LMFDB - Embedded newform 102.4.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [102,4,Mod(1,102)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(102, 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("102.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 102 = 2 \cdot 3 \cdot 17 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	102.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:	$6.01819482059$
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	$\chi$	$=$	102.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -5.00000 q^{5} -6.00000 q^{6} -32.0000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$

\(q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -5.00000 q^{5} -6.00000 q^{6} -32.0000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +10.0000 q^{10} +27.0000 q^{11} +12.0000 q^{12} -69.0000 q^{13} +64.0000 q^{14} -15.0000 q^{15} +16.0000 q^{16} -17.0000 q^{17} -18.0000 q^{18} -83.0000 q^{19} -20.0000 q^{20} -96.0000 q^{21} -54.0000 q^{22} -117.000 q^{23} -24.0000 q^{24} -100.000 q^{25} +138.000 q^{26} +27.0000 q^{27} -128.000 q^{28} +94.0000 q^{29} +30.0000 q^{30} +198.000 q^{31} -32.0000 q^{32} +81.0000 q^{33} +34.0000 q^{34} +160.000 q^{35} +36.0000 q^{36} -244.000 q^{37} +166.000 q^{38} -207.000 q^{39} +40.0000 q^{40} +169.000 q^{41} +192.000 q^{42} +227.000 q^{43} +108.000 q^{44} -45.0000 q^{45} +234.000 q^{46} -382.000 q^{47} +48.0000 q^{48} +681.000 q^{49} +200.000 q^{50} -51.0000 q^{51} -276.000 q^{52} +686.000 q^{53} -54.0000 q^{54} -135.000 q^{55} +256.000 q^{56} -249.000 q^{57} -188.000 q^{58} +450.000 q^{59} -60.0000 q^{60} -700.000 q^{61} -396.000 q^{62} -288.000 q^{63} +64.0000 q^{64} +345.000 q^{65} -162.000 q^{66} +540.000 q^{67} -68.0000 q^{68} -351.000 q^{69} -320.000 q^{70} -276.000 q^{71} -72.0000 q^{72} -298.000 q^{73} +488.000 q^{74} -300.000 q^{75} -332.000 q^{76} -864.000 q^{77} +414.000 q^{78} -182.000 q^{79} -80.0000 q^{80} +81.0000 q^{81} -338.000 q^{82} +282.000 q^{83} -384.000 q^{84} +85.0000 q^{85} -454.000 q^{86} +282.000 q^{87} -216.000 q^{88} -1468.00 q^{89} +90.0000 q^{90} +2208.00 q^{91} -468.000 q^{92} +594.000 q^{93} +764.000 q^{94} +415.000 q^{95} -96.0000 q^{96} -1140.00 q^{97} -1362.00 q^{98} +243.000 q^{99} +O(q^{100})\)

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 $

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	4.00000	0.500000
$5$	−5.00000	−0.447214	−0.223607	−	0.974679i	$-0.571783\pi$
$5$	−5.00000	−0.447214	−0.223607	+	0.974679i	$0.571783\pi$
$6$	−6.00000	−0.408248
$7$	−32.0000	−1.72784	−0.863919	−	0.503631i	$-0.831997\pi$
$7$	−32.0000	−1.72784	−0.863919	+	0.503631i	$0.831997\pi$
$8$	−8.00000	−0.353553
$9$	9.00000	0.333333
$10$	10.0000	0.316228
$11$	27.0000	0.740073	0.370037	−	0.929017i	$-0.379345\pi$
$11$	27.0000	0.740073	0.370037	+	0.929017i	$0.379345\pi$
$12$	12.0000	0.288675
$13$	−69.0000	−1.47209	−0.736044	−	0.676933i	$-0.763309\pi$
$13$	−69.0000	−1.47209	−0.736044	+	0.676933i	$0.763309\pi$
$14$	64.0000	1.22177
$15$	−15.0000	−0.258199
$16$	16.0000	0.250000
$17$	−17.0000	−0.242536
$17$	−17.0000	−0.242536
$18$	−18.0000	−0.235702
$19$	−83.0000	−1.00218	−0.501092	−	0.865394i	$-0.667068\pi$
$19$	−83.0000	−1.00218	−0.501092	+	0.865394i	$0.667068\pi$
$20$	−20.0000	−0.223607
$21$	−96.0000	−0.997567
$22$	−54.0000	−0.523311
$23$	−117.000	−1.06070	−0.530352	−	0.847778i	$-0.677940\pi$
$23$	−117.000	−1.06070	−0.530352	+	0.847778i	$0.677940\pi$
$24$	−24.0000	−0.204124
$25$	−100.000	−0.800000
$26$	138.000	1.04092
$27$	27.0000	0.192450
$28$	−128.000	−0.863919
$29$	94.0000	0.601909	0.300955	−	0.953638i	$-0.402695\pi$
$29$	94.0000	0.601909	0.300955	+	0.953638i	$0.402695\pi$
$30$	30.0000	0.182574
$31$	198.000	1.14716	0.573578	−	0.819151i	$-0.305555\pi$
$31$	198.000	1.14716	0.573578	+	0.819151i	$0.305555\pi$
$32$	−32.0000	−0.176777
$33$	81.0000	0.427282
$34$	34.0000	0.171499
$35$	160.000	0.772712
$36$	36.0000	0.166667
$37$	−244.000	−1.08414	−0.542072	−	0.840332i	$-0.682360\pi$
$37$	−244.000	−1.08414	−0.542072	+	0.840332i	$0.682360\pi$
$38$	166.000	0.708651
$39$	−207.000	−0.849911
$40$	40.0000	0.158114
$41$	169.000	0.643741	0.321870	−	0.946784i	$-0.395688\pi$
$41$	169.000	0.643741	0.321870	+	0.946784i	$0.395688\pi$
$42$	192.000	0.705387
$43$	227.000	0.805051	0.402525	−	0.915409i	$-0.368133\pi$
$43$	227.000	0.805051	0.402525	+	0.915409i	$0.368133\pi$
$44$	108.000	0.370037
$45$	−45.0000	−0.149071
$46$	234.000	0.750031
$47$	−382.000	−1.18554	−0.592770	−	0.805371i	$-0.701966\pi$
$47$	−382.000	−1.18554	−0.592770	+	0.805371i	$0.701966\pi$
$48$	48.0000	0.144338
$49$	681.000	1.98542
$50$	200.000	0.565685
$51$	−51.0000	−0.140028
$52$	−276.000	−0.736044
$53$	686.000	1.77791	0.888956	−	0.457992i	$-0.151431\pi$
$53$	686.000	1.77791	0.888956	+	0.457992i	$0.151431\pi$
$54$	−54.0000	−0.136083
$55$	−135.000	−0.330971
$56$	256.000	0.610883
$57$	−249.000	−0.578612
$58$	−188.000	−0.425614
$59$	450.000	0.992966	0.496483	−	0.868046i	$-0.334624\pi$
$59$	450.000	0.992966	0.496483	+	0.868046i	$0.334624\pi$
$60$	−60.0000	−0.129099
$61$	−700.000	−1.46928	−0.734638	−	0.678459i	$-0.762648\pi$
$61$	−700.000	−1.46928	−0.734638	+	0.678459i	$0.762648\pi$
$62$	−396.000	−0.811162
$63$	−288.000	−0.575946
$64$	64.0000	0.125000
$65$	345.000	0.658338
$66$	−162.000	−0.302134
$67$	540.000	0.984649	0.492325	−	0.870412i	$-0.336147\pi$
$67$	540.000	0.984649	0.492325	+	0.870412i	$0.336147\pi$
