LMFDB - Embedded newform 1014.4.b.h.337.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1014,4,Mod(337,1014)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1014.337"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1014, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 4, names="a")

Level:	$N$	$=$	$1014 = 2 \cdot 3 \cdot 13^{2}$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	1014.b (of order $2$ , degree $1$ , not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,6,-8,0,0,0,0,18,40] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(10)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$59.8279367458$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} + 1$ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$2$
Twist minimal:	no (minimal twist has level 78)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	1014.337
Dual form			1014.4.b.h.337.2

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000i q^{2} +3.00000 q^{3} -4.00000 q^{4} +10.0000i q^{5} -6.00000i q^{6} +8.00000i q^{7} +8.00000i q^{8} +9.00000 q^{9} +20.0000 q^{10} -40.0000i q^{11} -12.0000 q^{12} +16.0000 q^{14} +30.0000i q^{15} +16.0000 q^{16} -130.000 q^{17} -18.0000i q^{18} -20.0000i q^{19} -40.0000i q^{20} +24.0000i q^{21} -80.0000 q^{22} +24.0000i q^{24} +25.0000 q^{25} +27.0000 q^{27} -32.0000i q^{28} -18.0000 q^{29} +60.0000 q^{30} -184.000i q^{31} -32.0000i q^{32} -120.000i q^{33} +260.000i q^{34} -80.0000 q^{35} -36.0000 q^{36} +74.0000i q^{37} -40.0000 q^{38} -80.0000 q^{40} -362.000i q^{41} +48.0000 q^{42} -76.0000 q^{43} +160.000i q^{44} +90.0000i q^{45} +452.000i q^{47} +48.0000 q^{48} +279.000 q^{49} -50.0000i q^{50} -390.000 q^{51} +382.000 q^{53} -54.0000i q^{54} +400.000 q^{55} -64.0000 q^{56} -60.0000i q^{57} +36.0000i q^{58} -464.000i q^{59} -120.000i q^{60} +358.000 q^{61} -368.000 q^{62} +72.0000i q^{63} -64.0000 q^{64} -240.000 q^{66} -700.000i q^{67} +520.000 q^{68} +160.000i q^{70} -748.000i q^{71} +72.0000i q^{72} -1058.00i q^{73} +148.000 q^{74} +75.0000 q^{75} +80.0000i q^{76} +320.000 q^{77} -976.000 q^{79} +160.000i q^{80} +81.0000 q^{81} -724.000 q^{82} -1008.00i q^{83} -96.0000i q^{84} -1300.00i q^{85} +152.000i q^{86} -54.0000 q^{87} +320.000 q^{88} +386.000i q^{89} +180.000 q^{90} -552.000i q^{93} +904.000 q^{94} +200.000 q^{95} -96.0000i q^{96} -614.000i q^{97} -558.000i q^{98} -360.000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$2 q + 6 q^{3} - 8 q^{4} + 18 q^{9} + 40 q^{10} - 24 q^{12} + 32 q^{14} + 32 q^{16} - 260 q^{17} - 160 q^{22} + 50 q^{25} + 54 q^{27} - 36 q^{29} + 120 q^{30} - 160 q^{35} - 72 q^{36} - 80 q^{38} - 160 q^{40}+ \cdots + 400 q^{95}+O(q^{100})$ 2 * q + 6 * q^3 - 8 * q^4 + 18 * q^9 + 40 * q^10 - 24 * q^12 + 32 * q^14 + 32 * q^16 - 260 * q^17 - 160 * q^22 + 50 * q^25 + 54 * q^27 - 36 * q^29 + 120 * q^30 - 160 * q^35 - 72 * q^36 - 80 * q^38 - 160 * q^40 + 96 * q^42 - 152 * q^43 + 96 * q^48 + 558 * q^49 - 780 * q^51 + 764 * q^53 + 800 * q^55 - 128 * q^56 + 716 * q^61 - 736 * q^62 - 128 * q^64 - 480 * q^66 + 1040 * q^68 + 296 * q^74 + 150 * q^75 + 640 * q^77 - 1952 * q^79 + 162 * q^81 - 1448 * q^82 - 108 * q^87 + 640 * q^88 + 360 * q^90 + 1808 * q^94 + 400 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1014\mathbb{Z}\right)^\times$ .

$n$	$677$	$847$
$\chi(n)$	$1$	$-1$

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

)

