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Properties

Label 63.2.i.b
Level $63$
Weight $2$
Character orbit 63.i
Analytic conductor $0.503$
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] = [63,2,Mod(5,63)]
 
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([5, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("63.5");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 63 = 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 63.i (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: \(0.503057532734\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{6})\)
Coefficient field: 10.0.288778218147.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - x^{9} + 7x^{8} - 4x^{7} + 34x^{6} - 19x^{5} + 64x^{4} - x^{3} + 64x^{2} - 21x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{5} - \beta_{3}) q^{2} + ( - \beta_{7} + \beta_1) q^{3} + ( - \beta_{5} - \beta_{4} - 2 \beta_{2} - 1) q^{4} + ( - \beta_{9} - \beta_{5} + \cdots + \beta_1) q^{5}+ \cdots + ( - \beta_{9} + 2 \beta_{7} + \cdots + \beta_{3}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{5} - \beta_{3}) q^{2} + ( - \beta_{7} + \beta_1) q^{3} + ( - \beta_{5} - \beta_{4} - 2 \beta_{2} - 1) q^{4} + ( - \beta_{9} - \beta_{5} + \cdots + \beta_1) q^{5}+ \cdots + (5 \beta_{9} + 3 \beta_{8} - \beta_{7} + \cdots + 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 3 q^{3} - 8 q^{4} + 12 q^{6} - 6 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 3 q^{3} - 8 q^{4} + 12 q^{6} - 6 q^{7} + 3 q^{9} - 15 q^{10} - 12 q^{11} - 12 q^{12} - 6 q^{13} + 12 q^{14} - 3 q^{15} + 12 q^{16} + 12 q^{17} + 24 q^{18} + 3 q^{19} + 3 q^{20} - 9 q^{21} + 5 q^{22} - 15 q^{23} + 7 q^{25} - 3 q^{26} - 27 q^{27} + 2 q^{28} - 15 q^{29} + 6 q^{30} - 3 q^{34} + 15 q^{35} - 18 q^{36} + 6 q^{37} + 18 q^{38} + 18 q^{39} + 15 q^{40} + 9 q^{41} - 12 q^{42} + 3 q^{43} - 24 q^{44} + 30 q^{45} - 13 q^{46} + 30 q^{47} + 15 q^{48} + 4 q^{49} + 3 q^{50} + 21 q^{51} - 12 q^{52} + 9 q^{53} + 9 q^{54} - 30 q^{56} - 36 q^{57} + 8 q^{58} - 36 q^{59} - 48 q^{60} - 12 q^{62} - 15 q^{63} + 6 q^{64} - 39 q^{66} + 20 q^{67} - 27 q^{68} + 3 q^{69} + 6 q^{70} - 30 q^{72} + 3 q^{73} - 30 q^{74} + 6 q^{75} - 9 q^{76} + 39 q^{77} + 24 q^{78} - 40 q^{79} + 30 q^{80} + 15 q^{81} + 9 q^{82} + 15 q^{83} + 93 q^{84} + 18 q^{85} + 54 q^{86} + 6 q^{87} - 8 q^{88} - 24 q^{89} - 24 q^{90} - 24 q^{91} + 39 q^{92} + 36 q^{93} + 33 q^{96} - 6 q^{97} - 45 q^{98} + 21 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - x^{9} + 7x^{8} - 4x^{7} + 34x^{6} - 19x^{5} + 64x^{4} - x^{3} + 64x^{2} - 21x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 339 \nu^{9} + 1348 \nu^{8} - 4381 \nu^{7} + 7882 \nu^{6} - 19883 \nu^{5} + 36059 \nu^{4} + \cdots + 29709 ) / 72795 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 658 \nu^{9} + 2394 \nu^{8} + 