LMFDB - Embedded newform 2240.2.g.m.449.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2240, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))

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

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

chi := DirichletCharacter("2240.449");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$17.8864900528$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.5161984.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 4x^{3} + 25x^{2} - 20x + 8 $ x^6 - 4x^3 + 25x^2 - 20*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 280)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.3
Root			$-1.75233 + 1.75233i$ of defining polynomial
Character	$\chi$	$=$	2240.449
Dual form			2240.2.g.m.449.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.363328i q^{3} +(1.75233 - 1.38900i) q^{5} -1.00000i q^{7} +2.86799 q^{9} +O(q^{10})$	$q-0.363328i q^{3} +(1.75233 - 1.38900i) q^{5} -1.00000i q^{7} +2.86799 q^{9} +5.14134 q^{11} -4.64600i q^{13} +(-0.504664 - 0.636672i) q^{15} +3.86799i q^{17} +0.778008 q^{19} -0.363328 q^{21} +5.00933i q^{23} +(1.14134 - 4.86799i) q^{25} -2.13201i q^{27} +9.42401 q^{29} -4.72666 q^{31} -1.86799i q^{33} +(-1.38900 - 1.75233i) q^{35} -6.00000i q^{37} -1.68802 q^{39} -1.00933 q^{41} +7.00933i q^{43} +(5.02568 - 3.98365i) q^{45} +11.4240i q^{47} -1.00000 q^{49} +1.40535 q^{51} +7.55602i q^{53} +(9.00933 - 7.14134i) q^{55} -0.282672i q^{57} -12.5140 q^{59} -11.5047 q^{61} -2.86799i q^{63} +(-6.45331 - 8.14134i) q^{65} -11.7360i q^{67} +1.82003 q^{69} -2.72666 q^{71} -5.00933i q^{73} +(-1.76868 - 0.414680i) q^{75} -5.14134i q^{77} +5.68802 q^{79} +7.82936 q^{81} +4.67531i q^{83} +(5.37266 + 6.77801i) q^{85} -3.42401i q^{87} +2.82936 q^{89} -4.64600 q^{91} +1.71733i q^{93} +(1.36333 - 1.08066i) q^{95} -1.58532i q^{97} +14.7453 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 8 q^{9}+O(q^{10}) $ 6 * q - 8 * q^9	$ 6 q - 8 q^{9} + 14 q^{11} + 18 q^{15} - 8 q^{19} + 2 q^{21} - 10 q^{25} + 6 q^{29} - 20 q^{31} - 2 q^{35} - 10 q^{39} + 36 q^{41} + 28 q^{45} - 6 q^{49} + 42 q^{51} + 12 q^{55} - 12 q^{59} - 48 q^{61} - 22 q^{65} + 36 q^{69} - 8 q^{71} - 40 q^{75} + 34 q^{79} + 30 q^{81} - 14 q^{85} + 10 q^{91} + 4 q^{95} - 4 q^{99}+O(q^{100}) $ 6 * q - 8 * q^9 + 14 * q^11 + 18 * q^15 - 8 * q^19 + 2 * q^21 - 10 * q^25 + 6 * q^29 - 20 * q^31 - 2 * q^35 - 10 * q^39 + 36 * q^41 + 28 * q^45 - 6 * q^49 + 42 * q^51 + 12 * q^55 - 12 * q^59 - 48 * q^61 - 22 * q^65 + 36 * q^69 - 8 * q^71 - 40 * q^75 + 34 * q^79 + 30 * q^81 - 14 * q^85 + 10 * q^91 + 4 * q^95 - 4 * q^99

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
$3$		−	0.363328i		−	0.209768i	−0.994484	−	0.104884i	$-0.966553\pi$
$3$		−	0.363328i		−	0.209768i	0.994484	−	0.104884i	$-0.0334471\pi$
$4$		0			0
$5$	1.75233	−	1.38900i	0.783667	−	0.621181i
$5$	1.75233	−	1.38900i	0.783667	−	0.621181i
$6$		0			0
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$		0			0
$9$	2.86799			0.955998
$10$		0			0
$11$	5.14134			1.55017			0.775086	−	0.631856i	$-0.217707\pi$
$11$	5.14134			1.55017			0.775086	+	0.631856i	$0.217707\pi$
$12$		0			0
$13$		−	4.64600i		−	1.28857i	−0.764786	−	0.644284i	$-0.777155\pi$
$13$		−	4.64600i		−	1.28857i	0.764786	−	0.644284i	$-0.222845\pi$
$14$		0			0
$15$	−0.504664	−	0.636672i	−0.130304	−	0.164388i
$16$		0			0
$17$			3.86799i			0.938126i	0.883165	+	0.469063i	$0.155408\pi$
$17$			3.86799i			0.938126i	−0.883165	+	0.469063i	$0.844592\pi$
$18$		0			0
$19$	0.778008			0.178487			0.0892436	−	0.996010i	$-0.471555\pi$
$19$	0.778008			0.178487			0.0892436	+	0.996010i	$0.471555\pi$
$20$		0			0
$21$	−0.363328			−0.0792847
$22$		0			0
$23$			5.00933i			1.04452i	0.852787	+	0.522259i	$0.174910\pi$
$23$			5.00933i			1.04452i	−0.852787	+	0.522259i	$0.825090\pi$
$24$		0			0
$25$	1.14134	−	4.86799i	0.228267	−	0.973599i
$26$		0			0
$27$		−	2.13201i		−	0.410305i
$28$		0			0
$29$	9.42401			1.74999			0.874997	−	0.484128i	$-0.160863\pi$
$29$	9.42401			1.74999			0.874997	+	0.484128i	$0.160863\pi$
$30$		0			0
$31$	−4.72666			−0.848933			−0.424466	−	0.905444i	$-0.639538\pi$
$31$	−4.72666			−0.848933			−0.424466	+	0.905444i	$0.639538\pi$
$32$		0			0
$33$		−	1.86799i		−	0.325176i
$34$		0			0
$35$	−1.38900	−	1.75233i	−0.234785	−	0.296198i
$36$		0			0
$37$		−	6.00000i		−	0.986394i	−0.869918	−	0.493197i	$-0.835828\pi$
$37$		−	6.00000i		−	0.986394i	0.869918	−	0.493197i	$-0.164172\pi$
$38$		0			0
$39$	−1.68802			−0.270300
$40$		0			0
$41$	−1.00933			−0.157631			−0.0788153	−	0.996889i	$-0.525114\pi$
$41$	−1.00933			−0.157631			−0.0788153	+	0.996889i	$0.525114\pi$
$42$		0			0
$43$			7.00933i			1.06891i	0.845196	+	0.534456i	$0.179484\pi$
$43$			7.00933i			1.06891i	−0.845196	+	0.534456i	$0.820516\pi$
$44$		0			0
$45$	5.02568	−	3.98365i	0.749184	−	0.593848i
$46$		0			0
$47$			11.4240i			1.66636i	0.553000	+	0.833181i	$0.313483\pi$
$47$			11.4240i			1.66636i	−0.553000	+	0.833181i	$0.686517\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	1.40535			0.196788
$52$		0			0
$53$			7.55602i			1.03790i	0.854805	+	0.518949i	$0.173677\pi$
$53$			7.55602i			1.03790i	−0.854805	+	0.518949i	$0.826323\pi$
$54$		0			0
$55$	9.00933	−	7.14134i	1.21482	−	0.962938i
$56$		0			0
$57$		−	0.282672i		−	0.0374409i
$58$		0			0
$59$	−12.5140			−1.62918			−0.814592	−	0.580035i	$-0.803039\pi$
$59$	−12.5140			−1.62918			−0.814592	+	0.580035i	$0.803039\pi$
$60$		0			0
$61$	−11.5047			−1.47302			−0.736511	−	0.676426i	$-0.763528\pi$
$61$	−11.5047			−1.47302			−0.736511	+	0.676426i	$0.763528\pi$
$62$		0			0
$63$		−	2.86799i		−	0.361333i
$64$		0			0
$65$	−6.45331	−	8.14134i	−0.800435	−	1.00981i
$66$		0			0
$67$		−	11.7360i		−	1.43378i	−0.697187	−	0.716889i	$-0.745565\pi$