$68$	−68.0000	−0.121268
$69$	−351.000	−0.612398
$70$	−320.000	−0.546390
$71$	−276.000	−0.461340	−0.230670	−	0.973032i	$-0.574092\pi$
$71$	−276.000	−0.461340	−0.230670	+	0.973032i	$0.574092\pi$
$72$	−72.0000	−0.117851
$73$	−298.000	−0.477784	−0.238892	−	0.971046i	$-0.576784\pi$
$73$	−298.000	−0.477784	−0.238892	+	0.971046i	$0.576784\pi$
$74$	488.000	0.766606
$75$	−300.000	−0.461880
$76$	−332.000	−0.501092
$77$	−864.000	−1.27873
$78$	414.000	0.600978
$79$	−182.000	−0.259197	−0.129599	−	0.991567i	$-0.541369\pi$
$79$	−182.000	−0.259197	−0.129599	+	0.991567i	$0.541369\pi$
$80$	−80.0000	−0.111803
$81$	81.0000	0.111111
$82$	−338.000	−0.455193
$83$	282.000	0.372934	0.186467	−	0.982461i	$-0.440296\pi$
$83$	282.000	0.372934	0.186467	+	0.982461i	$0.440296\pi$
$84$	−384.000	−0.498784
$85$	85.0000	0.108465
$86$	−454.000	−0.569257
$87$	282.000	0.347512
$88$	−216.000	−0.261655
$89$	−1468.00	−1.74840	−0.874200	−	0.485565i	$-0.838614\pi$
$89$	−1468.00	−1.74840	−0.874200	+	0.485565i	$0.838614\pi$
$90$	90.0000	0.105409
$91$	2208.00	2.54353
$92$	−468.000	−0.530352
$93$	594.000	0.662311
$94$	764.000	0.838304
$95$	415.000	0.448191
$96$	−96.0000	−0.102062
$97$	−1140.00	−1.19329	−0.596647	−	0.802504i	$-0.703501\pi$
$97$	−1140.00	−1.19329	−0.596647	+	0.802504i	$0.703501\pi$
$98$	−1362.00	−1.40391
$99$	243.000	0.246691
$100$	−400.000	−0.400000
$101$	−1400.00	−1.37926	−0.689630	−	0.724162i	$-0.742227\pi$
$101$	−1400.00	−1.37926	−0.689630	+	0.724162i	$0.742227\pi$
$102$	102.000	0.0990148
$103$	1939.00	1.85491	0.927453	−	0.373939i	$-0.121993\pi$
$103$	1939.00	1.85491	0.927453	+	0.373939i	$0.121993\pi$
$104$	552.000	0.520462
$105$	480.000	0.446126
$106$	−1372.00	−1.25717
$107$	39.0000	0.0352362	0.0176181	−	0.999845i	$-0.494392\pi$
$107$	39.0000	0.0352362	0.0176181	+	0.999845i	$0.494392\pi$
$108$	108.000	0.0962250
$109$	−428.000	−0.376101	−0.188050	−	0.982159i	$-0.560217\pi$
$109$	−428.000	−0.376101	−0.188050	+	0.982159i	$0.560217\pi$
$110$	270.000	0.234032
$111$	−732.000	−0.625931
$112$	−512.000	−0.431959
$113$	715.000	0.595235	0.297617	−	0.954685i	$-0.403808\pi$
$113$	715.000	0.595235	0.297617	+	0.954685i	$0.403808\pi$
$114$	498.000	0.409140
$115$	585.000	0.474361
$116$	376.000	0.300955
$117$	−621.000	−0.490696
$118$	−900.000	−0.702133
$119$	544.000	0.419062
$120$	120.000	0.0912871
$121$	−602.000	−0.452292
$122$	1400.00	1.03893
$123$	507.000	0.371664
$124$	792.000	0.573578
$125$	1125.00	0.804984
$126$	576.000	0.407255
$127$	797.000	0.556869	0.278434	−	0.960455i	$-0.410185\pi$
$127$	797.000	0.556869	0.278434	+	0.960455i	$0.410185\pi$
$128$	−128.000	−0.0883883
$129$	681.000	0.464796
$130$	−690.000	−0.465515
$131$	373.000	0.248772	0.124386	−	0.992234i	$-0.460304\pi$
$131$	373.000	0.248772	0.124386	+	0.992234i	$0.460304\pi$
$132$	324.000	0.213641
$133$	2656.00	1.73161
$134$	−1080.00	−0.696252
$135$	−135.000	−0.0860663
$136$	136.000	0.0857493
$137$	−2706.00	−1.68751	−0.843756	−	0.536727i	$-0.819661\pi$
$137$	−2706.00	−1.68751	−0.843756	+	0.536727i	$0.819661\pi$
$138$	702.000	0.433030
$139$	−2086.00	−1.27289	−0.636447	−	0.771321i	$-0.719597\pi$
$139$	−2086.00	−1.27289	−0.636447	+	0.771321i	$0.719597\pi$
$140$	640.000	0.386356
$141$	−1146.00	−0.684472
$142$	552.000	0.326217
$143$	−1863.00	−1.08945
$144$	144.000	0.0833333
$145$	−470.000	−0.269182
$146$	596.000	0.337845
$147$	2043.00	1.14628
$148$	−976.000	−0.542072
$149$	−1446.00	−0.795040	−0.397520	−	0.917594i	$-0.630129\pi$
$149$	−1446.00	−0.795040	−0.397520	+	0.917594i	$0.630129\pi$
$150$	600.000	0.326599
$151$	−744.000	−0.400966	−0.200483	−	0.979697i	$-0.564251\pi$
$151$	−744.000	−0.400966	−0.200483	+	0.979697i	$0.564251\pi$
$152$	664.000	0.354326
$153$	−153.000	−0.0808452
$154$	1728.00	0.904196
$155$	−990.000	−0.513024
$156$	−828.000	−0.424955
$157$	207.000	0.105225	0.0526127	−	0.998615i	$-0.483245\pi$
$157$	207.000	0.105225	0.0526127	+	0.998615i	$0.483245\pi$
$158$	364.000	0.183280
$159$	2058.00	1.02648
$160$	160.000	0.0790569
$161$	3744.00	1.83272
$162$	−162.000	−0.0785674
$163$	−1746.00	−0.839002	−0.419501	−	0.907755i	$-0.637795\pi$
$163$	−1746.00	−0.839002	−0.419501	+	0.907755i	$0.637795\pi$
$164$	676.000	0.321870
$165$	−405.000	−0.191086
$166$	−564.000	−0.263704
$167$	−1153.00	−0.534262	−0.267131	−	0.963660i	$-0.586076\pi$
$167$	−1153.00	−0.534262	−0.267131	+	0.963660i	$0.586076\pi$
$168$	768.000	0.352693
$169$	2564.00	1.16705
$170$	−170.000	−0.0766965
$171$	−747.000	−0.334062
$172$	908.000	0.402525
$173$	−3521.00	−1.54738	−0.773690	−	0.633565i	$-0.781591\pi$
$173$	−3521.00	−1.54738	−0.773690	+	0.633565i	$0.781591\pi$
$174$	−564.000	−0.245728
$175$	3200.00	1.38227
$176$	432.000	0.185018
$177$	1350.00	0.573289
$178$	2936.00	1.23631
$179$	2646.00	1.10487	0.552434	−	0.833557i	$-0.313699\pi$
$179$	2646.00	1.10487	0.552434	+	0.833557i	$0.313699\pi$
$180$	−180.000	−0.0745356
$181$	−1454.00	−0.597099	−0.298550	−	0.954394i	$-0.596503\pi$
$181$	−1454.00	−0.597099	−0.298550	+	0.954394i	$0.596503\pi$
$182$	−4416.00	−1.79855
$183$	−2100.00	−0.848287
$184$	936.000	0.375015
$185$	1220.00	0.484844
$186$	−1188.00	−0.468325
$187$	−459.000	−0.179494
$188$	−1528.00	−0.592770
$189$	−864.000	−0.332522
$190$	−830.000	−0.316919
$191$	3730.00	1.41305	0.706527	−	0.707686i	$-0.250261\pi$
$191$	3730.00	1.41305	0.706527	+	0.707686i	$0.250261\pi$
$192$	192.000	0.0721688
$193$	−2478.00	−0.924199	−0.462099	−	0.886828i	$-0.652904\pi$
