LMFDB - Embedded newform 1050.3.h.c.349.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1050,3,Mod(349,1050)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1050, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1050.349");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1050.h (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:	$28.6104277578$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			349.14
Character	$\chi$	$=$	1050.349
Dual form			1050.3.h.c.349.13

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421i q^{2} +1.73205 q^{3} -2.00000 q^{4} +2.44949i q^{6} +(1.07086 + 6.91761i) q^{7} -2.82843i q^{8} +3.00000 q^{9} +O(q^{10})$	\(q+1.41421i q^{2} +1.73205 q^{3} -2.00000 q^{4} +2.44949i q^{6} +(1.07086 + 6.91761i) q^{7} -2.82843i q^{8} +3.00000 q^{9} -9.92899 q^{11} -3.46410 q^{12} +20.2629 q^{13} +(-9.78297 + 1.51442i) q^{14} +4.00000 q^{16} -18.9405 q^{17} +4.24264i q^{18} +4.68861i q^{19} +(1.85478 + 11.9816i) q^{21} -14.0417i q^{22} +25.4738i q^{23} -4.89898i q^{24} +28.6560i q^{26} +5.19615 q^{27} +(-2.14171 - 13.8352i) q^{28} -15.9850 q^{29} +20.9499i q^{31} +5.65685i q^{32} -17.1975 q^{33} -26.7859i q^{34} -6.00000 q^{36} +30.1846i q^{37} -6.63069 q^{38} +35.0963 q^{39} +42.2896i q^{41} +(-16.9446 + 2.62305i) q^{42} -49.7705i q^{43} +19.8580 q^{44} -36.0253 q^{46} +2.91549 q^{47} +6.92820 q^{48} +(-46.7065 + 14.8155i) q^{49} -32.8059 q^{51} -40.5258 q^{52} -11.1921i q^{53} +7.34847i q^{54} +(19.5659 - 3.02884i) q^{56} +8.12091i q^{57} -22.6062i q^{58} -63.6910i q^{59} +89.2830i q^{61} -29.6277 q^{62} +(3.21257 + 20.7528i) q^{63} -8.00000 q^{64} -24.3210i q^{66} +13.5923i q^{67} +37.8810 q^{68} +44.1218i q^{69} -50.9899 q^{71} -8.48528i q^{72} -78.9059 q^{73} -42.6875 q^{74} -9.37722i q^{76} +(-10.6325 - 68.6848i) q^{77} +49.6337i q^{78} +69.8667 q^{79} +9.00000 q^{81} -59.8066 q^{82} -68.8876 q^{83} +(-3.70956 - 23.9633i) q^{84} +70.3860 q^{86} -27.6868 q^{87} +28.0834i q^{88} +156.083i q^{89} +(21.6987 + 140.171i) q^{91} -50.9475i q^{92} +36.2863i q^{93} +4.12313i q^{94} +9.79796i q^{96} -154.698 q^{97} +(-20.9523 - 66.0530i) q^{98} -29.7870 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 48 q^{4} + 72 q^{9}+O(q^{10}) $ 24 * q - 48 * q^4 + 72 * q^9	$ 24 q - 48 q^{4} + 72 q^{9} - 32 q^{11} - 16 q^{14} + 96 q^{16} - 12 q^{21} - 96 q^{29} - 144 q^{36} + 24 q^{39} + 64 q^{44} + 160 q^{46} - 236 q^{49} + 144 q^{51} + 32 q^{56} - 192 q^{64} + 496 q^{71} + 128 q^{74} + 416 q^{79} + 216 q^{81} + 24 q^{84} + 256 q^{86} - 316 q^{91} - 96 q^{99}+O(q^{100}) $ 24 * q - 48 * q^4 + 72 * q^9 - 32 * q^11 - 16 * q^14 + 96 * q^16 - 12 * q^21 - 96 * q^29 - 144 * q^36 + 24 * q^39 + 64 * q^44 + 160 * q^46 - 236 * q^49 + 144 * q^51 + 32 * q^56 - 192 * q^64 + 496 * q^71 + 128 * q^74 + 416 * q^79 + 216 * q^81 + 24 * q^84 + 256 * q^86 - 316 * q^91 - 96 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1050\mathbb{Z}\right)^\times$.

$n$	$127$	$451$	$701$
$\chi(n)$	$-1$	$-1$	$1$

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 $

$2$			1.41421i			0.707107i
$2$			1.41421i			0.707107i
$3$	1.73205			0.577350
$3$	1.73205			0.577350
$4$	−2.00000			−0.500000
$5$		0			0
$5$		0			0
$6$			2.44949i			0.408248i
$7$	1.07086	+	6.91761i	0.152980	+	0.988229i
$7$	1.07086	+	6.91761i	0.152980	+	0.988229i
$8$		−	2.82843i		−	0.353553i
$9$	3.00000			0.333333
$10$		0			0
$11$	−9.92899			−0.902635			−0.451318	−	0.892363i	$-0.649046\pi$
$11$	−9.92899			−0.902635			−0.451318	+	0.892363i	$0.649046\pi$
$12$	−3.46410			−0.288675
$13$	20.2629			1.55868			0.779342	−	0.626599i	$-0.215554\pi$
$13$	20.2629			1.55868			0.779342	+	0.626599i	$0.215554\pi$
$14$	−9.78297	+	1.51442i	−0.698784	+	0.108173i
$15$		0			0
$16$	4.00000			0.250000
$17$	−18.9405			−1.11415			−0.557073	−	0.830463i	$-0.688076\pi$
$17$	−18.9405			−1.11415			−0.557073	+	0.830463i	$0.688076\pi$
$18$			4.24264i			0.235702i
$19$			4.68861i			0.246769i	0.992359	+	0.123384i	$0.0393748\pi$
$19$			4.68861i			0.246769i	−0.992359	+	0.123384i	$0.960625\pi$
$20$		0			0
$21$	1.85478	+	11.9816i	0.0883228	+	0.570554i
$22$		−	14.0417i		−	0.638259i
$23$			25.4738i			1.10755i	0.832665	+	0.553777i	$0.186814\pi$
$23$			25.4738i			1.10755i	−0.832665	+	0.553777i	$0.813186\pi$
$24$		−	4.89898i		−	0.204124i
$25$		0			0
$26$			28.6560i			1.10216i
$27$	5.19615			0.192450
$28$	−2.14171	−	13.8352i	−0.0764898	−	0.494115i
$29$	−15.9850			−0.551207			−0.275604	−	0.961271i	$-0.588878\pi$
$29$	−15.9850			−0.551207			−0.275604	+	0.961271i	$0.588878\pi$
$30$		0			0
$31$			20.9499i			0.675804i	0.941181	+	0.337902i	$0.109717\pi$
$31$			20.9499i			0.675804i	−0.941181	+	0.337902i	$0.890283\pi$
$32$			5.65685i			0.176777i
$33$	−17.1975			−0.521137
$34$		−	26.7859i		−	0.787821i
$35$		0			0
$36$	−6.00000			−0.166667
$37$			30.1846i			0.815800i	0.913027	+	0.407900i	$0.133739\pi$
$37$			30.1846i			0.815800i	−0.913027	+	0.407900i	$0.866261\pi$
$38$	−6.63069			−0.174492
$39$	35.0963			0.899906
$40$		0			0
$41$			42.2896i			1.03145i	0.856753	+	0.515727i	$0.172478\pi$
$41$			42.2896i			1.03145i	−0.856753	+	0.515727i	$0.827522\pi$
$42$	−16.9446	+	2.62305i	−0.403443	+	0.0624537i
$43$		−	49.7705i		−	1.15745i	−0.815522	−	0.578726i	$-0.803550\pi$
$43$		−	49.7705i		−	1.15745i	0.815522	−	0.578726i	$-0.196450\pi$
$44$	19.8580			0.451318
$45$		0			0
$46$	−36.0253			−0.783159
$47$	2.91549			0.0620318			0.0310159	−	0.999519i	$-0.490126\pi$
$47$	2.91549			0.0620318			0.0310159	+	0.999519i	$0.490126\pi$
$48$	6.92820			0.144338
$49$	−46.7065	+	14.8155i	−0.953194	+	0.302358i
$50$		0			0
$51$	−32.8059			−0.643253
$52$	−40.5258			−0.779342
$53$		−	11.1921i		−	0.211171i	−0.994410	−	0.105586i	$-0.966328\pi$
$53$		−	11.1921i		−	0.211171i	0.994410	−	0.105586i	$-0.0336717\pi$
$54$			7.34847i			0.136083i
$55$		0			0
$56$	19.5659	−	3.02884i	0.349392	−	0.0540865i
$57$			8.12091i			0.142472i
$58$		−	22.6062i		−	0.389762i
$59$		−	63.6910i		−	1.07951i	−0.841823	−	0.539754i	$-0.818517\pi$
$59$		−	63.6910i		−	1.07951i	0.841823	−	0.539754i	$-0.181483\pi$
$60$		0			0
$61$			89.2830i			1.46365i	0.681490	+	0.731827i	$0.261332\pi$
$61$			89.2830i			1.46365i	−0.681490	+	0.731827i	$0.738668\pi$
$62$	−29.6277			−0.477865
$63$	3.21257	+	20.7528i	0.0509932	+	0.329410i
$64$	−8.00000			−0.125000
$65$		0			0
$66$		−	24.3210i		−	0.368499i
