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Properties

Label 374.2.h.b
Level $374$
Weight $2$
Character orbit 374.h
Analytic conductor $2.986$
Analytic rank $0$
Dimension $24$
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] = [374,2,Mod(111,374)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(374, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("374.111");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 374 = 2 \cdot 11 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 374.h (of order \(8\), degree \(4\), 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: \(2.98640503560\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(6\) over \(\Q(\zeta_{8})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 16 q^{5} - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 16 q^{5} - 8 q^{7} + 16 q^{10} - 8 q^{14} + 16 q^{15} - 24 q^{16} + 16 q^{17} + 32 q^{18} - 8 q^{19} - 8 q^{23} + 32 q^{25} + 8 q^{26} - 8 q^{28} - 16 q^{29} - 8 q^{31} - 8 q^{33} - 8 q^{34} - 16 q^{35} + 64 q^{39} + 16 q^{41} - 32 q^{42} - 16 q^{43} - 48 q^{46} - 32 q^{49} - 32 q^{51} + 8 q^{52} + 48 q^{54} - 8 q^{56} - 64 q^{57} + 16 q^{58} - 48 q^{59} - 16 q^{60} + 24 q^{61} - 8 q^{62} + 64 q^{63} - 16 q^{65} + 48 q^{67} - 16 q^{69} + 24 q^{70} + 40 q^{73} + 8 q^{76} + 16 q^{77} - 64 q^{78} - 48 q^{79} + 16 q^{80} + 8 q^{82} - 16 q^{84} + 48 q^{85} - 8 q^{86} - 88 q^{87} - 48 q^{90} - 8 q^{91} + 48 q^{92} + 16 q^{93} + 32 q^{94} + 24 q^{95} - 16 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
111.1 0.707107 − 0.707107i −2.16576 − 0.897089i − 1.00000i −0.980198 + 2.36641i −2.16576 + 0.897089i −0.970957 − 2.34410i −0.707107 − 0.707107i 1.76445 + 1.76445i 0.980198 + 2.36641i
111.2 0.707107 − 0.707107i −1.97516 − 0.818138i − 1.00000i −0.587320 + 1.41791i −1.97516 + 0.818138i 1.90762 + 4.60540i −0.707107 − 0.707107i 1.11059 + 1.11059i 0.587320 + 1.41791i
111.3 0.707107 − 0.707107i −0.794970 − 0.329287i − 1.00000i −0.789477 + 1.90597i −0.794970 + 0.329287i −1.03744 − 2.50459i −0.707107 − 0.707107i −1.59777 − 1.59777i 0.789477 + 1.90597i
111.4 0.707107 − 0.707107i −0.00227657 0.000942988i − 1.00000i 0.633269 − 1.52885i −0.00227657 0.000942988i −0.564628 − 1.36313i −0.707107 − 0.707107i −2.12132 − 2.12132i −0.633269 − 1.52885i
111.5 0.707107 − 0.707107i 1.84625 + 0.764742i − 1.00000i 0.632498 − 1.52698i 1.84625 − 0.764742i 0.360259 + 0.869742i −0.707107 − 0.707107i 0.702489 + 0.702489i −0.632498 − 1.52698i
111.6 0.707107 − 0.707107i 3.09192 + 1.28072i − 1.00000i −0.0803445 + 0.193969i 3.09192 − 1.28072i −1.69486 − 4.09175i −0.707107 − 0.707107i 5.79842 + 5.79842i 0.0803445 + 0.193969i
155.1 0.707107 + 0.707107i −2.16576 + 0.897089i 1.00000i −0.980198 − 2.36641i −2.16576 − 0.897089i −0.970957 + 2.34410i −0.707107 + 0.707107i 1.76445 − 1.76445i 0.980198 − 2.36641i
155.2 0.707107 + 0.707107i −1.97516 + 0.818138i 1.00000i −0.587320 − 1.41791i −1.97516 − 0.818138i 1.90762 − 4.60540i −0.707107 + 0.707107i 1.11059 − 1.11059i 0.587320 − 1.41791i
