LMFDB - Embedded newform 99.4.d.c.98.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [99,4,Mod(98,99)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(99, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("99.98"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 99 = 3^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	99.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0] 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:	no
Analytic conductor:	$5.84118909057$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.12261951429820416.4
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 62x^{6} + 1113x^{4} + 5786x^{2} + 5776 $ x^8 + 62x^6 + 1113x^4 + 5786*x^2 + 5776
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			98.2
Root			$-1.14042i$ of defining polynomial
Character	$\chi$	$=$	99.98
Dual form			99.4.d.c.98.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-5.00866 q^{2} +17.0866 q^{4} +7.07107i q^{5} -10.4716i q^{7} -45.5118 q^{8} -35.4165i q^{10} +(-13.0640 - 34.0636i) q^{11} +43.7271i q^{13} +52.4484i q^{14} +91.2599 q^{16} +115.416 q^{17} +89.3084i q^{19} +120.821i q^{20} +(65.4332 + 170.613i) q^{22} +213.357i q^{23} +75.0000 q^{25} -219.014i q^{26} -178.924i q^{28} +124.132 q^{29} -149.040 q^{31} -92.9950 q^{32} -578.079 q^{34} +74.0451 q^{35} +161.733 q^{37} -447.315i q^{38} -321.817i q^{40} -172.049 q^{41} +258.081i q^{43} +(-223.220 - 582.033i) q^{44} -1068.63i q^{46} +240.416i q^{47} +233.347 q^{49} -375.649 q^{50} +747.149i q^{52} +334.790i q^{53} +(240.866 - 92.3765i) q^{55} +476.579i q^{56} -621.733 q^{58} -382.685i q^{59} +609.752i q^{61} +746.488 q^{62} -264.299 q^{64} -309.197 q^{65} +260.000 q^{67} +1972.07 q^{68} -370.866 q^{70} -17.3487i q^{71} -787.834i q^{73} -810.063 q^{74} +1525.98i q^{76} +(-356.699 + 136.801i) q^{77} -1000.93i q^{79} +645.305i q^{80} +861.733 q^{82} +183.820 q^{83} +816.115i q^{85} -1292.64i q^{86} +(594.567 + 1550.30i) q^{88} +11.1459i q^{89} +457.891 q^{91} +3645.56i q^{92} -1204.16i q^{94} -631.506 q^{95} +1795.20 q^{97} -1168.75 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 44 q^{4} + 452 q^{16} + 60 q^{22} + 600 q^{25} - 80 q^{31} - 2400 q^{34} - 560 q^{37} + 1496 q^{49} + 1000 q^{55} - 3120 q^{58} - 724 q^{64} + 2080 q^{67} - 2040 q^{70} + 5040 q^{82} + 5220 q^{88}+ \cdots + 8800 q^{97}+O(q^{100}) $ 8 * q + 44 * q^4 + 452 * q^16 + 60 * q^22 + 600 * q^25 - 80 * q^31 - 2400 * q^34 - 560 * q^37 + 1496 * q^49 + 1000 * q^55 - 3120 * q^58 - 724 * q^64 + 2080 * q^67 - 2040 * q^70 + 5040 * q^82 + 5220 * q^88 - 2640 * q^91 + 8800 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/99\mathbb{Z}\right)^\times$.

$n$	$46$	$56$
$\chi(n)$	$-1$	$-1$

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 $

$2$	−5.00866			−1.77083			−0.885414	−	0.464804i	$-0.846125\pi$
$2$	−5.00866			−1.77083			−0.885414	+	0.464804i	$0.846125\pi$
$3$		0			0
$3$		0			0
$4$	17.0866			2.13583
$5$			7.07107i			0.632456i	0.948683	+	0.316228i	$0.102416\pi$
$5$			7.07107i			0.632456i	−0.948683	+	0.316228i	$0.897584\pi$
$6$		0			0
$7$		−	10.4716i		−	0.565411i	−0.959207	−	0.282705i	$-0.908768\pi$
$7$		−	10.4716i		−	0.565411i	0.959207	−	0.282705i	$-0.0912319\pi$
$8$	−45.5118			−2.01136
$9$		0			0
$10$		−	35.4165i		−	1.11997i
$11$	−13.0640	−	34.0636i	−0.358086	−	0.933689i
$11$	−13.0640	−	34.0636i	−0.358086	−	0.933689i
$12$		0			0
$13$			43.7271i			0.932901i	0.884547	+	0.466451i	$0.154467\pi$
$13$			43.7271i			0.932901i	−0.884547	+	0.466451i	$0.845533\pi$
$14$			52.4484i			1.00124i
$15$		0			0
$16$	91.2599			1.42594
$17$	115.416			1.64662			0.823309	−	0.567594i	$-0.192126\pi$
$17$	115.416			1.64662			0.823309	+	0.567594i	$0.192126\pi$
$18$		0			0
$19$			89.3084i			1.07836i	0.842192	+	0.539178i	$0.181265\pi$
$19$			89.3084i			1.07836i	−0.842192	+	0.539178i	$0.818735\pi$
$20$			120.821i			1.35082i
$21$		0			0
$22$	65.4332	+	170.613i	0.634109	+	1.65340i
$23$			213.357i			1.93426i	0.254277	+	0.967131i	$0.418162\pi$
$23$			213.357i			1.93426i	−0.254277	+	0.967131i	$0.581838\pi$
$24$		0			0
$25$	75.0000			0.600000
$26$		−	219.014i		−	1.65201i
$27$		0			0
$28$		−	178.924i		−	1.20762i
$29$	124.132			0.794851			0.397425	−	0.917635i	$-0.369904\pi$
$29$	124.132			0.794851			0.397425	+	0.917635i	$0.369904\pi$
$30$		0			0
$31$	−149.040			−0.863493			−0.431747	−	0.901995i	$-0.642103\pi$
$31$	−149.040			−0.863493			−0.431747	+	0.901995i	$0.642103\pi$
$32$	−92.9950			−0.513729
$33$		0			0
$34$	−578.079			−2.91588
$35$	74.0451			0.357597
$36$		0			0
$37$	161.733			0.718613			0.359306	−	0.933220i	$-0.383013\pi$
$37$	161.733			0.718613			0.359306	+	0.933220i	$0.383013\pi$
$38$		−	447.315i		−	1.90958i
$39$		0			0
$40$		−	321.817i		−	1.27209i
$41$	−172.049			−0.655353			−0.327677	−	0.944790i	$-0.606266\pi$
$41$	−172.049			−0.655353			−0.327677	+	0.944790i	$0.606266\pi$
$42$		0			0
$43$			258.081i			0.915277i	0.889138	+	0.457639i	$0.151305\pi$
$43$			258.081i			0.915277i	−0.889138	+	0.457639i	$0.848695\pi$
$44$	−223.220	−	582.033i	−0.764811	−	1.99420i
$45$		0			0
$46$		−	1068.63i		−	3.42525i
$47$			240.416i			0.746135i	0.927804	+	0.373067i	$0.121694\pi$
$47$			240.416i			0.746135i	−0.927804	+	0.373067i	$0.878306\pi$
$48$		0			0
$49$	233.347			0.680311
$50$	−375.649			−1.06250
$51$		0			0
$52$			747.149i			1.99252i
$53$			334.790i			0.867679i	0.900990	+	0.433840i	$0.142842\pi$
$53$			334.790i			0.867679i	−0.900990	+	0.433840i	$0.857158\pi$
$54$		0			0
$55$	240.866	−	92.3765i	0.590516	−	0.226474i
$56$			476.579i			1.13724i
$57$		0			0
$58$	−621.733			−1.40754
$59$		−	382.685i		−	0.844429i	−0.906496	−	0.422214i	$-0.861253\pi$
$59$		−	382.685i		−	0.844429i	0.906496	−	0.422214i	$-0.138747\pi$
$60$		0			0
$61$			609.752i			1.27985i	0.768438	+	0.639924i	$0.221034\pi$
$61$			609.752i			1.27985i	−0.768438	+	0.639924i	$0.778966\pi$
