LMFDB - Embedded newform 294.4.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [294,4,Mod(1,294)]

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

lf = mfeigenbasis(mf)

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")

//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: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$17.3465615417$
Analytic rank:	$0$
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} +15.0000 q^{5} -6.00000 q^{6} -8.00000 q^{8} +9.00000 q^{9} -30.0000 q^{10} -9.00000 q^{11} +12.0000 q^{12} +88.0000 q^{13} +45.0000 q^{15} +16.0000 q^{16} +84.0000 q^{17} -18.0000 q^{18} -104.000 q^{19} +60.0000 q^{20} +18.0000 q^{22} -84.0000 q^{23} -24.0000 q^{24} +100.000 q^{25} -176.000 q^{26} +27.0000 q^{27} +51.0000 q^{29} -90.0000 q^{30} -185.000 q^{31} -32.0000 q^{32} -27.0000 q^{33} -168.000 q^{34} +36.0000 q^{36} +44.0000 q^{37} +208.000 q^{38} +264.000 q^{39} -120.000 q^{40} +168.000 q^{41} +326.000 q^{43} -36.0000 q^{44} +135.000 q^{45} +168.000 q^{46} +138.000 q^{47} +48.0000 q^{48} -200.000 q^{50} +252.000 q^{51} +352.000 q^{52} +639.000 q^{53} -54.0000 q^{54} -135.000 q^{55} -312.000 q^{57} -102.000 q^{58} -159.000 q^{59} +180.000 q^{60} -722.000 q^{61} +370.000 q^{62} +64.0000 q^{64} +1320.00 q^{65} +54.0000 q^{66} -166.000 q^{67} +336.000 q^{68} -252.000 q^{69} +1086.00 q^{71} -72.0000 q^{72} -218.000 q^{73} -88.0000 q^{74} +300.000 q^{75} -416.000 q^{76} -528.000 q^{78} -583.000 q^{79} +240.000 q^{80} +81.0000 q^{81} -336.000 q^{82} +597.000 q^{83} +1260.00 q^{85} -652.000 q^{86} +153.000 q^{87} +72.0000 q^{88} +1038.00 q^{89} -270.000 q^{90} -336.000 q^{92} -555.000 q^{93} -276.000 q^{94} -1560.00 q^{95} -96.0000 q^{96} +169.000 q^{97} -81.0000 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
$5$	15.0000	1.34164	0.670820	−	0.741620i	$-0.265942\pi$
$5$	15.0000	1.34164	0.670820	+	0.741620i	$0.265942\pi$
$6$	−6.00000	−0.408248
$7$	0	0
$7$	0	0
$8$	−8.00000	−0.353553
$9$	9.00000	0.333333
$10$	−30.0000	−0.948683
$11$	−9.00000	−0.246691	−0.123346	−	0.992364i	$-0.539362\pi$
$11$	−9.00000	−0.246691	−0.123346	+	0.992364i	$0.539362\pi$
$12$	12.0000	0.288675
$13$	88.0000	1.87745	0.938723	−	0.344671i	$-0.112010\pi$
$13$	88.0000	1.87745	0.938723	+	0.344671i	$0.112010\pi$
$14$	0	0
$15$	45.0000	0.774597
$16$	16.0000	0.250000
$17$	84.0000	1.19841	0.599206	−	0.800595i	$-0.295483\pi$
$17$	84.0000	1.19841	0.599206	+	0.800595i	$0.295483\pi$
$18$	−18.0000	−0.235702
$19$	−104.000	−1.25575	−0.627875	−	0.778314i	$-0.716075\pi$
$19$	−104.000	−1.25575	−0.627875	+	0.778314i	$0.716075\pi$
$20$	60.0000	0.670820
$21$	0	0
$22$	18.0000	0.174437
$23$	−84.0000	−0.761531	−0.380765	−	0.924672i	$-0.624339\pi$
$23$	−84.0000	−0.761531	−0.380765	+	0.924672i	$0.624339\pi$
$24$	−24.0000	−0.204124
$25$	100.000	0.800000
$26$	−176.000	−1.32756
$27$	27.0000	0.192450
$28$	0	0
$29$	51.0000	0.326568	0.163284	−	0.986579i	$-0.447791\pi$
$29$	51.0000	0.326568	0.163284	+	0.986579i	$0.447791\pi$
$30$	−90.0000	−0.547723
$31$	−185.000	−1.07184	−0.535919	−	0.844269i	$-0.680035\pi$
$31$	−185.000	−1.07184	−0.535919	+	0.844269i	$0.680035\pi$
$32$	−32.0000	−0.176777
$33$	−27.0000	−0.142427
$34$	−168.000	−0.847405
$35$	0	0
$36$	36.0000	0.166667
$37$	44.0000	0.195501	0.0977507	−	0.995211i	$-0.468835\pi$
$37$	44.0000	0.195501	0.0977507	+	0.995211i	$0.468835\pi$
$38$	208.000	0.887949
$39$	264.000	1.08394
$40$	−120.000	−0.474342
$41$	168.000	0.639932	0.319966	−	0.947429i	$-0.396329\pi$
$41$	168.000	0.639932	0.319966	+	0.947429i	$0.396329\pi$
$42$	0	0
$43$	326.000	1.15615	0.578076	−	0.815983i	$-0.303804\pi$
$43$	326.000	1.15615	0.578076	+	0.815983i	$0.303804\pi$
$44$	−36.0000	−0.123346
$45$	135.000	0.447214
$46$	168.000	0.538484
$47$	138.000	0.428284	0.214142	−	0.976802i	$-0.431304\pi$
$47$	138.000	0.428284	0.214142	+	0.976802i	$0.431304\pi$
$48$	48.0000	0.144338
$49$	0	0
$50$	−200.000	−0.565685
$51$	252.000	0.691903
$52$	352.000	0.938723
$53$	639.000	1.65610	0.828051	−	0.560653i	$-0.189450\pi$
$53$	639.000	1.65610	0.828051	+	0.560653i	$0.189450\pi$
$54$	−54.0000	−0.136083
$55$	−135.000	−0.330971
$56$	0	0
$57$	−312.000	−0.725007
$58$	−102.000	−0.230918
$59$	−159.000	−0.350848	−0.175424	−	0.984493i	$-0.556130\pi$
$59$	−159.000	−0.350848	−0.175424	+	0.984493i	$0.556130\pi$
$60$	180.000	0.387298
$61$	−722.000	−1.51545	−0.757726	−	0.652572i	$-0.773690\pi$
$61$	−722.000	−1.51545	−0.757726	+	0.652572i	$0.773690\pi$
$62$	370.000	0.757904
$63$	0	0
$64$	64.0000	0.125000
$65$	1320.00	2.51886
$66$	54.0000	0.100711
$67$	−166.000	−0.302688	−0.151344	−	0.988481i	$-0.548360\pi$
$67$	−166.000	−0.302688	−0.151344	+	0.988481i	$0.548360\pi$
$68$	336.000	0.599206
$69$	−252.000	−0.439670
$70$	0	0
$71$	1086.00	1.81527	0.907637	−	0.419755i	$-0.137884\pi$
$71$	1086.00	1.81527	0.907637	+	0.419755i	$0.137884\pi$
$72$	−72.0000	−0.117851
$73$	−218.000	−0.349520	−0.174760	−	0.984611i	$-0.555915\pi$
$73$	−218.000	−0.349520	−0.174760	+	0.984611i	$0.555915\pi$
$74$	−88.0000	−0.138240
$75$	300.000	0.461880
$76$	−416.000	−0.627875
$77$	0	0
$78$	−528.000	−0.766464
$79$	−583.000	−0.830286	−0.415143	−	0.909756i	$-0.636269\pi$