$67$			13.5923i			0.202871i	0.994842	+	0.101435i	$0.0323435\pi$
$67$			13.5923i			0.202871i	−0.994842	+	0.101435i	$0.967656\pi$
$68$	37.8810			0.557073
$69$			44.1218i			0.639447i
$70$		0			0
$71$	−50.9899			−0.718168			−0.359084	−	0.933305i	$-0.616911\pi$
$71$	−50.9899			−0.718168			−0.359084	+	0.933305i	$0.616911\pi$
$72$		−	8.48528i		−	0.117851i
$73$	−78.9059			−1.08090			−0.540451	−	0.841375i	$-0.681746\pi$
$73$	−78.9059			−1.08090			−0.540451	+	0.841375i	$0.681746\pi$
$74$	−42.6875			−0.576858
$75$		0			0
$76$		−	9.37722i		−	0.123384i
$77$	−10.6325	−	68.6848i	−0.138085	−	0.892011i
$78$			49.6337i			0.636330i
$79$	69.8667			0.884388			0.442194	−	0.896919i	$-0.354200\pi$
$79$	69.8667			0.884388			0.442194	+	0.896919i	$0.354200\pi$
$80$		0			0
$81$	9.00000			0.111111
$82$	−59.8066			−0.729348
$83$	−68.8876			−0.829971			−0.414985	−	0.909828i	$-0.636213\pi$
$83$	−68.8876			−0.829971			−0.414985	+	0.909828i	$0.636213\pi$
$84$	−3.70956	−	23.9633i	−0.0441614	−	0.285277i
$85$		0			0
$86$	70.3860			0.818442
$87$	−27.6868			−0.318240
$88$			28.0834i			0.319130i
$89$			156.083i			1.75374i	0.480726	+	0.876871i	$0.340373\pi$
$89$			156.083i			1.75374i	−0.480726	+	0.876871i	$0.659627\pi$
$90$		0			0
$91$	21.6987	+	140.171i	0.238447	+	1.54034i
$92$		−	50.9475i		−	0.553777i
$93$			36.2863i			0.390175i
$94$			4.12313i			0.0438631i
$95$		0			0
$96$			9.79796i			0.102062i
$97$	−154.698			−1.59482			−0.797410	−	0.603437i	$-0.793797\pi$
$97$	−154.698			−1.59482			−0.797410	+	0.603437i	$0.793797\pi$
$98$	−20.9523	−	66.0530i	−0.213799	−	0.674010i
$99$	−29.7870			−0.300878
$100$		0			0
$101$			133.558i			1.32236i	0.750228	+	0.661179i	$0.229943\pi$
$101$			133.558i			1.32236i	−0.750228	+	0.661179i	$0.770057\pi$
$102$		−	46.3946i		−	0.454849i
$103$	−81.7942			−0.794118			−0.397059	−	0.917793i	$-0.629969\pi$
$103$	−81.7942			−0.794118			−0.397059	+	0.917793i	$0.629969\pi$
$104$		−	57.3121i		−	0.551078i
$105$		0			0
$106$	15.8280			0.149321
$107$		−	160.090i		−	1.49617i	−0.663604	−	0.748084i	$-0.730974\pi$
$107$		−	160.090i		−	1.49617i	0.663604	−	0.748084i	$-0.269026\pi$
$108$	−10.3923			−0.0962250
$109$	183.616			1.68455			0.842276	−	0.539047i	$-0.181215\pi$
$109$	183.616			1.68455			0.842276	+	0.539047i	$0.181215\pi$
$110$		0			0
$111$			52.2812i			0.471002i
$112$	4.28343	+	27.6704i	0.0382449	+	0.247057i
$113$		−	14.6971i		−	0.130063i	−0.997883	−	0.0650314i	$-0.979285\pi$
$113$		−	14.6971i		−	0.130063i	0.997883	−	0.0650314i	$-0.0207147\pi$
$114$	−11.4847			−0.100743
$115$		0			0
$116$	31.9700			0.275604
$117$	60.7887			0.519561
$118$	90.0727			0.763328
$119$	−20.2826	−	131.023i	−0.170442	−	1.10103i
$120$		0			0
$121$	−22.4152			−0.185250
$122$	−126.265			−1.03496
$123$			73.2478i			0.595511i
$124$		−	41.8998i		−	0.337902i
$125$		0			0
$126$	−29.3489	+	4.54326i	−0.232928	+	0.0360576i
$127$			193.465i			1.52335i	0.647962	+	0.761673i	$0.275622\pi$
$127$			193.465i			1.52335i	−0.647962	+	0.761673i	$0.724378\pi$
$128$		−	11.3137i		−	0.0883883i
$129$		−	86.2050i		−	0.668255i
$130$		0			0
$131$			98.8368i			0.754480i	0.926116	+	0.377240i	$0.123127\pi$
$131$			98.8368i			0.754480i	−0.926116	+	0.377240i	$0.876873\pi$
$132$	34.3950			0.260568
$133$	−32.4339	+	5.02083i	−0.243864	+	0.0377506i
$134$	−19.2225			−0.143451
$135$		0			0
$136$			53.5718i			0.393910i
$137$			40.3738i			0.294699i	0.989084	+	0.147350i	$0.0470742\pi$
$137$			40.3738i			0.294699i	−0.989084	+	0.147350i	$0.952926\pi$
$138$	−62.3977			−0.452157
$139$		−	35.3525i		−	0.254334i	−0.991881	−	0.127167i	$-0.959412\pi$
$139$		−	35.3525i		−	0.254334i	0.991881	−	0.127167i	$-0.0405885\pi$
$140$		0			0
$141$	5.04978			0.0358141
$142$		−	72.1106i		−	0.507821i
$143$	−201.190			−1.40692
$144$	12.0000			0.0833333
$145$		0			0
$146$		−	111.590i		−	0.764313i
$147$	−80.8981	+	25.6613i	−0.550327	+	0.174566i
$148$		−	60.3692i		−	0.407900i
$149$	−274.171			−1.84007			−0.920035	−	0.391836i	$-0.871840\pi$
$149$	−274.171			−1.84007			−0.920035	+	0.391836i	$0.871840\pi$
$150$		0			0
$151$	160.545			1.06322			0.531608	−	0.846991i	$-0.321588\pi$
$151$	160.545			1.06322			0.531608	+	0.846991i	$0.321588\pi$
$152$	13.2614			0.0872460
$153$	−56.8215			−0.371382
$154$	97.1350	−	15.0367i	0.630747	−	0.0976407i
$155$		0			0
$156$	−70.1927			−0.449953
$157$	266.232			1.69574			0.847871	−	0.530202i	$-0.177884\pi$
$157$	266.232			1.69574			0.847871	+	0.530202i	$0.177884\pi$
$158$			98.8064i			0.625357i
$159$		−	19.3852i		−	0.121920i
$160$		0			0
$161$	−176.217	+	27.2788i	−1.09452	+	0.169433i
$162$			12.7279i			0.0785674i
$163$		−	187.398i		−	1.14968i	−0.818265	−	0.574842i	$-0.805064\pi$
$163$		−	187.398i		−	1.14968i	0.818265	−	0.574842i	$-0.194936\pi$
$164$		−	84.5793i		−	0.515727i
$165$		0			0
$166$		−	97.4217i		−	0.586878i
$167$	262.937			1.57447			0.787237	−	0.616650i	$-0.211511\pi$
$167$	262.937			1.57447			0.787237	+	0.616650i	$0.211511\pi$
$168$	33.8892	−	5.24611i	0.201721	−	0.0312268i
$169$	241.584			1.42949
$170$		0			0
$171$			14.0658i			0.0822563i
$172$			99.5409i			0.578726i
$173$	−0.130787			−0.000755997			−0.000377998	−	1.00000i	$-0.500120\pi$
$173$	−0.130787			−0.000755997			−0.000377998		1.00000i	$0.500120\pi$
$174$		−	39.1551i		−	0.225029i
$175$		0			0
$176$	−39.7159			−0.225659
$177$		−	110.316i		−	0.623255i
$178$	−220.735			−1.24008
$179$	−143.431			−0.801293			−0.400646	−	0.916233i	$-0.631214\pi$
$179$	−143.431			−0.801293			−0.400646	+	0.916233i	$0.631214\pi$
$180$		0			0
$181$		−	264.455i		−	1.46108i	−0.682871	−	0.730539i	$-0.739269\pi$
$181$		−	264.455i		−	1.46108i	0.682871	−	0.730539i	$-0.260731\pi$
$182$	−198.231	+	30.6865i	−1.08918	+	0.168607i
$183$			154.643i			0.845042i
$184$	72.0507			0.391580
$185$		0			0
$186$	−51.3166			−0.275896
$187$	188.060			1.00567
$188$	−5.83099			−0.0310159
$189$	5.56434	+	35.9449i	0.0294409	+	0.190185i
$190$		0			0
$191$	218.722			1.14514			0.572571	−	0.819855i	$-0.305946\pi$
$191$	218.722			1.14514			0.572571	+	0.819855i	$0.305946\pi$
$192$	−13.8564			−0.0721688
$193$		−	162.709i		−	0.843052i	−0.906816	−	0.421526i	$-0.861495\pi$
$193$		−	162.709i		−	0.843052i	0.906816	−	0.421526i	$-0.138505\pi$
$194$		−	218.775i		−	1.12771i
$195$		0			0
$196$	93.4131	−	29.6311i	0.476597	−	0.151179i
$197$		−	175.537i		−	0.891052i	−0.895269	−	0.445526i	$-0.853017\pi$
$197$		−	175.537i		−	0.891052i	0.895269	−	0.445526i	$-0.146983\pi$
