LMFDB - Embedded newform 201.4.j.a.119.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [201,4,Mod(5,201)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(201, base_ring=CyclotomicField(22))

chi = DirichletCharacter(H, H._module([11, 5]))

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

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

chi := DirichletCharacter("201.5");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 201 = 3 \cdot 67 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	201.j (of order $22$, degree $10$, 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:	$11.8593839112$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$2$ over $\Q(\zeta_{22})$
Coefficient field:	$\Q(\zeta_{33})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - x^{19} + x^{17} - x^{16} + x^{14} - x^{13} + x^{11} - x^{10} + x^{9} - x^{7} + x^{6} - x^{4} + x^{3} - x + 1 $ x^20 - x^19 + x^17 - x^16 + x^14 - x^13 + x^11 - x^10 + x^9 - x^7 + x^6 - x^4 + x^3 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 2^{10}\cdot 3^{10} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{22}]$

Embedding invariants

Embedding label			119.1
Root			$0.580057 - 0.814576i$ of defining polynomial
Character	$\chi$	$=$	201.119
Dual form			201.4.j.a.125.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.46393 - 4.98567i) q^{3} +(-3.32332 + 7.27706i) q^{4} +(-2.83365 - 4.40925i) q^{7} +(-22.7138 + 14.5973i) q^{9} +O(q^{10})$	$q+(-1.46393 - 4.98567i) q^{3} +(-3.32332 + 7.27706i) q^{4} +(-2.83365 - 4.40925i) q^{7} +(-22.7138 + 14.5973i) q^{9} +(41.1461 + 5.91592i) q^{12} +(89.3811 + 12.8511i) q^{13} +(-41.9111 - 48.3680i) q^{16} +(82.1479 + 52.7932i) q^{19} +(-17.8348 + 20.5825i) q^{21} +(17.7894 - 123.728i) q^{25} +(106.029 + 91.8744i) q^{27} +(41.5035 - 5.96730i) q^{28} +(331.445 - 47.6546i) q^{31} +(-30.7400 - 213.801i) q^{36} +447.689 q^{37} +(-66.7761 - 464.438i) q^{39} +(-496.446 + 226.719i) q^{43} +(-179.792 + 279.762i) q^{48} +(131.075 - 287.015i) q^{49} +(-390.560 + 607.723i) q^{52} +(142.951 - 486.848i) q^{57} +(716.977 + 621.264i) q^{61} +(128.726 + 58.7873i) q^{63} +(491.260 - 144.247i) q^{64} +(-386.644 + 388.934i) q^{67} +(-695.480 + 802.627i) q^{73} +(-642.908 + 92.4362i) q^{75} +(-657.183 + 422.346i) q^{76} +(-712.384 - 102.425i) q^{79} +(302.838 - 663.122i) q^{81} +(-90.5090 - 198.187i) q^{84} +(-196.611 - 430.519i) q^{91} +(-722.801 - 1582.71i) q^{93} -1690.92i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 16 q^{4} + 54 q^{9}+O(q^{10}) $ 20 * q + 16 * q^4 + 54 * q^9	$ 20 q + 16 q^{4} + 54 q^{9} - 128 q^{16} + 112 q^{19} + 324 q^{21} + 250 q^{25} - 432 q^{36} + 220 q^{37} - 648 q^{39} + 1258 q^{49} - 6160 q^{52} + 8910 q^{57} + 5940 q^{63} + 1024 q^{64} - 880 q^{67} - 10710 q^{73} - 896 q^{76} - 9724 q^{79} - 1458 q^{81} + 11664 q^{84} - 3888 q^{91} - 1620 q^{93}+O(q^{100}) $ 20 * q + 16 * q^4 + 54 * q^9 - 128 * q^16 + 112 * q^19 + 324 * q^21 + 250 * q^25 - 432 * q^36 + 220 * q^37 - 648 * q^39 + 1258 * q^49 - 6160 * q^52 + 8910 * q^57 + 5940 * q^63 + 1024 * q^64 - 880 * q^67 - 10710 * q^73 - 896 * q^76 - 9724 * q^79 - 1458 * q^81 + 11664 * q^84 - 3888 * q^91 - 1620 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/201\mathbb{Z}\right)^\times$.

$n$	$68$	$136$
$\chi(n)$	$-1$	$e\left(\frac{7}{22}\right)$

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 $

$2$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$2$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$3$	−1.46393	−	4.98567i	−0.281733	−	0.959493i
$3$	−1.46393	−	4.98567i	−0.281733	−	0.959493i
$4$	−3.32332	+	7.27706i	−0.415415	+	0.909632i
$5$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$5$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$6$		0			0
$7$	−2.83365	−	4.40925i	−0.153003	−	0.238077i	0.756288	−	0.654239i	$-0.227011\pi$
$7$	−2.83365	−	4.40925i	−0.153003	−	0.238077i	−0.909290	+	0.416162i	$0.863375\pi$
$8$		0			0
$9$	−22.7138	+	14.5973i	−0.841254	+	0.540641i
$10$		0			0
$11$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$11$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$12$	41.1461	+	5.91592i	0.989821	+	0.142315i
$13$	89.3811	+	12.8511i	1.90691	+	0.274173i	0.991677	−	0.128748i	$-0.0410958\pi$
$13$	89.3811	+	12.8511i	1.90691	+	0.274173i	0.915235	+	0.402921i	$0.132005\pi$
$14$		0			0
$15$		0			0
$16$	−41.9111	−	48.3680i	−0.654861	−	0.755750i
$17$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$17$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$18$		0			0
$19$	82.1479	+	52.7932i	0.991896	+	0.637453i	0.932647	−	0.360791i	$-0.117493\pi$
$19$	82.1479	+	52.7932i	0.991896	+	0.637453i	0.0592487	+	0.998243i	$0.481129\pi$
$20$		0			0
$21$	−17.8348	+	20.5825i	−0.185327	+	0.213879i
$22$		0			0
$23$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$23$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$24$		0			0
$25$	17.7894	−	123.728i	0.142315	−	0.989821i
$26$		0			0
$27$	106.029	+	91.8744i	0.755750	+	0.654861i
$28$	41.5035	−	5.96730i	0.280122	−	0.0402755i
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$	331.445	−	47.6546i	1.92030	−	0.276098i	0.925566	−	0.378585i	$-0.123589\pi$
$31$	331.445	−	47.6546i	1.92030	−	0.276098i	0.994734	+	0.102488i	$0.0326803\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−30.7400	−	213.801i	−0.142315	−	0.989821i
