LMFDB - Embedded newform 1360.2.e.d.1089.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1360,2,Mod(1089,1360)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1360, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1360.1089");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1360 = 2^{4} \cdot 5 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1360.e (of order $2$, degree $1$, not 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:	$10.8596546749$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.619810816.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{5} + 14x^{4} - 8x^{3} + 2x^{2} + 2x + 1 $ x^8 - 2x^5 + 14x^4 - 8x^3 + 2x^2 + 2*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 85)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1089.8
Root			$-0.252709 + 0.252709i$ of defining polynomial
Character	$\chi$	$=$	1360.1089
Dual form			1360.2.e.d.1089.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.87228i q^{3} +(0.146426 - 2.23127i) q^{5} -1.42058i q^{7} -5.24997 q^{9} -0.0740063 q^{11} +5.70167i q^{13} +(6.40882 + 0.420575i) q^{15} -1.00000i q^{17} -4.90340 q^{19} +4.08029 q^{21} +3.88311i q^{23} +(-4.95712 - 0.653431i) q^{25} -6.46254i q^{27} -5.91424 q^{29} -0.388531 q^{31} -0.212567i q^{33} +(-3.16969 - 0.208009i) q^{35} +9.91424i q^{37} -16.3768 q^{39} -6.61055 q^{41} +6.94628i q^{43} +(-0.768731 + 11.7141i) q^{45} +5.70167i q^{47} +4.98197 q^{49} +2.87228 q^{51} +0.0216729i q^{53} +(-0.0108364 + 0.165128i) q^{55} -14.0839i q^{57} -2.00000 q^{59} -3.47337 q^{61} +7.45798i q^{63} +(12.7220 + 0.834872i) q^{65} +6.71251i q^{67} -11.1534 q^{69} -3.84023 q^{71} -13.5356i q^{73} +(1.87683 - 14.2382i) q^{75} +0.105132i q^{77} -1.06000 q^{79} +2.81228 q^{81} -11.8497i q^{83} +(-2.23127 - 0.146426i) q^{85} -16.9873i q^{87} +1.99452 q^{89} +8.09965 q^{91} -1.11597i q^{93} +(-0.717985 + 10.9408i) q^{95} +5.34109i q^{97} +0.388531 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 2 q^{5} - 8 q^{9} + 4 q^{11} + 8 q^{19} + 24 q^{21} - 12 q^{25} + 8 q^{29} + 24 q^{31} - 44 q^{39} - 12 q^{41} - 22 q^{45} - 16 q^{49} + 8 q^{51} + 8 q^{55} - 16 q^{59} + 12 q^{61} + 20 q^{65} - 8 q^{69}+ \cdots - 24 q^{99}+O(q^{100}) $ 8 * q - 2 * q^5 - 8 * q^9 + 4 * q^11 + 8 * q^19 + 24 * q^21 - 12 * q^25 + 8 * q^29 + 24 * q^31 - 44 * q^39 - 12 * q^41 - 22 * q^45 - 16 * q^49 + 8 * q^51 + 8 * q^55 - 16 * q^59 + 12 * q^61 + 20 * q^65 - 8 * q^69 + 20 * q^71 - 4 * q^75 - 24 * q^79 - 8 * q^81 - 2 * q^85 - 48 * q^89 - 36 * q^91 - 4 * q^95 - 24 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1360\mathbb{Z}\right)^\times$.

$n$	$241$	$341$	$511$	$817$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$			2.87228i			1.65831i	0.559019	+	0.829155i	$0.311178\pi$
$3$			2.87228i			1.65831i	−0.559019	+	0.829155i	$0.688822\pi$
$4$		0			0
$5$	0.146426	−	2.23127i	0.0654836	−	0.997854i
$5$	0.146426	−	2.23127i	0.0654836	−	0.997854i
$6$		0			0
$7$		−	1.42058i		−	0.536927i	−0.963290	−	0.268464i	$-0.913484\pi$
$7$		−	1.42058i		−	0.536927i	0.963290	−	0.268464i	$-0.0865159\pi$
$8$		0			0
$9$	−5.24997			−1.74999
$10$		0			0
$11$	−0.0740063			−0.0223137			−0.0111569	−	0.999938i	$-0.503551\pi$
$11$	−0.0740063			−0.0223137			−0.0111569	+	0.999938i	$0.503551\pi$
$12$		0			0
$13$			5.70167i			1.58136i	0.612230	+	0.790680i	$0.290273\pi$
$13$			5.70167i			1.58136i	−0.612230	+	0.790680i	$0.709727\pi$
$14$		0			0
$15$	6.40882	+	0.420575i	1.65475	+	0.108592i
$16$		0			0
$17$		−	1.00000i		−	0.242536i
$17$		−	1.00000i		−	0.242536i
$18$		0			0
$19$	−4.90340			−1.12492			−0.562459	−	0.826825i	$-0.690144\pi$
$19$	−4.90340			−1.12492			−0.562459	+	0.826825i	$0.690144\pi$
$20$		0			0
$21$	4.08029			0.890391
$22$		0			0
$23$			3.88311i			0.809685i	0.914386	+	0.404842i	$0.132674\pi$
$23$			3.88311i			0.809685i	−0.914386	+	0.404842i	$0.867326\pi$
$24$		0			0
$25$	−4.95712	−	0.653431i	−0.991424	−	0.130686i
$26$		0			0
$27$		−	6.46254i		−	1.24372i
$28$		0			0
$29$	−5.91424			−1.09825			−0.549123	−	0.835741i	$-0.685038\pi$
$29$	−5.91424			−1.09825			−0.549123	+	0.835741i	$0.685038\pi$
$30$		0			0
$31$	−0.388531			−0.0697822			−0.0348911	−	0.999391i	$-0.511108\pi$
$31$	−0.388531			−0.0697822			−0.0348911	+	0.999391i	$0.511108\pi$
$32$		0			0
$33$		−	0.212567i		−	0.0370031i
$34$		0			0
$35$	−3.16969	−	0.208009i	−0.535775	−	0.0351599i
$36$		0			0
$37$			9.91424i			1.62989i	0.579538	+	0.814945i	$0.303233\pi$
$37$			9.91424i			1.62989i	−0.579538	+	0.814945i	$0.696767\pi$
$38$		0			0
$39$	−16.3768			−2.62238
$40$		0			0
$41$	−6.61055			−1.03239			−0.516197	−	0.856470i	$-0.672653\pi$
$41$	−6.61055			−1.03239			−0.516197	+	0.856470i	$0.672653\pi$
$42$		0			0
$43$			6.94628i			1.05930i	0.848217	+	0.529649i	$0.177676\pi$
$43$			6.94628i			1.05930i	−0.848217	+	0.529649i	$0.822324\pi$
$44$		0			0
$45$	−0.768731	+	11.7141i	−0.114596	+	1.74623i
$46$		0			0
$47$			5.70167i			0.831674i	0.909439	+	0.415837i	$0.136511\pi$
$47$			5.70167i			0.831674i	−0.909439	+	0.415837i	$0.863489\pi$
$48$		0			0
$49$	4.98197			0.711709
$50$		0			0
$51$	2.87228			0.402199
$52$		0			0
$53$			0.0216729i			0.00297700i	0.999999	+	0.00148850i	$0.000473804\pi$
$53$			0.0216729i			0.00297700i	−0.999999	+	0.00148850i	$0.999526\pi$
$54$		0			0
$55$	−0.0108364	+	0.165128i	−0.00146118	+	0.0222658i
$56$		0			0
$57$		−	14.0839i		−	1.86546i
$58$		0			0
$59$	−2.00000			−0.260378			−0.130189	−	0.991489i	$-0.541558\pi$
$59$	−2.00000			−0.260378			−0.130189	+	0.991489i	$0.541558\pi$
$60$		0			0
$61$	−3.47337			−0.444720			−0.222360	−	0.974965i	$-0.571376\pi$
$61$	−3.47337			−0.444720			−0.222360	+	0.974965i	$0.571376\pi$
$62$		0			0
$63$			7.45798i			0.939617i
$64$		0			0
