Properties

Label 50.13.c.c
Level $50$
Weight $13$
Character orbit 50.c
Analytic conductor $45.700$
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] = [50,13,Mod(7,50)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(50, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 13, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("50.7");
 
S:= CuspForms(chi, 13);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 13 \)
Character orbit: \([\chi]\) \(=\) 50.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: \(45.6996908638\)
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: \( x^{6} + 43009x^{4} + 461169144x^{2} + 392422062096 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{6}\cdot 5^{6} \)
Twist minimal: no (minimal twist has level 10)
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.

\(f(q)\) \(=\) \( q + (32 \beta_1 - 32) q^{2} + (\beta_{3} - 49 \beta_1 - 49) q^{3} - 2048 \beta_1 q^{4} + ( - 32 \beta_{3} + 32 \beta_{2} + 3136) q^{6} + ( - 7 \beta_{5} - 38 \beta_{2} - 53699 \beta_1 + 53699) q^{7} + (65536 \beta_1 + 65536) q^{8} + (18 \beta_{5} - 18 \beta_{4} + 131 \beta_{3} + 131 \beta_{2} + 190337 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (32 \beta_1 - 32) q^{2} + (\beta_{3} - 49 \beta_1 - 49) q^{3} - 2048 \beta_1 q^{4} + ( - 32 \beta_{3} + 32 \beta_{2} + 3136) q^{6} + ( - 7 \beta_{5} - 38 \beta_{2} - 53699 \beta_1 + 53699) q^{7} + (65536 \beta_1 + 65536) q^{8} + (18 \beta_{5} - 18 \beta_{4} + 131 \beta_{3} + 131 \beta_{2} + 190337 \beta_1) q^{9} + ( - 41 \beta_{5} - 41 \beta_{4} - 980 \beta_{3} + 980 \beta_{2} + \cdots + 274826) q^{11}+ \cdots + (3402369 \beta_{5} - 3402369 \beta_{4} + \cdots + 302459212394 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 192 q^{2} - 296 q^{3} + 18944 q^{6} + 322104 q^{7} + 393216 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 192 q^{2} - 296 q^{3} + 18944 q^{6} + 322104 q^{7} + 393216 q^{8} + 1652712 q^{11} - 606208 q^{12} + 4646814 q^{13} - 25165824 q^{16} - 51200226 q^{17} - 36525632 q^{18} + 117123272 q^{21} - 52886784 q^{22} - 105826896 q^{23} - 297396096 q^{26} - 627050120 q^{27} - 659668992 q^{28} - 2667117168 q^{31} + 805306368 q^{32} - 4381479992 q^{33} + 2337640448 q^{36} - 1747956246 q^{37} + 2125152000 q^{38} + 22722098232 q^{41} - 3747944704 q^{42} + 15890524824 q^{43} + 6772921344 q^{46} + 18495531264 q^{47} + 1241513984 q^{48} + 114152506432 q^{51} + 9516675072 q^{52} + 88020413514 q^{53} + 42218815488 q^{56} + 174270786400 q^{57} + 107861859840 q^{58} + 291794891352 q^{61} + 85347749376 q^{62} + 297541783984 q^{63} + 280414719488 q^{66} + 3887251464 q^{67} + 104858062848 q^{68} + 929135015472 q^{71} - 74804494336 q^{72} - 12678070086 q^{73} - 136009728000 q^{76} + 436526954808 q^{77} + 80105703936 q^{78} + 833935849906 q^{81} - 727107143424 q^{82} - 68676615456 q^{83} - 1016993588736 q^{86} + 943420476880 q^{87} + 108312133632 q^{88} - 1619146872048 q^{91} - 216733483008 q^{92} - 2320495891112 q^{93} - 79456894976 q^{96} - 675735777846 q^{97} + 1327663144512 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 43009x^{4} + 461169144x^{2} + 392422062096 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{5} - 669445\nu^{3} - 13932675324\nu ) / 405892941840 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} - 1566090 \nu^{4} + 669445 \nu^{3} - 33678765450 \nu^{2} + 1028665029924 \nu + 1386948097080 ) / 202946470920 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{5} + 1566090 \nu^{4} + 669445 \nu^{3} + 33678765450 \nu^{2} + 1028665029924 \nu - 1386948097080 ) / 202946470920 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 2669821 \nu^{5} + 121632990 \nu^{4} - 96077728345 \nu^{3} + 4306938040950 \nu^{2} - 750930697792404 \nu + 24\!\cdots\!00 ) / 1217678825520 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2669821 \nu^{5} + 121632990 \nu^{4} + 96077728345 \nu^{3} + 4306938040950 \nu^{2} + 750930697792404 \nu + 24\!\cdots\!00 ) / 1217678825520 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} + 4\beta_1 ) / 10 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 18\beta_{5} + 18\beta_{4} - 233\beta_{3} + 233\beta_{2} - 716984 ) / 50 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -18\beta_{5} + 18\beta_{4} - 107753\beta_{3} - 107753\beta_{2} - 32468864\beta_1 ) / 50 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -77418\beta_{5} - 77418\beta_{4} + 1650073\beta_{3} - 1650073\beta_{2} + 3092604304 ) / 10 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 2410002\beta_{5} - 2410002\beta_{4} + 494266093\beta_{3} + 494266093\beta_{2} + 232563612400\beta_1 ) / 10 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/50\mathbb{Z}\right)^\times\).