$198$		−	42.1251i		−	0.212753i
$199$			61.4878i			0.308984i	0.987994	+	0.154492i	$0.0493741\pi$
$199$			61.4878i			0.308984i	−0.987994	+	0.154492i	$0.950626\pi$
$200$		0			0
$201$			23.5426i			0.117127i
$202$	−188.880			−0.935048
$203$	−17.1177	−	110.578i	−0.0843235	−	0.544719i
$204$	65.6118			0.321627
$205$		0			0
$206$		−	115.674i		−	0.561527i
$207$			76.4213i			0.369185i
$208$	81.0515			0.389671
$209$		−	46.5531i		−	0.222742i
$210$		0			0
$211$	126.033			0.597314			0.298657	−	0.954361i	$-0.403461\pi$
$211$	126.033			0.597314			0.298657	+	0.954361i	$0.403461\pi$
$212$			22.3842i			0.105586i
$213$	−88.3171			−0.414634
$214$	226.401			1.05795
$215$		0			0
$216$		−	14.6969i		−	0.0680414i
$217$	−144.923	+	22.4344i	−0.667849	+	0.103384i
$218$			259.672i			1.19116i
$219$	−136.669			−0.624059
$220$		0			0
$221$	−383.789			−1.73660
$222$	−73.9368			−0.333049
$223$	62.6219			0.280816			0.140408	−	0.990094i	$-0.455159\pi$
$223$	62.6219			0.280816			0.140408	+	0.990094i	$0.455159\pi$
$224$	−39.1319	+	6.05768i	−0.174696	+	0.0270432i
$225$		0			0
$226$	20.7848			0.0919683
$227$	220.557			0.971619			0.485809	−	0.874065i	$-0.338525\pi$
$227$	220.557			0.971619			0.485809	+	0.874065i	$0.338525\pi$
$228$		−	16.2418i		−	0.0712360i
$229$			206.797i			0.903043i	0.892260	+	0.451521i	$0.149119\pi$
$229$			206.797i			0.903043i	−0.892260	+	0.451521i	$0.850881\pi$
$230$		0			0
$231$	−18.4161	−	118.966i	−0.0797233	−	0.515003i
$232$			45.2124i			0.194881i
$233$		−	167.313i		−	0.718082i	−0.933322	−	0.359041i	$-0.883104\pi$
$233$		−	167.313i		−	0.718082i	0.933322	−	0.359041i	$-0.116896\pi$
$234$			85.9681i			0.367385i
$235$		0			0
$236$			127.382i			0.539754i
$237$	121.013			0.510602
$238$	185.294	−	28.6839i	0.778548	−	0.120521i
$239$	14.5321			0.0608036			0.0304018	−	0.999538i	$-0.490321\pi$
$239$	14.5321			0.0608036			0.0304018	+	0.999538i	$0.490321\pi$
$240$		0			0
$241$			337.752i			1.40146i	0.713427	+	0.700730i	$0.247142\pi$
$241$			337.752i			1.40146i	−0.713427	+	0.700730i	$0.752858\pi$
$242$		−	31.6999i		−	0.130991i
$243$	15.5885			0.0641500
$244$		−	178.566i		−	0.731827i
$245$		0			0
$246$	−103.588			−0.421090
$247$			95.0047i			0.384635i
$248$	59.2553			0.238933
$249$	−119.317			−0.479184
$250$		0			0
$251$		−	391.239i		−	1.55872i	−0.626576	−	0.779360i	$-0.715544\pi$
$251$		−	391.239i		−	1.55872i	0.626576	−	0.779360i	$-0.284456\pi$
$252$	−6.42514	−	41.5056i	−0.0254966	−	0.164705i
$253$		−	252.929i		−	0.999718i
$254$	−273.601			−1.07717
$255$		0			0
$256$	16.0000			0.0625000
$257$	275.467			1.07186			0.535928	−	0.844264i	$-0.319962\pi$
$257$	275.467			1.07186			0.535928	+	0.844264i	$0.319962\pi$
$258$	121.912			0.472528
$259$	−208.805	+	32.3234i	−0.806197	+	0.124801i
$260$		0			0
$261$	−47.9550			−0.183736
$262$	−139.776			−0.533498
$263$			352.033i			1.33853i	0.743025	+	0.669264i	$0.233390\pi$
$263$			352.033i			1.33853i	−0.743025	+	0.669264i	$0.766610\pi$
$264$			48.6419i			0.184250i
$265$		0			0
$266$	−7.10053	−	45.8685i	−0.0266937	−	0.172438i
$267$			270.344i			1.01252i
$268$		−	27.1847i		−	0.101435i
$269$			90.5828i			0.336739i	0.985724	+	0.168370i	$0.0538502\pi$
$269$			90.5828i			0.336739i	−0.985724	+	0.168370i	$0.946150\pi$
$270$		0			0
$271$			56.5234i			0.208573i	0.994547	+	0.104287i	$0.0332559\pi$
$271$			56.5234i			0.208573i	−0.994547	+	0.104287i	$0.966744\pi$
$272$	−75.7620			−0.278537
$273$	37.5832	+	242.783i	0.137667	+	0.889314i
$274$	−57.0971			−0.208384
$275$		0			0
$276$		−	88.2437i		−	0.319723i
$277$			158.702i			0.572931i	0.958091	+	0.286466i	$0.0924804\pi$
$277$			158.702i			0.572931i	−0.958091	+	0.286466i	$0.907520\pi$
$278$	49.9960			0.179842
$279$			62.8497i			0.225268i
$280$		0			0
$281$	−50.2344			−0.178770			−0.0893850	−	0.995997i	$-0.528490\pi$
$281$	−50.2344			−0.178770			−0.0893850	+	0.995997i	$0.528490\pi$
$282$			7.14147i			0.0253244i
$283$	104.925			0.370760			0.185380	−	0.982667i	$-0.440648\pi$
$283$	104.925			0.370760			0.185380	+	0.982667i	$0.440648\pi$
$284$	101.980			0.359084
$285$		0			0
$286$		−	284.525i		−	0.994844i
$287$	−292.543	+	45.2862i	−1.01931	+	0.157792i
$288$			16.9706i			0.0589256i
$289$	69.7425			0.241323
$290$		0			0
$291$	−267.944			−0.920770
$292$	157.812			0.540451
$293$	506.046			1.72712			0.863560	−	0.504247i	$-0.168230\pi$
$293$	506.046			1.72712			0.863560	+	0.504247i	$0.168230\pi$
$294$	−36.2905	−	114.407i	−0.123437	−	0.389140i
$295$		0			0
$296$	85.3749			0.288429
$297$	−51.5925			−0.173712
$298$		−	387.736i		−	1.30113i
$299$			516.172i			1.72633i
$300$		0			0
$301$	344.292	−	53.2971i	1.14383	−	0.177067i
$302$			227.046i			0.751807i
$303$			231.329i			0.763463i
$304$			18.7544i			0.0616922i
$305$		0			0
$306$		−	80.3577i		−	0.262607i
$307$	221.769			0.722376			0.361188	−	0.932493i	$-0.382371\pi$
$307$	221.769			0.722376			0.361188	+	0.932493i	$0.382371\pi$
$308$	21.2651	+	137.370i	0.0690424	+	0.446005i
$309$	−141.672			−0.458485
$310$		0			0
$311$		−	80.8561i		−	0.259988i	−0.991515	−	0.129994i	$-0.958504\pi$
$311$		−	80.8561i		−	0.259988i	0.991515	−	0.129994i	$-0.0414957\pi$
$312$		−	99.2675i		−	0.318165i
$313$	340.104			1.08659			0.543297	−	0.839541i	$-0.317176\pi$
$313$	340.104			1.08659			0.543297	+	0.839541i	$0.317176\pi$
$314$			376.508i			1.19907i
$315$		0			0
$316$	−139.733			−0.442194
$317$			304.527i			0.960654i	0.877090	+	0.480327i	$0.159482\pi$
$317$			304.527i			0.960654i	−0.877090	+	0.480327i	$0.840518\pi$
$318$	27.4149			0.0862103
$319$	158.715			0.497539
$320$		0			0
$321$		−	277.284i		−	0.863813i
$322$	−38.5780	−	249.209i	−0.119807	−	0.773941i
$323$		−	88.8046i		−	0.274937i
$324$	−18.0000			−0.0555556
$325$		0			0
$326$	265.021			0.812949
$327$	318.032			0.972576
$328$	119.613			0.364674
$329$	3.12208	+	20.1682i	0.00948960	+	0.0613016i
$330$		0			0
$331$	323.175			0.976361			0.488180	−	0.872743i	$-0.337661\pi$
$331$	323.175			0.976361			0.488180	+	0.872743i	$0.337661\pi$
$332$	137.775			0.414985
$333$			90.5538i			0.271933i
$334$			371.849i			1.11332i
$335$		0			0
$336$	7.41912	+	47.9266i	0.0220807	+	0.142639i
$337$		−	333.282i		−	0.988966i	−0.869187	−	0.494483i	$-0.835357\pi$
$337$		−	333.282i		−	0.988966i	0.869187	−	0.494483i	$-0.164643\pi$
$338$			341.652i			1.01080i
$339$		−	25.4561i		−	0.0750918i
$340$		0			0
$341$		−	208.011i		−	0.610004i
