LMFDB - Embedded newform 147.4.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [147,4,Mod(1,147)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(147, 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("147.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 147 = 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	147.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:	$8.67328077084$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 21)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	147.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-3.00000 q^{2} +3.00000 q^{3} +1.00000 q^{4} -3.00000 q^{5} -9.00000 q^{6} +21.0000 q^{8} +9.00000 q^{9} +O(q^{10})$

\(q-3.00000 q^{2} +3.00000 q^{3} +1.00000 q^{4} -3.00000 q^{5} -9.00000 q^{6} +21.0000 q^{8} +9.00000 q^{9} +9.00000 q^{10} -15.0000 q^{11} +3.00000 q^{12} -64.0000 q^{13} -9.00000 q^{15} -71.0000 q^{16} +84.0000 q^{17} -27.0000 q^{18} -16.0000 q^{19} -3.00000 q^{20} +45.0000 q^{22} -84.0000 q^{23} +63.0000 q^{24} -116.000 q^{25} +192.000 q^{26} +27.0000 q^{27} -297.000 q^{29} +27.0000 q^{30} -253.000 q^{31} +45.0000 q^{32} -45.0000 q^{33} -252.000 q^{34} +9.00000 q^{36} -316.000 q^{37} +48.0000 q^{38} -192.000 q^{39} -63.0000 q^{40} +360.000 q^{41} +26.0000 q^{43} -15.0000 q^{44} -27.0000 q^{45} +252.000 q^{46} -30.0000 q^{47} -213.000 q^{48} +348.000 q^{50} +252.000 q^{51} -64.0000 q^{52} +363.000 q^{53} -81.0000 q^{54} +45.0000 q^{55} -48.0000 q^{57} +891.000 q^{58} -15.0000 q^{59} -9.00000 q^{60} -118.000 q^{61} +759.000 q^{62} +433.000 q^{64} +192.000 q^{65} +135.000 q^{66} -370.000 q^{67} +84.0000 q^{68} -252.000 q^{69} -342.000 q^{71} +189.000 q^{72} +362.000 q^{73} +948.000 q^{74} -348.000 q^{75} -16.0000 q^{76} +576.000 q^{78} +467.000 q^{79} +213.000 q^{80} +81.0000 q^{81} -1080.00 q^{82} +477.000 q^{83} -252.000 q^{85} -78.0000 q^{86} -891.000 q^{87} -315.000 q^{88} +906.000 q^{89} +81.0000 q^{90} -84.0000 q^{92} -759.000 q^{93} +90.0000 q^{94} +48.0000 q^{95} +135.000 q^{96} +503.000 q^{97} -135.000 q^{99} +O(q^{100})\)

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 $

$2$	−3.00000	−1.06066	−0.530330	−	0.847791i	$-0.677932\pi$
$2$	−3.00000	−1.06066	−0.530330	+	0.847791i	$0.677932\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	1.00000	0.125000
$5$	−3.00000	−0.268328	−0.134164	−	0.990959i	$-0.542835\pi$
$5$	−3.00000	−0.268328	−0.134164	+	0.990959i	$0.542835\pi$
$6$	−9.00000	−0.612372
$7$	0	0
$7$	0	0
$8$	21.0000	0.928078
$9$	9.00000	0.333333
$10$	9.00000	0.284605
$11$	−15.0000	−0.411152	−0.205576	−	0.978641i	$-0.565907\pi$
$11$	−15.0000	−0.411152	−0.205576	+	0.978641i	$0.565907\pi$
$12$	3.00000	0.0721688
$13$	−64.0000	−1.36542	−0.682708	−	0.730691i	$-0.739198\pi$
$13$	−64.0000	−1.36542	−0.682708	+	0.730691i	$0.739198\pi$
$14$	0	0
$15$	−9.00000	−0.154919
$16$	−71.0000	−1.10938
$17$	84.0000	1.19841	0.599206	−	0.800595i	$-0.295483\pi$
$17$	84.0000	1.19841	0.599206	+	0.800595i	$0.295483\pi$
$18$	−27.0000	−0.353553
$19$	−16.0000	−0.193192	−0.0965961	−	0.995324i	$-0.530796\pi$
$19$	−16.0000	−0.193192	−0.0965961	+	0.995324i	$0.530796\pi$
$20$	−3.00000	−0.0335410
$21$	0	0
$22$	45.0000	0.436092
$23$	−84.0000	−0.761531	−0.380765	−	0.924672i	$-0.624339\pi$
$23$	−84.0000	−0.761531	−0.380765	+	0.924672i	$0.624339\pi$
$24$	63.0000	0.535826
$25$	−116.000	−0.928000
$26$	192.000	1.44824
$27$	27.0000	0.192450
$28$	0	0
$29$	−297.000	−1.90178	−0.950888	−	0.309535i	$-0.899827\pi$
$29$	−297.000	−1.90178	−0.950888	+	0.309535i	$0.899827\pi$
$30$	27.0000	0.164317
$31$	−253.000	−1.46581	−0.732906	−	0.680330i	$-0.761836\pi$
$31$	−253.000	−1.46581	−0.732906	+	0.680330i	$0.761836\pi$
$32$	45.0000	0.248592
$33$	−45.0000	−0.237379
$34$	−252.000	−1.27111
$35$	0	0
$36$	9.00000	0.0416667
$37$	−316.000	−1.40406	−0.702028	−	0.712149i	$-0.747722\pi$
$37$	−316.000	−1.40406	−0.702028	+	0.712149i	$0.747722\pi$
$38$	48.0000	0.204911
$39$	−192.000	−0.788323
$40$	−63.0000	−0.249029
$41$	360.000	1.37128	0.685641	−	0.727940i	$-0.259522\pi$
$41$	360.000	1.37128	0.685641	+	0.727940i	$0.259522\pi$
$42$	0	0
$43$	26.0000	0.0922084	0.0461042	−	0.998937i	$-0.485319\pi$
$43$	26.0000	0.0922084	0.0461042	+	0.998937i	$0.485319\pi$
$44$	−15.0000	−0.0513940
$45$	−27.0000	−0.0894427
$46$	252.000	0.807725
$47$	−30.0000	−0.0931053	−0.0465527	−	0.998916i	$-0.514824\pi$
$47$	−30.0000	−0.0931053	−0.0465527	+	0.998916i	$0.514824\pi$
$48$	−213.000	−0.640498
$49$	0	0
$50$	348.000	0.984293
$51$	252.000	0.691903
$52$	−64.0000	−0.170677
$53$	363.000	0.940790	0.470395	−	0.882456i	$-0.344111\pi$
$53$	363.000	0.940790	0.470395	+	0.882456i	$0.344111\pi$
$54$	−81.0000	−0.204124
$55$	45.0000	0.110324
$56$	0	0
$57$	−48.0000	−0.111540
$58$	891.000	2.01714
$59$	−15.0000	−0.0330989	−0.0165494	−	0.999863i	$-0.505268\pi$
$59$	−15.0000	−0.0330989	−0.0165494	+	0.999863i	$0.505268\pi$
$60$	−9.00000	−0.0193649
$61$	−118.000	−0.247678	−0.123839	−	0.992302i	$-0.539521\pi$
$61$	−118.000	−0.247678	−0.123839	+	0.992302i	$0.539521\pi$
$62$	759.000	1.55473
$63$	0	0
$64$	433.000	0.845703
$65$	192.000	0.366380
$66$	135.000	0.251778
$67$	−370.000	−0.674667	−0.337334	−	0.941385i	$-0.609525\pi$
$67$	−370.000	−0.674667	−0.337334	+	0.941385i	$0.609525\pi$
