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Properties

Label 47.7.b.b
Level $47$
Weight $7$
Character orbit 47.b
Analytic conductor $10.813$
Analytic rank $0$
Dimension $18$
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] = [47,7,Mod(46,47)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(47, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("47.46");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 47 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 47.b (of order \(2\), degree \(1\), 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: \(10.8125419301\)
Analytic rank: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} + 194688 x^{16} + 16254958200 x^{14} + 763408746012000 x^{12} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{14}\cdot 3^{3}\cdot 5^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{2} + (\beta_{2} - 2) q^{3} + (\beta_{5} - \beta_{3} + \beta_{2} + 29) q^{4} + \beta_1 q^{5} + ( - \beta_{6} - 4 \beta_{2} - 41) q^{6} + ( - \beta_{4} + 8 \beta_{3} - 3 \beta_{2} + 7) q^{7} + ( - \beta_{8} - \beta_{6} + 4 \beta_{5} + \cdots + 35) q^{8}+ \cdots + (\beta_{7} + \beta_{4} + 33 \beta_{3} + \cdots - 54) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{2} + (\beta_{2} - 2) q^{3} + (\beta_{5} - \beta_{3} + \beta_{2} + 29) q^{4} + \beta_1 q^{5} + ( - \beta_{6} - 4 \beta_{2} - 41) q^{6} + ( - \beta_{4} + 8 \beta_{3} - 3 \beta_{2} + 7) q^{7} + ( - \beta_{8} - \beta_{6} + 4 \beta_{5} + \cdots + 35) q^{8}+ \cdots + (10 \beta_{17} - 47 \beta_{16} + \cdots + 1948 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 2 q^{2} - 34 q^{3} + 514 q^{4} - 740 q^{6} + 142 q^{7} + 566 q^{8} - 928 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q - 2 q^{2} - 34 q^{3} + 514 q^{4} - 740 q^{6} + 142 q^{7} + 566 q^{8} - 928 q^{9} + 5892 q^{12} - 11520 q^{14} + 962 q^{16} + 142 q^{17} - 46958 q^{18} - 35132 q^{21} + 2660 q^{24} - 108126 q^{25} - 60748 q^{27} + 12144 q^{28} + 335302 q^{32} + 219536 q^{34} - 167586 q^{36} + 15694 q^{37} + 435608 q^{42} + 385522 q^{47} + 244972 q^{48} - 419336 q^{49} - 259334 q^{50} - 109718 q^{51} + 377566 q^{53} + 535720 q^{54} - 208464 q^{55} - 476920 q^{56} - 977570 q^{59} - 500906 q^{61} - 1385528 q^{63} - 711246 q^{64} + 1048584 q^{65} + 2498824 q^{68} + 656614 q^{71} - 1000270 q^{72} - 188524 q^{74} - 1628938 q^{75} + 638686 q^{79} - 1594474 q^{81} - 921604 q^{83} + 452880 q^{84} + 769078 q^{89} - 3122438 q^{94} - 3636648 q^{95} + 4403700 q^{96} + 1013110 q^{97} + 889554 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} + 194688 x^{16} + 16254958200 x^{14} + 763408746012000 x^{12} + \cdots + 22\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 23\!\cdots\!63 \nu^{16} + \cdots + 10\!\cdots\!00 ) / 17\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 14\!\cdots\!01 \nu^{16} + \cdots - 16\!\cdots\!00 ) / 86\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 12\!\cdots\!87 \nu^{16} + \cdots - 87\!\cdots\!00 ) / 17\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 33\!\cdots\!51 \nu^{16} + \cdots - 30\!\cdots\!00 ) / 34\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 16\!\cdots\!09 \nu^{16} + \cdots + 16\!\cdots\!00 ) / 86\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 35\!\cdots\!39 \nu^{16} + \cdots + 48\!