LMFDB - Embedded newform 136.4.s.a.27.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [136,4,Mod(3,136)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(136, base_ring=CyclotomicField(16))

chi = DirichletCharacter(H, H._module([8, 8, 1]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("136.3");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 136 = 2^{3} \cdot 17 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	136.s (of order $16$, degree $8$, 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:	$8.02425976078$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{16})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 1 $ x^8 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{16}]$

Embedding invariants

Embedding label			27.1
Root			$0.382683 + 0.923880i$ of defining polynomial
Character	$\chi$	$=$	136.27
Dual form			136.4.s.a.131.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.08239 + 2.61313i) q^{2} +(-3.92868 - 2.62506i) q^{3} +(-5.65685 - 5.65685i) q^{4} +(11.1120 - 7.42479i) q^{6} +(20.9050 - 8.65914i) q^{8} +(-1.78887 - 4.31871i) q^{9} +O(q^{10})$	\(q+(-1.08239 + 2.61313i) q^{2} +(-3.92868 - 2.62506i) q^{3} +(-5.65685 - 5.65685i) q^{4} +(11.1120 - 7.42479i) q^{6} +(20.9050 - 8.65914i) q^{8} +(-1.78887 - 4.31871i) q^{9} +(13.5105 + 20.2199i) q^{11} +(7.37439 + 37.0735i) q^{12} +64.0000i q^{16} +(32.4286 + 62.1400i) q^{17} +13.2216 q^{18} +(-63.1249 + 152.397i) q^{19} +(-67.4609 + 13.4188i) q^{22} +(-104.860 - 20.8579i) q^{24} +(115.485 - 47.8354i) q^{25} +(-29.1975 + 146.786i) q^{27} +(-167.240 - 69.2731i) q^{32} -114.904i q^{33} +(-197.480 + 17.4802i) q^{34} +(-14.3109 + 34.5497i) q^{36} +(-329.907 - 329.907i) q^{38} +(330.567 + 65.7538i) q^{41} +(128.530 + 310.300i) q^{43} +(37.9541 - 190.808i) q^{44} +(168.004 - 251.436i) q^{48} +(316.891 + 131.260i) q^{49} +353.553i q^{50} +(35.7196 - 329.255i) q^{51} +(-351.967 - 235.177i) q^{54} +(648.049 - 433.012i) q^{57} +(21.4716 - 8.89383i) q^{59} +(362.039 - 362.039i) q^{64} +(300.257 + 124.371i) q^{66} +354.214i q^{67} +(168.073 - 534.961i) q^{68} +(-74.7926 - 74.7926i) q^{72} +(-287.393 + 57.1660i) q^{73} +(-579.274 - 115.225i) q^{75} +(1219.18 - 504.999i) q^{76} +(410.784 - 410.784i) q^{81} +(-529.625 + 792.641i) q^{82} +(-911.297 - 377.472i) q^{83} -949.973 q^{86} +(457.525 + 305.708i) q^{88} +(-1146.08 - 1146.08i) q^{89} +(475.187 + 711.167i) q^{96} +(363.525 + 1827.56i) q^{97} +(-686.000 + 686.000i) q^{98} +(63.1554 - 94.5188i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{3} + 80 q^{6} - 80 q^{9}+O(q^{10}) $ 8 * q - 8 * q^3 + 80 * q^6 - 80 * q^9	$ 8 q - 8 q^{3} + 80 q^{6} - 80 q^{9} + 320 q^{12} + 144 q^{22} - 256 q^{24} + 496 q^{27} - 720 q^{34} - 640 q^{36} - 1440 q^{38} + 160 q^{41} + 208 q^{43} - 1600 q^{44} + 512 q^{48} + 1880 q^{51} - 176 q^{54} + 2840 q^{57} - 920 q^{59} + 5232 q^{66} + 3456 q^{72} - 3312 q^{73} - 2632 q^{81} - 3472 q^{83} - 5120 q^{96} - 5488 q^{98} - 1536 q^{99}+O(q^{100}) $ 8 * q - 8 * q^3 + 80 * q^6 - 80 * q^9 + 320 * q^12 + 144 * q^22 - 256 * q^24 + 496 * q^27 - 720 * q^34 - 640 * q^36 - 1440 * q^38 + 160 * q^41 + 208 * q^43 - 1600 * q^44 + 512 * q^48 + 1880 * q^51 - 176 * q^54 + 2840 * q^57 - 920 * q^59 + 5232 * q^66 + 3456 * q^72 - 3312 * q^73 - 2632 * q^81 - 3472 * q^83 - 5120 * q^96 - 5488 * q^98 - 1536 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/136\mathbb{Z}\right)^\times$.

$n$	$69$	$103$	$105$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{3}{16}\right)$

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 $

$2$	−1.08239	+	2.61313i	−0.382683	+	0.923880i
$2$	−1.08239	+	2.61313i	−0.382683	+	0.923880i
$3$	−3.92868	−	2.62506i	−0.756075	−	0.505193i	0.116789	−	0.993157i	$-0.462740\pi$
$3$	−3.92868	−	2.62506i	−0.756075	−	0.505193i	−0.872864	+	0.487964i	$0.837740\pi$
$4$	−5.65685	−	5.65685i	−0.707107	−	0.707107i
$5$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$5$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$6$	11.1120	−	7.42479i	0.756075	−	0.505193i
$7$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$7$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$8$	20.9050	−	8.65914i	0.923880	−	0.382683i
$9$	−1.78887	−	4.31871i	−0.0662544	−	0.159952i
$10$		0			0
$11$	13.5105	+	20.2199i	0.370325	+	0.554231i	0.969094	−	0.246691i	$-0.0793433\pi$
$11$	13.5105	+	20.2199i	0.370325	+	0.554231i	−0.598769	+	0.800922i	$0.704343\pi$
$12$	7.37439	+	37.0735i	0.177400	+	0.891851i
$13$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$13$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$14$		0			0
$15$		0			0
$16$			64.0000i			1.00000i
$17$	32.4286	+	62.1400i	0.462653	+	0.886539i
$17$	32.4286	+	62.1400i	0.462653	+	0.886539i
$18$	13.2216			0.173131
$19$	−63.1249	+	152.397i	−0.762202	+	1.84012i	−0.297175	+	0.954823i	$0.596045\pi$
$19$	−63.1249	+	152.397i	−0.762202	+	1.84012i	−0.465027	+	0.885296i	$0.653955\pi$
$20$		0			0
$21$		0			0
$22$	−67.4609	+	13.4188i	−0.653760	+	0.130041i
$23$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$23$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$24$	−104.860	−	20.8579i	−0.891851	−	0.177400i
$25$	115.485	−	47.8354i	0.923880	−	0.382683i
$26$		0			0
$27$	−29.1975	+	146.786i	−0.208114	+	1.04626i
$28$		0			0
$29$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$29$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$30$		0			0
$31$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$31$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$32$	−167.240	−	69.2731i	−0.923880	−	0.382683i
$33$		−	114.904i		−	0.606125i
$34$	−197.480	+	17.4802i	−0.996105	+	0.0881716i
$35$		0			0
$36$	−14.3109	+	34.5497i	−0.0662544	+	0.159952i
$37$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$37$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$38$	−329.907	−	329.907i	−1.40837	−	1.40837i
$39$		0			0
$40$		0			0
$41$	330.567	+	65.7538i	1.25917	+	0.250464i	0.779173	−	0.626809i	$-0.215639\pi$
