LMFDB - Embedded newform 2240.2.h.f.671.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2240,2,Mod(671,2240)]

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([1, 1, 0, 1]))

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

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

chi := DirichletCharacter("2240.671");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2240 = 2^{6} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2240.h (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$17.8864900528$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			671.16
Character	$\chi$	$=$	2240.671
Dual form			2240.2.h.f.671.15

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.85258i q^{3} +1.00000 q^{5} +(-1.29892 + 2.30495i) q^{7} -0.432061 q^{9} +O(q^{10})$	$q+1.85258i q^{3} +1.00000 q^{5} +(-1.29892 + 2.30495i) q^{7} -0.432061 q^{9} -2.01905 q^{11} +1.82816 q^{13} +1.85258i q^{15} -1.64627i q^{17} +1.26022i q^{19} +(-4.27011 - 2.40636i) q^{21} +3.19054i q^{23} +1.00000 q^{25} +4.75732i q^{27} +4.57459i q^{29} +0.834131 q^{31} -3.74046i q^{33} +(-1.29892 + 2.30495i) q^{35} +7.82567i q^{37} +3.38681i q^{39} -2.75676i q^{41} -6.74801 q^{43} -0.432061 q^{45} -10.8378 q^{47} +(-3.62559 - 5.98791i) q^{49} +3.04985 q^{51} +8.65980i q^{53} -2.01905 q^{55} -2.33465 q^{57} -3.88971i q^{59} -0.285800 q^{61} +(0.561215 - 0.995879i) q^{63} +1.82816 q^{65} +3.94648 q^{67} -5.91073 q^{69} +11.1095i q^{71} -13.4725i q^{73} +1.85258i q^{75} +(2.62260 - 4.65382i) q^{77} -12.8586i q^{79} -10.1095 q^{81} +7.74578i q^{83} -1.64627i q^{85} -8.47481 q^{87} -8.39967i q^{89} +(-2.37464 + 4.21381i) q^{91} +1.54530i q^{93} +1.26022i q^{95} +9.32004i q^{97} +0.872354 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 24 q^{5} - 36 q^{9}+O(q^{10}) $ 24 * q + 24 * q^5 - 36 * q^9	$ 24 q + 24 q^{5} - 36 q^{9} + 4 q^{13} + 8 q^{21} + 24 q^{25} - 36 q^{45} + 24 q^{57} + 56 q^{61} + 4 q^{65} + 96 q^{69} - 12 q^{77} + 24 q^{81}+O(q^{100}) $ 24 * q + 24 * q^5 - 36 * q^9 + 4 * q^13 + 8 * q^21 + 24 * q^25 - 36 * q^45 + 24 * q^57 + 56 * q^61 + 4 * q^65 + 96 * q^69 - 12 * q^77 + 24 * q^81

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
$3$			1.85258i			1.06959i	0.844982	+	0.534794i	$0.179611\pi$
$3$			1.85258i			1.06959i	−0.844982	+	0.534794i	$0.820389\pi$
$4$		0			0
$5$	1.00000			0.447214
$5$	1.00000			0.447214
$6$		0			0
$7$	−1.29892	+	2.30495i	−0.490947	+	0.871189i
$7$	−1.29892	+	2.30495i	−0.490947	+	0.871189i
$8$		0			0
$9$	−0.432061			−0.144020
$10$		0			0
$11$	−2.01905			−0.608767			−0.304384	−	0.952550i	$-0.598450\pi$
$11$	−2.01905			−0.608767			−0.304384	+	0.952550i	$0.598450\pi$
$12$		0			0
$13$	1.82816			0.507039			0.253519	−	0.967330i	$-0.418412\pi$
$13$	1.82816			0.507039			0.253519	+	0.967330i	$0.418412\pi$
$14$		0			0
$15$			1.85258i			0.478335i
$16$		0			0
$17$		−	1.64627i		−	0.399279i	−0.979869	−	0.199640i	$-0.936023\pi$
$17$		−	1.64627i		−	0.399279i	0.979869	−	0.199640i	$-0.0639772\pi$
$18$		0			0
$19$			1.26022i			0.289113i	0.989497	+	0.144557i	$0.0461756\pi$
$19$			1.26022i			0.289113i	−0.989497	+	0.144557i	$0.953824\pi$
$20$		0			0
$21$	−4.27011	−	2.40636i	−0.931814	−	0.525112i
$22$		0			0
$23$			3.19054i			0.665273i	0.943055	+	0.332636i	$0.107938\pi$
$23$			3.19054i			0.665273i	−0.943055	+	0.332636i	$0.892062\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$			4.75732i			0.915546i
$28$		0			0
$29$			4.57459i			0.849480i	0.905315	+	0.424740i	$0.139635\pi$
$29$			4.57459i			0.849480i	−0.905315	+	0.424740i	$0.860365\pi$
$30$		0			0
$31$	0.834131			0.149814			0.0749072	−	0.997191i	$-0.476134\pi$
$31$	0.834131			0.149814			0.0749072	+	0.997191i	$0.476134\pi$
$32$		0			0
$33$		−	3.74046i		−	0.651131i
$34$		0			0
$35$	−1.29892	+	2.30495i	−0.219558	+	0.389608i
$36$		0			0
$37$			7.82567i			1.28653i	0.765643	+	0.643266i	$0.222421\pi$
$37$			7.82567i			1.28653i	−0.765643	+	0.643266i	$0.777579\pi$
$38$		0			0
$39$			3.38681i			0.542323i
$40$		0			0
$41$		−	2.75676i		−	0.430534i	−0.976555	−	0.215267i	$-0.930938\pi$
$41$		−	2.75676i		−	0.430534i	0.976555	−	0.215267i	$-0.0690622\pi$
$42$		0			0
$43$	−6.74801			−1.02906			−0.514531	−	0.857472i	$-0.672034\pi$
$43$	−6.74801			−1.02906			−0.514531	+	0.857472i	$0.672034\pi$
$44$		0			0
$45$	−0.432061			−0.0644079
$46$		0			0
$47$	−10.8378			−1.58085			−0.790425	−	0.612558i	$-0.790140\pi$
$47$	−10.8378			−1.58085			−0.790425	+	0.612558i	$0.790140\pi$
$48$		0			0
$49$	−3.62559	−	5.98791i	−0.517941	−	0.855416i
$50$		0			0
$51$	3.04985			0.427065
$52$		0			0
$53$			8.65980i			1.18951i	0.803905	+	0.594757i	$0.202752\pi$
$53$			8.65980i			1.18951i	−0.803905	+	0.594757i	$0.797248\pi$
$54$		0			0
$55$	−2.01905			−0.272249
$56$		0			0
$57$	−2.33465			−0.309233
$58$		0			0
$59$		−	3.88971i		−	0.506397i	−0.967414	−	0.253198i	$-0.918517\pi$
$59$		−	3.88971i		−	0.506397i	0.967414	−	0.253198i	$-0.0814825\pi$
$60$		0			0
$61$	−0.285800			−0.0365929			−0.0182965	−	0.999833i	$-0.505824\pi$
$61$	−0.285800			−0.0365929			−0.0182965	+	0.999833i	$0.505824\pi$
$62$		0			0
$63$	0.561215	−	0.995879i	0.0707064	−	0.125469i
$64$		0			0
$65$	1.82816			0.226755
$66$		0			0
$67$	3.94648			0.482139			0.241069	−	0.970508i	$-0.422502\pi$
$67$	3.94648			0.482139			0.241069	+	0.970508i	$0.422502\pi$
$68$		0			0
$69$	−5.91073			−0.711568
$70$		0			0
$71$			11.1095i			1.31846i	0.751943	+	0.659228i	$0.229117\pi$
$71$			11.1095i			1.31846i	−0.751943	+	0.659228i	$0.770883\pi$
