LMFDB - Embedded newform 105.4.a.b.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [105,4,Mod(1,105)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("105.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(105, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 105 = 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	105.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$6.19520055060$
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	$\chi$	$=$	105.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

\(q+5.00000 q^{2} -3.00000 q^{3} +17.0000 q^{4} +5.00000 q^{5} -15.0000 q^{6} +7.00000 q^{7} +45.0000 q^{8} +9.00000 q^{9} +25.0000 q^{10} +12.0000 q^{11} -51.0000 q^{12} +30.0000 q^{13} +35.0000 q^{14} -15.0000 q^{15} +89.0000 q^{16} -134.000 q^{17} +45.0000 q^{18} -92.0000 q^{19} +85.0000 q^{20} -21.0000 q^{21} +60.0000 q^{22} +112.000 q^{23} -135.000 q^{24} +25.0000 q^{25} +150.000 q^{26} -27.0000 q^{27} +119.000 q^{28} -58.0000 q^{29} -75.0000 q^{30} -224.000 q^{31} +85.0000 q^{32} -36.0000 q^{33} -670.000 q^{34} +35.0000 q^{35} +153.000 q^{36} -146.000 q^{37} -460.000 q^{38} -90.0000 q^{39} +225.000 q^{40} +18.0000 q^{41} -105.000 q^{42} +340.000 q^{43} +204.000 q^{44} +45.0000 q^{45} +560.000 q^{46} +208.000 q^{47} -267.000 q^{48} +49.0000 q^{49} +125.000 q^{50} +402.000 q^{51} +510.000 q^{52} -754.000 q^{53} -135.000 q^{54} +60.0000 q^{55} +315.000 q^{56} +276.000 q^{57} -290.000 q^{58} +380.000 q^{59} -255.000 q^{60} +718.000 q^{61} -1120.00 q^{62} +63.0000 q^{63} -287.000 q^{64} +150.000 q^{65} -180.000 q^{66} +412.000 q^{67} -2278.00 q^{68} -336.000 q^{69} +175.000 q^{70} -960.000 q^{71} +405.000 q^{72} +1066.00 q^{73} -730.000 q^{74} -75.0000 q^{75} -1564.00 q^{76} +84.0000 q^{77} -450.000 q^{78} +896.000 q^{79} +445.000 q^{80} +81.0000 q^{81} +90.0000 q^{82} +436.000 q^{83} -357.000 q^{84} -670.000 q^{85} +1700.00 q^{86} +174.000 q^{87} +540.000 q^{88} -1038.00 q^{89} +225.000 q^{90} +210.000 q^{91} +1904.00 q^{92} +672.000 q^{93} +1040.00 q^{94} -460.000 q^{95} -255.000 q^{96} -702.000 q^{97} +245.000 q^{98} +108.000 q^{99} +O(q^{100})\)

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 $

$2$	5.00000	1.76777	0.883883	−	0.467707i	$-0.154920\pi$
$2$	5.00000	1.76777	0.883883	+	0.467707i	$0.154920\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	17.0000	2.12500
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	−15.0000	−1.02062
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	45.0000	1.98874
$9$	9.00000	0.333333
$10$	25.0000	0.790569
$11$	12.0000	0.328921	0.164461	−	0.986384i	$-0.447412\pi$
$11$	12.0000	0.328921	0.164461	+	0.986384i	$0.447412\pi$
$12$	−51.0000	−1.22687
$13$	30.0000	0.640039	0.320019	−	0.947411i	$-0.396311\pi$
$13$	30.0000	0.640039	0.320019	+	0.947411i	$0.396311\pi$
$14$	35.0000	0.668153
$15$	−15.0000	−0.258199
$16$	89.0000	1.39062
$17$	−134.000	−1.91175	−0.955876	−	0.293771i	$-0.905090\pi$
$17$	−134.000	−1.91175	−0.955876	+	0.293771i	$0.905090\pi$
$18$	45.0000	0.589256
$19$	−92.0000	−1.11086	−0.555428	−	0.831565i	$-0.687445\pi$
$19$	−92.0000	−1.11086	−0.555428	+	0.831565i	$0.687445\pi$
$20$	85.0000	0.950329
$21$	−21.0000	−0.218218
$22$	60.0000	0.581456
$23$	112.000	1.01537	0.507687	−	0.861541i	$-0.330501\pi$
$23$	112.000	1.01537	0.507687	+	0.861541i	$0.330501\pi$
$24$	−135.000	−1.14820
$25$	25.0000	0.200000
$26$	150.000	1.13144
$27$	−27.0000	−0.192450
$28$	119.000	0.803175
$29$	−58.0000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−58.0000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	−75.0000	−0.456435
$31$	−224.000	−1.29779	−0.648897	−	0.760877i	$-0.724769\pi$
$31$	−224.000	−1.29779	−0.648897	+	0.760877i	$0.724769\pi$
$32$	85.0000	0.469563
$33$	−36.0000	−0.189903
$34$	−670.000	−3.37953
$35$	35.0000	0.169031
$36$	153.000	0.708333
$37$	−146.000	−0.648710	−0.324355	−	0.945936i	$-0.605147\pi$
$37$	−146.000	−0.648710	−0.324355	+	0.945936i	$0.605147\pi$
$38$	−460.000	−1.96373
$39$	−90.0000	−0.369527
$40$	225.000	0.889391
$41$	18.0000	0.0685641	0.0342820	−	0.999412i	$-0.489086\pi$
$41$	18.0000	0.0685641	0.0342820	+	0.999412i	$0.489086\pi$
$42$	−105.000	−0.385758
$43$	340.000	1.20580	0.602901	−	0.797816i	$-0.294011\pi$
$43$	340.000	1.20580	0.602901	+	0.797816i	$0.294011\pi$
$44$	204.000	0.698958
$45$	45.0000	0.149071
$46$	560.000	1.79495
$47$	208.000	0.645530	0.322765	−	0.946479i	$-0.395388\pi$
$47$	208.000	0.645530	0.322765	+	0.946479i	$0.395388\pi$
$48$	−267.000	−0.802878
$49$	49.0000	0.142857
$50$	125.000	0.353553
$51$	402.000	1.10375
$52$	510.000	1.36008
$53$	−754.000	−1.95415	−0.977074	−	0.212899i	$-0.931709\pi$
$53$	−754.000	−1.95415	−0.977074	+	0.212899i	$0.931709\pi$
$54$	−135.000	−0.340207
$55$	60.0000	0.147098
$56$	315.000	0.751672
$57$	276.000	0.641353
$58$	−290.000	−0.656532
$59$	380.000	0.838505	0.419252	−	0.907870i	$-0.362292\pi$
$59$	380.000	0.838505	0.419252	+	0.907870i	$0.362292\pi$
$60$	−255.000	−0.548673
$61$	718.000	1.50706	0.753529	−	0.657415i	$-0.228350\pi$
$61$	718.000	1.50706	0.753529	+	0.657415i	$0.228350\pi$
$62$	−1120.00	−2.29420
$63$	63.0000	0.125988
$64$	−287.000	−0.560547
$65$	150.000	0.286234
$66$	−180.000	−0.335704
$67$	412.000	0.751251	0.375625	−	0.926772i	$-0.377428\pi$
$67$	412.000	0.751251	0.375625	+	0.926772i	$0.377428\pi$
$68$	−2278.00	−4.06247
$69$	−336.000	−0.586227
$70$	175.000	0.298807
