LMFDB - Embedded newform 3648.2.k.l.2431.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3648,2,Mod(2431,3648)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3648, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("3648.2431");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3648 = 2^{6} \cdot 3 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3648.k (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$29.1294266574$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + 52 x^{18} + 986 x^{16} + 8824 x^{14} + 42757 x^{12} + 118964 x^{10} + 190576 x^{8} + 166880 x^{6} + \cdots + 784 $ x^20 + 52x^18 + 986x^16 + 8824x^14 + 42757x^12 + 118964x^10 + 190576x^8 + 166880x^6 + 70964x^4 + 12912*x^2 + 784
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{27} $
Twist minimal:	no (minimal twist has level 1824)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2431.13
Root			$1.52988i$ of defining polynomial
Character	$\chi$	$=$	3648.2431
Dual form			3648.2.k.l.2431.14

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +1.49782 q^{5} -0.665758i q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.49782 q^{5} -0.665758i q^{7} +1.00000 q^{9} -0.592107i q^{11} -1.37538i q^{13} +1.49782 q^{15} -3.18220 q^{17} +(-4.33389 - 0.466292i) q^{19} -0.665758i q^{21} -8.03453i q^{23} -2.75654 q^{25} +1.00000 q^{27} -3.06928i q^{29} +0.0945089 q^{31} -0.592107i q^{33} -0.997183i q^{35} -3.62946i q^{37} -1.37538i q^{39} +4.18397i q^{41} -10.7263i q^{43} +1.49782 q^{45} -6.22510i q^{47} +6.55677 q^{49} -3.18220 q^{51} -4.18397i q^{53} -0.886868i q^{55} +(-4.33389 - 0.466292i) q^{57} -7.83326 q^{59} -4.17521 q^{61} -0.665758i q^{63} -2.06006i q^{65} -3.03026 q^{67} -8.03453i q^{69} +11.4897 q^{71} +1.21010 q^{73} -2.75654 q^{75} -0.394200 q^{77} +10.6176 q^{79} +1.00000 q^{81} +1.84334i q^{83} -4.76635 q^{85} -3.06928i q^{87} +6.73140i q^{89} -0.915667 q^{91} +0.0945089 q^{93} +(-6.49137 - 0.698421i) q^{95} +12.7628i q^{97} -0.592107i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 20 q^{3} + 20 q^{9}+O(q^{10}) $ 20 * q + 20 * q^3 + 20 * q^9	$ 20 q + 20 q^{3} + 20 q^{9} + 4 q^{19} + 20 q^{25} + 20 q^{27} + 8 q^{31} - 44 q^{49} + 4 q^{57} + 8 q^{61} + 8 q^{67} - 64 q^{71} + 8 q^{73} + 20 q^{75} - 40 q^{77} + 16 q^{79} + 20 q^{81} - 40 q^{85} + 8 q^{93} + 8 q^{95}+O(q^{100}) $ 20 * q + 20 * q^3 + 20 * q^9 + 4 * q^19 + 20 * q^25 + 20 * q^27 + 8 * q^31 - 44 * q^49 + 4 * q^57 + 8 * q^61 + 8 * q^67 - 64 * q^71 + 8 * q^73 + 20 * q^75 - 40 * q^77 + 16 * q^79 + 20 * q^81 - 40 * q^85 + 8 * q^93 + 8 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3648\mathbb{Z}\right)^\times$.

$n$	$1217$	$1921$	$2053$	$2623$
$\chi(n)$	$1$	$-1$	$1$	$-1$

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
$3$	1.00000			0.577350
$3$	1.00000			0.577350
$4$		0			0
$5$	1.49782			0.669844			0.334922	−	0.942246i	$-0.391290\pi$
$5$	1.49782			0.669844			0.334922	+	0.942246i	$0.391290\pi$
$6$		0			0
$7$		−	0.665758i		−	0.251633i	−0.992054	−	0.125816i	$-0.959845\pi$
$7$		−	0.665758i		−	0.251633i	0.992054	−	0.125816i	$-0.0401550\pi$
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$		−	0.592107i		−	0.178527i	−0.996008	−	0.0892635i	$-0.971549\pi$
$11$		−	0.592107i		−	0.178527i	0.996008	−	0.0892635i	$-0.0284513\pi$
$12$		0			0
$13$		−	1.37538i		−	0.381461i	−0.981642	−	0.190730i	$-0.938914\pi$
$13$		−	1.37538i		−	0.381461i	0.981642	−	0.190730i	$-0.0610856\pi$
$14$		0			0
$15$	1.49782			0.386735
$16$		0			0
$17$	−3.18220			−0.771796			−0.385898	−	0.922541i	$-0.626108\pi$
$17$	−3.18220			−0.771796			−0.385898	+	0.922541i	$0.626108\pi$
$18$		0			0
$19$	−4.33389	−	0.466292i	−0.994262	−	0.106975i
$19$	−4.33389	−	0.466292i	−0.994262	−	0.106975i
$20$		0			0
$21$		−	0.665758i		−	0.145280i
$22$		0			0
$23$		−	8.03453i		−	1.67532i	−0.546195	−	0.837658i	$-0.683924\pi$
$23$		−	8.03453i		−	1.67532i	0.546195	−	0.837658i	$-0.316076\pi$
$24$		0			0
$25$	−2.75654			−0.551309
$26$		0			0
$27$	1.00000			0.192450
$28$		0			0
$29$		−	3.06928i		−	0.569952i	−0.958535	−	0.284976i	$-0.908014\pi$
$29$		−	3.06928i		−	0.569952i	0.958535	−	0.284976i	$-0.0919856\pi$
$30$		0			0
$31$	0.0945089			0.0169743			0.00848715	−	0.999964i	$-0.497298\pi$
$31$	0.0945089			0.0169743			0.00848715	+	0.999964i	$0.497298\pi$
$32$		0			0
$33$		−	0.592107i		−	0.103073i
$34$		0			0
$35$		−	0.997183i		−	0.168555i
$36$		0			0
$37$		−	3.62946i		−	0.596679i	−0.954460	−	0.298339i	$-0.903567\pi$
$37$		−	3.62946i		−	0.596679i	0.954460	−	0.298339i	$-0.0964327\pi$
$38$		0			0
$39$		−	1.37538i		−	0.220236i
$40$		0			0
$41$			4.18397i			0.653426i	0.945124	+	0.326713i	$0.105941\pi$
$41$			4.18397i			0.653426i	−0.945124	+	0.326713i	$0.894059\pi$
$42$		0			0
$43$		−	10.7263i		−	1.63574i	−0.575401	−	0.817872i	$-0.695154\pi$
$43$		−	10.7263i		−	1.63574i	0.575401	−	0.817872i	$-0.304846\pi$
$44$		0			0
$45$	1.49782			0.223281
$46$		0			0
$47$		−	6.22510i		−	0.908024i	−0.890996	−	0.454012i	$-0.849992\pi$
$47$		−	6.22510i		−	0.908024i	0.890996	−	0.454012i	$-0.150008\pi$
$48$		0			0
$49$	6.55677			0.936681
$50$		0			0
$51$	−3.18220			−0.445597
$52$		0			0
$53$		−	4.18397i		−	0.574712i	−0.957824	−	0.287356i	$-0.907224\pi$
$53$		−	4.18397i		−	0.574712i	0.957824	−	0.287356i	$-0.0927763\pi$
$54$		0			0
$55$		−	0.886868i		−	0.119585i
$56$		0			0
$57$	−4.33389	−	0.466292i	−0.574037	−	0.0617619i
$58$		0			0
$59$	−7.83326			−1.01980			−0.509902	−	0.860233i	$-0.670318\pi$
$59$	−7.83326			−1.01980			−0.509902	+	0.860233i	$0.670318\pi$
