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Properties

Label 52.5.k.a
Level $52$
Weight $5$
Character orbit 52.k
Analytic conductor $5.375$
Analytic rank $0$
Dimension $20$
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] = [52,5,Mod(33,52)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(52, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 11]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("52.33");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 52 = 2^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 52.k (of order \(12\), degree \(4\), 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: \(5.37523808036\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(5\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 10 x^{19} + 1171 x^{18} - 10254 x^{17} + 544443 x^{16} - 4124412 x^{15} + 127792914 x^{14} + \cdots + 66\!\cdots\!01 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{4}\cdot 13^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{8} - \beta_{7}) q^{3} + ( - \beta_{12} - \beta_{9} - \beta_{8} + \cdots - 1) q^{5}+ \cdots + (\beta_{19} - \beta_{16} - \beta_{14} + \cdots - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{8} - \beta_{7}) q^{3} + ( - \beta_{12} - \beta_{9} - \beta_{8} + \cdots - 1) q^{5}+ \cdots + ( - 84 \beta_{19} - 76 \beta_{18} + \cdots - 1904) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 12 q^{5} + 28 q^{7} - 324 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 12 q^{5} + 28 q^{7} - 324 q^{9} - 384 q^{11} - 102 q^{13} + 1020 q^{15} - 450 q^{17} - 470 q^{19} + 1806 q^{21} + 1008 q^{23} - 1764 q^{27} - 480 q^{29} - 3232 q^{31} - 1506 q^{33} + 2406 q^{35} + 854 q^{37} - 1944 q^{39} + 5340 q^{41} + 7086 q^{43} + 6234 q^{45} + 144 q^{47} - 2790 q^{49} - 3828 q^{53} - 7974 q^{55} - 18846 q^{57} - 11208 q^{59} + 3696 q^{61} - 31452 q^{63} - 18300 q^{65} + 33406 q^{67} + 45042 q^{69} + 17202 q^{71} + 32236 q^{73} + 42840 q^{75} - 5856 q^{79} - 54606 q^{81} - 66792 q^{83} - 31764 q^{85} + 2844 q^{87} - 37680 q^{89} - 31256 q^{91} + 89562 q^{93} + 77688 q^{95} + 12356 q^{97} + 34272 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 10 x^{19} + 1171 x^{18} - 10254 x^{17} + 544443 x^{16} - 4124412 x^{15} + 127792914 x^{14} + \cdots + 66\!\cdots\!01 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 30\!\cdots\!53 \nu^{19} + \cdots + 33\!\cdots\!36 ) / 18\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 61\!\cdots\!06 \nu^{19} + \cdots + 77\!\cdots\!70 ) / 18\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 22\!\cdots\!13 \nu^{19} + \cdots + 16\!\cdots\!16 ) / 52\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 63\!\cdots\!56 \nu^{19} + \cdots - 20\!\cdots\!93 ) / 14\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 63\!\cdots\!56 \nu^{19} + \cdots - 34\!\cdots\!47 ) / 14\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 49\!\cdots\!36 \nu^{19} + \cdots + 64\!\cdots\!72 ) / 11\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 53\!\cdots\!18 \nu^{19} + \cdots + 73\!\cdots\!09 ) / 14\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 10\!\cdots\!72 \nu^{19} + \cdots - 35\!\cdots\!39 ) / 15\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 10\!\cdots\!33 \nu^{19} + \cdots + 20\!\cdots\!86 ) / 15\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 85\!\cdots\!77 \nu^{19} + \cdots + 45\!\cdots\!38 ) / 12\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 85\!\cdots\!77 \nu^{19} + \cdots - 91\!\cdots\!92 ) / 12\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 15\!\cdots\!70 \nu^{19} + \cdots + 28\!\cdots\!00 ) / 15\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 12\!\cdots\!90 \nu^{19} + \cdots + 18\!\cdots\!97 ) / 12\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 11\!\cdots\!85 \nu^{19} + \cdots + 61\!\cdots\!70 ) / 52\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 88\!\cdots\!39 \nu^{19} + \cdots - 34\!