LMFDB - Embedded newform 2898.2.h.b.827.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2898,2,Mod(827,2898)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2898, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("2898.827");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2898 = 2 \cdot 3^{2} \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2898.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:	$23.1406465058$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			827.10
Character	$\chi$	$=$	2898.827
Dual form			2898.2.h.b.827.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{4} -0.526990 q^{5} -1.00000i q^{7} -1.00000i q^{8} +O(q^{10})$	\(q+1.00000i q^{2} -1.00000 q^{4} -0.526990 q^{5} -1.00000i q^{7} -1.00000i q^{8} -0.526990i q^{10} -5.12748 q^{11} -2.15150 q^{13} +1.00000 q^{14} +1.00000 q^{16} +4.57331 q^{17} -3.59685i q^{19} +0.526990 q^{20} -5.12748i q^{22} +(4.07953 + 2.52140i) q^{23} -4.72228 q^{25} -2.15150i q^{26} +1.00000i q^{28} +0.655814i q^{29} +7.07786 q^{31} +1.00000i q^{32} +4.57331i q^{34} +0.526990i q^{35} +2.56927i q^{37} +3.59685 q^{38} +0.526990i q^{40} +1.75581i q^{41} -0.776183i q^{43} +5.12748 q^{44} +(-2.52140 + 4.07953i) q^{46} +4.99360i q^{47} -1.00000 q^{49} -4.72228i q^{50} +2.15150 q^{52} +1.84124 q^{53} +2.70213 q^{55} -1.00000 q^{56} -0.655814 q^{58} +12.1238i q^{59} -4.36713i q^{61} +7.07786i q^{62} -1.00000 q^{64} +1.13382 q^{65} -0.255016i q^{67} -4.57331 q^{68} -0.526990 q^{70} +10.7154i q^{71} +12.3335 q^{73} -2.56927 q^{74} +3.59685i q^{76} +5.12748i q^{77} +1.40471i q^{79} -0.526990 q^{80} -1.75581 q^{82} -8.46436 q^{83} -2.41009 q^{85} +0.776183 q^{86} +5.12748i q^{88} +9.95446 q^{89} +2.15150i q^{91} +(-4.07953 - 2.52140i) q^{92} -4.99360 q^{94} +1.89551i q^{95} -0.822861i q^{97} -1.00000i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 24 q^{4} + 8 q^{5}+O(q^{10}) $ 24 * q - 24 * q^4 + 8 * q^5	$ 24 q - 24 q^{4} + 8 q^{5} - 16 q^{13} + 24 q^{14} + 24 q^{16} - 16 q^{17} - 8 q^{20} - 8 q^{23} + 48 q^{25} - 16 q^{31} + 8 q^{46} - 24 q^{49} + 16 q^{52} - 48 q^{53} + 16 q^{55} - 24 q^{56} - 8 q^{58} - 24 q^{64} + 48 q^{65} + 16 q^{68} + 8 q^{70} + 64 q^{73} + 32 q^{74} + 8 q^{80} + 16 q^{82} + 16 q^{83} + 32 q^{85} - 32 q^{86} - 48 q^{89} + 8 q^{92} - 16 q^{94}+O(q^{100}) $ 24 * q - 24 * q^4 + 8 * q^5 - 16 * q^13 + 24 * q^14 + 24 * q^16 - 16 * q^17 - 8 * q^20 - 8 * q^23 + 48 * q^25 - 16 * q^31 + 8 * q^46 - 24 * q^49 + 16 * q^52 - 48 * q^53 + 16 * q^55 - 24 * q^56 - 8 * q^58 - 24 * q^64 + 48 * q^65 + 16 * q^68 + 8 * q^70 + 64 * q^73 + 32 * q^74 + 8 * q^80 + 16 * q^82 + 16 * q^83 + 32 * q^85 - 32 * q^86 - 48 * q^89 + 8 * q^92 - 16 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2898\mathbb{Z}\right)^\times$.

$n$	$829$	$1289$	$1891$
$\chi(n)$	$1$	$-1$	$-1$

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 $

$2$			1.00000i			0.707107i
$2$			1.00000i			0.707107i
$3$		0			0
$3$		0			0
$4$	−1.00000			−0.500000
$5$	−0.526990			−0.235677			−0.117838	−	0.993033i	$-0.537597\pi$
$5$	−0.526990			−0.235677			−0.117838	+	0.993033i	$0.537597\pi$
$6$		0			0
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$		−	1.00000i		−	0.353553i
$9$		0			0
$10$		−	0.526990i		−	0.166649i
$11$	−5.12748			−1.54599			−0.772997	−	0.634410i	$-0.781243\pi$
$11$	−5.12748			−1.54599			−0.772997	+	0.634410i	$0.781243\pi$
$12$		0			0
$13$	−2.15150			−0.596719			−0.298360	−	0.954454i	$-0.596439\pi$
$13$	−2.15150			−0.596719			−0.298360	+	0.954454i	$0.596439\pi$
$14$	1.00000			0.267261
$15$		0			0
$16$	1.00000			0.250000
$17$	4.57331			1.10919			0.554595	−	0.832120i	$-0.312873\pi$
$17$	4.57331			1.10919			0.554595	+	0.832120i	$0.312873\pi$
$18$		0			0
$19$		−	3.59685i		−	0.825175i	−0.910918	−	0.412587i	$-0.864625\pi$
$19$		−	3.59685i		−	0.825175i	0.910918	−	0.412587i	$-0.135375\pi$
$20$	0.526990			0.117838
$21$		0			0
$22$		−	5.12748i		−	1.09318i
$23$	4.07953	+	2.52140i	0.850640	+	0.525748i
$23$	4.07953	+	2.52140i	0.850640	+	0.525748i
$24$		0			0
$25$	−4.72228			−0.944456
$26$		−	2.15150i		−	0.421944i
$27$		0			0
$28$			1.00000i			0.188982i
$29$			0.655814i			0.121782i	0.998144	+	0.0608908i	$0.0193942\pi$
$29$			0.655814i			0.121782i	−0.998144	+	0.0608908i	$0.980606\pi$
$30$		0			0
$31$	7.07786			1.27122			0.635611	−	0.772010i	$-0.280748\pi$
$31$	7.07786			1.27122			0.635611	+	0.772010i	$0.280748\pi$
$32$			1.00000i			0.176777i
$33$		0			0
$34$			4.57331i			0.784316i
$35$			0.526990i			0.0890775i
$36$		0			0
$37$			2.56927i			0.422386i	0.977444	+	0.211193i	$0.0677349\pi$
$37$			2.56927i			0.422386i	−0.977444	+	0.211193i	$0.932265\pi$
$38$	3.59685			0.583487
$39$		0			0
$40$			0.526990i			0.0833244i
$41$			1.75581i			0.274211i	0.990556	+	0.137106i	$0.0437800\pi$
$41$			1.75581i			0.274211i	−0.990556	+	0.137106i	$0.956220\pi$
$42$		0			0
$43$		−	0.776183i		−	0.118367i	−0.998247	−	0.0591834i	$-0.981150\pi$
$43$		−	0.776183i		−	0.118367i	0.998247	−	0.0591834i	$-0.0188497\pi$
$44$	5.12748			0.772997
$45$		0			0
$46$	−2.52140	+	4.07953i	−0.371760	+	0.601494i
$47$			4.99360i			0.728391i	0.931323	+	0.364195i	$0.118656\pi$
$47$			4.99360i			0.728391i	−0.931323	+	0.364195i	$0.881344\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		−	4.72228i		−	0.667832i
$51$		0			0
$52$	2.15150			0.298360
$53$	1.84124			0.252914			0.126457	−	0.991972i	$-0.459639\pi$
$53$	1.84124			0.252914			0.126457	+	0.991972i	$0.459639\pi$
$54$		0			0
$55$	2.70213			0.364355
$56$	−1.00000			−0.133631
$57$		0			0
$58$	−0.655814			−0.0861127
$59$			12.1238i			1.57839i	0.614145	+	0.789193i	$0.289501\pi$
$59$			12.1238i			1.57839i	−0.614145	+	0.789193i	$0.710499\pi$
$60$		0			0
$61$		−	4.36713i		−	0.559154i	−0.960123	−	0.279577i	$-0.909806\pi$
$61$		−	4.36713i		−	0.559154i	0.960123	−	0.279577i	$-0.0901942\pi$
$62$			7.07786i			0.898890i