$65$	12.7220	+	0.834872i	1.57796	+	0.103553i
$66$		0			0
$67$			6.71251i			0.820063i	0.912071	+	0.410032i	$0.134482\pi$
$67$			6.71251i			0.820063i	−0.912071	+	0.410032i	$0.865518\pi$
$68$		0			0
$69$	−11.1534			−1.34271
$70$		0			0
$71$	−3.84023			−0.455752			−0.227876	−	0.973690i	$-0.573178\pi$
$71$	−3.84023			−0.455752			−0.227876	+	0.973690i	$0.573178\pi$
$72$		0			0
$73$		−	13.5356i		−	1.58422i	−0.610375	−	0.792112i	$-0.708981\pi$
$73$		−	13.5356i		−	1.58422i	0.610375	−	0.792112i	$-0.291019\pi$
$74$		0			0
$75$	1.87683	−	14.2382i	0.216718	−	1.64409i
$76$		0			0
$77$			0.105132i			0.0119808i
$78$		0			0
$79$	−1.06000			−0.119259			−0.0596295	−	0.998221i	$-0.518992\pi$
$79$	−1.06000			−0.119259			−0.0596295	+	0.998221i	$0.518992\pi$
$80$		0			0
$81$	2.81228			0.312476
$82$		0			0
$83$		−	11.8497i		−	1.30067i	−0.759647	−	0.650336i	$-0.774628\pi$
$83$		−	11.8497i		−	1.30067i	0.759647	−	0.650336i	$-0.225372\pi$
$84$		0			0
$85$	−2.23127	−	0.146426i	−0.242015	−	0.0158821i
$86$		0			0
$87$		−	16.9873i		−	1.82123i
$88$		0			0
$89$	1.99452			0.211419			0.105710	−	0.994397i	$-0.466289\pi$
$89$	1.99452			0.211419			0.105710	+	0.994397i	$0.466289\pi$
$90$		0			0
$91$	8.09965			0.849075
$92$		0			0
$93$		−	1.11597i		−	0.115720i
$94$		0			0
$95$	−0.717985	+	10.9408i	−0.0736637	+	1.12250i
$96$		0			0
$97$			5.34109i			0.542306i	0.962536	+	0.271153i	$0.0874049\pi$
$97$			5.34109i			0.542306i	−0.962536	+	0.271153i	$0.912595\pi$
$98$		0			0
$99$	0.388531			0.0390488
$100$		0			0
$101$	11.4305			1.13738			0.568688	−	0.822553i	$-0.307451\pi$
$101$	11.4305			1.13738			0.568688	+	0.822553i	$0.307451\pi$
$102$		0			0
$103$			8.21257i			0.809208i	0.914492	+	0.404604i	$0.132591\pi$
$103$			8.21257i			0.809208i	−0.914492	+	0.404604i	$0.867409\pi$
$104$		0			0
$105$	0.597459	−	9.10421i	0.0583061	−	0.888480i
$106$		0			0
$107$		−	6.34882i		−	0.613764i	−0.951748	−	0.306882i	$-0.900714\pi$
$107$		−	6.34882i		−	0.613764i	0.951748	−	0.306882i	$-0.0992857\pi$
$108$		0			0
$109$	−0.292852			−0.0280501			−0.0140251	−	0.999902i	$-0.504464\pi$
$109$	−0.292852			−0.0280501			−0.0140251	+	0.999902i	$0.504464\pi$
$110$		0			0
$111$	−28.4764			−2.70286
$112$		0			0
$113$		−	8.62311i		−	0.811194i	−0.914052	−	0.405597i	$-0.867064\pi$
$113$		−	8.62311i		−	0.811194i	0.914052	−	0.405597i	$-0.132936\pi$
$114$		0			0
$115$	8.66427	+	0.568588i	0.807947	+	0.0530211i
$116$		0			0
$117$		−	29.9336i		−	2.76736i
$118$		0			0
$119$	−1.42058			−0.130224
$120$		0			0
$121$	−10.9945			−0.999502
$122$		0			0
$123$		−	18.9873i		−	1.71203i
$124$		0			0
$125$	−2.18383	+	10.9650i	−0.195328	+	0.980738i
$126$		0			0
$127$		−	17.2640i		−	1.53193i	−0.642882	−	0.765965i	$-0.722261\pi$
$127$		−	17.2640i		−	1.53193i	0.642882	−	0.765965i	$-0.277739\pi$
$128$		0			0
$129$	−19.9516			−1.75664
$130$		0			0
$131$	14.7188			1.28599			0.642993	−	0.765872i	$-0.277692\pi$
$131$	14.7188			1.28599			0.642993	+	0.765872i	$0.277692\pi$
$132$		0			0
$133$			6.96565i			0.603999i
$134$		0			0
$135$	−14.4197	−	0.946283i	−1.24105	−	0.0814430i
$136$		0			0
$137$		−	1.36230i		−	0.116389i	−0.998305	−	0.0581946i	$-0.981466\pi$
$137$		−	1.36230i		−	0.116389i	0.998305	−	0.0581946i	$-0.0185344\pi$
$138$		0			0
$139$	−13.7454			−1.16587			−0.582933	−	0.812520i	$-0.698095\pi$
$139$	−13.7454			−1.16587			−0.582933	+	0.812520i	$0.698095\pi$
$140$		0			0
$141$	−16.3768			−1.37917
$142$		0			0
$143$		−	0.421960i		−	0.0352860i
$144$		0			0
$145$	−0.865997	+	13.1963i	−0.0719172	+	1.09589i
$146$		0			0
$147$			14.3096i			1.18023i
$148$		0			0
$149$	17.3628			1.42241			0.711207	−	0.702983i	$-0.248149\pi$
$149$	17.3628			1.42241			0.711207	+	0.702983i	$0.248149\pi$
$150$		0			0
$151$	3.59654			0.292682			0.146341	−	0.989234i	$-0.453250\pi$
$151$	3.59654			0.292682			0.146341	+	0.989234i	$0.453250\pi$
$152$		0			0
$153$			5.24997i			0.424435i
$154$		0			0
$155$	−0.0568910	+	0.866917i	−0.00456959	+	0.0696324i
$156$		0			0
$157$			5.19308i			0.414453i	0.978293	+	0.207226i	$0.0664437\pi$
$157$			5.19308i			0.414453i	−0.978293	+	0.207226i	$0.933556\pi$
$158$		0			0
$159$	−0.0622505			−0.00493678
$160$		0			0
$161$	5.51625			0.434742
$162$		0			0
$163$			6.56687i			0.514357i	0.966364	+	0.257178i	$0.0827928\pi$
$163$			6.56687i			0.514357i	−0.966364	+	0.257178i	$0.917207\pi$
$164$		0			0
$165$	−0.474293	−	0.0311252i	−0.0369237	−	0.00242310i
$166$		0			0
$167$			0.897123i			0.0694214i	0.999397	+	0.0347107i	$0.0110510\pi$
$167$			0.897123i			0.0694214i	−0.999397	+	0.0347107i	$0.988949\pi$
$168$		0			0
$169$	−19.5091			−1.50070
$170$		0			0
$171$	25.7427			1.96859
$172$		0			0
$173$			3.83031i			0.291213i	0.989343	+	0.145607i	$0.0465134\pi$
$173$			3.83031i			0.291213i	−0.989343	+	0.145607i	$0.953487\pi$
$174$		0			0
$175$	−0.928248	+	7.04196i	−0.0701689	+	0.532322i
$176$		0			0
$177$		−	5.74455i		−	0.431787i
$178$		0			0
$179$	9.74455			0.728342			0.364171	−	0.931332i	$-0.381352\pi$
$179$	9.74455			0.728342			0.364171	+	0.931332i	$0.381352\pi$
$180$		0			0
$181$	22.4764			1.67066			0.835330	−	0.549749i	$-0.185277\pi$
$181$	22.4764			1.67066			0.835330	+	0.549749i	$0.185277\pi$
$182$		0			0
$183$		−	9.97649i		−	0.737483i
$184$		0			0
$185$	22.1213	+	1.45170i	1.62639	+	0.106731i
$186$		0			0
$187$			0.0740063i			0.00541188i
$188$		0			0
$189$	−9.18052			−0.667785
$190$		0			0
$191$	5.47655			0.396269			0.198135	−	0.980175i	$-0.436512\pi$
$191$	5.47655			0.396269			0.198135	+	0.980175i	$0.436512\pi$
$192$		0			0
$193$		−	2.14484i		−	0.154389i	−0.997016	−	0.0771944i	$-0.975404\pi$
$193$		−	2.14484i		−	0.154389i	0.997016	−	0.0771944i	$-0.0245962\pi$
$194$		0			0
$195$	−2.39798	+	36.5410i	−0.171723	+	2.61675i
$196$		0			0