$62$	746.488			1.52910
$63$		0			0
$64$	−264.299			−0.516210
$65$	−309.197			−0.590019
$66$		0			0
$67$	260.000			0.474090			0.237045	−	0.971499i	$-0.423821\pi$
$67$	260.000			0.474090			0.237045	+	0.971499i	$0.423821\pi$
$68$	1972.07			3.51689
$69$		0			0
$70$	−370.866			−0.633243
$71$		−	17.3487i		−	0.0289988i	−0.999895	−	0.0144994i	$-0.995385\pi$
$71$		−	17.3487i		−	0.0289988i	0.999895	−	0.0144994i	$-0.00461546\pi$
$72$		0			0
$73$		−	787.834i		−	1.26314i	−0.775320	−	0.631569i	$-0.782411\pi$
$73$		−	787.834i		−	1.26314i	0.775320	−	0.631569i	$-0.217589\pi$
$74$	−810.063			−1.27254
$75$		0			0
$76$			1525.98i			2.30318i
$77$	−356.699	+	136.801i	−0.527918	+	0.202466i
$78$		0			0
$79$		−	1000.93i		−	1.42549i	−0.701422	−	0.712746i	$-0.747451\pi$
$79$		−	1000.93i		−	1.42549i	0.701422	−	0.712746i	$-0.252549\pi$
$80$			645.305i			0.901841i
$81$		0			0
$82$	861.733			1.16052
$83$	183.820			0.243095			0.121548	−	0.992586i	$-0.461214\pi$
$83$	183.820			0.243095			0.121548	+	0.992586i	$0.461214\pi$
$84$		0			0
$85$			816.115i			1.04141i
$86$		−	1292.64i		−	1.62080i
$87$		0			0
$88$	594.567	+	1550.30i	0.720239	+	1.87798i
$89$			11.1459i			0.0132748i	0.999978	+	0.00663741i	$0.00211277\pi$
$89$			11.1459i			0.0132748i	−0.999978	+	0.00663741i	$0.997887\pi$
$90$		0			0
$91$	457.891			0.527473
$92$			3645.56i			4.13125i
$93$		0			0
$94$		−	1204.16i		−	1.32128i
$95$	−631.506			−0.682012
$96$		0			0
$97$	1795.20			1.87912			0.939560	−	0.342383i	$-0.111234\pi$
$97$	1795.20			1.87912			0.939560	+	0.342383i	$0.111234\pi$
$98$	−1168.75			−1.20471
$99$		0			0
$100$	1281.50			1.28150
$101$	−346.173			−0.341044			−0.170522	−	0.985354i	$-0.554545\pi$
$101$	−346.173			−0.341044			−0.170522	+	0.985354i	$0.554545\pi$
$102$		0			0
$103$	−940.000			−0.899233			−0.449616	−	0.893222i	$-0.648439\pi$
$103$	−940.000			−0.899233			−0.449616	+	0.893222i	$0.648439\pi$
$104$		−	1990.10i		−	1.87640i
$105$		0			0
$106$		−	1676.85i		−	1.53651i
$107$	−1826.66			−1.65037			−0.825186	−	0.564861i	$-0.808930\pi$
$107$	−1826.66			−1.65037			−0.825186	+	0.564861i	$0.808930\pi$
$108$		0			0
$109$		−	1419.73i		−	1.24757i	−0.781595	−	0.623787i	$-0.785593\pi$
$109$		−	1419.73i		−	1.24757i	0.781595	−	0.623787i	$-0.214407\pi$
$110$	−1206.42	+	462.682i	−1.04570	+	0.401046i
$111$		0			0
$112$		−	955.633i		−	0.806240i
$113$		−	763.955i		−	0.635990i	−0.948092	−	0.317995i	$-0.896991\pi$
$113$		−	763.955i		−	0.635990i	0.948092	−	0.317995i	$-0.103009\pi$
$114$		0			0
$115$	−1508.66			−1.22334
$116$	2120.99			1.69766
$117$		0			0
$118$			1916.74i			1.49534i
$119$		−	1208.59i		−	0.931015i
$120$		0			0
$121$	−989.663	+	890.016i	−0.743548	+	0.668682i
$122$		−	3054.04i		−	2.26639i
$123$		0			0
$124$	−2546.58			−1.84427
$125$			1414.21i			1.01193i
$126$		0			0
$127$		−	453.826i		−	0.317091i	−0.987352	−	0.158545i	$-0.949320\pi$
$127$		−	453.826i		−	0.317091i	0.987352	−	0.158545i	$-0.0506804\pi$
$128$	2067.74			1.42785
$129$		0			0
$130$	1548.66			1.04482
$131$	2674.02			1.78344			0.891718	−	0.452591i	$-0.149500\pi$
$131$	2674.02			1.78344			0.891718	+	0.452591i	$0.149500\pi$
$132$		0			0
$133$	935.198			0.609714
$134$	−1302.25			−0.839532
$135$		0			0
$136$	−5252.79			−3.31193
$137$			1209.71i			0.754399i	0.926132	+	0.377200i	$0.123113\pi$
$137$			1209.71i			0.754399i	−0.926132	+	0.377200i	$0.876887\pi$
$138$		0			0
$139$			175.615i			0.107162i	0.998564	+	0.0535809i	$0.0170635\pi$
$139$			175.615i			0.107162i	−0.998564	+	0.0535809i	$0.982937\pi$
$140$	1265.18			0.763766
$141$		0			0
$142$			86.8937i			0.0513518i
$143$	1489.50	−	571.252i	0.871039	−	0.334059i
$144$		0			0
$145$			877.743i			0.502708i
$146$			3945.99i			2.23680i
$147$		0			0
$148$	2763.47			1.53483
$149$	−1578.81			−0.868062			−0.434031	−	0.900898i	$-0.642909\pi$
$149$	−1578.81			−0.868062			−0.434031	+	0.900898i	$0.642909\pi$
$150$		0			0
$151$			1333.62i			0.718733i	0.933197	+	0.359366i	$0.117007\pi$
$151$			1333.62i			0.718733i	−0.933197	+	0.359366i	$0.882993\pi$
$152$		−	4064.59i		−	2.16896i
$153$		0			0
$154$	1786.58	−	685.187i	0.934851	−	0.358532i
$155$		−	1053.87i		−	0.546121i
$156$		0			0
$157$	−970.000			−0.493086			−0.246543	−	0.969132i	$-0.579295\pi$
$157$	−970.000			−0.493086			−0.246543	+	0.969132i	$0.579295\pi$
$158$			5013.33i			2.52430i
$159$		0			0
$160$		−	657.574i		−	0.324911i
$161$	2234.18			1.09365
$162$		0			0
$163$	−2499.06			−1.20087			−0.600434	−	0.799675i	$-0.705005\pi$
$163$	−2499.06			−1.20087			−0.600434	+	0.799675i	$0.705005\pi$
$164$	−2939.73			−1.39972
$165$		0			0
$166$	−920.693			−0.430480
$167$	2314.23			1.07234			0.536168	−	0.844111i	$-0.319871\pi$
$167$	2314.23			1.07234			0.536168	+	0.844111i	$0.319871\pi$
$168$		0			0
$169$	284.940			0.129695
$170$		−	4087.64i		−	1.84416i
$171$		0			0
$172$			4409.73i			1.95488i
$173$	929.968			0.408695			0.204347	−	0.978898i	$-0.434493\pi$
$173$	929.968			0.408695			0.204347	+	0.978898i	$0.434493\pi$
$174$		0			0
$175$		−	785.367i		−	0.339247i
$176$	−1192.22	−	3108.64i	−0.510608	−	1.33138i
$177$		0			0
$178$		−	55.8258i		−	0.0235074i
$179$		−	973.917i		−	0.406670i	−0.979109	−	0.203335i	$-0.934822\pi$
$179$		−	973.917i		−	0.406670i	0.979109	−	0.203335i	$-0.0651781\pi$
$180$		0			0
$181$	560.377			0.230124			0.115062	−	0.993358i	$-0.463293\pi$
$181$	560.377			0.230124			0.115062	+	0.993358i	$0.463293\pi$
$182$	−2293.42			−0.934063
$183$		0			0
$184$		−	9710.27i		−	3.89049i
$185$			1143.62i			0.454491i
$186$		0			0
$187$	−1507.80	−	3931.49i	−0.589631	−	1.53743i
$188$			4107.90i			1.59362i
$189$		0			0
$190$	3162.99			1.20772
$191$		−	2226.87i		−	0.843615i	−0.906685	−	0.421808i	$-0.861396\pi$
$191$		−	2226.87i		−	0.843615i	0.906685	−	0.421808i	$-0.138604\pi$
$192$		0			0
$193$			4036.71i			1.50554i	0.658284	+	0.752769i	$0.271282\pi$
