LMFDB - Embedded newform 400.2.y.c.369.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [400,2,Mod(129,400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(400, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([0, 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("400.129");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 400 = 2^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	400.y (of order $10$, degree $4$, 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:	$3.19401608085$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$2$ over $\Q(\zeta_{10})$
Coefficient field:	8.0.58140625.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} + 4x^{6} - 7x^{5} + 11x^{4} + 5x^{3} - 10x^{2} - 25x + 25 $ x^8 - 3x^7 + 4x^6 - 7x^5 + 11x^4 + 5x^3 - 10x^2 - 25*x + 25
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 25)
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			369.2
Root			$-0.983224 + 0.644389i$ of defining polynomial
Character	$\chi$	$=$	400.369
Dual form			400.2.y.c.129.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.865190 + 1.19083i) q^{3} +(0.107666 + 2.23347i) q^{5} +3.26086i q^{7} +(0.257524 - 0.792578i) q^{9} +O(q^{10})$	\(q+(0.865190 + 1.19083i) q^{3} +(0.107666 + 2.23347i) q^{5} +3.26086i q^{7} +(0.257524 - 0.792578i) q^{9} +(-0.618034 - 1.90211i) q^{11} +(0.281873 + 0.0915860i) q^{13} +(-2.56654 + 2.06059i) q^{15} +(-3.03472 + 4.17693i) q^{17} +(-1.39991 - 1.01709i) q^{19} +(-3.88313 + 2.82126i) q^{21} +(-0.836161 + 0.271685i) q^{23} +(-4.97682 + 0.480938i) q^{25} +(5.36635 - 1.74363i) q^{27} +(4.78304 - 3.47508i) q^{29} +(4.93462 + 3.58521i) q^{31} +(1.73038 - 2.38166i) q^{33} +(-7.28304 + 0.351083i) q^{35} +(7.69215 + 2.49933i) q^{37} +(0.134810 + 0.414902i) q^{39} +(-0.313697 + 0.965461i) q^{41} -3.24199i q^{43} +(1.79793 + 0.489840i) q^{45} +(2.48043 + 3.41402i) q^{47} -3.63318 q^{49} -7.59963 q^{51} +(-4.76148 - 6.55362i) q^{53} +(4.18178 - 1.58516i) q^{55} -2.54703i q^{57} +(1.83443 - 5.64581i) q^{59} +(0.282941 + 0.870802i) q^{61} +(2.58448 + 0.839749i) q^{63} +(-0.174207 + 0.639416i) q^{65} +(-4.04870 + 5.57255i) q^{67} +(-1.04697 - 0.760668i) q^{69} +(-4.82884 + 3.50836i) q^{71} +(8.40107 - 2.72967i) q^{73} +(-4.87861 - 5.51045i) q^{75} +(6.20252 - 2.01532i) q^{77} +(6.27851 - 4.56161i) q^{79} +(4.69667 + 3.41233i) q^{81} +(8.53192 - 11.7432i) q^{83} +(-9.65580 - 6.32825i) q^{85} +(8.27647 + 2.68919i) q^{87} +(-2.32579 - 7.15805i) q^{89} +(-0.298649 + 0.919147i) q^{91} +8.97820i q^{93} +(2.12093 - 3.23616i) q^{95} +(3.95373 + 5.44184i) q^{97} -1.66673 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 5 q^{3} + q^{9}+O(q^{10}) $ 8 * q + 5 * q^3 + q^9	$ 8 q + 5 q^{3} + q^{9} + 4 q^{11} - 5 q^{13} - 15 q^{15} - 10 q^{17} + 5 q^{19} - 4 q^{21} - 5 q^{23} - 10 q^{25} + 5 q^{27} - 5 q^{29} + 9 q^{31} + 10 q^{33} - 15 q^{35} + 30 q^{37} + 3 q^{39} - 4 q^{41} - 15 q^{45} + 14 q^{49} + 4 q^{51} - 10 q^{53} + 10 q^{55} - 9 q^{61} - 10 q^{63} + 5 q^{65} - 20 q^{67} + 17 q^{69} - 6 q^{71} + 15 q^{73} + 10 q^{75} + 10 q^{77} - 15 q^{79} + 28 q^{81} + 45 q^{83} - 15 q^{85} + 20 q^{87} - 25 q^{89} - 6 q^{91} - 15 q^{95} - 60 q^{97} + 8 q^{99}+O(q^{100}) $ 8 * q + 5 * q^3 + q^9 + 4 * q^11 - 5 * q^13 - 15 * q^15 - 10 * q^17 + 5 * q^19 - 4 * q^21 - 5 * q^23 - 10 * q^25 + 5 * q^27 - 5 * q^29 + 9 * q^31 + 10 * q^33 - 15 * q^35 + 30 * q^37 + 3 * q^39 - 4 * q^41 - 15 * q^45 + 14 * q^49 + 4 * q^51 - 10 * q^53 + 10 * q^55 - 9 * q^61 - 10 * q^63 + 5 * q^65 - 20 * q^67 + 17 * q^69 - 6 * q^71 + 15 * q^73 + 10 * q^75 + 10 * q^77 - 15 * q^79 + 28 * q^81 + 45 * q^83 - 15 * q^85 + 20 * q^87 - 25 * q^89 - 6 * q^91 - 15 * q^95 - 60 * q^97 + 8 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/400\mathbb{Z}\right)^\times$.

$n$	$101$	$177$	$351$
$\chi(n)$	$1$	$e\left(\frac{9}{10}\right)$	$1$

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
$3$	0.865190	+	1.19083i	0.499518	+	0.687527i	0.982108	−	0.188319i	$-0.0603039\pi$
$3$	0.865190	+	1.19083i	0.499518	+	0.687527i	−0.482590	+	0.875846i	$0.660304\pi$
$4$		0			0
$5$	0.107666	+	2.23347i	0.0481496	+	0.998840i
$5$	0.107666	+	2.23347i	0.0481496	+	0.998840i
$6$		0			0
$7$			3.26086i			1.23249i	0.787555	+	0.616244i	$0.211346\pi$
$7$			3.26086i			1.23249i	−0.787555	+	0.616244i	$0.788654\pi$
$8$		0			0
$9$	0.257524	−	0.792578i	0.0858414	−	0.264193i
$10$		0			0
$11$	−0.618034	−	1.90211i	−0.186344	−	0.573509i	0.813625	−	0.581390i	$-0.197491\pi$
$11$	−0.618034	−	1.90211i	−0.186344	−	0.573509i	−0.999969	+	0.00788181i	$0.997491\pi$
$12$		0			0
$13$	0.281873	+	0.0915860i	0.0781775	+	0.0254014i	0.347845	−	0.937552i	$-0.386914\pi$
$13$	0.281873	+	0.0915860i	0.0781775	+	0.0254014i	−0.269667	+	0.962954i	$0.586914\pi$
$14$		0			0
$15$	−2.56654	+	2.06059i	−0.662678	+	0.532042i
$16$		0			0
$17$	−3.03472	+	4.17693i	−0.736027	+	1.01305i	0.262810	+	0.964847i	$0.415351\pi$
$17$	−3.03472	+	4.17693i	−0.736027	+	1.01305i	−0.998837	+	0.0482067i	$0.984649\pi$
$18$		0			0
$19$	−1.39991	−	1.01709i	−0.321161	−	0.233337i	0.415510	−	0.909589i	$-0.363603\pi$
$19$	−1.39991	−	1.01709i	−0.321161	−	0.233337i	−0.736671	+	0.676252i	$0.763603\pi$
$20$		0			0
$21$	−3.88313	+	2.82126i	−0.847369	+	0.615649i
$22$		0			0
$23$	−0.836161	+	0.271685i	−0.174352	+	0.0566503i	−0.394892	−	0.918728i	$-0.629218\pi$
$23$	−0.836161	+	0.271685i	−0.174352	+	0.0566503i	0.220540	+	0.975378i	$0.429218\pi$
$24$		0			0
$25$	−4.97682	+	0.480938i	−0.995363	+	0.0961876i
$26$		0			0
$27$	5.36635	−	1.74363i	1.03276	−	0.335563i
$28$		0			0
$29$	4.78304	−	3.47508i	0.888188	−	0.645306i	−0.0472171	−	0.998885i	$-0.515035\pi$
$29$	4.78304	−	3.47508i	0.888188	−	0.645306i	0.935405	+	0.353578i	$0.115035\pi$
