LMFDB - Embedded newform 440.2.b.a.89.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [440,2,Mod(89,440)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(440, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("440.89");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 440 = 2^{3} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	440.b (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:	$3.51341768894$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			89.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	440.89
Dual form			440.2.b.a.89.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000i q^{3} +(-1.00000 - 2.00000i) q^{5} -2.00000i q^{7} -1.00000 q^{9} +O(q^{10})$	$q-2.00000i q^{3} +(-1.00000 - 2.00000i) q^{5} -2.00000i q^{7} -1.00000 q^{9} -1.00000 q^{11} +(-4.00000 + 2.00000i) q^{15} +4.00000i q^{17} -4.00000 q^{19} -4.00000 q^{21} -6.00000i q^{23} +(-3.00000 + 4.00000i) q^{25} -4.00000i q^{27} -2.00000 q^{29} +8.00000 q^{31} +2.00000i q^{33} +(-4.00000 + 2.00000i) q^{35} +4.00000i q^{37} -6.00000 q^{41} -6.00000i q^{43} +(1.00000 + 2.00000i) q^{45} +2.00000i q^{47} +3.00000 q^{49} +8.00000 q^{51} -12.0000i q^{53} +(1.00000 + 2.00000i) q^{55} +8.00000i q^{57} -4.00000 q^{59} +14.0000 q^{61} +2.00000i q^{63} -10.0000i q^{67} -12.0000 q^{69} +8.00000 q^{71} +4.00000i q^{73} +(8.00000 + 6.00000i) q^{75} +2.00000i q^{77} +8.00000 q^{79} -11.0000 q^{81} +2.00000i q^{83} +(8.00000 - 4.00000i) q^{85} +4.00000i q^{87} +10.0000 q^{89} -16.0000i q^{93} +(4.00000 + 8.00000i) q^{95} +8.00000i q^{97} +1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^5 - 2 * q^9	$ 2 q - 2 q^{5} - 2 q^{9} - 2 q^{11} - 8 q^{15} - 8 q^{19} - 8 q^{21} - 6 q^{25} - 4 q^{29} + 16 q^{31} - 8 q^{35} - 12 q^{41} + 2 q^{45} + 6 q^{49} + 16 q^{51} + 2 q^{55} - 8 q^{59} + 28 q^{61} - 24 q^{69} + 16 q^{71} + 16 q^{75} + 16 q^{79} - 22 q^{81} + 16 q^{85} + 20 q^{89} + 8 q^{95} + 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^5 - 2 * q^9 - 2 * q^11 - 8 * q^15 - 8 * q^19 - 8 * q^21 - 6 * q^25 - 4 * q^29 + 16 * q^31 - 8 * q^35 - 12 * q^41 + 2 * q^45 + 6 * q^49 + 16 * q^51 + 2 * q^55 - 8 * q^59 + 28 * q^61 - 24 * q^69 + 16 * q^71 + 16 * q^75 + 16 * q^79 - 22 * q^81 + 16 * q^85 + 20 * q^89 + 8 * q^95 + 2 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/440\mathbb{Z}\right)^\times$.

$n$	$111$	$177$	$221$	$321$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$3$		−	2.00000i		−	1.15470i	−0.816497	−	0.577350i	$-0.804087\pi$
$3$		−	2.00000i		−	1.15470i	0.816497	−	0.577350i	$-0.195913\pi$
$4$		0			0
$5$	−1.00000	−	2.00000i	−0.447214	−	0.894427i
$5$	−1.00000	−	2.00000i	−0.447214	−	0.894427i
$6$		0			0
$7$		−	2.00000i		−	0.755929i	−0.925820	−	0.377964i	$-0.876624\pi$
$7$		−	2.00000i		−	0.755929i	0.925820	−	0.377964i	$-0.123376\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	−1.00000			−0.301511
$11$	−1.00000			−0.301511
$12$		0			0
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$		0			0
$15$	−4.00000	+	2.00000i	−1.03280	+	0.516398i
$16$		0			0
$17$			4.00000i			0.970143i	0.874475	+	0.485071i	$0.161206\pi$
$17$			4.00000i			0.970143i	−0.874475	+	0.485071i	$0.838794\pi$
$18$		0			0
$19$	−4.00000			−0.917663			−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000			−0.917663			−0.458831	+	0.888523i	$0.651732\pi$
$20$		0			0
$21$	−4.00000			−0.872872
$22$		0			0
$23$		−	6.00000i		−	1.25109i	−0.780189	−	0.625543i	$-0.784877\pi$
$23$		−	6.00000i		−	1.25109i	0.780189	−	0.625543i	$-0.215123\pi$
$24$		0			0
$25$	−3.00000	+	4.00000i	−0.600000	+	0.800000i
$26$		0			0
$27$		−	4.00000i		−	0.769800i
$28$		0			0
$29$	−2.00000			−0.371391			−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000			−0.371391			−0.185695	+	0.982607i	$0.559454\pi$
$30$		0			0
$31$	8.00000			1.43684			0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000			1.43684			0.718421	+	0.695608i	$0.244865\pi$
$32$		0			0
$33$			2.00000i			0.348155i
$34$		0			0
$35$	−4.00000	+	2.00000i	−0.676123	+	0.338062i
$36$		0			0
$37$			4.00000i			0.657596i	0.944400	+	0.328798i	$0.106644\pi$
$37$			4.00000i			0.657596i	−0.944400	+	0.328798i	$0.893356\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−6.00000			−0.937043			−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000			−0.937043			−0.468521	+	0.883452i	$0.655213\pi$
$42$		0			0
$43$		−	6.00000i		−	0.914991i	−0.889212	−	0.457496i	$-0.848747\pi$
$43$		−	6.00000i		−	0.914991i	0.889212	−	0.457496i	$-0.151253\pi$
$44$		0			0
$45$	1.00000	+	2.00000i	0.149071	+	0.298142i
$46$		0			0
$47$			2.00000i			0.291730i	0.989305	+	0.145865i	$0.0465965\pi$
$47$			2.00000i			0.291730i	−0.989305	+	0.145865i	$0.953403\pi$
$48$		0			0
$49$	3.00000			0.428571
$50$		0			0
$51$	8.00000			1.12022
$52$		0			0
$53$		−	12.0000i		−	1.64833i	−0.566352	−	0.824163i	$-0.691646\pi$
$53$		−	12.0000i		−	1.64833i	0.566352	−	0.824163i	$-0.308354\pi$
$54$		0			0
$55$	1.00000	+	2.00000i	0.134840	+	0.269680i
$56$		0			0
$57$			8.00000i			1.05963i
$58$		0			0
$59$	−4.00000			−0.520756			−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000			−0.520756			−0.260378	+	0.965507i	$0.583847\pi$
$60$		0			0
$61$	14.0000			1.79252			0.896258	−	0.443533i	$-0.146275\pi$
$61$	14.0000			1.79252			0.896258	+	0.443533i	$0.146275\pi$
$62$		0			0
$63$			2.00000i			0.251976i
$64$		0			0
$65$		0			0
$66$		0			0
$67$		−	10.0000i		−	1.22169i	−0.791748	−	0.610847i	$-0.790829\pi$
$67$		−	10.0000i		−	1.22169i	0.791748	−	0.610847i	$-0.209171\pi$