$193$			4036.71i			1.50554i	−0.658284	+	0.752769i	$0.728718\pi$
$194$	−8991.53			−3.32760
$195$		0			0
$196$	3987.11			1.45303
$197$	−1369.56			−0.495316			−0.247658	−	0.968848i	$-0.579661\pi$
$197$	−1369.56			−0.495316			−0.247658	+	0.968848i	$0.579661\pi$
$198$		0			0
$199$	−2363.47			−0.841918			−0.420959	−	0.907080i	$-0.638306\pi$
$199$	−2363.47			−0.841918			−0.420959	+	0.907080i	$0.638306\pi$
$200$	−3413.39			−1.20681
$201$		0			0
$202$	1733.86			0.603931
$203$		−	1299.85i		−	0.449417i
$204$		0			0
$205$		−	1216.57i		−	0.414482i
$206$	4708.14			1.59239
$207$		0			0
$208$			3990.53i			1.33026i
$209$	3042.17	−	1166.73i	1.00685	−	0.386144i
$210$		0			0
$211$			3156.87i			1.02999i	0.857193	+	0.514996i	$0.172206\pi$
$211$			3156.87i			1.02999i	−0.857193	+	0.514996i	$0.827794\pi$
$212$			5720.44i			1.85321i
$213$		0			0
$214$	9149.11			2.92252
$215$	−1824.91			−0.578872
$216$		0			0
$217$			1560.68i			0.488229i
$218$			7110.94i			2.20924i
$219$		0			0
$220$	4115.59	−	1578.40i	1.26124	−	0.483709i
$221$			5046.81i			1.53613i
$222$		0			0
$223$	−1572.13			−0.472096			−0.236048	−	0.971741i	$-0.575852\pi$
$223$	−1572.13			−0.472096			−0.236048	+	0.971741i	$0.575852\pi$
$224$			973.802i			0.290468i
$225$		0			0
$226$			3826.39i			1.12623i
$227$	1189.02			0.347657			0.173829	−	0.984776i	$-0.444386\pi$
$227$	1189.02			0.347657			0.173829	+	0.984776i	$0.444386\pi$
$228$		0			0
$229$	2068.27			0.596834			0.298417	−	0.954436i	$-0.403541\pi$
$229$	2068.27			0.596834			0.298417	+	0.954436i	$0.403541\pi$
$230$	7556.37			2.16632
$231$		0			0
$232$	−5649.45			−1.59873
$233$	−2563.83			−0.720867			−0.360433	−	0.932785i	$-0.617371\pi$
$233$	−2563.83			−0.720867			−0.360433	+	0.932785i	$0.617371\pi$
$234$		0			0
$235$	−1700.00			−0.471897
$236$		−	6538.79i		−	1.80356i
$237$		0			0
$238$			6053.39i			1.64867i
$239$	−2556.68			−0.691957			−0.345979	−	0.938242i	$-0.612453\pi$
$239$	−2556.68			−0.691957			−0.345979	+	0.938242i	$0.612453\pi$
$240$		0			0
$241$		−	5996.13i		−	1.60268i	−0.598212	−	0.801338i	$-0.704122\pi$
$241$		−	5996.13i		−	1.60268i	0.598212	−	0.801338i	$-0.295878\pi$
$242$	4956.88	−	4457.78i	1.31670	−	1.18412i
$243$		0			0
$244$			10418.6i			2.73353i
$245$			1650.01i			0.430266i
$246$		0			0
$247$	−3905.20			−1.00600
$248$	6783.06			1.73679
$249$		0			0
$250$		−	7083.31i		−	1.79195i
$251$		−	1959.54i		−	0.492769i	−0.969172	−	0.246385i	$-0.920757\pi$
$251$		−	1959.54i		−	0.492769i	0.969172	−	0.246385i	$-0.0792427\pi$
$252$		0			0
$253$	7267.72	−	2787.30i	1.80600	−	0.692633i
$254$			2273.06i			0.561513i
$255$		0			0
$256$	−8242.22			−2.01226
$257$		−	5441.22i		−	1.32068i	−0.750968	−	0.660339i	$-0.770413\pi$
$257$		−	5441.22i		−	1.32068i	0.750968	−	0.660339i	$-0.229587\pi$
$258$		0			0
$259$		−	1693.59i		−	0.406312i
$260$	−5283.14			−1.26018
$261$		0			0
$262$	−13393.2			−3.15816
$263$	4960.72			1.16308			0.581542	−	0.813516i	$-0.302450\pi$
$263$	4960.72			1.16308			0.581542	+	0.813516i	$0.302450\pi$
$264$		0			0
$265$	−2367.33			−0.548769
$266$	−4684.08			−1.07970
$267$		0			0
$268$	4442.52			1.01258
$269$			3662.16i			0.830058i	0.909808	+	0.415029i	$0.136229\pi$
$269$			3662.16i			0.830058i	−0.909808	+	0.415029i	$0.863771\pi$
$270$		0			0
$271$		−	3106.99i		−	0.696443i	−0.937412	−	0.348221i	$-0.886786\pi$
$271$		−	3106.99i		−	0.696443i	0.937412	−	0.348221i	$-0.113214\pi$
$272$	10532.9			2.34797
$273$		0			0
$274$		−	6059.03i		−	1.33591i
$275$	−979.801	−	2554.77i	−0.214852	−	0.560213i
$276$		0			0
$277$			4313.35i			0.935610i	0.883832	+	0.467805i	$0.154955\pi$
$277$			4313.35i			0.935610i	−0.883832	+	0.467805i	$0.845045\pi$
$278$		−	879.595i		−	0.189765i
$279$		0			0
$280$	−3369.92			−0.719255
$281$	−6300.34			−1.33753			−0.668767	−	0.743472i	$-0.733177\pi$
$281$	−6300.34			−1.33753			−0.668767	+	0.743472i	$0.733177\pi$
$282$		0			0
$283$			5388.72i			1.13189i	0.824442	+	0.565947i	$0.191489\pi$
$283$			5388.72i			1.13189i	−0.824442	+	0.565947i	$0.808511\pi$
$284$		−	296.431i		−	0.0619364i
$285$		0			0
$286$	−7460.41	+	2861.20i	−1.54246	+	0.591561i
$287$			1801.62i			0.370544i
$288$		0			0
$289$	8407.86			1.71135
$290$		−	4396.31i		−	0.890208i
$291$		0			0
$292$		−	13461.4i		−	2.69785i
$293$	−1569.12			−0.312862			−0.156431	−	0.987689i	$-0.549999\pi$
$293$	−1569.12			−0.312862			−0.156431	+	0.987689i	$0.549999\pi$
$294$		0			0
$295$	2705.99			0.534064
$296$	−7360.74			−1.44539
$297$		0			0
$298$	7907.72			1.53719
$299$	−9329.49			−1.80448
$300$		0			0
$301$	2702.50			0.517508
$302$		−	6679.66i		−	1.27275i
$303$		0			0
$304$			8150.27i			1.53767i
$305$	−4311.59			−0.809447
$306$		0			0
$307$		−	8427.74i		−	1.56676i	−0.621541	−	0.783382i	$-0.713493\pi$
$307$		−	8427.74i		−	1.56676i	0.621541	−	0.783382i	$-0.286507\pi$
$308$	−6094.79	+	2337.46i	−1.12754	+	0.432432i
$309$		0			0
$310$			5278.47i			0.967086i
$311$		−	4650.78i		−	0.847979i	−0.905667	−	0.423990i	$-0.860629\pi$
$311$		−	4650.78i		−	0.847979i	0.905667	−	0.423990i	$-0.139371\pi$
$312$		0			0
$313$	3019.06			0.545199			0.272599	−	0.962128i	$-0.412117\pi$
$313$	3019.06			0.545199			0.272599	+	0.962128i	$0.412117\pi$
$314$	4858.40			0.873169
$315$		0			0
$316$		−	17102.6i		−	3.04461i
$317$		−	4790.12i		−	0.848707i	−0.905497	−	0.424353i	$-0.860501\pi$
$317$		−	4790.12i		−	0.848707i	0.905497	−	0.424353i	$-0.139499\pi$
$318$		0			0
$319$	−1621.66	−	4228.38i	−0.284625	−	0.742143i
$320$		−	1868.88i		−	0.326480i
$321$		0			0
$322$	−11190.2			−1.93667
$323$			10307.6i			1.77564i
$324$		0			0
$325$			3279.53i			0.559741i
$326$	12516.9			2.12653
$327$		0			0
$328$	7830.25			1.31815
$329$	2517.53			0.421873
$330$		0			0
$331$	−2129.04			−0.353543			−0.176771	−	0.984252i	$-0.556565\pi$
$331$	−2129.04			−0.353543			−0.176771	+	0.984252i	$0.556565\pi$
$332$	3140.87			0.519210
$333$		0			0
$334$	−11591.2			−1.89892
