LMFDB - Embedded newform 231.4.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [231,4,Mod(1,231)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(231, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("231.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 231 = 3 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	231.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:	$13.6294412113$
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	$\chi$	$=$	231.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

\(q-2.00000 q^{2} +3.00000 q^{3} -4.00000 q^{4} +1.00000 q^{5} -6.00000 q^{6} -7.00000 q^{7} +24.0000 q^{8} +9.00000 q^{9} -2.00000 q^{10} -11.0000 q^{11} -12.0000 q^{12} +7.00000 q^{13} +14.0000 q^{14} +3.00000 q^{15} -16.0000 q^{16} -14.0000 q^{17} -18.0000 q^{18} -45.0000 q^{19} -4.00000 q^{20} -21.0000 q^{21} +22.0000 q^{22} -88.0000 q^{23} +72.0000 q^{24} -124.000 q^{25} -14.0000 q^{26} +27.0000 q^{27} +28.0000 q^{28} -69.0000 q^{29} -6.00000 q^{30} +22.0000 q^{31} -160.000 q^{32} -33.0000 q^{33} +28.0000 q^{34} -7.00000 q^{35} -36.0000 q^{36} +57.0000 q^{37} +90.0000 q^{38} +21.0000 q^{39} +24.0000 q^{40} -380.000 q^{41} +42.0000 q^{42} +48.0000 q^{43} +44.0000 q^{44} +9.00000 q^{45} +176.000 q^{46} -385.000 q^{47} -48.0000 q^{48} +49.0000 q^{49} +248.000 q^{50} -42.0000 q^{51} -28.0000 q^{52} -672.000 q^{53} -54.0000 q^{54} -11.0000 q^{55} -168.000 q^{56} -135.000 q^{57} +138.000 q^{58} -469.000 q^{59} -12.0000 q^{60} -342.000 q^{61} -44.0000 q^{62} -63.0000 q^{63} +448.000 q^{64} +7.00000 q^{65} +66.0000 q^{66} -139.000 q^{67} +56.0000 q^{68} -264.000 q^{69} +14.0000 q^{70} +132.000 q^{71} +216.000 q^{72} +145.000 q^{73} -114.000 q^{74} -372.000 q^{75} +180.000 q^{76} +77.0000 q^{77} -42.0000 q^{78} +1244.00 q^{79} -16.0000 q^{80} +81.0000 q^{81} +760.000 q^{82} +522.000 q^{83} +84.0000 q^{84} -14.0000 q^{85} -96.0000 q^{86} -207.000 q^{87} -264.000 q^{88} +822.000 q^{89} -18.0000 q^{90} -49.0000 q^{91} +352.000 q^{92} +66.0000 q^{93} +770.000 q^{94} -45.0000 q^{95} -480.000 q^{96} +272.000 q^{97} -98.0000 q^{98} -99.0000 q^{99} +O(q^{100})\)

Coefficient data

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

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

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

$2$	−2.00000	−0.707107	−0.353553	−	0.935414i	$-0.615027\pi$
$2$	−2.00000	−0.707107	−0.353553	+	0.935414i	$0.615027\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	−4.00000	−0.500000
$5$	1.00000	0.0894427	0.0447214	−	0.998999i	$-0.485760\pi$
$5$	1.00000	0.0894427	0.0447214	+	0.998999i	$0.485760\pi$
$6$	−6.00000	−0.408248
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	24.0000	1.06066
$9$	9.00000	0.333333
$10$	−2.00000	−0.0632456
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	−12.0000	−0.288675
$13$	7.00000	0.149342	0.0746712	−	0.997208i	$-0.476209\pi$
$13$	7.00000	0.149342	0.0746712	+	0.997208i	$0.476209\pi$
$14$	14.0000	0.267261
$15$	3.00000	0.0516398
$16$	−16.0000	−0.250000
$17$	−14.0000	−0.199735	−0.0998676	−	0.995001i	$-0.531842\pi$
$17$	−14.0000	−0.199735	−0.0998676	+	0.995001i	$0.531842\pi$
$18$	−18.0000	−0.235702
$19$	−45.0000	−0.543353	−0.271677	−	0.962389i	$-0.587578\pi$
$19$	−45.0000	−0.543353	−0.271677	+	0.962389i	$0.587578\pi$
$20$	−4.00000	−0.0447214
$21$	−21.0000	−0.218218
$22$	22.0000	0.213201
$23$	−88.0000	−0.797794	−0.398897	−	0.916996i	$-0.630607\pi$
$23$	−88.0000	−0.797794	−0.398897	+	0.916996i	$0.630607\pi$
$24$	72.0000	0.612372
$25$	−124.000	−0.992000
$26$	−14.0000	−0.105601
$27$	27.0000	0.192450
$28$	28.0000	0.188982
$29$	−69.0000	−0.441827	−0.220913	−	0.975293i	$-0.570904\pi$
$29$	−69.0000	−0.441827	−0.220913	+	0.975293i	$0.570904\pi$
$30$	−6.00000	−0.0365148
$31$	22.0000	0.127462	0.0637309	−	0.997967i	$-0.479700\pi$
$31$	22.0000	0.127462	0.0637309	+	0.997967i	$0.479700\pi$
$32$	−160.000	−0.883883
$33$	−33.0000	−0.174078
$34$	28.0000	0.141234
$35$	−7.00000	−0.0338062
$36$	−36.0000	−0.166667
$37$	57.0000	0.253263	0.126632	−	0.991950i	$-0.459583\pi$
$37$	57.0000	0.253263	0.126632	+	0.991950i	$0.459583\pi$
$38$	90.0000	0.384209
$39$	21.0000	0.0862229
$40$	24.0000	0.0948683
$41$	−380.000	−1.44746	−0.723732	−	0.690081i	$-0.757575\pi$
$41$	−380.000	−1.44746	−0.723732	+	0.690081i	$0.757575\pi$
$42$	42.0000	0.154303
$43$	48.0000	0.170231	0.0851155	−	0.996371i	$-0.472874\pi$
$43$	48.0000	0.170231	0.0851155	+	0.996371i	$0.472874\pi$
$44$	44.0000	0.150756
$45$	9.00000	0.0298142
$46$	176.000	0.564126
$47$	−385.000	−1.19485	−0.597426	−	0.801924i	$-0.703810\pi$
$47$	−385.000	−1.19485	−0.597426	+	0.801924i	$0.703810\pi$
$48$	−48.0000	−0.144338
$49$	49.0000	0.142857
$50$	248.000	0.701450
$51$	−42.0000	−0.115317
$52$	−28.0000	−0.0746712
$53$	−672.000	−1.74163	−0.870814	−	0.491612i	$-0.836408\pi$
$53$	−672.000	−1.74163	−0.870814	+	0.491612i	$0.836408\pi$
$54$	−54.0000	−0.136083
$55$	−11.0000	−0.0269680
$56$	−168.000	−0.400892
$57$	−135.000	−0.313705
$58$	138.000	0.312419
$59$	−469.000	−1.03489	−0.517446	−	0.855716i	$-0.673117\pi$
$59$	−469.000	−1.03489	−0.517446	+	0.855716i	$0.673117\pi$
$60$	−12.0000	−0.0258199
$61$	−342.000	−0.717846	−0.358923	−	0.933367i	$-0.616856\pi$
$61$	−342.000	−0.717846	−0.358923	+	0.933367i	$0.616856\pi$
$62$	−44.0000	−0.0901291
$63$	−63.0000	−0.125988
$64$	448.000	0.875000
$65$	7.00000	0.0133576
$66$	66.0000	0.123091
$67$	−139.000	−0.253456	−0.126728	−	0.991938i	$-0.540448\pi$