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

$2$	− 2.00000i	− 0.707107i
$2$	− 2.00000i	− 0.707107i
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	−4.00000	−0.500000
$5$	10.0000i	0.894427i	0.894427	+	0.447214i	$0.147584\pi$
$5$	10.0000i	0.894427i	−0.894427	+	0.447214i	$0.852416\pi$
$6$	− 6.00000i	− 0.408248i
$7$	8.00000i	0.431959i	0.976398	+	0.215980i	$0.0692945\pi$
$7$	8.00000i	0.431959i	−0.976398	+	0.215980i	$0.930705\pi$
$8$	8.00000i	0.353553i
$9$	9.00000	0.333333
$10$	20.0000	0.632456
$11$	− 40.0000i	− 1.09640i	−0.836346	−	0.548202i	$-0.815312\pi$
$11$	− 40.0000i	− 1.09640i	0.836346	−	0.548202i	$-0.184688\pi$
$12$	−12.0000	−0.288675
$13$	0	0
$13$	0	0
$14$	16.0000	0.305441
$15$	30.0000i	0.516398i
$16$	16.0000	0.250000
$17$	−130.000	−1.85468	−0.927342	−	0.374215i	$-0.877912\pi$
$17$	−130.000	−1.85468	−0.927342	+	0.374215i	$0.877912\pi$
$18$	− 18.0000i	− 0.235702i
$19$	− 20.0000i	− 0.241490i	−0.992684	−	0.120745i	$-0.961472\pi$
$19$	− 20.0000i	− 0.241490i	0.992684	−	0.120745i	$-0.0385284\pi$
$20$	− 40.0000i	− 0.447214i
$21$	24.0000i	0.249392i
$22$	−80.0000	−0.775275
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	24.0000i	0.204124i
$25$	25.0000	0.200000
$26$	0	0
$27$	27.0000	0.192450
$28$	− 32.0000i	− 0.215980i
$29$	−18.0000	−0.115259	−0.0576296	−	0.998338i	$-0.518354\pi$
$29$	−18.0000	−0.115259	−0.0576296	+	0.998338i	$0.518354\pi$
$30$	60.0000	0.365148
$31$	− 184.000i	− 1.06604i	−0.846101	−	0.533022i	$-0.821056\pi$
$31$	− 184.000i	− 1.06604i	0.846101	−	0.533022i	$-0.178944\pi$
$32$	− 32.0000i	− 0.176777i
$33$	− 120.000i	− 0.633010i
$34$	260.000i	1.31146i
$35$	−80.0000	−0.386356
$36$	−36.0000	−0.166667
$37$	74.0000i	0.328798i	0.986394	+	0.164399i	$0.0525685\pi$
$37$	74.0000i	0.328798i	−0.986394	+	0.164399i	$0.947432\pi$
$38$	−40.0000	−0.170759
$39$	0	0
$40$	−80.0000	−0.316228
$41$	− 362.000i	− 1.37890i	−0.724333	−	0.689450i	$-0.757852\pi$
$41$	− 362.000i	− 1.37890i	0.724333	−	0.689450i	$-0.242148\pi$
$42$	48.0000	0.176347
$43$	−76.0000	−0.269532	−0.134766	−	0.990877i	$-0.543028\pi$
$43$	−76.0000	−0.269532	−0.134766	+	0.990877i	$0.543028\pi$
$44$	160.000i	0.548202i
$45$	90.0000i	0.298142i
$46$	0	0
$47$	452.000i	1.40279i	0.712774	+	0.701393i	$0.247438\pi$
$47$	452.000i	1.40279i	−0.712774	+	0.701393i	$0.752562\pi$
$48$	48.0000	0.144338
$49$	279.000	0.813411
$50$	− 50.0000i	− 0.141421i
$51$	−390.000	−1.07080
$52$	0	0
$53$	382.000	0.990033	0.495016	−	0.868884i	$-0.335162\pi$
$53$	382.000	0.990033	0.495016	+	0.868884i	$0.335162\pi$
$54$	− 54.0000i	− 0.136083i
$55$	400.000	0.980654
$56$	−64.0000	−0.152721
$57$	− 60.0000i	− 0.139424i
$58$	36.0000i	0.0815005i
$59$	− 464.000i	− 1.02386i	−0.859028	−	0.511929i	$-0.828931\pi$
$59$	− 464.000i	− 1.02386i	0.859028	−	0.511929i	$-0.171069\pi$
$60$	− 120.000i	− 0.258199i
$61$	358.000	0.751430	0.375715	−	0.926735i	$-0.377397\pi$
$61$	358.000	0.751430	0.375715	+	0.926735i	$0.377397\pi$
$62$	−368.000	−0.753807
$63$	72.0000i	0.143986i
$64$	−64.0000	−0.125000
$65$	0	0
$66$	−240.000	−0.447605
$67$	− 700.000i	− 1.27640i	−0.769872	−	0.638199i	$-0.779680\pi$
$67$	− 700.000i	− 1.27640i	0.769872	−	0.638199i	$-0.220320\pi$
$68$	520.000	0.927342
$69$	0	0
$70$	160.000i	0.273195i
$71$	− 748.000i	− 1.25030i	−0.780505	−	0.625150i	$-0.785038\pi$
$71$	− 748.000i	− 1.25030i	0.780505	−	0.625150i	$-0.214962\pi$
$72$	72.0000i	0.117851i
$73$	− 1058.00i	− 1.69629i	−0.529760	−	0.848147i	$-0.677718\pi$
$73$	− 1058.00i	− 1.69629i	0.529760	−	0.848147i	$-0.322282\pi$
$74$	148.000	0.232495
$75$	75.0000	0.115470
$76$	80.0000i	0.120745i
$77$	320.000	0.473602
$78$	0	0
$79$	−976.000	−1.38998	−0.694991	−	0.719018i	$-0.744592\pi$
$79$	−976.000	−1.38998	−0.694991	+	0.719018i	$0.744592\pi$
$80$	160.000i	0.223607i
$81$	81.0000	0.111111
$82$	−724.000	−0.975030
$83$	− 1008.00i	− 1.33304i	−0.745487	−	0.666520i	$-0.767783\pi$
$83$	− 1008.00i	− 1.33304i	0.745487	−	0.666520i	$-0.232217\pi$
$84$	− 96.0000i	− 0.124696i
$85$	− 1300.00i	− 1.65888i
$86$	152.000i	0.190588i
$87$	−54.0000	−0.0665449
$88$	320.000	0.387638
$89$	386.000i	0.459729i	0.973223	+	0.229865i	$0.0738284\pi$
$89$	386.000i	0.459729i	−0.973223	+	0.229865i	$0.926172\pi$
$90$	180.000	0.210819
$91$	0	0
$92$	0	0
$93$	− 552.000i	− 0.615481i
$94$	904.000	0.991920
$95$	200.000	0.215995
$96$	− 96.0000i	− 0.102062i
$97$	− 614.000i	− 0.642704i	−0.946960	−	0.321352i	$-0.895863\pi$
$97$	− 614.000i	− 0.642704i	0.946960	−	0.321352i	$-0.104137\pi$
$98$	− 558.000i	− 0.575168i
$99$	− 360.000i	− 0.365468i
$100$	−100.000	−0.100000
$101$	−518.000	−0.510326	−0.255163	−	0.966898i	$-0.582129\pi$
$101$	−518.000	−0.510326	−0.255163	+	0.966898i	$0.582129\pi$
$102$	780.000i	0.757172i
$103$	−112.000	−0.107143	−0.0535713	−	0.998564i	$-0.517060\pi$
$103$	−112.000	−0.107143	−0.0535713	+	0.998564i	$0.517060\pi$
$104$	0	0
$105$	−240.000	−0.223063
$106$	− 764.000i	− 0.700059i
$107$	−372.000	−0.336099	−0.168050	−	0.985779i	$-0.553747\pi$
$107$	−372.000	−0.336099	−0.168050	+	0.985779i	$0.553747\pi$
$108$	−108.000	−0.0962250
$109$	934.000i	0.820743i	0.911918	+	0.410371i	$0.134601\pi$
$109$	934.000i	0.820743i	−0.911918	+	0.410371i	$0.865399\pi$
$110$	− 800.000i	− 0.693427i
$111$	222.000i	0.189832i
$112$	128.000i	0.107990i
$113$	1914.00	1.59340	0.796699	−	0.604376i	$-0.206578\pi$
$113$	1914.00	1.59340	0.796699	+	0.604376i	$0.206578\pi$
$114$	−120.000	−0.0985880
$115$	0	0
$116$	72.0000	0.0576296
$117$	0	0
$118$	−928.000	−0.723977
$119$	− 1040.00i	− 0.801148i
$120$	−240.000	−0.182574
$121$	−269.000	−0.202104
$122$	− 716.000i	− 0.531341i
$123$	− 1086.00i	− 0.796108i
$124$	736.000i	0.533022i
$125$	1500.00i	1.07331i
$126$	144.000	0.101814
$127$	−1296.00	−0.905523	−0.452761	−	0.891632i	$-0.649561\pi$
$127$	−1296.00	−0.905523	−0.452761	+	0.891632i	$0.649561\pi$
$128$	128.000i	0.0883883i
$129$	−228.000	−0.155615
$130$	0	0
$131$	−892.000	−0.594919	−0.297460	−	0.954734i	$-0.596139\pi$
$131$	−892.000	−0.594919	−0.297460	+	0.954734i	$0.596139\pi$
$132$	480.000i	0.316505i
$133$	160.000	0.104314
$134$	−1400.00	−0.902549
$135$	270.000i	0.172133i
$136$	− 1040.00i	− 0.655730i
$137$	− 2326.00i	− 1.45054i	−0.688466	−	0.725269i	$-0.741716\pi$
$137$	− 2326.00i	− 1.45054i	0.688466	−	0.725269i	$-0.258284\pi$
$138$	0	0
$139$	1932.00	1.17892	0.589461	−	0.807797i	$-0.299340\pi$
$139$	1932.00	1.17892	0.589461	+	0.807797i	$0.299340\pi$
$140$	320.000	0.193178
$141$	1356.00i	0.809899i
$142$	−1496.00	−0.884095
$143$	0	0
$144$	144.000	0.0833333
$145$	− 180.000i	− 0.103091i
$146$	−2116.00	−1.19946
$147$	837.000	0.469623
$148$	− 296.000i	− 0.164399i
$149$	882.000i	0.484941i	0.970159	+	0.242471i	$0.0779578\pi$
$149$	882.000i	0.484941i	−0.970159	+	0.242471i	$0.922042\pi$
$150$	− 150.000i	− 0.0816497i
