LMFDB - Embedded newform 2240.2.k.b.1791.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2240,2,Mod(1791,2240)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2240.1791"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2240, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2240 = 2^{6} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2240.k (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	no
Analytic conductor:	$17.8864900528$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 140)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1791.1
Root			$0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	2240.1791
Dual form			2240.2.k.b.1791.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205 q^{3} -1.00000i q^{5} +(1.73205 - 2.00000i) q^{7} +0.267949i q^{11} -0.464102i q^{13} +1.73205i q^{15} +6.46410i q^{17} +6.00000 q^{19} +(-3.00000 + 3.46410i) q^{21} -1.46410i q^{23} -1.00000 q^{25} +5.19615 q^{27} -7.92820 q^{29} +6.00000 q^{31} -0.464102i q^{33} +(-2.00000 - 1.73205i) q^{35} +9.46410 q^{37} +0.803848i q^{39} +3.46410i q^{41} +2.00000i q^{43} -1.73205 q^{47} +(-1.00000 - 6.92820i) q^{49} -11.1962i q^{51} -2.00000 q^{53} +0.267949 q^{55} -10.3923 q^{57} -3.46410 q^{59} +9.46410i q^{61} -0.464102 q^{65} -3.46410i q^{67} +2.53590i q^{69} -7.46410i q^{71} -12.9282i q^{73} +1.73205 q^{75} +(0.535898 + 0.464102i) q^{77} -14.6603i q^{79} -9.00000 q^{81} +15.4641 q^{83} +6.46410 q^{85} +13.7321 q^{87} -2.53590i q^{89} +(-0.928203 - 0.803848i) q^{91} -10.3923 q^{93} -6.00000i q^{95} +13.3923i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 24 q^{19} - 12 q^{21} - 4 q^{25} - 4 q^{29} + 24 q^{31} - 8 q^{35} + 24 q^{37} - 4 q^{49} - 8 q^{53} + 8 q^{55} + 12 q^{65} + 16 q^{77} - 36 q^{81} + 48 q^{83} + 12 q^{85} + 48 q^{87} + 24 q^{91}+O(q^{100}) $ 4 * q + 24 * q^19 - 12 * q^21 - 4 * q^25 - 4 * q^29 + 24 * q^31 - 8 * q^35 + 24 * q^37 - 4 * q^49 - 8 * q^53 + 8 * q^55 + 12 * q^65 + 16 * q^77 - 36 * q^81 + 48 * q^83 + 12 * q^85 + 48 * q^87 + 24 * q^91

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
$3$	−1.73205			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$3$	−1.73205			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$4$		0			0
$5$		−	1.00000i		−	0.447214i
$5$		−	1.00000i		−	0.447214i
$6$		0			0
$7$	1.73205	−	2.00000i	0.654654	−	0.755929i
$7$	1.73205	−	2.00000i	0.654654	−	0.755929i
$8$		0			0
$9$		0			0
$10$		0			0
$11$			0.267949i			0.0807897i	0.999184	+	0.0403949i	$0.0128616\pi$
$11$			0.267949i			0.0807897i	−0.999184	+	0.0403949i	$0.987138\pi$
$12$		0			0
$13$		−	0.464102i		−	0.128719i	−0.997927	−	0.0643593i	$-0.979500\pi$
$13$		−	0.464102i		−	0.128719i	0.997927	−	0.0643593i	$-0.0205004\pi$
$14$		0			0
$15$			1.73205i			0.447214i
$16$		0			0
$17$			6.46410i			1.56777i	0.620903	+	0.783887i	$0.286766\pi$
$17$			6.46410i			1.56777i	−0.620903	+	0.783887i	$0.713234\pi$
$18$		0			0
$19$	6.00000			1.37649			0.688247	−	0.725476i	$-0.258380\pi$
$19$	6.00000			1.37649			0.688247	+	0.725476i	$0.258380\pi$
$20$		0			0
$21$	−3.00000	+	3.46410i	−0.654654	+	0.755929i
$22$		0			0
$23$		−	1.46410i		−	0.305286i	−0.988281	−	0.152643i	$-0.951221\pi$
$23$		−	1.46410i		−	0.305286i	0.988281	−	0.152643i	$-0.0487785\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$	5.19615			1.00000
$28$		0			0
$29$	−7.92820			−1.47223			−0.736115	−	0.676856i	$-0.763342\pi$
$29$	−7.92820			−1.47223			−0.736115	+	0.676856i	$0.763342\pi$
$30$		0			0
$31$	6.00000			1.07763			0.538816	−	0.842424i	$-0.318872\pi$
$31$	6.00000			1.07763			0.538816	+	0.842424i	$0.318872\pi$
$32$		0			0
$33$		−	0.464102i		−	0.0807897i
$34$		0			0
$35$	−2.00000	−	1.73205i	−0.338062	−	0.292770i
$36$		0			0
$37$	9.46410			1.55589			0.777944	−	0.628333i	$-0.216263\pi$
$37$	9.46410			1.55589			0.777944	+	0.628333i	$0.216263\pi$
$38$		0			0
$39$			0.803848i			0.128719i
$40$		0			0
$41$			3.46410i			0.541002i	0.962720	+	0.270501i	$0.0871893\pi$
$41$			3.46410i			0.541002i	−0.962720	+	0.270501i	$0.912811\pi$
$42$		0			0
$43$			2.00000i			0.304997i	0.988304	+	0.152499i	$0.0487319\pi$
$43$			2.00000i			0.304997i	−0.988304	+	0.152499i	$0.951268\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−1.73205			−0.252646			−0.126323	−	0.991989i	$-0.540318\pi$
$47$	−1.73205			−0.252646			−0.126323	+	0.991989i	$0.540318\pi$
$48$		0			0
$49$	−1.00000	−	6.92820i	−0.142857	−	0.989743i
$50$		0			0
$51$		−	11.1962i		−	1.56777i
$52$		0			0
$53$	−2.00000			−0.274721			−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000			−0.274721			−0.137361	+	0.990521i	$0.543862\pi$
$54$		0			0
$55$	0.267949			0.0361303
$56$		0			0
$57$	−10.3923			−1.37649
$58$		0			0
$59$	−3.46410			−0.450988			−0.225494	−	0.974245i	$-0.572400\pi$
$59$	−3.46410			−0.450988			−0.225494	+	0.974245i	$0.572400\pi$
$60$		0			0
$61$			9.46410i			1.21175i	0.795558	+	0.605877i	$0.207178\pi$
$61$			9.46410i			1.21175i	−0.795558	+	0.605877i	$0.792822\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−0.464102			−0.0575647
$66$		0			0
$67$		−	3.46410i		−	0.423207i	−0.977356	−	0.211604i	$-0.932131\pi$
$67$		−	3.46410i		−	0.423207i	0.977356	−	0.211604i	$-0.0678686\pi$
$68$		0			0
$69$			2.53590i			0.305286i
$70$		0			0
$71$		−	7.46410i		−	0.885826i	−0.896565	−	0.442913i	$-0.853945\pi$
$71$		−	7.46410i		−	0.885826i	0.896565	−	0.442913i	$-0.146055\pi$
$72$		0			0
$73$		−	12.9282i		−	1.51313i	−0.653917	−	0.756566i	$-0.726876\pi$
$73$		−	12.9282i		−	1.51313i	0.653917	−	0.756566i	$-0.273124\pi$
$74$		0			0
$75$	1.73205			0.200000
$76$		0			0
