Properties

Label 4016.2.a.i
Level 4016
Weight 2
Character orbit 4016.a
Self dual yes
Analytic conductor 32.068
Analytic rank 1
Dimension 12
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4016 = 2^{4} \cdot 251 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4016.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.0679214517\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2008)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{3} + \beta_{8} q^{5} -\beta_{5} q^{7} + ( \beta_{1} + \beta_{2} ) q^{9} +O(q^{10})\) \( q -\beta_{1} q^{3} + \beta_{8} q^{5} -\beta_{5} q^{7} + ( \beta_{1} + \beta_{2} ) q^{9} + ( -1 - \beta_{4} ) q^{11} + ( 1 + \beta_{1} + \beta_{4} - \beta_{8} + \beta_{11} ) q^{13} + ( -1 + \beta_{4} - \beta_{6} + \beta_{10} ) q^{15} + ( \beta_{5} - \beta_{7} ) q^{17} + ( -1 + \beta_{6} - \beta_{8} ) q^{19} + ( -\beta_{2} - \beta_{4} + \beta_{5} - \beta_{10} - \beta_{11} ) q^{21} + ( -1 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} ) q^{23} + ( -1 - \beta_{2} + \beta_{7} + \beta_{8} - \beta_{11} ) q^{25} + ( -1 - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{9} - \beta_{10} ) q^{27} + ( \beta_{1} + \beta_{7} - \beta_{8} + \beta_{10} ) q^{29} + ( -1 - \beta_{2} - \beta_{3} + \beta_{5} + \beta_{7} - \beta_{8} - \beta_{10} ) q^{31} + ( \beta_{1} - \beta_{2} - \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} ) q^{33} + ( -1 + \beta_{1} + \beta_{2} + \beta_{4} - \beta_{5} - \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} ) q^{35} + ( -1 - \beta_{4} + \beta_{8} + \beta_{10} - \beta_{11} ) q^{37} + ( -2 - \beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} + \beta_{9} + \beta_{11} ) q^{39} + ( \beta_{1} - \beta_{2} - \beta_{6} + \beta_{7} + \beta_{10} + \beta_{11} ) q^{41} + ( -2 + 2 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{9} - \beta_{11} ) q^{43} + ( \beta_{1} + \beta_{2} - \beta_{3} + \beta_{6} - \beta_{7} ) q^{45} + ( -3 + \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{9} + \beta_{11} ) q^{47} + ( -\beta_{1} + \beta_{5} - 2 \beta_{7} + \beta_{9} - \beta_{10} + \beta_{11} ) q^{49} + ( -1 + \beta_{1} - \beta_{2} + 2 \beta_{4} - \beta_{5} - 2 \beta_{9} + \beta_{10} ) q^{51} + ( 3 + \beta_{2} + \beta_{3} - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} - 2 \beta_{10} - \beta_{11} ) q^{53} + ( -2 + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{7} - \beta_{8} + \beta_{9} ) q^{55} + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} - \beta_{8} - \beta_{9} - \beta_{10} ) q^{57} + ( -2 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{6} + \beta_{7} - 3 \beta_{9} - \beta_{10} - 3 \beta_{11} ) q^{59} + ( \beta_{1} + \beta_{2} + \beta_{4} - \beta_{6} - \beta_{7} - \beta_{9} + \beta_{10} ) q^{61} + ( -2 + \beta_{1} + \beta_{3} + \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} ) q^{63} + ( 2 - 2 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{6} - 2 \beta_{8} - \beta_{9} - 2 \beta_{10} ) q^{65} + ( -2 + 3 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - \beta_{6} - \beta_{8} - 2 \beta_{9} ) q^{67} + ( 2 + 2 \beta_{1} + 2 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - 2 \beta_{8} - \beta_{9} ) q^{69} + ( -3 - \beta_{1} - \beta_{3} - \beta_{6} - \beta_{7} + 3 \beta_{8} + 2 \beta_{9} ) q^{71} + ( -2 + 2 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{6} + \beta_{8} + \beta_{10} ) q^{73} + ( -2 