$197$			9.78513i			0.697162i	0.937279	+	0.348581i	$0.113336\pi$
$197$			9.78513i			0.697162i	−0.937279	+	0.348581i	$0.886664\pi$
$198$		0			0
$199$	−1.47257			−0.104388			−0.0521939	−	0.998637i	$-0.516621\pi$
$199$	−1.47257			−0.104388			−0.0521939	+	0.998637i	$0.516621\pi$
$200$		0			0
$201$	−19.2802			−1.35992
$202$		0			0
$203$			8.40162i			0.589678i
$204$		0			0
$205$	−0.967955	+	14.7499i	−0.0676049	+	1.03018i
$206$		0			0
$207$		−	20.3862i		−	1.41694i
$208$		0			0
$209$	0.362883			0.0251011
$210$		0			0
$211$	4.87591			0.335672			0.167836	−	0.985815i	$-0.446322\pi$
$211$	4.87591			0.335672			0.167836	+	0.985815i	$0.446322\pi$
$212$		0			0
$213$		−	11.0302i		−	0.755777i
$214$		0			0
$215$	15.4990	+	1.01712i	1.05702	+	0.0693667i
$216$		0			0
$217$			0.551937i			0.0374680i
$218$		0			0
$219$	38.8781			2.62714
$220$		0			0
$221$	5.70167			0.383536
$222$		0			0
$223$			20.0536i			1.34289i	0.741055	+	0.671444i	$0.234326\pi$
$223$			20.0536i			1.34289i	−0.741055	+	0.671444i	$0.765674\pi$
$224$		0			0
$225$	26.0247	+	3.43049i	1.73498	+	0.228700i
$226$		0			0
$227$			14.1308i			0.937893i	0.883227	+	0.468946i	$0.155366\pi$
$227$			14.1308i			0.937893i	−0.883227	+	0.468946i	$0.844634\pi$
$228$		0			0
$229$	−11.2156			−0.741149			−0.370575	−	0.928803i	$-0.620839\pi$
$229$	−11.2156			−0.741149			−0.370575	+	0.928803i	$0.620839\pi$
$230$		0			0
$231$	−0.301967			−0.0198680
$232$		0			0
$233$			13.8285i			0.905934i	0.891527	+	0.452967i	$0.149634\pi$
$233$			13.8285i			0.905934i	−0.891527	+	0.452967i	$0.850366\pi$
$234$		0			0
$235$	12.7220	+	0.834872i	0.829889	+	0.0544610i
$236$		0			0
$237$		−	3.04460i		−	0.197768i
$238$		0			0
$239$	−16.2851			−1.05339			−0.526697	−	0.850053i	$-0.676570\pi$
$239$	−16.2851			−1.05339			−0.526697	+	0.850053i	$0.676570\pi$
$240$		0			0
$241$	4.94081			0.318265			0.159133	−	0.987257i	$-0.449130\pi$
$241$	4.94081			0.318265			0.159133	+	0.987257i	$0.449130\pi$
$242$		0			0
$243$		−	11.3100i		−	0.725535i
$244$		0			0
$245$	0.729489	−	11.1161i	0.0466053	−	0.710182i
$246$		0			0
$247$		−	27.9576i		−	1.77890i
$248$		0			0
$249$	34.0356			2.15692
$250$		0			0
$251$	−12.9160			−0.815248			−0.407624	−	0.913150i	$-0.633643\pi$
$251$	−12.9160			−0.815248			−0.407624	+	0.913150i	$0.633643\pi$
$252$		0			0
$253$		−	0.287375i		−	0.0180671i
$254$		0			0
$255$	0.420575	−	6.40882i	0.0263375	−	0.401336i
$256$		0			0
$257$			13.6159i			0.849337i	0.905349	+	0.424669i	$0.139609\pi$
$257$			13.6159i			0.849337i	−0.905349	+	0.424669i	$0.860391\pi$
$258$		0			0
$259$	14.0839			0.875132
$260$		0			0
$261$	31.0496			1.92192
$262$		0			0
$263$			12.6719i			0.781385i	0.920521	+	0.390692i	$0.127764\pi$
$263$			12.6719i			0.781385i	−0.920521	+	0.390692i	$0.872236\pi$
$264$		0			0
$265$	0.0483580	+	0.00317347i	0.00297061	+	0.000194945i
$266$		0			0
$267$			5.72882i			0.350598i
$268$		0			0
$269$	25.8967			1.57895			0.789474	−	0.613784i	$-0.210354\pi$
$269$	25.8967			1.57895			0.789474	+	0.613784i	$0.210354\pi$
$270$		0			0
$271$	−21.2211			−1.28909			−0.644545	−	0.764566i	$-0.722953\pi$
$271$	−21.2211			−1.28909			−0.644545	+	0.764566i	$0.722953\pi$
$272$		0			0
$273$			23.2644i			1.40803i
$274$		0			0
$275$	0.366858	+	0.0483580i	0.0221224	+	0.00291610i
$276$		0			0
$277$		−	2.05908i		−	0.123718i	−0.998085	−	0.0618590i	$-0.980297\pi$
$277$		−	2.05908i		−	0.123718i	0.998085	−	0.0618590i	$-0.0197029\pi$
$278$		0			0
$279$	2.03978			0.122118
$280$		0			0
$281$	−21.7336			−1.29652			−0.648259	−	0.761420i	$-0.724503\pi$
$281$	−21.7336			−1.29652			−0.648259	+	0.761420i	$0.724503\pi$
$282$		0			0
$283$			5.32226i			0.316375i	0.987409	+	0.158188i	$0.0505651\pi$
$283$			5.32226i			0.316375i	−0.987409	+	0.158188i	$0.949435\pi$
$284$		0			0
$285$	−31.4250	−	2.06225i	−1.86146	−	0.122157i
$286$		0			0
$287$			9.39078i			0.554321i
$288$		0			0
$289$	−1.00000			−0.0588235
$290$		0			0
$291$	−15.3411			−0.899311
$292$		0			0
$293$			23.3285i			1.36287i	0.731880	+	0.681434i	$0.238643\pi$
$293$			23.3285i			1.36287i	−0.731880	+	0.681434i	$0.761357\pi$
$294$		0			0
$295$	−0.292852	+	4.46254i	−0.0170505	+	0.259819i
$296$		0			0
$297$			0.478268i			0.0277519i
$298$		0			0
$299$	−22.1402			−1.28040
$300$		0			0
$301$	9.86772			0.568766
$302$		0			0
$303$			32.8315i			1.88612i
$304$		0			0
$305$	−0.508592	+	7.75003i	−0.0291219	+	0.443765i
$306$		0			0
$307$			17.4588i			0.996425i	0.867055	+	0.498213i	$0.166010\pi$
$307$			17.4588i			0.996425i	−0.867055	+	0.498213i	$0.833990\pi$
$308$		0			0
$309$	−23.5888			−1.34192
$310$		0			0
$311$	−2.98997			−0.169545			−0.0847727	−	0.996400i	$-0.527016\pi$
$311$	−2.98997			−0.169545			−0.0847727	+	0.996400i	$0.527016\pi$
$312$		0			0
$313$			7.76028i			0.438637i	0.975653	+	0.219319i	$0.0703834\pi$
$313$			7.76028i			0.438637i	−0.975653	+	0.219319i	$0.929617\pi$
$314$		0			0
$315$	16.6408	+	1.09204i	0.937600	+	0.0615295i
$316$		0			0
$317$		−	6.88951i		−	0.386953i	−0.981105	−	0.193477i	$-0.938024\pi$
$317$		−	6.88951i		−	0.386953i	0.981105	−	0.193477i	$-0.0619764\pi$
$318$		0			0
$319$	0.437691			0.0245060
$320$		0			0
$321$	18.2356			1.01781
$322$		0			0
$323$			4.90340i			0.272833i
$324$		0			0
$325$	3.72565	−	28.2639i	0.206662	−	1.56780i
$326$		0			0
$327$		−	0.841151i		−	0.0465158i
$328$		0			0
$329$	8.09965			0.446548
$330$		0			0
$331$	−15.8565			−0.871552			−0.435776	−	0.900055i	$-0.643526\pi$
$331$	−15.8565			−0.871552			−0.435776	+	0.900055i	$0.643526\pi$
$332$		0			0
$333$		−	52.0495i		−	2.85229i
$334$		0			0
$335$	14.9774	+	0.982885i	0.818303	+	0.0537007i
$336$		0			0
$337$		−	19.1787i		−	1.04473i	−0.852722	−	0.522365i	$-0.825050\pi$
$337$		−	19.1787i		−	1.04473i	0.852722	−	0.522365i	$-0.174950\pi$
