LMFDB - Embedded newform 325.2.c.c.51.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [325,2,Mod(51,325)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(325, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("325.51");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 325 = 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	325.c (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$2.59513806569$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			51.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	325.51
Dual form			325.2.c.c.51.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000i q^{2} -1.00000 q^{3} -2.00000 q^{4} -2.00000i q^{6} +2.00000i q^{7} -2.00000 q^{9} +O(q^{10})$	\(q+2.00000i q^{2} -1.00000 q^{3} -2.00000 q^{4} -2.00000i q^{6} +2.00000i q^{7} -2.00000 q^{9} +2.00000 q^{12} +(-3.00000 + 2.00000i) q^{13} -4.00000 q^{14} -4.00000 q^{16} -2.00000 q^{17} -4.00000i q^{18} -4.00000i q^{19} -2.00000i q^{21} -1.00000 q^{23} +(-4.00000 - 6.00000i) q^{26} +5.00000 q^{27} -4.00000i q^{28} +5.00000 q^{29} +10.0000i q^{31} -8.00000i q^{32} -4.00000i q^{34} +4.00000 q^{36} +2.00000i q^{37} +8.00000 q^{38} +(3.00000 - 2.00000i) q^{39} +10.0000i q^{41} +4.00000 q^{42} -11.0000 q^{43} -2.00000i q^{46} -8.00000i q^{47} +4.00000 q^{48} +3.00000 q^{49} +2.00000 q^{51} +(6.00000 - 4.00000i) q^{52} +9.00000 q^{53} +10.0000i q^{54} +4.00000i q^{57} +10.0000i q^{58} +6.00000i q^{59} +7.00000 q^{61} -20.0000 q^{62} -4.00000i q^{63} +8.00000 q^{64} +12.0000i q^{67} +4.00000 q^{68} +1.00000 q^{69} -10.0000i q^{71} +14.0000i q^{73} -4.00000 q^{74} +8.00000i q^{76} +(4.00000 + 6.00000i) q^{78} -5.00000 q^{79} +1.00000 q^{81} -20.0000 q^{82} -6.00000i q^{83} +4.00000i q^{84} -22.0000i q^{86} -5.00000 q^{87} +6.00000i q^{89} +(-4.00000 - 6.00000i) q^{91} +2.00000 q^{92} -10.0000i q^{93} +16.0000 q^{94} +8.00000i q^{96} +2.00000i q^{97} +6.00000i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 4 q^{4} - 4 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 - 4 * q^4 - 4 * q^9	$ 2 q - 2 q^{3} - 4 q^{4} - 4 q^{9} + 4 q^{12} - 6 q^{13} - 8 q^{14} - 8 q^{16} - 4 q^{17} - 2 q^{23} - 8 q^{26} + 10 q^{27} + 10 q^{29} + 8 q^{36} + 16 q^{38} + 6 q^{39} + 8 q^{42} - 22 q^{43} + 8 q^{48} + 6 q^{49} + 4 q^{51} + 12 q^{52} + 18 q^{53} + 14 q^{61} - 40 q^{62} + 16 q^{64} + 8 q^{68} + 2 q^{69} - 8 q^{74} + 8 q^{78} - 10 q^{79} + 2 q^{81} - 40 q^{82} - 10 q^{87} - 8 q^{91} + 4 q^{92} + 32 q^{94}+O(q^{100}) $ 2 * q - 2 * q^3 - 4 * q^4 - 4 * q^9 + 4 * q^12 - 6 * q^13 - 8 * q^14 - 8 * q^16 - 4 * q^17 - 2 * q^23 - 8 * q^26 + 10 * q^27 + 10 * q^29 + 8 * q^36 + 16 * q^38 + 6 * q^39 + 8 * q^42 - 22 * q^43 + 8 * q^48 + 6 * q^49 + 4 * q^51 + 12 * q^52 + 18 * q^53 + 14 * q^61 - 40 * q^62 + 16 * q^64 + 8 * q^68 + 2 * q^69 - 8 * q^74 + 8 * q^78 - 10 * q^79 + 2 * q^81 - 40 * q^82 - 10 * q^87 - 8 * q^91 + 4 * q^92 + 32 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/325\mathbb{Z}\right)^\times$.

$n$	$27$	$301$
$\chi(n)$	$1$	$-1$

Coefficient data

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

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

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

$2$			2.00000i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$2$			2.00000i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$3$	−1.00000			−0.577350			−0.288675	−	0.957427i	$-0.593215\pi$
$3$	−1.00000			−0.577350			−0.288675	+	0.957427i	$0.593215\pi$
$4$	−2.00000			−1.00000
$5$		0			0
$5$		0			0
$6$		−	2.00000i		−	0.816497i
$7$			2.00000i			0.755929i	0.925820	+	0.377964i	$0.123376\pi$
$7$			2.00000i			0.755929i	−0.925820	+	0.377964i	$0.876624\pi$
$8$		0			0
$9$	−2.00000			−0.666667
$10$		0			0
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$	2.00000			0.577350
$13$	−3.00000	+	2.00000i	−0.832050	+	0.554700i
$13$	−3.00000	+	2.00000i	−0.832050	+	0.554700i
$14$	−4.00000			−1.06904
$15$		0			0
$16$	−4.00000			−1.00000
$17$	−2.00000			−0.485071			−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000			−0.485071			−0.242536	+	0.970143i	$0.577979\pi$
$18$		−	4.00000i		−	0.942809i
$19$		−	4.00000i		−	0.917663i	−0.888523	−	0.458831i	$-0.848268\pi$
$19$		−	4.00000i		−	0.917663i	0.888523	−	0.458831i	$-0.151732\pi$
$20$		0			0
$21$		−	2.00000i		−	0.436436i
$22$		0			0
$23$	−1.00000			−0.208514			−0.104257	−	0.994550i	$-0.533247\pi$
$23$	−1.00000			−0.208514			−0.104257	+	0.994550i	$0.533247\pi$
$24$		0			0
$25$		0			0
$26$	−4.00000	−	6.00000i	−0.784465	−	1.17670i
$27$	5.00000			0.962250
$28$		−	4.00000i		−	0.755929i
$29$	5.00000			0.928477			0.464238	−	0.885710i	$-0.346328\pi$
$29$	5.00000			0.928477			0.464238	+	0.885710i	$0.346328\pi$
$30$		0			0
$31$			10.0000i			1.79605i	0.439941	+	0.898027i	$0.354999\pi$
$31$			10.0000i			1.79605i	−0.439941	+	0.898027i	$0.645001\pi$
$32$		−	8.00000i		−	1.41421i
$33$		0			0
$34$		−	4.00000i		−	0.685994i
$35$		0			0
$36$	4.00000			0.666667
$37$			2.00000i			0.328798i	0.986394	+	0.164399i	$0.0525685\pi$
$37$			2.00000i			0.328798i	−0.986394	+	0.164399i	$0.947432\pi$
$38$	8.00000			1.29777
$39$	3.00000	−	2.00000i	0.480384	−	0.320256i
$40$		0			0
$41$			10.0000i			1.56174i	0.624695	+	0.780869i	$0.285223\pi$
$41$			10.0000i			1.56174i	−0.624695	+	0.780869i	$0.714777\pi$
$42$	4.00000			0.617213
$43$	−11.0000			−1.67748			−0.838742	−	0.544529i	$-0.816708\pi$
$43$	−11.0000			−1.67748			−0.838742	+	0.544529i	$0.816708\pi$
$44$		0			0
$45$		0			0
$46$		−	2.00000i		−	0.294884i
$47$		−	8.00000i		−	1.16692i	−0.812142	−	0.583460i	$-0.801699\pi$
$47$		−	8.00000i		−	1.16692i	0.812142	−	0.583460i	$-0.198301\pi$
$48$	4.00000			0.577350
$49$	3.00000			0.428571
$50$		0			0
$51$	2.00000			0.280056
$52$	6.00000	−	4.00000i	0.832050	−	0.554700i
$53$	9.00000			1.23625			0.618123	−	0.786082i	$-0.287894\pi$
$53$	9.00000			1.23625			0.618123	+	0.786082i	$0.287894\pi$
$54$			10.0000i			1.36083i
$55$		0			0
$56$		0			0
$57$			4.00000i			0.529813i
$58$			10.0000i			1.31306i
