LMFDB - Embedded newform 1216.3.e.g.1025.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1216,3,Mod(1025,1216)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1216, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1216.1025");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1216 = 2^{6} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1216.e (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:	$33.1336001462$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-13}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 13 $ x^2 + 13
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 19)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1025.1
Root			$-3.60555i$ of defining polynomial
Character	$\chi$	$=$	1216.1025
Dual form			1216.3.e.g.1025.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.60555i q^{3} -4.00000 q^{5} -5.00000 q^{7} -4.00000 q^{9} +O(q^{10})$	$q-3.60555i q^{3} -4.00000 q^{5} -5.00000 q^{7} -4.00000 q^{9} +10.0000 q^{11} +3.60555i q^{13} +14.4222i q^{15} +15.0000 q^{17} +(6.00000 + 18.0278i) q^{19} +18.0278i q^{21} +35.0000 q^{23} -9.00000 q^{25} -18.0278i q^{27} +18.0278i q^{29} +36.0555i q^{31} -36.0555i q^{33} +20.0000 q^{35} -21.6333i q^{37} +13.0000 q^{39} -36.0555i q^{41} +20.0000 q^{43} +16.0000 q^{45} +10.0000 q^{47} -24.0000 q^{49} -54.0833i q^{51} -75.7166i q^{53} -40.0000 q^{55} +(65.0000 - 21.6333i) q^{57} +18.0278i q^{59} +40.0000 q^{61} +20.0000 q^{63} -14.4222i q^{65} +39.6611i q^{67} -126.194i q^{69} -108.167i q^{71} +105.000 q^{73} +32.4500i q^{75} -50.0000 q^{77} +36.0555i q^{79} -101.000 q^{81} +40.0000 q^{83} -60.0000 q^{85} +65.0000 q^{87} -18.0278i q^{91} +130.000 q^{93} +(-24.0000 - 72.1110i) q^{95} -122.589i q^{97} -40.0000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{5} - 10 q^{7} - 8 q^{9}+O(q^{10}) $ 2 * q - 8 * q^5 - 10 * q^7 - 8 * q^9	$ 2 q - 8 q^{5} - 10 q^{7} - 8 q^{9} + 20 q^{11} + 30 q^{17} + 12 q^{19} + 70 q^{23} - 18 q^{25} + 40 q^{35} + 26 q^{39} + 40 q^{43} + 32 q^{45} + 20 q^{47} - 48 q^{49} - 80 q^{55} + 130 q^{57} + 80 q^{61} + 40 q^{63} + 210 q^{73} - 100 q^{77} - 202 q^{81} + 80 q^{83} - 120 q^{85} + 130 q^{87} + 260 q^{93} - 48 q^{95} - 80 q^{99}+O(q^{100}) $ 2 * q - 8 * q^5 - 10 * q^7 - 8 * q^9 + 20 * q^11 + 30 * q^17 + 12 * q^19 + 70 * q^23 - 18 * q^25 + 40 * q^35 + 26 * q^39 + 40 * q^43 + 32 * q^45 + 20 * q^47 - 48 * q^49 - 80 * q^55 + 130 * q^57 + 80 * q^61 + 40 * q^63 + 210 * q^73 - 100 * q^77 - 202 * q^81 + 80 * q^83 - 120 * q^85 + 130 * q^87 + 260 * q^93 - 48 * q^95 - 80 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times$.

$n$	$191$	$705$	$837$
$\chi(n)$	$1$	$-1$	$1$

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 $

$2$		0			0
$2$		0			0
$3$		−	3.60555i		−	1.20185i	−0.799305	−	0.600925i	$-0.794799\pi$
$3$		−	3.60555i		−	1.20185i	0.799305	−	0.600925i	$-0.205201\pi$
$4$		0			0
$5$	−4.00000			−0.800000			−0.400000	−	0.916515i	$-0.630990\pi$
$5$	−4.00000			−0.800000			−0.400000	+	0.916515i	$0.630990\pi$
$6$		0			0
$7$	−5.00000			−0.714286			−0.357143	−	0.934050i	$-0.616249\pi$
$7$	−5.00000			−0.714286			−0.357143	+	0.934050i	$0.616249\pi$
$8$		0			0
$9$	−4.00000			−0.444444
$10$		0			0
$11$	10.0000			0.909091			0.454545	−	0.890724i	$-0.349802\pi$
$11$	10.0000			0.909091			0.454545	+	0.890724i	$0.349802\pi$
$12$		0			0
$13$			3.60555i			0.277350i	0.990338	+	0.138675i	$0.0442844\pi$
$13$			3.60555i			0.277350i	−0.990338	+	0.138675i	$0.955716\pi$
$14$		0			0
$15$			14.4222i			0.961480i
$16$		0			0
$17$	15.0000			0.882353			0.441176	−	0.897420i	$-0.354561\pi$
$17$	15.0000			0.882353			0.441176	+	0.897420i	$0.354561\pi$
$18$		0			0
$19$	6.00000	+	18.0278i	0.315789	+	0.948829i
$19$	6.00000	+	18.0278i	0.315789	+	0.948829i
$20$		0			0
$21$			18.0278i			0.858465i
$22$		0			0
$23$	35.0000			1.52174			0.760870	−	0.648905i	$-0.224773\pi$
$23$	35.0000			1.52174			0.760870	+	0.648905i	$0.224773\pi$
$24$		0			0
$25$	−9.00000			−0.360000
$26$		0			0
$27$		−	18.0278i		−	0.667695i
$28$		0			0
$29$			18.0278i			0.621647i	0.950468	+	0.310823i	$0.100605\pi$
$29$			18.0278i			0.621647i	−0.950468	+	0.310823i	$0.899395\pi$
$30$		0			0
$31$			36.0555i			1.16308i	0.813517	+	0.581541i	$0.197550\pi$
$31$			36.0555i			1.16308i	−0.813517	+	0.581541i	$0.802450\pi$
$32$		0			0
$33$		−	36.0555i		−	1.09259i
$34$		0			0
$35$	20.0000			0.571429
$36$		0			0
$37$		−	21.6333i		−	0.584684i	−0.956314	−	0.292342i	$-0.905565\pi$
$37$		−	21.6333i		−	0.584684i	0.956314	−	0.292342i	$-0.0944346\pi$
$38$		0			0
$39$	13.0000			0.333333
$40$		0			0
$41$		−	36.0555i		−	0.879403i	−0.898144	−	0.439701i	$-0.855084\pi$
$41$		−	36.0555i		−	0.879403i	0.898144	−	0.439701i	$-0.144916\pi$
$42$		0			0
$43$	20.0000			0.465116			0.232558	−	0.972582i	$-0.425290\pi$
$43$	20.0000			0.465116			0.232558	+	0.972582i	$0.425290\pi$
$44$		0			0
$45$	16.0000			0.355556
$46$		0			0
$47$	10.0000			0.212766			0.106383	−	0.994325i	$-0.466073\pi$
$47$	10.0000			0.212766			0.106383	+	0.994325i	$0.466073\pi$
$48$		0			0
$49$	−24.0000			−0.489796
$50$		0			0
$51$		−	54.0833i		−	1.06046i
$52$		0			0
$53$		−	75.7166i		−	1.42861i	−0.699832	−	0.714307i	$-0.746742\pi$
$53$		−	75.7166i		−	1.42861i	0.699832	−	0.714307i	$-0.253258\pi$
$54$		0			0
$55$	−40.0000			−0.727273
$56$		0			0
$57$	65.0000	−	21.6333i	1.14035	−	0.379532i
$58$		0			0
$59$			18.0278i			0.305555i	0.988261	+	0.152778i	$0.0488218\pi$
$59$			18.0278i			0.305555i	−0.988261	+	0.152778i	$0.951178\pi$
$60$		0			0
$61$	40.0000			0.655738			0.327869	−	0.944723i	$-0.393670\pi$
$61$	40.0000			0.655738			0.327869	+	0.944723i	$0.393670\pi$
$62$		0			0
$63$	20.0000			0.317460
$64$		0			0
