pi$
$233$	12.0920			0.792176			0.396088	+	0.918213i	$0.370368\pi$
$234$		0			0
$235$	−7.28252			−0.475059
$236$		0			0
$237$	−42.1429			−2.73748
$238$		0			0
$239$		−	2.22195i		−	0.143726i	−0.997415	−	0.0718631i	$-0.977106\pi$
$239$		−	2.22195i		−	0.143726i	0.997415	−	0.0718631i	$-0.0228945\pi$
$240$		0			0
$241$		−	3.68414i		−	0.237316i	−0.992935	−	0.118658i	$-0.962141\pi$
$241$		−	3.68414i		−	0.237316i	0.992935	−	0.118658i	$-0.0378593\pi$
$242$		0			0
$243$	−55.6051			−3.56707
$244$		0			0
$245$		−	32.1673i		−	2.05510i
$246$		0			0
$247$	0.814126	−	20.8583i	0.0518016	−	1.32718i
$248$		0			0
$249$			38.3730i			2.43179i
$250$		0			0
$251$	−15.3405			−0.968286			−0.484143	−	0.874989i	$-0.660869\pi$
$251$	−15.3405			−0.968286			−0.484143	+	0.874989i	$0.660869\pi$
$252$		0			0
$253$		−	1.58384i		−	0.0995749i
$254$		0			0
$255$		−	20.8235i		−	1.30402i
$256$		0			0
$257$	7.57083			0.472255			0.236128	−	0.971722i	$-0.424122\pi$
$257$	7.57083			0.472255			0.236128	+	0.971722i	$0.424122\pi$
$258$		0			0
$259$	32.0129			1.98919
$260$		0			0
$261$	−11.6938			−0.723825
$262$		0			0
$263$	−21.1806			−1.30605			−0.653026	−	0.757335i	$-0.726501\pi$
$263$	−21.1806			−1.30605			−0.653026	+	0.757335i	$0.726501\pi$
$264$		0			0
$265$			32.3748i			1.98877i
$266$		0			0
$267$			21.8456i			1.33693i
$268$		0			0
$269$	−1.69928			−0.103607			−0.0518033	−	0.998657i	$-0.516497\pi$
$269$	−1.69928			−0.103607			−0.0518033	+	0.998657i	$0.516497\pi$
$270$		0			0
$271$		−	21.5804i		−	1.31091i	−0.755233	−	0.655457i	$-0.772476\pi$
$271$		−	21.5804i		−	1.31091i	0.755233	−	0.655457i	$-0.227524\pi$
$272$		0			0
$273$	2.02467	−	51.8731i	0.122539	−	3.13950i
$274$		0			0
$275$			2.52641i			0.152348i
$276$		0			0
$277$	−10.2562			−0.616238			−0.308119	−	0.951348i	$-0.599699\pi$
$277$	−10.2562			−0.616238			−0.308119	+	0.951348i	$0.599699\pi$
$278$		0			0
$279$			70.5656i			4.22465i
$280$		0			0
$281$		−	19.0216i		−	1.13473i	−0.823465	−	0.567367i	$-0.807962\pi$
$281$		−	19.0216i		−	1.13473i	0.823465	−	0.567367i	$-0.192038\pi$
$282$		0			0
$283$	−30.2448			−1.79787			−0.898934	−	0.438083i	$-0.855658\pi$
$283$	−30.2448			−1.79787			−0.898934	+	0.438083i	$0.855658\pi$
$284$		0			0
$285$	52.8468			3.13037
$286$		0			0
$287$	18.7746			1.10823
$288$		0			0
$289$	−11.7959			−0.693878
$290$		0			0
$291$		−	17.4900i		−	1.02528i
$292$		0			0
$293$		−	12.4881i		−	0.729565i	−0.931093	−	0.364782i	$-0.881143\pi$
$293$		−	12.4881i		−	0.729565i	0.931093	−	0.364782i	$-0.118857\pi$
$294$		0			0
$295$	−2.70702			−0.157609
$296$		0			0
$297$			16.8716i			0.978987i
$298$		0			0
$299$	−5.70625	−	0.222723i	−0.330001	−	0.0128804i
$300$		0			0
$301$			14.9244i			0.860226i
$302$		0			0
$303$	19.9636			1.14688
$304$		0			0
$305$		−	14.9487i		−	0.855961i
$306$		0			0
$307$		−	15.4603i		−	0.882367i	−0.897417	−	0.441184i	$-0.854559\pi$
$307$		−	15.4603i		−	0.882367i	0.897417	−	0.441184i	$-0.145441\pi$
$308$		0			0
$309$	8.71715			0.495901
$310$		0			0
$311$	−27.6989			−1.57066			−0.785332	−	0.619075i	$-0.787508\pi$
$311$	−27.6989			−1.57066			−0.785332	+	0.619075i	$0.787508\pi$
$312$		0			0
$313$	−16.5841			−0.937389			−0.468694	−	0.883360i	$-0.655275\pi$
$313$	−16.5841			−0.937389			−0.468694	+	0.883360i	$0.655275\pi$
$314$		0			0
$315$	95.8117			5.39838
$316$		0			0
$317$			10.4372i			0.586211i	0.956080	+	0.293105i	$0.0946887\pi$
$317$			10.4372i			0.586211i	−0.956080	+	0.293105i	$0.905311\pi$
$318$		0			0
$319$			1.44892i			0.0811237i
$320$		0			0
$321$	−9.76811			−0.545202
$322$		0			0
$323$			13.2072i			0.734866i
$324$		0			0
$325$	9.10217	+	0.355270i	0.504898	+	0.0197068i
$326$		0			0
$327$			2.87835i			0.159173i
$328$		0			0
$329$	11.4869			0.633291
$330$		0			0
$331$		−	1.07877i		−	0.0592946i	−0.999560	−	0.0296473i	$-0.990562\pi$
$331$		−	1.07877i		−	0.0592946i	0.999560	−	0.0296473i	$-0.00943842\pi$
$332$		0			0