$59$			6.00000i			0.781133i	0.920575	+	0.390567i	$0.127721\pi$
$59$			6.00000i			0.781133i	−0.920575	+	0.390567i	$0.872279\pi$
$60$		0			0
$61$	7.00000			0.896258			0.448129	−	0.893969i	$-0.352090\pi$
$61$	7.00000			0.896258			0.448129	+	0.893969i	$0.352090\pi$
$62$	−20.0000			−2.54000
$63$		−	4.00000i		−	0.503953i
$64$	8.00000			1.00000
$65$		0			0
$66$		0			0
$67$			12.0000i			1.46603i	0.680211	+	0.733017i	$0.261888\pi$
$67$			12.0000i			1.46603i	−0.680211	+	0.733017i	$0.738112\pi$
$68$	4.00000			0.485071
$69$	1.00000			0.120386
$70$		0			0
$71$		−	10.0000i		−	1.18678i	−0.804914	−	0.593391i	$-0.797789\pi$
$71$		−	10.0000i		−	1.18678i	0.804914	−	0.593391i	$-0.202211\pi$
$72$		0			0
$73$			14.0000i			1.63858i	0.573382	+	0.819288i	$0.305631\pi$
$73$			14.0000i			1.63858i	−0.573382	+	0.819288i	$0.694369\pi$
$74$	−4.00000			−0.464991
$75$		0			0
$76$			8.00000i			0.917663i
$77$		0			0
$78$	4.00000	+	6.00000i	0.452911	+	0.679366i
$79$	−5.00000			−0.562544			−0.281272	−	0.959628i	$-0.590756\pi$
$79$	−5.00000			−0.562544			−0.281272	+	0.959628i	$0.590756\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$	−20.0000			−2.20863
$83$		−	6.00000i		−	0.658586i	−0.944228	−	0.329293i	$-0.893190\pi$
$83$		−	6.00000i		−	0.658586i	0.944228	−	0.329293i	$-0.106810\pi$
$84$			4.00000i			0.436436i
$85$		0			0
$86$		−	22.0000i		−	2.37232i
$87$	−5.00000			−0.536056
$88$		0			0
$89$			6.00000i			0.635999i	0.948091	+	0.317999i	$0.103011\pi$
$89$			6.00000i			0.635999i	−0.948091	+	0.317999i	$0.896989\pi$
$90$		0			0
$91$	−4.00000	−	6.00000i	−0.419314	−	0.628971i
$92$	2.00000			0.208514
$93$		−	10.0000i		−	1.03695i
$94$	16.0000			1.65027
$95$		0			0
$96$			8.00000i			0.816497i
$97$			2.00000i			0.203069i	0.994832	+	0.101535i	$0.0323753\pi$
$97$			2.00000i			0.203069i	−0.994832	+	0.101535i	$0.967625\pi$
$98$			6.00000i			0.606092i
$99$		0			0
$100$		0			0
$101$	7.00000			0.696526			0.348263	−	0.937397i	$-0.386772\pi$
$101$	7.00000			0.696526			0.348263	+	0.937397i	$0.386772\pi$
$102$			4.00000i			0.396059i
$103$	−1.00000			−0.0985329			−0.0492665	−	0.998786i	$-0.515688\pi$
$103$	−1.00000			−0.0985329			−0.0492665	+	0.998786i	$0.515688\pi$
$104$		0			0
$105$		0			0
$106$			18.0000i			1.74831i
$107$	13.0000			1.25676			0.628379	−	0.777908i	$-0.283719\pi$
$107$	13.0000			1.25676			0.628379	+	0.777908i	$0.283719\pi$
$108$	−10.0000			−0.962250
$109$			16.0000i			1.53252i	0.642529	+	0.766261i	$0.277885\pi$
$109$			16.0000i			1.53252i	−0.642529	+	0.766261i	$0.722115\pi$
$110$		0			0
$111$		−	2.00000i		−	0.189832i
$112$		−	8.00000i		−	0.755929i
$113$	−1.00000			−0.0940721			−0.0470360	−	0.998893i	$-0.514978\pi$
$113$	−1.00000			−0.0940721			−0.0470360	+	0.998893i	$0.514978\pi$
$114$	−8.00000			−0.749269
$115$		0			0
$116$	−10.0000			−0.928477
$117$	6.00000	−	4.00000i	0.554700	−	0.369800i
$118$	−12.0000			−1.10469
$119$		−	4.00000i		−	0.366679i
$120$		0			0
$121$	11.0000			1.00000
$122$			14.0000i			1.26750i
$123$		−	10.0000i		−	0.901670i
$124$		−	20.0000i		−	1.79605i
$125$		0			0
$126$	8.00000			0.712697
$127$	3.00000			0.266207			0.133103	−	0.991102i	$-0.457506\pi$
$127$	3.00000			0.266207			0.133103	+	0.991102i	$0.457506\pi$
$128$		0			0
$129$	11.0000			0.968496
$130$		0			0
$131$	−3.00000			−0.262111			−0.131056	−	0.991375i	$-0.541837\pi$
$131$	−3.00000			−0.262111			−0.131056	+	0.991375i	$0.541837\pi$
$132$		0			0
$133$	8.00000			0.693688
$134$	−24.0000			−2.07328
$135$		0			0
$136$		0			0
$137$		−	18.0000i		−	1.53784i	−0.639343	−	0.768922i	$-0.720793\pi$
$137$		−	18.0000i		−	1.53784i	0.639343	−	0.768922i	$-0.279207\pi$
$138$			2.00000i			0.170251i
$139$	−5.00000			−0.424094			−0.212047	−	0.977259i	$-0.568013\pi$
$139$	−5.00000			−0.424094			−0.212047	+	0.977259i	$0.568013\pi$
$140$		0			0
$141$			8.00000i			0.673722i
$142$	20.0000			1.67836
$143$		0			0
$144$	8.00000			0.666667
$145$		0			0
$146$	−28.0000			−2.31730
$147$	−3.00000			−0.247436
$148$		−	4.00000i		−	0.328798i
$149$		−	14.0000i		−	1.14692i	−0.819232	−	0.573462i	$-0.805600\pi$
$149$		−	14.0000i		−	1.14692i	0.819232	−	0.573462i	$-0.194400\pi$
$150$		0			0
$151$		−	10.0000i		−	0.813788i	−0.913475	−	0.406894i	$-0.866612\pi$
$151$		−	10.0000i		−	0.813788i	0.913475	−	0.406894i	$-0.133388\pi$
$152$		0			0
$153$	4.00000			0.323381
$154$		0			0
$155$		0			0
$156$	−6.00000	+	4.00000i	−0.480384	+	0.320256i
$157$	−2.00000			−0.159617			−0.0798087	−	0.996810i	$-0.525431\pi$
$157$	−2.00000			−0.159617			−0.0798087	+	0.996810i	$0.525431\pi$
$158$		−	10.0000i		−	0.795557i
$159$	−9.00000			−0.713746
$160$		0			0
$161$		−	2.00000i		−	0.157622i
$162$			2.00000i			0.157135i
$163$		−	6.00000i		−	0.469956i	−0.972001	−	0.234978i	$-0.924498\pi$
$163$		−	6.00000i		−	0.469956i	0.972001	−	0.234978i	$-0.0755019\pi$
$164$		−	20.0000i		−	1.56174i
$165$		0			0
$166$	12.0000			0.931381
$167$			2.00000i			0.154765i	0.997001	+	0.0773823i	$0.0246562\pi$
$167$			2.00000i			0.154765i	−0.997001	+	0.0773823i	$0.975344\pi$
$168$		0			0
$169$	5.00000	−	12.0000i	0.384615	−	0.923077i
$170$		0			0
$171$			8.00000i			0.611775i
$172$	22.0000			1.67748
$173$	14.0000			1.06440			0.532200	−	0.846619i	$-0.321365\pi$
$173$	14.0000			1.06440			0.532200	+	0.846619i	$0.321365\pi$
$174$		−	10.0000i		−	0.758098i
$175$		0			0
$176$		0			0
$177$		−	6.00000i		−	0.450988i
$178$	−12.0000			−0.899438
$179$	−15.0000			−1.12115			−0.560576	−	0.828103i	$-0.689420\pi$
$179$	−15.0000			−1.12115			−0.560576	+	0.828103i	$0.689420\pi$
$180$		0			0
$181$	−18.0000			−1.33793			−0.668965	−	0.743294i	$-0.733262\pi$
$181$	−18.0000			−1.33793			−0.668965	+	0.743294i	$0.733262\pi$
$182$	12.0000	−	8.00000i	0.889499	−	0.592999i
$183$	−7.00000			−0.517455
$184$		0			0
$185$		0			0
$186$	20.0000			1.46647
$187$		0			0
$188$			16.0000i			1.16692i
$189$			10.0000i			0.727393i
$190$		0			0
$191$	−23.0000			−1.66422			−0.832111	−	0.554609i	$-0.812868\pi$
$191$	−23.0000			−1.66422			−0.832111	+	0.554609i	$0.812868\pi$
$192$	−8.00000			−0.577350