$68$	84.0000	0.149801
$69$	−252.000	−0.439670
$70$	0	0
$71$	−342.000	−0.571661	−0.285831	−	0.958280i	$-0.592269\pi$
$71$	−342.000	−0.571661	−0.285831	+	0.958280i	$0.592269\pi$
$72$	189.000	0.309359
$73$	362.000	0.580396	0.290198	−	0.956967i	$-0.406279\pi$
$73$	362.000	0.580396	0.290198	+	0.956967i	$0.406279\pi$
$74$	948.000	1.48923
$75$	−348.000	−0.535781
$76$	−16.0000	−0.0241490
$77$	0	0
$78$	576.000	0.836143
$79$	467.000	0.665084	0.332542	−	0.943089i	$-0.392094\pi$
$79$	467.000	0.665084	0.332542	+	0.943089i	$0.392094\pi$
$80$	213.000	0.297677
$81$	81.0000	0.111111
$82$	−1080.00	−1.45446
$83$	477.000	0.630814	0.315407	−	0.948957i	$-0.397859\pi$
$83$	477.000	0.630814	0.315407	+	0.948957i	$0.397859\pi$
$84$	0	0
$85$	−252.000	−0.321568
$86$	−78.0000	−0.0978018
$87$	−891.000	−1.09799
$88$	−315.000	−0.381581
$89$	906.000	1.07905	0.539527	−	0.841968i	$-0.318603\pi$
$89$	906.000	1.07905	0.539527	+	0.841968i	$0.318603\pi$
$90$	81.0000	0.0948683
$91$	0	0
$92$	−84.0000	−0.0951914
$93$	−759.000	−0.846286
$94$	90.0000	0.0987531
$95$	48.0000	0.0518389
$96$	135.000	0.143525
$97$	503.000	0.526515	0.263257	−	0.964726i	$-0.415203\pi$
$97$	503.000	0.526515	0.263257	+	0.964726i	$0.415203\pi$
$98$	0	0
$99$	−135.000	−0.137051
$100$	−116.000	−0.116000
$101$	−1086.00	−1.06991	−0.534956	−	0.844880i	$-0.679672\pi$
$101$	−1086.00	−1.06991	−0.534956	+	0.844880i	$0.679672\pi$
$102$	−756.000	−0.733874
$103$	1736.00	1.66071	0.830355	−	0.557235i	$-0.188138\pi$
$103$	1736.00	1.66071	0.830355	+	0.557235i	$0.188138\pi$
$104$	−1344.00	−1.26721
$105$	0	0
$106$	−1089.00	−0.997859
$107$	−1353.00	−1.22242	−0.611212	−	0.791467i	$-0.709318\pi$
$107$	−1353.00	−1.22242	−0.611212	+	0.791467i	$0.709318\pi$
$108$	27.0000	0.0240563
$109$	−370.000	−0.325134	−0.162567	−	0.986698i	$-0.551977\pi$
$109$	−370.000	−0.325134	−0.162567	+	0.986698i	$0.551977\pi$
$110$	−135.000	−0.117016
$111$	−948.000	−0.810632
$112$	0	0
$113$	−648.000	−0.539458	−0.269729	−	0.962936i	$-0.586934\pi$
$113$	−648.000	−0.539458	−0.269729	+	0.962936i	$0.586934\pi$
$114$	144.000	0.118306
$115$	252.000	0.204340
$116$	−297.000	−0.237722
$117$	−576.000	−0.455139
$118$	45.0000	0.0351067
$119$	0	0
$120$	−189.000	−0.143777
$121$	−1106.00	−0.830954
$122$	354.000	0.262702
$123$	1080.00	0.791710
$124$	−253.000	−0.183226
$125$	723.000	0.517337
$126$	0	0
$127$	377.000	0.263412	0.131706	−	0.991289i	$-0.457954\pi$
$127$	377.000	0.263412	0.131706	+	0.991289i	$0.457954\pi$
$128$	−1659.00	−1.14560
$129$	78.0000	0.0532366
$130$	−576.000	−0.388604
$131$	−651.000	−0.434184	−0.217092	−	0.976151i	$-0.569657\pi$
$131$	−651.000	−0.434184	−0.217092	+	0.976151i	$0.569657\pi$
$132$	−45.0000	−0.0296723
$133$	0	0
$134$	1110.00	0.715593
$135$	−81.0000	−0.0516398
$136$	1764.00	1.11222
$137$	−1770.00	−1.10381	−0.551903	−	0.833909i	$-0.686098\pi$
$137$	−1770.00	−1.10381	−0.551903	+	0.833909i	$0.686098\pi$
$138$	756.000	0.466341
$139$	−1558.00	−0.950704	−0.475352	−	0.879796i	$-0.657679\pi$
$139$	−1558.00	−0.950704	−0.475352	+	0.879796i	$0.657679\pi$
$140$	0	0
$141$	−90.0000	−0.0537544
$142$	1026.00	0.606338
$143$	960.000	0.561393
$144$	−639.000	−0.369792
$145$	891.000	0.510300
$146$	−1086.00	−0.615603
$147$	0	0
$148$	−316.000	−0.175507
$149$	2454.00	1.34926	0.674629	−	0.738157i	$-0.264304\pi$
$149$	2454.00	1.34926	0.674629	+	0.738157i	$0.264304\pi$
$150$	1044.00	0.568282
$151$	1259.00	0.678516	0.339258	−	0.940693i	$-0.389824\pi$
$151$	1259.00	0.678516	0.339258	+	0.940693i	$0.389824\pi$
$152$	−336.000	−0.179297
$153$	756.000	0.399470
$154$	0	0
$155$	759.000	0.393318
$156$	−192.000	−0.0985404
$157$	−196.000	−0.0996338	−0.0498169	−	0.998758i	$-0.515864\pi$
$157$	−196.000	−0.0996338	−0.0498169	+	0.998758i	$0.515864\pi$
$158$	−1401.00	−0.705428
$159$	1089.00	0.543166
$160$	−135.000	−0.0667043
$161$	0	0
$162$	−243.000	−0.117851
$163$	−1252.00	−0.601621	−0.300810	−	0.953684i	$-0.597257\pi$
$163$	−1252.00	−0.601621	−0.300810	+	0.953684i	$0.597257\pi$
$164$	360.000	0.171410
$165$	135.000	0.0636954
$166$	−1431.00	−0.669079
$167$	−2646.00	−1.22607	−0.613035	−	0.790056i	$-0.710051\pi$
$167$	−2646.00	−1.22607	−0.613035	+	0.790056i	$0.710051\pi$
$168$	0	0
$169$	1899.00	0.864360
$170$	756.000	0.341074
$171$	−144.000	−0.0643974
$172$	26.0000	0.0115261
$173$	−786.000	−0.345425	−0.172712	−	0.984972i	$-0.555253\pi$
$173$	−786.000	−0.345425	−0.172712	+	0.984972i	$0.555253\pi$
$174$	2673.00	1.16460
$175$	0	0
$176$	1065.00	0.456122
$177$	−45.0000	−0.0191096
$178$	−2718.00	−1.14451
$179$	2892.00	1.20759	0.603794	−	0.797140i	$-0.293655\pi$
$179$	2892.00	1.20759	0.603794	+	0.797140i	$0.293655\pi$
$180$	−27.0000	−0.0111803
$181$	1352.00	0.555212	0.277606	−	0.960695i	$-0.410459\pi$
$181$	1352.00	0.555212	0.277606	+	0.960695i	$0.410459\pi$
$182$	0	0
$183$	−354.000	−0.142997
$184$	−1764.00	−0.706760
$185$	948.000	0.376748
$186$	2277.00	0.897622
$187$	−1260.00	−0.492729
$188$	−30.0000	−0.0116382
$189$	0	0
$190$	−144.000	−0.0549835
$191$	3912.00	1.48200	0.741001	−	0.671504i	$-0.234351\pi$
$191$	3912.00	1.48200	0.741001	+	0.671504i	$0.234351\pi$
$192$	1299.00	0.488267
$193$	1493.00	0.556832	0.278416	−	0.960461i	$-0.410191\pi$
$193$	1493.00	0.556832	0.278416	+	0.960461i	$0.410191\pi$
$194$	−1509.00	−0.558453