$79$	−583.000	−0.830286	−0.415143	+	0.909756i	$0.636269\pi$
$80$	240.000	0.335410
$81$	81.0000	0.111111
$82$	−336.000	−0.452500
$83$	597.000	0.789509	0.394755	−	0.918787i	$-0.370830\pi$
$83$	597.000	0.789509	0.394755	+	0.918787i	$0.370830\pi$
$84$	0	0
$85$	1260.00	1.60784
$86$	−652.000	−0.817523
$87$	153.000	0.188544
$88$	72.0000	0.0872185
$89$	1038.00	1.23627	0.618134	−	0.786073i	$-0.287889\pi$
$89$	1038.00	1.23627	0.618134	+	0.786073i	$0.287889\pi$
$90$	−270.000	−0.316228
$91$	0	0
$92$	−336.000	−0.380765
$93$	−555.000	−0.618826
$94$	−276.000	−0.302843
$95$	−1560.00	−1.68476
$96$	−96.0000	−0.102062
$97$	169.000	0.176901	0.0884503	−	0.996081i	$-0.471809\pi$
$97$	169.000	0.176901	0.0884503	+	0.996081i	$0.471809\pi$
$98$	0	0
$99$	−81.0000	−0.0822304
$100$	400.000	0.400000
$101$	−642.000	−0.632489	−0.316244	−	0.948678i	$-0.602422\pi$
$101$	−642.000	−0.632489	−0.316244	+	0.948678i	$0.602422\pi$
$102$	−504.000	−0.489249
$103$	−464.000	−0.443876	−0.221938	−	0.975061i	$-0.571238\pi$
$103$	−464.000	−0.443876	−0.221938	+	0.975061i	$0.571238\pi$
$104$	−704.000	−0.663778
$105$	0	0
$106$	−1278.00	−1.17104
$107$	393.000	0.355072	0.177536	−	0.984114i	$-0.443187\pi$
$107$	393.000	0.355072	0.177536	+	0.984114i	$0.443187\pi$
$108$	108.000	0.0962250
$109$	14.0000	0.0123024	0.00615118	−	0.999981i	$-0.498042\pi$
$109$	14.0000	0.0123024	0.00615118	+	0.999981i	$0.498042\pi$
$110$	270.000	0.234032
$111$	132.000	0.112873
$112$	0	0
$113$	−2184.00	−1.81817	−0.909086	−	0.416608i	$-0.863219\pi$
$113$	−2184.00	−1.81817	−0.909086	+	0.416608i	$0.863219\pi$
$114$	624.000	0.512657
$115$	−1260.00	−1.02170
$116$	204.000	0.163284
$117$	792.000	0.625816
$118$	318.000	0.248087
$119$	0	0
$120$	−360.000	−0.273861
$121$	−1250.00	−0.939144
$122$	1444.00	1.07159
$123$	504.000	0.369465
$124$	−740.000	−0.535919
$125$	−375.000	−0.268328
$126$	0	0
$127$	−373.000	−0.260617	−0.130309	−	0.991473i	$-0.541597\pi$
$127$	−373.000	−0.260617	−0.130309	+	0.991473i	$0.541597\pi$
$128$	−128.000	−0.0883883
$129$	978.000	0.667505
$130$	−2640.00	−1.78110
$131$	1173.00	0.782332	0.391166	−	0.920320i	$-0.372072\pi$
$131$	1173.00	0.782332	0.391166	+	0.920320i	$0.372072\pi$
$132$	−108.000	−0.0712136
$133$	0	0
$134$	332.000	0.214033
$135$	405.000	0.258199
$136$	−672.000	−0.423702
$137$	30.0000	0.0187086	0.00935428	−	0.999956i	$-0.497022\pi$
$137$	30.0000	0.0187086	0.00935428	+	0.999956i	$0.497022\pi$
$138$	504.000	0.310894
$139$	82.0000	0.0500370	0.0250185	−	0.999687i	$-0.492036\pi$
$139$	82.0000	0.0500370	0.0250185	+	0.999687i	$0.492036\pi$
$140$	0	0
$141$	414.000	0.247270
$142$	−2172.00	−1.28359
$143$	−792.000	−0.463149
$144$	144.000	0.0833333
$145$	765.000	0.438136
$146$	436.000	0.247148
$147$	0	0
$148$	176.000	0.0977507
$149$	−1434.00	−0.788442	−0.394221	−	0.919016i	$-0.628986\pi$
$149$	−1434.00	−0.788442	−0.394221	+	0.919016i	$0.628986\pi$
$150$	−600.000	−0.326599
$151$	−2671.00	−1.43949	−0.719745	−	0.694239i	$-0.755741\pi$
$151$	−2671.00	−1.43949	−0.719745	+	0.694239i	$0.755741\pi$
$152$	832.000	0.443974
$153$	756.000	0.399470
$154$	0	0
$155$	−2775.00	−1.43802
$156$	1056.00	0.541972
$157$	−2252.00	−1.14477	−0.572386	−	0.819984i	$-0.693982\pi$
$157$	−2252.00	−1.14477	−0.572386	+	0.819984i	$0.693982\pi$
$158$	1166.00	0.587101
$159$	1917.00	0.956151
$160$	−480.000	−0.237171
$161$	0	0
$162$	−162.000	−0.0785674
$163$	1676.00	0.805365	0.402682	−	0.915340i	$-0.368078\pi$
$163$	1676.00	0.805365	0.402682	+	0.915340i	$0.368078\pi$
$164$	672.000	0.319966
$165$	−405.000	−0.191086
$166$	−1194.00	−0.558267
$167$	−3030.00	−1.40400	−0.702001	−	0.712176i	$-0.747710\pi$
$167$	−3030.00	−1.40400	−0.702001	+	0.712176i	$0.747710\pi$
$168$	0	0
$169$	5547.00	2.52481
$170$	−2520.00	−1.13691
$171$	−936.000	−0.418583
$172$	1304.00	0.578076
$173$	−3438.00	−1.51090	−0.755452	−	0.655204i	$-0.772583\pi$
$173$	−3438.00	−1.51090	−0.755452	+	0.655204i	$0.772583\pi$
$174$	−306.000	−0.133321
$175$	0	0
$176$	−144.000	−0.0616728
$177$	−477.000	−0.202562
$178$	−2076.00	−0.874173
$179$	−1212.00	−0.506085	−0.253042	−	0.967455i	$-0.581431\pi$
$179$	−1212.00	−0.506085	−0.253042	+	0.967455i	$0.581431\pi$
$180$	540.000	0.223607
$181$	−3032.00	−1.24512	−0.622560	−	0.782572i	$-0.713907\pi$
$181$	−3032.00	−1.24512	−0.622560	+	0.782572i	$0.713907\pi$
$182$	0	0
$183$	−2166.00	−0.874947
$184$	672.000	0.269242
$185$	660.000	0.262293
$186$	1110.00	0.437576
$187$	−756.000	−0.295637
$188$	552.000	0.214142
$189$	0	0
$190$	3120.00	1.19131
$191$	2520.00	0.954664	0.477332	−	0.878723i	$-0.341604\pi$
$191$	2520.00	0.954664	0.477332	+	0.878723i	$0.341604\pi$
$192$	192.000	0.0721688
$193$	365.000	0.136131	0.0680655	−	0.997681i	$-0.478317\pi$
$193$	365.000	0.136131	0.0680655	+	0.997681i	$0.478317\pi$
$194$	−338.000	−0.125088
$195$	3960.00	1.45426
$196$	0	0
$197$	−1590.00	−0.575040	−0.287520	−	0.957775i	$-0.592831\pi$
$197$	−1590.00	−0.575040	−0.287520	+	0.957775i	$0.592831\pi$
$198$	162.000	0.0581456
$199$	5380.00	1.91647	0.958236	−	0.285977i	$-0.0923182\pi$
$199$	5380.00	1.91647	0.958236	+	0.285977i	$0.0923182\pi$
$200$	−800.000	−0.282843
$201$	−498.000	−0.174757
$202$	1284.00	0.447237
$203$	0	0
$204$	1008.00	0.345952
$205$	2520.00	0.858558
