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Properties

Label 975.2.bw.a
Level $975$
Weight $2$
Character orbit 975.bw
Analytic conductor $7.785$
Analytic rank $0$
Dimension $288$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [975,2,Mod(16,975)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(975, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([0, 6, 10]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("975.16");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 975.bw (of order \(15\), degree \(8\), 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: \(7.78541419707\)
Analytic rank: \(0\)
Dimension: \(288\)
Relative dimension: \(36\) over \(\Q(\zeta_{15})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 288 q + 2 q^{2} - 36 q^{3} + 44 q^{4} + 12 q^{5} - 2 q^{6} - 12 q^{8} + 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 288 q + 2 q^{2} - 36 q^{3} + 44 q^{4} + 12 q^{5} - 2 q^{6} - 12 q^{8} + 36 q^{9} + 9 q^{10} + 2 q^{11} + 68 q^{12} - 20 q^{14} - 9 q^{15} + 28 q^{16} - 16 q^{17} + 16 q^{18} - 6 q^{19} + 30 q^{20} + 17 q^{22} - 17 q^{23} + 24 q^{24} - 4 q^{25} - 94 q^{26} + 72 q^{27} - 15 q^{29} + 11 q^{30} - 56 q^{32} + 3 q^{33} - 130 q^{34} - 9 q^{35} + 34 q^{36} + 18 q^{37} + 52 q^{38} - 6 q^{39} - 16 q^{40} + 11 q^{41} - 10 q^{42} + 58 q^{43} - 12 q^{44} - q^{45} - 60 q^{46} + 30 q^{47} - 28 q^{48} - 168 q^{49} + 36 q^{50} + 48 q^{51} - 70 q^{52} + 28 q^{53} - 2 q^{54} - 7 q^{55} + 14 q^{56} + 8 q^{57} + 24 q^{58} - 21 q^{59} + 10 q^{60} + 4 q^{61} - 24 q^{62} - 32 q^{64} - 57 q^{65} - 56 q^{66} + 62 q^{68} + 2 q^{69} + 56 q^{70} + 90 q^{71} + 6 q^{72} + 80 q^{73} - 52 q^{74} + 3 q^{75} - 90 q^{76} + 4 q^{77} - 8 q^{78} + 8 q^{79} + 17 q^{80} + 36 q^{81} + 62 q^{82} + 76 q^{83} - 9 q^{85} + 8 q^{86} - 10 q^{87} - 54 q^{88} - 78 q^{89} + 2 q^{90} + 134 q^{91} + 128 q^{92} + 18 q^{94} - 57 q^{95} + 28 q^{96} - 6 q^{97} - 58 q^{98} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
16.1 −0.287984 + 2.73999i −0.669131 + 0.743145i −5.46831 − 1.16233i −2.21899 + 0.275871i −1.84351 − 2.04742i −2.06247 + 3.57230i 3.05681 − 9.40790i −0.104528 − 0.994522i −0.116849 − 6.15944i
16.2 −0.284011 + 2.70218i −0.669131 + 0.743145i −5.26482 − 1.11907i 1.39196 + 1.74998i −1.81807 − 2.01917i 1.09292 − 1.89299i 2.83996 − 8.74050i −0.104528 − 0.994522i −5.12410 + 3.26433i
16.3 −0.260934 + 2.48263i −0.669131 + 0.743145i −4.13905 − 0.879782i 1.33127 − 1.79658i −1.67035 − 1.85511i 0.776807 − 1.34547i 1.72139 − 5.29789i −0.104528 − 0.994522i 4.11287 + 3.77384i
16.4 −0.252846 + 2.40567i −0.669131 + 0.743145i −3.76701 − 0.800703i −1.33518 − 1.79368i −1.61857 − 1.79761i 0.736661 − 1.27593i 1.38372 − 4.25866i −0.104528 − 0.994522i 4.65260 − 2.75847i
16.5 −0.242144 + 2.30385i −0.669131 + 0.743145i −3.29277 − 0.699901i 0.00109140 + 2.23607i −1.55006 − 1.72152i 0.326543 − 0.565589i 0.978091 − 3.01026i −0.104528 − 0.994522i −5.15182 − 0.538936i
16.6 −0.224635 + 2.13726i −0.669131 + 0.743145i −2.56111 − 0.544381i 0.306075 − 2.21502i −1.43798 − 1.59704i −1.63488 + 2.83169i 0.410622 − 1.26376i −0.104528 − 0.994522i 4.66531 + 1.15173i
16.7 −0.212818 + 2.02483i −0.669131 + 0.743145i −2.09833 − 0.446014i 2.08053 + 0.819387i −1.36234 − 1.51303i −2.45007 + 4.24364i 0.0913590 − 0.281174i −0.104528 − 0.994522i −2.10189 + 4.03833i
