Properties

Label 4598.2.a.ca
Level $4598$
Weight $2$
Character orbit 4598.a
Self dual yes
Analytic conductor $36.715$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4598 = 2 \cdot 11^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4598.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(36.7152148494\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \( x^{8} - 2x^{7} - 10x^{6} + 16x^{5} + 26x^{4} - 32x^{3} - 16x^{2} + 20x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 418)
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.

\(f(q)\) \(=\) \( q + q^{2} + ( - \beta_{6} + \beta_{4} + \beta_{2}) q^{3} + q^{4} - \beta_{3} q^{5} + ( - \beta_{6} + \beta_{4} + \beta_{2}) q^{6} + (2 \beta_{6} - \beta_{3} - \beta_1 + 2) q^{7} + q^{8} + (2 \beta_{6} - \beta_{4} + \beta_{3} - 2 \beta_1 + 4) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + ( - \beta_{6} + \beta_{4} + \beta_{2}) q^{3} + q^{4} - \beta_{3} q^{5} + ( - \beta_{6} + \beta_{4} + \beta_{2}) q^{6} + (2 \beta_{6} - \beta_{3} - \beta_1 + 2) q^{7} + q^{8} + (2 \beta_{6} - \beta_{4} + \beta_{3} - 2 \beta_1 + 4) q^{9} - \beta_{3} q^{10} + ( - \beta_{6} + \beta_{4} + \beta_{2}) q^{12} + (\beta_{6} + 2 \beta_{5} + \beta_{3} - \beta_{2} + \beta_1 + 3) q^{13} + (2 \beta_{6} - \beta_{3} - \beta_1 + 2) q^{14} + ( - 2 \beta_{4} + \beta_{2} + 1) q^{15} + q^{16} + ( - \beta_{6} - 2 \beta_{5} - 2 \beta_{3} - 1) q^{17} + (2 \beta_{6} - \beta_{4} + \beta_{3} - 2 \beta_1 + 4) q^{18} - q^{19} - \beta_{3} q^{20} + (\beta_{7} - \beta_{6} - \beta_{5} + \beta_{3} + 4 \beta_{2} - \beta_1 + 2) q^{21} + ( - \beta_{7} - \beta_{5} - \beta_{3} - 2 \beta_{2} + 1) q^{23} + ( - \beta_{6} + \beta_{4} + \beta_{2}) q^{24} + (\beta_{7} - \beta_{6} + \beta_{2} + \beta_1 - 1) q^{25} + (\beta_{6} + 2 \beta_{5} + \beta_{3} - \beta_{2} + \beta_1 + 3) q^{26} + (\beta_{7} + \beta_{6} + \beta_{5} + 3 \beta_{4} + \beta_{2} + \beta_1 - 2) q^{27} + (2 \beta_{6} - \beta_{3} - \beta_1 + 2) q^{28} + ( - 2 \beta_{6} + \beta_{4} + \beta_{3} + \beta_{2} - 2 \beta_1 + 2) q^{29} + ( - 2 \beta_{4} + \beta_{2} + 1) q^{30} + (2 \beta_{6} + 2 \beta_{4} - \beta_{2} + 1) q^{31} + q^{32} + ( - \beta_{6} - 2 \beta_{5} - 2 \beta_{3} - 1) q^{34} + ( - \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} + 2 \beta_1 + 5) q^{35} + (2 \beta_{6} - \beta_{4} + \beta_{3} - 2 \beta_1 + 4) q^{36} + (2 \beta_{7} + \beta_{4} - \beta_{3} - \beta_{2} - 4) q^{37} - q^{38} + ( - 8 \beta_{6} + 2 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} - 4 \beta_1 - 2) q^{39} - \beta_{3} q^{40} + ( - 3 \beta_{7} + \beta_{6} + 2 \beta_{5} - \beta_{3} - \beta_{2} + 3) q^{41} + (\beta_{7} - \beta_{6} - \beta_{5} + \beta_{3} + 4 \beta_{2} - \beta_1 + 2) q^{42} + (\beta_{7} - 4 \beta_{6} - 2 \beta_{5} + 2 \beta_{4} + \beta_{3} + 3 \beta_{2} + 2) q^{43} + ( - 3 \beta_{7} + \beta_{6} + 2 \beta_{5} - 2 \beta_{3} - 2 \beta_{2} - 2) q^{45} + ( - \beta_{7} - \beta_{5} - \beta_{3} - 2 \beta_{2} + 1) q^{46} + ( - \beta_{7} - 2 \beta_{6} - \beta_{4} - 3 \beta_{2} + \beta_1 - 1) q^{47} + ( - \beta_{6} + \beta_{4} + \beta_{2}) q^{48} + ( - 3 \beta_{7} + 3 \beta_{6} + 2 \beta_{5} + \beta_{4} - \beta_{3} - 3 \beta_{2} + 3 \beta_1 + 6) q^{49} + (\beta_{7} - \beta_{6} + \beta_{2} + \beta_1 - 1) q^{50} + (3 \beta_{6} - \beta_{5} - \beta_{4} - 2 \beta_{3} + \beta_{2} + 5 \beta_1 - 2) q^{51} + (\beta_{6} + 2 \beta_{5} + \beta_{3} - \beta_{2} + \beta_1 + 3) q^{52} + ( - 4 \beta_{6} + 3 \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 - 4) q^{53} + (\beta_{7} + \beta_{6} + \beta_{5} + 3 \beta_{4} + \beta_{2} + \beta_1 - 2) q^{54} + (2 \beta_{6} - \beta_{3} - \beta_1 + 2) q^{56} + (\beta_{6} - \beta_{4} - \beta_{2}) q^{57} + ( - 2 \beta_{6} + \beta_{4} + \beta_{3} + \beta_{2} - 2 \beta_1 + 2) q^{58} + (\beta_{6} - 2 \beta_{4} - \beta_{3} - \beta_{2} + \beta_1 + 5) q^{59} + ( - 2 \beta_{4} + \beta_{2} + 1) q^{60} + (2 \beta_{7} - 6 \beta_{6} - 2 \beta_{5} + 4 \beta_{4} + \beta_{3} + 3 \beta_{2} + 2 \beta_1 - 6) q^{61} + (2 \beta_{6} + 2 \beta_{4} - \beta_{2} + 1) q^{62} + ( - 2 \beta_{7} + \beta_{5} + 3 \beta_{4} - \beta_{2} - 2 \beta_1 + 6) q^{63} + q^{64} + ( - \beta_{7} + 3 \beta_{6} + 3 \beta_{5} - \beta_{4} - 2 \beta_{3} - 3 \beta_{2} - 3 \beta_1) q^{65} + (3 \beta_{7} - \beta_{6} + 3 \beta_{3} + 3 \beta_{2} - \beta_1 - 3) q^{67} + ( - \beta_{6} - 2 \beta_{5} - 2 \beta_{3} - 1) q^{68} + (\beta_{7} - 2 \beta_{6} - \beta_{5} + 2 \beta_{4} + 3 \beta_{2} + 7 \beta_1 - 7) q^{69} + ( - \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} + 2 \beta_1 + 5) q^{70} + ( - \beta_{7} - \beta_{6} - 3 \beta_{4} + 2 \beta_{2} - \beta_1 + 1) q^{71} + (2 \beta_{6} - \beta_{4} + \beta_{3} - 2 \beta_1 + 4) q^{72} + ( - \beta_{7} - 2 \beta_{5} + 3 \beta_{4} - \beta_{3} - \beta_{2} + \beta_1 + 2) q^{73} + (2 \beta_{7} + \beta_{4} - \beta_{3} - \beta_{2} - 4) q^{74} + ( - \beta_{7} - \beta_{4} - \beta_{3} - 2 \beta_{2} - 3 \beta_1 + 1) q^{75} - q^{76} + ( - 8 \beta_{6} + 2 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} - 4 \beta_1 - 2) q^{78} + (2 \beta_{7} - 9 \beta_{6} + \beta_{5} + 2 \beta_{4} + 3 \beta_{3} + 3 \beta_{2} + \beta_1 - 3) q^{79} - \beta_{3} q^{80} + (2 \beta_{7} - 6 \beta_{6} - 4 \beta_{5} - \beta_{4} + 3 \beta_{3} - 4 \beta_1) q^{81} + ( - 3 \beta_{7} + \beta_{6} + 2 \beta_{5} - \beta_{3} - \beta_{2} + 3) q^{82} + (\beta_{7} - \beta_{6} - \beta_{5} - 2 \beta_{4} - \beta_1) q^{83} + (\beta_{7} - \beta_{6} - \beta_{5} + \beta_{3} + 4 \beta_{2} - \beta_1 + 2) q^{84} + (3 \beta_{7} - 3 \beta_{6} - 4 \beta_{5} + 2 \beta_{4} - \beta_{3} + 3 \beta_{2} + 4 \beta_1 + 2) q^{85} + (\beta_{7} - 4 \beta_{6} - 2 \beta_{5} + 2 \beta_{4} + \beta_{3} + 3 \beta_{2} + 2) q^{86} + (2 \beta_{7} + 2 \beta_{6} + 3 \beta_{5} + 3 \beta_{4} + 3 \beta_{3} + 3 \beta_{2} + \cdots + 8) q^{87}+ \cdots + ( - 3 \beta_{7} + 3 \beta_{6} + 2 \beta_{5} + \beta_{4} - \beta_{3} - 3 \beta_{2} + 3 \beta_1 + 6) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{2} + 8 q^{4} + 2 q^{5} + 8 q^{7} + 8 q^{8} + 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{2} + 8 q^{4} + 2 q^{5} + 8 q^{7} + 8 q^{8} + 20 q^{9} + 2 q^{10} + 18 q^{13} + 8 q^{14} + 10 q^{15} + 8 q^{16} + 4 q^{17} + 20 q^{18} - 8 