$71$	−960.000	−1.60466	−0.802331	−	0.596879i	$-0.796407\pi$
$71$	−960.000	−1.60466	−0.802331	+	0.596879i	$0.796407\pi$
$72$	405.000	0.662913
$73$	1066.00	1.70912	0.854561	−	0.519352i	$-0.173826\pi$
$73$	1066.00	1.70912	0.854561	+	0.519352i	$0.173826\pi$
$74$	−730.000	−1.14677
$75$	−75.0000	−0.115470
$76$	−1564.00	−2.36057
$77$	84.0000	0.124321
$78$	−450.000	−0.653237
$79$	896.000	1.27605	0.638025	−	0.770016i	$-0.279752\pi$
$79$	896.000	1.27605	0.638025	+	0.770016i	$0.279752\pi$
$80$	445.000	0.621906
$81$	81.0000	0.111111
$82$	90.0000	0.121205
$83$	436.000	0.576593	0.288296	−	0.957541i	$-0.406911\pi$
$83$	436.000	0.576593	0.288296	+	0.957541i	$0.406911\pi$
$84$	−357.000	−0.463713
$85$	−670.000	−0.854961
$86$	1700.00	2.13158
$87$	174.000	0.214423
$88$	540.000	0.654139
$89$	−1038.00	−1.23627	−0.618134	−	0.786073i	$-0.712111\pi$
$89$	−1038.00	−1.23627	−0.618134	+	0.786073i	$0.712111\pi$
$90$	225.000	0.263523
$91$	210.000	0.241912
$92$	1904.00	2.15767
$93$	672.000	0.749281
$94$	1040.00	1.14115
$95$	−460.000	−0.496790
$96$	−255.000	−0.271102
$97$	−702.000	−0.734818	−0.367409	−	0.930060i	$-0.619755\pi$
$97$	−702.000	−0.734818	−0.367409	+	0.930060i	$0.619755\pi$
$98$	245.000	0.252538
$99$	108.000	0.109640
$100$	425.000	0.425000
$101$	46.0000	0.0453185	0.0226593	−	0.999743i	$-0.492787\pi$
$101$	46.0000	0.0453185	0.0226593	+	0.999743i	$0.492787\pi$
$102$	2010.00	1.95117
$103$	1880.00	1.79847	0.899233	−	0.437471i	$-0.144126\pi$
$103$	1880.00	1.79847	0.899233	+	0.437471i	$0.144126\pi$
$104$	1350.00	1.27287
$105$	−105.000	−0.0975900
$106$	−3770.00	−3.45448
$107$	732.000	0.661356	0.330678	−	0.943744i	$-0.392723\pi$
$107$	732.000	0.661356	0.330678	+	0.943744i	$0.392723\pi$
$108$	−459.000	−0.408956
$109$	−378.000	−0.332164	−0.166082	−	0.986112i	$-0.553112\pi$
$109$	−378.000	−0.332164	−0.166082	+	0.986112i	$0.553112\pi$
$110$	300.000	0.260035
$111$	438.000	0.374533
$112$	623.000	0.525607
$113$	1458.00	1.21378	0.606890	−	0.794786i	$-0.292417\pi$
$113$	1458.00	1.21378	0.606890	+	0.794786i	$0.292417\pi$
$114$	1380.00	1.13376
$115$	560.000	0.454089
$116$	−986.000	−0.789205
$117$	270.000	0.213346
$118$	1900.00	1.48228
$119$	−938.000	−0.722574
$120$	−675.000	−0.513490
$121$	−1187.00	−0.891811
$122$	3590.00	2.66413
$123$	−54.0000	−0.0395855
$124$	−3808.00	−2.75781
$125$	125.000	0.0894427
$126$	315.000	0.222718
$127$	608.000	0.424813	0.212407	−	0.977181i	$-0.431870\pi$
$127$	608.000	0.424813	0.212407	+	0.977181i	$0.431870\pi$
$128$	−2115.00	−1.46048
$129$	−1020.00	−0.696170
$130$	750.000	0.505995
$131$	−956.000	−0.637604	−0.318802	−	0.947821i	$-0.603280\pi$
$131$	−956.000	−0.637604	−0.318802	+	0.947821i	$0.603280\pi$
$132$	−612.000	−0.403544
$133$	−644.000	−0.419864
$134$	2060.00	1.32804
$135$	−135.000	−0.0860663
$136$	−6030.00	−3.80197
$137$	−374.000	−0.233233	−0.116617	−	0.993177i	$-0.537205\pi$
$137$	−374.000	−0.233233	−0.116617	+	0.993177i	$0.537205\pi$
$138$	−1680.00	−1.03631
$139$	396.000	0.241642	0.120821	−	0.992674i	$-0.461447\pi$
$139$	396.000	0.241642	0.120821	+	0.992674i	$0.461447\pi$
$140$	595.000	0.359191
$141$	−624.000	−0.372697
$142$	−4800.00	−2.83667
$143$	360.000	0.210522
$144$	801.000	0.463542
$145$	−290.000	−0.166091
$146$	5330.00	3.02133
$147$	−147.000	−0.0824786
$148$	−2482.00	−1.37851
$149$	−1874.00	−1.03036	−0.515181	−	0.857081i	$-0.672275\pi$
$149$	−1874.00	−1.03036	−0.515181	+	0.857081i	$0.672275\pi$
$150$	−375.000	−0.204124
$151$	−1096.00	−0.590670	−0.295335	−	0.955394i	$-0.595431\pi$
$151$	−1096.00	−0.590670	−0.295335	+	0.955394i	$0.595431\pi$
$152$	−4140.00	−2.20920
$153$	−1206.00	−0.637250
$154$	420.000	0.219770
$155$	−1120.00	−0.580391
$156$	−1530.00	−0.785244
$157$	1918.00	0.974988	0.487494	−	0.873126i	$-0.337911\pi$
$157$	1918.00	0.974988	0.487494	+	0.873126i	$0.337911\pi$
$158$	4480.00	2.25576
$159$	2262.00	1.12823
$160$	425.000	0.209995
$161$	784.000	0.383776
$162$	405.000	0.196419
$163$	2316.00	1.11290	0.556451	−	0.830880i	$-0.312163\pi$
$163$	2316.00	1.11290	0.556451	+	0.830880i	$0.312163\pi$
$164$	306.000	0.145699
$165$	−180.000	−0.0849272
$166$	2180.00	1.01928
$167$	−1736.00	−0.804405	−0.402203	−	0.915551i	$-0.631755\pi$
$167$	−1736.00	−0.804405	−0.402203	+	0.915551i	$0.631755\pi$
$168$	−945.000	−0.433978
$169$	−1297.00	−0.590350
$170$	−3350.00	−1.51137
$171$	−828.000	−0.370285
$172$	5780.00	2.56233
$173$	−2442.00	−1.07319	−0.536595	−	0.843840i	$-0.680290\pi$
$173$	−2442.00	−1.07319	−0.536595	+	0.843840i	$0.680290\pi$
$174$	870.000	0.379049
$175$	175.000	0.0755929
$176$	1068.00	0.457406
$177$	−1140.00	−0.484111
$178$	−5190.00	−2.18543
$179$	−4092.00	−1.70866	−0.854331	−	0.519730i	$-0.826033\pi$
$179$	−4092.00	−1.70866	−0.854331	+	0.519730i	$0.826033\pi$
$180$	765.000	0.316776
$181$	1270.00	0.521538	0.260769	−	0.965401i	$-0.416024\pi$
$181$	1270.00	0.521538	0.260769	+	0.965401i	$0.416024\pi$
$182$	1050.00	0.427644
$183$	−2154.00	−0.870100
$184$	5040.00	2.01931
$185$	−730.000	−0.290112
$186$	3360.00	1.32455
$187$	−1608.00	−0.628816
$188$	3536.00	1.37175
$189$	−189.000	−0.0727393
$190$	−2300.00	−0.878208
$191$	4904.00	1.85781	0.928903	−	0.370323i	$-0.120753\pi$
$191$	4904.00	1.85781	0.928903	+	0.370323i	$0.120753\pi$
$192$	861.000	0.323632
$193$	2178.00	0.812310	0.406155	−	0.913804i	$-0.366869\pi$
$193$	2178.00	0.812310	0.406155	+	0.913804i	$0.366869\pi$
$194$	−3510.00	−1.29899
$195$	−450.000	−0.165257
