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Properties

Label 390.2.bb.c
Level $390$
Weight $2$
Character orbit 390.bb
Analytic conductor $3.114$
Analytic rank $0$
Dimension $8$
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] = [390,2,Mod(121,390)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(390, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("390.121");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 390 = 2 \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 390.bb (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: \(3.11416567883\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.17284886784.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + 2x^{6} + 30x^{5} + 185x^{4} + 36x^{3} + 8x^{2} + 208x + 2704 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{5} q^{2} - \beta_{6} q^{3} + ( - \beta_{6} + 1) q^{4} - \beta_{2} q^{5} + ( - \beta_{5} - \beta_{2}) q^{6} + (\beta_{4} - \beta_1) q^{7} - \beta_{2} q^{8} + (\beta_{6} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{5} q^{2} - \beta_{6} q^{3} + ( - \beta_{6} + 1) q^{4} - \beta_{2} q^{5} + ( - \beta_{5} - \beta_{2}) q^{6} + (\beta_{4} - \beta_1) q^{7} - \beta_{2} q^{8} + (\beta_{6} - 1) q^{9} - \beta_{6} q^{10} + ( - \beta_{7} + \beta_{4} - \beta_{2} + 1) q^{11} - q^{12} + (\beta_{7} - \beta_{5} - \beta_{2} - 2) q^{13} + ( - \beta_{7} - \beta_{5} + \beta_{4} - \beta_{2} - \beta_1 + 1) q^{14} - \beta_{5} q^{15} - \beta_{6} q^{16} + ( - 4 \beta_{6} + 4) q^{17} + \beta_{2} q^{18} + (\beta_{7} + 2 \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + 2 \beta_1 - 2) q^{19} + ( - \beta_{5} - \beta_{2}) q^{20} + ( - \beta_{7} + \beta_{3} - \beta_{2}) q^{21} + ( - \beta_{7} - \beta_{6} + \beta_{3} - \beta_{2} - \beta_1 + 1) q^{22} + (\beta_{7} + 2 \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 - 2) q^{23} - \beta_{5} q^{24} - q^{25} + ( - 2 \beta_{5} - \beta_{3} - 1) q^{26} + q^{27} + ( - \beta_{7} + \beta_{4} + \beta_{3} - \beta_{2} - \beta_1) q^{28} + (\beta_{7} - 2 \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} - \beta_1) q^{29} + (\beta_{6} - 1) q^{30} + (\beta_{7} + 6 \beta_{6} + \beta_{5} + \beta_{4} + \beta_{2} + \beta_1 - 4) q^{31} + ( - \beta_{5} - \beta_{2}) q^{32} + (\beta_{7} + \beta_{5} + \beta_{3} + \beta_1 - 2) q^{33} - 4 \beta_{2} q^{34} + ( - \beta_{7} + \beta_{4} + \beta_{3} - \beta_{2}) q^{35} + \beta_{6} q^{36} + (\beta_{7} - 3 \beta_{6} + 3 \beta_{5} - \beta_{4} + \beta_{2} + 5) q^{37} + (\beta_{7} - 2 \beta_{4} - \beta_{3} + \beta_{2} + 2 \beta_1) q^{38} + ( - \beta_{7} + \beta_{6} - \beta_{5} + \beta_{2} - \beta_1 + 1) q^{39} - q^{40} + (2 \beta_{7} + 2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{41} + (\beta_{5} + \beta_{3} + \beta_1 - 1) q^{42} + ( - \beta_{7} - 3 \beta_{6} - 4 \beta_{5} + \beta_{4} + \beta_{3} + 3 \beta_{2} + 3) q^{43} + (\beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 - 1) q^{44} + (\beta_{5} + \beta_{2}) q^{45} + ( - \beta_{7} - \beta_{6} - \beta_{4} - \beta_{3} + \beta_{2} + 1) q^{46} + (2 \beta_{7} + 4 \beta_{6} - 2 \beta_{3} - \beta_{2} - 2) q^{47} + (\beta_{6} - 1) q^{48} + (3 \beta_{6} + 3 \beta_{5} - \beta_{3} + 8 \beta_{2} - \beta_1 + 1) q^{49} - \beta_{5} q^{50} - 4 q^{51} + (\beta_{6} - 2 \beta_{5} - \beta_1 - 1) q^{52} + ( - \beta_{7} + 4 \beta_{5} + 2 \beta_{4} + \beta_{3} + \beta_{2} - 2 \beta_1 + 2) q^{53} + \beta_{5} q^{54} + ( - \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} - \beta_{2} + 1) q^{55} + ( - \beta_{7} + \beta_{4} + \beta_{3} - \beta_{2}) q^{56} + (\beta_{7} - \beta_{5} - \beta_{4} - 2 \beta_{3} + \beta_{2} - \beta_1 + 1) q^{57} + ( - \beta_{7} - \beta_{6} - 4 \beta_{5} + \beta_{4} - \beta_{3} - 3 \beta_{2} - 2 \beta_1 + 1) q^{58} + (2 \beta_{6} - \beta_{5} - \beta_{4} - \beta_{2} + \beta_1 + 2) q^{59} + \beta_{2} q^{60} + (4 \beta_{6} - 2 \beta_{5} + 2 \beta_{2} - 4) q^{61} + ( - \beta_{7} + \beta_{5} - \beta_{4} - \beta_{3} + 5 \beta_{2} - \beta_1 + 2) q^{62} + (\beta_{7} - \beta_{4} - \beta_{3} + \beta_{2} + \beta_1) q^{63} - q^{64} + (2 \beta_{6} + \beta_{4} + \beta_{2} - 2) q^{65} + ( - \beta_{5} - \beta_{4} - \beta_{3} + \beta_1) q^{66} + (2 \beta_{7} - 2 \beta_{6} + 4 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + \cdots + 4) q^{67}+ \cdots + ( - \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{3} + 4 q^{4} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{3} + 4 q^{4} - 4 q^{9} - 4 q^{10} + 6 q^{11} - 8 q^{12} - 12 q^{13} + 4 q^{14} - 4 q^{16} + 16 q^{17} - 6 q^{19} + 2 q^{22} + 4 q^{23} - 8 q^{25} - 12 q^{26} + 8 q^{27} - 8 q^{29} - 4 q^{30} - 6 q^{33} + 2 q^{35} + 4 q^{36} + 30 q^{37} + 6 q^{39} - 8 q^{40} - 2 q^{42} + 14 q^{43} - 6 q^{46} - 4 q^{48} + 14 q^{49} - 32 q^{51} - 6 q^{52} + 16 q^{53} - 2 q^{55} + 2 q^{56} - 6 q^{58} + 24 q^{59} - 16 q^{61} + 4 q^{62} - 8 q^{64} - 6 q^{65} - 4 q^{66} + 24 q^{67} - 16 q^{68} + 4 q^{69} - 12 q^{71} + 10 q^{74} + 4 q^{75} - 6 q^{76} + 16 q^{77} + 6 q^{78} - 20 q^{79} - 4 q^{81} + 4 q^{82} - 8 q^{87} - 2 q^{88} + 42 q^{89} + 8 q^{90} - 10 q^{91} + 8 q^{92} + 30 q^{93} - 8 q^{94} - 24 q^{97} + 48 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} + 2x^{6} + 30x^{5} + 185x^{4} + 36x^{3} + 8x^{2} + 208x + 2704 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 2225 \nu^{7} + 27304 \nu^{6} - 47714 \nu^{5} - 21770 \nu^{4} + 204731 \nu^{3} + 5488658 \nu^{2} + 258164 \nu - 320632 ) / 20561424 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1777 \nu^{7} - 125666 \nu^{6} + 800522 \nu^{5} - 2370014 \nu^{4} - 784473 \nu^{3} - 5277648 \nu^{2} + 34618460 \nu - 62031216 ) / 6853808 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 299 \nu^{7} + 7801 \nu^{6} - 39200 \nu^{5} + 84154 \nu^{4} + 80915 \nu^{3} + 360425 \nu^{2} - 1640788 \nu + 2274116 ) / 395412 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 43733 \nu^{7} - 71918 \nu^{6} - 318186 \nu^{5} + 3350390 \nu^{4} + 3714597 \nu^{3} - 2633192 \nu^{2} - 18392236 \nu + 94417440 ) / 20561424 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -151\nu^{7} + 822\nu^{6} - 2018\nu^{5} - 2554\nu^{4} - 7135\nu^{3} + 37828\nu^{2} - 