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Properties

Label 304.3.l.a
Level $304$
Weight $3$
Character orbit 304.l
Analytic conductor $8.283$
Analytic rank $0$
Dimension $144$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [304,3,Mod(115,304)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(304, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("304.115");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 304 = 2^{4} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 304.l (of order \(4\), 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: \(8.28340003655\)
Analytic rank: \(0\)
Dimension: \(144\)
Relative dimension: \(72\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 144 q + 12 q^{4} + 12 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 144 q + 12 q^{4} + 12 q^{6} + 32 q^{11} - 108 q^{12} - 44 q^{14} + 20 q^{16} + 160 q^{18} + 80 q^{20} + 100 q^{22} - 128 q^{23} - 156 q^{24} - 48 q^{26} - 96 q^{27} - 120 q^{28} + 32 q^{29} + 104 q^{30} - 180 q^{32} + 160 q^{34} + 68 q^{36} - 96 q^{37} + 384 q^{39} - 264 q^{40} + 100 q^{42} + 224 q^{43} + 228 q^{44} - 236 q^{48} + 1008 q^{49} + 296 q^{50} - 160 q^{51} - 364 q^{52} - 160 q^{53} + 56 q^{54} - 512 q^{55} - 196 q^{56} + 432 q^{58} - 416 q^{59} + 160 q^{60} - 84 q^{62} - 32 q^{65} + 92 q^{66} + 160 q^{67} + 216 q^{68} - 564 q^{70} + 512 q^{71} + 204 q^{72} + 88 q^{74} + 384 q^{75} + 224 q^{77} - 284 q^{78} - 472 q^{80} - 1296 q^{81} - 916 q^{82} - 480 q^{83} - 92 q^{84} - 764 q^{86} - 8 q^{88} + 1144 q^{90} - 576 q^{91} + 136 q^{92} + 96 q^{93} - 744 q^{94} - 328 q^{96} - 304 q^{98} + 608 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
115.1 −1.99891 + 0.0661024i −0.749402 + 0.749402i 3.99126 − 0.264265i −4.62753 + 4.62753i 1.44845 − 1.54752i −0.557993 −7.96069 + 0.792074i 7.87679i 8.94412 − 9.55590i
115.2 −1.98510 + 0.243651i −1.17323 + 1.17323i 3.88127 − 0.967346i 3.21562 − 3.21562i 2.04312 − 2.61484i −5.63419 −7.46902 + 2.86596i 6.24707i −5.59984 + 7.16682i
115.3 −1.97990 − 0.282828i 2.75234 − 2.75234i 3.84002 + 1.11994i 4.02339 − 4.02339i −6.22780 + 4.67093i −5.48220 −7.28610 − 3.30344i − 6.15077i −9.10383 + 6.82798i
115.4 −1.97844 + 0.292898i 3.98995 − 3.98995i 3.82842 − 1.15896i −0.180669 + 0.180669i −6.72521 + 9.06250i 9.82489 −7.23483 + 3.41427i − 22.8393i 0.304525 − 0.410360i
115.5 −1.97441 + 0.318936i −3.36638 + 3.36638i 3.79656 − 1.25942i 5.37031 − 5.37031i 5.57294 − 7.72025i 11.0487 −7.09428 + 3.69746i − 13.6650i −8.89038 + 12.3159i
115.6 −1.96718 − 0.360858i 2.76750 − 2.76750i 3.73956 + 1.41974i −3.38706 + 3.38706i −6.44284 + 4.44549i −8.76091 −6.84405 − 4.14233i − 6.31816i 7.88520 − 5.44070i
115.7 −1.89680 + 0.634148i 1.88633 − 1.88633i 3.19571 − 2.40571i −5.35492 + 5.35492i −2.38179 + 4.77422i 4.16803 −4.53606 + 6.58970i 1.88349i 6.76140 − 13.5530i
115.8 −1.89583 − 0.637057i 0.652274 − 0.652274i 3.18832 + 2.41550i 1.73638 − 1.73638i −1.65214 + 0.821063i 10.8838 −4.50568 − 6.61051i 8.14908i −4.39805 + 2.18570i
115.9 −1.89056 − 0.652512i −3.96874 + 3.96874i 3.14846 + 2.46723i −2.57117 + 2.57117i 10.0928 − 4.91351i −4.99221 −4.34246 − 6.71886i − 22.5019i 6.53869 − 3.18324i
115.10 −1.87606 − 0.693099i −1.41010 + 1.41010i 3.03923 + 2.60060i −1.30860 + 1.30860i 3.62278 − 1.66810i −0.871881 −3.89931 − 6.98537i 5.02323i 3.36199 − 1.54802i
115.11 −1.87095 + 0.706793i −3.17442 + 3.17442i 3.00089 − 2.64474i −4.41300 + 4.41300i 3.69552 − 8.18284i 11.1716 −3.74522 + 7.06918i − 11.1539i 5.13741 − 11.3756i
115.12 −1.81058 + 0.849596i 1.42166 − 1.42166i 2.55637 − 3.07652i 5.60916 − 5.60916i −1.36618 + 3.78185i 2.04687 −2.01471 + 7.74215i 4.95779i −5.39029 + 14.9213i
115.13 −1.77673 + 0.918278i −3.18694 + 3.18694i 2.31353 − 3.26306i −1.60761 + 1.60761i 2.73583 − 8.58883i −12.2886 −1.11412 + 7.92204i − 11.3132i 1.38006 − 4.33253i
115.14 −1.64678 + 1.13495i 2.61941 − 2.61941i 1.42376 − 3.73804i 1.10631 − 1.10631i −1.34068 + 7.28651i −10.9216 1.89788 + 7.77162i − 4.72265i −0.566235 + 3.07745i
115.15 −1.55804 + 1.25400i 0.281196 − 0.281196i 0.854956 − 3.90756i −0.510671 + 0.510671i −0.0854933 + 0.790735i 7.47988 3.56804 + 7.16024i 8.84186i 0.155261 − 1.43603i
115.16 −1.55516 − 1.25757i 0.808692 − 0.808692i 0.837034 + 3.91144i 1.99901 − 1.99901i −2.27463 + 0.240657i −10.1668 3.61719 − 7.13554i 7.69203i −5.62268 + 0.594882i
115.17 −1.53523 − 1.28183i −2.29618 + 2.29618i 0.713844 + 3.93579i 5.95759 − 5.95759i 6.46847 − 0.581855i −5.61715 3.94908 − 6.95735i − 1.54492i −16.7829 + 1.50966i
115.18 −1.50417 + 1.31813i −1.57317 + 1.57317i 0.525077 − 3.96539i 0.368050 − 0.368050i 0.292681 − 4.43996i 0.284809 4.43708 + 6.65675i 4.05028i −0.0684740 + 1.03875i
115.19 −1.40499 − 1.42338i −2.55784 + 2.55784i −0.0520112 + 3.99966i −0.294292 + 0.294292i 7.23452 + 0.0470365i 9.36042 5.76611 − 5.54545i − 4.08513i 0.832366 + 0.00541177i
115.20 −1.37235 + 1.45487i 2.50866 − 2.50866i −0.233315 − 3.99319i −3.34962 + 3.34962i 0.207026 + 7.09254i −2.87503 6.12978 + 5.14061i − 3.58674i −0.276426 − 9.47012i
See next 80 embeddings (of 144 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 115.72
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.f odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 304.3.l.a ✓ 144
16.f odd 4 1 inner 304.3.l.a ✓ 144
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
304.3.l.a ✓ 144 1.a even 1 1 trivial
304.3.l.a ✓ 144 16.f odd 4 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(304, [\chi])\).