$196$	833.000	0.303571
$197$	−2850.00	−1.03073	−0.515366	−	0.856970i	$-0.672344\pi$
$197$	−2850.00	−1.03073	−0.515366	+	0.856970i	$0.672344\pi$
$198$	540.000	0.193819
$199$	−1144.00	−0.407518	−0.203759	−	0.979021i	$-0.565316\pi$
$199$	−1144.00	−0.407518	−0.203759	+	0.979021i	$0.565316\pi$
$200$	1125.00	0.397748
$201$	−1236.00	−0.433735
$202$	230.000	0.0801126
$203$	−406.000	−0.140372
$204$	6834.00	2.34547
$205$	90.0000	0.0306628
$206$	9400.00	3.17927
$207$	1008.00	0.338458
$208$	2670.00	0.890054
$209$	−1104.00	−0.365384
$210$	−525.000	−0.172516
$211$	412.000	0.134423	0.0672115	−	0.997739i	$-0.478590\pi$
$211$	412.000	0.134423	0.0672115	+	0.997739i	$0.478590\pi$
$212$	−12818.0	−4.15257
$213$	2880.00	0.926452
$214$	3660.00	1.16912
$215$	1700.00	0.539251
$216$	−1215.00	−0.382733
$217$	−1568.00	−0.490520
$218$	−1890.00	−0.587188
$219$	−3198.00	−0.986762
$220$	1020.00	0.312584
$221$	−4020.00	−1.22359
$222$	2190.00	0.662086
$223$	−1632.00	−0.490075	−0.245038	−	0.969514i	$-0.578800\pi$
$223$	−1632.00	−0.490075	−0.245038	+	0.969514i	$0.578800\pi$
$224$	595.000	0.177478
$225$	225.000	0.0666667
$226$	7290.00	2.14568
$227$	4084.00	1.19412	0.597059	−	0.802198i	$-0.296336\pi$
$227$	4084.00	1.19412	0.597059	+	0.802198i	$0.296336\pi$
$228$	4692.00	1.36287
$229$	−3386.00	−0.977088	−0.488544	−	0.872539i	$-0.662472\pi$
$229$	−3386.00	−0.977088	−0.488544	+	0.872539i	$0.662472\pi$
$230$	2800.00	0.802724
$231$	−252.000	−0.0717765
$232$	−2610.00	−0.738599
$233$	5322.00	1.49638	0.748188	−	0.663486i	$-0.230924\pi$
$233$	5322.00	1.49638	0.748188	+	0.663486i	$0.230924\pi$
$234$	1350.00	0.377146
$235$	1040.00	0.288690
$236$	6460.00	1.78182
$237$	−2688.00	−0.736727
$238$	−4690.00	−1.27734
$239$	3736.00	1.01114	0.505569	−	0.862786i	$-0.331283\pi$
$239$	3736.00	1.01114	0.505569	+	0.862786i	$0.331283\pi$
$240$	−1335.00	−0.359058
$241$	210.000	0.0561298	0.0280649	−	0.999606i	$-0.491065\pi$
$241$	210.000	0.0561298	0.0280649	+	0.999606i	$0.491065\pi$
$242$	−5935.00	−1.57651
$243$	−243.000	−0.0641500
$244$	12206.0	3.20250
$245$	245.000	0.0638877
$246$	−270.000	−0.0699779
$247$	−2760.00	−0.710990
$248$	−10080.0	−2.58097
$249$	−1308.00	−0.332896
$250$	625.000	0.158114
$251$	−4212.00	−1.05920	−0.529600	−	0.848248i	$-0.677658\pi$
$251$	−4212.00	−1.05920	−0.529600	+	0.848248i	$0.677658\pi$
$252$	1071.00	0.267725
$253$	1344.00	0.333978
$254$	3040.00	0.750971
$255$	2010.00	0.493612
$256$	−8279.00	−2.02124
$257$	5130.00	1.24514	0.622569	−	0.782565i	$-0.286089\pi$
$257$	5130.00	1.24514	0.622569	+	0.782565i	$0.286089\pi$
$258$	−5100.00	−1.23067
$259$	−1022.00	−0.245189
$260$	2550.00	0.608247
$261$	−522.000	−0.123797
$262$	−4780.00	−1.12714
$263$	848.000	0.198821	0.0994105	−	0.995047i	$-0.468304\pi$
$263$	848.000	0.198821	0.0994105	+	0.995047i	$0.468304\pi$
$264$	−1620.00	−0.377667
$265$	−3770.00	−0.873922
$266$	−3220.00	−0.742221
$267$	3114.00	0.713759
$268$	7004.00	1.59641
$269$	−1274.00	−0.288763	−0.144381	−	0.989522i	$-0.546119\pi$
$269$	−1274.00	−0.288763	−0.144381	+	0.989522i	$0.546119\pi$
$270$	−675.000	−0.152145
$271$	864.000	0.193669	0.0968344	−	0.995301i	$-0.469128\pi$
$271$	864.000	0.193669	0.0968344	+	0.995301i	$0.469128\pi$
$272$	−11926.0	−2.65853
$273$	−630.000	−0.139668
$274$	−1870.00	−0.412302
$275$	300.000	0.0657843
$276$	−5712.00	−1.24573
$277$	−8530.00	−1.85025	−0.925123	−	0.379668i	$-0.876038\pi$
$277$	−8530.00	−1.85025	−0.925123	+	0.379668i	$0.876038\pi$
$278$	1980.00	0.427167
$279$	−2016.00	−0.432598
$280$	1575.00	0.336158
$281$	−5382.00	−1.14257	−0.571287	−	0.820750i	$-0.693556\pi$
$281$	−5382.00	−1.14257	−0.571287	+	0.820750i	$0.693556\pi$
$282$	−3120.00	−0.658841
$283$	6236.00	1.30986	0.654932	−	0.755687i	$-0.272697\pi$
$283$	6236.00	1.30986	0.654932	+	0.755687i	$0.272697\pi$
$284$	−16320.0	−3.40991
$285$	1380.00	0.286822
$286$	1800.00	0.372155
$287$	126.000	0.0259148
$288$	765.000	0.156521
$289$	13043.0	2.65479
$290$	−1450.00	−0.293610
$291$	2106.00	0.424247
$292$	18122.0	3.63188
$293$	−818.000	−0.163099	−0.0815496	−	0.996669i	$-0.525987\pi$
$293$	−818.000	−0.163099	−0.0815496	+	0.996669i	$0.525987\pi$
$294$	−735.000	−0.145803
$295$	1900.00	0.374991
$296$	−6570.00	−1.29011
$297$	−324.000	−0.0633010
$298$	−9370.00	−1.82144
$299$	3360.00	0.649879
$300$	−1275.00	−0.245374
$301$	2380.00	0.455751
$302$	−5480.00	−1.04417
$303$	−138.000	−0.0261647
$304$	−8188.00	−1.54478
$305$	3590.00	0.673976
$306$	−6030.00	−1.12651
$307$	−2268.00	−0.421634	−0.210817	−	0.977526i	$-0.567612\pi$
$307$	−2268.00	−0.421634	−0.210817	+	0.977526i	$0.567612\pi$
$308$	1428.00	0.264181
$309$	−5640.00	−1.03834
$310$	−5600.00	−1.02600
$311$	6648.00	1.21213	0.606067	−	0.795414i	$-0.292746\pi$
$311$	6648.00	1.21213	0.606067	+	0.795414i	$0.292746\pi$
$312$	−4050.00	−0.734891
$313$	9818.00	1.77299	0.886495	−	0.462737i	$-0.153133\pi$
$313$	9818.00	1.77299	0.886495	+	0.462737i	$0.153133\pi$
$314$	9590.00	1.72355
$315$	315.000	0.0563436
$316$	15232.0	2.71160
$317$	934.000	0.165485	0.0827424	−	0.996571i	$-0.473632\pi$
$317$	934.000	0.165485	0.0827424	+	0.996571i	$0.473632\pi$
$318$	11310.0	1.99444
$319$	−696.000	−0.122158
$320$	−1435.00	−0.250684
$321$	−2196.00	−0.381834
$322$	3920.00	0.678426
$323$	12328.0	2.12368
$324$	1377.00	0.236111
$325$	750.000	0.128008
$326$	11580.0	1.96735
$327$	1134.00	0.191775
$328$	810.000	0.136356
$329$	1456.00	0.243987
$330$	−900.000	−0.150131