$151$	1776.00i	0.957145i	0.878048	+	0.478572i	$0.158846\pi$
$151$	1776.00i	0.957145i	−0.878048	+	0.478572i	$0.841154\pi$
$152$	160.000	0.0853797
$153$	−1170.00	−0.618228
$154$	− 640.000i	− 0.334887i
$155$	1840.00	0.953499
$156$	0	0
$157$	−2410.00	−1.22509	−0.612544	−	0.790436i	$-0.709854\pi$
$157$	−2410.00	−1.22509	−0.612544	+	0.790436i	$0.709854\pi$
$158$	1952.00i	0.982866i
$159$	1146.00	0.571596
$160$	320.000	0.158114
$161$	0	0
$162$	− 162.000i	− 0.0785674i
$163$	− 3212.00i	− 1.54346i	−0.635953	−	0.771728i	$-0.719393\pi$
$163$	− 3212.00i	− 1.54346i	0.635953	−	0.771728i	$-0.280607\pi$
$164$	1448.00i	0.689450i
$165$	1200.00	0.566181
$166$	−2016.00	−0.942602
$167$	− 1668.00i	− 0.772896i	−0.922311	−	0.386448i	$-0.873702\pi$
$167$	− 1668.00i	− 0.772896i	0.922311	−	0.386448i	$-0.126298\pi$
$168$	−192.000	−0.0881733
$169$	0	0
$170$	−2600.00	−1.17301
$171$	− 180.000i	− 0.0804967i
$172$	304.000	0.134766
$173$	−3598.00	−1.58122	−0.790609	−	0.612321i	$-0.790236\pi$
$173$	−3598.00	−1.58122	−0.790609	+	0.612321i	$0.790236\pi$
$174$	108.000i	0.0470544i
$175$	200.000i	0.0863919i
$176$	− 640.000i	− 0.274101i
$177$	− 1392.00i	− 0.591125i
$178$	772.000	0.325078
$179$	−1068.00	−0.445956	−0.222978	−	0.974824i	$-0.571578\pi$
$179$	−1068.00	−0.445956	−0.222978	+	0.974824i	$0.571578\pi$
$180$	− 360.000i	− 0.149071i
$181$	4786.00	1.96542	0.982709	−	0.185158i	$-0.0592797\pi$
$181$	4786.00	1.96542	0.982709	+	0.185158i	$0.0592797\pi$
$182$	0	0
$183$	1074.00	0.433838
$184$	0	0
$185$	−740.000	−0.294086
$186$	−1104.00	−0.435211
$187$	5200.00i	2.03348i
$188$	− 1808.00i	− 0.701393i
$189$	216.000i	0.0831306i
$190$	− 400.000i	− 0.152732i
$191$	−1312.00	−0.497031	−0.248516	−	0.968628i	$-0.579943\pi$
$191$	−1312.00	−0.497031	−0.248516	+	0.968628i	$0.579943\pi$
$192$	−192.000	−0.0721688
$193$	350.000i	0.130537i	0.997868	+	0.0652683i	$0.0207903\pi$
$193$	350.000i	0.130537i	−0.997868	+	0.0652683i	$0.979210\pi$
$194$	−1228.00	−0.454460
$195$	0	0
$196$	−1116.00	−0.406706
$197$	− 342.000i	− 0.123688i	−0.998086	−	0.0618439i	$-0.980302\pi$
$197$	− 342.000i	− 0.123688i	0.998086	−	0.0618439i	$-0.0196981\pi$
$198$	−720.000	−0.258425
$199$	3368.00	1.19975	0.599877	−	0.800092i	$-0.295216\pi$
$199$	3368.00	1.19975	0.599877	+	0.800092i	$0.295216\pi$
$200$	200.000i	0.0707107i
$201$	− 2100.00i	− 0.736928i
$202$	1036.00i	0.360855i
$203$	− 144.000i	− 0.0497873i
$204$	1560.00	0.535401
$205$	3620.00	1.23333
$206$	224.000i	0.0757613i
$207$	0	0
$208$	0	0
$209$	−800.000	−0.264771
$210$	480.000i	0.157729i
$211$	−2004.00	−0.653844	−0.326922	−	0.945051i	$-0.606011\pi$
$211$	−2004.00	−0.653844	−0.326922	+	0.945051i	$0.606011\pi$
$212$	−1528.00	−0.495016
$213$	− 2244.00i	− 0.721861i
$214$	744.000i	0.237658i
$215$	− 760.000i	− 0.241077i
$216$	216.000i	0.0680414i
$217$	1472.00	0.460488
$218$	1868.00	0.580353
$219$	− 3174.00i	− 0.979356i
$220$	−1600.00	−0.490327
$221$	0	0
$222$	444.000	0.134231
$223$	− 5608.00i	− 1.68403i	−0.539451	−	0.842017i	$-0.681368\pi$
$223$	− 5608.00i	− 1.68403i	0.539451	−	0.842017i	$-0.318632\pi$
$224$	256.000	0.0763604
$225$	225.000	0.0666667
$226$	− 3828.00i	− 1.12670i
$227$	− 1928.00i	− 0.563726i	−0.959455	−	0.281863i	$-0.909048\pi$
$227$	− 1928.00i	− 0.563726i	0.959455	−	0.281863i	$-0.0909524\pi$
$228$	240.000i	0.0697122i
$229$	3938.00i	1.13638i	0.822898	+	0.568189i	$0.192356\pi$
$229$	3938.00i	1.13638i	−0.822898	+	0.568189i	$0.807644\pi$
$230$	0	0
$231$	960.000	0.273434
$232$	− 144.000i	− 0.0407503i
$233$	−2562.00	−0.720353	−0.360176	−	0.932884i	$-0.617283\pi$
$233$	−2562.00	−0.720353	−0.360176	+	0.932884i	$0.617283\pi$
$234$	0	0
$235$	−4520.00	−1.25469
$236$	1856.00i	0.511929i
$237$	−2928.00	−0.802506
$238$	−2080.00	−0.566497
$239$	7164.00i	1.93891i	0.245260	+	0.969457i	$0.421127\pi$
$239$	7164.00i	1.93891i	−0.245260	+	0.969457i	$0.578873\pi$
$240$	480.000i	0.129099i
$241$	6182.00i	1.65236i	0.563410	+	0.826178i	$0.309489\pi$
$241$	6182.00i	1.65236i	−0.563410	+	0.826178i	$0.690511\pi$
$242$	538.000i	0.142909i
$243$	243.000	0.0641500
$244$	−1432.00	−0.375715
$245$	2790.00i	0.727537i
$246$	−2172.00	−0.562934
$247$	0	0
$248$	1472.00	0.376904
$249$	− 3024.00i	− 0.769631i
$250$	3000.00	0.758947
$251$	1396.00	0.351055	0.175527	−	0.984475i	$-0.443837\pi$
$251$	1396.00	0.351055	0.175527	+	0.984475i	$0.443837\pi$
$252$	− 288.000i	− 0.0719932i
$253$	0	0
$254$	2592.00i	0.640301i
$255$	− 3900.00i	− 0.957755i
$256$	256.000	0.0625000
$257$	−6906.00	−1.67620	−0.838102	−	0.545514i	$-0.816335\pi$
$257$	−6906.00	−1.67620	−0.838102	+	0.545514i	$0.816335\pi$
$258$	456.000i	0.110036i
$259$	−592.000	−0.142027
$260$	0	0
$261$	−162.000	−0.0384197
$262$	1784.00i	0.420671i
$263$	−6848.00	−1.60557	−0.802787	−	0.596266i	$-0.796650\pi$
$263$	−6848.00	−1.60557	−0.802787	+	0.596266i	$0.796650\pi$
$264$	960.000	0.223803
$265$	3820.00i	0.885512i
$266$	− 320.000i	− 0.0737611i
$267$	1158.00i	0.265425i
$268$	2800.00i	0.638199i
$269$	−6034.00	−1.36766	−0.683828	−	0.729643i	$-0.739686\pi$
$269$	−6034.00	−1.36766	−0.683828	+	0.729643i	$0.739686\pi$
$270$	540.000	0.121716
$271$	− 4832.00i	− 1.08311i	−0.840665	−	0.541556i	$-0.817836\pi$
$271$	− 4832.00i	− 1.08311i	0.840665	−	0.541556i	$-0.182164\pi$
$272$	−2080.00	−0.463671
$273$	0	0
$274$	−4652.00	−1.02568
$275$	− 1000.00i	− 0.219281i
$276$	0	0
$277$	4082.00	0.885428	0.442714	−	0.896663i	$-0.354016\pi$
$277$	4082.00	0.885428	0.442714	+	0.896663i	$0.354016\pi$
$278$	− 3864.00i	− 0.833623i
$279$	− 1656.00i	− 0.355348i
$280$	− 640.000i	− 0.136598i
$281$	− 3350.00i	− 0.711189i	−0.934640	−	0.355595i	$-0.884278\pi$
$281$	− 3350.00i	− 0.711189i	0.934640	−	0.355595i	$-0.115722\pi$
$282$	2712.00	0.572685
$283$	−7796.00	−1.63754	−0.818770	−	0.574121i	$-0.805344\pi$
$283$	−7796.00	−1.63754	−0.818770	+	0.574121i	$0.805344\pi$
$284$	2992.00i	0.625150i
$285$	600.000	0.124705
$286$	0	0
$287$	2896.00	0.595629
$288$	− 288.000i	− 0.0589256i
$289$	11987.0	2.43985
$290$	−360.000	−0.0728963
$291$	− 1842.00i	− 0.371065i
$292$	4232.00i	0.848147i
$293$	− 3922.00i	− 0.781999i	−0.920391	−	0.390999i	$-0.872129\pi$
$293$	− 3922.00i	− 0.781999i	0.920391	−	0.390999i	$-0.127871\pi$
$294$	− 1674.00i	− 0.332074i
$295$	4640.00	0.915767
$296$	−592.000	−0.116248
$297$	− 1080.00i	− 0.211003i
$298$	1764.00	0.342905
$299$	0	0
$300$	−300.000	−0.0577350
$301$	− 608.000i	− 0.116427i
$302$	3552.00	0.676803
$303$	−1554.00	−0.294637
$304$	− 320.000i	− 0.0603726i
$305$	3580.00i	0.672099i
$306$	2340.00i	0.437153i
$307$	− 5956.00i	− 1.10725i	−0.832765	−	0.553627i	$-0.813243\pi$
$307$	− 5956.00i	− 1.10725i	0.832765	−	0.553627i	$-0.186757\pi$
$308$	−1280.00	−0.236801
$309$	−336.000	−0.0618588
$310$	− 3680.00i	− 0.674226i
$311$	−2352.00	−0.428841	−0.214421	−	0.976741i	$-0.568786\pi$
