LMFDB - Embedded newform 675.2.e.c.226.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [675,2,Mod(226,675)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(675, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([2, 0]))

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

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

chi := DirichletCharacter("675.226");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 675 = 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	675.e (of order $3$, degree $2$, 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:	$5.38990213644$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{3})$
Coefficient field:	8.0.1223810289.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} + 8x^{6} - 2x^{5} + 23x^{4} - 8x^{3} + 37x^{2} + 15x + 9 $ x^8 - 2x^7 + 8x^6 - 2x^5 + 23x^4 - 8x^3 + 37x^2 + 15*x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 225)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			226.2
Root			$0.736627 + 1.27588i$ of defining polynomial
Character	$\chi$	$=$	675.226
Dual form			675.2.e.c.451.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.736627 - 1.27588i) q^{2} +(-0.0852394 + 0.147639i) q^{4} +(1.93291 + 3.34791i) q^{7} -2.69535 q^{8} +O(q^{10})$	\(q+(-0.736627 - 1.27588i) q^{2} +(-0.0852394 + 0.147639i) q^{4} +(1.93291 + 3.34791i) q^{7} -2.69535 q^{8} +(0.130139 + 0.225407i) q^{11} +(-2.03940 + 3.53235i) q^{13} +(2.84768 - 4.93232i) q^{14} +(2.15595 + 3.73421i) q^{16} +3.26028 q^{17} +4.24928 q^{19} +(0.191728 - 0.332082i) q^{22} +(-4.34768 + 7.53039i) q^{23} +6.00912 q^{26} -0.659042 q^{28} +(2.11105 + 3.65644i) q^{29} +(-1.32643 + 2.29744i) q^{31} +(0.480909 - 0.832959i) q^{32} +(-2.40161 - 4.15971i) q^{34} +2.27559 q^{37} +(-3.13014 - 5.42156i) q^{38} +(2.82093 - 4.88599i) q^{41} +(-4.53631 - 7.85712i) q^{43} -0.0443719 q^{44} +12.8105 q^{46} +(0.714441 + 1.23745i) q^{47} +(-3.97232 + 6.88026i) q^{49} +(-0.347675 - 0.602191i) q^{52} +11.3816 q^{53} +(-5.20988 - 9.02378i) q^{56} +(3.11011 - 5.38687i) q^{58} +(-3.56212 + 6.16977i) q^{59} +(-1.26244 - 2.18660i) q^{61} +3.90833 q^{62} +7.20679 q^{64} +(5.64280 - 9.77361i) q^{67} +(-0.277904 + 0.481344i) q^{68} +8.38158 q^{71} -0.403568 q^{73} +(-1.67626 - 2.90337i) q^{74} +(-0.362207 + 0.627360i) q^{76} +(-0.503095 + 0.871386i) q^{77} +(1.52125 + 2.63488i) q^{79} -8.31189 q^{82} +(-2.29012 - 3.96660i) q^{83} +(-6.68314 + 11.5755i) q^{86} +(-0.350770 - 0.607551i) q^{88} -7.17772 q^{89} -15.7680 q^{91} +(-0.741187 - 1.28377i) q^{92} +(1.05255 - 1.82308i) q^{94} +(1.55756 + 2.69777i) q^{97} +11.7045 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 2 q^{2} - 4 q^{4} - q^{7} + 18 q^{8}+O(q^{10}) $ 8 * q - 2 * q^2 - 4 * q^4 - q^7 + 18 * q^8	$ 8 q - 2 q^{2} - 4 q^{4} - q^{7} + 18 q^{8} - q^{11} + 2 q^{13} + 3 q^{14} - 4 q^{16} + 22 q^{17} + 4 q^{19} + 3 q^{22} - 15 q^{23} + 20 q^{26} + 8 q^{28} + q^{29} + 4 q^{31} - 10 q^{32} - 9 q^{34} + 2 q^{37} - 23 q^{38} - 5 q^{41} - 10 q^{43} - 44 q^{44} - 20 q^{47} + 3 q^{49} + 17 q^{52} + 40 q^{53} - 30 q^{56} - 18 q^{58} + 17 q^{59} + 13 q^{61} - 12 q^{62} + 38 q^{64} + 17 q^{67} - 34 q^{68} + 16 q^{71} - 4 q^{73} + 40 q^{74} - 11 q^{76} - 12 q^{77} + 7 q^{79} - 24 q^{82} - 30 q^{83} - 34 q^{86} + 9 q^{88} + 18 q^{89} - 34 q^{91} + 12 q^{92} - 3 q^{94} - 19 q^{97} + 26 q^{98}+O(q^{100}) $ 8 * q - 2 * q^2 - 4 * q^4 - q^7 + 18 * q^8 - q^11 + 2 * q^13 + 3 * q^14 - 4 * q^16 + 22 * q^17 + 4 * q^19 + 3 * q^22 - 15 * q^23 + 20 * q^26 + 8 * q^28 + q^29 + 4 * q^31 - 10 * q^32 - 9 * q^34 + 2 * q^37 - 23 * q^38 - 5 * q^41 - 10 * q^43 - 44 * q^44 - 20 * q^47 + 3 * q^49 + 17 * q^52 + 40 * q^53 - 30 * q^56 - 18 * q^58 + 17 * q^59 + 13 * q^61 - 12 * q^62 + 38 * q^64 + 17 * q^67 - 34 * q^68 + 16 * q^71 - 4 * q^73 + 40 * q^74 - 11 * q^76 - 12 * q^77 + 7 * q^79 - 24 * q^82 - 30 * q^83 - 34 * q^86 + 9 * q^88 + 18 * q^89 - 34 * q^91 + 12 * q^92 - 3 * q^94 - 19 * q^97 + 26 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/675\mathbb{Z}\right)^\times$.

$n$	$326$	$352$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$1$

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 $

$2$	−0.736627	−	1.27588i	−0.520874	−	0.902180i	−0.999705	−	0.0242735i	$-0.992273\pi$
$2$	−0.736627	−	1.27588i	−0.520874	−	0.902180i	0.478831	−	0.877907i	$-0.341061\pi$
$3$		0			0
$3$		0			0
$4$	−0.0852394	+	0.147639i	−0.0426197	+	0.0738195i
$5$		0			0
$5$		0			0
$6$		0			0
$7$	1.93291	+	3.34791i	0.730573	+	1.26539i	0.956639	+	0.291278i	$0.0940803\pi$
$7$	1.93291	+	3.34791i	0.730573	+	1.26539i	−0.226066	+	0.974112i	$0.572586\pi$
$8$	−2.69535			−0.952950
$9$		0			0
$10$		0			0
$11$	0.130139	+	0.225407i	0.0392384	+	0.0679628i	0.884978	−	0.465634i	$-0.154174\pi$
$11$	0.130139	+	0.225407i	0.0392384	+	0.0679628i	−0.845739	+	0.533596i	$0.820840\pi$
$12$		0			0
$13$	−2.03940	+	3.53235i	−0.565629	+	0.979697i	0.431362	+	0.902179i	$0.358033\pi$
$13$	−2.03940	+	3.53235i	−0.565629	+	0.979697i	−0.996991	+	0.0775187i	$0.975300\pi$
$14$	2.84768	−	4.93232i	0.761073	−	1.31822i
$15$		0			0
$16$	2.15595	+	3.73421i	0.538987	+	0.933553i
$17$	3.26028			0.790734			0.395367	−	0.918523i	$-0.370617\pi$
$17$	3.26028			0.790734			0.395367	+	0.918523i	$0.370617\pi$
$18$		0			0
$19$	4.24928			0.974853			0.487426	−	0.873164i	$-0.337936\pi$
$19$	4.24928			0.974853			0.487426	+	0.873164i	$0.337936\pi$
$20$		0			0
$21$		0			0
$22$	0.191728	−	0.332082i	0.0408765	−	0.0708002i
$23$	−4.34768	+	7.53039i	−0.906553	+	1.57020i	−0.0877339	+	0.996144i	$0.527963\pi$
$23$	−4.34768	+	7.53039i	−0.906553	+	1.57020i	−0.818819	+	0.574052i	$0.805371\pi$
