LMFDB - Embedded newform 156.3.g.b.77.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [156,3,Mod(77,156)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("156.77"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(156, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 156 = 2^{2} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	156.g (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,-10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	no
Analytic conductor:	$4.25069212402$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{11})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} + 9 $ x^4 - 5*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			77.4
Root			$-1.65831 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	156.77
Dual form			156.3.g.b.77.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-2.50000 + 1.65831i) q^{3} +3.31662 q^{5} +7.00000i q^{7} +(3.50000 - 8.29156i) q^{9} -19.8997 q^{11} +13.0000i q^{13} +(-8.29156 + 5.50000i) q^{15} +23.2164i q^{17} +30.0000i q^{19} +(-11.6082 - 17.5000i) q^{21} -26.5330i q^{23} -14.0000 q^{25} +(5.00000 + 26.5330i) q^{27} -33.1662i q^{29} +10.0000i q^{31} +(49.7494 - 33.0000i) q^{33} +23.2164i q^{35} -13.0000i q^{37} +(-21.5581 - 32.5000i) q^{39} +46.4327 q^{41} +55.0000 q^{43} +(11.6082 - 27.5000i) q^{45} +49.7494 q^{47} +(-38.5000 - 58.0409i) q^{51} +39.7995i q^{53} -66.0000 q^{55} +(-49.7494 - 75.0000i) q^{57} -92.8655 q^{59} +14.0000 q^{61} +(58.0409 + 24.5000i) q^{63} +43.1161i q^{65} +16.0000i q^{67} +(44.0000 + 66.3325i) q^{69} +29.8496 q^{71} -4.00000i q^{73} +(35.0000 - 23.2164i) q^{75} -139.298i q^{77} +54.0000 q^{79} +(-56.5000 - 58.0409i) q^{81} +33.1662 q^{83} +77.0000i q^{85} +(55.0000 + 82.9156i) q^{87} -19.8997 q^{89} -91.0000 q^{91} +(-16.5831 - 25.0000i) q^{93} +99.4987i q^{95} -78.0000i q^{97} +(-69.6491 + 165.000i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 10 q^{3} + 14 q^{9} - 56 q^{25} + 20 q^{27} + 220 q^{43} - 154 q^{51} - 264 q^{55} + 56 q^{61} + 176 q^{69} + 140 q^{75} + 216 q^{79} - 226 q^{81} + 220 q^{87} - 364 q^{91}+O(q^{100}) $ 4 * q - 10 * q^3 + 14 * q^9 - 56 * q^25 + 20 * q^27 + 220 * q^43 - 154 * q^51 - 264 * q^55 + 56 * q^61 + 176 * q^69 + 140 * q^75 + 216 * q^79 - 226 * q^81 + 220 * q^87 - 364 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/156\mathbb{Z}\right)^\times$.

$n$	$53$	$79$	$145$
$\chi(n)$	$-1$	$1$	$-1$

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 $

$2$		0			0
$2$		0			0
$3$	−2.50000	+	1.65831i	−0.833333	+	0.552771i
$3$	−2.50000	+	1.65831i	−0.833333	+	0.552771i
$4$		0			0
$5$	3.31662			0.663325			0.331662	−	0.943398i	$-0.392390\pi$
$5$	3.31662			0.663325			0.331662	+	0.943398i	$0.392390\pi$
$6$		0			0
$7$			7.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$7$			7.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$8$		0			0
$9$	3.50000	−	8.29156i	0.388889	−	0.921285i
$10$		0			0
$11$	−19.8997			−1.80907			−0.904534	−	0.426401i	$-0.859781\pi$
$11$	−19.8997			−1.80907			−0.904534	+	0.426401i	$0.859781\pi$
$12$		0			0
$13$			13.0000i			1.00000i
$13$			13.0000i			1.00000i
$14$		0			0
$15$	−8.29156	+	5.50000i	−0.552771	+	0.366667i
$16$		0			0
$17$			23.2164i			1.36567i	0.730573	+	0.682835i	$0.239253\pi$
$17$			23.2164i			1.36567i	−0.730573	+	0.682835i	$0.760747\pi$
$18$		0			0
$19$			30.0000i			1.57895i	0.613784	+	0.789474i	$0.289646\pi$
$19$			30.0000i			1.57895i	−0.613784	+	0.789474i	$0.710354\pi$
$20$		0			0
$21$	−11.6082	−	17.5000i	−0.552771	−	0.833333i
$22$		0			0
$23$		−	26.5330i		−	1.15361i	−0.816882	−	0.576804i	$-0.804300\pi$
$23$		−	26.5330i		−	1.15361i	0.816882	−	0.576804i	$-0.195700\pi$
$24$		0			0
$25$	−14.0000			−0.560000
$26$		0			0
$27$	5.00000	+	26.5330i	0.185185	+	0.982704i
$28$		0			0
$29$		−	33.1662i		−	1.14366i	−0.820371	−	0.571832i	$-0.806233\pi$
$29$		−	33.1662i		−	1.14366i	0.820371	−	0.571832i	$-0.193767\pi$
$30$		0			0
$31$			10.0000i			0.322581i	0.986907	+	0.161290i	$0.0515656\pi$
$31$			10.0000i			0.322581i	−0.986907	+	0.161290i	$0.948434\pi$
$32$		0			0
$33$	49.7494	−	33.0000i	1.50756	−	1.00000i
$34$		0			0
$35$			23.2164i			0.663325i
$36$		0			0
$37$		−	13.0000i		−	0.351351i	−0.984448	−	0.175676i	$-0.943789\pi$
$37$		−	13.0000i		−	0.351351i	0.984448	−	0.175676i	$-0.0562110\pi$
$38$		0			0
$39$	−21.5581	−	32.5000i	−0.552771	−	0.833333i
$40$		0			0
$41$	46.4327			1.13251			0.566253	−	0.824231i	$-0.308393\pi$
$41$	46.4327			1.13251			0.566253	+	0.824231i	$0.308393\pi$
$42$		0			0
$43$	55.0000			1.27907			0.639535	−	0.768762i	$-0.279127\pi$
$43$	55.0000			1.27907			0.639535	+	0.768762i	$0.279127\pi$
$44$		0			0
$45$	11.6082	−	27.5000i	0.257960	−	0.611111i
$46$		0			0
$47$	49.7494			1.05850			0.529249	−	0.848467i	$-0.322474\pi$
$47$	49.7494			1.05850			0.529249	+	0.848467i	$0.322474\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	−38.5000	−	58.0409i	−0.754902	−	1.13806i
$52$		0			0
$53$			39.7995i			0.750934i	0.926836	+	0.375467i	$0.122518\pi$
$53$			39.7995i			0.750934i	−0.926836	+	0.375467i	$0.877482\pi$
$54$		0			0
$55$	−66.0000			−1.20000
$56$		0			0
$57$	−49.7494	−	75.0000i	−0.872796	−	1.31579i
$58$		0			0
$59$	−92.8655			−1.57399			−0.786996	−	0.616958i	$-0.788365\pi$
$59$	−92.8655			−1.57399			−0.786996	+	0.616958i	$0.788365\pi$
$60$		0			0
$61$	14.0000			0.229508			0.114754	−	0.993394i	$-0.463392\pi$
$61$	14.0000			0.229508			0.114754	+	0.993394i	$0.463392\pi$
$62$		0			0
$63$	58.0409	+	24.5000i	0.921285	+	0.388889i
$64$		0			0
$65$			43.1161i			0.663325i
$66$		0			0
$67$			16.0000i			0.238806i	0.992846	+	0.119403i	$0.0380980\pi$
$67$			16.0000i			0.238806i	−0.992846	+	0.119403i	$0.961902\pi$
$68$		0			0
