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Properties

Label 880.3.n.a
Level $880$
Weight $3$
Character orbit 880.n
Analytic conductor $23.978$
Analytic rank $0$
Dimension $20$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [880,3,Mod(639,880)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(880, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1, 0])) N = Newforms(chi, 3, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("880.639"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Level: \( N \) \(=\) \( 880 = 2^{4} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 880.n (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [20] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(23.9782632637\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 10 x^{19} - 35 x^{18} + 600 x^{17} + 281 x^{16} - 17140 x^{15} + 16590 x^{14} + \cdots + 10140467375 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{20}\cdot 5^{4}\cdot 11 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{10} q^{3} - \beta_{3} q^{5} + (\beta_{12} - \beta_{10}) q^{7} + ( - \beta_1 + 2) q^{9} + \beta_{11} q^{11} - \beta_{4} q^{13} + ( - \beta_{16} - \beta_{14} + \cdots - \beta_{11}) q^{15} + (\beta_{7} + \beta_{5} - \beta_{3}) q^{17}+ \cdots + ( - \beta_{17} + \beta_{16} + \cdots + 2 \beta_{11}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 10 q^{5} + 40 q^{9} + 184 q^{21} - 18 q^{25} + 16 q^{29} + 144 q^{41} + 16 q^{45} + 36 q^{49} - 384 q^{61} - 32 q^{65} - 244 q^{69} + 236 q^{81} - 348 q^{85} - 100 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 10 x^{19} - 35 x^{18} + 600 x^{17} + 281 x^{16} - 17140 x^{15} + 16590 x^{14} + \cdots + 10140467375 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 1081703611281 \nu^{18} - 9735332501529 \nu^{17} - 43096714616065 \nu^{16} + \cdots + 20\!\cdots\!75 ) / 19\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 990330923125237 \nu^{18} + \cdots - 18\!\cdots\!25 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 311641633881841 \nu^{18} + \cdots + 74\!\cdots\!50 ) / 29\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 17\!\cdots\!49 \nu^{18} + \cdots - 10\!\cdots\!50 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 57\!\cdots\!87 \nu^{18} + \cdots + 60\!\cdots\!25 ) / 23\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 642243997345056 \nu^{18} + \cdots - 25\!\cdots\!25 ) / 23\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 10\!\cdots\!91 \nu^{18} + \cdots - 15\!\cdots\!75 ) / 23\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 585685694136534 \nu^{18} + \cdots + 15\!\cdots\!75 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 19\!\cdots\!03 \nu^{18} + \cdots - 10\!\cdots\!25 ) / 23\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 64\!\cdots\!78 \nu^{19} + \cdots - 81\!\cdots\!25 ) / 36\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 64\!\cdots\!78 \nu^{19} + \cdots - 17\!\cdots\!75 ) / 18\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 28\!\cdots\!74 \nu^{19} + \cdots + 67\!\cdots\!75 ) / 42\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 60\!\cdots\!42 \nu^{19} + \cdots + 73\!\cdots\!00 ) / 71\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 10\!\cdots\!66 \nu^{19} + \cdots + 81\!\cdots\!75 ) / 85\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 54\!\cdots\!34 \nu^{19} + \cdots - 88\!\cdots\!75 ) / 42\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 35\!\cdots\!88 \nu^{19} + \cdots + 36\!\cdots\!25 ) / 21\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 43\!\cdots\!82 \nu^{19} + \cdots - 57\!\cdots\!75 ) / 85\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 17\!\cdots\!14 \nu^{19} + \cdots + 17\!\cdots\!25 ) / 28\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 19\!\cdots\!14 \nu^{19} + \cdots - 11\!