$72$		0			0
$73$		−	13.4725i		−	1.57684i	−0.615137	−	0.788420i	$-0.710899\pi$
$73$		−	13.4725i		−	1.57684i	0.615137	−	0.788420i	$-0.289101\pi$
$74$		0			0
$75$			1.85258i			0.213918i
$76$		0			0
$77$	2.62260	−	4.65382i	0.298873	−	0.530352i
$78$		0			0
$79$		−	12.8586i		−	1.44670i	−0.690480	−	0.723351i	$-0.742601\pi$
$79$		−	12.8586i		−	1.44670i	0.690480	−	0.723351i	$-0.257399\pi$
$80$		0			0
$81$	−10.1095			−1.12328
$82$		0			0
$83$			7.74578i			0.850210i	0.905144	+	0.425105i	$0.139763\pi$
$83$			7.74578i			0.850210i	−0.905144	+	0.425105i	$0.860237\pi$
$84$		0			0
$85$		−	1.64627i		−	0.178563i
$86$		0			0
$87$	−8.47481			−0.908595
$88$		0			0
$89$		−	8.39967i		−	0.890363i	−0.895440	−	0.445182i	$-0.853139\pi$
$89$		−	8.39967i		−	0.890363i	0.895440	−	0.445182i	$-0.146861\pi$
$90$		0			0
$91$	−2.37464	+	4.21381i	−0.248929	+	0.441727i
$92$		0			0
$93$			1.54530i			0.160240i
$94$		0			0
$95$			1.26022i			0.129295i
$96$		0			0
$97$			9.32004i			0.946306i	0.880980	+	0.473153i	$0.156884\pi$
$97$			9.32004i			0.946306i	−0.880980	+	0.473153i	$0.843116\pi$
$98$		0			0
$99$	0.872354			0.0876749
$100$		0			0
$101$	−14.5289			−1.44568			−0.722838	−	0.691017i	$-0.757163\pi$
$101$	−14.5289			−1.44568			−0.722838	+	0.691017i	$0.757163\pi$
$102$		0			0
$103$	4.72895			0.465958			0.232979	−	0.972482i	$-0.425153\pi$
$103$	4.72895			0.465958			0.232979	+	0.972482i	$0.425153\pi$
$104$		0			0
$105$	−4.27011	−	2.40636i	−0.416720	−	0.234837i
$106$		0			0
$107$	14.4021			1.39230			0.696152	−	0.717894i	$-0.254894\pi$
$107$	14.4021			1.39230			0.696152	+	0.717894i	$0.254894\pi$
$108$		0			0
$109$			4.27624i			0.409590i	0.978805	+	0.204795i	$0.0656527\pi$
$109$			4.27624i			0.409590i	−0.978805	+	0.204795i	$0.934347\pi$
$110$		0			0
$111$	−14.4977			−1.37606
$112$		0			0
$113$	−8.66456			−0.815093			−0.407547	−	0.913184i	$-0.633616\pi$
$113$	−8.66456			−0.815093			−0.407547	+	0.913184i	$0.633616\pi$
$114$		0			0
$115$			3.19054i			0.297519i
$116$		0			0
$117$	−0.789875			−0.0730240
$118$		0			0
$119$	3.79457	+	2.13838i	0.347848	+	0.196025i
$120$		0			0
$121$	−6.92343			−0.629402
$122$		0			0
$123$	5.10713			0.460494
$124$		0			0
$125$	1.00000			0.0894427
$126$		0			0
$127$			1.61570i			0.143370i	0.997427	+	0.0716850i	$0.0228376\pi$
$127$			1.61570i			0.143370i	−0.997427	+	0.0716850i	$0.977162\pi$
$128$		0			0
$129$		−	12.5012i		−	1.10067i
$130$		0			0
$131$			0.173193i			0.0151319i	0.999971	+	0.00756597i	$0.00240835\pi$
$131$			0.173193i			0.0151319i	−0.999971	+	0.00756597i	$0.997592\pi$
$132$		0			0
$133$	−2.90474	−	1.63693i	−0.251873	−	0.141939i
$134$		0			0
$135$			4.75732i			0.409445i
$136$		0			0
$137$	17.7750			1.51862			0.759312	−	0.650727i	$-0.225536\pi$
$137$	17.7750			1.51862			0.759312	+	0.650727i	$0.225536\pi$
$138$		0			0
$139$		−	6.19653i		−	0.525583i	−0.964853	−	0.262791i	$-0.915357\pi$
$139$		−	6.19653i		−	0.525583i	0.964853	−	0.262791i	$-0.0846431\pi$
$140$		0			0
$141$		−	20.0779i		−	1.69086i
$142$		0			0
$143$	−3.69114			−0.308669
$144$		0			0
$145$			4.57459i			0.379899i
$146$		0			0
$147$	11.0931	−	6.71670i	0.914943	−	0.553984i
$148$		0			0
$149$			9.49776i			0.778087i	0.921219	+	0.389043i	$0.127194\pi$
$149$			9.49776i			0.778087i	−0.921219	+	0.389043i	$0.872806\pi$
$150$		0			0
$151$			1.49710i			0.121832i	0.998143	+	0.0609162i	$0.0194022\pi$
$151$			1.49710i			0.121832i	−0.998143	+	0.0609162i	$0.980598\pi$
$152$		0			0
$153$			0.711290i			0.0575044i
$154$		0			0
$155$	0.834131			0.0669990
$156$		0			0
$157$	7.26866			0.580102			0.290051	−	0.957011i	$-0.406328\pi$
$157$	7.26866			0.580102			0.290051	+	0.957011i	$0.406328\pi$
$158$		0			0
$159$	−16.0430			−1.27229
$160$		0			0
$161$	−7.35403	−	4.14426i	−0.579578	−	0.326614i
$162$		0			0
$163$	17.2922			1.35443			0.677215	−	0.735785i	$-0.263187\pi$
$163$	17.2922			1.35443			0.677215	+	0.735785i	$0.263187\pi$
$164$		0			0
$165$		−	3.74046i		−	0.291195i
$166$		0			0
$167$	−20.8910			−1.61659			−0.808295	−	0.588777i	$-0.799610\pi$
$167$	−20.8910			−1.61659			−0.808295	+	0.588777i	$0.799610\pi$
$168$		0			0
$169$	−9.65785			−0.742911
$170$		0			0
$171$		−	0.544491i		−	0.0416382i
$172$		0			0
$173$	1.87701			0.142706			0.0713532	−	0.997451i	$-0.477268\pi$
$173$	1.87701			0.142706			0.0713532	+	0.997451i	$0.477268\pi$
$174$		0			0
$175$	−1.29892	+	2.30495i	−0.0981895	+	0.174238i
$176$		0			0
$177$	7.20600			0.541636
$178$		0			0
$179$	5.47135			0.408948			0.204474	−	0.978872i	$-0.434452\pi$
$179$	5.47135			0.408948			0.204474	+	0.978872i	$0.434452\pi$
$180$		0			0
$181$	−7.67879			−0.570760			−0.285380	−	0.958414i	$-0.592120\pi$
$181$	−7.67879			−0.570760			−0.285380	+	0.958414i	$0.592120\pi$
$182$		0			0
$183$		−	0.529468i		−	0.0391394i
$184$		0			0
$185$			7.82567i			0.575355i
$186$		0			0
$187$			3.32391i			0.243068i
$188$		0			0
$189$	−10.9654	−	6.17940i	−0.797614	−	0.449485i
$190$		0			0
$191$		−	3.63878i		−	0.263293i	−0.991297	−	0.131646i	$-0.957974\pi$
$191$		−	3.63878i		−	0.263293i	0.991297	−	0.131646i	$-0.0420263\pi$
$192$		0			0
$193$	−0.555242			−0.0399672			−0.0199836	−	0.999800i	$-0.506361\pi$
$193$	−0.555242			−0.0399672			−0.0199836	+	0.999800i	$0.506361\pi$
$194$		0			0
$195$			3.38681i			0.242534i
$196$		0			0
$197$		−	3.44936i		−	0.245757i	−0.992422	−	0.122878i	$-0.960787\pi$
$197$		−	3.44936i		−	0.245757i	0.992422	−	0.122878i	$-0.0392125\pi$
$198$		0			0
$199$	23.8014			1.68723			0.843617	−	0.536945i	$-0.180422\pi$