$65$		−	14.4222i		−	0.221880i
$66$		0			0
$67$			39.6611i			0.591956i	0.955195	+	0.295978i	$0.0956455\pi$
$67$			39.6611i			0.591956i	−0.955195	+	0.295978i	$0.904354\pi$
$68$		0			0
$69$		−	126.194i		−	1.82890i
$70$		0			0
$71$		−	108.167i		−	1.52347i	−0.647887	−	0.761736i	$-0.724347\pi$
$71$		−	108.167i		−	1.52347i	0.647887	−	0.761736i	$-0.275653\pi$
$72$		0			0
$73$	105.000			1.43836			0.719178	−	0.694826i	$-0.244519\pi$
$73$	105.000			1.43836			0.719178	+	0.694826i	$0.244519\pi$
$74$		0			0
$75$			32.4500i			0.432666i
$76$		0			0
$77$	−50.0000			−0.649351
$78$		0			0
$79$			36.0555i			0.456399i	0.973614	+	0.228199i	$0.0732838\pi$
$79$			36.0555i			0.456399i	−0.973614	+	0.228199i	$0.926716\pi$
$80$		0			0
$81$	−101.000			−1.24691
$82$		0			0
$83$	40.0000			0.481928			0.240964	−	0.970534i	$-0.422536\pi$
$83$	40.0000			0.481928			0.240964	+	0.970534i	$0.422536\pi$
$84$		0			0
$85$	−60.0000			−0.705882
$86$		0			0
$87$	65.0000			0.747126
$88$		0			0
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$		0			0
$91$		−	18.0278i		−	0.198107i
$92$		0			0
$93$	130.000			1.39785
$94$		0			0
$95$	−24.0000	−	72.1110i	−0.252632	−	0.759063i
$96$		0			0
$97$		−	122.589i		−	1.26380i	−0.775049	−	0.631901i	$-0.782275\pi$
$97$		−	122.589i		−	1.26380i	0.775049	−	0.631901i	$-0.217725\pi$
$98$		0			0
$99$	−40.0000			−0.404040
$100$		0			0
$101$	50.0000			0.495050			0.247525	−	0.968882i	$-0.420383\pi$
$101$	50.0000			0.495050			0.247525	+	0.968882i	$0.420383\pi$
$102$		0			0
$103$		−	57.6888i		−	0.560086i	−0.959988	−	0.280043i	$-0.909651\pi$
$103$		−	57.6888i		−	0.560086i	0.959988	−	0.280043i	$-0.0903487\pi$
$104$		0			0
$105$		−	72.1110i		−	0.686772i
$106$		0			0
$107$			75.7166i			0.707632i	0.935315	+	0.353816i	$0.115116\pi$
$107$			75.7166i			0.707632i	−0.935315	+	0.353816i	$0.884884\pi$
$108$		0			0
$109$		−	198.305i		−	1.81931i	−0.415359	−	0.909657i	$-0.636344\pi$
$109$		−	198.305i		−	1.81931i	0.415359	−	0.909657i	$-0.363656\pi$
$110$		0			0
$111$	−78.0000			−0.702703
$112$		0			0
$113$			122.589i			1.08486i	0.840102	+	0.542428i	$0.182495\pi$
$113$			122.589i			1.08486i	−0.840102	+	0.542428i	$0.817505\pi$
$114$		0			0
$115$	−140.000			−1.21739
$116$		0			0
$117$		−	14.4222i		−	0.123267i
$118$		0			0
$119$	−75.0000			−0.630252
$120$		0			0
$121$	−21.0000			−0.173554
$122$		0			0
$123$	−130.000			−1.05691
$124$		0			0
$125$	136.000			1.08800
$126$		0			0
$127$			129.800i			1.02205i	0.859567	+	0.511023i	$0.170733\pi$
$127$			129.800i			1.02205i	−0.859567	+	0.511023i	$0.829267\pi$
$128$		0			0
$129$		−	72.1110i		−	0.559000i
$130$		0			0
$131$	−112.000			−0.854962			−0.427481	−	0.904024i	$-0.640599\pi$
$131$	−112.000			−0.854962			−0.427481	+	0.904024i	$0.640599\pi$
$132$		0			0
$133$	−30.0000	−	90.1388i	−0.225564	−	0.677735i
$134$		0			0
$135$			72.1110i			0.534156i
$136$		0			0
$137$	125.000			0.912409			0.456204	−	0.889875i	$-0.349209\pi$
$137$	125.000			0.912409			0.456204	+	0.889875i	$0.349209\pi$
$138$		0			0
$139$	−50.0000			−0.359712			−0.179856	−	0.983693i	$-0.557563\pi$
$139$	−50.0000			−0.359712			−0.179856	+	0.983693i	$0.557563\pi$
$140$		0			0
$141$		−	36.0555i		−	0.255713i
$142$		0			0
$143$			36.0555i			0.252136i
$144$		0			0
$145$		−	72.1110i		−	0.497317i
$146$		0			0
$147$			86.5332i			0.588661i
$148$		0			0
$149$	−70.0000			−0.469799			−0.234899	−	0.972020i	$-0.575476\pi$
$149$	−70.0000			−0.469799			−0.234899	+	0.972020i	$0.575476\pi$
$150$		0			0
$151$		−	36.0555i		−	0.238778i	−0.992848	−	0.119389i	$-0.961906\pi$
$151$		−	36.0555i		−	0.238778i	0.992848	−	0.119389i	$-0.0380936\pi$
$152$		0			0
$153$	−60.0000			−0.392157
$154$		0			0
$155$		−	144.222i		−	0.930465i
$156$		0			0
$157$	−10.0000			−0.0636943			−0.0318471	−	0.999493i	$-0.510139\pi$
$157$	−10.0000			−0.0636943			−0.0318471	+	0.999493i	$0.510139\pi$
$158$		0			0
$159$	−273.000			−1.71698
$160$		0			0
$161$	−175.000			−1.08696
$162$		0			0
$163$	270.000			1.65644			0.828221	−	0.560402i	$-0.189353\pi$
$163$	270.000			1.65644			0.828221	+	0.560402i	$0.189353\pi$
$164$		0			0
$165$			144.222i			0.874073i
$166$		0			0
$167$		−	122.589i		−	0.734064i	−0.930208	−	0.367032i	$-0.880374\pi$
$167$		−	122.589i		−	0.734064i	0.930208	−	0.367032i	$-0.119626\pi$
$168$		0			0
$169$	156.000			0.923077
$170$		0			0
$171$	−24.0000	−	72.1110i	−0.140351	−	0.421702i
$172$		0			0
$173$		−	122.589i		−	0.708605i	−0.935131	−	0.354303i	$-0.884718\pi$
$173$		−	122.589i		−	0.708605i	0.935131	−	0.354303i	$-0.115282\pi$
$174$		0			0
$175$	45.0000			0.257143
$176$		0			0
$177$	65.0000			0.367232
$178$		0			0
$179$			36.0555i			0.201427i	0.994915	+	0.100714i	$0.0321126\pi$
$179$			36.0555i			0.201427i	−0.994915	+	0.100714i	$0.967887\pi$
$180$		0			0
$181$			108.167i			0.597605i	0.954315	+	0.298803i	$0.0965872\pi$
$181$			108.167i			0.597605i	−0.954315	+	0.298803i	$0.903413\pi$
$182$		0			0
$183$		−	144.222i		−	0.788099i
$184$		0			0
$185$			86.5332i			0.467747i
$186$		0			0
$187$	150.000			0.802139
$188$		0			0
$189$			90.1388i			0.476925i
$190$		0			0
$191$	193.000			1.01047			0.505236	−	0.862981i	$-0.331406\pi$
$191$	193.000			1.01047			0.505236	+	0.862981i	$0.331406\pi$
$192$		0			0
$193$		−	266.811i		−	1.38244i	−0.722645	−	0.691220i	$-0.757074\pi$
$193$		−	266.811i		−	1.38244i	0.722645	−	0.691220i	$-0.242926\pi$
$194$		0			0
$195$	−52.0000			−0.266667
$196$		0			0
$197$	−90.0000			−0.456853			−0.228426	−	0.973561i	$-0.573358\pi$
$197$	−90.0000			−0.456853			−0.228426	+	0.973561i	$0.573358\pi$
$198$		0			0
$199$	123.000			0.618090			0.309045	−	0.951047i	$-0.399991\pi$
$199$	123.000			0.618090			0.309045	+	0.951047i	$0.399991\pi$
