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Properties

Label 294.2.p.a
Level $294$
Weight $2$
Character orbit 294.p
Analytic conductor $2.348$
Analytic rank $0$
Dimension $216$
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] = [294,2,Mod(5,294)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(294, base_ring=CyclotomicField(42))
 
chi = DirichletCharacter(H, H._module([21, 29]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("294.5");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 294 = 2 \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 294.p (of order \(42\), degree \(12\), 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: \(2.34760181943\)
Analytic rank: \(0\)
Dimension: \(216\)
Relative dimension: \(18\) over \(\Q(\zeta_{42})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{42}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 216 q - 18 q^{4} + 14 q^{6} + 10 q^{7} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 216 q - 18 q^{4} + 14 q^{6} + 10 q^{7} + 16 q^{9} + 6 q^{10} - 12 q^{15} + 18 q^{16} - 4 q^{18} - 12 q^{19} + 10 q^{22} - 6 q^{24} + 16 q^{25} - 42 q^{27} + 2 q^{28} + 4 q^{30} + 6 q^{31} + 18 q^{33} - 10 q^{36} - 36 q^{37} + 4 q^{39} - 22 q^{40} + 20 q^{42} + 40 q^{43} - 14 q^{45} - 156 q^{46} - 134 q^{49} - 12 q^{51} + 16 q^{52} - 18 q^{54} - 322 q^{55} - 34 q^{57} - 164 q^{58} - 6 q^{60} + 56 q^{61} - 24 q^{63} + 36 q^{64} - 28 q^{69} - 4 q^{70} + 4 q^{72} - 24 q^{73} + 84 q^{75} + 16 q^{78} + 22 q^{79} - 52 q^{81} + 24 q^{82} + 42 q^{84} + 32 q^{85} - 94 q^{87} - 2 q^{88} + 42 q^{90} - 100 q^{91} - 266 q^{93} - 88 q^{94} - 6 q^{96} + 224 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 −0.149042 − 0.988831i −1.68431 − 0.403867i −0.955573 + 0.294755i 0.188368 − 2.51360i −0.148323 + 1.72569i 1.37468 − 2.26059i 0.433884 + 0.900969i 2.67378 + 1.36047i −2.51360 + 0.188368i
5.2 −0.149042 − 0.988831i −1.35483 + 1.07909i −0.955573 + 0.294755i 0.133215 − 1.77763i 1.26896 + 1.17887i 0.839187 + 2.50914i 0.433884 + 0.900969i 0.671134 − 2.92397i −1.77763 + 0.133215i
5.3 −0.149042 − 0.988831i −1.06987 − 1.36212i −0.955573 + 0.294755i −0.248265 + 3.31287i −1.18746 + 1.26093i 2.60990 + 0.434057i 0.433884 + 0.900969i −0.710768 + 2.91459i 3.31287 − 0.248265i
5.4 −0.149042 − 0.988831i −0.406691 + 1.68363i −0.955573 + 0.294755i −0.0790178 + 1.05442i 1.72544 + 0.151217i −2.58461 − 0.565480i 0.433884 + 0.900969i −2.66920 − 1.36943i 1.05442 − 0.0790178i
5.5 −0.149042 − 0.988831i 0.531113 − 1.64861i −0.955573 + 0.294755i −0.0742306 + 0.990539i −1.70936 − 0.279468i −1.28065 − 2.31515i 0.433884 + 0.900969i −2.43584 − 1.75120i 0.990539 − 0.0742306i
5.6 −0.149042 − 0.988831i 0.982113 + 1.42669i −0.955573 + 0.294755i −0.100922 + 1.34671i 1.26438 − 1.18378i 1.03285 + 2.43582i 0.433884 + 0.900969i −1.07091 + 2.80235i 1.34671 − 0.100922i
5.7 −0.149042 − 0.988831i 1.28858 + 1.15740i −0.955573 + 0.294755i 0.0213134 − 0.284407i 0.952416 − 1.44669i 1.20478 − 2.35552i 0.433884 + 0.900969i 0.320868 + 2.98279i −0.284407 + 0.0213134i
