Properties

Label 4005.2.a.l
Level 4005
Weight 2
Character orbit 4005.a
Self dual yes
Analytic conductor 31.980
Analytic rank 0
Dimension 4
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4005 = 3^{2} \cdot 5 \cdot 89 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4005.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(31.9800860095\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.725.1
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 445)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + \beta_{1} + \beta_{2} ) q^{2} + ( \beta_{1} + \beta_{3} ) q^{4} + q^{5} + ( 1 + 2 \beta_{1} - 2 \beta_{3} ) q^{7} + ( 2 + \beta_{1} ) q^{8} +O(q^{10})\) \( q + ( -1 + \beta_{1} + \beta_{2} ) q^{2} + ( \beta_{1} + \beta_{3} ) q^{4} + q^{5} + ( 1 + 2 \beta_{1} - 2 \beta_{3} ) q^{7} + ( 2 + \beta_{1} ) q^{8} + ( -1 + \beta_{1} + \beta_{2} ) q^{10} + ( 3 + \beta_{2} ) q^{11} + ( -3 - \beta_{1} + \beta_{2} + 3 \beta_{3} ) q^{13} + ( -1 - \beta_{1} + \beta_{2} + 4 \beta_{3} ) q^{14} + ( -2 + \beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{16} + ( 2 - \beta_{1} + \beta_{2} - 3 \beta_{3} ) q^{17} + ( -1 - 3 \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{19} + ( \beta_{1} + \beta_{3} ) q^{20} + ( -2 + 4 \beta_{1} + 3 \beta_{2} ) q^{22} + ( -\beta_{1} - \beta_{2} + 3 \beta_{3} ) q^{23} + q^{25} + ( 4 + 3 \beta_{1} - \beta_{2} - 4 \beta_{3} ) q^{26} + ( 3 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{28} + ( 2 + 4 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{29} + ( -3 + 5 \beta_{1} - 2 \beta_{3} ) q^{31} + ( 1 - 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{32} + ( -1 - 4 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{34} + ( 1 + 2 \beta_{1} - 2 \beta_{3} ) q^{35} + ( 2 - 3 \beta_{1} - 4 \beta_{2} + 3 \beta_{3} ) q^{37} + ( 3 - 3 \beta_{2} - 4 \beta_{3} ) q^{38} + ( 2 + \beta_{1} ) q^{40} + ( -3 \beta_{1} - 2 \beta_{2} + 5 \beta_{3} ) q^{41} + ( 3 - 3 \beta_{1} + \beta_{2} - \beta_{3} ) q^{43} + ( -1 + 5 \beta_{1} + 4 \beta_{3} ) q^{44} + ( -1 + 4 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} ) q^{46} + ( 6 + 4 \beta_{1} + \beta_{2} - 3 \beta_{3} ) q^{47} + ( 2 - 4 \beta_{2} ) q^{49} + ( -1 + \beta_{1} + \beta_{2} ) q^{50} + ( 1 + \beta_{2} + \beta_{3} ) q^{52} + ( -5 + \beta_{1} + 3 \beta_{2} + 5 \beta_{3} ) q^{53} + ( 3 + \beta_{2} ) q^{55} + ( 4 + 5 \beta_{1} - 4 \beta_{3} ) q^{56} + ( -1 + 3 \beta_{1} + 4 \beta_{2} + 6 \beta_{3} ) q^{58} + ( 7 + 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{59} + ( 3 + 3 \beta_{1} + \beta_{2} - 10 \beta_{3} ) q^{61} + ( 3 - 2 \beta_{1} + 7 \beta_{3} ) q^{62} + ( 1 - \beta_{1} - 5 \beta_{2} - 2 \beta_{3} ) q^{64} + ( -3 - \beta_{1} + \beta_{2} + 3 \beta_{3} ) q^{65} + ( -\beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{67} + ( -5 - \beta_{1} - 5 \beta_{2} ) q^{68} + ( -1 - \beta_{1} + \beta_{2} + 4 \beta_{3} ) q^{70} + ( -1 - 2 \beta_{2} - 4 \beta_{3} ) q^{71} + ( 3 + 5 \beta_{1} - \beta_{2} - 6 \beta_{3} ) q^{73} + ( -6 + \beta_{1} + 2 \beta_{2} - 6 \beta_{3} ) q^{74} + ( -4 - 2 \beta_{1} - 5 \beta_{2} + 2 \beta_{3} ) q^{76} + ( 1 + 6 \beta_{1} + \beta_{2} - 4 \beta_{3} ) q^{77} + ( -7 - 5 \beta_{1} - 3 \beta_{2} + 6 \beta_{3} ) q^{79} + ( -2 + \beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{80} + ( -2 + 5 \beta_{1} + 2 \beta_{2} - 8 \beta_{3} ) q^{82} + ( 2 - 4 \beta_{1} - 6 \beta_{2} + \beta_{3} ) q^{83} + ( 2 - \beta_{1} + \beta_{2} - 3 \beta_{3} ) q^{85} + ( -2 - \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{86} + ( 5 + 4 \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{88} + q^{89} + ( -13 - \beta_{1} + 7 \beta_{2} + 5 \beta_{3} ) q^{91} + ( 3 - \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{92} + ( -5 + 5 \beta_{1} + 7 \beta_{2} + 7 \beta_{3} ) q^{94} + ( -1 - 3 \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{95} + ( 9 - 5 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} ) q^{97} + ( -6 - 2 \beta_{1} + 2 \beta_{2} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - q^{2} + 3q^{4} + 4q^{5} + 2q^{7} + 9q^{8} + O(q^{10}) \) \( 4q - q^{2} + 3q^{4} + 4q^{5} + 2q^{7} + 9q^{8} - q^{10} + 14q^{11} - 5q^{13} + 5q^{14} - 3q^{16} + 3q^{17} - q^{19} + 3q^{20} + 2q^{22} + 3q^{23} + 4q^{25} + 9q^{26} + 5q^{28} + 10q^{29} - 11q^{31} + 2q^{32} - 8q^{34} + 2q^{35} + 3q^{37} - 2q^{38} + 9q^{40} + 3q^{41} + 9q^{43} + 9q^{44} - 4q^{46} + 24q^{47} - q^{50} + 8q^{52} - 3q^{53} + 14q^{55} + 13q^{56} + 19q^{58} + 22q^{59} - 3q^{61} + 24q^{62} - 11q^{64} - 5q^{65} - 9q^{67} - 31q^{68} + 5q^{70} - 16q^{71} + 3q^{73} - 31q^{74} - 24q^{76} + 4q^{77} - 27q^{79} - 3q^{80} - 15q^{82} - 6q^{83} + 3q^{85} - 15q^{86} + 30q^{88} + 4q^{89} - 29q^{91} + 17q^{92} + 13q^{94} - q^{95} + 41q^{97} - 22q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 3 x^{2} + x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 1 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - \nu^{2} - 2 \nu + 1 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 1\)
\(\nu^{3}\)\(=\)\(\beta_{3} + \beta_{2} + 3 \beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−0.477260
0.737640
−1.35567
2.09529
−1.77222 0 1.14077 1.00000 0 −3.19059 1.52274 0 −1.77222
1.2 −1.45589 0 0.119606 1.00000 0 3.71135 2.73764 0 −1.45589
1.3 −0.162147 0 −1.97371 1.00000 0 −0.475281 0.644326 0 −0.162147
1.4 2.39026 0 3.71333 1.00000 0 1.95452 4.09529 0 2.39026
\(n\): 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 4005.2.a.l 4
3.b odd 2 1 445.2.a.d 4
12.b even 2 1 7120.2.a.bc 4
15.d odd 2 1 2225.2.a.i 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
445.2.a.d 4 3.b odd 2 1
2225.2.a.i 4 15.d odd 2 1
4005.2.a.l 4 1.a even 1 1 trivial
7120.2.a.bc 4 12.b even 2 1

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(-1\)
\(89\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4005))\):

