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Properties

Label 385.2.bd.a
Level $385$
Weight $2$
Character orbit 385.bd
Analytic conductor $3.074$
Analytic rank $0$
Dimension $80$
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] = [385,2,Mod(12,385)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(385, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([3, 10, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("385.12");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 385 = 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 385.bd (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: \(3.07424047782\)
Analytic rank: \(0\)
Dimension: \(80\)
Relative dimension: \(20\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 80 q - 36 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 80 q - 36 q^{8} - 40 q^{11} + 44 q^{15} + 28 q^{16} - 42 q^{17} + 22 q^{18} - 40 q^{21} - 4 q^{23} - 8 q^{25} - 12 q^{26} - 16 q^{28} - 8 q^{30} + 60 q^{31} + 56 q^{32} - 18 q^{35} - 64 q^{36} + 28 q^{37} + 6 q^{38} - 24 q^{40} + 24 q^{42} + 8 q^{43} + 18 q^{45} - 20 q^{46} - 12 q^{47} + 76 q^{50} - 44 q^{51} - 126 q^{52} + 4 q^{53} + 60 q^{56} - 84 q^{57} + 4 q^{58} - 40 q^{60} - 36 q^{61} + 14 q^{63} + 48 q^{65} + 12 q^{66} + 20 q^{67} - 144 q^{68} + 2 q^{70} + 24 q^{71} + 134 q^{72} + 48 q^{73} + 108 q^{75} - 124 q^{78} + 54 q^{80} - 8 q^{81} + 228 q^{82} - 124 q^{86} - 198 q^{87} + 18 q^{88} - 40 q^{91} - 100 q^{92} - 74 q^{93} - 26 q^{95} - 36 q^{96} + 148 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
12.1 −2.68329 + 0.718984i 0.726767 − 2.71233i 4.95103 − 2.85848i 0.0327717 + 2.23583i 7.80049i −1.07817 − 2.41610i −7.30123 + 7.30123i −4.23048 − 2.44247i −1.69546 − 5.97580i
12.2 −2.46992 + 0.661813i −0.702152 + 2.62047i 3.93046 − 2.26925i −2.08838 + 0.799155i − 6.93704i 0.481975 + 2.60148i −4.58988 + 4.58988i −3.77576 − 2.17994i 4.62925 − 3.35597i
12.3 −2.36396 + 0.633420i −0.0988601 + 0.368951i 3.45502 − 1.99476i 0.182608 − 2.22860i − 0.934805i 2.17919 − 1.50037i −3.44293 + 3.44293i 2.47172 + 1.42705i 0.979963 + 5.38398i
12.4 −1.76785 + 0.473694i 0.595902 − 2.22394i 1.16886 − 0.674842i −2.01922 + 0.960606i 4.21387i 1.96453 + 1.77218i 0.841610 − 0.841610i −1.99272 − 1.15050i 3.11464 − 2.65470i
12.5 −1.60942 + 0.431242i −0.499555 + 1.86436i 0.672203 − 0.388097i 1.38558 − 1.75504i − 3.21597i −0.506630 + 2.59679i 1.44186 − 1.44186i −0.628224 − 0.362705i −1.47312 + 3.42212i
12.6 −1.45012 + 0.388559i 0.193933 − 0.723768i 0.219821 − 0.126914i 2.23268 + 0.122972i 1.12491i −2.29666 − 1.31353i 1.85367 − 1.85367i 2.11185 + 1.21928i −3.28544 + 0.689204i
12.7 −0.845927 + 0.226665i 0.526357 − 1.96439i −1.06784 + 0.616516i −1.01825 − 1.99077i 1.78104i −2.06656 + 1.65207i 2.00209 − 2.00209i −0.983701 − 0.567940i 1.31260 + 1.45325i
12.8 −0.731624 + 0.196038i −0.214138 + 0.799175i −1.23521 + 0.713147i 1.43324 + 1.71634i − 0.626675i 2.52260 + 0.797791i 1.83508 − 1.83508i 2.00525 + 1.15773i −1.38506 − 0.974750i
