Properties

Label 10.13.c.a
Level $10$
Weight $13$
Character orbit 10.c
Analytic conductor $9.140$
Analytic rank $0$
Dimension $6$
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] = [10,13,Mod(3,10)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(10, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([3]))
 
N = Newforms(chi, 13, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("10.3");
 
S:= CuspForms(chi, 13);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 10 = 2 \cdot 5 \)
Weight: \( k \) \(=\) \( 13 \)
Character orbit: \([\chi]\) \(=\) 10.c (of order \(4\), 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: \(9.13993817276\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 2385x^{4} + 1422264x^{2} + 490000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{5}\cdot 3^{2}\cdot 5^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 32 \beta_1 - 32) q^{2} + (\beta_{3} + 156 \beta_1 - 156) q^{3} + 2048 \beta_1 q^{4} + (2 \beta_{5} + \beta_{4} + 7 \beta_{3} + 9 \beta_{2} + 1905 \beta_1 - 2710) q^{5} + ( - 32 \beta_{3} - 32 \beta_{2} + 9984) q^{6} + (\beta_{5} + 7 \beta_{4} + \beta_{3} - 56 \beta_{2} - 7556 \beta_1 - 7556) q^{7} + ( - 65536 \beta_1 + 65536) q^{8} + ( - 32 \beta_{5} - 24 \beta_{4} - 589 \beta_{3} + 581 \beta_{2} - 45331 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 32 \beta_1 - 32) q^{2} + (\beta_{3} + 156 \beta_1 - 156) q^{3} + 2048 \beta_1 q^{4} + (2 \beta_{5} + \beta_{4} + 7 \beta_{3} + 9 \beta_{2} + 1905 \beta_1 - 2710) q^{5} + ( - 32 \beta_{3} - 32 \beta_{2} + 9984) q^{6} + (\beta_{5} + 7 \beta_{4} + \beta_{3} - 56 \beta_{2} - 7556 \beta_1 - 7556) q^{7} + ( - 65536 \beta_1 + 65536) q^{8} + ( - 32 \beta_{5} - 24 \beta_{4} - 589 \beta_{3} + 581 \beta_{2} - 45331 \beta_1) q^{9} + ( - 96 \beta_{5} + 32 \beta_{4} + 64 \beta_{3} - 512 \beta_{2} + \cdots + 147680) q^{10}+ \cdots + (160041464 \beta_{5} + 120031098 \beta_{4} + \cdots + 755875762708 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 192 q^{2} - 936 q^{3} - 16260 q^{5} + 59904 q^{6} - 45336 q^{7} + 393216 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 192 q^{2} - 936 q^{3} - 16260 q^{5} + 59904 q^{6} - 45336 q^{7} + 393216 q^{8} + 886080 q^{10} - 3418008 q^{11} - 1916928 q^{12} + 8106834 q^{13} + 26693880 q^{15} - 25165824 q^{16} + 10772514 q^{17} - 8703552 q^{18} - 23408640 q^{20} - 173036568 q^{21} + 109376256 q^{22} + 241146864 q^{23} - 520941150 q^{25} - 518837376 q^{26} + 1437591240 q^{27} + 92848128 q^{28} - 1764983040 q^{30} - 1709108208 q^{31} + 805306368 q^{32} + 6415914648 q^{33} - 3823952160 q^{35} + 557027328 q^{36} - 742614906 q^{37} + 3927847680 q^{38} - 316538880 q^{40} - 24748899528 q^{41} + 5537170176 q^{42} + 14532965304 q^{43} + 17091495930 q^{45} - 15433399296 q^{46} + 8427287424 q^{47} + 3925868544 q^{48} + 4302859200 q^{50} - 82983967968 q^{51} + 16602796032 q^{52} + 1297014534 q^{53} + 75773173080 q^{55} - 5942280192 q^{56} + 30123821760 q^{57} + 576729600 q^{58} + 58289848320 q^{60} - 201276821928 q^{61} + 54691462656 q^{62} + 149855233584 q^{63} + 146156357430 q^{65} - 410618537472 q^{66} + 108303977544 q^{67} - 22062108672 q^{68} + 132252737280 q^{70} - 570401667408 q^{71} - 17824874496 q^{72} + 710242228134 q^{73} - 180009328800 q^{75} - 251382251520 q^{76} + 9201523848 q^{77} + 422419616256 q^{78} + 68199383040 q^{80} - 754940669454 q^{81} + 791964784896 q^{82} + 727867488144 q^{83} - 1459335929310 q^{85} - 930109779456 q^{86} + 610413099600 q^{87} - 224002572288 q^{88} + 536586058560 q^{90} + 1841466543792 q^{91} + 493868777472 q^{92} + 503663914248 q^{93} - 972100512600 q^{95} - 251255586816 q^{96} - 2207840630346 q^{97} - 1960900383168 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 2385x^{4} + 1422264x^{2} + 490000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{5} + 1685\nu^{3} + 587164\nu ) / 344400 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 17\nu^{5} - 2450\nu^{4} + 86045\nu^{3} - 2865450\nu^{2} + 77197188\nu + 44319800 ) / 91840 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -17\nu^{5} - 2450\nu^{4} - 86045\nu^{3} - 2865450\nu^{2} - 77197188\nu + 44319800 ) / 91840 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 103\nu^{5} + 22750\nu^{4} + 575355\nu^{3} + 27657350\nu^{2} + 547516892\nu + 422891000 ) / 91840 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 79\nu^{5} - 9870\nu^{4} + 443075\nu^{3} - 12015990\nu^{2} + 422574076\nu - 196948920 ) / 55104 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 4\beta_{5} + 3\beta_{4} + 29\beta_{3} - 28\beta_{2} + 500\beta_1 ) / 1500 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -21\beta_{5} + 28\beta_{4} + 204\beta_{3} + 197\beta_{2} - 397500 ) / 500 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -4684\beta_{5} - 3513\beta_{4} - 35159\beta_{3} + 33988\beta_{2} - 738500\beta_1 ) / 1500 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 24561\beta_{5} - 32748\beta_{4} - 247964\beta_{3} - 239777\beta_{2} + 473949500 ) / 500 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 5543884\beta_{5} + 4157913\beta_{4} + 42215159\beta_{3} - 40829188\beta_{2} + 1467390500\beta_1 ) / 1500 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/10\mathbb{Z}\right)^\times\).