$67$	−139.000	−0.253456	−0.126728	+	0.991938i	$0.540448\pi$
$68$	56.0000	0.0998676
$69$	−264.000	−0.460607
$70$	14.0000	0.0239046
$71$	132.000	0.220641	0.110321	−	0.993896i	$-0.464812\pi$
$71$	132.000	0.220641	0.110321	+	0.993896i	$0.464812\pi$
$72$	216.000	0.353553
$73$	145.000	0.232479	0.116239	−	0.993221i	$-0.462916\pi$
$73$	145.000	0.232479	0.116239	+	0.993221i	$0.462916\pi$
$74$	−114.000	−0.179084
$75$	−372.000	−0.572731
$76$	180.000	0.271677
$77$	77.0000	0.113961
$78$	−42.0000	−0.0609688
$79$	1244.00	1.77166	0.885829	−	0.464012i	$-0.153591\pi$
$79$	1244.00	1.77166	0.885829	+	0.464012i	$0.153591\pi$
$80$	−16.0000	−0.0223607
$81$	81.0000	0.111111
$82$	760.000	1.02351
$83$	522.000	0.690325	0.345162	−	0.938543i	$-0.387824\pi$
$83$	522.000	0.690325	0.345162	+	0.938543i	$0.387824\pi$
$84$	84.0000	0.109109
$85$	−14.0000	−0.0178649
$86$	−96.0000	−0.120371
$87$	−207.000	−0.255089
$88$	−264.000	−0.319801
$89$	822.000	0.979009	0.489505	−	0.872001i	$-0.337178\pi$
$89$	822.000	0.979009	0.489505	+	0.872001i	$0.337178\pi$
$90$	−18.0000	−0.0210819
$91$	−49.0000	−0.0564461
$92$	352.000	0.398897
$93$	66.0000	0.0735901
$94$	770.000	0.844888
$95$	−45.0000	−0.0485990
$96$	−480.000	−0.510310
$97$	272.000	0.284716	0.142358	−	0.989815i	$-0.454532\pi$
$97$	272.000	0.284716	0.142358	+	0.989815i	$0.454532\pi$
$98$	−98.0000	−0.101015
$99$	−99.0000	−0.100504
$100$	496.000	0.496000
$101$	−874.000	−0.861052	−0.430526	−	0.902578i	$-0.641672\pi$
$101$	−874.000	−0.861052	−0.430526	+	0.902578i	$0.641672\pi$
$102$	84.0000	0.0815416
$103$	−826.000	−0.790177	−0.395088	−	0.918643i	$-0.629286\pi$
$103$	−826.000	−0.790177	−0.395088	+	0.918643i	$0.629286\pi$
$104$	168.000	0.158401
$105$	−21.0000	−0.0195180
$106$	1344.00	1.23152
$107$	219.000	0.197865	0.0989324	−	0.995094i	$-0.468457\pi$
$107$	219.000	0.197865	0.0989324	+	0.995094i	$0.468457\pi$
$108$	−108.000	−0.0962250
$109$	1426.00	1.25308	0.626541	−	0.779388i	$-0.284470\pi$
$109$	1426.00	1.25308	0.626541	+	0.779388i	$0.284470\pi$
$110$	22.0000	0.0190693
$111$	171.000	0.146222
$112$	112.000	0.0944911
$113$	882.000	0.734262	0.367131	−	0.930169i	$-0.380340\pi$
$113$	882.000	0.734262	0.367131	+	0.930169i	$0.380340\pi$
$114$	270.000	0.221823
$115$	−88.0000	−0.0713569
$116$	276.000	0.220913
$117$	63.0000	0.0497808
$118$	938.000	0.731779
$119$	98.0000	0.0754928
$120$	72.0000	0.0547723
$121$	121.000	0.0909091
$122$	684.000	0.507594
$123$	−1140.00	−0.835694
$124$	−88.0000	−0.0637309
$125$	−249.000	−0.178170
$126$	126.000	0.0890871
$127$	1826.00	1.27584	0.637918	−	0.770104i	$-0.279796\pi$
$127$	1826.00	1.27584	0.637918	+	0.770104i	$0.279796\pi$
$128$	384.000	0.265165
$129$	144.000	0.0982829
$130$	−14.0000	−0.00944524
$131$	84.0000	0.0560238	0.0280119	−	0.999608i	$-0.491082\pi$
$131$	84.0000	0.0560238	0.0280119	+	0.999608i	$0.491082\pi$
$132$	132.000	0.0870388
$133$	315.000	0.205368
$134$	278.000	0.179220
$135$	27.0000	0.0172133
$136$	−336.000	−0.211851
$137$	−1834.00	−1.14372	−0.571858	−	0.820352i	$-0.693777\pi$
$137$	−1834.00	−1.14372	−0.571858	+	0.820352i	$0.693777\pi$
$138$	528.000	0.325698
$139$	2416.00	1.47426	0.737131	−	0.675750i	$-0.236180\pi$
$139$	2416.00	1.47426	0.737131	+	0.675750i	$0.236180\pi$
$140$	28.0000	0.0169031
$141$	−1155.00	−0.689848
$142$	−264.000	−0.156017
$143$	−77.0000	−0.0450284
$144$	−144.000	−0.0833333
$145$	−69.0000	−0.0395182
$146$	−290.000	−0.164387
$147$	147.000	0.0824786
$148$	−228.000	−0.126632
$149$	1895.00	1.04191	0.520955	−	0.853584i	$-0.325576\pi$
$149$	1895.00	1.04191	0.520955	+	0.853584i	$0.325576\pi$
$150$	744.000	0.404982
$151$	−3478.00	−1.87441	−0.937204	−	0.348782i	$-0.886596\pi$
$151$	−3478.00	−1.87441	−0.937204	+	0.348782i	$0.886596\pi$
$152$	−1080.00	−0.576313
$153$	−126.000	−0.0665784
$154$	−154.000	−0.0805823
$155$	22.0000	0.0114005
$156$	−84.0000	−0.0431114
$157$	−952.000	−0.483935	−0.241968	−	0.970284i	$-0.577793\pi$
$157$	−952.000	−0.483935	−0.241968	+	0.970284i	$0.577793\pi$
$158$	−2488.00	−1.25275
$159$	−2016.00	−1.00553
$160$	−160.000	−0.0790569
$161$	616.000	0.301538
$162$	−162.000	−0.0785674
$163$	−2483.00	−1.19315	−0.596575	−	0.802557i	$-0.703472\pi$
$163$	−2483.00	−1.19315	−0.596575	+	0.802557i	$0.703472\pi$
$164$	1520.00	0.723732
$165$	−33.0000	−0.0155700
$166$	−1044.00	−0.488133
$167$	468.000	0.216856	0.108428	−	0.994104i	$-0.465418\pi$
$167$	468.000	0.216856	0.108428	+	0.994104i	$0.465418\pi$
$168$	−504.000	−0.231455
$169$	−2148.00	−0.977697
$170$	28.0000	0.0126324
$171$	−405.000	−0.181118
$172$	−192.000	−0.0851155
$173$	−2148.00	−0.943985	−0.471993	−	0.881603i	$-0.656465\pi$
$173$	−2148.00	−0.943985	−0.471993	+	0.881603i	$0.656465\pi$
$174$	414.000	0.180375
$175$	868.000	0.374941
$176$	176.000	0.0753778
$177$	−1407.00	−0.597495
$178$	−1644.00	−0.692264
$179$	−1464.00	−0.611310	−0.305655	−	0.952142i	$-0.598875\pi$
$179$	−1464.00	−0.611310	−0.305655	+	0.952142i	$0.598875\pi$
$180$	−36.0000	−0.0149071
$181$	1432.00	0.588065	0.294032	−	0.955795i	$-0.405003\pi$
$181$	1432.00	0.588065	0.294032	+	0.955795i	$0.405003\pi$
$182$	98.0000	0.0399134
$183$	−1026.00	−0.414449
$184$	−2112.00	−0.846189
$185$	57.0000	0.0226526
$186$	−132.000	−0.0520361
$187$	154.000	0.0602224
$188$	1540.00	0.597426
$189$	−189.000	−0.0727393
$190$	90.0000	0.0343647
$191$	−1330.00	−0.503850	−0.251925	−	0.967747i	$-0.581064\pi$
$191$	−1330.00	−0.503850	−0.251925	+	0.967747i	$0.581064\pi$
$192$	1344.00	0.505181
