LMFDB - Embedded newform 102.4.a.a.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)

$f(q)$

$=$

$q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -3.00000 q^{5} +6.00000 q^{6} +20.0000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$

\(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -3.00000 q^{5} +6.00000 q^{6} +20.0000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +6.00000 q^{10} -51.0000 q^{11} -12.0000 q^{12} -61.0000 q^{13} -40.0000 q^{14} +9.00000 q^{15} +16.0000 q^{16} +17.0000 q^{17} -18.0000 q^{18} -43.0000 q^{19} -12.0000 q^{20} -60.0000 q^{21} +102.000 q^{22} -219.000 q^{23} +24.0000 q^{24} -116.000 q^{25} +122.000 q^{26} -27.0000 q^{27} +80.0000 q^{28} -150.000 q^{29} -18.0000 q^{30} +290.000 q^{31} -32.0000 q^{32} +153.000 q^{33} -34.0000 q^{34} -60.0000 q^{35} +36.0000 q^{36} +56.0000 q^{37} +86.0000 q^{38} +183.000 q^{39} +24.0000 q^{40} +15.0000 q^{41} +120.000 q^{42} +83.0000 q^{43} -204.000 q^{44} -27.0000 q^{45} +438.000 q^{46} +426.000 q^{47} -48.0000 q^{48} +57.0000 q^{49} +232.000 q^{50} -51.0000 q^{51} -244.000 q^{52} -378.000 q^{53} +54.0000 q^{54} +153.000 q^{55} -160.000 q^{56} +129.000 q^{57} +300.000 q^{58} -210.000 q^{59} +36.0000 q^{60} -448.000 q^{61} -580.000 q^{62} +180.000 q^{63} +64.0000 q^{64} +183.000 q^{65} -306.000 q^{66} -124.000 q^{67} +68.0000 q^{68} +657.000 q^{69} +120.000 q^{70} +900.000 q^{71} -72.0000 q^{72} -1078.00 q^{73} -112.000 q^{74} +348.000 q^{75} -172.000 q^{76} -1020.00 q^{77} -366.000 q^{78} +722.000 q^{79} -48.0000 q^{80} +81.0000 q^{81} -30.0000 q^{82} -78.0000 q^{83} -240.000 q^{84} -51.0000 q^{85} -166.000 q^{86} +450.000 q^{87} +408.000 q^{88} -144.000 q^{89} +54.0000 q^{90} -1220.00 q^{91} -876.000 q^{92} -870.000 q^{93} -852.000 q^{94} +129.000 q^{95} +96.0000 q^{96} -268.000 q^{97} -114.000 q^{98} -459.000 q^{99} +O(q^{100})\)

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
$5$	−3.00000	−0.268328	−0.134164	−	0.990959i	$-0.542835\pi$
$5$	−3.00000	−0.268328	−0.134164	+	0.990959i	$0.542835\pi$
$6$	6.00000	0.408248
$7$	20.0000	1.07990	0.539949	−	0.841698i	$-0.318443\pi$
$7$	20.0000	1.07990	0.539949	+	0.841698i	$0.318443\pi$
$8$	−8.00000	−0.353553
$9$	9.00000	0.333333
$10$	6.00000	0.189737
$11$	−51.0000	−1.39792	−0.698958	−	0.715163i	$-0.746353\pi$
$11$	−51.0000	−1.39792	−0.698958	+	0.715163i	$0.746353\pi$
$12$	−12.0000	−0.288675
$13$	−61.0000	−1.30141	−0.650706	−	0.759330i	$-0.725527\pi$
$13$	−61.0000	−1.30141	−0.650706	+	0.759330i	$0.725527\pi$
$14$	−40.0000	−0.763604
$15$	9.00000	0.154919
$16$	16.0000	0.250000
$17$	17.0000	0.242536
$17$	17.0000	0.242536
$18$	−18.0000	−0.235702
$19$	−43.0000	−0.519204	−0.259602	−	0.965716i	$-0.583591\pi$
$19$	−43.0000	−0.519204	−0.259602	+	0.965716i	$0.583591\pi$
$20$	−12.0000	−0.134164
$21$	−60.0000	−0.623480
$22$	102.000	0.988476
$23$	−219.000	−1.98542	−0.992710	−	0.120528i	$-0.961541\pi$
$23$	−219.000	−1.98542	−0.992710	+	0.120528i	$0.961541\pi$
$24$	24.0000	0.204124
$25$	−116.000	−0.928000
$26$	122.000	0.920237
$27$	−27.0000	−0.192450
$28$	80.0000	0.539949
$29$	−150.000	−0.960493	−0.480247	−	0.877134i	$-0.659453\pi$
$29$	−150.000	−0.960493	−0.480247	+	0.877134i	$0.659453\pi$
$30$	−18.0000	−0.109545
$31$	290.000	1.68018	0.840089	−	0.542448i	$-0.182502\pi$
$31$	290.000	1.68018	0.840089	+	0.542448i	$0.182502\pi$
$32$	−32.0000	−0.176777
$33$	153.000	0.807087
$34$	−34.0000	−0.171499
$35$	−60.0000	−0.289767
$36$	36.0000	0.166667
$37$	56.0000	0.248820	0.124410	−	0.992231i	$-0.460296\pi$
$37$	56.0000	0.248820	0.124410	+	0.992231i	$0.460296\pi$
$38$	86.0000	0.367133
$39$	183.000	0.751371
$40$	24.0000	0.0948683
$41$	15.0000	0.0571367	0.0285684	−	0.999592i	$-0.490905\pi$
$41$	15.0000	0.0571367	0.0285684	+	0.999592i	$0.490905\pi$
$42$	120.000	0.440867
$43$	83.0000	0.294358	0.147179	−	0.989110i	$-0.452981\pi$
$43$	83.0000	0.294358	0.147179	+	0.989110i	$0.452981\pi$
$44$	−204.000	−0.698958
$45$	−27.0000	−0.0894427
$46$	438.000	1.40390
$47$	426.000	1.32210	0.661048	−	0.750344i	$-0.270112\pi$
$47$	426.000	1.32210	0.661048	+	0.750344i	$0.270112\pi$
$48$	−48.0000	−0.144338
$49$	57.0000	0.166181
$50$	232.000	0.656195
$51$	−51.0000	−0.140028
$52$	−244.000	−0.650706
$53$	−378.000	−0.979666	−0.489833	−	0.871816i	$-0.662942\pi$
$53$	−378.000	−0.979666	−0.489833	+	0.871816i	$0.662942\pi$
$54$	54.0000	0.136083
$55$	153.000	0.375100
$56$	−160.000	−0.381802
$57$	129.000	0.299763
$58$	300.000	0.679171
$59$	−210.000	−0.463384	−0.231692	−	0.972789i	$-0.574426\pi$
$59$	−210.000	−0.463384	−0.231692	+	0.972789i	$0.574426\pi$
$60$	36.0000	0.0774597
$61$	−448.000	−0.940336	−0.470168	−	0.882577i	$-0.655807\pi$
$61$	−448.000	−0.940336	−0.470168	+	0.882577i	$0.655807\pi$
$62$	−580.000	−1.18807
$63$	180.000	0.359966
$64$	64.0000	0.125000
$65$	183.000	0.349205
$66$	−306.000	−0.570697
$67$	−124.000	−0.226105	−0.113052	−	0.993589i	$-0.536063\pi$
$67$	−124.000	−0.226105	−0.113052	+	0.993589i	$0.536063\pi$
$68$	68.0000	0.121268
$69$	657.000	1.14628
$70$	120.000	0.204896
$71$	900.000	1.50437	0.752186	−	0.658951i	$-0.229000\pi$
$71$	900.000	1.50437	0.752186	+	0.658951i	$0.229000\pi$
$72$	−72.0000	−0.117851
$73$	−1078.00	−1.72836	−0.864181	−	0.503182i	$-0.832163\pi$
$73$	−1078.00	−1.72836	−0.864181	+	0.503182i	$0.832163\pi$
$74$	−112.000	−0.175942