$333$			59.7067i			3.27191i
$334$		0			0
$335$	0.496399			0.0271212
$336$		0			0
$337$	−20.1300			−1.09655			−0.548275	−	0.836298i	$-0.684715\pi$
$337$	−20.1300			−1.09655			−0.548275	+	0.836298i	$0.684715\pi$
$338$		0			0
$339$	24.8502			1.34968
$340$		0			0
$341$	8.74343			0.473483
$342$		0			0
$343$			20.4473i			1.10405i
$344$		0			0
$345$		−	14.4574i		−	0.778362i
$346$		0			0
$347$	13.4489			0.721976			0.360988	−	0.932571i	$-0.382440\pi$
$347$	13.4489			0.721976			0.360988	+	0.932571i	$0.382440\pi$
$348$		0			0
$349$			23.5822i			1.26233i	0.775649	+	0.631164i	$0.217422\pi$
$349$			23.5822i			1.26233i	−0.775649	+	0.631164i	$0.782578\pi$
$350$		0			0
$351$	60.7850	+	2.37252i	3.24446	+	0.126636i
$352$		0			0
$353$		−	33.9240i		−	1.80559i	−0.430066	−	0.902797i	$-0.641510\pi$
$353$		−	33.9240i		−	1.80559i	0.430066	−	0.902797i	$-0.358490\pi$
$354$		0			0
$355$	−41.5029			−2.20275
$356$		0			0
$357$			32.8453i			1.73836i
$358$		0			0
$359$			0.273169i			0.0144173i	0.999974	+	0.00720866i	$0.00229461\pi$
$359$			0.273169i			0.0144173i	−0.999974	+	0.00720866i	$0.997705\pi$
$360$		0			0
$361$	−14.5177			−0.764092
$362$		0			0
$363$	3.32727			0.174636
$364$		0			0
$365$	−18.4502			−0.965727
$366$		0			0
$367$	−15.4915			−0.808648			−0.404324	−	0.914616i	$-0.632493\pi$
$367$	−15.4915			−0.808648			−0.404324	+	0.914616i	$0.632493\pi$
$368$		0			0
$369$			35.0161i			1.82286i
$370$		0			0
$371$		−	51.0654i		−	2.65119i
$372$		0			0
$373$	−10.6052			−0.549115			−0.274558	−	0.961571i	$-0.588531\pi$
$373$	−10.6052			−0.549115			−0.274558	+	0.961571i	$0.588531\pi$
$374$		0			0
$375$		−	22.5792i		−	1.16599i
$376$		0			0
$377$	5.22016	+	0.203750i	0.268852	+	0.0104936i
$378$		0			0
$379$		−	21.6542i		−	1.11230i	−0.831081	−	0.556151i	$-0.812278\pi$
$379$		−	21.6542i		−	1.11230i	0.831081	−	0.556151i	$-0.187722\pi$
$380$		0			0
$381$	47.9500			2.45655
$382$		0			0
$383$		−	10.5001i		−	0.536532i	−0.963345	−	0.268266i	$-0.913549\pi$
$383$		−	10.5001i		−	0.536532i	0.963345	−	0.268266i	$-0.0864506\pi$
$384$		0			0
$385$		−	11.8716i		−	0.605030i
$386$		0			0
$387$	−27.8351			−1.41494
$388$		0			0
$389$	−9.98180			−0.506097			−0.253049	−	0.967454i	$-0.581433\pi$
$389$	−9.98180			−0.506097			−0.253049	+	0.967454i	$0.581433\pi$
$390$		0			0
$391$	3.61312			0.182723
$392$		0			0
$393$	−35.9209			−1.81197
$394$		0			0
$395$			34.7481i			1.74837i
$396$		0			0
$397$			15.4603i			0.775931i	0.921674	+	0.387966i	$0.126822\pi$
$397$			15.4603i			0.775931i	−0.921674	+	0.387966i	$0.873178\pi$
$398$		0			0
$399$	−83.3563			−4.17303
$400$		0			0
$401$			4.02281i			0.200889i	0.994943	+	0.100445i	$0.0320266\pi$
$401$			4.02281i			0.200889i	−0.994943	+	0.100445i	$0.967973\pi$
$402$		0			0
$403$	1.22952	−	31.5009i	0.0612468	−	1.56917i
$404$		0			0
$405$			87.5814i			4.35195i
$406$		0			0
$407$	7.39796			0.366703
$408$		0			0
$409$		−	4.41123i		−	0.218121i	−0.994035	−	0.109061i	$-0.965216\pi$
$409$		−	4.41123i		−	0.218121i	0.994035	−	0.109061i	$-0.0347843\pi$
$410$		0			0
$411$		−	53.4868i		−	2.63831i
$412$		0			0
$413$	4.26984			0.210105
$414$		0			0
$415$	31.6397			1.55313
$416$		0			0
$417$	−12.8817			−0.630819
$418$		0			0
$419$	26.2482			1.28231			0.641153	−	0.767413i	$-0.278456\pi$
$419$	26.2482			1.28231			0.641153	+	0.767413i	$0.278456\pi$
$420$		0			0
$421$			14.2828i			0.696100i	0.937476	+	0.348050i	$0.113156\pi$
$421$			14.2828i			0.696100i	−0.937476	+	0.348050i	$0.886844\pi$
$422$		0			0
$423$			21.4239i			1.04167i
$424$		0			0
$425$	−5.76336			−0.279564
$426$		0			0
$427$			23.5789i			1.14106i
$428$		0			0
$429$	0.467888	−	11.9875i	0.0225898	−	0.578762i
$430$		0			0
$431$			5.86822i			0.282662i	0.989962	+	0.141331i	$0.0451382\pi$
$431$			5.86822i			0.282662i	−0.989962	+	0.141331i	$0.954862\pi$
$432$		0			0
$433$	−29.2216			−1.40430			−0.702151	−	0.712028i	$-0.747777\pi$
