LMFDB - Embedded newform 2352.2.h.n.2255.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2352,2,Mod(2255,2352)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2352, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2352.2255");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2352 = 2^{4} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2352.h (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:	$18.7808145554$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.8275904784.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} + 5x^{6} - 6x^{5} + 4x^{4} - 18x^{3} + 45x^{2} - 81x + 81 $ x^8 - 3x^7 + 5x^6 - 6x^5 + 4x^4 - 18x^3 + 45x^2 - 81*x + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 336)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2255.6
Root			$1.73142 + 0.0465589i$ of defining polynomial
Character	$\chi$	$=$	2352.2255
Dual form			2352.2.h.n.2255.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.825391 + 1.52274i) q^{3} -3.94333i q^{5} +(-1.63746 + 2.51371i) q^{9} +O(q^{10})$	$q+(0.825391 + 1.52274i) q^{3} -3.94333i q^{5} +(-1.63746 + 2.51371i) q^{9} -3.52848 q^{11} +2.00000 q^{13} +(6.00465 - 3.25479i) q^{15} -5.02742i q^{17} +3.04547i q^{19} -1.65078 q^{23} -10.5498 q^{25} +(-5.17926 - 0.418627i) q^{27} -6.80257i q^{29} +1.73205i q^{31} +(-2.91238 - 5.37295i) q^{33} +1.27492 q^{37} +(1.65078 + 3.04547i) q^{39} -2.16818i q^{41} +0.837253i q^{43} +(9.91238 + 6.45704i) q^{45} +4.95235 q^{47} +(7.65544 - 4.14959i) q^{51} -9.66181i q^{53} +13.9140i q^{55} +(-4.63746 + 2.51371i) q^{57} -10.1316 q^{59} -11.2749 q^{61} -7.88666i q^{65} -16.0646i q^{67} +(-1.36254 - 2.51371i) q^{69} -10.3585 q^{71} -7.27492 q^{73} +(-8.70774 - 16.0646i) q^{75} -5.19615i q^{79} +(-3.63746 - 8.23219i) q^{81} -5.17926 q^{83} -19.8248 q^{85} +(10.3585 - 5.61478i) q^{87} +7.19559i q^{89} +(-2.63746 + 1.42962i) q^{93} +12.0093 q^{95} +4.27492 q^{97} +(5.77774 - 8.86957i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 2 q^{9}+O(q^{10}) $ 8 * q + 2 * q^9	$ 8 q + 2 q^{9} + 16 q^{13} - 24 q^{25} + 22 q^{33} - 20 q^{37} + 34 q^{45} - 22 q^{57} - 60 q^{61} - 26 q^{69} - 28 q^{73} - 14 q^{81} - 68 q^{85} - 6 q^{93} + 4 q^{97}+O(q^{100}) $ 8 * q + 2 * q^9 + 16 * q^13 - 24 * q^25 + 22 * q^33 - 20 * q^37 + 34 * q^45 - 22 * q^57 - 60 * q^61 - 26 * q^69 - 28 * q^73 - 14 * q^81 - 68 * q^85 - 6 * q^93 + 4 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2352\mathbb{Z}\right)^\times$.

$n$	$785$	$1471$	$1765$	$2257$
$\chi(n)$	$-1$	$-1$	$1$	$1$

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
$3$	0.825391	+	1.52274i	0.476540	+	0.879153i
$3$	0.825391	+	1.52274i	0.476540	+	0.879153i
$4$		0			0
$5$		−	3.94333i		−	1.76351i	−0.471708	−	0.881755i	$-0.656362\pi$
$5$		−	3.94333i		−	1.76351i	0.471708	−	0.881755i	$-0.343638\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	−1.63746	+	2.51371i	−0.545820	+	0.837903i
$10$		0			0
$11$	−3.52848			−1.06388			−0.531938	−	0.846783i	$-0.678536\pi$
$11$	−3.52848			−1.06388			−0.531938	+	0.846783i	$0.678536\pi$
$12$		0			0
$13$	2.00000			0.554700			0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000			0.554700			0.277350	+	0.960769i	$0.410544\pi$
$14$		0			0
$15$	6.00465	−	3.25479i	1.55039	−	0.840383i
$16$		0			0
$17$		−	5.02742i		−	1.21933i	−0.792660	−	0.609664i	$-0.791304\pi$
$17$		−	5.02742i		−	1.21933i	0.792660	−	0.609664i	$-0.208696\pi$
$18$		0			0
$19$			3.04547i			0.698680i	0.936996	+	0.349340i	$0.113594\pi$
$19$			3.04547i			0.698680i	−0.936996	+	0.349340i	$0.886406\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−1.65078			−0.344212			−0.172106	−	0.985078i	$-0.555057\pi$
$23$	−1.65078			−0.344212			−0.172106	+	0.985078i	$0.555057\pi$
$24$		0			0
$25$	−10.5498			−2.10997
$26$		0			0
$27$	−5.17926	−	0.418627i	−0.996749	−	0.0805648i
$28$		0			0
$29$		−	6.80257i		−	1.26320i	−0.775292	−	0.631602i	$-0.782397\pi$
$29$		−	6.80257i		−	1.26320i	0.775292	−	0.631602i	$-0.217603\pi$
$30$		0			0
$31$			1.73205i			0.311086i	0.987829	+	0.155543i	$0.0497126\pi$
$31$			1.73205i			0.311086i	−0.987829	+	0.155543i	$0.950287\pi$
$32$		0			0
$33$	−2.91238	−	5.37295i	−0.506980	−	0.935310i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	1.27492			0.209595			0.104798	−	0.994494i	$-0.466581\pi$
$37$	1.27492			0.209595			0.104798	+	0.994494i	$0.466581\pi$
$38$		0			0
$39$	1.65078	+	3.04547i	0.264337	+	0.487666i
$40$		0			0
$41$		−	2.16818i		−	0.338612i	−0.985563	−	0.169306i	$-0.945847\pi$
$41$		−	2.16818i		−	0.338612i	0.985563	−	0.169306i	$-0.0541527\pi$
$42$		0			0
$43$			0.837253i			0.127680i	0.997960	+	0.0638400i	$0.0203347\pi$
$43$			0.837253i			0.127680i	−0.997960	+	0.0638400i	$0.979665\pi$
$44$		0			0
$45$	9.91238	+	6.45704i	1.47765	+	0.962558i
$46$		0			0
$47$	4.95235			0.722374			0.361187	−	0.932493i	$-0.382372\pi$
$47$	4.95235			0.722374			0.361187	+	0.932493i	$0.382372\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	7.65544	−	4.14959i	1.07198	−	0.581058i
$52$		0			0
$53$		−	9.66181i		−	1.32715i	−0.748109	−	0.663576i	$-0.769038\pi$
$53$		−	9.66181i		−	1.32715i	0.748109	−	0.663576i	$-0.230962\pi$
$54$		0			0
$55$			13.9140i			1.87616i
$56$		0			0
$57$	−4.63746	+	2.51371i	−0.614246	+	0.332949i
$58$		0			0
$59$	−10.1316			−1.31902			−0.659512	−	0.751694i	$-0.729237\pi$
$59$	−10.1316			−1.31902			−0.659512	+	0.751694i	$0.729237\pi$
$60$		0			0
$61$	−11.2749			−1.44361			−0.721803	−	0.692099i	$-0.756686\pi$
$61$	−11.2749			−1.44361			−0.721803	+	0.692099i	$0.756686\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	7.88666i		−	0.978219i
$66$		0			0