$75$	348.000	0.535781
$76$	−172.000	−0.259602
$77$	−1020.00	−1.50961
$78$	−366.000	−0.531299
$79$	722.000	1.02824	0.514122	−	0.857717i	$-0.328118\pi$
$79$	722.000	1.02824	0.514122	+	0.857717i	$0.328118\pi$
$80$	−48.0000	−0.0670820
$81$	81.0000	0.111111
$82$	−30.0000	−0.0404018
$83$	−78.0000	−0.103152	−0.0515760	−	0.998669i	$-0.516424\pi$
$83$	−78.0000	−0.103152	−0.0515760	+	0.998669i	$0.516424\pi$
$84$	−240.000	−0.311740
$85$	−51.0000	−0.0650791
$86$	−166.000	−0.208142
$87$	450.000	0.554541
$88$	408.000	0.494238
$89$	−144.000	−0.171505	−0.0857526	−	0.996316i	$-0.527329\pi$
$89$	−144.000	−0.171505	−0.0857526	+	0.996316i	$0.527329\pi$
$90$	54.0000	0.0632456
$91$	−1220.00	−1.40539
$92$	−876.000	−0.992710
$93$	−870.000	−0.970052
$94$	−852.000	−0.934863
$95$	129.000	0.139317
$96$	96.0000	0.102062
$97$	−268.000	−0.280529	−0.140264	−	0.990114i	$-0.544795\pi$
$97$	−268.000	−0.280529	−0.140264	+	0.990114i	$0.544795\pi$
$98$	−114.000	−0.117508
$99$	−459.000	−0.465972
$100$	−464.000	−0.464000
$101$	768.000	0.756622	0.378311	−	0.925678i	$-0.376505\pi$
$101$	768.000	0.756622	0.378311	+	0.925678i	$0.376505\pi$
$102$	102.000	0.0990148
$103$	−1645.00	−1.57366	−0.786828	−	0.617172i	$-0.788278\pi$
$103$	−1645.00	−1.57366	−0.786828	+	0.617172i	$0.788278\pi$
$104$	488.000	0.460119
$105$	180.000	0.167297
$106$	756.000	0.692728
$107$	1377.00	1.24411	0.622054	−	0.782974i	$-0.286298\pi$
$107$	1377.00	1.24411	0.622054	+	0.782974i	$0.286298\pi$
$108$	−108.000	−0.0962250
$109$	1676.00	1.47277	0.736384	−	0.676564i	$-0.236532\pi$
$109$	1676.00	1.47277	0.736384	+	0.676564i	$0.236532\pi$
$110$	−306.000	−0.265236
$111$	−168.000	−0.143656
$112$	320.000	0.269975
$113$	141.000	0.117382	0.0586910	−	0.998276i	$-0.481307\pi$
$113$	141.000	0.117382	0.0586910	+	0.998276i	$0.481307\pi$
$114$	−258.000	−0.211964
$115$	657.000	0.532744
$116$	−600.000	−0.480247
$117$	−549.000	−0.433804
$118$	420.000	0.327662
$119$	340.000	0.261914
$120$	−72.0000	−0.0547723
$121$	1270.00	0.954170
$122$	896.000	0.664918
$123$	−45.0000	−0.0329879
$124$	1160.00	0.840089
$125$	723.000	0.517337
$126$	−360.000	−0.254535
$127$	−1483.00	−1.03618	−0.518090	−	0.855326i	$-0.673357\pi$
$127$	−1483.00	−1.03618	−0.518090	+	0.855326i	$0.673357\pi$
$128$	−128.000	−0.0883883
$129$	−249.000	−0.169948
$130$	−366.000	−0.246926
$131$	1491.00	0.994422	0.497211	−	0.867630i	$-0.334357\pi$
$131$	1491.00	0.994422	0.497211	+	0.867630i	$0.334357\pi$
$132$	612.000	0.403544
$133$	−860.000	−0.560688
$134$	248.000	0.159880
$135$	81.0000	0.0516398
$136$	−136.000	−0.0857493
$137$	−6.00000	−0.00374171	−0.00187086	−	0.999998i	$-0.500596\pi$
$137$	−6.00000	−0.00374171	−0.00187086	+	0.999998i	$0.500596\pi$
$138$	−1314.00	−0.810544
$139$	1010.00	0.616310	0.308155	−	0.951336i	$-0.400288\pi$
$139$	1010.00	0.616310	0.308155	+	0.951336i	$0.400288\pi$
$140$	−240.000	−0.144884
$141$	−1278.00	−0.763312
$142$	−1800.00	−1.06375
$143$	3111.00	1.81926
$144$	144.000	0.0833333
$145$	450.000	0.257727
$146$	2156.00	1.22214
$147$	−171.000	−0.0959445
$148$	224.000	0.124410
$149$	942.000	0.517931	0.258965	−	0.965887i	$-0.416618\pi$
$149$	942.000	0.517931	0.258965	+	0.965887i	$0.416618\pi$
$150$	−696.000	−0.378854
$151$	−88.0000	−0.0474261	−0.0237130	−	0.999719i	$-0.507549\pi$
$151$	−88.0000	−0.0474261	−0.0237130	+	0.999719i	$0.507549\pi$
$152$	344.000	0.183566
$153$	153.000	0.0808452
$154$	2040.00	1.06745
$155$	−870.000	−0.450839
$156$	732.000	0.375685
$157$	−1897.00	−0.964313	−0.482156	−	0.876085i	$-0.660146\pi$
$157$	−1897.00	−0.964313	−0.482156	+	0.876085i	$0.660146\pi$
$158$	−1444.00	−0.727079
$159$	1134.00	0.565610
$160$	96.0000	0.0474342
$161$	−4380.00	−2.14405
$162$	−162.000	−0.0785674
$163$	2738.00	1.31569	0.657843	−	0.753155i	$-0.271469\pi$
$163$	2738.00	1.31569	0.657843	+	0.753155i	$0.271469\pi$
$164$	60.0000	0.0285684
$165$	−459.000	−0.216564
$166$	156.000	0.0729394
$167$	−3207.00	−1.48602	−0.743009	−	0.669281i	$-0.766602\pi$
$167$	−3207.00	−1.48602	−0.743009	+	0.669281i	$0.766602\pi$
$168$	480.000	0.220433
$169$	1524.00	0.693673
$170$	102.000	0.0460179
$171$	−387.000	−0.173068
$172$	332.000	0.147179
$173$	345.000	0.151618	0.0758089	−	0.997122i	$-0.475846\pi$
$173$	345.000	0.151618	0.0758089	+	0.997122i	$0.475846\pi$
$174$	−900.000	−0.392120
$175$	−2320.00	−1.00215
$176$	−816.000	−0.349479
$177$	630.000	0.267535
$178$	288.000	0.121273
$179$	1422.00	0.593772	0.296886	−	0.954913i	$-0.404052\pi$
$179$	1422.00	0.593772	0.296886	+	0.954913i	$0.404052\pi$
$180$	−108.000	−0.0447214
$181$	−2158.00	−0.886204	−0.443102	−	0.896471i	$-0.646122\pi$
$181$	−2158.00	−0.886204	−0.443102	+	0.896471i	$0.646122\pi$
$182$	2440.00	0.993763
$183$	1344.00	0.542903
$184$	1752.00	0.701952
$185$	−168.000	−0.0667654
$186$	1740.00	0.685930
$187$	−867.000	−0.339044
$188$	1704.00	0.661048
$189$	−540.000	−0.207827
$190$	−258.000	−0.0985120
$191$	4710.00	1.78431	0.892156	−	0.451727i	$-0.149192\pi$
$191$	4710.00	1.78431	0.892156	+	0.451727i	$0.149192\pi$
$192$	−192.000	−0.0721688
$193$	−4282.00	−1.59702	−0.798511	−	0.601981i	$-0.794378\pi$
$193$	−4282.00	−1.59702	−0.798511	+	0.601981i	$0.794378\pi$
$194$	536.000	0.198364
$195$	−549.000	−0.201614
$196$	228.000	0.0830904
$197$	−2691.00	−0.973227	−0.486614	−	0.873617i	$-0.661768\pi$
$197$	−2691.00	−0.973227	−0.486614	+	0.873617i	$0.661768\pi$
