LMFDB - Embedded newform 2240.2.g.p.449.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2240,2,Mod(449,2240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2240, 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("2240.449");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2240 = 2^{6} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2240.g (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:	$17.8864900528$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 20x^{10} + 148x^{8} + 494x^{6} + 708x^{4} + 304x^{2} + 9 $ x^12 + 20x^10 + 148x^8 + 494x^6 + 708x^4 + 304*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 1120)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.3
Root			$0.810591i$ of defining polynomial
Character	$\chi$	$=$	2240.449
Dual form			2240.2.g.p.449.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.17741i q^{3} +(1.45926 - 1.69427i) q^{5} +1.00000i q^{7} -1.74111 q^{9} +O(q^{10})$	$q-2.17741i q^{3} +(1.45926 - 1.69427i) q^{5} +1.00000i q^{7} -1.74111 q^{9} +2.13290 q^{11} +5.52144i q^{13} +(-3.68912 - 3.17741i) q^{15} +5.24535i q^{17} +6.50100 q^{19} +2.17741 q^{21} +6.35482i q^{23} +(-0.741113 - 4.94477i) q^{25} -2.74111i q^{27} +9.09593 q^{29} +6.77709 q^{31} -4.64419i q^{33} +(1.69427 + 1.45926i) q^{35} +9.88954i q^{37} +12.0224 q^{39} -2.35482 q^{41} -0.354819i q^{43} +(-2.54074 + 2.94992i) q^{45} -8.57816i q^{47} -1.00000 q^{49} +11.4213 q^{51} +7.37825i q^{53} +(3.11245 - 3.61371i) q^{55} -14.1553i q^{57} -6.50100 q^{59} -5.43630 q^{61} -1.74111i q^{63} +(9.35482 + 8.05722i) q^{65} -5.48223i q^{67} +13.8370 q^{69} -3.11245 q^{71} +7.37825i q^{73} +(-10.7668 + 1.61371i) q^{75} +2.13290i q^{77} -12.0224 q^{79} -11.1919 q^{81} -11.1104i q^{83} +(8.88705 + 7.65434i) q^{85} -19.8056i q^{87} -6.00000 q^{89} -5.52144 q^{91} -14.7565i q^{93} +(9.48665 - 11.0145i) q^{95} -0.979557i q^{97} -3.71361 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 16 q^{9}+O(q^{10}) $ 12 * q - 16 * q^9	$ 12 q - 16 q^{9} - 4 q^{21} - 4 q^{25} + 44 q^{29} + 32 q^{41} - 48 q^{45} - 12 q^{49} - 40 q^{61} + 52 q^{65} + 96 q^{69} - 4 q^{81} - 44 q^{85} - 72 q^{89}+O(q^{100}) $ 12 * q - 16 * q^9 - 4 * q^21 - 4 * q^25 + 44 * q^29 + 32 * q^41 - 48 * q^45 - 12 * q^49 - 40 * q^61 + 52 * q^65 + 96 * q^69 - 4 * q^81 - 44 * q^85 - 72 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2240\mathbb{Z}\right)^\times$.

$n$	$897$	$1471$	$1541$	$1921$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$3$		−	2.17741i		−	1.25713i	−0.777758	−	0.628564i	$-0.783643\pi$
$3$		−	2.17741i		−	1.25713i	0.777758	−	0.628564i	$-0.216357\pi$
$4$		0			0
$5$	1.45926	−	1.69427i	0.652601	−	0.757701i
$5$	1.45926	−	1.69427i	0.652601	−	0.757701i
$6$		0			0
$7$			1.00000i			0.377964i
$7$			1.00000i			0.377964i
$8$		0			0
$9$	−1.74111			−0.580371
$10$		0			0
$11$	2.13290			0.643093			0.321546	−	0.946894i	$-0.395797\pi$
$11$	2.13290			0.643093			0.321546	+	0.946894i	$0.395797\pi$
$12$		0			0
$13$			5.52144i			1.53137i	0.643215	+	0.765686i	$0.277600\pi$
$13$			5.52144i			1.53137i	−0.643215	+	0.765686i	$0.722400\pi$
$14$		0			0
$15$	−3.68912	−	3.17741i	−0.952528	−	0.820404i
$16$		0			0
$17$			5.24535i			1.27218i	0.771613	+	0.636092i	$0.219450\pi$
$17$			5.24535i			1.27218i	−0.771613	+	0.636092i	$0.780550\pi$
$18$		0			0
$19$	6.50100			1.49143			0.745715	−	0.666265i	$-0.232108\pi$
$19$	6.50100			1.49143			0.745715	+	0.666265i	$0.232108\pi$
$20$		0			0
$21$	2.17741			0.475150
$22$		0			0
$23$			6.35482i			1.32507i	0.749030	+	0.662536i	$0.230520\pi$
$23$			6.35482i			1.32507i	−0.749030	+	0.662536i	$0.769480\pi$
$24$		0			0
$25$	−0.741113	−	4.94477i	−0.148223	−	0.988954i
$26$		0			0
$27$		−	2.74111i		−	0.527527i
$28$		0			0
$29$	9.09593			1.68907			0.844536	−	0.535499i	$-0.179876\pi$
$29$	9.09593			1.68907			0.844536	+	0.535499i	$0.179876\pi$
$30$		0			0
$31$	6.77709			1.21720			0.608600	−	0.793477i	$-0.291731\pi$
$31$	6.77709			1.21720			0.608600	+	0.793477i	$0.291731\pi$
$32$		0			0
$33$		−	4.64419i		−	0.808450i
$34$		0			0
$35$	1.69427	+	1.45926i	0.286384	+	0.246660i
$36$		0			0
$37$			9.88954i			1.62583i	0.582382	+	0.812915i	$0.302121\pi$
$37$			9.88954i			1.62583i	−0.582382	+	0.812915i	$0.697879\pi$
$38$		0			0
$39$	12.0224			1.92513
$40$		0			0
$41$	−2.35482			−0.367761			−0.183880	−	0.982949i	$-0.558866\pi$
$41$	−2.35482			−0.367761			−0.183880	+	0.982949i	$0.558866\pi$
$42$		0			0
$43$		−	0.354819i		−	0.0541094i	−0.999634	−	0.0270547i	$-0.991387\pi$
$43$		−	0.354819i		−	0.0541094i	0.999634	−	0.0270547i	$-0.00861284\pi$
$44$		0			0
$45$	−2.54074	+	2.94992i	−0.378751	+	0.439748i
$46$		0			0
$47$		−	8.57816i		−	1.25125i	−0.780123	−	0.625626i	$-0.784843\pi$
$47$		−	8.57816i		−	1.25125i	0.780123	−	0.625626i	$-0.215157\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	11.4213			1.59930
$52$		0			0
$53$			7.37825i			1.01348i	0.862099	+	0.506740i	$0.169150\pi$
$53$			7.37825i			1.01348i	−0.862099	+	0.506740i	$0.830850\pi$
$54$		0			0
$55$	3.11245	−	3.61371i	0.419683	−	0.487272i
$56$		0			0
$57$		−	14.1553i		−	1.87492i
$58$		0			0
$59$	−6.50100			−0.846358			−0.423179	−	0.906046i	$-0.639086\pi$
$59$	−6.50100			−0.846358			−0.423179	+	0.906046i	$0.639086\pi$
$60$		0			0
$61$	−5.43630			−0.696046			−0.348023	−	0.937486i	$-0.613147\pi$
$61$	−5.43630			−0.696046			−0.348023	+	0.937486i	$0.613147\pi$
$62$		0			0
$63$		−	1.74111i		−	0.219360i
$64$		0			0
$65$	9.35482	+	8.05722i	1.16032	+	0.999376i
$66$		0			0
$67$		−	5.48223i		−	0.669760i	−0.942261	−	0.334880i	$-0.891304\pi$
$67$		−	5.48223i		−	0.669760i	0.942261	−	0.334880i	$-0.108696\pi$