$199$	23.8014			1.68723			0.843617	+	0.536945i	$0.180422\pi$
$200$		0			0
$201$			7.31117i			0.515690i
$202$		0			0
$203$	−10.5442	−	5.94205i	−0.740058	−	0.417050i
$204$		0			0
$205$		−	2.75676i		−	0.192541i
$206$		0			0
$207$		−	1.37851i		−	0.0958128i
$208$		0			0
$209$		−	2.54444i		−	0.176003i
$210$		0			0
$211$	−25.3284			−1.74368			−0.871841	−	0.489789i	$-0.837074\pi$
$211$	−25.3284			−1.74368			−0.871841	+	0.489789i	$0.837074\pi$
$212$		0			0
$213$	−20.5813			−1.41021
$214$		0			0
$215$	−6.74801			−0.460210
$216$		0			0
$217$	−1.08347	+	1.92263i	−0.0735510	+	0.130517i
$218$		0			0
$219$	24.9590			1.68657
$220$		0			0
$221$		−	3.00964i		−	0.202450i
$222$		0			0
$223$	−13.4743			−0.902305			−0.451152	−	0.892447i	$-0.648987\pi$
$223$	−13.4743			−0.902305			−0.451152	+	0.892447i	$0.648987\pi$
$224$		0			0
$225$	−0.432061			−0.0288041
$226$		0			0
$227$			1.28098i			0.0850218i	0.999096	+	0.0425109i	$0.0135357\pi$
$227$			1.28098i			0.0850218i	−0.999096	+	0.0425109i	$0.986464\pi$
$228$		0			0
$229$	−12.0084			−0.793540			−0.396770	−	0.917918i	$-0.629869\pi$
$229$	−12.0084			−0.793540			−0.396770	+	0.917918i	$0.629869\pi$
$230$		0			0
$231$	8.62158	+	4.85858i	0.567258	+	0.319671i
$232$		0			0
$233$	8.35568			0.547399			0.273699	−	0.961815i	$-0.411753\pi$
$233$	8.35568			0.547399			0.273699	+	0.961815i	$0.411753\pi$
$234$		0			0
$235$	−10.8378			−0.706978
$236$		0			0
$237$	23.8216			1.54738
$238$		0			0
$239$			0.0313201i			0.00202593i	0.999999	+	0.00101296i	$0.000322437\pi$
$239$			0.0313201i			0.00202593i	−0.999999	+	0.00101296i	$0.999678\pi$
$240$		0			0
$241$		−	6.13884i		−	0.395437i	−0.980259	−	0.197719i	$-0.936647\pi$
$241$		−	6.13884i		−	0.395437i	0.980259	−	0.197719i	$-0.0633533\pi$
$242$		0			0
$243$		−	4.45674i		−	0.285900i
$244$		0			0
$245$	−3.62559	−	5.98791i	−0.231630	−	0.382554i
$246$		0			0
$247$			2.30387i			0.146592i
$248$		0			0
$249$	−14.3497			−0.909375
$250$		0			0
$251$			29.7162i			1.87567i	0.347084	+	0.937834i	$0.387172\pi$
$251$			29.7162i			1.87567i	−0.347084	+	0.937834i	$0.612828\pi$
$252$		0			0
$253$		−	6.44186i		−	0.404996i
$254$		0			0
$255$	3.04985			0.190989
$256$		0			0
$257$			3.91131i			0.243981i	0.992531	+	0.121991i	$0.0389277\pi$
$257$			3.91131i			0.243981i	−0.992531	+	0.121991i	$0.961072\pi$
$258$		0			0
$259$	−18.0378	−	10.1650i	−1.12081	−	0.631619i
$260$		0			0
$261$		−	1.97650i		−	0.122343i
$262$		0			0
$263$			4.11255i			0.253591i	0.991929	+	0.126795i	$0.0404691\pi$
$263$			4.11255i			0.253591i	−0.991929	+	0.126795i	$0.959531\pi$
$264$		0			0
$265$			8.65980i			0.531967i
$266$		0			0
$267$	15.5611			0.952323
$268$		0			0
$269$	10.2052			0.622223			0.311111	−	0.950373i	$-0.399299\pi$
$269$	10.2052			0.622223			0.311111	+	0.950373i	$0.399299\pi$
$270$		0			0
$271$	23.2208			1.41056			0.705282	−	0.708926i	$-0.250820\pi$
$271$	23.2208			1.41056			0.705282	+	0.708926i	$0.250820\pi$
$272$		0			0
$273$	−7.80642	−	4.39921i	−0.472466	−	0.266252i
$274$		0			0
$275$	−2.01905			−0.121753
$276$		0			0
$277$			18.5201i			1.11277i	0.830925	+	0.556384i	$0.187812\pi$
$277$			18.5201i			1.11277i	−0.830925	+	0.556384i	$0.812188\pi$
$278$		0			0
$279$	−0.360396			−0.0215763
$280$		0			0
$281$	−20.0391			−1.19543			−0.597717	−	0.801707i	$-0.703925\pi$
$281$	−20.0391			−1.19543			−0.597717	+	0.801707i	$0.703925\pi$
$282$		0			0
$283$		−	14.0081i		−	0.832693i	−0.909206	−	0.416346i	$-0.863310\pi$
$283$		−	14.0081i		−	0.832693i	0.909206	−	0.416346i	$-0.136690\pi$
$284$		0			0
$285$	−2.33465			−0.138293
$286$		0			0
$287$	6.35420	+	3.58083i	0.375077	+	0.211369i
$288$		0			0
$289$	14.2898			0.840576
$290$		0			0
$291$	−17.2661			−1.01216
$292$		0			0
$293$	8.98711			0.525033			0.262516	−	0.964928i	$-0.415448\pi$
$293$	8.98711			0.525033			0.262516	+	0.964928i	$0.415448\pi$
$294$		0			0
$295$		−	3.88971i		−	0.226467i
$296$		0			0
$297$		−	9.60528i		−	0.557355i
$298$		0			0
$299$			5.83279i			0.337319i
$300$		0			0
$301$	8.76515	−	15.5538i	0.505215	−	0.896507i
$302$		0			0
$303$		−	26.9159i		−	1.54628i
$304$		0			0
$305$	−0.285800			−0.0163649
$306$		0			0
$307$		−	1.00496i		−	0.0573559i	−0.999589	−	0.0286779i	$-0.990870\pi$
$307$		−	1.00496i		−	0.0573559i	0.999589	−	0.0286779i	$-0.00912972\pi$
$308$		0			0
$309$			8.76077i			0.498383i
$310$		0			0
$311$	11.9507			0.677663			0.338831	−	0.940847i	$-0.389968\pi$
$311$	11.9507			0.677663			0.338831	+	0.940847i	$0.389968\pi$
$312$		0			0
$313$			10.4336i			0.589745i	0.955537	+	0.294872i	$0.0952771\pi$
$313$			10.4336i			0.589745i	−0.955537	+	0.294872i	$0.904723\pi$
$314$		0			0
$315$	0.561215	−	0.995879i	0.0316209	−	0.0561115i
$316$		0			0
$317$			31.6214i			1.77604i	0.459809	+	0.888018i	$0.347918\pi$
$317$			31.6214i			1.77604i	−0.459809	+	0.888018i	$0.652082\pi$
$318$		0			0
$319$		−	9.23634i		−	0.517136i
$320$		0			0
$321$			26.6811i			1.48919i
$322$		0			0
$323$	2.07466			0.115437
$324$		0			0
$325$	1.82816			0.101408
$326$		0			0
$327$	−7.92209			−0.438093
$328$		0			0
$329$	14.0774	−	24.9805i	0.776114	−	1.37722i
$330$		0			0
$331$	−18.5944			−1.02204			−0.511020	−	0.859569i	$-0.670732\pi$
$331$	−18.5944			−1.02204			−0.511020	+	0.859569i	$0.670732\pi$
$332$		0			0
$333$		−	3.38117i		−	0.185287i
$334$		0			0
$335$	3.94648			0.215619
$336$		0			0
$337$	23.9730			1.30589			0.652947	−	0.757403i	$-0.273532\pi$
$337$	23.9730			1.30589			0.652947	+	0.757403i	$0.273532\pi$
$338$		0			0
$339$		−	16.0518i		−	0.871815i
$340$		0			0
