# Properties

 Label 315.4.a.f Level $315$ Weight $4$ Character orbit 315.a Self dual yes Analytic conductor $18.586$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$315 = 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 315.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$18.5856016518$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 35) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -4 + \beta ) q^{2} + ( 10 - 8 \beta ) q^{4} + 5 q^{5} -7 q^{7} + ( -24 + 34 \beta ) q^{8} +O(q^{10})$$ $$q + ( -4 + \beta ) q^{2} + ( 10 - 8 \beta ) q^{4} + 5 q^{5} -7 q^{7} + ( -24 + 34 \beta ) q^{8} + ( -20 + 5 \beta ) q^{10} + ( 7 - 32 \beta ) q^{11} + ( 25 - 4 \beta ) q^{13} + ( 28 - 7 \beta ) q^{14} + ( 84 - 96 \beta ) q^{16} + ( 25 + 44 \beta ) q^{17} + ( 18 - 44 \beta ) q^{19} + ( 50 - 40 \beta ) q^{20} + ( -92 + 135 \beta ) q^{22} + ( -122 - 68 \beta ) q^{23} + 25 q^{25} + ( -108 + 41 \beta ) q^{26} + ( -70 + 56 \beta ) q^{28} + ( 13 + 24 \beta ) q^{29} + ( -60 + 180 \beta ) q^{31} + ( -336 + 196 \beta ) q^{32} + ( -12 - 151 \beta ) q^{34} -35 q^{35} + ( 282 + 60 \beta ) q^{37} + ( -160 + 194 \beta ) q^{38} + ( -120 + 170 \beta ) q^{40} + ( 164 + 124 \beta ) q^{41} + ( -130 - 68 \beta ) q^{43} + ( 582 - 376 \beta ) q^{44} + ( 352 + 150 \beta ) q^{46} + ( 175 - 132 \beta ) q^{47} + 49 q^{49} + ( -100 + 25 \beta ) q^{50} + ( 314 - 240 \beta ) q^{52} + ( 28 + 128 \beta ) q^{53} + ( 35 - 160 \beta ) q^{55} + ( 168 - 238 \beta ) q^{56} + ( -4 - 83 \beta ) q^{58} + 616 q^{59} + ( 168 + 108 \beta ) q^{61} + ( 600 - 780 \beta ) q^{62} + ( 1064 - 352 \beta ) q^{64} + ( 125 - 20 \beta ) q^{65} + ( -76 + 64 \beta ) q^{67} + ( -454 + 240 \beta ) q^{68} + ( 140 - 35 \beta ) q^{70} + 952 q^{71} + ( 338 + 344 \beta ) q^{73} + ( -1008 + 42 \beta ) q^{74} + ( 884 - 584 \beta ) q^{76} + ( -49 + 224 \beta ) q^{77} + ( 507 - 248 \beta ) q^{79} + ( 420 - 480 \beta ) q^{80} + ( -408 - 332 \beta ) q^{82} + ( 188 + 600 \beta ) q^{83} + ( 125 + 220 \beta ) q^{85} + ( 384 + 142 \beta ) q^{86} + ( -2344 + 1006 \beta ) q^{88} + ( 108 + 44 \beta ) q^{89} + ( -175 + 28 \beta ) q^{91} + ( -132 + 296 \beta ) q^{92} + ( -964 + 703 \beta ) q^{94} + ( 90 - 220 \beta ) q^{95} + ( 1371 - 220 \beta ) q^{97} + ( -196 + 49 \beta ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 8 q^{2} + 20 q^{4} + 10 q^{5} - 14 q^{7} - 48 q^{8} + O(q^{10})$$ $$2 q - 8 q^{2} + 20 q^{4} + 10 q^{5} - 14 q^{7} - 48 q^{8} - 40 q^{10} + 14 q^{11} + 50 q^{13} + 56 q^{14} + 168 q^{16} + 50 q^{17} + 36 q^{19} + 100 q^{20} - 184 q^{22} - 244 q^{23} + 50 q^{25} - 216 q^{26} - 140 q^{28} + 26 q^{29} - 120 q^{31} - 672 q^{32} - 24 q^{34} - 70 q^{35} + 564 q^{37} - 320 q^{38} - 240 q^{40} + 328 q^{41} - 260 q^{43} + 1164 q^{44} + 704 q^{46} + 350 q^{47} + 98 q^{49} - 200 q^{50} + 628 q^{52} + 56 q^{53} + 70 q^{55} + 336 q^{56} - 8 q^{58} + 1232 q^{59} + 336 q^{61} + 1200 q^{62} + 2128 q^{64} + 250 q^{65} - 152 q^{67} - 908 q^{68} + 280 q^{70} + 1904 q^{71} + 676 q^{73} - 2016 q^{74} + 1768 q^{76} - 98 q^{77} + 1014 q^{79} + 840 q^{80} - 816 q^{82} + 376 q^{83} + 250 q^{85} + 768 q^{86} - 4688 q^{88} + 216 q^{89} - 350 q^{91} - 264 q^{92} - 1928 q^{94} + 180 q^{95} + 2742 q^{97} - 392 q^{98} + O(q^{100})$$

