LMFDB - Embedded newform 325.2.d.d.324.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [325,2,Mod(324,325)]

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([1, 1]))

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

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

chi := DirichletCharacter("325.324");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 325 = 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	325.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$2.59513806569$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			324.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	325.324
Dual form			325.2.d.d.324.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000 q^{2} +1.00000i q^{3} +2.00000 q^{4} +2.00000i q^{6} +2.00000 q^{7} +2.00000 q^{9} +O(q^{10})$	\(q+2.00000 q^{2} +1.00000i q^{3} +2.00000 q^{4} +2.00000i q^{6} +2.00000 q^{7} +2.00000 q^{9} +2.00000i q^{12} +(-2.00000 + 3.00000i) q^{13} +4.00000 q^{14} -4.00000 q^{16} -2.00000i q^{17} +4.00000 q^{18} -4.00000i q^{19} +2.00000i q^{21} +1.00000i q^{23} +(-4.00000 + 6.00000i) q^{26} +5.00000i q^{27} +4.00000 q^{28} -5.00000 q^{29} -10.0000i q^{31} -8.00000 q^{32} -4.00000i q^{34} +4.00000 q^{36} +2.00000 q^{37} -8.00000i q^{38} +(-3.00000 - 2.00000i) q^{39} -10.0000i q^{41} +4.00000i q^{42} +11.0000i q^{43} +2.00000i q^{46} -8.00000 q^{47} -4.00000i q^{48} -3.00000 q^{49} +2.00000 q^{51} +(-4.00000 + 6.00000i) q^{52} -9.00000i q^{53} +10.0000i q^{54} +4.00000 q^{57} -10.0000 q^{58} +6.00000i q^{59} +7.00000 q^{61} -20.0000i q^{62} +4.00000 q^{63} -8.00000 q^{64} +12.0000 q^{67} -4.00000i q^{68} -1.00000 q^{69} +10.0000i q^{71} -14.0000 q^{73} +4.00000 q^{74} -8.00000i q^{76} +(-6.00000 - 4.00000i) q^{78} +5.00000 q^{79} +1.00000 q^{81} -20.0000i q^{82} +6.00000 q^{83} +4.00000i q^{84} +22.0000i q^{86} -5.00000i q^{87} +6.00000i q^{89} +(-4.00000 + 6.00000i) q^{91} +2.00000i q^{92} +10.0000 q^{93} -16.0000 q^{94} -8.00000i q^{96} +2.00000 q^{97} -6.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{2} + 4 q^{4} + 4 q^{7} + 4 q^{9}+O(q^{10}) $ 2 * q + 4 * q^2 + 4 * q^4 + 4 * q^7 + 4 * q^9	$ 2 q + 4 q^{2} + 4 q^{4} + 4 q^{7} + 4 q^{9} - 4 q^{13} + 8 q^{14} - 8 q^{16} + 8 q^{18} - 8 q^{26} + 8 q^{28} - 10 q^{29} - 16 q^{32} + 8 q^{36} + 4 q^{37} - 6 q^{39} - 16 q^{47} - 6 q^{49} + 4 q^{51} - 8 q^{52} + 8 q^{57} - 20 q^{58} + 14 q^{61} + 8 q^{63} - 16 q^{64} + 24 q^{67} - 2 q^{69} - 28 q^{73} + 8 q^{74} - 12 q^{78} + 10 q^{79} + 2 q^{81} + 12 q^{83} - 8 q^{91} + 20 q^{93} - 32 q^{94} + 4 q^{97} - 12 q^{98}+O(q^{100}) $ 2 * q + 4 * q^2 + 4 * q^4 + 4 * q^7 + 4 * q^9 - 4 * q^13 + 8 * q^14 - 8 * q^16 + 8 * q^18 - 8 * q^26 + 8 * q^28 - 10 * q^29 - 16 * q^32 + 8 * q^36 + 4 * q^37 - 6 * q^39 - 16 * q^47 - 6 * q^49 + 4 * q^51 - 8 * q^52 + 8 * q^57 - 20 * q^58 + 14 * q^61 + 8 * q^63 - 16 * q^64 + 24 * q^67 - 2 * q^69 - 28 * q^73 + 8 * q^74 - 12 * q^78 + 10 * q^79 + 2 * q^81 + 12 * q^83 - 8 * q^91 + 20 * q^93 - 32 * q^94 + 4 * q^97 - 12 * q^98

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 $

$2$	2.00000			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$2$	2.00000			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$3$			1.00000i			0.577350i	0.957427	+	0.288675i	$0.0932147\pi$
$3$			1.00000i			0.577350i	−0.957427	+	0.288675i	$0.906785\pi$
$4$	2.00000			1.00000
$5$		0			0
$5$		0			0
$6$			2.00000i			0.816497i
$7$	2.00000			0.755929			0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000			0.755929			0.377964	+	0.925820i	$0.376624\pi$
$8$		0			0
$9$	2.00000			0.666667
$10$		0			0
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$			2.00000i			0.577350i
$13$	−2.00000	+	3.00000i	−0.554700	+	0.832050i
$13$	−2.00000	+	3.00000i	−0.554700	+	0.832050i
$14$	4.00000			1.06904
$15$		0			0
$16$	−4.00000			−1.00000
$17$		−	2.00000i		−	0.485071i	−0.970143	−	0.242536i	$-0.922021\pi$
$17$		−	2.00000i		−	0.485071i	0.970143	−	0.242536i	$-0.0779791\pi$
$18$	4.00000			0.942809
$19$		−	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
$23$			1.00000i			0.208514i	0.994550	+	0.104257i	$0.0332465\pi$
$23$			1.00000i			0.208514i	−0.994550	+	0.104257i	$0.966753\pi$
$24$		0			0
$25$		0			0
$26$	−4.00000	+	6.00000i	−0.784465	+	1.17670i
$27$			5.00000i			0.962250i
$28$	4.00000			0.755929
$29$	−5.00000			−0.928477			−0.464238	−	0.885710i	$-0.653672\pi$
$29$	−5.00000			−0.928477			−0.464238	+	0.885710i	$0.653672\pi$
$30$		0			0
$31$		−	10.0000i		−	1.79605i	−0.439941	−	0.898027i	$-0.645001\pi$
$31$		−	10.0000i		−	1.79605i	0.439941	−	0.898027i	$-0.354999\pi$
$32$	−8.00000			−1.41421
$33$		0			0
$34$		−	4.00000i		−	0.685994i
$35$		0			0
$36$	4.00000			0.666667
$37$	2.00000			0.328798			0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000			0.328798			0.164399	+	0.986394i	$0.447432\pi$
$38$		−	8.00000i		−	1.29777i
$39$	−3.00000	−	2.00000i	−0.480384	−	0.320256i
$40$		0			0
$41$		−	10.0000i		−	1.56174i	−0.624695	−	0.780869i	$-0.714777\pi$
$41$		−	10.0000i		−	1.56174i	0.624695	−	0.780869i	$-0.285223\pi$
$42$			4.00000i			0.617213i
$43$			11.0000i			1.67748i	0.544529	+	0.838742i	$0.316708\pi$
$43$			11.0000i			1.67748i	−0.544529	+	0.838742i	$0.683292\pi$
$44$		0			0
$45$		0			0
$46$			2.00000i			0.294884i
$47$	−8.00000			−1.16692			−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000			−1.16692			−0.583460	+	0.812142i	$0.698301\pi$
