LMFDB - Embedded newform 88.5.h.a.65.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [88,5,Mod(65,88)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(88, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 5, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("88.65");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 88 = 2^{3} \cdot 11 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	88.h (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$9.09655675138$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 378x^{10} + 49709x^{8} + 2770310x^{6} + 62444900x^{4} + 470757120x^{2} + 33918976 $ x^12 + 378x^10 + 49709x^8 + 2770310x^6 + 62444900x^4 + 470757120*x^2 + 33918976
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{32}\cdot 5^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			65.6
Root			$-12.8952i$ of defining polynomial
Character	$\chi$	$=$	88.65
Dual form			88.5.h.a.65.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.95596 q^{3} +6.72383 q^{5} +35.8576i q^{7} -77.1742 q^{9} +O(q^{10})$	$q-1.95596 q^{3} +6.72383 q^{5} +35.8576i q^{7} -77.1742 q^{9} +(120.912 + 4.60990i) q^{11} +138.170i q^{13} -13.1515 q^{15} +364.891i q^{17} +570.563i q^{19} -70.1358i q^{21} +70.8767 q^{23} -579.790 q^{25} +309.382 q^{27} +317.406i q^{29} +1159.17 q^{31} +(-236.499 - 9.01676i) q^{33} +241.100i q^{35} -1361.35 q^{37} -270.255i q^{39} -1336.82i q^{41} -2158.22i q^{43} -518.906 q^{45} +1416.39 q^{47} +1115.24 q^{49} -713.710i q^{51} -3013.55 q^{53} +(812.993 + 30.9962i) q^{55} -1116.00i q^{57} +1663.07 q^{59} -5396.77i q^{61} -2767.28i q^{63} +929.033i q^{65} -588.504 q^{67} -138.632 q^{69} -4021.12 q^{71} +2259.81i q^{73} +1134.04 q^{75} +(-165.300 + 4335.61i) q^{77} +6315.63i q^{79} +5645.98 q^{81} +777.535i q^{83} +2453.46i q^{85} -620.833i q^{87} +4643.68 q^{89} -4954.44 q^{91} -2267.28 q^{93} +3836.37i q^{95} +6327.31 q^{97} +(-9331.30 - 355.766i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 8 q^{3} + 492 q^{9}+O(q^{10}) $ 12 * q + 8 * q^3 + 492 * q^9	$ 12 q + 8 q^{3} + 492 q^{9} + 12 q^{11} + 480 q^{15} + 1608 q^{23} + 1484 q^{25} + 2048 q^{27} - 2136 q^{31} + 1568 q^{33} - 288 q^{37} - 8264 q^{45} - 6024 q^{47} - 9940 q^{49} + 3576 q^{53} - 8344 q^{55} - 7704 q^{59} + 10312 q^{67} - 616 q^{69} + 8904 q^{71} - 3544 q^{75} + 9120 q^{77} + 15340 q^{81} + 4224 q^{89} + 23904 q^{91} + 23480 q^{93} - 21280 q^{97} + 14420 q^{99}+O(q^{100}) $ 12 * q + 8 * q^3 + 492 * q^9 + 12 * q^11 + 480 * q^15 + 1608 * q^23 + 1484 * q^25 + 2048 * q^27 - 2136 * q^31 + 1568 * q^33 - 288 * q^37 - 8264 * q^45 - 6024 * q^47 - 9940 * q^49 + 3576 * q^53 - 8344 * q^55 - 7704 * q^59 + 10312 * q^67 - 616 * q^69 + 8904 * q^71 - 3544 * q^75 + 9120 * q^77 + 15340 * q^81 + 4224 * q^89 + 23904 * q^91 + 23480 * q^93 - 21280 * q^97 + 14420 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/88\mathbb{Z}\right)^\times$.

$n$	$23$	$45$	$57$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$		0			0
$2$		0			0
$3$	−1.95596			−0.217328			−0.108664	−	0.994079i	$-0.534657\pi$
$3$	−1.95596			−0.217328			−0.108664	+	0.994079i	$0.534657\pi$
$4$		0			0
$5$	6.72383			0.268953			0.134477	−	0.990917i	$-0.457065\pi$
$5$	6.72383			0.268953			0.134477	+	0.990917i	$0.457065\pi$
$6$		0			0
$7$			35.8576i			0.731787i	0.930657	+	0.365893i	$0.119237\pi$
$7$			35.8576i			0.731787i	−0.930657	+	0.365893i	$0.880763\pi$
$8$		0			0
$9$	−77.1742			−0.952768
$10$		0			0
$11$	120.912	+	4.60990i	0.999274	+	0.0380984i
$11$	120.912	+	4.60990i	0.999274	+	0.0380984i
$12$		0			0
$13$			138.170i			0.817575i	0.912630	+	0.408788i	$0.134048\pi$
$13$			138.170i			0.817575i	−0.912630	+	0.408788i	$0.865952\pi$
$14$		0			0
$15$	−13.1515			−0.0584512
$16$		0			0
$17$			364.891i			1.26260i	0.775540	+	0.631299i	$0.217478\pi$
$17$			364.891i			1.26260i	−0.775540	+	0.631299i	$0.782522\pi$
$18$		0			0
$19$			570.563i			1.58051i	0.612780	+	0.790253i	$0.290051\pi$
$19$			570.563i			1.58051i	−0.612780	+	0.790253i	$0.709949\pi$
$20$		0			0
$21$		−	70.1358i		−	0.159038i
$22$		0			0
$23$	70.8767			0.133982			0.0669912	−	0.997754i	$-0.478660\pi$
$23$	70.8767			0.133982			0.0669912	+	0.997754i	$0.478660\pi$
$24$		0			0
$25$	−579.790			−0.927664
$26$		0			0
$27$	309.382			0.424392
$28$		0			0
$29$			317.406i			0.377416i	0.982033	+	0.188708i	$0.0604299\pi$
$29$			317.406i			0.377416i	−0.982033	+	0.188708i	$0.939570\pi$
$30$		0			0
$31$	1159.17			1.20621			0.603105	−	0.797662i	$-0.293930\pi$
$31$	1159.17			1.20621			0.603105	+	0.797662i	$0.293930\pi$
$32$		0			0
$33$	−236.499	−	9.01676i	−0.217171	−	0.00827986i
$34$		0			0
$35$			241.100i			0.196816i
$36$		0			0
$37$	−1361.35			−0.994412			−0.497206	−	0.867632i	$-0.665641\pi$
$37$	−1361.35			−0.994412			−0.497206	+	0.867632i	$0.665641\pi$
$38$		0			0
$39$		−	270.255i		−	0.177682i
$40$		0			0
$41$		−	1336.82i		−	0.795251i	−0.917548	−	0.397626i	$-0.869834\pi$
$41$		−	1336.82i		−	0.795251i	0.917548	−	0.397626i	$-0.130166\pi$
$42$		0			0
$43$		−	2158.22i		−	1.16724i	−0.812028	−	0.583619i	$-0.801636\pi$
$43$		−	2158.22i		−	1.16724i	0.812028	−	0.583619i	$-0.198364\pi$
$44$		0			0
$45$	−518.906			−0.256250
$46$		0			0
$47$	1416.39			0.641189			0.320595	−	0.947217i	$-0.396117\pi$
$47$	1416.39			0.641189			0.320595	+	0.947217i	$0.396117\pi$
$48$		0			0
$49$	1115.24			0.464488
$50$		0			0
$51$		−	713.710i		−	0.274398i
$52$		0			0
$53$	−3013.55			−1.07282			−0.536410	−	0.843957i	$-0.680220\pi$
$53$	−3013.55			−1.07282			−0.536410	+	0.843957i	$0.680220\pi$
$54$		0			0
$55$	812.993	+	30.9962i	0.268758	+	0.0102467i