$311$	−2352.00	−0.428841	−0.214421	+	0.976741i	$0.568786\pi$
$312$	0	0
$313$	8442.00	1.52450	0.762252	−	0.647280i	$-0.224093\pi$
$313$	8442.00	1.52450	0.762252	+	0.647280i	$0.224093\pi$
$314$	4820.00i	0.866269i
$315$	−720.000	−0.128785
$316$	3904.00	0.694991
$317$	− 5550.00i	− 0.983341i	−0.870781	−	0.491670i	$-0.836386\pi$
$317$	− 5550.00i	− 0.983341i	0.870781	−	0.491670i	$-0.163614\pi$
$318$	− 2292.00i	− 0.404179i
$319$	720.000i	0.126371i
$320$	− 640.000i	− 0.111803i
$321$	−1116.00	−0.194047
$322$	0	0
$323$	2600.00i	0.447888i
$324$	−324.000	−0.0555556
$325$	0	0
$326$	−6424.00	−1.09139
$327$	2802.00i	0.473856i
$328$	2896.00	0.487515
$329$	−3616.00	−0.605947
$330$	− 2400.00i	− 0.400350i
$331$	140.000i	0.0232480i	0.999932	+	0.0116240i	$0.00370012\pi$
$331$	140.000i	0.0232480i	−0.999932	+	0.0116240i	$0.996300\pi$
$332$	4032.00i	0.666520i
$333$	666.000i	0.109599i
$334$	−3336.00	−0.546520
$335$	7000.00	1.14164
$336$	384.000i	0.0623480i
$337$	6174.00	0.997980	0.498990	−	0.866608i	$-0.333704\pi$
$337$	6174.00	0.997980	0.498990	+	0.866608i	$0.333704\pi$
$338$	0	0
$339$	5742.00	0.919949
$340$	5200.00i	0.829440i
$341$	−7360.00	−1.16882
$342$	−360.000	−0.0569198
$343$	4976.00i	0.783320i
$344$	− 608.000i	− 0.0952941i
$345$	0	0
$346$	7196.00i	1.11809i
$347$	−2988.00	−0.462260	−0.231130	−	0.972923i	$-0.574242\pi$
$347$	−2988.00	−0.462260	−0.231130	+	0.972923i	$0.574242\pi$
$348$	216.000	0.0332725
$349$	162.000i	0.0248472i	0.999923	+	0.0124236i	$0.00395465\pi$
$349$	162.000i	0.0248472i	−0.999923	+	0.0124236i	$0.996045\pi$
$350$	400.000	0.0610883
$351$	0	0
$352$	−1280.00	−0.193819
$353$	− 10754.0i	− 1.62147i	−0.585416	−	0.810733i	$-0.699069\pi$
$353$	− 10754.0i	− 1.62147i	0.585416	−	0.810733i	$-0.300931\pi$
$354$	−2784.00	−0.417989
$355$	7480.00	1.11830
$356$	− 1544.00i	− 0.229865i
$357$	− 3120.00i	− 0.462543i
$358$	2136.00i	0.315338i
$359$	− 3588.00i	− 0.527486i	−0.964593	−	0.263743i	$-0.915043\pi$
$359$	− 3588.00i	− 0.527486i	0.964593	−	0.263743i	$-0.0849570\pi$
$360$	−720.000	−0.105409
$361$	6459.00	0.941682
$362$	− 9572.00i	− 1.38976i
$363$	−807.000	−0.116685
$364$	0	0
$365$	10580.0	1.51721
$366$	− 2148.00i	− 0.306770i
$367$	11272.0	1.60325	0.801626	−	0.597826i	$-0.203968\pi$
$367$	11272.0	1.60325	0.801626	+	0.597826i	$0.203968\pi$
$368$	0	0
$369$	− 3258.00i	− 0.459633i
$370$	1480.00i	0.207950i
$371$	3056.00i	0.427654i
$372$	2208.00i	0.307741i
$373$	−10914.0	−1.51503	−0.757514	−	0.652819i	$-0.773586\pi$
$373$	−10914.0	−1.51503	−0.757514	+	0.652819i	$0.773586\pi$
$374$	10400.0	1.43789
$375$	4500.00i	0.619677i
$376$	−3616.00	−0.495960
$377$	0	0
$378$	432.000	0.0587822
$379$	8100.00i	1.09781i	0.835886	+	0.548904i	$0.184955\pi$
$379$	8100.00i	1.09781i	−0.835886	+	0.548904i	$0.815045\pi$
$380$	−800.000	−0.107998
$381$	−3888.00	−0.522804
$382$	2624.00i	0.351454i
$383$	6180.00i	0.824499i	0.911071	+	0.412250i	$0.135257\pi$
$383$	6180.00i	0.824499i	−0.911071	+	0.412250i	$0.864743\pi$
$384$	384.000i	0.0510310i
$385$	3200.00i	0.423603i
$386$	700.000	0.0923033
$387$	−684.000	−0.0898441
$388$	2456.00i	0.321352i
$389$	7522.00	0.980413	0.490206	−	0.871606i	$-0.336921\pi$
$389$	7522.00	0.980413	0.490206	+	0.871606i	$0.336921\pi$
$390$	0	0
$391$	0	0
$392$	2232.00i	0.287584i
$393$	−2676.00	−0.343477
$394$	−684.000	−0.0874605
$395$	− 9760.00i	− 1.24324i
$396$	1440.00i	0.182734i
$397$	− 6078.00i	− 0.768378i	−0.923254	−	0.384189i	$-0.874481\pi$
$397$	− 6078.00i	− 0.768378i	0.923254	−	0.384189i	$-0.125519\pi$
$398$	− 6736.00i	− 0.848355i
$399$	480.000	0.0602257
$400$	400.000	0.0500000
$401$	− 1830.00i	− 0.227895i	−0.993487	−	0.113947i	$-0.963650\pi$
$401$	− 1830.00i	− 0.227895i	0.993487	−	0.113947i	$-0.0363495\pi$
$402$	−4200.00	−0.521087
$403$	0	0
$404$	2072.00	0.255163
$405$	810.000i	0.0993808i
$406$	−288.000	−0.0352049
$407$	2960.00	0.360496
$408$	− 3120.00i	− 0.378586i
$409$	12434.0i	1.50323i	0.659601	+	0.751616i	$0.270725\pi$
$409$	12434.0i	1.50323i	−0.659601	+	0.751616i	$0.729275\pi$
$410$	− 7240.00i	− 0.872093i
$411$	− 6978.00i	− 0.837468i
$412$	448.000	0.0535713
$413$	3712.00	0.442265
$414$	0	0
$415$	10080.0	1.19231
$416$	0	0
$417$	5796.00	0.680651
$418$	1600.00i	0.187221i
$419$	−14188.0	−1.65425	−0.827123	−	0.562021i	$-0.810024\pi$
$419$	−14188.0	−1.65425	−0.827123	+	0.562021i	$0.810024\pi$
$420$	960.000	0.111531
$421$	8638.00i	0.999977i	0.866032	+	0.499989i	$0.166662\pi$
$421$	8638.00i	0.999977i	−0.866032	+	0.499989i	$0.833338\pi$
$422$	4008.00i	0.462337i
$423$	4068.00i	0.467596i
$424$	3056.00i	0.350029i
$425$	−3250.00	−0.370937
$426$	−4488.00	−0.510433
$427$	2864.00i	0.324587i
$428$	1488.00	0.168050
$429$	0	0
$430$	−1520.00	−0.170467
$431$	4292.00i	0.479671i	0.970813	+	0.239836i	$0.0770936\pi$
$431$	4292.00i	0.479671i	−0.970813	+	0.239836i	$0.922906\pi$
$432$	432.000	0.0481125
$433$	5982.00	0.663918	0.331959	−	0.943294i	$-0.392290\pi$
$433$	5982.00	0.663918	0.331959	+	0.943294i	$0.392290\pi$
$434$	− 2944.00i	− 0.325614i
$435$	− 540.000i	− 0.0595196i
$436$	− 3736.00i	− 0.410371i
$437$	0	0
$438$	−6348.00	−0.692510
$439$	−256.000	−0.0278319	−0.0139160	−	0.999903i	$-0.504430\pi$
$439$	−256.000	−0.0278319	−0.0139160	+	0.999903i	$0.504430\pi$
$440$	3200.00i	0.346714i
$441$	2511.00	0.271137
$442$	0	0
$443$	12556.0	1.34662	0.673311	−	0.739359i	$-0.264872\pi$
$443$	12556.0	1.34662	0.673311	+	0.739359i	$0.264872\pi$
$444$	− 888.000i	− 0.0949158i
$445$	−3860.00	−0.411194
$446$	−11216.0	−1.19079
$447$	2646.00i	0.279981i
$448$	− 512.000i	− 0.0539949i
$449$	− 5574.00i	− 0.585865i	−0.956133	−	0.292932i	$-0.905369\pi$
$449$	− 5574.00i	− 0.585865i	0.956133	−	0.292932i	$-0.0946311\pi$
$450$	− 450.000i	− 0.0471405i
$451$	−14480.0	−1.51183
$452$	−7656.00	−0.796699
$453$	5328.00i	0.552608i
$454$	−3856.00	−0.398615
$455$	0	0
$456$	480.000	0.0492940
$457$	1266.00i	0.129586i	0.997899	+	0.0647932i	$0.0206388\pi$
$457$	1266.00i	0.129586i	−0.997899	+	0.0647932i	$0.979361\pi$
$458$	7876.00	0.803540
$459$	−3510.00	−0.356934
$460$	0	0
$461$	7554.00i	0.763178i	0.924332	+	0.381589i	$0.124623\pi$
$461$	7554.00i	0.763178i	−0.924332	+	0.381589i	$0.875377\pi$
$462$	− 1920.00i	− 0.193347i
$463$	6752.00i	0.677737i	0.940834	+	0.338868i	$0.110044\pi$
$463$	6752.00i	0.677737i	−0.940834	+	0.338868i	$0.889956\pi$
$464$	−288.000	−0.0288148
$465$	5520.00	0.550503
$466$	5124.00i	0.509366i
$467$	−7924.00	−0.785180	−0.392590	−	0.919714i	$-0.628421\pi$
$467$	−7924.00	−0.785180	−0.392590	+	0.919714i	$0.628421\pi$
$468$	0	0
$469$	5600.00	0.551352
$470$	9040.00i	0.887200i
$471$	−7230.00	−0.707305
$472$	3712.00	0.361989
$473$	3040.00i	0.295517i
$474$	5856.00i	0.567458i
$475$	− 500.000i	− 0.0482980i
$476$	4160.00i	0.400574i
$477$	3438.00	0.330011
$478$	14328.0	1.37102
$479$	11084.0i	1.05729i	0.848844	+	0.528644i	$0.177299\pi$