$200$		0			0
$201$	143.000			0.711443
$202$		0			0
$203$		−	90.1388i		−	0.444033i
$204$		0			0
$205$			144.222i			0.703522i
$206$		0			0
$207$	−140.000			−0.676329
$208$		0			0
$209$	60.0000	+	180.278i	0.287081	+	0.862572i
$210$		0			0
$211$		−	234.361i		−	1.11071i	−0.831612	−	0.555357i	$-0.812581\pi$
$211$		−	234.361i		−	1.11071i	0.831612	−	0.555357i	$-0.187419\pi$
$212$		0			0
$213$	−390.000			−1.83099
$214$		0			0
$215$	−80.0000			−0.372093
$216$		0			0
$217$		−	180.278i		−	0.830772i
$218$		0			0
$219$		−	378.583i		−	1.72869i
$220$		0			0
$221$			54.0833i			0.244721i
$222$		0			0
$223$			201.911i			0.905430i	0.891655	+	0.452715i	$0.149544\pi$
$223$			201.911i			0.905430i	−0.891655	+	0.452715i	$0.850456\pi$
$224$		0			0
$225$	36.0000			0.160000
$226$		0			0
$227$			255.994i			1.12773i	0.825868	+	0.563864i	$0.190686\pi$
$227$			255.994i			1.12773i	−0.825868	+	0.563864i	$0.809314\pi$
$228$		0			0
$229$	160.000			0.698690			0.349345	−	0.936994i	$-0.386404\pi$
$229$	160.000			0.698690			0.349345	+	0.936994i	$0.386404\pi$
$230$		0			0
$231$			180.278i			0.780422i
$232$		0			0
$233$	−270.000			−1.15880			−0.579399	−	0.815044i	$-0.696713\pi$
$233$	−270.000			−1.15880			−0.579399	+	0.815044i	$0.696713\pi$
$234$		0			0
$235$	−40.0000			−0.170213
$236$		0			0
$237$	130.000			0.548523
$238$		0			0
$239$	197.000			0.824268			0.412134	−	0.911123i	$-0.364784\pi$
$239$	197.000			0.824268			0.412134	+	0.911123i	$0.364784\pi$
$240$		0			0
$241$		−	396.611i		−	1.64569i	−0.568268	−	0.822844i	$-0.692386\pi$
$241$		−	396.611i		−	1.64569i	0.568268	−	0.822844i	$-0.307614\pi$
$242$		0			0
$243$			201.911i			0.830909i
$244$		0			0
$245$	96.0000			0.391837
$246$		0			0
$247$	−65.0000	+	21.6333i	−0.263158	+	0.0875842i
$248$		0			0
$249$		−	144.222i		−	0.579205i
$250$		0			0
$251$	402.000			1.60159			0.800797	−	0.598936i	$-0.204410\pi$
$251$	402.000			1.60159			0.800797	+	0.598936i	$0.204410\pi$
$252$		0			0
$253$	350.000			1.38340
$254$		0			0
$255$			216.333i			0.848365i
$256$		0			0
$257$			418.244i			1.62741i	0.581279	+	0.813704i	$0.302552\pi$
$257$			418.244i			1.62741i	−0.581279	+	0.813704i	$0.697448\pi$
$258$		0			0
$259$			108.167i			0.417631i
$260$		0			0
$261$		−	72.1110i		−	0.276287i
$262$		0			0
$263$	310.000			1.17871			0.589354	−	0.807875i	$-0.299382\pi$
$263$	310.000			1.17871			0.589354	+	0.807875i	$0.299382\pi$
$264$		0			0
$265$			302.866i			1.14289i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			108.167i			0.402106i	0.979580	+	0.201053i	$0.0644364\pi$
$269$			108.167i			0.402106i	−0.979580	+	0.201053i	$0.935564\pi$
$270$		0			0
$271$	105.000			0.387454			0.193727	−	0.981055i	$-0.437942\pi$
$271$	105.000			0.387454			0.193727	+	0.981055i	$0.437942\pi$
$272$		0			0
$273$	−65.0000			−0.238095
$274$		0			0
$275$	−90.0000			−0.327273
$276$		0			0
$277$	50.0000			0.180505			0.0902527	−	0.995919i	$-0.471233\pi$
$277$	50.0000			0.180505			0.0902527	+	0.995919i	$0.471233\pi$
$278$		0			0
$279$		−	144.222i		−	0.516925i
$280$		0			0
$281$			288.444i			1.02649i	0.858242	+	0.513246i	$0.171557\pi$
$281$			288.444i			1.02649i	−0.858242	+	0.513246i	$0.828443\pi$
$282$		0			0
$283$	320.000			1.13074			0.565371	−	0.824837i	$-0.308733\pi$
$283$	320.000			1.13074			0.565371	+	0.824837i	$0.308733\pi$
$284$		0			0
$285$	−260.000	+	86.5332i	−0.912281	+	0.303625i
$286$		0			0
$287$			180.278i			0.628145i
$288$		0			0
$289$	−64.0000			−0.221453
$290$		0			0
$291$	−442.000			−1.51890
$292$		0			0
$293$		−	219.939i		−	0.750644i	−0.926895	−	0.375322i	$-0.877532\pi$
$293$		−	219.939i		−	0.750644i	0.926895	−	0.375322i	$-0.122468\pi$
$294$		0			0
$295$		−	72.1110i		−	0.244444i
$296$		0			0
$297$		−	180.278i		−	0.606995i
$298$		0			0
$299$			126.194i			0.422054i
$300$		0			0
$301$	−100.000			−0.332226
$302$		0			0
$303$		−	180.278i		−	0.594975i
$304$		0			0
$305$	−160.000			−0.524590
$306$		0			0
$307$			237.966i			0.775135i	0.921841	+	0.387567i	$0.126685\pi$
$307$			237.966i			0.775135i	−0.921841	+	0.387567i	$0.873315\pi$
$308$		0			0
$309$	−208.000			−0.673139
$310$		0			0
$311$	395.000			1.27010			0.635048	−	0.772472i	$-0.280980\pi$
$311$	395.000			1.27010			0.635048	+	0.772472i	$0.280980\pi$
$312$		0			0
$313$	125.000			0.399361			0.199681	−	0.979861i	$-0.436010\pi$
$313$	125.000			0.399361			0.199681	+	0.979861i	$0.436010\pi$
$314$		0			0
$315$	−80.0000			−0.253968
$316$		0			0
$317$			3.60555i			0.0113740i	0.999984	+	0.00568699i	$0.00181023\pi$
$317$			3.60555i			0.0113740i	−0.999984	+	0.00568699i	$0.998190\pi$
$318$		0			0
$319$			180.278i			0.565133i
$320$		0			0
$321$	273.000			0.850467
$322$		0			0
$323$	90.0000	+	270.416i	0.278638	+	0.837202i
$324$		0			0
$325$		−	32.4500i		−	0.0998460i
$326$		0			0
$327$	−715.000			−2.18654
$328$		0			0
$329$	−50.0000			−0.151976
$330$		0			0
$331$		−	198.305i		−	0.599110i	−0.954079	−	0.299555i	$-0.903162\pi$
$331$		−	198.305i		−	0.599110i	0.954079	−	0.299555i	$-0.0968382\pi$
$332$		0			0
$333$			86.5332i			0.259860i
$334$		0			0
$335$		−	158.644i		−	0.473565i
$336$		0			0
$337$		−	57.6888i		−	0.171183i	−0.996330	−	0.0855917i	$-0.972722\pi$
$337$		−	57.6888i		−	0.171183i	0.996330	−	0.0855917i	$-0.0272781\pi$
$338$		0			0
$339$	442.000			1.30383
$340$		0			0
$341$			360.555i			1.05735i
$342$		0			0
$343$	365.000			1.06414
$344$		0			0
$345$			504.777i			1.46312i
$346$		0			0
$347$	40.0000			0.115274			0.0576369	−	0.998338i	$-0.481643\pi$
$347$	40.0000			0.115274			0.0576369	+	0.998338i	$0.481643\pi$
$348$		0			0
$349$	−98.0000			−0.280802			−0.140401	−	0.990095i	$-0.544839\pi$
$349$	−98.0000			−0.280802			−0.140401	+	0.990095i	$0.544839\pi$
$350$		0			0