$41$	330.567	+	65.7538i	1.25917	+	0.250464i	0.479993	+	0.877272i	$0.340639\pi$
$42$		0			0
$43$	128.530	+	310.300i	0.455831	+	1.10047i	0.970070	+	0.242826i	$0.0780743\pi$
$43$	128.530	+	310.300i	0.455831	+	1.10047i	−0.514239	+	0.857647i	$0.671926\pi$
$44$	37.9541	−	190.808i	0.130041	−	0.653760i
$45$		0			0
$46$		0			0
$47$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$47$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$48$	168.004	−	251.436i	0.505193	−	0.756075i
$49$	316.891	+	131.260i	0.923880	+	0.382683i
$50$			353.553i			1.00000i
$51$	35.7196	−	329.255i	0.0980733	−	0.904019i
$52$		0			0
$53$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$53$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$54$	−351.967	−	235.177i	−0.886975	−	0.592658i
$55$		0			0
$56$		0			0
$57$	648.049	−	433.012i	1.50590	−	1.00621i
$58$		0			0
$59$	21.4716	−	8.89383i	0.0473791	−	0.0196251i	−0.358868	−	0.933388i	$-0.616837\pi$
$59$	21.4716	−	8.89383i	0.0473791	−	0.0196251i	0.406247	+	0.913763i	$0.366837\pi$
$60$		0			0
$61$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$61$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$62$		0			0
$63$		0			0
$64$	362.039	−	362.039i	0.707107	−	0.707107i
$65$		0			0
$66$	300.257	+	124.371i	0.559987	+	0.231954i
$67$			354.214i			0.645883i	0.946419	+	0.322941i	$0.104672\pi$
$67$			354.214i			0.645883i	−0.946419	+	0.322941i	$0.895328\pi$
$68$	168.073	−	534.961i	0.299733	−	0.954023i
$69$		0			0
$70$		0			0
$71$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$71$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$72$	−74.7926	−	74.7926i	−0.122422	−	0.122422i
$73$	−287.393	+	57.1660i	−0.460778	+	0.0916545i	−0.420021	−	0.907515i	$-0.637977\pi$
$73$	−287.393	+	57.1660i	−0.460778	+	0.0916545i	−0.0407577	+	0.999169i	$0.512977\pi$
$74$		0			0
$75$	−579.274	−	115.225i	−0.891851	−	0.177400i
$76$	1219.18	−	504.999i	1.84012	−	0.762202i
$77$		0			0
$78$		0			0
$79$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$79$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$80$		0			0
$81$	410.784	−	410.784i	0.563490	−	0.563490i
$82$	−529.625	+	792.641i	−0.713260	+	1.06747i
$83$	−911.297	−	377.472i	−1.20515	−	0.499191i	−0.312494	−	0.949920i	$-0.601164\pi$
$83$	−911.297	−	377.472i	−1.20515	−	0.499191i	−0.892661	+	0.450728i	$0.851164\pi$
$84$		0			0
$85$		0			0
$86$	−949.973			−1.19114
$87$		0			0
$88$	457.525	+	305.708i	0.554231	+	0.370325i
$89$	−1146.08	−	1146.08i	−1.36499	−	1.36499i	−0.867426	−	0.497565i	$-0.834227\pi$
$89$	−1146.08	−	1146.08i	−1.36499	−	1.36499i	−0.497565	−	0.867426i	$-0.665773\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$	475.187	+	711.167i	0.505193	+	0.756075i
$97$	363.525	+	1827.56i	0.380519	+	1.91300i	0.407165	+	0.913355i	$0.366517\pi$
$97$	363.525	+	1827.56i	0.380519	+	1.91300i	−0.0266459	+	0.999645i	$0.508483\pi$
$98$	−686.000	+	686.000i	−0.707107	+	0.707107i
$99$	63.1554	−	94.5188i	0.0641147	−	0.0959545i
$100$	−923.880	−	382.683i	−0.923880	−	0.382683i
$101$		0			0			−	1.00000i	$-0.5\pi$
$101$		0			0				1.00000i	$0.5\pi$
$102$	821.723	+	449.723i	0.797674	+	0.436561i
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−1913.89	+	380.697i	−1.72919	+	0.343956i	−0.956700	−	0.291076i	$-0.905987\pi$
$107$	−1913.89	+	380.697i	−1.72919	+	0.343956i	−0.772486	+	0.635032i	$0.780987\pi$
$108$	995.513	−	665.180i	0.886975	−	0.592658i
$109$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$109$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	111.617	+	167.047i	0.0929211	+	0.139066i	0.875018	−	0.484091i	$-0.160850\pi$
$113$	111.617	+	167.047i	0.0929211	+	0.139066i	−0.782097	+	0.623157i	$0.785850\pi$
$114$	430.073	+	2162.12i	0.353333	+	1.77633i
$115$		0			0
$116$		0			0
$117$		0			0
$118$			65.7346i			0.0512827i
$119$		0			0
$120$		0			0
$121$	283.041	−	683.320i	0.212653	−	0.513389i
$122$		0			0
$123$	−1126.08	−	1126.08i	−0.825491	−	0.825491i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$127$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$128$	554.185	+	1337.92i	0.382683	+	0.923880i
$129$	309.601	−	1556.47i	0.211309	−	1.06232i
$130$		0			0
$131$	577.362	+	2902.59i	0.385071	+	1.93588i	0.350740	+	0.936473i	$0.385930\pi$
$131$	577.362	+	2902.59i	0.385071	+	1.93588i	0.0343312	+	0.999411i	$0.489070\pi$
$132$	−649.993	+	649.993i	−0.428595	+	0.428595i
$133$		0			0
$134$	−925.606	−	383.399i	−0.596718	−	0.247169i
$135$		0			0
$136$	1216.00	+	1018.23i	0.766700	+	0.642006i
$137$	25.0097			0.0155965			0.00779827	−	0.999970i	$-0.497518\pi$
$137$	25.0097			0.0155965			0.00779827	+	0.999970i	$0.497518\pi$
$138$		0			0
$139$	1676.90	+	1120.47i	1.02326	+	0.683718i	0.949567	−	0.313563i	$-0.101523\pi$
$139$	1676.90	+	1120.47i	1.02326	+	0.683718i	0.0736895	+	0.997281i	$0.476523\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	276.397	−	114.488i	0.159952	−	0.0662544i
$145$		0			0
$146$	161.690	−	812.871i	0.0916545	−	0.460778i
$147$	−900.396	−	1347.54i	−0.505193	−	0.756075i
$148$		0			0
$149$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$149$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$150$	928.099	−	1389.00i	0.505193	−	0.756075i
$151$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$151$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$152$			3732.47i			1.99173i
$153$	210.354	−	251.210i	0.111151	−	0.132739i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$157$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	628.801	+	1518.06i	0.304958	+	0.736235i
$163$	−31.5535	+	158.630i	−0.0151623	+	0.0762261i	−0.987626	−	0.156830i	$-0.949873\pi$
$163$	−31.5535	+	158.630i	−0.0151623	+	0.0762261i	0.972463	+	0.233056i	$0.0748726\pi$
$164$	−1498.01	−	2241.93i	−0.713260	−	1.06747i
$165$		0			0
$166$	1972.76	−	1972.76i	0.922386	−	0.922386i
$167$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$167$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$168$		0			0
$169$		−	2197.00i		−	1.00000i