$193$	−2540.00	−0.947322	−0.473661	−	0.880707i	$-0.657068\pi$
$193$	−2540.00	−0.947322	−0.473661	+	0.880707i	$0.657068\pi$
$194$	−544.000	−0.201324
$195$	21.0000	0.00771201
$196$	−196.000	−0.0714286
$197$	1606.00	0.580826	0.290413	−	0.956901i	$-0.406207\pi$
$197$	1606.00	0.580826	0.290413	+	0.956901i	$0.406207\pi$
$198$	198.000	0.0710669
$199$	288.000	0.102592	0.0512959	−	0.998683i	$-0.483665\pi$
$199$	288.000	0.102592	0.0512959	+	0.998683i	$0.483665\pi$
$200$	−2976.00	−1.05217
$201$	−417.000	−0.146333
$202$	1748.00	0.608856
$203$	483.000	0.166995
$204$	168.000	0.0576586
$205$	−380.000	−0.129465
$206$	1652.00	0.558739
$207$	−792.000	−0.265931
$208$	−112.000	−0.0373356
$209$	495.000	0.163827
$210$	42.0000	0.0138013
$211$	1482.00	0.483531	0.241766	−	0.970335i	$-0.422273\pi$
$211$	1482.00	0.483531	0.241766	+	0.970335i	$0.422273\pi$
$212$	2688.00	0.870814
$213$	396.000	0.127387
$214$	−438.000	−0.139912
$215$	48.0000	0.0152259
$216$	648.000	0.204124
$217$	−154.000	−0.0481760
$218$	−2852.00	−0.886063
$219$	435.000	0.134222
$220$	44.0000	0.0134840
$221$	−98.0000	−0.0298289
$222$	−342.000	−0.103394
$223$	3502.00	1.05162	0.525810	−	0.850602i	$-0.323762\pi$
$223$	3502.00	1.05162	0.525810	+	0.850602i	$0.323762\pi$
$224$	1120.00	0.334077
$225$	−1116.00	−0.330667
$226$	−1764.00	−0.519201
$227$	1830.00	0.535072	0.267536	−	0.963548i	$-0.413791\pi$
$227$	1830.00	0.535072	0.267536	+	0.963548i	$0.413791\pi$
$228$	540.000	0.156853
$229$	−2268.00	−0.654470	−0.327235	−	0.944943i	$-0.606117\pi$
$229$	−2268.00	−0.654470	−0.327235	+	0.944943i	$0.606117\pi$
$230$	176.000	0.0504569
$231$	231.000	0.0657952
$232$	−1656.00	−0.468628
$233$	2430.00	0.683239	0.341619	−	0.939838i	$-0.389025\pi$
$233$	2430.00	0.683239	0.341619	+	0.939838i	$0.389025\pi$
$234$	−126.000	−0.0352003
$235$	−385.000	−0.106871
$236$	1876.00	0.517446
$237$	3732.00	1.02287
$238$	−196.000	−0.0533815
$239$	−2151.00	−0.582162	−0.291081	−	0.956698i	$-0.594015\pi$
$239$	−2151.00	−0.582162	−0.291081	+	0.956698i	$0.594015\pi$
$240$	−48.0000	−0.0129099
$241$	2803.00	0.749200	0.374600	−	0.927187i	$-0.377780\pi$
$241$	2803.00	0.749200	0.374600	+	0.927187i	$0.377780\pi$
$242$	−242.000	−0.0642824
$243$	243.000	0.0641500
$244$	1368.00	0.358923
$245$	49.0000	0.0127775
$246$	2280.00	0.590925
$247$	−315.000	−0.0811456
$248$	528.000	0.135194
$249$	1566.00	0.398559
$250$	498.000	0.125985
$251$	−5339.00	−1.34261	−0.671304	−	0.741182i	$-0.734266\pi$
$251$	−5339.00	−1.34261	−0.671304	+	0.741182i	$0.734266\pi$
$252$	252.000	0.0629941
$253$	968.000	0.240544
$254$	−3652.00	−0.902153
$255$	−42.0000	−0.0103143
$256$	−4352.00	−1.06250
$257$	−4439.00	−1.07742	−0.538711	−	0.842491i	$-0.681088\pi$
$257$	−4439.00	−1.07742	−0.538711	+	0.842491i	$0.681088\pi$
$258$	−288.000	−0.0694965
$259$	−399.000	−0.0957245
$260$	−28.0000	−0.00667879
$261$	−621.000	−0.147276
$262$	−168.000	−0.0396148
$263$	1271.00	0.297997	0.148999	−	0.988837i	$-0.452395\pi$
$263$	1271.00	0.297997	0.148999	+	0.988837i	$0.452395\pi$
$264$	−792.000	−0.184637
$265$	−672.000	−0.155776
$266$	−630.000	−0.145217
$267$	2466.00	0.565231
$268$	556.000	0.126728
$269$	3682.00	0.834556	0.417278	−	0.908779i	$-0.362984\pi$
$269$	3682.00	0.834556	0.417278	+	0.908779i	$0.362984\pi$
$270$	−54.0000	−0.0121716
$271$	4127.00	0.925083	0.462541	−	0.886598i	$-0.346938\pi$
$271$	4127.00	0.925083	0.462541	+	0.886598i	$0.346938\pi$
$272$	224.000	0.0499338
$273$	−147.000	−0.0325892
$274$	3668.00	0.808730
$275$	1364.00	0.299099
$276$	1056.00	0.230303
$277$	−44.0000	−0.00954406	−0.00477203	−	0.999989i	$-0.501519\pi$
$277$	−44.0000	−0.00954406	−0.00477203	+	0.999989i	$0.501519\pi$
$278$	−4832.00	−1.04246
$279$	198.000	0.0424873
$280$	−168.000	−0.0358569
$281$	−561.000	−0.119098	−0.0595489	−	0.998225i	$-0.518966\pi$
$281$	−561.000	−0.119098	−0.0595489	+	0.998225i	$0.518966\pi$
$282$	2310.00	0.487796
$283$	−3845.00	−0.807638	−0.403819	−	0.914839i	$-0.632317\pi$
$283$	−3845.00	−0.807638	−0.403819	+	0.914839i	$0.632317\pi$
$284$	−528.000	−0.110321
$285$	−135.000	−0.0280586
$286$	154.000	0.0318399
$287$	2660.00	0.547090
$288$	−1440.00	−0.294628
$289$	−4717.00	−0.960106
$290$	138.000	0.0279436
$291$	816.000	0.164381
$292$	−580.000	−0.116239
$293$	912.000	0.181842	0.0909208	−	0.995858i	$-0.471019\pi$
$293$	912.000	0.181842	0.0909208	+	0.995858i	$0.471019\pi$
$294$	−294.000	−0.0583212
$295$	−469.000	−0.0925635
$296$	1368.00	0.268626
$297$	−297.000	−0.0580259
$298$	−3790.00	−0.736741
$299$	−616.000	−0.119144
$300$	1488.00	0.286366
$301$	−336.000	−0.0643413
$302$	6956.00	1.32541
$303$	−2622.00	−0.497129
$304$	720.000	0.135838
$305$	−342.000	−0.0642061
$306$	252.000	0.0470780
$307$	6860.00	1.27531	0.637656	−	0.770321i	$-0.279904\pi$
$307$	6860.00	1.27531	0.637656	+	0.770321i	$0.279904\pi$
$308$	−308.000	−0.0569803
$309$	−2478.00	−0.456209
$310$	−44.0000	−0.00806139
$311$	5472.00	0.997713	0.498856	−	0.866685i	$-0.333754\pi$
$311$	5472.00	0.997713	0.498856	+	0.866685i	$0.333754\pi$
$312$	504.000	0.0914531
$313$	6422.00	1.15972	0.579861	−	0.814716i	$-0.303107\pi$
$313$	6422.00	1.15972	0.579861	+	0.814716i	$0.303107\pi$
$314$	1904.00	0.342194
$315$	−63.0000	−0.0112687
$316$	−4976.00	−0.885829
$317$	−6476.00	−1.14741	−0.573704	−	0.819063i	$-0.694494\pi$
$317$	−6476.00	−1.14741	−0.573704	+	0.819063i	$0.694494\pi$
$318$	4032.00	0.711017
$319$	759.000	0.133216
$320$	448.000	0.0782624
$321$	657.000	0.114237
$322$	−1232.00	−0.213219
