LMFDB - Embedded newform 2240.2.g.p.449.1

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			449.1
Root			$1.67709i$ of defining polynomial
Character	$\chi$	$=$	2240.449
Dual form			2240.2.g.p.449.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.85577i q^{3} +(0.649832 - 2.13956i) q^{5} -1.00000i q^{7} -5.15544 q^{9} +O(q^{10})$	$q-2.85577i q^{3} +(0.649832 - 2.13956i) q^{5} -1.00000i q^{7} -5.15544 q^{9} +4.61169 q^{11} -0.332566i q^{13} +(-6.11010 - 1.85577i) q^{15} -7.60851i q^{17} -1.28230 q^{19} -2.85577 q^{21} +3.71155i q^{23} +(-4.15544 - 2.78071i) q^{25} +6.15544i q^{27} +2.44389 q^{29} -8.55824 q^{31} -13.1699i q^{33} +(-2.13956 - 0.649832i) q^{35} +5.56142i q^{37} -0.949733 q^{39} +7.71155 q^{41} -9.71155i q^{43} +(-3.35017 + 11.0304i) q^{45} +8.75476i q^{47} -1.00000 q^{49} -21.7282 q^{51} -12.2202i q^{53} +(2.99682 - 9.86698i) q^{55} +3.66195i q^{57} +1.28230 q^{59} +3.01121 q^{61} +5.15544i q^{63} +(-0.711545 - 0.216112i) q^{65} +12.3109i q^{67} +10.5993 q^{69} -2.99682 q^{71} -12.2202i q^{73} +(-7.94107 + 11.8670i) q^{75} -4.61169i q^{77} +0.949733 q^{79} +2.11222 q^{81} -3.81255i q^{83} +(-16.2789 - 4.94425i) q^{85} -6.97920i q^{87} -6.00000 q^{89} -0.332566 q^{91} +24.4404i q^{93} +(-0.833279 + 2.74356i) q^{95} -1.61486i q^{97} -23.7753 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$		−	2.85577i		−	1.64878i	−0.566021	−	0.824391i	$-0.691518\pi$
$3$		−	2.85577i		−	1.64878i	0.566021	−	0.824391i	$-0.308482\pi$
$4$		0			0
$5$	0.649832	−	2.13956i	0.290614	−	0.956840i
$5$	0.649832	−	2.13956i	0.290614	−	0.956840i
$6$		0			0
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$		0			0
$9$	−5.15544			−1.71848
$10$		0			0
$11$	4.61169			1.39048			0.695238	−	0.718780i	$-0.255299\pi$
$11$	4.61169			1.39048			0.695238	+	0.718780i	$0.255299\pi$
$12$		0			0
$13$		−	0.332566i		−	0.0922372i	−0.998936	−	0.0461186i	$-0.985315\pi$
$13$		−	0.332566i		−	0.0922372i	0.998936	−	0.0461186i	$-0.0146852\pi$
$14$		0			0
$15$	−6.11010	−	1.85577i	−1.57762	−	0.479158i
$16$		0			0
$17$		−	7.60851i		−	1.84533i	−0.385597	−	0.922667i	$-0.626005\pi$
$17$		−	7.60851i		−	1.84533i	0.385597	−	0.922667i	$-0.373995\pi$
$18$		0			0
$19$	−1.28230			−0.294180			−0.147090	−	0.989123i	$-0.546991\pi$
$19$	−1.28230			−0.294180			−0.147090	+	0.989123i	$0.546991\pi$
$20$		0			0
$21$	−2.85577			−0.623181
$22$		0			0
$23$			3.71155i			0.773911i	0.922098	+	0.386955i	$0.126473\pi$
$23$			3.71155i			0.773911i	−0.922098	+	0.386955i	$0.873527\pi$
$24$		0			0
$25$	−4.15544	−	2.78071i	−0.831087	−	0.556142i
$26$		0			0
$27$			6.15544i			1.18461i
$28$		0			0
$29$	2.44389			0.453819			0.226910	−	0.973916i	$-0.427138\pi$
$29$	2.44389			0.453819			0.226910	+	0.973916i	$0.427138\pi$
$30$		0			0
$31$	−8.55824			−1.53711			−0.768553	−	0.639786i	$-0.779023\pi$
$31$	−8.55824			−1.53711			−0.768553	+	0.639786i	$0.779023\pi$
$32$		0			0
$33$		−	13.1699i		−	2.29259i
$34$		0			0
$35$	−2.13956	−	0.649832i	−0.361652	−	0.109842i
$36$		0			0
$37$			5.56142i			0.914292i	0.889392	+	0.457146i	$0.151128\pi$
$37$			5.56142i			0.914292i	−0.889392	+	0.457146i	$0.848872\pi$
$38$		0			0
$39$	−0.949733			−0.152079
$40$		0			0
$41$	7.71155			1.20434			0.602170	−	0.798368i	$-0.294303\pi$
$41$	7.71155			1.20434			0.602170	+	0.798368i	$0.294303\pi$
$42$		0			0
$43$		−	9.71155i		−	1.48100i	−0.672058	−	0.740498i	$-0.734590\pi$
$43$		−	9.71155i		−	1.48100i	0.672058	−	0.740498i	$-0.265410\pi$
$44$		0			0
$45$	−3.35017	+	11.0304i	−0.499414	+	1.64431i
$46$		0			0
$47$			8.75476i			1.27701i	0.769616	+	0.638507i	$0.220448\pi$
$47$			8.75476i			1.27701i	−0.769616	+	0.638507i	$0.779552\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	−21.7282			−3.04255
$52$		0			0
$53$		−	12.2202i		−	1.67857i	−0.543690	−	0.839286i	$-0.682973\pi$
$53$		−	12.2202i		−	1.67857i	0.543690	−	0.839286i	$-0.317027\pi$
$54$		0			0
$55$	2.99682	−	9.86698i	0.404091	−	1.33046i
$56$		0			0
$57$			3.66195i			0.485038i
$58$		0			0
$59$	1.28230			0.166941			0.0834705	−	0.996510i	$-0.473400\pi$
$59$	1.28230			0.166941			0.0834705	+	0.996510i	$0.473400\pi$
$60$		0			0
$61$	3.01121			0.385546			0.192773	−	0.981243i	$-0.438252\pi$
$61$	3.01121			0.385546			0.192773	+	0.981243i	$0.438252\pi$
$62$		0			0
$63$			5.15544i			0.649524i
$64$		0			0
$65$	−0.711545	−	0.216112i	−0.0882563	−	0.0268054i
$66$		0			0
$67$			12.3109i			1.50401i	0.659156	+	0.752006i	$0.270914\pi$
$67$			12.3109i			1.50401i	−0.659156	+	0.752006i	$0.729086\pi$
$68$		0			0
$69$	10.5993			1.27601
$70$		0			0
$71$	−2.99682			−0.355657			−0.177829	−	0.984061i	$-0.556907\pi$
$71$	−2.99682			−0.355657			−0.177829	+	0.984061i	$0.556907\pi$
$72$		0			0
$73$		−	12.2202i		−	1.43027i	−0.698989	−	0.715133i	$-0.746366\pi$
$73$		−	12.2202i		−	1.43027i	0.698989	−	0.715133i	$-0.253634\pi$
$74$		0			0
$75$	−7.94107	+	11.8670i	−0.916956	+	1.37028i
$76$		0			0
$77$		−	4.61169i		−	0.525550i
$78$		0			0
$79$	0.949733			0.106853			0.0534266	−	0.998572i	$-0.482986\pi$
$79$	0.949733			0.106853			0.0534266	+	0.998572i	$0.482986\pi$
$80$		0			0
$81$	2.11222			0.234691
$82$		0			0
$83$		−	3.81255i		−	0.418482i	−0.977864	−	0.209241i	$-0.932901\pi$
$83$		−	3.81255i		−	0.418482i	0.977864	−	0.209241i	$-0.0670993\pi$
$84$		0			0
$85$	−16.2789	−	4.94425i	−1.76569	−	0.536279i
$86$		0			0
$87$		−	6.97920i		−	0.748249i
$88$		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
