LMFDB - Embedded newform 567.4.a.i.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [567,4,Mod(1,567)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(567, 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("567.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 567 = 3^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	567.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:	$33.4540829733$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} - 49x^{6} + 138x^{5} + 708x^{4} - 1941x^{3} - 2506x^{2} + 8592x - 4616 $ x^8 - 3x^7 - 49x^6 + 138x^5 + 708x^4 - 1941x^3 - 2506x^2 + 8592*x - 4616
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2\cdot 3^{5} $
Twist minimal:	no (minimal twist has level 63)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-3.93417$ of defining polynomial
Character	$\chi$	$=$	567.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.93417 q^{2} +7.47771 q^{4} -2.43142 q^{5} -7.00000 q^{7} +2.05480 q^{8} +O(q^{10})$	\(q-3.93417 q^{2} +7.47771 q^{4} -2.43142 q^{5} -7.00000 q^{7} +2.05480 q^{8} +9.56561 q^{10} +42.1723 q^{11} -39.5731 q^{13} +27.5392 q^{14} -67.9056 q^{16} -2.07697 q^{17} +96.1013 q^{19} -18.1814 q^{20} -165.913 q^{22} +73.7508 q^{23} -119.088 q^{25} +155.688 q^{26} -52.3439 q^{28} +19.2076 q^{29} -239.497 q^{31} +250.714 q^{32} +8.17114 q^{34} +17.0199 q^{35} -144.310 q^{37} -378.079 q^{38} -4.99606 q^{40} -72.2818 q^{41} +480.549 q^{43} +315.352 q^{44} -290.148 q^{46} +294.707 q^{47} +49.0000 q^{49} +468.513 q^{50} -295.916 q^{52} -627.210 q^{53} -102.538 q^{55} -14.3836 q^{56} -75.5659 q^{58} +149.619 q^{59} -630.716 q^{61} +942.221 q^{62} -443.106 q^{64} +96.2188 q^{65} +4.04248 q^{67} -15.5309 q^{68} -66.9592 q^{70} -798.373 q^{71} +444.022 q^{73} +567.742 q^{74} +718.617 q^{76} -295.206 q^{77} -575.539 q^{79} +165.107 q^{80} +284.369 q^{82} +1290.23 q^{83} +5.04997 q^{85} -1890.56 q^{86} +86.6556 q^{88} +750.701 q^{89} +277.012 q^{91} +551.487 q^{92} -1159.43 q^{94} -233.662 q^{95} +419.548 q^{97} -192.774 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 3 q^{2} + 43 q^{4} + 30 q^{5} - 56 q^{7} + 6 q^{8}+O(q^{10}) $ 8 * q + 3 * q^2 + 43 * q^4 + 30 * q^5 - 56 * q^7 + 6 * q^8	$ 8 q + 3 q^{2} + 43 q^{4} + 30 q^{5} - 56 q^{7} + 6 q^{8} - 14 q^{10} + 24 q^{11} + 68 q^{13} - 21 q^{14} + 103 q^{16} + 168 q^{17} + 176 q^{19} + 330 q^{20} + 151 q^{22} + 228 q^{23} + 244 q^{25} + 795 q^{26} - 301 q^{28} + 618 q^{29} + 72 q^{31} + 786 q^{32} - 261 q^{34} - 210 q^{35} + 210 q^{37} + 1032 q^{38} - 375 q^{40} + 420 q^{41} - 2 q^{43} + 387 q^{44} + 402 q^{46} + 570 q^{47} + 392 q^{49} + 1110 q^{50} - 431 q^{52} + 528 q^{53} - 838 q^{55} - 42 q^{56} + 37 q^{58} - 150 q^{59} + 578 q^{61} + 1170 q^{62} - 112 q^{64} - 366 q^{65} - 898 q^{67} + 2526 q^{68} + 98 q^{70} + 882 q^{71} + 972 q^{73} - 222 q^{74} + 1423 q^{76} - 168 q^{77} - 158 q^{79} + 2475 q^{80} - 211 q^{82} + 2958 q^{83} - 774 q^{85} - 114 q^{86} + 1317 q^{88} + 4380 q^{89} - 476 q^{91} + 4629 q^{92} - 3234 q^{94} + 930 q^{95} - 60 q^{97} + 147 q^{98}+O(q^{100}) $ 8 * q + 3 * q^2 + 43 * q^4 + 30 * q^5 - 56 * q^7 + 6 * q^8 - 14 * q^10 + 24 * q^11 + 68 * q^13 - 21 * q^14 + 103 * q^16 + 168 * q^17 + 176 * q^19 + 330 * q^20 + 151 * q^22 + 228 * q^23 + 244 * q^25 + 795 * q^26 - 301 * q^28 + 618 * q^29 + 72 * q^31 + 786 * q^32 - 261 * q^34 - 210 * q^35 + 210 * q^37 + 1032 * q^38 - 375 * q^40 + 420 * q^41 - 2 * q^43 + 387 * q^44 + 402 * q^46 + 570 * q^47 + 392 * q^49 + 1110 * q^50 - 431 * q^52 + 528 * q^53 - 838 * q^55 - 42 * q^56 + 37 * q^58 - 150 * q^59 + 578 * q^61 + 1170 * q^62 - 112 * q^64 - 366 * q^65 - 898 * q^67 + 2526 * q^68 + 98 * q^70 + 882 * q^71 + 972 * q^73 - 222 * q^74 + 1423 * q^76 - 168 * q^77 - 158 * q^79 + 2475 * q^80 - 211 * q^82 + 2958 * q^83 - 774 * q^85 - 114 * q^86 + 1317 * q^88 + 4380 * q^89 - 476 * q^91 + 4629 * q^92 - 3234 * q^94 + 930 * q^95 - 60 * q^97 + 147 * 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 $

$2$	−3.93417	−1.39094	−0.695470	−	0.718555i	$-0.744804\pi$
$2$	−3.93417	−1.39094	−0.695470	+	0.718555i	$0.744804\pi$
$3$	0	0
$3$	0	0
$4$	7.47771	0.934713
$5$	−2.43142	−0.217472	−0.108736	−	0.994071i	$-0.534680\pi$
$5$	−2.43142	−0.217472	−0.108736	+	0.994071i	$0.534680\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	2.05480	0.0908100
$9$	0	0
$10$	9.56561	0.302491
$11$	42.1723	1.15595	0.577975	−	0.816055i	$-0.303843\pi$
$11$	42.1723	1.15595	0.577975	+	0.816055i	$0.303843\pi$
$12$	0	0
$13$	−39.5731	−0.844278	−0.422139	−	0.906531i	$-0.638721\pi$
$13$	−39.5731	−0.844278	−0.422139	+	0.906531i	$0.638721\pi$
$14$	27.5392	0.525726
$15$	0	0
$16$	−67.9056	−1.06102
$17$	−2.07697	−0.0296317	−0.0148158	−	0.999890i	$-0.504716\pi$
$17$	−2.07697	−0.0296317	−0.0148158	+	0.999890i	$0.504716\pi$
$18$	0	0
$19$	96.1013	1.16038	0.580188	−	0.814483i	$-0.302979\pi$
$19$	96.1013	1.16038	0.580188	+	0.814483i	$0.302979\pi$
$20$	−18.1814	−0.203274
$21$	0	0
$22$	−165.913	−1.60786
$23$	73.7508	0.668614	0.334307	−	0.942464i	$-0.391498\pi$
$23$	73.7508	0.668614	0.334307	+	0.942464i	$0.391498\pi$
$24$	0	0
$25$	−119.088	−0.952706
$26$	155.688	1.17434
$27$	0	0
$28$	−52.3439	−0.353288
$29$	19.2076	0.122992	0.0614958	−	0.998107i	$-0.480413\pi$
$29$	19.2076	0.122992	0.0614958	+	0.998107i	$0.480413\pi$
$30$	0	0
$31$	−239.497	−1.38758	−0.693788	−	0.720179i	$-0.744060\pi$
$31$	−239.497	−1.38758	−0.693788	+	0.720179i	$0.744060\pi$
