LMFDB - Embedded newform 25.3.c.a.18.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [25,3,Mod(7,25)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(25, base_ring=CyclotomicField(4))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("25.7");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 25 = 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	25.c (of order $4$, degree $2$, 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:	$0.681200660901$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(i, \sqrt{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9 $ x^4 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			18.2
Root			$1.22474 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	25.18
Dual form			25.3.c.a.7.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.22474 - 1.22474i) q^{2} +(-1.22474 - 1.22474i) q^{3} +1.00000i q^{4} -3.00000 q^{6} +(-4.89898 + 4.89898i) q^{7} +(6.12372 + 6.12372i) q^{8} -6.00000i q^{9} +O(q^{10})$	\(q+(1.22474 - 1.22474i) q^{2} +(-1.22474 - 1.22474i) q^{3} +1.00000i q^{4} -3.00000 q^{6} +(-4.89898 + 4.89898i) q^{7} +(6.12372 + 6.12372i) q^{8} -6.00000i q^{9} -3.00000 q^{11} +(1.22474 - 1.22474i) q^{12} +(-7.34847 - 7.34847i) q^{13} +12.0000i q^{14} +11.0000 q^{16} +(13.4722 - 13.4722i) q^{17} +(-7.34847 - 7.34847i) q^{18} -5.00000i q^{19} +12.0000 q^{21} +(-3.67423 + 3.67423i) q^{22} +(17.1464 + 17.1464i) q^{23} -15.0000i q^{24} -18.0000 q^{26} +(-18.3712 + 18.3712i) q^{27} +(-4.89898 - 4.89898i) q^{28} +30.0000i q^{29} -38.0000 q^{31} +(-11.0227 + 11.0227i) q^{32} +(3.67423 + 3.67423i) q^{33} -33.0000i q^{34} +6.00000 q^{36} +(19.5959 - 19.5959i) q^{37} +(-6.12372 - 6.12372i) q^{38} +18.0000i q^{39} +57.0000 q^{41} +(14.6969 - 14.6969i) q^{42} +(4.89898 + 4.89898i) q^{43} -3.00000i q^{44} +42.0000 q^{46} +(7.34847 - 7.34847i) q^{47} +(-13.4722 - 13.4722i) q^{48} +1.00000i q^{49} -33.0000 q^{51} +(7.34847 - 7.34847i) q^{52} +(-31.8434 - 31.8434i) q^{53} +45.0000i q^{54} -60.0000 q^{56} +(-6.12372 + 6.12372i) q^{57} +(36.7423 + 36.7423i) q^{58} -90.0000i q^{59} -28.0000 q^{61} +(-46.5403 + 46.5403i) q^{62} +(29.3939 + 29.3939i) q^{63} +71.0000i q^{64} +9.00000 q^{66} +(-47.7650 + 47.7650i) q^{67} +(13.4722 + 13.4722i) q^{68} -42.0000i q^{69} +42.0000 q^{71} +(36.7423 - 36.7423i) q^{72} +(-13.4722 - 13.4722i) q^{73} -48.0000i q^{74} +5.00000 q^{76} +(14.6969 - 14.6969i) q^{77} +(22.0454 + 22.0454i) q^{78} +80.0000i q^{79} -9.00000 q^{81} +(69.8105 - 69.8105i) q^{82} +(-111.452 - 111.452i) q^{83} +12.0000i q^{84} +12.0000 q^{86} +(36.7423 - 36.7423i) q^{87} +(-18.3712 - 18.3712i) q^{88} +15.0000i q^{89} +72.0000 q^{91} +(-17.1464 + 17.1464i) q^{92} +(46.5403 + 46.5403i) q^{93} -18.0000i q^{94} +27.0000 q^{96} +(-53.8888 + 53.8888i) q^{97} +(1.22474 + 1.22474i) q^{98} +18.0000i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 12 q^{6}+O(q^{10}) $ 4 * q - 12 * q^6	$ 4 q - 12 q^{6} - 12 q^{11} + 44 q^{16} + 48 q^{21} - 72 q^{26} - 152 q^{31} + 24 q^{36} + 228 q^{41} + 168 q^{46} - 132 q^{51} - 240 q^{56} - 112 q^{61} + 36 q^{66} + 168 q^{71} + 20 q^{76} - 36 q^{81} + 48 q^{86} + 288 q^{91} + 108 q^{96}+O(q^{100}) $ 4 * q - 12 * q^6 - 12 * q^11 + 44 * q^16 + 48 * q^21 - 72 * q^26 - 152 * q^31 + 24 * q^36 + 228 * q^41 + 168 * q^46 - 132 * q^51 - 240 * q^56 - 112 * q^61 + 36 * q^66 + 168 * q^71 + 20 * q^76 - 36 * q^81 + 48 * q^86 + 288 * q^91 + 108 * q^96

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/25\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$e\left(\frac{3}{4}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	1.22474	−	1.22474i	0.612372	−	0.612372i	−0.331191	−	0.943564i	$-0.607451\pi$
$2$	1.22474	−	1.22474i	0.612372	−	0.612372i	0.943564	+	0.331191i	$0.107451\pi$
$3$	−1.22474	−	1.22474i	−0.408248	−	0.408248i	0.472879	−	0.881127i	$-0.343215\pi$
$3$	−1.22474	−	1.22474i	−0.408248	−	0.408248i	−0.881127	+	0.472879i	$0.843215\pi$
$4$			1.00000i			0.250000i
$5$		0			0
$5$		0			0
$6$	−3.00000			−0.500000
$7$	−4.89898	+	4.89898i	−0.699854	+	0.699854i	−0.964379	−	0.264525i	$-0.914785\pi$
$7$	−4.89898	+	4.89898i	−0.699854	+	0.699854i	0.264525	+	0.964379i	$0.414785\pi$
$8$	6.12372	+	6.12372i	0.765466	+	0.765466i
$9$		−	6.00000i		−	0.666667i
$10$		0			0
$11$	−3.00000			−0.272727			−0.136364	−	0.990659i	$-0.543542\pi$
$11$	−3.00000			−0.272727			−0.136364	+	0.990659i	$0.543542\pi$
$12$	1.22474	−	1.22474i	0.102062	−	0.102062i
$13$	−7.34847	−	7.34847i	−0.565267	−	0.565267i	0.365532	−	0.930799i	$-0.380887\pi$
$13$	−7.34847	−	7.34847i	−0.565267	−	0.565267i	−0.930799	+	0.365532i	$0.880887\pi$
$14$			12.0000i			0.857143i
$15$		0			0
$16$	11.0000			0.687500
$17$	13.4722	−	13.4722i	0.792482	−	0.792482i	−0.189415	−	0.981897i	$-0.560659\pi$
$17$	13.4722	−	13.4722i	0.792482	−	0.792482i	0.981897	+	0.189415i	$0.0606592\pi$
$18$	−7.34847	−	7.34847i	−0.408248	−	0.408248i
$19$		−	5.00000i		−	0.263158i	−0.991306	−	0.131579i	$-0.957995\pi$
$19$		−	5.00000i		−	0.263158i	0.991306	−	0.131579i	$-0.0420047\pi$
$20$		0			0
$21$	12.0000			0.571429
$22$	−3.67423	+	3.67423i	−0.167011	+	0.167011i
$23$	17.1464	+	17.1464i	0.745497	+	0.745497i	0.973630	−	0.228133i	$-0.0732621\pi$
$23$	17.1464	+	17.1464i	0.745497	+	0.745497i	−0.228133	+	0.973630i	$0.573262\pi$
$24$		−	15.0000i		−	0.625000i
$25$		0			0
$26$	−18.0000			−0.692308
$27$	−18.3712	+	18.3712i	−0.680414	+	0.680414i
$28$	−4.89898	−	4.89898i	−0.174964	−	0.174964i
$29$			30.0000i			1.03448i	0.855840	+	0.517241i	$0.173041\pi$
$29$			30.0000i			1.03448i	−0.855840	+	0.517241i	$0.826959\pi$
$30$		0			0
$31$	−38.0000			−1.22581			−0.612903	−	0.790158i	$-0.709998\pi$
$31$	−38.0000			−1.22581			−0.612903	+	0.790158i	$0.709998\pi$
$32$	−11.0227	+	11.0227i	−0.344459	+	0.344459i
$33$	3.67423	+	3.67423i	0.111340	+	0.111340i
$34$		−	33.0000i		−	0.970588i
$35$		0			0
$36$	6.00000			0.166667
$37$	19.5959	−	19.5959i	0.529619	−	0.529619i	−0.390839	−	0.920459i	$-0.627815\pi$
