LMFDB - Embedded newform 1062.4.a.n.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1062,4,Mod(1,1062)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1062, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1062.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1062 = 2 \cdot 3^{2} \cdot 59 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1062.a (trivial)

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:	yes
Analytic conductor:	$62.6600284261$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 44x^{2} + 19x + 3 $ x^4 - x^3 - 44x^2 + 19x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 354)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$6.93810$ of defining polynomial
Character	$\chi$	$=$	1062.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{2} +4.00000 q^{4} -17.3705 q^{5} +29.1373 q^{7} -8.00000 q^{8} +O(q^{10})$	\(q-2.00000 q^{2} +4.00000 q^{4} -17.3705 q^{5} +29.1373 q^{7} -8.00000 q^{8} +34.7410 q^{10} -34.2581 q^{11} -37.9155 q^{13} -58.2746 q^{14} +16.0000 q^{16} +84.9002 q^{17} -8.58000 q^{19} -69.4820 q^{20} +68.5162 q^{22} +103.119 q^{23} +176.734 q^{25} +75.8310 q^{26} +116.549 q^{28} -169.530 q^{29} +117.912 q^{31} -32.0000 q^{32} -169.800 q^{34} -506.129 q^{35} -173.663 q^{37} +17.1600 q^{38} +138.964 q^{40} -346.519 q^{41} +26.7692 q^{43} -137.032 q^{44} -206.238 q^{46} +472.136 q^{47} +505.981 q^{49} -353.468 q^{50} -151.662 q^{52} +514.004 q^{53} +595.081 q^{55} -233.098 q^{56} +339.059 q^{58} -59.0000 q^{59} +133.802 q^{61} -235.823 q^{62} +64.0000 q^{64} +658.611 q^{65} -266.749 q^{67} +339.601 q^{68} +1012.26 q^{70} -222.827 q^{71} +950.821 q^{73} +347.325 q^{74} -34.3200 q^{76} -998.189 q^{77} -1216.98 q^{79} -277.928 q^{80} +693.037 q^{82} -183.540 q^{83} -1474.76 q^{85} -53.5385 q^{86} +274.065 q^{88} -654.925 q^{89} -1104.75 q^{91} +412.476 q^{92} -944.272 q^{94} +149.039 q^{95} -422.708 q^{97} -1011.96 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{2} + 16 q^{4} - 22 q^{5} + 13 q^{7} - 32 q^{8}+O(q^{10}) $ 4 * q - 8 * q^2 + 16 * q^4 - 22 * q^5 + 13 * q^7 - 32 * q^8	$ 4 q - 8 q^{2} + 16 q^{4} - 22 q^{5} + 13 q^{7} - 32 q^{8} + 44 q^{10} - 24 q^{11} + 20 q^{13} - 26 q^{14} + 64 q^{16} - 91 q^{17} + 141 q^{19} - 88 q^{20} + 48 q^{22} - 13 q^{23} + 278 q^{25} - 40 q^{26} + 52 q^{28} - 295 q^{29} + 311 q^{31} - 128 q^{32} + 182 q^{34} - 551 q^{35} + 609 q^{37} - 282 q^{38} + 176 q^{40} - 677 q^{41} + 170 q^{43} - 96 q^{44} + 26 q^{46} - 17 q^{47} + 651 q^{49} - 556 q^{50} + 80 q^{52} - 166 q^{53} + 108 q^{55} - 104 q^{56} + 590 q^{58} - 236 q^{59} + 651 q^{61} - 622 q^{62} + 256 q^{64} - 700 q^{65} - 894 q^{67} - 364 q^{68} + 1102 q^{70} - 298 q^{71} + 887 q^{73} - 1218 q^{74} + 564 q^{76} + 79 q^{77} - 784 q^{79} - 352 q^{80} + 1354 q^{82} - 971 q^{83} - 799 q^{85} - 340 q^{86} + 192 q^{88} - 1321 q^{89} - 2673 q^{91} - 52 q^{92} + 34 q^{94} + 3133 q^{95} - 1922 q^{97} - 1302 q^{98}+O(q^{100}) $ 4 * q - 8 * q^2 + 16 * q^4 - 22 * q^5 + 13 * q^7 - 32 * q^8 + 44 * q^10 - 24 * q^11 + 20 * q^13 - 26 * q^14 + 64 * q^16 - 91 * q^17 + 141 * q^19 - 88 * q^20 + 48 * q^22 - 13 * q^23 + 278 * q^25 - 40 * q^26 + 52 * q^28 - 295 * q^29 + 311 * q^31 - 128 * q^32 + 182 * q^34 - 551 * q^35 + 609 * q^37 - 282 * q^38 + 176 * q^40 - 677 * q^41 + 170 * q^43 - 96 * q^44 + 26 * q^46 - 17 * q^47 + 651 * q^49 - 556 * q^50 + 80 * q^52 - 166 * q^53 + 108 * q^55 - 104 * q^56 + 590 * q^58 - 236 * q^59 + 651 * q^61 - 622 * q^62 + 256 * q^64 - 700 * q^65 - 894 * q^67 - 364 * q^68 + 1102 * q^70 - 298 * q^71 + 887 * q^73 - 1218 * q^74 + 564 * q^76 + 79 * q^77 - 784 * q^79 - 352 * q^80 + 1354 * q^82 - 971 * q^83 - 799 * q^85 - 340 * q^86 + 192 * q^88 - 1321 * q^89 - 2673 * q^91 - 52 * q^92 + 34 * q^94 + 3133 * q^95 - 1922 * q^97 - 1302 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	4.00000	0.500000
$5$	−17.3705	−1.55366	−0.776832	−	0.629708i	$-0.783175\pi$
$5$	−17.3705	−1.55366	−0.776832	+	0.629708i	$0.783175\pi$
$6$	0	0
$7$	29.1373	1.57327	0.786633	−	0.617421i	$-0.211823\pi$
$7$	29.1373	1.57327	0.786633	+	0.617421i	$0.211823\pi$
$8$	−8.00000	−0.353553
$9$	0	0
$10$	34.7410	1.09861
$11$	−34.2581	−0.939019	−0.469510	−	0.882927i	$-0.655569\pi$
$11$	−34.2581	−0.939019	−0.469510	+	0.882927i	$0.655569\pi$
$12$	0	0
$13$	−37.9155	−0.808913	−0.404456	−	0.914557i	$-0.632539\pi$
$13$	−37.9155	−0.808913	−0.404456	+	0.914557i	$0.632539\pi$
$14$	−58.2746	−1.11247
$15$	0	0
$16$	16.0000	0.250000
$17$	84.9002	1.21125	0.605627	−	0.795748i	$-0.292922\pi$
$17$	84.9002	1.21125	0.605627	+	0.795748i	$0.292922\pi$
$18$	0	0
$19$	−8.58000	−0.103599	−0.0517997	−	0.998657i	$-0.516496\pi$
$19$	−8.58000	−0.103599	−0.0517997	+	0.998657i	$0.516496\pi$
$20$	−69.4820	−0.776832
$21$	0	0
$22$	68.5162	0.663987
$23$	103.119	0.934862	0.467431	−	0.884030i	$-0.345180\pi$
$23$	103.119	0.934862	0.467431	+	0.884030i	$0.345180\pi$
$24$	0	0
$25$	176.734	1.41387
$26$	75.8310	0.571988
$27$	0	0
$28$	116.549	0.786633
$29$	−169.530	−1.08555	−0.542774	−	0.839879i	$-0.682626\pi$
$29$	−169.530	−1.08555	−0.542774	+	0.839879i	$0.682626\pi$
$30$	0	0
$31$	117.912	0.683147	0.341574	−	0.939855i	$-0.389040\pi$
$31$	117.912	0.683147	0.341574	+	0.939855i	$0.389040\pi$
$32$	−32.0000	−0.176777
$33$	0	0
$34$	−169.800	−0.856486
$35$	−506.129	−2.44433
$36$	0	0
$37$	−173.663	−0.771621	−0.385811	−	0.922578i	$-0.626078\pi$