$206$	928.000	0.313868
$207$	−756.000	−0.253844
$208$	1408.00	0.469362
$209$	936.000	0.309782
$210$	0	0
$211$	−5362.00	−1.74946	−0.874728	−	0.484614i	$-0.838960\pi$
$211$	−5362.00	−1.74946	−0.874728	+	0.484614i	$0.838960\pi$
$212$	2556.00	0.828051
$213$	3258.00	1.04805
$214$	−786.000	−0.251074
$215$	4890.00	1.55114
$216$	−216.000	−0.0680414
$217$	0	0
$218$	−28.0000	−0.00869908
$219$	−654.000	−0.201796
$220$	−540.000	−0.165485
$221$	7392.00	2.24995
$222$	−264.000	−0.0798132
$223$	1573.00	0.472358	0.236179	−	0.971710i	$-0.424105\pi$
$223$	1573.00	0.472358	0.236179	+	0.971710i	$0.424105\pi$
$224$	0	0
$225$	900.000	0.266667
$226$	4368.00	1.28564
$227$	−921.000	−0.269290	−0.134645	−	0.990894i	$-0.542989\pi$
$227$	−921.000	−0.269290	−0.134645	+	0.990894i	$0.542989\pi$
$228$	−1248.00	−0.362504
$229$	−4052.00	−1.16927	−0.584637	−	0.811295i	$-0.698763\pi$
$229$	−4052.00	−1.16927	−0.584637	+	0.811295i	$0.698763\pi$
$230$	2520.00	0.722452
$231$	0	0
$232$	−408.000	−0.115459
$233$	−468.000	−0.131587	−0.0657933	−	0.997833i	$-0.520958\pi$
$233$	−468.000	−0.131587	−0.0657933	+	0.997833i	$0.520958\pi$
$234$	−1584.00	−0.442518
$235$	2070.00	0.574604
$236$	−636.000	−0.175424
$237$	−1749.00	−0.479366
$238$	0	0
$239$	4932.00	1.33483	0.667415	−	0.744686i	$-0.267401\pi$
$239$	4932.00	1.33483	0.667415	+	0.744686i	$0.267401\pi$
$240$	720.000	0.193649
$241$	1537.00	0.410817	0.205408	−	0.978676i	$-0.434148\pi$
$241$	1537.00	0.410817	0.205408	+	0.978676i	$0.434148\pi$
$242$	2500.00	0.664075
$243$	243.000	0.0641500
$244$	−2888.00	−0.757726
$245$	0	0
$246$	−1008.00	−0.261251
$247$	−9152.00	−2.35760
$248$	1480.00	0.378952
$249$	1791.00	0.455823
$250$	750.000	0.189737
$251$	−5319.00	−1.33758	−0.668789	−	0.743452i	$-0.733187\pi$
$251$	−5319.00	−1.33758	−0.668789	+	0.743452i	$0.733187\pi$
$252$	0	0
$253$	756.000	0.187863
$254$	746.000	0.184284
$255$	3780.00	0.928285
$256$	256.000	0.0625000
$257$	−5346.00	−1.29757	−0.648783	−	0.760974i	$-0.724722\pi$
$257$	−5346.00	−1.29757	−0.648783	+	0.760974i	$0.724722\pi$
$258$	−1956.00	−0.471997
$259$	0	0
$260$	5280.00	1.25943
$261$	459.000	0.108856
$262$	−2346.00	−0.553192
$263$	774.000	0.181471	0.0907355	−	0.995875i	$-0.471078\pi$
$263$	774.000	0.181471	0.0907355	+	0.995875i	$0.471078\pi$
$264$	216.000	0.0503556
$265$	9585.00	2.22189
$266$	0	0
$267$	3114.00	0.713759
$268$	−664.000	−0.151344
$269$	−2415.00	−0.547380	−0.273690	−	0.961818i	$-0.588244\pi$
$269$	−2415.00	−0.547380	−0.273690	+	0.961818i	$0.588244\pi$
$270$	−810.000	−0.182574
$271$	475.000	0.106473	0.0532365	−	0.998582i	$-0.483046\pi$
$271$	475.000	0.106473	0.0532365	+	0.998582i	$0.483046\pi$
$272$	1344.00	0.299603
$273$	0	0
$274$	−60.0000	−0.0132290
$275$	−900.000	−0.197353
$276$	−1008.00	−0.219835
$277$	3728.00	0.808642	0.404321	−	0.914617i	$-0.367508\pi$
$277$	3728.00	0.808642	0.404321	+	0.914617i	$0.367508\pi$
$278$	−164.000	−0.0353815
$279$	−1665.00	−0.357279
$280$	0	0
$281$	1602.00	0.340097	0.170049	−	0.985436i	$-0.445608\pi$
$281$	1602.00	0.340097	0.170049	+	0.985436i	$0.445608\pi$
$282$	−828.000	−0.174846
$283$	−686.000	−0.144094	−0.0720468	−	0.997401i	$-0.522953\pi$
$283$	−686.000	−0.144094	−0.0720468	+	0.997401i	$0.522953\pi$
$284$	4344.00	0.907637
$285$	−4680.00	−0.972699
$286$	1584.00	0.327496
$287$	0	0
$288$	−288.000	−0.0589256
$289$	2143.00	0.436190
$290$	−1530.00	−0.309809
$291$	507.000	0.102134
$292$	−872.000	−0.174760
$293$	1101.00	0.219526	0.109763	−	0.993958i	$-0.464991\pi$
$293$	1101.00	0.219526	0.109763	+	0.993958i	$0.464991\pi$
$294$	0	0
$295$	−2385.00	−0.470712
$296$	−352.000	−0.0691202
$297$	−243.000	−0.0474757
$298$	2868.00	0.557513
$299$	−7392.00	−1.42973
$300$	1200.00	0.230940
$301$	0	0
$302$	5342.00	1.01787
$303$	−1926.00	−0.365168
$304$	−1664.00	−0.313937
$305$	−10830.0	−2.03319
$306$	−1512.00	−0.282468
$307$	−2780.00	−0.516818	−0.258409	−	0.966036i	$-0.583198\pi$
$307$	−2780.00	−0.516818	−0.258409	+	0.966036i	$0.583198\pi$
$308$	0	0
$309$	−1392.00	−0.256272
$310$	5550.00	1.01683
$311$	−4296.00	−0.783292	−0.391646	−	0.920116i	$-0.628094\pi$
$311$	−4296.00	−0.783292	−0.391646	+	0.920116i	$0.628094\pi$
$312$	−2112.00	−0.383232
$313$	−5489.00	−0.991235	−0.495618	−	0.868541i	$-0.665058\pi$
$313$	−5489.00	−0.991235	−0.495618	+	0.868541i	$0.665058\pi$
$314$	4504.00	0.809476
$315$	0	0
$316$	−2332.00	−0.415143
$317$	4491.00	0.795709	0.397854	−	0.917449i	$-0.369755\pi$
$317$	4491.00	0.795709	0.397854	+	0.917449i	$0.369755\pi$
$318$	−3834.00	−0.676101
$319$	−459.000	−0.0805613
$320$	960.000	0.167705
$321$	1179.00	0.205001
$322$	0	0
$323$	−8736.00	−1.50490
$324$	324.000	0.0555556
$325$	8800.00	1.50196
$326$	−3352.00	−0.569479
$327$	42.0000	0.00710277
$328$	−1344.00	−0.226250
$329$	0	0
$330$	810.000	0.135118
$331$	−3964.00	−0.658251	−0.329126	−	0.944286i	$-0.606754\pi$
$331$	−3964.00	−0.658251	−0.329126	+	0.944286i	$0.606754\pi$
$332$	2388.00	0.394755
$333$	396.000	0.0651672
$334$	6060.00	0.992780
$335$	−2490.00	−0.406099
$336$	0	0
$337$	161.000	0.0260244	0.0130122	−	0.999915i	$-0.495858\pi$
$337$	161.000	0.0260244	0.0130122	+	0.999915i	$0.495858\pi$
$338$	−11094.0	−1.78531
$339$	−6552.00	−1.04972
$340$	5040.00	0.803919
