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Properties

Label 50.4.d.a
Level $50$
Weight $4$
Character orbit 50.d
Analytic conductor $2.950$
Analytic rank $0$
Dimension $12$
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] = [50,4,Mod(11,50)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(50, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([8]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("50.11");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 50.d (of order \(5\), degree \(4\), 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: \(2.95009550029\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(3\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 6 x^{11} + 78 x^{10} - 335 x^{9} + 1991 x^{8} - 6020 x^{7} + 20827 x^{6} - 42752 x^{5} + \cdots + 11005 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 5^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 \beta_{5} q^{2} + (\beta_{5} - \beta_{4} + \cdots - \beta_{2}) q^{3}+ \cdots + ( - \beta_{10} + \beta_{9} + \cdots - 5 \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 \beta_{5} q^{2} + (\beta_{5} - \beta_{4} + \cdots - \beta_{2}) q^{3}+ \cdots + ( - 13 \beta_{11} + 3 \beta_{10} + \cdots - 114) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6 q^{2} + q^{3} - 12 q^{4} + 20 q^{5} - 2 q^{6} + 58 q^{7} + 24 q^{8} - 26 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 6 q^{2} + q^{3} - 12 q^{4} + 20 q^{5} - 2 q^{6} + 58 q^{7} + 24 q^{8} - 26 q^{9} + 40 q^{10} - 46 q^{11} - 36 q^{12} - 19 q^{13} + 4 q^{14} - 110 q^{15} - 48 q^{16} + 178 q^{17} + 72 q^{18} + 130 q^{19} + 60 q^{20} - 221 q^{21} - 158 q^{22} - 259 q^{23} - 128 q^{24} - 50 q^{25} + 128 q^{26} - 215 q^{27} - 108 q^{28} + 15 q^{29} + 60 q^{30} + 309 q^{31} - 384 q^{32} - 608 q^{33} + 304 q^{34} + 1090 q^{35} - 104 q^{36} - 557 q^{37} + 110 q^{38} + 1158 q^{39} - 596 q^{41} + 442 q^{42} + 746 q^{43} + 316 q^{44} + 90 q^{45} - 292 q^{46} - 1442 q^{47} - 144 q^{48} - 834 q^{49} + 500 q^{50} + 2724 q^{51} - 76 q^{52} + 96 q^{53} - 510 q^{54} - 1610 q^{55} + 216 q^{56} + 1280 q^{57} - 30 q^{58} + 975 q^{59} + 1100 q^{60} - 2016 q^{61} + 592 q^{62} + 1206 q^{63} - 192 q^{64} - 1230 q^{65} - 1014 q^{66} + 68 q^{67} - 208 q^{68} + 783 q^{69} - 610 q^{70} + 1384 q^{71} - 352 q^{72} - 3834 q^{73} - 2656 q^{74} - 1225 q^{75} - 600 q^{76} - 1059 q^{77} + 664 q^{78} - 4550 q^{79} - 240 q^{80} + 1227 q^{81} + 1012 q^{82} + 1211 q^{83} + 1236 q^{84} + 2165 q^{85} - 382 q^{86} + 1625 q^{87} + 368 q^{88} + 1685 q^{89} - 1370 q^{90} - 3271 q^{91} + 584 q^{92} + 4552 q^{93} + 2884 q^{94} + 4220 q^{95} - 32 q^{96} - 4247 q^{97} - 1952 q^{98} - 1202 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 6 x^{11} + 78 x^{10} - 335 x^{9} + 1991 x^{8} - 6020 x^{7} + 20827 x^{6} - 42752 x^{5} + \cdots + 11005 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 12401 \nu^{10} - 62005 \nu^{9} + 741378 \nu^{8} - 2593482 \nu^{7} + 12545334 \nu^{6} + \cdots + 41708794 ) / 52069204 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 8508892 \nu^{11} - 65966879 \nu^{10} + 732806306 \nu^{9} - 3759260966 \nu^{8} + \cdots - 229364886066 ) / 52537826836 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 8508892 \nu^{11} + 27630933 \nu^{10} - 541126576 \nu^{9} + 1271453413 \nu^{8} + \cdots + 99615655939 ) / 52537826836 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 11490438 \nu^{11} - 135415575 \nu^{10} + 1106497120 \nu^{9} - 7534097393 \nu^{8} + \cdots - 136422080159 ) / 52537826836 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 15355070 \nu^{11} + 103620858 \nu^{10} - 1188674703 \nu^{9} + 5528263910 \nu^{8} + \cdots + 214847522154 ) / 52537826836 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 16654665 \nu^{11} - 140727354 \nu^{10} + 1466421602 \nu^{9} - 7948463175 \nu^{8} + \cdots - 325767764375 ) / 