$195$	576.000	0.211529
$196$	0	0
$197$	−4086.00	−1.47774	−0.738872	−	0.673846i	$-0.764641\pi$
$197$	−4086.00	−1.47774	−0.738872	+	0.673846i	$0.764641\pi$
$198$	405.000	0.145364
$199$	−3556.00	−1.26672	−0.633362	−	0.773855i	$-0.718326\pi$
$199$	−3556.00	−1.26672	−0.633362	+	0.773855i	$0.718326\pi$
$200$	−2436.00	−0.861256
$201$	−1110.00	−0.389519
$202$	3258.00	1.13481
$203$	0	0
$204$	252.000	0.0864879
$205$	−1080.00	−0.367954
$206$	−5208.00	−1.76145
$207$	−756.000	−0.253844
$208$	4544.00	1.51476
$209$	240.000	0.0794313
$210$	0	0
$211$	1250.00	0.407837	0.203918	−	0.978988i	$-0.434632\pi$
$211$	1250.00	0.407837	0.203918	+	0.978988i	$0.434632\pi$
$212$	363.000	0.117599
$213$	−1026.00	−0.330049
$214$	4059.00	1.29658
$215$	−78.0000	−0.0247421
$216$	567.000	0.178609
$217$	0	0
$218$	1110.00	0.344856
$219$	1086.00	0.335092
$220$	45.0000	0.0137905
$221$	−5376.00	−1.63633
$222$	2844.00	0.859805
$223$	425.000	0.127624	0.0638119	−	0.997962i	$-0.479674\pi$
$223$	425.000	0.127624	0.0638119	+	0.997962i	$0.479674\pi$
$224$	0	0
$225$	−1044.00	−0.309333
$226$	1944.00	0.572181
$227$	3855.00	1.12716	0.563580	−	0.826061i	$-0.309424\pi$
$227$	3855.00	1.12716	0.563580	+	0.826061i	$0.309424\pi$
$228$	−48.0000	−0.0139424
$229$	−2188.00	−0.631385	−0.315692	−	0.948862i	$-0.602237\pi$
$229$	−2188.00	−0.631385	−0.315692	+	0.948862i	$0.602237\pi$
$230$	−756.000	−0.216735
$231$	0	0
$232$	−6237.00	−1.76500
$233$	852.000	0.239555	0.119778	−	0.992801i	$-0.461782\pi$
$233$	852.000	0.239555	0.119778	+	0.992801i	$0.461782\pi$
$234$	1728.00	0.482747
$235$	90.0000	0.0249828
$236$	−15.0000	−0.00413736
$237$	1401.00	0.383986
$238$	0	0
$239$	5508.00	1.49072	0.745362	−	0.666660i	$-0.232277\pi$
$239$	5508.00	1.49072	0.745362	+	0.666660i	$0.232277\pi$
$240$	639.000	0.171864
$241$	791.000	0.211422	0.105711	−	0.994397i	$-0.466288\pi$
$241$	791.000	0.211422	0.105711	+	0.994397i	$0.466288\pi$
$242$	3318.00	0.881360
$243$	243.000	0.0641500
$244$	−118.000	−0.0309597
$245$	0	0
$246$	−3240.00	−0.839735
$247$	1024.00	0.263788
$248$	−5313.00	−1.36039
$249$	1431.00	0.364201
$250$	−2169.00	−0.548718
$251$	5265.00	1.32400	0.662000	−	0.749504i	$-0.269708\pi$
$251$	5265.00	1.32400	0.662000	+	0.749504i	$0.269708\pi$
$252$	0	0
$253$	1260.00	0.313105
$254$	−1131.00	−0.279391
$255$	−756.000	−0.185657
$256$	1513.00	0.369385
$257$	−6870.00	−1.66747	−0.833733	−	0.552168i	$-0.813801\pi$
$257$	−6870.00	−1.66747	−0.833733	+	0.552168i	$0.813801\pi$
$258$	−234.000	−0.0564659
$259$	0	0
$260$	192.000	0.0457974
$261$	−2673.00	−0.633925
$262$	1953.00	0.460522
$263$	−222.000	−0.0520498	−0.0260249	−	0.999661i	$-0.508285\pi$
$263$	−222.000	−0.0520498	−0.0260249	+	0.999661i	$0.508285\pi$
$264$	−945.000	−0.220306
$265$	−1089.00	−0.252441
$266$	0	0
$267$	2718.00	0.622992
$268$	−370.000	−0.0843334
$269$	7851.00	1.77949	0.889747	−	0.456454i	$-0.150881\pi$
$269$	7851.00	1.77949	0.889747	+	0.456454i	$0.150881\pi$
$270$	243.000	0.0547723
$271$	5183.00	1.16179	0.580895	−	0.813979i	$-0.302703\pi$
$271$	5183.00	1.16179	0.580895	+	0.813979i	$0.302703\pi$
$272$	−5964.00	−1.32949
$273$	0	0
$274$	5310.00	1.17076
$275$	1740.00	0.381549
$276$	−252.000	−0.0549588
$277$	−4960.00	−1.07588	−0.537938	−	0.842985i	$-0.680796\pi$
$277$	−4960.00	−1.07588	−0.537938	+	0.842985i	$0.680796\pi$
$278$	4674.00	1.00837
$279$	−2277.00	−0.488604
$280$	0	0
$281$	−774.000	−0.164317	−0.0821583	−	0.996619i	$-0.526181\pi$
$281$	−774.000	−0.164317	−0.0821583	+	0.996619i	$0.526181\pi$
$282$	270.000	0.0570151
$283$	3698.00	0.776761	0.388380	−	0.921499i	$-0.373035\pi$
$283$	3698.00	0.776761	0.388380	+	0.921499i	$0.373035\pi$
$284$	−342.000	−0.0714576
$285$	144.000	0.0299292
$286$	−2880.00	−0.595447
$287$	0	0
$288$	405.000	0.0828641
$289$	2143.00	0.436190
$290$	−2673.00	−0.541255
$291$	1509.00	0.303983
$292$	362.000	0.0725495
$293$	−6273.00	−1.25076	−0.625380	−	0.780321i	$-0.715056\pi$
$293$	−6273.00	−1.25076	−0.625380	+	0.780321i	$0.715056\pi$
$294$	0	0
$295$	45.0000	0.00888136
$296$	−6636.00	−1.30307
$297$	−405.000	−0.0791262
$298$	−7362.00	−1.43110
$299$	5376.00	1.03981
$300$	−348.000	−0.0669726
$301$	0	0
$302$	−3777.00	−0.719675
$303$	−3258.00	−0.617714
$304$	1136.00	0.214323
$305$	354.000	0.0664590
$306$	−2268.00	−0.423702
$307$	−1684.00	−0.313065	−0.156533	−	0.987673i	$-0.550032\pi$
$307$	−1684.00	−0.313065	−0.156533	+	0.987673i	$0.550032\pi$
$308$	0	0
$309$	5208.00	0.958812
$310$	−2277.00	−0.417177
$311$	−1320.00	−0.240676	−0.120338	−	0.992733i	$-0.538398\pi$
$311$	−1320.00	−0.240676	−0.120338	+	0.992733i	$0.538398\pi$
$312$	−4032.00	−0.731625
$313$	−8503.00	−1.53552	−0.767760	−	0.640737i	$-0.778629\pi$
$313$	−8503.00	−1.53552	−0.767760	+	0.640737i	$0.778629\pi$
$314$	588.000	0.105678
$315$	0	0
$316$	467.000	0.0831355
$317$	−2577.00	−0.456589	−0.228295	−	0.973592i	$-0.573315\pi$
$317$	−2577.00	−0.456589	−0.228295	+	0.973592i	$0.573315\pi$
$318$	−3267.00	−0.576114
$319$	4455.00	0.781919
$320$	−1299.00	−0.226926
$321$	−4059.00	−0.705767
$322$	0	0
$323$	−1344.00	−0.231524
$324$	81.0000	0.0138889
$325$	7424.00	1.26711
$326$	3756.00	0.638115
$327$	−1110.00	−0.187716
$328$	7560.00	1.27266
$329$	0	0
$330$	−405.000	−0.0675591
$331$	−484.000	−0.0803717	−0.0401859	−	0.999192i	$-0.512795\pi$