$342$	−19.8921			−0.0581640
$343$	−152.504	−	307.232i	−0.444618	−	0.895720i
$344$	−140.772			−0.409221
$345$		0			0
$346$		−	0.184961i		−	0.000534571i
$347$		−	257.512i		−	0.742110i	−0.928611	−	0.371055i	$-0.878996\pi$
$347$		−	257.512i		−	0.742110i	0.928611	−	0.371055i	$-0.121004\pi$
$348$	55.3737			0.159120
$349$			306.979i			0.879596i	0.898097	+	0.439798i	$0.144950\pi$
$349$			306.979i			0.879596i	−0.898097	+	0.439798i	$0.855050\pi$
$350$		0			0
$351$	105.289			0.299969
$352$		−	56.1668i		−	0.159565i
$353$	248.442			0.703802			0.351901	−	0.936037i	$-0.385535\pi$
$353$	248.442			0.703802			0.351901	+	0.936037i	$0.385535\pi$
$354$	156.010			0.440708
$355$		0			0
$356$		−	312.166i		−	0.876871i
$357$	−35.1304	−	226.938i	−0.0984046	−	0.635682i
$358$		−	202.843i		−	0.566600i
$359$	348.412			0.970507			0.485253	−	0.874374i	$-0.338727\pi$
$359$	348.412			0.970507			0.485253	+	0.874374i	$0.338727\pi$
$360$		0			0
$361$	339.017			0.939105
$362$	373.996			1.03314
$363$	−38.8243			−0.106954
$364$	−43.3973	−	280.341i	−0.119223	−	0.770168i
$365$		0			0
$366$	−218.698			−0.597535
$367$	73.9213			0.201420			0.100710	−	0.994916i	$-0.467888\pi$
$367$	73.9213			0.201420			0.100710	+	0.994916i	$0.467888\pi$
$368$			101.895i			0.276889i
$369$			126.869i			0.343818i
$370$		0			0
$371$	77.4224	−	11.9851i	0.208686	−	0.0323049i
$372$		−	72.5726i		−	0.195088i
$373$		−	52.9503i		−	0.141958i	−0.997478	−	0.0709789i	$-0.977388\pi$
$373$		−	52.9503i		−	0.141958i	0.997478	−	0.0709789i	$-0.0226123\pi$
$374$			265.957i			0.711115i
$375$		0			0
$376$		−	8.24626i		−	0.0219315i
$377$	−323.902			−0.859157
$378$	−50.8338	+	7.86916i	−0.134481	+	0.0208179i
$379$	−112.721			−0.297417			−0.148709	−	0.988881i	$-0.547512\pi$
$379$	−112.721			−0.297417			−0.148709	+	0.988881i	$0.547512\pi$
$380$		0			0
$381$			335.091i			0.879504i
$382$			309.320i			0.809737i
$383$	−524.061			−1.36830			−0.684152	−	0.729339i	$-0.739828\pi$
$383$	−524.061			−1.36830			−0.684152	+	0.729339i	$0.739828\pi$
$384$		−	19.5959i		−	0.0510310i
$385$		0			0
$386$	230.105			0.596128
$387$		−	149.311i		−	0.385817i
$388$	309.395			0.797410
$389$	114.232			0.293655			0.146827	−	0.989162i	$-0.453094\pi$
$389$	114.232			0.293655			0.146827	+	0.989162i	$0.453094\pi$
$390$		0			0
$391$		−	482.486i		−	1.23398i
$392$	41.9047	+	132.106i	0.106900	+	0.337005i
$393$			171.190i			0.435599i
$394$	248.247			0.630069
$395$		0			0
$396$	59.5739			0.150439
$397$	657.463			1.65608			0.828039	−	0.560670i	$-0.189456\pi$
$397$	657.463			1.65608			0.828039	+	0.560670i	$0.189456\pi$
$398$	−86.9569			−0.218485
$399$	−56.1772	+	8.69634i	−0.140795	+	0.0217953i
$400$		0			0
$401$	−614.317			−1.53196			−0.765981	−	0.642863i	$-0.777747\pi$
$401$	−614.317			−1.53196			−0.765981	+	0.642863i	$0.777747\pi$
$402$	−33.2943			−0.0828216
$403$			424.506i			1.05336i
$404$		−	267.116i		−	0.661179i
$405$		0			0
$406$	156.381	−	24.2080i	0.385175	−	0.0596257i
$407$		−	299.702i		−	0.736369i
$408$			92.7891i			0.227424i
$409$		−	195.617i		−	0.478282i	−0.970985	−	0.239141i	$-0.923134\pi$
$409$		−	195.617i		−	0.478282i	0.970985	−	0.239141i	$-0.0768658\pi$
$410$		0			0
$411$			69.9294i			0.170145i
$412$	163.588			0.397059
$413$	440.589	−	68.2040i	1.06680	−	0.165143i
$414$	−108.076			−0.261053
$415$		0			0
$416$			114.624i			0.275539i
$417$		−	61.2323i		−	0.146840i
$418$	65.8361			0.157503
$419$		−	320.586i		−	0.765121i	−0.923930	−	0.382561i	$-0.875042\pi$
$419$		−	320.586i		−	0.765121i	0.923930	−	0.382561i	$-0.124958\pi$
$420$		0			0
$421$	394.568			0.937217			0.468609	−	0.883406i	$-0.344755\pi$
$421$	394.568			0.937217			0.468609	+	0.883406i	$0.344755\pi$
$422$			178.238i			0.422365i
$423$	8.74648			0.0206773
$424$	−31.6560			−0.0746603
$425$		0			0
$426$		−	124.899i		−	0.293191i
$427$	−617.624	+	95.6093i	−1.44643	+	0.223909i
$428$			320.180i			0.748084i
$429$	−348.471			−0.812287
$430$		0			0
$431$	−687.598			−1.59535			−0.797677	−	0.603084i	$-0.793938\pi$
$431$	−687.598			−1.59535			−0.797677	+	0.603084i	$0.793938\pi$
$432$	20.7846			0.0481125
$433$	738.775			1.70618			0.853089	−	0.521766i	$-0.174727\pi$
$433$	738.775			1.70618			0.853089	+	0.521766i	$0.174727\pi$
$434$	−31.7270	−	204.952i	−0.0731037	−	0.472241i
$435$		0			0
$436$	−367.232			−0.842276
$437$	−119.436			−0.273310
$438$		−	193.279i		−	0.441277i
$439$			323.601i			0.737133i	0.929601	+	0.368567i	$0.120151\pi$
$439$			323.601i			0.737133i	−0.929601	+	0.368567i	$0.879849\pi$
$440$		0			0
$441$	−140.120	+	44.4466i	−0.317731	+	0.100786i
$442$		−	542.760i		−	1.22796i
$443$		−	246.877i		−	0.557285i	−0.960395	−	0.278642i	$-0.910116\pi$
$443$		−	246.877i		−	0.557285i	0.960395	−	0.278642i	$-0.0898844\pi$
$444$		−	104.562i		−	0.235501i
$445$		0			0
$446$			88.5608i			0.198567i
$447$	−474.877			−1.06237
$448$	−8.56686	−	55.3408i	−0.0191225	−	0.123529i
$449$	472.482			1.05230			0.526150	−	0.850392i	$-0.323635\pi$
$449$	472.482			1.05230			0.526150	+	0.850392i	$0.323635\pi$
$450$		0			0
$451$		−	419.893i		−	0.931027i
$452$			29.3942i			0.0650314i
$453$	278.073			0.613848
$454$			311.915i			0.687038i
$455$		0			0
$456$	22.9694			0.0503715
$457$			601.310i			1.31578i	0.753115	+	0.657889i	$0.228550\pi$
$457$			601.310i			1.31578i	−0.753115	+	0.657889i	$0.771450\pi$
$458$	−292.455			−0.638548
$459$	−98.4177			−0.214418
$460$		0			0
$461$		−	270.297i		−	0.586328i	−0.956062	−	0.293164i	$-0.905292\pi$
$461$		−	270.297i		−	0.586328i	0.956062	−	0.293164i	$-0.0947081\pi$
$462$	168.243	−	26.0443i	0.364162	−	0.0563729i
$463$		−	579.731i		−	1.25212i	−0.779775	−	0.626060i	$-0.784667\pi$
$463$		−	579.731i		−	1.25212i	0.779775	−	0.626060i	$-0.215333\pi$
$464$	−63.9400			−0.137802
$465$		0			0
$466$	236.616			0.507761
$467$	895.295			1.91712			0.958560	−	0.284891i	$-0.0919575\pi$
$467$	895.295			1.91712			0.958560	+	0.284891i	$0.0919575\pi$
$468$	−121.577			−0.259781
$469$	−94.0264	+	14.5555i	−0.200483	+	0.0310351i
$470$		0			0
$471$	461.127			0.979037
$472$	−180.145			−0.381664
$473$			494.170i			1.04476i
$474$			171.138i			0.361050i
$475$		0			0
$476$	40.5651	+	262.046i	0.0852209	+	0.550516i
$477$		−	33.5762i		−	0.0703904i
$478$			20.5515i			0.0429947i
$479$		−	252.755i		−	0.527673i	−0.964567	−	0.263836i	$-0.915012\pi$
$479$		−	252.755i		−	0.527673i	0.964567	−	0.263836i	$-0.0849879\pi$
$480$		0			0
$481$			611.627i			1.27157i