$60$		0			0
$61$	−4.17521			−0.534581			−0.267290	−	0.963616i	$-0.586128\pi$
$61$	−4.17521			−0.534581			−0.267290	+	0.963616i	$0.586128\pi$
$62$		0			0
$63$		−	0.665758i		−	0.0838776i
$64$		0			0
$65$		−	2.06006i		−	0.255519i
$66$		0			0
$67$	−3.03026			−0.370205			−0.185102	−	0.982719i	$-0.559262\pi$
$67$	−3.03026			−0.370205			−0.185102	+	0.982719i	$0.559262\pi$
$68$		0			0
$69$		−	8.03453i		−	0.967244i
$70$		0			0
$71$	11.4897			1.36358			0.681789	−	0.731549i	$-0.261202\pi$
$71$	11.4897			1.36358			0.681789	+	0.731549i	$0.261202\pi$
$72$		0			0
$73$	1.21010			0.141631			0.0708155	−	0.997489i	$-0.477440\pi$
$73$	1.21010			0.141631			0.0708155	+	0.997489i	$0.477440\pi$
$74$		0			0
$75$	−2.75654			−0.318298
$76$		0			0
$77$	−0.394200			−0.0449233
$78$		0			0
$79$	10.6176			1.19457			0.597286	−	0.802028i	$-0.296246\pi$
$79$	10.6176			1.19457			0.597286	+	0.802028i	$0.296246\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$			1.84334i			0.202333i	0.994870	+	0.101167i	$0.0322575\pi$
$83$			1.84334i			0.202333i	−0.994870	+	0.101167i	$0.967743\pi$
$84$		0			0
$85$	−4.76635			−0.516983
$86$		0			0
$87$		−	3.06928i		−	0.329062i
$88$		0			0
$89$			6.73140i			0.713527i	0.934195	+	0.356764i	$0.116120\pi$
$89$			6.73140i			0.713527i	−0.934195	+	0.356764i	$0.883880\pi$
$90$		0			0
$91$	−0.915667			−0.0959880
$92$		0			0
$93$	0.0945089			0.00980012
$94$		0			0
$95$	−6.49137	−	0.698421i	−0.666000	−	0.0716564i
$96$		0			0
$97$			12.7628i			1.29587i	0.761697	+	0.647933i	$0.224366\pi$
$97$			12.7628i			1.29587i	−0.761697	+	0.647933i	$0.775634\pi$
$98$		0			0
$99$		−	0.592107i		−	0.0595090i
$100$		0			0
$101$	9.66838			0.962040			0.481020	−	0.876710i	$-0.340266\pi$
$101$	9.66838			0.962040			0.481020	+	0.876710i	$0.340266\pi$
$102$		0			0
$103$	4.35468			0.429079			0.214539	−	0.976715i	$-0.431175\pi$
$103$	4.35468			0.429079			0.214539	+	0.976715i	$0.431175\pi$
$104$		0			0
$105$		−	0.997183i		−	0.0973151i
$106$		0			0
$107$	−10.6610			−1.03064			−0.515320	−	0.856998i	$-0.672327\pi$
$107$	−10.6610			−1.03064			−0.515320	+	0.856998i	$0.672327\pi$
$108$		0			0
$109$		−	10.7354i		−	1.02827i	−0.857711	−	0.514133i	$-0.828114\pi$
$109$		−	10.7354i		−	1.02827i	0.857711	−	0.514133i	$-0.171886\pi$
$110$		0			0
$111$		−	3.62946i		−	0.344493i
$112$		0			0
$113$			11.4769i			1.07966i	0.841776	+	0.539828i	$0.181511\pi$
$113$			11.4769i			1.07966i	−0.841776	+	0.539828i	$0.818489\pi$
$114$		0			0
$115$		−	12.0343i		−	1.12220i
$116$		0			0
$117$		−	1.37538i		−	0.127154i
$118$		0			0
$119$			2.11857i			0.194209i
$120$		0			0
$121$	10.6494			0.968128
$122$		0			0
$123$			4.18397i			0.377255i
$124$		0			0
$125$	−11.6179			−1.03914
$126$		0			0
$127$	−6.18341			−0.548689			−0.274344	−	0.961631i	$-0.588461\pi$
$127$	−6.18341			−0.548689			−0.274344	+	0.961631i	$0.588461\pi$
$128$		0			0
$129$		−	10.7263i		−	0.944397i
$130$		0			0
$131$		−	6.71162i		−	0.586397i	−0.956052	−	0.293198i	$-0.905280\pi$
$131$		−	6.71162i		−	0.586397i	0.956052	−	0.293198i	$-0.0947197\pi$
$132$		0			0
$133$	−0.310438	+	2.88532i	−0.0269184	+	0.250189i
$134$		0			0
$135$	1.49782			0.128912
$136$		0			0
$137$	−5.72463			−0.489088			−0.244544	−	0.969638i	$-0.578638\pi$
$137$	−5.72463			−0.489088			−0.244544	+	0.969638i	$0.578638\pi$
$138$		0			0
$139$		−	4.12405i		−	0.349797i	−0.984586	−	0.174899i	$-0.944040\pi$
$139$		−	4.12405i		−	0.349797i	0.984586	−	0.174899i	$-0.0559598\pi$
$140$		0			0
$141$		−	6.22510i		−	0.524248i
$142$		0			0
$143$	−0.814370			−0.0681010
$144$		0			0
$145$		−	4.59723i		−	0.381779i
$146$		0			0
$147$	6.55677			0.540793
$148$		0			0
$149$	9.71640			0.795999			0.397999	−	0.917386i	$-0.369705\pi$
$149$	9.71640			0.795999			0.397999	+	0.917386i	$0.369705\pi$
$150$		0			0
$151$	9.39047			0.764185			0.382093	−	0.924124i	$-0.375204\pi$
$151$	9.39047			0.764185			0.382093	+	0.924124i	$0.375204\pi$
$152$		0			0
$153$	−3.18220			−0.257265
$154$		0			0
$155$	0.141557			0.0113701
$156$		0			0
$157$	−22.4441			−1.79124			−0.895619	−	0.444823i	$-0.853267\pi$
$157$	−22.4441			−1.79124			−0.895619	+	0.444823i	$0.853267\pi$
$158$		0			0
$159$		−	4.18397i		−	0.331810i
$160$		0			0
$161$	−5.34905			−0.421564
$162$		0			0
$163$		−	14.5987i		−	1.14346i	−0.820442	−	0.571729i	$-0.806273\pi$
$163$		−	14.5987i		−	1.14346i	0.820442	−	0.571729i	$-0.193727\pi$
$164$		0			0
$165$		−	0.886868i		−	0.0690426i
$166$		0			0
$167$	−2.68237			−0.207568			−0.103784	−	0.994600i	$-0.533095\pi$
$167$	−2.68237			−0.207568			−0.103784	+	0.994600i	$0.533095\pi$
$168$		0			0
$169$	11.1083			0.854488
$170$		0			0
$171$	−4.33389	−	0.466292i	−0.331421	−	0.0356583i
$172$		0			0
$173$			12.6910i			0.964876i	0.875930	+	0.482438i	$0.160249\pi$
$173$			12.6910i			0.964876i	−0.875930	+	0.482438i	$0.839751\pi$
$174$		0			0
$175$			1.83519i			0.138727i
$176$		0			0
$177$	−7.83326			−0.588784
$178$		0			0
$179$	13.0063			0.972135			0.486067	−	0.873921i	$-0.338431\pi$
$179$	13.0063			0.972135			0.486067	+	0.873921i	$0.338431\pi$
$180$		0			0
$181$			7.88661i			0.586207i	0.956081	+	0.293103i	$0.0946880\pi$
$181$			7.88661i			0.586207i	−0.956081	+	0.293103i	$0.905312\pi$
$182$		0			0
$183$	−4.17521			−0.308640
$184$		0			0
$185$		−	5.43626i		−	0.399682i
$186$		0			0
$187$			1.88420i			0.137786i
$188$		0			0
$189$		−	0.665758i		−	0.0484268i
$190$		0			0
$191$		−	4.06106i		−	0.293848i	−0.989148	−	0.146924i	$-0.953063\pi$
$191$		−	4.06106i		−	0.293848i	0.989148	−	0.146924i	$-0.0469373\pi$
$192$		0			0