$63$		0			0
$64$	−1.00000			−0.125000
$65$	1.13382			0.140633
$66$		0			0
$67$		−	0.255016i		−	0.0311551i	−0.999879	−	0.0155776i	$-0.995041\pi$
$67$		−	0.255016i		−	0.0311551i	0.999879	−	0.0155776i	$-0.00495869\pi$
$68$	−4.57331			−0.554595
$69$		0			0
$70$	−0.526990			−0.0629873
$71$			10.7154i			1.27169i	0.771817	+	0.635844i	$0.219348\pi$
$71$			10.7154i			1.27169i	−0.771817	+	0.635844i	$0.780652\pi$
$72$		0			0
$73$	12.3335			1.44353			0.721765	−	0.692138i	$-0.243331\pi$
$73$	12.3335			1.44353			0.721765	+	0.692138i	$0.243331\pi$
$74$	−2.56927			−0.298672
$75$		0			0
$76$			3.59685i			0.412587i
$77$			5.12748i			0.584330i
$78$		0			0
$79$			1.40471i			0.158042i	0.996873	+	0.0790208i	$0.0251794\pi$
$79$			1.40471i			0.158042i	−0.996873	+	0.0790208i	$0.974821\pi$
$80$	−0.526990			−0.0589192
$81$		0			0
$82$	−1.75581			−0.193897
$83$	−8.46436			−0.929085			−0.464542	−	0.885551i	$-0.653781\pi$
$83$	−8.46436			−0.929085			−0.464542	+	0.885551i	$0.653781\pi$
$84$		0			0
$85$	−2.41009			−0.261411
$86$	0.776183			0.0836980
$87$		0			0
$88$			5.12748i			0.546591i
$89$	9.95446			1.05517			0.527585	−	0.849502i	$-0.323098\pi$
$89$	9.95446			1.05517			0.527585	+	0.849502i	$0.323098\pi$
$90$		0			0
$91$			2.15150i			0.225539i
$92$	−4.07953	−	2.52140i	−0.425320	−	0.262874i
$93$		0			0
$94$	−4.99360			−0.515050
$95$			1.89551i			0.194475i
$96$		0			0
$97$		−	0.822861i		−	0.0835489i	−0.999127	−	0.0417744i	$-0.986699\pi$
$97$		−	0.822861i		−	0.0835489i	0.999127	−	0.0417744i	$-0.0133011\pi$
$98$		−	1.00000i		−	0.101015i
$99$		0			0
$100$	4.72228			0.472228
$101$			8.77864i			0.873508i	0.899581	+	0.436754i	$0.143872\pi$
$101$			8.77864i			0.873508i	−0.899581	+	0.436754i	$0.856128\pi$
$102$		0			0
$103$			2.22382i			0.219119i	0.993980	+	0.109560i	$0.0349441\pi$
$103$			2.22382i			0.219119i	−0.993980	+	0.109560i	$0.965056\pi$
$104$			2.15150i			0.210972i
$105$		0			0
$106$			1.84124i			0.178837i
$107$	16.6980			1.61425			0.807126	−	0.590379i	$-0.201022\pi$
$107$	16.6980			1.61425			0.807126	+	0.590379i	$0.201022\pi$
$108$		0			0
$109$			3.89564i			0.373135i	0.982442	+	0.186568i	$0.0597363\pi$
$109$			3.89564i			0.373135i	−0.982442	+	0.186568i	$0.940264\pi$
$110$			2.70213i			0.257638i
$111$		0			0
$112$		−	1.00000i		−	0.0944911i
$113$	11.5946			1.09073			0.545363	−	0.838200i	$-0.316392\pi$
$113$	11.5946			1.09073			0.545363	+	0.838200i	$0.316392\pi$
$114$		0			0
$115$	−2.14987	−	1.32875i	−0.200476	−	0.123907i
$116$		−	0.655814i		−	0.0608908i
$117$		0			0
$118$	−12.1238			−1.11609
$119$		−	4.57331i		−	0.419234i
$120$		0			0
$121$	15.2910			1.39009
$122$	4.36713			0.395381
$123$		0			0
$124$	−7.07786			−0.635611
$125$	5.12354			0.458264
$126$		0			0
$127$	−15.4104			−1.36745			−0.683727	−	0.729738i	$-0.739642\pi$
$127$	−15.4104			−1.36745			−0.683727	+	0.729738i	$0.739642\pi$
$128$		−	1.00000i		−	0.0883883i
$129$		0			0
$130$			1.13382i			0.0994426i
$131$			12.7723i			1.11592i	0.829867	+	0.557961i	$0.188416\pi$
$131$			12.7723i			1.11592i	−0.829867	+	0.557961i	$0.811584\pi$
$132$		0			0
$133$	−3.59685			−0.311887
$134$	0.255016			0.0220300
$135$		0			0
$136$		−	4.57331i		−	0.392158i
$137$	15.8875			1.35736			0.678678	−	0.734436i	$-0.262553\pi$
$137$	15.8875			1.35736			0.678678	+	0.734436i	$0.262553\pi$
$138$		0			0
$139$	9.49560			0.805406			0.402703	−	0.915331i	$-0.368071\pi$
$139$	9.49560			0.805406			0.402703	+	0.915331i	$0.368071\pi$
$140$		−	0.526990i		−	0.0445388i
$141$		0			0
$142$	−10.7154			−0.899220
$143$	11.0318			0.922524
$144$		0			0
$145$		−	0.345607i		−	0.0287011i
$146$			12.3335i			1.02073i
$147$		0			0
$148$		−	2.56927i		−	0.211193i
$149$	1.39243			0.114072			0.0570361	−	0.998372i	$-0.481835\pi$
$149$	1.39243			0.114072			0.0570361	+	0.998372i	$0.481835\pi$
$150$		0			0
$151$	0.954983			0.0777154			0.0388577	−	0.999245i	$-0.487628\pi$
$151$	0.954983			0.0777154			0.0388577	+	0.999245i	$0.487628\pi$
$152$	−3.59685			−0.291743
$153$		0			0
$154$	−5.12748			−0.413184
$155$	−3.72996			−0.299598
$156$		0			0
$157$		−	1.75548i		−	0.140103i	−0.997543	−	0.0700514i	$-0.977684\pi$
$157$		−	1.75548i		−	0.140103i	0.997543	−	0.0700514i	$-0.0223163\pi$
$158$	−1.40471			−0.111752
$159$		0			0
$160$		−	0.526990i		−	0.0416622i
$161$	2.52140	−	4.07953i	0.198714	−	0.321512i
$162$		0			0
$163$	−15.7795			−1.23595			−0.617973	−	0.786199i	$-0.712046\pi$
$163$	−15.7795			−1.23595			−0.617973	+	0.786199i	$0.712046\pi$
$164$		−	1.75581i		−	0.137106i
$165$		0			0
$166$		−	8.46436i		−	0.656962i
$167$		−	11.4094i		−	0.882885i	−0.897290	−	0.441442i	$-0.854467\pi$
$167$		−	11.4094i		−	0.882885i	0.897290	−	0.441442i	$-0.145533\pi$
$168$		0			0
$169$	−8.37104			−0.643926
$170$		−	2.41009i		−	0.184845i
$171$		0			0
$172$			0.776183i			0.0591834i
$173$			6.71819i			0.510774i	0.966839	+	0.255387i	$0.0822029\pi$
$173$			6.71819i			0.510774i	−0.966839	+	0.255387i	$0.917797\pi$
$174$		0			0
$175$			4.72228i			0.356971i
$176$	−5.12748			−0.386498
$177$		0			0
$178$			9.95446i			0.746118i
$179$			0.160686i			0.0120103i	0.999982	+	0.00600513i	$0.00191151\pi$
$179$			0.160686i			0.0120103i	−0.999982	+	0.00600513i	$0.998088\pi$
$180$		0			0
$181$			3.86580i			0.287343i	0.989625	+	0.143671i	$0.0458909\pi$
$181$			3.86580i			0.287343i	−0.989625	+	0.143671i	$0.954109\pi$
$182$	−2.15150			−0.159480
$183$		0			0
$184$	2.52140	−	4.07953i	0.185880	−	0.300747i
$185$		−	1.35398i		−	0.0995467i
$186$		0			0
$187$	−23.4495			−1.71480
$188$		−	4.99360i		−	0.364195i
$189$		0			0
$190$	−1.89551			−0.137514
$191$	−15.6539			−1.13268			−0.566340	−	0.824172i	$-0.691641\pi$
$191$	−15.6539			−1.13268			−0.566340	+	0.824172i	$0.691641\pi$
$192$		0			0
$193$	18.7029			1.34626			0.673131	−	0.739523i	$-0.264949\pi$
$193$	18.7029			1.34626			0.673131	+	0.739523i	$0.264949\pi$
$194$	0.822861			0.0590780