$91$	−0.332566			−0.0348624
$92$		0			0
$93$			24.4404i			2.53435i
$94$		0			0
$95$	−0.833279	+	2.74356i	−0.0854926	+	0.281483i
$96$		0			0
$97$		−	1.61486i		−	0.163965i	−0.996634	−	0.0819823i	$-0.973875\pi$
$97$		−	1.61486i		−	0.163965i	0.996634	−	0.0819823i	$-0.0261251\pi$
$98$		0			0
$99$	−23.7753			−2.38950
$100$		0			0
$101$	14.4343			1.43627			0.718133	−	0.695906i	$-0.244997\pi$
$101$	14.4343			1.43627			0.718133	+	0.695906i	$0.244997\pi$
$102$		0			0
$103$			3.86698i			0.381025i	0.981685	+	0.190513i	$0.0610150\pi$
$103$			3.86698i			0.381025i	−0.981685	+	0.190513i	$0.938985\pi$
$104$		0			0
$105$	−1.85577	+	6.11010i	−0.181105	+	0.596284i
$106$		0			0
$107$			16.6217i			1.60688i	0.595382	+	0.803442i	$0.297001\pi$
$107$			16.6217i			1.60688i	−0.595382	+	0.803442i	$0.702999\pi$
$108$		0			0
$109$	−1.86698			−0.178824			−0.0894122	−	0.995995i	$-0.528499\pi$
$109$	−1.86698			−0.178824			−0.0894122	+	0.995995i	$0.528499\pi$
$110$		0			0
$111$	15.8821			1.50747
$112$		0			0
$113$			11.5551i			1.08701i	0.839406	+	0.543504i	$0.182903\pi$
$113$			11.5551i			1.08701i	−0.839406	+	0.543504i	$0.817097\pi$
$114$		0			0
$115$	7.94107	+	2.41188i	0.740509	+	0.224909i
$116$		0			0
$117$			1.71452i			0.158508i
$118$		0			0
$119$	−7.60851			−0.697471
$120$		0			0
$121$	10.2677			0.933423
$122$		0			0
$123$		−	22.0224i		−	1.98569i
$124$		0			0
$125$	−8.64983	+	7.08381i	−0.773664	+	0.633595i
$126$		0			0
$127$			12.0224i			1.06682i	0.845858	+	0.533409i	$0.179089\pi$
$127$			12.0224i			1.06682i	−0.845858	+	0.533409i	$0.820911\pi$
$128$		0			0
$129$	−27.7340			−2.44184
$130$		0			0
$131$	10.5057			0.917885			0.458942	−	0.888466i	$-0.348228\pi$
$131$	10.5057			0.917885			0.458942	+	0.888466i	$0.348228\pi$
$132$		0			0
$133$			1.28230i			0.111189i
$134$		0			0
$135$	13.1699	+	4.00000i	1.13349	+	0.344265i
$136$		0			0
$137$			9.22337i			0.788006i	0.919109	+	0.394003i	$0.128910\pi$
$137$			9.22337i			0.788006i	−0.919109	+	0.394003i	$0.871090\pi$
$138$		0			0
$139$	17.1644			1.45587			0.727935	−	0.685646i	$-0.240480\pi$
$139$	17.1644			1.45587			0.727935	+	0.685646i	$0.240480\pi$
$140$		0			0
$141$	25.0016			2.10552
$142$		0			0
$143$		−	1.53369i		−	0.128254i
$144$		0			0
$145$	1.58812	−	5.22885i	0.131886	−	0.434233i
$146$		0			0
$147$			2.85577i			0.235540i
$148$		0			0
$149$	0.801344			0.0656486			0.0328243	−	0.999461i	$-0.489550\pi$
$149$	0.801344			0.0656486			0.0328243	+	0.999461i	$0.489550\pi$
$150$		0			0
$151$	−5.27682			−0.429421			−0.214711	−	0.976678i	$-0.568881\pi$
$151$	−5.27682			−0.429421			−0.214711	+	0.976678i	$0.568881\pi$
$152$		0			0
$153$			39.2252i			3.17117i
$154$		0			0
$155$	−5.56142	+	18.3109i	−0.446704	+	1.47076i
$156$		0			0
$157$			20.1613i			1.60904i	0.593923	+	0.804522i	$0.297579\pi$
$157$			20.1613i			1.60904i	−0.593923	+	0.804522i	$0.702421\pi$
$158$		0			0
$159$	−34.8981			−2.76760
$160$		0			0
$161$	3.71155			0.292511
$162$		0			0
$163$		−	4.51289i		−	0.353477i	−0.984258	−	0.176738i	$-0.943445\pi$
$163$		−	4.51289i		−	0.353477i	0.984258	−	0.176738i	$-0.0565546\pi$
$164$		0			0
$165$	−28.1779	−	8.55824i	−2.19364	−	0.666258i
$166$		0			0
$167$			7.55611i			0.584709i	0.956310	+	0.292355i	$0.0944388\pi$
$167$			7.55611i			0.584709i	−0.956310	+	0.292355i	$0.905561\pi$
$168$		0			0
$169$	12.8894			0.991492
$170$		0			0
$171$	6.61081			0.505541
$172$		0			0
$173$		−	16.7839i		−	1.27606i	−0.770013	−	0.638029i	$-0.779750\pi$
$173$		−	16.7839i		−	1.27606i	0.770013	−	0.638029i	$-0.220250\pi$
$174$		0			0
$175$	−2.78071	+	4.15544i	−0.210202	+	0.314121i
$176$		0			0
$177$		−	3.66195i		−	0.275249i
$178$		0			0
$179$	5.99364			0.447986			0.223993	−	0.974591i	$-0.428091\pi$
$179$	5.99364			0.447986			0.223993	+	0.974591i	$0.428091\pi$
$180$		0			0
$181$	−4.41188			−0.327933			−0.163966	−	0.986466i	$-0.552429\pi$
$181$	−4.41188			−0.327933			−0.163966	+	0.986466i	$0.552429\pi$
$182$		0			0
$183$		−	8.59933i		−	0.635681i
$184$		0			0
$185$	11.8990	+	3.61399i	0.874831	+	0.265706i
$186$		0			0
$187$		−	35.0881i		−	2.56589i
$188$		0			0
$189$	6.15544			0.447742
$190$		0			0
$191$	−17.4970			−1.26604			−0.633020	−	0.774136i	$-0.718185\pi$
$191$	−17.4970			−1.26604			−0.633020	+	0.774136i	$0.718185\pi$
$192$		0			0
$193$		−	2.33169i		−	0.167839i	−0.996473	−	0.0839194i	$-0.973256\pi$
$193$		−	2.33169i		−	0.167839i	0.996473	−	0.0839194i	$-0.0267438\pi$
$194$		0			0
$195$	−0.617167	+	2.03201i	−0.0441962	+	0.145515i
$196$		0			0
$197$		−	20.1133i		−	1.43301i	−0.697580	−	0.716507i	$-0.745740\pi$
$197$		−	20.1133i		−	1.43301i	0.697580	−	0.716507i	$-0.254260\pi$
$198$		0			0
$199$	0.665132			0.0471500			0.0235750	−	0.999722i	$-0.492495\pi$
$199$	0.665132			0.0471500			0.0235750	+	0.999722i	$0.492495\pi$
$200$		0			0
$201$	35.1571			2.47979
$202$		0			0
$203$		−	2.44389i		−	0.171528i
$204$		0			0
$205$	5.01121	−	16.4993i	0.349998	−	1.15236i
$206$		0			0
$207$		−	19.1346i		−	1.32995i
$208$		0			0
$209$	−5.91356			−0.409050
$210$		0			0
$211$	−13.1699			−0.906655			−0.453328	−	0.891344i	$-0.649763\pi$
$211$	−13.1699			−0.906655			−0.453328	+	0.891344i	$0.649763\pi$
$212$		0			0
$213$			8.55824i			0.586401i
$214$		0			0
$215$	−20.7784	−	6.31087i	−1.41708	−	0.430398i
$216$		0			0
$217$			8.55824i			0.580971i
$218$		0			0
$219$	−34.8981			−2.35819
$220$		0			0
$221$	−2.53033			−0.170208
$222$		0			0
$223$		−	13.5785i		−	0.909285i	−0.890674	−	0.454643i	$-0.849767\pi$