$68$		0			0
$69$	13.8370			1.66578
$70$		0			0
$71$	−3.11245			−0.369380			−0.184690	−	0.982797i	$-0.559128\pi$
$71$	−3.11245			−0.369380			−0.184690	+	0.982797i	$0.559128\pi$
$72$		0			0
$73$			7.37825i			0.863558i	0.901979	+	0.431779i	$0.142114\pi$
$73$			7.37825i			0.863558i	−0.901979	+	0.431779i	$0.857886\pi$
$74$		0			0
$75$	−10.7668	+	1.61371i	−1.24324	+	0.186335i
$76$		0			0
$77$			2.13290i			0.243066i
$78$		0			0
$79$	−12.0224			−1.35263			−0.676315	−	0.736613i	$-0.736424\pi$
$79$	−12.0224			−1.35263			−0.676315	+	0.736613i	$0.736424\pi$
$80$		0			0
$81$	−11.1919			−1.24354
$82$		0			0
$83$		−	11.1104i		−	1.21952i	−0.792585	−	0.609762i	$-0.791265\pi$
$83$		−	11.1104i		−	1.21952i	0.792585	−	0.609762i	$-0.208735\pi$
$84$		0			0
$85$	8.88705	+	7.65434i	0.963936	+	0.830229i
$86$		0			0
$87$		−	19.8056i		−	2.12338i
$88$		0			0
$89$	−6.00000			−0.635999			−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000			−0.635999			−0.317999	+	0.948091i	$0.603011\pi$
$90$		0			0
$91$	−5.52144			−0.578804
$92$		0			0
$93$		−	14.7565i		−	1.53018i
$94$		0			0
$95$	9.48665	−	11.0145i	0.973310	−	1.13006i
$96$		0			0
$97$		−	0.979557i		−	0.0994589i	−0.998763	−	0.0497295i	$-0.984164\pi$
$97$		−	0.979557i		−	0.0994589i	0.998763	−	0.0497295i	$-0.0158359\pi$
$98$		0			0
$99$	−3.71361			−0.373232
$100$		0			0
$101$	−14.1459			−1.40757			−0.703787	−	0.710411i	$-0.748509\pi$
$101$	−14.1459			−1.40757			−0.703787	+	0.710411i	$0.748509\pi$
$102$		0			0
$103$			9.61371i			0.947267i	0.880722	+	0.473633i	$0.157058\pi$
$103$			9.61371i			0.947267i	−0.880722	+	0.473633i	$0.842942\pi$
$104$		0			0
$105$	3.17741	−	3.68912i	0.310083	−	0.360022i
$106$		0			0
$107$		−	2.96445i		−	0.286584i	−0.989680	−	0.143292i	$-0.954231\pi$
$107$		−	2.96445i		−	0.286584i	0.989680	−	0.143292i	$-0.0457689\pi$
$108$		0			0
$109$	11.6137			1.11239			0.556196	−	0.831051i	$-0.312261\pi$
$109$	11.6137			1.11239			0.556196	+	0.831051i	$0.312261\pi$
$110$		0			0
$111$	21.5336			2.04388
$112$		0			0
$113$			3.66463i			0.344740i	0.985032	+	0.172370i	$0.0551424\pi$
$113$			3.66463i			0.344740i	−0.985032	+	0.172370i	$0.944858\pi$
$114$		0			0
$115$	10.7668	+	9.27334i	1.00401	+	0.864744i
$116$		0			0
$117$		−	9.61345i		−	0.888764i
$118$		0			0
$119$	−5.24535			−0.480840
$120$		0			0
$121$	−6.45075			−0.586432
$122$		0			0
$123$			5.12741i			0.462323i
$124$		0			0
$125$	−9.45926	−	5.96007i	−0.846062	−	0.533084i
$126$		0			0
$127$			4.87259i			0.432373i	0.976352	+	0.216186i	$0.0693619\pi$
$127$			4.87259i			0.432373i	−0.976352	+	0.216186i	$0.930638\pi$
$128$		0			0
$129$	−0.772587			−0.0680225
$130$		0			0
$131$	−2.23520			−0.195291			−0.0976453	−	0.995221i	$-0.531131\pi$
$131$	−2.23520			−0.195291			−0.0976453	+	0.995221i	$0.531131\pi$
$132$		0			0
$133$			6.50100i			0.563708i
$134$		0			0
$135$	−4.64419	−	4.00000i	−0.399708	−	0.344265i
$136$		0			0
$137$		−	4.26579i		−	0.364451i	−0.983257	−	0.182226i	$-0.941670\pi$
$137$		−	4.26579i		−	0.364451i	0.983257	−	0.182226i	$-0.0583301\pi$
$138$		0			0
$139$	15.0326			1.27505			0.637524	−	0.770431i	$-0.279959\pi$
$139$	15.0326			1.27505			0.637524	+	0.770431i	$0.279959\pi$
$140$		0			0
$141$	−18.6782			−1.57299
$142$		0			0
$143$			11.7767i			0.984814i
$144$		0			0
$145$	13.2733	−	15.4110i	1.10229	−	1.27981i
$146$		0			0
$147$			2.17741i			0.179590i
$148$		0			0
$149$	−5.67409			−0.464840			−0.232420	−	0.972616i	$-0.574664\pi$
$149$	−5.67409			−0.464840			−0.232420	+	0.972616i	$0.574664\pi$
$150$		0			0
$151$	−13.1758			−1.07223			−0.536115	−	0.844145i	$-0.680109\pi$
$151$	−13.1758			−1.07223			−0.536115	+	0.844145i	$0.680109\pi$
$152$		0			0
$153$		−	9.13275i		−	0.738339i
$154$		0			0
$155$	9.88954	−	11.4822i	0.794347	−	0.922275i
$156$		0			0
$157$		−	18.1450i		−	1.44813i	−0.689731	−	0.724066i	$-0.742271\pi$
$157$		−	18.1450i		−	1.44813i	0.689731	−	0.724066i	$-0.257729\pi$
$158$		0			0
$159$	16.0655			1.27407
$160$		0			0
$161$	−6.35482			−0.500830
$162$		0			0
$163$		−	12.0289i		−	0.942177i	−0.882086	−	0.471088i	$-0.843861\pi$
$163$		−	12.0289i		−	0.942177i	0.882086	−	0.471088i	$-0.156139\pi$
$164$		0			0
$165$	−7.86852	−	6.77709i	−0.612563	−	0.527595i
$166$		0			0
$167$		−	0.904068i		−	0.0699589i	−0.999388	−	0.0349794i	$-0.988863\pi$
$167$		−	0.904068i		−	0.0699589i	0.999388	−	0.0349794i	$-0.0111366\pi$
$168$		0			0
$169$	−17.4863			−1.34510
$170$		0			0
$171$	−11.3190			−0.865583
$172$		0			0
$173$		−	19.0756i		−	1.45029i	−0.688595	−	0.725146i	$-0.741772\pi$
$173$		−	19.0756i		−	1.45029i	0.688595	−	0.725146i	$-0.258228\pi$
$174$		0			0
$175$	4.94477	−	0.741113i	0.373789	−	0.0560229i
$176$		0			0
$177$			14.1553i			1.06398i
$178$		0			0
$179$	6.22491			0.465271			0.232636	−	0.972564i	$-0.425265\pi$
$179$	6.22491			0.465271			0.232636	+	0.972564i	$0.425265\pi$
$180$		0			0
$181$	7.27334			0.540623			0.270312	−	0.962773i	$-0.412873\pi$
$181$	7.27334			0.540623			0.270312	+	0.962773i	$0.412873\pi$
$182$		0			0
$183$			11.8370i			0.875020i
$184$		0			0
$185$	16.7556	+	14.4314i	1.23189	+	1.06102i
$186$		0			0
$187$			11.1878i			0.818132i
$188$		0			0
$189$	2.74111			0.199387
$190$		0			0
$191$	−20.5540			−1.48724			−0.743618	−	0.668604i	$-0.766892\pi$
$191$	−20.5540			−1.48724			−0.743618	+	0.668604i	$0.766892\pi$
$192$		0			0
$193$		−	7.93043i		−	0.570845i	−0.958402	−	0.285422i	$-0.907866\pi$
$193$		−	7.93043i		−	0.570845i	0.958402	−	0.285422i	$-0.0921338\pi$
$194$		0			0
$195$	17.5439	−	20.3693i	1.25634	−	1.45867i
$196$		0			0
$197$		−	10.4417i		−	0.743942i	−0.928244	−	0.371971i	$-0.878682\pi$
$197$		−	10.4417i		−	0.743942i	0.928244	−	0.371971i	$-0.121318\pi$