$323$	630.000	0.108527
$324$	−324.000	−0.0555556
$325$	−868.000	−0.148148
$326$	4966.00	0.843685
$327$	4278.00	0.723468
$328$	−9120.00	−1.53527
$329$	2695.00	0.451611
$330$	66.0000	0.0110096
$331$	−11840.0	−1.96612	−0.983059	−	0.183288i	$-0.941326\pi$
$331$	−11840.0	−1.96612	−0.983059	+	0.183288i	$0.941326\pi$
$332$	−2088.00	−0.345162
$333$	513.000	0.0844211
$334$	−936.000	−0.153340
$335$	−139.000	−0.0226698
$336$	336.000	0.0545545
$337$	1640.00	0.265093	0.132547	−	0.991177i	$-0.457685\pi$
$337$	1640.00	0.265093	0.132547	+	0.991177i	$0.457685\pi$
$338$	4296.00	0.691336
$339$	2646.00	0.423926
$340$	56.0000	0.00893243
$341$	−242.000	−0.0384312
$342$	810.000	0.128070
$343$	−343.000	−0.0539949
$344$	1152.00	0.180557
$345$	−264.000	−0.0411979
$346$	4296.00	0.667498
$347$	2032.00	0.314362	0.157181	−	0.987570i	$-0.449759\pi$
$347$	2032.00	0.314362	0.157181	+	0.987570i	$0.449759\pi$
$348$	828.000	0.127544
$349$	1755.00	0.269178	0.134589	−	0.990902i	$-0.457029\pi$
$349$	1755.00	0.269178	0.134589	+	0.990902i	$0.457029\pi$
$350$	−1736.00	−0.265123
$351$	189.000	0.0287410
$352$	1760.00	0.266501
$353$	−1101.00	−0.166007	−0.0830033	−	0.996549i	$-0.526451\pi$
$353$	−1101.00	−0.166007	−0.0830033	+	0.996549i	$0.526451\pi$
$354$	2814.00	0.422493
$355$	132.000	0.0197347
$356$	−3288.00	−0.489505
$357$	294.000	0.0435858
$358$	2928.00	0.432261
$359$	5020.00	0.738010	0.369005	−	0.929427i	$-0.379699\pi$
$359$	5020.00	0.738010	0.369005	+	0.929427i	$0.379699\pi$
$360$	216.000	0.0316228
$361$	−4834.00	−0.704767
$362$	−2864.00	−0.415825
$363$	363.000	0.0524864
$364$	196.000	0.0282231
$365$	145.000	0.0207936
$366$	2052.00	0.293059
$367$	11638.0	1.65531	0.827655	−	0.561237i	$-0.189675\pi$
$367$	11638.0	1.65531	0.827655	+	0.561237i	$0.189675\pi$
$368$	1408.00	0.199449
$369$	−3420.00	−0.482488
$370$	−114.000	−0.0160178
$371$	4704.00	0.658274
$372$	−264.000	−0.0367951
$373$	−8834.00	−1.22629	−0.613146	−	0.789969i	$-0.710096\pi$
$373$	−8834.00	−1.22629	−0.613146	+	0.789969i	$0.710096\pi$
$374$	−308.000	−0.0425837
$375$	−747.000	−0.102866
$376$	−9240.00	−1.26733
$377$	−483.000	−0.0659835
$378$	378.000	0.0514344
$379$	7507.00	1.01744	0.508719	−	0.860933i	$-0.330119\pi$
$379$	7507.00	1.01744	0.508719	+	0.860933i	$0.330119\pi$
$380$	180.000	0.0242995
$381$	5478.00	0.736605
$382$	2660.00	0.356276
$383$	3608.00	0.481358	0.240679	−	0.970605i	$-0.422630\pi$
$383$	3608.00	0.481358	0.240679	+	0.970605i	$0.422630\pi$
$384$	1152.00	0.153093
$385$	77.0000	0.0101929
$386$	5080.00	0.669858
$387$	432.000	0.0567437
$388$	−1088.00	−0.142358
$389$	−7896.00	−1.02916	−0.514580	−	0.857442i	$-0.672052\pi$
$389$	−7896.00	−1.02916	−0.514580	+	0.857442i	$0.672052\pi$
$390$	−42.0000	−0.00545321
$391$	1232.00	0.159348
$392$	1176.00	0.151523
$393$	252.000	0.0323453
$394$	−3212.00	−0.410706
$395$	1244.00	0.158462
$396$	396.000	0.0502519
$397$	6934.00	0.876593	0.438297	−	0.898830i	$-0.355582\pi$
$397$	6934.00	0.876593	0.438297	+	0.898830i	$0.355582\pi$
$398$	−576.000	−0.0725434
$399$	945.000	0.118569
$400$	1984.00	0.248000
$401$	−700.000	−0.0871729	−0.0435864	−	0.999050i	$-0.513878\pi$
$401$	−700.000	−0.0871729	−0.0435864	+	0.999050i	$0.513878\pi$
$402$	834.000	0.103473
$403$	154.000	0.0190355
$404$	3496.00	0.430526
$405$	81.0000	0.00993808
$406$	−966.000	−0.118083
$407$	−627.000	−0.0763618
$408$	−1008.00	−0.122312
$409$	−11550.0	−1.39636	−0.698179	−	0.715923i	$-0.746006\pi$
$409$	−11550.0	−1.39636	−0.698179	+	0.715923i	$0.746006\pi$
$410$	760.000	0.0915457
$411$	−5502.00	−0.660325
$412$	3304.00	0.395088
$413$	3283.00	0.391152
$414$	1584.00	0.188042
$415$	522.000	0.0617445
$416$	−1120.00	−0.132001
$417$	7248.00	0.851166
$418$	−990.000	−0.115843
$419$	−4555.00	−0.531089	−0.265545	−	0.964099i	$-0.585552\pi$
$419$	−4555.00	−0.531089	−0.265545	+	0.964099i	$0.585552\pi$
$420$	84.0000	0.00975900
$421$	−161.000	−0.0186381	−0.00931907	−	0.999957i	$-0.502966\pi$
$421$	−161.000	−0.0186381	−0.00931907	+	0.999957i	$0.502966\pi$
$422$	−2964.00	−0.341908
$423$	−3465.00	−0.398284
$424$	−16128.0	−1.84728
$425$	1736.00	0.198137
$426$	−792.000	−0.0900764
$427$	2394.00	0.271320
$428$	−876.000	−0.0989324
$429$	−231.000	−0.0259972
$430$	−96.0000	−0.0107664
$431$	−3249.00	−0.363106	−0.181553	−	0.983381i	$-0.558112\pi$
$431$	−3249.00	−0.363106	−0.181553	+	0.983381i	$0.558112\pi$
$432$	−432.000	−0.0481125
$433$	1792.00	0.198887	0.0994434	−	0.995043i	$-0.468294\pi$
$433$	1792.00	0.198887	0.0994434	+	0.995043i	$0.468294\pi$
$434$	308.000	0.0340656
$435$	−207.000	−0.0228158
$436$	−5704.00	−0.626541
$437$	3960.00	0.433484
$438$	−870.000	−0.0949092
$439$	9319.00	1.01315	0.506574	−	0.862197i	$-0.330912\pi$
$439$	9319.00	1.01315	0.506574	+	0.862197i	$0.330912\pi$
$440$	−264.000	−0.0286039
$441$	441.000	0.0476190
$442$	196.000	0.0210922
$443$	5608.00	0.601454	0.300727	−	0.953710i	$-0.402771\pi$
$443$	5608.00	0.601454	0.300727	+	0.953710i	$0.402771\pi$
$444$	−684.000	−0.0731108
$445$	822.000	0.0875653
$446$	−7004.00	−0.743608
$447$	5685.00	0.601546
$448$	−3136.00	−0.330719
$449$	−4384.00	−0.460788	−0.230394	−	0.973097i	$-0.574001\pi$
$449$	−4384.00	−0.460788	−0.230394	+	0.973097i	$0.574001\pi$
$450$	2232.00	0.233817
$451$	4180.00	0.436427
$452$	−3528.00	−0.367131
$453$	−10434.0	−1.08219
$454$	−3660.00	−0.378353
$455$	−49.0000	−0.00504869
$456$	−3240.00	−0.332734
$457$	324.000	0.0331643	0.0165821	−	0.999863i	$-0.494721\pi$
$457$	324.000	0.0331643	0.0165821	+	0.999863i	$0.494721\pi$