$30$		0			0
$31$	4.93462	+	3.58521i	0.886285	+	0.643923i	0.934907	−	0.354894i	$-0.115483\pi$
$31$	4.93462	+	3.58521i	0.886285	+	0.643923i	−0.0486220	+	0.998817i	$0.515483\pi$
$32$		0			0
$33$	1.73038	−	2.38166i	0.301220	−	0.414594i
$34$		0			0
$35$	−7.28304	+	0.351083i	−1.23106	+	0.0593438i
$36$		0			0
$37$	7.69215	+	2.49933i	1.26458	+	0.410887i	0.863125	−	0.504991i	$-0.168504\pi$
$37$	7.69215	+	2.49933i	1.26458	+	0.410887i	0.401457	+	0.915878i	$0.368504\pi$
$38$		0			0
$39$	0.134810	+	0.414902i	0.0215869	+	0.0664376i
$40$		0			0
$41$	−0.313697	+	0.965461i	−0.0489913	+	0.150780i	−0.972559	−	0.232655i	$-0.925259\pi$
$41$	−0.313697	+	0.965461i	−0.0489913	+	0.150780i	0.923568	+	0.383434i	$0.125259\pi$
$42$		0			0
$43$		−	3.24199i		−	0.494399i	−0.968965	−	0.247200i	$-0.920490\pi$
$43$		−	3.24199i		−	0.494399i	0.968965	−	0.247200i	$-0.0795103\pi$
$44$		0			0
$45$	1.79793	+	0.489840i	0.268019	+	0.0730210i
$46$		0			0
$47$	2.48043	+	3.41402i	0.361808	+	0.497986i	0.950651	−	0.310261i	$-0.100417\pi$
$47$	2.48043	+	3.41402i	0.361808	+	0.497986i	−0.588844	+	0.808247i	$0.700417\pi$
$48$		0			0
$49$	−3.63318			−0.519026
$50$		0			0
$51$	−7.59963			−1.06416
$52$		0			0
$53$	−4.76148	−	6.55362i	−0.654040	−	0.900209i	0.345226	−	0.938520i	$-0.387802\pi$
$53$	−4.76148	−	6.55362i	−0.654040	−	0.900209i	−0.999266	+	0.0383106i	$0.987802\pi$
$54$		0			0
$55$	4.18178	−	1.58516i	0.563871	−	0.213742i
$56$		0			0
$57$		−	2.54703i		−	0.337363i
$58$		0			0
$59$	1.83443	−	5.64581i	0.238823	−	0.735021i	−0.757768	−	0.652524i	$-0.773710\pi$
$59$	1.83443	−	5.64581i	0.238823	−	0.735021i	0.996591	−	0.0824976i	$-0.0262897\pi$
$60$		0			0
$61$	0.282941	+	0.870802i	0.0362268	+	0.111495i	0.967535	−	0.252738i	$-0.0813312\pi$
$61$	0.282941	+	0.870802i	0.0362268	+	0.111495i	−0.931308	+	0.364233i	$0.881331\pi$
$62$		0			0
$63$	2.58448	+	0.839749i	0.325614	+	0.105798i
$64$		0			0
$65$	−0.174207	+	0.639416i	−0.0216077	+	0.0793099i
$66$		0			0
$67$	−4.04870	+	5.57255i	−0.494627	+	0.680796i	−0.981233	−	0.192826i	$-0.938235\pi$
$67$	−4.04870	+	5.57255i	−0.494627	+	0.680796i	0.486606	+	0.873622i	$0.338235\pi$
$68$		0			0
$69$	−1.04697	−	0.760668i	−0.126040	−	0.0915737i
$70$		0			0
$71$	−4.82884	+	3.50836i	−0.573078	+	0.416366i	−0.836222	−	0.548391i	$-0.815241\pi$
$71$	−4.82884	+	3.50836i	−0.573078	+	0.416366i	0.263144	+	0.964757i	$0.415241\pi$
$72$		0			0
$73$	8.40107	−	2.72967i	0.983271	−	0.319484i	0.227110	−	0.973869i	$-0.427072\pi$
$73$	8.40107	−	2.72967i	0.983271	−	0.319484i	0.756161	+	0.654385i	$0.227072\pi$
$74$		0			0
$75$	−4.87861	−	5.51045i	−0.563333	−	0.636292i
$76$		0			0
$77$	6.20252	−	2.01532i	0.706842	−	0.229667i
$78$		0			0
$79$	6.27851	−	4.56161i	0.706388	−	0.513221i	−0.175618	−	0.984458i	$-0.556192\pi$
$79$	6.27851	−	4.56161i	0.706388	−	0.513221i	0.882006	+	0.471237i	$0.156192\pi$
$80$		0			0
$81$	4.69667	+	3.41233i	0.521852	+	0.379148i
$82$		0			0
$83$	8.53192	−	11.7432i	0.936500	−	1.28898i	−0.0207694	−	0.999784i	$-0.506612\pi$
$83$	8.53192	−	11.7432i	0.936500	−	1.28898i	0.957269	−	0.289197i	$-0.0933884\pi$
$84$		0			0
$85$	−9.65580	−	6.32825i	−1.04732	−	0.686395i
$86$		0			0
$87$	8.27647	+	2.68919i	0.887331	+	0.288311i
$88$		0			0
$89$	−2.32579	−	7.15805i	−0.246534	−	0.758752i	−0.995380	−	0.0960092i	$-0.969392\pi$
$89$	−2.32579	−	7.15805i	−0.246534	−	0.758752i	0.748847	−	0.662743i	$-0.230608\pi$
$90$		0			0
$91$	−0.298649	+	0.919147i	−0.0313069	+	0.0963528i
$92$		0			0
$93$			8.97820i			0.930996i
$94$		0			0
$95$	2.12093	−	3.23616i	0.217602	−	0.332023i
$96$		0			0
$97$	3.95373	+	5.44184i	0.401440	+	0.552535i	0.961105	−	0.276184i	$-0.0890699\pi$
$97$	3.95373	+	5.44184i	0.401440	+	0.552535i	−0.559664	+	0.828719i	$0.689070\pi$
$98$		0			0
$99$	−1.66673			−0.167513
$100$		0			0
$101$	−12.1955			−1.21350			−0.606748	−	0.794894i	$-0.707526\pi$
$101$	−12.1955			−1.21350			−0.606748	+	0.794894i	$0.707526\pi$
$102$		0			0
$103$	−0.811969	−	1.11758i	−0.0800057	−	0.110118i	0.767138	−	0.641482i	$-0.221680\pi$
$103$	−0.811969	−	1.11758i	−0.0800057	−	0.110118i	−0.847144	+	0.531363i	$0.821680\pi$
$104$		0			0
$105$	−6.71929	−	8.36912i	−0.655736	−	0.816742i
$106$		0			0
$107$		−	15.8285i		−	1.53020i	−0.643911	−	0.765101i	$-0.722689\pi$
$107$		−	15.8285i		−	1.53020i	0.643911	−	0.765101i	$-0.277311\pi$
$108$		0			0
$109$	0.619199	−	1.90570i	0.0593085	−	0.182533i	−0.917013	−	0.398857i	$-0.869407\pi$
$109$	0.619199	−	1.90570i	0.0593085	−	0.182533i	0.976322	+	0.216324i	$0.0694069\pi$
$110$		0			0
$111$	3.67889	+	11.3225i	0.349185	+	1.07468i
$112$		0			0
$113$	9.91713	+	3.22227i	0.932925	+	0.303126i	0.735758	−	0.677244i	$-0.236826\pi$
$113$	9.91713	+	3.22227i	0.932925	+	0.303126i	0.197167	+	0.980370i	$0.436826\pi$
$114$		0			0
$115$	−0.696828	−	1.83829i	−0.0649795	−	0.171422i
$116$		0			0
$117$	0.145178	−	0.199821i	0.0134217	−	0.0184734i
$118$		0			0
$119$	−13.6204	−	9.89577i	−1.24858	−	0.907144i
$120$		0			0
$121$	5.66312	−	4.11450i	0.514829	−	0.374045i
$122$		0			0
$123$	−1.42111	+	0.461746i	−0.128137	+	0.0416343i
$124$		0			0
$125$	−1.61000	−	11.0638i	−0.144002	−	0.989577i
$126$		0			0
$127$	−5.56375	+	1.80777i	−0.493703	+	0.160414i	−0.545278	−	0.838255i	$-0.683576\pi$
$127$	−5.56375	+	1.80777i	−0.493703	+	0.160414i	0.0515752	+	0.998669i	$0.483576\pi$
$128$		0			0
$129$	3.86067	−	2.80494i	0.339913	−	0.246961i
$130$		0			0
$131$	−1.21081	−	0.879704i	−0.105789	−	0.0768601i	0.533633	−	0.845716i	$-0.320826\pi$
$131$	−1.21081	−	0.879704i	−0.105789	−	0.0768601i	−0.639422	+	0.768856i	$0.720826\pi$
$132$		0			0
$133$	3.31659	−	4.56489i	0.287585	−	0.395827i
$134$		0			0
$135$	4.47214	+	11.7979i	0.384900	+	1.01540i
$136$		0			0
$137$	−7.46472	−	2.42543i	−0.637754	−	0.207219i	−0.0277472	−	0.999615i	$-0.508833\pi$
