LMFDB - Embedded newform 294.4.a.d.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [294,4,Mod(1,294)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(294, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("294.1"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$N$	$=$	$294 = 2 \cdot 3 \cdot 7^{2}$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	294.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,-2,3,4,-6,-6,0,-8,9,12,-30] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$17.3465615417$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 42)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	294.1

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

f(q)

=

q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -6.00000 q^{5} -6.00000 q^{6} -8.00000 q^{8} +9.00000 q^{9} +12.0000 q^{10} -30.0000 q^{11} +12.0000 q^{12} +53.0000 q^{13} -18.0000 q^{15} +16.0000 q^{16} -84.0000 q^{17} -18.0000 q^{18} -97.0000 q^{19} -24.0000 q^{20} +60.0000 q^{22} +84.0000 q^{23} -24.0000 q^{24} -89.0000 q^{25} -106.000 q^{26} +27.0000 q^{27} -180.000 q^{29} +36.0000 q^{30} +179.000 q^{31} -32.0000 q^{32} -90.0000 q^{33} +168.000 q^{34} +36.0000 q^{36} -145.000 q^{37} +194.000 q^{38} +159.000 q^{39} +48.0000 q^{40} +126.000 q^{41} -325.000 q^{43} -120.000 q^{44} -54.0000 q^{45} -168.000 q^{46} -366.000 q^{47} +48.0000 q^{48} +178.000 q^{50} -252.000 q^{51} +212.000 q^{52} -768.000 q^{53} -54.0000 q^{54} +180.000 q^{55} -291.000 q^{57} +360.000 q^{58} -264.000 q^{59} -72.0000 q^{60} +818.000 q^{61} -358.000 q^{62} +64.0000 q^{64} -318.000 q^{65} +180.000 q^{66} -523.000 q^{67} -336.000 q^{68} +252.000 q^{69} -342.000 q^{71} -72.0000 q^{72} -43.0000 q^{73} +290.000 q^{74} -267.000 q^{75} -388.000 q^{76} -318.000 q^{78} -1171.00 q^{79} -96.0000 q^{80} +81.0000 q^{81} -252.000 q^{82} -810.000 q^{83} +504.000 q^{85} +650.000 q^{86} -540.000 q^{87} +240.000 q^{88} -600.000 q^{89} +108.000 q^{90} +336.000 q^{92} +537.000 q^{93} +732.000 q^{94} +582.000 q^{95} -96.0000 q^{96} +386.000 q^{97} -270.000 q^{99} +O(q^{100})

Coefficient data

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

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

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

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	4.00000	0.500000
$5$	−6.00000	−0.536656	−0.268328	−	0.963328i	$-0.586471\pi$
$5$	−6.00000	−0.536656	−0.268328	+	0.963328i	$0.586471\pi$
$6$	−6.00000	−0.408248
$7$	0	0
$7$	0	0
$8$	−8.00000	−0.353553
$9$	9.00000	0.333333
$10$	12.0000	0.379473
$11$	−30.0000	−0.822304	−0.411152	−	0.911567i	$-0.634873\pi$
$11$	−30.0000	−0.822304	−0.411152	+	0.911567i	$0.634873\pi$
$12$	12.0000	0.288675
$13$	53.0000	1.13074	0.565368	−	0.824839i	$-0.308734\pi$
$13$	53.0000	1.13074	0.565368	+	0.824839i	$0.308734\pi$
$14$	0	0
$15$	−18.0000	−0.309839
$16$	16.0000	0.250000
$17$	−84.0000	−1.19841	−0.599206	−	0.800595i	$-0.704517\pi$
$17$	−84.0000	−1.19841	−0.599206	+	0.800595i	$0.704517\pi$
$18$	−18.0000	−0.235702
$19$	−97.0000	−1.17123	−0.585614	−	0.810590i	$-0.699146\pi$
$19$	−97.0000	−1.17123	−0.585614	+	0.810590i	$0.699146\pi$
$20$	−24.0000	−0.268328
$21$	0	0
$22$	60.0000	0.581456
$23$	84.0000	0.761531	0.380765	−	0.924672i	$-0.375661\pi$
$23$	84.0000	0.761531	0.380765	+	0.924672i	$0.375661\pi$
$24$	−24.0000	−0.204124
$25$	−89.0000	−0.712000
$26$	−106.000	−0.799550
$27$	27.0000	0.192450
$28$	0	0
$29$	−180.000	−1.15259	−0.576296	−	0.817241i	$-0.695502\pi$
$29$	−180.000	−1.15259	−0.576296	+	0.817241i	$0.695502\pi$
$30$	36.0000	0.219089
$31$	179.000	1.03708	0.518538	−	0.855055i	$-0.326477\pi$
$31$	179.000	1.03708	0.518538	+	0.855055i	$0.326477\pi$
$32$	−32.0000	−0.176777
$33$	−90.0000	−0.474757
$34$	168.000	0.847405
$35$	0	0
$36$	36.0000	0.166667
$37$	−145.000	−0.644266	−0.322133	−	0.946694i	$-0.604400\pi$
$37$	−145.000	−0.644266	−0.322133	+	0.946694i	$0.604400\pi$
$38$	194.000	0.828183
$39$	159.000	0.652830
$40$	48.0000	0.189737
$41$	126.000	0.479949	0.239974	−	0.970779i	$-0.422861\pi$
$41$	126.000	0.479949	0.239974	+	0.970779i	$0.422861\pi$
$42$	0	0
$43$	−325.000	−1.15261	−0.576303	−	0.817236i	$-0.695505\pi$
$43$	−325.000	−1.15261	−0.576303	+	0.817236i	$0.695505\pi$
$44$	−120.000	−0.411152
$45$	−54.0000	−0.178885
$46$	−168.000	−0.538484
$47$	−366.000	−1.13588	−0.567942	−	0.823068i	$-0.692260\pi$
$47$	−366.000	−1.13588	−0.567942	+	0.823068i	$0.692260\pi$
$48$	48.0000	0.144338
$49$	0	0
$50$	178.000	0.503460
$51$	−252.000	−0.691903
$52$	212.000	0.565368
$53$	−768.000	−1.99043	−0.995216	−	0.0976975i	$-0.968852\pi$
$53$	−768.000	−1.99043	−0.995216	+	0.0976975i	$0.968852\pi$
$54$	−54.0000	−0.136083
$55$	180.000	0.441294
$56$	0	0
$57$	−291.000	−0.676209
$58$	360.000	0.815005
$59$	−264.000	−0.582540	−0.291270	−	0.956641i	$-0.594078\pi$
$59$	−264.000	−0.582540	−0.291270	+	0.956641i	$0.594078\pi$
$60$	−72.0000	−0.154919
$61$	818.000	1.71695	0.858477	−	0.512852i	$-0.171411\pi$
$61$	818.000	1.71695	0.858477	+	0.512852i	$0.171411\pi$
$62$	−358.000	−0.733323
$63$	0	0
$64$	64.0000	0.125000
$65$	−318.000	−0.606816
$66$	180.000	0.335704
$67$	−523.000	−0.953651	−0.476826	−	0.878998i	$-0.658213\pi$
$67$	−523.000	−0.953651	−0.476826	+	0.878998i	$0.658213\pi$