$69$	44.0000	+	66.3325i	0.637681	+	0.961341i
$70$		0			0
$71$	29.8496			0.420417			0.210209	−	0.977657i	$-0.432586\pi$
$71$	29.8496			0.420417			0.210209	+	0.977657i	$0.432586\pi$
$72$		0			0
$73$		−	4.00000i		−	0.0547945i	−0.999625	−	0.0273973i	$-0.991278\pi$
$73$		−	4.00000i		−	0.0547945i	0.999625	−	0.0273973i	$-0.00872191\pi$
$74$		0			0
$75$	35.0000	−	23.2164i	0.466667	−	0.309552i
$76$		0			0
$77$		−	139.298i		−	1.80907i
$78$		0			0
$79$	54.0000			0.683544			0.341772	−	0.939783i	$-0.388973\pi$
$79$	54.0000			0.683544			0.341772	+	0.939783i	$0.388973\pi$
$80$		0			0
$81$	−56.5000	−	58.0409i	−0.697531	−	0.716555i
$82$		0			0
$83$	33.1662			0.399593			0.199797	−	0.979837i	$-0.435972\pi$
$83$	33.1662			0.399593			0.199797	+	0.979837i	$0.435972\pi$
$84$		0			0
$85$			77.0000i			0.905882i
$86$		0			0
$87$	55.0000	+	82.9156i	0.632184	+	0.953053i
$88$		0			0
$89$	−19.8997			−0.223593			−0.111796	−	0.993731i	$-0.535660\pi$
$89$	−19.8997			−0.223593			−0.111796	+	0.993731i	$0.535660\pi$
$90$		0			0
$91$	−91.0000			−1.00000
$92$		0			0
$93$	−16.5831	−	25.0000i	−0.178313	−	0.268817i
$94$		0			0
$95$			99.4987i			1.04736i
$96$		0			0
$97$		−	78.0000i		−	0.804124i	−0.915613	−	0.402062i	$-0.868294\pi$
$97$		−	78.0000i		−	0.804124i	0.915613	−	0.402062i	$-0.131706\pi$
$98$		0			0
$99$	−69.6491	+	165.000i	−0.703526	+	1.66667i
$100$		0			0
$101$			66.3325i			0.656757i	0.944546	+	0.328379i	$0.106502\pi$
$101$			66.3325i			0.656757i	−0.944546	+	0.328379i	$0.893498\pi$
$102$		0			0
$103$	−70.0000			−0.679612			−0.339806	−	0.940496i	$-0.610361\pi$
$103$	−70.0000			−0.679612			−0.339806	+	0.940496i	$0.610361\pi$
$104$		0			0
$105$	−38.5000	−	58.0409i	−0.366667	−	0.552771i
$106$		0			0
$107$		−	92.8655i		−	0.867902i	−0.900937	−	0.433951i	$-0.857119\pi$
$107$		−	92.8655i		−	0.867902i	0.900937	−	0.433951i	$-0.142881\pi$
$108$		0			0
$109$			175.000i			1.60550i	0.596313	+	0.802752i	$0.296632\pi$
$109$			175.000i			1.60550i	−0.596313	+	0.802752i	$0.703368\pi$
$110$		0			0
$111$	21.5581	+	32.5000i	0.194217	+	0.292793i
$112$		0			0
$113$		−	13.2665i		−	0.117403i	−0.998276	−	0.0587013i	$-0.981304\pi$
$113$		−	13.2665i		−	0.117403i	0.998276	−	0.0587013i	$-0.0186960\pi$
$114$		0			0
$115$		−	88.0000i		−	0.765217i
$116$		0			0
$117$	107.790	+	45.5000i	0.921285	+	0.388889i
$118$		0			0
$119$	−162.515			−1.36567
$120$		0			0
$121$	275.000			2.27273
$122$		0			0
$123$	−116.082	+	77.0000i	−0.943755	+	0.626016i
$124$		0			0
$125$	−129.348			−1.03479
$126$		0			0
$127$	140.000			1.10236			0.551181	−	0.834386i	$-0.314177\pi$
$127$	140.000			1.10236			0.551181	+	0.834386i	$0.314177\pi$
$128$		0			0
$129$	−137.500	+	91.2072i	−1.06589	+	0.707032i
$130$		0			0
$131$			82.9156i			0.632944i	0.948602	+	0.316472i	$0.102498\pi$
$131$			82.9156i			0.632944i	−0.948602	+	0.316472i	$0.897502\pi$
$132$		0			0
$133$	−210.000			−1.57895
$134$		0			0
$135$	16.5831	+	88.0000i	0.122838	+	0.651852i
$136$		0			0
$137$	66.3325			0.484179			0.242089	−	0.970254i	$-0.422167\pi$
$137$	66.3325			0.484179			0.242089	+	0.970254i	$0.422167\pi$
$138$		0			0
$139$	−11.0000			−0.0791367			−0.0395683	−	0.999217i	$-0.512598\pi$
$139$	−11.0000			−0.0791367			−0.0395683	+	0.999217i	$0.512598\pi$
$140$		0			0
$141$	−124.373	+	82.5000i	−0.882081	+	0.585106i
$142$		0			0
$143$		−	258.697i		−	1.80907i
$144$		0			0
$145$		−	110.000i		−	0.758621i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	13.2665			0.0890369			0.0445185	−	0.999009i	$-0.485825\pi$
$149$	13.2665			0.0890369			0.0445185	+	0.999009i	$0.485825\pi$
$150$		0			0
$151$		−	215.000i		−	1.42384i	−0.702260	−	0.711921i	$-0.747826\pi$
$151$		−	215.000i		−	1.42384i	0.702260	−	0.711921i	$-0.252174\pi$
$152$		0			0
$153$	192.500	+	81.2573i	1.25817	+	0.531094i
$154$		0			0
$155$			33.1662i			0.213976i
$156$		0			0
$157$	−140.000			−0.891720			−0.445860	−	0.895103i	$-0.647102\pi$
$157$	−140.000			−0.891720			−0.445860	+	0.895103i	$0.647102\pi$
$158$		0			0
$159$	−66.0000	−	99.4987i	−0.415094	−	0.625778i
$160$		0			0
$161$	185.731			1.15361
$162$		0			0
$163$		−	188.000i		−	1.15337i	−0.816965	−	0.576687i	$-0.804345\pi$
$163$		−	188.000i		−	1.15337i	0.816965	−	0.576687i	$-0.195655\pi$
$164$		0			0
$165$	165.000	−	109.449i	1.00000	−	0.663325i
$166$		0			0
$167$	198.997			1.19160			0.595801	−	0.803132i	$-0.296835\pi$
$167$	198.997			1.19160			0.595801	+	0.803132i	$0.296835\pi$
$168$		0			0
$169$	−169.000			−1.00000
$170$		0			0
$171$	248.747	+	105.000i	1.45466	+	0.614035i
$172$		0			0
$173$			218.897i			1.26530i	0.774437	+	0.632651i	$0.218033\pi$
$173$			218.897i			1.26530i	−0.774437	+	0.632651i	$0.781967\pi$
$174$		0			0
$175$		−	98.0000i		−	0.560000i
$176$		0			0
$177$	232.164	−	154.000i	1.31166	−	0.870056i
$178$		0			0
$179$			248.747i			1.38965i	0.719180	+	0.694824i	$0.244518\pi$
$179$			248.747i			1.38965i	−0.719180	+	0.694824i	$0.755482\pi$
$180$		0			0
$181$	94.0000			0.519337			0.259669	−	0.965698i	$-0.416387\pi$
$181$	94.0000			0.519337			0.259669	+	0.965698i	$0.416387\pi$
$182$		0			0
$183$	−35.0000	+	23.2164i	−0.191257	+	0.126865i
$184$		0			0
$185$		−	43.1161i		−	0.233060i
$186$		0			0
$187$		−	462.000i		−	2.47059i
$188$		0			0
$189$	−185.731	+	35.0000i	−0.982704	+	0.185185i
$190$		0			0
$191$			99.4987i			0.520936i	0.965483	+	0.260468i	$0.0838768\pi$
$191$			99.4987i			0.520936i	−0.965483	+	0.260468i	$0.916123\pi$
$192$		0			0
$193$			82.0000i			0.424870i	0.977175	+	0.212435i	$0.0681395\pi$
$193$			82.0000i			0.424870i	−0.977175	+	0.212435i	$0.931861\pi$
$194$		0			0
$195$	−71.5000	−	107.790i	−0.366667	−	0.552771i
$196$		0			0
$197$	−116.082			−0.589248			−0.294624	−	0.955613i	$-0.595194\pi$
