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Properties

Label 252.8.j.b
Level $252$
Weight $8$
Character orbit 252.j
Analytic conductor $78.721$
Analytic rank $0$
Dimension $42$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [252,8,Mod(85,252)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(252, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("252.85");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 252.j (of order \(3\), 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: \(78.7210264220\)
Analytic rank: \(0\)
Dimension: \(42\)
Relative dimension: \(21\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 42 q + 82 q^{3} + 179 q^{5} + 7203 q^{7} + 890 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 42 q + 82 q^{3} + 179 q^{5} + 7203 q^{7} + 890 q^{9} + 3958 q^{11} - 6177 q^{13} + 20077 q^{15} + 22036 q^{17} + 2094 q^{19} + 686 q^{21} + 91390 q^{23} - 284160 q^{25} - 16247 q^{27} + 180179 q^{29} + 99327 q^{31} + 175070 q^{33} + 122794 q^{35} - 151782 q^{37} - 8736 q^{39} - 59940 q^{41} + 360321 q^{43} + 357467 q^{45} + 1348899 q^{47} - 2470629 q^{49} - 1657207 q^{51} + 1010028 q^{53} - 1105080 q^{55} - 1937515 q^{57} - 458041 q^{59} - 1085874 q^{61} - 949424 q^{63} - 1175811 q^{65} + 384222 q^{67} + 3999311 q^{69} - 4838366 q^{71} - 1822650 q^{73} - 1387237 q^{75} - 1357594 q^{77} + 238365 q^{79} - 17003890 q^{81} + 1228174 q^{83} + 356127 q^{85} + 33054031 q^{87} + 34786068 q^{89} - 4237422 q^{91} - 1689051 q^{93} - 12321596 q^{95} + 926001 q^{97} - 61176680 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
85.1 0 −46.3084 + 6.52189i 0 −45.8869 + 79.4784i 0 171.500 + 297.047i 0 2101.93 − 604.036i 0
85.2 0 −42.6661 + 19.1470i 0 199.603 − 345.722i 0 171.500 + 297.047i 0 1453.79 − 1633.85i 0
85.3 0 −41.3339 − 21.8748i 0 28.6503 − 49.6238i 0 171.500 + 297.047i 0 1229.99 + 1808.34i 0
85.4 0 −39.4677 + 25.0858i 0 −137.863 + 238.785i 0 171.500 + 297.047i 0 928.402 − 1980.16i 0
85.5 0 −39.1578 − 25.5669i 0 185.330 − 321.000i 0 171.500 + 297.047i 0 879.665 + 2002.29i 0
85.6 0 −34.4001 − 31.6802i 0 −211.363 + 366.092i 0 171.500 + 297.047i 0 179.735 + 2179.60i 0
85.7 0 −18.2767 + 43.0461i 0 65.2686 − 113.049i 0 171.500 + 297.047i 0 −1518.93 − 1573.48i 0
85.8 0 −17.6514 − 43.3062i 0 −131.455 + 227.687i 0 171.500 + 297.047i 0 −1563.85 + 1528.83i 0
85.9 0 −10.5399 + 45.5622i 0 −255.827 + 443.105i 0 171.500 + 297.047i 0 −1964.82 − 960.442i 0
85.10 0 −6.63534 + 46.2922i 0 117.741 − 203.933i 0 171.500 + 297.047i 0 −2098.94 − 614.329i 0
85.11 0 5.39849 − 46.4527i 0 97.1422 − 168.255i 0 171.500 + 297.047i 0 −2128.71 − 501.549i 0
85.12 0 5.48874 − 46.4422i 0 61.8162 − 107.069i 0 171.500 + 297.047i 0 −2126.75 − 509.818i 0
85.13 0 26.4041 + 38.5982i 0 24.2601 − 42.0197i 0 171.500 + 297.047i 0 −792.648 + 2038.30i 0
85.14 0 27.2294 + 38.0205i 0 −62.2878 + 107.886i 0 171.500 + 297.047i 0 −704.122 + 2070.55i 0
85.15 0 29.1840 − 36.5417i 0 −220.853 + 382.528i 0 171.500 + 297.047i 0 −483.588 − 2132.86i 0
85.16 0 31.2103 + 34.8270i 0 229.083 − 396.783i 0 171.500 + 297.047i 0 −238.838 + 2173.92i 0
85.17 0 35.1126 − 30.8886i 0 29.1202 − 50.4377i 0 171.500 + 297.047i 0 278.795 − 2169.16i 0
85.18 0 42.6230 + 19.2426i 0 −72.0794 + 124.845i 0 171.500 + 297.047i 0 1446.45 + 1640.35i 0
85.19 0 42.6693 − 19.1397i 0 242.229 − 419.553i 0 171.500 + 297.047i 0 1454.35 − 1633.35i 0
85.20 0 45.4460 + 11.0301i 0 −191.548 + 331.771i 0 171.500 + 297.047i 0 1943.68 + 1002.54i 0
See all 42 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 85.21
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 252.8.j.b ✓ 42
9.c even 3 1 inner 252.8.j.b ✓ 42
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.8.j.b ✓ 42 1.a even 1 1 trivial
252.8.j.b ✓ 42 9.c even 3 1 inner