$198$		0			0
$199$	11.0429			0.782809			0.391405	−	0.920219i	$-0.371989\pi$
$199$	11.0429			0.782809			0.391405	+	0.920219i	$0.371989\pi$
$200$		0			0
$201$	−11.9371			−0.841975
$202$		0			0
$203$			9.09593i			0.638409i
$204$		0			0
$205$	−3.43630	+	3.98970i	−0.240001	+	0.278653i
$206$		0			0
$207$		−	11.0645i		−	0.769033i
$208$		0			0
$209$	13.8660			0.959128
$210$		0			0
$211$	4.64419			0.319719			0.159860	−	0.987140i	$-0.448896\pi$
$211$	4.64419			0.319719			0.159860	+	0.987140i	$0.448896\pi$
$212$		0			0
$213$			6.77709i			0.464358i
$214$		0			0
$215$	−0.601160	−	0.517774i	−0.0409988	−	0.0353119i
$216$		0			0
$217$			6.77709i			0.460059i
$218$		0			0
$219$	16.0655			1.08560
$220$		0			0
$221$	−28.9619			−1.94819
$222$		0			0
$223$		−	9.96853i		−	0.667542i	−0.942654	−	0.333771i	$-0.891679\pi$
$223$		−	9.96853i		−	0.667542i	0.942654	−	0.333771i	$-0.108321\pi$
$224$		0			0
$225$	1.29036	+	8.60940i	0.0860241	+	0.573960i
$226$		0			0
$227$		−	23.2419i		−	1.54262i	−0.636461	−	0.771308i	$-0.719603\pi$
$227$		−	23.2419i		−	1.54262i	0.636461	−	0.771308i	$-0.280397\pi$
$228$		0			0
$229$	0.371839			0.0245718			0.0122859	−	0.999925i	$-0.496089\pi$
$229$	0.371839			0.0245718			0.0122859	+	0.999925i	$0.496089\pi$
$230$		0			0
$231$	4.64419			0.305565
$232$		0			0
$233$		−	6.22491i		−	0.407807i	−0.978991	−	0.203904i	$-0.934637\pi$
$233$		−	6.22491i		−	0.407807i	0.978991	−	0.203904i	$-0.0653629\pi$
$234$		0			0
$235$	−14.5337	−	12.5178i	−0.948076	−	0.816570i
$236$		0			0
$237$			26.1778i			1.70043i
$238$		0			0
$239$	−8.90998			−0.576339			−0.288169	−	0.957579i	$-0.593047\pi$
$239$	−8.90998			−0.576339			−0.288169	+	0.957579i	$0.593047\pi$
$240$		0			0
$241$	30.0289			1.93433			0.967166	−	0.254145i	$-0.0817941\pi$
$241$	30.0289			1.93433			0.967166	+	0.254145i	$0.0817941\pi$
$242$		0			0
$243$			16.1459i			1.03576i
$244$		0			0
$245$	−1.45926	+	1.69427i	−0.0932288	+	0.108243i
$246$		0			0
$247$			35.8949i			2.28394i
$248$		0			0
$249$	−24.1919			−1.53310
$250$		0			0
$251$	28.0346			1.76953			0.884763	−	0.466040i	$-0.154320\pi$
$251$	28.0346			1.76953			0.884763	+	0.466040i	$0.154320\pi$
$252$		0			0
$253$			13.5542i			0.852144i
$254$		0			0
$255$	16.6666	−	19.3507i	1.04370	−	1.21179i
$256$		0			0
$257$			1.15334i			0.0719434i	0.999353	+	0.0359717i	$0.0114526\pi$
$257$			1.15334i			0.0719434i	−0.999353	+	0.0359717i	$0.988547\pi$
$258$		0			0
$259$	−9.88954			−0.614506
$260$		0			0
$261$	−15.8370			−0.980288
$262$		0			0
$263$		−	11.6741i		−	0.719855i	−0.932980	−	0.359928i	$-0.882801\pi$
$263$		−	11.6741i		−	0.719855i	0.932980	−	0.359928i	$-0.117199\pi$
$264$		0			0
$265$	12.5008	+	10.7668i	0.767915	+	0.661399i
$266$		0			0
$267$			13.0645i			0.799532i
$268$		0			0
$269$	15.1815			0.925631			0.462816	−	0.886455i	$-0.346839\pi$
$269$	15.1815			0.925631			0.462816	+	0.886455i	$0.346839\pi$
$270$		0			0
$271$	−1.95911			−0.119008			−0.0595038	−	0.998228i	$-0.518952\pi$
$271$	−1.95911			−0.119008			−0.0595038	+	0.998228i	$0.518952\pi$
$272$		0			0
$273$			12.0224i			0.727631i
$274$		0			0
$275$	−1.58072	−	10.5467i	−0.0953208	−	0.635989i
$276$		0			0
$277$		−	14.7075i		−	0.883689i	−0.897092	−	0.441844i	$-0.854324\pi$
$277$		−	14.7075i		−	0.883689i	0.897092	−	0.441844i	$-0.145676\pi$
$278$		0			0
$279$	−11.7997			−0.706428
$280$		0			0
$281$	24.3523			1.45273			0.726367	−	0.687307i	$-0.241207\pi$
$281$	24.3523			1.45273			0.726367	+	0.687307i	$0.241207\pi$
$282$		0			0
$283$			24.3982i			1.45032i	0.688580	+	0.725161i	$0.258235\pi$
$283$			24.3982i			1.45032i	−0.688580	+	0.725161i	$0.741765\pi$
$284$		0			0
$285$	−23.9830	−	20.6563i	−1.42063	−	1.22358i
$286$		0			0
$287$		−	2.35482i		−	0.139001i
$288$		0			0
$289$	−10.5137			−0.618453
$290$		0			0
$291$	−2.13290			−0.125033
$292$		0			0
$293$		−	18.5234i		−	1.08215i	−0.840974	−	0.541075i	$-0.818017\pi$
$293$		−	18.5234i		−	1.08215i	0.840974	−	0.541075i	$-0.181983\pi$
$294$		0			0
$295$	−9.48665	+	11.0145i	−0.552334	+	0.641286i
$296$		0			0
$297$		−	5.84651i		−	0.339249i
$298$		0			0
$299$	−35.0878			−2.02918
$300$		0			0
$301$	0.354819			0.0204514
$302$		0			0
$303$			30.8015i			1.76950i
$304$		0			0
$305$	−7.93298	+	9.21056i	−0.454241	+	0.527395i
$306$		0			0
$307$		−	9.05000i		−	0.516511i	−0.966077	−	0.258255i	$-0.916852\pi$
$307$		−	9.05000i		−	0.516511i	0.966077	−	0.258255i	$-0.0831476\pi$
$308$		0			0
$309$	20.9330			1.19084
$310$		0			0
$311$	10.4907			0.594873			0.297437	−	0.954742i	$-0.403868\pi$
$311$	10.4907			0.594873			0.297437	+	0.954742i	$0.403868\pi$
$312$		0			0
$313$			5.24535i			0.296485i	0.988951	+	0.148242i	$0.0473616\pi$
$313$			5.24535i			0.296485i	−0.988951	+	0.148242i	$0.952638\pi$
$314$		0			0
$315$	−2.94992	−	2.54074i	−0.166209	−	0.143154i
$316$		0			0
$317$		−	8.13504i		−	0.456909i	−0.973555	−	0.228455i	$-0.926633\pi$
$317$		−	8.13504i		−	0.456909i	0.973555	−	0.228455i	$-0.0733673\pi$
$318$		0			0
$319$	19.4007			1.08623
$320$		0			0
$321$	−6.45483			−0.360273
$322$		0			0
$323$			34.1000i			1.89738i
$324$		0			0
$325$	27.3023	−	4.09201i	1.51446	−	0.226984i
$326$		0			0
$327$		−	25.2878i		−	1.39842i
$328$		0			0
$329$	8.57816			0.472929
$330$		0			0
$331$	−27.7585			−1.52574			−0.762872	−	0.646549i	$-0.776211\pi$
$331$	−27.7585			−1.52574			−0.762872	+	0.646549i	$0.776211\pi$
$332$		0			0
$333$		−	17.2188i		−	0.943585i
$334$		0			0
$335$	−9.28838	−	8.00000i	−0.507478	−	0.437087i
$336$		0			0
$337$			11.8487i			0.645437i	0.946495	+	0.322719i	$0.104597\pi$
$337$			11.8487i			0.645437i	−0.946495	+	0.322719i	$0.895403\pi$
$338$		0			0
