LMFDB - Embedded newform 1150.3.c.c.1149.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1150,3,Mod(1149,1150)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1150, 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("1150.1149");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1150 = 2 \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1150.c (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:	$31.3352304014$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	no (minimal twist has level 230)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1149.11
Character	$\chi$	$=$	1150.1149
Dual form			1150.3.c.c.1149.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421i q^{2} -2.34854i q^{3} -2.00000 q^{4} -3.32134 q^{6} +7.61815 q^{7} +2.82843i q^{8} +3.48436 q^{9} +O(q^{10})$	\(q-1.41421i q^{2} -2.34854i q^{3} -2.00000 q^{4} -3.32134 q^{6} +7.61815 q^{7} +2.82843i q^{8} +3.48436 q^{9} -12.3764i q^{11} +4.69708i q^{12} +13.0302i q^{13} -10.7737i q^{14} +4.00000 q^{16} -9.13040 q^{17} -4.92764i q^{18} +14.4549i q^{19} -17.8915i q^{21} -17.5029 q^{22} +(4.51289 - 22.5529i) q^{23} +6.64267 q^{24} +18.4275 q^{26} -29.3200i q^{27} -15.2363 q^{28} +21.2813 q^{29} +36.8428 q^{31} -5.65685i q^{32} -29.0666 q^{33} +12.9123i q^{34} -6.96873 q^{36} +56.9603 q^{37} +20.4424 q^{38} +30.6019 q^{39} +70.7680 q^{41} -25.3024 q^{42} -70.0086 q^{43} +24.7529i q^{44} +(-31.8946 - 6.38219i) q^{46} -66.2614i q^{47} -9.39416i q^{48} +9.03623 q^{49} +21.4431i q^{51} -26.0604i q^{52} +77.4364 q^{53} -41.4648 q^{54} +21.5474i q^{56} +33.9480 q^{57} -30.0963i q^{58} -82.7923 q^{59} -23.9941i q^{61} -52.1036i q^{62} +26.5444 q^{63} -8.00000 q^{64} +41.1063i q^{66} -118.512 q^{67} +18.2608 q^{68} +(-52.9664 - 10.5987i) q^{69} +69.0263 q^{71} +9.85527i q^{72} -25.9840i q^{73} -80.5540i q^{74} -28.9099i q^{76} -94.2857i q^{77} -43.2777i q^{78} -28.8543i q^{79} -37.4999 q^{81} -100.081i q^{82} +69.3871 q^{83} +35.7831i q^{84} +99.0071i q^{86} -49.9800i q^{87} +35.0059 q^{88} -45.4428i q^{89} +99.2661i q^{91} +(-9.02579 + 45.1058i) q^{92} -86.5269i q^{93} -93.7077 q^{94} -13.2853 q^{96} -74.4458 q^{97} -12.7792i q^{98} -43.1241i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 64 q^{4} - 16 q^{6} - 128 q^{9}+O(q^{10}) $ 32 * q - 64 * q^4 - 16 * q^6 - 128 * q^9	$ 32 q - 64 q^{4} - 16 q^{6} - 128 q^{9} + 128 q^{16} + 32 q^{24} + 192 q^{26} + 216 q^{29} - 232 q^{31} + 256 q^{36} - 496 q^{39} - 312 q^{41} - 248 q^{46} + 56 q^{49} - 448 q^{54} - 408 q^{59} - 256 q^{64} + 536 q^{69} + 472 q^{71} - 272 q^{81} + 432 q^{94} - 64 q^{96}+O(q^{100}) $ 32 * q - 64 * q^4 - 16 * q^6 - 128 * q^9 + 128 * q^16 + 32 * q^24 + 192 * q^26 + 216 * q^29 - 232 * q^31 + 256 * q^36 - 496 * q^39 - 312 * q^41 - 248 * q^46 + 56 * q^49 - 448 * q^54 - 408 * q^59 - 256 * q^64 + 536 * q^69 + 472 * q^71 - 272 * q^81 + 432 * q^94 - 64 * q^96

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1150\mathbb{Z}\right)^\times$.

$n$	$51$	$277$
$\chi(n)$	$-1$	$-1$

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 $

$2$		−	1.41421i		−	0.707107i
$2$		−	1.41421i		−	0.707107i
$3$		−	2.34854i		−	0.782846i	−0.920211	−	0.391423i	$-0.871983\pi$
$3$		−	2.34854i		−	0.782846i	0.920211	−	0.391423i	$-0.128017\pi$
$4$	−2.00000			−0.500000
$5$		0			0
$5$		0			0
$6$	−3.32134			−0.553556
$7$	7.61815			1.08831			0.544154	−	0.838986i	$-0.316851\pi$
$7$	7.61815			1.08831			0.544154	+	0.838986i	$0.316851\pi$
$8$			2.82843i			0.353553i
$9$	3.48436			0.387152
$10$		0			0
$11$		−	12.3764i		−	1.12513i	−0.826752	−	0.562566i	$-0.809814\pi$
$11$		−	12.3764i		−	1.12513i	0.826752	−	0.562566i	$-0.190186\pi$
$12$			4.69708i			0.391423i
$13$			13.0302i			1.00232i	0.865354	+	0.501162i	$0.167094\pi$
$13$			13.0302i			1.00232i	−0.865354	+	0.501162i	$0.832906\pi$
$14$		−	10.7737i		−	0.769549i
$15$		0			0
$16$	4.00000			0.250000
$17$	−9.13040			−0.537083			−0.268541	−	0.963268i	$-0.586542\pi$
$17$	−9.13040			−0.537083			−0.268541	+	0.963268i	$0.586542\pi$
$18$		−	4.92764i		−	0.273758i
$19$			14.4549i			0.760787i	0.924825	+	0.380393i	$0.124211\pi$
$19$			14.4549i			0.760787i	−0.924825	+	0.380393i	$0.875789\pi$
$20$		0			0
$21$		−	17.8915i		−	0.851977i
$22$	−17.5029			−0.795588
$23$	4.51289	−	22.5529i	0.196213	−	0.980561i
$23$	4.51289	−	22.5529i	0.196213	−	0.980561i
$24$	6.64267			0.276778
$25$		0			0
$26$	18.4275			0.708750
$27$		−	29.3200i		−	1.08593i
$28$	−15.2363			−0.544154
$29$	21.2813			0.733839			0.366919	−	0.930253i	$-0.380412\pi$
$29$	21.2813			0.733839			0.366919	+	0.930253i	$0.380412\pi$
$30$		0			0
$31$	36.8428			1.18848			0.594239	−	0.804288i	$-0.297453\pi$
$31$	36.8428			1.18848			0.594239	+	0.804288i	$0.297453\pi$
$32$		−	5.65685i		−	0.176777i
$33$	−29.0666			−0.880805
$34$			12.9123i			0.379775i
$35$		0			0
$36$	−6.96873			−0.193576
$37$	56.9603			1.53947			0.769734	−	0.638365i	$-0.220389\pi$
$37$	56.9603			1.53947			0.769734	+	0.638365i	$0.220389\pi$
$38$	20.4424			0.537957
$39$	30.6019			0.784665
$40$		0			0
$41$	70.7680			1.72605			0.863024	−	0.505163i	$-0.168567\pi$
$41$	70.7680			1.72605			0.863024	+	0.505163i	$0.168567\pi$
$42$	−25.3024			−0.602439
$43$	−70.0086			−1.62811			−0.814053	−	0.580790i	$-0.802744\pi$
$43$	−70.0086			−1.62811			−0.814053	+	0.580790i	$0.802744\pi$
$44$			24.7529i			0.562566i
$45$		0			0
$46$	−31.8946	−	6.38219i	−0.693362	−	0.138743i
$47$		−	66.2614i		−	1.40982i	−0.709299	−	0.704908i	$-0.750988\pi$
$47$		−	66.2614i		−	1.40982i	0.709299	−	0.704908i	$-0.249012\pi$
$48$		−	9.39416i		−	0.195712i
$49$	9.03623			0.184413
$50$		0			0
$51$			21.4431i			0.420453i
$52$		−	26.0604i		−	0.501162i
$53$	77.4364			1.46106			0.730532	−	0.682879i	$-0.239272\pi$
$53$	77.4364			1.46106			0.730532	+	0.682879i	$0.239272\pi$
$54$	−41.4648			−0.767866
$55$		0			0
$56$			21.5474i			0.384775i
$57$	33.9480			0.595579
$58$		−	30.0963i		−	0.518902i
$59$	−82.7923			−1.40326			−0.701630	−	0.712541i	$-0.747544\pi$
$59$	−82.7923			−1.40326			−0.701630	+	0.712541i	$0.747544\pi$
$60$		0			0
$61$		−	23.9941i		−	0.393346i	−0.980469	−	0.196673i	$-0.936986\pi$
$61$		−	23.9941i		−	0.393346i	0.980469	−	0.196673i	$-0.0630137\pi$
$62$		−	52.1036i		−	0.840381i