$195$		0			0
$196$	1.00000			0.0714286
$197$			24.4524i			1.74216i	0.491138	+	0.871082i	$0.336581\pi$
$197$			24.4524i			1.74216i	−0.491138	+	0.871082i	$0.663419\pi$
$198$		0			0
$199$			1.21079i			0.0858308i	0.999079	+	0.0429154i	$0.0136646\pi$
$199$			1.21079i			0.0858308i	−0.999079	+	0.0429154i	$0.986335\pi$
$200$			4.72228i			0.333916i
$201$		0			0
$202$	−8.77864			−0.617663
$203$	0.655814			0.0460291
$204$		0			0
$205$		−	0.925293i		−	0.0646252i
$206$	−2.22382			−0.154941
$207$		0			0
$208$	−2.15150			−0.149180
$209$			18.4428i			1.27571i
$210$		0			0
$211$	11.2833			0.776773			0.388386	−	0.921497i	$-0.373033\pi$
$211$	11.2833			0.776773			0.388386	+	0.921497i	$0.373033\pi$
$212$	−1.84124			−0.126457
$213$		0			0
$214$			16.6980i			1.14145i
$215$			0.409040i			0.0278963i
$216$		0			0
$217$		−	7.07786i		−	0.480477i
$218$	−3.89564			−0.263846
$219$		0			0
$220$	−2.70213			−0.182177
$221$	−9.83948			−0.661875
$222$		0			0
$223$	−4.34331			−0.290850			−0.145425	−	0.989369i	$-0.546455\pi$
$223$	−4.34331			−0.290850			−0.145425	+	0.989369i	$0.546455\pi$
$224$	1.00000			0.0668153
$225$		0			0
$226$			11.5946i			0.771260i
$227$	1.61928			0.107475			0.0537376	−	0.998555i	$-0.482887\pi$
$227$	1.61928			0.107475			0.0537376	+	0.998555i	$0.482887\pi$
$228$		0			0
$229$		−	9.58977i		−	0.633710i	−0.948474	−	0.316855i	$-0.897373\pi$
$229$		−	9.58977i		−	0.633710i	0.948474	−	0.316855i	$-0.102627\pi$
$230$	1.32875	−	2.14987i	0.0876153	−	0.141758i
$231$		0			0
$232$	0.655814			0.0430563
$233$			5.15119i			0.337466i	0.985662	+	0.168733i	$0.0539675\pi$
$233$			5.15119i			0.337466i	−0.985662	+	0.168733i	$0.946033\pi$
$234$		0			0
$235$		−	2.63157i		−	0.171665i
$236$		−	12.1238i		−	0.789193i
$237$		0			0
$238$	4.57331			0.296444
$239$			9.24437i			0.597969i	0.954258	+	0.298984i	$0.0966478\pi$
$239$			9.24437i			0.597969i	−0.954258	+	0.298984i	$0.903352\pi$
$240$		0			0
$241$			22.7431i			1.46501i	0.680761	+	0.732505i	$0.261649\pi$
$241$			22.7431i			1.46501i	−0.680761	+	0.732505i	$0.738351\pi$
$242$			15.2910i			0.982945i
$243$		0			0
$244$			4.36713i			0.279577i
$245$	0.526990			0.0336681
$246$		0			0
$247$			7.73864i			0.492398i
$248$		−	7.07786i		−	0.449445i
$249$		0			0
$250$			5.12354i			0.324041i
$251$	−23.8981			−1.50843			−0.754217	−	0.656625i	$-0.771983\pi$
$251$	−23.8981			−1.50843			−0.754217	+	0.656625i	$0.771983\pi$
$252$		0			0
$253$	−20.9177	−	12.9284i	−1.31508	−	0.812803i
$254$		−	15.4104i		−	0.966936i
$255$		0			0
$256$	1.00000			0.0625000
$257$			3.18628i			0.198755i	0.995050	+	0.0993774i	$0.0316851\pi$
$257$			3.18628i			0.198755i	−0.995050	+	0.0993774i	$0.968315\pi$
$258$		0			0
$259$	2.56927			0.159647
$260$	−1.13382			−0.0703165
$261$		0			0
$262$	−12.7723			−0.789077
$263$	14.5545			0.897469			0.448735	−	0.893665i	$-0.351875\pi$
$263$	14.5545			0.897469			0.448735	+	0.893665i	$0.351875\pi$
$264$		0			0
$265$	−0.970316			−0.0596060
$266$		−	3.59685i		−	0.220537i
$267$		0			0
$268$			0.255016i			0.0155776i
$269$			9.06507i			0.552707i	0.961056	+	0.276354i	$0.0891261\pi$
$269$			9.06507i			0.552707i	−0.961056	+	0.276354i	$0.910874\pi$
$270$		0			0
$271$	0.633897			0.0385065			0.0192532	−	0.999815i	$-0.493871\pi$
$271$	0.633897			0.0385065			0.0192532	+	0.999815i	$0.493871\pi$
$272$	4.57331			0.277298
$273$		0			0
$274$			15.8875i			0.959796i
$275$	24.2134			1.46012
$276$		0			0
$277$	7.94055			0.477101			0.238551	−	0.971130i	$-0.423328\pi$
$277$	7.94055			0.477101			0.238551	+	0.971130i	$0.423328\pi$
$278$			9.49560i			0.569508i
$279$		0			0
$280$	0.526990			0.0314937
$281$	5.25468			0.313468			0.156734	−	0.987641i	$-0.449903\pi$
$281$	5.25468			0.313468			0.156734	+	0.987641i	$0.449903\pi$
$282$		0			0
$283$			23.9112i			1.42138i	0.703507	+	0.710688i	$0.251616\pi$
$283$			23.9112i			1.42138i	−0.703507	+	0.710688i	$0.748384\pi$
$284$		−	10.7154i		−	0.635844i
$285$		0			0
$286$			11.0318i			0.652323i
$287$	1.75581			0.103642
$288$		0			0
$289$	3.91515			0.230303
$290$	0.345607			0.0202948
$291$		0			0
$292$	−12.3335			−0.721765
$293$	−14.3300			−0.837169			−0.418584	−	0.908178i	$-0.637474\pi$
$293$	−14.3300			−0.837169			−0.418584	+	0.908178i	$0.637474\pi$
$294$		0			0
$295$		−	6.38912i		−	0.371989i
$296$	2.56927			0.149336
$297$		0			0
$298$			1.39243i			0.0806613i
$299$	−8.77711	−	5.42480i	−0.507594	−	0.313724i
$300$		0			0
$301$	−0.776183			−0.0447384
$302$			0.954983i			0.0549531i
$303$		0			0
$304$		−	3.59685i		−	0.206294i
$305$			2.30143i			0.131780i
$306$		0			0
$307$	1.56254			0.0891787			0.0445893	−	0.999005i	$-0.485802\pi$
$307$	1.56254			0.0891787			0.0445893	+	0.999005i	$0.485802\pi$
$308$		−	5.12748i		−	0.292165i
$309$		0			0
$310$		−	3.72996i		−	0.211848i
$311$		−	20.1013i		−	1.13984i	−0.821699	−	0.569921i	$-0.806974\pi$
$311$		−	20.1013i		−	1.13984i	0.821699	−	0.569921i	$-0.193026\pi$
$312$		0			0
$313$		−	21.4396i		−	1.21184i	−0.795526	−	0.605919i	$-0.792806\pi$
$313$		−	21.4396i		−	1.21184i	0.795526	−	0.605919i	$-0.207194\pi$
$314$	1.75548			0.0990677
$315$		0			0
$316$		−	1.40471i		−	0.0790208i
$317$		−	9.77680i		−	0.549120i	−0.961570	−	0.274560i	$-0.911468\pi$
$317$		−	9.77680i		−	0.549120i	0.961570	−	0.274560i	$-0.0885322\pi$
$318$		0			0
$319$		−	3.36267i		−	0.188274i
$320$	0.526990			0.0294596
$321$		0			0
$322$	4.07953	+	2.52140i	0.227343	+	0.140512i
$323$		−	16.4495i		−	0.915276i
$324$		0			0
$325$	10.1600			0.563575
$326$		−	15.7795i		−	0.873946i
$327$		0			0
$328$	1.75581			0.0969483
$329$	4.99360			0.275306
$330$		0			0
$331$	−8.82923			−0.485298			−0.242649	−	0.970114i	$-0.578016\pi$
$331$	−8.82923			−0.485298			−0.242649	+	0.970114i	$0.578016\pi$
$332$	8.46436			0.464542
$333$		0			0
$334$	11.4094			0.624294
$335$			0.134391i			0.00734254i
$336$		0			0
$337$		−	18.6898i		−	1.01810i	−0.860737	−	0.509050i	$-0.829997\pi$