$223$		−	13.5785i		−	0.909285i	0.890674	−	0.454643i	$-0.150233\pi$
$224$		0			0
$225$	21.4231	+	14.3358i	1.42821	+	0.955718i
$226$		0			0
$227$		−	11.9904i		−	0.795831i	−0.917422	−	0.397916i	$-0.869734\pi$
$227$		−	11.9904i		−	0.795831i	0.917422	−	0.397916i	$-0.130266\pi$
$228$		0			0
$229$	22.1234			1.46196			0.730979	−	0.682400i	$-0.239064\pi$
$229$	22.1234			1.46196			0.730979	+	0.682400i	$0.239064\pi$
$230$		0			0
$231$	−13.1699			−0.866518
$232$		0			0
$233$			5.99364i			0.392657i	0.980538	+	0.196328i	$0.0629018\pi$
$233$			5.99364i			0.392657i	−0.980538	+	0.196328i	$0.937098\pi$
$234$		0			0
$235$	18.7313	+	5.68913i	1.22190	+	0.371118i
$236$		0			0
$237$		−	2.71222i		−	0.176178i
$238$		0			0
$239$	3.94655			0.255281			0.127641	−	0.991820i	$-0.459260\pi$
$239$	3.94655			0.255281			0.127641	+	0.991820i	$0.459260\pi$
$240$		0			0
$241$	13.4871			0.868781			0.434391	−	0.900725i	$-0.356964\pi$
$241$	13.4871			0.868781			0.434391	+	0.900725i	$0.356964\pi$
$242$		0			0
$243$			12.4343i			0.797661i
$244$		0			0
$245$	−0.649832	+	2.13956i	−0.0415162	+	0.136691i
$246$		0			0
$247$			0.426449i			0.0271343i
$248$		0			0
$249$	−10.8878			−0.689985
$250$		0			0
$251$	14.5998			0.921534			0.460767	−	0.887521i	$-0.347574\pi$
$251$	14.5998			0.921534			0.460767	+	0.887521i	$0.347574\pi$
$252$		0			0
$253$			17.1165i			1.07610i
$254$		0			0
$255$	−14.1197	+	46.4887i	−0.884207	+	2.91124i
$256$		0			0
$257$		−	6.22655i		−	0.388402i	−0.980962	−	0.194201i	$-0.937789\pi$
$257$		−	6.22655i		−	0.388402i	0.980962	−	0.194201i	$-0.0622113\pi$
$258$		0			0
$259$	5.56142			0.345570
$260$		0			0
$261$	−12.5993			−0.779879
$262$		0			0
$263$			5.19866i			0.320563i	0.987071	+	0.160281i	$0.0512402\pi$
$263$			5.19866i			0.320563i	−0.987071	+	0.160281i	$0.948760\pi$
$264$		0			0
$265$	−26.1458	−	7.94107i	−1.60613	−	0.487816i
$266$		0			0
$267$			17.1346i			1.04862i
$268$		0			0
$269$	−27.0560			−1.64964			−0.824818	−	0.565398i	$-0.808723\pi$
$269$	−27.0560			−1.64964			−0.824818	+	0.565398i	$0.808723\pi$
$270$		0			0
$271$	3.22973			0.196192			0.0980961	−	0.995177i	$-0.468725\pi$
$271$	3.22973			0.196192			0.0980961	+	0.995177i	$0.468725\pi$
$272$		0			0
$273$			0.949733i			0.0574804i
$274$		0			0
$275$	−19.1636	−	12.8238i	−1.15561	−	0.773302i
$276$		0			0
$277$		−	10.8899i		−	0.654313i	−0.944970	−	0.327156i	$-0.893910\pi$
$277$		−	10.8899i		−	0.654313i	0.944970	−	0.327156i	$-0.106090\pi$
$278$		0			0
$279$	44.1215			2.64148
$280$		0			0
$281$	−25.8030			−1.53928			−0.769638	−	0.638481i	$-0.779563\pi$
$281$	−25.8030			−1.53928			−0.769638	+	0.638481i	$0.779563\pi$
$282$		0			0
$283$			10.4809i			0.623024i	0.950242	+	0.311512i	$0.100835\pi$
$283$			10.4809i			0.623024i	−0.950242	+	0.311512i	$0.899165\pi$
$284$		0			0
$285$	7.83497	+	2.37965i	0.464104	+	0.140959i
$286$		0			0
$287$		−	7.71155i		−	0.455198i
$288$		0			0
$289$	−40.8894			−2.40526
$290$		0			0
$291$	−4.61169			−0.270342
$292$		0			0
$293$		−	2.23203i		−	0.130397i	−0.997872	−	0.0651983i	$-0.979232\pi$
$293$		−	2.23203i		−	0.130397i	0.997872	−	0.0651983i	$-0.0207680\pi$
$294$		0			0
$295$	0.833279	−	2.74356i	0.0485154	−	0.159736i
$296$		0			0
$297$			28.3869i			1.64718i
$298$		0			0
$299$	1.23433			0.0713834
$300$		0			0
$301$	−9.71155			−0.559764
$302$		0			0
$303$		−	41.2211i		−	2.36809i
$304$		0			0
$305$	1.95678	−	6.44266i	0.112045	−	0.368906i
$306$		0			0
$307$		−	12.8782i		−	0.734997i	−0.930024	−	0.367499i	$-0.880214\pi$
$307$		−	12.8782i		−	0.734997i	0.930024	−	0.367499i	$-0.119786\pi$
$308$		0			0
$309$	11.0432			0.628227
$310$		0			0
$311$	15.2170			0.862878			0.431439	−	0.902142i	$-0.358006\pi$
$311$	15.2170			0.862878			0.431439	+	0.902142i	$0.358006\pi$
$312$		0			0
$313$		−	7.60851i		−	0.430058i	−0.976608	−	0.215029i	$-0.931015\pi$
$313$		−	7.60851i		−	0.430058i	0.976608	−	0.215029i	$-0.0689847\pi$
$314$		0			0
$315$	11.0304	+	3.35017i	0.621491	+	0.188761i
$316$		0			0
$317$		−	32.5664i		−	1.82911i	−0.404460	−	0.914556i	$-0.632540\pi$
$317$		−	32.5664i		−	1.82911i	0.404460	−	0.914556i	$-0.367460\pi$
$318$		0			0
$319$	11.2705			0.631025
$320$		0			0
$321$	47.4679			2.64940
$322$		0			0
$323$			9.75638i			0.542860i
$324$		0			0
$325$	−0.924770	+	1.38196i	−0.0512970	+	0.0766572i
$326$		0			0
$327$			5.33167i			0.294842i
$328$		0			0
$329$	8.75476			0.482666
$330$		0			0
$331$	−21.8758			−1.20240			−0.601201	−	0.799098i	$-0.705311\pi$
$331$	−21.8758			−1.20240			−0.601201	+	0.799098i	$0.705311\pi$
$332$		0			0
$333$		−	28.6715i		−	1.57119i
$334$		0			0
$335$	26.3399	+	8.00000i	1.43910	+	0.437087i
$336$		0			0
$337$			8.79115i			0.478884i	0.970911	+	0.239442i	$0.0769646\pi$
$337$			8.79115i			0.478884i	−0.970911	+	0.239442i	$0.923035\pi$
$338$		0			0
$339$	32.9986			1.79224
$340$		0			0
$341$	−39.4679			−2.13731
$342$		0			0
$343$			1.00000i			0.0539949i
$344$		0			0
$345$	6.88778	−	22.6779i	0.370826	−	1.22094i
$346$		0			0
$347$		−	26.6442i		−	1.43033i	−0.698954	−	0.715167i	$-0.746351\pi$
$347$		−	26.6442i		−	1.43033i	0.698954	−	0.715167i	$-0.253649\pi$
$348$		0			0
$349$	9.61054			0.514440			0.257220	−	0.966353i	$-0.417193\pi$
$349$	9.61054			0.514440			0.257220	+	0.966353i	$0.417193\pi$
$350$		0			0
$351$	2.04709			0.109266
$352$		0			0
$353$			27.2896i			1.45248i	0.687442	+	0.726239i	$0.258733\pi$
$353$			27.2896i			1.45248i	−0.687442	+	0.726239i	$0.741267\pi$
