Properties

Label 3025.2.a.bj
Level $3025$
Weight $2$
Character orbit 3025.a
Self dual yes
Analytic conductor $24.155$
Analytic rank $1$
Dimension $8$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 3025 = 5^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3025.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(24.1547466114\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \( x^{8} - 3x^{7} - 7x^{6} + 18x^{5} + 16x^{4} - 30x^{3} - 12x^{2} + 12x + 4 \) 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 275)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{2} - \beta_{5} q^{3} + (\beta_{4} + \beta_{3} - \beta_1 + 1) q^{4} + ( - \beta_{7} + \beta_{5}) q^{6} + ( - \beta_{7} - 1) q^{7} + (\beta_{7} + \beta_{6} - \beta_{3} - \beta_{2} + 2 \beta_1 - 1) q^{8} + (\beta_{7} - \beta_{6} - \beta_{3} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{2} - \beta_{5} q^{3} + (\beta_{4} + \beta_{3} - \beta_1 + 1) q^{4} + ( - \beta_{7} + \beta_{5}) q^{6} + ( - \beta_{7} - 1) q^{7} + (\beta_{7} + \beta_{6} - \beta_{3} - \beta_{2} + 2 \beta_1 - 1) q^{8} + (\beta_{7} - \beta_{6} - \beta_{3} + 1) q^{9} + (2 \beta_{7} + \beta_{6} - \beta_{4} + 1) q^{12} + ( - \beta_{4} + \beta_{3} - \beta_1 - 1) q^{13} + (\beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} - \beta_1 + 2) q^{14} + ( - \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 2) q^{16} + ( - \beta_{7} - \beta_{5} - \beta_{4} - \beta_{3} - 2) q^{17} + ( - \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} + \beta_{2} - \beta_1 - 2) q^{18} + (2 \beta_{6} - \beta_{2} - 2 \beta_1 + 2) q^{19} + (2 \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} + 2 \beta_1 - 1) q^{21} + ( - 2 \beta_{6} + \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 - 2) q^{23} + ( - \beta_{7} - 5 \beta_{6} + 3 \beta_{4} - 3) q^{24} + ( - \beta_{7} - 3 \beta_{6} - \beta_{3} - \beta_{2} - 1) q^{26} + ( - \beta_{6} - \beta_{5} + 2 \beta_{4} - 2 \beta_{2} - \beta_1) q^{27} + ( - \beta_{7} - 4 \beta_{6} + 2 \beta_{5} + \beta_{4} - \beta_{3} + \beta_1 - 3) q^{28} + (\beta_{7} + \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 - 2) q^{29} + (2 \beta_{6} + 2 \beta_{5} + \beta_{3} - 2 \beta_1 + 1) q^{31} + (\beta_{7} + \beta_{6} - 2 \beta_{5} - 3 \beta_{4} - 2 \beta_{3} - \beta_{2} + 3 \beta_1 - 3) q^{32} + ( - \beta_{7} - 2 \beta_{4} + \beta_{2} - 5 \beta_1 + 3) q^{34} + ( - \beta_{7} - 2 \beta_{5} - \beta_{4} - \beta_{2} - 2 \beta_1 - 1) q^{36} + ( - 2 \beta_{3} + \beta_{2} + \beta_1 - 1) q^{37} + ( - 2 \beta_{6} - \beta_{3} + \beta_{2} + \beta_1 - 6) q^{38} + (\beta_{6} + 2 \beta_{5} - \beta_{4} + 2 \beta_{3} - 2 \beta_1 + 1) q^{39} + (\beta_{7} - \beta_{6} + \beta_{5} - 2 \beta_{4} - 2 \beta_{2} - \beta_1 + 1) q^{41} + (\beta_{7} - \beta_{6} - 2 \beta_{5} + \beta_{4} + \beta_{3} - 3 \beta_1 + 5) q^{42} + (\beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + 2 \beta_{2} - 2) q^{43} + (\beta_{7} + 3 \beta_{6} - 2 \beta_{2} + 2 \beta_1 + 4) q^{46} + ( - \beta_{7} + \beta_{6} + 2 \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} - 2 \beta_1 + 3) q^{47} + (10 \beta_{6} - \beta_{5} - 4 \beta_{4} + 2) q^{48} + (2 \beta_{7} - \beta_{6} + \beta_{2} - \beta_1 - 2) q^{49} + ( - 2 \beta_{6} + 