↑

Properties

Label 304.3.z.d
Level $304$
Weight $3$
Character orbit 304.z
Analytic conductor $8.283$
Analytic rank $0$
Dimension $60$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [304,3,Mod(33,304)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(304, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 0, 7]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("304.33");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 304 = 2^{4} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 304.z (of order \(18\), degree \(6\), not 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: \(8.28340003655\)
Analytic rank: \(0\)
Dimension: \(60\)
Relative dimension: \(10\) over \(\Q(\zeta_{18})\)
Twist minimal: no (minimal twist has level 152)
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 60 q - 6 q^{3} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 60 q - 6 q^{3} + 18 q^{9} + 12 q^{13} - 24 q^{15} + 24 q^{17} - 48 q^{19} - 36 q^{21} + 72 q^{23} - 48 q^{25} - 90 q^{27} + 96 q^{29} - 108 q^{31} + 234 q^{33} + 12 q^{35} + 72 q^{39} - 30 q^{41} + 336 q^{43} - 144 q^{45} + 144 q^{47} - 282 q^{49} + 126 q^{51} - 36 q^{53} + 144 q^{55} - 336 q^{57} - 246 q^{59} - 60 q^{61} + 360 q^{63} - 540 q^{65} + 606 q^{67} + 180 q^{71} + 24 q^{73} + 72 q^{77} - 288 q^{79} + 642 q^{81} - 432 q^{83} - 120 q^{85} - 684 q^{87} + 324 q^{89} - 720 q^{91} + 648 q^{93} - 1080 q^{95} + 6 q^{97} - 246 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
33.1 0 −5.82403 − 1.02693i 0 −0.172982 − 0.145149i 0 2.43984 + 4.22593i 0 24.4075 + 8.88359i 0
33.2 0 −4.02296 − 0.709356i 0 −2.52114 − 2.11548i 0 −5.99493 − 10.3835i 0 7.22375 + 2.62923i 0
33.3 0 −3.25724 − 0.574339i 0 6.70213 + 5.62375i 0 1.51839 + 2.62993i 0 1.82249 + 0.663334i 0
33.4 0 −1.86146 − 0.328226i 0 −4.53951 − 3.80910i 0 6.78466 + 11.7514i 0 −5.09993 − 1.85622i 0
33.5 0 −1.30624 − 0.230325i 0 3.62531 + 3.04199i 0 −3.54542 − 6.14085i 0 −6.80402 − 2.47646i 0
33.6 0 0.220911 + 0.0389526i 0 −2.46245 − 2.06624i 0 2.40895 + 4.17242i 0 −8.40995 − 3.06097i 0
33.7 0 1.86549 + 0.328936i 0 −2.42977 − 2.03882i 0 −2.67900 − 4.64016i 0 −5.08538 − 1.85093i 0
33.8 0 3.78874 + 0.668057i 0 4.65845 + 3.90890i 0 4.65601 + 8.06445i 0 5.45102 + 1.98401i 0
33.9 0 3.98504 + 0.702671i 0 2.78829 + 2.33966i 0 0.472142 + 0.817775i 0 6.92959 + 2.52216i 0
33.10 0 5.06444 + 0.892998i 0 −5.64832 − 4.73951i 0 −4.67145 − 8.09119i 0 16.3939 + 5.96689i 0
97.1 0 −3.03181 + 3.61317i 0 −7.17869 + 2.61283i 0 4.52270 + 7.83354i 0 −2.30029 − 13.0456i 0
97.2 0 −2.83093 + 3.37377i 0 3.67927 − 1.33914i 0 5.35922 + 9.28244i 0 −1.80534 − 10.2386i 0
97.3 0 −2.53149 + 3.01691i 0 5.65617 − 2.05868i 0 −3.07945 − 5.33377i 0 −1.13047 − 6.41124i 0
97.4 0 −1.49370 + 1.78012i 0 2.52682 − 0.919689i 0 −4.33556 − 7.50941i 0 0.625138 + 3.54533i 0
97.5 0 −0.830926 + 0.990259i 0 −8.20651 + 2.98693i 0 −1.97279 − 3.41697i 0 1.27266 + 7.21761i 0
97.6 0 0.177431 − 0.211454i 0 −2.97941 + 1.08442i 0 −2.37180 − 4.10808i 0 1.54960 + 8.78823i 0
97.7 0 1.25860 − 1.49995i 0 7.30467 − 2.65868i 0 4.19477 + 7.26556i 0 0.897080 + 5.08759i 0
97.8 0 1.40512 − 1.67455i 0 −2.66910 + 0.971472i 0 6.08508 + 10.5397i 0 0.733060 + 4.15739i 0
97.9 0 2.16869 − 2.58455i 0 4.66951 − 1.69956i 0 −1.81441 − 3.14265i 0 −0.413821 − 2.34690i 0
97.10 0 3.17692 − 3.78611i 0 −2.80274 + 1.02011i 0 −0.459408 − 0.795718i 0 −2.67895 − 15.1931i 0
See all 60 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 33.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.f odd 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 304.3.z.d 60
4.b odd 2 1 152.3.r.a ✓ 60
19.f odd 18 1 inner 304.3.z.d 60
76.k even 18 1 152.3.r.a ✓ 60
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.3.r.a ✓ 60 4.b odd 2 1
152.3.r.a ✓ 60 76.k even 18 1
304.3.z.d 60 1.a even 1 1 trivial
304.3.z.d 60 19.f odd 18 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{60} + 6 T_{3}^{59} + 9 T_{3}^{58} + 12 T_{3}^{57} - 291 T_{3}^{56} - 2286 T_{3}^{55} + \cdots + 67\!\cdots\!21 \) acting on \(S_{3}^{\mathrm{new}}(304, [\chi])\). Copy content Toggle raw display