$137$	−7.46472	−	2.42543i	−0.637754	−	0.207219i	−0.610007	+	0.792396i	$0.708833\pi$
$138$		0			0
$139$	1.66607	+	5.12764i	0.141314	+	0.434921i	0.996519	−	0.0833702i	$-0.0265684\pi$
$139$	1.66607	+	5.12764i	0.141314	+	0.434921i	−0.855204	+	0.518291i	$0.826568\pi$
$140$		0			0
$141$	−1.91948	+	5.90755i	−0.161649	+	0.497505i
$142$		0			0
$143$		−	0.592757i		−	0.0495689i
$144$		0			0
$145$	8.27647	+	10.3086i	0.687324	+	0.856086i
$146$		0			0
$147$	−3.14339	−	4.32651i	−0.259262	−	0.356844i
$148$		0			0
$149$	−18.8229			−1.54203			−0.771015	−	0.636817i	$-0.780251\pi$
$149$	−18.8229			−1.54203			−0.771015	+	0.636817i	$0.780251\pi$
$150$		0			0
$151$	3.88797			0.316398			0.158199	−	0.987407i	$-0.449431\pi$
$151$	3.88797			0.316398			0.158199	+	0.987407i	$0.449431\pi$
$152$		0			0
$153$	2.52903	+	3.48091i	0.204460	+	0.281415i
$154$		0			0
$155$	−7.47619	+	11.4074i	−0.600502	+	0.916261i
$156$		0			0
$157$			4.28378i			0.341883i	0.985281	+	0.170941i	$0.0546808\pi$
$157$			4.28378i			0.341883i	−0.985281	+	0.170941i	$0.945319\pi$
$158$		0			0
$159$	3.68467	−	11.3403i	0.292214	−	0.899341i
$160$		0			0
$161$	−0.885926	−	2.72660i	−0.0698208	−	0.214886i
$162$		0			0
$163$	14.9566	+	4.85970i	1.17149	+	0.380641i	0.829200	−	0.558952i	$-0.188796\pi$
$163$	14.9566	+	4.85970i	1.17149	+	0.380641i	0.342293	+	0.939593i	$0.388796\pi$
$164$		0			0
$165$	5.50569	+	3.60834i	0.428617	+	0.280909i
$166$		0			0
$167$	−12.3629	+	17.0161i	−0.956670	+	1.31674i	−0.00816967	+	0.999967i	$0.502601\pi$
$167$	−12.3629	+	17.0161i	−0.956670	+	1.31674i	−0.948500	+	0.316777i	$0.897399\pi$
$168$		0			0
$169$	−10.4462	−	7.58958i	−0.803551	−	0.583814i
$170$		0			0
$171$	−1.16663	+	0.847609i	−0.0892148	+	0.0648183i
$172$		0			0
$173$	6.81587	−	2.21461i	0.518201	−	0.168374i	−0.0382277	−	0.999269i	$-0.512171\pi$
$173$	6.81587	−	2.21461i	0.518201	−	0.168374i	0.556429	+	0.830895i	$0.312171\pi$
$174$		0			0
$175$	−1.56827	−	16.2287i	−0.118550	−	1.22677i
$176$		0			0
$177$	8.31034	−	2.70019i	0.624643	−	0.202959i
$178$		0			0
$179$	−6.50396	+	4.72540i	−0.486129	+	0.353193i	−0.803694	−	0.595043i	$-0.797135\pi$
$179$	−6.50396	+	4.72540i	−0.486129	+	0.353193i	0.317565	+	0.948237i	$0.397135\pi$
$180$		0			0
$181$	16.6796	+	12.1184i	1.23978	+	0.900756i	0.997584	−	0.0694707i	$-0.0221310\pi$
$181$	16.6796	+	12.1184i	1.23978	+	0.900756i	0.242200	+	0.970226i	$0.422131\pi$
$182$		0			0
$183$	−0.792181	+	1.09034i	−0.0585597	+	0.0806005i
$184$		0			0
$185$	−4.75401	+	17.4493i	−0.349522	+	1.28290i
$186$		0			0
$187$	9.82055	+	3.19089i	0.718150	+	0.233341i
$188$		0			0
$189$	5.68574	+	17.4989i	0.413577	+	1.27286i
$190$		0			0
$191$	−5.57167	+	17.1478i	−0.403152	+	1.24077i	0.519277	+	0.854606i	$0.326201\pi$
$191$	−5.57167	+	17.1478i	−0.403152	+	1.24077i	−0.922429	+	0.386167i	$0.873799\pi$
$192$		0			0
$193$		−	6.78859i		−	0.488653i	−0.969693	−	0.244327i	$-0.921433\pi$
$193$		−	6.78859i		−	0.488653i	0.969693	−	0.244327i	$-0.0785669\pi$
$194$		0			0
$195$	−0.912160	+	0.345766i	−0.0653211	+	0.0247608i
$196$		0			0
$197$	−4.69956	−	6.46839i	−0.334830	−	0.460854i	0.608092	−	0.793866i	$-0.291935\pi$
$197$	−4.69956	−	6.46839i	−0.334830	−	0.460854i	−0.942923	+	0.333012i	$0.891935\pi$
$198$		0			0
$199$	−5.20485			−0.368962			−0.184481	−	0.982836i	$-0.559060\pi$
$199$	−5.20485			−0.368962			−0.184481	+	0.982836i	$0.559060\pi$
$200$		0			0
$201$	−10.1389			−0.715141
$202$		0			0
$203$	11.3317	+	15.5968i	0.795332	+	1.09468i
$204$		0			0
$205$	−2.19011	−	0.596687i	−0.152964	−	0.0416745i
$206$		0			0
$207$			0.732688i			0.0509254i
$208$		0			0
$209$	−1.06943	+	3.29138i	−0.0739743	+	0.227669i
$210$		0			0
$211$	−5.13029	−	15.7894i	−0.353184	−	1.08699i	−0.957055	−	0.289906i	$-0.906376\pi$
$211$	−5.13029	−	15.7894i	−0.353184	−	1.08699i	0.603872	−	0.797082i	$-0.293624\pi$
$212$		0			0
$213$	−8.35573	−	2.71494i	−0.572525	−	0.186025i
$214$		0			0
$215$	7.24091	−	0.349052i	0.493826	−	0.0238051i
$216$		0			0
$217$	−11.6909	+	16.0911i	−0.793628	+	1.09233i
$218$		0			0
$219$	10.5191	+	7.64258i	0.710815	+	0.516438i
$220$		0			0
$221$	−1.23795	+	0.899425i	−0.0832737	+	0.0605019i
$222$		0			0
$223$	−6.30368	+	2.04819i	−0.422126	+	0.137157i	−0.512374	−	0.858762i	$-0.671234\pi$
$223$	−6.30368	+	2.04819i	−0.422126	+	0.137157i	0.0902485	+	0.995919i	$0.471234\pi$
$224$		0			0
$225$	−0.900470	+	4.06837i	−0.0600313	+	0.271224i
$226$		0			0
$227$	12.7365	−	4.13833i	0.845350	−	0.274671i	0.145853	−	0.989306i	$-0.453407\pi$
$227$	12.7365	−	4.13833i	0.845350	−	0.274671i	0.699497	+	0.714636i	$0.253407\pi$
$228$		0			0
$229$	8.16032	−	5.92882i	0.539249	−	0.391788i	−0.284557	−	0.958659i	$-0.591846\pi$
$229$	8.16032	−	5.92882i	0.539249	−	0.391788i	0.823806	+	0.566872i	$0.191846\pi$
$230$		0			0
$231$	7.76626	+	5.64252i	0.510983	+	0.371251i
$232$		0			0
$233$	−12.9345	+	17.8028i	−0.847368	+	1.16630i	0.137069	+	0.990562i	$0.456232\pi$
$233$	−12.9345	+	17.8028i	−0.847368	+	1.16630i	−0.984437	+	0.175740i	$0.943768\pi$
$234$		0			0
$235$	−7.35806	+	5.90755i	−0.479987	+	0.385366i
$236$		0			0
$237$	10.8642	+	3.53000i	0.705707	+	0.229298i
$238$		0			0
$239$	2.33626	+	7.19026i	0.151120	+	0.465099i	0.997747	−	0.0670870i	$-0.0213705\pi$
$239$	2.33626	+	7.19026i	0.151120	+	0.465099i	−0.846627	+	0.532187i	$0.821371\pi$
$240$		0			0
$241$	−6.30226	+	19.3964i	−0.405964	+	1.24943i	0.514123	+	0.857716i	$0.328117\pi$
$241$	−6.30226	+	19.3964i	−0.405964	+	1.24943i	−0.920087	+	0.391713i	$0.871883\pi$
$242$		0			0
$243$		−	8.38230i		−	0.537725i
$244$		0			0
$245$	−0.391169	−	8.11461i	−0.0249909	−	0.518424i
$246$		0			0
$247$	−0.301444	−	0.414902i	−0.0191804	−	0.0263996i
$248$		0			0
$249$	21.3659			1.35401
$250$		0			0