$198$	918.000	0.329492
$199$	3908.00	1.39211	0.696057	−	0.717986i	$-0.254936\pi$
$199$	3908.00	1.39211	0.696057	+	0.717986i	$0.254936\pi$
$200$	928.000	0.328098
$201$	372.000	0.130542
$202$	−1536.00	−0.535013
$203$	−3000.00	−1.03724
$204$	−204.000	−0.0700140
$205$	−45.0000	−0.0153314
$206$	3290.00	1.11274
$207$	−1971.00	−0.661807
$208$	−976.000	−0.325353
$209$	2193.00	0.725804
$210$	−360.000	−0.118297
$211$	−4534.00	−1.47931	−0.739653	−	0.672989i	$-0.765010\pi$
$211$	−4534.00	−1.47931	−0.739653	+	0.672989i	$0.765010\pi$
$212$	−1512.00	−0.489833
$213$	−2700.00	−0.868549
$214$	−2754.00	−0.879718
$215$	−249.000	−0.0789845
$216$	216.000	0.0680414
$217$	5800.00	1.81442
$218$	−3352.00	−1.04140
$219$	3234.00	0.997870
$220$	612.000	0.187550
$221$	−1037.00	−0.315639
$222$	336.000	0.101580
$223$	−4615.00	−1.38584	−0.692922	−	0.721012i	$-0.743677\pi$
$223$	−4615.00	−1.38584	−0.692922	+	0.721012i	$0.743677\pi$
$224$	−640.000	−0.190901
$225$	−1044.00	−0.309333
$226$	−282.000	−0.0830016
$227$	−2373.00	−0.693839	−0.346920	−	0.937895i	$-0.612772\pi$
$227$	−2373.00	−0.693839	−0.346920	+	0.937895i	$0.612772\pi$
$228$	516.000	0.149881
$229$	−5218.00	−1.50574	−0.752871	−	0.658168i	$-0.771332\pi$
$229$	−5218.00	−1.50574	−0.752871	+	0.658168i	$0.771332\pi$
$230$	−1314.00	−0.376707
$231$	3060.00	0.871572
$232$	1200.00	0.339586
$233$	4671.00	1.31334	0.656668	−	0.754180i	$-0.271965\pi$
$233$	4671.00	1.31334	0.656668	+	0.754180i	$0.271965\pi$
$234$	1098.00	0.306746
$235$	−1278.00	−0.354755
$236$	−840.000	−0.231692
$237$	−2166.00	−0.593657
$238$	−680.000	−0.185201
$239$	−4440.00	−1.20167	−0.600836	−	0.799372i	$-0.705166\pi$
$239$	−4440.00	−1.20167	−0.600836	+	0.799372i	$0.705166\pi$
$240$	144.000	0.0387298
$241$	2360.00	0.630792	0.315396	−	0.948960i	$-0.397863\pi$
$241$	2360.00	0.630792	0.315396	+	0.948960i	$0.397863\pi$
$242$	−2540.00	−0.674700
$243$	−243.000	−0.0641500
$244$	−1792.00	−0.470168
$245$	−171.000	−0.0445910
$246$	90.0000	0.0233260
$247$	2623.00	0.675698
$248$	−2320.00	−0.594033
$249$	234.000	0.0595548
$250$	−1446.00	−0.365812
$251$	−5148.00	−1.29458	−0.647289	−	0.762245i	$-0.724097\pi$
$251$	−5148.00	−1.29458	−0.647289	+	0.762245i	$0.724097\pi$
$252$	720.000	0.179983
$253$	11169.0	2.77545
$254$	2966.00	0.732691
$255$	153.000	0.0375735
$256$	256.000	0.0625000
$257$	−4044.00	−0.981548	−0.490774	−	0.871287i	$-0.663286\pi$
$257$	−4044.00	−0.981548	−0.490774	+	0.871287i	$0.663286\pi$
$258$	498.000	0.120171
$259$	1120.00	0.268700
$260$	732.000	0.174603
$261$	−1350.00	−0.320164
$262$	−2982.00	−0.703163
$263$	−2544.00	−0.596463	−0.298231	−	0.954494i	$-0.596397\pi$
$263$	−2544.00	−0.596463	−0.298231	+	0.954494i	$0.596397\pi$
$264$	−1224.00	−0.285348
$265$	1134.00	0.262872
$266$	1720.00	0.396466
$267$	432.000	0.0990186
$268$	−496.000	−0.113052
$269$	−4401.00	−0.997523	−0.498762	−	0.866739i	$-0.666212\pi$
$269$	−4401.00	−0.997523	−0.498762	+	0.866739i	$0.666212\pi$
$270$	−162.000	−0.0365148
$271$	2549.00	0.571368	0.285684	−	0.958324i	$-0.407779\pi$
$271$	2549.00	0.571368	0.285684	+	0.958324i	$0.407779\pi$
$272$	272.000	0.0606339
$273$	3660.00	0.811404
$274$	12.0000	0.00264579
$275$	5916.00	1.29727
$276$	2628.00	0.573141
$277$	8426.00	1.82769	0.913843	−	0.406067i	$-0.133100\pi$
$277$	8426.00	1.82769	0.913843	+	0.406067i	$0.133100\pi$
$278$	−2020.00	−0.435797
$279$	2610.00	0.560060
$280$	480.000	0.102448
$281$	−2508.00	−0.532437	−0.266218	−	0.963913i	$-0.585774\pi$
$281$	−2508.00	−0.532437	−0.266218	+	0.963913i	$0.585774\pi$
$282$	2556.00	0.539743
$283$	2018.00	0.423879	0.211939	−	0.977283i	$-0.432022\pi$
$283$	2018.00	0.423879	0.211939	+	0.977283i	$0.432022\pi$
$284$	3600.00	0.752186
$285$	−387.000	−0.0804347
$286$	−6222.00	−1.28641
$287$	300.000	0.0617019
$288$	−288.000	−0.0589256
$289$	289.000	0.0588235
$290$	−900.000	−0.182241
$291$	804.000	0.161963
$292$	−4312.00	−0.864181
$293$	−5772.00	−1.15087	−0.575433	−	0.817849i	$-0.695167\pi$
$293$	−5772.00	−1.15087	−0.575433	+	0.817849i	$0.695167\pi$
$294$	342.000	0.0678430
$295$	630.000	0.124339
$296$	−448.000	−0.0879712
$297$	1377.00	0.269029
$298$	−1884.00	−0.366232
$299$	13359.0	2.58385
$300$	1392.00	0.267891
$301$	1660.00	0.317876
$302$	176.000	0.0335353
$303$	−2304.00	−0.436836
$304$	−688.000	−0.129801
$305$	1344.00	0.252319
$306$	−306.000	−0.0571662
$307$	1892.00	0.351733	0.175867	−	0.984414i	$-0.443727\pi$
$307$	1892.00	0.351733	0.175867	+	0.984414i	$0.443727\pi$
$308$	−4080.00	−0.754804
$309$	4935.00	0.908551
$310$	1740.00	0.318791
$311$	456.000	0.0831427	0.0415714	−	0.999136i	$-0.486764\pi$
$311$	456.000	0.0831427	0.0415714	+	0.999136i	$0.486764\pi$
$312$	−1464.00	−0.265650
$313$	5564.00	1.00478	0.502390	−	0.864641i	$-0.332454\pi$
$313$	5564.00	1.00478	0.502390	+	0.864641i	$0.332454\pi$
$314$	3794.00	0.681872
$315$	−540.000	−0.0965891
$316$	2888.00	0.514122
$317$	1830.00	0.324237	0.162118	−	0.986771i	$-0.448167\pi$
$317$	1830.00	0.324237	0.162118	+	0.986771i	$0.448167\pi$
$318$	−2268.00	−0.399947
$319$	7650.00	1.34269
$320$	−192.000	−0.0335410
$321$	−4131.00	−0.718286
$322$	8760.00	1.51607
$323$	−731.000	−0.125925
$324$	324.000	0.0555556
$325$	7076.00	1.20771
$326$	−5476.00	−0.930330
$327$	−5028.00	−0.850303
$328$	−120.000	−0.0202009
$329$	8520.00	1.42773
$330$	918.000	0.153134
$331$	−11383.0	−1.89023	−0.945115	−	0.326737i	$-0.894051\pi$