$67$		−	11.7360i		−	1.43378i	0.697187	−	0.716889i	$-0.254435\pi$
$68$		0			0
$69$	1.82003			0.219106
$70$		0			0
$71$	−2.72666			−0.323595			−0.161797	−	0.986824i	$-0.551729\pi$
$71$	−2.72666			−0.323595			−0.161797	+	0.986824i	$0.551729\pi$
$72$		0			0
$73$		−	5.00933i		−	0.586298i	−0.956067	−	0.293149i	$-0.905297\pi$
$73$		−	5.00933i		−	0.586298i	0.956067	−	0.293149i	$-0.0947031\pi$
$74$		0			0
$75$	−1.76868	−	0.414680i	−0.204229	−	0.0478831i
$76$		0			0
$77$		−	5.14134i		−	0.585910i
$78$		0			0
$79$	5.68802			0.639953			0.319976	−	0.947426i	$-0.396325\pi$
$79$	5.68802			0.639953			0.319976	+	0.947426i	$0.396325\pi$
$80$		0			0
$81$	7.82936			0.869929
$82$		0			0
$83$			4.67531i			0.513181i	0.966520	+	0.256591i	$0.0825992\pi$
$83$			4.67531i			0.513181i	−0.966520	+	0.256591i	$0.917401\pi$
$84$		0			0
$85$	5.37266	+	6.77801i	0.582746	+	0.735178i
$86$		0			0
$87$		−	3.42401i		−	0.367092i
$88$		0			0
$89$	2.82936			0.299911			0.149956	−	0.988693i	$-0.452087\pi$
$89$	2.82936			0.299911			0.149956	+	0.988693i	$0.452087\pi$
$90$		0			0
$91$	−4.64600			−0.487033
$92$		0			0
$93$			1.71733i			0.178079i
$94$		0			0
$95$	1.36333	−	1.08066i	0.139875	−	0.110873i
$96$		0			0
$97$		−	1.58532i		−	0.160965i	−0.996756	−	0.0804824i	$-0.974354\pi$
$97$		−	1.58532i		−	0.160965i	0.996756	−	0.0804824i	$-0.0256461\pi$
$98$		0			0
$99$	14.7453			1.48196
$100$		0			0
$101$	9.06068			0.901571			0.450786	−	0.892632i	$-0.351144\pi$
$101$	9.06068			0.901571			0.450786	+	0.892632i	$0.351144\pi$
$102$		0			0
$103$			5.14134i			0.506591i	0.967389	+	0.253295i	$0.0815145\pi$
$103$			5.14134i			0.506591i	−0.967389	+	0.253295i	$0.918486\pi$
$104$		0			0
$105$	−0.636672	+	0.504664i	−0.0621328	+	0.0492502i
$106$		0			0
$107$		0			0		1.00000			$0$
$107$		0			0		−1.00000			$\pi$
$108$		0			0
$109$	−3.97070			−0.380324			−0.190162	−	0.981753i	$-0.560901\pi$
$109$	−3.97070			−0.380324			−0.190162	+	0.981753i	$0.560901\pi$
$110$		0			0
$111$	−2.17997			−0.206914
$112$		0			0
$113$		−	4.54669i		−	0.427716i	−0.976865	−	0.213858i	$-0.931397\pi$
$113$		−	4.54669i		−	0.427716i	0.976865	−	0.213858i	$-0.0686030\pi$
$114$		0			0
$115$	6.95798	+	8.77801i	0.648835	+	0.818553i
$116$		0			0
$117$		−	13.3247i		−	1.23187i
$118$		0			0
$119$	3.86799			0.354578
$120$		0			0
$121$	15.4333			1.40303
$122$		0			0
$123$			0.366718i			0.0330658i
$124$		0			0
$125$	−4.76166	−	10.1157i	−0.425896	−	0.904772i
$126$		0			0
$127$		−	16.1214i		−	1.43054i	−0.698849	−	0.715270i	$-0.746304\pi$
$127$		−	16.1214i		−	1.43054i	0.698849	−	0.715270i	$-0.253696\pi$
$128$		0			0
$129$	2.54669			0.224223
$130$		0			0
$131$	0.0513514			0.00448659			0.00224330	−	0.999997i	$-0.499286\pi$
$131$	0.0513514			0.00448659			0.00224330	+	0.999997i	$0.499286\pi$
$132$		0			0
$133$		−	0.778008i		−	0.0674618i
$134$		0			0
$135$	−2.96137	−	3.73599i	−0.254874	−	0.321542i
$136$		0			0
$137$		−	14.0187i		−	1.19769i	−0.800863	−	0.598847i	$-0.795626\pi$
$137$		−	14.0187i		−	1.19769i	0.800863	−	0.598847i	$-0.204374\pi$
$138$		0			0
$139$	−12.6167			−1.07013			−0.535067	−	0.844810i	$-0.679714\pi$
$139$	−12.6167			−1.07013			−0.535067	+	0.844810i	$0.679714\pi$
$140$		0			0
$141$	4.15066			0.349549
$142$		0			0
$143$		−	23.8867i		−	1.99750i
$144$		0			0
$145$	16.5140	−	13.0900i	1.37141	−	1.08706i
$146$		0			0
$147$			0.363328i			0.0299668i
$148$		0			0
$149$	−11.4533			−0.938292			−0.469146	−	0.883121i	$-0.655438\pi$
$149$	−11.4533			−0.938292			−0.469146	+	0.883121i	$0.655438\pi$
$150$		0			0
$151$	−5.86799			−0.477530			−0.238765	−	0.971077i	$-0.576743\pi$
$151$	−5.86799			−0.477530			−0.238765	+	0.971077i	$0.576743\pi$
$152$		0			0
$153$			11.0934i			0.896846i
$154$		0			0
$155$	−8.28267	+	6.56534i	−0.665280	+	0.527341i
$156$		0			0
$157$			5.78734i			0.461880i	0.972968	+	0.230940i	$0.0741801\pi$
$157$			5.78734i			0.461880i	−0.972968	+	0.230940i	$0.925820\pi$
$158$		0			0
$159$	2.74531			0.217718
$160$		0			0
$161$	5.00933			0.394790
$162$		0			0
$163$		−	3.27334i		−	0.256388i	−0.991749	−	0.128194i	$-0.959082\pi$
$163$		−	3.27334i		−	0.256388i	0.991749	−	0.128194i	$-0.0409180\pi$
$164$		0			0
$165$	−2.59465	−	3.27334i	−0.201993	−	0.254829i
$166$		0			0
$167$			24.8773i			1.92506i	0.271166	+	0.962532i	$0.412591\pi$
$167$			24.8773i			1.92506i	−0.271166	+	0.962532i	$0.587409\pi$
$168$		0			0
$169$	−8.58532			−0.660409
$170$		0			0
$171$	2.23132			0.170633
$172$		0			0
$173$			6.62734i			0.503868i	0.967744	+	0.251934i	$0.0810665\pi$
$173$			6.62734i			0.503868i	−0.967744	+	0.251934i	$0.918933\pi$
$174$		0			0
$175$	−4.86799	−	1.14134i	−0.367986	−	0.0862769i
$176$		0			0
$177$			4.54669i			0.341750i
$178$		0			0
$179$	10.0187			0.748830			0.374415	−	0.927261i	$-0.377844\pi$
$179$	10.0187			0.748830			0.374415	+	0.927261i	$0.377844\pi$
$180$		0			0
$181$	−1.78734			−0.132852			−0.0664258	−	0.997791i	$-0.521160\pi$
$181$	−1.78734			−0.132852			−0.0664258	+	0.997791i	$0.521160\pi$
$182$		0			0
$183$			4.17997i			0.308992i
$184$		0			0
$185$	−8.33402	−	10.5140i	−0.612730	−	0.773004i
$186$		0			0
$187$			19.8867i			1.45426i
$188$		0			0
$189$	−2.13201			−0.155081
$190$		0			0
$191$	8.59465			0.621887			0.310943	−	0.950428i	$-0.399355\pi$
$191$	8.59465			0.621887			0.310943	+	0.950428i	$0.399355\pi$
$192$		0			0
$193$			5.17064i			0.372191i	0.982532	+	0.186095i	$0.0595834\pi$
$193$			5.17064i			0.372191i	−0.982532	+	0.186095i	$0.940417\pi$
$194$		0			0
$195$	−2.95798	+	2.34467i	−0.211825	+	0.167905i
$196$		0			0
$197$		−	23.9160i		−	1.70394i	−0.523590	−	0.851971i	$-0.675407\pi$
$197$		−	23.9160i		−	1.70394i	0.523590	−	0.851971i	$-0.324593\pi$
$198$		0			0
$199$	−15.3107			−1.08534			−0.542672	−	0.839945i	$-0.682587\pi$
