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Properties

Label 63.11.m.b
Level $63$
Weight $11$
Character orbit 63.m
Analytic conductor $40.028$
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] = [63,11,Mod(10,63)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(63, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 11, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("63.10");
 
S:= CuspForms(chi, 11);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 63 = 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 63.m (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: \(40.0275069184\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{11} + 3773 x^{10} + 44516 x^{9} + 11068388 x^{8} + 100480832 x^{7} + 11177140432 x^{6} + \cdots + 30\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{7}\cdot 7^{7} \)
Twist minimal: no (minimal twist has level 21)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 2 \beta_{2} + \beta_1) q^{2} + (\beta_{7} + \beta_{5} + 5 \beta_{3} + \cdots - 236) q^{4}+ \cdots + (2 \beta_{11} + 7 \beta_{10} + \cdots - 8272) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \beta_{2} + \beta_1) q^{2} + (\beta_{7} + \beta_{5} + 5 \beta_{3} + \cdots - 236) q^{4}+ \cdots + ( - 235774 \beta_{11} + \cdots - 2758816823) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 11 q^{2} - 1421 q^{4} + 1287 q^{5} + 20090 q^{7} - 99310 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 11 q^{2} - 1421 q^{4} + 1287 q^{5} + 20090 q^{7} - 99310 q^{8} + 359661 q^{10} - 236165 q^{11} - 223832 q^{14} + 1350895 q^{16} + 2038782 q^{17} + 1012389 q^{19} + 29162074 q^{22} + 4341928 q^{23} + 12365439 q^{25} - 4398486 q^{26} + 37097165 q^{28} - 95444234 q^{29} + 9658932 q^{31} + 49606359 q^{32} + 77592186 q^{35} - 87545045 q^{37} + 319584648 q^{38} - 607773201 q^{40} + 700816306 q^{43} - 133692537 q^{44} - 74981552 q^{46} + 181842702 q^{47} + 333171090 q^{49} - 3398506976 q^{50} - 1573473588 q^{52} + 695152867 q^{53} - 1977142699 q^{56} - 1023653321 q^{58} - 1373785545 q^{59} - 2524633584 q^{61} - 482439838 q^{64} - 2692260666 q^{65} - 2255105709 q^{67} - 4313617758 q^{68} - 650925555 q^{70} + 3878911780 q^{71} - 1888675383 q^{73} + 445804820 q^{74} + 7759715803 q^{77} + 8260659900 q^{79} + 5477579577 q^{80} + 14912212206 q^{82} + 15563062356 q^{85} + 257575928 q^{86} - 10015831801 q^{88} - 9451951530 q^{89} - 3182715375 q^{91} - 20533179072 q^{92} - 54481927140 q^{94} + 33375267288 q^{95} + 3841749499 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - x^{11} + 3773 x^{10} + 44516 x^{9} + 11068388 x^{8} + 100480832 x^{7} + 11177140432 x^{6} + \cdots + 30\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 13\!\cdots\!85 \nu^{11} + \cdots + 55\!\cdots\!60 ) / 54\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 19\!\cdots\!86 \nu^{11} + \cdots - 24\!\cdots\!00 ) / 32\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 40\!\cdots\!73 \nu^{11} + \cdots + 25\!\cdots\!60 ) / 81\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 11\!\cdots\!23 \nu^{11} + \cdots + 12\!\cdots\!20 ) / 10\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 38\!\cdots\!59 \nu^{11} + \cdots - 41\!\cdots\!20 ) / 16\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 51\!\cdots\!55 \nu^{11} + \cdots - 21\!\cdots\!20 ) / 19\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 32\!\cdots\!91 \nu^{11} + \cdots - 74\!\cdots\!20 ) / 33\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 19\!\cdots\!31 \nu^{11} + \cdots - 19\!\cdots\!60 ) / 16\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 41\!\cdots\!91 \nu^{11} + \cdots + 76\!\cdots\!00 ) / 27\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 27\!\cdots\!63 \nu^{11} + \cdots - 54\!\cdots\!00 ) / 81\!\cdots\!40 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} + \beta_{5} + 9\beta_{3} + 1256\beta_{2} - 1256 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2 \beta_{11} + 7 \beta_{10} + 6 \beta_{9} - \beta_{8} + 2 \beta_{7} + 2 \beta_{6} + 19 \beta_{5} + \cdots - 11720 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 49 \beta_{11} + 92 \beta_{10} + 100 \beta_{9} + 49 \beta_{8} - 3041 \beta_{7} + 89 \beta_{6} + 46 \beta_{5} + \cdots - 46 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 4985 \beta_{11} - 11807 \beta_{10} - 18490 \beta_{9} + 9970 \beta_{8} - 85383 \beta_{7} + \cdots + 57905494 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 420370 \beta_{11} - 873419 \beta_{10} - 285046 \beta_{9} + 210185 \beta_{8} - 221078 \beta_{7} + \cdots + 6929260332 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 16872609 \beta_{11} - 41224652 \beta_{10} - 6142124 \beta_{9} - 16872609 \beta_{8} + 296313649 \beta_{7} + \cdots + 20612326 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 740063673 \beta_{11} + 1554437295 \beta_{10} + 165376074 \beta_{9} - 1480127346 \beta_{8} + \cdots - 19677039076598 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 105427992482 \beta_{11} + 238563843427 \beta_{10} + 158675910438 \beta_{9} - 52713996241 \beta_{8} + \cdots - 772687374238988 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 2468471109865 \beta_{11} + 5532921404972 \beta_{10} + 3298579732876 \beta_{9} + 2468471109865 \beta_{8} + \cdots - 2766460702486 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 162499451784257 \beta_{11} - 350726880646247 \beta_{10} - 357292586394394 \beta_{9} + \cdots + 25\!\cdots\!18 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/63\mathbb{Z}\right)^\times\).