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} + \beta_{8} + \beta_{10} ) q^{75} + ( 2 - \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} ) q^{77} + ( -5 + 3 \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} ) q^{79} + ( -2 + \beta_{1} + \beta_{2} - 3 \beta_{5} + \beta_{6} + \beta_{8} + 2 \beta_{10} + \beta_{11} ) q^{81} + ( -3 + \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} - 2 \beta_{7} + \beta_{8} + 2 \beta_{9} + \beta_{10} + 2 \beta_{11} ) q^{83} + ( -\beta_{1} + 2 \beta_{2} - 2 \beta_{4} + \beta_{5} - \beta_{7} + \beta_{9} - \beta_{10} + \beta_{11} ) q^{85} + ( -1 - 2 \beta_{1} - 2 \beta_{4} + \beta_{6} + \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} ) q^{87} + ( 2 + \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{6} - 2 \beta_{8} + \beta_{9} + \beta_{11} ) q^{89} + ( -2 + \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{5} - 2 \beta_{6} - \beta_{7} + 3 \beta_{8} + 3 \beta_{9} + \beta_{11} ) q^{91} + ( 1 + 2 \beta_{1} + 4 \beta_{2} - \beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{8} + 2 \beta_{9} + \beta_{10} + 2 \beta_{11} ) q^{93} + ( -2 + 2 \beta_{1} + \beta_{2} - \beta_{5} + \beta_{6} - 2 \beta_{8} - \beta_{9} + \beta_{11} ) q^{95} + ( -2 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{6} + \beta_{8} + \beta_{10} + \beta_{11} ) q^{97} + ( -\beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{8} - \beta_{10} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 3q^{3} + 5q^{5} - 5q^{7} + 7q^{9} + O(q^{10}) \) \( 12q - 3q^{3} + 5q^{5} - 5q^{7} + 7q^{9} - 10q^{11} + 3q^{13} - 11q^{15} + 2q^{17} - 15q^{19} + 3q^{21} - 20q^{23} - 3q^{25} - 15q^{27} + 6q^{29} - 14q^{31} - 6q^{33} - 16q^{35} + 5q^{37} - 21q^{39} - 21q^{43} + 10q^{45} - 27q^{47} - 13q^{49} - 19q^{51} + 22q^{53} - 24q^{55} + q^{57} - 23q^{59} + 4q^{61} - 21q^{63} - q^{65} - 26q^{67} + 10q^{69} - 23q^{71} - 8q^{73} - 16q^{75} + 22q^{77} - 37q^{79} - 20q^{81} - 30q^{83} + 2q^{85} - 16q^{87} + 3q^{89} - 8q^{91} + 20q^{93} - 33q^{95} - 4q^{97} - 15q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 3 x^{11} - 17 x^{10} + 49 x^{9} + 106 x^{8} - 277 x^{7} - 317 x^{6} + 644 x^{5} + 537 x^{4} - 601 x^{3} - 462 x^{2} + 136 x + 104\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 3 \)
\(\beta_{3}\)\(=\)\((\)\( 5 \nu^{11} - 13 \nu^{10} - 72 \nu^{9} + 159 \nu^{8} + 331 \nu^{7} - 431 \nu^{6} - 551 \nu^{5} - 669 \nu^{4} + 457 \nu^{3} + 2253 \nu^{2} + 133 \nu - 728 \)\()/26\)
\(\beta_{4}\)\(=\)\((\)\( -15 \nu^{11} - 13 \nu^{10} + 385 \nu^{9} + 251 \nu^{8} - 3424 \nu^{7} - 1801 \nu^{6} + 12521 \nu^{5} + 5894 \nu^{4} - 16503 \nu^{3} - 8631 \nu^{2} + 4424 \nu + 2132 \)\()/52\)
\(\beta_{5}\)\(=\)\((\)\( 32 \nu^{11} - 117 \nu^{10} - 427 \nu^{9} + 1743 \nu^{8} + 1601 \nu^{7} - 8390 \nu^{6} - 1033 \nu^{5} + 14389 \nu^{4} - 710 \nu^{3} - 8999 \nu^{2} + 209 \nu + 1612 \)\()/26\)
\(\beta_{6}\)\(=\)\((\)\( -29 \nu^{11} - 13 \nu^{10} + 701 \nu^{9} + 331 \nu^{8} - 5994 \nu^{7} - 3007 \nu^{6} + 21401 \nu^{5} + 11688 \nu^{4} - 27933 \nu^{3} - 18023 \nu^{2} + 6974 \nu + 4680 \)\()/52\)
\(\beta_{7}\)\(=\)\((\)\( -113 \nu^{11} + 91 \nu^{10} + 2363 \nu^{9} - 1009 \nu^{8} - 18008 \nu^{7} + 1249 \nu^{6} + 59185 \nu^{5} + 14758 \nu^{4} - 74725 \nu^{3} - 37559 \nu^{2} + 18504 \nu + 10556 \)\()/52\)
\(\beta_{8}\)\(=\)\((\)\( -74 \nu^{11} + 91 \nu^{10} + 1466 \nu^{9} - 1165 \nu^{8} - 10624 \nu^{7} + 3628 \nu^{6} + 33549 \nu^{5} + 3058 \nu^{4} - 41796 \nu^{3} - 17565 \nu^{2} + 10470 \nu + 5304 \)\()/26\)