$197$	−116.082			−0.589248			−0.294624	+	0.955613i	$0.595194\pi$
$198$		0			0
$199$	252.000			1.26633			0.633166	−	0.774016i	$-0.281755\pi$
$199$	252.000			1.26633			0.633166	+	0.774016i	$0.281755\pi$
$200$		0			0
$201$	−26.5330	−	40.0000i	−0.132005	−	0.199005i
$202$		0			0
$203$	232.164			1.14366
$204$		0			0
$205$	154.000			0.751220
$206$		0			0
$207$	−220.000	−	92.8655i	−1.06280	−	0.448626i
$208$		0			0
$209$		−	596.992i		−	2.85642i
$210$		0			0
$211$	−153.000			−0.725118			−0.362559	−	0.931961i	$-0.618097\pi$
$211$	−153.000			−0.725118			−0.362559	+	0.931961i	$0.618097\pi$
$212$		0			0
$213$	−74.6241	+	49.5000i	−0.350348	+	0.232394i
$214$		0			0
$215$	182.414			0.848439
$216$		0			0
$217$	−70.0000			−0.322581
$218$		0			0
$219$	6.63325	+	10.0000i	0.0302888	+	0.0456621i
$220$		0			0
$221$	−301.813			−1.36567
$222$		0			0
$223$		−	29.0000i		−	0.130045i	−0.997884	−	0.0650224i	$-0.979288\pi$
$223$		−	29.0000i		−	0.130045i	0.997884	−	0.0650224i	$-0.0207119\pi$
$224$		0			0
$225$	−49.0000	+	116.082i	−0.217778	+	0.515919i
$226$		0			0
$227$	33.1662			0.146107			0.0730534	−	0.997328i	$-0.476726\pi$
$227$	33.1662			0.146107			0.0730534	+	0.997328i	$0.476726\pi$
$228$		0			0
$229$			385.000i			1.68122i	0.541639	+	0.840611i	$0.317804\pi$
$229$			385.000i			1.68122i	−0.541639	+	0.840611i	$0.682196\pi$
$230$		0			0
$231$	231.000	+	348.246i	1.00000	+	1.50756i
$232$		0			0
$233$			189.048i			0.811363i	0.914014	+	0.405682i	$0.132966\pi$
$233$			189.048i			0.811363i	−0.914014	+	0.405682i	$0.867034\pi$
$234$		0			0
$235$	165.000			0.702128
$236$		0			0
$237$	−135.000	+	89.5489i	−0.569620	+	0.377843i
$238$		0			0
$239$	−434.478			−1.81790			−0.908949	−	0.416906i	$-0.863114\pi$
$239$	−434.478			−1.81790			−0.908949	+	0.416906i	$0.863114\pi$
$240$		0			0
$241$		−	130.000i		−	0.539419i	−0.962942	−	0.269710i	$-0.913072\pi$
$241$		−	130.000i		−	0.539419i	0.962942	−	0.269710i	$-0.0869277\pi$
$242$		0			0
$243$	237.500	+	51.4077i	0.977366	+	0.211554i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−390.000			−1.57895
$248$		0			0
$249$	−82.9156	+	55.0000i	−0.332994	+	0.220884i
$250$		0			0
$251$		−	331.662i		−	1.32136i	−0.750666	−	0.660682i	$-0.770267\pi$
$251$		−	331.662i		−	1.32136i	0.750666	−	0.660682i	$-0.229733\pi$
$252$		0			0
$253$			528.000i			2.08696i
$254$		0			0
$255$	−127.690	−	192.500i	−0.500745	−	0.754902i
$256$		0			0
$257$			268.647i			1.04532i	0.852542	+	0.522659i	$0.175060\pi$
$257$			268.647i			1.04532i	−0.852542	+	0.522659i	$0.824940\pi$
$258$		0			0
$259$	91.0000			0.351351
$260$		0			0
$261$	−275.000	−	116.082i	−1.05364	−	0.444758i
$262$		0			0
$263$		−	179.098i		−	0.680980i	−0.940248	−	0.340490i	$-0.889407\pi$
$263$		−	179.098i		−	0.680980i	0.940248	−	0.340490i	$-0.110593\pi$
$264$		0			0
$265$			132.000i			0.498113i
$266$		0			0
$267$	49.7494	−	33.0000i	0.186327	−	0.123596i
$268$		0			0
$269$			165.831i			0.616473i	0.951310	+	0.308237i	$0.0997388\pi$
$269$			165.831i			0.616473i	−0.951310	+	0.308237i	$0.900261\pi$
$270$		0			0
$271$			245.000i			0.904059i	0.892003	+	0.452030i	$0.149300\pi$
$271$			245.000i			0.904059i	−0.892003	+	0.452030i	$0.850700\pi$
$272$		0			0
$273$	227.500	−	150.906i	0.833333	−	0.552771i
$274$		0			0
$275$	278.596			1.01308
$276$		0			0
$277$	430.000			1.55235			0.776173	−	0.630520i	$-0.217158\pi$
$277$	430.000			1.55235			0.776173	+	0.630520i	$0.217158\pi$
$278$		0			0
$279$	82.9156	+	35.0000i	0.297189	+	0.125448i
$280$		0			0
$281$	−325.029			−1.15669			−0.578344	−	0.815793i	$-0.696301\pi$
$281$	−325.029			−1.15669			−0.578344	+	0.815793i	$0.696301\pi$
$282$		0			0
$283$	70.0000			0.247350			0.123675	−	0.992323i	$-0.460532\pi$
$283$	70.0000			0.247350			0.123675	+	0.992323i	$0.460532\pi$
$284$		0			0
$285$	−165.000	−	248.747i	−0.578947	−	0.872796i
$286$		0			0
$287$			325.029i			1.13251i
$288$		0			0
$289$	−250.000			−0.865052
$290$		0			0
$291$	129.348	+	195.000i	0.444496	+	0.670103i
$292$		0			0
$293$	−82.9156			−0.282988			−0.141494	−	0.989939i	$-0.545191\pi$
$293$	−82.9156			−0.282988			−0.141494	+	0.989939i	$0.545191\pi$
$294$		0			0
$295$	−308.000			−1.04407
$296$		0			0
$297$	−99.4987	−	528.000i	−0.335013	−	1.77778i
$298$		0			0
$299$	344.929			1.15361
$300$		0			0
$301$			385.000i			1.27907i
$302$		0			0
$303$	−110.000	−	165.831i	−0.363036	−	0.547298i
$304$		0			0
$305$	46.4327			0.152239
$306$		0			0
$307$			232.000i			0.755700i	0.925867	+	0.377850i	$0.123337\pi$
$307$			232.000i			0.755700i	−0.925867	+	0.377850i	$0.876663\pi$
$308$		0			0
$309$	175.000	−	116.082i	0.566343	−	0.375669i
$310$		0			0
$311$		−	232.164i		−	0.746507i	−0.927729	−	0.373254i	$-0.878242\pi$
$311$		−	232.164i		−	0.746507i	0.927729	−	0.373254i	$-0.121758\pi$
$312$		0			0
$313$	−155.000			−0.495208			−0.247604	−	0.968861i	$-0.579643\pi$
$313$	−155.000			−0.495208			−0.247604	+	0.968861i	$0.579643\pi$
$314$		0			0
$315$	192.500	+	81.2573i	0.611111	+	0.257960i
$316$		0			0
$317$	198.997			0.627752			0.313876	−	0.949464i	$-0.398372\pi$
$317$	198.997			0.627752			0.313876	+	0.949464i	$0.398372\pi$
$318$		0			0
$319$			660.000i			2.06897i
$320$		0			0
$321$	154.000	+	232.164i	0.479751	+	0.723252i
$322$		0			0
$323$	−696.491			−2.15632
$324$		0			0
$325$		−	182.000i		−	0.560000i
$326$		0			0
$327$	−290.205	−	437.500i	−0.887476	−	1.33792i
$328$		0			0
$329$			348.246i			1.05850i
$330$		0			0
$331$		−	490.000i		−	1.48036i	−0.672407	−	0.740181i	$-0.734740\pi$
$331$		−	490.000i		−	1.48036i	0.672407	−	0.740181i	$-0.265260\pi$
$332$		0			0
$333$	−107.790	−	45.5000i	−0.323695	−	0.136637i
$334$		0			0
$335$			53.0660i			0.158406i
$336$		0			0
$337$	475.000			1.40950			0.704748	−	0.709458i	$-0.251060\pi$
