LMFDB - Embedded newform 57.4.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [57,4,Mod(1,57)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(57, 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("57.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 57 = 3 \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	57.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:	$3.36310887033$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	57.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} +3.00000 q^{3} -7.00000 q^{4} -12.0000 q^{5} -3.00000 q^{6} -20.0000 q^{7} +15.0000 q^{8} +9.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} +3.00000 q^{3} -7.00000 q^{4} -12.0000 q^{5} -3.00000 q^{6} -20.0000 q^{7} +15.0000 q^{8} +9.00000 q^{9} +12.0000 q^{10} -4.00000 q^{11} -21.0000 q^{12} -76.0000 q^{13} +20.0000 q^{14} -36.0000 q^{15} +41.0000 q^{16} +22.0000 q^{17} -9.00000 q^{18} -19.0000 q^{19} +84.0000 q^{20} -60.0000 q^{21} +4.00000 q^{22} +82.0000 q^{23} +45.0000 q^{24} +19.0000 q^{25} +76.0000 q^{26} +27.0000 q^{27} +140.000 q^{28} +242.000 q^{29} +36.0000 q^{30} -126.000 q^{31} -161.000 q^{32} -12.0000 q^{33} -22.0000 q^{34} +240.000 q^{35} -63.0000 q^{36} -180.000 q^{37} +19.0000 q^{38} -228.000 q^{39} -180.000 q^{40} -390.000 q^{41} +60.0000 q^{42} +308.000 q^{43} +28.0000 q^{44} -108.000 q^{45} -82.0000 q^{46} -522.000 q^{47} +123.000 q^{48} +57.0000 q^{49} -19.0000 q^{50} +66.0000 q^{51} +532.000 q^{52} -70.0000 q^{53} -27.0000 q^{54} +48.0000 q^{55} -300.000 q^{56} -57.0000 q^{57} -242.000 q^{58} +188.000 q^{59} +252.000 q^{60} -706.000 q^{61} +126.000 q^{62} -180.000 q^{63} -167.000 q^{64} +912.000 q^{65} +12.0000 q^{66} +104.000 q^{67} -154.000 q^{68} +246.000 q^{69} -240.000 q^{70} -432.000 q^{71} +135.000 q^{72} +718.000 q^{73} +180.000 q^{74} +57.0000 q^{75} +133.000 q^{76} +80.0000 q^{77} +228.000 q^{78} +94.0000 q^{79} -492.000 q^{80} +81.0000 q^{81} +390.000 q^{82} -1296.00 q^{83} +420.000 q^{84} -264.000 q^{85} -308.000 q^{86} +726.000 q^{87} -60.0000 q^{88} +846.000 q^{89} +108.000 q^{90} +1520.00 q^{91} -574.000 q^{92} -378.000 q^{93} +522.000 q^{94} +228.000 q^{95} -483.000 q^{96} +830.000 q^{97} -57.0000 q^{98} -36.0000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

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

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

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

$2$	−1.00000	−0.353553	−0.176777	−	0.984251i	$-0.556567\pi$
$2$	−1.00000	−0.353553	−0.176777	+	0.984251i	$0.556567\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	−7.00000	−0.875000
$5$	−12.0000	−1.07331	−0.536656	−	0.843801i	$-0.680313\pi$
$5$	−12.0000	−1.07331	−0.536656	+	0.843801i	$0.680313\pi$
$6$	−3.00000	−0.204124
$7$	−20.0000	−1.07990	−0.539949	−	0.841698i	$-0.681557\pi$
$7$	−20.0000	−1.07990	−0.539949	+	0.841698i	$0.681557\pi$
$8$	15.0000	0.662913
$9$	9.00000	0.333333
$10$	12.0000	0.379473
$11$	−4.00000	−0.109640	−0.0548202	−	0.998496i	$-0.517459\pi$
$11$	−4.00000	−0.109640	−0.0548202	+	0.998496i	$0.517459\pi$
$12$	−21.0000	−0.505181
$13$	−76.0000	−1.62143	−0.810716	−	0.585440i	$-0.800922\pi$
$13$	−76.0000	−1.62143	−0.810716	+	0.585440i	$0.800922\pi$
$14$	20.0000	0.381802
$15$	−36.0000	−0.619677
$16$	41.0000	0.640625
$17$	22.0000	0.313870	0.156935	−	0.987609i	$-0.449839\pi$
$17$	22.0000	0.313870	0.156935	+	0.987609i	$0.449839\pi$
$18$	−9.00000	−0.117851
$19$	−19.0000	−0.229416
$19$	−19.0000	−0.229416
$20$	84.0000	0.939149
$21$	−60.0000	−0.623480
$22$	4.00000	0.0387638
$23$	82.0000	0.743399	0.371700	−	0.928353i	$-0.378775\pi$
$23$	82.0000	0.743399	0.371700	+	0.928353i	$0.378775\pi$
$24$	45.0000	0.382733
$25$	19.0000	0.152000
$26$	76.0000	0.573263
$27$	27.0000	0.192450
$28$	140.000	0.944911
$29$	242.000	1.54960	0.774798	−	0.632209i	$-0.217852\pi$
$29$	242.000	1.54960	0.774798	+	0.632209i	$0.217852\pi$
$30$	36.0000	0.219089
$31$	−126.000	−0.730009	−0.365004	−	0.931006i	$-0.618932\pi$
$31$	−126.000	−0.730009	−0.365004	+	0.931006i	$0.618932\pi$
$32$	−161.000	−0.889408
$33$	−12.0000	−0.0633010
$34$	−22.0000	−0.110970
$35$	240.000	1.15907
$36$	−63.0000	−0.291667
$37$	−180.000	−0.799779	−0.399889	−	0.916563i	$-0.630951\pi$
$37$	−180.000	−0.799779	−0.399889	+	0.916563i	$0.630951\pi$
$38$	19.0000	0.0811107
$39$	−228.000	−0.936134
$40$	−180.000	−0.711512
$41$	−390.000	−1.48556	−0.742778	−	0.669538i	$-0.766492\pi$
$41$	−390.000	−1.48556	−0.742778	+	0.669538i	$0.766492\pi$
$42$	60.0000	0.220433
$43$	308.000	1.09232	0.546158	−	0.837682i	$-0.316090\pi$
$43$	308.000	1.09232	0.546158	+	0.837682i	$0.316090\pi$
$44$	28.0000	0.0959354
$45$	−108.000	−0.357771
$46$	−82.0000	−0.262831
$47$	−522.000	−1.62003	−0.810016	−	0.586407i	$-0.800542\pi$
$47$	−522.000	−1.62003	−0.810016	+	0.586407i	$0.800542\pi$
$48$	123.000	0.369865
$49$	57.0000	0.166181
$50$	−19.0000	−0.0537401
$51$	66.0000	0.181213
$52$	532.000	1.41875
$53$	−70.0000	−0.181420	−0.0907098	−	0.995877i	$-0.528914\pi$
$53$	−70.0000	−0.181420	−0.0907098	+	0.995877i	$0.528914\pi$
$54$	−27.0000	−0.0680414
$55$	48.0000	0.117679
$56$	−300.000	−0.715878
$57$	−57.0000	−0.132453
$58$	−242.000	−0.547865
$59$	188.000	0.414839	0.207420	−	0.978252i	$-0.433493\pi$
$59$	188.000	0.414839	0.207420	+	0.978252i	$0.433493\pi$
$60$	252.000	0.542218
$61$	−706.000	−1.48187	−0.740935	−	0.671577i	$-0.765617\pi$
$61$	−706.000	−1.48187	−0.740935	+	0.671577i	$0.765617\pi$