$433$	−29.2216			−1.40430			−0.702151	+	0.712028i	$0.747777\pi$
$434$		0			0
$435$			13.2259i			0.634132i
$436$		0			0
$437$			9.16954i			0.438638i
$438$		0			0
$439$	35.0408			1.67240			0.836202	−	0.548421i	$-0.184771\pi$
$439$	35.0408			1.67240			0.836202	+	0.548421i	$0.184771\pi$
$440$		0			0
$441$	−94.6307			−4.50623
$442$		0			0
$443$	25.5740			1.21506			0.607528	−	0.794298i	$-0.292161\pi$
$443$	25.5740			1.21506			0.607528	+	0.794298i	$0.292161\pi$
$444$		0			0
$445$	18.0124			0.853867
$446$		0			0
$447$			31.5922i			1.49426i
$448$		0			0
$449$		−	30.2958i		−	1.42975i	−0.699254	−	0.714873i	$-0.746484\pi$
$449$		−	30.2958i		−	1.42975i	0.699254	−	0.714873i	$-0.253516\pi$
$450$		0			0
$451$	4.33867			0.204300
$452$		0			0
$453$		−	50.0774i		−	2.35284i
$454$		0			0
$455$	−42.7709	−	1.66941i	−2.00513	−	0.0782629i
$456$		0			0
$457$			27.8929i			1.30477i	0.757886	+	0.652387i	$0.226232\pi$
$457$			27.8929i			1.30477i	−0.757886	+	0.652387i	$0.773768\pi$
$458$		0			0
$459$	−38.4881			−1.79647
$460$		0			0
$461$			19.3782i			0.902534i	0.892389	+	0.451267i	$0.149028\pi$
$461$			19.3782i			0.902534i	−0.892389	+	0.451267i	$0.850972\pi$
$462$		0			0
$463$		−	23.6542i		−	1.09930i	−0.835394	−	0.549652i	$-0.814760\pi$
$463$		−	23.6542i		−	1.09930i	0.835394	−	0.549652i	$-0.185240\pi$
$464$		0			0
$465$	79.8111			3.70115
$466$		0			0
$467$	−6.34514			−0.293618			−0.146809	−	0.989165i	$-0.546900\pi$
$467$	−6.34514			−0.293618			−0.146809	+	0.989165i	$0.546900\pi$
$468$		0			0
$469$	−0.782980			−0.0361546
$470$		0			0
$471$	−44.7696			−2.06288
$472$		0			0
$473$			3.44892i			0.158581i
$474$		0			0
$475$		−	14.6265i		−	0.671111i
$476$		0			0
$477$	95.2411			4.36079
$478$		0			0
$479$			25.5918i			1.16932i	0.811278	+	0.584661i	$0.198772\pi$
$479$			25.5918i			1.16932i	−0.811278	+	0.584661i	$0.801228\pi$
$480$		0			0
$481$	1.04032	−	26.6534i	0.0474344	−	1.21529i
$482$		0			0
$483$			22.8040i			1.03762i
$484$		0			0
$485$	−14.4210			−0.654825
$486$		0			0
$487$			21.2202i			0.961577i	0.876836	+	0.480789i	$0.159650\pi$
$487$			21.2202i			0.961577i	−0.876836	+	0.480789i	$0.840350\pi$
$488$		0			0
$489$		−	8.52608i		−	0.385563i
$490$		0			0
$491$	32.6811			1.47488			0.737438	−	0.675415i	$-0.236035\pi$
$491$	32.6811			1.47488			0.737438	+	0.675415i	$0.236035\pi$
$492$		0			0
$493$	−3.30533			−0.148865
$494$		0			0
$495$	22.1414			0.995182
$496$		0			0
$497$	65.4634			2.93643
$498$		0			0
$499$			14.7466i			0.660147i	0.943955	+	0.330074i	$0.107074\pi$
$499$			14.7466i			0.660147i	−0.943955	+	0.330074i	$0.892926\pi$
$500$		0			0
$501$		−	33.8900i		−	1.51409i
$502$		0			0
$503$	34.7731			1.55046			0.775228	−	0.631681i	$-0.217635\pi$
$503$	34.7731			1.55046			0.775228	+	0.631681i	$0.217635\pi$
$504$		0			0
$505$		−	16.4606i		−	0.732486i
$506$		0			0
$507$	−43.1229	−	3.37142i	−1.91515	−	0.149730i
$508$		0			0
$509$		−	24.4637i		−	1.08434i	−0.840270	−	0.542168i	$-0.817604\pi$
$509$		−	24.4637i		−	1.08434i	0.840270	−	0.542168i	$-0.182396\pi$
$510$		0			0
$511$	29.1018			1.28739
$512$		0			0
$513$		−	97.6770i		−	4.31255i
$514$		0			0
$515$		−	7.18755i		−	0.316721i
$516$		0			0
$517$	2.65453			0.116746
$518$		0			0
$519$	−63.3106			−2.77903
$520$		0			0
$521$	8.70881			0.381540			0.190770	−	0.981635i	$-0.438902\pi$
$521$	8.70881			0.381540			0.190770	+	0.981635i	$0.438902\pi$
$522$		0			0
$523$	−33.1531			−1.44968			−0.724841	−	0.688916i	$-0.758087\pi$
$523$	−33.1531			−1.44968			−0.724841	+	0.688916i	$0.758087\pi$
$524$		0			0
$525$		−	36.3752i		−	1.58754i
$526$		0			0
$527$			19.9459i			0.868857i
$528$		0			0
$529$	−20.4915			−0.890933
$530$		0			0
$531$			7.96359i			0.345591i
$532$		0			0
$533$	0.610114	−	15.6314i	0.0264270	−	0.677071i
$534$		0			0
$535$			8.05410i			0.348209i
$536$		0			0
$537$	−28.1180			−1.21338
$538$		0			0
$539$			11.7252i			0.505041i
$540$		0			0
$541$			30.3782i			1.30606i	0.757333	+	0.653029i	$0.226502\pi$
