Properties

Label 625.2.e.i
Level $625$
Weight $2$
Character orbit 625.e
Analytic conductor $4.991$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 625 = 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 625.e (of order \(10\), degree \(4\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.99065012633\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{10})\)
Coefficient field: 8.0.58140625.2
Defining polynomial: \( x^{8} - 3x^{7} + 4x^{6} - 7x^{5} + 11x^{4} + 5x^{3} - 10x^{2} - 25x + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{5} - \beta_{4}) q^{2} + ( - \beta_{5} - \beta_{2} - \beta_1 + 1) q^{3} + (\beta_{7} + \beta_{6} + \beta_{5} + \beta_{2} + \beta_1) q^{4} + (\beta_{6} - \beta_{5} - 2 \beta_{4} - \beta_{2} + 1) q^{6} + ( - \beta_{6} - \beta_{4} - 2 \beta_{2} + 1) q^{7} + (\beta_{4} - 2 \beta_{3} - 2) q^{8} + (\beta_{6} - \beta_{3} - \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{5} - \beta_{4}) q^{2} + ( - \beta_{5} - \beta_{2} - \beta_1 + 1) q^{3} + (\beta_{7} + \beta_{6} + \beta_{5} + \beta_{2} + \beta_1) q^{4} + (\beta_{6} - \beta_{5} - 2 \beta_{4} - \beta_{2} + 1) q^{6} + ( - \beta_{6} - \beta_{4} - 2 \beta_{2} + 1) q^{7} + (\beta_{4} - 2 \beta_{3} - 2) q^{8} + (\beta_{6} - \beta_{3} - \beta_1) q^{9} - 2 \beta_{2} q^{11} + (\beta_{7} + \beta_{5} - 2 \beta_{4} + \beta_{3} - 2 \beta_{2} + \beta_1) q^{12} + ( - \beta_{6} + \beta_{4} - \beta_1 + 1) q^{13} + ( - \beta_{7} + 2 \beta_{6} - 2 \beta_{4} + \beta_{3} + \beta_{2} - 1) q^{14} + (\beta_{7} - \beta_{5} - \beta_1) q^{16} + ( - \beta_{4} + 2 \beta_{3} - 3 \beta_{2} - \beta_1) q^{17} + (\beta_{7} + \beta_{4} - 3 \beta_{3} - \beta_{2} + \beta_1 - 1) q^{18} + (\beta_{6} - \beta_{5} - \beta_{2} + 1) q^{19} + (\beta_{7} + \beta_{6} + \beta_{4} - \beta_{3} + 3 \beta_{2} + 2 \beta_1 - 2) q^{21} + ( - 2 \beta_{7} + 2 \beta_{6} - 2) q^{22} + ( - \beta_{7} + 2 \beta_{4} - \beta_{3} + \beta_{2} - 3) q^{23} + (\beta_{7} + 2 \beta_{5} + \beta_{4} - 3 \beta_{3} + 3 \beta_{2} + \beta_1 - 4) q^{24} + ( - \beta_{6} - 2 \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} - 2 \beta_1 + 1) q^{26} + ( - \beta_{7} + 2 \beta_{5} - \beta_{3} - \beta_{2} + 1) q^{27} + (\beta_{7} - \beta_{6} + 3 \beta_{5} + 3 \beta_{4} - 3 \beta_{3} + 3 \beta_1 + 1) q^{28} + ( - 2 \beta_{7} - 2 \beta_{6} - \beta_{5} - 2 \beta_{4} + 5 \beta_{3} - 5 \beta_{2} + \cdots + 6) q^{29}+ \cdots + (2 \beta_{7} + 4 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 5 q^{3} + 4 q^{4} + 6 q^{6} - 10 q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 5 q^{3} + 4 q^{4} + 6 q^{6} - 10 q^{8} + q^{9} - 4 q^{11} - 10 q^{12} + 5 q^{13} - 7 q^{14} - 2 q^{16} - 15 q^{17} + 10 q^{19} + q^{21} - 10 q^{22} - 15 q^{23} - 20 q^{24} + 6 q^{26} + 5 q^{27} + 20 q^{28} + 15 q^{29} + q^{31} - 10 q^{33} - 12 q^{34} - 17 q^{36} + 5 q^{37} + 12 q^{39} - 9 q^{41} - 5 q^{42} + 8 q^{44} + 16 q^{46} + 15 q^{47} + 5 q^{48} + 14 q^{49} - 4 q^{51} + 20 q^{52} - 35 q^{53} - 10 q^{54} - 15 q^{56} + 20 q^{58} + 15 q^{59} + 6 q^{61} - 45 q^{62} + 20 q^{63} - 26 q^{64} - 18 q^{66} - 13 q^{69} - 29 q^{71} - 5 q^{72} - 10 q^{73} - 12 q^{74} - 20 q^{76} - 20 q^{77} + 25 q^{78} - 10 q^{79} - 12 q^{81} - 15 q^{83} - 27 q^{84} + 16 q^{86} + 55 q^{87} - 20 q^{88} + 40 q^{89} + q^{91} + 5 q^{92} - 7 q^{94} + 11 q^{96} + 10 q^{97} + 40 q^{98} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 406\nu^{7} - 714\nu^{6} + 