$68$	−336.000	−0.599206
$69$	252.000	0.439670
$70$	0	0
$71$	−342.000	−0.571661	−0.285831	−	0.958280i	$-0.592269\pi$
$71$	−342.000	−0.571661	−0.285831	+	0.958280i	$0.592269\pi$
$72$	−72.0000	−0.117851
$73$	−43.0000	−0.0689420	−0.0344710	−	0.999406i	$-0.510975\pi$
$73$	−43.0000	−0.0689420	−0.0344710	+	0.999406i	$0.510975\pi$
$74$	290.000	0.455565
$75$	−267.000	−0.411073
$76$	−388.000	−0.585614
$77$	0	0
$78$	−318.000	−0.461621
$79$	−1171.00	−1.66769	−0.833847	−	0.551996i	$-0.813866\pi$
$79$	−1171.00	−1.66769	−0.833847	+	0.551996i	$0.813866\pi$
$80$	−96.0000	−0.134164
$81$	81.0000	0.111111
$82$	−252.000	−0.339375
$83$	−810.000	−1.07119	−0.535597	−	0.844474i	$-0.679913\pi$
$83$	−810.000	−1.07119	−0.535597	+	0.844474i	$0.679913\pi$
$84$	0	0
$85$	504.000	0.643135
$86$	650.000	0.815015
$87$	−540.000	−0.665449
$88$	240.000	0.290728
$89$	−600.000	−0.714605	−0.357303	−	0.933989i	$-0.616304\pi$
$89$	−600.000	−0.714605	−0.357303	+	0.933989i	$0.616304\pi$
$90$	108.000	0.126491
$91$	0	0
$92$	336.000	0.380765
$93$	537.000	0.598756
$94$	732.000	0.803192
$95$	582.000	0.628547
$96$	−96.0000	−0.102062
$97$	386.000	0.404045	0.202022	−	0.979381i	$-0.435249\pi$
$97$	386.000	0.404045	0.202022	+	0.979381i	$0.435249\pi$
$98$	0	0
$99$	−270.000	−0.274101
$100$	−356.000	−0.356000
$101$	618.000	0.608845	0.304422	−	0.952537i	$-0.401537\pi$
$101$	618.000	0.608845	0.304422	+	0.952537i	$0.401537\pi$
$102$	504.000	0.489249
$103$	1475.00	1.41103	0.705515	−	0.708695i	$-0.250716\pi$
$103$	1475.00	1.41103	0.705515	+	0.708695i	$0.250716\pi$
$104$	−424.000	−0.399775
$105$	0	0
$106$	1536.00	1.40745
$107$	1884.00	1.70218	0.851090	−	0.525021i	$-0.175942\pi$
$107$	1884.00	1.70218	0.851090	+	0.525021i	$0.175942\pi$
$108$	108.000	0.0962250
$109$	413.000	0.362920	0.181460	−	0.983398i	$-0.441918\pi$
$109$	413.000	0.362920	0.181460	+	0.983398i	$0.441918\pi$
$110$	−360.000	−0.312042
$111$	−435.000	−0.371967
$112$	0	0
$113$	−882.000	−0.734262	−0.367131	−	0.930169i	$-0.619660\pi$
$113$	−882.000	−0.734262	−0.367131	+	0.930169i	$0.619660\pi$
$114$	582.000	0.478152
$115$	−504.000	−0.408680
$116$	−720.000	−0.576296
$117$	477.000	0.376912
$118$	528.000	0.411918
$119$	0	0
$120$	144.000	0.109545
$121$	−431.000	−0.323817
$122$	−1636.00	−1.21407
$123$	378.000	0.277098
$124$	716.000	0.518538
$125$	1284.00	0.918756
$126$	0	0
$127$	2483.00	1.73489	0.867443	−	0.497536i	$-0.165762\pi$
$127$	2483.00	1.73489	0.867443	+	0.497536i	$0.165762\pi$
$128$	−128.000	−0.0883883
$129$	−975.000	−0.665457
$130$	636.000	0.429084
$131$	2118.00	1.41260	0.706300	−	0.707913i	$-0.250363\pi$
$131$	2118.00	1.41260	0.706300	+	0.707913i	$0.250363\pi$
$132$	−360.000	−0.237379
$133$	0	0
$134$	1046.00	0.674333
$135$	−162.000	−0.103280
$136$	672.000	0.423702
$137$	3012.00	1.87834	0.939170	−	0.343453i	$-0.111597\pi$
$137$	3012.00	1.87834	0.939170	+	0.343453i	$0.111597\pi$
$138$	−504.000	−0.310894
$139$	−37.0000	−0.0225777	−0.0112888	−	0.999936i	$-0.503593\pi$
$139$	−37.0000	−0.0225777	−0.0112888	+	0.999936i	$0.503593\pi$
$140$	0	0
$141$	−1098.00	−0.655803
$142$	684.000	0.404225
$143$	−1590.00	−0.929808
$144$	144.000	0.0833333
$145$	1080.00	0.618546
$146$	86.0000	0.0487494
$147$	0	0
$148$	−580.000	−0.322133
$149$	−1644.00	−0.903904	−0.451952	−	0.892042i	$-0.649272\pi$
$149$	−1644.00	−0.903904	−0.451952	+	0.892042i	$0.649272\pi$
$150$	534.000	0.290673
$151$	1088.00	0.586359	0.293179	−	0.956057i	$-0.405287\pi$
$151$	1088.00	0.586359	0.293179	+	0.956057i	$0.405287\pi$
$152$	776.000	0.414092
$153$	−756.000	−0.399470
$154$	0	0
$155$	−1074.00	−0.556553
$156$	636.000	0.326415
$157$	506.000	0.257218	0.128609	−	0.991695i	$-0.458949\pi$
$157$	506.000	0.257218	0.128609	+	0.991695i	$0.458949\pi$
$158$	2342.00	1.17924
$159$	−2304.00	−1.14918
$160$	192.000	0.0948683
$161$	0	0
$162$	−162.000	−0.0785674
$163$	1844.00	0.886093	0.443047	−	0.896499i	$-0.353898\pi$
$163$	1844.00	0.886093	0.443047	+	0.896499i	$0.353898\pi$
$164$	504.000	0.239974
$165$	540.000	0.254781
$166$	1620.00	0.757448
$167$	162.000	0.0750655	0.0375327	−	0.999295i	$-0.488050\pi$
$167$	162.000	0.0750655	0.0375327	+	0.999295i	$0.488050\pi$
$168$	0	0
$169$	612.000	0.278562
$170$	−1008.00	−0.454765
$171$	−873.000	−0.390409
$172$	−1300.00	−0.576303
$173$	−2724.00	−1.19712	−0.598560	−	0.801078i	$-0.704260\pi$
$173$	−2724.00	−1.19712	−0.598560	+	0.801078i	$0.704260\pi$
$174$	1080.00	0.470544
$175$	0	0
$176$	−480.000	−0.205576
$177$	−792.000	−0.336330
$178$	1200.00	0.505302
$179$	−1254.00	−0.523622	−0.261811	−	0.965119i	$-0.584320\pi$
$179$	−1254.00	−0.523622	−0.261811	+	0.965119i	$0.584320\pi$
$180$	−216.000	−0.0894427
$181$	−1807.00	−0.742062	−0.371031	−	0.928620i	$-0.620996\pi$
$181$	−1807.00	−0.742062	−0.371031	+	0.928620i	$0.620996\pi$
$182$	0	0
$183$	2454.00	0.991284
$184$	−672.000	−0.269242
$185$	870.000	0.345750
$186$	−1074.00	−0.423384
$187$	2520.00	0.985458
$188$	−1464.00	−0.567942
$189$	0	0
$190$	−1164.00	−0.444450
$191$	714.000	0.270488	0.135244	−	0.990812i	$-0.456818\pi$
$191$	714.000	0.270488	0.135244	+	0.990812i	$0.456818\pi$
$192$	192.000	0.0721688
$193$	−3709.00	−1.38331	−0.691657	−	0.722226i	$-0.743119\pi$
$193$	−3709.00	−1.38331	−0.691657	+	0.722226i	$0.743119\pi$
$194$	−772.000	−0.285703
$195$	−954.000	−0.350345