$541$			30.3782i			1.30606i	−0.757333	+	0.653029i	$0.773498\pi$
$542$		0			0
$543$	−50.2643			−2.15705
$544$		0			0
$545$	2.37329			0.101660
$546$		0			0
$547$	−9.89809			−0.423212			−0.211606	−	0.977355i	$-0.567869\pi$
$547$	−9.89809			−0.423212			−0.211606	+	0.977355i	$0.567869\pi$
$548$		0			0
$549$	−43.9765			−1.87687
$550$		0			0
$551$		−	8.38842i		−	0.357359i
$552$		0			0
$553$		−	54.8089i		−	2.33071i
$554$		0			0
$555$	67.5295			2.86647
$556$		0			0
$557$		−	19.0197i		−	0.805892i	−0.915224	−	0.402946i	$-0.867986\pi$
$557$		−	19.0197i		−	0.805892i	0.915224	−	0.402946i	$-0.132014\pi$
$558$		0			0
$559$	12.4258	+	0.484995i	0.525554	+	0.0205131i
$560$		0			0
$561$			7.59031i			0.320463i
$562$		0			0
$563$	38.5239			1.62359			0.811794	−	0.583944i	$-0.198491\pi$
$563$	38.5239			1.62359			0.811794	+	0.583944i	$0.198491\pi$
$564$		0			0
$565$		−	20.4897i		−	0.862010i
$566$		0			0
$567$		−	138.144i		−	5.80149i
$568$		0			0
$569$	18.8323			0.789492			0.394746	−	0.918790i	$-0.370833\pi$
$569$	18.8323			0.789492			0.394746	+	0.918790i	$0.370833\pi$
$570$		0			0
$571$	24.2334			1.01414			0.507069	−	0.861906i	$-0.330729\pi$
$571$	24.2334			1.01414			0.507069	+	0.861906i	$0.330729\pi$
$572$		0			0
$573$	39.5604			1.65266
$574$		0			0
$575$	−4.00142			−0.166871
$576$		0			0
$577$		−	31.4473i		−	1.30917i	−0.755989	−	0.654584i	$-0.772844\pi$
$577$		−	31.4473i		−	1.30917i	0.755989	−	0.654584i	$-0.227156\pi$
$578$		0			0
$579$			30.6943i			1.27561i
$580$		0			0
$581$	−49.9058			−2.07044
$582$		0			0
$583$		−	11.8009i		−	0.488742i
$584$		0			0
$585$	3.11357	−	79.7712i	0.128730	−	3.29813i
$586$		0			0
$587$		−	42.8880i		−	1.77018i	−0.465424	−	0.885088i	$-0.654098\pi$
$587$		−	42.8880i		−	1.77018i	0.465424	−	0.885088i	$-0.345902\pi$
$588$		0			0
$589$	−50.6197			−2.08575
$590$		0			0
$591$			59.6986i			2.45567i
$592$		0			0
$593$			8.10678i			0.332905i	0.986049	+	0.166453i	$0.0532313\pi$
$593$			8.10678i			0.332905i	−0.986049	+	0.166453i	$0.946769\pi$
$594$		0			0
$595$	27.0819			1.11025
$596$		0			0
$597$	−5.68568			−0.232699
$598$		0			0
$599$	14.4554			0.590631			0.295316	−	0.955400i	$-0.404575\pi$
$599$	14.4554			0.590631			0.295316	+	0.955400i	$0.404575\pi$
$600$		0			0
$601$	29.7203			1.21232			0.606158	−	0.795344i	$-0.292710\pi$
$601$	29.7203			1.21232			0.606158	+	0.795344i	$0.292710\pi$
$602$		0			0
$603$		−	1.46032i		−	0.0594688i
$604$		0			0
$605$		−	2.74343i		−	0.111536i
$606$		0			0
$607$	9.67094			0.392532			0.196266	−	0.980551i	$-0.437118\pi$
$607$	9.67094			0.392532			0.196266	+	0.980551i	$0.437118\pi$
$608$		0			0
$609$		−	20.8614i		−	0.845348i
$610$		0			0
$611$	0.373286	−	9.56377i	0.0151015	−	0.386909i
$612$		0			0
$613$		−	1.72395i		−	0.0696297i	−0.999394	−	0.0348149i	$-0.988916\pi$
$613$		−	1.72395i		−	0.0696297i	0.999394	−	0.0348149i	$-0.0110842\pi$
$614$		0			0
$615$	39.6039			1.59698
$616$		0			0
$617$			11.0886i			0.446409i	0.974772	+	0.223205i	$0.0716518\pi$
$617$			11.0886i			0.446409i	−0.974772	+	0.223205i	$0.928348\pi$
$618$		0			0
$619$		−	26.6577i		−	1.07146i	−0.844389	−	0.535731i	$-0.820036\pi$
$619$		−	26.6577i		−	1.07146i	0.844389	−	0.535731i	$-0.179964\pi$
$620$		0			0
$621$	−26.7218			−1.07231
$622$		0			0
$623$	−28.4112			−1.13827
$624$		0			0
$625$	−31.2493			−1.24997
$626$		0			0
$627$	−19.2630			−0.769292
$628$		0			0
$629$			16.8766i			0.672913i
$630$		0			0
$631$		−	0.371683i		−	0.0147965i	−0.999973	−	0.00739823i	$-0.997645\pi$
$631$		−	0.371683i		−	0.0147965i	0.999973	−	0.00739823i	$-0.00235495\pi$
$632$		0			0
$633$	26.7823			1.06450
$634$		0			0
$635$		−	39.5362i		−	1.56895i
$636$		0			0
$637$	42.2437	+	1.64883i	1.67376	+	0.0653290i
$638$		0			0
$639$			122.094i			4.82998i
$640$		0			0
$641$	19.3671			0.764954			0.382477	−	0.923965i	$-0.375071\pi$
$641$	19.3671			0.764954			0.382477	+	0.923965i	$0.375071\pi$
$642$		0			0