$337$	475.000			1.40950			0.704748	+	0.709458i	$0.251060\pi$
$338$		0			0
$339$	22.0000	+	33.1662i	0.0648968	+	0.0978355i
$340$		0			0
$341$		−	198.997i		−	0.583570i
$342$		0			0
$343$			343.000i			1.00000i
$344$		0			0
$345$	145.931	+	220.000i	0.422990	+	0.637681i
$346$		0			0
$347$		−	208.947i		−	0.602154i	−0.953600	−	0.301077i	$-0.902654\pi$
$347$		−	208.947i		−	0.602154i	0.953600	−	0.301077i	$-0.0973461\pi$
$348$		0			0
$349$		−	85.0000i		−	0.243553i	−0.992558	−	0.121777i	$-0.961141\pi$
$349$		−	85.0000i		−	0.243553i	0.992558	−	0.121777i	$-0.0388591\pi$
$350$		0			0
$351$	−344.929	+	65.0000i	−0.982704	+	0.185185i
$352$		0			0
$353$	232.164			0.657688			0.328844	−	0.944384i	$-0.393341\pi$
$353$	232.164			0.657688			0.328844	+	0.944384i	$0.393341\pi$
$354$		0			0
$355$	99.0000			0.278873
$356$		0			0
$357$	406.287	−	269.500i	1.13806	−	0.754902i
$358$		0			0
$359$	−92.8655			−0.258678			−0.129339	−	0.991600i	$-0.541286\pi$
$359$	−92.8655			−0.258678			−0.129339	+	0.991600i	$0.541286\pi$
$360$		0			0
$361$	−539.000			−1.49307
$362$		0			0
$363$	−687.500	+	456.036i	−1.89394	+	1.25630i
$364$		0			0
$365$		−	13.2665i		−	0.0363466i
$366$		0			0
$367$	140.000			0.381471			0.190736	−	0.981641i	$-0.438913\pi$
$367$	140.000			0.381471			0.190736	+	0.981641i	$0.438913\pi$
$368$		0			0
$369$	162.515	−	385.000i	0.440419	−	1.04336i
$370$		0			0
$371$	−278.596			−0.750934
$372$		0			0
$373$	−60.0000			−0.160858			−0.0804290	−	0.996760i	$-0.525629\pi$
$373$	−60.0000			−0.160858			−0.0804290	+	0.996760i	$0.525629\pi$
$374$		0			0
$375$	323.371	−	214.500i	0.862322	−	0.572000i
$376$		0			0
$377$	431.161			1.14366
$378$		0			0
$379$			220.000i			0.580475i	0.956955	+	0.290237i	$0.0937343\pi$
$379$			220.000i			0.580475i	−0.956955	+	0.290237i	$0.906266\pi$
$380$		0			0
$381$	−350.000	+	232.164i	−0.918635	+	0.609354i
$382$		0			0
$383$	−116.082			−0.303086			−0.151543	−	0.988451i	$-0.548424\pi$
$383$	−116.082			−0.303086			−0.151543	+	0.988451i	$0.548424\pi$
$384$		0			0
$385$		−	462.000i		−	1.20000i
$386$		0			0
$387$	192.500	−	456.036i	0.497416	−	1.17839i
$388$		0			0
$389$		0			0		1.00000			$0$
$389$		0			0		−1.00000			$\pi$
$390$		0			0
$391$	616.000			1.57545
$392$		0			0
$393$	−137.500	−	207.289i	−0.349873	−	0.527453i
$394$		0			0
$395$	179.098			0.453412
$396$		0			0
$397$		−	478.000i		−	1.20403i	−0.798485	−	0.602015i	$-0.794365\pi$
$397$		−	478.000i		−	1.20403i	0.798485	−	0.602015i	$-0.205635\pi$
$398$		0			0
$399$	525.000	−	348.246i	1.31579	−	0.872796i
$400$		0			0
$401$	636.792			1.58801			0.794005	−	0.607911i	$-0.207992\pi$
$401$	636.792			1.58801			0.794005	+	0.607911i	$0.207992\pi$
$402$		0			0
$403$	−130.000			−0.322581
$404$		0			0
$405$	−187.389	−	192.500i	−0.462690	−	0.475309i
$406$		0			0
$407$			258.697i			0.635619i
$408$		0			0
$409$		−	200.000i		−	0.488998i	−0.969650	−	0.244499i	$-0.921377\pi$
$409$		−	200.000i		−	0.488998i	0.969650	−	0.244499i	$-0.0786234\pi$
$410$		0			0
$411$	−165.831	+	110.000i	−0.403482	+	0.267640i
$412$		0			0
$413$		−	650.058i		−	1.57399i
$414$		0			0
$415$	110.000			0.265060
$416$		0			0
$417$	27.5000	−	18.2414i	0.0659472	−	0.0437445i
$418$		0			0
$419$			480.911i			1.14776i	0.818940	+	0.573879i	$0.194562\pi$
$419$			480.911i			1.14776i	−0.818940	+	0.573879i	$0.805438\pi$
$420$		0			0
$421$		−	385.000i		−	0.914489i	−0.889341	−	0.457245i	$-0.848836\pi$
$421$		−	385.000i		−	0.914489i	0.889341	−	0.457245i	$-0.151164\pi$
$422$		0			0
$423$	174.123	−	412.500i	0.411638	−	0.975177i
$424$		0			0
$425$		−	325.029i		−	0.764775i
$426$		0			0
$427$			98.0000i			0.229508i
$428$		0			0
$429$	429.000	+	646.742i	1.00000	+	1.50756i
$430$		0			0
$431$	255.380			0.592529			0.296265	−	0.955106i	$-0.404259\pi$
$431$	255.380			0.592529			0.296265	+	0.955106i	$0.404259\pi$
$432$		0			0
$433$	235.000			0.542725			0.271363	−	0.962477i	$-0.412526\pi$
$433$	235.000			0.542725			0.271363	+	0.962477i	$0.412526\pi$
$434$		0			0
$435$	182.414	+	275.000i	0.419343	+	0.632184i
$436$		0			0
$437$	795.990			1.82149
$438$		0			0
$439$	−336.000			−0.765376			−0.382688	−	0.923878i	$-0.625002\pi$
$439$	−336.000			−0.765376			−0.382688	+	0.923878i	$0.625002\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			653.375i			1.47489i	0.675409	+	0.737444i	$0.263967\pi$
$443$			653.375i			1.47489i	−0.675409	+	0.737444i	$0.736033\pi$
$444$		0			0
$445$	−66.0000			−0.148315
$446$		0			0
$447$	−33.1662	+	22.0000i	−0.0741974	+	0.0492170i
$448$		0			0
$449$	179.098			0.398881			0.199441	−	0.979910i	$-0.436087\pi$
$449$	179.098			0.398881			0.199441	+	0.979910i	$0.436087\pi$
$450$		0			0
$451$	−924.000			−2.04878
$452$		0			0
$453$	356.537	+	537.500i	0.787058	+	1.18653i
$454$		0			0
$455$	−301.813			−0.663325
$456$		0			0
$457$		−	518.000i		−	1.13348i	−0.823897	−	0.566740i	$-0.808205\pi$
$457$		−	518.000i		−	1.13348i	0.823897	−	0.566740i	$-0.191795\pi$
$458$		0			0
$459$	−616.000	+	116.082i	−1.34205	+	0.252902i
$460$		0			0
$461$	−766.140			−1.66191			−0.830955	−	0.556340i	$-0.812205\pi$
$461$	−766.140			−1.66191			−0.830955	+	0.556340i	$0.812205\pi$
$462$		0			0
$463$			462.000i			0.997840i	0.866648	+	0.498920i	$0.166270\pi$
$463$			462.000i			0.997840i	−0.866648	+	0.498920i	$0.833730\pi$
$464$		0			0
$465$	−55.0000	−	82.9156i	−0.118280	−	0.178313i
$466$		0			0
$467$		−	557.193i		−	1.19313i	−0.802564	−	0.596566i	$-0.796531\pi$
$467$		−	557.193i		−	1.19313i	0.802564	−	0.596566i	$-0.203469\pi$
$468$		0			0
$469$	−112.000			−0.238806
$470$		0			0
$471$	350.000	−	232.164i	0.743100	−	0.492917i
$472$		0			0
$473$	−1094.49			−2.31392
$474$		0			0
$475$		−	420.000i		−	0.884211i
$476$		0			0
$477$	330.000	+	139.298i	0.691824	+	0.292030i