$196$	0	0
$197$	−1044.00	−0.377573	−0.188787	−	0.982018i	$-0.560455\pi$
$197$	−1044.00	−0.377573	−0.188787	+	0.982018i	$0.560455\pi$
$198$	540.000	0.193819
$199$	−136.000	−0.0484462	−0.0242231	−	0.999707i	$-0.507711\pi$
$199$	−136.000	−0.0484462	−0.0242231	+	0.999707i	$0.507711\pi$
$200$	712.000	0.251730
$201$	−1569.00	−0.550591
$202$	−1236.00	−0.430518
$203$	0	0
$204$	−1008.00	−0.345952
$205$	−756.000	−0.257567
$206$	−2950.00	−0.997749
$207$	756.000	0.253844
$208$	848.000	0.282684
$209$	2910.00	0.963105
$210$	0	0
$211$	1484.00	0.484184	0.242092	−	0.970253i	$-0.422166\pi$
$211$	1484.00	0.484184	0.242092	+	0.970253i	$0.422166\pi$
$212$	−3072.00	−0.995216
$213$	−1026.00	−0.330049
$214$	−3768.00	−1.20362
$215$	1950.00	0.618553
$216$	−216.000	−0.0680414
$217$	0	0
$218$	−826.000	−0.256623
$219$	−129.000	−0.0398037
$220$	720.000	0.220647
$221$	−4452.00	−1.35509
$222$	870.000	0.263021
$223$	−2032.00	−0.610192	−0.305096	−	0.952322i	$-0.598689\pi$
$223$	−2032.00	−0.610192	−0.305096	+	0.952322i	$0.598689\pi$
$224$	0	0
$225$	−801.000	−0.237333
$226$	1764.00	0.519201
$227$	6198.00	1.81223	0.906114	−	0.423034i	$-0.139035\pi$
$227$	6198.00	1.81223	0.906114	+	0.423034i	$0.139035\pi$
$228$	−1164.00	−0.338104
$229$	−4591.00	−1.32481	−0.662406	−	0.749145i	$-0.730464\pi$
$229$	−4591.00	−1.32481	−0.662406	+	0.749145i	$0.730464\pi$
$230$	1008.00	0.288981
$231$	0	0
$232$	1440.00	0.407503
$233$	4530.00	1.27369	0.636846	−	0.770991i	$-0.280239\pi$
$233$	4530.00	1.27369	0.636846	+	0.770991i	$0.280239\pi$
$234$	−954.000	−0.266517
$235$	2196.00	0.609580
$236$	−1056.00	−0.291270
$237$	−3513.00	−0.962843
$238$	0	0
$239$	1530.00	0.414090	0.207045	−	0.978331i	$-0.433615\pi$
$239$	1530.00	0.414090	0.207045	+	0.978331i	$0.433615\pi$
$240$	−288.000	−0.0774597
$241$	5534.00	1.47915	0.739577	−	0.673072i	$-0.235025\pi$
$241$	5534.00	1.47915	0.739577	+	0.673072i	$0.235025\pi$
$242$	862.000	0.228973
$243$	243.000	0.0641500
$244$	3272.00	0.858477
$245$	0	0
$246$	−756.000	−0.195938
$247$	−5141.00	−1.32435
$248$	−1432.00	−0.366662
$249$	−2430.00	−0.618454
$250$	−2568.00	−0.649658
$251$	−468.000	−0.117689	−0.0588444	−	0.998267i	$-0.518742\pi$
$251$	−468.000	−0.117689	−0.0588444	+	0.998267i	$0.518742\pi$
$252$	0	0
$253$	−2520.00	−0.626210
$254$	−4966.00	−1.22675
$255$	1512.00	0.371314
$256$	256.000	0.0625000
$257$	−2490.00	−0.604365	−0.302183	−	0.953250i	$-0.597715\pi$
$257$	−2490.00	−0.604365	−0.302183	+	0.953250i	$0.597715\pi$
$258$	1950.00	0.470549
$259$	0	0
$260$	−1272.00	−0.303408
$261$	−1620.00	−0.384197
$262$	−4236.00	−0.998859
$263$	1572.00	0.368569	0.184285	−	0.982873i	$-0.441003\pi$
$263$	1572.00	0.368569	0.184285	+	0.982873i	$0.441003\pi$
$264$	720.000	0.167852
$265$	4608.00	1.06818
$266$	0	0
$267$	−1800.00	−0.412578
$268$	−2092.00	−0.476826
$269$	1806.00	0.409345	0.204672	−	0.978831i	$-0.434387\pi$
$269$	1806.00	0.409345	0.204672	+	0.978831i	$0.434387\pi$
$270$	324.000	0.0730297
$271$	−6112.00	−1.37003	−0.685014	−	0.728530i	$-0.740204\pi$
$271$	−6112.00	−1.37003	−0.685014	+	0.728530i	$0.740204\pi$
$272$	−1344.00	−0.299603
$273$	0	0
$274$	−6024.00	−1.32819
$275$	2670.00	0.585480
$276$	1008.00	0.219835
$277$	−4231.00	−0.917748	−0.458874	−	0.888501i	$-0.651747\pi$
$277$	−4231.00	−0.917748	−0.458874	+	0.888501i	$0.651747\pi$
$278$	74.0000	0.0159648
$279$	1611.00	0.345692
$280$	0	0
$281$	−3816.00	−0.810119	−0.405060	−	0.914290i	$-0.632749\pi$
$281$	−3816.00	−0.810119	−0.405060	+	0.914290i	$0.632749\pi$
$282$	2196.00	0.463723
$283$	−3997.00	−0.839565	−0.419783	−	0.907625i	$-0.637894\pi$
$283$	−3997.00	−0.839565	−0.419783	+	0.907625i	$0.637894\pi$
$284$	−1368.00	−0.285831
$285$	1746.00	0.362892
$286$	3180.00	0.657473
$287$	0	0
$288$	−288.000	−0.0589256
$289$	2143.00	0.436190
$290$	−2160.00	−0.437378
$291$	1158.00	0.233275
$292$	−172.000	−0.0344710
$293$	4608.00	0.918779	0.459389	−	0.888235i	$-0.348068\pi$
$293$	4608.00	0.918779	0.459389	+	0.888235i	$0.348068\pi$
$294$	0	0
$295$	1584.00	0.312624
$296$	1160.00	0.227783
$297$	−810.000	−0.158252
$298$	3288.00	0.639157
$299$	4452.00	0.861090
$300$	−1068.00	−0.205537
$301$	0	0
$302$	−2176.00	−0.414618
$303$	1854.00	0.351517
$304$	−1552.00	−0.292807
$305$	−4908.00	−0.921414
$306$	1512.00	0.282468
$307$	−631.000	−0.117306	−0.0586532	−	0.998278i	$-0.518681\pi$
$307$	−631.000	−0.117306	−0.0586532	+	0.998278i	$0.518681\pi$
$308$	0	0
$309$	4425.00	0.814658
$310$	2148.00	0.393543
$311$	3894.00	0.709995	0.354998	−	0.934867i	$-0.384482\pi$
$311$	3894.00	0.709995	0.354998	+	0.934867i	$0.384482\pi$
$312$	−1272.00	−0.230810
$313$	−2185.00	−0.394580	−0.197290	−	0.980345i	$-0.563214\pi$
$313$	−2185.00	−0.394580	−0.197290	+	0.980345i	$0.563214\pi$
$314$	−1012.00	−0.181880
$315$	0	0
$316$	−4684.00	−0.833847
$317$	3504.00	0.620834	0.310417	−	0.950601i	$-0.399531\pi$
$317$	3504.00	0.620834	0.310417	+	0.950601i	$0.399531\pi$
$318$	4608.00	0.812591
$319$	5400.00	0.947780
$320$	−384.000	−0.0670820
$321$	5652.00	0.982754
$322$	0	0
$323$	8148.00	1.40361
$324$	324.000	0.0555556
$325$	−4717.00	−0.805083
$326$	−3688.00	−0.626563
$327$	1239.00	0.209532
$328$	−1008.00	−0.169687
$329$	0	0
$330$	−1080.00	−0.180158
$331$	2945.00	0.489039	0.244519	−	0.969644i	$-0.421370\pi$
