Properties

Label 12.8.b.a
Level 12
Weight 8
Character orbit 12.b
Analytic conductor 3.749
Analytic rank 0
Dimension 4
CM No
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 12 = 2^{2} \cdot 3 \)
Weight: \( k \) = \( 8 \)
Character orbit: \([\chi]\) = 12.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(3.74862030581\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{3}, \sqrt{-5})\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{5}\cdot 3^{2} \)
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.

\(f(q)\) \(=\) \( q\) \( + \beta_{3} q^{2} \) \( + ( -3 \beta_{2} + 3 \beta_{3} ) q^{3} \) \( + ( 88 + 4 \beta_{1} + 8 \beta_{2} ) q^{4} \) \( + 23 \beta_{1} q^{5} \) \( + ( 324 + 54 \beta_{1} + 12 \beta_{2} + 15 \beta_{3} ) q^{6} \) \( + ( 43 \beta_{1} + 86 \beta_{2} ) q^{7} \) \( + ( -128 \beta_{1} + 48 \beta_{3} ) q^{8} \) \( + ( -243 + 243 \beta_{1} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + \beta_{3} q^{2} \) \( + ( -3 \beta_{2} + 3 \beta_{3} ) q^{3} \) \( + ( 88 + 4 \beta_{1} + 8 \beta_{2} ) q^{4} \) \( + 23 \beta_{1} q^{5} \) \( + ( 324 + 54 \beta_{1} + 12 \beta_{2} + 15 \beta_{3} ) q^{6} \) \( + ( 43 \beta_{1} + 86 \beta_{2} ) q^{7} \) \( + ( -128 \beta_{1} + 48 \beta_{3} ) q^{8} \) \( + ( -243 + 243 \beta_{1} ) q^{9} \) \( + ( 920 - 92 \beta_{1} - 184 \beta_{2} ) q^{10} \) \( + ( -95 \beta_{1} - 190 \beta_{3} ) q^{11} \) \( + ( 3240 - 324 \beta_{1} - 264 \beta_{2} + 264 \beta_{3} ) q^{12} \) \( -12730 q^{13} \) \( + ( -1376 \beta_{1} - 430 \beta_{3} ) q^{14} \) \( + ( -621 \beta_{1} - 552 \beta_{2} - 690 \beta_{3} ) q^{15} \) \( + ( -896 + 704 \beta_{1} + 1408 \beta_{2} ) q^{16} \) \( + 2108 \beta_{1} q^{17} \) \( + ( 9720 - 972 \beta_{1} - 1944 \beta_{2} - 243 \beta_{3} ) q^{18} \) \( + ( 1029 \beta_{1} + 2058 \beta_{2} ) q^{19} \) \( + ( 2944 \beta_{1} + 1840 \beta_{3} ) q^{20} \) \( + ( 34830 - 3483 \beta_{1} ) q^{21} \) \( + ( -20520 - 380 \beta_{1} - 760 \beta_{2} ) q^{22} \) \( + ( 3482 \beta_{1} + 6964 \beta_{3} ) q^{23} \) \( + ( 15552 + 6048 \beta_{1} + 3648 \beta_{2} + 4560 \beta_{3} ) q^{24} \) \( + 35805 q^{25} \) \( -12730 \beta_{3} q^{26} \) \( + ( -6561 \beta_{1} - 5103 \beta_{2} - 8019 \beta_{3} ) q^{27} \) \( + ( -92880 + 3784 \beta_{1} + 7568 \beta_{2} ) q^{28} \) \( -11863 \beta_{1} q^{29} \) \( + ( -74520 + 7452 \beta_{1} - 2760 \beta_{2} + 2760 \beta_{3} ) q^{30} \) \( + ( -3889 \beta_{1} - 7778 \beta_{2} ) q^{31} \) \( + ( -22528 \beta_{1} - 7936 \beta_{3} ) q^{32} \) \( + ( -61560 - 7695 \beta_{1} ) q^{33} \) \( + ( 84320 - 8432 \beta_{1} - 16864 \beta_{2} ) q^{34} \) \( + ( 9890 \beta_{1} + 19780 \beta_{3} ) q^{35} \) \( + ( -21384 + 30132 \beta_{1} - 1944 \beta_{2} + 19440 \beta_{3} ) q^{36} \) \( + 43310 q^{37} \) \( + ( -32928 \beta_{1} - 10290 \beta_{3} ) q^{38} \) \( + ( 38190 \beta_{2} - 38190 \beta_{3} ) q^{39} \) \( + ( 279680 - 4416 \beta_{1} - 8832 \beta_{2} ) q^{40} \) \( + 87854 \beta_{1} q^{41} \) \( + ( -139320 + 13932 \beta_{1} + 27864 \beta_{2} + 34830 \beta_{3} ) q^{42} \) \( + ( -3047 \beta_{1} - 6094 \beta_{2} ) q^{43} \) \( + ( 12160 \beta_{1} - 16720 \beta_{3} ) q^{44} \) \( + ( -447120 - 5589 \beta_{1} ) q^{45} \) \( + ( 752112 + 13928 \beta_{1} + 27856 \beta_{2} ) q^{46} \) \( + ( 27124 \beta_{1} + 54248 \beta_{3} ) q^{47} \) \( + ( 570240 - 57024 \beta_{1} + 2688 \beta_{2} - 2688 \beta_{3} ) q^{48} \) \( -174917 q^{49} \) \( + 35805 \beta_{3} q^{50} \) \( + ( -56916 \beta_{1} - 50592 \beta_{2} - 63240 \beta_{3} ) q^{51} \) \( + ( -1120240 - 50920 \beta_{1} - 101840 \beta_{2} ) q^{52} \) \( -39013 \beta_{1} q^{53} \) \( + ( -866052 + 65610 \beta_{1} - 32076 \beta_{2} + 25515 \beta_{3} ) q^{54} \) \( + ( 17480 \beta_{1} + 34960 \beta_{2} ) q^{55} \) \( + ( -121088 \beta_{1} - 130720 \beta_{3} ) q^{56} \) \( + ( 833490 - 83349 \beta_{1} ) q^{57} \) \( + ( -474520 + 47452 \beta_{1} + 94904 \beta_{2} ) q^{58} \) \( + ( -36475 \beta_{1} - 72950 \beta_{3} ) q^{59} \) \( + ( 596160 + 19872 \beta_{1} - 48576 \beta_{2} - 60720 \beta_{3} ) q^{60} \) \( + 314198 q^{61} \) \( + ( 124448 \beta_{1} + 38890 \beta_{3} ) q^{62} \) \( + ( 94041 \beta_{1} - 20898 \beta_{2} + 208980 \beta_{3} ) q^{63} \) \( + ( -1599488 + 58368 \beta_{1} + 116736 \beta_{2} ) q^{64} \) \( -292790 \beta_{1} q^{65} \) \( + ( -307800 + 30780 \beta_{1} + 61560 \beta_{2} - 61560 \beta_{3} ) q^{66} \) \( + ( -34915 \beta_{1} - 69830 \beta_{2} ) q^{67} \) \( + ( 269824 \beta_{1} + 168640 \beta_{3} ) q^{68} \) \( + ( 2256336 + 282042 \beta_{1} ) q^{69} \) \( + ( 2136240 + 39560 \beta_{1} + 79120 \beta_{2} ) q^{70} \) \( + ( 12750 \beta_{1} + 25500 \beta_{3} ) q^{71} \) \( + ( 2954880 - 15552 \beta_{1} - 93312 \beta_{2} - 11664 \beta_{3} ) q^{72} \) \( -259270 q^{73} \) \( + 43310 \beta_{3} q^{74} \) \( + ( -107415 \beta_{2} + 107415 \beta_{3} ) q^{75} \) \( + ( -2222640 + 90552 \beta_{1} + 181104 \beta_{2} ) q^{76} \) \( + 220590 \beta_{1} q^{77} \) \( + ( -4124520 - 687420 \beta_{1} - 152760 \beta_{2} - 190950 \beta_{3} ) q^{78} \) \( + ( 225271 \beta_{1} + 450542 \beta_{2} ) q^{79} \) \( + ( 141312 \beta_{1} + 323840 \beta_{3} ) q^{80} \) \( + ( -4664871 - 118098 \beta_{1} ) q^{81} \) \( + ( 3514160 - 351416 \beta_{1} - 702832 \beta_{2} ) q^{82} \) \( + ( -486017 \beta_{1} - 972034 \beta_{3} ) q^{83} \) \( + ( 3065040 - 306504 \beta_{1} + 278640 \beta_{2} - 278640 \beta_{3} ) q^{84} \) \( -3878720 q^{85} \) \( + ( 97504 \beta_{1} + 30470 \beta_{3} ) q^{86} \) \( + ( 320301 \beta_{1} + 284712 \beta_{2} + 355890 \beta_{3} ) q^{87} \) \( + ( -984960 - 115520 \beta_{1} - 231040 \beta_{2} ) q^{88} \) \( + 434306 \beta_{1} q^{89} \) \( + ( -223560 + 22356 \beta_{1} + 44712 \beta_{2} - 447120 \beta_{3} ) q^{90} \) \( + ( -547390 \beta_{1} - 1094780 \beta_{2} ) q^{91} \) \( + ( -445696 \beta_{1} + 612832 \beta_{3} ) q^{92} \) \( + ( -3150090 + 315009 \beta_{1} ) q^{93} \) \( + ( 5858784 + 108496 \beta_{1} + 216992 \beta_{2} ) q^{94} \) \( + ( 236670 \beta_{1} + 473340 \beta_{3} ) q^{95} \) \( + ( -2571264 + 179712 \beta_{1} + 445440 \beta_{2} + 556800 \beta_{3} ) q^{96} \) \( + 7243010 q^{97} \) \( -174917 \beta_{3} q^{98} \) \( + ( 207765 \beta_{1} + 369360 \beta_{2} + 46170 \beta_{3} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(4q \) \(\mathstrut +\mathstrut 352q^{4} \) \(\mathstrut +\mathstrut 1296q^{6} \) \(\mathstrut -\mathstrut 972q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut +\mathstrut 352q^{4} \) \(\mathstrut +\mathstrut 1296q^{6} \) \(\mathstrut -\mathstrut 972q^{9} \) \(\mathstrut +\mathstrut 3680q^{10} \) \(\mathstrut +\mathstrut 12960q^{12} \) \(\mathstrut -\mathstrut 50920q^{13} \) \(\mathstrut -\mathstrut 3584q^{16} \) \(\mathstrut +\mathstrut 38880q^{18} \) \(\mathstrut +\mathstrut 139320q^{21} \) \(\mathstrut -\mathstrut 82080q^{22} \) \(\mathstrut +\mathstrut 62208q^{24} \) \(\mathstrut +\mathstrut 143220q^{25} \) \(\mathstrut -\mathstrut 371520q^{28} \) \(\mathstrut -\mathstrut 298080q^{30} \) \(\mathstrut -\mathstrut 246240q^{33} \) \(\mathstrut +\mathstrut 337280q^{34} \) \(\mathstrut -\mathstrut 85536q^{36} \) \(\mathstrut +\mathstrut 173240q^{37} \) \(\mathstrut +\mathstrut 1118720q^{40} \) \(\mathstrut -\mathstrut 557280q^{42} \) \(\mathstrut -\mathstrut 1788480q^{45} \) \(\mathstrut +\mathstrut 3008448q^{46} \) \(\mathstrut +\mathstrut 2280960q^{48} \) \(\mathstrut -\mathstrut 699668q^{49} \) \(\mathstrut -\mathstrut 4480960q^{52} \) \(\mathstrut -\mathstrut 3464208q^{54} \) \(\mathstrut +\mathstrut 3333960q^{57} \) \(\mathstrut -\mathstrut 1898080q^{58} \) \(\mathstrut +\mathstrut 2384640q^{60} \) \(\mathstrut +\mathstrut 1256792q^{61} \) \(\mathstrut -\mathstrut 6397952q^{64} \) \(\mathstrut -\mathstrut 1231200q^{66} \) \(\mathstrut +\mathstrut 9025344q^{69} \) \(\mathstrut +\mathstrut 8544960q^{70} \) \(\mathstrut +\mathstrut 11819520q^{72} \) \(\mathstrut -\mathstrut 1037080q^{73} \) \(\mathstrut -\mathstrut 8890560q^{76} \) \(\mathstrut -\mathstrut 16498080q^{78} \) \(\mathstrut -\mathstrut 18659484q^{81} \) \(\mathstrut +\mathstrut 14056640q^{82} \) \(\mathstrut +\mathstrut 12260160q^{84} \) \(\mathstrut -\mathstrut 15514880q^{85} \) \(\mathstrut -\mathstrut 3939840q^{88} \) \(\mathstrut -\mathstrut 894240q^{90} \) \(\mathstrut -\mathstrut 12600360q^{93} \) \(\mathstrut +\mathstrut 23435136q^{94} \) \(\mathstrut -\mathstrut 10285056q^{96} \) \(\mathstrut +\mathstrut 28972040q^{97} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( -2 \nu^{3} - 6 \nu \)
\(\beta_{2}\)\(=\)\( \nu^{3} + 6 \nu^{2} + 3 \nu + 3 \)
\(\beta_{3}\)\(=\)\( -2 \nu^{3} + 6 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3}\mathstrut -\mathstrut \) \(\beta_{1}\)\()/12\)
\(\nu^{2}\)\(=\)\((\)\(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut -\mathstrut \) \(6\)\()/12\)
\(\nu^{3}\)\(=\)\((\)\(-\)\(\beta_{3}\mathstrut -\mathstrut \) \(\beta_{1}\)\()/4\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1
−0.866025 1.11803i
−0.866025 + 1.11803i
0.866025 1.11803i
0.866025 + 1.11803i
−10.3923 4.47214i −31.1769 34.8569i 88.0000 + 92.9516i 205.718i 168.115 + 501.670i 999.230i −498.831 1359.53i −243.000 + 2173.46i 920.000 2137.89i
11.2 −10.3923 + 4.47214i −31.1769 + 34.8569i 88.0000 92.9516i 205.718i 168.115 501.670i 999.230i −498.831 + 1359.53i −243.000 2173.46i 920.000 + 2137.89i
11.3 10.3923 4.47214i 31.1769 + 34.8569i 88.0000 92.9516i 205.718i 479.885 + 222.816i 999.230i 498.831 1359.53i −243.000 + 2173.46i 920.000 + 2137.89i
11.4 10.3923 + 4.47214i 31.1769 34.8569i 88.0000 + 92.9516i 205.718i 479.885 222.816i 999.230i 498.831 + 1359.53i −243.000 2173.46i 920.000 2137.89i
\(n\): 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
4.b Odd 1 yes
12.b Even 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{5}^{2} \) \(\mathstrut +\mathstrut 42320 \) acting on \(S_{8}^{\mathrm{new}}(12, [\chi])\).