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Properties

Label 62.7.b.a
Level $62$
Weight $7$
Character orbit 62.b
Analytic conductor $14.263$
Analytic rank $0$
Dimension $16$
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] = [62,7,Mod(61,62)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(62, 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("62.61");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 62 = 2 \cdot 31 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 62.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: \(14.2633531844\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 3728 x^{14} + 5344136 x^{12} + 3791594256 x^{10} + 1424839419639 x^{8} + 283023018795852 x^{6} + \cdots + 54\!\cdots\!68 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{27}\cdot 3^{2} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + \beta_{8} q^{3} + 32 q^{4} + (\beta_{3} - 3 \beta_{2} + 18) q^{5} + (\beta_{9} - \beta_{8}) q^{6} + ( - \beta_{6} + 6 \beta_{2} - 20) q^{7} + 32 \beta_{2} q^{8} + (\beta_{4} + \beta_{3} + 22 \beta_{2} - 203) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + \beta_{8} q^{3} + 32 q^{4} + (\beta_{3} - 3 \beta_{2} + 18) q^{5} + (\beta_{9} - \beta_{8}) q^{6} + ( - \beta_{6} + 6 \beta_{2} - 20) q^{7} + 32 \beta_{2} q^{8} + (\beta_{4} + \beta_{3} + 22 \beta_{2} - 203) q^{9} + (\beta_{5} + 18 \beta_{2} - 84) q^{10} + (\beta_{12} - 12 \beta_{8}) q^{11} + 32 \beta_{8} q^{12} + ( - \beta_{13} + \beta_{9} + 2 \beta_{8}) q^{13} + (\beta_{7} + \beta_{5} + 2 \beta_{3} + \cdots + 189) q^{14}+ \cdots + ( - 78 \beta_{15} + \cdots + 22946 \beta_{8}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 512 q^{4} + 288 q^{5} - 328 q^{7} - 3248 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 512 q^{4} + 288 q^{5} - 328 q^{7} - 3248 q^{9} - 1344 q^{10} + 3040 q^{14} + 16384 q^{16} + 11200 q^{18} - 2912 q^{19} + 9216 q^{20} + 8408 q^{25} - 10496 q^{28} - 9424 q^{31} + 175560 q^{33} + 63272 q^{35} - 103936 q^{36} - 117024 q^{38} - 34720 q^{39} - 43008 q^{40} + 190576 q^{41} + 151832 q^{45} - 326200 q^{47} + 172464 q^{49} - 68736 q^{50} - 423352 q^{51} + 97280 q^{56} - 1246280 q^{59} + 261152 q^{62} + 600688 q^{63} + 524288 q^{64} - 293888 q^{66} - 857192 q^{67} + 576824 q^{69} + 656928 q^{70} - 869848 q^{71} + 358400 q^{72} - 93184 q^{76} - 555008 q^{78} + 294912 q^{80} + 1568208 q^{81} - 860544 q^{82} - 778400 q^{87} - 2870528 q^{90} + 4950280 q^{93} + 710208 q^{94} + 3082632 q^{95} - 2183416 q^{97} + 1252032 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 3728 x^{14} + 5344136 x^{12} + 3791594256 x^{10} + 1424839419639 x^{8} + 283023018795852 x^{6} + \cdots + 54\!\cdots\!68 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 40\!\cdots\!41 \nu^{14} + \cdots - 20\!\cdots\!46 ) / 29\!\cdots\!86 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 20\!\cdots\!51 \nu^{14} + \cdots + 12\!\cdots\!44 ) / 38\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 94\!\cdots\!29 \nu^{14} + \cdots - 44\!\cdots\!84 ) / 17\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 29\!\cdots\!47 \nu^{14} + \cdots + 19\!\cdots\!36 ) / 17\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 98\!\cdots\!23 \nu^{14} + \cdots - 22\!\cdots\!56 ) / 17\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 26\!\cdots\!77 \nu^{14} + \cdots - 11\!\cdots\!56 ) / 19\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 52\!\cdots\!25 \nu^{14} + \cdots + 16\!\cdots\!68 ) / 11\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 20\!\cdots\!51 \nu^{15} + \cdots + 12\!\cdots\!44 \nu ) / 15\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 20\!\cdots\!51 \nu^{15} + \cdots + 13\!\cdots\!20 \nu ) / 15\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 18\!\cdots\!77 \nu^{15} + \cdots + 41\!\cdots\!56 \nu ) / 15\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 61\!\cdots\!97 \nu^{15} + \cdots - 17\!\cdots\!64 \nu ) / 75\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 89\!\cdots\!87 \nu^{15} + \cdots + 43\!\cdots\!52 \nu ) / 37\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 29\!\cdots\!75 \nu^{15} + \cdots - 13\!\cdots\!48 \nu ) / 75\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 27\!\cdots\!61 \nu^{15} + \cdots - 10\!\cdots\!52 \nu ) / 50\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 28\!\cdots\!69 \nu^{15} + \cdots - 33\!\cdots\!64 \nu ) / 50\!\cdots\!56 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{9} - \beta_{8} ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{4} + \beta_{3} + 22\beta_{2} - 932 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3 \beta_{15} + 62 \beta_{14} - 70 \beta_{13} + 12 \beta_{12} + 34 \beta_{11} + 40 \beta_{10} + \cdots + 3009 \beta_{8} ) / 16 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 58\beta_{7} - 114\beta_{6} + 69\beta_{5} - 1171\beta_{4} - 2733\beta_{3} - 37111\beta_{2} + 192\beta _1 + 802552 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 8787 \beta_{15} - 91574 \beta_{14} + 110426 \beta_{13} - 6380 \beta_{12} - 48318 \beta_{11} + \cdots - 5355637 \beta_{8} ) / 16 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 232717 \beta_{7} + 338448 \beta_{6} - 296682 \beta_{5} + 2717958 \beta_{4} + 8584148 \beta_{3} + \cdots - 1709276729 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 15376617 \beta_{15} + 121198458 \beta_{14} - 152378294 \beta_{13} - 308180 \beta_{12} + \cdots + 8538103799 \beta_{8} ) / 16 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 185439203 \beta_{7} - 216444474 \beta_{6} + 226015977 \beta_{5} - 1652713759 \beta_{4} + \cdots + 1006228057323 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 23492764335 \beta_{15} - 159383256398 \beta_{14} + 205741868538 \beta_{13} + \cdots - 12618699473057 \beta_{8} ) / 16 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 542721834651 \beta_{7} + 546003701256 \beta_{6} - 631671188058 \beta_{5} + 4179215416798 \beta_{4} + \cdots - 25\!