$62$	126.000	0.258097
$63$	−180.000	−0.359966
$64$	−167.000	−0.326172
$65$	912.000	1.74030
$66$	12.0000	0.0223803
$67$	104.000	0.189636	0.0948181	−	0.995495i	$-0.469773\pi$
$67$	104.000	0.189636	0.0948181	+	0.995495i	$0.469773\pi$
$68$	−154.000	−0.274636
$69$	246.000	0.429202
$70$	−240.000	−0.409793
$71$	−432.000	−0.722098	−0.361049	−	0.932547i	$-0.617581\pi$
$71$	−432.000	−0.722098	−0.361049	+	0.932547i	$0.617581\pi$
$72$	135.000	0.220971
$73$	718.000	1.15117	0.575586	−	0.817741i	$-0.304774\pi$
$73$	718.000	1.15117	0.575586	+	0.817741i	$0.304774\pi$
$74$	180.000	0.282765
$75$	57.0000	0.0877572
$76$	133.000	0.200739
$77$	80.0000	0.118401
$78$	228.000	0.330973
$79$	94.0000	0.133871	0.0669356	−	0.997757i	$-0.478678\pi$
$79$	94.0000	0.133871	0.0669356	+	0.997757i	$0.478678\pi$
$80$	−492.000	−0.687591
$81$	81.0000	0.111111
$82$	390.000	0.525223
$83$	−1296.00	−1.71391	−0.856955	−	0.515392i	$-0.827646\pi$
$83$	−1296.00	−1.71391	−0.856955	+	0.515392i	$0.827646\pi$
$84$	420.000	0.545545
$85$	−264.000	−0.336880
$86$	−308.000	−0.386192
$87$	726.000	0.894659
$88$	−60.0000	−0.0726821
$89$	846.000	1.00759	0.503797	−	0.863822i	$-0.331936\pi$
$89$	846.000	1.00759	0.503797	+	0.863822i	$0.331936\pi$
$90$	108.000	0.126491
$91$	1520.00	1.75098
$92$	−574.000	−0.650474
$93$	−378.000	−0.421471
$94$	522.000	0.572768
$95$	228.000	0.246235
$96$	−483.000	−0.513500
$97$	830.000	0.868801	0.434401	−	0.900720i	$-0.356960\pi$
$97$	830.000	0.868801	0.434401	+	0.900720i	$0.356960\pi$
$98$	−57.0000	−0.0587538
$99$	−36.0000	−0.0365468
$100$	−133.000	−0.133000
$101$	1612.00	1.58812	0.794059	−	0.607840i	$-0.207964\pi$
$101$	1612.00	1.58812	0.794059	+	0.607840i	$0.207964\pi$
$102$	−66.0000	−0.0640684
$103$	−1874.00	−1.79273	−0.896363	−	0.443322i	$-0.853800\pi$
$103$	−1874.00	−1.79273	−0.896363	+	0.443322i	$0.853800\pi$
$104$	−1140.00	−1.07487
$105$	720.000	0.669189
$106$	70.0000	0.0641415
$107$	−1932.00	−1.74555	−0.872773	−	0.488126i	$-0.837681\pi$
$107$	−1932.00	−1.74555	−0.872773	+	0.488126i	$0.837681\pi$
$108$	−189.000	−0.168394
$109$	1096.00	0.963099	0.481549	−	0.876419i	$-0.340074\pi$
$109$	1096.00	0.963099	0.481549	+	0.876419i	$0.340074\pi$
$110$	−48.0000	−0.0416056
$111$	−540.000	−0.461753
$112$	−820.000	−0.691810
$113$	1474.00	1.22710	0.613550	−	0.789656i	$-0.289741\pi$
$113$	1474.00	1.22710	0.613550	+	0.789656i	$0.289741\pi$
$114$	57.0000	0.0468293
$115$	−984.000	−0.797900
$116$	−1694.00	−1.35590
$117$	−684.000	−0.540477
$118$	−188.000	−0.146668
$119$	−440.000	−0.338947
$120$	−540.000	−0.410792
$121$	−1315.00	−0.987979
$122$	706.000	0.523920
$123$	−1170.00	−0.857686
$124$	882.000	0.638758
$125$	1272.00	0.910169
$126$	180.000	0.127267
$127$	−1166.00	−0.814691	−0.407346	−	0.913274i	$-0.633546\pi$
$127$	−1166.00	−0.814691	−0.407346	+	0.913274i	$0.633546\pi$
$128$	1455.00	1.00473
$129$	924.000	0.630649
$130$	−912.000	−0.615290
$131$	2192.00	1.46195	0.730977	−	0.682402i	$-0.239065\pi$
$131$	2192.00	1.46195	0.730977	+	0.682402i	$0.239065\pi$
$132$	84.0000	0.0553883
$133$	380.000	0.247746
$134$	−104.000	−0.0670465
$135$	−324.000	−0.206559
$136$	330.000	0.208068
$137$	558.000	0.347979	0.173990	−	0.984747i	$-0.444334\pi$
$137$	558.000	0.347979	0.173990	+	0.984747i	$0.444334\pi$
$138$	−246.000	−0.151746
$139$	68.0000	0.0414941	0.0207471	−	0.999785i	$-0.493396\pi$
$139$	68.0000	0.0414941	0.0207471	+	0.999785i	$0.493396\pi$
$140$	−1680.00	−1.01419
$141$	−1566.00	−0.935326
$142$	432.000	0.255300
$143$	304.000	0.177775
$144$	369.000	0.213542
$145$	−2904.00	−1.66320
$146$	−718.000	−0.407001
$147$	171.000	0.0959445
$148$	1260.00	0.699807
$149$	576.000	0.316696	0.158348	−	0.987383i	$-0.449383\pi$
$149$	576.000	0.316696	0.158348	+	0.987383i	$0.449383\pi$
$150$	−57.0000	−0.0310269
$151$	990.000	0.533543	0.266772	−	0.963760i	$-0.414043\pi$
$151$	990.000	0.533543	0.266772	+	0.963760i	$0.414043\pi$
$152$	−285.000	−0.152083
$153$	198.000	0.104623
$154$	−80.0000	−0.0418609
$155$	1512.00	0.783528
$156$	1596.00	0.819117
$157$	−654.000	−0.332451	−0.166226	−	0.986088i	$-0.553158\pi$
$157$	−654.000	−0.332451	−0.166226	+	0.986088i	$0.553158\pi$
$158$	−94.0000	−0.0473306
$159$	−210.000	−0.104743
$160$	1932.00	0.954613
$161$	−1640.00	−0.802796
$162$	−81.0000	−0.0392837
$163$	−900.000	−0.432475	−0.216238	−	0.976341i	$-0.569379\pi$
$163$	−900.000	−0.432475	−0.216238	+	0.976341i	$0.569379\pi$
$164$	2730.00	1.29986
$165$	144.000	0.0679417
$166$	1296.00	0.605958
$167$	740.000	0.342892	0.171446	−	0.985194i	$-0.445156\pi$
$167$	740.000	0.342892	0.171446	+	0.985194i	$0.445156\pi$
$168$	−900.000	−0.413313
$169$	3579.00	1.62904
$170$	264.000	0.119105
$171$	−171.000	−0.0764719
$172$	−2156.00	−0.955776
$173$	582.000	0.255772	0.127886	−	0.991789i	$-0.459181\pi$
$173$	582.000	0.255772	0.127886	+	0.991789i	$0.459181\pi$
$174$	−726.000	−0.316310
$175$	−380.000	−0.164145
$176$	−164.000	−0.0702384
$177$	564.000	0.239508
$178$	−846.000	−0.356238
$179$	2748.00	1.14746	0.573730	−	0.819045i	$-0.305496\pi$
$179$	2748.00	1.14746	0.573730	+	0.819045i	$0.305496\pi$
$180$	756.000	0.313050
$181$	1336.00	0.548641	0.274321	−	0.961638i	$-0.411547\pi$
$181$	1336.00	0.548641	0.274321	+	0.961638i	$0.411547\pi$
$182$	−1520.00	−0.619065
$183$	−2118.00	−0.855558
$184$	1230.00	0.492809
$185$	2160.00	0.858413
$186$	378.000	0.149012
$187$	−88.0000	−0.0344128
$188$	3654.00	1.41753
$189$	−540.000	−0.207827
$190$	−228.000	−0.0870572