$331$	2945.00	0.489039	0.244519	+	0.969644i	$0.421370\pi$
$332$	−3240.00	−0.535597
$333$	−1305.00	−0.214755
$334$	−324.000	−0.0530793
$335$	3138.00	0.511783
$336$	0	0
$337$	4277.00	0.691344	0.345672	−	0.938355i	$-0.387651\pi$
$337$	4277.00	0.691344	0.345672	+	0.938355i	$0.387651\pi$
$338$	−1224.00	−0.196973
$339$	−2646.00	−0.423926
$340$	2016.00	0.321568
$341$	−5370.00	−0.852791
$342$	1746.00	0.276061
$343$	0	0
$344$	2600.00	0.407508
$345$	−1512.00	−0.235952
$346$	5448.00	0.846492
$347$	7188.00	1.11202	0.556012	−	0.831175i	$-0.312331\pi$
$347$	7188.00	1.11202	0.556012	+	0.831175i	$0.312331\pi$
$348$	−2160.00	−0.332725
$349$	−9406.00	−1.44267	−0.721335	−	0.692587i	$-0.756471\pi$
$349$	−9406.00	−1.44267	−0.721335	+	0.692587i	$0.756471\pi$
$350$	0	0
$351$	1431.00	0.217610
$352$	960.000	0.145364
$353$	3390.00	0.511137	0.255569	−	0.966791i	$-0.417737\pi$
$353$	3390.00	0.511137	0.255569	+	0.966791i	$0.417737\pi$
$354$	1584.00	0.237821
$355$	2052.00	0.306785
$356$	−2400.00	−0.357303
$357$	0	0
$358$	2508.00	0.370257
$359$	−4812.00	−0.707431	−0.353715	−	0.935353i	$-0.615082\pi$
$359$	−4812.00	−0.707431	−0.353715	+	0.935353i	$0.615082\pi$
$360$	432.000	0.0632456
$361$	2550.00	0.371774
$362$	3614.00	0.524717
$363$	−1293.00	−0.186956
$364$	0	0
$365$	258.000	0.0369982
$366$	−4908.00	−0.700943
$367$	−7099.00	−1.00971	−0.504857	−	0.863203i	$-0.668455\pi$
$367$	−7099.00	−1.00971	−0.504857	+	0.863203i	$0.668455\pi$
$368$	1344.00	0.190383
$369$	1134.00	0.159983
$370$	−1740.00	−0.244482
$371$	0	0
$372$	2148.00	0.299378
$373$	2963.00	0.411309	0.205655	−	0.978625i	$-0.434068\pi$
$373$	2963.00	0.411309	0.205655	+	0.978625i	$0.434068\pi$
$374$	−5040.00	−0.696824
$375$	3852.00	0.530444
$376$	2928.00	0.401596
$377$	−9540.00	−1.30328
$378$	0	0
$379$	−11899.0	−1.61269	−0.806346	−	0.591444i	$-0.798558\pi$
$379$	−11899.0	−1.61269	−0.806346	+	0.591444i	$0.798558\pi$
$380$	2328.00	0.314273
$381$	7449.00	1.00164
$382$	−1428.00	−0.191264
$383$	2568.00	0.342607	0.171304	−	0.985218i	$-0.445202\pi$
$383$	2568.00	0.342607	0.171304	+	0.985218i	$0.445202\pi$
$384$	−384.000	−0.0510310
$385$	0	0
$386$	7418.00	0.978151
$387$	−2925.00	−0.384202
$388$	1544.00	0.202022
$389$	−10146.0	−1.32242	−0.661212	−	0.750199i	$-0.729957\pi$
$389$	−10146.0	−1.32242	−0.661212	+	0.750199i	$0.729957\pi$
$390$	1908.00	0.247732
$391$	−7056.00	−0.912627
$392$	0	0
$393$	6354.00	0.815565
$394$	2088.00	0.266985
$395$	7026.00	0.894978
$396$	−1080.00	−0.137051
$397$	−6229.00	−0.787467	−0.393734	−	0.919225i	$-0.628817\pi$
$397$	−6229.00	−0.787467	−0.393734	+	0.919225i	$0.628817\pi$
$398$	272.000	0.0342566
$399$	0	0
$400$	−1424.00	−0.178000
$401$	−2472.00	−0.307845	−0.153922	−	0.988083i	$-0.549191\pi$
$401$	−2472.00	−0.307845	−0.153922	+	0.988083i	$0.549191\pi$
$402$	3138.00	0.389326
$403$	9487.00	1.17266
$404$	2472.00	0.304422
$405$	−486.000	−0.0596285
$406$	0	0
$407$	4350.00	0.529783
$408$	2016.00	0.244625
$409$	−7075.00	−0.855345	−0.427673	−	0.903934i	$-0.640666\pi$
$409$	−7075.00	−0.855345	−0.427673	+	0.903934i	$0.640666\pi$
$410$	1512.00	0.182128
$411$	9036.00	1.08446
$412$	5900.00	0.705515
$413$	0	0
$414$	−1512.00	−0.179495
$415$	4860.00	0.574863
$416$	−1696.00	−0.199888
$417$	−111.000	−0.0130352
$418$	−5820.00	−0.681018
$419$	−4158.00	−0.484801	−0.242400	−	0.970176i	$-0.577935\pi$
$419$	−4158.00	−0.484801	−0.242400	+	0.970176i	$0.577935\pi$
$420$	0	0
$421$	−6595.00	−0.763469	−0.381735	−	0.924272i	$-0.624673\pi$
$421$	−6595.00	−0.763469	−0.381735	+	0.924272i	$0.624673\pi$
$422$	−2968.00	−0.342370
$423$	−3294.00	−0.378628
$424$	6144.00	0.703724
$425$	7476.00	0.853269
$426$	2052.00	0.233380
$427$	0	0
$428$	7536.00	0.851090
$429$	−4770.00	−0.536825
$430$	−3900.00	−0.437383
$431$	1518.00	0.169651	0.0848254	−	0.996396i	$-0.472967\pi$
$431$	1518.00	0.169651	0.0848254	+	0.996396i	$0.472967\pi$
$432$	432.000	0.0481125
$433$	8567.00	0.950817	0.475408	−	0.879765i	$-0.342300\pi$
$433$	8567.00	0.950817	0.475408	+	0.879765i	$0.342300\pi$
$434$	0	0
$435$	3240.00	0.357117
$436$	1652.00	0.181460
$437$	−8148.00	−0.891926
$438$	258.000	0.0281455
$439$	10640.0	1.15676	0.578382	−	0.815766i	$-0.303684\pi$
$439$	10640.0	1.15676	0.578382	+	0.815766i	$0.303684\pi$
$440$	−1440.00	−0.156021
$441$	0	0
$442$	8904.00	0.958190
$443$	7032.00	0.754177	0.377088	−	0.926177i	$-0.376925\pi$
$443$	7032.00	0.754177	0.377088	+	0.926177i	$0.376925\pi$
$444$	−1740.00	−0.185984
$445$	3600.00	0.383497
$446$	4064.00	0.431471
$447$	−4932.00	−0.521869
$448$	0	0
$449$	−14814.0	−1.55705	−0.778525	−	0.627613i	$-0.784032\pi$
$449$	−14814.0	−1.55705	−0.778525	+	0.627613i	$0.784032\pi$
$450$	1602.00	0.167820
$451$	−3780.00	−0.394664
$452$	−3528.00	−0.367131
$453$	3264.00	0.338534
$454$	−12396.0	−1.28144
$455$	0	0
$456$	2328.00	0.239076
$457$	−11251.0	−1.15164	−0.575820	−	0.817576i	$-0.695317\pi$
$457$	−11251.0	−1.15164	−0.575820	+	0.817576i	$0.695317\pi$
$458$	9182.00	0.936783
$459$	−2268.00	−0.230634
$460$	−2016.00	−0.204340
$461$	−3852.00	−0.389166	−0.194583	−	0.980886i	$-0.562335\pi$
$461$	−3852.00	−0.389166	−0.194583	+	0.980886i	$0.562335\pi$
$462$	0	0
$463$	−475.000	−0.0476784	−0.0238392	−	0.999716i	$-0.507589\pi$
$463$	−475.000	−0.0476784	−0.0238392	+	0.999716i	$0.507589\pi$