$37$	−173.663	−0.771621	−0.385811	+	0.922578i	$0.626078\pi$
$38$	17.1600	0.0732558
$39$	0	0
$40$	138.964	0.549303
$41$	−346.519	−1.31993	−0.659965	−	0.751296i	$-0.729429\pi$
$41$	−346.519	−1.31993	−0.659965	+	0.751296i	$0.729429\pi$
$42$	0	0
$43$	26.7692	0.0949365	0.0474683	−	0.998873i	$-0.484885\pi$
$43$	26.7692	0.0949365	0.0474683	+	0.998873i	$0.484885\pi$
$44$	−137.032	−0.469510
$45$	0	0
$46$	−206.238	−0.661047
$47$	472.136	1.46528	0.732639	−	0.680617i	$-0.238288\pi$
$47$	472.136	1.46528	0.732639	+	0.680617i	$0.238288\pi$
$48$	0	0
$49$	505.981	1.47516
$50$	−353.468	−0.999760
$51$	0	0
$52$	−151.662	−0.404456
$53$	514.004	1.33215	0.666074	−	0.745885i	$-0.267973\pi$
$53$	514.004	1.33215	0.666074	+	0.745885i	$0.267973\pi$
$54$	0	0
$55$	595.081	1.45892
$56$	−233.098	−0.556233
$57$	0	0
$58$	339.059	0.767598
$59$	−59.0000	−0.130189
$59$	−59.0000	−0.130189
$60$	0	0
$61$	133.802	0.280846	0.140423	−	0.990092i	$-0.455154\pi$
$61$	133.802	0.280846	0.140423	+	0.990092i	$0.455154\pi$
$62$	−235.823	−0.483058
$63$	0	0
$64$	64.0000	0.125000
$65$	658.611	1.25678
$66$	0	0
$67$	−266.749	−0.486397	−0.243198	−	0.969977i	$-0.578197\pi$
$67$	−266.749	−0.486397	−0.243198	+	0.969977i	$0.578197\pi$
$68$	339.601	0.605627
$69$	0	0
$70$	1012.26	1.72840
$71$	−222.827	−0.372461	−0.186230	−	0.982506i	$-0.559627\pi$
$71$	−222.827	−0.372461	−0.186230	+	0.982506i	$0.559627\pi$
$72$	0	0
$73$	950.821	1.52445	0.762227	−	0.647310i	$-0.224106\pi$
$73$	950.821	1.52445	0.762227	+	0.647310i	$0.224106\pi$
$74$	347.325	0.545618
$75$	0	0
$76$	−34.3200	−0.0517997
$77$	−998.189	−1.47733
$78$	0	0
$79$	−1216.98	−1.73317	−0.866586	−	0.499028i	$-0.833690\pi$
$79$	−1216.98	−1.73317	−0.866586	+	0.499028i	$0.833690\pi$
$80$	−277.928	−0.388416
$81$	0	0
$82$	693.037	0.933332
$83$	−183.540	−0.242724	−0.121362	−	0.992608i	$-0.538726\pi$
$83$	−183.540	−0.242724	−0.121362	+	0.992608i	$0.538726\pi$
$84$	0	0
$85$	−1474.76	−1.88188
$86$	−53.5385	−0.0671303
$87$	0	0
$88$	274.065	0.331993
$89$	−654.925	−0.780022	−0.390011	−	0.920810i	$-0.627529\pi$
$89$	−654.925	−0.780022	−0.390011	+	0.920810i	$0.627529\pi$
$90$	0	0
$91$	−1104.75	−1.27263
$92$	412.476	0.467431
$93$	0	0
$94$	−944.272	−1.03611
$95$	149.039	0.160959
$96$	0	0
$97$	−422.708	−0.442469	−0.221235	−	0.975221i	$-0.571009\pi$
$97$	−422.708	−0.442469	−0.221235	+	0.975221i	$0.571009\pi$
$98$	−1011.96	−1.04310
$99$	0	0
$100$	706.937	0.706937
$101$	149.751	0.147532	0.0737662	−	0.997276i	$-0.476498\pi$
$101$	149.751	0.147532	0.0737662	+	0.997276i	$0.476498\pi$
$102$	0	0
$103$	−596.202	−0.570345	−0.285173	−	0.958476i	$-0.592051\pi$
$103$	−596.202	−0.570345	−0.285173	+	0.958476i	$0.592051\pi$
$104$	303.324	0.285994
$105$	0	0
$106$	−1028.01	−0.941971
$107$	2113.18	1.90924	0.954622	−	0.297820i	$-0.0962595\pi$
$107$	2113.18	1.90924	0.954622	+	0.297820i	$0.0962595\pi$
$108$	0	0
$109$	989.673	0.869665	0.434833	−	0.900511i	$-0.356808\pi$
$109$	989.673	0.869665	0.434833	+	0.900511i	$0.356808\pi$
$110$	−1190.16	−1.03161
$111$	0	0
$112$	466.197	0.393316
$113$	−1390.37	−1.15748	−0.578738	−	0.815514i	$-0.696455\pi$
$113$	−1390.37	−1.15748	−0.578738	+	0.815514i	$0.696455\pi$
$114$	0	0
$115$	−1791.23	−1.45246
$116$	−678.119	−0.542774
$117$	0	0
$118$	118.000	0.0920575
$119$	2473.76	1.90563
$120$	0	0
$121$	−157.381	−0.118243
$122$	−267.604	−0.198588
$123$	0	0
$124$	471.647	0.341574
$125$	−898.649	−0.643021
$126$	0	0
$127$	−703.565	−0.491585	−0.245792	−	0.969322i	$-0.579048\pi$
$127$	−703.565	−0.491585	−0.245792	+	0.969322i	$0.579048\pi$
$128$	−128.000	−0.0883883
$129$	0	0
$130$	−1317.22	−0.888677
$131$	−2588.74	−1.72656	−0.863281	−	0.504723i	$-0.831595\pi$
$131$	−2588.74	−1.72656	−0.863281	+	0.504723i	$0.831595\pi$
$132$	0	0
$133$	−249.998	−0.162989
$134$	533.498	0.343934
$135$	0	0
$136$	−679.202	−0.428243
$137$	1459.48	0.910161	0.455080	−	0.890450i	$-0.349610\pi$
$137$	1459.48	0.910161	0.455080	+	0.890450i	$0.349610\pi$
$138$	0	0
$139$	−2841.33	−1.73380	−0.866902	−	0.498479i	$-0.833892\pi$
$139$	−2841.33	−1.73380	−0.866902	+	0.498479i	$0.833892\pi$
$140$	−2024.52	−1.22216
$141$	0	0
$142$	445.655	0.263370
$143$	1298.91	0.759584
$144$	0	0
$145$	2944.82	1.68658
$146$	−1901.64	−1.07795
$147$	0	0
$148$	−694.651	−0.385811
$149$	−1815.14	−0.998003	−0.499001	−	0.866601i	$-0.666300\pi$
$149$	−1815.14	−0.998003	−0.499001	+	0.866601i	$0.666300\pi$
$150$	0	0
$151$	−242.349	−0.130610	−0.0653049	−	0.997865i	$-0.520802\pi$
$151$	−242.349	−0.130610	−0.0653049	+	0.997865i	$0.520802\pi$
$152$	68.6400	0.0366279
$153$	0	0
$154$	1996.38	1.04463
$155$	−2048.19	−1.06138
$156$	0	0
$157$	387.440	0.196950	0.0984748	−	0.995140i	$-0.468604\pi$
$157$	387.440	0.196950	0.0984748	+	0.995140i	$0.468604\pi$
$158$	2433.95	1.22554
$159$	0	0
$160$	555.856	0.274652
$161$	3004.61	1.47079
$162$	0	0
$163$	−2658.41	−1.27744	−0.638721	−	0.769438i	$-0.720536\pi$
$163$	−2658.41	−1.27744	−0.638721	+	0.769438i	$0.720536\pi$
$164$	−1386.07	−0.659965
$165$	0	0
$166$	367.079	0.171632
$167$	−2489.17	−1.15340	−0.576700	−	0.816956i	$-0.695660\pi$
$167$	−2489.17	−1.15340	−0.576700	+	0.816956i	$0.695660\pi$
$168$	0	0
$169$	−759.416	−0.345661
$170$	2949.52	1.33069
$171$	0	0
$172$	107.077	0.0474683
$173$	−3249.10	−1.42789	−0.713945	−	0.700202i	$-0.753093\pi$
$173$	−3249.10	−1.42789	−0.713945	+	0.700202i	$0.753093\pi$
$174$	0	0