$37$	19.5959	−	19.5959i	0.529619	−	0.529619i	0.920459	+	0.390839i	$0.127815\pi$
$38$	−6.12372	−	6.12372i	−0.161151	−	0.161151i
$39$			18.0000i			0.461538i
$40$		0			0
$41$	57.0000			1.39024			0.695122	−	0.718892i	$-0.255350\pi$
$41$	57.0000			1.39024			0.695122	+	0.718892i	$0.255350\pi$
$42$	14.6969	−	14.6969i	0.349927	−	0.349927i
$43$	4.89898	+	4.89898i	0.113930	+	0.113930i	0.761773	−	0.647844i	$-0.224329\pi$
$43$	4.89898	+	4.89898i	0.113930	+	0.113930i	−0.647844	+	0.761773i	$0.724329\pi$
$44$		−	3.00000i		−	0.0681818i
$45$		0			0
$46$	42.0000			0.913043
$47$	7.34847	−	7.34847i	0.156350	−	0.156350i	−0.624597	−	0.780947i	$-0.714737\pi$
$47$	7.34847	−	7.34847i	0.156350	−	0.156350i	0.780947	+	0.624597i	$0.214737\pi$
$48$	−13.4722	−	13.4722i	−0.280671	−	0.280671i
$49$			1.00000i			0.0204082i
$50$		0			0
$51$	−33.0000			−0.647059
$52$	7.34847	−	7.34847i	0.141317	−	0.141317i
$53$	−31.8434	−	31.8434i	−0.600818	−	0.600818i	0.339711	−	0.940530i	$-0.389671\pi$
$53$	−31.8434	−	31.8434i	−0.600818	−	0.600818i	−0.940530	+	0.339711i	$0.889671\pi$
$54$			45.0000i			0.833333i
$55$		0			0
$56$	−60.0000			−1.07143
$57$	−6.12372	+	6.12372i	−0.107434	+	0.107434i
$58$	36.7423	+	36.7423i	0.633489	+	0.633489i
$59$		−	90.0000i		−	1.52542i	−0.646738	−	0.762712i	$-0.723867\pi$
$59$		−	90.0000i		−	1.52542i	0.646738	−	0.762712i	$-0.276133\pi$
$60$		0			0
$61$	−28.0000			−0.459016			−0.229508	−	0.973307i	$-0.573712\pi$
$61$	−28.0000			−0.459016			−0.229508	+	0.973307i	$0.573712\pi$
$62$	−46.5403	+	46.5403i	−0.750650	+	0.750650i
$63$	29.3939	+	29.3939i	0.466569	+	0.466569i
$64$			71.0000i			1.10938i
$65$		0			0
$66$	9.00000			0.136364
$67$	−47.7650	+	47.7650i	−0.712911	+	0.712911i	−0.967143	−	0.254232i	$-0.918177\pi$
$67$	−47.7650	+	47.7650i	−0.712911	+	0.712911i	0.254232	+	0.967143i	$0.418177\pi$
$68$	13.4722	+	13.4722i	0.198120	+	0.198120i
$69$		−	42.0000i		−	0.608696i
$70$		0			0
$71$	42.0000			0.591549			0.295775	−	0.955258i	$-0.404422\pi$
$71$	42.0000			0.591549			0.295775	+	0.955258i	$0.404422\pi$
$72$	36.7423	−	36.7423i	0.510310	−	0.510310i
$73$	−13.4722	−	13.4722i	−0.184551	−	0.184551i	0.608785	−	0.793335i	$-0.291657\pi$
$73$	−13.4722	−	13.4722i	−0.184551	−	0.184551i	−0.793335	+	0.608785i	$0.791657\pi$
$74$		−	48.0000i		−	0.648649i
$75$		0			0
$76$	5.00000			0.0657895
$77$	14.6969	−	14.6969i	0.190869	−	0.190869i
$78$	22.0454	+	22.0454i	0.282633	+	0.282633i
$79$			80.0000i			1.01266i	0.862340	+	0.506329i	$0.168998\pi$
$79$			80.0000i			1.01266i	−0.862340	+	0.506329i	$0.831002\pi$
$80$		0			0
$81$	−9.00000			−0.111111
$82$	69.8105	−	69.8105i	0.851347	−	0.851347i
$83$	−111.452	−	111.452i	−1.34279	−	1.34279i	−0.893268	−	0.449525i	$-0.851593\pi$
$83$	−111.452	−	111.452i	−1.34279	−	1.34279i	−0.449525	−	0.893268i	$-0.648407\pi$
$84$			12.0000i			0.142857i
$85$		0			0
$86$	12.0000			0.139535
$87$	36.7423	−	36.7423i	0.422326	−	0.422326i
$88$	−18.3712	−	18.3712i	−0.208763	−	0.208763i
$89$			15.0000i			0.168539i	0.996443	+	0.0842697i	$0.0268557\pi$
$89$			15.0000i			0.168539i	−0.996443	+	0.0842697i	$0.973144\pi$
$90$		0			0
$91$	72.0000			0.791209
$92$	−17.1464	+	17.1464i	−0.186374	+	0.186374i
$93$	46.5403	+	46.5403i	0.500433	+	0.500433i
$94$		−	18.0000i		−	0.191489i
$95$		0			0
$96$	27.0000			0.281250
$97$	−53.8888	+	53.8888i	−0.555554	+	0.555554i	−0.928039	−	0.372484i	$-0.878506\pi$
$97$	−53.8888	+	53.8888i	−0.555554	+	0.555554i	0.372484	+	0.928039i	$0.378506\pi$
$98$	1.22474	+	1.22474i	0.0124974	+	0.0124974i
$99$			18.0000i			0.181818i
$100$		0			0
$101$	−48.0000			−0.475248			−0.237624	−	0.971357i	$-0.576369\pi$
$101$	−48.0000			−0.475248			−0.237624	+	0.971357i	$0.576369\pi$
$102$	−40.4166	+	40.4166i	−0.396241	+	0.396241i
$103$	90.6311	+	90.6311i	0.879914	+	0.879914i	0.993525	−	0.113611i	$-0.0362419\pi$
$103$	90.6311	+	90.6311i	0.879914	+	0.879914i	−0.113611	+	0.993525i	$0.536242\pi$
$104$		−	90.0000i		−	0.865385i
$105$		0			0
$106$	−78.0000			−0.735849
$107$	25.7196	−	25.7196i	0.240370	−	0.240370i	−0.576633	−	0.817003i	$-0.695634\pi$
$107$	25.7196	−	25.7196i	0.240370	−	0.240370i	0.817003	+	0.576633i	$0.195634\pi$
$108$	−18.3712	−	18.3712i	−0.170103	−	0.170103i
$109$		−	40.0000i		−	0.366972i	−0.983022	−	0.183486i	$-0.941262\pi$
$109$		−	40.0000i		−	0.366972i	0.983022	−	0.183486i	$-0.0587383\pi$
$110$		0			0
$111$	−48.0000			−0.432432
$112$	−53.8888	+	53.8888i	−0.481150	+	0.481150i
$113$	84.5074	+	84.5074i	0.747853	+	0.747853i	0.974076	−	0.226223i	$-0.0726377\pi$
$113$	84.5074	+	84.5074i	0.747853	+	0.747853i	−0.226223	+	0.974076i	$0.572638\pi$
$114$			15.0000i			0.131579i
$115$		0			0
$116$	−30.0000			−0.258621
$117$	−44.0908	+	44.0908i	−0.376845	+	0.376845i
$118$	−110.227	−	110.227i	−0.934127	−	0.934127i
$119$			132.000i			1.10924i
$120$		0			0
$121$	−112.000			−0.925620
$122$	−34.2929	+	34.2929i	−0.281089	+	0.281089i
$123$	−69.8105	−	69.8105i	−0.567565	−	0.567565i
$124$		−	38.0000i		−	0.306452i
$125$		0			0
$126$	72.0000			0.571429
$127$	154.318	−	154.318i	1.21510	−	1.21510i	0.245774	−	0.969327i	$-0.420958\pi$
$127$	154.318	−	154.318i	1.21510	−	1.21510i	0.969327	−	0.245774i	$-0.0790421\pi$
$128$	42.8661	+	42.8661i	0.334891	+	0.334891i
$129$		−	12.0000i		−	0.0930233i
$130$		0			0
$131$	162.000			1.23664			0.618321	−	0.785926i	$-0.287813\pi$
$131$	162.000			1.23664			0.618321	+	0.785926i	$0.287813\pi$
$132$	−3.67423	+	3.67423i	−0.0278351	+	0.0278351i
$133$	24.4949	+	24.4949i	0.184172	+	0.184172i
$134$			117.000i			0.873134i
$135$		0			0
$136$	165.000			1.21324
$137$	50.2145	−	50.2145i	0.366529	−	0.366529i	−0.499680	−	0.866210i	$-0.666549\pi$
$137$	50.2145	−	50.2145i	0.366529	−	0.366529i	0.866210	+	0.499680i	$0.166549\pi$
$138$	−51.4393	−	51.4393i	−0.372748	−	0.372748i
$139$		−	185.000i		−	1.33094i	−0.746427	−	0.665468i	$-0.768232\pi$
$139$		−	185.000i		−	1.33094i	0.746427	−	0.665468i	$-0.231768\pi$
$140$		0			0
$141$	−18.0000			−0.127660
$142$	51.4393	−	51.4393i	0.362248	−	0.362248i