$251$	−10.5717			−0.667278			−0.333639	−	0.942701i	$-0.608277\pi$
$251$	−10.5717			−0.667278			−0.333639	+	0.942701i	$0.608277\pi$
$252$		0			0
$253$	1.03355	+	1.42256i	0.0649789	+	0.0894357i
$254$		0			0
$255$	−0.818220	−	16.9736i	−0.0512389	−	1.06293i
$256$		0			0
$257$		−	20.2700i		−	1.26441i	−0.774801	−	0.632205i	$-0.782150\pi$
$257$		−	20.2700i		−	1.26441i	0.774801	−	0.632205i	$-0.217850\pi$
$258$		0			0
$259$	−8.14996	+	25.0830i	−0.506414	+	1.55858i
$260$		0			0
$261$	−1.52252	−	4.68585i	−0.0942419	−	0.290047i
$262$		0			0
$263$	−26.7160	−	8.68056i	−1.64738	−	0.535267i	−0.669211	−	0.743072i	$-0.733368\pi$
$263$	−26.7160	−	8.68056i	−1.64738	−	0.535267i	−0.978170	+	0.207806i	$0.933368\pi$
$264$		0			0
$265$	14.1247	−	11.3403i	0.867673	−	0.696626i
$266$		0			0
$267$	6.51179	−	8.96271i	0.398515	−	0.548509i
$268$		0			0
$269$	16.4416	+	11.9455i	1.00246	+	0.728333i	0.962615	−	0.270873i	$-0.0873123\pi$
$269$	16.4416	+	11.9455i	1.00246	+	0.728333i	0.0398490	+	0.999206i	$0.487312\pi$
$270$		0			0
$271$	25.4409	−	18.4839i	1.54543	−	1.12282i	0.598610	−	0.801041i	$-0.295720\pi$
$271$	25.4409	−	18.4839i	1.54543	−	1.12282i	0.946816	−	0.321777i	$-0.104280\pi$
$272$		0			0
$273$	−1.35294	+	0.439596i	−0.0818835	+	0.0266056i
$274$		0			0
$275$	3.99064	+	9.16923i	0.240645	+	0.552925i
$276$		0			0
$277$	−13.2487	+	4.30475i	−0.796035	+	0.258648i	−0.678672	−	0.734441i	$-0.737444\pi$
$277$	−13.2487	+	4.30475i	−0.796035	+	0.258648i	−0.117363	+	0.993089i	$0.537444\pi$
$278$		0			0
$279$	4.11235	−	2.98779i	0.246200	−	0.178875i
$280$		0			0
$281$	−20.9355	−	15.2105i	−1.24891	−	0.907383i	−0.250748	−	0.968052i	$-0.580677\pi$
$281$	−20.9355	−	15.2105i	−1.24891	−	0.907383i	−0.998158	+	0.0606690i	$0.980677\pi$
$282$		0			0
$283$	−13.9491	+	19.1993i	−0.829188	+	1.14128i	0.158885	+	0.987297i	$0.449210\pi$
$283$	−13.9491	+	19.1993i	−0.829188	+	1.14128i	−0.988074	+	0.153983i	$0.950790\pi$
$284$		0			0
$285$	5.68873	−	0.274228i	0.336971	−	0.0162439i
$286$		0			0
$287$	−3.14823	−	1.02292i	−0.185834	−	0.0603811i
$288$		0			0
$289$	−2.98394	−	9.18363i	−0.175526	−	0.540214i
$290$		0			0
$291$	−3.05959	+	9.41645i	−0.179356	+	0.552002i
$292$		0			0
$293$		−	12.3029i		−	0.718742i	−0.933195	−	0.359371i	$-0.882991\pi$
$293$		−	12.3029i		−	0.718742i	0.933195	−	0.359371i	$-0.117009\pi$
$294$		0			0
$295$	12.8073	+	3.48930i	0.745668	+	0.203155i
$296$		0			0
$297$	−6.63318	−	9.12979i	−0.384896	−	0.529764i
$298$		0			0
$299$	−0.260574			−0.0150694
$300$		0			0
$301$	10.5717			0.609341
$302$		0			0
$303$	−10.5514	−	14.5228i	−0.606163	−	0.834311i
$304$		0			0
$305$	−1.91445	+	0.725696i	−0.109621	+	0.0415532i
$306$		0			0
$307$			4.28249i			0.244415i	0.992505	+	0.122207i	$0.0389973\pi$
$307$			4.28249i			0.244415i	−0.992505	+	0.122207i	$0.961003\pi$
$308$		0			0
$309$	0.628342	−	1.93384i	0.0357451	−	0.110012i
$310$		0			0
$311$	−7.92526	−	24.3915i	−0.449400	−	1.38311i	−0.877585	−	0.479421i	$-0.840847\pi$
$311$	−7.92526	−	24.3915i	−0.449400	−	1.38311i	0.428185	−	0.903691i	$-0.359153\pi$
$312$		0			0
$313$	−21.2573	−	6.90692i	−1.20153	−	0.390402i	−0.361209	−	0.932485i	$-0.617636\pi$
$313$	−21.2573	−	6.90692i	−1.20153	−	0.390402i	−0.840325	+	0.542083i	$0.817636\pi$
$314$		0			0
$315$	−1.59730	+	5.86279i	−0.0899975	+	0.330331i
$316$		0			0
$317$	−12.8859	+	17.7360i	−0.723746	+	0.996151i	0.275645	+	0.961259i	$0.411108\pi$
$317$	−12.8859	+	17.7360i	−0.723746	+	0.996151i	−0.999391	+	0.0348911i	$0.988892\pi$
$318$		0			0
$319$	−9.56608	−	6.95016i	−0.535597	−	0.389134i
$320$		0			0
$321$	18.8491	−	13.6947i	1.05205	−	0.764363i
$322$		0			0
$323$	8.49664	−	2.76073i	0.472766	−	0.153611i
$324$		0			0
$325$	−1.44688	−	0.320244i	−0.0802583	−	0.0177639i
$326$		0			0
$327$	2.80509	−	0.911429i	0.155122	−	0.0504021i
$328$		0			0
$329$	−11.1326	+	8.08832i	−0.613761	+	0.445923i
$330$		0			0
$331$	−7.25121	−	5.26831i	−0.398563	−	0.289573i	0.370393	−	0.928875i	$-0.379223\pi$
$331$	−7.25121	−	5.26831i	−0.398563	−	0.289573i	−0.768955	+	0.639303i	$0.779223\pi$
$332$		0			0
$333$	3.96183	−	5.45299i	0.217107	−	0.298822i
$334$		0			0
$335$	−12.8821	−	8.44269i	−0.703822	−	0.461273i
$336$		0			0
$337$	27.6601	+	8.98731i	1.50674	+	0.489570i	0.941976	−	0.335681i	$-0.108966\pi$
$337$	27.6601	+	8.98731i	1.50674	+	0.489570i	0.564766	+	0.825251i	$0.308966\pi$
$338$		0			0
$339$	4.74302	+	14.5975i	0.257605	+	0.792828i
$340$		0			0
$341$	3.76972	−	11.6020i	0.204142	−	0.628283i
$342$		0			0
$343$			10.9787i			0.592795i
$344$		0			0
$345$	1.58621	−	2.42028i	0.0853987	−	0.130303i
$346$		0			0
$347$	−8.40368	−	11.5667i	−0.451133	−	0.620931i	0.521508	−	0.853247i	$-0.325370\pi$
$347$	−8.40368	−	11.5667i	−0.451133	−	0.620931i	−0.972641	+	0.232315i	$0.925370\pi$
$348$		0			0
$349$	−5.62382			−0.301036			−0.150518	−	0.988607i	$-0.548094\pi$
$349$	−5.62382			−0.301036			−0.150518	+	0.988607i	$0.548094\pi$
$350$		0			0
$351$	1.67232			0.0892620
$352$		0			0
$353$	−1.12265	−	1.54520i	−0.0597529	−	0.0822427i	0.778095	−	0.628147i	$-0.216186\pi$
$353$	−1.12265	−	1.54520i	−0.0597529	−	0.0822427i	−0.837848	+	0.545904i	$0.816186\pi$
$354$		0			0
$355$	−8.35573	−	10.4074i	−0.443476	−	0.552366i
$356$		0			0
$357$		−	24.7813i		−	1.31156i
$358$		0			0
$359$	6.86161	−	21.1179i	0.362142	−	1.11456i	−0.589609	−	0.807689i	$-0.700718\pi$
$359$	6.86161	−	21.1179i	0.362142	−	1.11456i	0.951751	−	0.306870i	$-0.0992818\pi$
$360$		0			0
$361$	−4.94606	−	15.2224i	−0.260319	−	0.801179i
$362$		0			0
$363$	9.79935	+	3.18400i	0.514332	+	0.167117i
$364$		0			0
$365$	7.00116	+	18.4697i	0.366458	+	0.966748i
$366$		0			0
$367$	−12.6050	+	17.3493i	−0.657978	+	0.905629i	−0.999412	−	0.0342768i	$-0.989087\pi$
$367$	−12.6050	+	17.3493i	−0.657978	+	0.905629i	0.341435	+	0.939906i	$0.389087\pi$
