LMFDB - Embedded newform 102.4.a.d.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:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	102.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +5.00000 q^{5} -6.00000 q^{6} +12.0000 q^{7} +8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$

\(q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +5.00000 q^{5} -6.00000 q^{6} +12.0000 q^{7} +8.00000 q^{8} +9.00000 q^{9} +10.0000 q^{10} +37.0000 q^{11} -12.0000 q^{12} +19.0000 q^{13} +24.0000 q^{14} -15.0000 q^{15} +16.0000 q^{16} +17.0000 q^{17} +18.0000 q^{18} +37.0000 q^{19} +20.0000 q^{20} -36.0000 q^{21} +74.0000 q^{22} -3.00000 q^{23} -24.0000 q^{24} -100.000 q^{25} +38.0000 q^{26} -27.0000 q^{27} +48.0000 q^{28} -86.0000 q^{29} -30.0000 q^{30} -142.000 q^{31} +32.0000 q^{32} -111.000 q^{33} +34.0000 q^{34} +60.0000 q^{35} +36.0000 q^{36} -296.000 q^{37} +74.0000 q^{38} -57.0000 q^{39} +40.0000 q^{40} -121.000 q^{41} -72.0000 q^{42} +3.00000 q^{43} +148.000 q^{44} +45.0000 q^{45} -6.00000 q^{46} +402.000 q^{47} -48.0000 q^{48} -199.000 q^{49} -200.000 q^{50} -51.0000 q^{51} +76.0000 q^{52} +174.000 q^{53} -54.0000 q^{54} +185.000 q^{55} +96.0000 q^{56} -111.000 q^{57} -172.000 q^{58} +270.000 q^{59} -60.0000 q^{60} -520.000 q^{61} -284.000 q^{62} +108.000 q^{63} +64.0000 q^{64} +95.0000 q^{65} -222.000 q^{66} -780.000 q^{67} +68.0000 q^{68} +9.00000 q^{69} +120.000 q^{70} +84.0000 q^{71} +72.0000 q^{72} -302.000 q^{73} -592.000 q^{74} +300.000 q^{75} +148.000 q^{76} +444.000 q^{77} -114.000 q^{78} +178.000 q^{79} +80.0000 q^{80} +81.0000 q^{81} -242.000 q^{82} +698.000 q^{83} -144.000 q^{84} +85.0000 q^{85} +6.00000 q^{86} +258.000 q^{87} +296.000 q^{88} +1512.00 q^{89} +90.0000 q^{90} +228.000 q^{91} -12.0000 q^{92} +426.000 q^{93} +804.000 q^{94} +185.000 q^{95} -96.0000 q^{96} -500.000 q^{97} -398.000 q^{98} +333.000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	2.00000	0.707107
$2$	2.00000	0.707107
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	4.00000	0.500000
$5$	5.00000	0.447214	0.223607	−	0.974679i	$-0.428217\pi$
$5$	5.00000	0.447214	0.223607	+	0.974679i	$0.428217\pi$
$6$	−6.00000	−0.408248
$7$	12.0000	0.647939	0.323970	−	0.946068i	$-0.394982\pi$
$7$	12.0000	0.647939	0.323970	+	0.946068i	$0.394982\pi$
$8$	8.00000	0.353553
$9$	9.00000	0.333333
$10$	10.0000	0.316228
$11$	37.0000	1.01417	0.507087	−	0.861895i	$-0.330722\pi$
$11$	37.0000	1.01417	0.507087	+	0.861895i	$0.330722\pi$
$12$	−12.0000	−0.288675
$13$	19.0000	0.405358	0.202679	−	0.979245i	$-0.435035\pi$
$13$	19.0000	0.405358	0.202679	+	0.979245i	$0.435035\pi$
$14$	24.0000	0.458162
$15$	−15.0000	−0.258199
$16$	16.0000	0.250000
$17$	17.0000	0.242536
$17$	17.0000	0.242536
$18$	18.0000	0.235702
$19$	37.0000	0.446757	0.223378	−	0.974732i	$-0.428291\pi$
$19$	37.0000	0.446757	0.223378	+	0.974732i	$0.428291\pi$
$20$	20.0000	0.223607
$21$	−36.0000	−0.374088
$22$	74.0000	0.717130
$23$	−3.00000	−0.0271975	−0.0135988	−	0.999908i	$-0.504329\pi$
$23$	−3.00000	−0.0271975	−0.0135988	+	0.999908i	$0.504329\pi$
$24$	−24.0000	−0.204124
$25$	−100.000	−0.800000
$26$	38.0000	0.286631
$27$	−27.0000	−0.192450
$28$	48.0000	0.323970
$29$	−86.0000	−0.550683	−0.275341	−	0.961347i	$-0.588791\pi$
$29$	−86.0000	−0.550683	−0.275341	+	0.961347i	$0.588791\pi$
$30$	−30.0000	−0.182574
$31$	−142.000	−0.822708	−0.411354	−	0.911476i	$-0.634944\pi$
$31$	−142.000	−0.822708	−0.411354	+	0.911476i	$0.634944\pi$
$32$	32.0000	0.176777
$33$	−111.000	−0.585534
$34$	34.0000	0.171499
$35$	60.0000	0.289767
$36$	36.0000	0.166667
$37$	−296.000	−1.31519	−0.657596	−	0.753371i	$-0.728427\pi$
$37$	−296.000	−1.31519	−0.657596	+	0.753371i	$0.728427\pi$
$38$	74.0000	0.315905
$39$	−57.0000	−0.234033
$40$	40.0000	0.158114
$41$	−121.000	−0.460903	−0.230452	−	0.973084i	$-0.574020\pi$
$41$	−121.000	−0.460903	−0.230452	+	0.973084i	$0.574020\pi$
$42$	−72.0000	−0.264520
$43$	3.00000	0.0106394	0.00531972	−	0.999986i	$-0.498307\pi$
$43$	3.00000	0.0106394	0.00531972	+	0.999986i	$0.498307\pi$
$44$	148.000	0.507087
$45$	45.0000	0.149071
$46$	−6.00000	−0.0192316
$47$	402.000	1.24761	0.623806	−	0.781580i	$-0.285586\pi$
$47$	402.000	1.24761	0.623806	+	0.781580i	$0.285586\pi$
$48$	−48.0000	−0.144338
$49$	−199.000	−0.580175
$50$	−200.000	−0.565685
$51$	−51.0000	−0.140028
$52$	76.0000	0.202679
$53$	174.000	0.450957	0.225479	−	0.974248i	$-0.427605\pi$
$53$	174.000	0.450957	0.225479	+	0.974248i	$0.427605\pi$
$54$	−54.0000	−0.136083
$55$	185.000	0.453553
$56$	96.0000	0.229081
$57$	−111.000	−0.257935
$58$	−172.000	−0.389391
$59$	270.000	0.595780	0.297890	−	0.954600i	$-0.403717\pi$
$59$	270.000	0.595780	0.297890	+	0.954600i	$0.403717\pi$
$60$	−60.0000	−0.129099
$61$	−520.000	−1.09146	−0.545731	−	0.837960i	$-0.683748\pi$
$61$	−520.000	−1.09146	−0.545731	+	0.837960i	$0.683748\pi$
$62$	−284.000	−0.581743
$63$	108.000	0.215980
$64$	64.0000	0.125000
$65$	95.0000	0.181282
$66$	−222.000	−0.414035
$67$	−780.000	−1.42227	−0.711136	−	0.703055i	$-0.751819\pi$
$67$	−780.000	−1.42227	−0.711136	+	0.703055i	$0.751819\pi$
$68$	68.0000	0.121268
$69$	9.00000	0.0157025
$70$	120.000	0.204896
$71$	84.0000	0.140408	0.0702040	−	0.997533i	$-0.477635\pi$
$71$	84.0000	0.140408	0.0702040	+	0.997533i	$0.477635\pi$
$72$	72.0000	0.117851