$341$	−1.68415			−0.0912021
$342$		0			0
$343$	18.5112	−	0.578959i	0.999511	−	0.0312609i
$344$		0			0
$345$	−5.91073			−0.318223
$346$		0			0
$347$	7.56248			0.405975			0.202988	−	0.979181i	$-0.434935\pi$
$347$	7.56248			0.405975			0.202988	+	0.979181i	$0.434935\pi$
$348$		0			0
$349$	33.5581			1.79633			0.898163	−	0.439662i	$-0.144902\pi$
$349$	33.5581			1.79633			0.898163	+	0.439662i	$0.144902\pi$
$350$		0			0
$351$			8.69712i			0.464218i
$352$		0			0
$353$		−	14.1431i		−	0.752763i	−0.926465	−	0.376382i	$-0.877168\pi$
$353$		−	14.1431i		−	0.752763i	0.926465	−	0.376382i	$-0.122832\pi$
$354$		0			0
$355$			11.1095i			0.589631i
$356$		0			0
$357$	−3.96153	+	7.02976i	−0.209666	+	0.372054i
$358$		0			0
$359$		−	10.5868i		−	0.558748i	−0.960182	−	0.279374i	$-0.909873\pi$
$359$		−	10.5868i		−	0.558748i	0.960182	−	0.279374i	$-0.0901269\pi$
$360$		0			0
$361$	17.4119			0.916413
$362$		0			0
$363$		−	12.8262i		−	0.673202i
$364$		0			0
$365$		−	13.4725i		−	0.705184i
$366$		0			0
$367$	21.2730			1.11044			0.555222	−	0.831703i	$-0.312633\pi$
$367$	21.2730			1.11044			0.555222	+	0.831703i	$0.312633\pi$
$368$		0			0
$369$			1.19109i			0.0620057i
$370$		0			0
$371$	−19.9604	−	11.2484i	−1.03629	−	0.583989i
$372$		0			0
$373$			8.97368i			0.464639i	0.972639	+	0.232320i	$0.0746315\pi$
$373$			8.97368i			0.464639i	−0.972639	+	0.232320i	$0.925368\pi$
$374$		0			0
$375$			1.85258i			0.0956669i
$376$		0			0
$377$			8.36306i			0.430720i
$378$		0			0
$379$	−17.1082			−0.878789			−0.439394	−	0.898294i	$-0.644807\pi$
$379$	−17.1082			−0.878789			−0.439394	+	0.898294i	$0.644807\pi$
$380$		0			0
$381$	−2.99321			−0.153347
$382$		0			0
$383$	15.0052			0.766729			0.383364	−	0.923597i	$-0.374765\pi$
$383$	15.0052			0.766729			0.383364	+	0.923597i	$0.374765\pi$
$384$		0			0
$385$	2.62260	−	4.65382i	0.133660	−	0.237180i
$386$		0			0
$387$	2.91555			0.148206
$388$		0			0
$389$			22.1486i			1.12298i	0.827485	+	0.561488i	$0.189771\pi$
$389$			22.1486i			1.12298i	−0.827485	+	0.561488i	$0.810229\pi$
$390$		0			0
$391$	5.25249			0.265630
$392$		0			0
$393$	−0.320854			−0.0161850
$394$		0			0
$395$		−	12.8586i		−	0.646985i
$396$		0			0
$397$	32.6284			1.63757			0.818787	−	0.574097i	$-0.194647\pi$
$397$	32.6284			1.63757			0.818787	+	0.574097i	$0.194647\pi$
$398$		0			0
$399$	3.03254	−	5.38126i	0.151817	−	0.269400i
$400$		0			0
$401$	4.98039			0.248709			0.124354	−	0.992238i	$-0.460314\pi$
$401$	4.98039			0.248709			0.124354	+	0.992238i	$0.460314\pi$
$402$		0			0
$403$	1.52492			0.0759617
$404$		0			0
$405$	−10.1095			−0.502345
$406$		0			0
$407$		−	15.8004i		−	0.783199i
$408$		0			0
$409$			37.0802i			1.83350i	0.399467	+	0.916748i	$0.369195\pi$
$409$			37.0802i			1.83350i	−0.399467	+	0.916748i	$0.630805\pi$
$410$		0			0
$411$			32.9297i			1.62430i
$412$		0			0
$413$	8.96558	+	5.05243i	0.441167	+	0.248614i
$414$		0			0
$415$			7.74578i			0.380225i
$416$		0			0
$417$	11.4796			0.562158
$418$		0			0
$419$			5.49215i			0.268309i	0.990960	+	0.134155i	$0.0428319\pi$
$419$			5.49215i			0.268309i	−0.990960	+	0.134155i	$0.957168\pi$
$420$		0			0
$421$			4.32022i			0.210555i	0.994443	+	0.105277i	$0.0335731\pi$
$421$			4.32022i			0.210555i	−0.994443	+	0.105277i	$0.966427\pi$
$422$		0			0
$423$	4.68258			0.227675
$424$		0			0
$425$		−	1.64627i		−	0.0798559i
$426$		0			0
$427$	0.371233	−	0.658755i	0.0179652	−	0.0318794i
$428$		0			0
$429$		−	6.83814i		−	0.330149i
$430$		0			0
$431$			19.2143i			0.925518i	0.886484	+	0.462759i	$0.153140\pi$
$431$			19.2143i			0.925518i	−0.886484	+	0.462759i	$0.846860\pi$
$432$		0			0
$433$			38.8480i			1.86691i	0.358689	+	0.933457i	$0.383224\pi$
$433$			38.8480i			1.86691i	−0.358689	+	0.933457i	$0.616776\pi$
$434$		0			0
$435$	−8.47481			−0.406336
$436$		0			0
$437$	−4.02077			−0.192339
$438$		0			0
$439$	−5.10197			−0.243504			−0.121752	−	0.992561i	$-0.538851\pi$
$439$	−5.10197			−0.243504			−0.121752	+	0.992561i	$0.538851\pi$
$440$		0			0
$441$	1.56648	+	2.58714i	0.0745941	+	0.123197i
$442$		0			0
$443$	−7.06583			−0.335708			−0.167854	−	0.985812i	$-0.553684\pi$
$443$	−7.06583			−0.335708			−0.167854	+	0.985812i	$0.553684\pi$
$444$		0			0
$445$		−	8.39967i		−	0.398183i
$446$		0			0
$447$	−17.5954			−0.832233
$448$		0			0
$449$	38.3486			1.80978			0.904892	−	0.425641i	$-0.139952\pi$
$449$	38.3486			1.80978			0.904892	+	0.425641i	$0.139952\pi$
$450$		0			0
$451$			5.56605i			0.262095i
$452$		0			0
$453$	−2.77350			−0.130311
$454$		0			0
$455$	−2.37464	+	4.21381i	−0.111325	+	0.197546i
$456$		0			0
$457$	33.1807			1.55213			0.776064	−	0.630654i	$-0.217213\pi$
$457$	33.1807			1.55213			0.776064	+	0.630654i	$0.217213\pi$
$458$		0			0
$459$	7.83184			0.365559
$460$		0			0
$461$	10.2266			0.476299			0.238150	−	0.971228i	$-0.423459\pi$
$461$	10.2266			0.476299			0.238150	+	0.971228i	$0.423459\pi$
$462$		0			0
$463$			17.6829i			0.821793i	0.911682	+	0.410897i	$0.134784\pi$
$463$			17.6829i			0.821793i	−0.911682	+	0.410897i	$0.865216\pi$
$464$		0			0
$465$			1.54530i			0.0716614i
$466$		0			0
$467$		−	22.6870i		−	1.04983i	−0.851154	−	0.524916i	$-0.824097\pi$
$467$		−	22.6870i		−	1.04983i	0.851154	−	0.524916i	$-0.175903\pi$
$468$		0			0
$469$	−5.12617	+	9.09643i	−0.236705	+	0.420034i
$470$		0			0
$471$			13.4658i			0.620470i
$472$		0			0
$473$	13.6246			0.626459
$474$		0			0
$475$			1.26022i			0.0578227i
$476$		0			0
$477$		−	3.74156i		−	0.171314i
$478$		0			0
$479$	32.2749			1.47468			0.737338	−	0.675524i	$-0.236082\pi$
