LMFDB - Embedded newform 240.3.j.b.79.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [240,3,Mod(79,240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(240, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("240.79");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 240 = 2^{4} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	240.j (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$6.53952634465$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{3}, \sqrt{-7})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - x^{2} + 2x + 22 $ x^4 - 2x^3 - x^2 + 2x + 22
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2}\cdot 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			79.3
Root			$2.23205 + 1.32288i$ of defining polynomial
Character	$\chi$	$=$	240.79
Dual form			240.3.j.b.79.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205 q^{3} +(-2.00000 - 4.58258i) q^{5} +3.46410 q^{7} +3.00000 q^{9} +O(q^{10})$	$q+1.73205 q^{3} +(-2.00000 - 4.58258i) q^{5} +3.46410 q^{7} +3.00000 q^{9} -15.8745i q^{11} +9.16515i q^{13} +(-3.46410 - 7.93725i) q^{15} -9.16515i q^{17} -31.7490i q^{19} +6.00000 q^{21} +27.7128 q^{23} +(-17.0000 + 18.3303i) q^{25} +5.19615 q^{27} -8.00000 q^{29} -27.4955i q^{33} +(-6.92820 - 15.8745i) q^{35} +45.8258i q^{37} +15.8745i q^{39} +50.0000 q^{41} +62.3538 q^{43} +(-6.00000 - 13.7477i) q^{45} -48.4974 q^{47} -37.0000 q^{49} -15.8745i q^{51} -27.4955i q^{53} +(-72.7461 + 31.7490i) q^{55} -54.9909i q^{57} +15.8745i q^{59} -26.0000 q^{61} +10.3923 q^{63} +(42.0000 - 18.3303i) q^{65} -55.4256 q^{67} +48.0000 q^{69} +95.2470i q^{71} +128.312i q^{73} +(-29.4449 + 31.7490i) q^{75} -54.9909i q^{77} +126.996i q^{79} +9.00000 q^{81} +131.636 q^{83} +(-42.0000 + 18.3303i) q^{85} -13.8564 q^{87} -86.0000 q^{89} +31.7490i q^{91} +(-145.492 + 63.4980i) q^{95} +109.982i q^{97} -47.6235i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{5} + 12 q^{9}+O(q^{10}) $ 4 * q - 8 * q^5 + 12 * q^9	$ 4 q - 8 q^{5} + 12 q^{9} + 24 q^{21} - 68 q^{25} - 32 q^{29} + 200 q^{41} - 24 q^{45} - 148 q^{49} - 104 q^{61} + 168 q^{65} + 192 q^{69} + 36 q^{81} - 168 q^{85} - 344 q^{89}+O(q^{100}) $ 4 * q - 8 * q^5 + 12 * q^9 + 24 * q^21 - 68 * q^25 - 32 * q^29 + 200 * q^41 - 24 * q^45 - 148 * q^49 - 104 * q^61 + 168 * q^65 + 192 * q^69 + 36 * q^81 - 168 * q^85 - 344 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/240\mathbb{Z}\right)^\times$.

$n$	$31$	$97$	$161$	$181$
$\chi(n)$	$-1$	$-1$	$1$	$1$

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			0.577350
$3$	1.73205			0.577350
$4$		0			0
$5$	−2.00000	−	4.58258i	−0.400000	−	0.916515i
$5$	−2.00000	−	4.58258i	−0.400000	−	0.916515i
$6$		0			0
$7$	3.46410			0.494872			0.247436	−	0.968904i	$-0.420412\pi$
$7$	3.46410			0.494872			0.247436	+	0.968904i	$0.420412\pi$
$8$		0			0
$9$	3.00000			0.333333
$10$		0			0
$11$		−	15.8745i		−	1.44314i	−0.692343	−	0.721569i	$-0.743421\pi$
$11$		−	15.8745i		−	1.44314i	0.692343	−	0.721569i	$-0.256579\pi$
$12$		0			0
$13$			9.16515i			0.705012i	0.935810	+	0.352506i	$0.114670\pi$
$13$			9.16515i			0.705012i	−0.935810	+	0.352506i	$0.885330\pi$
$14$		0			0
$15$	−3.46410	−	7.93725i	−0.230940	−	0.529150i
$16$		0			0
$17$		−	9.16515i		−	0.539127i	−0.962983	−	0.269563i	$-0.913121\pi$
$17$		−	9.16515i		−	0.539127i	0.962983	−	0.269563i	$-0.0868793\pi$
$18$		0			0
$19$		−	31.7490i		−	1.67100i	−0.549490	−	0.835500i	$-0.685178\pi$
$19$		−	31.7490i		−	1.67100i	0.549490	−	0.835500i	$-0.314822\pi$
$20$		0			0
$21$	6.00000			0.285714
$22$		0			0
$23$	27.7128			1.20490			0.602452	−	0.798155i	$-0.294190\pi$
$23$	27.7128			1.20490			0.602452	+	0.798155i	$0.294190\pi$
$24$		0			0
$25$	−17.0000	+	18.3303i	−0.680000	+	0.733212i
$26$		0			0
$27$	5.19615			0.192450
$28$		0			0
$29$	−8.00000			−0.275862			−0.137931	−	0.990442i	$-0.544045\pi$
$29$	−8.00000			−0.275862			−0.137931	+	0.990442i	$0.544045\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$		0			0
$33$		−	27.4955i		−	0.833196i
$34$		0			0
$35$	−6.92820	−	15.8745i	−0.197949	−	0.453557i
$36$		0			0
$37$			45.8258i			1.23853i	0.785180	+	0.619267i	$0.212570\pi$
$37$			45.8258i			1.23853i	−0.785180	+	0.619267i	$0.787430\pi$
$38$		0			0
$39$			15.8745i			0.407039i
$40$		0			0
$41$	50.0000			1.21951			0.609756	−	0.792589i	$-0.291267\pi$
$41$	50.0000			1.21951			0.609756	+	0.792589i	$0.291267\pi$
$42$		0			0
$43$	62.3538			1.45009			0.725045	−	0.688702i	$-0.241819\pi$
$43$	62.3538			1.45009			0.725045	+	0.688702i	$0.241819\pi$
$44$		0			0
$45$	−6.00000	−	13.7477i	−0.133333	−	0.305505i
$46$		0			0
$47$	−48.4974			−1.03186			−0.515930	−	0.856631i	$-0.672554\pi$
$47$	−48.4974			−1.03186			−0.515930	+	0.856631i	$0.672554\pi$
$48$		0			0
$49$	−37.0000			−0.755102
$50$		0			0
$51$		−	15.8745i		−	0.311265i
$52$		0			0
$53$		−	27.4955i		−	0.518782i	−0.965772	−	0.259391i	$-0.916478\pi$
$53$		−	27.4955i		−	0.518782i	0.965772	−	0.259391i	$-0.0835219\pi$
$54$		0			0
$55$	−72.7461	+	31.7490i	−1.32266	+	0.577255i
$56$		0			0
$57$		−	54.9909i		−	0.964753i
$58$		0			0
$59$			15.8745i			0.269059i	0.990910	+	0.134530i	$0.0429524\pi$
$59$			15.8745i			0.269059i	−0.990910	+	0.134530i	$0.957048\pi$
$60$		0			0
$61$	−26.0000			−0.426230			−0.213115	−	0.977027i	$-0.568361\pi$
$61$	−26.0000			−0.426230			−0.213115	+	0.977027i	$0.568361\pi$
$62$		0			0
$63$	10.3923			0.164957
$64$		0			0
$65$	42.0000	−	18.3303i	0.646154	−	0.282005i
$66$		0			0
$67$	−55.4256			−0.827248			−0.413624	−	0.910448i	$-0.635737\pi$
$67$	−55.4256			−0.827248			−0.413624	+	0.910448i	$0.635737\pi$
$68$		0			0
$69$	48.0000			0.695652
$70$		0			0