$338$		0			0
$339$	24.7679			1.34521
$340$		0			0
$341$	0.0287537			0.00155710
$342$		0			0
$343$		−	17.0213i		−	0.919063i
$344$		0			0
$345$	−1.63314	+	24.8862i	−0.0879254	+	1.33983i
$346$		0			0
$347$		−	33.5775i		−	1.80253i	−0.433265	−	0.901266i	$-0.642639\pi$
$347$		−	33.5775i		−	1.80253i	0.433265	−	0.901266i	$-0.357361\pi$
$348$		0			0
$349$	−8.61649			−0.461230			−0.230615	−	0.973045i	$-0.574074\pi$
$349$	−8.61649			−0.461230			−0.230615	+	0.973045i	$0.574074\pi$
$350$		0			0
$351$	36.8473			1.96676
$352$		0			0
$353$			0.596657i			0.0317569i	0.999874	+	0.0158784i	$0.00505447\pi$
$353$			0.596657i			0.0317569i	−0.999874	+	0.0158784i	$0.994946\pi$
$354$		0			0
$355$	−0.562309	+	8.56859i	−0.0298443	+	0.454773i
$356$		0			0
$357$		−	4.08029i		−	0.215952i
$358$		0			0
$359$	31.5514			1.66522			0.832608	−	0.553862i	$-0.186847\pi$
$359$	31.5514			1.66522			0.832608	+	0.553862i	$0.186847\pi$
$360$		0			0
$361$	5.04335			0.265439
$362$		0			0
$363$		−	31.5793i		−	1.65748i
$364$		0			0
$365$	−30.2016	−	1.98197i	−1.58082	−	0.103741i
$366$		0			0
$367$		−	1.37723i		−	0.0718908i	−0.999354	−	0.0359454i	$-0.988556\pi$
$367$		−	1.37723i		−	0.0718908i	0.999354	−	0.0359454i	$-0.0114442\pi$
$368$		0			0
$369$	34.7052			1.80668
$370$		0			0
$371$	0.0307879			0.00159843
$372$		0			0
$373$			28.5627i			1.47892i	0.673201	+	0.739459i	$0.264919\pi$
$373$			28.5627i			1.47892i	−0.673201	+	0.739459i	$0.735081\pi$
$374$		0			0
$375$	−31.4945	−	6.27256i	−1.62637	−	0.323914i
$376$		0			0
$377$		−	33.7210i		−	1.73672i
$378$		0			0
$379$	−24.2405			−1.24515			−0.622576	−	0.782559i	$-0.713914\pi$
$379$	−24.2405			−1.24515			−0.622576	+	0.782559i	$0.713914\pi$
$380$		0			0
$381$	49.5869			2.54041
$382$		0			0
$383$		−	24.4678i		−	1.25025i	−0.780527	−	0.625123i	$-0.785049\pi$
$383$		−	24.4678i		−	1.25025i	0.780527	−	0.625123i	$-0.214951\pi$
$384$		0			0
$385$	0.234577	+	0.0153940i	0.0119551	+	0.000784550i
$386$		0			0
$387$		−	36.4678i		−	1.85376i
$388$		0			0
$389$	10.9711			0.556258			0.278129	−	0.960544i	$-0.410286\pi$
$389$	10.9711			0.556258			0.278129	+	0.960544i	$0.410286\pi$
$390$		0			0
$391$	3.88311			0.196377
$392$		0			0
$393$			42.2764i			2.13256i
$394$		0			0
$395$	−0.155211	+	2.36514i	−0.00780951	+	0.119003i
$396$		0			0
$397$		−	26.8550i		−	1.34782i	−0.738815	−	0.673908i	$-0.764614\pi$
$397$		−	26.8550i		−	1.34782i	0.738815	−	0.673908i	$-0.235386\pi$
$398$		0			0
$399$	−20.0073			−1.00162
$400$		0			0
$401$	−17.5080			−0.874308			−0.437154	−	0.899387i	$-0.644014\pi$
$401$	−17.5080			−0.874308			−0.437154	+	0.899387i	$0.644014\pi$
$402$		0			0
$403$		−	2.21528i		−	0.110351i
$404$		0			0
$405$	0.411790	−	6.27495i	0.0204620	−	0.311805i
$406$		0			0
$407$		−	0.733716i		−	0.0363690i
$408$		0			0
$409$	17.8285			0.881561			0.440781	−	0.897615i	$-0.354702\pi$
$409$	17.8285			0.881561			0.440781	+	0.897615i	$0.354702\pi$
$410$		0			0
$411$	3.91290			0.193009
$412$		0			0
$413$			2.84115i			0.139804i
$414$		0			0
$415$	−26.4398	−	1.73510i	−1.29788	−	0.0851727i
$416$		0			0
$417$		−	39.4805i		−	1.93337i
$418$		0			0
$419$	34.7826			1.69924			0.849622	−	0.527393i	$-0.176830\pi$
$419$	34.7826			1.69924			0.849622	+	0.527393i	$0.176830\pi$
$420$		0			0
$421$	−17.0676			−0.831824			−0.415912	−	0.909405i	$-0.636538\pi$
$421$	−17.0676			−0.831824			−0.415912	+	0.909405i	$0.636538\pi$
$422$		0			0
$423$		−	29.9336i		−	1.45542i
$424$		0			0
$425$	−0.653431	+	4.95712i	−0.0316960	+	0.240456i
$426$		0			0
$427$			4.93419i			0.238782i
$428$		0			0
$429$	1.21198			0.0585152
$430$		0			0
$431$	22.5970			1.08846			0.544229	−	0.838937i	$-0.316822\pi$
$431$	22.5970			1.08846			0.544229	+	0.838937i	$0.316822\pi$
$432$		0			0
$433$		−	13.3513i		−	0.641625i	−0.947143	−	0.320812i	$-0.896044\pi$
$433$		−	13.3513i		−	0.641625i	0.947143	−	0.320812i	$-0.103956\pi$
$434$		0			0
$435$	−37.9033	−	2.48738i	−1.81732	−	0.119261i
$436$		0			0
$437$		−	19.0405i		−	0.910829i
$438$		0			0
$439$	5.73280			0.273611			0.136806	−	0.990598i	$-0.456316\pi$
$439$	5.73280			0.273611			0.136806	+	0.990598i	$0.456316\pi$
$440$		0			0
$441$	−26.1552			−1.24548
$442$		0			0
$443$		−	38.8153i		−	1.84417i	−0.386985	−	0.922086i	$-0.626483\pi$
$443$		−	38.8153i		−	1.84417i	0.386985	−	0.922086i	$-0.373517\pi$
$444$		0			0
$445$	0.292050	−	4.45032i	0.0138445	−	0.210965i
$446$		0			0
$447$			49.8707i			2.35880i
$448$		0			0
$449$	25.3848			1.19798			0.598992	−	0.800755i	$-0.295568\pi$
$449$	25.3848			1.19798			0.598992	+	0.800755i	$0.295568\pi$
$450$		0			0
$451$	0.489222			0.0230366
$452$		0			0
$453$			10.3303i			0.485358i
$454$		0			0
$455$	1.18600	−	18.0725i	0.0556005	−	0.847252i
$456$		0			0
$457$			8.90110i			0.416376i	0.978089	+	0.208188i	$0.0667566\pi$
$457$			8.90110i			0.416376i	−0.978089	+	0.208188i	$0.933243\pi$
$458$		0			0
$459$	−6.46254			−0.301645
$460$		0			0
$461$	−13.9124			−0.647965			−0.323983	−	0.946063i	$-0.605022\pi$
$461$	−13.9124			−0.647965			−0.323983	+	0.946063i	$0.605022\pi$
$462$		0			0
$463$			37.5085i			1.74317i	0.490247	+	0.871583i	$0.336906\pi$
$463$			37.5085i			1.74317i	−0.490247	+	0.871583i	$0.663094\pi$
$464$		0			0
$465$	−2.49002	−	0.163407i	−0.115472	−	0.00757780i
$466$		0			0
$467$			11.2108i			0.518776i	0.965773	+	0.259388i	$0.0835208\pi$
$467$			11.2108i			0.518776i	−0.965773	+	0.259388i	$0.916479\pi$
$468$		0			0
$469$	9.53562			0.440314
$470$		0			0
$471$	−14.9160			−0.687291
$472$		0			0
$473$		−	0.514069i		−	0.0236369i
$474$		0			0
$475$	24.3067	+	3.20403i	1.11527	+	0.147011i
$476$		0			0
$477$		−	0.113782i		−	0.00520972i
$478$		0			0
$479$	5.16559			0.236022			0.118011	−	0.993012i	$-0.462348\pi$
$479$	5.16559			0.236022			0.118011	+	0.993012i	$0.462348\pi$