$77$	0.535898	+	0.464102i	0.0610713	+	0.0528893i
$78$		0			0
$79$		−	14.6603i		−	1.64941i	−0.565565	−	0.824704i	$-0.691342\pi$
$79$		−	14.6603i		−	1.64941i	0.565565	−	0.824704i	$-0.308658\pi$
$80$		0			0
$81$	−9.00000			−1.00000
$82$		0			0
$83$	15.4641			1.69741			0.848703	−	0.528870i	$-0.177384\pi$
$83$	15.4641			1.69741			0.848703	+	0.528870i	$0.177384\pi$
$84$		0			0
$85$	6.46410			0.701130
$86$		0			0
$87$	13.7321			1.47223
$88$		0			0
$89$		−	2.53590i		−	0.268805i	−0.990927	−	0.134402i	$-0.957089\pi$
$89$		−	2.53590i		−	0.268805i	0.990927	−	0.134402i	$-0.0429115\pi$
$90$		0			0
$91$	−0.928203	−	0.803848i	−0.0973021	−	0.0842661i
$92$		0			0
$93$	−10.3923			−1.07763
$94$		0			0
$95$		−	6.00000i		−	0.615587i
$96$		0			0
$97$			13.3923i			1.35978i	0.733313	+	0.679891i	$0.237973\pi$
$97$			13.3923i			1.35978i	−0.733313	+	0.679891i	$0.762027\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	15.4641i		−	1.53874i	−0.638806	−	0.769368i	$-0.720571\pi$
$101$		−	15.4641i		−	1.53874i	0.638806	−	0.769368i	$-0.279429\pi$
$102$		0			0
$103$	6.80385			0.670403			0.335202	−	0.942146i	$-0.391196\pi$
$103$	6.80385			0.670403			0.335202	+	0.942146i	$0.391196\pi$
$104$		0			0
$105$	3.46410	+	3.00000i	0.338062	+	0.292770i
$106$		0			0
$107$		−	2.39230i		−	0.231273i	−0.993292	−	0.115636i	$-0.963109\pi$
$107$		−	2.39230i		−	0.231273i	0.993292	−	0.115636i	$-0.0368907\pi$
$108$		0			0
$109$	−2.07180			−0.198442			−0.0992211	−	0.995065i	$-0.531635\pi$
$109$	−2.07180			−0.198442			−0.0992211	+	0.995065i	$0.531635\pi$
$110$		0			0
$111$	−16.3923			−1.55589
$112$		0			0
$113$	5.46410			0.514019			0.257010	−	0.966409i	$-0.417263\pi$
$113$	5.46410			0.514019			0.257010	+	0.966409i	$0.417263\pi$
$114$		0			0
$115$	−1.46410			−0.136528
$116$		0			0
$117$		0			0
$118$		0			0
$119$	12.9282	+	11.1962i	1.18513	+	1.02635i
$120$		0			0
$121$	10.9282			0.993473
$122$		0			0
$123$		−	6.00000i		−	0.541002i
$124$		0			0
$125$			1.00000i			0.0894427i
$126$		0			0
$127$			15.4641i			1.37222i	0.727499	+	0.686109i	$0.240683\pi$
$127$			15.4641i			1.37222i	−0.727499	+	0.686109i	$0.759317\pi$
$128$		0			0
$129$		−	3.46410i		−	0.304997i
$130$		0			0
$131$	2.53590			0.221562			0.110781	−	0.993845i	$-0.464665\pi$
$131$	2.53590			0.221562			0.110781	+	0.993845i	$0.464665\pi$
$132$		0			0
$133$	10.3923	−	12.0000i	0.901127	−	1.04053i
$134$		0			0
$135$		−	5.19615i		−	0.447214i
$136$		0			0
$137$	20.3923			1.74223			0.871116	−	0.491077i	$-0.163397\pi$
$137$	20.3923			1.74223			0.871116	+	0.491077i	$0.163397\pi$
$138$		0			0
$139$	6.92820			0.587643			0.293821	−	0.955860i	$-0.405073\pi$
$139$	6.92820			0.587643			0.293821	+	0.955860i	$0.405073\pi$
$140$		0			0
$141$	3.00000			0.252646
$142$		0			0
$143$	0.124356			0.0103991
$144$		0			0
$145$			7.92820i			0.658401i
$146$		0			0
$147$	1.73205	+	12.0000i	0.142857	+	0.989743i
$148$		0			0
$149$	−3.07180			−0.251651			−0.125826	−	0.992052i	$-0.540158\pi$
$149$	−3.07180			−0.251651			−0.125826	+	0.992052i	$0.540158\pi$
$150$		0			0
$151$		−	15.1962i		−	1.23665i	−0.785924	−	0.618323i	$-0.787812\pi$
$151$		−	15.1962i		−	1.23665i	0.785924	−	0.618323i	$-0.212188\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	6.00000i		−	0.481932i
$156$		0			0
$157$			7.85641i			0.627009i	0.949587	+	0.313505i	$0.101503\pi$
$157$			7.85641i			0.627009i	−0.949587	+	0.313505i	$0.898497\pi$
$158$		0			0
$159$	3.46410			0.274721
$160$		0			0
$161$	−2.92820	−	2.53590i	−0.230775	−	0.199857i
$162$		0			0
$163$		−	20.7846i		−	1.62798i	−0.580881	−	0.813988i	$-0.697292\pi$
$163$		−	20.7846i		−	1.62798i	0.580881	−	0.813988i	$-0.302708\pi$
$164$		0			0
$165$	−0.464102			−0.0361303
$166$		0			0
$167$	5.19615			0.402090			0.201045	−	0.979582i	$-0.435566\pi$
$167$	5.19615			0.402090			0.201045	+	0.979582i	$0.435566\pi$
$168$		0			0
$169$	12.7846			0.983432
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	14.3205i		−	1.08877i	−0.838836	−	0.544384i	$-0.816763\pi$
$173$		−	14.3205i		−	1.08877i	0.838836	−	0.544384i	$-0.183237\pi$
$174$		0			0
$175$	−1.73205	+	2.00000i	−0.130931	+	0.151186i
$176$		0			0
$177$	6.00000			0.450988
$178$		0			0
$179$		−	6.39230i		−	0.477783i	−0.971046	−	0.238892i	$-0.923216\pi$
$179$		−	6.39230i		−	0.477783i	0.971046	−	0.238892i	$-0.0767841\pi$
$180$		0			0
$181$		−	0.928203i		−	0.0689928i	−0.999405	−	0.0344964i	$-0.989017\pi$
$181$		−	0.928203i		−	0.0689928i	0.999405	−	0.0344964i	$-0.0109827\pi$
$182$		0			0
$183$		−	16.3923i		−	1.21175i
$184$		0			0
$185$		−	9.46410i		−	0.695815i
$186$		0			0
$187$	−1.73205			−0.126660
$188$		0			0
$189$	9.00000	−	10.3923i	0.654654	−	0.755929i
$190$		0			0
$191$		−	7.19615i		−	0.520695i	−0.965515	−	0.260348i	$-0.916163\pi$
$191$		−	7.19615i		−	0.520695i	0.965515	−	0.260348i	$-0.0838372\pi$
$192$		0			0
$193$	−9.46410			−0.681241			−0.340620	−	0.940201i	$-0.610637\pi$
$193$	−9.46410			−0.681241			−0.340620	+	0.940201i	$0.610637\pi$
$194$		0			0
$195$	0.803848			0.0575647
$196$		0			0
$197$	13.3205			0.949047			0.474523	−	0.880243i	$-0.342620\pi$
$197$	13.3205			0.949047			0.474523	+	0.880243i	$0.342620\pi$
$198$		0			0
$199$	−3.46410			−0.245564			−0.122782	−	0.992434i	$-0.539182\pi$
$199$	−3.46410			−0.245564			−0.122782	+	0.992434i	$0.539182\pi$
$200$		0			0
$201$			6.00000i			0.423207i
$202$		0			0
$203$	−13.7321	+	15.8564i	−0.963801	+	1.11290i
$204$		0			0
$205$	3.46410			0.241943
$206$		0			0
$207$		0			0
$208$		0			0
$209$			1.60770i			0.111207i
$210$		0			0
$211$			3.19615i			0.220032i	0.993930	+	0.110016i	$0.0350902\pi$