$71$			95.2470i			1.34151i	0.741680	+	0.670754i	$0.234029\pi$
$71$			95.2470i			1.34151i	−0.741680	+	0.670754i	$0.765971\pi$
$72$		0			0
$73$			128.312i			1.75770i	0.477098	+	0.878850i	$0.341689\pi$
$73$			128.312i			1.75770i	−0.477098	+	0.878850i	$0.658311\pi$
$74$		0			0
$75$	−29.4449	+	31.7490i	−0.392598	+	0.423320i
$76$		0			0
$77$		−	54.9909i		−	0.714168i
$78$		0			0
$79$			126.996i			1.60755i	0.594937	+	0.803773i	$0.297177\pi$
$79$			126.996i			1.60755i	−0.594937	+	0.803773i	$0.702823\pi$
$80$		0			0
$81$	9.00000			0.111111
$82$		0			0
$83$	131.636			1.58597			0.792987	−	0.609238i	$-0.208525\pi$
$83$	131.636			1.58597			0.792987	+	0.609238i	$0.208525\pi$
$84$		0			0
$85$	−42.0000	+	18.3303i	−0.494118	+	0.215651i
$86$		0			0
$87$	−13.8564			−0.159269
$88$		0			0
$89$	−86.0000			−0.966292			−0.483146	−	0.875540i	$-0.660506\pi$
$89$	−86.0000			−0.966292			−0.483146	+	0.875540i	$0.660506\pi$
$90$		0			0
$91$			31.7490i			0.348890i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−145.492	+	63.4980i	−1.53150	+	0.668400i
$96$		0			0
$97$			109.982i			1.13383i	0.823775	+	0.566917i	$0.191864\pi$
$97$			109.982i			1.13383i	−0.823775	+	0.566917i	$0.808136\pi$
$98$		0			0
$99$		−	47.6235i		−	0.481046i
$100$		0			0
$101$	−32.0000			−0.316832			−0.158416	−	0.987372i	$-0.550639\pi$
$101$	−32.0000			−0.316832			−0.158416	+	0.987372i	$0.550639\pi$
$102$		0			0
$103$	93.5307			0.908065			0.454033	−	0.890985i	$-0.349985\pi$
$103$	93.5307			0.908065			0.454033	+	0.890985i	$0.349985\pi$
$104$		0			0
$105$	−12.0000	−	27.4955i	−0.114286	−	0.261861i
$106$		0			0
$107$	−76.2102			−0.712245			−0.356123	−	0.934439i	$-0.615901\pi$
$107$	−76.2102			−0.712245			−0.356123	+	0.934439i	$0.615901\pi$
$108$		0			0
$109$	130.000			1.19266			0.596330	−	0.802739i	$-0.296625\pi$
$109$	130.000			1.19266			0.596330	+	0.802739i	$0.296625\pi$
$110$		0			0
$111$			79.3725i			0.715068i
$112$		0			0
$113$		−	119.147i		−	1.05440i	−0.849742	−	0.527199i	$-0.823242\pi$
$113$		−	119.147i		−	1.05440i	0.849742	−	0.527199i	$-0.176758\pi$
$114$		0			0
$115$	−55.4256	−	126.996i	−0.481962	−	1.10431i
$116$		0			0
$117$			27.4955i			0.235004i
$118$		0			0
$119$		−	31.7490i		−	0.266798i
$120$		0			0
$121$	−131.000			−1.08264
$122$		0			0
$123$	86.6025			0.704086
$124$		0			0
$125$	118.000	+	41.2432i	0.944000	+	0.329945i
$126$		0			0
$127$	−24.2487			−0.190935			−0.0954674	−	0.995433i	$-0.530435\pi$
$127$	−24.2487			−0.190935			−0.0954674	+	0.995433i	$0.530435\pi$
$128$		0			0
$129$	108.000			0.837209
$130$		0			0
$131$		−	174.620i		−	1.33297i	−0.745517	−	0.666487i	$-0.767797\pi$
$131$		−	174.620i		−	1.33297i	0.745517	−	0.666487i	$-0.232203\pi$
$132$		0			0
$133$		−	109.982i		−	0.826931i
$134$		0			0
$135$	−10.3923	−	23.8118i	−0.0769800	−	0.176383i
$136$		0			0
$137$			45.8258i			0.334495i	0.985915	+	0.167247i	$0.0534878\pi$
$137$			45.8258i			0.334495i	−0.985915	+	0.167247i	$0.946512\pi$
$138$		0			0
$139$		−	63.4980i		−	0.456820i	−0.973565	−	0.228410i	$-0.926647\pi$
$139$		−	63.4980i		−	0.456820i	0.973565	−	0.228410i	$-0.0733527\pi$
$140$		0			0
$141$	−84.0000			−0.595745
$142$		0			0
$143$	145.492			1.01743
$144$		0			0
$145$	16.0000	+	36.6606i	0.110345	+	0.252832i
$146$		0			0
$147$	−64.0859			−0.435958
$148$		0			0
$149$	−124.000			−0.832215			−0.416107	−	0.909315i	$-0.636606\pi$
$149$	−124.000			−0.832215			−0.416107	+	0.909315i	$0.636606\pi$
$150$		0			0
$151$		−	126.996i		−	0.841034i	−0.907285	−	0.420517i	$-0.861849\pi$
$151$		−	126.996i		−	0.841034i	0.907285	−	0.420517i	$-0.138151\pi$
$152$		0			0
$153$		−	27.4955i		−	0.179709i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		−	265.789i		−	1.69293i	−0.532448	−	0.846463i	$-0.678728\pi$
$157$		−	265.789i		−	1.69293i	0.532448	−	0.846463i	$-0.321272\pi$
$158$		0			0
$159$		−	47.6235i		−	0.299519i
$160$		0			0
$161$	96.0000			0.596273
$162$		0			0
$163$	290.985			1.78518			0.892591	−	0.450868i	$-0.148886\pi$
$163$	290.985			1.78518			0.892591	+	0.450868i	$0.148886\pi$
$164$		0			0
$165$	−126.000	+	54.9909i	−0.763636	+	0.333278i
$166$		0			0
$167$	180.133			1.07864			0.539321	−	0.842100i	$-0.318681\pi$
$167$	180.133			1.07864			0.539321	+	0.842100i	$0.318681\pi$
$168$		0			0
$169$	85.0000			0.502959
$170$		0			0
$171$		−	95.2470i		−	0.557000i
$172$		0			0
$173$			320.780i			1.85422i	0.374788	+	0.927111i	$0.377716\pi$
$173$			320.780i			1.85422i	−0.374788	+	0.927111i	$0.622284\pi$
$174$		0			0
$175$	−58.8897	+	63.4980i	−0.336513	+	0.362846i
$176$		0			0
$177$			27.4955i			0.155342i
$178$		0			0
$179$		−	111.122i		−	0.620791i	−0.950608	−	0.310395i	$-0.899539\pi$
$179$		−	111.122i		−	0.620791i	0.950608	−	0.310395i	$-0.100461\pi$
$180$		0			0
$181$	62.0000			0.342541			0.171271	−	0.985224i	$-0.445213\pi$
$181$	62.0000			0.342541			0.171271	+	0.985224i	$0.445213\pi$
$182$		0			0
$183$	−45.0333			−0.246084
$184$		0			0
$185$	210.000	−	91.6515i	1.13514	−	0.495414i
$186$		0			0
$187$	−145.492			−0.778034
$188$		0			0
$189$	18.0000			0.0952381
$190$		0			0
$191$		−	31.7490i		−	0.166225i	−0.996540	−	0.0831126i	$-0.973514\pi$
$191$		−	31.7490i		−	0.166225i	0.996540	−	0.0831126i	$-0.0264861\pi$
$192$		0			0
$193$		−	91.6515i		−	0.474878i	−0.971402	−	0.237439i	$-0.923692\pi$
$193$		−	91.6515i		−	0.474878i	0.971402	−	0.237439i	$-0.0763080\pi$
$194$		0			0
$195$	72.7461	−	31.7490i	0.373057	−	0.162815i
$196$		0			0
$197$			174.138i			0.883949i	0.897028	+	0.441974i	$0.145722\pi$
$197$			174.138i			0.883949i	−0.897028	+	0.441974i	$0.854278\pi$
$198$		0			0
$199$		−	253.992i		−	1.27634i	−0.769895	−	0.638171i	$-0.779691\pi$
$199$		−	253.992i		−	1.27634i	0.769895	−	0.638171i	$-0.220309\pi$