$193$		−	16.0000i		−	1.15171i	−0.817554	−	0.575853i	$-0.804670\pi$
$193$		−	16.0000i		−	1.15171i	0.817554	−	0.575853i	$-0.195330\pi$
$194$	−4.00000			−0.287183
$195$		0			0
$196$	−6.00000			−0.428571
$197$			2.00000i			0.142494i	0.997459	+	0.0712470i	$0.0226979\pi$
$197$			2.00000i			0.142494i	−0.997459	+	0.0712470i	$0.977302\pi$
$198$		0			0
$199$	−15.0000			−1.06332			−0.531661	−	0.846957i	$-0.678432\pi$
$199$	−15.0000			−1.06332			−0.531661	+	0.846957i	$0.678432\pi$
$200$		0			0
$201$		−	12.0000i		−	0.846415i
$202$			14.0000i			0.985037i
$203$			10.0000i			0.701862i
$204$	−4.00000			−0.280056
$205$		0			0
$206$		−	2.00000i		−	0.139347i
$207$	2.00000			0.139010
$208$	12.0000	−	8.00000i	0.832050	−	0.554700i
$209$		0			0
$210$		0			0
$211$	12.0000			0.826114			0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000			0.826114			0.413057	+	0.910705i	$0.364461\pi$
$212$	−18.0000			−1.23625
$213$			10.0000i			0.685189i
$214$			26.0000i			1.77732i
$215$		0			0
$216$		0			0
$217$	−20.0000			−1.35769
$218$	−32.0000			−2.16731
$219$		−	14.0000i		−	0.946032i
$220$		0			0
$221$	6.00000	−	4.00000i	0.403604	−	0.269069i
$222$	4.00000			0.268462
$223$			4.00000i			0.267860i	0.990991	+	0.133930i	$0.0427597\pi$
$223$			4.00000i			0.267860i	−0.990991	+	0.133930i	$0.957240\pi$
$224$	16.0000			1.06904
$225$		0			0
$226$		−	2.00000i		−	0.133038i
$227$			12.0000i			0.796468i	0.917284	+	0.398234i	$0.130377\pi$
$227$			12.0000i			0.796468i	−0.917284	+	0.398234i	$0.869623\pi$
$228$		−	8.00000i		−	0.529813i
$229$			26.0000i			1.71813i	0.511868	+	0.859064i	$0.328954\pi$
$229$			26.0000i			1.71813i	−0.511868	+	0.859064i	$0.671046\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−11.0000			−0.720634			−0.360317	−	0.932830i	$-0.617331\pi$
$233$	−11.0000			−0.720634			−0.360317	+	0.932830i	$0.617331\pi$
$234$	8.00000	+	12.0000i	0.522976	+	0.784465i
$235$		0			0
$236$		−	12.0000i		−	0.781133i
$237$	5.00000			0.324785
$238$	8.00000			0.518563
$239$		−	4.00000i		−	0.258738i	−0.991596	−	0.129369i	$-0.958705\pi$
$239$		−	4.00000i		−	0.258738i	0.991596	−	0.129369i	$-0.0412952\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$			22.0000i			1.41421i
$243$	−16.0000			−1.02640
$244$	−14.0000			−0.896258
$245$		0			0
$246$	20.0000			1.27515
$247$	8.00000	+	12.0000i	0.509028	+	0.763542i
$248$		0			0
$249$			6.00000i			0.380235i
$250$		0			0
$251$	12.0000			0.757433			0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000			0.757433			0.378717	+	0.925513i	$0.376365\pi$
$252$			8.00000i			0.503953i
$253$		0			0
$254$			6.00000i			0.376473i
$255$		0			0
$256$	16.0000			1.00000
$257$	−27.0000			−1.68421			−0.842107	−	0.539311i	$-0.818685\pi$
$257$	−27.0000			−1.68421			−0.842107	+	0.539311i	$0.818685\pi$
$258$			22.0000i			1.36966i
$259$	−4.00000			−0.248548
$260$		0			0
$261$	−10.0000			−0.618984
$262$		−	6.00000i		−	0.370681i
$263$	24.0000			1.47990			0.739952	−	0.672660i	$-0.234848\pi$
$263$	24.0000			1.47990			0.739952	+	0.672660i	$0.234848\pi$
$264$		0			0
$265$		0			0
$266$			16.0000i			0.981023i
$267$		−	6.00000i		−	0.367194i
$268$		−	24.0000i		−	1.46603i
$269$	25.0000			1.52428			0.762138	−	0.647414i	$-0.224150\pi$
$269$	25.0000			1.52428			0.762138	+	0.647414i	$0.224150\pi$
$270$		0			0
$271$		−	20.0000i		−	1.21491i	−0.794353	−	0.607457i	$-0.792190\pi$
$271$		−	20.0000i		−	1.21491i	0.794353	−	0.607457i	$-0.207810\pi$
$272$	8.00000			0.485071
$273$	4.00000	+	6.00000i	0.242091	+	0.363137i
$274$	36.0000			2.17484
$275$		0			0
$276$	−2.00000			−0.120386
$277$	−7.00000			−0.420589			−0.210295	−	0.977638i	$-0.567442\pi$
$277$	−7.00000			−0.420589			−0.210295	+	0.977638i	$0.567442\pi$
$278$		−	10.0000i		−	0.599760i
$279$		−	20.0000i		−	1.19737i
$280$		0			0
$281$			30.0000i			1.78965i	0.446417	+	0.894825i	$0.352700\pi$
$281$			30.0000i			1.78965i	−0.446417	+	0.894825i	$0.647300\pi$
$282$	−16.0000			−0.952786
$283$	4.00000			0.237775			0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000			0.237775			0.118888	+	0.992908i	$0.462067\pi$
$284$			20.0000i			1.18678i
$285$		0			0
$286$		0			0
$287$	−20.0000			−1.18056
$288$			16.0000i			0.942809i
$289$	−13.0000			−0.764706
$290$		0			0
$291$		−	2.00000i		−	0.117242i
$292$		−	28.0000i		−	1.63858i
$293$			24.0000i			1.40209i	0.713115	+	0.701047i	$0.247284\pi$
$293$			24.0000i			1.40209i	−0.713115	+	0.701047i	$0.752716\pi$
$294$		−	6.00000i		−	0.349927i
$295$		0			0
$296$		0			0
$297$		0			0
$298$	28.0000			1.62200
$299$	3.00000	−	2.00000i	0.173494	−	0.115663i
$300$		0			0
$301$		−	22.0000i		−	1.26806i
$302$	20.0000			1.15087
$303$	−7.00000			−0.402139
$304$			16.0000i			0.917663i
$305$		0			0
$306$			8.00000i			0.457330i
$307$		−	8.00000i		−	0.456584i	−0.973593	−	0.228292i	$-0.926686\pi$
$307$		−	8.00000i		−	0.456584i	0.973593	−	0.228292i	$-0.0733141\pi$
$308$		0			0
$309$	1.00000			0.0568880
$310$		0			0
$311$	−23.0000			−1.30421			−0.652105	−	0.758129i	$-0.726114\pi$
$311$	−23.0000			−1.30421			−0.652105	+	0.758129i	$0.726114\pi$
$312$		0			0
$313$	19.0000			1.07394			0.536972	−	0.843600i	$-0.319568\pi$
$313$	19.0000			1.07394			0.536972	+	0.843600i	$0.319568\pi$
$314$		−	4.00000i		−	0.225733i
$315$		0			0
$316$	10.0000			0.562544
$317$		−	28.0000i		−	1.57264i	−0.617822	−	0.786318i	$-0.711985\pi$
$317$		−	28.0000i		−	1.57264i	0.617822	−	0.786318i	$-0.288015\pi$
$318$		−	18.0000i		−	1.00939i
$319$		0			0
$320$		0			0
$321$	−13.0000			−0.725589
$322$	4.00000			0.222911
$323$			8.00000i			0.445132i
$324$	−2.00000			−0.111111
$325$		0			0
$326$	12.0000			0.664619
$327$		−	16.0000i		−	0.884802i
$328$		0			0
$329$	16.0000			0.882109
$330$		0			0
$331$			10.0000i			0.549650i	0.961494	+	0.274825i	$0.0886199\pi$
$331$			10.0000i			0.549650i	−0.961494	+	0.274825i	$0.911380\pi$
$332$			12.0000i			0.658586i
$333$		−	4.00000i		−	0.219199i
$334$	−4.00000			−0.218870
$335$		0			0
$336$			8.00000i			0.436436i
$337$	3.00000			0.163420			0.0817102	−	0.996656i	$-0.473962\pi$