\cdots\!93 ) / 15\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 88\!\cdots\!39 \nu^{19} + \cdots + 29\!\cdots\!67 ) / 15\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 96\!\cdots\!33 \nu^{19} + \cdots - 32\!\cdots\!24 ) / 15\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 96\!\cdots\!33 \nu^{19} + \cdots - 15\!\cdots\!37 ) / 15\!\cdots\!08 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{19} + \beta_{18} + \beta_{17} - \beta_{16} + \beta_{12} - \beta_{11} + \beta_{10} + \beta_{9} + \cdots - 116 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 5 \beta_{19} - 2 \beta_{18} + 4 \beta_{17} + \beta_{16} + 3 \beta_{14} + 17 \beta_{13} + 19 \beta_{12} + \cdots - 92 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 232 \beta_{19} - 246 \beta_{18} - 263 \beta_{17} + 273 \beta_{16} + 20 \beta_{15} + 6 \beta_{14} + \cdots + 26919 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 1672 \beta_{19} + 472 \beta_{18} - 1561 \beta_{17} - 216 \beta_{16} + 74 \beta_{15} - 882 \beta_{14} + \cdots + 53157 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 47435 \beta_{19} + 53902 \beta_{18} + 61291 \beta_{17} - 66647 \beta_{16} - 9609 \beta_{15} + \cdots - 6828987 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 477848 \beta_{19} - 118965 \beta_{18} + 500923 \beta_{17} + 48429 \beta_{16} - 33253 \beta_{15} + \cdots - 20999760 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 9209162 \beta_{19} - 11626626 \beta_{18} - 14082443 \beta_{17} + 16304869 \beta_{16} + \cdots + 1770351866 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 130677816 \beta_{19} + 34758426 \beta_{18} - 150058761 \beta_{17} - 10595238 \beta_{16} + \cdots + 7333123271 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 1670274654 \beta_{19} + 2515632183 \beta_{18} + 3238784688 \beta_{17} - 4058760420 \beta_{16} + \cdots - 464208013251 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 34970815493 \beta_{19} - 11065105885 \beta_{18} + 43452650771 \beta_{17} + 2032744804 \beta_{16} + \cdots - 2415839800516 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 265221873506 \beta_{19} - 550776365254 \beta_{18} - 747244675996 \beta_{17} + 1029213570524 \beta_{16} + \cdots + 122763825096251 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 9225575095562 \beta_{19} + 3608511525444 \beta_{18} - 12351324303946 \beta_{17} + \cdots + 770754185506008 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 28906340362765 \beta_{19} + 123104236761287 \beta_{18} + 172885961225911 \beta_{17} + \cdots - 32\!\cdots\!40 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 24\!\cdots\!35 \beta_{19} + \cdots - 24\!\cdots\!08 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 21\!\cdots\!30 \beta_{19} + \cdots + 87\!\cdots\!41 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 62\!\cdots\!40 \beta_{19} + \cdots + 74\!\cdots\!93 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 29\!\cdots\!55 \beta_{19} + \cdots - 23\!\cdots\!05 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 15\!\cdots\!34 \beta_{19} + \cdots - 22\!\cdots\!20 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/52\mathbb{Z}\right)^\times\).

\(n\) \(27\) \(41\)
\(\chi(n)\) \(1\) \(\beta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
33.1
0.500000 − 16.8270i
0.500000 − 2.80248i
0.500000 − 2.59124i
0.500000 + 6.23193i
0.500000 + 16.8548i
0.500000 − 14.4254i
0.500000 − 6.17953i
0.500000 − 0.680052i
0.500000 + 5.32836i
0.500000 + 15.0906i
0.500000 + 16.8270i
0.500000 + 2.80248i
0.500000 + 2.59124i
0.500000 − 6.23193i
0.500000 − 16.8548i
0.500000 + 14.4254i
0.500000 + 6.17953i
0.500000 + 0.680052i
0.500000 − 5.32836i
0.500000 − 15.0906i
0 −7.98050 + 13.8226i 0 −5.17458 − 5.17458i 0 −10.1947 − 2.73167i 0 −86.8766 − 150.475i 0
33.2 0 −0.968229 + 1.67702i 0 20.8824 + 20.8824i 0 −51.8388 − 13.8902i 0 38.6251 + 66.9006i 0
33.3 0 −0.862610 + 1.49408i 0 −4.42596 − 4.42596i 0 80.4552 + 21.5579i 0 39.0118 + 67.5704i 0