$479$	11084.0i	1.05729i	−0.848844	+	0.528644i	$0.822701\pi$
$480$	960.000	0.0912871
$481$	0	0
$482$	12364.0	1.16839
$483$	0	0
$484$	1076.00	0.101052
$485$	6140.00	0.574852
$486$	− 486.000i	− 0.0453609i
$487$	4432.00i	0.412388i	0.978511	+	0.206194i	$0.0661078\pi$
$487$	4432.00i	0.412388i	−0.978511	+	0.206194i	$0.933892\pi$
$488$	2864.00i	0.265670i
$489$	− 9636.00i	− 0.891114i
$490$	5580.00	0.514446
$491$	1140.00	0.104781	0.0523905	−	0.998627i	$-0.483316\pi$
$491$	1140.00	0.104781	0.0523905	+	0.998627i	$0.483316\pi$
$492$	4344.00i	0.398054i
$493$	2340.00	0.213769
$494$	0	0
$495$	3600.00	0.326885
$496$	− 2944.00i	− 0.266511i
$497$	5984.00	0.540079
$498$	−6048.00	−0.544212
$499$	1764.00i	0.158251i	0.996865	+	0.0791257i	$0.0252129\pi$
$499$	1764.00i	0.158251i	−0.996865	+	0.0791257i	$0.974787\pi$
$500$	− 6000.00i	− 0.536656i
$501$	− 5004.00i	− 0.446232i
$502$	− 2792.00i	− 0.248233i
$503$	16976.0	1.50482	0.752408	−	0.658697i	$-0.228892\pi$
$503$	16976.0	1.50482	0.752408	+	0.658697i	$0.228892\pi$
$504$	−576.000	−0.0509069
$505$	− 5180.00i	− 0.456449i
$506$	0	0
$507$	0	0
$508$	5184.00	0.452761
$509$	9474.00i	0.825005i	0.910956	+	0.412503i	$0.135345\pi$
$509$	9474.00i	0.825005i	−0.910956	+	0.412503i	$0.864655\pi$
$510$	−7800.00	−0.677235
$511$	8464.00	0.732731
$512$	− 512.000i	− 0.0441942i
$513$	− 540.000i	− 0.0464748i
$514$	13812.0i	1.18526i
$515$	− 1120.00i	− 0.0958313i
$516$	912.000	0.0778073
$517$	18080.0	1.53802
$518$	1184.00i	0.100429i
$519$	−10794.0	−0.912917
$520$	0	0
$521$	14114.0	1.18684	0.593422	−	0.804892i	$-0.297777\pi$
$521$	14114.0	1.18684	0.593422	+	0.804892i	$0.297777\pi$
$522$	324.000i	0.0271668i
$523$	20284.0	1.69590	0.847952	−	0.530074i	$-0.177836\pi$
$523$	20284.0	1.69590	0.847952	+	0.530074i	$0.177836\pi$
$524$	3568.00	0.297460
$525$	600.000i	0.0498784i
$526$	13696.0i	1.13531i
$527$	23920.0i	1.97718i
$528$	− 1920.00i	− 0.158252i
$529$	−12167.0	−1.00000
$530$	7640.00	0.626152
$531$	− 4176.00i	− 0.341286i
$532$	−640.000	−0.0521570
$533$	0	0
$534$	2316.00	0.187684
$535$	− 3720.00i	− 0.300616i
$536$	5600.00	0.451275
$537$	−3204.00	−0.257473
$538$	12068.0i	0.967079i
$539$	− 11160.0i	− 0.891828i
$540$	− 1080.00i	− 0.0860663i
$541$	14362.0i	1.14135i	0.821176	+	0.570675i	$0.193318\pi$
$541$	14362.0i	1.14135i	−0.821176	+	0.570675i	$0.806682\pi$
$542$	−9664.00	−0.765875
$543$	14358.0	1.13473
$544$	4160.00i	0.327865i
$545$	−9340.00	−0.734095
$546$	0	0
$547$	−20956.0	−1.63805	−0.819025	−	0.573757i	$-0.805485\pi$
$547$	−20956.0	−1.63805	−0.819025	+	0.573757i	$0.805485\pi$
$548$	9304.00i	0.725269i
$549$	3222.00	0.250477
$550$	−2000.00	−0.155055
$551$	360.000i	0.0278340i
$552$	0	0
$553$	− 7808.00i	− 0.600416i
$554$	− 8164.00i	− 0.626092i
$555$	−2220.00	−0.169791
$556$	−7728.00	−0.589461
$557$	4134.00i	0.314476i	0.987561	+	0.157238i	$0.0502590\pi$
$557$	4134.00i	0.314476i	−0.987561	+	0.157238i	$0.949741\pi$
$558$	−3312.00	−0.251269
$559$	0	0
$560$	−1280.00	−0.0965891
$561$	15600.0i	1.17403i
$562$	−6700.00	−0.502887
$563$	16228.0	1.21479	0.607397	−	0.794399i	$-0.292214\pi$
$563$	16228.0	1.21479	0.607397	+	0.794399i	$0.292214\pi$
$564$	− 5424.00i	− 0.404950i
$565$	19140.0i	1.42518i
$566$	15592.0i	1.15792i
$567$	648.000i	0.0479955i
$568$	5984.00	0.442048
$569$	−2514.00	−0.185224	−0.0926119	−	0.995702i	$-0.529522\pi$
$569$	−2514.00	−0.185224	−0.0926119	+	0.995702i	$0.529522\pi$
$570$	− 1200.00i	− 0.0881798i
$571$	11612.0	0.851046	0.425523	−	0.904948i	$-0.360090\pi$
$571$	11612.0	0.851046	0.425523	+	0.904948i	$0.360090\pi$
$572$	0	0
$573$	−3936.00	−0.286961
$574$	− 5792.00i	− 0.421173i
$575$	0	0
$576$	−576.000	−0.0416667
$577$	6354.00i	0.458441i	0.973375	+	0.229221i	$0.0736177\pi$
$577$	6354.00i	0.458441i	−0.973375	+	0.229221i	$0.926382\pi$
$578$	− 23974.0i	− 1.72524i
$579$	1050.00i	0.0753653i
$580$	720.000i	0.0515455i
$581$	8064.00	0.575819
$582$	−3684.00	−0.262383
$583$	− 15280.0i	− 1.08548i
$584$	8464.00	0.599731
$585$	0	0
$586$	−7844.00	−0.552957
$587$	− 13240.0i	− 0.930960i	−0.885059	−	0.465480i	$-0.845882\pi$
$587$	− 13240.0i	− 0.930960i	0.885059	−	0.465480i	$-0.154118\pi$
$588$	−3348.00	−0.234812
$589$	−3680.00	−0.257439
$590$	− 9280.00i	− 0.647545i
$591$	− 1026.00i	− 0.0714112i
$592$	1184.00i	0.0821995i
$593$	1146.00i	0.0793602i	0.999212	+	0.0396801i	$0.0126339\pi$
$593$	1146.00i	0.0793602i	−0.999212	+	0.0396801i	$0.987366\pi$
$594$	−2160.00	−0.149202
$595$	10400.0	0.716569
$596$	− 3528.00i	− 0.242471i
$597$	10104.0	0.692679
$598$	0	0
$599$	10464.0	0.713769	0.356884	−	0.934149i	$-0.383839\pi$
$599$	10464.0	0.713769	0.356884	+	0.934149i	$0.383839\pi$
$600$	600.000i	0.0408248i
$601$	6650.00	0.451346	0.225673	−	0.974203i	$-0.427542\pi$
$601$	6650.00	0.451346	0.225673	+	0.974203i	$0.427542\pi$
$602$	−1216.00	−0.0823263
$603$	− 6300.00i	− 0.425466i
$604$	− 7104.00i	− 0.478572i
$605$	− 2690.00i	− 0.180767i
$606$	3108.00i	0.208340i
$607$	−6664.00	−0.445607	−0.222803	−	0.974863i	$-0.571521\pi$
$607$	−6664.00	−0.445607	−0.222803	+	0.974863i	$0.571521\pi$
$608$	−640.000	−0.0426898
$609$	− 432.000i	− 0.0287447i
$610$	7160.00	0.475246
$611$	0	0
$612$	4680.00	0.309114
$613$	2134.00i	0.140606i	0.997526	+	0.0703030i	$0.0223966\pi$
$613$	2134.00i	0.140606i	−0.997526	+	0.0703030i	$0.977603\pi$
$614$	−11912.0	−0.782947
$615$	10860.0	0.712061
$616$	2560.00i	0.167444i
$617$	− 714.000i	− 0.0465876i	−0.999729	−	0.0232938i	$-0.992585\pi$
$617$	− 714.000i	− 0.0465876i	0.999729	−	0.0232938i	$-0.00741532\pi$
$618$	672.000i	0.0437408i
$619$	− 29228.0i	− 1.89786i	−0.315494	−	0.948928i	$-0.602170\pi$
$619$	− 29228.0i	− 1.89786i	0.315494	−	0.948928i	$-0.397830\pi$
$620$	−7360.00	−0.476750
$621$	0	0
$622$	4704.00i	0.303237i
$623$	−3088.00	−0.198584
$624$	0	0
$625$	−11875.0	−0.760000
$626$	− 16884.0i	− 1.07799i
$627$	−2400.00	−0.152866
$628$	9640.00	0.612544
$629$	− 9620.00i	− 0.609816i
$630$	1440.00i	0.0910650i
$631$	13536.0i	0.853977i	0.904257	+	0.426989i	$0.140426\pi$
$631$	13536.0i	0.853977i	−0.904257	+	0.426989i	$0.859574\pi$
$632$	− 7808.00i	− 0.491433i
$633$	−6012.00	−0.377497
$634$	−11100.0	−0.695327
$635$	− 12960.0i	− 0.809924i
$636$	−4584.00	−0.285798
$637$	0	0
$638$	1440.00	0.0893576
$639$	− 6732.00i	− 0.416767i
$640$	−1280.00	−0.0790569
$641$	−17218.0	−1.06095	−0.530476	−	0.847700i	$-0.677987\pi$
$641$	−17218.0	−1.06095	−0.530476	+	0.847700i	$0.677987\pi$
$642$	2232.00i	0.137212i
$643$	15044.0i	0.922671i	0.887226	+	0.461335i	$0.152630\pi$
$643$	15044.0i	0.922671i	−0.887226	+	0.461335i	$0.847370\pi$
$644$	0	0
$645$	− 2280.00i	− 0.139186i
$646$	5200.00	0.316705
$647$	−25176.0	−1.52978	−0.764892	−	0.644158i	$-0.777208\pi$
$647$	−25176.0	−1.52978	−0.764892	+	0.644158i	$0.777208\pi$
$648$	648.000i	0.0392837i
$649$	−18560.0	−1.12256
