# Properties

 Label 126.4.g.a Level $126$ Weight $4$ Character orbit 126.g Analytic conductor $7.434$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$126 = 2 \cdot 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 126.g (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.43424066072$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 42) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -2 + 2 \zeta_{6} ) q^{2} -4 \zeta_{6} q^{4} + ( -15 + 15 \zeta_{6} ) q^{5} + ( 14 + 7 \zeta_{6} ) q^{7} + 8 q^{8} +O(q^{10})$$ $$q + ( -2 + 2 \zeta_{6} ) q^{2} -4 \zeta_{6} q^{4} + ( -15 + 15 \zeta_{6} ) q^{5} + ( 14 + 7 \zeta_{6} ) q^{7} + 8 q^{8} -30 \zeta_{6} q^{10} -9 \zeta_{6} q^{11} -88 q^{13} + ( -42 + 28 \zeta_{6} ) q^{14} + ( -16 + 16 \zeta_{6} ) q^{16} -84 \zeta_{6} q^{17} + ( -104 + 104 \zeta_{6} ) q^{19} + 60 q^{20} + 18 q^{22} + ( -84 + 84 \zeta_{6} ) q^{23} -100 \zeta_{6} q^{25} + ( 176 - 176 \zeta_{6} ) q^{26} + ( 28 - 84 \zeta_{6} ) q^{28} -51 q^{29} -185 \zeta_{6} q^{31} -32 \zeta_{6} q^{32} + 168 q^{34} + ( -315 + 210 \zeta_{6} ) q^{35} + ( -44 + 44 \zeta_{6} ) q^{37} -208 \zeta_{6} q^{38} + ( -120 + 120 \zeta_{6} ) q^{40} + 168 q^{41} + 326 q^{43} + ( -36 + 36 \zeta_{6} ) q^{44} -168 \zeta_{6} q^{46} + ( -138 + 138 \zeta_{6} ) q^{47} + ( 147 + 245 \zeta_{6} ) q^{49} + 200 q^{50} + 352 \zeta_{6} q^{52} + 639 \zeta_{6} q^{53} + 135 q^{55} + ( 112 + 56 \zeta_{6} ) q^{56} + ( 102 - 102 \zeta_{6} ) q^{58} + 159 \zeta_{6} q^{59} + ( -722 + 722 \zeta_{6} ) q^{61} + 370 q^{62} + 64 q^{64} + ( 1320 - 1320 \zeta_{6} ) q^{65} + 166 \zeta_{6} q^{67} + ( -336 + 336 \zeta_{6} ) q^{68} + ( 210 - 630 \zeta_{6} ) q^{70} -1086 q^{71} -218 \zeta_{6} q^{73} -88 \zeta_{6} q^{74} + 416 q^{76} + ( 63 - 189 \zeta_{6} ) q^{77} + ( 583 - 583 \zeta_{6} ) q^{79} -240 \zeta_{6} q^{80} + ( -336 + 336 \zeta_{6} ) q^{82} + 597 q^{83} + 1260 q^{85} + ( -652 + 652 \zeta_{6} ) q^{86} -72 \zeta_{6} q^{88} + ( -1038 + 1038 \zeta_{6} ) q^{89} + ( -1232 - 616 \zeta_{6} ) q^{91} + 336 q^{92} -276 \zeta_{6} q^{94} -1560 \zeta_{6} q^{95} -169 q^{97} + ( -784 + 294 \zeta_{6} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} - 4q^{4} - 15q^{5} + 35q^{7} + 16q^{8} + O(q^{10})$$ $$2q - 2q^{2} - 4q^{4} - 15q^{5} + 35q^{7} + 16q^{8} - 30q^{10} - 9q^{11} - 176q^{13} - 56q^{14} - 16q^{16} - 84q^{17} - 104q^{19} + 120q^{20} + 36q^{22} - 84q^{23} - 100q^{25} + 176q^{26} - 28q^{28} - 102q^{29} - 185q^{31} - 32q^{32} + 336q^{34} - 420q^{35} - 44q^{37} - 208q^{38} - 120q^{40} + 336q^{41} + 652q^{43} - 36q^{44} - 168q^{46} - 138q^{47} + 539q^{49} + 400q^{50} + 352q^{52} + 639q^{53} + 270q^{55} + 280q^{56} + 102q^{58} + 159q^{59} - 722q^{61} + 740q^{62} + 128q^{64} + 1320q^{65} + 166q^{67} - 336q^{68} - 210q^{70} - 2172q^{71} - 218q^{73} - 88q^{74} + 832q^{76} - 63q^{77} + 583q^{79} - 240q^{80} - 336q^{82} + 1194q^{83} + 2520q^{85} - 652q^{86} - 72q^{88} - 1038q^{89} - 3080q^{91} + 672q^{92} - 276q^{94} - 1560q^{95} - 338q^{97} - 1274q^{98} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/126\mathbb{Z}\right)^\times$$.

