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Properties

Label 325.3.bj.a
Level $325$
Weight $3$
Character orbit 325.bj
Analytic conductor $8.856$
Analytic rank $0$
Dimension $1088$
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] = [325,3,Mod(19,325)] 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("325.19"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(325, base_ring=CyclotomicField(60)) chi = DirichletCharacter(H, H._module([54, 25])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 325 = 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 325.bj (of order \(60\), degree \(16\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(8.85560859171\)
Analytic rank: \(0\)
Dimension: \(1088\)
Relative dimension: \(68\) over \(\Q(\zeta_{60})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{60}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 1088 q - 20 q^{2} - 10 q^{3} - 18 q^{4} - 18 q^{5} + 4 q^{6} - 20 q^{8} - 390 q^{9} + 6 q^{10} - 12 q^{11} - 20 q^{13} - 24 q^{14} - 86 q^{15} - 622 q^{16} - 30 q^{17} + 48 q^{19} + 302 q^{20} - 66 q^{21}+ \cdots - 720 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −2.13793 − 3.29212i 4.42665 − 0.465260i −4.64036 + 10.4224i 4.81603 − 1.34383i −10.9955 − 13.5784i −0.665892 − 0.178425i 28.7243 − 4.54948i 10.5755 − 2.24789i −14.7204 − 12.9819i
19.2 −2.13736 − 3.29125i −4.50226 + 0.473206i −4.63705 + 10.4150i −0.221393 − 4.99510i 11.1804 + 13.8066i 1.66192 + 0.445310i 28.6852 − 4.54329i 11.2431 − 2.38979i −15.9669 + 11.4050i
19.3 −2.06482 − 3.17954i −1.15209 + 0.121089i −4.21907 + 9.47618i −3.80579 + 3.24283i 2.76386 + 3.41308i −11.8498 − 3.17515i 23.8635 − 3.77961i −7.49069 + 1.59219i 18.1690 + 5.40482i
19.4 −2.02337 − 3.11572i 2.72002 − 0.285886i −3.98674 + 8.95437i −4.31233 − 2.53057i −6.39437 − 7.89639i 5.26272 + 1.41014i 21.2887 − 3.37179i −1.48653 + 0.315972i 0.840911 + 18.5563i
19.5 −1.93032 − 2.97244i −2.85898 + 0.300491i −3.48229 + 7.82135i 4.01013 + 2.98645i 6.41195 + 7.91810i 6.36766 + 1.70621i 15.9680 − 2.52909i −0.719837 + 0.153006i 1.13617 − 17.6847i
19.6 −1.81398 − 2.79329i 4.17381 − 0.438685i −2.88499 + 6.47980i −0.786356 + 4.93778i −8.79660 − 10.8629i −6.35792 − 1.70360i 10.1748 − 1.61154i 8.42491 − 1.79077i 15.2191 − 6.76053i
19.7 −1.78126 − 2.74290i −1.47868 + 0.155415i −2.72367 + 6.11747i 4.95940 + 0.635859i 3.06020 + 3.77903i −9.32440 − 2.49847i 8.71011 − 1.37955i −6.64100 + 1.41159i −7.08989 − 14.7358i
19.8 −1.74450 − 2.68629i −0.562532 + 0.0591245i −2.54595 + 5.71829i −4.74315 − 1.58194i 1.14016 + 1.40798i 4.46695 + 1.19691i 7.14800 − 1.13213i −8.49038 + 1.80469i 4.02486 + 15.5012i
19.9 −1.73303 − 2.66863i 1.19636 − 0.125742i −2.49124 + 5.59543i 3.14700 − 3.88540i −2.40888 − 2.97472i 3.94651 + 1.05746i 6.67832 − 1.05774i −7.38786 + 1.57034i −15.8225 − 1.66467i