$331$	2292.00	0.380603	0.190302	−	0.981726i	$-0.439053\pi$
$331$	2292.00	0.380603	0.190302	+	0.981726i	$0.439053\pi$
$332$	7412.00	1.22526
$333$	−1314.00	−0.216237
$334$	−8680.00	−1.42200
$335$	2060.00	0.335970
$336$	−1869.00	−0.303459
$337$	−6062.00	−0.979876	−0.489938	−	0.871757i	$-0.662981\pi$
$337$	−6062.00	−0.979876	−0.489938	+	0.871757i	$0.662981\pi$
$338$	−6485.00	−1.04360
$339$	−4374.00	−0.700776
$340$	−11390.0	−1.81679
$341$	−2688.00	−0.426872
$342$	−4140.00	−0.654578
$343$	343.000	0.0539949
$344$	15300.0	2.39803
$345$	−1680.00	−0.262169
$346$	−12210.0	−1.89715
$347$	1484.00	0.229583	0.114791	−	0.993390i	$-0.463380\pi$
$347$	1484.00	0.229583	0.114791	+	0.993390i	$0.463380\pi$
$348$	2958.00	0.455648
$349$	254.000	0.0389579	0.0194790	−	0.999810i	$-0.493799\pi$
$349$	254.000	0.0389579	0.0194790	+	0.999810i	$0.493799\pi$
$350$	875.000	0.133631
$351$	−810.000	−0.123176
$352$	1020.00	0.154449
$353$	−10950.0	−1.65102	−0.825509	−	0.564388i	$-0.809112\pi$
$353$	−10950.0	−1.65102	−0.825509	+	0.564388i	$0.809112\pi$
$354$	−5700.00	−0.855795
$355$	−4800.00	−0.717627
$356$	−17646.0	−2.62707
$357$	2814.00	0.417178
$358$	−20460.0	−3.02052
$359$	11376.0	1.67243	0.836215	−	0.548402i	$-0.184764\pi$
$359$	11376.0	1.67243	0.836215	+	0.548402i	$0.184764\pi$
$360$	2025.00	0.296464
$361$	1605.00	0.233999
$362$	6350.00	0.921957
$363$	3561.00	0.514887
$364$	3570.00	0.514063
$365$	5330.00	0.764342
$366$	−10770.0	−1.53813
$367$	−1136.00	−0.161577	−0.0807884	−	0.996731i	$-0.525744\pi$
$367$	−1136.00	−0.161577	−0.0807884	+	0.996731i	$0.525744\pi$
$368$	9968.00	1.41201
$369$	162.000	0.0228547
$370$	−3650.00	−0.512850
$371$	−5278.00	−0.738599
$372$	11424.0	1.59222
$373$	−8242.00	−1.14411	−0.572057	−	0.820214i	$-0.693854\pi$
$373$	−8242.00	−1.14411	−0.572057	+	0.820214i	$0.693854\pi$
$374$	−8040.00	−1.11160
$375$	−375.000	−0.0516398
$376$	9360.00	1.28379
$377$	−1740.00	−0.237704
$378$	−945.000	−0.128586
$379$	3620.00	0.490625	0.245313	−	0.969444i	$-0.421109\pi$
$379$	3620.00	0.490625	0.245313	+	0.969444i	$0.421109\pi$
$380$	−7820.00	−1.05568
$381$	−1824.00	−0.245266
$382$	24520.0	3.28417
$383$	−8464.00	−1.12922	−0.564609	−	0.825359i	$-0.690973\pi$
$383$	−8464.00	−1.12922	−0.564609	+	0.825359i	$0.690973\pi$
$384$	6345.00	0.843208
$385$	420.000	0.0555979
$386$	10890.0	1.43598
$387$	3060.00	0.401934
$388$	−11934.0	−1.56149
$389$	3678.00	0.479388	0.239694	−	0.970848i	$-0.422953\pi$
$389$	3678.00	0.479388	0.239694	+	0.970848i	$0.422953\pi$
$390$	−2250.00	−0.292136
$391$	−15008.0	−1.94114
$392$	2205.00	0.284105
$393$	2868.00	0.368121
$394$	−14250.0	−1.82209
$395$	4480.00	0.570666
$396$	1836.00	0.232986
$397$	12590.0	1.59162	0.795811	−	0.605545i	$-0.207045\pi$
$397$	12590.0	1.59162	0.795811	+	0.605545i	$0.207045\pi$
$398$	−5720.00	−0.720396
$399$	1932.00	0.242408
$400$	2225.00	0.278125
$401$	2850.00	0.354918	0.177459	−	0.984128i	$-0.443212\pi$
$401$	2850.00	0.354918	0.177459	+	0.984128i	$0.443212\pi$
$402$	−6180.00	−0.766742
$403$	−6720.00	−0.830638
$404$	782.000	0.0963019
$405$	405.000	0.0496904
$406$	−2030.00	−0.248146
$407$	−1752.00	−0.213374
$408$	18090.0	2.19507
$409$	1226.00	0.148220	0.0741098	−	0.997250i	$-0.476388\pi$
$409$	1226.00	0.148220	0.0741098	+	0.997250i	$0.476388\pi$
$410$	450.000	0.0542047
$411$	1122.00	0.134657
$412$	31960.0	3.82174
$413$	2660.00	0.316925
$414$	5040.00	0.598315
$415$	2180.00	0.257860
$416$	2550.00	0.300539
$417$	−1188.00	−0.139512
$418$	−5520.00	−0.645914
$419$	612.000	0.0713560	0.0356780	−	0.999363i	$-0.488641\pi$
$419$	612.000	0.0713560	0.0356780	+	0.999363i	$0.488641\pi$
$420$	−1785.00	−0.207379
$421$	5182.00	0.599894	0.299947	−	0.953956i	$-0.403031\pi$
$421$	5182.00	0.599894	0.299947	+	0.953956i	$0.403031\pi$
$422$	2060.00	0.237629
$423$	1872.00	0.215177
$424$	−33930.0	−3.88629
$425$	−3350.00	−0.382350
$426$	14400.0	1.63775
$427$	5026.00	0.569614
$428$	12444.0	1.40538
$429$	−1080.00	−0.121545
$430$	8500.00	0.953271
$431$	−4984.00	−0.557009	−0.278504	−	0.960435i	$-0.589839\pi$
$431$	−4984.00	−0.557009	−0.278504	+	0.960435i	$0.589839\pi$
$432$	−2403.00	−0.267626
$433$	−1694.00	−0.188010	−0.0940051	−	0.995572i	$-0.529967\pi$
$433$	−1694.00	−0.188010	−0.0940051	+	0.995572i	$0.529967\pi$
$434$	−7840.00	−0.867125
$435$	870.000	0.0958927
$436$	−6426.00	−0.705848
$437$	−10304.0	−1.12793
$438$	−15990.0	−1.74436
$439$	13864.0	1.50727	0.753636	−	0.657292i	$-0.228298\pi$
$439$	13864.0	1.50727	0.753636	+	0.657292i	$0.228298\pi$
$440$	2700.00	0.292540
$441$	441.000	0.0476190
$442$	−20100.0	−2.16303
$443$	−4644.00	−0.498066	−0.249033	−	0.968495i	$-0.580113\pi$
$443$	−4644.00	−0.498066	−0.249033	+	0.968495i	$0.580113\pi$
$444$	7446.00	0.795882
$445$	−5190.00	−0.552875
$446$	−8160.00	−0.866339
$447$	5622.00	0.594880
$448$	−2009.00	−0.211867
$449$	−4926.00	−0.517756	−0.258878	−	0.965910i	$-0.583353\pi$
$449$	−4926.00	−0.517756	−0.258878	+	0.965910i	$0.583353\pi$
$450$	1125.00	0.117851
$451$	216.000	0.0225522
$452$	24786.0	2.57928
$453$	3288.00	0.341024
$454$	20420.0	2.11092
$455$	1050.00	0.108186
$456$	12420.0	1.27548
$457$	−14694.0	−1.50406	−0.752031	−	0.659128i	$-0.770926\pi$
$457$	−14694.0	−1.50406	−0.752031	+	0.659128i	$0.770926\pi$
$458$	−16930.0	−1.72726
$459$	3618.00	0.367917
$460$	9520.00	0.964940
$461$	2006.00	0.202665	0.101333	−	0.994853i	$-0.467689\pi$
$461$	2006.00	0.202665	0.101333	+	0.994853i	$0.467689\pi$