$211$			3.19615i			0.220032i	−0.993930	+	0.110016i	$0.964910\pi$
$212$		0			0
$213$			12.9282i			0.885826i
$214$		0			0
$215$	2.00000			0.136399
$216$		0			0
$217$	10.3923	−	12.0000i	0.705476	−	0.814613i
$218$		0			0
$219$			22.3923i			1.51313i
$220$		0			0
$221$	3.00000			0.201802
$222$		0			0
$223$	13.7321			0.919566			0.459783	−	0.888031i	$-0.347927\pi$
$223$	13.7321			0.919566			0.459783	+	0.888031i	$0.347927\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	20.6603			1.37127			0.685635	−	0.727946i	$-0.259525\pi$
$227$	20.6603			1.37127			0.685635	+	0.727946i	$0.259525\pi$
$228$		0			0
$229$		−	8.53590i		−	0.564068i	−0.959404	−	0.282034i	$-0.908991\pi$
$229$		−	8.53590i		−	0.564068i	0.959404	−	0.282034i	$-0.0910091\pi$
$230$		0			0
$231$	−0.928203	−	0.803848i	−0.0610713	−	0.0528893i
$232$		0			0
$233$	−9.07180			−0.594313			−0.297157	−	0.954829i	$-0.596038\pi$
$233$	−9.07180			−0.594313			−0.297157	+	0.954829i	$0.596038\pi$
$234$		0			0
$235$			1.73205i			0.112987i
$236$		0			0
$237$			25.3923i			1.64941i
$238$		0			0
$239$			23.9808i			1.55119i	0.631233	+	0.775593i	$0.282549\pi$
$239$			23.9808i			1.55119i	−0.631233	+	0.775593i	$0.717451\pi$
$240$		0			0
$241$		−	16.3923i		−	1.05592i	−0.849269	−	0.527961i	$-0.822957\pi$
$241$		−	16.3923i		−	1.05592i	0.849269	−	0.527961i	$-0.177043\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−6.92820	+	1.00000i	−0.442627	+	0.0638877i
$246$		0			0
$247$		−	2.78461i		−	0.177180i
$248$		0			0
$249$	−26.7846			−1.69741
$250$		0			0
$251$	−25.8564			−1.63204			−0.816021	−	0.578022i	$-0.803825\pi$
$251$	−25.8564			−1.63204			−0.816021	+	0.578022i	$0.803825\pi$
$252$		0			0
$253$	0.392305			0.0246640
$254$		0			0
$255$	−11.1962			−0.701130
$256$		0			0
$257$		−	6.00000i		−	0.374270i	−0.982334	−	0.187135i	$-0.940080\pi$
$257$		−	6.00000i		−	0.374270i	0.982334	−	0.187135i	$-0.0599201\pi$
$258$		0			0
$259$	16.3923	−	18.9282i	1.01857	−	1.17614i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			11.4641i			0.706907i	0.935452	+	0.353453i	$0.114993\pi$
$263$			11.4641i			0.706907i	−0.935452	+	0.353453i	$0.885007\pi$
$264$		0			0
$265$			2.00000i			0.122859i
$266$		0			0
$267$			4.39230i			0.268805i
$268$		0			0
$269$		−	12.0000i		−	0.731653i	−0.930683	−	0.365826i	$-0.880786\pi$
$269$		−	12.0000i		−	0.731653i	0.930683	−	0.365826i	$-0.119214\pi$
$270$		0			0
$271$	−9.46410			−0.574903			−0.287452	−	0.957795i	$-0.592808\pi$
$271$	−9.46410			−0.574903			−0.287452	+	0.957795i	$0.592808\pi$
$272$		0			0
$273$	1.60770	+	1.39230i	0.0973021	+	0.0842661i
$274$		0			0
$275$		−	0.267949i		−	0.0161579i
$276$		0			0
$277$	−16.7846			−1.00849			−0.504245	−	0.863561i	$-0.668229\pi$
$277$	−16.7846			−1.00849			−0.504245	+	0.863561i	$0.668229\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−7.92820			−0.472957			−0.236478	−	0.971637i	$-0.575993\pi$
$281$	−7.92820			−0.472957			−0.236478	+	0.971637i	$0.575993\pi$
$282$		0			0
$283$	12.1244			0.720718			0.360359	−	0.932814i	$-0.382654\pi$
$283$	12.1244			0.720718			0.360359	+	0.932814i	$0.382654\pi$
$284$		0			0
$285$			10.3923i			0.615587i
$286$		0			0
$287$	6.92820	+	6.00000i	0.408959	+	0.354169i
$288$		0			0
$289$	−24.7846			−1.45792
$290$		0			0
$291$		−	23.1962i		−	1.35978i
$292$		0			0
$293$			20.3205i			1.18714i	0.804784	+	0.593568i	$0.202281\pi$
$293$			20.3205i			1.18714i	−0.804784	+	0.593568i	$0.797719\pi$
$294$		0			0
$295$			3.46410i			0.201688i
$296$		0			0
$297$			1.39230i			0.0807897i
$298$		0			0
$299$	−0.679492			−0.0392960
$300$		0			0
$301$	4.00000	+	3.46410i	0.230556	+	0.199667i
$302$		0			0
$303$			26.7846i			1.53874i
$304$		0			0
$305$	9.46410			0.541913
$306$		0			0
$307$	1.73205			0.0988534			0.0494267	−	0.998778i	$-0.484261\pi$
$307$	1.73205			0.0988534			0.0494267	+	0.998778i	$0.484261\pi$
$308$		0			0
$309$	−11.7846			−0.670403
$310$		0			0
$311$	−7.85641			−0.445496			−0.222748	−	0.974876i	$-0.571503\pi$
$311$	−7.85641			−0.445496			−0.222748	+	0.974876i	$0.571503\pi$
$312$		0			0
$313$			17.5359i			0.991188i	0.868554	+	0.495594i	$0.165050\pi$
$313$			17.5359i			0.991188i	−0.868554	+	0.495594i	$0.834950\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	3.07180			0.172529			0.0862646	−	0.996272i	$-0.472507\pi$
$317$	3.07180			0.172529			0.0862646	+	0.996272i	$0.472507\pi$
$318$		0			0
$319$		−	2.12436i		−	0.118941i
$320$		0			0
$321$			4.14359i			0.231273i
$322$		0			0
$323$			38.7846i			2.15803i
$324$		0			0
$325$			0.464102i			0.0257437i
$326$		0			0
$327$	3.58846			0.198442
$328$		0			0
$329$	−3.00000	+	3.46410i	−0.165395	+	0.190982i
$330$		0			0
$331$		−	26.3923i		−	1.45065i	−0.688405	−	0.725326i	$-0.741689\pi$
$331$		−	26.3923i		−	1.45065i	0.688405	−	0.725326i	$-0.258311\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−3.46410			−0.189264
$336$		0			0
$337$		0			0			−	1.00000i	$-0.5\pi$
$337$		0			0				1.00000i	$0.5\pi$
$338$		0			0
$339$	−9.46410			−0.514019
$340$		0			0
$341$			1.60770i			0.0870616i
$342$		0			0
$343$	−15.5885	−	10.0000i	−0.841698	−	0.539949i
$344$		0			0
$345$	2.53590			0.136528
$346$		0			0
$347$			34.2487i			1.83857i	0.393596	+	0.919284i	$0.371231\pi$
$347$			34.2487i			1.83857i	−0.393596	+	0.919284i	$0.628769\pi$
$348$		0			0
$349$		−	5.32051i		−	0.284800i	−0.989809	−	0.142400i	$-0.954518\pi$
$349$		−	5.32051i		−	0.284800i	0.989809	−	0.142400i	$-0.0454820\pi$
$350$		0			0
$351$		−	2.41154i		−	0.128719i
$352$		0			0
$353$			7.39230i			0.393453i	0.980458	+	0.196726i	$0.0630311\pi$
$353$			7.39230i			0.393453i	−0.980458	+	0.196726i	$0.936969\pi$