$37$	447.689			1.98918			0.994589	−	0.103890i	$-0.0331289\pi$
$37$	447.689			1.98918			0.994589	+	0.103890i	$0.0331289\pi$
$38$		0			0
$39$	−66.7761	−	464.438i	−0.274173	−	1.90691i
$40$		0			0
$41$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$41$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$42$		0			0
$43$	−496.446	+	226.719i	−1.76063	+	0.804055i	−0.775706	+	0.631094i	$0.782606\pi$
$43$	−496.446	+	226.719i	−1.76063	+	0.804055i	−0.984929	+	0.172961i	$0.944667\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$47$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$48$	−179.792	+	279.762i	−0.540641	+	0.841254i
$49$	131.075	−	287.015i	0.382144	−	0.836779i
$50$		0			0
$51$		0			0
$52$	−390.560	+	607.723i	−1.04156	+	1.62069i
$53$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$53$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	142.951	−	486.848i	0.332182	−	1.13131i
$58$		0			0
$59$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$59$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$60$		0			0
$61$	716.977	+	621.264i	1.50491	+	1.30401i	0.813538	+	0.581512i	$0.197539\pi$
$61$	716.977	+	621.264i	1.50491	+	1.30401i	0.691371	+	0.722500i	$0.257007\pi$
$62$		0			0
$63$	128.726	+	58.7873i	0.257428	+	0.117564i
$64$	491.260	−	144.247i	0.959493	−	0.281733i
$65$		0			0
$66$		0			0
$67$	−386.644	+	388.934i	−0.705016	+	0.709191i
$67$	−386.644	+	388.934i	−0.705016	+	0.709191i
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$71$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$72$		0			0
$73$	−695.480	+	802.627i	−1.11507	+	1.28685i	−0.161100	+	0.986938i	$0.551504\pi$
$73$	−695.480	+	802.627i	−1.11507	+	1.28685i	−0.953966	+	0.299916i	$0.903041\pi$
$74$		0			0
$75$	−642.908	+	92.4362i	−0.989821	+	0.142315i
$76$	−657.183	+	422.346i	−0.991896	+	0.637453i
$77$		0			0
$78$		0			0
$79$	−712.384	−	102.425i	−1.01455	−	0.145870i	−0.385070	−	0.922887i	$-0.625823\pi$
$79$	−712.384	−	102.425i	−1.01455	−	0.145870i	−0.629480	+	0.777017i	$0.716732\pi$
$80$		0			0
$81$	302.838	−	663.122i	0.415415	−	0.909632i
$82$		0			0
$83$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$83$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$84$	−90.5090	−	198.187i	−0.117564	−	0.257428i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$89$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$90$		0			0
$91$	−196.611	−	430.519i	−0.226489	−	0.495941i
$92$		0			0
$93$	−722.801	−	1582.71i	−0.805925	−	1.76473i
$94$		0			0
$95$		0			0
$96$		0			0
$97$		−	1690.92i		−	1.76997i	−0.465620	−	0.884985i	$-0.654169\pi$
$97$		−	1690.92i		−	1.76997i	0.465620	−	0.884985i	$-0.345831\pi$
$98$		0			0
$99$		0			0
$100$	841.254	+	540.641i	0.841254	+	0.540641i
$101$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$101$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$102$		0			0
$103$	297.056	+	2066.07i	0.284172	+	1.97646i	0.198167	+	0.980168i	$0.436501\pi$
$103$	297.056	+	2066.07i	0.284172	+	1.97646i	0.0860053	+	0.996295i	$0.472590\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$107$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$108$	−1020.94	+	466.249i	−0.909632	+	0.415415i
$109$	−931.366	−	133.910i	−0.818428	−	0.117672i	−0.279631	−	0.960108i	$-0.590212\pi$
$109$	−931.366	−	133.910i	−0.818428	−	0.117672i	−0.538797	+	0.842435i	$0.681121\pi$
$110$		0			0
$111$	−655.383	−	2232.03i	−0.560416	−	1.90860i
$112$	−94.5049	+	321.854i	−0.0797310	+	0.271539i
$113$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$113$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−2217.78	+	1012.83i	−1.75243	+	0.800306i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	189.421	−	1317.45i	0.142315	−	0.989821i
$122$		0			0
$123$		0			0
$124$	−754.713	+	2570.32i	−0.546575	+	1.86146i
$125$		0			0
$126$		0			0
$127$	2407.92	−	1547.48i	1.68243	−	1.08123i	0.835996	−	0.548736i	$-0.184891\pi$
$127$	2407.92	−	1547.48i	1.68243	−	1.08123i	0.846433	−	0.532496i	$-0.178746\pi$
$128$		0			0
$129$	1857.11	+	2143.22i	1.26751	+	1.46279i
$130$		0			0
$131$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$131$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$132$		0			0
$133$		−	511.808i		−	0.333679i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$137$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$138$		0			0
$139$	921.009	−	798.058i	0.562007	−	0.486982i	−0.326907	−	0.945057i	$-0.606006\pi$
$139$	921.009	−	798.058i	0.562007	−	0.486982i	0.888914	+	0.458075i	$0.151461\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	1658.00	+	486.834i	0.959493	+	0.281733i
$145$		0			0
$146$		0			0
$147$	−1622.85	−	233.330i	−0.910546	−	0.130917i
$148$	−1487.81	+	3257.86i	−0.826334	+	1.80942i
$149$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$149$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$150$		0			0
$151$	552.214	+	1209.18i	0.297606	+	0.651667i	0.998075	−	0.0620161i	$-0.0197530\pi$
$151$	552.214	+	1209.18i	0.297606	+	0.651667i	−0.700469	+	0.713683i	$0.747026\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	3601.66	+	1057.54i	1.84848	+	0.542764i
$157$	3528.12	−	1035.95i	1.79347	−	0.526610i	0.796517	−	0.604616i	$-0.206673\pi$
