# Properties

 Label 1225.4.a.m Level $1225$ Weight $4$ Character orbit 1225.a Self dual yes Analytic conductor $72.277$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1225 = 5^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1225.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$72.2773397570$$ Analytic rank: $$1$$ 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.

 $$f(q)$$ $$=$$ $$q + ( -4 + \beta ) q^{2} + ( 1 + 4 \beta ) q^{3} + ( 10 - 8 \beta ) q^{4} + ( 4 - 15 \beta ) q^{6} + ( -24 + 34 \beta ) q^{8} + ( 6 + 8 \beta ) q^{9} +O(q^{10})$$ $$q + ( -4 + \beta ) q^{2} + ( 1 + 4 \beta ) q^{3} + ( 10 - 8 \beta ) q^{4} + ( 4 - 15 \beta ) q^{6} + ( -24 + 34 \beta ) q^{8} + ( 6 + 8 \beta ) q^{9} + ( -7 + 32 \beta ) q^{11} + ( -54 + 32 \beta ) q^{12} + ( 25 - 4 \beta ) q^{13} + ( 84 - 96 \beta ) q^{16} + ( -25 - 44 \beta ) q^{17} + ( -8 - 26 \beta ) q^{18} + ( -18 + 44 \beta ) q^{19} + ( 92 - 135 \beta ) q^{22} + ( -122 - 68 \beta ) q^{23} + ( 248 - 62 \beta ) q^{24} + ( -108 + 41 \beta ) q^{26} + ( 43 - 76 \beta ) q^{27} + ( -13 - 24 \beta ) q^{29} + ( 60 - 180 \beta ) q^{31} + ( -336 + 196 \beta ) q^{32} + ( 249 + 4 \beta ) q^{33} + ( 12 + 151 \beta ) q^{34} + ( -68 + 32 \beta ) q^{36} + ( -282 - 60 \beta ) q^{37} + ( 160 - 194 \beta ) q^{38} + ( -7 + 96 \beta ) q^{39} + ( 164 + 124 \beta ) q^{41} + ( 130 + 68 \beta ) q^{43} + ( -582 + 376 \beta ) q^{44} + ( 352 + 150 \beta ) q^{46} + ( -175 + 132 \beta ) q^{47} + ( -684 + 240 \beta ) q^{48} + ( -377 - 144 \beta ) q^{51} + ( 314 - 240 \beta ) q^{52} + ( 28 + 128 \beta ) q^{53} + ( -324 + 347 \beta ) q^{54} + ( 334 - 28 \beta ) q^{57} + ( 4 + 83 \beta ) q^{58} + 616 q^{59} + ( -168 - 108 \beta ) q^{61} + ( -600 + 780 \beta ) q^{62} + ( 1064 - 352 \beta ) q^{64} + ( -988 + 233 \beta ) q^{66} + ( 76 - 64 \beta ) q^{67} + ( 454 - 240 \beta ) q^{68} + ( -666 - 556 \beta ) q^{69} -952 q^{71} + ( 400 + 12 \beta ) q^{72} + ( 338 + 344 \beta ) q^{73} + ( 1008 - 42 \beta ) q^{74} + ( -884 + 584 \beta ) q^{76} + ( 220 - 391 \beta ) q^{78} + ( 507 - 248 \beta ) q^{79} + ( -727 - 120 \beta ) q^{81} + ( -408 - 332 \beta ) q^{82} + ( -188 - 600 \beta ) q^{83} + ( -384 - 142 \beta ) q^{86} + ( -205 - 76 \beta ) q^{87} + ( 2344 - 1006 \beta ) q^{88} + ( 108 + 44 \beta ) q^{89} + ( -132 + 296 \beta ) q^{92} + ( -1380 + 60 \beta ) q^{93} + ( 964 - 703 \beta ) q^{94} + ( 1232 - 1148 \beta ) q^{96} + ( 1371 - 220 \beta ) q^{97} + ( 470 + 136 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 8 q^{2} + 2 q^{3} + 20 q^{4} + 8 q^{6} - 48 q^{8} + 12 q^{9} + O(q^{10})$$ $$2 q - 8 q^{2} + 2 q^{3} + 20 q^{4} + 8 q^{6} - 48 q^{8} + 12 q^{9} - 14 q^{11} - 108 q^{12} + 50 q^{13} + 168 q^{16} - 50 q^{17} - 16 q^{18} - 36 q^{19} + 184 q^{22} - 244 q^{23} + 496 q^{24} - 216 q^{26} + 86 q^{27} - 26 q^{29} + 120 q^{31} - 672 q^{32} + 498 q^{33} + 24 q^{34} - 136 q^{36} - 564 q^{37} + 320 q^{38} - 14 q^{39} + 328 q^{41} + 260 q^{43} - 1164 q^{44} + 704 q^{46} - 350 q^{47} - 1368 q^{48} - 754 q^{51} + 628 q^{52} + 56 q^{53} - 648 q^{54} + 668 q^{57} + 8 q^{58} + 1232 q^{59} - 336 q^{61} - 1200 q^{62} + 2128 q^{64} - 1976 q^{66} + 152 q^{67} + 908 q^{68} - 1332 q^{69} - 1904 q^{71} + 800 q^{72} + 676 q^{73} + 2016 q^{74} - 1768 q^{76} + 440 q^{78} + 1014 q^{79} - 1454 q^{81} - 816 q^{82} - 376 q^{83} - 768 q^{86} - 410 q^{87} + 4688 q^{88} + 216 q^{89} - 264 q^{92} - 2760 q^{93} + 1928 q^{94} + 2464 q^{96} + 2742 q^{97} + 940 q^{99} + O(q^{100})$$

## 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
 −1.41421 1.41421
−5.41421 −4.65685 21.3137 0 25.2132 0 −72.0833 −5.31371 0
1.2 −2.58579 6.65685 −1.31371 0 −17.2132 0 24.0833 17.3137 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
$$5$$ $$1$$
$$7$$ $$-1$$

## 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 1225.4.a.m 2
5.b even 2 1 245.4.a.k 2
7.b odd 2 1 175.4.a.c 2
15.d odd 2 1 2205.4.a.u 2
21.c even 2 1 1575.4.a.z 2
35.c odd 2 1 35.4.a.b 2
35.f even 4 2 175.4.b.c 4
35.i odd 6 2 245.4.e.h 4
35.j even 6 2 245.4.e.i 4
105.g even 2 1 315.4.a.f 2
140.c even 2 1 560.4.a.r 2
280.c odd 2 1 2240.4.a.bn 2
280.n 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 35.c odd 2 1
175.4.a.c 2 7.b odd 2 1
175.4.b.c 4 35.f even 4 2
245.4.a.k 2 5.b even 2 1
245.4.e.h 4 35.i odd 6 2
245.4.e.i 4 35.j even 6 2
315.4.a.f 2 105.g even 2 1
560.4.a.r 2 140.c even 2 1
1225.4.a.m 2 1.a even 1 1 trivial
1575.4.a.z 2 21.c even 2 1
2205.4.a.u 2 15.d odd 2 1
2240.4.a.bn 2 280.c odd 2 1
2240.4.a.bo 2 280.n even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1225))$$:

 $$T_{2}^{2} + 8 T_{2} + 14$$ $$T_{3}^{2} - 2 T_{3} - 31$$ $$T_{19}^{2} + 36 T_{19} - 3548$$

## Hecke characteristic polynomials

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