# Properties

 Label 49.4.c.c Level $49$ Weight $4$ Character orbit 49.c Analytic conductor $2.891$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$49 = 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 49.c (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.89109359028$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 7) 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.

 $$f(q)$$ $$=$$ $$q + \zeta_{6} q^{2} + ( 2 - 2 \zeta_{6} ) q^{3} + ( 7 - 7 \zeta_{6} ) q^{4} -16 \zeta_{6} q^{5} + 2 q^{6} + 15 q^{8} + 23 \zeta_{6} q^{9} +O(q^{10})$$ $$q + \zeta_{6} q^{2} + ( 2 - 2 \zeta_{6} ) q^{3} + ( 7 - 7 \zeta_{6} ) q^{4} -16 \zeta_{6} q^{5} + 2 q^{6} + 15 q^{8} + 23 \zeta_{6} q^{9} + ( 16 - 16 \zeta_{6} ) q^{10} + ( 8 - 8 \zeta_{6} ) q^{11} -14 \zeta_{6} q^{12} + 28 q^{13} -32 q^{15} -41 \zeta_{6} q^{16} + ( -54 + 54 \zeta_{6} ) q^{17} + ( -23 + 23 \zeta_{6} ) q^{18} + 110 \zeta_{6} q^{19} -112 q^{20} + 8 q^{22} -48 \zeta_{6} q^{23} + ( 30 - 30 \zeta_{6} ) q^{24} + ( -131 + 131 \zeta_{6} ) q^{25} + 28 \zeta_{6} q^{26} + 100 q^{27} -110 q^{29} -32 \zeta_{6} q^{30} + ( -12 + 12 \zeta_{6} ) q^{31} + ( 161 - 161 \zeta_{6} ) q^{32} -16 \zeta_{6} q^{33} -54 q^{34} + 161 q^{36} + 246 \zeta_{6} q^{37} + ( -110 + 110 \zeta_{6} ) q^{38} + ( 56 - 56 \zeta_{6} ) q^{39} -240 \zeta_{6} q^{40} + 182 q^{41} + 128 q^{43} -56 \zeta_{6} q^{44} + ( 368 - 368 \zeta_{6} ) q^{45} + ( 48 - 48 \zeta_{6} ) q^{46} -324 \zeta_{6} q^{47} -82 q^{48} -131 q^{50} + 108 \zeta_{6} q^{51} + ( 196 - 196 \zeta_{6} ) q^{52} + ( 162 - 162 \zeta_{6} ) q^{53} + 100 \zeta_{6} q^{54} -128 q^{55} + 220 q^{57} -110 \zeta_{6} q^{58} + ( -810 + 810 \zeta_{6} ) q^{59} + ( -224 + 224 \zeta_{6} ) q^{60} + 488 \zeta_{6} q^{61} -12 q^{62} -167 q^{64} -448 \zeta_{6} q^{65} + ( 16 - 16 \zeta_{6} ) q^{66} + ( -244 + 244 \zeta_{6} ) q^{67} + 378 \zeta_{6} q^{68} -96 q^{69} -768 q^{71} + 345 \zeta_{6} q^{72} + ( 702 - 702 \zeta_{6} ) q^{73} + ( -246 + 246 \zeta_{6} ) q^{74} + 262 \zeta_{6} q^{75} + 770 q^{76} + 56 q^{78} -440 \zeta_{6} q^{79} + ( -656 + 656 \zeta_{6} ) q^{80} + ( -421 + 421 \zeta_{6} ) q^{81} + 182 \zeta_{6} q^{82} -1302 q^{83} + 864 q^{85} + 128 \zeta_{6} q^{86} + ( -220 + 220 \zeta_{6} ) q^{87} + ( 120 - 120 \zeta_{6} ) q^{88} -730 \zeta_{6} q^{89} + 368 q^{90} -336 q^{92} + 24 \zeta_{6} q^{93} + ( 324 - 324 \zeta_{6} ) q^{94} + ( 1760 - 1760 \zeta_{6} ) q^{95} -322 \zeta_{6} q^{96} + 294 q^{97} + 184 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{2} + 2q^{3} + 7q^{4} - 16q^{5} + 4q^{6} + 30q^{8} + 23q^{9} + O(q^{10})$$ $$2q + q^{2} + 2q^{3} + 7q^{4} - 16q^{5} + 4q^{6} + 30q^{8} + 23q^{9} + 16q^{10} + 8q^{11} - 14q^{12} + 56q^{13} - 64q^{15} - 41q^{16} - 54q^{17} - 23q^{18} + 110q^{19} - 224q^{20} + 16q^{22} - 48q^{23} + 30q^{24} - 131q^{25} + 28q^{26} + 200q^{27} - 220q^{29} - 32q^{30} - 12q^{31} + 161q^{32} - 16q^{33} - 108q^{34} + 322q^{36} + 246q^{37} - 110q^{38} + 56q^{39} - 240q^{40} + 364q^{41} + 256q^{43} - 56q^{44} + 368q^{45} + 48q^{46} - 324q^{47} - 164q^{48} - 262q^{50} + 108q^{51} + 196q^{52} + 162q^{53} + 100q^{54} - 256q^{55} + 440q^{57} - 110q^{58} - 810q^{59} - 224q^{60} + 488q^{61} - 24q^{62} - 334q^{64} - 448q^{65} + 16q^{66} - 244q^{67} + 378q^{68} - 192q^{69} - 1536q^{71} + 345q^{72} + 702q^{73} - 246q^{74} + 262q^{75} + 1540q^{76} + 112q^{78} - 440q^{79} - 656q^{80} - 421q^{81} + 182q^{82} - 2604q^{83} + 1728q^{85} + 128q^{86} - 220q^{87} + 120q^{88} - 730q^{89} + 736q^{90} - 672q^{92} + 24q^{93} + 324q^{94} + 1760q^{95} - 322q^{96} + 588q^{97} + 368q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/49\mathbb{Z}\right)^\times$$.

