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Properties

Label 735.2.j.e.197.9
Level $735$
Weight $2$
Character 735.197
Analytic conductor $5.869$
Analytic rank $0$
Dimension $24$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [735,2,Mod(197,735)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("735.197"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(735, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 735.j (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [24,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.86900454856\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 105)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 197.9
Character \(\chi\) \(=\) 735.197
Dual form 735.2.j.e.638.9

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.929340 - 0.929340i) q^{2} +(-1.37531 - 1.05286i) q^{3} +0.272655i q^{4} +(0.980304 - 2.00973i) q^{5} +(-2.25660 + 0.299670i) q^{6} +(2.11207 + 2.11207i) q^{8} +(0.782976 + 2.89602i) q^{9} +(-0.956684 - 2.77876i) q^{10} -3.90548i q^{11} +(0.287068 - 0.374987i) q^{12} +(1.56642 - 1.56642i) q^{13} +(-3.46419 + 1.73188i) q^{15} +3.38035 q^{16} +(1.89349 - 1.89349i) q^{17} +(3.41904 + 1.96374i) q^{18} -1.86019i q^{19} +(0.547963 + 0.267285i) q^{20} +(-3.62951 - 3.62951i) q^{22} +(-1.74459 - 1.74459i) q^{23} +(-0.681047 - 5.12847i) q^{24} +(-3.07801 - 3.94029i) q^{25} -2.91148i q^{26} +(1.97227 - 4.80730i) q^{27} -0.513153 q^{29} +(-1.60990 + 4.82891i) q^{30} -8.58277 q^{31} +(-1.08265 + 1.08265i) q^{32} +(-4.11191 + 5.37125i) q^{33} -3.51939i q^{34} +(-0.789616 + 0.213483i) q^{36} +(4.83665 + 4.83665i) q^{37} +(-1.72875 - 1.72875i) q^{38} +(-3.80355 + 0.505101i) q^{39} +(6.31515 - 2.17421i) q^{40} +0.308469i q^{41} +(7.60892 - 7.60892i) q^{43} +1.06485 q^{44} +(6.58777 + 1.26542i) q^{45} -3.24263 q^{46} +(-3.74074 + 3.74074i) q^{47} +(-4.64904 - 3.55903i) q^{48} +(-6.52238 - 0.801352i) q^{50} +(-4.59772 + 0.610565i) q^{51} +(0.427094 + 0.427094i) q^{52} +(-1.36127 - 1.36127i) q^{53} +(-2.63471 - 6.30052i) q^{54} +(-7.84894 - 3.82855i) q^{55} +(-1.95852 + 2.55835i) q^{57} +(-0.476893 + 0.476893i) q^{58} -0.518229 q^{59} +(-0.472207 - 0.944529i) q^{60} -5.10902 q^{61} +(-7.97631 + 7.97631i) q^{62} +8.77299i q^{64} +(-1.61251 - 4.68365i) q^{65} +(1.17035 + 8.81309i) q^{66} +(6.40207 + 6.40207i) q^{67} +(0.516270 + 0.516270i) q^{68} +(0.562551 + 4.23616i) q^{69} -15.3749i q^{71} +(-4.46290 + 7.77030i) q^{72} +(2.04880 - 2.04880i) q^{73} +8.98978 q^{74} +(0.0846572 + 8.65984i) q^{75} +0.507191 q^{76} +(-3.06538 + 4.00420i) q^{78} -5.05241i q^{79} +(3.31377 - 6.79358i) q^{80} +(-7.77390 + 4.53503i) q^{81} +(0.286673 + 0.286673i) q^{82} +(9.16088 + 9.16088i) q^{83} +(-1.94920 - 5.66159i) q^{85} -14.1425i q^{86} +(0.705746 + 0.540277i) q^{87} +(8.24863 - 8.24863i) q^{88} +11.3504 q^{89} +(7.29828 - 4.94628i) q^{90} +(0.475671 - 0.475671i) q^{92} +(11.8040 + 9.03644i) q^{93} +6.95283i q^{94} +(-3.73848 - 1.82355i) q^{95} +(2.62885 - 0.349104i) q^{96} +(6.81964 + 6.81964i) q^{97} +(11.3103 - 3.05789i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q - 2 q^{3} + 12 q^{6} - 8 q^{10} - 10 q^{12} + 8 q^{13} + 2 q^{15} + 8 q^{16} - 14 q^{18} - 4 q^{22} - 4 q^{25} - 20 q^{27} - 40 q^{30} - 24 q^{31} - 4 q^{33} + 4 q^{36} - 4 q^{37} - 16 q^{40} + 8 q^{43}+ \cdots + 60 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/735\mathbb{Z}\right)^\times\).

