# Properties

 Label 49.4.c.a 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: $$\Z[a_1, a_2, a_3]$$ 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.

 $$f(q)$$ $$=$$ $$q -2 \zeta_{6} q^{2} + ( 7 - 7 \zeta_{6} ) q^{3} + ( 4 - 4 \zeta_{6} ) q^{4} + 7 \zeta_{6} q^{5} -14 q^{6} -24 q^{8} -22 \zeta_{6} q^{9} +O(q^{10})$$ $$q -2 \zeta_{6} q^{2} + ( 7 - 7 \zeta_{6} ) q^{3} + ( 4 - 4 \zeta_{6} ) q^{4} + 7 \zeta_{6} q^{5} -14 q^{6} -24 q^{8} -22 \zeta_{6} q^{9} + ( 14 - 14 \zeta_{6} ) q^{10} + ( 5 - 5 \zeta_{6} ) q^{11} -28 \zeta_{6} q^{12} + 14 q^{13} + 49 q^{15} + 16 \zeta_{6} q^{16} + ( -21 + 21 \zeta_{6} ) q^{17} + ( -44 + 44 \zeta_{6} ) q^{18} + 49 \zeta_{6} q^{19} + 28 q^{20} -10 q^{22} + 159 \zeta_{6} q^{23} + ( -168 + 168 \zeta_{6} ) q^{24} + ( 76 - 76 \zeta_{6} ) q^{25} -28 \zeta_{6} q^{26} + 35 q^{27} + 58 q^{29} -98 \zeta_{6} q^{30} + ( 147 - 147 \zeta_{6} ) q^{31} + ( -160 + 160 \zeta_{6} ) q^{32} -35 \zeta_{6} q^{33} + 42 q^{34} -88 q^{36} -219 \zeta_{6} q^{37} + ( 98 - 98 \zeta_{6} ) q^{38} + ( 98 - 98 \zeta_{6} ) q^{39} -168 \zeta_{6} q^{40} -350 q^{41} -124 q^{43} -20 \zeta_{6} q^{44} + ( 154 - 154 \zeta_{6} ) q^{45} + ( 318 - 318 \zeta_{6} ) q^{46} + 525 \zeta_{6} q^{47} + 112 q^{48} -152 q^{50} + 147 \zeta_{6} q^{51} + ( 56 - 56 \zeta_{6} ) q^{52} + ( -303 + 303 \zeta_{6} ) q^{53} -70 \zeta_{6} q^{54} + 35 q^{55} + 343 q^{57} -116 \zeta_{6} q^{58} + ( -105 + 105 \zeta_{6} ) q^{59} + ( 196 - 196 \zeta_{6} ) q^{60} -413 \zeta_{6} q^{61} -294 q^{62} + 448 q^{64} + 98 \zeta_{6} q^{65} + ( -70 + 70 \zeta_{6} ) q^{66} + ( -415 + 415 \zeta_{6} ) q^{67} + 84 \zeta_{6} q^{68} + 1113 q^{69} -432 q^{71} + 528 \zeta_{6} q^{72} + ( -1113 + 1113 \zeta_{6} ) q^{73} + ( -438 + 438 \zeta_{6} ) q^{74} -532 \zeta_{6} q^{75} + 196 q^{76} -196 q^{78} + 103 \zeta_{6} q^{79} + ( -112 + 112 \zeta_{6} ) q^{80} + ( 839 - 839 \zeta_{6} ) q^{81} + 700 \zeta_{6} q^{82} -1092 q^{83} -147 q^{85} + 248 \zeta_{6} q^{86} + ( 406 - 406 \zeta_{6} ) q^{87} + ( -120 + 120 \zeta_{6} ) q^{88} -329 \zeta_{6} q^{89} -308 q^{90} + 636 q^{92} -1029 \zeta_{6} q^{93} + ( 1050 - 1050 \zeta_{6} ) q^{94} + ( -343 + 343 \zeta_{6} ) q^{95} + 1120 \zeta_{6} q^{96} + 882 q^{97} -110 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} + 7q^{3} + 4q^{4} + 7q^{5} - 28q^{6} - 48q^{8} - 22q^{9} + O(q^{10})$$ $$2q - 2q^{2} + 7q^{3} + 4q^{4} + 7q^{5} - 28q^{6} - 48q^{8} - 22q^{9} + 14q^{10} + 5q^{11} - 28q^{12} + 28q^{13} + 98q^{15} + 16q^{16} - 21q^{17} - 44q^{18} + 49q^{19} + 56q^{20} - 20q^{22} + 159q^{23} - 168q^{24} + 76q^{25} - 28q^{26} + 70q^{27} + 116q^{29} - 98q^{30} + 147q^{31} - 160q^{32} - 35q^{33} + 84q^{34} - 176q^{36} - 219q^{37} + 98q^{38} + 98q^{39} - 168q^{40} - 700q^{41} - 248q^{43} - 20q^{44} + 154q^{45} + 318q^{46} + 525q^{47} + 224q^{48} - 304q^{50} + 147q^{51} + 56q^{52} - 303q^{53} - 70q^{54} + 70q^{55} + 686q^{57} - 116q^{58} - 105q^{59} + 196q^{60} - 413q^{61} - 588q^{62} + 896q^{64} + 98q^{65} - 70q^{66} - 415q^{67} + 84q^{68} + 2226q^{69} - 864q^{71} + 528q^{72} - 1113q^{73} - 438q^{74} - 532q^{75} + 392q^{76} - 392q^{78} + 103q^{79} - 112q^{80} + 839q^{81} + 700q^{82} - 2184q^{83} - 294q^{85} + 248q^{86} + 406q^{87} - 120q^{88} - 329q^{89} - 616q^{90} + 1272q^{92} - 1029q^{93} + 1050q^{94} - 343q^{95} + 1120q^{96} + 1764q^{97} - 220q^{99} + O(q^{100})$$