 $$n$$ $$3$$ $$\chi(n)$$ $$-\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
18.1
 0.5 + 0.866025i 0.5 − 0.866025i
0.500000 + 0.866025i 1.00000 1.73205i 3.50000 6.06218i −8.00000 13.8564i 2.00000 0 15.0000 11.5000 + 19.9186i 8.00000 13.8564i
30.1 0.500000 0.866025i 1.00000 + 1.73205i 3.50000 + 6.06218i −8.00000 + 13.8564i 2.00000 0 15.0000 11.5000 19.9186i 8.00000 + 13.8564i
 $$n$$: 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 49.4.c.c 2
3.b odd 2 1 441.4.e.h 2
7.b odd 2 1 49.4.c.b 2
7.c even 3 1 7.4.a.a 1
7.c even 3 1 inner 49.4.c.c 2
7.d odd 6 1 49.4.a.b 1
7.d odd 6 1 49.4.c.b 2
21.c even 2 1 441.4.e.e 2
21.g even 6 1 441.4.a.i 1
21.g even 6 1 441.4.e.e 2
21.h odd 6 1 63.4.a.b 1
21.h odd 6 1 441.4.e.h 2
28.f even 6 1 784.4.a.g 1
28.g odd 6 1 112.4.a.f 1
35.i odd 6 1 1225.4.a.j 1
35.j even 6 1 175.4.a.b 1
35.l odd 12 2 175.4.b.b 2
56.k odd 6 1 448.4.a.e 1
56.p even 6 1 448.4.a.i 1
77.h odd 6 1 847.4.a.b 1
84.n even 6 1 1008.4.a.c 1
91.r even 6 1 1183.4.a.b 1
105.o odd 6 1 1575.4.a.e 1
119.j even 6 1 2023.4.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.4.a.a 1 7.c even 3 1
49.4.a.b 1 7.d odd 6 1
49.4.c.b 2 7.b odd 2 1
49.4.c.b 2 7.d odd 6 1
49.4.c.c 2 1.a even 1 1 trivial
49.4.c.c 2 7.c even 3 1 inner
63.4.a.b 1 21.h odd 6 1
112.4.a.f 1 28.g odd 6 1
175.4.a.b 1 35.j even 6 1
175.4.b.b 2 35.l odd 12 2
441.4.a.i 1 21.g even 6 1
441.4.e.e 2 21.c even 2 1
441.4.e.e 2 21.g even 6 1
441.4.e.h 2 3.b odd 2 1
441.4.e.h 2 21.h odd 6 1
448.4.a.e 1 56.k odd 6 1
448.4.a.i 1 56.p even 6 1
784.4.a.g 1 28.f even 6 1
847.4.a.b 1 77.h odd 6 1
1008.4.a.c 1 84.n even 6 1
1183.4.a.b 1 91.r even 6 1
1225.4.a.j 1 35.i odd 6 1
1575.4.a.e 1 105.o odd 6 1
2023.4.a.a 1 119.j even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(49, [\chi])$$:

 $$T_{2}^{2} - T_{2} + 1$$ $$T_{3}^{2} - 2 T_{3} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T + T^{2}$$
$3$ $$4 - 2 T + T^{2}$$
$5$ $$256 + 16 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$64 - 8 T + T^{2}$$
$13$ $$( -28 + T )^{2}$$
$17$ $$2916 + 54 T + T^{2}$$
$19$ $$12100 - 110 T + T^{2}$$
$23$ $$2304 + 48 T + T^{2}$$
$29$ $$( 110 + T )^{2}$$
$31$ $$144 + 12 T + T^{2}$$
$37$ $$60516 - 246 T + T^{2}$$
$41$ $$( -182 + T )^{2}$$
$43$ $$( -128 + T )^{2}$$
$47$ $$104976 + 324 T + T^{2}$$
$53$ $$26244 - 162 T + T^{2}$$
$59$ $$656100 + 810 T + T^{2}$$
$61$ $$238144 - 488 T + T^{2}$$
$67$ $$59536 + 244 T + T^{2}$$
$71$ $$( 768 + T )^{2}$$
$73$ $$492804 - 702 T + T^{2}$$
$79$ $$193600 + 440 T + T^{2}$$
$83$ $$( 1302 + T )^{2}$$
$89$ $$532900 + 730 T + T^{2}$$
$97$ $$( -294 + T )^{2}$$