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Properties

Label 52.6.h.a
Level $52$
Weight $6$
Character orbit 52.h
Analytic conductor $8.340$
Analytic rank $0$
Dimension $10$
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] = [52,6,Mod(17,52)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(52, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("52.17");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 52 = 2^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 52.h (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: \(8.33995863027\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2 x^{9} - 311 x^{8} - 7364 x^{7} + 10751 x^{6} + 888384 x^{5} + 18980275 x^{4} + \cdots + 19967455989 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{10}\cdot 3\cdot 13^{2} \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{4} + 2 \beta_1) q^{3} + (\beta_{8} - \beta_{2} + 12 \beta_1 - 6) q^{5} + (\beta_{9} - \beta_{4} - \beta_{3} + \cdots - 5) q^{7}+ \cdots + ( - \beta_{8} + \beta_{7} - \beta_{5} + \cdots - 10) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{4} + 2 \beta_1) q^{3} + (\beta_{8} - \beta_{2} + 12 \beta_1 - 6) q^{5} + (\beta_{9} - \beta_{4} - \beta_{3} + \cdots - 5) q^{7}+ \cdots + (286 \beta_{8} - 183 \beta_{7} + \cdots + 17408) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 9 q^{3} - 87 q^{7} - 52 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 9 q^{3} - 87 q^{7} - 52 q^{9} - 297 q^{11} - 780 q^{13} - 1056 q^{15} - 827 q^{17} - 111 q^{19} + 1801 q^{23} - 3338 q^{25} + 3114 q^{27} + 2789 q^{29} - 4875 q^{33} + 12424 q^{35} + 3627 q^{37} - 10439 q^{39} + 3 q^{41} - 1925 q^{43} + 24750 q^{45} + 14982 q^{49} - 4646 q^{51} + 40632 q^{53} + 15960 q^{55} - 71175 q^{59} + 7901 q^{61} - 128502 q^{63} - 45890 q^{65} - 10821 q^{67} - 61211 q^{69} + 77973 q^{71} - 50683 q^{75} + 114750 q^{77} + 150792 q^{79} + 143387 q^{81} - 169002 q^{85} + 63995 q^{87} - 426687 q^{89} + 68237 q^{91} - 31470 q^{93} - 52800 q^{95} + 273609 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 2 x^{9} - 311 x^{8} - 7364 x^{7} + 10751 x^{6} + 888384 x^{5} + 18980275 x^{4} + \cdots + 19967455989 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 30773149710427 \nu^{9} + 205402957780069 \nu^{8} + \cdots + 17\!\cdots\!76 ) / 30\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 93\!\cdots\!97 \nu^{9} + \cdots - 39\!\cdots\!21 ) / 12\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 29\!\cdots\!95 \nu^{9} + \cdots - 87\!\cdots\!77 ) / 34\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 43\!\cdots\!51 \nu^{9} + \cdots - 11\!\cdots\!20 ) / 34\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 47\!\cdots\!01 \nu^{9} + \cdots + 24\!\cdots\!12 ) / 34\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 58\!\cdots\!45 \nu^{9} + \cdots + 46\!\cdots\!12 ) / 17\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 13\!\cdots\!13 \nu^{9} + \cdots - 19\!\cdots\!25 ) / 34\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 93\!\cdots\!18 \nu^{9} + \cdots + 82\!\cdots\!61 ) / 12\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 74\!\cdots\!90 \nu^{9} + \cdots - 12\!\cdots\!71 ) / 34\!\cdots\!64 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2 \beta_{9} - 2 \beta_{8} - 3 \beta_{7} - \beta_{6} + 2 \beta_{5} - 32 \beta_{4} - 10 \beta_{3} + \cdots + 2 ) / 52 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 24 \beta_{9} + 45 \beta_{8} + 19 \beta_{7} - 42 \beta_{6} + 7 \beta_{5} - 296 \beta_{4} - 520 \beta_{3} + \cdots + 3794 ) / 52 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 200 \beta_{9} - 13 \beta_{8} - 163 \beta_{7} - 118 \beta_{6} + 260 \beta_{5} - 4247 \beta_{4} + \cdots + 54176 ) / 26 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 12572 \beta_{9} - 8012 \beta_{8} - 3075 \beta_{7} - 13355 \beta_{6} + 11782 \beta_{5} - 156732 \beta_{4} + \cdots + 1035464 ) / 52 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 201394 \beta_{9} + 184805 \beta_{8} + 33367 \beta_{7} - 207058 \beta_{6} + 97035 \beta_{5} + \cdots + 40642132 ) / 52 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 1049953 \beta_{9} - 702131 \beta_{8} - 349775 \beta_{7} - 995226 \beta_{6} + 