$354$		0			0
$355$	−1.94743	+	6.41188i	−0.103359	+	0.340307i
$356$		0			0
$357$			21.7282i			1.14998i
$358$		0			0
$359$	23.1101			1.21971			0.609853	−	0.792515i	$-0.291229\pi$
$359$	23.1101			1.21971			0.609853	+	0.792515i	$0.291229\pi$
$360$		0			0
$361$	−17.3557			−0.913458
$362$		0			0
$363$		−	29.3221i		−	1.53901i
$364$		0			0
$365$	−26.1458	−	7.94107i	−1.36854	−	0.415655i
$366$		0			0
$367$		−	5.26765i		−	0.274969i	−0.990504	−	0.137485i	$-0.956098\pi$
$367$		−	5.26765i		−	0.274969i	0.990504	−	0.137485i	$-0.0439018\pi$
$368$		0			0
$369$	−39.7564			−2.06963
$370$		0			0
$371$	−12.2202			−0.634441
$372$		0			0
$373$		−	29.3367i		−	1.51900i	−0.650510	−	0.759498i	$-0.725445\pi$
$373$		−	29.3367i		−	1.51900i	0.650510	−	0.759498i	$-0.274555\pi$
$374$		0			0
$375$	20.2298	+	24.7020i	1.04466	+	1.27560i
$376$		0			0
$377$		−	0.812755i		−	0.0418590i
$378$		0			0
$379$	31.6684			1.62669			0.813347	−	0.581778i	$-0.197643\pi$
$379$	31.6684			1.62669			0.813347	+	0.581778i	$0.197643\pi$
$380$		0			0
$381$	34.3333			1.75895
$382$		0			0
$383$		−	36.2469i		−	1.85213i	−0.377367	−	0.926064i	$-0.623170\pi$
$383$		−	36.2469i		−	1.85213i	0.377367	−	0.926064i	$-0.376830\pi$
$384$		0			0
$385$	−9.86698	−	2.99682i	−0.502868	−	0.152732i
$386$		0			0
$387$			50.0673i			2.54506i
$388$		0			0
$389$	32.4023			1.64286			0.821431	−	0.570308i	$-0.193176\pi$
$389$	32.4023			1.64286			0.821431	+	0.570308i	$0.193176\pi$
$390$		0			0
$391$	28.2393			1.42812
$392$		0			0
$393$		−	30.0018i		−	1.51339i
$394$		0			0
$395$	0.617167	−	2.03201i	0.0310530	−	0.102242i
$396$		0			0
$397$		−	30.7666i		−	1.54413i	−0.635543	−	0.772066i	$-0.719224\pi$
$397$		−	30.7666i		−	1.54413i	0.635543	−	0.772066i	$-0.280776\pi$
$398$		0			0
$399$	3.66195			0.183327
$400$		0			0
$401$	17.8894			0.893354			0.446677	−	0.894695i	$-0.352607\pi$
$401$	17.8894			0.893354			0.446677	+	0.894695i	$0.352607\pi$
$402$		0			0
$403$			2.84618i			0.141778i
$404$		0			0
$405$	1.37259	−	4.51922i	0.0682044	−	0.224562i
$406$		0			0
$407$			25.6475i			1.27130i
$408$		0			0
$409$	7.13464			0.352785			0.176392	−	0.984320i	$-0.443557\pi$
$409$	7.13464			0.352785			0.176392	+	0.984320i	$0.443557\pi$
$410$		0			0
$411$	26.3399			1.29925
$412$		0			0
$413$		−	1.28230i		−	0.0630978i
$414$		0			0
$415$	−8.15719	−	2.47752i	−0.400421	−	0.121617i
$416$		0			0
$417$		−	49.0178i		−	2.40041i
$418$		0			0
$419$	−34.2809			−1.67473			−0.837367	−	0.546642i	$-0.815906\pi$
$419$	−34.2809			−1.67473			−0.837367	+	0.546642i	$0.815906\pi$
$420$		0			0
$421$	−1.30926			−0.0638093			−0.0319046	−	0.999491i	$-0.510157\pi$
$421$	−1.30926			−0.0638093			−0.0319046	+	0.999491i	$0.510157\pi$
$422$		0			0
$423$		−	45.1346i		−	2.19452i
$424$		0			0
$425$	−21.1571	+	31.6167i	−1.02627	+	1.53363i
$426$		0			0
$427$		−	3.01121i		−	0.145723i
$428$		0			0
$429$	−4.37987			−0.211462
$430$		0			0
$431$	−11.8397			−0.570297			−0.285148	−	0.958483i	$-0.592043\pi$
$431$	−11.8397			−0.570297			−0.285148	+	0.958483i	$0.592043\pi$
$432$		0			0
$433$		−	33.2315i		−	1.59701i	−0.601991	−	0.798503i	$-0.705626\pi$
$433$		−	33.2315i		−	1.59701i	0.601991	−	0.798503i	$-0.294374\pi$
$434$		0			0
$435$	−14.9324	−	4.53531i	−0.715955	−	0.217451i
$436$		0			0
$437$		−	4.75931i		−	0.227669i
$438$		0			0
$439$	25.0096			1.19364			0.596821	−	0.802374i	$-0.296430\pi$
$439$	25.0096			1.19364			0.596821	+	0.802374i	$0.296430\pi$
$440$		0			0
$441$	5.15544			0.245497
$442$		0			0
$443$			15.7340i			0.747543i	0.927521	+	0.373772i	$0.121936\pi$
$443$			15.7340i			0.747543i	−0.927521	+	0.373772i	$0.878064\pi$
$444$		0			0
$445$	−3.89899	+	12.8374i	−0.184830	+	0.608549i
$446$		0			0
$447$		−	2.28845i		−	0.108240i
$448$		0			0
$449$	21.3541			1.00776			0.503881	−	0.863773i	$-0.331905\pi$
$449$	21.3541			1.00776			0.503881	+	0.863773i	$0.331905\pi$
$450$		0			0
$451$	35.5632			1.67461
$452$		0			0
$453$			15.0694i			0.708022i
$454$		0			0
$455$	−0.216112	+	0.711545i	−0.0101315	+	0.0333577i
$456$		0			0
$457$		−	28.6715i		−	1.34120i	−0.741820	−	0.670599i	$-0.766037\pi$
$457$		−	28.6715i		−	1.34120i	0.741820	−	0.670599i	$-0.233963\pi$
$458$		0			0
$459$	46.8337			2.18601
$460$		0			0
$461$	−6.94719			−0.323563			−0.161781	−	0.986827i	$-0.551724\pi$
$461$	−6.94719			−0.323563			−0.161781	+	0.986827i	$0.551724\pi$
$462$		0			0
$463$			3.68913i			0.171448i	0.996319	+	0.0857241i	$0.0273204\pi$
$463$			3.68913i			0.171448i	−0.996319	+	0.0857241i	$0.972680\pi$
$464$		0			0
$465$	52.2917	+	15.8821i	2.42497	+	0.736517i
$466$		0			0
$467$			1.72114i			0.0796447i	0.999207	+	0.0398224i	$0.0126792\pi$
$467$			1.72114i			0.0796447i	−0.999207	+	0.0398224i	$0.987321\pi$
$468$		0			0
$469$	12.3109			0.568463
$470$		0			0
$471$	57.5760			2.65296
$472$		0			0
$473$		−	44.7866i		−	2.05929i
$474$		0			0
$475$	5.32851	+	3.56570i	0.244489	+	0.163606i
$476$		0			0
$477$			63.0004i			2.88459i
$478$		0			0
$479$	−32.9986			−1.50775			−0.753873	−	0.657020i	$-0.771817\pi$
$479$	−32.9986			−1.50775			−0.753873	+	0.657020i	$0.771817\pi$
$480$		0			0
$481$	1.84954			0.0843317
$482$		0			0
$483$		−	10.5993i		−	0.482286i
$484$		0			0
$485$	−3.45510	−	1.04939i	−0.156888	−	0.0476504i
$486$		0			0
$487$		−	0.599328i		−	0.0271582i	−0.999908	−	0.0135791i	$-0.995678\pi$
$487$		−	0.599328i		−	0.0271582i	0.999908	−	0.0135791i	$-0.00432249\pi$
$488$		0			0
$489$	−12.8878			−0.582806