$32$	250.714	1.38501
$33$	0	0
$34$	8.17114	0.0412159
$35$	17.0199	0.0821968
$36$	0	0
$37$	−144.310	−0.641202	−0.320601	−	0.947214i	$-0.603885\pi$
$37$	−144.310	−0.641202	−0.320601	+	0.947214i	$0.603885\pi$
$38$	−378.079	−1.61401
$39$	0	0
$40$	−4.99606	−0.0197487
$41$	−72.2818	−0.275330	−0.137665	−	0.990479i	$-0.543960\pi$
$41$	−72.2818	−0.275330	−0.137665	+	0.990479i	$0.543960\pi$
$42$	0	0
$43$	480.549	1.70426	0.852128	−	0.523334i	$-0.175312\pi$
$43$	480.549	1.70426	0.852128	+	0.523334i	$0.175312\pi$
$44$	315.352	1.08048
$45$	0	0
$46$	−290.148	−0.930001
$47$	294.707	0.914627	0.457313	−	0.889306i	$-0.348812\pi$
$47$	294.707	0.914627	0.457313	+	0.889306i	$0.348812\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	468.513	1.32516
$51$	0	0
$52$	−295.916	−0.789158
$53$	−627.210	−1.62555	−0.812773	−	0.582580i	$-0.802043\pi$
$53$	−627.210	−1.62555	−0.812773	+	0.582580i	$0.802043\pi$
$54$	0	0
$55$	−102.538	−0.251387
$56$	−14.3836	−0.0343230
$57$	0	0
$58$	−75.5659	−0.171074
$59$	149.619	0.330147	0.165074	−	0.986281i	$-0.447214\pi$
$59$	149.619	0.330147	0.165074	+	0.986281i	$0.447214\pi$
$60$	0	0
$61$	−630.716	−1.32385	−0.661926	−	0.749569i	$-0.730261\pi$
$61$	−630.716	−1.32385	−0.661926	+	0.749569i	$0.730261\pi$
$62$	942.221	1.93004
$63$	0	0
$64$	−443.106	−0.865442
$65$	96.2188	0.183607
$66$	0	0
$67$	4.04248	0.00737115	0.00368558	−	0.999993i	$-0.498827\pi$
$67$	4.04248	0.00737115	0.00368558	+	0.999993i	$0.498827\pi$
$68$	−15.5309	−0.0276971
$69$	0	0
$70$	−66.9592	−0.114331
$71$	−798.373	−1.33450	−0.667249	−	0.744835i	$-0.732528\pi$
$71$	−798.373	−1.33450	−0.667249	+	0.744835i	$0.732528\pi$
$72$	0	0
$73$	444.022	0.711902	0.355951	−	0.934505i	$-0.384157\pi$
$73$	444.022	0.711902	0.355951	+	0.934505i	$0.384157\pi$
$74$	567.742	0.891874
$75$	0	0
$76$	718.617	1.08462
$77$	−295.206	−0.436908
$78$	0	0
$79$	−575.539	−0.819660	−0.409830	−	0.912162i	$-0.634412\pi$
$79$	−575.539	−0.819660	−0.409830	+	0.912162i	$0.634412\pi$
$80$	165.107	0.230744
$81$	0	0
$82$	284.369	0.382967
$83$	1290.23	1.70629	0.853143	−	0.521678i	$-0.174694\pi$
$83$	1290.23	1.70629	0.853143	+	0.521678i	$0.174694\pi$
$84$	0	0
$85$	5.04997	0.00644407
$86$	−1890.56	−2.37052
$87$	0	0
$88$	86.6556	0.104972
$89$	750.701	0.894091	0.447046	−	0.894511i	$-0.352476\pi$
$89$	750.701	0.894091	0.447046	+	0.894511i	$0.352476\pi$
$90$	0	0
$91$	277.012	0.319107
$92$	551.487	0.624962
$93$	0	0
$94$	−1159.43	−1.27219
$95$	−233.662	−0.252350
$96$	0	0
$97$	419.548	0.439161	0.219581	−	0.975594i	$-0.429531\pi$
$97$	419.548	0.439161	0.219581	+	0.975594i	$0.429531\pi$
$98$	−192.774	−0.198706
$99$	0	0
$100$	−890.507	−0.890507
$101$	905.623	0.892207	0.446103	−	0.894981i	$-0.352811\pi$
$101$	905.623	0.892207	0.446103	+	0.894981i	$0.352811\pi$
$102$	0	0
$103$	2067.74	1.97806	0.989030	−	0.147714i	$-0.0471916\pi$
$103$	2067.74	1.97806	0.989030	+	0.147714i	$0.0471916\pi$
$104$	−81.3147	−0.0766689
$105$	0	0
$106$	2467.55	2.26104
$107$	1728.25	1.56146	0.780730	−	0.624869i	$-0.214848\pi$
$107$	1728.25	1.56146	0.780730	+	0.624869i	$0.214848\pi$
$108$	0	0
$109$	925.211	0.813020	0.406510	−	0.913646i	$-0.366746\pi$
$109$	925.211	0.813020	0.406510	+	0.913646i	$0.366746\pi$
$110$	403.404	0.349664
$111$	0	0
$112$	475.339	0.401030
$113$	2187.21	1.82085	0.910424	−	0.413677i	$-0.135756\pi$
$113$	2187.21	1.82085	0.910424	+	0.413677i	$0.135756\pi$
$114$	0	0
$115$	−179.319	−0.145405
$116$	143.629	0.114962
$117$	0	0
$118$	−588.625	−0.459215
$119$	14.5388	0.0111997
$120$	0	0
$121$	447.507	0.336218
$122$	2481.35	1.84140
$123$	0	0
$124$	−1790.89	−1.29699
$125$	593.480	0.424660
$126$	0	0
$127$	898.130	0.627529	0.313764	−	0.949501i	$-0.398410\pi$
$127$	898.130	0.627529	0.313764	+	0.949501i	$0.398410\pi$
$128$	−262.453	−0.181233
$129$	0	0
$130$	−378.541	−0.255387
$131$	2134.53	1.42362	0.711812	−	0.702370i	$-0.247875\pi$
$131$	2134.53	1.42362	0.711812	+	0.702370i	$0.247875\pi$
$132$	0	0
$133$	−672.709	−0.438581
$134$	−15.9038	−0.0102528
$135$	0	0
$136$	−4.26774	−0.00269085
$137$	−2161.26	−1.34780	−0.673901	−	0.738821i	$-0.735383\pi$
$137$	−2161.26	−1.34780	−0.673901	+	0.738821i	$0.735383\pi$
$138$	0	0
$139$	2300.16	1.40358	0.701789	−	0.712385i	$-0.252385\pi$
$139$	2300.16	1.40358	0.701789	+	0.712385i	$0.252385\pi$
$140$	127.270	0.0768305
$141$	0	0
$142$	3140.93	1.85621
$143$	−1668.89	−0.975943
$144$	0	0
$145$	−46.7016	−0.0267473
$146$	−1746.86	−0.990213
$147$	0	0
$148$	−1079.11	−0.599340
$149$	−1183.54	−0.650732	−0.325366	−	0.945588i	$-0.605488\pi$
$149$	−1183.54	−0.650732	−0.325366	+	0.945588i	$0.605488\pi$
$150$	0	0
$151$	3437.62	1.85264	0.926322	−	0.376733i	$-0.122953\pi$
$151$	3437.62	1.85264	0.926322	+	0.376733i	$0.122953\pi$
$152$	197.469	0.105374
$153$	0	0
$154$	1161.39	0.607712
$155$	582.316	0.301760
$156$	0	0
$157$	−20.8614	−0.0106046	−0.00530229	−	0.999986i	$-0.501688\pi$
$157$	−20.8614	−0.0106046	−0.00530229	+	0.999986i	$0.501688\pi$
$158$	2264.27	1.14010
$159$	0	0
$160$	−609.589	−0.301202
$161$	−516.256	−0.252712
$162$	0	0
$163$	−1654.46	−0.795015	−0.397507	−	0.917599i	$-0.630125\pi$
$163$	−1654.46	−0.795015	−0.397507	+	0.917599i	$0.630125\pi$
$164$	−540.502	−0.257354
$165$	0	0
$166$	−5076.00	−2.37334
$167$	−1961.22	−0.908766	−0.454383	−	0.890807i	$-0.650140\pi$
$167$	−1961.22	−0.908766	−0.454383	+	0.890807i	$0.650140\pi$
$168$	0	0
$169$	−630.966	−0.287194
$170$	−19.8674	−0.00896331
$171$	0	0
$172$	3593.40	1.59299