$48$		−	4.00000i		−	0.577350i
$49$	−3.00000			−0.428571
$50$		0			0
$51$	2.00000			0.280056
$52$	−4.00000	+	6.00000i	−0.554700	+	0.832050i
$53$		−	9.00000i		−	1.23625i	−0.786082	−	0.618123i	$-0.787894\pi$
$53$		−	9.00000i		−	1.23625i	0.786082	−	0.618123i	$-0.212106\pi$
$54$			10.0000i			1.36083i
$55$		0			0
$56$		0			0
$57$	4.00000			0.529813
$58$	−10.0000			−1.31306
$59$			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	$0.352090\pi$
$62$		−	20.0000i		−	2.54000i
$63$	4.00000			0.503953
$64$	−8.00000			−1.00000
$65$		0			0
$66$		0			0
$67$	12.0000			1.46603			0.733017	−	0.680211i	$-0.238112\pi$
$67$	12.0000			1.46603			0.733017	+	0.680211i	$0.238112\pi$
$68$		−	4.00000i		−	0.485071i
$69$	−1.00000			−0.120386
$70$		0			0
$71$			10.0000i			1.18678i	0.804914	+	0.593391i	$0.202211\pi$
$71$			10.0000i			1.18678i	−0.804914	+	0.593391i	$0.797789\pi$
$72$		0			0
$73$	−14.0000			−1.63858			−0.819288	−	0.573382i	$-0.805631\pi$
$73$	−14.0000			−1.63858			−0.819288	+	0.573382i	$0.805631\pi$
$74$	4.00000			0.464991
$75$		0			0
$76$		−	8.00000i		−	0.917663i
$77$		0			0
$78$	−6.00000	−	4.00000i	−0.679366	−	0.452911i
$79$	5.00000			0.562544			0.281272	−	0.959628i	$-0.409244\pi$
$79$	5.00000			0.562544			0.281272	+	0.959628i	$0.409244\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		−	20.0000i		−	2.20863i
$83$	6.00000			0.658586			0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000			0.658586			0.329293	+	0.944228i	$0.393190\pi$
$84$			4.00000i			0.436436i
$85$		0			0
$86$			22.0000i			2.37232i
$87$		−	5.00000i		−	0.536056i
$88$		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
$92$			2.00000i			0.208514i
$93$	10.0000			1.03695
$94$	−16.0000			−1.65027
$95$		0			0
$96$		−	8.00000i		−	0.816497i
$97$	2.00000			0.203069			0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000			0.203069			0.101535	+	0.994832i	$0.467625\pi$
$98$	−6.00000			−0.606092
$99$		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	$0.386772\pi$
$102$	4.00000			0.396059
$103$			1.00000i			0.0985329i	0.998786	+	0.0492665i	$0.0156884\pi$
$103$			1.00000i			0.0985329i	−0.998786	+	0.0492665i	$0.984312\pi$
$104$		0			0
$105$		0			0
$106$		−	18.0000i		−	1.74831i
$107$			13.0000i			1.25676i	0.777908	+	0.628379i	$0.216281\pi$
$107$			13.0000i			1.25676i	−0.777908	+	0.628379i	$0.783719\pi$
$108$			10.0000i			0.962250i
$109$			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
$112$	−8.00000			−0.755929
$113$			1.00000i			0.0940721i	0.998893	+	0.0470360i	$0.0149776\pi$
$113$			1.00000i			0.0940721i	−0.998893	+	0.0470360i	$0.985022\pi$
$114$	8.00000			0.749269
$115$		0			0
$116$	−10.0000			−0.928477
$117$	−4.00000	+	6.00000i	−0.369800	+	0.554700i
$118$			12.0000i			1.10469i
$119$		−	4.00000i		−	0.366679i
$120$		0			0
$121$	11.0000			1.00000
$122$	14.0000			1.26750
$123$	10.0000			0.901670
$124$		−	20.0000i		−	1.79605i
$125$		0			0
$126$	8.00000			0.712697
$127$			3.00000i			0.266207i	0.991102	+	0.133103i	$0.0424943\pi$
$127$			3.00000i			0.266207i	−0.991102	+	0.133103i	$0.957506\pi$
$128$		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
$133$		−	8.00000i		−	0.693688i
$134$	24.0000			2.07328
$135$		0			0
$136$		0			0
$137$	−18.0000			−1.53784			−0.768922	−	0.639343i	$-0.779207\pi$
$137$	−18.0000			−1.53784			−0.768922	+	0.639343i	$0.779207\pi$
$138$	−2.00000			−0.170251
$139$	5.00000			0.424094			0.212047	−	0.977259i	$-0.431987\pi$
$139$	5.00000			0.424094			0.212047	+	0.977259i	$0.431987\pi$
$140$		0			0
$141$		−	8.00000i		−	0.673722i
$142$			20.0000i			1.67836i
$143$		0			0
$144$	−8.00000			−0.666667
$145$		0			0
$146$	−28.0000			−2.31730
$147$		−	3.00000i		−	0.247436i
$148$	4.00000			0.328798
$149$		−	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	$0.133388\pi$
$151$			10.0000i			0.813788i	−0.913475	+	0.406894i	$0.866612\pi$
$152$		0			0
$153$		−	4.00000i		−	0.323381i
$154$		0			0
$155$		0			0
$156$	−6.00000	−	4.00000i	−0.480384	−	0.320256i
$157$		−	2.00000i		−	0.159617i	−0.996810	−	0.0798087i	$-0.974569\pi$
$157$		−	2.00000i		−	0.159617i	0.996810	−	0.0798087i	$-0.0254309\pi$
$158$	10.0000			0.795557
$159$	9.00000			0.713746
$160$		0			0
$161$			2.00000i			0.157622i
$162$	2.00000			0.157135
$163$	6.00000			0.469956			0.234978	−	0.972001i	$-0.424498\pi$
$163$	6.00000			0.469956			0.234978	+	0.972001i	$0.424498\pi$
$164$		−	20.0000i		−	1.56174i
$165$		0			0
$166$	12.0000			0.931381
$167$	2.00000			0.154765			0.0773823	−	0.997001i	$-0.475344\pi$
$167$	2.00000			0.154765			0.0773823	+	0.997001i	$0.475344\pi$
$168$		0			0
$169$	−5.00000	−	12.0000i	−0.384615	−	0.923077i
$170$		0			0
$171$		−	8.00000i		−	0.611775i
$172$			22.0000i			1.67748i
$173$		−	14.0000i		−	1.06440i	−0.846619	−	0.532200i	$-0.821365\pi$
$173$		−	14.0000i		−	1.06440i	0.846619	−	0.532200i	$-0.178635\pi$
$174$		−	10.0000i		−	0.758098i
$175$		0			0
$176$		0			0
$177$	−6.00000			−0.450988
$178$			12.0000i			0.899438i
$179$	15.0000			1.12115			0.560576	−	0.828103i	$-0.310580\pi$
$179$	15.0000			1.12115			0.560576	+	0.828103i	$0.310580\pi$
$180$		0			0
$181$	−18.0000			−1.33793			−0.668965	−	0.743294i	$-0.733262\pi$
$181$	−18.0000			−1.33793			−0.668965	+	0.743294i	$0.733262\pi$
$182$	−8.00000	+	12.0000i	−0.592999	+	0.889499i
$183$			7.00000i			0.517455i
$184$		0			0
$185$		0			0
$186$	20.0000			1.46647
$187$		0			0
$188$	−16.0000			−1.16692