16.8 −0.211029 + 2.00781i −0.669131 + 0.743145i −2.03046 − 0.431589i −1.95268 + 1.08952i −1.35089 − 1.50031i 1.08740 − 1.88342i 0.0473054 − 0.145591i −0.104528 − 0.994522i −1.77548 − 4.15052i
16.9 −0.167165 + 1.59047i −0.669131 + 0.743145i −0.545345 − 0.115917i 0.519099 + 2.17498i −1.07009 − 1.18846i −0.580779 + 1.00594i −0.712853 + 2.19394i −0.104528 − 0.994522i −3.54601 + 0.462030i
16.10 −0.161419 + 1.53580i −0.669131 + 0.743145i −0.376315 − 0.0799883i 1.88972 − 1.19540i −1.03331 − 1.14760i 1.38218 − 2.39400i −0.770812 + 2.37232i −0.104528 − 0.994522i 1.53085 + 3.09518i
16.11 −0.152978 + 1.45549i −0.669131 + 0.743145i −0.138742 − 0.0294906i −2.18801 − 0.461086i −0.979275 − 1.08759i 1.77019 − 3.06606i −0.840347 + 2.58632i −0.104528 − 0.994522i 1.00582 − 3.11409i
16.12 −0.140383 + 1.33565i −0.669131 + 0.743145i 0.192033 + 0.0408178i 2.03080 + 0.935869i −0.898649 − 0.998051i 0.459456 − 0.795801i −0.911503 + 2.80532i −0.104528 − 0.994522i −1.53509 + 2.58107i
16.13 −0.111234 + 1.05832i −0.669131 + 0.743145i 0.848624 + 0.180381i −1.79778 + 1.32966i −0.712056 − 0.790818i −2.14070 + 3.70780i −0.942978 + 2.90219i −0.104528 − 0.994522i −1.20723 − 2.05053i
16.14 −0.0833104 + 0.792645i −0.669131 + 0.743145i 1.33495 + 0.283752i 1.38715 − 1.75380i −0.533305 − 0.592295i −1.19638 + 2.07220i −0.828710 + 2.55051i −0.104528 − 0.994522i 1.27458 + 1.24563i
16.15 −0.0784571 + 0.746469i −0.669131 + 0.743145i 1.40523 + 0.298692i −1.78049 − 1.35273i −0.502237 − 0.557790i −0.345656 + 0.598694i −0.797099 + 2.45322i −0.104528 − 0.994522i 1.14946 − 1.22295i
16.16 −0.0493707 + 0.469731i −0.669131 + 0.743145i 1.73809 + 0.369441i −0.432432 + 2.19386i −0.316043 − 0.351001i 1.12760 − 1.95306i −0.551257 + 1.69660i −0.104528 − 0.994522i −1.00917 − 0.311439i
16.17 −0.0414933 + 0.394782i −0.669131 + 0.743145i 1.80216 + 0.383062i −1.26170 − 1.84611i −0.265616 − 0.294996i 0.621018 − 1.07564i −0.471336 + 1.45062i −0.104528 − 0.994522i 0.781163 − 0.421494i
16.18 −0.0375076 + 0.356861i −0.669131 + 0.743145i 1.83035 + 0.389053i 1.92596 + 1.13607i −0.240102 − 0.266660i 2.06316 − 3.57350i −0.429257 + 1.32112i −0.104528 − 0.994522i −0.477658 + 0.644690i
16.19 0.000460196 − 0.00437848i −0.669131 + 0.743145i 1.95628 + 0.415819i 2.21702 + 0.291247i 0.00294591 + 0.00327176i −1.84177 + 3.19004i 0.00544188 − 0.0167484i −0.104528 − 0.994522i 0.00229548 − 0.00957313i
16.20 0.0345987 − 0.329184i −0.669131 + 0.743145i 1.84913 + 0.393045i −2.23542 − 0.0538330i 0.221481 + 0.245979i −1.12801 + 1.95378i 0.397929 − 1.22470i −0.104528 − 0.994522i −0.0950635 + 0.734003i
See next 80 embeddings (of 288 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 16.36
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner
25.d even 5 1 inner
325.y even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 975.2.bw.a ✓ 288
13.c even 3 1 inner 975.2.bw.a ✓ 288
25.d even 5 1 inner 975.2.bw.a ✓ 288
325.y even 15 1 inner 975.2.bw.a ✓ 288
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
975.2.bw.a ✓ 288 1.a even 1 1 trivial
975.2.bw.a ✓ 288 13.c even 3 1 inner
975.2.bw.a ✓ 288 25.d even 5 1 inner
975.2.bw.a ✓ 288 325.y even 15 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{288} - 2 T_{2}^{287} - 56 T_{2}^{286} + 124 T_{2}^{285} + 1418 T_{2}^{284} + \cdots + 22\!\cdots\!25 \) acting on \(S_{2}^{\mathrm{new}}(975, [\chi])\). Copy content Toggle raw display