52537826836 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 16654665 \nu^{11} + 42473961 \nu^{10} - 975154637 \nu^{9} + 1664620038 \nu^{8} + \cdots + 911884719970 ) / 52537826836 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 37898175 \nu^{11} - 221735051 \nu^{10} + 3291499581 \nu^{9} - 13620748459 \nu^{8} + \cdots - 1255256691195 ) / 52537826836 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 41759035 \nu^{11} - 295636554 \nu^{10} + 3141390924 \nu^{9} - 15209281535 \nu^{8} + \cdots - 163007245871 ) / 52537826836 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 41759035 \nu^{11} - 163712831 \nu^{10} + 2481772309 \nu^{9} - 6649832626 \nu^{8} + \cdots + 47775379656 ) / 52537826836 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 54552840 \nu^{11} - 237618835 \nu^{10} + 4133703333 \nu^{9} - 13942489388 \nu^{8} + \cdots - 2194741915000 ) / 52537826836 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{10} - 3\beta_{9} - 3\beta_{5} + 4\beta_{4} - \beta_{3} - 2\beta_{2} + 2\beta _1 + 4 ) / 5 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2 \beta_{11} - 2 \beta_{10} - 2 \beta_{8} + 4 \beta_{7} + 2 \beta_{6} - 3 \beta_{5} + 4 \beta_{4} + \cdots - 40 ) / 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 4 \beta_{11} - 7 \beta_{10} + 67 \beta_{9} - 2 \beta_{8} + 10 \beta_{7} + 185 \beta_{5} - 65 \beta_{4} + \cdots - 156 ) / 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 52 \beta_{11} + 155 \beta_{10} - 33 \beta_{9} + 56 \beta_{8} - 50 \beta_{7} - 10 \beta_{6} + \cdots + 489 ) / 5 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 204 \beta_{11} + 205 \beta_{10} - 1975 \beta_{9} + 76 \beta_{8} - 356 \beta_{7} + 122 \beta_{6} + \cdots + 4830 ) / 5 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 1270 \beta_{11} - 5905 \beta_{10} + 289 \beta_{9} - 1664 \beta_{8} + 486 \beta_{7} + 68 \beta_{6} + \cdots - 4732 ) / 5 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 7874 \beta_{11} - 11592 \beta_{10} + 61452 \beta_{9} - 3382 \beta_{8} + 10740 \beta_{7} - 5260 \beta_{6} + \cdots - 144141 ) / 5 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 29706 \beta_{11} + 187164 \beta_{10} + 38770 \beta_{9} + 49522 \beta_{8} + 2324 \beta_{7} + \cdots - 48095 ) / 5 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 276800 \beta_{11} + 553586 \beta_{10} - 1878648 \beta_{9} + 148450 \beta_{8} - 315230 \beta_{7} + \cdots + 4256064 ) / 5 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 618378 \beta_{11} - 5417333 \beta_{10} - 2942407 \beta_{9} - 1411526 \beta_{8} - 442882 \beta_{7} + \cdots + 6158501 ) / 5 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 9215204 \beta_{11} - 22803892 \beta_{10} + 55487882 \beta_{9} - 5974322 \beta_{8} + 9091070 \beta_{7} + \cdots - 123338956 ) / 5 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(27\)
\(\chi(n)\) \(-\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1
0.500000 − 3.48876i
0.500000 + 0.215168i
0.500000 + 2.91032i
0.500000 + 3.03905i
0.500000 + 0.958970i
0.500000 − 5.53686i
0.500000 − 3.03905i
0.500000 − 0.958970i
0.500000 + 5.53686i
0.500000 + 3.48876i
0.500000 − 0.215168i
0.500000 − 2.91032i
1.61803 − 1.17557i −1.33712 + 4.11522i 1.23607 − 3.80423i 10.6469 + 3.41218i 2.67423 + 8.23044i 17.3088 −2.47214 − 7.60845i 6.69632 + 4.86516i 21.2383 − 6.99519i
11.2 1.61803 − 1.17557i 0.839998 − 2.58525i 1.23607 − 3.80423i −5.08411 − 9.95750i −1.68000 − 5.17050i 4.23839 −2.47214 − 7.60845i 15.8655 + 11.5270i −19.9320 − 10.1348i
11.3 1.61803 − 1.17557i 2.42417 − 7.46082i 1.23607 − 3.80423i 5.02735 + 9.98628i −4.84834 − 14.9216i −12.6374 −2.47214 − 7.60845i −27.9438 − 20.3024i 19.8740 + 10.2481i
21.1 −0.618034 + 1.90211i −3.85384 − 2.79998i −3.23607 − 2.35114i 10.3421 − 4.24747i 7.70767 − 5.59995i 29.7799 6.47214 − 4.70228i −1.33127 − 4.09724i 1.68741 + 22.2969i
