Properties

 Label 7350.2.a.cw Level $7350$ Weight $2$ Character orbit 7350.a Self dual yes Analytic conductor $58.690$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$7350 = 2 \cdot 3 \cdot 5^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7350.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$58.6900454856$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 42) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{3} + q^{4} + q^{6} + q^{8} + q^{9} + O(q^{10})$$ $$q + q^{2} + q^{3} + q^{4} + q^{6} + q^{8} + q^{9} + 3q^{11} + q^{12} - 4q^{13} + q^{16} + q^{18} + 4q^{19} + 3q^{22} + q^{24} - 4q^{26} + q^{27} + 9q^{29} + q^{31} + q^{32} + 3q^{33} + q^{36} - 8q^{37} + 4q^{38} - 4q^{39} + 10q^{43} + 3q^{44} - 6q^{47} + q^{48} - 4q^{52} + 3q^{53} + q^{54} + 4q^{57} + 9q^{58} - 3q^{59} + 10q^{61} + q^{62} + q^{64} + 3q^{66} + 10q^{67} - 6q^{71} + q^{72} + 2q^{73} - 8q^{74} + 4q^{76} - 4q^{78} - q^{79} + q^{81} - 9q^{83} + 10q^{86} + 9q^{87} + 3q^{88} - 6q^{89} + q^{93} - 6q^{94} + q^{96} - q^{97} + 3q^{99} + O(q^{100})$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
1.00000 1.00000 1.00000 0 1.00000 0 1.00000 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$5$$ $$1$$
$$7$$ $$-1$$

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7350.2.a.cw 1
5.b even 2 1 294.2.a.a 1
7.b odd 2 1 7350.2.a.ce 1
7.d odd 6 2 1050.2.i.e 2
15.d odd 2 1 882.2.a.k 1
20.d odd 2 1 2352.2.a.n 1
35.c odd 2 1 294.2.a.d 1
35.i odd 6 2 42.2.e.b 2
35.j even 6 2 294.2.e.f 2
35.k even 12 4 1050.2.o.b 4
40.e odd 2 1 9408.2.a.bm 1
40.f even 2 1 9408.2.a.db 1
60.h even 2 1 7056.2.a.bz 1
105.g even 2 1 882.2.a.g 1
105.o odd 6 2 882.2.g.b 2
105.p even 6 2 126.2.g.b 2
140.c even 2 1 2352.2.a.m 1
140.p odd 6 2 2352.2.q.m 2
140.s even 6 2 336.2.q.d 2
280.c odd 2 1 9408.2.a.d 1
280.n even 2 1 9408.2.a.bu 1
280.ba even 6 2 1344.2.q.j 2
280.bk odd 6 2 1344.2.q.v 2
315.q odd 6 2 1134.2.e.a 2
315.u even 6 2 1134.2.h.a 2
315.bn odd 6 2 1134.2.h.p 2
315.bq even 6 2 1134.2.e.p 2
420.o odd 2 1 7056.2.a.g 1
420.be odd 6 2 1008.2.s.n 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.2.e.b 2 35.i odd 6 2
126.2.g.b 2 105.p even 6 2
294.2.a.a 1 5.b even 2 1
294.2.a.d 1 35.c odd 2 1
294.2.e.f 2 35.j even 6 2
336.2.q.d 2 140.s even 6 2
882.2.a.g 1 105.g even 2 1
882.2.a.k 1 15.d odd 2 1
882.2.g.b 2 105.o odd 6 2
1008.2.s.n 2 420.be odd 6 2
1050.2.i.e 2 7.d odd 6 2
1050.2.o.b 4 35.k even 12 4
1134.2.e.a 2 315.q odd 6 2
1134.2.e.p 2 315.bq even 6 2
1134.2.h.a 2 315.u even 6 2
1134.2.h.p 2 315.bn odd 6 2
1344.2.q.j 2 280.ba even 6 2
1344.2.q.v 2 280.bk odd 6 2
2352.2.a.m 1 140.c even 2 1
2352.2.a.n 1 20.d odd 2 1
2352.2.q.m 2 140.p odd 6 2
7056.2.a.g 1 420.o odd 2 1
7056.2.a.bz 1 60.h even 2 1
7350.2.a.ce 1 7.b odd 2 1
7350.2.a.cw 1 1.a even 1 1 trivial
9408.2.a.d 1 280.c odd 2 1
9408.2.a.bm 1 40.e odd 2 1
9408.2.a.bu 1 280.n even 2 1
9408.2.a.db 1 40.f even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(7350))$$:

 $$T_{11} - 3$$ $$T_{13} + 4$$ $$T_{17}$$ $$T_{19} - 4$$ $$T_{23}$$ $$T_{31} - 1$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T$$
$3$ $$1 - T$$
$5$ 1
$7$ 1
$11$ $$1 - 3 T + 11 T^{2}$$
$13$ $$1 + 4 T + 13 T^{2}$$
$17$ $$1 + 17 T^{2}$$
$19$ $$1 - 4 T + 19 T^{2}$$
$23$ $$1 + 23 T^{2}$$
$29$ $$1 - 9 T + 29 T^{2}$$
$31$ $$1 - T + 31 T^{2}$$
$37$ $$1 + 8 T + 37 T^{2}$$
$41$ $$1 + 41 T^{2}$$
$43$ $$1 - 10 T + 43 T^{2}$$
$47$ $$1 + 6 T + 47 T^{2}$$
$53$ $$1 - 3 T + 53 T^{2}$$
$59$ $$1 + 3 T + 59 T^{2}$$
$61$ $$1 - 10 T + 61 T^{2}$$
$67$ $$1 - 10 T + 67 T^{2}$$
$71$ $$1 + 6 T + 71 T^{2}$$
$73$ $$1 - 2 T + 73 T^{2}$$
$79$ $$1 + T + 79 T^{2}$$
$83$ $$1 + 9 T + 83 T^{2}$$
$89$ $$1 + 6 T + 89 T^{2}$$
$97$ $$1 + T + 97 T^{2}$$