$200$		0			0
$201$	−96.0000			−0.477612
$202$		0			0
$203$	−27.7128			−0.136516
$204$		0			0
$205$	−100.000	−	229.129i	−0.487805	−	1.11770i
$206$		0			0
$207$	83.1384			0.401635
$208$		0			0
$209$	−504.000			−2.41148
$210$		0			0
$211$			63.4980i			0.300939i	0.988615	+	0.150469i	$0.0480785\pi$
$211$			63.4980i			0.300939i	−0.988615	+	0.150469i	$0.951922\pi$
$212$		0			0
$213$			164.973i			0.774520i
$214$		0			0
$215$	−124.708	−	285.741i	−0.580036	−	1.32903i
$216$		0			0
$217$		0			0
$218$		0			0
$219$			222.243i			1.01481i
$220$		0			0
$221$	84.0000			0.380090
$222$		0			0
$223$	−169.741			−0.761170			−0.380585	−	0.924746i	$-0.624277\pi$
$223$	−169.741			−0.761170			−0.380585	+	0.924746i	$0.624277\pi$
$224$		0			0
$225$	−51.0000	+	54.9909i	−0.226667	+	0.244404i
$226$		0			0
$227$	−145.492			−0.640935			−0.320468	−	0.947259i	$-0.603840\pi$
$227$	−145.492			−0.640935			−0.320468	+	0.947259i	$0.603840\pi$
$228$		0			0
$229$	326.000			1.42358			0.711790	−	0.702392i	$-0.247885\pi$
$229$	326.000			1.42358			0.711790	+	0.702392i	$0.247885\pi$
$230$		0			0
$231$		−	95.2470i		−	0.412325i
$232$		0			0
$233$			45.8258i			0.196677i	0.995153	+	0.0983385i	$0.0313528\pi$
$233$			45.8258i			0.196677i	−0.995153	+	0.0983385i	$0.968647\pi$
$234$		0			0
$235$	96.9948	+	222.243i	0.412744	+	0.945715i
$236$		0			0
$237$			219.964i			0.928117i
$238$		0			0
$239$			222.243i			0.929887i	0.885340	+	0.464944i	$0.153925\pi$
$239$			222.243i			0.929887i	−0.885340	+	0.464944i	$0.846075\pi$
$240$		0			0
$241$	118.000			0.489627			0.244813	−	0.969570i	$-0.421273\pi$
$241$	118.000			0.489627			0.244813	+	0.969570i	$0.421273\pi$
$242$		0			0
$243$	15.5885			0.0641500
$244$		0			0
$245$	74.0000	+	169.555i	0.302041	+	0.692062i
$246$		0			0
$247$	290.985			1.17808
$248$		0			0
$249$	228.000			0.915663
$250$		0			0
$251$			460.361i			1.83411i	0.398765	+	0.917053i	$0.369439\pi$
$251$			460.361i			1.83411i	−0.398765	+	0.917053i	$0.630561\pi$
$252$		0			0
$253$		−	439.927i		−	1.73884i
$254$		0			0
$255$	−72.7461	+	31.7490i	−0.285279	+	0.124506i
$256$		0			0
$257$		−	284.120i		−	1.10552i	−0.833339	−	0.552762i	$-0.813574\pi$
$257$		−	284.120i		−	1.10552i	0.833339	−	0.552762i	$-0.186426\pi$
$258$		0			0
$259$			158.745i			0.612915i
$260$		0			0
$261$	−24.0000			−0.0919540
$262$		0			0
$263$	−249.415			−0.948347			−0.474174	−	0.880431i	$-0.657253\pi$
$263$	−249.415			−0.948347			−0.474174	+	0.880431i	$0.657253\pi$
$264$		0			0
$265$	−126.000	+	54.9909i	−0.475472	+	0.207513i
$266$		0			0
$267$	−148.956			−0.557889
$268$		0			0
$269$	−472.000			−1.75465			−0.877323	−	0.479900i	$-0.840673\pi$
$269$	−472.000			−1.75465			−0.877323	+	0.479900i	$0.840673\pi$
$270$		0			0
$271$			190.494i			0.702930i	0.936201	+	0.351465i	$0.114316\pi$
$271$			190.494i			0.702930i	−0.936201	+	0.351465i	$0.885684\pi$
$272$		0			0
$273$			54.9909i			0.201432i
$274$		0			0
$275$	290.985	+	269.867i	1.05813	+	0.981333i
$276$		0			0
$277$		−	247.459i		−	0.893354i	−0.894695	−	0.446677i	$-0.852607\pi$
$277$		−	247.459i		−	0.893354i	0.894695	−	0.446677i	$-0.147393\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	278.000			0.989324			0.494662	−	0.869085i	$-0.335292\pi$
$281$	278.000			0.989324			0.494662	+	0.869085i	$0.335292\pi$
$282$		0			0
$283$	−484.974			−1.71369			−0.856845	−	0.515574i	$-0.827579\pi$
$283$	−484.974			−1.71369			−0.856845	+	0.515574i	$0.827579\pi$
$284$		0			0
$285$	−252.000	+	109.982i	−0.884211	+	0.385901i
$286$		0			0
$287$	173.205			0.603502
$288$		0			0
$289$	205.000			0.709343
$290$		0			0
$291$			190.494i			0.654619i
$292$		0			0
$293$		−	320.780i		−	1.09481i	−0.836867	−	0.547407i	$-0.815615\pi$
$293$		−	320.780i		−	1.09481i	0.836867	−	0.547407i	$-0.184385\pi$
$294$		0			0
$295$	72.7461	−	31.7490i	0.246597	−	0.107624i
$296$		0			0
$297$		−	82.4864i		−	0.277732i
$298$		0			0
$299$			253.992i			0.849472i
$300$		0			0
$301$	216.000			0.717608
$302$		0			0
$303$	−55.4256			−0.182923
$304$		0			0
$305$	52.0000	+	119.147i	0.170492	+	0.390646i
$306$		0			0
$307$	−256.344			−0.834995			−0.417498	−	0.908678i	$-0.637093\pi$
$307$	−256.344			−0.834995			−0.417498	+	0.908678i	$0.637093\pi$
$308$		0			0
$309$	162.000			0.524272
$310$		0			0
$311$		−	412.737i		−	1.32713i	−0.748119	−	0.663565i	$-0.769043\pi$
$311$		−	412.737i		−	1.32713i	0.748119	−	0.663565i	$-0.230957\pi$
$312$		0			0
$313$			201.633i			0.644196i	0.946706	+	0.322098i	$0.104388\pi$
$313$			201.633i			0.644196i	−0.946706	+	0.322098i	$0.895612\pi$
$314$		0			0
$315$	−20.7846	−	47.6235i	−0.0659829	−	0.151186i
$316$		0			0
$317$			559.074i			1.76364i	0.471585	+	0.881821i	$0.343682\pi$
$317$			559.074i			1.76364i	−0.471585	+	0.881821i	$0.656318\pi$
$318$		0			0
$319$			126.996i			0.398107i
$320$		0			0
$321$	−132.000			−0.411215
$322$		0			0
$323$	−290.985			−0.900881
$324$		0			0
$325$	−168.000	−	155.808i	−0.516923	−	0.479408i
$326$		0			0
$327$	225.167			0.688583
$328$		0			0
$329$	−168.000			−0.510638
$330$		0			0
$331$			95.2470i			0.287755i	0.989595	+	0.143878i	$0.0459572\pi$
$331$			95.2470i			0.287755i	−0.989595	+	0.143878i	$0.954043\pi$
$332$		0			0
$333$			137.477i			0.412845i
$334$		0			0
$335$	110.851	+	253.992i	0.330899	+	0.758185i
$336$		0			0
$337$			54.9909i			0.163178i	0.996666	+	0.0815889i	$0.0259994\pi$
$337$			54.9909i			0.163178i	−0.996666	+	0.0815889i	$0.974001\pi$
$338$		0			0
$339$		−	206.369i		−	0.608757i
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−297.913			−0.868550
$344$		0			0
$345$	−96.0000	−	219.964i	−0.278261	−	0.637576i
$346$		0			0
$347$	−394.908			−1.13806			−0.569031	−	0.822316i	$-0.692682\pi$