$67$		−	16.0646i		−	1.96261i	−0.192468	−	0.981303i	$-0.561649\pi$
$67$		−	16.0646i		−	1.96261i	0.192468	−	0.981303i	$-0.438351\pi$
$68$		0			0
$69$	−1.36254	−	2.51371i	−0.164031	−	0.302615i
$70$		0			0
$71$	−10.3585			−1.22933			−0.614665	−	0.788788i	$-0.710709\pi$
$71$	−10.3585			−1.22933			−0.614665	+	0.788788i	$0.710709\pi$
$72$		0			0
$73$	−7.27492			−0.851465			−0.425732	−	0.904849i	$-0.639983\pi$
$73$	−7.27492			−0.851465			−0.425732	+	0.904849i	$0.639983\pi$
$74$		0			0
$75$	−8.70774	−	16.0646i	−1.00548	−	1.85498i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		−	5.19615i		−	0.584613i	−0.956325	−	0.292306i	$-0.905577\pi$
$79$		−	5.19615i		−	0.584613i	0.956325	−	0.292306i	$-0.0944227\pi$
$80$		0			0
$81$	−3.63746	−	8.23219i	−0.404162	−	0.914687i
$82$		0			0
$83$	−5.17926			−0.568498			−0.284249	−	0.958751i	$-0.591744\pi$
$83$	−5.17926			−0.568498			−0.284249	+	0.958751i	$0.591744\pi$
$84$		0			0
$85$	−19.8248			−2.15030
$86$		0			0
$87$	10.3585	−	5.61478i	1.11055	−	0.601968i
$88$		0			0
$89$			7.19559i			0.762731i	0.924424	+	0.381366i	$0.124546\pi$
$89$			7.19559i			0.762731i	−0.924424	+	0.381366i	$0.875454\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−2.63746	+	1.42962i	−0.273492	+	0.148245i
$94$		0			0
$95$	12.0093			1.23213
$96$		0			0
$97$	4.27492			0.434052			0.217026	−	0.976166i	$-0.430364\pi$
$97$	4.27492			0.434052			0.217026	+	0.976166i	$0.430364\pi$
$98$		0			0
$99$	5.77774	−	8.86957i	0.580685	−	0.891425i
$100$		0			0
$101$			12.9141i			1.28500i	0.766286	+	0.642499i	$0.222102\pi$
$101$			12.9141i			1.28500i	−0.766286	+	0.642499i	$0.777898\pi$
$102$		0			0
$103$		−	4.83507i		−	0.476414i	−0.971214	−	0.238207i	$-0.923440\pi$
$103$		−	4.83507i		−	0.476414i	0.971214	−	0.238207i	$-0.0765596\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−0.226914			−0.0219366			−0.0109683	−	0.999940i	$-0.503491\pi$
$107$	−0.226914			−0.0219366			−0.0109683	+	0.999940i	$0.503491\pi$
$108$		0			0
$109$	−7.82475			−0.749475			−0.374738	−	0.927131i	$-0.622267\pi$
$109$	−7.82475			−0.749475			−0.374738	+	0.927131i	$0.622267\pi$
$110$		0			0
$111$	1.05231	+	1.94136i	0.0998804	+	0.184266i
$112$		0			0
$113$			2.16818i			0.203965i	0.994786	+	0.101982i	$0.0325186\pi$
$113$			2.16818i			0.203965i	−0.994786	+	0.101982i	$0.967481\pi$
$114$		0			0
$115$			6.50958i			0.607021i
$116$		0			0
$117$	−3.27492	+	5.02742i	−0.302766	+	0.464785i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	1.45017			0.131833
$122$		0			0
$123$	3.30156	−	1.78959i	0.297692	−	0.161362i
$124$		0			0
$125$			21.8848i			1.95744i
$126$		0			0
$127$			4.77753i			0.423937i	0.977277	+	0.211968i	$0.0679874\pi$
$127$			4.77753i			0.423937i	−0.977277	+	0.211968i	$0.932013\pi$
$128$		0			0
$129$	−1.27492	+	0.691062i	−0.112250	+	0.0608446i
$130$		0			0
$131$	3.52848			0.308285			0.154142	−	0.988049i	$-0.450739\pi$
$131$	3.52848			0.308285			0.154142	+	0.988049i	$0.450739\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	−1.65078	+	20.4235i	−0.142077	+	1.75778i
$136$		0			0
$137$		−	0.691062i		−	0.0590414i	−0.999564	−	0.0295207i	$-0.990602\pi$
$137$		−	0.691062i		−	0.0590414i	0.999564	−	0.0295207i	$-0.00939809\pi$
$138$		0			0
$139$		−	20.8997i		−	1.77269i	−0.463026	−	0.886345i	$-0.653236\pi$
$139$		−	20.8997i		−	1.77269i	0.463026	−	0.886345i	$-0.346764\pi$
$140$		0			0
$141$	4.08762	+	7.54112i	0.344240	+	0.635077i
$142$		0			0
$143$	−7.05696			−0.590133
$144$		0			0
$145$	−26.8248			−2.22767
$146$		0			0
$147$		0			0
$148$		0			0
$149$			7.19559i			0.589486i	0.955577	+	0.294743i	$0.0952340\pi$
$149$			7.19559i			0.589486i	−0.955577	+	0.294743i	$0.904766\pi$
$150$		0			0
$151$		−	0.894797i		−	0.0728176i	−0.999337	−	0.0364088i	$-0.988408\pi$
$151$		−	0.894797i		−	0.0728176i	0.999337	−	0.0364088i	$-0.0115918\pi$
$152$		0			0
$153$	12.6375	+	8.23219i	1.02168	+	0.665533i
$154$		0			0
$155$	6.83004			0.548602
$156$		0			0
$157$	14.3746			1.14722			0.573608	−	0.819130i	$-0.305543\pi$
$157$	14.3746			1.14722			0.573608	+	0.819130i	$0.305543\pi$
$158$		0			0
$159$	14.7124	−	7.97477i	1.16677	−	0.632440i
$160$		0			0
$161$		0			0
$162$		0			0
$163$		−	6.50958i		−	0.509869i	−0.966958	−	0.254935i	$-0.917946\pi$
$163$		−	6.50958i		−	0.509869i	0.966958	−	0.254935i	$-0.0820540\pi$
$164$		0			0
$165$	−21.1873	+	11.4845i	−1.64943	+	0.894063i
$166$		0			0
$167$	14.1139			1.09217			0.546084	−	0.837731i	$-0.316118\pi$
$167$	14.1139			1.09217			0.546084	+	0.837731i	$0.316118\pi$
$168$		0			0
$169$	−9.00000			−0.692308
$170$		0			0
$171$	−7.65544	−	4.98684i	−0.585426	−	0.381353i
$172$		0			0
$173$		−	8.57772i		−	0.652152i	−0.945344	−	0.326076i	$-0.894273\pi$
$173$		−	8.57772i		−	0.652152i	0.945344	−	0.326076i	$-0.105727\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−8.36254	−	15.4278i	−0.628567	−	1.15962i
$178$		0			0
$179$	8.25391			0.616926			0.308463	−	0.951236i	$-0.400185\pi$
$179$	8.25391			0.616926			0.308463	+	0.951236i	$0.400185\pi$
$180$		0			0
$181$	15.0997			1.12235			0.561175	−	0.827697i	$-0.310350\pi$
$181$	15.0997			1.12235			0.561175	+	0.827697i	$0.310350\pi$
$182$		0			0
$183$	−9.30622	−	17.1687i	−0.687935	−	1.26915i
$184$		0			0
$185$		−	5.02742i		−	0.369623i
$186$		0			0
$187$			17.7391i			1.29721i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−18.6124			−1.34675			−0.673374	−	0.739302i	$-0.735156\pi$
$191$	−18.6124			−1.34675			−0.673374	+	0.739302i	$0.735156\pi$
$192$		0			0
$193$	12.0997			0.870953			0.435477	−	0.900200i	$-0.356580\pi$
$193$	12.0997			0.870953			0.435477	+	0.900200i	$0.356580\pi$
$194$		0			0
$195$	12.0093	−	6.50958i	0.860004	−	0.466160i
$196$		0			0