$331$	−484.000	−0.0803717	−0.0401859	+	0.999192i	$0.512795\pi$
$332$	477.000	0.0788517
$333$	−2844.00	−0.468019
$334$	7938.00	1.30044
$335$	1110.00	0.181032
$336$	0	0
$337$	−8359.00	−1.35117	−0.675584	−	0.737283i	$-0.736109\pi$
$337$	−8359.00	−1.35117	−0.675584	+	0.737283i	$0.736109\pi$
$338$	−5697.00	−0.916793
$339$	−1944.00	−0.311456
$340$	−252.000	−0.0401959
$341$	3795.00	0.602671
$342$	432.000	0.0683038
$343$	0	0
$344$	546.000	0.0855766
$345$	756.000	0.117976
$346$	2358.00	0.366378
$347$	−1860.00	−0.287752	−0.143876	−	0.989596i	$-0.545957\pi$
$347$	−1860.00	−0.287752	−0.143876	+	0.989596i	$0.545957\pi$
$348$	−891.000	−0.137249
$349$	−1918.00	−0.294178	−0.147089	−	0.989123i	$-0.546990\pi$
$349$	−1918.00	−0.294178	−0.147089	+	0.989123i	$0.546990\pi$
$350$	0	0
$351$	−1728.00	−0.262774
$352$	−675.000	−0.102209
$353$	−3048.00	−0.459571	−0.229786	−	0.973241i	$-0.573803\pi$
$353$	−3048.00	−0.459571	−0.229786	+	0.973241i	$0.573803\pi$
$354$	135.000	0.0202688
$355$	1026.00	0.153393
$356$	906.000	0.134882
$357$	0	0
$358$	−8676.00	−1.28084
$359$	−30.0000	−0.00441042	−0.00220521	−	0.999998i	$-0.500702\pi$
$359$	−30.0000	−0.00441042	−0.00220521	+	0.999998i	$0.500702\pi$
$360$	−567.000	−0.0830098
$361$	−6603.00	−0.962677
$362$	−4056.00	−0.588891
$363$	−3318.00	−0.479752
$364$	0	0
$365$	−1086.00	−0.155737
$366$	1062.00	0.151671
$367$	−11311.0	−1.60880	−0.804400	−	0.594088i	$-0.797513\pi$
$367$	−11311.0	−1.60880	−0.804400	+	0.594088i	$0.797513\pi$
$368$	5964.00	0.844823
$369$	3240.00	0.457094
$370$	−2844.00	−0.399601
$371$	0	0
$372$	−759.000	−0.105786
$373$	1208.00	0.167689	0.0838443	−	0.996479i	$-0.473280\pi$
$373$	1208.00	0.167689	0.0838443	+	0.996479i	$0.473280\pi$
$374$	3780.00	0.522618
$375$	2169.00	0.298684
$376$	−630.000	−0.0864090
$377$	19008.0	2.59672
$378$	0	0
$379$	7640.00	1.03546	0.517731	−	0.855543i	$-0.326777\pi$
$379$	7640.00	1.03546	0.517731	+	0.855543i	$0.326777\pi$
$380$	48.0000	0.00647986
$381$	1131.00	0.152081
$382$	−11736.0	−1.57190
$383$	12750.0	1.70103	0.850515	−	0.525951i	$-0.176290\pi$
$383$	12750.0	1.70103	0.850515	+	0.525951i	$0.176290\pi$
$384$	−4977.00	−0.661410
$385$	0	0
$386$	−4479.00	−0.590609
$387$	234.000	0.0307361
$388$	503.000	0.0658143
$389$	3126.00	0.407441	0.203720	−	0.979029i	$-0.434697\pi$
$389$	3126.00	0.407441	0.203720	+	0.979029i	$0.434697\pi$
$390$	−1728.00	−0.224361
$391$	−7056.00	−0.912627
$392$	0	0
$393$	−1953.00	−0.250676
$394$	12258.0	1.56738
$395$	−1401.00	−0.178461
$396$	−135.000	−0.0171313
$397$	−5932.00	−0.749921	−0.374960	−	0.927041i	$-0.622344\pi$
$397$	−5932.00	−0.749921	−0.374960	+	0.927041i	$0.622344\pi$
$398$	10668.0	1.34356
$399$	0	0
$400$	8236.00	1.02950
$401$	1608.00	0.200249	0.100124	−	0.994975i	$-0.468076\pi$
$401$	1608.00	0.200249	0.100124	+	0.994975i	$0.468076\pi$
$402$	3330.00	0.413148
$403$	16192.0	2.00144
$404$	−1086.00	−0.133739
$405$	−243.000	−0.0298142
$406$	0	0
$407$	4740.00	0.577280
$408$	5292.00	0.642140
$409$	−4465.00	−0.539805	−0.269902	−	0.962888i	$-0.586991\pi$
$409$	−4465.00	−0.539805	−0.269902	+	0.962888i	$0.586991\pi$
$410$	3240.00	0.390274
$411$	−5310.00	−0.637282
$412$	1736.00	0.207589
$413$	0	0
$414$	2268.00	0.269242
$415$	−1431.00	−0.169265
$416$	−2880.00	−0.339432
$417$	−4674.00	−0.548889
$418$	−720.000	−0.0842496
$419$	−1584.00	−0.184686	−0.0923430	−	0.995727i	$-0.529436\pi$
$419$	−1584.00	−0.184686	−0.0923430	+	0.995727i	$0.529436\pi$
$420$	0	0
$421$	−1330.00	−0.153967	−0.0769837	−	0.997032i	$-0.524529\pi$
$421$	−1330.00	−0.153967	−0.0769837	+	0.997032i	$0.524529\pi$
$422$	−3750.00	−0.432576
$423$	−270.000	−0.0310351
$424$	7623.00	0.873126
$425$	−9744.00	−1.11213
$426$	3078.00	0.350069
$427$	0	0
$428$	−1353.00	−0.152803
$429$	2880.00	0.324121
$430$	234.000	0.0262430
$431$	9588.00	1.07155	0.535775	−	0.844361i	$-0.320020\pi$
$431$	9588.00	1.07155	0.535775	+	0.844361i	$0.320020\pi$
$432$	−1917.00	−0.213499
$433$	494.000	0.0548271	0.0274135	−	0.999624i	$-0.491273\pi$
$433$	494.000	0.0548271	0.0274135	+	0.999624i	$0.491273\pi$
$434$	0	0
$435$	2673.00	0.294622
$436$	−370.000	−0.0406417
$437$	1344.00	0.147122
$438$	−3258.00	−0.355418
$439$	−16009.0	−1.74047	−0.870237	−	0.492634i	$-0.836034\pi$
$439$	−16009.0	−1.74047	−0.870237	+	0.492634i	$0.836034\pi$
$440$	945.000	0.102389
$441$	0	0
$442$	16128.0	1.73559
$443$	7773.00	0.833649	0.416824	−	0.908987i	$-0.363143\pi$
$443$	7773.00	0.833649	0.416824	+	0.908987i	$0.363143\pi$
$444$	−948.000	−0.101329
$445$	−2718.00	−0.289541
$446$	−1275.00	−0.135365
$447$	7362.00	0.778995
$448$	0	0
$449$	864.000	0.0908122	0.0454061	−	0.998969i	$-0.485542\pi$
$449$	864.000	0.0908122	0.0454061	+	0.998969i	$0.485542\pi$
$450$	3132.00	0.328098
$451$	−5400.00	−0.563805
$452$	−648.000	−0.0674322
$453$	3777.00	0.391742
$454$	−11565.0	−1.19553
$455$	0	0
$456$	−1008.00	−0.103517
$457$	2519.00	0.257842	0.128921	−	0.991655i	$-0.458849\pi$
$457$	2519.00	0.257842	0.128921	+	0.991655i	$0.458849\pi$
$458$	6564.00	0.669685
$459$	2268.00	0.230634
$460$	252.000	0.0255425
$461$	−342.000	−0.0345521	−0.0172761	−	0.999851i	$-0.505499\pi$
$461$	−342.000	−0.0345521	−0.0172761	+	0.999851i	$0.505499\pi$
$462$	0	0
$463$	−4336.00	−0.435229	−0.217614	−	0.976035i	$-0.569828\pi$
