LMFDB - Embedded newform 275.3.d.c.274.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [275,3,Mod(274,275)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(275, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("275.274");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$N$	$=$	$275 = 5^{2} \cdot 11$
Weight:	$k$	$=$	$3$
Character orbit:	$[\chi]$	$=$	275.d (of order $2$ , degree $1$ , not 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:	$7.49320726991$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{16} - 8 x^{15} + 32 x^{14} - 54 x^{13} + 51 x^{12} - 118 x^{11} + 770 x^{10} - 1222 x^{9} + \cdots + 256$ x^16 - 8x^15 + 32x^14 - 54x^13 + 51x^12 - 118x^11 + 770x^10 - 1222x^9 + 749x^8 + 724x^7 + 3920x^6 - 5016x^5 + 2916x^4 + 7088x^3 + 5408x^2 + 1664*x + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$2^{22}$
Twist minimal:	no (minimal twist has level 55)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			274.9
Root			$-0.230520 + 0.230520i$ of defining polynomial
Character	$\chi$	$=$	275.274
Dual form			275.3.d.c.274.10

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.46104 q^{2} -0.114554i q^{3} -1.86536 q^{4} -0.167368i q^{6} +4.56066 q^{7} -8.56953 q^{8} +8.98688 q^{9} +(5.89241 + 9.28868i) q^{11} +0.213685i q^{12} +16.5645 q^{13} +6.66330 q^{14} -5.05897 q^{16} +17.2939 q^{17} +13.1302 q^{18} -35.8838i q^{19} -0.522441i q^{21} +(8.60904 + 13.5711i) q^{22} +29.3902i q^{23} +0.981674i q^{24} +24.2014 q^{26} -2.06047i q^{27} -8.50728 q^{28} +8.51985i q^{29} -26.3476 q^{31} +26.8868 q^{32} +(1.06406 - 0.674999i) q^{33} +25.2670 q^{34} -16.7638 q^{36} +44.4227i q^{37} -52.4276i q^{38} -1.89753i q^{39} -52.2243i q^{41} -0.763308i q^{42} -6.77375 q^{43} +(-10.9915 - 17.3268i) q^{44} +42.9402i q^{46} +15.0434i q^{47} +0.579525i q^{48} -28.2004 q^{49} -1.98108i q^{51} -30.8989 q^{52} +33.1498i q^{53} -3.01043i q^{54} -39.0827 q^{56} -4.11063 q^{57} +12.4478i q^{58} -51.5447 q^{59} -23.1889i q^{61} -38.4948 q^{62} +40.9861 q^{63} +59.5185 q^{64} +(1.55463 - 0.986200i) q^{66} -113.668i q^{67} -32.2593 q^{68} +3.36676 q^{69} +8.00364 q^{71} -77.0133 q^{72} -32.5342 q^{73} +64.9034i q^{74} +66.9363i q^{76} +(26.8732 + 42.3625i) q^{77} -2.77237i q^{78} +52.0160i q^{79} +80.6459 q^{81} -76.3018i q^{82} +43.3699 q^{83} +0.974543i q^{84} -9.89672 q^{86} +0.975983 q^{87} +(-50.4951 - 79.5996i) q^{88} -73.8028 q^{89} +75.5451 q^{91} -54.8233i q^{92} +3.01822i q^{93} +21.9790i q^{94} -3.07999i q^{96} +22.0298i q^{97} -41.2019 q^{98} +(52.9543 + 83.4762i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$16 q + 56 q^{4} + 8 q^{9} + 16 q^{11} + 176 q^{16} - 200 q^{26} + 72 q^{31} - 160 q^{34} - 432 q^{36} - 24 q^{44} - 344 q^{49} - 160 q^{56} + 32 q^{59} + 1176 q^{64} + 360 q^{66} - 16 q^{69} + 552 q^{71}+ \cdots + 368 q^{99}+O(q^{100})$ 16 * q + 56 * q^4 + 8 * q^9 + 16 * q^11 + 176 * q^16 - 200 * q^26 + 72 * q^31 - 160 * q^34 - 432 * q^36 - 24 * q^44 - 344 * q^49 - 160 * q^56 + 32 * q^59 + 1176 * q^64 + 360 * q^66 - 16 * q^69 + 552 * q^71 - 640 * q^81 + 840 * q^86 - 888 * q^89 - 80 * q^91 + 368 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/275\mathbb{Z}\right)^\times$ .

$n$	$101$	$177$
$\chi(n)$	$-1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$ -expansion $a_n$ , the Satake parameters $\alpha_p$ , and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$ .

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$\alpha_n$			$\theta_n$
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$\alpha_p$			$\theta_p$

$2$	1.46104			0.730520			0.365260	−	0.930906i	$-0.380980\pi$
$2$	1.46104			0.730520			0.365260	+	0.930906i	$0.380980\pi$
$3$		−	0.114554i		−	0.0381847i	−0.999818	−	0.0190923i	$-0.993922\pi$
$3$		−	0.114554i		−	0.0381847i	0.999818	−	0.0190923i	$-0.00607765\pi$
$4$	−1.86536			−0.466341
$5$		0			0
$5$		0			0
$6$		−	0.167368i		−	0.0278947i
$7$	4.56066			0.651522			0.325761	−	0.945452i	$-0.394379\pi$
$7$	4.56066			0.651522			0.325761	+	0.945452i	$0.394379\pi$
$8$	−8.56953			−1.07119
$9$	8.98688			0.998542
$10$		0			0
$11$	5.89241	+	9.28868i	0.535673	+	0.844425i
$11$	5.89241	+	9.28868i	0.535673	+	0.844425i
$12$			0.213685i			0.0178071i
$13$	16.5645			1.27420			0.637098	−	0.770783i	$-0.280135\pi$
$13$	16.5645			1.27420			0.637098	+	0.770783i	$0.280135\pi$
$14$	6.66330			0.475950
$15$		0			0
$16$	−5.05897			−0.316185
$17$	17.2939			1.01729			0.508643	−	0.860978i	$-0.330147\pi$
$17$	17.2939			1.01729			0.508643	+	0.860978i	$0.330147\pi$
$18$	13.1302			0.729455
$19$		−	35.8838i		−	1.88862i	−0.329057	−	0.944310i	$-0.606731\pi$
$19$		−	35.8838i		−	1.88862i	0.329057	−	0.944310i	$-0.393269\pi$
$20$		0			0
$21$		−	0.522441i		−	0.0248782i
$22$	8.60904	+	13.5711i	0.391320	+	0.616869i
$23$			29.3902i			1.27783i	0.769276	+	0.638917i	$0.220617\pi$
$23$			29.3902i			1.27783i	−0.769276	+	0.638917i	$0.779383\pi$
$24$			0.981674i			0.0409031i
$25$		0			0
$26$	24.2014			0.930825
$27$		−	2.06047i		−	0.0763137i
$28$	−8.50728			−0.303831
$29$			8.51985i			0.293788i	0.989152	+	0.146894i	$0.0469276\pi$
$29$			8.51985i			0.293788i	−0.989152	+	0.146894i	$0.953072\pi$
$30$		0			0
$31$	−26.3476			−0.849921			−0.424961	−	0.905212i	$-0.639712\pi$
$31$	−26.3476			−0.849921			−0.424961	+	0.905212i	$0.639712\pi$
$32$	26.8868			0.840211
$33$	1.06406	−	0.674999i	0.0322441	−	0.0204545i
$34$	25.2670			0.743147
$35$		0			0
$36$	−16.7638			−0.465661
$37$			44.4227i			1.20061i	0.799769	+	0.600307i	$0.204955\pi$
$37$			44.4227i			1.20061i	−0.799769	+	0.600307i	$0.795045\pi$
$38$		−	52.4276i		−	1.37967i
$39$		−	1.89753i		−	0.0486547i
$40$		0			0
$41$		−	52.2243i		−	1.27376i	−0.770961	−	0.636882i	$-0.780224\pi$
$41$		−	52.2243i		−	1.27376i	0.770961	−	0.636882i	$-0.219776\pi$
$42$		−	0.763308i		−	0.0181740i
$43$	−6.77375			−0.157529			−0.0787646	−	0.996893i	$-0.525098\pi$
$43$	−6.77375			−0.157529			−0.0787646	+	0.996893i	$0.525098\pi$
$44$	−10.9915	−	17.3268i	−0.249806	−	0.393790i
$45$		0			0
$46$			42.9402i			0.933482i
$47$			15.0434i			0.320072i	0.987111	+	0.160036i	$0.0511611\pi$