$68$		0			0
$69$	−12.0000			−1.44463
$70$		0			0
$71$	8.00000			0.949425			0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000			0.949425			0.474713	+	0.880141i	$0.342552\pi$
$72$		0			0
$73$			4.00000i			0.468165i	0.972217	+	0.234082i	$0.0752085\pi$
$73$			4.00000i			0.468165i	−0.972217	+	0.234082i	$0.924791\pi$
$74$		0			0
$75$	8.00000	+	6.00000i	0.923760	+	0.692820i
$76$		0			0
$77$			2.00000i			0.227921i
$78$		0			0
$79$	8.00000			0.900070			0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000			0.900070			0.450035	+	0.893011i	$0.351411\pi$
$80$		0			0
$81$	−11.0000			−1.22222
$82$		0			0
$83$			2.00000i			0.219529i	0.993958	+	0.109764i	$0.0350096\pi$
$83$			2.00000i			0.219529i	−0.993958	+	0.109764i	$0.964990\pi$
$84$		0			0
$85$	8.00000	−	4.00000i	0.867722	−	0.433861i
$86$		0			0
$87$			4.00000i			0.428845i
$88$		0			0
$89$	10.0000			1.06000			0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000			1.06000			0.529999	+	0.847998i	$0.322192\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		−	16.0000i		−	1.65912i
$94$		0			0
$95$	4.00000	+	8.00000i	0.410391	+	0.820783i
$96$		0			0
$97$			8.00000i			0.812277i	0.913812	+	0.406138i	$0.133125\pi$
$97$			8.00000i			0.812277i	−0.913812	+	0.406138i	$0.866875\pi$
$98$		0			0
$99$	1.00000			0.100504
$100$		0			0
$101$	−10.0000			−0.995037			−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000			−0.995037			−0.497519	+	0.867453i	$0.665755\pi$
$102$		0			0
$103$		−	14.0000i		−	1.37946i	−0.724066	−	0.689730i	$-0.757729\pi$
$103$		−	14.0000i		−	1.37946i	0.724066	−	0.689730i	$-0.242271\pi$
$104$		0			0
$105$	4.00000	+	8.00000i	0.390360	+	0.780720i
$106$		0			0
$107$			2.00000i			0.193347i	0.995316	+	0.0966736i	$0.0308203\pi$
$107$			2.00000i			0.193347i	−0.995316	+	0.0966736i	$0.969180\pi$
$108$		0			0
$109$	14.0000			1.34096			0.670478	−	0.741929i	$-0.266089\pi$
$109$	14.0000			1.34096			0.670478	+	0.741929i	$0.266089\pi$
$110$		0			0
$111$	8.00000			0.759326
$112$		0			0
$113$		−	8.00000i		−	0.752577i	−0.926503	−	0.376288i	$-0.877200\pi$
$113$		−	8.00000i		−	0.752577i	0.926503	−	0.376288i	$-0.122800\pi$
$114$		0			0
$115$	−12.0000	+	6.00000i	−1.11901	+	0.559503i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	8.00000			0.733359
$120$		0			0
$121$	1.00000			0.0909091
$122$		0			0
$123$			12.0000i			1.08200i
$124$		0			0
$125$	11.0000	+	2.00000i	0.983870	+	0.178885i
$126$		0			0
$127$			14.0000i			1.24230i	0.783692	+	0.621150i	$0.213334\pi$
$127$			14.0000i			1.24230i	−0.783692	+	0.621150i	$0.786666\pi$
$128$		0			0
$129$	−12.0000			−1.05654
$130$		0			0
$131$	−12.0000			−1.04844			−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000			−1.04844			−0.524222	+	0.851581i	$0.675644\pi$
$132$		0			0
$133$			8.00000i			0.693688i
$134$		0			0
$135$	−8.00000	+	4.00000i	−0.688530	+	0.344265i
$136$		0			0
$137$		0			0		1.00000			$0$
$137$		0			0		−1.00000			$\pi$
$138$		0			0
$139$	12.0000			1.01783			0.508913	−	0.860818i	$-0.330047\pi$
$139$	12.0000			1.01783			0.508913	+	0.860818i	$0.330047\pi$
$140$		0			0
$141$	4.00000			0.336861
$142$		0			0
$143$		0			0
$144$		0			0
$145$	2.00000	+	4.00000i	0.166091	+	0.332182i
$146$		0			0
$147$		−	6.00000i		−	0.494872i
$148$		0			0
$149$	22.0000			1.80231			0.901155	−	0.433497i	$-0.142720\pi$
$149$	22.0000			1.80231			0.901155	+	0.433497i	$0.142720\pi$
$150$		0			0
$151$	−8.00000			−0.651031			−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000			−0.651031			−0.325515	+	0.945537i	$0.605538\pi$
$152$		0			0
$153$		−	4.00000i		−	0.323381i
$154$		0			0
$155$	−8.00000	−	16.0000i	−0.642575	−	1.28515i
$156$		0			0
$157$			4.00000i			0.319235i	0.987179	+	0.159617i	$0.0510260\pi$
$157$			4.00000i			0.319235i	−0.987179	+	0.159617i	$0.948974\pi$
$158$		0			0
$159$	−24.0000			−1.90332
$160$		0			0
$161$	−12.0000			−0.945732
$162$		0			0
$163$			14.0000i			1.09656i	0.836293	+	0.548282i	$0.184718\pi$
$163$			14.0000i			1.09656i	−0.836293	+	0.548282i	$0.815282\pi$
$164$		0			0
$165$	4.00000	−	2.00000i	0.311400	−	0.155700i
$166$		0			0
$167$			6.00000i			0.464294i	0.972681	+	0.232147i	$0.0745750\pi$
$167$			6.00000i			0.464294i	−0.972681	+	0.232147i	$0.925425\pi$
$168$		0			0
$169$	13.0000			1.00000
$170$		0			0
$171$	4.00000			0.305888
$172$		0			0
$173$		−	8.00000i		−	0.608229i	−0.952636	−	0.304114i	$-0.901639\pi$
$173$		−	8.00000i		−	0.608229i	0.952636	−	0.304114i	$-0.0983605\pi$
$174$		0			0
$175$	8.00000	+	6.00000i	0.604743	+	0.453557i
$176$		0			0
$177$			8.00000i			0.601317i
$178$		0			0
$179$	−20.0000			−1.49487			−0.747435	−	0.664335i	$-0.768715\pi$
$179$	−20.0000			−1.49487			−0.747435	+	0.664335i	$0.768715\pi$
$180$		0			0
$181$	−10.0000			−0.743294			−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000			−0.743294			−0.371647	+	0.928374i	$0.621207\pi$
$182$		0			0
$183$		−	28.0000i		−	2.06982i
$184$		0			0
$185$	8.00000	−	4.00000i	0.588172	−	0.294086i
$186$		0			0
$187$		−	4.00000i		−	0.292509i
$188$		0			0
$189$	−8.00000			−0.581914
$190$		0			0
$191$	−24.0000			−1.73658			−0.868290	−	0.496058i	$-0.834780\pi$
$191$	−24.0000			−1.73658			−0.868290	+	0.496058i	$0.834780\pi$
$192$		0			0
$193$		−	4.00000i		−	0.287926i	−0.989583	−	0.143963i	$-0.954015\pi$
$193$		−	4.00000i		−	0.287926i	0.989583	−	0.143963i	$-0.0459847\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	24.0000i		−	1.70993i	−0.518686	−	0.854965i	$-0.673579\pi$
$197$		−	24.0000i		−	1.70993i	0.518686	−	0.854965i	$-0.326421\pi$