$24$		0			0
$25$		0			0
$26$	6.00912			1.17849
$27$		0			0
$28$	−0.659042			−0.124547
$29$	2.11105	+	3.65644i	0.392012	+	0.678984i	0.992715	−	0.120488i	$-0.0384459\pi$
$29$	2.11105	+	3.65644i	0.392012	+	0.678984i	−0.600703	+	0.799472i	$0.705113\pi$
$30$		0			0
$31$	−1.32643	+	2.29744i	−0.238233	+	0.412632i	−0.960207	−	0.279288i	$-0.909902\pi$
$31$	−1.32643	+	2.29744i	−0.238233	+	0.412632i	0.721974	+	0.691920i	$0.243235\pi$
$32$	0.480909	−	0.832959i	0.0850135	−	0.147248i
$33$		0			0
$34$	−2.40161	−	4.15971i	−0.411873	−	0.713384i
$35$		0			0
$36$		0			0
$37$	2.27559			0.374104			0.187052	−	0.982350i	$-0.440107\pi$
$37$	2.27559			0.374104			0.187052	+	0.982350i	$0.440107\pi$
$38$	−3.13014	−	5.42156i	−0.507776	−	0.879493i
$39$		0			0
$40$		0			0
$41$	2.82093	−	4.88599i	0.440555	−	0.763064i	−0.557176	−	0.830395i	$-0.688115\pi$
$41$	2.82093	−	4.88599i	0.440555	−	0.763064i	0.997731	+	0.0673308i	$0.0214483\pi$
$42$		0			0
$43$	−4.53631	−	7.85712i	−0.691780	−	1.19820i	−0.971254	−	0.238046i	$-0.923493\pi$
$43$	−4.53631	−	7.85712i	−0.691780	−	1.19820i	0.279474	−	0.960153i	$-0.409840\pi$
$44$	−0.0443719			−0.00668931
$45$		0			0
$46$	12.8105			1.88880
$47$	0.714441	+	1.23745i	0.104212	+	0.180500i	0.913416	−	0.407027i	$-0.133435\pi$
$47$	0.714441	+	1.23745i	0.104212	+	0.180500i	−0.809204	+	0.587528i	$0.800101\pi$
$48$		0			0
$49$	−3.97232	+	6.88026i	−0.567474	+	0.982894i
$50$		0			0
$51$		0			0
$52$	−0.347675	−	0.602191i	−0.0482139	−	0.0835089i
$53$	11.3816			1.56338			0.781690	−	0.623667i	$-0.214358\pi$
$53$	11.3816			1.56338			0.781690	+	0.623667i	$0.214358\pi$
$54$		0			0
$55$		0			0
$56$	−5.20988	−	9.02378i	−0.696200	−	1.20585i
$57$		0			0
$58$	3.11011	−	5.38687i	0.408378	−	0.707331i
$59$	−3.56212	+	6.16977i	−0.463748	+	0.803235i	−0.999144	−	0.0413660i	$-0.986829\pi$
$59$	−3.56212	+	6.16977i	−0.463748	+	0.803235i	0.535396	+	0.844601i	$0.320162\pi$
$60$		0			0
$61$	−1.26244	−	2.18660i	−0.161638	−	0.279966i	0.773818	−	0.633408i	$-0.218344\pi$
$61$	−1.26244	−	2.18660i	−0.161638	−	0.279966i	−0.935456	+	0.353442i	$0.885011\pi$
$62$	3.90833			0.496358
$63$		0			0
$64$	7.20679			0.900848
$65$		0			0
$66$		0			0
$67$	5.64280	−	9.77361i	0.689377	−	1.19404i	−0.282662	−	0.959219i	$-0.591218\pi$
$67$	5.64280	−	9.77361i	0.689377	−	1.19404i	0.972040	−	0.234817i	$-0.0754491\pi$
$68$	−0.277904	+	0.481344i	−0.0337008	+	0.0583716i
$69$		0			0
$70$		0			0
$71$	8.38158			0.994711			0.497355	−	0.867547i	$-0.334305\pi$
$71$	8.38158			0.994711			0.497355	+	0.867547i	$0.334305\pi$
$72$		0			0
$73$	−0.403568			−0.0472340			−0.0236170	−	0.999721i	$-0.507518\pi$
$73$	−0.403568			−0.0472340			−0.0236170	+	0.999721i	$0.507518\pi$
$74$	−1.67626	−	2.90337i	−0.194861	−	0.337509i
$75$		0			0
$76$	−0.362207	+	0.627360i	−0.0415480	+	0.0719632i
$77$	−0.503095	+	0.871386i	−0.0573330	+	0.0993036i
$78$		0			0
$79$	1.52125	+	2.63488i	0.171154	+	0.296447i	0.938824	−	0.344399i	$-0.111917\pi$
$79$	1.52125	+	2.63488i	0.171154	+	0.296447i	−0.767670	+	0.640846i	$0.778584\pi$
$80$		0			0
$81$		0			0
$82$	−8.31189			−0.917895
$83$	−2.29012	−	3.96660i	−0.251373	−	0.435391i	0.712531	−	0.701641i	$-0.247549\pi$
$83$	−2.29012	−	3.96660i	−0.251373	−	0.435391i	−0.963904	+	0.266250i	$0.914215\pi$
$84$		0			0
$85$		0			0
$86$	−6.68314	+	11.5755i	−0.720661	+	1.24822i
$87$		0			0
$88$	−0.350770	−	0.607551i	−0.0373922	−	0.0647652i
$89$	−7.17772			−0.760837			−0.380419	−	0.924814i	$-0.624220\pi$
$89$	−7.17772			−0.760837			−0.380419	+	0.924814i	$0.624220\pi$
$90$		0			0
$91$	−15.7680			−1.65293
$92$	−0.741187	−	1.28377i	−0.0772741	−	0.133843i
$93$		0			0
$94$	1.05255	−	1.82308i	0.108563	−	0.188036i
$95$		0			0
$96$		0			0
$97$	1.55756	+	2.69777i	0.158146	+	0.273917i	0.934200	−	0.356749i	$-0.116115\pi$
$97$	1.55756	+	2.69777i	0.158146	+	0.273917i	−0.776054	+	0.630666i	$0.782782\pi$
$98$	11.7045			1.18233
$99$		0			0
$100$		0			0
$101$	1.92286	+	3.33049i	0.191332	+	0.331396i	0.945692	−	0.325065i	$-0.105386\pi$
$101$	1.92286	+	3.33049i	0.191332	+	0.331396i	−0.754360	+	0.656461i	$0.772053\pi$
$102$		0			0
$103$	−2.10339	+	3.64318i	−0.207254	+	0.358974i	−0.950848	−	0.309657i	$-0.899786\pi$
$103$	−2.10339	+	3.64318i	−0.207254	+	0.358974i	0.743595	+	0.668630i	$0.233119\pi$
$104$	5.49691	−	9.52092i	0.539016	−	0.933603i
$105$		0			0
$106$	−8.38398	−	14.5215i	−0.814324	−	1.41045i
$107$	−1.62655			−0.157245			−0.0786223	−	0.996904i	$-0.525052\pi$
$107$	−1.62655			−0.157245			−0.0786223	+	0.996904i	$0.525052\pi$
$108$		0			0
$109$	−12.9021			−1.23580			−0.617900	−	0.786256i	$-0.712016\pi$
$109$	−12.9021			−1.23580			−0.617900	+	0.786256i	$0.712016\pi$
$110$		0			0
$111$		0			0
$112$	−8.33452	+	14.4358i	−0.787539	+	1.36406i
$113$	−0.664539	+	1.15102i	−0.0625146	+	0.108278i	−0.895589	−	0.444883i	$-0.853245\pi$
$113$	−0.664539	+	1.15102i	−0.0625146	+	0.108278i	0.833074	+	0.553161i	$0.186579\pi$
$114$		0			0
$115$		0			0
$116$	−0.719778			−0.0668297
$117$		0			0
$118$	10.4958			0.966217
$119$	6.30184	+	10.9151i	0.577689	+	1.00059i
$120$		0			0
$121$	5.46613	−	9.46761i	0.496921	−	0.860692i
$122$	−1.85989	+	3.22142i	−0.168386	+	0.291654i
$123$		0			0
$124$	−0.226128	−	0.391665i	−0.0203069	−	0.0351725i
$125$		0			0
$126$		0			0
$127$	1.65285			0.146667			0.0733335	−	0.997307i	$-0.476636\pi$
$127$	1.65285			0.146667			0.0733335	+	0.997307i	$0.476636\pi$
$128$	−6.27053	−	10.8609i	−0.554242	−	0.959975i
$129$		0			0
$130$		0			0
$131$	−6.58886	+	11.4122i	−0.575672	+	0.997092i	0.420297	+	0.907387i	$0.361926\pi$
$131$	−6.58886	+	11.4122i	−0.575672	+	0.997092i	−0.995968	+	0.0897057i	$0.971407\pi$
$132$		0			0