$63$	26.5444			0.421340
$64$	−8.00000			−0.125000
$65$		0			0
$66$			41.1063i			0.622823i
$67$	−118.512			−1.76884			−0.884418	−	0.466695i	$-0.845445\pi$
$67$	−118.512			−1.76884			−0.884418	+	0.466695i	$0.845445\pi$
$68$	18.2608			0.268541
$69$	−52.9664	−	10.5987i	−0.767629	−	0.153604i
$70$		0			0
$71$	69.0263			0.972202			0.486101	−	0.873903i	$-0.338419\pi$
$71$	69.0263			0.972202			0.486101	+	0.873903i	$0.338419\pi$
$72$			9.85527i			0.136879i
$73$		−	25.9840i		−	0.355945i	−0.984035	−	0.177973i	$-0.943046\pi$
$73$		−	25.9840i		−	0.355945i	0.984035	−	0.177973i	$-0.0569539\pi$
$74$		−	80.5540i		−	1.08857i
$75$		0			0
$76$		−	28.9099i		−	0.380393i
$77$		−	94.2857i		−	1.22449i
$78$		−	43.2777i		−	0.554842i
$79$		−	28.8543i		−	0.365244i	−0.983183	−	0.182622i	$-0.941541\pi$
$79$		−	28.8543i		−	0.365244i	0.983183	−	0.182622i	$-0.0584585\pi$
$80$		0			0
$81$	−37.4999			−0.462962
$82$		−	100.081i		−	1.22050i
$83$	69.3871			0.835989			0.417995	−	0.908449i	$-0.362733\pi$
$83$	69.3871			0.835989			0.417995	+	0.908449i	$0.362733\pi$
$84$			35.7831i			0.425989i
$85$		0			0
$86$			99.0071i			1.15125i
$87$		−	49.9800i		−	0.574483i
$88$	35.0059			0.397794
$89$		−	45.4428i		−	0.510594i	−0.966863	−	0.255297i	$-0.917827\pi$
$89$		−	45.4428i		−	0.510594i	0.966863	−	0.255297i	$-0.0821732\pi$
$90$		0			0
$91$			99.2661i			1.09084i
$92$	−9.02579	+	45.1058i	−0.0981064	+	0.490281i
$93$		−	86.5269i		−	0.930396i
$94$	−93.7077			−0.996891
$95$		0			0
$96$	−13.2853			−0.138389
$97$	−74.4458			−0.767482			−0.383741	−	0.923441i	$-0.625364\pi$
$97$	−74.4458			−0.767482			−0.383741	+	0.923441i	$0.625364\pi$
$98$		−	12.7792i		−	0.130400i
$99$		−	43.1241i		−	0.435597i
$100$		0			0
$101$	−17.0563			−0.168874			−0.0844372	−	0.996429i	$-0.526909\pi$
$101$	−17.0563			−0.168874			−0.0844372	+	0.996429i	$0.526909\pi$
$102$	30.3251			0.297305
$103$	−153.952			−1.49468			−0.747340	−	0.664442i	$-0.768669\pi$
$103$	−153.952			−1.49468			−0.747340	+	0.664442i	$0.768669\pi$
$104$	−36.8550			−0.354375
$105$		0			0
$106$		−	109.512i		−	1.03313i
$107$	−112.821			−1.05441			−0.527203	−	0.849740i	$-0.676759\pi$
$107$	−112.821			−1.05441			−0.527203	+	0.849740i	$0.676759\pi$
$108$			58.6400i			0.542963i
$109$			97.6061i			0.895468i	0.894167	+	0.447734i	$0.147769\pi$
$109$			97.6061i			0.895468i	−0.894167	+	0.447734i	$0.852231\pi$
$110$		0			0
$111$		−	133.773i		−	1.20517i
$112$	30.4726			0.272077
$113$	−57.0620			−0.504974			−0.252487	−	0.967600i	$-0.581248\pi$
$113$	−57.0620			−0.504974			−0.252487	+	0.967600i	$0.581248\pi$
$114$		−	48.0097i		−	0.421138i
$115$		0			0
$116$	−42.5627			−0.366919
$117$			45.4020i			0.388051i
$118$			117.086i			0.992255i
$119$	−69.5568			−0.584511
$120$		0			0
$121$	−32.1765			−0.265921
$122$	−33.9328			−0.278137
$123$		−	166.201i		−	1.35123i
$124$	−73.6857			−0.594239
$125$		0			0
$126$		−	37.5395i		−	0.297932i
$127$		−	151.376i		−	1.19194i	−0.803007	−	0.595969i	$-0.796768\pi$
$127$		−	151.376i		−	1.19194i	0.803007	−	0.595969i	$-0.203232\pi$
$128$			11.3137i			0.0883883i
$129$			164.418i			1.27456i
$130$		0			0
$131$	46.7334			0.356743			0.178372	−	0.983963i	$-0.442917\pi$
$131$	46.7334			0.356743			0.178372	+	0.983963i	$0.442917\pi$
$132$	58.1331			0.440403
$133$			110.120i			0.827970i
$134$			167.601i			1.25076i
$135$		0			0
$136$		−	25.8247i		−	0.189887i
$137$	−31.1949			−0.227700			−0.113850	−	0.993498i	$-0.536318\pi$
$137$	−31.1949			−0.227700			−0.113850	+	0.993498i	$0.536318\pi$
$138$	−14.9888	+	74.9058i	−0.108615	+	0.542796i
$139$	−36.5517			−0.262962			−0.131481	−	0.991319i	$-0.541973\pi$
$139$	−36.5517			−0.262962			−0.131481	+	0.991319i	$0.541973\pi$
$140$		0			0
$141$	−155.617			−1.10367
$142$		−	97.6180i		−	0.687451i
$143$	161.268			1.12775
$144$	13.9375			0.0967879
$145$		0			0
$146$	−36.7470			−0.251691
$147$		−	21.2219i		−	0.144367i
$148$	−113.921			−0.769734
$149$			182.281i			1.22336i	0.791104	+	0.611681i	$0.209506\pi$
$149$			182.281i			1.22336i	−0.791104	+	0.611681i	$0.790494\pi$
$150$		0			0
$151$	40.9462			0.271167			0.135584	−	0.990766i	$-0.456709\pi$
$151$	40.9462			0.271167			0.135584	+	0.990766i	$0.456709\pi$
$152$	−40.8848			−0.268979
$153$	−31.8136			−0.207932
$154$	−133.340			−0.865845
$155$		0			0
$156$	−61.2039			−0.392333
$157$	252.396			1.60762			0.803810	−	0.594887i	$-0.202803\pi$
$157$	252.396			1.60762			0.803810	+	0.594887i	$0.202803\pi$
$158$	−40.8062			−0.258267
$159$		−	181.862i		−	1.14379i
$160$		0			0
$161$	34.3799	−	171.811i	0.213540	−	1.06715i
$162$			53.0329i			0.327364i
$163$			46.9686i			0.288151i	0.989567	+	0.144076i	$0.0460208\pi$
$163$			46.9686i			0.288151i	−0.989567	+	0.144076i	$0.953979\pi$
$164$	−141.536			−0.863024
$165$		0			0
$166$		−	98.1282i		−	0.591134i
$167$			78.4530i			0.469778i	0.972022	+	0.234889i	$0.0754727\pi$
$167$			78.4530i			0.469778i	−0.972022	+	0.234889i	$0.924527\pi$
$168$	50.6049			0.301220
$169$	−0.786229			−0.00465224
$170$		0			0
$171$			50.3663i			0.294540i
$172$	140.017			0.814053
$173$		−	137.080i		−	0.792369i	−0.918171	−	0.396184i	$-0.870334\pi$
$173$		−	137.080i		−	0.792369i	0.918171	−	0.396184i	$-0.129666\pi$
$174$	−70.6824			−0.406221
$175$		0			0
$176$		−	49.5058i		−	0.281283i
$177$			194.441i			1.09854i
$178$	−64.2659			−0.361044
$179$	55.6699			0.311005			0.155503	−	0.987835i	$-0.450300\pi$
$179$	55.6699			0.311005			0.155503	+	0.987835i	$0.450300\pi$
$180$		0			0
$181$			23.7317i			0.131114i	0.997849	+	0.0655572i	$0.0208825\pi$
$181$			23.7317i			0.131114i	−0.997849	+	0.0655572i	$0.979118\pi$
$182$	140.383			0.771337
$183$	−56.3511			−0.307929
$184$	63.7893	+	12.7644i	0.346681	+	0.0693717i
$185$		0			0
$186$	−122.367			−0.657890
$187$			113.002i			0.604289i
$188$			132.523i			0.704908i
$189$		−	223.364i		−	1.18182i
$190$		0			0
$191$			351.658i			1.84114i	0.390576	+	0.920571i	$0.372276\pi$
$191$			351.658i			1.84114i	−0.390576	+	0.920571i	$0.627724\pi$
$192$			18.7883i			0.0978558i
$193$			39.4808i			0.204564i	0.994755	+	0.102282i	$0.0326144\pi$
$193$			39.4808i			0.204564i	−0.994755	+	0.102282i	$0.967386\pi$
$194$			105.282i			0.542692i
$195$		0			0
$196$	−18.0725			−0.0922064