$47$			15.0434i			0.320072i	−0.987111	+	0.160036i	$0.948839\pi$
$48$			0.579525i			0.0120734i
$49$	−28.2004			−0.575519
$50$		0			0
$51$		−	1.98108i		−	0.0388447i
$52$	−30.8989			−0.594209
$53$			33.1498i			0.625468i	0.949841	+	0.312734i	$0.101245\pi$
$53$			33.1498i			0.625468i	−0.949841	+	0.312734i	$0.898755\pi$
$54$		−	3.01043i		−	0.0557486i
$55$		0			0
$56$	−39.0827			−0.697905
$57$	−4.11063			−0.0721163
$58$			12.4478i			0.214618i
$59$	−51.5447			−0.873639			−0.436819	−	0.899549i	$-0.643895\pi$
$59$	−51.5447			−0.873639			−0.436819	+	0.899549i	$0.643895\pi$
$60$		0			0
$61$		−	23.1889i		−	0.380146i	−0.981770	−	0.190073i	$-0.939127\pi$
$61$		−	23.1889i		−	0.380146i	0.981770	−	0.190073i	$-0.0608725\pi$
$62$	−38.4948			−0.620884
$63$	40.9861			0.650572
$64$	59.5185			0.929976
$65$		0			0
$66$	1.55463	−	0.986200i	0.0235550	−	0.0149424i
$67$		−	113.668i		−	1.69653i	−0.529572	−	0.848265i	$-0.677647\pi$
$67$		−	113.668i		−	1.69653i	0.529572	−	0.848265i	$-0.322353\pi$
$68$	−32.2593			−0.474402
$69$	3.36676			0.0487936
$70$		0			0
$71$	8.00364			0.112727			0.0563637	−	0.998410i	$-0.482049\pi$
$71$	8.00364			0.112727			0.0563637	+	0.998410i	$0.482049\pi$
$72$	−77.0133			−1.06963
$73$	−32.5342			−0.445674			−0.222837	−	0.974856i	$-0.571532\pi$
$73$	−32.5342			−0.445674			−0.222837	+	0.974856i	$0.571532\pi$
$74$			64.9034i			0.877073i
$75$		0			0
$76$			66.9363i			0.880741i
$77$	26.8732	+	42.3625i	0.349003	+	0.550162i
$78$		−	2.77237i		−	0.0355432i
$79$			52.0160i			0.658431i	0.944255	+	0.329215i	$0.106784\pi$
$79$			52.0160i			0.658431i	−0.944255	+	0.329215i	$0.893216\pi$
$80$		0			0
$81$	80.6459			0.995628
$82$		−	76.3018i		−	0.930510i
$83$	43.3699			0.522529			0.261264	−	0.965267i	$-0.415861\pi$
$83$	43.3699			0.522529			0.261264	+	0.965267i	$0.415861\pi$
$84$			0.974543i			0.0116017i
$85$		0			0
$86$	−9.89672			−0.115078
$87$	0.975983			0.0112182
$88$	−50.4951	−	79.5996i	−0.573808	−	0.904541i
$89$	−73.8028			−0.829245			−0.414623	−	0.909993i	$-0.636086\pi$
$89$	−73.8028			−0.829245			−0.414623	+	0.909993i	$0.636086\pi$
$90$		0			0
$91$	75.5451			0.830166
$92$		−	54.8233i		−	0.595906i
$93$			3.01822i			0.0324540i
$94$			21.9790i			0.233819i
$95$		0			0
$96$		−	3.07999i		−	0.0320832i
$97$			22.0298i			0.227111i	0.993532	+	0.113556i	$0.0362240\pi$
$97$			22.0298i			0.227111i	−0.993532	+	0.113556i	$0.963776\pi$
$98$	−41.2019			−0.420428
$99$	52.9543	+	83.4762i	0.534892	+	0.843194i
$100$		0			0
$101$		−	163.013i		−	1.61399i	−0.590561	−	0.806993i	$-0.701094\pi$
$101$		−	163.013i		−	1.61399i	0.590561	−	0.806993i	$-0.298906\pi$
$102$		−	2.89444i		−	0.0283768i
$103$		−	42.6423i		−	0.414003i	−0.978341	−	0.207001i	$-0.933630\pi$
$103$		−	42.6423i		−	0.414003i	0.978341	−	0.207001i	$-0.0663705\pi$
$104$	−141.950			−1.36491
$105$		0			0
$106$			48.4332i			0.456917i
$107$	−131.098			−1.22522			−0.612608	−	0.790387i	$-0.709879\pi$
$107$	−131.098			−1.22522			−0.612608	+	0.790387i	$0.709879\pi$
$108$			3.84352i			0.0355882i
$109$			54.5201i			0.500184i	0.968222	+	0.250092i	$0.0804609\pi$
$109$			54.5201i			0.500184i	−0.968222	+	0.250092i	$0.919539\pi$
$110$		0			0
$111$	5.08880			0.0458451
$112$	−23.0722			−0.206002
$113$			117.985i			1.04412i	0.852910	+	0.522058i	$0.174836\pi$
$113$			117.985i			1.04412i	−0.852910	+	0.522058i	$0.825164\pi$
$114$	−6.00579			−0.0526824
$115$		0			0
$116$		−	15.8926i		−	0.137005i
$117$	148.863			1.27234
$118$	−75.3088			−0.638210
$119$	78.8713			0.662784
$120$		0			0
$121$	−51.5591	+	109.465i	−0.426108	+	0.904672i
$122$		−	33.8799i		−	0.277704i
$123$	−5.98251			−0.0486383
$124$	49.1478			0.396353
$125$		0			0
$126$	59.8822			0.475256
$127$	−113.764			−0.895782			−0.447891	−	0.894088i	$-0.647825\pi$
$127$	−113.764			−0.895782			−0.447891	+	0.894088i	$0.647825\pi$
$128$	−20.5882			−0.160845
$129$			0.775961i			0.00601520i
$130$		0			0
$131$			115.599i			0.882435i	0.897400	+	0.441218i	$0.145453\pi$
$131$			115.599i			0.882435i	−0.897400	+	0.441218i	$0.854547\pi$
$132$	−1.98485	+	1.25912i	−0.0150367	+	0.00953877i
$133$		−	163.654i		−	1.23048i
$134$		−	166.073i		−	1.23935i
$135$		0			0
$136$	−148.200			−1.08971
$137$		−	216.355i		−	1.57924i	−0.613598	−	0.789618i	$-0.710279\pi$
$137$		−	216.355i		−	1.57924i	0.613598	−	0.789618i	$-0.289721\pi$
$138$	4.91897			0.0356447
$139$		−	5.08013i		−	0.0365477i	−0.999833	−	0.0182738i	$-0.994183\pi$
$139$		−	5.08013i		−	0.0365477i	0.999833	−	0.0182738i	$-0.00581707\pi$
$140$		0			0
$141$	1.72328			0.0122219
$142$	11.6936			0.0823495
$143$	97.6050	+	153.863i	0.682552	+	1.07596i
$144$	−45.4643			−0.315724
$145$		0			0
$146$	−47.5338			−0.325574
$147$			3.23047i			0.0219760i
$148$		−	82.8646i		−	0.559896i
$149$		−	187.947i		−	1.26139i	−0.776030	−	0.630696i	$-0.782769\pi$
$149$		−	187.947i		−	1.26139i	0.776030	−	0.630696i	$-0.217231\pi$
$150$		0			0
$151$		−	136.741i		−	0.905571i	−0.891620	−	0.452785i	$-0.850430\pi$
$151$		−	136.741i		−	0.905571i	0.891620	−	0.452785i	$-0.149570\pi$
$152$			307.507i			2.02307i
$153$	155.418			1.01580
$154$	39.2629	+	61.8932i	0.254954	+	0.401904i
$155$		0			0
$156$			3.53959i			0.0226897i
$157$			256.057i			1.63094i	0.578800	+	0.815470i	$0.303521\pi$
$157$			256.057i			1.63094i	−0.578800	+	0.815470i	$0.696479\pi$
$158$			75.9975i			0.480997i
$159$	3.79744			0.0238833
$160$		0			0
$161$			134.038i			0.832537i
$162$	117.827			0.727326
$163$			25.1370i			0.154215i	0.997023	+	0.0771075i	$0.0245685\pi$
$163$			25.1370i			0.154215i	−0.997023	+	0.0771075i	$0.975432\pi$
$164$			97.4174i			0.594008i
$165$		0			0
$166$	63.3651			0.381718
$167$	−4.21038			−0.0252119			−0.0126059	−	0.999921i	$-0.504013\pi$
$167$	−4.21038			−0.0252119			−0.0126059	+	0.999921i	$0.504013\pi$
$168$			4.47708i			0.0266493i
$169$	105.384			0.623573
$170$		0			0
$171$		−	322.483i		−	1.88587i
$172$	12.6355			0.0734623
$173$	168.015			0.971185			0.485592	−	0.874185i	$-0.338604\pi$
$173$	168.015			0.971185			0.485592	+	0.874185i	$0.338604\pi$
$174$	1.42595			0.00819511
$175$		0			0
$176$	−29.8095	−	46.9911i	−0.169372	−	0.266995i
$177$			5.90465i			0.0333596i