46500\nu - 3328 ) / 51792 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 123974 \nu^{7} + 362933 \nu^{6} - 209286 \nu^{5} - 5793110 \nu^{4} - 10547568 \nu^{3} + 7562729 \nu^{2} + 3493156 \nu - 117816036 ) / 10280712 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} - 4\beta_{6} + \beta_{5} + \beta_{4} + 9\beta_{2} + \beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 6\beta_{7} - 10\beta_{6} + 26\beta_{5} + 10\beta_{4} + 10\beta_{3} + 18\beta_{2} + 3\beta _1 - 18 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 23\beta_{7} + 139\beta_{5} + 5\beta_{4} + 28\beta_{3} + 67\beta_{2} - 5\beta _1 - 149 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 250\beta_{6} + 322\beta_{5} - 72\beta_{4} + 72\beta_{3} - 76\beta_{2} - 149\beta _1 - 398 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -471\beta_{7} + 1748\beta_{6} - 619\beta_{5} - 619\beta_{4} - 148\beta_{3} - 2095\beta_{2} - 619\beta _1 - 255 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 2986 \beta_{7} + 5302 \beta_{6} - 11002 \beta_{5} - 2714 \beta_{4} - 2714 \beta_{3} - 10118 \beta_{2} - 1493 \beta _1 + 7530 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/390\mathbb{Z}\right)^\times\).

\(n\) \(131\) \(157\) \(301\)
\(\chi(n)\) \(1\) \(1\) \(1 - \beta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
121.1
3.17270 + 3.17270i
−1.80668 − 1.80668i
1.33404 − 1.33404i
−1.70006 + 1.70006i
3.17270 − 3.17270i
−1.80668 + 1.80668i
1.33404 + 1.33404i
−1.70006 − 1.70006i
−0.866025 − 0.500000i −0.500000 + 0.866025i 0.500000 + 0.866025i − 1.00000i 0.866025 − 0.500000i −2.01141 + 1.16129i − 1.00000i −0.500000 − 0.866025i −0.500000 + 0.866025i
121.2 −0.866025 − 0.500000i −0.500000 + 0.866025i 0.500000 + 0.866025i − 1.00000i 0.866025 − 0.500000i 1.14539 − 0.661290i − 1.00000i −0.500000 − 0.866025i −0.500000 + 0.866025i
121.3 0.866025 + 0.500000i −0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000i −0.866025 + 0.500000i −3.15637 + 1.82233i 1.00000i −0.500000 − 0.866025i −0.500000 + 0.866025i
121.4 0.866025 + 0.500000i −0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000i −0.866025 + 0.500000i 4.02239 − 2.32233i 1.00000i −0.500000 − 0.866025i −0.500000 + 0.866025i
361.1 −0.866025 + 0.500000i −0.500000 − 0.866025i 0.500000 − 0.866025i 1.00000i 0.866025 + 0.500000i −2.01141 − 1.16129i 1.00000i −0.500000 + 0.866025i −0.500000 − 0.866025i
361.2 −0.866025 + 0.500000i −0.500000 − 0.866025i 0.500000 − 0.866025i 1.00000i 0.866025 + 0.500000i 1.14539 + 0.661290i 1.00000i −0.500000 + 0.866025i −0.500000 − 0.866025i
361.3 0.866025 − 0.500000i −0.500000 − 0.866025i 0.500000 − 0.866025i − 1.00000i −0.866025 − 0.500000i −3.15637 − 1.82233i − 1.00000i −0.500000 + 0.866025i −0.500000 − 0.866025i
361.4 0.866025 − 0.500000i −0.500000 − 0.866025i 0.500000 − 0.866025i − 1.00000i −0.866025 − 0.500000i 4.02239 + 2.32233i − 1.00000i −0.500000 + 0.866025i −0.500000 − 0.866025i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 121.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 390.2.bb.c ✓ 8
3.b odd 2 1 1170.2.bs.f 8