$354$		0			0
$355$	−7.46410			−0.396153
$356$		0			0
$357$	−22.3923	−	19.3923i	−1.18513	−	1.02635i
$358$		0			0
$359$		−	25.3205i		−	1.33637i	−0.743997	−	0.668183i	$-0.767072\pi$
$359$		−	25.3205i		−	1.33637i	0.743997	−	0.668183i	$-0.232928\pi$
$360$		0			0
$361$	17.0000			0.894737
$362$		0			0
$363$	−18.9282			−0.993473
$364$		0			0
$365$	−12.9282			−0.676693
$366$		0			0
$367$	−11.8756			−0.619904			−0.309952	−	0.950752i	$-0.600313\pi$
$367$	−11.8756			−0.619904			−0.309952	+	0.950752i	$0.600313\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−3.46410	+	4.00000i	−0.179847	+	0.207670i
$372$		0			0
$373$	3.60770			0.186799			0.0933997	−	0.995629i	$-0.470227\pi$
$373$	3.60770			0.186799			0.0933997	+	0.995629i	$0.470227\pi$
$374$		0			0
$375$		−	1.73205i		−	0.0894427i
$376$		0			0
$377$			3.67949i			0.189503i
$378$		0			0
$379$			5.60770i			0.288048i	0.989574	+	0.144024i	$0.0460042\pi$
$379$			5.60770i			0.288048i	−0.989574	+	0.144024i	$0.953996\pi$
$380$		0			0
$381$		−	26.7846i		−	1.37222i
$382$		0			0
$383$	−27.4641			−1.40335			−0.701675	−	0.712497i	$-0.747564\pi$
$383$	−27.4641			−1.40335			−0.701675	+	0.712497i	$0.747564\pi$
$384$		0			0
$385$	0.464102	−	0.535898i	0.0236528	−	0.0273119i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	20.8564			1.05746			0.528731	−	0.848790i	$-0.322668\pi$
$389$	20.8564			1.05746			0.528731	+	0.848790i	$0.322668\pi$
$390$		0			0
$391$	9.46410			0.478620
$392$		0			0
$393$	−4.39230			−0.221562
$394$		0			0
$395$	−14.6603			−0.737637
$396$		0			0
$397$			12.4641i			0.625555i	0.949826	+	0.312778i	$0.101259\pi$
$397$			12.4641i			0.625555i	−0.949826	+	0.312778i	$0.898741\pi$
$398$		0			0
$399$	−18.0000	+	20.7846i	−0.901127	+	1.04053i
$400$		0			0
$401$	−10.0718			−0.502962			−0.251481	−	0.967862i	$-0.580918\pi$
$401$	−10.0718			−0.502962			−0.251481	+	0.967862i	$0.580918\pi$
$402$		0			0
$403$		−	2.78461i		−	0.138711i
$404$		0			0
$405$			9.00000i			0.447214i
$406$		0			0
$407$			2.53590i			0.125700i
$408$		0			0
$409$		−	4.14359i		−	0.204888i	−0.994739	−	0.102444i	$-0.967334\pi$
$409$		−	4.14359i		−	0.204888i	0.994739	−	0.102444i	$-0.0326662\pi$
$410$		0			0
$411$	−35.3205			−1.74223
$412$		0			0
$413$	−6.00000	+	6.92820i	−0.295241	+	0.340915i
$414$		0			0
$415$		−	15.4641i		−	0.759103i
$416$		0			0
$417$	−12.0000			−0.587643
$418$		0			0
$419$	−24.2487			−1.18463			−0.592314	−	0.805708i	$-0.701785\pi$
$419$	−24.2487			−1.18463			−0.592314	+	0.805708i	$0.701785\pi$
$420$		0			0
$421$	−19.0000			−0.926003			−0.463002	−	0.886357i	$-0.653228\pi$
$421$	−19.0000			−0.926003			−0.463002	+	0.886357i	$0.653228\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	6.46410i		−	0.313555i
$426$		0			0
$427$	18.9282	+	16.3923i	0.916000	+	0.793279i
$428$		0			0
$429$	−0.215390			−0.0103991
$430$		0			0
$431$			13.5885i			0.654533i	0.944932	+	0.327266i	$0.106127\pi$
$431$			13.5885i			0.654533i	−0.944932	+	0.327266i	$0.893873\pi$
$432$		0			0
$433$			31.8564i			1.53092i	0.643483	+	0.765461i	$0.277489\pi$
$433$			31.8564i			1.53092i	−0.643483	+	0.765461i	$0.722511\pi$
$434$		0			0
$435$		−	13.7321i		−	0.658401i
$436$		0			0
$437$		−	8.78461i		−	0.420225i
$438$		0			0
$439$	39.7128			1.89539			0.947695	−	0.319179i	$-0.103407\pi$
$439$	39.7128			1.89539			0.947695	+	0.319179i	$0.103407\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	26.0000i		−	1.23530i	−0.786454	−	0.617649i	$-0.788085\pi$
$443$		−	26.0000i		−	1.23530i	0.786454	−	0.617649i	$-0.211915\pi$
$444$		0			0
$445$	−2.53590			−0.120213
$446$		0			0
$447$	5.32051			0.251651
$448$		0			0
$449$	11.9282			0.562927			0.281463	−	0.959572i	$-0.409180\pi$
$449$	11.9282			0.562927			0.281463	+	0.959572i	$0.409180\pi$
$450$		0			0
$451$	−0.928203			−0.0437074
$452$		0			0
$453$			26.3205i			1.23665i
$454$		0			0
$455$	−0.803848	+	0.928203i	−0.0376850	+	0.0435148i
$456$		0			0
$457$	20.5359			0.960629			0.480314	−	0.877096i	$-0.340523\pi$
$457$	20.5359			0.960629			0.480314	+	0.877096i	$0.340523\pi$
$458$		0			0
$459$			33.5885i			1.56777i
$460$		0			0
$461$			27.7128i			1.29071i	0.763881	+	0.645357i	$0.223291\pi$
$461$			27.7128i			1.29071i	−0.763881	+	0.645357i	$0.776709\pi$
$462$		0			0
$463$			16.3923i			0.761815i	0.924613	+	0.380908i	$0.124388\pi$
$463$			16.3923i			0.761815i	−0.924613	+	0.380908i	$0.875612\pi$
$464$		0			0
$465$			10.3923i			0.481932i
$466$		0			0
$467$	22.5167			1.04195			0.520973	−	0.853573i	$-0.325569\pi$
$467$	22.5167			1.04195			0.520973	+	0.853573i	$0.325569\pi$
$468$		0			0
$469$	−6.92820	−	6.00000i	−0.319915	−	0.277054i
$470$		0			0
$471$		−	13.6077i		−	0.627009i
$472$		0			0
$473$	−0.535898			−0.0246406
$474$		0			0
$475$	−6.00000			−0.275299
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−25.1769			−1.15036			−0.575181	−	0.818026i	$-0.695068\pi$
$479$	−25.1769			−1.15036			−0.575181	+	0.818026i	$0.695068\pi$
$480$		0			0
$481$		−	4.39230i		−	0.200272i
$482$		0			0
$483$	5.07180	+	4.39230i	0.230775	+	0.199857i
$484$		0			0
$485$	13.3923			0.608113
$486$		0			0
$487$			12.7846i			0.579326i	0.957129	+	0.289663i	$0.0935432\pi$
$487$			12.7846i			0.579326i	−0.957129	+	0.289663i	$0.906457\pi$
$488$		0			0
$489$			36.0000i			1.62798i
$490$		0			0
$491$		−	9.87564i		−	0.445682i	−0.974855	−	0.222841i	$-0.928467\pi$
$491$		−	9.87564i		−	0.445682i	0.974855	−	0.222841i	$-0.0715330\pi$
$492$		0			0
$493$		−	51.2487i		−	2.30813i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−14.9282	−	12.9282i	−0.669621	−	0.579909i
$498$		0			0