$347$	−394.908			−1.13806			−0.569031	+	0.822316i	$0.692682\pi$
$348$		0			0
$349$	146.000			0.418338			0.209169	−	0.977879i	$-0.432924\pi$
$349$	146.000			0.418338			0.209169	+	0.977879i	$0.432924\pi$
$350$		0			0
$351$			47.6235i			0.135680i
$352$		0			0
$353$			302.450i			0.856799i	0.903589	+	0.428399i	$0.140922\pi$
$353$			302.450i			0.856799i	−0.903589	+	0.428399i	$0.859078\pi$
$354$		0			0
$355$	436.477	−	190.494i	1.22951	−	0.536603i
$356$		0			0
$357$		−	54.9909i		−	0.154036i
$358$		0			0
$359$		−	380.988i		−	1.06125i	−0.847607	−	0.530624i	$-0.821958\pi$
$359$		−	380.988i		−	1.06125i	0.847607	−	0.530624i	$-0.178042\pi$
$360$		0			0
$361$	−647.000			−1.79224
$362$		0			0
$363$	−226.899			−0.625065
$364$		0			0
$365$	588.000	−	256.624i	1.61096	−	0.703080i
$366$		0			0
$367$	613.146			1.67070			0.835349	−	0.549720i	$-0.185266\pi$
$367$	613.146			1.67070			0.835349	+	0.549720i	$0.185266\pi$
$368$		0			0
$369$	150.000			0.406504
$370$		0			0
$371$		−	95.2470i		−	0.256731i
$372$		0			0
$373$			302.450i			0.810858i	0.914127	+	0.405429i	$0.132878\pi$
$373$			302.450i			0.810858i	−0.914127	+	0.405429i	$0.867122\pi$
$374$		0			0
$375$	204.382	+	71.4353i	0.545019	+	0.190494i
$376$		0			0
$377$		−	73.3212i		−	0.194486i
$378$		0			0
$379$			380.988i			1.00525i	0.864506	+	0.502623i	$0.167632\pi$
$379$			380.988i			1.00525i	−0.864506	+	0.502623i	$0.832368\pi$
$380$		0			0
$381$	−42.0000			−0.110236
$382$		0			0
$383$	−311.769			−0.814019			−0.407009	−	0.913424i	$-0.633428\pi$
$383$	−311.769			−0.814019			−0.407009	+	0.913424i	$0.633428\pi$
$384$		0			0
$385$	−252.000	+	109.982i	−0.654545	+	0.285667i
$386$		0			0
$387$	187.061			0.483363
$388$		0			0
$389$	−484.000			−1.24422			−0.622108	−	0.782931i	$-0.713724\pi$
$389$	−484.000			−1.24422			−0.622108	+	0.782931i	$0.713724\pi$
$390$		0			0
$391$		−	253.992i		−	0.649596i
$392$		0			0
$393$		−	302.450i		−	0.769593i
$394$		0			0
$395$	581.969	−	253.992i	1.47334	−	0.643018i
$396$		0			0
$397$		−	100.817i		−	0.253946i	−0.991906	−	0.126973i	$-0.959474\pi$
$397$		−	100.817i		−	0.253946i	0.991906	−	0.126973i	$-0.0405262\pi$
$398$		0			0
$399$		−	190.494i		−	0.477429i
$400$		0			0
$401$	670.000			1.67082			0.835411	−	0.549625i	$-0.185229\pi$
$401$	670.000			1.67082			0.835411	+	0.549625i	$0.185229\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−18.0000	−	41.2432i	−0.0444444	−	0.101835i
$406$		0			0
$407$	727.461			1.78737
$408$		0			0
$409$	130.000			0.317848			0.158924	−	0.987291i	$-0.449197\pi$
$409$	130.000			0.317848			0.158924	+	0.987291i	$0.449197\pi$
$410$		0			0
$411$			79.3725i			0.193121i
$412$		0			0
$413$			54.9909i			0.133150i
$414$		0			0
$415$	−263.272	−	603.231i	−0.634390	−	1.45357i
$416$		0			0
$417$		−	109.982i		−	0.263745i
$418$		0			0
$419$		−	333.365i		−	0.795620i	−0.917468	−	0.397810i	$-0.869770\pi$
$419$		−	333.365i		−	0.795620i	0.917468	−	0.397810i	$-0.130230\pi$
$420$		0			0
$421$	106.000			0.251781			0.125891	−	0.992044i	$-0.459821\pi$
$421$	106.000			0.251781			0.125891	+	0.992044i	$0.459821\pi$
$422$		0			0
$423$	−145.492			−0.343953
$424$		0			0
$425$	168.000	+	155.808i	0.395294	+	0.366606i
$426$		0			0
$427$	−90.0666			−0.210929
$428$		0			0
$429$	252.000			0.587413
$430$		0			0
$431$			126.996i			0.294654i	0.989088	+	0.147327i	$0.0470670\pi$
$431$			126.996i			0.294654i	−0.989088	+	0.147327i	$0.952933\pi$
$432$		0			0
$433$			201.633i			0.465666i	0.972517	+	0.232833i	$0.0747995\pi$
$433$			201.633i			0.465666i	−0.972517	+	0.232833i	$0.925200\pi$
$434$		0			0
$435$	27.7128	+	63.4980i	0.0637076	+	0.145972i
$436$		0			0
$437$		−	879.855i		−	2.01340i
$438$		0			0
$439$		−	507.984i		−	1.15714i	−0.815633	−	0.578570i	$-0.803611\pi$
$439$		−	507.984i		−	1.15714i	0.815633	−	0.578570i	$-0.196389\pi$
$440$		0			0
$441$	−111.000			−0.251701
$442$		0			0
$443$	−214.774			−0.484818			−0.242409	−	0.970174i	$-0.577938\pi$
$443$	−214.774			−0.484818			−0.242409	+	0.970174i	$0.577938\pi$
$444$		0			0
$445$	172.000	+	394.102i	0.386517	+	0.885621i
$446$		0			0
$447$	−214.774			−0.480479
$448$		0			0
$449$	−286.000			−0.636971			−0.318486	−	0.947928i	$-0.603174\pi$
$449$	−286.000			−0.636971			−0.318486	+	0.947928i	$0.603174\pi$
$450$		0			0
$451$		−	793.725i		−	1.75992i
$452$		0			0
$453$		−	219.964i		−	0.485571i
$454$		0			0
$455$	145.492	−	63.4980i	0.319763	−	0.139556i
$456$		0			0
$457$			806.533i			1.76484i	0.470460	+	0.882422i	$0.344088\pi$
$457$			806.533i			1.76484i	−0.470460	+	0.882422i	$0.655912\pi$
$458$		0			0
$459$		−	47.6235i		−	0.103755i
$460$		0			0
$461$	−740.000			−1.60521			−0.802603	−	0.596514i	$-0.796552\pi$
$461$	−740.000			−1.60521			−0.802603	+	0.596514i	$0.796552\pi$
$462$		0			0
$463$	−426.084			−0.920269			−0.460134	−	0.887849i	$-0.652199\pi$
$463$	−426.084			−0.920269			−0.460134	+	0.887849i	$0.652199\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	20.7846			0.0445067			0.0222533	−	0.999752i	$-0.492916\pi$
$467$	20.7846			0.0445067			0.0222533	+	0.999752i	$0.492916\pi$
$468$		0			0
$469$	−192.000			−0.409382
$470$		0			0
$471$		−	460.361i		−	0.977411i
$472$		0			0
$473$		−	989.836i		−	2.09268i
$474$		0			0
$475$	581.969	+	539.733i	1.22520	+	1.13628i
$476$		0			0
$477$		−	82.4864i		−	0.172927i
$478$		0			0
$479$		−	285.741i		−	0.596537i	−0.954482	−	0.298268i	$-0.903591\pi$
$479$		−	285.741i		−	0.596537i	0.954482	−	0.298268i	$-0.0964091\pi$
$480$		0			0
$481$	−420.000			−0.873181
$482$		0			0
$483$	166.277			0.344259
$484$		0			0
$485$	504.000	−	219.964i	1.03918	−	0.453533i
$486$		0			0
$487$	509.223			1.04563			0.522816	−	0.852445i	$-0.324881\pi$