$133$	8.21350	+	14.2262i	0.712201	+	1.23357i
$134$	−16.6266			−1.43632
$135$		0			0
$136$	−8.78759			−0.753530
$137$	−10.1464	−	17.5741i	−0.866867	−	1.50146i	−0.865181	−	0.501459i	$-0.832797\pi$
$137$	−10.1464	−	17.5741i	−0.866867	−	1.50146i	−0.00168578	−	0.999999i	$-0.500537\pi$
$138$		0			0
$139$	−1.53440	+	2.65766i	−0.130146	+	0.225420i	−0.923733	−	0.383038i	$-0.874878\pi$
$139$	−1.53440	+	2.65766i	−0.130146	+	0.225420i	0.793587	+	0.608457i	$0.208211\pi$
$140$		0			0
$141$		0			0
$142$	−6.17410	−	10.6939i	−0.518119	−	0.897409i
$143$	−1.06162			−0.0887774
$144$		0			0
$145$		0			0
$146$	0.297279	+	0.514902i	0.0246030	+	0.0426136i
$147$		0			0
$148$	−0.193970	+	0.335965i	−0.0159442	+	0.0276162i
$149$	2.03081	−	3.51747i	0.166371	−	0.288162i	−0.770771	−	0.637113i	$-0.780129\pi$
$149$	2.03081	−	3.51747i	0.166371	−	0.288162i	0.937141	+	0.348951i	$0.113462\pi$
$150$		0			0
$151$	6.80994	+	11.7952i	0.554185	+	0.959876i	0.997966	+	0.0637412i	$0.0203032\pi$
$151$	6.80994	+	11.7952i	0.554185	+	0.959876i	−0.443782	+	0.896135i	$0.646363\pi$
$152$	−11.4533			−0.928986
$153$		0			0
$154$	1.48237			0.119453
$155$		0			0
$156$		0			0
$157$	1.03131	−	1.78627i	0.0823071	−	0.142560i	−0.821933	−	0.569584i	$-0.807104\pi$
$157$	1.03131	−	1.78627i	0.0823071	−	0.142560i	0.904241	+	0.427023i	$0.140438\pi$
$158$	2.24119	−	3.88185i	0.178299	−	0.308823i
$159$		0			0
$160$		0			0
$161$	−33.6147			−2.64921
$162$		0			0
$163$	3.50525			0.274552			0.137276	−	0.990533i	$-0.456165\pi$
$163$	3.50525			0.274552			0.137276	+	0.990533i	$0.456165\pi$
$164$	0.480909	+	0.832959i	0.0375527	+	0.0650431i
$165$		0			0
$166$	−3.37393	+	5.84381i	−0.261868	+	0.453568i
$167$	−10.2674	+	17.7837i	−0.794518	+	1.37615i	0.128626	+	0.991693i	$0.458943\pi$
$167$	−10.2674	+	17.7837i	−0.794518	+	1.37615i	−0.923145	+	0.384453i	$0.874390\pi$
$168$		0			0
$169$	−1.81833	−	3.14944i	−0.139871	−	0.242264i
$170$		0			0
$171$		0			0
$172$	1.54669			0.117934
$173$	3.87589	+	6.71323i	0.294678	+	0.510397i	0.974910	−	0.222600i	$-0.0714542\pi$
$173$	3.87589	+	6.71323i	0.294678	+	0.510397i	−0.680232	+	0.732997i	$0.738121\pi$
$174$		0			0
$175$		0			0
$176$	−0.561145	+	0.971932i	−0.0422979	+	0.0732622i
$177$		0			0
$178$	5.28731	+	9.15788i	0.396300	+	0.686412i
$179$	10.7632			0.804477			0.402238	−	0.915535i	$-0.368232\pi$
$179$	10.7632			0.804477			0.402238	+	0.915535i	$0.368232\pi$
$180$		0			0
$181$	−7.84572			−0.583168			−0.291584	−	0.956545i	$-0.594182\pi$
$181$	−7.84572			−0.583168			−0.291584	+	0.956545i	$0.594182\pi$
$182$	11.6151	+	20.1180i	0.860970	+	1.49124i
$183$		0			0
$184$	11.7185	−	20.2970i	0.863900	−	1.49632i
$185$		0			0
$186$		0			0
$187$	0.424289	+	0.734890i	0.0310271	+	0.0537405i
$188$	−0.243594			−0.0177659
$189$		0			0
$190$		0			0
$191$	−2.86627	−	4.96453i	−0.207396	−	0.359221i	0.743497	−	0.668739i	$-0.233166\pi$
$191$	−2.86627	−	4.96453i	−0.207396	−	0.359221i	−0.950894	+	0.309518i	$0.899832\pi$
$192$		0			0
$193$	−4.24119	+	7.34595i	−0.305287	+	0.528773i	−0.977325	−	0.211744i	$-0.932086\pi$
$193$	−4.24119	+	7.34595i	−0.305287	+	0.528773i	0.672038	+	0.740517i	$0.265419\pi$
$194$	2.29468	−	3.97450i	0.164748	−	0.285352i
$195$		0			0
$196$	−0.677196	−	1.17294i	−0.0483712	−	0.0837813i
$197$	10.6266			0.757110			0.378555	−	0.925579i	$-0.376421\pi$
$197$	10.6266			0.757110			0.378555	+	0.925579i	$0.376421\pi$
$198$		0			0
$199$	−18.5784			−1.31699			−0.658495	−	0.752585i	$-0.728807\pi$
$199$	−18.5784			−1.31699			−0.658495	+	0.752585i	$0.728807\pi$
$200$		0			0
$201$		0			0
$202$	2.83286	−	4.90666i	0.199319	−	0.345231i
$203$	−8.16095	+	14.1352i	−0.572786	+	0.992095i
$204$		0			0
$205$		0			0
$206$	6.19767			0.431812
$207$		0			0
$208$	−17.5874			−1.21947
$209$	0.552997	+	0.957820i	0.0382516	+	0.0662538i
$210$		0			0
$211$	−5.22666	+	9.05283i	−0.359818	+	0.623223i	−0.987930	−	0.154900i	$-0.950494\pi$
$211$	−5.22666	+	9.05283i	−0.359818	+	0.623223i	0.628112	+	0.778123i	$0.283828\pi$
$212$	−0.970160	+	1.68037i	−0.0666308	+	0.115408i
$213$		0			0
$214$	1.19816	+	2.07528i	0.0819046	+	0.141863i
$215$		0			0
$216$		0			0
$217$	−10.2555			−0.696187
$218$	9.50407	+	16.4615i	0.643697	+	1.11492i
$219$		0			0
$220$		0			0
$221$	−6.64902	+	11.5164i	−0.447261	+	0.774680i
$222$		0			0
$223$	1.96560	+	3.40452i	0.131626	+	0.227983i	0.924304	−	0.381658i	$-0.124647\pi$
$223$	1.96560	+	3.40452i	0.131626	+	0.227983i	−0.792677	+	0.609641i	$0.791313\pi$
$224$	3.71822			0.248434
$225$		0			0
$226$	1.95807			0.130249
$227$	−2.41570	−	4.18411i	−0.160335	−	0.277709i	0.774654	−	0.632386i	$-0.217924\pi$
$227$	−2.41570	−	4.18411i	−0.160335	−	0.277709i	−0.934989	+	0.354677i	$0.884591\pi$
$228$		0			0
$229$	9.42648	−	16.3271i	0.622919	−	1.07893i	−0.366020	−	0.930607i	$-0.619280\pi$
$229$	9.42648	−	16.3271i	0.622919	−	1.07893i	0.988939	−	0.148321i	$-0.0473869\pi$
$230$		0			0
$231$		0			0
$232$	−5.69001	−	9.85539i	−0.373568	−	0.647038i
$233$	−11.9021			−0.779735			−0.389867	−	0.920871i	$-0.627479\pi$
$233$	−11.9021			−0.779735			−0.389867	+	0.920871i	$0.627479\pi$
$234$		0			0
$235$		0			0
$236$	−0.607266	−	1.05181i	−0.0395296	−	0.0684673i
$237$		0			0
$238$	9.28421	−	16.0807i	0.601806	−	1.04236i
$239$	−10.8147	+	18.7317i	−0.699547	+	1.21165i	0.269076	+	0.963119i	$0.413282\pi$
$239$	−10.8147	+	18.7317i	−0.699547	+	1.21165i	−0.968624	+	0.248533i	$0.920052\pi$
$240$		0			0
$241$	−1.94916	−	3.37604i	−0.125556	−	0.217470i	0.796394	−	0.604778i	$-0.206738\pi$
$241$	−1.94916	−	3.37604i	−0.125556	−	0.217470i	−0.921950	+	0.387308i	$0.873405\pi$
$242$	−16.1060			−1.03533
$243$		0			0
$244$	0.430437			0.0275559
$245$		0			0
$246$		0			0
