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Properties

Label 50.15.c.e
Level $50$
Weight $15$
Character orbit 50.c
Analytic conductor $62.164$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [50,15,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, 15, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("50.7");
 
S:= CuspForms(chi, 15);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 15 \)
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: \(62.1644840760\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 513401x^{6} + 81983771116x^{4} + 4511941511282436x^{2} + 65023716741123799296 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{15}\cdot 3^{4}\cdot 5^{14} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (64 \beta_1 - 64) q^{2} + ( - \beta_{3} - 176 \beta_1 - 176) q^{3} - 8192 \beta_1 q^{4} + (64 \beta_{3} + 64 \beta_{2} + \cdots + 22528) q^{6}+ \cdots + (\beta_{7} - \beta_{6} + \cdots + 4060644 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (64 \beta_1 - 64) q^{2} + ( - \beta_{3} - 176 \beta_1 - 176) q^{3} - 8192 \beta_1 q^{4} + (64 \beta_{3} + 64 \beta_{2} + \cdots + 22528) q^{6}+ \cdots + (14441585 \beta_{7} + \cdots + 130186347684033 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 512 q^{2} - 1404 q^{3} + 179712 q^{6} - 1333276 q^{7} + 4194304 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 512 q^{2} - 1404 q^{3} + 179712 q^{6} - 1333276 q^{7} + 4194304 q^{8} + 8181256 q^{11} - 11501568 q^{12} + 10529136 q^{13} - 536870912 q^{16} + 1669855424 q^{17} - 2079049728 q^{18} + 192520776 q^{21} - 523600384 q^{22} + 4778918036 q^{23} - 1347729408 q^{26} + 20842498800 q^{27} + 10922196992 q^{28} + 35788345016 q^{31} + 34359738368 q^{32} + 130209466872 q^{33} + 266118365184 q^{36} + 31454650344 q^{37} + 100622935040 q^{38} + 94198718056 q^{41} - 12321329664 q^{42} + 942463128516 q^{43} - 611701508608 q^{46} + 1797815170164 q^{47} + 94220845056 q^{48} - 1982287069224 q^{51} + 86254682112 q^{52} - 540438256184 q^{53} - 1398041214976 q^{56} - 6485834218080 q^{57} + 841739223040 q^{58} - 19913995236984 q^{61} - 2290454081024 q^{62} - 25848827893644 q^{63} - 16666811759616 q^{66} - 13553624120956 q^{67} - 13679455633408 q^{68} - 51541518798664 q^{71} - 17031575371776 q^{72} - 44383738947944 q^{73} - 12879735685120 q^{76} - 74886492270632 q^{77} - 66217996893696 q^{78} - 322723412222112 q^{81} - 6028717955584 q^{82} - 134272999400364 q^{83} - 120635280450048 q^{86} - 68423417580480 q^{87} + 4289334345728 q^{88} - 234891728536584 q^{91} + 39148896550912 q^{92} - 66751748435208 q^{93} - 12060268167168 q^{96} - 167451300858216 q^{97} - 78801761769472 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 513401x^{6} + 81983771116x^{4} + 4511941511282436x^{2} + 65023716741123799296 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 9772303 \nu^{7} - 5013078268271 \nu^{5} + \cdots - 23\!\cdots\!52 \nu ) / 15\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 2207237675177 \nu^{7} - 142904296239672 \nu^{6} + \cdots - 58\!\cdots\!08 ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1104024388163 \nu^{7} - 71452148119836 \nu^{6} + \cdots - 29\!\cdots\!04 ) / 63\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 103166695694099 \nu^{7} + \cdots + 35\!\cdots\!56 ) / 31\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 412667593877545 \nu^{7} + \cdots + 14\!\cdots\!24 ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 745951821015979 \nu^{7} + \cdots + 69\!\cdots\!80 ) / 63\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 14\!\cdots\!07 \nu^{7} + \cdots + 13\!\cdots\!60 ) / 12\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{7} + \beta_{6} + 8\beta_{5} - 8\beta_{4} - 821\beta_{3} + 821\beta_{2} - 7500\beta_1 ) / 30000 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 19 \beta_{7} - 19 \beta_{6} + 2 \beta_{5} + 2 \beta_{4} - 128549 \beta_{3} - 128549 \beta_{2} + \cdots - 770230066 ) / 6000 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 124729 \beta_{7} - 124729 \beta_{6} - 1640582 \beta_{5} + 1640582 \beta_{4} + \cdots + 439753642500 \beta_1 ) / 30000 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 8759827 \beta_{7} + 8759827 \beta_{6} - 31702166 \beta_{5} - 31702166 \beta_{4} + \cdots + 149455770145978 ) / 6000 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 25830356041 \beta_{7} + 25830356041 \beta_{6} + 356978961578 \beta_{5} + \cdots - 18\!\cdots\!00 \beta_1 ) / 30000 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 2581755066787 \beta_{7} - 2581755066787 \beta_{6} + 14521892627846 \beta_{5} + \cdots - 33\!\cdots\!18 ) / 6000 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 64\!\cdots\!89 \beta_{7} + \cdots + 59\!\cdots\!00 \beta_1 ) / 30000 \) Copy content Toggle raw display