$198$		0			0
$199$	−8.00000			−0.567105			−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000			−0.567105			−0.283552	+	0.958957i	$0.591513\pi$
$200$		0			0
$201$	−20.0000			−1.41069
$202$		0			0
$203$			4.00000i			0.280745i
$204$		0			0
$205$	6.00000	+	12.0000i	0.419058	+	0.838116i
$206$		0			0
$207$			6.00000i			0.417029i
$208$		0			0
$209$	4.00000			0.276686
$210$		0			0
$211$	20.0000			1.37686			0.688428	−	0.725304i	$-0.258301\pi$
$211$	20.0000			1.37686			0.688428	+	0.725304i	$0.258301\pi$
$212$		0			0
$213$		−	16.0000i		−	1.09630i
$214$		0			0
$215$	−12.0000	+	6.00000i	−0.818393	+	0.409197i
$216$		0			0
$217$		−	16.0000i		−	1.08615i
$218$		0			0
$219$	8.00000			0.540590
$220$		0			0
$221$		0			0
$222$		0			0
$223$		−	14.0000i		−	0.937509i	−0.883328	−	0.468755i	$-0.844703\pi$
$223$		−	14.0000i		−	0.937509i	0.883328	−	0.468755i	$-0.155297\pi$
$224$		0			0
$225$	3.00000	−	4.00000i	0.200000	−	0.266667i
$226$		0			0
$227$			18.0000i			1.19470i	0.801980	+	0.597351i	$0.203780\pi$
$227$			18.0000i			1.19470i	−0.801980	+	0.597351i	$0.796220\pi$
$228$		0			0
$229$	6.00000			0.396491			0.198246	−	0.980152i	$-0.436476\pi$
$229$	6.00000			0.396491			0.198246	+	0.980152i	$0.436476\pi$
$230$		0			0
$231$	4.00000			0.263181
$232$		0			0
$233$		−	4.00000i		−	0.262049i	−0.991379	−	0.131024i	$-0.958173\pi$
$233$		−	4.00000i		−	0.262049i	0.991379	−	0.131024i	$-0.0418266\pi$
$234$		0			0
$235$	4.00000	−	2.00000i	0.260931	−	0.130466i
$236$		0			0
$237$		−	16.0000i		−	1.03931i
$238$		0			0
$239$	−8.00000			−0.517477			−0.258738	−	0.965947i	$-0.583307\pi$
$239$	−8.00000			−0.517477			−0.258738	+	0.965947i	$0.583307\pi$
$240$		0			0
$241$	−30.0000			−1.93247			−0.966235	−	0.257663i	$-0.917048\pi$
$241$	−30.0000			−1.93247			−0.966235	+	0.257663i	$0.917048\pi$
$242$		0			0
$243$			10.0000i			0.641500i
$244$		0			0
$245$	−3.00000	−	6.00000i	−0.191663	−	0.383326i
$246$		0			0
$247$		0			0
$248$		0			0
$249$	4.00000			0.253490
$250$		0			0
$251$	12.0000			0.757433			0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000			0.757433			0.378717	+	0.925513i	$0.376365\pi$
$252$		0			0
$253$			6.00000i			0.377217i
$254$		0			0
$255$	−8.00000	−	16.0000i	−0.500979	−	1.00196i
$256$		0			0
$257$		0			0		1.00000			$0$
$257$		0			0		−1.00000			$\pi$
$258$		0			0
$259$	8.00000			0.497096
$260$		0			0
$261$	2.00000			0.123797
$262$		0			0
$263$			6.00000i			0.369976i	0.982741	+	0.184988i	$0.0592246\pi$
$263$			6.00000i			0.369976i	−0.982741	+	0.184988i	$0.940775\pi$
$264$		0			0
$265$	−24.0000	+	12.0000i	−1.47431	+	0.737154i
$266$		0			0
$267$		−	20.0000i		−	1.22398i
$268$		0			0
$269$	−26.0000			−1.58525			−0.792624	−	0.609711i	$-0.791286\pi$
$269$	−26.0000			−1.58525			−0.792624	+	0.609711i	$0.791286\pi$
$270$		0			0
$271$		0			0			−	1.00000i	$-0.5\pi$
$271$		0			0				1.00000i	$0.5\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	3.00000	−	4.00000i	0.180907	−	0.241209i
$276$		0			0
$277$		−	32.0000i		−	1.92269i	−0.275340	−	0.961347i	$-0.588791\pi$
$277$		−	32.0000i		−	1.92269i	0.275340	−	0.961347i	$-0.411209\pi$
$278$		0			0
$279$	−8.00000			−0.478947
$280$		0			0
$281$	26.0000			1.55103			0.775515	−	0.631329i	$-0.217490\pi$
$281$	26.0000			1.55103			0.775515	+	0.631329i	$0.217490\pi$
$282$		0			0
$283$			26.0000i			1.54554i	0.634686	+	0.772770i	$0.281129\pi$
$283$			26.0000i			1.54554i	−0.634686	+	0.772770i	$0.718871\pi$
$284$		0			0
$285$	16.0000	−	8.00000i	0.947758	−	0.473879i
$286$		0			0
$287$			12.0000i			0.708338i
$288$		0			0
$289$	1.00000			0.0588235
$290$		0			0
$291$	16.0000			0.937937
$292$		0			0
$293$			8.00000i			0.467365i	0.972313	+	0.233682i	$0.0750776\pi$
$293$			8.00000i			0.467365i	−0.972313	+	0.233682i	$0.924922\pi$
$294$		0			0
$295$	4.00000	+	8.00000i	0.232889	+	0.465778i
$296$		0			0
$297$			4.00000i			0.232104i
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−12.0000			−0.691669
$302$		0			0
$303$			20.0000i			1.14897i
$304$		0			0
$305$	−14.0000	−	28.0000i	−0.801638	−	1.60328i
$306$		0			0
$307$			26.0000i			1.48390i	0.670456	+	0.741949i	$0.266098\pi$
$307$			26.0000i			1.48390i	−0.670456	+	0.741949i	$0.733902\pi$
$308$		0			0
$309$	−28.0000			−1.59286
$310$		0			0
$311$	8.00000			0.453638			0.226819	−	0.973937i	$-0.427167\pi$
$311$	8.00000			0.453638			0.226819	+	0.973937i	$0.427167\pi$
$312$		0			0
$313$			24.0000i			1.35656i	0.734803	+	0.678280i	$0.237274\pi$
$313$			24.0000i			1.35656i	−0.734803	+	0.678280i	$0.762726\pi$
$314$		0			0
$315$	4.00000	−	2.00000i	0.225374	−	0.112687i
$316$		0			0
$317$			20.0000i			1.12331i	0.827371	+	0.561656i	$0.189836\pi$
$317$			20.0000i			1.12331i	−0.827371	+	0.561656i	$0.810164\pi$
$318$		0			0
$319$	2.00000			0.111979
$320$		0			0
$321$	4.00000			0.223258
$322$		0			0
$323$		−	16.0000i		−	0.890264i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		−	28.0000i		−	1.54840i
$328$		0			0
$329$	4.00000			0.220527
$330$		0			0
$331$	20.0000			1.09930			0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000			1.09930			0.549650	+	0.835395i	$0.314761\pi$
$332$		0			0
$333$		−	4.00000i		−	0.219199i
$334$		0			0
$335$	−20.0000	+	10.0000i	−1.09272	+	0.546358i
$336$		0			0
$337$			20.0000i			1.08947i	0.838608	+	0.544735i	$0.183370\pi$
$337$			20.0000i			1.08947i	−0.838608	+	0.544735i	$0.816630\pi$
$338$		0			0
$339$	−16.0000			−0.869001
$340$		0			0
$341$	−8.00000			−0.433224
$342$		0			0
$343$		−	20.0000i		−	1.07990i
$344$		0			0
$345$	12.0000	+	24.0000i	0.646058	+	1.29212i