$341$	1665.00	0.264413
$342$	1872.00	0.295983
$343$	0	0
$344$	−2608.00	−0.408761
$345$	−3780.00	−0.589879
$346$	6876.00	1.06837
$347$	−5916.00	−0.915238	−0.457619	−	0.889148i	$-0.651298\pi$
$347$	−5916.00	−0.915238	−0.457619	+	0.889148i	$0.651298\pi$
$348$	612.000	0.0942720
$349$	142.000	0.0217796	0.0108898	−	0.999941i	$-0.496534\pi$
$349$	142.000	0.0217796	0.0108898	+	0.999941i	$0.496534\pi$
$350$	0	0
$351$	2376.00	0.361315
$352$	288.000	0.0436092
$353$	4440.00	0.669454	0.334727	−	0.942315i	$-0.391356\pi$
$353$	4440.00	0.669454	0.334727	+	0.942315i	$0.391356\pi$
$354$	954.000	0.143233
$355$	16290.0	2.43545
$356$	4152.00	0.618134
$357$	0	0
$358$	2424.00	0.357856
$359$	2286.00	0.336074	0.168037	−	0.985781i	$-0.446257\pi$
$359$	2286.00	0.336074	0.168037	+	0.985781i	$0.446257\pi$
$360$	−1080.00	−0.158114
$361$	3957.00	0.576906
$362$	6064.00	0.880433
$363$	−3750.00	−0.542215
$364$	0	0
$365$	−3270.00	−0.468930
$366$	4332.00	0.618681
$367$	2869.00	0.408067	0.204033	−	0.978964i	$-0.434595\pi$
$367$	2869.00	0.408067	0.204033	+	0.978964i	$0.434595\pi$
$368$	−1344.00	−0.190383
$369$	1512.00	0.213311
$370$	−1320.00	−0.185469
$371$	0	0
$372$	−2220.00	−0.309413
$373$	−3064.00	−0.425330	−0.212665	−	0.977125i	$-0.568214\pi$
$373$	−3064.00	−0.425330	−0.212665	+	0.977125i	$0.568214\pi$
$374$	1512.00	0.209047
$375$	−1125.00	−0.154919
$376$	−1104.00	−0.151421
$377$	4488.00	0.613113
$378$	0	0
$379$	−6040.00	−0.818612	−0.409306	−	0.912397i	$-0.634229\pi$
$379$	−6040.00	−0.818612	−0.409306	+	0.912397i	$0.634229\pi$
$380$	−6240.00	−0.842382
$381$	−1119.00	−0.150467
$382$	−5040.00	−0.675049
$383$	−1842.00	−0.245749	−0.122874	−	0.992422i	$-0.539211\pi$
$383$	−1842.00	−0.245749	−0.122874	+	0.992422i	$0.539211\pi$
$384$	−384.000	−0.0510310
$385$	0	0
$386$	−730.000	−0.0962591
$387$	2934.00	0.385384
$388$	676.000	0.0884503
$389$	7830.00	1.02056	0.510279	−	0.860009i	$-0.329542\pi$
$389$	7830.00	1.02056	0.510279	+	0.860009i	$0.329542\pi$
$390$	−7920.00	−1.02832
$391$	−7056.00	−0.912627
$392$	0	0
$393$	3519.00	0.451680
$394$	3180.00	0.406614
$395$	−8745.00	−1.11395
$396$	−324.000	−0.0411152
$397$	14764.0	1.86646	0.933229	−	0.359282i	$-0.116978\pi$
$397$	14764.0	1.86646	0.933229	+	0.359282i	$0.116978\pi$
$398$	−10760.0	−1.35515
$399$	0	0
$400$	1600.00	0.200000
$401$	6264.00	0.780073	0.390036	−	0.920799i	$-0.372462\pi$
$401$	6264.00	0.780073	0.390036	+	0.920799i	$0.372462\pi$
$402$	996.000	0.123572
$403$	−16280.0	−2.01232
$404$	−2568.00	−0.316244
$405$	1215.00	0.149071
$406$	0	0
$407$	−396.000	−0.0482285
$408$	−2016.00	−0.244625
$409$	−4751.00	−0.574381	−0.287191	−	0.957873i	$-0.592721\pi$
$409$	−4751.00	−0.574381	−0.287191	+	0.957873i	$0.592721\pi$
$410$	−5040.00	−0.607092
$411$	90.0000	0.0108014
$412$	−1856.00	−0.221938
$413$	0	0
$414$	1512.00	0.179495
$415$	8955.00	1.05924
$416$	−2816.00	−0.331889
$417$	246.000	0.0288889
$418$	−1872.00	−0.219049
$419$	4704.00	0.548462	0.274231	−	0.961664i	$-0.411577\pi$
$419$	4704.00	0.548462	0.274231	+	0.961664i	$0.411577\pi$
$420$	0	0
$421$	−4474.00	−0.517932	−0.258966	−	0.965886i	$-0.583382\pi$
$421$	−4474.00	−0.517932	−0.258966	+	0.965886i	$0.583382\pi$
$422$	10724.0	1.23705
$423$	1242.00	0.142761
$424$	−5112.00	−0.585520
$425$	8400.00	0.958729
$426$	−6516.00	−0.741083
$427$	0	0
$428$	1572.00	0.177536
$429$	−2376.00	−0.267399
$430$	−9780.00	−1.09682
$431$	−12804.0	−1.43097	−0.715484	−	0.698629i	$-0.753794\pi$
$431$	−12804.0	−1.43097	−0.715484	+	0.698629i	$0.753794\pi$
$432$	432.000	0.0481125
$433$	5074.00	0.563143	0.281571	−	0.959540i	$-0.409144\pi$
$433$	5074.00	0.563143	0.281571	+	0.959540i	$0.409144\pi$
$434$	0	0
$435$	2295.00	0.252958
$436$	56.0000	0.00615118
$437$	8736.00	0.956292
$438$	1308.00	0.142691
$439$	1267.00	0.137746	0.0688731	−	0.997625i	$-0.478060\pi$
$439$	1267.00	0.137746	0.0688731	+	0.997625i	$0.478060\pi$
$440$	1080.00	0.117016
$441$	0	0
$442$	−14784.0	−1.59096
$443$	−6933.00	−0.743559	−0.371780	−	0.928321i	$-0.621252\pi$
$443$	−6933.00	−0.743559	−0.371780	+	0.928321i	$0.621252\pi$
$444$	528.000	0.0564364
$445$	15570.0	1.65863
$446$	−3146.00	−0.334008
$447$	−4302.00	−0.455207
$448$	0	0
$449$	11688.0	1.22849	0.614244	−	0.789116i	$-0.289461\pi$
$449$	11688.0	1.22849	0.614244	+	0.789116i	$0.289461\pi$
$450$	−1800.00	−0.188562
$451$	−1512.00	−0.157865
$452$	−8736.00	−0.909086
$453$	−8013.00	−0.831090
$454$	1842.00	0.190417
$455$	0	0
$456$	2496.00	0.256329
$457$	551.000	0.0563998	0.0281999	−	0.999602i	$-0.491023\pi$
$457$	551.000	0.0563998	0.0281999	+	0.999602i	$0.491023\pi$
$458$	8104.00	0.826801
$459$	2268.00	0.230634
$460$	−5040.00	−0.510850
$461$	−13386.0	−1.35238	−0.676191	−	0.736726i	$-0.736371\pi$
$461$	−13386.0	−1.35238	−0.676191	+	0.736726i	$0.736371\pi$
$462$	0	0
$463$	−6376.00	−0.639995	−0.319998	−	0.947418i	$-0.603682\pi$
$463$	−6376.00	−0.639995	−0.319998	+	0.947418i	$0.603682\pi$
$464$	816.000	0.0816419
$465$	−8325.00	−0.830242
$466$	936.000	0.0930458
$467$	−5700.00	−0.564806	−0.282403	−	0.959296i	$-0.591132\pi$
$467$	−5700.00	−0.564806	−0.282403	+	0.959296i	$0.591132\pi$
$468$	3168.00	0.312908
$469$	0	0
$470$	−4140.00	−0.406306
$471$	−6756.00	−0.660934
$472$	1272.00	0.124044
