Properties

Label 400.2.y.c
Level $400$
Weight $2$
Character orbit 400.y
Analytic conductor $3.194$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 400.y (of order \(10\), degree \(4\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.19401608085\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{10})\)
Coefficient field: 8.0.58140625.2
Defining polynomial: \(x^{8} - 3 x^{7} + 4 x^{6} - 7 x^{5} + 11 x^{4} + 5 x^{3} - 10 x^{2} - 25 x + 25\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 25)
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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.

\(f(q)\) \(=\) \( q + ( 1 + \beta_{3} + \beta_{7} ) q^{3} + ( \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} ) q^{5} + ( -\beta_{2} - \beta_{3} - \beta_{4} + \beta_{6} ) q^{7} + ( 1 - \beta_{1} - \beta_{2} + \beta_{5} ) q^{9} +O(q^{10})\) \( q + ( 1 + \beta_{3} + \beta_{7} ) q^{3} + ( \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} ) q^{5} + ( -\beta_{2} - \beta_{3} - \beta_{4} + \beta_{6} ) q^{7} + ( 1 - \beta_{1} - \beta_{2} + \beta_{5} ) q^{9} + 2 \beta_{4} q^{11} + ( -\beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} ) q^{13} + ( -\beta_{1} - 3 \beta_{2} + 2 \beta_{3} - 4 \beta_{4} - \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{15} + ( \beta_{1} - 2 \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{7} ) q^{17} + ( 1 + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{19} + ( -2 + 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} + 2 \beta_{6} ) q^{21} + ( 2 - \beta_{1} - 3 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{6} ) q^{23} + ( -5 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{7} ) q^{25} + ( 2 - 3 \beta_{1} - \beta_{2} - \beta_{5} - \beta_{6} ) q^{27} + ( 3 - 4 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} - 3 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{29} + ( 2 + \beta_{1} + \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 3 \beta_{6} + 2 \beta_{7} ) q^{31} + ( 4 - 2 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} - 2 \beta_{6} ) q^{33} + ( -5 + 2 \beta_{1} + \beta_{2} - 4 \beta_{3} + 3 \beta_{4} + 2 \beta_{5} + 3 \beta_{6} - \beta_{7} ) q^{35} + ( 4 - \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{37} + ( 2 - \beta_{1} - 3 \beta_{2} - 3 \beta_{4} - 2 \beta_{7} ) q^{39} + ( -5 + 3 \beta_{1} + 6 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} ) q^{41} + ( -3 + \beta_{1} + 2 \beta_{2} - 4 \beta_{3} + 6 \beta_{4} - \beta_{6} + \beta_{7} ) q^{43} + ( -5 + 5 \beta_{2} - 5 \beta_{3} + 2 \beta_{6} - \beta_{7} ) q^{45} + ( \beta_{2} + 3 \beta_{3} - 3 \beta_{5} + 2 \beta_{7} ) q^{47} + ( 3 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 4 \beta_{5} + \beta_{6} + \beta_{7} ) q^{49} + ( -2 + 4 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 4 \beta_{5} + 2 \beta_{6} ) q^{51} + ( -2 + 4 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} - 2 \beta_{7} ) q^{53} + ( 4 - 4 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{7} ) q^{55} + ( -1 - \beta_{1} + 4 \beta_{2} + 2 \beta_{3} + 3 \beta_{6} - \beta_{7} ) q^{57} + ( -3 - \beta_{1} + 5 \beta_{2} - 5 \beta_{3} + 6 \beta_{4} + 3 \beta_{5} + \beta_{6} + \beta_{7} ) q^{59} + ( -1 + 3 \beta_{1} + 4 \beta_{2} + 2 \beta_{4} + 4 \beta_{5} - 4 \beta_{6} + 6 \beta_{7} ) q^{61} + ( 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{6} + 2 \beta_{7} ) q^{63} + ( 2 - \beta_{1} - 4 \beta_{2} - \beta_{3} - 2 \beta_{4} - 2 \beta_{7} ) q^{65} + ( -4 + 4 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} + 4 \beta_{5} + 4 \beta_{7} ) q^{67} + ( 2 + \beta_{1} - \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{69} + ( -4 + 6 \beta_{1} + 9 \beta_{2} + \beta_{3} - 2 \beta_{4} + 3 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{71} + ( 3 - 3 \beta_{1} - 2 \beta_{5} - 2 \beta_{6} ) q^{73} + ( -5 + 5 \beta_{1} + 5 \beta_{2} - 7 \beta_{3} + 7 \beta_{4} + 2 \beta_{5} - \beta_{7} ) q^{75} + ( -2 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} + 2 \beta_{4} ) q^{77} + ( 7 - 6 \beta_{1} - 9 \beta_{2} + 8 \beta_{3} - 4 \beta_{4} - 3 \beta_{5} - 8 \beta_{6} + \beta_{7} ) q^{79} + ( 4 + 2 \beta_{1} - \beta_{3} - 4 \beta_{4} - 2 \beta_{6} ) q^{81} + ( 6 - 5 \beta_{1} - 2 \beta_{2} + \beta_{3} + 4 \beta_{4} - 4 \beta_{5} - \beta_{6} - 4 \beta_{7} ) q^{83} + ( -5 + 3 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} + 4 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{85} + ( 1 - \beta_{1} + 5 \beta_{2} - 3 \beta_{3} - 6 \beta_{4} - \beta_{5} + 3 \beta_{6} - 3 \beta_{7} ) q^{87} + ( -7 + \beta_{1} + 8 \beta_{2} + 7 \beta_{4} + 2 \beta_{7} ) q^{89} + ( -3 + \beta_{1} + 3 \beta_{2} - 3 \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{91} + ( -1 + 5 \beta_{1} - \beta_{2} - 3 \beta_{3} + \beta_{4} - 4 \beta_{6} + 5 \beta_{7} ) q^{93} + ( -5 + 5 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - 3 \beta_{5} - \beta_{7} ) q^{95} + ( -12 + 8 \beta_{2} - 5 \beta_{3} + 3 \beta_{4} - 4 \beta_{5} + 4 \beta_{7} ) q^{97} + ( 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 4 \beta_{5} + 2 \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 5q^{3} + q^{9} + O(q^{10}) \) \( 8q + 5q^{3} + q^{9} + 4q^{11} - 5q^{13} - 15q^{15} - 10q^{17} + 5q^{19} - 4q^{21} - 5q^{23} - 10q^{25} + 5q^{27} - 5q^{29} + 9q^{31} + 10q^{33} - 15q^{35} + 30q^{37} + 3q^{39} - 4q^{41} - 15q^{45} + 14q^{49} + 4q^{51} - 10q^{53} + 10q^{55} - 9q^{61} - 10q^{63} + 5q^{65} - 20q^{67} + 17q^{69} - 6q^{71} + 15q^{73} + 10q^{75} + 10q^{77} - 15q^{79} + 28q^{81} + 45q^{83} - 15q^{85} + 20q^{87} - 25q^{89} - 6q^{91} - 15q^{95} - 60q^{97} + 8q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 3 x^{7} + 4 x^{6} - 7 x^{5} + 11 x^{4} + 5 x^{3} - 10 x^{2} - 25 x + 25\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 406 \nu^{7} - 714 \nu^{6} + 747 \nu^{5} - 1896 \nu^{4} + 2103 \nu^{3} + 4949 \nu^{2} + 1065 \nu - 7800 \)\()/1355\)
\(\beta_{3}\)\(=\)\((\)\( 420 \nu^{7} - 776 \nu^{6} + 698 \nu^{5} - 1924 \nu^{4} + 2297 \nu^{3} + 5129 \nu^{2} + 1055 \nu - 10265 \)\()/1355\)