$337$		−	18.6898i		−	1.01810i	0.860737	−	0.509050i	$-0.170003\pi$
$338$		−	8.37104i		−	0.455324i
$339$		0			0
$340$	2.41009			0.130705
$341$	−36.2916			−1.96530
$342$		0			0
$343$			1.00000i			0.0539949i
$344$	−0.776183			−0.0418490
$345$		0			0
$346$	−6.71819			−0.361172
$347$			5.97833i			0.320933i	0.987041	+	0.160467i	$0.0512999\pi$
$347$			5.97833i			0.320933i	−0.987041	+	0.160467i	$0.948700\pi$
$348$		0			0
$349$	14.7234			0.788128			0.394064	−	0.919083i	$-0.371069\pi$
$349$	14.7234			0.788128			0.394064	+	0.919083i	$0.371069\pi$
$350$	−4.72228			−0.252417
$351$		0			0
$352$		−	5.12748i		−	0.273296i
$353$		−	31.6859i		−	1.68647i	−0.537544	−	0.843236i	$-0.680648\pi$
$353$		−	31.6859i		−	1.68647i	0.537544	−	0.843236i	$-0.319352\pi$
$354$		0			0
$355$		−	5.64693i		−	0.299708i
$356$	−9.95446			−0.527585
$357$		0			0
$358$	−0.160686			−0.00849254
$359$	9.11846			0.481254			0.240627	−	0.970618i	$-0.422647\pi$
$359$	9.11846			0.481254			0.240627	+	0.970618i	$0.422647\pi$
$360$		0			0
$361$	6.06264			0.319086
$362$	−3.86580			−0.203182
$363$		0			0
$364$		−	2.15150i		−	0.112769i
$365$	−6.49964			−0.340207
$366$		0			0
$367$			26.8563i			1.40189i	0.713215	+	0.700945i	$0.247238\pi$
$367$			26.8563i			1.40189i	−0.713215	+	0.700945i	$0.752762\pi$
$368$	4.07953	+	2.52140i	0.212660	+	0.131437i
$369$		0			0
$370$	1.35398			0.0703901
$371$		−	1.84124i		−	0.0955926i
$372$		0			0
$373$		−	26.5566i		−	1.37505i	−0.726161	−	0.687525i	$-0.758697\pi$
$373$		−	26.5566i		−	1.37505i	0.726161	−	0.687525i	$-0.241303\pi$
$374$		−	23.4495i		−	1.21255i
$375$		0			0
$376$	4.99360			0.257525
$377$		−	1.41099i		−	0.0726695i
$378$		0			0
$379$		−	26.1367i		−	1.34255i	−0.741207	−	0.671276i	$-0.765746\pi$
$379$		−	26.1367i		−	1.34255i	0.741207	−	0.671276i	$-0.234254\pi$
$380$		−	1.89551i		−	0.0972374i
$381$		0			0
$382$		−	15.6539i		−	0.800925i
$383$	11.4557			0.585358			0.292679	−	0.956211i	$-0.405453\pi$
$383$	11.4557			0.585358			0.292679	+	0.956211i	$0.405453\pi$
$384$		0			0
$385$		−	2.70213i		−	0.137713i
$386$			18.7029i			0.951951i
$387$		0			0
$388$			0.822861i			0.0417744i
$389$	2.49239			0.126369			0.0631846	−	0.998002i	$-0.479874\pi$
$389$	2.49239			0.126369			0.0631846	+	0.998002i	$0.479874\pi$
$390$		0			0
$391$	18.6569	+	11.5311i	0.943522	+	0.583155i
$392$			1.00000i			0.0505076i
$393$		0			0
$394$	−24.4524			−1.23190
$395$		−	0.740265i		−	0.0372468i
$396$		0			0
$397$	16.2536			0.815743			0.407871	−	0.913039i	$-0.366271\pi$
$397$	16.2536			0.815743			0.407871	+	0.913039i	$0.366271\pi$
$398$	−1.21079			−0.0606915
$399$		0			0
$400$	−4.72228			−0.236114
$401$	−1.63283			−0.0815396			−0.0407698	−	0.999169i	$-0.512981\pi$
$401$	−1.63283			−0.0815396			−0.0407698	+	0.999169i	$0.512981\pi$
$402$		0			0
$403$	−15.2280			−0.758563
$404$		−	8.77864i		−	0.436754i
$405$		0			0
$406$			0.655814i			0.0325475i
$407$		−	13.1739i		−	0.653006i
$408$		0			0
$409$	−5.97232			−0.295312			−0.147656	−	0.989039i	$-0.547173\pi$
$409$	−5.97232			−0.295312			−0.147656	+	0.989039i	$0.547173\pi$
$410$	0.925293			0.0456969
$411$		0			0
$412$		−	2.22382i		−	0.109560i
$413$	12.1238			0.596574
$414$		0			0
$415$	4.46063			0.218964
$416$		−	2.15150i		−	0.105486i
$417$		0			0
$418$	−18.4428			−0.902067
$419$	1.33706			0.0653199			0.0326599	−	0.999467i	$-0.489602\pi$
$419$	1.33706			0.0653199			0.0326599	+	0.999467i	$0.489602\pi$
$420$		0			0
$421$		−	22.7739i		−	1.10993i	−0.831874	−	0.554965i	$-0.812732\pi$
$421$		−	22.7739i		−	1.10993i	0.831874	−	0.554965i	$-0.187268\pi$
$422$			11.2833i			0.549261i
$423$		0			0
$424$		−	1.84124i		−	0.0894187i
$425$	−21.5965			−1.04758
$426$		0			0
$427$	−4.36713			−0.211340
$428$	−16.6980			−0.807126
$429$		0			0
$430$	−0.409040			−0.0197257
$431$	17.0470			0.821127			0.410564	−	0.911832i	$-0.365332\pi$
$431$	17.0470			0.821127			0.410564	+	0.911832i	$0.365332\pi$
$432$		0			0
$433$			10.6128i			0.510017i	0.966939	+	0.255008i	$0.0820782\pi$
$433$			10.6128i			0.510017i	−0.966939	+	0.255008i	$0.917922\pi$
$434$	7.07786			0.339748
$435$		0			0
$436$		−	3.89564i		−	0.186568i
$437$	9.06911	−	14.6735i	0.433834	−	0.701927i
$438$		0			0
$439$	−17.9680			−0.857563			−0.428782	−	0.903408i	$-0.641057\pi$
$439$	−17.9680			−0.857563			−0.428782	+	0.903408i	$0.641057\pi$
$440$		−	2.70213i		−	0.128819i
$441$		0			0
$442$		−	9.83948i		−	0.468017i
$443$		−	29.9812i		−	1.42445i	−0.701952	−	0.712225i	$-0.747688\pi$
$443$		−	29.9812i		−	1.42445i	0.701952	−	0.712225i	$-0.252312\pi$
$444$		0			0
$445$	−5.24590			−0.248679
$446$		−	4.34331i		−	0.205662i
$447$		0			0
$448$			1.00000i			0.0472456i
$449$			5.47792i			0.258519i	0.991611	+	0.129260i	$0.0412601\pi$
$449$			5.47792i			0.258519i	−0.991611	+	0.129260i	$0.958740\pi$
$450$		0			0
$451$		−	9.00287i		−	0.423928i
$452$	−11.5946			−0.545363
$453$		0			0
$454$			1.61928i			0.0759964i
$455$		−	1.13382i		−	0.0531543i
$456$		0			0
$457$			34.3133i			1.60511i	0.596580	+	0.802553i	$0.296526\pi$
$457$			34.3133i			1.60511i	−0.596580	+	0.802553i	$0.703474\pi$
$458$	9.58977			0.448101
$459$		0			0
$460$	2.14987	+	1.32875i	0.100238	+	0.0619534i
$461$			30.6915i			1.42945i	0.699407	+	0.714723i	$0.253447\pi$
$461$			30.6915i			1.42945i	−0.699407	+	0.714723i	$0.746553\pi$
$462$		0			0
$463$	41.8755			1.94612			0.973059	−	0.230554i	$-0.0740539\pi$
$463$	41.8755			1.94612			0.973059	+	0.230554i	$0.0740539\pi$
$464$			0.655814i			0.0304454i
$465$		0			0
$466$	−5.15119			−0.238624
$467$	−29.4863			−1.36446			−0.682231	−	0.731137i	$-0.738990\pi$
$467$	−29.4863			−1.36446			−0.682231	+	0.731137i	$0.738990\pi$
$468$		0			0
$469$	−0.255016			−0.0117755
$470$	2.63157			0.121385
$471$		0			0
$472$	12.1238			0.558044
$473$			3.97986i			0.182994i
$474$		0			0
$475$			16.9854i			0.779342i
$476$			4.57331i			0.209617i