$191$	−606.000	−0.229574	−0.114787	−	0.993390i	$-0.536619\pi$
$191$	−606.000	−0.229574	−0.114787	+	0.993390i	$0.536619\pi$
$192$	−501.000	−0.188315
$193$	−3002.00	−1.11963	−0.559815	−	0.828617i	$-0.689128\pi$
$193$	−3002.00	−1.11963	−0.559815	+	0.828617i	$0.689128\pi$
$194$	−830.000	−0.307168
$195$	2736.00	1.00476
$196$	−399.000	−0.145408
$197$	−4456.00	−1.61156	−0.805779	−	0.592217i	$-0.798253\pi$
$197$	−4456.00	−1.61156	−0.805779	+	0.592217i	$0.798253\pi$
$198$	36.0000	0.0129213
$199$	−2844.00	−1.01309	−0.506547	−	0.862212i	$-0.669078\pi$
$199$	−2844.00	−1.01309	−0.506547	+	0.862212i	$0.669078\pi$
$200$	285.000	0.100763
$201$	312.000	0.109486
$202$	−1612.00	−0.561485
$203$	−4840.00	−1.67341
$204$	−462.000	−0.158561
$205$	4680.00	1.59447
$206$	1874.00	0.633824
$207$	738.000	0.247800
$208$	−3116.00	−1.03873
$209$	76.0000	0.0251533
$210$	−720.000	−0.236594
$211$	−3108.00	−1.01405	−0.507023	−	0.861933i	$-0.669254\pi$
$211$	−3108.00	−1.01405	−0.507023	+	0.861933i	$0.669254\pi$
$212$	490.000	0.158742
$213$	−1296.00	−0.416904
$214$	1932.00	0.617144
$215$	−3696.00	−1.17240
$216$	405.000	0.127578
$217$	2520.00	0.788335
$218$	−1096.00	−0.340507
$219$	2154.00	0.664629
$220$	−336.000	−0.102969
$221$	−1672.00	−0.508918
$222$	540.000	0.163254
$223$	−4686.00	−1.40716	−0.703582	−	0.710614i	$-0.748417\pi$
$223$	−4686.00	−1.40716	−0.703582	+	0.710614i	$0.748417\pi$
$224$	3220.00	0.960470
$225$	171.000	0.0506667
$226$	−1474.00	−0.433845
$227$	−3036.00	−0.887693	−0.443847	−	0.896103i	$-0.646387\pi$
$227$	−3036.00	−0.887693	−0.443847	+	0.896103i	$0.646387\pi$
$228$	399.000	0.115897
$229$	−4970.00	−1.43418	−0.717089	−	0.696981i	$-0.754526\pi$
$229$	−4970.00	−1.43418	−0.717089	+	0.696981i	$0.754526\pi$
$230$	984.000	0.282100
$231$	240.000	0.0683586
$232$	3630.00	1.02725
$233$	−2982.00	−0.838443	−0.419222	−	0.907884i	$-0.637697\pi$
$233$	−2982.00	−0.838443	−0.419222	+	0.907884i	$0.637697\pi$
$234$	684.000	0.191088
$235$	6264.00	1.73880
$236$	−1316.00	−0.362984
$237$	282.000	0.0772906
$238$	440.000	0.119836
$239$	−522.000	−0.141278	−0.0706389	−	0.997502i	$-0.522504\pi$
$239$	−522.000	−0.141278	−0.0706389	+	0.997502i	$0.522504\pi$
$240$	−1476.00	−0.396981
$241$	3350.00	0.895404	0.447702	−	0.894183i	$-0.352242\pi$
$241$	3350.00	0.895404	0.447702	+	0.894183i	$0.352242\pi$
$242$	1315.00	0.349303
$243$	243.000	0.0641500
$244$	4942.00	1.29664
$245$	−684.000	−0.178364
$246$	1170.00	0.303238
$247$	1444.00	0.371982
$248$	−1890.00	−0.483932
$249$	−3888.00	−0.989526
$250$	−1272.00	−0.321793
$251$	−2968.00	−0.746369	−0.373184	−	0.927757i	$-0.621734\pi$
$251$	−2968.00	−0.746369	−0.373184	+	0.927757i	$0.621734\pi$
$252$	1260.00	0.314970
$253$	−328.000	−0.0815067
$254$	1166.00	0.288037
$255$	−792.000	−0.194498
$256$	−119.000	−0.0290527
$257$	−1234.00	−0.299513	−0.149756	−	0.988723i	$-0.547849\pi$
$257$	−1234.00	−0.299513	−0.149756	+	0.988723i	$0.547849\pi$
$258$	−924.000	−0.222968
$259$	3600.00	0.863680
$260$	−6384.00	−1.52276
$261$	2178.00	0.516532
$262$	−2192.00	−0.516879
$263$	−1994.00	−0.467511	−0.233755	−	0.972295i	$-0.575101\pi$
$263$	−1994.00	−0.467511	−0.233755	+	0.972295i	$0.575101\pi$
$264$	−180.000	−0.0419630
$265$	840.000	0.194720
$266$	−380.000	−0.0875913
$267$	2538.00	0.581734
$268$	−728.000	−0.165932
$269$	7214.00	1.63511	0.817556	−	0.575849i	$-0.195328\pi$
$269$	7214.00	1.63511	0.817556	+	0.575849i	$0.195328\pi$
$270$	324.000	0.0730297
$271$	7572.00	1.69729	0.848646	−	0.528961i	$-0.177418\pi$
$271$	7572.00	1.69729	0.848646	+	0.528961i	$0.177418\pi$
$272$	902.000	0.201073
$273$	4560.00	1.01093
$274$	−558.000	−0.123029
$275$	−76.0000	−0.0166654
$276$	−1722.00	−0.375552
$277$	6262.00	1.35829	0.679146	−	0.734003i	$-0.262350\pi$
$277$	6262.00	1.35829	0.679146	+	0.734003i	$0.262350\pi$
$278$	−68.0000	−0.0146704
$279$	−1134.00	−0.243336
$280$	3600.00	0.768361
$281$	−2710.00	−0.575320	−0.287660	−	0.957733i	$-0.592877\pi$
$281$	−2710.00	−0.575320	−0.287660	+	0.957733i	$0.592877\pi$
$282$	1566.00	0.330688
$283$	−556.000	−0.116787	−0.0583936	−	0.998294i	$-0.518598\pi$
$283$	−556.000	−0.116787	−0.0583936	+	0.998294i	$0.518598\pi$
$284$	3024.00	0.631836
$285$	684.000	0.142164
$286$	−304.000	−0.0628528
$287$	7800.00	1.60425
$288$	−1449.00	−0.296469
$289$	−4429.00	−0.901486
$290$	2904.00	0.588030
$291$	2490.00	0.501603
$292$	−5026.00	−1.00728
$293$	3694.00	0.736539	0.368269	−	0.929719i	$-0.379950\pi$
$293$	3694.00	0.736539	0.368269	+	0.929719i	$0.379950\pi$
$294$	−171.000	−0.0339215
$295$	−2256.00	−0.445252
$296$	−2700.00	−0.530183
$297$	−108.000	−0.0211003
$298$	−576.000	−0.111969
$299$	−6232.00	−1.20537
$300$	−399.000	−0.0767876
$301$	−6160.00	−1.17959
$302$	−990.000	−0.188636
$303$	4836.00	0.916901
$304$	−779.000	−0.146969
$305$	8472.00	1.59051
$306$	−198.000	−0.0369899
$307$	3384.00	0.629104	0.314552	−	0.949240i	$-0.398146\pi$
$307$	3384.00	0.629104	0.314552	+	0.949240i	$0.398146\pi$
$308$	−560.000	−0.103601
$309$	−5622.00	−1.03503
$310$	−1512.00	−0.277019
$311$	−9666.00	−1.76241	−0.881203	−	0.472737i	$-0.843266\pi$
$311$	−9666.00	−1.76241	−0.881203	+	0.472737i	$0.843266\pi$
$312$	−3420.00	−0.620575
$313$	−6794.00	−1.22690	−0.613450	−	0.789734i	$-0.710219\pi$
$313$	−6794.00	−1.22690	−0.613450	+	0.789734i	$0.710219\pi$
$314$	654.000	0.117539
$315$	2160.00	0.386356
$316$	−658.000	−0.117137
$317$	−3242.00	−0.574413	−0.287206	−	0.957869i	$-0.592727\pi$
$317$	−3242.00	−0.574413	−0.287206	+	0.957869i	$0.592727\pi$