$643$			12.4242i			0.489964i	0.969528	+	0.244982i	$0.0787821\pi$
$643$			12.4242i			0.489964i	−0.969528	+	0.244982i	$0.921218\pi$
$644$		0			0
$645$			31.4821i			1.23961i
$646$		0			0
$647$	3.10678			0.122140			0.0610700	−	0.998133i	$-0.480549\pi$
$647$	3.10678			0.122140			0.0610700	+	0.998133i	$0.480549\pi$
$648$		0			0
$649$	0.986729			0.0387325
$650$		0			0
$651$	−125.888			−4.93392
$652$		0			0
$653$	8.47732			0.331743			0.165872	−	0.986147i	$-0.446956\pi$
$653$	8.47732			0.331743			0.165872	+	0.986147i	$0.446956\pi$
$654$		0			0
$655$			29.6179i			1.15727i
$656$		0			0
$657$			54.2773i			2.11756i
$658$		0			0
$659$	−14.3239			−0.557981			−0.278991	−	0.960294i	$-0.590000\pi$
$659$	−14.3239			−0.557981			−0.278991	+	0.960294i	$0.590000\pi$
$660$		0			0
$661$			32.8620i			1.27818i	0.769130	+	0.639092i	$0.220690\pi$
$661$			32.8620i			1.27818i	−0.769130	+	0.639092i	$0.779310\pi$
$662$		0			0
$663$	27.3464	+	1.06737i	1.06205	+	0.0414531i
$664$		0			0
$665$			68.7298i			2.66523i
$666$		0			0
$667$	−2.29484			−0.0888567
$668$		0			0
$669$		−	70.4631i		−	2.72426i
$670$		0			0
$671$			5.44892i			0.210353i
$672$		0			0
$673$	−22.6561			−0.873327			−0.436664	−	0.899625i	$-0.643840\pi$
$673$	−22.6561			−0.873327			−0.436664	+	0.899625i	$0.643840\pi$
$674$		0			0
$675$	42.6245			1.64062
$676$		0			0
$677$	−30.7694			−1.18256			−0.591282	−	0.806465i	$-0.701378\pi$
$677$	−30.7694			−1.18256			−0.591282	+	0.806465i	$0.701378\pi$
$678$		0			0
$679$	22.7466			0.872933
$680$		0			0
$681$		−	54.2430i		−	2.07860i
$682$		0			0
$683$			49.0886i			1.87832i	0.343477	+	0.939161i	$0.388395\pi$
$683$			49.0886i			1.87832i	−0.343477	+	0.939161i	$0.611605\pi$
$684$		0			0
$685$	−44.1015			−1.68503
$686$		0			0
$687$		−	9.21643i		−	0.351629i
$688$		0			0
$689$	−42.5162	−	1.65946i	−1.61974	−	0.0632205i
$690$		0			0
$691$		−	25.6542i		−	0.975932i	−0.872863	−	0.487966i	$-0.837739\pi$
$691$		−	25.6542i		−	0.975932i	0.872863	−	0.487966i	$-0.162261\pi$
$692$		0			0
$693$	−34.9240			−1.32665
$694$		0			0
$695$			10.6213i			0.402890i
$696$		0			0
$697$			9.89757i			0.374897i
$698$		0			0
$699$	−40.2334			−1.52177
$700$		0			0
$701$	16.0784			0.607274			0.303637	−	0.952788i	$-0.401799\pi$
$701$	16.0784			0.607274			0.303637	+	0.952788i	$0.401799\pi$
$702$		0			0
$703$	−42.8301			−1.61537
$704$		0			0
$705$	24.2309			0.912588
$706$		0			0
$707$			25.9636i			0.976461i
$708$		0			0
$709$		−	33.2066i		−	1.24710i	−0.781784	−	0.623549i	$-0.785690\pi$
$709$		−	33.2066i		−	1.24710i	0.781784	−	0.623549i	$-0.214310\pi$
$710$		0			0
$711$	102.223			3.83366
$712$		0			0
$713$			13.8482i			0.518617i
$714$		0			0
$715$	−9.88405	−	0.385788i	−0.369643	−	0.0144276i
$716$		0			0
$717$			7.39303i			0.276098i
$718$		0			0
$719$	−20.4508			−0.762685			−0.381343	−	0.924434i	$-0.624538\pi$
$719$	−20.4508			−0.762685			−0.381343	+	0.924434i	$0.624538\pi$
$720$		0			0
$721$			11.3371i			0.422214i
$722$		0			0
$723$			12.2581i			0.455884i
$724$		0			0
$725$	3.66055			0.135950
$726$		0			0
$727$	37.5968			1.39439			0.697194	−	0.716883i	$-0.254432\pi$
$727$	37.5968			1.39439			0.697194	+	0.716883i	$0.254432\pi$
$728$		0			0
$729$	89.2409			3.30522
$730$		0			0
$731$	−7.86782			−0.291002
$732$		0			0
$733$			25.9305i			0.957765i	0.877879	+	0.478883i	$0.158958\pi$
$733$			25.9305i			0.957765i	−0.877879	+	0.478883i	$0.841042\pi$
$734$		0			0
$735$			107.029i			3.94783i
$736$		0			0
$737$	−0.180941			−0.00666505
$738$		0			0
$739$			14.1760i			0.521473i	0.965410	+	0.260736i	$0.0839654\pi$
$739$			14.1760i			0.521473i	−0.965410	+	0.260736i	$0.916035\pi$
$740$		0			0
$741$	−2.70881	+	69.4010i	−0.0995108	+	2.54951i
$742$		0			0
$743$			39.1328i			1.43564i	0.696227	+	0.717822i	$0.254861\pi$
$743$			39.1328i			1.43564i	−0.696227	+	0.717822i	$0.745139\pi$
$744$		0			0
$745$	26.0487			0.954350
$746$		0			0
$747$		−	93.0784i		−	3.40556i