$56$		0			0
$57$		−	1116.00i		−	0.343489i
$58$		0			0
$59$	1663.07			0.477756			0.238878	−	0.971050i	$-0.423220\pi$
$59$	1663.07			0.477756			0.238878	+	0.971050i	$0.423220\pi$
$60$		0			0
$61$		−	5396.77i		−	1.45036i	−0.688562	−	0.725178i	$-0.741758\pi$
$61$		−	5396.77i		−	1.45036i	0.688562	−	0.725178i	$-0.258242\pi$
$62$		0			0
$63$		−	2767.28i		−	0.697223i
$64$		0			0
$65$			929.033i			0.219889i
$66$		0			0
$67$	−588.504			−0.131099			−0.0655496	−	0.997849i	$-0.520880\pi$
$67$	−588.504			−0.131099			−0.0655496	+	0.997849i	$0.520880\pi$
$68$		0			0
$69$	−138.632			−0.0291182
$70$		0			0
$71$	−4021.12			−0.797682			−0.398841	−	0.917020i	$-0.630588\pi$
$71$	−4021.12			−0.797682			−0.398841	+	0.917020i	$0.630588\pi$
$72$		0			0
$73$			2259.81i			0.424058i	0.977263	+	0.212029i	$0.0680072\pi$
$73$			2259.81i			0.424058i	−0.977263	+	0.212029i	$0.931993\pi$
$74$		0			0
$75$	1134.04			0.201608
$76$		0			0
$77$	−165.300	+	4335.61i	−0.0278799	+	0.731255i
$78$		0			0
$79$			6315.63i			1.01196i	0.862546	+	0.505979i	$0.168869\pi$
$79$			6315.63i			1.01196i	−0.862546	+	0.505979i	$0.831131\pi$
$80$		0			0
$81$	5645.98			0.860536
$82$		0			0
$83$			777.535i			0.112866i	0.998406	+	0.0564331i	$0.0179728\pi$
$83$			777.535i			0.112866i	−0.998406	+	0.0564331i	$0.982027\pi$
$84$		0			0
$85$			2453.46i			0.339580i
$86$		0			0
$87$		−	620.833i		−	0.0820231i
$88$		0			0
$89$	4643.68			0.586249			0.293125	−	0.956074i	$-0.405305\pi$
$89$	4643.68			0.586249			0.293125	+	0.956074i	$0.405305\pi$
$90$		0			0
$91$	−4954.44			−0.598291
$92$		0			0
$93$	−2267.28			−0.262144
$94$		0			0
$95$			3836.37i			0.425082i
$96$		0			0
$97$	6327.31			0.672474			0.336237	−	0.941777i	$-0.390846\pi$
$97$	6327.31			0.672474			0.336237	+	0.941777i	$0.390846\pi$
$98$		0			0
$99$	−9331.30	−	355.766i	−0.952077	−	0.0362989i
$100$		0			0
$101$			19278.4i			1.88985i	0.327281	+	0.944927i	$0.393868\pi$
$101$			19278.4i			1.88985i	−0.327281	+	0.944927i	$0.606132\pi$
$102$		0			0
$103$	−3753.94			−0.353845			−0.176922	−	0.984225i	$-0.556614\pi$
$103$	−3753.94			−0.353845			−0.176922	+	0.984225i	$0.556614\pi$
$104$		0			0
$105$		−	471.581i		−	0.0427738i
$106$		0			0
$107$			18411.8i			1.60815i	0.594525	+	0.804077i	$0.297340\pi$
$107$			18411.8i			1.60815i	−0.594525	+	0.804077i	$0.702660\pi$
$108$		0			0
$109$		−	9835.23i		−	0.827812i	−0.910320	−	0.413906i	$-0.864164\pi$
$109$		−	9835.23i		−	0.827812i	0.910320	−	0.413906i	$-0.135836\pi$
$110$		0			0
$111$	2662.74			0.216114
$112$		0			0
$113$	13378.6			1.04774			0.523870	−	0.851798i	$-0.324488\pi$
$113$	13378.6			1.04774			0.523870	+	0.851798i	$0.324488\pi$
$114$		0			0
$115$	476.563			0.0360350
$116$		0			0
$117$		−	10663.2i		−	0.778960i
$118$		0			0
$119$	−13084.1			−0.923952
$120$		0			0
$121$	14598.5	+	1114.79i	0.997097	+	0.0761414i
$122$		0			0
$123$			2614.75i			0.172831i
$124$		0			0
$125$	−8100.80			−0.518451
$126$		0			0
$127$		−	24171.9i		−	1.49866i	−0.662198	−	0.749329i	$-0.730376\pi$
$127$		−	24171.9i		−	1.49866i	0.662198	−	0.749329i	$-0.269624\pi$
$128$		0			0
$129$			4221.39i			0.253674i
$130$		0			0
$131$		−	10327.1i		−	0.601776i	−0.953660	−	0.300888i	$-0.902717\pi$
$131$		−	10327.1i		−	0.601776i	0.953660	−	0.300888i	$-0.0972830\pi$
$132$		0			0
$133$	−20459.0			−1.15659
$134$		0			0
$135$	2080.23			0.114142
$136$		0			0
$137$	18041.0			0.961211			0.480605	−	0.876937i	$-0.340417\pi$
$137$	18041.0			0.961211			0.480605	+	0.876937i	$0.340417\pi$
$138$		0			0
$139$		−	18076.1i		−	0.935567i	−0.883843	−	0.467783i	$-0.845053\pi$
$139$		−	18076.1i		−	0.935567i	0.883843	−	0.467783i	$-0.154947\pi$
$140$		0			0
$141$	−2770.39			−0.139349
$142$		0			0
$143$	−636.951	+	16706.5i	−0.0311483	+	0.816981i
$144$		0			0
$145$			2134.19i			0.101507i
$146$		0			0
$147$	−2181.35			−0.100946
$148$		0			0
$149$		−	26839.5i		−	1.20893i	−0.796630	−	0.604467i	$-0.793386\pi$
$149$		−	26839.5i		−	1.20893i	0.796630	−	0.604467i	$-0.206614\pi$
$150$		0			0
$151$		−	15391.1i		−	0.675018i	−0.941322	−	0.337509i	$-0.890416\pi$
$151$		−	15391.1i		−	0.675018i	0.941322	−	0.337509i	$-0.109584\pi$
$152$		0			0
$153$		−	28160.2i		−	1.20296i
$154$		0			0
$155$	7794.04			0.324414
$156$		0			0
$157$	−34362.6			−1.39408			−0.697038	−	0.717034i	$-0.745499\pi$
$157$	−34362.6			−1.39408			−0.697038	+	0.717034i	$0.745499\pi$
$158$		0			0
$159$	5894.38			0.233154
$160$		0			0
$161$			2541.46i			0.0980465i
$162$		0			0
$163$	34043.9			1.28134			0.640670	−	0.767817i	$-0.278657\pi$
$163$	34043.9			1.28134			0.640670	+	0.767817i	$0.278657\pi$
$164$		0			0
$165$	−1590.18	−	60.6272i	−0.0584087	−	0.00222689i
$166$		0			0
$167$			34873.6i			1.25044i	0.780447	+	0.625222i	$0.214991\pi$
$167$			34873.6i			1.25044i	−0.780447	+	0.625222i	$0.785009\pi$
$168$		0			0
$169$	9470.00			0.331571
$170$		0			0
$171$		−	44032.8i		−	1.50586i
$172$		0			0
$173$			29337.5i			0.980238i	0.871655	+	0.490119i	$0.163047\pi$
$173$			29337.5i			0.980238i	−0.871655	+	0.490119i	$0.836953\pi$
$174$		0			0
$175$		−	20789.9i		−	0.678852i
$176$		0			0
$177$	−3252.89			−0.103830
$178$		0			0
$179$	2401.98			0.0749660			0.0374830	−	0.999297i	$-0.488066\pi$
$179$	2401.98			0.0749660			0.0374830	+	0.999297i	$0.488066\pi$
$180$		0			0
$181$	−28001.5			−0.854721			−0.427360	−	0.904081i	$-0.640556\pi$
$181$	−28001.5			−0.854721			−0.427360	+	0.904081i	$0.640556\pi$
$182$		0			0
$183$			10555.9i			0.315204i
$184$		0			0
$185$	−9153.49			−0.267450
$186$		0			0
$187$	−1682.11	+	44119.7i	−0.0481029	+	1.26168i
$188$		0			0
$189$			11093.7i			0.310564i
$190$		0			0