$173$	−27.3427	−0.0120163	−0.00600816	−	0.999982i	$-0.501912\pi$
$173$	−27.3427	−0.0120163	−0.00600816	+	0.999982i	$0.501912\pi$
$174$	0	0
$175$	833.618	0.360089
$176$	−2863.74	−1.22649
$177$	0	0
$178$	−2953.38	−1.24363
$179$	130.832	0.0546302	0.0273151	−	0.999627i	$-0.491304\pi$
$179$	130.832	0.0546302	0.0273151	+	0.999627i	$0.491304\pi$
$180$	0	0
$181$	2460.11	1.01027	0.505134	−	0.863041i	$-0.331443\pi$
$181$	2460.11	1.01027	0.505134	+	0.863041i	$0.331443\pi$
$182$	−1089.81	−0.443859
$183$	0	0
$184$	151.543	0.0607168
$185$	350.879	0.139444
$186$	0	0
$187$	−87.5905	−0.0342527
$188$	2203.73	0.854914
$189$	0	0
$190$	919.267	0.351003
$191$	−1309.41	−0.496049	−0.248024	−	0.968754i	$-0.579781\pi$
$191$	−1309.41	−0.496049	−0.248024	+	0.968754i	$0.579781\pi$
$192$	0	0
$193$	863.735	0.322140	0.161070	−	0.986943i	$-0.448506\pi$
$193$	863.735	0.322140	0.161070	+	0.986943i	$0.448506\pi$
$194$	−1650.57	−0.610847
$195$	0	0
$196$	366.408	0.133530
$197$	2958.90	1.07012	0.535058	−	0.844815i	$-0.320290\pi$
$197$	2958.90	1.07012	0.535058	+	0.844815i	$0.320290\pi$
$198$	0	0
$199$	−1559.83	−0.555646	−0.277823	−	0.960632i	$-0.589613\pi$
$199$	−1559.83	−0.555646	−0.277823	+	0.960632i	$0.589613\pi$
$200$	−244.702	−0.0865152
$201$	0	0
$202$	−3562.88	−1.24101
$203$	−134.453	−0.0464865
$204$	0	0
$205$	175.747	0.0598767
$206$	−8134.83	−2.75136
$207$	0	0
$208$	2687.24	0.895800
$209$	4052.82	1.34134
$210$	0	0
$211$	2506.39	0.817760	0.408880	−	0.912588i	$-0.365919\pi$
$211$	2506.39	0.817760	0.408880	+	0.912588i	$0.365919\pi$
$212$	−4690.10	−1.51942
$213$	0	0
$214$	−6799.23	−2.17190
$215$	−1168.41	−0.370628
$216$	0	0
$217$	1676.48	0.524455
$218$	−3639.94	−1.13086
$219$	0	0
$220$	−766.753	−0.234975
$221$	82.1921	0.0250174
$222$	0	0
$223$	4048.41	1.21570	0.607851	−	0.794051i	$-0.292032\pi$
$223$	4048.41	1.21570	0.607851	+	0.794051i	$0.292032\pi$
$224$	−1755.00	−0.523485
$225$	0	0
$226$	−8604.87	−2.53269
$227$	−627.469	−0.183465	−0.0917325	−	0.995784i	$-0.529240\pi$
$227$	−627.469	−0.183465	−0.0917325	+	0.995784i	$0.529240\pi$
$228$	0	0
$229$	−989.126	−0.285429	−0.142715	−	0.989764i	$-0.545583\pi$
$229$	−989.126	−0.285429	−0.142715	+	0.989764i	$0.545583\pi$
$230$	705.472	0.202250
$231$	0	0
$232$	39.4676	0.0111689
$233$	−3125.53	−0.878799	−0.439399	−	0.898292i	$-0.644809\pi$
$233$	−3125.53	−0.878799	−0.439399	+	0.898292i	$0.644809\pi$
$234$	0	0
$235$	−716.555	−0.198906
$236$	1118.80	0.308593
$237$	0	0
$238$	−57.1980	−0.0155781
$239$	−3025.03	−0.818714	−0.409357	−	0.912374i	$-0.634247\pi$
$239$	−3025.03	−0.818714	−0.409357	+	0.912374i	$0.634247\pi$
$240$	0	0
$241$	−5598.85	−1.49649	−0.748245	−	0.663423i	$-0.769103\pi$
$241$	−5598.85	−1.49649	−0.748245	+	0.663423i	$0.769103\pi$
$242$	−1760.57	−0.467659
$243$	0	0
$244$	−4716.31	−1.23742
$245$	−119.139	−0.0310675
$246$	0	0
$247$	−3803.03	−0.979680
$248$	−492.117	−0.126006
$249$	0	0
$250$	−2334.85	−0.590676
$251$	2960.32	0.744438	0.372219	−	0.928145i	$-0.378597\pi$
$251$	2960.32	0.744438	0.372219	+	0.928145i	$0.378597\pi$
$252$	0	0
$253$	3110.25	0.772883
$254$	−3533.40	−0.872855
$255$	0	0
$256$	4577.39	1.11753
$257$	5995.68	1.45525	0.727627	−	0.685973i	$-0.240623\pi$
$257$	5995.68	1.45525	0.727627	+	0.685973i	$0.240623\pi$
$258$	0	0
$259$	1010.17	0.242352
$260$	719.496	0.171620
$261$	0	0
$262$	−8397.61	−1.98017
$263$	1783.69	0.418201	0.209100	−	0.977894i	$-0.432946\pi$
$263$	1783.69	0.418201	0.209100	+	0.977894i	$0.432946\pi$
$264$	0	0
$265$	1525.01	0.353512
$266$	2646.55	0.610040
$267$	0	0
$268$	30.2285	0.00688991
$269$	6348.32	1.43890	0.719450	−	0.694545i	$-0.244394\pi$
$269$	6348.32	1.43890	0.719450	+	0.694545i	$0.244394\pi$
$270$	0	0
$271$	−6712.35	−1.50460	−0.752299	−	0.658822i	$-0.771055\pi$
$271$	−6712.35	−1.50460	−0.752299	+	0.658822i	$0.771055\pi$
$272$	141.038	0.0314399
$273$	0	0
$274$	8502.77	1.87471
$275$	−5022.23	−1.10128
$276$	0	0
$277$	2276.57	0.493811	0.246905	−	0.969040i	$-0.420586\pi$
$277$	2276.57	0.493811	0.246905	+	0.969040i	$0.420586\pi$
$278$	−9049.24	−1.95229
$279$	0	0
$280$	34.9724	0.00746430
$281$	−1485.99	−0.315468	−0.157734	−	0.987482i	$-0.550419\pi$
$281$	−1485.99	−0.315468	−0.157734	+	0.987482i	$0.550419\pi$
$282$	0	0
$283$	−9010.19	−1.89258	−0.946291	−	0.323317i	$-0.895202\pi$
$283$	−9010.19	−1.89258	−0.946291	+	0.323317i	$0.895202\pi$
$284$	−5969.99	−1.24737
$285$	0	0
$286$	6565.71	1.35748
$287$	505.973	0.104065
$288$	0	0
$289$	−4908.69	−0.999122
$290$	183.732	0.0372039
$291$	0	0
$292$	3320.27	0.665424
$293$	7535.12	1.50241	0.751205	−	0.660069i	$-0.229473\pi$
$293$	7535.12	1.50241	0.751205	+	0.660069i	$0.229473\pi$
$294$	0	0
$295$	−363.785	−0.0717979
$296$	−296.529	−0.0582276
$297$	0	0
$298$	4656.24	0.905129
$299$	−2918.55	−0.564496
$300$	0	0
$301$	−3363.84	−0.644148
$302$	−13524.2	−2.57692
$303$	0	0
$304$	−6525.81	−1.23119
$305$	1533.53	0.287901
$306$	0	0
$307$	2765.48	0.514119	0.257059	−	0.966396i	$-0.417246\pi$
$307$	2765.48	0.514119	0.257059	+	0.966396i	$0.417246\pi$
$308$	−2207.47	−0.408383
$309$	0	0
$310$	−2290.93	−0.419729
$311$	−3014.21	−0.549583	−0.274792	−	0.961504i	$-0.588609\pi$
$311$	−3014.21	−0.549583	−0.274792	+	0.961504i	$0.588609\pi$
$312$	0	0
$313$	2748.14	0.496275	0.248138	−	0.968725i	$-0.420181\pi$
$313$	2748.14	0.496275	0.248138	+	0.968725i	$0.420181\pi$
$314$	82.0722	0.0147503
$315$	0	0
$316$	−4303.71	−0.766147
$317$	3933.40	0.696913	0.348457	−	0.937325i	$-0.386706\pi$