$193$	−2478.00	−0.924199	−0.462099	+	0.886828i	$0.652904\pi$
$194$	2280.00	0.843786
$195$	1035.00	0.380092
$196$	2724.00	0.992711
$197$	−2141.00	−0.774314	−0.387157	−	0.922014i	$-0.626543\pi$
$197$	−2141.00	−0.774314	−0.387157	+	0.922014i	$0.626543\pi$
$198$	−486.000	−0.174437
$199$	−580.000	−0.206609	−0.103304	−	0.994650i	$-0.532942\pi$
$199$	−580.000	−0.206609	−0.103304	+	0.994650i	$0.532942\pi$
$200$	800.000	0.282843
$201$	1620.00	0.568488
$202$	2800.00	0.975284
$203$	−3008.00	−1.04000
$204$	−204.000	−0.0700140
$205$	−845.000	−0.287890
$206$	−3878.00	−1.31162
$207$	−1053.00	−0.353568
$208$	−1104.00	−0.368022
$209$	−2241.00	−0.741690
$210$	−960.000	−0.315459
$211$	−3826.00	−1.24831	−0.624153	−	0.781302i	$-0.714556\pi$
$211$	−3826.00	−1.24831	−0.624153	+	0.781302i	$0.714556\pi$
$212$	2744.00	0.888956
$213$	−828.000	−0.266355
$214$	−78.0000	−0.0249157
$215$	−1135.00	−0.360030
$216$	−216.000	−0.0680414
$217$	−6336.00	−1.98210
$218$	856.000	0.265943
$219$	−894.000	−0.275849
$220$	−540.000	−0.165485
$221$	1173.00	0.357034
$222$	1464.00	0.442600
$223$	−1759.00	−0.528212	−0.264106	−	0.964494i	$-0.585077\pi$
$223$	−1759.00	−0.528212	−0.264106	+	0.964494i	$0.585077\pi$
$224$	1024.00	0.305441
$225$	−900.000	−0.266667
$226$	−1430.00	−0.420895
$227$	341.000	0.0997047	0.0498523	−	0.998757i	$-0.484125\pi$
$227$	341.000	0.0997047	0.0498523	+	0.998757i	$0.484125\pi$
$228$	−996.000	−0.289306
$229$	3958.00	1.14215	0.571074	−	0.820898i	$-0.306527\pi$
$229$	3958.00	1.14215	0.571074	+	0.820898i	$0.306527\pi$
$230$	−1170.00	−0.335424
$231$	−2592.00	−0.738273
$232$	−752.000	−0.212807
$233$	3009.00	0.846035	0.423017	−	0.906122i	$-0.360971\pi$
$233$	3009.00	0.846035	0.423017	+	0.906122i	$0.360971\pi$
$234$	1242.00	0.346975
$235$	1910.00	0.530190
$236$	1800.00	0.496483
$237$	−546.000	−0.149648
$238$	−1088.00	−0.296322
$239$	3464.00	0.937521	0.468761	−	0.883325i	$-0.344701\pi$
$239$	3464.00	0.937521	0.468761	+	0.883325i	$0.344701\pi$
$240$	−240.000	−0.0645497
$241$	156.000	0.0416964	0.0208482	−	0.999783i	$-0.493363\pi$
$241$	156.000	0.0416964	0.0208482	+	0.999783i	$0.493363\pi$
$242$	1204.00	0.319818
$243$	243.000	0.0641500
$244$	−2800.00	−0.734638
$245$	−3405.00	−0.887908
$246$	−1014.00	−0.262806
$247$	5727.00	1.47530
$248$	−1584.00	−0.405581
$249$	846.000	0.215314
$250$	−2250.00	−0.569210
$251$	−4464.00	−1.12257	−0.561285	−	0.827622i	$-0.689693\pi$
$251$	−4464.00	−1.12257	−0.561285	+	0.827622i	$0.689693\pi$
$252$	−1152.00	−0.287973
$253$	−3159.00	−0.784999
$254$	−1594.00	−0.393766
$255$	255.000	0.0626224
$256$	256.000	0.0625000
$257$	5544.00	1.34562	0.672812	−	0.739814i	$-0.265086\pi$
$257$	5544.00	1.34562	0.672812	+	0.739814i	$0.265086\pi$
$258$	−1362.00	−0.328661
$259$	7808.00	1.87323
$260$	1380.00	0.329169
$261$	846.000	0.200636
$262$	−746.000	−0.175909
$263$	612.000	0.143489	0.0717444	−	0.997423i	$-0.477143\pi$
$263$	612.000	0.143489	0.0717444	+	0.997423i	$0.477143\pi$
$264$	−648.000	−0.151067
$265$	−3430.00	−0.795107
$266$	−5312.00	−1.22443
$267$	−4404.00	−1.00944
$268$	2160.00	0.492325
$269$	5601.00	1.26951	0.634757	−	0.772712i	$-0.281100\pi$
$269$	5601.00	1.26951	0.634757	+	0.772712i	$0.281100\pi$
$270$	270.000	0.0608581
$271$	−5851.00	−1.31152	−0.655762	−	0.754968i	$-0.727652\pi$
$271$	−5851.00	−1.31152	−0.655762	+	0.754968i	$0.727652\pi$
$272$	−272.000	−0.0606339
$273$	6624.00	1.46851
$274$	5412.00	1.19325
$275$	−2700.00	−0.592059
$276$	−1404.00	−0.306199
$277$	−5138.00	−1.11449	−0.557243	−	0.830350i	$-0.688141\pi$
$277$	−5138.00	−1.11449	−0.557243	+	0.830350i	$0.688141\pi$
$278$	4172.00	0.900072
$279$	1782.00	0.382385
$280$	−1280.00	−0.273195
$281$	7608.00	1.61514	0.807572	−	0.589770i	$-0.200781\pi$
$281$	7608.00	1.61514	0.807572	+	0.589770i	$0.200781\pi$
$282$	2292.00	0.483995
$283$	8474.00	1.77995	0.889977	−	0.456005i	$-0.150720\pi$
$283$	8474.00	1.77995	0.889977	+	0.456005i	$0.150720\pi$
$284$	−1104.00	−0.230670
$285$	1245.00	0.258763
$286$	3726.00	0.770360
$287$	−5408.00	−1.11228
$288$	−288.000	−0.0589256
$289$	289.000	0.0588235
$290$	940.000	0.190340
$291$	−3420.00	−0.688948
$292$	−1192.00	−0.238892
$293$	7672.00	1.52970	0.764852	−	0.644207i	$-0.222812\pi$
$293$	7672.00	1.52970	0.764852	+	0.644207i	$0.222812\pi$
$294$	−4086.00	−0.810545
$295$	−2250.00	−0.444068
$296$	1952.00	0.383303
$297$	729.000	0.142427
$298$	2892.00	0.562178
$299$	8073.00	1.56145
$300$	−1200.00	−0.230940
$301$	−7264.00	−1.39100
$302$	1488.00	0.283526
$303$	−4200.00	−0.796316
$304$	−1328.00	−0.250546
$305$	3500.00	0.657080
$306$	306.000	0.0571662
$307$	−3396.00	−0.631335	−0.315668	−	0.948870i	$-0.602228\pi$
$307$	−3396.00	−0.631335	−0.315668	+	0.948870i	$0.602228\pi$
$308$	−3456.00	−0.639363
$309$	5817.00	1.07093
$310$	1980.00	0.362763
$311$	6480.00	1.18150	0.590751	−	0.806854i	$-0.298832\pi$
$311$	6480.00	1.18150	0.590751	+	0.806854i	$0.298832\pi$
$312$	1656.00	0.300489
$313$	−4940.00	−0.892094	−0.446047	−	0.895010i	$-0.647169\pi$
$313$	−4940.00	−0.892094	−0.446047	+	0.895010i	$0.647169\pi$
$314$	−414.000	−0.0744056
$315$	1440.00	0.257571
$316$	−728.000	−0.129599
$317$	−3998.00	−0.708360	−0.354180	−	0.935177i	$-0.615240\pi$
$317$	−3998.00	−0.708360	−0.354180	+	0.935177i	$0.615240\pi$
$318$	−4116.00	−0.725830
$319$	2538.00	0.445457
$320$	−320.000	−0.0559017
$321$	117.000	0.0203436
$322$	−7488.00	−1.29593
$323$	1411.00	0.243065
$324$	324.000	0.0555556