$458$	4536.00	0.462780
$459$	−378.000	−0.0384391
$460$	352.000	0.0356784
$461$	−18360.0	−1.85490	−0.927452	−	0.373943i	$-0.878006\pi$
$461$	−18360.0	−1.85490	−0.927452	+	0.373943i	$0.878006\pi$
$462$	−462.000	−0.0465242
$463$	−1667.00	−0.167326	−0.0836631	−	0.996494i	$-0.526662\pi$
$463$	−1667.00	−0.167326	−0.0836631	+	0.996494i	$0.526662\pi$
$464$	1104.00	0.110457
$465$	66.0000	0.00658210
$466$	−4860.00	−0.483123
$467$	10839.0	1.07402	0.537012	−	0.843575i	$-0.319553\pi$
$467$	10839.0	1.07402	0.537012	+	0.843575i	$0.319553\pi$
$468$	−252.000	−0.0248904
$469$	973.000	0.0957974
$470$	770.000	0.0755690
$471$	−2856.00	−0.279400
$472$	−11256.0	−1.09767
$473$	−528.000	−0.0513266
$474$	−7464.00	−0.723276
$475$	5580.00	0.539006
$476$	−392.000	−0.0377464
$477$	−6048.00	−0.580543
$478$	4302.00	0.411650
$479$	−910.000	−0.0868037	−0.0434018	−	0.999058i	$-0.513820\pi$
$479$	−910.000	−0.0868037	−0.0434018	+	0.999058i	$0.513820\pi$
$480$	−480.000	−0.0456435
$481$	399.000	0.0378229
$482$	−5606.00	−0.529764
$483$	1848.00	0.174093
$484$	−484.000	−0.0454545
$485$	272.000	0.0254657
$486$	−486.000	−0.0453609
$487$	12032.0	1.11955	0.559776	−	0.828644i	$-0.310887\pi$
$487$	12032.0	1.11955	0.559776	+	0.828644i	$0.310887\pi$
$488$	−8208.00	−0.761391
$489$	−7449.00	−0.688866
$490$	−98.0000	−0.00903508
$491$	4243.00	0.389988	0.194994	−	0.980804i	$-0.437531\pi$
$491$	4243.00	0.389988	0.194994	+	0.980804i	$0.437531\pi$
$492$	4560.00	0.417847
$493$	966.000	0.0882484
$494$	630.000	0.0573786
$495$	−99.0000	−0.00898933
$496$	−352.000	−0.0318655
$497$	−924.000	−0.0833945
$498$	−3132.00	−0.281824
$499$	−5869.00	−0.526518	−0.263259	−	0.964725i	$-0.584797\pi$
$499$	−5869.00	−0.526518	−0.263259	+	0.964725i	$0.584797\pi$
$500$	996.000	0.0890849
$501$	1404.00	0.125202
$502$	10678.0	0.949367
$503$	148.000	0.0131193	0.00655964	−	0.999978i	$-0.497912\pi$
$503$	148.000	0.0131193	0.00655964	+	0.999978i	$0.497912\pi$
$504$	−1512.00	−0.133631
$505$	−874.000	−0.0770148
$506$	−1936.00	−0.170090
$507$	−6444.00	−0.564474
$508$	−7304.00	−0.637918
$509$	12114.0	1.05490	0.527450	−	0.849586i	$-0.323148\pi$
$509$	12114.0	1.05490	0.527450	+	0.849586i	$0.323148\pi$
$510$	84.0000	0.00729330
$511$	−1015.00	−0.0878688
$512$	5632.00	0.486136
$513$	−1215.00	−0.104568
$514$	8878.00	0.761852
$515$	−826.000	−0.0706756
$516$	−576.000	−0.0491414
$517$	4235.00	0.360261
$518$	798.000	0.0676875
$519$	−6444.00	−0.545010
$520$	168.000	0.0141679
$521$	−14943.0	−1.25655	−0.628277	−	0.777990i	$-0.716240\pi$
$521$	−14943.0	−1.25655	−0.628277	+	0.777990i	$0.716240\pi$
$522$	1242.00	0.104140
$523$	16589.0	1.38697	0.693486	−	0.720470i	$-0.256074\pi$
$523$	16589.0	1.38697	0.693486	+	0.720470i	$0.256074\pi$
$524$	−336.000	−0.0280119
$525$	2604.00	0.216472
$526$	−2542.00	−0.210716
$527$	−308.000	−0.0254586
$528$	528.000	0.0435194
$529$	−4423.00	−0.363524
$530$	1344.00	0.110150
$531$	−4221.00	−0.344964
$532$	−1260.00	−0.102684
$533$	−2660.00	−0.216168
$534$	−4932.00	−0.399679
$535$	219.000	0.0176976
$536$	−3336.00	−0.268831
$537$	−4392.00	−0.352940
$538$	−7364.00	−0.590120
$539$	−539.000	−0.0430730
$540$	−108.000	−0.00860663
$541$	3232.00	0.256848	0.128424	−	0.991719i	$-0.459008\pi$
$541$	3232.00	0.256848	0.128424	+	0.991719i	$0.459008\pi$
$542$	−8254.00	−0.654132
$543$	4296.00	0.339519
$544$	2240.00	0.176543
$545$	1426.00	0.112079
$546$	294.000	0.0230440
$547$	−16468.0	−1.28724	−0.643621	−	0.765345i	$-0.722568\pi$
$547$	−16468.0	−1.28724	−0.643621	+	0.765345i	$0.722568\pi$
$548$	7336.00	0.571858
$549$	−3078.00	−0.239282
$550$	−2728.00	−0.211495
$551$	3105.00	0.240068
$552$	−6336.00	−0.488547
$553$	−8708.00	−0.669624
$554$	88.0000	0.00674867
$555$	171.000	0.0130785
$556$	−9664.00	−0.737131
$557$	14143.0	1.07587	0.537934	−	0.842987i	$-0.319205\pi$
$557$	14143.0	1.07587	0.537934	+	0.842987i	$0.319205\pi$
$558$	−396.000	−0.0300430
$559$	336.000	0.0254227
$560$	112.000	0.00845154
$561$	462.000	0.0347694
$562$	1122.00	0.0842148
$563$	−22916.0	−1.71544	−0.857721	−	0.514115i	$-0.828121\pi$
$563$	−22916.0	−1.71544	−0.857721	+	0.514115i	$0.828121\pi$
$564$	4620.00	0.344924
$565$	882.000	0.0656744
$566$	7690.00	0.571086
$567$	−567.000	−0.0419961
$568$	3168.00	0.234025
$569$	−294.000	−0.0216610	−0.0108305	−	0.999941i	$-0.503448\pi$
$569$	−294.000	−0.0216610	−0.0108305	+	0.999941i	$0.503448\pi$
$570$	270.000	0.0198404
$571$	13292.0	0.974173	0.487087	−	0.873354i	$-0.338060\pi$
$571$	13292.0	0.974173	0.487087	+	0.873354i	$0.338060\pi$
$572$	308.000	0.0225142
$573$	−3990.00	−0.290898
$574$	−5320.00	−0.386851
$575$	10912.0	0.791412
$576$	4032.00	0.291667
$577$	9630.00	0.694804	0.347402	−	0.937716i	$-0.387064\pi$
$577$	9630.00	0.694804	0.347402	+	0.937716i	$0.387064\pi$
$578$	9434.00	0.678897
$579$	−7620.00	−0.546937
$580$	276.000	0.0197591
$581$	−3654.00	−0.260918
$582$	−1632.00	−0.116235
$583$	7392.00	0.525121
$584$	3480.00	0.246581
$585$	63.0000	0.00445253
$586$	−1824.00	−0.128581
$587$	−17015.0	−1.19640	−0.598198	−	0.801348i	$-0.704116\pi$
$587$	−17015.0	−1.19640	−0.598198	+	0.801348i	$0.704116\pi$
$588$	−588.000	−0.0412393
$589$	−990.000	−0.0692568
$590$	938.000	0.0654523
$591$	4818.00	0.335340
$592$	−912.000	−0.0633158
$593$	−12028.0	−0.832936	−0.416468	−	0.909150i	$-0.636732\pi$
$593$	−12028.0	−0.832936	−0.416468	+	0.909150i	$0.636732\pi$
$594$	594.000	0.0410305
$595$	98.0000	0.00675228
$596$	−7580.00	−0.520955
$597$	864.000	0.0592314