\(n\) \(10\) \(29\)
\(\chi(n)\) \(\beta_{2}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1
−22.8056 − 39.5005i
−15.0682 − 26.0989i
−10.2102 − 17.6846i
5.98749 + 10.3706i
14.5243 + 25.1569i
28.0722 + 48.6225i
−22.8056 + 39.5005i
−15.0682 + 26.0989i
−10.2102 + 17.6846i
5.98749 − 10.3706i
14.5243 − 25.1569i
28.0722 − 48.6225i
−23.8056 − 41.2326i 0 −621.416 + 1076.32i −4637.22 + 2677.30i 0 −5755.42 − 15790.8i 10418.9 0 220784. + 127470.i
10.2 −16.0682 − 27.8309i 0 −4.37327 + 7.57472i 4433.83 − 2559.87i 0 10835.6 − 12847.8i −32626.6 0 −142487. − 82265.1i
10.3 −11.2102 − 19.4166i 0 260.663 − 451.481i 354.602 − 204.730i 0 −12291.7 + 11462.5i −34646.8 0 −7950.32 − 4590.12i
10.4 4.98749 + 8.63859i 0 462.250 − 800.640i −2618.43 + 1511.75i 0 13450.1 + 10078.2i 19436.3 0 −26118.8 − 15079.7i
10.5 13.5243 + 23.4248i 0 146.185 − 253.199i 1211.43 − 699.418i 0 −12945.1 − 10719.1i 35606.0 0 32767.5 + 18918.3i
10.6 27.0722 + 46.8904i 0 −953.808 + 1652.04i 1899.28 − 1096.55i 0 16751.6 − 1363.71i −47842.8 0 102836. + 59372.1i
19.1 −23.8056 + 41.2326i 0 −621.416 − 1076.32i −4637.22 − 2677.30i 0 −5755.42 + 15790.8i 10418.9 0 220784. − 127470.i
19.2 −16.0682 + 27.8309i 0 −4.37327 − 7.57472i 4433.83 + 2559.87i 0 10835.6 + 12847.8i −32626.6 0 −142487. + 82265.1i
19.3 −11.2102 + 19.4166i 0 260.663 + 451.481i 354.602 + 204.730i 0 −12291.7 − 11462.5i −34646.8 0 −7950.32 + 4590.12i
19.4 4.98749 − 8.63859i 0 462.250 + 800.640i −2618.43 − 1511.75i 0 13450.1 − 10078.2i 19436.3 0 −26118.8 + 15079.7i
19.5 13.5243 − 23.4248i 0 146.185 + 253.199i 1211.43 + 699.418i 0 −12945.1 + 10719.1i 35606.0 0 32767.5 − 18918.3i
19.6 27.0722 − 46.8904i 0 −953.808 − 1652.04i 1899.28 + 1096.55i 0 16751.6 + 1363.71i −47842.8 0 102836. − 59372.1i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 10.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 63.11.m.b 12
3.b odd 2 1 21.11.f.a ✓ 12
7.d odd 6 1 inner 63.11.m.b 12
21.g even 6 1 21.11.f.a ✓ 12
21.g even 6 1 147.11.d.a 12
21.h odd 6 1 147.11.d.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.11.f.a ✓ 12 3.b odd 2 1
21.11.f.a ✓ 12 21.g even 6 1
63.11.m.b 12 1.a even 1 1 trivial
63.11.m.b 12 7.d odd 6 1 inner
147.11.d.a 12 21.g even 6 1
147.11.d.a 12 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} + 11 T_{2}^{11} + 3843 T_{2}^{10} + 59914 T_{2}^{9} + 11481368 T_{2}^{8} + \cdots + 25\!\cdots\!00 \) acting on \(S_{11}^{\mathrm{new}}(63, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 50\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 73\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 62\!\cdots\!24 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 61\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots - 47\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 28\!\cdots\!49 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 14\!\cdots\!84 \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots - 26\!\cdots\!24)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 35\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 39\!\cdots\!64 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 95\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 86\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots + 36\!\cdots\!44)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 10\!\cdots\!64 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 42\!\cdots\!25 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 78\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 17\!\cdots\!44 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 28\!\cdots\!00 \) Copy content Toggle raw display
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