21.2 −0.618034 + 1.90211i −1.87556 − 1.36267i −3.23607 − 2.35114i −9.42241 − 6.01815i 3.75112 − 2.72535i −15.5675 6.47214 − 4.70228i −6.68261 − 20.5670i 17.2706 − 14.2031i
21.3 −0.618034 + 1.90211i 4.30234 + 3.12584i −3.23607 − 2.35114i −1.50985 + 11.0779i −8.60469 + 6.25167i 5.87773 6.47214 − 4.70228i 0.395855 + 1.21832i −20.1383 − 9.71844i
31.1 −0.618034 − 1.90211i −3.85384 + 2.79998i −3.23607 + 2.35114i 10.3421 + 4.24747i 7.70767 + 5.59995i 29.7799 6.47214 + 4.70228i −1.33127 + 4.09724i 1.68741 − 22.2969i
31.2 −0.618034 − 1.90211i −1.87556 + 1.36267i −3.23607 + 2.35114i −9.42241 + 6.01815i 3.75112 + 2.72535i −15.5675 6.47214 + 4.70228i −6.68261 + 20.5670i 17.2706 + 14.2031i
31.3 −0.618034 − 1.90211i 4.30234 − 3.12584i −3.23607 + 2.35114i −1.50985 − 11.0779i −8.60469 − 6.25167i 5.87773 6.47214 + 4.70228i 0.395855 − 1.21832i −20.1383 + 9.71844i
41.1 1.61803 + 1.17557i −1.33712 − 4.11522i 1.23607 + 3.80423i 10.6469 − 3.41218i 2.67423 − 8.23044i 17.3088 −2.47214 + 7.60845i 6.69632 − 4.86516i 21.2383 + 6.99519i
41.2 1.61803 + 1.17557i 0.839998 + 2.58525i 1.23607 + 3.80423i −5.08411 + 9.95750i −1.68000 + 5.17050i 4.23839 −2.47214 + 7.60845i 15.8655 − 11.5270i −19.9320 + 10.1348i
41.3 1.61803 + 1.17557i 2.42417 + 7.46082i 1.23607 + 3.80423i 5.02735 − 9.98628i −4.84834 + 14.9216i −12.6374 −2.47214 + 7.60845i −27.9438 + 20.3024i 19.8740 − 10.2481i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 50.4.d.a ✓ 12
5.b even 2 1 250.4.d.a 12
5.c odd 4 2 250.4.e.a 24
25.d even 5 1 inner 50.4.d.a ✓ 12
25.d even 5 1 1250.4.a.d 6
25.e even 10 1 250.4.d.a 12
25.e even 10 1 1250.4.a.e 6
25.f odd 20 2 250.4.e.a 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
50.4.d.a ✓ 12 1.a even 1 1 trivial
50.4.d.a ✓ 12 25.d even 5 1 inner
250.4.d.a 12 5.b even 2 1
250.4.d.a 12 25.e even 10 1
250.4.e.a 24 5.c odd 4 2
250.4.e.a 24 25.f odd 20 2
1250.4.a.d 6 25.d even 5 1
1250.4.a.e 6 25.e even 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} - T_{3}^{11} + 54 T_{3}^{10} + 180 T_{3}^{9} + 1120 T_{3}^{8} + 639 T_{3}^{7} + \cdots + 29365561 \) acting on \(S_{4}^{\mathrm{new}}(50, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 2 T^{3} + 4 T^{2} + \cdots + 16)^{3} \) Copy content Toggle raw display
$3$ \( T^{12} - T^{11} + \cdots + 29365561 \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 3814697265625 \) Copy content Toggle raw display
$7$ \( (T^{6} - 29 T^{5} + \cdots + 2526256)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 749325684496 \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 489228086487001 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 47\!\cdots\!96 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 25\!\cdots\!25 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 26\!\cdots\!61 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 71\!\cdots\!25 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 12\!\cdots\!61 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 31\!\cdots\!61 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 25\!\cdots\!56 \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots + 541993135311104)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 15\!\cdots\!21 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 13\!\cdots\!56 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 47\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 50\!\cdots\!21 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 14\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 49\!\cdots\!41 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 17\!\cdots\!36 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 30\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 10\!\cdots\!01 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 12\!\cdots\!61 \) Copy content Toggle raw display
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