$157$	3528.12	−	1035.95i	1.79347	−	0.526610i	0.996953	+	0.0780055i	$0.0248552\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−3596.09			−1.72802			−0.864012	−	0.503472i	$-0.832056\pi$
$163$	−3596.09			−1.72802			−0.864012	+	0.503472i	$0.832056\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$167$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$168$		0			0
$169$	5715.83	+	1678.32i	2.60165	+	0.763914i
$170$		0			0
$171$	−2636.53			−1.17907
$172$		−	4366.13i		−	1.93555i
$173$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$173$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$174$		0			0
$175$	−595.955	+	272.163i	−0.257428	+	0.117564i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$179$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$180$		0			0
$181$	2256.47	−	662.561i	0.926643	−	0.272087i	0.216612	−	0.976258i	$-0.430499\pi$
$181$	2256.47	−	662.561i	0.926643	−	0.272087i	0.710031	+	0.704171i	$0.248681\pi$
$182$		0			0
$183$	2047.82	−	4484.09i	0.827208	−	1.81133i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	104.649	−	727.847i	0.0402755	−	0.280122i
$190$		0			0
$191$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$191$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$192$	−1438.34	−	2238.10i	−0.540641	−	0.841254i
$193$	−265.315	−	1845.31i	−0.0989524	−	0.688229i	−0.977556	−	0.210677i	$-0.932433\pi$
$193$	−265.315	−	1845.31i	−0.0989524	−	0.688229i	0.878603	−	0.477552i	$-0.158476\pi$
$194$		0			0
$195$		0			0
$196$	1653.02	+	1907.69i	0.602413	+	0.695221i
$197$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$197$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$198$		0			0
$199$	279.081	−	179.354i	0.0994147	−	0.0638900i	−0.489989	−	0.871729i	$-0.662999\pi$
$199$	279.081	−	179.354i	0.0994147	−	0.0638900i	0.589404	+	0.807839i	$0.299363\pi$
$200$		0			0
$201$	2505.11	+	1358.31i	0.879090	+	0.476656i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	−3124.48	−	4861.79i	−1.04156	−	1.62069i
$209$		0			0
$210$		0			0
$211$	−2998.85	−	880.542i	−0.978433	−	0.287294i	−0.246855	−	0.969052i	$-0.579397\pi$
$211$	−2998.85	−	880.542i	−0.978433	−	0.287294i	−0.731577	+	0.681759i	$0.761215\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−1149.32	−	1326.39i	−0.359544	−	0.414936i
$218$		0			0
$219$	5019.77	+	2292.45i	1.54888	+	0.707349i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−4540.41	−	1333.18i	−1.36345	−	0.400344i	−0.483469	−	0.875362i	$-0.660623\pi$
$223$	−4540.41	−	1333.18i	−1.36345	−	0.400344i	−0.879976	+	0.475018i	$0.842442\pi$
$224$		0			0
$225$	1402.03	+	3070.01i	0.415415	+	0.909632i
$226$		0			0
$227$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$227$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$228$	3067.74	+	2658.22i	0.891080	+	0.772126i
$229$	3628.95	−	521.764i	1.04719	−	0.150564i	0.402824	−	0.915277i	$-0.368029\pi$
$229$	3628.95	−	521.764i	1.04719	−	0.150564i	0.644370	+	0.764714i	$0.277120\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$233$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	532.218	+	3701.65i	0.145870	+	1.01455i
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	−1495.95	+	1726.42i	−0.399845	+	0.461446i	−0.919592	−	0.392874i	$-0.871481\pi$
$241$	−1495.95	+	1726.42i	−0.399845	+	0.461446i	0.519747	+	0.854320i	$0.326026\pi$
$242$		0			0
$243$	−3749.44	−	539.088i	−0.989821	−	0.142315i
$244$	−6903.72	+	3152.82i	−1.81133	+	0.827208i
$245$		0			0
$246$		0			0
$247$	6664.02	+	5774.41i	1.71669	+	1.48752i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$251$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$252$	−855.597	+	741.379i	−0.213879	+	0.185327i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−582.922	+	4054.31i	−0.142315	+	0.989821i
$257$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$257$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$258$		0			0
$259$	−1268.59	−	1973.97i	−0.304350	−	0.473577i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$263$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−1545.35	−	4106.18i	−0.352228	−	0.935914i
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$	−1982.24	−	6750.89i	−0.444326	−	1.51324i	−0.812210	−	0.583365i	$-0.801736\pi$
$271$	−1982.24	−	6750.89i	−0.444326	−	1.51324i	0.367883	−	0.929872i	$-0.380083\pi$
$272$		0			0
$273$	−1858.60	+	1610.49i	−0.412043	+	0.357037i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−4937.75	+	3173.30i	−1.07105	+	0.688321i	−0.952473	−	0.304624i	$-0.901469\pi$
$277$	−4937.75	+	3173.30i	−1.07105	+	0.688321i	−0.118576	+	0.992945i	$0.537833\pi$
$278$		0			0
$279$	−6832.77	+	5920.63i	−1.46619	+	1.27046i
$280$		0			0
$281$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$281$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$282$		0			0
$283$	1810.86	+	1163.77i	0.380370	+	0.244449i	0.716829	−	0.697249i	$-0.245593\pi$
$283$	1810.86	+	1163.77i	0.380370	+	0.244449i	−0.336459	+	0.941698i	$0.609229\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−3217.33	+	3713.00i	−0.654861	+	0.755750i
$290$		0			0
$291$	−8430.38	+	2475.38i	−1.69827	+	0.498658i
$292$	−3529.46	−	7728.43i	−0.707349	−	1.54888i
$293$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$293$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	1463.93	−	4985.67i	0.281733	−	0.959493i