$170$		0			0
$171$	771.080			0.344830
$172$	1028.24	−	2482.40i	0.455831	−	1.10047i
$173$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$173$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$174$		0			0
$175$		0			0
$176$	−1294.08	+	864.673i	−0.554231	+	0.370325i
$177$	−107.702	−	21.4232i	−0.0457366	−	0.00909757i
$178$	4235.36	−	1754.34i	1.78345	−	0.738728i
$179$	−1814.32	−	4380.16i	−0.757590	−	1.82898i	−0.510161	−	0.860079i	$-0.670414\pi$
$179$	−1814.32	−	4380.16i	−0.757590	−	1.82898i	−0.247429	−	0.968906i	$-0.579586\pi$
$180$		0			0
$181$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$181$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−818.339	+	1495.25i	−0.320015	+	0.584724i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$191$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$192$	−2372.71	+	471.961i	−0.891851	+	0.177400i
$193$	−3688.24	+	2464.40i	−1.37557	+	0.919127i	−0.999971	−	0.00765530i	$-0.997563\pi$
$193$	−3688.24	+	2464.40i	−1.37557	+	0.919127i	−0.375600	+	0.926782i	$0.622563\pi$
$194$	−5169.13	−	1028.20i	−1.91300	−	0.380519i
$195$		0			0
$196$	−1050.08	−	2535.13i	−0.382683	−	0.923880i
$197$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$197$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$198$	178.631	+	267.339i	0.0641147	+	0.0959545i
$199$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$199$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$200$	2000.00	−	2000.00i	0.707107	−	0.707107i
$201$	929.834	−	1391.59i	0.326296	−	0.488336i
$202$		0			0
$203$		0			0
$204$	−2064.61	+	1660.49i	−0.708586	+	0.569890i
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−3934.31	+	782.582i	−1.30211	+	0.259006i
$210$		0			0
$211$	5197.25	+	1033.80i	1.69570	+	0.337296i	0.945923	−	0.324391i	$-0.105159\pi$
$211$	5197.25	+	1033.80i	1.69570	+	0.337296i	0.749780	+	0.661687i	$0.230159\pi$
$212$		0			0
$213$		0			0
$214$	1076.77	−	5413.30i	0.343956	−	1.72919i
$215$		0			0
$216$	660.665	+	3321.39i	0.208114	+	1.04626i
$217$		0			0
$218$		0			0
$219$	1279.14	+	529.837i	0.394686	+	0.163484i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$223$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$224$		0			0
$225$	−413.175	−	413.175i	−0.122422	−	0.122422i
$226$	−557.330	+	110.860i	−0.164040	+	0.0326296i
$227$	5666.57	−	3786.28i	1.65684	−	1.10707i	0.780889	−	0.624669i	$-0.214766\pi$
$227$	5666.57	−	3786.28i	1.65684	−	1.10707i	0.875953	−	0.482397i	$-0.160234\pi$
$228$	−6115.41	−	1216.43i	−1.77633	−	0.353333i
$229$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$229$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	951.459	+	4783.31i	0.267520	+	1.34491i	0.847722	+	0.530441i	$0.177974\pi$
$233$	951.459	+	4783.31i	0.267520	+	1.34491i	−0.580202	+	0.814473i	$0.697026\pi$
$234$		0			0
$235$		0			0
$236$	−171.773	−	71.1507i	−0.0473791	−	0.0196251i
$237$		0			0
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$	4539.13	+	3032.95i	1.21324	+	0.810661i	0.986575	−	0.163311i	$-0.0522174\pi$
$241$	4539.13	+	3032.95i	1.21324	+	0.810661i	0.226666	+	0.973973i	$0.427217\pi$
$242$	1479.24	+	1479.24i	0.392931	+	0.392931i
$243$	1271.05	−	252.827i	0.335547	−	0.0667444i
$244$		0			0
$245$		0			0
$246$	4161.46	−	1723.73i	1.07856	−	0.446753i
$247$		0			0
$248$		0			0
$249$	2589.31	+	3875.17i	0.658999	+	0.986262i
$250$		0			0
$251$	4620.51	−	4620.51i	1.16193	−	1.16193i	0.177875	−	0.984053i	$-0.443078\pi$
$251$	4620.51	−	4620.51i	1.16193	−	1.16193i	0.984053	−	0.177875i	$-0.0569223\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−4096.00			−1.00000
$257$	−2433.61	+	5875.25i	−0.590678	+	1.42602i	0.292170	+	0.956366i	$0.405623\pi$
$257$	−2433.61	+	5875.25i	−0.590678	+	1.42602i	−0.882849	+	0.469658i	$0.844377\pi$
$258$	3732.14	+	2493.74i	0.900593	+	0.601757i
$259$		0			0
$260$		0			0
$261$		0			0
$262$	−8209.77	−	1633.03i	−1.93588	−	0.385071i
$263$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$263$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$264$	−994.965	−	2402.06i	−0.231954	−	0.559987i
$265$		0			0
$266$		0			0
$267$	1494.05	+	7511.11i	0.342452	+	1.72162i
$268$	2003.74	−	2003.74i	0.456708	−	0.456708i
$269$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$269$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$270$		0			0
$271$		0			0			−	1.00000i	$-0.5\pi$
$271$		0			0				1.00000i	$0.5\pi$
$272$	−3976.96	+	2075.43i	−0.886539	+	0.462653i
$273$		0			0
$274$	−27.0704	+	65.3536i	−0.00596854	+	0.0144093i
$275$	2527.49	+	1688.82i	0.554231	+	0.370325i
$276$		0			0
$277$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$277$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$278$	−4742.99	+	3169.16i	−1.02326	+	0.683718i
$279$		0			0
$280$		0			0
$281$	−2198.82	−	5308.42i	−0.466799	−	1.12695i	−0.965552	−	0.260209i	$-0.916209\pi$
$281$	−2198.82	−	5308.42i	−0.466799	−	1.12695i	0.498753	−	0.866744i	$-0.333791\pi$
$282$		0			0
$283$	114.954	+	172.040i	0.0241459	+	0.0361369i	0.843346	−	0.537371i	$-0.180582\pi$
$283$	114.954	+	172.040i	0.0241459	+	0.0361369i	−0.819200	+	0.573507i	$0.805582\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$			846.181i			0.173131i
$289$	−2809.77	+	4030.23i	−0.571905	+	0.820320i
$290$		0			0
$291$	3369.29	−	8134.19i	0.678733	−	1.63861i
$292$	1949.12	+	1302.36i	0.390629	+	0.261010i
$293$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$293$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$294$	4495.87	−	894.283i	0.891851	−	0.177400i
$295$		0			0
$296$		0			0
$297$	−3362.47	+	1392.78i	−0.656938	+	0.272113i
$298$		0			0
$299$		0			0
$300$	2625.06	+	3928.68i	0.505193	+	0.756075i
$301$		0			0
$302$		0			0
$303$		0			0
$304$	−9753.41	−	4039.99i	−1.84012	−	0.762202i
$305$		0			0
$306$	428.758	+	821.590i	0.0800996	+	0.153487i
$307$	7204.00			1.33926			0.669632	−	0.742693i	$-0.266452\pi$
$307$	7204.00			1.33926			0.669632	+	0.742693i	$0.266452\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$311$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$312$		0			0