33.4 0 3.54898 − 6.14701i 0 −20.6301 − 20.6301i 0 −46.4829 − 12.4551i 0 15.3095 + 26.5169i 0
33.5 0 8.86044 − 15.3467i 0 27.9367 + 27.9367i 0 35.9272 + 9.62668i 0 −116.515 − 201.809i 0
37.1 0 −7.64572 − 13.2428i 0 −16.1206 − 16.1206i 0 21.7640 + 81.2244i 0 −76.4142 + 132.353i 0
37.2 0 −3.52278 − 6.10163i 0 14.9357 + 14.9357i 0 −20.0515 − 74.8331i 0 15.6801 − 27.1587i 0
37.3 0 −0.773038 − 1.33894i 0 12.9496 + 12.9496i 0 9.74878 + 36.3829i 0 39.3048 − 68.0780i 0
37.4 0 2.23117 + 3.86449i 0 −32.6499 − 32.6499i 0 −6.89945 − 25.7491i 0 30.5438 − 52.9034i 0
37.5 0 7.11230 + 12.3189i 0 8.29678 + 8.29678i 0 1.57208 + 5.86709i 0 −60.6696 + 105.083i 0
41.1 0 −7.98050 − 13.8226i 0 −5.17458 + 5.17458i 0 −10.1947 + 2.73167i 0 −86.8766 + 150.475i 0
41.2 0 −0.968229 − 1.67702i 0 20.8824 − 20.8824i 0 −51.8388 + 13.8902i 0 38.6251 − 66.9006i 0
41.3 0 −0.862610 − 1.49408i 0 −4.42596 + 4.42596i 0 80.4552 − 21.5579i 0 39.0118 − 67.5704i 0
41.4 0 3.54898 + 6.14701i 0 −20.6301 + 20.6301i 0 −46.4829 + 12.4551i 0 15.3095 − 26.5169i 0
41.5 0 8.86044 + 15.3467i 0 27.9367 − 27.9367i 0 35.9272 − 9.62668i 0 −116.515 + 201.809i 0
45.1 0 −7.64572 + 13.2428i 0 −16.1206 + 16.1206i 0 21.7640 − 81.2244i 0 −76.4142 − 132.353i 0
45.2 0 −3.52278 + 6.10163i 0 14.9357 − 14.9357i 0 −20.0515 + 74.8331i 0 15.6801 + 27.1587i 0
45.3 0 −0.773038 + 1.33894i 0 12.9496 − 12.9496i 0 9.74878 − 36.3829i 0 39.3048 + 68.0780i 0
45.4 0 2.23117 − 3.86449i 0 −32.6499 + 32.6499i 0 −6.89945 + 25.7491i 0 30.5438 + 52.9034i 0
45.5 0 7.11230 − 12.3189i 0 8.29678 − 8.29678i 0 1.57208 − 5.86709i 0 −60.6696 − 105.083i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 33.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.f odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 52.5.k.a ✓ 20
3.b odd 2 1 468.5.cd.d 20
13.f odd 12 1 inner 52.5.k.a ✓ 20
39.k even 12 1 468.5.cd.d 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
52.5.k.a ✓ 20 1.a even 1 1 trivial
52.5.k.a ✓ 20 13.f odd 12 1 inner
468.5.cd.d 20 3.b odd 2 1
468.5.cd.d 20 39.k even 12 1

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(52, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( T^{20} + \cdots + 50\!\cdots\!24 \) Copy content Toggle raw display
$5$ \( T^{20} + \cdots + 55\!\cdots\!16 \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots + 11\!\cdots\!24 \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 29\!\cdots\!24 \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 36\!\cdots\!01 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 21\!\cdots\!29 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 64\!\cdots\!36 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 56\!\cdots\!04 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 75\!\cdots\!01 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 43\!\cdots\!76 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 24\!\cdots\!89 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 26\!\cdots\!25 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 28\!\cdots\!16 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 18\!\cdots\!44 \) Copy content Toggle raw display
$53$ \( (T^{10} + \cdots + 26\!\cdots\!28)^{2} \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 46\!\cdots\!64 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 17\!\cdots\!69 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 16\!\cdots\!44 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 59\!\cdots\!56 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 95\!\cdots\!04 \) Copy content Toggle raw display
$79$ \( (T^{10} + \cdots - 38\!\cdots\!32)^{2} \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 10\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 22\!\cdots\!56 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 14\!\cdots\!36 \) Copy content Toggle raw display
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