$477$		0			0
$478$	−9.24437			−0.422828
$479$	19.1778			0.876256			0.438128	−	0.898913i	$-0.355642\pi$
$479$	19.1778			0.876256			0.438128	+	0.898913i	$0.355642\pi$
$480$		0			0
$481$		−	5.52780i		−	0.252046i
$482$	−22.7431			−1.03592
$483$		0			0
$484$	−15.2910			−0.695047
$485$			0.433639i			0.0196905i
$486$		0			0
$487$	32.5320			1.47417			0.737083	−	0.675802i	$-0.236203\pi$
$487$	32.5320			1.47417			0.737083	+	0.675802i	$0.236203\pi$
$488$	−4.36713			−0.197691
$489$		0			0
$490$			0.526990i			0.0238070i
$491$		−	18.1334i		−	0.818348i	−0.912456	−	0.409174i	$-0.865817\pi$
$491$		−	18.1334i		−	0.818348i	0.912456	−	0.409174i	$-0.134183\pi$
$492$		0			0
$493$			2.99924i			0.135079i
$494$	−7.73864			−0.348178
$495$		0			0
$496$	7.07786			0.317805
$497$	10.7154			0.480653
$498$		0			0
$499$	−11.7545			−0.526206			−0.263103	−	0.964768i	$-0.584746\pi$
$499$	−11.7545			−0.526206			−0.263103	+	0.964768i	$0.584746\pi$
$500$	−5.12354			−0.229132
$501$		0			0
$502$		−	23.8981i		−	1.06662i
$503$	−32.2657			−1.43866			−0.719328	−	0.694670i	$-0.755550\pi$
$503$	−32.2657			−1.43866			−0.719328	+	0.694670i	$0.755550\pi$
$504$		0			0
$505$		−	4.62626i		−	0.205866i
$506$	12.9284	−	20.9177i	0.574738	−	0.929905i
$507$		0			0
$508$	15.4104			0.683727
$509$			24.6680i			1.09339i	0.837332	+	0.546695i	$0.184114\pi$
$509$			24.6680i			1.09339i	−0.837332	+	0.546695i	$0.815886\pi$
$510$		0			0
$511$		−	12.3335i		−	0.545603i
$512$			1.00000i			0.0441942i
$513$		0			0
$514$	−3.18628			−0.140541
$515$		−	1.17193i		−	0.0516414i
$516$		0			0
$517$		−	25.6046i		−	1.12609i
$518$			2.56927i			0.112887i
$519$		0			0
$520$		−	1.13382i		−	0.0497213i
$521$	−14.2561			−0.624571			−0.312285	−	0.949988i	$-0.601094\pi$
$521$	−14.2561			−0.624571			−0.312285	+	0.949988i	$0.601094\pi$
$522$		0			0
$523$			24.5683i			1.07430i	0.843488	+	0.537148i	$0.180498\pi$
$523$			24.5683i			1.07430i	−0.843488	+	0.537148i	$0.819502\pi$
$524$		−	12.7723i		−	0.557961i
$525$		0			0
$526$			14.5545i			0.634607i
$527$	32.3693			1.41003
$528$		0			0
$529$	10.2851	+	20.5722i	0.447178	+	0.894445i
$530$		−	0.970316i		−	0.0421478i
$531$		0			0
$532$	3.59685			0.155943
$533$		−	3.77762i		−	0.163627i
$534$		0			0
$535$	−8.79965			−0.380442
$536$	−0.255016			−0.0110150
$537$		0			0
$538$	−9.06507			−0.390823
$539$	5.12748			0.220856
$540$		0			0
$541$	40.7886			1.75364			0.876819	−	0.480820i	$-0.159661\pi$
$541$	40.7886			1.75364			0.876819	+	0.480820i	$0.159661\pi$
$542$			0.633897i			0.0272282i
$543$		0			0
$544$			4.57331i			0.196079i
$545$		−	2.05296i		−	0.0879393i
$546$		0			0
$547$	9.29041			0.397229			0.198615	−	0.980078i	$-0.436356\pi$
$547$	9.29041			0.397229			0.198615	+	0.980078i	$0.436356\pi$
$548$	−15.8875			−0.678678
$549$		0			0
$550$			24.2134i			1.03246i
$551$	2.35887			0.100491
$552$		0			0
$553$	1.40471			0.0597341
$554$			7.94055i			0.337361i
$555$		0			0
$556$	−9.49560			−0.402703
$557$	9.90092			0.419515			0.209758	−	0.977753i	$-0.432732\pi$
$557$	9.90092			0.419515			0.209758	+	0.977753i	$0.432732\pi$
$558$		0			0
$559$			1.66996i			0.0706318i
$560$			0.526990i			0.0222694i
$561$		0			0
$562$			5.25468i			0.221655i
$563$	−38.8541			−1.63751			−0.818753	−	0.574145i	$-0.805334\pi$
$563$	−38.8541			−1.63751			−0.818753	+	0.574145i	$0.805334\pi$
$564$		0			0
$565$	−6.11023			−0.257059
$566$	−23.9112			−1.00506
$567$		0			0
$568$	10.7154			0.449610
$569$	−1.25796			−0.0527363			−0.0263681	−	0.999652i	$-0.508394\pi$
$569$	−1.25796			−0.0527363			−0.0263681	+	0.999652i	$0.508394\pi$
$570$		0			0
$571$			31.5460i			1.32016i	0.751195	+	0.660080i	$0.229478\pi$
$571$			31.5460i			1.32016i	−0.751195	+	0.660080i	$0.770522\pi$
$572$	−11.0318			−0.461262
$573$		0			0
$574$			1.75581i			0.0732860i
$575$	−19.2647	−	11.9068i	−0.803393	−	0.496546i
$576$		0			0
$577$	30.5038			1.26989			0.634946	−	0.772557i	$-0.281023\pi$
$577$	30.5038			1.26989			0.634946	+	0.772557i	$0.281023\pi$
$578$			3.91515i			0.162849i
$579$		0			0
$580$			0.345607i			0.0143506i
$581$			8.46436i			0.351161i
$582$		0			0
$583$	−9.44093			−0.391004
$584$		−	12.3335i		−	0.510365i
$585$		0			0
$586$		−	14.3300i		−	0.591968i
$587$		−	18.4415i		−	0.761163i	−0.924747	−	0.380582i	$-0.875724\pi$
$587$		−	18.4415i		−	0.761163i	0.924747	−	0.380582i	$-0.124276\pi$
$588$		0			0
$589$		−	25.4580i		−	1.04898i
$590$	6.38912			0.263036
$591$		0			0
$592$			2.56927i			0.105597i
$593$		−	2.58090i		−	0.105985i	−0.998595	−	0.0529924i	$-0.983124\pi$
$593$		−	2.58090i		−	0.105985i	0.998595	−	0.0529924i	$-0.0168759\pi$
$594$		0			0
$595$			2.41009i			0.0988039i
$596$	−1.39243			−0.0570361
$597$		0			0
$598$	5.42480	−	8.77711i	0.221836	−	0.358923i
$599$		−	39.1201i		−	1.59840i	−0.601063	−	0.799201i	$-0.705256\pi$
$599$		−	39.1201i		−	1.59840i	0.601063	−	0.799201i	$-0.294744\pi$
$600$		0			0
$601$	8.70675			0.355156			0.177578	−	0.984107i	$-0.443174\pi$
$601$	8.70675			0.355156			0.177578	+	0.984107i	$0.443174\pi$
$602$		−	0.776183i		−	0.0316349i
$603$		0			0
$604$	−0.954983			−0.0388577
$605$	−8.05822			−0.327613
$606$		0			0
$607$	6.90097			0.280102			0.140051	−	0.990144i	$-0.455273\pi$
$607$	6.90097			0.280102			0.140051	+	0.990144i	$0.455273\pi$
$608$	3.59685			0.145872
$609$		0			0
$610$	−2.30143			−0.0931823
$611$		−	10.7437i		−	0.434645i
$612$		0			0
$613$			21.9071i			0.884820i	0.896813	+	0.442410i	$0.145876\pi$
$613$			21.9071i			0.884820i	−0.896813	+	0.442410i	$0.854124\pi$
$614$			1.56254i			0.0630588i
$615$		0			0
$616$	5.12748			0.206592
$617$	−21.8084			−0.877975			−0.438987	−	0.898493i	$-0.644663\pi$
$617$	−21.8084			−0.877975			−0.438987	+	0.898493i	$0.644663\pi$
$618$		0			0
$619$			30.8810i			1.24121i	0.784123	+	0.620606i	$0.213113\pi$
$619$			30.8810i			1.24121i	−0.784123	+	0.620606i	$0.786887\pi$
$620$	3.72996			0.149799
$621$		0			0