$473$	−2934.00	−0.285212
$474$	3498.00	0.338963
$475$	−10400.0	−1.00460
$476$	0	0
$477$	5751.00	0.552034
$478$	−9864.00	−0.943868
$479$	−19794.0	−1.88812	−0.944062	−	0.329769i	$-0.893029\pi$
$479$	−19794.0	−1.88812	−0.944062	+	0.329769i	$0.893029\pi$
$480$	−1440.00	−0.136931
$481$	3872.00	0.367044
$482$	−3074.00	−0.290491
$483$	0	0
$484$	−5000.00	−0.469572
$485$	2535.00	0.237337
$486$	−486.000	−0.0453609
$487$	15935.0	1.48272	0.741359	−	0.671109i	$-0.234182\pi$
$487$	15935.0	1.48272	0.741359	+	0.671109i	$0.234182\pi$
$488$	5776.00	0.535794
$489$	5028.00	0.464978
$490$	0	0
$491$	9963.00	0.915731	0.457865	−	0.889021i	$-0.348614\pi$
$491$	9963.00	0.915731	0.457865	+	0.889021i	$0.348614\pi$
$492$	2016.00	0.184732
$493$	4284.00	0.391362
$494$	18304.0	1.66708
$495$	−1215.00	−0.110324
$496$	−2960.00	−0.267960
$497$	0	0
$498$	−3582.00	−0.322316
$499$	19142.0	1.71726	0.858631	−	0.512594i	$-0.171316\pi$
$499$	19142.0	1.71726	0.858631	+	0.512594i	$0.171316\pi$
$500$	−1500.00	−0.134164
$501$	−9090.00	−0.810601
$502$	10638.0	0.945811
$503$	12192.0	1.08074	0.540372	−	0.841426i	$-0.318283\pi$
$503$	12192.0	1.08074	0.540372	+	0.841426i	$0.318283\pi$
$504$	0	0
$505$	−9630.00	−0.848573
$506$	−1512.00	−0.132839
$507$	16641.0	1.45770
$508$	−1492.00	−0.130309
$509$	19809.0	1.72499	0.862494	−	0.506068i	$-0.168902\pi$
$509$	19809.0	1.72499	0.862494	+	0.506068i	$0.168902\pi$
$510$	−7560.00	−0.656397
$511$	0	0
$512$	−512.000	−0.0441942
$513$	−2808.00	−0.241669
$514$	10692.0	0.917517
$515$	−6960.00	−0.595523
$516$	3912.00	0.333752
$517$	−1242.00	−0.105654
$518$	0	0
$519$	−10314.0	−0.872321
$520$	−10560.0	−0.890551
$521$	1794.00	0.150857	0.0754286	−	0.997151i	$-0.475968\pi$
$521$	1794.00	0.150857	0.0754286	+	0.997151i	$0.475968\pi$
$522$	−918.000	−0.0769727
$523$	6448.00	0.539104	0.269552	−	0.962986i	$-0.413124\pi$
$523$	6448.00	0.539104	0.269552	+	0.962986i	$0.413124\pi$
$524$	4692.00	0.391166
$525$	0	0
$526$	−1548.00	−0.128319
$527$	−15540.0	−1.28450
$528$	−432.000	−0.0356068
$529$	−5111.00	−0.420071
$530$	−19170.0	−1.57112
$531$	−1431.00	−0.116949
$532$	0	0
$533$	14784.0	1.20144
$534$	−6228.00	−0.504704
$535$	5895.00	0.476380
$536$	1328.00	0.107017
$537$	−3636.00	−0.292188
$538$	4830.00	0.387056
$539$	0	0
$540$	1620.00	0.129099
$541$	7262.00	0.577112	0.288556	−	0.957463i	$-0.406825\pi$
$541$	7262.00	0.577112	0.288556	+	0.957463i	$0.406825\pi$
$542$	−950.000	−0.0752878
$543$	−9096.00	−0.718871
$544$	−2688.00	−0.211851
$545$	210.000	0.0165053
$546$	0	0
$547$	14204.0	1.11027	0.555136	−	0.831759i	$-0.312666\pi$
$547$	14204.0	1.11027	0.555136	+	0.831759i	$0.312666\pi$
$548$	120.000	0.00935428
$549$	−6498.00	−0.505151
$550$	1800.00	0.139550
$551$	−5304.00	−0.410087
$552$	2016.00	0.155447
$553$	0	0
$554$	−7456.00	−0.571796
$555$	1980.00	0.151435
$556$	328.000	0.0250185
$557$	15825.0	1.20382	0.601909	−	0.798565i	$-0.294407\pi$
$557$	15825.0	1.20382	0.601909	+	0.798565i	$0.294407\pi$
$558$	3330.00	0.252635
$559$	28688.0	2.17061
$560$	0	0
$561$	−2268.00	−0.170686
$562$	−3204.00	−0.240485
$563$	−1059.00	−0.0792745	−0.0396372	−	0.999214i	$-0.512620\pi$
$563$	−1059.00	−0.0792745	−0.0396372	+	0.999214i	$0.512620\pi$
$564$	1656.00	0.123635
$565$	−32760.0	−2.43933
$566$	1372.00	0.101890
$567$	0	0
$568$	−8688.00	−0.641796
$569$	3960.00	0.291761	0.145880	−	0.989302i	$-0.453399\pi$
$569$	3960.00	0.291761	0.145880	+	0.989302i	$0.453399\pi$
$570$	9360.00	0.687802
$571$	−2530.00	−0.185424	−0.0927121	−	0.995693i	$-0.529554\pi$
$571$	−2530.00	−0.185424	−0.0927121	+	0.995693i	$0.529554\pi$
$572$	−3168.00	−0.231575
$573$	7560.00	0.551175
$574$	0	0
$575$	−8400.00	−0.609225
$576$	576.000	0.0416667
$577$	−11831.0	−0.853607	−0.426803	−	0.904344i	$-0.640360\pi$
$577$	−11831.0	−0.853607	−0.426803	+	0.904344i	$0.640360\pi$
$578$	−4286.00	−0.308433
$579$	1095.00	0.0785952
$580$	3060.00	0.219068
$581$	0	0
$582$	−1014.00	−0.0722193
$583$	−5751.00	−0.408546
$584$	1744.00	0.123574
$585$	11880.0	0.839620
$586$	−2202.00	−0.155228
$587$	−4809.00	−0.338141	−0.169070	−	0.985604i	$-0.554077\pi$
$587$	−4809.00	−0.338141	−0.169070	+	0.985604i	$0.554077\pi$
$588$	0	0
$589$	19240.0	1.34596
$590$	4770.00	0.332844
$591$	−4770.00	−0.331999
$592$	704.000	0.0488754
$593$	−21804.0	−1.50992	−0.754960	−	0.655770i	$-0.772344\pi$
$593$	−21804.0	−1.50992	−0.754960	+	0.655770i	$0.772344\pi$
$594$	486.000	0.0335704
$595$	0	0
$596$	−5736.00	−0.394221
$597$	16140.0	1.10648
$598$	14784.0	1.01097
$599$	14166.0	0.966289	0.483144	−	0.875541i	$-0.339495\pi$
$599$	14166.0	0.966289	0.483144	+	0.875541i	$0.339495\pi$
$600$	−2400.00	−0.163299
$601$	−5891.00	−0.399832	−0.199916	−	0.979813i	$-0.564067\pi$
$601$	−5891.00	−0.399832	−0.199916	+	0.979813i	$0.564067\pi$
$602$	0	0
$603$	−1494.00	−0.100896
$604$	−10684.0	−0.719745
$605$	−18750.0	−1.25999
$606$	3852.00	0.258213
$607$	2737.00	0.183017	0.0915086	−	0.995804i	$-0.470831\pi$
$607$	2737.00	0.183017	0.0915086	+	0.995804i	$0.470831\pi$
$608$	3328.00	0.221987
$609$	0	0
$610$	21660.0	1.43768
$611$	12144.0	0.804081
$612$	3024.00	0.199735
$613$	−26188.0	−1.72549	−0.862743	−	0.505642i	$-0.831256\pi$
$613$	−26188.0	−1.72549	−0.862743	+	0.505642i	$0.831256\pi$