747\nu^{5} - 1896\nu^{4} + 2103\nu^{3} + 4949\nu^{2} + 1065\nu - 7800 ) / 1355 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 420\nu^{7} - 776\nu^{6} + 698\nu^{5} - 1924\nu^{4} + 2297\nu^{3} + 5129\nu^{2} + 1055\nu - 10265 ) / 1355 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 728\nu^{7} - 1327\nu^{6} + 1246\nu^{5} - 3353\nu^{4} + 3584\nu^{3} + 8547\nu^{2} + 2190\nu - 15715 ) / 1355 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -857\nu^{7} + 1666\nu^{6} - 1743\nu^{5} + 4424\nu^{4} - 4907\nu^{3} - 9470\nu^{2} - 2485\nu + 18200 ) / 1355 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 891\nu^{7} - 1623\nu^{6} + 1624\nu^{5} - 4492\nu^{4} + 4991\nu^{3} + 9520\nu^{2} + 3235\nu - 18960 ) / 1355 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 955\nu^{7} - 1829\nu^{6} + 1942\nu^{5} - 4891\nu^{4} + 5723\nu^{3} + 9646\nu^{2} + 2415\nu - 20550 ) / 1355 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{7} - \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + \beta_{5} - 3\beta_{4} + 4\beta_{3} + \beta_{2} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 5\beta_{7} - 5\beta_{6} + 4\beta_{5} + 2\beta_{4} + 2\beta_{3} + 2\beta_{2} + 4\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4\beta_{7} - 6\beta_{6} - 7\beta_{3} + 11\beta_{2} + 4\beta _1 - 13 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 7\beta_{6} + 7\beta_{5} - 8\beta_{4} - 7\beta_{3} + 21\beta_{2} - 2\beta _1 - 21 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 23\beta_{7} + 38\beta_{5} + 23\beta_{4} - 23\beta_{3} + 12\beta_{2} \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/625\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(1 - \beta_{2} + \beta_{3} - \beta_{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
124.1
−0.357358 1.86824i
1.66637 + 0.917186i
1.17421 + 0.0566033i
−0.983224 0.644389i
1.17421 0.0566033i
−0.983224 + 0.644389i
−0.357358 + 1.86824i
1.66637 0.917186i
−2.19625 0.713605i 0.279141 0.384204i 2.69625 + 1.95894i 0 −0.887234 + 0.644613i 3.03582i −1.80902 2.48990i 0.857358 + 2.63868i 0
124.2 1.07822 + 0.350334i 1.52988 2.10569i −0.578217 0.420099i 0 2.38723 1.73443i 0.407162i −1.80902 2.48990i −1.16637 3.58973i 0
249.1 −0.107666 + 0.148189i −1.39991 + 0.454857i 0.607666 + 1.87020i 0 0.0833172 0.256424i 3.26086i −0.690983 0.224514i −0.674207 + 0.489840i 0
249.2 1.22570 1.68703i 2.09089 0.679371i −0.725700 2.23347i 0 1.41668 4.36010i 0.992398i −0.690983 0.224514i 1.48322 1.07763i 0
374.1 −0.107666 0.148189i −1.39991 0.454857i 0.607666 1.87020i 0 0.0833172 + 0.256424i 3.26086i −0.690983 + 0.224514i −0.674207 0.489840i 0
374.2 1.22570 + 1.68703i 2.09089 + 0.679371i −0.725700 + 2.23347i 0 1.41668 + 4.36010i 0.992398i −0.690983 + 0.224514i 1.48322 + 1.07763i 0
499.1 −2.19625 + 0.713605i 0.279141 + 0.384204i 2.69625 1.95894i 0 −0.887234 0.644613i 3.03582i −1.80902 + 2.48990i 0.857358 2.63868i 0
499.2 1.07822 0.350334i 1.52988 + 2.10569i −0.578217 + 0.420099i 0 2.38723 + 1.73443i 0.407162i −1.80902 + 2.48990i −1.16637 + 3.58973i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 499.2
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 625.2.e.i 8
5.b even 2 1 625.2.e.a 8
5.c odd 4 2 625.2.d.o 16
25.d even 5 1 25.2.e.a 8
25.d even 5 1 125.2.e.b 8
25.d even 5 1 625.2.b.c 8
25.d even 5 1 625.2.e.a 8
25.e even 10 1 25.2.e.a 8
25.e even 10 1 125.2.e.b 8
25.e even 10 1 625.2.b.c 8