$463$	−4336.00	−0.435229	−0.217614	+	0.976035i	$0.569828\pi$
$464$	21087.0	2.10978
$465$	2277.00	0.227082
$466$	−2556.00	−0.254087
$467$	18636.0	1.84662	0.923310	−	0.384056i	$-0.125473\pi$
$467$	18636.0	1.84662	0.923310	+	0.384056i	$0.125473\pi$
$468$	−576.000	−0.0568923
$469$	0	0
$470$	−270.000	−0.0264982
$471$	−588.000	−0.0575236
$472$	−315.000	−0.0307183
$473$	−390.000	−0.0379117
$474$	−4203.00	−0.407279
$475$	1856.00	0.179282
$476$	0	0
$477$	3267.00	0.313597
$478$	−16524.0	−1.58115
$479$	15078.0	1.43827	0.719135	−	0.694870i	$-0.244538\pi$
$479$	15078.0	1.43827	0.719135	+	0.694870i	$0.244538\pi$
$480$	−405.000	−0.0385117
$481$	20224.0	1.91712
$482$	−2373.00	−0.224247
$483$	0	0
$484$	−1106.00	−0.103869
$485$	−1509.00	−0.141279
$486$	−729.000	−0.0680414
$487$	6221.00	0.578851	0.289425	−	0.957201i	$-0.406536\pi$
$487$	6221.00	0.578851	0.289425	+	0.957201i	$0.406536\pi$
$488$	−2478.00	−0.229864
$489$	−3756.00	−0.347346
$490$	0	0
$491$	−7371.00	−0.677492	−0.338746	−	0.940878i	$-0.610003\pi$
$491$	−7371.00	−0.677492	−0.338746	+	0.940878i	$0.610003\pi$
$492$	1080.00	0.0989637
$493$	−24948.0	−2.27911
$494$	−3072.00	−0.279789
$495$	405.000	0.0367745
$496$	17963.0	1.62613
$497$	0	0
$498$	−4293.00	−0.386293
$499$	4274.00	0.383428	0.191714	−	0.981451i	$-0.438595\pi$
$499$	4274.00	0.383428	0.191714	+	0.981451i	$0.438595\pi$
$500$	723.000	0.0646671
$501$	−7938.00	−0.707872
$502$	−15795.0	−1.40431
$503$	−2520.00	−0.223382	−0.111691	−	0.993743i	$-0.535627\pi$
$503$	−2520.00	−0.223382	−0.111691	+	0.993743i	$0.535627\pi$
$504$	0	0
$505$	3258.00	0.287087
$506$	−3780.00	−0.332098
$507$	5697.00	0.499039
$508$	377.000	0.0329265
$509$	−14277.0	−1.24326	−0.621628	−	0.783313i	$-0.713528\pi$
$509$	−14277.0	−1.24326	−0.621628	+	0.783313i	$0.713528\pi$
$510$	2268.00	0.196919
$511$	0	0
$512$	8733.00	0.753804
$513$	−432.000	−0.0371799
$514$	20610.0	1.76862
$515$	−5208.00	−0.445615
$516$	78.0000	0.00665457
$517$	450.000	0.0382804
$518$	0	0
$519$	−2358.00	−0.199431
$520$	4032.00	0.340029
$521$	−6306.00	−0.530270	−0.265135	−	0.964211i	$-0.585417\pi$
$521$	−6306.00	−0.530270	−0.265135	+	0.964211i	$0.585417\pi$
$522$	8019.00	0.672379
$523$	8072.00	0.674883	0.337442	−	0.941346i	$-0.390438\pi$
$523$	8072.00	0.674883	0.337442	+	0.941346i	$0.390438\pi$
$524$	−651.000	−0.0542730
$525$	0	0
$526$	666.000	0.0552072
$527$	−21252.0	−1.75664
$528$	3195.00	0.263342
$529$	−5111.00	−0.420071
$530$	3267.00	0.267754
$531$	−135.000	−0.0110330
$532$	0	0
$533$	−23040.0	−1.87237
$534$	−8154.00	−0.660783
$535$	4059.00	0.328011
$536$	−7770.00	−0.626143
$537$	8676.00	0.697201
$538$	−23553.0	−1.88744
$539$	0	0
$540$	−81.0000	−0.00645497
$541$	−22858.0	−1.81653	−0.908264	−	0.418396i	$-0.862592\pi$
$541$	−22858.0	−1.81653	−0.908264	+	0.418396i	$0.862592\pi$
$542$	−15549.0	−1.23226
$543$	4056.00	0.320552
$544$	3780.00	0.297916
$545$	1110.00	0.0872425
$546$	0	0
$547$	−24724.0	−1.93258	−0.966291	−	0.257454i	$-0.917116\pi$
$547$	−24724.0	−1.93258	−0.966291	+	0.257454i	$0.917116\pi$
$548$	−1770.00	−0.137976
$549$	−1062.00	−0.0825593
$550$	−5220.00	−0.404694
$551$	4752.00	0.367408
$552$	−5292.00	−0.408048
$553$	0	0
$554$	14880.0	1.14114
$555$	2844.00	0.217515
$556$	−1558.00	−0.118838
$557$	−9843.00	−0.748764	−0.374382	−	0.927275i	$-0.622145\pi$
$557$	−9843.00	−0.748764	−0.374382	+	0.927275i	$0.622145\pi$
$558$	6831.00	0.518242
$559$	−1664.00	−0.125903
$560$	0	0
$561$	−3780.00	−0.284477
$562$	2322.00	0.174284
$563$	−13371.0	−1.00092	−0.500462	−	0.865758i	$-0.666837\pi$
$563$	−13371.0	−1.00092	−0.500462	+	0.865758i	$0.666837\pi$
$564$	−90.0000	−0.00671930
$565$	1944.00	0.144752
$566$	−11094.0	−0.823879
$567$	0	0
$568$	−7182.00	−0.530546
$569$	−5232.00	−0.385478	−0.192739	−	0.981250i	$-0.561737\pi$
$569$	−5232.00	−0.385478	−0.192739	+	0.981250i	$0.561737\pi$
$570$	−432.000	−0.0317447
$571$	−14398.0	−1.05523	−0.527616	−	0.849483i	$-0.676914\pi$
$571$	−14398.0	−1.05523	−0.527616	+	0.849483i	$0.676914\pi$
$572$	960.000	0.0701742
$573$	11736.0	0.855634
$574$	0	0
$575$	9744.00	0.706701
$576$	3897.00	0.281901
$577$	19871.0	1.43369	0.716846	−	0.697231i	$-0.245585\pi$
$577$	19871.0	1.43369	0.716846	+	0.697231i	$0.245585\pi$
$578$	−6429.00	−0.462649
$579$	4479.00	0.321487
$580$	891.000	0.0637875
$581$	0	0
$582$	−4527.00	−0.322423
$583$	−5445.00	−0.386808
$584$	7602.00	0.538652
$585$	1728.00	0.122127
$586$	18819.0	1.32663
$587$	−16137.0	−1.13466	−0.567330	−	0.823491i	$-0.692024\pi$
$587$	−16137.0	−1.13466	−0.567330	+	0.823491i	$0.692024\pi$
$588$	0	0
$589$	4048.00	0.283183
$590$	−135.000	−0.00942011
$591$	−12258.0	−0.853176
$592$	22436.0	1.55762
$593$	−21324.0	−1.47668	−0.738340	−	0.674428i	$-0.764390\pi$
$593$	−21324.0	−1.47668	−0.738340	+	0.674428i	$0.764390\pi$
$594$	1215.00	0.0839260
$595$	0	0
$596$	2454.00	0.168657
$597$	−10668.0	−0.731344
$598$	−16128.0	−1.10288
$599$	−8646.00	−0.589760	−0.294880	−	0.955534i	$-0.595280\pi$
$599$	−8646.00	−0.589760	−0.294880	+	0.955534i	$0.595280\pi$
$600$	−7308.00	−0.497246
$601$	11195.0	0.759823	0.379911	−	0.925023i	$-0.375954\pi$
$601$	11195.0	0.759823	0.379911	+	0.925023i	$0.375954\pi$
$602$	0	0
$603$	−3330.00	−0.224889
$604$	1259.00	0.0848145
$605$	3318.00	0.222968
$606$	9774.00	0.655184