$748$		0			0
$749$		−	12.7039i		−	0.464190i
$750$		0			0
$751$	12.4585			0.454618			0.227309	−	0.973823i	$-0.427007\pi$
$751$	12.4585			0.454618			0.227309	+	0.973823i	$0.427007\pi$
$752$		0			0
$753$	51.0420			1.86008
$754$		0			0
$755$	−41.2903			−1.50271
$756$		0			0
$757$	35.1460			1.27740			0.638701	−	0.769455i	$-0.279472\pi$
$757$	35.1460			1.27740			0.638701	+	0.769455i	$0.279472\pi$
$758$		0			0
$759$			5.26984i			0.191283i
$760$		0			0
$761$		−	41.7626i		−	1.51389i	−0.653477	−	0.756946i	$-0.726690\pi$
$761$		−	41.7626i		−	1.51389i	0.653477	−	0.756946i	$-0.273310\pi$
$762$		0			0
$763$	−3.74343			−0.135521
$764$		0			0
$765$			50.5099i			1.82619i
$766$		0			0
$767$	0.138756	−	3.55500i	0.00501020	−	0.128363i
$768$		0			0
$769$			33.8407i			1.22033i	0.792276	+	0.610163i	$0.208896\pi$
$769$			33.8407i			1.22033i	−0.792276	+	0.610163i	$0.791104\pi$
$770$		0			0
$771$	−25.1902			−0.907201
$772$		0			0
$773$		−	13.3584i		−	0.480469i	−0.970715	−	0.240234i	$-0.922776\pi$
$773$		−	13.3584i		−	0.480469i	0.970715	−	0.240234i	$-0.0772243\pi$
$774$		0			0
$775$		−	22.0895i		−	0.793478i
$776$		0			0
$777$	−106.516			−3.82122
$778$		0			0
$779$	−25.1185			−0.899964
$780$		0			0
$781$	15.1281			0.541327
$782$		0			0
$783$	24.4454			0.873609
$784$		0			0
$785$			36.9139i			1.31751i
$786$		0			0
$787$		−	45.4424i		−	1.61985i	−0.586536	−	0.809923i	$-0.699509\pi$
$787$		−	45.4424i		−	1.61985i	0.586536	−	0.809923i	$-0.300491\pi$
$788$		0			0
$789$	70.4735			2.50892
$790$		0			0
$791$			32.3189i			1.14913i
$792$		0			0
$793$	19.6314	+	0.766239i	0.697131	+	0.0272100i
$794$		0			0
$795$		−	107.720i		−	3.82042i
$796$		0			0
$797$	−56.1445			−1.98874			−0.994371	−	0.105957i	$-0.966209\pi$
$797$	−56.1445			−1.98874			−0.994371	+	0.105957i	$0.966209\pi$
$798$		0			0
$799$			6.05563i			0.214233i
$800$		0			0
$801$		−	52.9892i		−	1.87228i
$802$		0			0
$803$	6.72523			0.237328
$804$		0			0
$805$	18.8026			0.662704
$806$		0			0
$807$	5.65394			0.199028
$808$		0			0
$809$	8.75323			0.307747			0.153874	−	0.988091i	$-0.450825\pi$
$809$	8.75323			0.307747			0.153874	+	0.988091i	$0.450825\pi$
$810$		0			0
$811$		−	12.5850i		−	0.441921i	−0.975283	−	0.220960i	$-0.929081\pi$
$811$		−	12.5850i		−	0.441921i	0.975283	−	0.220960i	$-0.0709191\pi$
$812$		0			0
$813$			71.8036i			2.51826i
$814$		0			0
$815$	−7.03001			−0.246251
$816$		0			0
$817$		−	19.9673i		−	0.698568i
$818$		0			0
$819$	−4.91110	+	125.825i	−0.171608	+	4.39667i
$820$		0			0
$821$		−	1.41251i		−	0.0492969i	−0.999696	−	0.0246484i	$-0.992153\pi$
$821$		−	1.41251i		−	0.0492969i	0.999696	−	0.0246484i	$-0.00784664\pi$
$822$		0			0
$823$	5.30413			0.184890			0.0924452	−	0.995718i	$-0.470532\pi$
$823$	5.30413			0.184890			0.0924452	+	0.995718i	$0.470532\pi$
$824$		0			0
$825$		−	8.40604i		−	0.292661i
$826$		0			0
$827$			28.2994i			0.984066i	0.870577	+	0.492033i	$0.163746\pi$
$827$			28.2994i			0.984066i	−0.870577	+	0.492033i	$0.836254\pi$
$828$		0			0
$829$	−8.61319			−0.299148			−0.149574	−	0.988750i	$-0.547790\pi$
$829$	−8.61319			−0.299148			−0.149574	+	0.988750i	$0.547790\pi$
$830$		0			0
$831$	34.1252			1.18379
$832$		0			0
$833$	−26.7481			−0.926767
$834$		0			0
$835$	−27.9433			−0.967019
$836$		0			0
$837$		−	147.515i		−	5.09887i
$838$		0			0
$839$			27.3745i			0.945073i	0.881311	+	0.472536i	$0.156661\pi$
$839$			27.3745i			0.945073i	−0.881311	+	0.472536i	$0.843339\pi$
$840$		0			0
$841$	−26.9006			−0.927608
$842$		0			0
$843$			63.2899i			2.17982i
$844$		0			0
$845$	−2.77984	+	35.5561i	−0.0956293	+	1.22317i
$846$		0			0
$847$			4.32727i			0.148687i
$848$		0			0
$849$	100.633			3.45370
$850$		0			0
$851$			11.7172i			0.401659i
$852$		0			0
$853$			39.2940i			1.34540i	0.739915	+	0.672700i	$0.234866\pi$
$853$			39.2940i			1.34540i	−0.739915	+	0.672700i	$0.765134\pi$
$854$		0			0
$855$	−128.187			−4.38389
$856$		0			0