$339$	7.97941			0.433382
$340$		0			0
$341$	14.4548			0.782773
$342$		0			0
$343$		−	1.00000i		−	0.0539949i
$344$		0			0
$345$	20.1919	−	23.4437i	1.08709	−	1.26217i
$346$		0			0
$347$		−	3.90814i		−	0.209800i	−0.994483	−	0.104900i	$-0.966548\pi$
$347$		−	3.90814i		−	0.209800i	0.994483	−	0.104900i	$-0.0334523\pi$
$348$		0			0
$349$	4.40075			0.235567			0.117783	−	0.993039i	$-0.462421\pi$
$349$	4.40075			0.235567			0.117783	+	0.993039i	$0.462421\pi$
$350$		0			0
$351$	15.1349			0.807841
$352$		0			0
$353$			21.3108i			1.13426i	0.823628	+	0.567130i	$0.191946\pi$
$353$			21.3108i			1.13426i	−0.823628	+	0.567130i	$0.808054\pi$
$354$		0			0
$355$	−4.54188	+	5.27334i	−0.241058	+	0.279880i
$356$		0			0
$357$			11.4213i			0.604478i
$358$		0			0
$359$	−7.32927			−0.386824			−0.193412	−	0.981118i	$-0.561955\pi$
$359$	−7.32927			−0.386824			−0.193412	+	0.981118i	$0.561955\pi$
$360$		0			0
$361$	23.2630			1.22437
$362$		0			0
$363$			14.0459i			0.737220i
$364$		0			0
$365$	12.5008	+	10.7668i	0.654319	+	0.563560i
$366$		0			0
$367$		−	11.4508i		−	0.597724i	−0.954296	−	0.298862i	$-0.903393\pi$
$367$		−	11.4508i		−	0.597724i	0.954296	−	0.298862i	$-0.0966071\pi$
$368$		0			0
$369$	4.10001			0.213438
$370$		0			0
$371$	−7.37825			−0.383059
$372$		0			0
$373$		−	6.17593i		−	0.319777i	−0.987135	−	0.159889i	$-0.948886\pi$
$373$		−	6.17593i		−	0.319777i	0.987135	−	0.159889i	$-0.0511135\pi$
$374$		0			0
$375$	−12.9775	+	20.5967i	−0.670155	+	1.06361i
$376$		0			0
$377$			50.2226i			2.58660i
$378$		0			0
$379$	−14.1064			−0.724595			−0.362297	−	0.932063i	$-0.618008\pi$
$379$	−14.1064			−0.724595			−0.362297	+	0.932063i	$0.618008\pi$
$380$		0			0
$381$	10.6096			0.543548
$382$		0			0
$383$		−	7.25632i		−	0.370781i	−0.982665	−	0.185390i	$-0.940645\pi$
$383$		−	7.25632i		−	0.370781i	0.982665	−	0.185390i	$-0.0593550\pi$
$384$		0			0
$385$	3.61371	+	3.11245i	0.184172	+	0.158625i
$386$		0			0
$387$			0.617781i			0.0314036i
$388$		0			0
$389$	−14.5152			−0.735950			−0.367975	−	0.929836i	$-0.619949\pi$
$389$	−14.5152			−0.735950			−0.367975	+	0.929836i	$0.619949\pi$
$390$		0			0
$391$	−33.3333			−1.68574
$392$		0			0
$393$			4.86695i			0.245505i
$394$		0			0
$395$	−17.5439	+	20.3693i	−0.882728	+	1.02489i
$396$		0			0
$397$			26.5028i			1.33014i	0.746781	+	0.665070i	$0.231598\pi$
$397$			26.5028i			1.33014i	−0.746781	+	0.665070i	$0.768402\pi$
$398$		0			0
$399$	14.1553			0.708653
$400$		0			0
$401$	−12.4863			−0.623536			−0.311768	−	0.950158i	$-0.600921\pi$
$401$	−12.4863			−0.623536			−0.311768	+	0.950158i	$0.600921\pi$
$402$		0			0
$403$			37.4193i			1.86399i
$404$		0			0
$405$	−16.3319	+	18.9621i	−0.811536	+	0.942232i
$406$		0			0
$407$			21.0934i			1.04556i
$408$		0			0
$409$	−23.0645			−1.14046			−0.570232	−	0.821484i	$-0.693147\pi$
$409$	−23.0645			−1.14046			−0.570232	+	0.821484i	$0.693147\pi$
$410$		0			0
$411$	−9.28838			−0.458162
$412$		0			0
$413$		−	6.50100i		−	0.319893i
$414$		0			0
$415$	−18.8240	−	16.2130i	−0.924035	−	0.795863i
$416$		0			0
$417$		−	32.7321i		−	1.60290i
$418$		0			0
$419$	−1.47841			−0.0722251			−0.0361125	−	0.999348i	$-0.511497\pi$
$419$	−1.47841			−0.0722251			−0.0361125	+	0.999348i	$0.511497\pi$
$420$		0			0
$421$	−38.1604			−1.85982			−0.929912	−	0.367783i	$-0.880117\pi$
$421$	−38.1604			−1.85982			−0.929912	+	0.367783i	$0.880117\pi$
$422$		0			0
$423$			14.9355i			0.726191i
$424$		0			0
$425$	25.9371	−	3.88740i	1.25813	−	0.188566i
$426$		0			0
$427$		−	5.43630i		−	0.263081i
$428$		0			0
$429$	25.6426			1.23804
$430$		0			0
$431$	26.7300			1.28754			0.643768	−	0.765221i	$-0.277370\pi$
$431$	26.7300			1.28754			0.643768	+	0.765221i	$0.277370\pi$
$432$		0			0
$433$			2.90784i			0.139742i	0.997556	+	0.0698709i	$0.0222587\pi$
$433$			2.90784i			0.139742i	−0.997556	+	0.0698709i	$0.977741\pi$
$434$		0			0
$435$	−33.5560	−	28.9015i	−1.60889	−	1.38572i
$436$		0			0
$437$			41.3127i			1.97625i
$438$		0			0
$439$	−31.3741			−1.49741			−0.748703	−	0.662906i	$-0.769323\pi$
$439$	−31.3741			−1.49741			−0.748703	+	0.662906i	$0.769323\pi$
$440$		0			0
$441$	1.74111			0.0829101
$442$		0			0
$443$			11.2274i			0.533430i	0.963775	+	0.266715i	$0.0859383\pi$
$443$			11.2274i			0.533430i	−0.963775	+	0.266715i	$0.914062\pi$
$444$		0			0
$445$	−8.75557	+	10.1656i	−0.415054	+	0.481897i
$446$		0			0
$447$			12.3548i			0.584363i
$448$		0			0
$449$	24.4152			1.15222			0.576112	−	0.817371i	$-0.304569\pi$
$449$	24.4152			1.15222			0.576112	+	0.817371i	$0.304569\pi$
$450$		0			0
$451$	−5.02259			−0.236504
$452$		0			0
$453$			28.6891i			1.34793i
$454$		0			0
$455$	−8.05722	+	9.35482i	−0.377728	+	0.438561i
$456$		0			0
$457$		−	17.2188i		−	0.805462i	−0.915318	−	0.402731i	$-0.868061\pi$
$457$		−	17.2188i		−	0.805462i	0.915318	−	0.402731i	$-0.131939\pi$
$458$		0			0
$459$	14.3781			0.671112
$460$		0			0
$461$	38.1748			1.77798			0.888990	−	0.457927i	$-0.151408\pi$
$461$	38.1748			1.77798			0.888990	+	0.457927i	$0.151408\pi$
$462$		0			0
$463$		−	10.5178i		−	0.488802i	−0.969674	−	0.244401i	$-0.921409\pi$
$463$		−	10.5178i		−	0.488802i	0.969674	−	0.244401i	$-0.0785914\pi$
$464$		0			0
$465$	−25.0015	−	21.5336i	−1.15942	−	0.998596i
$466$		0			0
$467$		−	26.8870i		−	1.24418i	−0.782944	−	0.622092i	$-0.786283\pi$
$467$		−	26.8870i		−	1.24418i	0.782944	−	0.622092i	$-0.213717\pi$
$468$		0			0
$469$	5.48223			0.253146
$470$		0			0
$471$	−39.5092			−1.82049
$472$		0			0
$473$		−	0.756793i		−	0.0347974i
$474$		0			0
$475$	−4.81797	−	32.1459i	−0.221064	−	1.47496i
$476$		0			0
$477$		−	12.8464i		−	0.588194i
$478$		0			0
$479$	−7.97941			−0.364588			−0.182294	−	0.983244i	$-0.558352\pi$