$73$	−302.000	−0.484198	−0.242099	−	0.970252i	$-0.577836\pi$
$73$	−302.000	−0.484198	−0.242099	+	0.970252i	$0.577836\pi$
$74$	−592.000	−0.929981
$75$	300.000	0.461880
$76$	148.000	0.223378
$77$	444.000	0.657123
$78$	−114.000	−0.165487
$79$	178.000	0.253501	0.126750	−	0.991935i	$-0.459545\pi$
$79$	178.000	0.253501	0.126750	+	0.991935i	$0.459545\pi$
$80$	80.0000	0.111803
$81$	81.0000	0.111111
$82$	−242.000	−0.325908
$83$	698.000	0.923078	0.461539	−	0.887120i	$-0.347297\pi$
$83$	698.000	0.923078	0.461539	+	0.887120i	$0.347297\pi$
$84$	−144.000	−0.187044
$85$	85.0000	0.108465
$86$	6.00000	0.00752322
$87$	258.000	0.317937
$88$	296.000	0.358565
$89$	1512.00	1.80081	0.900403	−	0.435057i	$-0.143272\pi$
$89$	1512.00	1.80081	0.900403	+	0.435057i	$0.143272\pi$
$90$	90.0000	0.105409
$91$	228.000	0.262647
$92$	−12.0000	−0.0135988
$93$	426.000	0.474991
$94$	804.000	0.882194
$95$	185.000	0.199796
$96$	−96.0000	−0.102062
$97$	−500.000	−0.523374	−0.261687	−	0.965153i	$-0.584279\pi$
$97$	−500.000	−0.523374	−0.261687	+	0.965153i	$0.584279\pi$
$98$	−398.000	−0.410246
$99$	333.000	0.338058
$100$	−400.000	−0.400000
$101$	−1040.00	−1.02459	−0.512296	−	0.858809i	$-0.671205\pi$
$101$	−1040.00	−1.02459	−0.512296	+	0.858809i	$0.671205\pi$
$102$	−102.000	−0.0990148
$103$	1891.00	1.80899	0.904494	−	0.426486i	$-0.140249\pi$
$103$	1891.00	1.80899	0.904494	+	0.426486i	$0.140249\pi$
$104$	152.000	0.143316
$105$	−180.000	−0.167297
$106$	348.000	0.318875
$107$	−599.000	−0.541192	−0.270596	−	0.962693i	$-0.587221\pi$
$107$	−599.000	−0.541192	−0.270596	+	0.962693i	$0.587221\pi$
$108$	−108.000	−0.0962250
$109$	812.000	0.713537	0.356768	−	0.934193i	$-0.383879\pi$
$109$	812.000	0.713537	0.356768	+	0.934193i	$0.383879\pi$
$110$	370.000	0.320710
$111$	888.000	0.759326
$112$	192.000	0.161985
$113$	−635.000	−0.528635	−0.264318	−	0.964436i	$-0.585147\pi$
$113$	−635.000	−0.528635	−0.264318	+	0.964436i	$0.585147\pi$
$114$	−222.000	−0.182388
$115$	−15.0000	−0.0121631
$116$	−344.000	−0.275341
$117$	171.000	0.135119
$118$	540.000	0.421280
$119$	204.000	0.157148
$120$	−120.000	−0.0912871
$121$	38.0000	0.0285500
$122$	−1040.00	−0.771780
$123$	363.000	0.266103
$124$	−568.000	−0.411354
$125$	−1125.00	−0.804984
$126$	216.000	0.152721
$127$	−27.0000	−0.0188651	−0.00943253	−	0.999956i	$-0.503003\pi$
$127$	−27.0000	−0.0188651	−0.00943253	+	0.999956i	$0.503003\pi$
$128$	128.000	0.0883883
$129$	−9.00000	−0.00614268
$130$	190.000	0.128185
$131$	−2037.00	−1.35858	−0.679288	−	0.733872i	$-0.737711\pi$
$131$	−2037.00	−1.35858	−0.679288	+	0.733872i	$0.737711\pi$
$132$	−444.000	−0.292767
$133$	444.000	0.289471
$134$	−1560.00	−1.00570
$135$	−135.000	−0.0860663
$136$	136.000	0.0857493
$137$	−1294.00	−0.806963	−0.403481	−	0.914988i	$-0.632200\pi$
$137$	−1294.00	−0.806963	−0.403481	+	0.914988i	$0.632200\pi$
$138$	18.0000	0.0111033
$139$	−646.000	−0.394194	−0.197097	−	0.980384i	$-0.563151\pi$
$139$	−646.000	−0.394194	−0.197097	+	0.980384i	$0.563151\pi$
$140$	240.000	0.144884
$141$	−1206.00	−0.720309
$142$	168.000	0.0992834
$143$	703.000	0.411104
$144$	144.000	0.0833333
$145$	−430.000	−0.246273
$146$	−604.000	−0.342379
$147$	597.000	0.334964
$148$	−1184.00	−0.657596
$149$	−3386.00	−1.86169	−0.930845	−	0.365413i	$-0.880928\pi$
$149$	−3386.00	−1.86169	−0.930845	+	0.365413i	$0.880928\pi$
$150$	600.000	0.326599
$151$	−984.000	−0.530310	−0.265155	−	0.964206i	$-0.585423\pi$
$151$	−984.000	−0.530310	−0.265155	+	0.964206i	$0.585423\pi$
$152$	296.000	0.157952
$153$	153.000	0.0808452
$154$	888.000	0.464656
$155$	−710.000	−0.367926
$156$	−228.000	−0.117017
$157$	1783.00	0.906362	0.453181	−	0.891418i	$-0.350289\pi$
$157$	1783.00	0.906362	0.453181	+	0.891418i	$0.350289\pi$
$158$	356.000	0.179252
$159$	−522.000	−0.260360
$160$	160.000	0.0790569
$161$	−36.0000	−0.0176223
$162$	162.000	0.0785674
$163$	−1214.00	−0.583361	−0.291680	−	0.956516i	$-0.594214\pi$
$163$	−1214.00	−0.583361	−0.291680	+	0.956516i	$0.594214\pi$
$164$	−484.000	−0.230452
$165$	−555.000	−0.261859
$166$	1396.00	0.652715
$167$	1073.00	0.497193	0.248597	−	0.968607i	$-0.420031\pi$
$167$	1073.00	0.497193	0.248597	+	0.968607i	$0.420031\pi$
$168$	−288.000	−0.132260
$169$	−1836.00	−0.835685
$170$	170.000	0.0766965
$171$	333.000	0.148919
$172$	12.0000	0.00531972
$173$	801.000	0.352017	0.176008	−	0.984389i	$-0.443681\pi$
$173$	801.000	0.352017	0.176008	+	0.984389i	$0.443681\pi$
$174$	516.000	0.224815
$175$	−1200.00	−0.518351
$176$	592.000	0.253544
$177$	−810.000	−0.343974
$178$	3024.00	1.27336
$179$	2526.00	1.05476	0.527380	−	0.849629i	$-0.323174\pi$
$179$	2526.00	1.05476	0.527380	+	0.849629i	$0.323174\pi$
$180$	180.000	0.0745356
$181$	−1654.00	−0.679231	−0.339616	−	0.940564i	$-0.610297\pi$
$181$	−1654.00	−0.679231	−0.339616	+	0.940564i	$0.610297\pi$
$182$	456.000	0.185720
$183$	1560.00	0.630156
$184$	−24.0000	−0.00961578
$185$	−1480.00	−0.588172
$186$	852.000	0.335869
$187$	629.000	0.245973
$188$	1608.00	0.623806
$189$	−324.000	−0.124696
$190$	370.000	0.141277
$191$	4030.00	1.52670	0.763352	−	0.645982i	$-0.223552\pi$
$191$	4030.00	1.52670	0.763352	+	0.645982i	$0.223552\pi$
$192$	−192.000	−0.0721688
$193$	5238.00	1.95357	0.976786	−	0.214216i	$-0.0687197\pi$
$193$	5238.00	1.95357	0.976786	+	0.214216i	$0.0687197\pi$
$194$	−1000.00	−0.370082
$195$	−285.000	−0.104663
$196$	−796.000	−0.290087