$325$	6900.00	1.17767
$326$	3492.00	0.593264
$327$	−1284.00	−0.217142
$328$	−1352.00	−0.227597
$329$	12224.0	2.04842
$330$	810.000	0.135118
$331$	2105.00	0.349551	0.174775	−	0.984608i	$-0.444080\pi$
$331$	2105.00	0.349551	0.174775	+	0.984608i	$0.444080\pi$
$332$	1128.00	0.186467
$333$	−2196.00	−0.361382
$334$	2306.00	0.377781
$335$	−2700.00	−0.440349
$336$	−1536.00	−0.249392
$337$	7818.00	1.26372	0.631860	−	0.775083i	$-0.282292\pi$
$337$	7818.00	1.26372	0.631860	+	0.775083i	$0.282292\pi$
$338$	−5128.00	−0.825226
$339$	2145.00	0.343659
$340$	340.000	0.0542326
$341$	5346.00	0.848980
$342$	1494.00	0.236217
$343$	−10816.0	−1.70265
$344$	−1816.00	−0.284628
$345$	1755.00	0.273873
$346$	7042.00	1.09416
$347$	−1388.00	−0.214731	−0.107366	−	0.994220i	$-0.534242\pi$
$347$	−1388.00	−0.214731	−0.107366	+	0.994220i	$0.534242\pi$
$348$	1128.00	0.173756
$349$	−9591.00	−1.47104	−0.735522	−	0.677501i	$-0.763063\pi$
$349$	−9591.00	−1.47104	−0.735522	+	0.677501i	$0.763063\pi$
$350$	−6400.00	−0.977413
$351$	−1863.00	−0.283304
$352$	−864.000	−0.130828
$353$	−1442.00	−0.217422	−0.108711	−	0.994073i	$-0.534672\pi$
$353$	−1442.00	−0.217422	−0.108711	+	0.994073i	$0.534672\pi$
$354$	−2700.00	−0.405377
$355$	1380.00	0.206318
$356$	−5872.00	−0.874200
$357$	1632.00	0.241946
$358$	−5292.00	−0.781259
$359$	4104.00	0.603345	0.301672	−	0.953412i	$-0.402455\pi$
$359$	4104.00	0.603345	0.301672	+	0.953412i	$0.402455\pi$
$360$	360.000	0.0527046
$361$	30.0000	0.00437382
$362$	2908.00	0.422213
$363$	−1806.00	−0.261131
$364$	8832.00	1.27177
$365$	1490.00	0.213672
$366$	4200.00	0.599829
$367$	3904.00	0.555278	0.277639	−	0.960685i	$-0.410448\pi$
$367$	3904.00	0.555278	0.277639	+	0.960685i	$0.410448\pi$
$368$	−1872.00	−0.265176
$369$	1521.00	0.214580
$370$	−2440.00	−0.342837
$371$	−21952.0	−3.07194
$372$	2376.00	0.331156
$373$	4882.00	0.677695	0.338848	−	0.940841i	$-0.389963\pi$
$373$	4882.00	0.677695	0.338848	+	0.940841i	$0.389963\pi$
$374$	918.000	0.126922
$375$	3375.00	0.464758
$376$	3056.00	0.419152
$377$	−6486.00	−0.886064
$378$	1728.00	0.235129
$379$	−10324.0	−1.39923	−0.699615	−	0.714520i	$-0.746645\pi$
$379$	−10324.0	−1.39923	−0.699615	+	0.714520i	$0.746645\pi$
$380$	1660.00	0.224095
$381$	2391.00	0.321508
$382$	−7460.00	−0.999180
$383$	−10480.0	−1.39818	−0.699090	−	0.715034i	$-0.746411\pi$
$383$	−10480.0	−1.39818	−0.699090	+	0.715034i	$0.746411\pi$
$384$	−384.000	−0.0510310
$385$	4320.00	0.571864
$386$	4956.00	0.653507
$387$	2043.00	0.268350
$388$	−4560.00	−0.596647
$389$	−9724.00	−1.26742	−0.633710	−	0.773571i	$-0.718469\pi$
$389$	−9724.00	−1.26742	−0.633710	+	0.773571i	$0.718469\pi$
$390$	−2070.00	−0.268765
$391$	1989.00	0.257258
$392$	−5448.00	−0.701953
$393$	1119.00	0.143629
$394$	4282.00	0.547523
$395$	910.000	0.115917
$396$	972.000	0.123346
$397$	−14632.0	−1.84977	−0.924885	−	0.380246i	$-0.875839\pi$
$397$	−14632.0	−1.84977	−0.924885	+	0.380246i	$0.875839\pi$
$398$	1160.00	0.146094
$399$	7968.00	0.999747
$400$	−1600.00	−0.200000
$401$	−8907.00	−1.10921	−0.554606	−	0.832113i	$-0.687131\pi$
$401$	−8907.00	−1.10921	−0.554606	+	0.832113i	$0.687131\pi$
$402$	−3240.00	−0.401981
$403$	−13662.0	−1.68872
$404$	−5600.00	−0.689630
$405$	−405.000	−0.0496904
$406$	6016.00	0.735392
$407$	−6588.00	−0.802347
$408$	408.000	0.0495074
$409$	−4859.00	−0.587438	−0.293719	−	0.955892i	$-0.594893\pi$
$409$	−4859.00	−0.587438	−0.293719	+	0.955892i	$0.594893\pi$
$410$	1690.00	0.203569
$411$	−8118.00	−0.974286
$412$	7756.00	0.927453
$413$	−14400.0	−1.71568
$414$	2106.00	0.250010
$415$	−1410.00	−0.166781
$416$	2208.00	0.260231
$417$	−6258.00	−0.734905
$418$	4482.00	0.524454
$419$	−444.000	−0.0517681	−0.0258840	−	0.999665i	$-0.508240\pi$
$419$	−444.000	−0.0517681	−0.0258840	+	0.999665i	$0.508240\pi$
$420$	1920.00	0.223063
$421$	14731.0	1.70533	0.852666	−	0.522456i	$-0.174984\pi$
$421$	14731.0	1.70533	0.852666	+	0.522456i	$0.174984\pi$
$422$	7652.00	0.882686
$423$	−3438.00	−0.395180
$424$	−5488.00	−0.628587
$425$	1700.00	0.194029
$426$	1656.00	0.188341
$427$	22400.0	2.53867
$428$	156.000	0.0176181
$429$	−5589.00	−0.628996
$430$	2270.00	0.254579
$431$	−9968.00	−1.11402	−0.557009	−	0.830507i	$-0.688051\pi$
$431$	−9968.00	−1.11402	−0.557009	+	0.830507i	$0.688051\pi$
$432$	432.000	0.0481125
$433$	1167.00	0.129521	0.0647603	−	0.997901i	$-0.479372\pi$
$433$	1167.00	0.129521	0.0647603	+	0.997901i	$0.479372\pi$
$434$	12672.0	1.40156
$435$	−1410.00	−0.155412
$436$	−1712.00	−0.188050
$437$	9711.00	1.06302
$438$	1788.00	0.195055
$439$	16532.0	1.79733	0.898667	−	0.438632i	$-0.144537\pi$
$439$	16532.0	1.79733	0.898667	+	0.438632i	$0.144537\pi$
$440$	1080.00	0.117016
$441$	6129.00	0.661808
$442$	−2346.00	−0.252461
$443$	10398.0	1.11518	0.557589	−	0.830117i	$-0.311727\pi$
$443$	10398.0	1.11518	0.557589	+	0.830117i	$0.311727\pi$
$444$	−2928.00	−0.312966
$445$	7340.00	0.781909
$446$	3518.00	0.373503
$447$	−4338.00	−0.459016
$448$	−2048.00	−0.215980
$449$	4922.00	0.517335	0.258668	−	0.965966i	$-0.416717\pi$
$449$	4922.00	0.517335	0.258668	+	0.965966i	$0.416717\pi$
$450$	1800.00	0.188562
$451$	4563.00	0.476415
$452$	2860.00	0.297617
$453$	−2232.00	−0.231498
$454$	−682.000	−0.0705019
$455$	−11040.0	−1.13750
$456$	1992.00	0.204570
$457$	4573.00	0.468087	0.234044	−	0.972226i	$-0.424804\pi$
$457$	4573.00	0.468087	0.234044	+	0.972226i	$0.424804\pi$
$458$	−7916.00	−0.807621
$459$	−459.000	−0.0466760
