Properties

Label 12.5.d.a
Level 12
Weight 5
Character orbit 12.d
Analytic conductor 1.240
Analytic rank 0
Dimension 4
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 12 = 2^{2} \cdot 3 \)
Weight: \( k \) = \( 5 \)
Character orbit: \([\chi]\) = 12.d (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(1.24043955701\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{13})\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4}\cdot 3 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q\) \( + ( 2 - \beta_{1} ) q^{2} \) \( -\beta_{2} q^{3} \) \( + ( -4 - 3 \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{4} \) \( + ( 6 + 4 \beta_{1} - 4 \beta_{3} ) q^{5} \) \( + ( -6 - 2 \beta_{2} + 3 \beta_{3} ) q^{6} \) \( + ( -8 + 12 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} ) q^{7} \) \( + ( 2 \beta_{1} + 12 \beta_{2} - 2 \beta_{3} ) q^{8} \) \( -27 q^{9} \) \(+O(q^{10})\) \( q\) \( + ( 2 - \beta_{1} ) q^{2} \) \( -\beta_{2} q^{3} \) \( + ( -4 - 3 \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{4} \) \( + ( 6 + 4 \beta_{1} - 4 \beta_{3} ) q^{5} \) \( + ( -6 - 2 \beta_{2} + 3 \beta_{3} ) q^{6} \) \( + ( -8 + 12 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} ) q^{7} \) \( + ( 2 \beta_{1} + 12 \beta_{2} - 2 \beta_{3} ) q^{8} \) \( -27 q^{9} \) \( + ( -36 - 6 \beta_{1} - 16 \beta_{2} - 8 \beta_{3} ) q^{10} \) \( + ( 16 - 24 \beta_{1} + 12 \beta_{2} - 8 \beta_{3} ) q^{11} \) \( + ( 36 + 9 \beta_{1} + 2 \beta_{2} + 9 \beta_{3} ) q^{12} \) \( + ( 74 - 24 \beta_{1} + 24 \beta_{3} ) q^{13} \) \( + ( 136 + 24 \beta_{1} - 24 \beta_{2} + 4 \beta_{3} ) q^{14} \) \( + ( 24 - 36 \beta_{1} + 2 \beta_{2} - 12 \beta_{3} ) q^{15} \) \( + ( 48 + 16 \beta_{2} - 40 \beta_{3} ) q^{16} \) \( + ( -150 - 8 \beta_{1} + 8 \beta_{3} ) q^{17} \) \( + ( -54 + 27 \beta_{1} ) q^{18} \) \( + ( -48 + 72 \beta_{1} + 36 \beta_{2} + 24 \beta_{3} ) q^{19} \) \( + ( -376 + 22 \beta_{1} - 36 \beta_{2} + 46 \beta_{3} ) q^{20} \) \( + ( -36 + 36 \beta_{1} - 36 \beta_{3} ) q^{21} \) \( + ( -248 - 48 \beta_{1} + 56 \beta_{2} - 20 \beta_{3} ) q^{22} \) \( + ( -16 + 24 \beta_{1} - 96 \beta_{2} + 8 \beta_{3} ) q^{23} \) \( + ( 336 - 18 \beta_{1} + 4 \beta_{2} - 6 \beta_{3} ) q^{24} \) \( + ( 243 + 48 \beta_{1} - 48 \beta_{3} ) q^{25} \) \( + ( 436 - 74 \beta_{1} + 96 \beta_{2} + 48 \beta_{3} ) q^{26} \) \( + 27 \beta_{2} q^{27} \) \( + ( 400 - 108 \beta_{1} - 88 \beta_{2} + 52 \beta_{3} ) q^{28} \) \( + ( 222 + 28 \beta_{1} - 28 \beta_{3} ) q^{29} \) \( + ( -468 - 72 \beta_{1} + 52 \beta_{2} + 18 \beta_{3} ) q^{30} \) \( + ( 56 - 84 \beta_{1} - 140 \beta_{2} - 28 \beta_{3} ) q^{31} \) \( + ( -608 - 88 \beta_{1} - 48 \beta_{2} - 88 \beta_{3} ) q^{32} \) \( + ( 180 - 72 \beta_{1} + 72 \beta_{3} ) q^{33} \) \( + ( -204 + 150 \beta_{1} + 32 \beta_{2} + 16 \beta_{3} ) q^{34} \) \( + ( -16 + 24 \beta_{1} + 264 \beta_{2} + 8 \beta_{3} ) q^{35} \) \( + ( 108 + 81 \beta_{1} - 