$482$	−477.653			−0.990981
$483$	−305.217	+	47.2482i	−0.631920	+	0.0978224i
$484$	44.8305			0.0926249
$485$		0			0
$486$			22.0454i			0.0453609i
$487$			421.873i			0.866269i	0.901329	+	0.433135i	$0.142593\pi$
$487$			421.873i			0.866269i	−0.901329	+	0.433135i	$0.857407\pi$
$488$	252.530			0.517480
$489$		−	324.584i		−	0.663770i
$490$		0			0
$491$	−232.477			−0.473476			−0.236738	−	0.971574i	$-0.576078\pi$
$491$	−232.477			−0.473476			−0.236738	+	0.971574i	$0.576078\pi$
$492$		−	146.496i		−	0.297755i
$493$	302.764			0.614126
$494$	−134.357			−0.271978
$495$		0			0
$496$			83.7997i			0.168951i
$497$	−54.6029	−	352.728i	−0.109865	−	0.709714i
$498$		−	168.739i		−	0.338834i
$499$	528.736			1.05959			0.529796	−	0.848125i	$-0.322269\pi$
$499$	528.736			1.05959			0.529796	+	0.848125i	$0.322269\pi$
$500$		0			0
$501$	455.421			0.909023
$502$	553.295			1.10218
$503$	865.962			1.72160			0.860798	−	0.508948i	$-0.169965\pi$
$503$	865.962			1.72160			0.860798	+	0.508948i	$0.169965\pi$
$504$	58.6978	−	9.08653i	0.116464	−	0.0180288i
$505$		0			0
$506$	357.695			0.706907
$507$	418.437			0.825319
$508$		−	386.930i		−	0.761673i
$509$			614.157i			1.20659i	0.797516	+	0.603297i	$0.206147\pi$
$509$			614.157i			1.20659i	−0.797516	+	0.603297i	$0.793853\pi$
$510$		0			0
$511$	−84.4970	−	545.840i	−0.165356	−	1.06818i
$512$			22.6274i			0.0441942i
$513$			24.3627i			0.0474907i
$514$			389.569i			0.757917i
$515$		0			0
$516$			172.410i			0.334128i
$517$	−28.9479			−0.0559921
$518$	−45.7122	−	295.295i	−0.0882475	−	0.570068i
$519$	−0.226531			−0.000436475
$520$		0			0
$521$			428.487i			0.822432i	0.911538	+	0.411216i	$0.134896\pi$
$521$			428.487i			0.822432i	−0.911538	+	0.411216i	$0.865104\pi$
$522$		−	67.8186i		−	0.129921i
$523$	−1029.66			−1.96876			−0.984379	−	0.176060i	$-0.943665\pi$
$523$	−1029.66			−1.96876			−0.984379	+	0.176060i	$0.943665\pi$
$524$		−	197.674i		−	0.377240i
$525$		0			0
$526$	−497.850			−0.946482
$527$		−	396.802i		−	0.752945i
$528$	−68.7900			−0.130284
$529$	−119.912			−0.226677
$530$		0			0
$531$		−	191.073i		−	0.359836i
$532$	64.8679	−	10.0417i	0.121932	−	0.0188753i
$533$			856.910i			1.60771i
$534$	−382.324			−0.715962
$535$		0			0
$536$	38.4449			0.0717256
$537$	−248.431			−0.462627
$538$	−128.103			−0.238111
$539$	463.748	−	147.103i	0.860387	−	0.272919i
$540$		0			0
$541$	−533.849			−0.986782			−0.493391	−	0.869808i	$-0.664243\pi$
$541$	−533.849			−0.986782			−0.493391	+	0.869808i	$0.664243\pi$
$542$	−79.9361			−0.147484
$543$		−	458.050i		−	0.843554i
$544$		−	107.144i		−	0.196955i
$545$		0			0
$546$	−343.347	+	53.1506i	−0.628840	+	0.0973455i
$547$			349.653i			0.639220i	0.947549	+	0.319610i	$0.103552\pi$
$547$			349.653i			0.639220i	−0.947549	+	0.319610i	$0.896448\pi$
$548$		−	80.7476i		−	0.147350i
$549$			267.849i			0.487885i
$550$		0			0
$551$		−	74.9474i		−	0.136021i
$552$	124.795			0.226079
$553$	74.8172	+	483.310i	0.135293	+	0.873978i
$554$	−224.438			−0.405124
$555$		0			0
$556$			70.7050i			0.127167i
$557$		−	193.971i		−	0.348243i	−0.984724	−	0.174122i	$-0.944291\pi$
$557$		−	193.971i		−	0.348243i	0.984724	−	0.174122i	$-0.0557086\pi$
$558$	−88.8830			−0.159288
$559$		−	1008.49i		−	1.80410i
$560$		0			0
$561$	325.729			0.580623
$562$		−	71.0421i		−	0.126410i
$563$	−388.914			−0.690788			−0.345394	−	0.938458i	$-0.612255\pi$
$563$	−388.914			−0.690788			−0.345394	+	0.938458i	$0.612255\pi$
$564$	−10.0996			−0.0179070
$565$		0			0
$566$			148.386i			0.262167i
$567$	9.63772	+	62.2584i	0.0169977	+	0.109803i
$568$			144.221i			0.253911i
$569$	470.383			0.826683			0.413342	−	0.910576i	$-0.364362\pi$
$569$	470.383			0.826683			0.413342	+	0.910576i	$0.364362\pi$
$570$		0			0
$571$	887.703			1.55465			0.777323	−	0.629101i	$-0.216577\pi$
$571$	887.703			1.55465			0.777323	+	0.629101i	$0.216577\pi$
$572$	402.380			0.703461
$573$	378.838			0.661148
$574$	−64.0443	−	413.718i	−0.111575	−	0.720764i
$575$		0			0
$576$	−24.0000			−0.0416667
$577$	−927.661			−1.60773			−0.803866	−	0.594811i	$-0.797227\pi$
$577$	−927.661			−1.60773			−0.803866	+	0.594811i	$0.797227\pi$
$578$			98.6307i			0.170641i
$579$		−	281.820i		−	0.486736i
$580$		0			0
$581$	−73.7688	−	476.537i	−0.126969	−	0.820201i
$582$		−	378.930i		−	0.651083i
$583$			111.126i			0.190611i
$584$			223.180i			0.382157i
$585$		0			0
$586$			715.657i			1.22126i
$587$	−504.818			−0.859996			−0.429998	−	0.902830i	$-0.641486\pi$
$587$	−504.818			−0.859996			−0.429998	+	0.902830i	$0.641486\pi$
$588$	161.796	−	51.3225i	0.275164	−	0.0872832i
$589$	−98.2260			−0.166767
$590$		0			0
$591$		−	304.039i		−	0.514449i
$592$			120.738i			0.203950i
$593$	−931.909			−1.57152			−0.785758	−	0.618534i	$-0.787727\pi$
$593$	−931.909			−1.57152			−0.785758	+	0.618534i	$0.787727\pi$
$594$		−	72.9629i		−	0.122833i
$595$		0			0
$596$	548.341			0.920035
$597$			106.500i			0.178392i
$598$	−729.977			−1.22070
$599$	−841.411			−1.40469			−0.702347	−	0.711835i	$-0.747864\pi$
$599$	−841.411			−1.40469			−0.702347	+	0.711835i	$0.747864\pi$
$600$		0			0
$601$			116.550i			0.193926i	0.995288	+	0.0969632i	$0.0309129\pi$
$601$			116.550i			0.193926i	−0.995288	+	0.0969632i	$0.969087\pi$
$602$	75.3734	+	486.903i	0.125205	+	0.808809i
$603$			40.7770i			0.0676236i
$604$	−321.091			−0.531608
$605$		0			0
$606$	−327.149			−0.539850
$607$	−6.57481			−0.0108316			−0.00541582	−	0.999985i	$-0.501724\pi$
$607$	−6.57481			−0.0108316			−0.00541582	+	0.999985i	$0.501724\pi$
$608$	−26.5228			−0.0436230
$609$	−29.6487	−	191.527i	−0.0486842	−	0.314494i
$610$		0			0
$611$	59.0763			0.0966879
$612$	113.643			0.185691
$613$		−	51.9420i		−	0.0847340i	−0.999102	−	0.0423670i	$-0.986510\pi$
$613$		−	51.9420i		−	0.0847340i	0.999102	−	0.0423670i	$-0.0134899\pi$
$614$			313.629i			0.510797i
$615$		0			0
$616$	−194.270	+	30.0733i	−0.315373	+	0.0488203i
$617$			1064.09i			1.72462i	0.506383	+	0.862309i	$0.330982\pi$
$617$			1064.09i			1.72462i	−0.506383	+	0.862309i	$0.669018\pi$
$618$		−	200.354i		−	0.324198i
$619$		−	585.402i		−	0.945722i	−0.881137	−	0.472861i	$-0.843221\pi$
$619$		−	585.402i		−	0.945722i	0.881137	−	0.472861i	$-0.156779\pi$
$620$		0			0
$621$			132.366i			0.213149i
$622$	114.348			0.183839
$623$	−1079.72	+	167.143i	−1.73310	+	0.268287i
$624$	140.385			0.224977
$625$		0			0
$626$			480.980i			0.768338i
$627$		−	80.6324i		−	0.128600i