$480$		0			0
$481$	−56.5277			−2.57744
$482$		0			0
$483$			15.8442i			0.720936i
$484$		0			0
$485$	11.9174	+	0.782074i	0.541142	+	0.0355122i
$486$		0			0
$487$		−	34.0560i		−	1.54322i	−0.636094	−	0.771612i	$-0.719451\pi$
$487$		−	34.0560i		−	1.54322i	0.636094	−	0.771612i	$-0.280549\pi$
$488$		0			0
$489$	−18.8619			−0.852963
$490$		0			0
$491$	18.3051			0.826099			0.413050	−	0.910708i	$-0.364464\pi$
$491$	18.3051			0.826099			0.413050	+	0.910708i	$0.364464\pi$
$492$		0			0
$493$			5.91424i			0.266364i
$494$		0			0
$495$	0.0568910	−	0.866917i	0.00255706	−	0.0389650i
$496$		0			0
$497$			5.45534i			0.244705i
$498$		0			0
$499$	40.7792			1.82553			0.912764	−	0.408488i	$-0.133944\pi$
$499$	40.7792			1.82553			0.912764	+	0.408488i	$0.133944\pi$
$500$		0			0
$501$	−2.57678			−0.115122
$502$		0			0
$503$			10.7274i			0.478313i	0.970981	+	0.239156i	$0.0768709\pi$
$503$			10.7274i			0.478313i	−0.970981	+	0.239156i	$0.923129\pi$
$504$		0			0
$505$	1.67372	−	25.5045i	0.0744795	−	1.13494i
$506$		0			0
$507$		−	56.0354i		−	2.48862i
$508$		0			0
$509$	21.5278			0.954205			0.477102	−	0.878848i	$-0.341687\pi$
$509$	21.5278			0.954205			0.477102	+	0.878848i	$0.341687\pi$
$510$		0			0
$511$	−19.2284			−0.850613
$512$		0			0
$513$			31.6884i			1.39908i
$514$		0			0
$515$	18.3244	+	1.20253i	0.807471	+	0.0529899i
$516$		0			0
$517$		−	0.421960i		−	0.0185578i
$518$		0			0
$519$	−11.0017			−0.482922
$520$		0			0
$521$	−20.5451			−0.900098			−0.450049	−	0.893004i	$-0.648594\pi$
$521$	−20.5451			−0.900098			−0.450049	+	0.893004i	$0.648594\pi$
$522$		0			0
$523$			20.0508i			0.876762i	0.898789	+	0.438381i	$0.144448\pi$
$523$			20.0508i			0.876762i	−0.898789	+	0.438381i	$0.855552\pi$
$524$		0			0
$525$	−20.2265	−	2.66618i	−0.882755	−	0.116362i
$526$		0			0
$527$			0.388531i			0.0169247i
$528$		0			0
$529$	7.92144			0.344410
$530$		0			0
$531$	10.4999			0.455659
$532$		0			0
$533$		−	37.6912i		−	1.63259i
$534$		0			0
$535$	−14.1659	−	0.929632i	−0.612447	−	0.0401915i
$536$		0			0
$537$			27.9890i			1.20782i
$538$		0			0
$539$	−0.368697			−0.0158809
$540$		0			0
$541$	−8.55768			−0.367924			−0.183962	−	0.982933i	$-0.558892\pi$
$541$	−8.55768			−0.367924			−0.183962	+	0.982933i	$0.558892\pi$
$542$		0			0
$543$			64.5585i			2.77047i
$544$		0			0
$545$	−0.0428811	+	0.653431i	−0.00183682	+	0.0279899i
$546$		0			0
$547$		−	11.2911i		−	0.482771i	−0.970429	−	0.241385i	$-0.922398\pi$
$547$		−	11.2911i		−	0.482771i	0.970429	−	0.241385i	$-0.0776018\pi$
$548$		0			0
$549$	18.2351			0.778256
$550$		0			0
$551$	28.9999			1.23544
$552$		0			0
$553$			1.50580i			0.0640333i
$554$		0			0
$555$	−4.16969	+	63.5386i	−0.176993	+	2.69706i
$556$		0			0
$557$		−	22.4662i		−	0.951922i	−0.879466	−	0.475961i	$-0.842100\pi$
$557$		−	22.4662i		−	0.951922i	0.879466	−	0.475961i	$-0.157900\pi$
$558$		0			0
$559$	−39.6054			−1.67513
$560$		0			0
$561$	−0.212567			−0.00897457
$562$		0			0
$563$			6.32795i			0.266691i	0.991070	+	0.133346i	$0.0425721\pi$
$563$			6.32795i			0.266691i	−0.991070	+	0.133346i	$0.957428\pi$
$564$		0			0
$565$	−19.2405	−	1.26265i	−0.809453	−	0.0531199i
$566$		0			0
$567$		−	3.99506i		−	0.167777i
$568$		0			0
$569$	16.6885			0.699620			0.349810	−	0.936821i	$-0.386246\pi$
$569$	16.6885			0.699620			0.349810	+	0.936821i	$0.386246\pi$
$570$		0			0
$571$	6.20629			0.259725			0.129863	−	0.991532i	$-0.458546\pi$
$571$	6.20629			0.259725			0.129863	+	0.991532i	$0.458546\pi$
$572$		0			0
$573$			15.7302i			0.657137i
$574$		0			0
$575$	2.53735	−	19.2491i	0.105815	−	0.802741i
$576$		0			0
$577$			23.9273i			0.996105i	0.867147	+	0.498052i	$0.165951\pi$
$577$			23.9273i			0.996105i	−0.867147	+	0.498052i	$0.834049\pi$
$578$		0			0
$579$	6.16057			0.256025
$580$		0			0
$581$	−16.8334			−0.698366
$582$		0			0
$583$		−	0.00160393i		−	6.64279e-5i
$584$		0			0
$585$	−66.7899	−	4.38305i	−2.76142	−	0.181217i
$586$		0			0
$587$			26.1140i			1.07784i	0.842357	+	0.538920i	$0.181168\pi$
$587$			26.1140i			1.07784i	−0.842357	+	0.538920i	$0.818832\pi$
$588$		0			0
$589$	1.90512			0.0784992
$590$		0			0
$591$	−28.1056			−1.15611
$592$		0			0
$593$		−	29.8969i		−	1.22772i	−0.789415	−	0.613860i	$-0.789616\pi$
$593$		−	29.8969i		−	1.22772i	0.789415	−	0.613860i	$-0.210384\pi$
$594$		0			0
$595$	−0.208009	+	3.16969i	−0.00852754	+	0.129944i
$596$		0			0
$597$		−	4.22963i		−	0.173107i
$598$		0			0
$599$	15.8194			0.646362			0.323181	−	0.946337i	$-0.395248\pi$
$599$	15.8194			0.646362			0.323181	+	0.946337i	$0.395248\pi$
$600$		0			0
$601$	32.4048			1.32182			0.660910	−	0.750466i	$-0.270171\pi$
$601$	32.4048			1.32182			0.660910	+	0.750466i	$0.270171\pi$
$602$		0			0
$603$		−	35.2405i		−	1.43510i
$604$		0			0
$605$	−1.60988	+	24.5317i	−0.0654510	+	0.997357i
$606$		0			0
$607$			20.4218i			0.828895i	0.910073	+	0.414447i	$0.136025\pi$
$607$			20.4218i			0.828895i	−0.910073	+	0.414447i	$0.863975\pi$
$608$		0			0
$609$	−24.1318			−0.977869
$610$		0			0
$611$	−32.5091			−1.31518
$612$		0			0
$613$		−	12.0930i		−	0.488433i	−0.969721	−	0.244217i	$-0.921469\pi$
$613$		−	12.0930i		−	0.488433i	0.969721	−	0.244217i	$-0.0785308\pi$
$614$		0			0
$615$	−42.3658	−	2.78024i	−1.70835	−	0.112110i
$616$		0			0
$617$		−	3.79742i		−	0.152878i	−0.997074	−	0.0764392i	$-0.975645\pi$
$617$		−	3.79742i		−	0.152878i	0.997074	−	0.0764392i	$-0.0243551\pi$
$618$		0			0
$619$	−0.765306			−0.0307602			−0.0153801	−	0.999882i	$-0.504896\pi$
$619$	−0.765306			−0.0307602			−0.0153801	+	0.999882i	$0.504896\pi$
$620$		0			0
$621$	25.0948			1.00702
$622$		0			0
$623$		−	2.83337i		−	0.113517i
$624$		0			0
$625$	24.1461	+	6.47827i	0.965842	+	0.259131i
$626$		0			0
$627$			1.04230i			0.0416254i