$598$	1232.00	0.0842479
$599$	1252.00	0.0854012	0.0427006	−	0.999088i	$-0.486404\pi$
$599$	1252.00	0.0854012	0.0427006	+	0.999088i	$0.486404\pi$
$600$	−8928.00	−0.607473
$601$	−19699.0	−1.33700	−0.668502	−	0.743711i	$-0.733064\pi$
$601$	−19699.0	−1.33700	−0.668502	+	0.743711i	$0.733064\pi$
$602$	672.000	0.0454961
$603$	−1251.00	−0.0844853
$604$	13912.0	0.937204
$605$	121.000	0.00813116
$606$	5244.00	0.351523
$607$	29855.0	1.99634	0.998169	−	0.0604881i	$-0.0192657\pi$
$607$	29855.0	1.99634	0.998169	+	0.0604881i	$0.0192657\pi$
$608$	7200.00	0.480261
$609$	1449.00	0.0964145
$610$	684.000	0.0454006
$611$	−2695.00	−0.178442
$612$	504.000	0.0332892
$613$	−3178.00	−0.209393	−0.104697	−	0.994504i	$-0.533387\pi$
$613$	−3178.00	−0.209393	−0.104697	+	0.994504i	$0.533387\pi$
$614$	−13720.0	−0.901782
$615$	−1140.00	−0.0747467
$616$	1848.00	0.120873
$617$	−14394.0	−0.939191	−0.469595	−	0.882882i	$-0.655600\pi$
$617$	−14394.0	−0.939191	−0.469595	+	0.882882i	$0.655600\pi$
$618$	4956.00	0.322588
$619$	3878.00	0.251809	0.125905	−	0.992042i	$-0.459817\pi$
$619$	3878.00	0.251809	0.125905	+	0.992042i	$0.459817\pi$
$620$	−88.0000	−0.00570027
$621$	−2376.00	−0.153536
$622$	−10944.0	−0.705489
$623$	−5754.00	−0.370031
$624$	−336.000	−0.0215557
$625$	15251.0	0.976064
$626$	−12844.0	−0.820047
$627$	1485.00	0.0945856
$628$	3808.00	0.241968
$629$	−798.000	−0.0505856
$630$	126.000	0.00796819
$631$	18100.0	1.14192	0.570958	−	0.820979i	$-0.306572\pi$
$631$	18100.0	1.14192	0.570958	+	0.820979i	$0.306572\pi$
$632$	29856.0	1.87913
$633$	4446.00	0.279167
$634$	12952.0	0.811340
$635$	1826.00	0.114114
$636$	8064.00	0.502765
$637$	343.000	0.0213346
$638$	−1518.00	−0.0941978
$639$	1188.00	0.0735470
$640$	384.000	0.0237171
$641$	−7470.00	−0.460292	−0.230146	−	0.973156i	$-0.573920\pi$
$641$	−7470.00	−0.460292	−0.230146	+	0.973156i	$0.573920\pi$
$642$	−1314.00	−0.0807779
$643$	25400.0	1.55782	0.778910	−	0.627136i	$-0.215773\pi$
$643$	25400.0	1.55782	0.778910	+	0.627136i	$0.215773\pi$
$644$	−2464.00	−0.150769
$645$	144.000	0.00879069
$646$	−1260.00	−0.0767400
$647$	25061.0	1.52280	0.761398	−	0.648284i	$-0.224513\pi$
$647$	25061.0	1.52280	0.761398	+	0.648284i	$0.224513\pi$
$648$	1944.00	0.117851
$649$	5159.00	0.312032
$650$	1736.00	0.104756
$651$	−462.000	−0.0278145
$652$	9932.00	0.596575
$653$	−1832.00	−0.109788	−0.0548941	−	0.998492i	$-0.517482\pi$
$653$	−1832.00	−0.109788	−0.0548941	+	0.998492i	$0.517482\pi$
$654$	−8556.00	−0.511569
$655$	84.0000	0.00501092
$656$	6080.00	0.361866
$657$	1305.00	0.0774930
$658$	−5390.00	−0.319338
$659$	−27809.0	−1.64383	−0.821916	−	0.569609i	$-0.807095\pi$
$659$	−27809.0	−1.64383	−0.821916	+	0.569609i	$0.807095\pi$
$660$	132.000	0.00778499
$661$	−29612.0	−1.74247	−0.871235	−	0.490865i	$-0.836681\pi$
$661$	−29612.0	−1.74247	−0.871235	+	0.490865i	$0.836681\pi$
$662$	23680.0	1.39026
$663$	−294.000	−0.0172217
$664$	12528.0	0.732200
$665$	315.000	0.0183687
$666$	−1026.00	−0.0596947
$667$	6072.00	0.352487
$668$	−1872.00	−0.108428
$669$	10506.0	0.607153
$670$	278.000	0.0160300
$671$	3762.00	0.216439
$672$	3360.00	0.192879
$673$	−3892.00	−0.222921	−0.111460	−	0.993769i	$-0.535553\pi$
$673$	−3892.00	−0.222921	−0.111460	+	0.993769i	$0.535553\pi$
$674$	−3280.00	−0.187449
$675$	−3348.00	−0.190910
$676$	8592.00	0.488848
$677$	10518.0	0.597104	0.298552	−	0.954393i	$-0.403496\pi$
$677$	10518.0	0.597104	0.298552	+	0.954393i	$0.403496\pi$
$678$	−5292.00	−0.299761
$679$	−1904.00	−0.107612
$680$	−336.000	−0.0189485
$681$	5490.00	0.308924
$682$	484.000	0.0271750
$683$	12902.0	0.722813	0.361407	−	0.932408i	$-0.382297\pi$
$683$	12902.0	0.722813	0.361407	+	0.932408i	$0.382297\pi$
$684$	1620.00	0.0905588
$685$	−1834.00	−0.102297
$686$	686.000	0.0381802
$687$	−6804.00	−0.377859
$688$	−768.000	−0.0425577
$689$	−4704.00	−0.260099
$690$	528.000	0.0291313
$691$	30386.0	1.67285	0.836424	−	0.548083i	$-0.184642\pi$
$691$	30386.0	1.67285	0.836424	+	0.548083i	$0.184642\pi$
$692$	8592.00	0.471993
$693$	693.000	0.0379869
$694$	−4064.00	−0.222287
$695$	2416.00	0.131862
$696$	−4968.00	−0.270563
$697$	5320.00	0.289110
$698$	−3510.00	−0.190337
$699$	7290.00	0.394468
$700$	−3472.00	−0.187470
$701$	−26842.0	−1.44623	−0.723116	−	0.690727i	$-0.757291\pi$
$701$	−26842.0	−1.44623	−0.723116	+	0.690727i	$0.757291\pi$
$702$	−378.000	−0.0203229
$703$	−2565.00	−0.137611
$704$	−4928.00	−0.263822
$705$	−1155.00	−0.0617019
$706$	2202.00	0.117384
$707$	6118.00	0.325447
$708$	5628.00	0.298747
$709$	−5701.00	−0.301982	−0.150991	−	0.988535i	$-0.548247\pi$
$709$	−5701.00	−0.301982	−0.150991	+	0.988535i	$0.548247\pi$
$710$	−264.000	−0.0139546
$711$	11196.0	0.590552
$712$	19728.0	1.03840
$713$	−1936.00	−0.101688
$714$	−588.000	−0.0308198
$715$	−77.0000	−0.00402746
$716$	5856.00	0.305655
$717$	−6453.00	−0.336111
$718$	−10040.0	−0.521852
$719$	−23919.0	−1.24065	−0.620326	−	0.784344i	$-0.713000\pi$
$719$	−23919.0	−1.24065	−0.620326	+	0.784344i	$0.713000\pi$
$720$	−144.000	−0.00745356
$721$	5782.00	0.298659
$722$	9668.00	0.498346
$723$	8409.00	0.432551
$724$	−5728.00	−0.294032
$725$	8556.00	0.438292
$726$	−726.000	−0.0371135
$727$	19188.0	0.978877	0.489438	−	0.872038i	$-0.337202\pi$
$727$	19188.0	0.978877	0.489438	+	0.872038i	$0.337202\pi$
$728$	−1176.00	−0.0598701
$729$	729.000	0.0370370
$730$	−290.000	−0.0147033
$731$	−672.000	−0.0340011
$732$	4104.00	0.207224
$733$	−5670.00	−0.285711	−0.142856	−	0.989744i	$-0.545628\pi$