$191$	−17399.1			−0.476935			−0.238467	−	0.971151i	$-0.576645\pi$
$191$	−17399.1			−0.476935			−0.238467	+	0.971151i	$0.576645\pi$
$192$		0			0
$193$		−	37593.4i		−	1.00925i	−0.863340	−	0.504623i	$-0.831632\pi$
$193$		−	37593.4i		−	1.00925i	0.863340	−	0.504623i	$-0.168368\pi$
$194$		0			0
$195$		−	1817.15i		−	0.0477882i
$196$		0			0
$197$			19628.9i			0.505781i	0.967495	+	0.252891i	$0.0813813\pi$
$197$			19628.9i			0.505781i	−0.967495	+	0.252891i	$0.918619\pi$
$198$		0			0
$199$	63712.2			1.60885			0.804427	−	0.594052i	$-0.202472\pi$
$199$	63712.2			1.60885			0.804427	+	0.594052i	$0.202472\pi$
$200$		0			0
$201$	1151.09			0.0284916
$202$		0			0
$203$	−11381.4			−0.276188
$204$		0			0
$205$		−	8988.53i		−	0.213885i
$206$		0			0
$207$	−5469.85			−0.127654
$208$		0			0
$209$	−2630.24	+	68988.0i	−0.0602147	+	1.57936i
$210$		0			0
$211$			42840.1i			0.962245i	0.876653	+	0.481123i	$0.159771\pi$
$211$			42840.1i			0.962245i	−0.876653	+	0.481123i	$0.840229\pi$
$212$		0			0
$213$	7865.13			0.173359
$214$		0			0
$215$		−	14511.5i		−	0.313932i
$216$		0			0
$217$			41564.9i			0.882688i
$218$		0			0
$219$		−	4420.08i		−	0.0921599i
$220$		0			0
$221$	−50417.0			−1.03227
$222$		0			0
$223$	59661.7			1.19974			0.599868	−	0.800099i	$-0.295220\pi$
$223$	59661.7			1.19974			0.599868	+	0.800099i	$0.295220\pi$
$224$		0			0
$225$	44744.9			0.883849
$226$		0			0
$227$			24125.2i			0.468187i	0.972214	+	0.234094i	$0.0752122\pi$
$227$			24125.2i			0.468187i	−0.972214	+	0.234094i	$0.924788\pi$
$228$		0			0
$229$	−71548.3			−1.36436			−0.682179	−	0.731185i	$-0.738967\pi$
$229$	−71548.3			−1.36436			−0.682179	+	0.731185i	$0.738967\pi$
$230$		0			0
$231$	323.319	−	8480.27i	0.00605909	−	0.158923i
$232$		0			0
$233$			76741.8i			1.41358i	0.707424	+	0.706790i	$0.249857\pi$
$233$			76741.8i			1.41358i	−0.707424	+	0.706790i	$0.750143\pi$
$234$		0			0
$235$	9523.54			0.172450
$236$		0			0
$237$		−	12353.1i		−	0.219927i
$238$		0			0
$239$			97651.3i			1.70955i	0.518997	+	0.854776i	$0.326305\pi$
$239$			97651.3i			1.70955i	−0.518997	+	0.854776i	$0.673695\pi$
$240$		0			0
$241$		−	109461.i		−	1.88463i	−0.334722	−	0.942317i	$-0.608642\pi$
$241$		−	109461.i		−	1.88463i	0.334722	−	0.942317i	$-0.391358\pi$
$242$		0			0
$243$	−36103.2			−0.611411
$244$		0			0
$245$	7498.66			0.124926
$246$		0			0
$247$	−78834.8			−1.29218
$248$		0			0
$249$		−	1520.82i		−	0.0245290i
$250$		0			0
$251$	54758.2			0.869164			0.434582	−	0.900632i	$-0.356896\pi$
$251$	54758.2			0.869164			0.434582	+	0.900632i	$0.356896\pi$
$252$		0			0
$253$	8569.85	+	326.734i	0.133885	+	0.00510451i
$254$		0			0
$255$		−	4798.86i		−	0.0738003i
$256$		0			0
$257$	92386.3			1.39875			0.699377	−	0.714753i	$-0.253461\pi$
$257$	92386.3			1.39875			0.699377	+	0.714753i	$0.253461\pi$
$258$		0			0
$259$		−	48814.7i		−	0.727698i
$260$		0			0
$261$		−	24495.6i		−	0.359590i
$262$		0			0
$263$		−	77399.6i		−	1.11899i	−0.828833	−	0.559496i	$-0.810995\pi$
$263$		−	77399.6i		−	1.11899i	0.828833	−	0.559496i	$-0.189005\pi$
$264$		0			0
$265$	−20262.6			−0.288538
$266$		0			0
$267$	−9082.83			−0.127409
$268$		0			0
$269$	66027.2			0.912470			0.456235	−	0.889859i	$-0.349198\pi$
$269$	66027.2			0.912470			0.456235	+	0.889859i	$0.349198\pi$
$270$		0			0
$271$		−	73926.8i		−	1.00661i	−0.864108	−	0.503307i	$-0.832116\pi$
$271$		−	73926.8i		−	1.00661i	0.864108	−	0.503307i	$-0.167884\pi$
$272$		0			0
$273$	9690.67			0.130026
$274$		0			0
$275$	−70103.7	−	2672.78i	−0.926991	−	0.0353425i
$276$		0			0
$277$			64374.6i			0.838987i	0.907758	+	0.419493i	$0.137792\pi$
$277$			64374.6i			0.838987i	−0.907758	+	0.419493i	$0.862208\pi$
$278$		0			0
$279$	−89457.8			−1.14924
$280$		0			0
$281$		−	104033.i		−	1.31752i	−0.752354	−	0.658759i	$-0.771081\pi$
$281$		−	104033.i		−	1.31752i	0.752354	−	0.658759i	$-0.228919\pi$
$282$		0			0
$283$		−	3909.45i		−	0.0488139i	−0.999702	−	0.0244069i	$-0.992230\pi$
$283$		−	3909.45i		−	0.0488139i	0.999702	−	0.0244069i	$-0.00776974\pi$
$284$		0			0
$285$		−	7503.77i		−	0.0923825i
$286$		0			0
$287$	47935.0			0.581954
$288$		0			0
$289$	−49624.1			−0.594151
$290$		0			0
$291$	−12375.9			−0.146148
$292$		0			0
$293$		−	64081.9i		−	0.746449i	−0.927741	−	0.373224i	$-0.878252\pi$
$293$		−	64081.9i		−	0.746449i	0.927741	−	0.373224i	$-0.121748\pi$
$294$		0			0
$295$	11182.2			0.128494
$296$		0			0
$297$	37408.0	+	1426.22i	0.424084	+	0.0161686i
$298$		0			0
$299$			9793.04i			0.109541i
$300$		0			0
$301$	77388.6			0.854169
$302$		0			0
$303$		−	37707.7i		−	0.410719i
$304$		0			0
$305$		−	36287.0i		−	0.390078i
$306$		0			0
$307$		−	52705.6i		−	0.559216i	−0.960114	−	0.279608i	$-0.909795\pi$
$307$		−	52705.6i		−	0.559216i	0.960114	−	0.279608i	$-0.0902045\pi$
$308$		0			0
$309$	7342.54			0.0769005
$310$		0			0
$311$	−80464.5			−0.831924			−0.415962	−	0.909382i	$-0.636555\pi$
$311$	−80464.5			−0.831924			−0.415962	+	0.909382i	$0.636555\pi$
$312$		0			0
$313$	−19122.0			−0.195184			−0.0975920	−	0.995227i	$-0.531114\pi$
$313$	−19122.0			−0.195184			−0.0975920	+	0.995227i	$0.531114\pi$
$314$		0			0
$315$		−	18606.7i		−	0.187520i
$316$		0			0
$317$	50197.6			0.499533			0.249766	−	0.968306i	$-0.419646\pi$
$317$	50197.6			0.499533			0.249766	+	0.968306i	$0.419646\pi$
$318$		0			0
$319$	−1463.21	+	38378.3i	−0.0143789	+	0.377142i
$320$		0			0
$321$		−	36012.6i		−	0.349498i
$322$		0			0
$323$	−208193.			−1.99554
$324$		0			0
$325$		−	80109.7i		−	0.758435i
$326$		0			0
$327$			19237.3i			0.179907i
$328$		0			0
$329$			50788.2i			0.469214i
$330$		0			0
$331$	170531.			1.55649			0.778247	−	0.627958i	$-0.216109\pi$