\cdots\!00 ) / 57\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 41\!\cdots\!23 \nu^{16} + \cdots + 55\!\cdots\!00 ) / 57\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 41\!\cdots\!37 \nu^{16} + \cdots - 28\!\cdots\!00 ) / 43\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 12\!\cdots\!69 \nu^{17} + \cdots + 32\!\cdots\!00 \nu ) / 25\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 19\!\cdots\!19 \nu^{17} + \cdots + 14\!\cdots\!00 \nu ) / 25\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 23\!\cdots\!63 \nu^{17} + \cdots - 10\!\cdots\!00 \nu ) / 17\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 14\!\cdots\!01 \nu^{17} + \cdots + 16\!\cdots\!00 \nu ) / 86\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 13\!\cdots\!53 \nu^{17} + \cdots + 11\!\cdots\!00 \nu ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 28\!\cdots\!09 \nu^{17} + \cdots - 33\!\cdots\!00 \nu ) / 25\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 16\!\cdots\!69 \nu^{17} + \cdots + 20\!\cdots\!00 \nu ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 44\!\cdots\!97 \nu^{17} + \cdots + 48\!\cdots\!00 \nu ) / 25\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{9} + 11\beta_{8} - 17\beta_{7} + 31\beta_{6} - 52\beta_{5} + 240\beta_{3} - 178\beta_{2} - 21647 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - 34 \beta_{17} + 45 \beta_{16} - 26 \beta_{15} + 83 \beta_{14} - 453 \beta_{13} + 125 \beta_{12} + \cdots - 28073 \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 96636 \beta_{9} - 559268 \beta_{8} + 707086 \beta_{7} - 1309188 \beta_{6} + 4046506 \beta_{5} + \cdots + 601867226 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 1327442 \beta_{17} - 2077960 \beta_{16} + 1256038 \beta_{15} - 5312534 \beta_{14} + \cdots + 923594874 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 4929581068 \beta_{9} + 25433029484 \beta_{8} - 26396420868 \beta_{7} + 51363518344 \beta_{6} + \cdots - 19683087646188 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 44839015096 \beta_{17} + 96214538880 \beta_{16} - 43353490444 \beta_{15} + \cdots - 33883841778712 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 252525021427584 \beta_{9} + \cdots + 71\!\cdots\!44 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 15\!\cdots\!48 \beta_{17} + \cdots + 13\!\cdots\!56 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 12\!\cdots\!92 \beta_{9} + \cdots - 28\!\cdots\!72 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 57\!\cdots\!24 \beta_{17} + \cdots - 55\!\cdots\!28 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 61\!\cdots\!96 \beta_{9} + \cdots + 11\!\cdots\!36 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 23\!\cdots\!12 \beta_{17} + \cdots + 23\!\cdots\!64 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 29\!\cdots\!48 \beta_{9} + \cdots - 49\!\cdots\!68 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 10\!\cdots\!56 \beta_{17} + \cdots - 10\!\cdots\!32 \beta_1 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 13\!\cdots\!24 \beta_{9} + \cdots + 21\!\cdots\!84 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 44\!\cdots\!28 \beta_{17} + \cdots + 45\!\cdots\!16 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/47\mathbb{Z}\right)^\times\).