$197$	−4619.00	−1.67051	−0.835254	−	0.549864i	$-0.814679\pi$
$197$	−4619.00	−1.67051	−0.835254	+	0.549864i	$0.814679\pi$
$198$	666.000	0.239043
$199$	−4740.00	−1.68849	−0.844245	−	0.535957i	$-0.819951\pi$
$199$	−4740.00	−1.68849	−0.844245	+	0.535957i	$0.819951\pi$
$200$	−800.000	−0.282843
$201$	2340.00	0.821149
$202$	−2080.00	−0.724496
$203$	−1032.00	−0.356809
$204$	−204.000	−0.0700140
$205$	−605.000	−0.206122
$206$	3782.00	1.27915
$207$	−27.0000	−0.00906584
$208$	304.000	0.101339
$209$	1369.00	0.453090
$210$	−360.000	−0.118297
$211$	1754.00	0.572276	0.286138	−	0.958188i	$-0.407628\pi$
$211$	1754.00	0.572276	0.286138	+	0.958188i	$0.407628\pi$
$212$	696.000	0.225479
$213$	−252.000	−0.0810646
$214$	−1198.00	−0.382680
$215$	15.0000	0.00475810
$216$	−216.000	−0.0680414
$217$	−1704.00	−0.533065
$218$	1624.00	0.504547
$219$	906.000	0.279552
$220$	740.000	0.226776
$221$	323.000	0.0983137
$222$	1776.00	0.536925
$223$	2569.00	0.771448	0.385724	−	0.922614i	$-0.373952\pi$
$223$	2569.00	0.771448	0.385724	+	0.922614i	$0.373952\pi$
$224$	384.000	0.114541
$225$	−900.000	−0.266667
$226$	−1270.00	−0.373802
$227$	5459.00	1.59615	0.798076	−	0.602557i	$-0.205851\pi$
$227$	5459.00	1.59615	0.798076	+	0.602557i	$0.205851\pi$
$228$	−444.000	−0.128968
$229$	3838.00	1.10752	0.553760	−	0.832676i	$-0.313192\pi$
$229$	3838.00	1.10752	0.553760	+	0.832676i	$0.313192\pi$
$230$	−30.0000	−0.00860061
$231$	−1332.00	−0.379390
$232$	−688.000	−0.194696
$233$	631.000	0.177417	0.0887086	−	0.996058i	$-0.471726\pi$
$233$	631.000	0.177417	0.0887086	+	0.996058i	$0.471726\pi$
$234$	342.000	0.0955438
$235$	2010.00	0.557949
$236$	1080.00	0.297890
$237$	−534.000	−0.146359
$238$	408.000	0.111121
$239$	−3216.00	−0.870401	−0.435200	−	0.900334i	$-0.643322\pi$
$239$	−3216.00	−0.870401	−0.435200	+	0.900334i	$0.643322\pi$
$240$	−240.000	−0.0645497
$241$	2336.00	0.624378	0.312189	−	0.950020i	$-0.398938\pi$
$241$	2336.00	0.624378	0.312189	+	0.950020i	$0.398938\pi$
$242$	76.0000	0.0201879
$243$	−243.000	−0.0641500
$244$	−2080.00	−0.545731
$245$	−995.000	−0.259462
$246$	726.000	0.188163
$247$	703.000	0.181096
$248$	−1136.00	−0.290871
$249$	−2094.00	−0.532939
$250$	−2250.00	−0.569210
$251$	−2284.00	−0.574362	−0.287181	−	0.957876i	$-0.592718\pi$
$251$	−2284.00	−0.574362	−0.287181	+	0.957876i	$0.592718\pi$
$252$	432.000	0.107990
$253$	−111.000	−0.0275830
$254$	−54.0000	−0.0133396
$255$	−255.000	−0.0626224
$256$	256.000	0.0625000
$257$	2236.00	0.542715	0.271358	−	0.962479i	$-0.412527\pi$
$257$	2236.00	0.542715	0.271358	+	0.962479i	$0.412527\pi$
$258$	−18.0000	−0.00434353
$259$	−3552.00	−0.852164
$260$	380.000	0.0906408
$261$	−774.000	−0.183561
$262$	−4074.00	−0.960659
$263$	5168.00	1.21168	0.605841	−	0.795586i	$-0.292837\pi$
$263$	5168.00	1.21168	0.605841	+	0.795586i	$0.292837\pi$
$264$	−888.000	−0.207018
$265$	870.000	0.201674
$266$	888.000	0.204687
$267$	−4536.00	−1.03970
$268$	−3120.00	−0.711136
$269$	4551.00	1.03152	0.515761	−	0.856733i	$-0.327509\pi$
$269$	4551.00	1.03152	0.515761	+	0.856733i	$0.327509\pi$
$270$	−270.000	−0.0608581
$271$	−6971.00	−1.56258	−0.781288	−	0.624171i	$-0.785437\pi$
$271$	−6971.00	−1.56258	−0.781288	+	0.624171i	$0.785437\pi$
$272$	272.000	0.0606339
$273$	−684.000	−0.151639
$274$	−2588.00	−0.570609
$275$	−3700.00	−0.811340
$276$	36.0000	0.00785125
$277$	−462.000	−0.100213	−0.0501063	−	0.998744i	$-0.515956\pi$
$277$	−462.000	−0.100213	−0.0501063	+	0.998744i	$0.515956\pi$
$278$	−1292.00	−0.278737
$279$	−1278.00	−0.274236
$280$	480.000	0.102448
$281$	7068.00	1.50050	0.750252	−	0.661152i	$-0.229932\pi$
$281$	7068.00	1.50050	0.750252	+	0.661152i	$0.229932\pi$
$282$	−2412.00	−0.509335
$283$	786.000	0.165098	0.0825492	−	0.996587i	$-0.473694\pi$
$283$	786.000	0.165098	0.0825492	+	0.996587i	$0.473694\pi$
$284$	336.000	0.0702040
$285$	−555.000	−0.115352
$286$	1406.00	0.290694
$287$	−1452.00	−0.298637
$288$	288.000	0.0589256
$289$	289.000	0.0588235
$290$	−860.000	−0.174141
$291$	1500.00	0.302170
$292$	−1208.00	−0.242099
$293$	−172.000	−0.0342947	−0.0171474	−	0.999853i	$-0.505458\pi$
$293$	−172.000	−0.0342947	−0.0171474	+	0.999853i	$0.505458\pi$
$294$	1194.00	0.236855
$295$	1350.00	0.266441
$296$	−2368.00	−0.464991
$297$	−999.000	−0.195178
$298$	−6772.00	−1.31641
$299$	−57.0000	−0.0110247
$300$	1200.00	0.230940
$301$	36.0000	0.00689371
$302$	−1968.00	−0.374986
$303$	3120.00	0.591549
$304$	592.000	0.111689
$305$	−2600.00	−0.488117
$306$	306.000	0.0571662
$307$	−1404.00	−0.261011	−0.130506	−	0.991448i	$-0.541660\pi$
$307$	−1404.00	−0.261011	−0.130506	+	0.991448i	$0.541660\pi$
$308$	1776.00	0.328562
$309$	−5673.00	−1.04442
$310$	−1420.00	−0.260163
$311$	5400.00	0.984585	0.492292	−	0.870430i	$-0.336159\pi$
$311$	5400.00	0.984585	0.492292	+	0.870430i	$0.336159\pi$
$312$	−456.000	−0.0827433
$313$	10660.0	1.92504	0.962522	−	0.271203i	$-0.0874216\pi$
$313$	10660.0	1.92504	0.962522	+	0.271203i	$0.0874216\pi$
$314$	3566.00	0.640895
$315$	540.000	0.0965891
$316$	712.000	0.126750
$317$	−7002.00	−1.24060	−0.620302	−	0.784363i	$-0.712990\pi$
$317$	−7002.00	−1.24060	−0.620302	+	0.784363i	$0.712990\pi$
$318$	−1044.00	−0.184103
$319$	−3182.00	−0.558488
$320$	320.000	0.0559017
$321$	1797.00	0.312457
$322$	−72.0000	−0.0124609
$323$	629.000	0.108354
$324$	324.000	0.0555556
$325$	−1900.00	−0.324286
