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Properties

Label 325.2.bf.a
Level $325$
Weight $2$
Character orbit 325.bf
Analytic conductor $2.595$
Analytic rank $0$
Dimension $272$
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] = [325,2,Mod(9,325)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(325, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([21, 20]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("325.9");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 325 = 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 325.bf (of order \(30\), 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: \(2.59513806569\)
Analytic rank: \(0\)
Dimension: \(272\)
Relative dimension: \(34\) over \(\Q(\zeta_{30})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{30}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 272 q - 5 q^{2} - 5 q^{3} - 37 q^{4} - 14 q^{5} + 5 q^{6} - 20 q^{8} - 37 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 272 q - 5 q^{2} - 5 q^{3} - 37 q^{4} - 14 q^{5} + 5 q^{6} - 20 q^{8} - 37 q^{9} - 12 q^{10} - 3 q^{11} - 20 q^{12} - 10 q^{13} - 8 q^{14} - 41 q^{15} + 15 q^{16} - 5 q^{17} + 9 q^{19} - 14 q^{20} - 30 q^{21} + 20 q^{22} + 5 q^{23} - 34 q^{24} - 38 q^{25} - 36 q^{26} - 20 q^{27} - 45 q^{28} - 11 q^{29} - 9 q^{30} - 12 q^{31} - 65 q^{33} - 4 q^{34} + 10 q^{35} + 19 q^{36} + 30 q^{37} - 20 q^{38} + 32 q^{39} + 40 q^{40} + 4 q^{41} - 65 q^{42} - 20 q^{44} - 36 q^{45} - 15 q^{46} - 70 q^{47} - 50 q^{48} + 112 q^{49} - 19 q^{50} + 104 q^{51} - 5 q^{52} + 30 q^{53} - 49 q^{54} + 36 q^{55} + 40 q^{56} + 55 q^{58} + 3 q^{59} + 76 q^{60} - q^{61} + 105 q^{62} - 120 q^{63} - 68 q^{64} - 43 q^{65} - 64 q^{66} - 5 q^{67} - 24 q^{69} - 30 q^{70} + 13 q^{71} + 45 q^{72} + 140 q^{73} + 84 q^{74} + 68 q^{75} - 6 q^{76} - 210 q^{77} + 50 q^{78} + 36 q^{79} + 57 q^{80} + 77 q^{81} + 50 q^{83} - 18 q^{84} + 4 q^{85} + 5 q^{87} - 25 q^{88} + 12 q^{89} - 42 q^{90} + 31 q^{91} + 40 q^{92} + 41 q^{94} + 55 q^{95} - 146 q^{96} - 5 q^{97} + 230 q^{98} - 96 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1 −1.08614 − 2.43951i −0.0836430 + 0.393509i −3.43326 + 3.81302i −2.23371 + 0.102674i 1.05082 − 0.223359i −1.60880 + 0.928839i 7.95157 + 2.58362i 2.59278 + 1.15438i 2.67660 + 5.33765i
9.2 −1.07578 − 2.41624i −0.630159 + 2.96467i −3.34265 + 3.71238i 1.44256 + 1.70851i 7.84125 − 1.66671i −0.813962 + 0.469941i 7.53505 + 2.44829i −5.65151 − 2.51622i 2.57630 − 5.32355i
9.3 −0.980733 − 2.20276i −0.207114 + 0.974395i −2.55207 + 2.83436i 0.958926 − 2.02002i 2.34948 − 0.499398i 0.928062 − 0.535817i 4.15989 + 1.35163i 1.83409 + 0.816588i −5.39007 − 0.131190i
9.4 −0.935144 − 2.10037i 0.583028 − 2.74293i −2.19879 + 2.44200i −1.72131 − 1.42727i −6.30638 + 1.34046i 2.27250 − 1.31203i 2.81206 + 0.913694i −4.44311 − 1.97820i −1.38811 + 4.95008i
9.5 −0.838603 − 1.88353i 0.277555 − 1.30579i −1.50618 + 1.67278i 2.14650 + 0.626517i −2.69226 + 0.572258i 2.30500 − 1.33079i 0.492082 + 0.159887i 1.11258 + 0.495352i −0.619999 − 4.56841i
9.6 −0.818254 − 1.83783i −0.465132 + 2.18828i −1.36981 + 1.52133i −2.22668 − 0.204699i 4.40227 − 0.935732i 2.93634 − 1.69530i 0.0902186 + 0.0293138i −1.83157 − 0.815467i 1.44579 + 4.25975i