$368$		0			0
$369$	0.684418	+	0.497259i	0.0356294	+	0.0258863i
$370$		0			0
$371$	21.3704	−	15.5265i	1.10950	−	0.806096i
$372$		0			0
$373$	−22.0074	+	7.15063i	−1.13950	+	0.370245i	−0.817178	−	0.576385i	$-0.804463\pi$
$373$	−22.0074	+	7.15063i	−1.13950	+	0.370245i	−0.322320	+	0.946631i	$0.604463\pi$
$374$		0			0
$375$	11.7822	−	11.4895i	0.608430	−	0.593317i
$376$		0			0
$377$	1.66648	−	0.541471i	0.0858279	−	0.0278872i
$378$		0			0
$379$	17.5153	−	12.7256i	0.899702	−	0.653672i	−0.0386872	−	0.999251i	$-0.512318\pi$
$379$	17.5153	−	12.7256i	0.899702	−	0.653672i	0.938390	+	0.345579i	$0.112318\pi$
$380$		0			0
$381$	−6.96645	−	5.06142i	−0.356902	−	0.259304i
$382$		0			0
$383$	−3.32381	+	4.57484i	−0.169839	+	0.233763i	−0.885449	−	0.464737i	$-0.846149\pi$
$383$	−3.32381	+	4.57484i	−0.169839	+	0.233763i	0.715610	+	0.698500i	$0.246149\pi$
$384$		0			0
$385$	5.16896	+	13.6362i	0.263435	+	0.694964i
$386$		0			0
$387$	−2.56953	−	0.834891i	−0.130617	−	0.0424399i
$388$		0			0
$389$	2.51109	+	7.72833i	0.127317	+	0.391842i	0.994316	−	0.106468i	$-0.0339541\pi$
$389$	2.51109	+	7.72833i	0.127317	+	0.391842i	−0.866999	+	0.498310i	$0.833954\pi$
$390$		0			0
$391$	1.40270	−	4.31707i	0.0709377	−	0.218324i
$392$		0			0
$393$		−	2.20298i		−	0.111126i
$394$		0			0
$395$	10.8642	+	13.5318i	0.546638	+	0.680857i
$396$		0			0
$397$	−12.2076	−	16.8024i	−0.612684	−	0.843287i	0.384111	−	0.923287i	$-0.374508\pi$
$397$	−12.2076	−	16.8024i	−0.612684	−	0.843287i	−0.996795	+	0.0799998i	$0.974508\pi$
$398$		0			0
$399$	8.30550			0.415795
$400$		0			0
$401$	30.1195			1.50410			0.752049	−	0.659107i	$-0.229066\pi$
$401$	30.1195			1.50410			0.752049	+	0.659107i	$0.229066\pi$
$402$		0			0
$403$	1.06258	+	1.46252i	0.0529309	+	0.0728532i
$404$		0			0
$405$	−7.11568	+	10.8573i	−0.353581	+	0.539503i
$406$		0			0
$407$		−	16.1760i		−	0.801815i
$408$		0			0
$409$	−3.41317	+	10.5046i	−0.168770	+	0.519421i	−0.999294	−	0.0375613i	$-0.988041\pi$
$409$	−3.41317	+	10.5046i	−0.168770	+	0.519421i	0.830524	+	0.556983i	$0.188041\pi$
$410$		0			0
$411$	−3.57012	−	10.9877i	−0.176101	−	0.541983i
$412$		0			0
$413$	18.4102	+	5.98182i	0.905905	+	0.294346i
$414$		0			0
$415$	27.1467	+	17.7915i	1.33258	+	0.873350i
$416$		0			0
$417$	−4.66469	+	6.42039i	−0.228431	+	0.314408i
$418$		0			0
$419$	26.6338	+	19.3506i	1.30115	+	0.945337i	0.999966	−	0.00823011i	$-0.00261975\pi$
$419$	26.6338	+	19.3506i	1.30115	+	0.945337i	0.301179	+	0.953568i	$0.402620\pi$
$420$		0			0
$421$	16.3945	−	11.9113i	0.799019	−	0.580522i	−0.111607	−	0.993752i	$-0.535600\pi$
$421$	16.3945	−	11.9113i	0.799019	−	0.580522i	0.910626	+	0.413231i	$0.135600\pi$
$422$		0			0
$423$	3.34464	−	1.08674i	0.162622	−	0.0528392i
$424$		0			0
$425$	13.0944	−	22.2473i	0.635171	−	1.07915i
$426$		0			0
$427$	−2.83956	+	0.922629i	−0.137416	+	0.0446491i
$428$		0			0
$429$	0.705874	−	0.512848i	0.0340799	−	0.0247605i
$430$		0			0
$431$	5.78873	+	4.20576i	0.278833	+	0.202584i	0.718408	−	0.695622i	$-0.244871\pi$
$431$	5.78873	+	4.20576i	0.278833	+	0.202584i	−0.439575	+	0.898206i	$0.644871\pi$
$432$		0			0
$433$	−3.22262	+	4.43555i	−0.154869	+	0.213159i	−0.879400	−	0.476083i	$-0.842056\pi$
$433$	−3.22262	+	4.43555i	−0.154869	+	0.213159i	0.724531	+	0.689242i	$0.242056\pi$
$434$		0			0
$435$	−5.11514	+	18.7748i	−0.245252	+	0.900184i
$436$		0			0
$437$	1.44688	+	0.470119i	0.0692135	+	0.0224888i
$438$		0			0
$439$	−9.50415	−	29.2508i	−0.453609	−	1.39606i	−0.872761	−	0.488148i	$-0.837673\pi$
$439$	−9.50415	−	29.2508i	−0.453609	−	1.39606i	0.419153	−	0.907916i	$-0.362327\pi$
$440$		0			0
$441$	−0.935631	+	2.87958i	−0.0445539	+	0.137123i
$442$		0			0
$443$		−	11.3527i		−	0.539381i	−0.962947	−	0.269691i	$-0.913079\pi$
$443$		−	11.3527i		−	0.539381i	0.962947	−	0.269691i	$-0.0869214\pi$
$444$		0			0
$445$	15.7369	−	5.96528i	0.746002	−	0.282781i
$446$		0			0
$447$	−16.2854	−	22.4149i	−0.770271	−	1.06019i
$448$		0			0
$449$	15.7661			0.744050			0.372025	−	0.928223i	$-0.378664\pi$
$449$	15.7661			0.744050			0.372025	+	0.928223i	$0.378664\pi$
$450$		0			0
$451$	2.03029			0.0956027
$452$		0			0
$453$	3.36383	+	4.62992i	0.158047	+	0.217532i
$454$		0			0
$455$	−2.08504	−	0.568064i	−0.0977484	−	0.0266312i
$456$		0			0
$457$			4.16714i			0.194931i	0.995239	+	0.0974653i	$0.0310735\pi$
$457$			4.16714i			0.194931i	−0.995239	+	0.0974653i	$0.968926\pi$
$458$		0			0
$459$	−9.00233	+	27.7063i	−0.420193	+	1.29322i
$460$		0			0
$461$	−7.40758	−	22.7982i	−0.345005	−	1.06182i	−0.961581	−	0.274521i	$-0.911481\pi$
$461$	−7.40758	−	22.7982i	−0.345005	−	1.06182i	0.616576	−	0.787296i	$-0.288519\pi$
$462$		0			0
$463$	−39.3021	−	12.7700i	−1.82652	−	0.593473i	−0.999510	−	0.0312899i	$-0.990038\pi$
$463$	−39.3021	−	12.7700i	−1.82652	−	0.593473i	−0.827013	−	0.562183i	$-0.809962\pi$
$464$		0			0
$465$	−20.0526	+	0.966645i	−0.929916	+	0.0448271i
$466$		0			0
$467$	−6.11096	+	8.41102i	−0.282782	+	0.389216i	−0.926653	−	0.375918i	$-0.877327\pi$
$467$	−6.11096	+	8.41102i	−0.282782	+	0.389216i	0.643871	+	0.765134i	$0.277327\pi$
$468$		0			0
$469$	−18.1713	−	13.2022i	−0.839072	−	0.609622i
$470$		0			0
$471$	−5.10126	+	3.70628i	−0.235054	+	0.170776i
$472$		0			0
$473$	−6.16663	+	2.00366i	−0.283542	+	0.0921284i
$474$		0			0
$475$	7.45624	+	4.38861i	0.342116	+	0.201363i
$476$		0			0
$477$	−6.42045	+	2.08613i	−0.293972	+	0.0955174i
$478$		0			0
$479$	−29.1312	+	21.1650i	−1.33104	+	0.967055i	−0.331314	+	0.943520i	$0.607492\pi$
$479$	−29.1312	+	21.1650i	−1.33104	+	0.967055i	−0.999723	+	0.0235349i	$0.992508\pi$
$480$		0			0
$481$	1.93930	+	1.40899i	0.0884246	+	0.0642443i
$482$		0			0
$483$	2.48043	−	3.41402i	0.112863	−	0.155343i
$484$		0			0
