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Properties

Label 312.2.ba.a
Level $312$
Weight $2$
Character orbit 312.ba
Analytic conductor $2.491$
Analytic rank $0$
Dimension $104$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [312,2,Mod(179,312)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(312, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 3, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("312.179");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 312 = 2^{3} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 312.ba (of order \(6\), degree \(2\), 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: \(2.49133254306\)
Analytic rank: \(0\)
Dimension: \(104\)
Relative dimension: \(52\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 104 q - 2 q^{3} - 2 q^{4} - 12 q^{6} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 104 q - 2 q^{3} - 2 q^{4} - 12 q^{6} - 2 q^{9} + 6 q^{10} + 8 q^{12} - 6 q^{16} - 12 q^{19} + 8 q^{22} - 36 q^{24} - 88 q^{25} + 16 q^{27} + 12 q^{28} + 14 q^{30} - 6 q^{33} + 22 q^{36} + 4 q^{40} + 26 q^{42} - 20 q^{43} - 36 q^{46} + 14 q^{48} - 24 q^{49} - 20 q^{51} - 16 q^{52} - 36 q^{54} - 66 q^{58} + 28 q^{64} + 32 q^{66} - 12 q^{67} - 66 q^{72} + 2 q^{75} - 60 q^{76} - 14 q^{78} - 2 q^{81} + 50 q^{82} - 90 q^{84} + 24 q^{88} + 8 q^{90} + 44 q^{91} + 72 q^{94} + 12 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
179.1 −1.40512 − 0.160093i −1.23972 − 1.20959i 1.94874 + 0.449901i − 3.72006i 1.54831 + 1.89809i 1.77405 − 3.07274i −2.66619 − 0.944145i 0.0737946 + 2.99909i −0.595556 + 5.22715i
179.2 −1.39956 + 0.203080i −0.556132 − 1.64034i 1.91752 − 0.568443i 2.03571i 1.11146 + 2.18281i −0.0498708 + 0.0863787i −2.56823 + 1.18498i −2.38143 + 1.82449i −0.413412 − 2.84910i
179.3 −1.39357 + 0.240739i 0.544462 + 1.64425i 1.88409 − 0.670974i 2.53880i −1.15458 − 2.16031i −1.43882 + 2.49211i −2.46409 + 1.38862i −2.40712 + 1.79046i −0.611187 − 3.53800i
179.4 −1.39203 − 0.249521i 1.72978 + 0.0887168i 1.87548 + 0.694679i − 0.672388i −2.38576 − 0.555111i −2.06840 + 3.58258i −2.43738 − 1.43498i 2.98426 + 0.306920i −0.167775 + 0.935983i
179.5 −1.39055 − 0.257614i 1.60008 − 0.663134i 1.86727 + 0.716452i 2.98497i −2.39583 + 0.509920i 2.08540 − 3.61202i −2.41197 − 1.47730i 2.12051 − 2.12213i 0.768971 − 4.15076i
179.6 −1.29488 + 0.568578i 1.66200 + 0.487588i 1.35344 − 1.47248i − 3.32967i −2.42933 + 0.313610i 0.774008 − 1.34062i −0.915323 + 2.67623i 2.52452 + 1.62075i 1.89318 + 4.31153i
179.7 −1.29342 − 0.571888i −1.54136 + 0.790061i 1.34589 + 1.47939i − 1.36930i 2.44546 − 0.140396i −0.577755 + 1.00070i −0.894761 − 2.68317i 1.75161 − 2.43554i −0.783088 + 1.77109i
179.8 −1.27374 + 0.614471i −1.72431 + 0.163538i 1.24485 − 1.56536i − 0.119305i 2.09584 − 1.26785i −0.682422 + 1.18199i −0.623752 + 2.75879i 2.94651 − 0.563982i 0.0733092 + 0.151963i
179.9 −1.24853 − 0.664211i 0.249885 − 1.71393i 1.11765 + 1.65857i − 0.0944148i −1.45040 + 1.97392i −1.24893 + 2.16322i −0.293774 − 2.81313i −2.87512 − 0.856571i −0.0627114 + 0.117880i