$143$	22.0454	+	22.0454i	0.154164	+	0.154164i
$144$		−	66.0000i		−	0.458333i
$145$		0			0
$146$	−33.0000			−0.226027
$147$	1.22474	−	1.22474i	0.00833160	−	0.00833160i
$148$	19.5959	+	19.5959i	0.132405	+	0.132405i
$149$			150.000i			1.00671i	0.864079	+	0.503356i	$0.167901\pi$
$149$			150.000i			1.00671i	−0.864079	+	0.503356i	$0.832099\pi$
$150$		0			0
$151$	52.0000			0.344371			0.172185	−	0.985065i	$-0.444917\pi$
$151$	52.0000			0.344371			0.172185	+	0.985065i	$0.444917\pi$
$152$	30.6186	−	30.6186i	0.201438	−	0.201438i
$153$	−80.8332	−	80.8332i	−0.528321	−	0.528321i
$154$		−	36.0000i		−	0.233766i
$155$		0			0
$156$	−18.0000			−0.115385
$157$	−188.611	+	188.611i	−1.20134	+	1.20134i	−0.227584	+	0.973759i	$0.573082\pi$
$157$	−188.611	+	188.611i	−1.20134	+	1.20134i	−0.973759	+	0.227584i	$0.926918\pi$
$158$	97.9796	+	97.9796i	0.620124	+	0.620124i
$159$			78.0000i			0.490566i
$160$		0			0
$161$	−168.000			−1.04348
$162$	−11.0227	+	11.0227i	−0.0680414	+	0.0680414i
$163$	−99.2043	−	99.2043i	−0.608616	−	0.608616i	0.333969	−	0.942584i	$-0.391612\pi$
$163$	−99.2043	−	99.2043i	−0.608616	−	0.608616i	−0.942584	+	0.333969i	$0.891612\pi$
$164$			57.0000i			0.347561i
$165$		0			0
$166$	−273.000			−1.64458
$167$	−17.1464	+	17.1464i	−0.102673	+	0.102673i	−0.756577	−	0.653904i	$-0.773130\pi$
$167$	−17.1464	+	17.1464i	−0.102673	+	0.102673i	0.653904	+	0.756577i	$0.273130\pi$
$168$	73.4847	+	73.4847i	0.437409	+	0.437409i
$169$		−	61.0000i		−	0.360947i
$170$		0			0
$171$	−30.0000			−0.175439
$172$	−4.89898	+	4.89898i	−0.0284824	+	0.0284824i
$173$	78.3837	+	78.3837i	0.453085	+	0.453085i	0.896377	−	0.443292i	$-0.146190\pi$
$173$	78.3837	+	78.3837i	0.453085	+	0.453085i	−0.443292	+	0.896377i	$0.646190\pi$
$174$		−	90.0000i		−	0.517241i
$175$		0			0
$176$	−33.0000			−0.187500
$177$	−110.227	+	110.227i	−0.622752	+	0.622752i
$178$	18.3712	+	18.3712i	0.103209	+	0.103209i
$179$		−	195.000i		−	1.08939i	−0.838636	−	0.544693i	$-0.816646\pi$
$179$		−	195.000i		−	1.08939i	0.838636	−	0.544693i	$-0.183354\pi$
$180$		0			0
$181$	262.000			1.44751			0.723757	−	0.690055i	$-0.242414\pi$
$181$	262.000			1.44751			0.723757	+	0.690055i	$0.242414\pi$
$182$	88.1816	−	88.1816i	0.484514	−	0.484514i
$183$	34.2929	+	34.2929i	0.187393	+	0.187393i
$184$			210.000i			1.14130i
$185$		0			0
$186$	114.000			0.612903
$187$	−40.4166	+	40.4166i	−0.216131	+	0.216131i
$188$	7.34847	+	7.34847i	0.0390876	+	0.0390876i
$189$		−	180.000i		−	0.952381i
$190$		0			0
$191$	−18.0000			−0.0942408			−0.0471204	−	0.998889i	$-0.515004\pi$
$191$	−18.0000			−0.0942408			−0.0471204	+	0.998889i	$0.515004\pi$
$192$	86.9569	−	86.9569i	0.452900	−	0.452900i
$193$	−86.9569	−	86.9569i	−0.450554	−	0.450554i	0.444984	−	0.895538i	$-0.353209\pi$
$193$	−86.9569	−	86.9569i	−0.450554	−	0.450554i	−0.895538	+	0.444984i	$0.853209\pi$
$194$			132.000i			0.680412i
$195$		0			0
$196$	−1.00000			−0.00510204
$197$	129.823	−	129.823i	0.659000	−	0.659000i	−0.296144	−	0.955143i	$-0.595701\pi$
$197$	129.823	−	129.823i	0.659000	−	0.659000i	0.955143	+	0.296144i	$0.0957007\pi$
$198$	22.0454	+	22.0454i	0.111340	+	0.111340i
$199$			200.000i			1.00503i	0.864570	+	0.502513i	$0.167591\pi$
$199$			200.000i			1.00503i	−0.864570	+	0.502513i	$0.832409\pi$
$200$		0			0
$201$	117.000			0.582090
$202$	−58.7878	+	58.7878i	−0.291028	+	0.291028i
$203$	−146.969	−	146.969i	−0.723987	−	0.723987i
$204$		−	33.0000i		−	0.161765i
$205$		0			0
$206$	222.000			1.07767
$207$	102.879	−	102.879i	0.496998	−	0.496998i
$208$	−80.8332	−	80.8332i	−0.388621	−	0.388621i
$209$			15.0000i			0.0717703i
$210$		0			0
$211$	−103.000			−0.488152			−0.244076	−	0.969756i	$-0.578485\pi$
$211$	−103.000			−0.488152			−0.244076	+	0.969756i	$0.578485\pi$
$212$	31.8434	−	31.8434i	0.150205	−	0.150205i
$213$	−51.4393	−	51.4393i	−0.241499	−	0.241499i
$214$		−	63.0000i		−	0.294393i
$215$		0			0
$216$	−225.000			−1.04167
$217$	186.161	−	186.161i	0.857886	−	0.857886i
$218$	−48.9898	−	48.9898i	−0.224724	−	0.224724i
$219$			33.0000i			0.150685i
$220$		0			0
$221$	−198.000			−0.895928
$222$	−58.7878	+	58.7878i	−0.264810	+	0.264810i
$223$	200.858	+	200.858i	0.900709	+	0.900709i	0.995497	−	0.0947882i	$-0.0302174\pi$
$223$	200.858	+	200.858i	0.900709	+	0.900709i	−0.0947882	+	0.995497i	$0.530217\pi$
$224$		−	108.000i		−	0.482143i
$225$		0			0
$226$	207.000			0.915929
$227$	−274.343	+	274.343i	−1.20856	+	1.20856i	−0.237065	+	0.971494i	$0.576185\pi$
$227$	−274.343	+	274.343i	−1.20856	+	1.20856i	−0.971494	+	0.237065i	$0.923815\pi$
$228$	−6.12372	−	6.12372i	−0.0268584	−	0.0268584i
$229$		−	20.0000i		−	0.0873362i	−0.999046	−	0.0436681i	$-0.986096\pi$
$229$		−	20.0000i		−	0.0873362i	0.999046	−	0.0436681i	$-0.0139044\pi$
$230$		0			0
$231$	−36.0000			−0.155844
$232$	−183.712	+	183.712i	−0.791861	+	0.791861i
$233$	102.879	+	102.879i	0.441539	+	0.441539i	0.892529	−	0.450990i	$-0.148929\pi$
$233$	102.879	+	102.879i	0.441539	+	0.441539i	−0.450990	+	0.892529i	$0.648929\pi$
$234$			108.000i			0.461538i
$235$		0			0
$236$	90.0000			0.381356
$237$	97.9796	−	97.9796i	0.413416	−	0.413416i
$238$	161.666	+	161.666i	0.679270	+	0.679270i
$239$		−	210.000i		−	0.878661i	−0.898325	−	0.439331i	$-0.855216\pi$
$239$		−	210.000i		−	0.878661i	0.898325	−	0.439331i	$-0.144784\pi$
$240$		0			0
$241$	−43.0000			−0.178423			−0.0892116	−	0.996013i	$-0.528435\pi$
$241$	−43.0000			−0.178423			−0.0892116	+	0.996013i	$0.528435\pi$
$242$	−137.171	+	137.171i	−0.566824	+	0.566824i
$243$	176.363	+	176.363i	0.725775	+	0.725775i
$244$		−	28.0000i		−	0.114754i
$245$		0			0
$246$	−171.000			−0.695122
$247$	−36.7423	+	36.7423i	−0.148754	+	0.148754i
$248$	−232.702	−	232.702i	−0.938313	−	0.938313i
$249$			273.000i			1.09639i
$250$		0			0
$251$	−123.000			−0.490040			−0.245020	−	0.969518i	$-0.578795\pi$
$251$	−123.000			−0.490040			−0.245020	+	0.969518i	$0.578795\pi$
$252$	−29.3939	+	29.3939i	−0.116642	+	0.116642i
$253$	−51.4393	−	51.4393i	−0.203317	−	0.203317i