$628$		0			0
$629$	9.91424			0.395307
$630$		0			0
$631$	−41.7318			−1.66132			−0.830658	−	0.556784i	$-0.812035\pi$
$631$	−41.7318			−1.66132			−0.830658	+	0.556784i	$0.812035\pi$
$632$		0			0
$633$			14.0050i			0.556648i
$634$		0			0
$635$	−38.5206	−	2.52789i	−1.52864	−	0.100316i
$636$		0			0
$637$			28.4055i			1.12547i
$638$		0			0
$639$	20.1611			0.797561
$640$		0			0
$641$	22.6429			0.894342			0.447171	−	0.894448i	$-0.352431\pi$
$641$	22.6429			0.894342			0.447171	+	0.894448i	$0.352431\pi$
$642$		0			0
$643$		−	24.8814i		−	0.981226i	−0.871377	−	0.490613i	$-0.836773\pi$
$643$		−	24.8814i		−	0.981226i	0.871377	−	0.490613i	$-0.163227\pi$
$644$		0			0
$645$	−2.92144	+	44.5175i	−0.115031	+	1.75287i
$646$		0			0
$647$			35.6376i			1.40106i	0.713624	+	0.700529i	$0.247053\pi$
$647$			35.6376i			1.40106i	−0.713624	+	0.700529i	$0.752947\pi$
$648$		0			0
$649$	0.148013			0.00581000
$650$		0			0
$651$	−1.58532			−0.0621335
$652$		0			0
$653$			8.31930i			0.325559i	0.986662	+	0.162780i	$0.0520460\pi$
$653$			8.31930i			0.325559i	−0.986662	+	0.162780i	$0.947954\pi$
$654$		0			0
$655$	2.15521	−	32.8416i	0.0842111	−	1.28323i
$656$		0			0
$657$			71.0616i			2.77238i
$658$		0			0
$659$	−3.44669			−0.134264			−0.0671320	−	0.997744i	$-0.521385\pi$
$659$	−3.44669			−0.134264			−0.0671320	+	0.997744i	$0.521385\pi$
$660$		0			0
$661$	−36.6867			−1.42695			−0.713473	−	0.700682i	$-0.752879\pi$
$661$	−36.6867			−1.42695			−0.713473	+	0.700682i	$0.752879\pi$
$662$		0			0
$663$			16.3768i			0.636021i
$664$		0			0
$665$	15.5422	+	1.01995i	0.602702	+	0.0395520i
$666$		0			0
$667$		−	22.9657i		−	0.889234i
$668$		0			0
$669$	−57.5995			−2.22692
$670$		0			0
$671$	0.257052			0.00992336
$672$		0			0
$673$		−	41.5075i		−	1.60000i	−0.600002	−	0.799998i	$-0.704834\pi$
$673$		−	41.5075i		−	1.60000i	0.600002	−	0.799998i	$-0.295166\pi$
$674$		0			0
$675$	−4.22282	+	32.0356i	−0.162536	+	1.23305i
$676$		0			0
$677$			26.7378i			1.02762i	0.857905	+	0.513809i	$0.171766\pi$
$677$			26.7378i			1.02762i	−0.857905	+	0.513809i	$0.828234\pi$
$678$		0			0
$679$	7.58742			0.291179
$680$		0			0
$681$	−40.5875			−1.55532
$682$		0			0
$683$			37.8270i			1.44741i	0.690110	+	0.723704i	$0.257562\pi$
$683$			37.8270i			1.44741i	−0.690110	+	0.723704i	$0.742438\pi$
$684$		0			0
$685$	−3.03966	−	0.199476i	−0.116139	−	0.00762159i
$686$		0			0
$687$		−	32.2144i		−	1.22905i
$688$		0			0
$689$	−0.123572			−0.00470770
$690$		0			0
$691$	−33.0739			−1.25819			−0.629095	−	0.777328i	$-0.716574\pi$
$691$	−33.0739			−1.25819			−0.629095	+	0.777328i	$0.716574\pi$
$692$		0			0
$693$		−	0.551937i		−	0.0209664i
$694$		0			0
$695$	−2.01268	+	30.6696i	−0.0763451	+	1.16336i
$696$		0			0
$697$			6.61055i			0.250392i
$698$		0			0
$699$	−39.7192			−1.50232
$700$		0			0
$701$	−24.7292			−0.934008			−0.467004	−	0.884255i	$-0.654667\pi$
$701$	−24.7292			−0.934008			−0.467004	+	0.884255i	$0.654667\pi$
$702$		0			0
$703$		−	48.6135i		−	1.83349i
$704$		0			0
$705$	−2.39798	+	36.5410i	−0.0903133	+	1.37621i
$706$		0			0
$707$		−	16.2379i		−	0.610688i
$708$		0			0
$709$	−40.5902			−1.52440			−0.762199	−	0.647343i	$-0.775880\pi$
$709$	−40.5902			−1.52440			−0.762199	+	0.647343i	$0.775880\pi$
$710$		0			0
$711$	5.56495			0.208702
$712$		0			0
$713$		−	1.50871i		−	0.0565016i
$714$		0			0
$715$	−0.941505	−	0.0617858i	−0.0352103	−	0.00231066i
$716$		0			0
$717$		−	46.7752i		−	1.74685i
$718$		0			0
$719$	32.1253			1.19807			0.599036	−	0.800722i	$-0.295551\pi$
$719$	32.1253			1.19807			0.599036	+	0.800722i	$0.295551\pi$
$720$		0			0
$721$	11.6666			0.434486
$722$		0			0
$723$			14.1914i			0.527782i
$724$		0			0
$725$	29.3176	+	3.86455i	1.08883	+	0.143526i
$726$		0			0
$727$			11.3089i			0.419425i	0.977763	+	0.209713i	$0.0672528\pi$
$727$			11.3089i			0.419425i	−0.977763	+	0.209713i	$0.932747\pi$
$728$		0			0
$729$	40.9222			1.51564
$730$		0			0
$731$	6.94628			0.256918
$732$		0			0
$733$			32.5450i			1.20208i	0.799220	+	0.601039i	$0.205246\pi$
$733$			32.5450i			1.20208i	−0.799220	+	0.601039i	$0.794754\pi$
$734$		0			0
$735$	31.9285	+	2.09529i	1.17770	+	0.0772860i
$736$		0			0
$737$		−	0.496768i		−	0.0182987i
$738$		0			0
$739$	−29.6821			−1.09187			−0.545936	−	0.837827i	$-0.683826\pi$
$739$	−29.6821			−1.09187			−0.545936	+	0.837827i	$0.683826\pi$
$740$		0			0
$741$	80.3019			2.94996
$742$		0			0
$743$		−	27.9705i		−	1.02614i	−0.858348	−	0.513069i	$-0.828509\pi$
$743$		−	27.9705i		−	1.02614i	0.858348	−	0.513069i	$-0.171491\pi$
$744$		0			0
$745$	2.54236	−	38.7410i	0.0931448	−	1.41936i
$746$		0			0
$747$			62.2105i			2.27616i
$748$		0			0
$749$	−9.01898			−0.329546
$750$		0			0
$751$	52.7167			1.92366			0.961829	−	0.273650i	$-0.0882309\pi$
$751$	52.7167			1.92366			0.961829	+	0.273650i	$0.0882309\pi$
$752$		0			0
$753$		−	37.0982i		−	1.35193i
$754$		0			0
$755$	0.526626	−	8.02485i	0.0191659	−	0.292054i
$756$		0			0
$757$		−	7.80911i		−	0.283827i	−0.989879	−	0.141913i	$-0.954675\pi$
$757$		−	7.80911i		−	0.283827i	0.989879	−	0.141913i	$-0.0453255\pi$
$758$		0			0
$759$	0.825420			0.0299608
$760$		0			0
$761$	19.2266			0.696963			0.348481	−	0.937316i	$-0.386697\pi$
$761$	19.2266			0.696963			0.348481	+	0.937316i	$0.386697\pi$
$762$		0			0
$763$			0.416018i			0.0150609i
$764$		0			0
$765$	11.7141	+	0.768731i	0.423524	+	0.0277935i
$766$		0			0
$767$		−	11.4033i		−	0.411751i
$768$		0			0
$769$	1.45851			0.0525953			0.0262977	−	0.999654i	$-0.491628\pi$
$769$	1.45851			0.0525953			0.0262977	+	0.999654i	$0.491628\pi$
$770$		0			0
$771$	−39.1087			−1.40846
$772$		0			0
$773$			18.8055i			0.676388i	0.941076	+	0.338194i	$0.109816\pi$
$773$			18.8055i			0.676388i	−0.941076	+	0.338194i	$0.890184\pi$
$774$		0			0
