Properties

Label 4004.2.a.e
Level 4004
Weight 2
Character orbit 4004.a
Self dual yes
Analytic conductor 31.972
Analytic rank 1
Dimension 4
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4004.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(31.9721009693\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{15})^+\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a root \(\beta\) of the polynomial \(x^{4} - x^{3} - 4 x^{2} + 4 x + 1\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 2 + 3 \beta - \beta^{2} - \beta^{3} ) q^{3} + ( 1 - 4 \beta + \beta^{3} ) q^{5} - q^{7} + ( -1 + \beta ) q^{9} +O(q^{10})\) \( q + ( 2 + 3 \beta - \beta^{2} - \beta^{3} ) q^{3} + ( 1 - 4 \beta + \beta^{3} ) q^{5} - q^{7} + ( -1 + \beta ) q^{9} + q^{11} - q^{13} + ( 1 - 7 \beta + 2 \beta^{3} ) q^{15} + ( -5 + 4 \beta + 2 \beta^{2} - \beta^{3} ) q^{17} + ( -4 + 6 \beta + \beta^{2} - 2 \beta^{3} ) q^{19} + ( -2 - 3 \beta + \beta^{2} + \beta^{3} ) q^{21} + ( 2 - \beta^{2} ) q^{23} + ( -1 + 3 \beta - \beta^{2} - \beta^{3} ) q^{25} + ( -7 - 6 \beta + 3 \beta^{2} + 2 \beta^{3} ) q^{27} + 3 \beta q^{29} + ( 1 + 2 \beta ) q^{31} + ( 2 + 3 \beta - \beta^{2} - \beta^{3} ) q^{33} + ( -1 + 4 \beta - \beta^{3} ) q^{35} + ( -6 + 2 \beta + 3 \beta^{2} - \beta^{3} ) q^{37} + ( -2 - 3 \beta + \beta^{2} + \beta^{3} ) q^{39} + ( 7 - 9 \beta - 3 \beta^{2} + 2 \beta^{3} ) q^{41} + ( -3 - 13 \beta + 2 \beta^{2} + 4 \beta^{3} ) q^{43} + ( -2 + \beta ) q^{45} + ( -6 + 10 \beta + \beta^{2} - 2 \beta^{3} ) q^{47} + q^{49} + ( -5 + 13 \beta - 4 \beta^{3} ) q^{51} + ( -9 - \beta + 2 \beta^{2} - \beta^{3} ) q^{53} + ( 1 - 4 \beta + \beta^{3} ) q^{55} + ( -6 + 5 \beta + 2 \beta^{2} - \beta^{3} ) q^{57} + ( -13 - 19 \beta + 6 \beta^{2} + 6 \beta^{3} ) q^{59} + ( -1 - 8 \beta - \beta^{2} + 3 \beta^{3} ) q^{61} + ( 1 - \beta ) q^{63} + ( -1 + 4 \beta - \beta^{3} ) q^{65} + ( -5 + 4 \beta ) q^{67} + ( 2 - 3 \beta + \beta^{3} ) q^{69} + ( -6 + 12 \beta - 4 \beta^{3} ) q^{71} + ( 13 - 4 \beta - 5 \beta^{2} + \beta^{3} ) q^{73} + ( -4 - 8 \beta + 3 \beta^{2} + 3 \beta^{3} ) q^{75} - q^{77} + ( 2 - 24 \beta + 2 \beta^{2} + 7 \beta^{3} ) q^{79} + ( -5 - 5 \beta + \beta^{2} ) q^{81} + ( -3 - 16 \beta + \beta^{2} + 5 \beta^{3} ) q^{83} + ( -10 + 3 \beta + 3 \beta^{2} - \beta^{3} ) q^{85} + ( 3 + 18 \beta - 3 \beta^{2} - 6 \beta^{3} ) q^{87} + ( -4 + 28 \beta - 3 \beta^{2} - 8 \beta^{3} ) q^{89} + q^{91} + ( 4 + 15 \beta - 3 \beta^{2} - 5 \beta^{3} ) q^{93} + ( -9 + 3 \beta + 3 \beta^{2} - \beta^{3} ) q^{95} + ( 6 - 29 \beta + \beta^{2} + 10 \beta^{3} ) q^{97} + ( -1 + \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + q^{3} + q^{5} - 4q^{7} - 3q^{9} + O(q^{10}) \) \( 4q + q^{3} + q^{5} - 4q^{7} - 3q^{9} + 4q^{11} - 4q^{13} - q^{15} + q^{17} - 3q^{19} - q^{21} - q^{23} - 11q^{25} - 5q^{27} + 3q^{29} + 6q^{31} + q^{33} - q^{35} + 4q^{37} - q^{39} - 6q^{41} - 3q^{43} - 7q^{45} - 7q^{47} + 4q^{49} - 11q^{51} - 20q^{53} + q^{55} - 2q^{57} - 11q^{59} - 18q^{61} + 3q^{63} - q^{65} - 16q^{67} + 6q^{69} - 16q^{71} + 4q^{73} + 6q^{75} - 4q^{77} + 9q^{79} - 16q^{81} - 14q^{83} - 11q^{85} - 3q^{87} - 23q^{89} + 4q^{91} - q^{93} - 7q^{95} + 14q^{97} - 3q^{99} + O(q^{100}) \)