$460$	2340.00	0.237181
$461$	−13586.0	−1.37259	−0.686294	−	0.727324i	$-0.740764\pi$
$461$	−13586.0	−1.37259	−0.686294	+	0.727324i	$0.740764\pi$
$462$	5184.00	0.522038
$463$	−4120.00	−0.413548	−0.206774	−	0.978389i	$-0.566296\pi$
$463$	−4120.00	−0.413548	−0.206774	+	0.978389i	$0.566296\pi$
$464$	1504.00	0.150477
$465$	−2970.00	−0.296195
$466$	−6018.00	−0.598237
$467$	−174.000	−0.0172415	−0.00862073	−	0.999963i	$-0.502744\pi$
$467$	−174.000	−0.0172415	−0.00862073	+	0.999963i	$0.502744\pi$
$468$	−2484.00	−0.245348
$469$	−17280.0	−1.70131
$470$	−3820.00	−0.374901
$471$	621.000	0.0607520
$472$	−3600.00	−0.351067
$473$	6129.00	0.595796
$474$	1092.00	0.105817
$475$	8300.00	0.801748
$476$	2176.00	0.209531
$477$	6174.00	0.592637
$478$	−6928.00	−0.662927
$479$	18311.0	1.74666	0.873331	−	0.487128i	$-0.161955\pi$
$479$	18311.0	1.74666	0.873331	+	0.487128i	$0.161955\pi$
$480$	480.000	0.0456435
$481$	16836.0	1.59596
$482$	−312.000	−0.0294838
$483$	11232.0	1.05812
$484$	−2408.00	−0.226146
$485$	5700.00	0.533657
$486$	−486.000	−0.0453609
$487$	6766.00	0.629562	0.314781	−	0.949164i	$-0.398069\pi$
$487$	6766.00	0.629562	0.314781	+	0.949164i	$0.398069\pi$
$488$	5600.00	0.519467
$489$	−5238.00	−0.484398
$490$	6810.00	0.627846
$491$	−13846.0	−1.27263	−0.636315	−	0.771429i	$-0.719542\pi$
$491$	−13846.0	−1.27263	−0.636315	+	0.771429i	$0.719542\pi$
$492$	2028.00	0.185832
$493$	−1598.00	−0.145984
$494$	−11454.0	−1.04320
$495$	−1215.00	−0.110324
$496$	3168.00	0.286789
$497$	8832.00	0.797121
$498$	−1692.00	−0.152250
$499$	−5854.00	−0.525172	−0.262586	−	0.964909i	$-0.584575\pi$
$499$	−5854.00	−0.525172	−0.262586	+	0.964909i	$0.584575\pi$
$500$	4500.00	0.402492
$501$	−3459.00	−0.308457
$502$	8928.00	0.793777
$503$	−883.000	−0.0782724	−0.0391362	−	0.999234i	$-0.512461\pi$
$503$	−883.000	−0.0782724	−0.0391362	+	0.999234i	$0.512461\pi$
$504$	2304.00	0.203628
$505$	7000.00	0.616824
$506$	6318.00	0.555078
$507$	7692.00	0.673794
$508$	3188.00	0.278434
$509$	6748.00	0.587622	0.293811	−	0.955863i	$-0.405076\pi$
$509$	6748.00	0.587622	0.293811	+	0.955863i	$0.405076\pi$
$510$	−510.000	−0.0442807
$511$	9536.00	0.825534
$512$	−512.000	−0.0441942
$513$	−2241.00	−0.192871
$514$	−11088.0	−0.951499
$515$	−9695.00	−0.829539
$516$	2724.00	0.232398
$517$	−10314.0	−0.877387
$518$	−15616.0	−1.32457
$519$	−10563.0	−0.893380
$520$	−2760.00	−0.232758
$521$	−9263.00	−0.778924	−0.389462	−	0.921043i	$-0.627339\pi$
$521$	−9263.00	−0.778924	−0.389462	+	0.921043i	$0.627339\pi$
$522$	−1692.00	−0.141871
$523$	3868.00	0.323395	0.161698	−	0.986840i	$-0.448303\pi$
$523$	3868.00	0.323395	0.161698	+	0.986840i	$0.448303\pi$
$524$	1492.00	0.124386
$525$	9600.00	0.798054
$526$	−1224.00	−0.101462
$527$	−3366.00	−0.278226
$528$	1296.00	0.106820
$529$	1522.00	0.125092
$530$	6860.00	0.562225
$531$	4050.00	0.330989
$532$	10624.0	0.865806
$533$	−11661.0	−0.947643
$534$	8808.00	0.713782
$535$	−195.000	−0.0157581
$536$	−4320.00	−0.348126
$537$	7938.00	0.637896
$538$	−11202.0	−0.897681
$539$	18387.0	1.46936
$540$	−540.000	−0.0430331
$541$	−6840.00	−0.543576	−0.271788	−	0.962357i	$-0.587615\pi$
$541$	−6840.00	−0.543576	−0.271788	+	0.962357i	$0.587615\pi$
$542$	11702.0	0.927387
$543$	−4362.00	−0.344735
$544$	544.000	0.0428746
$545$	2140.00	0.168197
$546$	−13248.0	−1.03839
$547$	−5564.00	−0.434917	−0.217458	−	0.976070i	$-0.569777\pi$
$547$	−5564.00	−0.434917	−0.217458	+	0.976070i	$0.569777\pi$
$548$	−10824.0	−0.843756
$549$	−6300.00	−0.489759
$550$	5400.00	0.418649
$551$	−7802.00	−0.603224
$552$	2808.00	0.216515
$553$	5824.00	0.447851
$554$	10276.0	0.788060
$555$	3660.00	0.279925
$556$	−8344.00	−0.636447
$557$	1618.00	0.123082	0.0615412	−	0.998105i	$-0.480398\pi$
$557$	1618.00	0.123082	0.0615412	+	0.998105i	$0.480398\pi$
$558$	−3564.00	−0.270387
$559$	−15663.0	−1.18511
$560$	2560.00	0.193178
$561$	−1377.00	−0.103631
$562$	−15216.0	−1.14208
$563$	−21050.0	−1.57576	−0.787879	−	0.615830i	$-0.788821\pi$
$563$	−21050.0	−1.57576	−0.787879	+	0.615830i	$0.788821\pi$
$564$	−4584.00	−0.342236
$565$	−3575.00	−0.266197
$566$	−16948.0	−1.25862
$567$	−2592.00	−0.191982
$568$	2208.00	0.163108
$569$	26484.0	1.95126	0.975630	−	0.219422i	$-0.0704171\pi$
$569$	26484.0	1.95126	0.975630	+	0.219422i	$0.0704171\pi$
$570$	−2490.00	−0.182973
$571$	−10996.0	−0.805899	−0.402949	−	0.915222i	$-0.632015\pi$
$571$	−10996.0	−0.805899	−0.402949	+	0.915222i	$0.632015\pi$
$572$	−7452.00	−0.544727
$573$	11190.0	0.815827
$574$	10816.0	0.786500
$575$	11700.0	0.848563
$576$	576.000	0.0416667
$577$	16089.0	1.16082	0.580411	−	0.814324i	$-0.302892\pi$
$577$	16089.0	1.16082	0.580411	+	0.814324i	$0.302892\pi$
$578$	−578.000	−0.0415945
$579$	−7434.00	−0.533586
$580$	−1880.00	−0.134591
$581$	−9024.00	−0.644369
$582$	6840.00	0.487160
$583$	18522.0	1.31579
$584$	2384.00	0.168922
$585$	3105.00	0.219446
$586$	−15344.0	−1.08166
$587$	5700.00	0.400791	0.200395	−	0.979715i	$-0.435777\pi$
$587$	5700.00	0.400791	0.200395	+	0.979715i	$0.435777\pi$
$588$	8172.00	0.573142
$589$	−16434.0	−1.14966
$590$	4500.00	0.314004
$591$	−6423.00	−0.447051
$592$	−3904.00	−0.271036
$593$	3474.00	0.240573	0.120287	−	0.992739i	$-0.461619\pi$
$593$	3474.00	0.240573	0.120287	+	0.992739i	$0.461619\pi$
$594$	−1458.00	−0.100711
$595$	−2720.00	−0.187410
$596$	−5784.00	−0.397520
$597$	−1740.00	−0.119286
$598$	−16146.0	−1.10411
$599$	−12710.0	−0.866972	−0.433486	−	0.901160i	$-0.642717\pi$