\(\beta_{4}\)\(=\)\((\)\( 728 \nu^{7} - 1327 \nu^{6} + 1246 \nu^{5} - 3353 \nu^{4} + 3584 \nu^{3} + 8547 \nu^{2} + 2190 \nu - 15715 \)\()/1355\)
\(\beta_{5}\)\(=\)\((\)\( -857 \nu^{7} + 1666 \nu^{6} - 1743 \nu^{5} + 4424 \nu^{4} - 4907 \nu^{3} - 9470 \nu^{2} - 2485 \nu + 18200 \)\()/1355\)
\(\beta_{6}\)\(=\)\((\)\( 891 \nu^{7} - 1623 \nu^{6} + 1624 \nu^{5} - 4492 \nu^{4} + 4991 \nu^{3} + 9520 \nu^{2} + 3235 \nu - 18960 \)\()/1355\)
\(\beta_{7}\)\(=\)\((\)\( 955 \nu^{7} - 1829 \nu^{6} + 1942 \nu^{5} - 4891 \nu^{4} + 5723 \nu^{3} + 9646 \nu^{2} + 2415 \nu - 20550 \)\()/1355\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{7} - \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2}\)
\(\nu^{3}\)\(=\)\(\beta_{7} + \beta_{5} - 3 \beta_{4} + 4 \beta_{3} + \beta_{2} + \beta_{1} + 3\)
\(\nu^{4}\)\(=\)\(5 \beta_{7} - 5 \beta_{6} + 4 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + 4 \beta_{1} + 2\)
\(\nu^{5}\)\(=\)\(4 \beta_{7} - 6 \beta_{6} - 7 \beta_{3} + 11 \beta_{2} + 4 \beta_{1} - 13\)
\(\nu^{6}\)\(=\)\(7 \beta_{6} + 7 \beta_{5} - 8 \beta_{4} - 7 \beta_{3} + 21 \beta_{2} - 2 \beta_{1} - 21\)
\(\nu^{7}\)\(=\)\(23 \beta_{7} + 38 \beta_{5} + 23 \beta_{4} - 23 \beta_{3} + 12 \beta_{2}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/400\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\) \(351\)
\(\chi(n)\) \(1\) \(\beta_{2}\) \(1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
129.1
1.17421 + 0.0566033i
−0.983224 0.644389i
1.66637 0.917186i
−0.357358 + 1.86824i
1.66637 + 0.917186i
−0.357358 1.86824i
1.17421 0.0566033i
−0.983224 + 0.644389i
0 −1.29224 + 1.77862i 0 −1.22570 + 1.87020i 0 0.992398i 0 −0.566541 1.74363i 0
129.2 0 0.865190 1.19083i 0 0.107666 2.23347i 0 3.26086i 0 0.257524 + 0.792578i 0
209.1 0 0.451659 0.146753i 0 2.19625 + 0.420099i 0 3.03582i 0 −2.24459 + 1.63079i 0
209.2 0 2.47539 0.804303i 0 −1.07822 1.95894i 0 0.407162i 0 3.05361 2.21858i 0
289.1 0 0.451659 + 0.146753i 0 2.19625 0.420099i 0 3.03582i 0 −2.24459 1.63079i 0
289.2 0 2.47539 + 0.804303i 0 −1.07822 + 1.95894i 0 0.407162i 0 3.05361 + 2.21858i 0
369.1 0 −1.29224 1.77862i 0 −1.22570 1.87020i 0 0.992398i 0 −0.566541 + 1.74363i 0
369.2 0 0.865190 + 1.19083i 0 0.107666 + 2.23347i 0 3.26086i 0 0.257524 0.792578i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 369.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.e even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 400.2.y.c 8
4.b odd 2 1 25.2.e.a 8
12.b even 2 1 225.2.m.a 8
20.d odd 2 1 125.2.e.b 8
20.e even 4 2 125.2.d.b 16
25.e even 10 1 inner 400.2.y.c 8
25.f odd 20 2 10000.2.a.bj 8
100.h odd 10 1 25.2.e.a 8
100.h odd 10 1 625.2.b.c 8
100.h odd 10 1 625.2.e.a 8
100.h odd 10 1 625.2.e.i 8
100.j odd 10 1 125.2.e.b 8
100.j odd 10 1 625.2.b.c 8
100.j odd 10 1 625.2.e.a 8
100.j odd 10 1 625.2.e.i 8
100.l even 20 2 125.2.d.b 16
100.l even 20 2 625.2.a.f 8
100.l even 20 4 625.2.d.o 16
300.r even 10 1 225.2.m.a 8
300.u odd 20 2 5625.2.a.x 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.2.e.a 8 4.b odd 2 1
25.2.e.a 8 100.h odd 10 1
125.2.d.b 16 20.e even 4 2
125.2.d.b 16 100.l even 20 2