$331$	170531.			1.55649			0.778247	+	0.627958i	$0.216109\pi$
$332$		0			0
$333$	105061.			0.947444
$334$		0			0
$335$	−3957.00			−0.0352595
$336$		0			0
$337$			131558.i			1.15840i	0.815185	+	0.579200i	$0.196635\pi$
$337$			131558.i			1.15840i	−0.815185	+	0.579200i	$0.803365\pi$
$338$		0			0
$339$	−26167.9			−0.227704
$340$		0			0
$341$	140157.	+	5343.65i	1.20533	+	0.0459546i
$342$		0			0
$343$			126084.i			1.07169i
$344$		0			0
$345$	−932.135			−0.00783142
$346$		0			0
$347$			25233.1i			0.209562i	0.994495	+	0.104781i	$0.0334141\pi$
$347$			25233.1i			0.209562i	−0.994495	+	0.104781i	$0.966586\pi$
$348$		0			0
$349$			10590.3i			0.0869473i	0.999055	+	0.0434736i	$0.0138425\pi$
$349$			10590.3i			0.0869473i	−0.999055	+	0.0434736i	$0.986158\pi$
$350$		0			0
$351$			42747.3i			0.346972i
$352$		0			0
$353$	−184997.			−1.48462			−0.742311	−	0.670055i	$-0.766270\pi$
$353$	−184997.			−1.48462			−0.742311	+	0.670055i	$0.766270\pi$
$354$		0			0
$355$	−27037.3			−0.214539
$356$		0			0
$357$	25591.9			0.200801
$358$		0			0
$359$			2401.44i			0.0186330i	0.999957	+	0.00931651i	$0.00296558\pi$
$359$			2401.44i			0.0186330i	−0.999957	+	0.00931651i	$0.997034\pi$
$360$		0			0
$361$	−195221.			−1.49800
$362$		0			0
$363$	−28554.0	−	2180.47i	−0.216698	−	0.0165477i
$364$		0			0
$365$			15194.6i			0.114052i
$366$		0			0
$367$	43796.3			0.325166			0.162583	−	0.986695i	$-0.448017\pi$
$367$	43796.3			0.325166			0.162583	+	0.986695i	$0.448017\pi$
$368$		0			0
$369$			103168.i			0.757690i
$370$		0			0
$371$		−	108059.i		−	0.785076i
$372$		0			0
$373$			116581.i			0.837931i	0.908002	+	0.418966i	$0.137607\pi$
$373$			116581.i			0.837931i	−0.908002	+	0.418966i	$0.862393\pi$
$374$		0			0
$375$	15844.8			0.112674
$376$		0			0
$377$	−43856.1			−0.308566
$378$		0			0
$379$	145996.			1.01640			0.508199	−	0.861240i	$-0.330312\pi$
$379$	145996.			1.01640			0.508199	+	0.861240i	$0.330312\pi$
$380$		0			0
$381$			47279.1i			0.325701i
$382$		0			0
$383$	173807.			1.18486			0.592432	−	0.805620i	$-0.298168\pi$
$383$	173807.			1.18486			0.592432	+	0.805620i	$0.298168\pi$
$384$		0			0
$385$	−1111.45	+	29151.9i	−0.00749838	+	0.196673i
$386$		0			0
$387$			166559.i			1.11211i
$388$		0			0
$389$	172951.			1.14294			0.571471	−	0.820622i	$-0.306373\pi$
$389$	172951.			1.14294			0.571471	+	0.820622i	$0.306373\pi$
$390$		0			0
$391$			25862.2i			0.169166i
$392$		0			0
$393$			20199.3i			0.130783i
$394$		0			0
$395$			42465.2i			0.272169i
$396$		0			0
$397$	−205854.			−1.30611			−0.653053	−	0.757312i	$-0.726512\pi$
$397$	−205854.			−1.30611			−0.653053	+	0.757312i	$0.726512\pi$
$398$		0			0
$399$	40016.9			0.251361
$400$		0			0
$401$	−29545.2			−0.183738			−0.0918689	−	0.995771i	$-0.529284\pi$
$401$	−29545.2			−0.183738			−0.0918689	+	0.995771i	$0.529284\pi$
$402$		0			0
$403$			160162.i			0.986166i
$404$		0			0
$405$	37962.6			0.231444
$406$		0			0
$407$	−164604.	−	6275.69i	−0.993690	−	0.0378855i
$408$		0			0
$409$		−	103104.i		−	0.616349i	−0.951330	−	0.308175i	$-0.900282\pi$
$409$		−	103104.i		−	0.616349i	0.951330	−	0.308175i	$-0.0997181\pi$
$410$		0			0
$411$	−35287.3			−0.208898
$412$		0			0
$413$			59633.6i			0.349616i
$414$		0			0
$415$			5228.01i			0.0303557i
$416$		0			0
$417$			35356.0i			0.203325i
$418$		0			0
$419$	123880.			0.705621			0.352811	−	0.935695i	$-0.385226\pi$
$419$	123880.			0.705621			0.352811	+	0.935695i	$0.385226\pi$
$420$		0			0
$421$	−72192.8			−0.407314			−0.203657	−	0.979042i	$-0.565283\pi$
$421$	−72192.8			−0.407314			−0.203657	+	0.979042i	$0.565283\pi$
$422$		0			0
$423$	−109309.			−0.610905
$424$		0			0
$425$		−	211560.i		−	1.17127i
$426$		0			0
$427$	193515.			1.06135
$428$		0			0
$429$	1245.85	−	32677.1i	0.00676940	−	0.177553i
$430$		0			0
$431$		−	370816.i		−	1.99620i	−0.0616150	−	0.998100i	$-0.519625\pi$
$431$		−	370816.i		−	1.99620i	0.0616150	−	0.998100i	$-0.480375\pi$
$432$		0			0
$433$	−67611.3			−0.360615			−0.180307	−	0.983610i	$-0.557709\pi$
$433$	−67611.3			−0.360615			−0.180307	+	0.983610i	$0.557709\pi$
$434$		0			0
$435$		−	4174.38i		−	0.0220604i
$436$		0			0
$437$			40439.6i			0.211760i
$438$		0			0
$439$			61838.3i			0.320869i	0.987046	+	0.160435i	$0.0512896\pi$
$439$			61838.3i			0.320869i	−0.987046	+	0.160435i	$0.948710\pi$
$440$		0			0
$441$	−86067.5			−0.442550
$442$		0			0
$443$	225163.			1.14733			0.573665	−	0.819090i	$-0.305521\pi$
$443$	225163.			1.14733			0.573665	+	0.819090i	$0.305521\pi$
$444$		0			0
$445$	31223.3			0.157674
$446$		0			0
$447$			52496.9i			0.262736i
$448$		0			0
$449$	−348050.			−1.72643			−0.863214	−	0.504838i	$-0.831552\pi$
$449$	−348050.			−1.72643			−0.863214	+	0.504838i	$0.831552\pi$
$450$		0			0
$451$	6162.59	−	161637.i	0.0302978	−	0.794674i
$452$		0			0
$453$			30104.3i			0.146701i
$454$		0			0
$455$	−33312.8			−0.160912
$456$		0			0
$457$			367325.i			1.75881i	0.476077	+	0.879404i	$0.342058\pi$
$457$			367325.i			1.75881i	−0.476077	+	0.879404i	$0.657942\pi$
$458$		0			0
$459$			112891.i			0.535836i
$460$		0			0
$461$		−	373933.i		−	1.75951i	−0.475425	−	0.879756i	$-0.657706\pi$
$461$		−	373933.i		−	1.75951i	0.475425	−	0.879756i	$-0.342294\pi$
$462$		0			0
$463$	381923.			1.78161			0.890807	−	0.454382i	$-0.150140\pi$
$463$	381923.			1.78161			0.890807	+	0.454382i	$0.150140\pi$
$464$		0			0
$465$	−15244.8			−0.0705043
$466$		0			0
$467$	−198666.			−0.910940			−0.455470	−	0.890251i	$-0.650529\pi$
$467$	−198666.			−0.910940			−0.455470	+	0.890251i	$0.650529\pi$
$468$		0			0
$469$		−	21102.3i		−	0.0959366i
$470$		0			0
$471$	67211.7			0.302972
$472$		0			0
$473$	9949.20	−	260955.i	0.0444699	−	1.16639i