\cdots\!25 ) / 28\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{11} + 2\beta_{10} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{11} + 2\beta_{10} + 2\beta_{5} - 2\beta_{3} + 2\beta_{2} - 2\beta _1 + 17 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2 \beta_{18} - 3 \beta_{17} + 5 \beta_{16} + \beta_{15} + 9 \beta_{14} - 4 \beta_{13} - 2 \beta_{12} + \cdots + 25 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 4 \beta_{18} - 6 \beta_{17} + 10 \beta_{16} + 2 \beta_{15} + 18 \beta_{14} - 8 \beta_{13} - 4 \beta_{12} + \cdots + 135 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 20 \beta_{19} + 12 \beta_{18} - 95 \beta_{17} + 169 \beta_{16} + 121 \beta_{15} + 365 \beta_{14} + \cdots + 296 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 30 \beta_{19} + 13 \beta_{18} - 135 \beta_{17} + 241 \beta_{16} + 179 \beta_{15} + 525 \beta_{14} + \cdots - 2474 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 980 \beta_{19} - 1272 \beta_{18} - 2394 \beta_{17} + 3296 \beta_{16} + 5916 \beta_{15} + 9562 \beta_{14} + \cdots - 18325 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 3640 \beta_{19} - 5200 \beta_{18} - 8330 \beta_{17} + 10958 \beta_{16} + 21998 \beta_{15} + \cdots - 380017 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 26532 \beta_{19} - 80304 \beta_{18} - 39885 \beta_{17} - 13309 \beta_{16} + 200947 \beta_{15} + \cdots - 1598933 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 105780 \beta_{19} - 362358 \beta_{18} - 138810 \beta_{17} - 145406 \beta_{16} + 842246 \beta_{15} + \cdots - 15650773 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 192940 \beta_{19} - 3144166 \beta_{18} + 243485 \beta_{17} - 4648853 \beta_{16} + 4693931 \beta_{15} + \cdots - 71627060 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 26400 \beta_{19} - 7482682 \beta_{18} + 1428108 \beta_{17} - 13061642 \beta_{16} + 9627002 \beta_{15} + \cdots - 238512747 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 24538176 \beta_{19} - 89937220 \beta_{18} + 57129800 \beta_{17} - 254419992 \beta_{16} + \cdots - 2206860967 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 169292760 \beta_{19} - 414302432 \beta_{18} + 352526720 \beta_{17} - 1390324524 \beta_{16} + \cdots - 10212600103 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 1737140520 \beta_{19} - 1563896070 \beta_{18} + 3029998209 \beta_{17} - 9312730527 \beta_{16} + \cdots - 41046920755 ) / 2 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( - 10526709320 \beta_{19} - 5066173808 \beta_{18} + 17385782282 \beta_{17} - 48248029046 \beta_{16} + \cdots - 55278524161 ) / 2 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( - 72686343284 \beta_{19} + 15927287824 \beta_{18} + 111042747525 \beta_{17} - 238865214795 \beta_{16} + \cdots + 305289659404 ) / 2 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 201447347490 \beta_{19} + 118188837927 \beta_{18} + 295283227965 \beta_{17} - 524987494121 \beta_{16} + \cdots + 4295753864986 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( - 2130411234140 \beta_{19} + 2863679103036 \beta_{18} + 2955273209598 \beta_{17} + \cdots + 68871433256275 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/880\mathbb{Z}\right)^\times\).

\(n\) \(111\) \(177\) \(321\) \(661\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
639.1
5.71178 − 1.65831i
5.71178 + 1.65831i
4.86992 − 1.65831i
4.86992 + 1.65831i
2.79160 + 1.65831i
2.79160 − 1.65831i
2.21475 − 1.65831i
2.21475 + 1.65831i
1.24119 + 1.65831i
1.24119 − 1.65831i
−0.241189 − 1.65831i
−0.241189 + 1.65831i
−1.21475 + 1.65831i
−1.21475 − 1.65831i
−1.79160 − 1.65831i
−1.79160 + 1.65831i
−3.86992 + 1.65831i
−3.86992 − 1.65831i
−4.71178 + 1.65831i
−4.71178 − 1.65831i
0 −5.21178 0 2.33138 − 4.42320i 0 −8.94099 0 18.1626 0
639.2 0 −5.21178 0 2.33138 + 4.42320i 0 −8.94099 0 18.1626 0