$351$	65.0000			0.185185
$352$		0			0
$353$	−185.000			−0.524079			−0.262040	−	0.965057i	$-0.584395\pi$
$353$	−185.000			−0.524079			−0.262040	+	0.965057i	$0.584395\pi$
$354$		0			0
$355$			432.666i			1.21878i
$356$		0			0
$357$			270.416i			0.757469i
$358$		0			0
$359$	−225.000			−0.626741			−0.313370	−	0.949631i	$-0.601458\pi$
$359$	−225.000			−0.626741			−0.313370	+	0.949631i	$0.601458\pi$
$360$		0			0
$361$	−289.000	+	216.333i	−0.800554	+	0.599261i
$362$		0			0
$363$			75.7166i			0.208586i
$364$		0			0
$365$	−420.000			−1.15068
$366$		0			0
$367$	50.0000			0.136240			0.0681199	−	0.997677i	$-0.478300\pi$
$367$	50.0000			0.136240			0.0681199	+	0.997677i	$0.478300\pi$
$368$		0			0
$369$			144.222i			0.390846i
$370$		0			0
$371$			378.583i			1.02044i
$372$		0			0
$373$		−	436.272i		−	1.16963i	−0.811167	−	0.584815i	$-0.801167\pi$
$373$		−	436.272i		−	1.16963i	0.811167	−	0.584815i	$-0.198833\pi$
$374$		0			0
$375$		−	490.355i		−	1.30761i
$376$		0			0
$377$	−65.0000			−0.172414
$378$		0			0
$379$		−	486.749i		−	1.28430i	−0.766579	−	0.642150i	$-0.778043\pi$
$379$		−	486.749i		−	1.28430i	0.766579	−	0.642150i	$-0.221957\pi$
$380$		0			0
$381$	468.000			1.22835
$382$		0			0
$383$		−	201.911i		−	0.527182i	−0.964634	−	0.263591i	$-0.915093\pi$
$383$		−	201.911i		−	0.527182i	0.964634	−	0.263591i	$-0.0849070\pi$
$384$		0			0
$385$	200.000			0.519481
$386$		0			0
$387$	−80.0000			−0.206718
$388$		0			0
$389$	478.000			1.22879			0.614396	−	0.788998i	$-0.289400\pi$
$389$	478.000			1.22879			0.614396	+	0.788998i	$0.289400\pi$
$390$		0			0
$391$	525.000			1.34271
$392$		0			0
$393$			403.822i			1.02754i
$394$		0			0
$395$		−	144.222i		−	0.365119i
$396$		0			0
$397$	−750.000			−1.88917			−0.944584	−	0.328269i	$-0.893535\pi$
$397$	−750.000			−1.88917			−0.944584	+	0.328269i	$0.893535\pi$
$398$		0			0
$399$	−325.000	+	108.167i	−0.814536	+	0.271094i
$400$		0			0
$401$			288.444i			0.719312i	0.933085	+	0.359656i	$0.117106\pi$
$401$			288.444i			0.719312i	−0.933085	+	0.359656i	$0.882894\pi$
$402$		0			0
$403$	−130.000			−0.322581
$404$		0			0
$405$	404.000			0.997531
$406$		0			0
$407$		−	216.333i		−	0.531531i
$408$		0			0
$409$			36.0555i			0.0881553i	0.999028	+	0.0440776i	$0.0140349\pi$
$409$			36.0555i			0.0881553i	−0.999028	+	0.0440776i	$0.985965\pi$
$410$		0			0
$411$		−	450.694i		−	1.09658i
$412$		0			0
$413$		−	90.1388i		−	0.218254i
$414$		0			0
$415$	−160.000			−0.385542
$416$		0			0
$417$			180.278i			0.432320i
$418$		0			0
$419$	−112.000			−0.267303			−0.133652	−	0.991028i	$-0.542670\pi$
$419$	−112.000			−0.267303			−0.133652	+	0.991028i	$0.542670\pi$
$420$		0			0
$421$			630.971i			1.49874i	0.662149	+	0.749372i	$0.269645\pi$
$421$			630.971i			1.49874i	−0.662149	+	0.749372i	$0.730355\pi$
$422$		0			0
$423$	−40.0000			−0.0945626
$424$		0			0
$425$	−135.000			−0.317647
$426$		0			0
$427$	−200.000			−0.468384
$428$		0			0
$429$	130.000			0.303030
$430$		0			0
$431$			432.666i			1.00387i	0.864907	+	0.501933i	$0.167378\pi$
$431$			432.666i			1.00387i	−0.864907	+	0.501933i	$0.832622\pi$
$432$		0			0
$433$			735.532i			1.69869i	0.527839	+	0.849345i	$0.323003\pi$
$433$			735.532i			1.69869i	−0.527839	+	0.849345i	$0.676997\pi$
$434$		0			0
$435$	−260.000			−0.597701
$436$		0			0
$437$	210.000	+	630.971i	0.480549	+	1.44387i
$438$		0			0
$439$		−	793.221i		−	1.80688i	−0.428712	−	0.903441i	$-0.641033\pi$
$439$		−	793.221i		−	1.80688i	0.428712	−	0.903441i	$-0.358967\pi$
$440$		0			0
$441$	96.0000			0.217687
$442$		0			0
$443$	−670.000			−1.51242			−0.756208	−	0.654332i	$-0.772950\pi$
$443$	−670.000			−1.51242			−0.756208	+	0.654332i	$0.772950\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$			252.389i			0.564628i
$448$		0			0
$449$			36.0555i			0.0803018i	0.999194	+	0.0401509i	$0.0127839\pi$
$449$			36.0555i			0.0803018i	−0.999194	+	0.0401509i	$0.987216\pi$
$450$		0			0
$451$		−	360.555i		−	0.799457i
$452$		0			0
$453$	−130.000			−0.286976
$454$		0			0
$455$			72.1110i			0.158486i
$456$		0			0
$457$	−755.000			−1.65208			−0.826039	−	0.563612i	$-0.809411\pi$
$457$	−755.000			−1.65208			−0.826039	+	0.563612i	$0.809411\pi$
$458$		0			0
$459$		−	270.416i		−	0.589142i
$460$		0			0
$461$	−772.000			−1.67462			−0.837310	−	0.546728i	$-0.815873\pi$
$461$	−772.000			−1.67462			−0.837310	+	0.546728i	$0.815873\pi$
$462$		0			0
$463$	−350.000			−0.755940			−0.377970	−	0.925818i	$-0.623378\pi$
$463$	−350.000			−0.755940			−0.377970	+	0.925818i	$0.623378\pi$
$464$		0			0
$465$	−520.000			−1.11828
$466$		0			0
$467$	−70.0000			−0.149893			−0.0749465	−	0.997188i	$-0.523879\pi$
$467$	−70.0000			−0.149893			−0.0749465	+	0.997188i	$0.523879\pi$
$468$		0			0
$469$		−	198.305i		−	0.422826i
$470$		0			0
$471$			36.0555i			0.0765510i
$472$		0			0
$473$	200.000			0.422833
$474$		0			0
$475$	−54.0000	−	162.250i	−0.113684	−	0.341579i
$476$		0			0
$477$			302.866i			0.634940i
$478$		0			0
$479$	−370.000			−0.772443			−0.386221	−	0.922406i	$-0.626220\pi$
$479$	−370.000			−0.772443			−0.386221	+	0.922406i	$0.626220\pi$
$480$		0			0
$481$	78.0000			0.162162
$482$		0			0
$483$			630.971i			1.30636i
$484$		0			0
$485$			490.355i			1.01104i
$486$		0			0
$487$			519.199i			1.06612i	0.846078	+	0.533059i	$0.178958\pi$
$487$			519.199i			1.06612i	−0.846078	+	0.533059i	$0.821042\pi$
$488$		0			0
$489$		−	973.499i		−	1.99080i
$490$		0			0
$491$	632.000			1.28717			0.643585	−	0.765375i	$-0.277446\pi$
$491$	632.000			1.28717			0.643585	+	0.765375i	$0.277446\pi$
$492$		0			0
$493$			270.416i			0.548512i
$494$		0			0
$495$	160.000			0.323232
$496$		0			0
$497$			540.833i			1.08819i
$498$		0			0
$499$	−380.000			−0.761523			−0.380762	−	0.924673i	$-0.624338\pi$