$247$	−8.66600	+	15.0100i	−0.551405	+	0.955061i
$248$	3.57518	−	6.19240i	0.227024	−	0.393218i
$249$		0			0
$250$		0			0
$251$	30.1033			1.90010			0.950052	−	0.312092i	$-0.101030\pi$
$251$	30.1033			1.90010			0.950052	+	0.312092i	$0.101030\pi$
$252$		0			0
$253$	−2.26321			−0.142287
$254$	−1.21754	−	2.10883i	−0.0763950	−	0.132320i
$255$		0			0
$256$	−2.03131	+	3.51832i	−0.126957	+	0.219895i
$257$	8.20707	−	14.2151i	0.511943	−	0.886711i	−0.487961	−	0.872865i	$-0.662259\pi$
$257$	8.20707	−	14.2151i	0.511943	−	0.886711i	0.999904	−	0.0138459i	$-0.00440744\pi$
$258$		0			0
$259$	4.39851	+	7.61845i	0.273310	+	0.473388i
$260$		0			0
$261$		0			0
$262$	19.4141			1.19941
$263$	−12.9036	−	22.3497i	−0.795670	−	1.37814i	−0.922413	−	0.386206i	$-0.873786\pi$
$263$	−12.9036	−	22.3497i	−0.795670	−	1.37814i	0.126743	−	0.991936i	$-0.459548\pi$
$264$		0			0
$265$		0			0
$266$	12.1006	−	20.9588i	0.741934	−	1.28507i
$267$		0			0
$268$	0.961978	+	1.66619i	0.0587621	+	0.101779i
$269$	12.5206			0.763392			0.381696	−	0.924288i	$-0.375340\pi$
$269$	12.5206			0.763392			0.381696	+	0.924288i	$0.375340\pi$
$270$		0			0
$271$	19.6462			1.19342			0.596710	−	0.802457i	$-0.296474\pi$
$271$	19.6462			1.19342			0.596710	+	0.802457i	$0.296474\pi$
$272$	7.02899	+	12.1746i	0.426195	+	0.738191i
$273$		0			0
$274$	−14.9483	+	25.8911i	−0.903057	+	1.56414i
$275$		0			0
$276$		0			0
$277$	−10.4150	−	18.0394i	−0.625779	−	1.08388i	−0.988390	−	0.151941i	$-0.951448\pi$
$277$	−10.4150	−	18.0394i	−0.625779	−	1.08388i	0.362610	−	0.931941i	$-0.381886\pi$
$278$	4.52112			0.271159
$279$		0			0
$280$		0			0
$281$	2.36221	+	4.09146i	0.140917	+	0.244076i	0.927842	−	0.372973i	$-0.121661\pi$
$281$	2.36221	+	4.09146i	0.140917	+	0.244076i	−0.786925	+	0.617049i	$0.788328\pi$
$282$		0			0
$283$	11.5762	−	20.0506i	0.688136	−	1.19189i	−0.284304	−	0.958734i	$-0.591763\pi$
$283$	11.5762	−	20.0506i	0.688136	−	1.19189i	0.972440	−	0.233152i	$-0.0749041\pi$
$284$	−0.714441	+	1.23745i	−0.0423943	+	0.0734291i
$285$		0			0
$286$	0.782020	+	1.35450i	0.0462418	+	0.0800932i
$287$	21.8105			1.28743
$288$		0			0
$289$	−6.37059			−0.374740
$290$		0			0
$291$		0			0
$292$	0.0343999	−	0.0595824i	0.00201310	−	0.00348679i
$293$	8.43963	−	14.6179i	0.493049	−	0.853985i	−0.506919	−	0.861993i	$-0.669216\pi$
$293$	8.43963	−	14.6179i	0.493049	−	0.853985i	0.999968	+	0.00800832i	$0.00254915\pi$
$294$		0			0
$295$		0			0
$296$	−6.13350			−0.356503
$297$		0			0
$298$	−5.98380			−0.346632
$299$	−17.7333	−	30.7150i	−1.02554	−	1.77630i
$300$		0			0
$301$	17.5366	−	30.3743i	1.01079	−	1.75074i
$302$	10.0328	−	17.3773i	0.577321	−	0.999949i
$303$		0			0
$304$	9.16123	+	15.8677i	0.525433	+	0.910076i
$305$		0			0
$306$		0			0
$307$	22.7177			1.29657			0.648285	−	0.761398i	$-0.275487\pi$
$307$	22.7177			1.29657			0.648285	+	0.761398i	$0.275487\pi$
$308$	−0.0857671	−	0.148553i	−0.00488703	−	0.00846459i
$309$		0			0
$310$		0			0
$311$	15.7968	−	27.3608i	0.895754	−	1.55149i	0.0628843	−	0.998021i	$-0.479970\pi$
$311$	15.7968	−	27.3608i	0.895754	−	1.55149i	0.832869	−	0.553470i	$-0.186697\pi$
$312$		0			0
$313$	−15.2498	−	26.4134i	−0.861970	−	1.49298i	−0.870024	−	0.493009i	$-0.835897\pi$
$313$	−15.2498	−	26.4134i	−0.861970	−	1.49298i	0.00805392	−	0.999968i	$-0.497436\pi$
$314$	−3.03875			−0.171487
$315$		0			0
$316$	−0.518682			−0.0291781
$317$	−11.0445	−	19.1296i	−0.620320	−	1.07443i	−0.989426	−	0.145039i	$-0.953669\pi$
$317$	−11.0445	−	19.1296i	−0.620320	−	1.07443i	0.369106	−	0.929387i	$-0.379664\pi$
$318$		0			0
$319$	−0.549459	+	0.951691i	−0.0307638	+	0.0532845i
$320$		0			0
$321$		0			0
$322$	24.7615	+	42.8882i	1.37991	+	2.39007i
$323$	13.8538			0.770849
$324$		0			0
$325$		0			0
$326$	−2.58206	−	4.47226i	−0.143007	−	0.247696i
$327$		0			0
$328$	−7.60339	+	13.1695i	−0.419827	+	0.727162i
$329$	−2.76191	+	4.78377i	−0.152269	+	0.263738i
$330$		0			0
$331$	14.8024	+	25.6385i	0.813612	+	1.40922i	0.910321	+	0.413904i	$0.135835\pi$
$331$	14.8024	+	25.6385i	0.813612	+	1.40922i	−0.0967089	+	0.995313i	$0.530832\pi$
$332$	0.780834			0.0428538
$333$		0			0
$334$	30.2531			1.65538
$335$		0			0
$336$		0			0
$337$	6.26553	−	10.8522i	0.341305	−	0.591158i	−0.643370	−	0.765555i	$-0.722464\pi$
$337$	6.26553	−	10.8522i	0.341305	−	0.591158i	0.984675	+	0.174397i	$0.0557977\pi$
$338$	−2.67886	+	4.63992i	−0.145711	+	0.252379i
$339$		0			0
$340$		0			0
$341$	−0.690479			−0.0373915
$342$		0			0
$343$	−3.65180			−0.197179
$344$	12.2269	+	21.1777i	0.659232	+	1.14182i
$345$		0			0
$346$	5.71017	−	9.89030i	0.306980	−	0.531706i
$347$	8.54872	−	14.8068i	0.458919	−	0.794872i	−0.539985	−	0.841675i	$-0.681570\pi$
$347$	8.54872	−	14.8068i	0.458919	−	0.794872i	0.998904	+	0.0468031i	$0.0149033\pi$
$348$		0			0
$349$	9.20231	+	15.9389i	0.492588	+	0.853188i	0.999964	−	0.00853709i	$-0.00271747\pi$
$349$	9.20231	+	15.9389i	0.492588	+	0.853188i	−0.507375	+	0.861725i	$0.669384\pi$
$350$		0			0
$351$		0			0
$352$	0.250340			0.0133432
$353$	15.8594	+	27.4693i	0.844110	+	1.46204i	0.886391	+	0.462936i	$0.153204\pi$
$353$	15.8594	+	27.4693i	0.844110	+	1.46204i	−0.0422810	+	0.999106i	$0.513462\pi$
$354$		0			0
$355$		0			0
$356$	0.611825	−	1.05971i	0.0324267	−	0.0561646i
$357$		0			0
$358$	−7.92844	−	13.7325i	−0.419031	−	0.725783i
$359$	11.4533			0.604483			0.302241	−	0.953231i	$-0.402265\pi$
$359$	11.4533			0.604483			0.302241	+	0.953231i	$0.402265\pi$
$360$		0			0
$361$	−0.943580			−0.0496621
$362$	5.77937	+	10.0102i	0.303757	+	0.526122i
$363$		0			0
$364$	1.34405	−	2.32797i	0.0704475	−	0.122019i
$365$		0			0
$366$		0			0
$367$	1.24619	+	2.15846i	0.0650506	+	0.112671i	0.896716	−	0.442606i	$-0.145946\pi$