$326$	−2428.00	−0.412498
$327$	−2436.00	−0.411961
$328$	−968.000	−0.162954
$329$	4824.00	0.808376
$330$	−1110.00	−0.185162
$331$	6025.00	1.00050	0.500248	−	0.865882i	$-0.333242\pi$
$331$	6025.00	1.00050	0.500248	+	0.865882i	$0.333242\pi$
$332$	2792.00	0.461539
$333$	−2664.00	−0.438397
$334$	2146.00	0.351569
$335$	−3900.00	−0.636059
$336$	−576.000	−0.0935220
$337$	10522.0	1.70080	0.850400	−	0.526136i	$-0.176360\pi$
$337$	10522.0	1.70080	0.850400	+	0.526136i	$0.176360\pi$
$338$	−3672.00	−0.590919
$339$	1905.00	0.305208
$340$	340.000	0.0542326
$341$	−5254.00	−0.834370
$342$	666.000	0.105302
$343$	−6504.00	−1.02386
$344$	24.0000	0.00376161
$345$	45.0000	0.00702237
$346$	1602.00	0.248913
$347$	−4612.00	−0.713502	−0.356751	−	0.934200i	$-0.616116\pi$
$347$	−4612.00	−0.713502	−0.356751	+	0.934200i	$0.616116\pi$
$348$	1032.00	0.158968
$349$	3089.00	0.473783	0.236892	−	0.971536i	$-0.423871\pi$
$349$	3089.00	0.473783	0.236892	+	0.971536i	$0.423871\pi$
$350$	−2400.00	−0.366530
$351$	−513.000	−0.0780112
$352$	1184.00	0.179282
$353$	3282.00	0.494853	0.247427	−	0.968907i	$-0.420415\pi$
$353$	3282.00	0.494853	0.247427	+	0.968907i	$0.420415\pi$
$354$	−1620.00	−0.243226
$355$	420.000	0.0627924
$356$	6048.00	0.900403
$357$	−612.000	−0.0907296
$358$	5052.00	0.745828
$359$	−6876.00	−1.01087	−0.505434	−	0.862865i	$-0.668667\pi$
$359$	−6876.00	−1.01087	−0.505434	+	0.862865i	$0.668667\pi$
$360$	360.000	0.0527046
$361$	−5490.00	−0.800408
$362$	−3308.00	−0.480289
$363$	−114.000	−0.0164833
$364$	912.000	0.131324
$365$	−1510.00	−0.216540
$366$	3120.00	0.445587
$367$	7956.00	1.13161	0.565804	−	0.824540i	$-0.308566\pi$
$367$	7956.00	1.13161	0.565804	+	0.824540i	$0.308566\pi$
$368$	−48.0000	−0.00679938
$369$	−1089.00	−0.153634
$370$	−2960.00	−0.415900
$371$	2088.00	0.292193
$372$	1704.00	0.237495
$373$	−1422.00	−0.197395	−0.0986975	−	0.995117i	$-0.531468\pi$
$373$	−1422.00	−0.197395	−0.0986975	+	0.995117i	$0.531468\pi$
$374$	1258.00	0.173929
$375$	3375.00	0.464758
$376$	3216.00	0.441097
$377$	−1634.00	−0.223224
$378$	−648.000	−0.0881733
$379$	−10504.0	−1.42363	−0.711813	−	0.702369i	$-0.752126\pi$
$379$	−10504.0	−1.42363	−0.711813	+	0.702369i	$0.752126\pi$
$380$	740.000	0.0998979
$381$	81.0000	0.0108917
$382$	8060.00	1.07954
$383$	5200.00	0.693753	0.346877	−	0.937911i	$-0.387242\pi$
$383$	5200.00	0.693753	0.346877	+	0.937911i	$0.387242\pi$
$384$	−384.000	−0.0510310
$385$	2220.00	0.293874
$386$	10476.0	1.38138
$387$	27.0000	0.00354648
$388$	−2000.00	−0.261687
$389$	4376.00	0.570365	0.285183	−	0.958473i	$-0.407946\pi$
$389$	4376.00	0.570365	0.285183	+	0.958473i	$0.407946\pi$
$390$	−570.000	−0.0740079
$391$	−51.0000	−0.00659637
$392$	−1592.00	−0.205123
$393$	6111.00	0.784374
$394$	−9238.00	−1.18123
$395$	890.000	0.113369
$396$	1332.00	0.169029
$397$	4792.00	0.605802	0.302901	−	0.953022i	$-0.402045\pi$
$397$	4792.00	0.605802	0.302901	+	0.953022i	$0.402045\pi$
$398$	−9480.00	−1.19394
$399$	−1332.00	−0.167126
$400$	−1600.00	−0.200000
$401$	5963.00	0.742589	0.371294	−	0.928515i	$-0.378914\pi$
$401$	5963.00	0.742589	0.371294	+	0.928515i	$0.378914\pi$
$402$	4680.00	0.580640
$403$	−2698.00	−0.333491
$404$	−4160.00	−0.512296
$405$	405.000	0.0496904
$406$	−2064.00	−0.252302
$407$	−10952.0	−1.33383
$408$	−408.000	−0.0495074
$409$	10781.0	1.30339	0.651695	−	0.758482i	$-0.274058\pi$
$409$	10781.0	1.30339	0.651695	+	0.758482i	$0.274058\pi$
$410$	−1210.00	−0.145750
$411$	3882.00	0.465900
$412$	7564.00	0.904494
$413$	3240.00	0.386029
$414$	−54.0000	−0.00641052
$415$	3490.00	0.412813
$416$	608.000	0.0716578
$417$	1938.00	0.227588
$418$	2738.00	0.320383
$419$	−7444.00	−0.867931	−0.433966	−	0.900929i	$-0.642886\pi$
$419$	−7444.00	−0.867931	−0.433966	+	0.900929i	$0.642886\pi$
$420$	−720.000	−0.0836486
$421$	−11069.0	−1.28140	−0.640701	−	0.767791i	$-0.721356\pi$
$421$	−11069.0	−1.28140	−0.640701	+	0.767791i	$0.721356\pi$
$422$	3508.00	0.404661
$423$	3618.00	0.415870
$424$	1392.00	0.159437
$425$	−1700.00	−0.194029
$426$	−504.000	−0.0573213
$427$	−6240.00	−0.707201
$428$	−2396.00	−0.270596
$429$	−2109.00	−0.237351
$430$	30.0000	0.00336448
$431$	5552.00	0.620488	0.310244	−	0.950657i	$-0.399589\pi$
$431$	5552.00	0.620488	0.310244	+	0.950657i	$0.399589\pi$
$432$	−432.000	−0.0481125
$433$	16423.0	1.82272	0.911361	−	0.411607i	$-0.135032\pi$
$433$	16423.0	1.82272	0.911361	+	0.411607i	$0.135032\pi$
$434$	−3408.00	−0.376934
$435$	1290.00	0.142186
$436$	3248.00	0.356768
$437$	−111.000	−0.0121507
$438$	1812.00	0.197673
$439$	−17228.0	−1.87300	−0.936501	−	0.350666i	$-0.885955\pi$
$439$	−17228.0	−1.87300	−0.936501	+	0.350666i	$0.885955\pi$
$440$	1480.00	0.160355
$441$	−1791.00	−0.193392
$442$	646.000	0.0695183
$443$	−3698.00	−0.396608	−0.198304	−	0.980141i	$-0.563543\pi$
$443$	−3698.00	−0.396608	−0.198304	+	0.980141i	$0.563543\pi$
$444$	3552.00	0.379663
$445$	7560.00	0.805345
$446$	5138.00	0.545496
$447$	10158.0	1.07485
$448$	768.000	0.0809924
$449$	−15338.0	−1.61213	−0.806063	−	0.591829i	$-0.798406\pi$
$449$	−15338.0	−1.61213	−0.806063	+	0.591829i	$0.798406\pi$
$450$	−1800.00	−0.188562
$451$	−4477.00	−0.467436
$452$	−2540.00	−0.264318
$453$	2952.00	0.306175
$454$	10918.0	1.12865
$455$	1140.00	0.117459
$456$	−888.000	−0.0911939
$457$	12397.0	1.26894	0.634472	−	0.772946i	$-0.281218\pi$