$490$		0			0
$491$	16.3997			0.740106			0.370053	−	0.929011i	$-0.379339\pi$
$491$	16.3997			0.740106			0.370053	+	0.929011i	$0.379339\pi$
$492$		0			0
$493$		−	18.5944i		−	0.837448i
$494$		0			0
$495$	−15.4499	+	50.8686i	−0.694422	+	2.28637i
$496$		0			0
$497$			2.99682i			0.134426i
$498$		0			0
$499$	−0.716825			−0.0320895			−0.0160447	−	0.999871i	$-0.505107\pi$
$499$	−0.716825			−0.0320895			−0.0160447	+	0.999871i	$0.505107\pi$
$500$		0			0
$501$	21.5785			0.964057
$502$		0			0
$503$		−	27.6650i		−	1.23352i	−0.787151	−	0.616760i	$-0.788445\pi$
$503$		−	27.6650i		−	1.23352i	0.787151	−	0.616760i	$-0.211555\pi$
$504$		0			0
$505$	9.37987	−	30.8831i	0.417399	−	1.37428i
$506$		0			0
$507$		−	36.8092i		−	1.63475i
$508$		0			0
$509$	−26.2323			−1.16273			−0.581363	−	0.813644i	$-0.697480\pi$
$509$	−26.2323			−1.16273			−0.581363	+	0.813644i	$0.697480\pi$
$510$		0			0
$511$	−12.2202			−0.540590
$512$		0			0
$513$		−	7.89311i		−	0.348489i
$514$		0			0
$515$	8.27364	+	2.51289i	0.364580	+	0.110731i
$516$		0			0
$517$			40.3742i			1.77566i
$518$		0			0
$519$	−47.9310			−2.10394
$520$		0			0
$521$	−30.2469			−1.32514			−0.662569	−	0.749001i	$-0.730534\pi$
$521$	−30.2469			−1.32514			−0.662569	+	0.749001i	$0.730534\pi$
$522$		0			0
$523$		−	1.38610i		−	0.0606101i	−0.999541	−	0.0303050i	$-0.990352\pi$
$523$		−	1.38610i		−	0.0606101i	0.999541	−	0.0303050i	$-0.00964787\pi$
$524$		0			0
$525$	11.8670	+	7.94107i	0.517918	+	0.346577i
$526$		0			0
$527$			65.1155i			2.83647i
$528$		0			0
$529$	9.22443			0.401062
$530$		0			0
$531$	−6.61081			−0.286885
$532$		0			0
$533$		−	2.56460i		−	0.111085i
$534$		0			0
$535$	35.5632	+	10.8013i	1.53753	+	0.466983i
$536$		0			0
$537$		−	17.1165i		−	0.738631i
$538$		0			0
$539$	−4.61169			−0.198639
$540$		0			0
$541$	−25.4921			−1.09599			−0.547995	−	0.836481i	$-0.684609\pi$
$541$	−25.4921			−1.09599			−0.547995	+	0.836481i	$0.684609\pi$
$542$		0			0
$543$			12.5993i			0.540689i
$544$		0			0
$545$	−1.21322	+	3.99452i	−0.0519688	+	0.171106i
$546$		0			0
$547$			14.2885i			0.610930i	0.952203	+	0.305465i	$0.0988119\pi$
$547$			14.2885i			0.610930i	−0.952203	+	0.305465i	$0.901188\pi$
$548$		0			0
$549$	−15.5241			−0.662552
$550$		0			0
$551$	−3.13380			−0.133504
$552$		0			0
$553$		−	0.949733i		−	0.0403867i
$554$		0			0
$555$	10.3207	−	33.9808i	0.438091	−	1.44241i
$556$		0			0
$557$		−	25.2425i		−	1.06956i	−0.844992	−	0.534780i	$-0.820395\pi$
$557$		−	25.2425i		−	1.06956i	0.844992	−	0.534780i	$-0.179605\pi$
$558$		0			0
$559$	−3.22973			−0.136603
$560$		0			0
$561$	−100.204			−4.23060
$562$		0			0
$563$			14.9888i			0.631702i	0.948809	+	0.315851i	$0.102290\pi$
$563$			14.9888i			0.631702i	−0.948809	+	0.315851i	$0.897710\pi$
$564$		0			0
$565$	24.7228	+	7.50885i	1.04009	+	0.315900i
$566$		0			0
$567$		−	2.11222i		−	0.0887048i
$568$		0			0
$569$	22.5353			0.944729			0.472365	−	0.881403i	$-0.343401\pi$
$569$	22.5353			0.944729			0.472365	+	0.881403i	$0.343401\pi$
$570$		0			0
$571$	13.2216			0.553308			0.276654	−	0.960970i	$-0.410774\pi$
$571$	13.2216			0.553308			0.276654	+	0.960970i	$0.410774\pi$
$572$		0			0
$573$			49.9675i			2.08742i
$574$		0			0
$575$	10.3207	−	15.4231i	0.430404	−	0.643187i
$576$		0			0
$577$			23.4907i			0.977929i	0.872304	+	0.488964i	$0.162625\pi$
$577$			23.4907i			0.977929i	−0.872304	+	0.488964i	$0.837375\pi$
$578$		0			0
$579$	−6.65878			−0.276729
$580$		0			0
$581$	−3.81255			−0.158171
$582$		0			0
$583$		−	56.3557i		−	2.33402i
$584$		0			0
$585$	3.66833	+	1.11415i	0.151667	+	0.0460645i
$586$		0			0
$587$			33.6330i			1.38818i	0.719888	+	0.694090i	$0.244193\pi$
$587$			33.6330i			1.38818i	−0.719888	+	0.694090i	$0.755807\pi$
$588$		0			0
$589$	10.9742			0.452185
$590$		0			0
$591$	−57.4390			−2.36273
$592$		0			0
$593$			20.7267i			0.851145i	0.904924	+	0.425573i	$0.139927\pi$
$593$			20.7267i			0.851145i	−0.904924	+	0.425573i	$0.860073\pi$
$594$		0			0
$595$	−4.94425	+	16.2789i	−0.202695	+	0.667368i
$596$		0			0
$597$		−	1.89947i		−	0.0777400i
$598$		0			0
$599$	−14.2673			−0.582945			−0.291473	−	0.956579i	$-0.594145\pi$
$599$	−14.2673			−0.582945			−0.291473	+	0.956579i	$0.594145\pi$
$600$		0			0
$601$	14.2660			0.581924			0.290962	−	0.956735i	$-0.406025\pi$
$601$	14.2660			0.581924			0.290962	+	0.956735i	$0.406025\pi$
$602$		0			0
$603$		−	63.4679i		−	2.58461i
$604$		0			0
$605$	6.67225	−	21.9683i	0.271266	−	0.893137i
$606$		0			0
$607$		−	10.1554i		−	0.412196i	−0.978531	−	0.206098i	$-0.933923\pi$
$607$		−	10.1554i		−	0.412196i	0.978531	−	0.206098i	$-0.0660766\pi$
$608$		0			0
$609$	−6.97920			−0.282811
$610$		0			0
$611$	2.91154			0.117788
$612$		0			0
$613$			30.8663i			1.24668i	0.781952	+	0.623338i	$0.214224\pi$
$613$			30.8663i			1.24668i	−0.781952	+	0.623338i	$0.785776\pi$
$614$		0			0
$615$	−47.1183	−	14.3109i	−1.89999	−	0.577070i
$616$		0			0
$617$		−	8.79115i		−	0.353918i	−0.984218	−	0.176959i	$-0.943374\pi$
$617$		−	8.79115i		−	0.353918i	0.984218	−	0.176959i	$-0.0566261\pi$
$618$		0			0
$619$	7.84514			0.315323			0.157661	−	0.987493i	$-0.449605\pi$
$619$	7.84514			0.315323			0.157661	+	0.987493i	$0.449605\pi$
$620$		0			0
$621$	−22.8462			−0.916786
$622$		0			0
$623$			6.00000i			0.240385i
$624$		0			0
$625$	9.53531	+	23.1101i	0.381412	+	0.924405i
$626$		0			0
$627$			16.8878i			0.674433i
$628$		0			0
$629$	42.3141			1.68717
$630$		0			0