$367$	1.24619	+	2.15846i	0.0650506	+	0.112671i	−0.831666	+	0.555276i	$0.812612\pi$
$368$	−37.4934			−1.95448
$369$		0			0
$370$		0			0
$371$	21.9996	+	38.1045i	1.14216	+	1.97829i
$372$		0			0
$373$	−7.51868	+	13.0227i	−0.389303	+	0.674292i	−0.992356	−	0.123409i	$-0.960617\pi$
$373$	−7.51868	+	13.0227i	−0.389303	+	0.674292i	0.603053	+	0.797701i	$0.293951\pi$
$374$	0.625086	−	1.08268i	0.0323224	−	0.0559841i
$375$		0			0
$376$	−1.92567	−	3.33536i	−0.0993088	−	0.172008i
$377$	−17.2211			−0.886932
$378$		0			0
$379$	6.27273			0.322208			0.161104	−	0.986937i	$-0.448495\pi$
$379$	6.27273			0.322208			0.161104	+	0.986937i	$0.448495\pi$
$380$		0			0
$381$		0			0
$382$	−4.22275	+	7.31402i	−0.216055	+	0.374218i
$383$	11.0944	−	19.2161i	0.566897	−	0.981894i	−0.429973	−	0.902842i	$-0.641477\pi$
$383$	11.0944	−	19.2161i	0.566897	−	0.981894i	0.996870	−	0.0790528i	$-0.0251896\pi$
$384$		0			0
$385$		0			0
$386$	12.4967			0.636065
$387$		0			0
$388$	−0.531061			−0.0269605
$389$	−15.0461	−	26.0606i	−0.762869	−	1.32133i	−0.941366	−	0.337387i	$-0.890457\pi$
$389$	−15.0461	−	26.0606i	−0.762869	−	1.32133i	0.178498	−	0.983940i	$-0.442876\pi$
$390$		0			0
$391$	−14.1746	+	24.5512i	−0.716842	+	1.24161i
$392$	10.7068	−	18.5447i	0.540774	−	0.936649i
$393$		0			0
$394$	−7.82781	−	13.5582i	−0.394359	−	0.683050i
$395$		0			0
$396$		0			0
$397$	−29.2313			−1.46708			−0.733538	−	0.679648i	$-0.762132\pi$
$397$	−29.2313			−1.46708			−0.733538	+	0.679648i	$0.762132\pi$
$398$	13.6854	+	23.7038i	0.685986	+	1.18816i
$399$		0			0
$400$		0			0
$401$	12.1171	−	20.9874i	0.605098	−	1.04806i	−0.386938	−	0.922106i	$-0.626467\pi$
$401$	12.1171	−	20.9874i	0.605098	−	1.04806i	0.992036	−	0.125954i	$-0.0401993\pi$
$402$		0			0
$403$	−5.41024	−	9.37080i	−0.269503	−	0.466793i
$404$	−0.655614			−0.0326180
$405$		0			0
$406$	24.0463			1.19340
$407$	0.296142	+	0.512934i	0.0146792	+	0.0254252i
$408$		0			0
$409$	−1.16995	+	2.02642i	−0.0578504	+	0.100200i	−0.893500	−	0.449063i	$-0.851758\pi$
$409$	−1.16995	+	2.02642i	−0.0578504	+	0.100200i	0.835650	+	0.549263i	$0.185091\pi$
$410$		0			0
$411$		0			0
$412$	−0.358584	−	0.621086i	−0.0176662	−	0.0305987i
$413$	−27.5411			−1.35521
$414$		0			0
$415$		0			0
$416$	1.96153	+	3.39748i	0.0961721	+	0.166575i
$417$		0			0
$418$	0.814706	−	1.41111i	0.0398486	−	0.0690197i
$419$	11.4212	−	19.7821i	0.557964	−	0.966421i	−0.439703	−	0.898143i	$-0.644916\pi$
$419$	11.4212	−	19.7821i	0.557964	−	0.966421i	0.997666	−	0.0682778i	$-0.0217504\pi$
$420$		0			0
$421$	−5.93792	−	10.2848i	−0.289396	−	0.501249i	0.684269	−	0.729229i	$-0.260121\pi$
$421$	−5.93792	−	10.2848i	−0.289396	−	0.501249i	−0.973666	+	0.227980i	$0.926788\pi$
$422$	15.4004			0.749679
$423$		0			0
$424$	−30.6773			−1.48982
$425$		0			0
$426$		0			0
$427$	4.88036	−	8.45303i	0.236177	−	0.409071i
$428$	0.138646	−	0.240142i	0.00670172	−	0.0116077i
$429$		0			0
$430$		0			0
$431$	−8.86916			−0.427212			−0.213606	−	0.976920i	$-0.568521\pi$
$431$	−8.86916			−0.427212			−0.213606	+	0.976920i	$0.568521\pi$
$432$		0			0
$433$	−9.37059			−0.450322			−0.225161	−	0.974322i	$-0.572291\pi$
$433$	−9.37059			−0.450322			−0.225161	+	0.974322i	$0.572291\pi$
$434$	7.55446	+	13.0847i	0.362626	+	0.628086i
$435$		0			0
$436$	1.09977	−	1.90486i	0.0526695	−	0.0912262i
$437$	−18.4745	+	31.9988i	−0.883756	+	1.53071i
$438$		0			0
$439$	−9.71155	−	16.8209i	−0.463507	−	0.802817i	0.535626	−	0.844455i	$-0.320076\pi$
$439$	−9.71155	−	16.8209i	−0.463507	−	0.802817i	−0.999133	+	0.0416380i	$0.986742\pi$
$440$		0			0
$441$		0			0
$442$	19.5914			0.931868
$443$	−5.43963	−	9.42172i	−0.258445	−	0.447639i	0.707381	−	0.706833i	$-0.249877\pi$
$443$	−5.43963	−	9.42172i	−0.258445	−	0.447639i	−0.965825	+	0.259193i	$0.916543\pi$
$444$		0			0
$445$		0			0
$446$	2.89583	−	5.01572i	0.137121	−	0.237501i
$447$		0			0
$448$	13.9301	+	24.1276i	0.658136	+	1.13992i
$449$	−1.34014			−0.0632451			−0.0316225	−	0.999500i	$-0.510067\pi$
$449$	−1.34014			−0.0632451			−0.0316225	+	0.999500i	$0.510067\pi$
$450$		0			0
$451$	1.46845			0.0691467
$452$	−0.113290	−	0.196224i	−0.00532871	−	0.00922959i
$453$		0			0
$454$	−3.55894	+	6.16426i	−0.167029	+	0.289303i
$455$		0			0
$456$		0			0
$457$	10.0556	+	17.4169i	0.470383	+	0.814728i	0.999426	−	0.0338671i	$-0.0107823\pi$
$457$	10.0556	+	17.4169i	0.470383	+	0.814728i	−0.529043	+	0.848595i	$0.677449\pi$
$458$	−27.7752			−1.29785
$459$		0			0
$460$		0			0
$461$	−16.8766	−	29.2312i	−0.786024	−	1.36143i	−0.928386	−	0.371618i	$-0.878803\pi$
$461$	−16.8766	−	29.2312i	−0.786024	−	1.36143i	0.142362	−	0.989815i	$-0.454530\pi$
$462$		0			0
$463$	−2.62268	+	4.54262i	−0.121886	+	0.211114i	−0.920512	−	0.390715i	$-0.872228\pi$
$463$	−2.62268	+	4.54262i	−0.121886	+	0.211114i	0.798625	+	0.601829i	$0.205561\pi$
$464$	−9.10262	+	15.7662i	−0.422578	+	0.731927i
$465$		0			0
$466$	8.76744	+	15.1856i	0.406144	+	0.703462i
$467$	14.2120			0.657652			0.328826	−	0.944390i	$-0.393347\pi$
$467$	14.2120			0.657652			0.328826	+	0.944390i	$0.393347\pi$
$468$		0			0
$469$	43.6282			2.01456
$470$		0			0
$471$		0			0
$472$	9.60115	−	16.6297i	0.441929	−	0.765443i
$473$	1.18070	−	2.04503i	0.0542887	−	0.0940307i
$474$		0			0
$475$		0			0
$476$	−2.14866			−0.0984837
$477$		0			0
$478$	31.8657			1.45750
$479$	10.2417	+	17.7391i	0.467954	+	0.810519i	0.999329	−	0.0366168i	$-0.0116581\pi$
$479$	10.2417	+	17.7391i	0.467954	+	0.810519i	−0.531376	+	0.847136i	$0.678325\pi$
$480$		0			0
$481$	−4.64084	+	8.03817i	−0.211604	+	0.366509i
$482$	−2.87161	+	4.97377i	−0.130798	+	0.226549i
$483$		0			0
$484$	0.931859	+	1.61403i	0.0423572	+	0.0733649i