$487$	509.223			1.04563			0.522816	+	0.852445i	$0.324881\pi$
$488$		0			0
$489$	504.000			1.03067
$490$		0			0
$491$			79.3725i			0.161655i	0.996728	+	0.0808274i	$0.0257563\pi$
$491$			79.3725i			0.161655i	−0.996728	+	0.0808274i	$0.974244\pi$
$492$		0			0
$493$			73.3212i			0.148725i
$494$		0			0
$495$	−218.238	+	95.2470i	−0.440886	+	0.192418i
$496$		0			0
$497$			329.945i			0.663874i
$498$		0			0
$499$		−	158.745i		−	0.318126i	−0.987268	−	0.159063i	$-0.949153\pi$
$499$		−	158.745i		−	0.318126i	0.987268	−	0.159063i	$-0.0508474\pi$
$500$		0			0
$501$	312.000			0.622754
$502$		0			0
$503$	−422.620			−0.840200			−0.420100	−	0.907478i	$-0.638005\pi$
$503$	−422.620			−0.840200			−0.420100	+	0.907478i	$0.638005\pi$
$504$		0			0
$505$	64.0000	+	146.642i	0.126733	+	0.290381i
$506$		0			0
$507$	147.224			0.290383
$508$		0			0
$509$	40.0000			0.0785855			0.0392927	−	0.999228i	$-0.487490\pi$
$509$	40.0000			0.0785855			0.0392927	+	0.999228i	$0.487490\pi$
$510$		0			0
$511$			444.486i			0.869836i
$512$		0			0
$513$		−	164.973i		−	0.321584i
$514$		0			0
$515$	−187.061	−	428.612i	−0.363226	−	0.832256i
$516$		0			0
$517$			769.873i			1.48912i
$518$		0			0
$519$			555.608i			1.07054i
$520$		0			0
$521$	190.000			0.364683			0.182342	−	0.983235i	$-0.441632\pi$
$521$	190.000			0.364683			0.182342	+	0.983235i	$0.441632\pi$
$522$		0			0
$523$	−96.9948			−0.185459			−0.0927293	−	0.995691i	$-0.529559\pi$
$523$	−96.9948			−0.185459			−0.0927293	+	0.995691i	$0.529559\pi$
$524$		0			0
$525$	−102.000	+	109.982i	−0.194286	+	0.209489i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	239.000			0.451796
$530$		0			0
$531$			47.6235i			0.0896865i
$532$		0			0
$533$			458.258i			0.859770i
$534$		0			0
$535$	152.420	+	349.239i	0.284898	+	0.652784i
$536$		0			0
$537$		−	192.468i		−	0.358414i
$538$		0			0
$539$			587.357i			1.08972i
$540$		0			0
$541$	−562.000			−1.03882			−0.519409	−	0.854526i	$-0.673848\pi$
$541$	−562.000			−1.03882			−0.519409	+	0.854526i	$0.673848\pi$
$542$		0			0
$543$	107.387			0.197766
$544$		0			0
$545$	−260.000	−	595.735i	−0.477064	−	1.09309i
$546$		0			0
$547$	−547.328			−1.00060			−0.500300	−	0.865852i	$-0.666777\pi$
$547$	−547.328			−1.00060			−0.500300	+	0.865852i	$0.666777\pi$
$548$		0			0
$549$	−78.0000			−0.142077
$550$		0			0
$551$			253.992i			0.460966i
$552$		0			0
$553$			439.927i			0.795529i
$554$		0			0
$555$	363.731	−	158.745i	0.655371	−	0.286027i
$556$		0			0
$557$		−	119.147i		−	0.213908i	−0.994264	−	0.106954i	$-0.965890\pi$
$557$		−	119.147i		−	0.213908i	0.994264	−	0.106954i	$-0.0341098\pi$
$558$		0			0
$559$			571.482i			1.02233i
$560$		0			0
$561$	−252.000			−0.449198
$562$		0			0
$563$	575.041			1.02139			0.510693	−	0.859763i	$-0.329389\pi$
$563$	575.041			1.02139			0.510693	+	0.859763i	$0.329389\pi$
$564$		0			0
$565$	−546.000	+	238.294i	−0.966372	+	0.421759i
$566$		0			0
$567$	31.1769			0.0549857
$568$		0			0
$569$	−214.000			−0.376098			−0.188049	−	0.982160i	$-0.560216\pi$
$569$	−214.000			−0.376098			−0.188049	+	0.982160i	$0.560216\pi$
$570$		0			0
$571$			603.231i			1.05645i	0.849105	+	0.528224i	$0.177142\pi$
$571$			603.231i			1.05645i	−0.849105	+	0.528224i	$0.822858\pi$
$572$		0			0
$573$		−	54.9909i		−	0.0959702i
$574$		0			0
$575$	−471.118	+	507.984i	−0.819335	+	0.883451i
$576$		0			0
$577$		−	293.285i		−	0.508293i	−0.967166	−	0.254146i	$-0.918206\pi$
$577$		−	293.285i		−	0.508293i	0.967166	−	0.254146i	$-0.0817945\pi$
$578$		0			0
$579$		−	158.745i		−	0.274171i
$580$		0			0
$581$	456.000			0.784854
$582$		0			0
$583$	−436.477			−0.748674
$584$		0			0
$585$	126.000	−	54.9909i	0.215385	−	0.0940016i
$586$		0			0
$587$	−242.487			−0.413096			−0.206548	−	0.978437i	$-0.566223\pi$
$587$	−242.487			−0.413096			−0.206548	+	0.978437i	$0.566223\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$			301.616i			0.510348i
$592$		0			0
$593$			614.065i			1.03552i	0.855525	+	0.517762i	$0.173235\pi$
$593$			614.065i			1.03552i	−0.855525	+	0.517762i	$0.826765\pi$
$594$		0			0
$595$	−145.492	+	63.4980i	−0.244525	+	0.106719i
$596$		0			0
$597$		−	439.927i		−	0.736897i
$598$		0			0
$599$			888.972i			1.48409i	0.670348	+	0.742047i	$0.266145\pi$
$599$			888.972i			1.48409i	−0.670348	+	0.742047i	$0.733855\pi$
$600$		0			0
$601$	250.000			0.415973			0.207987	−	0.978132i	$-0.433309\pi$
$601$	250.000			0.415973			0.207987	+	0.978132i	$0.433309\pi$
$602$		0			0
$603$	−166.277			−0.275749
$604$		0			0
$605$	262.000	+	600.317i	0.433058	+	0.992260i
$606$		0			0
$607$	−336.018			−0.553571			−0.276786	−	0.960932i	$-0.589269\pi$
$607$	−336.018			−0.553571			−0.276786	+	0.960932i	$0.589269\pi$
$608$		0			0
$609$	−48.0000			−0.0788177
$610$		0			0
$611$		−	444.486i		−	0.727473i
$612$		0			0
$613$			467.423i			0.762517i	0.924469	+	0.381258i	$0.124509\pi$
$613$			467.423i			0.762517i	−0.924469	+	0.381258i	$0.875491\pi$
$614$		0			0
$615$	−173.205	−	396.863i	−0.281634	−	0.645305i
$616$		0			0
$617$		−	705.717i		−	1.14379i	−0.820328	−	0.571894i	$-0.806209\pi$
$617$		−	705.717i		−	1.14379i	0.820328	−	0.571894i	$-0.193791\pi$
$618$		0			0
$619$		−	1015.97i		−	1.64131i	−0.571427	−	0.820653i	$-0.693610\pi$
$619$		−	1015.97i		−	1.64131i	0.571427	−	0.820653i	$-0.306390\pi$
$620$		0			0
$621$	144.000			0.231884
$622$		0			0
$623$	−297.913			−0.478191
$624$		0			0
$625$	−47.0000	−	623.230i	−0.0752000	−	0.997168i
$626$		0			0
$627$	−872.954			−1.39227
$628$		0			0
$629$	420.000			0.667727
$630$		0			0
$631$			888.972i			1.40883i	0.709788	+	0.704416i	$0.248791\pi$
$631$			888.972i			1.40883i	−0.709788	+	0.704416i	$0.751209\pi$
$632$		0			0