$607$	−8971.00	−0.599871	−0.299935	−	0.953959i	$-0.596965\pi$
$607$	−8971.00	−0.599871	−0.299935	+	0.953959i	$0.596965\pi$
$608$	−720.000	−0.0480261
$609$	0	0
$610$	−1062.00	−0.0704904
$611$	1920.00	0.127127
$612$	756.000	0.0499338
$613$	−12772.0	−0.841527	−0.420764	−	0.907170i	$-0.638238\pi$
$613$	−12772.0	−0.841527	−0.420764	+	0.907170i	$0.638238\pi$
$614$	5052.00	0.332056
$615$	−3240.00	−0.212438
$616$	0	0
$617$	12762.0	0.832705	0.416352	−	0.909203i	$-0.363308\pi$
$617$	12762.0	0.832705	0.416352	+	0.909203i	$0.363308\pi$
$618$	−15624.0	−1.01697
$619$	12842.0	0.833867	0.416933	−	0.908937i	$-0.363105\pi$
$619$	12842.0	0.833867	0.416933	+	0.908937i	$0.363105\pi$
$620$	759.000	0.0491648
$621$	−2268.00	−0.146557
$622$	3960.00	0.255276
$623$	0	0
$624$	13632.0	0.874546
$625$	12331.0	0.789184
$626$	25509.0	1.62867
$627$	720.000	0.0458597
$628$	−196.000	−0.0124542
$629$	−26544.0	−1.68264
$630$	0	0
$631$	21365.0	1.34790	0.673952	−	0.738775i	$-0.264596\pi$
$631$	21365.0	1.34790	0.673952	+	0.738775i	$0.264596\pi$
$632$	9807.00	0.617249
$633$	3750.00	0.235465
$634$	7731.00	0.484286
$635$	−1131.00	−0.0706809
$636$	1089.00	0.0678957
$637$	0	0
$638$	−13365.0	−0.829350
$639$	−3078.00	−0.190554
$640$	4977.00	0.307396
$641$	8274.00	0.509834	0.254917	−	0.966963i	$-0.417952\pi$
$641$	8274.00	0.509834	0.254917	+	0.966963i	$0.417952\pi$
$642$	12177.0	0.748579
$643$	27998.0	1.71716	0.858580	−	0.512680i	$-0.171347\pi$
$643$	27998.0	1.71716	0.858580	+	0.512680i	$0.171347\pi$
$644$	0	0
$645$	−234.000	−0.0142849
$646$	4032.00	0.245568
$647$	−17466.0	−1.06130	−0.530649	−	0.847592i	$-0.678052\pi$
$647$	−17466.0	−1.06130	−0.530649	+	0.847592i	$0.678052\pi$
$648$	1701.00	0.103120
$649$	225.000	0.0136087
$650$	−22272.0	−1.34397
$651$	0	0
$652$	−1252.00	−0.0752026
$653$	2157.00	0.129265	0.0646324	−	0.997909i	$-0.479413\pi$
$653$	2157.00	0.129265	0.0646324	+	0.997909i	$0.479413\pi$
$654$	3330.00	0.199103
$655$	1953.00	0.116504
$656$	−25560.0	−1.52127
$657$	3258.00	0.193465
$658$	0	0
$659$	19944.0	1.17892	0.589460	−	0.807798i	$-0.299341\pi$
$659$	19944.0	1.17892	0.589460	+	0.807798i	$0.299341\pi$
$660$	135.000	0.00796192
$661$	27506.0	1.61855	0.809273	−	0.587432i	$-0.199861\pi$
$661$	27506.0	1.61855	0.809273	+	0.587432i	$0.199861\pi$
$662$	1452.00	0.0852471
$663$	−16128.0	−0.944735
$664$	10017.0	0.585444
$665$	0	0
$666$	8532.00	0.496409
$667$	24948.0	1.44826
$668$	−2646.00	−0.153259
$669$	1275.00	0.0736836
$670$	−3330.00	−0.192014
$671$	1770.00	0.101833
$672$	0	0
$673$	−19123.0	−1.09530	−0.547650	−	0.836707i	$-0.684478\pi$
$673$	−19123.0	−1.09530	−0.547650	+	0.836707i	$0.684478\pi$
$674$	25077.0	1.43313
$675$	−3132.00	−0.178594
$676$	1899.00	0.108045
$677$	13857.0	0.786658	0.393329	−	0.919398i	$-0.371323\pi$
$677$	13857.0	0.786658	0.393329	+	0.919398i	$0.371323\pi$
$678$	5832.00	0.330349
$679$	0	0
$680$	−5292.00	−0.298440
$681$	11565.0	0.650766
$682$	−11385.0	−0.639229
$683$	−22245.0	−1.24624	−0.623120	−	0.782127i	$-0.714135\pi$
$683$	−22245.0	−1.24624	−0.623120	+	0.782127i	$0.714135\pi$
$684$	−144.000	−0.00804967
$685$	5310.00	0.296182
$686$	0	0
$687$	−6564.00	−0.364530
$688$	−1846.00	−0.102294
$689$	−23232.0	−1.28457
$690$	−2268.00	−0.125132
$691$	−640.000	−0.0352341	−0.0176170	−	0.999845i	$-0.505608\pi$
$691$	−640.000	−0.0352341	−0.0176170	+	0.999845i	$0.505608\pi$
$692$	−786.000	−0.0431781
$693$	0	0
$694$	5580.00	0.305207
$695$	4674.00	0.255101
$696$	−18711.0	−1.01902
$697$	30240.0	1.64336
$698$	5754.00	0.312023
$699$	2556.00	0.138307
$700$	0	0
$701$	−15561.0	−0.838418	−0.419209	−	0.907890i	$-0.637693\pi$
$701$	−15561.0	−0.838418	−0.419209	+	0.907890i	$0.637693\pi$
$702$	5184.00	0.278714
$703$	5056.00	0.271253
$704$	−6495.00	−0.347712
$705$	270.000	0.0144238
$706$	9144.00	0.487449
$707$	0	0
$708$	−45.0000	−0.00238871
$709$	5534.00	0.293136	0.146568	−	0.989201i	$-0.453177\pi$
$709$	5534.00	0.293136	0.146568	+	0.989201i	$0.453177\pi$
$710$	−3078.00	−0.162698
$711$	4203.00	0.221695
$712$	19026.0	1.00145
$713$	21252.0	1.11626
$714$	0	0
$715$	−2880.00	−0.150638
$716$	2892.00	0.150948
$717$	16524.0	0.860670
$718$	90.0000	0.00467795
$719$	−21846.0	−1.13313	−0.566564	−	0.824018i	$-0.691727\pi$
$719$	−21846.0	−1.13313	−0.566564	+	0.824018i	$0.691727\pi$
$720$	1917.00	0.0992255
$721$	0	0
$722$	19809.0	1.02107
$723$	2373.00	0.122065
$724$	1352.00	0.0694015
$725$	34452.0	1.76485
$726$	9954.00	0.508853
$727$	−11089.0	−0.565706	−0.282853	−	0.959163i	$-0.591281\pi$
$727$	−11089.0	−0.565706	−0.282853	+	0.959163i	$0.591281\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	3258.00	0.165184
$731$	2184.00	0.110504
$732$	−354.000	−0.0178746
$733$	11762.0	0.592687	0.296343	−	0.955081i	$-0.404233\pi$
$733$	11762.0	0.592687	0.296343	+	0.955081i	$0.404233\pi$
$734$	33933.0	1.70639
$735$	0	0
$736$	−3780.00	−0.189311
$737$	5550.00	0.277391
$738$	−9720.00	−0.484821
$739$	−22726.0	−1.13124	−0.565622	−	0.824665i	$-0.691364\pi$
$739$	−22726.0	−1.13124	−0.565622	+	0.824665i	$0.691364\pi$
$740$	948.000	0.0470935
$741$	3072.00	0.152298
$742$	0	0
$743$	6678.00	0.329734	0.164867	−	0.986316i	$-0.447281\pi$
$743$	6678.00	0.329734	0.164867	+	0.986316i	$0.447281\pi$
$744$	−15939.0	−0.785419
$745$	−7362.00	−0.362044
$746$	−3624.00	−0.177861