$175$	5149.55	2.22440
$176$	−548.130	−0.234755
$177$	0	0
$178$	1309.85	0.551559
$179$	−2940.15	−1.22769	−0.613847	−	0.789425i	$-0.710379\pi$
$179$	−2940.15	−1.22769	−0.613847	+	0.789425i	$0.710379\pi$
$180$	0	0
$181$	−1789.70	−0.734958	−0.367479	−	0.930032i	$-0.619779\pi$
$181$	−1789.70	−0.734958	−0.367479	+	0.930032i	$0.619779\pi$
$182$	2209.51	0.899888
$183$	0	0
$184$	−824.953	−0.330523
$185$	3016.61	1.19884
$186$	0	0
$187$	−2908.52	−1.13739
$188$	1888.54	0.732639
$189$	0	0
$190$	−298.078	−0.113815
$191$	534.945	0.202656	0.101328	−	0.994853i	$-0.467691\pi$
$191$	534.945	0.202656	0.101328	+	0.994853i	$0.467691\pi$
$192$	0	0
$193$	2365.19	0.882124	0.441062	−	0.897477i	$-0.354602\pi$
$193$	2365.19	0.882124	0.441062	+	0.897477i	$0.354602\pi$
$194$	845.416	0.312873
$195$	0	0
$196$	2023.92	0.737582
$197$	2812.15	1.01704	0.508522	−	0.861049i	$-0.330192\pi$
$197$	2812.15	1.01704	0.508522	+	0.861049i	$0.330192\pi$
$198$	0	0
$199$	−2808.88	−1.00059	−0.500293	−	0.865856i	$-0.666774\pi$
$199$	−2808.88	−1.00059	−0.500293	+	0.865856i	$0.666774\pi$
$200$	−1413.87	−0.499880
$201$	0	0
$202$	−299.502	−0.104321
$203$	−4939.64	−1.70785
$204$	0	0
$205$	6019.20	2.05073
$206$	1192.40	0.403295
$207$	0	0
$208$	−606.648	−0.202228
$209$	293.935	0.0972818
$210$	0	0
$211$	−630.688	−0.205774	−0.102887	−	0.994693i	$-0.532808\pi$
$211$	−630.688	−0.205774	−0.102887	+	0.994693i	$0.532808\pi$
$212$	2056.02	0.666074
$213$	0	0
$214$	−4226.37	−1.35004
$215$	−464.995	−0.147500
$216$	0	0
$217$	3435.63	1.07477
$218$	−1979.35	−0.614946
$219$	0	0
$220$	2380.32	0.729461
$221$	−3219.03	−0.979799
$222$	0	0
$223$	−6344.93	−1.90533	−0.952664	−	0.304026i	$-0.901669\pi$
$223$	−6344.93	−1.90533	−0.952664	+	0.304026i	$0.901669\pi$
$224$	−932.393	−0.278117
$225$	0	0
$226$	2780.73	0.818459
$227$	3146.10	0.919885	0.459942	−	0.887949i	$-0.347870\pi$
$227$	3146.10	0.919885	0.459942	+	0.887949i	$0.347870\pi$
$228$	0	0
$229$	−5682.84	−1.63988	−0.819940	−	0.572450i	$-0.805993\pi$
$229$	−5682.84	−1.63988	−0.819940	+	0.572450i	$0.805993\pi$
$230$	3582.46	1.02705
$231$	0	0
$232$	1356.24	0.383799
$233$	−4997.43	−1.40512	−0.702560	−	0.711625i	$-0.747960\pi$
$233$	−4997.43	−1.40512	−0.702560	+	0.711625i	$0.747960\pi$
$234$	0	0
$235$	−8201.23	−2.27655
$236$	−236.000	−0.0650945
$237$	0	0
$238$	−4947.52	−1.34748
$239$	5519.90	1.49394	0.746972	−	0.664856i	$-0.231507\pi$
$239$	5519.90	1.49394	0.746972	+	0.664856i	$0.231507\pi$
$240$	0	0
$241$	−4940.83	−1.32061	−0.660304	−	0.750998i	$-0.729573\pi$
$241$	−4940.83	−1.32061	−0.660304	+	0.750998i	$0.729573\pi$
$242$	314.762	0.0836102
$243$	0	0
$244$	535.208	0.140423
$245$	−8789.15	−2.29191
$246$	0	0
$247$	325.315	0.0838028
$248$	−943.294	−0.241529
$249$	0	0
$250$	1797.30	0.454684
$251$	−2797.43	−0.703474	−0.351737	−	0.936099i	$-0.614409\pi$
$251$	−2797.43	−0.703474	−0.351737	+	0.936099i	$0.614409\pi$
$252$	0	0
$253$	−3532.67	−0.877853
$254$	1407.13	0.347603
$255$	0	0
$256$	256.000	0.0625000
$257$	3940.27	0.956370	0.478185	−	0.878259i	$-0.341295\pi$
$257$	3940.27	0.956370	0.478185	+	0.878259i	$0.341295\pi$
$258$	0	0
$259$	−5060.06	−1.21396
$260$	2634.44	0.628389
$261$	0	0
$262$	5177.49	1.22086
$263$	−6130.23	−1.43728	−0.718642	−	0.695380i	$-0.755236\pi$
$263$	−6130.23	−1.43728	−0.718642	+	0.695380i	$0.755236\pi$
$264$	0	0
$265$	−8928.50	−2.06971
$266$	499.996	0.115251
$267$	0	0
$268$	−1067.00	−0.243198
$269$	2577.21	0.584146	0.292073	−	0.956396i	$-0.405655\pi$
$269$	2577.21	0.584146	0.292073	+	0.956396i	$0.405655\pi$
$270$	0	0
$271$	632.343	0.141742	0.0708710	−	0.997485i	$-0.477422\pi$
$271$	632.343	0.141742	0.0708710	+	0.997485i	$0.477422\pi$
$272$	1358.40	0.302814
$273$	0	0
$274$	−2918.97	−0.643581
$275$	−6054.58	−1.32765
$276$	0	0
$277$	4444.11	0.963973	0.481986	−	0.876179i	$-0.339916\pi$
$277$	4444.11	0.963973	0.481986	+	0.876179i	$0.339916\pi$
$278$	5682.67	1.22598
$279$	0	0
$280$	4049.03	0.864200
$281$	702.072	0.149047	0.0745233	−	0.997219i	$-0.476256\pi$
$281$	702.072	0.149047	0.0745233	+	0.997219i	$0.476256\pi$
$282$	0	0
$283$	8489.01	1.78311	0.891553	−	0.452916i	$-0.149616\pi$
$283$	8489.01	1.78311	0.891553	+	0.452916i	$0.149616\pi$
$284$	−891.309	−0.186230
$285$	0	0
$286$	−2597.83	−0.537107
$287$	−10096.6	−2.07660
$288$	0	0
$289$	2295.05	0.467138
$290$	−5889.63	−1.19259
$291$	0	0
$292$	3803.28	0.762227
$293$	−5446.22	−1.08591	−0.542955	−	0.839762i	$-0.682694\pi$
$293$	−5446.22	−1.08591	−0.542955	+	0.839762i	$0.682694\pi$
$294$	0	0
$295$	1024.86	0.202270
$296$	1389.30	0.272809
$297$	0	0
$298$	3630.29	0.705695
$299$	−3909.81	−0.756221
$300$	0	0
$301$	779.983	0.149360
$302$	484.698	0.0923551
$303$	0	0
$304$	−137.280	−0.0258998
$305$	−2324.21	−0.436340
$306$	0	0
$307$	−3753.63	−0.697822	−0.348911	−	0.937156i	$-0.613448\pi$
$307$	−3753.63	−0.697822	−0.348911	+	0.937156i	$0.613448\pi$
$308$	−3992.75	−0.738663
$309$	0	0
$310$	4096.37	0.750510
$311$	−9088.46	−1.65710	−0.828552	−	0.559912i	$-0.810835\pi$
$311$	−9088.46	−1.65710	−0.828552	+	0.559912i	$0.810835\pi$
$312$	0	0
$313$	8278.14	1.49491	0.747457	−	0.664310i	$-0.231275\pi$
$313$	8278.14	1.49491	0.747457	+	0.664310i	$0.231275\pi$
$314$	−774.881	−0.139264
$315$	0	0
$316$	−4867.91	−0.866586
$317$	8949.11	1.58559	0.792795	−	0.609488i	$-0.208625\pi$
$317$	8949.11	1.58559	0.792795	+	0.609488i	$0.208625\pi$
$318$	0	0
$319$	5807.77	1.01935