$633$			109.982i			0.173747i
$634$		0			0
$635$	48.4974	+	111.122i	0.0763739	+	0.174995i
$636$		0			0
$637$		−	339.111i		−	0.532356i
$638$		0			0
$639$			285.741i			0.447169i
$640$		0			0
$641$	242.000			0.377535			0.188768	−	0.982022i	$-0.439551\pi$
$641$	242.000			0.377535			0.188768	+	0.982022i	$0.439551\pi$
$642$		0			0
$643$	775.959			1.20678			0.603389	−	0.797447i	$-0.293816\pi$
$643$	775.959			1.20678			0.603389	+	0.797447i	$0.293816\pi$
$644$		0			0
$645$	−216.000	−	494.918i	−0.334884	−	0.767315i
$646$		0			0
$647$	200.918			0.310538			0.155269	−	0.987872i	$-0.450376\pi$
$647$	200.918			0.310538			0.155269	+	0.987872i	$0.450376\pi$
$648$		0			0
$649$	252.000			0.388290
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	742.377i		−	1.13687i	−0.822728	−	0.568436i	$-0.807549\pi$
$653$		−	742.377i		−	1.13687i	0.822728	−	0.568436i	$-0.192451\pi$
$654$		0			0
$655$	−800.207	+	349.239i	−1.22169	+	0.533190i
$656$		0			0
$657$			384.936i			0.585900i
$658$		0			0
$659$		−	15.8745i		−	0.0240888i	−0.999927	−	0.0120444i	$-0.996166\pi$
$659$		−	15.8745i		−	0.0240888i	0.999927	−	0.0120444i	$-0.00383394\pi$
$660$		0			0
$661$	382.000			0.577912			0.288956	−	0.957342i	$-0.406692\pi$
$661$	382.000			0.577912			0.288956	+	0.957342i	$0.406692\pi$
$662$		0			0
$663$	145.492			0.219445
$664$		0			0
$665$	−504.000	+	219.964i	−0.757895	+	0.330772i
$666$		0			0
$667$	−221.703			−0.332388
$668$		0			0
$669$	−294.000			−0.439462
$670$		0			0
$671$			412.737i			0.615108i
$672$		0			0
$673$		−	641.561i		−	0.953285i	−0.879097	−	0.476642i	$-0.841854\pi$
$673$		−	641.561i		−	0.953285i	0.879097	−	0.476642i	$-0.158146\pi$
$674$		0			0
$675$	−88.3346	+	95.2470i	−0.130866	+	0.141107i
$676$		0			0
$677$			962.341i			1.42148i	0.703456	+	0.710739i	$0.251639\pi$
$677$			962.341i			1.42148i	−0.703456	+	0.710739i	$0.748361\pi$
$678$		0			0
$679$			380.988i			0.561102i
$680$		0			0
$681$	−252.000			−0.370044
$682$		0			0
$683$	394.908			0.578196			0.289098	−	0.957300i	$-0.406645\pi$
$683$	394.908			0.578196			0.289098	+	0.957300i	$0.406645\pi$
$684$		0			0
$685$	210.000	−	91.6515i	0.306569	−	0.133798i
$686$		0			0
$687$	564.649			0.821905
$688$		0			0
$689$	252.000			0.365747
$690$		0			0
$691$		−	222.243i		−	0.321625i	−0.986985	−	0.160813i	$-0.948589\pi$
$691$		−	222.243i		−	0.321625i	0.986985	−	0.160813i	$-0.0514115\pi$
$692$		0			0
$693$		−	164.973i		−	0.238056i
$694$		0			0
$695$	−290.985	+	126.996i	−0.418683	+	0.182728i
$696$		0			0
$697$		−	458.258i		−	0.657471i
$698$		0			0
$699$			79.3725i			0.113552i
$700$		0			0
$701$	−328.000			−0.467903			−0.233951	−	0.972248i	$-0.575166\pi$
$701$	−328.000			−0.467903			−0.233951	+	0.972248i	$0.575166\pi$
$702$		0			0
$703$	1454.92			2.06959
$704$		0			0
$705$	168.000	+	384.936i	0.238298	+	0.546009i
$706$		0			0
$707$	−110.851			−0.156791
$708$		0			0
$709$	−1234.00			−1.74048			−0.870240	−	0.492628i	$-0.836036\pi$
$709$	−1234.00			−1.74048			−0.870240	+	0.492628i	$0.836036\pi$
$710$		0			0
$711$			380.988i			0.535848i
$712$		0			0
$713$		0			0
$714$		0			0
$715$	−290.985	−	666.729i	−0.406971	−	0.932489i
$716$		0			0
$717$			384.936i			0.536871i
$718$		0			0
$719$			63.4980i			0.0883144i	0.999025	+	0.0441572i	$0.0140602\pi$
$719$			63.4980i			0.0883144i	−0.999025	+	0.0441572i	$0.985940\pi$
$720$		0			0
$721$	324.000			0.449376
$722$		0			0
$723$	204.382			0.282686
$724$		0			0
$725$	136.000	−	146.642i	0.187586	−	0.202265i
$726$		0			0
$727$	−377.587			−0.519377			−0.259688	−	0.965692i	$-0.583620\pi$
$727$	−377.587			−0.519377			−0.259688	+	0.965692i	$0.583620\pi$
$728$		0			0
$729$	27.0000			0.0370370
$730$		0			0
$731$		−	571.482i		−	0.781782i
$732$		0			0
$733$		−	742.377i		−	1.01279i	−0.862301	−	0.506396i	$-0.830977\pi$
$733$		−	742.377i		−	1.01279i	0.862301	−	0.506396i	$-0.169023\pi$
$734$		0			0
$735$	128.172	+	293.678i	0.174383	+	0.399562i
$736$		0			0
$737$			879.855i			1.19383i
$738$		0			0
$739$			158.745i			0.214811i	0.994215	+	0.107405i	$0.0342543\pi$
$739$			158.745i			0.214811i	−0.994215	+	0.107405i	$0.965746\pi$
$740$		0			0
$741$	504.000			0.680162
$742$		0			0
$743$	131.636			0.177168			0.0885840	−	0.996069i	$-0.471766\pi$
$743$	131.636			0.177168			0.0885840	+	0.996069i	$0.471766\pi$
$744$		0			0
$745$	248.000	+	568.239i	0.332886	+	0.762737i
$746$		0			0
$747$	394.908			0.528658
$748$		0			0
$749$	−264.000			−0.352470
$750$		0			0
$751$		−	1206.46i		−	1.60647i	−0.595659	−	0.803237i	$-0.703109\pi$
$751$		−	1206.46i		−	1.60647i	0.595659	−	0.803237i	$-0.296891\pi$
$752$		0			0
$753$			797.368i			1.05892i
$754$		0			0
$755$	−581.969	+	253.992i	−0.770820	+	0.336413i
$756$		0			0
$757$			412.432i			0.544824i	0.962181	+	0.272412i	$0.0878214\pi$
$757$			412.432i			0.544824i	−0.962181	+	0.272412i	$0.912179\pi$
$758$		0			0
$759$		−	761.976i		−	1.00392i
$760$		0			0
$761$	214.000			0.281209			0.140604	−	0.990066i	$-0.455095\pi$
$761$	214.000			0.281209			0.140604	+	0.990066i	$0.455095\pi$
$762$		0			0
$763$	450.333			0.590214
$764$		0			0
$765$	−126.000	+	54.9909i	−0.164706	+	0.0718835i
$766$		0			0
$767$	−145.492			−0.189690
$768$		0			0
$769$	254.000			0.330299			0.165150	−	0.986269i	$-0.447189\pi$
$769$	254.000			0.330299			0.165150	+	0.986269i	$0.447189\pi$
$770$		0			0
$771$		−	492.110i		−	0.638275i
$772$		0			0
$773$		−	265.789i		−	0.343841i	−0.985111	−	0.171921i	$-0.945003\pi$
$773$		−	265.789i		−	0.343841i	0.985111	−	0.171921i	$-0.0549973\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$			274.955i			0.353867i
$778$		0			0
$779$		−	1587.45i		−	2.03781i
$780$		0			0
$781$	1512.00			1.93598
$782$		0			0