$747$	4293.00	0.210271
$748$	−1260.00	−0.0615911
$749$	0	0
$750$	−6507.00	−0.316803
$751$	−19987.0	−0.971153	−0.485577	−	0.874194i	$-0.661390\pi$
$751$	−19987.0	−0.971153	−0.485577	+	0.874194i	$0.661390\pi$
$752$	2130.00	0.103289
$753$	15795.0	0.764411
$754$	−57024.0	−2.75423
$755$	−3777.00	−0.182065
$756$	0	0
$757$	314.000	0.0150760	0.00753799	−	0.999972i	$-0.497601\pi$
$757$	314.000	0.0150760	0.00753799	+	0.999972i	$0.497601\pi$
$758$	−22920.0	−1.09827
$759$	3780.00	0.180771
$760$	1008.00	0.0481105
$761$	−11496.0	−0.547608	−0.273804	−	0.961786i	$-0.588282\pi$
$761$	−11496.0	−0.547608	−0.273804	+	0.961786i	$0.588282\pi$
$762$	−3393.00	−0.161306
$763$	0	0
$764$	3912.00	0.185250
$765$	−2268.00	−0.107189
$766$	−38250.0	−1.80421
$767$	960.000	0.0451937
$768$	4539.00	0.213264
$769$	2765.00	0.129660	0.0648299	−	0.997896i	$-0.479350\pi$
$769$	2765.00	0.129660	0.0648299	+	0.997896i	$0.479350\pi$
$770$	0	0
$771$	−20610.0	−0.962712
$772$	1493.00	0.0696039
$773$	−14046.0	−0.653557	−0.326778	−	0.945101i	$-0.605963\pi$
$773$	−14046.0	−0.653557	−0.326778	+	0.945101i	$0.605963\pi$
$774$	−702.000	−0.0326006
$775$	29348.0	1.36027
$776$	10563.0	0.488646
$777$	0	0
$778$	−9378.00	−0.432156
$779$	−5760.00	−0.264921
$780$	576.000	0.0264412
$781$	5130.00	0.235039
$782$	21168.0	0.967987
$783$	−8019.00	−0.365997
$784$	0	0
$785$	588.000	0.0267345
$786$	5859.00	0.265882
$787$	−18514.0	−0.838568	−0.419284	−	0.907855i	$-0.637719\pi$
$787$	−18514.0	−0.838568	−0.419284	+	0.907855i	$0.637719\pi$
$788$	−4086.00	−0.184718
$789$	−666.000	−0.0300510
$790$	4203.00	0.189286
$791$	0	0
$792$	−2835.00	−0.127194
$793$	7552.00	0.338183
$794$	17796.0	0.795411
$795$	−3267.00	−0.145747
$796$	−3556.00	−0.158341
$797$	−27495.0	−1.22199	−0.610993	−	0.791636i	$-0.709230\pi$
$797$	−27495.0	−1.22199	−0.610993	+	0.791636i	$0.709230\pi$
$798$	0	0
$799$	−2520.00	−0.111578
$800$	−5220.00	−0.230694
$801$	8154.00	0.359685
$802$	−4824.00	−0.212396
$803$	−5430.00	−0.238631
$804$	−1110.00	−0.0486899
$805$	0	0
$806$	−48576.0	−2.12285
$807$	23553.0	1.02739
$808$	−22806.0	−0.992961
$809$	−7944.00	−0.345236	−0.172618	−	0.984989i	$-0.555223\pi$
$809$	−7944.00	−0.345236	−0.172618	+	0.984989i	$0.555223\pi$
$810$	729.000	0.0316228
$811$	−28942.0	−1.25313	−0.626567	−	0.779368i	$-0.715540\pi$
$811$	−28942.0	−1.25313	−0.626567	+	0.779368i	$0.715540\pi$
$812$	0	0
$813$	15549.0	0.670759
$814$	−14220.0	−0.612298
$815$	3756.00	0.161432
$816$	−17892.0	−0.767580
$817$	−416.000	−0.0178140
$818$	13395.0	0.572549
$819$	0	0
$820$	−1080.00	−0.0459942
$821$	8187.00	0.348025	0.174012	−	0.984743i	$-0.444327\pi$
$821$	8187.00	0.348025	0.174012	+	0.984743i	$0.444327\pi$
$822$	15930.0	0.675940
$823$	−280.000	−0.0118593	−0.00592964	−	0.999982i	$-0.501887\pi$
$823$	−280.000	−0.0118593	−0.00592964	+	0.999982i	$0.501887\pi$
$824$	36456.0	1.54127
$825$	5220.00	0.220287
$826$	0	0
$827$	25317.0	1.06452	0.532260	−	0.846581i	$-0.321343\pi$
$827$	25317.0	1.06452	0.532260	+	0.846581i	$0.321343\pi$
$828$	−756.000	−0.0317305
$829$	15320.0	0.641840	0.320920	−	0.947106i	$-0.396008\pi$
$829$	15320.0	0.641840	0.320920	+	0.947106i	$0.396008\pi$
$830$	4293.00	0.179533
$831$	−14880.0	−0.621157
$832$	−27712.0	−1.15474
$833$	0	0
$834$	14022.0	0.582185
$835$	7938.00	0.328989
$836$	240.000	0.00992892
$837$	−6831.00	−0.282095
$838$	4752.00	0.195889
$839$	34092.0	1.40284	0.701422	−	0.712746i	$-0.252549\pi$
$839$	34092.0	1.40284	0.701422	+	0.712746i	$0.252549\pi$
$840$	0	0
$841$	63820.0	2.61675
$842$	3990.00	0.163307
$843$	−2322.00	−0.0948682
$844$	1250.00	0.0509796
$845$	−5697.00	−0.231932
$846$	810.000	0.0329177
$847$	0	0
$848$	−25773.0	−1.04369
$849$	11094.0	0.448463
$850$	29232.0	1.17959
$851$	26544.0	1.06923
$852$	−1026.00	−0.0412561
$853$	−7378.00	−0.296152	−0.148076	−	0.988976i	$-0.547308\pi$
$853$	−7378.00	−0.296152	−0.148076	+	0.988976i	$0.547308\pi$
$854$	0	0
$855$	432.000	0.0172796
$856$	−28413.0	−1.13451
$857$	−15594.0	−0.621565	−0.310782	−	0.950481i	$-0.600591\pi$
$857$	−15594.0	−0.621565	−0.310782	+	0.950481i	$0.600591\pi$
$858$	−8640.00	−0.343782
$859$	−30538.0	−1.21297	−0.606486	−	0.795094i	$-0.707421\pi$
$859$	−30538.0	−1.21297	−0.606486	+	0.795094i	$0.707421\pi$
$860$	−78.0000	−0.00309277
$861$	0	0
$862$	−28764.0	−1.13655
$863$	−822.000	−0.0324232	−0.0162116	−	0.999869i	$-0.505161\pi$
$863$	−822.000	−0.0324232	−0.0162116	+	0.999869i	$0.505161\pi$
$864$	1215.00	0.0478416
$865$	2358.00	0.0926872
$866$	−1482.00	−0.0581529
$867$	6429.00	0.251834
$868$	0	0
$869$	−7005.00	−0.273450
$870$	−8019.00	−0.312494
$871$	23680.0	0.921201
$872$	−7770.00	−0.301749
$873$	4527.00	0.175505
$874$	−4032.00	−0.156046
$875$	0	0
$876$	1086.00	0.0418865
$877$	−41824.0	−1.61037	−0.805186	−	0.593022i	$-0.797935\pi$
$877$	−41824.0	−1.61037	−0.805186	+	0.593022i	$0.797935\pi$
$878$	48027.0	1.84605
$879$	−18819.0	−0.722126
$880$	−3195.00	−0.122390
$881$	−46098.0	−1.76286	−0.881431	−	0.472313i	$-0.843419\pi$
$881$	−46098.0	−1.76286	−0.881431	+	0.472313i	$0.843419\pi$
$882$	0	0
$883$	21008.0	0.800652	0.400326	−	0.916373i	$-0.368897\pi$
$883$	21008.0	0.800652	0.400326	+	0.916373i	$0.368897\pi$
$884$	−5376.00	−0.204541
$885$	135.000	0.00512766
$886$	−23319.0	−0.884218