$733$	−5670.00	−0.285711	−0.142856	+	0.989744i	$0.545628\pi$
$734$	−23276.0	−1.17048
$735$	147.000	0.00737711
$736$	14080.0	0.705157
$737$	1529.00	0.0764199
$738$	6840.00	0.341171
$739$	17276.0	0.859957	0.429978	−	0.902839i	$-0.358521\pi$
$739$	17276.0	0.859957	0.429978	+	0.902839i	$0.358521\pi$
$740$	−228.000	−0.0113263
$741$	−945.000	−0.0468494
$742$	−9408.00	−0.465470
$743$	15367.0	0.758763	0.379381	−	0.925240i	$-0.376137\pi$
$743$	15367.0	0.758763	0.379381	+	0.925240i	$0.376137\pi$
$744$	1584.00	0.0780541
$745$	1895.00	0.0931912
$746$	17668.0	0.867120
$747$	4698.00	0.230108
$748$	−616.000	−0.0301112
$749$	−1533.00	−0.0747858
$750$	1494.00	0.0727376
$751$	−38123.0	−1.85237	−0.926184	−	0.377072i	$-0.876931\pi$
$751$	−38123.0	−1.85237	−0.926184	+	0.377072i	$0.876931\pi$
$752$	6160.00	0.298713
$753$	−16017.0	−0.775155
$754$	966.000	0.0466574
$755$	−3478.00	−0.167652
$756$	756.000	0.0363696
$757$	23417.0	1.12431	0.562157	−	0.827031i	$-0.309972\pi$
$757$	23417.0	1.12431	0.562157	+	0.827031i	$0.309972\pi$
$758$	−15014.0	−0.719437
$759$	2904.00	0.138878
$760$	−1080.00	−0.0515470
$761$	−16332.0	−0.777969	−0.388985	−	0.921244i	$-0.627174\pi$
$761$	−16332.0	−0.777969	−0.388985	+	0.921244i	$0.627174\pi$
$762$	−10956.0	−0.520858
$763$	−9982.00	−0.473621
$764$	5320.00	0.251925
$765$	−126.000	−0.00595495
$766$	−7216.00	−0.340372
$767$	−3283.00	−0.154553
$768$	−13056.0	−0.613435
$769$	19523.0	0.915497	0.457749	−	0.889082i	$-0.348656\pi$
$769$	19523.0	0.915497	0.457749	+	0.889082i	$0.348656\pi$
$770$	−154.000	−0.00720750
$771$	−13317.0	−0.622049
$772$	10160.0	0.473661
$773$	21525.0	1.00155	0.500776	−	0.865577i	$-0.333048\pi$
$773$	21525.0	1.00155	0.500776	+	0.865577i	$0.333048\pi$
$774$	−864.000	−0.0401238
$775$	−2728.00	−0.126442
$776$	6528.00	0.301987
$777$	−1197.00	−0.0552666
$778$	15792.0	0.727726
$779$	17100.0	0.786484
$780$	−84.0000	−0.00385600
$781$	−1452.00	−0.0665258
$782$	−2464.00	−0.112676
$783$	−1863.00	−0.0850296
$784$	−784.000	−0.0357143
$785$	−952.000	−0.0432845
$786$	−504.000	−0.0228716
$787$	−3707.00	−0.167904	−0.0839519	−	0.996470i	$-0.526754\pi$
$787$	−3707.00	−0.167904	−0.0839519	+	0.996470i	$0.526754\pi$
$788$	−6424.00	−0.290413
$789$	3813.00	0.172049
$790$	−2488.00	−0.112049
$791$	−6174.00	−0.277525
$792$	−2376.00	−0.106600
$793$	−2394.00	−0.107205
$794$	−13868.0	−0.619845
$795$	−2016.00	−0.0899373
$796$	−1152.00	−0.0512959
$797$	−7621.00	−0.338707	−0.169354	−	0.985555i	$-0.554168\pi$
$797$	−7621.00	−0.338707	−0.169354	+	0.985555i	$0.554168\pi$
$798$	−1890.00	−0.0838412
$799$	5390.00	0.238654
$800$	19840.0	0.876812
$801$	7398.00	0.326336
$802$	1400.00	0.0616405
$803$	−1595.00	−0.0700951
$804$	1668.00	0.0731664
$805$	616.000	0.0269704
$806$	−308.000	−0.0134601
$807$	11046.0	0.481831
$808$	−20976.0	−0.913284
$809$	23451.0	1.01915	0.509576	−	0.860426i	$-0.329802\pi$
$809$	23451.0	1.01915	0.509576	+	0.860426i	$0.329802\pi$
$810$	−162.000	−0.00702728
$811$	−4499.00	−0.194798	−0.0973990	−	0.995245i	$-0.531052\pi$
$811$	−4499.00	−0.194798	−0.0973990	+	0.995245i	$0.531052\pi$
$812$	−1932.00	−0.0834974
$813$	12381.0	0.534097
$814$	1254.00	0.0539959
$815$	−2483.00	−0.106719
$816$	672.000	0.0288293
$817$	−2160.00	−0.0924955
$818$	23100.0	0.987375
$819$	−441.000	−0.0188154
$820$	1520.00	0.0647326
$821$	−30215.0	−1.28442	−0.642211	−	0.766528i	$-0.721983\pi$
$821$	−30215.0	−1.28442	−0.642211	+	0.766528i	$0.721983\pi$
$822$	11004.0	0.466920
$823$	−19947.0	−0.844847	−0.422423	−	0.906399i	$-0.638820\pi$
$823$	−19947.0	−0.844847	−0.422423	+	0.906399i	$0.638820\pi$
$824$	−19824.0	−0.838109
$825$	4092.00	0.172685
$826$	−6566.00	−0.276586
$827$	−20341.0	−0.855291	−0.427646	−	0.903946i	$-0.640657\pi$
$827$	−20341.0	−0.855291	−0.427646	+	0.903946i	$0.640657\pi$
$828$	3168.00	0.132966
$829$	−14024.0	−0.587544	−0.293772	−	0.955876i	$-0.594911\pi$
$829$	−14024.0	−0.587544	−0.293772	+	0.955876i	$0.594911\pi$
$830$	−1044.00	−0.0436600
$831$	−132.000	−0.00551026
$832$	3136.00	0.130675
$833$	−686.000	−0.0285336
$834$	−14496.0	−0.601865
$835$	468.000	0.0193962
$836$	−1980.00	−0.0819136
$837$	594.000	0.0245300
$838$	9110.00	0.375537
$839$	−37193.0	−1.53045	−0.765223	−	0.643765i	$-0.777372\pi$
$839$	−37193.0	−1.53045	−0.765223	+	0.643765i	$0.777372\pi$
$840$	−504.000	−0.0207020
$841$	−19628.0	−0.804789
$842$	322.000	0.0131792
$843$	−1683.00	−0.0687611
$844$	−5928.00	−0.241766
$845$	−2148.00	−0.0874479
$846$	6930.00	0.281629
$847$	−847.000	−0.0343604
$848$	10752.0	0.435407
$849$	−11535.0	−0.466290
$850$	−3472.00	−0.140104
$851$	−5016.00	−0.202052
$852$	−1584.00	−0.0636936
$853$	−37042.0	−1.48686	−0.743431	−	0.668812i	$-0.766803\pi$
$853$	−37042.0	−1.48686	−0.743431	+	0.668812i	$0.766803\pi$
$854$	−4788.00	−0.191852
$855$	−405.000	−0.0161997
$856$	5256.00	0.209867
$857$	−41962.0	−1.67257	−0.836286	−	0.548293i	$-0.815278\pi$
$857$	−41962.0	−1.67257	−0.836286	+	0.548293i	$0.815278\pi$
$858$	462.000	0.0183828
$859$	−9372.00	−0.372257	−0.186128	−	0.982525i	$-0.559594\pi$
$859$	−9372.00	−0.372257	−0.186128	+	0.982525i	$0.559594\pi$
$860$	−192.000	−0.00761296
$861$	7980.00	0.315863
$862$	6498.00	0.256755
$863$	18096.0	0.713783	0.356892	−	0.934146i	$-0.383837\pi$
$863$	18096.0	0.713783	0.356892	+	0.934146i	$0.383837\pi$
$864$	−4320.00	−0.170103
$865$	−2148.00	−0.0844326
$866$	−3584.00	−0.140634
$867$	−14151.0	−0.554317
$868$	616.000	0.0240880
$869$	−13684.0	−0.534175
$870$	414.000	0.0161332
$871$	−973.000	−0.0378517