$331$	−11383.0	−1.89023	−0.945115	+	0.326737i	$0.894051\pi$
$332$	−312.000	−0.0515760
$333$	504.000	0.0829400
$334$	6414.00	1.05077
$335$	372.000	0.0606702
$336$	−960.000	−0.155870
$337$	5186.00	0.838277	0.419139	−	0.907922i	$-0.362332\pi$
$337$	5186.00	0.838277	0.419139	+	0.907922i	$0.362332\pi$
$338$	−3048.00	−0.490501
$339$	−423.000	−0.0677705
$340$	−204.000	−0.0325396
$341$	−14790.0	−2.34875
$342$	774.000	0.122378
$343$	−5720.00	−0.900440
$344$	−664.000	−0.104071
$345$	−1971.00	−0.307580
$346$	−690.000	−0.107210
$347$	−6420.00	−0.993209	−0.496605	−	0.867977i	$-0.665420\pi$
$347$	−6420.00	−0.993209	−0.496605	+	0.867977i	$0.665420\pi$
$348$	1800.00	0.277270
$349$	−3823.00	−0.586362	−0.293181	−	0.956057i	$-0.594714\pi$
$349$	−3823.00	−0.586362	−0.293181	+	0.956057i	$0.594714\pi$
$350$	4640.00	0.708624
$351$	1647.00	0.250457
$352$	1632.00	0.247119
$353$	5826.00	0.878433	0.439216	−	0.898381i	$-0.355256\pi$
$353$	5826.00	0.878433	0.439216	+	0.898381i	$0.355256\pi$
$354$	−1260.00	−0.189176
$355$	−2700.00	−0.403665
$356$	−576.000	−0.0857526
$357$	−1020.00	−0.151216
$358$	−2844.00	−0.419861
$359$	−6732.00	−0.989697	−0.494849	−	0.868979i	$-0.664777\pi$
$359$	−6732.00	−0.989697	−0.494849	+	0.868979i	$0.664777\pi$
$360$	216.000	0.0316228
$361$	−5010.00	−0.730427
$362$	4316.00	0.626641
$363$	−3810.00	−0.550890
$364$	−4880.00	−0.702696
$365$	3234.00	0.463768
$366$	−2688.00	−0.383891
$367$	−10780.0	−1.53327	−0.766637	−	0.642081i	$-0.778071\pi$
$367$	−10780.0	−1.53327	−0.766637	+	0.642081i	$0.778071\pi$
$368$	−3504.00	−0.496355
$369$	135.000	0.0190456
$370$	336.000	0.0472103
$371$	−7560.00	−1.05794
$372$	−3480.00	−0.485026
$373$	−5902.00	−0.819287	−0.409643	−	0.912246i	$-0.634347\pi$
$373$	−5902.00	−0.819287	−0.409643	+	0.912246i	$0.634347\pi$
$374$	1734.00	0.239741
$375$	−2169.00	−0.298684
$376$	−3408.00	−0.467431
$377$	9150.00	1.25000
$378$	1080.00	0.146956
$379$	8480.00	1.14931	0.574655	−	0.818396i	$-0.305136\pi$
$379$	8480.00	1.14931	0.574655	+	0.818396i	$0.305136\pi$
$380$	516.000	0.0696585
$381$	4449.00	0.598239
$382$	−9420.00	−1.26170
$383$	2904.00	0.387435	0.193717	−	0.981057i	$-0.437946\pi$
$383$	2904.00	0.387435	0.193717	+	0.981057i	$0.437946\pi$
$384$	384.000	0.0510310
$385$	3060.00	0.405070
$386$	8564.00	1.12926
$387$	747.000	0.0981192
$388$	−1072.00	−0.140264
$389$	−2760.00	−0.359737	−0.179868	−	0.983691i	$-0.557567\pi$
$389$	−2760.00	−0.359737	−0.179868	+	0.983691i	$0.557567\pi$
$390$	1098.00	0.142563
$391$	−3723.00	−0.481535
$392$	−456.000	−0.0587538
$393$	−4473.00	−0.574130
$394$	5382.00	0.688176
$395$	−2166.00	−0.275907
$396$	−1836.00	−0.232986
$397$	9776.00	1.23588	0.617939	−	0.786226i	$-0.287968\pi$
$397$	9776.00	1.23588	0.617939	+	0.786226i	$0.287968\pi$
$398$	−7816.00	−0.984374
$399$	2580.00	0.323713
$400$	−1856.00	−0.232000
$401$	6675.00	0.831256	0.415628	−	0.909535i	$-0.363562\pi$
$401$	6675.00	0.831256	0.415628	+	0.909535i	$0.363562\pi$
$402$	−744.000	−0.0923068
$403$	−17690.0	−2.18660
$404$	3072.00	0.378311
$405$	−243.000	−0.0298142
$406$	6000.00	0.733436
$407$	−2856.00	−0.347830
$408$	408.000	0.0495074
$409$	1181.00	0.142779	0.0713896	−	0.997449i	$-0.477257\pi$
$409$	1181.00	0.142779	0.0713896	+	0.997449i	$0.477257\pi$
$410$	90.0000	0.0108409
$411$	18.0000	0.00216028
$412$	−6580.00	−0.786828
$413$	−4200.00	−0.500408
$414$	3942.00	0.467968
$415$	234.000	0.0276786
$416$	1952.00	0.230059
$417$	−3030.00	−0.355827
$418$	−4386.00	−0.513221
$419$	−10740.0	−1.25223	−0.626114	−	0.779732i	$-0.715355\pi$
$419$	−10740.0	−1.25223	−0.626114	+	0.779732i	$0.715355\pi$
$420$	720.000	0.0836486
$421$	−10285.0	−1.19064	−0.595321	−	0.803488i	$-0.702975\pi$
$421$	−10285.0	−1.19064	−0.595321	+	0.803488i	$0.702975\pi$
$422$	9068.00	1.04603
$423$	3834.00	0.440698
$424$	3024.00	0.346364
$425$	−1972.00	−0.225073
$426$	5400.00	0.614157
$427$	−8960.00	−1.01547
$428$	5508.00	0.622054
$429$	−9333.00	−1.05035
$430$	498.000	0.0558504
$431$	5184.00	0.579361	0.289680	−	0.957123i	$-0.406451\pi$
$431$	5184.00	0.579361	0.289680	+	0.957123i	$0.406451\pi$
$432$	−432.000	−0.0481125
$433$	−25.0000	−0.00277465	−0.00138732	−	0.999999i	$-0.500442\pi$
$433$	−25.0000	−0.00277465	−0.00138732	+	0.999999i	$0.500442\pi$
$434$	−11600.0	−1.28299
$435$	−1350.00	−0.148799
$436$	6704.00	0.736384
$437$	9417.00	1.03084
$438$	−6468.00	−0.705600
$439$	4484.00	0.487493	0.243747	−	0.969839i	$-0.421623\pi$
$439$	4484.00	0.487493	0.243747	+	0.969839i	$0.421623\pi$
$440$	−1224.00	−0.132618
$441$	513.000	0.0553936
$442$	2074.00	0.223190
$443$	16230.0	1.74066	0.870328	−	0.492473i	$-0.163907\pi$
$443$	16230.0	1.74066	0.870328	+	0.492473i	$0.163907\pi$
$444$	−672.000	−0.0718282
$445$	432.000	0.0460197
$446$	9230.00	0.979940
$447$	−2826.00	−0.299027
$448$	1280.00	0.134987
$449$	14166.0	1.48894	0.744471	−	0.667655i	$-0.232702\pi$
$449$	14166.0	1.48894	0.744471	+	0.667655i	$0.232702\pi$
$450$	2088.00	0.218732
$451$	−765.000	−0.0798724
$452$	564.000	0.0586910
$453$	264.000	0.0273815
$454$	4746.00	0.490619
$455$	3660.00	0.377106
$456$	−1032.00	−0.105982
$457$	11261.0	1.15266	0.576332	−	0.817216i	$-0.304484\pi$
$457$	11261.0	1.15266	0.576332	+	0.817216i	$0.304484\pi$
$458$	10436.0	1.06472
$459$	−459.000	−0.0466760
$460$	2628.00	0.266372
$461$	−6990.00	−0.706197	−0.353098	−	0.935586i	$-0.614872\pi$