$650$	0	0
$651$	4416.00	0.265863
$652$	12848.0i	0.771728i
$653$	−16034.0	−0.960887	−0.480443	−	0.877026i	$-0.659524\pi$
$653$	−16034.0	−0.960887	−0.480443	+	0.877026i	$0.659524\pi$
$654$	5604.00	0.335067
$655$	− 8920.00i	− 0.532112i
$656$	− 5792.00i	− 0.344725i
$657$	− 9522.00i	− 0.565432i
$658$	7232.00i	0.428469i
$659$	25356.0	1.49883	0.749415	−	0.662100i	$-0.230335\pi$
$659$	25356.0	1.49883	0.749415	+	0.662100i	$0.230335\pi$
$660$	−4800.00	−0.283091
$661$	− 18310.0i	− 1.07742i	−0.842490	−	0.538711i	$-0.818911\pi$
$661$	− 18310.0i	− 1.07742i	0.842490	−	0.538711i	$-0.181089\pi$
$662$	280.000	0.0164388
$663$	0	0
$664$	8064.00	0.471301
$665$	1600.00i	0.0933013i
$666$	1332.00	0.0774984
$667$	0	0
$668$	6672.00i	0.386448i
$669$	− 16824.0i	− 0.972277i
$670$	− 14000.0i	− 0.807264i
$671$	− 14320.0i	− 0.823871i
$672$	768.000	0.0440867
$673$	−24802.0	−1.42057	−0.710287	−	0.703912i	$-0.751435\pi$
$673$	−24802.0	−1.42057	−0.710287	+	0.703912i	$0.751435\pi$
$674$	− 12348.0i	− 0.705678i
$675$	675.000	0.0384900
$676$	0	0
$677$	−22706.0	−1.28901	−0.644507	−	0.764598i	$-0.722937\pi$
$677$	−22706.0	−1.28901	−0.644507	+	0.764598i	$0.722937\pi$
$678$	− 11484.0i	− 0.650502i
$679$	4912.00	0.277622
$680$	10400.0	0.586503
$681$	− 5784.00i	− 0.325467i
$682$	14720.0i	0.826478i
$683$	14792.0i	0.828697i	0.910118	+	0.414349i	$0.135991\pi$
$683$	14792.0i	0.828697i	−0.910118	+	0.414349i	$0.864009\pi$
$684$	720.000i	0.0402484i
$685$	23260.0	1.29740
$686$	9952.00	0.553891
$687$	11814.0i	0.656088i
$688$	−1216.00	−0.0673831
$689$	0	0
$690$	0	0
$691$	− 1148.00i	− 0.0632011i	−0.999501	−	0.0316006i	$-0.989940\pi$
$691$	− 1148.00i	− 0.0632011i	0.999501	−	0.0316006i	$-0.0100604\pi$
$692$	14392.0	0.790609
$693$	2880.00	0.157867
$694$	5976.00i	0.326867i
$695$	19320.0i	1.05446i
$696$	− 432.000i	− 0.0235272i
$697$	47060.0i	2.55742i
$698$	324.000	0.0175696
$699$	−7686.00	−0.415896
$700$	− 800.000i	− 0.0431959i
$701$	−14870.0	−0.801187	−0.400594	−	0.916256i	$-0.631196\pi$
$701$	−14870.0	−0.801187	−0.400594	+	0.916256i	$0.631196\pi$
$702$	0	0
$703$	1480.00	0.0794015
$704$	2560.00i	0.137051i
$705$	−13560.0	−0.724396
$706$	−21508.0	−1.14655
$707$	− 4144.00i	− 0.220440i
$708$	5568.00i	0.295563i
$709$	6354.00i	0.336572i	0.985738	+	0.168286i	$0.0538232\pi$
$709$	6354.00i	0.336572i	−0.985738	+	0.168286i	$0.946177\pi$
$710$	− 14960.0i	− 0.790759i
$711$	−8784.00	−0.463327
$712$	−3088.00	−0.162539
$713$	0	0
$714$	−6240.00	−0.327067
$715$	0	0
$716$	4272.00	0.222978
$717$	21492.0i	1.11943i
$718$	−7176.00	−0.372989
$719$	−9288.00	−0.481758	−0.240879	−	0.970555i	$-0.577436\pi$
$719$	−9288.00	−0.481758	−0.240879	+	0.970555i	$0.577436\pi$
$720$	1440.00i	0.0745356i
$721$	− 896.000i	− 0.0462813i
$722$	− 12918.0i	− 0.665870i
$723$	18546.0i	0.953988i
$724$	−19144.0	−0.982709
$725$	−450.000	−0.0230518
$726$	1614.00i	0.0825085i
$727$	21544.0	1.09907	0.549534	−	0.835471i	$-0.314805\pi$
$727$	21544.0	1.09907	0.549534	+	0.835471i	$0.314805\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	− 21160.0i	− 1.07283i
$731$	9880.00	0.499897
$732$	−4296.00	−0.216919
$733$	19990.0i	1.00730i	0.863909	+	0.503648i	$0.168009\pi$
$733$	19990.0i	1.00730i	−0.863909	+	0.503648i	$0.831991\pi$
$734$	− 22544.0i	− 1.13367i
$735$	8370.00i	0.420044i
$736$	0	0
$737$	−28000.0	−1.39945
$738$	−6516.00	−0.325010
$739$	− 532.000i	− 0.0264816i	−0.999912	−	0.0132408i	$-0.995785\pi$
$739$	− 532.000i	− 0.0264816i	0.999912	−	0.0132408i	$-0.00421481\pi$
$740$	2960.00	0.147043
$741$	0	0
$742$	6112.00	0.302397
$743$	− 25452.0i	− 1.25672i	−0.777922	−	0.628360i	$-0.783726\pi$
$743$	− 25452.0i	− 1.25672i	0.777922	−	0.628360i	$-0.216274\pi$
$744$	4416.00	0.217605
$745$	−8820.00	−0.433745
$746$	21828.0i	1.07129i
$747$	− 9072.00i	− 0.444347i
$748$	− 20800.0i	− 1.01674i
$749$	− 2976.00i	− 0.145181i
$750$	9000.00	0.438178
$751$	−6440.00	−0.312915	−0.156457	−	0.987685i	$-0.550007\pi$
$751$	−6440.00	−0.312915	−0.156457	+	0.987685i	$0.550007\pi$
$752$	7232.00i	0.350697i
$753$	4188.00	0.202682
$754$	0	0
$755$	−17760.0	−0.856096
$756$	− 864.000i	− 0.0415653i
$757$	−786.000	−0.0377380	−0.0188690	−	0.999822i	$-0.506007\pi$
$757$	−786.000	−0.0377380	−0.0188690	+	0.999822i	$0.506007\pi$
$758$	16200.0	0.776267
$759$	0	0
$760$	1600.00i	0.0763659i
$761$	1498.00i	0.0713567i	0.999363	+	0.0356784i	$0.0113592\pi$
$761$	1498.00i	0.0713567i	−0.999363	+	0.0356784i	$0.988641\pi$
$762$	7776.00i	0.369678i
$763$	−7472.00	−0.354528
$764$	5248.00	0.248516
$765$	− 11700.0i	− 0.552960i
$766$	12360.0	0.583009
$767$	0	0
$768$	768.000	0.0360844
$769$	14738.0i	0.691113i	0.938398	+	0.345556i	$0.112310\pi$
$769$	14738.0i	0.691113i	−0.938398	+	0.345556i	$0.887690\pi$
$770$	6400.00	0.299532
$771$	−20718.0	−0.967757
$772$	− 1400.00i	− 0.0652683i
$773$	− 3822.00i	− 0.177837i	−0.996039	−	0.0889184i	$-0.971659\pi$
$773$	− 3822.00i	− 0.177837i	0.996039	−	0.0889184i	$-0.0283410\pi$
$774$	1368.00i	0.0635294i
$775$	− 4600.00i	− 0.213209i
$776$	4912.00	0.227230
$777$	−1776.00	−0.0819995
$778$	− 15044.0i	− 0.693256i
$779$	−7240.00	−0.332991
$780$	0	0
$781$	−29920.0	−1.37083
$782$	0	0
$783$	−486.000	−0.0221816
$784$	4464.00	0.203353
$785$	− 24100.0i	− 1.09575i
$786$	5352.00i	0.242875i
$787$	11900.0i	0.538995i	0.963001	+	0.269498i	$0.0868576\pi$
$787$	11900.0i	0.538995i	−0.963001	+	0.269498i	$0.913142\pi$
$788$	1368.00i	0.0618439i
$789$	−20544.0	−0.926978
$790$	−19520.0	−0.879102
$791$	15312.0i	0.688283i
$792$	2880.00	0.129213
$793$	0	0
$794$	−12156.0	−0.543325
$795$	11460.0i	0.511251i
$796$	−13472.0	−0.599877
$797$	21274.0	0.945500	0.472750	−	0.881197i	$-0.343261\pi$
$797$	21274.0	0.945500	0.472750	+	0.881197i	$0.343261\pi$
$798$	− 960.000i	− 0.0425860i
$799$	− 58760.0i	− 2.60173i
$800$	− 800.000i	− 0.0353553i
$801$	3474.00i	0.153243i
$802$	−3660.00	−0.161146
$803$	−42320.0	−1.85983
$804$	8400.00i	0.368464i
$805$	0	0
$806$	0	0
$807$	−18102.0	−0.789617
$808$	− 4144.00i	− 0.180427i
$809$	−27566.0	−1.19798	−0.598992	−	0.800755i	$-0.704432\pi$
$809$	−27566.0	−1.19798	−0.598992	+	0.800755i	$0.704432\pi$
$810$	1620.00	0.0702728
$811$	− 11244.0i	− 0.486844i	−0.969921	−	0.243422i	$-0.921730\pi$
$811$	− 11244.0i	− 0.486844i	0.969921	−	0.243422i	$-0.0782699\pi$
$812$	576.000i	0.0248936i
$813$	− 14496.0i	− 0.625334i
$814$	− 5920.00i	− 0.254909i
$815$	32120.0	1.38051
$816$	−6240.00	−0.267701
$817$	1520.00i	0.0650894i
$818$	24868.0	1.06295
$819$	0	0
$820$	−14480.0	−0.616663
$821$	13554.0i	0.576173i	0.957604	+	0.288086i	$0.0930190\pi$
$821$	13554.0i	0.576173i	−0.957604	+	0.288086i	$0.906981\pi$
$822$	−13956.0	−0.592179
$823$	−14384.0	−0.609228	−0.304614	−	0.952476i	$-0.598527\pi$
$823$	−14384.0	−0.609228	−0.304614	+	0.952476i	$0.598527\pi$
$824$	− 896.000i	− 0.0378806i