$346$		0			0
$347$			18.0000i			0.966291i	0.875540	+	0.483145i	$0.160506\pi$
$347$			18.0000i			0.966291i	−0.875540	+	0.483145i	$0.839494\pi$
$348$		0			0
$349$	14.0000			0.749403			0.374701	−	0.927146i	$-0.377745\pi$
$349$	14.0000			0.749403			0.374701	+	0.927146i	$0.377745\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			32.0000i			1.70319i	0.524202	+	0.851594i	$0.324364\pi$
$353$			32.0000i			1.70319i	−0.524202	+	0.851594i	$0.675636\pi$
$354$		0			0
$355$	−8.00000	−	16.0000i	−0.424596	−	0.849192i
$356$		0			0
$357$		−	16.0000i		−	0.846810i
$358$		0			0
$359$	16.0000			0.844448			0.422224	−	0.906492i	$-0.361250\pi$
$359$	16.0000			0.844448			0.422224	+	0.906492i	$0.361250\pi$
$360$		0			0
$361$	−3.00000			−0.157895
$362$		0			0
$363$		−	2.00000i		−	0.104973i
$364$		0			0
$365$	8.00000	−	4.00000i	0.418739	−	0.209370i
$366$		0			0
$367$			18.0000i			0.939592i	0.882775	+	0.469796i	$0.155673\pi$
$367$			18.0000i			0.939592i	−0.882775	+	0.469796i	$0.844327\pi$
$368$		0			0
$369$	6.00000			0.312348
$370$		0			0
$371$	−24.0000			−1.24602
$372$		0			0
$373$			32.0000i			1.65690i	0.560065	+	0.828449i	$0.310776\pi$
$373$			32.0000i			1.65690i	−0.560065	+	0.828449i	$0.689224\pi$
$374$		0			0
$375$	4.00000	−	22.0000i	0.206559	−	1.13608i
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−20.0000			−1.02733			−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000			−1.02733			−0.513665	+	0.857991i	$0.671713\pi$
$380$		0			0
$381$	28.0000			1.43448
$382$		0			0
$383$		−	6.00000i		−	0.306586i	−0.988181	−	0.153293i	$-0.951012\pi$
$383$		−	6.00000i		−	0.306586i	0.988181	−	0.153293i	$-0.0489878\pi$
$384$		0			0
$385$	4.00000	−	2.00000i	0.203859	−	0.101929i
$386$		0			0
$387$			6.00000i			0.304997i
$388$		0			0
$389$	−18.0000			−0.912636			−0.456318	−	0.889817i	$-0.650832\pi$
$389$	−18.0000			−0.912636			−0.456318	+	0.889817i	$0.650832\pi$
$390$		0			0
$391$	24.0000			1.21373
$392$		0			0
$393$			24.0000i			1.21064i
$394$		0			0
$395$	−8.00000	−	16.0000i	−0.402524	−	0.805047i
$396$		0			0
$397$			4.00000i			0.200754i	0.994949	+	0.100377i	$0.0320049\pi$
$397$			4.00000i			0.200754i	−0.994949	+	0.100377i	$0.967995\pi$
$398$		0			0
$399$	16.0000			0.801002
$400$		0			0
$401$	34.0000			1.69788			0.848939	−	0.528490i	$-0.177242\pi$
$401$	34.0000			1.69788			0.848939	+	0.528490i	$0.177242\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	11.0000	+	22.0000i	0.546594	+	1.09319i
$406$		0			0
$407$		−	4.00000i		−	0.198273i
$408$		0			0
$409$	26.0000			1.28562			0.642809	−	0.766027i	$-0.277769\pi$
$409$	26.0000			1.28562			0.642809	+	0.766027i	$0.277769\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			8.00000i			0.393654i
$414$		0			0
$415$	4.00000	−	2.00000i	0.196352	−	0.0981761i
$416$		0			0
$417$		−	24.0000i		−	1.17529i
$418$		0			0
$419$	28.0000			1.36789			0.683945	−	0.729534i	$-0.260263\pi$
$419$	28.0000			1.36789			0.683945	+	0.729534i	$0.260263\pi$
$420$		0			0
$421$	30.0000			1.46211			0.731055	−	0.682318i	$-0.239028\pi$
$421$	30.0000			1.46211			0.731055	+	0.682318i	$0.239028\pi$
$422$		0			0
$423$		−	2.00000i		−	0.0972433i
$424$		0			0
$425$	−16.0000	−	12.0000i	−0.776114	−	0.582086i
$426$		0			0
$427$		−	28.0000i		−	1.35501i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	24.0000			1.15604			0.578020	−	0.816023i	$-0.303826\pi$
$431$	24.0000			1.15604			0.578020	+	0.816023i	$0.303826\pi$
$432$		0			0
$433$		−	8.00000i		−	0.384455i	−0.981350	−	0.192228i	$-0.938429\pi$
$433$		−	8.00000i		−	0.384455i	0.981350	−	0.192228i	$-0.0615712\pi$
$434$		0			0
$435$	8.00000	−	4.00000i	0.383571	−	0.191785i
$436$		0			0
$437$			24.0000i			1.14808i
$438$		0			0
$439$	−8.00000			−0.381819			−0.190910	−	0.981608i	$-0.561144\pi$
$439$	−8.00000			−0.381819			−0.190910	+	0.981608i	$0.561144\pi$
$440$		0			0
$441$	−3.00000			−0.142857
$442$		0			0
$443$		−	26.0000i		−	1.23530i	−0.786454	−	0.617649i	$-0.788085\pi$
$443$		−	26.0000i		−	1.23530i	0.786454	−	0.617649i	$-0.211915\pi$
$444$		0			0
$445$	−10.0000	−	20.0000i	−0.474045	−	0.948091i
$446$		0			0
$447$		−	44.0000i		−	2.08113i
$448$		0			0
$449$	26.0000			1.22702			0.613508	−	0.789689i	$-0.289758\pi$
$449$	26.0000			1.22702			0.613508	+	0.789689i	$0.289758\pi$
$450$		0			0
$451$	6.00000			0.282529
$452$		0			0
$453$			16.0000i			0.751746i
$454$		0			0
$455$		0			0
$456$		0			0
$457$		−	28.0000i		−	1.30978i	−0.755722	−	0.654892i	$-0.772714\pi$
$457$		−	28.0000i		−	1.30978i	0.755722	−	0.654892i	$-0.227286\pi$
$458$		0			0
$459$	16.0000			0.746816
$460$		0			0
$461$	−18.0000			−0.838344			−0.419172	−	0.907907i	$-0.637680\pi$
$461$	−18.0000			−0.838344			−0.419172	+	0.907907i	$0.637680\pi$
$462$		0			0
$463$			18.0000i			0.836531i	0.908325	+	0.418265i	$0.137362\pi$
$463$			18.0000i			0.836531i	−0.908325	+	0.418265i	$0.862638\pi$
$464$		0			0
$465$	−32.0000	+	16.0000i	−1.48396	+	0.741982i
$466$		0			0
$467$			22.0000i			1.01804i	0.860755	+	0.509019i	$0.169992\pi$
$467$			22.0000i			1.01804i	−0.860755	+	0.509019i	$0.830008\pi$
$468$		0			0
$469$	−20.0000			−0.923514
$470$		0			0
$471$	8.00000			0.368621
$472$		0			0
$473$			6.00000i			0.275880i
$474$		0			0
$475$	12.0000	−	16.0000i	0.550598	−	0.734130i
$476$		0			0
$477$			12.0000i			0.549442i
$478$		0			0
$479$	−8.00000			−0.365529			−0.182765	−	0.983157i	$-0.558505\pi$
$479$	−8.00000			−0.365529			−0.182765	+	0.983157i	$0.558505\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$			24.0000i			1.09204i
