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Properties

Label 430.3.m.a
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.

\(\operatorname{Tr}(f)(q) = \) \( 88 q - 88 q^{2} - 2 q^{3} - 4 q^{5} + 4 q^{6} - 6 q^{7} + 176 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 88 q - 88 q^{2} - 2 q^{3} - 4 q^{5} + 4 q^{6} - 6 q^{7} + 176 q^{8} + 4 q^{10} - 4 q^{12} + 8 q^{13} + 6 q^{15} - 352 q^{16} - 22 q^{17} + 140 q^{18} + 168 q^{21} + 32 q^{23} + 60 q^{25} - 16 q^{26} - 164 q^{27} + 12 q^{28} - 36 q^{30} - 148 q^{31} + 352 q^{32} + 192 q^{33} + 92 q^{35} - 280 q^{36} + 16 q^{37} + 80 q^{38} - 8 q^{40} - 296 q^{41} - 168 q^{42} - 42 q^{43} - 148 q^{45} - 64 q^{46} + 128 q^{47} + 8 q^{48} - 320 q^{51} + 16 q^{52} - 238 q^{53} + 68 q^{55} - 24 q^{56} - 6 q^{57} - 16 q^{58} + 60 q^{60} - 188 q^{61} + 148 q^{62} - 172 q^{63} + 428 q^{65} - 384 q^{66} + 6 q^{67} + 44 q^{68} + 280 q^{70} - 120 q^{71} + 280 q^{72} + 370 q^{73} - 560 q^{75} - 160 q^{76} + 56 q^{77} - 440 q^{78} + 16 q^{80} + 336 q^{81} + 296 q^{82} + 120 q^{83} + 584 q^{85} + 84 q^{86} + 16 q^{87} + 160 q^{90} + 316 q^{91} + 64 q^{92} - 128 q^{93} - 136 q^{95} - 16 q^{96} + 16 q^{97} - 608 q^{98}+O(q^{100}) \) 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 −1.39144 − 5.19293i 2.00000i 4.93680 + 0.792466i −3.80149 + 6.58437i −2.56474 + 9.57173i 2.00000 − 2.00000i −17.2362 + 9.95132i −4.14433 − 5.72927i
307.2 −1.00000 − 1.00000i −1.32063 − 4.92867i 2.00000i 0.608264 − 4.96286i −3.60803 + 6.24930i 0.786313 − 2.93456i 2.00000 − 2.00000i −14.7535 + 8.51791i −5.57113 + 4.35460i
307.3 −1.00000 − 1.00000i −1.11498 − 4.16115i 2.00000i −4.96914 − 0.554620i −3.04618 + 5.27613i −2.14481 + 8.00456i 2.00000 − 2.00000i −8.27779 + 4.77918i 4.41452 + 5.52376i
307.4 −1.00000 − 1.00000i −1.03099 − 3.84772i 2.00000i −2.69968 + 4.20853i −2.81673 + 4.87872i 3.23509 − 12.0735i 2.00000 − 2.00000i −5.94779 + 3.43396i 6.90821 − 1.50886i
307.5 −1.00000 − 1.00000i −0.972517 − 3.62948i 2.00000i 3.02511 + 3.98104i −2.65697 + 4.60200i −0.0775077 + 0.289263i 2.00000 − 2.00000i −4.43314 + 2.55947i 0.955936 − 7.00615i
307.6 −1.00000 − 1.00000i −0.863242 − 3.22166i 2.00000i 0.931512 + 4.91246i −2.35842 + 4.08491i 0.379433 − 1.41606i 2.00000 − 2.00000i −1.83970 + 1.06215i 3.98095 − 5.84397i
307.7 −1.00000 − 1.00000i −0.750813 − 2.80207i 2.00000i −4.50633 − 2.16633i −2.05126 + 3.55288i 2.65871 − 9.92243i 2.00000 − 2.00000i 0.506344 − 0.292338i 2.34000 + 6.67266i
307.8 −1.00000 − 1.00000i −0.441795 − 1.64880i 2.00000i 3.70003 − 3.36300i −1.20701 + 2.09059i 2.15209 − 8.03170i 2.00000 − 2.00000i 5.27087 − 3.04314i −7.06303 − 0.337029i