54 \beta_{2} - 27 \beta_{3} ) q^{36} \) \( + ( -1102 - 48 \beta_{1} + 48 \beta_{3} ) q^{37} \) \( + ( 1176 + 144 \beta_{1} - 24 \beta_{2} - 156 \beta_{3} ) q^{38} \) \( + ( -144 + 216 \beta_{1} - 122 \beta_{2} + 72 \beta_{3} ) q^{39} \) \( + ( 128 + 444 \beta_{1} - 24 \beta_{2} + 132 \beta_{3} ) q^{40} \) \( + ( 138 - 56 \beta_{1} + 56 \beta_{3} ) q^{41} \) \( + ( -504 + 36 \beta_{1} - 144 \beta_{2} - 72 \beta_{3} ) q^{42} \) \( + ( 304 - 456 \beta_{1} + 332 \beta_{2} - 152 \beta_{3} ) q^{43} \) \( + ( -944 + 180 \beta_{1} + 168 \beta_{2} - 140 \beta_{3} ) q^{44} \) \( + ( -162 - 108 \beta_{1} + 108 \beta_{3} ) q^{45} \) \( + ( -256 + 48 \beta_{1} - 224 \beta_{2} + 272 \beta_{3} ) q^{46} \) \( + ( 16 - 24 \beta_{1} - 216 \beta_{2} - 8 \beta_{3} ) q^{47} \) \( + ( 432 - 360 \beta_{1} + 32 \beta_{2} ) q^{48} \) \( + ( -143 + 96 \beta_{1} - 96 \beta_{3} ) q^{49} \) \( + ( -90 - 243 \beta_{1} - 192 \beta_{2} - 96 \beta_{3} ) q^{50} \) \( + ( -48 + 72 \beta_{1} + 134 \beta_{2} + 24 \beta_{3} ) q^{51} \) \( + ( 1816 - 462 \beta_{1} + 436 \beta_{2} - 166 \beta_{3} ) q^{52} \) \( + ( 1278 - 68 \beta_{1} + 68 \beta_{3} ) q^{53} \) \( + ( 162 + 54 \beta_{2} - 81 \beta_{3} ) q^{54} \) \( + ( -64 + 96 \beta_{1} - 536 \beta_{2} + 32 \beta_{3} ) q^{55} \) \( + ( 448 - 456 \beta_{1} + 144 \beta_{2} + 424 \beta_{3} ) q^{56} \) \( + ( 1404 + 216 \beta_{1} - 216 \beta_{3} ) q^{57} \) \( + ( 108 - 222 \beta_{1} - 112 \beta_{2} - 56 \beta_{3} ) q^{58} \) \( + ( -448 + 672 \beta_{1} - 108 \beta_{2} + 224 \beta_{3} ) q^{59} \) \( + ( -840 + 414 \beta_{1} + 284 \beta_{2} - 66 \beta_{3} ) q^{60} \) \( + ( 1058 + 480 \beta_{1} - 480 \beta_{3} ) q^{61} \) \( + ( -1960 - 168 \beta_{1} - 168 \beta_{2} + 476 \beta_{3} ) q^{62} \) \( + ( 216 - 324 \beta_{1} + 108 \beta_{2} - 108 \beta_{3} ) q^{63} \) \( + ( -3968 + 432 \beta_{1} - 96 \beta_{2} + 144 \beta_{3} ) q^{64} \) \( + ( -4548 + 152 \beta_{1} - 152 \beta_{3} ) q^{65} \) \( + ( 1224 - 180 \beta_{1} + 288 \beta_{2} + 144 \beta_{3} ) q^{66} \) \( + ( -448 + 672 \beta_{1} + 316 \beta_{2} + 224 \beta_{3} ) q^{67} \) \( + ( 1304 + 370 \beta_{1} - 204 \beta_{2} - 230 \beta_{3} ) q^{68} \) \( + ( -2448 + 72 \beta_{1} - 72 \beta_{3} ) q^{69} \) \( + ( 1904 + 48 \beta_{1} + 496 \beta_{2} - 808 \beta_{3} ) q^{70} \) \( + ( 432 - 648 \beta_{1} - 240 \beta_{2} - 216 \beta_{3} ) q^{71} \) \( + ( -54 \beta_{1} - 324 \beta_{2} + 54 \beta_{3} ) q^{72} \) \( + ( 2210 - 1056 \beta_{1} + 1056 \beta_{3} ) q^{73} \) \( + ( -1628 + 1102 \beta_{1} + 192 \beta_{2} + 96 \beta_{3} ) q^{74} \) \( + ( 288 - 432 \beta_{1} - 147 \beta_{2} - 144 \beta_{3} ) q^{75} \) \( + ( 240 - 1188 \beta_{1} - 648 \beta_{2} - 228 \beta_{3} ) q^{76} \) \( + ( 5232 - 336 \beta_{1} + 336 \beta_{3} ) q^{77} \) \( + ( 2148 + 432 \beta_{1} - 532 \beta_{2} + 222 \beta_{3} ) q^{78} \) \( + ( -296 + 444 \beta_{1} + 980 \beta_{2} + 148 \beta_{3} ) q^{79} \) \( + ( 6304 + 448 \beta_{1} - 672 \beta_{2} - 240 \beta_{3} ) q^{80} \) \( + 729 q^{81} \) \( + ( 948 - 138 \beta_{1} + 224 \beta_{2} + 112 \beta_{3} ) q^{82} \) \( + ( 1552 - 2328 \beta_{1} + 660 \beta_{2} - 776 \beta_{3} ) q^{83} \) \( + ( -3024 + 468 \beta_{1} - 504 \beta_{2} + 324 \beta_{3} ) q^{84} \) \( + ( -2564 - 648 \beta_{1} + 648 \beta_{3} ) q^{85} \) \( + ( -4088 - 912 \beta_{1} + 1272 \beta_{2} - 692 \beta_{3} ) q^{86} \) \( + ( 168 - 252 \beta_{1} - 166 \beta_{2} - 84 \beta_{3} ) q^{87} \) \( + ( -2240 + 984 \beta_{1} - 304 \beta_{2} - 824 \beta_{3} ) q^{88} \) \( + ( -6270 + 784 \beta_{1} - 784 \beta_{3} ) q^{89} \) \( + ( 972 + 162 \beta_{1} + 432 \beta_{2} + 216 \beta_{3} ) q^{90} \) \( + ( -784 + 1176 \beta_{1} - 2024 \beta_{2} + 392 \beta_{3} ) q^{91} \) \( + ( 3968 + 576 \beta_{1} + 896 \beta_{3} ) q^{92} \) \( + ( -4284 - 252 \beta_{1} + 252 \beta_{3} ) q^{93} \) \( + ( -1616 - 48 \beta_{1} - 400 \beta_{2} + 664 \beta_{3} ) q^{94} \) \( + ( -1536 + 2304 \beta_{1} + 1464 \beta_{2} + 768 \beta_{3} ) q^{95} \) \( + ( -1824 - 792 \beta_{1} + 784 \beta_{2} + 264 \beta_{3} ) q^{96} \) \( + ( 5762 + 1104 \beta_{1} - 1104 \beta_{3} ) q^{97} \) \( + ( -1438 + 143 \beta_{1} - 384 \beta_{2} - 192 \beta_{3} ) q^{98} \) \( + ( -432 + 648 \beta_{1} - 324 \beta_{2} + 216 \beta_{3} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(4q \) \(\mathstrut +\mathstrut 6q^{2} \) \(\mathstrut -\mathstrut 20q^{4} \) \(\mathstrut +\mathstrut 24q^{5} \) \(\mathstrut -\mathstrut 18q^{6} \) \(\mathstrut -\mathstrut 108q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut +\mathstrut 6q^{2} \) \(\mathstrut -\mathstrut 20q^{4} \) \(\mathstrut +\mathstrut 24q^{5} \) \(\mathstrut -\mathstrut 18q^{6} \) \(\mathstrut -\mathstrut 108q^{9} \) \(\mathstrut -\mathstrut 172q^{10} \) \(\mathstrut +\mathstrut 180q^{12} \) \(\mathstrut +\mathstrut 296q^{13} \) \(\mathstrut +\mathstrut 600q^{14} \) \(\mathstrut +\mathstrut 112q^{16} \) \(\mathstrut -\mathstrut 600q^{17} \) \(\mathstrut -\mathstrut 162q^{18} \) \(\mathstrut -\mathstrut 1368q^{20} \) \(\mathstrut -\mathstrut 144q^{21} \) \(\mathstrut -\mathstrut 1128q^{22} \) \(\mathstrut +\mathstrut 1296q^{24} \) \(\mathstrut +\mathstrut 972q^{25} \) \(\mathstrut +\mathstrut 1692q^{26} \) \(\mathstrut +\mathstrut 1488q^{28} \) \(\mathstrut +\mathstrut 888q^{29} \) \(\mathstrut -\mathstrut 1980q^{30} \) \(\mathstrut -\mathstrut 2784q^{32} \) \(\mathstrut +\mathstrut 720q^{33} \) \(\mathstrut -\mathstrut 484q^{34} \) \(\mathstrut +\mathstrut 540q^{36} \) \(\mathstrut -\mathstrut 4408q^{37} \) \(\mathstrut +\mathstrut 4680q^{38} \) \(\mathstrut +\mathstrut 1664q^{40} \) \(\mathstrut +\mathstrut 552q^{41} \) \(\mathstrut -\mathstrut 2088q^{42} \) \(\mathstrut -\mathstrut 3696q^{44} \) \(\mathstrut -\mathstrut 648q^{45} \) \(\mathstrut -\mathstrut 384q^{46} \) \(\mathstrut +\mathstrut 1008q^{48} \) \(\mathstrut -\mathstrut 