$478$		0			0
$479$	29.8496			0.0623165			0.0311583	−	0.999514i	$-0.490080\pi$
$479$	29.8496			0.0623165			0.0311583	+	0.999514i	$0.490080\pi$
$480$		0			0
$481$	169.000			0.351351
$482$		0			0
$483$	−464.327	+	308.000i	−0.961341	+	0.637681i
$484$		0			0
$485$		−	258.697i		−	0.533395i
$486$		0			0
$487$			462.000i			0.948665i	0.880346	+	0.474333i	$0.157311\pi$
$487$			462.000i			0.948665i	−0.880346	+	0.474333i	$0.842689\pi$
$488$		0			0
$489$	311.763	+	470.000i	0.637552	+	0.961145i
$490$		0			0
$491$		−	580.409i		−	1.18210i	−0.806636	−	0.591048i	$-0.798714\pi$
$491$		−	580.409i		−	1.18210i	0.806636	−	0.591048i	$-0.201286\pi$
$492$		0			0
$493$	770.000			1.56187
$494$		0			0
$495$	−231.000	+	547.243i	−0.466667	+	1.10554i
$496$		0			0
$497$			208.947i			0.420417i
$498$		0			0
$499$			350.000i			0.701403i	0.936487	+	0.350701i	$0.114057\pi$
$499$			350.000i			0.701403i	−0.936487	+	0.350701i	$0.885943\pi$
$500$		0			0
$501$	−497.494	+	330.000i	−0.993001	+	0.658683i
$502$		0			0
$503$			417.895i			0.830805i	0.909638	+	0.415402i	$0.136359\pi$
$503$			417.895i			0.830805i	−0.909638	+	0.415402i	$0.863641\pi$
$504$		0			0
$505$			220.000i			0.435644i
$506$		0			0
$507$	422.500	−	280.255i	0.833333	−	0.552771i
$508$		0			0
$509$	−119.398			−0.234575			−0.117287	−	0.993098i	$-0.537420\pi$
$509$	−119.398			−0.234575			−0.117287	+	0.993098i	$0.537420\pi$
$510$		0			0
$511$	28.0000			0.0547945
$512$		0			0
$513$	−795.990	+	150.000i	−1.55164	+	0.292398i
$514$		0			0
$515$	−232.164			−0.450803
$516$		0			0
$517$	−990.000			−1.91489
$518$		0			0
$519$	−363.000	−	547.243i	−0.699422	−	1.05442i
$520$		0			0
$521$			348.246i			0.668418i	0.942499	+	0.334209i	$0.108469\pi$
$521$			348.246i			0.668418i	−0.942499	+	0.334209i	$0.891531\pi$
$522$		0			0
$523$	910.000			1.73996			0.869981	−	0.493085i	$-0.164131\pi$
$523$	910.000			1.73996			0.869981	+	0.493085i	$0.164131\pi$
$524$		0			0
$525$	162.515	+	245.000i	0.309552	+	0.466667i
$526$		0			0
$527$	−232.164			−0.440538
$528$		0			0
$529$	−175.000			−0.330813
$530$		0			0
$531$	−325.029	+	770.000i	−0.612108	+	1.45009i
$532$		0			0
$533$			603.626i			1.13251i
$534$		0			0
$535$		−	308.000i		−	0.575701i
$536$		0			0
$537$	−412.500	−	621.867i	−0.768156	−	1.15804i
$538$		0			0
$539$		0			0
$540$		0			0
$541$		−	755.000i		−	1.39556i	−0.716310	−	0.697782i	$-0.754170\pi$
$541$		−	755.000i		−	1.39556i	0.716310	−	0.697782i	$-0.245830\pi$
$542$		0			0
$543$	−235.000	+	155.881i	−0.432781	+	0.287074i
$544$		0			0
$545$			580.409i			1.06497i
$546$		0			0
$547$	715.000			1.30713			0.653565	−	0.756870i	$-0.273273\pi$
$547$	715.000			1.30713			0.653565	+	0.756870i	$0.273273\pi$
$548$		0			0
$549$	49.0000	−	116.082i	0.0892532	−	0.211442i
$550$		0			0
$551$	994.987			1.80578
$552$		0			0
$553$			378.000i			0.683544i
$554$		0			0
$555$	71.5000	+	107.790i	0.128829	+	0.194217i
$556$		0			0
$557$	−348.246			−0.625217			−0.312608	−	0.949882i	$-0.601203\pi$
$557$	−348.246			−0.625217			−0.312608	+	0.949882i	$0.601203\pi$
$558$		0			0
$559$			715.000i			1.27907i
$560$		0			0
$561$	766.140	+	1155.00i	1.36567	+	2.05882i
$562$		0			0
$563$		−	739.607i		−	1.31369i	−0.754026	−	0.656845i	$-0.771891\pi$
$563$		−	739.607i		−	1.31369i	0.754026	−	0.656845i	$-0.228109\pi$
$564$		0			0
$565$		−	44.0000i		−	0.0778761i
$566$		0			0
$567$	406.287	−	395.500i	0.716555	−	0.697531i
$568$		0			0
$569$		−	514.077i		−	0.903474i	−0.892151	−	0.451737i	$-0.850804\pi$
$569$		−	514.077i		−	0.903474i	0.892151	−	0.451737i	$-0.149196\pi$
$570$		0			0
$571$	479.000			0.838879			0.419440	−	0.907783i	$-0.362227\pi$
$571$	479.000			0.838879			0.419440	+	0.907783i	$0.362227\pi$
$572$		0			0
$573$	−165.000	−	248.747i	−0.287958	−	0.434113i
$574$		0			0
$575$			371.462i			0.646021i
$576$		0			0
$577$		−	448.000i		−	0.776430i	−0.921569	−	0.388215i	$-0.873092\pi$
$577$		−	448.000i		−	0.776430i	0.921569	−	0.388215i	$-0.126908\pi$
$578$		0			0
$579$	−135.982	−	205.000i	−0.234856	−	0.354059i
$580$		0			0
$581$			232.164i			0.399593i
$582$		0			0
$583$		−	792.000i		−	1.35849i
$584$		0			0
$585$	357.500	+	150.906i	0.611111	+	0.257960i
$586$		0			0
$587$	−729.657			−1.24303			−0.621514	−	0.783403i	$-0.713482\pi$
$587$	−729.657			−1.24303			−0.621514	+	0.783403i	$0.713482\pi$
$588$		0			0
$589$	−300.000			−0.509338
$590$		0			0
$591$	290.205	−	192.500i	0.491040	−	0.325719i
$592$		0			0
$593$	−762.824			−1.28638			−0.643190	−	0.765706i	$-0.722390\pi$
$593$	−762.824			−1.28638			−0.643190	+	0.765706i	$0.722390\pi$
$594$		0			0
$595$	−539.000			−0.905882
$596$		0			0
$597$	−630.000	+	417.895i	−1.05528	+	0.699991i
$598$		0			0
$599$		−	795.990i		−	1.32886i	−0.747348	−	0.664432i	$-0.768673\pi$
$599$		−	795.990i		−	1.32886i	0.747348	−	0.664432i	$-0.231327\pi$
$600$		0			0
$601$	−101.000			−0.168053			−0.0840266	−	0.996464i	$-0.526778\pi$
$601$	−101.000			−0.168053			−0.0840266	+	0.996464i	$0.526778\pi$
$602$		0			0
$603$	132.665	+	56.0000i	0.220008	+	0.0928690i
$604$		0			0
$605$	912.072			1.50756
$606$		0			0
$607$	−1100.00			−1.81219			−0.906096	−	0.423073i	$-0.860951\pi$
$607$	−1100.00			−1.81219			−0.906096	+	0.423073i	$0.860951\pi$
$608$		0			0
$609$	−580.409	+	385.000i	−0.953053	+	0.632184i
$610$		0			0
$611$			646.742i			1.05850i
$612$		0			0
$613$		−	338.000i		−	0.551387i	−0.961246	−	0.275693i	$-0.911093\pi$
$613$		−	338.000i		−	0.551387i	0.961246	−	0.275693i	$-0.0889074\pi$
$614$		0			0
$615$	−385.000	+	255.380i	−0.626016	+	0.415252i
$616$		0			0
$617$	928.655			1.50511			0.752557	−	0.658527i	$-0.228820\pi$
$617$	928.655			1.50511			0.752557	+	0.658527i	$0.228820\pi$
$618$		0			0
$619$		−	770.000i		−	1.24394i	−0.783040	−	0.621971i	$-0.786332\pi$