$485$	−11.7285	+	9.41645i	−0.532565	+	0.427579i
$486$		0			0
$487$	10.1172	+	3.28726i	0.458452	+	0.148960i	0.529133	−	0.848539i	$-0.322517\pi$
$487$	10.1172	+	3.28726i	0.458452	+	0.148960i	−0.0706809	+	0.997499i	$0.522517\pi$
$488$		0			0
$489$	7.15323	+	22.0154i	0.323480	+	0.995570i
$490$		0			0
$491$	5.46010	−	16.8045i	0.246411	−	0.758375i	−0.748990	−	0.662581i	$-0.769461\pi$
$491$	5.46010	−	16.8045i	0.246411	−	0.758375i	0.995401	−	0.0957938i	$-0.0305389\pi$
$492$		0			0
$493$			30.5243i			1.37475i
$494$		0			0
$495$	−0.179450	−	3.72260i	−0.00806568	−	0.167319i
$496$		0			0
$497$	−11.4403	−	15.7462i	−0.513166	−	0.706312i
$498$		0			0
$499$	−9.41734			−0.421578			−0.210789	−	0.977532i	$-0.567603\pi$
$499$	−9.41734			−0.421578			−0.210789	+	0.977532i	$0.567603\pi$
$500$		0			0
$501$	−30.9595			−1.38317
$502$		0			0
$503$	−10.5879	−	14.5730i	−0.472093	−	0.649780i	0.504869	−	0.863196i	$-0.331541\pi$
$503$	−10.5879	−	14.5730i	−0.472093	−	0.649780i	−0.976961	+	0.213416i	$0.931541\pi$
$504$		0			0
$505$	−1.31304	−	27.2383i	−0.0584294	−	1.21209i
$506$		0			0
$507$		−	19.0060i		−	0.844088i
$508$		0			0
$509$	4.95926	−	15.2630i	0.219815	−	0.676522i	−0.778961	−	0.627072i	$-0.784253\pi$
$509$	4.95926	−	15.2630i	0.219815	−	0.676522i	0.998777	−	0.0494500i	$-0.0157469\pi$
$510$		0			0
$511$	8.90107	+	27.3947i	0.393760	+	1.21187i
$512$		0			0
$513$	−9.28583	−	3.01715i	−0.409980	−	0.133210i
$514$		0			0
$515$	2.40867	−	1.93384i	0.106138	−	0.0852151i
$516$		0			0
$517$	4.96086	−	6.82803i	0.218178	−	0.300297i
$518$		0			0
$519$	8.53425	+	6.20050i	0.374612	+	0.272172i
$520$		0			0
$521$	1.78040	−	1.29354i	0.0780007	−	0.0566708i	−0.548102	−	0.836412i	$-0.684649\pi$
$521$	1.78040	−	1.29354i	0.0780007	−	0.0566708i	0.626102	+	0.779741i	$0.284649\pi$
$522$		0			0
$523$	−7.07194	+	2.29781i	−0.309234	+	0.100476i	−0.459523	−	0.888166i	$-0.651980\pi$
$523$	−7.07194	+	2.29781i	−0.309234	+	0.100476i	0.150289	+	0.988642i	$0.451980\pi$
$524$		0			0
$525$	17.9688	−	15.9084i	0.784222	−	0.694301i
$526$		0			0
$527$	−29.9504	+	9.73147i	−1.30466	+	0.423909i
$528$		0			0
$529$	−17.9820	+	13.0647i	−0.781828	+	0.568031i
$530$		0			0
$531$	−4.00233	−	2.90786i	−0.173686	−	0.126190i
$532$		0			0
$533$	−0.176845	+	0.243407i	−0.00766003	+	0.0105431i
$534$		0			0
$535$	35.3526	−	1.70419i	1.52843	−	0.0736786i
$536$		0			0
$537$	−11.2543	−	3.65675i	−0.485660	−	0.157800i
$538$		0			0
$539$	2.24543	+	6.91072i	0.0967174	+	0.297666i
$540$		0			0
$541$	6.38040	−	19.6368i	0.274315	−	0.844254i	−0.715085	−	0.699037i	$-0.753612\pi$
$541$	6.38040	−	19.6368i	0.274315	−	0.844254i	0.989400	−	0.145216i	$-0.0463878\pi$
$542$		0			0
$543$			30.3473i			1.30233i
$544$		0			0
$545$	4.32299	+	1.17779i	0.185177	+	0.0504508i
$546$		0			0
$547$	−18.4424	−	25.3839i	−0.788542	−	1.08534i	−0.994288	−	0.106730i	$-0.965962\pi$
$547$	−18.4424	−	25.3839i	−0.788542	−	1.08534i	0.205746	−	0.978605i	$-0.434038\pi$
$548$		0			0
$549$	0.763042			0.0325658
$550$		0			0
$551$	−10.2303			−0.435825
$552$		0			0
$553$	14.8747	+	20.4733i	0.632538	+	0.870615i
$554$		0			0
$555$	−24.8923	+	9.43574i	−1.05662	+	0.400525i
$556$		0			0
$557$			22.3515i			0.947064i	0.880776	+	0.473532i	$0.157021\pi$
$557$			22.3515i			0.947064i	−0.880776	+	0.473532i	$0.842979\pi$
$558$		0			0
$559$	0.296921	−	0.913829i	0.0125584	−	0.0386509i
$560$		0			0
$561$	4.69683	+	14.4554i	0.198300	+	0.610305i
$562$		0			0
$563$	32.6843	+	10.6198i	1.37748	+	0.447570i	0.901840	−	0.432070i	$-0.142217\pi$
$563$	32.6843	+	10.6198i	1.37748	+	0.447570i	0.475639	+	0.879640i	$0.342217\pi$
$564$		0			0
$565$	−6.12912	+	22.4966i	−0.257854	+	0.946438i
$566$		0			0
$567$	−11.1271	+	15.3152i	−0.467295	+	0.643177i
$568$		0			0
$569$	26.0230	+	18.9068i	1.09094	+	0.792615i	0.979558	−	0.201162i	$-0.0644718\pi$
$569$	26.0230	+	18.9068i	1.09094	+	0.792615i	0.111383	+	0.993778i	$0.464472\pi$
$570$		0			0
$571$	−21.9784	+	15.9683i	−0.919768	+	0.668251i	−0.943466	−	0.331468i	$-0.892456\pi$
$571$	−21.9784	+	15.9683i	−0.919768	+	0.668251i	0.0236979	+	0.999719i	$0.492456\pi$
$572$		0			0
$573$	−25.2407	+	8.20121i	−1.05445	+	0.342610i
$574$		0			0
$575$	4.03076	−	1.75427i	0.168094	−	0.0731581i
$576$		0			0
$577$	13.1724	−	4.27998i	0.548375	−	0.178178i	−0.0217089	−	0.999764i	$-0.506911\pi$
$577$	13.1724	−	4.27998i	0.548375	−	0.178178i	0.570084	+	0.821586i	$0.306911\pi$
$578$		0			0
$579$	8.08407	−	5.87342i	0.335963	−	0.244091i
$580$		0			0
$581$	38.2928	+	27.8214i	1.58865	+	1.15422i
$582$		0			0
$583$	−9.52297	+	13.1072i	−0.394401	+	0.542847i
$584$		0			0
$585$	0.461925	+	0.302738i	0.0190982	+	0.0125167i
$586$		0			0
$587$	41.9890	+	13.6431i	1.73307	+	0.563110i	0.993888	−	0.110392i	$-0.0352107\pi$
$587$	41.9890	+	13.6431i	1.73307	+	0.563110i	0.739185	+	0.673502i	$0.235211\pi$
$588$		0			0
$589$	−3.26152	−	10.0379i	−0.134389	−	0.413606i
$590$		0			0
$591$	3.63675	−	11.1928i	0.149596	−	0.460409i
$592$		0			0
$593$		−	16.2531i		−	0.667437i	−0.942673	−	0.333718i	$-0.891697\pi$
$593$		−	16.2531i		−	0.667437i	0.942673	−	0.333718i	$-0.108303\pi$
$594$		0			0
$595$	20.6355	−	31.4862i	0.845973	−	1.29081i
$596$		0			0
$597$	−4.50318	−	6.19810i	−0.184303	−	0.253671i
$598$		0			0
$599$	30.4822			1.24547			0.622734	−	0.782433i	$-0.286022\pi$
$599$	30.4822			1.24547			0.622734	+	0.782433i	$0.286022\pi$
$600$		0			0
$601$	−28.9162			−1.17952			−0.589758	−	0.807580i	$-0.700777\pi$
$601$	−28.9162			−1.17952			−0.589758	+	0.807580i	$0.700777\pi$
$602$		0			0
$603$	3.37405	+	4.64398i	0.137402	+	0.189117i
$604$		0			0
$605$	9.79935	+	12.2054i	0.398400	+	0.496222i
$606$		0			0
$607$			8.23276i			0.334157i	0.985944	+	0.167079i	$0.0534334\pi$
$607$			8.23276i			0.334157i	−0.985944	+	0.167079i	$0.946567\pi$
