# 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

# Related objects

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 $$S_{8}^{\mathrm{new}}(12, [\chi])$$.