$197$		−	225.017i		−	1.14222i	−0.820875	−	0.571108i	$-0.806514\pi$
$197$		−	225.017i		−	1.14222i	0.820875	−	0.571108i	$-0.193486\pi$
$198$	−60.9866			−0.308013
$199$		−	37.0872i		−	0.186368i	−0.995649	−	0.0931840i	$-0.970296\pi$
$199$		−	37.0872i		−	0.186368i	0.995649	−	0.0931840i	$-0.0297045\pi$
$200$		0			0
$201$			278.330i			1.38473i
$202$			24.1213i			0.119412i
$203$	162.124			0.798642
$204$		−	42.8862i		−	0.210227i
$205$		0			0
$206$			217.721i			1.05690i
$207$	15.7246	−	78.5826i	0.0759641	−	0.379626i
$208$			52.1208i			0.250581i
$209$	178.901			0.855985
$210$		0			0
$211$	217.064			1.02874			0.514369	−	0.857569i	$-0.328026\pi$
$211$	217.064			1.02874			0.514369	+	0.857569i	$0.328026\pi$
$212$	−154.873			−0.730532
$213$		−	162.111i		−	0.761085i
$214$			159.554i			0.745577i
$215$		0			0
$216$	82.9295			0.383933
$217$	280.674			1.29343
$218$	138.036			0.633192
$219$	−61.0245			−0.278651
$220$		0			0
$221$		−	118.971i		−	0.538330i
$222$	−189.184			−0.852181
$223$		−	269.398i		−	1.20806i	−0.796961	−	0.604030i	$-0.793561\pi$
$223$		−	269.398i		−	1.20806i	0.796961	−	0.604030i	$-0.206439\pi$
$224$		−	43.0948i		−	0.192387i
$225$		0			0
$226$			80.6979i			0.357070i
$227$	−175.997			−0.775319			−0.387659	−	0.921803i	$-0.626716\pi$
$227$	−175.997			−0.775319			−0.387659	+	0.921803i	$0.626716\pi$
$228$	−67.8960			−0.297789
$229$		−	63.0669i		−	0.275401i	−0.990474	−	0.137701i	$-0.956029\pi$
$229$		−	63.0669i		−	0.275401i	0.990474	−	0.137701i	$-0.0439712\pi$
$230$		0			0
$231$	−221.434			−0.958587
$232$			60.1927i			0.259451i
$233$		−	4.48041i		−	0.0192292i	−0.999954	−	0.00961461i	$-0.996940\pi$
$233$		−	4.48041i		−	0.0192292i	0.999954	−	0.00961461i	$-0.00306047\pi$
$234$	64.2081			0.274394
$235$		0			0
$236$	165.585			0.701630
$237$	−67.7655			−0.285930
$238$			98.3682i			0.413312i
$239$	−85.5205			−0.357826			−0.178913	−	0.983865i	$-0.557258\pi$
$239$	−85.5205			−0.357826			−0.178913	+	0.983865i	$0.557258\pi$
$240$		0			0
$241$		−	257.515i		−	1.06853i	−0.845318	−	0.534263i	$-0.820589\pi$
$241$		−	257.515i		−	1.06853i	0.845318	−	0.534263i	$-0.179411\pi$
$242$			45.5044i			0.188035i
$243$		−	175.810i		−	0.723498i
$244$			47.9882i			0.196673i
$245$		0			0
$246$	−235.044			−0.955464
$247$	−188.351			−0.762554
$248$			104.207i			0.420191i
$249$		−	162.958i		−	0.654451i
$250$		0			0
$251$		−	121.438i		−	0.483816i	−0.970299	−	0.241908i	$-0.922227\pi$
$251$		−	121.438i		−	0.483816i	0.970299	−	0.241908i	$-0.0777733\pi$
$252$	−53.0888			−0.210670
$253$	−279.125	−	55.8536i	−1.10326	−	0.220765i
$254$	−214.078			−0.842827
$255$		0			0
$256$	16.0000			0.0625000
$257$			192.295i			0.748230i	0.927382	+	0.374115i	$0.122053\pi$
$257$			192.295i			0.748230i	−0.927382	+	0.374115i	$0.877947\pi$
$258$	232.522			0.901248
$259$	433.932			1.67541
$260$		0			0
$261$	74.1519			0.284107
$262$		−	66.0910i		−	0.252256i
$263$	−417.798			−1.58858			−0.794292	−	0.607536i	$-0.792158\pi$
$263$	−417.798			−1.58858			−0.794292	+	0.607536i	$0.792158\pi$
$264$		−	82.2127i		−	0.311412i
$265$		0			0
$266$	155.733			0.585463
$267$	−106.724			−0.399716
$268$	237.024			0.884418
$269$	474.878			1.76534			0.882672	−	0.469989i	$-0.155742\pi$
$269$	474.878			1.76534			0.882672	+	0.469989i	$0.155742\pi$
$270$		0			0
$271$	−32.0762			−0.118362			−0.0591812	−	0.998247i	$-0.518849\pi$
$271$	−32.0762			−0.118362			−0.0591812	+	0.998247i	$0.518849\pi$
$272$	−36.5216			−0.134271
$273$	233.130			0.853957
$274$			44.1162i			0.161008i
$275$		0			0
$276$	105.933	+	21.1974i	0.383814	+	0.0768022i
$277$			459.038i			1.65718i	0.559858	+	0.828589i	$0.310856\pi$
$277$			459.038i			1.65718i	−0.559858	+	0.828589i	$0.689144\pi$
$278$			51.6919i			0.185942i
$279$	128.374			0.460121
$280$		0			0
$281$		−	321.543i		−	1.14428i	−0.820156	−	0.572140i	$-0.806113\pi$
$281$		−	321.543i		−	1.14428i	0.820156	−	0.572140i	$-0.193887\pi$
$282$			220.076i			0.780412i
$283$	428.712			1.51488			0.757441	−	0.652904i	$-0.226449\pi$
$283$	428.712			1.51488			0.757441	+	0.652904i	$0.226449\pi$
$284$	−138.053			−0.486101
$285$		0			0
$286$		−	228.067i		−	0.797437i
$287$	539.121			1.87847
$288$		−	19.7105i		−	0.0684394i
$289$	−205.636			−0.711542
$290$		0			0
$291$			174.839i			0.600821i
$292$			51.9680i			0.177973i
$293$	−359.840			−1.22812			−0.614062	−	0.789258i	$-0.710466\pi$
$293$	−359.840			−1.22812			−0.614062	+	0.789258i	$0.710466\pi$
$294$	−30.0123			−0.102083
$295$		0			0
$296$			161.108i			0.544284i
$297$	−362.878			−1.22181
$298$	257.784			0.865048
$299$	293.869	+	58.8039i	0.982840	+	0.196669i
$300$		0			0
$301$	−533.336			−1.77188
$302$		−	57.9067i		−	0.191744i
$303$			40.0574i			0.132203i
$304$			57.8198i			0.190197i
$305$		0			0
$306$			44.9913i			0.147030i
$307$		−	448.843i		−	1.46203i	−0.682362	−	0.731014i	$-0.739047\pi$
$307$		−	448.843i		−	1.46203i	0.682362	−	0.731014i	$-0.260953\pi$
$308$			188.571i			0.612245i
$309$			361.562i			1.17010i
$310$		0			0
$311$	187.562			0.603094			0.301547	−	0.953451i	$-0.402497\pi$
$311$	187.562			0.603094			0.301547	+	0.953451i	$0.402497\pi$
$312$			86.5554i			0.277421i
$313$	93.4265			0.298487			0.149244	−	0.988800i	$-0.452316\pi$
$313$	93.4265			0.298487			0.149244	+	0.988800i	$0.452316\pi$
$314$		−	356.942i		−	1.13676i
$315$		0			0
$316$			57.7086i			0.182622i
$317$			453.069i			1.42924i	0.699513	+	0.714620i	$0.253400\pi$
$317$			453.069i			1.42924i	−0.699513	+	0.714620i	$0.746600\pi$
$318$	−257.192			−0.808780
$319$		−	263.387i		−	0.825665i
$320$		0			0
$321$			264.965i			0.825437i
$322$	−242.978	−	48.6205i	−0.754590	−	0.150995i
$323$		−	131.979i		−	0.408605i
$324$	74.9999			0.231481
$325$		0			0
$326$	66.4237			0.203754
$327$	229.232			0.701014
$328$			200.162i			0.610250i
$329$		−	504.789i		−	1.53431i
$330$		0			0
$331$	−178.326			−0.538749			−0.269374	−	0.963036i	$-0.586817\pi$
$331$	−178.326			−0.538749			−0.269374	+	0.963036i	$0.586817\pi$
$332$	−138.774			−0.417995
$333$	198.470			0.596007
$334$	110.949			0.332183
$335$		0			0
$336$		−	71.5661i		−	0.212994i
$337$	85.7271			0.254383			0.127192	−	0.991878i	$-0.459404\pi$
$337$	85.7271			0.254383			0.127192	+	0.991878i	$0.459404\pi$
$338$			1.11190i			0.00328963i
$339$			134.012i			0.395317i