$631$	35.1827			1.40060			0.700301	−	0.713848i	$-0.253049\pi$
$631$	35.1827			1.40060			0.700301	+	0.713848i	$0.253049\pi$
$632$		0			0
$633$			37.6103i			1.49488i
$634$		0			0
$635$	25.7227	+	7.81255i	1.02077	+	0.310032i
$636$		0			0
$637$			0.332566i			0.0131767i
$638$		0			0
$639$	15.4499			0.611190
$640$		0			0
$641$	−8.31087			−0.328260			−0.164130	−	0.986439i	$-0.552482\pi$
$641$	−8.31087			−0.328260			−0.164130	+	0.986439i	$0.552482\pi$
$642$		0			0
$643$		−	23.4327i		−	0.924095i	−0.886855	−	0.462047i	$-0.847115\pi$
$643$		−	23.4327i		−	0.924095i	0.886855	−	0.462047i	$-0.152885\pi$
$644$		0			0
$645$	−18.0224	+	59.3385i	−0.709632	+	2.33645i
$646$		0			0
$647$			23.1762i			0.911152i	0.890197	+	0.455576i	$0.150567\pi$
$647$			23.1762i			0.911152i	−0.890197	+	0.455576i	$0.849433\pi$
$648$		0			0
$649$	5.91356			0.232128
$650$		0			0
$651$	24.4404			0.957894
$652$		0			0
$653$		−	10.9859i		−	0.429910i	−0.976624	−	0.214955i	$-0.931040\pi$
$653$		−	10.9859i		−	0.429910i	0.976624	−	0.214955i	$-0.0689605\pi$
$654$		0			0
$655$	6.82692	−	22.4775i	0.266750	−	0.878269i
$656$		0			0
$657$			63.0004i			2.45788i
$658$		0			0
$659$	−14.4043			−0.561110			−0.280555	−	0.959838i	$-0.590519\pi$
$659$	−14.4043			−0.561110			−0.280555	+	0.959838i	$0.590519\pi$
$660$		0			0
$661$	11.6970			0.454960			0.227480	−	0.973783i	$-0.426951\pi$
$661$	11.6970			0.454960			0.227480	+	0.973783i	$0.426951\pi$
$662$		0			0
$663$			7.22605i			0.280637i
$664$		0			0
$665$	2.74356	+	0.833279i	0.106391	+	0.0323132i
$666$		0			0
$667$			9.07061i			0.351216i
$668$		0			0
$669$	−38.7772			−1.49921
$670$		0			0
$671$	13.8868			0.536092
$672$		0			0
$673$			43.4563i			1.67512i	0.546346	+	0.837559i	$0.316018\pi$
$673$			43.4563i			1.67512i	−0.546346	+	0.837559i	$0.683982\pi$
$674$		0			0
$675$	17.1165	−	25.5785i	0.658814	−	0.984518i
$676$		0			0
$677$			27.3376i			1.05067i	0.850896	+	0.525334i	$0.176060\pi$
$677$			27.3376i			1.05067i	−0.850896	+	0.525334i	$0.823940\pi$
$678$		0			0
$679$	−1.61486			−0.0619728
$680$		0			0
$681$	−34.2419			−1.31215
$682$		0			0
$683$		−	5.46469i		−	0.209101i	−0.994520	−	0.104550i	$-0.966660\pi$
$683$		−	5.46469i		−	0.209101i	0.994520	−	0.104550i	$-0.0333403\pi$
$684$		0			0
$685$	19.7340	+	5.99364i	0.753996	+	0.229005i
$686$		0			0
$687$		−	63.1795i		−	2.41045i
$688$		0			0
$689$	−4.06402			−0.154827
$690$		0			0
$691$	22.4930			0.855672			0.427836	−	0.903856i	$-0.359276\pi$
$691$	22.4930			0.855672			0.427836	+	0.903856i	$0.359276\pi$
$692$		0			0
$693$			23.7753i			0.903147i
$694$		0			0
$695$	11.1540	−	36.7244i	0.423096	−	1.39303i
$696$		0			0
$697$		−	58.6734i		−	2.22241i
$698$		0			0
$699$	17.1165			0.647405
$700$		0			0
$701$	37.0240			1.39838			0.699189	−	0.714937i	$-0.253544\pi$
$701$	37.0240			1.39838			0.699189	+	0.714937i	$0.253544\pi$
$702$		0			0
$703$		−	7.13140i		−	0.268966i
$704$		0			0
$705$	16.2469	−	53.4925i	0.611892	−	2.01464i
$706$		0			0
$707$		−	14.4343i		−	0.542858i
$708$		0			0
$709$	10.8928			0.409086			0.204543	−	0.978858i	$-0.434429\pi$
$709$	10.8928			0.409086			0.204543	+	0.978858i	$0.434429\pi$
$710$		0			0
$711$	−4.89629			−0.183625
$712$		0			0
$713$		−	31.7643i		−	1.18958i
$714$		0			0
$715$	−3.28142	−	0.996641i	−0.122718	−	0.0372723i
$716$		0			0
$717$		−	11.2705i		−	0.420903i
$718$		0			0
$719$	9.88851			0.368779			0.184390	−	0.982853i	$-0.440969\pi$
$719$	9.88851			0.368779			0.184390	+	0.982853i	$0.440969\pi$
$720$		0			0
$721$	3.86698			0.144014
$722$		0			0
$723$		−	38.5161i		−	1.43243i
$724$		0			0
$725$	−10.1554	−	6.79575i	−0.377163	−	0.252388i
$726$		0			0
$727$			27.0482i			1.00316i	0.865111	+	0.501581i	$0.167248\pi$
$727$			27.0482i			1.00316i	−0.865111	+	0.501581i	$0.832752\pi$
$728$		0			0
$729$	41.8462			1.54986
$730$		0			0
$731$	−73.8904			−2.73293
$732$		0			0
$733$		−	28.5719i		−	1.05533i	−0.849454	−	0.527663i	$-0.823068\pi$
$733$		−	28.5719i		−	1.05533i	0.849454	−	0.527663i	$-0.176932\pi$
$734$		0			0
$735$	6.11010	+	1.85577i	0.225374	+	0.0684512i
$736$		0			0
$737$			56.7739i			2.09129i
$738$		0			0
$739$	−41.0394			−1.50966			−0.754829	−	0.655922i	$-0.772280\pi$
$739$	−41.0394			−1.50966			−0.754829	+	0.655922i	$0.772280\pi$
$740$		0			0
$741$	1.21784			0.0447385
$742$		0			0
$743$		−	2.82376i		−	0.103594i	−0.998658	−	0.0517969i	$-0.983505\pi$
$743$		−	2.82376i		−	0.103594i	0.998658	−	0.0517969i	$-0.0164948\pi$
$744$		0			0
$745$	0.520739	−	1.71452i	0.0190784	−	0.0628153i
$746$		0			0
$747$			19.6554i			0.719153i
$748$		0			0
$749$	16.6217			0.607345
$750$		0			0
$751$	8.04073			0.293410			0.146705	−	0.989180i	$-0.453133\pi$
$751$	8.04073			0.293410			0.146705	+	0.989180i	$0.453133\pi$
$752$		0			0
$753$		−	41.6939i		−	1.51941i
$754$		0			0
$755$	−3.42905	+	11.2901i	−0.124796	+	0.410888i
$756$		0			0
$757$			15.3540i			0.558050i	0.960284	+	0.279025i	$0.0900113\pi$
$757$			15.3540i			0.558050i	−0.960284	+	0.279025i	$0.909989\pi$
$758$		0			0
$759$	48.8808			1.77426
$760$		0			0
$761$	−10.5129			−0.381092			−0.190546	−	0.981678i	$-0.561026\pi$
$761$	−10.5129			−0.381092			−0.190546	+	0.981678i	$0.561026\pi$
$762$		0			0
$763$			1.86698i			0.0675893i
$764$		0			0
$765$	83.9246	+	25.4898i	3.03430	+	0.921585i
$766$		0			0
$767$		−	0.426449i		−	0.0153982i
$768$		0			0
$769$	−9.93598			−0.358301			−0.179150	−	0.983822i	$-0.557335\pi$
$769$	−9.93598			−0.358301			−0.179150	+	0.983822i	$0.557335\pi$