$254$		−	378.000i		−	1.48819i
$255$		0			0
$256$	−179.000			−0.699219
$257$	−4.89898	+	4.89898i	−0.0190622	+	0.0190622i	−0.716574	−	0.697511i	$-0.754291\pi$
$257$	−4.89898	+	4.89898i	−0.0190622	+	0.0190622i	0.697511	+	0.716574i	$0.254291\pi$
$258$	−14.6969	−	14.6969i	−0.0569649	−	0.0569649i
$259$			192.000i			0.741313i
$260$		0			0
$261$	180.000			0.689655
$262$	198.409	−	198.409i	0.757285	−	0.757285i
$263$	−7.34847	−	7.34847i	−0.0279409	−	0.0279409i	0.692998	−	0.720939i	$-0.256289\pi$
$263$	−7.34847	−	7.34847i	−0.0279409	−	0.0279409i	−0.720939	+	0.692998i	$0.756289\pi$
$264$			45.0000i			0.170455i
$265$		0			0
$266$	60.0000			0.225564
$267$	18.3712	−	18.3712i	0.0688059	−	0.0688059i
$268$	−47.7650	−	47.7650i	−0.178228	−	0.178228i
$269$			120.000i			0.446097i	0.974807	+	0.223048i	$0.0716008\pi$
$269$			120.000i			0.446097i	−0.974807	+	0.223048i	$0.928399\pi$
$270$		0			0
$271$	−58.0000			−0.214022			−0.107011	−	0.994258i	$-0.534128\pi$
$271$	−58.0000			−0.214022			−0.107011	+	0.994258i	$0.534128\pi$
$272$	148.194	−	148.194i	0.544831	−	0.544831i
$273$	−88.1816	−	88.1816i	−0.323010	−	0.323010i
$274$		−	123.000i		−	0.448905i
$275$		0			0
$276$	42.0000			0.152174
$277$	276.792	−	276.792i	0.999250	−	0.999250i	−0.000749391	−	1.00000i	$-0.500239\pi$
$277$	276.792	−	276.792i	0.999250	−	0.999250i	1.00000		0.000749391i	$0.000238539\pi$
$278$	−226.578	−	226.578i	−0.815028	−	0.815028i
$279$			228.000i			0.817204i
$280$		0			0
$281$	462.000			1.64413			0.822064	−	0.569395i	$-0.192822\pi$
$281$	462.000			1.64413			0.822064	+	0.569395i	$0.192822\pi$
$282$	−22.0454	+	22.0454i	−0.0781752	+	0.0781752i
$283$	72.2599	+	72.2599i	0.255336	+	0.255336i	0.823154	−	0.567818i	$-0.192212\pi$
$283$	72.2599	+	72.2599i	0.255336	+	0.255336i	−0.567818	+	0.823154i	$0.692212\pi$
$284$			42.0000i			0.147887i
$285$		0			0
$286$	54.0000			0.188811
$287$	−279.242	+	279.242i	−0.972968	+	0.972968i
$288$	66.1362	+	66.1362i	0.229640	+	0.229640i
$289$		−	74.0000i		−	0.256055i
$290$		0			0
$291$	132.000			0.453608
$292$	13.4722	−	13.4722i	0.0461376	−	0.0461376i
$293$	4.89898	+	4.89898i	0.0167201	+	0.0167201i	0.715417	−	0.698697i	$-0.246237\pi$
$293$	4.89898	+	4.89898i	0.0167201	+	0.0167201i	−0.698697	+	0.715417i	$0.746237\pi$
$294$		−	3.00000i		−	0.0102041i
$295$		0			0
$296$	240.000			0.810811
$297$	55.1135	−	55.1135i	0.185567	−	0.185567i
$298$	183.712	+	183.712i	0.616482	+	0.616482i
$299$		−	252.000i		−	0.842809i
$300$		0			0
$301$	−48.0000			−0.159468
$302$	63.6867	−	63.6867i	0.210883	−	0.210883i
$303$	58.7878	+	58.7878i	0.194019	+	0.194019i
$304$		−	55.0000i		−	0.180921i
$305$		0			0
$306$	−198.000			−0.647059
$307$	86.9569	−	86.9569i	0.283247	−	0.283247i	−0.551155	−	0.834403i	$-0.685813\pi$
$307$	86.9569	−	86.9569i	0.283247	−	0.283247i	0.834403	+	0.551155i	$0.185813\pi$
$308$	14.6969	+	14.6969i	0.0477173	+	0.0477173i
$309$		−	222.000i		−	0.718447i
$310$		0			0
$311$	−528.000			−1.69775			−0.848875	−	0.528594i	$-0.822719\pi$
$311$	−528.000			−1.69775			−0.848875	+	0.528594i	$0.822719\pi$
$312$	−110.227	+	110.227i	−0.353292	+	0.353292i
$313$	−191.060	−	191.060i	−0.610416	−	0.610416i	0.332638	−	0.943054i	$-0.392061\pi$
$313$	−191.060	−	191.060i	−0.610416	−	0.610416i	−0.943054	+	0.332638i	$0.892061\pi$
$314$			462.000i			1.47134i
$315$		0			0
$316$	−80.0000			−0.253165
$317$	−200.858	+	200.858i	−0.633622	+	0.633622i	−0.948975	−	0.315353i	$-0.897877\pi$
$317$	−200.858	+	200.858i	−0.633622	+	0.633622i	0.315353	+	0.948975i	$0.397877\pi$
$318$	95.5301	+	95.5301i	0.300409	+	0.300409i
$319$		−	90.0000i		−	0.282132i
$320$		0			0
$321$	−63.0000			−0.196262
$322$	−205.757	+	205.757i	−0.638997	+	0.638997i
$323$	−67.3610	−	67.3610i	−0.208548	−	0.208548i
$324$		−	9.00000i		−	0.0277778i
$325$		0			0
$326$	−243.000			−0.745399
$327$	−48.9898	+	48.9898i	−0.149816	+	0.149816i
$328$	349.052	+	349.052i	1.06418	+	1.06418i
$329$			72.0000i			0.218845i
$330$		0			0
$331$	−313.000			−0.945619			−0.472810	−	0.881165i	$-0.656760\pi$
$331$	−313.000			−0.945619			−0.472810	+	0.881165i	$0.656760\pi$
$332$	111.452	−	111.452i	0.335698	−	0.335698i
$333$	−117.576	−	117.576i	−0.353080	−	0.353080i
$334$			42.0000i			0.125749i
$335$		0			0
$336$	132.000			0.392857
$337$	−194.734	+	194.734i	−0.577847	+	0.577847i	−0.934310	−	0.356463i	$-0.883983\pi$
$337$	−194.734	+	194.734i	−0.577847	+	0.577847i	0.356463	+	0.934310i	$0.383983\pi$
$338$	−74.7094	−	74.7094i	−0.221034	−	0.221034i
$339$		−	207.000i		−	0.610619i
$340$		0			0
$341$	114.000			0.334311
$342$	−36.7423	+	36.7423i	−0.107434	+	0.107434i
$343$	−244.949	−	244.949i	−0.714137	−	0.714137i
$344$			60.0000i			0.174419i
$345$		0			0
$346$	192.000			0.554913
$347$	99.2043	−	99.2043i	0.285891	−	0.285891i	−0.549562	−	0.835453i	$-0.685205\pi$
$347$	99.2043	−	99.2043i	0.285891	−	0.285891i	0.835453	+	0.549562i	$0.185205\pi$
$348$	36.7423	+	36.7423i	0.105581	+	0.105581i
$349$			100.000i			0.286533i	0.989684	+	0.143266i	$0.0457606\pi$
$349$			100.000i			0.286533i	−0.989684	+	0.143266i	$0.954239\pi$
$350$		0			0
$351$	270.000			0.769231
$352$	33.0681	−	33.0681i	0.0939435	−	0.0939435i
$353$	396.817	+	396.817i	1.12413	+	1.12413i	0.991114	+	0.133014i	$0.0424656\pi$
$353$	396.817	+	396.817i	1.12413	+	1.12413i	0.133014	+	0.991114i	$0.457534\pi$
$354$			270.000i			0.762712i
$355$		0			0
$356$	−15.0000			−0.0421348
$357$	161.666	−	161.666i	0.452847	−	0.452847i
$358$	−238.825	−	238.825i	−0.667110	−	0.667110i
$359$		−	540.000i		−	1.50418i	−0.659061	−	0.752089i	$-0.729046\pi$
$359$		−	540.000i		−	1.50418i	0.659061	−	0.752089i	$-0.270954\pi$
$360$		0			0
$361$	336.000			0.930748
$362$	320.883	−	320.883i	0.886418	−	0.886418i
$363$	137.171	+	137.171i	0.377883	+	0.377883i
$364$			72.0000i			0.197802i
$365$		0			0
$366$	84.0000			0.229508
$367$	44.0908	−	44.0908i	0.120138	−	0.120138i	−0.644481	−	0.764620i	$-0.722927\pi$
$367$	44.0908	−	44.0908i	0.120138	−	0.120138i	0.764620	+	0.644481i	$0.222927\pi$