$340$		0			0
$341$		−	455.984i		−	1.33720i
$342$	71.2287			0.208271
$343$	−304.450			−0.887610
$344$		−	198.014i		−	0.575623i
$345$		0			0
$346$	−193.860			−0.560289
$347$			216.730i			0.624582i	0.949986	+	0.312291i	$0.101096\pi$
$347$			216.730i			0.624582i	−0.949986	+	0.312291i	$0.898904\pi$
$348$			99.9601i			0.287242i
$349$	528.822			1.51525			0.757624	−	0.652691i	$-0.226360\pi$
$349$	528.822			1.51525			0.757624	+	0.652691i	$0.226360\pi$
$350$		0			0
$351$	382.046			1.08845
$352$	−70.0118			−0.198897
$353$			28.7915i			0.0815623i	0.999168	+	0.0407811i	$0.0129846\pi$
$353$			28.7915i			0.0815623i	−0.999168	+	0.0407811i	$0.987015\pi$
$354$	274.981			0.776783
$355$		0			0
$356$			90.8857i			0.255297i
$357$			163.357i			0.457582i
$358$		−	78.7292i		−	0.219914i
$359$		−	55.5671i		−	0.154783i	−0.997001	−	0.0773914i	$-0.975341\pi$
$359$		−	55.5671i		−	0.154783i	0.997001	−	0.0773914i	$-0.0246591\pi$
$360$		0			0
$361$	152.055			0.421204
$362$	33.5617			0.0927119
$363$			75.5677i			0.208176i
$364$		−	198.532i		−	0.545418i
$365$		0			0
$366$			79.6924i			0.217739i
$367$	164.288			0.447652			0.223826	−	0.974629i	$-0.428145\pi$
$367$	164.288			0.447652			0.223826	+	0.974629i	$0.428145\pi$
$368$	18.0516	−	90.2116i	0.0490532	−	0.245140i
$369$	246.581			0.668242
$370$		0			0
$371$	589.922			1.59009
$372$			173.054i			0.465198i
$373$	−25.1481			−0.0674212			−0.0337106	−	0.999432i	$-0.510732\pi$
$373$	−25.1481			−0.0674212			−0.0337106	+	0.999432i	$0.510732\pi$
$374$	159.809			0.427297
$375$		0			0
$376$	187.415			0.498445
$377$			277.300i			0.735544i
$378$	−315.885			−0.835674
$379$			456.740i			1.20512i	0.798074	+	0.602560i	$0.205852\pi$
$379$			456.740i			1.20512i	−0.798074	+	0.602560i	$0.794148\pi$
$380$		0			0
$381$	−355.513			−0.933104
$382$	497.319			1.30188
$383$	371.017			0.968712			0.484356	−	0.874871i	$-0.339054\pi$
$383$	371.017			0.968712			0.484356	+	0.874871i	$0.339054\pi$
$384$	26.5707			0.0691945
$385$		0			0
$386$	55.8343			0.144648
$387$	−243.935			−0.630324
$388$	148.892			0.383741
$389$			400.013i			1.02831i	0.857697	+	0.514155i	$0.171895\pi$
$389$			400.013i			1.02831i	−0.857697	+	0.514155i	$0.828105\pi$
$390$		0			0
$391$	−41.2045	+	205.917i	−0.105382	+	0.526642i
$392$			25.5583i			0.0651998i
$393$		−	109.755i		−	0.279275i
$394$	−318.222			−0.807669
$395$		0			0
$396$			86.2481i			0.217798i
$397$			264.267i			0.665659i	0.942987	+	0.332830i	$0.108003\pi$
$397$			264.267i			0.665659i	−0.942987	+	0.332830i	$0.891997\pi$
$398$	−52.4493			−0.131782
$399$	258.621			0.648173
$400$		0			0
$401$			425.319i			1.06065i	0.847796	+	0.530323i	$0.177929\pi$
$401$			425.319i			1.06065i	−0.847796	+	0.530323i	$0.822071\pi$
$402$	393.618			0.979150
$403$			480.070i			1.19124i
$404$	34.1126			0.0844372
$405$		0			0
$406$		−	229.278i		−	0.564725i
$407$		−	704.966i		−	1.73210i
$408$	−60.6503			−0.148653
$409$	241.558			0.590607			0.295304	−	0.955403i	$-0.404579\pi$
$409$	241.558			0.590607			0.295304	+	0.955403i	$0.404579\pi$
$410$		0			0
$411$			73.2624i			0.178254i
$412$	307.904			0.747340
$413$	−630.725			−1.52718
$414$	−111.133	−	22.2379i	−0.268436	−	0.0537147i
$415$		0			0
$416$	73.7100			0.177187
$417$			85.8431i			0.205859i
$418$		−	253.004i		−	0.605273i
$419$			358.352i			0.855256i	0.903955	+	0.427628i	$0.140651\pi$
$419$			358.352i			0.855256i	−0.903955	+	0.427628i	$0.859349\pi$
$420$		0			0
$421$			672.051i			1.59632i	0.602446	+	0.798160i	$0.294193\pi$
$421$			672.051i			1.59632i	−0.602446	+	0.798160i	$0.705807\pi$
$422$		−	306.975i		−	0.727428i
$423$		−	230.879i		−	0.545813i
$424$			219.023i			0.516564i
$425$		0			0
$426$	−229.260			−0.538168
$427$		−	182.791i		−	0.428081i
$428$	225.643			0.527203
$429$		−	378.743i		−	0.882852i
$430$		0			0
$431$		−	819.926i		−	1.90238i	−0.308604	−	0.951191i	$-0.599862\pi$
$431$		−	819.926i		−	1.90238i	0.308604	−	0.951191i	$-0.400138\pi$
$432$		−	117.280i		−	0.271482i
$433$	316.231			0.730326			0.365163	−	0.930944i	$-0.381013\pi$
$433$	316.231			0.730326			0.365163	+	0.930944i	$0.381013\pi$
$434$		−	396.933i		−	0.914593i
$435$		0			0
$436$		−	195.212i		−	0.447734i
$437$	326.001	+	65.2336i	0.745998	+	0.149276i
$438$			86.3017i			0.197036i
$439$	−208.878			−0.475805			−0.237903	−	0.971289i	$-0.576460\pi$
$439$	−208.878			−0.475805			−0.237903	+	0.971289i	$0.576460\pi$
$440$		0			0
$441$	31.4855			0.0713957
$442$	−168.250			−0.380657
$443$			706.981i			1.59589i	0.602728	+	0.797947i	$0.294081\pi$
$443$			706.981i			1.59589i	−0.602728	+	0.797947i	$0.705919\pi$
$444$			267.547i			0.602583i
$445$		0			0
$446$	−380.986			−0.854228
$447$	428.094			0.957705
$448$	−60.9452			−0.136038
$449$	524.552			1.16827			0.584134	−	0.811658i	$-0.301434\pi$
$449$	524.552			1.16827			0.584134	+	0.811658i	$0.301434\pi$
$450$		0			0
$451$		−	875.856i		−	1.94203i
$452$	114.124			0.252487
$453$		−	96.1639i		−	0.212282i
$454$			248.898i			0.548233i
$455$		0			0
$456$			96.0194i			0.210569i
$457$	−786.926			−1.72194			−0.860969	−	0.508657i	$-0.830142\pi$
$457$	−786.926			−1.72194			−0.860969	+	0.508657i	$0.830142\pi$
$458$	−89.1901			−0.194738
$459$			267.704i			0.583232i
$460$		0			0
$461$	−38.2459			−0.0829629			−0.0414814	−	0.999139i	$-0.513208\pi$
$461$	−38.2459			−0.0829629			−0.0414814	+	0.999139i	$0.513208\pi$
$462$			313.154i			0.677823i
$463$			525.945i			1.13595i	0.823045	+	0.567976i	$0.192273\pi$
$463$			525.945i			1.13595i	−0.823045	+	0.567976i	$0.807727\pi$
$464$	85.1253			0.183460
$465$		0			0
$466$	−6.33625			−0.0135971
$467$	−166.631			−0.356812			−0.178406	−	0.983957i	$-0.557094\pi$
$467$	−166.631			−0.356812			−0.178406	+	0.983957i	$0.557094\pi$
$468$		−	90.8040i		−	0.194026i
$469$	−902.843			−1.92504
$470$		0			0
$471$		−	592.762i		−	1.25852i
$472$		−	234.172i		−	0.496127i
$473$			866.458i			1.83183i
$474$			95.8349i			0.202183i
$475$		0			0
$476$	139.114			0.292255
$477$	269.817			0.565653
$478$			120.944i			0.253021i
$479$		−	32.0653i		−	0.0669422i	−0.999440	−	0.0334711i	$-0.989344\pi$
$479$		−	32.0653i		−	0.0669422i	0.999440	−	0.0334711i	$-0.0106562\pi$
$480$		0			0
$481$			742.204i			1.54304i
$482$	−364.181			−0.755562
$483$	−403.506	−	80.7425i	−0.835416	−	0.167169i
