# Properties

 Label 325.4.a.f Level $325$ Weight $4$ Character orbit 325.a Self dual yes Analytic conductor $19.176$ 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] = [325,4,Mod(1,325)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(325, 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("325.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$325 = 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 325.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: $$19.1756207519$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 4$$ x^2 - x - 4 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 13) 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 $$\beta = \frac{1}{2}(1 + \sqrt{17})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta q^{2} + (3 \beta - 4) q^{3} + (\beta - 4) q^{4} + (\beta - 12) q^{6} + ( - 11 \beta + 10) q^{7} + (11 \beta - 4) q^{8} + ( - 15 \beta + 25) q^{9} +O(q^{10})$$ q - b * q^2 + (3*b - 4) * q^3 + (b - 4) * q^4 + (b - 12) * q^6 + (-11*b + 10) * q^7 + (11*b - 4) * q^8 + (-15*b + 25) * q^9 $$q - \beta q^{2} + (3 \beta - 4) q^{3} + (\beta - 4) q^{4} + (\beta - 12) q^{6} + ( - 11 \beta + 10) q^{7} + (11 \beta - 4) q^{8} + ( - 15 \beta + 25) q^{9} + (12 \beta + 34) q^{11} + ( - 13 \beta + 28) q^{12} + 13 q^{13} + (\beta + 44) q^{14} + ( - 15 \beta - 12) q^{16} + (17 \beta - 18) q^{17} + ( - 10 \beta + 60) q^{18} + ( - 32 \beta - 26) q^{19} + (41 \beta - 172) q^{21} + ( - 46 \beta - 48) q^{22} + (12 \beta - 104) q^{23} + ( - 23 \beta + 148) q^{24} - 13 \beta q^{26} + (9 \beta - 172) q^{27} + (43 \beta - 84) q^{28} + (96 \beta - 70) q^{29} + ( - 34 \beta - 26) q^{31} + ( - 61 \beta + 92) q^{32} + (90 \beta + 8) q^{33} + (\beta - 68) q^{34} + (70 \beta - 160) q^{36} + ( - 5 \beta - 102) q^{37} + (58 \beta + 128) q^{38} + (39 \beta - 52) q^{39} + (22 \beta - 126) q^{41} + (131 \beta - 164) q^{42} + ( - 143 \beta - 72) q^{43} + ( - 2 \beta - 88) q^{44} + (92 \beta - 48) q^{46} + (121 \beta - 278) q^{47} + ( - 21 \beta - 132) q^{48} + ( - 99 \beta + 241) q^{49} + ( - 71 \beta + 276) q^{51} + (13 \beta - 52) q^{52} + ( - 30 \beta + 74) q^{53} + (163 \beta - 36) q^{54} + (33 \beta - 524) q^{56} + ( - 46 \beta - 280) q^{57} + ( - 26 \beta - 384) q^{58} + (124 \beta - 246) q^{59} + ( - 190 \beta - 434) q^{61} + (60 \beta + 136) q^{62} + ( - 260 \beta + 910) q^{63} + (89 \beta + 340) q^{64} + ( - 98 \beta - 360) q^{66} + (232 \beta - 150) q^{67} + ( - 69 \beta + 140) q^{68} + ( - 324 \beta + 560) q^{69} + ( - 231 \beta + 50) q^{71} + (170 \beta - 760) q^{72} + ( - 260 \beta - 98) q^{73} + (107 \beta + 20) q^{74} + (70 \beta - 24) q^{76} + ( - 386 \beta - 188) q^{77} + (13 \beta - 156) q^{78} + (40 \beta - 524) q^{79} + ( - 120 \beta + 121) q^{81} + (104 \beta - 88) q^{82} + (182 \beta - 1070) q^{83} + ( - 295 \beta + 852) q^{84} + (215 \beta + 572) q^{86} + ( - 306 \beta + 1432) q^{87} + (458 \beta + 392) q^{88} + ( - 388 \beta - 166) q^{89} + ( - 143 \beta + 