\(n\) \(346\) \(442\) \(491\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{4}\right)\) \(-1\)

Coefficient data

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



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.929340 − 0.929340i 0.657142 − 0.657142i −0.297561 − 0.954703i \(-0.596173\pi\)
0.954703 + 0.297561i \(0.0961730\pi\)
\(3\) −1.37531 − 1.05286i −0.794038 − 0.607868i
\(4\) 0.272655i 0.136328i
\(5\) 0.980304 − 2.00973i 0.438405 − 0.898777i
\(6\) −2.25660 + 0.299670i −0.921252 + 0.122340i
\(7\) 0 0
\(8\) 2.11207 + 2.11207i 0.746729 + 0.746729i
\(9\) 0.782976 + 2.89602i 0.260992 + 0.965341i
\(10\) −0.956684 − 2.77876i −0.302530 − 0.878719i
\(11\) − 3.90548i − 1.17755i −0.808299 − 0.588773i \(-0.799611\pi\)
0.808299 − 0.588773i \(-0.200389\pi\)
\(12\) 0.287068 − 0.374987i 0.0828693 − 0.108249i
\(13\) 1.56642 − 1.56642i 0.434448 − 0.434448i −0.455691 − 0.890138i \(-0.650608\pi\)
0.890138 + 0.455691i \(0.150608\pi\)
\(14\) 0 0
\(15\) −3.46419 + 1.73188i −0.894449 + 0.447170i
\(16\) 3.38035 0.845087
\(17\) 1.89349 − 1.89349i 0.459239 − 0.459239i −0.439167 − 0.898406i \(-0.644726\pi\)
0.898406 + 0.439167i \(0.144726\pi\)
\(18\) 3.41904 + 1.96374i 0.805875 + 0.462858i
\(19\) − 1.86019i − 0.426757i −0.976970 − 0.213379i \(-0.931553\pi\)
0.976970 − 0.213379i \(-0.0684468\pi\)
\(20\) 0.547963 + 0.267285i 0.122528 + 0.0597668i
\(21\) 0 0
\(22\) −3.62951 − 3.62951i −0.773815 − 0.773815i
\(23\) −1.74459 − 1.74459i −0.363772 − 0.363772i 0.501428 − 0.865200i \(-0.332808\pi\)
−0.865200 + 0.501428i \(0.832808\pi\)
\(24\) −0.681047 − 5.12847i −0.139018 − 1.04684i
\(25\) −3.07801 − 3.94029i −0.615601 − 0.788058i
\(26\) − 2.91148i − 0.570988i
\(27\) 1.97227 − 4.80730i 0.379563 − 0.925166i
\(28\) 0 0
\(29\) −0.513153 −0.0952901 −0.0476450 − 0.998864i \(-0.515172\pi\)
−0.0476450 + 0.998864i \(0.515172\pi\)
\(30\) −1.60990 + 4.82891i −0.293926 + 0.881635i
\(31\) −8.58277 −1.54151 −0.770755 − 0.637131i \(-0.780121\pi\)
−0.770755 + 0.637131i \(0.780121\pi\)
\(32\) −1.08265 + 1.08265i −0.191387 + 0.191387i
\(33\) −4.11191 + 5.37125i −0.715793 + 0.935015i
\(34\) − 3.51939i − 0.603570i
\(35\) 0 0
\(36\) −0.789616 + 0.213483i −0.131603 + 0.0355804i
\(37\) 4.83665 + 4.83665i 0.795140 + 0.795140i 0.982325 − 0.187185i \(-0.0599363\pi\)
−0.187185 + 0.982325i \(0.559936\pi\)
\(38\) −1.72875 − 1.72875i −0.280440 − 0.280440i
\(39\) −3.80355 + 0.505101i −0.609055 + 0.0808808i
\(40\) 6.31515 − 2.17421i 0.998513 − 0.343773i
\(41\) 0.308469i 0.0481748i 0.999710 + 0.0240874i \(0.00766800\pi\)
−0.999710 + 0.0240874i \(0.992332\pi\)
\(42\) 0 0
\(43\) 7.60892 − 7.60892i 1.16035 − 1.16035i 0.175950 − 0.984399i \(-0.443700\pi\)