$499$			25.5885i			1.14550i	0.819731	+	0.572748i	$0.194123\pi$
$499$			25.5885i			1.14550i	−0.819731	+	0.572748i	$0.805877\pi$
$500$		0			0
$501$	−9.00000			−0.402090
$502$		0			0
$503$	15.5885			0.695055			0.347527	−	0.937670i	$-0.387021\pi$
$503$	15.5885			0.695055			0.347527	+	0.937670i	$0.387021\pi$
$504$		0			0
$505$	−15.4641			−0.688143
$506$		0			0
$507$	−22.1436			−0.983432
$508$		0			0
$509$			25.8564i			1.14607i	0.819533	+	0.573033i	$0.194233\pi$
$509$			25.8564i			1.14607i	−0.819533	+	0.573033i	$0.805767\pi$
$510$		0			0
$511$	−25.8564	−	22.3923i	−1.14382	−	0.990577i
$512$		0			0
$513$	31.1769			1.37649
$514$		0			0
$515$		−	6.80385i		−	0.299813i
$516$		0			0
$517$		−	0.464102i		−	0.0204112i
$518$		0			0
$519$			24.8038i			1.08877i
$520$		0			0
$521$		−	13.6077i		−	0.596164i	−0.954540	−	0.298082i	$-0.903653\pi$
$521$		−	13.6077i		−	0.596164i	0.954540	−	0.298082i	$-0.0963469\pi$
$522$		0			0
$523$	−24.2487			−1.06032			−0.530161	−	0.847897i	$-0.677869\pi$
$523$	−24.2487			−1.06032			−0.530161	+	0.847897i	$0.677869\pi$
$524$		0			0
$525$	3.00000	−	3.46410i	0.130931	−	0.151186i
$526$		0			0
$527$			38.7846i			1.68948i
$528$		0			0
$529$	20.8564			0.906800
$530$		0			0
$531$		0			0
$532$		0			0
$533$	1.60770			0.0696370
$534$		0			0
$535$	−2.39230			−0.103428
$536$		0			0
$537$			11.0718i			0.477783i
$538$		0			0
$539$	1.85641	−	0.267949i	0.0799611	−	0.0115414i
$540$		0			0
$541$	−7.78461			−0.334687			−0.167343	−	0.985899i	$-0.553519\pi$
$541$	−7.78461			−0.334687			−0.167343	+	0.985899i	$0.553519\pi$
$542$		0			0
$543$			1.60770i			0.0689928i
$544$		0			0
$545$			2.07180i			0.0887460i
$546$		0			0
$547$			21.4641i			0.917739i	0.888504	+	0.458869i	$0.151745\pi$
$547$			21.4641i			0.917739i	−0.888504	+	0.458869i	$0.848255\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−47.5692			−2.02652
$552$		0			0
$553$	−29.3205	−	25.3923i	−1.24683	−	1.07979i
$554$		0			0
$555$			16.3923i			0.695815i
$556$		0			0
$557$	21.8564			0.926086			0.463043	−	0.886336i	$-0.346758\pi$
$557$	21.8564			0.926086			0.463043	+	0.886336i	$0.346758\pi$
$558$		0			0
$559$	0.928203			0.0392588
$560$		0			0
$561$	3.00000			0.126660
$562$		0			0
$563$	19.1769			0.808211			0.404105	−	0.914712i	$-0.367583\pi$
$563$	19.1769			0.808211			0.404105	+	0.914712i	$0.367583\pi$
$564$		0			0
$565$		−	5.46410i		−	0.229876i
$566$		0			0
$567$	−15.5885	+	18.0000i	−0.654654	+	0.755929i
$568$		0			0
$569$	7.07180			0.296465			0.148233	−	0.988953i	$-0.452642\pi$
$569$	7.07180			0.296465			0.148233	+	0.988953i	$0.452642\pi$
$570$		0			0
$571$		−	6.67949i		−	0.279528i	−0.990185	−	0.139764i	$-0.955366\pi$
$571$		−	6.67949i		−	0.279528i	0.990185	−	0.139764i	$-0.0446344\pi$
$572$		0			0
$573$			12.4641i			0.520695i
$574$		0			0
$575$			1.46410i			0.0610573i
$576$		0			0
$577$		−	19.3923i		−	0.807312i	−0.914911	−	0.403656i	$-0.867739\pi$
$577$		−	19.3923i		−	0.807312i	0.914911	−	0.403656i	$-0.132261\pi$
$578$		0			0
$579$	16.3923			0.681241
$580$		0			0
$581$	26.7846	−	30.9282i	1.11121	−	1.28312i
$582$		0			0
$583$		−	0.535898i		−	0.0221946i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−20.5359			−0.847607			−0.423804	−	0.905754i	$-0.639305\pi$
$587$	−20.5359			−0.847607			−0.423804	+	0.905754i	$0.639305\pi$
$588$		0			0
$589$	36.0000			1.48335
$590$		0			0
$591$	−23.0718			−0.949047
$592$		0			0
$593$		−	30.4641i		−	1.25101i	−0.780220	−	0.625505i	$-0.784893\pi$
$593$		−	30.4641i		−	1.25101i	0.780220	−	0.625505i	$-0.215107\pi$
$594$		0			0
$595$	11.1962	−	12.9282i	0.458997	−	0.530005i
$596$		0			0
$597$	6.00000			0.245564
$598$		0			0
$599$			10.1244i			0.413670i	0.978376	+	0.206835i	$0.0663163\pi$
$599$			10.1244i			0.413670i	−0.978376	+	0.206835i	$0.933684\pi$
$600$		0			0
$601$			14.7846i			0.603077i	0.953454	+	0.301538i	$0.0975001\pi$
$601$			14.7846i			0.603077i	−0.953454	+	0.301538i	$0.902500\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		−	10.9282i		−	0.444295i
$606$		0			0
$607$	29.1962			1.18504			0.592518	−	0.805557i	$-0.298134\pi$
$607$	29.1962			1.18504			0.592518	+	0.805557i	$0.298134\pi$
$608$		0			0
$609$	23.7846	−	27.4641i	0.963801	−	1.11290i
$610$		0			0
$611$			0.803848i			0.0325202i
$612$		0			0
$613$	10.0000			0.403896			0.201948	−	0.979396i	$-0.435273\pi$
$613$	10.0000			0.403896			0.201948	+	0.979396i	$0.435273\pi$
$614$		0			0
$615$	−6.00000			−0.241943
$616$		0			0
$617$	−22.9282			−0.923055			−0.461527	−	0.887126i	$-0.652698\pi$
$617$	−22.9282			−0.923055			−0.461527	+	0.887126i	$0.652698\pi$
$618$		0			0
$619$	0.679492			0.0273111			0.0136555	−	0.999907i	$-0.495653\pi$
$619$	0.679492			0.0273111			0.0136555	+	0.999907i	$0.495653\pi$
$620$		0			0
$621$		−	7.60770i		−	0.305286i
$622$		0			0
$623$	−5.07180	−	4.39230i	−0.203197	−	0.175974i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$		−	2.78461i		−	0.111207i
$628$		0			0
$629$			61.1769i			2.43928i
$630$		0			0
$631$		−	38.9090i		−	1.54894i	−0.632610	−	0.774471i	$-0.718016\pi$
$631$		−	38.9090i		−	1.54894i	0.632610	−	0.774471i	$-0.281984\pi$
$632$		0			0
$633$		−	5.53590i		−	0.220032i
$634$		0			0
$635$	15.4641			0.613674
$636$		0			0
$637$	−3.21539	+	0.464102i	−0.127398	+	0.0183884i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	8.92820			0.352643			0.176321	−	0.984333i	$-0.443580\pi$
$641$	8.92820			0.352643			0.176321	+	0.984333i	$0.443580\pi$
$642$		0			0
$643$	7.05256			0.278126			0.139063	−	0.990284i	$-0.455591\pi$
$643$	7.05256			0.278126			0.139063	+	0.990284i	$0.455591\pi$
$644$		0			0
$645$	−3.46410			−0.136399
$646$		0			0