$622$	20.1013			0.805990
$623$		−	9.95446i		−	0.398817i
$624$		0			0
$625$	20.9114			0.836454
$626$	21.4396			0.856899
$627$		0			0
$628$			1.75548i			0.0700514i
$629$			11.7501i			0.468506i
$630$		0			0
$631$		−	40.4572i		−	1.61058i	−0.592884	−	0.805288i	$-0.702011\pi$
$631$		−	40.4572i		−	1.61058i	0.592884	−	0.805288i	$-0.297989\pi$
$632$	1.40471			0.0558762
$633$		0			0
$634$	9.77680			0.388286
$635$	8.12113			0.322277
$636$		0			0
$637$	2.15150			0.0852456
$638$	3.36267			0.133130
$639$		0			0
$640$			0.526990i			0.0208311i
$641$	9.71732			0.383811			0.191906	−	0.981413i	$-0.438533\pi$
$641$	9.71732			0.383811			0.191906	+	0.981413i	$0.438533\pi$
$642$		0			0
$643$		−	27.6750i		−	1.09140i	−0.837982	−	0.545698i	$-0.816264\pi$
$643$		−	27.6750i		−	1.09140i	0.837982	−	0.545698i	$-0.183736\pi$
$644$	−2.52140	+	4.07953i	−0.0993571	+	0.160756i
$645$		0			0
$646$	16.4495			0.647198
$647$			34.2325i			1.34582i	0.739725	+	0.672909i	$0.234955\pi$
$647$			34.2325i			1.34582i	−0.739725	+	0.672909i	$0.765045\pi$
$648$		0			0
$649$		−	62.1646i		−	2.44017i
$650$			10.1600i			0.398508i
$651$		0			0
$652$	15.7795			0.617973
$653$		−	7.14382i		−	0.279559i	−0.990183	−	0.139780i	$-0.955361\pi$
$653$		−	7.14382i		−	0.279559i	0.990183	−	0.139780i	$-0.0446394\pi$
$654$		0			0
$655$		−	6.73088i		−	0.262997i
$656$			1.75581i			0.0685528i
$657$		0			0
$658$			4.99360i			0.194671i
$659$	−31.2157			−1.21599			−0.607996	−	0.793940i	$-0.708026\pi$
$659$	−31.2157			−1.21599			−0.607996	+	0.793940i	$0.708026\pi$
$660$		0			0
$661$			5.37980i			0.209250i	0.994512	+	0.104625i	$0.0333642\pi$
$661$			5.37980i			0.209250i	−0.994512	+	0.104625i	$0.966636\pi$
$662$		−	8.82923i		−	0.343158i
$663$		0			0
$664$			8.46436i			0.328481i
$665$	1.89551			0.0735045
$666$		0			0
$667$	−1.65357	+	2.67541i	−0.0640265	+	0.103592i
$668$			11.4094i			0.441442i
$669$		0			0
$670$	−0.134391			−0.00519196
$671$			22.3924i			0.864448i
$672$		0			0
$673$	−15.9289			−0.614014			−0.307007	−	0.951707i	$-0.599328\pi$
$673$	−15.9289			−0.614014			−0.307007	+	0.951707i	$0.599328\pi$
$674$	18.6898			0.719906
$675$		0			0
$676$	8.37104			0.321963
$677$	−19.6210			−0.754097			−0.377049	−	0.926193i	$-0.623061\pi$
$677$	−19.6210			−0.754097			−0.377049	+	0.926193i	$0.623061\pi$
$678$		0			0
$679$	−0.822861			−0.0315785
$680$			2.41009i			0.0924226i
$681$		0			0
$682$		−	36.2916i		−	1.38968i
$683$		−	31.6284i		−	1.21023i	−0.796140	−	0.605113i	$-0.793128\pi$
$683$		−	31.6284i		−	1.21023i	0.796140	−	0.605113i	$-0.206872\pi$
$684$		0			0
$685$	−8.37252			−0.319898
$686$	−1.00000			−0.0381802
$687$		0			0
$688$		−	0.776183i		−	0.0295917i
$689$	−3.96144			−0.150919
$690$		0			0
$691$	−16.1749			−0.615321			−0.307660	−	0.951496i	$-0.599546\pi$
$691$	−16.1749			−0.615321			−0.307660	+	0.951496i	$0.599546\pi$
$692$		−	6.71819i		−	0.255387i
$693$		0			0
$694$	−5.97833			−0.226934
$695$	−5.00408			−0.189816
$696$		0			0
$697$			8.02985i			0.304152i
$698$			14.7234i			0.557290i
$699$		0			0
$700$		−	4.72228i		−	0.178485i
$701$	19.2653			0.727641			0.363821	−	0.931469i	$-0.381472\pi$
$701$	19.2653			0.727641			0.363821	+	0.931469i	$0.381472\pi$
$702$		0			0
$703$	9.24130			0.348542
$704$	5.12748			0.193249
$705$		0			0
$706$	31.6859			1.19252
$707$	8.77864			0.330155
$708$		0			0
$709$		−	6.27004i		−	0.235476i	−0.993045	−	0.117738i	$-0.962436\pi$
$709$		−	6.27004i		−	0.235476i	0.993045	−	0.117738i	$-0.0375643\pi$
$710$	5.64693			0.211925
$711$		0			0
$712$		−	9.95446i		−	0.373059i
$713$	28.8743	+	17.8461i	1.08135	+	0.668342i
$714$		0			0
$715$	−5.81364			−0.217418
$716$		−	0.160686i		−	0.00600513i
$717$		0			0
$718$			9.11846i			0.340298i
$719$			36.7355i			1.37000i	0.728542	+	0.685002i	$0.240199\pi$
$719$			36.7355i			1.37000i	−0.728542	+	0.685002i	$0.759801\pi$
$720$		0			0
$721$	2.22382			0.0828193
$722$			6.06264i			0.225628i
$723$		0			0
$724$		−	3.86580i		−	0.143671i
$725$		−	3.09694i		−	0.115017i
$726$		0			0
$727$		−	10.0026i		−	0.370978i	−0.982646	−	0.185489i	$-0.940613\pi$
$727$		−	10.0026i		−	0.370978i	0.982646	−	0.185489i	$-0.0593869\pi$
$728$	2.15150			0.0797400
$729$		0			0
$730$		−	6.49964i		−	0.240563i
$731$		−	3.54972i		−	0.131291i
$732$		0			0
$733$		−	31.2296i		−	1.15349i	−0.816923	−	0.576746i	$-0.804322\pi$
$733$		−	31.2296i		−	1.15349i	0.816923	−	0.576746i	$-0.195678\pi$
$734$	−26.8563			−0.991286
$735$		0			0
$736$	−2.52140	+	4.07953i	−0.0929400	+	0.150373i
$737$			1.30759i			0.0481656i
$738$		0			0
$739$	−29.4000			−1.08150			−0.540749	−	0.841184i	$-0.681859\pi$
$739$	−29.4000			−1.08150			−0.540749	+	0.841184i	$0.681859\pi$
$740$			1.35398i			0.0497733i
$741$		0			0
$742$	1.84124			0.0675942
$743$	−20.0566			−0.735804			−0.367902	−	0.929865i	$-0.619924\pi$
$743$	−20.0566			−0.735804			−0.367902	+	0.929865i	$0.619924\pi$
$744$		0			0
$745$	−0.733796			−0.0268842
$746$	26.5566			0.972307
$747$		0			0
$748$	23.4495			0.857400
$749$		−	16.6980i		−	0.610130i
$750$		0			0
$751$		−	30.1038i		−	1.09850i	−0.835657	−	0.549251i	$-0.814913\pi$
$751$		−	30.1038i		−	1.09850i	0.835657	−	0.549251i	$-0.185087\pi$
$752$			4.99360i			0.182098i
$753$		0			0
$754$	1.41099			0.0513851
$755$	−0.503266			−0.0183157
$756$		0			0
$757$			45.2284i			1.64386i	0.569592	+	0.821928i	$0.307101\pi$
$757$			45.2284i			1.64386i	−0.569592	+	0.821928i	$0.692899\pi$
$758$	26.1367			0.949328
$759$		0			0
$760$	1.89551			0.0687572
$761$		−	39.3273i		−	1.42561i	−0.701360	−	0.712807i	$-0.747423\pi$
$761$		−	39.3273i		−	1.42561i	0.701360	−	0.712807i	$-0.252577\pi$
$762$		0			0
$763$	3.89564			0.141032
$764$	15.6539			0.566340
$765$		0			0
$766$			11.4557i			0.413911i
$767$		−	26.0844i		−	0.941854i
$768$		0			0
$769$		−	9.40634i		−	0.339201i	−0.985513	−	0.169601i	$-0.945752\pi$
$769$		−	9.40634i		−	0.339201i	0.985513	−	0.169601i	$-0.0542478\pi$