$199$	−15.3107			−1.08534			−0.542672	+	0.839945i	$0.682587\pi$
$200$		0			0
$201$	−4.26401			−0.300760
$202$		0			0
$203$		−	9.42401i		−	0.661436i
$204$		0			0
$205$	−1.76868	+	1.40196i	−0.123530	+	0.0979172i
$206$		0			0
$207$			14.3667i			0.998556i
$208$		0			0
$209$	4.00000			0.276686
$210$		0			0
$211$	−1.03863			−0.0715025			−0.0357512	−	0.999361i	$-0.511382\pi$
$211$	−1.03863			−0.0715025			−0.0357512	+	0.999361i	$0.511382\pi$
$212$		0			0
$213$			0.990671i			0.0678797i
$214$		0			0
$215$	9.73599	+	12.2827i	0.663989	+	0.837671i
$216$		0			0
$217$			4.72666i			0.320866i
$218$		0			0
$219$	−1.82003			−0.122986
$220$		0			0
$221$	17.9707			1.20884
$222$		0			0
$223$		−	9.86799i		−	0.660810i	−0.943839	−	0.330405i	$-0.892815\pi$
$223$		−	9.86799i		−	0.660810i	0.943839	−	0.330405i	$-0.107185\pi$
$224$		0			0
$225$	3.27334	−	13.9614i	0.218223	−	0.930758i
$226$		0			0
$227$		−	21.6553i		−	1.43731i	−0.695364	−	0.718657i	$-0.744757\pi$
$227$		−	21.6553i		−	1.43731i	0.695364	−	0.718657i	$-0.255243\pi$
$228$		0			0
$229$	20.2313			1.33692			0.668462	−	0.743747i	$-0.266953\pi$
$229$	20.2313			1.33692			0.668462	+	0.743747i	$0.266953\pi$
$230$		0			0
$231$	−1.86799			−0.122905
$232$		0			0
$233$			5.45331i			0.357258i	0.983916	+	0.178629i	$0.0571663\pi$
$233$			5.45331i			0.357258i	−0.983916	+	0.178629i	$0.942834\pi$
$234$		0			0
$235$	15.8680	+	20.0187i	1.03511	+	1.30587i
$236$		0			0
$237$		−	2.06662i		−	0.134241i
$238$		0			0
$239$	−11.5853			−0.749392			−0.374696	−	0.927148i	$-0.622253\pi$
$239$	−11.5853			−0.749392			−0.374696	+	0.927148i	$0.622253\pi$
$240$		0			0
$241$	−5.00933			−0.322679			−0.161340	−	0.986899i	$-0.551581\pi$
$241$	−5.00933			−0.322679			−0.161340	+	0.986899i	$0.551581\pi$
$242$		0			0
$243$		−	9.24065i		−	0.592788i
$244$		0			0
$245$	−1.75233	+	1.38900i	−0.111952	+	0.0887402i
$246$		0			0
$247$		−	3.61462i		−	0.229993i
$248$		0			0
$249$	1.69867			0.107649
$250$		0			0
$251$	−23.4206			−1.47830			−0.739148	−	0.673543i	$-0.764772\pi$
$251$	−23.4206			−1.47830			−0.739148	+	0.673543i	$0.764772\pi$
$252$		0			0
$253$			25.7546i			1.61918i
$254$		0			0
$255$	2.46264	−	1.95204i	0.154217	−	0.122241i
$256$		0			0
$257$			13.0093i			0.811500i	0.913984	+	0.405750i	$0.132990\pi$
$257$			13.0093i			0.811500i	−0.913984	+	0.405750i	$0.867010\pi$
$258$		0			0
$259$	−6.00000			−0.372822
$260$		0			0
$261$	27.0280			1.67299
$262$		0			0
$263$			6.01866i			0.371126i	0.982632	+	0.185563i	$0.0594109\pi$
$263$			6.01866i			0.371126i	−0.982632	+	0.185563i	$0.940589\pi$
$264$		0			0
$265$	10.4953	+	13.2406i	0.644723	+	0.813367i
$266$		0			0
$267$		−	1.02799i		−	0.0629117i
$268$		0			0
$269$	3.24065			0.197586			0.0987929	−	0.995108i	$-0.468502\pi$
$269$	3.24065			0.197586			0.0987929	+	0.995108i	$0.468502\pi$
$270$		0			0
$271$	29.1307			1.76956			0.884782	−	0.466006i	$-0.154307\pi$
$271$	29.1307			1.76956			0.884782	+	0.466006i	$0.154307\pi$
$272$		0			0
$273$			1.68802i			0.102164i
$274$		0			0
$275$	5.86799	−	25.0280i	0.353853	−	1.50924i
$276$		0			0
$277$			4.44398i			0.267013i	0.991048	+	0.133507i	$0.0426237\pi$
$277$			4.44398i			0.267013i	−0.991048	+	0.133507i	$0.957376\pi$
$278$		0			0
$279$	−13.5560			−0.811577
$280$		0			0
$281$	−24.6974			−1.47332			−0.736660	−	0.676263i	$-0.763598\pi$
$281$	−24.6974			−1.47332			−0.736660	+	0.676263i	$0.763598\pi$
$282$		0			0
$283$		−	7.73937i		−	0.460058i	−0.973184	−	0.230029i	$-0.926118\pi$
$283$		−	7.73937i		−	0.460058i	0.973184	−	0.230029i	$-0.0738821\pi$
$284$		0			0
$285$	−0.392633	−	0.495336i	−0.0232576	−	0.0293412i
$286$		0			0
$287$			1.00933i			0.0595788i
$288$		0			0
$289$	2.03863			0.119920
$290$		0			0
$291$	−0.575992			−0.0337652
$292$		0			0
$293$			25.1086i			1.46686i	0.679764	+	0.733431i	$0.262082\pi$
$293$			25.1086i			1.46686i	−0.679764	+	0.733431i	$0.737918\pi$
$294$		0			0
$295$	−21.9287	+	17.3820i	−1.27674	+	1.01202i
$296$		0			0
$297$		−	10.9614i		−	0.636043i
$298$		0			0
$299$	23.2733			1.34593
$300$		0			0
$301$	7.00933			0.404011
$302$		0			0
$303$		−	3.29200i		−	0.189121i
$304$		0			0
$305$	−20.1600	+	15.9800i	−1.15436	+	0.915014i
$306$		0			0
$307$		−	33.9193i		−	1.93588i	−0.251183	−	0.967940i	$-0.580820\pi$
$307$		−	33.9193i		−	1.93588i	0.251183	−	0.967940i	$-0.419180\pi$
$308$		0			0
$309$	1.86799			0.106266
$310$		0			0
$311$	−0.565344			−0.0320577			−0.0160289	−	0.999872i	$-0.505102\pi$
$311$	−0.565344			−0.0320577			−0.0160289	+	0.999872i	$0.505102\pi$
$312$		0			0
$313$			27.3400i			1.54535i	0.634804	+	0.772673i	$0.281081\pi$
$313$			27.3400i			1.54535i	−0.634804	+	0.772673i	$0.718919\pi$
$314$		0			0
$315$	−3.98365	−	5.02568i	−0.224453	−	0.283165i
$316$		0			0
$317$		−	22.7267i		−	1.27646i	−0.769847	−	0.638228i	$-0.779668\pi$
$317$		−	22.7267i		−	1.27646i	0.769847	−	0.638228i	$-0.220332\pi$
$318$		0			0
$319$	48.4520			2.71279
$320$		0			0
$321$		0			0
$322$		0			0
$323$			3.00933i			0.167444i
$324$		0			0
$325$	−22.6167	−	5.30265i	−1.25455	−	0.294138i
$326$		0			0
$327$			1.44267i			0.0797796i
$328$		0			0
$329$	11.4240			0.629826
$330$		0			0
$331$	0.462642			0.0254291			0.0127145	−	0.999919i	$-0.495953\pi$
$331$	0.462642			0.0254291			0.0127145	+	0.999919i	$0.495953\pi$
$332$		0			0
$333$		−	17.2080i		−	0.942990i
$334$		0			0
$335$	−16.3013	−	20.5653i	−0.890637	−	1.12360i
$336$		0			0
$337$		−	10.5653i		−	0.575531i	−0.957701	−	0.287765i	$-0.907088\pi$
$337$		−	10.5653i		−	0.575531i	0.957701	−	0.287765i	$-0.0929124\pi$
$338$		0			0
$339$	−1.65194			−0.0897211
$340$		0			0
$341$	−24.3013			−1.31599
$342$		0			0
$343$			1.00000i			0.0539949i
$344$		0			0
$345$	3.18930	−	2.52803i	0.171706	−	0.136105i
$346$		0			0