$193$			1.77237i			0.127578i	0.997963	+	0.0637888i	$0.0203184\pi$
$193$			1.77237i			0.127578i	−0.997963	+	0.0637888i	$0.979682\pi$
$194$		0			0
$195$		−	2.06006i		−	0.147524i
$196$		0			0
$197$	0.851084			0.0606372			0.0303186	−	0.999540i	$-0.490348\pi$
$197$	0.851084			0.0606372			0.0303186	+	0.999540i	$0.490348\pi$
$198$		0			0
$199$		−	1.78610i		−	0.126613i	−0.997994	−	0.0633067i	$-0.979835\pi$
$199$		−	1.78610i		−	0.126613i	0.997994	−	0.0633067i	$-0.0201646\pi$
$200$		0			0
$201$	−3.03026			−0.213738
$202$		0			0
$203$	−2.04340			−0.143419
$204$		0			0
$205$			6.26681i			0.437693i
$206$		0			0
$207$		−	8.03453i		−	0.558439i
$208$		0			0
$209$	−0.276095	+	2.56613i	−0.0190979	+	0.177503i
$210$		0			0
$211$	−5.23063			−0.360091			−0.180046	−	0.983658i	$-0.557625\pi$
$211$	−5.23063			−0.360091			−0.180046	+	0.983658i	$0.557625\pi$
$212$		0			0
$213$	11.4897			0.787262
$214$		0			0
$215$		−	16.0660i		−	1.09569i
$216$		0			0
$217$		−	0.0629201i		−	0.00427129i
$218$		0			0
$219$	1.21010			0.0817707
$220$		0			0
$221$			4.37672i			0.294410i
$222$		0			0
$223$	−18.6398			−1.24821			−0.624106	−	0.781340i	$-0.714536\pi$
$223$	−18.6398			−1.24821			−0.624106	+	0.781340i	$0.714536\pi$
$224$		0			0
$225$	−2.75654			−0.183770
$226$		0			0
$227$	9.12713			0.605789			0.302894	−	0.953024i	$-0.402047\pi$
$227$	9.12713			0.605789			0.302894	+	0.953024i	$0.402047\pi$
$228$		0			0
$229$	5.58646			0.369164			0.184582	−	0.982817i	$-0.440907\pi$
$229$	5.58646			0.369164			0.184582	+	0.982817i	$0.440907\pi$
$230$		0			0
$231$	−0.394200			−0.0259365
$232$		0			0
$233$	26.1728			1.71464			0.857318	−	0.514787i	$-0.172129\pi$
$233$	26.1728			1.71464			0.857318	+	0.514787i	$0.172129\pi$
$234$		0			0
$235$		−	9.32406i		−	0.608235i
$236$		0			0
$237$	10.6176			0.689686
$238$		0			0
$239$		−	3.84844i		−	0.248935i	−0.992224	−	0.124467i	$-0.960278\pi$
$239$		−	3.84844i		−	0.248935i	0.992224	−	0.124467i	$-0.0397222\pi$
$240$		0			0
$241$		−	16.1576i		−	1.04080i	−0.853922	−	0.520400i	$-0.825783\pi$
$241$		−	16.1576i		−	1.04080i	0.853922	−	0.520400i	$-0.174217\pi$
$242$		0			0
$243$	1.00000			0.0641500
$244$		0			0
$245$	9.82084			0.627430
$246$		0			0
$247$	−0.641327	+	5.96072i	−0.0408067	+	0.379272i
$248$		0			0
$249$			1.84334i			0.116817i
$250$		0			0
$251$		−	11.9799i		−	0.756166i	−0.925772	−	0.378083i	$-0.876583\pi$
$251$		−	11.9799i		−	0.756166i	0.925772	−	0.378083i	$-0.123417\pi$
$252$		0			0
$253$	−4.75730			−0.299089
$254$		0			0
$255$	−4.76635			−0.298480
$256$		0			0
$257$			21.9312i			1.36803i	0.729468	+	0.684015i	$0.239768\pi$
$257$			21.9312i			1.36803i	−0.729468	+	0.684015i	$0.760232\pi$
$258$		0			0
$259$	−2.41634			−0.150144
$260$		0			0
$261$		−	3.06928i		−	0.189984i
$262$		0			0
$263$		−	21.6496i		−	1.33497i	−0.744623	−	0.667485i	$-0.767371\pi$
$263$		−	21.6496i		−	1.33497i	0.744623	−	0.667485i	$-0.232629\pi$
$264$		0			0
$265$		−	6.26681i		−	0.384967i
$266$		0			0
$267$			6.73140i			0.411955i
$268$		0			0
$269$			8.05506i			0.491126i	0.969381	+	0.245563i	$0.0789728\pi$
$269$			8.05506i			0.491126i	−0.969381	+	0.245563i	$0.921027\pi$
$270$		0			0
$271$			5.68805i			0.345524i	0.984964	+	0.172762i	$0.0552692\pi$
$271$			5.68805i			0.345524i	−0.984964	+	0.172762i	$0.944731\pi$
$272$		0			0
$273$	−0.915667			−0.0554187
$274$		0			0
$275$			1.63217i			0.0984235i
$276$		0			0
$277$	12.2360			0.735190			0.367595	−	0.929986i	$-0.380181\pi$
$277$	12.2360			0.735190			0.367595	+	0.929986i	$0.380181\pi$
$278$		0			0
$279$	0.0945089			0.00565810
$280$		0			0
$281$			9.66307i			0.576451i	0.957563	+	0.288225i	$0.0930652\pi$
$281$			9.66307i			0.576451i	−0.957563	+	0.288225i	$0.906935\pi$
$282$		0			0
$283$		−	12.2714i		−	0.729456i	−0.931114	−	0.364728i	$-0.881162\pi$
$283$		−	12.2714i		−	0.729456i	0.931114	−	0.364728i	$-0.118838\pi$
$284$		0			0
$285$	−6.49137	−	0.698421i	−0.384515	−	0.0413709i
$286$		0			0
$287$	2.78551			0.164423
$288$		0			0
$289$	−6.87362			−0.404330
$290$		0			0
$291$			12.7628i			0.748169i
$292$		0			0
$293$			32.9593i			1.92550i	0.270386	+	0.962752i	$0.412849\pi$
$293$			32.9593i			1.92550i	−0.270386	+	0.962752i	$0.587151\pi$
$294$		0			0
$295$	−11.7328			−0.683109
$296$		0			0
$297$		−	0.592107i		−	0.0343575i
$298$		0			0
$299$	−11.0505			−0.639067
$300$		0			0
$301$	−7.14111			−0.411607
$302$		0			0
$303$	9.66838			0.555434
$304$		0			0
$305$	−6.25370			−0.358086
$306$		0			0
$307$	−6.55235			−0.373962			−0.186981	−	0.982363i	$-0.559870\pi$
$307$	−6.55235			−0.373962			−0.186981	+	0.982363i	$0.559870\pi$
$308$		0			0
$309$	4.35468			0.247729
$310$		0			0
$311$			20.9039i			1.18535i	0.805442	+	0.592674i	$0.201928\pi$
$311$			20.9039i			1.18535i	−0.805442	+	0.592674i	$0.798072\pi$
$312$		0			0
$313$	−4.35540			−0.246182			−0.123091	−	0.992395i	$-0.539281\pi$
$313$	−4.35540			−0.246182			−0.123091	+	0.992395i	$0.539281\pi$
$314$		0			0
$315$		−	0.997183i		−	0.0561849i
$316$		0			0
$317$			15.3719i			0.863375i	0.902023	+	0.431687i	$0.142082\pi$
$317$			15.3719i			0.863375i	−0.902023	+	0.431687i	$0.857918\pi$
$318$		0			0
$319$	−1.81735			−0.101752
$320$		0			0
$321$	−10.6610			−0.595040
$322$		0			0
$323$	13.7913	+	1.48383i	0.767368	+	0.0825628i
$324$		0			0
$325$			3.79128i			0.210303i
$326$		0			0
$327$		−	10.7354i		−	0.593669i
$328$		0			0
$329$	−4.14441			−0.228489
$330$		0			0
$331$	11.6836			0.642188			0.321094	−	0.947047i	$-0.395950\pi$
$331$	11.6836			0.642188			0.321094	+	0.947047i	$0.395950\pi$
$332$		0			0
$333$		−	3.62946i		−	0.198893i