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
128.447i
30.4929i
159.940i
128.447i
30.4929i
159.940i
−32.0000 32.0000i −693.233 + 693.233i 2048.00i 0 44366.9 108582. + 108582.i 65536.0 65536.0i 429704.i 0
7.2 −32.0000 32.0000i −203.464 + 203.464i 2048.00i 0 13021.7 −68548.2 68548.2i 65536.0 65536.0i 448646.i 0
7.3 −32.0000 32.0000i 748.698 748.698i 2048.00i 0 −47916.6 121018. + 121018.i 65536.0 65536.0i 589655.i 0
43.1 −32.0000 + 32.0000i −693.233 693.233i 2048.00i 0 44366.9 108582. 108582.i 65536.0 + 65536.0i 429704.i 0
43.2 −32.0000 + 32.0000i −203.464 203.464i 2048.00i 0 13021.7 −68548.2 + 68548.2i 65536.0 + 65536.0i 448646.i 0
43.3 −32.0000 + 32.0000i 748.698 + 748.698i 2048.00i 0 −47916.6 121018. 121018.i 65536.0 + 65536.0i 589655.i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.3
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 50.13.c.c 6
5.b even 2 1 10.13.c.b 6
5.c odd 4 1 10.13.c.b 6
5.c odd 4 1 inner 50.13.c.c 6
15.d odd 2 1 90.13.g.a 6
15.e even 4 1 90.13.g.a 6
20.d odd 2 1 80.13.p.a 6
20.e even 4 1 80.13.p.a 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.13.c.b 6 5.b even 2 1
10.13.c.b 6 5.c odd 4 1
50.13.c.c 6 1.a even 1 1 trivial
50.13.c.c 6 5.c odd 4 1 inner
80.13.p.a 6 20.d odd 2 1
80.13.p.a 6 20.e even 4 1
90.13.g.a 6 15.d odd 2 1
90.13.g.a 6 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} + 296 T_{3}^{5} + 43808 T_{3}^{4} + 108468072 T_{3}^{3} + 1124902056996 T_{3}^{2} + 448013764587024 T_{3} + 89\!\cdots\!28 \) acting on \(S_{13}^{\mathrm{new}}(50, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 64 T + 2048)^{3} \) Copy content Toggle raw display
$3$ \( T^{6} + 296 T^{5} + \cdots + 89\!\cdots\!28 \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} - 322104 T^{5} + \cdots + 64\!\cdots\!28 \) Copy content Toggle raw display
$11$ \( (T^{3} - 826356 T^{2} + \cdots - 16\!\cdots\!08)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} - 4646814 T^{5} + \cdots + 20\!\cdots\!48 \) Copy content Toggle raw display
$17$ \( T^{6} + 51200226 T^{5} + \cdots + 20\!\cdots\!68 \) Copy content Toggle raw display
$19$ \( T^{6} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{6} + 105826896 T^{5} + \cdots + 50\!\cdots\!28 \) Copy content Toggle raw display
$29$ \( T^{6} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{3} + 1333558584 T^{2} + \cdots - 55\!\cdots\!48)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 1747956246 T^{5} + \cdots + 38\!\cdots\!28 \) Copy content Toggle raw display
$41$ \( (T^{3} - 11361049116 T^{2} + \cdots - 18\!\cdots\!48)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} - 15890524824 T^{5} + \cdots + 78\!\cdots\!68 \) Copy content Toggle raw display
$47$ \( T^{6} - 18495531264 T^{5} + \cdots + 57\!\cdots\!48 \) Copy content Toggle raw display
$53$ \( T^{6} - 88020413514 T^{5} + \cdots + 11\!\cdots\!48 \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{3} - 145897445676 T^{2} + \cdots + 10\!\cdots\!12)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} - 3887251464 T^{5} + \cdots + 11\!\cdots\!48 \) Copy content Toggle raw display
$71$ \( (T^{3} - 464567507736 T^{2} + \cdots - 28\!\cdots\!28)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 12678070086 T^{5} + \cdots + 81\!\cdots\!48 \) Copy content Toggle raw display
$79$ \( T^{6} + \cdots + 32\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{6} + 68676615456 T^{5} + \cdots + 27\!\cdots\!08 \) Copy content Toggle raw display
$89$ \( T^{6} + \cdots + 33\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{6} + 675735777846 T^{5} + \cdots + 44\!\cdots\!28 \) Copy content Toggle raw display
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