$337$	3.00000			0.163420			0.0817102	+	0.996656i	$0.473962\pi$
$338$	24.0000	+	10.0000i	1.30543	+	0.543928i
$339$	1.00000			0.0543125
$340$		0			0
$341$		0			0
$342$	−16.0000			−0.865181
$343$			20.0000i			1.07990i
$344$		0			0
$345$		0			0
$346$			28.0000i			1.50529i
$347$	23.0000			1.23470			0.617352	−	0.786687i	$-0.288205\pi$
$347$	23.0000			1.23470			0.617352	+	0.786687i	$0.288205\pi$
$348$	10.0000			0.536056
$349$		−	14.0000i		−	0.749403i	−0.927146	−	0.374701i	$-0.877745\pi$
$349$		−	14.0000i		−	0.749403i	0.927146	−	0.374701i	$-0.122255\pi$
$350$		0			0
$351$	−15.0000	+	10.0000i	−0.800641	+	0.533761i
$352$		0			0
$353$			24.0000i			1.27739i	0.769460	+	0.638696i	$0.220526\pi$
$353$			24.0000i			1.27739i	−0.769460	+	0.638696i	$0.779474\pi$
$354$	12.0000			0.637793
$355$		0			0
$356$		−	12.0000i		−	0.635999i
$357$			4.00000i			0.211702i
$358$		−	30.0000i		−	1.58555i
$359$			16.0000i			0.844448i	0.906492	+	0.422224i	$0.138750\pi$
$359$			16.0000i			0.844448i	−0.906492	+	0.422224i	$0.861250\pi$
$360$		0			0
$361$	3.00000			0.157895
$362$		−	36.0000i		−	1.89212i
$363$	−11.0000			−0.577350
$364$	8.00000	+	12.0000i	0.419314	+	0.628971i
$365$		0			0
$366$		−	14.0000i		−	0.731792i
$367$	3.00000			0.156599			0.0782994	−	0.996930i	$-0.475051\pi$
$367$	3.00000			0.156599			0.0782994	+	0.996930i	$0.475051\pi$
$368$	4.00000			0.208514
$369$		−	20.0000i		−	1.04116i
$370$		0			0
$371$			18.0000i			0.934513i
$372$			20.0000i			1.03695i
$373$	−1.00000			−0.0517780			−0.0258890	−	0.999665i	$-0.508242\pi$
$373$	−1.00000			−0.0517780			−0.0258890	+	0.999665i	$0.508242\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−15.0000	+	10.0000i	−0.772539	+	0.515026i
$378$	−20.0000			−1.02869
$379$		−	24.0000i		−	1.23280i	−0.787434	−	0.616399i	$-0.788591\pi$
$379$		−	24.0000i		−	1.23280i	0.787434	−	0.616399i	$-0.211409\pi$
$380$		0			0
$381$	−3.00000			−0.153695
$382$		−	46.0000i		−	2.35356i
$383$		−	16.0000i		−	0.817562i	−0.912633	−	0.408781i	$-0.865954\pi$
$383$		−	16.0000i		−	0.817562i	0.912633	−	0.408781i	$-0.134046\pi$
$384$		0			0
$385$		0			0
$386$	32.0000			1.62876
$387$	22.0000			1.11832
$388$		−	4.00000i		−	0.203069i
$389$	25.0000			1.26755			0.633775	−	0.773517i	$-0.281504\pi$
$389$	25.0000			1.26755			0.633775	+	0.773517i	$0.281504\pi$
$390$		0			0
$391$	2.00000			0.101144
$392$		0			0
$393$	3.00000			0.151330
$394$	−4.00000			−0.201517
$395$		0			0
$396$		0			0
$397$			22.0000i			1.10415i	0.833795	+	0.552074i	$0.186163\pi$
$397$			22.0000i			1.10415i	−0.833795	+	0.552074i	$0.813837\pi$
$398$		−	30.0000i		−	1.50376i
$399$	−8.00000			−0.400501
$400$		0			0
$401$		−	10.0000i		−	0.499376i	−0.968326	−	0.249688i	$-0.919672\pi$
$401$		−	10.0000i		−	0.499376i	0.968326	−	0.249688i	$-0.0803281\pi$
$402$	24.0000			1.19701
$403$	−20.0000	−	30.0000i	−0.996271	−	1.49441i
$404$	−14.0000			−0.696526
$405$		0			0
$406$	−20.0000			−0.992583
$407$		0			0
$408$		0			0
$409$			6.00000i			0.296681i	0.988936	+	0.148340i	$0.0473931\pi$
$409$			6.00000i			0.296681i	−0.988936	+	0.148340i	$0.952607\pi$
$410$		0			0
$411$			18.0000i			0.887875i
$412$	2.00000			0.0985329
$413$	−12.0000			−0.590481
$414$			4.00000i			0.196589i
$415$		0			0
$416$	16.0000	+	24.0000i	0.784465	+	1.17670i
$417$	5.00000			0.244851
$418$		0			0
$419$	15.0000			0.732798			0.366399	−	0.930458i	$-0.380591\pi$
$419$	15.0000			0.732798			0.366399	+	0.930458i	$0.380591\pi$
$420$		0			0
$421$		−	10.0000i		−	0.487370i	−0.969854	−	0.243685i	$-0.921644\pi$
$421$		−	10.0000i		−	0.487370i	0.969854	−	0.243685i	$-0.0783563\pi$
$422$			24.0000i			1.16830i
$423$			16.0000i			0.777947i
$424$		0			0
$425$		0			0
$426$	−20.0000			−0.969003
$427$			14.0000i			0.677507i
$428$	−26.0000			−1.25676
$429$		0			0
$430$		0			0
$431$		−	30.0000i		−	1.44505i	−0.691345	−	0.722525i	$-0.742982\pi$
$431$		−	30.0000i		−	1.44505i	0.691345	−	0.722525i	$-0.257018\pi$
$432$	−20.0000			−0.962250
$433$	−11.0000			−0.528626			−0.264313	−	0.964437i	$-0.585145\pi$
$433$	−11.0000			−0.528626			−0.264313	+	0.964437i	$0.585145\pi$
$434$		−	40.0000i		−	1.92006i
$435$		0			0
$436$		−	32.0000i		−	1.53252i
$437$			4.00000i			0.191346i
$438$	28.0000			1.33789
$439$	−20.0000			−0.954548			−0.477274	−	0.878755i	$-0.658375\pi$
$439$	−20.0000			−0.954548			−0.477274	+	0.878755i	$0.658375\pi$
$440$		0			0
$441$	−6.00000			−0.285714
$442$	8.00000	+	12.0000i	0.380521	+	0.570782i
$443$	24.0000			1.14027			0.570137	−	0.821549i	$-0.306890\pi$
$443$	24.0000			1.14027			0.570137	+	0.821549i	$0.306890\pi$
$444$			4.00000i			0.189832i
$445$		0			0
$446$	−8.00000			−0.378811
$447$			14.0000i			0.662177i
$448$			16.0000i			0.755929i
$449$			36.0000i			1.69895i	0.527633	+	0.849473i	$0.323080\pi$
$449$			36.0000i			1.69895i	−0.527633	+	0.849473i	$0.676920\pi$
$450$		0			0
$451$		0			0
$452$	2.00000			0.0940721
$453$			10.0000i			0.469841i
$454$	−24.0000			−1.12638
$455$		0			0
$456$		0			0
$457$		−	8.00000i		−	0.374224i	−0.982339	−	0.187112i	$-0.940087\pi$
$457$		−	8.00000i		−	0.374224i	0.982339	−	0.187112i	$-0.0599128\pi$
$458$	−52.0000			−2.42980
$459$	−10.0000			−0.466760
$460$		0			0
$461$			20.0000i			0.931493i	0.884918	+	0.465746i	$0.154214\pi$
$461$			20.0000i			0.931493i	−0.884918	+	0.465746i	$0.845786\pi$
$462$		0			0
$463$			4.00000i			0.185896i	0.995671	+	0.0929479i	$0.0296290\pi$
$463$			4.00000i			0.185896i	−0.995671	+	0.0929479i	$0.970371\pi$
$464$	−20.0000			−0.928477
$465$		0			0
$466$		−	22.0000i		−	1.01913i
$467$	−27.0000			−1.24941			−0.624705	−	0.780860i	$-0.714781\pi$
$467$	−27.0000			−1.24941			−0.624705	+	0.780860i	$0.714781\pi$
$468$	−12.0000	+	8.00000i	−0.554700	+	0.369800i
$469$	−24.0000			−1.10822
$470$		0			0
$471$	2.00000			0.0921551
$472$		0			0
$473$		0			0
$474$			10.0000i			0.459315i
$475$		0			0
$476$			8.00000i			0.366679i
$477$	−18.0000			−0.824163
$478$	8.00000			0.365911
$479$		−	24.0000i		−	1.09659i	−0.836286	−	0.548294i	$-0.815277\pi$
$479$		−	24.0000i		−	1.09659i	0.836286	−	0.548294i	$-0.184723\pi$
$480$		0			0