12.9 −0.642835 + 0.172247i −0.837695 + 3.12632i −1.34848 + 0.778547i 1.09827 − 1.94777i − 2.15400i 0.675309 − 2.55812i 1.67393 − 1.67393i −6.47406 − 3.73780i −0.370509 + 1.44127i
12.10 0.0116835 − 0.00313058i −0.0619911 + 0.231354i −1.73192 + 0.999927i −1.89929 − 1.18013i 0.00289709i 1.80468 − 1.93472i −0.0342104 + 0.0342104i 2.54839 + 1.47132i −0.0258848 − 0.00784214i
12.11 0.250950 − 0.0672418i 0.708668 − 2.64478i −1.67360 + 0.966251i 2.13961 − 0.649667i − 0.711360i 1.25326 − 2.33009i −0.722432 + 0.722432i −3.89460 − 2.24855i 0.493250 − 0.306905i
12.12 0.762590 − 0.204335i −0.436664 + 1.62965i −1.19226 + 0.688352i −0.921732 + 2.03726i 1.33198i −1.02511 − 2.43909i −1.88506 + 1.88506i 0.132988 + 0.0767804i −0.286620 + 1.74193i
12.13 0.930268 − 0.249264i 0.331947 − 1.23884i −0.928786 + 0.536235i 1.04045 + 1.97926i − 1.23520i 0.345519 + 2.62309i −2.09236 + 2.09236i 1.17354 + 0.677542i 1.46126 + 1.58189i
12.14 0.936626 − 0.250968i 0.737825 − 2.75360i −0.917768 + 0.529873i −2.19281 + 0.437720i − 2.76426i −2.62170 − 0.355902i −2.09794 + 2.09794i −4.43986 − 2.56335i −1.94399 + 0.960305i
12.15 1.55352 − 0.416264i −0.608935 + 2.27258i 0.508096 − 0.293350i 2.23458 − 0.0816547i 3.78397i −0.262634 + 2.63268i −1.60728 + 1.60728i −2.19573 − 1.26771i 3.43747 − 1.05703i
12.16 1.68639 − 0.451868i −0.796332 + 2.97195i 0.907691 − 0.524055i −1.95104 − 1.09246i 5.37172i −2.63476 + 0.240866i −1.17513 + 1.17513i −5.60028 − 3.23332i −3.78386 − 0.960700i
12.17 1.72738 − 0.462851i 0.424490 − 1.58422i 1.03758 − 0.599044i −0.601522 − 2.15364i − 2.93303i 2.61858 + 0.378173i −1.01404 + 1.01404i 0.268524 + 0.155032i −2.03588 − 3.44175i
12.18 1.97283 − 0.528618i −0.0146490 + 0.0546710i 1.88056 − 1.08574i 2.10447 − 0.755792i 0.115600i −1.59701 − 2.10940i 0.247666 − 0.247666i 2.59530 + 1.49840i 3.75223 − 2.60351i
12.19 2.25757 − 0.604915i −0.346695 + 1.29388i 2.99866 − 1.73128i −1.75715 + 1.38291i 3.13076i 2.60415 + 0.467310i 2.41710 − 2.41710i 1.04414 + 0.602832i −3.13035 + 4.18494i
12.20 2.47512 − 0.663207i 0.371779 − 1.38750i 3.95433 − 2.28304i 0.565127 + 2.16348i − 3.68079i −2.36056 + 1.19488i 4.64950 − 4.64950i 0.811145 + 0.468315i 2.83359 + 4.98007i
See all 80 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 12.20
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
7.d odd 6 1 inner
35.k even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 385.2.bd.a ✓ 80
5.c odd 4 1 inner 385.2.bd.a ✓ 80
7.d odd 6 1 inner 385.2.bd.a ✓ 80
35.k even 12 1 inner 385.2.bd.a ✓ 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
385.2.bd.a ✓ 80 1.a even 1 1 trivial
385.2.bd.a ✓ 80 5.c odd 4 1 inner
385.2.bd.a ✓ 80 7.d odd 6 1 inner
385.2.bd.a ✓ 80 35.k even 12 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{80} + 12 T_{2}^{77} - 141 T_{2}^{76} - 40 T_{2}^{75} + 72 T_{2}^{74} - 1440 T_{2}^{73} + \cdots + 625 \) acting on \(S_{2}^{\mathrm{new}}(385, [\chi])\). Copy content Toggle raw display