$464$	−2880.00	−0.288148
$465$	−3222.00	−0.321326
$466$	−9060.00	−0.900636
$467$	5934.00	0.587993	0.293997	−	0.955806i	$-0.405015\pi$
$467$	5934.00	0.587993	0.293997	+	0.955806i	$0.405015\pi$
$468$	1908.00	0.188456
$469$	0	0
$470$	−4392.00	−0.431038
$471$	1518.00	0.148505
$472$	2112.00	0.205959
$473$	9750.00	0.947792
$474$	7026.00	0.680833
$475$	8633.00	0.833914
$476$	0	0
$477$	−6912.00	−0.663477
$478$	−3060.00	−0.292806
$479$	−13368.0	−1.27516	−0.637578	−	0.770386i	$-0.720064\pi$
$479$	−13368.0	−1.27516	−0.637578	+	0.770386i	$0.720064\pi$
$480$	576.000	0.0547723
$481$	−7685.00	−0.728494
$482$	−11068.0	−1.04592
$483$	0	0
$484$	−1724.00	−0.161908
$485$	−2316.00	−0.216833
$486$	−486.000	−0.0453609
$487$	6653.00	0.619048	0.309524	−	0.950892i	$-0.399830\pi$
$487$	6653.00	0.619048	0.309524	+	0.950892i	$0.399830\pi$
$488$	−6544.00	−0.607035
$489$	5532.00	0.511586
$490$	0	0
$491$	15444.0	1.41951	0.709754	−	0.704450i	$-0.248806\pi$
$491$	15444.0	1.41951	0.709754	+	0.704450i	$0.248806\pi$
$492$	1512.00	0.138549
$493$	15120.0	1.38128
$494$	10282.0	0.936456
$495$	1620.00	0.147098
$496$	2864.00	0.259269
$497$	0	0
$498$	4860.00	0.437313
$499$	683.000	0.0612731	0.0306366	−	0.999531i	$-0.490247\pi$
$499$	683.000	0.0612731	0.0306366	+	0.999531i	$0.490247\pi$
$500$	5136.00	0.459378
$501$	486.000	0.0433391
$502$	936.000	0.0832186
$503$	9882.00	0.875977	0.437989	−	0.898980i	$-0.355691\pi$
$503$	9882.00	0.875977	0.437989	+	0.898980i	$0.355691\pi$
$504$	0	0
$505$	−3708.00	−0.326740
$506$	5040.00	0.442797
$507$	1836.00	0.160828
$508$	9932.00	0.867443
$509$	4206.00	0.366263	0.183131	−	0.983088i	$-0.441377\pi$
$509$	4206.00	0.366263	0.183131	+	0.983088i	$0.441377\pi$
$510$	−3024.00	−0.262559
$511$	0	0
$512$	−512.000	−0.0441942
$513$	−2619.00	−0.225403
$514$	4980.00	0.427351
$515$	−8850.00	−0.757238
$516$	−3900.00	−0.332729
$517$	10980.0	0.934042
$518$	0	0
$519$	−8172.00	−0.691158
$520$	2544.00	0.214542
$521$	9060.00	0.761854	0.380927	−	0.924605i	$-0.375605\pi$
$521$	9060.00	0.761854	0.380927	+	0.924605i	$0.375605\pi$
$522$	3240.00	0.271668
$523$	−15679.0	−1.31089	−0.655444	−	0.755243i	$-0.727519\pi$
$523$	−15679.0	−1.31089	−0.655444	+	0.755243i	$0.727519\pi$
$524$	8472.00	0.706300
$525$	0	0
$526$	−3144.00	−0.260618
$527$	−15036.0	−1.24284
$528$	−1440.00	−0.118689
$529$	−5111.00	−0.420071
$530$	−9216.00	−0.755316
$531$	−2376.00	−0.194180
$532$	0	0
$533$	6678.00	0.542695
$534$	3600.00	0.291736
$535$	−11304.0	−0.913485
$536$	4184.00	0.337167
$537$	−3762.00	−0.302313
$538$	−3612.00	−0.289451
$539$	0	0
$540$	−648.000	−0.0516398
$541$	−7711.00	−0.612794	−0.306397	−	0.951904i	$-0.599124\pi$
$541$	−7711.00	−0.612794	−0.306397	+	0.951904i	$0.599124\pi$
$542$	12224.0	0.968756
$543$	−5421.00	−0.428430
$544$	2688.00	0.211851
$545$	−2478.00	−0.194763
$546$	0	0
$547$	4292.00	0.335489	0.167745	−	0.985830i	$-0.446352\pi$
$547$	4292.00	0.335489	0.167745	+	0.985830i	$0.446352\pi$
$548$	12048.0	0.939170
$549$	7362.00	0.572318
$550$	−5340.00	−0.413997
$551$	17460.0	1.34995
$552$	−2016.00	−0.155447
$553$	0	0
$554$	8462.00	0.648946
$555$	2610.00	0.199619
$556$	−148.000	−0.0112888
$557$	−9858.00	−0.749905	−0.374952	−	0.927044i	$-0.622341\pi$
$557$	−9858.00	−0.749905	−0.374952	+	0.927044i	$0.622341\pi$
$558$	−3222.00	−0.244441
$559$	−17225.0	−1.30329
$560$	0	0
$561$	7560.00	0.568954
$562$	7632.00	0.572841
$563$	−13890.0	−1.03978	−0.519888	−	0.854235i	$-0.674026\pi$
$563$	−13890.0	−1.03978	−0.519888	+	0.854235i	$0.674026\pi$
$564$	−4392.00	−0.327902
$565$	5292.00	0.394046
$566$	7994.00	0.593662
$567$	0	0
$568$	2736.00	0.202113
$569$	19038.0	1.40266	0.701331	−	0.712836i	$-0.252590\pi$
$569$	19038.0	1.40266	0.701331	+	0.712836i	$0.252590\pi$
$570$	−3492.00	−0.256603
$571$	−8053.00	−0.590206	−0.295103	−	0.955465i	$-0.595354\pi$
$571$	−8053.00	−0.590206	−0.295103	+	0.955465i	$0.595354\pi$
$572$	−6360.00	−0.464904
$573$	2142.00	0.156166
$574$	0	0
$575$	−7476.00	−0.542210
$576$	576.000	0.0416667
$577$	−17137.0	−1.23643	−0.618217	−	0.786007i	$-0.712145\pi$
$577$	−17137.0	−1.23643	−0.618217	+	0.786007i	$0.712145\pi$
$578$	−4286.00	−0.308433
$579$	−11127.0	−0.798657
$580$	4320.00	0.309273
$581$	0	0
$582$	−2316.00	−0.164951
$583$	23040.0	1.63674
$584$	344.000	0.0243747
$585$	−2862.00	−0.202272
$586$	−9216.00	−0.649675
$587$	18144.0	1.27578	0.637890	−	0.770127i	$-0.279807\pi$
$587$	18144.0	1.27578	0.637890	+	0.770127i	$0.279807\pi$
$588$	0	0
$589$	−17363.0	−1.21465
$590$	−3168.00	−0.221058
$591$	−3132.00	−0.217992
$592$	−2320.00	−0.161067
$593$	−24702.0	−1.71061	−0.855303	−	0.518128i	$-0.826629\pi$
$593$	−24702.0	−1.71061	−0.855303	+	0.518128i	$0.826629\pi$
$594$	1620.00	0.111901
$595$	0	0
$596$	−6576.00	−0.451952
$597$	−408.000	−0.0279704
$598$	−8904.00	−0.608882
$599$	−2172.00	−0.148156	−0.0740781	−	0.997252i	$-0.523601\pi$
$599$	−2172.00	−0.148156	−0.0740781	+	0.997252i	$0.523601\pi$
$600$	2136.00	0.145336
$601$	4175.00	0.283364	0.141682	−	0.989912i	$-0.454749\pi$
$601$	4175.00	0.283364	0.141682	+	0.989912i	$0.454749\pi$
$602$	0	0
$603$	−4707.00	−0.317884
$604$	4352.00	0.293179
$605$	2586.00	0.173778
$606$	−3708.00	−0.248560
$607$	2261.00	0.151188	0.0755940	−	0.997139i	$-0.475915\pi$
$607$	2261.00	0.151188	0.0755940	+	0.997139i	$0.475915\pi$