$481$	−4.00000	−	6.00000i	−0.182384	−	0.273576i
$482$		0			0
$483$			2.00000i			0.0910032i
$484$	−22.0000			−1.00000
$485$		0			0
$486$		−	32.0000i		−	1.45155i
$487$			12.0000i			0.543772i	0.962329	+	0.271886i	$0.0876473\pi$
$487$			12.0000i			0.543772i	−0.962329	+	0.271886i	$0.912353\pi$
$488$		0			0
$489$			6.00000i			0.271329i
$490$		0			0
$491$	17.0000			0.767199			0.383600	−	0.923499i	$-0.374684\pi$
$491$	17.0000			0.767199			0.383600	+	0.923499i	$0.374684\pi$
$492$			20.0000i			0.901670i
$493$	−10.0000			−0.450377
$494$	−24.0000	+	16.0000i	−1.07981	+	0.719874i
$495$		0			0
$496$		−	40.0000i		−	1.79605i
$497$	20.0000			0.897123
$498$	−12.0000			−0.537733
$499$			26.0000i			1.16392i	0.813217	+	0.581960i	$0.197714\pi$
$499$			26.0000i			1.16392i	−0.813217	+	0.581960i	$0.802286\pi$
$500$		0			0
$501$		−	2.00000i		−	0.0893534i
$502$			24.0000i			1.07117i
$503$	19.0000			0.847168			0.423584	−	0.905857i	$-0.360772\pi$
$503$	19.0000			0.847168			0.423584	+	0.905857i	$0.360772\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−5.00000	+	12.0000i	−0.222058	+	0.532939i
$508$	−6.00000			−0.266207
$509$			6.00000i			0.265945i	0.991120	+	0.132973i	$0.0424523\pi$
$509$			6.00000i			0.265945i	−0.991120	+	0.132973i	$0.957548\pi$
$510$		0			0
$511$	−28.0000			−1.23865
$512$			32.0000i			1.41421i
$513$		−	20.0000i		−	0.883022i
$514$		−	54.0000i		−	2.38184i
$515$		0			0
$516$	−22.0000			−0.968496
$517$		0			0
$518$		−	8.00000i		−	0.351500i
$519$	−14.0000			−0.614532
$520$		0			0
$521$	37.0000			1.62100			0.810500	−	0.585739i	$-0.199196\pi$
$521$	37.0000			1.62100			0.810500	+	0.585739i	$0.199196\pi$
$522$		−	20.0000i		−	0.875376i
$523$	4.00000			0.174908			0.0874539	−	0.996169i	$-0.472127\pi$
$523$	4.00000			0.174908			0.0874539	+	0.996169i	$0.472127\pi$
$524$	6.00000			0.262111
$525$		0			0
$526$			48.0000i			2.09290i
$527$		−	20.0000i		−	0.871214i
$528$		0			0
$529$	−22.0000			−0.956522
$530$		0			0
$531$		−	12.0000i		−	0.520756i
$532$	−16.0000			−0.693688
$533$	−20.0000	−	30.0000i	−0.866296	−	1.29944i
$534$	12.0000			0.519291
$535$		0			0
$536$		0			0
$537$	15.0000			0.647298
$538$			50.0000i			2.15565i
$539$		0			0
$540$		0			0
$541$			10.0000i			0.429934i	0.976621	+	0.214967i	$0.0689643\pi$
$541$			10.0000i			0.429934i	−0.976621	+	0.214967i	$0.931036\pi$
$542$	40.0000			1.71815
$543$	18.0000			0.772454
$544$			16.0000i			0.685994i
$545$		0			0
$546$	−12.0000	+	8.00000i	−0.513553	+	0.342368i
$547$	28.0000			1.19719			0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000			1.19719			0.598597	+	0.801050i	$0.295725\pi$
$548$			36.0000i			1.53784i
$549$	−14.0000			−0.597505
$550$		0			0
$551$		−	20.0000i		−	0.852029i
$552$		0			0
$553$		−	10.0000i		−	0.425243i
$554$		−	14.0000i		−	0.594803i
$555$		0			0
$556$	10.0000			0.424094
$557$			12.0000i			0.508456i	0.967144	+	0.254228i	$0.0818214\pi$
$557$			12.0000i			0.508456i	−0.967144	+	0.254228i	$0.918179\pi$
$558$	40.0000			1.69334
$559$	33.0000	−	22.0000i	1.39575	−	0.930501i
$560$		0			0
$561$		0			0
$562$	−60.0000			−2.53095
$563$	−21.0000			−0.885044			−0.442522	−	0.896758i	$-0.645916\pi$
$563$	−21.0000			−0.885044			−0.442522	+	0.896758i	$0.645916\pi$
$564$		−	16.0000i		−	0.673722i
$565$		0			0
$566$			8.00000i			0.336265i
$567$			2.00000i			0.0839921i
$568$		0			0
$569$	15.0000			0.628833			0.314416	−	0.949285i	$-0.398191\pi$
$569$	15.0000			0.628833			0.314416	+	0.949285i	$0.398191\pi$
$570$		0			0
$571$	32.0000			1.33916			0.669579	−	0.742741i	$-0.266474\pi$
$571$	32.0000			1.33916			0.669579	+	0.742741i	$0.266474\pi$
$572$		0			0
$573$	23.0000			0.960839
$574$		−	40.0000i		−	1.66957i
$575$		0			0
$576$	−16.0000			−0.666667
$577$		−	28.0000i		−	1.16566i	−0.812596	−	0.582828i	$-0.801946\pi$
$577$		−	28.0000i		−	1.16566i	0.812596	−	0.582828i	$-0.198054\pi$
$578$		−	26.0000i		−	1.08146i
$579$			16.0000i			0.664937i
$580$		0			0
$581$	12.0000			0.497844
$582$	4.00000			0.165805
$583$		0			0
$584$		0			0
$585$		0			0
$586$	−48.0000			−1.98286
$587$		−	28.0000i		−	1.15568i	−0.816149	−	0.577842i	$-0.803895\pi$
$587$		−	28.0000i		−	1.15568i	0.816149	−	0.577842i	$-0.196105\pi$
$588$	6.00000			0.247436
$589$	40.0000			1.64817
$590$		0			0
$591$		−	2.00000i		−	0.0822690i
$592$		−	8.00000i		−	0.328798i
$593$		−	26.0000i		−	1.06769i	−0.845582	−	0.533846i	$-0.820746\pi$
$593$		−	26.0000i		−	1.06769i	0.845582	−	0.533846i	$-0.179254\pi$
$594$		0			0
$595$		0			0
$596$			28.0000i			1.14692i
$597$	15.0000			0.613909
$598$	4.00000	+	6.00000i	0.163572	+	0.245358i
$599$	−35.0000			−1.43006			−0.715031	−	0.699093i	$-0.753587\pi$
$599$	−35.0000			−1.43006			−0.715031	+	0.699093i	$0.753587\pi$
$600$		0			0
$601$	17.0000			0.693444			0.346722	−	0.937968i	$-0.387295\pi$
$601$	17.0000			0.693444			0.346722	+	0.937968i	$0.387295\pi$
$602$	44.0000			1.79331
$603$		−	24.0000i		−	0.977356i
$604$			20.0000i			0.813788i
$605$		0			0
$606$		−	14.0000i		−	0.568711i
$607$	8.00000			0.324710			0.162355	−	0.986732i	$-0.448091\pi$
$607$	8.00000			0.324710			0.162355	+	0.986732i	$0.448091\pi$
$608$	−32.0000			−1.29777
$609$		−	10.0000i		−	0.405220i
$610$		0			0
$611$	16.0000	+	24.0000i	0.647291	+	0.970936i
$612$	−8.00000			−0.323381
$613$		−	16.0000i		−	0.646234i	−0.946359	−	0.323117i	$-0.895269\pi$
$613$		−	16.0000i		−	0.646234i	0.946359	−	0.323117i	$-0.104731\pi$
$614$	16.0000			0.645707
$615$		0			0
$616$		0			0
$617$			22.0000i			0.885687i	0.896599	+	0.442843i	$0.146030\pi$
$617$			22.0000i			0.885687i	−0.896599	+	0.442843i	$0.853970\pi$
$618$			2.00000i			0.0804518i
$619$			26.0000i			1.04503i	0.852631	+	0.522514i	$0.175006\pi$
$619$			26.0000i			1.04503i	−0.852631	+	0.522514i	$0.824994\pi$
$620$		0			0
$621$	−5.00000			−0.200643
$622$		−	46.0000i		−	1.84443i
$623$	−12.0000			−0.480770
$624$	−12.0000	+	8.00000i	−0.480384	+	0.320256i
$625$		0			0
$626$			38.0000i			1.51879i
$627$		0			0
$628$	4.00000			0.159617
$629$		−	4.00000i		−	0.159490i
$630$		0			0