$318$	210.000	0.0370321
$319$	−968.000	−0.169898
$320$	2004.00	0.350084
$321$	−5796.00	−1.00779
$322$	1640.00	0.283831
$323$	−418.000	−0.0720066
$324$	−567.000	−0.0972222
$325$	−1444.00	−0.246458
$326$	900.000	0.152903
$327$	3288.00	0.556045
$328$	−5850.00	−0.984793
$329$	10440.0	1.74947
$330$	−144.000	−0.0240210
$331$	176.000	0.0292261	0.0146130	−	0.999893i	$-0.495348\pi$
$331$	176.000	0.0292261	0.0146130	+	0.999893i	$0.495348\pi$
$332$	9072.00	1.49967
$333$	−1620.00	−0.266593
$334$	−740.000	−0.121231
$335$	−1248.00	−0.203539
$336$	−2460.00	−0.399417
$337$	−4262.00	−0.688920	−0.344460	−	0.938801i	$-0.611938\pi$
$337$	−4262.00	−0.688920	−0.344460	+	0.938801i	$0.611938\pi$
$338$	−3579.00	−0.575952
$339$	4422.00	0.708466
$340$	1848.00	0.294770
$341$	504.000	0.0800385
$342$	171.000	0.0270369
$343$	5720.00	0.900440
$344$	4620.00	0.724110
$345$	−2952.00	−0.460668
$346$	−582.000	−0.0904292
$347$	7060.00	1.09222	0.546110	−	0.837713i	$-0.316108\pi$
$347$	7060.00	1.09222	0.546110	+	0.837713i	$0.316108\pi$
$348$	−5082.00	−0.782827
$349$	4746.00	0.727930	0.363965	−	0.931413i	$-0.381423\pi$
$349$	4746.00	0.727930	0.363965	+	0.931413i	$0.381423\pi$
$350$	380.000	0.0580339
$351$	−2052.00	−0.312045
$352$	644.000	0.0975151
$353$	2546.00	0.383881	0.191940	−	0.981407i	$-0.438522\pi$
$353$	2546.00	0.383881	0.191940	+	0.981407i	$0.438522\pi$
$354$	−564.000	−0.0846787
$355$	5184.00	0.775037
$356$	−5922.00	−0.881644
$357$	−1320.00	−0.195691
$358$	−2748.00	−0.405688
$359$	−1702.00	−0.250218	−0.125109	−	0.992143i	$-0.539928\pi$
$359$	−1702.00	−0.250218	−0.125109	+	0.992143i	$0.539928\pi$
$360$	−1620.00	−0.237171
$361$	361.000	0.0526316
$362$	−1336.00	−0.193974
$363$	−3945.00	−0.570410
$364$	−10640.0	−1.53211
$365$	−8616.00	−1.23557
$366$	2118.00	0.302485
$367$	−7844.00	−1.11568	−0.557839	−	0.829950i	$-0.688369\pi$
$367$	−7844.00	−1.11568	−0.557839	+	0.829950i	$0.688369\pi$
$368$	3362.00	0.476240
$369$	−3510.00	−0.495185
$370$	−2160.00	−0.303495
$371$	1400.00	0.195915
$372$	2646.00	0.368787
$373$	13612.0	1.88955	0.944776	−	0.327718i	$-0.106280\pi$
$373$	13612.0	1.88955	0.944776	+	0.327718i	$0.106280\pi$
$374$	88.0000	0.0121668
$375$	3816.00	0.525486
$376$	−7830.00	−1.07394
$377$	−18392.0	−2.51256
$378$	540.000	0.0734778
$379$	976.000	0.132279	0.0661395	−	0.997810i	$-0.478932\pi$
$379$	976.000	0.132279	0.0661395	+	0.997810i	$0.478932\pi$
$380$	−1596.00	−0.215455
$381$	−3498.00	−0.470362
$382$	606.000	0.0811666
$383$	2152.00	0.287107	0.143554	−	0.989643i	$-0.454147\pi$
$383$	2152.00	0.287107	0.143554	+	0.989643i	$0.454147\pi$
$384$	4365.00	0.580079
$385$	−960.000	−0.127081
$386$	3002.00	0.395849
$387$	2772.00	0.364105
$388$	−5810.00	−0.760201
$389$	10572.0	1.37795	0.688974	−	0.724786i	$-0.258061\pi$
$389$	10572.0	1.37795	0.688974	+	0.724786i	$0.258061\pi$
$390$	−2736.00	−0.355238
$391$	1804.00	0.233330
$392$	855.000	0.110163
$393$	6576.00	0.844059
$394$	4456.00	0.569772
$395$	−1128.00	−0.143686
$396$	252.000	0.0319785
$397$	−10910.0	−1.37924	−0.689619	−	0.724173i	$-0.742222\pi$
$397$	−10910.0	−1.37924	−0.689619	+	0.724173i	$0.742222\pi$
$398$	2844.00	0.358183
$399$	1140.00	0.143036
$400$	779.000	0.0973750
$401$	10146.0	1.26351	0.631754	−	0.775169i	$-0.282335\pi$
$401$	10146.0	1.26351	0.631754	+	0.775169i	$0.282335\pi$
$402$	−312.000	−0.0387093
$403$	9576.00	1.18366
$404$	−11284.0	−1.38960
$405$	−972.000	−0.119257
$406$	4840.00	0.591638
$407$	720.000	0.0876881
$408$	990.000	0.120128
$409$	−13706.0	−1.65701	−0.828506	−	0.559980i	$-0.810809\pi$
$409$	−13706.0	−1.65701	−0.828506	+	0.559980i	$0.810809\pi$
$410$	−4680.00	−0.563729
$411$	1674.00	0.200906
$412$	13118.0	1.56863
$413$	−3760.00	−0.447984
$414$	−738.000	−0.0876104
$415$	15552.0	1.83956
$416$	12236.0	1.44211
$417$	204.000	0.0239566
$418$	−76.0000	−0.00889302
$419$	6812.00	0.794243	0.397122	−	0.917766i	$-0.370009\pi$
$419$	6812.00	0.794243	0.397122	+	0.917766i	$0.370009\pi$
$420$	−5040.00	−0.585540
$421$	−6724.00	−0.778403	−0.389202	−	0.921153i	$-0.627249\pi$
$421$	−6724.00	−0.778403	−0.389202	+	0.921153i	$0.627249\pi$
$422$	3108.00	0.358519
$423$	−4698.00	−0.540011
$424$	−1050.00	−0.120265
$425$	418.000	0.0477082
$426$	1296.00	0.147398
$427$	14120.0	1.60027
$428$	13524.0	1.52735
$429$	912.000	0.102638
$430$	3696.00	0.414505
$431$	13876.0	1.55077	0.775387	−	0.631487i	$-0.217555\pi$
$431$	13876.0	1.55077	0.775387	+	0.631487i	$0.217555\pi$
$432$	1107.00	0.123288
$433$	342.000	0.0379572	0.0189786	−	0.999820i	$-0.493959\pi$
$433$	342.000	0.0379572	0.0189786	+	0.999820i	$0.493959\pi$
$434$	−2520.00	−0.278719
$435$	−8712.00	−0.960249
$436$	−7672.00	−0.842711
$437$	−1558.00	−0.170547
$438$	−2154.00	−0.234982
$439$	−6526.00	−0.709497	−0.354748	−	0.934962i	$-0.615433\pi$
$439$	−6526.00	−0.709497	−0.354748	+	0.934962i	$0.615433\pi$
$440$	720.000	0.0780106
$441$	513.000	0.0553936
$442$	1672.00	0.179930
$443$	−5020.00	−0.538391	−0.269196	−	0.963085i	$-0.586758\pi$
$443$	−5020.00	−0.538391	−0.269196	+	0.963085i	$0.586758\pi$
$444$	3780.00	0.404033
$445$	−10152.0	−1.08146
$446$	4686.00	0.497508
$447$	1728.00	0.182845
$448$	3340.00	0.352233
$449$	9486.00	0.997042	0.498521	−	0.866878i	$-0.333877\pi$
$449$	9486.00	0.997042	0.498521	+	0.866878i	$0.333877\pi$
$450$	−171.000	−0.0179134
$451$	1560.00	0.162877
$452$	−10318.0	−1.07371
$453$	2970.00	0.308041
$454$	3036.00	0.313847
$455$	−18240.0	−1.87935
$456$	−855.000	−0.0878049