$313$	−8559.66	−	1702.62i	−1.54575	−	0.307469i	−0.652769	−	0.757557i	$-0.726393\pi$
$313$	−8559.66	−	1702.62i	−1.54575	−	0.307469i	−0.892984	+	0.450088i	$0.851393\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$317$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	8518.42	+	3528.44i	1.48116	+	0.613516i
$322$		0			0
$323$	−11517.0	+	1019.44i	−1.98397	+	0.175614i
$324$	−4647.49			−0.796895
$325$		0			0
$326$	−380.367	−	254.153i	−0.0646214	−	0.0431786i
$327$		0			0
$328$	7479.87	−	1487.84i	1.25917	−	0.250464i
$329$		0			0
$330$		0			0
$331$	−10140.0	+	4200.12i	−1.68382	+	0.697461i	−0.999497	−	0.0317191i	$-0.989902\pi$
$331$	−10140.0	+	4200.12i	−1.68382	+	0.697461i	−0.684322	+	0.729180i	$0.739902\pi$
$332$	3019.77	+	7290.38i	0.499191	+	1.20515i
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−5698.55	+	8528.48i	−0.921127	+	1.37856i	0.00444445	+	0.999990i	$0.498585\pi$
$337$	−5698.55	+	8528.48i	−0.921127	+	1.37856i	−0.925571	+	0.378574i	$0.876415\pi$
$338$	5741.04	+	2378.02i	0.923880	+	0.382683i
$339$		−	949.278i		−	0.152088i
$340$		0			0
$341$		0			0
$342$	−834.611	+	2014.93i	−0.131961	+	0.318582i
$343$		0			0
$344$	5373.86	+	5373.86i	0.842265	+	0.842265i
$345$		0			0
$346$		0			0
$347$	−6039.72	−	1201.37i	−0.934377	−	0.185859i	−0.295654	−	0.955295i	$-0.595537\pi$
$347$	−6039.72	−	1201.37i	−0.934377	−	0.185859i	−0.638724	+	0.769436i	$0.720537\pi$
$348$		0			0
$349$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$349$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$350$		0			0
$351$		0			0
$352$	−858.804	−	4317.50i	−0.130041	−	0.653760i
$353$	9376.55	−	9376.55i	1.41378	−	1.41378i	0.689316	−	0.724461i	$-0.257911\pi$
$353$	9376.55	−	9376.55i	1.41378	−	1.41378i	0.724461	−	0.689316i	$-0.242089\pi$
$354$	172.557	−	258.250i	0.0259077	−	0.0387736i
$355$		0			0
$356$			12966.4i			1.93039i
$357$		0			0
$358$	13409.7			1.97968
$359$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$359$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$360$		0			0
$361$	−14390.0	−	14390.0i	−2.09798	−	2.09798i
$362$		0			0
$363$	−2905.73	+	1941.55i	−0.420142	+	0.280730i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$367$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$368$		0			0
$369$	−307.368	−	1545.25i	−0.0433630	−	0.218001i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0			−	1.00000i	$-0.5\pi$
$373$		0			0				1.00000i	$0.5\pi$
$374$	−3021.51	−	3756.87i	−0.417750	−	0.519420i
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−1012.72	+	201.443i	−0.137256	+	0.0273019i	−0.263240	−	0.964730i	$-0.584791\pi$
$379$	−1012.72	+	201.443i	−0.137256	+	0.0273019i	0.125984	+	0.992032i	$0.459791\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$383$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$384$	1334.91	−	6711.03i	0.177400	−	0.891851i
$385$		0			0
$386$	−2447.67	−	12305.3i	−0.322754	−	1.62260i
$387$	1110.17	−	1110.17i	0.145822	−	0.145822i
$388$	8281.85	−	12394.7i	1.08363	−	1.62176i
$389$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$389$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$390$		0			0
$391$		0			0
$392$	7761.20			1.00000
$393$	5351.21	−	12919.0i	0.686852	−	1.65821i
$394$		0			0
$395$		0			0
$396$	−891.940	+	177.418i	−0.113186	+	0.0225141i
$397$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$397$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$398$		0			0
$399$		0			0
$400$	3061.47	+	7391.04i	0.382683	+	0.923880i
$401$	3032.01	−	15242.9i	0.377584	−	1.89824i	−0.0584046	−	0.998293i	$-0.518601\pi$
$401$	3032.01	−	15242.9i	0.377584	−	1.89824i	0.435989	−	0.899952i	$-0.356399\pi$
$402$	2629.97	+	3936.02i	0.326296	+	0.488336i
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$	−2104.35	−	7192.39i	−0.255345	−	0.872736i
$409$	14127.6			1.70798			0.853990	−	0.520289i	$-0.174176\pi$
$409$	14127.6			1.70798			0.853990	+	0.520289i	$0.174176\pi$
$410$		0			0
$411$	−98.2553	−	65.6521i	−0.0117922	−	0.00787926i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−3646.70	−	8803.92i	−0.428249	−	1.03388i
$418$	2213.48	−	11127.9i	0.259006	−	1.30211i
$419$	−8772.36	−	13128.8i	−1.02281	−	1.53075i	−0.836285	−	0.548296i	$-0.815277\pi$
$419$	−8772.36	−	13128.8i	−1.02281	−	1.53075i	−0.186527	−	0.982450i	$-0.559723\pi$
$420$		0			0
$421$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$421$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$422$	−8326.90	+	12462.1i	−0.960539	+	1.43755i
$423$		0			0
$424$		0			0
$425$	6717.51	+	5625.00i	0.766700	+	0.642006i
$426$		0			0
$427$		0			0
$428$	12980.2	+	8673.06i	1.46593	+	0.979505i
$429$		0			0
$430$		0			0
$431$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$431$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$432$	−9394.30	−	1868.64i	−1.04626	−	0.208114i
$433$	−14808.1	+	6133.71i	−1.64349	+	0.680756i	−0.996644	−	0.0818641i	$-0.973913\pi$
$433$	−14808.1	+	6133.71i	−1.64349	+	0.680756i	−0.646847	+	0.762620i	$0.723913\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$	−2769.06	+	2769.06i	−0.302080	+	0.302080i
$439$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$439$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$440$		0			0
$441$		−	1603.37i		−	0.173131i
$442$		0			0
$443$	−3539.69			−0.379629			−0.189814	−	0.981820i	$-0.560789\pi$
$443$	−3539.69			−0.379629			−0.189814	+	0.981820i	$0.560789\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	15424.1	+	3068.04i	1.62117	+	0.322471i	0.920420	−	0.390932i	$-0.127847\pi$
$449$	15424.1	+	3068.04i	1.62117	+	0.322471i	0.700754	+	0.713403i	$0.252847\pi$
$450$	1526.89	−	632.460i	0.159952	−	0.0662544i
$451$	3136.59	+	7572.40i	0.327486	+	0.790621i
$452$	313.559	−	1576.37i	0.0326296	−	0.164040i
$453$		0			0
$454$	3760.57	+	18905.7i	0.388750	+	1.95438i
$455$		0			0
$456$	9797.95	−	14663.7i	1.00621	−	1.50590i