$461$	−6990.00	−0.706197	−0.353098	+	0.935586i	$0.614872\pi$
$462$	−6120.00	−0.616295
$463$	−17152.0	−1.72164	−0.860822	−	0.508906i	$-0.830050\pi$
$463$	−17152.0	−1.72164	−0.860822	+	0.508906i	$0.830050\pi$
$464$	−2400.00	−0.240123
$465$	2610.00	0.260292
$466$	−9342.00	−0.928669
$467$	4302.00	0.426280	0.213140	−	0.977022i	$-0.431631\pi$
$467$	4302.00	0.426280	0.213140	+	0.977022i	$0.431631\pi$
$468$	−2196.00	−0.216902
$469$	−2480.00	−0.244170
$470$	2556.00	0.250850
$471$	5691.00	0.556746
$472$	1680.00	0.163831
$473$	−4233.00	−0.411487
$474$	4332.00	0.419779
$475$	4988.00	0.481821
$476$	1360.00	0.130957
$477$	−3402.00	−0.326555
$478$	8880.00	0.849711
$479$	−3183.00	−0.303622	−0.151811	−	0.988410i	$-0.548511\pi$
$479$	−3183.00	−0.303622	−0.151811	+	0.988410i	$0.548511\pi$
$480$	−288.000	−0.0273861
$481$	−3416.00	−0.323817
$482$	−4720.00	−0.446038
$483$	13140.0	1.23787
$484$	5080.00	0.477085
$485$	804.000	0.0752737
$486$	486.000	0.0453609
$487$	−9394.00	−0.874092	−0.437046	−	0.899439i	$-0.643975\pi$
$487$	−9394.00	−0.874092	−0.437046	+	0.899439i	$0.643975\pi$
$488$	3584.00	0.332459
$489$	−8214.00	−0.759611
$490$	342.000	0.0315306
$491$	14082.0	1.29432	0.647161	−	0.762354i	$-0.275956\pi$
$491$	14082.0	1.29432	0.647161	+	0.762354i	$0.275956\pi$
$492$	−180.000	−0.0164940
$493$	−2550.00	−0.232954
$494$	−5246.00	−0.477791
$495$	1377.00	0.125033
$496$	4640.00	0.420045
$497$	18000.0	1.62457
$498$	−468.000	−0.0421116
$499$	−3922.00	−0.351849	−0.175925	−	0.984404i	$-0.556292\pi$
$499$	−3922.00	−0.351849	−0.175925	+	0.984404i	$0.556292\pi$
$500$	2892.00	0.258668
$501$	9621.00	0.857953
$502$	10296.0	0.915404
$503$	1899.00	0.168334	0.0841672	−	0.996452i	$-0.473177\pi$
$503$	1899.00	0.168334	0.0841672	+	0.996452i	$0.473177\pi$
$504$	−1440.00	−0.127267
$505$	−2304.00	−0.203023
$506$	−22338.0	−1.96254
$507$	−4572.00	−0.400492
$508$	−5932.00	−0.518090
$509$	1572.00	0.136891	0.0684457	−	0.997655i	$-0.478196\pi$
$509$	1572.00	0.136891	0.0684457	+	0.997655i	$0.478196\pi$
$510$	−306.000	−0.0265684
$511$	−21560.0	−1.86645
$512$	−512.000	−0.0441942
$513$	1161.00	0.0999209
$514$	8088.00	0.694059
$515$	4935.00	0.422256
$516$	−996.000	−0.0849738
$517$	−21726.0	−1.84818
$518$	−2240.00	−0.190000
$519$	−1035.00	−0.0875365
$520$	−1464.00	−0.123463
$521$	−18657.0	−1.56886	−0.784432	−	0.620215i	$-0.787045\pi$
$521$	−18657.0	−1.56886	−0.784432	+	0.620215i	$0.787045\pi$
$522$	2700.00	0.226390
$523$	17012.0	1.42234	0.711169	−	0.703021i	$-0.248166\pi$
$523$	17012.0	1.42234	0.711169	+	0.703021i	$0.248166\pi$
$524$	5964.00	0.497211
$525$	6960.00	0.578589
$526$	5088.00	0.421763
$527$	4930.00	0.407503
$528$	2448.00	0.201772
$529$	35794.0	2.94189
$530$	−2268.00	−0.185879
$531$	−1890.00	−0.154461
$532$	−3440.00	−0.280344
$533$	−915.000	−0.0743584
$534$	−864.000	−0.0700167
$535$	−4131.00	−0.333829
$536$	992.000	0.0799401
$537$	−4266.00	−0.342815
$538$	8802.00	0.705355
$539$	−2907.00	−0.232307
$540$	324.000	0.0258199
$541$	14996.0	1.19173	0.595867	−	0.803083i	$-0.296808\pi$
$541$	14996.0	1.19173	0.595867	+	0.803083i	$0.296808\pi$
$542$	−5098.00	−0.404018
$543$	6474.00	0.511650
$544$	−544.000	−0.0428746
$545$	−5028.00	−0.395185
$546$	−7320.00	−0.573749
$547$	−1384.00	−0.108182	−0.0540910	−	0.998536i	$-0.517226\pi$
$547$	−1384.00	−0.108182	−0.0540910	+	0.998536i	$0.517226\pi$
$548$	−24.0000	−0.00187086
$549$	−4032.00	−0.313445
$550$	−11832.0	−0.917306
$551$	6450.00	0.498692
$552$	−5256.00	−0.405272
$553$	14440.0	1.11040
$554$	−16852.0	−1.29237
$555$	504.000	0.0385470
$556$	4040.00	0.308155
$557$	−14778.0	−1.12417	−0.562086	−	0.827079i	$-0.690001\pi$
$557$	−14778.0	−1.12417	−0.562086	+	0.827079i	$0.690001\pi$
$558$	−5220.00	−0.396022
$559$	−5063.00	−0.383081
$560$	−960.000	−0.0724418
$561$	2601.00	0.195747
$562$	5016.00	0.376490
$563$	−5754.00	−0.430732	−0.215366	−	0.976533i	$-0.569094\pi$
$563$	−5754.00	−0.430732	−0.215366	+	0.976533i	$0.569094\pi$
$564$	−5112.00	−0.381656
$565$	−423.000	−0.0314969
$566$	−4036.00	−0.299727
$567$	1620.00	0.119989
$568$	−7200.00	−0.531876
$569$	−5964.00	−0.439409	−0.219705	−	0.975566i	$-0.570509\pi$
$569$	−5964.00	−0.439409	−0.219705	+	0.975566i	$0.570509\pi$
$570$	774.000	0.0568760
$571$	7580.00	0.555540	0.277770	−	0.960648i	$-0.410405\pi$
$571$	7580.00	0.555540	0.277770	+	0.960648i	$0.410405\pi$
$572$	12444.0	0.909632
$573$	−14130.0	−1.03017
$574$	−600.000	−0.0436298
$575$	25404.0	1.84247
$576$	576.000	0.0416667
$577$	−4543.00	−0.327777	−0.163889	−	0.986479i	$-0.552404\pi$
$577$	−4543.00	−0.327777	−0.163889	+	0.986479i	$0.552404\pi$
$578$	−578.000	−0.0415945
$579$	12846.0	0.922041
$580$	1800.00	0.128864
$581$	−1560.00	−0.111394
$582$	−1608.00	−0.114525
$583$	19278.0	1.36949
$584$	8624.00	0.611068
$585$	1647.00	0.116402
$586$	11544.0	0.813785
$587$	19092.0	1.34244	0.671219	−	0.741259i	$-0.265771\pi$
$587$	19092.0	1.34244	0.671219	+	0.741259i	$0.265771\pi$
$588$	−684.000	−0.0479723
$589$	−12470.0	−0.872356
$590$	−1260.00	−0.0879210
$591$	8073.00	0.561893
$592$	896.000	0.0622050
$593$	−10854.0	−0.751636	−0.375818	−	0.926693i	$-0.622638\pi$
$593$	−10854.0	−0.751636	−0.375818	+	0.926693i	$0.622638\pi$
$594$	−2754.00	−0.190232
$595$	−1020.00	−0.0702789
$596$	3768.00	0.258965
$597$	−11724.0	−0.803738
$598$	−26718.0	−1.82706