$317$	3933.40	0.696913	0.348457	+	0.937325i	$0.386706\pi$
$318$	0	0
$319$	810.028	0.142172
$320$	1077.38	0.188210
$321$	0	0
$322$	2031.04	0.351507
$323$	−199.599	−0.0343839
$324$	0	0
$325$	4712.70	0.804349
$326$	6508.93	1.10582
$327$	0	0
$328$	−148.524	−0.0250027
$329$	−2062.95	−0.345696
$330$	0	0
$331$	4604.42	0.764598	0.382299	−	0.924039i	$-0.375132\pi$
$331$	4604.42	0.764598	0.382299	+	0.924039i	$0.375132\pi$
$332$	9648.00	1.59489
$333$	0	0
$334$	7715.78	1.26404
$335$	−9.82894	−0.00160302
$336$	0	0
$337$	2772.74	0.448192	0.224096	−	0.974567i	$-0.428057\pi$
$337$	2772.74	0.448192	0.224096	+	0.974567i	$0.428057\pi$
$338$	2482.33	0.399470
$339$	0	0
$340$	37.7622	0.00602336
$341$	−10100.1	−1.60397
$342$	0	0
$343$	−343.000	−0.0539949
$344$	987.429	0.154763
$345$	0	0
$346$	107.571	0.0167140
$347$	8422.89	1.30307	0.651534	−	0.758619i	$-0.274126\pi$
$347$	8422.89	1.30307	0.651534	+	0.758619i	$0.274126\pi$
$348$	0	0
$349$	−3879.97	−0.595101	−0.297550	−	0.954706i	$-0.596170\pi$
$349$	−3879.97	−0.595101	−0.297550	+	0.954706i	$0.596170\pi$
$350$	−3279.59	−0.500862
$351$	0	0
$352$	10573.2	1.60100
$353$	3924.73	0.591762	0.295881	−	0.955225i	$-0.404387\pi$
$353$	3924.73	0.591762	0.295881	+	0.955225i	$0.404387\pi$
$354$	0	0
$355$	1941.18	0.290217
$356$	5613.52	0.835719
$357$	0	0
$358$	−514.714	−0.0759873
$359$	6623.52	0.973749	0.486874	−	0.873472i	$-0.338137\pi$
$359$	6623.52	0.973749	0.486874	+	0.873472i	$0.338137\pi$
$360$	0	0
$361$	2376.46	0.346473
$362$	−9678.49	−1.40522
$363$	0	0
$364$	2071.41	0.298274
$365$	−1079.60	−0.154819
$366$	0	0
$367$	−9211.94	−1.31024	−0.655122	−	0.755524i	$-0.727383\pi$
$367$	−9211.94	−1.31024	−0.655122	+	0.755524i	$0.727383\pi$
$368$	−5008.09	−0.709415
$369$	0	0
$370$	−1380.42	−0.193958
$371$	4390.47	0.614399
$372$	0	0
$373$	−7526.33	−1.04477	−0.522384	−	0.852710i	$-0.674957\pi$
$373$	−7526.33	−1.04477	−0.522384	+	0.852710i	$0.674957\pi$
$374$	344.596	0.0476434
$375$	0	0
$376$	605.563	0.0830572
$377$	−760.104	−0.103839
$378$	0	0
$379$	9542.86	1.29336	0.646681	−	0.762761i	$-0.276157\pi$
$379$	9542.86	1.29336	0.646681	+	0.762761i	$0.276157\pi$
$380$	−1747.26	−0.235875
$381$	0	0
$382$	5151.43	0.689974
$383$	−3793.28	−0.506078	−0.253039	−	0.967456i	$-0.581430\pi$
$383$	−3793.28	−0.506078	−0.253039	+	0.967456i	$0.581430\pi$
$384$	0	0
$385$	717.769	0.0950154
$386$	−3398.08	−0.448077
$387$	0	0
$388$	3137.26	0.410490
$389$	−6179.16	−0.805388	−0.402694	−	0.915335i	$-0.631926\pi$
$389$	−6179.16	−0.805388	−0.402694	+	0.915335i	$0.631926\pi$
$390$	0	0
$391$	−153.178	−0.0198121
$392$	100.685	0.0129729
$393$	0	0
$394$	−11640.8	−1.48847
$395$	1399.37	0.178253
$396$	0	0
$397$	179.912	0.0227444	0.0113722	−	0.999935i	$-0.496380\pi$
$397$	179.912	0.0227444	0.0113722	+	0.999935i	$0.496380\pi$
$398$	6136.64	0.772870
$399$	0	0
$400$	8086.75	1.01084
$401$	7303.46	0.909519	0.454760	−	0.890614i	$-0.349725\pi$
$401$	7303.46	0.909519	0.454760	+	0.890614i	$0.349725\pi$
$402$	0	0
$403$	9477.64	1.17150
$404$	6771.98	0.833957
$405$	0	0
$406$	528.961	0.0646599
$407$	−6085.91	−0.741197
$408$	0	0
$409$	−2111.96	−0.255329	−0.127664	−	0.991817i	$-0.540748\pi$
$409$	−2111.96	−0.255329	−0.127664	+	0.991817i	$0.540748\pi$
$410$	−691.420	−0.0832848
$411$	0	0
$412$	15461.9	1.84892
$413$	−1047.33	−0.124784
$414$	0	0
$415$	−3137.10	−0.371070
$416$	−9921.53	−1.16933
$417$	0	0
$418$	−15944.5	−1.86572
$419$	−5725.74	−0.667591	−0.333795	−	0.942646i	$-0.608329\pi$
$419$	−5725.74	−0.667591	−0.333795	+	0.942646i	$0.608329\pi$
$420$	0	0
$421$	5266.52	0.609678	0.304839	−	0.952404i	$-0.401397\pi$
$421$	5266.52	0.609678	0.304839	+	0.952404i	$0.401397\pi$
$422$	−9860.59	−1.13745
$423$	0	0
$424$	−1288.79	−0.147616
$425$	247.342	0.0282303
$426$	0	0
$427$	4415.01	0.500369
$428$	12923.3	1.45952
$429$	0	0
$430$	4596.74	0.515522
$431$	15827.4	1.76886	0.884430	−	0.466672i	$-0.154547\pi$
$431$	15827.4	1.76886	0.884430	+	0.466672i	$0.154547\pi$
$432$	0	0
$433$	4537.27	0.503574	0.251787	−	0.967783i	$-0.418982\pi$
$433$	4537.27	0.503574	0.251787	+	0.967783i	$0.418982\pi$
$434$	−6595.55	−0.729485
$435$	0	0
$436$	6918.46	0.759940
$437$	7087.55	0.775843
$438$	0	0
$439$	8926.82	0.970510	0.485255	−	0.874373i	$-0.338727\pi$
$439$	8926.82	0.970510	0.485255	+	0.874373i	$0.338727\pi$
$440$	−210.696	−0.0228285
$441$	0	0
$442$	−323.358	−0.0347977
$443$	−6444.96	−0.691217	−0.345609	−	0.938379i	$-0.612327\pi$
$443$	−6444.96	−0.691217	−0.345609	+	0.938379i	$0.612327\pi$
$444$	0	0
$445$	−1825.26	−0.194440
$446$	−15927.1	−1.69097
$447$	0	0
$448$	3101.75	0.327106
$449$	−6202.27	−0.651901	−0.325950	−	0.945387i	$-0.605684\pi$
$449$	−6202.27	−0.651901	−0.325950	+	0.945387i	$0.605684\pi$
$450$	0	0
$451$	−3048.29	−0.318267
$452$	16355.3	1.70197
$453$	0	0
$454$	2468.57	0.255189
$455$	−673.531	−0.0693970
$456$	0	0
$457$	−3100.93	−0.317408	−0.158704	−	0.987326i	$-0.550732\pi$
$457$	−3100.93	−0.317408	−0.158704	+	0.987326i	$0.550732\pi$
$458$	3891.39	0.397015
$459$	0	0
$460$	−1340.89	−0.135912
$461$	4349.02	0.439379	0.219690	−	0.975570i	$-0.429495\pi$
$461$	4349.02	0.439379	0.219690	+	0.975570i	$0.429495\pi$
$462$	0	0
$463$	−7445.51	−0.747348	−0.373674	−	0.927560i	$-0.621902\pi$
$463$	−7445.51	−0.747348	−0.373674	+	0.927560i	$0.621902\pi$
$464$	−1304.30	−0.130497
$465$	0	0
$466$	12296.4	1.22236
$467$	15424.1	1.52836	0.764178	−	0.645006i	$-0.223145\pi$
$467$	15424.1	1.52836	0.764178	+	0.645006i	$0.223145\pi$