$178$	−107.829			−0.605780
$179$	34.5542			0.193040			0.0965200	−	0.995331i	$-0.469229\pi$
$179$	34.5542			0.193040			0.0965200	+	0.995331i	$0.469229\pi$
$180$		0			0
$181$	−107.349			−0.593091			−0.296545	−	0.955019i	$-0.595835\pi$
$181$	−107.349			−0.593091			−0.296545	+	0.955019i	$0.595835\pi$
$182$	110.374			0.606453
$183$	−2.65638			−0.0145158
$184$		−	251.860i		−	1.36880i
$185$		0			0
$186$			4.40974i			0.0237083i
$187$	101.902	+	160.637i	0.544933	+	0.859022i
$188$		−	28.0614i		−	0.149263i
$189$		−	9.39709i		−	0.0497200i
$190$		0			0
$191$	−75.4470			−0.395011			−0.197505	−	0.980302i	$-0.563284\pi$
$191$	−75.4470			−0.395011			−0.197505	+	0.980302i	$0.563284\pi$
$192$		−	6.81808i		−	0.0355108i
$193$	−167.064			−0.865617			−0.432809	−	0.901486i	$-0.642477\pi$
$193$	−167.064			−0.865617			−0.432809	+	0.901486i	$0.642477\pi$
$194$			32.1864i			0.165909i
$195$		0			0
$196$	52.6040			0.268388
$197$	362.777			1.84151			0.920754	−	0.390145i	$-0.127575\pi$
$197$	362.777			1.84151			0.920754	+	0.390145i	$0.127575\pi$
$198$	77.3684	+	121.962i	0.390749	+	0.615970i
$199$	−337.735			−1.69716			−0.848580	−	0.529068i	$-0.822542\pi$
$199$	−337.735			−1.69716			−0.848580	+	0.529068i	$0.822542\pi$
$200$		0			0
$201$	−13.0211			−0.0647814
$202$		−	238.168i		−	1.17905i
$203$			38.8561i			0.191409i
$204$			3.69543i			0.0181149i
$205$		0			0
$206$		−	62.3021i		−	0.302437i
$207$			264.126i			1.27597i
$208$	−83.7994			−0.402882
$209$	333.313	−	211.442i	1.59480	−	1.01168i
$210$		0			0
$211$			317.915i			1.50670i	0.657617	+	0.753352i	$0.271564\pi$
$211$			317.915i			1.50670i	−0.657617	+	0.753352i	$0.728436\pi$
$212$		−	61.8364i		−	0.291681i
$213$		−	0.916849i		−	0.00430446i
$214$	−191.539			−0.895044
$215$		0			0
$216$			17.6572i			0.0817465i
$217$	−120.162			−0.553743
$218$			79.6560i			0.365395i
$219$			3.72692i			0.0170179i
$220$		0			0
$221$	286.465			1.29622
$222$	7.43494			0.0334907
$223$		−	292.643i		−	1.31230i	−0.754630	−	0.656151i	$-0.772183\pi$
$223$		−	292.643i		−	1.31230i	0.754630	−	0.656151i	$-0.227817\pi$
$224$	122.621			0.547416
$225$		0			0
$226$			172.381i			0.762747i
$227$	9.54908			0.0420664			0.0210332	−	0.999779i	$-0.493304\pi$
$227$	9.54908			0.0420664			0.0210332	+	0.999779i	$0.493304\pi$
$228$	7.66782			0.0336308
$229$	355.912			1.55420			0.777100	−	0.629377i	$-0.216690\pi$
$229$	355.912			1.55420			0.777100	+	0.629377i	$0.216690\pi$
$230$		0			0
$231$	4.85279	−	3.07844i	0.0210077	−	0.0133266i
$232$		−	73.0111i		−	0.314703i
$233$	270.961			1.16292			0.581462	−	0.813573i	$-0.302481\pi$
$233$	270.961			1.16292			0.581462	+	0.813573i	$0.302481\pi$
$234$	217.495			0.929467
$235$		0			0
$236$	96.1496			0.407413
$237$	5.95864			0.0251420
$238$	115.234			0.484177
$239$			79.1374i			0.331119i	0.986200	+	0.165560i	$0.0529430\pi$
$239$			79.1374i			0.331119i	−0.986200	+	0.165560i	$0.947057\pi$
$240$		0			0
$241$			146.567i			0.608164i	0.952646	+	0.304082i	$0.0983496\pi$
$241$			146.567i			0.608164i	−0.952646	+	0.304082i	$0.901650\pi$
$242$	−75.3299	+	159.933i	−0.311281	+	0.660881i
$243$		−	27.7825i		−	0.114331i
$244$			43.2558i			0.177278i
$245$		0			0
$246$	−8.74068			−0.0355312
$247$		−	594.398i		−	2.40647i
$248$	225.786			0.910428
$249$		−	4.96819i		−	0.0199526i
$250$		0			0
$251$	52.7485			0.210154			0.105077	−	0.994464i	$-0.466491\pi$
$251$	52.7485			0.210154			0.105077	+	0.994464i	$0.466491\pi$
$252$	−76.4539			−0.303388
$253$	−272.996	+	173.179i	−1.07903	+	0.684501i
$254$	−166.214			−0.654386
$255$		0			0
$256$	−268.154			−1.04748
$257$			41.8668i			0.162906i	0.996677	+	0.0814529i	$0.0259560\pi$
$257$			41.8668i			0.162906i	−0.996677	+	0.0814529i	$0.974044\pi$
$258$			1.13371i			0.00439422i
$259$			202.597i			0.782227i
$260$		0			0
$261$			76.5668i			0.293360i
$262$			168.895i			0.644636i
$263$	−289.448			−1.10056			−0.550282	−	0.834979i	$-0.685480\pi$
$263$	−289.448			−1.10056			−0.550282	+	0.834979i	$0.685480\pi$
$264$	−9.11845	+	5.78442i	−0.0345396	+	0.0219107i
$265$		0			0
$266$		−	239.104i		−	0.898888i
$267$			8.45441i			0.0316645i
$268$			212.031i			0.791161i
$269$	−297.640			−1.10647			−0.553234	−	0.833026i	$-0.686607\pi$
$269$	−297.640			−1.10647			−0.553234	+	0.833026i	$0.686607\pi$
$270$		0			0
$271$		−	277.488i		−	1.02394i	−0.859003	−	0.511970i	$-0.828916\pi$
$271$		−	277.488i		−	1.02394i	0.859003	−	0.511970i	$-0.171084\pi$
$272$	−87.4890			−0.321651
$273$		−	8.65400i		−	0.0316996i
$274$		−	316.104i		−	1.15366i
$275$		0			0
$276$	−6.28023			−0.0227545
$277$	22.6740			0.0818556			0.0409278	−	0.999162i	$-0.486969\pi$
$277$	22.6740			0.0818556			0.0409278	+	0.999162i	$0.486969\pi$
$278$		−	7.42227i		−	0.0266988i
$279$	−236.782			−0.848682
$280$		0			0
$281$			182.887i			0.650845i	0.945569	+	0.325422i	$0.105506\pi$
$281$			182.887i			0.650845i	−0.945569	+	0.325422i	$0.894494\pi$
$282$	2.51778			0.00892831
$283$	−367.233			−1.29764			−0.648821	−	0.760941i	$-0.724738\pi$
$283$	−367.233			−1.29764			−0.648821	+	0.760941i	$0.724738\pi$
$284$	−14.9297			−0.0525694
$285$		0			0
$286$	142.605	+	224.799i	0.498618	+	0.786012i
$287$		−	238.177i		−	0.829886i
$288$	241.628			0.838986
$289$	10.0774			0.0348699
$290$		0			0
$291$	2.52360			0.00867218
$292$	60.6881			0.207836
$293$	10.3180			0.0352150			0.0176075	−	0.999845i	$-0.494395\pi$
$293$	10.3180			0.0352150			0.0176075	+	0.999845i	$0.494395\pi$
$294$			4.71985i			0.0160539i
$295$		0			0
$296$		−	380.682i		−	1.28609i
$297$	19.1390	−	12.1411i	0.0644412	−	0.0408792i
$298$		−	274.599i		−	0.921472i
$299$			486.834i			1.62821i
$300$		0			0
$301$	−30.8928			−0.102634
$302$		−	199.784i		−	0.661537i
$303$	−18.6737			−0.0616295
$304$			181.535i			0.597154i
$305$		0			0
$306$	227.071			0.742064
$307$	−459.671			−1.49730			−0.748649	−	0.662966i	$-0.769297\pi$
$307$	−459.671			−1.49730			−0.748649	+	0.662966i	$0.769297\pi$
$308$	−50.1283	−	79.0214i	−0.162754	−	0.256563i
$309$	−4.88484			−0.0158086
$310$		0			0
$311$	101.131			0.325181			0.162590	−	0.986694i	$-0.448015\pi$
$311$	101.131			0.325181			0.162590	+	0.986694i	$0.448015\pi$
$312$			16.2610i			0.0521185i