$320$	−1111.71	−0.194208
$321$	0	0
$322$	−6009.22	−1.04000
$323$	−728.444	−0.125485
$324$	0	0
$325$	−6700.96	−1.14370
$326$	5316.83	0.903288
$327$	0	0
$328$	2772.15	0.466666
$329$	13756.8	2.30527
$330$	0	0
$331$	3316.62	0.550749	0.275374	−	0.961337i	$-0.411198\pi$
$331$	3316.62	0.550749	0.275374	+	0.961337i	$0.411198\pi$
$332$	−734.158	−0.121362
$333$	0	0
$334$	4978.34	0.815577
$335$	4633.56	0.755697
$336$	0	0
$337$	−7369.02	−1.19115	−0.595573	−	0.803301i	$-0.703075\pi$
$337$	−7369.02	−1.19115	−0.595573	+	0.803301i	$0.703075\pi$
$338$	1518.83	0.244419
$339$	0	0
$340$	−5899.04	−0.940942
$341$	−4039.43	−0.641489
$342$	0	0
$343$	4748.83	0.747559
$344$	−214.154	−0.0335651
$345$	0	0
$346$	6498.21	1.00967
$347$	4795.39	0.741873	0.370936	−	0.928658i	$-0.379037\pi$
$347$	4795.39	0.741873	0.370936	+	0.928658i	$0.379037\pi$
$348$	0	0
$349$	4348.08	0.666897	0.333449	−	0.942768i	$-0.391788\pi$
$349$	4348.08	0.666897	0.333449	+	0.942768i	$0.391788\pi$
$350$	−10299.1	−1.57289
$351$	0	0
$352$	1096.26	0.165997
$353$	5661.33	0.853605	0.426802	−	0.904345i	$-0.359640\pi$
$353$	5661.33	0.853605	0.426802	+	0.904345i	$0.359640\pi$
$354$	0	0
$355$	3870.62	0.578679
$356$	−2619.70	−0.390011
$357$	0	0
$358$	5880.30	0.868110
$359$	−4745.70	−0.697683	−0.348842	−	0.937182i	$-0.613425\pi$
$359$	−4745.70	−0.697683	−0.348842	+	0.937182i	$0.613425\pi$
$360$	0	0
$361$	−6785.38	−0.989267
$362$	3579.40	0.519693
$363$	0	0
$364$	−4419.02	−0.636317
$365$	−16516.2	−2.36849
$366$	0	0
$367$	−6047.76	−0.860192	−0.430096	−	0.902783i	$-0.641520\pi$
$367$	−6047.76	−0.860192	−0.430096	+	0.902783i	$0.641520\pi$
$368$	1649.91	0.233715
$369$	0	0
$370$	−6033.22	−0.847708
$371$	14976.7	2.09582
$372$	0	0
$373$	−1108.71	−0.153906	−0.0769529	−	0.997035i	$-0.524519\pi$
$373$	−1108.71	−0.153906	−0.0769529	+	0.997035i	$0.524519\pi$
$374$	5817.05	0.804257
$375$	0	0
$376$	−3777.09	−0.518054
$377$	6427.80	0.878113
$378$	0	0
$379$	−7872.26	−1.06694	−0.533471	−	0.845818i	$-0.679113\pi$
$379$	−7872.26	−1.06694	−0.533471	+	0.845818i	$0.679113\pi$
$380$	596.156	0.0804793
$381$	0	0
$382$	−1069.89	−0.143299
$383$	−9968.18	−1.32990	−0.664948	−	0.746890i	$-0.731546\pi$
$383$	−9968.18	−1.32990	−0.664948	+	0.746890i	$0.731546\pi$
$384$	0	0
$385$	17339.0	2.29527
$386$	−4730.37	−0.623756
$387$	0	0
$388$	−1690.83	−0.221235
$389$	−14035.8	−1.82942	−0.914708	−	0.404115i	$-0.867579\pi$
$389$	−14035.8	−1.82942	−0.914708	+	0.404115i	$0.867579\pi$
$390$	0	0
$391$	8754.83	1.13236
$392$	−4047.85	−0.521549
$393$	0	0
$394$	−5624.31	−0.719158
$395$	21139.5	2.69277
$396$	0	0
$397$	13638.0	1.72411	0.862056	−	0.506813i	$-0.169176\pi$
$397$	13638.0	1.72411	0.862056	+	0.506813i	$0.169176\pi$
$398$	5617.77	0.707520
$399$	0	0
$400$	2827.75	0.353468
$401$	4100.61	0.510660	0.255330	−	0.966854i	$-0.417816\pi$
$401$	4100.61	0.510660	0.255330	+	0.966854i	$0.417816\pi$
$402$	0	0
$403$	−4470.68	−0.552606
$404$	599.004	0.0737662
$405$	0	0
$406$	9879.27	1.20764
$407$	5949.36	0.724567
$408$	0	0
$409$	336.584	0.0406920	0.0203460	−	0.999793i	$-0.493523\pi$
$409$	336.584	0.0406920	0.0203460	+	0.999793i	$0.493523\pi$
$410$	−12038.4	−1.45008
$411$	0	0
$412$	−2384.81	−0.285173
$413$	−1719.10	−0.204822
$414$	0	0
$415$	3188.17	0.377112
$416$	1213.30	0.142997
$417$	0	0
$418$	−587.869	−0.0687886
$419$	−13335.7	−1.55488	−0.777439	−	0.628959i	$-0.783481\pi$
$419$	−13335.7	−1.55488	−0.777439	+	0.628959i	$0.783481\pi$
$420$	0	0
$421$	2903.76	0.336154	0.168077	−	0.985774i	$-0.446244\pi$
$421$	2903.76	0.336154	0.168077	+	0.985774i	$0.446244\pi$
$422$	1261.38	0.145504
$423$	0	0
$424$	−4112.03	−0.470986
$425$	15004.8	1.71256
$426$	0	0
$427$	3898.62	0.441845
$428$	8452.73	0.954622
$429$	0	0
$430$	929.990	0.104298
$431$	9141.46	1.02164	0.510822	−	0.859686i	$-0.329341\pi$
$431$	9141.46	1.02164	0.510822	+	0.859686i	$0.329341\pi$
$432$	0	0
$433$	1104.62	0.122597	0.0612985	−	0.998119i	$-0.480476\pi$
$433$	1104.62	0.122597	0.0612985	+	0.998119i	$0.480476\pi$
$434$	−6871.25	−0.759979
$435$	0	0
$436$	3958.69	0.434833
$437$	−884.762	−0.0968510
$438$	0	0
$439$	5075.02	0.551748	0.275874	−	0.961194i	$-0.411033\pi$
$439$	5075.02	0.551748	0.275874	+	0.961194i	$0.411033\pi$
$440$	−4760.65	−0.515807
$441$	0	0
$442$	6438.07	0.692823
$443$	−7830.48	−0.839814	−0.419907	−	0.907567i	$-0.637937\pi$
$443$	−7830.48	−0.839814	−0.419907	+	0.907567i	$0.637937\pi$
$444$	0	0
$445$	11376.4	1.21189
$446$	12689.9	1.34727
$447$	0	0
$448$	1864.79	0.196658
$449$	14983.6	1.57488	0.787438	−	0.616393i	$-0.211407\pi$
$449$	14983.6	1.57488	0.787438	+	0.616393i	$0.211407\pi$
$450$	0	0
$451$	11871.1	1.23944
$452$	−5561.47	−0.578738
$453$	0	0
$454$	−6292.19	−0.650457
$455$	19190.1	1.97725
$456$	0	0
$457$	6129.96	0.627456	0.313728	−	0.949513i	$-0.398422\pi$
$457$	6129.96	0.627456	0.313728	+	0.949513i	$0.398422\pi$
$458$	11365.7	1.15957
$459$	0	0
$460$	−7164.92	−0.726231
$461$	−45.9092	−0.00463818	−0.00231909	−	0.999997i	$-0.500738\pi$
$461$	−45.9092	−0.00463818	−0.00231909	+	0.999997i	$0.500738\pi$
$462$	0	0
$463$	12471.8	1.25186	0.625931	−	0.779878i	$-0.284719\pi$
$463$	12471.8	1.25186	0.625931	+	0.779878i	$0.284719\pi$
$464$	−2712.48	−0.271387
$465$	0	0
$466$	9994.87	0.993569
$467$	16130.4	1.59835	0.799173	−	0.601100i	$-0.205271\pi$
$467$	16130.4	1.59835	0.799173	+	0.601100i	$0.205271\pi$
$468$	0	0
$469$	−7772.34	−0.765231