$189$			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	$0.812868\pi$
$192$		−	8.00000i		−	0.577350i
$193$	16.0000			1.15171			0.575853	−	0.817554i	$-0.304670\pi$
$193$	16.0000			1.15171			0.575853	+	0.817554i	$0.304670\pi$
$194$	4.00000			0.287183
$195$		0			0
$196$	−6.00000			−0.428571
$197$	2.00000			0.142494			0.0712470	−	0.997459i	$-0.477302\pi$
$197$	2.00000			0.142494			0.0712470	+	0.997459i	$0.477302\pi$
$198$		0			0
$199$	15.0000			1.06332			0.531661	−	0.846957i	$-0.321568\pi$
$199$	15.0000			1.06332			0.531661	+	0.846957i	$0.321568\pi$
$200$		0			0
$201$			12.0000i			0.846415i
$202$	14.0000			0.985037
$203$	−10.0000			−0.701862
$204$	4.00000			0.280056
$205$		0			0
$206$			2.00000i			0.139347i
$207$			2.00000i			0.139010i
$208$	8.00000	−	12.0000i	0.554700	−	0.832050i
$209$		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	$0.364461\pi$
$212$		−	18.0000i		−	1.23625i
$213$	−10.0000			−0.685189
$214$			26.0000i			1.77732i
$215$		0			0
$216$		0			0
$217$		−	20.0000i		−	1.35769i
$218$			32.0000i			2.16731i
$219$		−	14.0000i		−	0.946032i
$220$		0			0
$221$	6.00000	+	4.00000i	0.403604	+	0.269069i
$222$			4.00000i			0.268462i
$223$	−4.00000			−0.267860			−0.133930	−	0.990991i	$-0.542760\pi$
$223$	−4.00000			−0.267860			−0.133930	+	0.990991i	$0.542760\pi$
$224$	−16.0000			−1.06904
$225$		0			0
$226$			2.00000i			0.133038i
$227$	12.0000			0.796468			0.398234	−	0.917284i	$-0.369623\pi$
$227$	12.0000			0.796468			0.398234	+	0.917284i	$0.369623\pi$
$228$	8.00000			0.529813
$229$			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
$233$			11.0000i			0.720634i	0.932830	+	0.360317i	$0.117331\pi$
$233$			11.0000i			0.720634i	−0.932830	+	0.360317i	$0.882669\pi$
$234$	−8.00000	+	12.0000i	−0.522976	+	0.784465i
$235$		0			0
$236$			12.0000i			0.781133i
$237$			5.00000i			0.324785i
$238$		−	8.00000i		−	0.518563i
$239$		−	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			$\pi$
$242$	22.0000			1.41421
$243$			16.0000i			1.02640i
$244$	14.0000			0.896258
$245$		0			0
$246$	20.0000			1.27515
$247$	12.0000	+	8.00000i	0.763542	+	0.509028i
$248$		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	$0.376365\pi$
$252$	8.00000			0.503953
$253$		0			0
$254$			6.00000i			0.376473i
$255$		0			0
$256$	16.0000			1.00000
$257$		−	27.0000i		−	1.68421i	−0.539311	−	0.842107i	$-0.681315\pi$
$257$		−	27.0000i		−	1.68421i	0.539311	−	0.842107i	$-0.318685\pi$
$258$	−22.0000			−1.36966
$259$	4.00000			0.248548
$260$		0			0
$261$	−10.0000			−0.618984
$262$	−6.00000			−0.370681
$263$		−	24.0000i		−	1.47990i	−0.672660	−	0.739952i	$-0.734848\pi$
$263$		−	24.0000i		−	1.47990i	0.672660	−	0.739952i	$-0.265152\pi$
$264$		0			0
$265$		0			0
$266$		−	16.0000i		−	0.981023i
$267$	−6.00000			−0.367194
$268$	24.0000			1.46603
$269$	−25.0000			−1.52428			−0.762138	−	0.647414i	$-0.775850\pi$
$269$	−25.0000			−1.52428			−0.762138	+	0.647414i	$0.775850\pi$
$270$		0			0
$271$			20.0000i			1.21491i	0.794353	+	0.607457i	$0.207810\pi$
$271$			20.0000i			1.21491i	−0.794353	+	0.607457i	$0.792190\pi$
$272$			8.00000i			0.485071i
$273$	−6.00000	−	4.00000i	−0.363137	−	0.242091i
$274$	−36.0000			−2.17484
$275$		0			0
$276$	−2.00000			−0.120386
$277$		−	7.00000i		−	0.420589i	−0.977638	−	0.210295i	$-0.932558\pi$
$277$		−	7.00000i		−	0.420589i	0.977638	−	0.210295i	$-0.0674423\pi$
$278$	10.0000			0.599760
$279$		−	20.0000i		−	1.19737i
$280$		0			0
$281$		−	30.0000i		−	1.78965i	−0.446417	−	0.894825i	$-0.647300\pi$
$281$		−	30.0000i		−	1.78965i	0.446417	−	0.894825i	$-0.352700\pi$
$282$		−	16.0000i		−	0.952786i
$283$		−	4.00000i		−	0.237775i	−0.992908	−	0.118888i	$-0.962067\pi$
$283$		−	4.00000i		−	0.237775i	0.992908	−	0.118888i	$-0.0379328\pi$
$284$			20.0000i			1.18678i
$285$		0			0
$286$		0			0
$287$		−	20.0000i		−	1.18056i
$288$	−16.0000			−0.942809
$289$	13.0000			0.764706
$290$		0			0
$291$			2.00000i			0.117242i
$292$	−28.0000			−1.63858
$293$	−24.0000			−1.40209			−0.701047	−	0.713115i	$-0.747284\pi$
$293$	−24.0000			−1.40209			−0.701047	+	0.713115i	$0.747284\pi$
$294$		−	6.00000i		−	0.349927i
$295$		0			0
$296$		0			0
$297$		0			0
$298$		−	28.0000i		−	1.62200i
$299$	−3.00000	−	2.00000i	−0.173494	−	0.115663i
$300$		0			0
$301$			22.0000i			1.26806i
$302$			20.0000i			1.15087i
$303$			7.00000i			0.402139i
$304$			16.0000i			0.917663i
$305$		0			0
$306$		−	8.00000i		−	0.457330i
$307$	−8.00000			−0.456584			−0.228292	−	0.973593i	$-0.573314\pi$
$307$	−8.00000			−0.456584			−0.228292	+	0.973593i	$0.573314\pi$
$308$		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
$313$		−	19.0000i		−	1.07394i	−0.843600	−	0.536972i	$-0.819568\pi$
$313$		−	19.0000i		−	1.07394i	0.843600	−	0.536972i	$-0.180432\pi$
$314$		−	4.00000i		−	0.225733i
$315$		0			0
$316$	10.0000			0.562544
$317$	−28.0000			−1.57264			−0.786318	−	0.617822i	$-0.788015\pi$
$317$	−28.0000			−1.57264			−0.786318	+	0.617822i	$0.788015\pi$
$318$	18.0000			1.00939
$319$		0			0
$320$		0			0
$321$	−13.0000			−0.725589
$322$			4.00000i			0.222911i
$323$	−8.00000			−0.445132
$324$	2.00000			0.111111
$325$		0			0
$326$	12.0000			0.664619
$327$	−16.0000			−0.884802
$328$		0			0
$329$	−16.0000			−0.882109
$330$		0			0
$331$		−	10.0000i		−	0.549650i	−0.961494	−	0.274825i	$-0.911380\pi$
$331$		−	10.0000i		−	0.549650i	0.961494	−	0.274825i	$-0.0886199\pi$
$332$	12.0000			0.658586
$333$	4.00000			0.219199