$479$	−7.97941			−0.364588			−0.182294	+	0.983244i	$0.558352\pi$
$480$		0			0
$481$	−54.6045			−2.48975
$482$		0			0
$483$			13.8370i			0.629607i
$484$		0			0
$485$	−1.65964	−	1.42943i	−0.0753602	−	0.0649070i
$486$		0			0
$487$			3.83705i			0.173873i	0.996214	+	0.0869366i	$0.0277077\pi$
$487$			3.83705i			0.173873i	−0.996214	+	0.0869366i	$0.972292\pi$
$488$		0			0
$489$	−26.1919			−1.18444
$490$		0			0
$491$	−6.60330			−0.298003			−0.149001	−	0.988837i	$-0.547606\pi$
$491$	−6.60330			−0.298003			−0.149001	+	0.988837i	$0.547606\pi$
$492$		0			0
$493$			47.7114i			2.14881i
$494$		0			0
$495$	−5.41913	+	6.29187i	−0.243572	+	0.282799i
$496$		0			0
$497$		−	3.11245i		−	0.139613i
$498$		0			0
$499$	6.95087			0.311164			0.155582	−	0.987823i	$-0.450275\pi$
$499$	6.95087			0.311164			0.155582	+	0.987823i	$0.450275\pi$
$500$		0			0
$501$	−1.96853			−0.0879472
$502$		0			0
$503$			23.8974i			1.06553i	0.846262	+	0.532767i	$0.178848\pi$
$503$			23.8974i			1.06553i	−0.846262	+	0.532767i	$0.821152\pi$
$504$		0			0
$505$	−20.6426	+	23.9671i	−0.918584	+	1.06652i
$506$		0			0
$507$			38.0748i			1.69096i
$508$		0			0
$509$	−7.36520			−0.326457			−0.163228	−	0.986588i	$-0.552191\pi$
$509$	−7.36520			−0.326457			−0.163228	+	0.986588i	$0.552191\pi$
$510$		0			0
$511$	−7.37825			−0.326394
$512$		0			0
$513$		−	17.8200i		−	0.786771i
$514$		0			0
$515$	16.2882	+	14.0289i	0.717745	+	0.618188i
$516$		0			0
$517$		−	18.2963i		−	0.804671i
$518$		0			0
$519$	−41.5354			−1.82320
$520$		0			0
$521$	13.2563			0.580770			0.290385	−	0.956910i	$-0.406217\pi$
$521$	13.2563			0.580770			0.290385	+	0.956910i	$0.406217\pi$
$522$		0			0
$523$			22.7845i			0.996296i	0.867092	+	0.498148i	$0.165986\pi$
$523$			22.7845i			0.996296i	−0.867092	+	0.498148i	$0.834014\pi$
$524$		0			0
$525$	−1.61371	−	10.7668i	−0.0704279	−	0.469901i
$526$		0			0
$527$			35.5482i			1.54850i
$528$		0			0
$529$	−17.3837			−0.755814
$530$		0			0
$531$	11.3190			0.491201
$532$		0			0
$533$		−	13.0020i		−	0.563179i
$534$		0			0
$535$	−5.02259	−	4.32591i	−0.217145	−	0.187025i
$536$		0			0
$537$		−	13.5542i		−	0.584906i
$538$		0			0
$539$	−2.13290			−0.0918704
$540$		0			0
$541$	17.8345			0.766764			0.383382	−	0.923590i	$-0.374759\pi$
$541$	17.8345			0.766764			0.383382	+	0.923590i	$0.374759\pi$
$542$		0			0
$543$		−	15.8370i		−	0.679633i
$544$		0			0
$545$	16.9474	−	19.6768i	0.725948	−	0.842860i
$546$		0			0
$547$		−	24.3548i		−	1.04134i	−0.853759	−	0.520668i	$-0.825683\pi$
$547$		−	24.3548i		−	1.04134i	0.853759	−	0.520668i	$-0.174317\pi$
$548$		0			0
$549$	9.46521			0.403965
$550$		0			0
$551$	59.1326			2.51913
$552$		0			0
$553$		−	12.0224i		−	0.511246i
$554$		0			0
$555$	31.4231	−	36.4837i	1.33384	−	1.54865i
$556$		0			0
$557$		−	36.4457i		−	1.54425i	−0.635468	−	0.772127i	$-0.719193\pi$
$557$		−	36.4457i		−	1.54425i	0.635468	−	0.772127i	$-0.280807\pi$
$558$		0			0
$559$	1.95911			0.0828617
$560$		0			0
$561$	24.3604			1.02850
$562$		0			0
$563$		−	23.4363i		−	0.987722i	−0.869541	−	0.493861i	$-0.835585\pi$
$563$		−	23.4363i		−	0.987722i	0.869541	−	0.493861i	$-0.164415\pi$
$564$		0			0
$565$	6.20888	+	5.34766i	0.261210	+	0.224978i
$566$		0			0
$567$		−	11.1919i		−	0.470014i
$568$		0			0
$569$	−10.9015			−0.457015			−0.228507	−	0.973542i	$-0.573385\pi$
$569$	−10.9015			−0.457015			−0.228507	+	0.973542i	$0.573385\pi$
$570$		0			0
$571$	−22.6379			−0.947368			−0.473684	−	0.880695i	$-0.657076\pi$
$571$	−22.6379			−0.947368			−0.473684	+	0.880695i	$0.657076\pi$
$572$		0			0
$573$			44.7545i			1.86965i
$574$		0			0
$575$	31.4231	−	4.70964i	1.31043	−	0.196406i
$576$		0			0
$577$		−	26.7789i		−	1.11482i	−0.830237	−	0.557411i	$-0.811795\pi$
$577$		−	26.7789i		−	1.11482i	0.830237	−	0.557411i	$-0.188205\pi$
$578$		0			0
$579$	−17.2678			−0.717625
$580$		0			0
$581$	11.1104			0.460937
$582$		0			0
$583$			15.7370i			0.651762i
$584$		0			0
$585$	−16.2878	−	14.0285i	−0.673418	−	0.580009i
$586$		0			0
$587$		−	11.5282i		−	0.475818i	−0.971287	−	0.237909i	$-0.923538\pi$
$587$		−	11.5282i		−	0.475818i	0.971287	−	0.237909i	$-0.0764620\pi$
$588$		0			0
$589$	44.0578			1.81537
$590$		0			0
$591$	−22.7359			−0.935230
$592$		0			0
$593$		−	18.5949i		−	0.763601i	−0.924245	−	0.381801i	$-0.875304\pi$
$593$		−	18.5949i		−	0.763601i	0.924245	−	0.381801i	$-0.124696\pi$
$594$		0			0
$595$	−7.65434	+	8.88705i	−0.313797	+	0.364333i
$596$		0			0
$597$		−	24.0449i		−	0.984091i
$598$		0			0
$599$	−22.5131			−0.919862			−0.459931	−	0.887955i	$-0.652126\pi$
$599$	−22.5131			−0.919862			−0.459931	+	0.887955i	$0.652126\pi$
$600$		0			0
$601$	41.2274			1.68170			0.840851	−	0.541267i	$-0.182055\pi$
$601$	41.2274			1.68170			0.840851	+	0.541267i	$0.182055\pi$
$602$		0			0
$603$			9.54517i			0.388710i
$604$		0			0
$605$	−9.41333	+	10.9293i	−0.382706	+	0.444340i
$606$		0			0
$607$			6.74111i			0.273613i	0.990598	+	0.136807i	$0.0436839\pi$
$607$			6.74111i			0.273613i	−0.990598	+	0.136807i	$0.956316\pi$
$608$		0			0
$609$	19.8056			0.802562
$610$		0			0
$611$	47.3638			1.91613
$612$		0			0
$613$		−	37.0958i		−	1.49829i	−0.662408	−	0.749144i	$-0.730465\pi$
$613$		−	37.0958i		−	1.49829i	0.662408	−	0.749144i	$-0.269535\pi$
$614$		0			0
$615$	8.68722	+	7.48223i	0.350302	+	0.301712i
$616$		0			0
$617$		−	11.8487i		−	0.477009i	−0.971141	−	0.238504i	$-0.923343\pi$
$617$		−	11.8487i		−	0.477009i	0.971141	−	0.238504i	$-0.0766572\pi$
$618$		0			0
$619$	−46.4067			−1.86524			−0.932622	−	0.360855i	$-0.882485\pi$
$619$	−46.4067			−1.86524			−0.932622	+	0.360855i	$0.882485\pi$
$620$		0			0
$621$	17.4193			0.699012
$622$		0			0