$313$		−	288.760i		−	0.922556i	−0.887256	−	0.461278i	$-0.847391\pi$
$313$		−	288.760i		−	0.922556i	0.887256	−	0.461278i	$-0.152609\pi$
$314$			374.110i			1.19143i
$315$		0			0
$316$		−	97.0288i		−	0.307053i
$317$			242.929i			0.766338i	0.923678	+	0.383169i	$0.125167\pi$
$317$			242.929i			0.766338i	−0.923678	+	0.383169i	$0.874833\pi$
$318$	5.54821			0.0174472
$319$	−79.1381	+	50.2024i	−0.248082	+	0.157374i
$320$		0			0
$321$			15.0178i			0.0467845i
$322$			195.835i			0.608185i
$323$		−	620.569i		−	1.92127i
$324$	−150.434			−0.464302
$325$		0			0
$326$			36.7262i			0.112657i
$327$	6.24550			0.0190994
$328$			447.538i			1.36445i
$329$			68.6078i			0.208534i
$330$		0			0
$331$	−241.395			−0.729291			−0.364646	−	0.931146i	$-0.618810\pi$
$331$	−241.395			−0.729291			−0.364646	+	0.931146i	$0.618810\pi$
$332$	−80.9006			−0.243676
$333$			399.222i			1.19886i
$334$	−6.15154			−0.0184178
$335$		0			0
$336$			2.64301i			0.00786611i
$337$	−527.662			−1.56576			−0.782881	−	0.622171i	$-0.786251\pi$
$337$	−527.662			−1.56576			−0.782881	+	0.622171i	$0.786251\pi$
$338$	153.970			0.455532
$339$	13.5157			0.0398692
$340$		0			0
$341$	−155.251	−	244.734i	−0.455280	−	0.717695i
$342$		−	471.161i		−	1.37766i
$343$	−352.085			−1.02649
$344$	58.0479			0.168744
$345$		0			0
$346$	245.476			0.709470
$347$	−214.531			−0.618244			−0.309122	−	0.951022i	$-0.600035\pi$
$347$	−214.531			−0.618244			−0.309122	+	0.951022i	$0.600035\pi$
$348$	−1.82056			−0.00523150
$349$			76.4945i			0.219182i	0.993977	+	0.109591i	$0.0349541\pi$
$349$			76.4945i			0.219182i	−0.993977	+	0.109591i	$0.965046\pi$
$350$		0			0
$351$		−	34.1307i		−	0.0972385i
$352$	158.428	+	249.743i	0.450079	+	0.709496i
$353$			176.689i			0.500535i	0.968177	+	0.250268i	$0.0805186\pi$
$353$			176.689i			0.500535i	−0.968177	+	0.250268i	$0.919481\pi$
$354$			8.62693i			0.0243699i
$355$		0			0
$356$	137.669			0.386711
$357$		−	9.03503i		−	0.0253082i
$358$	50.4850			0.141020
$359$		−	357.810i		−	0.996686i	−0.866980	−	0.498343i	$-0.833942\pi$
$359$		−	357.810i		−	0.996686i	0.866980	−	0.498343i	$-0.166058\pi$
$360$		0			0
$361$	−926.646			−2.56689
$362$	−156.842			−0.433265
$363$	12.5397	+	5.90630i	0.0345446	+	0.0162708i
$364$	−140.919			−0.387140
$365$		0			0
$366$	−3.88108			−0.0106041
$367$		−	314.687i		−	0.857458i	−0.903433	−	0.428729i	$-0.858962\pi$
$367$		−	314.687i		−	0.857458i	0.903433	−	0.428729i	$-0.141038\pi$
$368$		−	148.684i		−	0.404032i
$369$		−	469.334i		−	1.27191i
$370$		0			0
$371$			151.185i			0.407506i
$372$		−	5.63007i		−	0.0151346i
$373$	420.645			1.12773			0.563867	−	0.825865i	$-0.309313\pi$
$373$	420.645			1.12773			0.563867	+	0.825865i	$0.309313\pi$
$374$	148.883	+	234.697i	0.398084	+	0.627532i
$375$		0			0
$376$		−	128.915i		−	0.342859i
$377$			141.127i			0.374343i
$378$		−	13.7295i		−	0.0363215i
$379$	−401.667			−1.05981			−0.529904	−	0.848058i	$-0.677772\pi$
$379$	−401.667			−1.05981			−0.529904	+	0.848058i	$0.677772\pi$
$380$		0			0
$381$			13.0322i			0.0342051i
$382$	−110.231			−0.288563
$383$		−	60.0864i		−	0.156884i	−0.996919	−	0.0784418i	$-0.975006\pi$
$383$		−	60.0864i		−	0.156884i	0.996919	−	0.0784418i	$-0.0249945\pi$
$384$			2.35846i			0.00614182i
$385$		0			0
$386$	−244.087			−0.632350
$387$	−60.8749			−0.157299
$388$		−	41.0936i		−	0.105911i
$389$	−411.464			−1.05775			−0.528874	−	0.848700i	$-0.677386\pi$
$389$	−411.464			−1.05775			−0.528874	+	0.848700i	$0.677386\pi$
$390$		0			0
$391$			508.269i			1.29992i
$392$	241.664			0.616491
$393$	13.2423			0.0336955
$394$	530.031			1.34526
$395$		0			0
$396$	−98.7791	−	155.713i	−0.249442	−	0.393216i
$397$		−	237.517i		−	0.598279i	−0.954209	−	0.299140i	$-0.903300\pi$
$397$		−	237.517i		−	0.598279i	0.954209	−	0.299140i	$-0.0966997\pi$
$398$	−493.444			−1.23981
$399$	−18.7472			−0.0469854
$400$		0			0
$401$	101.501			0.253120			0.126560	−	0.991959i	$-0.459606\pi$
$401$	101.501			0.253120			0.126560	+	0.991959i	$0.459606\pi$
$402$	−19.0243			−0.0473241
$403$	−436.435			−1.08297
$404$			304.078i			0.752667i
$405$		0			0
$406$			56.7703i			0.139828i
$407$	−412.629	+	261.757i	−1.01383	+	0.643137i
$408$			16.9769i			0.0416101i
$409$		−	694.176i		−	1.69725i	−0.528993	−	0.848626i	$-0.677430\pi$
$409$		−	694.176i		−	1.69725i	0.528993	−	0.848626i	$-0.322570\pi$
$410$		0			0
$411$	−24.7844			−0.0603026
$412$			79.5433i			0.193066i
$413$	−235.078			−0.569195
$414$			385.898i			0.932121i
$415$		0			0
$416$	445.367			1.07059
$417$	−0.581949			−0.00139556
$418$	486.983	−	308.925i	1.16503	−	0.739055i
$419$	416.354			0.993686			0.496843	−	0.867841i	$-0.334493\pi$
$419$	416.354			0.993686			0.496843	+	0.867841i	$0.334493\pi$
$420$		0			0
$421$	539.341			1.28110			0.640548	−	0.767918i	$-0.278707\pi$
$421$	539.341			1.28110			0.640548	+	0.767918i	$0.278707\pi$
$422$			464.486i			1.10068i
$423$			135.193i			0.319606i
$424$		−	284.078i		−	0.669995i
$425$		0			0
$426$		−	1.33955i		−	0.00314449i
$427$		−	105.757i		−	0.247674i
$428$	244.546			0.571368
$429$	17.6256	−	11.1810i	0.0410853	−	0.0260630i
$430$		0			0
$431$			343.493i			0.796967i	0.917176	+	0.398483i	$0.130463\pi$
$431$			343.493i			0.796967i	−0.917176	+	0.398483i	$0.869537\pi$
$432$			10.4238i			0.0241293i
$433$		−	18.0353i		−	0.0416520i	−0.999783	−	0.0208260i	$-0.993370\pi$
$433$		−	18.0353i		−	0.0416520i	0.999783	−	0.0208260i	$-0.00662960\pi$
$434$	−175.562			−0.404520
$435$		0			0
$436$		−	101.700i		−	0.233256i
$437$	1054.63			2.41334
$438$			5.44518i			0.0124319i
$439$			693.839i			1.58050i	0.612786	+	0.790249i	$0.290049\pi$
$439$			693.839i			1.58050i	−0.612786	+	0.790249i	$0.709951\pi$
$440$		0			0
$441$	−253.434			−0.574680
$442$	418.536			0.946914
$443$			359.314i			0.811092i	0.914075	+	0.405546i	$0.132919\pi$
$443$			359.314i			0.811092i	−0.914075	+	0.405546i	$0.867081\pi$
$444$	−9.49247			−0.0213794
$445$		0			0
$446$		−	427.564i		−	0.958663i
$447$	−21.5301			−0.0481658
$448$	271.443			0.605900
$449$	567.882			1.26477			0.632385	−	0.774654i	$-0.282076\pi$
$449$	567.882			1.26477			0.632385	+	0.774654i	$0.282076\pi$
$450$		0			0