$614$	5560.00	0.365445
$615$	7560.00	0.495689
$616$	0	0
$617$	−2358.00	−0.153857	−0.0769283	−	0.997037i	$-0.524511\pi$
$617$	−2358.00	−0.153857	−0.0769283	+	0.997037i	$0.524511\pi$
$618$	2784.00	0.181212
$619$	−13766.0	−0.893865	−0.446932	−	0.894568i	$-0.647484\pi$
$619$	−13766.0	−0.893865	−0.446932	+	0.894568i	$0.647484\pi$
$620$	−11100.0	−0.719011
$621$	−2268.00	−0.146557
$622$	8592.00	0.553871
$623$	0	0
$624$	4224.00	0.270986
$625$	−18125.0	−1.16000
$626$	10978.0	0.700909
$627$	2808.00	0.178853
$628$	−9008.00	−0.572386
$629$	3696.00	0.234291
$630$	0	0
$631$	21287.0	1.34298	0.671491	−	0.741012i	$-0.265654\pi$
$631$	21287.0	1.34298	0.671491	+	0.741012i	$0.265654\pi$
$632$	4664.00	0.293551
$633$	−16086.0	−1.01005
$634$	−8982.00	−0.562651
$635$	−5595.00	−0.349655
$636$	7668.00	0.478075
$637$	0	0
$638$	918.000	0.0569655
$639$	9774.00	0.605091
$640$	−1920.00	−0.118585
$641$	21426.0	1.32024	0.660122	−	0.751159i	$-0.270505\pi$
$641$	21426.0	1.32024	0.660122	+	0.751159i	$0.270505\pi$
$642$	−2358.00	−0.144958
$643$	−9962.00	−0.610984	−0.305492	−	0.952195i	$-0.598821\pi$
$643$	−9962.00	−0.610984	−0.305492	+	0.952195i	$0.598821\pi$
$644$	0	0
$645$	14670.0	0.895551
$646$	17472.0	1.06413
$647$	18174.0	1.10432	0.552159	−	0.833739i	$-0.313804\pi$
$647$	18174.0	1.10432	0.552159	+	0.833739i	$0.313804\pi$
$648$	−648.000	−0.0392837
$649$	1431.00	0.0865511
$650$	−17600.0	−1.06204
$651$	0	0
$652$	6704.00	0.402682
$653$	−19167.0	−1.14864	−0.574321	−	0.818630i	$-0.694734\pi$
$653$	−19167.0	−1.14864	−0.574321	+	0.818630i	$0.694734\pi$
$654$	−84.0000	−0.00502242
$655$	17595.0	1.04961
$656$	2688.00	0.159983
$657$	−1962.00	−0.116507
$658$	0	0
$659$	13080.0	0.773178	0.386589	−	0.922252i	$-0.373653\pi$
$659$	13080.0	0.773178	0.386589	+	0.922252i	$0.373653\pi$
$660$	−1620.00	−0.0955431
$661$	15190.0	0.893831	0.446916	−	0.894576i	$-0.352522\pi$
$661$	15190.0	0.893831	0.446916	+	0.894576i	$0.352522\pi$
$662$	7928.00	0.465454
$663$	22176.0	1.29901
$664$	−4776.00	−0.279134
$665$	0	0
$666$	−792.000	−0.0460801
$667$	−4284.00	−0.248691
$668$	−12120.0	−0.702001
$669$	4719.00	0.272716
$670$	4980.00	0.287155
$671$	6498.00	0.373849
$672$	0	0
$673$	4397.00	0.251845	0.125923	−	0.992040i	$-0.459811\pi$
$673$	4397.00	0.251845	0.125923	+	0.992040i	$0.459811\pi$
$674$	−322.000	−0.0184020
$675$	2700.00	0.153960
$676$	22188.0	1.26240
$677$	−4029.00	−0.228725	−0.114363	−	0.993439i	$-0.536483\pi$
$677$	−4029.00	−0.228725	−0.114363	+	0.993439i	$0.536483\pi$
$678$	13104.0	0.742266
$679$	0	0
$680$	−10080.0	−0.568456
$681$	−2763.00	−0.155475
$682$	−3330.00	−0.186968
$683$	15021.0	0.841526	0.420763	−	0.907170i	$-0.361762\pi$
$683$	15021.0	0.841526	0.420763	+	0.907170i	$0.361762\pi$
$684$	−3744.00	−0.209292
$685$	450.000	0.0251002
$686$	0	0
$687$	−12156.0	−0.675081
$688$	5216.00	0.289038
$689$	56232.0	3.10924
$690$	7560.00	0.417108
$691$	13984.0	0.769865	0.384932	−	0.922945i	$-0.374225\pi$
$691$	13984.0	0.769865	0.384932	+	0.922945i	$0.374225\pi$
$692$	−13752.0	−0.755452
$693$	0	0
$694$	11832.0	0.647171
$695$	1230.00	0.0671317
$696$	−1224.00	−0.0666603
$697$	14112.0	0.766901
$698$	−284.000	−0.0154005
$699$	−1404.00	−0.0759716
$700$	0	0
$701$	−31053.0	−1.67312	−0.836559	−	0.547877i	$-0.815436\pi$
$701$	−31053.0	−1.67312	−0.836559	+	0.547877i	$0.815436\pi$
$702$	−4752.00	−0.255488
$703$	−4576.00	−0.245501
$704$	−576.000	−0.0308364
$705$	6210.00	0.331748
$706$	−8880.00	−0.473376
$707$	0	0
$708$	−1908.00	−0.101281
$709$	15086.0	0.799107	0.399553	−	0.916710i	$-0.369165\pi$
$709$	15086.0	0.799107	0.399553	+	0.916710i	$0.369165\pi$
$710$	−32580.0	−1.72212
$711$	−5247.00	−0.276762
$712$	−8304.00	−0.437086
$713$	15540.0	0.816238
$714$	0	0
$715$	−11880.0	−0.621380
$716$	−4848.00	−0.253042
$717$	14796.0	0.770665
$718$	−4572.00	−0.237640
$719$	6378.00	0.330820	0.165410	−	0.986225i	$-0.447105\pi$
$719$	6378.00	0.330820	0.165410	+	0.986225i	$0.447105\pi$
$720$	2160.00	0.111803
$721$	0	0
$722$	−7914.00	−0.407934
$723$	4611.00	0.237185
$724$	−12128.0	−0.622560
$725$	5100.00	0.261254
$726$	7500.00	0.383404
$727$	7363.00	0.375624	0.187812	−	0.982205i	$-0.439860\pi$
$727$	7363.00	0.375624	0.187812	+	0.982205i	$0.439860\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	6540.00	0.331584
$731$	27384.0	1.38555
$732$	−8664.00	−0.437474
$733$	−32810.0	−1.65329	−0.826647	−	0.562720i	$-0.809755\pi$
$733$	−32810.0	−1.65329	−0.826647	+	0.562720i	$0.809755\pi$
$734$	−5738.00	−0.288547
$735$	0	0
$736$	2688.00	0.134621
$737$	1494.00	0.0746706
$738$	−3024.00	−0.150833
$739$	−24034.0	−1.19635	−0.598177	−	0.801364i	$-0.704108\pi$
$739$	−24034.0	−1.19635	−0.598177	+	0.801364i	$0.704108\pi$
$740$	2640.00	0.131146
$741$	−27456.0	−1.36116
$742$	0	0
$743$	−8022.00	−0.396095	−0.198048	−	0.980192i	$-0.563460\pi$
$743$	−8022.00	−0.396095	−0.198048	+	0.980192i	$0.563460\pi$
$744$	4440.00	0.218788
$745$	−21510.0	−1.05781
$746$	6128.00	0.300753
$747$	5373.00	0.263170
$748$	−3024.00	−0.147819
$749$	0	0
$750$	2250.00	0.109545
$751$	29519.0	1.43431	0.717153	−	0.696916i	$-0.245445\pi$
$751$	29519.0	1.43431	0.717153	+	0.696916i	$0.245445\pi$
$752$	2208.00	0.107071
$753$	−15957.0	−0.772252