130) q^{91} + ( - 140 \beta + 464) q^{92} + ( - 44 \beta - 304) q^{93} + (157 \beta - 484) q^{94} + (337 \beta - 1100) q^{96} + ( - 508 \beta + 718) q^{97} + ( - 142 \beta + 396) q^{98} + ( - 390 \beta + 130) q^{99} +O(q^{100})$$ q - b * q^2 + (3*b - 4) * q^3 + (b - 4) * q^4 + (b - 12) * q^6 + (-11*b + 10) * q^7 + (11*b - 4) * q^8 + (-15*b + 25) * q^9 + (12*b + 34) * q^11 + (-13*b + 28) * q^12 + 13 * q^13 + (b + 44) * q^14 + (-15*b - 12) * q^16 + (17*b - 18) * q^17 + (-10*b + 60) * q^18 + (-32*b - 26) * q^19 + (41*b - 172) * q^21 + (-46*b - 48) * q^22 + (12*b - 104) * q^23 + (-23*b + 148) * q^24 - 13*b * q^26 + (9*b - 172) * q^27 + (43*b - 84) * q^28 + (96*b - 70) * q^29 + (-34*b - 26) * q^31 + (-61*b + 92) * q^32 + (90*b + 8) * q^33 + (b - 68) * q^34 + (70*b - 160) * q^36 + (-5*b - 102) * q^37 + (58*b + 128) * q^38 + (39*b - 52) * q^39 + (22*b - 126) * q^41 + (131*b - 164) * q^42 + (-143*b - 72) * q^43 + (-2*b - 88) * q^44 + (92*b - 48) * q^46 + (121*b - 278) * q^47 + (-21*b - 132) * q^48 + (-99*b + 241) * q^49 + (-71*b + 276) * q^51 + (13*b - 52) * q^52 + (-30*b + 74) * q^53 + (163*b - 36) * q^54 + (33*b - 524) * q^56 + (-46*b - 280) * q^57 + (-26*b - 384) * q^58 + (124*b - 246) * q^59 + (-190*b - 434) * q^61 + (60*b + 136) * q^62 + (-260*b + 910) * q^63 + (89*b + 340) * q^64 + (-98*b - 360) * q^66 + (232*b - 150) * q^67 + (-69*b + 140) * q^68 + (-324*b + 560) * q^69 + (-231*b + 50) * q^71 + (170*b - 760) * q^72 + (-260*b - 98) * q^73 + (107*b + 20) * q^74 + (70*b - 24) * q^76 + (-386*b - 188) * q^77 + (13*b - 156) * q^78 + (40*b - 524) * q^79 + (-120*b + 121) * q^81 + (104*b - 88) * q^82 + (182*b - 1070) * q^83 + (-295*b + 852) * q^84 + (215*b + 572) * q^86 + (-306*b + 1432) * q^87 + (458*b + 392) * q^88 + (-388*b - 166) * q^89 + (-143*b + 130) * q^91 + (-140*b + 464) * q^92 + (-44*b - 304) * q^93 + (157*b - 484) * q^94 + (337*b - 1100) * q^96 + (-508*b + 718) * q^97 + (-142*b + 396) * q^98 + (-390*b + 130) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} - 5 q^{3} - 7 q^{4} - 23 q^{6} + 9 q^{7} + 3 q^{8} + 35 q^{9}+O(q^{10})$$ 2 * q - q^2 - 5 * q^3 - 7 * q^4 - 23 * q^6 + 9 * q^7 + 3 * q^8 + 35 * q^9 $$2 q - q^{2} - 5 q^{3} - 7 q^{4} - 23 q^{6} + 9 q^{7} + 3 q^{8} + 35 q^{9} + 80 q^{11} + 43 q^{12} + 26 q^{13} + 89 q^{14} - 39 q^{16} - 19 q^{17} + 110 q^{18} - 84 q^{19} - 303 q^{21} - 142 q^{22} - 196 q^{23} + 273 q^{24} - 13 q^{26} - 335 q^{27} - 125 q^{28} - 44 q^{29} - 86 q^{31} + 123 q^{32} + 106 q^{33} - 135 q^{34} - 250 q^{36} - 209 q^{37} + 314 q^{38} - 65 q^{39} - 230 q^{41} - 197 q^{42} - 287 q^{43} - 178 q^{44} - 4 q^{46} - 435 q^{47} - 285 q^{48} + 383 q^{49} + 481 q^{51} - 91 q^{52} + 118 q^{53} + 91 q^{54} - 1015 q^{56} - 606 q^{57} - 794 q^{58} - 368 q^{59} - 1058 q^{61} + 332 q^{62} + 1560 q^{63} + 769 q^{64} - 818 q^{66} - 68 q^{67} + 211 q^{68} + 796 q^{69} - 131 q^{71} - 1350 q^{72} - 456 q^{73} + 147 q^{74} + 22 q^{76} - 762 q^{77} - 299 q^{78} - 1008 q^{79} + 122 q^{81} - 72 q^{82} - 