$783$	−41.5692			−0.0530897
$784$		0			0
$785$	−1218.00	+	531.579i	−1.55159	+	0.677170i
$786$		0			0
$787$	623.538			0.792298			0.396149	−	0.918186i	$-0.370346\pi$
$787$	623.538			0.792298			0.396149	+	0.918186i	$0.370346\pi$
$788$		0			0
$789$	−432.000			−0.547529
$790$		0			0
$791$		−	412.737i		−	0.521792i
$792$		0			0
$793$		−	238.294i		−	0.300497i
$794$		0			0
$795$	−218.238	+	95.2470i	−0.274514	+	0.119808i
$796$		0			0
$797$		−	760.708i		−	0.954464i	−0.878777	−	0.477232i	$-0.841640\pi$
$797$		−	760.708i		−	0.954464i	0.878777	−	0.477232i	$-0.158360\pi$
$798$		0			0
$799$			444.486i			0.556303i
$800$		0			0
$801$	−258.000			−0.322097
$802$		0			0
$803$	2036.89			2.53660
$804$		0			0
$805$	−192.000	−	439.927i	−0.238509	−	0.546493i
$806$		0			0
$807$	−817.528			−1.01305
$808$		0			0
$809$	−746.000			−0.922126			−0.461063	−	0.887367i	$-0.652532\pi$
$809$	−746.000			−0.922126			−0.461063	+	0.887367i	$0.652532\pi$
$810$		0			0
$811$		−	698.478i		−	0.861256i	−0.902530	−	0.430628i	$-0.858292\pi$
$811$		−	698.478i		−	0.861256i	0.902530	−	0.430628i	$-0.141708\pi$
$812$		0			0
$813$			329.945i			0.405837i
$814$		0			0
$815$	−581.969	−	1333.46i	−0.714072	−	1.63615i
$816$		0			0
$817$		−	1979.67i		−	2.42310i
$818$		0			0
$819$			95.2470i			0.116297i
$820$		0			0
$821$	−928.000			−1.13033			−0.565164	−	0.824978i	$-0.691187\pi$
$821$	−928.000			−1.13033			−0.565164	+	0.824978i	$0.691187\pi$
$822$		0			0
$823$	−65.8179			−0.0799732			−0.0399866	−	0.999200i	$-0.512732\pi$
$823$	−65.8179			−0.0799732			−0.0399866	+	0.999200i	$0.512732\pi$
$824$		0			0
$825$	504.000	+	467.423i	0.610909	+	0.566573i
$826$		0			0
$827$	−713.605			−0.862884			−0.431442	−	0.902141i	$-0.641995\pi$
$827$	−713.605			−0.862884			−0.431442	+	0.902141i	$0.641995\pi$
$828$		0			0
$829$	874.000			1.05428			0.527141	−	0.849778i	$-0.323264\pi$
$829$	874.000			1.05428			0.527141	+	0.849778i	$0.323264\pi$
$830$		0			0
$831$		−	428.612i		−	0.515778i
$832$		0			0
$833$			339.111i			0.407096i
$834$		0			0
$835$	−360.267	−	825.474i	−0.431457	−	0.988592i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	571.482i		−	0.681147i	−0.940218	−	0.340573i	$-0.889379\pi$
$839$		−	571.482i		−	0.681147i	0.940218	−	0.340573i	$-0.110621\pi$
$840$		0			0
$841$	−777.000			−0.923900
$842$		0			0
$843$	481.510			0.571186
$844$		0			0
$845$	−170.000	−	389.519i	−0.201183	−	0.460969i
$846$		0			0
$847$	−453.797			−0.535770
$848$		0			0
$849$	−840.000			−0.989399
$850$		0			0
$851$			1269.96i			1.49232i
$852$		0			0
$853$			577.405i			0.676910i	0.940983	+	0.338455i	$0.109904\pi$
$853$			577.405i			0.676910i	−0.940983	+	0.338455i	$0.890096\pi$
$854$		0			0
$855$	−436.477	+	190.494i	−0.510499	+	0.222800i
$856$		0			0
$857$			504.083i			0.588195i	0.955775	+	0.294098i	$0.0950191\pi$
$857$			504.083i			0.588195i	−0.955775	+	0.294098i	$0.904981\pi$
$858$		0			0
$859$			253.992i			0.295683i	0.989011	+	0.147842i	$0.0472326\pi$
$859$			253.992i			0.295683i	−0.989011	+	0.147842i	$0.952767\pi$
$860$		0			0
$861$	300.000			0.348432
$862$		0			0
$863$	1718.19			1.99096			0.995478	−	0.0949962i	$-0.0302839\pi$
$863$	1718.19			1.99096			0.995478	+	0.0949962i	$0.0302839\pi$
$864$		0			0
$865$	1470.00	−	641.561i	1.69942	−	0.741689i
$866$		0			0
$867$	355.070			0.409539
$868$		0			0
$869$	2016.00			2.31991
$870$		0			0
$871$		−	507.984i		−	0.583220i
$872$		0			0
$873$			329.945i			0.377944i
$874$		0			0
$875$	408.764	+	142.871i	0.467159	+	0.163281i
$876$		0			0
$877$			100.817i			0.114956i	0.998347	+	0.0574781i	$0.0183060\pi$
$877$			100.817i			0.114956i	−0.998347	+	0.0574781i	$0.981694\pi$
$878$		0			0
$879$		−	555.608i		−	0.632091i
$880$		0			0
$881$	134.000			0.152100			0.0760499	−	0.997104i	$-0.475769\pi$
$881$	134.000			0.152100			0.0760499	+	0.997104i	$0.475769\pi$
$882$		0			0
$883$	193.990			0.219694			0.109847	−	0.993949i	$-0.464964\pi$
$883$	193.990			0.219694			0.109847	+	0.993949i	$0.464964\pi$
$884$		0			0
$885$	126.000	−	54.9909i	0.142373	−	0.0621366i
$886$		0			0
$887$	124.708			0.140595			0.0702974	−	0.997526i	$-0.477605\pi$
$887$	124.708			0.140595			0.0702974	+	0.997526i	$0.477605\pi$
$888$		0			0
$889$	−84.0000			−0.0944882
$890$		0			0
$891$		−	142.871i		−	0.160349i
$892$		0			0
$893$			1539.75i			1.72424i
$894$		0			0
$895$	−509.223	+	222.243i	−0.568964	+	0.248316i
$896$		0			0
$897$			439.927i			0.490443i
$898$		0			0
$899$		0			0
$900$		0			0
$901$	−252.000			−0.279689
$902$		0			0
$903$	374.123			0.414311
$904$		0			0
$905$	−124.000	−	284.120i	−0.137017	−	0.313944i
$906$		0			0
$907$	297.913			0.328459			0.164230	−	0.986422i	$-0.447486\pi$
$907$	297.913			0.328459			0.164230	+	0.986422i	$0.447486\pi$
$908$		0			0
$909$	−96.0000			−0.105611
$910$		0			0
$911$			507.984i			0.557612i	0.960347	+	0.278806i	$0.0899386\pi$
$911$			507.984i			0.557612i	−0.960347	+	0.278806i	$0.910061\pi$
$912$		0			0
$913$		−	2089.65i		−	2.28878i
$914$		0			0
$915$	90.0666	+	206.369i	0.0984335	+	0.225539i
$916$		0			0
$917$		−	604.900i		−	0.659651i
$918$		0			0
$919$		−	698.478i		−	0.760042i	−0.924978	−	0.380021i	$-0.875917\pi$
$919$		−	698.478i		−	0.760042i	0.924978	−	0.380021i	$-0.124083\pi$
$920$		0			0
$921$	−444.000			−0.482085
$922$		0			0
$923$	−872.954			−0.945779
$924$		0			0
$925$	−840.000	−	779.038i	−0.908108	−	0.842203i
$926$		0			0
$927$	280.592			0.302688
$928$		0			0
$929$	1010.00			1.08719			0.543595	−	0.839347i	$-0.317063\pi$
$929$	1010.00			1.08719			0.543595	+	0.839347i	$0.317063\pi$
$930$		0			0
$931$			1174.71i			1.26178i
$932$		0			0
$933$		−	714.882i		−	0.766218i
$934$		0			0
$935$	290.985	+	666.729i	0.311213	+	0.713079i