$334$	4.00000			0.218870
$335$		0			0
$336$		−	8.00000i		−	0.436436i
$337$			3.00000i			0.163420i	0.996656	+	0.0817102i	$0.0260382\pi$
$337$			3.00000i			0.163420i	−0.996656	+	0.0817102i	$0.973962\pi$
$338$	−10.0000	−	24.0000i	−0.543928	−	1.30543i
$339$	−1.00000			−0.0543125
$340$		0			0
$341$		0			0
$342$		−	16.0000i		−	0.865181i
$343$	−20.0000			−1.07990
$344$		0			0
$345$		0			0
$346$		−	28.0000i		−	1.50529i
$347$			23.0000i			1.23470i	0.786687	+	0.617352i	$0.211795\pi$
$347$			23.0000i			1.23470i	−0.786687	+	0.617352i	$0.788205\pi$
$348$		−	10.0000i		−	0.536056i
$349$		−	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
$353$	−24.0000			−1.27739			−0.638696	−	0.769460i	$-0.720526\pi$
$353$	−24.0000			−1.27739			−0.638696	+	0.769460i	$0.720526\pi$
$354$	−12.0000			−0.637793
$355$		0			0
$356$			12.0000i			0.635999i
$357$	4.00000			0.211702
$358$	30.0000			1.58555
$359$			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
$362$	−36.0000			−1.89212
$363$			11.0000i			0.577350i
$364$	−8.00000	+	12.0000i	−0.419314	+	0.628971i
$365$		0			0
$366$			14.0000i			0.731792i
$367$			3.00000i			0.156599i	0.996930	+	0.0782994i	$0.0249490\pi$
$367$			3.00000i			0.156599i	−0.996930	+	0.0782994i	$0.975051\pi$
$368$		−	4.00000i		−	0.208514i
$369$		−	20.0000i		−	1.04116i
$370$		0			0
$371$		−	18.0000i		−	0.934513i
$372$	20.0000			1.03695
$373$			1.00000i			0.0517780i	0.999665	+	0.0258890i	$0.00824165\pi$
$373$			1.00000i			0.0517780i	−0.999665	+	0.0258890i	$0.991758\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	10.0000	−	15.0000i	0.515026	−	0.772539i
$378$			20.0000i			1.02869i
$379$		−	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
$382$	−46.0000			−2.35356
$383$	16.0000			0.817562			0.408781	−	0.912633i	$-0.365954\pi$
$383$	16.0000			0.817562			0.408781	+	0.912633i	$0.365954\pi$
$384$		0			0
$385$		0			0
$386$	32.0000			1.62876
$387$			22.0000i			1.11832i
$388$	4.00000			0.203069
$389$	−25.0000			−1.26755			−0.633775	−	0.773517i	$-0.718496\pi$
$389$	−25.0000			−1.26755			−0.633775	+	0.773517i	$0.718496\pi$
$390$		0			0
$391$	2.00000			0.101144
$392$		0			0
$393$		−	3.00000i		−	0.151330i
$394$	4.00000			0.201517
$395$		0			0
$396$		0			0
$397$	22.0000			1.10415			0.552074	−	0.833795i	$-0.313837\pi$
$397$	22.0000			1.10415			0.552074	+	0.833795i	$0.313837\pi$
$398$	30.0000			1.50376
$399$	8.00000			0.400501
$400$		0			0
$401$			10.0000i			0.499376i	0.968326	+	0.249688i	$0.0803281\pi$
$401$			10.0000i			0.499376i	−0.968326	+	0.249688i	$0.919672\pi$
$402$			24.0000i			1.19701i
$403$	30.0000	+	20.0000i	1.49441	+	0.996271i
$404$	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
$412$			2.00000i			0.0985329i
$413$			12.0000i			0.590481i
$414$			4.00000i			0.196589i
$415$		0			0
$416$	16.0000	−	24.0000i	0.784465	−	1.17670i
$417$			5.00000i			0.244851i
$418$		0			0
$419$	−15.0000			−0.732798			−0.366399	−	0.930458i	$-0.619409\pi$
$419$	−15.0000			−0.732798			−0.366399	+	0.930458i	$0.619409\pi$
$420$		0			0
$421$			10.0000i			0.487370i	0.969854	+	0.243685i	$0.0783563\pi$
$421$			10.0000i			0.487370i	−0.969854	+	0.243685i	$0.921644\pi$
$422$	24.0000			1.16830
$423$	−16.0000			−0.777947
$424$		0			0
$425$		0			0
$426$	−20.0000			−0.969003
$427$	14.0000			0.677507
$428$			26.0000i			1.25676i
$429$		0			0
$430$		0			0
$431$			30.0000i			1.44505i	0.691345	+	0.722525i	$0.257018\pi$
$431$			30.0000i			1.44505i	−0.691345	+	0.722525i	$0.742982\pi$
$432$		−	20.0000i		−	0.962250i
$433$			11.0000i			0.528626i	0.964437	+	0.264313i	$0.0851452\pi$
$433$			11.0000i			0.528626i	−0.964437	+	0.264313i	$0.914855\pi$
$434$		−	40.0000i		−	1.92006i
$435$		0			0
$436$			32.0000i			1.53252i
$437$	4.00000			0.191346
$438$		−	28.0000i		−	1.33789i
$439$	20.0000			0.954548			0.477274	−	0.878755i	$-0.341625\pi$
$439$	20.0000			0.954548			0.477274	+	0.878755i	$0.341625\pi$
$440$		0			0
$441$	−6.00000			−0.285714
$442$	12.0000	+	8.00000i	0.570782	+	0.380521i
$443$		−	24.0000i		−	1.14027i	−0.821549	−	0.570137i	$-0.806890\pi$
$443$		−	24.0000i		−	1.14027i	0.821549	−	0.570137i	$-0.193110\pi$
$444$			4.00000i			0.189832i
$445$		0			0
$446$	−8.00000			−0.378811
$447$	14.0000			0.662177
$448$	−16.0000			−0.755929
$449$			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
$452$			2.00000i			0.0940721i
$453$	−10.0000			−0.469841
$454$	24.0000			1.12638
$455$		0			0
$456$		0			0
$457$	−8.00000			−0.374224			−0.187112	−	0.982339i	$-0.559913\pi$
$457$	−8.00000			−0.374224			−0.187112	+	0.982339i	$0.559913\pi$
$458$			52.0000i			2.42980i
$459$	10.0000			0.466760
$460$		0			0
$461$		−	20.0000i		−	0.931493i	−0.884918	−	0.465746i	$-0.845786\pi$
$461$		−	20.0000i		−	0.931493i	0.884918	−	0.465746i	$-0.154214\pi$
$462$		0			0
$463$	−4.00000			−0.185896			−0.0929479	−	0.995671i	$-0.529629\pi$
$463$	−4.00000			−0.185896			−0.0929479	+	0.995671i	$0.529629\pi$
$464$	20.0000			0.928477
$465$		0			0
$466$			22.0000i			1.01913i
$467$		−	27.0000i		−	1.24941i	−0.780860	−	0.624705i	$-0.785219\pi$
$467$		−	27.0000i		−	1.24941i	0.780860	−	0.624705i	$-0.214781\pi$
$468$	−8.00000	+	12.0000i	−0.369800	+	0.554700i
$469$	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
$477$		−	18.0000i		−	0.824163i
$478$		−	8.00000i		−	0.365911i
$479$		−	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
$483$	−2.00000			−0.0910032
$484$	22.0000			1.00000