$499$	−380.000			−0.761523			−0.380762	+	0.924673i	$0.624338\pi$
$500$		0			0
$501$	−442.000			−0.882236
$502$		0			0
$503$	−45.0000			−0.0894632			−0.0447316	−	0.998999i	$-0.514243\pi$
$503$	−45.0000			−0.0894632			−0.0447316	+	0.998999i	$0.514243\pi$
$504$		0			0
$505$	−200.000			−0.396040
$506$		0			0
$507$		−	562.466i		−	1.10940i
$508$		0			0
$509$			829.277i			1.62923i	0.580004	+	0.814614i	$0.303051\pi$
$509$			829.277i			1.62923i	−0.580004	+	0.814614i	$0.696949\pi$
$510$		0			0
$511$	−525.000			−1.02740
$512$		0			0
$513$	325.000	−	108.167i	0.633528	−	0.210851i
$514$		0			0
$515$			230.755i			0.448069i
$516$		0			0
$517$	100.000			0.193424
$518$		0			0
$519$	−442.000			−0.851638
$520$		0			0
$521$		−	612.944i		−	1.17648i	−0.808688	−	0.588238i	$-0.799822\pi$
$521$		−	612.944i		−	1.17648i	0.808688	−	0.588238i	$-0.200178\pi$
$522$		0			0
$523$			465.116i			0.889323i	0.895699	+	0.444662i	$0.146676\pi$
$523$			465.116i			0.889323i	−0.895699	+	0.444662i	$0.853324\pi$
$524$		0			0
$525$		−	162.250i		−	0.309047i
$526$		0			0
$527$			540.833i			1.02625i
$528$		0			0
$529$	696.000			1.31569
$530$		0			0
$531$		−	72.1110i		−	0.135802i
$532$		0			0
$533$	130.000			0.243902
$534$		0			0
$535$		−	302.866i		−	0.566105i
$536$		0			0
$537$	130.000			0.242086
$538$		0			0
$539$	−240.000			−0.445269
$540$		0			0
$541$	600.000			1.10906			0.554529	−	0.832165i	$-0.312899\pi$
$541$	600.000			1.10906			0.554529	+	0.832165i	$0.312899\pi$
$542$		0			0
$543$	390.000			0.718232
$544$		0			0
$545$			793.221i			1.45545i
$546$		0			0
$547$		−	598.522i		−	1.09419i	−0.837071	−	0.547095i	$-0.815734\pi$
$547$		−	598.522i		−	1.09419i	0.837071	−	0.547095i	$-0.184266\pi$
$548$		0			0
$549$	−160.000			−0.291439
$550$		0			0
$551$	−325.000	+	108.167i	−0.589837	+	0.196310i
$552$		0			0
$553$		−	180.278i		−	0.325999i
$554$		0			0
$555$	312.000			0.562162
$556$		0			0
$557$	−380.000			−0.682226			−0.341113	−	0.940022i	$-0.610804\pi$
$557$	−380.000			−0.682226			−0.341113	+	0.940022i	$0.610804\pi$
$558$		0			0
$559$			72.1110i			0.129000i
$560$		0			0
$561$		−	540.833i		−	0.964051i
$562$		0			0
$563$			122.589i			0.217742i	0.994056	+	0.108871i	$0.0347235\pi$
$563$			122.589i			0.217742i	−0.994056	+	0.108871i	$0.965276\pi$
$564$		0			0
$565$		−	490.355i		−	0.867885i
$566$		0			0
$567$	505.000			0.890653
$568$		0			0
$569$		−	36.0555i		−	0.0633665i	−0.999498	−	0.0316832i	$-0.989913\pi$
$569$		−	36.0555i		−	0.0633665i	0.999498	−	0.0316832i	$-0.0100868\pi$
$570$		0			0
$571$	790.000			1.38354			0.691769	−	0.722119i	$-0.256832\pi$
$571$	790.000			1.38354			0.691769	+	0.722119i	$0.256832\pi$
$572$		0			0
$573$		−	695.871i		−	1.21444i
$574$		0			0
$575$	−315.000			−0.547826
$576$		0			0
$577$	675.000			1.16984			0.584922	−	0.811090i	$-0.301125\pi$
$577$	675.000			1.16984			0.584922	+	0.811090i	$0.301125\pi$
$578$		0			0
$579$	−962.000			−1.66149
$580$		0			0
$581$	−200.000			−0.344234
$582$		0			0
$583$		−	757.166i		−	1.29874i
$584$		0			0
$585$			57.6888i			0.0986134i
$586$		0			0
$587$	280.000			0.477002			0.238501	−	0.971142i	$-0.423344\pi$
$587$	280.000			0.477002			0.238501	+	0.971142i	$0.423344\pi$
$588$		0			0
$589$	−650.000	+	216.333i	−1.10357	+	0.367289i
$590$		0			0
$591$			324.500i			0.549069i
$592$		0			0
$593$	750.000			1.26476			0.632378	−	0.774660i	$-0.282079\pi$
$593$	750.000			1.26476			0.632378	+	0.774660i	$0.282079\pi$
$594$		0			0
$595$	300.000			0.504202
$596$		0			0
$597$		−	443.483i		−	0.742852i
$598$		0			0
$599$		−	504.777i		−	0.842700i	−0.906898	−	0.421350i	$-0.861556\pi$
$599$		−	504.777i		−	0.842700i	0.906898	−	0.421350i	$-0.138444\pi$
$600$		0			0
$601$		−	612.944i		−	1.01987i	−0.860212	−	0.509937i	$-0.829669\pi$
$601$		−	612.944i		−	1.01987i	0.860212	−	0.509937i	$-0.170331\pi$
$602$		0			0
$603$		−	158.644i		−	0.263092i
$604$		0			0
$605$	84.0000			0.138843
$606$		0			0
$607$			987.921i			1.62755i	0.581182	+	0.813774i	$0.302590\pi$
$607$			987.921i			1.62755i	−0.581182	+	0.813774i	$0.697410\pi$
$608$		0			0
$609$	−325.000			−0.533662
$610$		0			0
$611$			36.0555i			0.0590107i
$612$		0			0
$613$	1200.00			1.95759			0.978793	−	0.204853i	$-0.0656715\pi$
$613$	1200.00			1.95759			0.978793	+	0.204853i	$0.0656715\pi$
$614$		0			0
$615$	520.000			0.845528
$616$		0			0
$617$	−350.000			−0.567261			−0.283630	−	0.958934i	$-0.591539\pi$
$617$	−350.000			−0.567261			−0.283630	+	0.958934i	$0.591539\pi$
$618$		0			0
$619$	−560.000			−0.904685			−0.452342	−	0.891844i	$-0.649412\pi$
$619$	−560.000			−0.904685			−0.452342	+	0.891844i	$0.649412\pi$
$620$		0			0
$621$		−	630.971i		−	1.01606i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−319.000			−0.510400
$626$		0			0
$627$	650.000	−	216.333i	1.03668	−	0.345029i
$628$		0			0
$629$		−	324.500i		−	0.515898i
$630$		0			0
$631$	−1050.00			−1.66403			−0.832013	−	0.554757i	$-0.812811\pi$
$631$	−1050.00			−1.66403			−0.832013	+	0.554757i	$0.812811\pi$
$632$		0			0
$633$	−845.000			−1.33491
$634$		0			0
$635$		−	519.199i		−	0.817637i
$636$		0			0
$637$		−	86.5332i		−	0.135845i
$638$		0			0
$639$			432.666i			0.677099i
$640$		0			0
$641$		−	1225.89i		−	1.91246i	−0.292615	−	0.956230i	$-0.594525\pi$
$641$		−	1225.89i		−	1.91246i	0.292615	−	0.956230i	$-0.405475\pi$
$642$		0			0
$643$	1030.00			1.60187			0.800933	−	0.598754i	$-0.204337\pi$
$643$	1030.00			1.60187			0.800933	+	0.598754i	$0.204337\pi$
$644$		0			0
$645$			288.444i			0.447200i
$646$		0			0
$647$	555.000			0.857805			0.428903	−	0.903351i	$-0.358900\pi$
$647$	555.000			0.857805			0.428903	+	0.903351i	$0.358900\pi$
$648$		0			0
$649$			180.278i			0.277777i
$650$		0			0