$457$	−7262.00	−0.743330	−0.371665	−	0.928367i	$-0.621213\pi$
$457$	−7262.00	−0.743330	−0.371665	+	0.928367i	$0.621213\pi$
$458$	4970.00	0.507059
$459$	594.000	0.0604042
$460$	6888.00	0.698162
$461$	−13968.0	−1.41118	−0.705591	−	0.708620i	$-0.749318\pi$
$461$	−13968.0	−1.41118	−0.705591	+	0.708620i	$0.749318\pi$
$462$	−240.000	−0.0241684
$463$	4604.00	0.462130	0.231065	−	0.972938i	$-0.425779\pi$
$463$	4604.00	0.462130	0.231065	+	0.972938i	$0.425779\pi$
$464$	9922.00	0.992710
$465$	4536.00	0.452370
$466$	2982.00	0.296435
$467$	−19480.0	−1.93025	−0.965125	−	0.261789i	$-0.915688\pi$
$467$	−19480.0	−1.93025	−0.965125	+	0.261789i	$0.915688\pi$
$468$	4788.00	0.472917
$469$	−2080.00	−0.204788
$470$	−6264.00	−0.614759
$471$	−1962.00	−0.191941
$472$	2820.00	0.275002
$473$	−1232.00	−0.119762
$474$	−282.000	−0.0273263
$475$	−361.000	−0.0348712
$476$	3080.00	0.296579
$477$	−630.000	−0.0604732
$478$	522.000	0.0499492
$479$	−12134.0	−1.15745	−0.578723	−	0.815524i	$-0.696449\pi$
$479$	−12134.0	−1.15745	−0.578723	+	0.815524i	$0.696449\pi$
$480$	5796.00	0.551146
$481$	13680.0	1.29679
$482$	−3350.00	−0.316573
$483$	−4920.00	−0.463494
$484$	9205.00	0.864482
$485$	−9960.00	−0.932495
$486$	−243.000	−0.0226805
$487$	−15658.0	−1.45694	−0.728472	−	0.685076i	$-0.759769\pi$
$487$	−15658.0	−1.45694	−0.728472	+	0.685076i	$0.759769\pi$
$488$	−10590.0	−0.982350
$489$	−2700.00	−0.249690
$490$	684.000	0.0630612
$491$	2520.00	0.231621	0.115811	−	0.993271i	$-0.463053\pi$
$491$	2520.00	0.231621	0.115811	+	0.993271i	$0.463053\pi$
$492$	8190.00	0.750475
$493$	5324.00	0.486371
$494$	−1444.00	−0.131515
$495$	432.000	0.0392262
$496$	−5166.00	−0.467662
$497$	8640.00	0.779793
$498$	3888.00	0.349850
$499$	−9460.00	−0.848673	−0.424336	−	0.905505i	$-0.639493\pi$
$499$	−9460.00	−0.848673	−0.424336	+	0.905505i	$0.639493\pi$
$500$	−8904.00	−0.796398
$501$	2220.00	0.197969
$502$	2968.00	0.263881
$503$	−12178.0	−1.07950	−0.539752	−	0.841824i	$-0.681482\pi$
$503$	−12178.0	−1.07950	−0.539752	+	0.841824i	$0.681482\pi$
$504$	−2700.00	−0.238626
$505$	−19344.0	−1.70455
$506$	328.000	0.0288170
$507$	10737.0	0.940526
$508$	8162.00	0.712855
$509$	−4746.00	−0.413286	−0.206643	−	0.978416i	$-0.566254\pi$
$509$	−4746.00	−0.413286	−0.206643	+	0.978416i	$0.566254\pi$
$510$	792.000	0.0687654
$511$	−14360.0	−1.24315
$512$	−11521.0	−0.994455
$513$	−513.000	−0.0441511
$514$	1234.00	0.105894
$515$	22488.0	1.92415
$516$	−6468.00	−0.551817
$517$	2088.00	0.177621
$518$	−3600.00	−0.305357
$519$	1746.00	0.147670
$520$	13680.0	1.15367
$521$	4326.00	0.363773	0.181886	−	0.983320i	$-0.441780\pi$
$521$	4326.00	0.363773	0.181886	+	0.983320i	$0.441780\pi$
$522$	−2178.00	−0.182622
$523$	−6328.00	−0.529071	−0.264535	−	0.964376i	$-0.585219\pi$
$523$	−6328.00	−0.529071	−0.264535	+	0.964376i	$0.585219\pi$
$524$	−15344.0	−1.27921
$525$	−1140.00	−0.0947689
$526$	1994.00	0.165290
$527$	−2772.00	−0.229128
$528$	−492.000	−0.0405522
$529$	−5443.00	−0.447358
$530$	−840.000	−0.0688439
$531$	1692.00	0.138280
$532$	−2660.00	−0.216777
$533$	29640.0	2.40873
$534$	−2538.00	−0.205674
$535$	23184.0	1.87352
$536$	1560.00	0.125712
$537$	8244.00	0.662486
$538$	−7214.00	−0.578100
$539$	−228.000	−0.0182201
$540$	2268.00	0.180739
$541$	9378.00	0.745271	0.372636	−	0.927978i	$-0.378454\pi$
$541$	9378.00	0.745271	0.372636	+	0.927978i	$0.378454\pi$
$542$	−7572.00	−0.600083
$543$	4008.00	0.316758
$544$	−3542.00	−0.279158
$545$	−13152.0	−1.03371
$546$	−4560.00	−0.357418
$547$	−5048.00	−0.394583	−0.197291	−	0.980345i	$-0.563215\pi$
$547$	−5048.00	−0.394583	−0.197291	+	0.980345i	$0.563215\pi$
$548$	−3906.00	−0.304482
$549$	−6354.00	−0.493956
$550$	76.0000	0.00589209
$551$	−4598.00	−0.355502
$552$	3690.00	0.284523
$553$	−1880.00	−0.144567
$554$	−6262.00	−0.480229
$555$	6480.00	0.495605
$556$	−476.000	−0.0363074
$557$	752.000	0.0572051	0.0286026	−	0.999591i	$-0.490894\pi$
$557$	752.000	0.0572051	0.0286026	+	0.999591i	$0.490894\pi$
$558$	1134.00	0.0860323
$559$	−23408.0	−1.77111
$560$	9840.00	0.742528
$561$	−264.000	−0.0198683
$562$	2710.00	0.203406
$563$	18156.0	1.35912	0.679560	−	0.733620i	$-0.262171\pi$
$563$	18156.0	1.35912	0.679560	+	0.733620i	$0.262171\pi$
$564$	10962.0	0.818410
$565$	−17688.0	−1.31706
$566$	556.000	0.0412905
$567$	−1620.00	−0.119989
$568$	−6480.00	−0.478688
$569$	1398.00	0.103000	0.0515002	−	0.998673i	$-0.483600\pi$
$569$	1398.00	0.103000	0.0515002	+	0.998673i	$0.483600\pi$
$570$	−684.000	−0.0502625
$571$	21180.0	1.55229	0.776143	−	0.630557i	$-0.217173\pi$
$571$	21180.0	1.55229	0.776143	+	0.630557i	$0.217173\pi$
$572$	−2128.00	−0.155553
$573$	−1818.00	−0.132545
$574$	−7800.00	−0.567188
$575$	1558.00	0.112997
$576$	−1503.00	−0.108724
$577$	27186.0	1.96147	0.980735	−	0.195344i	$-0.0625823\pi$
$577$	27186.0	1.96147	0.980735	+	0.195344i	$0.0625823\pi$
$578$	4429.00	0.318723
$579$	−9006.00	−0.646419
$580$	20328.0	1.45530
$581$	25920.0	1.85085
$582$	−2490.00	−0.177343
$583$	280.000	0.0198909
$584$	10770.0	0.763126
$585$	8208.00	0.580101
$586$	−3694.00	−0.260406
$587$	−10204.0	−0.717486	−0.358743	−	0.933436i	$-0.616795\pi$
$587$	−10204.0	−0.717486	−0.358743	+	0.933436i	$0.616795\pi$
$588$	−1197.00	−0.0839514
$589$	2394.00	0.167475
$590$	2256.00	0.157420
$591$	−13368.0	−0.930433
$592$	−7380.00	−0.512358
$593$	9978.00	0.690974	0.345487	−	0.938424i	$-0.387714\pi$
$593$	9978.00	0.690974	0.345487	+	0.938424i	$0.387714\pi$
$594$	108.000	0.00746009
$595$	5280.00	0.363796