$474$		0			0
$475$		−	330807.i		−	1.46618i
$476$		0			0
$477$	232569.			1.02215
$478$		0			0
$479$			94150.7i			0.410348i	0.978725	+	0.205174i	$0.0657761\pi$
$479$			94150.7i			0.410348i	−0.978725	+	0.205174i	$0.934224\pi$
$480$		0			0
$481$		−	188098.i		−	0.813007i
$482$		0			0
$483$		−	4970.99i		−	0.0213083i
$484$		0			0
$485$	42543.8			0.180864
$486$		0			0
$487$	−86151.1			−0.363248			−0.181624	−	0.983368i	$-0.558135\pi$
$487$	−86151.1			−0.363248			−0.181624	+	0.983368i	$0.558135\pi$
$488$		0			0
$489$	−66588.4			−0.278471
$490$		0			0
$491$		−	257966.i		−	1.07004i	−0.844840	−	0.535019i	$-0.820304\pi$
$491$		−	257966.i		−	1.07004i	0.844840	−	0.535019i	$-0.179696\pi$
$492$		0			0
$493$	−115819.			−0.476524
$494$		0			0
$495$	−62742.1	−	2392.11i	−0.256064	−	0.00976271i
$496$		0			0
$497$		−	144187.i		−	0.583733i
$498$		0			0
$499$	−310264.			−1.24604			−0.623018	−	0.782208i	$-0.714093\pi$
$499$	−310264.			−1.24604			−0.623018	+	0.782208i	$0.714093\pi$
$500$		0			0
$501$		−	68211.3i		−	0.271757i
$502$		0			0
$503$		−	423920.i		−	1.67552i	−0.546042	−	0.837758i	$-0.683866\pi$
$503$		−	423920.i		−	1.67552i	0.546042	−	0.837758i	$-0.316134\pi$
$504$		0			0
$505$			129625.i			0.508282i
$506$		0			0
$507$	−18522.9			−0.0720598
$508$		0			0
$509$	157612.			0.608350			0.304175	−	0.952616i	$-0.401619\pi$
$509$	157612.			0.608350			0.304175	+	0.952616i	$0.401619\pi$
$510$		0			0
$511$	−81031.1			−0.310320
$512$		0			0
$513$			176522.i			0.670755i
$514$		0			0
$515$	−25240.8			−0.0951676
$516$		0			0
$517$	171258.	+	6529.40i	0.640724	+	0.0244283i
$518$		0			0
$519$		−	57382.9i		−	0.213034i
$520$		0			0
$521$	282422.			1.04046			0.520228	−	0.854028i	$-0.325847\pi$
$521$	282422.			1.04046			0.520228	+	0.854028i	$0.325847\pi$
$522$		0			0
$523$			423736.i			1.54914i	0.632486	+	0.774572i	$0.282035\pi$
$523$			423736.i			1.54914i	−0.632486	+	0.774572i	$0.717965\pi$
$524$		0			0
$525$			40664.0i			0.147534i
$526$		0			0
$527$			422969.i			1.52296i
$528$		0			0
$529$	−274817.			−0.982049
$530$		0			0
$531$	−128346.			−0.455191
$532$		0			0
$533$	184708.			0.650177
$534$		0			0
$535$			123798.i			0.432518i
$536$		0			0
$537$	−4698.18			−0.0162922
$538$		0			0
$539$	134846.	+	5141.13i	0.464151	+	0.0176962i
$540$		0			0
$541$		−	335320.i		−	1.14568i	−0.819666	−	0.572841i	$-0.805841\pi$
$541$		−	335320.i		−	1.14568i	0.819666	−	0.572841i	$-0.194159\pi$
$542$		0			0
$543$	54769.7			0.185755
$544$		0			0
$545$		−	66130.4i		−	0.222643i
$546$		0			0
$547$			62911.8i			0.210260i	0.994458	+	0.105130i	$0.0335259\pi$
$547$			62911.8i			0.210260i	−0.994458	+	0.105130i	$0.966474\pi$
$548$		0			0
$549$			416492.i			1.38185i
$550$		0			0
$551$	−181100.			−0.596508
$552$		0			0
$553$	−226463.			−0.740538
$554$		0			0
$555$	17903.8			0.0581245
$556$		0			0
$557$			168355.i			0.542646i	0.962488	+	0.271323i	$0.0874612\pi$
$557$			168355.i			0.542646i	−0.962488	+	0.271323i	$0.912539\pi$
$558$		0			0
$559$	298202.			0.954305
$560$		0			0
$561$	3290.13	−	86296.2i	0.0104541	−	0.274199i
$562$		0			0
$563$		−	199140.i		−	0.628265i	−0.949379	−	0.314132i	$-0.898286\pi$
$563$		−	199140.i		−	0.628265i	0.949379	−	0.314132i	$-0.101714\pi$
$564$		0			0
$565$	89955.3			0.281793
$566$		0			0
$567$			202451.i			0.629729i
$568$		0			0
$569$			190357.i			0.587956i	0.955812	+	0.293978i	$0.0949792\pi$
$569$			190357.i			0.587956i	−0.955812	+	0.293978i	$0.905021\pi$
$570$		0			0
$571$			141142.i			0.432896i	0.976294	+	0.216448i	$0.0694472\pi$
$571$			141142.i			0.432896i	−0.976294	+	0.216448i	$0.930553\pi$
$572$		0			0
$573$	34031.8			0.103651
$574$		0			0
$575$	−41093.6			−0.124291
$576$		0			0
$577$	629490.			1.89076			0.945381	−	0.325967i	$-0.105690\pi$
$577$	629490.			1.89076			0.945381	+	0.325967i	$0.105690\pi$
$578$		0			0
$579$			73531.0i			0.219338i
$580$		0			0
$581$	−27880.5			−0.0825940
$582$		0			0
$583$	−364375.	−	13892.2i	−1.07204	−	0.0408727i
$584$		0			0
$585$		−	71697.4i		−	0.209504i
$586$		0			0
$587$	−459745.			−1.33426			−0.667130	−	0.744941i	$-0.732478\pi$
$587$	−459745.			−1.33426			−0.667130	+	0.744941i	$0.732478\pi$
$588$		0			0
$589$			661378.i			1.90642i
$590$		0			0
$591$		−	38393.2i		−	0.109921i
$592$		0			0
$593$			511611.i			1.45489i	0.686166	+	0.727445i	$0.259292\pi$
$593$			511611.i			1.45489i	−0.686166	+	0.727445i	$0.740708\pi$
$594$		0			0
$595$	−87975.1			−0.248500
$596$		0			0
$597$	−124618.			−0.349650
$598$		0			0
$599$	−266789.			−0.743556			−0.371778	−	0.928322i	$-0.621252\pi$
$599$	−266789.			−0.743556			−0.371778	+	0.928322i	$0.621252\pi$
$600$		0			0
$601$		−	76615.0i		−	0.212112i	−0.994360	−	0.106056i	$-0.966178\pi$
$601$		−	76615.0i		−	0.212112i	0.994360	−	0.106056i	$-0.0338223\pi$
$602$		0			0
$603$	45417.3			0.124907
$604$		0			0
$605$	98157.8	+	7495.63i	0.268172	+	0.0204785i
$606$		0			0
$607$			155813.i			0.422888i	0.977390	+	0.211444i	$0.0678166\pi$
$607$			155813.i			0.422888i	−0.977390	+	0.211444i	$0.932183\pi$
$608$		0			0
$609$	22261.5			0.0600234
$610$		0			0
$611$			195702.i			0.524220i
$612$		0			0
$613$		−	423746.i		−	1.12768i	−0.825885	−	0.563838i	$-0.809324\pi$
$613$		−	423746.i		−	1.12768i	0.825885	−	0.563838i	$-0.190676\pi$
$614$		0			0
$615$			17581.2i			0.0464834i
$616$		0			0
$617$	190437.			0.500243			0.250122	−	0.968214i	$-0.419529\pi$
$617$	190437.			0.500243			0.250122	+	0.968214i	$0.419529\pi$
$618$		0			0
$619$	287818.			0.751168			0.375584	−	0.926788i	$-0.377442\pi$
$619$	287818.			0.751168			0.375584	+	0.926788i	$0.377442\pi$
$620$		0			0
$621$	21927.9			0.0568610
$622$		0			0