$457$	16660.7	+	6901.09i	1.70537	+	0.706388i	0.999997	−	0.00244611i	$-0.000778620\pi$
$457$	16660.7	+	6901.09i	1.70537	+	0.706388i	0.705375	+	0.708834i	$0.250779\pi$
$458$		0			0
$459$	−10068.1	+	2945.73i	−1.02383	+	0.299553i
$460$		0			0
$461$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$461$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$462$		0			0
$463$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$463$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$464$		0			0
$465$		0			0
$466$	−13529.2	−	2691.13i	−1.34491	−	0.267520i
$467$	−1017.94	+	421.646i	−0.100867	+	0.0417804i	−0.432546	−	0.901612i	$-0.642385\pi$
$467$	−1017.94	+	421.646i	−0.100867	+	0.0417804i	0.331679	+	0.943392i	$0.392385\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$	371.851	−	371.851i	0.0362624	−	0.0362624i
$473$	−4537.73	+	6791.19i	−0.441110	+	0.660168i
$474$		0			0
$475$			20619.2i			1.99173i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$479$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$480$		0			0
$481$		0			0
$482$	−12838.6	+	8578.47i	−1.21324	+	0.810661i
$483$		0			0
$484$	−5466.56	+	2264.32i	−0.513389	+	0.212653i
$485$		0			0
$486$	−715.104	+	3595.07i	−0.0667444	+	0.335547i
$487$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$487$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$488$		0			0
$489$	540.377	−	540.377i	0.0499727	−	0.0499727i
$490$		0			0
$491$	−3776.58	−	1564.31i	−0.347117	−	0.143781i	0.202311	−	0.979321i	$-0.435155\pi$
$491$	−3776.58	−	1564.31i	−0.347117	−	0.143781i	−0.549429	+	0.835540i	$0.685155\pi$
$492$			12740.2i			1.16742i
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$	−12929.0	+	2571.73i	−1.16338	+	0.231410i
$499$	17301.1	−	11560.2i	1.55211	−	1.03709i	0.576631	−	0.817005i	$-0.304367\pi$
$499$	17301.1	−	11560.2i	1.55211	−	1.03709i	0.975481	−	0.220084i	$-0.0706330\pi$
$500$		0			0
$501$		0			0
$502$	7072.77	+	17075.2i	0.628831	+	1.51813i
$503$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$503$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−5767.26	+	8631.31i	−0.505193	+	0.756075i
$508$		0			0
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$		0			0
$511$		0			0
$512$	4433.48	−	10703.4i	0.382683	−	0.923880i
$513$	−20526.6	−	13715.5i	−1.76661	−	1.18041i
$514$	−12718.7	−	12718.7i	−1.09143	−	1.09143i
$515$		0			0
$516$	−10556.1	+	7053.35i	−0.900593	+	0.601757i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	11006.9	+	16473.0i	0.925569	+	1.38521i	0.922827	+	0.385215i	$0.125873\pi$
$521$	11006.9	+	16473.0i	0.925569	+	1.38521i	0.00274220	+	0.999996i	$0.499127\pi$
$522$		0			0
$523$	−16601.4	+	16601.4i	−1.38801	+	1.38801i	−0.558517	+	0.829493i	$0.688629\pi$
$523$	−16601.4	+	16601.4i	−1.38801	+	1.38801i	−0.829493	+	0.558517i	$0.811371\pi$
$524$	13153.5	−	19685.6i	1.09659	−	1.64116i
$525$		0			0
$526$		0			0
$527$		0			0
$528$	7353.83			0.606125
$529$	4656.11	−	11240.8i	0.382683	−	0.923880i
$530$		0			0
$531$	−76.8197	−	76.8197i	−0.00627814	−	0.00627814i
$532$		0			0
$533$		0			0
$534$	−21244.6	−	4225.82i	−1.72162	−	0.342452i
$535$		0			0
$536$	3067.19	+	7404.85i	0.247169	+	0.596718i
$537$	−4370.29	+	21970.9i	−0.351195	+	1.76558i
$538$		0			0
$539$	1627.28	+	8180.90i	0.130041	+	0.653760i
$540$		0			0
$541$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$541$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$542$		0			0
$543$		0			0
$544$	−1118.73	−	12638.7i	−0.0881716	−	0.996105i
$545$		0			0
$546$		0			0
$547$	−15658.9	−	10463.0i	−1.22400	−	0.817850i	−0.235917	−	0.971773i	$-0.575809\pi$
$547$	−15658.9	−	10463.0i	−1.22400	−	0.817850i	−0.988083	+	0.153923i	$0.950809\pi$
$548$	−141.476	−	141.476i	−0.0110284	−	0.0110284i
$549$		0			0
$550$	−7148.82	+	4776.69i	−0.554231	+	0.370325i
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	−3147.65	−	15824.3i	−0.240090	−	1.20701i
$557$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$557$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	7140.11	−	3726.16i	0.537354	−	0.280426i
$562$	16251.6			1.21980
$563$	10183.3	−	24584.6i	0.762298	−	1.84035i	0.298504	−	0.954408i	$-0.403512\pi$
$563$	10183.3	−	24584.6i	0.762298	−	1.84035i	0.463795	−	0.885943i	$-0.346488\pi$
$564$		0			0
$565$		0			0
$566$	−573.988	+	114.173i	−0.0426263	+	0.00847890i
$567$		0			0
$568$		0			0
$569$	7516.59	−	3113.47i	0.553800	−	0.229391i	−0.0881913	−	0.996104i	$-0.528109\pi$
$569$	7516.59	−	3113.47i	0.553800	−	0.229391i	0.641991	+	0.766712i	$0.278109\pi$
$570$		0			0
$571$	−3985.81	+	20038.0i	−0.292121	+	1.46859i	0.504131	+	0.863627i	$0.331813\pi$
$571$	−3985.81	+	20038.0i	−0.292121	+	1.46859i	−0.796252	+	0.604965i	$0.793187\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−2211.18	−	915.900i	−0.159952	−	0.0662544i
$577$			19651.9i			1.41789i	0.705266	+	0.708943i	$0.250827\pi$
$577$			19651.9i			1.41789i	−0.705266	+	0.708943i	$0.749173\pi$
$578$	−7490.24	−	11704.6i	−0.539019	−	0.842294i
$579$	20959.1			1.50437
$580$		0			0
$581$		0			0
$582$	17608.8	+	17608.8i	1.25413	+	1.25413i
$583$		0			0
$584$	−5512.95	+	3683.63i	−0.390629	+	0.261010i
$585$		0			0
$586$		0			0
$587$	−3960.77	−	9562.15i	−0.278498	−	0.672355i	0.721296	−	0.692627i	$-0.243547\pi$
$587$	−3960.77	−	9562.15i	−0.278498	−	0.672355i	−0.999795	+	0.0202721i	$0.993547\pi$
$588$	−2529.41	+	12716.2i	−0.177400	+	0.891851i
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	26658.6	+	11042.3i	1.84610	+	0.764679i	0.939273	+	0.343171i	$0.111501\pi$
$593$	26658.6	+	11042.3i	1.84610	+	0.764679i	0.906825	+	0.421507i	$0.138499\pi$
$594$		−	10294.1i		−	0.711064i
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$599$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$600$	−13107.5	+	2607.24i	−0.891851	+	0.177400i
$601$	−18631.1	+	12448.9i	−1.26453	+	0.844929i	−0.993070	−	0.117522i	$-0.962505\pi$
$601$	−18631.1	+	12448.9i	−1.26453	+	0.844929i	−0.271455	+	0.962451i	$0.587505\pi$
$602$		0			0