$647$	10.3923			0.408564			0.204282	−	0.978912i	$-0.434514\pi$
$647$	10.3923			0.408564			0.204282	+	0.978912i	$0.434514\pi$
$648$		0			0
$649$		−	0.928203i		−	0.0364352i
$650$		0			0
$651$	−18.0000	+	20.7846i	−0.705476	+	0.814613i
$652$		0			0
$653$	−17.6077			−0.689042			−0.344521	−	0.938779i	$-0.611959\pi$
$653$	−17.6077			−0.689042			−0.344521	+	0.938779i	$0.611959\pi$
$654$		0			0
$655$		−	2.53590i		−	0.0990857i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	31.1962i		−	1.21523i	−0.794232	−	0.607615i	$-0.792126\pi$
$659$		−	31.1962i		−	1.21523i	0.794232	−	0.607615i	$-0.207874\pi$
$660$		0			0
$661$			39.7128i			1.54465i	0.635228	+	0.772325i	$0.280906\pi$
$661$			39.7128i			1.54465i	−0.635228	+	0.772325i	$0.719094\pi$
$662$		0			0
$663$	−5.19615			−0.201802
$664$		0			0
$665$	−12.0000	−	10.3923i	−0.465340	−	0.402996i
$666$		0			0
$667$			11.6077i			0.449452i
$668$		0			0
$669$	−23.7846			−0.919566
$670$		0			0
$671$	−2.53590			−0.0978973
$672$		0			0
$673$	−13.1769			−0.507933			−0.253966	−	0.967213i	$-0.581735\pi$
$673$	−13.1769			−0.507933			−0.253966	+	0.967213i	$0.581735\pi$
$674$		0			0
$675$	−5.19615			−0.200000
$676$		0			0
$677$		−	25.3923i		−	0.975906i	−0.872870	−	0.487953i	$-0.837744\pi$
$677$		−	25.3923i		−	0.975906i	0.872870	−	0.487953i	$-0.162256\pi$
$678$		0			0
$679$	26.7846	+	23.1962i	1.02790	+	0.890187i
$680$		0			0
$681$	−35.7846			−1.37127
$682$		0			0
$683$			7.32051i			0.280111i	0.990144	+	0.140056i	$0.0447282\pi$
$683$			7.32051i			0.280111i	−0.990144	+	0.140056i	$0.955272\pi$
$684$		0			0
$685$		−	20.3923i		−	0.779150i
$686$		0			0
$687$			14.7846i			0.564068i
$688$		0			0
$689$			0.928203i			0.0353617i
$690$		0			0
$691$	−22.6410			−0.861305			−0.430652	−	0.902518i	$-0.641717\pi$
$691$	−22.6410			−0.861305			−0.430652	+	0.902518i	$0.641717\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	6.92820i		−	0.262802i
$696$		0			0
$697$	−22.3923			−0.848169
$698$		0			0
$699$	15.7128			0.594313
$700$		0			0
$701$	37.7846			1.42711			0.713553	−	0.700602i	$-0.247085\pi$
$701$	37.7846			1.42711			0.713553	+	0.700602i	$0.247085\pi$
$702$		0			0
$703$	56.7846			2.14167
$704$		0			0
$705$		−	3.00000i		−	0.112987i
$706$		0			0
$707$	−30.9282	−	26.7846i	−1.16317	−	1.00734i
$708$		0			0
$709$	−21.0000			−0.788672			−0.394336	−	0.918966i	$-0.629025\pi$
$709$	−21.0000			−0.788672			−0.394336	+	0.918966i	$0.629025\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	8.78461i		−	0.328986i
$714$		0			0
$715$		−	0.124356i		−	0.00465064i
$716$		0			0
$717$		−	41.5359i		−	1.55119i
$718$		0			0
$719$	−38.5359			−1.43715			−0.718573	−	0.695451i	$-0.755205\pi$
$719$	−38.5359			−1.43715			−0.718573	+	0.695451i	$0.755205\pi$
$720$		0			0
$721$	11.7846	−	13.6077i	0.438882	−	0.506777i
$722$		0			0
$723$			28.3923i			1.05592i
$724$		0			0
$725$	7.92820			0.294446
$726$		0			0
$727$	13.6077			0.504681			0.252341	−	0.967638i	$-0.418800\pi$
$727$	13.6077			0.504681			0.252341	+	0.967638i	$0.418800\pi$
$728$		0			0
$729$	27.0000			1.00000
$730$		0			0
$731$	−12.9282			−0.478167
$732$		0			0
$733$			11.5359i			0.426088i	0.977043	+	0.213044i	$0.0683378\pi$
$733$			11.5359i			0.426088i	−0.977043	+	0.213044i	$0.931662\pi$
$734$		0			0
$735$	12.0000	−	1.73205i	0.442627	−	0.0638877i
$736$		0			0
$737$	0.928203			0.0341908
$738$		0			0
$739$		−	4.26795i		−	0.156999i	−0.996914	−	0.0784995i	$-0.974987\pi$
$739$		−	4.26795i		−	0.156999i	0.996914	−	0.0784995i	$-0.0250129\pi$
$740$		0			0
$741$			4.82309i			0.177180i
$742$		0			0
$743$		−	30.3923i		−	1.11499i	−0.830182	−	0.557493i	$-0.811763\pi$
$743$		−	30.3923i		−	1.11499i	0.830182	−	0.557493i	$-0.188237\pi$
$744$		0			0
$745$			3.07180i			0.112542i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−4.78461	−	4.14359i	−0.174826	−	0.151404i
$750$		0			0
$751$			25.5885i			0.933736i	0.884327	+	0.466868i	$0.154618\pi$
$751$			25.5885i			0.933736i	−0.884327	+	0.466868i	$0.845382\pi$
$752$		0			0
$753$	44.7846			1.63204
$754$		0			0
$755$	−15.1962			−0.553045
$756$		0			0
$757$	−37.8564			−1.37591			−0.687957	−	0.725751i	$-0.741492\pi$
$757$	−37.8564			−1.37591			−0.687957	+	0.725751i	$0.741492\pi$
$758$		0			0
$759$	−0.679492			−0.0246640
$760$		0			0
$761$			42.2487i			1.53151i	0.643130	+	0.765757i	$0.277636\pi$
$761$			42.2487i			1.53151i	−0.643130	+	0.765757i	$0.722364\pi$
$762$		0			0
$763$	−3.58846	+	4.14359i	−0.129911	+	0.150008i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			1.60770i			0.0580505i
$768$		0			0
$769$		−	18.0000i		−	0.649097i	−0.945869	−	0.324548i	$-0.894788\pi$
$769$		−	18.0000i		−	0.649097i	0.945869	−	0.324548i	$-0.105212\pi$
$770$		0			0
$771$			10.3923i			0.374270i
$772$		0			0
$773$		−	5.53590i		−	0.199112i	−0.995032	−	0.0995562i	$-0.968258\pi$
$773$		−	5.53590i		−	0.199112i	0.995032	−	0.0995562i	$-0.0317423\pi$
$774$		0			0
$775$	−6.00000			−0.215526
$776$		0			0
$777$	−28.3923	+	32.7846i	−1.01857	+	1.17614i
$778$		0			0
$779$			20.7846i			0.744686i
$780$		0			0
$781$	2.00000			0.0715656
$782$		0			0
$783$	−41.1962			−1.47223
$784$		0			0
$785$	7.85641			0.280407
$786$		0			0
$787$	32.6603			1.16421			0.582106	−	0.813113i	$-0.302229\pi$
$787$	32.6603			1.16421			0.582106	+	0.813113i	$0.302229\pi$
$788$		0			0
$789$		−	19.8564i		−	0.706907i
$790$		0			0
$791$	9.46410	−	10.9282i	0.336505	−	0.388562i
$792$		0			0
$793$	4.39230			0.155975
$794$		0			0
$795$		−	3.46410i		−	0.122859i
$796$		0			0
$797$		−	22.1769i		−	0.785547i	−0.919635	−	0.392773i	$-0.871516\pi$
$797$		−	22.1769i		−	0.785547i	0.919635	−	0.392773i	$-0.128484\pi$