$335$			1838.48i			0.299841i
$336$		0			0
$337$			4473.34i			0.723082i	0.932356	+	0.361541i	$0.117749\pi$
$337$			4473.34i			0.723082i	−0.932356	+	0.361541i	$0.882251\pi$
$338$	−1427.17			−0.229667
$339$		0			0
$340$			13944.6i			2.22428i
$341$	1947.06	+	5076.83i	0.309205	+	0.806234i
$342$		0			0
$343$		−	6035.24i		−	0.950066i
$344$		−	11745.7i		−	1.84095i
$345$		0			0
$346$	−4657.89			−0.723728
$347$	3349.52			0.518189			0.259094	−	0.965852i	$-0.416576\pi$
$347$	3349.52			0.518189			0.259094	+	0.965852i	$0.416576\pi$
$348$		0			0
$349$			11357.7i			1.74202i	0.491268	+	0.871009i	$0.336534\pi$
$349$			11357.7i			1.74202i	−0.491268	+	0.871009i	$0.663466\pi$
$350$			3933.63i			0.600747i
$351$		0			0
$352$	1214.89	+	3167.75i	0.183959	+	0.479663i
$353$		−	3609.53i		−	0.544238i	−0.962264	−	0.272119i	$-0.912276\pi$
$353$		−	3609.53i		−	0.544238i	0.962264	−	0.272119i	$-0.0877245\pi$
$354$		0			0
$355$	122.674			0.0183404
$356$			190.445i			0.0283527i
$357$		0			0
$358$			4878.01i			0.720142i
$359$	−199.970			−0.0293984			−0.0146992	−	0.999892i	$-0.504679\pi$
$359$	−199.970			−0.0293984			−0.0146992	+	0.999892i	$0.504679\pi$
$360$		0			0
$361$	−1116.99			−0.162850
$362$	−2806.73			−0.407510
$363$		0			0
$364$	7823.81			1.12659
$365$	5570.83			0.798878
$366$		0			0
$367$	7160.00			1.01839			0.509195	−	0.860651i	$-0.329943\pi$
$367$	7160.00			1.01839			0.509195	+	0.860651i	$0.329943\pi$
$368$			19471.0i			2.75813i
$369$		0			0
$370$		−	5728.01i		−	0.804824i
$371$	3505.78			0.490595
$372$		0			0
$373$		−	28.9446i		−	0.00401794i	−0.999998	−	0.00200897i	$-0.999361\pi$
$373$		−	28.9446i		−	0.00401794i	0.999998	−	0.00200897i	$-0.000639476\pi$
$374$	7552.03	+	19691.5i	1.04413	+	2.72252i
$375$		0			0
$376$		−	10941.8i		−	1.50074i
$377$			5427.92i			0.741517i
$378$		0			0
$379$	−4354.80			−0.590214			−0.295107	−	0.955464i	$-0.595355\pi$
$379$	−4354.80			−0.590214			−0.295107	+	0.955464i	$0.595355\pi$
$380$	−10790.3			−1.45666
$381$		0			0
$382$			11153.6i			1.49390i
$383$		−	2755.16i		−	0.367577i	−0.982966	−	0.183789i	$-0.941164\pi$
$383$		−	2755.16i		−	0.367577i	0.982966	−	0.183789i	$-0.0588362\pi$
$384$		0			0
$385$	−967.326	−	2522.24i	−0.128051	−	0.333884i
$386$		−	20218.5i		−	2.66605i
$387$		0			0
$388$	30673.9			4.01348
$389$			8038.64i			1.04775i	0.851795	+	0.523876i	$0.175514\pi$
$389$			8038.64i			1.04775i	−0.851795	+	0.523876i	$0.824486\pi$
$390$		0			0
$391$			24624.8i			3.18499i
$392$	−10620.0			−1.36835
$393$		0			0
$394$	6859.66			0.877119
$395$	7077.67			0.901560
$396$		0			0
$397$	10575.3			1.33692			0.668462	−	0.743746i	$-0.266953\pi$
$397$	10575.3			1.33692			0.668462	+	0.743746i	$0.266953\pi$
$398$	11837.8			1.49089
$399$		0			0
$400$	6844.49			0.855561
$401$			14007.8i			1.74443i	0.489125	+	0.872214i	$0.337316\pi$
$401$			14007.8i			1.74443i	−0.489125	+	0.872214i	$0.662684\pi$
$402$		0			0
$403$		−	6517.07i		−	0.805554i
$404$	−5914.93			−0.728413
$405$		0			0
$406$			6510.51i			0.795840i
$407$	−2112.88	−	5509.20i	−0.257325	−	0.670961i
$408$		0			0
$409$		−	7098.52i		−	0.858188i	−0.903260	−	0.429094i	$-0.858833\pi$
$409$		−	7098.52i		−	0.858188i	0.903260	−	0.429094i	$-0.141167\pi$
$410$			6093.37i			0.733976i
$411$		0			0
$412$	−16061.4			−1.92061
$413$	−4007.30			−0.477449
$414$		0			0
$415$			1299.81i			0.153747i
$416$		−	4066.40i		−	0.479259i
$417$		0			0
$418$	−15237.2	+	5843.73i	−1.78295	+	0.683795i
$419$		−	1753.62i		−	0.204463i	−0.994761	−	0.102232i	$-0.967402\pi$
$419$		−	1753.62i		−	0.204463i	0.994761	−	0.102232i	$-0.0325983\pi$
$420$		0			0
$421$	−7738.53			−0.895850			−0.447925	−	0.894071i	$-0.647837\pi$
$421$	−7738.53			−0.895850			−0.447925	+	0.894071i	$0.647837\pi$
$422$		−	15811.7i		−	1.82394i
$423$		0			0
$424$		−	15236.9i		−	1.74521i
$425$	8656.20			0.987971
$426$		0			0
$427$	6385.05			0.723640
$428$	−31211.5			−3.52491
$429$		0			0
$430$	9140.32			1.02508
$431$	7260.66			0.811447			0.405724	−	0.913996i	$-0.367020\pi$
$431$	7260.66			0.811447			0.405724	+	0.913996i	$0.367020\pi$
$432$		0			0
$433$	−13808.4			−1.53253			−0.766267	−	0.642522i	$-0.777888\pi$
$433$	−13808.4			−1.53253			−0.766267	+	0.642522i	$0.777888\pi$
$434$		−	7816.89i		−	0.864568i
$435$		0			0
$436$		−	24258.4i		−	2.66460i
$437$	−19054.6			−2.08582
$438$		0			0
$439$			2417.86i			0.262866i	0.991325	+	0.131433i	$0.0419578\pi$
$439$			2417.86i			0.262866i	−0.991325	+	0.131433i	$0.958042\pi$
$440$	−10962.3	+	4204.22i	−1.18774	+	0.455519i
$441$		0			0
$442$		−	25277.7i		−	2.72022i
$443$		−	7612.14i		−	0.816397i	−0.912893	−	0.408199i	$-0.866157\pi$
$443$		−	7612.14i		−	0.816397i	0.912893	−	0.408199i	$-0.133843\pi$
$444$		0			0
$445$	−78.8131			−0.00839573
$446$	7874.25			0.836001
$447$		0			0
$448$			2767.63i			0.291871i
$449$		−	9823.10i		−	1.03247i	−0.856446	−	0.516237i	$-0.827332\pi$
$449$		−	9823.10i		−	1.03247i	0.856446	−	0.516237i	$-0.172668\pi$
$450$		0			0
$451$	2247.65	+	5860.60i	0.234673	+	0.611896i
$452$		−	13053.4i		−	1.35837i
$453$		0			0
$454$	−5955.41			−0.615641
$455$			3237.78i			0.333603i
$456$		0			0
$457$		−	11281.5i		−	1.15477i	−0.816473	−	0.577383i	$-0.804074\pi$
$457$		−	11281.5i		−	1.15477i	0.816473	−	0.577383i	$-0.195926\pi$
$458$	−10359.2			−1.05689
$459$		0			0
$460$	−25778.0			−2.61283
$461$	12836.5			1.29687			0.648433	−	0.761272i	$-0.275425\pi$
$461$	12836.5			1.29687			0.648433	+	0.761272i	$0.275425\pi$
$462$		0			0
$463$	−10788.5			−1.08290			−0.541452	−	0.840732i	$-0.682125\pi$
$463$	−10788.5			−1.08290			−0.541452	+	0.840732i	$0.682125\pi$
$464$	11328.2			1.13341
$465$		0			0
$466$	12841.3			1.27653
$467$			9953.72i			0.986302i	0.869944	+	0.493151i	$0.164155\pi$
$467$			9953.72i			0.986302i	−0.869944	+	0.493151i	$0.835845\pi$
$468$		0			0
$469$		−	2722.60i		−	0.268056i
$470$	8514.71			0.835648
$471$		0			0
$472$			17416.7i			1.69845i
