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Properties

Label 735.2.j.e.638.4
Level $735$
Weight $2$
Character 735.638
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 638.4
Character \(\chi\) \(=\) 735.638
Dual form 735.2.j.e.197.4

$q$-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} +(1.05286 - 1.37531i) 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} +(-0.374987 - 0.287068i) q^{12} +(1.56642 + 1.56642i) q^{13} +(-3.79613 - 0.767733i) q^{15} +3.38035 q^{16} +(-1.89349 - 1.89349i) q^{17} +(-1.96374 + 3.41904i) 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} +(-4.80730 - 1.97227i) q^{27} +0.513153 q^{29} +(2.81441 + 4.24138i) q^{30} -8.58277 q^{31} +(1.08265 + 1.08265i) q^{32} +(-5.37125 - 4.11191i) 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} +(-5.05266 + 4.41255i) q^{45} -3.24263 q^{46} +(3.74074 + 3.74074i) q^{47} +(3.55903 - 4.64904i) 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} +(2.55835 + 1.95852i) q^{57} +(-0.476893 - 0.476893i) q^{58} +0.518229 q^{59} +(-0.209327 + 1.03503i) 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} +(7.77030 + 4.46290i) q^{72} +(2.04880 + 2.04880i) q^{73} -8.98978 q^{74} +(2.17843 + 8.38179i) q^{75} +0.507191 q^{76} +(-4.00420 - 3.06538i) 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} +(0.540277 - 0.705746i) q^{87} +(8.24863 + 8.24863i) q^{88} -11.3504 q^{89} +(8.79640 + 0.594879i) q^{90} +(-0.475671 - 0.475671i) q^{92} +(-9.03644 + 11.8040i) 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 \(\left(\mathbb{Z}/735\mathbb{Z}\right)^\times\).

\(n\) \(346\) \(442\) \(491\)
\(\chi(n)\) \(1\) \(e\left(\frac{3}{4}\right)\) \(-1\)