$347$			31.8760i			1.71119i	0.517643	+	0.855597i	$0.326810\pi$
$347$			31.8760i			1.71119i	−0.517643	+	0.855597i	$0.673190\pi$
$348$		0			0
$349$	−9.62602			−0.515269			−0.257635	−	0.966242i	$-0.582943\pi$
$349$	−9.62602			−0.515269			−0.257635	+	0.966242i	$0.582943\pi$
$350$		0			0
$351$	−9.90531			−0.528706
$352$		0			0
$353$			26.2534i			1.39733i	0.715451	+	0.698663i	$0.246221\pi$
$353$			26.2534i			1.39733i	−0.715451	+	0.698663i	$0.753779\pi$
$354$		0			0
$355$	−4.77801	+	3.78734i	−0.253590	+	0.201011i
$356$		0			0
$357$		−	1.40535i		−	0.0743791i
$358$		0			0
$359$	−4.56534			−0.240950			−0.120475	−	0.992716i	$-0.538442\pi$
$359$	−4.56534			−0.240950			−0.120475	+	0.992716i	$0.538442\pi$
$360$		0			0
$361$	−18.3947			−0.968142
$362$		0			0
$363$		−	5.60737i		−	0.294310i
$364$		0			0
$365$	−6.95798	−	8.77801i	−0.364197	−	0.459462i
$366$		0			0
$367$		−	3.79073i		−	0.197874i	−0.995094	−	0.0989371i	$-0.968456\pi$
$367$		−	3.79073i		−	0.197874i	0.995094	−	0.0989371i	$-0.0315443\pi$
$368$		0			0
$369$	−2.89475			−0.150695
$370$		0			0
$371$	7.55602			0.392289
$372$		0			0
$373$			20.7453i			1.07415i	0.843534	+	0.537076i	$0.180471\pi$
$373$			20.7453i			1.07415i	−0.843534	+	0.537076i	$0.819529\pi$
$374$		0			0
$375$	−3.67531	+	1.73005i	−0.189792	+	0.0893392i
$376$		0			0
$377$		−	43.7839i		−	2.25499i
$378$		0			0
$379$	−3.00933			−0.154579			−0.0772894	−	0.997009i	$-0.524627\pi$
$379$	−3.00933			−0.154579			−0.0772894	+	0.997009i	$0.524627\pi$
$380$		0			0
$381$	−5.85735			−0.300081
$382$		0			0
$383$			7.00933i			0.358160i	0.983835	+	0.179080i	$0.0573121\pi$
$383$			7.00933i			0.358160i	−0.983835	+	0.179080i	$0.942688\pi$
$384$		0			0
$385$	−7.14134	−	9.00933i	−0.363956	−	0.459158i
$386$		0			0
$387$			20.1027i			1.02188i
$388$		0			0
$389$	−19.7653			−1.00214			−0.501070	−	0.865407i	$-0.667060\pi$
$389$	−19.7653			−1.00214			−0.501070	+	0.865407i	$0.667060\pi$
$390$		0			0
$391$	−19.3760			−0.979889
$392$		0			0
$393$		−	0.0186574i		−	0.000941142i
$394$		0			0
$395$	9.96731	−	7.90069i	0.501510	−	0.397527i
$396$		0			0
$397$			9.37266i			0.470400i	0.971947	+	0.235200i	$0.0755745\pi$
$397$			9.37266i			0.470400i	−0.971947	+	0.235200i	$0.924425\pi$
$398$		0			0
$399$	−0.282672			−0.0141513
$400$		0			0
$401$	−16.1507			−0.806526			−0.403263	−	0.915084i	$-0.632124\pi$
$401$	−16.1507			−0.806526			−0.403263	+	0.915084i	$0.632124\pi$
$402$		0			0
$403$			21.9600i			1.09391i
$404$		0			0
$405$	13.7196	−	10.8750i	0.681734	−	0.540384i
$406$		0			0
$407$		−	30.8480i		−	1.52908i
$408$		0			0
$409$	−15.2920			−0.756141			−0.378070	−	0.925777i	$-0.623412\pi$
$409$	−15.2920			−0.756141			−0.378070	+	0.925777i	$0.623412\pi$
$410$		0			0
$411$	−5.09337			−0.251238
$412$		0			0
$413$			12.5140i			0.615773i
$414$		0			0
$415$	6.49402	+	8.19269i	0.318779	+	0.402163i
$416$		0			0
$417$			4.58400i			0.224480i
$418$		0			0
$419$	33.1820			1.62105			0.810524	−	0.585705i	$-0.199182\pi$
$419$	33.1820			1.62105			0.810524	+	0.585705i	$0.199182\pi$
$420$		0			0
$421$	−27.8094			−1.35535			−0.677673	−	0.735363i	$-0.737012\pi$
$421$	−27.8094			−1.35535			−0.677673	+	0.735363i	$0.737012\pi$
$422$		0			0
$423$			32.7640i			1.59304i
$424$		0			0
$425$	18.8294	+	4.41468i	0.913358	+	0.214143i
$426$		0			0
$427$			11.5047i			0.556750i
$428$		0			0
$429$	−8.67869			−0.419011
$430$		0			0
$431$	21.5454			1.03780			0.518902	−	0.854834i	$-0.326341\pi$
$431$	21.5454			1.03780			0.518902	+	0.854834i	$0.326341\pi$
$432$		0			0
$433$			36.0187i			1.73095i	0.500955	+	0.865473i	$0.332982\pi$
$433$			36.0187i			1.73095i	−0.500955	+	0.865473i	$0.667018\pi$
$434$		0			0
$435$	−4.75596	−	6.00000i	−0.228031	−	0.287678i
$436$		0			0
$437$			3.89730i			0.186433i
$438$		0			0
$439$	−21.1893			−1.01131			−0.505655	−	0.862736i	$-0.668749\pi$
$439$	−21.1893			−1.01131			−0.505655	+	0.862736i	$0.668749\pi$
$440$		0			0
$441$	−2.86799			−0.136571
$442$		0			0
$443$			11.1120i			0.527949i	0.964530	+	0.263974i	$0.0850334\pi$
$443$			11.1120i			0.527949i	−0.964530	+	0.263974i	$0.914967\pi$
$444$		0			0
$445$	4.95798	−	3.92999i	0.235031	−	0.186299i
$446$		0			0
$447$			4.16131i			0.196823i
$448$		0			0
$449$	−0.961367			−0.0453697			−0.0226848	−	0.999743i	$-0.507221\pi$
$449$	−0.961367			−0.0453697			−0.0226848	+	0.999743i	$0.507221\pi$
$450$		0			0
$451$	−5.18930			−0.244355
$452$		0			0
$453$			2.13201i			0.100170i
$454$		0			0
$455$	−8.14134	+	6.45331i	−0.381672	+	0.302536i
$456$		0			0
$457$			28.2241i			1.32027i	0.751149	+	0.660133i	$0.229500\pi$
$457$			28.2241i			1.32027i	−0.751149	+	0.660133i	$0.770500\pi$
$458$		0			0
$459$	8.24659			0.384918
$460$		0			0
$461$	−26.0887			−1.21507			−0.607535	−	0.794293i	$-0.707842\pi$
$461$	−26.0887			−1.21507			−0.607535	+	0.794293i	$0.707842\pi$
$462$		0			0
$463$		−	2.90663i		−	0.135082i	−0.997716	−	0.0675412i	$-0.978485\pi$
$463$		−	2.90663i		−	0.135082i	0.997716	−	0.0675412i	$-0.0215154\pi$
$464$		0			0
$465$	2.38538	+	3.00933i	0.110619	+	0.139554i
$466$		0			0
$467$			12.2020i			0.564642i	0.959320	+	0.282321i	$0.0911043\pi$
$467$			12.2020i			0.564642i	−0.959320	+	0.282321i	$0.908896\pi$
$468$		0			0
$469$	−11.7360			−0.541917
$470$		0			0
$471$	2.10270			0.0968874
$472$		0			0
$473$			36.0373i			1.65700i
$474$		0			0
$475$	0.887968	−	3.78734i	0.0407428	−	0.173775i
$476$		0			0
$477$			21.6706i			0.992228i
$478$		0			0
$479$	−17.6519			−0.806538			−0.403269	−	0.915082i	$-0.632126\pi$
$479$	−17.6519			−0.806538			−0.403269	+	0.915082i	$0.632126\pi$
$480$		0			0
$481$	−27.8760			−1.27104
$482$		0			0
$483$		−	1.82003i		−	0.0828143i
$484$		0			0
$485$	−2.20202	−	2.77801i	−0.0999884	−	0.126143i
$486$		0			0
$487$			1.57467i			0.0713553i	0.999363	+	0.0356776i	$0.0113590\pi$