$451$	485.095	−	307.727i	1.07560	−	0.682322i
$452$		−	220.085i		−	0.486914i
$453$	−15.6643			−0.0345789
$454$	13.9516			0.0307303
$455$		0			0
$456$	35.2262			0.0772504
$457$	797.218			1.74446			0.872230	−	0.489096i	$-0.162673\pi$
$457$	797.218			1.74446			0.872230	+	0.489096i	$0.162673\pi$
$458$	520.001			1.13537
$459$		−	35.6335i		−	0.0776328i
$460$		0			0
$461$		−	87.9144i		−	0.190704i	−0.995444	−	0.0953518i	$-0.969602\pi$
$461$		−	87.9144i		−	0.190704i	0.995444	−	0.0953518i	$-0.0303976\pi$
$462$	7.09012	−	4.49772i	0.0153466	−	0.00973532i
$463$			50.9282i			0.109996i	0.998486	+	0.0549981i	$0.0175153\pi$
$463$			50.9282i			0.109996i	−0.998486	+	0.0549981i	$0.982485\pi$
$464$		−	43.1016i		−	0.0928914i
$465$		0			0
$466$	395.885			0.849539
$467$		−	463.064i		−	0.991572i	−0.868445	−	0.495786i	$-0.834880\pi$
$467$		−	463.064i		−	0.991572i	0.868445	−	0.495786i	$-0.165120\pi$
$468$	−277.684			−0.593343
$469$		−	518.398i		−	1.10533i
$470$		0			0
$471$	29.3324			0.0622769
$472$	441.714			0.935834
$473$	−39.9137	−	62.9192i	−0.0843841	−	0.133022i
$474$	8.70581			0.0183667
$475$		0			0
$476$	−147.124			−0.309083
$477$			297.913i			0.624556i
$478$			115.623i			0.241889i
$479$			182.184i			0.380343i	0.981751	+	0.190171i	$0.0609044\pi$
$479$			182.184i			0.380343i	−0.981751	+	0.190171i	$0.939096\pi$
$480$		0			0
$481$			735.842i			1.52982i
$482$			214.141i			0.444276i
$483$	15.3546			0.0317901
$484$	96.1765	−	204.193i	0.198712	−	0.421886i
$485$		0			0
$486$		−	40.5914i		−	0.0835213i
$487$			366.810i			0.753203i	0.926375	+	0.376602i	$0.122907\pi$
$487$			366.810i			0.753203i	−0.926375	+	0.376602i	$0.877093\pi$
$488$			198.718i			0.407209i
$489$	2.87955			0.00588865
$490$		0			0
$491$		−	42.0181i		−	0.0855766i	−0.999084	−	0.0427883i	$-0.986376\pi$
$491$		−	42.0181i		−	0.0855766i	0.999084	−	0.0427883i	$-0.0136241\pi$
$492$	11.1596			0.0226820
$493$			147.341i			0.298866i
$494$		−	868.439i		−	1.75797i
$495$		0			0
$496$	133.291			0.268733
$497$	36.5018			0.0734444
$498$		−	7.25873i		−	0.0145758i
$499$	229.963			0.460848			0.230424	−	0.973090i	$-0.425989\pi$
$499$	229.963			0.460848			0.230424	+	0.973090i	$0.425989\pi$
$500$		0			0
$501$			0.482316i			0.000962707i
$502$	77.0677			0.153521
$503$	−9.38400			−0.0186561			−0.00932803	−	0.999956i	$-0.502969\pi$
$503$	−9.38400			−0.0186561			−0.00932803	+	0.999956i	$0.502969\pi$
$504$	−351.231			−0.696887
$505$		0			0
$506$	−398.858	+	253.021i	−0.788256	+	0.500042i
$507$		−	12.0721i		−	0.0238109i
$508$	212.212			0.417740
$509$	584.355			1.14804			0.574022	−	0.818840i	$-0.305382\pi$
$509$	584.355			1.14804			0.574022	+	0.818840i	$0.305382\pi$
$510$		0			0
$511$	−148.377			−0.290367
$512$	−309.431			−0.604357
$513$	−73.9374			−0.144128
$514$			61.1691i			0.119006i
$515$		0			0
$516$		−	1.44745i		−	0.00280513i
$517$	−139.733	+	88.6418i	−0.270277	+	0.171454i
$518$			296.002i			0.571432i
$519$		−	19.2468i		−	0.0370844i
$520$		0			0
$521$	121.060			0.232361			0.116180	−	0.993228i	$-0.462935\pi$
$521$	121.060			0.232361			0.116180	+	0.993228i	$0.462935\pi$
$522$			111.867i			0.214305i
$523$	446.636			0.853988			0.426994	−	0.904254i	$-0.359572\pi$
$523$	446.636			0.853988			0.426994	+	0.904254i	$0.359572\pi$
$524$		−	215.634i		−	0.411516i
$525$		0			0
$526$	−422.896			−0.803984
$527$	−455.651			−0.864613
$528$	−5.38302	+	3.41480i	−0.0101951	+	0.00646742i
$529$	−334.782			−0.632858
$530$		0			0
$531$	−463.226			−0.872365
$532$			305.273i			0.573822i
$533$		−	865.072i		−	1.62302i
$534$			12.3522i			0.0231315i
$535$		0			0
$536$			974.077i			1.81731i
$537$		−	3.95832i		−	0.00737117i
$538$	−434.864			−0.808297
$539$	−166.168	−	261.945i	−0.308290	−	0.485983i
$540$		0			0
$541$		−	472.750i		−	0.873846i	−0.899499	−	0.436923i	$-0.856068\pi$
$541$		−	472.750i		−	0.873846i	0.899499	−	0.436923i	$-0.143932\pi$
$542$		−	405.421i		−	0.748009i
$543$			12.2973i			0.0226470i
$544$	464.976			0.854735
$545$		0			0
$546$		−	12.6438i		−	0.0231572i
$547$	24.2342			0.0443039			0.0221520	−	0.999755i	$-0.492948\pi$
$547$	24.2342			0.0443039			0.0221520	+	0.999755i	$0.492948\pi$
$548$			403.582i			0.736463i
$549$		−	208.396i		−	0.379592i
$550$		0			0
$551$	305.724			0.554854
$552$	−28.8516			−0.0522673
$553$			237.227i			0.428982i
$554$	33.1276			0.0597972
$555$		0			0
$556$			9.47629i			0.0170437i
$557$	597.483			1.07268			0.536340	−	0.844002i	$-0.319807\pi$
$557$	597.483			1.07268			0.536340	+	0.844002i	$0.319807\pi$
$558$	−345.948			−0.619979
$559$	−112.204			−0.200723
$560$		0			0
$561$	18.4016	−	11.6733i	0.0328015	−	0.0208081i
$562$			267.206i			0.475455i
$563$	963.269			1.71096			0.855479	−	0.517838i	$-0.173263\pi$
$563$	963.269			1.71096			0.855479	+	0.517838i	$0.173263\pi$
$564$	−3.21455			−0.00569955
$565$		0			0
$566$	−536.541			−0.947953
$567$	367.798			0.648674
$568$	−68.5874			−0.120753
$569$			136.737i			0.240310i	0.992755	+	0.120155i	$0.0383392\pi$
$569$			136.737i			0.240310i	−0.992755	+	0.120155i	$0.961661\pi$
$570$		0			0
$571$		−	829.845i		−	1.45332i	−0.686998	−	0.726659i	$-0.741072\pi$
$571$		−	829.845i		−	1.45332i	0.686998	−	0.726659i	$-0.258928\pi$
$572$	−182.069	−	287.010i	−0.318302	−	0.501765i
$573$			8.64276i			0.0150833i
$574$		−	347.986i		−	0.606248i
$575$		0			0
$576$	534.885			0.928620
$577$			636.305i			1.10278i	0.834247	+	0.551391i	$0.185903\pi$
$577$			636.305i			1.10278i	−0.834247	+	0.551391i	$0.814097\pi$
$578$	14.7235			0.0254732
$579$			19.1379i			0.0330533i
$580$		0			0
$581$	197.795			0.340439
$582$	3.68708			0.00633520
$583$	−307.918	+	195.332i	−0.528161	+	0.335046i
$584$	278.803			0.477402
$585$		0			0
$586$	15.0750			0.0257253
$587$		−	455.242i		−	0.775540i	−0.921756	−	0.387770i	$-0.873246\pi$
$587$		−	455.242i		−	0.775540i	0.921756	−	0.387770i	$-0.126754\pi$
$588$		−	6.02600i		−	0.0102483i
$589$			945.450i			1.60518i
$590$		0			0
$591$		−	41.5576i		−	0.0703173i
$592$		−	224.733i		−	0.379617i
$593$	−219.359			−0.369914			−0.184957	−	0.982747i	$-0.559215\pi$
$593$	−219.359			−0.369914			−0.184957	+	0.982747i	$0.559215\pi$
$594$	27.9629	−	17.7387i	0.0470756	−	0.0298631i