 $$n$$ $$29$$ $$73$$ $$\chi(n)$$ $$1$$ $$-1 + \zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
37.1
 0.5 + 0.866025i 0.5 − 0.866025i
−1.00000 + 1.73205i 0 −2.00000 3.46410i −7.50000 + 12.9904i 0 17.5000 + 6.06218i 8.00000 0 −15.0000 25.9808i
109.1 −1.00000 1.73205i 0 −2.00000 + 3.46410i −7.50000 12.9904i 0 17.5000 6.06218i 8.00000 0 −15.0000 + 25.9808i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 126.4.g.a 2
3.b odd 2 1 42.4.e.b 2
7.b odd 2 1 882.4.g.l 2
7.c even 3 1 inner 126.4.g.a 2
7.c even 3 1 882.4.a.r 1
7.d odd 6 1 882.4.a.h 1
7.d odd 6 1 882.4.g.l 2
12.b even 2 1 336.4.q.d 2
21.c even 2 1 294.4.e.e 2
21.g even 6 1 294.4.a.g 1
21.g even 6 1 294.4.e.e 2
21.h odd 6 1 42.4.e.b 2
21.h odd 6 1 294.4.a.a 1
84.j odd 6 1 2352.4.a.q 1
84.n even 6 1 336.4.q.d 2
84.n even 6 1 2352.4.a.u 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.4.e.b 2 3.b odd 2 1
42.4.e.b 2 21.h odd 6 1
126.4.g.a 2 1.a even 1 1 trivial
126.4.g.a 2 7.c even 3 1 inner
294.4.a.a 1 21.h odd 6 1
294.4.a.g 1 21.g even 6 1
294.4.e.e 2 21.c even 2 1
294.4.e.e 2 21.g even 6 1
336.4.q.d 2 12.b even 2 1
336.4.q.d 2 84.n even 6 1
882.4.a.h 1 7.d odd 6 1
882.4.a.r 1 7.c even 3 1
882.4.g.l 2 7.b odd 2 1
882.4.g.l 2 7.d odd 6 1
2352.4.a.q 1 84.j odd 6 1
2352.4.a.u 1 84.n even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 15 T_{5} + 225$$ acting on $$S_{4}^{\mathrm{new}}(126, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 + 2 T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$225 + 15 T + T^{2}$$
$7$ $$343 - 35 T + T^{2}$$
$11$ $$81 + 9 T + T^{2}$$
$13$ $$( 88 + T )^{2}$$
$17$ $$7056 + 84 T + T^{2}$$
$19$ $$10816 + 104 T + T^{2}$$
$23$ $$7056 + 84 T + T^{2}$$
$29$ $$( 51 + T )^{2}$$
$31$ $$34225 + 185 T + T^{2}$$
$37$ $$1936 + 44 T + T^{2}$$
$41$ $$( -168 + T )^{2}$$
$43$ $$( -326 + T )^{2}$$
$47$ $$19044 + 138 T + T^{2}$$
$53$ $$408321 - 639 T + T^{2}$$
$59$ $$25281 - 159 T + T^{2}$$
$61$ $$521284 + 722 T + T^{2}$$
$67$ $$27556 - 166 T + T^{2}$$
$71$ $$( 1086 + T )^{2}$$
$73$ $$47524 + 218 T + T^{2}$$
$79$ $$339889 - 583 T + T^{2}$$
$83$ $$( -597 + T )^{2}$$
$89$ $$1077444 + 1038 T + T^{2}$$
$97$ $$( 169 + T )^{2}$$