$487$			1.57467i			0.0713553i	−0.999363	+	0.0356776i	$0.988641\pi$
$488$		0			0
$489$	−1.18930			−0.0537819
$490$		0			0
$491$	−3.26270			−0.147243			−0.0736217	−	0.997286i	$-0.523456\pi$
$491$	−3.26270			−0.147243			−0.0736217	+	0.997286i	$0.523456\pi$
$492$		0			0
$493$			36.4520i			1.64172i
$494$		0			0
$495$	25.8387	−	20.4813i	1.16136	−	0.920566i
$496$		0			0
$497$			2.72666i			0.122307i
$498$		0			0
$499$	−42.5293			−1.90387			−0.951936	−	0.306298i	$-0.900910\pi$
$499$	−42.5293			−1.90387			−0.951936	+	0.306298i	$0.900910\pi$
$500$		0			0
$501$	9.03863			0.403816
$502$		0			0
$503$		−	18.6974i		−	0.833674i	−0.908981	−	0.416837i	$-0.863139\pi$
$503$		−	18.6974i		−	0.833674i	0.908981	−	0.416837i	$-0.136861\pi$
$504$		0			0
$505$	15.8773	−	12.5853i	0.706532	−	0.560039i
$506$		0			0
$507$			3.11929i			0.138533i
$508$		0			0
$509$	20.2313			0.896738			0.448369	−	0.893849i	$-0.352005\pi$
$509$	20.2313			0.896738			0.448369	+	0.893849i	$0.352005\pi$
$510$		0			0
$511$	−5.00933			−0.221600
$512$		0			0
$513$		−	1.65872i		−	0.0732342i
$514$		0			0
$515$	7.14134	+	9.00933i	0.314685	+	0.396998i
$516$		0			0
$517$			58.7347i			2.58315i
$518$		0			0
$519$	2.40790			0.105695
$520$		0			0
$521$	21.8387			0.956770			0.478385	−	0.878150i	$-0.341222\pi$
$521$	21.8387			0.956770			0.478385	+	0.878150i	$0.341222\pi$
$522$		0			0
$523$			20.4554i			0.894451i	0.894421	+	0.447226i	$0.147588\pi$
$523$			20.4554i			0.894451i	−0.894421	+	0.447226i	$0.852412\pi$
$524$		0			0
$525$	−0.414680	+	1.76868i	−0.0180981	+	0.0771915i
$526$		0			0
$527$		−	18.2827i		−	0.796406i
$528$		0			0
$529$	−2.09337			−0.0910163
$530$		0			0
$531$	−35.8900			−1.55750
$532$		0			0
$533$			4.68934i			0.203118i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		−	3.64006i		−	0.157080i
$538$		0			0
$539$	−5.14134			−0.221453
$540$		0			0
$541$	−27.3400			−1.17544			−0.587718	−	0.809066i	$-0.699974\pi$
$541$	−27.3400			−1.17544			−0.587718	+	0.809066i	$0.699974\pi$
$542$		0			0
$543$			0.649390i			0.0278680i
$544$		0			0
$545$	−6.95798	+	5.51531i	−0.298047	+	0.236250i
$546$		0			0
$547$		−	22.6867i		−	0.970013i	−0.874510	−	0.485007i	$-0.838817\pi$
$547$		−	22.6867i		−	0.970013i	0.874510	−	0.485007i	$-0.161183\pi$
$548$		0			0
$549$	−32.9953			−1.40820
$550$		0			0
$551$	7.33195			0.312352
$552$		0			0
$553$		−	5.68802i		−	0.241879i
$554$		0			0
$555$	−3.82003	+	3.02799i	−0.162151	+	0.128531i
$556$		0			0
$557$			12.7453i			0.540036i	0.962855	+	0.270018i	$0.0870297\pi$
$557$			12.7453i			0.540036i	−0.962855	+	0.270018i	$0.912970\pi$
$558$		0			0
$559$	32.5653			1.37737
$560$		0			0
$561$	7.22538			0.305056
$562$		0			0
$563$			5.20333i			0.219294i	0.993971	+	0.109647i	$0.0349721\pi$
$563$			5.20333i			0.219294i	−0.993971	+	0.109647i	$0.965028\pi$
$564$		0			0
$565$	−6.31537	−	7.96731i	−0.265689	−	0.335187i
$566$		0			0
$567$		−	7.82936i		−	0.328802i
$568$		0			0
$569$	7.37605			0.309220			0.154610	−	0.987976i	$-0.450588\pi$
$569$	7.37605			0.309220			0.154610	+	0.987976i	$0.450588\pi$
$570$		0			0
$571$	15.8973			0.665281			0.332641	−	0.943054i	$-0.392060\pi$
$571$	15.8973			0.665281			0.332641	+	0.943054i	$0.392060\pi$
$572$		0			0
$573$		−	3.12268i		−	0.130452i
$574$		0			0
$575$	24.3854	+	5.71733i	1.01694	+	0.238429i
$576$		0			0
$577$		−	19.9707i		−	0.831391i	−0.909504	−	0.415695i	$-0.863538\pi$
$577$		−	19.9707i		−	0.831391i	0.909504	−	0.415695i	$-0.136462\pi$
$578$		0			0
$579$	1.87864			0.0780736
$580$		0			0
$581$	4.67531			0.193964
$582$		0			0
$583$			38.8480i			1.60892i
$584$		0			0
$585$	−18.5081	−	23.3493i	−0.765214	−	0.965374i
$586$		0			0
$587$		−	13.8060i		−	0.569834i	−0.958552	−	0.284917i	$-0.908034\pi$
$587$		−	13.8060i		−	0.569834i	0.958552	−	0.284917i	$-0.0919661\pi$
$588$		0			0
$589$	−3.67738			−0.151524
$590$		0			0
$591$	−8.68934			−0.357432
$592$		0			0
$593$		−	20.0666i		−	0.824037i	−0.911175	−	0.412019i	$-0.864824\pi$
$593$		−	20.0666i		−	0.824037i	0.911175	−	0.412019i	$-0.135176\pi$
$594$		0			0
$595$	6.77801	−	5.37266i	0.277871	−	0.220257i
$596$		0			0
$597$			5.56279i			0.227670i
$598$		0			0
$599$	46.0267			1.88060			0.940299	−	0.340349i	$-0.110545\pi$
$599$	46.0267			1.88060			0.940299	+	0.340349i	$0.110545\pi$
$600$		0			0
$601$	−1.37605			−0.0561301			−0.0280651	−	0.999606i	$-0.508935\pi$
$601$	−1.37605			−0.0561301			−0.0280651	+	0.999606i	$0.508935\pi$
$602$		0			0
$603$		−	33.6587i		−	1.37069i
$604$		0			0
$605$	27.0443	−	21.4370i	1.09951	−	0.871537i
$606$		0			0
$607$		−	18.7933i		−	0.762796i	−0.924411	−	0.381398i	$-0.875443\pi$
$607$		−	18.7933i		−	0.762796i	0.924411	−	0.381398i	$-0.124557\pi$
$608$		0			0
$609$	−3.42401			−0.138748
$610$		0			0
$611$	53.0759			2.14722
$612$		0			0
$613$			24.8480i			1.00360i	0.864983	+	0.501801i	$0.167329\pi$
$613$			24.8480i			1.00360i	−0.864983	+	0.501801i	$0.832671\pi$
$614$		0			0
$615$	0.509372	+	0.642611i	0.0205399	+	0.0259126i
$616$		0			0
$617$			43.1680i			1.73788i	0.494919	+	0.868939i	$0.335198\pi$
$617$			43.1680i			1.73788i	−0.494919	+	0.868939i	$0.664802\pi$
$618$		0			0
$619$	−31.9486			−1.28412			−0.642062	−	0.766652i	$-0.721921\pi$
$619$	−31.9486			−1.28412			−0.642062	+	0.766652i	$0.721921\pi$
$620$		0			0
$621$	10.6799			0.428571
$622$		0			0
$623$		−	2.82936i		−	0.113356i
$624$		0			0
$625$	−22.3947	−	11.1120i	−0.895788	−	0.444481i
$626$		0			0
$627$		−	1.45331i		−	0.0580397i
$628$		0			0
$629$	23.2080			0.925362
$630$		0			0
$631$	8.59465			0.342148			0.171074	−	0.985258i	$-0.445276\pi$
$631$	8.59465			0.342148			0.171074	+	0.985258i	$0.445276\pi$
$632$		0			0
$633$			0.377365i			0.0149989i
$634$		0			0
$635$	−22.3926	−	28.2500i	−0.888625	−	1.12107i
$636$		0			0