$798$		0			0
$799$		−	11.1962i		−	0.396091i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	3.46410			0.122245
$804$		0			0
$805$	−2.53590	+	2.92820i	−0.0893787	+	0.103206i
$806$		0			0
$807$			20.7846i			0.731653i
$808$		0			0
$809$	7.92820			0.278741			0.139370	−	0.990240i	$-0.455492\pi$
$809$	7.92820			0.278741			0.139370	+	0.990240i	$0.455492\pi$
$810$		0			0
$811$	29.0718			1.02085			0.510424	−	0.859923i	$-0.329488\pi$
$811$	29.0718			1.02085			0.510424	+	0.859923i	$0.329488\pi$
$812$		0			0
$813$	16.3923			0.574903
$814$		0			0
$815$	−20.7846			−0.728053
$816$		0			0
$817$			12.0000i			0.419827i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−27.7846			−0.969690			−0.484845	−	0.874600i	$-0.661124\pi$
$821$	−27.7846			−0.969690			−0.484845	+	0.874600i	$0.661124\pi$
$822$		0			0
$823$		−	11.0718i		−	0.385939i	−0.981205	−	0.192969i	$-0.938188\pi$
$823$		−	11.0718i		−	0.385939i	0.981205	−	0.192969i	$-0.0618118\pi$
$824$		0			0
$825$			0.464102i			0.0161579i
$826$		0			0
$827$			14.1436i			0.491821i	0.969293	+	0.245910i	$0.0790869\pi$
$827$			14.1436i			0.491821i	−0.969293	+	0.245910i	$0.920913\pi$
$828$		0			0
$829$			0.248711i			0.00863810i	0.999991	+	0.00431905i	$0.00137480\pi$
$829$			0.248711i			0.00863810i	−0.999991	+	0.00431905i	$0.998625\pi$
$830$		0			0
$831$	29.0718			1.00849
$832$		0			0
$833$	44.7846	−	6.46410i	1.55169	−	0.223968i
$834$		0			0
$835$		−	5.19615i		−	0.179820i
$836$		0			0
$837$	31.1769			1.07763
$838$		0			0
$839$	−4.39230			−0.151639			−0.0758196	−	0.997122i	$-0.524157\pi$
$839$	−4.39230			−0.151639			−0.0758196	+	0.997122i	$0.524157\pi$
$840$		0			0
$841$	33.8564			1.16746
$842$		0			0
$843$	13.7321			0.472957
$844$		0			0
$845$		−	12.7846i		−	0.439804i
$846$		0			0
$847$	18.9282	−	21.8564i	0.650381	−	0.750995i
$848$		0			0
$849$	−21.0000			−0.720718
$850$		0			0
$851$		−	13.8564i		−	0.474991i
$852$		0			0
$853$		−	2.78461i		−	0.0953432i	−0.998863	−	0.0476716i	$-0.984820\pi$
$853$		−	2.78461i		−	0.0953432i	0.998863	−	0.0476716i	$-0.0151801\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	57.7128i		−	1.97143i	−0.168416	−	0.985716i	$-0.553865\pi$
$857$		−	57.7128i		−	1.97143i	0.168416	−	0.985716i	$-0.446135\pi$
$858$		0			0
$859$	39.7128			1.35498			0.677492	−	0.735530i	$-0.263067\pi$
$859$	39.7128			1.35498			0.677492	+	0.735530i	$0.263067\pi$
$860$		0			0
$861$	−12.0000	−	10.3923i	−0.408959	−	0.354169i
$862$		0			0
$863$			39.8564i			1.35673i	0.734726	+	0.678364i	$0.237311\pi$
$863$			39.8564i			1.35673i	−0.734726	+	0.678364i	$0.762689\pi$
$864$		0			0
$865$	−14.3205			−0.486912
$866$		0			0
$867$	42.9282			1.45792
$868$		0			0
$869$	3.92820			0.133255
$870$		0			0
$871$	−1.60770			−0.0544747
$872$		0			0
$873$		0			0
$874$		0			0
$875$	2.00000	+	1.73205i	0.0676123	+	0.0585540i
$876$		0			0
$877$	18.3923			0.621064			0.310532	−	0.950563i	$-0.399493\pi$
$877$	18.3923			0.621064			0.310532	+	0.950563i	$0.399493\pi$
$878$		0			0
$879$		−	35.1962i		−	1.18714i
$880$		0			0
$881$			29.0718i			0.979454i	0.871876	+	0.489727i	$0.162904\pi$
$881$			29.0718i			0.979454i	−0.871876	+	0.489727i	$0.837096\pi$
$882$		0			0
$883$		−	23.6077i		−	0.794462i	−0.917719	−	0.397231i	$-0.869971\pi$
$883$		−	23.6077i		−	0.794462i	0.917719	−	0.397231i	$-0.130029\pi$
$884$		0			0
$885$		−	6.00000i		−	0.201688i
$886$		0			0
$887$	8.53590			0.286607			0.143304	−	0.989679i	$-0.454227\pi$
$887$	8.53590			0.286607			0.143304	+	0.989679i	$0.454227\pi$
$888$		0			0
$889$	30.9282	+	26.7846i	1.03730	+	0.898327i
$890$		0			0
$891$		−	2.41154i		−	0.0807897i
$892$		0			0
$893$	−10.3923			−0.347765
$894$		0			0
$895$	−6.39230			−0.213671
$896$		0			0
$897$	1.17691			0.0392960
$898$		0			0
$899$	−47.5692			−1.58652
$900$		0			0
$901$		−	12.9282i		−	0.430701i
$902$		0			0
$903$	−6.92820	−	6.00000i	−0.230556	−	0.199667i
$904$		0			0
$905$	−0.928203			−0.0308545
$906$		0			0
$907$		−	32.3923i		−	1.07557i	−0.843082	−	0.537784i	$-0.819261\pi$
$907$		−	32.3923i		−	1.07557i	0.843082	−	0.537784i	$-0.180739\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			20.2487i			0.670870i	0.942063	+	0.335435i	$0.108883\pi$
$911$			20.2487i			0.670870i	−0.942063	+	0.335435i	$0.891117\pi$
$912$		0			0
$913$			4.14359i			0.137133i
$914$		0			0
$915$	−16.3923			−0.541913
$916$		0			0
$917$	4.39230	−	5.07180i	0.145047	−	0.167485i
$918$		0			0
$919$			6.41154i			0.211497i	0.994393	+	0.105749i	$0.0337239\pi$
$919$			6.41154i			0.211497i	−0.994393	+	0.105749i	$0.966276\pi$
$920$		0			0
$921$	−3.00000			−0.0988534
$922$		0			0
$923$	−3.46410			−0.114022
$924$		0			0
$925$	−9.46410			−0.311178
$926$		0			0
$927$		0			0
$928$		0			0
$929$			54.4974i			1.78800i	0.448065	+	0.894001i	$0.352113\pi$
$929$			54.4974i			1.78800i	−0.448065	+	0.894001i	$0.647887\pi$
$930$		0			0
$931$	−6.00000	−	41.5692i	−0.196642	−	1.36238i
$932$		0			0
$933$	13.6077			0.445496
$934$		0			0
$935$			1.73205i			0.0566441i
$936$		0			0
$937$			25.3923i			0.829530i	0.909928	+	0.414765i	$0.136136\pi$
$937$			25.3923i			0.829530i	−0.909928	+	0.414765i	$0.863864\pi$
$938$		0			0
$939$		−	30.3731i		−	0.991188i
$940$		0			0
$941$			39.0333i			1.27245i	0.771504	+	0.636225i	$0.219505\pi$
$941$			39.0333i			1.27245i	−0.771504	+	0.636225i	$0.780495\pi$
$942$		0			0
$943$	5.07180			0.165160
$944$		0			0
$945$	−10.3923	−	9.00000i	−0.338062	−	0.292770i
$946$		0			0
$947$			22.7846i			0.740400i	0.928952	+	0.370200i	$0.120711\pi$
$947$			22.7846i			0.740400i	−0.928952	+	0.370200i	$0.879289\pi$
$948$		0			0