$754$	−8976.00	−0.433537
$755$	−40065.0	−1.93128
$756$	0	0
$757$	−3742.00	−0.179664	−0.0898318	−	0.995957i	$-0.528633\pi$
$757$	−3742.00	−0.179664	−0.0898318	+	0.995957i	$0.528633\pi$
$758$	12080.0	0.578846
$759$	2268.00	0.108463
$760$	12480.0	0.595654
$761$	10896.0	0.519027	0.259514	−	0.965739i	$-0.416438\pi$
$761$	10896.0	0.519027	0.259514	+	0.965739i	$0.416438\pi$
$762$	2238.00	0.106397
$763$	0	0
$764$	10080.0	0.477332
$765$	11340.0	0.535946
$766$	3684.00	0.173771
$767$	−13992.0	−0.658699
$768$	768.000	0.0360844
$769$	−17285.0	−0.810550	−0.405275	−	0.914195i	$-0.632824\pi$
$769$	−17285.0	−0.810550	−0.405275	+	0.914195i	$0.632824\pi$
$770$	0	0
$771$	−16038.0	−0.749150
$772$	1460.00	0.0680655
$773$	−11826.0	−0.550261	−0.275130	−	0.961407i	$-0.588721\pi$
$773$	−11826.0	−0.550261	−0.275130	+	0.961407i	$0.588721\pi$
$774$	−5868.00	−0.272508
$775$	−18500.0	−0.857470
$776$	−1352.00	−0.0625438
$777$	0	0
$778$	−15660.0	−0.721643
$779$	−17472.0	−0.803594
$780$	15840.0	0.727132
$781$	−9774.00	−0.447812
$782$	14112.0	0.645325
$783$	1377.00	0.0628480
$784$	0	0
$785$	−33780.0	−1.53587
$786$	−7038.00	−0.319386
$787$	−17714.0	−0.802333	−0.401166	−	0.916005i	$-0.631395\pi$
$787$	−17714.0	−0.802333	−0.401166	+	0.916005i	$0.631395\pi$
$788$	−6360.00	−0.287520
$789$	2322.00	0.104772
$790$	17490.0	0.787679
$791$	0	0
$792$	648.000	0.0290728
$793$	−63536.0	−2.84518
$794$	−29528.0	−1.31979
$795$	28755.0	1.28281
$796$	21520.0	0.958236
$797$	24939.0	1.10839	0.554194	−	0.832388i	$-0.313027\pi$
$797$	24939.0	1.10839	0.554194	+	0.832388i	$0.313027\pi$
$798$	0	0
$799$	11592.0	0.513261
$800$	−3200.00	−0.141421
$801$	9342.00	0.412089
$802$	−12528.0	−0.551595
$803$	1962.00	0.0862235
$804$	−1992.00	−0.0873786
$805$	0	0
$806$	32560.0	1.42292
$807$	−7245.00	−0.316030
$808$	5136.00	0.223619
$809$	−29064.0	−1.26309	−0.631543	−	0.775341i	$-0.717578\pi$
$809$	−29064.0	−1.26309	−0.631543	+	0.775341i	$0.717578\pi$
$810$	−2430.00	−0.105409
$811$	15370.0	0.665492	0.332746	−	0.943017i	$-0.392025\pi$
$811$	15370.0	0.665492	0.332746	+	0.943017i	$0.392025\pi$
$812$	0	0
$813$	1425.00	0.0614722
$814$	792.000	0.0341027
$815$	25140.0	1.08051
$816$	4032.00	0.172976
$817$	−33904.0	−1.45184
$818$	9502.00	0.406149
$819$	0	0
$820$	10080.0	0.429279
$821$	44031.0	1.87173	0.935866	−	0.352355i	$-0.114619\pi$
$821$	44031.0	1.87173	0.935866	+	0.352355i	$0.114619\pi$
$822$	−180.000	−0.00763774
$823$	−4192.00	−0.177550	−0.0887752	−	0.996052i	$-0.528295\pi$
$823$	−4192.00	−0.177550	−0.0887752	+	0.996052i	$0.528295\pi$
$824$	3712.00	0.156934
$825$	−2700.00	−0.113942
$826$	0	0
$827$	33195.0	1.39577	0.697886	−	0.716209i	$-0.254124\pi$
$827$	33195.0	1.39577	0.697886	+	0.716209i	$0.254124\pi$
$828$	−3024.00	−0.126922
$829$	−16448.0	−0.689098	−0.344549	−	0.938768i	$-0.611968\pi$
$829$	−16448.0	−0.689098	−0.344549	+	0.938768i	$0.611968\pi$
$830$	−17910.0	−0.748994
$831$	11184.0	0.466870
$832$	5632.00	0.234681
$833$	0	0
$834$	−492.000	−0.0204275
$835$	−45450.0	−1.88367
$836$	3744.00	0.154891
$837$	−4995.00	−0.206275
$838$	−9408.00	−0.387821
$839$	−16860.0	−0.693769	−0.346884	−	0.937908i	$-0.612760\pi$
$839$	−16860.0	−0.693769	−0.346884	+	0.937908i	$0.612760\pi$
$840$	0	0
$841$	−21788.0	−0.893354
$842$	8948.00	0.366233
$843$	4806.00	0.196355
$844$	−21448.0	−0.874728
$845$	83205.0	3.38738
$846$	−2484.00	−0.100948
$847$	0	0
$848$	10224.0	0.414025
$849$	−2058.00	−0.0831924
$850$	−16800.0	−0.677924
$851$	−3696.00	−0.148880
$852$	13032.0	0.524025
$853$	−29054.0	−1.16623	−0.583113	−	0.812391i	$-0.698165\pi$
$853$	−29054.0	−1.16623	−0.583113	+	0.812391i	$0.698165\pi$
$854$	0	0
$855$	−14040.0	−0.561588
$856$	−3144.00	−0.125537
$857$	−41958.0	−1.67241	−0.836207	−	0.548415i	$-0.815232\pi$
$857$	−41958.0	−1.67241	−0.836207	+	0.548415i	$0.815232\pi$
$858$	4752.00	0.189080
$859$	−5546.00	−0.220288	−0.110144	−	0.993916i	$-0.535131\pi$
$859$	−5546.00	−0.220288	−0.110144	+	0.993916i	$0.535131\pi$
$860$	19560.0	0.775570
$861$	0	0
$862$	25608.0	1.01185
$863$	−32538.0	−1.28344	−0.641719	−	0.766940i	$-0.721778\pi$
$863$	−32538.0	−1.28344	−0.641719	+	0.766940i	$0.721778\pi$
$864$	−864.000	−0.0340207
$865$	−51570.0	−2.02709
$866$	−10148.0	−0.398202
$867$	6429.00	0.251834
$868$	0	0
$869$	5247.00	0.204824
$870$	−4590.00	−0.178868
$871$	−14608.0	−0.568282
$872$	−112.000	−0.00434954
$873$	1521.00	0.0589668
$874$	−17472.0	−0.676200
$875$	0	0
$876$	−2616.00	−0.100898
$877$	32096.0	1.23581	0.617905	−	0.786253i	$-0.287982\pi$
$877$	32096.0	1.23581	0.617905	+	0.786253i	$0.287982\pi$
$878$	−2534.00	−0.0974013
$879$	3303.00	0.126743
$880$	−2160.00	−0.0827427
$881$	8490.00	0.324671	0.162336	−	0.986736i	$-0.448097\pi$
$881$	8490.00	0.324671	0.162336	+	0.986736i	$0.448097\pi$
$882$	0	0
$883$	−48352.0	−1.84278	−0.921390	−	0.388640i	$-0.872945\pi$
$883$	−48352.0	−1.84278	−0.921390	+	0.388640i	$0.872945\pi$
$884$	29568.0	1.12498
$885$	−7155.00	−0.271766
$886$	13866.0	0.525776
$887$	−15492.0	−0.586438	−0.293219	−	0.956045i	$-0.594726\pi$
$887$	−15492.0	−0.586438	−0.293219	+	0.956045i	$0.594726\pi$
$888$	−1056.00	−0.0399066
$889$	0	0
$890$	−31140.0	−1.17283
$891$	−729.000	−0.0274101
$892$	6292.00	0.236179