$595$		0			0
$596$			350.590i			0.588239i
$597$			38.6889i			0.0648055i
$598$			711.284i			1.18944i
$599$	260.846			0.435470			0.217735	−	0.976008i	$-0.430133\pi$
$599$	260.846			0.435470			0.217735	+	0.976008i	$0.430133\pi$
$600$		0			0
$601$		−	314.178i		−	0.522759i	−0.965236	−	0.261379i	$-0.915823\pi$
$601$		−	314.178i		−	0.522759i	0.965236	−	0.261379i	$-0.0841773\pi$
$602$	−45.1355			−0.0749760
$603$		−	1021.52i		−	1.69406i
$604$			255.072i			0.422305i
$605$		0			0
$606$	−27.2831			−0.0450216
$607$	−423.627			−0.697903			−0.348951	−	0.937141i	$-0.613462\pi$
$607$	−423.627			−0.697903			−0.348951	+	0.937141i	$0.613462\pi$
$608$		−	964.799i		−	1.58684i
$609$	4.45112			0.00730890
$610$		0			0
$611$			249.187i			0.407835i
$612$	−289.911			−0.473710
$613$	144.866			0.236323			0.118162	−	0.992994i	$-0.462300\pi$
$613$	144.866			0.236323			0.118162	+	0.992994i	$0.462300\pi$
$614$	−671.597			−1.09381
$615$		0			0
$616$	−230.291	−	363.026i	−0.373849	−	0.589328i
$617$			988.195i			1.60161i	0.598923	+	0.800806i	$0.295595\pi$
$617$			988.195i			1.60161i	−0.598923	+	0.800806i	$0.704405\pi$
$618$	−7.13695			−0.0115485
$619$	−822.930			−1.32945			−0.664725	−	0.747088i	$-0.731451\pi$
$619$	−822.930			−1.32945			−0.664725	+	0.747088i	$0.731451\pi$
$620$		0			0
$621$	60.5575			0.0975161
$622$	147.757			0.237551
$623$	−336.589			−0.540272
$624$			9.59956i			0.0153839i
$625$		0			0
$626$		−	421.890i		−	0.673946i
$627$	−24.2215	−	38.1823i	−0.0386308	−	0.0608969i
$628$		−	477.640i		−	0.760574i
$629$			768.240i			1.22137i
$630$		0			0
$631$	−854.486			−1.35418			−0.677089	−	0.735902i	$-0.736759\pi$
$631$	−854.486			−1.35418			−0.677089	+	0.735902i	$0.736759\pi$
$632$		−	445.753i		−	0.705305i
$633$	36.4184			0.0575330
$634$			354.929i			0.559825i
$635$		0			0
$636$	−7.08361			−0.0111377
$637$	−467.127			−0.733323
$638$	−115.624	+	73.3477i	−0.181229	+	0.114965i
$639$	71.9277			0.112563
$640$		0			0
$641$	−70.2095			−0.109531			−0.0547656	−	0.998499i	$-0.517441\pi$
$641$	−70.2095			−0.109531			−0.0547656	+	0.998499i	$0.517441\pi$
$642$			21.9416i			0.0341770i
$643$			560.759i			0.872097i	0.899923	+	0.436049i	$0.143622\pi$
$643$			560.759i			0.872097i	−0.899923	+	0.436049i	$0.856378\pi$
$644$		−	250.030i		−	0.388246i
$645$		0			0
$646$		−	906.676i		−	1.40352i
$647$			73.2119i			0.113156i	0.998398	+	0.0565780i	$0.0180189\pi$
$647$			73.2119i			0.113156i	−0.998398	+	0.0565780i	$0.981981\pi$
$648$	−691.097			−1.06651
$649$	−303.722	−	478.782i	−0.467985	−	0.737723i
$650$		0			0
$651$			13.7651i			0.0211445i
$652$		−	46.8897i		−	0.0719167i
$653$			52.7951i			0.0808500i	0.999183	+	0.0404250i	$0.0128712\pi$
$653$			52.7951i			0.0808500i	−0.999183	+	0.0404250i	$0.987129\pi$
$654$	9.12492			0.0139525
$655$		0			0
$656$			264.201i			0.402746i
$657$	−292.381			−0.445024
$658$			100.239i			0.152338i
$659$		−	466.110i		−	0.707299i	−0.935378	−	0.353649i	$-0.884941\pi$
$659$		−	466.110i		−	0.707299i	0.935378	−	0.353649i	$-0.115059\pi$
$660$		0			0
$661$	759.575			1.14913			0.574565	−	0.818459i	$-0.305171\pi$
$661$	759.575			1.14913			0.574565	+	0.818459i	$0.305171\pi$
$662$	−352.688			−0.532762
$663$		−	32.8157i		−	0.0494957i
$664$	−371.659			−0.559728
$665$		0			0
$666$			583.279i			0.875794i
$667$	−250.400			−0.375412
$668$	7.85389			0.0117573
$669$	−33.5235			−0.0501098
$670$		0			0
$671$	215.394	−	136.639i	0.321005	−	0.203634i
$672$		−	14.0468i		−	0.0209029i
$673$	−608.412			−0.904030			−0.452015	−	0.892010i	$-0.649295\pi$
$673$	−608.412			−0.904030			−0.452015	+	0.892010i	$0.649295\pi$
$674$	−770.935			−1.14382
$675$		0			0
$676$	−196.579			−0.290798
$677$	−557.900			−0.824077			−0.412038	−	0.911166i	$-0.635183\pi$
$677$	−557.900			−0.824077			−0.412038	+	0.911166i	$0.635183\pi$
$678$	19.7469			0.0291253
$679$			100.470i			0.147968i
$680$		0			0
$681$		−	1.09388i		−	0.00160629i
$682$	−226.827	−	357.566i	−0.332591	−	0.524290i
$683$			785.901i			1.15066i	0.817921	+	0.575330i	$0.195126\pi$
$683$			785.901i			1.15066i	−0.817921	+	0.575330i	$0.804874\pi$
$684$			601.548i			0.879457i
$685$		0			0
$686$	−514.409			−0.749868
$687$		−	40.7711i		−	0.0593466i
$688$	34.2682			0.0498084
$689$			549.111i			0.796968i
$690$		0			0
$691$	480.448			0.695293			0.347647	−	0.937626i	$-0.386981\pi$
$691$	480.448			0.695293			0.347647	+	0.937626i	$0.386981\pi$
$692$	−313.409			−0.452903
$693$	241.506	+	380.706i	0.348494	+	0.549360i
$694$	−313.438			−0.451640
$695$		0			0
$696$	−8.36371			−0.0120168
$697$		−	903.160i		−	1.29578i
$698$			111.762i			0.160117i
$699$		−	31.0397i		−	0.0444059i
$700$		0			0
$701$		−	284.947i		−	0.406486i	−0.979128	−	0.203243i	$-0.934852\pi$
$701$		−	284.947i		−	0.406486i	0.979128	−	0.203243i	$-0.0651481\pi$
$702$		−	49.8663i		−	0.0710346i
$703$	1594.06			2.26750
$704$	350.707	+	552.848i	0.498163	+	0.785296i
$705$		0			0
$706$			258.150i			0.365651i
$707$		−	743.444i		−	1.05155i
$708$		−	11.0143i		−	0.0155570i
$709$	313.418			0.442057			0.221028	−	0.975267i	$-0.429059\pi$
$709$	313.418			0.442057			0.221028	+	0.975267i	$0.429059\pi$
$710$		0			0
$711$			467.462i			0.657471i
$712$	632.455			0.888280
$713$		−	774.359i		−	1.08606i
$714$		−	13.2005i		−	0.0184881i
$715$		0			0
$716$	−64.4561			−0.0900224
$717$	9.06551			0.0126437
$718$		−	522.775i		−	0.728099i
$719$	1198.70			1.66718			0.833590	−	0.552384i	$-0.186282\pi$
$719$	1198.70			1.66718			0.833590	+	0.552384i	$0.186282\pi$
$720$		0			0
$721$		−	194.477i		−	0.269732i
$722$	−1353.87			−1.87516
$723$	16.7899			0.0232225
$724$	200.246			0.276583
$725$		0			0
$726$	18.3210	+	8.62934i	0.0252355	+	0.0118861i
$727$			152.540i			0.209821i	0.994482	+	0.104910i	$0.0334555\pi$
$727$			152.540i			0.209821i	−0.994482	+	0.104910i	$0.966544\pi$
$728$	−647.386			−0.889267
$729$	722.630			0.991262
$730$		0			0
$731$	−117.144			−0.160252
$732$	4.95512			0.00676929
$733$	1107.11			1.51039			0.755194	−	0.655501i	$-0.227543\pi$
$733$	1107.11			1.51039			0.755194	+	0.655501i	$0.227543\pi$
$734$		−	459.770i		−	0.626390i
$735$		0			0
$736$			790.206i			1.07365i