$623$			166511.i			0.429009i
$624$		0			0
$625$	307900.			0.788225
$626$		0			0
$627$	5144.63	−	134937.i	0.0130864	−	0.343240i
$628$		0			0
$629$		−	496744.i		−	1.25554i
$630$		0			0
$631$	−95505.3			−0.239866			−0.119933	−	0.992782i	$-0.538268\pi$
$631$	−95505.3			−0.239866			−0.119933	+	0.992782i	$0.538268\pi$
$632$		0			0
$633$		−	83793.4i		−	0.209123i
$634$		0			0
$635$		−	162527.i		−	0.403069i
$636$		0			0
$637$			154092.i			0.379754i
$638$		0			0
$639$	310327.			0.760007
$640$		0			0
$641$	212740.			0.517765			0.258883	−	0.965909i	$-0.416646\pi$
$641$	212740.			0.517765			0.258883	+	0.965909i	$0.416646\pi$
$642$		0			0
$643$	−127119.			−0.307460			−0.153730	−	0.988113i	$-0.549129\pi$
$643$	−127119.			−0.307460			−0.153730	+	0.988113i	$0.549129\pi$
$644$		0			0
$645$			28383.9i			0.0682264i
$646$		0			0
$647$	−322480.			−0.770361			−0.385181	−	0.922841i	$-0.625861\pi$
$647$	−322480.			−0.770361			−0.385181	+	0.922841i	$0.625861\pi$
$648$		0			0
$649$	201085.	+	7666.58i	0.477409	+	0.0182017i
$650$		0			0
$651$		−	81299.1i		−	0.191833i
$652$		0			0
$653$	−427102.			−1.00163			−0.500813	−	0.865555i	$-0.666966\pi$
$653$	−427102.			−1.00163			−0.500813	+	0.865555i	$0.666966\pi$
$654$		0			0
$655$		−	69437.5i		−	0.161850i
$656$		0			0
$657$		−	174399.i		−	0.404029i
$658$		0			0
$659$			339223.i			0.781114i	0.920579	+	0.390557i	$0.127718\pi$
$659$			339223.i			0.781114i	−0.920579	+	0.390557i	$0.872282\pi$
$660$		0			0
$661$	−518565.			−1.18686			−0.593431	−	0.804885i	$-0.702227\pi$
$661$	−518565.			−1.18686			−0.593431	+	0.804885i	$0.702227\pi$
$662$		0			0
$663$	98613.4			0.224341
$664$		0			0
$665$	−137563.			−0.311070
$666$		0			0
$667$			22496.7i			0.0505670i
$668$		0			0
$669$	−116696.			−0.260737
$670$		0			0
$671$	24878.6	−	652536.i	0.0552562	−	1.44930i
$672$		0			0
$673$		−	37229.5i		−	0.0821972i	−0.999155	−	0.0410986i	$-0.986914\pi$
$673$		−	37229.5i		−	0.0821972i	0.999155	−	0.0410986i	$-0.0130858\pi$
$674$		0			0
$675$	−179377.			−0.393693
$676$		0			0
$677$			431005.i			0.940383i	0.882564	+	0.470192i	$0.155815\pi$
$677$			431005.i			0.940383i	−0.882564	+	0.470192i	$0.844185\pi$
$678$		0			0
$679$			226882.i			0.492108i
$680$		0			0
$681$		−	47187.9i		−	0.101750i
$682$		0			0
$683$	788957.			1.69127			0.845633	−	0.533765i	$-0.179223\pi$
$683$	788957.			1.69127			0.845633	+	0.533765i	$0.179223\pi$
$684$		0			0
$685$	121304.			0.258521
$686$		0			0
$687$	139945.			0.296514
$688$		0			0
$689$		−	416383.i		−	0.877111i
$690$		0			0
$691$	−281540.			−0.589637			−0.294818	−	0.955553i	$-0.595259\pi$
$691$	−281540.			−0.589637			−0.294818	+	0.955553i	$0.595259\pi$
$692$		0			0
$693$	12756.9	−	334598.i	0.0265631	−	0.696717i
$694$		0			0
$695$		−	121541.i		−	0.251624i
$696$		0			0
$697$	487792.			1.00408
$698$		0			0
$699$		−	150104.i		−	0.307211i
$700$		0			0
$701$		−	225633.i		−	0.459162i	−0.973290	−	0.229581i	$-0.926264\pi$
$701$		−	225633.i		−	0.459162i	0.973290	−	0.229581i	$-0.0737355\pi$
$702$		0			0
$703$		−	776736.i		−	1.57168i
$704$		0			0
$705$	−18627.6			−0.0374783
$706$		0			0
$707$	−691276.			−1.38297
$708$		0			0
$709$	−750874.			−1.49374			−0.746869	−	0.664971i	$-0.768444\pi$
$709$	−750874.			−1.49374			−0.746869	+	0.664971i	$0.768444\pi$
$710$		0			0
$711$		−	487404.i		−	0.964162i
$712$		0			0
$713$	82157.9			0.161611
$714$		0			0
$715$	−4282.75	+	112331.i	−0.00837743	+	0.219730i
$716$		0			0
$717$		−	191002.i		−	0.371534i
$718$		0			0
$719$	888683.			1.71905			0.859526	−	0.511092i	$-0.170759\pi$
$719$	888683.			1.71905			0.859526	+	0.511092i	$0.170759\pi$
$720$		0			0
$721$		−	134607.i		−	0.258939i
$722$		0			0
$723$			214102.i			0.409585i
$724$		0			0
$725$		−	184029.i		−	0.350115i
$726$		0			0
$727$	−361266.			−0.683530			−0.341765	−	0.939785i	$-0.611025\pi$
$727$	−361266.			−0.683530			−0.341765	+	0.939785i	$0.611025\pi$
$728$		0			0
$729$	−386708.			−0.727659
$730$		0			0
$731$	787515.			1.47375
$732$		0			0
$733$			178651.i			0.332504i	0.986083	+	0.166252i	$0.0531666\pi$
$733$			178651.i			0.332504i	−0.986083	+	0.166252i	$0.946833\pi$
$734$		0			0
$735$	−14667.0			−0.0271499
$736$		0			0
$737$	−71157.3	−	2712.95i	−0.131004	−	0.00499466i
$738$		0			0
$739$		−	791792.i		−	1.44985i	−0.688829	−	0.724924i	$-0.741875\pi$
$739$		−	791792.i		−	1.44985i	0.688829	−	0.724924i	$-0.258125\pi$
$740$		0			0
$741$	154197.			0.280828
$742$		0			0
$743$			774816.i			1.40353i	0.712409	+	0.701764i	$0.247604\pi$
$743$			774816.i			1.40353i	−0.712409	+	0.701764i	$0.752396\pi$
$744$		0			0
$745$		−	180464.i		−	0.325146i
$746$		0			0
$747$		−	60005.7i		−	0.107535i
$748$		0			0
$749$	−660201.			−1.17683
$750$		0			0
$751$	905890.			1.60618			0.803092	−	0.595854i	$-0.203186\pi$
$751$	905890.			1.60618			0.803092	+	0.595854i	$0.203186\pi$
$752$		0			0
$753$	−107105.			−0.188894
$754$		0			0
$755$		−	103487.i		−	0.181548i
$756$		0			0
$757$	−37734.7			−0.0658489			−0.0329245	−	0.999458i	$-0.510482\pi$
$757$	−37734.7			−0.0658489			−0.0329245	+	0.999458i	$0.510482\pi$
$758$		0			0
$759$	−16762.2	−	639.078i	−0.0290970	−	0.00110935i
$760$		0			0
$761$			793663.i			1.37046i	0.728326	+	0.685231i	$0.240299\pi$
$761$			793663.i			1.37046i	−0.728326	+	0.685231i	$0.759701\pi$
$762$		0			0
$763$	352667.			0.605782
$764$		0			0
$765$		−	189344.i		−	0.323541i
$766$		0			0
$767$			229787.i			0.390601i
$768$		0			0
$769$		−	248952.i		−	0.420982i	−0.977596	−	0.210491i	$-0.932494\pi$
$769$		−	248952.i		−	0.420982i	0.977596	−	0.210491i	$-0.0675062\pi$
$770$		0			0
$771$	−180704.			−0.303989
$772$		0			0