0.984399 − 0.175950i \(-0.0562999\pi\)
\(44\) 1.06485 0.160532
\(45\) 6.58777 + 1.26542i 0.982047 + 0.188637i
\(46\) −3.24263 −0.478100
\(47\) −3.74074 + 3.74074i −0.545642 + 0.545642i −0.925177 − 0.379535i \(-0.876084\pi\)
0.379535 + 0.925177i \(0.376084\pi\)
\(48\) −4.64904 − 3.55903i −0.671031 − 0.513702i
\(49\) 0 0
\(50\) −6.52238 − 0.801352i −0.922404 − 0.113328i
\(51\) −4.59772 + 0.610565i −0.643810 + 0.0854962i
\(52\) 0.427094 + 0.427094i 0.0592273 + 0.0592273i
\(53\) −1.36127 − 1.36127i −0.186985 − 0.186985i 0.607407 − 0.794391i \(-0.292210\pi\)
−0.794391 + 0.607407i \(0.792210\pi\)
\(54\) −2.63471 − 6.30052i −0.358539 − 0.857393i
\(55\) −7.84894 − 3.82855i −1.05835 − 0.516242i
\(56\) 0 0
\(57\) −1.95852 + 2.55835i −0.259412 + 0.338861i
\(58\) −0.476893 + 0.476893i −0.0626191 + 0.0626191i
\(59\) −0.518229 −0.0674676 −0.0337338 − 0.999431i \(-0.510740\pi\)
−0.0337338 + 0.999431i \(0.510740\pi\)
\(60\) −0.472207 − 0.944529i −0.0609617 − 0.121938i
\(61\) −5.10902 −0.654143 −0.327071 − 0.945000i \(-0.606062\pi\)
−0.327071 + 0.945000i \(0.606062\pi\)
\(62\) −7.97631 + 7.97631i −1.01299 + 1.01299i
\(63\) 0 0
\(64\) 8.77299i 1.09662i
\(65\) −1.61251 − 4.68365i −0.200007 − 0.580936i
\(66\) 1.17035 + 8.81309i 0.144061 + 1.08482i
\(67\) 6.40207 + 6.40207i 0.782138 + 0.782138i 0.980191 − 0.198054i \(-0.0634620\pi\)
−0.198054 + 0.980191i \(0.563462\pi\)
\(68\) 0.516270 + 0.516270i 0.0626070 + 0.0626070i
\(69\) 0.562551 + 4.23616i 0.0677232 + 0.509974i
\(70\) 0 0
\(71\) − 15.3749i − 1.82467i −0.409448 − 0.912333i \(-0.634279\pi\)
0.409448 − 0.912333i \(-0.365721\pi\)
\(72\) −4.46290 + 7.77030i −0.525958 + 0.915739i
\(73\) 2.04880 − 2.04880i 0.239794 − 0.239794i −0.576971 − 0.816765i \(-0.695765\pi\)
0.816765 + 0.576971i \(0.195765\pi\)
\(74\) 8.98978 1.04504
\(75\) 0.0846572 + 8.65984i 0.00977538 + 0.999952i
\(76\) 0.507191 0.0581788
\(77\) 0 0
\(78\) −3.06538 + 4.00420i −0.347086 + 0.453386i
\(79\) − 5.05241i − 0.568440i −0.958759 − 0.284220i \(-0.908265\pi\)
0.958759 − 0.284220i \(-0.0917347\pi\)
\(80\) 3.31377 − 6.79358i 0.370491 − 0.759545i
\(81\) −7.77390 + 4.53503i −0.863766 + 0.503893i
\(82\) 0.286673 + 0.286673i 0.0316577 + 0.0316577i
\(83\) 9.16088 + 9.16088i 1.00554 + 1.00554i 0.999985 + 0.00555287i \(0.00176754\pi\)
0.00555287 + 0.999985i \(0.498232\pi\)
\(84\) 0 0
\(85\) −1.94920 − 5.66159i −0.211421 − 0.614086i
\(86\) − 14.1425i − 1.52503i
\(87\) 0.705746 + 0.540277i 0.0756639 + 0.0579238i
\(88\) 8.24863 − 8.24863i 0.879307 − 0.879307i
\(89\) 11.3504 1.20314 0.601569 − 0.798821i \(-0.294542\pi\)
0.601569 + 0.798821i \(0.294542\pi\)
\(90\) 7.29828 − 4.94628i 0.769306 − 0.521383i
