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Properties

Label 684.2.bo.e
Level $684$
Weight $2$
Character orbit 684.bo
Analytic conductor $5.462$
Analytic rank $0$
Dimension $12$
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] = [684,2,Mod(73,684)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(684, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("684.73");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 684.bo (of order \(9\), degree \(6\), 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: \(5.46176749826\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{9})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 3 x^{11} + 27 x^{10} + 309 x^{8} + 42 x^{7} + 2059 x^{6} + 1245 x^{5} + 8226 x^{4} + \cdots + 16129 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 228)
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{10} + \beta_{8} + \beta_{2}) q^{5} + ( - \beta_{11} - \beta_{10} - \beta_{7} + \cdots + 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{10} + \beta_{8} + \beta_{2}) q^{5} + ( - \beta_{11} - \beta_{10} - \beta_{7} + \cdots + 1) q^{7}+ \cdots + ( - \beta_{11} - \beta_{10} + \cdots - \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6 q^{5} - 9 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 6 q^{5} - 9 q^{7} + 9 q^{11} - 3 q^{13} - 12 q^{17} + 9 q^{19} - 15 q^{23} + 12 q^{25} + 24 q^{29} - 6 q^{31} + 42 q^{35} + 12 q^{37} - 6 q^{41} - 39 q^{43} + 3 q^{47} - 21 q^{49} - 18 q^{53} + 45 q^{55} + 33 q^{61} + 33 q^{65} - 27 q^{67} - 6 q^{71} - 24 q^{73} + 18 q^{79} - 3 q^{83} + 39 q^{85} + 15 q^{89} + 18 q^{91} + 30 q^{95} - 15 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 3 x^{11} + 27 x^{10} + 309 x^{8} + 42 x^{7} + 2059 x^{6} + 1245 x^{5} + 8226 x^{4} + \cdots + 16129 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 51745067377 \nu^{11} - 42818012909494 \nu^{10} + 226125175033092 \nu^{9} + \cdots - 10\!\cdots\!48 ) / 69\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 29143066782 \nu^{11} + 133962654523 \nu^{10} - 872369837459 \nu^{9} + \cdots + 653351222762683 ) / 578389509037947 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 366405151 \nu^{11} + 673283735 \nu^{10} - 8299336511 \nu^{9} - 10899297887 \nu^{8} + \cdots - 3701169481314 ) / 4554248102661 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1573262129557 \nu^{11} - 25128247953302 \nu^{10} + 138133742168674 \nu^{9} + \cdots - 53\!\cdots\!84 ) / 69\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 1587657721179 \nu^{11} - 52742673155649 \nu^{10} + 272867905929940 \nu^{9} + \cdots - 19\!\cdots\!07 ) / 69\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2654928838606 \nu^{11} + 3826956506059 \nu^{10} - 9593166249832 \nu^{9} + \cdots + 56\!\cdots\!37 ) / 69\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 6707861328234 \nu^{11} + 15322141714987 \nu^{10} - 121627592574834 \nu^{9} + \cdots - 64\!\cdots\!19 ) / 69\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 7962362120451 \nu^{11} + 31870238085137 \nu^{10} - 211008602156696 \nu^{9} + \cdots - 36\!\cdots\!37 ) / 69\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 11442026220493 \nu^{11} + 79668693037918 \nu^{10} - 491070869140128 \nu^{9} + \cdots + 41\!\cdots\!42 ) / 69\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 6755245953217 \nu^{11} - 20512021150675 \nu^{10} + 151528939024997 \nu^{9} + \cdots + 41\!\cdots\!19 ) / 34\!\cdots\!82 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{10} - 2\beta_{8} - 3\beta_{7} - \beta_{6} + 4\beta_{5} - \beta_{4} + 7\beta_{3} - 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - 5 \beta_{11} + 5 \beta_{10} - 9 \beta_{9} - 9 \beta_{8} - 10 \beta_{7} - 2 \beta_{6} + 6 \beta_{5} + \cdots - 15 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 15 \beta_{11} + 13 \beta_{10} - 58 \beta_{9} + 3 \beta_{8} + 10 \beta_{7} + 13 \beta_{6} - 86 \beta_{5} + \cdots - 13 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 86 \beta_{11} - 51 \beta_{10} - 10 \beta_{9} + 274 \beta_{8} + 401 \beta_{7} + 137 \beta_{6} + \cdots + 332 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 684 \beta_{11} - 684 \beta_{10} + 1347 \beta_{9} + 1347 \beta_{8} + 1731 \beta_{7} + 273 \beta_{6} + \cdots + 2194 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 1188 \beta_{11} - 2142 \beta_{10} + 7068 \beta_{9} - 528 \beta_{8} - 1614 \beta_{7} - 2142 \beta_{6} + \cdots + 2142 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 