$334$		0			0
$335$	−4.53877			−0.247980
$336$		0			0
$337$		−	18.1137i		−	0.986714i	−0.869827	−	0.493357i	$-0.835770\pi$
$337$		−	18.1137i		−	0.986714i	0.869827	−	0.493357i	$-0.164230\pi$
$338$		0			0
$339$			11.4769i			0.623339i
$340$		0			0
$341$		−	0.0559594i		−	0.00303037i
$342$		0			0
$343$		−	9.02552i		−	0.487332i
$344$		0			0
$345$		−	12.0343i		−	0.647903i
$346$		0			0
$347$		−	14.5013i		−	0.778469i	−0.921139	−	0.389234i	$-0.872740\pi$
$347$		−	14.5013i		−	0.778469i	0.921139	−	0.389234i	$-0.127260\pi$
$348$		0			0
$349$	−23.7298			−1.27023			−0.635115	−	0.772418i	$-0.719047\pi$
$349$	−23.7298			−1.27023			−0.635115	+	0.772418i	$0.719047\pi$
$350$		0			0
$351$		−	1.37538i		−	0.0734121i
$352$		0			0
$353$	−25.9203			−1.37960			−0.689799	−	0.724001i	$-0.742301\pi$
$353$	−25.9203			−1.37960			−0.689799	+	0.724001i	$0.742301\pi$
$354$		0			0
$355$	17.2095			0.913385
$356$		0			0
$357$			2.11857i			0.112127i
$358$		0			0
$359$		−	11.2720i		−	0.594913i	−0.954735	−	0.297457i	$-0.903862\pi$
$359$		−	11.2720i		−	0.594913i	0.954735	−	0.297457i	$-0.0961383\pi$
$360$		0			0
$361$	18.5651	+	4.04172i	0.977113	+	0.212722i
$362$		0			0
$363$	10.6494			0.558949
$364$		0			0
$365$	1.81250			0.0948707
$366$		0			0
$367$		−	22.0342i		−	1.15018i	−0.818091	−	0.575089i	$-0.804967\pi$
$367$		−	22.0342i		−	1.15018i	0.818091	−	0.575089i	$-0.195033\pi$
$368$		0			0
$369$			4.18397i			0.217809i
$370$		0			0
$371$	−2.78551			−0.144616
$372$		0			0
$373$		−	30.1677i		−	1.56203i	−0.624514	−	0.781013i	$-0.714703\pi$
$373$		−	30.1677i		−	1.56203i	0.624514	−	0.781013i	$-0.285297\pi$
$374$		0			0
$375$	−11.6179			−0.599945
$376$		0			0
$377$	−4.22142			−0.217414
$378$		0			0
$379$	2.03179			0.104366			0.0521831	−	0.998638i	$-0.483382\pi$
$379$	2.03179			0.104366			0.0521831	+	0.998638i	$0.483382\pi$
$380$		0			0
$381$	−6.18341			−0.316786
$382$		0			0
$383$	−17.7769			−0.908355			−0.454178	−	0.890911i	$-0.650067\pi$
$383$	−17.7769			−0.908355			−0.454178	+	0.890911i	$0.650067\pi$
$384$		0			0
$385$	−0.590439			−0.0300916
$386$		0			0
$387$		−	10.7263i		−	0.545248i
$388$		0			0
$389$	17.2699			0.875620			0.437810	−	0.899068i	$-0.355754\pi$
$389$	17.2699			0.875620			0.437810	+	0.899068i	$0.355754\pi$
$390$		0			0
$391$			25.5675i			1.29300i
$392$		0			0
$393$		−	6.71162i		−	0.338556i
$394$		0			0
$395$	15.9032			0.800177
$396$		0			0
$397$	14.0904			0.707178			0.353589	−	0.935401i	$-0.384961\pi$
$397$	14.0904			0.707178			0.353589	+	0.935401i	$0.384961\pi$
$398$		0			0
$399$	−0.310438	+	2.88532i	−0.0155413	+	0.144447i
$400$		0			0
$401$			34.9525i			1.74544i	0.488218	+	0.872722i	$0.337647\pi$
$401$			34.9525i			1.74544i	−0.488218	+	0.872722i	$0.662353\pi$
$402$		0			0
$403$		−	0.129985i		−	0.00647503i
$404$		0			0
$405$	1.49782			0.0744271
$406$		0			0
$407$	−2.14903			−0.106523
$408$		0			0
$409$		−	24.5798i		−	1.21539i	−0.794169	−	0.607696i	$-0.792094\pi$
$409$		−	24.5798i		−	1.21539i	0.794169	−	0.607696i	$-0.207906\pi$
$410$		0			0
$411$	−5.72463			−0.282375
$412$		0			0
$413$			5.21505i			0.256616i
$414$		0			0
$415$			2.76099i			0.135532i
$416$		0			0
$417$		−	4.12405i		−	0.201956i
$418$		0			0
$419$			7.26287i			0.354814i	0.984138	+	0.177407i	$0.0567710\pi$
$419$			7.26287i			0.354814i	−0.984138	+	0.177407i	$0.943229\pi$
$420$		0			0
$421$		−	23.5379i		−	1.14717i	−0.819147	−	0.573583i	$-0.805553\pi$
$421$		−	23.5379i		−	1.14717i	0.819147	−	0.573583i	$-0.194447\pi$
$422$		0			0
$423$		−	6.22510i		−	0.302675i
$424$		0			0
$425$	8.77187			0.425498
$426$		0			0
$427$			2.77968i			0.134518i
$428$		0			0
$429$	−0.814370			−0.0393181
$430$		0			0
$431$	−12.5328			−0.603685			−0.301842	−	0.953358i	$-0.597602\pi$
$431$	−12.5328			−0.603685			−0.301842	+	0.953358i	$0.597602\pi$
$432$		0			0
$433$		−	30.4897i		−	1.46524i	−0.680637	−	0.732621i	$-0.738297\pi$
$433$		−	30.4897i		−	1.46524i	0.680637	−	0.732621i	$-0.261703\pi$
$434$		0			0
$435$		−	4.59723i		−	0.220420i
$436$		0			0
$437$	−3.74644	+	34.8207i	−0.179217	+	1.66570i
$438$		0			0
$439$	3.93198			0.187663			0.0938315	−	0.995588i	$-0.470089\pi$
$439$	3.93198			0.187663			0.0938315	+	0.995588i	$0.470089\pi$
$440$		0			0
$441$	6.55677			0.312227
$442$		0			0
$443$			13.2210i			0.628150i	0.949398	+	0.314075i	$0.101694\pi$
$443$			13.2210i			0.628150i	−0.949398	+	0.314075i	$0.898306\pi$
$444$		0			0
$445$			10.0824i			0.477952i
$446$		0			0
$447$	9.71640			0.459570
$448$		0			0
$449$		−	15.5602i		−	0.734332i	−0.930155	−	0.367166i	$-0.880328\pi$
$449$		−	15.5602i		−	0.734332i	0.930155	−	0.367166i	$-0.119672\pi$
$450$		0			0
$451$	2.47736			0.116654
$452$		0			0
$453$	9.39047			0.441203
$454$		0			0
$455$	−1.37150			−0.0642970
$456$		0			0
$457$	26.1845			1.22486			0.612430	−	0.790525i	$-0.290192\pi$
$457$	26.1845			1.22486			0.612430	+	0.790525i	$0.290192\pi$
$458$		0			0
$459$	−3.18220			−0.148532
$460$		0			0
$461$	−12.3583			−0.575585			−0.287792	−	0.957693i	$-0.592921\pi$
$461$	−12.3583			−0.575585			−0.287792	+	0.957693i	$0.592921\pi$
$462$		0			0
$463$			35.7355i			1.66077i	0.557190	+	0.830385i	$0.311879\pi$
$463$			35.7355i			1.66077i	−0.557190	+	0.830385i	$0.688121\pi$
$464$		0			0
$465$	0.141557			0.00656455
$466$		0			0
$467$		−	0.827129i		−	0.0382750i	−0.999817	−	0.0191375i	$-0.993908\pi$
$467$		−	0.827129i		−	0.0382750i	0.999817	−	0.0191375i	$-0.00609202\pi$
$468$		0			0
$469$			2.01742i			0.0931557i
$470$		0			0
$471$	−22.4441			−1.03417
$472$		0			0
$473$	−6.35111			−0.292024