$462$	−1260.00	−0.126884
$463$	4896.00	0.491439	0.245720	−	0.969341i	$-0.420976\pi$
$463$	4896.00	0.491439	0.245720	+	0.969341i	$0.420976\pi$
$464$	−5162.00	−0.516465
$465$	3360.00	0.335089
$466$	26610.0	2.64525
$467$	2660.00	0.263576	0.131788	−	0.991278i	$-0.457928\pi$
$467$	2660.00	0.263576	0.131788	+	0.991278i	$0.457928\pi$
$468$	4590.00	0.453361
$469$	2884.00	0.283946
$470$	5200.00	0.510336
$471$	−5754.00	−0.562909
$472$	17100.0	1.66757
$473$	4080.00	0.396614
$474$	−13440.0	−1.30236
$475$	−2300.00	−0.222171
$476$	−15946.0	−1.53547
$477$	−6786.00	−0.651383
$478$	18680.0	1.78745
$479$	−5600.00	−0.534176	−0.267088	−	0.963672i	$-0.586062\pi$
$479$	−5600.00	−0.534176	−0.267088	+	0.963672i	$0.586062\pi$
$480$	−1275.00	−0.121241
$481$	−4380.00	−0.415199
$482$	1050.00	0.0992245
$483$	−2352.00	−0.221573
$484$	−20179.0	−1.89510
$485$	−3510.00	−0.328620
$486$	−1215.00	−0.113402
$487$	−6424.00	−0.597740	−0.298870	−	0.954294i	$-0.596610\pi$
$487$	−6424.00	−0.597740	−0.298870	+	0.954294i	$0.596610\pi$
$488$	32310.0	2.99714
$489$	−6948.00	−0.642535
$490$	1225.00	0.112938
$491$	−18900.0	−1.73716	−0.868579	−	0.495550i	$-0.834967\pi$
$491$	−18900.0	−1.73716	−0.868579	+	0.495550i	$0.834967\pi$
$492$	−918.000	−0.0841192
$493$	7772.00	0.710007
$494$	−13800.0	−1.25687
$495$	540.000	0.0490327
$496$	−19936.0	−1.80474
$497$	−6720.00	−0.606505
$498$	−6540.00	−0.588483
$499$	−15364.0	−1.37833	−0.689165	−	0.724604i	$-0.742023\pi$
$499$	−15364.0	−1.37833	−0.689165	+	0.724604i	$0.742023\pi$
$500$	2125.00	0.190066
$501$	5208.00	0.464424
$502$	−21060.0	−1.87242
$503$	2216.00	0.196435	0.0982173	−	0.995165i	$-0.468686\pi$
$503$	2216.00	0.196435	0.0982173	+	0.995165i	$0.468686\pi$
$504$	2835.00	0.250557
$505$	230.000	0.0202671
$506$	6720.00	0.590396
$507$	3891.00	0.340839
$508$	10336.0	0.902728
$509$	−3754.00	−0.326902	−0.163451	−	0.986551i	$-0.552263\pi$
$509$	−3754.00	−0.326902	−0.163451	+	0.986551i	$0.552263\pi$
$510$	10050.0	0.872591
$511$	7462.00	0.645987
$512$	−24475.0	−2.11260
$513$	2484.00	0.213784
$514$	25650.0	2.20111
$515$	9400.00	0.804298
$516$	−17340.0	−1.47936
$517$	2496.00	0.212329
$518$	−5110.00	−0.433437
$519$	7326.00	0.619606
$520$	6750.00	0.569244
$521$	−4702.00	−0.395390	−0.197695	−	0.980264i	$-0.563346\pi$
$521$	−4702.00	−0.395390	−0.197695	+	0.980264i	$0.563346\pi$
$522$	−2610.00	−0.218844
$523$	−22660.0	−1.89456	−0.947278	−	0.320413i	$-0.896178\pi$
$523$	−22660.0	−1.89456	−0.947278	+	0.320413i	$0.896178\pi$
$524$	−16252.0	−1.35491
$525$	−525.000	−0.0436436
$526$	4240.00	0.351469
$527$	30016.0	2.48106
$528$	−3204.00	−0.264084
$529$	377.000	0.0309855
$530$	−18850.0	−1.54489
$531$	3420.00	0.279502
$532$	−10948.0	−0.892211
$533$	540.000	0.0438837
$534$	15570.0	1.26176
$535$	3660.00	0.295767
$536$	18540.0	1.49404
$537$	12276.0	0.986496
$538$	−6370.00	−0.510465
$539$	588.000	0.0469888
$540$	−2295.00	−0.182891
$541$	−8634.00	−0.686145	−0.343073	−	0.939309i	$-0.611468\pi$
$541$	−8634.00	−0.686145	−0.343073	+	0.939309i	$0.611468\pi$
$542$	4320.00	0.342361
$543$	−3810.00	−0.301110
$544$	−11390.0	−0.897688
$545$	−1890.00	−0.148548
$546$	−3150.00	−0.246900
$547$	−19284.0	−1.50736	−0.753679	−	0.657243i	$-0.771722\pi$
$547$	−19284.0	−1.50736	−0.753679	+	0.657243i	$0.771722\pi$
$548$	−6358.00	−0.495621
$549$	6462.00	0.502352
$550$	1500.00	0.116291
$551$	5336.00	0.412561
$552$	−15120.0	−1.16585
$553$	6272.00	0.482301
$554$	−42650.0	−3.27080
$555$	2190.00	0.167496
$556$	6732.00	0.513490
$557$	−19658.0	−1.49540	−0.747699	−	0.664038i	$-0.768841\pi$
$557$	−19658.0	−1.49540	−0.747699	+	0.664038i	$0.768841\pi$
$558$	−10080.0	−0.764732
$559$	10200.0	0.771760
$560$	3115.00	0.235059
$561$	4824.00	0.363047
$562$	−26910.0	−2.01980
$563$	−25612.0	−1.91726	−0.958630	−	0.284656i	$-0.908121\pi$
$563$	−25612.0	−1.91726	−0.958630	+	0.284656i	$0.908121\pi$
$564$	−10608.0	−0.791981
$565$	7290.00	0.542819
$566$	31180.0	2.31554
$567$	567.000	0.0419961
$568$	−43200.0	−3.19125
$569$	7002.00	0.515886	0.257943	−	0.966160i	$-0.416955\pi$
$569$	7002.00	0.515886	0.257943	+	0.966160i	$0.416955\pi$
$570$	6900.00	0.507034
$571$	−4524.00	−0.331565	−0.165782	−	0.986162i	$-0.553015\pi$
$571$	−4524.00	−0.331565	−0.165782	+	0.986162i	$0.553015\pi$
$572$	6120.00	0.447360
$573$	−14712.0	−1.07260
$574$	630.000	0.0458113
$575$	2800.00	0.203075
$576$	−2583.00	−0.186849
$577$	−6014.00	−0.433910	−0.216955	−	0.976182i	$-0.569612\pi$
$577$	−6014.00	−0.433910	−0.216955	+	0.976182i	$0.569612\pi$
$578$	65215.0	4.69306
$579$	−6534.00	−0.468988
$580$	−4930.00	−0.352943
$581$	3052.00	0.217932
$582$	10530.0	0.749970
$583$	−9048.00	−0.642761
$584$	47970.0	3.39899
$585$	1350.00	0.0954113
$586$	−4090.00	−0.288321
$587$	−11748.0	−0.826051	−0.413025	−	0.910719i	$-0.635528\pi$
$587$	−11748.0	−0.826051	−0.413025	+	0.910719i	$0.635528\pi$
$588$	−2499.00	−0.175267
$589$	20608.0	1.44166
$590$	9500.00	0.662896
$591$	8550.00	0.595093
$592$	−12994.0	−0.902112
$593$	−9462.00	−0.655241	−0.327620	−	0.944809i	$-0.606247\pi$
$593$	−9462.00	−0.655241	−0.327620	+	0.944809i	$0.606247\pi$
$594$	−1620.00	−0.111901
$595$	−4690.00	−0.323145
$596$	−31858.0	−2.18952
$597$	3432.00	0.235280
$598$	16800.0	1.14883
$599$	2320.00	0.158251	0.0791257	−	0.996865i	$-0.474787\pi$
$599$	2320.00	0.158251	0.0791257	+	0.996865i	$0.474787\pi$
$600$	−3375.00	−0.229640
$601$	4650.00	0.315603	0.157802	−	0.987471i	$-0.449559\pi$
