# 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

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1225,4,Mod(1,1225)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1225, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1225.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

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

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$72.2773397570$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ 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

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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

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

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

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 $$S_{4}^{\mathrm{new}}(\Gamma_0(1225))$$:

 $$T_{2}^{2} + 8T_{2} + 14$$ T2^2 + 8*T2 + 14 $$T_{3}^{2} - 2T_{3} - 31$$ T3^2 - 2*T3 - 31 $$T_{19}^{2} + 36T_{19} - 3548$$ T19^2 + 36*T19 - 3548

## Hecke characteristic polynomials

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