307.9 −1.00000 − 1.00000i −0.326812 − 1.21968i 2.00000i −2.79291 + 4.14725i −0.892866 + 1.54649i −3.25412 + 12.1445i 2.00000 − 2.00000i 6.41342 − 3.70279i 6.94016 − 1.35433i
307.10 −1.00000 − 1.00000i −0.306247 − 1.14293i 2.00000i −2.16397 − 4.50746i −0.836682 + 1.44918i −0.982621 + 3.66719i 2.00000 − 2.00000i 6.58173 − 3.79996i −2.34349 + 6.67144i
307.11 −1.00000 − 1.00000i −0.00152980 − 0.00570930i 2.00000i 3.33552 − 3.72482i −0.00417950 + 0.00723910i −3.41450 + 12.7431i 2.00000 − 2.00000i 7.79420 − 4.49998i −7.06034 + 0.389298i
307.12 −1.00000 − 1.00000i 0.234043 + 0.873460i 2.00000i −4.91659 − 0.909499i 0.639417 − 1.10750i −0.254207 + 0.948712i 2.00000 − 2.00000i 7.08607 − 4.09115i 4.00709 + 5.82608i
307.13 −1.00000 − 1.00000i 0.280545 + 1.04701i 2.00000i 4.65359 + 1.82869i 0.766463 − 1.32755i 2.63196 − 9.82260i 2.00000 − 2.00000i 6.77671 − 3.91253i −2.82491 − 6.48228i
307.14 −1.00000 − 1.00000i 0.350390 + 1.30767i 2.00000i −4.11666 + 2.83779i 0.957283 − 1.65806i 1.54303 − 5.75868i 2.00000 − 2.00000i 6.20699 − 3.58361i 6.95446 + 1.27887i
307.15 −1.00000 − 1.00000i 0.383357 + 1.43071i 2.00000i 4.17777 + 2.74704i 1.04735 − 1.81407i −0.843181 + 3.14679i 2.00000 − 2.00000i 5.89426 − 3.40305i −1.43073 − 6.92481i
307.16 −1.00000 − 1.00000i 0.593539 + 2.21512i 2.00000i 0.187326 + 4.99649i 1.62158 − 2.80866i −0.195409 + 0.729277i 2.00000 − 2.00000i 3.23977 − 1.87048i 4.80916 − 5.18382i
307.17 −1.00000 − 1.00000i 0.857246 + 3.19929i 2.00000i 1.02890 − 4.89299i 2.34204 − 4.05653i 1.44988 − 5.41104i 2.00000 − 2.00000i −1.70633 + 0.985148i −5.92189 + 3.86409i
307.18 −1.00000 − 1.00000i 0.961094 + 3.58685i 2.00000i 1.96170 − 4.59910i 2.62576 − 4.54795i −0.203097 + 0.757967i 2.00000 − 2.00000i −4.14758 + 2.39461i −6.56080 + 2.63741i
307.19 −1.00000 − 1.00000i 1.09551 + 4.08848i 2.00000i −4.03419 − 2.95387i 2.99298 − 5.18399i −2.38004 + 8.88241i 2.00000 − 2.00000i −7.72133 + 4.45791i 1.08032 + 6.98805i
307.20 −1.00000 − 1.00000i 1.31591 + 4.91103i 2.00000i 4.86371 + 1.15943i 3.59512 − 6.22693i −2.31505 + 8.63988i 2.00000 − 2.00000i −14.5924 + 8.42490i −3.70428 − 6.02315i
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.a ✓ 88
5.c odd 4 1 inner 430.3.m.a ✓ 88
43.c even 3 1 inner 430.3.m.a ✓ 88
215.m odd 12 1 inner 430.3.m.a ✓ 88
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
430.3.m.a ✓ 88 1.a even 1 1 trivial
430.3.m.a ✓ 88 5.c odd 4 1 inner
430.3.m.a ✓ 88 43.c even 3 1 inner
430.3.m.a ✓ 88 215.m odd 12 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{88} + 2 T_{3}^{87} + 2 T_{3}^{86} + 80 T_{3}^{85} - 2653 T_{3}^{84} - 6538 T_{3}^{83} + \cdots + 68\!\cdots\!25 \) acting on \(S_{3}^{\mathrm{new}}(430, [\chi])\). Copy content Toggle raw display