$857$	−32.9601			−1.12590			−0.562948	−	0.826492i	$-0.690333\pi$
$857$	−32.9601			−1.12590			−0.562948	+	0.826492i	$0.690333\pi$
$858$		0			0
$859$	5.21241			0.177845			0.0889227	−	0.996039i	$-0.471658\pi$
$859$	5.21241			0.177845			0.0889227	+	0.996039i	$0.471658\pi$
$860$		0			0
$861$	−62.4680			−2.12890
$862$		0			0
$863$		−	26.8614i		−	0.914373i	−0.889371	−	0.457187i	$-0.848857\pi$
$863$		−	26.8614i		−	0.914373i	0.889371	−	0.457187i	$-0.151143\pi$
$864$		0			0
$865$			52.2015i			1.77490i
$866$		0			0
$867$	39.2482			1.33294
$868$		0			0
$869$		−	12.6659i		−	0.429662i
$870$		0			0
$871$	−0.0254443	+	0.651896i	−0.000862149	+	0.0220886i
$872$		0			0
$873$			42.4242i			1.43584i
$874$		0			0
$875$	29.3654			0.992730
$876$		0			0
$877$		−	5.04288i		−	0.170286i	−0.996369	−	0.0851430i	$-0.972865\pi$
$877$		−	5.04288i		−	0.170286i	0.996369	−	0.0851430i	$-0.0271347\pi$
$878$		0			0
$879$			41.5513i			1.40149i
$880$		0			0
$881$	−3.12666			−0.105340			−0.0526699	−	0.998612i	$-0.516773\pi$
$881$	−3.12666			−0.105340			−0.0526699	+	0.998612i	$0.516773\pi$
$882$		0			0
$883$	35.8602			1.20679			0.603396	−	0.797441i	$-0.293814\pi$
$883$	35.8602			1.20679			0.603396	+	0.797441i	$0.293814\pi$
$884$		0			0
$885$	9.00699			0.302767
$886$		0			0
$887$	36.0572			1.21068			0.605341	−	0.795966i	$-0.293037\pi$
$887$	36.0572			1.21068			0.605341	+	0.795966i	$0.293037\pi$
$888$		0			0
$889$			62.3612i			2.09153i
$890$		0			0
$891$		−	31.9240i		−	1.06950i
$892$		0			0
$893$	−15.3683			−0.514280
$894$		0			0
$895$			23.1841i			0.774960i
$896$		0			0
$897$	18.9862	+	0.741057i	0.633932	+	0.0247432i
$898$		0			0
$899$		−	12.6685i		−	0.422518i
$900$		0			0
$901$	26.9206			0.896857
$902$		0			0
$903$		−	49.6573i		−	1.65249i
$904$		0			0
$905$			41.4444i			1.37766i
$906$		0			0
$907$	−6.35007			−0.210851			−0.105425	−	0.994427i	$-0.533620\pi$
$907$	−6.35007			−0.210851			−0.105425	+	0.994427i	$0.533620\pi$
$908$		0			0
$909$	−48.4242			−1.60613
$910$		0			0
$911$	19.7382			0.653957			0.326978	−	0.945032i	$-0.393970\pi$
$911$	19.7382			0.653957			0.326978	+	0.945032i	$0.393970\pi$
$912$		0			0
$913$	−11.5329			−0.381683
$914$		0			0
$915$			49.7384i			1.64430i
$916$		0			0
$917$		−	46.7168i		−	1.54273i
$918$		0			0
$919$	5.90950			0.194936			0.0974682	−	0.995239i	$-0.468926\pi$
$919$	5.90950			0.194936			0.0974682	+	0.995239i	$0.468926\pi$
$920$		0			0
$921$			51.4406i			1.69503i
$922$		0			0
$923$	2.12735	−	54.5037i	0.0700226	−	1.79401i
$924$		0			0
$925$		−	18.6903i		−	0.614533i
$926$		0			0
$927$	−21.1445			−0.694478
$928$		0			0
$929$		−	12.4502i		−	0.408478i	−0.978921	−	0.204239i	$-0.934528\pi$
$929$		−	12.4502i		−	0.408478i	0.978921	−	0.204239i	$-0.0654719\pi$
$930$		0			0
$931$		−	67.8826i		−	2.22476i
$932$		0			0
$933$	92.1618			3.01724
$934$		0			0
$935$	6.25844			0.204673
$936$		0			0
$937$	43.5289			1.42203			0.711014	−	0.703178i	$-0.248236\pi$
$937$	43.5289			1.42203			0.711014	+	0.703178i	$0.248236\pi$
$938$		0			0
$939$	55.1797			1.80072
$940$		0			0
$941$		−	28.4896i		−	0.928734i	−0.885643	−	0.464367i	$-0.846282\pi$
$941$		−	28.4896i		−	0.928734i	0.885643	−	0.464367i	$-0.153718\pi$
$942$		0			0
$943$			6.87174i			0.223775i
$944$		0			0
$945$	−200.291			−6.51548
$946$		0			0
$947$			18.4773i			0.600432i	0.953871	+	0.300216i	$0.0970588\pi$
$947$			18.4773i			0.600432i	−0.953871	+	0.300216i	$0.902941\pi$
$948$		0			0
$949$	0.945717	−	24.2297i	0.0306993	−	0.786529i
$950$		0			0
$951$		−	34.7273i		−	1.12611i
$952$		0			0
$953$	26.6384			0.862902			0.431451	−	0.902136i	$-0.358002\pi$
$953$	26.6384			0.862902			0.431451	+	0.902136i	$0.358002\pi$
$954$		0			0
$955$		−	32.6187i		−	1.05552i
$956$		0			0
$957$		−	4.82093i		−	0.155838i
$958$		0			0
$959$	69.5621			2.24628
$960$		0			0
$961$	−45.4476			−1.46605
$962$		0			0
$963$	23.6938			0.763521
$964$		0			0
$965$	25.3084			0.814706