\(n\) \(5\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
46.1
− 105.481i
105.481i
− 180.589i
180.589i
− 156.592i
156.592i
− 128.510i
128.510i
− 141.877i
141.877i
− 159.725i
159.725i
− 120.736i
120.736i
− 67.0172i
67.0172i
− 212.613i
212.613i
−13.1407 15.9829 108.677 − 105.481i −210.026 328.023 −587.086 −473.548 1386.09i
46.2 −13.1407 15.9829 108.677 105.481i −210.026 328.023 −587.086 −473.548 − 1386.09i
46.3 −10.8998 39.0739 54.8052 − 180.589i −425.896 −360.691 100.221 797.766 1968.38i
46.4 −10.8998 39.0739 54.8052 180.589i −425.896 −360.691 100.221 797.766 − 1968.38i
46.5 −10.6935 −38.8036 50.3519 − 156.592i 414.948 204.814 145.947 776.717 1674.52i
46.6 −10.6935 −38.8036 50.3519 156.592i 414.948 204.814 145.947 776.717 − 1674.52i
46.7 −4.36238 1.37302 −44.9696 − 128.510i −5.98965 204.386 475.367 −727.115 560.608i
46.8 −4.36238 1.37302 −44.9696 128.510i −5.98965 204.386 475.367 −727.115 − 560.608i
46.9 1.43377 −39.1840 −61.9443 − 141.877i −56.1806 −25.2326 −180.575 806.384 − 203.418i
46.10 1.43377 −39.1840 −61.9443 141.877i −56.1806 −25.2326 −180.575 806.384 203.418i
46.11 3.61110 24.3746 −50.9600 − 159.725i 88.0192 −379.174 −415.132 −134.878 − 576.783i
46.12 3.61110 24.3746 −50.9600 159.725i 88.0192 −379.174 −415.132 −134.878 576.783i
46.13 7.30818 −2.59761 −10.5905 − 120.736i −18.9838 503.172 −545.121 −722.252 − 882.359i
46.14 7.30818 −2.59761 −10.5905 120.736i −18.9838 503.172 −545.121 −722.252 882.359i
46.15 10.8381 −24.7755 53.4641 − 67.0172i −268.520 −357.767 −114.189 −115.172 − 726.338i
46.16 10.8381 −24.7755 53.4641 67.0172i −268.520 −357.767 −114.189 −115.172 726.338i
46.17 14.9052 7.55632 158.166 − 212.613i 112.629 −46.5315 1403.57 −671.902 − 3169.04i
46.18 14.9052 7.55632 158.166 212.613i 112.629 −46.5315 1403.57 −671.902 3169.04i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 46.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
47.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 47.7.b.b ✓ 18
47.b odd 2 1 inner 47.7.b.b ✓ 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
47.7.b.b ✓ 18 1.a even 1 1 trivial
47.7.b.b ✓ 18 47.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{9} + T_{2}^{8} - 416 T_{2}^{7} - 468 T_{2}^{6} + 55588 T_{2}^{5} + 33348 T_{2}^{4} + \cdots - 40841216 \) acting on \(S_{7}^{\mathrm{new}}(47, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{9} + T^{8} + \cdots - 40841216)^{2} \) Copy content Toggle raw display
$3$ \( (T^{9} + 17 T^{8} + \cdots - 15454134504)^{2} \) Copy content Toggle raw display
$5$ \( T^{18} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T^{9} + \cdots + 39\!\cdots\!64)^{2} \) Copy content Toggle raw display
$11$ \( T^{18} + \cdots + 98\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{18} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( (T^{9} + \cdots - 71\!\cdots\!88)^{2} \) Copy content Toggle raw display
$19$ \( T^{18} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots + 46\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{18} + \cdots + 99\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{9} + \cdots - 38\!\cdots\!40)^{2} \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots + 61\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 38\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 19\!\cdots\!69 \) Copy content Toggle raw display
$53$ \( (T^{9} + \cdots + 67\!\cdots\!16)^{2} \) Copy content Toggle raw display
$59$ \( (T^{9} + \cdots - 20\!\cdots\!80)^{2} \) Copy content Toggle raw display
$61$ \( (T^{9} + \cdots + 55\!\cdots\!32)^{2} \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 83\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{9} + \cdots - 14\!\cdots\!16)^{2} \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{9} + \cdots - 10\!\cdots\!16)^{2} \) Copy content Toggle raw display
$83$ \( (T^{9} + \cdots + 92\!\cdots\!12)^{2} \) Copy content Toggle raw display
$89$ \( (T^{9} + \cdots - 64\!\cdots\!20)^{2} \) Copy content Toggle raw display
$97$ \( (T^{9} + \cdots - 67\!\cdots\!12)^{2} \) Copy content Toggle raw display
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