9.7 −0.787637 − 1.76906i 0.410310 − 1.93036i −1.17094 + 1.30046i −1.46856 + 1.68622i −3.73809 + 0.794556i −1.08320 + 0.625384i −0.460522 − 0.149633i −0.817281 − 0.363877i 4.13972 + 1.26984i
9.8 −0.760282 − 1.70762i −0.0416230 + 0.195821i −0.999682 + 1.11026i 0.716290 + 2.11824i 0.366034 − 0.0778028i −2.91192 + 1.68120i −0.899533 − 0.292276i 2.70402 + 1.20391i 3.07257 − 2.83361i
9.9 −0.571394 − 1.28337i −0.251022 + 1.18096i 0.0177072 − 0.0196658i −0.969290 − 2.01506i 1.65905 − 0.352642i −3.34800 + 1.93297i −2.70749 − 0.879718i 1.40897 + 0.627315i −2.03223 + 2.39536i
9.10 −0.569875 − 1.27996i 0.467794 − 2.20080i 0.0247209 − 0.0274553i 0.982627 − 2.00859i −3.08352 + 0.655422i −1.46113 + 0.843583i −2.71426 − 0.881917i −1.88405 − 0.838833i −3.13089 − 0.113077i
9.11 −0.518278 − 1.16407i −0.663127 + 3.11977i 0.251810 − 0.279663i 1.44987 − 1.70231i 3.97532 − 0.844980i 1.34935 − 0.779050i −2.87980 − 0.935702i −6.55258 − 2.91740i −2.73305 − 0.805485i
9.12 −0.379237 − 0.851781i −0.281208 + 1.32298i 0.756552 − 0.840236i 2.09537 + 0.780656i 1.23353 − 0.262195i 2.93782 − 1.69615i −2.77612 − 0.902016i 1.06944 + 0.476147i −0.129695 − 2.08085i
9.13 −0.309830 − 0.695889i −0.585324 + 2.75373i 0.949994 − 1.05508i −0.914574 + 2.04048i 2.09764 − 0.445868i −2.45008 + 1.41456i −2.47748 − 0.804981i −4.49981 − 2.00345i 1.70331 + 0.00424101i
9.14 −0.299023 − 0.671617i 0.00655474 − 0.0308376i 0.976607 − 1.08463i −1.55681 + 1.60510i −0.0226711 + 0.00481889i 2.29832 − 1.32693i −2.41887 − 0.785939i 2.73973 + 1.21981i 1.54353 + 0.565616i
9.15 −0.218809 − 0.491453i 0.655521 − 3.08398i 1.14461 − 1.27122i 0.259951 + 2.22091i −1.65907 + 0.352645i 2.89397 − 1.67083i −1.89846 − 0.616847i −6.34061 − 2.82302i 1.03459 − 0.613708i
9.16 −0.0907185 − 0.203757i 0.199975 − 0.940809i 1.30497 − 1.44932i 1.02896 − 1.98526i −0.209838 + 0.0446024i 0.973893 − 0.562277i −0.837941 − 0.272264i 1.89550 + 0.843933i −0.497855 − 0.0295585i
9.17 −0.0545824 − 0.122594i 0.544042 − 2.55952i 1.32621 − 1.47291i −2.22926 − 0.174390i −0.343476 + 0.0730082i −4.27875 + 2.47034i −0.508213 − 0.165128i −3.51451 − 1.56476i 0.100299 + 0.282812i
9.18 −0.0113474 − 0.0254867i 0.0637096 − 0.299730i 1.33774 − 1.48571i 2.23124 − 0.146843i −0.00836208 + 0.00177741i −2.24318 + 1.29510i −0.106112 − 0.0344780i 2.65486 + 1.18202i −0.0290614 − 0.0552007i
9.19 0.0883829 + 0.198511i −0.329204 + 1.54878i 1.30667 − 1.45120i −1.42622 − 1.72217i −0.336547 + 0.0715353i −0.193612 + 0.111782i 0.816891 + 0.265424i 0.450280 + 0.200477i 0.215817 − 0.435332i
9.20 0.259276 + 0.582343i 0.429006 − 2.01831i 1.06636 − 1.18431i −1.91547 − 1.15367i 1.28658 − 0.273471i 1.99853 − 1.15385i 2.17867 + 0.707892i −1.14891 − 0.511528i 0.175197 − 1.41458i
See next 80 embeddings (of 272 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.34
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner
25.e even 10 1 inner
325.bf even 30 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 325.2.bf.a ✓ 272
13.c even 3 1 inner 325.2.bf.a ✓ 272
25.e even 10 1 inner 325.2.bf.a ✓ 272
325.bf even 30 1 inner 325.2.bf.a ✓ 272
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
325.2.bf.a ✓ 272 1.a even 1 1 trivial
325.2.bf.a ✓ 272 13.c even 3 1 inner
325.2.bf.a ✓ 272 25.e even 10 1 inner
325.2.bf.a ✓ 272 325.bf even 30 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(325, [\chi])\).