$603$	1529.75	−	633.642i	0.103310	−	0.0427926i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$607$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$608$	21114.0	−	21114.0i	1.40837	−	1.40837i
$609$		0			0
$610$		0			0
$611$		0			0
$612$	−2611.00	+	231.116i	−0.172457	+	0.0152652i
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$	−7797.56	+	18825.0i	−0.512514	+	1.23732i
$615$		0			0
$616$		0			0
$617$	24684.5	−	4910.04i	1.61063	−	0.320374i	0.693958	−	0.720015i	$-0.255865\pi$
$617$	24684.5	−	4910.04i	1.61063	−	0.320374i	0.916672	+	0.399641i	$0.130865\pi$
$618$		0			0
$619$	30000.0	+	5967.38i	1.94799	+	0.387479i	0.996913	+	0.0785136i	$0.0250174\pi$
$619$	30000.0	+	5967.38i	1.94799	+	0.387479i	0.951073	+	0.308965i	$0.0999826\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	11048.5	−	11048.5i	0.707107	−	0.707107i
$626$	13714.1	−	20524.6i	0.875599	−	1.31043i
$627$	17511.0	+	7253.27i	1.11534	+	0.461990i
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$631$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$632$		0			0
$633$	−17704.5	−	17704.5i	−1.11168	−	1.11168i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−2359.45	−	11861.8i	−0.145387	−	0.730907i	−0.982849	−	0.184413i	$-0.940962\pi$
$641$	−2359.45	−	11861.8i	−0.145387	−	0.730907i	0.837462	−	0.546495i	$-0.184038\pi$
$642$	−18440.5	+	18440.5i	−1.13363	+	1.13363i
$643$	−18052.0	+	27016.7i	−1.10715	+	1.65697i	−0.483205	+	0.875507i	$0.660528\pi$
$643$	−18052.0	+	27016.7i	−1.10715	+	1.65697i	−0.623948	+	0.781466i	$0.714472\pi$
$644$		0			0
$645$		0			0
$646$	9801.99	−	31198.8i	0.596988	−	1.90016i
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$	5030.41	−	12144.5i	0.304958	−	0.736235i
$649$	469.925	+	313.994i	0.0284225	+	0.0189913i
$650$		0			0
$651$		0			0
$652$	1075.84	−	718.853i	0.0646214	−	0.0431786i
$653$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$653$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$654$		0			0
$655$		0			0
$656$	−4208.24	+	21156.3i	−0.250464	+	1.25917i
$657$	760.992	+	1138.90i	0.0451889	+	0.0676300i
$658$		0			0
$659$	21039.3	−	21039.3i	1.24366	−	1.24366i	0.285191	−	0.958471i	$-0.407943\pi$
$659$	21039.3	−	21039.3i	1.24366	−	1.24366i	0.958471	−	0.285191i	$-0.0920570\pi$
$660$		0			0
$661$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$661$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$662$		−	31043.2i		−	1.82255i
$663$		0			0
$664$	−22319.3			−1.30445
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	6812.04	−	34246.4i	0.390171	−	1.96152i	0.188024	−	0.982165i	$-0.439792\pi$
$673$	6812.04	−	34246.4i	0.390171	−	1.96152i	0.202147	−	0.979355i	$-0.435208\pi$
$674$	−16117.9	−	24122.2i	−0.921127	−	1.37856i
$675$	3649.69	+	18348.2i	0.208114	+	1.04626i
$676$	−12428.1	+	12428.1i	−0.707107	+	0.707107i
$677$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$677$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$678$	2480.58	+	1027.49i	0.140511	+	0.0582014i
$679$		0			0
$680$		0			0
$681$	−32201.3			−1.81198
$682$		0			0
$683$	−17722.4	−	11841.7i	−0.992866	−	0.663412i	−0.0507540	−	0.998711i	$-0.516162\pi$
$683$	−17722.4	−	11841.7i	−0.992866	−	0.663412i	−0.942112	+	0.335300i	$0.891162\pi$
$684$	−4361.89	−	4361.89i	−0.243832	−	0.243832i
$685$		0			0
$686$		0			0
$687$		0			0
$688$	−19859.2	+	8225.95i	−1.10047	+	0.455831i
$689$		0			0
$690$		0			0
$691$	−16146.1	−	24164.4i	−0.888898	−	1.33033i	−0.943345	−	0.331813i	$-0.892340\pi$
$691$	−16146.1	−	24164.4i	−0.888898	−	1.33033i	0.0544477	−	0.998517i	$-0.482660\pi$
$692$		0			0
$693$		0			0
$694$	9676.68	−	14482.2i	0.529282	−	0.792127i
$695$		0			0
$696$		0			0
$697$	6633.88	+	22673.7i	0.360511	+	1.23218i
$698$		0			0
$699$	8818.49	−	21289.7i	0.477176	−	1.15200i
$700$		0			0
$701$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$701$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$702$		0			0
$703$		0			0
$704$	12211.7	+	2429.06i	0.653760	+	0.130041i
$705$		0			0
$706$	14353.0	+	34651.2i	0.765130	+	1.84719i
$707$		0			0
$708$	488.066	+	730.442i	0.0259077	+	0.0387736i
$709$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$709$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$710$		0			0
$711$		0			0
$712$	−33882.9	−	14034.8i	−1.78345	−	0.738728i
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−14514.6	+	35041.3i	−0.757590	+	1.82898i
$717$		0			0
$718$		0			0
$719$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$719$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$720$		0			0
$721$		0			0
$722$	53178.7	−	22027.3i	2.74114	−	1.13542i
$723$	−9871.11	−	23831.0i	−0.507760	−	1.22584i
$724$		0			0
$725$		0			0
$726$	−1928.37	−	9694.56i	−0.0985791	−	0.495591i
$727$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$727$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$728$		0			0
$729$	−20148.5	−	8345.80i	−1.02365	−	0.424010i
$730$		0			0
$731$	−15114.0	+	18049.5i	−0.764721	+	0.913249i
$732$		0			0
$733$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$733$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−7162.18	+	4785.62i	−0.357968	+	0.239187i
$738$	4370.61	+	869.369i	0.218001	+	0.0433630i
$739$	16345.1	−	6770.36i	0.813619	−	0.337012i	0.0632220	−	0.997999i	$-0.479862\pi$
$739$	16345.1	−	6770.36i	0.813619	−	0.337012i	0.750397	+	0.660988i	$0.229862\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$743$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$			4610.87i			0.225841i
$748$	13087.6	−	3829.18i	0.639747	−	0.187177i
$749$		0			0
$750$		0			0
$751$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$751$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$752$		0			0
$753$	−30281.6	+	6023.39i	−1.46550	+	0.291507i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$757$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$758$	569.767	−	2864.41i	0.0273019	−	0.137256i
$759$		0			0
$760$		0			0
$761$	−4900.48	+	4900.48i	−0.233433	+	0.233433i	−0.814124	−	0.580691i	$-0.802782\pi$