$473$	8791.16	−	3371.57i	0.854584	−	0.327748i
$474$		0			0
$475$			6698.13i			0.647013i
$476$		−	20650.7i		−	1.98849i
$477$		0			0
$478$	12805.5			1.22534
$479$	−2034.49			−0.194068			−0.0970338	−	0.995281i	$-0.530935\pi$
$479$	−2034.49			−0.194068			−0.0970338	+	0.995281i	$0.530935\pi$
$480$		0			0
$481$			7072.10i			0.670395i
$482$			30032.6i			2.83806i
$483$		0			0
$484$	−16910.0	+	15207.4i	−1.58809	+	1.42819i
$485$			12694.0i			1.18846i
$486$		0			0
$487$	8290.55			0.771418			0.385709	−	0.922621i	$-0.373957\pi$
$487$	8290.55			0.771418			0.385709	+	0.922621i	$0.373957\pi$
$488$		−	27750.9i		−	2.57423i
$489$		0			0
$490$		−	8264.33i		−	0.761927i
$491$	15107.5			1.38858			0.694288	−	0.719698i	$-0.255720\pi$
$491$	15107.5			1.38858			0.694288	+	0.719698i	$0.255720\pi$
$492$		0			0
$493$	14326.8			1.30881
$494$	19559.8			1.78145
$495$		0			0
$496$	−13601.3			−1.23129
$497$	−181.668			−0.0163962
$498$		0			0
$499$	5310.01			0.476370			0.238185	−	0.971220i	$-0.423448\pi$
$499$	5310.01			0.476370			0.238185	+	0.971220i	$0.423448\pi$
$500$			24164.1i			2.16131i
$501$		0			0
$502$			9814.66i			0.872609i
$503$	−12434.1			−1.10221			−0.551104	−	0.834437i	$-0.685793\pi$
$503$	−12434.1			−1.10221			−0.551104	+	0.834437i	$0.685793\pi$
$504$		0			0
$505$		−	2447.81i		−	0.215695i
$506$	−36401.5	+	13960.6i	−3.19811	+	1.22653i
$507$		0			0
$508$		−	7754.35i		−	0.677252i
$509$		−	15549.5i		−	1.35406i	−0.735954	−	0.677031i	$-0.763266\pi$
$509$		−	15549.5i		−	1.35406i	0.735954	−	0.677031i	$-0.236734\pi$
$510$		0			0
$511$	−8249.85			−0.714192
$512$	24740.5			2.13552
$513$		0			0
$514$			27253.2i			2.33869i
$515$		−	6646.80i		−	0.568725i
$516$		0			0
$517$	8189.45	−	3140.80i	0.696657	−	0.267181i
$518$			8482.62i			0.719507i
$519$		0			0
$520$	14072.1			1.18674
$521$		−	13129.1i		−	1.10402i	−0.833837	−	0.552011i	$-0.813861\pi$
$521$		−	13129.1i		−	1.10402i	0.833837	−	0.552011i	$-0.186139\pi$
$522$		0			0
$523$		−	7777.97i		−	0.650300i	−0.945662	−	0.325150i	$-0.894585\pi$
$523$		−	7777.97i		−	0.650300i	0.945662	−	0.325150i	$-0.105415\pi$
$524$	45690.0			3.80911
$525$		0			0
$526$	−24846.5			−2.05962
$527$	−17201.6			−1.42184
$528$		0			0
$529$	−33354.3			−2.74137
$530$	11857.1			0.971774
$531$		0			0
$532$	15979.4			1.30224
$533$		−	7523.19i		−	0.611380i
$534$		0			0
$535$		−	12916.4i		−	1.04379i
$536$	−11833.1			−0.953565
$537$		0			0
$538$		−	18342.5i		−	1.46989i
$539$	−3048.44	−	7948.63i	−0.243610	−	0.635198i
$540$		0			0
$541$			7276.87i			0.578294i	0.957285	+	0.289147i	$0.0933716\pi$
$541$			7276.87i			0.578294i	−0.957285	+	0.289147i	$0.906628\pi$
$542$			15561.8i			1.23328i
$543$		0			0
$544$	−10733.1			−0.845916
$545$	10039.0			0.789034
$546$		0			0
$547$			5405.93i			0.422561i	0.977426	+	0.211280i	$0.0677633\pi$
$547$			5405.93i			0.422561i	−0.977426	+	0.211280i	$0.932237\pi$
$548$			20669.9i			1.61127i
$549$		0			0
$550$	4907.49	+	12796.0i	0.380465	+	0.992041i
$551$			11086.0i			0.857131i
$552$		0			0
$553$	−10481.3			−0.805989
$554$		−	21604.1i		−	1.65680i
$555$		0			0
$556$			3000.67i			0.228879i
$557$	−19620.6			−1.49255			−0.746275	−	0.665638i	$-0.768160\pi$
$557$	−19620.6			−1.49255			−0.746275	+	0.665638i	$0.768160\pi$
$558$		0			0
$559$	−11285.1			−0.853863
$560$	6757.35			0.509911
$561$		0			0
$562$	31556.2			2.36854
$563$	−9812.61			−0.734551			−0.367276	−	0.930112i	$-0.619709\pi$
$563$	−9812.61			−0.734551			−0.367276	+	0.930112i	$0.619709\pi$
$564$		0			0
$565$	5401.98			0.402235
$566$		−	26990.2i		−	2.00439i
$567$		0			0
$568$			789.571i			0.0583269i
$569$	7357.91			0.542109			0.271054	−	0.962564i	$-0.412628\pi$
$569$	7357.91			0.542109			0.271054	+	0.962564i	$0.412628\pi$
$570$		0			0
$571$		−	12130.9i		−	0.889078i	−0.895759	−	0.444539i	$-0.853367\pi$
$571$		−	12130.9i		−	0.889078i	0.895759	−	0.444539i	$-0.146633\pi$
$572$	25450.6	−	9760.76i	1.86039	−	0.713493i
$573$		0			0
$574$		−	9023.68i		−	0.656169i
$575$			16001.8i			1.16056i
$576$		0			0
$577$	−13644.7			−0.984462			−0.492231	−	0.870465i	$-0.663818\pi$
$577$	−13644.7			−0.984462			−0.492231	+	0.870465i	$0.663818\pi$
$578$	−42112.1			−3.03050
$579$		0			0
$580$			14997.7i			1.07370i
$581$		−	1924.89i		−	0.137449i
$582$		0			0
$583$	11404.2	−	4373.71i	0.810142	−	0.310704i
$584$			35855.8i			2.54062i
$585$		0			0
$586$	7859.16			0.554025
$587$		−	8951.09i		−	0.629389i	−0.949193	−	0.314694i	$-0.898098\pi$
$587$		−	8951.09i		−	0.629389i	0.949193	−	0.314694i	$-0.101902\pi$
$588$		0			0
$589$		−	13310.5i		−	0.931153i
$590$	−13553.4			−0.945734
$591$		0			0
$592$	14759.7			1.02470
$593$	−11208.6			−0.776189			−0.388095	−	0.921620i	$-0.626867\pi$
$593$	−11208.6			−0.776189			−0.388095	+	0.921620i	$0.626867\pi$
$594$		0			0
$595$	8545.99			0.588826
$596$	−26976.6			−1.85403
$597$		0			0
$598$	46728.2			3.19542
$599$		−	4110.14i		−	0.280360i	−0.990126	−	0.140180i	$-0.955232\pi$
$599$		−	4110.14i		−	0.280360i	0.990126	−	0.140180i	$-0.0447682\pi$
$600$		0			0
$601$		−	16087.0i		−	1.09185i	−0.837834	−	0.545925i	$-0.816178\pi$
$601$		−	16087.0i		−	1.09185i	0.837834	−	0.545925i	$-0.183822\pi$
$602$	−13535.9			−0.916416
$603$		0			0
$604$			22787.1i			1.53509i
$605$	−6293.36	−	6997.97i	−0.422912	−	0.470261i
$606$		0			0
$607$		−	15611.8i		−	1.04393i	−0.852967	−	0.521965i	$-0.825199\pi$
$607$		−	15611.8i		−	1.04393i	0.852967	−	0.521965i	$-0.174801\pi$
$608$		−	8305.23i		−	0.553983i
$609$		0			0
$610$	21595.3			1.43339
$611$	−10512.7			−0.696070
$612$		0			0
$613$		−	6206.87i		−	0.408961i	−0.978871	−	0.204481i	$-0.934449\pi$
$613$		−	6206.87i		−	0.408961i	0.978871	−	0.204481i	$-0.0655505\pi$
$614$			42211.6i			2.77447i
$615$		0			0
$616$	16234.0	−	6226.04i	1.06183	−	0.407231i
$617$		−	8516.64i		−	0.555700i	−0.960624	−	0.277850i	$-0.910378\pi$
$617$		−	8516.64i		−	0.555700i	0.960624	−	0.277850i	$-0.0896219\pi$