$457$	12397.0	1.26894	0.634472	+	0.772946i	$0.281218\pi$
$458$	7676.00	0.783135
$459$	−459.000	−0.0466760
$460$	−60.0000	−0.00608155
$461$	354.000	0.0357645	0.0178822	−	0.999840i	$-0.494308\pi$
$461$	354.000	0.0357645	0.0178822	+	0.999840i	$0.494308\pi$
$462$	−2664.00	−0.268269
$463$	−13120.0	−1.31693	−0.658464	−	0.752612i	$-0.728794\pi$
$463$	−13120.0	−1.31693	−0.658464	+	0.752612i	$0.728794\pi$
$464$	−1376.00	−0.137671
$465$	2130.00	0.212422
$466$	1262.00	0.125453
$467$	−7866.00	−0.779433	−0.389716	−	0.920935i	$-0.627427\pi$
$467$	−7866.00	−0.779433	−0.389716	+	0.920935i	$0.627427\pi$
$468$	684.000	0.0675596
$469$	−9360.00	−0.921545
$470$	4020.00	0.394529
$471$	−5349.00	−0.523289
$472$	2160.00	0.210640
$473$	111.000	0.0107902
$474$	−1068.00	−0.103491
$475$	−3700.00	−0.357406
$476$	816.000	0.0785742
$477$	1566.00	0.150319
$478$	−6432.00	−0.615466
$479$	681.000	0.0649597	0.0324798	−	0.999472i	$-0.489660\pi$
$479$	681.000	0.0649597	0.0324798	+	0.999472i	$0.489660\pi$
$480$	−480.000	−0.0456435
$481$	−5624.00	−0.533123
$482$	4672.00	0.441502
$483$	108.000	0.0101743
$484$	152.000	0.0142750
$485$	−2500.00	−0.234060
$486$	−486.000	−0.0453609
$487$	−12226.0	−1.13760	−0.568802	−	0.822475i	$-0.692593\pi$
$487$	−12226.0	−1.13760	−0.568802	+	0.822475i	$0.692593\pi$
$488$	−4160.00	−0.385890
$489$	3642.00	0.336804
$490$	−1990.00	−0.183467
$491$	−5046.00	−0.463794	−0.231897	−	0.972740i	$-0.574493\pi$
$491$	−5046.00	−0.463794	−0.231897	+	0.972740i	$0.574493\pi$
$492$	1452.00	0.133051
$493$	−1462.00	−0.133560
$494$	1406.00	0.128055
$495$	1665.00	0.151184
$496$	−2272.00	−0.205677
$497$	1008.00	0.0909758
$498$	−4188.00	−0.376845
$499$	406.000	0.0364230	0.0182115	−	0.999834i	$-0.494203\pi$
$499$	406.000	0.0364230	0.0182115	+	0.999834i	$0.494203\pi$
$500$	−4500.00	−0.402492
$501$	−3219.00	−0.287055
$502$	−4568.00	−0.406135
$503$	12243.0	1.08527	0.542633	−	0.839970i	$-0.317428\pi$
$503$	12243.0	1.08527	0.542633	+	0.839970i	$0.317428\pi$
$504$	864.000	0.0763604
$505$	−5200.00	−0.458212
$506$	−222.000	−0.0195042
$507$	5508.00	0.482483
$508$	−108.000	−0.00943253
$509$	−15852.0	−1.38041	−0.690204	−	0.723615i	$-0.742479\pi$
$509$	−15852.0	−1.38041	−0.690204	+	0.723615i	$0.742479\pi$
$510$	−510.000	−0.0442807
$511$	−3624.00	−0.313731
$512$	512.000	0.0441942
$513$	−999.000	−0.0859784
$514$	4472.00	0.383758
$515$	9455.00	0.809004
$516$	−36.0000	−0.00307134
$517$	14874.0	1.26530
$518$	−7104.00	−0.602571
$519$	−2403.00	−0.203237
$520$	760.000	0.0640927
$521$	−2633.00	−0.221408	−0.110704	−	0.993853i	$-0.535311\pi$
$521$	−2633.00	−0.221408	−0.110704	+	0.993853i	$0.535311\pi$
$522$	−1548.00	−0.129797
$523$	212.000	0.0177249	0.00886244	−	0.999961i	$-0.497179\pi$
$523$	212.000	0.0177249	0.00886244	+	0.999961i	$0.497179\pi$
$524$	−8148.00	−0.679288
$525$	3600.00	0.299270
$526$	10336.0	0.856789
$527$	−2414.00	−0.199536
$528$	−1776.00	−0.146383
$529$	−12158.0	−0.999260
$530$	1740.00	0.142605
$531$	2430.00	0.198593
$532$	1776.00	0.144736
$533$	−2299.00	−0.186831
$534$	−9072.00	−0.735176
$535$	−2995.00	−0.242028
$536$	−6240.00	−0.502849
$537$	−7578.00	−0.608966
$538$	9102.00	0.729396
$539$	−7363.00	−0.588399
$540$	−540.000	−0.0430331
$541$	−20380.0	−1.61960	−0.809801	−	0.586705i	$-0.800425\pi$
$541$	−20380.0	−1.61960	−0.809801	+	0.586705i	$0.800425\pi$
$542$	−13942.0	−1.10491
$543$	4962.00	0.392154
$544$	544.000	0.0428746
$545$	4060.00	0.319103
$546$	−1368.00	−0.107225
$547$	10224.0	0.799171	0.399586	−	0.916696i	$-0.369154\pi$
$547$	10224.0	0.799171	0.399586	+	0.916696i	$0.369154\pi$
$548$	−5176.00	−0.403481
$549$	−4680.00	−0.363821
$550$	−7400.00	−0.573704
$551$	−3182.00	−0.246021
$552$	72.0000	0.00555167
$553$	2136.00	0.164253
$554$	−924.000	−0.0708610
$555$	4440.00	0.339581
$556$	−2584.00	−0.197097
$557$	4822.00	0.366813	0.183406	−	0.983037i	$-0.441288\pi$
$557$	4822.00	0.366813	0.183406	+	0.983037i	$0.441288\pi$
$558$	−2556.00	−0.193914
$559$	57.0000	0.00431278
$560$	960.000	0.0724418
$561$	−1887.00	−0.142013
$562$	14136.0	1.06102
$563$	14310.0	1.07122	0.535608	−	0.844467i	$-0.320082\pi$
$563$	14310.0	1.07122	0.535608	+	0.844467i	$0.320082\pi$
$564$	−4824.00	−0.360154
$565$	−3175.00	−0.236413
$566$	1572.00	0.116742
$567$	972.000	0.0719932
$568$	672.000	0.0496417
$569$	−8676.00	−0.639221	−0.319611	−	0.947549i	$-0.603552\pi$
$569$	−8676.00	−0.639221	−0.319611	+	0.947549i	$0.603552\pi$
$570$	−1110.00	−0.0815663
$571$	24484.0	1.79444	0.897218	−	0.441587i	$-0.145584\pi$
$571$	24484.0	1.79444	0.897218	+	0.441587i	$0.145584\pi$
$572$	2812.00	0.205552
$573$	−12090.0	−0.881443
$574$	−2904.00	−0.211168
$575$	300.000	0.0217580
$576$	576.000	0.0416667
$577$	4961.00	0.357936	0.178968	−	0.983855i	$-0.442724\pi$
$577$	4961.00	0.357936	0.178968	+	0.983855i	$0.442724\pi$
$578$	578.000	0.0415945
$579$	−15714.0	−1.12790
$580$	−1720.00	−0.123136
$581$	8376.00	0.598098
$582$	3000.00	0.213667
$583$	6438.00	0.457349
$584$	−2416.00	−0.171190
$585$	855.000	0.0604272
$586$	−344.000	−0.0242500
$587$	22580.0	1.58769	0.793847	−	0.608118i	$-0.208075\pi$
$587$	22580.0	1.58769	0.793847	+	0.608118i	$0.208075\pi$
$588$	2388.00	0.167482
$589$	−5254.00	−0.367551
$590$	2700.00	0.188402
$591$	13857.0	0.964468
$592$	−4736.00	−0.328798
$593$	13586.0	0.940827	0.470413	−	0.882446i	$-0.344105\pi$
$593$	13586.0	0.940827	0.470413	+	0.882446i	$0.344105\pi$