$637$			4.64600i			0.184081i
$638$		0			0
$639$	−7.82003			−0.309356
$640$		0			0
$641$	33.2266			1.31237			0.656186	−	0.754599i	$-0.272169\pi$
$641$	33.2266			1.31237			0.656186	+	0.754599i	$0.272169\pi$
$642$		0			0
$643$		−	17.4940i		−	0.689897i	−0.938622	−	0.344948i	$-0.887896\pi$
$643$		−	17.4940i		−	0.689897i	0.938622	−	0.344948i	$-0.112104\pi$
$644$		0			0
$645$	4.46264	−	3.53736i	0.175716	−	0.139283i
$646$		0			0
$647$		−	9.91595i		−	0.389836i	−0.980820	−	0.194918i	$-0.937556\pi$
$647$		−	9.91595i		−	0.389836i	0.980820	−	0.194918i	$-0.0624441\pi$
$648$		0			0
$649$	−64.3386			−2.52551
$650$		0			0
$651$	1.71733			0.0673074
$652$		0			0
$653$			14.6240i			0.572280i	0.958188	+	0.286140i	$0.0923722\pi$
$653$			14.6240i			0.572280i	−0.958188	+	0.286140i	$0.907628\pi$
$654$		0			0
$655$	0.0899847	−	0.0713273i	0.00351599	−	0.00278699i
$656$		0			0
$657$		−	14.3667i		−	0.560499i
$658$		0			0
$659$	−20.2534			−0.788959			−0.394480	−	0.918905i	$-0.629075\pi$
$659$	−20.2534			−0.788959			−0.394480	+	0.918905i	$0.629075\pi$
$660$		0			0
$661$	12.4299			0.483469			0.241734	−	0.970342i	$-0.422284\pi$
$661$	12.4299			0.483469			0.241734	+	0.970342i	$0.422284\pi$
$662$		0			0
$663$		−	6.52926i		−	0.253575i
$664$		0			0
$665$	−1.08066	−	1.36333i	−0.0419060	−	0.0528676i
$666$		0			0
$667$			47.2080i			1.82790i
$668$		0			0
$669$	−3.58532			−0.138616
$670$		0			0
$671$	−59.1493			−2.28344
$672$		0			0
$673$			47.6774i			1.83783i	0.394458	+	0.918914i	$0.370932\pi$
$673$			47.6774i			1.83783i	−0.394458	+	0.918914i	$0.629068\pi$
$674$		0			0
$675$	−10.3786	−	2.43334i	−0.399472	−	0.0936592i
$676$		0			0
$677$		−	6.00339i		−	0.230729i	−0.993323	−	0.115364i	$-0.963196\pi$
$677$		−	6.00339i		−	0.230729i	0.993323	−	0.115364i	$-0.0368036\pi$
$678$		0			0
$679$	−1.58532			−0.0608390
$680$		0			0
$681$	−7.86799			−0.301502
$682$		0			0
$683$			28.5653i			1.09302i	0.837452	+	0.546511i	$0.184044\pi$
$683$			28.5653i			1.09302i	−0.837452	+	0.546511i	$0.815956\pi$
$684$		0			0
$685$	−19.4720	−	24.5653i	−0.743986	−	0.938594i
$686$		0			0
$687$		−	7.35061i		−	0.280443i
$688$		0			0
$689$	35.1053			1.33740
$690$		0			0
$691$	47.8247			1.81934			0.909668	−	0.415337i	$-0.136336\pi$
$691$	47.8247			1.81934			0.909668	+	0.415337i	$0.136336\pi$
$692$		0			0
$693$		−	14.7453i		−	0.560128i
$694$		0			0
$695$	−22.1086	+	17.5246i	−0.838629	+	0.664747i
$696$		0			0
$697$		−	3.90408i		−	0.147877i
$698$		0			0
$699$	1.98134			0.0749413
$700$		0			0
$701$	4.80005			0.181296			0.0906478	−	0.995883i	$-0.471106\pi$
$701$	4.80005			0.181296			0.0906478	+	0.995883i	$0.471106\pi$
$702$		0			0
$703$		−	4.66805i		−	0.176059i
$704$		0			0
$705$	7.27334	−	5.76529i	0.273930	−	0.217133i
$706$		0			0
$707$		−	9.06068i		−	0.340762i
$708$		0			0
$709$	11.7653			0.441855			0.220927	−	0.975290i	$-0.429092\pi$
$709$	11.7653			0.441855			0.220927	+	0.975290i	$0.429092\pi$
$710$		0			0
$711$	16.3132			0.611793
$712$		0			0
$713$		−	23.6774i		−	0.886725i
$714$		0			0
$715$	−33.1787	−	41.8573i	−1.24081	−	1.56538i
$716$		0			0
$717$			4.20927i			0.157198i
$718$		0			0
$719$	−40.7640			−1.52024			−0.760120	−	0.649783i	$-0.774860\pi$
$719$	−40.7640			−1.52024			−0.760120	+	0.649783i	$0.774860\pi$
$720$		0			0
$721$	5.14134			0.191473
$722$		0			0
$723$			1.82003i			0.0676877i
$724$		0			0
$725$	10.7560	−	45.8760i	0.399466	−	1.70379i
$726$		0			0
$727$		−	22.1214i		−	0.820436i	−0.911988	−	0.410218i	$-0.865453\pi$
$727$		−	22.1214i		−	0.820436i	0.911988	−	0.410218i	$-0.134547\pi$
$728$		0			0
$729$	20.1307			0.745581
$730$		0			0
$731$	−27.1120			−1.00277
$732$		0			0
$733$			13.0500i			0.482014i	0.970523	+	0.241007i	$0.0774777\pi$
$733$			13.0500i			0.482014i	−0.970523	+	0.241007i	$0.922522\pi$
$734$		0			0
$735$	0.504664	+	0.636672i	0.0186148	+	0.0234840i
$736$		0			0
$737$		−	60.3386i		−	2.22260i
$738$		0			0
$739$	34.5734			1.27180			0.635901	−	0.771771i	$-0.280629\pi$
$739$	34.5734			1.27180			0.635901	+	0.771771i	$0.280629\pi$
$740$		0			0
$741$	−1.31330			−0.0482451
$742$		0			0
$743$		−	3.55602i		−	0.130458i	−0.997870	−	0.0652288i	$-0.979222\pi$
$743$		−	3.55602i		−	0.130458i	0.997870	−	0.0652288i	$-0.0207777\pi$
$744$		0			0
$745$	−20.0700	+	15.9087i	−0.735308	+	0.582850i
$746$		0			0
$747$			13.4087i			0.490600i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	23.6226			0.862002			0.431001	−	0.902351i	$-0.358161\pi$
$751$	23.6226			0.862002			0.431001	+	0.902351i	$0.358161\pi$
$752$		0			0
$753$			8.50937i			0.310099i
$754$		0			0
$755$	−10.2827	+	8.15066i	−0.374225	+	0.296633i
$756$		0			0
$757$		−	15.1893i		−	0.552064i	−0.961148	−	0.276032i	$-0.910980\pi$
$757$		−	15.1893i		−	0.552064i	0.961148	−	0.276032i	$-0.0890196\pi$
$758$		0			0
$759$	9.35739			0.339652
$760$		0			0
$761$	34.4626			1.24927			0.624635	−	0.780917i	$-0.285248\pi$
$761$	34.4626			1.24927			0.624635	+	0.780917i	$0.285248\pi$
$762$		0			0
$763$			3.97070i			0.143749i
$764$		0			0
$765$	15.4087	+	19.4393i	0.557104	+	0.702829i
$766$		0			0
$767$			58.1400i			2.09931i
$768$		0			0
$769$	−40.3854			−1.45633			−0.728167	−	0.685400i	$-0.759627\pi$
$769$	−40.3854			−1.45633			−0.728167	+	0.685400i	$0.759627\pi$
$770$		0			0
$771$	4.72666			0.170226
$772$		0			0
$773$		−	10.7674i		−	0.387275i	−0.981073	−	0.193638i	$-0.937971\pi$
$773$		−	10.7674i		−	0.387275i	0.981073	−	0.193638i	$-0.0620286\pi$
$774$		0			0
$775$	−5.39470	+	23.0093i	−0.193783	+	0.826519i
$776$		0			0
$777$			2.17997i			0.0782060i
$778$		0			0
$779$	−0.785266			−0.0281351
$780$		0			0
$781$	−14.0187			−0.501627
$782$		0			0
$783$		−	20.0921i		−	0.718031i
$784$		0			0
$785$	8.03863	+	10.1413i	0.286911	+	0.361960i
$786$		0			0