$368$	188.611	+	188.611i	0.512529	+	0.512529i
$369$		−	342.000i		−	0.926829i
$370$		0			0
$371$	312.000			0.840970
$372$	−46.5403	+	46.5403i	−0.125108	+	0.125108i
$373$	−350.277	−	350.277i	−0.939081	−	0.939081i	0.0591675	−	0.998248i	$-0.481155\pi$
$373$	−350.277	−	350.277i	−0.939081	−	0.939081i	−0.998248	+	0.0591675i	$0.981155\pi$
$374$			99.0000i			0.264706i
$375$		0			0
$376$	90.0000			0.239362
$377$	220.454	−	220.454i	0.584759	−	0.584759i
$378$	−220.454	−	220.454i	−0.583212	−	0.583212i
$379$			505.000i			1.33245i	0.745749	+	0.666227i	$0.232092\pi$
$379$			505.000i			1.33245i	−0.745749	+	0.666227i	$0.767908\pi$
$380$		0			0
$381$	−378.000			−0.992126
$382$	−22.0454	+	22.0454i	−0.0577105	+	0.0577105i
$383$	−142.070	−	142.070i	−0.370941	−	0.370941i	0.496879	−	0.867820i	$-0.334479\pi$
$383$	−142.070	−	142.070i	−0.370941	−	0.370941i	−0.867820	+	0.496879i	$0.834479\pi$
$384$		−	105.000i		−	0.273438i
$385$		0			0
$386$	−213.000			−0.551813
$387$	29.3939	−	29.3939i	0.0759532	−	0.0759532i
$388$	−53.8888	−	53.8888i	−0.138889	−	0.138889i
$389$			690.000i			1.77378i	0.461982	+	0.886889i	$0.347139\pi$
$389$			690.000i			1.77378i	−0.461982	+	0.886889i	$0.652861\pi$
$390$		0			0
$391$	462.000			1.18159
$392$	−6.12372	+	6.12372i	−0.0156217	+	0.0156217i
$393$	−198.409	−	198.409i	−0.504857	−	0.504857i
$394$		−	318.000i		−	0.807107i
$395$		0			0
$396$	−18.0000			−0.0454545
$397$	−421.312	+	421.312i	−1.06124	+	1.06124i	−0.0632416	+	0.997998i	$0.520144\pi$
$397$	−421.312	+	421.312i	−1.06124	+	1.06124i	−0.997998	+	0.0632416i	$0.979856\pi$
$398$	244.949	+	244.949i	0.615450	+	0.615450i
$399$		−	60.0000i		−	0.150376i
$400$		0			0
$401$	−573.000			−1.42893			−0.714464	−	0.699672i	$-0.753329\pi$
$401$	−573.000			−1.42893			−0.714464	+	0.699672i	$0.753329\pi$
$402$	143.295	−	143.295i	0.356456	−	0.356456i
$403$	279.242	+	279.242i	0.692908	+	0.692908i
$404$		−	48.0000i		−	0.118812i
$405$		0			0
$406$	−360.000			−0.886700
$407$	−58.7878	+	58.7878i	−0.144442	+	0.144442i
$408$	−202.083	−	202.083i	−0.495301	−	0.495301i
$409$		−	365.000i		−	0.892421i	−0.894928	−	0.446210i	$-0.852773\pi$
$409$		−	365.000i		−	0.892421i	0.894928	−	0.446210i	$-0.147227\pi$
$410$		0			0
$411$	−123.000			−0.299270
$412$	−90.6311	+	90.6311i	−0.219978	+	0.219978i
$413$	440.908	+	440.908i	1.06757	+	1.06757i
$414$		−	252.000i		−	0.608696i
$415$		0			0
$416$	162.000			0.389423
$417$	−226.578	+	226.578i	−0.543352	+	0.543352i
$418$	18.3712	+	18.3712i	0.0439502	+	0.0439502i
$419$			645.000i			1.53938i	0.638418	+	0.769690i	$0.279589\pi$
$419$			645.000i			1.53938i	−0.638418	+	0.769690i	$0.720411\pi$
$420$		0			0
$421$	−8.00000			−0.0190024			−0.00950119	−	0.999955i	$-0.503024\pi$
$421$	−8.00000			−0.0190024			−0.00950119	+	0.999955i	$0.503024\pi$
$422$	−126.149	+	126.149i	−0.298931	+	0.298931i
$423$	−44.0908	−	44.0908i	−0.104234	−	0.104234i
$424$		−	390.000i		−	0.919811i
$425$		0			0
$426$	−126.000			−0.295775
$427$	137.171	−	137.171i	0.321245	−	0.321245i
$428$	25.7196	+	25.7196i	0.0600926	+	0.0600926i
$429$		−	54.0000i		−	0.125874i
$430$		0			0
$431$	12.0000			0.0278422			0.0139211	−	0.999903i	$-0.495569\pi$
$431$	12.0000			0.0278422			0.0139211	+	0.999903i	$0.495569\pi$
$432$	−202.083	+	202.083i	−0.467784	+	0.467784i
$433$	439.683	+	439.683i	1.01544	+	1.01544i	0.999879	+	0.0155561i	$0.00495187\pi$
$433$	439.683	+	439.683i	1.01544	+	1.01544i	0.0155561	+	0.999879i	$0.495048\pi$
$434$		−	456.000i		−	1.05069i
$435$		0			0
$436$	40.0000			0.0917431
$437$	85.7321	−	85.7321i	0.196183	−	0.196183i
$438$	40.4166	+	40.4166i	0.0922753	+	0.0922753i
$439$		−	10.0000i		−	0.0227790i	−0.999935	−	0.0113895i	$-0.996375\pi$
$439$		−	10.0000i		−	0.0227790i	0.999935	−	0.0113895i	$-0.00362548\pi$
$440$		0			0
$441$	6.00000			0.0136054
$442$	−242.499	+	242.499i	−0.548641	+	0.548641i
$443$	−86.9569	−	86.9569i	−0.196291	−	0.196291i	0.602117	−	0.798408i	$-0.294324\pi$
$443$	−86.9569	−	86.9569i	−0.196291	−	0.196291i	−0.798408	+	0.602117i	$0.794324\pi$
$444$		−	48.0000i		−	0.108108i
$445$		0			0
$446$	492.000			1.10314
$447$	183.712	−	183.712i	0.410988	−	0.410988i
$448$	−347.828	−	347.828i	−0.776401	−	0.776401i
$449$			75.0000i			0.167038i	0.996506	+	0.0835189i	$0.0266159\pi$
$449$			75.0000i			0.167038i	−0.996506	+	0.0835189i	$0.973384\pi$
$450$		0			0
$451$	−171.000			−0.379157
$452$	−84.5074	+	84.5074i	−0.186963	+	0.186963i
$453$	−63.6867	−	63.6867i	−0.140589	−	0.140589i
$454$			672.000i			1.48018i
$455$		0			0
$456$	−75.0000			−0.164474
$457$	209.431	−	209.431i	0.458274	−	0.458274i	−0.439814	−	0.898089i	$-0.644956\pi$
$457$	209.431	−	209.431i	0.458274	−	0.458274i	0.898089	+	0.439814i	$0.144956\pi$
$458$	−24.4949	−	24.4949i	−0.0534823	−	0.0534823i
$459$			495.000i			1.07843i
$460$		0			0
$461$	−228.000			−0.494577			−0.247289	−	0.968942i	$-0.579540\pi$
$461$	−228.000			−0.494577			−0.247289	+	0.968942i	$0.579540\pi$
$462$	−44.0908	+	44.0908i	−0.0954347	+	0.0954347i
$463$	−436.009	−	436.009i	−0.941704	−	0.941704i	0.0566875	−	0.998392i	$-0.481946\pi$
$463$	−436.009	−	436.009i	−0.941704	−	0.941704i	−0.998392	+	0.0566875i	$0.981946\pi$
$464$			330.000i			0.711207i
$465$		0			0
$466$	252.000			0.540773
$467$	533.989	−	533.989i	1.14344	−	1.14344i	0.155629	−	0.987816i	$-0.450259\pi$
$467$	533.989	−	533.989i	1.14344	−	1.14344i	0.987816	−	0.155629i	$-0.0497406\pi$
$468$	−44.0908	−	44.0908i	−0.0942111	−	0.0942111i
$469$		−	468.000i		−	0.997868i
$470$		0			0
$471$	462.000			0.980892
$472$	551.135	−	551.135i	1.16766	−	1.16766i
$473$	−14.6969	−	14.6969i	−0.0310718	−	0.0310718i
$474$		−	240.000i		−	0.506329i
$475$		0			0
$476$	−132.000			−0.277311
$477$	−191.060	+	191.060i	−0.400545	+	0.400545i
$478$	−257.196	−	257.196i	−0.538068	−	0.538068i
$479$		−	270.000i		−	0.563674i	−0.959462	−	0.281837i	$-0.909056\pi$
$479$		−	270.000i		−	0.563674i	0.959462	−	0.281837i	$-0.0909438\pi$
$480$		0			0
$481$	−288.000			−0.598753
$482$	−52.6640	+	52.6640i	−0.109261	+	0.109261i