$936$		0			0
$937$			183.303i			0.195628i	0.995205	+	0.0978138i	$0.0311850\pi$
$937$			183.303i			0.195628i	−0.995205	+	0.0978138i	$0.968815\pi$
$938$		0			0
$939$			349.239i			0.371927i
$940$		0			0
$941$	−1532.00			−1.62806			−0.814028	−	0.580826i	$-0.802730\pi$
$941$	−1532.00			−1.62806			−0.814028	+	0.580826i	$0.802730\pi$
$942$		0			0
$943$	1385.64			1.46940
$944$		0			0
$945$	−36.0000	−	82.4864i	−0.0380952	−	0.0872872i
$946$		0			0
$947$	−1046.16			−1.10471			−0.552354	−	0.833610i	$-0.686270\pi$
$947$	−1046.16			−1.10471			−0.552354	+	0.833610i	$0.686270\pi$
$948$		0			0
$949$	−1176.00			−1.23920
$950$		0			0
$951$			968.345i			1.01824i
$952$		0			0
$953$			1053.99i			1.10597i	0.833190	+	0.552987i	$0.186512\pi$
$953$			1053.99i			1.10597i	−0.833190	+	0.552987i	$0.813488\pi$
$954$		0			0
$955$	−145.492	+	63.4980i	−0.152348	+	0.0664901i
$956$		0			0
$957$			219.964i			0.229847i
$958$		0			0
$959$			158.745i			0.165532i
$960$		0			0
$961$	961.000			1.00000
$962$		0			0
$963$	−228.631			−0.237415
$964$		0			0
$965$	−420.000	+	183.303i	−0.435233	+	0.189951i
$966$		0			0
$967$	−204.382			−0.211357			−0.105678	−	0.994400i	$-0.533701\pi$
$967$	−204.382			−0.211357			−0.105678	+	0.994400i	$0.533701\pi$
$968$		0			0
$969$	−504.000			−0.520124
$970$		0			0
$971$		−	1698.57i		−	1.74930i	−0.484753	−	0.874651i	$-0.661091\pi$
$971$		−	1698.57i		−	1.74930i	0.484753	−	0.874651i	$-0.338909\pi$
$972$		0			0
$973$		−	219.964i		−	0.226067i
$974$		0			0
$975$	−290.985	−	269.867i	−0.298446	−	0.276786i
$976$		0			0
$977$			687.386i			0.703568i	0.936081	+	0.351784i	$0.114425\pi$
$977$			687.386i			0.703568i	−0.936081	+	0.351784i	$0.885575\pi$
$978$		0			0
$979$			1365.21i			1.39449i
$980$		0			0
$981$	390.000			0.397554
$982$		0			0
$983$	−491.902			−0.500409			−0.250205	−	0.968193i	$-0.580498\pi$
$983$	−491.902			−0.500409			−0.250205	+	0.968193i	$0.580498\pi$
$984$		0			0
$985$	798.000	−	348.276i	0.810152	−	0.353579i
$986$		0			0
$987$	−290.985			−0.294817
$988$		0			0
$989$	1728.00			1.74722
$990$		0			0
$991$		−	63.4980i		−	0.0640747i	−0.999487	−	0.0320374i	$-0.989800\pi$
$991$		−	63.4980i		−	0.0640747i	0.999487	−	0.0320374i	$-0.0101996\pi$
$992$		0			0
$993$			164.973i			0.166136i
$994$		0			0
$995$	−1163.94	+	507.984i	−1.16979	+	0.510537i
$996$		0			0
$997$			430.762i			0.432058i	0.976387	+	0.216029i	$0.0693106\pi$
$997$			430.762i			0.432058i	−0.976387	+	0.216029i	$0.930689\pi$
$998$		0			0
$999$			238.118i			0.238356i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	240.3.j.b.79.3	yes	4
3.2	odd	2		720.3.j.f.559.4		4
4.3	odd	2	inner	240.3.j.b.79.1	✓	4
5.2	odd	4		1200.3.e.l.751.1		4
5.3	odd	4		1200.3.e.l.751.3		4
5.4	even	2	inner	240.3.j.b.79.2	yes	4
8.3	odd	2		960.3.j.c.319.4		4
8.5	even	2		960.3.j.c.319.2		4
12.11	even	2		720.3.j.f.559.3		4
15.2	even	4		3600.3.e.be.3151.4		4
15.8	even	4		3600.3.e.be.3151.2		4
15.14	odd	2		720.3.j.f.559.1		4
20.3	even	4		1200.3.e.l.751.2		4
20.7	even	4		1200.3.e.l.751.4		4
20.19	odd	2	inner	240.3.j.b.79.4	yes	4
40.19	odd	2		960.3.j.c.319.1		4
40.29	even	2		960.3.j.c.319.3		4
60.23	odd	4		3600.3.e.be.3151.3		4
60.47	odd	4		3600.3.e.be.3151.1		4
60.59	even	2		720.3.j.f.559.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
240.3.j.b.79.1	✓	4	4.3	odd	2	inner
240.3.j.b.79.2	yes	4	5.4	even	2	inner
240.3.j.b.79.3	yes	4	1.1	even	1	trivial
240.3.j.b.79.4	yes	4	20.19	odd	2	inner
720.3.j.f.559.1		4	15.14	odd	2
720.3.j.f.559.2		4	60.59	even	2
720.3.j.f.559.3		4	12.11	even	2
720.3.j.f.559.4		4	3.2	odd	2
960.3.j.c.319.1		4	40.19	odd	2
960.3.j.c.319.2		4	8.5	even	2
960.3.j.c.319.3		4	40.29	even	2
960.3.j.c.319.4		4	8.3	odd	2
1200.3.e.l.751.1		4	5.2	odd	4
1200.3.e.l.751.2		4	20.3	even	4
1200.3.e.l.751.3		4	5.3	odd	4
1200.3.e.l.751.4		4	20.7	even	4
3600.3.e.be.3151.1		4	60.47	odd	4
3600.3.e.be.3151.2		4	15.8	even	4
3600.3.e.be.3151.3		4	60.23	odd	4
3600.3.e.be.3151.4		4	15.2	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	240.j (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.73205 q^{3} +(-2.00000 - 4.58258i) q^{5} +3.46410 q^{7} +3.00000 q^{9} +O(q^{10})\)	\(q+1.73205 q^{3} +(-2.00000 - 4.58258i) q^{5} +3.46410 q^{7} +3.00000 q^{9} -15.8745i q^{11} +9.16515i q^{13} +(-3.46410 - 7.93725i) q^{15} -9.16515i q^{17} -31.7490i q^{19} +6.00000 q^{21} +27.7128 q^{23} +(-17.0000 + 18.3303i) q^{25} +5.19615 q^{27} -8.00000 q^{29} -27.4955i q^{33} +(-6.92820 - 15.8745i) q^{35} +45.8258i q^{37} +15.8745i q^{39} +50.0000 q^{41} +62.3538 q^{43} +(-6.00000 - 13.7477i) q^{45} -48.4974 q^{47} -37.0000 q^{49} -15.8745i q^{51} -27.4955i q^{53} +(-72.7461 + 31.7490i) q^{55} -54.9909i q^{57} +15.8745i q^{59} -26.0000 q^{61} +10.3923 q^{63} +(42.0000 - 18.3303i) q^{65} -55.4256 q^{67} +48.0000 q^{69} +95.2470i q^{71} +128.312i q^{73} +(-29.4449 + 31.7490i) q^{75} -54.9909i q^{77} +126.996i q^{79} +9.00000 q^{81} +131.636 q^{83} +(-42.0000 + 18.3303i) q^{85} -13.8564 q^{87} -86.0000 q^{89} +31.7490i q^{91} +(-145.492 + 63.4980i) q^{95} +109.982i q^{97} -47.6235i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{5} + 12 q^{9}+O(q^{10}) \) 4 * q - 8 * q^5 + 12 * q^9	\( 4 q - 8 q^{5} + 12 q^{9} + 24 q^{21} - 68 q^{25} - 32 q^{29} + 200 q^{41} - 24 q^{45} - 148 q^{49} - 104 q^{61} + 168 q^{65} + 192 q^{69} + 36 q^{81} - 168 q^{85} - 344 q^{89}+O(q^{100}) \) 4 * q - 8 * q^5 + 12 * q^9 + 24 * q^21 - 68 * q^25 - 32 * q^29 + 200 * q^41 - 24 * q^45 - 148 * q^49 - 104 * q^61 + 168 * q^65 + 192 * q^69 + 36 * q^81 - 168 * q^85 - 344 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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