19.10 −1.72578 − 2.65747i −5.21686 + 0.548314i −2.45688 + 5.51824i −3.01939 + 3.98539i 10.4603 + 12.9174i 3.45210 + 0.924988i 6.38594 − 1.01143i 18.1116 − 3.84975i 15.8018 + 1.14602i
19.11 −1.64461 − 2.53248i 2.69538 − 0.283296i −2.08176 + 4.67572i 2.08139 + 4.54619i −5.15030 − 6.36009i 3.97646 + 1.06549i 3.33501 − 0.528214i −1.61851 + 0.344025i 8.09006 − 12.7478i
19.12 −1.53468 − 2.36320i 5.60305 − 0.588904i −1.60252 + 3.59932i −4.70921 + 1.68027i −9.99058 − 12.3373i 8.90969 + 2.38734i −0.167142 + 0.0264727i 22.2440 − 4.72811i 11.1979 + 8.55013i
19.13 −1.52106 − 2.34222i 2.44488 − 0.256967i −1.54545 + 3.47113i −1.02744 − 4.89330i −4.32067 − 5.33559i −9.19047 − 2.46258i −0.552715 + 0.0875415i −2.89193 + 0.614700i −9.89841 + 9.84947i
19.14 −1.40324 − 2.16080i −2.45841 + 0.258389i −1.07303 + 2.41007i 1.78834 − 4.66924i 4.00807 + 4.94955i 12.3603 + 3.31194i −3.46555 + 0.548889i −2.82632 + 0.600753i −12.5988 + 2.68782i
19.15 −1.39254 − 2.14432i −3.90949 + 0.410904i −1.03200 + 2.31790i −3.35137 − 3.71057i 6.32522 + 7.81100i −6.77619 − 1.81567i −3.69390 + 0.585056i 6.31197 − 1.34165i −3.28974 + 12.3535i
19.16 −1.36802 − 2.10656i −1.64555 + 0.172955i −0.939185 + 2.10944i −3.92098 + 3.10257i 2.61548 + 3.22985i 1.06984 + 0.286663i −4.19495 + 0.664415i −6.12540 + 1.30199i 11.8997 + 4.01541i
19.17 −1.35842 − 2.09178i −5.36756 + 0.564154i −0.903303 + 2.02885i 4.27070 − 2.60021i 8.47149 + 10.4614i −8.03283 − 2.15239i −4.38285 + 0.694175i 19.6891 − 4.18506i −11.2405 − 5.40121i
19.18 −1.18444 − 1.82388i 5.72442 − 0.601661i −0.296697 + 0.666392i 1.52298 − 4.76241i −7.87761 − 9.72804i −0.608182 − 0.162962i −7.02499 + 1.11265i 23.6037 − 5.01711i −10.4900 + 2.86307i
19.19 −1.10993 − 1.70915i −3.48925 + 0.366735i −0.0622835 + 0.139891i 2.22150 + 4.47939i 4.49963 + 5.55658i 0.456377 + 0.122286i −7.74311 + 1.22639i 3.23702 − 0.688050i 5.19022 − 8.76869i
19.20 −1.08611 − 1.67246i 3.10040 − 0.325866i 0.00944822 − 0.0212211i 4.95826 + 0.644727i −3.91238 − 4.83139i 9.63613 + 2.58199i −7.92428 + 1.25508i 0.702988 − 0.149425i −4.30693 − 8.99275i
See next 80 embeddings (of 1088 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.68
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.f odd 12 1 inner
25.e even 10 1 inner
325.bj odd 60 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 325.3.bj.a ✓ 1088
13.f odd 12 1 inner 325.3.bj.a ✓ 1088
25.e even 10 1 inner 325.3.bj.a ✓ 1088
325.bj odd 60 1 inner 325.3.bj.a ✓ 1088
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
325.3.bj.a ✓ 1088 1.a even 1 1 trivial
325.3.bj.a ✓ 1088 13.f odd 12 1 inner
325.3.bj.a ✓ 1088 25.e even 10 1 inner
325.3.bj.a ✓ 1088 325.bj odd 60 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(325, [\chi])\).