25.e even 10 1 inner 625.2.e.i 8
25.f odd 20 4 125.2.d.b 16
25.f odd 20 2 625.2.a.f 8
25.f odd 20 2 625.2.d.o 16
75.h odd 10 1 225.2.m.a 8
75.j odd 10 1 225.2.m.a 8
75.l even 20 2 5625.2.a.x 8
100.h odd 10 1 400.2.y.c 8
100.j odd 10 1 400.2.y.c 8
100.l even 20 2 10000.2.a.bj 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.2.e.a 8 25.d even 5 1
25.2.e.a 8 25.e even 10 1
125.2.d.b 16 25.f odd 20 4
125.2.e.b 8 25.d even 5 1
125.2.e.b 8 25.e even 10 1
225.2.m.a 8 75.h odd 10 1
225.2.m.a 8 75.j odd 10 1
400.2.y.c 8 100.h odd 10 1
400.2.y.c 8 100.j odd 10 1
625.2.a.f 8 25.f odd 20 2
625.2.b.c 8 25.d even 5 1
625.2.b.c 8 25.e even 10 1
625.2.d.o 16 5.c odd 4 2
625.2.d.o 16 25.f odd 20 2
625.2.e.a 8 5.b even 2 1
625.2.e.a 8 25.d even 5 1
625.2.e.i 8 1.a even 1 1 trivial
625.2.e.i 8 25.e even 10 1 inner
5625.2.a.x 8 75.l even 20 2
10000.2.a.bj 8 100.l even 20 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(625, [\chi])\):

\( T_{2}^{8} - 4T_{2}^{6} + 10T_{2}^{5} + 11T_{2}^{4} - 40T_{2}^{3} + 21T_{2}^{2} + 5T_{2} + 1 \) Copy content Toggle raw display
\( T_{3}^{8} - 5T_{3}^{7} + 9T_{3}^{6} + 5T_{3}^{5} - 39T_{3}^{4} + 20T_{3}^{3} + 64T_{3}^{2} - 40T_{3} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 4 T^{6} + 10 T^{5} + 11 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{8} - 5 T^{7} + 9 T^{6} + 5 T^{5} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 21 T^{6} + 121 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$11$ \( (T^{4} + 2 T^{3} + 4 T^{2} + 8 T + 16)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} - 5 T^{7} + 14 T^{6} - 5 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( T^{8} + 15 T^{7} + 86 T^{6} + \cdots + 1936 \) Copy content Toggle raw display
$19$ \( T^{8} - 10 T^{7} + 50 T^{6} + \cdots + 400 \) Copy content Toggle raw display
$23$ \( T^{8} + 15 T^{7} + 109 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$29$ \( T^{8} - 15 T^{7} + 150 T^{6} + \cdots + 483025 \) Copy content Toggle raw display
$31$ \( T^{8} - T^{7} - 3 T^{6} + 67 T^{5} + \cdots + 1936 \) Copy content Toggle raw display
$37$ \( T^{8} - 5 T^{7} + T^{6} - 380 T^{5} + \cdots + 116281 \) Copy content Toggle raw display
$41$ \( T^{8} + 9 T^{7} + 62 T^{6} + \cdots + 13456 \) Copy content Toggle raw display
$43$ \( T^{8} + 129 T^{6} + 4421 T^{4} + \cdots + 246016 \) Copy content Toggle raw display
$47$ \( T^{8} - 15 T^{7} + 26 T^{6} + \cdots + 65536 \) Copy content Toggle raw display
$53$ \( T^{8} + 35 T^{7} + 549 T^{6} + \cdots + 8755681 \) Copy content Toggle raw display
$59$ \( T^{8} - 15 T^{7} + 220 T^{6} + \cdots + 4080400 \) Copy content Toggle raw display
$61$ \( T^{8} - 6 T^{7} + 252 T^{6} + \cdots + 116281 \) Copy content Toggle raw display
$67$ \( T^{8} - 4 T^{6} - 880 T^{5} + \cdots + 246016 \) Copy content Toggle raw display
$71$ \( T^{8} + 29 T^{7} + 462 T^{6} + \cdots + 24245776 \) Copy content Toggle raw display
$73$ \( T^{8} + 10 T^{7} + 74 T^{6} - 30 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$79$ \( T^{8} + 10 T^{7} + 270 T^{6} + \cdots + 33408400 \) Copy content Toggle raw display
$83$ \( T^{8} + 15 T^{7} - 11 T^{6} + \cdots + 99856 \) Copy content Toggle raw display
$89$ \( T^{8} - 40 T^{7} + 740 T^{6} + \cdots + 1392400 \) Copy content Toggle raw display
$97$ \( T^{8} - 10 T^{7} + \cdots + 301334881 \) Copy content Toggle raw display
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