155.3 0.707107 + 0.707107i −0.794970 + 0.329287i 1.00000i −0.789477 − 1.90597i −0.794970 − 0.329287i −1.03744 + 2.50459i −0.707107 + 0.707107i −1.59777 + 1.59777i 0.789477 − 1.90597i
155.4 0.707107 + 0.707107i −0.00227657 0.000942988i 1.00000i 0.633269 + 1.52885i −0.00227657 0.000942988i −0.564628 + 1.36313i −0.707107 + 0.707107i −2.12132 + 2.12132i −0.633269 + 1.52885i
155.5 0.707107 + 0.707107i 1.84625 − 0.764742i 1.00000i 0.632498 + 1.52698i 1.84625 + 0.764742i 0.360259 − 0.869742i −0.707107 + 0.707107i 0.702489 − 0.702489i −0.632498 + 1.52698i
155.6 0.707107 + 0.707107i 3.09192 − 1.28072i 1.00000i −0.0803445 − 0.193969i 3.09192 + 1.28072i −1.69486 + 4.09175i −0.707107 + 0.707107i 5.79842 − 5.79842i 0.0803445 − 0.193969i
287.1 −0.707107 − 0.707107i −1.16623 − 2.81552i 1.00000i −1.80165 + 0.746270i −1.16623 + 2.81552i −1.23865 − 0.513067i 0.707107 − 0.707107i −4.44575 + 4.44575i 1.80165 + 0.746270i
287.2 −0.707107 − 0.707107i −0.564522 − 1.36288i 1.00000i −2.51247 + 1.04070i −0.564522 + 1.36288i 4.18860 + 1.73498i 0.707107 − 0.707107i 0.582575 − 0.582575i 2.51247 + 1.04070i
287.3 −0.707107 − 0.707107i −0.351229 − 0.847941i 1.00000i 2.94870 − 1.22139i −0.351229 + 0.847941i −2.86069 − 1.18494i 0.707107 − 0.707107i 1.52568 − 1.52568i −2.94870 − 1.22139i
287.4 −0.707107 − 0.707107i 0.312459 + 0.754343i 1.00000i −3.79252 + 1.57091i 0.312459 − 0.754343i −0.684183 − 0.283398i 0.707107 − 0.707107i 1.64992 − 1.64992i 3.79252 + 1.57091i
287.5 −0.707107 − 0.707107i 0.471084 + 1.13730i 1.00000i −0.0504796 + 0.0209093i 0.471084 − 1.13730i 0.764323 + 0.316593i 0.707107 − 0.707107i 1.04980 − 1.04980i 0.0504796 + 0.0209093i
287.6 −0.707107 − 0.707107i 1.29843 + 3.13470i 1.00000i −1.61999 + 0.671024i 1.29843 − 3.13470i −2.16940 − 0.898594i 0.707107 − 0.707107i −6.01907 + 6.01907i 1.61999 + 0.671024i
331.1 −0.707107 + 0.707107i −1.16623 + 2.81552i − 1.00000i −1.80165 − 0.746270i −1.16623 − 2.81552i −1.23865 + 0.513067i 0.707107 + 0.707107i −4.44575 − 4.44575i 1.80165 − 0.746270i
331.2 −0.707107 + 0.707107i −0.564522 + 1.36288i − 1.00000i −2.51247 − 1.04070i −0.564522 − 1.36288i 4.18860 − 1.73498i 0.707107 + 0.707107i 0.582575 + 0.582575i 2.51247 − 1.04070i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 111.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.d even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 374.2.h.b ✓ 24
17.d even 8 1 inner 374.2.h.b ✓ 24
17.e odd 16 1 6358.2.a.bl 12
17.e odd 16 1 6358.2.a.bo 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
374.2.h.b ✓ 24 1.a even 1 1 trivial
374.2.h.b ✓ 24 17.d even 8 1 inner
6358.2.a.bl 12 17.e odd 16 1
6358.2.a.bo 12 17.e odd 16 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{24} - 16 T_{3}^{19} - 144 T_{3}^{18} + 1416 T_{3}^{17} + 10593 T_{3}^{16} + 13424 T_{3}^{15} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(374, [\chi])\). Copy content Toggle raw display