4352 \nu^{7} + 10326 \nu^{6} + 25351 \nu^{5} + 51907 \nu^{4} + \cdots + 98232 ) / 72795 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 4192 \nu^{9} - 796 \nu^{8} - 21678 \nu^{7} - 20279 \nu^{6} - 85319 \nu^{5} - 118353 \nu^{4} + \cdots - 398583 ) / 218385 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 8236 \nu^{9} - 9272 \nu^{8} + 54399 \nu^{7} - 28438 \nu^{6} + 233822 \nu^{5} - 150966 \nu^{4} + \cdots - 336546 ) / 218385 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3301 \nu^{9} - 2962 \nu^{8} + 21759 \nu^{7} - 8823 \nu^{6} + 104352 \nu^{5} - 42836 \nu^{4} + \cdots - 54156 ) / 72795 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 840 \nu^{9} + 248 \nu^{8} - 5659 \nu^{7} - 998 \nu^{6} - 27923 \nu^{5} - 3072 \nu^{4} + \cdots + 11514 ) / 14559 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 3085 \nu^{9} - 1373 \nu^{8} + 17808 \nu^{7} + 1181 \nu^{6} + 84554 \nu^{5} + 5736 \nu^{4} + \cdots + 17856 ) / 43677 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 18476 \nu^{9} + 18997 \nu^{8} - 128469 \nu^{7} + 65033 \nu^{6} - 601717 \nu^{5} + 295851 \nu^{4} + \cdots + 178101 ) / 218385 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} - \beta_{8} + 3\beta_{6} - \beta_{4} - \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} + \beta_{8} - 3\beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -5\beta_{9} + 5\beta_{8} + \beta_{7} - 12\beta_{6} - 5\beta_{5} - \beta_{3} - 5\beta_{2} + 5\beta _1 - 12 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -5\beta_{9} - \beta_{8} - \beta_{7} - 7\beta_{5} + 4\beta_{4} - 2\beta_{3} + 11\beta_{2} - 11\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 6\beta_{9} - 8\beta_{8} - 14\beta_{7} + 16\beta_{5} + 9\beta_{4} - 7\beta_{3} + 22\beta_{2} + 51 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( \beta_{9} - 31\beta_{8} - 8\beta_{7} + 31\beta_{5} - 30\beta_{4} + 8\beta_{3} + \beta_{2} + 43\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 75 \beta_{9} - 66 \beta_{8} + 38 \beta_{7} + 222 \beta_{6} + 47 \beta_{5} - 37 \beta_{4} + \cdots - 112 \beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 95\beta_{9} + 189\beta_{8} + 94\beta_{7} + 37\beta_{5} + 84\beta_{4} + 47\beta_{3} - 194\beta_{2} - 3 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(10\) \(29\)
\(\chi(n)\) \(1 + \beta_{6}\) \(1 + \beta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
0.827154 − 1.43267i
−1.04536 + 1.81062i
−0.539982 + 0.935277i
0.187540 − 0.324828i
1.07065 − 1.85442i
1.07065 + 1.85442i
0.187540 + 0.324828i
−0.539982 − 0.935277i
−1.04536 − 1.81062i
0.827154 + 1.43267i
− 2.09548i −1.72861 − 0.109097i −2.39104 −1.04492 − 1.80985i −0.228612 + 3.62227i 2.60068 + 0.486271i 0.819421i 2.97620 + 0.377174i −3.79250 + 2.18960i