\cdots\!87 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 33856634154633 \beta_{15} + 211110905810218 \beta_{14} - 277015899171830 \beta_{13} + \cdots + 17\!\cdots\!75 \beta_{8} ) / 16 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 381685492175162 \beta_{7} - 348365727838938 \beta_{6} + 430188616002549 \beta_{5} + \cdots + 16\!\cdots\!88 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 47\!\cdots\!83 \beta_{15} + \cdots - 24\!\cdots\!09 \beta_{8} ) / 16 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 10\!\cdots\!05 \beta_{7} + \cdots - 42\!\cdots\!21 ) / 4 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 65\!\cdots\!57 \beta_{15} + \cdots + 34\!\cdots\!63 \beta_{8} ) / 16 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/62\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
61.1
36.7341i
18.1070i
16.6146i
12.5772i
− 12.5772i
− 16.6146i
− 18.1070i
− 36.7341i
− 27.9265i
− 27.4255i
− 8.84772i
− 2.48719i
2.48719i
8.84772i
27.4255i
27.9265i
−5.65685 − 51.9499i 32.0000 −84.5087 293.873i −99.5159 −181.019 −1969.79 478.053
61.2 −5.65685 − 25.6071i 32.0000 230.375 144.856i 133.046 −181.019 73.2749 −1303.20
61.3 −5.65685 − 23.4966i 32.0000 53.2650 132.917i −571.905 −181.019 176.909 −301.312
61.4 −5.65685 − 17.7868i 32.0000 −67.7348 100.617i 322.025 −181.019 412.629 383.166
61.5 −5.65685 17.7868i 32.0000 −67.7348 − 100.617i 322.025 −181.019 412.629 383.166
61.6 −5.65685 23.4966i 32.0000 53.2650 − 132.917i −571.905 −181.019 176.909 −301.312
61.7 −5.65685 25.6071i 32.0000 230.375 − 144.856i 133.046 −181.019 73.2749 −1303.20
61.8 −5.65685 51.9499i 32.0000 −84.5087 − 293.873i −99.5159 −181.019 −1969.79 478.053
61.9 5.65685 − 39.4940i 32.0000 119.256 − 223.412i 493.111 181.019 −830.776 674.616
61.10 5.65685 − 38.7855i 32.0000 −70.2512 − 219.404i −515.728 181.019 −775.313 −397.401
61.11 5.65685 − 12.5126i 32.0000 126.271 − 70.7817i −131.484 181.019 572.436 714.297
61.12 5.65685 − 3.51742i 32.0000 −162.673 − 19.8976i 206.452 181.019 716.628 −920.218
61.13 5.65685 3.51742i 32.0000 −162.673 19.8976i 206.452 181.019 716.628 −920.218
61.14 5.65685 12.5126i 32.0000 126.271 70.7817i −131.484 181.019 572.436 714.297
61.15 5.65685 38.7855i 32.0000 −70.2512 219.404i −515.728 181.019 −775.313 −397.401
61.16 5.65685 39.4940i 32.0000 119.256 223.412i 493.111 181.019 −830.776 674.616
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 61.16
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 62.7.b.a ✓ 16
3.b odd 2 1 558.7.d.a 16
4.b odd 2 1 496.7.e.e 16
31.b odd 2 1 inner 62.7.b.a ✓ 16
93.c even 2 1 558.7.d.a 16
124.d even 2 1 496.7.e.e 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
62.7.b.a ✓ 16 1.a even 1 1 trivial
62.7.b.a ✓ 16 31.b odd 2 1 inner
496.7.e.e 16 4.b odd 2 1
496.7.e.e 16 124.d even 2 1
558.7.d.a 16 3.b odd 2 1
558.7.d.a 16 93.c even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(62, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 32)^{8} \) Copy content Toggle raw display
$3$ \( T^{16} + \cdots + 14\!\cdots\!08 \) Copy content Toggle raw display
$5$ \( (T^{8} + \cdots + 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
$7$ \( (T^{8} + \cdots + 16\!\cdots\!96)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 44\!\cdots\!28 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 24\!\cdots\!32 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 36\!\cdots\!12 \) Copy content Toggle raw display
$19$ \( (T^{8} + \cdots - 46\!\cdots\!52)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 34\!\cdots\!28 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 24\!\cdots\!88 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 38\!\cdots\!41 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 74\!\cdots\!48 \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots - 42\!\cdots\!76)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 30\!\cdots\!32 \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots - 16\!\cdots\!88)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 30\!\cdots\!92 \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots - 15\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 18\!\cdots\!88 \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 27\!\cdots\!76)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 37\!\cdots\!88)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 18\!\cdots\!92 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 18\!\cdots\!28 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 26\!\cdots\!52 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 57\!\cdots\!08 \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 42\!\cdots\!48)^{2} \) Copy content Toggle raw display
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