$787$		−	9.19269i		−	0.327684i	−0.986487	−	0.163842i	$-0.947611\pi$
$787$		−	9.19269i		−	0.327684i	0.986487	−	0.163842i	$-0.0523887\pi$
$788$		0			0
$789$	2.18675			0.0778503
$790$		0			0
$791$	−4.54669			−0.161662
$792$		0			0
$793$			53.4507i			1.89809i
$794$		0			0
$795$	4.81070	−	3.81325i	0.170618	−	0.135242i
$796$		0			0
$797$		−	21.1086i		−	0.747706i	−0.927488	−	0.373853i	$-0.878036\pi$
$797$		−	21.1086i		−	0.747706i	0.927488	−	0.373853i	$-0.121964\pi$
$798$		0			0
$799$	−44.1880			−1.56326
$800$		0			0
$801$	8.11458			0.286715
$802$		0			0
$803$		−	25.7546i		−	0.908862i
$804$		0			0
$805$	8.77801	−	6.95798i	0.309384	−	0.245236i
$806$		0			0
$807$		−	1.17742i		−	0.0414471i
$808$		0			0
$809$	27.8280			0.978382			0.489191	−	0.872177i	$-0.337292\pi$
$809$	27.8280			0.978382			0.489191	+	0.872177i	$0.337292\pi$
$810$		0			0
$811$	−11.0607			−0.388393			−0.194197	−	0.980963i	$-0.562210\pi$
$811$	−11.0607			−0.388393			−0.194197	+	0.980963i	$0.562210\pi$
$812$		0			0
$813$		−	10.5840i		−	0.371197i
$814$		0			0
$815$	−4.54669	−	5.73599i	−0.159264	−	0.200923i
$816$		0			0
$817$			5.45331i			0.190787i
$818$		0			0
$819$	−13.3247			−0.465603
$820$		0			0
$821$	17.8867			0.624248			0.312124	−	0.950041i	$-0.398959\pi$
$821$	17.8867			0.624248			0.312124	+	0.950041i	$0.398959\pi$
$822$		0			0
$823$			52.8667i			1.84282i	0.388596	+	0.921408i	$0.372960\pi$
$823$			52.8667i			1.84282i	−0.388596	+	0.921408i	$0.627040\pi$
$824$		0			0
$825$	−9.09337	−	2.13201i	−0.316591	−	0.0742270i
$826$		0			0
$827$			36.3013i			1.26232i	0.775652	+	0.631160i	$0.217421\pi$
$827$			36.3013i			1.26232i	−0.775652	+	0.631160i	$0.782579\pi$
$828$		0			0
$829$	−0.759350			−0.0263733			−0.0131867	−	0.999913i	$-0.504198\pi$
$829$	−0.759350			−0.0263733			−0.0131867	+	0.999913i	$0.504198\pi$
$830$		0			0
$831$	1.61462			0.0560107
$832$		0			0
$833$		−	3.86799i		−	0.134018i
$834$		0			0
$835$	34.5547	+	43.5933i	1.19581	+	1.50861i
$836$		0			0
$837$			10.0773i			0.348321i
$838$		0			0
$839$	−43.3107			−1.49525			−0.747625	−	0.664121i	$-0.768806\pi$
$839$	−43.3107			−1.49525			−0.747625	+	0.664121i	$0.768806\pi$
$840$		0			0
$841$	59.8119			2.06248
$842$		0			0
$843$			8.97325i			0.309055i
$844$		0			0
$845$	−15.0443	+	11.9250i	−0.517541	+	0.410234i
$846$		0			0
$847$		−	15.4333i		−	0.530296i
$848$		0			0
$849$	−2.81193			−0.0965053
$850$		0			0
$851$	30.0560			1.03031
$852$		0			0
$853$		−	40.3713i		−	1.38229i	−0.722717	−	0.691144i	$-0.757107\pi$
$853$		−	40.3713i		−	1.38229i	0.722717	−	0.691144i	$-0.242893\pi$
$854$		0			0
$855$	3.91002	−	3.09931i	0.133720	−	0.105994i
$856$		0			0
$857$			13.4720i			0.460194i	0.973168	+	0.230097i	$0.0739043\pi$
$857$			13.4720i			0.460194i	−0.973168	+	0.230097i	$0.926096\pi$
$858$		0			0
$859$	34.7313			1.18502			0.592508	−	0.805565i	$-0.298138\pi$
$859$	34.7313			1.18502			0.592508	+	0.805565i	$0.298138\pi$
$860$		0			0
$861$	0.366718			0.0124977
$862$		0			0
$863$		−	14.2454i		−	0.484918i	−0.970162	−	0.242459i	$-0.922046\pi$
$863$		−	14.2454i		−	0.484918i	0.970162	−	0.242459i	$-0.0779539\pi$
$864$		0			0
$865$	9.20541	+	11.6133i	0.312993	+	0.394864i
$866$		0			0
$867$		−	0.740693i		−	0.0251553i
$868$		0			0
$869$	29.2440			0.992036
$870$		0			0
$871$	−54.5254			−1.84752
$872$		0			0
$873$		−	4.54669i		−	0.153882i
$874$		0			0
$875$	−10.1157	+	4.76166i	−0.341972	+	0.160974i
$876$		0			0
$877$		−	19.7546i		−	0.667067i	−0.942738	−	0.333533i	$-0.891759\pi$
$877$		−	19.7546i		−	0.667067i	0.942738	−	0.333533i	$-0.108241\pi$
$878$		0			0
$879$	9.12268			0.307700
$880$		0			0
$881$	−9.53058			−0.321093			−0.160547	−	0.987028i	$-0.551326\pi$
$881$	−9.53058			−0.321093			−0.160547	+	0.987028i	$0.551326\pi$
$882$		0			0
$883$		−	51.1867i		−	1.72257i	−0.508124	−	0.861284i	$-0.669661\pi$
$883$		−	51.1867i		−	1.72257i	0.508124	−	0.861284i	$-0.330339\pi$
$884$		0			0
$885$	6.31537	+	7.96731i	0.212289	+	0.267818i
$886$		0			0
$887$		−	24.4626i		−	0.821375i	−0.911776	−	0.410688i	$-0.865289\pi$
$887$		−	24.4626i		−	0.821375i	0.911776	−	0.410688i	$-0.134711\pi$
$888$		0			0
$889$	−16.1214			−0.540693
$890$		0			0
$891$	40.2534			1.34854
$892$		0			0
$893$			8.88797i			0.297425i
$894$		0			0
$895$	17.5560	−	13.9160i	0.586833	−	0.465159i
$896$		0			0
$897$		−	8.45586i		−	0.282333i
$898$		0			0
$899$	−44.5441			−1.48563
$900$		0			0
$901$	−29.2266			−0.973680
$902$		0			0
$903$		−	2.54669i		−	0.0847484i
$904$		0			0
$905$	−3.13201	+	2.48262i	−0.104111	+	0.0825250i
$906$		0			0
$907$		−	8.09592i		−	0.268821i	−0.990926	−	0.134410i	$-0.957086\pi$
$907$		−	8.09592i		−	0.268821i	0.990926	−	0.134410i	$-0.0429140\pi$
$908$		0			0
$909$	25.9860			0.861900
$910$		0			0
$911$	59.5093			1.97163			0.985815	−	0.167834i	$-0.0536772\pi$
$911$	59.5093			1.97163			0.985815	+	0.167834i	$0.0536772\pi$
$912$		0			0
$913$			24.0373i			0.795519i
$914$		0			0
$915$	5.80599	+	7.32469i	0.191940	+	0.242147i
$916$		0			0
$917$		−	0.0513514i		−	0.00169577i
$918$		0			0
$919$	51.7067			1.70565			0.852823	−	0.522200i	$-0.174889\pi$
$919$	51.7067			1.70565			0.852823	+	0.522200i	$0.174889\pi$
$920$		0			0
$921$	−12.3239			−0.406085
$922$		0			0
$923$			12.6680i			0.416974i
$924$		0			0
$925$	−29.2080	−	6.84802i	−0.960352	−	0.225161i
$926$		0			0
$927$			14.7453i			0.484300i
$928$		0			0
$929$	3.76142			0.123408			0.0617041	−	0.998094i	$-0.480346\pi$
$929$	3.76142			0.123408			0.0617041	+	0.998094i	$0.480346\pi$
$930$		0			0
$931$	−0.778008			−0.0254982
$932$		0			0
$933$			0.205406i			0.00672468i
$934$		0			0
$935$	27.6226	+	34.8480i	0.903357	+	1.13965i
$936$		0			0
$937$			21.1014i			0.689352i	0.938722	+	0.344676i	$0.112011\pi$
$937$			21.1014i			0.689352i	−0.938722	+	0.344676i	$0.887989\pi$
$938$		0			0