$893$	−14352.0	−0.537818
$894$	8604.00	0.321880
$895$	−18180.0	−0.678984
$896$	0	0
$897$	−22176.0	−0.825457
$898$	−23376.0	−0.868672
$899$	−9435.00	−0.350028
$900$	3600.00	0.133333
$901$	53676.0	1.98469
$902$	3024.00	0.111628
$903$	0	0
$904$	17472.0	0.642821
$905$	−45480.0	−1.67050
$906$	16026.0	0.587669
$907$	−8116.00	−0.297119	−0.148560	−	0.988903i	$-0.547464\pi$
$907$	−8116.00	−0.297119	−0.148560	+	0.988903i	$0.547464\pi$
$908$	−3684.00	−0.134645
$909$	−5778.00	−0.210830
$910$	0	0
$911$	−4446.00	−0.161693	−0.0808466	−	0.996727i	$-0.525762\pi$
$911$	−4446.00	−0.161693	−0.0808466	+	0.996727i	$0.525762\pi$
$912$	−4992.00	−0.181252
$913$	−5373.00	−0.194765
$914$	−1102.00	−0.0398807
$915$	−32490.0	−1.17386
$916$	−16208.0	−0.584637
$917$	0	0
$918$	−4536.00	−0.163083
$919$	26504.0	0.951345	0.475673	−	0.879622i	$-0.342205\pi$
$919$	26504.0	0.951345	0.475673	+	0.879622i	$0.342205\pi$
$920$	10080.0	0.361226
$921$	−8340.00	−0.298385
$922$	26772.0	0.956279
$923$	95568.0	3.40808
$924$	0	0
$925$	4400.00	0.156401
$926$	12752.0	0.452545
$927$	−4176.00	−0.147959
$928$	−1632.00	−0.0577296
$929$	5430.00	0.191768	0.0958840	−	0.995393i	$-0.469432\pi$
$929$	5430.00	0.191768	0.0958840	+	0.995393i	$0.469432\pi$
$930$	16650.0	0.587070
$931$	0	0
$932$	−1872.00	−0.0657933
$933$	−12888.0	−0.452234
$934$	11400.0	0.399378
$935$	−11340.0	−0.396639
$936$	−6336.00	−0.221259
$937$	−33803.0	−1.17854	−0.589272	−	0.807935i	$-0.700585\pi$
$937$	−33803.0	−1.17854	−0.589272	+	0.807935i	$0.700585\pi$
$938$	0	0
$939$	−16467.0	−0.572290
$940$	8280.00	0.287302
$941$	48483.0	1.67960	0.839798	−	0.542898i	$-0.182673\pi$
$941$	48483.0	1.67960	0.839798	+	0.542898i	$0.182673\pi$
$942$	13512.0	0.467351
$943$	−14112.0	−0.487328
$944$	−2544.00	−0.0877120
$945$	0	0
$946$	5868.00	0.201676
$947$	37296.0	1.27979	0.639893	−	0.768464i	$-0.278979\pi$
$947$	37296.0	1.27979	0.639893	+	0.768464i	$0.278979\pi$
$948$	−6996.00	−0.239683
$949$	−19184.0	−0.656205
$950$	20800.0	0.710359
$951$	13473.0	0.459403
$952$	0	0
$953$	−38478.0	−1.30790	−0.653948	−	0.756540i	$-0.726888\pi$
$953$	−38478.0	−1.30790	−0.653948	+	0.756540i	$0.726888\pi$
$954$	−11502.0	−0.390347
$955$	37800.0	1.28082
$956$	19728.0	0.667415
$957$	−1377.00	−0.0465121
$958$	39588.0	1.33510
$959$	0	0
$960$	2880.00	0.0968246
$961$	4434.00	0.148837
$962$	−7744.00	−0.259539
$963$	3537.00	0.118357
$964$	6148.00	0.205408
$965$	5475.00	0.182639
$966$	0	0
$967$	27257.0	0.906438	0.453219	−	0.891399i	$-0.350275\pi$
$967$	27257.0	0.906438	0.453219	+	0.891399i	$0.350275\pi$
$968$	10000.0	0.332037
$969$	−26208.0	−0.868857
$970$	−5070.00	−0.167823
$971$	−34341.0	−1.13497	−0.567485	−	0.823384i	$-0.692083\pi$
$971$	−34341.0	−1.13497	−0.567485	+	0.823384i	$0.692083\pi$
$972$	972.000	0.0320750
$973$	0	0
$974$	−31870.0	−1.04844
$975$	26400.0	0.867156
$976$	−11552.0	−0.378863
$977$	27426.0	0.898092	0.449046	−	0.893509i	$-0.351764\pi$
$977$	27426.0	0.898092	0.449046	+	0.893509i	$0.351764\pi$
$978$	−10056.0	−0.328789
$979$	−9342.00	−0.304976
$980$	0	0
$981$	126.000	0.00410079
$982$	−19926.0	−0.647520
$983$	−12324.0	−0.399872	−0.199936	−	0.979809i	$-0.564073\pi$
$983$	−12324.0	−0.399872	−0.199936	+	0.979809i	$0.564073\pi$
$984$	−4032.00	−0.130625
$985$	−23850.0	−0.771497
$986$	−8568.00	−0.276735
$987$	0	0
$988$	−36608.0	−1.17880
$989$	−27384.0	−0.880445
$990$	2430.00	0.0780106
$991$	47597.0	1.52570	0.762850	−	0.646576i	$-0.223800\pi$
$991$	47597.0	1.52570	0.762850	+	0.646576i	$0.223800\pi$
$992$	5920.00	0.189476
$993$	−11892.0	−0.380042
$994$	0	0
$995$	80700.0	2.57122
$996$	7164.00	0.227912
$997$	11242.0	0.357109	0.178555	−	0.983930i	$-0.442858\pi$
$997$	11242.0	0.357109	0.178555	+	0.983930i	$0.442858\pi$
$998$	−38284.0	−1.21429
$999$	1188.00	0.0376243

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.g.1.1		1
3.2	odd	2		882.4.a.h.1.1		1
4.3	odd	2		2352.4.a.q.1.1		1
7.2	even	3		294.4.e.e.67.1		2
7.3	odd	6		42.4.e.b.37.1	yes	2
7.4	even	3		294.4.e.e.79.1		2
7.5	odd	6		42.4.e.b.25.1	✓	2
7.6	odd	2		294.4.a.a.1.1		1
21.2	odd	6		882.4.g.l.361.1		2
21.5	even	6		126.4.g.a.109.1		2
21.11	odd	6		882.4.g.l.667.1		2
21.17	even	6		126.4.g.a.37.1		2
21.20	even	2		882.4.a.r.1.1		1
28.3	even	6		336.4.q.d.289.1		2
28.19	even	6		336.4.q.d.193.1		2
28.27	even	2		2352.4.a.u.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
42.4.e.b.25.1	✓	2	7.5	odd	6
42.4.e.b.37.1	yes	2	7.3	odd	6
126.4.g.a.37.1		2	21.17	even	6
126.4.g.a.109.1		2	21.5	even	6
294.4.a.a.1.1		1	7.6	odd	2
294.4.a.g.1.1		1	1.1	even	1	trivial
294.4.e.e.67.1		2	7.2	even	3
294.4.e.e.79.1		2	7.4	even	3
336.4.q.d.193.1		2	28.19	even	6
336.4.q.d.289.1		2	28.3	even	6
882.4.a.h.1.1		1	3.2	odd	2
882.4.a.r.1.1		1	21.20	even	2
882.4.g.l.361.1		2	21.2	odd	6
882.4.g.l.667.1		2	21.11	odd	6
2352.4.a.q.1.1		1	4.3	odd	2
2352.4.a.u.1.1		1	28.27	even	2

Overview	Random
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Classical	Maass
Hilbert	Bianchi

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

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.g.1.1

Level	$294$
Weight	$4$
Character	294.1
Self dual	yes
Analytic conductor	$17.347$
Analytic rank	$0$
Dimension	$1$
CM	no
Inner twists	$1$