$596$	−4032.00	−0.277109
$597$	−8532.00	−0.584910
$598$	6232.00	0.426163
$599$	−11100.0	−0.757151	−0.378576	−	0.925570i	$-0.623586\pi$
$599$	−11100.0	−0.757151	−0.378576	+	0.925570i	$0.623586\pi$
$600$	855.000	0.0581754
$601$	−3030.00	−0.205651	−0.102826	−	0.994699i	$-0.532788\pi$
$601$	−3030.00	−0.205651	−0.102826	+	0.994699i	$0.532788\pi$
$602$	6160.00	0.417048
$603$	936.000	0.0632121
$604$	−6930.00	−0.466850
$605$	15780.0	1.06041
$606$	−4836.00	−0.324173
$607$	10478.0	0.700641	0.350320	−	0.936630i	$-0.386073\pi$
$607$	10478.0	0.700641	0.350320	+	0.936630i	$0.386073\pi$
$608$	3059.00	0.204044
$609$	−14520.0	−0.966141
$610$	−8472.00	−0.562330
$611$	39672.0	2.62677
$612$	−1386.00	−0.0915453
$613$	2706.00	0.178294	0.0891471	−	0.996018i	$-0.471586\pi$
$613$	2706.00	0.178294	0.0891471	+	0.996018i	$0.471586\pi$
$614$	−3384.00	−0.222422
$615$	14040.0	0.920565
$616$	1200.00	0.0784892
$617$	−19734.0	−1.28762	−0.643810	−	0.765186i	$-0.722647\pi$
$617$	−19734.0	−1.28762	−0.643810	+	0.765186i	$0.722647\pi$
$618$	5622.00	0.365939
$619$	21196.0	1.37632	0.688158	−	0.725561i	$-0.258420\pi$
$619$	21196.0	1.37632	0.688158	+	0.725561i	$0.258420\pi$
$620$	−10584.0	−0.685587
$621$	2214.00	0.143067
$622$	9666.00	0.623105
$623$	−16920.0	−1.08810
$624$	−9348.00	−0.599711
$625$	−17639.0	−1.12890
$626$	6794.00	0.433775
$627$	228.000	0.0145222
$628$	4578.00	0.290895
$629$	−3960.00	−0.251026
$630$	−2160.00	−0.136598
$631$	5040.00	0.317970	0.158985	−	0.987281i	$-0.449178\pi$
$631$	5040.00	0.317970	0.158985	+	0.987281i	$0.449178\pi$
$632$	1410.00	0.0887449
$633$	−9324.00	−0.585459
$634$	3242.00	0.203086
$635$	13992.0	0.874418
$636$	1470.00	0.0916498
$637$	−4332.00	−0.269451
$638$	968.000	0.0600682
$639$	−3888.00	−0.240699
$640$	−17460.0	−1.07839
$641$	−13602.0	−0.838138	−0.419069	−	0.907954i	$-0.637644\pi$
$641$	−13602.0	−0.838138	−0.419069	+	0.907954i	$0.637644\pi$
$642$	5796.00	0.356308
$643$	4628.00	0.283842	0.141921	−	0.989878i	$-0.454672\pi$
$643$	4628.00	0.283842	0.141921	+	0.989878i	$0.454672\pi$
$644$	11480.0	0.702446
$645$	−11088.0	−0.676883
$646$	418.000	0.0254582
$647$	−14142.0	−0.859319	−0.429659	−	0.902991i	$-0.641366\pi$
$647$	−14142.0	−0.859319	−0.429659	+	0.902991i	$0.641366\pi$
$648$	1215.00	0.0736570
$649$	−752.000	−0.0454832
$650$	1444.00	0.0871359
$651$	7560.00	0.455146
$652$	6300.00	0.378416
$653$	−14424.0	−0.864402	−0.432201	−	0.901777i	$-0.642263\pi$
$653$	−14424.0	−0.864402	−0.432201	+	0.901777i	$0.642263\pi$
$654$	−3288.00	−0.196592
$655$	−26304.0	−1.56913
$656$	−15990.0	−0.951684
$657$	6462.00	0.383724
$658$	−10440.0	−0.618531
$659$	24044.0	1.42128	0.710638	−	0.703558i	$-0.248406\pi$
$659$	24044.0	1.42128	0.710638	+	0.703558i	$0.248406\pi$
$660$	−1008.00	−0.0594490
$661$	10092.0	0.593848	0.296924	−	0.954901i	$-0.404039\pi$
$661$	10092.0	0.593848	0.296924	+	0.954901i	$0.404039\pi$
$662$	−176.000	−0.0103330
$663$	−5016.00	−0.293824
$664$	−19440.0	−1.13617
$665$	−4560.00	−0.265909
$666$	1620.00	0.0942548
$667$	19844.0	1.15197
$668$	−5180.00	−0.300030
$669$	−14058.0	−0.812427
$670$	1248.00	0.0719619
$671$	2824.00	0.162473
$672$	9660.00	0.554528
$673$	−7098.00	−0.406549	−0.203275	−	0.979122i	$-0.565158\pi$
$673$	−7098.00	−0.406549	−0.203275	+	0.979122i	$0.565158\pi$
$674$	4262.00	0.243570
$675$	513.000	0.0292524
$676$	−25053.0	−1.42541
$677$	29762.0	1.68958	0.844791	−	0.535097i	$-0.179725\pi$
$677$	29762.0	1.68958	0.844791	+	0.535097i	$0.179725\pi$
$678$	−4422.00	−0.250481
$679$	−16600.0	−0.938217
$680$	−3960.00	−0.223322
$681$	−9108.00	−0.512510
$682$	−504.000	−0.0282979
$683$	11748.0	0.658162	0.329081	−	0.944302i	$-0.393261\pi$
$683$	11748.0	0.658162	0.329081	+	0.944302i	$0.393261\pi$
$684$	1197.00	0.0669129
$685$	−6696.00	−0.373491
$686$	−5720.00	−0.318354
$687$	−14910.0	−0.828023
$688$	12628.0	0.699765
$689$	5320.00	0.294159
$690$	2952.00	0.162871
$691$	−30676.0	−1.68881	−0.844407	−	0.535703i	$-0.820047\pi$
$691$	−30676.0	−1.68881	−0.844407	+	0.535703i	$0.820047\pi$
$692$	−4074.00	−0.223801
$693$	720.000	0.0394669
$694$	−7060.00	−0.386158
$695$	−816.000	−0.0445362
$696$	10890.0	0.593081
$697$	−8580.00	−0.466271
$698$	−4746.00	−0.257362
$699$	−8946.00	−0.484076
$700$	2660.00	0.143626
$701$	31228.0	1.68255	0.841273	−	0.540610i	$-0.181806\pi$
$701$	31228.0	1.68255	0.841273	+	0.540610i	$0.181806\pi$
$702$	2052.00	0.110324
$703$	3420.00	0.183482
$704$	668.000	0.0357616
$705$	18792.0	1.00390
$706$	−2546.00	−0.135722
$707$	−32240.0	−1.71501
$708$	−3948.00	−0.209569
$709$	−14658.0	−0.776435	−0.388218	−	0.921568i	$-0.626909\pi$
$709$	−14658.0	−0.776435	−0.388218	+	0.921568i	$0.626909\pi$
$710$	−5184.00	−0.274017
$711$	846.000	0.0446237
$712$	12690.0	0.667946
$713$	−10332.0	−0.542688
$714$	1320.00	0.0691873
$715$	−3648.00	−0.190808
$716$	−19236.0	−1.00403
$717$	−1566.00	−0.0815667
$718$	1702.00	0.0884653
$719$	5502.00	0.285382	0.142691	−	0.989767i	$-0.454424\pi$
$719$	5502.00	0.285382	0.142691	+	0.989767i	$0.454424\pi$
$720$	−4428.00	−0.229197
$721$	37480.0	1.93596
$722$	−361.000	−0.0186081
$723$	10050.0	0.516962
$724$	−9352.00	−0.480061
$725$	4598.00	0.235539
$726$	3945.00	0.201670
$727$	6136.00	0.313028	0.156514	−	0.987676i	$-0.449974\pi$
$727$	6136.00	0.313028	0.156514	+	0.987676i	$0.449974\pi$
$728$	22800.0	1.16075
$729$	729.000	0.0370370
$730$	8616.00	0.436839
$731$	6776.00	0.342845
$732$	14826.0	0.748613
$733$	24442.0	1.23163	0.615816	−	0.787890i	$-0.288827\pi$