5.2 − 1.51009i 0.811070 + 1.53041i −0.280386 −0.387938 − 0.671929i 2.31107 − 1.22479i −2.46849 + 0.952131i − 2.59678i −1.68433 + 2.48254i −1.01468 + 0.585823i
5.3 0.293869i −1.65249 + 0.518912i 1.91364 1.53014 + 2.65027i −0.152492 − 0.485617i −1.41763 − 2.23391i 1.15010i 2.46146 − 1.71499i −0.778834 + 0.449660i
5.4 0.718167i −0.271473 − 1.71064i 1.48424 −0.723774 − 1.25361i 1.22853 − 0.194963i 0.182786 + 2.63943i 2.50226i −2.85261 + 0.928786i 0.900304 − 0.519791i
5.5 2.59354i 1.34151 − 1.09561i −4.72645 0.626493 + 1.08512i 2.84151 + 3.47925i −1.89735 − 1.84393i − 7.07116i 0.599280 − 2.93953i −2.81429 + 1.62483i
38.1 − 2.59354i 1.34151 + 1.09561i −4.72645 0.626493 − 1.08512i 2.84151 − 3.47925i −1.89735 + 1.84393i 7.07116i 0.599280 + 2.93953i −2.81429 − 1.62483i
38.2 − 0.718167i −0.271473 + 1.71064i 1.48424 −0.723774 + 1.25361i 1.22853 + 0.194963i 0.182786 − 2.63943i − 2.50226i −2.85261 − 0.928786i 0.900304 + 0.519791i
38.3 − 0.293869i −1.65249 − 0.518912i 1.91364 1.53014 − 2.65027i −0.152492 + 0.485617i −1.41763 + 2.23391i − 1.15010i 2.46146 + 1.71499i −0.778834 − 0.449660i
38.4 1.51009i 0.811070 − 1.53041i −0.280386 −0.387938 + 0.671929i 2.31107 + 1.22479i −2.46849 − 0.952131i 2.59678i −1.68433 − 2.48254i −1.01468 − 0.585823i
38.5 2.09548i −1.72861 + 0.109097i −2.39104 −1.04492 + 1.80985i −0.228612 − 3.62227i 2.60068 − 0.486271i − 0.819421i 2.97620 − 0.377174i −3.79250 − 2.18960i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.i even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 63.2.i.b ✓ 10
3.b odd 2 1 189.2.i.b 10
4.b odd 2 1 1008.2.ca.b 10
7.b odd 2 1 441.2.i.b 10
7.c even 3 1 441.2.o.d 10
7.c even 3 1 441.2.s.b 10
7.d odd 6 1 63.2.s.b yes 10
7.d odd 6 1 441.2.o.c 10
9.c even 3 1 189.2.s.b 10
9.c even 3 1 567.2.p.d 10
9.d odd 6 1 63.2.s.b yes 10
9.d odd 6 1 567.2.p.c 10
12.b even 2 1 3024.2.ca.b 10
21.c even 2 1 1323.2.i.b 10
21.g even 6 1 189.2.s.b 10
21.g even 6 1 1323.2.o.d 10
21.h odd 6 1 1323.2.o.c 10
21.h odd 6 1 1323.2.s.b 10
28.f even 6 1 1008.2.df.b 10
36.f odd 6 1 3024.2.df.b 10
36.h even 6 1 1008.2.df.b 10
63.g even 3 1 1323.2.o.d 10
63.h even 3 1 1323.2.i.b 10
63.i even 6 1 inner 63.2.i.b ✓ 10
63.j odd 6 1 441.2.i.b 10
63.k odd 6 1 567.2.p.c 10
63.k odd 6 1 1323.2.o.c 10
63.l odd 6 1 1323.2.s.b 10
63.n odd 6 1 441.2.o.c 10
63.o even 6 1 441.2.s.b 10
63.s even 6 1 441.2.o.d 10
63.s even 6 1 567.2.p.d 10
63.t odd 6 1 189.2.i.b 10
84.j odd 6 1 3024.2.df.b 10
252.r odd 6 1 1008.2.ca.b 10
252.bj even 6 1 3024.2.ca.b 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.i.b ✓ 10 1.a even 1 1 trivial
63.2.i.b ✓ 10 63.i even 6 1 inner
63.2.s.b yes 10 7.d odd 6 1
63.2.s.b yes 10 9.d odd 6 1
189.2.i.b 10 3.b odd 2 1
189.2.i.b 10 63.t odd 6 1
189.2.s.b 10 9.c even 3 1
189.2.s.b 10 21.g even 6 1
441.2.i.b 10 7.b odd 2 1
441.2.i.b 10 63.j odd 6 1
441.2.o.c 10 7.d odd 6 1
441.2.o.c 10 63.n odd 6 1
441.2.o.d 10 7.c even 3 1