$628$	−532.463			−0.847871
$629$		−	571.711i		−	0.908921i
$630$		0			0
$631$	−827.999			−1.31220			−0.656101	−	0.754673i	$-0.727795\pi$
$631$	−827.999			−1.31220			−0.656101	+	0.754673i	$0.727795\pi$
$632$		−	197.613i		−	0.312678i
$633$	218.296			0.344859
$634$	−430.667			−0.679285
$635$		0			0
$636$			38.7705i			0.0609599i
$637$	−946.409	+	300.206i	−1.48573	+	0.471280i
$638$			224.457i			0.351813i
$639$	−152.970			−0.239389
$640$		0			0
$641$	−806.971			−1.25893			−0.629463	−	0.777031i	$-0.716725\pi$
$641$	−806.971			−1.25893			−0.629463	+	0.777031i	$0.716725\pi$
$642$	392.139			0.610808
$643$	−588.657			−0.915485			−0.457743	−	0.889085i	$-0.651342\pi$
$643$	−588.657			−0.915485			−0.457743	+	0.889085i	$0.651342\pi$
$644$	352.435	−	54.5575i	0.547259	−	0.0847167i
$645$		0			0
$646$	125.589			0.194410
$647$	51.4386			0.0795032			0.0397516	−	0.999210i	$-0.487343\pi$
$647$	51.4386			0.0795032			0.0397516	+	0.999210i	$0.487343\pi$
$648$		−	25.4558i		−	0.0392837i
$649$			632.387i			0.974403i
$650$		0			0
$651$	−251.014	+	38.8575i	−0.385583	+	0.0596889i
$652$			374.797i			0.574842i
$653$		−	843.699i		−	1.29204i	−0.763322	−	0.646018i	$-0.776433\pi$
$653$		−	843.699i		−	1.29204i	0.763322	−	0.646018i	$-0.223567\pi$
$654$			449.766i			0.687715i
$655$		0			0
$656$			169.159i			0.257864i
$657$	−236.718			−0.360301
$658$	−28.5222	+	4.41529i	−0.0433468	+	0.00671016i
$659$	203.437			0.308705			0.154352	−	0.988016i	$-0.450671\pi$
$659$	203.437			0.308705			0.154352	+	0.988016i	$0.450671\pi$
$660$		0			0
$661$			1141.47i			1.72688i	0.504454	+	0.863439i	$0.331694\pi$
$661$			1141.47i			1.72688i	−0.504454	+	0.863439i	$0.668306\pi$
$662$			457.039i			0.690391i
$663$	−664.742			−1.00263
$664$			194.843i			0.293439i
$665$		0			0
$666$	−128.062			−0.192286
$667$		−	407.198i		−	0.610492i
$668$	−525.874			−0.787237
$669$	108.464			0.162129
$670$		0			0
$671$		−	886.489i		−	1.32115i
$672$	−67.7784	+	10.4922i	−0.100861	+	0.0156134i
$673$		−	1259.69i		−	1.87175i	−0.352327	−	0.935877i	$-0.614609\pi$
$673$		−	1259.69i		−	1.87175i	0.352327	−	0.935877i	$-0.385391\pi$
$674$	471.331			0.699305
$675$		0			0
$676$	−483.169			−0.714747
$677$	−563.250			−0.831979			−0.415989	−	0.909369i	$-0.636565\pi$
$677$	−563.250			−0.831979			−0.415989	+	0.909369i	$0.636565\pi$
$678$	36.0004			0.0530979
$679$	−165.659	−	1070.14i	−0.243975	−	1.57605i
$680$		0			0
$681$	382.017			0.560964
$682$	294.173			0.431338
$683$		−	881.725i		−	1.29096i	−0.763778	−	0.645480i	$-0.776658\pi$
$683$		−	881.725i		−	1.29096i	0.763778	−	0.645480i	$-0.223342\pi$
$684$		−	28.1317i		−	0.0411281i
$685$		0			0
$686$	434.492	−	215.673i	0.633370	−	0.314393i
$687$			358.183i			0.521372i
$688$		−	199.082i		−	0.289363i
$689$		−	226.784i		−	0.329149i
$690$		0			0
$691$		−	33.6567i		−	0.0487072i	−0.999703	−	0.0243536i	$-0.992247\pi$
$691$		−	33.6567i		−	0.0487072i	0.999703	−	0.0243536i	$-0.00775276\pi$
$692$	0.261575			0.000377998
$693$	−31.8976	−	206.054i	−0.0460283	−	0.297337i
$694$	364.177			0.524751
$695$		0			0
$696$			78.3102i			0.112515i
$697$		−	800.987i		−	1.14919i
$698$	−434.134			−0.621968
$699$		−	289.795i		−	0.414585i
$700$		0			0
$701$	123.366			0.175986			0.0879928	−	0.996121i	$-0.471955\pi$
$701$	123.366			0.175986			0.0879928	+	0.996121i	$0.471955\pi$
$702$			148.901i			0.212110i
$703$	−141.524			−0.201314
$704$	79.4319			0.112829
$705$		0			0
$706$			351.350i			0.497663i
$707$	−923.902	+	143.022i	−1.30679	+	0.202294i
$708$			220.632i			0.311627i
$709$	−1271.96			−1.79401			−0.897007	−	0.442016i	$-0.854264\pi$
$709$	−1271.96			−1.79401			−0.897007	+	0.442016i	$0.854264\pi$
$710$		0			0
$711$	209.600			0.294796
$712$	441.469			0.620041
$713$	−533.673			−0.748489
$714$	320.939	−	49.6820i	0.449495	−	0.0695826i
$715$		0			0
$716$	286.863			0.400646
$717$	25.1703			0.0351050
$718$			492.729i			0.686252i
$719$		−	482.510i		−	0.671085i	−0.942025	−	0.335542i	$-0.891080\pi$
$719$		−	482.510i		−	0.671085i	0.942025	−	0.335542i	$-0.108920\pi$
$720$		0			0
$721$	−87.5899	−	565.820i	−0.121484	−	0.784771i
$722$			479.442i			0.664048i
$723$			585.003i			0.809133i
$724$			528.910i			0.730539i
$725$		0			0
$726$		−	54.9059i		−	0.0756279i
$727$	−1033.97			−1.42224			−0.711119	−	0.703071i	$-0.751811\pi$
$727$	−1033.97			−1.42224			−0.711119	+	0.703071i	$0.751811\pi$
$728$	396.462	−	61.3731i	0.544591	−	0.0843037i
$729$	27.0000			0.0370370
$730$		0			0
$731$			942.677i			1.28957i
$732$		−	309.285i		−	0.422521i
$733$	−199.138			−0.271676			−0.135838	−	0.990731i	$-0.543373\pi$
$733$	−199.138			−0.271676			−0.135838	+	0.990731i	$0.543373\pi$
$734$			104.541i			0.142426i
$735$		0			0
$736$	−144.101			−0.195790
$737$		−	134.958i		−	0.183118i
$738$	−179.420			−0.243116
$739$	102.937			0.139292			0.0696460	−	0.997572i	$-0.477813\pi$
$739$	102.937			0.139292			0.0696460	+	0.997572i	$0.477813\pi$
$740$		0			0
$741$			164.553i			0.222069i
$742$	16.9495	+	109.492i	0.0228430	+	0.147563i
$743$		−	366.082i		−	0.492707i	−0.969180	−	0.246354i	$-0.920768\pi$
$743$		−	366.082i		−	0.492707i	0.969180	−	0.246354i	$-0.0792325\pi$
$744$	102.633			0.137948
$745$		0			0
$746$	74.8830			0.100379
$747$	−206.663			−0.276657
$748$	−376.120			−0.502834
$749$	1107.44	−	171.434i	1.47856	−	0.228883i
$750$		0			0
$751$	−230.728			−0.307227			−0.153614	−	0.988131i	$-0.549091\pi$
$751$	−230.728			−0.307227			−0.153614	+	0.988131i	$0.549091\pi$
$752$	11.6620			0.0155079
$753$		−	677.646i		−	0.899928i
$754$		−	458.067i		−	0.607516i
$755$		0			0
$756$	−11.1287	−	71.8899i	−0.0147205	−	0.0950924i
$757$		−	1340.08i		−	1.77026i	−0.465348	−	0.885128i	$-0.654071\pi$
$757$		−	1340.08i		−	1.77026i	0.465348	−	0.885128i	$-0.345929\pi$
$758$		−	159.412i		−	0.210306i
$759$		−	438.085i		−	0.577187i
$760$		0			0
$761$			1256.80i			1.65151i	0.564027	+	0.825756i	$0.309251\pi$
$761$			1256.80i			1.65151i	−0.564027	+	0.825756i	$0.690749\pi$
$762$	−473.890			−0.621903
$763$	196.627	+	1270.18i	0.257702	+	1.66472i
$764$	−437.444			−0.572571
$765$		0			0
$766$		−	741.134i		−	0.967537i
$767$		−	1290.56i		−	1.68261i
$768$	27.7128			0.0360844
$769$			361.861i			0.470560i	0.971928	+	0.235280i	$0.0756007\pi$
$769$			361.861i			0.470560i	−0.971928	+	0.235280i	$0.924399\pi$
$770$		0			0
$771$	477.123			0.618836
$772$			325.418i			0.421526i
$773$	746.551			0.965783			0.482892	−	0.875680i	$-0.339586\pi$
