# Properties

 Label 124.1.i.a Level 124 Weight 1 Character orbit 124.i Analytic conductor 0.062 Analytic rank 0 Dimension 4 Projective image $$A_{4}$$ CM/RM No Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$124 = 2^{2} \cdot 31$$ Weight: $$k$$ = $$1$$ Character orbit: $$[\chi]$$ = 124.i (of order $$6$$ and degree $$2$$)

## Newform invariants

 Self dual: No Analytic conductor: $$0.0618840615665$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Projective image $$A_{4}$$ Projective field Galois closure of 4.0.15376.1

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q$$ $$-\zeta_{12}^{3} q^{2}$$ $$-\zeta_{12} q^{3}$$ $$- q^{4}$$ $$-\zeta_{12}^{2} q^{5}$$ $$+ \zeta_{12}^{4} q^{6}$$ $$+ \zeta_{12} q^{7}$$ $$+ \zeta_{12}^{3} q^{8}$$ $$+O(q^{10})$$ $$q$$ $$-\zeta_{12}^{3} q^{2}$$ $$-\zeta_{12} q^{3}$$ $$- q^{4}$$ $$-\zeta_{12}^{2} q^{5}$$ $$+ \zeta_{12}^{4} q^{6}$$ $$+ \zeta_{12} q^{7}$$ $$+ \zeta_{12}^{3} q^{8}$$ $$+ \zeta_{12}^{5} q^{10}$$ $$-\zeta_{12}^{5} q^{11}$$ $$+ \zeta_{12} q^{12}$$ $$+ \zeta_{12}^{2} q^{13}$$ $$-\zeta_{12}^{4} q^{14}$$ $$+ \zeta_{12}^{3} q^{15}$$ $$+ q^{16}$$ $$+ \zeta_{12}^{4} q^{17}$$ $$-\zeta_{12} q^{19}$$ $$+ \zeta_{12}^{2} q^{20}$$ $$-\zeta_{12}^{2} q^{21}$$ $$-\zeta_{12}^{2} q^{22}$$ $$-\zeta_{12}^{4} q^{24}$$ $$-\zeta_{12}^{5} q^{26}$$ $$+ \zeta_{12}^{3} q^{27}$$ $$-\zeta_{12} q^{28}$$ $$+ q^{30}$$ $$+ \zeta_{12}^{3} q^{31}$$ $$-\zeta_{12}^{3} q^{32}$$ $$- q^{33}$$ $$+ \zeta_{12} q^{34}$$ $$-\zeta_{12}^{3} q^{35}$$ $$+ \zeta_{12}^{4} q^{37}$$ $$+ \zeta_{12}^{4} q^{38}$$ $$-\zeta_{12}^{3} q^{39}$$ $$-\zeta_{12}^{5} q^{40}$$ $$-\zeta_{12}^{2} q^{41}$$ $$+ \zeta_{12}^{5} q^{42}$$ $$-\zeta_{12} q^{43}$$ $$+ \zeta_{12}^{5} q^{44}$$ $$-\zeta_{12} q^{48}$$ $$-\zeta_{12}^{5} q^{51}$$ $$-\zeta_{12}^{2} q^{52}$$ $$+ \zeta_{12}^{2} q^{53}$$ $$+ q^{54}$$ $$-\zeta_{12} q^{55}$$ $$+ \zeta_{12}^{4} q^{56}$$ $$+ \zeta_{12}^{2} q^{57}$$ $$+ \zeta_{12} q^{59}$$ $$-\zeta_{12}^{3} q^{60}$$ $$+ q^{62}$$ $$- q^{64}$$ $$-\zeta_{12}^{4} q^{65}$$ $$+ \zeta_{12}^{3} q^{66}$$ $$-\zeta_{12}^{5} q^{67}$$ $$-\zeta_{12}^{4} q^{68}$$ $$- q^{70}$$ $$+ \zeta_{12}^{5} q^{71}$$ $$-\zeta_{12}^{2} q^{73}$$ $$+ \zeta_{12} q^{74}$$ $$+ \zeta_{12} q^{76}$$ $$+ q^{77}$$ $$- q^{78}$$ $$+ \zeta_{12} q^{79}$$ $$-\zeta_{12}^{2} q^{80}$$ $$-\zeta_{12}^{4} q^{81}$$ $$+ \zeta_{12}^{5} q^{82}$$ $$+ \zeta_{12}^{5} q^{83}$$ $$+ \zeta_{12}^{2} q^{84}$$ $$+ q^{85}$$ $$+ \zeta_{12}^{4} q^{86}$$ $$+ \zeta_{12}^{2} q^{88}$$ $$+ \zeta_{12}^{3} q^{91}$$ $$-\zeta_{12}^{4} q^{93}$$ $$+ \zeta_{12}^{3} q^{95}$$ $$+ \zeta_{12}^{4} q^{96}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q$$ $$\mathstrut -\mathstrut 4q^{4}$$ $$\mathstrut -\mathstrut 2q^{5}$$ $$\mathstrut -\mathstrut 2q^{6}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$4q$$ $$\mathstrut -\mathstrut 4q^{4}$$ $$\mathstrut -\mathstrut 2q^{5}$$ $$\mathstrut -\mathstrut 2q^{6}$$ $$\mathstrut +\mathstrut 2q^{13}$$ $$\mathstrut +\mathstrut 2q^{14}$$ $$\mathstrut +\mathstrut 4q^{16}$$ $$\mathstrut -\mathstrut 2q^{17}$$ $$\mathstrut +\mathstrut 2q^{20}$$ $$\mathstrut -\mathstrut 2q^{21}$$ $$\mathstrut -\mathstrut 2q^{22}$$ $$\mathstrut +\mathstrut 2q^{24}$$ $$\mathstrut +\mathstrut 4q^{30}$$ $$\mathstrut -\mathstrut 4q^{33}$$ $$\mathstrut -\mathstrut 2q^{37}$$ $$\mathstrut -\mathstrut 2q^{38}$$ $$\mathstrut -\mathstrut 2q^{41}$$ $$\mathstrut -\mathstrut 2q^{52}$$ $$\mathstrut +\mathstrut 2q^{53}$$ $$\mathstrut +\mathstrut 4q^{54}$$ $$\mathstrut -\mathstrut 2q^{56}$$ $$\mathstrut +\mathstrut 2q^{57}$$ $$\mathstrut +\mathstrut 4q^{62}$$ $$\mathstrut -\mathstrut 4q^{64}$$ $$\mathstrut +\mathstrut 2q^{65}$$ $$\mathstrut +\mathstrut 2q^{68}$$ $$\mathstrut -\mathstrut 4q^{70}$$ $$\mathstrut -\mathstrut 2q^{73}$$ $$\mathstrut +\mathstrut 4q^{77}$$ $$\mathstrut -\mathstrut 4q^{78}$$ $$\mathstrut -\mathstrut 2q^{80}$$ $$\mathstrut +\mathstrut 2q^{81}$$ $$\mathstrut +\mathstrut 2q^{84}$$ $$\mathstrut +\mathstrut 4q^{85}$$ $$\mathstrut -\mathstrut 2q^{86}$$ $$\mathstrut +\mathstrut 2q^{88}$$ $$\mathstrut +\mathstrut 2q^{93}$$ $$\mathstrut -\mathstrut 2q^{96}$$ $$\mathstrut +\mathstrut O(q^{100})$$