$485$		0			0
$486$		0			0
$487$	−31.3554			−1.42085			−0.710425	−	0.703772i	$-0.751497\pi$
$487$	−31.3554			−1.42085			−0.710425	+	0.703772i	$0.751497\pi$
$488$	3.40271	+	5.89366i	0.154033	+	0.266793i
$489$		0			0
$490$		0			0
$491$	−5.19604	+	8.99980i	−0.234494	+	0.406155i	−0.959125	−	0.282981i	$-0.908677\pi$
$491$	−5.19604	+	8.99980i	−0.234494	+	0.406155i	0.724632	+	0.689136i	$0.242010\pi$
$492$		0			0
$493$	6.88260	+	11.9210i	0.309977	+	0.536896i
$494$	25.5345			1.14885
$495$		0			0
$496$	−11.4388			−0.513618
$497$	16.2009	+	28.0607i	0.726709	+	1.25870i
$498$		0			0
$499$	1.91285	−	3.31316i	0.0856310	−	0.148317i	−0.820029	−	0.572322i	$-0.806043\pi$
$499$	1.91285	−	3.31316i	0.0856310	−	0.148317i	0.905660	+	0.424005i	$0.139376\pi$
$500$		0			0
$501$		0			0
$502$	−22.1749	−	38.4081i	−0.989715	−	1.71424i
$503$	1.00236			0.0446931			0.0223466	−	0.999750i	$-0.492886\pi$
$503$	1.00236			0.0446931			0.0223466	+	0.999750i	$0.492886\pi$
$504$		0			0
$505$		0			0
$506$	1.66714	+	2.88757i	0.0741134	+	0.128368i
$507$		0			0
$508$	−0.140888	+	0.244026i	−0.00625090	+	0.0108269i
$509$	2.28161	−	3.95187i	0.101131	−	0.175163i	−0.811020	−	0.585018i	$-0.801087\pi$
$509$	2.28161	−	3.95187i	0.101131	−	0.175163i	0.912151	+	0.409855i	$0.134421\pi$
$510$		0			0
$511$	−0.780062	−	1.35111i	−0.0345079	−	0.0597695i
$512$	−19.0969			−0.843971
$513$		0			0
$514$	−24.1822			−1.06663
$515$		0			0
$516$		0			0
$517$	−0.185953	+	0.322080i	−0.00817822	+	0.0141651i
$518$	6.48013	−	11.2239i	0.284721	−	0.493151i
$519$		0			0
$520$		0			0
$521$	−39.3708			−1.72486			−0.862432	−	0.506173i	$-0.831060\pi$
$521$	−39.3708			−1.72486			−0.862432	+	0.506173i	$0.831060\pi$
$522$		0			0
$523$	10.3998			0.454749			0.227375	−	0.973807i	$-0.426986\pi$
$523$	10.3998			0.454749			0.227375	+	0.973807i	$0.426986\pi$
$524$	−1.12326	−	1.94555i	−0.0490699	−	0.0849916i
$525$		0			0
$526$	−19.0103	+	32.9268i	−0.828888	+	1.43568i
$527$	−4.32452	+	7.49029i	−0.188379	+	0.326282i
$528$		0			0
$529$	−26.3046	−	45.5608i	−1.14368	−	1.98091i
$530$		0			0
$531$		0			0
$532$	−2.80046			−0.121415
$533$	11.5060	+	19.9290i	0.498381	+	0.863222i
$534$		0			0
$535$		0			0
$536$	−15.2093	+	26.3433i	−0.656942	+	1.13786i
$537$		0			0
$538$	−9.22298	−	15.9747i	−0.397631	−	0.688717i
$539$	−2.06781			−0.0890670
$540$		0			0
$541$	13.7093			0.589408			0.294704	−	0.955589i	$-0.404779\pi$
$541$	13.7093			0.589408			0.294704	+	0.955589i	$0.404779\pi$
$542$	−14.4719	−	25.0661i	−0.621622	−	1.07668i
$543$		0			0
$544$	1.56790	−	2.71568i	0.0672230	−	0.116434i
$545$		0			0
$546$		0			0
$547$	11.3924	+	19.7322i	0.487102	+	0.843686i	0.999890	−	0.0148294i	$-0.00472053\pi$
$547$	11.3924	+	19.7322i	0.487102	+	0.843686i	−0.512788	+	0.858515i	$0.671387\pi$
$548$	3.45950			0.147783
$549$		0			0
$550$		0			0
$551$	8.97044	+	15.5373i	0.382154	+	0.661910i
$552$		0			0
$553$	−5.88089	+	10.1860i	−0.250081	+	0.433153i
$554$	−15.3440	+	26.5766i	−0.651905	+	1.12913i
$555$		0			0
$556$	−0.261583	−	0.453075i	−0.0110936	−	0.0192146i
$557$	−18.2341			−0.772605			−0.386303	−	0.922372i	$-0.626248\pi$
$557$	−18.2341			−0.772605			−0.386303	+	0.922372i	$0.626248\pi$
$558$		0			0
$559$	37.0054			1.56516
$560$		0			0
$561$		0			0
$562$	3.48013	−	6.02776i	0.146800	−	0.254266i
$563$	−12.0556	+	20.8809i	−0.508083	+	0.880025i	0.491873	+	0.870667i	$0.336312\pi$
$563$	−12.0556	+	20.8809i	−0.508083	+	0.880025i	−0.999956	+	0.00935862i	$0.997021\pi$
$564$		0			0
$565$		0			0
$566$	−34.1095			−1.43373
$567$		0			0
$568$	−22.5913			−0.947910
$569$	−16.0024	−	27.7170i	−0.670857	−	1.16196i	−0.977661	−	0.210186i	$-0.932593\pi$
$569$	−16.0024	−	27.7170i	−0.670857	−	1.16196i	0.306804	−	0.951773i	$-0.400740\pi$
$570$		0			0
$571$	9.89042	−	17.1307i	0.413901	−	0.716898i	−0.581411	−	0.813610i	$-0.697499\pi$
$571$	9.89042	−	17.1307i	0.413901	−	0.716898i	0.995312	+	0.0967121i	$0.0308326\pi$
$572$	0.0904921	−	0.156737i	0.00378367	−	0.00655350i
$573$		0			0
$574$	−16.0662	−	27.8274i	−0.670589	−	1.16150i
$575$		0			0
$576$		0			0
$577$	35.4119			1.47422			0.737108	−	0.675775i	$-0.236191\pi$
$577$	35.4119			1.47422			0.737108	+	0.675775i	$0.236191\pi$
$578$	4.69275	+	8.12808i	0.195193	+	0.338084i
$579$		0			0
$580$		0			0
$581$	8.85321	−	15.3342i	0.367293	−	0.636170i
$582$		0			0
$583$	1.48119	+	2.56549i	0.0613445	+	0.106252i
$584$	1.08776			0.0450117
$585$		0			0
$586$	−24.8675			−1.02726
$587$	16.1925	+	28.0463i	0.668338	+	1.15759i	0.978369	+	0.206868i	$0.0663272\pi$
$587$	16.1925	+	28.0463i	0.668338	+	1.15759i	−0.310031	+	0.950726i	$0.600339\pi$
$588$		0			0
$589$	−5.63636	+	9.76247i	−0.232242	+	0.402255i
$590$		0			0
$591$		0			0
$592$	4.90605	+	8.49752i	0.201637	+	0.349246i
$593$	29.2504			1.20117			0.600585	−	0.799561i	$-0.294934\pi$
$593$	29.2504			1.20117			0.600585	+	0.799561i	$0.294934\pi$
$594$		0			0
$595$		0			0
$596$	0.346210	+	0.599654i	0.0141813	+	0.0245628i
$597$		0			0
$598$	−26.1257	+	45.2510i	−1.06836	+	1.85045i
$599$	2.03081	−	3.51747i	0.0829767	−	0.143720i	−0.821551	−	0.570136i	$-0.806891\pi$
$599$	2.03081	−	3.51747i	0.0829767	−	0.143720i	0.904527	+	0.426416i	$0.140224\pi$
$600$		0			0
$601$	−23.4538	−	40.6232i	−0.956700	−	1.65705i	−0.730429	−	0.682989i	$-0.760680\pi$
$601$	−23.4538	−	40.6232i	−0.956700	−	1.65705i	−0.226271	−	0.974064i	$-0.572653\pi$
$602$	−51.6717			−2.10598
$603$		0			0
$604$	−2.32190			−0.0944768
$605$		0			0
$606$		0			0
$607$	−20.4733	+	35.4608i	−0.830987	+	1.43931i	0.0662702	+	0.997802i	$0.478890\pi$
$607$	−20.4733	+	35.4608i	−0.830987	+	1.43931i	−0.897257	+	0.441509i	$0.854443\pi$
$608$	2.04352	−	3.53948i	0.0828756	−	0.143545i