$949$	−6.00000			−0.194768
$950$		0			0
$951$	−5.32051			−0.172529
$952$		0			0
$953$	21.3205			0.690639			0.345320	−	0.938485i	$-0.387771\pi$
$953$	21.3205			0.690639			0.345320	+	0.938485i	$0.387771\pi$
$954$		0			0
$955$	−7.19615			−0.232862
$956$		0			0
$957$			3.67949i			0.118941i
$958$		0			0
$959$	35.3205	−	40.7846i	1.14056	−	1.31700i
$960$		0			0
$961$	5.00000			0.161290
$962$		0			0
$963$		0			0
$964$		0			0
$965$			9.46410i			0.304660i
$966$		0			0
$967$		−	31.1769i		−	1.00258i	−0.865279	−	0.501291i	$-0.832859\pi$
$967$		−	31.1769i		−	1.00258i	0.865279	−	0.501291i	$-0.167141\pi$
$968$		0			0
$969$		−	67.1769i		−	2.15803i
$970$		0			0
$971$	14.7846			0.474461			0.237230	−	0.971453i	$-0.423760\pi$
$971$	14.7846			0.474461			0.237230	+	0.971453i	$0.423760\pi$
$972$		0			0
$973$	12.0000	−	13.8564i	0.384702	−	0.444216i
$974$		0			0
$975$		−	0.803848i		−	0.0257437i
$976$		0			0
$977$	−4.24871			−0.135928			−0.0679642	−	0.997688i	$-0.521650\pi$
$977$	−4.24871			−0.135928			−0.0679642	+	0.997688i	$0.521650\pi$
$978$		0			0
$979$	0.679492			0.0217167
$980$		0			0
$981$		0			0
$982$		0			0
$983$	27.5885			0.879935			0.439968	−	0.898014i	$-0.354990\pi$
$983$	27.5885			0.879935			0.439968	+	0.898014i	$0.354990\pi$
$984$		0			0
$985$		−	13.3205i		−	0.424427i
$986$		0			0
$987$	5.19615	−	6.00000i	0.165395	−	0.190982i
$988$		0			0
$989$	2.92820			0.0931114
$990$		0			0
$991$		−	47.1769i		−	1.49862i	−0.662217	−	0.749312i	$-0.730384\pi$
$991$		−	47.1769i		−	1.49862i	0.662217	−	0.749312i	$-0.269616\pi$
$992$		0			0
$993$			45.7128i			1.45065i
$994$		0			0
$995$			3.46410i			0.109819i
$996$		0			0
$997$			5.53590i			0.175324i	0.996150	+	0.0876618i	$0.0279395\pi$
$997$			5.53590i			0.175324i	−0.996150	+	0.0876618i	$0.972061\pi$
$998$		0			0
$999$	49.1769			1.55589

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2240.2.k.b.1791.1		4
4.3	odd	2		2240.2.k.a.1791.3		4
7.6	odd	2		2240.2.k.a.1791.4		4
8.3	odd	2		140.2.g.a.111.4	yes	4
8.5	even	2		140.2.g.b.111.3	yes	4
24.5	odd	2		1260.2.c.b.811.2		4
24.11	even	2		1260.2.c.a.811.1		4
28.27	even	2	inner	2240.2.k.b.1791.2		4
40.3	even	4		700.2.c.e.699.3		4
40.13	odd	4		700.2.c.c.699.1		4
40.19	odd	2		700.2.g.f.251.1		4
40.27	even	4		700.2.c.b.699.2		4
40.29	even	2		700.2.g.g.251.2		4
40.37	odd	4		700.2.c.f.699.4		4
56.3	even	6		980.2.o.b.411.1		4
56.5	odd	6		980.2.o.a.31.1		4
56.11	odd	6		980.2.o.a.411.1		4
56.13	odd	2		140.2.g.a.111.3	✓	4
56.19	even	6		980.2.o.d.31.1		4
56.27	even	2		140.2.g.b.111.4	yes	4
56.37	even	6		980.2.o.b.31.1		4
56.45	odd	6		980.2.o.c.411.2		4
56.51	odd	6		980.2.o.c.31.1		4
56.53	even	6		980.2.o.d.411.2		4
168.83	odd	2		1260.2.c.b.811.1		4
168.125	even	2		1260.2.c.a.811.2		4
280.13	even	4		700.2.c.b.699.1		4
280.27	odd	4		700.2.c.c.699.2		4
280.69	odd	2		700.2.g.f.251.2		4
280.83	odd	4		700.2.c.f.699.3		4
280.139	even	2		700.2.g.g.251.1		4
280.237	even	4		700.2.c.e.699.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
140.2.g.a.111.3	✓	4	56.13	odd	2
140.2.g.a.111.4	yes	4	8.3	odd	2
140.2.g.b.111.3	yes	4	8.5	even	2
140.2.g.b.111.4	yes	4	56.27	even	2
700.2.c.b.699.1		4	280.13	even	4
700.2.c.b.699.2		4	40.27	even	4
700.2.c.c.699.1		4	40.13	odd	4
700.2.c.c.699.2		4	280.27	odd	4
700.2.c.e.699.3		4	40.3	even	4
700.2.c.e.699.4		4	280.237	even	4
700.2.c.f.699.3		4	280.83	odd	4
700.2.c.f.699.4		4	40.37	odd	4
700.2.g.f.251.1		4	40.19	odd	2
700.2.g.f.251.2		4	280.69	odd	2
700.2.g.g.251.1		4	280.139	even	2
700.2.g.g.251.2		4	40.29	even	2
980.2.o.a.31.1		4	56.5	odd	6
980.2.o.a.411.1		4	56.11	odd	6
980.2.o.b.31.1		4	56.37	even	6
980.2.o.b.411.1		4	56.3	even	6
980.2.o.c.31.1		4	56.51	odd	6
980.2.o.c.411.2		4	56.45	odd	6
980.2.o.d.31.1		4	56.19	even	6
980.2.o.d.411.2		4	56.53	even	6
1260.2.c.a.811.1		4	24.11	even	2
1260.2.c.a.811.2		4	168.125	even	2
1260.2.c.b.811.1		4	168.83	odd	2
1260.2.c.b.811.2		4	24.5	odd	2
2240.2.k.a.1791.3		4	4.3	odd	2
2240.2.k.a.1791.4		4	7.6	odd	2
2240.2.k.b.1791.1		4	1.1	even	1	trivial
2240.2.k.b.1791.2		4	28.27	even	2	inner

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2240.k (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.73205 q^{3} -1.00000i q^{5} +(1.73205 - 2.00000i) q^{7} +0.267949i q^{11} -0.464102i q^{13} +1.73205i q^{15} +6.46410i q^{17} +6.00000 q^{19} +(-3.00000 + 3.46410i) q^{21} -1.46410i q^{23} -1.00000 q^{25} +5.19615 q^{27} -7.92820 q^{29} +6.00000 q^{31} -0.464102i q^{33} +(-2.00000 - 1.73205i) q^{35} +9.46410 q^{37} +0.803848i q^{39} +3.46410i q^{41} +2.00000i q^{43} -1.73205 q^{47} +(-1.00000 - 6.92820i) q^{49} -11.1962i q^{51} -2.00000 q^{53} +0.267949 q^{55} -10.3923 q^{57} -3.46410 q^{59} +9.46410i q^{61} -0.464102 q^{65} -3.46410i q^{67} +2.53590i q^{69} -7.46410i q^{71} -12.9282i q^{73} +1.73205 q^{75} +(0.535898 + 0.464102i) q^{77} -14.6603i q^{79} -9.00000 q^{81} +15.4641 q^{83} +6.46410 q^{85} +13.7321 q^{87} -2.53590i q^{89} +(-0.928203 - 0.803848i) q^{91} -10.3923 q^{93} -6.00000i q^{95} +13.3923i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 24 q^{19} - 12 q^{21} - 4 q^{25} - 4 q^{29} + 24 q^{31} - 8 q^{35} + 24 q^{37} - 4 q^{49} - 8 q^{53} + 8 q^{55} + 12 q^{65} + 16 q^{77} - 36 q^{81} + 48 q^{83} + 12 q^{85} + 48 q^{87} + 24 q^{91}+O(q^{100}) \) 4 * q + 24 * q^19 - 12 * q^21 - 4 * q^25 - 4 * q^29 + 24 * q^31 - 8 * q^35 + 24 * q^37 - 4 * q^49 - 8 * q^53 + 8 * q^55 + 12 * q^65 + 16 * q^77 - 36 * q^81 + 48 * q^83 + 12 * q^85 + 48 * q^87 + 24 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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