$825$	− 3000.00i	− 0.126602i
$826$	− 7424.00i	− 0.312729i
$827$	2488.00i	0.104615i	0.998631	+	0.0523073i	$0.0166575\pi$
$827$	2488.00i	0.104615i	−0.998631	+	0.0523073i	$0.983342\pi$
$828$	0	0
$829$	20858.0	0.873858	0.436929	−	0.899496i	$-0.356066\pi$
$829$	20858.0	0.873858	0.436929	+	0.899496i	$0.356066\pi$
$830$	− 20160.0i	− 0.843089i
$831$	12246.0	0.511202
$832$	0	0
$833$	−36270.0	−1.50862
$834$	− 11592.0i	− 0.481293i
$835$	16680.0	0.691300
$836$	3200.00	0.132386
$837$	− 4968.00i	− 0.205160i
$838$	28376.0i	1.16973i
$839$	− 23116.0i	− 0.951195i	−0.879663	−	0.475598i	$-0.842232\pi$
$839$	− 23116.0i	− 0.951195i	0.879663	−	0.475598i	$-0.157768\pi$
$840$	− 1920.00i	− 0.0788646i
$841$	−24065.0	−0.986715
$842$	17276.0	0.707091
$843$	− 10050.0i	− 0.410605i
$844$	8016.00	0.326922
$845$	0	0
$846$	8136.00	0.330640
$847$	− 2152.00i	− 0.0873006i
$848$	6112.00	0.247508
$849$	−23388.0	−0.945435
$850$	6500.00i	0.262292i
$851$	0	0
$852$	8976.00i	0.360930i
$853$	− 934.000i	− 0.0374907i	−0.999824	−	0.0187453i	$-0.994033\pi$
$853$	− 934.000i	− 0.0374907i	0.999824	−	0.0187453i	$-0.00596718\pi$
$854$	5728.00	0.229518
$855$	1800.00	0.0719985
$856$	− 2976.00i	− 0.118829i
$857$	−12642.0	−0.503900	−0.251950	−	0.967740i	$-0.581072\pi$
$857$	−12642.0	−0.503900	−0.251950	+	0.967740i	$0.581072\pi$
$858$	0	0
$859$	−22796.0	−0.905459	−0.452730	−	0.891648i	$-0.649550\pi$
$859$	−22796.0	−0.905459	−0.452730	+	0.891648i	$0.649550\pi$
$860$	3040.00i	0.120539i
$861$	8688.00	0.343886
$862$	8584.00	0.339179
$863$	− 76.0000i	− 0.00299776i	−0.999999	−	0.00149888i	$-0.999523\pi$
$863$	− 76.0000i	− 0.00299776i	0.999999	−	0.00149888i	$-0.000477109\pi$
$864$	− 864.000i	− 0.0340207i
$865$	− 35980.0i	− 1.41429i
$866$	− 11964.0i	− 0.469461i
$867$	35961.0	1.40865
$868$	−5888.00	−0.230244
$869$	39040.0i	1.52398i
$870$	−1080.00	−0.0420867
$871$	0	0
$872$	−7472.00	−0.290176
$873$	− 5526.00i	− 0.214235i
$874$	0	0
$875$	−12000.0	−0.463627
$876$	12696.0i	0.489678i
$877$	− 46130.0i	− 1.77617i	−0.459681	−	0.888084i	$-0.652036\pi$
$877$	− 46130.0i	− 1.77617i	0.459681	−	0.888084i	$-0.347964\pi$
$878$	512.000i	0.0196801i
$879$	− 11766.0i	− 0.451487i
$880$	6400.00	0.245164
$881$	−6682.00	−0.255530	−0.127765	−	0.991804i	$-0.540780\pi$
$881$	−6682.00	−0.255530	−0.127765	+	0.991804i	$0.540780\pi$
$882$	− 5022.00i	− 0.191723i
$883$	−47404.0	−1.80665	−0.903325	−	0.428957i	$-0.858881\pi$
$883$	−47404.0	−1.80665	−0.903325	+	0.428957i	$0.858881\pi$
$884$	0	0
$885$	13920.0	0.528718
$886$	− 25112.0i	− 0.952206i
$887$	33672.0	1.27463	0.637314	−	0.770604i	$-0.280045\pi$
$887$	33672.0	1.27463	0.637314	+	0.770604i	$0.280045\pi$
$888$	−1776.00	−0.0671156
$889$	− 10368.0i	− 0.391149i
$890$	7720.00i	0.290758i
$891$	− 3240.00i	− 0.121823i
$892$	22432.0i	0.842017i
$893$	9040.00	0.338759
$894$	5292.00	0.197976
$895$	− 10680.0i	− 0.398875i
$896$	−1024.00	−0.0381802
$897$	0	0
$898$	−11148.0	−0.414269
$899$	3312.00i	0.122871i
$900$	−900.000	−0.0333333
$901$	−49660.0	−1.83620
$902$	28960.0i	1.06903i
$903$	− 1824.00i	− 0.0672192i
$904$	15312.0i	0.563351i
$905$	47860.0i	1.75792i
$906$	10656.0	0.390753
$907$	14540.0	0.532296	0.266148	−	0.963932i	$-0.414249\pi$
$907$	14540.0	0.532296	0.266148	+	0.963932i	$0.414249\pi$
$908$	7712.00i	0.281863i
$909$	−4662.00	−0.170109
$910$	0	0
$911$	−7840.00	−0.285127	−0.142564	−	0.989786i	$-0.545535\pi$
$911$	−7840.00	−0.285127	−0.142564	+	0.989786i	$0.545535\pi$
$912$	− 960.000i	− 0.0348561i
$913$	−40320.0	−1.46155
$914$	2532.00	0.0916314
$915$	10740.0i	0.388037i
$916$	− 15752.0i	− 0.568189i
$917$	− 7136.00i	− 0.256981i
$918$	7020.00i	0.252391i
$919$	47720.0	1.71288	0.856440	−	0.516246i	$-0.172671\pi$
$919$	47720.0	1.71288	0.856440	+	0.516246i	$0.172671\pi$
$920$	0	0
$921$	− 17868.0i	− 0.639273i
$922$	15108.0	0.539648
$923$	0	0
$924$	−3840.00	−0.136717
$925$	1850.00i	0.0657596i
$926$	13504.0	0.479232
$927$	−1008.00	−0.0357142
$928$	576.000i	0.0203751i
$929$	7502.00i	0.264944i	0.991187	+	0.132472i	$0.0422914\pi$
$929$	7502.00i	0.264944i	−0.991187	+	0.132472i	$0.957709\pi$
$930$	− 11040.0i	− 0.389264i
$931$	− 5580.00i	− 0.196431i
$932$	10248.0	0.360176
$933$	−7056.00	−0.247592
$934$	15848.0i	0.555206i
$935$	−52000.0	−1.81880
$936$	0	0
$937$	22058.0	0.769054	0.384527	−	0.923114i	$-0.374365\pi$
$937$	22058.0	0.769054	0.384527	+	0.923114i	$0.374365\pi$
$938$	− 11200.0i	− 0.389865i
$939$	25326.0	0.880173
$940$	18080.0	0.627345
$941$	23338.0i	0.808498i	0.914649	+	0.404249i	$0.132467\pi$
$941$	23338.0i	0.808498i	−0.914649	+	0.404249i	$0.867533\pi$
$942$	14460.0i	0.500140i
$943$	0	0
$944$	− 7424.00i	− 0.255965i
$945$	−2160.00	−0.0743543
$946$	6080.00	0.208962
$947$	30488.0i	1.04617i	0.852279	+	0.523087i	$0.175220\pi$
$947$	30488.0i	1.04617i	−0.852279	+	0.523087i	$0.824780\pi$
$948$	11712.0	0.401253
$949$	0	0
$950$	−1000.00	−0.0341519
$951$	− 16650.0i	− 0.567732i
$952$	8320.00	0.283249
$953$	−9522.00	−0.323660	−0.161830	−	0.986819i	$-0.551740\pi$
$953$	−9522.00	−0.323660	−0.161830	+	0.986819i	$0.551740\pi$
$954$	− 6876.00i	− 0.233353i
$955$	− 13120.0i	− 0.444558i
$956$	− 28656.0i	− 0.969457i
$957$	2160.00i	0.0729602i
$958$	22168.0	0.747615
$959$	18608.0	0.626573
$960$	− 1920.00i	− 0.0645497i
$961$	−4065.00	−0.136451
$962$	0	0
$963$	−3348.00	−0.112033
$964$	− 24728.0i	− 0.826178i
$965$	−3500.00	−0.116755
$966$	0	0
$967$	− 7616.00i	− 0.253272i	−0.991949	−	0.126636i	$-0.959582\pi$
$967$	− 7616.00i	− 0.253272i	0.991949	−	0.126636i	$-0.0404180\pi$
$968$	− 2152.00i	− 0.0714544i
$969$	7800.00i	0.258588i
$970$	− 12280.0i	− 0.406481i
$971$	51316.0	1.69599	0.847996	−	0.530002i	$-0.177809\pi$
$971$	51316.0	1.69599	0.847996	+	0.530002i	$0.177809\pi$
$972$	−972.000	−0.0320750
$973$	15456.0i	0.509246i
$974$	8864.00	0.291603
$975$	0	0
$976$	5728.00	0.187857
$977$	− 48666.0i	− 1.59362i	−0.604232	−	0.796808i	$-0.706520\pi$
$977$	− 48666.0i	− 1.59362i	0.604232	−	0.796808i	$-0.293480\pi$
$978$	−19272.0	−0.630113
$979$	15440.0	0.504050
$980$	− 11160.0i	− 0.363768i
$981$	8406.00i	0.273581i
$982$	− 2280.00i	− 0.0740914i
$983$	− 17388.0i	− 0.564182i	−0.959388	−	0.282091i	$-0.908972\pi$
$983$	− 17388.0i	− 0.564182i	0.959388	−	0.282091i	$-0.0910280\pi$
$984$	8688.00	0.281467
$985$	3420.00	0.110630
$986$	− 4680.00i	− 0.151158i
$987$	−10848.0	−0.349844
$988$	0	0
$989$	0	0
$990$	− 7200.00i	− 0.231142i
$991$	11496.0	0.368499	0.184249	−	0.982880i	$-0.441015\pi$
$991$	11496.0	0.368499	0.184249	+	0.982880i	$0.441015\pi$
$992$	−5888.00	−0.188452
$993$	420.000i	0.0134223i
$994$	− 11968.0i	− 0.381893i
$995$	33680.0i	1.07309i
$996$	12096.0i	0.384816i
$997$	48862.0	1.55213	0.776066	−	0.630652i	$-0.217212\pi$
$997$	48862.0	1.55213	0.776066	+	0.630652i	$0.217212\pi$
$998$	3528.00	0.111901
$999$	1998.00i	0.0632772i