q^{19} + 2 q^{20} + 14 q^{21} + 12 q^{23} + 18 q^{26} - 24 q^{27} + 8 q^{28} + 14 q^{29} + 10 q^{30} - 2 q^{31} + 8 q^{32} + 4 q^{34} + 40 q^{35} + 20 q^{36} - 22 q^{37} - 8 q^{38} - 4 q^{39} + 2 q^{40} + 8 q^{41} + 14 q^{42} + 28 q^{43} - 28 q^{45} + 12 q^{46} + 6 q^{47} + 32 q^{49} - 12 q^{51} + 18 q^{52} - 24 q^{53} - 24 q^{54} + 8 q^{56} + 14 q^{58} + 46 q^{59} + 10 q^{60} - 24 q^{61} - 2 q^{62} + 30 q^{63} + 8 q^{64} - 16 q^{65} - 22 q^{67} + 4 q^{68} - 38 q^{69} + 40 q^{70} + 8 q^{71} + 20 q^{72} + 16 q^{73} - 22 q^{74} + 6 q^{75} - 8 q^{76} - 4 q^{78} + 4 q^{79} + 2 q^{80} + 28 q^{81} + 8 q^{82} + 12 q^{83} + 14 q^{84} + 48 q^{85} + 28 q^{86} + 42 q^{87} - 28 q^{89} - 28 q^{90} - 12 q^{91} + 12 q^{92} + 22 q^{93} + 6 q^{94} - 2 q^{95} - 22 q^{97} + 32 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} - 10x^{6} + 16x^{5} + 26x^{4} - 32x^{3} - 16x^{2} + 20x - 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} - 2\nu^{4} - 8\nu^{3} + 12\nu^{2} + 12\nu - 10 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{6} - 3\nu^{5} - 7\nu^{4} + 22\nu^{3} + 6\nu^{2} - 28\nu + 4 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{6} - 3\nu^{5} - 7\nu^{4} + 22\nu^{3} + 4\nu^{2} - 28\nu + 10 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} - 3\nu^{6} - 7\nu^{5} + 22\nu^{4} + 6\nu^{3} - 30\nu^{2} + 2\nu + 4 ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{7} - \nu^{6} - 12\nu^{5} + 7\nu^{4} + 40\nu^{3} - 14\nu^{2} - 34\nu + 12 ) / 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -2\nu^{7} + 3\nu^{6} + 21\nu^{5} - 20\nu^{4} - 58\nu^{3} + 24\nu^{2} + 38\nu - 8 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{4} + \beta_{3} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} + 2\beta_{3} + \beta_{2} + 5\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{7} + 4\beta_{6} - 10\beta_{4} + 8\beta_{3} + 2\beta _1 + 18 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 12\beta_{7} + 16\beta_{6} + 8\beta_{5} - 16\beta_{4} + 20\beta_{3} + 10\beta_{2} + 32\beta _1 + 26 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 28\beta_{7} + 54\beta_{6} + 2\beta_{5} - 90\beta_{4} + 68\beta_{3} + 8\beta_{2} + 28\beta _1 + 138 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 118\beta_{7} + 180\beta_{6} + 58\beta_{5} - 186\beta_{4} + 186\beta_{3} + 88\beta_{2} + 232\beta _1 + 274 \) 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
−2.40747
3.04921
0.313242
0.488861
2.03422
1.07616
−1.35168
−1.20254
1.00000 −3.37266 1.00000 −0.689994 −3.37266 0.481411 1.00000 8.37481 −0.689994
1.2 1.00000 −2.90436 1.00000 −2.98197 −2.90436 −2.79511 1.00000 5.43531 −2.98197
1.3 1.00000 −2.16972 1.00000 1.79068 −2.16972 4.71351 1.00000 1.70770 1.79068
1.4 1.00000 0.158731 1.00000 3.07692 0.158731 1.35199 1.00000 −2.97480 3.07692
1.5 1.00000 0.912987 1.00000 −1.77724 0.912987 −5.04753 1.00000 −2.16645 −1.77724
1.6 1.00000 2.05958 1.00000 1.96511 2.05958 4.12502 1.00000 1.24187 1.96511
1.7 1.00000 2.30094 1.00000 2.62639 2.30094 2.74200 1.00000 2.29431 2.62639
1.8 1.00000 3.01451 1.00000 −2.00989 3.01451 2.42872 1.00000 6.08725 −2.00989