5.8 −0.149042 − 0.988831i 1.65617 − 0.507046i −0.955573 + 0.294755i −0.273625 + 3.65128i −0.748222 − 1.56210i −1.75726 + 1.97789i 0.433884 + 0.900969i 2.48581 − 1.67951i 3.65128 − 0.273625i
5.9 −0.149042 − 0.988831i 1.70558 − 0.301667i −0.955573 + 0.294755i 0.294036 − 3.92364i −0.552501 − 1.64157i −2.61113 − 0.426617i 0.433884 + 0.900969i 2.81799 − 1.02903i −3.92364 + 0.294036i
5.10 0.149042 + 0.988831i −1.68688 − 0.392993i −0.955573 + 0.294755i 0.100922 − 1.34671i 0.137188 − 1.72661i 1.03285 + 2.43582i −0.433884 − 0.900969i 2.69111 + 1.32586i 1.34671 − 0.100922i
5.11 0.149042 + 0.988831i −1.54816 − 0.776659i −0.955573 + 0.294755i −0.0213134 + 0.284407i 0.537243 − 1.64662i 1.20478 − 2.35552i −0.433884 − 0.900969i 1.79360 + 2.40479i −0.284407 + 0.0213134i
5.12 0.149042 + 0.988831i −1.41866 + 0.993676i −0.955573 + 0.294755i 0.0790178 − 1.05442i −1.19402 − 1.25472i −2.58461 − 0.565480i −0.433884 − 0.900969i 1.02522 − 2.81939i 1.05442 − 0.0790178i
5.13 0.149042 + 0.988831i −0.509520 + 1.65541i −0.955573 + 0.294755i −0.133215 + 1.77763i −1.71286 − 0.257103i 0.839187 + 2.50914i −0.433884 − 0.900969i −2.48078 − 1.68693i −1.77763 + 0.133215i
5.14 0.149042 + 0.988831i −0.342303 − 1.69789i −0.955573 + 0.294755i −0.294036 + 3.92364i 1.62791 − 0.591537i −2.61113 − 0.426617i −0.433884 − 0.900969i −2.76566 + 1.16239i −3.92364 + 0.294036i
5.15 0.149042 + 0.988831i −0.133072 − 1.72693i −0.955573 + 0.294755i 0.273625 − 3.65128i 1.68781 − 0.388971i −1.75726 + 1.97789i −0.433884 − 0.900969i −2.96458 + 0.459611i 3.65128 − 0.273625i
5.16 0.149042 + 0.988831i 0.991296 + 1.42033i −0.955573 + 0.294755i −0.188368 + 2.51360i −1.25672 + 1.19191i 1.37468 − 2.26059i −0.433884 − 0.900969i −1.03467 + 2.81593i −2.51360 + 0.188368i
5.17 0.149042 + 0.988831i 1.34061 − 1.09670i −0.955573 + 0.294755i 0.0742306 − 0.990539i 1.28426 + 1.16218i −1.28065 − 2.31515i −0.433884 − 0.900969i 0.594480 − 2.94051i 0.990539 − 0.0742306i
5.18 0.149042 + 0.988831i 1.65883 + 0.498271i −0.955573 + 0.294755i 0.248265 − 3.31287i −0.245470 + 1.71457i 2.60990 + 0.434057i −0.433884 − 0.900969i 2.50345 + 1.65310i 3.31287 − 0.248265i
17.1 −0.997204 + 0.0747301i −1.67956 − 0.423173i 0.988831 − 0.149042i 0.523430 − 0.485672i 1.70649 + 0.296476i −1.79210 + 1.94638i −0.974928 + 0.222521i 2.64185 + 1.42149i −0.485672 + 0.523430i
17.2 −0.997204 + 0.0747301i −1.62662 + 0.595065i 0.988831 − 0.149042i −0.806451 + 0.748277i 1.57760 − 0.714958i 1.34190 − 2.28020i −0.974928 + 0.222521i 2.29180 − 1.93589i 0.748277 − 0.806451i
See next 80 embeddings (of 216 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
49.h odd 42 1 inner
147.o even 42 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 294.2.p.a ✓ 216
3.b odd 2 1 inner 294.2.p.a ✓ 216
49.h odd 42 1 inner 294.2.p.a ✓ 216
147.o even 42 1 inner 294.2.p.a ✓ 216
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
294.2.p.a ✓ 216 1.a even 1 1 trivial
294.2.p.a ✓ 216 3.b odd 2 1 inner
294.2.p.a ✓ 216 49.h odd 42 1 inner
294.2.p.a ✓ 216 147.o even 42 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(294, [\chi])\).