$484$	64.3530			0.132961
$485$		0			0
$486$	−248.633			−0.511591
$487$			685.808i			1.40823i	0.710085	+	0.704115i	$0.248656\pi$
$487$			685.808i			1.40823i	−0.710085	+	0.704115i	$0.751344\pi$
$488$	67.8655			0.139069
$489$	110.308			0.225578
$490$		0			0
$491$	775.805			1.58005			0.790026	−	0.613074i	$-0.210067\pi$
$491$	775.805			1.58005			0.790026	+	0.613074i	$0.210067\pi$
$492$			332.403i			0.675615i
$493$	−194.307			−0.394132
$494$			266.368i			0.539207i
$495$		0			0
$496$	147.371			0.297120
$497$	525.853			1.05805
$498$	−230.458			−0.462767
$499$	−392.467			−0.786507			−0.393253	−	0.919430i	$-0.628650\pi$
$499$	−392.467			−0.786507			−0.393253	+	0.919430i	$0.628650\pi$
$500$		0			0
$501$	184.250			0.367764
$502$	−171.739			−0.342110
$503$	633.449			1.25934			0.629671	−	0.776862i	$-0.283190\pi$
$503$	633.449			1.25934			0.629671	+	0.776862i	$0.283190\pi$
$504$			75.0789i			0.148966i
$505$		0			0
$506$	−78.9889	+	394.742i	−0.156105	+	0.780123i
$507$			1.84649i			0.00364199i
$508$			302.752i			0.595969i
$509$	−540.009			−1.06092			−0.530460	−	0.847710i	$-0.677981\pi$
$509$	−540.009			−1.06092			−0.530460	+	0.847710i	$0.677981\pi$
$510$		0			0
$511$		−	197.950i		−	0.387378i
$512$		−	22.6274i		−	0.0441942i
$513$	423.819			0.826158
$514$	271.946			0.529078
$515$		0			0
$516$		−	328.836i		−	0.637279i
$517$	−820.080			−1.58623
$518$		−	613.673i		−	1.18470i
$519$	−321.937			−0.620303
$520$		0			0
$521$			396.259i			0.760574i	0.924869	+	0.380287i	$0.124175\pi$
$521$			396.259i			0.760574i	−0.924869	+	0.380287i	$0.875825\pi$
$522$		−	104.867i		−	0.200894i
$523$	−27.4737			−0.0525309			−0.0262655	−	0.999655i	$-0.508362\pi$
$523$	−27.4737			−0.0525309			−0.0262655	+	0.999655i	$0.508362\pi$
$524$	−93.4668			−0.178372
$525$		0			0
$526$			590.855i			1.12330i
$527$	−336.390			−0.638311
$528$	−116.266			−0.220201
$529$	−488.268	−	203.558i	−0.923001	−	0.384797i
$530$		0			0
$531$	−288.479			−0.543274
$532$		−	220.240i		−	0.413985i
$533$			922.121i			1.73006i
$534$			150.931i			0.282642i
$535$		0			0
$536$		−	335.203i		−	0.625378i
$537$		−	130.743i		−	0.243469i
$538$		−	671.578i		−	1.24829i
$539$		−	111.836i		−	0.207489i
$540$		0			0
$541$	−802.725			−1.48378			−0.741890	−	0.670522i	$-0.766070\pi$
$541$	−802.725			−1.48378			−0.741890	+	0.670522i	$0.766070\pi$
$542$			45.3626i			0.0836948i
$543$	55.7349			0.102643
$544$			51.6494i			0.0949437i
$545$		0			0
$546$		−	329.696i		−	0.603839i
$547$		−	439.126i		−	0.802790i	−0.915905	−	0.401395i	$-0.868525\pi$
$547$		−	439.126i		−	0.802790i	0.915905	−	0.401395i	$-0.131475\pi$
$548$	62.3897			0.113850
$549$		−	83.6042i		−	0.152284i
$550$		0			0
$551$			307.620i			0.558295i
$552$	29.9777	−	149.812i	0.0543074	−	0.271398i
$553$		−	219.817i		−	0.397498i
$554$	649.178			1.17180
$555$		0			0
$556$	73.1034			0.131481
$557$	795.405			1.42802			0.714008	−	0.700137i	$-0.246878\pi$
$557$	795.405			1.42802			0.714008	+	0.700137i	$0.246878\pi$
$558$		−	181.548i		−	0.325355i
$559$		−	912.226i		−	1.63189i
$560$		0			0
$561$	265.390			0.473065
$562$	−454.730			−0.809128
$563$	−214.573			−0.381124			−0.190562	−	0.981675i	$-0.561031\pi$
$563$	−214.573			−0.381124			−0.190562	+	0.981675i	$0.561031\pi$
$564$	311.235			0.551835
$565$		0			0
$566$		−	606.290i		−	1.07118i
$567$	−285.680			−0.503845
$568$			195.236i			0.343725i
$569$		−	1101.54i		−	1.93592i	−0.251114	−	0.967958i	$-0.580797\pi$
$569$		−	1101.54i		−	1.93592i	0.251114	−	0.967958i	$-0.419203\pi$
$570$		0			0
$571$			560.543i			0.981686i	0.871248	+	0.490843i	$0.163311\pi$
$571$			560.543i			0.981686i	−0.871248	+	0.490843i	$0.836689\pi$
$572$	−322.535			−0.563873
$573$	825.882			1.44133
$574$		−	762.433i		−	1.32828i
$575$		0			0
$576$	−27.8749			−0.0483939
$577$			795.419i			1.37854i	0.724504	+	0.689271i	$0.242069\pi$
$577$			795.419i			1.37854i	−0.724504	+	0.689271i	$0.757931\pi$
$578$			290.813i			0.503136i
$579$	92.7222			0.160142
$580$		0			0
$581$	528.602			0.909813
$582$	247.259			0.424844
$583$		−	958.387i		−	1.64389i
$584$	73.4939			0.125846
$585$		0			0
$586$			508.891i			0.868415i
$587$			579.454i			0.987144i	0.869705	+	0.493572i	$0.164309\pi$
$587$			579.454i			0.987144i	−0.869705	+	0.493572i	$0.835691\pi$
$588$			42.4439i			0.0721834i
$589$			532.561i			0.904179i
$590$		0			0
$591$	−528.460			−0.894180
$592$	227.841			0.384867
$593$			175.139i			0.295344i	0.989036	+	0.147672i	$0.0471779\pi$
$593$			175.139i			0.295344i	−0.989036	+	0.147672i	$0.952822\pi$
$594$			513.187i			0.863950i
$595$		0			0
$596$		−	364.562i		−	0.611681i
$597$	−87.1009			−0.145898
$598$	83.1613	−	415.594i	0.139066	−	0.694973i
$599$	512.672			0.855879			0.427940	−	0.903807i	$-0.359240\pi$
$599$	512.672			0.855879			0.427940	+	0.903807i	$0.359240\pi$
$600$		0			0
$601$	−175.570			−0.292130			−0.146065	−	0.989275i	$-0.546661\pi$
$601$	−175.570			−0.292130			−0.146065	+	0.989275i	$0.546661\pi$
$602$			754.251i			1.25291i
$603$	−412.939			−0.684808
$604$	−81.8925			−0.135584
$605$		0			0
$606$	56.6497			0.0934814
$607$		−	173.873i		−	0.286447i	−0.989690	−	0.143223i	$-0.954253\pi$
$607$		−	173.873i		−	0.286447i	0.989690	−	0.143223i	$-0.0457467\pi$
$608$	81.7695			0.134489
$609$		−	380.755i		−	0.625214i
$610$		0			0
$611$	863.399			1.41309
$612$	63.6273			0.103966
$613$	560.182			0.913837			0.456919	−	0.889509i	$-0.348953\pi$
$613$	560.182			0.913837			0.456919	+	0.889509i	$0.348953\pi$
$614$	−634.760			−1.03381
$615$		0			0
$616$	266.680			0.432922
$617$	−308.240			−0.499578			−0.249789	−	0.968300i	$-0.580361\pi$
$617$	−308.240			−0.499578			−0.249789	+	0.968300i	$0.580361\pi$
$618$	511.326			0.827389
$619$		−	424.114i		−	0.685161i	−0.939489	−	0.342580i	$-0.888699\pi$
$619$		−	424.114i		−	0.685161i	0.939489	−	0.342580i	$-0.111301\pi$
$620$		0			0
$621$	−661.252	−	132.318i	−1.06482	−	0.213073i
$622$		−	265.253i		−	0.426452i
$623$		−	346.190i		−	0.555683i
$624$	122.408			0.196166
$625$		0			0
$626$		−	132.125i		−	0.211062i
$627$		−	420.156i		−	0.670105i
$628$	−504.792			−0.803810
$629$	−520.070			−0.826821
$630$		0			0
$631$		−	172.251i		−	0.272981i	−0.990641	−	0.136491i	$-0.956418\pi$
$631$		−	172.251i		−	0.272981i	0.990641	−	0.136491i	$-0.0435823\pi$