572q^{49} \) \(\mathstrut -\mathstrut 1038q^{50} \) \(\mathstrut +\mathstrut 6008q^{52} \) \(\mathstrut +\mathstrut 5112q^{53} \) \(\mathstrut +\mathstrut 486q^{54} \) \(\mathstrut +\mathstrut 1728q^{56} \) \(\mathstrut +\mathstrut 5616q^{57} \) \(\mathstrut -\mathstrut 124q^{58} \) \(\mathstrut -\mathstrut 2664q^{60} \) \(\mathstrut +\mathstrut 4232q^{61} \) \(\mathstrut -\mathstrut 7224q^{62} \) \(\mathstrut -\mathstrut 14720q^{64} \) \(\mathstrut -\mathstrut 18192q^{65} \) \(\mathstrut +\mathstrut 4824q^{66} \) \(\mathstrut +\mathstrut 5496q^{68} \) \(\mathstrut -\mathstrut 9792q^{69} \) \(\mathstrut +\mathstrut 6096q^{70} \) \(\mathstrut +\mathstrut 8840q^{73} \) \(\mathstrut -\mathstrut 4116q^{74} \) \(\mathstrut -\mathstrut 1872q^{76} \) \(\mathstrut +\mathstrut 20928q^{77} \) \(\mathstrut +\mathstrut 9900q^{78} \) \(\mathstrut +\mathstrut 25632q^{80} \) \(\mathstrut +\mathstrut 2916q^{81} \) \(\mathstrut +\mathstrut 3740q^{82} \) \(\mathstrut -\mathstrut 10512q^{84} \) \(\mathstrut -\mathstrut 10256q^{85} \) \(\mathstrut -\mathstrut 19560q^{86} \) \(\mathstrut -\mathstrut 8640q^{88} \) \(\mathstrut -\mathstrut 25080q^{89} \) \(\mathstrut +\mathstrut 4644q^{90} \) \(\mathstrut +\mathstrut 18816q^{92} \) \(\mathstrut -\mathstrut 17136q^{93} \) \(\mathstrut -\mathstrut 5232q^{94} \) \(\mathstrut -\mathstrut 8352q^{96} \) \(\mathstrut +\mathstrut 23048q^{97} \) \(\mathstrut -\mathstrut 5850q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4}\mathstrut -\mathstrut \) \(x^{3}\mathstrut +\mathstrut \) \(4\) \(x^{2}\mathstrut +\mathstrut \) \(3\) \(x\mathstrut +\mathstrut \) \(9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{3} + 4 \nu^{2} - 4 \nu + 3 \)\()/2\)
\(\beta_{3}\)\(=\)\( \nu^{3} + 2 \nu + 5 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3}\mathstrut +\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut -\mathstrut \) \(8\)\()/4\)
\(\nu^{3}\)\(=\)\(\beta_{3}\mathstrut -\mathstrut \) \(\beta_{1}\mathstrut -\mathstrut \) \(5\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/12\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(7\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
1.15139 + 1.99426i
1.15139 1.99426i
−0.651388 + 1.12824i
−0.651388 1.12824i
−0.302776 3.98852i 5.19615i −15.8167 + 2.41526i 34.8444 −20.7250 + 1.57327i 43.0318i 14.4222 + 62.3538i −27.0000 −10.5500 138.978i
7.2 −0.302776 + 3.98852i 5.19615i −15.8167 2.41526i 34.8444 −20.7250 1.57327i 43.0318i 14.4222 62.3538i −27.0000 −10.5500 + 138.978i
7.3 3.30278 2.25647i 5.19615i 5.81665 14.9053i −22.8444 11.7250 + 17.1617i 56.8882i −14.4222 62.3538i −27.0000 −75.4500 + 51.5478i
7.4 3.30278 + 2.25647i 5.19615i 5.81665 + 14.9053i −22.8444 11.7250 17.1617i 56.8882i −14.4222 + 62.3538i −27.0000 −75.4500 51.5478i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
4.b Odd 1 yes

Hecke kernels

There are no other newforms in \(S_{5}^{\mathrm{new}}(12, [\chi])\).