$887$	24036.0	0.909865	0.454932	−	0.890526i	$-0.349663\pi$
$887$	24036.0	0.909865	0.454932	+	0.890526i	$0.349663\pi$
$888$	−19908.0	−0.752330
$889$	0	0
$890$	8154.00	0.307104
$891$	−1215.00	−0.0456835
$892$	425.000	0.0159530
$893$	480.000	0.0179872
$894$	−22086.0	−0.826249
$895$	−8676.00	−0.324030
$896$	0	0
$897$	16128.0	0.600332
$898$	−2592.00	−0.0963209
$899$	75141.0	2.78764
$900$	−1044.00	−0.0386667
$901$	30492.0	1.12745
$902$	16200.0	0.598006
$903$	0	0
$904$	−13608.0	−0.500659
$905$	−4056.00	−0.148979
$906$	−11331.0	−0.415505
$907$	13292.0	0.486608	0.243304	−	0.969950i	$-0.421769\pi$
$907$	13292.0	0.486608	0.243304	+	0.969950i	$0.421769\pi$
$908$	3855.00	0.140895
$909$	−9774.00	−0.356637
$910$	0	0
$911$	−9306.00	−0.338443	−0.169221	−	0.985578i	$-0.554125\pi$
$911$	−9306.00	−0.338443	−0.169221	+	0.985578i	$0.554125\pi$
$912$	3408.00	0.123739
$913$	−7155.00	−0.259360
$914$	−7557.00	−0.273483
$915$	1062.00	0.0383701
$916$	−2188.00	−0.0789231
$917$	0	0
$918$	−6804.00	−0.244625
$919$	16496.0	0.592114	0.296057	−	0.955170i	$-0.404328\pi$
$919$	16496.0	0.592114	0.296057	+	0.955170i	$0.404328\pi$
$920$	5292.00	0.189644
$921$	−5052.00	−0.180748
$922$	1026.00	0.0366481
$923$	21888.0	0.780555
$924$	0	0
$925$	36656.0	1.30296
$926$	13008.0	0.461630
$927$	15624.0	0.553570
$928$	−13365.0	−0.472767
$929$	14154.0	0.499868	0.249934	−	0.968263i	$-0.419591\pi$
$929$	14154.0	0.499868	0.249934	+	0.968263i	$0.419591\pi$
$930$	−6831.00	−0.240857
$931$	0	0
$932$	852.000	0.0299444
$933$	−3960.00	−0.138955
$934$	−55908.0	−1.95864
$935$	3780.00	0.132213
$936$	−12096.0	−0.422404
$937$	−3781.00	−0.131825	−0.0659124	−	0.997825i	$-0.520996\pi$
$937$	−3781.00	−0.131825	−0.0659124	+	0.997825i	$0.520996\pi$
$938$	0	0
$939$	−25509.0	−0.886533
$940$	90.0000	0.00312285
$941$	−25863.0	−0.895972	−0.447986	−	0.894041i	$-0.647859\pi$
$941$	−25863.0	−0.895972	−0.447986	+	0.894041i	$0.647859\pi$
$942$	1764.00	0.0610130
$943$	−30240.0	−1.04427
$944$	1065.00	0.0367191
$945$	0	0
$946$	1170.00	0.0402114
$947$	−42384.0	−1.45438	−0.727188	−	0.686438i	$-0.759173\pi$
$947$	−42384.0	−1.45438	−0.727188	+	0.686438i	$0.759173\pi$
$948$	1401.00	0.0479983
$949$	−23168.0	−0.792482
$950$	−5568.00	−0.190158
$951$	−7731.00	−0.263612
$952$	0	0
$953$	10530.0	0.357923	0.178961	−	0.983856i	$-0.442726\pi$
$953$	10530.0	0.357923	0.178961	+	0.983856i	$0.442726\pi$
$954$	−9801.00	−0.332620
$955$	−11736.0	−0.397663
$956$	5508.00	0.186340
$957$	13365.0	0.451441
$958$	−45234.0	−1.52552
$959$	0	0
$960$	−3897.00	−0.131016
$961$	34218.0	1.14860
$962$	−60672.0	−2.03341
$963$	−12177.0	−0.407475
$964$	791.000	0.0264278
$965$	−4479.00	−0.149414
$966$	0	0
$967$	−38341.0	−1.27504	−0.637520	−	0.770434i	$-0.720040\pi$
$967$	−38341.0	−1.27504	−0.637520	+	0.770434i	$0.720040\pi$
$968$	−23226.0	−0.771190
$969$	−4032.00	−0.133670
$970$	4527.00	0.149849
$971$	1923.00	0.0635551	0.0317776	−	0.999495i	$-0.489883\pi$
$971$	1923.00	0.0635551	0.0317776	+	0.999495i	$0.489883\pi$
$972$	243.000	0.00801875
$973$	0	0
$974$	−18663.0	−0.613964
$975$	22272.0	0.731564
$976$	8378.00	0.274768
$977$	57090.0	1.86947	0.934734	−	0.355347i	$-0.115637\pi$
$977$	57090.0	1.86947	0.934734	+	0.355347i	$0.115637\pi$
$978$	11268.0	0.368416
$979$	−13590.0	−0.443655
$980$	0	0
$981$	−3330.00	−0.108378
$982$	22113.0	0.718589
$983$	5484.00	0.177937	0.0889687	−	0.996034i	$-0.471643\pi$
$983$	5484.00	0.177937	0.0889687	+	0.996034i	$0.471643\pi$
$984$	22680.0	0.734768
$985$	12258.0	0.396520
$986$	74844.0	2.41736
$987$	0	0
$988$	1024.00	0.0329735
$989$	−2184.00	−0.0702196
$990$	−1215.00	−0.0390053
$991$	−22465.0	−0.720105	−0.360053	−	0.932932i	$-0.617241\pi$
$991$	−22465.0	−0.720105	−0.360053	+	0.932932i	$0.617241\pi$
$992$	−11385.0	−0.364389
$993$	−1452.00	−0.0464026
$994$	0	0
$995$	10668.0	0.339898
$996$	1431.00	0.0455251
$997$	29366.0	0.932829	0.466415	−	0.884566i	$-0.345546\pi$
$997$	29366.0	0.932829	0.466415	+	0.884566i	$0.345546\pi$
$998$	−12822.0	−0.406687
$999$	−8532.00	−0.270211

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	147.4.a.b.1.1		1
3.2	odd	2		441.4.a.l.1.1		1
4.3	odd	2		2352.4.a.i.1.1		1
7.2	even	3		21.4.e.a.4.1	✓	2
7.3	odd	6		147.4.e.h.79.1		2
7.4	even	3		21.4.e.a.16.1	yes	2
7.5	odd	6		147.4.e.h.67.1		2
7.6	odd	2		147.4.a.a.1.1		1
21.2	odd	6		63.4.e.a.46.1		2
21.5	even	6		441.4.e.c.361.1		2
21.11	odd	6		63.4.e.a.37.1		2
21.17	even	6		441.4.e.c.226.1		2
21.20	even	2		441.4.a.k.1.1		1
28.11	odd	6		336.4.q.e.289.1		2
28.23	odd	6		336.4.q.e.193.1		2
28.27	even	2		2352.4.a.bd.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.4.e.a.4.1	✓	2	7.2	even	3
21.4.e.a.16.1	yes	2	7.4	even	3
63.4.e.a.37.1		2	21.11	odd	6
63.4.e.a.46.1		2	21.2	odd	6
147.4.a.a.1.1		1	7.6	odd	2
147.4.a.b.1.1		1	1.1	even	1	trivial
147.4.e.h.67.1		2	7.5	odd	6
147.4.e.h.79.1		2	7.3	odd	6
336.4.q.e.193.1		2	28.23	odd	6
336.4.q.e.289.1		2	28.11	odd	6
441.4.a.k.1.1		1	21.20	even	2
441.4.a.l.1.1		1	3.2	odd	2
441.4.e.c.226.1		2	21.17	even	6
441.4.e.c.361.1		2	21.5	even	6
2352.4.a.i.1.1		1	4.3	odd	2
2352.4.a.bd.1.1		1	28.27	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:	\( N \)	\(=\)	\( 147 = 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	147.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more