$608$	3104.00	0.207046
$609$	0	0
$610$	9816.00	0.651538
$611$	−19398.0	−1.28438
$612$	−3024.00	−0.199735
$613$	−16318.0	−1.07517	−0.537584	−	0.843210i	$-0.680663\pi$
$613$	−16318.0	−1.07517	−0.537584	+	0.843210i	$0.680663\pi$
$614$	1262.00	0.0829482
$615$	−2268.00	−0.148707
$616$	0	0
$617$	−26550.0	−1.73235	−0.866177	−	0.499737i	$-0.833430\pi$
$617$	−26550.0	−1.73235	−0.866177	+	0.499737i	$0.833430\pi$
$618$	−8850.00	−0.576051
$619$	19925.0	1.29379	0.646893	−	0.762581i	$-0.276068\pi$
$619$	19925.0	1.29379	0.646893	+	0.762581i	$0.276068\pi$
$620$	−4296.00	−0.278277
$621$	2268.00	0.146557
$622$	−7788.00	−0.502042
$623$	0	0
$624$	2544.00	0.163208
$625$	3421.00	0.218944
$626$	4370.00	0.279010
$627$	8730.00	0.556049
$628$	2024.00	0.128609
$629$	12180.0	0.772096
$630$	0	0
$631$	−6832.00	−0.431026	−0.215513	−	0.976501i	$-0.569142\pi$
$631$	−6832.00	−0.431026	−0.215513	+	0.976501i	$0.569142\pi$
$632$	9368.00	0.589619
$633$	4452.00	0.279544
$634$	−7008.00	−0.438996
$635$	−14898.0	−0.931038
$636$	−9216.00	−0.574588
$637$	0	0
$638$	−10800.0	−0.670182
$639$	−3078.00	−0.190554
$640$	768.000	0.0474342
$641$	10212.0	0.629251	0.314625	−	0.949216i	$-0.398121\pi$
$641$	10212.0	0.629251	0.314625	+	0.949216i	$0.398121\pi$
$642$	−11304.0	−0.694912
$643$	3779.00	0.231772	0.115886	−	0.993263i	$-0.463029\pi$
$643$	3779.00	0.231772	0.115886	+	0.993263i	$0.463029\pi$
$644$	0	0
$645$	5850.00	0.357122
$646$	−16296.0	−0.992504
$647$	16998.0	1.03286	0.516430	−	0.856329i	$-0.327261\pi$
$647$	16998.0	1.03286	0.516430	+	0.856329i	$0.327261\pi$
$648$	−648.000	−0.0392837
$649$	7920.00	0.479025
$650$	9434.00	0.569280
$651$	0	0
$652$	7376.00	0.443047
$653$	−21750.0	−1.30344	−0.651718	−	0.758462i	$-0.725951\pi$
$653$	−21750.0	−1.30344	−0.651718	+	0.758462i	$0.725951\pi$
$654$	−2478.00	−0.148161
$655$	−12708.0	−0.758080
$656$	2016.00	0.119987
$657$	−387.000	−0.0229807
$658$	0	0
$659$	−10944.0	−0.646916	−0.323458	−	0.946243i	$-0.604845\pi$
$659$	−10944.0	−0.646916	−0.323458	+	0.946243i	$0.604845\pi$
$660$	2160.00	0.127391
$661$	10955.0	0.644630	0.322315	−	0.946633i	$-0.395539\pi$
$661$	10955.0	0.644630	0.322315	+	0.946633i	$0.395539\pi$
$662$	−5890.00	−0.345803
$663$	−13356.0	−0.782359
$664$	6480.00	0.378724
$665$	0	0
$666$	2610.00	0.151855
$667$	−15120.0	−0.877734
$668$	648.000	0.0375327
$669$	−6096.00	−0.352294
$670$	−6276.00	−0.361885
$671$	−24540.0	−1.41186
$672$	0	0
$673$	25103.0	1.43782	0.718908	−	0.695106i	$-0.244642\pi$
$673$	25103.0	1.43782	0.718908	+	0.695106i	$0.244642\pi$
$674$	−8554.00	−0.488854
$675$	−2403.00	−0.137024
$676$	2448.00	0.139281
$677$	−5604.00	−0.318138	−0.159069	−	0.987267i	$-0.550849\pi$
$677$	−5604.00	−0.318138	−0.159069	+	0.987267i	$0.550849\pi$
$678$	5292.00	0.299761
$679$	0	0
$680$	−4032.00	−0.227383
$681$	18594.0	1.04629
$682$	10740.0	0.603014
$683$	10968.0	0.614464	0.307232	−	0.951635i	$-0.400597\pi$
$683$	10968.0	0.614464	0.307232	+	0.951635i	$0.400597\pi$
$684$	−3492.00	−0.195205
$685$	−18072.0	−1.00802
$686$	0	0
$687$	−13773.0	−0.764880
$688$	−5200.00	−0.288151
$689$	−40704.0	−2.25065
$690$	3024.00	0.166843
$691$	8405.00	0.462723	0.231361	−	0.972868i	$-0.425682\pi$
$691$	8405.00	0.462723	0.231361	+	0.972868i	$0.425682\pi$
$692$	−10896.0	−0.598560
$693$	0	0
$694$	−14376.0	−0.786319
$695$	222.000	0.0121165
$696$	4320.00	0.235272
$697$	−10584.0	−0.575176
$698$	18812.0	1.02012
$699$	13590.0	0.735366
$700$	0	0
$701$	468.000	0.0252156	0.0126078	−	0.999921i	$-0.495987\pi$
$701$	468.000	0.0252156	0.0126078	+	0.999921i	$0.495987\pi$
$702$	−2862.00	−0.153874
$703$	14065.0	0.754583
$704$	−1920.00	−0.102788
$705$	6588.00	0.351941
$706$	−6780.00	−0.361429
$707$	0	0
$708$	−3168.00	−0.168165
$709$	−25066.0	−1.32775	−0.663874	−	0.747844i	$-0.731089\pi$
$709$	−25066.0	−1.32775	−0.663874	+	0.747844i	$0.731089\pi$
$710$	−4104.00	−0.216930
$711$	−10539.0	−0.555898
$712$	4800.00	0.252651
$713$	15036.0	0.789765
$714$	0	0
$715$	9540.00	0.498987
$716$	−5016.00	−0.261811
$717$	4590.00	0.239075
$718$	9624.00	0.500229
$719$	11082.0	0.574811	0.287405	−	0.957809i	$-0.407207\pi$
$719$	11082.0	0.574811	0.287405	+	0.957809i	$0.407207\pi$
$720$	−864.000	−0.0447214
$721$	0	0
$722$	−5100.00	−0.262884
$723$	16602.0	0.853990
$724$	−7228.00	−0.371031
$725$	16020.0	0.820645
$726$	2586.00	0.132198
$727$	13481.0	0.687734	0.343867	−	0.939018i	$-0.388263\pi$
$727$	13481.0	0.687734	0.343867	+	0.939018i	$0.388263\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	−516.000	−0.0261617
$731$	27300.0	1.38130
$732$	9816.00	0.495642
$733$	24317.0	1.22533	0.612666	−	0.790342i	$-0.290097\pi$
$733$	24317.0	1.22533	0.612666	+	0.790342i	$0.290097\pi$
$734$	14198.0	0.713975
$735$	0	0
$736$	−2688.00	−0.134621
$737$	15690.0	0.784191
$738$	−2268.00	−0.113125
$739$	−18217.0	−0.906797	−0.453399	−	0.891308i	$-0.649789\pi$
$739$	−18217.0	−0.906797	−0.453399	+	0.891308i	$0.649789\pi$
$740$	3480.00	0.172875
$741$	−15423.0	−0.764613
$742$	0	0
$743$	19782.0	0.976758	0.488379	−	0.872632i	$-0.337588\pi$
$743$	19782.0	0.976758	0.488379	+	0.872632i	$0.337588\pi$
$744$	−4296.00	−0.211692
$745$	9864.00	0.485086
$746$	−5926.00	−0.290840
$747$	−7290.00	−0.357064
$748$	10080.0	0.492729
$749$	0	0
$750$	−7704.00	−0.375080