$631$			20.0000i			0.796187i	0.917345	+	0.398094i	$0.130328\pi$
$631$			20.0000i			0.796187i	−0.917345	+	0.398094i	$0.869672\pi$
$632$		0			0
$633$	−12.0000			−0.476957
$634$	56.0000			2.22404
$635$		0			0
$636$	18.0000			0.713746
$637$	−9.00000	+	6.00000i	−0.356593	+	0.237729i
$638$		0			0
$639$			20.0000i			0.791188i
$640$		0			0
$641$	−3.00000			−0.118493			−0.0592464	−	0.998243i	$-0.518870\pi$
$641$	−3.00000			−0.118493			−0.0592464	+	0.998243i	$0.518870\pi$
$642$		−	26.0000i		−	1.02614i
$643$			14.0000i			0.552106i	0.961142	+	0.276053i	$0.0890266\pi$
$643$			14.0000i			0.552106i	−0.961142	+	0.276053i	$0.910973\pi$
$644$			4.00000i			0.157622i
$645$		0			0
$646$	−16.0000			−0.629512
$647$	8.00000			0.314512			0.157256	−	0.987558i	$-0.449735\pi$
$647$	8.00000			0.314512			0.157256	+	0.987558i	$0.449735\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	20.0000			0.783862
$652$			12.0000i			0.469956i
$653$	14.0000			0.547862			0.273931	−	0.961749i	$-0.411676\pi$
$653$	14.0000			0.547862			0.273931	+	0.961749i	$0.411676\pi$
$654$	32.0000			1.25130
$655$		0			0
$656$		−	40.0000i		−	1.56174i
$657$		−	28.0000i		−	1.09238i
$658$			32.0000i			1.24749i
$659$	45.0000			1.75295			0.876476	−	0.481446i	$-0.159888\pi$
$659$	45.0000			1.75295			0.876476	+	0.481446i	$0.159888\pi$
$660$		0			0
$661$			20.0000i			0.777910i	0.921257	+	0.388955i	$0.127164\pi$
$661$			20.0000i			0.777910i	−0.921257	+	0.388955i	$0.872836\pi$
$662$	−20.0000			−0.777322
$663$	−6.00000	+	4.00000i	−0.233021	+	0.155347i
$664$		0			0
$665$		0			0
$666$	8.00000			0.309994
$667$	−5.00000			−0.193601
$668$		−	4.00000i		−	0.154765i
$669$		−	4.00000i		−	0.154649i
$670$		0			0
$671$		0			0
$672$	−16.0000			−0.617213
$673$	14.0000			0.539660			0.269830	−	0.962908i	$-0.413032\pi$
$673$	14.0000			0.539660			0.269830	+	0.962908i	$0.413032\pi$
$674$			6.00000i			0.231111i
$675$		0			0
$676$	−10.0000	+	24.0000i	−0.384615	+	0.923077i
$677$	33.0000			1.26829			0.634147	−	0.773213i	$-0.281352\pi$
$677$	33.0000			1.26829			0.634147	+	0.773213i	$0.281352\pi$
$678$			2.00000i			0.0768095i
$679$	−4.00000			−0.153506
$680$		0			0
$681$		−	12.0000i		−	0.459841i
$682$		0			0
$683$			4.00000i			0.153056i	0.997067	+	0.0765279i	$0.0243834\pi$
$683$			4.00000i			0.153056i	−0.997067	+	0.0765279i	$0.975617\pi$
$684$		−	16.0000i		−	0.611775i
$685$		0			0
$686$	−40.0000			−1.52721
$687$		−	26.0000i		−	0.991962i
$688$	44.0000			1.67748
$689$	−27.0000	+	18.0000i	−1.02862	+	0.685745i
$690$		0			0
$691$		−	10.0000i		−	0.380418i	−0.981744	−	0.190209i	$-0.939083\pi$
$691$		−	10.0000i		−	0.380418i	0.981744	−	0.190209i	$-0.0609166\pi$
$692$	−28.0000			−1.06440
$693$		0			0
$694$			46.0000i			1.74614i
$695$		0			0
$696$		0			0
$697$		−	20.0000i		−	0.757554i
$698$	28.0000			1.05982
$699$	11.0000			0.416058
$700$		0			0
$701$	−3.00000			−0.113308			−0.0566542	−	0.998394i	$-0.518043\pi$
$701$	−3.00000			−0.113308			−0.0566542	+	0.998394i	$0.518043\pi$
$702$	−20.0000	−	30.0000i	−0.754851	−	1.13228i
$703$	8.00000			0.301726
$704$		0			0
$705$		0			0
$706$	−48.0000			−1.80650
$707$			14.0000i			0.526524i
$708$			12.0000i			0.450988i
$709$		−	44.0000i		−	1.65245i	−0.563337	−	0.826227i	$-0.690483\pi$
$709$		−	44.0000i		−	1.65245i	0.563337	−	0.826227i	$-0.309517\pi$
$710$		0			0
$711$	10.0000			0.375029
$712$		0			0
$713$		−	10.0000i		−	0.374503i
$714$	−8.00000			−0.299392
$715$		0			0
$716$	30.0000			1.12115
$717$			4.00000i			0.149383i
$718$	−32.0000			−1.19423
$719$	−15.0000			−0.559406			−0.279703	−	0.960087i	$-0.590236\pi$
$719$	−15.0000			−0.559406			−0.279703	+	0.960087i	$0.590236\pi$
$720$		0			0
$721$		−	2.00000i		−	0.0744839i
$722$			6.00000i			0.223297i
$723$		0			0
$724$	36.0000			1.33793
$725$		0			0
$726$		−	22.0000i		−	0.816497i
$727$	−47.0000			−1.74313			−0.871567	−	0.490277i	$-0.836896\pi$
$727$	−47.0000			−1.74313			−0.871567	+	0.490277i	$0.836896\pi$
$728$		0			0
$729$	13.0000			0.481481
$730$		0			0
$731$	22.0000			0.813699
$732$	14.0000			0.517455
$733$			24.0000i			0.886460i	0.896408	+	0.443230i	$0.146168\pi$
$733$			24.0000i			0.886460i	−0.896408	+	0.443230i	$0.853832\pi$
$734$			6.00000i			0.221464i
$735$		0			0
$736$			8.00000i			0.294884i
$737$		0			0
$738$	40.0000			1.47242
$739$			6.00000i			0.220714i	0.993892	+	0.110357i	$0.0351994\pi$
$739$			6.00000i			0.220714i	−0.993892	+	0.110357i	$0.964801\pi$
$740$		0			0
$741$	−8.00000	−	12.0000i	−0.293887	−	0.440831i
$742$	−36.0000			−1.32160
$743$		−	16.0000i		−	0.586983i	−0.955962	−	0.293492i	$-0.905183\pi$
$743$		−	16.0000i		−	0.586983i	0.955962	−	0.293492i	$-0.0948173\pi$
$744$		0			0
$745$		0			0
$746$		−	2.00000i		−	0.0732252i
$747$			12.0000i			0.439057i
$748$		0			0
$749$			26.0000i			0.950019i
$750$		0			0
$751$	−23.0000			−0.839282			−0.419641	−	0.907690i	$-0.637844\pi$
$751$	−23.0000			−0.839282			−0.419641	+	0.907690i	$0.637844\pi$
$752$			32.0000i			1.16692i
$753$	−12.0000			−0.437304
$754$	−20.0000	−	30.0000i	−0.728357	−	1.09254i
$755$		0			0
$756$		−	20.0000i		−	0.727393i
$757$	−2.00000			−0.0726912			−0.0363456	−	0.999339i	$-0.511572\pi$
$757$	−2.00000			−0.0726912			−0.0363456	+	0.999339i	$0.511572\pi$
$758$	48.0000			1.74344
$759$		0			0
$760$		0			0
$761$			10.0000i			0.362500i	0.983437	+	0.181250i	$0.0580143\pi$
$761$			10.0000i			0.362500i	−0.983437	+	0.181250i	$0.941986\pi$
$762$		−	6.00000i		−	0.217357i
$763$	−32.0000			−1.15848
$764$	46.0000			1.66422
$765$		0			0
$766$	32.0000			1.15621
$767$	−12.0000	−	18.0000i	−0.433295	−	0.649942i
$768$	−16.0000			−0.577350
$769$		−	14.0000i		−	0.504853i	−0.967616	−	0.252426i	$-0.918771\pi$
$769$		−	14.0000i		−	0.504853i	0.967616	−	0.252426i	$-0.0812286\pi$
$770$		0			0
$771$	27.0000			0.972381
$772$			32.0000i			1.15171i
$773$		−	6.00000i		−	0.215805i	−0.994161	−	0.107903i	$-0.965587\pi$
$773$		−	6.00000i		−	0.215805i	0.994161	−	0.107903i	$-0.0344134\pi$
$774$			44.0000i			1.58155i
$775$		0			0
$776$		0			0
$777$	4.00000			0.143499
$778$			50.0000i			1.79259i
$779$	40.0000			1.43315