$939$	9.93338			0.324164
$940$		0			0
$941$	1.72873			0.0563549			0.0281775	−	0.999603i	$-0.491030\pi$
$941$	1.72873			0.0563549			0.0281775	+	0.999603i	$0.491030\pi$
$942$		0			0
$943$		−	5.05606i		−	0.164648i
$944$		0			0
$945$	−3.73599	+	2.96137i	−0.121532	+	0.0963333i
$946$		0			0
$947$		−	24.7267i		−	0.803508i	−0.915748	−	0.401754i	$-0.868401\pi$
$947$		−	24.7267i		−	0.803508i	0.915748	−	0.401754i	$-0.131599\pi$
$948$		0			0
$949$	−23.2733			−0.755485
$950$		0			0
$951$	−8.25724			−0.267759
$952$		0			0
$953$		−	1.18930i		−	0.0385251i	−0.999814	−	0.0192626i	$-0.993868\pi$
$953$		−	1.18930i		−	0.0385251i	0.999814	−	0.0192626i	$-0.00613185\pi$
$954$		0			0
$955$	15.0607	−	11.9380i	0.487352	−	0.386305i
$956$		0			0
$957$		−	17.6040i		−	0.569056i
$958$		0			0
$959$	−14.0187			−0.452686
$960$		0			0
$961$	−8.65872			−0.279314
$962$		0			0
$963$		0			0
$964$		0			0
$965$	7.18204	+	9.06068i	0.231198	+	0.291674i
$966$		0			0
$967$			23.3293i			0.750220i	0.926980	+	0.375110i	$0.122395\pi$
$967$			23.3293i			0.750220i	−0.926980	+	0.375110i	$0.877605\pi$
$968$		0			0
$969$	1.09337			0.0351242
$970$		0			0
$971$	−4.61670			−0.148157			−0.0740784	−	0.997252i	$-0.523602\pi$
$971$	−4.61670			−0.148157			−0.0740784	+	0.997252i	$0.523602\pi$
$972$		0			0
$973$			12.6167i			0.404473i
$974$		0			0
$975$	−1.92660	+	8.21728i	−0.0617006	+	0.263164i
$976$		0			0
$977$		−	19.1120i		−	0.611448i	−0.952120	−	0.305724i	$-0.901102\pi$
$977$		−	19.1120i		−	0.611448i	0.952120	−	0.305724i	$-0.0988985\pi$
$978$		0			0
$979$	14.5467			0.464914
$980$		0			0
$981$	−11.3879			−0.363588
$982$		0			0
$983$		−	52.6506i		−	1.67929i	−0.543132	−	0.839647i	$-0.682762\pi$
$983$		−	52.6506i		−	1.67929i	0.543132	−	0.839647i	$-0.317238\pi$
$984$		0			0
$985$	−33.2194	−	41.9087i	−1.05846	−	1.33532i
$986$		0			0
$987$		−	4.15066i		−	0.132117i
$988$		0			0
$989$	−35.1120			−1.11650
$990$		0			0
$991$	−52.4227			−1.66526			−0.832631	−	0.553828i	$-0.813166\pi$
$991$	−52.4227			−1.66526			−0.832631	+	0.553828i	$0.813166\pi$
$992$		0			0
$993$		−	0.168091i		−	0.00533420i
$994$		0			0
$995$	−26.8294	+	21.2666i	−0.850548	+	0.674195i
$996$		0			0
$997$		−	19.7580i		−	0.625743i	−0.949795	−	0.312872i	$-0.898709\pi$
$997$		−	19.7580i		−	0.625743i	0.949795	−	0.312872i	$-0.101291\pi$
$998$		0			0
$999$	−12.7920			−0.404722

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2240.2.g.m.449.3		6
4.3	odd	2		2240.2.g.l.449.4		6
5.4	even	2	inner	2240.2.g.m.449.4		6
8.3	odd	2		280.2.g.b.169.3	✓	6
8.5	even	2		560.2.g.f.449.4		6
20.19	odd	2		2240.2.g.l.449.3		6
24.5	odd	2		5040.2.t.y.1009.5		6
24.11	even	2		2520.2.t.g.1009.5		6
40.3	even	4		1400.2.a.s.1.2		3
40.13	odd	4		2800.2.a.br.1.2		3
40.19	odd	2		280.2.g.b.169.4	yes	6
40.27	even	4		1400.2.a.t.1.2		3
40.29	even	2		560.2.g.f.449.3		6
40.37	odd	4		2800.2.a.bq.1.2		3
56.27	even	2		1960.2.g.c.1569.4		6
120.29	odd	2		5040.2.t.y.1009.6		6
120.59	even	2		2520.2.t.g.1009.6		6
280.27	odd	4		9800.2.a.cd.1.2		3
280.83	odd	4		9800.2.a.cg.1.2		3
280.139	even	2		1960.2.g.c.1569.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.2.g.b.169.3	✓	6	8.3	odd	2
280.2.g.b.169.4	yes	6	40.19	odd	2
560.2.g.f.449.3		6	40.29	even	2
560.2.g.f.449.4		6	8.5	even	2
1400.2.a.s.1.2		3	40.3	even	4
1400.2.a.t.1.2		3	40.27	even	4
1960.2.g.c.1569.3		6	280.139	even	2
1960.2.g.c.1569.4		6	56.27	even	2
2240.2.g.l.449.3		6	20.19	odd	2
2240.2.g.l.449.4		6	4.3	odd	2
2240.2.g.m.449.3		6	1.1	even	1	trivial
2240.2.g.m.449.4		6	5.4	even	2	inner
2520.2.t.g.1009.5		6	24.11	even	2
2520.2.t.g.1009.6		6	120.59	even	2
2800.2.a.bq.1.2		3	40.37	odd	4
2800.2.a.br.1.2		3	40.13	odd	4
5040.2.t.y.1009.5		6	24.5	odd	2
5040.2.t.y.1009.6		6	120.29	odd	2
9800.2.a.cd.1.2		3	280.27	odd	4
9800.2.a.cg.1.2		3	280.83	odd	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)

\(f(q)\)	\(=\)	\(q-0.363328i q^{3} +(1.75233 - 1.38900i) q^{5} -1.00000i q^{7} +2.86799 q^{9} +O(q^{10})\)	\(q-0.363328i q^{3} +(1.75233 - 1.38900i) q^{5} -1.00000i q^{7} +2.86799 q^{9} +5.14134 q^{11} -4.64600i q^{13} +(-0.504664 - 0.636672i) q^{15} +3.86799i q^{17} +0.778008 q^{19} -0.363328 q^{21} +5.00933i q^{23} +(1.14134 - 4.86799i) q^{25} -2.13201i q^{27} +9.42401 q^{29} -4.72666 q^{31} -1.86799i q^{33} +(-1.38900 - 1.75233i) q^{35} -6.00000i q^{37} -1.68802 q^{39} -1.00933 q^{41} +7.00933i q^{43} +(5.02568 - 3.98365i) q^{45} +11.4240i q^{47} -1.00000 q^{49} +1.40535 q^{51} +7.55602i q^{53} +(9.00933 - 7.14134i) q^{55} -0.282672i q^{57} -12.5140 q^{59} -11.5047 q^{61} -2.86799i q^{63} +(-6.45331 - 8.14134i) q^{65} -11.7360i q^{67} +1.82003 q^{69} -2.72666 q^{71} -5.00933i q^{73} +(-1.76868 - 0.414680i) q^{75} -5.14134i q^{77} +5.68802 q^{79} +7.82936 q^{81} +4.67531i q^{83} +(5.37266 + 6.77801i) q^{85} -3.42401i q^{87} +2.82936 q^{89} -4.64600 q^{91} +1.71733i q^{93} +(1.36333 - 1.08066i) q^{95} -1.58532i q^{97} +14.7453 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 8 q^{9}+O(q^{10}) \) 6 * q - 8 * q^9	\( 6 q - 8 q^{9} + 14 q^{11} + 18 q^{15} - 8 q^{19} + 2 q^{21} - 10 q^{25} + 6 q^{29} - 20 q^{31} - 2 q^{35} - 10 q^{39} + 36 q^{41} + 28 q^{45} - 6 q^{49} + 42 q^{51} + 12 q^{55} - 12 q^{59} - 48 q^{61} - 22 q^{65} + 36 q^{69} - 8 q^{71} - 40 q^{75} + 34 q^{79} + 30 q^{81} - 14 q^{85} + 10 q^{91} + 4 q^{95} - 4 q^{99}+O(q^{100}) \) 6 * q - 8 * q^9 + 14 * q^11 + 18 * q^15 - 8 * q^19 + 2 * q^21 - 10 * q^25 + 6 * q^29 - 20 * q^31 - 2 * q^35 - 10 * q^39 + 36 * q^41 + 28 * q^45 - 6 * q^49 + 42 * q^51 + 12 * q^55 - 12 * q^59 - 48 * q^61 - 22 * q^65 + 36 * q^69 - 8 * q^71 - 40 * q^75 + 34 * q^79 + 30 * q^81 - 14 * q^85 + 10 * q^91 + 4 * q^95 - 4 * q^99

Introduction

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