$751$	−4921.00	−0.239108	−0.119554	−	0.992828i	$-0.538146\pi$
$751$	−4921.00	−0.239108	−0.119554	+	0.992828i	$0.538146\pi$
$752$	−5856.00	−0.283971
$753$	−1404.00	−0.0679477
$754$	19080.0	0.921555
$755$	−6528.00	−0.314673
$756$	0	0
$757$	18098.0	0.868934	0.434467	−	0.900688i	$-0.356937\pi$
$757$	18098.0	0.868934	0.434467	+	0.900688i	$0.356937\pi$
$758$	23798.0	1.14035
$759$	−7560.00	−0.361542
$760$	−4656.00	−0.222225
$761$	−24468.0	−1.16552	−0.582762	−	0.812643i	$-0.698028\pi$
$761$	−24468.0	−1.16552	−0.582762	+	0.812643i	$0.698028\pi$
$762$	−14898.0	−0.708265
$763$	0	0
$764$	2856.00	0.135244
$765$	4536.00	0.214378
$766$	−5136.00	−0.242260
$767$	−13992.0	−0.658699
$768$	768.000	0.0360844
$769$	21719.0	1.01847	0.509237	−	0.860626i	$-0.329928\pi$
$769$	21719.0	1.01847	0.509237	+	0.860626i	$0.329928\pi$
$770$	0	0
$771$	−7470.00	−0.348931
$772$	−14836.0	−0.691657
$773$	−30306.0	−1.41013	−0.705065	−	0.709142i	$-0.749082\pi$
$773$	−30306.0	−1.41013	−0.705065	+	0.709142i	$0.749082\pi$
$774$	5850.00	0.271672
$775$	−15931.0	−0.738398
$776$	−3088.00	−0.142851
$777$	0	0
$778$	20292.0	0.935094
$779$	−12222.0	−0.562129
$780$	−3816.00	−0.175173
$781$	10260.0	0.470079
$782$	14112.0	0.645325
$783$	−4860.00	−0.221816
$784$	0	0
$785$	−3036.00	−0.138038
$786$	−12708.0	−0.576691
$787$	27296.0	1.23634	0.618169	−	0.786046i	$-0.287875\pi$
$787$	27296.0	1.23634	0.618169	+	0.786046i	$0.287875\pi$
$788$	−4176.00	−0.188787
$789$	4716.00	0.212793
$790$	−14052.0	−0.632845
$791$	0	0
$792$	2160.00	0.0969094
$793$	43354.0	1.94142
$794$	12458.0	0.556824
$795$	13824.0	0.616713
$796$	−544.000	−0.0242231
$797$	−35100.0	−1.55998	−0.779991	−	0.625791i	$-0.784776\pi$
$797$	−35100.0	−1.55998	−0.779991	+	0.625791i	$0.784776\pi$
$798$	0	0
$799$	30744.0	1.36126
$800$	2848.00	0.125865
$801$	−5400.00	−0.238202
$802$	4944.00	0.217679
$803$	1290.00	0.0566913
$804$	−6276.00	−0.275295
$805$	0	0
$806$	−18974.0	−0.829194
$807$	5418.00	0.236335
$808$	−4944.00	−0.215259
$809$	44394.0	1.92931	0.964654	−	0.263520i	$-0.0848836\pi$
$809$	44394.0	1.92931	0.964654	+	0.263520i	$0.0848836\pi$
$810$	972.000	0.0421637
$811$	−8584.00	−0.371671	−0.185835	−	0.982581i	$-0.559499\pi$
$811$	−8584.00	−0.371671	−0.185835	+	0.982581i	$0.559499\pi$
$812$	0	0
$813$	−18336.0	−0.790986
$814$	−8700.00	−0.374613
$815$	−11064.0	−0.475528
$816$	−4032.00	−0.172976
$817$	31525.0	1.34996
$818$	14150.0	0.604820
$819$	0	0
$820$	−3024.00	−0.128784
$821$	−9834.00	−0.418038	−0.209019	−	0.977912i	$-0.567027\pi$
$821$	−9834.00	−0.418038	−0.209019	+	0.977912i	$0.567027\pi$
$822$	−18072.0	−0.766829
$823$	43856.0	1.85750	0.928751	−	0.370704i	$-0.120884\pi$
$823$	43856.0	1.85750	0.928751	+	0.370704i	$0.120884\pi$
$824$	−11800.0	−0.498874
$825$	8010.00	0.338027
$826$	0	0
$827$	13266.0	0.557804	0.278902	−	0.960320i	$-0.410030\pi$
$827$	13266.0	0.557804	0.278902	+	0.960320i	$0.410030\pi$
$828$	3024.00	0.126922
$829$	17453.0	0.731204	0.365602	−	0.930771i	$-0.380863\pi$
$829$	17453.0	0.731204	0.365602	+	0.930771i	$0.380863\pi$
$830$	−9720.00	−0.406489
$831$	−12693.0	−0.529862
$832$	3392.00	0.141342
$833$	0	0
$834$	222.000	0.00921730
$835$	−972.000	−0.0402844
$836$	11640.0	0.481552
$837$	4833.00	0.199585
$838$	8316.00	0.342806
$839$	−35172.0	−1.44729	−0.723643	−	0.690175i	$-0.757534\pi$
$839$	−35172.0	−1.44729	−0.723643	+	0.690175i	$0.757534\pi$
$840$	0	0
$841$	8011.00	0.328468
$842$	13190.0	0.539854
$843$	−11448.0	−0.467722
$844$	5936.00	0.242092
$845$	−3672.00	−0.149492
$846$	6588.00	0.267731
$847$	0	0
$848$	−12288.0	−0.497608
$849$	−11991.0	−0.484723
$850$	−14952.0	−0.603352
$851$	−12180.0	−0.490629
$852$	−4104.00	−0.165024
$853$	3503.00	0.140610	0.0703051	−	0.997526i	$-0.477603\pi$
$853$	3503.00	0.140610	0.0703051	+	0.997526i	$0.477603\pi$
$854$	0	0
$855$	5238.00	0.209516
$856$	−15072.0	−0.601811
$857$	22848.0	0.910703	0.455352	−	0.890312i	$-0.349514\pi$
$857$	22848.0	0.910703	0.455352	+	0.890312i	$0.349514\pi$
$858$	9540.00	0.379592
$859$	−13456.0	−0.534474	−0.267237	−	0.963631i	$-0.586111\pi$
$859$	−13456.0	−0.534474	−0.267237	+	0.963631i	$0.586111\pi$
$860$	7800.00	0.309277
$861$	0	0
$862$	−3036.00	−0.119961
$863$	40710.0	1.60578	0.802888	−	0.596130i	$-0.203296\pi$
$863$	40710.0	1.60578	0.802888	+	0.596130i	$0.203296\pi$
$864$	−864.000	−0.0340207
$865$	16344.0	0.642442
$866$	−17134.0	−0.672329
$867$	6429.00	0.251834
$868$	0	0
$869$	35130.0	1.37135
$870$	−6480.00	−0.252520
$871$	−27719.0	−1.07833
$872$	−3304.00	−0.128311
$873$	3474.00	0.134682
$874$	16296.0	0.630687
$875$	0	0
$876$	−516.000	−0.0199019
$877$	2906.00	0.111891	0.0559456	−	0.998434i	$-0.482183\pi$
$877$	2906.00	0.111891	0.0559456	+	0.998434i	$0.482183\pi$
$878$	−21280.0	−0.817956
$879$	13824.0	0.530457
$880$	2880.00	0.110324
$881$	−19188.0	−0.733780	−0.366890	−	0.930264i	$-0.619577\pi$
$881$	−19188.0	−0.733780	−0.366890	+	0.930264i	$0.619577\pi$
$882$	0	0
$883$	−17251.0	−0.657466	−0.328733	−	0.944423i	$-0.606622\pi$
$883$	−17251.0	−0.657466	−0.328733	+	0.944423i	$0.606622\pi$
$884$	−17808.0	−0.677543
$885$	4752.00	0.180493
$886$	−14064.0	−0.533284
$887$	−2094.00	−0.0792668	−0.0396334	−	0.999214i	$-0.512619\pi$
$887$	−2094.00	−0.0792668	−0.0396334	+	0.999214i	$0.512619\pi$
$888$	3480.00	0.131510
$889$	0	0