$479$	32.2749			1.47468			0.737338	+	0.675524i	$0.236082\pi$
$480$		0			0
$481$			14.3065i			0.652322i
$482$		0			0
$483$	7.67759	−	13.6239i	0.349343	−	0.619911i
$484$		0			0
$485$			9.32004i			0.423201i
$486$		0			0
$487$		−	41.7770i		−	1.89309i	−0.322565	−	0.946547i	$-0.604545\pi$
$487$		−	41.7770i		−	1.89309i	0.322565	−	0.946547i	$-0.395455\pi$
$488$		0			0
$489$			32.0352i			1.44868i
$490$		0			0
$491$	−21.2165			−0.957488			−0.478744	−	0.877955i	$-0.658908\pi$
$491$	−21.2165			−0.957488			−0.478744	+	0.877955i	$0.658908\pi$
$492$		0			0
$493$	7.53102			0.339180
$494$		0			0
$495$	0.872354			0.0392094
$496$		0			0
$497$	−25.6069	−	14.4304i	−1.14862	−	0.647292i
$498$		0			0
$499$	10.0845			0.451443			0.225722	−	0.974192i	$-0.427526\pi$
$499$	10.0845			0.451443			0.225722	+	0.974192i	$0.427526\pi$
$500$		0			0
$501$		−	38.7022i		−	1.72909i
$502$		0			0
$503$	−27.7345			−1.23662			−0.618311	−	0.785933i	$-0.712183\pi$
$503$	−27.7345			−1.23662			−0.618311	+	0.785933i	$0.712183\pi$
$504$		0			0
$505$	−14.5289			−0.646526
$506$		0			0
$507$		−	17.8920i		−	0.794610i
$508$		0			0
$509$	30.4848			1.35121			0.675607	−	0.737262i	$-0.263882\pi$
$509$	30.4848			1.35121			0.675607	+	0.737262i	$0.263882\pi$
$510$		0			0
$511$	31.0535	+	17.4998i	1.37373	+	0.774145i
$512$		0			0
$513$	−5.99525			−0.264697
$514$		0			0
$515$	4.72895			0.208383
$516$		0			0
$517$	21.8820			0.962370
$518$		0			0
$519$			3.47731i			0.152637i
$520$		0			0
$521$		−	25.7727i		−	1.12912i	−0.825391	−	0.564562i	$-0.809045\pi$
$521$		−	25.7727i		−	1.12912i	0.825391	−	0.564562i	$-0.190955\pi$
$522$		0			0
$523$		−	33.7050i		−	1.47382i	−0.675992	−	0.736909i	$-0.736285\pi$
$523$		−	33.7050i		−	1.47382i	0.675992	−	0.736909i	$-0.263715\pi$
$524$		0			0
$525$	−4.27011	−	2.40636i	−0.186363	−	0.105022i
$526$		0			0
$527$		−	1.37321i		−	0.0598178i
$528$		0			0
$529$	12.8205			0.557412
$530$		0			0
$531$			1.68059i			0.0729314i
$532$		0			0
$533$		−	5.03979i		−	0.218297i
$534$		0			0
$535$	14.4021			0.622658
$536$		0			0
$537$			10.1361i			0.437406i
$538$		0			0
$539$	7.32026	+	12.0899i	0.315306	+	0.520749i
$540$		0			0
$541$		−	30.0220i		−	1.29075i	−0.763867	−	0.645374i	$-0.776702\pi$
$541$		−	30.0220i		−	1.29075i	0.763867	−	0.645374i	$-0.223298\pi$
$542$		0			0
$543$		−	14.2256i		−	0.610479i
$544$		0			0
$545$			4.27624i			0.183174i
$546$		0			0
$547$	12.6879			0.542493			0.271247	−	0.962510i	$-0.412564\pi$
$547$	12.6879			0.542493			0.271247	+	0.962510i	$0.412564\pi$
$548$		0			0
$549$	0.123483			0.00527013
$550$		0			0
$551$	−5.76498			−0.245596
$552$		0			0
$553$	29.6384	+	16.7023i	1.26035	+	0.710255i
$554$		0			0
$555$	−14.4977			−0.615393
$556$		0			0
$557$			5.08838i			0.215602i	0.994173	+	0.107801i	$0.0343809\pi$
$557$			5.08838i			0.215602i	−0.994173	+	0.107801i	$0.965619\pi$
$558$		0			0
$559$	−12.3364			−0.521774
$560$		0			0
$561$	−6.15781			−0.259983
$562$		0			0
$563$			40.0160i			1.68647i	0.537544	+	0.843236i	$0.319352\pi$
$563$			40.0160i			1.68647i	−0.537544	+	0.843236i	$0.680648\pi$
$564$		0			0
$565$	−8.66456			−0.364521
$566$		0			0
$567$	13.1315	−	23.3019i	0.551471	−	0.978588i
$568$		0			0
$569$	−21.9994			−0.922263			−0.461131	−	0.887332i	$-0.652556\pi$
$569$	−21.9994			−0.922263			−0.461131	+	0.887332i	$0.652556\pi$
$570$		0			0
$571$	7.72411			0.323244			0.161622	−	0.986853i	$-0.448327\pi$
$571$	7.72411			0.323244			0.161622	+	0.986853i	$0.448327\pi$
$572$		0			0
$573$	6.74113			0.281615
$574$		0			0
$575$			3.19054i			0.133055i
$576$		0			0
$577$		−	21.0415i		−	0.875969i	−0.898983	−	0.437984i	$-0.855693\pi$
$577$		−	21.0415i		−	0.875969i	0.898983	−	0.437984i	$-0.144307\pi$
$578$		0			0
$579$		−	1.02863i		−	0.0427485i
$580$		0			0
$581$	−17.8536	−	10.0612i	−0.740693	−	0.417408i
$582$		0			0
$583$		−	17.4846i		−	0.724138i
$584$		0			0
$585$	−0.789875			−0.0326573
$586$		0			0
$587$		−	14.5780i		−	0.601700i	−0.953671	−	0.300850i	$-0.902730\pi$
$587$		−	14.5780i		−	0.601700i	0.953671	−	0.300850i	$-0.0972704\pi$
$588$		0			0
$589$			1.05119i			0.0433134i
$590$		0			0
$591$	6.39022			0.262859
$592$		0			0
$593$			40.6323i			1.66857i	0.551335	+	0.834284i	$0.314119\pi$
$593$			40.6323i			1.66857i	−0.551335	+	0.834284i	$0.685881\pi$
$594$		0			0
$595$	3.79457	+	2.13838i	0.155562	+	0.0876651i
$596$		0			0
$597$			44.0940i			1.80465i
$598$		0			0
$599$		−	6.14993i		−	0.251279i	−0.992076	−	0.125640i	$-0.959902\pi$
$599$		−	6.14993i		−	0.251279i	0.992076	−	0.125640i	$-0.0400983\pi$
$600$		0			0
$601$			41.5665i			1.69553i	0.530370	+	0.847766i	$0.322053\pi$
$601$			41.5665i			1.69553i	−0.530370	+	0.847766i	$0.677947\pi$
$602$		0			0
$603$	−1.70512			−0.0694378
$604$		0			0
$605$	−6.92343			−0.281477
$606$		0			0
$607$	33.7351			1.36927			0.684633	−	0.728888i	$-0.259962\pi$
$607$	33.7351			1.36927			0.684633	+	0.728888i	$0.259962\pi$
$608$		0			0
$609$	11.0081	−	19.5340i	0.446072	−	0.791558i
$610$		0			0
$611$	−19.8131			−0.801553
$612$		0			0
$613$		−	4.12505i		−	0.166609i	−0.996524	−	0.0833045i	$-0.973453\pi$
$613$		−	4.12505i		−	0.166609i	0.996524	−	0.0833045i	$-0.0265474\pi$
$614$		0			0
$615$	5.10713			0.205939
$616$		0			0
$617$	−23.2572			−0.936298			−0.468149	−	0.883649i	$-0.655079\pi$
$617$	−23.2572			−0.936298			−0.468149	+	0.883649i	$0.655079\pi$
$618$		0			0
$619$		−	49.3241i		−	1.98250i	−0.131981	−	0.991252i	$-0.542134\pi$
$619$		−	49.3241i		−	1.98250i	0.131981	−	0.991252i	$-0.457866\pi$
$620$		0			0