$197$		−	17.9415i		−	1.27828i	−0.769091	−	0.639139i	$-0.779291\pi$
$197$		−	17.9415i		−	1.27828i	0.769091	−	0.639139i	$-0.220709\pi$
$198$		0			0
$199$			21.2032i			1.50306i	0.659700	+	0.751529i	$0.270683\pi$
$199$			21.2032i			1.50306i	−0.659700	+	0.751529i	$0.729317\pi$
$200$		0			0
$201$	24.4622	−	13.2596i	1.72543	−	0.935260i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−8.54983			−0.597146
$206$		0			0
$207$	2.70309	−	4.14959i	0.187878	−	0.288416i
$208$		0			0
$209$		−	10.7459i		−	0.743309i
$210$		0			0
$211$		−	1.78959i		−	0.123201i	−0.998101	−	0.0616004i	$-0.980380\pi$
$211$		−	1.78959i		−	0.123201i	0.998101	−	0.0616004i	$-0.0196204\pi$
$212$		0			0
$213$	−8.54983	−	15.7733i	−0.585825	−	1.08077i
$214$		0			0
$215$	3.30156			0.225165
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−6.00465	−	11.0778i	−0.405757	−	0.748568i
$220$		0			0
$221$		−	10.0548i		−	0.676361i
$222$		0			0
$223$			10.0312i			0.671740i	0.941908	+	0.335870i	$0.109030\pi$
$223$			10.0312i			0.671740i	−0.941908	+	0.335870i	$0.890970\pi$
$224$		0			0
$225$	17.2749	−	26.5192i	1.15166	−	1.76795i
$226$		0			0
$227$	27.5471			1.82836			0.914182	−	0.405303i	$-0.132834\pi$
$227$	27.5471			1.82836			0.914182	+	0.405303i	$0.132834\pi$
$228$		0			0
$229$	8.72508			0.576570			0.288285	−	0.957545i	$-0.406915\pi$
$229$	8.72508			0.576570			0.288285	+	0.957545i	$0.406915\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			0.691062i			0.0452730i	0.999744	+	0.0226365i	$0.00720603\pi$
$233$			0.691062i			0.0452730i	−0.999744	+	0.0226365i	$0.992794\pi$
$234$		0			0
$235$		−	19.5287i		−	1.27391i
$236$		0			0
$237$	7.91238	−	4.28886i	0.513964	−	0.278591i
$238$		0			0
$239$	16.9617			1.09716			0.548579	−	0.836099i	$-0.315169\pi$
$239$	16.9617			1.09716			0.548579	+	0.836099i	$0.315169\pi$
$240$		0			0
$241$	7.00000			0.450910			0.225455	−	0.974254i	$-0.427613\pi$
$241$	7.00000			0.450910			0.225455	+	0.974254i	$0.427613\pi$
$242$		0			0
$243$	9.53313	−	12.3337i	0.611551	−	0.791205i
$244$		0			0
$245$		0			0
$246$		0			0
$247$			6.09095i			0.387558i
$248$		0			0
$249$	−4.27492	−	7.88666i	−0.270912	−	0.499796i
$250$		0			0
$251$	11.7824			0.743698			0.371849	−	0.928293i	$-0.378724\pi$
$251$	11.7824			0.743698			0.371849	+	0.928293i	$0.378724\pi$
$252$		0			0
$253$	5.82475			0.366199
$254$		0			0
$255$	−16.3632	−	30.1879i	−1.02470	−	1.89044i
$256$		0			0
$257$		−	17.2504i		−	1.07605i	−0.842928	−	0.538026i	$-0.819170\pi$
$257$		−	17.2504i		−	1.07605i	0.842928	−	0.538026i	$-0.180830\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	17.0997	+	11.1389i	1.05844	+	0.689482i
$262$		0			0
$263$	−21.9140			−1.35128			−0.675638	−	0.737234i	$-0.736132\pi$
$263$	−21.9140			−1.35128			−0.675638	+	0.737234i	$0.736132\pi$
$264$		0			0
$265$	−38.0997			−2.34044
$266$		0			0
$267$	−10.9570	+	5.93918i	−0.670558	+	0.363472i
$268$		0			0
$269$			3.94333i			0.240429i	0.992748	+	0.120214i	$0.0383582\pi$
$269$			3.94333i			0.240429i	−0.992748	+	0.120214i	$0.961642\pi$
$270$		0			0
$271$			3.40656i			0.206934i	0.994633	+	0.103467i	$0.0329936\pi$
$271$			3.40656i			0.206934i	−0.994633	+	0.103467i	$0.967006\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	37.2249			2.24474
$276$		0			0
$277$	13.2749			0.797612			0.398806	−	0.917035i	$-0.369425\pi$
$277$	13.2749			0.797612			0.398806	+	0.917035i	$0.369425\pi$
$278$		0			0
$279$	−4.35387	−	2.83616i	−0.260659	−	0.169797i
$280$		0			0
$281$			15.7733i			0.940957i	0.882411	+	0.470478i	$0.155919\pi$
$281$			15.7733i			0.940957i	−0.882411	+	0.470478i	$0.844081\pi$
$282$		0			0
$283$			20.4811i			1.21747i	0.793372	+	0.608737i	$0.208323\pi$
$283$			20.4811i			1.21747i	−0.793372	+	0.608737i	$0.791677\pi$
$284$		0			0
$285$	9.91238	+	18.2870i	0.587158	+	1.08323i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−8.27492			−0.486760
$290$		0			0
$291$	3.52848	+	6.50958i	0.206843	+	0.381598i
$292$		0			0
$293$		−	13.3071i		−	0.777409i	−0.921362	−	0.388705i	$-0.872923\pi$
$293$		−	13.3071i		−	0.777409i	0.921362	−	0.388705i	$-0.127077\pi$
$294$		0			0
$295$			39.9523i			2.32611i
$296$		0			0
$297$	18.2749	+	1.47712i	1.06042	+	0.0857109i
$298$		0			0
$299$	−3.30156			−0.190934
$300$		0			0
$301$		0			0
$302$		0			0
$303$	−19.6647	+	10.6592i	−1.12971	+	0.612353i
$304$		0			0
$305$			44.4607i			2.54581i
$306$		0			0
$307$			17.3205i			0.988534i	0.869310	+	0.494267i	$0.164563\pi$
$307$			17.3205i			0.988534i	−0.869310	+	0.494267i	$0.835437\pi$
$308$		0			0
$309$	7.36254	−	3.99082i	0.418840	−	0.227030i
$310$		0			0
$311$	22.3678			1.26836			0.634182	−	0.773184i	$-0.281337\pi$
$311$	22.3678			1.26836			0.634182	+	0.773184i	$0.281337\pi$
$312$		0			0
$313$	−4.45017			−0.251538			−0.125769	−	0.992060i	$-0.540140\pi$
$313$	−4.45017			−0.251538			−0.125769	+	0.992060i	$0.540140\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	6.11151i		−	0.343256i	−0.985162	−	0.171628i	$-0.945097\pi$
$317$		−	6.11151i		−	0.343256i	0.985162	−	0.171628i	$-0.0549028\pi$
$318$		0			0
$319$			24.0027i			1.34389i
$320$		0			0
$321$	−0.187293	−	0.345531i	−0.0104537	−	0.0192857i
$322$		0			0
$323$	15.3109			0.851920
$324$		0			0
$325$	−21.0997			−1.17040
$326$		0			0
$327$	−6.45848	−	11.9150i	−0.357155	−	0.658903i
$328$		0			0
$329$		0			0
$330$		0			0
$331$		−	28.2465i		−	1.55257i	−0.630382	−	0.776285i	$-0.717102\pi$
$331$		−	28.2465i		−	1.55257i	0.630382	−	0.776285i	$-0.282898\pi$
$332$		0			0
$333$	−2.08762	+	3.20477i	−0.114401	+	0.175620i
$334$		0			0
$335$	−63.3481			−3.46108
$336$		0			0
$337$	−20.8248			−1.13440			−0.567198	−	0.823581i	$-0.691973\pi$
$337$	−20.8248			−1.13440			−0.567198	+	0.823581i	$0.691973\pi$