$301$	2406.42	+	1546.51i	0.460809	+	0.296144i
$302$		0			0
$303$		0			0
$304$	−889.405	−	6185.95i	−0.167799	−	1.16707i
$305$		0			0
$306$		0			0
$307$	−834.869	−	5806.64i	−0.155207	−	1.07949i	−0.907316	−	0.420449i	$-0.861873\pi$
$307$	−834.869	−	5806.64i	−0.155207	−	1.07949i	0.752110	−	0.659038i	$-0.229036\pi$
$308$		0			0
$309$	9865.87	−	4505.59i	1.81634	−	0.829495i
$310$		0			0
$311$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$311$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$312$		0			0
$313$	−2647.84	+	9017.72i	−0.478163	+	1.62847i	0.268508	+	0.963278i	$0.413470\pi$
$313$	−2647.84	+	9017.72i	−0.478163	+	1.62847i	−0.746670	+	0.665194i	$0.768349\pi$
$314$		0			0
$315$		0			0
$316$	3112.83	−	4843.66i	0.554147	−	0.862270i
$317$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$317$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	3819.15	+	4407.53i	0.654861	+	0.755750i
$325$	3180.06	−	10830.3i	0.542764	−	1.84848i
$326$		0			0
$327$	695.818	+	4839.52i	0.117672	+	0.818428i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−10221.9	−	4668.19i	−1.69742	−	0.775188i	−0.998180	−	0.0603052i	$-0.980793\pi$
$331$	−10221.9	−	4668.19i	−1.69742	−	0.775188i	−0.699245	−	0.714883i	$-0.746480\pi$
$332$		0			0
$333$	−10168.7	+	6535.05i	−1.67340	+	1.07543i
$334$		0			0
$335$		0			0
$336$	1743.01			0.283003
$337$	−3511.39	−	5463.83i	−0.567589	−	0.883186i	0.432238	−	0.901759i	$-0.357724\pi$
$337$	−3511.39	−	5463.83i	−0.567589	−	0.883186i	−0.999827	+	0.0185734i	$0.994088\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−3416.40	+	491.205i	−0.537809	+	0.0773253i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$347$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$348$		0			0
$349$	2973.23	−	6510.47i	0.456027	−	0.998560i	−0.532348	−	0.846526i	$-0.678690\pi$
$349$	2973.23	−	6510.47i	0.456027	−	0.998560i	0.988375	−	0.152035i	$-0.0485825\pi$
$350$		0			0
$351$	8296.28	+	9574.42i	1.26160	+	1.45597i
$352$		0			0
$353$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$353$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$359$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$360$		0			0
$361$	1111.82	+	2434.54i	0.162096	+	0.354941i
$362$		0			0
$363$	−6845.68	+	984.261i	−0.989821	+	0.142315i
$364$	3786.31			0.545211
$365$		0			0
$366$		0			0
$367$	1543.54	−	5256.81i	0.219543	−	0.747693i	−0.773895	−	0.633314i	$-0.781694\pi$
$367$	1543.54	−	5256.81i	0.219543	−	0.747693i	0.993437	−	0.114379i	$-0.0364878\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	13919.6			1.94005
$373$			12919.1i			1.79337i	0.442672	+	0.896683i	$0.354031\pi$
$373$			12919.1i			1.79337i	−0.442672	+	0.896683i	$0.645969\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	4152.32	+	14141.5i	0.562772	+	1.91662i	0.328978	+	0.944338i	$0.393296\pi$
$379$	4152.32	+	14141.5i	0.562772	+	1.91662i	0.233794	+	0.972286i	$0.424886\pi$
$380$		0			0
$381$	−11240.2	−	9739.71i	−1.51143	−	1.30966i
$382$		0			0
$383$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$383$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	7966.71	−	12396.4i	1.04644	−	1.62829i
$388$	12304.9	+	5619.47i	1.61002	+	0.735272i
$389$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$389$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	9720.92	+	11218.5i	1.22891	+	1.41824i	0.875808	+	0.482660i	$0.160329\pi$
$397$	9720.92	+	11218.5i	1.22891	+	1.41824i	0.353107	+	0.935583i	$0.385125\pi$
$398$		0			0
$399$	−2551.71	+	749.248i	−0.320163	+	0.0940084i
$400$	−6730.03	+	4325.13i	−0.841254	+	0.540641i
$401$		0			0			−	1.00000i	$-0.5\pi$
$401$		0			0				1.00000i	$0.5\pi$
$402$		0			0
$403$	30237.4			3.73754
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	1708.26	+	2658.11i	0.206523	+	0.321357i	0.929028	−	0.370009i	$-0.120646\pi$
$409$	1708.26	+	2658.11i	0.206523	+	0.321357i	−0.722505	+	0.691366i	$0.757009\pi$
$410$		0			0
$411$		0			0
$412$	−16022.1	−	4704.51i	−1.91590	−	0.562560i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−5327.14	−	3423.55i	−0.625591	−	0.402043i
$418$		0			0
$419$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$419$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$420$		0			0
$421$	234.559	+	150.742i	0.0271537	+	0.0174506i	0.554147	−	0.832419i	$-0.313044\pi$
$421$	234.559	+	150.742i	0.0271537	+	0.0174506i	−0.526994	+	0.849869i	$0.676681\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	707.644	−	4921.77i	0.0801998	−	0.557802i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$		−	8978.95i		−	1.00000i
$433$	25.4476	−	3.65881i	0.00282432	−	0.000406076i	−0.140903	−	0.990023i	$-0.545000\pi$
$433$	25.4476	−	3.65881i	0.00282432	−	0.000406076i	0.143727	+	0.989617i	$0.454091\pi$
$434$		0			0
$435$		0			0
$436$	4069.70	−	6332.58i	0.447026	−	0.695586i
$437$		0			0
$438$		0			0
$439$	−15983.7			−1.73772			−0.868861	−	0.495056i	$-0.835147\pi$
$439$	−15983.7			−1.73772			−0.868861	+	0.495056i	$0.835147\pi$
$440$		0			0
$441$	1212.42	+	8432.57i	0.130917	+	0.910546i
$442$		0			0
$443$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$443$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$444$	18420.6	+	2648.49i	1.96893	+	0.283089i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−2028.08	−	1757.34i	−0.213879	−	0.185327i