$737$	1055.82	−	669.775i	1.43259	−	0.908786i
$738$		−	685.715i		−	0.929153i
$739$			438.677i			0.593609i	0.954938	+	0.296805i	$0.0959210\pi$
$739$			438.677i			0.593609i	−0.954938	+	0.296805i	$0.904079\pi$
$740$		0			0
$741$	−68.0907			−0.0918903
$742$			220.887i			0.297691i
$743$	−764.284			−1.02865			−0.514323	−	0.857597i	$-0.671957\pi$
$743$	−764.284			−1.02865			−0.514323	+	0.857597i	$0.671957\pi$
$744$		−	25.8647i		−	0.0347644i
$745$		0			0
$746$	614.579			0.823832
$747$	389.760			0.521767
$748$	−190.085	−	299.646i	−0.254124	−	0.400597i
$749$	−597.893			−0.798255
$750$		0			0
$751$	1224.32			1.63025			0.815126	−	0.579283i	$-0.196668\pi$
$751$	1224.32			1.63025			0.815126	+	0.579283i	$0.196668\pi$
$752$		−	76.1041i		−	0.101202i
$753$		−	6.04256i		−	0.00802464i
$754$			206.193i			0.273465i
$755$		0			0
$756$			17.5290i			0.0231865i
$757$		−	1346.79i		−	1.77911i	−0.456828	−	0.889555i	$-0.651014\pi$
$757$		−	1346.79i		−	1.77911i	0.456828	−	0.889555i	$-0.348986\pi$
$758$	−586.852			−0.774211
$759$	19.8383	+	31.2728i	0.0261374	+	0.0412026i
$760$		0			0
$761$			642.215i			0.843909i	0.906617	+	0.421954i	$0.138656\pi$
$761$			642.215i			0.843909i	−0.906617	+	0.421954i	$0.861344\pi$
$762$			19.0405i			0.0249875i
$763$			248.647i			0.325881i
$764$	140.736			0.184210
$765$		0			0
$766$		−	87.7886i		−	0.114607i
$767$	−853.814			−1.11319
$768$			30.7181i			0.0399976i
$769$			786.000i			1.02211i	0.859549	+	0.511053i	$0.170744\pi$
$769$			786.000i			1.02211i	−0.859549	+	0.511053i	$0.829256\pi$
$770$		0			0
$771$	4.79601			0.00622051
$772$	311.635			0.403673
$773$			1311.57i			1.69672i	0.529419	+	0.848360i	$0.322410\pi$
$773$			1311.57i			1.69672i	−0.529419	+	0.848360i	$0.677590\pi$
$774$	−88.9406			−0.114910
$775$		0			0
$776$		−	188.785i		−	0.243280i
$777$	23.2083			0.0298691
$778$	−601.166			−0.772706
$779$	−1874.01			−2.40566
$780$		0			0
$781$	47.1607	+	74.3432i	0.0603850	+	0.0951898i
$782$			742.601i			0.949618i
$783$	17.5549			0.0224200
$784$	142.665			0.181971
$785$		0			0
$786$	19.3476			0.0246152
$787$	−658.660			−0.836925			−0.418462	−	0.908234i	$-0.637431\pi$
$787$	−658.660			−0.836925			−0.418462	+	0.908234i	$0.637431\pi$
$788$	−676.711			−0.858770
$789$			33.1575i			0.0420247i
$790$		0			0
$791$			538.089i			0.680265i
$792$	−453.794	−	715.352i	−0.572972	−	0.903222i
$793$		−	384.114i		−	0.484381i
$794$		−	347.022i		−	0.437055i
$795$		0			0
$796$	629.998			0.791455
$797$			1499.40i			1.88130i	0.339376	+	0.940651i	$0.389784\pi$
$797$			1499.40i			1.88130i	−0.339376	+	0.940651i	$0.610216\pi$
$798$	−27.3904			−0.0343238
$799$			260.158i			0.325605i
$800$		0			0
$801$	−663.257			−0.828036
$802$	148.297			0.184909
$803$	−191.705	−	302.200i	−0.238736	−	0.376339i
$804$	24.2890			0.0302102
$805$		0			0
$806$	−637.649			−0.791128
$807$			34.0959i			0.0422501i
$808$			1396.94i			1.72889i
$809$		−	1400.12i		−	1.73068i	−0.501186	−	0.865340i	$-0.667103\pi$
$809$		−	1400.12i		−	1.73068i	0.501186	−	0.865340i	$-0.332897\pi$
$810$		0			0
$811$		−	611.795i		−	0.754371i	−0.926138	−	0.377186i	$-0.876892\pi$
$811$		−	611.795i		−	0.754371i	0.926138	−	0.377186i	$-0.123108\pi$
$812$		−	72.4807i		−	0.0892620i
$813$	−31.7873			−0.0390988
$814$	−602.867	+	382.437i	−0.740622	+	0.469824i
$815$		0			0
$816$			10.0222i			0.0122821i
$817$			243.068i			0.297513i
$818$		−	1014.22i		−	1.23988i
$819$	678.915			0.828956
$820$		0			0
$821$		−	1249.80i		−	1.52230i	−0.648579	−	0.761148i	$-0.724636\pi$
$821$		−	1249.80i		−	1.52230i	0.648579	−	0.761148i	$-0.275364\pi$
$822$	−36.2110			−0.0440523
$823$		−	899.845i		−	1.09337i	−0.837338	−	0.546686i	$-0.815889\pi$
$823$		−	899.845i		−	1.09337i	0.837338	−	0.546686i	$-0.184111\pi$
$824$			365.424i			0.443476i
$825$		0			0
$826$	−343.458			−0.415808
$827$	1331.57			1.61012			0.805060	−	0.593193i	$-0.202133\pi$
$827$	1331.57			1.61012			0.805060	+	0.593193i	$0.202133\pi$
$828$		−	492.691i		−	0.595037i
$829$	578.195			0.697461			0.348730	−	0.937223i	$-0.386613\pi$
$829$	578.195			0.697461			0.348730	+	0.937223i	$0.386613\pi$
$830$		0			0
$831$		−	2.59740i		−	0.00312563i
$832$	985.896			1.18497
$833$	−487.694			−0.585467
$834$	−0.850251			−0.00101949
$835$		0			0
$836$	−621.750	+	394.416i	−0.743720	+	0.471789i
$837$			54.2883i			0.0648606i
$838$	608.310			0.725907
$839$	−779.704			−0.929325			−0.464663	−	0.885488i	$-0.653824\pi$
$839$	−779.704			−0.929325			−0.464663	+	0.885488i	$0.653824\pi$
$840$		0			0
$841$	768.412			0.913689
$842$	787.999			0.935866
$843$	20.9505			0.0248523
$844$		−	593.026i		−	0.702638i
$845$		0			0
$846$			197.523i			0.233478i
$847$	−235.143	+	499.234i	−0.277619	+	0.589414i
$848$		−	167.704i		−	0.197764i
$849$			42.0680i			0.0495500i
$850$		0			0
$851$	−1305.59			−1.53419
$852$			1.71026i			0.00200734i
$853$	1330.04			1.55925			0.779623	−	0.626250i	$-0.215411\pi$
$853$	1330.04			1.55925			0.779623	+	0.626250i	$0.215411\pi$
$854$		−	154.515i		−	0.180931i
$855$		0			0
$856$	1123.45			1.31244
$857$	223.662			0.260983			0.130491	−	0.991449i	$-0.458345\pi$
$857$	223.662			0.260983			0.130491	+	0.991449i	$0.458345\pi$
$858$	25.7517	−	16.3359i	0.0300136	−	0.0190396i
$859$	1512.14			1.76035			0.880174	−	0.474651i	$-0.157426\pi$
$859$	1512.14			1.76035			0.880174	+	0.474651i	$0.157426\pi$
$860$		0			0
$861$	−27.2842			−0.0316889
$862$			501.857i			0.582200i
$863$			815.242i			0.944661i	0.881422	+	0.472330i	$0.156587\pi$
$863$			815.242i			0.944661i	−0.881422	+	0.472330i	$0.843413\pi$
$864$		−	55.3993i		−	0.0641196i
$865$		0			0
$866$		−	26.3503i		−	0.0304276i
$867$		−	1.15441i		−	0.00133150i
$868$	224.146			0.258233
$869$	−483.160	+	306.500i	−0.555996	+	0.352704i
$870$		0			0
$871$		−	1882.85i		−	2.16171i
$872$		−	467.211i		−	0.535793i
$873$			197.979i			0.226780i
$874$	1540.86			1.76299
$875$		0			0
$876$		−	6.95207i		−	0.00793615i
$877$	−138.555			−0.157987			−0.0789935	−	0.996875i	$-0.525171\pi$
$877$	−138.555			−0.157987			−0.0789935	+	0.996875i	$0.525171\pi$
$878$			1013.73i			1.15459i
$879$		−	1.18197i		−	0.00134467i
$880$		0			0
$881$	415.019			0.471077			0.235538	−	0.971865i	$-0.424315\pi$
