Properties

Label 12.7.c.a
Level 12
Weight 7
Character orbit 12.c
Analytic conductor 2.761
Analytic rank 0
Dimension 2
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 12 = 2^{2} \cdot 3 \)
Weight: \( k \) = \( 7 \)
Character orbit: \([\chi]\) = 12.c (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(2.76064900344\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-5}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 12\sqrt{-5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( -3 + \beta ) q^{3} \) \( + 6 \beta q^{5} \) \( + 242 q^{7} \) \( + ( -711 - 6 \beta ) q^{9} \) \(+O(q^{10})\) \( q\) \( + ( -3 + \beta ) q^{3} \) \( + 6 \beta q^{5} \) \( + 242 q^{7} \) \( + ( -711 - 6 \beta ) q^{9} \) \( -66 \beta q^{11} \) \( + 2618 q^{13} \) \( + ( -4320 - 18 \beta ) q^{15} \) \( + 264 \beta q^{17} \) \( + 5786 q^{19} \) \( + ( -726 + 242 \beta ) q^{21} \) \( -348 \beta q^{23} \) \( -10295 q^{25} \) \( + ( 6453 - 693 \beta ) q^{27} \) \( -462 \beta q^{29} \) \( -20446 q^{31} \) \( + ( 47520 + 198 \beta ) q^{33} \) \( + 1452 \beta q^{35} \) \( -46774 q^{37} \) \( + ( -7854 + 2618 \beta ) q^{39} \) \( + 132 \beta q^{41} \) \( + 68618 q^{43} \) \( + ( 25920 - 4266 \beta ) q^{45} \) \( -792 \beta q^{47} \) \( -59085 q^{49} \) \( + ( -190080 - 792 \beta ) q^{51} \) \( -6402 \beta q^{53} \) \( + 285120 q^{55} \) \( + ( -17358 + 5786 \beta ) q^{57} \) \( + 5574 \beta q^{59} \) \( + 24794 q^{61} \) \( + ( -172062 - 1452 \beta ) q^{63} \) \( + 15708 \beta q^{65} \) \( -84358 q^{67} \) \( + ( 250560 + 1044 \beta ) q^{69} \) \( -12084 \beta q^{71} \) \( -113806 q^{73} \) \( + ( 30885 - 10295 \beta ) q^{75} \) \( -15972 \beta q^{77} \) \( -159742 q^{79} \) \( + ( 479601 + 8532 \beta ) q^{81} \) \( -19206 \beta q^{83} \) \( -1140480 q^{85} \) \( + ( 332640 + 1386 \beta ) q^{87} \) \( + 46812 \beta q^{89} \) \( + 633556 q^{91} \) \( + ( 61338 - 20446 \beta ) q^{93} \) \( + 34716 \beta q^{95} \) \( + 899522 q^{97} \) \( + ( -285120 + 46926 \beta ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut 6q^{3} \) \(\mathstrut +\mathstrut 484q^{7} \) \(\mathstrut -\mathstrut 1422q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut 6q^{3} \) \(\mathstrut +\mathstrut 484q^{7} \) \(\mathstrut -\mathstrut 1422q^{9} \) \(\mathstrut +\mathstrut 5236q^{13} \) \(\mathstrut -\mathstrut 8640q^{15} \) \(\mathstrut +\mathstrut 11572q^{19} \) \(\mathstrut -\mathstrut 1452q^{21} \) \(\mathstrut -\mathstrut 20590q^{25} \) \(\mathstrut +\mathstrut 12906q^{27} \) \(\mathstrut -\mathstrut 40892q^{31} \) \(\mathstrut +\mathstrut 95040q^{33} \) \(\mathstrut -\mathstrut 93548q^{37} \) \(\mathstrut -\mathstrut 15708q^{39} \) \(\mathstrut +\mathstrut 137236q^{43} \) \(\mathstrut +\mathstrut 51840q^{45} \) \(\mathstrut -\mathstrut 118170q^{49} \) \(\mathstrut -\mathstrut 380160q^{51} \) \(\mathstrut +\mathstrut 570240q^{55} \) \(\mathstrut -\mathstrut 34716q^{57} \) \(\mathstrut +\mathstrut 49588q^{61} \) \(\mathstrut -\mathstrut 344124q^{63} \) \(\mathstrut -\mathstrut 168716q^{67} \) \(\mathstrut +\mathstrut 501120q^{69} \) \(\mathstrut -\mathstrut 227612q^{73} \) \(\mathstrut +\mathstrut 61770q^{75} \) \(\mathstrut -\mathstrut 319484q^{79} \) \(\mathstrut +\mathstrut 959202q^{81} \) \(\mathstrut -\mathstrut 2280960q^{85} \) \(\mathstrut +\mathstrut 665280q^{87} \) \(\mathstrut +\mathstrut 1267112q^{91} \) \(\mathstrut +\mathstrut 122676q^{93} \) \(\mathstrut +\mathstrut 1799044q^{97} \) \(\mathstrut -\mathstrut 570240q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
2.23607i
2.23607i
0 −3.00000 26.8328i 0 160.997i 0 242.000 0 −711.000 + 160.997i 0
5.2 0 −3.00000 + 26.8328i 0 160.997i 0 242.000 0 −711.000 160.997i 0
\(n\): 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
3.b Odd 1 yes

Hecke kernels

There are no other newforms in \(S_{7}^{\mathrm{new}}(12, [\chi])\).