1050583 \beta_{5} + \cdots + 141488264 ) / 13 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 89296498 \beta_{9} - 5940538 \beta_{8} + 14961877 \beta_{7} - 105731361 \beta_{6} + 67509526 \beta_{5} + \cdots + 11729533106 ) / 52 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 1695918588 \beta_{9} + 4116669 \beta_{8} + 180637527 \beta_{7} - 1669804654 \beta_{6} + \cdots + 266371180166 ) / 52 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 17529730778 \beta_{9} - 10073767423 \beta_{8} + 1940946971 \beta_{7} - 19869296450 \beta_{6} + \cdots + 2109166195566 ) / 26 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
−7.07349 + 8.45179i
−0.803404 + 7.00003i
−10.7537 − 10.4807i
−0.672765 − 5.92603i
20.3033 − 0.777135i
−7.07349 − 8.45179i
−0.803404 − 7.00003i
−10.7537 + 10.4807i
−0.672765 + 5.92603i
20.3033 + 0.777135i
0 −10.3562 + 17.9375i 0 34.7520i 0 −17.9058 + 10.3379i 0 −93.0022 − 161.085i 0
17.2 0 −5.96390 + 10.3298i 0 − 50.8897i 0 91.2918 − 52.7074i 0 50.3637 + 87.2326i 0
17.3 0 4.19971 − 7.27411i 0 61.3828i 0 −63.6555 + 36.7515i 0 86.2248 + 149.346i 0
17.4 0 5.29571 − 9.17244i 0 − 98.6239i 0 −200.021 + 115.482i 0 65.4109 + 113.295i 0
17.5 0 11.3247 − 19.6149i 0 1.41734i 0 146.791 − 84.7496i 0 −134.997 − 233.822i 0
49.1 0 −10.3562 − 17.9375i 0 − 34.7520i 0 −17.9058 − 10.3379i 0 −93.0022 + 161.085i 0
49.2 0 −5.96390 − 10.3298i 0 50.8897i 0 91.2918 + 52.7074i 0 50.3637 − 87.2326i 0
49.3 0 4.19971 + 7.27411i 0 − 61.3828i 0 −63.6555 − 36.7515i 0 86.2248 − 149.346i 0
49.4 0 5.29571 + 9.17244i 0 98.6239i 0 −200.021 − 115.482i 0 65.4109 − 113.295i 0
49.5 0 11.3247 + 19.6149i 0 − 1.41734i 0 146.791 + 84.7496i 0 −134.997 + 233.822i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.5
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 52.6.h.a ✓ 10
3.b odd 2 1 468.6.t.b 10
4.b odd 2 1 208.6.w.a 10
13.c even 3 1 676.6.d.e 10
13.e even 6 1 inner 52.6.h.a ✓ 10
13.e even 6 1 676.6.d.e 10
13.f odd 12 2 676.6.a.h 10
39.h odd 6 1 468.6.t.b 10
52.i odd 6 1 208.6.w.a 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
52.6.h.a ✓ 10 1.a even 1 1 trivial
52.6.h.a ✓ 10 13.e even 6 1 inner
208.6.w.a 10 4.b odd 2 1
208.6.w.a 10 52.i odd 6 1
468.6.t.b 10 3.b odd 2 1
468.6.t.b 10 39.h odd 6 1
676.6.a.h 10 13.f odd 12 2
676.6.d.e 10 13.c even 3 1
676.6.d.e 10 13.e even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(52, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} + \cdots + 247800857616 \) Copy content Toggle raw display
$5$ \( T^{10} + \cdots + 230263260364800 \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots + 39\!\cdots\!52 \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots + 10\!\cdots\!72 \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots + 70\!\cdots\!93 \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 84\!\cdots\!21 \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots + 51\!\cdots\!08 \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 39\!\cdots\!56 \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots + 12\!\cdots\!61 \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots + 27\!\cdots\!88 \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 23\!\cdots\!03 \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots + 46\!\cdots\!07 \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 35\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 31\!\cdots\!08 \) Copy content Toggle raw display
$53$ \( (T^{5} + \cdots - 99\!\cdots\!44)^{2} \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 51\!\cdots\!68 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots + 12\!\cdots\!25 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 13\!\cdots\!88 \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots + 20\!\cdots\!08 \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 13\!\cdots\!68 \) Copy content Toggle raw display
$79$ \( (T^{5} + \cdots + 43\!\cdots\!12)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 36\!\cdots\!88 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 53\!\cdots\!52 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 84\!\cdots\!48 \) Copy content Toggle raw display
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