$338$		0			0
$339$	−3.30156	+	1.78959i	−0.179316	+	0.0971974i
$340$		0			0
$341$		−	6.11151i		−	0.330957i
$342$		0			0
$343$		0			0
$344$		0			0
$345$	−9.91238	+	5.37295i	−0.533664	+	0.289270i
$346$		0			0
$347$	22.3678			1.20077			0.600384	−	0.799712i	$-0.295014\pi$
$347$	22.3678			1.20077			0.600384	+	0.799712i	$0.295014\pi$
$348$		0			0
$349$	5.45017			0.291741			0.145870	−	0.989304i	$-0.453402\pi$
$349$	5.45017			0.291741			0.145870	+	0.989304i	$0.453402\pi$
$350$		0			0
$351$	−10.3585	−	0.837253i	−0.552897	−	0.0446893i
$352$		0			0
$353$		−	5.02742i		−	0.267582i	−0.991010	−	0.133791i	$-0.957285\pi$
$353$		−	5.02742i		−	0.267582i	0.991010	−	0.133791i	$-0.0427151\pi$
$354$		0			0
$355$			40.8471i			2.16794i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	22.8217			1.20448			0.602241	−	0.798315i	$-0.294275\pi$
$359$	22.8217			1.20448			0.602241	+	0.798315i	$0.294275\pi$
$360$		0			0
$361$	9.72508			0.511846
$362$		0			0
$363$	1.19695	+	2.20822i	0.0628238	+	0.115902i
$364$		0			0
$365$			28.6874i			1.50157i
$366$		0			0
$367$		−	29.5600i		−	1.54302i	−0.636219	−	0.771508i	$-0.719503\pi$
$367$		−	29.5600i		−	1.54302i	0.636219	−	0.771508i	$-0.280497\pi$
$368$		0			0
$369$	5.45017	+	3.55030i	0.283724	+	0.184821i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	29.8248			1.54427			0.772134	−	0.635460i	$-0.219190\pi$
$373$	29.8248			1.54427			0.772134	+	0.635460i	$0.219190\pi$
$374$		0			0
$375$	−33.3248	+	18.0635i	−1.72089	+	0.932797i
$376$		0			0
$377$		−	13.6051i		−	0.700700i
$378$		0			0
$379$			10.3923i			0.533817i	0.963722	+	0.266908i	$0.0860021\pi$
$379$			10.3923i			0.533817i	−0.963722	+	0.266908i	$0.913998\pi$
$380$		0			0
$381$	−7.27492	+	3.94333i	−0.372705	+	0.202023i
$382$		0			0
$383$	−28.9710			−1.48035			−0.740173	−	0.672416i	$-0.765257\pi$
$383$	−28.9710			−1.48035			−0.740173	+	0.672416i	$0.765257\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−2.10461	−	1.37097i	−0.106983	−	0.0696902i
$388$		0			0
$389$		−	22.9689i		−	1.16457i	−0.812985	−	0.582285i	$-0.802159\pi$
$389$		−	22.9689i		−	1.16457i	0.812985	−	0.582285i	$-0.197841\pi$
$390$		0			0
$391$			8.29917i			0.419707i
$392$		0			0
$393$	2.91238	+	5.37295i	0.146910	+	0.271029i
$394$		0			0
$395$	−20.4901			−1.03097
$396$		0			0
$397$	6.37459			0.319931			0.159966	−	0.987123i	$-0.448862\pi$
$397$	6.37459			0.319931			0.159966	+	0.987123i	$0.448862\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			36.5740i			1.82642i	0.407489	+	0.913210i	$0.366405\pi$
$401$			36.5740i			1.82642i	−0.407489	+	0.913210i	$0.633595\pi$
$402$		0			0
$403$			3.46410i			0.172559i
$404$		0			0
$405$	−32.4622	+	14.3437i	−1.61306	+	0.712744i
$406$		0			0
$407$	−4.49852			−0.222983
$408$		0			0
$409$	−16.4502			−0.813408			−0.406704	−	0.913560i	$-0.633322\pi$
$409$	−16.4502			−0.813408			−0.406704	+	0.913560i	$0.633322\pi$
$410$		0			0
$411$	1.05231	−	0.570396i	0.0519064	−	0.0281356i
$412$		0			0
$413$		0			0
$414$		0			0
$415$			20.4235i			1.00255i
$416$		0			0
$417$	31.8248	−	17.2504i	1.55846	−	0.844757i
$418$		0			0
$419$	30.1679			1.47380			0.736899	−	0.676002i	$-0.236289\pi$
$419$	30.1679			1.47380			0.736899	+	0.676002i	$0.236289\pi$
$420$		0			0
$421$	11.0997			0.540965			0.270482	−	0.962725i	$-0.412817\pi$
$421$	11.0997			0.540965			0.270482	+	0.962725i	$0.412817\pi$
$422$		0			0
$423$	−8.10926	+	12.4488i	−0.394286	+	0.605279i
$424$		0			0
$425$			53.0384i			2.57274i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	−5.82475	−	10.7459i	−0.281222	−	0.518817i
$430$		0			0
$431$	−25.6694			−1.23645			−0.618226	−	0.786001i	$-0.712148\pi$
$431$	−25.6694			−1.23645			−0.618226	+	0.786001i	$0.712148\pi$
$432$		0			0
$433$	−2.00000			−0.0961139			−0.0480569	−	0.998845i	$-0.515303\pi$
$433$	−2.00000			−0.0961139			−0.0480569	+	0.998845i	$0.515303\pi$
$434$		0			0
$435$	−22.1409	−	40.8471i	−1.06158	−	1.95847i
$436$		0			0
$437$		−	5.02742i		−	0.240494i
$438$		0			0
$439$			9.61260i			0.458784i	0.973334	+	0.229392i	$0.0736738\pi$
$439$			9.61260i			0.458784i	−0.973334	+	0.229392i	$0.926326\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−34.1502			−1.62253			−0.811263	−	0.584681i	$-0.801220\pi$
$443$	−34.1502			−1.62253			−0.811263	+	0.584681i	$0.801220\pi$
$444$		0			0
$445$	28.3746			1.34508
$446$		0			0
$447$	−10.9570	+	5.93918i	−0.518248	+	0.280914i
$448$		0			0
$449$			29.3784i			1.38645i	0.720719	+	0.693227i	$0.243812\pi$
$449$			29.3784i			1.38645i	−0.720719	+	0.693227i	$0.756188\pi$
$450$		0			0
$451$			7.65037i			0.360242i
$452$		0			0
$453$	1.36254	−	0.738558i	0.0640178	−	0.0347005i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	28.6495			1.34017			0.670084	−	0.742286i	$-0.266258\pi$
$457$	28.6495			1.34017			0.670084	+	0.742286i	$0.266258\pi$
$458$		0			0
$459$	−2.10461	+	26.0383i	−0.0982348	+	1.21536i
$460$		0			0
$461$		−	13.6051i		−	0.633654i	−0.948483	−	0.316827i	$-0.897382\pi$
$461$		−	13.6051i		−	0.633654i	0.948483	−	0.316827i	$-0.102618\pi$
$462$		0			0
$463$		−	13.9715i		−	0.649310i	−0.945832	−	0.324655i	$-0.894752\pi$
$463$		−	13.9715i		−	0.649310i	0.945832	−	0.324655i	$-0.105248\pi$
$464$		0			0
$465$	5.63746	+	10.4004i	0.261431	+	0.482305i
$466$		0			0
$467$	−28.9710			−1.34062			−0.670308	−	0.742083i	$-0.733838\pi$
$467$	−28.9710			−1.34062			−0.670308	+	0.742083i	$0.733838\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	11.8647	+	21.8887i	0.546695	+	1.00858i
$472$		0			0
$473$		−	2.95423i		−	0.135836i
$474$		0			0
$475$		−	32.1293i		−	1.47419i
$476$		0			0
$477$	24.2870	+	15.8208i	1.11202	+	0.724385i
$478$		0			0