$632$	81.6123			0.129133
$633$		−	509.783i		−	0.805344i
$634$	640.737			1.01063
$635$		0			0
$636$			363.725i			0.571894i
$637$			117.744i			0.184841i
$638$	−372.486			−0.583834
$639$	240.513			0.376390
$640$		0			0
$641$			365.954i			0.570912i	0.958392	+	0.285456i	$0.0921450\pi$
$641$			365.954i			0.570912i	−0.958392	+	0.285456i	$0.907855\pi$
$642$	374.718			0.583672
$643$	823.317			1.28043			0.640215	−	0.768195i	$-0.278845\pi$
$643$	823.317			1.28043			0.640215	+	0.768195i	$0.278845\pi$
$644$	−68.7598	+	343.623i	−0.106770	+	0.533576i
$645$		0			0
$646$	−186.647			−0.288927
$647$		−	828.654i		−	1.28076i	−0.768057	−	0.640381i	$-0.778776\pi$
$647$		−	828.654i		−	1.28076i	0.768057	−	0.640381i	$-0.221224\pi$
$648$		−	106.066i		−	0.163682i
$649$			1024.68i			1.57885i
$650$		0			0
$651$		−	659.175i		−	1.01256i
$652$		−	93.9372i		−	0.144076i
$653$			912.811i			1.39787i	0.715183	+	0.698937i	$0.246343\pi$
$653$			912.811i			1.39787i	−0.715183	+	0.698937i	$0.753657\pi$
$654$		−	324.182i		−	0.495692i
$655$		0			0
$656$	283.072			0.431512
$657$		−	90.5378i		−	0.137805i
$658$	−713.880			−1.08492
$659$			247.106i			0.374971i	0.982267	+	0.187486i	$0.0600338\pi$
$659$			247.106i			0.374971i	−0.982267	+	0.187486i	$0.939966\pi$
$660$		0			0
$661$			937.016i			1.41757i	0.705423	+	0.708787i	$0.250757\pi$
$661$			937.016i			1.41757i	−0.705423	+	0.708787i	$0.749243\pi$
$662$			252.191i			0.380953i
$663$	−279.408			−0.421430
$664$			196.256i			0.295567i
$665$		0			0
$666$		−	280.680i		−	0.421441i
$667$	96.0404	−	479.956i	0.143989	−	0.719574i
$668$		−	156.906i		−	0.234889i
$669$	−632.691			−0.945726
$670$		0			0
$671$	−296.962			−0.442566
$672$	−101.210			−0.150610
$673$		−	1174.75i		−	1.74554i	−0.488134	−	0.872769i	$-0.662322\pi$
$673$		−	1174.75i		−	1.74554i	0.488134	−	0.872769i	$-0.337678\pi$
$674$		−	121.236i		−	0.179876i
$675$		0			0
$676$	1.57246			0.00232612
$677$	266.022			0.392943			0.196471	−	0.980510i	$-0.437052\pi$
$677$	266.022			0.392943			0.196471	+	0.980510i	$0.437052\pi$
$678$	189.522			0.279531
$679$	−567.139			−0.835256
$680$		0			0
$681$			413.337i			0.606955i
$682$	−644.858			−0.945540
$683$			214.808i			0.314507i	0.987558	+	0.157254i	$0.0502640\pi$
$683$			214.808i			0.314507i	−0.987558	+	0.157254i	$0.949736\pi$
$684$		−	100.733i		−	0.147270i
$685$		0			0
$686$			430.557i			0.627635i
$687$	−148.115			−0.215597
$688$	−280.034			−0.407027
$689$			1009.01i			1.46446i
$690$		0			0
$691$	−998.531			−1.44505			−0.722526	−	0.691344i	$-0.757019\pi$
$691$	−998.531			−1.44505			−0.722526	+	0.691344i	$0.757019\pi$
$692$			274.160i			0.396184i
$693$		−	328.526i		−	0.474063i
$694$	306.502			0.441646
$695$		0			0
$696$	141.365			0.203110
$697$	−646.140			−0.927030
$698$		−	747.867i		−	1.07144i
$699$	−10.5224			−0.0150535
$700$		0			0
$701$		−	1089.04i		−	1.55356i	−0.629773	−	0.776779i	$-0.716852\pi$
$701$		−	1089.04i		−	1.55356i	0.629773	−	0.776779i	$-0.283148\pi$
$702$		−	540.294i		−	0.769650i
$703$			823.358i			1.17121i
$704$			99.0116i			0.140641i
$705$		0			0
$706$	40.7173			0.0576732
$707$	−129.938			−0.183787
$708$		−	388.882i		−	0.549268i
$709$		−	779.102i		−	1.09887i	−0.835535	−	0.549437i	$-0.814842\pi$
$709$		−	779.102i		−	1.09887i	0.835535	−	0.549437i	$-0.185158\pi$
$710$		0			0
$711$		−	100.539i		−	0.141405i
$712$	128.532			0.180522
$713$	166.268	−	830.913i	0.233195	−	1.16538i
$714$	231.021			0.323559
$715$		0			0
$716$	−111.340			−0.155503
$717$			200.848i			0.280123i
$718$	−78.5837			−0.109448
$719$	802.454			1.11607			0.558035	−	0.829817i	$-0.311555\pi$
$719$	802.454			1.11607			0.558035	+	0.829817i	$0.311555\pi$
$720$		0			0
$721$	−1172.83			−1.62667
$722$		−	215.038i		−	0.297836i
$723$	−604.784			−0.836492
$724$		−	47.4634i		−	0.0655572i
$725$		0			0
$726$	106.869			0.147202
$727$	19.7852			0.0272149			0.0136075	−	0.999907i	$-0.495668\pi$
$727$	19.7852			0.0272149			0.0136075	+	0.999907i	$0.495668\pi$
$728$	−280.767			−0.385669
$729$	−750.396			−1.02935
$730$		0			0
$731$	639.207			0.874428
$732$	112.702			0.153965
$733$	−609.029			−0.830871			−0.415436	−	0.909623i	$-0.636371\pi$
$733$	−609.029			−0.830871			−0.415436	+	0.909623i	$0.636371\pi$
$734$		−	232.339i		−	0.316538i
$735$		0			0
$736$	−127.579	−	25.5288i	−0.173340	−	0.0346858i
$737$			1466.76i			1.99017i
$738$		−	348.719i		−	0.472519i
$739$	−780.692			−1.05642			−0.528209	−	0.849115i	$-0.677136\pi$
$739$	−780.692			−1.05642			−0.528209	+	0.849115i	$0.677136\pi$
$740$		0			0
$741$			442.349i			0.596963i
$742$		−	834.276i		−	1.12436i
$743$	−46.4819			−0.0625598			−0.0312799	−	0.999511i	$-0.509958\pi$
$743$	−46.4819			−0.0625598			−0.0312799	+	0.999511i	$0.509958\pi$
$744$	244.735			0.328945
$745$		0			0
$746$			35.5648i			0.0476740i
$747$	241.770			0.323655
$748$		−	226.004i		−	0.302144i
$749$	−859.490			−1.14752
$750$		0			0
$751$			1258.62i			1.67592i	0.545728	+	0.837962i	$0.316253\pi$
$751$			1258.62i			1.67592i	−0.545728	+	0.837962i	$0.683747\pi$
$752$		−	265.045i		−	0.352454i
$753$	−285.202			−0.378754
$754$	392.161			0.520108
$755$		0			0
$756$			446.729i			0.590911i
$757$	615.435			0.812992			0.406496	−	0.913653i	$-0.366751\pi$
$757$	615.435			0.812992			0.406496	+	0.913653i	$0.366751\pi$
$758$	645.928			0.852148
$759$	−131.174	+	655.536i	−0.172825	+	0.863684i
$760$		0			0
$761$	−795.461			−1.04528			−0.522642	−	0.852552i	$-0.675053\pi$
$761$	−795.461			−1.04528			−0.522642	+	0.852552i	$0.675053\pi$
$762$			502.771i			0.659804i
$763$			743.578i			0.974545i
$764$		−	703.316i		−	0.920571i
$765$		0			0
$766$		−	524.697i		−	0.684983i
$767$		−	1078.80i		−	1.40652i
$768$		−	37.5766i		−	0.0489279i
$769$			1031.00i			1.34070i	0.742046	+	0.670349i	$0.233855\pi$
$769$			1031.00i			1.34070i	−0.742046	+	0.670349i	$0.766145\pi$
$770$		0			0
$771$	451.613			0.585749
$772$		−	78.9616i		−	0.102282i
$773$	−1281.23			−1.65747			−0.828737	−	0.559638i	$-0.810940\pi$
$773$	−1281.23			−1.65747			−0.828737	+	0.559638i	$0.810940\pi$
$774$			344.977i			0.445706i
$775$		0			0
$776$		−	210.564i		−	0.271346i
$777$		−	1019.11i		−	1.31159i
$778$	565.704			0.727126
$779$			1022.95i			1.31315i
$780$		0			0