$621$	−15.1784			−0.609088
$622$		0			0
$623$	19.3608	+	10.9105i	0.775675	+	0.437122i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	4.71379			0.188251
$628$		0			0
$629$	12.8832			0.513686
$630$		0			0
$631$		−	18.0133i		−	0.717097i	−0.933511	−	0.358548i	$-0.883272\pi$
$631$		−	18.0133i		−	0.717097i	0.933511	−	0.358548i	$-0.116728\pi$
$632$		0			0
$633$		−	46.9230i		−	1.86502i
$634$		0			0
$635$			1.61570i			0.0641170i
$636$		0			0
$637$	−6.62814	−	10.9468i	−0.262617	−	0.433729i
$638$		0			0
$639$		−	4.79999i		−	0.189885i
$640$		0			0
$641$	−39.4333			−1.55752			−0.778761	−	0.627320i	$-0.784152\pi$
$641$	−39.4333			−1.55752			−0.778761	+	0.627320i	$0.784152\pi$
$642$		0			0
$643$			39.7995i			1.56954i	0.619789	+	0.784769i	$0.287218\pi$
$643$			39.7995i			1.56954i	−0.619789	+	0.784769i	$0.712782\pi$
$644$		0			0
$645$		−	12.5012i		−	0.492236i
$646$		0			0
$647$	9.70931			0.381712			0.190856	−	0.981618i	$-0.438874\pi$
$647$	9.70931			0.381712			0.190856	+	0.981618i	$0.438874\pi$
$648$		0			0
$649$			7.85352i			0.308278i
$650$		0			0
$651$	−3.56183	−	2.00722i	−0.139599	−	0.0786693i
$652$		0			0
$653$		−	29.6501i		−	1.16030i	−0.814511	−	0.580148i	$-0.802995\pi$
$653$		−	29.6501i		−	1.16030i	0.814511	−	0.580148i	$-0.197005\pi$
$654$		0			0
$655$			0.173193i			0.00676721i
$656$		0			0
$657$			5.82096i			0.227097i
$658$		0			0
$659$	15.5744			0.606692			0.303346	−	0.952880i	$-0.401896\pi$
$659$	15.5744			0.606692			0.303346	+	0.952880i	$0.401896\pi$
$660$		0			0
$661$	−25.0366			−0.973811			−0.486905	−	0.873455i	$-0.661874\pi$
$661$	−25.0366			−0.973811			−0.486905	+	0.873455i	$0.661874\pi$
$662$		0			0
$663$	5.57560			0.216539
$664$		0			0
$665$	−2.90474	−	1.63693i	−0.112641	−	0.0634773i
$666$		0			0
$667$	−14.5954			−0.565136
$668$		0			0
$669$		−	24.9622i		−	0.965095i
$670$		0			0
$671$	0.577045			0.0222766
$672$		0			0
$673$	33.5600			1.29364			0.646821	−	0.762642i	$-0.276098\pi$
$673$	33.5600			1.29364			0.646821	+	0.762642i	$0.276098\pi$
$674$		0			0
$675$			4.75732i			0.183109i
$676$		0			0
$677$	−15.9868			−0.614424			−0.307212	−	0.951641i	$-0.599396\pi$
$677$	−15.9868			−0.614424			−0.307212	+	0.951641i	$0.599396\pi$
$678$		0			0
$679$	−21.4822	−	12.1060i	−0.824412	−	0.464587i
$680$		0			0
$681$	−2.37313			−0.0909384
$682$		0			0
$683$	−2.96348			−0.113395			−0.0566973	−	0.998391i	$-0.518057\pi$
$683$	−2.96348			−0.113395			−0.0566973	+	0.998391i	$0.518057\pi$
$684$		0			0
$685$	17.7750			0.679149
$686$		0			0
$687$		−	22.2466i		−	0.848762i
$688$		0			0
$689$			15.8315i			0.603130i
$690$		0			0
$691$		−	40.5268i		−	1.54171i	−0.637009	−	0.770856i	$-0.719829\pi$
$691$		−	40.5268i		−	1.54171i	0.637009	−	0.770856i	$-0.280171\pi$
$692$		0			0
$693$	−1.13312	+	2.01073i	−0.0430438	+	0.0763814i
$694$		0			0
$695$		−	6.19653i		−	0.235048i
$696$		0			0
$697$	−4.53838			−0.171903
$698$		0			0
$699$			15.4796i			0.585492i
$700$		0			0
$701$		−	14.5734i		−	0.550431i	−0.961383	−	0.275216i	$-0.911251\pi$
$701$		−	14.5734i		−	0.550431i	0.961383	−	0.275216i	$-0.0887492\pi$
$702$		0			0
$703$	−9.86203			−0.371954
$704$		0			0
$705$		−	20.0779i		−	0.756176i
$706$		0			0
$707$	18.8719	−	33.4883i	0.709751	−	1.25946i
$708$		0			0
$709$		−	32.4879i		−	1.22011i	−0.792360	−	0.610054i	$-0.791148\pi$
$709$		−	32.4879i		−	1.22011i	0.792360	−	0.610054i	$-0.208852\pi$
$710$		0			0
$711$			5.55569i			0.208355i
$712$		0			0
$713$			2.66133i			0.0996674i
$714$		0			0
$715$	−3.69114			−0.138041
$716$		0			0
$717$	−0.0580231			−0.00216691
$718$		0			0
$719$	−47.7021			−1.77899			−0.889494	−	0.456946i	$-0.848943\pi$
$719$	−47.7021			−1.77899			−0.889494	+	0.456946i	$0.848943\pi$
$720$		0			0
$721$	−6.14255	+	10.9000i	−0.228761	+	0.405937i
$722$		0			0
$723$	11.3727			0.422955
$724$		0			0
$725$			4.57459i			0.169896i
$726$		0			0
$727$	0.196227			0.00727767			0.00363883	−	0.999993i	$-0.498842\pi$
$727$	0.196227			0.00727767			0.00363883	+	0.999993i	$0.498842\pi$
$728$		0			0
$729$	−22.0720			−0.817483
$730$		0			0
$731$			11.1090i			0.410883i
$732$		0			0
$733$	43.2927			1.59905			0.799527	−	0.600630i	$-0.205084\pi$
$733$	43.2927			1.59905			0.799527	+	0.600630i	$0.205084\pi$
$734$		0			0
$735$	11.0931	−	6.71670i	0.409175	−	0.247749i
$736$		0			0
$737$	−7.96814			−0.293510
$738$		0			0
$739$	39.7629			1.46270			0.731350	−	0.682002i	$-0.238890\pi$
$739$	39.7629			1.46270			0.731350	+	0.682002i	$0.238890\pi$
$740$		0			0
$741$	−4.26811			−0.156793
$742$		0			0
$743$		−	19.0536i		−	0.699009i	−0.936935	−	0.349505i	$-0.886350\pi$
$743$		−	19.0536i		−	0.699009i	0.936935	−	0.349505i	$-0.113650\pi$
$744$		0			0
$745$			9.49776i			0.347971i
$746$		0			0
$747$		−	3.34665i		−	0.122448i
$748$		0			0
$749$	−18.7073	+	33.1962i	−0.683548	+	1.21296i
$750$		0			0
$751$			10.3244i			0.376741i	0.982098	+	0.188371i	$0.0603206\pi$
$751$			10.3244i			0.376741i	−0.982098	+	0.188371i	$0.939679\pi$
$752$		0			0
$753$	−55.0517			−2.00619
$754$		0			0
$755$			1.49710i			0.0544851i
$756$		0			0
$757$			11.2488i			0.408846i	0.978883	+	0.204423i	$0.0655317\pi$
$757$			11.2488i			0.408846i	−0.978883	+	0.204423i	$0.934468\pi$
$758$		0			0
$759$	11.9341			0.433179
$760$		0			0
$761$			15.7291i			0.570180i	0.958501	+	0.285090i	$0.0920235\pi$
$761$			15.7291i			0.570180i	−0.958501	+	0.285090i	$0.907977\pi$
$762$		0			0
$763$	−9.85652	−	5.55452i	−0.356830	−	0.201087i
$764$		0			0
$765$			0.711290i			0.0257167i
$766$		0			0
$767$		−	7.11099i		−	0.256763i