$619$		−	770.000i		−	1.24394i	0.783040	−	0.621971i	$-0.213668\pi$
$620$		0			0
$621$	704.000	−	132.665i	1.13366	−	0.213631i
$622$		0			0
$623$		−	139.298i		−	0.223593i
$624$		0			0
$625$	−79.0000			−0.126400
$626$		0			0
$627$	990.000	+	1492.48i	1.57895	+	2.38035i
$628$		0			0
$629$	301.813			0.479830
$630$		0			0
$631$			385.000i			0.610143i	0.952330	+	0.305071i	$0.0986803\pi$
$631$			385.000i			0.610143i	−0.952330	+	0.305071i	$0.901320\pi$
$632$		0			0
$633$	382.500	−	253.722i	0.604265	−	0.400824i
$634$		0			0
$635$	464.327			0.731224
$636$		0			0
$637$		0			0
$638$		0			0
$639$	104.474	−	247.500i	0.163496	−	0.387324i
$640$		0			0
$641$		−	596.992i		−	0.931345i	−0.884957	−	0.465673i	$-0.845812\pi$
$641$		−	596.992i		−	0.931345i	0.884957	−	0.465673i	$-0.154188\pi$
$642$		0			0
$643$		−	168.000i		−	0.261275i	−0.991430	−	0.130638i	$-0.958298\pi$
$643$		−	168.000i		−	0.261275i	0.991430	−	0.130638i	$-0.0417024\pi$
$644$		0			0
$645$	−456.036	+	302.500i	−0.707032	+	0.468992i
$646$		0			0
$647$			371.462i			0.574130i	0.957911	+	0.287065i	$0.0926795\pi$
$647$			371.462i			0.574130i	−0.957911	+	0.287065i	$0.907320\pi$
$648$		0			0
$649$	1848.00			2.84746
$650$		0			0
$651$	175.000	−	116.082i	0.268817	−	0.178313i
$652$		0			0
$653$		−	742.924i		−	1.13771i	−0.822438	−	0.568854i	$-0.807387\pi$
$653$		−	742.924i		−	1.13771i	0.822438	−	0.568854i	$-0.192613\pi$
$654$		0			0
$655$			275.000i			0.419847i
$656$		0			0
$657$	−33.1662	−	14.0000i	−0.0504814	−	0.0213090i
$658$		0			0
$659$		−	729.657i		−	1.10722i	−0.832776	−	0.553610i	$-0.813250\pi$
$659$		−	729.657i		−	1.10722i	0.832776	−	0.553610i	$-0.186750\pi$
$660$		0			0
$661$		−	1270.00i		−	1.92133i	−0.277708	−	0.960666i	$-0.589575\pi$
$661$		−	1270.00i		−	1.92133i	0.277708	−	0.960666i	$-0.410425\pi$
$662$		0			0
$663$	754.532	−	500.500i	1.13806	−	0.754902i
$664$		0			0
$665$	−696.491			−1.04736
$666$		0			0
$667$	−880.000			−1.31934
$668$		0			0
$669$	48.0911	+	72.5000i	0.0718850	+	0.108371i
$670$		0			0
$671$	−278.596			−0.415196
$672$		0			0
$673$	−1035.00			−1.53789			−0.768945	−	0.639315i	$-0.779218\pi$
$673$	−1035.00			−1.53789			−0.768945	+	0.639315i	$0.779218\pi$
$674$		0			0
$675$	−70.0000	−	371.462i	−0.103704	−	0.550314i
$676$		0			0
$677$			139.298i			0.205758i	0.994694	+	0.102879i	$0.0328055\pi$
$677$			139.298i			0.205758i	−0.994694	+	0.102879i	$0.967195\pi$
$678$		0			0
$679$	546.000			0.804124
$680$		0			0
$681$	−82.9156	+	55.0000i	−0.121756	+	0.0807636i
$682$		0			0
$683$	696.491			1.01975			0.509876	−	0.860248i	$-0.329691\pi$
$683$	696.491			1.01975			0.509876	+	0.860248i	$0.329691\pi$
$684$		0			0
$685$	220.000			0.321168
$686$		0			0
$687$	−638.450	−	962.500i	−0.929331	−	1.40102i
$688$		0			0
$689$	−517.393			−0.750934
$690$		0			0
$691$		−	100.000i		−	0.144718i	−0.997379	−	0.0723589i	$-0.976947\pi$
$691$		−	100.000i		−	0.144718i	0.997379	−	0.0723589i	$-0.0230527\pi$
$692$		0			0
$693$	−1155.00	−	487.544i	−1.66667	−	0.703526i
$694$		0			0
$695$	−36.4829			−0.0524933
$696$		0			0
$697$			1078.00i			1.54663i
$698$		0			0
$699$	−313.500	−	472.619i	−0.448498	−	0.676136i
$700$		0			0
$701$			1127.65i			1.60863i	0.594200	+	0.804317i	$0.297469\pi$
$701$			1127.65i			1.60863i	−0.594200	+	0.804317i	$0.702531\pi$
$702$		0			0
$703$	390.000			0.554765
$704$		0			0
$705$	−412.500	+	273.622i	−0.585106	+	0.388116i
$706$		0			0
$707$	−464.327			−0.656757
$708$		0			0
$709$		−	330.000i		−	0.465444i	−0.972543	−	0.232722i	$-0.925237\pi$
$709$		−	330.000i		−	0.465444i	0.972543	−	0.232722i	$-0.0747633\pi$
$710$		0			0
$711$	189.000	−	447.744i	0.265823	−	0.629739i
$712$		0			0
$713$	265.330			0.372132
$714$		0			0
$715$		−	858.000i		−	1.20000i
$716$		0			0
$717$	1086.19	−	720.500i	1.51492	−	1.00488i
$718$		0			0
$719$			464.327i			0.645796i	0.946434	+	0.322898i	$0.104657\pi$
$719$			464.327i			0.645796i	−0.946434	+	0.322898i	$0.895343\pi$
$720$		0			0
$721$		−	490.000i		−	0.679612i
$722$		0			0
$723$	215.581	+	325.000i	0.298175	+	0.449516i
$724$		0			0
$725$			464.327i			0.640452i
$726$		0			0
$727$	−130.000			−0.178817			−0.0894085	−	0.995995i	$-0.528498\pi$
$727$	−130.000			−0.178817			−0.0894085	+	0.995995i	$0.528498\pi$
$728$		0			0
$729$	−679.000	+	265.330i	−0.931413	+	0.363964i
$730$		0			0
$731$			1276.90i			1.74679i
$732$		0			0
$733$			817.000i			1.11460i	0.830312	+	0.557299i	$0.188162\pi$
$733$			817.000i			1.11460i	−0.830312	+	0.557299i	$0.811838\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	318.396i		−	0.432016i
$738$		0			0
$739$			310.000i			0.419486i	0.977757	+	0.209743i	$0.0672627\pi$
$739$			310.000i			0.419486i	−0.977757	+	0.209743i	$0.932737\pi$
$740$		0			0
$741$	975.000	−	646.742i	1.31579	−	0.872796i
$742$		0			0
$743$	1409.57			1.89713			0.948564	−	0.316587i	$-0.102537\pi$
$743$	1409.57			1.89713			0.948564	+	0.316587i	$0.102537\pi$
$744$		0			0
$745$	44.0000			0.0590604
$746$		0			0
$747$	116.082	−	275.000i	0.155397	−	0.368139i
$748$		0			0
$749$	650.058			0.867902
$750$		0			0
$751$	674.000			0.897470			0.448735	−	0.893665i	$-0.351875\pi$
$751$	674.000			0.897470			0.448735	+	0.893665i	$0.351875\pi$
$752$		0			0
$753$	550.000	+	829.156i	0.730412	+	1.10114i
$754$		0			0
$755$		−	713.074i		−	0.944469i
$756$		0			0
$757$	860.000			1.13606			0.568032	−	0.823007i	$-0.307705\pi$
$757$	860.000			1.13606			0.568032	+	0.823007i	$0.307705\pi$
$758$		0			0
$759$	−875.589	−	1320.00i	−1.15361	−	1.73913i
$760$		0			0
$761$	−955.188			−1.25517			−0.627587	−	0.778546i	$-0.715957\pi$
$761$	−955.188			−1.25517			−0.627587	+	0.778546i	$0.715957\pi$
$762$		0			0
$763$	−1225.00			−1.60550
$764$		0			0
$765$	638.450	+	269.500i	0.834576	+	0.352288i