441.2.o.d 10 63.s even 6 1
441.2.s.b 10 7.c even 3 1
441.2.s.b 10 63.o even 6 1
567.2.p.c 10 9.d odd 6 1
567.2.p.c 10 63.k odd 6 1
567.2.p.d 10 9.c even 3 1
567.2.p.d 10 63.s even 6 1
1008.2.ca.b 10 4.b odd 2 1
1008.2.ca.b 10 252.r odd 6 1
1008.2.df.b 10 28.f even 6 1
1008.2.df.b 10 36.h even 6 1
1323.2.i.b 10 21.c even 2 1
1323.2.i.b 10 63.h even 3 1
1323.2.o.c 10 21.h odd 6 1
1323.2.o.c 10 63.k odd 6 1
1323.2.o.d 10 21.g even 6 1
1323.2.o.d 10 63.g even 3 1
1323.2.s.b 10 21.h odd 6 1
1323.2.s.b 10 63.l odd 6 1
3024.2.ca.b 10 12.b even 2 1
3024.2.ca.b 10 252.bj even 6 1
3024.2.df.b 10 36.f odd 6 1
3024.2.df.b 10 84.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{10} + 14T_{2}^{8} + 63T_{2}^{6} + 101T_{2}^{4} + 43T_{2}^{2} + 3 \) acting on \(S_{2}^{\mathrm{new}}(63, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + 14 T^{8} + \cdots + 3 \) Copy content Toggle raw display
$3$ \( T^{10} + 3 T^{9} + \cdots + 243 \) Copy content Toggle raw display
$5$ \( T^{10} + 9 T^{8} + \cdots + 81 \) Copy content Toggle raw display
$7$ \( T^{10} + 6 T^{9} + \cdots + 16807 \) Copy content Toggle raw display
$11$ \( T^{10} + 12 T^{9} + \cdots + 2883 \) Copy content Toggle raw display
$13$ \( T^{10} + 6 T^{9} + \cdots + 3267 \) Copy content Toggle raw display
$17$ \( T^{10} - 12 T^{9} + \cdots + 263169 \) Copy content Toggle raw display
$19$ \( T^{10} - 3 T^{9} + \cdots + 2187 \) Copy content Toggle raw display
$23$ \( T^{10} + 15 T^{9} + \cdots + 27 \) Copy content Toggle raw display
$29$ \( T^{10} + 15 T^{9} + \cdots + 186003 \) Copy content Toggle raw display
$31$ \( T^{10} + 93 T^{8} + \cdots + 16875 \) Copy content Toggle raw display
$37$ \( T^{10} - 6 T^{9} + \cdots + 369664 \) Copy content Toggle raw display
$41$ \( T^{10} - 9 T^{9} + \cdots + 40487769 \) Copy content Toggle raw display
$43$ \( T^{10} - 3 T^{9} + \cdots + 12243001 \) Copy content Toggle raw display
$47$ \( (T^{5} - 15 T^{4} + \cdots + 567)^{2} \) Copy content Toggle raw display
$53$ \( T^{10} - 9 T^{9} + \cdots + 871563 \) Copy content Toggle raw display
$59$ \( (T^{5} + 18 T^{4} + \cdots - 2025)^{2} \) Copy content Toggle raw display
$61$ \( T^{10} + 252 T^{8} + \cdots + 826875 \) Copy content Toggle raw display
$67$ \( (T^{5} - 10 T^{4} + \cdots + 19)^{2} \) Copy content Toggle raw display
$71$ \( T^{10} + 359 T^{8} + \cdots + 46216875 \) Copy content Toggle raw display
$73$ \( T^{10} - 3 T^{9} + \cdots + 789507 \) Copy content Toggle raw display
$79$ \( (T^{5} + 20 T^{4} + \cdots - 32675)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 340734681 \) Copy content Toggle raw display
$89$ \( T^{10} + 24 T^{9} + \cdots + 32455809 \) Copy content Toggle raw display
$97$ \( T^{10} + 6 T^{9} + \cdots + 9687627 \) Copy content Toggle raw display
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