$485$		0			0
$486$			32.0000i			1.45155i
$487$	12.0000			0.543772			0.271886	−	0.962329i	$-0.412353\pi$
$487$	12.0000			0.543772			0.271886	+	0.962329i	$0.412353\pi$
$488$		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	$0.374684\pi$
$492$	20.0000			0.901670
$493$			10.0000i			0.450377i
$494$	24.0000	+	16.0000i	1.07981	+	0.719874i
$495$		0			0
$496$			40.0000i			1.79605i
$497$			20.0000i			0.897123i
$498$			12.0000i			0.537733i
$499$			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
$502$	24.0000			1.07117
$503$		−	19.0000i		−	0.847168i	−0.905857	−	0.423584i	$-0.860772\pi$
$503$		−	19.0000i		−	0.847168i	0.905857	−	0.423584i	$-0.139228\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	12.0000	−	5.00000i	0.532939	−	0.222058i
$508$			6.00000i			0.266207i
$509$			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
$512$	32.0000			1.41421
$513$	20.0000			0.883022
$514$		−	54.0000i		−	2.38184i
$515$		0			0
$516$	−22.0000			−0.968496
$517$		0			0
$518$	8.00000			0.351500
$519$	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	$0.199196\pi$
$522$	−20.0000			−0.875376
$523$		−	4.00000i		−	0.174908i	−0.996169	−	0.0874539i	$-0.972127\pi$
$523$		−	4.00000i		−	0.174908i	0.996169	−	0.0874539i	$-0.0278730\pi$
$524$	−6.00000			−0.262111
$525$		0			0
$526$		−	48.0000i		−	2.09290i
$527$	−20.0000			−0.871214
$528$		0			0
$529$	22.0000			0.956522
$530$		0			0
$531$			12.0000i			0.520756i
$532$		−	16.0000i		−	0.693688i
$533$	30.0000	+	20.0000i	1.29944	+	0.866296i
$534$	−12.0000			−0.519291
$535$		0			0
$536$		0			0
$537$			15.0000i			0.647298i
$538$	−50.0000			−2.15565
$539$		0			0
$540$		0			0
$541$		−	10.0000i		−	0.429934i	−0.976621	−	0.214967i	$-0.931036\pi$
$541$		−	10.0000i		−	0.429934i	0.976621	−	0.214967i	$-0.0689643\pi$
$542$			40.0000i			1.71815i
$543$		−	18.0000i		−	0.772454i
$544$			16.0000i			0.685994i
$545$		0			0
$546$	−12.0000	−	8.00000i	−0.513553	−	0.342368i
$547$			28.0000i			1.19719i	0.801050	+	0.598597i	$0.204275\pi$
$547$			28.0000i			1.19719i	−0.801050	+	0.598597i	$0.795725\pi$
$548$	−36.0000			−1.53784
$549$	14.0000			0.597505
$550$		0			0
$551$			20.0000i			0.852029i
$552$		0			0
$553$	10.0000			0.425243
$554$		−	14.0000i		−	0.594803i
$555$		0			0
$556$	10.0000			0.424094
$557$	12.0000			0.508456			0.254228	−	0.967144i	$-0.418179\pi$
$557$	12.0000			0.508456			0.254228	+	0.967144i	$0.418179\pi$
$558$		−	40.0000i		−	1.69334i
$559$	−33.0000	−	22.0000i	−1.39575	−	0.930501i
$560$		0			0
$561$		0			0
$562$		−	60.0000i		−	2.53095i
$563$			21.0000i			0.885044i	0.896758	+	0.442522i	$0.145916\pi$
$563$			21.0000i			0.885044i	−0.896758	+	0.442522i	$0.854084\pi$
$564$		−	16.0000i		−	0.673722i
$565$		0			0
$566$		−	8.00000i		−	0.336265i
$567$	2.00000			0.0839921
$568$		0			0
$569$	−15.0000			−0.628833			−0.314416	−	0.949285i	$-0.601809\pi$
$569$	−15.0000			−0.628833			−0.314416	+	0.949285i	$0.601809\pi$
$570$		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
$573$		−	23.0000i		−	0.960839i
$574$		−	40.0000i		−	1.66957i
$575$		0			0
$576$	−16.0000			−0.666667
$577$	−28.0000			−1.16566			−0.582828	−	0.812596i	$-0.698054\pi$
$577$	−28.0000			−1.16566			−0.582828	+	0.812596i	$0.698054\pi$
$578$	26.0000			1.08146
$579$			16.0000i			0.664937i
$580$		0			0
$581$	12.0000			0.497844
$582$			4.00000i			0.165805i
$583$		0			0
$584$		0			0
$585$		0			0
$586$	−48.0000			−1.98286
$587$	−28.0000			−1.15568			−0.577842	−	0.816149i	$-0.696105\pi$
$587$	−28.0000			−1.15568			−0.577842	+	0.816149i	$0.696105\pi$
$588$		−	6.00000i		−	0.247436i
$589$	−40.0000			−1.64817
$590$		0			0
$591$			2.00000i			0.0822690i
$592$	−8.00000			−0.328798
$593$	26.0000			1.06769			0.533846	−	0.845582i	$-0.320746\pi$
$593$	26.0000			1.06769			0.533846	+	0.845582i	$0.320746\pi$
$594$		0			0
$595$		0			0
$596$		−	28.0000i		−	1.14692i
$597$			15.0000i			0.613909i
$598$	−6.00000	−	4.00000i	−0.245358	−	0.163572i
$599$	35.0000			1.43006			0.715031	−	0.699093i	$-0.246413\pi$
$599$	35.0000			1.43006			0.715031	+	0.699093i	$0.246413\pi$
$600$		0			0
$601$	17.0000			0.693444			0.346722	−	0.937968i	$-0.387295\pi$
$601$	17.0000			0.693444			0.346722	+	0.937968i	$0.387295\pi$
$602$			44.0000i			1.79331i
$603$	24.0000			0.977356
$604$			20.0000i			0.813788i
$605$		0			0
$606$			14.0000i			0.568711i
$607$			8.00000i			0.324710i	0.986732	+	0.162355i	$0.0519090\pi$
$607$			8.00000i			0.324710i	−0.986732	+	0.162355i	$0.948091\pi$
$608$			32.0000i			1.29777i
$609$		−	10.0000i		−	0.405220i
$610$		0			0
$611$	16.0000	−	24.0000i	0.647291	−	0.970936i
$612$		−	8.00000i		−	0.323381i
$613$	16.0000			0.646234			0.323117	−	0.946359i	$-0.395269\pi$
$613$	16.0000			0.646234			0.323117	+	0.946359i	$0.395269\pi$
$614$	−16.0000			−0.645707
$615$		0			0
$616$		0			0
$617$	22.0000			0.885687			0.442843	−	0.896599i	$-0.353970\pi$
$617$	22.0000			0.885687			0.442843	+	0.896599i	$0.353970\pi$
$618$	−2.00000			−0.0804518
$619$			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
$622$	−46.0000			−1.84443
$623$			12.0000i			0.480770i
$624$	12.0000	+	8.00000i	0.480384	+	0.320256i
$625$		0			0
$626$		−	38.0000i		−	1.51879i
$627$		0			0
$628$		−	4.00000i		−	0.159617i
$629$		−	4.00000i		−	0.159490i
$630$		0			0
$631$		−	20.0000i		−	0.796187i	−0.917345	−	0.398094i	$-0.869672\pi$
$631$		−	20.0000i		−	0.796187i	0.917345	−	0.398094i	$-0.130328\pi$
$632$		0			0