$609$		0			0
$610$		0			0
$611$	−5.82813			−0.235781
$612$		0			0
$613$	33.3827			1.34831			0.674157	−	0.738588i	$-0.264507\pi$
$613$	33.3827			1.34831			0.674157	+	0.738588i	$0.264507\pi$
$614$	−16.7345	−	28.9850i	−0.675350	−	1.16974i
$615$		0			0
$616$	1.35602	−	2.34869i	0.0546355	−	0.0946314i
$617$	−1.12440	+	1.94752i	−0.0452666	+	0.0784040i	−0.887771	−	0.460285i	$-0.847747\pi$
$617$	−1.12440	+	1.94752i	−0.0452666	+	0.0784040i	0.842504	+	0.538689i	$0.181080\pi$
$618$		0			0
$619$	17.1467	+	29.6990i	0.689184	+	1.19370i	0.972102	+	0.234558i	$0.0753642\pi$
$619$	17.1467	+	29.6990i	0.689184	+	1.19370i	−0.282918	+	0.959144i	$0.591302\pi$
$620$		0			0
$621$		0			0
$622$	−46.5454			−1.86630
$623$	−13.8739	−	24.0303i	−0.555847	−	0.962756i
$624$		0			0
$625$		0			0
$626$	−22.4669	+	38.9137i	−0.897956	+	1.55531i
$627$		0			0
$628$	0.175816	+	0.304522i	0.00701581	+	0.0121517i
$629$	7.41904			0.295817
$630$		0			0
$631$	−18.7552			−0.746633			−0.373316	−	0.927704i	$-0.621779\pi$
$631$	−18.7552			−0.746633			−0.373316	+	0.927704i	$0.621779\pi$
$632$	−4.10030	−	7.10193i	−0.163101	−	0.282499i
$633$		0			0
$634$	−16.2713	+	28.1828i	−0.646218	+	1.11928i
$635$		0			0
$636$		0			0
$637$	−16.2023	−	28.0632i	−0.641959	−	1.11191i
$638$	1.61899			0.0640963
$639$		0			0
$640$		0			0
$641$	10.3175	+	17.8704i	0.407517	+	0.705840i	0.994611	−	0.103679i	$-0.0330614\pi$
$641$	10.3175	+	17.8704i	0.407517	+	0.705840i	−0.587094	+	0.809519i	$0.699728\pi$
$642$		0			0
$643$	13.5970	−	23.5506i	0.536212	−	0.928746i	−0.462892	−	0.886415i	$-0.653188\pi$
$643$	13.5970	−	23.5506i	0.536212	−	0.928746i	0.999104	−	0.0423312i	$-0.0134785\pi$
$644$	2.86530	−	4.96285i	0.112909	−	0.195564i
$645$		0			0
$646$	−10.2051	−	17.6758i	−0.401515	−	0.695445i
$647$	16.7316			0.657787			0.328893	−	0.944367i	$-0.393324\pi$
$647$	16.7316			0.657787			0.328893	+	0.944367i	$0.393324\pi$
$648$		0			0
$649$	−1.85428			−0.0727869
$650$		0			0
$651$		0			0
$652$	−0.298785	+	0.517511i	−0.0117013	+	0.0202673i
$653$	22.8666	−	39.6060i	0.894837	−	1.54990i	0.0608319	−	0.998148i	$-0.480625\pi$
$653$	22.8666	−	39.6060i	0.894837	−	1.54990i	0.834006	−	0.551756i	$-0.186042\pi$
$654$		0			0
$655$		0			0
$656$	24.3271			0.949814
$657$		0			0
$658$	8.13799			0.317252
$659$	9.30543	+	16.1175i	0.362488	+	0.627848i	0.988370	−	0.152070i	$-0.0485941\pi$
$659$	9.30543	+	16.1175i	0.362488	+	0.627848i	−0.625882	+	0.779918i	$0.715261\pi$
$660$		0			0
$661$	−8.39799	+	14.5457i	−0.326644	+	0.565764i	−0.981844	−	0.189692i	$-0.939251\pi$
$661$	−8.39799	+	14.5457i	−0.326644	+	0.565764i	0.655200	+	0.755456i	$0.272584\pi$
$662$	21.8077	−	37.7720i	0.847579	−	1.46805i
$663$		0			0
$664$	6.17267	+	10.6914i	0.239546	+	0.414906i
$665$		0			0
$666$		0			0
$667$	−36.7126			−1.42152
$668$	−1.75038	−	3.03175i	−0.0677243	−	0.117302i
$669$		0			0
$670$		0			0
$671$	0.328584	−	0.569124i	0.0126848	−	0.0219708i
$672$		0			0
$673$	24.9740	+	43.2562i	0.962676	+	1.66740i	0.715733	+	0.698374i	$0.246093\pi$
$673$	24.9740	+	43.2562i	0.962676	+	1.66740i	0.246944	+	0.969030i	$0.420574\pi$
$674$	−18.4614			−0.711108
$675$		0			0
$676$	0.619973			0.0238451
$677$	−5.41553	−	9.37998i	−0.208136	−	0.360502i	0.742991	−	0.669301i	$-0.233406\pi$
$677$	−5.41553	−	9.37998i	−0.208136	−	0.360502i	−0.951127	+	0.308799i	$0.900073\pi$
$678$		0			0
$679$	−6.02125	+	10.4291i	−0.231074	+	0.400232i
$680$		0			0
$681$		0			0
$682$	0.508625	+	0.880965i	0.0194763	+	0.0337339i
$683$	0.429870			0.0164485			0.00822426	−	0.999966i	$-0.497382\pi$
$683$	0.429870			0.0164485			0.00822426	+	0.999966i	$0.497382\pi$
$684$		0			0
$685$		0			0
$686$	2.69001	+	4.65924i	0.102705	+	0.177891i
$687$		0			0
$688$	19.5601	−	33.8791i	0.745721	−	1.29163i
$689$	−23.2116	+	40.2037i	−0.884293	+	1.53164i
$690$		0			0
$691$	−17.3518	−	30.0542i	−0.660093	−	1.14331i	−0.980591	−	0.196065i	$-0.937184\pi$
$691$	−17.3518	−	30.0542i	−0.660093	−	1.14331i	0.320498	−	0.947249i	$-0.396150\pi$
$692$	−1.32151			−0.0502364
$693$		0			0
$694$	−25.1889			−0.956157
$695$		0			0
$696$		0			0
$697$	9.19701	−	15.9297i	0.348362	−	0.603380i
$698$	13.5573	−	23.4820i	0.513153	−	0.888807i
$699$		0			0
$700$		0			0
$701$	−1.84808			−0.0698010			−0.0349005	−	0.999391i	$-0.511111\pi$
$701$	−1.84808			−0.0698010			−0.0349005	+	0.999391i	$0.511111\pi$
$702$		0			0
$703$	9.66962			0.364696
$704$	0.937884	+	1.62446i	0.0353478	+	0.0612242i
$705$		0			0
$706$	23.3649	−	40.4692i	0.879351	−	1.52308i
$707$	−7.43344	+	12.8751i	−0.279563	+	0.484218i
$708$		0			0
$709$	3.15338	+	5.46181i	0.118428	+	0.205123i	0.919145	−	0.393920i	$-0.128881\pi$
$709$	3.15338	+	5.46181i	0.118428	+	0.205123i	−0.800717	+	0.599043i	$0.795548\pi$
$710$		0			0
$711$		0			0
$712$	19.3465			0.725040
$713$	−11.5337	−	19.9770i	−0.431942	−	0.748145i
$714$		0			0
$715$		0			0
$716$	−0.917446	+	1.58906i	−0.0342866	+	0.0593861i
$717$		0			0
$718$	−8.43682	−	14.6130i	−0.314859	−	0.545352i
$719$	−18.0129			−0.671770			−0.335885	−	0.941903i	$-0.609035\pi$
$719$	−18.0129			−0.671770			−0.335885	+	0.941903i	$0.609035\pi$
$720$		0			0
$721$	−16.2627			−0.605655
$722$	0.695067	+	1.20389i	0.0258677	+	0.0448042i
$723$		0			0
$724$	0.668765	−	1.15833i	0.0248544	−	0.0430491i
$725$		0			0
$726$		0			0
$727$	13.1412	+	22.7612i	0.487379	+	0.844165i	0.999895	−	0.0145126i	$-0.00461966\pi$
$727$	13.1412	+	22.7612i	0.487379	+	0.844165i	−0.512516	+	0.858678i	$0.671286\pi$
$728$	42.5002			1.57516
$729$		0			0
$730$		0			0
$731$	−14.7896	−	25.6164i	−0.547014	−	0.947456i
$732$		0			0
$733$	23.8317	−	41.2777i	0.880243	−	1.52462i	0.0291714	−	0.999574i	$-0.490713\pi$