$594$	−1998.00	−0.138012
$595$	1020.00	0.0702789
$596$	−13544.0	−0.930845
$597$	14220.0	0.974851
$598$	−114.000	−0.00779566
$599$	22910.0	1.56273	0.781367	−	0.624072i	$-0.214523\pi$
$599$	22910.0	1.56273	0.781367	+	0.624072i	$0.214523\pi$
$600$	2400.00	0.163299
$601$	666.000	0.0452025	0.0226013	−	0.999745i	$-0.492805\pi$
$601$	666.000	0.0452025	0.0226013	+	0.999745i	$0.492805\pi$
$602$	72.0000	0.00487459
$603$	−7020.00	−0.474090
$604$	−3936.00	−0.265155
$605$	190.000	0.0127679
$606$	6240.00	0.418288
$607$	12798.0	0.855774	0.427887	−	0.903832i	$-0.359258\pi$
$607$	12798.0	0.855774	0.427887	+	0.903832i	$0.359258\pi$
$608$	1184.00	0.0789762
$609$	3096.00	0.206004
$610$	−5200.00	−0.345151
$611$	7638.00	0.505729
$612$	612.000	0.0404226
$613$	5087.00	0.335175	0.167587	−	0.985857i	$-0.446402\pi$
$613$	5087.00	0.335175	0.167587	+	0.985857i	$0.446402\pi$
$614$	−2808.00	−0.184563
$615$	1815.00	0.119005
$616$	3552.00	0.232328
$617$	10878.0	0.709776	0.354888	−	0.934909i	$-0.384519\pi$
$617$	10878.0	0.709776	0.354888	+	0.934909i	$0.384519\pi$
$618$	−11346.0	−0.738516
$619$	26890.0	1.74604	0.873021	−	0.487682i	$-0.162157\pi$
$619$	26890.0	1.74604	0.873021	+	0.487682i	$0.162157\pi$
$620$	−2840.00	−0.183963
$621$	81.0000	0.00523417
$622$	10800.0	0.696207
$623$	18144.0	1.16681
$624$	−912.000	−0.0585084
$625$	6875.00	0.440000
$626$	21320.0	1.36121
$627$	−4107.00	−0.261591
$628$	7132.00	0.453181
$629$	−5032.00	−0.318981
$630$	1080.00	0.0682988
$631$	−24867.0	−1.56884	−0.784421	−	0.620228i	$-0.787040\pi$
$631$	−24867.0	−1.56884	−0.784421	+	0.620228i	$0.787040\pi$
$632$	1424.00	0.0896261
$633$	−5262.00	−0.330404
$634$	−14004.0	−0.877240
$635$	−135.000	−0.00843671
$636$	−2088.00	−0.130180
$637$	−3781.00	−0.235178
$638$	−6364.00	−0.394911
$639$	756.000	0.0468027
$640$	640.000	0.0395285
$641$	23485.0	1.44712	0.723558	−	0.690263i	$-0.242505\pi$
$641$	23485.0	1.44712	0.723558	+	0.690263i	$0.242505\pi$
$642$	3594.00	0.220941
$643$	13148.0	0.806386	0.403193	−	0.915115i	$-0.367900\pi$
$643$	13148.0	0.806386	0.403193	+	0.915115i	$0.367900\pi$
$644$	−144.000	−0.00881117
$645$	−45.0000	−0.00274709
$646$	1258.00	0.0766182
$647$	−13994.0	−0.850326	−0.425163	−	0.905117i	$-0.639783\pi$
$647$	−13994.0	−0.850326	−0.425163	+	0.905117i	$0.639783\pi$
$648$	648.000	0.0392837
$649$	9990.00	0.604225
$650$	−3800.00	−0.229305
$651$	5112.00	0.307765
$652$	−4856.00	−0.291680
$653$	10701.0	0.641290	0.320645	−	0.947199i	$-0.396100\pi$
$653$	10701.0	0.641290	0.320645	+	0.947199i	$0.396100\pi$
$654$	−4872.00	−0.291300
$655$	−10185.0	−0.607574
$656$	−1936.00	−0.115226
$657$	−2718.00	−0.161399
$658$	9648.00	0.571608
$659$	17130.0	1.01258	0.506290	−	0.862363i	$-0.331017\pi$
$659$	17130.0	1.01258	0.506290	+	0.862363i	$0.331017\pi$
$660$	−2220.00	−0.130929
$661$	−16995.0	−1.00004	−0.500022	−	0.866013i	$-0.666675\pi$
$661$	−16995.0	−1.00004	−0.500022	+	0.866013i	$0.666675\pi$
$662$	12050.0	0.707457
$663$	−969.000	−0.0567615
$664$	5584.00	0.326357
$665$	2220.00	0.129456
$666$	−5328.00	−0.309994
$667$	258.000	0.0149772
$668$	4292.00	0.248597
$669$	−7707.00	−0.445396
$670$	−7800.00	−0.449762
$671$	−19240.0	−1.10693
$672$	−1152.00	−0.0661300
$673$	−18286.0	−1.04736	−0.523680	−	0.851915i	$-0.675441\pi$
$673$	−18286.0	−1.04736	−0.523680	+	0.851915i	$0.675441\pi$
$674$	21044.0	1.20265
$675$	2700.00	0.153960
$676$	−7344.00	−0.417843
$677$	−28621.0	−1.62481	−0.812404	−	0.583096i	$-0.801841\pi$
$677$	−28621.0	−1.62481	−0.812404	+	0.583096i	$0.801841\pi$
$678$	3810.00	0.215814
$679$	−6000.00	−0.339115
$680$	680.000	0.0383482
$681$	−16377.0	−0.921539
$682$	−10508.0	−0.589988
$683$	14363.0	0.804663	0.402332	−	0.915494i	$-0.368200\pi$
$683$	14363.0	0.804663	0.402332	+	0.915494i	$0.368200\pi$
$684$	1332.00	0.0744595
$685$	−6470.00	−0.360885
$686$	−13008.0	−0.723976
$687$	−11514.0	−0.639427
$688$	48.0000	0.00265986
$689$	3306.00	0.182799
$690$	90.0000	0.00496557
$691$	−33316.0	−1.83415	−0.917077	−	0.398710i	$-0.869458\pi$
$691$	−33316.0	−1.83415	−0.917077	+	0.398710i	$0.869458\pi$
$692$	3204.00	0.176008
$693$	3996.00	0.219041
$694$	−9224.00	−0.504522
$695$	−3230.00	−0.176289
$696$	2064.00	0.112408
$697$	−2057.00	−0.111785
$698$	6178.00	0.335015
$699$	−1893.00	−0.102432
$700$	−4800.00	−0.259176
$701$	−11368.0	−0.612501	−0.306251	−	0.951951i	$-0.599075\pi$
$701$	−11368.0	−0.612501	−0.306251	+	0.951951i	$0.599075\pi$
$702$	−1026.00	−0.0551622
$703$	−10952.0	−0.587571
$704$	2368.00	0.126772
$705$	−6030.00	−0.322132
$706$	6564.00	0.349914
$707$	−12480.0	−0.663874
$708$	−3240.00	−0.171987
$709$	12814.0	0.678759	0.339379	−	0.940650i	$-0.389783\pi$
$709$	12814.0	0.678759	0.339379	+	0.940650i	$0.389783\pi$
$710$	840.000	0.0444009
$711$	1602.00	0.0845003
$712$	12096.0	0.636681
$713$	426.000	0.0223756
$714$	−1224.00	−0.0641555
$715$	3515.00	0.183851
$716$	10104.0	0.527380
$717$	9648.00	0.502526
$718$	−13752.0	−0.714791
$719$	5543.00	0.287509	0.143755	−	0.989613i	$-0.454082\pi$
$719$	5543.00	0.287509	0.143755	+	0.989613i	$0.454082\pi$
$720$	720.000	0.0372678
$721$	22692.0	1.17211
$722$	−10980.0	−0.565974
$723$	−7008.00	−0.360485
$724$	−6616.00	−0.339616
$725$	8600.00	0.440546
$726$	−228.000	−0.0116555
$727$	11968.0	0.610548	0.305274	−	0.952265i	$-0.401252\pi$
$727$	11968.0	0.610548	0.305274	+	0.952265i	$0.401252\pi$