$599$	−23898.0	−1.63013	−0.815063	−	0.579372i	$-0.803298\pi$
$599$	−23898.0	−1.63013	−0.815063	+	0.579372i	$0.803298\pi$
$600$	−2784.00	−0.189427
$601$	−8422.00	−0.571615	−0.285807	−	0.958287i	$-0.592262\pi$
$601$	−8422.00	−0.571615	−0.285807	+	0.958287i	$0.592262\pi$
$602$	−3320.00	−0.224773
$603$	−1116.00	−0.0753682
$604$	−352.000	−0.0237130
$605$	−3810.00	−0.256031
$606$	4608.00	0.308890
$607$	−24154.0	−1.61512	−0.807562	−	0.589782i	$-0.799214\pi$
$607$	−24154.0	−1.61512	−0.807562	+	0.589782i	$0.799214\pi$
$608$	1376.00	0.0917832
$609$	9000.00	0.598848
$610$	−2688.00	−0.178416
$611$	−25986.0	−1.72059
$612$	612.000	0.0404226
$613$	−14641.0	−0.964673	−0.482336	−	0.875986i	$-0.660212\pi$
$613$	−14641.0	−0.964673	−0.482336	+	0.875986i	$0.660212\pi$
$614$	−3784.00	−0.248713
$615$	135.000	0.00885159
$616$	8160.00	0.533727
$617$	−5538.00	−0.361348	−0.180674	−	0.983543i	$-0.557828\pi$
$617$	−5538.00	−0.361348	−0.180674	+	0.983543i	$0.557828\pi$
$618$	−9870.00	−0.642443
$619$	12746.0	0.827633	0.413817	−	0.910360i	$-0.364195\pi$
$619$	12746.0	0.827633	0.413817	+	0.910360i	$0.364195\pi$
$620$	−3480.00	−0.225420
$621$	5913.00	0.382094
$622$	−912.000	−0.0587908
$623$	−2880.00	−0.185208
$624$	2928.00	0.187843
$625$	12331.0	0.789184
$626$	−11128.0	−0.710486
$627$	−6579.00	−0.419043
$628$	−7588.00	−0.482156
$629$	952.000	0.0603477
$630$	1080.00	0.0682988
$631$	18893.0	1.19195	0.595973	−	0.803004i	$-0.296766\pi$
$631$	18893.0	1.19195	0.595973	+	0.803004i	$0.296766\pi$
$632$	−5776.00	−0.363539
$633$	13602.0	0.854077
$634$	−3660.00	−0.229270
$635$	4449.00	0.278037
$636$	4536.00	0.282805
$637$	−3477.00	−0.216270
$638$	−15300.0	−0.949424
$639$	8100.00	0.501457
$640$	384.000	0.0237171
$641$	10533.0	0.649030	0.324515	−	0.945880i	$-0.394799\pi$
$641$	10533.0	0.649030	0.324515	+	0.945880i	$0.394799\pi$
$642$	8262.00	0.507905
$643$	19028.0	1.16702	0.583508	−	0.812108i	$-0.301680\pi$
$643$	19028.0	1.16702	0.583508	+	0.812108i	$0.301680\pi$
$644$	−17520.0	−1.07203
$645$	747.000	0.0456017
$646$	1462.00	0.0890428
$647$	11214.0	0.681403	0.340702	−	0.940171i	$-0.389335\pi$
$647$	11214.0	0.681403	0.340702	+	0.940171i	$0.389335\pi$
$648$	−648.000	−0.0392837
$649$	10710.0	0.647772
$650$	−14152.0	−0.853980
$651$	−17400.0	−1.04756
$652$	10952.0	0.657843
$653$	−795.000	−0.0476428	−0.0238214	−	0.999716i	$-0.507583\pi$
$653$	−795.000	−0.0476428	−0.0238214	+	0.999716i	$0.507583\pi$
$654$	10056.0	0.601255
$655$	−4473.00	−0.266831
$656$	240.000	0.0142842
$657$	−9702.00	−0.576120
$658$	−17040.0	−1.00956
$659$	−25854.0	−1.52827	−0.764134	−	0.645057i	$-0.776833\pi$
$659$	−25854.0	−1.52827	−0.764134	+	0.645057i	$0.776833\pi$
$660$	−1836.00	−0.108282
$661$	−26611.0	−1.56588	−0.782941	−	0.622096i	$-0.786281\pi$
$661$	−26611.0	−1.56588	−0.782941	+	0.622096i	$0.786281\pi$
$662$	22766.0	1.33659
$663$	3111.00	0.182234
$664$	624.000	0.0364697
$665$	2580.00	0.150448
$666$	−1008.00	−0.0586475
$667$	32850.0	1.90698
$668$	−12828.0	−0.743009
$669$	13845.0	0.800118
$670$	−744.000	−0.0429003
$671$	22848.0	1.31451
$672$	1920.00	0.110217
$673$	−6550.00	−0.375162	−0.187581	−	0.982249i	$-0.560065\pi$
$673$	−6550.00	−0.375162	−0.187581	+	0.982249i	$0.560065\pi$
$674$	−10372.0	−0.592752
$675$	3132.00	0.178594
$676$	6096.00	0.346837
$677$	−32613.0	−1.85143	−0.925716	−	0.378219i	$-0.876536\pi$
$677$	−32613.0	−1.85143	−0.925716	+	0.378219i	$0.876536\pi$
$678$	846.000	0.0479210
$679$	−5360.00	−0.302942
$680$	408.000	0.0230089
$681$	7119.00	0.400588
$682$	29580.0	1.66082
$683$	−20925.0	−1.17229	−0.586144	−	0.810207i	$-0.699355\pi$
$683$	−20925.0	−1.17229	−0.586144	+	0.810207i	$0.699355\pi$
$684$	−1548.00	−0.0865340
$685$	18.0000	0.00100401
$686$	11440.0	0.636707
$687$	15654.0	0.869341
$688$	1328.00	0.0735894
$689$	23058.0	1.27495
$690$	3942.00	0.217492
$691$	7724.00	0.425231	0.212616	−	0.977136i	$-0.431802\pi$
$691$	7724.00	0.425231	0.212616	+	0.977136i	$0.431802\pi$
$692$	1380.00	0.0758089
$693$	−9180.00	−0.503203
$694$	12840.0	0.702305
$695$	−3030.00	−0.165373
$696$	−3600.00	−0.196060
$697$	255.000	0.0138577
$698$	7646.00	0.414621
$699$	−14013.0	−0.758255
$700$	−9280.00	−0.501073
$701$	−72.0000	−0.00387932	−0.00193966	−	0.999998i	$-0.500617\pi$
$701$	−72.0000	−0.00387932	−0.00193966	+	0.999998i	$0.500617\pi$
$702$	−3294.00	−0.177100
$703$	−2408.00	−0.129188
$704$	−3264.00	−0.174740
$705$	3834.00	0.204818
$706$	−11652.0	−0.621146
$707$	15360.0	0.817075
$708$	2520.00	0.133768
$709$	−12058.0	−0.638713	−0.319357	−	0.947635i	$-0.603467\pi$
$709$	−12058.0	−0.638713	−0.319357	+	0.947635i	$0.603467\pi$
$710$	5400.00	0.285434
$711$	6498.00	0.342748
$712$	1152.00	0.0606363
$713$	−63510.0	−3.33586
$714$	2040.00	0.106926
$715$	−9333.00	−0.488160
$716$	5688.00	0.296886
$717$	13320.0	0.693786
$718$	13464.0	0.699822
$719$	−5553.00	−0.288028	−0.144014	−	0.989576i	$-0.546001\pi$
$719$	−5553.00	−0.288028	−0.144014	+	0.989576i	$0.546001\pi$
$720$	−432.000	−0.0223607
$721$	−32900.0	−1.69939
$722$	10020.0	0.516490
$723$	−7080.00	−0.364188
$724$	−8632.00	−0.443102
$725$	17400.0	0.891338
$726$	7620.00	0.389538
$727$	−4048.00	−0.206509	−0.103254	−	0.994655i	$-0.532926\pi$
$727$	−4048.00	−0.206509	−0.103254	+	0.994655i	$0.532926\pi$
$728$	9760.00	0.496881
$729$	729.000	0.0370370
$730$	−6468.00	−0.327933
$731$	1411.00	0.0713922
$732$	5376.00	0.271452