$775$	1.92599	+	0.253878i	0.0691837	+	0.00911957i
$776$		0			0
$777$			40.4529i			1.45124i
$778$		0			0
$779$	32.4142			1.16136
$780$		0			0
$781$	0.284201			0.0101695
$782$		0			0
$783$			38.2210i			1.36591i
$784$		0			0
$785$	11.5872	+	0.760401i	0.413563	+	0.0271399i
$786$		0			0
$787$		−	19.0449i		−	0.678877i	−0.940628	−	0.339439i	$-0.889763\pi$
$787$		−	19.0449i		−	0.678877i	0.940628	−	0.339439i	$-0.110237\pi$
$788$		0			0
$789$	−36.3973			−1.29578
$790$		0			0
$791$	−12.2498			−0.435552
$792$		0			0
$793$		−	19.8040i		−	0.703262i
$794$		0			0
$795$	−0.00911508	+	0.138898i	−0.000323278	+	0.00492619i
$796$		0			0
$797$			49.6667i			1.75929i	0.475634	+	0.879643i	$0.342219\pi$
$797$			49.6667i			1.75929i	−0.475634	+	0.879643i	$0.657781\pi$
$798$		0			0
$799$	5.70167			0.201711
$800$		0			0
$801$	−10.4712			−0.369981
$802$		0			0
$803$			1.00172i			0.0353500i
$804$		0			0
$805$	0.807722	−	12.3082i	0.0284685	−	0.433809i
$806$		0			0
$807$			74.3824i			2.61838i
$808$		0			0
$809$	−38.2625			−1.34524			−0.672619	−	0.739989i	$-0.734831\pi$
$809$	−38.2625			−1.34524			−0.672619	+	0.739989i	$0.734831\pi$
$810$		0			0
$811$	31.4356			1.10385			0.551927	−	0.833893i	$-0.313893\pi$
$811$	31.4356			1.10385			0.551927	+	0.833893i	$0.313893\pi$
$812$		0			0
$813$		−	60.9529i		−	2.13771i
$814$		0			0
$815$	14.6524	+	0.961559i	0.513253	+	0.0336819i
$816$		0			0
$817$		−	34.0604i		−	1.19162i
$818$		0			0
$819$	−42.5229			−1.48587
$820$		0			0
$821$	42.9887			1.50031			0.750157	−	0.661259i	$-0.229978\pi$
$821$	42.9887			1.50031			0.750157	+	0.661259i	$0.229978\pi$
$822$		0			0
$823$			12.7450i			0.444263i	0.975017	+	0.222132i	$0.0713014\pi$
$823$			12.7450i			0.444263i	−0.975017	+	0.222132i	$0.928699\pi$
$824$		0			0
$825$	−0.138898	+	1.05372i	−0.00483579	+	0.0366857i
$826$		0			0
$827$			12.8443i			0.446639i	0.974745	+	0.223319i	$0.0716893\pi$
$827$			12.8443i			0.446639i	−0.974745	+	0.223319i	$0.928311\pi$
$828$		0			0
$829$	17.2491			0.599087			0.299543	−	0.954083i	$-0.403166\pi$
$829$	17.2491			0.599087			0.299543	+	0.954083i	$0.403166\pi$
$830$		0			0
$831$	5.91424			0.205163
$832$		0			0
$833$		−	4.98197i		−	0.172615i
$834$		0			0
$835$	2.00172	+	0.131362i	0.0692724	+	0.00454597i
$836$		0			0
$837$			2.51090i			0.0867892i
$838$		0			0
$839$	−17.7454			−0.612638			−0.306319	−	0.951929i	$-0.599097\pi$
$839$	−17.7454			−0.612638			−0.306319	+	0.951929i	$0.599097\pi$
$840$		0			0
$841$	5.97821			0.206145
$842$		0			0
$843$		−	62.4249i		−	2.15003i
$844$		0			0
$845$	−2.85663	+	43.5299i	−0.0982711	+	1.49748i
$846$		0			0
$847$			15.6185i			0.536660i
$848$		0			0
$849$	−15.2870			−0.524648
$850$		0			0
$851$	−38.4981			−1.31970
$852$		0			0
$853$			45.1570i			1.54615i	0.634317	+	0.773073i	$0.281282\pi$
$853$			45.1570i			1.54615i	−0.634317	+	0.773073i	$0.718718\pi$
$854$		0			0
$855$	3.76940	−	57.4389i	0.128911	−	1.96437i
$856$		0			0
$857$		−	40.5102i		−	1.38380i	−0.721993	−	0.691900i	$-0.756774\pi$
$857$		−	40.5102i		−	1.38380i	0.721993	−	0.691900i	$-0.243226\pi$
$858$		0			0
$859$	−25.4439			−0.868135			−0.434068	−	0.900880i	$-0.642922\pi$
$859$	−25.4439			−0.868135			−0.434068	+	0.900880i	$0.642922\pi$
$860$		0			0
$861$	−26.9729			−0.919235
$862$		0			0
$863$			21.6953i			0.738517i	0.929327	+	0.369259i	$0.120388\pi$
$863$			21.6953i			0.738517i	−0.929327	+	0.369259i	$0.879612\pi$
$864$		0			0
$865$	8.54646	+	0.560857i	0.290588	+	0.0190697i
$866$		0			0
$867$		−	2.87228i		−	0.0975476i
$868$		0			0
$869$	0.0784464			0.00266111
$870$		0			0
$871$	−38.2725			−1.29681
$872$		0			0
$873$		−	28.0406i		−	0.949030i
$874$		0			0
$875$	15.5766	+	3.10230i	0.526585	+	0.104877i
$876$		0			0
$877$			15.4643i			0.522191i	0.965313	+	0.261095i	$0.0840837\pi$
$877$			15.4643i			0.522191i	−0.965313	+	0.261095i	$0.915916\pi$
$878$		0			0
$879$	−67.0060			−2.26006
$880$		0			0
$881$	−13.2839			−0.447544			−0.223772	−	0.974641i	$-0.571837\pi$
$881$	−13.2839			−0.447544			−0.223772	+	0.974641i	$0.571837\pi$
$882$		0			0
$883$		−	30.1936i		−	1.01609i	−0.861329	−	0.508047i	$-0.830368\pi$
$883$		−	30.1936i		−	1.01609i	0.861329	−	0.508047i	$-0.169632\pi$
$884$		0			0
$885$	−12.8176	−	0.841151i	−0.430860	−	0.0282750i
$886$		0			0
$887$		−	39.7258i		−	1.33386i	−0.745120	−	0.666930i	$-0.767608\pi$
$887$		−	39.7258i		−	1.33386i	0.745120	−	0.666930i	$-0.232392\pi$
$888$		0			0
$889$	−24.5248			−0.822535
$890$		0			0
$891$	−0.208126			−0.00697250
$892$		0			0
$893$		−	27.9576i		−	0.935565i
$894$		0			0
$895$	1.42685	−	21.7427i	0.0476945	−	0.726779i
$896$		0			0
$897$		−	63.5929i		−	2.12330i
$898$		0			0
$899$	2.29786			0.0766381
$900$		0			0
$901$	0.0216729			0.000722028
$902$		0			0
$903$			28.3428i			0.943190i
$904$		0			0
$905$	3.29113	−	50.1510i	0.109401	−	1.66707i
$906$		0			0
$907$			20.4051i			0.677541i	0.940869	+	0.338771i	$0.110011\pi$
$907$			20.4051i			0.677541i	−0.940869	+	0.338771i	$0.889989\pi$
$908$		0			0
$909$	−60.0098			−1.99040
$910$		0			0
$911$	44.1137			1.46155			0.730776	−	0.682618i	$-0.239159\pi$
$911$	44.1137			1.46155			0.730776	+	0.682618i	$0.239159\pi$
$912$		0			0
$913$			0.876951i			0.0290228i
$914$		0			0
$915$	−22.2602	−	1.46082i	−0.735900	−	0.0482931i
$916$		0			0
$917$		−	20.9091i		−	0.690481i
$918$		0			0
$919$	38.1261			1.25766			0.628832	−	0.777541i	$-0.283533\pi$
$919$	38.1261			1.25766			0.628832	+	0.777541i	$0.283533\pi$
$920$		0			0
$921$	−50.1464			−1.65238
$922$		0			0
$923$		−	21.8957i		−	0.720707i
$924$		0			0
$925$	6.47827	−	49.1461i	0.213004	−	1.61591i
$926$		0			0
$927$		−	43.1157i		−	1.41611i
$928$		0			0
$929$	−11.6168			−0.381134			−0.190567	−	0.981674i	$-0.561033\pi$
$929$	−11.6168			−0.381134			−0.190567	+	0.981674i	$0.561033\pi$