$761$	−4900.48	+	4900.48i	−0.233433	+	0.233433i	0.580691	+	0.814124i	$0.302782\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	16091.9	+	10752.2i	0.756075	+	0.505193i
$769$	−12798.1	−	12798.1i	−0.600144	−	0.600144i	0.340207	−	0.940351i	$-0.389503\pi$
$769$	−12798.1	−	12798.1i	−0.600144	−	0.600144i	−0.940351	+	0.340207i	$0.889503\pi$
$770$		0			0
$771$	24983.8	−	16693.6i	1.16701	−	0.779774i
$772$	34804.6	+	6923.06i	1.62260	+	0.322754i
$773$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$773$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$774$	1699.38	+	4102.66i	0.0789184	+	0.190526i
$775$		0			0
$776$	23424.6	+	35057.4i	1.08363	+	1.62176i
$777$		0			0
$778$		0			0
$779$	−30887.7	+	46226.6i	−1.42062	+	2.12611i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−8400.67	+	20281.0i	−0.382683	+	0.923880i
$785$		0			0
$786$	27966.8	+	27966.8i	1.26914	+	1.26914i
$787$	43276.0	−	8608.14i	1.96013	−	0.389895i	0.972597	−	0.232499i	$-0.0746902\pi$
$787$	43276.0	−	8608.14i	1.96013	−	0.389895i	0.987536	−	0.157396i	$-0.0503098\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$	501.814	−	2522.79i	0.0225141	−	0.113186i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$797$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$798$		0			0
$799$		0			0
$800$	−22627.4			−1.00000
$801$	−2899.40	+	6999.77i	−0.127897	+	0.308770i
$802$	36549.9	+	24421.9i	1.60925	+	1.07527i
$803$	−5038.72	−	5038.72i	−0.221435	−	0.221435i
$804$	−13132.0	+	2612.11i	−0.576031	+	0.114580i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−7820.88	+	39318.2i	−0.339886	+	1.70872i	0.311730	+	0.950171i	$0.399092\pi$
$809$	−7820.88	+	39318.2i	−0.339886	+	1.70872i	−0.651615	+	0.758550i	$0.725908\pi$
$810$		0			0
$811$	−3708.28	−	18642.8i	−0.160561	−	0.807196i	−0.974176	−	0.225791i	$-0.927503\pi$
$811$	−3708.28	−	18642.8i	−0.160561	−	0.807196i	0.813614	−	0.581405i	$-0.197497\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$	21072.3	+	2286.05i	0.904019	+	0.0980733i
$817$	−55402.3			−2.37244
$818$	−15291.6	+	36917.2i	−0.653616	+	1.57797i
$819$		0			0
$820$		0			0
$821$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$821$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$822$	277.908	−	185.692i	0.0117922	−	0.00787926i
$823$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$823$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$824$		0			0
$825$	−5496.46	−	13269.6i	−0.231954	−	0.559987i
$826$		0			0
$827$	−25082.9	−	37539.2i	−1.05468	−	1.57844i	−0.789019	−	0.614369i	$-0.789411\pi$
$827$	−25082.9	−	37539.2i	−1.05468	−	1.57844i	−0.265658	−	0.964067i	$-0.585589\pi$
$828$		0			0
$829$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$829$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	2119.81	+	23948.2i	0.0881716	+	0.996105i
$834$	26952.9			1.11907
$835$		0			0
$836$	26682.7	+	17828.8i	1.10388	+	0.737588i
$837$		0			0
$838$	43802.3	−	8712.81i	1.80564	−	0.359164i
$839$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$839$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$840$		0			0
$841$	−22532.5	+	9333.27i	−0.923880	+	0.382683i
$842$		0			0
$843$	−5296.46	+	26627.1i	−0.216394	+	1.08788i
$844$	−23552.0	−	35248.1i	−0.960539	−	1.43755i
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$		−	977.651i		−	0.0395205i
$850$	−21969.8	+	11465.3i	−0.886539	+	0.462653i
$851$		0			0
$852$		0			0
$853$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$853$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$854$		0			0
$855$		0			0
$856$	−36713.4	+	24531.1i	−1.46593	+	0.979505i
$857$	40579.2	+	8071.71i	1.61746	+	0.321732i	0.919102	−	0.394020i	$-0.128916\pi$
$857$	40579.2	+	8071.71i	1.61746	+	0.321732i	0.698354	+	0.715752i	$0.253916\pi$
$858$		0			0
$859$	830.298	+	2004.52i	0.0329795	+	0.0796196i	0.939511	−	0.342518i	$-0.111280\pi$
$859$	830.298	+	2004.52i	0.0329795	+	0.0796196i	−0.906532	+	0.422138i	$0.861280\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$863$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$864$	15051.3	−	22525.9i	0.592658	−	0.886975i
$865$		0			0
$866$		−	45334.5i		−	1.77890i
$867$	21618.3	−	8457.69i	0.846823	−	0.331301i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	7242.42	−	4839.23i	0.280777	−	0.187609i
$874$		0			0
$875$		0			0
$876$	−4238.70	−	10233.1i	−0.163484	−	0.394686i
$877$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$877$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−9556.88	+	14302.9i	−0.365470	+	0.546965i	−0.967942	−	0.251176i	$-0.919183\pi$
$881$	−9556.88	+	14302.9i	−0.365470	+	0.546965i	0.602471	+	0.798141i	$0.294183\pi$
$882$	4189.80	+	1735.47i	0.159952	+	0.0662544i
$883$		−	50260.1i		−	1.91550i	−0.287602	−	0.957750i	$-0.592858\pi$
$883$		−	50260.1i		−	1.91550i	0.287602	−	0.957750i	$-0.407142\pi$
$884$		0			0
$885$		0			0
$886$	3831.33	−	9249.65i	0.145278	−	0.350731i
$887$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$887$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	13855.9	+	2756.11i	0.520977	+	0.103629i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	−24712.1	+	36984.2i	−0.918321	+	1.37436i
$899$		0			0
$900$			4674.54i			0.173131i
$901$		0			0
$902$	−23182.6			−0.855762
$903$		0			0
$904$	3779.85	+	2525.61i	0.139066	+	0.0929211i
$905$		0			0
$906$		0			0
$907$	40262.4	−	26902.5i	1.47397	−	0.984875i	0.479762	−	0.877399i	$-0.340723\pi$
$907$	40262.4	−	26902.5i	1.47397	−	0.984875i	0.994208	−	0.107477i	$-0.0342771\pi$
$908$	−53473.4	−	10636.5i	−1.95438	−	0.388750i
$909$		0			0
$910$		0			0
$911$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$911$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$912$	27712.8	+	41475.1i	1.00621	+	1.50590i
$913$	−4679.65	−	23526.2i	−0.169632	−	0.852797i
$914$	−36066.8	+	36066.8i	−1.30524	+	1.30524i
$915$		0			0
$916$		0			0
$917$		0			0
$918$	3200.08	−	29497.7i	0.115053	−	1.06053i
