Properties

Label 304.5.q.b
Level $304$
Weight $5$
Character orbit 304.q
Analytic conductor $31.424$
Analytic rank $0$
Dimension $28$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [304,5,Mod(159,304)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(304, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 2]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("304.159");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 304 = 2^{4} \cdot 19 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 304.q (of order \(6\), degree \(2\), 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: \(31.4244687775\)
Analytic rank: \(0\)
Dimension: \(28\)
Relative dimension: \(14\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 28 q - 18 q^{3} - 9 q^{5} + 376 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q - 18 q^{3} - 9 q^{5} + 376 q^{9} + 19 q^{13} + 117 q^{15} - 249 q^{17} + 129 q^{19} - 906 q^{21} - 585 q^{23} - 2231 q^{25} + 15 q^{29} + 1521 q^{33} - 3456 q^{35} - 900 q^{37} + 1680 q^{41} + 1551 q^{43} - 1828 q^{45} - 12519 q^{47} - 9892 q^{49} + 1101 q^{51} - 693 q^{53} + 14190 q^{55} + 1131 q^{57} - 11268 q^{59} - 1189 q^{61} + 26316 q^{63} - 606 q^{65} - 15864 q^{67} - 2330 q^{69} + 24813 q^{71} - 5310 q^{73} + 1596 q^{77} + 4167 q^{79} + 134 q^{81} + 5707 q^{85} - 16629 q^{89} + 5964 q^{91} + 13700 q^{93} - 45531 q^{95} - 14872 q^{97} - 52914 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
159.1 0 −14.5223 + 8.38447i 0 14.2737 + 24.7229i 0 68.5423i 0 100.099 173.376i 0
159.2 0 −11.6173 + 6.70724i 0 −20.2503 35.0746i 0 70.8433i 0 49.4742 85.6918i 0
159.3 0 −10.9589 + 6.32715i 0 −14.7491 25.5461i 0 60.9439i 0 39.5656 68.5296i 0
159.4 0 −9.53297 + 5.50386i 0 0.698568 + 1.20995i 0 40.9545i 0 20.0850 34.7882i 0
159.5 0 −7.34804 + 4.24239i 0 6.30754 + 10.9250i 0 23.4029i 0 −4.50418 + 7.80147i 0
159.6 0 −5.08005 + 2.93297i 0 21.4753 + 37.1962i 0 1.12615i 0 −23.2954 + 40.3488i 0
159.7 0 −1.32793 + 0.766679i 0 −3.49962 6.06153i 0 1.93586i 0 −39.3244 + 68.1119i 0
159.8 0 1.59371 0.920131i 0 2.29460 + 3.97436i 0 89.4704i 0 −38.8067 + 67.2152i 0
159.9 0 2.94613 1.70095i 0 −15.4219 26.7115i 0 25.4325i 0 −34.7135 + 60.1256i 0
159.10 0 6.36283 3.67358i 0 11.8247 + 20.4810i 0 89.1614i 0 −13.5096 + 23.3993i 0
159.11 0 6.47147 3.73630i 0 −22.5885 39.1244i 0 61.0442i 0 −12.5801 + 21.7893i 0
159.12 0 9.74583 5.62676i 0 2.35475 + 4.07855i 0 19.2049i 0 22.8209 39.5269i 0
159.13 0 9.78032 5.64667i 0 20.8784 + 36.1624i 0 40.9622i 0 23.2698 40.3044i 0
159.14 0 14.4872 8.36421i 0 −8.09816 14.0264i 0 22.9000i 0 99.4200 172.200i 0
239.1 0 −14.5223 8.38447i 0 14.2737 24.7229i 0 68.5423i 0 100.099 + 173.376i 0
239.2 0 −11.6173 6.70724i 0 −20.2503 + 35.0746i 0 70.8433i 0 49.4742 + 85.6918i 0
239.3 0 −10.9589 6.32715i 0 −14.7491 + 25.5461i 0 60.9439i 0 39.5656 + 68.5296i 0
239.4 0 −9.53297 5.50386i 0 0.698568 1.20995i 0 40.9545i 0 20.0850 + 34.7882i 0
239.5 0 −7.34804 4.24239i 0 6.30754 10.9250i 0 23.4029i 0 −4.50418 7.80147i 0
239.6 0 −5.08005 2.93297i 0 21.4753 37.1962i 0 1.12615i 0 −23.2954 40.3488i 0
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 159.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
76.g odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 304.5.q.b 28
4.b odd 2 1 304.5.q.c yes 28
19.c even 3 1 304.5.q.c yes 28
76.g odd 6 1 inner 304.5.q.b 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
304.5.q.b 28 1.a even 1 1 trivial
304.5.q.b 28 76.g odd 6 1 inner
304.5.q.c yes 28 4.b odd 2 1
304.5.q.c yes 28 19.c even 3 1