$728$	1824.00	0.0928598
$729$	729.000	0.0370370
$730$	−3020.00	−0.153117
$731$	51.0000	0.00258044
$732$	6240.00	0.315078
$733$	−35630.0	−1.79539	−0.897697	−	0.440613i	$-0.854761\pi$
$733$	−35630.0	−1.79539	−0.897697	+	0.440613i	$0.854761\pi$
$734$	15912.0	0.800167
$735$	2985.00	0.149801
$736$	−96.0000	−0.00480789
$737$	−28860.0	−1.44243
$738$	−2178.00	−0.108636
$739$	−10043.0	−0.499916	−0.249958	−	0.968257i	$-0.580417\pi$
$739$	−10043.0	−0.499916	−0.249958	+	0.968257i	$0.580417\pi$
$740$	−5920.00	−0.294086
$741$	−2109.00	−0.104556
$742$	4176.00	0.206612
$743$	5104.00	0.252016	0.126008	−	0.992029i	$-0.459784\pi$
$743$	5104.00	0.252016	0.126008	+	0.992029i	$0.459784\pi$
$744$	3408.00	0.167935
$745$	−16930.0	−0.832573
$746$	−2844.00	−0.139579
$747$	6282.00	0.307693
$748$	2516.00	0.122987
$749$	−7188.00	−0.350659
$750$	6750.00	0.328634
$751$	8882.00	0.431570	0.215785	−	0.976441i	$-0.430769\pi$
$751$	8882.00	0.431570	0.215785	+	0.976441i	$0.430769\pi$
$752$	6432.00	0.311903
$753$	6852.00	0.331608
$754$	−3268.00	−0.157843
$755$	−4920.00	−0.237162
$756$	−1296.00	−0.0623480
$757$	−28591.0	−1.37273	−0.686366	−	0.727257i	$-0.740795\pi$
$757$	−28591.0	−1.37273	−0.686366	+	0.727257i	$0.740795\pi$
$758$	−21008.0	−1.00666
$759$	333.000	0.0159251
$760$	1480.00	0.0706385
$761$	−2874.00	−0.136902	−0.0684510	−	0.997654i	$-0.521806\pi$
$761$	−2874.00	−0.136902	−0.0684510	+	0.997654i	$0.521806\pi$
$762$	162.000	0.00770163
$763$	9744.00	0.462328
$764$	16120.0	0.763352
$765$	765.000	0.0361551
$766$	10400.0	0.490558
$767$	5130.00	0.241504
$768$	−768.000	−0.0360844
$769$	3897.00	0.182743	0.0913715	−	0.995817i	$-0.470875\pi$
$769$	3897.00	0.182743	0.0913715	+	0.995817i	$0.470875\pi$
$770$	4440.00	0.207801
$771$	−6708.00	−0.313337
$772$	20952.0	0.976786
$773$	−27620.0	−1.28515	−0.642576	−	0.766222i	$-0.722134\pi$
$773$	−27620.0	−1.28515	−0.642576	+	0.766222i	$0.722134\pi$
$774$	54.0000	0.00250774
$775$	14200.0	0.658167
$776$	−4000.00	−0.185041
$777$	10656.0	0.491997
$778$	8752.00	0.403309
$779$	−4477.00	−0.205912
$780$	−1140.00	−0.0523315
$781$	3108.00	0.142398
$782$	−102.000	−0.00466434
$783$	2322.00	0.105979
$784$	−3184.00	−0.145044
$785$	8915.00	0.405338
$786$	12222.0	0.554637
$787$	−21116.0	−0.956422	−0.478211	−	0.878245i	$-0.658715\pi$
$787$	−21116.0	−0.956422	−0.478211	+	0.878245i	$0.658715\pi$
$788$	−18476.0	−0.835254
$789$	−15504.0	−0.699565
$790$	1780.00	0.0801640
$791$	−7620.00	−0.342523
$792$	2664.00	0.119522
$793$	−9880.00	−0.442433
$794$	9584.00	0.428367
$795$	−2610.00	−0.116437
$796$	−18960.0	−0.844245
$797$	12352.0	0.548972	0.274486	−	0.961591i	$-0.411492\pi$
$797$	12352.0	0.548972	0.274486	+	0.961591i	$0.411492\pi$
$798$	−2664.00	−0.118176
$799$	6834.00	0.302590
$800$	−3200.00	−0.141421
$801$	13608.0	0.600268
$802$	11926.0	0.525089
$803$	−11174.0	−0.491061
$804$	9360.00	0.410574
$805$	−180.000	−0.00788095
$806$	−5396.00	−0.235814
$807$	−13653.0	−0.595549
$808$	−8320.00	−0.362248
$809$	3583.00	0.155713	0.0778563	−	0.996965i	$-0.475192\pi$
$809$	3583.00	0.155713	0.0778563	+	0.996965i	$0.475192\pi$
$810$	810.000	0.0351364
$811$	37162.0	1.60904	0.804522	−	0.593923i	$-0.202422\pi$
$811$	37162.0	1.60904	0.804522	+	0.593923i	$0.202422\pi$
$812$	−4128.00	−0.178404
$813$	20913.0	0.902154
$814$	−21904.0	−0.943163
$815$	−6070.00	−0.260887
$816$	−816.000	−0.0350070
$817$	111.000	0.00475324
$818$	21562.0	0.921635
$819$	2052.00	0.0875491
$820$	−2420.00	−0.103061
$821$	−15443.0	−0.656473	−0.328237	−	0.944596i	$-0.606454\pi$
$821$	−15443.0	−0.656473	−0.328237	+	0.944596i	$0.606454\pi$
$822$	7764.00	0.329441
$823$	41456.0	1.75585	0.877925	−	0.478797i	$-0.158927\pi$
$823$	41456.0	1.75585	0.877925	+	0.478797i	$0.158927\pi$
$824$	15128.0	0.639574
$825$	11100.0	0.468427
$826$	6480.00	0.272964
$827$	−35869.0	−1.50821	−0.754104	−	0.656755i	$-0.771928\pi$
$827$	−35869.0	−1.50821	−0.754104	+	0.656755i	$0.771928\pi$
$828$	−108.000	−0.00453292
$829$	1646.00	0.0689601	0.0344801	−	0.999405i	$-0.489022\pi$
$829$	1646.00	0.0689601	0.0344801	+	0.999405i	$0.489022\pi$
$830$	6980.00	0.291903
$831$	1386.00	0.0578578
$832$	1216.00	0.0506697
$833$	−3383.00	−0.140713
$834$	3876.00	0.160929
$835$	5365.00	0.222351
$836$	5476.00	0.226545
$837$	3834.00	0.158330
$838$	−14888.0	−0.613720
$839$	−9477.00	−0.389967	−0.194984	−	0.980807i	$-0.562465\pi$
$839$	−9477.00	−0.389967	−0.194984	+	0.980807i	$0.562465\pi$
$840$	−1440.00	−0.0591485
$841$	−16993.0	−0.696749
$842$	−22138.0	−0.906088
$843$	−21204.0	−0.866316
$844$	7016.00	0.286138
$845$	−9180.00	−0.373730
$846$	7236.00	0.294065
$847$	456.000	0.0184986
$848$	2784.00	0.112739
$849$	−2358.00	−0.0953196
$850$	−3400.00	−0.137199
$851$	888.000	0.0357700
$852$	−1008.00	−0.0405323
$853$	28978.0	1.16317	0.581587	−	0.813484i	$-0.302432\pi$
$853$	28978.0	1.16317	0.581587	+	0.813484i	$0.302432\pi$
$854$	−12480.0	−0.500067
$855$	1665.00	0.0665986
$856$	−4792.00	−0.191340
$857$	3666.00	0.146124	0.0730619	−	0.997327i	$-0.476723\pi$
$857$	3666.00	0.146124	0.0730619	+	0.997327i	$0.476723\pi$
$858$	−4218.00	−0.167832
$859$	−15156.0	−0.601998	−0.300999	−	0.953624i	$-0.597320\pi$
$859$	−15156.0	−0.601998	−0.300999	+	0.953624i	$0.597320\pi$
$860$	60.0000	0.00237905
$861$	4356.00	0.172418
$862$	11104.0	0.438751