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 \(-0.403827\pi\)
−0.954703 + 0.297561i \(0.903827\pi\)
\(3\) 1.05286 − 1.37531i 0.607868 − 0.794038i
\(4\) − 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 \(-0.200389\pi\)
\(12\) −0.374987 − 0.287068i −0.108249 − 0.0828693i
\(13\) 1.56642 + 1.56642i 0.434448 + 0.434448i 0.890138 − 0.455691i \(-0.150608\pi\)
−0.455691 + 0.890138i \(0.650608\pi\)
\(14\) 0 0
\(15\) −3.79613 − 0.767733i −0.980156 − 0.198228i
\(16\) 3.38035 0.845087
\(17\) −1.89349 − 1.89349i −0.459239 − 0.459239i 0.439167 − 0.898406i \(-0.355274\pi\)
−0.898406 + 0.439167i \(0.855274\pi\)
\(18\) −1.96374 + 3.41904i −0.462858 + 0.805875i
\(19\) 1.86019i 0.426757i 0.976970 + 0.213379i \(0.0684468\pi\)
−0.976970 + 0.213379i \(0.931553\pi\)
\(20\) −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 \(-0.667192\pi\)
0.865200 + 0.501428i \(0.167192\pi\)
\(24\) 0.681047 + 5.12847i 0.139018 + 1.04684i
\(25\) −3.07801 + 3.94029i −0.615601 + 0.788058i
\(26\) − 2.91148i − 0.570988i
\(27\) −4.80730 − 1.97227i −0.925166 − 0.379563i
\(28\) 0 0
\(29\) 0.513153 0.0952901 0.0476450 − 0.998864i \(-0.484828\pi\)
0.0476450 + 0.998864i \(0.484828\pi\)
\(30\) 2.81441 + 4.24138i 0.513838 + 0.774366i
\(31\) −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
\(33\) −5.37125 − 4.11191i −0.935015 − 0.715793i
\(34\) 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 −0.187185 − 0.982325i \(-0.559936\pi\)
0.982325 + 0.187185i \(0.0599363\pi\)
\(38\) 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 0.984399 + 0.175950i \(0.0562999\pi\)
0.175950 + 0.984399i \(0.443700\pi\)
\(44\) −1.06485 −0.160532
\(45\) −5.05266 + 4.41255i −0.753206 + 0.657784i
\(46\) −3.24263 −0.478100
\(47\) 3.74074 + 3.74074i 0.545642 + 0.545642i 0.925177 − 0.379535i \(-0.123916\pi\)
−0.379535 + 0.925177i \(0.623916\pi\)
\(48\) 3.55903 − 4.64904i 0.513702 − 0.671031i
\(49\) 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 \(-0.707790\pi\)
0.794391 + 0.607407i \(0.207790\pi\)
\(54\) 2.63471 + 6.30052i 0.358539 + 0.857393i
\(55\) −7.84894 + 3.82855i −1.05835 + 0.516242i
\(56\) 0 0
\(57\) 2.55835 + 1.95852i 0.338861 + 0.259412i
\(58\) −0.476893 − 0.476893i −0.0626191 − 0.0626191i
\(59\) 0.518229 0.0674676 0.0337338 − 0.999431i \(-0.489260\pi\)
0.0337338 + 0.999431i \(0.489260\pi\)
\(60\) −0.209327 + 1.03503i −0.0270240 + 0.133622i
\(61\) −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 −0.198054 − 0.980191i \(-0.563462\pi\)
0.980191 + 0.198054i \(0.0634620\pi\)
\(68\) −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 \(-0.365721\pi\)
\(72\) 7.77030 + 4.46290i 0.915739 + 0.525958i
\(73\) 2.04880 + 2.04880i 0.239794 + 0.239794i 0.816765 − 0.576971i \(-0.195765\pi\)
−0.576971 + 0.816765i \(0.695765\pi\)
\(74\) −8.98978 −1.04504
\(75\) 2.17843 + 8.38179i 0.251543 + 0.967846i
\(76\) 0.507191 0.0581788
\(77\) 0 0
\(78\) −4.00420 − 3.06538i −0.453386 − 0.347086i
\(79\) 5.05241i 0.568440i 0.958759 + 0.284220i \(0.0917347\pi\)
−0.958759 + 0.284220i \(0.908265\pi\)
\(80\) −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 −0.00555287 + 0.999985i \(0.501768\pi\)
−0.999985 + 0.00555287i \(0.998232\pi\)
\(84\) 0 0
\(85\) −1.94920 + 5.66159i −0.211421 + 0.614086i
\(86\) − 14.1425i − 1.52503i
\(87\) 0.540277 − 0.705746i 0.0579238 − 0.0756639i
\(88\) 8.24863 + 8.24863i 0.879307 + 0.879307i
\(89\) −11.3504 −1.20314 −0.601569 − 0.798821i \(-0.705458\pi\)
−0.601569 + 0.798821i \(0.705458\pi\)
\(90\) 8.79640 + 0.594879i 0.927222 + 0.0627058i
\(91\) 0 0
\(92\) −0.475671 − 0.475671i −0.0495922 − 0.0495922i
\(93\) −9.03644 + 11.8040i −0.937036 + 1.22402i
\(94\) − 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 −0.270336 − 0.962766i \(-0.587135\pi\)
0.962766 + 0.270336i \(0.0871349\pi\)
\(98\) 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.638.4 24
3.2 odd 2 inner 735.2.j.e.638.9 24
5.2 odd 4 inner 735.2.j.e.197.9 24
7.2 even 3 735.2.y.i.263.4 48
7.3 odd 6 105.2.x.a.23.9 yes 48
7.4 even 3 735.2.y.i.128.9 48
7.5 odd 6 105.2.x.a.53.4 yes 48
7.6 odd 2 735.2.j.g.638.4 24
15.2 even 4 inner 735.2.j.e.197.4 24
21.2 odd 6 735.2.y.i.263.9 48
21.5 even 6 105.2.x.a.53.9 yes 48
21.11 odd 6 735.2.y.i.128.4 48
21.17 even 6 105.2.x.a.23.4 yes 48
21.20 even 2 735.2.j.g.638.9 24
35.2 odd 12 735.2.y.i.557.4 48
35.3 even 12 525.2.bf.f.107.4 48
35.12 even 12 105.2.x.a.32.4 yes 48
35.17 even 12 105.2.x.a.2.9 yes 48
35.19 odd 6 525.2.bf.f.368.9 48
35.24 odd 6 525.2.bf.f.443.4 48
35.27 even 4 735.2.j.g.197.9 24
35.32 odd 12 735.2.y.i.422.9 48
35.33 even 12 525.2.bf.f.32.9 48
105.2 even 12 735.2.y.i.557.9 48
105.17 odd 12 105.2.x.a.2.4 ✓ 48
105.32 even 12 735.2.y.i.422.4 48
105.38 odd 12 525.2.bf.f.107.9 48
105.47 odd 12 105.2.x.a.32.9 yes 48
105.59 even 6 525.2.bf.f.443.9 48
105.62 odd 4 735.2.j.g.197.4 24
105.68 odd 12 525.2.bf.f.32.4 48
105.89 even 6 525.2.bf.f.368.4 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
105.2.x.a.2.4 ✓ 48 105.17 odd 12
105.2.x.a.2.9 yes 48 35.17 even 12
105.2.x.a.23.4 yes 48 21.17 even 6
105.2.x.a.23.9 yes 48 7.3 odd 6
105.2.x.a.32.4 yes 48 35.12 even 12
105.2.x.a.32.9 yes 48 105.47 odd 12
105.2.x.a.53.4 yes 48 7.5 odd 6
105.2.x.a.53.9 yes 48 21.5 even 6
525.2.bf.f.32.4 48 105.68 odd 12
525.2.bf.f.32.9 48 35.33 even 12
525.2.bf.f.107.4 48 35.3 even 12
525.2.bf.f.107.9 48 105.38 odd 12
525.2.bf.f.368.4 48 105.89 even 6
525.2.bf.f.368.9 48 35.19 odd 6
525.2.bf.f.443.4 48 35.24 odd 6
525.2.bf.f.443.9 48 105.59 even 6
735.2.j.e.197.4 24 15.2 even 4 inner
735.2.j.e.197.9 24 5.2 odd 4 inner
735.2.j.e.638.4 24 1.1 even 1 trivial
735.2.j.e.638.9 24 3.2 odd 2 inner
735.2.j.g.197.4 24 105.62 odd 4
735.2.j.g.197.9 24 35.27 even 4
735.2.j.g.638.4 24 7.6 odd 2
735.2.j.g.638.9 24 21.20 even 2
735.2.y.i.128.4 48 21.11 odd 6
735.2.y.i.128.9 48 7.4 even 3
735.2.y.i.263.4 48 7.2 even 3
735.2.y.i.263.9 48 21.2 odd 6
735.2.y.i.422.4 48 105.32 even 12
735.2.y.i.422.9 48 35.32 odd 12
735.2.y.i.557.4 48 35.2 odd 12
735.2.y.i.557.9 48 105.2 even 12