$781$		−	854.301i		−	1.09386i
$782$	291.211	+	58.2720i	0.372392	+	0.0745166i
$783$		−	623.969i		−	0.796895i
$784$	36.1449			0.0461032
$785$		0			0
$786$	−155.217			−0.197477
$787$	1496.92			1.90206			0.951031	−	0.309096i	$-0.100027\pi$
$787$	1496.92			1.90206			0.951031	+	0.309096i	$0.100027\pi$
$788$			450.033i			0.571108i
$789$			981.215i			1.24362i
$790$		0			0
$791$	−434.707			−0.549567
$792$	121.973			0.154007
$793$	312.648			0.394260
$794$	373.730			0.470692
$795$		0			0
$796$			74.1745i			0.0931840i
$797$	−87.1739			−0.109378			−0.0546888	−	0.998503i	$-0.517417\pi$
$797$	−87.1739			−0.109378			−0.0546888	+	0.998503i	$0.517417\pi$
$798$		−	365.745i		−	0.458327i
$799$			604.993i			0.757188i
$800$		0			0
$801$		−	158.339i		−	0.197677i
$802$	601.492			0.749990
$803$	−321.590			−0.400486
$804$		−	556.660i		−	0.692364i
$805$		0			0
$806$	678.921			0.842334
$807$		−	1115.27i		−	1.38199i
$808$		−	48.2425i		−	0.0597061i
$809$	−602.054			−0.744196			−0.372098	−	0.928194i	$-0.621361\pi$
$809$	−602.054			−0.744196			−0.372098	+	0.928194i	$0.621361\pi$
$810$		0			0
$811$	1198.57			1.47790			0.738948	−	0.673763i	$-0.235323\pi$
$811$	1198.57			1.47790			0.738948	+	0.673763i	$0.235323\pi$
$812$	−324.249			−0.399321
$813$			75.3322i			0.0926595i
$814$	−996.973			−1.22478
$815$		0			0
$816$			85.7724i			0.105113i
$817$		−	1011.97i		−	1.23864i
$818$		−	341.615i		−	0.417622i
$819$			345.879i			0.422319i
$820$		0			0
$821$	−816.712			−0.994777			−0.497388	−	0.867528i	$-0.665708\pi$
$821$	−816.712			−0.994777			−0.497388	+	0.867528i	$0.665708\pi$
$822$	103.609			0.126045
$823$		−	143.774i		−	0.174694i	−0.996178	−	0.0873472i	$-0.972161\pi$
$823$		−	143.774i		−	0.174694i	0.996178	−	0.0873472i	$-0.0278390\pi$
$824$		−	435.442i		−	0.528449i
$825$		0			0
$826$			891.979i			1.07988i
$827$	−523.617			−0.633153			−0.316576	−	0.948567i	$-0.602533\pi$
$827$	−523.617			−0.633153			−0.316576	+	0.948567i	$0.602533\pi$
$828$	−31.4491	+	157.165i	−0.0379820	+	0.189813i
$829$	−735.461			−0.887167			−0.443583	−	0.896233i	$-0.646293\pi$
$829$	−735.461			−0.887167			−0.443583	+	0.896233i	$0.646293\pi$
$830$		0			0
$831$	1078.07			1.29732
$832$		−	104.242i		−	0.125290i
$833$	−82.5044			−0.0990449
$834$	121.400			0.145564
$835$		0			0
$836$	−357.802			−0.427993
$837$		−	1080.23i		−	1.29060i
$838$	506.786			0.604757
$839$		−	752.262i		−	0.896617i	−0.893879	−	0.448309i	$-0.852027\pi$
$839$		−	752.262i		−	0.896617i	0.893879	−	0.448309i	$-0.147973\pi$
$840$		0			0
$841$	−388.105			−0.461481
$842$	950.423			1.12877
$843$	−755.156			−0.895796
$844$	−434.128			−0.514369
$845$		0			0
$846$	−326.512			−0.385948
$847$	−245.125			−0.289404
$848$	309.745			0.365266
$849$		−	1006.85i		−	1.18592i
$850$		0			0
$851$	257.056	−	1284.62i	0.302063	−	1.50954i
$852$			324.222i			0.380542i
$853$			463.137i			0.542951i	0.962445	+	0.271475i	$0.0875116\pi$
$853$			463.137i			0.542951i	−0.962445	+	0.271475i	$0.912488\pi$
$854$	−258.505			−0.302699
$855$		0			0
$856$		−	319.107i		−	0.372789i
$857$		−	284.684i		−	0.332186i	−0.986110	−	0.166093i	$-0.946885\pi$
$857$		−	284.684i		−	0.332186i	0.986110	−	0.166093i	$-0.0531152\pi$
$858$	−535.624			−0.624270
$859$	−625.746			−0.728459			−0.364230	−	0.931309i	$-0.618668\pi$
$859$	−625.746			−0.728459			−0.364230	+	0.931309i	$0.618668\pi$
$860$		0			0
$861$		−	1266.15i		−	1.47055i
$862$	−1159.55			−1.34519
$863$		−	110.774i		−	0.128359i	−0.997938	−	0.0641793i	$-0.979557\pi$
$863$		−	110.774i		−	0.128359i	0.997938	−	0.0641793i	$-0.0204430\pi$
$864$	−165.859			−0.191967
$865$		0			0
$866$		−	447.219i		−	0.516419i
$867$			482.944i			0.557028i
$868$	−561.349			−0.646715
$869$	−357.114			−0.410948
$870$		0			0
$871$		−	1544.24i		−	1.77295i
$872$	−276.072			−0.316596
$873$	−259.396			−0.297132
$874$	92.2543	−	461.035i	0.105554	−	0.527500i
$875$		0			0
$876$	122.049			0.139325
$877$		−	336.486i		−	0.383678i	−0.981426	−	0.191839i	$-0.938555\pi$
$877$		−	336.486i		−	0.383678i	0.981426	−	0.191839i	$-0.0614452\pi$
$878$			295.399i			0.336445i
$879$			845.099i			0.961433i
$880$		0			0
$881$		−	508.337i		−	0.577000i	−0.957480	−	0.288500i	$-0.906843\pi$
$881$		−	508.337i		−	0.577000i	0.957480	−	0.288500i	$-0.0931565\pi$
$882$		−	44.5272i		−	0.0504844i
$883$			334.663i			0.379006i	0.981880	+	0.189503i	$0.0606877\pi$
$883$			334.663i			0.379006i	−0.981880	+	0.189503i	$0.939312\pi$
$884$			237.942i			0.269165i
$885$		0			0
$886$	999.822			1.12847
$887$		−	278.810i		−	0.314330i	−0.987572	−	0.157165i	$-0.949765\pi$
$887$		−	278.810i		−	0.314330i	0.987572	−	0.157165i	$-0.0502354\pi$
$888$	378.369			0.426091
$889$		−	1153.21i		−	1.29719i
$890$		0			0
$891$			464.116i			0.520893i
$892$			538.795i			0.604030i
$893$	957.804			1.07257
$894$		−	605.416i		−	0.677200i
$895$		0			0
$896$			86.1895i			0.0961937i
$897$	138.103	−	690.163i	0.153961	−	0.769412i
$898$		−	741.828i		−	0.826090i
$899$	784.065			0.872152
$900$		0			0
$901$	−707.025			−0.784712
$902$	−1238.65			−1.37322
$903$			1252.56i			1.38711i
$904$		−	161.396i		−	0.178535i
$905$		0			0
$906$	−135.996			−0.150106
$907$	−547.165			−0.603269			−0.301634	−	0.953424i	$-0.597532\pi$
$907$	−547.165			−0.603269			−0.301634	+	0.953424i	$0.597532\pi$
$908$	351.995			0.387659
$909$	−59.4304			−0.0653800
$910$		0			0
$911$			1702.27i			1.86857i	0.356527	+	0.934285i	$0.383961\pi$
$911$			1702.27i			1.86857i	−0.356527	+	0.934285i	$0.616039\pi$
$912$	135.792			0.148895
$913$		−	858.766i		−	0.940598i
$914$			1112.88i			1.21759i
$915$		0			0
$916$			126.134i			0.137701i
$917$	356.022			0.388247
$918$	378.590			0.412407
$919$		−	230.872i		−	0.251220i	−0.992080	−	0.125610i	$-0.959911\pi$
$919$		−	230.872i		−	0.251220i	0.992080	−	0.125610i	$-0.0400889\pi$
$920$		0			0
$921$	−1054.12			−1.14454
$922$			54.0879i			0.0586636i
$923$			899.427i			0.974461i
$924$	442.867			0.479293
$925$		0			0
$926$	743.799			0.803239
$927$	−536.425			−0.578668
$928$		−	120.385i		−	0.129726i
$929$	−1312.19			−1.41247			−0.706237	−	0.707975i	$-0.749609\pi$
$929$	−1312.19			−1.41247			−0.706237	+	0.707975i	$0.749609\pi$
$930$		0			0
$931$			130.618i			0.140299i
$932$			8.96082i			0.00961461i
$933$		−	440.497i		−	0.472130i