$474$		0			0
$475$	11.9466	+	1.28536i	0.548145	+	0.0589762i
$476$		0			0
$477$		−	4.18397i		−	0.191571i
$478$		0			0
$479$		−	16.3306i		−	0.746164i	−0.927798	−	0.373082i	$-0.878301\pi$
$479$		−	16.3306i		−	0.746164i	0.927798	−	0.373082i	$-0.121699\pi$
$480$		0			0
$481$	−4.99187			−0.227610
$482$		0			0
$483$	−5.34905			−0.243390
$484$		0			0
$485$			19.1163i			0.868028i
$486$		0			0
$487$	17.6701			0.800707			0.400354	−	0.916361i	$-0.368887\pi$
$487$	17.6701			0.800707			0.400354	+	0.916361i	$0.368887\pi$
$488$		0			0
$489$		−	14.5987i		−	0.660176i
$490$		0			0
$491$			29.9356i			1.35098i	0.737371	+	0.675488i	$0.236067\pi$
$491$			29.9356i			1.35098i	−0.737371	+	0.675488i	$0.763933\pi$
$492$		0			0
$493$			9.76707i			0.439887i
$494$		0			0
$495$		−	0.886868i		−	0.0398618i
$496$		0			0
$497$		−	7.64937i		−	0.343121i
$498$		0			0
$499$		−	33.5155i		−	1.50036i	−0.661233	−	0.750181i	$-0.729967\pi$
$499$		−	33.5155i		−	1.50036i	0.661233	−	0.750181i	$-0.270033\pi$
$500$		0			0
$501$	−2.68237			−0.119839
$502$		0			0
$503$			3.57287i			0.159307i	0.996823	+	0.0796533i	$0.0253813\pi$
$503$			3.57287i			0.159307i	−0.996823	+	0.0796533i	$0.974619\pi$
$504$		0			0
$505$	14.4815			0.644417
$506$		0			0
$507$	11.1083			0.493339
$508$		0			0
$509$		−	0.547009i		−	0.0242457i	−0.999927	−	0.0121229i	$-0.996141\pi$
$509$		−	0.547009i		−	0.0242457i	0.999927	−	0.0121229i	$-0.00385892\pi$
$510$		0			0
$511$		−	0.805631i		−	0.0356390i
$512$		0			0
$513$	−4.33389	−	0.466292i	−0.191346	−	0.0205873i
$514$		0			0
$515$	6.52251			0.287416
$516$		0			0
$517$	−3.68593			−0.162107
$518$		0			0
$519$			12.6910i			0.557071i
$520$		0			0
$521$			3.45971i			0.151573i	0.997124	+	0.0757863i	$0.0241467\pi$
$521$			3.45971i			0.151573i	−0.997124	+	0.0757863i	$0.975853\pi$
$522$		0			0
$523$	24.7456			1.08205			0.541025	−	0.841006i	$-0.318036\pi$
$523$	24.7456			1.08205			0.541025	+	0.841006i	$0.318036\pi$
$524$		0			0
$525$			1.83519i			0.0800943i
$526$		0			0
$527$	−0.300746			−0.0131007
$528$		0			0
$529$	−41.5537			−1.80668
$530$		0			0
$531$	−7.83326			−0.339934
$532$		0			0
$533$	5.75453			0.249256
$534$		0			0
$535$	−15.9683			−0.690368
$536$		0			0
$537$	13.0063			0.561262
$538$		0			0
$539$		−	3.88231i		−	0.167223i
$540$		0			0
$541$	−11.0020			−0.473014			−0.236507	−	0.971630i	$-0.576003\pi$
$541$	−11.0020			−0.473014			−0.236507	+	0.971630i	$0.576003\pi$
$542$		0			0
$543$			7.88661i			0.338447i
$544$		0			0
$545$		−	16.0797i		−	0.688777i
$546$		0			0
$547$	−23.5573			−1.00724			−0.503619	−	0.863926i	$-0.667998\pi$
$547$	−23.5573			−1.00724			−0.503619	+	0.863926i	$0.667998\pi$
$548$		0			0
$549$	−4.17521			−0.178194
$550$		0			0
$551$	−1.43118	+	13.3019i	−0.0609705	+	0.566681i
$552$		0			0
$553$		−	7.06874i		−	0.300593i
$554$		0			0
$555$		−	5.43626i		−	0.230756i
$556$		0			0
$557$	9.31667			0.394760			0.197380	−	0.980327i	$-0.436757\pi$
$557$	9.31667			0.394760			0.197380	+	0.980327i	$0.436757\pi$
$558$		0			0
$559$	−14.7527			−0.623972
$560$		0			0
$561$			1.88420i			0.0795511i
$562$		0			0
$563$	31.7074			1.33631			0.668154	−	0.744023i	$-0.267085\pi$
$563$	31.7074			1.33631			0.668154	+	0.744023i	$0.267085\pi$
$564$		0			0
$565$			17.1903i			0.723201i
$566$		0			0
$567$		−	0.665758i		−	0.0279592i
$568$		0			0
$569$			19.2592i			0.807387i	0.914894	+	0.403694i	$0.132274\pi$
$569$			19.2592i			0.807387i	−0.914894	+	0.403694i	$0.867726\pi$
$570$		0			0
$571$		−	19.5757i		−	0.819217i	−0.912261	−	0.409609i	$-0.865665\pi$
$571$		−	19.5757i		−	0.819217i	0.912261	−	0.409609i	$-0.134335\pi$
$572$		0			0
$573$		−	4.06106i		−	0.169653i
$574$		0			0
$575$			22.1475i			0.923617i
$576$		0			0
$577$	4.06979			0.169428			0.0847138	−	0.996405i	$-0.473002\pi$
$577$	4.06979			0.169428			0.0847138	+	0.996405i	$0.473002\pi$
$578$		0			0
$579$			1.77237i			0.0736570i
$580$		0			0
$581$	1.22722			0.0509136
$582$		0			0
$583$	−2.47736			−0.102602
$584$		0			0
$585$		−	2.06006i		−	0.0851730i
$586$		0			0
$587$			26.6158i			1.09855i	0.835642	+	0.549275i	$0.185096\pi$
$587$			26.6158i			1.09855i	−0.835642	+	0.549275i	$0.814904\pi$
$588$		0			0
$589$	−0.409591	−	0.0440688i	−0.0168769	−	0.00181582i
$590$		0			0
$591$	0.851084			0.0350089
$592$		0			0
$593$	5.23541			0.214993			0.107496	−	0.994205i	$-0.465717\pi$
$593$	5.23541			0.214993			0.107496	+	0.994205i	$0.465717\pi$
$594$		0			0
$595$			3.17323i			0.130090i
$596$		0			0
$597$		−	1.78610i		−	0.0731002i
$598$		0			0
$599$	20.7600			0.848230			0.424115	−	0.905608i	$-0.360585\pi$
$599$	20.7600			0.848230			0.424115	+	0.905608i	$0.360585\pi$
$600$		0			0
$601$		−	11.0547i		−	0.450931i	−0.974251	−	0.225465i	$-0.927610\pi$
$601$		−	11.0547i		−	0.450931i	0.974251	−	0.225465i	$-0.0723902\pi$
$602$		0			0
$603$	−3.03026			−0.123402
$604$		0			0
$605$	15.9509			0.648495
$606$		0			0
$607$	−11.8896			−0.482584			−0.241292	−	0.970453i	$-0.577571\pi$
$607$	−11.8896			−0.482584			−0.241292	+	0.970453i	$0.577571\pi$
$608$		0			0
$609$	−2.04340			−0.0828027
$610$		0			0
$611$	−8.56185			−0.346375
$612$		0			0
$613$	−29.8139			−1.20417			−0.602085	−	0.798432i	$-0.705663\pi$
$613$	−29.8139			−1.20417			−0.602085	+	0.798432i	$0.705663\pi$
$614$		0			0
$615$			6.26681i			0.252702i
$616$		0			0
$617$	11.8426			0.476765			0.238382	−	0.971171i	$-0.423383\pi$
$617$	11.8426			0.476765			0.238382	+	0.971171i	$0.423383\pi$
$618$		0			0
$619$			43.5947i			1.75222i	0.482112	+	0.876110i	$0.339870\pi$
$619$			43.5947i			1.75222i	−0.482112	+	0.876110i	$0.660130\pi$