\(n\) \(7\)
\(\chi(n)\) \(\beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
34.7359i
34.3230i
0.587129i
34.7359i
34.3230i
0.587129i
−32.0000 + 32.0000i −864.774 864.774i 2048.00i −14634.8 5473.81i 55345.5 43339.1 43339.1i 65536.0 + 65536.0i 964226.i 643476. 293152.i
3.2 −32.0000 + 32.0000i 59.4453 + 59.4453i 2048.00i 3818.66 + 15151.2i −3804.50 −59325.1 + 59325.1i 65536.0 + 65536.0i 524374.i −607035. 362641.i
3.3 −32.0000 + 32.0000i 337.328 + 337.328i 2048.00i 2686.16 15392.4i −21589.0 −6681.98 + 6681.98i 65536.0 + 65536.0i 303860.i 406599. + 578513.i
7.1 −32.0000 32.0000i −864.774 + 864.774i 2048.00i −14634.8 + 5473.81i 55345.5 43339.1 + 43339.1i 65536.0 65536.0i 964226.i 643476. + 293152.i
7.2 −32.0000 32.0000i 59.4453 59.4453i 2048.00i 3818.66 15151.2i −3804.50 −59325.1 59325.1i 65536.0 65536.0i 524374.i −607035. + 362641.i
7.3 −32.0000 32.0000i 337.328 337.328i 2048.00i 2686.16 + 15392.4i −21589.0 −6681.98 6681.98i 65536.0 65536.0i 303860.i 406599. 578513.i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 10.13.c.a 6
3.b odd 2 1 90.13.g.b 6
4.b odd 2 1 80.13.p.b 6
5.b even 2 1 50.13.c.d 6
5.c odd 4 1 inner 10.13.c.a 6
5.c odd 4 1 50.13.c.d 6
15.e even 4 1 90.13.g.b 6
20.e even 4 1 80.13.p.b 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.13.c.a 6 1.a even 1 1 trivial
10.13.c.a 6 5.c odd 4 1 inner
50.13.c.d 6 5.b even 2 1
50.13.c.d 6 5.c odd 4 1
80.13.p.b 6 4.b odd 2 1
80.13.p.b 6 20.e even 4 1
90.13.g.b 6 3.b odd 2 1
90.13.g.b 6 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} + 936 T_{3}^{5} + 438048 T_{3}^{4} - 674145288 T_{3}^{3} + 417489145956 T_{3}^{2} - 44818350901776 T_{3} + 24\!\cdots\!48 \) acting on \(S_{13}^{\mathrm{new}}(10, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 64 T + 2048)^{3} \) Copy content Toggle raw display
$3$ \( T^{6} + 936 T^{5} + \cdots + 24\!\cdots\!48 \) Copy content Toggle raw display
$5$ \( T^{6} + 16260 T^{5} + \cdots + 14\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{6} + 45336 T^{5} + \cdots + 23\!\cdots\!48 \) Copy content Toggle raw display
$11$ \( (T^{3} + 1709004 T^{2} + \cdots - 19\!\cdots\!68)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} - 8106834 T^{5} + \cdots + 55\!\cdots\!88 \) Copy content Toggle raw display
$17$ \( T^{6} - 10772514 T^{5} + \cdots + 18\!\cdots\!48 \) Copy content Toggle raw display
$19$ \( T^{6} + \cdots + 76\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{6} - 241146864 T^{5} + \cdots + 11\!\cdots\!48 \) Copy content Toggle raw display
$29$ \( T^{6} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{3} + 854554104 T^{2} + \cdots - 11\!\cdots\!68)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 742614906 T^{5} + \cdots + 34\!\cdots\!08 \) Copy content Toggle raw display
$41$ \( (T^{3} + 12374449764 T^{2} + \cdots - 19\!\cdots\!28)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} - 14532965304 T^{5} + \cdots + 65\!\cdots\!28 \) Copy content Toggle raw display
$47$ \( T^{6} - 8427287424 T^{5} + \cdots + 36\!\cdots\!68 \) Copy content Toggle raw display
$53$ \( T^{6} - 1297014534 T^{5} + \cdots + 10\!\cdots\!88 \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{3} + 100638410964 T^{2} + \cdots + 23\!\cdots\!72)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} - 108303977544 T^{5} + \cdots + 55\!\cdots\!08 \) Copy content Toggle raw display
$71$ \( (T^{3} + 285200833704 T^{2} + \cdots + 36\!\cdots\!32)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} - 710242228134 T^{5} + \cdots + 18\!\cdots\!88 \) Copy content Toggle raw display
$79$ \( T^{6} + \cdots + 43\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{6} - 727867488144 T^{5} + \cdots + 54\!\cdots\!08 \) Copy content Toggle raw display
$89$ \( T^{6} + \cdots + 97\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{6} + 2207840630346 T^{5} + \cdots + 51\!\cdots\!28 \) Copy content Toggle raw display
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