$601$	4650.00	0.315603	0.157802	+	0.987471i	$0.449559\pi$
$602$	11900.0	0.805661
$603$	3708.00	0.250417
$604$	−18632.0	−1.25517
$605$	−5935.00	−0.398830
$606$	−690.000	−0.0462530
$607$	−14656.0	−0.980014	−0.490007	−	0.871718i	$-0.663006\pi$
$607$	−14656.0	−0.980014	−0.490007	+	0.871718i	$0.663006\pi$
$608$	−7820.00	−0.521617
$609$	1218.00	0.0810441
$610$	17950.0	1.19143
$611$	6240.00	0.413164
$612$	−20502.0	−1.35416
$613$	29166.0	1.92170	0.960851	−	0.277065i	$-0.0893616\pi$
$613$	29166.0	1.92170	0.960851	+	0.277065i	$0.0893616\pi$
$614$	−11340.0	−0.745350
$615$	−270.000	−0.0177032
$616$	3780.00	0.247241
$617$	28554.0	1.86311	0.931557	−	0.363597i	$-0.118451\pi$
$617$	28554.0	1.86311	0.931557	+	0.363597i	$0.118451\pi$
$618$	−28200.0	−1.83555
$619$	−3876.00	−0.251679	−0.125840	−	0.992051i	$-0.540163\pi$
$619$	−3876.00	−0.251679	−0.125840	+	0.992051i	$0.540163\pi$
$620$	−19040.0	−1.23333
$621$	−3024.00	−0.195409
$622$	33240.0	2.14277
$623$	−7266.00	−0.467265
$624$	−8010.00	−0.513873
$625$	625.000	0.0400000
$626$	49090.0	3.13423
$627$	3312.00	0.210955
$628$	32606.0	2.07185
$629$	19564.0	1.24017
$630$	1575.00	0.0996024
$631$	2904.00	0.183211	0.0916057	−	0.995795i	$-0.470800\pi$
$631$	2904.00	0.183211	0.0916057	+	0.995795i	$0.470800\pi$
$632$	40320.0	2.53773
$633$	−1236.00	−0.0776091
$634$	4670.00	0.292538
$635$	3040.00	0.189982
$636$	38454.0	2.39748
$637$	1470.00	0.0914341
$638$	−3480.00	−0.215948
$639$	−8640.00	−0.534888
$640$	−10575.0	−0.653146
$641$	9330.00	0.574903	0.287452	−	0.957795i	$-0.407192\pi$
$641$	9330.00	0.574903	0.287452	+	0.957795i	$0.407192\pi$
$642$	−10980.0	−0.674994
$643$	−18332.0	−1.12433	−0.562164	−	0.827025i	$-0.690031\pi$
$643$	−18332.0	−1.12433	−0.562164	+	0.827025i	$0.690031\pi$
$644$	13328.0	0.815523
$645$	−5100.00	−0.311337
$646$	61640.0	3.75417
$647$	−2088.00	−0.126874	−0.0634372	−	0.997986i	$-0.520206\pi$
$647$	−2088.00	−0.126874	−0.0634372	+	0.997986i	$0.520206\pi$
$648$	3645.00	0.220971
$649$	4560.00	0.275802
$650$	3750.00	0.226288
$651$	4704.00	0.283202
$652$	39372.0	2.36492
$653$	22.0000	0.00131842	0.000659209	−	1.00000i	$-0.499790\pi$
$653$	22.0000	0.00131842	0.000659209		1.00000i	$0.499790\pi$
$654$	5670.00	0.339013
$655$	−4780.00	−0.285145
$656$	1602.00	0.0953469
$657$	9594.00	0.569707
$658$	7280.00	0.431313
$659$	16260.0	0.961153	0.480576	−	0.876953i	$-0.340427\pi$
$659$	16260.0	0.961153	0.480576	+	0.876953i	$0.340427\pi$
$660$	−3060.00	−0.180470
$661$	−23818.0	−1.40153	−0.700766	−	0.713391i	$-0.747158\pi$
$661$	−23818.0	−1.40153	−0.700766	+	0.713391i	$0.747158\pi$
$662$	11460.0	0.672818
$663$	12060.0	0.706443
$664$	19620.0	1.14669
$665$	−3220.00	−0.187769
$666$	−6570.00	−0.382256
$667$	−6496.00	−0.377101
$668$	−29512.0	−1.70936
$669$	4896.00	0.282945
$670$	10300.0	0.593916
$671$	8616.00	0.495703
$672$	−1785.00	−0.102467
$673$	31106.0	1.78165	0.890823	−	0.454350i	$-0.150128\pi$
$673$	31106.0	1.78165	0.890823	+	0.454350i	$0.150128\pi$
$674$	−30310.0	−1.73219
$675$	−675.000	−0.0384900
$676$	−22049.0	−1.25449
$677$	−1090.00	−0.0618790	−0.0309395	−	0.999521i	$-0.509850\pi$
$677$	−1090.00	−0.0618790	−0.0309395	+	0.999521i	$0.509850\pi$
$678$	−21870.0	−1.23881
$679$	−4914.00	−0.277735
$680$	−30150.0	−1.70029
$681$	−12252.0	−0.689424
$682$	−13440.0	−0.754610
$683$	−12372.0	−0.693121	−0.346560	−	0.938028i	$-0.612650\pi$
$683$	−12372.0	−0.693121	−0.346560	+	0.938028i	$0.612650\pi$
$684$	−14076.0	−0.786856
$685$	−1870.00	−0.104305
$686$	1715.00	0.0954504
$687$	10158.0	0.564122
$688$	30260.0	1.67682
$689$	−22620.0	−1.25073
$690$	−8400.00	−0.463453
$691$	3252.00	0.179033	0.0895166	−	0.995985i	$-0.471468\pi$
$691$	3252.00	0.179033	0.0895166	+	0.995985i	$0.471468\pi$
$692$	−41514.0	−2.28053
$693$	756.000	0.0414402
$694$	7420.00	0.405849
$695$	1980.00	0.108066
$696$	7830.00	0.426430
$697$	−2412.00	−0.131077
$698$	1270.00	0.0688685
$699$	−15966.0	−0.863934
$700$	2975.00	0.160635
$701$	−5434.00	−0.292781	−0.146390	−	0.989227i	$-0.546766\pi$
$701$	−5434.00	−0.292781	−0.146390	+	0.989227i	$0.546766\pi$
$702$	−4050.00	−0.217746
$703$	13432.0	0.720622
$704$	−3444.00	−0.184376
$705$	−3120.00	−0.166675
$706$	−54750.0	−2.91862
$707$	322.000	0.0171288
$708$	−19380.0	−1.02874
$709$	−5330.00	−0.282331	−0.141165	−	0.989986i	$-0.545085\pi$
$709$	−5330.00	−0.282331	−0.141165	+	0.989986i	$0.545085\pi$
$710$	−24000.0	−1.26860
$711$	8064.00	0.425350
$712$	−46710.0	−2.45861
$713$	−25088.0	−1.31775
$714$	14070.0	0.737474
$715$	1800.00	0.0941485
$716$	−69564.0	−3.63091
$717$	−11208.0	−0.583780
$718$	56880.0	2.95647
$719$	−7520.00	−0.390054	−0.195027	−	0.980798i	$-0.562479\pi$
$719$	−7520.00	−0.390054	−0.195027	+	0.980798i	$0.562479\pi$
$720$	4005.00	0.207302
$721$	13160.0	0.679756
$722$	8025.00	0.413656
$723$	−630.000	−0.0324066
$724$	21590.0	1.10827
$725$	−1450.00	−0.0742781
$726$	17805.0	0.910200
$727$	19336.0	0.986427	0.493214	−	0.869908i	$-0.335822\pi$
$727$	19336.0	0.986427	0.493214	+	0.869908i	$0.335822\pi$
$728$	9450.00	0.481099
$729$	729.000	0.0370370
$730$	26650.0	1.35118
$731$	−45560.0	−2.30519
$732$	−36618.0	−1.84896
$733$	−22498.0	−1.13367	−0.566837	−	0.823830i	$-0.691833\pi$
$733$	−22498.0	−1.13367	−0.566837	+	0.823830i	$0.691833\pi$
$734$	−5680.00	−0.285630
$735$	−735.000	−0.0368856
$736$	9520.00	0.476782
$737$	4944.00	0.247103
$738$	810.000	0.0404018
$739$	−18292.0	−0.910531	−0.455265	−	0.890356i	$-0.650456\pi$