$881$	415.019			0.471077			0.235538	+	0.971865i	$0.424315\pi$
$882$	−370.277			−0.419815
$883$			285.257i			0.323055i	0.986868	+	0.161527i	$0.0516420\pi$
$883$			285.257i			0.323055i	−0.986868	+	0.161527i	$0.948358\pi$
$884$	−534.361			−0.604480
$885$		0			0
$886$			524.972i			0.592519i
$887$	405.942			0.457657			0.228829	−	0.973467i	$-0.426510\pi$
$887$	405.942			0.457657			0.228829	+	0.973467i	$0.426510\pi$
$888$	−43.6086			−0.0491088
$889$	−518.840			−0.583622
$890$		0			0
$891$	475.198	+	749.093i	0.533331	+	0.840733i
$892$			545.886i			0.611980i
$893$	539.814			0.604495
$894$	−31.4564			−0.0351861
$895$		0			0
$896$	−93.8956			−0.104794
$897$	55.7688			0.0621726
$898$	829.697			0.923939
$899$		−	224.477i		−	0.249697i
$900$		0			0
$901$			573.288i			0.636279i
$902$	708.743	−	449.601i	0.785746	−	0.498449i
$903$			3.53889i			0.00391903i
$904$		−	1011.08i		−	1.11845i
$905$		0			0
$906$	−22.8861			−0.0252606
$907$			1577.64i			1.73940i	0.493581	+	0.869700i	$0.335688\pi$
$907$			1577.64i			1.73940i	−0.493581	+	0.869700i	$0.664312\pi$
$908$	−17.8125			−0.0196173
$909$		−	1464.97i		−	1.61163i
$910$		0			0
$911$	−574.512			−0.630639			−0.315319	−	0.948986i	$-0.602112\pi$
$911$	−574.512			−0.630639			−0.315319	+	0.948986i	$0.602112\pi$
$912$	20.7955			0.0228021
$913$	255.553	+	402.849i	0.279905	+	0.441236i
$914$	1164.77			1.27436
$915$		0			0
$916$	−663.905			−0.724787
$917$			527.207i			0.574926i
$918$		−	52.0619i		−	0.0567123i
$919$			354.240i			0.385462i	0.981252	+	0.192731i	$0.0617345\pi$
$919$			354.240i			0.385462i	−0.981252	+	0.192731i	$0.938266\pi$
$920$		0			0
$921$			52.6571i			0.0571739i
$922$		−	128.446i		−	0.139313i
$923$	132.577			0.143637
$924$	−9.05222	+	5.74240i	−0.00979677	+	0.00621472i
$925$		0			0
$926$			74.4082i			0.0803544i
$927$		−	383.221i		−	0.413399i
$928$			229.071i			0.246844i
$929$	230.873			0.248518			0.124259	−	0.992250i	$-0.460345\pi$
$929$	230.873			0.248518			0.124259	+	0.992250i	$0.460345\pi$
$930$		0			0
$931$			1011.94i			1.08694i
$932$	−505.442			−0.542319
$933$		−	11.5850i		−	0.0124169i
$934$		−	676.555i		−	0.724363i
$935$		0			0
$936$	−1275.69			−1.36292
$937$	−414.814			−0.442704			−0.221352	−	0.975194i	$-0.571047\pi$
$937$	−414.814			−0.442704			−0.221352	+	0.975194i	$0.571047\pi$
$938$		−	757.400i		−	0.807463i
$939$	−33.0786			−0.0352275
$940$		0			0
$941$		−	1428.08i		−	1.51762i	−0.651315	−	0.758808i	$-0.725782\pi$
$941$		−	1428.08i		−	1.51762i	0.651315	−	0.758808i	$-0.274218\pi$
$942$	42.8558			0.0454945
$943$	1534.88			1.62766
$944$	260.763			0.276232
$945$		0			0
$946$	−58.3155	−	91.9275i	−0.0616443	−	0.0971749i
$947$			976.020i			1.03064i	0.856997	+	0.515322i	$0.172328\pi$
$947$			976.020i			1.03064i	−0.856997	+	0.515322i	$0.827672\pi$
$948$	−11.1150			−0.0117247
$949$	−538.914			−0.567876
$950$		0			0
$951$	27.8285			0.0292624
$952$	−675.890			−0.709968
$953$	−1273.90			−1.33673			−0.668364	−	0.743834i	$-0.733005\pi$
$953$	−1273.90			−1.33673			−0.668364	+	0.743834i	$0.733005\pi$
$954$			435.263i			0.456250i
$955$		0			0
$956$		−	147.620i		−	0.154414i
$957$	5.75089	+	9.06559i	0.00600929	+	0.00947293i
$958$			266.178i			0.277848i
$959$		−	986.723i		−	1.02891i
$960$		0			0
$961$	−266.806			−0.277634
$962$			1075.09i			1.11756i
$963$	−1178.16			−1.22343
$964$		−	273.402i		−	0.283612i
$965$		0			0
$966$	22.4337			0.0232233
$967$	1586.32			1.64045			0.820227	−	0.572038i	$-0.193847\pi$
$967$	1586.32			1.64045			0.820227	+	0.572038i	$0.193847\pi$
$968$	441.837	−	938.066i	0.456443	−	0.969077i
$969$	−71.0887			−0.0733629
$970$		0			0
$971$	662.523			0.682310			0.341155	−	0.940007i	$-0.389182\pi$
$971$	662.523			0.682310			0.341155	+	0.940007i	$0.389182\pi$
$972$			51.8245i			0.0533174i
$973$		−	23.1687i		−	0.0238116i
$974$			535.924i			0.550230i
$975$		0			0
$976$			117.312i			0.120197i
$977$		−	1552.47i		−	1.58902i	−0.607254	−	0.794508i	$-0.707729\pi$
$977$		−	1552.47i		−	1.58902i	0.607254	−	0.794508i	$-0.292271\pi$
$978$	4.20713			0.00430177
$979$	−434.876	−	685.531i	−0.444204	−	0.700236i
$980$		0			0
$981$			489.965i			0.499455i
$982$		−	61.3901i		−	0.0625154i
$983$		−	862.208i		−	0.877119i	−0.898702	−	0.438560i	$-0.855489\pi$
$983$		−	862.208i		−	0.877119i	0.898702	−	0.438560i	$-0.144511\pi$
$984$	51.2673			0.0521009
$985$		0			0
$986$			215.271i			0.218328i
$987$	7.85930			0.00796281
$988$			1108.77i			1.12224i
$989$		−	199.082i		−	0.201296i
$990$		0			0
$991$	−1075.74			−1.08551			−0.542755	−	0.839891i	$-0.682619\pi$
$991$	−1075.74			−1.08551			−0.542755	+	0.839891i	$0.682619\pi$
$992$	−708.401			−0.714113
$993$			27.6528i			0.0278477i
$994$	53.3306			0.0536526
$995$		0			0
$996$			9.26749i			0.00930470i
$997$	872.880			0.875507			0.437753	−	0.899095i	$-0.355774\pi$
$997$	872.880			0.875507			0.437753	+	0.899095i	$0.355774\pi$
$998$	335.985			0.336658
$999$	91.5317			0.0916233

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	275.3.d.c.274.9		16
5.2	odd	4		275.3.c.f.76.5		8
5.3	odd	4		55.3.c.a.21.4	✓	8
5.4	even	2	inner	275.3.d.c.274.8		16
11.10	odd	2	inner	275.3.d.c.274.7		16
15.8	even	4		495.3.b.a.406.5		8
20.3	even	4		880.3.j.a.241.6		8
55.32	even	4		275.3.c.f.76.4		8
55.43	even	4		55.3.c.a.21.5	yes	8
55.54	odd	2	inner	275.3.d.c.274.10		16
165.98	odd	4		495.3.b.a.406.4		8
220.43	odd	4		880.3.j.a.241.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
55.3.c.a.21.4	✓	8	5.3	odd	4
55.3.c.a.21.5	yes	8	55.43	even	4
275.3.c.f.76.4		8	55.32	even	4
275.3.c.f.76.5		8	5.2	odd	4
275.3.d.c.274.7		16	11.10	odd	2	inner
275.3.d.c.274.8		16	5.4	even	2	inner
275.3.d.c.274.9		16	1.1	even	1	trivial
275.3.d.c.274.10		16	55.54	odd	2	inner
495.3.b.a.406.4		8	165.98	odd	4
495.3.b.a.406.5		8	15.8	even	4
880.3.j.a.241.5		8	220.43	odd	4
880.3.j.a.241.6		8	20.3	even	4

Overview	Random
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Classical	Maass
Hilbert	Bianchi

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	275.3.d.c.274.9

Level	$275$
Weight	$3$
Character	275.274
Analytic conductor	$7.493$
Analytic rank	$0$
Dimension	$16$
Inner twists	$4$