$863$	36920.0	1.45628	0.728141	−	0.685427i	$-0.240385\pi$
$863$	36920.0	1.45628	0.728141	+	0.685427i	$0.240385\pi$
$864$	−864.000	−0.0340207
$865$	4005.00	0.157427
$866$	32846.0	1.28886
$867$	−867.000	−0.0339618
$868$	−6816.00	−0.266532
$869$	6586.00	0.257094
$870$	2580.00	0.100540
$871$	−14820.0	−0.576529
$872$	6496.00	0.252273
$873$	−4500.00	−0.174458
$874$	−222.000	−0.00859183
$875$	−13500.0	−0.521581
$876$	3624.00	0.139776
$877$	18298.0	0.704538	0.352269	−	0.935899i	$-0.385410\pi$
$877$	18298.0	0.704538	0.352269	+	0.935899i	$0.385410\pi$
$878$	−34456.0	−1.32441
$879$	516.000	0.0198001
$880$	2960.00	0.113388
$881$	−29762.0	−1.13815	−0.569073	−	0.822287i	$-0.692698\pi$
$881$	−29762.0	−1.13815	−0.569073	+	0.822287i	$0.692698\pi$
$882$	−3582.00	−0.136749
$883$	32385.0	1.23425	0.617125	−	0.786865i	$-0.288297\pi$
$883$	32385.0	1.23425	0.617125	+	0.786865i	$0.288297\pi$
$884$	1292.00	0.0491569
$885$	−4050.00	−0.153830
$886$	−7396.00	−0.280444
$887$	−30525.0	−1.15550	−0.577750	−	0.816214i	$-0.696069\pi$
$887$	−30525.0	−1.15550	−0.577750	+	0.816214i	$0.696069\pi$
$888$	7104.00	0.268462
$889$	−324.000	−0.0122234
$890$	15120.0	0.569465
$891$	2997.00	0.112686
$892$	10276.0	0.385724
$893$	14874.0	0.557379
$894$	20316.0	0.760032
$895$	12630.0	0.471703
$896$	1536.00	0.0572703
$897$	171.000	0.00636513
$898$	−30676.0	−1.13995
$899$	12212.0	0.453051
$900$	−3600.00	−0.133333
$901$	2958.00	0.109373
$902$	−8954.00	−0.330527
$903$	−108.000	−0.00398008
$904$	−5080.00	−0.186901
$905$	−8270.00	−0.303761
$906$	5904.00	0.216498
$907$	18994.0	0.695353	0.347677	−	0.937615i	$-0.386971\pi$
$907$	18994.0	0.695353	0.347677	+	0.937615i	$0.386971\pi$
$908$	21836.0	0.798076
$909$	−9360.00	−0.341531
$910$	2280.00	0.0830563
$911$	24601.0	0.894695	0.447348	−	0.894360i	$-0.352369\pi$
$911$	24601.0	0.894695	0.447348	+	0.894360i	$0.352369\pi$
$912$	−1776.00	−0.0644838
$913$	25826.0	0.936162
$914$	24794.0	0.897279
$915$	7800.00	0.281814
$916$	15352.0	0.553760
$917$	−24444.0	−0.880275
$918$	−918.000	−0.0330049
$919$	20997.0	0.753675	0.376837	−	0.926279i	$-0.377012\pi$
$919$	20997.0	0.753675	0.376837	+	0.926279i	$0.377012\pi$
$920$	−120.000	−0.00430031
$921$	4212.00	0.150695
$922$	708.000	0.0252893
$923$	1596.00	0.0569155
$924$	−5328.00	−0.189695
$925$	29600.0	1.05215
$926$	−26240.0	−0.931209
$927$	17019.0	0.602996
$928$	−2752.00	−0.0973479
$929$	53725.0	1.89737	0.948687	−	0.316217i	$-0.102413\pi$
$929$	53725.0	1.89737	0.948687	+	0.316217i	$0.102413\pi$
$930$	4260.00	0.150205
$931$	−7363.00	−0.259197
$932$	2524.00	0.0887086
$933$	−16200.0	−0.568450
$934$	−15732.0	−0.551142
$935$	3145.00	0.110003
$936$	1368.00	0.0477719
$937$	16514.0	0.575762	0.287881	−	0.957666i	$-0.407049\pi$
$937$	16514.0	0.575762	0.287881	+	0.957666i	$0.407049\pi$
$938$	−18720.0	−0.651631
$939$	−31980.0	−1.11142
$940$	8040.00	0.278974
$941$	−6218.00	−0.215410	−0.107705	−	0.994183i	$-0.534350\pi$
$941$	−6218.00	−0.215410	−0.107705	+	0.994183i	$0.534350\pi$
$942$	−10698.0	−0.370021
$943$	363.000	0.0125354
$944$	4320.00	0.148945
$945$	−1620.00	−0.0557657
$946$	222.000	0.00762985
$947$	−49892.0	−1.71201	−0.856004	−	0.516969i	$-0.827060\pi$
$947$	−49892.0	−1.71201	−0.856004	+	0.516969i	$0.827060\pi$
$948$	−2136.00	−0.0731794
$949$	−5738.00	−0.196273
$950$	−7400.00	−0.252724
$951$	21006.0	0.716263
$952$	1632.00	0.0555603
$953$	2964.00	0.100749	0.0503743	−	0.998730i	$-0.483959\pi$
$953$	2964.00	0.100749	0.0503743	+	0.998730i	$0.483959\pi$
$954$	3132.00	0.106292
$955$	20150.0	0.682763
$956$	−12864.0	−0.435200
$957$	9546.00	0.322443
$958$	1362.00	0.0459334
$959$	−15528.0	−0.522863
$960$	−960.000	−0.0322749
$961$	−9627.00	−0.323151
$962$	−11248.0	−0.376975
$963$	−5391.00	−0.180397
$964$	9344.00	0.312189
$965$	26190.0	0.873664
$966$	216.000	0.00719429
$967$	−6257.00	−0.208078	−0.104039	−	0.994573i	$-0.533177\pi$
$967$	−6257.00	−0.208078	−0.104039	+	0.994573i	$0.533177\pi$
$968$	304.000	0.0100939
$969$	−1887.00	−0.0625585
$970$	−5000.00	−0.165505
$971$	−17750.0	−0.586637	−0.293319	−	0.956015i	$-0.594760\pi$
$971$	−17750.0	−0.586637	−0.293319	+	0.956015i	$0.594760\pi$
$972$	−972.000	−0.0320750
$973$	−7752.00	−0.255414
$974$	−24452.0	−0.804407
$975$	5700.00	0.187227
$976$	−8320.00	−0.272865
$977$	−28634.0	−0.937649	−0.468824	−	0.883291i	$-0.655322\pi$
$977$	−28634.0	−0.937649	−0.468824	+	0.883291i	$0.655322\pi$
$978$	7284.00	0.238156
$979$	55944.0	1.82633
$980$	−3980.00	−0.129731
$981$	7308.00	0.237846
$982$	−10092.0	−0.327952
$983$	4659.00	0.151169	0.0755844	−	0.997139i	$-0.475918\pi$
$983$	4659.00	0.151169	0.0755844	+	0.997139i	$0.475918\pi$
$984$	2904.00	0.0940814
$985$	−23095.0	−0.747074
$986$	−2924.00	−0.0944413
$987$	−14472.0	−0.466716
$988$	2812.00	0.0905482
$989$	−9.00000	−0.000289366 0
$990$	3330.00	0.106903
$991$	−4744.00	−0.152067	−0.0760334	−	0.997105i	$-0.524226\pi$
$991$	−4744.00	−0.152067	−0.0760334	+	0.997105i	$0.524226\pi$
$992$	−4544.00	−0.145436
$993$	−18075.0	−0.577636
$994$	2016.00	0.0643296
$995$	−23700.0	−0.755116
$996$	−8376.00	−0.266470
$997$	1150.00	0.0365305	0.0182652	−	0.999833i	$-0.494186\pi$
$997$	1150.00	0.0365305	0.0182652	+	0.999833i	$0.494186\pi$
$998$	812.000	0.0257549
$999$	7992.00	0.253109

Twists

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

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