$733$	23.8317	−	41.2777i	0.880243	−	1.52462i	0.851071	−	0.525050i	$-0.175954\pi$
$734$	1.83595	−	3.17997i	0.0677663	−	0.117375i
$735$		0			0
$736$	4.18167	+	7.24287i	0.154138	+	0.266976i
$737$	2.93739			0.108200
$738$		0			0
$739$	−10.0273			−0.368859			−0.184429	−	0.982846i	$-0.559044\pi$
$739$	−10.0273			−0.368859			−0.184429	+	0.982846i	$0.559044\pi$
$740$		0			0
$741$		0			0
$742$	32.4110	−	56.1376i	1.18985	−	2.06088i
$743$	−4.13633	+	7.16433i	−0.151747	+	0.262834i	−0.931870	−	0.362793i	$-0.881823\pi$
$743$	−4.13633	+	7.16433i	−0.151747	+	0.262834i	0.780123	+	0.625627i	$0.215157\pi$
$744$		0			0
$745$		0			0
$746$	22.1539			0.811111
$747$		0			0
$748$	−0.144665			−0.00528946
$749$	−3.14398	−	5.44554i	−0.114879	−	0.198976i
$750$		0			0
$751$	2.89880	−	5.02087i	0.105779	−	0.183214i	−0.808277	−	0.588802i	$-0.799600\pi$
$751$	2.89880	−	5.02087i	0.105779	−	0.183214i	0.914056	+	0.405588i	$0.132933\pi$
$752$	−3.08060	+	5.33575i	−0.112338	+	0.194575i
$753$		0			0
$754$	12.6855	+	21.9720i	0.461980	+	0.800173i
$755$		0			0
$756$		0			0
$757$	25.2804			0.918830			0.459415	−	0.888222i	$-0.348059\pi$
$757$	25.2804			0.918830			0.459415	+	0.888222i	$0.348059\pi$
$758$	−4.62066	−	8.00322i	−0.167830	−	0.290690i
$759$		0			0
$760$		0			0
$761$	9.73190	−	16.8561i	0.352781	−	0.611035i	−0.633954	−	0.773370i	$-0.718569\pi$
$761$	9.73190	−	16.8561i	0.352781	−	0.611035i	0.986736	+	0.162335i	$0.0519027\pi$
$762$		0			0
$763$	−24.9387	−	43.1951i	−0.902843	−	1.56377i
$764$	0.977278			0.0353567
$765$		0			0
$766$	−32.6897			−1.18113
$767$	−14.5292	−	25.1653i	−0.524618	−	0.908666i
$768$		0			0
$769$	24.6715	−	42.7324i	0.889678	−	1.54097i	0.0494224	−	0.998778i	$-0.484262\pi$
$769$	24.6715	−	42.7324i	0.889678	−	1.54097i	0.840256	−	0.542190i	$-0.182405\pi$
$770$		0			0
$771$		0			0
$772$	−0.723033	−	1.25233i	−0.0260225	−	0.0450723i
$773$	20.8502			0.749930			0.374965	−	0.927039i	$-0.377655\pi$
$773$	20.8502			0.749930			0.374965	+	0.927039i	$0.377655\pi$
$774$		0			0
$775$		0			0
$776$	−4.19816	−	7.27143i	−0.150705	−	0.261029i
$777$		0			0
$778$	−22.1668	+	38.3940i	−0.794717	+	1.37649i
$779$	11.9869	−	20.7620i	0.429476	−	0.743875i
$780$		0			0
$781$	1.09077	+	1.88927i	0.0390308	+	0.0676034i
$782$	41.7657			1.49354
$783$		0			0
$784$	−34.2564			−1.22344
$785$		0			0
$786$		0			0
$787$	−22.0941	+	38.2682i	−0.787571	+	1.36411i	0.139880	+	0.990169i	$0.455328\pi$
$787$	−22.0941	+	38.2682i	−0.787571	+	1.36411i	−0.927451	+	0.373945i	$0.878005\pi$
$788$	−0.905801	+	1.56889i	−0.0322678	+	0.0558895i
$789$		0			0
$790$

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:	\( N \)	\(=\)	\( 675 = 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	675.e (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.736627 - 1.27588i) q^{2} +(-0.0852394 + 0.147639i) q^{4} +(1.93291 + 3.34791i) q^{7} -2.69535 q^{8} +O(q^{10})\)	\(q+(-0.736627 - 1.27588i) q^{2} +(-0.0852394 + 0.147639i) q^{4} +(1.93291 + 3.34791i) q^{7} -2.69535 q^{8} +(0.130139 + 0.225407i) q^{11} +(-2.03940 + 3.53235i) q^{13} +(2.84768 - 4.93232i) q^{14} +(2.15595 + 3.73421i) q^{16} +3.26028 q^{17} +4.24928 q^{19} +(0.191728 - 0.332082i) q^{22} +(-4.34768 + 7.53039i) q^{23} +6.00912 q^{26} -0.659042 q^{28} +(2.11105 + 3.65644i) q^{29} +(-1.32643 + 2.29744i) q^{31} +(0.480909 - 0.832959i) q^{32} +(-2.40161 - 4.15971i) q^{34} +2.27559 q^{37} +(-3.13014 - 5.42156i) q^{38} +(2.82093 - 4.88599i) q^{41} +(-4.53631 - 7.85712i) q^{43} -0.0443719 q^{44} +12.8105 q^{46} +(0.714441 + 1.23745i) q^{47} +(-3.97232 + 6.88026i) q^{49} +(-0.347675 - 0.602191i) q^{52} +11.3816 q^{53} +(-5.20988 - 9.02378i) q^{56} +(3.11011 - 5.38687i) q^{58} +(-3.56212 + 6.16977i) q^{59} +(-1.26244 - 2.18660i) q^{61} +3.90833 q^{62} +7.20679 q^{64} +(5.64280 - 9.77361i) q^{67} +(-0.277904 + 0.481344i) q^{68} +8.38158 q^{71} -0.403568 q^{73} +(-1.67626 - 2.90337i) q^{74} +(-0.362207 + 0.627360i) q^{76} +(-0.503095 + 0.871386i) q^{77} +(1.52125 + 2.63488i) q^{79} -8.31189 q^{82} +(-2.29012 - 3.96660i) q^{83} +(-6.68314 + 11.5755i) q^{86} +(-0.350770 - 0.607551i) q^{88} -7.17772 q^{89} -15.7680 q^{91} +(-0.741187 - 1.28377i) q^{92} +(1.05255 - 1.82308i) q^{94} +(1.55756 + 2.69777i) q^{97} +11.7045 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{2} - 4 q^{4} - q^{7} + 18 q^{8}+O(q^{10}) \) 8 * q - 2 * q^2 - 4 * q^4 - q^7 + 18 * q^8	\( 8 q - 2 q^{2} - 4 q^{4} - q^{7} + 18 q^{8} - q^{11} + 2 q^{13} + 3 q^{14} - 4 q^{16} + 22 q^{17} + 4 q^{19} + 3 q^{22} - 15 q^{23} + 20 q^{26} + 8 q^{28} + q^{29} + 4 q^{31} - 10 q^{32} - 9 q^{34} + 2 q^{37} - 23 q^{38} - 5 q^{41} - 10 q^{43} - 44 q^{44} - 20 q^{47} + 3 q^{49} + 17 q^{52} + 40 q^{53} - 30 q^{56} - 18 q^{58} + 17 q^{59} + 13 q^{61} - 12 q^{62} + 38 q^{64} + 17 q^{67} - 34 q^{68} + 16 q^{71} - 4 q^{73} + 40 q^{74} - 11 q^{76} - 12 q^{77} + 7 q^{79} - 24 q^{82} - 30 q^{83} - 34 q^{86} + 9 q^{88} + 18 q^{89} - 34 q^{91} + 12 q^{92} - 3 q^{94} - 19 q^{97} + 26 q^{98}+O(q^{100}) \) 8 * q - 2 * q^2 - 4 * q^4 - q^7 + 18 * q^8 - q^11 + 2 * q^13 + 3 * q^14 - 4 * q^16 + 22 * q^17 + 4 * q^19 + 3 * q^22 - 15 * q^23 + 20 * q^26 + 8 * q^28 + q^29 + 4 * q^31 - 10 * q^32 - 9 * q^34 + 2 * q^37 - 23 * q^38 - 5 * q^41 - 10 * q^43 - 44 * q^44 - 20 * q^47 + 3 * q^49 + 17 * q^52 + 40 * q^53 - 30 * q^56 - 18 * q^58 + 17 * q^59 + 13 * q^61 - 12 * q^62 + 38 * q^64 + 17 * q^67 - 34 * q^68 + 16 * q^71 - 4 * q^73 + 40 * q^74 - 11 * q^76 - 12 * q^77 + 7 * q^79 - 24 * q^82 - 30 * q^83 - 34 * q^86 + 9 * q^88 + 18 * q^89 - 34 * q^91 + 12 * q^92 - 3 * q^94 - 19 * q^97 + 26 * q^98

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