$470$	16402.5	1.60976
$471$	0	0
$472$	472.000	0.0460287
$473$	−917.064	−0.0891472
$474$	0	0
$475$	−1516.38	−0.146476
$476$	9895.05	0.952813
$477$	0	0
$478$	−11039.8	−1.05638
$479$	20631.2	1.96798	0.983992	−	0.178214i	$-0.0570319\pi$
$479$	20631.2	1.96798	0.983992	+	0.178214i	$0.0570319\pi$
$480$	0	0
$481$	6584.51	0.624174
$482$	9881.65	0.933811
$483$	0	0
$484$	−629.524	−0.0591213
$485$	7342.65	0.687449
$486$	0	0
$487$	−8705.95	−0.810070	−0.405035	−	0.914301i	$-0.632741\pi$
$487$	−8705.95	−0.810070	−0.405035	+	0.914301i	$0.632741\pi$
$488$	−1070.42	−0.0992939
$489$	0	0
$490$	17578.3	1.62062
$491$	12174.3	1.11898	0.559490	−	0.828837i	$-0.310997\pi$
$491$	12174.3	1.11898	0.559490	+	0.828837i	$0.310997\pi$
$492$	0	0
$493$	−14393.1	−1.31487
$494$	−650.630	−0.0592575
$495$	0	0
$496$	1886.59	0.170787
$497$	−6492.58	−0.585980
$498$	0	0
$499$	84.8554	0.00761253	0.00380626	−	0.999993i	$-0.498788\pi$
$499$	84.8554	0.00761253	0.00380626	+	0.999993i	$0.498788\pi$
$500$	−3594.60	−0.321510
$501$	0	0
$502$	5594.85	0.497431
$503$	10415.0	0.923220	0.461610	−	0.887083i	$-0.347272\pi$
$503$	10415.0	0.923220	0.461610	+	0.887083i	$0.347272\pi$
$504$	0	0
$505$	−2601.25	−0.229216
$506$	7065.33	0.620736
$507$	0	0
$508$	−2814.26	−0.245792
$509$	−14229.7	−1.23914	−0.619568	−	0.784943i	$-0.712692\pi$
$509$	−14229.7	−1.23914	−0.619568	+	0.784943i	$0.712692\pi$
$510$	0	0
$511$	27704.3	2.39837
$512$	−512.000	−0.0441942
$513$	0	0
$514$	−7880.54	−0.676256
$515$	10356.3	0.886125
$516$	0	0
$517$	−16174.5	−1.37592
$518$	10120.1	0.858403
$519$	0	0
$520$	−5268.89	−0.444338
$521$	−7782.48	−0.654427	−0.327213	−	0.944950i	$-0.606110\pi$
$521$	−7782.48	−0.654427	−0.327213	+	0.944950i	$0.606110\pi$
$522$	0	0
$523$	−8333.56	−0.696752	−0.348376	−	0.937355i	$-0.613267\pi$
$523$	−8333.56	−0.696752	−0.348376	+	0.937355i	$0.613267\pi$
$524$	−10355.0	−0.863281
$525$	0	0
$526$	12260.5	1.01631
$527$	10010.7	0.827466
$528$	0	0
$529$	−1533.46	−0.126034
$530$	17857.0	1.46351
$531$	0	0
$532$	−999.992	−0.0814946
$533$	13138.4	1.06771
$534$	0	0
$535$	−36707.0	−2.96632
$536$	2133.99	0.171967
$537$	0	0
$538$	−5154.42	−0.413054
$539$	−17334.0	−1.38521
$540$	0	0
$541$	−21941.3	−1.74368	−0.871840	−	0.489791i	$-0.837073\pi$
$541$	−21941.3	−1.74368	−0.871840	+	0.489791i	$0.837073\pi$
$542$	−1264.69	−0.100227
$543$	0	0
$544$	−2716.81	−0.214122
$545$	−17191.1	−1.35117
$546$	0	0
$547$	−5699.52	−0.445510	−0.222755	−	0.974874i	$-0.571505\pi$
$547$	−5699.52	−0.445510	−0.222755	+	0.974874i	$0.571505\pi$
$548$	5837.93	0.455080
$549$	0	0
$550$	12109.2	0.938794
$551$	1454.57	0.112462
$552$	0	0
$553$	−35459.4	−2.72674
$554$	−8888.21	−0.681632
$555$	0	0
$556$	−11365.3	−0.866902
$557$	−15408.0	−1.17210	−0.586048	−	0.810277i	$-0.699317\pi$
$557$	−15408.0	−1.17210	−0.586048	+	0.810277i	$0.699317\pi$
$558$	0	0
$559$	−1014.97	−0.0767953
$560$	−8098.07	−0.611082
$561$	0	0
$562$	−1404.14	−0.105392
$563$	3619.15	0.270922	0.135461	−	0.990783i	$-0.456748\pi$
$563$	3619.15	0.270922	0.135461	+	0.990783i	$0.456748\pi$
$564$	0	0
$565$	24151.4	1.79833
$566$	−16978.0	−1.26085
$567$	0	0
$568$	1782.62	0.131685
$569$	−1437.31	−0.105896	−0.0529481	−	0.998597i	$-0.516862\pi$
$569$	−1437.31	−0.105896	−0.0529481	+	0.998597i	$0.516862\pi$
$570$	0	0
$571$	−10974.4	−0.804315	−0.402157	−	0.915571i	$-0.631740\pi$
$571$	−10974.4	−0.804315	−0.402157	+	0.915571i	$0.631740\pi$
$572$	5195.65	0.379792
$573$	0	0
$574$	20193.2	1.46838
$575$	18224.7	1.32178
$576$	0	0
$577$	20053.9	1.44689	0.723446	−	0.690381i	$-0.242557\pi$
$577$	20053.9	1.44689	0.723446	+	0.690381i	$0.242557\pi$
$578$	−4590.10	−0.330317
$579$	0	0
$580$	11779.3	0.843288
$581$	−5347.84	−0.381869
$582$	0	0
$583$	−17608.8	−1.25091
$584$	−7606.57	−0.538976
$585$	0	0
$586$	10892.4	0.767854
$587$	15322.1	1.07736	0.538681	−	0.842510i	$-0.318923\pi$
$587$	15322.1	1.07736	0.538681	+	0.842510i	$0.318923\pi$
$588$	0	0
$589$	−1011.68	−0.0707736
$590$	−2049.72	−0.143026
$591$	0	0
$592$	−2778.60	−0.192905
$593$	−15281.1	−1.05821	−0.529105	−	0.848556i	$-0.677472\pi$
$593$	−15281.1	−1.05821	−0.529105	+	0.848556i	$0.677472\pi$
$594$	0	0
$595$	−42970.5	−2.96070
$596$	−7260.58	−0.499001
$597$	0	0
$598$	7819.62	0.534729
$599$	5682.42	0.387608	0.193804	−	0.981040i	$-0.437917\pi$
$599$	5682.42	0.387608	0.193804	+	0.981040i	$0.437917\pi$
$600$	0	0
$601$	−1423.57	−0.0966202	−0.0483101	−	0.998832i	$-0.515384\pi$
$601$	−1423.57	−0.0966202	−0.0483101	+	0.998832i	$0.515384\pi$
$602$	−1559.97	−0.105614
$603$	0	0
$604$	−969.396	−0.0653049
$605$	2733.79	0.183709
$606$	0	0
$607$	8125.76	0.543352	0.271676	−	0.962389i	$-0.412422\pi$
$607$	8125.76	0.543352	0.271676	+	0.962389i	$0.412422\pi$
$608$	274.560	0.0183139
$609$	0	0
$610$	4648.41	0.308539
$611$	−17901.3	−1.18528
$612$	0	0
$613$	19879.4	1.30982	0.654910	−	0.755707i	$-0.272706\pi$
$613$	19879.4	1.30982	0.654910	+	0.755707i	$0.272706\pi$
$614$	7507.27	0.493434
$615$	0	0
$616$	7985.51	0.522314
$617$	−18716.4	−1.22122	−0.610611	−	0.791931i	$-0.709076\pi$
$617$	−18716.4	−1.22122	−0.610611	+	0.791931i	$0.709076\pi$
$618$	0	0
$619$	24093.4	1.56445	0.782225	−	0.622996i	$-0.214085\pi$
$619$	24093.4	1.56445	0.782225	+	0.622996i	$0.214085\pi$
$620$	−8192.74	−0.530691
$621$	0	0
$622$	18176.9	1.17175
$623$	−19082.7	−1.22718
$624$	0	0
$625$	−6481.79	−0.414835
$626$	−16556.3	−1.05706
$627$	0	0
$628$	1549.76	0.0984748