Display

a_p

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p

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a_n

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a_n

instead) (See

a_n

instead) Display

a_n

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up to: 50 250 1000 (See only

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a_p

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1014.4.b.h.337.1		2
13.5	odd	4		78.4.a.c.1.1	✓	1
13.8	odd	4		1014.4.a.j.1.1		1
13.12	even	2	inner	1014.4.b.h.337.2		2
39.5	even	4		234.4.a.h.1.1		1
52.31	even	4		624.4.a.d.1.1		1
65.44	odd	4		1950.4.a.l.1.1		1
104.5	odd	4		2496.4.a.a.1.1		1
104.83	even	4		2496.4.a.j.1.1		1
156.83	odd	4		1872.4.a.d.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
78.4.a.c.1.1	✓	1	13.5	odd	4
234.4.a.h.1.1		1	39.5	even	4
624.4.a.d.1.1		1	52.31	even	4
1014.4.a.j.1.1		1	13.8	odd	4
1014.4.b.h.337.1		2	1.1	even	1	trivial
1014.4.b.h.337.2		2	13.12	even	2	inner
1872.4.a.d.1.1		1	156.83	odd	4
1950.4.a.l.1.1		1	65.44	odd	4
2496.4.a.a.1.1		1	104.5	odd	4
2496.4.a.j.1.1		1	104.83	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 $\F_{q}$
Belyi maps

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

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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	1014.4.b.h.337.1

Level	$1014$
Weight	$4$
Character	1014.337
Analytic conductor	$59.828$
Analytic rank	$0$
Dimension	$2$
Inner twists	$2$