$483$	205.757	+	205.757i	0.425998	+	0.425998i
$484$		−	112.000i		−	0.231405i
$485$		0			0
$486$	432.000			0.888889
$487$	80.8332	−	80.8332i	0.165982	−	0.165982i	−0.619229	−	0.785211i	$-0.712555\pi$
$487$	80.8332	−	80.8332i	0.165982	−	0.165982i	0.785211	+	0.619229i	$0.212555\pi$
$488$	−171.464	−	171.464i	−0.351361	−	0.351361i
$489$			243.000i			0.496933i
$490$		0			0
$491$	582.000			1.18534			0.592668	−	0.805447i	$-0.298075\pi$
$491$	582.000			1.18534			0.592668	+	0.805447i	$0.298075\pi$
$492$	69.8105	−	69.8105i	0.141891	−	0.141891i
$493$	404.166	+	404.166i	0.819809	+	0.819809i
$494$			90.0000i			0.182186i
$495$		0			0
$496$	−418.000			−0.842742
$497$	−205.757	+	205.757i	−0.413998	+	0.413998i
$498$	334.355	+	334.355i	0.671396	+	0.671396i
$499$		−	250.000i		−	0.501002i	−0.968116	−	0.250501i	$-0.919405\pi$
$499$		−	250.000i		−	0.501002i	0.968116	−	0.250501i	$-0.0805953\pi$
$500$		0			0
$501$	42.0000			0.0838323
$502$	−150.644	+	150.644i	−0.300087	+	0.300087i
$503$	−460.504	−	460.504i	−0.915515	−	0.915515i	0.0811841	−	0.996699i	$-0.474130\pi$
$503$	−460.504	−	460.504i	−0.915515	−	0.915515i	−0.996699	+	0.0811841i	$0.974130\pi$
$504$			360.000i			0.714286i
$505$		0			0
$506$	−126.000			−0.249012
$507$	−74.7094	+	74.7094i	−0.147356	+	0.147356i
$508$	154.318	+	154.318i	0.303775	+	0.303775i
$509$		−	390.000i		−	0.766208i	−0.923705	−	0.383104i	$-0.874855\pi$
$509$		−	390.000i		−	0.766208i	0.923705	−	0.383104i	$-0.125145\pi$
$510$		0			0
$511$	132.000			0.258317
$512$	−390.694	+	390.694i	−0.763073	+	0.763073i
$513$	91.8559	+	91.8559i	0.179056	+	0.179056i
$514$			12.0000i			0.0233463i
$515$		0			0
$516$	12.0000			0.0232558
$517$	−22.0454	+	22.0454i	−0.0426410	+	0.0426410i
$518$	235.151	+	235.151i	0.453959	+	0.453959i
$519$		−	192.000i		−	0.369942i
$520$		0			0
$521$	−183.000			−0.351248			−0.175624	−	0.984457i	$-0.556194\pi$
$521$	−183.000			−0.351248			−0.175624	+	0.984457i	$0.556194\pi$
$522$	220.454	−	220.454i	0.422326	−	0.422326i
$523$	476.426	+	476.426i	0.910948	+	0.910948i	0.996347	−	0.0853989i	$-0.0272165\pi$
$523$	476.426	+	476.426i	0.910948	+	0.910948i	−0.0853989	+	0.996347i	$0.527216\pi$
$524$			162.000i			0.309160i
$525$		0			0
$526$	−18.0000			−0.0342205
$527$	−511.943	+	511.943i	−0.971430	+	0.971430i
$528$	40.4166	+	40.4166i	0.0765466	+	0.0765466i
$529$			59.0000i			0.111531i
$530$		0			0
$531$	−540.000			−1.01695
$532$	−24.4949	+	24.4949i	−0.0460430	+	0.0460430i
$533$	−418.863	−	418.863i	−0.785859	−	0.785859i
$534$		−	45.0000i		−	0.0842697i
$535$		0			0
$536$	−585.000			−1.09142
$537$	−238.825	+	238.825i	−0.444740	+	0.444740i
$538$	146.969	+	146.969i	0.273177	+	0.273177i
$539$		−	3.00000i		−	0.00556586i
$540$		0			0
$541$	−568.000			−1.04991			−0.524954	−	0.851131i	$-0.675917\pi$
$541$	−568.000			−1.04991			−0.524954	+	0.851131i	$0.675917\pi$
$542$	−71.0352	+	71.0352i	−0.131061	+	0.131061i
$543$	−320.883	−	320.883i	−0.590945	−	0.590945i
$544$			297.000i			0.545956i
$545$		0			0
$546$	−216.000			−0.395604
$547$	160.442	−	160.442i	0.293312	−	0.293312i	−0.545075	−	0.838387i	$-0.683499\pi$
$547$	160.442	−	160.442i	0.293312	−	0.293312i	0.838387	+	0.545075i	$0.183499\pi$
$548$	50.2145	+	50.2145i	0.0916324	+	0.0916324i
$549$			168.000i			0.306011i
$550$		0			0
$551$	150.000			0.272232
$552$	257.196	−	257.196i	0.465936	−	0.465936i
$553$	−391.918	−	391.918i	−0.708713	−	0.708713i
$554$		−	678.000i		−	1.22383i
$555$		0			0
$556$	185.000			0.332734
$557$	−66.1362	+	66.1362i	−0.118736	+	0.118736i	−0.763978	−	0.645242i	$-0.776757\pi$
$557$	−66.1362	+	66.1362i	−0.118736	+	0.118736i	0.645242	+	0.763978i	$0.276757\pi$
$558$	279.242	+	279.242i	0.500433	+	0.500433i
$559$		−	72.0000i		−	0.128801i
$560$		0			0
$561$	99.0000			0.176471
$562$	565.832	−	565.832i	1.00682	−	1.00682i
$563$	−191.060	−	191.060i	−0.339361	−	0.339361i	0.516766	−	0.856127i	$-0.327136\pi$
$563$	−191.060	−	191.060i	−0.339361	−	0.339361i	−0.856127	+	0.516766i	$0.827136\pi$
$564$		−	18.0000i		−	0.0319149i
$565$		0			0
$566$	177.000			0.312721
$567$	44.0908	−	44.0908i	0.0777616	−	0.0777616i
$568$	257.196	+	257.196i	0.452811	+	0.452811i
$569$			45.0000i			0.0790861i	0.999218	+	0.0395431i	$0.0125902\pi$
$569$			45.0000i			0.0790861i	−0.999218	+	0.0395431i	$0.987410\pi$
$570$		0			0
$571$	542.000			0.949212			0.474606	−	0.880198i	$-0.342591\pi$
$571$	542.000			0.949212			0.474606	+	0.880198i	$0.342591\pi$
$572$	−22.0454	+	22.0454i	−0.0385409	+	0.0385409i
$573$	22.0454	+	22.0454i	0.0384737	+	0.0384737i
$574$			684.000i			1.19164i
$575$		0			0
$576$	426.000			0.739583
$577$	−549.910	+	549.910i	−0.953051	+	0.953051i	−0.998946	−	0.0458952i	$-0.985386\pi$
$577$	−549.910	+	549.910i	−0.953051	+	0.953051i	0.0458952	+	0.998946i	$0.485386\pi$
$578$	−90.6311	−	90.6311i	−0.156801	−	0.156801i
$579$			213.000i			0.367876i
$580$		0			0
$581$	1092.00			1.87952
$582$	161.666	−	161.666i	0.277777	−	0.277777i
$583$	95.5301	+	95.5301i	0.163860	+	0.163860i
$584$		−	165.000i		−	0.282534i
$585$		0			0
$586$	12.0000			0.0204778
$587$	417.638	−	417.638i	0.711479	−	0.711479i	−0.255366	−	0.966845i	$-0.582196\pi$
$587$	417.638	−	417.638i	0.711479	−	0.711479i	0.966845	+	0.255366i	$0.0821959\pi$
$588$	1.22474	+	1.22474i	0.00208290	+	0.00208290i
$589$			190.000i			0.322581i
$590$		0			0
$591$	−318.000			−0.538071
$592$	215.555	−	215.555i	0.364113	−	0.364113i
$593$	−331.906	−	331.906i	−0.559706	−	0.559706i	0.369517	−	0.929224i	$-0.379523\pi$
$593$	−331.906	−	331.906i	−0.559706	−	0.559706i	−0.929224	+	0.369517i	$0.879523\pi$
$594$		−	135.000i		−	0.227273i
$595$		0			0
$596$	−150.000			−0.251678
$597$	244.949	−	244.949i	0.410300	−	0.410300i
$598$	−308.636	−	308.636i	−0.516113	−	0.516113i
$599$		−	900.000i		−	1.50250i	−0.660015	−	0.751252i	$-0.729450\pi$
$599$		−	900.000i		−	1.50250i	0.660015	−	0.751252i	$-0.270550\pi$
$600$		0			0
$601$	577.000			0.960067			0.480033	−	0.877250i	$-0.340625\pi$
$601$	577.000			0.960067			0.480033	+	0.877250i	$0.340625\pi$