$773$	−696432.			−1.16552			−0.582760	−	0.812644i	$-0.698027\pi$
$773$	−696432.			−1.16552			−0.582760	+	0.812644i	$0.698027\pi$
$774$		0			0
$775$	−672074.			−1.11896
$776$		0			0
$777$			95479.4i			0.158149i
$778$		0			0
$779$	762738.			1.25690
$780$		0			0
$781$	−486202.	−	18537.0i	−0.797103	−	0.0303904i
$782$		0			0
$783$			98199.8i			0.160172i
$784$		0			0
$785$	−231048.			−0.374941
$786$		0			0
$787$		−	107990.i		−	0.174355i	−0.996193	−	0.0871776i	$-0.972215\pi$
$787$		−	107990.i		−	0.174355i	0.996193	−	0.0871776i	$-0.0277848\pi$
$788$		0			0
$789$			151390.i			0.243189i
$790$		0			0
$791$			479723.i			0.766722i
$792$		0			0
$793$	745673.			1.18577
$794$		0			0
$795$	39632.8			0.0627076
$796$		0			0
$797$	−79943.4			−0.125854			−0.0629268	−	0.998018i	$-0.520043\pi$
$797$	−79943.4			−0.125854			−0.0629268	+	0.998018i	$0.520043\pi$
$798$		0			0
$799$			516826.i			0.809563i
$800$		0			0
$801$	−358372.			−0.558560
$802$		0			0
$803$	−10417.5	+	273238.i	−0.0161559	+	0.423751i
$804$		0			0
$805$			17088.4i			0.0263699i
$806$		0			0
$807$	−129146.			−0.198306
$808$		0			0
$809$		−	140244.i		−	0.214282i	−0.994244	−	0.107141i	$-0.965830\pi$
$809$		−	140244.i		−	0.214282i	0.994244	−	0.107141i	$-0.0341697\pi$
$810$		0			0
$811$		−	796857.i		−	1.21154i	−0.795639	−	0.605771i	$-0.792865\pi$
$811$		−	796857.i		−	1.21154i	0.795639	−	0.605771i	$-0.207135\pi$
$812$		0			0
$813$			144597.i			0.218766i
$814$		0			0
$815$	228905.			0.344620
$816$		0			0
$817$	1.23140e6			1.84483
$818$		0			0
$819$	382355.			0.570032
$820$		0			0
$821$			1.03290e6i			1.53240i	0.642601	+	0.766201i	$0.277855\pi$
$821$			1.03290e6i			1.53240i	−0.642601	+	0.766201i	$0.722145\pi$
$822$		0			0
$823$	−640524.			−0.945662			−0.472831	−	0.881153i	$-0.656768\pi$
$823$	−640524.			−0.945662			−0.472831	+	0.881153i	$0.656768\pi$
$824$		0			0
$825$	137120.	+	5227.83i	0.201461	+	0.00768093i
$826$		0			0
$827$			1.26963e6i			1.85637i	0.372114	+	0.928187i	$0.378633\pi$
$827$			1.26963e6i			1.85637i	−0.372114	+	0.928187i	$0.621367\pi$
$828$		0			0
$829$	−975440.			−1.41936			−0.709678	−	0.704526i	$-0.751160\pi$
$829$	−975440.			−1.41936			−0.709678	+	0.704526i	$0.751160\pi$
$830$		0			0
$831$		−	125914.i		−	0.182336i
$832$		0			0
$833$			406939.i			0.586461i
$834$		0			0
$835$			234484.i			0.336311i
$836$		0			0
$837$	358625.			0.511906
$838$		0			0
$839$	376637.			0.535055			0.267528	−	0.963550i	$-0.413793\pi$
$839$	376637.			0.535055			0.267528	+	0.963550i	$0.413793\pi$
$840$		0			0
$841$	606534.			0.857558
$842$		0			0
$843$			203483.i			0.286334i
$844$		0			0
$845$	63674.7			0.0891771
$846$		0			0
$847$	−39973.5	+	523466.i	−0.0557193	+	0.729662i
$848$		0			0
$849$			7646.72i			0.0106086i
$850$		0			0
$851$	−96488.0			−0.133234
$852$		0			0
$853$		−	362817.i		−	0.498643i	−0.968421	−	0.249322i	$-0.919792\pi$
$853$		−	362817.i		−	0.498643i	0.968421	−	0.249322i	$-0.0802076\pi$
$854$		0			0
$855$		−	296069.i		−	0.405005i
$856$		0			0
$857$			73874.1i			0.100584i	0.998735	+	0.0502922i	$0.0160153\pi$
$857$			73874.1i			0.100584i	−0.998735	+	0.0502922i	$0.983985\pi$
$858$		0			0
$859$	−706662.			−0.957691			−0.478845	−	0.877899i	$-0.658945\pi$
$859$	−706662.			−0.957691			−0.478845	+	0.877899i	$0.658945\pi$
$860$		0			0
$861$	−93758.7			−0.126475
$862$		0			0
$863$	−327867.			−0.440227			−0.220113	−	0.975474i	$-0.570643\pi$
$863$	−327867.			−0.440227			−0.220113	+	0.975474i	$0.570643\pi$
$864$		0			0
$865$			197261.i			0.263638i
$866$		0			0
$867$	97062.6			0.129126
$868$		0			0
$869$	−29114.4	+	763636.i	−0.0385539	+	1.01122i
$870$		0			0
$871$		−	81313.7i		−	0.107183i
$872$		0			0
$873$	−488305.			−0.640712
$874$		0			0
$875$		−	290475.i		−	0.379396i
$876$		0			0
$877$			661821.i			0.860481i	0.902714	+	0.430240i	$0.141571\pi$
$877$			661821.i			0.860481i	−0.902714	+	0.430240i	$0.858429\pi$
$878$		0			0
$879$			125341.i			0.162224i
$880$		0			0
$881$	185765.			0.239338			0.119669	−	0.992814i	$-0.461817\pi$
$881$	185765.			0.239338			0.119669	+	0.992814i	$0.461817\pi$
$882$		0			0
$883$	−1.04690e6			−1.34272			−0.671359	−	0.741132i	$-0.734289\pi$
$883$	−1.04690e6			−1.34272			−0.671359	+	0.741132i	$0.734289\pi$
$884$		0			0
$885$	−21871.9			−0.0279254
$886$		0			0
$887$		−	355686.i		−	0.452085i	−0.974117	−	0.226042i	$-0.927421\pi$
$887$		−	355686.i		−	0.452085i	0.974117	−	0.226042i	$-0.0725788\pi$
$888$		0			0
$889$	866744.			1.09670
$890$		0			0
$891$	682667.	+	26027.4i	0.859911	+	0.0327850i
$892$		0			0
$893$			808138.i			1.01340i
$894$		0			0
$895$	16150.5			0.0201623
$896$		0			0
$897$		−	19154.8i		−	0.0238063i
$898$		0			0
$899$			367927.i			0.455242i
$900$		0			0
$901$		−	1.09962e6i		−	1.35454i
$902$		0			0
$903$	−151369.			−0.185635
$904$		0			0
$905$	−188277.			−0.229880
$906$		0			0
$907$	139907.			0.170069			0.0850347	−	0.996378i	$-0.472900\pi$
$907$	139907.			0.170069			0.0850347	+	0.996378i	$0.472900\pi$
$908$		0			0
$909$		−	1.48780e6i		−	1.80059i
$910$		0			0
$911$	−366506.			−0.441616			−0.220808	−	0.975317i	$-0.570869\pi$
$911$	−366506.			−0.441616			−0.220808	+	0.975317i	$0.570869\pi$
$912$		0			0
$913$	−3584.36	+	94013.5i	−0.00430002	+	0.112784i
$914$		0			0
$915$			70975.7i			0.0847750i
$916$		0			0
$917$	370304.			0.440372
$918$		0			0
$919$		−	752164.i		−	0.890598i	−0.895382	−	0.445299i	$-0.853097\pi$
$919$		−	752164.i		−	0.890598i	0.895382	−	0.445299i	$-0.146903\pi$
$920$		0			0
$921$			103090.i			0.121534i
$922$		0			0
$923$		−	555598.i		−	0.652165i
$924$		0			0
$925$	789297.			0.922480
$926$		0			0
$927$	289707.			0.337132
$928$		0			0