$733$	17138.0	0.863583	0.431792	−	0.901973i	$-0.357882\pi$
$733$	17138.0	0.863583	0.431792	+	0.901973i	$0.357882\pi$
$734$	21560.0	1.08419
$735$	513.000	0.0257446
$736$	7008.00	0.350976
$737$	6324.00	0.316075
$738$	−270.000	−0.0134673
$739$	−9259.00	−0.460890	−0.230445	−	0.973085i	$-0.574018\pi$
$739$	−9259.00	−0.460890	−0.230445	+	0.973085i	$0.574018\pi$
$740$	−672.000	−0.0333827
$741$	−7869.00	−0.390115
$742$	15120.0	0.748076
$743$	−25248.0	−1.24665	−0.623324	−	0.781964i	$-0.714218\pi$
$743$	−25248.0	−1.24665	−0.623324	+	0.781964i	$0.714218\pi$
$744$	6960.00	0.342965
$745$	−2826.00	−0.138975
$746$	11804.0	0.579323
$747$	−702.000	−0.0343840
$748$	−3468.00	−0.169522
$749$	27540.0	1.34351
$750$	4338.00	0.211202
$751$	27506.0	1.33650	0.668248	−	0.743939i	$-0.267045\pi$
$751$	27506.0	1.33650	0.668248	+	0.743939i	$0.267045\pi$
$752$	6816.00	0.330524
$753$	15444.0	0.747424
$754$	−18300.0	−0.883882
$755$	264.000	0.0127258
$756$	−2160.00	−0.103913
$757$	21665.0	1.04020	0.520098	−	0.854107i	$-0.325896\pi$
$757$	21665.0	1.04020	0.520098	+	0.854107i	$0.325896\pi$
$758$	−16960.0	−0.812685
$759$	−33507.0	−1.60241
$760$	−1032.00	−0.0492560
$761$	13254.0	0.631350	0.315675	−	0.948867i	$-0.397769\pi$
$761$	13254.0	0.631350	0.315675	+	0.948867i	$0.397769\pi$
$762$	−8898.00	−0.423019
$763$	33520.0	1.59044
$764$	18840.0	0.892156
$765$	−459.000	−0.0216930
$766$	−5808.00	−0.273958
$767$	12810.0	0.603054
$768$	−768.000	−0.0360844
$769$	7193.00	0.337303	0.168652	−	0.985676i	$-0.446059\pi$
$769$	7193.00	0.337303	0.168652	+	0.985676i	$0.446059\pi$
$770$	−6120.00	−0.286428
$771$	12132.0	0.566697
$772$	−17128.0	−0.798511
$773$	16452.0	0.765508	0.382754	−	0.923850i	$-0.374976\pi$
$773$	16452.0	0.765508	0.382754	+	0.923850i	$0.374976\pi$
$774$	−1494.00	−0.0693808
$775$	−33640.0	−1.55921
$776$	2144.00	0.0991818
$777$	−3360.00	−0.155134
$778$	5520.00	0.254372
$779$	−645.000	−0.0296656
$780$	−2196.00	−0.100807
$781$	−45900.0	−2.10298
$782$	7446.00	0.340497
$783$	4050.00	0.184847
$784$	912.000	0.0415452
$785$	5691.00	0.258752
$786$	8946.00	0.405971
$787$	−7684.00	−0.348037	−0.174018	−	0.984742i	$-0.555675\pi$
$787$	−7684.00	−0.348037	−0.174018	+	0.984742i	$0.555675\pi$
$788$	−10764.0	−0.486614
$789$	7632.00	0.344368
$790$	4332.00	0.195096
$791$	2820.00	0.126761
$792$	3672.00	0.164746
$793$	27328.0	1.22377
$794$	−19552.0	−0.873897
$795$	−3402.00	−0.151769
$796$	15632.0	0.696057
$797$	7728.00	0.343463	0.171731	−	0.985144i	$-0.445064\pi$
$797$	7728.00	0.343463	0.171731	+	0.985144i	$0.445064\pi$
$798$	−5160.00	−0.228900
$799$	7242.00	0.320655
$800$	3712.00	0.164049
$801$	−1296.00	−0.0571684
$802$	−13350.0	−0.587787
$803$	54978.0	2.41610
$804$	1488.00	0.0652708
$805$	13140.0	0.575309
$806$	35380.0	1.54616
$807$	13203.0	0.575920
$808$	−6144.00	−0.267506
$809$	−32985.0	−1.43349	−0.716743	−	0.697337i	$-0.754368\pi$
$809$	−32985.0	−1.43349	−0.716743	+	0.697337i	$0.754368\pi$
$810$	486.000	0.0210819
$811$	−28942.0	−1.25313	−0.626567	−	0.779368i	$-0.715540\pi$
$811$	−28942.0	−1.25313	−0.626567	+	0.779368i	$0.715540\pi$
$812$	−12000.0	−0.518618
$813$	−7647.00	−0.329879
$814$	5712.00	0.245953
$815$	−8214.00	−0.353035
$816$	−816.000	−0.0350070
$817$	−3569.00	−0.152832
$818$	−2362.00	−0.100960
$819$	−10980.0	−0.468464
$820$	−180.000	−0.00766570
$821$	25653.0	1.09049	0.545247	−	0.838275i	$-0.316436\pi$
$821$	25653.0	1.09049	0.545247	+	0.838275i	$0.316436\pi$
$822$	−36.0000	−0.00152755
$823$	−4192.00	−0.177550	−0.0887752	−	0.996052i	$-0.528295\pi$
$823$	−4192.00	−0.177550	−0.0887752	+	0.996052i	$0.528295\pi$
$824$	13160.0	0.556372
$825$	−17748.0	−0.748977
$826$	8400.00	0.353842
$827$	−13509.0	−0.568022	−0.284011	−	0.958821i	$-0.591665\pi$
$827$	−13509.0	−0.568022	−0.284011	+	0.958821i	$0.591665\pi$
$828$	−7884.00	−0.330903
$829$	−10258.0	−0.429765	−0.214882	−	0.976640i	$-0.568937\pi$
$829$	−10258.0	−0.429765	−0.214882	+	0.976640i	$0.568937\pi$
$830$	−468.000	−0.0195717
$831$	−25278.0	−1.05522
$832$	−3904.00	−0.162676
$833$	969.000	0.0403048
$834$	6060.00	0.251607
$835$	9621.00	0.398741
$836$	8772.00	0.362902
$837$	−7830.00	−0.323351
$838$	21480.0	0.885459
$839$	1299.00	0.0534523	0.0267261	−	0.999643i	$-0.491492\pi$
$839$	1299.00	0.0534523	0.0267261	+	0.999643i	$0.491492\pi$
$840$	−1440.00	−0.0591485
$841$	−1889.00	−0.0774530
$842$	20570.0	0.841911
$843$	7524.00	0.307403
$844$	−18136.0	−0.739653
$845$	−4572.00	−0.186132
$846$	−7668.00	−0.311621
$847$	25400.0	1.03041
$848$	−6048.00	−0.244916
$849$	−6054.00	−0.244726
$850$	3944.00	0.159151
$851$	−12264.0	−0.494012
$852$	−10800.0	−0.434275
$853$	−44350.0	−1.78021	−0.890103	−	0.455760i	$-0.849367\pi$
$853$	−44350.0	−1.78021	−0.890103	+	0.455760i	$0.849367\pi$
$854$	17920.0	0.718044
$855$	1161.00	0.0464390
$856$	−11016.0	−0.439859
$857$	49026.0	1.95414	0.977069	−	0.212923i	$-0.0682983\pi$
$857$	49026.0	1.95414	0.977069	+	0.212923i	$0.0682983\pi$
$858$	18666.0	0.742712
$859$	43292.0	1.71956	0.859781	−	0.510663i	$-0.170600\pi$
$859$	43292.0	1.71956	0.859781	+	0.510663i	$0.170600\pi$
$860$	−996.000	−0.0394922
$861$	−900.000	−0.0356236
$862$	−10368.0	−0.409670
$863$	15696.0	0.619117	0.309559	−	0.950880i	$-0.399819\pi$
$863$	15696.0	0.619117	0.309559	+	0.950880i	$0.399819\pi$
$864$	864.000	0.0340207
$865$	−1035.00	−0.0406833
$866$	50.0000	0.00196197