$633$			12.0000i			0.476957i
$634$	−56.0000			−2.22404
$635$		0			0
$636$	18.0000			0.713746
$637$	6.00000	−	9.00000i	0.237729	−	0.356593i
$638$		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	$0.518870\pi$
$642$	−26.0000			−1.02614
$643$	−14.0000			−0.552106			−0.276053	−	0.961142i	$-0.589027\pi$
$643$	−14.0000			−0.552106			−0.276053	+	0.961142i	$0.589027\pi$
$644$			4.00000i			0.157622i
$645$		0			0
$646$	−16.0000			−0.629512
$647$			8.00000i			0.314512i	0.987558	+	0.157256i	$0.0502649\pi$
$647$			8.00000i			0.314512i	−0.987558	+	0.157256i	$0.949735\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	20.0000			0.783862
$652$	12.0000			0.469956
$653$		−	14.0000i		−	0.547862i	−0.961749	−	0.273931i	$-0.911676\pi$
$653$		−	14.0000i		−	0.547862i	0.961749	−	0.273931i	$-0.0883240\pi$
$654$	−32.0000			−1.25130
$655$		0			0
$656$			40.0000i			1.56174i
$657$	−28.0000			−1.09238
$658$	−32.0000			−1.24749
$659$	−45.0000			−1.75295			−0.876476	−	0.481446i	$-0.840112\pi$
$659$	−45.0000			−1.75295			−0.876476	+	0.481446i	$0.840112\pi$
$660$		0			0
$661$		−	20.0000i		−	0.777910i	−0.921257	−	0.388955i	$-0.872836\pi$
$661$		−	20.0000i		−	0.777910i	0.921257	−	0.388955i	$-0.127164\pi$
$662$		−	20.0000i		−	0.777322i
$663$	−4.00000	+	6.00000i	−0.155347	+	0.233021i
$664$		0			0
$665$		0			0
$666$	8.00000			0.309994
$667$		−	5.00000i		−	0.193601i
$668$	4.00000			0.154765
$669$		−	4.00000i		−	0.154649i
$670$		0			0
$671$		0			0
$672$		−	16.0000i		−	0.617213i
$673$		−	14.0000i		−	0.539660i	−0.962908	−	0.269830i	$-0.913032\pi$
$673$		−	14.0000i		−	0.539660i	0.962908	−	0.269830i	$-0.0869676\pi$
$674$			6.00000i			0.231111i
$675$		0			0
$676$	−10.0000	−	24.0000i	−0.384615	−	0.923077i
$677$			33.0000i			1.26829i	0.773213	+	0.634147i	$0.218648\pi$
$677$			33.0000i			1.26829i	−0.773213	+	0.634147i	$0.781352\pi$
$678$	−2.00000			−0.0768095
$679$	4.00000			0.153506
$680$		0			0
$681$			12.0000i			0.459841i
$682$		0			0
$683$	−4.00000			−0.153056			−0.0765279	−	0.997067i	$-0.524383\pi$
$683$	−4.00000			−0.153056			−0.0765279	+	0.997067i	$0.524383\pi$
$684$		−	16.0000i		−	0.611775i
$685$		0			0
$686$	−40.0000			−1.52721
$687$	−26.0000			−0.991962
$688$		−	44.0000i		−	1.67748i
$689$	27.0000	+	18.0000i	1.02862	+	0.685745i
$690$		0			0
$691$			10.0000i			0.380418i	0.981744	+	0.190209i	$0.0609166\pi$
$691$			10.0000i			0.380418i	−0.981744	+	0.190209i	$0.939083\pi$
$692$		−	28.0000i		−	1.06440i
$693$		0			0
$694$			46.0000i			1.74614i
$695$		0			0
$696$		0			0
$697$	−20.0000			−0.757554
$698$		−	28.0000i		−	1.05982i
$699$	−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	$0.518043\pi$
$702$	−30.0000	−	20.0000i	−1.13228	−	0.754851i
$703$		−	8.00000i		−	0.301726i
$704$		0			0
$705$		0			0
$706$	−48.0000			−1.80650
$707$	14.0000			0.526524
$708$	−12.0000			−0.450988
$709$		−	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
$713$	10.0000			0.374503
$714$	8.00000			0.299392
$715$		0			0
$716$	30.0000			1.12115
$717$	4.00000			0.149383
$718$			32.0000i			1.19423i
$719$	15.0000			0.559406			0.279703	−	0.960087i	$-0.409764\pi$
$719$	15.0000			0.559406			0.279703	+	0.960087i	$0.409764\pi$
$720$		0			0
$721$			2.00000i			0.0744839i
$722$	6.00000			0.223297
$723$		0			0
$724$	−36.0000			−1.33793
$725$		0			0
$726$			22.0000i			0.816497i
$727$		−	47.0000i		−	1.74313i	−0.490277	−	0.871567i	$-0.663104\pi$
$727$		−	47.0000i		−	1.74313i	0.490277	−	0.871567i	$-0.336896\pi$
$728$		0			0
$729$	−13.0000			−0.481481
$730$		0			0
$731$	22.0000			0.813699
$732$			14.0000i			0.517455i
$733$	−24.0000			−0.886460			−0.443230	−	0.896408i	$-0.646168\pi$
$733$	−24.0000			−0.886460			−0.443230	+	0.896408i	$0.646168\pi$
$734$			6.00000i			0.221464i
$735$		0			0
$736$		−	8.00000i		−	0.294884i
$737$		0			0
$738$		−	40.0000i		−	1.47242i
$739$			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
$742$		−	36.0000i		−	1.32160i
$743$	16.0000			0.586983			0.293492	−	0.955962i	$-0.405183\pi$
$743$	16.0000			0.586983			0.293492	+	0.955962i	$0.405183\pi$
$744$		0			0
$745$		0			0
$746$			2.00000i			0.0732252i
$747$	12.0000			0.439057
$748$		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	$0.637844\pi$
$752$	32.0000			1.16692
$753$			12.0000i			0.437304i
$754$	20.0000	−	30.0000i	0.728357	−	1.09254i
$755$		0			0
$756$			20.0000i			0.727393i
$757$		−	2.00000i		−	0.0726912i	−0.999339	−	0.0363456i	$-0.988428\pi$
$757$		−	2.00000i		−	0.0726912i	0.999339	−	0.0363456i	$-0.0115717\pi$
$758$		−	48.0000i		−	1.74344i
$759$		0			0
$760$		0			0
$761$		−	10.0000i		−	0.362500i	−0.983437	−	0.181250i	$-0.941986\pi$
$761$		−	10.0000i		−	0.362500i	0.983437	−	0.181250i	$-0.0580143\pi$
$762$	−6.00000			−0.217357
$763$			32.0000i			1.15848i
$764$	−46.0000			−1.66422
$765$		0			0
$766$	32.0000			1.15621
$767$	−18.0000	−	12.0000i	−0.649942	−	0.433295i
$768$			16.0000i			0.577350i
$769$		−	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
$772$	32.0000			1.15171
$773$	6.00000			0.215805			0.107903	−	0.994161i	$-0.465587\pi$
$773$	6.00000			0.215805			0.107903	+	0.994161i	$0.465587\pi$
$774$			44.0000i			1.58155i
$775$		0			0
$776$		0			0
$777$			4.00000i			0.143499i
$778$	−50.0000			−1.79259
$779$	−40.0000			−1.43315
$780$		0			0
$781$		0			0
$782$	4.00000			0.143040
$783$		−	25.0000i		−	0.893427i
$784$	12.0000			0.428571
$785$		0			0
$786$		−	6.00000i		−	0.214013i