$599$	−12710.0	−0.866972	−0.433486	+	0.901160i	$0.642717\pi$
$600$	2400.00	0.163299
$601$	8806.00	0.597678	0.298839	−	0.954304i	$-0.403401\pi$
$601$	8806.00	0.597678	0.298839	+	0.954304i	$0.403401\pi$
$602$	14528.0	0.983583
$603$	4860.00	0.328216
$604$	−2976.00	−0.200483
$605$	3010.00	0.202271
$606$	8400.00	0.563080
$607$	6022.00	0.402678	0.201339	−	0.979522i	$-0.435471\pi$
$607$	6022.00	0.402678	0.201339	+	0.979522i	$0.435471\pi$
$608$	2656.00	0.177163
$609$	−9024.00	−0.600445
$610$	−7000.00	−0.464626
$611$	26358.0	1.74522
$612$	−612.000	−0.0404226
$613$	−2337.00	−0.153981	−0.0769907	−	0.997032i	$-0.524531\pi$
$613$	−2337.00	−0.153981	−0.0769907	+	0.997032i	$0.524531\pi$
$614$	6792.00	0.446422
$615$	−2535.00	−0.166213
$616$	6912.00	0.452098
$617$	23082.0	1.50607	0.753036	−	0.657979i	$-0.228589\pi$
$617$	23082.0	1.50607	0.753036	+	0.657979i	$0.228589\pi$
$618$	−11634.0	−0.757262
$619$	−20510.0	−1.33177	−0.665886	−	0.746054i	$-0.731946\pi$
$619$	−20510.0	−1.33177	−0.665886	+	0.746054i	$0.731946\pi$
$620$	−3960.00	−0.256512
$621$	−3159.00	−0.204133
$622$	−12960.0	−0.835448
$623$	46976.0	3.02095
$624$	−3312.00	−0.212478
$625$	6875.00	0.440000
$626$	9880.00	0.630805
$627$	−6723.00	−0.428215
$628$	828.000	0.0526127
$629$	4148.00	0.262944
$630$	−2880.00	−0.182130
$631$	29453.0	1.85817	0.929085	−	0.369866i	$-0.120596\pi$
$631$	29453.0	1.85817	0.929085	+	0.369866i	$0.120596\pi$
$632$	1456.00	0.0916401
$633$	−11478.0	−0.720710
$634$	7996.00	0.500886
$635$	−3985.00	−0.249039
$636$	8232.00	0.513239
$637$	−46989.0	−2.92272
$638$	−5076.00	−0.314986
$639$	−2484.00	−0.153780
$640$	640.000	0.0395285
$641$	−4045.00	−0.249248	−0.124624	−	0.992204i	$-0.539772\pi$
$641$	−4045.00	−0.249248	−0.124624	+	0.992204i	$0.539772\pi$
$642$	−234.000	−0.0143851
$643$	−30188.0	−1.85148	−0.925738	−	0.378167i	$-0.876555\pi$
$643$	−30188.0	−1.85148	−0.925738	+	0.378167i	$0.876555\pi$
$644$	14976.0	0.916362
$645$	−3405.00	−0.207863
$646$	−2822.00	−0.171873
$647$	−28246.0	−1.71633	−0.858164	−	0.513375i	$-0.828395\pi$
$647$	−28246.0	−1.71633	−0.858164	+	0.513375i	$0.828395\pi$
$648$	−648.000	−0.0392837
$649$	12150.0	0.734868
$650$	−13800.0	−0.832739
$651$	−19008.0	−1.14437
$652$	−6984.00	−0.419501
$653$	−1301.00	−0.0779664	−0.0389832	−	0.999240i	$-0.512412\pi$
$653$	−1301.00	−0.0779664	−0.0389832	+	0.999240i	$0.512412\pi$
$654$	2568.00	0.153542
$655$	−1865.00	−0.111254
$656$	2704.00	0.160935
$657$	−2682.00	−0.159261
$658$	−24448.0	−1.44845
$659$	−6450.00	−0.381269	−0.190635	−	0.981661i	$-0.561055\pi$
$659$	−6450.00	−0.381269	−0.190635	+	0.981661i	$0.561055\pi$
$660$	−1620.00	−0.0955431
$661$	12445.0	0.732306	0.366153	−	0.930555i	$-0.380675\pi$
$661$	12445.0	0.732306	0.366153	+	0.930555i	$0.380675\pi$
$662$	−4210.00	−0.247170
$663$	3519.00	0.206134
$664$	−2256.00	−0.131852
$665$	−13280.0	−0.774400
$666$	4392.00	0.255535
$667$	−10998.0	−0.638447
$668$	−4612.00	−0.267131
$669$	−5277.00	−0.304964
$670$	5400.00	0.311373
$671$	−18900.0	−1.08737
$672$	3072.00	0.176347
$673$	5566.00	0.318802	0.159401	−	0.987214i	$-0.449044\pi$
$673$	5566.00	0.318802	0.159401	+	0.987214i	$0.449044\pi$
$674$	−15636.0	−0.893585
$675$	−2700.00	−0.153960
$676$	10256.0	0.583523
$677$	−33539.0	−1.90400	−0.952000	−	0.306097i	$-0.900977\pi$
$677$	−33539.0	−1.90400	−0.952000	+	0.306097i	$0.900977\pi$
$678$	−4290.00	−0.243004
$679$	36480.0	2.06182
$680$	−680.000	−0.0383482
$681$	1023.00	0.0575645
$682$	−10692.0	−0.600319
$683$	29517.0	1.65364	0.826820	−	0.562466i	$-0.190147\pi$
$683$	29517.0	1.65364	0.826820	+	0.562466i	$0.190147\pi$
$684$	−2988.00	−0.167031
$685$	13530.0	0.754678
$686$	21632.0	1.20396
$687$	11874.0	0.659420
$688$	3632.00	0.201263
$689$	−47334.0	−2.61724
$690$	−3510.00	−0.193657
$691$	1684.00	0.0927097	0.0463548	−	0.998925i	$-0.485240\pi$
$691$	1684.00	0.0927097	0.0463548	+	0.998925i	$0.485240\pi$
$692$	−14084.0	−0.773690
$693$	−7776.00	−0.426242
$694$	2776.00	0.151838
$695$	10430.0	0.569255
$696$	−2256.00	−0.122864
$697$	−2873.00	−0.156130
$698$	19182.0	1.04019
$699$	9027.00	0.488459
$700$	12800.0	0.691135
$701$	34152.0	1.84009	0.920045	−	0.391812i	$-0.128152\pi$
$701$	34152.0	1.84009	0.920045	+	0.391812i	$0.128152\pi$
$702$	3726.00	0.200326
$703$	20252.0	1.08651
$704$	1728.00	0.0925092
$705$	5730.00	0.306105
$706$	2884.00	0.153740
$707$	44800.0	2.38314
$708$	5400.00	0.286645
$709$	7994.00	0.423443	0.211721	−	0.977330i	$-0.432093\pi$
$709$	7994.00	0.423443	0.211721	+	0.977330i	$0.432093\pi$
$710$	−2760.00	−0.145889
$711$	−1638.00	−0.0863992
$712$	11744.0	0.618153
$713$	−23166.0	−1.21679
$714$	−3264.00	−0.171081
$715$	9315.00	0.487219
$716$	10584.0	0.552434
$717$	10392.0	0.541278
$718$	−8208.00	−0.426629
$719$	22713.0	1.17810	0.589049	−	0.808098i	$-0.299503\pi$
$719$	22713.0	1.17810	0.589049	+	0.808098i	$0.299503\pi$
$720$	−720.000	−0.0372678
$721$	−62048.0	−3.20498
$722$	−60.0000	−0.00309275
$723$	468.000	0.0240735
$724$	−5816.00	−0.298550
$725$	−9400.00	−0.481527
$726$	3612.00	0.184647
$727$	21712.0	1.10764	0.553819	−	0.832637i	$-0.313170\pi$
$727$	21712.0	1.10764	0.553819	+	0.832637i	$0.313170\pi$
$728$	−17664.0	−0.899274
$729$	729.000	0.0370370
$730$	−2980.00	−0.151089
$731$	−3859.00	−0.195253
$732$	−8400.00	−0.424143
$733$	−5430.00	−0.273617	−0.136809	−	0.990597i	$-0.543685\pi$
$733$	−5430.00	−0.273617	−0.136809	+	0.990597i	$0.543685\pi$
$734$	−7808.00	−0.392641