$479$	4.49852			0.205543			0.102771	−	0.994705i	$-0.467229\pi$
$479$	4.49852			0.205543			0.102771	+	0.994705i	$0.467229\pi$
$480$		0			0
$481$	2.54983			0.116262
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	16.8574i		−	0.765455i
$486$		0			0
$487$		−	2.56930i		−	0.116426i	−0.998304	−	0.0582131i	$-0.981460\pi$
$487$		−	2.56930i		−	0.116426i	0.998304	−	0.0582131i	$-0.0185403\pi$
$488$		0			0
$489$	9.91238	−	5.37295i	0.448253	−	0.242973i
$490$		0			0
$491$	−22.1409			−0.999206			−0.499603	−	0.866255i	$-0.666521\pi$
$491$	−22.1409			−0.999206			−0.499603	+	0.866255i	$0.666521\pi$
$492$		0			0
$493$	−34.1993			−1.54026
$494$		0			0
$495$	−34.9756	−	22.7835i	−1.57204	−	1.02404i
$496$		0			0
$497$		0			0
$498$		0			0
$499$		−	25.6197i		−	1.14689i	−0.819243	−	0.573447i	$-0.805606\pi$
$499$		−	25.6197i		−	1.14689i	0.819243	−	0.573447i	$-0.194394\pi$
$500$		0			0
$501$	11.6495	+	21.4918i	0.520461	+	0.960182i
$502$		0			0
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$	50.9244			2.26611
$506$		0			0
$507$	−7.42852	−	13.7046i	−0.329912	−	0.608644i
$508$		0			0
$509$			9.66181i			0.428252i	0.976806	+	0.214126i	$0.0686903\pi$
$509$			9.66181i			0.428252i	−0.976806	+	0.214126i	$0.931310\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	1.27492	−	15.7733i	0.0562890	−	0.696409i
$514$		0			0
$515$	−19.0663			−0.840160
$516$		0			0
$517$	−17.4743			−0.768517
$518$		0			0
$519$	13.0616	−	7.07997i	0.573341	−	0.310776i
$520$		0			0
$521$			19.4186i			0.850745i	0.905018	+	0.425372i	$0.139857\pi$
$521$			19.4186i			0.850745i	−0.905018	+	0.425372i	$0.860143\pi$
$522$		0			0
$523$			24.7824i			1.08366i	0.840488	+	0.541830i	$0.182268\pi$
$523$			24.7824i			1.08366i	−0.840488	+	0.541830i	$0.817732\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	8.70774			0.379315
$528$		0			0
$529$	−20.2749			−0.881518
$530$		0			0
$531$	16.5901	−	25.4679i	0.719949	−	1.10521i
$532$		0			0
$533$		−	4.33635i		−	0.187828i
$534$		0			0
$535$			0.894797i			0.0386855i
$536$		0			0
$537$	6.81271	+	12.5685i	0.293990	+	0.542373i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−26.7251			−1.14900			−0.574501	−	0.818504i	$-0.694804\pi$
$541$	−26.7251			−1.14900			−0.574501	+	0.818504i	$0.694804\pi$
$542$		0			0
$543$	12.4631	+	22.9928i	0.534844	+	0.986717i
$544$		0			0
$545$			30.8556i			1.32171i
$546$		0			0
$547$			13.0192i			0.556659i	0.960486	+	0.278329i	$0.0897807\pi$
$547$			13.0192i			0.556659i	−0.960486	+	0.278329i	$0.910219\pi$
$548$		0			0
$549$	18.4622	−	28.3419i	0.787948	−	1.20960i
$550$		0			0
$551$	20.7170			0.882576
$552$		0			0
$553$		0			0
$554$		0			0
$555$	7.65544	−	4.14959i	0.324955	−	0.176140i
$556$		0			0
$557$			15.3803i			0.651684i	0.945424	+	0.325842i	$0.105648\pi$
$557$			15.3803i			0.651684i	−0.945424	+	0.325842i	$0.894352\pi$
$558$		0			0
$559$			1.67451i			0.0708241i
$560$		0			0
$561$	−27.0120	+	14.6417i	−1.14045	+	0.618174i
$562$		0			0
$563$	−28.0009			−1.18010			−0.590049	−	0.807367i	$-0.700892\pi$
$563$	−28.0009			−1.18010			−0.590049	+	0.807367i	$0.700892\pi$
$564$		0			0
$565$	8.54983			0.359694
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	37.9562i		−	1.59121i	−0.605819	−	0.795603i	$-0.707154\pi$
$569$		−	37.9562i		−	1.59121i	0.605819	−	0.795603i	$-0.292846\pi$
$570$		0			0
$571$			17.8542i			0.747176i	0.927595	+	0.373588i	$0.121873\pi$
$571$			17.8542i			0.747176i	−0.927595	+	0.373588i	$0.878127\pi$
$572$		0			0
$573$	−15.3625	−	28.3419i	−0.641779	−	1.18400i
$574$		0			0
$575$	17.4155			0.726276
$576$		0			0
$577$	−34.0997			−1.41959			−0.709794	−	0.704409i	$-0.751212\pi$
$577$	−34.0997			−1.41959			−0.709794	+	0.704409i	$0.751212\pi$
$578$		0			0
$579$	9.98696	+	18.4246i	0.415044	+	0.765701i
$580$		0			0
$581$		0			0
$582$		0			0
$583$			34.0915i			1.41192i
$584$		0			0
$585$	19.8248	+	12.9141i	0.819653	+	0.533931i
$586$		0			0
$587$	5.17926			0.213771			0.106886	−	0.994271i	$-0.465912\pi$
$587$	5.17926			0.213771			0.106886	+	0.994271i	$0.465912\pi$
$588$		0			0
$589$	−5.27492			−0.217349
$590$		0			0
$591$	27.3202	−	14.8087i	1.12380	−	0.609150i
$592$		0			0
$593$		−	12.9141i		−	0.530317i	−0.964205	−	0.265159i	$-0.914576\pi$
$593$		−	12.9141i		−	0.530317i	0.964205	−	0.265159i	$-0.0854243\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−32.2870	+	17.5010i	−1.32142	+	0.716267i
$598$		0			0
$599$	−1.19695			−0.0489062			−0.0244531	−	0.999701i	$-0.507784\pi$
$599$	−1.19695			−0.0489062			−0.0244531	+	0.999701i	$0.507784\pi$
$600$		0			0
$601$	17.9244			0.731152			0.365576	−	0.930781i	$-0.380872\pi$
$601$	17.9244			0.731152			0.365576	+	0.930781i	$0.380872\pi$
$602$		0			0
$603$	40.3818	+	26.3052i	1.64447	+	1.07123i
$604$		0			0
$605$		−	5.71848i		−	0.232489i
$606$		0			0
$607$			34.6986i			1.40837i	0.710016	+	0.704186i	$0.248688\pi$
$607$			34.6986i			1.40837i	−0.710016	+	0.704186i	$0.751312\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	9.90469			0.400701
$612$		0			0
$613$	37.8248			1.52773			0.763864	−	0.645378i	$-0.223300\pi$
$613$	37.8248			1.52773			0.763864	+	0.645378i	$0.223300\pi$
$614$		0			0
$615$	−7.05696	−	13.0192i	−0.284564	−	0.524983i
$616$		0			0
$617$		−	38.0512i		−	1.53188i	−0.642911	−	0.765941i	$-0.722273\pi$
$617$		−	38.0512i		−	1.53188i	0.642911	−	0.765941i	$-0.277727\pi$
$618$		0			0
$619$			10.9260i			0.439154i	0.975595	+	0.219577i	$0.0704677\pi$
$619$			10.9260i			0.439154i	−0.975595	+	0.219577i	$0.929532\pi$
$620$		0			0
$621$	8.54983	+	0.691062i	0.343093	+	0.0277314i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	33.5498			1.34199
$626$		0			0
$627$	16.3632	−	8.86957i	0.653482	−	0.354216i