$484$		0			0
$485$	16.0000	−	8.00000i	0.726523	−	0.363261i
$486$		0			0
$487$		−	6.00000i		−	0.271886i	−0.990717	−	0.135943i	$-0.956594\pi$
$487$		−	6.00000i		−	0.271886i	0.990717	−	0.135943i	$-0.0434064\pi$
$488$		0			0
$489$	28.0000			1.26620
$490$		0			0
$491$	−12.0000			−0.541552			−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000			−0.541552			−0.270776	+	0.962642i	$0.587280\pi$
$492$		0			0
$493$		−	8.00000i		−	0.360302i
$494$		0			0
$495$	−1.00000	−	2.00000i	−0.0449467	−	0.0898933i
$496$		0			0
$497$		−	16.0000i		−	0.717698i
$498$		0			0
$499$	20.0000			0.895323			0.447661	−	0.894203i	$-0.352257\pi$
$499$	20.0000			0.895323			0.447661	+	0.894203i	$0.352257\pi$
$500$		0			0
$501$	12.0000			0.536120
$502$		0			0
$503$			30.0000i			1.33763i	0.743427	+	0.668817i	$0.233199\pi$
$503$			30.0000i			1.33763i	−0.743427	+	0.668817i	$0.766801\pi$
$504$		0			0
$505$	10.0000	+	20.0000i	0.444994	+	0.889988i
$506$		0			0
$507$		−	26.0000i		−	1.15470i
$508$		0			0
$509$	−2.00000			−0.0886484			−0.0443242	−	0.999017i	$-0.514113\pi$
$509$	−2.00000			−0.0886484			−0.0443242	+	0.999017i	$0.514113\pi$
$510$		0			0
$511$	8.00000			0.353899
$512$		0			0
$513$			16.0000i			0.706417i
$514$		0			0
$515$	−28.0000	+	14.0000i	−1.23383	+	0.616914i
$516$		0			0
$517$		−	2.00000i		−	0.0879599i
$518$		0			0
$519$	−16.0000			−0.702322
$520$		0			0
$521$	−38.0000			−1.66481			−0.832405	−	0.554168i	$-0.813037\pi$
$521$	−38.0000			−1.66481			−0.832405	+	0.554168i	$0.813037\pi$
$522$		0			0
$523$		−	14.0000i		−	0.612177i	−0.952003	−	0.306089i	$-0.900980\pi$
$523$		−	14.0000i		−	0.612177i	0.952003	−	0.306089i	$-0.0990204\pi$
$524$		0			0
$525$	12.0000	−	16.0000i	0.523723	−	0.698297i
$526$		0			0
$527$			32.0000i			1.39394i
$528$		0			0
$529$	−13.0000			−0.565217
$530$		0			0
$531$	4.00000			0.173585
$532$		0			0
$533$		0			0
$534$		0			0
$535$	4.00000	−	2.00000i	0.172935	−	0.0864675i
$536$		0			0
$537$			40.0000i			1.72613i
$538$		0			0
$539$	−3.00000			−0.129219
$540$		0			0
$541$	−18.0000			−0.773880			−0.386940	−	0.922105i	$-0.626468\pi$
$541$	−18.0000			−0.773880			−0.386940	+	0.922105i	$0.626468\pi$
$542$		0			0
$543$			20.0000i			0.858282i
$544$		0			0
$545$	−14.0000	−	28.0000i	−0.599694	−	1.19939i
$546$		0			0
$547$		−	30.0000i		−	1.28271i	−0.767245	−	0.641354i	$-0.778373\pi$
$547$		−	30.0000i		−	1.28271i	0.767245	−	0.641354i	$-0.221627\pi$
$548$		0			0
$549$	−14.0000			−0.597505
$550$		0			0
$551$	8.00000			0.340811
$552$		0			0
$553$		−	16.0000i		−	0.680389i
$554$		0			0
$555$	−8.00000	−	16.0000i	−0.339581	−	0.679162i
$556$		0			0
$557$			32.0000i			1.35588i	0.735116	+	0.677942i	$0.237128\pi$
$557$			32.0000i			1.35588i	−0.735116	+	0.677942i	$0.762872\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	−8.00000			−0.337760
$562$		0			0
$563$			42.0000i			1.77009i	0.465506	+	0.885044i	$0.345872\pi$
$563$			42.0000i			1.77009i	−0.465506	+	0.885044i	$0.654128\pi$
$564$		0			0
$565$	−16.0000	+	8.00000i	−0.673125	+	0.336563i
$566$		0			0
$567$			22.0000i			0.923913i
$568$		0			0
$569$	−6.00000			−0.251533			−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000			−0.251533			−0.125767	+	0.992060i	$0.540139\pi$
$570$		0			0
$571$	−44.0000			−1.84134			−0.920671	−	0.390339i	$-0.872358\pi$
$571$	−44.0000			−1.84134			−0.920671	+	0.390339i	$0.872358\pi$
$572$		0			0
$573$			48.0000i			2.00523i
$574$		0			0
$575$	24.0000	+	18.0000i	1.00087	+	0.750652i
$576$		0			0
$577$		−	8.00000i		−	0.333044i	−0.986038	−	0.166522i	$-0.946746\pi$
$577$		−	8.00000i		−	0.333044i	0.986038	−	0.166522i	$-0.0532537\pi$
$578$		0			0
$579$	−8.00000			−0.332469
$580$		0			0
$581$	4.00000			0.165948
$582$		0			0
$583$			12.0000i			0.496989i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	18.0000i		−	0.742940i	−0.928445	−	0.371470i	$-0.878854\pi$
$587$		−	18.0000i		−	0.742940i	0.928445	−	0.371470i	$-0.121146\pi$
$588$		0			0
$589$	−32.0000			−1.31854
$590$		0			0
$591$	−48.0000			−1.97446
$592$		0			0
$593$		−	36.0000i		−	1.47834i	−0.673517	−	0.739171i	$-0.735217\pi$
$593$		−	36.0000i		−	1.47834i	0.673517	−	0.739171i	$-0.264783\pi$
$594$		0			0
$595$	−8.00000	−	16.0000i	−0.327968	−	0.655936i
$596$		0			0
$597$			16.0000i			0.654836i
$598$		0			0
$599$	32.0000			1.30748			0.653742	−	0.756717i	$-0.273198\pi$
$599$	32.0000			1.30748			0.653742	+	0.756717i	$0.273198\pi$
$600$		0			0
$601$	−22.0000			−0.897399			−0.448699	−	0.893683i	$-0.648113\pi$
$601$	−22.0000			−0.897399			−0.448699	+	0.893683i	$0.648113\pi$
$602$		0			0
$603$			10.0000i			0.407231i
$604$		0			0
$605$	−1.00000	−	2.00000i	−0.0406558	−	0.0813116i
$606$		0			0
$607$		−	18.0000i		−	0.730597i	−0.930890	−	0.365299i	$-0.880967\pi$
$607$		−	18.0000i		−	0.730597i	0.930890	−	0.365299i	$-0.119033\pi$
$608$		0			0
$609$	8.00000			0.324176
$610$		0			0
$611$		0			0
$612$		0			0
$613$			16.0000i			0.646234i	0.946359	+	0.323117i	$0.104731\pi$
$613$			16.0000i			0.646234i	−0.946359	+	0.323117i	$0.895269\pi$
$614$		0			0
$615$	24.0000	−	12.0000i	0.967773	−	0.483887i
$616$		0			0
$617$		−	32.0000i		−	1.28827i	−0.764911	−	0.644136i	$-0.777217\pi$
$617$		−	32.0000i		−	1.28827i	0.764911	−	0.644136i	$-0.222783\pi$
$618$		0			0
$619$	−12.0000			−0.482321			−0.241160	−	0.970485i	$-0.577528\pi$
$619$	−12.0000			−0.482321			−0.241160	+	0.970485i	$0.577528\pi$
$620$		0			0
$621$	−24.0000			−0.963087
$622$		0			0
$623$		−	20.0000i		−	0.801283i
$624$		0			0
$625$	−7.00000	−	24.0000i	−0.280000	−	0.960000i
