# Properties

 Label 12.9.c.b Level 12 Weight 9 Character orbit 12.c Analytic conductor 4.889 Analytic rank 0 Dimension 2 CM No Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$12 = 2^{2} \cdot 3$$ Weight: $$k$$ = $$9$$ Character orbit: $$[\chi]$$ = 12.c (of order $$2$$ and degree $$1$$)

## Newform invariants

 Self dual: No Analytic conductor: $$4.88854332073$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-110})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2\cdot 3$$ Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 6\sqrt{-110}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q$$ $$+ ( -51 + \beta ) q^{3}$$ $$-18 \beta q^{5}$$ $$-3094 q^{7}$$ $$+ ( -1359 - 102 \beta ) q^{9}$$ $$+O(q^{10})$$ $$q$$ $$+ ( -51 + \beta ) q^{3}$$ $$-18 \beta q^{5}$$ $$-3094 q^{7}$$ $$+ ( -1359 - 102 \beta ) q^{9}$$ $$-18 \beta q^{11}$$ $$-7294 q^{13}$$ $$+ ( 71280 + 918 \beta ) q^{15}$$ $$+ 936 \beta q^{17}$$ $$-80326 q^{19}$$ $$+ ( 157794 - 3094 \beta ) q^{21}$$ $$-1548 \beta q^{23}$$ $$-892415 q^{25}$$ $$+ ( 473229 + 3843 \beta ) q^{27}$$ $$-13734 \beta q^{29}$$ $$+ 435914 q^{31}$$ $$+ ( 71280 + 918 \beta ) q^{33}$$ $$+ 55692 \beta q^{35}$$ $$+ 1159298 q^{37}$$ $$+ ( 371994 - 7294 \beta ) q^{39}$$ $$-43164 \beta q^{41}$$ $$+ 990266 q^{43}$$ $$+ ( -7270560 + 24462 \beta ) q^{45}$$ $$-106488 \beta q^{47}$$ $$+ 3808035 q^{49}$$ $$+ ( -3706560 - 47736 \beta ) q^{51}$$ $$+ 160038 \beta q^{53}$$ $$-1283040 q^{55}$$ $$+ ( 4096626 - 80326 \beta ) q^{57}$$ $$+ 25398 \beta q^{59}$$ $$+ 19369154 q^{61}$$ $$+ ( 4204746 + 315588 \beta ) q^{63}$$ $$+ 131292 \beta q^{65}$$ $$-28024294 q^{67}$$ $$+ ( 6130080 + 78948 \beta ) q^{69}$$ $$-534852 \beta q^{71}$$ $$-25230142 q^{73}$$ $$+ ( 45513165 - 892415 \beta ) q^{75}$$ $$+ 55692 \beta q^{77}$$ $$-63401398 q^{79}$$ $$+ ( -39352959 + 277236 \beta ) q^{81}$$ $$+ 755514 \beta q^{83}$$ $$+ 66718080 q^{85}$$ $$+ ( 54386640 + 700434 \beta ) q^{87}$$ $$-1244196 \beta q^{89}$$ $$+ 22567636 q^{91}$$ $$+ ( -22231614 + 435914 \beta ) q^{93}$$ $$+ 1445868 \beta q^{95}$$ $$+ 19550306 q^{97}$$ $$+ ( -7270560 + 24462 \beta ) q^{99}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q$$ $$\mathstrut -\mathstrut 102q^{3}$$ $$\mathstrut -\mathstrut 6188q^{7}$$ $$\mathstrut -\mathstrut 2718q^{9}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$2q$$ $$\mathstrut -\mathstrut 102q^{3}$$ $$\mathstrut -\mathstrut 6188q^{7}$$ $$\mathstrut -\mathstrut 2718q^{9}$$ $$\mathstrut -\mathstrut 14588q^{13}$$ $$\mathstrut +\mathstrut 142560q^{15}$$ $$\mathstrut -\mathstrut 160652q^{19}$$ $$\mathstrut +\mathstrut 315588q^{21}$$ $$\mathstrut -\mathstrut 1784830q^{25}$$ $$\mathstrut +\mathstrut 946458q^{27}$$ $$\mathstrut +\mathstrut 871828q^{31}$$ $$\mathstrut +\mathstrut 142560q^{33}$$ $$\mathstrut +\mathstrut 2318596q^{37}$$ $$\mathstrut +\mathstrut 743988q^{39}$$ $$\mathstrut +\mathstrut 1980532q^{43}$$ $$\mathstrut -\mathstrut 14541120q^{45}$$ $$\mathstrut +\mathstrut 7616070q^{49}$$ $$\mathstrut -\mathstrut 7413120q^{51}$$ $$\mathstrut -\mathstrut 2566080q^{55}$$ $$\mathstrut +\mathstrut 8193252q^{57}$$ $$\mathstrut +\mathstrut 38738308q^{61}$$ $$\mathstrut +\mathstrut 8409492q^{63}$$ $$\mathstrut -\mathstrut 56048588q^{67}$$ $$\mathstrut +\mathstrut 12260160q^{69}$$ $$\mathstrut -\mathstrut 50460284q^{73}$$ $$\mathstrut +\mathstrut 91026330q^{75}$$ $$\mathstrut -\mathstrut 126802796q^{79}$$ $$\mathstrut -\mathstrut 78705918q^{81}$$ $$\mathstrut +\mathstrut 133436160q^{85}$$ $$\mathstrut +\mathstrut 108773280q^{87}$$ $$\mathstrut +\mathstrut 45135272q^{91}$$ $$\mathstrut -\mathstrut 44463228q^{93}$$ $$\mathstrut +\mathstrut 39100612q^{97}$$ $$\mathstrut -\mathstrut 14541120q^{99}$$ $$\mathstrut +\mathstrut O(q^{100})$$

## 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
 − 10.4881i 10.4881i
0 −51.0000 62.9285i 0 1132.71i 0 −3094.00 0 −1359.00 + 6418.71i 0
5.2 0 −51.0000 + 62.9285i 0 1132.71i 0 −3094.00 0 −1359.00 6418.71i 0
 $$n$$: 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

## Hecke kernels

This newform can be constructed as the kernel of the linear operator $$T_{5}^{2}$$ $$\mathstrut +\mathstrut 1283040$$ acting on $$S_{9}^{\mathrm{new}}(12, [\chi])$$.