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Properties

Label 430.3.m.b
Level $430$
Weight $3$
Character orbit 430.m
Analytic conductor $11.717$
Analytic rank $0$
Dimension $88$
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] = [430,3,Mod(307,430)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(430, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([3, 8]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("430.307");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 430 = 2 \cdot 5 \cdot 43 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 430.m (of order \(12\), 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: \(11.7166513675\)
Analytic rank: \(0\)
Dimension: \(88\)
Relative dimension: \(22\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 88 q + 88 q^{2} + 2 q^{3} - 4 q^{5} + 4 q^{6} + 18 q^{7} - 176 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 88 q + 88 q^{2} + 2 q^{3} - 4 q^{5} + 4 q^{6} + 18 q^{7} - 176 q^{8} - 4 q^{10} + 4 q^{12} + 8 q^{13} - 6 q^{15} - 352 q^{16} - 6 q^{17} - 140 q^{18} - 24 q^{21} - 24 q^{23} + 12 q^{25} + 16 q^{26} + 212 q^{27} - 36 q^{28} - 12 q^{30} - 12 q^{31} - 352 q^{32} - 32 q^{33} + 108 q^{35} - 280 q^{36} - 16 q^{37} + 96 q^{38} + 8 q^{40} - 104 q^{41} - 24 q^{42} - 198 q^{43} + 108 q^{45} - 48 q^{46} - 8 q^{47} - 8 q^{48} - 32 q^{50} + 16 q^{52} - 222 q^{53} + 44 q^{55} - 72 q^{56} - 134 q^{57} + 16 q^{58} - 12 q^{60} + 148 q^{61} - 12 q^{62} + 132 q^{63} + 28 q^{65} - 64 q^{66} - 134 q^{67} + 12 q^{68} + 8 q^{70} + 352 q^{71} - 280 q^{72} + 226 q^{73} + 456 q^{75} + 192 q^{76} + 168 q^{77} - 280 q^{78} + 16 q^{80} + 272 q^{81} - 104 q^{82} + 180 q^{83} - 632 q^{85} - 396 q^{86} - 424 q^{87} + 448 q^{90} + 212 q^{91} - 48 q^{92} - 192 q^{93} + 360 q^{95} - 16 q^{96} - 592 q^{97} + 208 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
307.1 1.00000 + 1.00000i −1.49081 − 5.56376i 2.00000i −3.43979 − 3.62875i 4.07296 − 7.05457i 0.744750 − 2.77944i −2.00000 + 2.00000i −20.9387 + 12.0890i 0.188964 − 7.06854i
307.2 1.00000 + 1.00000i −1.42587 − 5.32141i 2.00000i −1.67780 + 4.71010i 3.89554 − 6.74728i −0.0873114 + 0.325851i −2.00000 + 2.00000i −18.4901 + 10.6752i −6.38789 + 3.03230i
307.3 1.00000 + 1.00000i −1.22525 − 4.57269i 2.00000i 4.37707 + 2.41687i 3.34744 − 5.79793i −2.65913 + 9.92400i −2.00000 + 2.00000i −11.6140 + 6.70534i 1.96019 + 6.79394i
307.4 1.00000 + 1.00000i −1.00686 − 3.75767i 2.00000i 3.69254 − 3.37123i 2.75080 − 4.76453i 1.93592 − 7.22495i −2.00000 + 2.00000i −5.31205 + 3.06691i 7.06376 + 0.321313i
307.5 1.00000 + 1.00000i −0.896572 − 3.34605i 2.00000i 2.69493 − 4.21157i 2.44948 − 4.24263i −3.04094 + 11.3489i −2.00000 + 2.00000i −2.59800 + 1.49996i 6.90650 − 1.51664i
307.6 1.00000 + 1.00000i −0.855516 − 3.19283i 2.00000i −1.39603 + 4.80115i 2.33731 − 4.04835i 3.07379 − 11.4715i −2.00000 + 2.00000i −1.66803 + 0.963037i −6.19719 + 3.40512i
307.7 1.00000 + 1.00000i −0.744256 − 2.77760i 2.00000i −3.15656 − 3.87765i 2.03334 − 3.52186i −1.52139 + 5.67792i −2.00000 + 2.00000i 0.633086 − 0.365512i 0.721091 − 7.03420i