$626$		0			0
$627$		−	8.00000i		−	0.319489i
$628$		0			0
$629$	−16.0000			−0.637962
$630$		0			0
$631$	24.0000			0.955425			0.477712	−	0.878516i	$-0.341466\pi$
$631$	24.0000			0.955425			0.477712	+	0.878516i	$0.341466\pi$
$632$		0			0
$633$		−	40.0000i		−	1.58986i
$634$		0			0
$635$	28.0000	−	14.0000i	1.11115	−	0.555573i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	−8.00000			−0.316475
$640$		0			0
$641$	26.0000			1.02694			0.513469	−	0.858108i	$-0.328360\pi$
$641$	26.0000			1.02694			0.513469	+	0.858108i	$0.328360\pi$
$642$		0			0
$643$		−	2.00000i		−	0.0788723i	−0.999222	−	0.0394362i	$-0.987444\pi$
$643$		−	2.00000i		−	0.0788723i	0.999222	−	0.0394362i	$-0.0125562\pi$
$644$		0			0
$645$	12.0000	+	24.0000i	0.472500	+	0.944999i
$646$		0			0
$647$		−	30.0000i		−	1.17942i	−0.807614	−	0.589711i	$-0.799242\pi$
$647$		−	30.0000i		−	1.17942i	0.807614	−	0.589711i	$-0.200758\pi$
$648$		0			0
$649$	4.00000			0.157014
$650$		0			0
$651$	−32.0000			−1.25418
$652$		0			0
$653$		−	36.0000i		−	1.40879i	−0.709809	−	0.704394i	$-0.751219\pi$
$653$		−	36.0000i		−	1.40879i	0.709809	−	0.704394i	$-0.248781\pi$
$654$		0			0
$655$	12.0000	+	24.0000i	0.468879	+	0.937758i
$656$		0			0
$657$		−	4.00000i		−	0.156055i
$658$		0			0
$659$	−12.0000			−0.467454			−0.233727	−	0.972302i	$-0.575092\pi$
$659$	−12.0000			−0.467454			−0.233727	+	0.972302i	$0.575092\pi$
$660$		0			0
$661$	−2.00000			−0.0777910			−0.0388955	−	0.999243i	$-0.512384\pi$
$661$	−2.00000			−0.0777910			−0.0388955	+	0.999243i	$0.512384\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	16.0000	−	8.00000i	0.620453	−	0.310227i
$666$		0			0
$667$			12.0000i			0.464642i
$668$		0			0
$669$	−28.0000			−1.08254
$670$		0			0
$671$	−14.0000			−0.540464
$672$		0			0
$673$			44.0000i			1.69608i	0.529936	+	0.848038i	$0.322216\pi$
$673$			44.0000i			1.69608i	−0.529936	+	0.848038i	$0.677784\pi$
$674$		0			0
$675$	16.0000	+	12.0000i	0.615840	+	0.461880i
$676$		0			0
$677$		0			0		1.00000			$0$
$677$		0			0		−1.00000			$\pi$
$678$		0			0
$679$	16.0000			0.614024
$680$		0			0
$681$	36.0000			1.37952
$682$		0			0
$683$		−	18.0000i		−	0.688751i	−0.938832	−	0.344375i	$-0.888091\pi$
$683$		−	18.0000i		−	0.688751i	0.938832	−	0.344375i	$-0.111909\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		−	12.0000i		−	0.457829i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−44.0000			−1.67384			−0.836919	−	0.547326i	$-0.815646\pi$
$691$	−44.0000			−1.67384			−0.836919	+	0.547326i	$0.815646\pi$
$692$		0			0
$693$		−	2.00000i		−	0.0759737i
$694$		0			0
$695$	−12.0000	−	24.0000i	−0.455186	−	0.910372i
$696$		0			0
$697$		−	24.0000i		−	0.909065i
$698$		0			0
$699$	−8.00000			−0.302588
$700$		0			0
$701$	30.0000			1.13308			0.566542	−	0.824033i	$-0.308281\pi$
$701$	30.0000			1.13308			0.566542	+	0.824033i	$0.308281\pi$
$702$		0			0
$703$		−	16.0000i		−	0.603451i
$704$		0			0
$705$	−4.00000	−	8.00000i	−0.150649	−	0.301297i
$706$		0			0
$707$			20.0000i			0.752177i
$708$		0			0
$709$	6.00000			0.225335			0.112667	−	0.993633i	$-0.464061\pi$
$709$	6.00000			0.225335			0.112667	+	0.993633i	$0.464061\pi$
$710$		0			0
$711$	−8.00000			−0.300023
$712$		0			0
$713$		−	48.0000i		−	1.79761i
$714$		0			0
$715$		0			0
$716$		0			0
$717$			16.0000i			0.597531i
$718$		0			0
$719$	8.00000			0.298350			0.149175	−	0.988811i	$-0.452338\pi$
$719$	8.00000			0.298350			0.149175	+	0.988811i	$0.452338\pi$
$720$		0			0
$721$	−28.0000			−1.04277
$722$		0			0
$723$			60.0000i			2.23142i
$724$		0			0
$725$	6.00000	−	8.00000i	0.222834	−	0.297113i
$726$		0			0
$727$		−	30.0000i		−	1.11264i	−0.830969	−	0.556319i	$-0.812213\pi$
$727$		−	30.0000i		−	1.11264i	0.830969	−	0.556319i	$-0.187787\pi$
$728$		0			0
$729$	−13.0000			−0.481481
$730$		0			0
$731$	24.0000			0.887672
$732$		0			0
$733$		−	24.0000i		−	0.886460i	−0.896408	−	0.443230i	$-0.853832\pi$
$733$		−	24.0000i		−	0.886460i	0.896408	−	0.443230i	$-0.146168\pi$
$734$		0			0
$735$	−12.0000	+	6.00000i	−0.442627	+	0.221313i
$736$		0			0
$737$			10.0000i			0.368355i
$738$		0			0
$739$	−44.0000			−1.61857			−0.809283	−	0.587419i	$-0.800144\pi$
$739$	−44.0000			−1.61857			−0.809283	+	0.587419i	$0.800144\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			6.00000i			0.220119i	0.993925	+	0.110059i	$0.0351041\pi$
$743$			6.00000i			0.220119i	−0.993925	+	0.110059i	$0.964896\pi$
$744$		0			0
$745$	−22.0000	−	44.0000i	−0.806018	−	1.61204i
$746$		0			0
$747$		−	2.00000i		−	0.0731762i
$748$		0			0
$749$	4.00000			0.146157
$750$		0			0
$751$	−32.0000			−1.16770			−0.583848	−	0.811863i	$-0.698454\pi$
$751$	−32.0000			−1.16770			−0.583848	+	0.811863i	$0.698454\pi$
$752$		0			0
$753$		−	24.0000i		−	0.874609i
$754$		0			0
$755$	8.00000	+	16.0000i	0.291150	+	0.582300i
$756$		0			0
$757$		−	20.0000i		−	0.726912i	−0.931611	−	0.363456i	$-0.881597\pi$
$757$		−	20.0000i		−	0.726912i	0.931611	−	0.363456i	$-0.118403\pi$
$758$		0			0
$759$	12.0000			0.435572
$760$		0			0
$761$	10.0000			0.362500			0.181250	−	0.983437i	$-0.441986\pi$
$761$	10.0000			0.362500			0.181250	+	0.983437i	$0.441986\pi$
$762$		0			0
$763$		−	28.0000i		−	1.01367i
$764$		0			0
$765$	−8.00000	+	4.00000i	−0.289241	+	0.144620i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	34.0000			1.22607			0.613036	−	0.790055i	$-0.289948\pi$
$769$	34.0000			1.22607			0.613036	+	0.790055i	$0.289948\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			36.0000i			1.29483i	0.762138	+	0.647415i	$0.224150\pi$