$770$		0			0
$771$	−17.7816			−0.640389
$772$		0			0
$773$			33.9963i			1.22276i	0.791336	+	0.611381i	$0.209386\pi$
$773$			33.9963i			1.22276i	−0.791336	+	0.611381i	$0.790614\pi$
$774$		0			0
$775$	35.5632	+	23.7980i	1.27747	+	0.854849i
$776$		0			0
$777$		−	15.8821i		−	0.569769i
$778$		0			0
$779$	−9.88851			−0.354292
$780$		0			0
$781$	−13.8204			−0.494533
$782$		0			0
$783$			15.0432i			0.537601i
$784$		0			0
$785$	43.1363	+	13.1014i	1.53960	+	0.467610i
$786$		0			0
$787$		−	34.5897i		−	1.23299i	−0.787358	−	0.616495i	$-0.788552\pi$
$787$		−	34.5897i		−	1.23299i	0.787358	−	0.616495i	$-0.211448\pi$
$788$		0			0
$789$	14.8462			0.528538
$790$		0			0
$791$	11.5551			0.410851
$792$		0			0
$793$		−	1.00143i		−	0.0355617i
$794$		0			0
$795$	−22.6779	+	74.6666i	−0.804302	+	2.64815i
$796$		0			0
$797$			31.8016i			1.12647i	0.826296	+	0.563236i	$0.190444\pi$
$797$			31.8016i			1.12647i	−0.826296	+	0.563236i	$0.809556\pi$
$798$		0			0
$799$	66.6107			2.35652
$800$		0			0
$801$	30.9326			1.09295
$802$		0			0
$803$		−	56.3557i		−	1.98875i
$804$		0			0
$805$	2.41188	−	7.94107i	0.0850076	−	0.279886i
$806$		0			0
$807$			77.2659i			2.71989i
$808$		0			0
$809$	45.0016			1.58217			0.791086	−	0.611705i	$-0.209516\pi$
$809$	45.0016			1.58217			0.791086	+	0.611705i	$0.209516\pi$
$810$		0			0
$811$	5.08123			0.178426			0.0892131	−	0.996013i	$-0.471565\pi$
$811$	5.08123			0.178426			0.0892131	+	0.996013i	$0.471565\pi$
$812$		0			0
$813$		−	9.22337i		−	0.323478i
$814$		0			0
$815$	−9.65560	−	2.93262i	−0.338221	−	0.102725i
$816$		0			0
$817$			12.4531i			0.435679i
$818$		0			0
$819$	1.71452			0.0599103
$820$		0			0
$821$	−39.6234			−1.38286			−0.691432	−	0.722441i	$-0.743020\pi$
$821$	−39.6234			−1.38286			−0.691432	+	0.722441i	$0.743020\pi$
$822$		0			0
$823$		−	4.17288i		−	0.145457i	−0.997352	−	0.0727287i	$-0.976829\pi$
$823$		−	4.17288i		−	0.145457i	0.997352	−	0.0727287i	$-0.0231707\pi$
$824$		0			0
$825$	−36.6217	+	54.7268i	−1.27501	+	1.90534i
$826$		0			0
$827$			1.46469i			0.0509324i	0.999676	+	0.0254662i	$0.00810702\pi$
$827$			1.46469i			0.0509324i	−0.999676	+	0.0254662i	$0.991893\pi$
$828$		0			0
$829$	9.92141			0.344585			0.172292	−	0.985046i	$-0.444883\pi$
$829$	9.92141			0.344585			0.172292	+	0.985046i	$0.444883\pi$
$830$		0			0
$831$	−31.0992			−1.07882
$832$		0			0
$833$			7.60851i			0.263619i
$834$		0			0
$835$	16.1667	+	4.91020i	0.559473	+	0.169925i
$836$		0			0
$837$		−	52.6797i		−	1.82088i
$838$		0			0
$839$	−36.7976			−1.27039			−0.635196	−	0.772351i	$-0.719081\pi$
$839$	−36.7976			−1.27039			−0.635196	+	0.772351i	$0.719081\pi$
$840$		0			0
$841$	−23.0274			−0.794048
$842$		0			0
$843$			73.6874i			2.53793i
$844$		0			0
$845$	8.37595	−	27.5776i	0.288141	−	0.948700i
$846$		0			0
$847$		−	10.2677i		−	0.352801i
$848$		0			0
$849$	29.9310			1.02723
$850$		0			0
$851$	−20.6415			−0.707580
$852$		0			0
$853$			18.0625i			0.618448i	0.950989	+	0.309224i	$0.100069\pi$
$853$			18.0625i			0.618448i	−0.950989	+	0.309224i	$0.899931\pi$
$854$		0			0
$855$	4.29592	−	14.1442i	0.146917	−	0.483722i
$856$		0			0
$857$			24.5774i			0.839547i	0.907629	+	0.419773i	$0.137890\pi$
$857$			24.5774i			0.839547i	−0.907629	+	0.419773i	$0.862110\pi$
$858$		0			0
$859$	6.70674			0.228831			0.114416	−	0.993433i	$-0.463500\pi$
$859$	6.70674			0.228831			0.114416	+	0.993433i	$0.463500\pi$
$860$		0			0
$861$	−22.0224			−0.750522
$862$		0			0
$863$			5.82040i			0.198129i	0.995081	+	0.0990644i	$0.0315850\pi$
$863$			5.82040i			0.198129i	−0.995081	+	0.0990644i	$0.968415\pi$
$864$		0			0
$865$	−35.9102	−	10.9067i	−1.22098	−	0.370840i
$866$		0			0
$867$			116.771i			3.96575i
$868$		0			0
$869$	4.37987			0.148577
$870$		0			0
$871$	4.09418			0.138726
$872$		0			0
$873$			8.32533i			0.281770i
$874$		0			0
$875$	7.08381	+	8.64983i	0.239477	+	0.292418i
$876$		0			0
$877$			25.9076i			0.874839i	0.899258	+	0.437419i	$0.144107\pi$
$877$			25.9076i			0.874839i	−0.899258	+	0.437419i	$0.855893\pi$
$878$		0			0
$879$	−6.37417			−0.214996
$880$		0			0
$881$	−3.33665			−0.112415			−0.0562073	−	0.998419i	$-0.517901\pi$
$881$	−3.33665			−0.112415			−0.0562073	+	0.998419i	$0.517901\pi$
$882$		0			0
$883$		−	16.8878i		−	0.568319i	−0.958777	−	0.284160i	$-0.908285\pi$
$883$		−	16.8878i		−	0.568319i	0.958777	−	0.284160i	$-0.0917146\pi$
$884$		0			0
$885$	−7.83497	−	2.37965i	−0.263370	−	0.0799912i
$886$		0			0
$887$		−	52.3365i		−	1.75729i	−0.477477	−	0.878644i	$-0.658449\pi$
$887$		−	52.3365i		−	1.75729i	0.477477	−	0.878644i	$-0.341551\pi$
$888$		0			0
$889$	12.0224			0.403219
$890$		0			0
$891$	9.74088			0.326332
$892$		0			0
$893$		−	11.2262i		−	0.375671i
$894$		0			0
$895$	3.89486	−	12.8238i	0.130191	−	0.428651i
$896$		0			0
$897$		−	3.52498i		−	0.117696i
$898$		0			0
$899$	−20.9154			−0.697568
$900$		0			0
$901$	−92.9775			−3.09753
$902$		0			0
$903$			27.7340i			0.922929i
$904$		0			0
$905$	−2.86698	+	9.43949i	−0.0953017	+	0.313779i
$906$		0			0
$907$		−	42.2693i		−	1.40353i	−0.712409	−	0.701764i	$-0.752396\pi$
$907$		−	42.2693i		−	1.40353i	0.712409	−	0.701764i	$-0.247604\pi$
$908$		0			0
$909$	−74.4151			−2.46819
$910$		0			0
$911$	29.5696			0.979684			0.489842	−	0.871811i	$-0.337054\pi$
$911$	29.5696			0.979684			0.489842	+	0.871811i	$0.337054\pi$
$912$		0			0
$913$		−	17.5823i		−	0.581889i
$914$		0			0
$915$	−18.3988	−	5.58812i	−0.608245	−	0.184738i
$916$		0			0