$468$	0	0
$469$	−28.2973	−0.00278603
$470$	2819.05	0.276666
$471$	0	0
$472$	307.436	0.0299807
$473$	20265.9	1.97003
$474$	0	0
$475$	−11444.5	−1.10550
$476$	108.717	0.0104685
$477$	0	0
$478$	11901.0	1.13878
$479$	8650.42	0.825152	0.412576	−	0.910923i	$-0.364629\pi$
$479$	8650.42	0.825152	0.412576	+	0.910923i	$0.364629\pi$
$480$	0	0
$481$	5710.82	0.541353
$482$	22026.9	2.08153
$483$	0	0
$484$	3346.32	0.314268
$485$	−1020.10	−0.0955055
$486$	0	0
$487$	−13668.2	−1.27179	−0.635897	−	0.771774i	$-0.719370\pi$
$487$	−13668.2	−1.27179	−0.635897	+	0.771774i	$0.719370\pi$
$488$	−1295.99	−0.120219
$489$	0	0
$490$	468.715	0.0432130
$491$	−1178.61	−0.108330	−0.0541648	−	0.998532i	$-0.517250\pi$
$491$	−1178.61	−0.108330	−0.0541648	+	0.998532i	$0.517250\pi$
$492$	0	0
$493$	−39.8935	−0.00364445
$494$	14961.8	1.36268
$495$	0	0
$496$	16263.2	1.47225
$497$	5588.61	0.504393
$498$	0	0
$499$	−38.5072	−0.00345455	−0.00172727	−	0.999999i	$-0.500550\pi$
$499$	−38.5072	−0.00345455	−0.00172727	+	0.999999i	$0.500550\pi$
$500$	4437.87	0.396935
$501$	0	0
$502$	−11646.4	−1.03547
$503$	4489.37	0.397954	0.198977	−	0.980004i	$-0.436238\pi$
$503$	4489.37	0.397954	0.198977	+	0.980004i	$0.436238\pi$
$504$	0	0
$505$	−2201.95	−0.194030
$506$	−12236.2	−1.07503
$507$	0	0
$508$	6715.95	0.586560
$509$	−13156.5	−1.14568	−0.572842	−	0.819666i	$-0.694159\pi$
$509$	−13156.5	−1.14568	−0.572842	+	0.819666i	$0.694159\pi$
$510$	0	0
$511$	−3108.16	−0.269074
$512$	−15908.6	−1.37318
$513$	0	0
$514$	−23588.0	−2.02417
$515$	−5027.53	−0.430174
$516$	0	0
$517$	12428.5	1.05726
$518$	−3974.19	−0.337097
$519$	0	0
$520$	197.710	0.0166734
$521$	2198.40	0.184863	0.0924317	−	0.995719i	$-0.470536\pi$
$521$	2198.40	0.184863	0.0924317	+	0.995719i	$0.470536\pi$
$522$	0	0
$523$	−7784.24	−0.650824	−0.325412	−	0.945572i	$-0.605503\pi$
$523$	−7784.24	−0.650824	−0.325412	+	0.945572i	$0.605503\pi$
$524$	15961.4	1.33068
$525$	0	0
$526$	−7017.33	−0.581692
$527$	497.427	0.0411162
$528$	0	0
$529$	−6727.81	−0.552956
$530$	−5999.65	−0.491713
$531$	0	0
$532$	−5030.32	−0.409947
$533$	2860.42	0.232455
$534$	0	0
$535$	−4202.09	−0.339574
$536$	8.30647	0.000669374 0
$537$	0	0
$538$	−24975.4	−2.00142
$539$	2066.44	0.165136
$540$	0	0
$541$	−8493.26	−0.674961	−0.337480	−	0.941333i	$-0.609575\pi$
$541$	−8493.26	−0.674961	−0.337480	+	0.941333i	$0.609575\pi$
$542$	26407.5	2.09281
$543$	0	0
$544$	−520.724	−0.0410402
$545$	−2249.57	−0.176809
$546$	0	0
$547$	7326.76	0.572705	0.286352	−	0.958124i	$-0.407557\pi$
$547$	7326.76	0.572705	0.286352	+	0.958124i	$0.407557\pi$
$548$	−16161.3	−1.25981
$549$	0	0
$550$	19758.3	1.53181
$551$	1845.87	0.142717
$552$	0	0
$553$	4028.77	0.309802
$554$	−8956.40	−0.686861
$555$	0	0
$556$	17199.9	1.31194
$557$	7394.34	0.562492	0.281246	−	0.959636i	$-0.409252\pi$
$557$	7394.34	0.562492	0.281246	+	0.959636i	$0.409252\pi$
$558$	0	0
$559$	−19016.8	−1.43887
$560$	−1155.75	−0.0872129
$561$	0	0
$562$	5846.13	0.438797
$563$	−2874.36	−0.215169	−0.107584	−	0.994196i	$-0.534312\pi$
$563$	−2874.36	−0.215169	−0.107584	+	0.994196i	$0.534312\pi$
$564$	0	0
$565$	−5318.03	−0.395984
$566$	35447.6	2.63247
$567$	0	0
$568$	−1640.49	−0.121186
$569$	15729.8	1.15893	0.579463	−	0.814998i	$-0.303262\pi$
$569$	15729.8	1.15893	0.579463	+	0.814998i	$0.303262\pi$
$570$	0	0
$571$	−11319.1	−0.829576	−0.414788	−	0.909918i	$-0.636144\pi$
$571$	−11319.1	−0.829576	−0.414788	+	0.909918i	$0.636144\pi$
$572$	−12479.5	−0.912226
$573$	0	0
$574$	−1990.58	−0.144748
$575$	−8782.86	−0.636992
$576$	0	0
$577$	1090.45	0.0786758	0.0393379	−	0.999226i	$-0.487475\pi$
$577$	1090.45	0.0786758	0.0393379	+	0.999226i	$0.487475\pi$
$578$	19311.6	1.38972
$579$	0	0
$580$	−349.221	−0.0250010
$581$	−9031.64	−0.644915
$582$	0	0
$583$	−26450.9	−1.87905
$584$	912.375	0.0646478
$585$	0	0
$586$	−29644.4	−2.08976
$587$	26096.0	1.83492	0.917458	−	0.397832i	$-0.130237\pi$
$587$	26096.0	1.83492	0.917458	+	0.397832i	$0.130237\pi$
$588$	0	0
$589$	−23015.9	−1.61011
$590$	1431.19	0.0998665
$591$	0	0
$592$	9799.48	0.680332
$593$	−2675.40	−0.185271	−0.0926354	−	0.995700i	$-0.529529\pi$
$593$	−2675.40	−0.185271	−0.0926354	+	0.995700i	$0.529529\pi$
$594$	0	0
$595$	−35.3498	−0.00243563
$596$	−8850.14	−0.608248
$597$	0	0
$598$	11482.1	0.785180
$599$	−4967.73	−0.338858	−0.169429	−	0.985542i	$-0.554192\pi$
$599$	−4967.73	−0.338858	−0.169429	+	0.985542i	$0.554192\pi$
$600$	0	0
$601$	−18880.6	−1.28146	−0.640728	−	0.767768i	$-0.721368\pi$
$601$	−18880.6	−1.28146	−0.640728	+	0.767768i	$0.721368\pi$
$602$	13233.9	0.895971
$603$	0	0
$604$	25705.5	1.73169
$605$	−1088.07	−0.0731182
$606$	0	0
$607$	22802.1	1.52473	0.762364	−	0.647148i	$-0.224038\pi$
$607$	22802.1	1.52473	0.762364	+	0.647148i	$0.224038\pi$
$608$	24093.9	1.60713
$609$	0	0
$610$	−6033.18	−0.400453
$611$	−11662.5	−0.772199
$612$	0	0
$613$	−16589.5	−1.09306	−0.546529	−	0.837440i	$-0.684051\pi$
$613$	−16589.5	−1.09306	−0.546529	+	0.837440i	$0.684051\pi$
$614$	−10879.9	−0.715108
$615$	0	0
$616$	−606.589	−0.0396756
$617$	25190.6	1.64365	0.821827	−	0.569737i	$-0.192955\pi$
$617$	25190.6	1.64365	0.821827	+	0.569737i	$0.192955\pi$
$618$	0	0
$619$	12152.2	0.789077	0.394538	−	0.918879i	$-0.370904\pi$
$619$	12152.2	0.789077	0.394538	+	0.918879i	$0.370904\pi$
$620$	4354.39	0.282059
$621$	0	0
$622$	11858.4	0.764437
$623$	−5254.90	−0.337935
$624$	0	0
$625$	13443.0	0.860354
$626$	−10811.7	−0.690289
$627$	0	0
$628$	−155.995	−0.00991224