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(11\) \(1\)
\(19\) \(1\)

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 4598.2.a.ca 8
11.b odd 2 1 4598.2.a.bx 8
11.c even 5 2 418.2.f.f 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.f.f 16 11.c even 5 2
4598.2.a.bx 8 11.b odd 2 1
4598.2.a.ca 8 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4598))\):

\( T_{3}^{8} - 22T_{3}^{6} + 8T_{3}^{5} + 154T_{3}^{4} - 108T_{3}^{3} - 332T_{3}^{2} + 332T_{3} - 44 \) Copy content Toggle raw display
\( T_{5}^{8} - 2T_{5}^{7} - 18T_{5}^{6} + 32T_{5}^{5} + 107T_{5}^{4} - 152T_{5}^{3} - 256T_{5}^{2} + 224T_{5} + 209 \) Copy content Toggle raw display
\( T_{7}^{8} - 8T_{7}^{7} - 12T_{7}^{6} + 246T_{7}^{5} - 501T_{7}^{4} - 902T_{7}^{3} + 4018T_{7}^{2} - 4152T_{7} + 1189 \) Copy content Toggle raw display
\( T_{13}^{8} - 18T_{13}^{7} + 68T_{13}^{6} + 472T_{13}^{5} - 3496T_{13}^{4} + 608T_{13}^{3} + 34048T_{13}^{2} - 46080T_{13} - 33344 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 22 T^{6} + 8 T^{5} + 154 T^{4} + \cdots - 44 \) Copy content Toggle raw display
$5$ \( T^{8} - 2 T^{7} - 18 T^{6} + 32 T^{5} + \cdots + 209 \) Copy content Toggle raw display
$7$ \( T^{8} - 8 T^{7} - 12 T^{6} + \cdots + 1189 \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} - 18 T^{7} + 68 T^{6} + \cdots - 33344 \) Copy content Toggle raw display
$17$ \( T^{8} - 4 T^{7} - 90 T^{6} + \cdots - 4259 \) Copy content Toggle raw display
$19$ \( (T + 1)^{8} \) Copy content Toggle raw display
$23$ \( T^{8} - 12 T^{7} - 34 T^{6} + \cdots - 96331 \) Copy content Toggle raw display
$29$ \( T^{8} - 14 T^{7} + 6 T^{6} + \cdots + 1900 \) Copy content Toggle raw display
$31$ \( T^{8} + 2 T^{7} - 116 T^{6} + \cdots - 5036 \) Copy content Toggle raw display
$37$ \( T^{8} + 22 T^{7} + 14 T^{6} + \cdots - 810644 \) Copy content Toggle raw display
$41$ \( T^{8} - 8 T^{7} - 186 T^{6} + \cdots + 2695396 \) Copy content Toggle raw display
$43$ \( T^{8} - 28 T^{7} + 222 T^{6} + \cdots + 51869 \) Copy content Toggle raw display
$47$ \( T^{8} - 6 T^{7} - 170 T^{6} + \cdots + 59921 \) Copy content Toggle raw display
$53$ \( T^{8} + 24 T^{7} + 124 T^{6} + \cdots + 815956 \) Copy content Toggle raw display
$59$ \( T^{8} - 46 T^{7} + 844 T^{6} + \cdots - 3488320 \) Copy content Toggle raw display
$61$ \( T^{8} + 24 T^{7} + 12 T^{6} + \cdots - 4721299 \) Copy content Toggle raw display
$67$ \( T^{8} + 22 T^{7} - 62 T^{6} + \cdots + 10564 \) Copy content Toggle raw display
$71$ \( T^{8} - 8 T^{7} - 248 T^{6} + \cdots + 1691036 \) Copy content Toggle raw display
$73$ \( T^{8} - 16 T^{7} - 156 T^{6} + \cdots + 1356976 \) Copy content Toggle raw display
$79$ \( T^{8} - 4 T^{7} - 368 T^{6} + \cdots - 7215500 \) Copy content Toggle raw display
$83$ \( T^{8} - 12 T^{7} - 54 T^{6} + \cdots + 47849 \) Copy content Toggle raw display
$89$ \( T^{8} + 28 T^{7} + 36 T^{6} + \cdots - 31782080 \) Copy content Toggle raw display
$97$ \( T^{8} + 22 T^{7} - 166 T^{6} + \cdots + 19644956 \) Copy content Toggle raw display
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