$735$	−10215.0	−0.512634
$736$	3744.00	0.187508
$737$	14580.0	0.728713
$738$	−3042.00	−0.151731
$739$	−30723.0	−1.52932	−0.764658	−	0.644437i	$-0.777092\pi$
$739$	−30723.0	−1.52932	−0.764658	+	0.644437i	$0.777092\pi$
$740$	4880.00	0.242422
$741$	17181.0	0.851768
$742$	43904.0	2.17219
$743$	28096.0	1.38727	0.693635	−	0.720326i	$-0.256008\pi$
$743$	28096.0	1.38727	0.693635	+	0.720326i	$0.256008\pi$
$744$	−4752.00	−0.234162
$745$	7230.00	0.355553
$746$	−9764.00	−0.479203
$747$	2538.00	0.124311
$748$	−1836.00	−0.0897471
$749$	−1248.00	−0.0608824
$750$	−6750.00	−0.328634
$751$	−5458.00	−0.265200	−0.132600	−	0.991170i	$-0.542333\pi$
$751$	−5458.00	−0.265200	−0.132600	+	0.991170i	$0.542333\pi$
$752$	−6112.00	−0.296385
$753$	−13392.0	−0.648116
$754$	12972.0	0.626542
$755$	3720.00	0.179317
$756$	−3456.00	−0.166261
$757$	−24319.0	−1.16762	−0.583810	−	0.811890i	$-0.698439\pi$
$757$	−24319.0	−1.16762	−0.583810	+	0.811890i	$0.698439\pi$
$758$	20648.0	0.989405
$759$	−9477.00	−0.453219
$760$	−3320.00	−0.158459
$761$	2126.00	0.101271	0.0506356	−	0.998717i	$-0.483875\pi$
$761$	2126.00	0.101271	0.0506356	+	0.998717i	$0.483875\pi$
$762$	−4782.00	−0.227341
$763$	13696.0	0.649841
$764$	14920.0	0.706527
$765$	765.000	0.0361551
$766$	20960.0	0.988663
$767$	−31050.0	−1.46173
$768$	768.000	0.0360844
$769$	−2503.00	−0.117374	−0.0586869	−	0.998276i	$-0.518691\pi$
$769$	−2503.00	−0.117374	−0.0586869	+	0.998276i	$0.518691\pi$
$770$	−8640.00	−0.404369
$771$	16632.0	0.776896
$772$	−9912.00	−0.462099
$773$	36780.0	1.71136	0.855682	−	0.517502i	$-0.173138\pi$
$773$	36780.0	1.71136	0.855682	+	0.517502i	$0.173138\pi$
$774$	−4086.00	−0.189752
$775$	−19800.0	−0.917725
$776$	9120.00	0.421893
$777$	23424.0	1.08151
$778$	19448.0	0.896201
$779$	−14027.0	−0.645147
$780$	4140.00	0.190046
$781$	−7452.00	−0.341426
$782$	−3978.00	−0.181909
$783$	2538.00	0.115837
$784$	10896.0	0.496356
$785$	−1035.00	−0.0470583
$786$	−2238.00	−0.101561
$787$	7356.00	0.333181	0.166590	−	0.986026i	$-0.446724\pi$
$787$	7356.00	0.333181	0.166590	+	0.986026i	$0.446724\pi$
$788$	−8564.00	−0.387157
$789$	1836.00	0.0828433
$790$	−1820.00	−0.0819654
$791$	−22880.0	−1.02847
$792$	−1944.00	−0.0872185
$793$	48300.0	2.16290
$794$	29264.0	1.30799
$795$	−10290.0	−0.459055
$796$	−2320.00	−0.103304
$797$	−19672.0	−0.874301	−0.437151	−	0.899388i	$-0.644012\pi$
$797$	−19672.0	−0.874301	−0.437151	+	0.899388i	$0.644012\pi$
$798$	−15936.0	−0.706928
$799$	6494.00	0.287536
$800$	3200.00	0.141421
$801$	−13212.0	−0.582800
$802$	17814.0	0.784332
$803$	−8046.00	−0.353595
$804$	6480.00	0.284244
$805$	−18720.0	−0.819619
$806$	27324.0	1.19410
$807$	16803.0	0.732954
$808$	11200.0	0.487642
$809$	18713.0	0.813244	0.406622	−	0.913597i	$-0.366707\pi$
$809$	18713.0	0.813244	0.406622	+	0.913597i	$0.366707\pi$
$810$	810.000	0.0351364
$811$	16062.0	0.695454	0.347727	−	0.937596i	$-0.386954\pi$
$811$	16062.0	0.695454	0.347727	+	0.937596i	$0.386954\pi$
$812$	−12032.0	−0.520001
$813$	−17553.0	−0.757209
$814$	13176.0	0.567345
$815$	8730.00	0.375213
$816$	−816.000	−0.0350070
$817$	−18841.0	−0.806809
$818$	9718.00	0.415381
$819$	19872.0	0.847844
$820$	−3380.00	−0.143945
$821$	−19653.0	−0.835438	−0.417719	−	0.908576i	$-0.637170\pi$
$821$	−19653.0	−0.835438	−0.417719	+	0.908576i	$0.637170\pi$
$822$	16236.0	0.688924
$823$	−3276.00	−0.138754	−0.0693768	−	0.997591i	$-0.522101\pi$
$823$	−3276.00	−0.138754	−0.0693768	+	0.997591i	$0.522101\pi$
$824$	−15512.0	−0.655808
$825$	−8100.00	−0.341825
$826$	28800.0	1.21317
$827$	−11491.0	−0.483170	−0.241585	−	0.970380i	$-0.577667\pi$
$827$	−11491.0	−0.483170	−0.241585	+	0.970380i	$0.577667\pi$
$828$	−4212.00	−0.176784
$829$	3606.00	0.151075	0.0755377	−	0.997143i	$-0.475933\pi$
$829$	3606.00	0.151075	0.0755377	+	0.997143i	$0.475933\pi$
$830$	2820.00	0.117932
$831$	−15414.0	−0.643449
$832$	−4416.00	−0.184011
$833$	−11577.0	−0.481536
$834$	12516.0	0.519657
$835$	5765.00	0.238929
$836$	−8964.00	−0.370845
$837$	5346.00	0.220770
$838$	888.000	0.0366056
$839$	−36027.0	−1.48247	−0.741234	−	0.671247i	$-0.765759\pi$
$839$	−36027.0	−1.48247	−0.741234	+	0.671247i	$0.765759\pi$
$840$	−3840.00	−0.157729
$841$	−15553.0	−0.637706
$842$	−29462.0	−1.20585
$843$	22824.0	0.932503
$844$	−15304.0	−0.624153
$845$	−12820.0	−0.521919
$846$	6876.00	0.279435
$847$	19264.0	0.781486
$848$	10976.0	0.444478
$849$	25422.0	1.02766
$850$	−3400.00	−0.137199
$851$	28548.0	1.14996
$852$	−3312.00	−0.133178
$853$	−31078.0	−1.24747	−0.623734	−	0.781637i	$-0.714385\pi$
$853$	−31078.0	−1.24747	−0.623734	+	0.781637i	$0.714385\pi$
$854$	−44800.0	−1.79511
$855$	3735.00	0.149397
$856$	−312.000	−0.0124579
$857$	−22106.0	−0.881128	−0.440564	−	0.897721i	$-0.645221\pi$
$857$	−22106.0	−0.881128	−0.440564	+	0.897721i	$0.645221\pi$
$858$	11178.0	0.444768
$859$	23924.0	0.950263	0.475132	−	0.879915i	$-0.342400\pi$
$859$	23924.0	0.950263	0.475132	+	0.879915i	$0.342400\pi$
$860$	−4540.00	−0.180015
$861$	−16224.0	−0.642175
$862$	19936.0	0.787730
$863$	31860.0	1.25669	0.628347	−	0.777933i	$-0.283732\pi$
$863$	31860.0	1.25669	0.628347	+	0.777933i	$0.283732\pi$
$864$	−864.000	−0.0340207
$865$	17605.0	0.692009
$866$	−2334.00	−0.0915849
$867$	867.000	0.0339618
$868$	−25344.0	−0.991050
$869$	−4914.00	−0.191825
$870$	2820.00	0.109893
$871$	−37260.0	−1.44949
$872$	3424.00	0.132972
$873$	−10260.0	−0.397764
$874$	−19422.0	−0.751669
$875$	−36000.0	−1.39088
$876$	−3576.00	−0.137924