$872$	34224.0	1.32910
$873$	2448.00	0.0949052
$874$	−7920.00	−0.306519
$875$	1743.00	0.0673419
$876$	−1740.00	−0.0671109
$877$	−28848.0	−1.11075	−0.555375	−	0.831600i	$-0.687425\pi$
$877$	−28848.0	−1.11075	−0.555375	+	0.831600i	$0.687425\pi$
$878$	−18638.0	−0.716403
$879$	2736.00	0.104986
$880$	176.000	0.00674200
$881$	26273.0	1.00472	0.502361	−	0.864658i	$-0.332465\pi$
$881$	26273.0	1.00472	0.502361	+	0.864658i	$0.332465\pi$
$882$	−882.000	−0.0336718
$883$	−33821.0	−1.28898	−0.644489	−	0.764614i	$-0.722930\pi$
$883$	−33821.0	−1.28898	−0.644489	+	0.764614i	$0.722930\pi$
$884$	392.000	0.0149145
$885$	−1407.00	−0.0534416
$886$	−11216.0	−0.425292
$887$	−14134.0	−0.535032	−0.267516	−	0.963553i	$-0.586203\pi$
$887$	−14134.0	−0.535032	−0.267516	+	0.963553i	$0.586203\pi$
$888$	4104.00	0.155091
$889$	−12782.0	−0.482221
$890$	−1644.00	−0.0619180
$891$	−891.000	−0.0335013
$892$	−14008.0	−0.525810
$893$	17325.0	0.649226
$894$	−11370.0	−0.425358
$895$	−1464.00	−0.0546772
$896$	−2688.00	−0.100223
$897$	−1848.00	−0.0687881
$898$	8768.00	0.325826
$899$	−1518.00	−0.0563161
$900$	4464.00	0.165333
$901$	9408.00	0.347865
$902$	−8360.00	−0.308600
$903$	−1008.00	−0.0371474
$904$	21168.0	0.778802
$905$	1432.00	0.0525981
$906$	20868.0	0.765224
$907$	−35748.0	−1.30870	−0.654351	−	0.756191i	$-0.727058\pi$
$907$	−35748.0	−1.30870	−0.654351	+	0.756191i	$0.727058\pi$
$908$	−7320.00	−0.267536
$909$	−7866.00	−0.287017
$910$	98.0000	0.00356997
$911$	−9522.00	−0.346299	−0.173149	−	0.984896i	$-0.555394\pi$
$911$	−9522.00	−0.346299	−0.173149	+	0.984896i	$0.555394\pi$
$912$	2160.00	0.0784263
$913$	−5742.00	−0.208141
$914$	−648.000	−0.0234507
$915$	−1026.00	−0.0370694
$916$	9072.00	0.327235
$917$	−588.000	−0.0211750
$918$	756.000	0.0271805
$919$	−33260.0	−1.19385	−0.596924	−	0.802298i	$-0.703611\pi$
$919$	−33260.0	−1.19385	−0.596924	+	0.802298i	$0.703611\pi$
$920$	−2112.00	−0.0756854
$921$	20580.0	0.736302
$922$	36720.0	1.31161
$923$	924.000	0.0329511
$924$	−924.000	−0.0328976
$925$	−7068.00	−0.251237
$926$	3334.00	0.118318
$927$	−7434.00	−0.263392
$928$	11040.0	0.390523
$929$	−55101.0	−1.94597	−0.972984	−	0.230871i	$-0.925843\pi$
$929$	−55101.0	−1.94597	−0.972984	+	0.230871i	$0.925843\pi$
$930$	−132.000	−0.00465425
$931$	−2205.00	−0.0776219
$932$	−9720.00	−0.341619
$933$	16416.0	0.576030
$934$	−21678.0	−0.759449
$935$	154.000	0.00538646
$936$	1512.00	0.0528005
$937$	−19474.0	−0.678962	−0.339481	−	0.940613i	$-0.610252\pi$
$937$	−19474.0	−0.678962	−0.339481	+	0.940613i	$0.610252\pi$
$938$	−1946.00	−0.0677390
$939$	19266.0	0.669566
$940$	1540.00	0.0534354
$941$	52104.0	1.80504	0.902520	−	0.430649i	$-0.141715\pi$
$941$	52104.0	1.80504	0.902520	+	0.430649i	$0.141715\pi$
$942$	5712.00	0.197566
$943$	33440.0	1.15478
$944$	7504.00	0.258723
$945$	−189.000	−0.00650600
$946$	1056.00	0.0362934
$947$	16830.0	0.577510	0.288755	−	0.957403i	$-0.406759\pi$
$947$	16830.0	0.577510	0.288755	+	0.957403i	$0.406759\pi$
$948$	−14928.0	−0.511433
$949$	1015.00	0.0347190
$950$	−11160.0	−0.381135
$951$	−19428.0	−0.662456
$952$	2352.00	0.0800722
$953$	−13997.0	−0.475768	−0.237884	−	0.971294i	$-0.576454\pi$
$953$	−13997.0	−0.475768	−0.237884	+	0.971294i	$0.576454\pi$
$954$	12096.0	0.410506
$955$	−1330.00	−0.0450657
$956$	8604.00	0.291081
$957$	2277.00	0.0769122
$958$	1820.00	0.0613795
$959$	12838.0	0.432284
$960$	1344.00	0.0451848
$961$	−29307.0	−0.983753
$962$	−798.000	−0.0267449
$963$	1971.00	0.0659549
$964$	−11212.0	−0.374600
$965$	−2540.00	−0.0847311
$966$	−3696.00	−0.123102
$967$	9574.00	0.318386	0.159193	−	0.987247i	$-0.449111\pi$
$967$	9574.00	0.318386	0.159193	+	0.987247i	$0.449111\pi$
$968$	2904.00	0.0964237
$969$	1890.00	0.0626579
$970$	−544.000	−0.0180070
$971$	32585.0	1.07693	0.538467	−	0.842647i	$-0.319004\pi$
$971$	32585.0	1.07693	0.538467	+	0.842647i	$0.319004\pi$
$972$	−972.000	−0.0320750
$973$	−16912.0	−0.557219
$974$	−24064.0	−0.791643
$975$	−2604.00	−0.0855331
$976$	5472.00	0.179462
$977$	42036.0	1.37651	0.688255	−	0.725469i	$-0.258377\pi$
$977$	42036.0	1.37651	0.688255	+	0.725469i	$0.258377\pi$
$978$	14898.0	0.487102
$979$	−9042.00	−0.295182
$980$	−196.000	−0.00638877
$981$	12834.0	0.417694
$982$	−8486.00	−0.275763
$983$	42336.0	1.37366	0.686830	−	0.726818i	$-0.259001\pi$
$983$	42336.0	1.37366	0.686830	+	0.726818i	$0.259001\pi$
$984$	−27360.0	−0.886387
$985$	1606.00	0.0519507
$986$	−1932.00	−0.0624010
$987$	8085.00	0.260738
$988$	1260.00	0.0405728
$989$	−4224.00	−0.135809
$990$	198.000	0.00635642
$991$	−4691.00	−0.150368	−0.0751839	−	0.997170i	$-0.523954\pi$
$991$	−4691.00	−0.150368	−0.0751839	+	0.997170i	$0.523954\pi$
$992$	−3520.00	−0.112661
$993$	−35520.0	−1.13514
$994$	1848.00	0.0589688
$995$	288.000	0.00917609
$996$	−6264.00	−0.199280
$997$	18518.0	0.588236	0.294118	−	0.955769i	$-0.404974\pi$
$997$	18518.0	0.588236	0.294118	+	0.955769i	$0.404974\pi$
$998$	11738.0	0.372305
$999$	1539.00	0.0487405

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	231.4.a.b.1.1	✓	1
3.2	odd	2		693.4.a.e.1.1		1
7.6	odd	2		1617.4.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.4.a.b.1.1	✓	1	1.1	even	1	trivial
693.4.a.e.1.1		1	3.2	odd	2
1617.4.a.c.1.1		1	7.6	odd	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:	\( N \)	\(=\)	\( 231 = 3 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	231.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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