$623$		−	6.00000i		−	0.240385i
$624$		0			0
$625$	−23.9015	+	7.32927i	−0.956060	+	0.293171i
$626$		0			0
$627$		−	30.1919i		−	1.20575i
$628$		0			0
$629$	−51.8741			−2.06836
$630$		0			0
$631$	−39.1308			−1.55777			−0.778886	−	0.627166i	$-0.784215\pi$
$631$	−39.1308			−1.55777			−0.778886	+	0.627166i	$0.784215\pi$
$632$		0			0
$633$		−	10.1123i		−	0.401928i
$634$		0			0
$635$	8.25550	+	7.11039i	0.327609	+	0.282167i
$636$		0			0
$637$		−	5.52144i		−	0.218767i
$638$		0			0
$639$	5.41913			0.214378
$640$		0			0
$641$	−1.48223			−0.0585444			−0.0292722	−	0.999571i	$-0.509319\pi$
$641$	−1.48223			−0.0585444			−0.0292722	+	0.999571i	$0.509319\pi$
$642$		0			0
$643$			38.5322i			1.51956i	0.650179	+	0.759781i	$0.274694\pi$
$643$			38.5322i			1.51956i	−0.650179	+	0.759781i	$0.725306\pi$
$644$		0			0
$645$	−1.12741	+	1.30897i	−0.0443916	+	0.0515407i
$646$		0			0
$647$		−	46.5467i		−	1.82994i	−0.403523	−	0.914969i	$-0.632214\pi$
$647$		−	46.5467i		−	1.82994i	0.403523	−	0.914969i	$-0.367786\pi$
$648$		0			0
$649$	−13.8660			−0.544286
$650$		0			0
$651$	14.7565			0.578353
$652$		0			0
$653$			42.4660i			1.66182i	0.556405	+	0.830912i	$0.312180\pi$
$653$			42.4660i			1.66182i	−0.556405	+	0.830912i	$0.687820\pi$
$654$		0			0
$655$	−3.26175	+	3.78704i	−0.127447	+	0.147972i
$656$		0			0
$657$		−	12.8464i		−	0.501184i
$658$		0			0
$659$	39.7319			1.54774			0.773868	−	0.633346i	$-0.218319\pi$
$659$	39.7319			1.54774			0.773868	+	0.633346i	$0.218319\pi$
$660$		0			0
$661$	26.2667			1.02166			0.510828	−	0.859683i	$-0.329339\pi$
$661$	26.2667			1.02166			0.510828	+	0.859683i	$0.329339\pi$
$662$		0			0
$663$			63.0619i			2.44912i
$664$		0			0
$665$	11.0145	+	9.48665i	0.427122	+	0.367877i
$666$		0			0
$667$			57.8030i			2.23814i
$668$		0			0
$669$	−21.7056			−0.839186
$670$		0			0
$671$	−11.5951			−0.447622
$672$		0			0
$673$			22.8426i			0.880516i	0.897871	+	0.440258i	$0.145113\pi$
$673$			22.8426i			0.880516i	−0.897871	+	0.440258i	$0.854887\pi$
$674$		0			0
$675$	−13.5542	+	2.03147i	−0.521700	+	0.0781915i
$676$		0			0
$677$		−	7.27594i		−	0.279637i	−0.990177	−	0.139819i	$-0.955348\pi$
$677$		−	7.27594i		−	0.279637i	0.990177	−	0.139819i	$-0.0446519\pi$
$678$		0			0
$679$	0.979557			0.0375919
$680$		0			0
$681$	−50.6071			−1.93927
$682$		0			0
$683$			38.9015i			1.48852i	0.667887	+	0.744262i	$0.267199\pi$
$683$			38.9015i			1.48852i	−0.667887	+	0.744262i	$0.732801\pi$
$684$		0			0
$685$	−7.22741	−	6.22491i	−0.276145	−	0.237841i
$686$		0			0
$687$		−	0.809645i		−	0.0308899i
$688$		0			0
$689$	−40.7385			−1.55202
$690$		0			0
$691$	10.2146			0.388582			0.194291	−	0.980944i	$-0.437759\pi$
$691$	10.2146			0.388582			0.194291	+	0.980944i	$0.437759\pi$
$692$		0			0
$693$		−	3.71361i		−	0.141069i
$694$		0			0
$695$	21.9365	−	25.4693i	0.832098	−	0.966105i
$696$		0			0
$697$		−	12.3519i		−	0.467860i
$698$		0			0
$699$	−13.5542			−0.512666
$700$		0			0
$701$	−23.5508			−0.889500			−0.444750	−	0.895655i	$-0.646707\pi$
$701$	−23.5508			−0.889500			−0.444750	+	0.895655i	$0.646707\pi$
$702$		0			0
$703$			64.2919i			2.42481i
$704$		0			0
$705$	−27.2563	+	31.6459i	−1.02653	+	1.19185i
$706$		0			0
$707$		−	14.1459i		−	0.532013i
$708$		0			0
$709$	−35.6715			−1.33967			−0.669836	−	0.742509i	$-0.733636\pi$
$709$	−35.6715			−1.33967			−0.669836	+	0.742509i	$0.733636\pi$
$710$		0			0
$711$	20.9324			0.785027
$712$		0			0
$713$			43.0672i			1.61288i
$714$		0			0
$715$	19.9529	+	17.1852i	0.746195	+	0.642691i
$716$		0			0
$717$			19.4007i			0.724532i
$718$		0			0
$719$	15.3087			0.570917			0.285459	−	0.958391i	$-0.407854\pi$
$719$	15.3087			0.570917			0.285459	+	0.958391i	$0.407854\pi$
$720$		0			0
$721$	−9.61371			−0.358033
$722$		0			0
$723$		−	65.3852i		−	2.43170i
$724$		0			0
$725$	−6.74111	−	44.9773i	−0.250359	−	1.67041i
$726$		0			0
$727$			22.9304i			0.850442i	0.905090	+	0.425221i	$0.139804\pi$
$727$			22.9304i			0.850442i	−0.905090	+	0.425221i	$0.860196\pi$
$728$		0			0
$729$	1.58072			0.0585453
$730$		0			0
$731$	1.86115			0.0688372
$732$		0			0
$733$		−	27.8118i		−	1.02725i	−0.858014	−	0.513626i	$-0.828302\pi$
$733$		−	27.8118i		−	1.02725i	0.858014	−	0.513626i	$-0.171698\pi$
$734$		0			0
$735$	3.68912	+	3.17741i	0.136075	+	0.117201i
$736$		0			0
$737$		−	11.6930i		−	0.430718i
$738$		0			0
$739$	−29.3392			−1.07926			−0.539630	−	0.841902i	$-0.681436\pi$
$739$	−29.3392			−1.07926			−0.539630	+	0.841902i	$0.681436\pi$
$740$		0			0
$741$	78.1578			2.87120
$742$		0			0
$743$		−	20.5467i		−	0.753785i	−0.926257	−	0.376892i	$-0.876993\pi$
$743$		−	20.5467i		−	0.753785i	0.926257	−	0.376892i	$-0.123007\pi$
$744$		0			0
$745$	−8.27998	+	9.61345i	−0.303355	+	0.352210i
$746$		0			0
$747$			19.3444i			0.707776i
$748$		0			0
$749$	2.96445			0.108319
$750$		0			0
$751$	21.3598			0.779430			0.389715	−	0.920935i	$-0.372574\pi$
$751$	21.3598			0.779430			0.389715	+	0.920935i	$0.372574\pi$
$752$		0			0
$753$		−	61.0428i		−	2.22452i
$754$		0			0
$755$	−19.2269	+	22.3233i	−0.699739	+	0.812430i
$756$		0			0
$757$			51.7544i			1.88104i	0.339731	+	0.940522i	$0.389664\pi$
$757$			51.7544i			1.88104i	−0.339731	+	0.940522i	$0.610336\pi$
$758$		0			0
$759$	29.5130			1.07125
$760$		0			0
$761$	6.02891			0.218548			0.109274	−	0.994012i	$-0.465147\pi$
$761$	6.02891			0.218548			0.109274	+	0.994012i	$0.465147\pi$
$762$		0			0
$763$			11.6137i			0.420444i
$764$		0			0
$765$	−15.4734	−	13.3271i	−0.559440	−	0.481841i
$766$		0			0
$767$		−	35.8949i		−	1.29609i
$768$		0			0
$769$	26.7385			0.964217			0.482108	−	0.876112i	$-0.339871\pi$
$769$	26.7385			0.964217			0.482108	+	0.876112i	$0.339871\pi$