$629$	299.728	0.0189999
$630$	0	0
$631$	13761.8	0.868224	0.434112	−	0.900859i	$-0.357062\pi$
$631$	13761.8	0.868224	0.434112	+	0.900859i	$0.357062\pi$
$632$	−1182.61	−0.0744333
$633$	0	0
$634$	−15474.7	−0.969365
$635$	−2183.73	−0.136470
$636$	0	0
$637$	−1939.08	−0.120611
$638$	−3186.79	−0.197753
$639$	0	0
$640$	638.133	0.0394132
$641$	−382.378	−0.0235617	−0.0117808	−	0.999931i	$-0.503750\pi$
$641$	−382.378	−0.0235617	−0.0117808	+	0.999931i	$0.503750\pi$
$642$	0	0
$643$	−8017.97	−0.491754	−0.245877	−	0.969301i	$-0.579076\pi$
$643$	−8017.97	−0.491754	−0.245877	+	0.969301i	$0.579076\pi$
$644$	−3860.41	−0.236213
$645$	0	0
$646$	785.257	0.0478259
$647$	−548.646	−0.0333377	−0.0166689	−	0.999861i	$-0.505306\pi$
$647$	−548.646	−0.0333377	−0.0166689	+	0.999861i	$0.505306\pi$
$648$	0	0
$649$	6309.76	0.381633
$650$	−18540.6	−1.11880
$651$	0	0
$652$	−12371.6	−0.743111
$653$	6817.12	0.408537	0.204268	−	0.978915i	$-0.434518\pi$
$653$	6817.12	0.408537	0.204268	+	0.978915i	$0.434518\pi$
$654$	0	0
$655$	−5189.93	−0.309599
$656$	4908.34	0.292132
$657$	0	0
$658$	8116.00	0.480843
$659$	−31753.1	−1.87697	−0.938487	−	0.345314i	$-0.887772\pi$
$659$	−31753.1	−1.87697	−0.938487	+	0.345314i	$0.887772\pi$
$660$	0	0
$661$	−19081.6	−1.12283	−0.561413	−	0.827536i	$-0.689742\pi$
$661$	−19081.6	−1.12283	−0.561413	+	0.827536i	$0.689742\pi$
$662$	−18114.6	−1.06351
$663$	0	0
$664$	2651.17	0.154948
$665$	1635.64	0.0953793
$666$	0	0
$667$	1416.58	0.0822339
$668$	−14665.4	−0.849435
$669$	0	0
$670$	38.6687	0.00222971
$671$	−26598.8	−1.53030
$672$	0	0
$673$	−13271.8	−0.760164	−0.380082	−	0.924953i	$-0.624104\pi$
$673$	−13271.8	−0.760164	−0.380082	+	0.924953i	$0.624104\pi$
$674$	−10908.4	−0.623408
$675$	0	0
$676$	−4718.18	−0.268444
$677$	5335.45	0.302892	0.151446	−	0.988466i	$-0.451607\pi$
$677$	5335.45	0.302892	0.151446	+	0.988466i	$0.451607\pi$
$678$	0	0
$679$	−2936.84	−0.165987
$680$	10.3767	0.000585186 0
$681$	0	0
$682$	39735.7	2.23102
$683$	−26921.0	−1.50820	−0.754101	−	0.656758i	$-0.771927\pi$
$683$	−26921.0	−1.50820	−0.754101	+	0.656758i	$0.771927\pi$
$684$	0	0
$685$	5254.92	0.293110
$686$	1349.42	0.0751037
$687$	0	0
$688$	−32631.9	−1.80826
$689$	24820.7	1.37241
$690$	0	0
$691$	4617.29	0.254197	0.127099	−	0.991890i	$-0.459434\pi$
$691$	4617.29	0.254197	0.127099	+	0.991890i	$0.459434\pi$
$692$	−204.460	−0.0112318
$693$	0	0
$694$	−33137.1	−1.81249
$695$	−5592.65	−0.305239
$696$	0	0
$697$	150.127	0.00815848
$698$	15264.5	0.827749
$699$	0	0
$700$	6233.55	0.336580
$701$	−15895.3	−0.856430	−0.428215	−	0.903677i	$-0.640857\pi$
$701$	−15895.3	−0.856430	−0.428215	+	0.903677i	$0.640857\pi$
$702$	0	0
$703$	−13868.4	−0.744036
$704$	−18686.8	−1.00041
$705$	0	0
$706$	−15440.5	−0.823106
$707$	−6339.36	−0.337222
$708$	0	0
$709$	−3589.82	−0.190153	−0.0950766	−	0.995470i	$-0.530310\pi$
$709$	−3589.82	−0.190153	−0.0950766	+	0.995470i	$0.530310\pi$
$710$	−7636.92	−0.403674
$711$	0	0
$712$	1542.54	0.0811924
$713$	−17663.1	−0.927753
$714$	0	0
$715$	4057.77	0.212241
$716$	978.319	0.0510636
$717$	0	0
$718$	−26058.1	−1.35443
$719$	3759.41	0.194996	0.0974981	−	0.995236i	$-0.468916\pi$
$719$	3759.41	0.194996	0.0974981	+	0.995236i	$0.468916\pi$
$720$	0	0
$721$	−14474.2	−0.747636
$722$	−9349.39	−0.481923
$723$	0	0
$724$	18396.0	0.944310
$725$	−2287.40	−0.117175
$726$	0	0
$727$	25387.5	1.29514	0.647572	−	0.762004i	$-0.275785\pi$
$727$	25387.5	1.29514	0.647572	+	0.762004i	$0.275785\pi$
$728$	569.203	0.0289781
$729$	0	0
$730$	4247.34	0.215344
$731$	−998.083	−0.0504999
$732$	0	0
$733$	20195.1	1.01763	0.508815	−	0.860876i	$-0.330084\pi$
$733$	20195.1	1.01763	0.508815	+	0.860876i	$0.330084\pi$
$734$	36241.3	1.82247
$735$	0	0
$736$	18490.4	0.926037
$737$	170.481	0.00852068
$738$	0	0
$739$	−2153.33	−0.107187	−0.0535936	−	0.998563i	$-0.517068\pi$
$739$	−2153.33	−0.107187	−0.0535936	+	0.998563i	$0.517068\pi$
$740$	2623.77	0.130340
$741$	0	0
$742$	−17272.9	−0.854592
$743$	988.365	0.0488016	0.0244008	−	0.999702i	$-0.492232\pi$
$743$	988.365	0.0488016	0.0244008	+	0.999702i	$0.492232\pi$
$744$	0	0
$745$	2877.67	0.141516
$746$	29609.9	1.45321
$747$	0	0
$748$	−654.976	−0.0320165
$749$	−12097.7	−0.590176
$750$	0	0
$751$	17929.8	0.871194	0.435597	−	0.900142i	$-0.356537\pi$
$751$	17929.8	0.871194	0.435597	+	0.900142i	$0.356537\pi$
$752$	−20012.3	−0.970441
$753$	0	0
$754$	2990.38	0.144434
$755$	−8358.27	−0.402899
$756$	0	0
$757$	34271.4	1.64546	0.822732	−	0.568429i	$-0.192449\pi$
$757$	34271.4	1.64546	0.822732	+	0.568429i	$0.192449\pi$
$758$	−37543.3	−1.79899
$759$	0	0
$760$	−480.128	−0.0229159
$761$	−17359.9	−0.826933	−0.413466	−	0.910519i	$-0.635682\pi$
$761$	−17359.9	−0.826933	−0.413466	+	0.910519i	$0.635682\pi$
$762$	0	0
$763$	−6476.48	−0.307293
$764$	−9791.36	−0.463663
$765$	0	0
$766$	14923.4	0.703923
$767$	−5920.88	−0.278736
$768$	0	0
$769$	−4823.40	−0.226185	−0.113093	−	0.993584i	$-0.536076\pi$
$769$	−4823.40	−0.226185	−0.113093	+	0.993584i	$0.536076\pi$
$770$	−2823.83	−0.132161
$771$	0	0
$772$	6458.76	0.301108
$773$	4103.77	0.190947	0.0954736	−	0.995432i	$-0.469563\pi$
$773$	4103.77	0.190947	0.0954736	+	0.995432i	$0.469563\pi$
$774$	0	0
$775$	28521.2	1.32195
$776$	862.086	0.0398803
$777$	0	0
$778$	24309.9	1.12025
$779$	−6946.38	−0.319486
$780$	0	0
$781$	−33669.2	−1.54261
$782$	602.629	0.0275575
$783$	0	0
$784$	−3327.37	−0.151575
$785$	50.7227	0.00230620
$786$	0	0
$787$	24680.4	1.11787	0.558934	−	0.829212i	$-0.311211\pi$