$618$		0			0
$619$	19648.7			1.27584			0.637922	−	0.770101i	$-0.279794\pi$
$619$	19648.7			1.27584			0.637922	+	0.770101i	$0.279794\pi$
$620$		−	18007.1i		−	1.16642i
$621$		0			0
$622$			23294.2i			1.50163i
$623$	116.714			0.00750572
$624$		0			0
$625$	−625.000			−0.0400000
$626$	−15121.4			−0.965453
$627$		0			0
$628$	−16574.0			−1.05315
$629$	18666.5			1.18328
$630$		0			0
$631$	−2063.47			−0.130183			−0.0650913	−	0.997879i	$-0.520734\pi$
$631$	−2063.47			−0.130183			−0.0650913	+	0.997879i	$0.520734\pi$
$632$			45554.3i			2.86717i
$633$		0			0
$634$			23992.1i			1.50291i
$635$	3209.03			0.200546
$636$		0			0
$637$			10203.6i			0.634663i
$638$	8122.32	+	21178.5i	0.504022	+	1.31421i
$639$		0			0
$640$			14621.2i			0.903050i
$641$		−	14470.7i		−	0.891666i	−0.895116	−	0.445833i	$-0.852907\pi$
$641$		−	14470.7i		−	0.891666i	0.895116	−	0.445833i	$-0.147093\pi$
$642$		0			0
$643$	−4528.17			−0.277720			−0.138860	−	0.990312i	$-0.544344\pi$
$643$	−4528.17			−0.277720			−0.138860	+	0.990312i	$0.544344\pi$
$644$	38174.6			2.33586
$645$		0			0
$646$		−	51627.3i		−	3.14435i
$647$			3313.49i			0.201340i	0.994920	+	0.100670i	$0.0320986\pi$
$647$			3313.49i			0.201340i	−0.994920	+	0.100670i	$0.967901\pi$
$648$		0			0
$649$	−13035.6	+	4999.40i	−0.788433	+	0.302378i
$650$		−	16426.1i		−	0.991204i
$651$		0			0
$652$	−42700.5			−2.56485
$653$		−	3577.12i		−	0.214370i	−0.994239	−	0.107185i	$-0.965816\pi$
$653$		−	3577.12i		−	0.214370i	0.994239	−	0.107185i	$-0.0341837\pi$
$654$		0			0
$655$			18908.2i			1.12794i
$656$	−15701.1			−0.934492
$657$		0			0
$658$	−12609.5			−0.747063
$659$	1368.75			0.0809086			0.0404543	−	0.999181i	$-0.487119\pi$
$659$	1368.75			0.0809086			0.0404543	+	0.999181i	$0.487119\pi$
$660$		0			0
$661$	1907.63			0.112251			0.0561256	−	0.998424i	$-0.482125\pi$
$661$	1907.63			0.112251			0.0561256	+	0.998424i	$0.482125\pi$
$662$	10663.6			0.626063
$663$		0			0
$664$	−8366.00			−0.488951
$665$			6612.85i			0.385617i
$666$		0			0
$667$			26484.4i			1.53745i
$668$	39542.3			2.29033
$669$		0			0
$670$		−	9208.30i		−	0.530967i
$671$	20770.4	−	7965.80i	1.19498	−	0.458296i
$672$		0			0
$673$		−	2614.03i		−	0.149723i	−0.997194	−	0.0748613i	$-0.976149\pi$
$673$		−	2614.03i		−	0.149723i	0.997194	−	0.0748613i	$-0.0238514\pi$
$674$		−	22405.4i		−	1.28045i
$675$		0			0
$676$	4868.66			0.277006
$677$	−11895.5			−0.675307			−0.337654	−	0.941270i	$-0.609633\pi$
$677$	−11895.5			−0.675307			−0.337654	+	0.941270i	$0.609633\pi$
$678$		0			0
$679$		−	18798.5i		−	1.06248i
$680$		−	37142.8i		−	2.09465i
$681$		0			0
$682$	−9752.13	−	25428.1i	−0.547549	−	1.42770i
$683$			24116.5i			1.35109i	0.737321	+	0.675543i	$0.236091\pi$
$683$			24116.5i			1.35109i	−0.737321	+	0.675543i	$0.763909\pi$
$684$		0			0
$685$	−8553.96			−0.477124
$686$			30228.5i			1.68240i
$687$		0			0
$688$			23552.4i			1.30513i
$689$	−14639.4			−0.809459
$690$		0			0
$691$	−7279.98			−0.400786			−0.200393	−	0.979716i	$-0.564222\pi$
$691$	−7279.98			−0.400786			−0.200393	+	0.979716i	$0.564222\pi$
$692$	15890.0			0.872902
$693$		0			0
$694$	−16776.6			−0.917622
$695$	−1241.79			−0.0677750
$696$		0			0
$697$	−19857.2			−1.07912
$698$		−	56886.8i		−	3.08481i
$699$		0			0
$700$		−	13419.3i		−	0.724572i
$701$	32505.5			1.75138			0.875689	−	0.482875i	$-0.160407\pi$
$701$	32505.5			1.75138			0.875689	+	0.482875i	$0.160407\pi$
$702$		0			0
$703$			14444.1i			0.774920i
$704$	3452.81	+	9003.00i	0.184848	+	0.481979i
$705$		0			0
$706$			18078.9i			0.963752i
$707$			3624.97i			0.192830i
$708$		0			0
$709$	−3264.46			−0.172919			−0.0864595	−	0.996255i	$-0.527555\pi$
$709$	−3264.46			−0.172919			−0.0864595	+	0.996255i	$0.527555\pi$
$710$	−614.432			−0.0324778
$711$		0			0
$712$		−	507.268i		−	0.0267004i
$713$		−	31798.7i		−	1.67022i
$714$		0			0
$715$	4039.36	+	10532.4i	0.211278	+	0.550894i
$716$		−	16641.0i		−	0.868578i
$717$		0			0
$718$	1001.58			0.0520595
$719$			25591.6i			1.32741i	0.747995	+	0.663705i	$0.231017\pi$
$719$			25591.6i			1.32741i	−0.747995	+	0.663705i	$0.768983\pi$
$720$		0			0
$721$			9843.26i			0.508436i
$722$	5594.61			0.288379
$723$		0			0
$724$	9574.95			0.491506
$725$	9309.87			0.476910
$726$		0			0
$727$	33653.1			1.71682			0.858408	−	0.512968i	$-0.171454\pi$
$727$	33653.1			1.71682			0.858408	+	0.512968i	$0.171454\pi$
$728$	−20839.4			−1.06094
$729$		0			0
$730$	−27902.4			−1.41468
$731$			29786.6i			1.50711i
$732$		0			0
$733$			14383.4i			0.724778i	0.932027	+	0.362389i	$0.118039\pi$
$733$			14383.4i			0.724778i	−0.932027	+	0.362389i	$0.881961\pi$
$734$	−35862.0			−1.80339
$735$		0			0
$736$		−	19841.1i		−	0.993688i
$737$	−3396.64	−	8856.55i	−0.169765	−	0.442653i
$738$		0			0
$739$			20920.5i			1.04137i	0.853749	+	0.520686i	$0.174324\pi$
$739$			20920.5i			1.04137i	−0.853749	+	0.520686i	$0.825676\pi$
$740$			19540.6i			0.970714i
$741$		0			0
$742$	−17559.2			−0.868760
$743$	18892.5			0.932836			0.466418	−	0.884564i	$-0.345544\pi$
$743$	18892.5			0.932836			0.466418	+	0.884564i	$0.345544\pi$
$744$		0			0
$745$		−	11163.9i		−	0.549011i
$746$			144.973i			0.00711508i
$747$		0			0
$748$	−25763.2	−	67175.9i	−1.25935	−	3.28368i
$749$			19128.0i			0.933138i
$750$		0			0
$751$	28833.9			1.40102			0.700509	−	0.713643i	$-0.252956\pi$
$751$	28833.9			1.40102			0.700509	+	0.713643i	$0.252956\pi$
$752$			21940.4i			1.06394i
$753$		0			0
$754$		−	27186.6i		−	1.31310i
$755$	−9430.14			−0.454567
$756$		0			0
$757$	36338.9			1.74473			0.872365	−	0.488856i	$-0.162586\pi$
$757$	36338.9			1.74473			0.872365	+	0.488856i	$0.162586\pi$
$758$	21811.7			1.04517
$759$		0			0
$760$	28741.0			1.37177
$761$	−5978.88			−0.284802			−0.142401	−	0.989809i	$-0.545482\pi$
$761$	−5978.88			−0.284802			−0.142401	+	0.989809i	$0.545482\pi$
$762$		0			0
$763$	−14866.8			−0.705391
$764$		−	38049.7i		−	1.80182i