307.8 1.00000 + 1.00000i −0.622572 − 2.32347i 2.00000i −4.99921 + 0.0891501i 1.70090 − 2.94604i −0.635568 + 2.37197i −2.00000 + 2.00000i 2.78330 − 1.60694i −5.08836 − 4.91006i
307.9 1.00000 + 1.00000i −0.437160 − 1.63150i 2.00000i 3.71874 + 3.34230i 1.19434 − 2.06866i 1.13145 − 4.22262i −2.00000 + 2.00000i 5.32353 − 3.07354i 0.376446 + 7.06104i
307.10 1.00000 + 1.00000i −0.143853 − 0.536866i 2.00000i −1.04446 + 4.88969i 0.393014 − 0.680719i −0.875886 + 3.26885i −2.00000 + 2.00000i 7.52670 − 4.34554i −5.93416 + 3.84523i
307.11 1.00000 + 1.00000i −0.0478451 − 0.178560i 2.00000i 4.96029 + 0.628906i 0.130715 − 0.226405i −0.173418 + 0.647203i −2.00000 + 2.00000i 7.76463 − 4.48291i 4.33138 + 5.58920i
307.12 1.00000 + 1.00000i 0.143215 + 0.534485i 2.00000i 0.284897 − 4.99188i −0.391270 + 0.677700i 1.39566 − 5.20868i −2.00000 + 2.00000i 7.52906 − 4.34691i 5.27677 − 4.70698i
307.13 1.00000 + 1.00000i 0.223985 + 0.835923i 2.00000i −4.86388 − 1.15872i −0.611938 + 1.05991i 1.71055 − 6.38386i −2.00000 + 2.00000i 7.14563 − 4.12553i −3.70516 − 6.02261i
307.14 1.00000 + 1.00000i 0.331144 + 1.23585i 2.00000i −3.27590 + 3.77736i −0.904702 + 1.56699i −1.52062 + 5.67505i −2.00000 + 2.00000i 6.37657 − 3.68151i −7.05326 + 0.501462i
307.15 1.00000 + 1.00000i 0.548332 + 2.04640i 2.00000i −0.460770 − 4.97872i −1.49807 + 2.59474i −0.483314 + 1.80375i −2.00000 + 2.00000i 3.90713 − 2.25578i 4.51795 − 5.43949i
307.16 1.00000 + 1.00000i 0.613876 + 2.29102i 2.00000i 3.06713 + 3.94876i −1.67714 + 2.90489i −2.76738 + 10.3280i −2.00000 + 2.00000i 2.92232 − 1.68720i −0.881636 + 7.01589i
307.17 1.00000 + 1.00000i 0.819304 + 3.05768i 2.00000i 4.74130 − 1.58749i −2.23838 + 3.87699i 3.11403 − 11.6217i −2.00000 + 2.00000i −0.883948 + 0.510347i 6.32878 + 3.15381i
307.18 1.00000 + 1.00000i 0.952392 + 3.55438i 2.00000i −2.57964 − 4.28316i −2.60198 + 4.50677i −3.49746 + 13.0527i −2.00000 + 2.00000i −3.93232 + 2.27032i 1.70352 − 6.86280i
307.19 1.00000 + 1.00000i 1.11676 + 4.16779i 2.00000i −4.95422 − 0.675083i −3.05103 + 5.28454i 1.76713 − 6.59501i −2.00000 + 2.00000i −8.32908 + 4.80880i −4.27913 − 5.62930i
307.20 1.00000 + 1.00000i 1.15850 + 4.32358i 2.00000i 2.28931 + 4.44511i −3.16508 + 5.48209i 1.25707 − 4.69146i −2.00000 + 2.00000i −9.55703 + 5.51775i −2.15580 + 6.73443i
See all 88 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 307.22
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
43.c even 3 1 inner
215.m odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 430.3.m.b ✓ 88
5.c odd 4 1 inner 430.3.m.b ✓ 88
43.c even 3 1 inner 430.3.m.b ✓ 88
215.m odd 12 1 inner 430.3.m.b ✓ 88
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
430.3.m.b ✓ 88 1.a even 1 1 trivial
430.3.m.b ✓ 88 5.c odd 4 1 inner
430.3.m.b ✓ 88 43.c even 3 1 inner
430.3.m.b ✓ 88 215.m odd 12 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{88} - 2 T_{3}^{87} + 2 T_{3}^{86} - 96 T_{3}^{85} - 2605 T_{3}^{84} + 6614 T_{3}^{83} + \cdots + 40\!\cdots\!81 \) acting on \(S_{3}^{\mathrm{new}}(430, [\chi])\). Copy content Toggle raw display