10398 \beta_{11} + 5884 \beta_{10} + 2976 \beta_{9} - 35120 \beta_{8} - 48426 \beta_{7} + \cdots - 39838 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 79145 \beta_{11} + 79145 \beta_{10} - 156222 \beta_{9} - 156222 \beta_{8} - 198457 \beta_{7} + \cdots - 241293 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 135210 \beta_{11} + 249859 \beta_{10} - 834016 \beta_{9} + 72993 \beta_{8} + 176866 \beta_{7} + \cdots - 249859 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 1219085 \beta_{11} - 651669 \beta_{10} - 355393 \beta_{9} + 4052914 \beta_{8} + 5568275 \beta_{7} + \cdots + 4479362 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/684\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(343\) \(533\)
\(\chi(n)\) \(-\beta_{5} + \beta_{8}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
73.1
1.30338 + 2.25752i
−0.629732 − 1.09073i
1.30338 − 2.25752i
−0.629732 + 1.09073i
2.42841 − 4.20614i
−1.16237 + 2.01328i
−1.17108 − 2.02836i
0.731383 + 1.26679i
−1.17108 + 2.02836i
0.731383 − 1.26679i
2.42841 + 4.20614i
−1.16237 − 2.01328i
0 0 0 −0.392352 + 2.22514i 0 −2.18351 − 3.78195i 0 0 0
73.2 0 0 0 0.279011 − 1.58235i 0 1.44955 + 2.51070i 0 0 0
253.1 0 0 0 −0.392352 − 2.22514i 0 −2.18351 + 3.78195i 0 0 0
253.2 0 0 0 0.279011 + 1.58235i 0 1.44955 − 2.51070i 0 0 0
289.1 0 0 0 −2.54690 + 2.13710i 0 −0.596313 + 1.03284i 0 0 0
289.2 0 0 0 2.95450 − 2.47912i 0 −1.84338 + 3.19283i 0 0 0
397.1 0 0 0 −0.434858 + 0.158276i 0 −2.12054 − 3.67289i 0 0 0
397.2 0 0 0 3.14060 − 1.14308i 0 0.794193 + 1.37558i 0 0 0
541.1 0 0 0 −0.434858 − 0.158276i 0 −2.12054 + 3.67289i 0 0 0
541.2 0 0 0 3.14060 + 1.14308i 0 0.794193 − 1.37558i 0 0 0
613.1 0 0 0 −2.54690 − 2.13710i 0 −0.596313 − 1.03284i 0 0 0
613.2 0 0 0 2.95450 + 2.47912i 0 −1.84338 − 3.19283i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 73.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.e even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 684.2.bo.e 12
3.b odd 2 1 228.2.q.b ✓ 12
12.b even 2 1 912.2.bo.g 12
19.e even 9 1 inner 684.2.bo.e 12
57.j even 18 1 4332.2.a.u 6
57.l odd 18 1 228.2.q.b ✓ 12
57.l odd 18 1 4332.2.a.t 6
228.v even 18 1 912.2.bo.g 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
228.2.q.b ✓ 12 3.b odd 2 1
228.2.q.b ✓ 12 57.l odd 18 1
684.2.bo.e 12 1.a even 1 1 trivial
684.2.bo.e 12 19.e even 9 1 inner
912.2.bo.g 12 12.b even 2 1
912.2.bo.g 12 228.v even 18 1
4332.2.a.t 6 57.l odd 18 1
4332.2.a.u 6 57.j even 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} - 6 T_{5}^{11} + 12 T_{5}^{10} - 8 T_{5}^{9} + 150 T_{5}^{8} - 669 T_{5}^{7} + 1477 T_{5}^{6} + \cdots + 5184 \) acting on \(S_{2}^{\mathrm{new}}(684, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} - 6 T^{11} + \cdots + 5184 \) Copy content Toggle raw display
$7$ \( T^{12} + 9 T^{11} + \cdots + 140625 \) Copy content Toggle raw display
$11$ \( T^{12} - 9 T^{11} + \cdots + 23104 \) Copy content Toggle raw display
$13$ \( T^{12} + 3 T^{11} + \cdots + 3249 \) Copy content Toggle raw display
$17$ \( T^{12} + 12 T^{11} + \cdots + 576 \) Copy content Toggle raw display
$19$ \( T^{12} - 9 T^{11} + \cdots + 47045881 \) Copy content Toggle raw display
$23$ \( T^{12} + 15 T^{11} + \cdots + 341056 \) Copy content Toggle raw display
$29$ \( T^{12} - 24 T^{11} + \cdots + 4562496 \) Copy content Toggle raw display
$31$ \( T^{12} + 6 T^{11} + \cdots + 23357889 \) Copy content Toggle raw display
$37$ \( (T^{6} - 6 T^{5} + \cdots - 7757)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 66065449024 \) Copy content Toggle raw display
$43$ \( T^{12} + 39 T^{11} + \cdots + 14676561 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 22603318336 \) Copy content Toggle raw display
$53$ \( T^{12} + 18 T^{11} + \cdots + 64 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 101597737536 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 16350992641 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 1947986496 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 1030666816 \) Copy content Toggle raw display
$73$ \( T^{12} + 24 T^{11} + \cdots + 660969 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 563050634689 \) Copy content Toggle raw display
$83$ \( T^{12} + 3 T^{11} + \cdots + 576 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 6117742656 \) Copy content Toggle raw display
$97$ \( T^{12} + 15 T^{11} + \cdots + 78730129 \) Copy content Toggle raw display
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