$787$	32.0000			1.14068			0.570338	−	0.821410i	$-0.306812\pi$
$787$	32.0000			1.14068			0.570338	+	0.821410i	$0.306812\pi$
$788$	4.00000			0.142494
$789$	24.0000			0.854423
$790$		0			0
$791$			2.00000i			0.0711118i
$792$		0			0
$793$	−14.0000	+	21.0000i	−0.497155	+	0.745732i
$794$	44.0000			1.56150
$795$		0			0
$796$	30.0000			1.06332
$797$			23.0000i			0.814702i	0.913272	+	0.407351i	$0.133547\pi$
$797$			23.0000i			0.814702i	−0.913272	+	0.407351i	$0.866453\pi$
$798$	16.0000			0.566394
$799$			16.0000i			0.566039i
$800$		0			0
$801$			12.0000i			0.423999i
$802$			20.0000i			0.706225i
$803$		0			0
$804$			24.0000i			0.846415i
$805$		0			0
$806$	60.0000	+	40.0000i	2.11341	+	1.40894i
$807$		−	25.0000i		−	0.880042i
$808$		0			0
$809$	30.0000			1.05474			0.527372	−	0.849635i	$-0.323177\pi$
$809$	30.0000			1.05474			0.527372	+	0.849635i	$0.323177\pi$
$810$		0			0
$811$		−	40.0000i		−	1.40459i	−0.711886	−	0.702295i	$-0.752159\pi$
$811$		−	40.0000i		−	1.40459i	0.711886	−	0.702295i	$-0.247841\pi$
$812$	−20.0000			−0.701862
$813$	−20.0000			−0.701431
$814$		0			0
$815$		0			0
$816$	−8.00000			−0.280056
$817$	44.0000			1.53937
$818$			12.0000i			0.419570i
$819$	−8.00000	+	12.0000i	−0.279543	+	0.419314i
$820$		0			0
$821$		−	30.0000i		−	1.04701i	−0.852023	−	0.523504i	$-0.824625\pi$
$821$		−	30.0000i		−	1.04701i	0.852023	−	0.523504i	$-0.175375\pi$
$822$		−	36.0000i		−	1.25564i
$823$		−	9.00000i		−	0.313720i	−0.987621	−	0.156860i	$-0.949863\pi$
$823$		−	9.00000i		−	0.313720i	0.987621	−	0.156860i	$-0.0501372\pi$
$824$		0			0
$825$		0			0
$826$			24.0000i			0.835067i
$827$	22.0000			0.765015			0.382507	−	0.923952i	$-0.375061\pi$
$827$	22.0000			0.765015			0.382507	+	0.923952i	$0.375061\pi$
$828$			4.00000i			0.139010i
$829$	5.00000			0.173657			0.0868286	−	0.996223i	$-0.472327\pi$
$829$	5.00000			0.173657			0.0868286	+	0.996223i	$0.472327\pi$
$830$		0			0
$831$	7.00000			0.242827
$832$	16.0000	−	24.0000i	0.554700	−	0.832050i
$833$			6.00000i			0.207888i
$834$			10.0000i			0.346272i
$835$		0			0
$836$		0			0
$837$	50.0000			1.72825
$838$	−30.0000			−1.03633
$839$			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
$842$			20.0000i			0.689246i
$843$	30.0000			1.03325
$844$	24.0000			0.826114
$845$		0			0
$846$	−32.0000			−1.10018
$847$	22.0000			0.755929
$848$			36.0000i			1.23625i
$849$	4.00000			0.137280
$850$		0			0
$851$			2.00000i			0.0685591i
$852$	−20.0000			−0.685189
$853$	−24.0000			−0.821744			−0.410872	−	0.911693i	$-0.634776\pi$
$853$	−24.0000			−0.821744			−0.410872	+	0.911693i	$0.634776\pi$
$854$	28.0000			0.958140
$855$		0			0
$856$		0			0
$857$			13.0000i			0.444072i	0.975039	+	0.222036i	$0.0712702\pi$
$857$			13.0000i			0.444072i	−0.975039	+	0.222036i	$0.928730\pi$
$858$		0			0
$859$	−20.0000			−0.682391			−0.341196	−	0.939992i	$-0.610832\pi$
$859$	−20.0000			−0.682391			−0.341196	+	0.939992i	$0.610832\pi$
$860$		0			0
$861$	20.0000			0.681598
$862$			60.0000i			2.04361i
$863$	6.00000			0.204242			0.102121	−	0.994772i	$-0.467437\pi$
$863$	6.00000			0.204242			0.102121	+	0.994772i	$0.467437\pi$
$864$		−	40.0000i		−	1.36083i
$865$		0			0
$866$			22.0000i			0.747590i
$867$			13.0000i			0.441503i
$868$		−	40.0000i		−	1.35769i
$869$		0			0
$870$		0			0
$871$	−24.0000	+	36.0000i	−0.813209	+	1.21981i
$872$		0			0
$873$	4.00000			0.135379
$874$	8.00000			0.270604
$875$		0			0
$876$		−	28.0000i		−	0.946032i
$877$	−38.0000			−1.28317			−0.641584	−	0.767052i	$-0.721723\pi$
$877$	−38.0000			−1.28317			−0.641584	+	0.767052i	$0.721723\pi$
$878$	40.0000			1.34993
$879$		−	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	$0.349701\pi$
$882$	−12.0000			−0.404061
$883$			31.0000i			1.04323i	0.853180	+	0.521617i	$0.174671\pi$
$883$			31.0000i			1.04323i	−0.853180	+	0.521617i	$0.825329\pi$
$884$	12.0000	+	8.00000i	0.403604	+	0.269069i
$885$		0			0
$886$		−	48.0000i		−	1.61259i
$887$		−	32.0000i		−	1.07445i	−0.843437	−	0.537227i	$-0.819472\pi$
$887$		−	32.0000i		−	1.07445i	0.843437	−	0.537227i	$-0.180528\pi$
$888$		0			0
$889$			6.00000i			0.201234i
$890$		0			0
$891$		0			0
$892$	−8.00000			−0.267860
$893$			32.0000i			1.07084i
$894$	28.0000			0.936460
$895$		0			0
$896$		0			0
$897$	2.00000	−	3.00000i	0.0667781	−	0.100167i
$898$			72.0000i			2.40267i
$899$			50.0000i			1.66759i
$900$		0			0
$901$	−18.0000			−0.599667
$902$		0			0
$903$	−22.0000			−0.732114
$904$		0			0
$905$		0			0
$906$	−20.0000			−0.664455
$907$			53.0000i			1.75984i	0.475125	+	0.879918i	$0.342403\pi$
$907$			53.0000i			1.75984i	−0.475125	+	0.879918i	$0.657597\pi$
$908$	24.0000			0.796468
$909$	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	$0.569091\pi$
$912$	−16.0000			−0.529813
$913$		0			0
$914$	−16.0000			−0.529233
$915$		0			0
$916$			52.0000i			1.71813i
$917$	−6.00000			−0.198137
$918$	20.0000			0.660098
$919$	−40.0000			−1.31948			−0.659739	−	0.751495i	$-0.729333\pi$
$919$	−40.0000			−1.31948			−0.659739	+	0.751495i	$0.729333\pi$
$920$		0			0
$921$		−	8.00000i		−	0.263609i
$922$		−	40.0000i		−	1.31733i
$923$	−30.0000	−	20.0000i	−0.987462	−	0.658308i
$924$		0			0
$925$		0			0
$926$	−8.00000			−0.262896
$927$			2.00000i			0.0656886i
$928$	40.0000			1.31306
$929$		−	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
$932$			22.0000i			0.720634i
$933$		−	23.0000i		−	0.752986i
$934$		−	54.0000i		−	1.76693i
$935$		0			0
$936$		0			0