$773$			36.0000i			1.29483i	−0.762138	+	0.647415i	$0.775850\pi$
$774$		0			0
$775$	−24.0000	+	32.0000i	−0.862105	+	1.14947i
$776$		0			0
$777$		−	16.0000i		−	0.573997i
$778$		0			0
$779$	24.0000			0.859889
$780$		0			0
$781$	−8.00000			−0.286263
$782$		0			0
$783$			8.00000i			0.285897i
$784$		0			0
$785$	8.00000	−	4.00000i	0.285532	−	0.142766i
$786$		0			0
$787$		−	6.00000i		−	0.213877i	−0.994266	−	0.106938i	$-0.965895\pi$
$787$		−	6.00000i		−	0.213877i	0.994266	−	0.106938i	$-0.0341048\pi$
$788$		0			0
$789$	12.0000			0.427211
$790$		0			0
$791$	−16.0000			−0.568895
$792$		0			0
$793$		0			0
$794$		0			0
$795$	24.0000	+	48.0000i	0.851192	+	1.70238i
$796$		0			0
$797$			36.0000i			1.27519i	0.770374	+	0.637593i	$0.220070\pi$
$797$			36.0000i			1.27519i	−0.770374	+	0.637593i	$0.779930\pi$
$798$		0			0
$799$	−8.00000			−0.283020
$800$		0			0
$801$	−10.0000			−0.353333
$802$		0			0
$803$		−	4.00000i		−	0.141157i
$804$		0			0
$805$	12.0000	+	24.0000i	0.422944	+	0.845889i
$806$		0			0
$807$			52.0000i			1.83049i
$808$		0			0
$809$	−22.0000			−0.773479			−0.386739	−	0.922189i	$-0.626399\pi$
$809$	−22.0000			−0.773479			−0.386739	+	0.922189i	$0.626399\pi$
$810$		0			0
$811$	4.00000			0.140459			0.0702295	−	0.997531i	$-0.477627\pi$
$811$	4.00000			0.140459			0.0702295	+	0.997531i	$0.477627\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	28.0000	−	14.0000i	0.980797	−	0.490399i
$816$		0			0
$817$			24.0000i			0.839654i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	6.00000			0.209401			0.104701	−	0.994504i	$-0.466612\pi$
$821$	6.00000			0.209401			0.104701	+	0.994504i	$0.466612\pi$
$822$		0			0
$823$		−	14.0000i		−	0.488009i	−0.969774	−	0.244005i	$-0.921539\pi$
$823$		−	14.0000i		−	0.488009i	0.969774	−	0.244005i	$-0.0784612\pi$
$824$		0			0
$825$	−8.00000	−	6.00000i	−0.278524	−	0.208893i
$826$		0			0
$827$			34.0000i			1.18230i	0.806563	+	0.591148i	$0.201325\pi$
$827$			34.0000i			1.18230i	−0.806563	+	0.591148i	$0.798675\pi$
$828$		0			0
$829$	−10.0000			−0.347314			−0.173657	−	0.984806i	$-0.555558\pi$
$829$	−10.0000			−0.347314			−0.173657	+	0.984806i	$0.555558\pi$
$830$		0			0
$831$	−64.0000			−2.22014
$832$		0			0
$833$			12.0000i			0.415775i
$834$		0			0
$835$	12.0000	−	6.00000i	0.415277	−	0.207639i
$836$		0			0
$837$		−	32.0000i		−	1.10608i
$838$		0			0
$839$	24.0000			0.828572			0.414286	−	0.910147i	$-0.364031\pi$
$839$	24.0000			0.828572			0.414286	+	0.910147i	$0.364031\pi$
$840$		0			0
$841$	−25.0000			−0.862069
$842$		0			0
$843$		−	52.0000i		−	1.79098i
$844$		0			0
$845$	−13.0000	−	26.0000i	−0.447214	−	0.894427i
$846$		0			0
$847$		−	2.00000i		−	0.0687208i
$848$		0			0
$849$	52.0000			1.78464
$850$		0			0
$851$	24.0000			0.822709
$852$		0			0
$853$			16.0000i			0.547830i	0.961754	+	0.273915i	$0.0883186\pi$
$853$			16.0000i			0.547830i	−0.961754	+	0.273915i	$0.911681\pi$
$854$		0			0
$855$	−4.00000	−	8.00000i	−0.136797	−	0.273594i
$856$		0			0
$857$		−	4.00000i		−	0.136637i	−0.997664	−	0.0683187i	$-0.978237\pi$
$857$		−	4.00000i		−	0.136637i	0.997664	−	0.0683187i	$-0.0217635\pi$
$858$		0			0
$859$	4.00000			0.136478			0.0682391	−	0.997669i	$-0.478262\pi$
$859$	4.00000			0.136478			0.0682391	+	0.997669i	$0.478262\pi$
$860$		0			0
$861$	24.0000			0.817918
$862$		0			0
$863$			26.0000i			0.885050i	0.896756	+	0.442525i	$0.145917\pi$
$863$			26.0000i			0.885050i	−0.896756	+	0.442525i	$0.854083\pi$
$864$		0			0
$865$	−16.0000	+	8.00000i	−0.544016	+	0.272008i
$866$		0			0
$867$		−	2.00000i		−	0.0679236i
$868$		0			0
$869$	−8.00000			−0.271381
$870$		0			0
$871$		0			0
$872$		0			0
$873$		−	8.00000i		−	0.270759i
$874$		0			0
$875$	4.00000	−	22.0000i	0.135225	−	0.743736i
$876$		0			0
$877$			16.0000i			0.540282i	0.962821	+	0.270141i	$0.0870703\pi$
$877$			16.0000i			0.540282i	−0.962821	+	0.270141i	$0.912930\pi$
$878$		0			0
$879$	16.0000			0.539667
$880$		0			0
$881$	26.0000			0.875962			0.437981	−	0.898984i	$-0.355694\pi$
$881$	26.0000			0.875962			0.437981	+	0.898984i	$0.355694\pi$
$882$		0			0
$883$		−	2.00000i		−	0.0673054i	−0.999434	−	0.0336527i	$-0.989286\pi$
$883$		−	2.00000i		−	0.0673054i	0.999434	−	0.0336527i	$-0.0107140\pi$
$884$		0			0
$885$	16.0000	−	8.00000i	0.537834	−	0.268917i
$886$		0			0
$887$			6.00000i			0.201460i	0.994914	+	0.100730i	$0.0321179\pi$
$887$			6.00000i			0.201460i	−0.994914	+	0.100730i	$0.967882\pi$
$888$		0			0
$889$	28.0000			0.939090
$890$		0			0
$891$	11.0000			0.368514
$892$		0			0
$893$		−	8.00000i		−	0.267710i
$894$		0			0
$895$	20.0000	+	40.0000i	0.668526	+	1.33705i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−16.0000			−0.533630
$900$		0			0
$901$	48.0000			1.59911
$902$		0			0
$903$			24.0000i			0.798670i
$904$		0			0
$905$	10.0000	+	20.0000i	0.332411	+	0.664822i
$906$		0			0
$907$		−	2.00000i		−	0.0664089i	−0.999449	−	0.0332045i	$-0.989429\pi$
$907$		−	2.00000i		−	0.0664089i	0.999449	−	0.0332045i	$-0.0105712\pi$
$908$		0			0
$909$	10.0000			0.331679
$910$		0			0
$911$	−24.0000			−0.795155			−0.397578	−	0.917568i	$-0.630149\pi$
$911$	−24.0000			−0.795155			−0.397578	+	0.917568i	$0.630149\pi$
$912$		0			0
$913$		−	2.00000i		−	0.0661903i
$914$		0			0
$915$	−56.0000	+	28.0000i	−1.85130	+	0.925651i
$916$		0			0
$917$			24.0000i			0.792550i
$918$		0			0
$919$	−40.0000			−1.31948			−0.659739	−	0.751495i	$-0.729333\pi$
$919$	−40.0000			−1.31948			−0.659739	+	0.751495i	$0.729333\pi$
$920$		0			0
$921$	52.0000			1.71346
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−16.0000	−	12.0000i	−0.526077	−	0.394558i