$770$	2.70213			0.0973780
$771$		0			0
$772$	−18.7029			−0.673131
$773$	−21.0966			−0.758791			−0.379395	−	0.925235i	$-0.623868\pi$
$773$	−21.0966			−0.758791			−0.379395	+	0.925235i	$0.623868\pi$
$774$		0			0
$775$	−33.4237			−1.20061
$776$	−0.822861			−0.0295390
$777$		0			0
$778$			2.49239i			0.0893565i
$779$	6.31538			0.226272
$780$		0			0
$781$		−	54.9432i		−	1.96602i
$782$	−11.5311	+	18.6569i	−0.412353	+	0.667171i
$783$		0			0
$784$	−1.00000			−0.0357143
$785$			0.925122i			0.0330190i
$786$		0			0
$787$		−	45.9306i		−	1.63725i	−0.574328	−	0.818625i	$-0.694737\pi$
$787$		−	45.9306i		−	1.63725i	0.574328	−	0.818625i	$-0.305263\pi$
$788$		−	24.4524i		−	0.871082i
$789$		0			0
$790$	0.740265			0.0263375
$791$		−	11.5946i		−	0.412256i
$792$		0			0
$793$			9.39589i			0.333658i
$794$			16.2536i			0.576817i
$795$		0			0
$796$		−	1.21079i		−	0.0429154i
$797$	39.4823			1.39854			0.699268	−	0.714860i	$-0.253509\pi$
$797$	39.4823			1.39854			0.699268	+	0.714860i	$0.253509\pi$
$798$		0			0
$799$			22.8373i			0.807924i
$800$		−	4.72228i		−	0.166958i
$801$		0			0
$802$		−	1.63283i		−	0.0576572i
$803$	−63.2399			−2.23169
$804$		0			0
$805$	−1.32875	+	2.14987i	−0.0468323	+	0.0757729i
$806$		−	15.2280i		−	0.536385i
$807$		0			0
$808$	8.77864			0.308832
$809$			10.4506i			0.367424i	0.982980	+	0.183712i	$0.0588114\pi$
$809$			10.4506i			0.367424i	−0.982980	+	0.183712i	$0.941189\pi$
$810$		0			0
$811$	−24.3357			−0.854541			−0.427270	−	0.904124i	$-0.640525\pi$
$811$	−24.3357			−0.854541			−0.427270	+	0.904124i	$0.640525\pi$
$812$	−0.655814			−0.0230146
$813$		0			0
$814$	13.1739			0.461745
$815$	8.31563			0.291284
$816$		0			0
$817$	−2.79182			−0.0976733
$818$		−	5.97232i		−	0.208817i
$819$		0			0
$820$			0.925293i			0.0323126i
$821$		−	50.4575i		−	1.76098i	−0.474066	−	0.880489i	$-0.657214\pi$
$821$		−	50.4575i		−	1.76098i	0.474066	−	0.880489i	$-0.342786\pi$
$822$		0			0
$823$	2.44592			0.0852595			0.0426297	−	0.999091i	$-0.486426\pi$
$823$	2.44592			0.0852595			0.0426297	+	0.999091i	$0.486426\pi$
$824$	2.22382			0.0774704
$825$		0			0
$826$			12.1238i			0.421841i
$827$	50.0557			1.74061			0.870304	−	0.492514i	$-0.163922\pi$
$827$	50.0557			1.74061			0.870304	+	0.492514i	$0.163922\pi$
$828$		0			0
$829$	50.1866			1.74305			0.871526	−	0.490349i	$-0.163131\pi$
$829$	50.1866			1.74305			0.871526	+	0.490349i	$0.163131\pi$
$830$			4.46063i			0.154831i
$831$		0			0
$832$	2.15150			0.0745899
$833$	−4.57331			−0.158456
$834$		0			0
$835$			6.01263i			0.208076i
$836$		−	18.4428i		−	0.637857i
$837$		0			0
$838$			1.33706i			0.0461881i
$839$	−34.2092			−1.18103			−0.590516	−	0.807026i	$-0.701076\pi$
$839$	−34.2092			−1.18103			−0.590516	+	0.807026i	$0.701076\pi$
$840$		0			0
$841$	28.5699			0.985169
$842$	22.7739			0.784839
$843$		0			0
$844$	−11.2833			−0.388386
$845$	4.41145			0.151759
$846$		0			0
$847$		−	15.2910i		−	0.525406i
$848$	1.84124			0.0632285
$849$		0			0
$850$		−	21.5965i		−	0.740752i
$851$	−6.47817	+	10.4814i	−0.222069	+	0.359299i
$852$		0			0
$853$	−6.56914			−0.224923			−0.112462	−	0.993656i	$-0.535874\pi$
$853$	−6.56914			−0.224923			−0.112462	+	0.993656i	$0.535874\pi$
$854$		−	4.36713i		−	0.149440i
$855$		0			0
$856$		−	16.6980i		−	0.570725i
$857$			36.7073i			1.25390i	0.779060	+	0.626949i	$0.215697\pi$
$857$			36.7073i			1.25390i	−0.779060	+	0.626949i	$0.784303\pi$
$858$		0			0
$859$	36.8687			1.25794			0.628971	−	0.777429i	$-0.283476\pi$
$859$	36.8687			1.25794			0.628971	+	0.777429i	$0.283476\pi$
$860$		−	0.409040i		−	0.0139482i
$861$		0			0
$862$			17.0470i			0.580625i
$863$			8.13336i			0.276863i	0.990372	+	0.138431i	$0.0442061\pi$
$863$			8.13336i			0.276863i	−0.990372	+	0.138431i	$0.955794\pi$
$864$		0			0
$865$		−	3.54042i		−	0.120378i
$866$	−10.6128			−0.360636
$867$		0			0
$868$			7.07786i			0.240238i
$869$		−	7.20260i		−	0.244331i
$870$		0			0
$871$			0.548666i			0.0185909i
$872$	3.89564			0.131923
$873$		0			0
$874$	14.6735	+	9.06911i	0.496337	+	0.306767i
$875$		−	5.12354i		−	0.173207i
$876$		0			0
$877$	47.5745			1.60648			0.803239	−	0.595657i	$-0.203108\pi$
$877$	47.5745			1.60648			0.803239	+	0.595657i	$0.203108\pi$
$878$		−	17.9680i		−	0.606389i
$879$		0			0
$880$	2.70213			0.0910887
$881$	41.5563			1.40007			0.700033	−	0.714110i	$-0.253169\pi$
$881$	41.5563			1.40007			0.700033	+	0.714110i	$0.253169\pi$
$882$		0			0
$883$	−47.6745			−1.60437			−0.802187	−	0.597074i	$-0.796330\pi$
$883$	−47.6745			−1.60437			−0.802187	+	0.597074i	$0.796330\pi$
$884$	9.83948			0.330938
$885$		0			0
$886$	29.9812			1.00724
$887$			31.8201i			1.06841i	0.845354	+	0.534207i	$0.179390\pi$
$887$			31.8201i			1.06841i	−0.845354	+	0.534207i	$0.820610\pi$
$888$		0			0
$889$			15.4104i			0.516849i
$890$		−	5.24590i		−	0.175843i
$891$		0			0
$892$	4.34331			0.145425
$893$	17.9612			0.601050
$894$		0			0
$895$		−	0.0846801i		−	0.00283054i
$896$	−1.00000			−0.0334077
$897$		0			0
$898$	−5.47792			−0.182801
$899$			4.64176i			0.154812i
$900$		0			0
$901$	8.42057			0.280530
$902$	9.00287			0.299763
$903$		0			0
$904$		−	11.5946i		−	0.385630i
$905$		−	2.03724i		−	0.0677201i
$906$		0			0
$907$			25.0876i			0.833019i	0.909131	+	0.416510i	$0.136747\pi$
$907$			25.0876i			0.833019i	−0.909131	+	0.416510i	$0.863253\pi$
$908$	−1.61928			−0.0537376
$909$		0			0
$910$	1.13382			0.0375858
$911$	−32.7591			−1.08536			−0.542678	−	0.839941i	$-0.682590\pi$
$911$	−32.7591			−1.08536			−0.542678	+	0.839941i	$0.682590\pi$
$912$		0			0
$913$	43.4008			1.43636
$914$	−34.3133			−1.13498
$915$		0			0
$916$			9.58977i			0.316855i
$917$	12.7723			0.421779
$918$		0			0
$919$			11.6070i			0.382879i	0.981504	+	0.191439i	$0.0613155\pi$
$919$			11.6070i			0.382879i	−0.981504	+	0.191439i	$0.938684\pi$
$920$	−1.32875	+	2.14987i	−0.0438076	+	0.0708791i
$921$		0			0