$733$	24442.0	1.23163	0.615816	+	0.787890i	$0.288827\pi$
$734$	7844.00	0.394451
$735$	−2052.00	−0.102978
$736$	−13202.0	−0.661185
$737$	−416.000	−0.0207918
$738$	3510.00	0.175074
$739$	11980.0	0.596335	0.298167	−	0.954514i	$-0.403625\pi$
$739$	11980.0	0.596335	0.298167	+	0.954514i	$0.403625\pi$
$740$	−15120.0	−0.751111
$741$	4332.00	0.214764
$742$	−1400.00	−0.0692663
$743$	15524.0	0.766515	0.383257	−	0.923642i	$-0.374802\pi$
$743$	15524.0	0.766515	0.383257	+	0.923642i	$0.374802\pi$
$744$	−5670.00	−0.279398
$745$	−6912.00	−0.339914
$746$	−13612.0	−0.668057
$747$	−11664.0	−0.571303
$748$	616.000	0.0301112
$749$	38640.0	1.88501
$750$	−3816.00	−0.185787
$751$	10494.0	0.509895	0.254948	−	0.966955i	$-0.417942\pi$
$751$	10494.0	0.509895	0.254948	+	0.966955i	$0.417942\pi$
$752$	−21402.0	−1.03783
$753$	−8904.00	−0.430916
$754$	18392.0	0.888325
$755$	−11880.0	−0.572659
$756$	3780.00	0.181848
$757$	−24446.0	−1.17372	−0.586859	−	0.809689i	$-0.699636\pi$
$757$	−24446.0	−1.17372	−0.586859	+	0.809689i	$0.699636\pi$
$758$	−976.000	−0.0467677
$759$	−984.000	−0.0470579
$760$	3420.00	0.163232
$761$	−28650.0	−1.36473	−0.682366	−	0.731010i	$-0.739049\pi$
$761$	−28650.0	−1.36473	−0.682366	+	0.731010i	$0.739049\pi$
$762$	3498.00	0.166298
$763$	−21920.0	−1.04005
$764$	4242.00	0.200877
$765$	−2376.00	−0.112293
$766$	−2152.00	−0.101508
$767$	−14288.0	−0.672633
$768$	−357.000	−0.0167736
$769$	6974.00	0.327034	0.163517	−	0.986541i	$-0.447716\pi$
$769$	6974.00	0.327034	0.163517	+	0.986541i	$0.447716\pi$
$770$	960.000	0.0449299
$771$	−3702.00	−0.172924
$772$	21014.0	0.979677
$773$	−6170.00	−0.287089	−0.143544	−	0.989644i	$-0.545850\pi$
$773$	−6170.00	−0.287089	−0.143544	+	0.989644i	$0.545850\pi$
$774$	−2772.00	−0.128731
$775$	−2394.00	−0.110961
$776$	12450.0	0.575939
$777$	10800.0	0.498646
$778$	−10572.0	−0.487178
$779$	7410.00	0.340810
$780$	−19152.0	−0.879169
$781$	1728.00	0.0791712
$782$	−1804.00	−0.0824948
$783$	6534.00	0.298220
$784$	2337.00	0.106460
$785$	7848.00	0.356824
$786$	−6576.00	−0.298420
$787$	−17900.0	−0.810757	−0.405379	−	0.914149i	$-0.632860\pi$
$787$	−17900.0	−0.810757	−0.405379	+	0.914149i	$0.632860\pi$
$788$	31192.0	1.41011
$789$	−5982.00	−0.269917
$790$	1128.00	0.0508006
$791$	−29480.0	−1.32514
$792$	−540.000	−0.0242274
$793$	53656.0	2.40275
$794$	10910.0	0.487634
$795$	2520.00	0.112422
$796$	19908.0	0.886458
$797$	31358.0	1.39367	0.696836	−	0.717230i	$-0.254590\pi$
$797$	31358.0	1.39367	0.696836	+	0.717230i	$0.254590\pi$
$798$	−1140.00	−0.0505709
$799$	−11484.0	−0.508479
$800$	−3059.00	−0.135190
$801$	7614.00	0.335864
$802$	−10146.0	−0.446718
$803$	−2872.00	−0.126215
$804$	−2184.00	−0.0958007
$805$	19680.0	0.861651
$806$	−9576.00	−0.418487
$807$	21642.0	0.944033
$808$	24180.0	1.05278
$809$	−20210.0	−0.878301	−0.439151	−	0.898413i	$-0.644721\pi$
$809$	−20210.0	−0.878301	−0.439151	+	0.898413i	$0.644721\pi$
$810$	972.000	0.0421637
$811$	−8648.00	−0.374442	−0.187221	−	0.982318i	$-0.559948\pi$
$811$	−8648.00	−0.374442	−0.187221	+	0.982318i	$0.559948\pi$
$812$	33880.0	1.46423
$813$	22716.0	0.979932
$814$	−720.000	−0.0310024
$815$	10800.0	0.464181
$816$	2706.00	0.116089
$817$	−5852.00	−0.250594
$818$	13706.0	0.585842
$819$	13680.0	0.583660
$820$	−32760.0	−1.39516
$821$	−8940.00	−0.380034	−0.190017	−	0.981781i	$-0.560854\pi$
$821$	−8940.00	−0.380034	−0.190017	+	0.981781i	$0.560854\pi$
$822$	−1674.00	−0.0710310
$823$	−17504.0	−0.741374	−0.370687	−	0.928758i	$-0.620878\pi$
$823$	−17504.0	−0.741374	−0.370687	+	0.928758i	$0.620878\pi$
$824$	−28110.0	−1.18842
$825$	−228.000	−0.00962175
$826$	3760.00	0.158386
$827$	4356.00	0.183160	0.0915798	−	0.995798i	$-0.470808\pi$
$827$	4356.00	0.183160	0.0915798	+	0.995798i	$0.470808\pi$
$828$	−5166.00	−0.216825
$829$	1528.00	0.0640164	0.0320082	−	0.999488i	$-0.489810\pi$
$829$	1528.00	0.0640164	0.0320082	+	0.999488i	$0.489810\pi$
$830$	−15552.0	−0.650383
$831$	18786.0	0.784211
$832$	12692.0	0.528865
$833$	1254.00	0.0521591
$834$	−204.000	−0.00846995
$835$	−8880.00	−0.368030
$836$	−532.000	−0.0220091
$837$	−3402.00	−0.140490
$838$	−6812.00	−0.280807
$839$	−30204.0	−1.24286	−0.621429	−	0.783470i	$-0.713448\pi$
$839$	−30204.0	−1.24286	−0.621429	+	0.783470i	$0.713448\pi$
$840$	10800.0	0.443614
$841$	34175.0	1.40125
$842$	6724.00	0.275207
$843$	−8130.00	−0.332161
$844$	21756.0	0.887290
$845$	−42948.0	−1.74847
$846$	4698.00	0.190923
$847$	26300.0	1.06692
$848$	−2870.00	−0.116222
$849$	−1668.00	−0.0674271
$850$	−418.000	−0.0168674
$851$	−14760.0	−0.594555
$852$	9072.00	0.364791
$853$	−9218.00	−0.370010	−0.185005	−	0.982738i	$-0.559230\pi$
$853$	−9218.00	−0.370010	−0.185005	+	0.982738i	$0.559230\pi$
$854$	−14120.0	−0.565780
$855$	2052.00	0.0820783
$856$	−28980.0	−1.15714
$857$	−44554.0	−1.77589	−0.887944	−	0.459952i	$-0.847867\pi$
$857$	−44554.0	−1.77589	−0.887944	+	0.459952i	$0.847867\pi$
$858$	−912.000	−0.0362881
$859$	9828.00	0.390369	0.195185	−	0.980767i	$-0.437469\pi$
$859$	9828.00	0.390369	0.195185	+	0.980767i	$0.437469\pi$
$860$	25872.0	1.02585
$861$	23400.0	0.926214
$862$	−13876.0	−0.548281
$863$	−11668.0	−0.460236	−0.230118	−	0.973163i	$-0.573911\pi$
$863$	−11668.0	−0.460236	−0.230118	+	0.973163i	$0.573911\pi$
$864$	−4347.00	−0.171167
$865$	−6984.00	−0.274524
$866$	−342.000	−0.0134199
$867$	−13287.0	−0.520473
$868$	−17640.0	−0.689793
$869$	−376.000	−0.0146777
$870$	8712.00	0.339499
$871$	−7904.00	−0.307482
$872$	16440.0	0.638450