$934$			235.652i			0.252304i
$935$		0			0
$936$	−128.416			−0.137197
$937$	−933.379			−0.996136			−0.498068	−	0.867138i	$-0.665957\pi$
$937$	−933.379			−0.996136			−0.498068	+	0.867138i	$0.665957\pi$
$938$			1276.81i			1.36121i
$939$		−	219.416i		−	0.233670i
$940$		0			0
$941$		−	58.6896i		−	0.0623694i	−0.999514	−	0.0311847i	$-0.990072\pi$
$941$		−	58.6896i		−	0.0623694i	0.999514	−	0.0311847i	$-0.00992801\pi$
$942$	−838.293			−0.889907
$943$	319.368	−	1596.02i	0.338673	−	1.69250i
$944$	−331.169			−0.350815
$945$		0			0
$946$	1225.36			1.29530
$947$			1070.42i			1.13033i	0.824978	+	0.565165i	$0.191188\pi$
$947$			1070.42i			1.13033i	−0.824978	+	0.565165i	$0.808812\pi$
$948$	135.531			0.142965
$949$	338.577			0.356772
$950$		0			0
$951$	1064.05			1.11888
$952$		−	196.736i		−	0.206656i
$953$	1370.22			1.43780			0.718899	−	0.695114i	$-0.244646\pi$
$953$	1370.22			1.43780			0.718899	+	0.695114i	$0.244646\pi$
$954$		−	381.578i		−	0.399977i
$955$		0			0
$956$	171.041			0.178913
$957$	−618.575			−0.646369
$958$	−45.3472			−0.0473353
$959$	−237.647			−0.247807
$960$		0			0
$961$	396.395			0.412482
$962$	1049.64			1.09110
$963$	−393.111			−0.408215
$964$			515.030i			0.534263i
$965$		0			0
$966$	−114.187	+	570.644i	−0.118206	+	0.590728i
$967$			145.068i			0.150019i	0.997183	+	0.0750094i	$0.0238987\pi$
$967$			145.068i			0.150019i	−0.997183	+	0.0750094i	$0.976101\pi$
$968$		−	91.0088i		−	0.0940174i
$969$	−309.959			−0.319875
$970$		0			0
$971$			996.133i			1.02588i	0.858423	+	0.512942i	$0.171444\pi$
$971$			996.133i			1.02588i	−0.858423	+	0.512942i	$0.828556\pi$
$972$			351.620i			0.361749i
$973$	−278.456			−0.286183
$974$	969.880			0.995770
$975$		0			0
$976$		−	95.9764i		−	0.0983365i
$977$	−161.683			−0.165489			−0.0827444	−	0.996571i	$-0.526369\pi$
$977$	−161.683			−0.165489			−0.0827444	+	0.996571i	$0.526369\pi$
$978$		−	155.999i		−	0.159508i
$979$	−562.421			−0.574485
$980$		0			0
$981$			340.095i			0.346682i
$982$		−	1097.15i		−	1.11727i
$983$	1023.25			1.04095			0.520474	−	0.853878i	$-0.325755\pi$
$983$	1023.25			1.04095			0.520474	+	0.853878i	$0.325755\pi$
$984$	470.088			0.477732
$985$		0			0
$986$			274.792i			0.278693i
$987$	−1185.52			−1.20113
$988$	376.702			0.381277
$989$	−315.941	+	1578.90i	−0.319455	+	1.59646i
$990$		0			0
$991$	−518.210			−0.522917			−0.261458	−	0.965215i	$-0.584203\pi$
$991$	−518.210			−0.522917			−0.261458	+	0.965215i	$0.584203\pi$
$992$		−	208.415i		−	0.210095i
$993$			418.805i			0.421757i
$994$		−	743.669i		−	0.748158i
$995$		0			0
$996$			325.917i			0.327226i
$997$		−	1927.76i		−	1.93356i	−0.255611	−	0.966780i	$-0.582277\pi$
$997$		−	1927.76i		−	1.93356i	0.255611	−	0.966780i	$-0.417723\pi$
$998$			555.032i			0.556144i
$999$		−	1670.08i		−	1.67175i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1150.3.c.c.1149.11		32
5.2	odd	4		1150.3.d.b.551.12		16
5.3	odd	4		230.3.d.a.91.5	✓	16
5.4	even	2	inner	1150.3.c.c.1149.22		32
15.8	even	4		2070.3.c.a.91.13		16
20.3	even	4		1840.3.k.d.321.5		16
23.22	odd	2	inner	1150.3.c.c.1149.21		32
115.22	even	4		1150.3.d.b.551.11		16
115.68	even	4		230.3.d.a.91.6	yes	16
115.114	odd	2	inner	1150.3.c.c.1149.12		32
345.68	odd	4		2070.3.c.a.91.12		16
460.183	odd	4		1840.3.k.d.321.6		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.3.d.a.91.5	✓	16	5.3	odd	4
230.3.d.a.91.6	yes	16	115.68	even	4
1150.3.c.c.1149.11		32	1.1	even	1	trivial
1150.3.c.c.1149.12		32	115.114	odd	2	inner
1150.3.c.c.1149.21		32	23.22	odd	2	inner
1150.3.c.c.1149.22		32	5.4	even	2	inner
1150.3.d.b.551.11		16	115.22	even	4
1150.3.d.b.551.12		16	5.2	odd	4
1840.3.k.d.321.5		16	20.3	even	4
1840.3.k.d.321.6		16	460.183	odd	4
2070.3.c.a.91.12		16	345.68	odd	4
2070.3.c.a.91.13		16	15.8	even	4

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:	\( N \)	\(=\)	\( 1150 = 2 \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1150.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.41421i q^{2} -2.34854i q^{3} -2.00000 q^{4} -3.32134 q^{6} +7.61815 q^{7} +2.82843i q^{8} +3.48436 q^{9} +O(q^{10})\)	\(q-1.41421i q^{2} -2.34854i q^{3} -2.00000 q^{4} -3.32134 q^{6} +7.61815 q^{7} +2.82843i q^{8} +3.48436 q^{9} -12.3764i q^{11} +4.69708i q^{12} +13.0302i q^{13} -10.7737i q^{14} +4.00000 q^{16} -9.13040 q^{17} -4.92764i q^{18} +14.4549i q^{19} -17.8915i q^{21} -17.5029 q^{22} +(4.51289 - 22.5529i) q^{23} +6.64267 q^{24} +18.4275 q^{26} -29.3200i q^{27} -15.2363 q^{28} +21.2813 q^{29} +36.8428 q^{31} -5.65685i q^{32} -29.0666 q^{33} +12.9123i q^{34} -6.96873 q^{36} +56.9603 q^{37} +20.4424 q^{38} +30.6019 q^{39} +70.7680 q^{41} -25.3024 q^{42} -70.0086 q^{43} +24.7529i q^{44} +(-31.8946 - 6.38219i) q^{46} -66.2614i q^{47} -9.39416i q^{48} +9.03623 q^{49} +21.4431i q^{51} -26.0604i q^{52} +77.4364 q^{53} -41.4648 q^{54} +21.5474i q^{56} +33.9480 q^{57} -30.0963i q^{58} -82.7923 q^{59} -23.9941i q^{61} -52.1036i q^{62} +26.5444 q^{63} -8.00000 q^{64} +41.1063i q^{66} -118.512 q^{67} +18.2608 q^{68} +(-52.9664 - 10.5987i) q^{69} +69.0263 q^{71} +9.85527i q^{72} -25.9840i q^{73} -80.5540i q^{74} -28.9099i q^{76} -94.2857i q^{77} -43.2777i q^{78} -28.8543i q^{79} -37.4999 q^{81} -100.081i q^{82} +69.3871 q^{83} +35.7831i q^{84} +99.0071i q^{86} -49.9800i q^{87} +35.0059 q^{88} -45.4428i q^{89} +99.2661i q^{91} +(-9.02579 + 45.1058i) q^{92} -86.5269i q^{93} -93.7077 q^{94} -13.2853 q^{96} -74.4458 q^{97} -12.7792i q^{98} -43.1241i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 64 q^{4} - 16 q^{6} - 128 q^{9}+O(q^{10}) \) 32 * q - 64 * q^4 - 16 * q^6 - 128 * q^9	\( 32 q - 64 q^{4} - 16 q^{6} - 128 q^{9} + 128 q^{16} + 32 q^{24} + 192 q^{26} + 216 q^{29} - 232 q^{31} + 256 q^{36} - 496 q^{39} - 312 q^{41} - 248 q^{46} + 56 q^{49} - 448 q^{54} - 408 q^{59} - 256 q^{64} + 536 q^{69} + 472 q^{71} - 272 q^{81} + 432 q^{94} - 64 q^{96}+O(q^{100}) \) 32 * q - 64 * q^4 - 16 * q^6 - 128 * q^9 + 128 * q^16 + 32 * q^24 + 192 * q^26 + 216 * q^29 - 232 * q^31 + 256 * q^36 - 496 * q^39 - 312 * q^41 - 248 * q^46 + 56 * q^49 - 448 * q^54 - 408 * q^59 - 256 * q^64 + 536 * q^69 + 472 * q^71 - 272 * q^81 + 432 * q^94 - 64 * q^96

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more