## 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
−1.00000 1.73205i 3.50000 6.06218i 2.00000 3.46410i 3.50000 + 6.06218i −14.0000 0 −24.0000 −11.0000 19.0526i 7.00000 12.1244i
30.1 −1.00000 + 1.73205i 3.50000 + 6.06218i 2.00000 + 3.46410i 3.50000 6.06218i −14.0000 0 −24.0000 −11.0000 + 19.0526i 7.00000 + 12.1244i
 $$n$$: 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.a 2
3.b odd 2 1 441.4.e.k 2
7.b odd 2 1 7.4.c.a 2
7.c even 3 1 49.4.a.c 1
7.c even 3 1 inner 49.4.c.a 2
7.d odd 6 1 7.4.c.a 2
7.d odd 6 1 49.4.a.d 1
21.c even 2 1 63.4.e.b 2
21.g even 6 1 63.4.e.b 2
21.g even 6 1 441.4.a.d 1
21.h odd 6 1 441.4.a.e 1
21.h odd 6 1 441.4.e.k 2
28.d even 2 1 112.4.i.c 2
28.f even 6 1 112.4.i.c 2
28.f even 6 1 784.4.a.b 1
28.g odd 6 1 784.4.a.r 1
35.c odd 2 1 175.4.e.a 2
35.f even 4 2 175.4.k.a 4
35.i odd 6 1 175.4.e.a 2
35.i odd 6 1 1225.4.a.c 1
35.j even 6 1 1225.4.a.d 1
35.k even 12 2 175.4.k.a 4
56.e even 2 1 448.4.i.a 2
56.h odd 2 1 448.4.i.f 2
56.j odd 6 1 448.4.i.f 2
56.m even 6 1 448.4.i.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.4.c.a 2 7.b odd 2 1
7.4.c.a 2 7.d odd 6 1
49.4.a.c 1 7.c even 3 1
49.4.a.d 1 7.d odd 6 1
49.4.c.a 2 1.a even 1 1 trivial
49.4.c.a 2 7.c even 3 1 inner
63.4.e.b 2 21.c even 2 1
63.4.e.b 2 21.g even 6 1
112.4.i.c 2 28.d even 2 1
112.4.i.c 2 28.f even 6 1
175.4.e.a 2 35.c odd 2 1
175.4.e.a 2 35.i odd 6 1
175.4.k.a 4 35.f even 4 2
175.4.k.a 4 35.k even 12 2
441.4.a.d 1 21.g even 6 1
441.4.a.e 1 21.h odd 6 1
441.4.e.k 2 3.b odd 2 1
441.4.e.k 2 21.h odd 6 1
448.4.i.a 2 56.e even 2 1
448.4.i.a 2 56.m even 6 1
448.4.i.f 2 56.h odd 2 1
448.4.i.f 2 56.j odd 6 1
784.4.a.b 1 28.f even 6 1
784.4.a.r 1 28.g odd 6 1
1225.4.a.c 1 35.i odd 6 1
1225.4.a.d 1 35.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 $$S_{4}^{\mathrm{new}}(49, [\chi])$$:

 $$T_{2}^{2} + 2 T_{2} + 4$$ $$T_{3}^{2} - 7 T_{3} + 49$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 + 2 T + T^{2}$$
$3$ $$49 - 7 T + T^{2}$$
$5$ $$49 - 7 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$25 - 5 T + T^{2}$$
$13$ $$( -14 + T )^{2}$$
$17$ $$441 + 21 T + T^{2}$$
$19$ $$2401 - 49 T + T^{2}$$
$23$ $$25281 - 159 T + T^{2}$$
$29$ $$( -58 + T )^{2}$$
$31$ $$21609 - 147 T + T^{2}$$
$37$ $$47961 + 219 T + T^{2}$$
$41$ $$( 350 + T )^{2}$$
$43$ $$( 124 + T )^{2}$$
$47$ $$275625 - 525 T + T^{2}$$
$53$ $$91809 + 303 T + T^{2}$$
$59$ $$11025 + 105 T + T^{2}$$
$61$ $$170569 + 413 T + T^{2}$$
$67$ $$172225 + 415 T + T^{2}$$
$71$ $$( 432 + T )^{2}$$
$73$ $$1238769 + 1113 T + T^{2}$$
$79$ $$10609 - 103 T + T^{2}$$
$83$ $$( 1092 + T )^{2}$$
$89$ $$108241 + 329 T + T^{2}$$
$97$ $$( -882 + T )^{2}$$