$449$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$449$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	5220.18	−	4523.31i	0.541424	−	0.469147i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−2663.50	+	18525.1i	−0.272633	+	1.89621i	0.148021	+	0.988984i	$0.452710\pi$
$457$	−2663.50	+	18525.1i	−0.272633	+	1.89621i	−0.420655	+	0.907221i	$0.638200\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$461$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$462$		0			0
$463$	−9608.15	−	8325.51i	−0.964425	−	0.835679i	0.0220224	−	0.999757i	$-0.492989\pi$
$463$	−9608.15	−	8325.51i	−0.964425	−	0.835679i	−0.986447	+	0.164079i	$0.947535\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$467$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$468$		−	19504.9i		−	1.92652i
$469$	2810.52	+	602.707i	0.276711	+	0.0593399i
$470$		0			0
$471$	−10329.8	−	16073.5i	−1.01056	−	1.57246i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	7993.34	−	9224.81i	0.772126	−	0.891080i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$479$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$480$		0			0
$481$	40014.9	+	5753.28i	3.79319	+	0.545378i
$482$		0			0
$483$		0			0
$484$	8957.67	+	5756.74i	0.841254	+	0.540641i
$485$		0			0
$486$		0			0
$487$	−19190.0	−	8763.79i	−1.78559	−	0.815453i	−0.972351	−	0.233526i	$-0.924974\pi$
$487$	−19190.0	−	8763.79i	−1.78559	−	0.815453i	−0.813241	−	0.581926i	$-0.802299\pi$
$488$		0			0
$489$	5264.41	+	17928.9i	0.486840	+	1.65803i
$490$		0			0
$491$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$491$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−16196.2	−	14034.1i	−1.46619	−	1.27046i
$497$		0			0
$498$		0			0
$499$			12652.5i			1.13508i	0.823345	+	0.567541i	$0.192105\pi$
$499$			12652.5i			1.13508i	−0.823345	+	0.567541i	$0.807895\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$503$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		−	30954.2i		−	2.71149i
$508$	3258.78	+	22665.3i	0.284616	+	1.97955i
$509$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$509$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$510$		0			0
$511$	5509.73	+	792.179i	0.476978	+	0.0685792i
$512$		0			0
$513$	3859.69	+	13144.9i	0.332182	+	1.13131i
$514$		0			0
$515$		0			0
$516$	−21768.1	+	6391.68i	−1.85714	+	0.545306i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$521$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$522$		0			0
$523$	148.934	−	1035.86i	0.0124521	−	0.0866061i	−0.982648	−	0.185482i	$-0.940615\pi$
$523$	148.934	−	1035.86i	0.0124521	−	0.0866061i	0.995100	+	0.0988755i	$0.0315246\pi$
$524$		0			0
$525$	2229.35	+	2572.81i	0.185327	+	0.213879i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	10235.5	−	6577.98i	0.841254	−	0.540641i
$530$		0			0
$531$		0			0
$532$	3724.45	+	1700.90i	0.303525	+	0.138615i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−12164.3	+	10540.4i	−0.966698	+	0.837648i	−0.986766	−	0.162150i	$-0.948157\pi$
$541$	−12164.3	+	10540.4i	−0.966698	+	0.837648i	0.0200682	+	0.999799i	$0.493612\pi$
$542$		0			0
$543$	−6606.62	−	10280.1i	−0.522131	−	0.812452i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	6888.93	−	5969.30i	0.538482	−	0.466597i	−0.342653	−	0.939462i	$-0.611326\pi$
$547$	6888.93	−	5969.30i	0.538482	−	0.466597i	0.881135	+	0.472865i	$0.156780\pi$
$548$		0			0
$549$	−25354.1	−	3645.37i	−1.97101	−	0.283389i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	1567.03	+	3431.31i	0.120501	+	0.263859i
$554$		0			0
$555$		0			0
$556$	2746.71	+	9354.43i	0.209508	+	0.713519i
$557$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$557$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$558$		0			0
$559$	−47286.5	+	13884.6i	−3.57783	+	1.05054i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$563$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−3782.00	+	543.770i	−0.280122	+	0.0402755i
$568$		0			0
$569$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$569$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$570$		0			0
$571$	17627.0	+	5175.76i	1.29189	+	0.379332i	0.854270	−	0.519829i	$-0.174004\pi$
$571$	17627.0	+	5175.76i	1.29189	+	0.379332i	0.437617	+	0.899161i	$0.355822\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−9052.79	+	10447.5i	−0.654861	+	0.755750i
$577$	528.120	−	241.184i	0.0381039	−	0.0174014i	−0.396272	−	0.918133i	$-0.629696\pi$
$577$	528.120	−	241.184i	0.0381039	−	0.0174014i	0.434376	+	0.900732i	$0.356969\pi$
$578$		0			0
$579$	−8811.70	+	4024.17i	−0.632473	+	0.288841i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$587$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$588$	7091.20	−	11034.1i	0.497341	−	0.773877i
$589$	29743.4	+	13583.3i	2.08074	+	0.950241i
$590$		0			0
$591$		0			0
$592$	−18763.1	−	21653.8i	−1.30263	−	1.50332i
$593$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$593$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−1302.76	−	1128.84i	−0.0893103	−	0.0773878i
$598$		0			0
$599$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$599$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$600$		0			0
$601$	−18124.7	+	11648.1i	−1.23016	+	0.790573i	−0.983915	−	0.178636i	$-0.942832\pi$
$601$	−18124.7	+	11648.1i	−1.23016	+	0.790573i	−0.246241	+	0.969209i	$0.579195\pi$
$602$		0			0