$620$		0			0
$621$		−	8.03453i		−	0.322415i
$622$		0			0
$623$	4.48148			0.179547
$624$		0			0
$625$	−3.61874			−0.144750
$626$		0			0
$627$	−0.276095	+	2.56613i	−0.0110262	+	0.102481i
$628$		0			0
$629$			11.5496i			0.460515i
$630$		0			0
$631$			22.8947i			0.911422i	0.890128	+	0.455711i	$0.150615\pi$
$631$			22.8947i			0.911422i	−0.890128	+	0.455711i	$0.849385\pi$
$632$		0			0
$633$	−5.23063			−0.207899
$634$		0			0
$635$	−9.26162			−0.367536
$636$		0			0
$637$		−	9.01802i		−	0.357307i
$638$		0			0
$639$	11.4897			0.454526
$640$		0			0
$641$			29.8752i			1.18000i	0.807404	+	0.589999i	$0.200872\pi$
$641$			29.8752i			1.18000i	−0.807404	+	0.589999i	$0.799128\pi$
$642$		0			0
$643$			35.5576i			1.40226i	0.713035	+	0.701128i	$0.247320\pi$
$643$			35.5576i			1.40226i	−0.713035	+	0.701128i	$0.752680\pi$
$644$		0			0
$645$		−	16.0660i		−	0.632599i
$646$		0			0
$647$			47.4811i			1.86668i	0.358999	+	0.933338i	$0.383118\pi$
$647$			47.4811i			1.86668i	−0.358999	+	0.933338i	$0.616882\pi$
$648$		0			0
$649$			4.63813i			0.182062i
$650$		0			0
$651$		−	0.0629201i		−	0.00246603i
$652$		0			0
$653$	−18.1893			−0.711802			−0.355901	−	0.934524i	$-0.615826\pi$
$653$	−18.1893			−0.711802			−0.355901	+	0.934524i	$0.615826\pi$
$654$		0			0
$655$		−	10.0528i		−	0.392795i
$656$		0			0
$657$	1.21010			0.0472103
$658$		0			0
$659$	5.48997			0.213859			0.106929	−	0.994267i	$-0.465898\pi$
$659$	5.48997			0.213859			0.106929	+	0.994267i	$0.465898\pi$
$660$		0			0
$661$		−	43.0068i		−	1.67277i	−0.548142	−	0.836385i	$-0.684665\pi$
$661$		−	43.0068i		−	1.67277i	0.548142	−	0.836385i	$-0.315335\pi$
$662$		0			0
$663$			4.37672i			0.169978i
$664$		0			0
$665$	−0.464979	+	4.32168i	−0.0180311	+	0.167588i
$666$		0			0
$667$	−24.6603			−0.954849
$668$		0			0
$669$	−18.6398			−0.720655
$670$		0			0
$671$			2.47217i			0.0954371i
$672$		0			0
$673$		−	11.4270i		−	0.440478i	−0.975446	−	0.220239i	$-0.929316\pi$
$673$		−	11.4270i		−	0.440478i	0.975446	−	0.220239i	$-0.0706838\pi$
$674$		0			0
$675$	−2.75654			−0.106099
$676$		0			0
$677$		−	22.3404i		−	0.858612i	−0.903159	−	0.429306i	$-0.858758\pi$
$677$		−	22.3404i		−	0.858612i	0.903159	−	0.429306i	$-0.141242\pi$
$678$		0			0
$679$	8.49694			0.326082
$680$		0			0
$681$	9.12713			0.349752
$682$		0			0
$683$	−38.3763			−1.46843			−0.734214	−	0.678918i	$-0.762449\pi$
$683$	−38.3763			−1.46843			−0.734214	+	0.678918i	$0.762449\pi$
$684$		0			0
$685$	−8.57445			−0.327613
$686$		0			0
$687$	5.58646			0.213137
$688$		0			0
$689$	−5.75453			−0.219230
$690$		0			0
$691$			18.7524i			0.713376i	0.934224	+	0.356688i	$0.116094\pi$
$691$			18.7524i			0.713376i	−0.934224	+	0.356688i	$0.883906\pi$
$692$		0			0
$693$	−0.394200			−0.0149744
$694$		0			0
$695$		−	6.17707i		−	0.234310i
$696$		0			0
$697$		−	13.3142i		−	0.504312i
$698$		0			0
$699$	26.1728			0.989946
$700$		0			0
$701$	−11.1772			−0.422158			−0.211079	−	0.977469i	$-0.567698\pi$
$701$	−11.1772			−0.422158			−0.211079	+	0.977469i	$0.567698\pi$
$702$		0			0
$703$	−1.69239	+	15.7296i	−0.0638296	+	0.593255i
$704$		0			0
$705$		−	9.32406i		−	0.351164i
$706$		0			0
$707$		−	6.43680i		−	0.242081i
$708$		0			0
$709$	29.7173			1.11606			0.558029	−	0.829822i	$-0.311558\pi$
$709$	29.7173			1.11606			0.558029	+	0.829822i	$0.311558\pi$
$710$		0			0
$711$	10.6176			0.398191
$712$		0			0
$713$		−	0.759335i		−	0.0284373i
$714$		0			0
$715$	−1.21978			−0.0456171
$716$		0			0
$717$		−	3.84844i		−	0.143722i
$718$		0			0
$719$			25.0957i			0.935912i	0.883752	+	0.467956i	$0.155009\pi$
$719$			25.0957i			0.935912i	−0.883752	+	0.467956i	$0.844991\pi$
$720$		0			0
$721$		−	2.89916i		−	0.107970i
$722$		0			0
$723$		−	16.1576i		−	0.600907i
$724$		0			0
$725$			8.46062i			0.314220i
$726$		0			0
$727$			4.58125i			0.169909i	0.996385	+	0.0849546i	$0.0270745\pi$
$727$			4.58125i			0.169909i	−0.996385	+	0.0849546i	$0.972925\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$			34.1332i			1.26246i
$732$		0			0
$733$	−10.4938			−0.387596			−0.193798	−	0.981041i	$-0.562081\pi$
$733$	−10.4938			−0.387596			−0.193798	+	0.981041i	$0.562081\pi$
$734$		0			0
$735$	9.82084			0.362247
$736$		0			0
$737$			1.79424i			0.0660916i
$738$		0			0
$739$			39.0976i			1.43823i	0.694891	+	0.719115i	$0.255453\pi$
$739$			39.0976i			1.43823i	−0.694891	+	0.719115i	$0.744547\pi$
$740$		0			0
$741$	−0.641327	+	5.96072i	−0.0235597	+	0.218973i
$742$		0			0
$743$	12.0470			0.441960			0.220980	−	0.975278i	$-0.429074\pi$
$743$	12.0470			0.441960			0.220980	+	0.975278i	$0.429074\pi$
$744$		0			0
$745$	14.5534			0.533195
$746$		0			0
$747$			1.84334i			0.0674444i
$748$		0			0
$749$			7.09766i			0.259343i
$750$		0			0
$751$	35.0097			1.27752			0.638760	−	0.769406i	$-0.279448\pi$
$751$	35.0097			1.27752			0.638760	+	0.769406i	$0.279448\pi$
$752$		0			0
$753$		−	11.9799i		−	0.436573i
$754$		0			0
$755$	14.0652			0.511885
$756$		0			0
$757$	49.5970			1.80263			0.901316	−	0.433162i	$-0.142602\pi$
$757$	49.5970			1.80263			0.901316	+	0.433162i	$0.142602\pi$
$758$		0			0
$759$	−4.75730			−0.172679
$760$		0			0
$761$	24.2017			0.877312			0.438656	−	0.898655i	$-0.355455\pi$
$761$	24.2017			0.877312			0.438656	+	0.898655i	$0.355455\pi$
$762$		0			0
$763$	−7.14718			−0.258745
$764$		0			0
$765$	−4.76635			−0.172328
$766$		0			0
$767$			10.7737i			0.389015i
$768$		0			0
$769$	−31.7205			−1.14387			−0.571935	−	0.820299i	$-0.693807\pi$
$769$	−31.7205			−1.14387			−0.571935	+	0.820299i	$0.693807\pi$
$770$		0			0