2 \beta_{5} - 2 \beta_{3} + \beta_{2} + 2 \beta_1 + 2) q^{51} + (\beta_{7} + 5 \beta_{6} - \beta_{5} - 3 \beta_{4} - \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 4) q^{52} + (\beta_{7} - 3 \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} + 3 \beta_1 - 1) q^{53} + (\beta_{7} + 5 \beta_{6} + \beta_{5} - 2 \beta_{4} + \beta_{3} + 2 \beta_{2} - 2) q^{54} + (2 \beta_{7} + 6 \beta_{6} - \beta_{5} - 3 \beta_{4} + \beta_{3} + \beta_{2} - 2 \beta_1 + 2) q^{56} + (3 \beta_{7} - 4 \beta_{6} + \beta_{4} + 2 \beta_{2} + \beta_1 - 3) q^{57} + (\beta_{7} + 3 \beta_{4} - 2 \beta_{2} + 2 \beta_1 + 3) q^{58} + (\beta_{7} + \beta_{6} - \beta_{5} + 4 \beta_{4} - \beta_{3} + \beta_1 + 2) q^{59} + (2 \beta_{7} + 3 \beta_{6} + 2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + \beta_{2} - \beta_1 + 3) q^{61} + (2 \beta_{7} - 3 \beta_{6} - 2 \beta_{5} + \beta_{4} - 2 \beta_{3} - \beta_{2} + 3 \beta_1 - 5) q^{62} + ( - 3 \beta_{7} + 5 \beta_{6} + 2 \beta_{3} - \beta_1 - 3) q^{63} + ( - 4 \beta_{7} - 4 \beta_{6} + \beta_{5} + \beta_{4} + 2 \beta_{3} - \beta_{2} - 7 \beta_1 + 4) q^{64} + ( - 3 \beta_{7} + 3 \beta_{6} + \beta_{5} - 3 \beta_{4} + 2 \beta_{3} - \beta_{2} - 4 \beta_1 + 4) q^{67} + (\beta_{7} - 3 \beta_{6} + \beta_{5} - 4 \beta_{4} - 4 \beta_{3} - \beta_{2} + 2 \beta_1 - 8) q^{68} + ( - \beta_{7} + 5 \beta_{6} + \beta_{5} - 2 \beta_{4} - 2 \beta_{2} - \beta_1 + 4) q^{69} + (\beta_{7} + 4 \beta_{6} + \beta_{5} - \beta_{4} - 2 \beta_{3} + 3 \beta_1 + 4) q^{71} + ( - 3 \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 + 2) q^{72} + (\beta_{7} - \beta_{6} + 3 \beta_{4} - 2 \beta_{2} - \beta_1 - 4) q^{73} + (2 \beta_{6} - \beta_{4} + \beta_{2} - 4 \beta_1 + 3) q^{74} + ( - \beta_{6} - 2 \beta_{4} + 2 \beta_{2} - 3 \beta_1 + 4) q^{76} + (\beta_{7} - 5 \beta_{6} - 2 \beta_{5} + \beta_{4} - 2 \beta_{3} - 2 \beta_{2} + 4 \beta_1 - 5) q^{78} + (\beta_{7} - 4 \beta_{6} + \beta_{5} + 4 \beta_{4} + \beta_{3} - \beta_{2} + 2 \beta_1 - 6) q^{79} + (3 \beta_{7} - 6 \beta_{6} - \beta_{5} + 2 \beta_{4} - \beta_{2} + 4 \beta_1 - 5) q^{81} + ( - 2 \beta_{7} - 4 \beta_{6} - \beta_{4} + \beta_{3} + 2 \beta_{2} - 3 \beta_1 - 4) q^{82} + (2 \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} - 6) q^{83} + ( - 2 \beta_{7} + \beta_{6} - \beta_{5} - \beta_{3} - 3 \beta_{2} + 4 \beta_1 - 10) q^{84} + (\beta_{7} + 2 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} + \beta_1 + 1) q^{86} + ( - 2 \beta_{7} + 6 \beta_{6} + 2 \beta_{5} - \beta_{4} + 2 \beta_{3} - \beta_{2} - 3 \beta_1 + 1) q^{87} + (\beta_{7} + \beta_{6} - 2 \beta_{5} - 2 \beta_{4} - \beta_{3} - 3 \beta_{2} - \beta_1 + 1) q^{89} + (\beta_{7} - 4 \beta_{6} + \beta_{4} - \beta_{3} - 2 \beta_{2} + 3 \beta_1 - 3) q^{91} + ( - \beta_{7} + \beta_{5} + 4 \beta_{4} + 2 \beta_{3} + 3) q^{92} + (3 \beta_{6} + 2 \beta_{5} - \beta_{4} + 3 \beta_{3} + 2 \beta_{2} - \beta_1 - 7) q^{93} + (2 \beta_{7} - \beta_{6} - 3 \beta_{5} - 3 \beta_{4} - 3 \beta_{3} + \beta_1 - 6) q^{94} + ( - 3 \beta_{7} - 8 \beta_{6} + \beta_{5} + 4 \beta_{4} - 2 \beta_1 + 4) q^{96} + ( - \beta_{7} - 4 \beta_{6} + 2 \beta_{5} + \beta_{4} + 2 \beta_{2} - \beta_1 - 2) q^{97} + ( - 2 \beta_{7} - \beta_{6} + 2 \beta_{5} - 2 \beta_{3} - \beta_{2} - \beta_1 - 2) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 5 q^{2} + q^{3} + 9 q^{4} - q^{6} - 8 q^{7} - 12 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 5 q^{2} + q^{3} + 9 q^{4} - q^{6} - 8 q^{7} - 12 q^{8} + 9 q^{9} + 3 q^{12} - 9 q^{13} + 9 q^{14} + 23 q^{16} - 19 q^{17} - 22 q^{18} - q^{19} - 5 q^{21} + 2 q^{23} - q^{24} - 2 q^{26} - 2 q^{27} - 9 q^{28} - 7 q^{29} - 5 q^{31} - 29 q^{32} + 10 q^{34} - 16 q^{36} - 8 q^{37} - 37 q^{38} + q^{39} + 41 q^{42} - 14 q^{43} + 20 q^{46} + 11 q^{47} - 27 q^{48} - 12 q^{49} + 25 q^{51} + 7 q^{52} + 11 q^{53} - 30 q^{54} - 10 q^{56} + 2 q^{57} + 27 q^{58} + 17 q^{59} + 2 q^{61} - 25 q^{62} - 41 q^{63} + 30 q^{64} + 7 q^{67} - 66 q^{68} + 17 q^{71} + 19 q^{72} - 34 q^{73} + 6 q^{74} + 31 q^{76} - 17 q^{78} - 23 q^{79} - 4 q^{81} - 17 q^{82} - 41 q^{83} - 83 q^{84} + q^{86} - 25 q^{87} - 11 q^{89} - 7 q^{91} + 33 q^{92} - 59 q^{93} - 50 q^{94} + 61 q^{96} + 2 q^{97} - 26 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} - 7x^{6} + 18x^{5} + 16x^{4} - 30x^{3} - 12x^{2} + 12x + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} - 3\nu^{4} - 5\nu^{3} + 12\nu^{2} + 6\nu - 6 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{6} + 3\nu^{5} + 5\nu^{4} - 12\nu^{3} - 6\nu^{2} + 8\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{6} - 3\nu^{5} - 5\nu^{4} + 12\nu^{3} + 8\nu^{2} - 10\nu - 4 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{6} + 2\nu^{5} + 10\nu^{4} - 13\nu^{3} - 26\nu^{2} + 18\nu + 12 ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{7} - 3\nu^{6} - 6\nu^{5} + 15\nu^{4} + 11\nu^{3} - 18\nu^{2} - 4\nu + 2 ) / 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{7} + 2\nu^{6} + 10\nu^{5} - 13\nu^{4} - 26\nu^{3} + 18\nu^{2} + 12\nu ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} + \beta_{3} + \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + \beta_{6} + 3\beta_{4} + 2\beta_{3} - \beta_{2} + 6\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3\beta_{7} + 3\beta_{6} + \beta_{5} + 13\beta_{4} + 9\beta_{3} - 2\beta_{2} + 14\beta _1 + 11 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 14\beta_{7} + 14\beta_{6} + 3\beta_{5} + 42\beta_{4} + 25\beta_{3} - 9\beta_{2} + 54\beta _1 + 25 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 45\beta_{7} + 45\beta_{6} + 14\beta_{5} + 149\beta_{4} + 88\beta_{3} - 25\beta_{2} + 162\beta _1 + 94 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 163\beta_{7} + 165\beta_{6} + 45\beta_{5} + 489\beta_{4} + 275\beta_{3} - 88\beta_{2} + 556\beta _1 + 279 \) Copy content Toggle raw display

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.76541
−1.56247
−0.672032
−0.321622
0.860597
1.20828
1.95882
3.29384
−2.76541 −1.64150 5.64751 0 4.53942 1.89792 −10.0869 −0.305488 0
1.2 −2.56247 1.79260 4.56626 0 −4.59347 −3.80088 −6.57595 0.213399 0
1.3 −1.67203 3.10994 0.795692 0 −5.19992 −3.08998 2.01364 6.67173 0
1.4 −1.32162 −2.02642 −0.253315 0 2.67816 −0.348258 2.97803 1.10639 0
1.5 −0.139403 −2.98582 −1.98057 0 0.416233 −3.56959 0.554904 5.91512 0
1.6 0.208285 1.89427 −1.95662 0 0.394547 1.28881 −0.824104 0.588246 0
1.7 0.958815 0.899342 −1.08067 0 0.862303 0.761645 −2.95380 −2.19118 0