$768$		0			0
$769$			36.5658i			1.31860i	0.751881	+	0.659299i	$0.229147\pi$
$769$			36.5658i			1.31860i	−0.751881	+	0.659299i	$0.770853\pi$
$770$		0			0
$771$	−7.24603			−0.260959
$772$		0			0
$773$	−27.4432			−0.987063			−0.493532	−	0.869728i	$-0.664294\pi$
$773$	−27.4432			−0.987063			−0.493532	+	0.869728i	$0.664294\pi$
$774$		0			0
$775$	0.834131			0.0299629
$776$		0			0
$777$	18.8314	−	33.4165i	0.675573	−	1.19881i
$778$		0			0
$779$	3.47412			0.124473
$780$		0			0
$781$		−	22.4307i		−	0.802633i
$782$		0			0
$783$	−21.7628			−0.777739
$784$		0			0
$785$	7.26866			0.259429
$786$		0			0
$787$		−	44.2718i		−	1.57812i	−0.614316	−	0.789060i	$-0.710568\pi$
$787$		−	44.2718i		−	1.57812i	0.614316	−	0.789060i	$-0.289432\pi$
$788$		0			0
$789$	−7.61883			−0.271238
$790$		0			0
$791$	11.2546	−	19.9714i	0.400168	−	0.710100i
$792$		0			0
$793$	−0.522487			−0.0185540
$794$		0			0
$795$	−16.0430			−0.568986
$796$		0			0
$797$	−40.7235			−1.44250			−0.721249	−	0.692675i	$-0.756432\pi$
$797$	−40.7235			−1.44250			−0.721249	+	0.692675i	$0.756432\pi$
$798$		0			0
$799$			17.8419i			0.631201i
$800$		0			0
$801$			3.62917i			0.128230i
$802$		0			0
$803$			27.2017i			0.959929i
$804$		0			0
$805$	−7.35403	−	4.14426i	−0.259195	−	0.146066i
$806$		0			0
$807$			18.9060i			0.665523i
$808$		0			0
$809$	12.7995			0.450007			0.225004	−	0.974358i	$-0.427761\pi$
$809$	12.7995			0.450007			0.225004	+	0.974358i	$0.427761\pi$
$810$		0			0
$811$			7.24848i			0.254529i	0.991869	+	0.127264i	$0.0406197\pi$
$811$			7.24848i			0.254529i	−0.991869	+	0.127264i	$0.959380\pi$
$812$		0			0
$813$			43.0185i			1.50872i
$814$		0			0
$815$	17.2922			0.605720
$816$		0			0
$817$		−	8.50395i		−	0.297515i
$818$		0			0
$819$	1.02599	−	1.82062i	0.0358509	−	0.0636177i
$820$		0			0
$821$		−	42.0607i		−	1.46793i	−0.679188	−	0.733964i	$-0.737668\pi$
$821$		−	42.0607i		−	1.46793i	0.679188	−	0.733964i	$-0.262332\pi$
$822$		0			0
$823$			7.64831i			0.266603i	0.991076	+	0.133302i	$0.0425579\pi$
$823$			7.64831i			0.266603i	−0.991076	+	0.133302i	$0.957442\pi$
$824$		0			0
$825$		−	3.74046i		−	0.130226i
$826$		0			0
$827$	52.3388			1.82000			0.909999	−	0.414611i	$-0.136082\pi$
$827$	52.3388			1.82000			0.909999	+	0.414611i	$0.136082\pi$
$828$		0			0
$829$	23.8432			0.828109			0.414054	−	0.910252i	$-0.364112\pi$
$829$	23.8432			0.828109			0.414054	+	0.910252i	$0.364112\pi$
$830$		0			0
$831$	−34.3101			−1.19020
$832$		0			0
$833$	−9.85773	+	5.96871i	−0.341550	+	0.206803i
$834$		0			0
$835$	−20.8910			−0.722961
$836$		0			0
$837$			3.96823i			0.137162i
$838$		0			0
$839$	−37.3951			−1.29102			−0.645510	−	0.763751i	$-0.723355\pi$
$839$	−37.3951			−1.29102			−0.645510	+	0.763751i	$0.723355\pi$
$840$		0			0
$841$	8.07310			0.278383
$842$		0			0
$843$		−	37.1241i		−	1.27862i
$844$		0			0
$845$	−9.65785			−0.332240
$846$		0			0
$847$	8.99301	−	15.9582i	0.309003	−	0.548329i
$848$		0			0
$849$	25.9511			0.890639
$850$		0			0
$851$	−24.9681			−0.855895
$852$		0			0
$853$	9.61196			0.329107			0.164554	−	0.986368i	$-0.447382\pi$
$853$	9.61196			0.329107			0.164554	+	0.986368i	$0.447382\pi$
$854$		0			0
$855$		−	0.544491i		−	0.0186212i
$856$		0			0
$857$		−	34.6718i		−	1.18437i	−0.805804	−	0.592183i	$-0.798266\pi$
$857$		−	34.6718i		−	1.18437i	0.805804	−	0.592183i	$-0.201734\pi$
$858$		0			0
$859$			42.4621i			1.44879i	0.689386	+	0.724394i	$0.257880\pi$
$859$			42.4621i			1.44879i	−0.689386	+	0.724394i	$0.742120\pi$
$860$		0			0
$861$	−6.63377	+	11.7717i	−0.226078	+	0.401178i
$862$		0			0
$863$			53.8661i			1.83362i	0.399319	+	0.916812i	$0.369247\pi$
$863$			53.8661i			1.83362i	−0.399319	+	0.916812i	$0.630753\pi$
$864$		0			0
$865$	1.87701			0.0638202
$866$		0			0
$867$			26.4730i			0.899071i
$868$		0			0
$869$			25.9621i			0.880705i
$870$		0			0
$871$	7.21477			0.244463
$872$		0			0
$873$		−	4.02683i		−	0.136287i
$874$		0			0
$875$	−1.29892	+	2.30495i	−0.0439117	+	0.0779215i
$876$		0			0
$877$			11.8357i			0.399665i	0.979830	+	0.199832i	$0.0640398\pi$
$877$			11.8357i			0.399665i	−0.979830	+	0.199832i	$0.935960\pi$
$878$		0			0
$879$			16.6494i			0.561569i
$880$		0			0
$881$			4.53888i			0.152919i	0.997073	+	0.0764595i	$0.0243616\pi$
$881$			4.53888i			0.152919i	−0.997073	+	0.0764595i	$0.975638\pi$
$882$		0			0
$883$	29.4525			0.991156			0.495578	−	0.868563i	$-0.334956\pi$
$883$	29.4525			0.991156			0.495578	+	0.868563i	$0.334956\pi$
$884$		0			0
$885$	7.20600			0.242227
$886$		0			0
$887$	2.42071			0.0812795			0.0406398	−	0.999174i	$-0.487060\pi$
$887$	2.42071			0.0812795			0.0406398	+	0.999174i	$0.487060\pi$
$888$		0			0
$889$	−3.72410	−	2.09867i	−0.124902	−	0.0703871i
$890$		0			0
$891$	20.4116			0.683815
$892$		0			0
$893$		−	13.6579i		−	0.457045i
$894$		0			0
$895$	5.47135			0.182887
$896$		0			0
$897$	−10.8057			−0.360793
$898$		0			0
$899$			3.81581i			0.127264i
$900$		0			0
$901$	14.2564			0.474949
$902$		0			0
$903$	28.8147	+	16.2382i	0.958894	+	0.540372i
$904$		0			0
$905$	−7.67879			−0.255252
$906$		0			0
$907$	−7.19443			−0.238887			−0.119444	−	0.992841i	$-0.538111\pi$
$907$	−7.19443			−0.238887			−0.119444	+	0.992841i	$0.538111\pi$
$908$		0			0
$909$	6.27736			0.208207
$910$		0			0
$911$			8.87733i			0.294119i	0.989128	+	0.147059i	$0.0469809\pi$
$911$			8.87733i			0.294119i	−0.989128	+	0.147059i	$0.953019\pi$
$912$		0			0
$913$		−	15.6391i		−	0.517580i
$914$		0			0
$915$		−	0.529468i		−	0.0175037i
$916$		0			0