## 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
 −1.41421 1.41421
−5.41421 0 21.3137 5.00000 0 −7.00000 −72.0833 0 −27.0711
1.2 −2.58579 0 −1.31371 5.00000 0 −7.00000 24.0833 0 −12.9289
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

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

## 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 315.4.a.f 2
3.b odd 2 1 35.4.a.b 2
5.b even 2 1 1575.4.a.z 2
7.b odd 2 1 2205.4.a.u 2
12.b even 2 1 560.4.a.r 2
15.d odd 2 1 175.4.a.c 2
15.e even 4 2 175.4.b.c 4
21.c even 2 1 245.4.a.k 2
21.g even 6 2 245.4.e.i 4
21.h odd 6 2 245.4.e.h 4
24.f even 2 1 2240.4.a.bo 2
24.h odd 2 1 2240.4.a.bn 2
105.g even 2 1 1225.4.a.m 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.4.a.b 2 3.b odd 2 1
175.4.a.c 2 15.d odd 2 1
175.4.b.c 4 15.e even 4 2
245.4.a.k 2 21.c even 2 1
245.4.e.h 4 21.h odd 6 2
245.4.e.i 4 21.g even 6 2
315.4.a.f 2 1.a even 1 1 trivial
560.4.a.r 2 12.b even 2 1
1225.4.a.m 2 105.g even 2 1
1575.4.a.z 2 5.b even 2 1
2205.4.a.u 2 7.b odd 2 1
2240.4.a.bn 2 24.h odd 2 1
2240.4.a.bo 2 24.f even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 8 T_{2} + 14$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(315))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$14 + 8 T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$( -5 + T )^{2}$$
$7$ $$( 7 + T )^{2}$$
$11$ $$-1999 - 14 T + T^{2}$$
$13$ $$593 - 50 T + T^{2}$$
$17$ $$-3247 - 50 T + T^{2}$$
$19$ $$-3548 - 36 T + T^{2}$$
$23$ $$5636 + 244 T + T^{2}$$
$29$ $$-983 - 26 T + T^{2}$$
$31$ $$-61200 + 120 T + T^{2}$$
$37$ $$72324 - 564 T + T^{2}$$
$41$ $$-3856 - 328 T + T^{2}$$
$43$ $$7652 + 260 T + T^{2}$$
$47$ $$-4223 - 350 T + T^{2}$$
$53$ $$-31984 - 56 T + T^{2}$$
$59$ $$( -616 + T )^{2}$$
$61$ $$4896 - 336 T + T^{2}$$
$67$ $$-2416 + 152 T + T^{2}$$
$71$ $$( -952 + T )^{2}$$
$73$ $$-122428 - 676 T + T^{2}$$
$79$ $$134041 - 1014 T + T^{2}$$
$83$ $$-684656 - 376 T + T^{2}$$
$89$ $$7792 - 216 T + T^{2}$$
$97$ $$1782841 - 2742 T + T^{2}$$