\(91\) 0 0
\(92\) 0.475671 − 0.475671i 0.0495922 − 0.0495922i
\(93\) 11.8040 + 9.03644i 1.22402 + 0.937036i
\(94\) 6.95283i 0.717129i
\(95\) −3.73848 − 1.82355i −0.383560 − 0.187093i
\(96\) 2.62885 − 0.349104i 0.268306 − 0.0356303i
\(97\) 6.81964 + 6.81964i 0.692430 + 0.692430i 0.962766 − 0.270336i \(-0.0871349\pi\)
−0.270336 + 0.962766i \(0.587135\pi\)
\(98\) 0 0
\(99\) 11.3103 − 3.05789i 1.13673 − 0.307330i
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 735.2.j.e.197.9 24
3.2 odd 2 inner 735.2.j.e.197.4 24
5.3 odd 4 inner 735.2.j.e.638.4 24
7.2 even 3 735.2.y.i.557.4 48
7.3 odd 6 105.2.x.a.2.9 yes 48
7.4 even 3 735.2.y.i.422.9 48
7.5 odd 6 105.2.x.a.32.4 yes 48
7.6 odd 2 735.2.j.g.197.9 24
15.8 even 4 inner 735.2.j.e.638.9 24
21.2 odd 6 735.2.y.i.557.9 48
21.5 even 6 105.2.x.a.32.9 yes 48
21.11 odd 6 735.2.y.i.422.4 48
21.17 even 6 105.2.x.a.2.4 ✓ 48
21.20 even 2 735.2.j.g.197.4 24
35.3 even 12 105.2.x.a.23.9 yes 48
35.12 even 12 525.2.bf.f.368.9 48
35.13 even 4 735.2.j.g.638.4 24
35.17 even 12 525.2.bf.f.443.4 48
35.18 odd 12 735.2.y.i.128.9 48
35.19 odd 6 525.2.bf.f.32.9 48
35.23 odd 12 735.2.y.i.263.4 48
35.24 odd 6 525.2.bf.f.107.4 48
35.33 even 12 105.2.x.a.53.4 yes 48
105.17 odd 12 525.2.bf.f.443.9 48
105.23 even 12 735.2.y.i.263.9 48
105.38 odd 12 105.2.x.a.23.4 yes 48
105.47 odd 12 525.2.bf.f.368.4 48
105.53 even 12 735.2.y.i.128.4 48
105.59 even 6 525.2.bf.f.107.9 48
105.68 odd 12 105.2.x.a.53.9 yes 48
105.83 odd 4 735.2.j.g.638.9 24
105.89 even 6 525.2.bf.f.32.4 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
105.2.x.a.2.4 ✓ 48 21.17 even 6
105.2.x.a.2.9 yes 48 7.3 odd 6
105.2.x.a.23.4 yes 48 105.38 odd 12
105.2.x.a.23.9 yes 48 35.3 even 12
105.2.x.a.32.4 yes 48 7.5 odd 6
105.2.x.a.32.9 yes 48 21.5 even 6
105.2.x.a.53.4 yes 48 35.33 even 12
105.2.x.a.53.9 yes 48 105.68 odd 12
525.2.bf.f.32.4 48 105.89 even 6
525.2.bf.f.32.9 48 35.19 odd 6
525.2.bf.f.107.4 48 35.24 odd 6
525.2.bf.f.107.9 48 105.59 even 6
525.2.bf.f.368.4 48 105.47 odd 12
525.2.bf.f.368.9 48 35.12 even 12
525.2.bf.f.443.4 48 35.17 even 12
525.2.bf.f.443.9 48 105.17 odd 12
735.2.j.e.197.4 24 3.2 odd 2 inner
735.2.j.e.197.9 24 1.1 even 1 trivial
735.2.j.e.638.4 24 5.3 odd 4 inner
735.2.j.e.638.9 24 15.8 even 4 inner
735.2.j.g.197.4 24 21.20 even 2
735.2.j.g.197.9 24 7.6 odd 2
735.2.j.g.638.4 24 35.13 even 4
735.2.j.g.638.9 24 105.83 odd 4
735.2.y.i.128.4 48 105.53 even 12
735.2.y.i.128.9 48 35.18 odd 12
735.2.y.i.263.4 48 35.23 odd 12
735.2.y.i.263.9 48 105.23 even 12
735.2.y.i.422.4 48 21.11 odd 6
735.2.y.i.422.9 48 7.4 even 3
735.2.y.i.557.4 48 7.2 even 3
735.2.y.i.557.9 48 21.2 odd 6