$771$			21.9312i			0.789832i
$772$		0			0
$773$		−	40.0757i		−	1.44142i	−0.693236	−	0.720711i	$-0.743816\pi$
$773$		−	40.0757i		−	1.44142i	0.693236	−	0.720711i	$-0.256184\pi$
$774$		0			0
$775$	−0.260518			−0.00935809
$776$		0			0
$777$	−2.41634			−0.0866857
$778$		0			0
$779$	1.95095	−	18.1328i	0.0699001	−	0.649676i
$780$		0			0
$781$		−	6.80314i		−	0.243436i
$782$		0			0
$783$		−	3.06928i		−	0.109687i
$784$		0			0
$785$	−33.6172			−1.19985
$786$		0			0
$787$	−14.7243			−0.524865			−0.262433	−	0.964950i	$-0.584525\pi$
$787$	−14.7243			−0.524865			−0.262433	+	0.964950i	$0.584525\pi$
$788$		0			0
$789$		−	21.6496i		−	0.770746i
$790$		0			0
$791$	7.64083			0.271677
$792$		0			0
$793$			5.74248i			0.203922i
$794$		0			0
$795$		−	6.26681i		−	0.222261i
$796$		0			0
$797$		−	1.06094i		−	0.0375805i	−0.999823	−	0.0187903i	$-0.994019\pi$
$797$		−	1.06094i		−	0.0375805i	0.999823	−	0.0187903i	$-0.00598147\pi$
$798$		0			0
$799$			19.8095i			0.700810i
$800$		0			0
$801$			6.73140i			0.237842i
$802$		0			0
$803$		−	0.716506i		−	0.0252850i
$804$		0			0
$805$	−8.01190			−0.282382
$806$		0			0
$807$			8.05506i			0.283552i
$808$		0			0
$809$	44.9029			1.57870			0.789350	−	0.613943i	$-0.210417\pi$
$809$	44.9029			1.57870			0.789350	+	0.613943i	$0.210417\pi$
$810$		0			0
$811$	13.3679			0.469409			0.234704	−	0.972067i	$-0.424588\pi$
$811$	13.3679			0.469409			0.234704	+	0.972067i	$0.424588\pi$
$812$		0			0
$813$			5.68805i			0.199488i
$814$		0			0
$815$		−	21.8662i		−	0.765939i
$816$		0			0
$817$	−5.00159	+	46.4865i	−0.174983	+	1.62636i
$818$		0			0
$819$	−0.915667			−0.0319960
$820$		0			0
$821$	4.80262			0.167613			0.0838063	−	0.996482i	$-0.473292\pi$
$821$	4.80262			0.167613			0.0838063	+	0.996482i	$0.473292\pi$
$822$		0			0
$823$		−	18.0542i		−	0.629329i	−0.949203	−	0.314664i	$-0.898108\pi$
$823$		−	18.0542i		−	0.629329i	0.949203	−	0.314664i	$-0.101892\pi$
$824$		0			0
$825$			1.63217i			0.0568249i
$826$		0			0
$827$	−4.70731			−0.163689			−0.0818445	−	0.996645i	$-0.526081\pi$
$827$	−4.70731			−0.163689			−0.0818445	+	0.996645i	$0.526081\pi$
$828$		0			0
$829$		−	31.7736i		−	1.10354i	−0.833996	−	0.551771i	$-0.813952\pi$
$829$		−	31.7736i		−	1.10354i	0.833996	−	0.551771i	$-0.186048\pi$
$830$		0			0
$831$	12.2360			0.424462
$832$		0			0
$833$	−20.8649			−0.722927
$834$		0			0
$835$	−4.01770			−0.139038
$836$		0			0
$837$	0.0945089			0.00326671
$838$		0			0
$839$	−11.9648			−0.413070			−0.206535	−	0.978439i	$-0.566219\pi$
$839$	−11.9648			−0.413070			−0.206535	+	0.978439i	$0.566219\pi$
$840$		0			0
$841$	19.5795			0.675155
$842$		0			0
$843$			9.66307i			0.332814i
$844$		0			0
$845$	16.6383			0.572374
$846$		0			0
$847$		−	7.08993i		−	0.243613i
$848$		0			0
$849$		−	12.2714i		−	0.421152i
$850$		0			0
$851$	−29.1610			−0.999626
$852$		0			0
$853$	30.1716			1.03306			0.516528	−	0.856270i	$-0.327224\pi$
$853$	30.1716			1.03306			0.516528	+	0.856270i	$0.327224\pi$
$854$		0			0
$855$	−6.49137	−	0.698421i	−0.222000	−	0.0238855i
$856$		0			0
$857$			45.5018i			1.55431i	0.629309	+	0.777156i	$0.283338\pi$
$857$			45.5018i			1.55431i	−0.629309	+	0.777156i	$0.716662\pi$
$858$		0			0
$859$			28.7358i			0.980453i	0.871595	+	0.490226i	$0.163086\pi$
$859$			28.7358i			0.980453i	−0.871595	+	0.490226i	$0.836914\pi$
$860$		0			0
$861$	2.78551			0.0949298
$862$		0			0
$863$	−44.0133			−1.49823			−0.749115	−	0.662440i	$-0.769521\pi$
$863$	−44.0133			−1.49823			−0.749115	+	0.662440i	$0.769521\pi$
$864$		0			0
$865$			19.0087i			0.646316i
$866$		0			0
$867$	−6.87362			−0.233440
$868$		0			0
$869$		−	6.28675i		−	0.213263i
$870$		0			0
$871$			4.16774i			0.141219i
$872$		0			0
$873$			12.7628i			0.431955i
$874$		0			0
$875$			7.73470i			0.261480i
$876$		0			0
$877$		−	44.3517i		−	1.49765i	−0.662767	−	0.748826i	$-0.730618\pi$
$877$		−	44.3517i		−	1.49765i	0.662767	−	0.748826i	$-0.269382\pi$
$878$		0			0
$879$			32.9593i			1.11169i
$880$		0			0
$881$	39.1569			1.31923			0.659615	−	0.751604i	$-0.270719\pi$
$881$	39.1569			1.31923			0.659615	+	0.751604i	$0.270719\pi$
$882$		0			0
$883$			39.1304i			1.31684i	0.752650	+	0.658421i	$0.228775\pi$
$883$			39.1304i			1.31684i	−0.752650	+	0.658421i	$0.771225\pi$
$884$		0			0
$885$	−11.7328			−0.394393
$886$		0			0
$887$	42.4504			1.42535			0.712673	−	0.701496i	$-0.247484\pi$
$887$	42.4504			1.42535			0.712673	+	0.701496i	$0.247484\pi$
$888$		0			0
$889$			4.11665i			0.138068i
$890$		0			0
$891$		−	0.592107i		−	0.0198363i
$892$		0			0
$893$	−2.90272	+	26.9789i	−0.0971357	+	0.902814i
$894$		0			0
$895$	19.4810			0.651179
$896$		0			0
$897$	−11.0505			−0.368966
$898$		0			0
$899$		−	0.290075i		−	0.00967454i
$900$		0			0
$901$			13.3142i			0.443561i
$902$		0			0
$903$	−7.14111			−0.237641
$904$		0			0
$905$			11.8127i			0.392667i
$906$		0			0
$907$	15.4407			0.512698			0.256349	−	0.966584i	$-0.417480\pi$
$907$	15.4407			0.512698			0.256349	+	0.966584i	$0.417480\pi$
$908$		0			0
$909$	9.66838			0.320680
$910$		0			0
$911$	47.5096			1.57406			0.787032	−	0.616913i	$-0.211617\pi$
$911$	47.5096			1.57406			0.787032	+	0.616913i	$0.211617\pi$
$912$		0			0
$913$	1.09146			0.0361219
$914$		0			0
$915$	−6.25370			−0.206741
$916$		0			0
$917$	−4.46831			−0.147557
$918$		0			0
$919$		−	3.62653i		−	0.119628i	−0.998210	−	0.0598141i	$-0.980949\pi$
$919$		−	3.62653i		−	0.119628i	0.998210	−	0.0598141i	$-0.0190508\pi$
$920$		0			0
$921$	−6.55235			−0.215907
$922$		0			0
$923$		−	15.8027i		−	0.520151i
$924$		0			0
$925$			10.0048i			0.328954i
$926$		0			0