$739$	−18292.0	−0.910531	−0.455265	+	0.890356i	$0.650456\pi$
$740$	−12410.0	−0.616487
$741$	8280.00	0.410490
$742$	−26390.0	−1.30567
$743$	17904.0	0.884030	0.442015	−	0.897008i	$-0.354264\pi$
$743$	17904.0	0.884030	0.442015	+	0.897008i	$0.354264\pi$
$744$	30240.0	1.49012
$745$	−9370.00	−0.460792
$746$	−41210.0	−2.02253
$747$	3924.00	0.192198
$748$	−27336.0	−1.33623
$749$	5124.00	0.249969
$750$	−1875.00	−0.0912871
$751$	5408.00	0.262771	0.131385	−	0.991331i	$-0.458057\pi$
$751$	5408.00	0.262771	0.131385	+	0.991331i	$0.458057\pi$
$752$	18512.0	0.897690
$753$	12636.0	0.611529
$754$	−8700.00	−0.420206
$755$	−5480.00	−0.264156
$756$	−3213.00	−0.154571
$757$	8318.00	0.399370	0.199685	−	0.979860i	$-0.436008\pi$
$757$	8318.00	0.399370	0.199685	+	0.979860i	$0.436008\pi$
$758$	18100.0	0.867311
$759$	−4032.00	−0.192823
$760$	−20700.0	−0.987984
$761$	6690.00	0.318676	0.159338	−	0.987224i	$-0.449064\pi$
$761$	6690.00	0.318676	0.159338	+	0.987224i	$0.449064\pi$
$762$	−9120.00	−0.433573
$763$	−2646.00	−0.125546
$764$	83368.0	3.94784
$765$	−6030.00	−0.284987
$766$	−42320.0	−1.99619
$767$	11400.0	0.536676
$768$	24837.0	1.16696
$769$	9266.00	0.434513	0.217257	−	0.976115i	$-0.430289\pi$
$769$	9266.00	0.434513	0.217257	+	0.976115i	$0.430289\pi$
$770$	2100.00	0.0982841
$771$	−15390.0	−0.718881
$772$	37026.0	1.72616
$773$	9678.00	0.450315	0.225157	−	0.974322i	$-0.427710\pi$
$773$	9678.00	0.450315	0.225157	+	0.974322i	$0.427710\pi$
$774$	15300.0	0.710526
$775$	−5600.00	−0.259559
$776$	−31590.0	−1.46136
$777$	3066.00	0.141560
$778$	18390.0	0.847447
$779$	−1656.00	−0.0761648
$780$	−7650.00	−0.351172
$781$	−11520.0	−0.527808
$782$	−75040.0	−3.43149
$783$	1566.00	0.0714742
$784$	4361.00	0.198661
$785$	9590.00	0.436028
$786$	14340.0	0.650752
$787$	−6860.00	−0.310715	−0.155357	−	0.987858i	$-0.549653\pi$
$787$	−6860.00	−0.310715	−0.155357	+	0.987858i	$0.549653\pi$
$788$	−48450.0	−2.19030
$789$	−2544.00	−0.114789
$790$	22400.0	1.00881
$791$	10206.0	0.458766
$792$	4860.00	0.218046
$793$	21540.0	0.964575
$794$	62950.0	2.81362
$795$	11310.0	0.504559
$796$	−19448.0	−0.865975
$797$	10950.0	0.486661	0.243331	−	0.969943i	$-0.421760\pi$
$797$	10950.0	0.486661	0.243331	+	0.969943i	$0.421760\pi$
$798$	9660.00	0.428522
$799$	−27872.0	−1.23409
$800$	2125.00	0.0939126
$801$	−9342.00	−0.412089
$802$	14250.0	0.627413
$803$	12792.0	0.562167
$804$	−21012.0	−0.921687
$805$	3920.00	0.171630
$806$	−33600.0	−1.46837
$807$	3822.00	0.166717
$808$	2070.00	0.0901267
$809$	26010.0	1.13036	0.565181	−	0.824967i	$-0.308806\pi$
$809$	26010.0	1.13036	0.565181	+	0.824967i	$0.308806\pi$
$810$	2025.00	0.0878410
$811$	−14628.0	−0.633364	−0.316682	−	0.948532i	$-0.602569\pi$
$811$	−14628.0	−0.633364	−0.316682	+	0.948532i	$0.602569\pi$
$812$	−6902.00	−0.298292
$813$	−2592.00	−0.111815
$814$	−8760.00	−0.377196
$815$	11580.0	0.497705
$816$	35778.0	1.53490
$817$	−31280.0	−1.33947
$818$	6130.00	0.262018
$819$	1890.00	0.0806373
$820$	1530.00	0.0651584
$821$	8718.00	0.370597	0.185299	−	0.982682i	$-0.440675\pi$
$821$	8718.00	0.370597	0.185299	+	0.982682i	$0.440675\pi$
$822$	5610.00	0.238043
$823$	−7432.00	−0.314779	−0.157390	−	0.987537i	$-0.550308\pi$
$823$	−7432.00	−0.314779	−0.157390	+	0.987537i	$0.550308\pi$
$824$	84600.0	3.57668
$825$	−900.000	−0.0379806
$826$	13300.0	0.560250
$827$	17388.0	0.731125	0.365562	−	0.930787i	$-0.380877\pi$
$827$	17388.0	0.731125	0.365562	+	0.930787i	$0.380877\pi$
$828$	17136.0	0.719224
$829$	7902.00	0.331059	0.165529	−	0.986205i	$-0.447067\pi$
$829$	7902.00	0.331059	0.165529	+	0.986205i	$0.447067\pi$
$830$	10900.0	0.455837
$831$	25590.0	1.06824
$832$	−8610.00	−0.358772
$833$	−6566.00	−0.273107
$834$	−5940.00	−0.246625
$835$	−8680.00	−0.359741
$836$	−18768.0	−0.776441
$837$	6048.00	0.249760
$838$	3060.00	0.126141
$839$	−31848.0	−1.31051	−0.655253	−	0.755409i	$-0.727438\pi$
$839$	−31848.0	−1.31051	−0.655253	+	0.755409i	$0.727438\pi$
$840$	−4725.00	−0.194081
$841$	−21025.0	−0.862069
$842$	25910.0	1.06047
$843$	16146.0	0.659665
$844$	7004.00	0.285649
$845$	−6485.00	−0.264013
$846$	9360.00	0.380382
$847$	−8309.00	−0.337073
$848$	−67106.0	−2.71749
$849$	−18708.0	−0.756251
$850$	−16750.0	−0.675906
$851$	−16352.0	−0.658683
$852$	48960.0	1.96871
$853$	30150.0	1.21022	0.605109	−	0.796142i	$-0.293129\pi$
$853$	30150.0	1.21022	0.605109	+	0.796142i	$0.293129\pi$
$854$	25130.0	1.00694
$855$	−4140.00	−0.165597
$856$	32940.0	1.31526
$857$	−4350.00	−0.173388	−0.0866938	−	0.996235i	$-0.527630\pi$
$857$	−4350.00	−0.173388	−0.0866938	+	0.996235i	$0.527630\pi$
$858$	−5400.00	−0.214864
$859$	−30676.0	−1.21845	−0.609227	−	0.792996i	$-0.708520\pi$
$859$	−30676.0	−1.21845	−0.609227	+	0.792996i	$0.708520\pi$
$860$	28900.0	1.14591
$861$	−378.000	−0.0149619
$862$	−24920.0	−0.984662
$863$	−23688.0	−0.934356	−0.467178	−	0.884163i	$-0.654729\pi$
$863$	−23688.0	−0.934356	−0.467178	+	0.884163i	$0.654729\pi$
$864$	−2295.00	−0.0903675
$865$	−12210.0	−0.479945
$866$	−8470.00	−0.332358
$867$	−39129.0	−1.53275
$868$	−26656.0	−1.04235
$869$	10752.0	0.419720
$870$	4350.00	0.169516
$871$	12360.0	0.480830
$872$	−17010.0	−0.660586
$873$	−6318.00	−0.244939
$874$	−51520.0	−1.99392
$875$	875.000	0.0338062
$876$	−54366.0	−2.09687
$877$	31910.0	1.22865	0.614324	−	0.789054i	$-0.289429\pi$
$877$	31910.0	1.22865	0.614324	+	0.789054i	$0.289429\pi$
$878$	69320.0	2.66451
$879$	2454.00	0.0941654