125.2.e.b 8 20.d odd 2 1
125.2.e.b 8 100.j odd 10 1
225.2.m.a 8 12.b even 2 1
225.2.m.a 8 300.r even 10 1
400.2.y.c 8 1.a even 1 1 trivial
400.2.y.c 8 25.e even 10 1 inner
625.2.a.f 8 100.l even 20 2
625.2.b.c 8 100.h odd 10 1
625.2.b.c 8 100.j odd 10 1
625.2.d.o 16 100.l even 20 4
625.2.e.a 8 100.h odd 10 1
625.2.e.a 8 100.j odd 10 1
625.2.e.i 8 100.h odd 10 1
625.2.e.i 8 100.j odd 10 1
5625.2.a.x 8 300.u odd 20 2
10000.2.a.bj 8 25.f odd 20 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{3}^{8} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(400, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( 16 - 80 T + 144 T^{2} - 110 T^{3} + 51 T^{4} - 15 T^{5} + 9 T^{6} - 5 T^{7} + T^{8} \)
$5$ \( 625 + 125 T^{2} - 100 T^{3} + 5 T^{4} - 20 T^{5} + 5 T^{6} + T^{8} \)
$7$ \( 16 + 116 T^{2} + 121 T^{4} + 21 T^{6} + T^{8} \)
$11$ \( ( 16 - 8 T + 4 T^{2} - 2 T^{3} + T^{4} )^{2} \)
$13$ \( 1 - 5 T + 4 T^{2} + 5 T^{3} + 21 T^{4} - 5 T^{5} + 4 T^{6} + 5 T^{7} + T^{8} \)
$17$ \( 1936 + 880 T - 784 T^{2} - 550 T^{3} + 31 T^{4} + 125 T^{5} + 56 T^{6} + 10 T^{7} + T^{8} \)
$19$ \( 400 - 200 T + 400 T^{2} + 100 T^{3} - 15 T^{4} - 40 T^{5} + 30 T^{6} - 5 T^{7} + T^{8} \)
$23$ \( 256 + 320 T - 16 T^{2} + 60 T^{3} + 241 T^{4} + 15 T^{5} - T^{6} + 5 T^{7} + T^{8} \)
$29$ \( 483025 + 142475 T + 49350 T^{2} + 4525 T^{3} + 485 T^{4} - 5 T^{5} + 30 T^{6} + 5 T^{7} + T^{8} \)
$31$ \( 1936 - 23496 T + 110532 T^{2} - 31178 T^{3} + 6855 T^{4} - 917 T^{5} + 117 T^{6} - 9 T^{7} + T^{8} \)
$37$ \( 116281 - 192665 T + 141631 T^{2} - 61170 T^{3} + 17321 T^{4} - 3270 T^{5} + 406 T^{6} - 30 T^{7} + T^{8} \)
$41$ \( 13456 - 1624 T + 9492 T^{2} - 7622 T^{3} + 2655 T^{4} + 457 T^{5} + 52 T^{6} + 4 T^{7} + T^{8} \)
$43$ \( 246016 + 56784 T^{2} + 4421 T^{4} + 129 T^{6} + T^{8} \)
$47$ \( 65536 + 61440 T - 4864 T^{2} - 9840 T^{3} + 4101 T^{4} - 615 T^{5} + 16 T^{6} + T^{8} \)
$53$ \( 8755681 + 4157395 T + 722619 T^{2} + 29590 T^{3} - 8079 T^{4} - 1290 T^{5} - 6 T^{6} + 10 T^{7} + T^{8} \)
$59$ \( 4080400 - 1333200 T + 407000 T^{2} - 54150 T^{3} + 5635 T^{4} - 15 T^{5} + T^{8} \)
$61$ \( 116281 - 59334 T + 139032 T^{2} + 12978 T^{3} + 16405 T^{4} - 1068 T^{5} - 43 T^{6} + 9 T^{7} + T^{8} \)
$67$ \( 246016 - 198400 T + 74816 T^{2} - 6080 T^{3} - 2384 T^{4} - 80 T^{5} + 116 T^{6} + 20 T^{7} + T^{8} \)
$71$ \( 24245776 + 5889104 T + 966712 T^{2} + 53922 T^{3} + 3455 T^{4} - 297 T^{5} + 142 T^{6} + 6 T^{7} + T^{8} \)
$73$ \( 1 + 4 T^{2} + 30 T^{3} + 91 T^{4} + 120 T^{5} + 49 T^{6} - 15 T^{7} + T^{8} \)
$79$ \( 33408400 - 4913000 T + 1416100 T^{2} - 79050 T^{3} + 12185 T^{4} - 600 T^{5} + 100 T^{6} + 15 T^{7} + T^{8} \)
$83$ \( 99856 - 284400 T + 329344 T^{2} - 199950 T^{3} + 70651 T^{4} - 11175 T^{5} + 949 T^{6} - 45 T^{7} + T^{8} \)
$89$ \( 1392400 + 1640200 T + 888600 T^{2} + 258800 T^{3} + 47985 T^{4} + 5890 T^{5} + 520 T^{6} + 25 T^{7} + T^{8} \)
$97$ \( 301334881 + 63360350 T + 5529361 T^{2} + 971040 T^{3} + 213086 T^{4} + 24990 T^{5} + 1636 T^{6} + 60 T^{7} + T^{8} \)
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