$603$	3104.79	−	14478.1i	0.209680	−	0.977770i
$604$	−10634.5			−0.716407
$605$		0			0
$606$		0			0
$607$	−7838.31	+	17163.5i	−0.524131	+	1.14769i	0.443721	+	0.896165i	$0.353658\pi$
$607$	−7838.31	+	17163.5i	−0.524131	+	1.14769i	−0.967852	+	0.251521i	$0.919069\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−28645.6	−	8411.12i	−1.88742	−	0.554195i	−0.994601	−	0.103771i	$-0.966909\pi$
$613$	−28645.6	−	8411.12i	−1.88742	−	0.554195i	−0.892815	−	0.450424i	$-0.851273\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$617$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$618$		0			0
$619$	−3793.48	−	4377.91i	−0.246321	−	0.284270i	0.619103	−	0.785310i	$-0.287496\pi$
$619$	−3793.48	−	4377.91i	−0.246321	−	0.284270i	−0.865424	+	0.501040i	$0.832951\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	−19665.3	+	22694.9i	−1.26160	+	1.45597i
$625$	−14992.1	−	4402.07i	−0.959493	−	0.281733i
$626$		0			0
$627$		0			0
$628$	−4186.41	+	29117.1i	−0.266013	+	1.85016i
$629$		0			0
$630$		0			0
$631$	30581.3	−	4396.93i	1.92935	−	0.277399i	0.932804	−	0.360385i	$-0.117355\pi$
$631$	30581.3	−	4396.93i	1.92935	−	0.277399i	0.996551	+	0.0829861i	$0.0264457\pi$
$632$		0			0
$633$			16240.3i			1.01974i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	15404.1	−	23969.3i	0.958137	−	1.49089i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0			−	1.00000i	$-0.5\pi$
$641$		0			0				1.00000i	$0.5\pi$
$642$		0			0
$643$	21353.0	−	24642.7i	1.30961	−	1.51137i	0.644986	−	0.764194i	$-0.276863\pi$
$643$	21353.0	−	24642.7i	1.30961	−	1.51137i	0.664625	−	0.747177i	$-0.268591\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$647$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−4930.41	+	7671.87i	−0.296833	+	0.461881i
$652$	11951.0	−	26169.0i	0.717847	−	1.57187i
$653$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$653$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	4080.84	−	28382.9i	0.242327	−	1.68542i
$658$		0			0
$659$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$659$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$660$		0			0
$661$	3980.96	+	6194.49i	0.234253	+	0.364505i	0.938401	−	0.345549i	$-0.112307\pi$
$661$	3980.96	+	6194.49i	0.234253	+	0.364505i	−0.704148	+	0.710053i	$0.748671\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$			24588.7i			1.42101i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−9791.33	−	33346.2i	−0.560815	−	1.90996i	−0.373363	−	0.927685i	$-0.621796\pi$
$673$	−9791.33	−	33346.2i	−0.560815	−	1.90996i	−0.187451	−	0.982274i	$-0.560023\pi$
$674$		0			0
$675$	13253.6	−	11484.3i	0.755750	−	0.654861i
$676$	−31208.7	+	36016.8i	−1.77565	+	2.04920i
$677$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$677$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$678$		0			0
$679$	−7455.69	+	4791.48i	−0.421389	+	0.270810i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$683$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$684$	8762.04	−	19186.2i	0.489803	−	1.07252i
$685$		0			0
$686$		0			0
$687$	−7913.85	−	17328.9i	−0.439494	−	0.962357i
$688$	31772.5	+	14510.0i	1.76063	+	0.804055i
$689$		0			0
$690$		0			0
$691$	−19620.3	+	22643.0i	−1.08016	+	1.24657i	−0.112677	+	0.993632i	$0.535943\pi$
$691$	−19620.3	+	22643.0i	−1.08016	+	1.24657i	−0.967482	+	0.252939i	$0.918603\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		−	5241.28i		−	0.283003i
$701$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$701$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$702$		0			0
$703$	36776.7	+	23634.9i	1.97306	+	1.26801i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−4498.23	−	31285.9i	−0.238272	−	1.65722i	−0.660575	−	0.750760i	$-0.729687\pi$
$709$	−4498.23	−	31285.9i	−0.238272	−	1.65722i	0.422303	−	0.906455i	$-0.361222\pi$
$710$		0			0
$711$	17676.1	−	8072.41i	0.932357	−	0.425793i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$719$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$720$		0			0
$721$	8268.05	−	7164.31i	0.427071	−	0.370059i
$722$		0			0
$723$	10797.3	+	4930.98i	0.555404	+	0.253644i
$724$	−2677.50	+	18622.4i	−0.137442	+	0.955933i
$725$		0			0
$726$		0			0
$727$	−1648.80	+	5615.30i	−0.0841136	+	0.286465i	−0.990799	−	0.135343i	$-0.956786\pi$
$727$	−1648.80	+	5615.30i	−0.0841136	+	0.286465i	0.906685	+	0.421808i	$0.138604\pi$
$728$		0			0
$729$	2801.18	+	19482.7i	0.142315	+	0.989821i
$730$		0			0
$731$		0			0
$732$	25825.5	+	29804.2i	1.30401	+	1.50491i
$733$	20702.8	+	9454.67i	1.04322	+	0.476420i	0.861939	−	0.507012i	$-0.169250\pi$
$733$	20702.8	+	9454.67i	1.04322	+	0.476420i	0.181276	+	0.983432i	$0.441977\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−9793.58	−	15239.1i	−0.487500	−	0.758565i	0.507151	−	0.861857i	$-0.330699\pi$
$739$	−9793.58	−	15239.1i	−0.487500	−	0.758565i	−0.994651	+	0.103292i	$0.967062\pi$
$740$		0			0
$741$	19033.7	−	41677.9i	0.943616	−	2.06623i
$742$		0			0
$743$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$743$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	8735.25	−	19127.5i	0.424439	−	0.929392i	−0.569757	−	0.821813i	$-0.692963\pi$
$751$	8735.25	−	19127.5i	0.424439	−	0.929392i	0.994197	−	0.107579i	$-0.0343099\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	4948.80	+	3180.40i	0.238077	+	0.153003i