$787$	24680.4	1.11787	0.558934	+	0.829212i	$0.311211\pi$
$788$	22125.8	1.00025
$789$	0	0
$790$	−5505.38	−0.247940
$791$	−15310.5	−0.688216
$792$	0	0
$793$	24959.4	1.11770
$794$	−707.803	−0.0316360
$795$	0	0
$796$	−11664.0	−0.519369
$797$	10139.3	0.450632	0.225316	−	0.974286i	$-0.427658\pi$
$797$	10139.3	0.450632	0.225316	+	0.974286i	$0.427658\pi$
$798$	0	0
$799$	−612.097	−0.0271019
$800$	−29857.1	−1.31951
$801$	0	0
$802$	−28733.1	−1.26509
$803$	18725.5	0.822923
$804$	0	0
$805$	1255.23	0.0549579
$806$	−37286.6	−1.62949
$807$	0	0
$808$	1860.87	0.0810213
$809$	−25446.0	−1.10585	−0.552926	−	0.833230i	$-0.686489\pi$
$809$	−25446.0	−1.10585	−0.552926	+	0.833230i	$0.686489\pi$
$810$	0	0
$811$	22279.3	0.964652	0.482326	−	0.875992i	$-0.339792\pi$
$811$	22279.3	0.964652	0.482326	+	0.875992i	$0.339792\pi$
$812$	−1005.40	−0.0434515
$813$	0	0
$814$	23943.0	1.03096
$815$	4022.68	0.172894
$816$	0	0
$817$	46181.3	1.97758
$818$	8308.80	0.355147
$819$	0	0
$820$	1314.19	0.0559675
$821$	8806.80	0.374372	0.187186	−	0.982324i	$-0.440063\pi$
$821$	8806.80	0.374372	0.187186	+	0.982324i	$0.440063\pi$
$822$	0	0
$823$	−10072.3	−0.426609	−0.213304	−	0.976986i	$-0.568423\pi$
$823$	−10072.3	−0.426609	−0.213304	+	0.976986i	$0.568423\pi$
$824$	4248.78	0.179628
$825$	0	0
$826$	4120.38	0.173567
$827$	24609.0	1.03475	0.517375	−	0.855759i	$-0.326909\pi$
$827$	24609.0	1.03475	0.517375	+	0.855759i	$0.326909\pi$
$828$	0	0
$829$	40946.1	1.71546	0.857730	−	0.514101i	$-0.171874\pi$
$829$	40946.1	1.71546	0.857730	+	0.514101i	$0.171874\pi$
$830$	12341.9	0.516136
$831$	0	0
$832$	17535.1	0.730674
$833$	−101.771	−0.00423310
$834$	0	0
$835$	4768.54	0.197631
$836$	30305.8	1.25376
$837$	0	0
$838$	22526.0	0.928579
$839$	40878.9	1.68212	0.841058	−	0.540944i	$-0.181933\pi$
$839$	40878.9	1.68212	0.841058	+	0.540944i	$0.181933\pi$
$840$	0	0
$841$	−24020.1	−0.984873
$842$	−20719.4	−0.848026
$843$	0	0
$844$	18742.1	0.764371
$845$	1534.14	0.0624568
$846$	0	0
$847$	−3132.55	−0.127079
$848$	42591.1	1.72474
$849$	0	0
$850$	−973.087	−0.0392666
$851$	−10643.0	−0.428717
$852$	0	0
$853$	−2368.11	−0.0950558	−0.0475279	−	0.998870i	$-0.515134\pi$
$853$	−2368.11	−0.0950558	−0.0475279	+	0.998870i	$0.515134\pi$
$854$	−17369.4	−0.695983
$855$	0	0
$856$	3551.20	0.141796
$857$	−12190.4	−0.485899	−0.242949	−	0.970039i	$-0.578115\pi$
$857$	−12190.4	−0.485899	−0.242949	+	0.970039i	$0.578115\pi$
$858$	0	0
$859$	−18236.4	−0.724351	−0.362175	−	0.932110i	$-0.617966\pi$
$859$	−18236.4	−0.724351	−0.362175	+	0.932110i	$0.617966\pi$
$860$	−8737.05	−0.346431
$861$	0	0
$862$	−62267.7	−2.46038
$863$	−35518.1	−1.40099	−0.700493	−	0.713659i	$-0.747036\pi$
$863$	−35518.1	−1.40099	−0.700493	+	0.713659i	$0.747036\pi$
$864$	0	0
$865$	66.4814	0.00261322
$866$	−17850.4	−0.700441
$867$	0	0
$868$	12536.2	0.490215
$869$	−24271.8	−0.947485
$870$	0	0
$871$	−159.974	−0.00622330
$872$	1901.12	0.0738303
$873$	0	0
$874$	−27883.6	−1.07915
$875$	−4154.36	−0.160506
$876$	0	0
$877$	184.478	0.00710304	0.00355152	−	0.999994i	$-0.498870\pi$
$877$	184.478	0.00710304	0.00355152	+	0.999994i	$0.498870\pi$
$878$	−35119.6	−1.34992
$879$	0	0
$880$	6962.93	0.266728
$881$	16067.4	0.614444	0.307222	−	0.951638i	$-0.400601\pi$
$881$	16067.4	0.614444	0.307222	+	0.951638i	$0.400601\pi$
$882$	0	0
$883$	−591.975	−0.0225612	−0.0112806	−	0.999936i	$-0.503591\pi$
$883$	−591.975	−0.0225612	−0.0112806	+	0.999936i	$0.503591\pi$
$884$	614.608	0.0233841
$885$	0	0
$886$	25355.6	0.961441
$887$	−16855.0	−0.638034	−0.319017	−	0.947749i	$-0.603353\pi$
$887$	−16855.0	−0.638034	−0.319017	+	0.947749i	$0.603353\pi$
$888$	0	0
$889$	−6286.91	−0.237184
$890$	7180.91	0.270454
$891$	0	0
$892$	30272.8	1.13633
$893$	28321.7	1.06131
$894$	0	0
$895$	−318.106	−0.0118806
$896$	1837.17	0.0684996
$897$	0	0
$898$	24400.8	0.906755
$899$	−4600.15	−0.170660
$900$	0	0
$901$	1302.70	0.0481677
$902$	11992.5	0.442691
$903$	0	0
$904$	4494.28	0.165351
$905$	−5981.55	−0.219705
$906$	0	0
$907$	38689.6	1.41639	0.708195	−	0.706017i	$-0.249510\pi$
$907$	38689.6	1.41639	0.708195	+	0.706017i	$0.249510\pi$
$908$	−4692.03	−0.171487
$909$	0	0
$910$	2649.79	0.0965270
$911$	−23616.0	−0.858871	−0.429436	−	0.903098i	$-0.641287\pi$
$911$	−23616.0	−0.858871	−0.429436	+	0.903098i	$0.641287\pi$
$912$	0	0
$913$	54412.2	1.97238
$914$	12199.6	0.441495
$915$	0	0
$916$	−7396.39	−0.266794
$917$	−14941.7	−0.538079
$918$	0	0
$919$	−30560.7	−1.09696	−0.548478	−	0.836165i	$-0.684793\pi$
$919$	−30560.7	−1.09696	−0.548478	+	0.836165i	$0.684793\pi$
$920$	−368.464	−0.0132042
$921$	0	0
$922$	−17109.8	−0.611150
$923$	31594.1	1.12669
$924$	0	0
$925$	17185.7	0.610877
$926$	29291.9	1.03952
$927$	0	0
$928$	4815.60	0.170345
$929$	11684.5	0.412654	0.206327	−	0.978483i	$-0.433849\pi$
$929$	11684.5	0.412654	0.206327	+	0.978483i	$0.433849\pi$
$930$	0	0
$931$	4708.96	0.165768
$932$	−23371.8	−0.821425
$933$	0	0
$934$	−60681.0	−2.12585
$935$	212.969	0.00744902
$936$	0	0
$937$	−49115.3	−1.71241	−0.856204	−	0.516638i	$-0.827183\pi$
$937$	−49115.3	−1.71241	−0.856204	+	0.516638i	$0.827183\pi$
$938$	111.327	0.00387521
$939$	0	0
$940$	−5358.19	−0.185920
$941$	−23654.1	−0.819451	−0.409725	−	0.912209i	$-0.634375\pi$
$941$	−23654.1	−0.819451	−0.409725	+	0.912209i	$0.634375\pi$
$942$	0	0
$943$	−5330.85	−0.184089
$944$	−10159.9	−0.350294
$945$	0	0
$946$	−79729.4	−2.74020
$947$	−45583.7	−1.56417	−0.782086	−	0.623171i	$-0.785844\pi$