$629$	−14744.0	−0.934630
$630$	0	0
$631$	22921.0	1.44607	0.723036	−	0.690810i	$-0.242746\pi$
$631$	22921.0	1.44607	0.723036	+	0.690810i	$0.242746\pi$
$632$	9735.81	0.612769
$633$	0	0
$634$	−17898.2	−1.12118
$635$	12221.3	0.763758
$636$	0	0
$637$	−19184.5	−1.19328
$638$	−11615.5	−0.720789
$639$	0	0
$640$	2223.42	0.137326
$641$	−29002.5	−1.78710	−0.893550	−	0.448964i	$-0.851793\pi$
$641$	−29002.5	−1.78710	−0.893550	+	0.448964i	$0.851793\pi$
$642$	0	0
$643$	2024.75	0.124181	0.0620904	−	0.998071i	$-0.480223\pi$
$643$	2024.75	0.124181	0.0620904	+	0.998071i	$0.480223\pi$
$644$	12018.4	0.735393
$645$	0	0
$646$	1456.89	0.0887314
$647$	−8460.57	−0.514095	−0.257047	−	0.966399i	$-0.582750\pi$
$647$	−8460.57	−0.514095	−0.257047	+	0.966399i	$0.582750\pi$
$648$	0	0
$649$	2021.23	0.122250
$650$	13401.9	0.808718
$651$	0	0
$652$	−10633.7	−0.638721
$653$	15136.5	0.907103	0.453551	−	0.891230i	$-0.350157\pi$
$653$	15136.5	0.907103	0.453551	+	0.891230i	$0.350157\pi$
$654$	0	0
$655$	44967.8	2.68250
$656$	−5544.30	−0.329983
$657$	0	0
$658$	−27513.5	−1.63007
$659$	−7902.37	−0.467121	−0.233560	−	0.972342i	$-0.575038\pi$
$659$	−7902.37	−0.467121	−0.233560	+	0.972342i	$0.575038\pi$
$660$	0	0
$661$	6588.47	0.387688	0.193844	−	0.981032i	$-0.437904\pi$
$661$	6588.47	0.387688	0.193844	+	0.981032i	$0.437904\pi$
$662$	−6633.24	−0.389438
$663$	0	0
$664$	1468.32	0.0858159
$665$	4342.59	0.253231
$666$	0	0
$667$	−17481.8	−1.01484
$668$	−9956.68	−0.576700
$669$	0	0
$670$	−9267.13	−0.534359
$671$	−4583.80	−0.263719
$672$	0	0
$673$	−19281.7	−1.10439	−0.552194	−	0.833716i	$-0.686209\pi$
$673$	−19281.7	−1.10439	−0.552194	+	0.833716i	$0.686209\pi$
$674$	14738.0	0.842267
$675$	0	0
$676$	−3037.66	−0.172830
$677$	8752.86	0.496898	0.248449	−	0.968645i	$-0.420079\pi$
$677$	8752.86	0.496898	0.248449	+	0.968645i	$0.420079\pi$
$678$	0	0
$679$	−12316.6	−0.696121
$680$	11798.1	0.665346
$681$	0	0
$682$	8078.87	0.453601
$683$	−3472.38	−0.194534	−0.0972671	−	0.995258i	$-0.531010\pi$
$683$	−3472.38	−0.194534	−0.0972671	+	0.995258i	$0.531010\pi$
$684$	0	0
$685$	−25351.9	−1.41408
$686$	−9497.66	−0.528604
$687$	0	0
$688$	428.308	0.0237341
$689$	−19488.7	−1.07759
$690$	0	0
$691$	13653.8	0.751687	0.375843	−	0.926683i	$-0.377353\pi$
$691$	13653.8	0.751687	0.375843	+	0.926683i	$0.377353\pi$
$692$	−12996.4	−0.713945
$693$	0	0
$694$	−9590.77	−0.524583
$695$	49355.4	2.69375
$696$	0	0
$697$	−29419.5	−1.59877
$698$	−8696.15	−0.471568
$699$	0	0
$700$	20598.2	1.11220
$701$	17065.6	0.919483	0.459741	−	0.888053i	$-0.347942\pi$
$701$	17065.6	0.919483	0.459741	+	0.888053i	$0.347942\pi$
$702$	0	0
$703$	1490.03	0.0799394
$704$	−2192.52	−0.117377
$705$	0	0
$706$	−11322.7	−0.603590
$707$	4363.34	0.232108
$708$	0	0
$709$	19602.7	1.03836	0.519179	−	0.854666i	$-0.326238\pi$
$709$	19602.7	1.03836	0.519179	+	0.854666i	$0.326238\pi$
$710$	−7741.24	−0.409188
$711$	0	0
$712$	5239.40	0.275779
$713$	12158.9	0.638648
$714$	0	0
$715$	−22562.8	−1.18014
$716$	−11760.6	−0.613847
$717$	0	0
$718$	9491.40	0.493337
$719$	−28943.2	−1.50125	−0.750624	−	0.660729i	$-0.770247\pi$
$719$	−28943.2	−1.50125	−0.750624	+	0.660729i	$0.770247\pi$
$720$	0	0
$721$	−17371.7	−0.897304
$722$	13570.8	0.699518
$723$	0	0
$724$	−7158.80	−0.367479
$725$	−29961.7	−1.53483
$726$	0	0
$727$	−4459.41	−0.227497	−0.113749	−	0.993510i	$-0.536286\pi$
$727$	−4459.41	−0.227497	−0.113749	+	0.993510i	$0.536286\pi$
$728$	8838.03	0.449944
$729$	0	0
$730$	33032.5	1.67478
$731$	2272.71	0.114992
$732$	0	0
$733$	−2564.77	−0.129239	−0.0646194	−	0.997910i	$-0.520583\pi$
$733$	−2564.77	−0.129239	−0.0646194	+	0.997910i	$0.520583\pi$
$734$	12095.5	0.608248
$735$	0	0
$736$	−3299.81	−0.165262
$737$	9138.32	0.456736
$738$	0	0
$739$	9845.29	0.490074	0.245037	−	0.969514i	$-0.421200\pi$
$739$	9845.29	0.490074	0.245037	+	0.969514i	$0.421200\pi$
$740$	12066.4	0.599420
$741$	0	0
$742$	−29953.4	−1.48197
$743$	37299.7	1.84171	0.920856	−	0.389903i	$-0.127492\pi$
$743$	37299.7	1.84171	0.920856	+	0.389903i	$0.127492\pi$
$744$	0	0
$745$	31530.0	1.55056
$746$	2217.42	0.108828
$747$	0	0
$748$	−11634.1	−0.568696
$749$	61572.4	3.00375
$750$	0	0
$751$	−43.6748	−0.00212212	−0.00106106	−	0.999999i	$-0.500338\pi$
$751$	−43.6748	−0.00212212	−0.00106106	+	0.999999i	$0.500338\pi$
$752$	7554.17	0.366320
$753$	0	0
$754$	−12855.6	−0.620920
$755$	4209.72	0.202924
$756$	0	0
$757$	781.640	0.0375286	0.0187643	−	0.999824i	$-0.494027\pi$
$757$	781.640	0.0375286	0.0187643	+	0.999824i	$0.494027\pi$
$758$	15744.5	0.754442
$759$	0	0
$760$	−1192.31	−0.0569075
$761$	−26687.0	−1.27123	−0.635613	−	0.772008i	$-0.719253\pi$
$761$	−26687.0	−1.27123	−0.635613	+	0.772008i	$0.719253\pi$
$762$	0	0
$763$	28836.4	1.36821
$764$	2139.78	0.101328
$765$	0	0
$766$	19936.4	0.940378
$767$	2237.01	0.105311
$768$	0	0
$769$	25366.1	1.18950	0.594750	−	0.803910i	$-0.297251\pi$
$769$	25366.1	1.18950	0.594750	+	0.803910i	$0.297251\pi$
$770$	−34678.1	−1.62300
$771$	0	0
$772$	9460.75	0.441062
$773$	−34042.0	−1.58397	−0.791983	−	0.610543i	$-0.790951\pi$
$773$	−34042.0	−1.58397	−0.791983	+	0.610543i	$0.790951\pi$
$774$	0	0
$775$	20839.0	0.965884
$776$	3381.67	0.156436
$777$	0	0
$778$	28071.6	1.29359
$779$	2973.13	0.136744
$780$	0	0
$781$	7633.64	0.349748
$782$	−17509.7	−0.800696
$783$	0	0
$784$	8095.70	0.368791
$785$	−6730.03	−0.305994
$786$	0	0