\(\beta_{9}\)\(=\)\((\)\( 45 \nu^{11} - 52 \nu^{10} - 895 \nu^{9} + 638 \nu^{8} + 6515 \nu^{7} - 1669 \nu^{6} - 20676 \nu^{5} - 3603 \nu^{4} + 25875 \nu^{3} + 12802 \nu^{2} - 6382 \nu - 3757 \)\()/13\)
\(\beta_{10}\)\(=\)\((\)\( 124 \nu^{11} - 208 \nu^{10} - 2303 \nu^{9} + 2820 \nu^{8} + 15585 \nu^{7} - 10694 \nu^{6} - 46274 \nu^{5} + 4955 \nu^{4} + 56324 \nu^{3} + 19022 \nu^{2} - 14093 \nu - 6396 \)\()/26\)
\(\beta_{11}\)\(=\)\((\)\( -127 \nu^{11} - 39 \nu^{10} + 3017 \nu^{9} + 1177 \nu^{8} - 25492 \nu^{7} - 11813 \nu^{6} + 90399 \nu^{5} + 48762 \nu^{4} - 118031 \nu^{3} - 77449 \nu^{2} + 29712 \nu + 20696 \)\()/52\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{10} - \beta_{9} + \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + 6 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(\beta_{11} + 2 \beta_{10} + \beta_{8} + \beta_{6} - 3 \beta_{5} + 10 \beta_{2} + 10 \beta_{1} + 16\)
\(\nu^{5}\)\(=\)\(2 \beta_{11} + 11 \beta_{10} - 8 \beta_{9} + 12 \beta_{6} - 14 \beta_{5} - 8 \beta_{4} + 9 \beta_{3} + 15 \beta_{2} + 45 \beta_{1} + 11\)
\(\nu^{6}\)\(=\)\(12 \beta_{11} + 27 \beta_{10} - 2 \beta_{9} + 11 \beta_{8} + 20 \beta_{6} - 41 \beta_{5} + 2 \beta_{3} + 88 \beta_{2} + 92 \beta_{1} + 106\)
\(\nu^{7}\)\(=\)\(29 \beta_{11} + 106 \beta_{10} - 55 \beta_{9} + 5 \beta_{8} + 2 \beta_{7} + 125 \beta_{6} - 151 \beta_{5} - 58 \beta_{4} + 68 \beta_{3} + 168 \beta_{2} + 364 \beta_{1} + 111\)
\(\nu^{8}\)\(=\)\(120 \beta_{11} + 285 \beta_{10} - 25 \beta_{9} + 106 \beta_{8} - 3 \beta_{7} + 268 \beta_{6} - 441 \beta_{5} - 7 \beta_{4} + 30 \beta_{3} + 771 \beta_{2} + 839 \beta_{1} + 772\)
\(\nu^{9}\)\(=\)\(324 \beta_{11} + 986 \beta_{10} - 357 \beta_{9} + 110 \beta_{8} + 18 \beta_{7} + 1256 \beta_{6} - 1498 \beta_{5} - 434 \beta_{4} + 498 \beta_{3} + 1726 \beta_{2} + 3055 \beta_{1} + 1089\)
\(\nu^{10}\)\(=\)\(1156 \beta_{11} + 2790 \beta_{10} - 201 \beta_{9} + 1028 \beta_{8} - 77 \beta_{7} + 3092 \beta_{6} - 4403 \beta_{5} - 179 \beta_{4} + 329 \beta_{3} + 6869 \beta_{2} + 7647 \beta_{1} + 5946\)
\(\nu^{11}\)\(=\)\(3324 \beta_{11} + 9088 \beta_{10} - 2190 \beta_{9} + 1637 \beta_{8} + 22 \beta_{7} + 12417 \beta_{6} - 14386 \beta_{5} - 3443 \beta_{4} + 3649 \beta_{3} + 17109 \beta_{2} + 26299 \beta_{1} + 10436\)

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
3.07960
2.69680
2.20058
1.58102
1.57790
0.571724
−0.589492
−0.770730
−1.18772
−1.20604
−2.41194
−2.54171
0 −3.07960 0 2.30205 0 0.712294 0 6.48396 0
1.2 0 −2.69680 0 −1.00937 0 0.288703 0 4.27276 0
1.3 0 −2.20058 0 2.09881 0 −3.80569 0 1.84254 0
1.4 0 −1.58102 0 3.46315 0 −1.91212 0 −0.500388 0
1.5 0 −1.57790 0 −2.90606 0 −2.41450 0 −0.510222 0
1.6 0 −0.571724 0 0.953980 0 4.33542 0 −2.67313 0
1.7 0 0.589492 0 −1.64401 0 1.73430 0 −2.65250 0
1.8 0 0.770730 0 3.57480 0 −0.727506 0 −2.40598 0
1.9 0 1.18772 0 0.261605 0 −3.40145 0 −1.58932 0
1.10 0 1.20604 0 −1.79940 0 2.78727 0 −1.54547 0
1.11 0 2.41194 0 −1.89365 0 −0.462635 0 2.81747 0
1.12 0 2.54171 0 1.59809 0 −2.13408 0 3.46027 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
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 4016.2.a.i 12
4.b odd 2 1 2008.2.a.b 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2008.2.a.b 12 4.b odd 2 1
4016.2.a.i 12 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(251\) \(1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{3}^{12} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4016))\).