$773$	746.551			0.965783			0.482892	+	0.875680i	$0.339586\pi$
$774$	211.158			0.272814
$775$		0			0
$776$			437.551i			0.563854i
$777$	−361.661	+	55.9858i	−0.465458	+	0.0720537i
$778$			161.548i			0.207645i
$779$	−198.280			−0.254531
$780$		0			0
$781$	506.278			0.648243
$782$	682.338			0.872555
$783$	−83.0605			−0.106080
$784$	−186.826	+	59.2622i	−0.238299	+	0.0755895i
$785$		0			0
$786$	−242.100			−0.308015
$787$	1121.67			1.42524			0.712622	−	0.701548i	$-0.247507\pi$
$787$	1121.67			1.42524			0.712622	+	0.701548i	$0.247507\pi$
$788$			351.074i			0.445526i
$789$			609.739i			0.772799i
$790$		0			0
$791$	101.669	−	15.7385i	0.128532	−	0.0198970i
$792$			84.2502i			0.106377i
$793$			1809.13i			2.28137i
$794$			929.793i			1.17102i
$795$		0			0
$796$		−	122.976i		−	0.154492i
$797$	−1458.38			−1.82983			−0.914916	−	0.403644i	$-0.867743\pi$
$797$	−1458.38			−1.82983			−0.914916	+	0.403644i	$0.867743\pi$
$798$	−12.2985	−	79.4466i	−0.0154116	−	0.0995572i
$799$	−55.2209			−0.0691125
$800$		0			0
$801$			468.249i			0.584580i
$802$		−	868.775i		−	1.08326i
$803$	783.455			0.975661
$804$		−	47.0852i		−	0.0585637i
$805$		0			0
$806$	−600.342			−0.744841
$807$			156.894i			0.194416i
$808$	377.759			0.467524
$809$	384.044			0.474714			0.237357	−	0.971422i	$-0.423719\pi$
$809$	384.044			0.474714			0.237357	+	0.971422i	$0.423719\pi$
$810$		0			0
$811$			723.519i			0.892132i	0.895000	+	0.446066i	$0.147175\pi$
$811$			723.519i			0.892132i	−0.895000	+	0.446066i	$0.852825\pi$
$812$	34.2353	+	221.156i	0.0421617	+	0.272360i
$813$			97.9013i			0.120420i
$814$	423.843			0.520692
$815$		0			0
$816$	−131.224			−0.160813
$817$	233.354			0.285623
$818$	276.645			0.338196
$819$	65.0960	+	420.512i	0.0794823	+	0.513446i
$820$		0			0
$821$	77.1102			0.0939223			0.0469611	−	0.998897i	$-0.485046\pi$
$821$	77.1102			0.0939223			0.0469611	+	0.998897i	$0.485046\pi$
$822$	−98.8952			−0.120310
$823$		−	234.896i		−	0.285414i	−0.989765	−	0.142707i	$-0.954419\pi$
$823$		−	234.896i		−	0.285414i	0.989765	−	0.142707i	$-0.0455807\pi$
$824$			231.349i			0.280763i
$825$		0			0
$826$	96.4550	+	623.087i	0.116774	+	0.754343i
$827$			173.944i			0.210331i	0.994455	+	0.105166i	$0.0335373\pi$
$827$			173.944i			0.210331i	−0.994455	+	0.105166i	$0.966463\pi$
$828$		−	152.843i		−	0.184592i
$829$		−	800.748i		−	0.965920i	−0.875642	−	0.482960i	$-0.839562\pi$
$829$		−	800.748i		−	0.965920i	0.875642	−	0.482960i	$-0.160438\pi$
$830$		0			0
$831$			274.880i			0.330782i
$832$	−162.103			−0.194835
$833$	884.645	−	280.614i	1.06200	−	0.336871i
$834$	86.5955			0.103832
$835$		0			0
$836$			93.1063i			0.111371i
$837$			108.859i			0.130058i
$838$	453.377			0.541022
$839$		−	943.082i		−	1.12406i	−0.827118	−	0.562028i	$-0.810021\pi$
$839$		−	943.082i		−	1.12406i	0.827118	−	0.562028i	$-0.189979\pi$
$840$		0			0
$841$	−585.480			−0.696171
$842$			558.004i			0.662713i
$843$	−87.0085			−0.103213
$844$	−252.067			−0.298657
$845$		0			0
$846$			12.3694i			0.0146210i
$847$	−24.0035	−	155.060i	−0.0283395	−	0.183069i
$848$		−	44.7683i		−	0.0527928i
$849$	181.735			0.214058
$850$		0			0
$851$	−768.915			−0.903543
$852$	176.634			0.207317
$853$	225.297			0.264124			0.132062	−	0.991241i	$-0.457840\pi$
$853$	225.297			0.264124			0.132062	+	0.991241i	$0.457840\pi$
$854$	−135.212	−	873.453i	−0.158328	−	1.02278i
$855$		0			0
$856$	−452.803			−0.528975
$857$	899.165			1.04920			0.524600	−	0.851349i	$-0.324215\pi$
$857$	899.165			1.04920			0.524600	+	0.851349i	$0.324215\pi$
$858$		−	492.813i		−	0.574374i
$859$			785.611i			0.914564i	0.889322	+	0.457282i	$0.151177\pi$
$859$			785.611i			0.914564i	−0.889322	+	0.457282i	$0.848823\pi$
$860$		0			0
$861$	−506.699	+	78.4379i	−0.588501	+	0.0911010i
$862$		−	972.410i		−	1.12809i
$863$		−	694.379i		−	0.804610i	−0.915506	−	0.402305i	$-0.868209\pi$
$863$		−	694.379i		−	0.804610i	0.915506	−	0.402305i	$-0.131791\pi$
$864$			29.3939i			0.0340207i
$865$		0			0
$866$			1044.79i			1.20645i
$867$	120.797			0.139328
$868$	289.846	−	44.8687i	0.333925	−	0.0516921i
$869$	−693.705			−0.798280
$870$		0			0
$871$			275.420i			0.316211i
$872$		−	519.345i		−	0.595579i
$873$	−464.093			−0.531607
$874$		−	168.909i		−	0.193259i
$875$		0			0
$876$	273.338			0.312030
$877$		−	699.811i		−	0.797960i	−0.916960	−	0.398980i	$-0.869364\pi$
$877$		−	699.811i		−	0.797960i	0.916960	−	0.398980i	$-0.130636\pi$
$878$	−457.642			−0.521232
$879$	876.497			0.997153
$880$		0			0
$881$			1112.73i			1.26303i	0.775362	+	0.631517i	$0.217567\pi$
$881$			1112.73i			1.26303i	−0.775362	+	0.631517i	$0.782433\pi$
$882$	−62.8570	−	198.159i	−0.0712664	−	0.224670i
$883$			522.489i			0.591720i	0.955231	+	0.295860i	$0.0956062\pi$
$883$			522.489i			0.591720i	−0.955231	+	0.295860i	$0.904394\pi$
$884$	767.578			0.868301
$885$		0			0
$886$	349.137			0.394060
$887$	−149.877			−0.168971			−0.0844853	−	0.996425i	$-0.526925\pi$
$887$	−149.877			−0.168971			−0.0844853	+	0.996425i	$0.526925\pi$
$888$	147.874			0.166524
$889$	−1338.31	+	207.173i	−1.50541	+	0.233041i
$890$		0			0
$891$	−89.3609			−0.100293
$892$	−125.244			−0.140408
$893$			13.6696i			0.0153075i
$894$		−	671.578i		−	0.751206i
$895$		0			0
$896$	78.2638	−	12.1154i	0.0873480	−	0.0135216i
$897$			894.036i			0.996695i
$898$			668.191i			0.744088i
$899$		−	334.885i		−	0.372508i
$900$		0			0
$901$			211.983i			0.235276i
$902$	593.819			0.658336
$903$	596.332	−	92.3132i	0.660390	−	0.102229i
$904$	−41.5697			−0.0459841
$905$		0			0
$906$			393.254i			0.434056i
$907$			1544.51i			1.70288i	0.524452	+	0.851440i	$0.324270\pi$
$907$			1544.51i			1.70288i	−0.524452	+	0.851440i	$0.675730\pi$
$908$	−441.115			−0.485809
$909$			400.674i			0.440786i
$910$		0			0
$911$	1324.82			1.45425			0.727126	−	0.686504i	$-0.240856\pi$
$911$	1324.82			1.45425			0.727126	+	0.686504i	$0.240856\pi$
$912$			32.4836i			0.0356180i
$913$	683.984			0.749161
$914$	−850.381			−0.930395
$915$		0			0
$916$		−	413.594i		−	0.451521i
$917$	−683.714	+	105.840i	−0.745599	+	0.115420i
$918$		−	139.184i		−	0.151616i
$919$	1469.75			1.59930			0.799648	−	0.600470i	$-0.205020\pi$
$919$	1469.75			1.59930			0.799648	+	0.600470i	$0.205020\pi$
$920$		0			0
$921$	384.116			0.417064
$922$	382.258			0.414596
$923$	−1033.20			−1.11940
$924$	36.8322	+	237.931i	0.0398616	+	0.257501i
$925$		0			0
$926$	819.864			0.885382