$602$	−58.7878	+	58.7878i	−0.0976541	+	0.0976541i
$603$	286.590	+	286.590i	0.475274	+	0.475274i
$604$			52.0000i			0.0860927i
$605$		0			0
$606$	144.000			0.237624
$607$	−127.373	+	127.373i	−0.209841	+	0.209841i	−0.804200	−	0.594359i	$-0.797406\pi$
$607$	−127.373	+	127.373i	−0.209841	+	0.209841i	0.594359	+	0.804200i	$0.297406\pi$
$608$	55.1135	+	55.1135i	0.0906472	+	0.0906472i
$609$			360.000i			0.591133i
$610$		0			0
$611$	−108.000			−0.176759
$612$	80.8332	−	80.8332i	0.132080	−	0.132080i
$613$	237.601	+	237.601i	0.387603	+	0.387603i	0.873832	−	0.486229i	$-0.161628\pi$
$613$	237.601	+	237.601i	0.387603	+	0.387603i	−0.486229	+	0.873832i	$0.661628\pi$
$614$		−	213.000i		−	0.346906i
$615$		0			0
$616$	180.000			0.292208
$617$	656.463	−	656.463i	1.06396	−	1.06396i	0.0661502	−	0.997810i	$-0.478928\pi$
$617$	656.463	−	656.463i	1.06396	−	1.06396i	0.997810	−	0.0661502i	$-0.0210717\pi$
$618$	−271.893	−	271.893i	−0.439957	−	0.439957i
$619$			470.000i			0.759289i	0.925132	+	0.379645i	$0.123954\pi$
$619$			470.000i			0.759289i	−0.925132	+	0.379645i	$0.876046\pi$
$620$		0			0
$621$	−630.000			−1.01449
$622$	−646.665	+	646.665i	−1.03965	+	1.03965i
$623$	−73.4847	−	73.4847i	−0.117953	−	0.117953i
$624$			198.000i			0.317308i
$625$		0			0
$626$	−468.000			−0.747604
$627$	18.3712	−	18.3712i	0.0293001	−	0.0293001i
$628$	−188.611	−	188.611i	−0.300336	−	0.300336i
$629$		−	528.000i		−	0.839428i
$630$		0			0
$631$	−838.000			−1.32805			−0.664025	−	0.747710i	$-0.731153\pi$
$631$	−838.000			−1.32805			−0.664025	+	0.747710i	$0.731153\pi$
$632$	−489.898	+	489.898i	−0.775155	+	0.775155i
$633$	126.149	+	126.149i	0.199287	+	0.199287i
$634$			492.000i			0.776025i
$635$		0			0
$636$	−78.0000			−0.122642
$637$	7.34847	−	7.34847i	0.0115361	−	0.0115361i
$638$	−110.227	−	110.227i	−0.172770	−	0.172770i
$639$		−	252.000i		−	0.394366i
$640$		0			0
$641$	−618.000			−0.964119			−0.482059	−	0.876139i	$-0.660111\pi$
$641$	−618.000			−0.964119			−0.482059	+	0.876139i	$0.660111\pi$
$642$	−77.1589	+	77.1589i	−0.120185	+	0.120185i
$643$	249.848	+	249.848i	0.388566	+	0.388566i	0.874176	−	0.485610i	$-0.161402\pi$
$643$	249.848	+	249.848i	0.388566	+	0.388566i	−0.485610	+	0.874176i	$0.661402\pi$
$644$		−	168.000i		−	0.260870i
$645$		0			0
$646$	−165.000			−0.255418
$647$	−421.312	+	421.312i	−0.651178	+	0.651178i	−0.953277	−	0.302099i	$-0.902313\pi$
$647$	−421.312	+	421.312i	−0.651178	+	0.651178i	0.302099	+	0.953277i	$0.402313\pi$
$648$	−55.1135	−	55.1135i	−0.0850517	−	0.0850517i
$649$			270.000i			0.416025i
$650$		0			0
$651$	−456.000			−0.700461
$652$	99.2043	−	99.2043i	0.152154	−	0.152154i
$653$	519.292	+	519.292i	0.795240	+	0.795240i	0.982341	−	0.187101i	$-0.0599090\pi$
$653$	519.292	+	519.292i	0.795240	+	0.795240i	−0.187101	+	0.982341i	$0.559909\pi$
$654$			120.000i			0.183486i
$655$		0			0
$656$	627.000			0.955793
$657$	−80.8332	+	80.8332i	−0.123034	+	0.123034i
$658$	88.1816	+	88.1816i	0.134015	+	0.134015i
$659$			885.000i			1.34294i	0.741030	+	0.671472i	$0.234338\pi$
$659$			885.000i			1.34294i	−0.741030	+	0.671472i	$0.765662\pi$
$660$		0			0
$661$	922.000			1.39486			0.697428	−	0.716655i	$-0.254328\pi$
$661$	922.000			1.39486			0.697428	+	0.716655i	$0.254328\pi$
$662$	−383.345	+	383.345i	−0.579071	+	0.579071i
$663$	242.499	+	242.499i	0.365761	+	0.365761i
$664$		−	1365.00i		−	2.05572i
$665$		0			0
$666$	−288.000			−0.432432
$667$	−514.393	+	514.393i	−0.771204	+	0.771204i
$668$	−17.1464	−	17.1464i	−0.0256683	−	0.0256683i
$669$		−	492.000i		−	0.735426i
$670$		0			0
$671$	84.0000			0.125186
$672$	−132.272	+	132.272i	−0.196834	+	0.196834i
$673$	78.3837	+	78.3837i	0.116469	+	0.116469i	0.762939	−	0.646470i	$-0.223756\pi$
$673$	78.3837	+	78.3837i	0.116469	+	0.116469i	−0.646470	+	0.762939i	$0.723756\pi$
$674$			477.000i			0.707715i
$675$		0			0
$676$	61.0000			0.0902367
$677$	705.453	−	705.453i	1.04203	−	1.04203i	0.0429510	−	0.999077i	$-0.486324\pi$
$677$	705.453	−	705.453i	1.04203	−	1.04203i	0.999077	−	0.0429510i	$-0.0136759\pi$
$678$	−253.522	−	253.522i	−0.373927	−	0.373927i
$679$		−	528.000i		−	0.777614i
$680$		0			0
$681$	672.000			0.986784
$682$	139.621	−	139.621i	0.204723	−	0.204723i
$683$	133.497	+	133.497i	0.195457	+	0.195457i	0.798049	−	0.602592i	$-0.205865\pi$
$683$	133.497	+	133.497i	0.195457	+	0.195457i	−0.602592	+	0.798049i	$0.705865\pi$
$684$		−	30.0000i		−	0.0438596i
$685$		0			0
$686$	−600.000			−0.874636
$687$	−24.4949	+	24.4949i	−0.0356549	+	0.0356549i
$688$	53.8888	+	53.8888i	0.0783267	+	0.0783267i
$689$			468.000i			0.679245i
$690$		0			0
$691$	−893.000			−1.29233			−0.646165	−	0.763198i	$-0.723628\pi$
$691$	−893.000			−1.29233			−0.646165	+	0.763198i	$0.723628\pi$
$692$	−78.3837	+	78.3837i	−0.113271	+	0.113271i
$693$	−88.1816	−	88.1816i	−0.127246	−	0.127246i
$694$		−	243.000i		−	0.350144i
$695$		0			0
$696$	450.000			0.646552
$697$	767.915	−	767.915i	1.10174	−	1.10174i
$698$	122.474	+	122.474i	0.175465	+	0.175465i
$699$		−	252.000i		−	0.360515i
$700$		0			0
$701$	402.000			0.573466			0.286733	−	0.958010i	$-0.407431\pi$
$701$	402.000			0.573466			0.286733	+	0.958010i	$0.407431\pi$
$702$	330.681	−	330.681i	0.471056	−	0.471056i
$703$	−97.9796	−	97.9796i	−0.139374	−	0.139374i
$704$		−	213.000i		−	0.302557i
$705$		0			0
$706$	972.000			1.37677
$707$	235.151	−	235.151i	0.332604	−	0.332604i
$708$	−110.227	−	110.227i	−0.155688	−	0.155688i
$709$			1060.00i			1.49506i	0.664226	+	0.747532i	$0.268761\pi$
$709$			1060.00i			1.49506i	−0.664226	+	0.747532i	$0.731239\pi$
$710$		0			0
$711$	480.000			0.675105
$712$	−91.8559	+	91.8559i	−0.129011	+	0.129011i
$713$	−651.564	−	651.564i	−0.913835	−	0.913835i
$714$		−	396.000i		−	0.554622i
$715$		0			0
$716$	195.000			0.272346
$717$	−257.196	+	257.196i	−0.358712	+	0.358712i
$718$	−661.362	−	661.362i	−0.921117	−	0.921117i
$719$			1320.00i			1.83588i	0.396716	+	0.917942i	$0.370150\pi$
$719$			1320.00i			1.83588i	−0.396716	+	0.917942i	$0.629850\pi$
$720$		0			0