$628$		0			0
$629$		−	6.40954i		−	0.255565i
$630$		0			0
$631$			37.0219i			1.47382i	0.675992	+	0.736909i	$0.263715\pi$
$631$			37.0219i			1.47382i	−0.675992	+	0.736909i	$0.736285\pi$
$632$		0			0
$633$	2.72508	−	1.47712i	0.108312	−	0.0587101i
$634$		0			0
$635$	18.8394			0.747617
$636$		0			0
$637$		0			0
$638$		0			0
$639$	16.9617	−	26.0383i	0.670993	−	1.03006i
$640$		0			0
$641$		−	7.19559i		−	0.284209i	−0.989852	−	0.142104i	$-0.954613\pi$
$641$		−	7.19559i		−	0.284209i	0.989852	−	0.142104i	$-0.0453869\pi$
$642$		0			0
$643$			6.09095i			0.240204i	0.992762	+	0.120102i	$0.0383221\pi$
$643$			6.09095i			0.240204i	−0.992762	+	0.120102i	$0.961678\pi$
$644$		0			0
$645$	2.72508	+	5.02742i	0.107300	+	0.197954i
$646$		0			0
$647$	9.16157			0.360178			0.180089	−	0.983650i	$-0.442361\pi$
$647$	9.16157			0.360178			0.180089	+	0.983650i	$0.442361\pi$
$648$		0			0
$649$	35.7492			1.40328
$650$		0			0
$651$		0			0
$652$		0			0
$653$			21.0988i			0.825658i	0.910808	+	0.412829i	$0.135459\pi$
$653$			21.0988i			0.825658i	−0.910808	+	0.412829i	$0.864541\pi$
$654$		0			0
$655$		−	13.9140i		−	0.543663i
$656$		0			0
$657$	11.9124	−	18.2870i	0.464746	−	0.713445i
$658$		0			0
$659$	−45.1895			−1.76033			−0.880166	−	0.474666i	$-0.842569\pi$
$659$	−45.1895			−1.76033			−0.880166	+	0.474666i	$0.842569\pi$
$660$		0			0
$661$	29.2749			1.13866			0.569331	−	0.822108i	$-0.307202\pi$
$661$	29.2749			1.13866			0.569331	+	0.822108i	$0.307202\pi$
$662$		0			0
$663$	15.3109	−	8.29917i	0.594625	−	0.322313i
$664$		0			0
$665$		0			0
$666$		0			0
$667$			11.2296i			0.434810i
$668$		0			0
$669$	−15.2749	+	8.27968i	−0.590562	+	0.320111i
$670$		0			0
$671$	39.7833			1.53582
$672$		0			0
$673$	12.2749			0.473163			0.236582	−	0.971612i	$-0.423973\pi$
$673$	12.2749			0.473163			0.236582	+	0.971612i	$0.423973\pi$
$674$		0			0
$675$	54.6404	+	4.41644i	2.10311	+	0.169989i
$676$		0			0
$677$		−	12.6160i		−	0.484874i	−0.970167	−	0.242437i	$-0.922053\pi$
$677$		−	12.6160i		−	0.484874i	0.970167	−	0.242437i	$-0.0779467\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	22.7371	+	41.9470i	0.871289	+	1.60741i
$682$		0			0
$683$	24.6994			0.945095			0.472547	−	0.881305i	$-0.343335\pi$
$683$	24.6994			0.945095			0.472547	+	0.881305i	$0.343335\pi$
$684$		0			0
$685$	−2.72508			−0.104120
$686$		0			0
$687$	7.20161	+	13.2860i	0.274758	+	0.506893i
$688$		0			0
$689$		−	19.3236i		−	0.736171i
$690$		0			0
$691$		−	30.8734i		−	1.17448i	−0.809413	−	0.587239i	$-0.800215\pi$
$691$		−	30.8734i		−	1.17448i	0.809413	−	0.587239i	$-0.199785\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−82.4144			−3.12615
$696$		0			0
$697$	−10.9003			−0.412879
$698$		0			0
$699$	−1.05231	+	0.570396i	−0.0398018	+	0.0215744i
$700$		0			0
$701$		−	22.5759i		−	0.852679i	−0.904563	−	0.426340i	$-0.859803\pi$
$701$		−	22.5759i		−	0.852679i	0.904563	−	0.426340i	$-0.140197\pi$
$702$		0			0
$703$			3.88273i			0.146440i
$704$		0			0
$705$	29.7371	−	16.1188i	1.11997	−	0.607071i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−20.3746			−0.765184			−0.382592	−	0.923917i	$-0.624968\pi$
$709$	−20.3746			−0.765184			−0.382592	+	0.923917i	$0.624968\pi$
$710$		0			0
$711$	13.0616	+	8.50848i	0.489849	+	0.319093i
$712$		0			0
$713$		−	2.85924i		−	0.107079i
$714$		0			0
$715$			27.8279i			1.04070i
$716$		0			0
$717$	14.0000	+	25.8281i	0.522840	+	0.964570i
$718$		0			0
$719$	19.0663			0.711052			0.355526	−	0.934666i	$-0.384302\pi$
$719$	19.0663			0.711052			0.355526	+	0.934666i	$0.384302\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	5.77774	+	10.6592i	0.214876	+	0.396419i
$724$		0			0
$725$			71.7660i			2.66532i
$726$		0			0
$727$			10.7534i			0.398821i	0.979916	+	0.199411i	$0.0639027\pi$
$727$			10.7534i			0.398821i	−0.979916	+	0.199411i	$0.936097\pi$
$728$		0			0
$729$	26.6495	+	4.33635i	0.987019	+	0.160606i
$730$		0			0
$731$	4.20922			0.155684
$732$		0			0
$733$	33.2749			1.22904			0.614519	−	0.788902i	$-0.289350\pi$
$733$	33.2749			1.22904			0.614519	+	0.788902i	$0.289350\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			56.6837i			2.08797i
$738$		0			0
$739$		−	41.3808i		−	1.52222i	−0.648625	−	0.761108i	$-0.724656\pi$
$739$		−	41.3808i		−	1.52222i	0.648625	−	0.761108i	$-0.275344\pi$
$740$		0			0
$741$	−9.27492	+	5.02742i	−0.340723	+	0.184687i
$742$		0			0
$743$	−33.9233			−1.24453			−0.622263	−	0.782808i	$-0.713786\pi$
$743$	−33.9233			−1.24453			−0.622263	+	0.782808i	$0.713786\pi$
$744$		0			0
$745$	28.3746			1.03956
$746$		0			0
$747$	8.48083	−	13.0192i	0.310297	−	0.476346i
$748$		0			0
$749$		0			0
$750$		0			0
$751$		−	8.77534i		−	0.320217i	−0.987099	−	0.160108i	$-0.948816\pi$
$751$		−	8.77534i		−	0.320217i	0.987099	−	0.160108i	$-0.0511844\pi$
$752$		0			0
$753$	9.72508	+	17.9415i	0.354402	+	0.653824i
$754$		0			0
$755$	−3.52848			−0.128415
$756$		0			0
$757$	−15.0997			−0.548807			−0.274403	−	0.961615i	$-0.588480\pi$
$757$	−15.0997			−0.548807			−0.274403	+	0.961615i	$0.588480\pi$
$758$		0			0
$759$	4.80770	+	8.86957i	0.174508	+	0.321945i
$760$		0			0
$761$		−	4.24136i		−	0.153749i	−0.997041	−	0.0768746i	$-0.975506\pi$
$761$		−	4.24136i		−	0.153749i	0.997041	−	0.0768746i	$-0.0244941\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	32.4622	−	49.8336i	1.17367	−	1.80174i
$766$		0			0
$767$	−20.2632			−0.731662
$768$		0			0
$769$	−28.8248			−1.03945			−0.519724	−	0.854334i	$-0.673965\pi$
$769$	−28.8248			−1.03945			−0.519724	+	0.854334i	$0.673965\pi$
$770$		0			0
$771$	26.2679	−	14.2384i	0.946014	−	0.512782i
$772$		0			0
$773$		−	54.5155i		−	1.96079i	−0.197049	−	0.980394i	$-0.563136\pi$