$929$	161987.			0.187693			0.0938467	−	0.995587i	$-0.470084\pi$
$929$	161987.			0.187693			0.0938467	+	0.995587i	$0.470084\pi$
$930$		0			0
$931$			636312.i			0.734127i
$932$		0			0
$933$	157385.			0.180801
$934$		0			0
$935$	−11310.2	+	296653.i	−0.0129374	+	0.339333i
$936$		0			0
$937$		−	915780.i		−	1.04307i	−0.853231	−	0.521533i	$-0.825360\pi$
$937$		−	915780.i		−	1.04307i	0.853231	−	0.521533i	$-0.174640\pi$
$938$		0			0
$939$	37401.7			0.0424190
$940$		0			0
$941$		−	784090.i		−	0.885497i	−0.896646	−	0.442748i	$-0.854004\pi$
$941$		−	784090.i		−	0.885497i	0.896646	−	0.442748i	$-0.145996\pi$
$942$		0			0
$943$		−	94749.1i		−	0.106550i
$944$		0			0
$945$			74592.0i			0.0835273i
$946$		0			0
$947$	−1.47216e6			−1.64155			−0.820776	−	0.571250i	$-0.806459\pi$
$947$	−1.47216e6			−1.64155			−0.820776	+	0.571250i	$0.806459\pi$
$948$		0			0
$949$	−312238.			−0.346700
$950$		0			0
$951$	−98184.2			−0.108563
$952$		0			0
$953$		−	372692.i		−	0.410360i	−0.978724	−	0.205180i	$-0.934222\pi$
$953$		−	372692.i		−	0.410360i	0.978724	−	0.205180i	$-0.0657779\pi$
$954$		0			0
$955$	−116988.			−0.128273
$956$		0			0
$957$	2861.98	−	75066.2i	0.00312495	−	0.0819636i
$958$		0			0
$959$			646905.i			0.703401i
$960$		0			0
$961$	420147.			0.454940
$962$		0			0
$963$		−	1.42091e6i		−	1.53220i
$964$		0			0
$965$		−	252772.i		−	0.271440i
$966$		0			0
$967$		−	1.33645e6i		−	1.42922i	−0.699523	−	0.714610i	$-0.746604\pi$
$967$		−	1.33645e6i		−	1.42922i	0.699523	−	0.714610i	$-0.253396\pi$
$968$		0			0
$969$	407216.			0.433688
$970$		0			0
$971$	−1.43833e6			−1.52552			−0.762762	−	0.646679i	$-0.776157\pi$
$971$	−1.43833e6			−1.52552			−0.762762	+	0.646679i	$0.776157\pi$
$972$		0			0
$973$	648164.			0.684635
$974$		0			0
$975$			156691.i			0.164829i
$976$		0			0
$977$	−133419.			−0.139774			−0.0698872	−	0.997555i	$-0.522264\pi$
$977$	−133419.			−0.139774			−0.0698872	+	0.997555i	$0.522264\pi$
$978$		0			0
$979$	561477.	+	21406.9i	0.585824	+	0.0223351i
$980$		0			0
$981$			759026.i			0.788713i
$982$		0			0
$983$	1.04704e6			1.08357			0.541783	−	0.840518i	$-0.317749\pi$
$983$	1.04704e6			1.08357			0.541783	+	0.840518i	$0.317749\pi$
$984$		0			0
$985$			131981.i			0.136032i
$986$		0			0
$987$		−	99339.4i		−	0.101973i
$988$		0			0
$989$		−	152968.i		−	0.156389i
$990$		0			0
$991$	−869334.			−0.885196			−0.442598	−	0.896720i	$-0.645943\pi$
$991$	−869334.			−0.885196			−0.442598	+	0.896720i	$0.645943\pi$
$992$		0			0
$993$	−333551.			−0.338270
$994$		0			0
$995$	428390.			0.432706
$996$		0			0
$997$		−	706445.i		−	0.710703i	−0.934733	−	0.355351i	$-0.884361\pi$
$997$		−	706445.i		−	0.710703i	0.934733	−	0.355351i	$-0.115639\pi$
$998$		0			0
$999$	−421177.			−0.422021

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	88.5.h.a.65.6	yes	12
3.2	odd	2		792.5.j.a.505.6		12
4.3	odd	2		176.5.h.f.65.7		12
8.3	odd	2		704.5.h.l.65.5		12
8.5	even	2		704.5.h.k.65.8		12
11.10	odd	2	inner	88.5.h.a.65.5	✓	12
33.32	even	2		792.5.j.a.505.5		12
44.43	even	2		176.5.h.f.65.8		12
88.21	odd	2		704.5.h.k.65.7		12
88.43	even	2		704.5.h.l.65.6		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
88.5.h.a.65.5	✓	12	11.10	odd	2	inner
88.5.h.a.65.6	yes	12	1.1	even	1	trivial
176.5.h.f.65.7		12	4.3	odd	2
176.5.h.f.65.8		12	44.43	even	2
704.5.h.k.65.7		12	88.21	odd	2
704.5.h.k.65.8		12	8.5	even	2
704.5.h.l.65.5		12	8.3	odd	2
704.5.h.l.65.6		12	88.43	even	2
792.5.j.a.505.5		12	33.32	even	2
792.5.j.a.505.6		12	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 88 = 2^{3} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	88.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.95596 q^{3} +6.72383 q^{5} +35.8576i q^{7} -77.1742 q^{9} +O(q^{10})\)	\(q-1.95596 q^{3} +6.72383 q^{5} +35.8576i q^{7} -77.1742 q^{9} +(120.912 + 4.60990i) q^{11} +138.170i q^{13} -13.1515 q^{15} +364.891i q^{17} +570.563i q^{19} -70.1358i q^{21} +70.8767 q^{23} -579.790 q^{25} +309.382 q^{27} +317.406i q^{29} +1159.17 q^{31} +(-236.499 - 9.01676i) q^{33} +241.100i q^{35} -1361.35 q^{37} -270.255i q^{39} -1336.82i q^{41} -2158.22i q^{43} -518.906 q^{45} +1416.39 q^{47} +1115.24 q^{49} -713.710i q^{51} -3013.55 q^{53} +(812.993 + 30.9962i) q^{55} -1116.00i q^{57} +1663.07 q^{59} -5396.77i q^{61} -2767.28i q^{63} +929.033i q^{65} -588.504 q^{67} -138.632 q^{69} -4021.12 q^{71} +2259.81i q^{73} +1134.04 q^{75} +(-165.300 + 4335.61i) q^{77} +6315.63i q^{79} +5645.98 q^{81} +777.535i q^{83} +2453.46i q^{85} -620.833i q^{87} +4643.68 q^{89} -4954.44 q^{91} -2267.28 q^{93} +3836.37i q^{95} +6327.31 q^{97} +(-9331.30 - 355.766i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 8 q^{3} + 492 q^{9}+O(q^{10}) \) 12 * q + 8 * q^3 + 492 * q^9	\( 12 q + 8 q^{3} + 492 q^{9} + 12 q^{11} + 480 q^{15} + 1608 q^{23} + 1484 q^{25} + 2048 q^{27} - 2136 q^{31} + 1568 q^{33} - 288 q^{37} - 8264 q^{45} - 6024 q^{47} - 9940 q^{49} + 3576 q^{53} - 8344 q^{55} - 7704 q^{59} + 10312 q^{67} - 616 q^{69} + 8904 q^{71} - 3544 q^{75} + 9120 q^{77} + 15340 q^{81} + 4224 q^{89} + 23904 q^{91} + 23480 q^{93} - 21280 q^{97} + 14420 q^{99}+O(q^{100}) \) 12 * q + 8 * q^3 + 492 * q^9 + 12 * q^11 + 480 * q^15 + 1608 * q^23 + 1484 * q^25 + 2048 * q^27 - 2136 * q^31 + 1568 * q^33 - 288 * q^37 - 8264 * q^45 - 6024 * q^47 - 9940 * q^49 + 3576 * q^53 - 8344 * q^55 - 7704 * q^59 + 10312 * q^67 - 616 * q^69 + 8904 * q^71 - 3544 * q^75 + 9120 * q^77 + 15340 * q^81 + 4224 * q^89 + 23904 * q^91 + 23480 * q^93 - 21280 * q^97 + 14420 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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