$917$	−0.399201	−	0.224965i	−0.0131828	−	0.00742899i
$918$		0			0
$919$		−	5.15534i		−	0.170059i	−0.996378	−	0.0850295i	$-0.972902\pi$
$919$		−	5.15534i		−	0.170059i	0.996378	−	0.0850295i	$-0.0270985\pi$
$920$		0			0
$921$	1.86176			0.0613472
$922$		0			0
$923$			20.3099i			0.668509i
$924$		0			0
$925$			7.82567i			0.257306i
$926$		0			0
$927$	−2.04320			−0.0671074
$928$		0			0
$929$		−	54.1036i		−	1.77508i	−0.460729	−	0.887541i	$-0.652412\pi$
$929$		−	54.1036i		−	1.77508i	0.460729	−	0.887541i	$-0.347588\pi$
$930$		0			0
$931$	7.54606	−	4.56903i	0.247312	−	0.149744i
$932$		0			0
$933$			22.1397i			0.724821i
$934$		0			0
$935$			3.32391i			0.108703i
$936$		0			0
$937$			1.66847i			0.0545064i	0.999629	+	0.0272532i	$0.00867603\pi$
$937$			1.66847i			0.0545064i	−0.999629	+	0.0272532i	$0.991324\pi$
$938$		0			0
$939$	−19.3292			−0.630784
$940$		0			0
$941$	−48.0397			−1.56605			−0.783025	−	0.621990i	$-0.786325\pi$
$941$	−48.0397			−1.56605			−0.783025	+	0.621990i	$0.786325\pi$
$942$		0			0
$943$	8.79555			0.286422
$944$		0			0
$945$	−10.9654	−	6.17940i	−0.356704	−	0.201016i
$946$		0			0
$947$	−52.8737			−1.71816			−0.859081	−	0.511839i	$-0.828964\pi$
$947$	−52.8737			−1.71816			−0.859081	+	0.511839i	$0.828964\pi$
$948$		0			0
$949$		−	24.6299i		−	0.799519i
$950$		0			0
$951$	−58.5813			−1.89963
$952$		0			0
$953$	11.4029			0.369376			0.184688	−	0.982797i	$-0.440873\pi$
$953$	11.4029			0.369376			0.184688	+	0.982797i	$0.440873\pi$
$954$		0			0
$955$		−	3.63878i		−	0.117748i
$956$		0			0
$957$	17.1111			0.553123
$958$		0			0
$959$	−23.0884	+	40.9706i	−0.745565	+	1.32301i
$960$		0			0
$961$	−30.3042			−0.977556
$962$		0			0
$963$	−6.22259			−0.200520
$964$		0			0
$965$	−0.555242			−0.0178739
$966$		0			0
$967$		−	34.8940i		−	1.12212i	−0.827776	−	0.561058i	$-0.810394\pi$
$967$		−	34.8940i		−	1.12212i	0.827776	−	0.561058i	$-0.189606\pi$
$968$		0			0
$969$			3.84347i			0.123470i
$970$		0			0
$971$			27.3230i			0.876837i	0.898771	+	0.438418i	$0.144461\pi$
$971$			27.3230i			0.876837i	−0.898771	+	0.438418i	$0.855539\pi$
$972$		0			0
$973$	14.2827	+	8.04882i	0.457882	+	0.258033i
$974$		0			0
$975$			3.38681i			0.108465i
$976$		0			0
$977$	−0.876235			−0.0280332			−0.0140166	−	0.999902i	$-0.504462\pi$
$977$	−0.876235			−0.0280332			−0.0140166	+	0.999902i	$0.504462\pi$
$978$		0			0
$979$			16.9594i			0.542024i
$980$		0			0
$981$		−	1.84760i		−	0.0589893i
$982$		0			0
$983$	−43.4686			−1.38643			−0.693216	−	0.720730i	$-0.743807\pi$
$983$	−43.4686			−1.38643			−0.693216	+	0.720730i	$0.743807\pi$
$984$		0			0
$985$		−	3.44936i		−	0.109906i
$986$		0			0
$987$	46.2784	+	26.0796i	1.47306	+	0.830123i
$988$		0			0
$989$		−	21.5298i		−	0.684606i
$990$		0			0
$991$		−	18.8129i		−	0.597610i	−0.954314	−	0.298805i	$-0.903412\pi$
$991$		−	18.8129i		−	0.597610i	0.954314	−	0.298805i	$-0.0965880\pi$
$992$		0			0
$993$		−	34.4477i		−	1.09316i
$994$		0			0
$995$	23.8014			0.754554
$996$		0			0
$997$	−22.3046			−0.706394			−0.353197	−	0.935549i	$-0.614906\pi$
$997$	−22.3046			−0.706394			−0.353197	+	0.935549i	$0.614906\pi$
$998$		0			0
$999$	−37.2292			−1.17788

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2240.2.h.f.671.16	yes	24
4.3	odd	2	inner	2240.2.h.f.671.9	yes	24
7.6	odd	2		2240.2.h.e.671.9	✓	24
8.3	odd	2		2240.2.h.e.671.10	yes	24
8.5	even	2		2240.2.h.e.671.15	yes	24
28.27	even	2		2240.2.h.e.671.16	yes	24
56.13	odd	2	inner	2240.2.h.f.671.10	yes	24
56.27	even	2	inner	2240.2.h.f.671.15	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2240.2.h.e.671.9	✓	24	7.6	odd	2
2240.2.h.e.671.10	yes	24	8.3	odd	2
2240.2.h.e.671.15	yes	24	8.5	even	2
2240.2.h.e.671.16	yes	24	28.27	even	2
2240.2.h.f.671.9	yes	24	4.3	odd	2	inner
2240.2.h.f.671.10	yes	24	56.13	odd	2	inner
2240.2.h.f.671.15	yes	24	56.27	even	2	inner
2240.2.h.f.671.16	yes	24	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2240.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.85258i q^{3} +1.00000 q^{5} +(-1.29892 + 2.30495i) q^{7} -0.432061 q^{9} +O(q^{10})\)	\(q+1.85258i q^{3} +1.00000 q^{5} +(-1.29892 + 2.30495i) q^{7} -0.432061 q^{9} -2.01905 q^{11} +1.82816 q^{13} +1.85258i q^{15} -1.64627i q^{17} +1.26022i q^{19} +(-4.27011 - 2.40636i) q^{21} +3.19054i q^{23} +1.00000 q^{25} +4.75732i q^{27} +4.57459i q^{29} +0.834131 q^{31} -3.74046i q^{33} +(-1.29892 + 2.30495i) q^{35} +7.82567i q^{37} +3.38681i q^{39} -2.75676i q^{41} -6.74801 q^{43} -0.432061 q^{45} -10.8378 q^{47} +(-3.62559 - 5.98791i) q^{49} +3.04985 q^{51} +8.65980i q^{53} -2.01905 q^{55} -2.33465 q^{57} -3.88971i q^{59} -0.285800 q^{61} +(0.561215 - 0.995879i) q^{63} +1.82816 q^{65} +3.94648 q^{67} -5.91073 q^{69} +11.1095i q^{71} -13.4725i q^{73} +1.85258i q^{75} +(2.62260 - 4.65382i) q^{77} -12.8586i q^{79} -10.1095 q^{81} +7.74578i q^{83} -1.64627i q^{85} -8.47481 q^{87} -8.39967i q^{89} +(-2.37464 + 4.21381i) q^{91} +1.54530i q^{93} +1.26022i q^{95} +9.32004i q^{97} +0.872354 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 24 q^{5} - 36 q^{9}+O(q^{10}) \) 24 * q + 24 * q^5 - 36 * q^9	\( 24 q + 24 q^{5} - 36 q^{9} + 4 q^{13} + 8 q^{21} + 24 q^{25} - 36 q^{45} + 24 q^{57} + 56 q^{61} + 4 q^{65} + 96 q^{69} - 12 q^{77} + 24 q^{81}+O(q^{100}) \) 24 * q + 24 * q^5 - 36 * q^9 + 4 * q^13 + 8 * q^21 + 24 * q^25 - 36 * q^45 + 24 * q^57 + 56 * q^61 + 4 * q^65 + 96 * q^69 - 12 * q^77 + 24 * q^81

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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