$780$		0			0
$781$		0			0
$782$			4.00000i			0.143040i
$783$	25.0000			0.893427
$784$	−12.0000			−0.428571
$785$		0			0
$786$			6.00000i			0.214013i
$787$			32.0000i			1.14068i	0.821410	+	0.570338i	$0.193188\pi$
$787$			32.0000i			1.14068i	−0.821410	+	0.570338i	$0.806812\pi$
$788$		−	4.00000i		−	0.142494i
$789$	−24.0000			−0.854423
$790$		0			0
$791$		−	2.00000i		−	0.0711118i
$792$		0			0
$793$	−21.0000	+	14.0000i	−0.745732	+	0.497155i
$794$	−44.0000			−1.56150
$795$		0			0
$796$	30.0000			1.06332
$797$	23.0000			0.814702			0.407351	−	0.913272i	$-0.366453\pi$
$797$	23.0000			0.814702			0.407351	+	0.913272i	$0.366453\pi$
$798$		−	16.0000i		−	0.566394i
$799$			16.0000i			0.566039i
$800$		0			0
$801$		−	12.0000i		−	0.423999i
$802$	20.0000			0.706225
$803$		0			0
$804$			24.0000i			0.846415i
$805$		0			0
$806$	60.0000	−	40.0000i	2.11341	−	1.40894i
$807$	−25.0000			−0.880042
$808$		0			0
$809$	−30.0000			−1.05474			−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000			−1.05474			−0.527372	+	0.849635i	$0.676823\pi$
$810$		0			0
$811$			40.0000i			1.40459i	0.711886	+	0.702295i	$0.247841\pi$
$811$			40.0000i			1.40459i	−0.711886	+	0.702295i	$0.752159\pi$
$812$		−	20.0000i		−	0.701862i
$813$			20.0000i			0.701431i
$814$		0			0
$815$		0			0
$816$	−8.00000			−0.280056
$817$			44.0000i			1.53937i
$818$	−12.0000			−0.419570
$819$	8.00000	+	12.0000i	0.279543	+	0.419314i
$820$		0			0
$821$			30.0000i			1.04701i	0.852023	+	0.523504i	$0.175375\pi$
$821$			30.0000i			1.04701i	−0.852023	+	0.523504i	$0.824625\pi$
$822$	−36.0000			−1.25564
$823$	9.00000			0.313720			0.156860	−	0.987621i	$-0.449863\pi$
$823$	9.00000			0.313720			0.156860	+	0.987621i	$0.449863\pi$
$824$		0			0
$825$		0			0
$826$		−	24.0000i		−	0.835067i
$827$			22.0000i			0.765015i	0.923952	+	0.382507i	$0.124939\pi$
$827$			22.0000i			0.765015i	−0.923952	+	0.382507i	$0.875061\pi$
$828$	−4.00000			−0.139010
$829$	−5.00000			−0.173657			−0.0868286	−	0.996223i	$-0.527673\pi$
$829$	−5.00000			−0.173657			−0.0868286	+	0.996223i	$0.527673\pi$
$830$		0			0
$831$	7.00000			0.242827
$832$	−24.0000	+	16.0000i	−0.832050	+	0.554700i
$833$	−6.00000			−0.207888
$834$			10.0000i			0.346272i
$835$		0			0
$836$		0			0
$837$			50.0000i			1.72825i
$838$			30.0000i			1.03633i
$839$			6.00000i			0.207143i	0.994622	+	0.103572i	$0.0330271\pi$
$839$			6.00000i			0.207143i	−0.994622	+	0.103572i	$0.966973\pi$
$840$		0			0
$841$	−4.00000			−0.137931
$842$	20.0000			0.689246
$843$		−	30.0000i		−	1.03325i
$844$	−24.0000			−0.826114
$845$		0			0
$846$	−32.0000			−1.10018
$847$			22.0000i			0.755929i
$848$	−36.0000			−1.23625
$849$	−4.00000			−0.137280
$850$		0			0
$851$		−	2.00000i		−	0.0685591i
$852$		−	20.0000i		−	0.685189i
$853$			24.0000i			0.821744i	0.911693	+	0.410872i	$0.134776\pi$
$853$			24.0000i			0.821744i	−0.911693	+	0.410872i	$0.865224\pi$
$854$	−28.0000			−0.958140
$855$		0			0
$856$		0			0
$857$	13.0000			0.444072			0.222036	−	0.975039i	$-0.428730\pi$
$857$	13.0000			0.444072			0.222036	+	0.975039i	$0.428730\pi$
$858$		0			0
$859$	20.0000			0.682391			0.341196	−	0.939992i	$-0.389168\pi$
$859$	20.0000			0.682391			0.341196	+	0.939992i	$0.389168\pi$
$860$		0			0
$861$	20.0000			0.681598
$862$	60.0000			2.04361
$863$		−	6.00000i		−	0.204242i	−0.994772	−	0.102121i	$-0.967437\pi$
$863$		−	6.00000i		−	0.204242i	0.994772	−	0.102121i	$-0.0325630\pi$
$864$		−	40.0000i		−	1.36083i
$865$		0			0
$866$		−	22.0000i		−	0.747590i
$867$	13.0000			0.441503
$868$	40.0000			1.35769
$869$		0			0
$870$		0			0
$871$	−24.0000	−	36.0000i	−0.813209	−	1.21981i
$872$		0			0
$873$		−	4.00000i		−	0.135379i
$874$	−8.00000			−0.270604
$875$		0			0
$876$			28.0000i			0.946032i
$877$		−	38.0000i		−	1.28317i	−0.767052	−	0.641584i	$-0.778277\pi$
$877$		−	38.0000i		−	1.28317i	0.767052	−	0.641584i	$-0.221723\pi$
$878$		−	40.0000i		−	1.34993i
$879$		−	24.0000i		−	0.809500i
$880$		0			0
$881$	27.0000			0.909653			0.454827	−	0.890580i	$-0.349701\pi$
$881$	27.0000			0.909653			0.454827	+	0.890580i	$0.349701\pi$
$882$		−	12.0000i		−	0.404061i
$883$	−31.0000			−1.04323			−0.521617	−	0.853180i	$-0.674671\pi$
$883$	−31.0000			−1.04323			−0.521617	+	0.853180i	$0.674671\pi$
$884$	−12.0000	+	8.00000i	−0.403604	+	0.269069i
$885$		0			0
$886$			48.0000i			1.61259i
$887$	−32.0000			−1.07445			−0.537227	−	0.843437i	$-0.680528\pi$
$887$	−32.0000			−1.07445			−0.537227	+	0.843437i	$0.680528\pi$
$888$		0			0
$889$			6.00000i			0.201234i
$890$		0			0
$891$		0			0
$892$		−	8.00000i		−	0.267860i
$893$	−32.0000			−1.07084
$894$	−28.0000			−0.936460
$895$		0			0
$896$		0			0
$897$	−3.00000	+	2.00000i	−0.100167	+	0.0667781i
$898$	−72.0000			−2.40267
$899$			50.0000i			1.66759i
$900$		0			0
$901$	−18.0000			−0.599667
$902$		0			0
$903$			22.0000i			0.732114i
$904$		0			0
$905$		0			0
$906$	−20.0000			−0.664455
$907$	53.0000			1.75984			0.879918	−	0.475125i	$-0.157597\pi$
$907$	53.0000			1.75984			0.879918	+	0.475125i	$0.157597\pi$
$908$		−	24.0000i		−	0.796468i
$909$	−14.0000			−0.464351
$910$		0			0
$911$	−13.0000			−0.430709			−0.215355	−	0.976536i	$-0.569091\pi$
$911$	−13.0000			−0.430709			−0.215355	+	0.976536i	$0.569091\pi$
$912$		−	16.0000i		−	0.529813i
$913$		0			0
$914$	16.0000			0.529233
$915$		0			0
$916$		−	52.0000i		−	1.71813i
$917$		−	6.00000i		−	0.198137i
$918$		−	20.0000i		−	0.660098i
$919$	40.0000			1.31948			0.659739	−	0.751495i	$-0.270667\pi$
$919$	40.0000			1.31948			0.659739	+	0.751495i	$0.270667\pi$
$920$		0			0
$921$			8.00000i			0.263609i
$922$	−40.0000			−1.31733
$923$	20.0000	+	30.0000i	0.658308	+	0.987462i
$924$		0			0
$925$		0			0
$926$	−8.00000			−0.262896
$927$	2.00000			0.0656886
$928$		−	40.0000i		−	1.31306i
$929$		−	44.0000i		−	1.44359i	−0.692105	−	0.721797i	$-0.743317\pi$
$929$		−	44.0000i		−	1.44359i	0.692105	−	0.721797i	$-0.256683\pi$
$930$		0			0
$931$		−	12.0000i		−	0.393284i
$932$	22.0000			0.720634
$933$	23.0000			0.752986
$934$		−	54.0000i		−	1.76693i
$935$		0			0
$936$		0			0