$651$	−650.000			−0.998464
$652$		0			0
$653$	−50.0000			−0.0765697			−0.0382848	−	0.999267i	$-0.512189\pi$
$653$	−50.0000			−0.0765697			−0.0382848	+	0.999267i	$0.512189\pi$
$654$		0			0
$655$	448.000			0.683969
$656$		0			0
$657$	−420.000			−0.639269
$658$		0			0
$659$			198.305i			0.300919i	0.988616	+	0.150459i	$0.0480752\pi$
$659$			198.305i			0.300919i	−0.988616	+	0.150459i	$0.951925\pi$
$660$		0			0
$661$			198.305i			0.300008i	0.988685	+	0.150004i	$0.0479287\pi$
$661$			198.305i			0.300008i	−0.988685	+	0.150004i	$0.952071\pi$
$662$		0			0
$663$	195.000			0.294118
$664$		0			0
$665$	120.000	+	360.555i	0.180451	+	0.542188i
$666$		0			0
$667$			630.971i			0.945984i
$668$		0			0
$669$	728.000			1.08819
$670$		0			0
$671$	400.000			0.596125
$672$		0			0
$673$		−	598.522i		−	0.889334i	−0.895696	−	0.444667i	$-0.853322\pi$
$673$		−	598.522i		−	0.889334i	0.895696	−	0.444667i	$-0.146678\pi$
$674$		0			0
$675$			162.250i			0.240370i
$676$		0			0
$677$			68.5055i			0.101190i	0.998719	+	0.0505949i	$0.0161117\pi$
$677$			68.5055i			0.101190i	−0.998719	+	0.0505949i	$0.983888\pi$
$678$		0			0
$679$			612.944i			0.902715i
$680$		0			0
$681$	923.000			1.35536
$682$		0			0
$683$		−	237.966i		−	0.348413i	−0.984709	−	0.174207i	$-0.944264\pi$
$683$		−	237.966i		−	0.348413i	0.984709	−	0.174207i	$-0.0557361\pi$
$684$		0			0
$685$	−500.000			−0.729927
$686$		0			0
$687$		−	576.888i		−	0.839721i
$688$		0			0
$689$	273.000			0.396226
$690$		0			0
$691$	820.000			1.18669			0.593343	−	0.804950i	$-0.297808\pi$
$691$	820.000			1.18669			0.593343	+	0.804950i	$0.297808\pi$
$692$		0			0
$693$	200.000			0.288600
$694$		0			0
$695$	200.000			0.287770
$696$		0			0
$697$		−	540.833i		−	0.775944i
$698$		0			0
$699$			973.499i			1.39270i
$700$		0			0
$701$	540.000			0.770328			0.385164	−	0.922848i	$-0.374145\pi$
$701$	540.000			0.770328			0.385164	+	0.922848i	$0.374145\pi$
$702$		0			0
$703$	390.000	−	129.800i	0.554765	−	0.184637i
$704$		0			0
$705$			144.222i			0.204570i
$706$		0			0
$707$	−250.000			−0.353607
$708$		0			0
$709$	−268.000			−0.377997			−0.188999	−	0.981977i	$-0.560524\pi$
$709$	−268.000			−0.377997			−0.188999	+	0.981977i	$0.560524\pi$
$710$		0			0
$711$		−	144.222i		−	0.202844i
$712$		0			0
$713$			1261.94i			1.76991i
$714$		0			0
$715$		−	144.222i		−	0.201709i
$716$		0			0
$717$		−	710.294i		−	0.990647i
$718$		0			0
$719$	105.000			0.146036			0.0730181	−	0.997331i	$-0.476737\pi$
$719$	105.000			0.146036			0.0730181	+	0.997331i	$0.476737\pi$
$720$		0			0
$721$			288.444i			0.400061i
$722$		0			0
$723$	−1430.00			−1.97787
$724$		0			0
$725$		−	162.250i		−	0.223793i
$726$		0			0
$727$	695.000			0.955983			0.477992	−	0.878364i	$-0.341365\pi$
$727$	695.000			0.955983			0.477992	+	0.878364i	$0.341365\pi$
$728$		0			0
$729$	−181.000			−0.248285
$730$		0			0
$731$	300.000			0.410397
$732$		0			0
$733$	−160.000			−0.218281			−0.109141	−	0.994026i	$-0.534810\pi$
$733$	−160.000			−0.218281			−0.109141	+	0.994026i	$0.534810\pi$
$734$		0			0
$735$		−	346.133i		−	0.470929i
$736$		0			0
$737$			396.611i			0.538142i
$738$		0			0
$739$	−1028.00			−1.39107			−0.695535	−	0.718493i	$-0.744832\pi$
$739$	−1028.00			−1.39107			−0.695535	+	0.718493i	$0.744832\pi$
$740$		0			0
$741$	78.0000	+	234.361i	0.105263	+	0.316276i
$742$		0			0
$743$			526.410i			0.708493i	0.935152	+	0.354247i	$0.115263\pi$
$743$			526.410i			0.708493i	−0.935152	+	0.354247i	$0.884737\pi$
$744$		0			0
$745$	280.000			0.375839
$746$		0			0
$747$	−160.000			−0.214190
$748$		0			0
$749$		−	378.583i		−	0.505451i
$750$		0			0
$751$		−	36.0555i		−	0.0480100i	−0.999712	−	0.0240050i	$-0.992358\pi$
$751$		−	36.0555i		−	0.0480100i	0.999712	−	0.0240050i	$-0.00764176\pi$
$752$		0			0
$753$		−	1449.43i		−	1.92488i
$754$		0			0
$755$			144.222i			0.191023i
$756$		0			0
$757$	−60.0000			−0.0792602			−0.0396301	−	0.999214i	$-0.512618\pi$
$757$	−60.0000			−0.0792602			−0.0396301	+	0.999214i	$0.512618\pi$
$758$		0			0
$759$		−	1261.94i		−	1.66264i
$760$		0			0
$761$	−655.000			−0.860710			−0.430355	−	0.902660i	$-0.641612\pi$
$761$	−655.000			−0.860710			−0.430355	+	0.902660i	$0.641612\pi$
$762$		0			0
$763$			991.527i			1.29951i
$764$		0			0
$765$	240.000			0.313725
$766$		0			0
$767$	−65.0000			−0.0847458
$768$		0			0
$769$	−185.000			−0.240572			−0.120286	−	0.992739i	$-0.538381\pi$
$769$	−185.000			−0.240572			−0.120286	+	0.992739i	$0.538381\pi$
$770$		0			0
$771$	1508.00			1.95590
$772$		0			0
$773$		−	320.894i		−	0.415128i	−0.978221	−	0.207564i	$-0.933446\pi$
$773$		−	320.894i		−	0.415128i	0.978221	−	0.207564i	$-0.0665536\pi$
$774$		0			0
$775$		−	324.500i		−	0.418709i
$776$		0			0
$777$	390.000			0.501931
$778$		0			0
$779$	650.000	−	216.333i	0.834403	−	0.277706i
$780$		0			0
$781$		−	1081.67i		−	1.38497i
$782$		0			0
$783$	325.000			0.415070
$784$		0			0
$785$	40.0000			0.0509554
$786$		0			0
$787$			68.5055i			0.0870463i	0.999052	+	0.0435232i	$0.0138582\pi$
$787$			68.5055i			0.0870463i	−0.999052	+	0.0435232i	$0.986142\pi$
$788$		0			0
$789$		−	1117.72i		−	1.41663i
$790$		0			0
$791$		−	612.944i		−	0.774897i
$792$		0			0
$793$			144.222i			0.181869i
$794$		0			0
$795$	1092.00			1.37358
$796$		0			0
$797$			1258.34i			1.57884i	0.613852	+	0.789421i	$0.289619\pi$
$797$			1258.34i			1.57884i	−0.613852	+	0.789421i	$0.710381\pi$
$798$		0			0
$799$	150.000			0.187735
$800$		0			0
$801$		0			0
$802$		0			0
$803$	1050.00			1.30760
$804$		0			0
$805$	700.000			0.869565
$806$		0			0
$807$	390.000			0.483271
$808$		0			0
$809$	−7.00000			−0.00865266			−0.00432633	−	0.999991i	$-0.501377\pi$
$809$	−7.00000			−0.00865266			−0.00432633	+	0.999991i	$0.501377\pi$