$787$	29184.8	1.32189	0.660944	−	0.750435i	$-0.270156\pi$
$787$	29184.8	1.32189	0.660944	+	0.750435i	$0.270156\pi$
$788$	11248.6	0.508522
$789$	0	0
$790$	−42279.0	−1.90407
$791$	−40511.5	−1.82102
$792$	0	0
$793$	−5073.16	−0.227179
$794$	−27276.0	−1.21913
$795$	0	0
$796$	−11235.5	−0.500293
$797$	−34093.0	−1.51523	−0.757613	−	0.652704i	$-0.773634\pi$
$797$	−34093.0	−1.51523	−0.757613	+	0.652704i	$0.773634\pi$
$798$	0	0
$799$	40084.4	1.77483
$800$	−5655.49	−0.249940
$801$	0	0
$802$	−8201.22	−0.361091
$803$	−32573.3	−1.43149
$804$	0	0
$805$	−52191.6	−2.28511
$806$	8941.36	0.390752
$807$	0	0
$808$	−1198.01	−0.0521606
$809$	−3117.30	−0.135474	−0.0677369	−	0.997703i	$-0.521578\pi$
$809$	−3117.30	−0.135474	−0.0677369	+	0.997703i	$0.521578\pi$
$810$	0	0
$811$	−1122.19	−0.0485887	−0.0242944	−	0.999705i	$-0.507734\pi$
$811$	−1122.19	−0.0485887	−0.0242944	+	0.999705i	$0.507734\pi$
$812$	−19758.5	−0.853927
$813$	0	0
$814$	−11898.7	−0.512346
$815$	46178.0	1.98472
$816$	0	0
$817$	−229.680	−0.00983536
$818$	−673.168	−0.0287736
$819$	0	0
$820$	24076.8	1.02536
$821$	−29677.1	−1.26156	−0.630779	−	0.775962i	$-0.717265\pi$
$821$	−29677.1	−1.26156	−0.630779	+	0.775962i	$0.717265\pi$
$822$	0	0
$823$	−27566.6	−1.16757	−0.583786	−	0.811908i	$-0.698429\pi$
$823$	−27566.6	−1.16757	−0.583786	+	0.811908i	$0.698429\pi$
$824$	4769.62	0.201647
$825$	0	0
$826$	3438.20	0.144831
$827$	−17934.9	−0.754122	−0.377061	−	0.926188i	$-0.623065\pi$
$827$	−17934.9	−0.754122	−0.377061	+	0.926188i	$0.623065\pi$
$828$	0	0
$829$	−7735.33	−0.324076	−0.162038	−	0.986785i	$-0.551807\pi$
$829$	−7735.33	−0.324076	−0.162038	+	0.986785i	$0.551807\pi$
$830$	−6376.35	−0.266658
$831$	0	0
$832$	−2426.59	−0.101114
$833$	42957.9	1.78680
$834$	0	0
$835$	43238.1	1.79200
$836$	1175.74	0.0486409
$837$	0	0
$838$	26671.5	1.09946
$839$	46742.8	1.92341	0.961704	−	0.274091i	$-0.0883769\pi$
$839$	46742.8	1.92341	0.961704	+	0.274091i	$0.0883769\pi$
$840$	0	0
$841$	4351.33	0.178414
$842$	−5807.53	−0.237697
$843$	0	0
$844$	−2522.75	−0.102887
$845$	13191.4	0.537041
$846$	0	0
$847$	−4585.65	−0.186027
$848$	8224.06	0.333037
$849$	0	0
$850$	−30009.6	−1.21096
$851$	−17907.9	−0.721359
$852$	0	0
$853$	−33867.5	−1.35944	−0.679719	−	0.733472i	$-0.737898\pi$
$853$	−33867.5	−1.35944	−0.679719	+	0.733472i	$0.737898\pi$
$854$	−7797.25	−0.312431
$855$	0	0
$856$	−16905.5	−0.675020
$857$	51.2174	0.00204149	0.00102074	−	0.999999i	$-0.499675\pi$
$857$	51.2174	0.00204149	0.00102074	+	0.999999i	$0.499675\pi$
$858$	0	0
$859$	11206.0	0.445103	0.222551	−	0.974921i	$-0.428561\pi$
$859$	11206.0	0.445103	0.222551	+	0.974921i	$0.428561\pi$
$860$	−1859.98	−0.0737498
$861$	0	0
$862$	−18282.9	−0.722411
$863$	−37668.2	−1.48579	−0.742896	−	0.669406i	$-0.766549\pi$
$863$	−37668.2	−1.48579	−0.742896	+	0.669406i	$0.766549\pi$
$864$	0	0
$865$	56438.6	2.21846
$866$	−2209.23	−0.0866891
$867$	0	0
$868$	13742.5	0.537386
$869$	41691.3	1.62748
$870$	0	0
$871$	10113.9	0.393452
$872$	−7917.39	−0.307473
$873$	0	0
$874$	1769.52	0.0684840
$875$	−26184.2	−1.01164
$876$	0	0
$877$	−20656.9	−0.795363	−0.397681	−	0.917524i	$-0.630185\pi$
$877$	−20656.9	−0.795363	−0.397681	+	0.917524i	$0.630185\pi$
$878$	−10150.0	−0.390145
$879$	0	0
$880$	9521.29	0.364730
$881$	17503.6	0.669367	0.334684	−	0.942331i	$-0.391371\pi$
$881$	17503.6	0.669367	0.334684	+	0.942331i	$0.391371\pi$
$882$	0	0
$883$	35401.3	1.34921	0.674603	−	0.738181i	$-0.264315\pi$
$883$	35401.3	1.34921	0.674603	+	0.738181i	$0.264315\pi$
$884$	−12876.1	−0.489900
$885$	0	0
$886$	15661.0	0.593838
$887$	17056.3	0.645654	0.322827	−	0.946458i	$-0.395367\pi$
$887$	17056.3	0.645654	0.322827	+	0.946458i	$0.395367\pi$
$888$	0	0
$889$	−20500.0	−0.773394
$890$	−22752.8	−0.856937
$891$	0	0
$892$	−25379.7	−0.952664
$893$	−4050.92	−0.151802
$894$	0	0
$895$	51071.9	1.90742
$896$	−3729.57	−0.139058
$897$	0	0
$898$	−29967.2	−1.11361
$899$	−19989.5	−0.741589
$900$	0	0
$901$	43639.1	1.61357
$902$	−23742.2	−0.876416
$903$	0	0
$904$	11122.9	0.409229
$905$	31088.0	1.14188
$906$	0	0
$907$	13277.4	0.486075	0.243037	−	0.970017i	$-0.421856\pi$
$907$	13277.4	0.486075	0.243037	+	0.970017i	$0.421856\pi$
$908$	12584.4	0.459942
$909$	0	0
$910$	−38380.3	−1.39812
$911$	9407.42	0.342132	0.171066	−	0.985260i	$-0.445279\pi$
$911$	9407.42	0.342132	0.171066	+	0.985260i	$0.445279\pi$
$912$	0	0
$913$	6287.72	0.227922
$914$	−12259.9	−0.443679
$915$	0	0
$916$	−22731.3	−0.819940
$917$	−75429.0	−2.71634
$918$	0	0
$919$	−39183.0	−1.40645	−0.703224	−	0.710968i	$-0.748257\pi$
$919$	−39183.0	−1.40645	−0.703224	+	0.710968i	$0.748257\pi$
$920$	14329.8	0.513523
$921$	0	0
$922$	91.8183	0.00327969
$923$	8448.60	0.301288
$924$	0	0
$925$	−30692.1	−1.09097
$926$	−24943.5	−0.885200
$927$	0	0
$928$	5424.95	0.191900
$929$	13093.0	0.462396	0.231198	−	0.972907i	$-0.425735\pi$
$929$	13093.0	0.462396	0.231198	+	0.972907i	$0.425735\pi$
$930$	0	0
$931$	−4341.32	−0.152826
$932$	−19989.7	−0.702560
$933$	0	0
$934$	−32260.9	−1.13020
$935$	50522.5	1.76713
$936$	0	0
$937$	37433.0	1.30510	0.652552	−	0.757744i	$-0.273699\pi$
$937$	37433.0	1.30510	0.652552	+	0.757744i	$0.273699\pi$
$938$	15544.7	0.541100
$939$	0	0
$940$	−32804.9	−1.13828
$941$	22450.6	0.777756	0.388878	−	0.921289i	$-0.372863\pi$
$941$	22450.6	0.777756	0.388878	+	0.921289i	$0.372863\pi$
$942$	0	0
$943$	−35732.7	−1.23395
$944$	−944.000	−0.0325472
$945$	0	0
$946$	1834.13	0.0630366