$966$		0			0
$967$		−	38.4455i		−	1.23632i	−0.786050	−	0.618162i	$-0.787877\pi$
$967$		−	38.4455i		−	1.23632i	0.786050	−	0.618162i	$-0.212123\pi$
$968$		0			0
$969$		−	43.9437i		−	1.41168i
$970$		0			0
$971$	−14.1887			−0.455337			−0.227668	−	0.973739i	$-0.573110\pi$
$971$	−14.1887			−0.455337			−0.227668	+	0.973739i	$0.573110\pi$
$972$		0			0
$973$		−	16.7532i		−	0.537084i
$974$		0			0
$975$	−30.2853	−	1.18208i	−0.969907	−	0.0378567i
$976$		0			0
$977$			20.7298i			0.663206i	0.943419	+	0.331603i	$0.107589\pi$
$977$			20.7298i			0.663206i	−0.943419	+	0.331603i	$0.892411\pi$
$978$		0			0
$979$	−6.56563			−0.209839
$980$		0			0
$981$		−	6.98180i		−	0.222912i
$982$		0			0
$983$			15.7333i			0.501814i	0.968011	+	0.250907i	$0.0807289\pi$
$983$			15.7333i			0.501814i	−0.968011	+	0.250907i	$0.919271\pi$
$984$		0			0
$985$	49.2233			1.56838
$986$		0			0
$987$	−38.2198			−1.21655
$988$		0			0
$989$	−5.46251			−0.173698
$990$		0			0
$991$	17.1754			0.545595			0.272798	−	0.962071i	$-0.412051\pi$
$991$	17.1754			0.545595			0.272798	+	0.962071i	$0.412051\pi$
$992$		0			0
$993$			3.58936i			0.113905i
$994$		0			0
$995$			4.68801i			0.148620i
$996$		0			0
$997$	−17.6511			−0.559015			−0.279507	−	0.960144i	$-0.590171\pi$
$997$	−17.6511			−0.559015			−0.279507	+	0.960144i	$0.590171\pi$
$998$		0			0
$999$		−	124.815i		−	3.94897i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	572.2.f.c.441.2	yes	8
3.2	odd	2		5148.2.e.c.1585.3		8
4.3	odd	2		2288.2.j.i.1585.8		8
13.5	odd	4		7436.2.a.p.1.1		4
13.8	odd	4		7436.2.a.o.1.1		4
13.12	even	2	inner	572.2.f.c.441.1	✓	8
39.38	odd	2		5148.2.e.c.1585.6		8
52.51	odd	2		2288.2.j.i.1585.7		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
572.2.f.c.441.1	✓	8	13.12	even	2	inner
572.2.f.c.441.2	yes	8	1.1	even	1	trivial
2288.2.j.i.1585.7		8	52.51	odd	2
2288.2.j.i.1585.8		8	4.3	odd	2
5148.2.e.c.1585.3		8	3.2	odd	2
5148.2.e.c.1585.6		8	39.38	odd	2
7436.2.a.o.1.1		4	13.8	odd	4
7436.2.a.p.1.1		4	13.5	odd	4

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:	\( N \)	\(=\)	\( 572 = 2^{2} \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	572.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.32727 q^{3} +2.74343i q^{5} -4.32727i q^{7} +8.07070 q^{9} +O(q^{10})\)	\(q-3.32727 q^{3} +2.74343i q^{5} -4.32727i q^{7} +8.07070 q^{9} -1.00000i q^{11} +(-3.60281 - 0.140622i) q^{13} -9.12812i q^{15} +2.28124 q^{17} +5.78945i q^{19} +14.3980i q^{21} +1.58384 q^{23} -2.52641 q^{25} -16.8716 q^{27} -1.44892 q^{29} +8.74343i q^{31} +3.32727i q^{33} +11.8716 q^{35} +7.39796i q^{37} +(11.9875 + 0.467888i) q^{39} +4.33867i q^{41} -3.44892 q^{43} +22.1414i q^{45} +2.65453i q^{47} -11.7252 q^{49} -7.59031 q^{51} +11.8009 q^{53} +2.74343 q^{55} -19.2630i q^{57} +0.986729i q^{59} -5.44892 q^{61} -34.9240i q^{63} +(0.385788 - 9.88405i) q^{65} -0.180941i q^{67} -5.26984 q^{69} +15.1281i q^{71} +6.72523i q^{73} +8.40604 q^{75} -4.32727 q^{77} +12.6659 q^{79} +31.9240 q^{81} -11.5329i q^{83} +6.25844i q^{85} +4.82093 q^{87} -6.56563i q^{89} +(-0.608510 + 15.5903i) q^{91} -29.0917i q^{93} -15.8830 q^{95} +5.25657i q^{97} -8.07070i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{3} + 18 q^{9}+O(q^{10}) \) 8 * q - 2 * q^3 + 18 * q^9	\( 8 q - 2 q^{3} + 18 q^{9} - 8 q^{13} + 16 q^{17} + 10 q^{23} - 52 q^{25} - 32 q^{27} - 4 q^{29} - 8 q^{35} + 16 q^{39} - 20 q^{43} + 2 q^{49} + 40 q^{51} + 38 q^{53} - 36 q^{61} + 36 q^{65} - 52 q^{69} + 10 q^{75} - 10 q^{77} + 40 q^{79} + 32 q^{81} + 56 q^{87} + 22 q^{91} - 12 q^{95}+O(q^{100}) \) 8 * q - 2 * q^3 + 18 * q^9 - 8 * q^13 + 16 * q^17 + 10 * q^23 - 52 * q^25 - 32 * q^27 - 4 * q^29 - 8 * q^35 + 16 * q^39 - 20 * q^43 + 2 * q^49 + 40 * q^51 + 38 * q^53 - 36 * q^61 + 36 * q^65 - 52 * q^69 + 10 * q^75 - 10 * q^77 + 40 * q^79 + 32 * q^81 + 56 * q^87 + 22 * q^91 - 12 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more