$937$	13.0000			0.424691			0.212346	−	0.977195i	$-0.431890\pi$
$937$	13.0000			0.424691			0.212346	+	0.977195i	$0.431890\pi$
$938$		−	48.0000i		−	1.56726i
$939$	−19.0000			−0.620042
$940$		0			0
$941$			20.0000i			0.651981i	0.945373	+	0.325991i	$0.105698\pi$
$941$			20.0000i			0.651981i	−0.945373	+	0.325991i	$0.894302\pi$
$942$			4.00000i			0.130327i
$943$		−	10.0000i		−	0.325645i
$944$		−	24.0000i		−	0.781133i
$945$		0			0
$946$		0			0
$947$		−	18.0000i		−	0.584921i	−0.956278	−	0.292461i	$-0.905526\pi$
$947$		−	18.0000i		−	0.584921i	0.956278	−	0.292461i	$-0.0944741\pi$
$948$	−10.0000			−0.324785
$949$	−28.0000	−	42.0000i	−0.908918	−	1.36338i
$950$		0			0
$951$			28.0000i			0.907962i
$952$		0			0
$953$	54.0000			1.74923			0.874616	−	0.484817i	$-0.161114\pi$
$953$	54.0000			1.74923			0.874616	+	0.484817i	$0.161114\pi$
$954$		−	36.0000i		−	1.16554i
$955$		0			0
$956$			8.00000i			0.258738i
$957$		0			0
$958$	48.0000			1.55081
$959$	36.0000			1.16250
$960$		0			0
$961$	−69.0000			−2.22581
$962$	12.0000	−	8.00000i	0.386896	−	0.257930i
$963$	−26.0000			−0.837838
$964$		0			0
$965$		0			0
$966$	−4.00000			−0.128698
$967$		−	58.0000i		−	1.86515i	−0.360971	−	0.932577i	$-0.617555\pi$
$967$		−	58.0000i		−	1.86515i	0.360971	−	0.932577i	$-0.382445\pi$
$968$		0			0
$969$		−	8.00000i		−	0.256997i
$970$		0			0
$971$	−48.0000			−1.54039			−0.770197	−	0.637806i	$-0.779842\pi$
$971$	−48.0000			−1.54039			−0.770197	+	0.637806i	$0.779842\pi$
$972$	32.0000			1.02640
$973$		−	10.0000i		−	0.320585i
$974$	−24.0000			−0.769010
$975$		0			0
$976$	−28.0000			−0.896258
$977$			32.0000i			1.02377i	0.859054	+	0.511885i	$0.171053\pi$
$977$			32.0000i			1.02377i	−0.859054	+	0.511885i	$0.828947\pi$
$978$	−12.0000			−0.383718
$979$		0			0
$980$		0			0
$981$		−	32.0000i		−	1.02168i
$982$			34.0000i			1.08498i
$983$			34.0000i			1.08443i	0.840239	+	0.542216i	$0.182414\pi$
$983$			34.0000i			1.08443i	−0.840239	+	0.542216i	$0.817586\pi$
$984$		0			0
$985$		0			0
$986$		−	20.0000i		−	0.636930i
$987$	−16.0000			−0.509286
$988$	−16.0000	−	24.0000i	−0.509028	−	0.763542i
$989$	11.0000			0.349780
$990$		0			0
$991$	7.00000			0.222362			0.111181	−	0.993800i	$-0.464537\pi$
$991$	7.00000			0.222362			0.111181	+	0.993800i	$0.464537\pi$
$992$	80.0000			2.54000
$993$		−	10.0000i		−	0.317340i
$994$			40.0000i			1.26872i
$995$		0			0
$996$		−	12.0000i		−	0.380235i
$997$	−27.0000			−0.855099			−0.427549	−	0.903992i	$-0.640623\pi$
$997$	−27.0000			−0.855099			−0.427549	+	0.903992i	$0.640623\pi$
$998$	−52.0000			−1.64603
$999$			10.0000i			0.316386i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	325.2.c.c.51.2	yes	2
5.2	odd	4		325.2.d.a.324.2		2
5.3	odd	4		325.2.d.d.324.1		2
5.4	even	2		325.2.c.d.51.1	yes	2
13.5	odd	4		4225.2.a.o.1.1		1
13.8	odd	4		4225.2.a.a.1.1		1
13.12	even	2	inner	325.2.c.c.51.1	✓	2
65.12	odd	4		325.2.d.d.324.2		2
65.34	odd	4		4225.2.a.q.1.1		1
65.38	odd	4		325.2.d.a.324.1		2
65.44	odd	4		4225.2.a.c.1.1		1
65.64	even	2		325.2.c.d.51.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
325.2.c.c.51.1	✓	2	13.12	even	2	inner
325.2.c.c.51.2	yes	2	1.1	even	1	trivial
325.2.c.d.51.1	yes	2	5.4	even	2
325.2.c.d.51.2	yes	2	65.64	even	2
325.2.d.a.324.1		2	65.38	odd	4
325.2.d.a.324.2		2	5.2	odd	4
325.2.d.d.324.1		2	5.3	odd	4
325.2.d.d.324.2		2	65.12	odd	4
4225.2.a.a.1.1		1	13.8	odd	4
4225.2.a.c.1.1		1	65.44	odd	4
4225.2.a.o.1.1		1	13.5	odd	4
4225.2.a.q.1.1		1	65.34	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 325 = 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	325.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.00000i q^{2} -1.00000 q^{3} -2.00000 q^{4} -2.00000i q^{6} +2.00000i q^{7} -2.00000 q^{9} +O(q^{10})\)	\(q+2.00000i q^{2} -1.00000 q^{3} -2.00000 q^{4} -2.00000i q^{6} +2.00000i q^{7} -2.00000 q^{9} +2.00000 q^{12} +(-3.00000 + 2.00000i) q^{13} -4.00000 q^{14} -4.00000 q^{16} -2.00000 q^{17} -4.00000i q^{18} -4.00000i q^{19} -2.00000i q^{21} -1.00000 q^{23} +(-4.00000 - 6.00000i) q^{26} +5.00000 q^{27} -4.00000i q^{28} +5.00000 q^{29} +10.0000i q^{31} -8.00000i q^{32} -4.00000i q^{34} +4.00000 q^{36} +2.00000i q^{37} +8.00000 q^{38} +(3.00000 - 2.00000i) q^{39} +10.0000i q^{41} +4.00000 q^{42} -11.0000 q^{43} -2.00000i q^{46} -8.00000i q^{47} +4.00000 q^{48} +3.00000 q^{49} +2.00000 q^{51} +(6.00000 - 4.00000i) q^{52} +9.00000 q^{53} +10.0000i q^{54} +4.00000i q^{57} +10.0000i q^{58} +6.00000i q^{59} +7.00000 q^{61} -20.0000 q^{62} -4.00000i q^{63} +8.00000 q^{64} +12.0000i q^{67} +4.00000 q^{68} +1.00000 q^{69} -10.0000i q^{71} +14.0000i q^{73} -4.00000 q^{74} +8.00000i q^{76} +(4.00000 + 6.00000i) q^{78} -5.00000 q^{79} +1.00000 q^{81} -20.0000 q^{82} -6.00000i q^{83} +4.00000i q^{84} -22.0000i q^{86} -5.00000 q^{87} +6.00000i q^{89} +(-4.00000 - 6.00000i) q^{91} +2.00000 q^{92} -10.0000i q^{93} +16.0000 q^{94} +8.00000i q^{96} +2.00000i q^{97} +6.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 4 q^{4} - 4 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 - 4 * q^4 - 4 * q^9	\( 2 q - 2 q^{3} - 4 q^{4} - 4 q^{9} + 4 q^{12} - 6 q^{13} - 8 q^{14} - 8 q^{16} - 4 q^{17} - 2 q^{23} - 8 q^{26} + 10 q^{27} + 10 q^{29} + 8 q^{36} + 16 q^{38} + 6 q^{39} + 8 q^{42} - 22 q^{43} + 8 q^{48} + 6 q^{49} + 4 q^{51} + 12 q^{52} + 18 q^{53} + 14 q^{61} - 40 q^{62} + 16 q^{64} + 8 q^{68} + 2 q^{69} - 8 q^{74} + 8 q^{78} - 10 q^{79} + 2 q^{81} - 40 q^{82} - 10 q^{87} - 8 q^{91} + 4 q^{92} + 32 q^{94}+O(q^{100}) \) 2 * q - 2 * q^3 - 4 * q^4 - 4 * q^9 + 4 * q^12 - 6 * q^13 - 8 * q^14 - 8 * q^16 - 4 * q^17 - 2 * q^23 - 8 * q^26 + 10 * q^27 + 10 * q^29 + 8 * q^36 + 16 * q^38 + 6 * q^39 + 8 * q^42 - 22 * q^43 + 8 * q^48 + 6 * q^49 + 4 * q^51 + 12 * q^52 + 18 * q^53 + 14 * q^61 - 40 * q^62 + 16 * q^64 + 8 * q^68 + 2 * q^69 - 8 * q^74 + 8 * q^78 - 10 * q^79 + 2 * q^81 - 40 * q^82 - 10 * q^87 - 8 * q^91 + 4 * q^92 + 32 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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