$917$		−	10.5057i		−	0.346928i
$918$		0			0
$919$	−0.380531			−0.0125526			−0.00627628	−	0.999980i	$-0.501998\pi$
$919$	−0.380531			−0.0125526			−0.00627628	+	0.999980i	$0.501998\pi$
$920$		0			0
$921$	−36.7772			−1.21185
$922$		0			0
$923$			0.996641i			0.0328048i
$924$		0			0
$925$	15.4647	−	23.1101i	0.508476	−	0.759856i
$926$		0			0
$927$		−	19.9360i		−	0.654783i
$928$		0			0
$929$	22.8238			0.748823			0.374412	−	0.927263i	$-0.377845\pi$
$929$	22.8238			0.748823			0.374412	+	0.927263i	$0.377845\pi$
$930$		0			0
$931$	1.28230			0.0420256
$932$		0			0
$933$		−	43.4563i		−	1.42270i
$934$		0			0
$935$	−75.0730	−	22.8013i	−2.45515	−	0.745684i
$936$		0			0
$937$		−	8.84284i		−	0.288883i	−0.989513	−	0.144442i	$-0.953861\pi$
$937$		−	8.84284i		−	0.288883i	0.989513	−	0.144442i	$-0.0461386\pi$
$938$		0			0
$939$	−21.7282			−0.709072
$940$		0			0
$941$	−24.8316			−0.809487			−0.404744	−	0.914430i	$-0.632639\pi$
$941$	−24.8316			−0.809487			−0.404744	+	0.914430i	$0.632639\pi$
$942$		0			0
$943$			28.6217i			0.932052i
$944$		0			0
$945$	4.00000	−	13.1699i	0.130120	−	0.428418i
$946$		0			0
$947$			19.0482i			0.618983i	0.950902	+	0.309492i	$0.100159\pi$
$947$			19.0482i			0.618983i	−0.950902	+	0.309492i	$0.899841\pi$
$948$		0			0
$949$	−4.06402			−0.131924
$950$		0			0
$951$	−93.0023			−3.01580
$952$		0			0
$953$			30.8998i			1.00094i	0.865753	+	0.500472i	$0.166840\pi$
$953$			30.8998i			1.00094i	−0.865753	+	0.500472i	$0.833160\pi$
$954$		0			0
$955$	−11.3701	+	37.4359i	−0.367928	+	1.21140i
$956$		0			0
$957$		−	32.1859i		−	1.04042i
$958$		0			0
$959$	9.22337			0.297838
$960$		0			0
$961$	42.2435			1.36269
$962$		0			0
$963$		−	85.6924i		−	2.76140i
$964$		0			0
$965$	−4.98879	−	1.51521i	−0.160595	−	0.0487762i
$966$		0			0
$967$			4.91020i			0.157901i	0.996879	+	0.0789507i	$0.0251570\pi$
$967$			4.91020i			0.157901i	−0.996879	+	0.0789507i	$0.974843\pi$
$968$		0			0
$969$	27.8620			0.895057
$970$		0			0
$971$	−46.0689			−1.47842			−0.739211	−	0.673474i	$-0.764801\pi$
$971$	−46.0689			−1.47842			−0.739211	+	0.673474i	$0.764801\pi$
$972$		0			0
$973$		−	17.1644i		−	0.550267i
$974$		0			0
$975$	3.94655	+	2.64093i	0.126391	+	0.0845775i
$976$		0			0
$977$			9.22337i			0.295082i	0.989056	+	0.147541i	$0.0471358\pi$
$977$			9.22337i			0.295082i	−0.989056	+	0.147541i	$0.952864\pi$
$978$		0			0
$979$	−27.6701			−0.884341
$980$		0			0
$981$	9.62511			0.307306
$982$		0			0
$983$		−	47.8670i		−	1.52672i	−0.645974	−	0.763360i	$-0.723548\pi$
$983$		−	47.8670i		−	1.52672i	0.645974	−	0.763360i	$-0.276452\pi$
$984$		0			0
$985$	−43.0336	−	13.0703i	−1.37117	−	0.416453i
$986$		0			0
$987$		−	25.0016i		−	0.795810i
$988$		0			0
$989$	36.0448			1.14616
$990$		0			0
$991$	20.1133			0.638920			0.319460	−	0.947600i	$-0.396498\pi$
$991$	20.1133			0.638920			0.319460	+	0.947600i	$0.396498\pi$
$992$		0			0
$993$			62.4723i			1.98250i
$994$		0			0
$995$	0.432224	−	1.42309i	0.0137024	−	0.0451150i
$996$		0			0
$997$			0.531882i			0.0168449i	0.999965	+	0.00842244i	$0.00268098\pi$
$997$			0.531882i			0.0168449i	−0.999965	+	0.00842244i	$0.997319\pi$
$998$		0			0
$999$	−34.2330			−1.08308

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2240.2.g.p.449.1		12
4.3	odd	2	inner	2240.2.g.p.449.11		12
5.4	even	2	inner	2240.2.g.p.449.12		12
8.3	odd	2		1120.2.g.d.449.2	yes	12
8.5	even	2		1120.2.g.d.449.12	yes	12
20.19	odd	2	inner	2240.2.g.p.449.2		12
40.3	even	4		5600.2.a.bz.1.6		6
40.13	odd	4		5600.2.a.by.1.1		6
40.19	odd	2		1120.2.g.d.449.11	yes	12
40.27	even	4		5600.2.a.by.1.2		6
40.29	even	2		1120.2.g.d.449.1	✓	12
40.37	odd	4		5600.2.a.bz.1.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1120.2.g.d.449.1	✓	12	40.29	even	2
1120.2.g.d.449.2	yes	12	8.3	odd	2
1120.2.g.d.449.11	yes	12	40.19	odd	2
1120.2.g.d.449.12	yes	12	8.5	even	2
2240.2.g.p.449.1		12	1.1	even	1	trivial
2240.2.g.p.449.2		12	20.19	odd	2	inner
2240.2.g.p.449.11		12	4.3	odd	2	inner
2240.2.g.p.449.12		12	5.4	even	2	inner
5600.2.a.by.1.1		6	40.13	odd	4
5600.2.a.by.1.2		6	40.27	even	4
5600.2.a.bz.1.5		6	40.37	odd	4
5600.2.a.bz.1.6		6	40.3	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)

\(f(q)\)	\(=\)	\(q-2.85577i q^{3} +(0.649832 - 2.13956i) q^{5} -1.00000i q^{7} -5.15544 q^{9} +O(q^{10})\)	\(q-2.85577i q^{3} +(0.649832 - 2.13956i) q^{5} -1.00000i q^{7} -5.15544 q^{9} +4.61169 q^{11} -0.332566i q^{13} +(-6.11010 - 1.85577i) q^{15} -7.60851i q^{17} -1.28230 q^{19} -2.85577 q^{21} +3.71155i q^{23} +(-4.15544 - 2.78071i) q^{25} +6.15544i q^{27} +2.44389 q^{29} -8.55824 q^{31} -13.1699i q^{33} +(-2.13956 - 0.649832i) q^{35} +5.56142i q^{37} -0.949733 q^{39} +7.71155 q^{41} -9.71155i q^{43} +(-3.35017 + 11.0304i) q^{45} +8.75476i q^{47} -1.00000 q^{49} -21.7282 q^{51} -12.2202i q^{53} +(2.99682 - 9.86698i) q^{55} +3.66195i q^{57} +1.28230 q^{59} +3.01121 q^{61} +5.15544i q^{63} +(-0.711545 - 0.216112i) q^{65} +12.3109i q^{67} +10.5993 q^{69} -2.99682 q^{71} -12.2202i q^{73} +(-7.94107 + 11.8670i) q^{75} -4.61169i q^{77} +0.949733 q^{79} +2.11222 q^{81} -3.81255i q^{83} +(-16.2789 - 4.94425i) q^{85} -6.97920i q^{87} -6.00000 q^{89} -0.332566 q^{91} +24.4404i q^{93} +(-0.833279 + 2.74356i) q^{95} -1.61486i q^{97} -23.7753 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 16 q^{9}+O(q^{10}) \) 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

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