$927$	4.35468			0.143026
$928$		0			0
$929$	−28.4344			−0.932903			−0.466452	−	0.884547i	$-0.654468\pi$
$929$	−28.4344			−0.932903			−0.466452	+	0.884547i	$0.654468\pi$
$930$		0			0
$931$	−28.4163	−	3.05737i	−0.931306	−	0.100201i
$932$		0			0
$933$			20.9039i			0.684362i
$934$		0			0
$935$			2.82219i			0.0922955i
$936$		0			0
$937$	−26.5453			−0.867197			−0.433598	−	0.901106i	$-0.642756\pi$
$937$	−26.5453			−0.867197			−0.433598	+	0.901106i	$0.642756\pi$
$938$		0			0
$939$	−4.35540			−0.142133
$940$		0			0
$941$			8.39450i			0.273653i	0.990595	+	0.136826i	$0.0436903\pi$
$941$			8.39450i			0.273653i	−0.990595	+	0.136826i	$0.956310\pi$
$942$		0			0
$943$	33.6162			1.09469
$944$		0			0
$945$		−	0.997183i		−	0.0324384i
$946$		0			0
$947$			34.9126i			1.13451i	0.823543	+	0.567254i	$0.191994\pi$
$947$			34.9126i			1.13451i	−0.823543	+	0.567254i	$0.808006\pi$
$948$		0			0
$949$		−	1.66434i		−	0.0540266i
$950$		0			0
$951$			15.3719i			0.498470i
$952$		0			0
$953$		−	33.2442i		−	1.07689i	−0.842662	−	0.538443i	$-0.819013\pi$
$953$		−	33.2442i		−	1.07689i	0.842662	−	0.538443i	$-0.180987\pi$
$954$		0			0
$955$		−	6.08273i		−	0.196832i
$956$		0			0
$957$	−1.81735			−0.0587464
$958$		0			0
$959$			3.81122i			0.123071i
$960$		0			0
$961$	−30.9911			−0.999712
$962$		0			0
$963$	−10.6610			−0.343547
$964$		0			0
$965$			2.65468i			0.0854571i
$966$		0			0
$967$		−	51.6769i		−	1.66182i	−0.556409	−	0.830908i	$-0.687821\pi$
$967$		−	51.6769i		−	1.66182i	0.556409	−	0.830908i	$-0.312179\pi$
$968$		0			0
$969$	13.7913	+	1.48383i	0.443040	+	0.0476676i
$970$		0			0
$971$	−49.2613			−1.58087			−0.790436	−	0.612545i	$-0.790146\pi$
$971$	−49.2613			−1.58087			−0.790436	+	0.612545i	$0.790146\pi$
$972$		0			0
$973$	−2.74562			−0.0880205
$974$		0			0
$975$			3.79128i			0.121418i
$976$		0			0
$977$			49.1842i			1.57354i	0.617245	+	0.786771i	$0.288249\pi$
$977$			49.1842i			1.57354i	−0.617245	+	0.786771i	$0.711751\pi$
$978$		0			0
$979$	3.98571			0.127384
$980$		0			0
$981$		−	10.7354i		−	0.342755i
$982$		0			0
$983$	4.54411			0.144934			0.0724672	−	0.997371i	$-0.476913\pi$
$983$	4.54411			0.144934			0.0724672	+	0.997371i	$0.476913\pi$
$984$		0			0
$985$	1.27477			0.0406175
$986$		0			0
$987$	−4.14441			−0.131918
$988$		0			0
$989$	−86.1807			−2.74039
$990$		0			0
$991$	51.1756			1.62565			0.812823	−	0.582511i	$-0.197930\pi$
$991$	51.1756			1.62565			0.812823	+	0.582511i	$0.197930\pi$
$992$		0			0
$993$	11.6836			0.370767
$994$		0			0
$995$		−	2.67525i		−	0.0848112i
$996$		0			0
$997$	12.6620			0.401008			0.200504	−	0.979693i	$-0.435742\pi$
$997$	12.6620			0.401008			0.200504	+	0.979693i	$0.435742\pi$
$998$		0			0
$999$		−	3.62946i		−	0.114831i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3648.2.k.l.2431.13		20
4.3	odd	2		3648.2.k.k.2431.14		20
8.3	odd	2		1824.2.k.b.607.8	yes	20
8.5	even	2		1824.2.k.a.607.7	✓	20
19.18	odd	2		3648.2.k.k.2431.13		20
24.5	odd	2		5472.2.k.a.2431.13		20
24.11	even	2		5472.2.k.b.2431.14		20
76.75	even	2	inner	3648.2.k.l.2431.14		20
152.37	odd	2		1824.2.k.b.607.7	yes	20
152.75	even	2		1824.2.k.a.607.8	yes	20
456.227	odd	2		5472.2.k.a.2431.14		20
456.341	even	2		5472.2.k.b.2431.13		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1824.2.k.a.607.7	✓	20	8.5	even	2
1824.2.k.a.607.8	yes	20	152.75	even	2
1824.2.k.b.607.7	yes	20	152.37	odd	2
1824.2.k.b.607.8	yes	20	8.3	odd	2
3648.2.k.k.2431.13		20	19.18	odd	2
3648.2.k.k.2431.14		20	4.3	odd	2
3648.2.k.l.2431.13		20	1.1	even	1	trivial
3648.2.k.l.2431.14		20	76.75	even	2	inner
5472.2.k.a.2431.13		20	24.5	odd	2
5472.2.k.a.2431.14		20	456.227	odd	2
5472.2.k.b.2431.13		20	456.341	even	2
5472.2.k.b.2431.14		20	24.11	even	2

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:	\( N \)	\(=\)	\( 3648 = 2^{6} \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3648.k (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.49782 q^{5} -0.665758i q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.49782 q^{5} -0.665758i q^{7} +1.00000 q^{9} -0.592107i q^{11} -1.37538i q^{13} +1.49782 q^{15} -3.18220 q^{17} +(-4.33389 - 0.466292i) q^{19} -0.665758i q^{21} -8.03453i q^{23} -2.75654 q^{25} +1.00000 q^{27} -3.06928i q^{29} +0.0945089 q^{31} -0.592107i q^{33} -0.997183i q^{35} -3.62946i q^{37} -1.37538i q^{39} +4.18397i q^{41} -10.7263i q^{43} +1.49782 q^{45} -6.22510i q^{47} +6.55677 q^{49} -3.18220 q^{51} -4.18397i q^{53} -0.886868i q^{55} +(-4.33389 - 0.466292i) q^{57} -7.83326 q^{59} -4.17521 q^{61} -0.665758i q^{63} -2.06006i q^{65} -3.03026 q^{67} -8.03453i q^{69} +11.4897 q^{71} +1.21010 q^{73} -2.75654 q^{75} -0.394200 q^{77} +10.6176 q^{79} +1.00000 q^{81} +1.84334i q^{83} -4.76635 q^{85} -3.06928i q^{87} +6.73140i q^{89} -0.915667 q^{91} +0.0945089 q^{93} +(-6.49137 - 0.698421i) q^{95} +12.7628i q^{97} -0.592107i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 20 q^{3} + 20 q^{9}+O(q^{10}) \) 20 * q + 20 * q^3 + 20 * q^9	\( 20 q + 20 q^{3} + 20 q^{9} + 4 q^{19} + 20 q^{25} + 20 q^{27} + 8 q^{31} - 44 q^{49} + 4 q^{57} + 8 q^{61} + 8 q^{67} - 64 q^{71} + 8 q^{73} + 20 q^{75} - 40 q^{77} + 16 q^{79} + 20 q^{81} - 40 q^{85} + 8 q^{93} + 8 q^{95}+O(q^{100}) \) 20 * q + 20 * q^3 + 20 * q^9 + 4 * q^19 + 20 * q^25 + 20 * q^27 + 8 * q^31 - 44 * q^49 + 4 * q^57 + 8 * q^61 + 8 * q^67 - 64 * q^71 + 8 * q^73 + 20 * q^75 - 40 * q^77 + 16 * q^79 + 20 * q^81 - 40 * q^85 + 8 * q^93 + 8 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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