$867$	−867.000	−0.0339618
$868$	23200.0	0.907211
$869$	−36822.0	−1.43740
$870$	2700.00	0.105217
$871$	7564.00	0.294255
$872$	−13408.0	−0.520702
$873$	−2412.00	−0.0935095
$874$	−18834.0	−0.728913
$875$	14460.0	0.558671
$876$	12936.0	0.498935
$877$	−30670.0	−1.18090	−0.590452	−	0.807073i	$-0.701050\pi$
$877$	−30670.0	−1.18090	−0.590452	+	0.807073i	$0.701050\pi$
$878$	−8968.00	−0.344710
$879$	17316.0	0.664453
$880$	2448.00	0.0937751
$881$	−47682.0	−1.82344	−0.911718	−	0.410816i	$-0.865244\pi$
$881$	−47682.0	−1.82344	−0.911718	+	0.410816i	$0.865244\pi$
$882$	−1026.00	−0.0391692
$883$	42977.0	1.63793	0.818964	−	0.573844i	$-0.194549\pi$
$883$	42977.0	1.63793	0.818964	+	0.573844i	$0.194549\pi$
$884$	−4148.00	−0.157819
$885$	−1890.00	−0.0717872
$886$	−32460.0	−1.23083
$887$	−19173.0	−0.725779	−0.362890	−	0.931832i	$-0.618210\pi$
$887$	−19173.0	−0.725779	−0.362890	+	0.931832i	$0.618210\pi$
$888$	1344.00	0.0507902
$889$	−29660.0	−1.11897
$890$	−864.000	−0.0325408
$891$	−4131.00	−0.155324
$892$	−18460.0	−0.692922
$893$	−18318.0	−0.686437
$894$	5652.00	0.211444
$895$	−4266.00	−0.159326
$896$	−2560.00	−0.0954504
$897$	−40077.0	−1.49179
$898$	−28332.0	−1.05284
$899$	−43500.0	−1.61380
$900$	−4176.00	−0.154667
$901$	−6426.00	−0.237604
$902$	1530.00	0.0564783
$903$	−4980.00	−0.183526
$904$	−1128.00	−0.0415008
$905$	6474.00	0.237793
$906$	−528.000	−0.0193616
$907$	16634.0	0.608956	0.304478	−	0.952519i	$-0.401518\pi$
$907$	16634.0	0.608956	0.304478	+	0.952519i	$0.401518\pi$
$908$	−9492.00	−0.346920
$909$	6912.00	0.252207
$910$	−7320.00	−0.266655
$911$	21153.0	0.769298	0.384649	−	0.923063i	$-0.374323\pi$
$911$	21153.0	0.769298	0.384649	+	0.923063i	$0.374323\pi$
$912$	2064.00	0.0749406
$913$	3978.00	0.144198
$914$	−22522.0	−0.815056
$915$	−4032.00	−0.145676
$916$	−20872.0	−0.752871
$917$	29820.0	1.07387
$918$	918.000	0.0330049
$919$	−24379.0	−0.875070	−0.437535	−	0.899201i	$-0.644148\pi$
$919$	−24379.0	−0.875070	−0.437535	+	0.899201i	$0.644148\pi$
$920$	−5256.00	−0.188353
$921$	−5676.00	−0.203073
$922$	13980.0	0.499357
$923$	−54900.0	−1.95781
$924$	12240.0	0.435786
$925$	−6496.00	−0.230905
$926$	34304.0	1.21739
$927$	−14805.0	−0.524552
$928$	4800.00	0.169793
$929$	−6171.00	−0.217938	−0.108969	−	0.994045i	$-0.534755\pi$
$929$	−6171.00	−0.217938	−0.108969	+	0.994045i	$0.534755\pi$
$930$	−5220.00	−0.184054
$931$	−2451.00	−0.0862817
$932$	18684.0	0.656668
$933$	−1368.00	−0.0480025
$934$	−8604.00	−0.301426
$935$	2601.00	0.0909752
$936$	4392.00	0.153373
$937$	−19438.0	−0.677707	−0.338854	−	0.940839i	$-0.610039\pi$
$937$	−19438.0	−0.677707	−0.338854	+	0.940839i	$0.610039\pi$
$938$	4960.00	0.172654
$939$	−16692.0	−0.580110
$940$	−5112.00	−0.177378
$941$	−20010.0	−0.693207	−0.346603	−	0.938012i	$-0.612665\pi$
$941$	−20010.0	−0.693207	−0.346603	+	0.938012i	$0.612665\pi$
$942$	−11382.0	−0.393679
$943$	−3285.00	−0.113440
$944$	−3360.00	−0.115846
$945$	1620.00	0.0557657
$946$	8466.00	0.290966
$947$	−35988.0	−1.23490	−0.617451	−	0.786609i	$-0.711835\pi$
$947$	−35988.0	−1.23490	−0.617451	+	0.786609i	$0.711835\pi$
$948$	−8664.00	−0.296829
$949$	65758.0	2.24931
$950$	−9976.00	−0.340699
$951$	−5490.00	−0.187198
$952$	−2720.00	−0.0926005
$953$	40788.0	1.38641	0.693207	−	0.720738i	$-0.256197\pi$
$953$	40788.0	1.38641	0.693207	+	0.720738i	$0.256197\pi$
$954$	6804.00	0.230909
$955$	−14130.0	−0.478781
$956$	−17760.0	−0.600836
$957$	−22950.0	−0.775202
$958$	6366.00	0.214693
$959$	−120.000	−0.00404067
$960$	576.000	0.0193649
$961$	54309.0	1.82300
$962$	6832.00	0.228974
$963$	12393.0	0.414703
$964$	9440.00	0.315396
$965$	12846.0	0.428526
$966$	−26280.0	−0.875306
$967$	−28897.0	−0.960977	−0.480489	−	0.877001i	$-0.659541\pi$
$967$	−28897.0	−0.960977	−0.480489	+	0.877001i	$0.659541\pi$
$968$	−10160.0	−0.337350
$969$	2193.00	0.0727031
$970$	−1608.00	−0.0532266
$971$	−198.000	−0.00654390	−0.00327195	−	0.999995i	$-0.501041\pi$
$971$	−198.000	−0.00654390	−0.00327195	+	0.999995i	$0.501041\pi$
$972$	−972.000	−0.0320750
$973$	20200.0	0.665552
$974$	18788.0	0.618076
$975$	−21228.0	−0.697272
$976$	−7168.00	−0.235084
$977$	33846.0	1.10832	0.554160	−	0.832410i	$-0.313039\pi$
$977$	33846.0	1.10832	0.554160	+	0.832410i	$0.313039\pi$
$978$	16428.0	0.537126
$979$	7344.00	0.239750
$980$	−684.000	−0.0222955
$981$	15084.0	0.490923
$982$	−28164.0	−0.915223
$983$	59835.0	1.94144	0.970722	−	0.240204i	$-0.0772145\pi$
$983$	59835.0	1.94144	0.970722	+	0.240204i	$0.0772145\pi$
$984$	360.000	0.0116630
$985$	8073.00	0.261144
$986$	5100.00	0.164723
$987$	−25560.0	−0.824300
$988$	10492.0	0.337849
$989$	−18177.0	−0.584424
$990$	−2754.00	−0.0884120
$991$	44336.0	1.42117	0.710585	−	0.703611i	$-0.248430\pi$
$991$	44336.0	1.42117	0.710585	+	0.703611i	$0.248430\pi$
$992$	−9280.00	−0.297016
$993$	34149.0	1.09133
$994$	−36000.0	−1.14874
$995$	−11724.0	−0.373544
$996$	936.000	0.0297774
$997$	14366.0	0.456345	0.228172	−	0.973621i	$-0.426725\pi$
$997$	14366.0	0.456345	0.228172	+	0.973621i	$0.426725\pi$
$998$	7844.00	0.248795
$999$	−1512.00	−0.0478854

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
102.4.a.a.1.1	✓	1	1.1	even	1	trivial
306.4.a.f.1.1		1	3.2	odd	2
816.4.a.g.1.1		1	4.3	odd	2
1734.4.a.e.1.1		1	17.16	even	2
2448.4.a.h.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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