$721$	−888.000			−1.23162
$722$	411.514	−	411.514i	0.569964	−	0.569964i
$723$	52.6640	+	52.6640i	0.0728410	+	0.0728410i
$724$			262.000i			0.361878i
$725$		0			0
$726$	336.000			0.462810
$727$	31.8434	−	31.8434i	0.0438011	−	0.0438011i	−0.684867	−	0.728668i	$-0.740140\pi$
$727$	31.8434	−	31.8434i	0.0438011	−	0.0438011i	0.728668	+	0.684867i	$0.240140\pi$
$728$	440.908	+	440.908i	0.605643	+	0.605643i
$729$		−	351.000i		−	0.481481i
$730$		0			0
$731$	132.000			0.180575
$732$	−34.2929	+	34.2929i	−0.0468482	+	0.0468482i
$733$	−815.680	−	815.680i	−1.11280	−	1.11280i	−0.992771	−	0.120026i	$-0.961702\pi$
$733$	−815.680	−	815.680i	−1.11280	−	1.11280i	−0.120026	−	0.992771i	$-0.538298\pi$
$734$		−	108.000i		−	0.147139i
$735$		0			0
$736$	−378.000			−0.513587
$737$	143.295	−	143.295i	0.194430	−	0.194430i
$738$	−418.863	−	418.863i	−0.567565	−	0.567565i
$739$		−	710.000i		−	0.960758i	−0.877061	−	0.480379i	$-0.840499\pi$
$739$		−	710.000i		−	0.960758i	0.877061	−	0.480379i	$-0.159501\pi$
$740$		0			0
$741$	90.0000			0.121457
$742$	382.120	−	382.120i	0.514987	−	0.514987i
$743$	800.983	+	800.983i	1.07804	+	1.07804i	0.996685	+	0.0813540i	$0.0259244\pi$
$743$	800.983	+	800.983i	1.07804	+	1.07804i	0.0813540	+	0.996685i	$0.474076\pi$
$744$			570.000i			0.766129i
$745$		0			0
$746$	−858.000			−1.15013
$747$	−668.711	+	668.711i	−0.895195	+	0.895195i
$748$	−40.4166	−	40.4166i	−0.0540329	−	0.0540329i
$749$			252.000i			0.336449i
$750$		0			0
$751$	502.000			0.668442			0.334221	−	0.942495i	$-0.391527\pi$
$751$	502.000			0.668442			0.334221	+	0.942495i	$0.391527\pi$
$752$	80.8332	−	80.8332i	0.107491	−	0.107491i
$753$	150.644	+	150.644i	0.200058	+	0.200058i
$754$		−	540.000i		−	0.716180i
$755$		0			0
$756$	180.000			0.238095
$757$	−4.89898	+	4.89898i	−0.00647157	+	0.00647157i	−0.710335	−	0.703864i	$-0.751457\pi$
$757$	−4.89898	+	4.89898i	−0.00647157	+	0.00647157i	0.703864	+	0.710335i	$0.251457\pi$
$758$	618.496	+	618.496i	0.815958	+	0.815958i
$759$			126.000i			0.166008i
$760$		0			0
$761$	747.000			0.981603			0.490802	−	0.871271i	$-0.336704\pi$
$761$	747.000			0.981603			0.490802	+	0.871271i	$0.336704\pi$
$762$	−462.954	+	462.954i	−0.607551	+	0.607551i
$763$	195.959	+	195.959i	0.256827	+	0.256827i
$764$		−	18.0000i		−	0.0235602i
$765$		0			0
$766$	−348.000			−0.454308
$767$	−661.362	+	661.362i	−0.862271	+	0.862271i
$768$	219.229	+	219.229i	0.285455	+	0.285455i
$769$		−	1255.00i		−	1.63199i	−0.578059	−	0.815995i	$-0.696190\pi$
$769$		−	1255.00i		−	1.63199i	0.578059	−	0.815995i	$-0.303810\pi$
$770$		0			0
$771$	12.0000			0.0155642
$772$	86.9569	−	86.9569i	0.112638	−	0.112638i
$773$	17.1464	+	17.1464i	0.0221817	+	0.0221817i	0.718111	−	0.695929i	$-0.245007\pi$
$773$	17.1464	+	17.1464i	0.0221817	+	0.0221817i	−0.695929	+	0.718111i	$0.745007\pi$
$774$		−	72.0000i		−	0.0930233i
$775$		0			0
$776$	−660.000			−0.850515
$777$	235.151	−	235.151i	0.302640	−	0.302640i
$778$	845.074	+	845.074i	1.08621	+	1.08621i
$779$		−	285.000i		−	0.365854i
$780$		0			0
$781$	−126.000			−0.161332

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:	\( N \)	\(=\)	\( 25 = 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	25.c (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.22474 - 1.22474i) q^{2} +(-1.22474 - 1.22474i) q^{3} +1.00000i q^{4} -3.00000 q^{6} +(-4.89898 + 4.89898i) q^{7} +(6.12372 + 6.12372i) q^{8} -6.00000i q^{9} +O(q^{10})\)	\(q+(1.22474 - 1.22474i) q^{2} +(-1.22474 - 1.22474i) q^{3} +1.00000i q^{4} -3.00000 q^{6} +(-4.89898 + 4.89898i) q^{7} +(6.12372 + 6.12372i) q^{8} -6.00000i q^{9} -3.00000 q^{11} +(1.22474 - 1.22474i) q^{12} +(-7.34847 - 7.34847i) q^{13} +12.0000i q^{14} +11.0000 q^{16} +(13.4722 - 13.4722i) q^{17} +(-7.34847 - 7.34847i) q^{18} -5.00000i q^{19} +12.0000 q^{21} +(-3.67423 + 3.67423i) q^{22} +(17.1464 + 17.1464i) q^{23} -15.0000i q^{24} -18.0000 q^{26} +(-18.3712 + 18.3712i) q^{27} +(-4.89898 - 4.89898i) q^{28} +30.0000i q^{29} -38.0000 q^{31} +(-11.0227 + 11.0227i) q^{32} +(3.67423 + 3.67423i) q^{33} -33.0000i q^{34} +6.00000 q^{36} +(19.5959 - 19.5959i) q^{37} +(-6.12372 - 6.12372i) q^{38} +18.0000i q^{39} +57.0000 q^{41} +(14.6969 - 14.6969i) q^{42} +(4.89898 + 4.89898i) q^{43} -3.00000i q^{44} +42.0000 q^{46} +(7.34847 - 7.34847i) q^{47} +(-13.4722 - 13.4722i) q^{48} +1.00000i q^{49} -33.0000 q^{51} +(7.34847 - 7.34847i) q^{52} +(-31.8434 - 31.8434i) q^{53} +45.0000i q^{54} -60.0000 q^{56} +(-6.12372 + 6.12372i) q^{57} +(36.7423 + 36.7423i) q^{58} -90.0000i q^{59} -28.0000 q^{61} +(-46.5403 + 46.5403i) q^{62} +(29.3939 + 29.3939i) q^{63} +71.0000i q^{64} +9.00000 q^{66} +(-47.7650 + 47.7650i) q^{67} +(13.4722 + 13.4722i) q^{68} -42.0000i q^{69} +42.0000 q^{71} +(36.7423 - 36.7423i) q^{72} +(-13.4722 - 13.4722i) q^{73} -48.0000i q^{74} +5.00000 q^{76} +(14.6969 - 14.6969i) q^{77} +(22.0454 + 22.0454i) q^{78} +80.0000i q^{79} -9.00000 q^{81} +(69.8105 - 69.8105i) q^{82} +(-111.452 - 111.452i) q^{83} +12.0000i q^{84} +12.0000 q^{86} +(36.7423 - 36.7423i) q^{87} +(-18.3712 - 18.3712i) q^{88} +15.0000i q^{89} +72.0000 q^{91} +(-17.1464 + 17.1464i) q^{92} +(46.5403 + 46.5403i) q^{93} -18.0000i q^{94} +27.0000 q^{96} +(-53.8888 + 53.8888i) q^{97} +(1.22474 + 1.22474i) q^{98} +18.0000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{6}+O(q^{10}) \) 4 * q - 12 * q^6	\( 4 q - 12 q^{6} - 12 q^{11} + 44 q^{16} + 48 q^{21} - 72 q^{26} - 152 q^{31} + 24 q^{36} + 228 q^{41} + 168 q^{46} - 132 q^{51} - 240 q^{56} - 112 q^{61} + 36 q^{66} + 168 q^{71} + 20 q^{76} - 36 q^{81} + 48 q^{86} + 288 q^{91} + 108 q^{96}+O(q^{100}) \) 4 * q - 12 * q^6 - 12 * q^11 + 44 * q^16 + 48 * q^21 - 72 * q^26 - 152 * q^31 + 24 * q^36 + 228 * q^41 + 168 * q^46 - 132 * q^51 - 240 * q^56 - 112 * q^61 + 36 * q^66 + 168 * q^71 + 20 * q^76 - 36 * q^81 + 48 * q^86 + 288 * q^91 + 108 * q^96

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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