$927$	−245.383			−0.264706
$928$		−	90.4249i		−	0.0974406i
$929$			1613.82i			1.73715i	0.495554	+	0.868577i	$0.334965\pi$
$929$			1613.82i			1.73715i	−0.495554	+	0.868577i	$0.665035\pi$
$930$		0			0
$931$	−69.4643	−	218.989i	−0.0746125	−	0.235219i
$932$			334.626i			0.359041i
$933$		−	140.047i		−	0.150104i
$934$			1266.14i			1.35561i
$935$		0			0
$936$		−	171.936i		−	0.183693i
$937$	−1422.24			−1.51787			−0.758934	−	0.651167i	$-0.774280\pi$
$937$	−1422.24			−1.51787			−0.758934	+	0.651167i	$0.774280\pi$
$938$	−20.5845	−	132.973i	−0.0219451	−	0.141763i
$939$	589.077			0.627345
$940$		0			0
$941$		−	904.869i		−	0.961603i	−0.876829	−	0.480802i	$-0.840346\pi$
$941$		−	904.869i		−	0.961603i	0.876829	−	0.480802i	$-0.159654\pi$
$942$			652.131i			0.692284i
$943$	−1077.28			−1.14239
$944$		−	254.764i		−	0.269877i
$945$		0			0
$946$	−698.862			−0.738755
$947$			414.868i			0.438087i	0.975715	+	0.219043i	$0.0702936\pi$
$947$			414.868i			0.438087i	−0.975715	+	0.219043i	$0.929706\pi$
$948$	−242.025			−0.255301
$949$	−1598.86			−1.68478
$950$		0			0
$951$			527.457i			0.554634i
$952$	−370.589	+	57.3678i	−0.389274	+	0.0602603i
$953$			30.9413i			0.0324672i	0.999868	+	0.0162336i	$0.00516755\pi$
$953$			30.9413i			0.0324672i	−0.999868	+	0.0162336i	$0.994832\pi$
$954$	47.4840			0.0497735
$955$		0			0
$956$	−29.0641			−0.0304018
$957$	274.902			0.287254
$958$	357.450			0.373121
$959$	−279.290	+	43.2346i	−0.291230	+	0.0450830i
$960$		0			0
$961$	522.101			0.543289
$962$	−864.971			−0.899138
$963$		−	480.270i		−	0.498723i
$964$		−	675.503i		−	0.700730i
$965$		0			0
$966$	−66.8190	−	431.643i	−0.0691709	−	0.446835i
$967$		−	1662.25i		−	1.71897i	−0.511159	−	0.859486i	$-0.670784\pi$
$967$		−	1662.25i		−	1.71897i	0.511159	−	0.859486i	$-0.329216\pi$
$968$			63.3998i			0.0654957i
$969$		−	153.814i		−	0.158735i
$970$		0			0
$971$			5.10561i			0.00525810i	0.999997	+	0.00262905i	$0.000836853\pi$
$971$			5.10561i			0.00525810i	−0.999997	+	0.00262905i	$0.999163\pi$
$972$	−31.1769			−0.0320750
$973$	244.555	−	37.8575i	0.251341	−	0.0389080i
$974$	−596.619			−0.612545
$975$		0			0
$976$			357.132i			0.365914i
$977$		−	531.274i		−	0.543781i	−0.962328	−	0.271891i	$-0.912351\pi$
$977$		−	531.274i		−	0.543781i	0.962328	−	0.271891i	$-0.0876489\pi$
$978$	459.031			0.469356
$979$		−	1549.75i		−	1.58299i
$980$		0			0
$981$	550.848			0.561517
$982$		−	328.772i		−	0.334798i
$983$	1746.75			1.77696			0.888481	−	0.458913i	$-0.151761\pi$
$983$	1746.75			1.77696			0.888481	+	0.458913i	$0.151761\pi$
$984$	207.176			0.210545
$985$		0			0
$986$			428.173i			0.434252i
$987$	5.40760	+	34.9324i	0.00547882	+	0.0353925i
$988$		−	190.009i		−	0.192317i
$989$	1267.84			1.28194
$990$		0			0
$991$	301.944			0.304687			0.152343	−	0.988328i	$-0.451318\pi$
$991$	301.944			0.304687			0.152343	+	0.988328i	$0.451318\pi$
$992$	−118.511			−0.119466
$993$	559.756			0.563702
$994$	498.833	−	77.2202i	0.501844	−	0.0776863i
$995$		0			0
$996$	238.633			0.239592
$997$	521.642			0.523212			0.261606	−	0.965175i	$-0.415748\pi$
$997$	521.642			0.523212			0.261606	+	0.965175i	$0.415748\pi$
$998$			747.746i			0.749244i
$999$			156.844i			0.157001i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1050.3.h.c.349.14		24
5.2	odd	4		1050.3.f.c.601.1	✓	12
5.3	odd	4		1050.3.f.d.601.12	yes	12
5.4	even	2	inner	1050.3.h.c.349.11		24
7.6	odd	2	inner	1050.3.h.c.349.12		24
35.13	even	4		1050.3.f.d.601.9	yes	12
35.27	even	4		1050.3.f.c.601.4	yes	12
35.34	odd	2	inner	1050.3.h.c.349.13		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1050.3.f.c.601.1	✓	12	5.2	odd	4
1050.3.f.c.601.4	yes	12	35.27	even	4
1050.3.f.d.601.9	yes	12	35.13	even	4
1050.3.f.d.601.12	yes	12	5.3	odd	4
1050.3.h.c.349.11		24	5.4	even	2	inner
1050.3.h.c.349.12		24	7.6	odd	2	inner
1050.3.h.c.349.13		24	35.34	odd	2	inner
1050.3.h.c.349.14		24	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1050.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.41421i q^{2} +1.73205 q^{3} -2.00000 q^{4} +2.44949i q^{6} +(1.07086 + 6.91761i) q^{7} -2.82843i q^{8} +3.00000 q^{9} +O(q^{10})\)	\(q+1.41421i q^{2} +1.73205 q^{3} -2.00000 q^{4} +2.44949i q^{6} +(1.07086 + 6.91761i) q^{7} -2.82843i q^{8} +3.00000 q^{9} -9.92899 q^{11} -3.46410 q^{12} +20.2629 q^{13} +(-9.78297 + 1.51442i) q^{14} +4.00000 q^{16} -18.9405 q^{17} +4.24264i q^{18} +4.68861i q^{19} +(1.85478 + 11.9816i) q^{21} -14.0417i q^{22} +25.4738i q^{23} -4.89898i q^{24} +28.6560i q^{26} +5.19615 q^{27} +(-2.14171 - 13.8352i) q^{28} -15.9850 q^{29} +20.9499i q^{31} +5.65685i q^{32} -17.1975 q^{33} -26.7859i q^{34} -6.00000 q^{36} +30.1846i q^{37} -6.63069 q^{38} +35.0963 q^{39} +42.2896i q^{41} +(-16.9446 + 2.62305i) q^{42} -49.7705i q^{43} +19.8580 q^{44} -36.0253 q^{46} +2.91549 q^{47} +6.92820 q^{48} +(-46.7065 + 14.8155i) q^{49} -32.8059 q^{51} -40.5258 q^{52} -11.1921i q^{53} +7.34847i q^{54} +(19.5659 - 3.02884i) q^{56} +8.12091i q^{57} -22.6062i q^{58} -63.6910i q^{59} +89.2830i q^{61} -29.6277 q^{62} +(3.21257 + 20.7528i) q^{63} -8.00000 q^{64} -24.3210i q^{66} +13.5923i q^{67} +37.8810 q^{68} +44.1218i q^{69} -50.9899 q^{71} -8.48528i q^{72} -78.9059 q^{73} -42.6875 q^{74} -9.37722i q^{76} +(-10.6325 - 68.6848i) q^{77} +49.6337i q^{78} +69.8667 q^{79} +9.00000 q^{81} -59.8066 q^{82} -68.8876 q^{83} +(-3.70956 - 23.9633i) q^{84} +70.3860 q^{86} -27.6868 q^{87} +28.0834i q^{88} +156.083i q^{89} +(21.6987 + 140.171i) q^{91} -50.9475i q^{92} +36.2863i q^{93} +4.12313i q^{94} +9.79796i q^{96} -154.698 q^{97} +(-20.9523 - 66.0530i) q^{98} -29.7870 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 48 q^{4} + 72 q^{9}+O(q^{10}) \) 24 * q - 48 * q^4 + 72 * q^9	\( 24 q - 48 q^{4} + 72 q^{9} - 32 q^{11} - 16 q^{14} + 96 q^{16} - 12 q^{21} - 96 q^{29} - 144 q^{36} + 24 q^{39} + 64 q^{44} + 160 q^{46} - 236 q^{49} + 144 q^{51} + 32 q^{56} - 192 q^{64} + 496 q^{71} + 128 q^{74} + 416 q^{79} + 216 q^{81} + 24 q^{84} + 256 q^{86} - 316 q^{91} - 96 q^{99}+O(q^{100}) \) 24 * q - 48 * q^4 + 72 * q^9 - 32 * q^11 - 16 * q^14 + 96 * q^16 - 12 * q^21 - 96 * q^29 - 144 * q^36 + 24 * q^39 + 64 * q^44 + 160 * q^46 - 236 * q^49 + 144 * q^51 + 32 * q^56 - 192 * q^64 + 496 * q^71 + 128 * q^74 + 416 * q^79 + 216 * q^81 + 24 * q^84 + 256 * q^86 - 316 * q^91 - 96 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more