$930$		0			0
$931$	−24.4286			−0.800614
$932$		0			0
$933$		−	8.58801i		−	0.281159i
$934$		0			0
$935$	0.165128	+	0.0108364i	0.00540026	+	0.000354389i
$936$		0			0
$937$		−	39.8889i		−	1.30311i	−0.758600	−	0.651557i	$-0.774116\pi$
$937$		−	39.8889i		−	1.30311i	0.758600	−	0.651557i	$-0.225884\pi$
$938$		0			0
$939$	−22.2897			−0.727396
$940$		0			0
$941$	−21.8113			−0.711028			−0.355514	−	0.934671i	$-0.615694\pi$
$941$	−21.8113			−0.711028			−0.355514	+	0.934671i	$0.615694\pi$
$942$		0			0
$943$		−	25.6695i		−	0.835914i
$944$		0			0
$945$	−1.34427	+	20.4842i	−0.0437290	+	0.666351i
$946$		0			0
$947$		−	18.2678i		−	0.593625i	−0.954936	−	0.296812i	$-0.904076\pi$
$947$		−	18.2678i		−	0.593625i	0.954936	−	0.296812i	$-0.0959236\pi$
$948$		0			0
$949$	77.1757			2.50523
$950$		0			0
$951$	19.7886			0.641688
$952$		0			0
$953$			27.5942i			0.893865i	0.894568	+	0.446932i	$0.147484\pi$
$953$			27.5942i			0.893865i	−0.894568	+	0.446932i	$0.852516\pi$
$954$		0			0
$955$	0.801908	−	12.2196i	0.0259491	−	0.395419i
$956$		0			0
$957$			1.25717i			0.0406385i
$958$		0			0
$959$	−1.93525			−0.0624925
$960$		0			0
$961$	−30.8490			−0.995130
$962$		0			0
$963$			33.3311i			1.07408i
$964$		0			0
$965$	−4.78571	−	0.314060i	−0.154058	−	0.0101099i
$966$		0			0
$967$		−	36.3289i		−	1.16826i	−0.811661	−	0.584129i	$-0.801436\pi$
$967$		−	36.3289i		−	1.16826i	0.811661	−	0.584129i	$-0.198564\pi$
$968$		0			0
$969$	−14.0839			−0.452441
$970$		0			0
$971$	−45.2409			−1.45185			−0.725925	−	0.687774i	$-0.758588\pi$
$971$	−45.2409			−1.45185			−0.725925	+	0.687774i	$0.758588\pi$
$972$		0			0
$973$			19.5263i			0.625985i
$974$		0			0
$975$	81.1816	+	10.7011i	2.59989	+	0.342709i
$976$		0			0
$977$			10.8393i			0.346780i	0.984853	+	0.173390i	$0.0554722\pi$
$977$			10.8393i			0.346780i	−0.984853	+	0.173390i	$0.944528\pi$
$978$		0			0
$979$	−0.147607			−0.00471755
$980$		0			0
$981$	1.53746			0.0490874
$982$		0			0
$983$		−	3.69018i		−	0.117699i	−0.998267	−	0.0588493i	$-0.981257\pi$
$983$		−	3.69018i		−	0.117699i	0.998267	−	0.0588493i	$-0.0187431\pi$
$984$		0			0
$985$	21.8333	+	1.43280i	0.695665	+	0.0456527i
$986$		0			0
$987$			23.2644i			0.740515i
$988$		0			0
$989$	−26.9732			−0.857698
$990$		0			0
$991$	−36.6378			−1.16384			−0.581919	−	0.813247i	$-0.697698\pi$
$991$	−36.6378			−1.16384			−0.581919	+	0.813247i	$0.697698\pi$
$992$		0			0
$993$		−	45.5442i		−	1.44530i
$994$		0			0
$995$	−0.215623	+	3.28570i	−0.00683569	+	0.104164i
$996$		0			0
$997$			32.6021i			1.03252i	0.856432	+	0.516259i	$0.172676\pi$
$997$			32.6021i			1.03252i	−0.856432	+	0.516259i	$0.827324\pi$
$998$		0			0
$999$	64.0711			2.02712

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1360.2.e.d.1089.8		8
4.3	odd	2		85.2.b.a.69.6	yes	8
5.2	odd	4		6800.2.a.bw.1.4		4
5.3	odd	4		6800.2.a.bt.1.1		4
5.4	even	2	inner	1360.2.e.d.1089.1		8
12.11	even	2		765.2.b.c.154.3		8
20.3	even	4		425.2.a.h.1.3		4
20.7	even	4		425.2.a.g.1.2		4
20.19	odd	2		85.2.b.a.69.3	✓	8
60.23	odd	4		3825.2.a.bh.1.2		4
60.47	odd	4		3825.2.a.bj.1.3		4
60.59	even	2		765.2.b.c.154.6		8
68.67	odd	2		1445.2.b.e.579.6		8
340.67	even	4		7225.2.a.v.1.2		4
340.203	even	4		7225.2.a.w.1.3		4
340.339	odd	2		1445.2.b.e.579.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
85.2.b.a.69.3	✓	8	20.19	odd	2
85.2.b.a.69.6	yes	8	4.3	odd	2
425.2.a.g.1.2		4	20.7	even	4
425.2.a.h.1.3		4	20.3	even	4
765.2.b.c.154.3		8	12.11	even	2
765.2.b.c.154.6		8	60.59	even	2
1360.2.e.d.1089.1		8	5.4	even	2	inner
1360.2.e.d.1089.8		8	1.1	even	1	trivial
1445.2.b.e.579.3		8	340.339	odd	2
1445.2.b.e.579.6		8	68.67	odd	2
3825.2.a.bh.1.2		4	60.23	odd	4
3825.2.a.bj.1.3		4	60.47	odd	4
6800.2.a.bt.1.1		4	5.3	odd	4
6800.2.a.bw.1.4		4	5.2	odd	4
7225.2.a.v.1.2		4	340.67	even	4
7225.2.a.w.1.3		4	340.203	even	4

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:	\( N \)	\(=\)	\( 1360 = 2^{4} \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1360.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.87228i q^{3} +(0.146426 - 2.23127i) q^{5} -1.42058i q^{7} -5.24997 q^{9} -0.0740063 q^{11} +5.70167i q^{13} +(6.40882 + 0.420575i) q^{15} -1.00000i q^{17} -4.90340 q^{19} +4.08029 q^{21} +3.88311i q^{23} +(-4.95712 - 0.653431i) q^{25} -6.46254i q^{27} -5.91424 q^{29} -0.388531 q^{31} -0.212567i q^{33} +(-3.16969 - 0.208009i) q^{35} +9.91424i q^{37} -16.3768 q^{39} -6.61055 q^{41} +6.94628i q^{43} +(-0.768731 + 11.7141i) q^{45} +5.70167i q^{47} +4.98197 q^{49} +2.87228 q^{51} +0.0216729i q^{53} +(-0.0108364 + 0.165128i) q^{55} -14.0839i q^{57} -2.00000 q^{59} -3.47337 q^{61} +7.45798i q^{63} +(12.7220 + 0.834872i) q^{65} +6.71251i q^{67} -11.1534 q^{69} -3.84023 q^{71} -13.5356i q^{73} +(1.87683 - 14.2382i) q^{75} +0.105132i q^{77} -1.06000 q^{79} +2.81228 q^{81} -11.8497i q^{83} +(-2.23127 - 0.146426i) q^{85} -16.9873i q^{87} +1.99452 q^{89} +8.09965 q^{91} -1.11597i q^{93} +(-0.717985 + 10.9408i) q^{95} +5.34109i q^{97} +0.388531 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{5} - 8 q^{9} + 4 q^{11} + 8 q^{19} + 24 q^{21} - 12 q^{25} + 8 q^{29} + 24 q^{31} - 44 q^{39} - 12 q^{41} - 22 q^{45} - 16 q^{49} + 8 q^{51} + 8 q^{55} - 16 q^{59} + 12 q^{61} + 20 q^{65} - 8 q^{69}+ \cdots - 24 q^{99}+O(q^{100}) \) 8 * q - 2 * q^5 - 8 * q^9 + 4 * q^11 + 8 * q^19 + 24 * q^21 - 12 * q^25 + 8 * q^29 + 24 * q^31 - 44 * q^39 - 12 * q^41 - 22 * q^45 - 16 * q^49 + 8 * q^51 + 8 * q^55 - 16 * q^59 + 12 * q^61 + 20 * q^65 - 8 * q^69 + 20 * q^71 - 4 * q^75 - 24 * q^79 - 8 * q^81 - 2 * q^85 - 48 * q^89 - 36 * q^91 - 4 * q^95 - 24 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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