$810$		0			0
$811$			1027.58i			1.26706i	0.773720	+	0.633528i	$0.218394\pi$
$811$			1027.58i			1.26706i	−0.773720	+	0.633528i	$0.781606\pi$
$812$		0			0
$813$		−	378.583i		−	0.465662i
$814$		0			0
$815$	−1080.00			−1.32515
$816$		0			0
$817$	120.000	+	360.555i	0.146879	+	0.441316i
$818$		0			0
$819$			72.1110i			0.0880477i
$820$		0			0
$821$	−842.000			−1.02558			−0.512789	−	0.858515i	$-0.671388\pi$
$821$	−842.000			−1.02558			−0.512789	+	0.858515i	$0.671388\pi$
$822$		0			0
$823$	775.000			0.941677			0.470838	−	0.882219i	$-0.343951\pi$
$823$	775.000			0.941677			0.470838	+	0.882219i	$0.343951\pi$
$824$		0			0
$825$			324.500i			0.393333i
$826$		0			0
$827$		−	1265.55i		−	1.53029i	−0.643859	−	0.765144i	$-0.722668\pi$
$827$		−	1265.55i		−	1.53029i	0.643859	−	0.765144i	$-0.277332\pi$
$828$		0			0
$829$		−	1568.41i		−	1.89194i	−0.324260	−	0.945968i	$-0.605115\pi$
$829$		−	1568.41i		−	1.89194i	0.324260	−	0.945968i	$-0.394885\pi$
$830$		0			0
$831$		−	180.278i		−	0.216941i
$832$		0			0
$833$	−360.000			−0.432173
$834$		0			0
$835$			490.355i			0.587251i
$836$		0			0
$837$	650.000			0.776583
$838$		0			0
$839$			1153.78i			1.37518i	0.726099	+	0.687590i	$0.241331\pi$
$839$			1153.78i			1.37518i	−0.726099	+	0.687590i	$0.758669\pi$
$840$		0			0
$841$	516.000			0.613555
$842$		0			0
$843$	1040.00			1.23369
$844$		0			0
$845$	−624.000			−0.738462
$846$		0			0
$847$	105.000			0.123967
$848$		0			0
$849$		−	1153.78i		−	1.35898i
$850$		0			0
$851$		−	757.166i		−	0.889737i
$852$		0			0
$853$	−400.000			−0.468933			−0.234467	−	0.972124i	$-0.575334\pi$
$853$	−400.000			−0.468933			−0.234467	+	0.972124i	$0.575334\pi$
$854$		0			0
$855$	96.0000	+	288.444i	0.112281	+	0.337362i
$856$		0			0
$857$		−	158.644i		−	0.185116i	−0.995707	−	0.0925579i	$-0.970496\pi$
$857$		−	158.644i		−	0.185116i	0.995707	−	0.0925579i	$-0.0295043\pi$
$858$		0			0
$859$	−1432.00			−1.66705			−0.833527	−	0.552478i	$-0.813682\pi$
$859$	−1432.00			−1.66705			−0.833527	+	0.552478i	$0.813682\pi$
$860$		0			0
$861$	650.000			0.754936
$862$		0			0
$863$			129.800i			0.150405i	0.997168	+	0.0752027i	$0.0239604\pi$
$863$			129.800i			0.150405i	−0.997168	+	0.0752027i	$0.976040\pi$
$864$		0			0
$865$			490.355i			0.566884i
$866$		0			0
$867$			230.755i			0.266154i
$868$		0			0
$869$			360.555i			0.414908i
$870$		0			0
$871$	−143.000			−0.164179
$872$		0			0
$873$			490.355i			0.561690i
$874$		0			0
$875$	−680.000			−0.777143
$876$		0			0
$877$			393.005i			0.448124i	0.974575	+	0.224062i	$0.0719319\pi$
$877$			393.005i			0.448124i	−0.974575	+	0.224062i	$0.928068\pi$
$878$		0			0
$879$	−793.000			−0.902162
$880$		0			0
$881$	750.000			0.851305			0.425653	−	0.904887i	$-0.360045\pi$
$881$	750.000			0.851305			0.425653	+	0.904887i	$0.360045\pi$
$882$		0			0
$883$	−120.000			−0.135900			−0.0679502	−	0.997689i	$-0.521646\pi$
$883$	−120.000			−0.135900			−0.0679502	+	0.997689i	$0.521646\pi$
$884$		0			0
$885$	−260.000			−0.293785
$886$		0			0
$887$		−	850.910i		−	0.959312i	−0.877457	−	0.479656i	$-0.840761\pi$
$887$		−	850.910i		−	0.959312i	0.877457	−	0.479656i	$-0.159239\pi$
$888$		0			0
$889$		−	648.999i		−	0.730033i
$890$		0			0
$891$	−1010.00			−1.13356
$892$		0			0
$893$	60.0000	+	180.278i	0.0671892	+	0.201879i
$894$		0			0
$895$		−	144.222i		−	0.161142i
$896$		0			0
$897$	455.000			0.507246
$898$		0			0
$899$	−650.000			−0.723026
$900$		0			0
$901$		−	1135.75i		−	1.26054i
$902$		0			0
$903$			360.555i			0.399286i
$904$		0			0
$905$		−	432.666i		−	0.478084i
$906$		0			0
$907$		−	645.394i		−	0.711570i	−0.934568	−	0.355785i	$-0.884214\pi$
$907$		−	645.394i		−	0.711570i	0.934568	−	0.355785i	$-0.115786\pi$
$908$		0			0
$909$	−200.000			−0.220022
$910$		0			0
$911$			1225.89i			1.34565i	0.739802	+	0.672825i	$0.234919\pi$
$911$			1225.89i			1.34565i	−0.739802	+	0.672825i	$0.765081\pi$
$912$		0			0
$913$	400.000			0.438116
$914$		0			0
$915$			576.888i			0.630479i
$916$		0			0
$917$	560.000			0.610687
$918$		0			0
$919$	−513.000			−0.558215			−0.279108	−	0.960260i	$-0.590039\pi$
$919$	−513.000			−0.558215			−0.279108	+	0.960260i	$0.590039\pi$
$920$		0			0
$921$	858.000			0.931596
$922$		0			0
$923$	390.000			0.422535
$924$		0			0
$925$			194.700i			0.210486i
$926$		0			0
$927$			230.755i			0.248927i
$928$		0			0
$929$	−1217.00			−1.31001			−0.655005	−	0.755624i	$-0.727334\pi$
$929$	−1217.00			−1.31001			−0.655005	+	0.755624i	$0.727334\pi$
$930$		0			0
$931$	−144.000	−	432.666i	−0.154672	−	0.464733i
$932$		0			0
$933$		−	1424.19i		−	1.52647i
$934$		0			0
$935$	−600.000			−0.641711
$936$		0			0
$937$	−1095.00			−1.16862			−0.584312	−	0.811529i	$-0.698635\pi$
$937$	−1095.00			−1.16862			−0.584312	+	0.811529i	$0.698635\pi$
$938$		0			0
$939$		−	450.694i		−	0.479972i
$940$		0			0
$941$			1099.69i			1.16864i	0.811522	+	0.584322i	$0.198639\pi$
$941$			1099.69i			1.16864i	−0.811522	+	0.584322i	$0.801361\pi$
$942$		0			0
$943$		−	1261.94i		−	1.33822i
$944$		0			0
$945$		−	360.555i		−	0.381540i
$946$		0			0
$947$	−640.000			−0.675818			−0.337909	−	0.941179i	$-0.609720\pi$
$947$	−640.000			−0.675818			−0.337909	+	0.941179i	$0.609720\pi$
$948$		0			0
$949$			378.583i			0.398928i
$950$		0			0
$951$	13.0000			0.0136698
$952$		0			0
$953$		−	1168.20i		−	1.22581i	−0.790156	−	0.612906i	$-0.790000\pi$
$953$		−	1168.20i		−	1.22581i	0.790156	−	0.612906i	$-0.210000\pi$
$954$		0			0
$955$	−772.000			−0.808377
$956$		0			0
$957$	650.000			0.679206
$958$		0			0
$959$	−625.000			−0.651721
$960$		0			0
$961$	−339.000			−0.352758
$962$		0			0
$963$		−	302.866i		−	0.314503i
$964$		0			0
$965$			1067.24i			1.10595i
$966$		0			0