\( T_{2}^{4} + T_{2}^{3} - 5 T_{2}^{2} - 7 T_{2} - 1 \)
\( T_{7}^{4} - 2 T_{7}^{3} - 12 T_{7}^{2} + 18 T_{7} + 11 \)
\( T_{11}^{4} - 14 T_{11}^{3} + 70 T_{11}^{2} - 147 T_{11} + 109 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + 3 T^{2} - T^{3} + 3 T^{4} - 2 T^{5} + 12 T^{6} + 8 T^{7} + 16 T^{8} \)
$3$ \( \)
$5$ \( ( 1 - T )^{4} \)
$7$ \( 1 - 2 T + 16 T^{2} - 24 T^{3} + 137 T^{4} - 168 T^{5} + 784 T^{6} - 686 T^{7} + 2401 T^{8} \)
$11$ \( 1 - 14 T + 114 T^{2} - 609 T^{3} + 2375 T^{4} - 6699 T^{5} + 13794 T^{6} - 18634 T^{7} + 14641 T^{8} \)
$13$ \( 1 + 5 T + 38 T^{2} + 185 T^{3} + 669 T^{4} + 2405 T^{5} + 6422 T^{6} + 10985 T^{7} + 28561 T^{8} \)
$17$ \( 1 - 3 T + 33 T^{2} - 154 T^{3} + 683 T^{4} - 2618 T^{5} + 9537 T^{6} - 14739 T^{7} + 83521 T^{8} \)
$19$ \( 1 + T + 28 T^{2} - 9 T^{3} + 533 T^{4} - 171 T^{5} + 10108 T^{6} + 6859 T^{7} + 130321 T^{8} \)
$23$ \( 1 - 3 T + 75 T^{2} - 200 T^{3} + 2423 T^{4} - 4600 T^{5} + 39675 T^{6} - 36501 T^{7} + 279841 T^{8} \)
$29$ \( 1 - 10 T + 112 T^{2} - 725 T^{3} + 4813 T^{4} - 21025 T^{5} + 94192 T^{6} - 243890 T^{7} + 707281 T^{8} \)
$31$ \( 1 + 11 T + 100 T^{2} + 709 T^{3} + 4879 T^{4} + 21979 T^{5} + 96100 T^{6} + 327701 T^{7} + 923521 T^{8} \)
$37$ \( 1 - 3 T + 83 T^{2} - 354 T^{3} + 3943 T^{4} - 13098 T^{5} + 113627 T^{6} - 151959 T^{7} + 1874161 T^{8} \)
$41$ \( 1 - 3 T + 107 T^{2} - 512 T^{3} + 5333 T^{4} - 20992 T^{5} + 179867 T^{6} - 206763 T^{7} + 2825761 T^{8} \)
$43$ \( 1 - 9 T + 154 T^{2} - 1067 T^{3} + 9527 T^{4} - 45881 T^{5} + 284746 T^{6} - 715563 T^{7} + 3418801 T^{8} \)
$47$ \( 1 - 24 T + 360 T^{2} - 3655 T^{3} + 28913 T^{4} - 171785 T^{5} + 795240 T^{6} - 2491752 T^{7} + 4879681 T^{8} \)
$53$ \( 1 + 3 T + 110 T^{2} + 355 T^{3} + 6533 T^{4} + 18815 T^{5} + 308990 T^{6} + 446631 T^{7} + 7890481 T^{8} \)
$59$ \( 1 - 22 T + 384 T^{2} - 4256 T^{3} + 38601 T^{4} - 251104 T^{5} + 1336704 T^{6} - 4518338 T^{7} + 12117361 T^{8} \)
$61$ \( 1 + 3 T + 43 T^{2} + 616 T^{3} + 4025 T^{4} + 37576 T^{5} + 160003 T^{6} + 680943 T^{7} + 13845841 T^{8} \)
$67$ \( 1 + 9 T + 269 T^{2} + 1738 T^{3} + 26979 T^{4} + 116446 T^{5} + 1207541 T^{6} + 2706867 T^{7} + 20151121 T^{8} \)
$71$ \( 1 + 16 T + 326 T^{2} + 3152 T^{3} + 35431 T^{4} + 223792 T^{5} + 1643366 T^{6} + 5726576 T^{7} + 25411681 T^{8} \)
$73$ \( 1 - 3 T + 185 T^{2} - 620 T^{3} + 16923 T^{4} - 45260 T^{5} + 985865 T^{6} - 1167051 T^{7} + 28398241 T^{8} \)
$79$ \( 1 + 27 T + 481 T^{2} + 5558 T^{3} + 55595 T^{4} + 439082 T^{5} + 3001921 T^{6} + 13312053 T^{7} + 38950081 T^{8} \)
$83$ \( 1 + 6 T + 209 T^{2} + 1238 T^{3} + 24167 T^{4} + 102754 T^{5} + 1439801 T^{6} + 3430722 T^{7} + 47458321 T^{8} \)
$89$ \( ( 1 - T )^{4} \)
$97$ \( 1 - 41 T + 920 T^{2} - 14165 T^{3} + 161493 T^{4} - 1374005 T^{5} + 8656280 T^{6} - 37419593 T^{7} + 88529281 T^{8} \)
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