5.b even 2 1 1950.2.bc.g 8
5.c odd 4 1 1950.2.y.j 8
5.c odd 4 1 1950.2.y.k 8
13.c even 3 1 5070.2.b.ba 8
13.e even 6 1 inner 390.2.bb.c ✓ 8
13.e even 6 1 5070.2.b.ba 8
13.f odd 12 1 5070.2.a.bz 4
13.f odd 12 1 5070.2.a.ca 4
39.h odd 6 1 1170.2.bs.f 8
65.l even 6 1 1950.2.bc.g 8
65.r odd 12 1 1950.2.y.j 8
65.r odd 12 1 1950.2.y.k 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.bb.c ✓ 8 1.a even 1 1 trivial
390.2.bb.c ✓ 8 13.e even 6 1 inner
1170.2.bs.f 8 3.b odd 2 1
1170.2.bs.f 8 39.h odd 6 1
1950.2.y.j 8 5.c odd 4 1
1950.2.y.j 8 65.r odd 12 1
1950.2.y.k 8 5.c odd 4 1
1950.2.y.k 8 65.r odd 12 1
1950.2.bc.g 8 5.b even 2 1
1950.2.bc.g 8 65.l even 6 1
5070.2.a.bz 4 13.f odd 12 1
5070.2.a.ca 4 13.f odd 12 1
5070.2.b.ba 8 13.c even 3 1
5070.2.b.ba 8 13.e even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{8} - 21T_{7}^{6} + 389T_{7}^{4} + 504T_{7}^{3} - 900T_{7}^{2} - 1248T_{7} + 2704 \) acting on \(S_{2}^{\mathrm{new}}(390, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} + T + 1)^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{8} - 21 T^{6} + 389 T^{4} + \cdots + 2704 \) Copy content Toggle raw display
$11$ \( T^{8} - 6 T^{7} - 18 T^{6} + \cdots + 32761 \) Copy content Toggle raw display
$13$ \( T^{8} + 12 T^{7} + 45 T^{6} + \cdots + 28561 \) Copy content Toggle raw display
$17$ \( (T^{2} - 4 T + 16)^{4} \) Copy content Toggle raw display
$19$ \( T^{8} + 6 T^{7} - 45 T^{6} + \cdots + 219024 \) Copy content Toggle raw display
$23$ \( T^{8} - 4 T^{7} + 59 T^{6} + \cdots + 2704 \) Copy content Toggle raw display
$29$ \( T^{8} + 8 T^{7} + 103 T^{6} + \cdots + 141376 \) Copy content Toggle raw display
$31$ \( T^{8} + 174 T^{6} + 8777 T^{4} + \cdots + 913936 \) Copy content Toggle raw display
$37$ \( T^{8} - 30 T^{7} + 366 T^{6} + \cdots + 644809 \) Copy content Toggle raw display
$41$ \( T^{8} - 84 T^{6} + 6224 T^{4} + \cdots + 692224 \) Copy content Toggle raw display
$43$ \( T^{8} - 14 T^{7} + 239 T^{6} + \cdots + 327184 \) Copy content Toggle raw display
$47$ \( T^{8} + 228 T^{6} + 16934 T^{4} + \cdots + 1216609 \) Copy content Toggle raw display
$53$ \( (T^{4} - 8 T^{3} - 75 T^{2} + 4 T + 52)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} - 24 T^{7} + 239 T^{6} + \cdots + 80656 \) Copy content Toggle raw display
$61$ \( (T^{4} + 8 T^{3} + 60 T^{2} + 32 T + 16)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} - 24 T^{7} + 164 T^{6} + \cdots + 123904 \) Copy content Toggle raw display
$71$ \( T^{8} + 12 T^{7} - 156 T^{6} + \cdots + 25240576 \) Copy content Toggle raw display
$73$ \( T^{8} + 392 T^{6} + \cdots + 20647936 \) Copy content Toggle raw display
$79$ \( (T^{4} + 10 T^{3} - 147 T^{2} - 860 T + 3508)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 456 T^{6} + \cdots + 25240576 \) Copy content Toggle raw display
$89$ \( T^{8} - 42 T^{7} + 759 T^{6} + \cdots + 1008016 \) Copy content Toggle raw display
$97$ \( T^{8} + 24 T^{7} + 20 T^{6} + \cdots + 29246464 \) Copy content Toggle raw display
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