$757$	−8105.01	−	27603.1i	−0.389143	−	1.32530i	−0.888489	−	0.458898i	$-0.848244\pi$
$757$	−8105.01	−	27603.1i	−0.389143	−	1.32530i	0.499346	−	0.866403i	$-0.333574\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$761$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$762$		0			0
$763$	2048.72	+	4486.08i	0.0972067	+	0.212853i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	21066.8	−	3028.95i	0.989821	−	0.142315i
$769$	2984.49	−	10164.2i	0.139953	−	0.476635i	−0.859449	−	0.511221i	$-0.829193\pi$
$769$	2984.49	−	10164.2i	0.139953	−	0.476635i	0.999402	+	0.0345865i	$0.0110114\pi$
$770$		0			0
$771$		0			0
$772$	14310.1	+	4201.84i	0.667142	+	0.195890i
$773$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$773$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$774$		0			0
$775$		−	41856.7i		−	1.94005i
$776$		0			0
$777$	−7984.44	+	9214.54i	−0.368649	+	0.425444i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−19375.9	+	5689.27i	−0.882647	+	0.259168i
$785$		0			0
$786$		0			0
$787$	−38022.8	+	17364.4i	−1.72219	+	0.786500i	−0.727216	+	0.686409i	$0.759186\pi$
$787$	−38022.8	+	17364.4i	−1.72219	+	0.786500i	−0.994978	+	0.100091i	$0.968087\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	56100.3	+	64743.2i	2.51221	+	2.89924i
$794$		0			0
$795$		0			0
$796$	377.697	+	2626.94i	0.0168180	+	0.116972i
$797$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$797$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$	−18209.8	+	13715.7i	−0.798769	+	0.601638i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$809$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$810$		0			0
$811$	−19208.2	−	29888.6i	−0.831680	−	1.29412i	−0.953447	−	0.301561i	$-0.902492\pi$
$811$	−19208.2	−	29888.6i	−0.831680	−	1.29412i	0.121767	−	0.992559i	$-0.461144\pi$
$812$		0			0
$813$	−30755.9	+	19765.6i	−1.32676	+	0.852656i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−52751.2	−	7584.48i	−2.25891	−	0.324783i
$818$		0			0
$819$	10750.2	+	6908.74i	0.458660	+	0.294763i
$820$		0			0
$821$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$821$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$822$		0			0
$823$	−12096.8	−	7774.14i	−0.512355	−	0.329270i	0.258786	−	0.965935i	$-0.416677\pi$
$823$	−12096.8	−	7774.14i	−0.512355	−	0.329270i	−0.771141	+	0.636664i	$0.780314\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$827$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$828$		0			0
$829$	6780.39	−	47158.6i	0.284068	−	1.97574i	0.0798996	−	0.996803i	$-0.474540\pi$
$829$	6780.39	−	47158.6i	0.284068	−	1.97574i	0.204169	−	0.978936i	$-0.434551\pi$
$830$		0			0
$831$	23049.5	+	19972.5i	0.962188	+	0.833741i
$832$	45763.1	−	6579.74i	1.90691	−	0.274173i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	39521.0	+	25398.6i	1.63207	+	1.04887i
$838$		0			0
$839$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$839$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$840$		0			0
$841$	24389.0			1.00000
$842$		0			0
$843$		0			0
$844$	16373.9	−	18896.5i	0.667787	−	0.770668i
$845$		0			0
$846$		0			0
$847$	−6345.73	+	2898.00i	−0.257428	+	0.117564i
$848$		0			0
$849$	3151.21	−	10732.1i	0.127384	−	0.433832i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	9913.60	−	21707.8i	0.397931	−	0.871347i	−0.599545	−	0.800341i	$-0.704652\pi$
$853$	9913.60	−	21707.8i	0.397931	−	0.871347i	0.997476	−	0.0710062i	$-0.0226210\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$857$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$858$		0			0
$859$	−4921.58	+	34230.3i	−0.195486	+	1.35963i	0.621699	+	0.783256i	$0.286443\pi$
$859$	−4921.58	+	34230.3i	−0.195486	+	1.35963i	−0.817185	+	0.576376i	$0.804466\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$863$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	23221.7	+	10605.0i	0.909632	+	0.415415i
$868$	13471.8	−	3955.66i	0.526799	−	0.154682i
$869$		0			0
$870$		0			0
$871$	−39556.9	+	29794.5i	−1.53885	+	1.15907i
$872$		0			0
$873$	24682.9	+	38407.3i	0.956918	+	1.48899i
$874$		0			0
$875$		0			0
$876$	−33364.6	+	28910.6i	−1.28685	+	1.11507i
$877$	22837.3	−	26355.7i	0.879318	−	1.01479i	−0.120438	−	0.992721i	$-0.538430\pi$
$877$	22837.3	−	26355.7i	0.879318	−	1.01479i	0.999757	−	0.0220663i	$-0.00702451\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$881$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$882$		0			0
$883$	−40039.0	−	5756.75i	−1.52596	−	0.219400i	−0.672318	−	0.740263i	$-0.734701\pi$
$883$	−40039.0	−	5756.75i	−1.52596	−	0.219400i	−0.853640	+	0.520863i	$0.825610\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$887$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$888$		0			0
$889$	−13646.4	−	6232.11i	−0.514832	−	0.235116i
$890$		0			0
$891$		0			0
$892$	24790.9	−	28610.2i	0.930561	−	1.07392i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−27000.0			−1.00000
$901$		0			0
$902$		0			0
$903$	4187.58	−	14261.6i	0.154323	−	0.525576i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	822.360	+	5719.64i	0.0301058	+	0.209391i	0.999322	−	0.0368211i	$-0.0117232\pi$
$907$	822.360	+	5719.64i	0.0301058	+	0.209391i	−0.969216	+	0.246212i	$0.920814\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$911$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$912$	−29539.1	+	13490.0i	−1.07252	+	0.489803i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−8263.25	+	28142.0i	−0.298063	+	1.01511i