$765$		0			0
$766$			13799.6i			0.650916i
$767$	16733.7			0.787769
$768$		0			0
$769$		−	21501.2i		−	1.00826i	−0.863627	−	0.504132i	$-0.831813\pi$
$769$		−	21501.2i		−	1.00826i	0.863627	−	0.504132i	$-0.168187\pi$
$770$	4845.00	+	12633.1i	0.226756	+	0.591252i
$771$		0			0
$772$			68973.8i			3.21557i
$773$			398.575i			0.0185456i	0.999957	+	0.00927280i	$0.00295166\pi$
$773$			398.575i			0.0185456i	−0.999957	+	0.00927280i	$0.997048\pi$
$774$		0			0
$775$	−11178.0			−0.518096
$776$	−81702.7			−3.77958
$777$		0			0
$778$		−	40262.8i		−	1.85539i
$779$		−	15365.4i		−	0.706704i
$780$		0			0
$781$	−590.960	+	226.644i	−0.0270758	+	0.0103841i
$782$		−	123337.i		−	5.64007i
$783$		0			0
$784$	21295.2			0.970079
$785$		−	6858.94i		−	0.311855i
$786$		0			0
$787$		−	40678.9i		−	1.84250i	−0.388976	−	0.921248i	$-0.627171\pi$
$787$		−	40678.9i		−	1.84250i	0.388976	−	0.921248i	$-0.372829\pi$
$788$	−23401.2			−1.05791
$789$		0			0
$790$	−35449.6			−1.59651
$791$	−7999.80			−0.359596
$792$		0			0
$793$	−26662.7			−1.19397
$794$	−52968.0			−2.36746
$795$		0			0
$796$	−40383.7			−1.79819
$797$			44293.9i			1.96860i	0.176513	+	0.984298i	$0.443518\pi$
$797$			44293.9i			1.96860i	−0.176513	+	0.984298i	$0.556482\pi$
$798$		0			0
$799$			27747.9i			1.22860i
$800$	−6974.62			−0.308238
$801$		0			0
$802$		−	70160.2i		−	3.08908i
$803$	−26836.5	+	10292.3i	−1.17938	+	0.452312i
$804$		0			0
$805$			15798.0i			0.691687i
$806$			32641.8i			1.42650i
$807$		0			0
$808$	15755.0			0.685962
$809$	19692.0			0.855791			0.427896	−	0.903828i	$-0.359255\pi$
$809$	19692.0			0.855791			0.427896	+	0.903828i	$0.359255\pi$
$810$		0			0
$811$			4220.18i			0.182726i	0.995818	+	0.0913628i	$0.0291223\pi$
$811$			4220.18i			0.182726i	−0.995818	+	0.0913628i	$0.970878\pi$
$812$		−	22210.1i		−	0.959878i
$813$		0			0
$814$	10582.7	+	27593.7i	0.455679	+	1.18816i
$815$		−	17671.0i		−	0.759495i
$816$		0			0
$817$	−23048.8			−0.986994
$818$			35554.0i			1.51970i
$819$		0			0
$820$		−	20787.0i		−	0.885262i
$821$	32899.6			1.39854			0.699272	−	0.714856i	$-0.253508\pi$
$821$	32899.6			1.39854			0.699272	+	0.714856i	$0.253508\pi$
$822$		0			0
$823$	38387.4			1.62588			0.812941	−	0.582346i	$-0.197865\pi$
$823$	38387.4			1.62588			0.812941	+	0.582346i	$0.197865\pi$
$824$	42781.1			1.80868
$825$		0			0
$826$	20071.2			0.845480
$827$	−11091.3			−0.466362			−0.233181	−	0.972433i	$-0.574913\pi$
$827$	−11091.3			−0.466362			−0.233181	+	0.972433i	$0.574913\pi$
$828$		0			0
$829$	14819.2			0.620859			0.310429	−	0.950596i	$-0.399527\pi$
$829$	14819.2			0.620859			0.310429	+	0.950596i	$0.399527\pi$
$830$		−	6510.28i		−	0.272259i
$831$		0			0
$832$		−	11557.1i		−	0.481573i
$833$	26931.9			1.12021
$834$		0			0
$835$			16364.0i			0.678205i
$836$	51980.4	−	19935.4i	2.15045	−	0.824738i
$837$		0			0
$838$			8783.30i			0.362069i
$839$		−	21329.0i		−	0.877663i	−0.898569	−	0.438831i	$-0.855393\pi$
$839$		−	21329.0i		−	0.877663i	0.898569	−	0.438831i	$-0.144607\pi$
$840$		0			0
$841$	−8980.34			−0.368213
$842$	38759.6			1.58640
$843$		0			0
$844$			53940.3i			2.19989i
$845$			2014.83i			0.0820263i
$846$		0			0
$847$	9319.85	+	10363.3i	0.378080	+	0.420410i
$848$			30552.9i			1.23726i
$849$		0			0
$850$	−43355.9			−1.74953
$851$			34506.8i			1.38999i
$852$		0			0
$853$			17901.8i			0.718578i	0.933226	+	0.359289i	$0.116981\pi$
$853$			17901.8i			0.718578i	−0.933226	+	0.359289i	$0.883019\pi$
$854$	−31980.5			−1.28144
$855$		0			0
$856$	83134.6			3.31949
$857$	−6828.52			−0.272180			−0.136090	−	0.990697i	$-0.543454\pi$
$857$	−6828.52			−0.272180			−0.136090	+	0.990697i	$0.543454\pi$
$858$		0			0
$859$	8439.55			0.335220			0.167610	−	0.985853i	$-0.446395\pi$
$859$	8439.55			0.335220			0.167610	+	0.985853i	$0.446395\pi$
$860$	−31181.5			−1.23637
$861$		0			0
$862$	−36366.2			−1.43693
$863$			11754.0i			0.463629i	0.972760	+	0.231815i	$0.0744663\pi$
$863$			11754.0i			0.463629i	−0.972760	+	0.231815i	$0.925534\pi$
$864$		0			0
$865$			6575.87i			0.258481i
$866$	69161.3			2.71385
$867$		0			0
$868$			26666.7i			1.04277i
$869$	−34095.5	+	13076.2i	−1.33097	+	0.510449i
$870$		0			0
$871$			11369.0i			0.442280i
$872$			64614.5i			2.50931i
$873$		0			0
$874$	95437.9			3.69363
$875$	14809.0			0.572156
$876$		0			0
$877$		−	44064.0i		−	1.69662i	−0.529501	−	0.848309i	$-0.677621\pi$
$877$		−	44064.0i		−	1.69662i	0.529501	−	0.848309i	$-0.322379\pi$
$878$		−	12110.2i		−	0.465491i
$879$		0			0
$880$	21981.4	−	8430.27i	0.842039	−	0.322937i
$881$			1988.29i			0.0760352i	0.999277	+	0.0380176i	$0.0121043\pi$
$881$			1988.29i			0.0760352i	−0.999277	+	0.0380176i	$0.987896\pi$
$882$		0			0
$883$	1677.78			0.0639430			0.0319715	−	0.999489i	$-0.489821\pi$
$883$	1677.78			0.0639430			0.0319715	+	0.999489i	$0.489821\pi$
$884$			86233.0i			3.28091i
$885$		0			0
$886$			38126.6i			1.44570i
$887$	−45311.6			−1.71524			−0.857618	−	0.514287i	$-0.828057\pi$
$887$	−45311.6			−1.71524			−0.857618	+	0.514287i	$0.828057\pi$
$888$		0			0
$889$	−4752.26			−0.179287
$890$	394.748			0.0148674
$891$		0			0
$892$	−26862.4			−1.00832
$893$	−21471.2			−0.804598
$894$		0			0
$895$	6886.63			0.257201
$896$		−	21652.5i		−	0.807321i
$897$		0			0
$898$			49200.5i			1.82833i
$899$	−18500.5			−0.686348
$900$		0			0
$901$			38640.2i			1.42874i
$902$	−11257.7	−	29353.7i	−0.415565	−	1.08356i
$903$		0			0
$904$			34769.0i			1.27920i
$905$			3962.46i			0.145543i
$906$		0			0
$907$	5307.53			0.194304			0.0971520	−	0.995270i	$-0.469027\pi$
$907$	5307.53			0.194304			0.0971520	+	0.995270i	$0.469027\pi$
$908$	20316.4			0.742536
$909$		0			0
$910$		−	16216.9i		−	0.590753i
$911$			29019.6i			1.05539i	0.849433	+	0.527696i	$0.176944\pi$
$911$			29019.6i			1.05539i	−0.849433	+	0.527696i	$0.823056\pi$
$912$		0			0
$913$	−2401.43	−	6261.59i	−0.0870491	−	0.226975i
$914$			56505.3i			2.04489i