$947$	−39917.7	−1.36975	−0.684875	−	0.728661i	$-0.740143\pi$
$947$	−39917.7	−1.36975	−0.684875	+	0.728661i	$0.740143\pi$
$948$	0	0
$949$	−36050.8	−1.23315
$950$	3032.76	0.103574
$951$	0	0
$952$	−19790.1	−0.673740
$953$	−41081.5	−1.39639	−0.698195	−	0.715908i	$-0.746013\pi$
$953$	−41081.5	−1.39639	−0.698195	+	0.715908i	$0.746013\pi$
$954$	0	0
$955$	−9292.26	−0.314859
$956$	22079.6	0.746972
$957$	0	0
$958$	−41262.4	−1.39157
$959$	42525.4	1.43192
$960$	0	0
$961$	−15887.8	−0.533310
$962$	−13169.0	−0.441358
$963$	0	0
$964$	−19763.3	−0.660304
$965$	−41084.5	−1.37052
$966$	0	0
$967$	−28323.2	−0.941897	−0.470948	−	0.882161i	$-0.656088\pi$
$967$	−28323.2	−0.941897	−0.470948	+	0.882161i	$0.656088\pi$
$968$	1259.05	0.0418051
$969$	0	0
$970$	−14685.3	−0.486100
$971$	−32795.1	−1.08388	−0.541938	−	0.840418i	$-0.682309\pi$
$971$	−32795.1	−1.08388	−0.541938	+	0.840418i	$0.682309\pi$
$972$	0	0
$973$	−82788.7	−2.72773
$974$	17411.9	0.572806
$975$	0	0
$976$	2140.83	0.0702114
$977$	12694.6	0.415699	0.207849	−	0.978161i	$-0.433354\pi$
$977$	12694.6	0.415699	0.207849	+	0.978161i	$0.433354\pi$
$978$	0	0
$979$	22436.5	0.732456
$980$	−35156.6	−1.14595
$981$	0	0
$982$	−24348.6	−0.791238
$983$	−8087.36	−0.262408	−0.131204	−	0.991355i	$-0.541884\pi$
$983$	−8087.36	−0.262408	−0.131204	+	0.991355i	$0.541884\pi$
$984$	0	0
$985$	−48848.5	−1.58014
$986$	28786.2	0.929757
$987$	0	0
$988$	1301.26	0.0419014
$989$	2760.42	0.0887525
$990$	0	0
$991$	−25723.9	−0.824566	−0.412283	−	0.911056i	$-0.635269\pi$
$991$	−25723.9	−0.824566	−0.412283	+	0.911056i	$0.635269\pi$
$992$	−3773.17	−0.120765
$993$	0	0
$994$	12985.2	0.414350
$995$	48791.7	1.55457
$996$	0	0
$997$	33928.2	1.07775	0.538875	−	0.842386i	$-0.318849\pi$
$997$	33928.2	1.07775	0.538875	+	0.842386i	$0.318849\pi$
$998$	−169.711	−0.00538287
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1062.4.a.n.1.1		4
3.2	odd	2		354.4.a.h.1.4	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
354.4.a.h.1.4	✓	4	3.2	odd	2
1062.4.a.n.1.1		4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1062 = 2 \cdot 3^{2} \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1062.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +4.00000 q^{4} -17.3705 q^{5} +29.1373 q^{7} -8.00000 q^{8} +O(q^{10})\)	\(q-2.00000 q^{2} +4.00000 q^{4} -17.3705 q^{5} +29.1373 q^{7} -8.00000 q^{8} +34.7410 q^{10} -34.2581 q^{11} -37.9155 q^{13} -58.2746 q^{14} +16.0000 q^{16} +84.9002 q^{17} -8.58000 q^{19} -69.4820 q^{20} +68.5162 q^{22} +103.119 q^{23} +176.734 q^{25} +75.8310 q^{26} +116.549 q^{28} -169.530 q^{29} +117.912 q^{31} -32.0000 q^{32} -169.800 q^{34} -506.129 q^{35} -173.663 q^{37} +17.1600 q^{38} +138.964 q^{40} -346.519 q^{41} +26.7692 q^{43} -137.032 q^{44} -206.238 q^{46} +472.136 q^{47} +505.981 q^{49} -353.468 q^{50} -151.662 q^{52} +514.004 q^{53} +595.081 q^{55} -233.098 q^{56} +339.059 q^{58} -59.0000 q^{59} +133.802 q^{61} -235.823 q^{62} +64.0000 q^{64} +658.611 q^{65} -266.749 q^{67} +339.601 q^{68} +1012.26 q^{70} -222.827 q^{71} +950.821 q^{73} +347.325 q^{74} -34.3200 q^{76} -998.189 q^{77} -1216.98 q^{79} -277.928 q^{80} +693.037 q^{82} -183.540 q^{83} -1474.76 q^{85} -53.5385 q^{86} +274.065 q^{88} -654.925 q^{89} -1104.75 q^{91} +412.476 q^{92} -944.272 q^{94} +149.039 q^{95} -422.708 q^{97} -1011.96 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{2} + 16 q^{4} - 22 q^{5} + 13 q^{7} - 32 q^{8}+O(q^{10}) \) 4 * q - 8 * q^2 + 16 * q^4 - 22 * q^5 + 13 * q^7 - 32 * q^8	\( 4 q - 8 q^{2} + 16 q^{4} - 22 q^{5} + 13 q^{7} - 32 q^{8} + 44 q^{10} - 24 q^{11} + 20 q^{13} - 26 q^{14} + 64 q^{16} - 91 q^{17} + 141 q^{19} - 88 q^{20} + 48 q^{22} - 13 q^{23} + 278 q^{25} - 40 q^{26} + 52 q^{28} - 295 q^{29} + 311 q^{31} - 128 q^{32} + 182 q^{34} - 551 q^{35} + 609 q^{37} - 282 q^{38} + 176 q^{40} - 677 q^{41} + 170 q^{43} - 96 q^{44} + 26 q^{46} - 17 q^{47} + 651 q^{49} - 556 q^{50} + 80 q^{52} - 166 q^{53} + 108 q^{55} - 104 q^{56} + 590 q^{58} - 236 q^{59} + 651 q^{61} - 622 q^{62} + 256 q^{64} - 700 q^{65} - 894 q^{67} - 364 q^{68} + 1102 q^{70} - 298 q^{71} + 887 q^{73} - 1218 q^{74} + 564 q^{76} + 79 q^{77} - 784 q^{79} - 352 q^{80} + 1354 q^{82} - 971 q^{83} - 799 q^{85} - 340 q^{86} + 192 q^{88} - 1321 q^{89} - 2673 q^{91} - 52 q^{92} + 34 q^{94} + 3133 q^{95} - 1922 q^{97} - 1302 q^{98}+O(q^{100}) \) 4 * q - 8 * q^2 + 16 * q^4 - 22 * q^5 + 13 * q^7 - 32 * q^8 + 44 * q^10 - 24 * q^11 + 20 * q^13 - 26 * q^14 + 64 * q^16 - 91 * q^17 + 141 * q^19 - 88 * q^20 + 48 * q^22 - 13 * q^23 + 278 * q^25 - 40 * q^26 + 52 * q^28 - 295 * q^29 + 311 * q^31 - 128 * q^32 + 182 * q^34 - 551 * q^35 + 609 * q^37 - 282 * q^38 + 176 * q^40 - 677 * q^41 + 170 * q^43 - 96 * q^44 + 26 * q^46 - 17 * q^47 + 651 * q^49 - 556 * q^50 + 80 * q^52 - 166 * q^53 + 108 * q^55 - 104 * q^56 + 590 * q^58 - 236 * q^59 + 651 * q^61 - 622 * q^62 + 256 * q^64 - 700 * q^65 - 894 * q^67 - 364 * q^68 + 1102 * q^70 - 298 * q^71 + 887 * q^73 - 1218 * q^74 + 564 * q^76 + 79 * q^77 - 784 * q^79 - 352 * q^80 + 1354 * q^82 - 971 * q^83 - 799 * q^85 - 340 * q^86 + 192 * q^88 - 1321 * q^89 - 2673 * q^91 - 52 * q^92 + 34 * q^94 + 3133 * q^95 - 1922 * q^97 - 1302 * q^98

Introduction

L-functions

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Database

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