$770$		0			0
$771$	2.51129			0.0904420
$772$		0			0
$773$		−	24.5437i		−	0.882777i	−0.897316	−	0.441388i	$-0.854486\pi$
$773$		−	24.5437i		−	0.882777i	0.897316	−	0.441388i	$-0.145514\pi$
$774$		0			0
$775$	−5.02259	−	33.5111i	−0.180417	−	1.20376i
$776$		0			0
$777$			21.5336i			0.772513i
$778$		0			0
$779$	−15.3087			−0.548490
$780$		0			0
$781$	−6.63854			−0.237546
$782$		0			0
$783$		−	24.9330i		−	0.891032i
$784$		0			0
$785$	−30.7426	−	26.4784i	−1.09725	−	0.945053i
$786$		0			0
$787$			2.59518i			0.0925081i	0.998930	+	0.0462540i	$0.0147284\pi$
$787$			2.59518i			0.0925081i	−0.998930	+	0.0462540i	$0.985272\pi$
$788$		0			0
$789$	−25.4193			−0.904950
$790$		0			0
$791$	−3.66463			−0.130299
$792$		0			0
$793$		−	30.0162i		−	1.06591i
$794$		0			0
$795$	23.4437	−	27.2193i	0.831463	−	0.965368i
$796$		0			0
$797$			29.7709i			1.05454i	0.849698	+	0.527270i	$0.176784\pi$
$797$			29.7709i			1.05454i	−0.849698	+	0.527270i	$0.823216\pi$
$798$		0			0
$799$	44.9954			1.59182
$800$		0			0
$801$	10.4467			0.369115
$802$		0			0
$803$			15.7370i			0.555348i
$804$		0			0
$805$	−9.27334	+	10.7668i	−0.326842	+	0.379479i
$806$		0			0
$807$		−	33.0563i		−	1.16364i
$808$		0			0
$809$	1.32184			0.0464733			0.0232366	−	0.999730i	$-0.492603\pi$
$809$	1.32184			0.0464733			0.0232366	+	0.999730i	$0.492603\pi$
$810$		0			0
$811$	−54.5907			−1.91694			−0.958470	−	0.285193i	$-0.907942\pi$
$811$	−54.5907			−1.91694			−0.958470	+	0.285193i	$0.907942\pi$
$812$		0			0
$813$			4.26579i			0.149608i
$814$		0			0
$815$	−20.3802	−	17.5533i	−0.713889	−	0.614866i
$816$		0			0
$817$		−	2.30668i		−	0.0807005i
$818$		0			0
$819$	9.61345			0.335921
$820$		0			0
$821$	17.7137			0.618213			0.309106	−	0.951027i	$-0.399970\pi$
$821$	17.7137			0.618213			0.309106	+	0.951027i	$0.399970\pi$
$822$		0			0
$823$			43.7319i			1.52440i	0.647342	+	0.762199i	$0.275880\pi$
$823$			43.7319i			1.52440i	−0.647342	+	0.762199i	$0.724120\pi$
$824$		0			0
$825$	−22.9645	+	3.44187i	−0.799520	+	0.119831i
$826$		0			0
$827$		−	34.9015i		−	1.21364i	−0.794838	−	0.606822i	$-0.792444\pi$
$827$		−	34.9015i		−	1.21364i	0.794838	−	0.606822i	$-0.207556\pi$
$828$		0			0
$829$	−2.11703			−0.0735273			−0.0367637	−	0.999324i	$-0.511705\pi$
$829$	−2.11703			−0.0735273			−0.0367637	+	0.999324i	$0.511705\pi$
$830$		0			0
$831$	−32.0243			−1.11091
$832$		0			0
$833$		−	5.24535i		−	0.181741i
$834$		0			0
$835$	−1.53174	−	1.31927i	−0.0530079	−	0.0456553i
$836$		0			0
$837$		−	18.5768i		−	0.642107i
$838$		0			0
$839$	40.1103			1.38476			0.692381	−	0.721532i	$-0.256562\pi$
$839$	40.1103			1.38476			0.692381	+	0.721532i	$0.256562\pi$
$840$		0			0
$841$	53.7360			1.85296
$842$		0			0
$843$		−	53.0248i		−	1.82627i
$844$		0			0
$845$	−25.5171	+	29.6265i	−0.877814	+	1.01918i
$846$		0			0
$847$		−	6.45075i		−	0.221650i
$848$		0			0
$849$	53.1248			1.82324
$850$		0			0
$851$	−62.8462			−2.15434
$852$		0			0
$853$		−	21.0039i		−	0.719160i	−0.933114	−	0.359580i	$-0.882920\pi$
$853$		−	21.0039i		−	0.719160i	0.933114	−	0.359580i	$-0.117080\pi$
$854$		0			0
$855$	−16.5173	+	19.1774i	−0.564881	+	0.655854i
$856$		0			0
$857$			47.4886i			1.62218i	0.584922	+	0.811090i	$0.301125\pi$
$857$			47.4886i			1.62218i	−0.584922	+	0.811090i	$0.698875\pi$
$858$		0			0
$859$	45.8545			1.56454			0.782268	−	0.622941i	$-0.214063\pi$
$859$	45.8545			1.56454			0.782268	+	0.622941i	$0.214063\pi$
$860$		0			0
$861$	−5.12741			−0.174742
$862$		0			0
$863$			1.36146i			0.0463446i	0.999731	+	0.0231723i	$0.00737663\pi$
$863$			1.36146i			0.0463446i	−0.999731	+	0.0231723i	$0.992623\pi$
$864$		0			0
$865$	−32.3193	−	27.8363i	−1.09889	−	0.946463i
$866$		0			0
$867$			22.8926i			0.777475i
$868$		0			0
$869$	−25.6426			−0.869866
$870$		0			0
$871$	30.2698			1.02565
$872$		0			0
$873$			1.70552i			0.0577231i
$874$		0			0
$875$	5.96007	−	9.45926i	0.201487	−	0.319781i
$876$		0			0
$877$			25.4028i			0.857793i	0.903354	+	0.428896i	$0.141097\pi$
$877$			25.4028i			0.857793i	−0.903354	+	0.428896i	$0.858903\pi$
$878$		0			0
$879$	−40.3331			−1.36040
$880$		0			0
$881$	36.5756			1.23226			0.616132	−	0.787643i	$-0.288699\pi$
$881$	36.5756			1.23226			0.616132	+	0.787643i	$0.288699\pi$
$882$		0			0
$883$			30.1919i			1.01604i	0.861346	+	0.508019i	$0.169622\pi$
$883$			30.1919i			1.01604i	−0.861346	+	0.508019i	$0.830378\pi$
$884$		0			0
$885$	23.9830	+	20.6563i	0.806179	+	0.694355i
$886$		0			0
$887$		−	58.7467i		−	1.97252i	−0.165197	−	0.986261i	$-0.552826\pi$
$887$		−	58.7467i		−	1.97252i	0.165197	−	0.986261i	$-0.447174\pi$
$888$		0			0
$889$	−4.87259			−0.163422
$890$		0			0
$891$	−23.8711			−0.799712
$892$		0			0
$893$		−	55.7666i		−	1.86616i
$894$		0			0
$895$	9.08377	−	10.5467i	0.303637	−	0.352537i
$896$		0			0
$897$			76.4004i			2.55094i
$898$		0			0
$899$	61.6439			2.05594
$900$		0			0
$901$	−38.7015			−1.28933
$902$		0			0
$903$		−	0.772587i		−	0.0257101i
$904$		0			0
$905$	10.6137	−	12.3230i	0.352812	−	0.409631i
$906$		0			0
$907$		−	18.1289i		−	0.601961i	−0.953630	−	0.300980i	$-0.902686\pi$
$907$		−	18.1289i		−	0.601961i	0.953630	−	0.300980i	$-0.0973139\pi$
$908$		0			0
$909$	24.6297			0.816915
$910$		0			0
$911$	−11.2475			−0.372646			−0.186323	−	0.982489i	$-0.559657\pi$
$911$	−11.2475			−0.372646			−0.186323	+	0.982489i	$0.559657\pi$
$912$		0			0
$913$		−	23.6973i		−	0.784266i
$914$		0			0
$915$	20.0552	+	17.2733i	0.663003	+	0.571039i
$916$		0			0
$917$		−	2.23520i		−	0.0738129i
$918$		0			0
$919$	−34.1082			−1.12513			−0.562563	−	0.826755i	$-0.690185\pi$
$919$	−34.1082			−1.12513			−0.562563	+	0.826755i	$0.690185\pi$
$920$		0			0