$937$			13.0000i			0.424691i	0.977195	+	0.212346i	$0.0681103\pi$
$937$			13.0000i			0.424691i	−0.977195	+	0.212346i	$0.931890\pi$
$938$	48.0000			1.56726
$939$	19.0000			0.620042
$940$		0			0
$941$		−	20.0000i		−	0.651981i	−0.945373	−	0.325991i	$-0.894302\pi$
$941$		−	20.0000i		−	0.651981i	0.945373	−	0.325991i	$-0.105698\pi$
$942$	4.00000			0.130327
$943$	10.0000			0.325645
$944$		−	24.0000i		−	0.781133i
$945$		0			0
$946$		0			0
$947$	−18.0000			−0.584921			−0.292461	−	0.956278i	$-0.594474\pi$
$947$	−18.0000			−0.584921			−0.292461	+	0.956278i	$0.594474\pi$
$948$			10.0000i			0.324785i
$949$	28.0000	−	42.0000i	0.908918	−	1.36338i
$950$		0			0
$951$		−	28.0000i		−	0.907962i
$952$		0			0
$953$		−	54.0000i		−	1.74923i	−0.484817	−	0.874616i	$-0.661114\pi$
$953$		−	54.0000i		−	1.74923i	0.484817	−	0.874616i	$-0.338886\pi$
$954$		−	36.0000i		−	1.16554i
$955$		0			0
$956$		−	8.00000i		−	0.258738i
$957$		0			0
$958$		−	48.0000i		−	1.55081i
$959$	−36.0000			−1.16250
$960$		0			0
$961$	−69.0000			−2.22581
$962$	−8.00000	+	12.0000i	−0.257930	+	0.386896i
$963$			26.0000i			0.837838i
$964$		0			0
$965$		0			0
$966$	−4.00000			−0.128698
$967$	−58.0000			−1.86515			−0.932577	−	0.360971i	$-0.882445\pi$
$967$	−58.0000			−1.86515			−0.932577	+	0.360971i	$0.882445\pi$
$968$		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	$0.779842\pi$
$972$			32.0000i			1.02640i
$973$	10.0000			0.320585
$974$	24.0000			0.769010
$975$		0			0
$976$	−28.0000			−0.896258
$977$	32.0000			1.02377			0.511885	−	0.859054i	$-0.328947\pi$
$977$	32.0000			1.02377			0.511885	+	0.859054i	$0.328947\pi$
$978$			12.0000i			0.383718i
$979$		0			0
$980$		0			0
$981$			32.0000i			1.02168i
$982$	34.0000			1.08498
$983$	−34.0000			−1.08443			−0.542216	−	0.840239i	$-0.682414\pi$
$983$	−34.0000			−1.08443			−0.542216	+	0.840239i	$0.682414\pi$
$984$		0			0
$985$		0			0
$986$			20.0000i			0.636930i
$987$		−	16.0000i		−	0.509286i
$988$	24.0000	+	16.0000i	0.763542	+	0.509028i
$989$	−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	$0.464537\pi$
$992$			80.0000i			2.54000i
$993$	10.0000			0.317340
$994$			40.0000i			1.26872i
$995$		0			0
$996$			12.0000i			0.380235i
$997$		−	27.0000i		−	0.855099i	−0.903992	−	0.427549i	$-0.859377\pi$
$997$		−	27.0000i		−	0.855099i	0.903992	−	0.427549i	$-0.140623\pi$
$998$			52.0000i			1.64603i
$999$			10.0000i			0.316386i

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
325.2.c.c.51.1	✓	2	5.3	odd	4
325.2.c.c.51.2	yes	2	65.38	odd	4
325.2.c.d.51.1	yes	2	65.12	odd	4
325.2.c.d.51.2	yes	2	5.2	odd	4
325.2.d.a.324.1		2	5.4	even	2
325.2.d.a.324.2		2	13.12	even	2
325.2.d.d.324.1		2	65.64	even	2	inner
325.2.d.d.324.2		2	1.1	even	1	trivial
4225.2.a.a.1.1		1	65.18	even	4
4225.2.a.c.1.1		1	65.47	even	4
4225.2.a.o.1.1		1	65.8	even	4
4225.2.a.q.1.1		1	65.57	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 325 = 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	325.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} +1.00000i q^{3} +2.00000 q^{4} +2.00000i q^{6} +2.00000 q^{7} +2.00000 q^{9} +O(q^{10})\)	\(q+2.00000 q^{2} +1.00000i q^{3} +2.00000 q^{4} +2.00000i q^{6} +2.00000 q^{7} +2.00000 q^{9} +2.00000i q^{12} +(-2.00000 + 3.00000i) q^{13} +4.00000 q^{14} -4.00000 q^{16} -2.00000i q^{17} +4.00000 q^{18} -4.00000i q^{19} +2.00000i q^{21} +1.00000i q^{23} +(-4.00000 + 6.00000i) q^{26} +5.00000i q^{27} +4.00000 q^{28} -5.00000 q^{29} -10.0000i q^{31} -8.00000 q^{32} -4.00000i q^{34} +4.00000 q^{36} +2.00000 q^{37} -8.00000i q^{38} +(-3.00000 - 2.00000i) q^{39} -10.0000i q^{41} +4.00000i q^{42} +11.0000i q^{43} +2.00000i q^{46} -8.00000 q^{47} -4.00000i q^{48} -3.00000 q^{49} +2.00000 q^{51} +(-4.00000 + 6.00000i) q^{52} -9.00000i q^{53} +10.0000i q^{54} +4.00000 q^{57} -10.0000 q^{58} +6.00000i q^{59} +7.00000 q^{61} -20.0000i q^{62} +4.00000 q^{63} -8.00000 q^{64} +12.0000 q^{67} -4.00000i q^{68} -1.00000 q^{69} +10.0000i q^{71} -14.0000 q^{73} +4.00000 q^{74} -8.00000i q^{76} +(-6.00000 - 4.00000i) q^{78} +5.00000 q^{79} +1.00000 q^{81} -20.0000i q^{82} +6.00000 q^{83} +4.00000i q^{84} +22.0000i q^{86} -5.00000i q^{87} +6.00000i q^{89} +(-4.00000 + 6.00000i) q^{91} +2.00000i q^{92} +10.0000 q^{93} -16.0000 q^{94} -8.00000i q^{96} +2.00000 q^{97} -6.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{2} + 4 q^{4} + 4 q^{7} + 4 q^{9}+O(q^{10}) \) 2 * q + 4 * q^2 + 4 * q^4 + 4 * q^7 + 4 * q^9	\( 2 q + 4 q^{2} + 4 q^{4} + 4 q^{7} + 4 q^{9} - 4 q^{13} + 8 q^{14} - 8 q^{16} + 8 q^{18} - 8 q^{26} + 8 q^{28} - 10 q^{29} - 16 q^{32} + 8 q^{36} + 4 q^{37} - 6 q^{39} - 16 q^{47} - 6 q^{49} + 4 q^{51} - 8 q^{52} + 8 q^{57} - 20 q^{58} + 14 q^{61} + 8 q^{63} - 16 q^{64} + 24 q^{67} - 2 q^{69} - 28 q^{73} + 8 q^{74} - 12 q^{78} + 10 q^{79} + 2 q^{81} + 12 q^{83} - 8 q^{91} + 20 q^{93} - 32 q^{94} + 4 q^{97} - 12 q^{98}+O(q^{100}) \) 2 * q + 4 * q^2 + 4 * q^4 + 4 * q^7 + 4 * q^9 - 4 * q^13 + 8 * q^14 - 8 * q^16 + 8 * q^18 - 8 * q^26 + 8 * q^28 - 10 * q^29 - 16 * q^32 + 8 * q^36 + 4 * q^37 - 6 * q^39 - 16 * q^47 - 6 * q^49 + 4 * q^51 - 8 * q^52 + 8 * q^57 - 20 * q^58 + 14 * q^61 + 8 * q^63 - 16 * q^64 + 24 * q^67 - 2 * q^69 - 28 * q^73 + 8 * q^74 - 12 * q^78 + 10 * q^79 + 2 * q^81 + 12 * q^83 - 8 * q^91 + 20 * q^93 - 32 * q^94 + 4 * q^97 - 12 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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