$967$	−1010.00			−1.04447			−0.522234	−	0.852802i	$-0.674901\pi$
$967$	−1010.00			−1.04447			−0.522234	+	0.852802i	$0.674901\pi$
$968$		0			0
$969$	975.000	−	324.500i	1.00619	−	0.334881i
$970$		0			0
$971$		−	757.166i		−	0.779779i	−0.920862	−	0.389890i	$-0.872513\pi$
$971$		−	757.166i		−	0.779779i	0.920862	−	0.389890i	$-0.127487\pi$
$972$		0			0
$973$	250.000			0.256937
$974$		0			0
$975$	−117.000			−0.120000
$976$		0			0
$977$		−	598.522i		−	0.612612i	−0.951933	−	0.306306i	$-0.900907\pi$
$977$		−	598.522i		−	0.612612i	0.951933	−	0.306306i	$-0.0990930\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$			793.221i			0.808584i
$982$		0			0
$983$		−	57.6888i		−	0.0586865i	−0.999569	−	0.0293432i	$-0.990658\pi$
$983$		−	57.6888i		−	0.0586865i	0.999569	−	0.0293432i	$-0.00934159\pi$
$984$		0			0
$985$	360.000			0.365482
$986$		0			0
$987$			180.278i			0.182652i
$988$		0			0
$989$	700.000			0.707786
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$		0			0
$993$	−715.000			−0.720040
$994$		0			0
$995$	−492.000			−0.494472
$996$		0			0
$997$	170.000			0.170512			0.0852558	−	0.996359i	$-0.472829\pi$
$997$	170.000			0.170512			0.0852558	+	0.996359i	$0.472829\pi$
$998$		0			0
$999$	−390.000			−0.390390

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1216.3.e.g.1025.1		2
4.3	odd	2		1216.3.e.h.1025.2		2
8.3	odd	2		304.3.e.d.113.1		2
8.5	even	2		19.3.b.b.18.1	✓	2
19.18	odd	2	inner	1216.3.e.g.1025.2		2
24.5	odd	2		171.3.c.b.37.2		2
24.11	even	2		2736.3.o.d.721.1		2
40.13	odd	4		475.3.d.b.474.2		4
40.29	even	2		475.3.c.b.151.2		2
40.37	odd	4		475.3.d.b.474.3		4
76.75	even	2		1216.3.e.h.1025.1		2
152.5	even	18		361.3.f.d.127.1		12
152.13	odd	18		361.3.f.d.116.2		12
152.21	odd	18		361.3.f.d.262.2		12
152.29	odd	18		361.3.f.d.299.1		12
152.37	odd	2		19.3.b.b.18.2	yes	2
152.45	even	6		361.3.d.b.293.2		4
152.53	odd	18		361.3.f.d.307.1		12
152.61	even	18		361.3.f.d.307.2		12
152.69	odd	6		361.3.d.b.293.1		4
152.75	even	2		304.3.e.d.113.2		2
152.85	even	18		361.3.f.d.299.2		12
152.93	even	18		361.3.f.d.262.1		12
152.101	even	18		361.3.f.d.116.1		12
152.109	odd	18		361.3.f.d.127.2		12
152.117	odd	18		361.3.f.d.333.1		12
152.125	even	6		361.3.d.b.69.1		4
152.141	odd	6		361.3.d.b.69.2		4
152.149	even	18		361.3.f.d.333.2		12
456.227	odd	2		2736.3.o.d.721.2		2
456.341	even	2		171.3.c.b.37.1		2
760.37	even	4		475.3.d.b.474.1		4
760.189	odd	2		475.3.c.b.151.1		2
760.493	even	4		475.3.d.b.474.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.3.b.b.18.1	✓	2	8.5	even	2
19.3.b.b.18.2	yes	2	152.37	odd	2
171.3.c.b.37.1		2	456.341	even	2
171.3.c.b.37.2		2	24.5	odd	2
304.3.e.d.113.1		2	8.3	odd	2
304.3.e.d.113.2		2	152.75	even	2
361.3.d.b.69.1		4	152.125	even	6
361.3.d.b.69.2		4	152.141	odd	6
361.3.d.b.293.1		4	152.69	odd	6
361.3.d.b.293.2		4	152.45	even	6
361.3.f.d.116.1		12	152.101	even	18
361.3.f.d.116.2		12	152.13	odd	18
361.3.f.d.127.1		12	152.5	even	18
361.3.f.d.127.2		12	152.109	odd	18
361.3.f.d.262.1		12	152.93	even	18
361.3.f.d.262.2		12	152.21	odd	18
361.3.f.d.299.1		12	152.29	odd	18
361.3.f.d.299.2		12	152.85	even	18
361.3.f.d.307.1		12	152.53	odd	18
361.3.f.d.307.2		12	152.61	even	18
361.3.f.d.333.1		12	152.117	odd	18
361.3.f.d.333.2		12	152.149	even	18
475.3.c.b.151.1		2	760.189	odd	2
475.3.c.b.151.2		2	40.29	even	2
475.3.d.b.474.1		4	760.37	even	4
475.3.d.b.474.2		4	40.13	odd	4
475.3.d.b.474.3		4	40.37	odd	4
475.3.d.b.474.4		4	760.493	even	4
1216.3.e.g.1025.1		2	1.1	even	1	trivial
1216.3.e.g.1025.2		2	19.18	odd	2	inner
1216.3.e.h.1025.1		2	76.75	even	2
1216.3.e.h.1025.2		2	4.3	odd	2
2736.3.o.d.721.1		2	24.11	even	2
2736.3.o.d.721.2		2	456.227	odd	2

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:	\( N \)	\(=\)	\( 1216 = 2^{6} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1216.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.60555i q^{3} -4.00000 q^{5} -5.00000 q^{7} -4.00000 q^{9} +O(q^{10})\)	\(q-3.60555i q^{3} -4.00000 q^{5} -5.00000 q^{7} -4.00000 q^{9} +10.0000 q^{11} +3.60555i q^{13} +14.4222i q^{15} +15.0000 q^{17} +(6.00000 + 18.0278i) q^{19} +18.0278i q^{21} +35.0000 q^{23} -9.00000 q^{25} -18.0278i q^{27} +18.0278i q^{29} +36.0555i q^{31} -36.0555i q^{33} +20.0000 q^{35} -21.6333i q^{37} +13.0000 q^{39} -36.0555i q^{41} +20.0000 q^{43} +16.0000 q^{45} +10.0000 q^{47} -24.0000 q^{49} -54.0833i q^{51} -75.7166i q^{53} -40.0000 q^{55} +(65.0000 - 21.6333i) q^{57} +18.0278i q^{59} +40.0000 q^{61} +20.0000 q^{63} -14.4222i q^{65} +39.6611i q^{67} -126.194i q^{69} -108.167i q^{71} +105.000 q^{73} +32.4500i q^{75} -50.0000 q^{77} +36.0555i q^{79} -101.000 q^{81} +40.0000 q^{83} -60.0000 q^{85} +65.0000 q^{87} -18.0278i q^{91} +130.000 q^{93} +(-24.0000 - 72.1110i) q^{95} -122.589i q^{97} -40.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{5} - 10 q^{7} - 8 q^{9}+O(q^{10}) \) 2 * q - 8 * q^5 - 10 * q^7 - 8 * q^9	\( 2 q - 8 q^{5} - 10 q^{7} - 8 q^{9} + 20 q^{11} + 30 q^{17} + 12 q^{19} + 70 q^{23} - 18 q^{25} + 40 q^{35} + 26 q^{39} + 40 q^{43} + 32 q^{45} + 20 q^{47} - 48 q^{49} - 80 q^{55} + 130 q^{57} + 80 q^{61} + 40 q^{63} + 210 q^{73} - 100 q^{77} - 202 q^{81} + 80 q^{83} - 120 q^{85} + 130 q^{87} + 260 q^{93} - 48 q^{95} - 80 q^{99}+O(q^{100}) \) 2 * q - 8 * q^5 - 10 * q^7 - 8 * q^9 + 20 * q^11 + 30 * q^17 + 12 * q^19 + 70 * q^23 - 18 * q^25 + 40 * q^35 + 26 * q^39 + 40 * q^43 + 32 * q^45 + 20 * q^47 - 48 * q^49 - 80 * q^55 + 130 * q^57 + 80 * q^61 + 40 * q^63 + 210 * q^73 - 100 * q^77 - 202 * q^81 + 80 * q^83 - 120 * q^85 + 130 * q^87 + 260 * q^93 - 48 * q^95 - 80 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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