$947$	−45583.7	−1.56417	−0.782086	+	0.623171i	$0.785844\pi$
$948$	0	0
$949$	−17571.4	−0.601044
$950$	45024.7	1.53768
$951$	0	0
$952$	29.8742	0.00101705
$953$	44104.9	1.49916	0.749579	−	0.661915i	$-0.230256\pi$
$953$	44104.9	1.49916	0.749579	+	0.661915i	$0.230256\pi$
$954$	0	0
$955$	3183.71	0.107877
$956$	−22620.3	−0.765263
$957$	0	0
$958$	−34032.2	−1.14774
$959$	15128.8	0.509421
$960$	0	0
$961$	27567.7	0.925368
$962$	−22467.3	−0.752990
$963$	0	0
$964$	−41866.6	−1.39879
$965$	−2100.10	−0.0700565
$966$	0	0
$967$	41903.8	1.39352	0.696760	−	0.717304i	$-0.254624\pi$
$967$	41903.8	1.39352	0.696760	+	0.717304i	$0.254624\pi$
$968$	919.535	0.0305320
$969$	0	0
$970$	4013.23	0.132842
$971$	−21924.5	−0.724606	−0.362303	−	0.932060i	$-0.618009\pi$
$971$	−21924.5	−0.724606	−0.362303	+	0.932060i	$0.618009\pi$
$972$	0	0
$973$	−16101.1	−0.530503
$974$	53772.9	1.76899
$975$	0	0
$976$	42829.1	1.40464
$977$	−13777.9	−0.451171	−0.225586	−	0.974223i	$-0.572430\pi$
$977$	−13777.9	−0.451171	−0.225586	+	0.974223i	$0.572430\pi$
$978$	0	0
$979$	31658.8	1.03352
$980$	−890.889	−0.0290392
$981$	0	0
$982$	4636.85	0.150680
$983$	−13108.4	−0.425323	−0.212662	−	0.977126i	$-0.568213\pi$
$983$	−13108.4	−0.425323	−0.212662	+	0.977126i	$0.568213\pi$
$984$	0	0
$985$	−7194.31	−0.232721
$986$	156.948	0.00506921
$987$	0	0
$988$	−28437.9	−0.915720
$989$	35440.9	1.13949
$990$	0	0
$991$	−12950.1	−0.415109	−0.207555	−	0.978223i	$-0.566550\pi$
$991$	−12950.1	−0.415109	−0.207555	+	0.978223i	$0.566550\pi$
$992$	−60045.1	−1.92181
$993$	0	0
$994$	−21986.5	−0.701580
$995$	3792.60	0.120838
$996$	0	0
$997$	36119.7	1.14736	0.573682	−	0.819078i	$-0.305515\pi$
$997$	36119.7	1.14736	0.573682	+	0.819078i	$0.305515\pi$
$998$	151.494	0.00480507
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	567.4.a.i.1.2		8
3.2	odd	2		567.4.a.g.1.7		8
9.2	odd	6		189.4.f.b.64.2		16
9.4	even	3		63.4.f.b.43.7	yes	16
9.5	odd	6		189.4.f.b.127.2		16
9.7	even	3		63.4.f.b.22.7	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.4.f.b.22.7	✓	16	9.7	even	3
63.4.f.b.43.7	yes	16	9.4	even	3
189.4.f.b.64.2		16	9.2	odd	6
189.4.f.b.127.2		16	9.5	odd	6
567.4.a.g.1.7		8	3.2	odd	2
567.4.a.i.1.2		8	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:	\( N \)	\(=\)	\( 567 = 3^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	567.a (trivial)

\(f(q)\)	\(=\)	\(q-3.93417 q^{2} +7.47771 q^{4} -2.43142 q^{5} -7.00000 q^{7} +2.05480 q^{8} +O(q^{10})\)	\(q-3.93417 q^{2} +7.47771 q^{4} -2.43142 q^{5} -7.00000 q^{7} +2.05480 q^{8} +9.56561 q^{10} +42.1723 q^{11} -39.5731 q^{13} +27.5392 q^{14} -67.9056 q^{16} -2.07697 q^{17} +96.1013 q^{19} -18.1814 q^{20} -165.913 q^{22} +73.7508 q^{23} -119.088 q^{25} +155.688 q^{26} -52.3439 q^{28} +19.2076 q^{29} -239.497 q^{31} +250.714 q^{32} +8.17114 q^{34} +17.0199 q^{35} -144.310 q^{37} -378.079 q^{38} -4.99606 q^{40} -72.2818 q^{41} +480.549 q^{43} +315.352 q^{44} -290.148 q^{46} +294.707 q^{47} +49.0000 q^{49} +468.513 q^{50} -295.916 q^{52} -627.210 q^{53} -102.538 q^{55} -14.3836 q^{56} -75.5659 q^{58} +149.619 q^{59} -630.716 q^{61} +942.221 q^{62} -443.106 q^{64} +96.2188 q^{65} +4.04248 q^{67} -15.5309 q^{68} -66.9592 q^{70} -798.373 q^{71} +444.022 q^{73} +567.742 q^{74} +718.617 q^{76} -295.206 q^{77} -575.539 q^{79} +165.107 q^{80} +284.369 q^{82} +1290.23 q^{83} +5.04997 q^{85} -1890.56 q^{86} +86.6556 q^{88} +750.701 q^{89} +277.012 q^{91} +551.487 q^{92} -1159.43 q^{94} -233.662 q^{95} +419.548 q^{97} -192.774 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 3 q^{2} + 43 q^{4} + 30 q^{5} - 56 q^{7} + 6 q^{8}+O(q^{10}) \) 8 * q + 3 * q^2 + 43 * q^4 + 30 * q^5 - 56 * q^7 + 6 * q^8	\( 8 q + 3 q^{2} + 43 q^{4} + 30 q^{5} - 56 q^{7} + 6 q^{8} - 14 q^{10} + 24 q^{11} + 68 q^{13} - 21 q^{14} + 103 q^{16} + 168 q^{17} + 176 q^{19} + 330 q^{20} + 151 q^{22} + 228 q^{23} + 244 q^{25} + 795 q^{26} - 301 q^{28} + 618 q^{29} + 72 q^{31} + 786 q^{32} - 261 q^{34} - 210 q^{35} + 210 q^{37} + 1032 q^{38} - 375 q^{40} + 420 q^{41} - 2 q^{43} + 387 q^{44} + 402 q^{46} + 570 q^{47} + 392 q^{49} + 1110 q^{50} - 431 q^{52} + 528 q^{53} - 838 q^{55} - 42 q^{56} + 37 q^{58} - 150 q^{59} + 578 q^{61} + 1170 q^{62} - 112 q^{64} - 366 q^{65} - 898 q^{67} + 2526 q^{68} + 98 q^{70} + 882 q^{71} + 972 q^{73} - 222 q^{74} + 1423 q^{76} - 168 q^{77} - 158 q^{79} + 2475 q^{80} - 211 q^{82} + 2958 q^{83} - 774 q^{85} - 114 q^{86} + 1317 q^{88} + 4380 q^{89} - 476 q^{91} + 4629 q^{92} - 3234 q^{94} + 930 q^{95} - 60 q^{97} + 147 q^{98}+O(q^{100}) \) 8 * q + 3 * q^2 + 43 * q^4 + 30 * q^5 - 56 * q^7 + 6 * q^8 - 14 * q^10 + 24 * q^11 + 68 * q^13 - 21 * q^14 + 103 * q^16 + 168 * q^17 + 176 * q^19 + 330 * q^20 + 151 * q^22 + 228 * q^23 + 244 * q^25 + 795 * q^26 - 301 * q^28 + 618 * q^29 + 72 * q^31 + 786 * q^32 - 261 * q^34 - 210 * q^35 + 210 * q^37 + 1032 * q^38 - 375 * q^40 + 420 * q^41 - 2 * q^43 + 387 * q^44 + 402 * q^46 + 570 * q^47 + 392 * q^49 + 1110 * q^50 - 431 * q^52 + 528 * q^53 - 838 * q^55 - 42 * q^56 + 37 * q^58 - 150 * q^59 + 578 * q^61 + 1170 * q^62 - 112 * q^64 - 366 * q^65 - 898 * q^67 + 2526 * q^68 + 98 * q^70 + 882 * q^71 + 972 * q^73 - 222 * q^74 + 1423 * q^76 - 168 * q^77 - 158 * q^79 + 2475 * q^80 - 211 * q^82 + 2958 * q^83 - 774 * q^85 - 114 * q^86 + 1317 * q^88 + 4380 * q^89 - 476 * q^91 + 4629 * q^92 - 3234 * q^94 + 930 * q^95 - 60 * q^97 + 147 * q^98

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