$608$		0			0
$609$	−8.76906	+	26.9884i	−0.355340	+	1.09362i
$610$		0			0
$611$	0.386489	+	1.18949i	0.0156357	+	0.0481217i
$612$		0			0
$613$	4.56327	+	1.48270i	0.184309	+	0.0598856i	0.399717	−	0.916638i	$-0.369108\pi$
$613$	4.56327	+	1.48270i	0.184309	+	0.0598856i	−0.215408	+	0.976524i	$0.569108\pi$
$614$		0			0
$615$	−1.18430	−	3.12430i	−0.0477557	−	0.125984i
$616$		0			0
$617$	1.19428	−	1.64379i	0.0480800	−	0.0661765i	−0.784300	−	0.620382i	$-0.786978\pi$
$617$	1.19428	−	1.64379i	0.0480800	−	0.0661765i	0.832380	+	0.554205i	$0.186978\pi$
$618$		0			0
$619$	−6.58621	−	4.78516i	−0.264722	−	0.192332i	0.447504	−	0.894282i	$-0.352313\pi$
$619$	−6.58621	−	4.78516i	−0.264722	−	0.192332i	−0.712226	+	0.701950i	$0.752313\pi$
$620$		0			0
$621$	−4.01342	+	2.91592i	−0.161053	+	0.117012i
$622$		0			0
$623$	23.3414	−	7.58408i	0.935153	−	0.303850i
$624$		0			0
$625$	24.5374	−	4.78708i	0.981496	−	0.191483i
$626$		0			0
$627$	−4.84474	+	1.57415i	−0.193480	+	0.0628656i
$628$		0			0
$629$	−33.7830	+	24.5448i	−1.34702	+	0.978665i
$630$		0			0
$631$	26.9279	+	19.5643i	1.07198	+	0.778841i	0.976268	−	0.216567i	$-0.0694861\pi$
$631$	26.9279	+	19.5643i	1.07198	+	0.778841i	0.0957154	+	0.995409i	$0.469486\pi$
$632$		0			0
$633$	14.3638	−	19.7701i	0.570912	−	0.785793i
$634$		0			0
$635$	−4.63663	−	12.2318i	−0.183999	−	0.485406i
$636$		0			0
$637$	−1.02409	−	0.332749i	−0.0405761	−	0.0131840i
$638$		0			0
$639$	1.53710	+	4.73072i	0.0608069	+	0.187144i
$640$		0			0
$641$	−12.3755	+	38.0880i	−0.488804	+	1.50439i	0.337590	+	0.941293i	$0.390388\pi$
$641$	−12.3755	+	38.0880i	−0.488804	+	1.50439i	−0.826394	+	0.563092i	$0.809612\pi$
$642$		0			0
$643$			11.6870i			0.460890i	0.973085	+	0.230445i	$0.0740182\pi$
$643$			11.6870i			0.460890i	−0.973085	+	0.230445i	$0.925982\pi$
$644$		0			0
$645$	6.68042	+	8.32070i	0.263041	+	0.327627i
$646$		0			0
$647$	4.67252	+	6.43117i	0.183696	+	0.252835i	0.890927	−	0.454147i	$-0.150056\pi$
$647$	4.67252	+	6.43117i	0.183696	+	0.252835i	−0.707231	+	0.706982i	$0.750056\pi$
$648$		0			0
$649$	−11.8727			−0.466044
$650$		0			0
$651$	−29.2766			−1.14744
$652$		0			0
$653$	−1.96165	−	2.69998i	−0.0767653	−	0.105658i	0.768908	−	0.639360i	$-0.220801\pi$
$653$	−1.96165	−	2.69998i	−0.0767653	−	0.105658i	−0.845673	+	0.533702i	$0.820801\pi$
$654$		0			0
$655$	1.83443	−	2.79902i	0.0716772	−	0.109367i
$656$		0			0
$657$		−	7.36146i		−	0.287198i
$658$		0			0
$659$	−9.28621	+	28.5800i	−0.361739	+	1.11332i	0.590259	+	0.807214i	$0.299026\pi$
$659$	−9.28621	+	28.5800i	−0.361739	+	1.11332i	−0.951998	+	0.306105i	$0.900974\pi$
$660$		0			0
$661$	−2.03462	−	6.26192i	−0.0791375	−	0.243560i	0.903659	−	0.428253i	$-0.140871\pi$
$661$	−2.03462	−	6.26192i	−0.0791375	−	0.243560i	−0.982796	+	0.184693i	$0.940871\pi$
$662$		0			0
$663$	−2.14213	−	0.696020i	−0.0831934	−	0.0270312i
$664$		0			0
$665$	10.5527	+	6.91604i	0.409215	+	0.268192i
$666$		0			0
$667$	−3.05526	+	4.20521i	−0.118300	+	0.162826i
$668$		0			0
$669$	−7.89293	−	5.73455i	−0.305158	−	0.221711i
$670$		0			0
$671$	1.48150	−	1.07637i	0.0571925	−	0.0415528i
$672$		0			0
$673$	−6.40194	+	2.08012i	−0.246777	+	0.0801826i	−0.429794	−	0.902927i	$-0.641414\pi$
$673$	−6.40194	+	2.08012i	−0.246777	+	0.0801826i	0.183017	+	0.983110i	$0.441414\pi$
$674$		0			0
$675$	−25.8688	+	11.2586i	−0.995690	+	0.433345i
$676$		0			0
$677$	12.9799	−	4.21741i	0.498857	−	0.162088i	−0.0487718	−	0.998810i	$-0.515531\pi$
$677$	12.9799	−	4.21741i	0.498857	−	0.162088i	0.547629	+	0.836722i	$0.315531\pi$
$678$		0			0
$679$	−17.7451	+	12.8925i	−0.680993	+	0.494770i
$680$		0			0
$681$	15.9475	+	11.5866i	0.611111	+	0.443998i
$682$		0			0
$683$	−0.689001	+	0.948329i	−0.0263639	+	0.0362868i	−0.821995	−	0.569494i	$-0.807139\pi$
$683$	−0.689001	+	0.948329i	−0.0263639	+	0.0362868i	0.795632	+	0.605781i	$0.207139\pi$
$684$		0			0
$685$	4.61345	−	16.9334i	0.176271	−	0.646992i
$686$		0			0
$687$	14.1205	+	4.58802i	0.538729	+	0.175044i
$688$		0			0
$689$	−0.741913	−	2.28337i	−0.0282646	−	0.0869896i
$690$		0			0
$691$	−3.79083	+	11.6670i	−0.144210	+	0.443832i	−0.996909	−	0.0785709i	$-0.974964\pi$
$691$	−3.79083	+	11.6670i	−0.144210	+	0.443832i	0.852699	+	0.522403i	$0.174964\pi$
$692$		0			0
$693$		−	5.43497i		−	0.206457i
$694$		0			0
$695$	−11.2731	+	4.27320i	−0.427612	+	0.162092i
$696$		0			0
$697$	−3.08068	−	4.24019i	−0.116689	−	0.160609i
$698$		0			0
$699$	−32.3910			−1.22514
$700$		0			0
$701$	−20.0271			−0.756415			−0.378207	−	0.925721i	$-0.623459\pi$
$701$	−20.0271			−0.756415			−0.378207	+	0.925721i	$0.623459\pi$
$702$		0			0
$703$	−8.22624	−	11.3225i	−0.310259	−	0.427034i
$704$		0			0
$705$	−13.4010	−	3.65106i	−0.504711	−	0.137507i
$706$		0			0
$707$		−	39.7677i		−	1.49562i
$708$		0			0
$709$	1.35816	−	4.17998i	0.0510067	−	0.156983i	−0.922309	−	0.386454i	$-0.873700\pi$
$709$	1.35816	−	4.17998i	0.0510067	−	0.156983i	0.973315	+	0.229471i	$0.0736998\pi$
$710$		0			0
$711$	−1.99856	−	6.15094i	−0.0749519	−	0.230678i
$712$		0			0
$713$	−5.10019	−	1.65715i	−0.191004	−	0.0620608i
$714$		0			0
$715$	1.32391	−	0.0638197i	0.0495114	−	0.00238672i
$716$		0			0
$717$	−6.54109	+	9.00303i	−0.244281	+	0.336224i
$718$		0			0
$719$	8.84119	+	6.42350i	0.329721	+	0.239556i	0.740312	−	0.672263i	$-0.234678\pi$
$719$	8.84119	+	6.42350i	0.329721	+	0.239556i	−0.410591	+	0.911819i	$0.634678\pi$
$720$		0			0
$721$	3.64427	−	2.64772i	0.135720	−	0.0986061i
$722$		0			0
$723$	−28.5505	+	9.27661i	−1.06180	+	0.345001i
$724$		0			0
$725$	−22.1330	+	19.5952i	−0.821999	+	0.727747i
$726$		0			0
$727$	31.2928	−	10.1677i	1.16059	−	0.377097i	0.335465	−	0.942053i	$-0.391107\pi$
$727$	31.2928	−	10.1677i	1.16059	−	0.377097i	0.825122	+	0.564955i	$0.191107\pi$
$728$		0			0