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
1.82709
−1.95630
−0.209057
1.33826
0 −1.95630 0 −0.209057 0 −1.00000 0 0.827091 0
1.2 0 −0.209057 0 1.33826 0 −1.00000 0 −2.95630 0
1.3 0 1.33826 0 1.82709 0 −1.00000 0 −1.20906 0
1.4 0 1.82709 0 −1.95630 0 −1.00000 0 0.338261 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 4004.2.a.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4004.2.a.e 4 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(1\)
\(11\) \(-1\)
\(13\) \(1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - T_{3}^{3} - 4 T_{3}^{2} + 4 T_{3} + 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4004))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( \)
$3$ \( 1 - T + 8 T^{2} - 5 T^{3} + 31 T^{4} - 15 T^{5} + 72 T^{6} - 27 T^{7} + 81 T^{8} \)
$5$ \( 1 - T + 16 T^{2} - 11 T^{3} + 111 T^{4} - 55 T^{5} + 400 T^{6} - 125 T^{7} + 625 T^{8} \)
$7$ \( ( 1 + T )^{4} \)
$11$ \( ( 1 - T )^{4} \)
$13$ \( ( 1 + T )^{4} \)
$17$ \( 1 - T + 44 T^{2} + 23 T^{3} + 859 T^{4} + 391 T^{5} + 12716 T^{6} - 4913 T^{7} + 83521 T^{8} \)
$19$ \( 1 + 3 T + 60 T^{2} + 153 T^{3} + 1589 T^{4} + 2907 T^{5} + 21660 T^{6} + 20577 T^{7} + 130321 T^{8} \)
$23$ \( 1 + T + 88 T^{2} + 65 T^{3} + 2991 T^{4} + 1495 T^{5} + 46552 T^{6} + 12167 T^{7} + 279841 T^{8} \)
$29$ \( 1 - 3 T + 80 T^{2} - 153 T^{3} + 3039 T^{4} - 4437 T^{5} + 67280 T^{6} - 73167 T^{7} + 707281 T^{8} \)
$31$ \( 1 - 6 T + 120 T^{2} - 504 T^{3} + 5489 T^{4} - 15624 T^{5} + 115320 T^{6} - 178746 T^{7} + 923521 T^{8} \)
$37$ \( 1 - 4 T + 94 T^{2} - 373 T^{3} + 4249 T^{4} - 13801 T^{5} + 128686 T^{6} - 202612 T^{7} + 1874161 T^{8} \)
$41$ \( 1 + 6 T + 100 T^{2} + 249 T^{3} + 4179 T^{4} + 10209 T^{5} + 168100 T^{6} + 413526 T^{7} + 2825761 T^{8} \)
$43$ \( 1 + 3 T + 141 T^{2} + 384 T^{3} + 8429 T^{4} + 16512 T^{5} + 260709 T^{6} + 238521 T^{7} + 3418801 T^{8} \)
$47$ \( 1 + 7 T + 142 T^{2} + 785 T^{3} + 9741 T^{4} + 36895 T^{5} + 313678 T^{6} + 726761 T^{7} + 4879681 T^{8} \)
$53$ \( 1 + 20 T + 247 T^{2} + 1870 T^{3} + 13959 T^{4} + 99110 T^{5} + 693823 T^{6} + 2977540 T^{7} + 7890481 T^{8} \)
$59$ \( 1 + 11 T + 127 T^{2} + 1078 T^{3} + 10335 T^{4} + 63602 T^{5} + 442087 T^{6} + 2259169 T^{7} + 12117361 T^{8} \)
$61$ \( 1 + 18 T + 318 T^{2} + 3291 T^{3} + 31325 T^{4} + 200751 T^{5} + 1183278 T^{6} + 4085658 T^{7} + 13845841 T^{8} \)
$67$ \( 1 + 16 T + 294 T^{2} + 3032 T^{3} + 30479 T^{4} + 203144 T^{5} + 1319766 T^{6} + 4812208 T^{7} + 20151121 T^{8} \)
$71$ \( ( 1 + 8 T + 138 T^{2} + 568 T^{3} + 5041 T^{4} )^{2} \)
$73$ \( 1 - 4 T + 178 T^{2} - 1135 T^{3} + 15751 T^{4} - 82855 T^{5} + 948562 T^{6} - 1556068 T^{7} + 28398241 T^{8} \)
$79$ \( 1 - 9 T + 247 T^{2} - 1242 T^{3} + 24375 T^{4} - 98118 T^{5} + 1541527 T^{6} - 4437351 T^{7} + 38950081 T^{8} \)
$83$ \( 1 + 14 T + 358 T^{2} + 3385 T^{3} + 45681 T^{4} + 280955 T^{5} + 2466262 T^{6} + 8005018 T^{7} + 47458321 T^{8} \)
$89$ \( 1 + 23 T + 410 T^{2} + 4313 T^{3} + 46459 T^{4} + 383857 T^{5} + 3247610 T^{6} + 16214287 T^{7} + 62742241 T^{8} \)
$97$ \( 1 - 14 T + 204 T^{2} - 2593 T^{3} + 32969 T^{4} - 251521 T^{5} + 1919436 T^{6} - 12777422 T^{7} + 88529281 T^{8} \)
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