$880$	5340.00	0.204558
$881$	50250.0	1.92164	0.960820	−	0.277172i	$-0.0893971\pi$
$881$	50250.0	1.92164	0.960820	+	0.277172i	$0.0893971\pi$
$882$	2205.00	0.0841794
$883$	5980.00	0.227908	0.113954	−	0.993486i	$-0.463648\pi$
$883$	5980.00	0.227908	0.113954	+	0.993486i	$0.463648\pi$
$884$	−68340.0	−2.60014
$885$	−5700.00	−0.216501
$886$	−23220.0	−0.880464
$887$	−24568.0	−0.930003	−0.465002	−	0.885310i	$-0.653946\pi$
$887$	−24568.0	−0.930003	−0.465002	+	0.885310i	$0.653946\pi$
$888$	19710.0	0.744847
$889$	4256.00	0.160564
$890$	−25950.0	−0.977355
$891$	972.000	0.0365468
$892$	−27744.0	−1.04141
$893$	−19136.0	−0.717091
$894$	28110.0	1.05161
$895$	−20460.0	−0.764137
$896$	−14805.0	−0.552009
$897$	−10080.0	−0.375208
$898$	−24630.0	−0.915271
$899$	12992.0	0.481988
$900$	3825.00	0.141667
$901$	101036.	3.73585
$902$	1080.00	0.0398670
$903$	−7140.00	−0.263128
$904$	65610.0	2.41389
$905$	6350.00	0.233239
$906$	16440.0	0.602850
$907$	13252.0	0.485144	0.242572	−	0.970133i	$-0.422009\pi$
$907$	13252.0	0.485144	0.242572	+	0.970133i	$0.422009\pi$
$908$	69428.0	2.53750
$909$	414.000	0.0151062
$910$	5250.00	0.191248
$911$	−6744.00	−0.245267	−0.122634	−	0.992452i	$-0.539134\pi$
$911$	−6744.00	−0.245267	−0.122634	+	0.992452i	$0.539134\pi$
$912$	24564.0	0.891881
$913$	5232.00	0.189654
$914$	−73470.0	−2.65883
$915$	−10770.0	−0.389120
$916$	−57562.0	−2.07631
$917$	−6692.00	−0.240992
$918$	18090.0	0.650391
$919$	−45336.0	−1.62731	−0.813654	−	0.581349i	$-0.802525\pi$
$919$	−45336.0	−1.62731	−0.813654	+	0.581349i	$0.802525\pi$
$920$	25200.0	0.903065
$921$	6804.00	0.243430
$922$	10030.0	0.358265
$923$	−28800.0	−1.02705
$924$	−4284.00	−0.152525
$925$	−3650.00	−0.129742
$926$	24480.0	0.868750
$927$	16920.0	0.599488
$928$	−4930.00	−0.174391
$929$	30074.0	1.06211	0.531053	−	0.847339i	$-0.321797\pi$
$929$	30074.0	1.06211	0.531053	+	0.847339i	$0.321797\pi$
$930$	16800.0	0.592359
$931$	−4508.00	−0.158694
$932$	90474.0	3.17980
$933$	−19944.0	−0.699826
$934$	13300.0	0.465941
$935$	−8040.00	−0.281215
$936$	12150.0	0.424290
$937$	21754.0	0.758455	0.379227	−	0.925303i	$-0.376190\pi$
$937$	21754.0	0.758455	0.379227	+	0.925303i	$0.376190\pi$
$938$	14420.0	0.501951
$939$	−29454.0	−1.02364
$940$	17680.0	0.613466
$941$	14550.0	0.504056	0.252028	−	0.967720i	$-0.418903\pi$
$941$	14550.0	0.504056	0.252028	+	0.967720i	$0.418903\pi$
$942$	−28770.0	−0.995093
$943$	2016.00	0.0696182
$944$	33820.0	1.16605
$945$	−945.000	−0.0325300
$946$	20400.0	0.701122
$947$	46660.0	1.60110	0.800552	−	0.599263i	$-0.204540\pi$
$947$	46660.0	1.60110	0.800552	+	0.599263i	$0.204540\pi$
$948$	−45696.0	−1.56555
$949$	31980.0	1.09390
$950$	−11500.0	−0.392747
$951$	−2802.00	−0.0955427
$952$	−42210.0	−1.43701
$953$	20810.0	0.707347	0.353674	−	0.935369i	$-0.384932\pi$
$953$	20810.0	0.707347	0.353674	+	0.935369i	$0.384932\pi$
$954$	−33930.0	−1.15149
$955$	24520.0	0.830836
$956$	63512.0	2.14867
$957$	2088.00	0.0705282
$958$	−28000.0	−0.944300
$959$	−2618.00	−0.0881539
$960$	4305.00	0.144733
$961$	20385.0	0.684267
$962$	−21900.0	−0.733975
$963$	6588.00	0.220452
$964$	3570.00	0.119276
$965$	10890.0	0.363276
$966$	−11760.0	−0.391689
$967$	2776.00	0.0923166	0.0461583	−	0.998934i	$-0.485302\pi$
$967$	2776.00	0.0923166	0.0461583	+	0.998934i	$0.485302\pi$
$968$	−53415.0	−1.77358
$969$	−36984.0	−1.22611
$970$	−17550.0	−0.580924
$971$	27292.0	0.902000	0.451000	−	0.892524i	$-0.351067\pi$
$971$	27292.0	0.902000	0.451000	+	0.892524i	$0.351067\pi$
$972$	−4131.00	−0.136319
$973$	2772.00	0.0913322
$974$	−32120.0	−1.05666
$975$	−2250.00	−0.0739053
$976$	63902.0	2.09575
$977$	−62.0000	−0.00203025	−0.00101513	−	0.999999i	$-0.500323\pi$
$977$	−62.0000	−0.00203025	−0.00101513	+	0.999999i	$0.500323\pi$
$978$	−34740.0	−1.13585
$979$	−12456.0	−0.406635
$980$	4165.00	0.135761
$981$	−3402.00	−0.110721
$982$	−94500.0	−3.07089
$983$	37912.0	1.23012	0.615058	−	0.788481i	$-0.289132\pi$
$983$	37912.0	1.23012	0.615058	+	0.788481i	$0.289132\pi$
$984$	−2430.00	−0.0787252
$985$	−14250.0	−0.460957
$986$	38860.0	1.25513
$987$	−4368.00	−0.140866
$988$	−46920.0	−1.51085
$989$	38080.0	1.22434
$990$	2700.00	0.0866784
$991$	10656.0	0.341573	0.170787	−	0.985308i	$-0.445369\pi$
$991$	10656.0	0.341573	0.170787	+	0.985308i	$0.445369\pi$
$992$	−19040.0	−0.609396
$993$	−6876.00	−0.219741
$994$	−33600.0	−1.07216
$995$	−5720.00	−0.182247
$996$	−22236.0	−0.707404
$997$	−29434.0	−0.934989	−0.467495	−	0.883996i	$-0.654843\pi$
$997$	−29434.0	−0.934989	−0.467495	+	0.883996i	$0.654843\pi$
$998$	−76820.0	−2.43657
$999$	3942.00	0.124844

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.4.a.b.1.1	✓	1
3.2	odd	2		315.4.a.a.1.1		1
4.3	odd	2		1680.4.a.u.1.1		1
5.2	odd	4		525.4.d.a.274.2		2
5.3	odd	4		525.4.d.a.274.1		2
5.4	even	2		525.4.a.a.1.1		1
7.6	odd	2		735.4.a.j.1.1		1
15.14	odd	2		1575.4.a.l.1.1		1
21.20	even	2		2205.4.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.a.b.1.1	✓	1	1.1	even	1	trivial
315.4.a.a.1.1		1	3.2	odd	2
525.4.a.a.1.1		1	5.4	even	2
525.4.d.a.274.1		2	5.3	odd	4
525.4.d.a.274.2		2	5.2	odd	4
735.4.a.j.1.1		1	7.6	odd	2
1575.4.a.l.1.1		1	15.14	odd	2
1680.4.a.u.1.1		1	4.3	odd	2
2205.4.a.b.1.1		1	21.20	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 105 = 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	105.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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