$890$	−7200.00	−0.271174
$891$	−2430.00	−0.0913671
$892$	−8128.00	−0.305096
$893$	35502.0	1.33038
$894$	9864.00	0.369017
$895$	7524.00	0.281005
$896$	0	0
$897$	13356.0	0.497150
$898$	29628.0	1.10100
$899$	−32220.0	−1.19532
$900$	−3204.00	−0.118667
$901$	64512.0	2.38536
$902$	7560.00	0.279069
$903$	0	0
$904$	7056.00	0.259601
$905$	10842.0	0.398232
$906$	−6528.00	−0.239380
$907$	−40267.0	−1.47414	−0.737069	−	0.675817i	$-0.763791\pi$
$907$	−40267.0	−1.47414	−0.737069	+	0.675817i	$0.763791\pi$
$908$	24792.0	0.906114
$909$	5562.00	0.202948
$910$	0	0
$911$	17604.0	0.640227	0.320113	−	0.947379i	$-0.396279\pi$
$911$	17604.0	0.640227	0.320113	+	0.947379i	$0.396279\pi$
$912$	−4656.00	−0.169052
$913$	24300.0	0.880846
$914$	22502.0	0.814333
$915$	−14724.0	−0.531979
$916$	−18364.0	−0.662406
$917$	0	0
$918$	4536.00	0.163083
$919$	3509.00	0.125953	0.0629767	−	0.998015i	$-0.479941\pi$
$919$	3509.00	0.125953	0.0629767	+	0.998015i	$0.479941\pi$
$920$	4032.00	0.144490
$921$	−1893.00	−0.0677269
$922$	7704.00	0.275182
$923$	−18126.0	−0.646397
$924$	0	0
$925$	12905.0	0.458718
$926$	950.000	0.0337138
$927$	13275.0	0.470343
$928$	5760.00	0.203751
$929$	−34638.0	−1.22329	−0.611645	−	0.791133i	$-0.709492\pi$
$929$	−34638.0	−1.22329	−0.611645	+	0.791133i	$0.709492\pi$
$930$	6444.00	0.227212
$931$	0	0
$932$	18120.0	0.636846
$933$	11682.0	0.409916
$934$	−11868.0	−0.415774
$935$	−15120.0	−0.528852
$936$	−3816.00	−0.133258
$937$	−17353.0	−0.605014	−0.302507	−	0.953147i	$-0.597824\pi$
$937$	−17353.0	−0.605014	−0.302507	+	0.953147i	$0.597824\pi$
$938$	0	0
$939$	−6555.00	−0.227811
$940$	8784.00	0.304790
$941$	−46920.0	−1.62545	−0.812725	−	0.582648i	$-0.802017\pi$
$941$	−46920.0	−1.62545	−0.812725	+	0.582648i	$0.802017\pi$
$942$	−3036.00	−0.105009
$943$	10584.0	0.365496
$944$	−4224.00	−0.145635
$945$	0	0
$946$	−19500.0	−0.670190
$947$	18354.0	0.629804	0.314902	−	0.949124i	$-0.398028\pi$
$947$	18354.0	0.629804	0.314902	+	0.949124i	$0.398028\pi$
$948$	−14052.0	−0.481422
$949$	−2279.00	−0.0779552
$950$	−17266.0	−0.589666
$951$	10512.0	0.358438
$952$	0	0
$953$	35568.0	1.20898	0.604491	−	0.796612i	$-0.293376\pi$
$953$	35568.0	1.20898	0.604491	+	0.796612i	$0.293376\pi$
$954$	13824.0	0.469149
$955$	−4284.00	−0.145159
$956$	6120.00	0.207045
$957$	16200.0	0.547201
$958$	26736.0	0.901671
$959$	0	0
$960$	−1152.00	−0.0387298
$961$	2250.00	0.0755262
$962$	15370.0	0.515123
$963$	16956.0	0.567393
$964$	22136.0	0.739577
$965$	22254.0	0.742364
$966$	0	0
$967$	−27343.0	−0.909298	−0.454649	−	0.890671i	$-0.650235\pi$
$967$	−27343.0	−0.909298	−0.454649	+	0.890671i	$0.650235\pi$
$968$	3448.00	0.114486
$969$	24444.0	0.810376
$970$	4632.00	0.153324
$971$	51024.0	1.68634	0.843171	−	0.537645i	$-0.180686\pi$
$971$	51024.0	1.68634	0.843171	+	0.537645i	$0.180686\pi$
$972$	972.000	0.0320750
$973$	0	0
$974$	−13306.0	−0.437733
$975$	−14151.0	−0.464815
$976$	13088.0	0.429238
$977$	−2226.00	−0.0728926	−0.0364463	−	0.999336i	$-0.511604\pi$
$977$	−2226.00	−0.0728926	−0.0364463	+	0.999336i	$0.511604\pi$
$978$	−11064.0	−0.361746
$979$	18000.0	0.587623
$980$	0	0
$981$	3717.00	0.120973
$982$	−30888.0	−1.00374
$983$	35304.0	1.14550	0.572748	−	0.819731i	$-0.305877\pi$
$983$	35304.0	1.14550	0.572748	+	0.819731i	$0.305877\pi$
$984$	−3024.00	−0.0979691
$985$	6264.00	0.202627
$986$	−30240.0	−0.976712
$987$	0	0
$988$	−20564.0	−0.662174
$989$	−27300.0	−0.877745
$990$	−3240.00	−0.104014
$991$	−2341.00	−0.0750397	−0.0375198	−	0.999296i	$-0.511946\pi$
$991$	−2341.00	−0.0750397	−0.0375198	+	0.999296i	$0.511946\pi$
$992$	−5728.00	−0.183331
$993$	8835.00	0.282347
$994$	0	0
$995$	816.000	0.0259989
$996$	−9720.00	−0.309227
$997$	29015.0	0.921679	0.460840	−	0.887483i	$-0.347548\pi$
$997$	29015.0	0.921679	0.460840	+	0.887483i	$0.347548\pi$
$998$	−1366.00	−0.0433266
$999$	−3915.00	−0.123989

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	294.4.a.d.1.1		1
3.2	odd	2		882.4.a.o.1.1		1
4.3	odd	2		2352.4.a.f.1.1		1
7.2	even	3		42.4.e.a.25.1	✓	2
7.3	odd	6		294.4.e.i.79.1		2
7.4	even	3		42.4.e.a.37.1	yes	2
7.5	odd	6		294.4.e.i.67.1		2
7.6	odd	2		294.4.a.c.1.1		1
21.2	odd	6		126.4.g.b.109.1		2
21.5	even	6		882.4.g.g.361.1		2
21.11	odd	6		126.4.g.b.37.1		2
21.17	even	6		882.4.g.g.667.1		2
21.20	even	2		882.4.a.l.1.1		1
28.11	odd	6		336.4.q.f.289.1		2
28.23	odd	6		336.4.q.f.193.1		2
28.27	even	2		2352.4.a.bf.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
42.4.e.a.25.1	✓	2	7.2	even	3
42.4.e.a.37.1	yes	2	7.4	even	3
126.4.g.b.37.1		2	21.11	odd	6
126.4.g.b.109.1		2	21.2	odd	6
294.4.a.c.1.1		1	7.6	odd	2
294.4.a.d.1.1		1	1.1	even	1	trivial
294.4.e.i.67.1		2	7.5	odd	6
294.4.e.i.79.1		2	7.3	odd	6
336.4.q.f.193.1		2	28.23	odd	6
336.4.q.f.289.1		2	28.11	odd	6
882.4.a.l.1.1		1	21.20	even	2
882.4.a.o.1.1		1	3.2	odd	2
882.4.g.g.361.1		2	21.5	even	6
882.4.g.g.667.1		2	21.17	even	6
2352.4.a.f.1.1		1	4.3	odd	2
2352.4.a.bf.1.1		1	28.27	even	2

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Classical	Maass
Hilbert	Bianchi

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

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	294.4.a.d.1.1

Level	$294$
Weight	$4$
Character	294.1
Self dual	yes
Analytic conductor	$17.347$
Analytic rank	$1$
Dimension	$1$
CM	no
Inner twists	$1$