$877$	34122.0	1.31382	0.656909	−	0.753970i	$-0.271864\pi$
$877$	34122.0	1.31382	0.656909	+	0.753970i	$0.271864\pi$
$878$	−33064.0	−1.27091
$879$	23016.0	0.883175
$880$	−2160.00	−0.0827427
$881$	−48062.0	−1.83797	−0.918984	−	0.394295i	$-0.870989\pi$
$881$	−48062.0	−1.83797	−0.918984	+	0.394295i	$0.870989\pi$
$882$	−12258.0	−0.467969
$883$	6185.00	0.235721	0.117861	−	0.993030i	$-0.462396\pi$
$883$	6185.00	0.235721	0.117861	+	0.993030i	$0.462396\pi$
$884$	4692.00	0.178517
$885$	−6750.00	−0.256383
$886$	−20796.0	−0.788550
$887$	−41835.0	−1.58363	−0.791816	−	0.610760i	$-0.790864\pi$
$887$	−41835.0	−1.58363	−0.791816	+	0.610760i	$0.790864\pi$
$888$	5856.00	0.221300
$889$	−25504.0	−0.962179
$890$	−14680.0	−0.552893
$891$	2187.00	0.0822304
$892$	−7036.00	−0.264106
$893$	31706.0	1.18813
$894$	8676.00	0.324574
$895$	−13230.0	−0.494112
$896$	4096.00	0.152721
$897$	24219.0	0.901504
$898$	−9844.00	−0.365811
$899$	18612.0	0.690484
$900$	−3600.00	−0.133333
$901$	−11662.0	−0.431207
$902$	−9126.00	−0.336876
$903$	−21792.0	−0.803092
$904$	−5720.00	−0.210447
$905$	7270.00	0.267031
$906$	4464.00	0.163694
$907$	−12174.0	−0.445679	−0.222840	−	0.974855i	$-0.571533\pi$
$907$	−12174.0	−0.445679	−0.222840	+	0.974855i	$0.571533\pi$
$908$	1364.00	0.0498523
$909$	−12600.0	−0.459753
$910$	22080.0	0.804335
$911$	−24089.0	−0.876075	−0.438037	−	0.898957i	$-0.644326\pi$
$911$	−24089.0	−0.876075	−0.438037	+	0.898957i	$0.644326\pi$
$912$	−3984.00	−0.144653
$913$	7614.00	0.275998
$914$	−9146.00	−0.330988
$915$	10500.0	0.379365
$916$	15832.0	0.571074
$917$	−11936.0	−0.429838
$918$	918.000	0.0330049
$919$	−55163.0	−1.98004	−0.990021	−	0.140917i	$-0.954995\pi$
$919$	−55163.0	−1.98004	−0.990021	+	0.140917i	$0.954995\pi$
$920$	−4680.00	−0.167712
$921$	−10188.0	−0.364502
$922$	27172.0	0.970566
$923$	19044.0	0.679134
$924$	−10368.0	−0.369137
$925$	24400.0	0.867316
$926$	8240.00	0.292422
$927$	17451.0	0.618302
$928$	−3008.00	−0.106403
$929$	19555.0	0.690612	0.345306	−	0.938490i	$-0.387775\pi$
$929$	19555.0	0.690612	0.345306	+	0.938490i	$0.387775\pi$
$930$	5940.00	0.209441
$931$	−56523.0	−1.98976
$932$	12036.0	0.423017
$933$	19440.0	0.682140
$934$	348.000	0.0121916
$935$	2295.00	0.0802722
$936$	4968.00	0.173487
$937$	10066.0	0.350952	0.175476	−	0.984484i	$-0.443854\pi$
$937$	10066.0	0.350952	0.175476	+	0.984484i	$0.443854\pi$
$938$	34560.0	1.20301
$939$	−14820.0	−0.515051
$940$	7640.00	0.265095
$941$	−10998.0	−0.381004	−0.190502	−	0.981687i	$-0.561012\pi$
$941$	−10998.0	−0.381004	−0.190502	+	0.981687i	$0.561012\pi$
$942$	−1242.00	−0.0429581
$943$	−19773.0	−0.682818
$944$	7200.00	0.248242
$945$	4320.00	0.148709
$946$	−12258.0	−0.421292
$947$	−2268.00	−0.0778248	−0.0389124	−	0.999243i	$-0.512389\pi$
$947$	−2268.00	−0.0778248	−0.0389124	+	0.999243i	$0.512389\pi$
$948$	−2184.00	−0.0748239
$949$	20562.0	0.703341
$950$	−16600.0	−0.566921
$951$	−11994.0	−0.408972
$952$	−4352.00	−0.148161
$953$	23876.0	0.811563	0.405781	−	0.913970i	$-0.366999\pi$
$953$	23876.0	0.811563	0.405781	+	0.913970i	$0.366999\pi$
$954$	−12348.0	−0.419058
$955$	−18650.0	−0.631937
$956$	13856.0	0.468761
$957$	7614.00	0.257185
$958$	−36622.0	−1.23508
$959$	86592.0	2.91575
$960$	−960.000	−0.0322749
$961$	9413.00	0.315968
$962$	−33672.0	−1.12851
$963$	351.000	0.0117454
$964$	624.000	0.0208482
$965$	12390.0	0.413314
$966$	−22464.0	−0.748206
$967$	−12113.0	−0.402821	−0.201410	−	0.979507i	$-0.564553\pi$
$967$	−12113.0	−0.402821	−0.201410	+	0.979507i	$0.564553\pi$
$968$	4816.00	0.159909
$969$	4233.00	0.140334
$970$	−11400.0	−0.377353
$971$	−1190.00	−0.0393295	−0.0196647	−	0.999807i	$-0.506260\pi$
$971$	−1190.00	−0.0393295	−0.0196647	+	0.999807i	$0.506260\pi$
$972$	972.000	0.0320750
$973$	66752.0	2.19935
$974$	−13532.0	−0.445168
$975$	20700.0	0.679929
$976$	−11200.0	−0.367319
$977$	10474.0	0.342982	0.171491	−	0.985186i	$-0.445142\pi$
$977$	10474.0	0.342982	0.171491	+	0.985186i	$0.445142\pi$
$978$	10476.0	0.342521
$979$	−39636.0	−1.29394
$980$	−13620.0	−0.443954
$981$	−3852.00	−0.125367
$982$	27692.0	0.899885
$983$	−48259.0	−1.56584	−0.782921	−	0.622121i	$-0.786271\pi$
$983$	−48259.0	−1.56584	−0.782921	+	0.622121i	$0.786271\pi$
$984$	−4056.00	−0.131403
$985$	10705.0	0.346284
$986$	3196.00	0.103227
$987$	36672.0	1.18266
$988$	22908.0	0.737652
$989$	−26559.0	−0.853920
$990$	2430.00	0.0780106
$991$	37256.0	1.19422	0.597112	−	0.802158i	$-0.296315\pi$
$991$	37256.0	1.19422	0.597112	+	0.802158i	$0.296315\pi$
$992$	−6336.00	−0.202791
$993$	6315.00	0.201813
$994$	−17664.0	−0.563650
$995$	2900.00	0.0923982
$996$	3384.00	0.107657
$997$	17830.0	0.566381	0.283190	−	0.959064i	$-0.408607\pi$
$997$	17830.0	0.566381	0.283190	+	0.959064i	$0.408607\pi$
$998$	11708.0	0.371353
$999$	−6588.00	−0.208644

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	102.4.a.b.1.1	✓	1
3.2	odd	2		306.4.a.g.1.1		1
4.3	odd	2		816.4.a.c.1.1		1
12.11	even	2		2448.4.a.j.1.1		1
17.16	even	2		1734.4.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
102.4.a.b.1.1	✓	1	1.1	even	1	trivial
306.4.a.g.1.1		1	3.2	odd	2
816.4.a.c.1.1		1	4.3	odd	2
1734.4.a.a.1.1		1	17.16	even	2
2448.4.a.j.1.1		1	12.11	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:	\( N \)	\(=\)	\( 102 = 2 \cdot 3 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	102.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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