$915$		0			0
$916$	35339.7			1.27473
$917$		−	28001.1i		−	1.00837i
$918$		0			0
$919$		−	41537.5i		−	1.49096i	−0.666527	−	0.745481i	$-0.732220\pi$
$919$		−	41537.5i		−	1.49096i	0.666527	−	0.745481i	$-0.267780\pi$
$920$	68662.0			2.46056
$921$		0			0
$922$	−64293.6			−2.29653
$923$	758.609			0.0270530
$924$		0			0
$925$	12129.9			0.431168
$926$	54035.9			1.91764
$927$		0			0
$928$	−11543.6			−0.408338
$929$		−	48738.5i		−	1.72127i	−0.509223	−	0.860634i	$-0.670067\pi$
$929$		−	48738.5i		−	1.72127i	0.509223	−	0.860634i	$-0.329933\pi$
$930$		0			0
$931$			20839.8i			0.733616i
$932$	−43807.2			−1.53965
$933$		0			0
$934$		−	49854.8i		−	1.74657i
$935$	27799.8	−	10661.7i	0.972355	−	0.372915i
$936$		0			0
$937$		−	37129.8i		−	1.29453i	−0.762263	−	0.647267i	$-0.775912\pi$
$937$		−	37129.8i		−	1.29453i	0.762263	−	0.647267i	$-0.224088\pi$
$938$			13636.6i			0.474681i
$939$		0			0
$940$	−29047.3			−1.00789
$941$	2434.90			0.0843521			0.0421760	−	0.999110i	$-0.486571\pi$
$941$	2434.90			0.0843521			0.0421760	+	0.999110i	$0.486571\pi$
$942$		0			0
$943$		−	36707.8i		−	1.26763i
$944$		−	34923.8i		−	1.20410i
$945$		0			0
$946$	−44031.9	+	16887.0i	−1.51332	+	0.580385i
$947$			6355.49i			0.218084i	0.994037	+	0.109042i	$0.0347783\pi$
$947$			6355.49i			0.218084i	−0.994037	+	0.109042i	$0.965222\pi$
$948$		0			0
$949$	34449.7			1.17838
$950$		−	33548.6i		−	1.14575i
$951$		0			0
$952$			55004.9i			1.87260i
$953$	6746.81			0.229329			0.114664	−	0.993404i	$-0.463421\pi$
$953$	6746.81			0.229329			0.114664	+	0.993404i	$0.463421\pi$
$954$		0			0
$955$	15746.3			0.533549
$956$	−43685.0			−1.47790
$957$		0			0
$958$	10190.1			0.343660
$959$	12667.6			0.426545
$960$		0			0
$961$	−7578.21			−0.254379
$962$		−	35421.7i		−	1.18715i
$963$		0			0
$964$		−	102454.i		−	3.42304i
$965$	−28543.9			−0.952186
$966$		0			0
$967$			48052.2i			1.59799i	0.601339	+	0.798994i	$0.294634\pi$
$967$			48052.2i			1.59799i	−0.601339	+	0.798994i	$0.705366\pi$
$968$	45041.3	−	40506.2i	1.49554	−	1.34496i
$969$		0			0
$970$		−	63579.7i		−	2.10456i
$971$		−	22937.1i		−	0.758071i	−0.925382	−	0.379035i	$-0.876256\pi$
$971$		−	22937.1i		−	0.758071i	0.925382	−	0.379035i	$-0.123744\pi$
$972$		0			0
$973$	1838.96			0.0605904
$974$	−41524.5			−1.36605
$975$		0			0
$976$			55645.9i			1.82498i
$977$		−	9793.08i		−	0.320684i	−0.987062	−	0.160342i	$-0.948740\pi$
$977$		−	9793.08i		−	0.320684i	0.987062	−	0.160342i	$-0.0512597\pi$
$978$		0			0
$979$	379.668	−	145.610i	0.0123945	−	0.00475353i
$980$			28193.1i			0.918975i
$981$		0			0
$982$	−75668.1			−2.45893
$983$		−	16592.9i		−	0.538383i	−0.963087	−	0.269192i	$-0.913243\pi$
$983$		−	16592.9i		−	0.538383i	0.963087	−	0.269192i	$-0.0867565\pi$
$984$		0			0
$985$		−	9684.26i		−	0.313265i
$986$	−71757.9			−2.31768
$987$		0			0
$988$	−66726.7			−2.14864
$989$	−55063.3			−1.77039
$990$		0			0
$991$	−44600.6			−1.42965			−0.714825	−	0.699303i	$-0.753494\pi$
$991$	−44600.6			−1.42965			−0.714825	+	0.699303i	$0.753494\pi$
$992$	13859.9			0.443602
$993$		0			0
$994$	909.913			0.0290349
$995$		−	16712.2i		−	0.532475i
$996$		0			0
$997$		−	22478.3i		−	0.714037i	−0.934097	−	0.357019i	$-0.883793\pi$
$997$		−	22478.3i		−	0.714037i	0.934097	−	0.357019i	$-0.116207\pi$
$998$	−26596.0			−0.843569
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	99.4.d.c.98.2	yes	8
3.2	odd	2	inner	99.4.d.c.98.7	yes	8
4.3	odd	2		1584.4.b.g.593.7		8
11.10	odd	2	inner	99.4.d.c.98.8	yes	8
12.11	even	2		1584.4.b.g.593.3		8
33.32	even	2	inner	99.4.d.c.98.1	✓	8
44.43	even	2		1584.4.b.g.593.6		8
132.131	odd	2		1584.4.b.g.593.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
99.4.d.c.98.1	✓	8	33.32	even	2	inner
99.4.d.c.98.2	yes	8	1.1	even	1	trivial
99.4.d.c.98.7	yes	8	3.2	odd	2	inner
99.4.d.c.98.8	yes	8	11.10	odd	2	inner
1584.4.b.g.593.2		8	132.131	odd	2
1584.4.b.g.593.3		8	12.11	even	2
1584.4.b.g.593.6		8	44.43	even	2
1584.4.b.g.593.7		8	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 99 = 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	99.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-5.00866 q^{2} +17.0866 q^{4} +7.07107i q^{5} -10.4716i q^{7} -45.5118 q^{8} -35.4165i q^{10} +(-13.0640 - 34.0636i) q^{11} +43.7271i q^{13} +52.4484i q^{14} +91.2599 q^{16} +115.416 q^{17} +89.3084i q^{19} +120.821i q^{20} +(65.4332 + 170.613i) q^{22} +213.357i q^{23} +75.0000 q^{25} -219.014i q^{26} -178.924i q^{28} +124.132 q^{29} -149.040 q^{31} -92.9950 q^{32} -578.079 q^{34} +74.0451 q^{35} +161.733 q^{37} -447.315i q^{38} -321.817i q^{40} -172.049 q^{41} +258.081i q^{43} +(-223.220 - 582.033i) q^{44} -1068.63i q^{46} +240.416i q^{47} +233.347 q^{49} -375.649 q^{50} +747.149i q^{52} +334.790i q^{53} +(240.866 - 92.3765i) q^{55} +476.579i q^{56} -621.733 q^{58} -382.685i q^{59} +609.752i q^{61} +746.488 q^{62} -264.299 q^{64} -309.197 q^{65} +260.000 q^{67} +1972.07 q^{68} -370.866 q^{70} -17.3487i q^{71} -787.834i q^{73} -810.063 q^{74} +1525.98i q^{76} +(-356.699 + 136.801i) q^{77} -1000.93i q^{79} +645.305i q^{80} +861.733 q^{82} +183.820 q^{83} +816.115i q^{85} -1292.64i q^{86} +(594.567 + 1550.30i) q^{88} +11.1459i q^{89} +457.891 q^{91} +3645.56i q^{92} -1204.16i q^{94} -631.506 q^{95} +1795.20 q^{97} -1168.75 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 44 q^{4} + 452 q^{16} + 60 q^{22} + 600 q^{25} - 80 q^{31} - 2400 q^{34} - 560 q^{37} + 1496 q^{49} + 1000 q^{55} - 3120 q^{58} - 724 q^{64} + 2080 q^{67} - 2040 q^{70} + 5040 q^{82} + 5220 q^{88}+ \cdots + 8800 q^{97}+O(q^{100}) \) 8 * q + 44 * q^4 + 452 * q^16 + 60 * q^22 + 600 * q^25 - 80 * q^31 - 2400 * q^34 - 560 * q^37 + 1496 * q^49 + 1000 * q^55 - 3120 * q^58 - 724 * q^64 + 2080 * q^67 - 2040 * q^70 + 5040 * q^82 + 5220 * q^88 - 2640 * q^91 + 8800 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more