$766$		0			0
$767$		−	1207.25i		−	1.57399i
$768$		0			0
$769$			540.000i			0.702211i	0.936336	+	0.351105i	$0.114194\pi$
$769$			540.000i			0.702211i	−0.936336	+	0.351105i	$0.885806\pi$
$770$		0			0
$771$	−445.500	−	671.617i	−0.577821	−	0.871098i
$772$		0			0
$773$	1243.73			1.60897			0.804485	−	0.593973i	$-0.202441\pi$
$773$	1243.73			1.60897			0.804485	+	0.593973i	$0.202441\pi$
$774$		0			0
$775$		−	140.000i		−	0.180645i
$776$		0			0
$777$	−227.500	+	150.906i	−0.292793	+	0.194217i
$778$		0			0
$779$			1392.98i			1.78817i
$780$		0			0
$781$	−594.000			−0.760563
$782$		0			0
$783$	880.000	−	165.831i	1.12388	−	0.211790i
$784$		0			0
$785$	−464.327			−0.591500
$786$		0			0
$787$			626.000i			0.795426i	0.917510	+	0.397713i	$0.130196\pi$
$787$			626.000i			0.795426i	−0.917510	+	0.397713i	$0.869804\pi$
$788$		0			0
$789$	297.000	+	447.744i	0.376426	+	0.567483i
$790$		0			0
$791$	92.8655			0.117403
$792$		0			0
$793$			182.000i			0.229508i
$794$		0			0
$795$	−218.897	−	330.000i	−0.275342	−	0.415094i
$796$		0			0
$797$		−	1121.02i		−	1.40655i	−0.710919	−	0.703274i	$-0.751721\pi$
$797$		−	1121.02i		−	1.40655i	0.710919	−	0.703274i	$-0.248279\pi$
$798$		0			0
$799$			1155.00i			1.44556i
$800$		0			0
$801$	−69.6491	+	165.000i	−0.0869527	+	0.205993i
$802$		0			0
$803$			79.5990i			0.0991270i
$804$		0			0
$805$	616.000			0.765217
$806$		0			0
$807$	−275.000	−	414.578i	−0.340768	−	0.513728i
$808$		0			0
$809$		−	16.5831i		−	0.0204983i	−0.999947	−	0.0102491i	$-0.996738\pi$
$809$		−	16.5831i		−	0.0204983i	0.999947	−	0.0102491i	$-0.00326246\pi$
$810$		0			0
$811$		−	740.000i		−	0.912454i	−0.889863	−	0.456227i	$-0.849201\pi$
$811$		−	740.000i		−	0.912454i	0.889863	−	0.456227i	$-0.150799\pi$
$812$		0			0
$813$	−406.287	−	612.500i	−0.499737	−	0.753383i
$814$		0			0
$815$		−	623.525i		−	0.765062i
$816$		0			0
$817$			1650.00i			2.01958i
$818$		0			0
$819$	−318.500	+	754.532i	−0.388889	+	0.921285i
$820$		0			0
$821$	−965.138			−1.17556			−0.587782	−	0.809019i	$-0.699999\pi$
$821$	−965.138			−1.17556			−0.587782	+	0.809019i	$0.699999\pi$
$822$		0			0
$823$	420.000			0.510328			0.255164	−	0.966898i	$-0.417871\pi$
$823$	420.000			0.510328			0.255164	+	0.966898i	$0.417871\pi$
$824$		0			0
$825$	−696.491	+	462.000i	−0.844232	+	0.560000i
$826$		0			0
$827$	−99.4987			−0.120313			−0.0601564	−	0.998189i	$-0.519160\pi$
$827$	−99.4987			−0.120313			−0.0601564	+	0.998189i	$0.519160\pi$
$828$		0			0
$829$	−1008.00			−1.21592			−0.607961	−	0.793967i	$-0.708012\pi$
$829$	−1008.00			−1.21592			−0.607961	+	0.793967i	$0.708012\pi$
$830$		0			0
$831$	−1075.00	+	713.074i	−1.29362	+	0.858092i
$832$		0			0
$833$		0			0
$834$		0			0
$835$	660.000			0.790419
$836$		0			0
$837$	−265.330	+	50.0000i	−0.317001	+	0.0597372i
$838$		0			0
$839$	−1180.72			−1.40729			−0.703646	−	0.710550i	$-0.748446\pi$
$839$	−1180.72			−1.40729			−0.703646	+	0.710550i	$0.748446\pi$
$840$		0			0
$841$	−259.000			−0.307967
$842$		0			0
$843$	812.573	−	539.000i	0.963906	−	0.639383i
$844$		0			0
$845$	−560.510			−0.663325
$846$		0			0
$847$			1925.00i			2.27273i
$848$		0			0
$849$	−175.000	+	116.082i	−0.206125	+	0.136728i
$850$		0			0
$851$	−344.929			−0.405322
$852$		0			0
$853$			797.000i			0.934349i	0.884165	+	0.467175i	$0.154728\pi$
$853$			797.000i			0.934349i	−0.884165	+	0.467175i	$0.845272\pi$
$854$		0			0
$855$	825.000	+	348.246i	0.964912	+	0.407305i
$856$		0			0
$857$		−	92.8655i		−	0.108361i	−0.998531	−	0.0541806i	$-0.982745\pi$
$857$		−	92.8655i		−	0.108361i	0.998531	−	0.0541806i	$-0.0172547\pi$
$858$		0			0
$859$	34.0000			0.0395809			0.0197905	−	0.999804i	$-0.493700\pi$
$859$	34.0000			0.0395809			0.0197905	+	0.999804i	$0.493700\pi$
$860$		0			0
$861$	−539.000	−	812.573i	−0.626016	−	0.943755i
$862$		0			0
$863$	116.082			0.134510			0.0672548	−	0.997736i	$-0.478576\pi$
$863$	116.082			0.134510			0.0672548	+	0.997736i	$0.478576\pi$
$864$		0			0
$865$			726.000i			0.839306i
$866$		0			0
$867$	625.000	−	414.578i	0.720877	−	0.478175i
$868$		0			0
$869$	−1074.59			−1.23658
$870$		0			0
$871$	−208.000			−0.238806
$872$		0			0
$873$	−646.742	−	273.000i	−0.740827	−	0.312715i
$874$		0			0
$875$		−	905.439i		−	1.03479i
$876$		0			0
$877$		−	679.000i		−	0.774230i	−0.922031	−	0.387115i	$-0.873472\pi$
$877$		−	679.000i		−	0.774230i	0.922031	−	0.387115i	$-0.126528\pi$
$878$		0			0
$879$	207.289	−	137.500i	0.235824	−	0.156428i
$880$		0			0
$881$		−	447.744i		−	0.508223i	−0.967175	−	0.254111i	$-0.918217\pi$
$881$		−	447.744i		−	0.508223i	0.967175	−	0.254111i	$-0.0817830\pi$
$882$		0			0
$883$	525.000			0.594564			0.297282	−	0.954790i	$-0.403920\pi$
$883$	525.000			0.594564			0.297282	+	0.954790i	$0.403920\pi$
$884$		0			0
$885$	770.000	−	510.760i	0.870056	−	0.577130i
$886$		0			0
$887$			1333.28i			1.50314i	0.659655	+	0.751569i	$0.270703\pi$
$887$			1333.28i			1.50314i	−0.659655	+	0.751569i	$0.729297\pi$
$888$		0			0
$889$			980.000i			1.10236i
$890$		0			0
$891$	1124.34	+	1155.00i	1.26188	+	1.29630i
$892$		0			0
$893$			1492.48i			1.67131i
$894$		0			0
$895$			825.000i			0.921788i
$896$		0			0
$897$	−862.322	+	572.000i	−0.961341	+	0.637681i
$898$		0			0
$899$	331.662			0.368924
$900$		0			0
$901$	−924.000			−1.02553
$902$		0			0
$903$	−638.450	−	962.500i	−0.707032	−	1.06589i
$904$		0			0
$905$	311.763			0.344489
$906$		0			0
$907$	445.000			0.490628			0.245314	−	0.969444i	$-0.421109\pi$
$907$	445.000			0.490628			0.245314	+	0.969444i	$0.421109\pi$
$908$		0			0
$909$	550.000	+	232.164i	0.605061	+	0.255406i
$910$		0			0
$911$			497.494i			0.546096i	0.962000	+	0.273048i	$0.0880318\pi$
$911$			497.494i			0.546096i	−0.962000	+	0.273048i	$0.911968\pi$
$912$		0			0