1.8 2.29384 −0.0424059 3.26171 0 −0.0972724 −1.13968 2.89416 −2.99820 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)
\(11\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3025.2.a.bj 8
5.b even 2 1 3025.2.a.bm 8
11.b odd 2 1 3025.2.a.bn 8
11.d odd 10 2 275.2.h.e yes 16
55.d odd 2 1 3025.2.a.bi 8
55.h odd 10 2 275.2.h.c 16
55.l even 20 4 275.2.z.c 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
275.2.h.c 16 55.h odd 10 2
275.2.h.e yes 16 11.d odd 10 2
275.2.z.c 32 55.l even 20 4
3025.2.a.bi 8 55.d odd 2 1
3025.2.a.bj 8 1.a even 1 1 trivial
3025.2.a.bm 8 5.b even 2 1
3025.2.a.bn 8 11.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3025))\):

\( T_{2}^{8} + 5T_{2}^{7} - 31T_{2}^{5} - 34T_{2}^{4} + 25T_{2}^{3} + 34T_{2}^{2} - 3T_{2} - 1 \) Copy content Toggle raw display
\( T_{3}^{8} - T_{3}^{7} - 16T_{3}^{6} + 15T_{3}^{5} + 74T_{3}^{4} - 67T_{3}^{3} - 105T_{3}^{2} + 90T_{3} + 4 \) Copy content Toggle raw display
\( T_{19}^{8} + T_{19}^{7} - 64T_{19}^{6} - 119T_{19}^{5} + 1016T_{19}^{4} + 1735T_{19}^{3} - 5555T_{19}^{2} - 5520T_{19} + 10480 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 5 T^{7} - 31 T^{5} - 34 T^{4} + \cdots - 1 \) Copy content Toggle raw display
$3$ \( T^{8} - T^{7} - 16 T^{6} + 15 T^{5} + \cdots + 4 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 8 T^{7} + 10 T^{6} - 50 T^{5} + \cdots - 31 \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} + 9 T^{7} - 4 T^{6} - 241 T^{5} + \cdots + 3475 \) Copy content Toggle raw display
$17$ \( T^{8} + 19 T^{7} + 107 T^{6} + \cdots + 439 \) Copy content Toggle raw display
$19$ \( T^{8} + T^{7} - 64 T^{6} - 119 T^{5} + \cdots + 10480 \) Copy content Toggle raw display
$23$ \( T^{8} - 2 T^{7} - 62 T^{6} + 252 T^{5} + \cdots - 121 \) Copy content Toggle raw display
$29$ \( T^{8} + 7 T^{7} - 62 T^{6} + \cdots - 35495 \) Copy content Toggle raw display
$31$ \( T^{8} + 5 T^{7} - 100 T^{6} + \cdots + 33569 \) Copy content Toggle raw display
$37$ \( T^{8} + 8 T^{7} - 54 T^{6} + \cdots - 4451 \) Copy content Toggle raw display
$41$ \( T^{8} - 197 T^{6} + 396 T^{5} + \cdots + 442576 \) Copy content Toggle raw display
$43$ \( T^{8} + 14 T^{7} - 28 T^{6} + \cdots - 101 \) Copy content Toggle raw display
$47$ \( T^{8} - 11 T^{7} - 108 T^{6} + \cdots - 459469 \) Copy content Toggle raw display
$53$ \( T^{8} - 11 T^{7} - 216 T^{6} + \cdots - 48896 \) Copy content Toggle raw display
$59$ \( T^{8} - 17 T^{7} - 140 T^{6} + \cdots - 78605 \) Copy content Toggle raw display
$61$ \( T^{8} - 2 T^{7} - 282 T^{6} + \cdots + 3319739 \) Copy content Toggle raw display
$67$ \( T^{8} - 7 T^{7} - 323 T^{6} + \cdots + 23994961 \) Copy content Toggle raw display
$71$ \( T^{8} - 17 T^{7} - 99 T^{6} + \cdots + 960859 \) Copy content Toggle raw display
$73$ \( T^{8} + 34 T^{7} + 290 T^{6} + \cdots - 466721 \) Copy content Toggle raw display
$79$ \( T^{8} + 23 T^{7} - 42 T^{6} + \cdots - 524695 \) Copy content Toggle raw display
$83$ \( T^{8} + 41 T^{7} + 568 T^{6} + \cdots - 8329 \) Copy content Toggle raw display
$89$ \( T^{8} + 11 T^{7} - 258 T^{6} + \cdots - 278125 \) Copy content Toggle raw display
$97$ \( T^{8} - 2 T^{7} - 257 T^{6} + \cdots + 85616 \) Copy content Toggle raw display
show more
show less