$926$		0			0
$927$			14.0000i			0.459820i
$928$		0			0
$929$	42.0000			1.37798			0.688988	−	0.724773i	$-0.258055\pi$
$929$	42.0000			1.37798			0.688988	+	0.724773i	$0.258055\pi$
$930$		0			0
$931$	−12.0000			−0.393284
$932$		0			0
$933$		−	16.0000i		−	0.523816i
$934$		0			0
$935$	−8.00000	+	4.00000i	−0.261628	+	0.130814i
$936$		0			0
$937$			4.00000i			0.130674i	0.997863	+	0.0653372i	$0.0208123\pi$
$937$			4.00000i			0.130674i	−0.997863	+	0.0653372i	$0.979188\pi$
$938$		0			0
$939$	48.0000			1.56642
$940$		0			0
$941$	30.0000			0.977972			0.488986	−	0.872292i	$-0.337367\pi$
$941$	30.0000			0.977972			0.488986	+	0.872292i	$0.337367\pi$
$942$		0			0
$943$			36.0000i			1.17232i
$944$		0			0
$945$	8.00000	+	16.0000i	0.260240	+	0.520480i
$946$		0			0
$947$			22.0000i			0.714904i	0.933932	+	0.357452i	$0.116354\pi$
$947$			22.0000i			0.714904i	−0.933932	+	0.357452i	$0.883646\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$	40.0000			1.29709
$952$		0			0
$953$		−	36.0000i		−	1.16615i	−0.812417	−	0.583077i	$-0.801849\pi$
$953$		−	36.0000i		−	1.16615i	0.812417	−	0.583077i	$-0.198151\pi$
$954$		0			0
$955$	24.0000	+	48.0000i	0.776622	+	1.55324i
$956$		0			0
$957$		−	4.00000i		−	0.129302i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	33.0000			1.06452
$962$		0			0
$963$		−	2.00000i		−	0.0644491i
$964$		0			0
$965$	−8.00000	+	4.00000i	−0.257529	+	0.128765i
$966$		0			0
$967$		−	58.0000i		−	1.86515i	−0.360971	−	0.932577i	$-0.617555\pi$
$967$		−	58.0000i		−	1.86515i	0.360971	−	0.932577i	$-0.382445\pi$
$968$		0			0
$969$	−32.0000			−1.02799
$970$		0			0
$971$	−12.0000			−0.385098			−0.192549	−	0.981287i	$-0.561675\pi$
$971$	−12.0000			−0.385098			−0.192549	+	0.981287i	$0.561675\pi$
$972$		0			0
$973$		−	24.0000i		−	0.769405i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			24.0000i			0.767828i	0.923369	+	0.383914i	$0.125424\pi$
$977$			24.0000i			0.767828i	−0.923369	+	0.383914i	$0.874576\pi$
$978$		0			0
$979$	−10.0000			−0.319601
$980$		0			0
$981$	−14.0000			−0.446986
$982$		0			0
$983$		−	30.0000i		−	0.956851i	−0.878128	−	0.478426i	$-0.841208\pi$
$983$		−	30.0000i		−	0.956851i	0.878128	−	0.478426i	$-0.158792\pi$
$984$		0			0
$985$	−48.0000	+	24.0000i	−1.52941	+	0.764704i
$986$		0			0
$987$		−	8.00000i		−	0.254643i
$988$		0			0
$989$	−36.0000			−1.14473
$990$		0			0
$991$	−24.0000			−0.762385			−0.381193	−	0.924496i	$-0.624487\pi$
$991$	−24.0000			−0.762385			−0.381193	+	0.924496i	$0.624487\pi$
$992$		0			0
$993$		−	40.0000i		−	1.26936i
$994$		0			0
$995$	8.00000	+	16.0000i	0.253617	+	0.507234i
$996$		0			0
$997$		−	8.00000i		−	0.253363i	−0.991943	−	0.126681i	$-0.959567\pi$
$997$		−	8.00000i		−	0.253363i	0.991943	−	0.126681i	$-0.0404325\pi$
$998$		0			0
$999$	16.0000			0.506218

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	440.2.b.a.89.1	✓	2
3.2	odd	2		3960.2.d.c.3169.2		2
4.3	odd	2		880.2.b.b.529.2		2
5.2	odd	4		2200.2.a.c.1.1		1
5.3	odd	4		2200.2.a.i.1.1		1
5.4	even	2	inner	440.2.b.a.89.2	yes	2
15.14	odd	2		3960.2.d.c.3169.1		2
20.3	even	4		4400.2.a.h.1.1		1
20.7	even	4		4400.2.a.x.1.1		1
20.19	odd	2		880.2.b.b.529.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
440.2.b.a.89.1	✓	2	1.1	even	1	trivial
440.2.b.a.89.2	yes	2	5.4	even	2	inner
880.2.b.b.529.1		2	20.19	odd	2
880.2.b.b.529.2		2	4.3	odd	2
2200.2.a.c.1.1		1	5.2	odd	4
2200.2.a.i.1.1		1	5.3	odd	4
3960.2.d.c.3169.1		2	15.14	odd	2
3960.2.d.c.3169.2		2	3.2	odd	2
4400.2.a.h.1.1		1	20.3	even	4
4400.2.a.x.1.1		1	20.7	even	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:	\( N \)	\(=\)	\( 440 = 2^{3} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	440.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.00000i q^{3} +(-1.00000 - 2.00000i) q^{5} -2.00000i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-2.00000i q^{3} +(-1.00000 - 2.00000i) q^{5} -2.00000i q^{7} -1.00000 q^{9} -1.00000 q^{11} +(-4.00000 + 2.00000i) q^{15} +4.00000i q^{17} -4.00000 q^{19} -4.00000 q^{21} -6.00000i q^{23} +(-3.00000 + 4.00000i) q^{25} -4.00000i q^{27} -2.00000 q^{29} +8.00000 q^{31} +2.00000i q^{33} +(-4.00000 + 2.00000i) q^{35} +4.00000i q^{37} -6.00000 q^{41} -6.00000i q^{43} +(1.00000 + 2.00000i) q^{45} +2.00000i q^{47} +3.00000 q^{49} +8.00000 q^{51} -12.0000i q^{53} +(1.00000 + 2.00000i) q^{55} +8.00000i q^{57} -4.00000 q^{59} +14.0000 q^{61} +2.00000i q^{63} -10.0000i q^{67} -12.0000 q^{69} +8.00000 q^{71} +4.00000i q^{73} +(8.00000 + 6.00000i) q^{75} +2.00000i q^{77} +8.00000 q^{79} -11.0000 q^{81} +2.00000i q^{83} +(8.00000 - 4.00000i) q^{85} +4.00000i q^{87} +10.0000 q^{89} -16.0000i q^{93} +(4.00000 + 8.00000i) q^{95} +8.00000i q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^5 - 2 * q^9	\( 2 q - 2 q^{5} - 2 q^{9} - 2 q^{11} - 8 q^{15} - 8 q^{19} - 8 q^{21} - 6 q^{25} - 4 q^{29} + 16 q^{31} - 8 q^{35} - 12 q^{41} + 2 q^{45} + 6 q^{49} + 16 q^{51} + 2 q^{55} - 8 q^{59} + 28 q^{61} - 24 q^{69} + 16 q^{71} + 16 q^{75} + 16 q^{79} - 22 q^{81} + 16 q^{85} + 20 q^{89} + 8 q^{95} + 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^5 - 2 * q^9 - 2 * q^11 - 8 * q^15 - 8 * q^19 - 8 * q^21 - 6 * q^25 - 4 * q^29 + 16 * q^31 - 8 * q^35 - 12 * q^41 + 2 * q^45 + 6 * q^49 + 16 * q^51 + 2 * q^55 - 8 * q^59 + 28 * q^61 - 24 * q^69 + 16 * q^71 + 16 * q^75 + 16 * q^79 - 22 * q^81 + 16 * q^85 + 20 * q^89 + 8 * q^95 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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