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

Character values

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

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

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
514.022i
− 394.425i
− 267.359i
148.763i
− 514.022i
394.425i
267.359i
− 148.763i
−64.0000 − 64.0000i −3060.13 + 3060.13i 8192.00i 0 391697. −586054. − 586054.i 524288. − 524288.i − 1.39459e7i 0
7.2 −64.0000 − 64.0000i −803.510 + 803.510i 8192.00i 0 102849. 657264. + 657264.i 524288. − 524288.i 3.49171e6i 0
7.3 −64.0000 − 64.0000i 423.414 − 423.414i 8192.00i 0 −54197.0 −346999. − 346999.i 524288. − 524288.i 4.42441e6i 0
7.4 −64.0000 − 64.0000i 2738.23 − 2738.23i 8192.00i 0 −350493. −390848. − 390848.i 524288. − 524288.i − 1.02128e7i 0
43.1 −64.0000 + 64.0000i −3060.13 − 3060.13i − 8192.00i 0 391697. −586054. + 586054.i 524288. + 524288.i 1.39459e7i 0
43.2 −64.0000 + 64.0000i −803.510 − 803.510i − 8192.00i 0 102849. 657264. − 657264.i 524288. + 524288.i − 3.49171e6i 0
43.3 −64.0000 + 64.0000i 423.414 + 423.414i − 8192.00i 0 −54197.0 −346999. + 346999.i 524288. + 524288.i − 4.42441e6i 0
43.4 −64.0000 + 64.0000i 2738.23 + 2738.23i − 8192.00i 0 −350493. −390848. + 390848.i 524288. + 524288.i 1.02128e7i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.4
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 50.15.c.e 8
5.b even 2 1 10.15.c.b ✓ 8
5.c odd 4 1 10.15.c.b ✓ 8
5.c odd 4 1 inner 50.15.c.e 8
15.d odd 2 1 90.15.g.b 8
15.e even 4 1 90.15.g.b 8
20.d odd 2 1 80.15.p.b 8
20.e even 4 1 80.15.p.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.15.c.b ✓ 8 5.b even 2 1
10.15.c.b ✓ 8 5.c odd 4 1
50.15.c.e 8 1.a even 1 1 trivial
50.15.c.e 8 5.c odd 4 1 inner
80.15.p.b 8 20.d odd 2 1
80.15.p.b 8 20.e even 4 1
90.15.g.b 8 15.d odd 2 1
90.15.g.b 8 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 1404 T_{3}^{7} + 985608 T_{3}^{6} - 10963094040 T_{3}^{5} + 272841733510596 T_{3}^{4} + \cdots + 13\!\cdots\!36 \) acting on \(S_{15}^{\mathrm{new}}(50, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 128 T + 8192)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} + \cdots + 13\!\cdots\!36 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 43\!\cdots\!76 \) Copy content Toggle raw display
$11$ \( (T^{4} + \cdots - 11\!\cdots\!24)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 12\!\cdots\!56 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 77\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 77\!\cdots\!56 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 45\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{4} + \cdots + 17\!\cdots\!16)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 11\!\cdots\!16 \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots - 10\!\cdots\!24)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 12\!\cdots\!96 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 36\!\cdots\!56 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 10\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots + 29\!\cdots\!16)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 45\!\cdots\!16 \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots - 18\!\cdots\!04)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 20\!\cdots\!16 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 45\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 55\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 25\!\cdots\!96 \) Copy content Toggle raw display
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