$873$	7470.00	0.289600
$874$	1558.00	0.0602976
$875$	−25440.0	−0.982890
$876$	−15078.0	−0.581551
$877$	17292.0	0.665803	0.332902	−	0.942962i	$-0.391972\pi$
$877$	17292.0	0.665803	0.332902	+	0.942962i	$0.391972\pi$
$878$	6526.00	0.250845
$879$	11082.0	0.425241
$880$	1968.00	0.0753878
$881$	4618.00	0.176600	0.0882999	−	0.996094i	$-0.471857\pi$
$881$	4618.00	0.176600	0.0882999	+	0.996094i	$0.471857\pi$
$882$	−513.000	−0.0195846
$883$	−17740.0	−0.676103	−0.338051	−	0.941128i	$-0.609768\pi$
$883$	−17740.0	−0.676103	−0.338051	+	0.941128i	$0.609768\pi$
$884$	11704.0	0.445303
$885$	−6768.00	−0.257066
$886$	5020.00	0.190350
$887$	−24516.0	−0.928035	−0.464017	−	0.885826i	$-0.653592\pi$
$887$	−24516.0	−0.928035	−0.464017	+	0.885826i	$0.653592\pi$
$888$	−8100.00	−0.306102
$889$	23320.0	0.879784
$890$	10152.0	0.382355
$891$	−324.000	−0.0121823
$892$	32802.0	1.23127
$893$	9918.00	0.371661
$894$	−1728.00	−0.0646454
$895$	−32976.0	−1.23158
$896$	−29100.0	−1.08500
$897$	−18696.0	−0.695921
$898$	−9486.00	−0.352508
$899$	−30492.0	−1.13122
$900$	−1197.00	−0.0443333
$901$	−1540.00	−0.0569421
$902$	−1560.00	−0.0575857
$903$	−18480.0	−0.681036
$904$	22110.0	0.813460
$905$	−16032.0	−0.588864
$906$	−2970.00	−0.108909
$907$	8392.00	0.307224	0.153612	−	0.988131i	$-0.450909\pi$
$907$	8392.00	0.307224	0.153612	+	0.988131i	$0.450909\pi$
$908$	21252.0	0.776732
$909$	14508.0	0.529373
$910$	18240.0	0.664451
$911$	17004.0	0.618406	0.309203	−	0.950996i	$-0.399938\pi$
$911$	17004.0	0.618406	0.309203	+	0.950996i	$0.399938\pi$
$912$	−2337.00	−0.0848529
$913$	5184.00	0.187914
$914$	7262.00	0.262807
$915$	25416.0	0.918281
$916$	34790.0	1.25491
$917$	−43840.0	−1.57876
$918$	−594.000	−0.0213561
$919$	−46288.0	−1.66148	−0.830740	−	0.556661i	$-0.812082\pi$
$919$	−46288.0	−1.66148	−0.830740	+	0.556661i	$0.812082\pi$
$920$	−14760.0	−0.528938
$921$	10152.0	0.363214
$922$	13968.0	0.498928
$923$	32832.0	1.17083
$924$	−1680.00	−0.0598138
$925$	−3420.00	−0.121566
$926$	−4604.00	−0.163388
$927$	−16866.0	−0.597575
$928$	−38962.0	−1.37822
$929$	−4978.00	−0.175805	−0.0879025	−	0.996129i	$-0.528016\pi$
$929$	−4978.00	−0.175805	−0.0879025	+	0.996129i	$0.528016\pi$
$930$	−4536.00	−0.159937
$931$	−1083.00	−0.0381245
$932$	20874.0	0.733638
$933$	−28998.0	−1.01753
$934$	19480.0	0.682447
$935$	1056.00	0.0369357
$936$	−10260.0	−0.358289
$937$	39798.0	1.38756	0.693780	−	0.720187i	$-0.255944\pi$
$937$	39798.0	1.38756	0.693780	+	0.720187i	$0.255944\pi$
$938$	2080.00	0.0724034
$939$	−20382.0	−0.708351
$940$	−43848.0	−1.52145
$941$	31662.0	1.09687	0.548433	−	0.836194i	$-0.315224\pi$
$941$	31662.0	1.09687	0.548433	+	0.836194i	$0.315224\pi$
$942$	1962.00	0.0678614
$943$	−31980.0	−1.10436
$944$	7708.00	0.265756
$945$	6480.00	0.223063
$946$	1232.00	0.0423423
$947$	−44744.0	−1.53536	−0.767679	−	0.640834i	$-0.778589\pi$
$947$	−44744.0	−1.53536	−0.767679	+	0.640834i	$0.778589\pi$
$948$	−1974.00	−0.0676293
$949$	−54568.0	−1.86655
$950$	361.000	0.0123288
$951$	−9726.00	−0.331637
$952$	−6600.00	−0.224692
$953$	18626.0	0.633112	0.316556	−	0.948574i	$-0.397474\pi$
$953$	18626.0	0.633112	0.316556	+	0.948574i	$0.397474\pi$
$954$	630.000	0.0213805
$955$	7272.00	0.246405
$956$	3654.00	0.123618
$957$	−2904.00	−0.0980909
$958$	12134.0	0.409219
$959$	−11160.0	−0.375782
$960$	6012.00	0.202121
$961$	−13915.0	−0.467087
$962$	−13680.0	−0.458483
$963$	−17388.0	−0.581849
$964$	−23450.0	−0.783479
$965$	36024.0	1.20171
$966$	4920.00	0.163870
$967$	30244.0	1.00577	0.502886	−	0.864353i	$-0.332272\pi$
$967$	30244.0	1.00577	0.502886	+	0.864353i	$0.332272\pi$
$968$	−19725.0	−0.654944
$969$	−1254.00	−0.0415730
$970$	9960.00	0.329687
$971$	46572.0	1.53920	0.769602	−	0.638524i	$-0.220455\pi$
$971$	46572.0	1.53920	0.769602	+	0.638524i	$0.220455\pi$
$972$	−1701.00	−0.0561313
$973$	−1360.00	−0.0448095
$974$	15658.0	0.515107
$975$	−4332.00	−0.142292
$976$	−28946.0	−0.949323
$977$	30162.0	0.987685	0.493842	−	0.869551i	$-0.335592\pi$
$977$	30162.0	0.987685	0.493842	+	0.869551i	$0.335592\pi$
$978$	2700.00	0.0882786
$979$	−3384.00	−0.110473
$980$	4788.00	0.156068
$981$	9864.00	0.321033
$982$	−2520.00	−0.0818905
$983$	17792.0	0.577291	0.288645	−	0.957436i	$-0.406795\pi$
$983$	17792.0	0.577291	0.288645	+	0.957436i	$0.406795\pi$
$984$	−17550.0	−0.568571
$985$	53472.0	1.72971
$986$	−5324.00	−0.171958
$987$	31320.0	1.01006
$988$	−10108.0	−0.325484
$989$	25256.0	0.812026
$990$	−432.000	−0.0138685
$991$	9434.00	0.302403	0.151201	−	0.988503i	$-0.451686\pi$
$991$	9434.00	0.302403	0.151201	+	0.988503i	$0.451686\pi$
$992$	20286.0	0.649275
$993$	528.000	0.0168737
$994$	−8640.00	−0.275698
$995$	34128.0	1.08737
$996$	27216.0	0.865835
$997$	−61286.0	−1.94679	−0.973394	−	0.229139i	$-0.926409\pi$
$997$	−61286.0	−1.94679	−0.973394	+	0.229139i	$0.926409\pi$
$998$	9460.00	0.300051
$999$	−4860.00	−0.153918

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	57.4.a.a.1.1	✓	1
3.2	odd	2		171.4.a.b.1.1		1
4.3	odd	2		912.4.a.a.1.1		1
5.4	even	2		1425.4.a.c.1.1		1
19.18	odd	2		1083.4.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
57.4.a.a.1.1	✓	1	1.1	even	1	trivial
171.4.a.b.1.1		1	3.2	odd	2
912.4.a.a.1.1		1	4.3	odd	2
1083.4.a.a.1.1		1	19.18	odd	2
1425.4.a.c.1.1		1	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 57 = 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	57.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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