# Properties

 Label 1575.4.a.z Level $1575$ Weight $4$ Character orbit 1575.a Self dual yes Analytic conductor $92.928$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1575 = 3^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1575.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$92.9280082590$$ 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.

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

## 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
2.58579 0 −1.31371 0 0 7.00000 −24.0833 0 0
1.2 5.41421 0 21.3137 0 0 7.00000 72.0833 0 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
$$3$$ $$-1$$
$$5$$ $$1$$
$$7$$ $$-1$$

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

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.4.a.b 2 15.d odd 2 1
175.4.a.c 2 3.b odd 2 1
175.4.b.c 4 15.e even 4 2
245.4.a.k 2 105.g even 2 1
245.4.e.h 4 105.o odd 6 2
245.4.e.i 4 105.p even 6 2
315.4.a.f 2 5.b even 2 1
560.4.a.r 2 60.h even 2 1
1225.4.a.m 2 21.c even 2 1
1575.4.a.z 2 1.a even 1 1 trivial
2205.4.a.u 2 35.c odd 2 1
2240.4.a.bn 2 120.i odd 2 1
2240.4.a.bo 2 120.m even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1575))$$:

 $$T_{2}^{2} - 8 T_{2} + 14$$ $$T_{11}^{2} - 14 T_{11} - 1999$$ $$T_{13}^{2} + 50 T_{13} + 593$$

## Hecke characteristic polynomials

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