179.10 −1.16902 + 0.795859i 1.00378 − 1.41153i 0.733216 − 1.86075i − 0.119305i −0.0500653 + 2.44898i 0.682422 − 1.18199i 0.623752 + 2.75879i −0.984833 − 2.83374i 0.0949496 + 0.139469i
179.11 −1.13984 + 0.837112i −0.408739 + 1.68313i 0.598488 − 1.90835i − 3.32967i −0.943071 − 2.26067i −0.774008 + 1.34062i 0.915323 + 2.67623i −2.66587 − 1.37592i 2.78730 + 3.79530i
179.12 −1.06768 − 0.927391i 1.27233 + 1.17524i 0.279891 + 1.98032i − 0.964063i −0.268532 − 2.43473i 1.25912 − 2.18086i 1.53769 − 2.37392i 0.237622 + 2.99057i −0.894063 + 1.02931i
179.13 −1.03244 − 0.966472i −0.535453 + 1.64721i 0.131862 + 1.99565i 3.02009i 2.14480 − 1.18314i −0.0230459 + 0.0399166i 1.79260 − 2.18783i −2.42658 − 1.76400i 2.91884 − 3.11806i
179.14 −0.905272 + 1.08650i 1.15173 + 1.29364i −0.360965 − 1.96716i 2.53880i −2.44817 + 0.0802577i 1.43882 − 2.49211i 2.46409 + 1.38862i −0.347026 + 2.97986i −2.75840 − 2.29830i
179.15 −0.875650 + 1.11051i −1.14251 − 1.30180i −0.466472 − 1.94484i 2.03571i 2.44610 − 0.128853i 0.0498708 − 0.0863787i 2.56823 + 1.18498i −0.389341 + 2.97463i −2.26068 − 1.78257i
179.16 −0.855967 − 1.12575i 1.20400 − 1.24514i −0.534639 + 1.92722i − 4.00781i −2.43231 − 0.289611i 0.756552 − 1.31039i 2.62720 − 1.04776i −0.100751 − 2.99831i −4.51181 + 3.43056i
179.17 −0.794649 − 1.16984i −1.56819 − 0.735386i −0.737066 + 1.85923i 1.76397i 0.385871 + 2.41891i 2.29917 − 3.98227i 2.76072 − 0.615182i 1.91842 + 2.30644i 2.06356 − 1.40173i
179.18 −0.615790 − 1.27311i −1.56819 − 0.735386i −1.24161 + 1.56793i − 1.76397i 0.0294471 + 2.44931i −2.29917 + 3.98227i 2.76072 + 0.615182i 1.91842 + 2.30644i −2.24572 + 1.08623i
179.19 −0.563917 + 1.29692i −0.427675 − 1.67842i −1.36400 − 1.46271i − 3.72006i 2.41795 + 0.391829i −1.77405 + 3.07274i 2.66619 − 0.944145i −2.63419 + 1.43564i 4.82462 + 2.09781i
179.20 −0.546947 − 1.30417i 1.20400 − 1.24514i −1.40170 + 1.42662i 4.00781i −2.28240 − 0.889195i −0.756552 + 1.31039i 2.62720 + 1.04776i −0.100751 − 2.99831i 5.22685 − 2.19206i
See next 80 embeddings (of 104 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 179.52
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.d odd 2 1 inner
13.e even 6 1 inner
24.f even 2 1 inner
39.h odd 6 1 inner
104.p odd 6 1 inner
312.ba even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 312.2.ba.a ✓ 104
3.b odd 2 1 inner 312.2.ba.a ✓ 104
8.d odd 2 1 inner 312.2.ba.a ✓ 104
13.e even 6 1 inner 312.2.ba.a ✓ 104
24.f even 2 1 inner 312.2.ba.a ✓ 104
39.h odd 6 1 inner 312.2.ba.a ✓ 104
104.p odd 6 1 inner 312.2.ba.a ✓ 104
312.ba even 6 1 inner 312.2.ba.a ✓ 104
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
312.2.ba.a ✓ 104 1.a even 1 1 trivial
312.2.ba.a ✓ 104 3.b odd 2 1 inner
312.2.ba.a ✓ 104 8.d odd 2 1 inner
312.2.ba.a ✓ 104 13.e even 6 1 inner
312.2.ba.a ✓ 104 24.f even 2 1 inner
312.2.ba.a ✓ 104 39.h odd 6 1 inner
312.2.ba.a ✓ 104 104.p odd 6 1 inner
312.2.ba.a ✓ 104 312.ba even 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(312, [\chi])\).