$917$		0			0
$918$		0			0
$919$	−20828.9	+	32410.4i	−0.747642	+	1.16335i	0.233929	+	0.972254i	$0.424842\pi$
$919$	−20828.9	+	32410.4i	−0.747642	+	1.16335i	−0.981571	+	0.191100i	$0.938795\pi$
$920$		0			0
$921$	−27727.8	+	12662.9i	−0.992033	+	0.453046i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	7964.09	−	55391.5i	0.283089	−	1.96893i
$926$		0			0
$927$	−36906.3	−	42592.1i	−1.30762	−	1.50907i
$928$		0			0
$929$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$929$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$930$		0			0
$931$	25920.0	−	16657.8i	0.912454	−	0.586399i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		−	44484.2i		−	1.55095i	−0.631380	−	0.775473i	$-0.717511\pi$
$937$		−	44484.2i		−	1.55095i	0.631380	−	0.775473i	$-0.282489\pi$
$938$		0			0
$939$	48835.6			1.69722
$940$		0			0
$941$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$941$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$947$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$948$	−28705.9	−	8428.81i	−0.983464	−	0.288771i
$949$	−72477.4	+	62802.0i	−2.47915	+	2.14820i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$953$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	79000.7	−	23196.7i	2.65183	−	0.778648i
$962$		0			0
$963$		0			0
$964$	−7591.73	−	16623.6i	−0.253644	−	0.555404i
$965$		0			0
$966$		0			0
$967$	−38586.7			−1.28321			−0.641605	−	0.767036i	$-0.721731\pi$
$967$	−38586.7			−1.28321			−0.641605	+	0.767036i	$0.721731\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$971$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$972$	16383.6	−	25493.3i	0.540641	−	0.841254i
$973$	−6128.65	−	1799.53i	−0.201928	−	0.0592913i
$974$		0			0
$975$	−58651.7			−1.92652
$976$		−	60716.6i		−	1.99128i
$977$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$977$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	23109.6	−	10553.8i	0.752124	−	0.343484i
$982$		0			0
$983$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$983$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	−64167.3	+	29304.2i	−2.06623	+	0.943616i
$989$		0			0
$990$		0			0
$991$	6017.00	+	2747.87i	0.192872	+	0.0880818i	0.509510	−	0.860465i	$-0.329827\pi$
$991$	6017.00	+	2747.87i	0.192872	+	0.0880818i	−0.316638	+	0.948547i	$0.602554\pi$
$992$		0			0
$993$	−8309.96	+	57797.0i	−0.265567	+	1.84706i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−6190.22	−	43053.9i	−0.196636	−	1.36763i	−0.813958	−	0.580923i	$-0.802692\pi$
$997$	−6190.22	−	43053.9i	−0.196636	−	1.36763i	0.617322	−	0.786710i	$-0.288217\pi$
$998$		0			0
$999$	47467.9	+	41131.1i	1.50332	+	1.30263i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.4.j.a.119.1	✓	20
3.2	odd	2	CM	201.4.j.a.119.1	✓	20
67.58	odd	22	inner	201.4.j.a.125.1	yes	20
201.125	even	22	inner	201.4.j.a.125.1	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.4.j.a.119.1	✓	20	1.1	even	1	trivial
201.4.j.a.119.1	✓	20	3.2	odd	2	CM
201.4.j.a.125.1	yes	20	67.58	odd	22	inner
201.4.j.a.125.1	yes	20	201.125	even	22	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:	\( N \)	\(=\)	\( 201 = 3 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	201.j (of order \(22\), degree \(10\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.46393 - 4.98567i) q^{3} +(-3.32332 + 7.27706i) q^{4} +(-2.83365 - 4.40925i) q^{7} +(-22.7138 + 14.5973i) q^{9} +O(q^{10})\)	\(q+(-1.46393 - 4.98567i) q^{3} +(-3.32332 + 7.27706i) q^{4} +(-2.83365 - 4.40925i) q^{7} +(-22.7138 + 14.5973i) q^{9} +(41.1461 + 5.91592i) q^{12} +(89.3811 + 12.8511i) q^{13} +(-41.9111 - 48.3680i) q^{16} +(82.1479 + 52.7932i) q^{19} +(-17.8348 + 20.5825i) q^{21} +(17.7894 - 123.728i) q^{25} +(106.029 + 91.8744i) q^{27} +(41.5035 - 5.96730i) q^{28} +(331.445 - 47.6546i) q^{31} +(-30.7400 - 213.801i) q^{36} +447.689 q^{37} +(-66.7761 - 464.438i) q^{39} +(-496.446 + 226.719i) q^{43} +(-179.792 + 279.762i) q^{48} +(131.075 - 287.015i) q^{49} +(-390.560 + 607.723i) q^{52} +(142.951 - 486.848i) q^{57} +(716.977 + 621.264i) q^{61} +(128.726 + 58.7873i) q^{63} +(491.260 - 144.247i) q^{64} +(-386.644 + 388.934i) q^{67} +(-695.480 + 802.627i) q^{73} +(-642.908 + 92.4362i) q^{75} +(-657.183 + 422.346i) q^{76} +(-712.384 - 102.425i) q^{79} +(302.838 - 663.122i) q^{81} +(-90.5090 - 198.187i) q^{84} +(-196.611 - 430.519i) q^{91} +(-722.801 - 1582.71i) q^{93} -1690.92i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 16 q^{4} + 54 q^{9}+O(q^{10}) \) 20 * q + 16 * q^4 + 54 * q^9	\( 20 q + 16 q^{4} + 54 q^{9} - 128 q^{16} + 112 q^{19} + 324 q^{21} + 250 q^{25} - 432 q^{36} + 220 q^{37} - 648 q^{39} + 1258 q^{49} - 6160 q^{52} + 8910 q^{57} + 5940 q^{63} + 1024 q^{64} - 880 q^{67} - 10710 q^{73} - 896 q^{76} - 9724 q^{79} - 1458 q^{81} + 11664 q^{84} - 3888 q^{91} - 1620 q^{93}+O(q^{100}) \) 20 * q + 16 * q^4 + 54 * q^9 - 128 * q^16 + 112 * q^19 + 324 * q^21 + 250 * q^25 - 432 * q^36 + 220 * q^37 - 648 * q^39 + 1258 * q^49 - 6160 * q^52 + 8910 * q^57 + 5940 * q^63 + 1024 * q^64 - 880 * q^67 - 10710 * q^73 - 896 * q^76 - 9724 * q^79 - 1458 * q^81 + 11664 * q^84 - 3888 * q^91 - 1620 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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