$773$		−	54.5155i		−	1.96079i	0.197049	−	0.980394i	$-0.436864\pi$
$774$		0			0
$775$		−	18.2728i		−	0.656380i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	6.60313			0.236582
$780$		0			0
$781$	36.5498			1.30786
$782$		0			0
$783$	−2.84774	+	35.2323i	−0.101770	+	1.25910i
$784$		0			0
$785$		−	56.6837i		−	2.02313i
$786$		0			0
$787$		−	37.8016i		−	1.34748i	−0.738968	−	0.673740i	$-0.764687\pi$
$787$		−	37.8016i		−	1.34748i	0.738968	−	0.673740i	$-0.235313\pi$
$788$		0			0
$789$	−18.0876	−	33.3693i	−0.643936	−	1.18798i
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−22.5498			−0.800768
$794$		0			0
$795$	−31.4471	−	58.0158i	−1.11531	−	2.05761i
$796$		0			0
$797$			24.7441i			0.876479i	0.898858	+	0.438240i	$0.144398\pi$
$797$			24.7441i			0.876479i	−0.898858	+	0.438240i	$0.855602\pi$
$798$		0			0
$799$		−	24.8975i		−	0.880811i
$800$		0			0
$801$	−18.0876	−	11.7825i	−0.639095	−	0.416314i
$802$		0			0
$803$	25.6694			0.905853
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−6.00465	+	3.25479i	−0.211374	+	0.114574i
$808$		0			0
$809$		−	48.0110i		−	1.68798i	−0.536361	−	0.843988i	$-0.680202\pi$
$809$		−	48.0110i		−	1.68798i	0.536361	−	0.843988i	$-0.319798\pi$
$810$		0			0
$811$		−	4.41644i		−	0.155082i	−0.996989	−	0.0775411i	$-0.975293\pi$
$811$		−	4.41644i		−	0.155082i	0.996989	−	0.0775411i	$-0.0247069\pi$
$812$		0			0
$813$	−5.18729	+	2.81174i	−0.181926	+	0.0986121i
$814$		0			0
$815$	−25.6694			−0.899160
$816$		0			0
$817$	−2.54983			−0.0892074
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	39.8263i		−	1.38995i	−0.719035	−	0.694974i	$-0.755416\pi$
$821$		−	39.8263i		−	1.38995i	0.719035	−	0.694974i	$-0.244584\pi$
$822$		0			0
$823$			19.5287i			0.680729i	0.940294	+	0.340364i	$0.110550\pi$
$823$			19.5287i			0.680729i	−0.940294	+	0.340364i	$0.889450\pi$
$824$		0			0
$825$	30.7251	+	56.6837i	1.06971	+	1.97347i
$826$		0			0
$827$	−8.02700			−0.279126			−0.139563	−	0.990213i	$-0.544570\pi$
$827$	−8.02700			−0.279126			−0.139563	+	0.990213i	$0.544570\pi$
$828$		0			0
$829$	10.3746			0.360324			0.180162	−	0.983637i	$-0.442338\pi$
$829$	10.3746			0.360324			0.180162	+	0.983637i	$0.442338\pi$
$830$		0			0
$831$	10.9570	+	20.2142i	0.380094	+	0.701223i
$832$		0			0
$833$		0			0
$834$		0			0
$835$		−	55.6558i		−	1.92605i
$836$		0			0
$837$	0.725083	−	8.97074i	0.0250625	−	0.310074i
$838$		0			0
$839$	20.7170			0.715232			0.357616	−	0.933869i	$-0.383590\pi$
$839$	20.7170			0.715232			0.357616	+	0.933869i	$0.383590\pi$
$840$		0			0
$841$	−17.2749			−0.595687
$842$		0			0
$843$	−24.0186	+	13.0192i	−0.827245	+	0.448403i
$844$		0			0
$845$			35.4900i			1.22089i
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−31.1873	+	16.9049i	−1.07035	+	0.580175i
$850$		0			0
$851$	−2.10461			−0.0721451
$852$		0			0
$853$	38.5498			1.31992			0.659961	−	0.751300i	$-0.270573\pi$
$853$	38.5498			1.31992			0.659961	+	0.751300i	$0.270573\pi$
$854$		0			0
$855$	−19.6647	+	30.1879i	−0.672520	+	1.03240i
$856$		0			0
$857$			24.3510i			0.831815i	0.909407	+	0.415908i	$0.136536\pi$
$857$			24.3510i			0.831815i	−0.909407	+	0.415908i	$0.863464\pi$
$858$		0			0
$859$		−	40.4284i		−	1.37940i	−0.724095	−	0.689700i	$-0.757742\pi$
$859$		−	40.4284i		−	1.37940i	0.724095	−	0.689700i	$-0.242258\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−1.65078			−0.0561933			−0.0280966	−	0.999605i	$-0.508945\pi$
$863$	−1.65078			−0.0561933			−0.0280966	+	0.999605i	$0.508945\pi$
$864$		0			0
$865$	−33.8248			−1.15008
$866$		0			0
$867$	−6.83004	−	12.6005i	−0.231960	−	0.427936i
$868$		0			0
$869$			18.3345i			0.621956i
$870$		0			0
$871$		−	32.1293i		−	1.08866i
$872$		0			0
$873$	−7.00000	+	10.7459i	−0.236914	+	0.363693i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−11.8248			−0.399293			−0.199647	−	0.979868i	$-0.563979\pi$
$877$	−11.8248			−0.399293			−0.199647	+	0.979868i	$0.563979\pi$
$878$		0			0
$879$	20.2632	−	10.9836i	0.683461	−	0.370466i
$880$		0			0
$881$			22.8739i			0.770642i	0.922783	+	0.385321i	$0.125909\pi$
$881$			22.8739i			0.770642i	−0.922783	+	0.385321i	$0.874091\pi$
$882$		0			0
$883$		−	38.1051i		−	1.28234i	−0.767399	−	0.641170i	$-0.778449\pi$
$883$		−	38.1051i		−	1.28234i	0.767399	−	0.641170i	$-0.221551\pi$
$884$		0			0
$885$	−60.8368	+	32.9762i	−2.04501	+	1.10848i
$886$		0			0
$887$	25.6694			0.861894			0.430947	−	0.902377i	$-0.358180\pi$
$887$	25.6694			0.861894			0.430947	+	0.902377i	$0.358180\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	12.8347	+	29.0471i	0.429979	+	0.973114i
$892$		0			0
$893$			15.0822i			0.504708i
$894$		0			0
$895$		−	32.5479i		−	1.08796i
$896$		0			0
$897$	−2.72508	−	5.02742i	−0.0909879	−	0.167861i
$898$		0			0
$899$	11.7824			0.392965
$900$		0			0
$901$	−48.5739			−1.61823
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	59.5429i		−	1.97927i
$906$		0			0
$907$			43.8925i			1.45743i	0.684818	+	0.728714i	$0.259882\pi$
$907$			43.8925i			1.45743i	−0.684818	+	0.728714i	$0.740118\pi$
$908$		0			0
$909$	−32.4622	−	21.1463i	−1.07670	−	0.701377i
$910$		0			0
$911$	−17.8693			−0.592037			−0.296018	−	0.955182i	$-0.595659\pi$
$911$	−17.8693			−0.592037			−0.296018	+	0.955182i	$0.595659\pi$
$912$		0			0
$913$	18.2749			0.604811
$914$		0			0
$915$	−67.7020	+	36.6975i	−2.23816	+	1.21318i
$916$		0			0
$917$		0			0
$918$		0			0
$919$			35.0596i			1.15651i	0.815856	+	0.578255i	$0.196266\pi$
$919$			35.0596i			1.15651i	−0.815856	+	0.578255i	$0.803734\pi$
$920$		0			0
$921$	−26.3746	+	14.2962i	−0.869072	+	0.471076i
$922$		0			0
$923$	−20.7170			−0.681910
$924$		0			0
$925$	−13.4502			−0.442239
$926$		0			0
$927$	12.1540	+	7.91723i	0.399188	+	0.260036i