639.3 0 −4.36992 0 −4.68346 − 1.75078i 0 1.67515 0 10.0962 0
639.4 0 −4.36992 0 −4.68346 + 1.75078i 0 1.67515 0 10.0962 0
639.5 0 −2.29160 0 −0.507339 − 4.97419i 0 3.40751 0 −3.74858 0
639.6 0 −2.29160 0 −0.507339 + 4.97419i 0 3.40751 0 −3.74858 0
639.7 0 −1.71475 0 4.21443 − 2.69047i 0 −3.18929 0 −6.05965 0
639.8 0 −1.71475 0 4.21443 + 2.69047i 0 −3.18929 0 −6.05965 0
639.9 0 −0.741189 0 −3.85501 − 3.18416i 0 −12.2258 0 −8.45064 0
639.10 0 −0.741189 0 −3.85501 + 3.18416i 0 −12.2258 0 −8.45064 0
639.11 0 0.741189 0 −3.85501 − 3.18416i 0 12.2258 0 −8.45064 0
639.12 0 0.741189 0 −3.85501 + 3.18416i 0 12.2258 0 −8.45064 0
639.13 0 1.71475 0 4.21443 − 2.69047i 0 3.18929 0 −6.05965 0
639.14 0 1.71475 0 4.21443 + 2.69047i 0 3.18929 0 −6.05965 0
639.15 0 2.29160 0 −0.507339 − 4.97419i 0 −3.40751 0 −3.74858 0
639.16 0 2.29160 0 −0.507339 + 4.97419i 0 −3.40751 0 −3.74858 0
639.17 0 4.36992 0 −4.68346 − 1.75078i 0 −1.67515 0 10.0962 0
639.18 0 4.36992 0 −4.68346 + 1.75078i 0 −1.67515 0 10.0962 0
639.19 0 5.21178 0 2.33138 − 4.42320i 0 8.94099 0 18.1626 0
639.20 0 5.21178 0 2.33138 + 4.42320i 0 8.94099 0 18.1626 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 639.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
20.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 880.3.n.a ✓ 20
4.b odd 2 1 inner 880.3.n.a ✓ 20
5.b even 2 1 inner 880.3.n.a ✓ 20
20.d odd 2 1 inner 880.3.n.a ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
880.3.n.a ✓ 20 1.a even 1 1 trivial
880.3.n.a ✓ 20 4.b odd 2 1 inner
880.3.n.a ✓ 20 5.b even 2 1 inner
880.3.n.a ✓ 20 20.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{10} - 55T_{3}^{8} + 943T_{3}^{6} - 5465T_{3}^{4} + 10736T_{3}^{2} - 4400 \) acting on \(S_{3}^{\mathrm{new}}(880, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( (T^{10} - 55 T^{8} + \cdots - 4400)^{2} \) Copy content Toggle raw display
$5$ \( (T^{10} + 5 T^{9} + \cdots + 9765625)^{2} \) Copy content Toggle raw display
$7$ \( (T^{10} - 254 T^{8} + \cdots - 3960000)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 11)^{10} \) Copy content Toggle raw display
$13$ \( (T^{10} + 820 T^{8} + \cdots + 18270388224)^{2} \) Copy content Toggle raw display
$17$ \( (T^{10} + 2274 T^{8} + \cdots + 49529392704)^{2} \) Copy content Toggle raw display
$19$ \( (T^{10} + \cdots + 7122467303424)^{2} \) Copy content Toggle raw display
$23$ \( (T^{10} - 2441 T^{8} + \cdots - 17375440896)^{2} \) Copy content Toggle raw display
$29$ \( (T^{5} - 4 T^{4} + \cdots + 180400)^{4} \) Copy content Toggle raw display
$31$ \( (T^{10} + \cdots + 16\!\cdots\!00)^{2} \) Copy content Toggle raw display
$37$ \( (T^{10} + \cdots + 63894315665664)^{2} \) Copy content Toggle raw display
$41$ \( (T^{5} - 36 T^{4} + \cdots - 9662720)^{4} \) Copy content Toggle raw display
$43$ \( (T^{10} + \cdots - 2142319094784)^{2} \) Copy content Toggle raw display
$47$ \( (T^{10} + \cdots - 13\!\cdots\!84)^{2} \) Copy content Toggle raw display
$53$ \( (T^{10} + \cdots + 45\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( (T^{10} + \cdots + 96\!\cdots\!16)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} + 96 T^{4} + \cdots + 1705002736)^{4} \) Copy content Toggle raw display
$67$ \( (T^{10} + \cdots - 21\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( (T^{10} + \cdots + 90619033595904)^{2} \) Copy content Toggle raw display
$73$ \( (T^{10} + \cdots + 21\!\cdots\!96)^{2} \) Copy content Toggle raw display
$79$ \( (T^{10} + \cdots + 57\!\cdots\!84)^{2} \) Copy content Toggle raw display
$83$ \( (T^{10} + \cdots - 20\!\cdots\!44)^{2} \) Copy content Toggle raw display
$89$ \( (T^{5} + 25 T^{4} + \cdots - 261449340)^{4} \) Copy content Toggle raw display
$97$ \( (T^{10} + \cdots + 42\!\cdots\!00)^{2} \) Copy content Toggle raw display
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