$913$	−660.000			−0.722892
$914$		0			0
$915$	−116.082	+	77.0000i	−0.126865	+	0.0841530i
$916$		0			0
$917$	−580.409			−0.632944
$918$		0			0
$919$	−756.000			−0.822633			−0.411317	−	0.911493i	$-0.634931\pi$
$919$	−756.000			−0.822633			−0.411317	+	0.911493i	$0.634931\pi$
$920$		0			0
$921$	−384.728	−	580.000i	−0.417729	−	0.629750i
$922$		0			0
$923$			388.045i			0.420417i
$924$		0			0
$925$			182.000i			0.196757i
$926$		0			0
$927$	−245.000	+	580.409i	−0.264293	+	0.626116i
$928$		0			0
$929$	−716.391			−0.771142			−0.385571	−	0.922678i	$-0.625996\pi$
$929$	−716.391			−0.771142			−0.385571	+	0.922678i	$0.625996\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	385.000	+	580.409i	0.412647	+	0.622089i
$934$		0			0
$935$		−	1532.28i		−	1.63880i
$936$		0			0
$937$	10.0000			0.0106724			0.00533618	−	0.999986i	$-0.498301\pi$
$937$	10.0000			0.0106724			0.00533618	+	0.999986i	$0.498301\pi$
$938$		0			0
$939$	387.500	−	257.038i	0.412673	−	0.273736i
$940$		0			0
$941$	1456.00			1.54729			0.773644	−	0.633620i	$-0.218432\pi$
$941$	1456.00			1.54729			0.773644	+	0.633620i	$0.218432\pi$
$942$		0			0
$943$		−	1232.00i		−	1.30647i
$944$		0			0
$945$	−616.000	+	116.082i	−0.651852	+	0.122838i
$946$		0			0
$947$	431.161			0.455292			0.227646	−	0.973744i	$-0.426897\pi$
$947$	431.161			0.455292			0.227646	+	0.973744i	$0.426897\pi$
$948$		0			0
$949$	52.0000			0.0547945
$950$		0			0
$951$	−497.494	+	330.000i	−0.523127	+	0.347003i
$952$		0			0
$953$			288.546i			0.302777i	0.988474	+	0.151388i	$0.0483744\pi$
$953$			288.546i			0.302777i	−0.988474	+	0.151388i	$0.951626\pi$
$954$		0			0
$955$			330.000i			0.345550i
$956$		0			0
$957$	−1094.49	−	1650.00i	−1.14366	−	1.72414i
$958$		0			0
$959$			464.327i			0.484179i
$960$		0			0
$961$	861.000			0.895942
$962$		0			0
$963$	−770.000	−	325.029i	−0.799585	−	0.337517i
$964$		0			0
$965$			271.963i			0.281827i
$966$		0			0
$967$		−	109.000i		−	0.112720i	−0.998411	−	0.0563599i	$-0.982051\pi$
$967$		−	109.000i		−	0.112720i	0.998411	−	0.0563599i	$-0.0179494\pi$
$968$		0			0
$969$	1741.23	−	1155.00i	1.79693	−	1.19195i
$970$		0			0
$971$		−	1741.23i		−	1.79323i	−0.442809	−	0.896616i	$-0.646018\pi$
$971$		−	1741.23i		−	1.79323i	0.442809	−	0.896616i	$-0.353982\pi$
$972$		0			0
$973$		−	77.0000i		−	0.0791367i
$974$		0			0
$975$	301.813	+	455.000i	0.309552	+	0.466667i
$976$		0			0
$977$	397.995			0.407364			0.203682	−	0.979037i	$-0.434709\pi$
$977$	397.995			0.407364			0.203682	+	0.979037i	$0.434709\pi$
$978$		0			0
$979$	396.000			0.404494
$980$		0			0
$981$	1451.02	+	612.500i	1.47913	+	0.624363i
$982$		0			0
$983$	447.744			0.455488			0.227744	−	0.973721i	$-0.426865\pi$
$983$	447.744			0.455488			0.227744	+	0.973721i	$0.426865\pi$
$984$		0			0
$985$	−385.000			−0.390863
$986$		0			0
$987$	−577.500	−	870.614i	−0.585106	−	0.882081i
$988$		0			0
$989$		−	1459.31i		−	1.47555i
$990$		0			0
$991$	−266.000			−0.268416			−0.134208	−	0.990953i	$-0.542849\pi$
$991$	−266.000			−0.268416			−0.134208	+	0.990953i	$0.542849\pi$
$992$		0			0
$993$	812.573	+	1225.00i	0.818301	+	1.23364i
$994$		0			0
$995$	835.789			0.839989
$996$		0			0
$997$	1890.00			1.89569			0.947844	−	0.318736i	$-0.103258\pi$
$997$	1890.00			1.89569			0.947844	+	0.318736i	$0.103258\pi$
$998$		0			0
$999$	344.929	−	65.0000i	0.345274	−	0.0650651i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	156.3.g.b.77.4	yes	4
3.2	odd	2	inner	156.3.g.b.77.1	✓	4
4.3	odd	2		624.3.l.h.545.2		4
12.11	even	2		624.3.l.h.545.3		4
13.12	even	2	inner	156.3.g.b.77.3	yes	4
39.38	odd	2	inner	156.3.g.b.77.2	yes	4
52.51	odd	2		624.3.l.h.545.1		4
156.155	even	2		624.3.l.h.545.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
156.3.g.b.77.1	✓	4	3.2	odd	2	inner
156.3.g.b.77.2	yes	4	39.38	odd	2	inner
156.3.g.b.77.3	yes	4	13.12	even	2	inner
156.3.g.b.77.4	yes	4	1.1	even	1	trivial
624.3.l.h.545.1		4	52.51	odd	2
624.3.l.h.545.2		4	4.3	odd	2
624.3.l.h.545.3		4	12.11	even	2
624.3.l.h.545.4		4	156.155	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 156 = 2^{2} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	156.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-2.50000 + 1.65831i) q^{3} +3.31662 q^{5} +7.00000i q^{7} +(3.50000 - 8.29156i) q^{9} -19.8997 q^{11} +13.0000i q^{13} +(-8.29156 + 5.50000i) q^{15} +23.2164i q^{17} +30.0000i q^{19} +(-11.6082 - 17.5000i) q^{21} -26.5330i q^{23} -14.0000 q^{25} +(5.00000 + 26.5330i) q^{27} -33.1662i q^{29} +10.0000i q^{31} +(49.7494 - 33.0000i) q^{33} +23.2164i q^{35} -13.0000i q^{37} +(-21.5581 - 32.5000i) q^{39} +46.4327 q^{41} +55.0000 q^{43} +(11.6082 - 27.5000i) q^{45} +49.7494 q^{47} +(-38.5000 - 58.0409i) q^{51} +39.7995i q^{53} -66.0000 q^{55} +(-49.7494 - 75.0000i) q^{57} -92.8655 q^{59} +14.0000 q^{61} +(58.0409 + 24.5000i) q^{63} +43.1161i q^{65} +16.0000i q^{67} +(44.0000 + 66.3325i) q^{69} +29.8496 q^{71} -4.00000i q^{73} +(35.0000 - 23.2164i) q^{75} -139.298i q^{77} +54.0000 q^{79} +(-56.5000 - 58.0409i) q^{81} +33.1662 q^{83} +77.0000i q^{85} +(55.0000 + 82.9156i) q^{87} -19.8997 q^{89} -91.0000 q^{91} +(-16.5831 - 25.0000i) q^{93} +99.4987i q^{95} -78.0000i q^{97} +(-69.6491 + 165.000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 10 q^{3} + 14 q^{9} - 56 q^{25} + 20 q^{27} + 220 q^{43} - 154 q^{51} - 264 q^{55} + 56 q^{61} + 176 q^{69} + 140 q^{75} + 216 q^{79} - 226 q^{81} + 220 q^{87} - 364 q^{91}+O(q^{100}) \) 4 * q - 10 * q^3 + 14 * q^9 - 56 * q^25 + 20 * q^27 + 220 * q^43 - 154 * q^51 - 264 * q^55 + 56 * q^61 + 176 * q^69 + 140 * q^75 + 216 * q^79 - 226 * q^81 + 220 * q^87 - 364 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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