$928$		0			0
$929$		−	6.40954i		−	0.210290i	−0.994457	−	0.105145i	$-0.966469\pi$
$929$		−	6.40954i		−	0.210290i	0.994457	−	0.105145i	$-0.0335307\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	18.4622	+	34.0603i	0.604426	+	1.11509i
$934$		0			0
$935$	69.9512			2.28765
$936$		0			0
$937$	−27.1752			−0.887777			−0.443888	−	0.896082i	$-0.646401\pi$
$937$	−27.1752			−0.887777			−0.443888	+	0.896082i	$0.646401\pi$
$938$		0			0
$939$	−3.67313	−	6.77643i	−0.119868	−	0.221141i
$940$		0			0
$941$		−	0.393027i		−	0.0128123i	−0.999979	−	0.00640616i	$-0.997961\pi$
$941$		−	0.393027i		−	0.0128123i	0.999979	−	0.00640616i	$-0.00203916\pi$
$942$		0			0
$943$			3.57919i			0.116554i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	12.4631			0.404997			0.202499	−	0.979283i	$-0.435094\pi$
$947$	12.4631			0.404997			0.202499	+	0.979283i	$0.435094\pi$
$948$		0			0
$949$	−14.5498			−0.472308
$950$		0			0
$951$	9.30622	−	5.04438i	0.301775	−	0.163575i
$952$		0			0
$953$			60.3290i			1.95425i	0.212670	+	0.977124i	$0.431784\pi$
$953$			60.3290i			1.95425i	−0.212670	+	0.977124i	$0.568216\pi$
$954$		0			0
$955$			73.3949i			2.37500i
$956$		0			0
$957$	−36.5498	+	19.8116i	−1.18149	+	0.640419i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	28.0000			0.903226
$962$		0			0
$963$	0.371563	−	0.570396i	0.0119734	−	0.0183808i
$964$		0			0
$965$		−	47.7130i		−	1.53593i
$966$		0			0
$967$		−	30.8158i		−	0.990970i	−0.868616	−	0.495485i	$-0.834990\pi$
$967$		−	30.8158i		−	0.990970i	0.868616	−	0.495485i	$-0.165010\pi$
$968$		0			0
$969$	12.6375	+	23.3144i	0.405974	+	0.748968i
$970$		0			0
$971$	−27.5471			−0.884028			−0.442014	−	0.897008i	$-0.645736\pi$
$971$	−27.5471			−0.884028			−0.442014	+	0.897008i	$0.645736\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	−17.4155	−	32.1293i	−0.557742	−	1.02896i
$976$		0			0
$977$		−	27.9013i		−	0.892643i	−0.894873	−	0.446321i	$-0.852734\pi$
$977$		−	27.9013i		−	0.892643i	0.894873	−	0.446321i	$-0.147266\pi$
$978$		0			0
$979$		−	25.3895i		−	0.811452i
$980$		0			0
$981$	12.8127	−	19.6691i	0.409078	−	0.627987i
$982$		0			0
$983$	−33.1802			−1.05828			−0.529142	−	0.848533i	$-0.677486\pi$
$983$	−33.1802			−1.05828			−0.529142	+	0.848533i	$0.677486\pi$
$984$		0			0
$985$	−70.7492			−2.25426
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	1.38212i		−	0.0439490i
$990$		0			0
$991$			19.1676i			0.608880i	0.952531	+	0.304440i	$0.0984694\pi$
$991$			19.1676i			0.608880i	−0.952531	+	0.304440i	$0.901531\pi$
$992$		0			0
$993$	43.0120	−	23.3144i	1.36495	−	0.739861i
$994$		0			0
$995$	83.6113			2.65066
$996$		0			0
$997$	21.2749			0.673783			0.336892	−	0.941543i	$-0.390624\pi$
$997$	21.2749			0.673783			0.336892	+	0.941543i	$0.390624\pi$
$998$		0			0
$999$	−6.60313	−	0.533714i	−0.208914	−	0.0168860i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2352.2.h.n.2255.6		8
3.2	odd	2	inner	2352.2.h.n.2255.4		8
4.3	odd	2	inner	2352.2.h.n.2255.3		8
7.2	even	3		336.2.bj.e.95.1	✓	8
7.4	even	3		336.2.bj.g.191.3	yes	8
7.6	odd	2		2352.2.h.m.2255.3		8
12.11	even	2	inner	2352.2.h.n.2255.5		8
21.2	odd	6		336.2.bj.e.95.2	yes	8
21.11	odd	6		336.2.bj.g.191.4	yes	8
21.20	even	2		2352.2.h.m.2255.5		8
28.11	odd	6		336.2.bj.e.191.2	yes	8
28.23	odd	6		336.2.bj.g.95.4	yes	8
28.27	even	2		2352.2.h.m.2255.6		8
84.11	even	6		336.2.bj.e.191.1	yes	8
84.23	even	6		336.2.bj.g.95.3	yes	8
84.83	odd	2		2352.2.h.m.2255.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.2.bj.e.95.1	✓	8	7.2	even	3
336.2.bj.e.95.2	yes	8	21.2	odd	6
336.2.bj.e.191.1	yes	8	84.11	even	6
336.2.bj.e.191.2	yes	8	28.11	odd	6
336.2.bj.g.95.3	yes	8	84.23	even	6
336.2.bj.g.95.4	yes	8	28.23	odd	6
336.2.bj.g.191.3	yes	8	7.4	even	3
336.2.bj.g.191.4	yes	8	21.11	odd	6
2352.2.h.m.2255.3		8	7.6	odd	2
2352.2.h.m.2255.4		8	84.83	odd	2
2352.2.h.m.2255.5		8	21.20	even	2
2352.2.h.m.2255.6		8	28.27	even	2
2352.2.h.n.2255.3		8	4.3	odd	2	inner
2352.2.h.n.2255.4		8	3.2	odd	2	inner
2352.2.h.n.2255.5		8	12.11	even	2	inner
2352.2.h.n.2255.6		8	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2352.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(0.825391 + 1.52274i) q^{3} -3.94333i q^{5} +(-1.63746 + 2.51371i) q^{9} +O(q^{10})\)	\(q+(0.825391 + 1.52274i) q^{3} -3.94333i q^{5} +(-1.63746 + 2.51371i) q^{9} -3.52848 q^{11} +2.00000 q^{13} +(6.00465 - 3.25479i) q^{15} -5.02742i q^{17} +3.04547i q^{19} -1.65078 q^{23} -10.5498 q^{25} +(-5.17926 - 0.418627i) q^{27} -6.80257i q^{29} +1.73205i q^{31} +(-2.91238 - 5.37295i) q^{33} +1.27492 q^{37} +(1.65078 + 3.04547i) q^{39} -2.16818i q^{41} +0.837253i q^{43} +(9.91238 + 6.45704i) q^{45} +4.95235 q^{47} +(7.65544 - 4.14959i) q^{51} -9.66181i q^{53} +13.9140i q^{55} +(-4.63746 + 2.51371i) q^{57} -10.1316 q^{59} -11.2749 q^{61} -7.88666i q^{65} -16.0646i q^{67} +(-1.36254 - 2.51371i) q^{69} -10.3585 q^{71} -7.27492 q^{73} +(-8.70774 - 16.0646i) q^{75} -5.19615i q^{79} +(-3.63746 - 8.23219i) q^{81} -5.17926 q^{83} -19.8248 q^{85} +(10.3585 - 5.61478i) q^{87} +7.19559i q^{89} +(-2.63746 + 1.42962i) q^{93} +12.0093 q^{95} +4.27492 q^{97} +(5.77774 - 8.86957i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 2 q^{9}+O(q^{10}) \) 8 * q + 2 * q^9	\( 8 q + 2 q^{9} + 16 q^{13} - 24 q^{25} + 22 q^{33} - 20 q^{37} + 34 q^{45} - 22 q^{57} - 60 q^{61} - 26 q^{69} - 28 q^{73} - 14 q^{81} - 68 q^{85} - 6 q^{93} + 4 q^{97}+O(q^{100}) \) 8 * q + 2 * q^9 + 16 * q^13 - 24 * q^25 + 22 * q^33 - 20 * q^37 + 34 * q^45 - 22 * q^57 - 60 * q^61 - 26 * q^69 - 28 * q^73 - 14 * q^81 - 68 * q^85 - 6 * q^93 + 4 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more