$921$	−19.7056			−0.649320
$922$		0			0
$923$		−	17.1852i		−	0.565659i
$924$		0			0
$925$	48.9015	−	7.32927i	1.60787	−	0.240985i
$926$		0			0
$927$		−	16.7385i		−	0.549766i
$928$		0			0
$929$	−0.546684			−0.0179361			−0.00896806	−	0.999960i	$-0.502855\pi$
$929$	−0.546684			−0.0179361			−0.00896806	+	0.999960i	$0.502855\pi$
$930$		0			0
$931$	−6.50100			−0.213062
$932$		0			0
$933$		−	22.8426i		−	0.747832i
$934$		0			0
$935$	18.9552	+	16.3259i	0.619900	+	0.533914i
$936$		0			0
$937$		−	29.8424i		−	0.974909i	−0.873148	−	0.487454i	$-0.837926\pi$
$937$		−	29.8424i		−	0.974909i	0.873148	−	0.487454i	$-0.162074\pi$
$938$		0			0
$939$	11.4213			0.372719
$940$		0			0
$941$	−9.20225			−0.299985			−0.149992	−	0.988687i	$-0.547925\pi$
$941$	−9.20225			−0.299985			−0.149992	+	0.988687i	$0.547925\pi$
$942$		0			0
$943$		−	14.9645i		−	0.487310i
$944$		0			0
$945$	4.00000	−	4.64419i	0.130120	−	0.151076i
$946$		0			0
$947$			30.9304i			1.00510i	0.864547	+	0.502552i	$0.167605\pi$
$947$			30.9304i			1.00510i	−0.864547	+	0.502552i	$0.832395\pi$
$948$		0			0
$949$	−40.7385			−1.32243
$950$		0			0
$951$	−17.7133			−0.574394
$952$		0			0
$953$		−	10.8383i		−	0.351086i	−0.984472	−	0.175543i	$-0.943832\pi$
$953$		−	10.8383i		−	0.351086i	0.984472	−	0.175543i	$-0.0561681\pi$
$954$		0			0
$955$	−29.9937	+	34.8241i	−0.970573	+	1.12688i
$956$		0			0
$957$		−	42.2432i		−	1.36553i
$958$		0			0
$959$	4.26579			0.137750
$960$		0			0
$961$	14.9289			0.481578
$962$		0			0
$963$			5.16145i			0.166325i
$964$		0			0
$965$	−13.4363	−	11.5726i	−0.432530	−	0.372534i
$966$		0			0
$967$		−	1.31927i		−	0.0424249i	−0.999775	−	0.0212124i	$-0.993247\pi$
$967$		−	1.31927i		−	0.0424249i	0.999775	−	0.0212124i	$-0.00675264\pi$
$968$		0			0
$969$	74.2497			2.38524
$970$		0			0
$971$	7.25779			0.232914			0.116457	−	0.993196i	$-0.462846\pi$
$971$	7.25779			0.232914			0.116457	+	0.993196i	$0.462846\pi$
$972$		0			0
$973$			15.0326i			0.481923i
$974$		0			0
$975$	−8.90998	−	59.4482i	−0.285348	−	1.90387i
$976$		0			0
$977$		−	4.26579i		−	0.136475i	−0.997669	−	0.0682374i	$-0.978262\pi$
$977$		−	4.26579i		−	0.136475i	0.997669	−	0.0682374i	$-0.0217375\pi$
$978$		0			0
$979$	−12.7974			−0.409006
$980$		0			0
$981$	−20.2208			−0.645600
$982$		0			0
$983$			34.3863i			1.09675i	0.836232	+	0.548376i	$0.184754\pi$
$983$			34.3863i			1.09675i	−0.836232	+	0.548376i	$0.815246\pi$
$984$		0			0
$985$	−17.6911	−	15.2372i	−0.563686	−	0.485497i
$986$		0			0
$987$		−	18.6782i		−	0.594533i
$988$		0			0
$989$	2.25481			0.0716989
$990$		0			0
$991$	−10.4417			−0.331692			−0.165846	−	0.986152i	$-0.553035\pi$
$991$	−10.4417			−0.331692			−0.165846	+	0.986152i	$0.553035\pi$
$992$		0			0
$993$			60.4416i			1.91806i
$994$		0			0
$995$	16.1144	−	18.7096i	0.510862	−	0.593135i
$996$		0			0
$997$		−	26.7075i		−	0.845834i	−0.906169	−	0.422917i	$-0.861006\pi$
$997$		−	26.7075i		−	0.845834i	0.906169	−	0.422917i	$-0.138994\pi$
$998$		0			0
$999$	27.1083			0.857670

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2240.2.g.p.449.3		12
4.3	odd	2	inner	2240.2.g.p.449.9		12
5.4	even	2	inner	2240.2.g.p.449.10		12
8.3	odd	2		1120.2.g.d.449.4	yes	12
8.5	even	2		1120.2.g.d.449.10	yes	12
20.19	odd	2	inner	2240.2.g.p.449.4		12
40.3	even	4		5600.2.a.by.1.6		6
40.13	odd	4		5600.2.a.bz.1.1		6
40.19	odd	2		1120.2.g.d.449.9	yes	12
40.27	even	4		5600.2.a.bz.1.2		6
40.29	even	2		1120.2.g.d.449.3	✓	12
40.37	odd	4		5600.2.a.by.1.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1120.2.g.d.449.3	✓	12	40.29	even	2
1120.2.g.d.449.4	yes	12	8.3	odd	2
1120.2.g.d.449.9	yes	12	40.19	odd	2
1120.2.g.d.449.10	yes	12	8.5	even	2
2240.2.g.p.449.3		12	1.1	even	1	trivial
2240.2.g.p.449.4		12	20.19	odd	2	inner
2240.2.g.p.449.9		12	4.3	odd	2	inner
2240.2.g.p.449.10		12	5.4	even	2	inner
5600.2.a.by.1.5		6	40.37	odd	4
5600.2.a.by.1.6		6	40.3	even	4
5600.2.a.bz.1.1		6	40.13	odd	4
5600.2.a.bz.1.2		6	40.27	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:	\( N \)	\(=\)	\( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2240.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.17741i q^{3} +(1.45926 - 1.69427i) q^{5} +1.00000i q^{7} -1.74111 q^{9} +O(q^{10})\)	\(q-2.17741i q^{3} +(1.45926 - 1.69427i) q^{5} +1.00000i q^{7} -1.74111 q^{9} +2.13290 q^{11} +5.52144i q^{13} +(-3.68912 - 3.17741i) q^{15} +5.24535i q^{17} +6.50100 q^{19} +2.17741 q^{21} +6.35482i q^{23} +(-0.741113 - 4.94477i) q^{25} -2.74111i q^{27} +9.09593 q^{29} +6.77709 q^{31} -4.64419i q^{33} +(1.69427 + 1.45926i) q^{35} +9.88954i q^{37} +12.0224 q^{39} -2.35482 q^{41} -0.354819i q^{43} +(-2.54074 + 2.94992i) q^{45} -8.57816i q^{47} -1.00000 q^{49} +11.4213 q^{51} +7.37825i q^{53} +(3.11245 - 3.61371i) q^{55} -14.1553i q^{57} -6.50100 q^{59} -5.43630 q^{61} -1.74111i q^{63} +(9.35482 + 8.05722i) q^{65} -5.48223i q^{67} +13.8370 q^{69} -3.11245 q^{71} +7.37825i q^{73} +(-10.7668 + 1.61371i) q^{75} +2.13290i q^{77} -12.0224 q^{79} -11.1919 q^{81} -11.1104i q^{83} +(8.88705 + 7.65434i) q^{85} -19.8056i q^{87} -6.00000 q^{89} -5.52144 q^{91} -14.7565i q^{93} +(9.48665 - 11.0145i) q^{95} -0.979557i q^{97} -3.71361 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 16 q^{9}+O(q^{10}) \) 12 * q - 16 * q^9	\( 12 q - 16 q^{9} - 4 q^{21} - 4 q^{25} + 44 q^{29} + 32 q^{41} - 48 q^{45} - 12 q^{49} - 40 q^{61} + 52 q^{65} + 96 q^{69} - 4 q^{81} - 44 q^{85} - 72 q^{89}+O(q^{100}) \) 12 * q - 16 * q^9 - 4 * q^21 - 4 * q^25 + 44 * q^29 + 32 * q^41 - 48 * q^45 - 12 * q^49 - 40 * q^61 + 52 * q^65 + 96 * q^69 - 4 * q^81 - 44 * q^85 - 72 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more