$922$	−30.6915			−1.01077
$923$		−	23.0543i		−	0.758841i
$924$		0			0
$925$		−	12.1328i		−	0.398925i
$926$			41.8755i			1.37611i
$927$		0			0
$928$	−0.655814			−0.0215282
$929$			35.7478i			1.17285i	0.810004	+	0.586424i	$0.199465\pi$
$929$			35.7478i			1.17285i	−0.810004	+	0.586424i	$0.800535\pi$
$930$		0			0
$931$			3.59685i			0.117882i
$932$		−	5.15119i		−	0.168733i
$933$		0			0
$934$		−	29.4863i		−	0.964821i
$935$	12.3577			0.404139
$936$		0			0
$937$			41.3117i			1.34959i	0.738004	+	0.674797i	$0.235769\pi$
$937$			41.3117i			1.34959i	−0.738004	+	0.674797i	$0.764231\pi$
$938$		−	0.255016i		−	0.00832655i
$939$		0			0
$940$			2.63157i			0.0858325i
$941$	27.0643			0.882272			0.441136	−	0.897440i	$-0.354576\pi$
$941$	27.0643			0.882272			0.441136	+	0.897440i	$0.354576\pi$
$942$		0			0
$943$	−4.42709	+	7.16287i	−0.144166	+	0.233255i
$944$			12.1238i			0.394597i
$945$		0			0
$946$	−3.97986			−0.129396
$947$			0.0758069i			0.00246339i	0.999999	+	0.00123170i	$0.000392061\pi$
$947$			0.0758069i			0.00246339i	−0.999999	+	0.00123170i	$0.999608\pi$
$948$		0			0
$949$	−26.5356			−0.861382
$950$	−16.9854			−0.551078
$951$		0			0
$952$	−4.57331			−0.148222
$953$	−4.63705			−0.150209			−0.0751044	−	0.997176i	$-0.523929\pi$
$953$	−4.63705			−0.150209			−0.0751044	+	0.997176i	$0.523929\pi$
$954$		0			0
$955$	8.24946			0.266946
$956$		−	9.24437i		−	0.298984i
$957$		0			0
$958$			19.1778i			0.619607i
$959$		−	15.8875i		−	0.513033i
$960$		0			0
$961$	19.0961			0.616005
$962$	5.52780			0.178223
$963$		0			0
$964$		−	22.7431i		−	0.732505i
$965$	−9.85622			−0.317283
$966$		0			0
$967$	−58.1948			−1.87142			−0.935709	−	0.352772i	$-0.885239\pi$
$967$	−58.1948			−1.87142			−0.935709	+	0.352772i	$0.885239\pi$
$968$		−	15.2910i		−	0.491473i
$969$		0			0
$970$	−0.433639			−0.0139233
$971$	−60.2092			−1.93221			−0.966103	−	0.258156i	$-0.916885\pi$
$971$	−60.2092			−1.93221			−0.966103	+	0.258156i	$0.916885\pi$
$972$		0			0
$973$		−	9.49560i		−	0.304415i
$974$			32.5320i			1.04239i
$975$		0			0
$976$		−	4.36713i		−	0.139788i
$977$	3.20266			0.102462			0.0512311	−	0.998687i	$-0.483685\pi$
$977$	3.20266			0.102462			0.0512311	+	0.998687i	$0.483685\pi$
$978$		0			0
$979$	−51.0413			−1.63129
$980$	−0.526990			−0.0168341
$981$		0			0
$982$	18.1334			0.578659
$983$	2.75050			0.0877275			0.0438637	−	0.999038i	$-0.486033\pi$
$983$	2.75050			0.0877275			0.0438637	+	0.999038i	$0.486033\pi$
$984$		0			0
$985$		−	12.8862i		−	0.410588i
$986$	−2.99924			−0.0955153
$987$		0			0
$988$		−	7.73864i		−	0.246199i
$989$	1.95707	−	3.16646i	0.0622311	−	0.100688i
$990$		0			0
$991$	44.9008			1.42632			0.713160	−	0.701002i	$-0.247263\pi$
$991$	44.9008			1.42632			0.713160	+	0.701002i	$0.247263\pi$
$992$			7.07786i			0.224722i
$993$		0			0
$994$			10.7154i			0.339873i
$995$		−	0.638075i		−	0.0202283i
$996$		0			0
$997$	−19.3647			−0.613287			−0.306643	−	0.951824i	$-0.599206\pi$
$997$	−19.3647			−0.613287			−0.306643	+	0.951824i	$0.599206\pi$
$998$		−	11.7545i		−	0.372084i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2898.2.h.b.827.10	yes	24
3.2	odd	2		2898.2.h.a.827.1	✓	24
23.22	odd	2		2898.2.h.a.827.2	yes	24
69.68	even	2	inner	2898.2.h.b.827.9	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2898.2.h.a.827.1	✓	24	3.2	odd	2
2898.2.h.a.827.2	yes	24	23.22	odd	2
2898.2.h.b.827.9	yes	24	69.68	even	2	inner
2898.2.h.b.827.10	yes	24	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2898 = 2 \cdot 3^{2} \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2898.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} -0.526990 q^{5} -1.00000i q^{7} -1.00000i q^{8} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000 q^{4} -0.526990 q^{5} -1.00000i q^{7} -1.00000i q^{8} -0.526990i q^{10} -5.12748 q^{11} -2.15150 q^{13} +1.00000 q^{14} +1.00000 q^{16} +4.57331 q^{17} -3.59685i q^{19} +0.526990 q^{20} -5.12748i q^{22} +(4.07953 + 2.52140i) q^{23} -4.72228 q^{25} -2.15150i q^{26} +1.00000i q^{28} +0.655814i q^{29} +7.07786 q^{31} +1.00000i q^{32} +4.57331i q^{34} +0.526990i q^{35} +2.56927i q^{37} +3.59685 q^{38} +0.526990i q^{40} +1.75581i q^{41} -0.776183i q^{43} +5.12748 q^{44} +(-2.52140 + 4.07953i) q^{46} +4.99360i q^{47} -1.00000 q^{49} -4.72228i q^{50} +2.15150 q^{52} +1.84124 q^{53} +2.70213 q^{55} -1.00000 q^{56} -0.655814 q^{58} +12.1238i q^{59} -4.36713i q^{61} +7.07786i q^{62} -1.00000 q^{64} +1.13382 q^{65} -0.255016i q^{67} -4.57331 q^{68} -0.526990 q^{70} +10.7154i q^{71} +12.3335 q^{73} -2.56927 q^{74} +3.59685i q^{76} +5.12748i q^{77} +1.40471i q^{79} -0.526990 q^{80} -1.75581 q^{82} -8.46436 q^{83} -2.41009 q^{85} +0.776183 q^{86} +5.12748i q^{88} +9.95446 q^{89} +2.15150i q^{91} +(-4.07953 - 2.52140i) q^{92} -4.99360 q^{94} +1.89551i q^{95} -0.822861i q^{97} -1.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 24 q^{4} + 8 q^{5}+O(q^{10}) \) 24 * q - 24 * q^4 + 8 * q^5	\( 24 q - 24 q^{4} + 8 q^{5} - 16 q^{13} + 24 q^{14} + 24 q^{16} - 16 q^{17} - 8 q^{20} - 8 q^{23} + 48 q^{25} - 16 q^{31} + 8 q^{46} - 24 q^{49} + 16 q^{52} - 48 q^{53} + 16 q^{55} - 24 q^{56} - 8 q^{58} - 24 q^{64} + 48 q^{65} + 16 q^{68} + 8 q^{70} + 64 q^{73} + 32 q^{74} + 8 q^{80} + 16 q^{82} + 16 q^{83} + 32 q^{85} - 32 q^{86} - 48 q^{89} + 8 q^{92} - 16 q^{94}+O(q^{100}) \) 24 * q - 24 * q^4 + 8 * q^5 - 16 * q^13 + 24 * q^14 + 24 * q^16 - 16 * q^17 - 8 * q^20 - 8 * q^23 + 48 * q^25 - 16 * q^31 + 8 * q^46 - 24 * q^49 + 16 * q^52 - 48 * q^53 + 16 * q^55 - 24 * q^56 - 8 * q^58 - 24 * q^64 + 48 * q^65 + 16 * q^68 + 8 * q^70 + 64 * q^73 + 32 * q^74 + 8 * q^80 + 16 * q^82 + 16 * q^83 + 32 * q^85 - 32 * q^86 - 48 * q^89 + 8 * q^92 - 16 * q^94

Introduction

L-functions

Modular forms

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Database

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