$729$	24.0719	−	17.4893i	0.891553	−	0.647751i
$730$		0			0
$731$	13.5416	+	9.83853i	0.500853	+	0.363891i
$732$		0			0
$733$	−8.08190	+	11.1238i	−0.298512	+	0.410866i	−0.931756	−	0.363086i	$-0.881723\pi$
$733$	−8.08190	+	11.1238i	−0.298512	+	0.410866i	0.633244	+	0.773952i	$0.281723\pi$
$734$		0			0
$735$	9.32470	−	7.48650i	0.343947	−	0.276144i
$736$		0			0
$737$	13.1019	+	4.25705i	0.482613	+	0.156811i
$738$		0			0
$739$	−13.3462	−	41.0754i	−0.490949	−	1.51098i	−0.823177	−	0.567785i	$-0.807800\pi$
$739$	−13.3462	−	41.0754i	−0.490949	−	1.51098i	0.332228	−	0.943199i	$-0.392200\pi$
$740$		0			0
$741$	0.233273	−	0.717939i	0.00856948	−	0.0263741i
$742$		0			0
$743$			31.8479i			1.16838i	0.811615	+	0.584192i	$0.198589\pi$
$743$			31.8479i			1.16838i	−0.811615	+	0.584192i	$0.801411\pi$
$744$		0			0
$745$	−2.02658	−	42.0404i	−0.0742482	−	1.54024i
$746$		0			0
$747$	−7.11021	−	9.78637i	−0.260149	−	0.358064i
$748$		0			0
$749$	51.6145			1.88595
$750$		0			0
$751$	29.5952			1.07995			0.539973	−	0.841682i	$-0.318435\pi$
$751$	29.5952			1.07995			0.539973	+	0.841682i	$0.318435\pi$
$752$		0			0
$753$	−9.14650	−	12.5891i	−0.333317	−	0.458771i
$754$		0			0
$755$	0.418601	+	8.68368i	0.0152345	+	0.316031i
$756$		0			0
$757$		−	0.0984401i		−	0.00357786i	−0.999998	−	0.00178893i	$-0.999431\pi$
$757$		−	0.0984401i		−	0.00357786i	0.999998	−	0.00178893i	$-0.000569435\pi$
$758$		0			0
$759$	−0.799814	+	2.46157i	−0.0290314	+	0.0893495i
$760$		0			0
$761$	−1.09516	−	3.37056i	−0.0396996	−	0.122183i	0.929243	−	0.369470i	$-0.120461\pi$
$761$	−1.09516	−	3.37056i	−0.0396996	−	0.122183i	−0.968942	+	0.247287i	$0.920461\pi$
$762$		0			0
$763$	6.21421	+	2.01912i	0.224969	+	0.0730970i
$764$		0			0
$765$	−7.50223	+	6.02330i	−0.271244	+	0.217773i
$766$		0			0
$767$	1.03415	−	1.42339i	0.0373411	−	0.0513957i
$768$		0			0
$769$	1.15494	+	0.839116i	0.0416484	+	0.0302593i	0.608415	−	0.793619i	$-0.291806\pi$
$769$	1.15494	+	0.839116i	0.0416484	+	0.0302593i	−0.566766	+	0.823879i	$0.691806\pi$
$770$		0			0
$771$	24.1382	−	17.5374i	0.869316	−	0.631595i
$772$		0			0
$773$	32.1274	−	10.4388i	1.15554	−	0.375458i	0.332313	−	0.943169i	$-0.392171\pi$
$773$	32.1274	−	10.4388i	1.15554	−	0.375458i	0.823228	+	0.567711i	$0.192171\pi$
$774$		0			0
$775$	−26.2830	−	15.4697i	−0.944113	−	0.555688i
$776$		0			0
$777$	−36.9209	+	11.9963i	−1.32453	+	0.430366i
$778$		0			0
$779$	1.42111	−	1.03250i	0.0509165	−	0.0369930i
$780$		0			0
$781$	9.65769	+	7.01672i	0.345579	+	0.251078i
$782$		0			0
$783$	19.6082	−	26.9884i	0.700740	−	0.964486i
$784$		0			0
$785$	−9.56771	+	0.461216i	−0.341486	+	0.0164615i
$786$		0			0
$787$	−2.07358	−	0.673749i	−0.0739153	−	0.0240165i	0.271826	−	0.962346i	$-0.412373\pi$
$787$	−2.07358	−	0.673749i	−0.0739153	−	0.0240165i	−0.345741	+	0.938330i	$0.612373\pi$
$788$		0			0
$789$	−12.7773	−	39.3246i	−0.454886

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:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	400.y (of order \(10\), degree \(4\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.865190 + 1.19083i) q^{3} +(0.107666 + 2.23347i) q^{5} +3.26086i q^{7} +(0.257524 - 0.792578i) q^{9} +O(q^{10})\)	\(q+(0.865190 + 1.19083i) q^{3} +(0.107666 + 2.23347i) q^{5} +3.26086i q^{7} +(0.257524 - 0.792578i) q^{9} +(-0.618034 - 1.90211i) q^{11} +(0.281873 + 0.0915860i) q^{13} +(-2.56654 + 2.06059i) q^{15} +(-3.03472 + 4.17693i) q^{17} +(-1.39991 - 1.01709i) q^{19} +(-3.88313 + 2.82126i) q^{21} +(-0.836161 + 0.271685i) q^{23} +(-4.97682 + 0.480938i) q^{25} +(5.36635 - 1.74363i) q^{27} +(4.78304 - 3.47508i) q^{29} +(4.93462 + 3.58521i) q^{31} +(1.73038 - 2.38166i) q^{33} +(-7.28304 + 0.351083i) q^{35} +(7.69215 + 2.49933i) q^{37} +(0.134810 + 0.414902i) q^{39} +(-0.313697 + 0.965461i) q^{41} -3.24199i q^{43} +(1.79793 + 0.489840i) q^{45} +(2.48043 + 3.41402i) q^{47} -3.63318 q^{49} -7.59963 q^{51} +(-4.76148 - 6.55362i) q^{53} +(4.18178 - 1.58516i) q^{55} -2.54703i q^{57} +(1.83443 - 5.64581i) q^{59} +(0.282941 + 0.870802i) q^{61} +(2.58448 + 0.839749i) q^{63} +(-0.174207 + 0.639416i) q^{65} +(-4.04870 + 5.57255i) q^{67} +(-1.04697 - 0.760668i) q^{69} +(-4.82884 + 3.50836i) q^{71} +(8.40107 - 2.72967i) q^{73} +(-4.87861 - 5.51045i) q^{75} +(6.20252 - 2.01532i) q^{77} +(6.27851 - 4.56161i) q^{79} +(4.69667 + 3.41233i) q^{81} +(8.53192 - 11.7432i) q^{83} +(-9.65580 - 6.32825i) q^{85} +(8.27647 + 2.68919i) q^{87} +(-2.32579 - 7.15805i) q^{89} +(-0.298649 + 0.919147i) q^{91} +8.97820i q^{93} +(2.12093 - 3.23616i) q^{95} +(3.95373 + 5.44184i) q^{97} -1.66673 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 5 q^{3} + q^{9}+O(q^{10}) \) 8 * q + 5 * q^3 + q^9	\( 8 q + 5 q^{3} + q^{9} + 4 q^{11} - 5 q^{13} - 15 q^{15} - 10 q^{17} + 5 q^{19} - 4 q^{21} - 5 q^{23} - 10 q^{25} + 5 q^{27} - 5 q^{29} + 9 q^{31} + 10 q^{33} - 15 q^{35} + 30 q^{37} + 3 q^{39} - 4 q^{41} - 15 q^{45} + 14 q^{49} + 4 q^{51} - 10 q^{53} + 10 q^{55} - 9 q^{61} - 10 q^{63} + 5 q^{65} - 20 q^{67} + 17 q^{69} - 6 q^{71} + 15 q^{73} + 10 q^{75} + 10 q^{77} - 15 q^{79} + 28 q^{81} + 45 q^{83} - 15 q^{85} + 20 q^{87} - 25 q^{89} - 6 q^{91} - 15 q^{95} - 60 q^{97} + 8 q^{99}+O(q^{100}) \) 8 * q + 5 * q^3 + q^9 + 4 * q^11 - 5 * q^13 - 15 * q^15 - 10 * q^17 + 5 * q^19 - 4 * q^21 - 5 * q^23 - 10 * q^25 + 5 * q^27 - 5 * q^29 + 9 * q^31 + 10 * q^33 - 15 * q^35 + 30 * q^37 + 3 * q^39 - 4 * q^41 - 15 * q^45 + 14 * q^49 + 4 * q^51 - 10 * q^53 + 10 * q^55 - 9 * q^61 - 10 * q^63 + 5 * q^65 - 20 * q^67 + 17 * q^69 - 6 * q^71 + 15 * q^73 + 10 * q^75 + 10 * q^77 - 15 * q^79 + 28 * q^81 + 45 * q^83 - 15 * q^85 + 20 * q^87 - 25 * q^89 - 6 * q^91 - 15 * q^95 - 60 * q^97 + 8 * q^99

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