## Character Values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/124\mathbb{Z}\right)^\times$$.

 $$n$$ $$63$$ $$65$$ $$\chi(n)$$ $$-1$$ $$-\zeta_{12}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
67.1
 0.866025 + 0.500000i −0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i
1.00000i −0.866025 0.500000i −1.00000 −0.500000 0.866025i −0.500000 + 0.866025i 0.866025 + 0.500000i 1.00000i 0 −0.866025 + 0.500000i
67.2 1.00000i 0.866025 + 0.500000i −1.00000 −0.500000 0.866025i −0.500000 + 0.866025i −0.866025 0.500000i 1.00000i 0 0.866025 0.500000i
87.1 1.00000i 0.866025 0.500000i −1.00000 −0.500000 + 0.866025i −0.500000 0.866025i −0.866025 + 0.500000i 1.00000i 0 0.866025 + 0.500000i
87.2 1.00000i −0.866025 + 0.500000i −1.00000 −0.500000 + 0.866025i −0.500000 0.866025i 0.866025 0.500000i 1.00000i 0 −0.866025 0.500000i
 $$n$$: 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
4.b Odd 1 yes
31.c Even 1 yes
124.i Odd 1 yes

## Hecke kernels

There are no other newforms in $$S_{1}^{\mathrm{new}}(124, [\chi])$$.