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$	−28302.2	−	18910.9i	−1.01258	−	0.676587i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	29868.5	+	44701.4i	1.05485	+	1.57869i	0.788716	+	0.614758i	$0.210746\pi$
$929$	29868.5	+	44701.4i	1.05485	+	1.57869i	0.266134	+	0.963936i	$0.414254\pi$
$930$		0			0
$931$	−40007.4	+	40007.4i	−1.40837	+	1.40837i
$932$	21676.2	−	32440.7i	0.761832	−	1.14016i
$933$		0			0
$934$		−	3116.40i		−	0.109178i
$935$		0			0
$936$		0			0
$937$	4266.79	−	10300.9i	0.148762	−	0.359143i	−0.831879	−	0.554957i	$-0.812735\pi$
$937$	4266.79	−	10300.9i	0.148762	−	0.359143i	0.980641	+	0.195814i	$0.0627348\pi$
$938$		0			0
$939$	29158.7	+	29158.7i	1.01337	+	1.01337i
$940$		0			0
$941$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$941$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$942$		0			0
$943$		0			0
$944$	569.205	+	1374.18i	0.0196251	+	0.0473791i
$945$		0			0
$946$	−12834.6	−	19208.4i	−0.441110	−	0.660168i
$947$	−2699.54	−	13571.5i	−0.0926326	−	0.465696i	−0.999061	−	0.0433353i	$-0.986202\pi$
$947$	−2699.54	−	13571.5i	−0.0926326	−	0.465696i	0.906428	−	0.422361i	$-0.138798\pi$
$948$		0			0
$949$		0			0
$950$	−53880.5	−	22318.0i	−1.84012	−	0.762202i
$951$		0			0
$952$		0			0
$953$	−16308.3			−0.554332			−0.277166	−	0.960822i	$-0.589395\pi$
$953$	−16308.3			−0.554332			−0.277166	+	0.960822i	$0.589395\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−11400.5	−	27523.3i	−0.382683	−	0.923880i
$962$		0			0
$963$	5067.82	+	7584.52i	0.169583	+	0.253798i
$964$	−8520.24	−	42834.1i	−0.284666	−	1.43111i
$965$		0			0
$966$		0			0
$967$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$967$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$968$		−	16735.7i		−	0.555688i
$969$	47922.7	+	26227.8i	1.58875	+	0.869512i
$970$		0			0
$971$	−21371.3	+	51594.8i	−0.706320	+	1.70521i	0.00267705	+	0.999996i	$0.499148\pi$
$971$	−21371.3	+	51594.8i	−0.706320	+	1.70521i	−0.708997	+	0.705211i	$0.750852\pi$
$972$	−8620.35	−	5759.93i	−0.284463	−	0.190072i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−43837.8	+	18158.2i	−1.43551	+	0.594609i	−0.958706	−	0.284399i	$-0.908206\pi$
$977$	−43837.8	+	18158.2i	−1.43551	+	0.594609i	−0.476807	+	0.879008i	$0.658206\pi$
$978$	827.173	+	1996.97i	0.0270451	+	0.0652925i
$979$	7689.51	−	38657.8i	0.251030	−	1.26201i
$980$		0			0
$981$		0			0
$982$	8175.48	−	8175.48i	0.265672	−	0.265672i
$983$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$983$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$984$	−33291.7	−	13789.9i	−1.07856	−	0.446753i
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$991$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$992$		0			0
$993$	50862.3	+	10117.1i	1.62545	+	0.323321i
$994$		0			0
$995$		0			0
$996$	7273.95	−	36568.6i	0.231410	−	1.16338i
$997$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$997$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$998$	11481.8	+	57722.7i	0.364177	+	1.83084i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	136.4.s.a.27.1	✓	8
8.3	odd	2	CM	136.4.s.a.27.1	✓	8
17.12	odd	16	inner	136.4.s.a.131.1	yes	8
136.131	even	16	inner	136.4.s.a.131.1	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
136.4.s.a.27.1	✓	8	1.1	even	1	trivial
136.4.s.a.27.1	✓	8	8.3	odd	2	CM
136.4.s.a.131.1	yes	8	17.12	odd	16	inner
136.4.s.a.131.1	yes	8	136.131	even	16	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 136 = 2^{3} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	136.s (of order \(16\), degree \(8\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.08239 + 2.61313i) q^{2} +(-3.92868 - 2.62506i) q^{3} +(-5.65685 - 5.65685i) q^{4} +(11.1120 - 7.42479i) q^{6} +(20.9050 - 8.65914i) q^{8} +(-1.78887 - 4.31871i) q^{9} +O(q^{10})\)	\(q+(-1.08239 + 2.61313i) q^{2} +(-3.92868 - 2.62506i) q^{3} +(-5.65685 - 5.65685i) q^{4} +(11.1120 - 7.42479i) q^{6} +(20.9050 - 8.65914i) q^{8} +(-1.78887 - 4.31871i) q^{9} +(13.5105 + 20.2199i) q^{11} +(7.37439 + 37.0735i) q^{12} +64.0000i q^{16} +(32.4286 + 62.1400i) q^{17} +13.2216 q^{18} +(-63.1249 + 152.397i) q^{19} +(-67.4609 + 13.4188i) q^{22} +(-104.860 - 20.8579i) q^{24} +(115.485 - 47.8354i) q^{25} +(-29.1975 + 146.786i) q^{27} +(-167.240 - 69.2731i) q^{32} -114.904i q^{33} +(-197.480 + 17.4802i) q^{34} +(-14.3109 + 34.5497i) q^{36} +(-329.907 - 329.907i) q^{38} +(330.567 + 65.7538i) q^{41} +(128.530 + 310.300i) q^{43} +(37.9541 - 190.808i) q^{44} +(168.004 - 251.436i) q^{48} +(316.891 + 131.260i) q^{49} +353.553i q^{50} +(35.7196 - 329.255i) q^{51} +(-351.967 - 235.177i) q^{54} +(648.049 - 433.012i) q^{57} +(21.4716 - 8.89383i) q^{59} +(362.039 - 362.039i) q^{64} +(300.257 + 124.371i) q^{66} +354.214i q^{67} +(168.073 - 534.961i) q^{68} +(-74.7926 - 74.7926i) q^{72} +(-287.393 + 57.1660i) q^{73} +(-579.274 - 115.225i) q^{75} +(1219.18 - 504.999i) q^{76} +(410.784 - 410.784i) q^{81} +(-529.625 + 792.641i) q^{82} +(-911.297 - 377.472i) q^{83} -949.973 q^{86} +(457.525 + 305.708i) q^{88} +(-1146.08 - 1146.08i) q^{89} +(475.187 + 711.167i) q^{96} +(363.525 + 1827.56i) q^{97} +(-686.000 + 686.000i) q^{98} +(63.1554 - 94.5188i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{3} + 80 q^{6} - 80 q^{9}+O(q^{10}) \) 8 * q - 8 * q^3 + 80 * q^6 - 80 * q^9	\( 8 q - 8 q^{3} + 80 q^{6} - 80 q^{9} + 320 q^{12} + 144 q^{22} - 256 q^{24} + 496 q^{27} - 720 q^{34} - 640 q^{36} - 1440 q^{38} + 160 q^{41} + 208 q^{43} - 1600 q^{44} + 512 q^{48} + 1880 q^{51} - 176 q^{54} + 2840 q^{57} - 920 q^{59} + 5232 q^{66} + 3456 q^{72} - 3312 q^{73} - 2632 q^{81} - 3472 q^{83} - 5120 q^{96} - 5488 q^{98} - 1536 q^{99}+O(q^{100}) \) 8 * q - 8 * q^3 + 80 * q^6 - 80 * q^9 + 320 * q^12 + 144 * q^22 - 256 * q^24 + 496 * q^27 - 720 * q^34 - 640 * q^36 - 1440 * q^38 + 160 * q^41 + 208 * q^43 - 1600 * q^44 + 512 * q^48 + 1880 * q^51 - 176 * q^54 + 2840 * q^57 - 920 * q^59 + 5232 * q^66 + 3456 * q^72 - 3312 * q^73 - 2632 * q^81 - 3472 * q^83 - 5120 * q^96 - 5488 * q^98 - 1536 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more