1958 q^{83} + 1409 q^{84} + 1359 q^{86} + 2558 q^{87} + 1242 q^{88} - 720 q^{89} + 117 q^{91} + 788 q^{92} - 652 q^{93} - 811 q^{94} - 1863 q^{96} + 928 q^{97} + 650 q^{98} - 130 q^{99}+O(q^{100})$$ 2 * q - q^2 - 5 * q^3 - 7 * q^4 - 23 * q^6 + 9 * q^7 + 3 * q^8 + 35 * q^9 + 80 * q^11 + 43 * q^12 + 26 * q^13 + 89 * q^14 - 39 * q^16 - 19 * q^17 + 110 * q^18 - 84 * q^19 - 303 * q^21 - 142 * q^22 - 196 * q^23 + 273 * q^24 - 13 * q^26 - 335 * q^27 - 125 * q^28 - 44 * q^29 - 86 * q^31 + 123 * q^32 + 106 * q^33 - 135 * q^34 - 250 * q^36 - 209 * q^37 + 314 * q^38 - 65 * q^39 - 230 * q^41 - 197 * q^42 - 287 * q^43 - 178 * q^44 - 4 * q^46 - 435 * q^47 - 285 * q^48 + 383 * q^49 + 481 * q^51 - 91 * q^52 + 118 * q^53 + 91 * q^54 - 1015 * q^56 - 606 * q^57 - 794 * q^58 - 368 * q^59 - 1058 * q^61 + 332 * q^62 + 1560 * q^63 + 769 * q^64 - 818 * q^66 - 68 * q^67 + 211 * q^68 + 796 * q^69 - 131 * q^71 - 1350 * q^72 - 456 * q^73 + 147 * q^74 + 22 * q^76 - 762 * q^77 - 299 * q^78 - 1008 * q^79 + 122 * q^81 - 72 * q^82 - 1958 * q^83 + 1409 * q^84 + 1359 * q^86 + 2558 * q^87 + 1242 * q^88 - 720 * q^89 + 117 * q^91 + 788 * q^92 - 652 * q^93 - 811 * q^94 - 1863 * q^96 + 928 * q^97 + 650 * q^98 - 130 * 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
 2.56155 −1.56155
−2.56155 3.68466 −1.43845 0 −9.43845 −18.1771 24.1771 −13.4233 0
1.2 1.56155 −8.68466 −5.56155 0 −13.5616 27.1771 −21.1771 48.4233 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
$$5$$ $$+1$$
$$13$$ $$-1$$

## 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 325.4.a.f 2
5.b even 2 1 13.4.a.b 2
5.c odd 4 2 325.4.b.e 4
15.d odd 2 1 117.4.a.d 2
20.d odd 2 1 208.4.a.h 2
35.c odd 2 1 637.4.a.b 2
40.e odd 2 1 832.4.a.z 2
40.f even 2 1 832.4.a.s 2
55.d odd 2 1 1573.4.a.b 2
60.h even 2 1 1872.4.a.bb 2
65.d even 2 1 169.4.a.g 2
65.g odd 4 2 169.4.b.f 4
65.l even 6 2 169.4.c.j 4
65.n even 6 2 169.4.c.g 4
65.s odd 12 4 169.4.e.f 8
195.e odd 2 1 1521.4.a.r 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.a.b 2 5.b even 2 1
117.4.a.d 2 15.d odd 2 1
169.4.a.g 2 65.d even 2 1
169.4.b.f 4 65.g odd 4 2
169.4.c.g 4 65.n even 6 2
169.4.c.j 4 65.l even 6 2
169.4.e.f 8 65.s odd 12 4
208.4.a.h 2 20.d odd 2 1
325.4.a.f 2 1.a even 1 1 trivial
325.4.b.e 4 5.c odd 4 2
637.4.a.b 2 35.c odd 2 1
832.4.a.s 2 40.f even 2 1
832.4.a.z 2 40.e odd 2 1
1521.4.a.r 2 195.e odd 2 1
1573.4.a.b 2 55.d odd 2 1
1872.4.a.bb 2 60.h even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(325))$$:

 $$T_{2}^{2} + T_{2} - 4$$ T2^2 + T2 - 4 $$T_{3}^{2} + 5T_{3} - 32$$ T3^2 + 5*T3 - 32

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T - 4$$
$3$ $$T^{2} + 5T - 32$$
$5$ $$T^{2}$$
$7$ $$T^{2} - 9T - 494$$
$11$ $$T^{2} - 80T + 988$$
$13$ $$(T - 13)^{2}$$
$17$ $$T^{2} + 19T - 1138$$
$19$ $$T^{2} + 84T - 2588$$
$23$ $$T^{2} + 196T + 8992$$
$29$ $$T^{2} + 44T - 38684$$
$31$ $$T^{2} + 86T - 3064$$
$37$ $$T^{2} + 209T + 10814$$
$41$ $$T^{2} + 230T + 11168$$
$43$ $$T^{2} + 287T - 66316$$
$47$ $$T^{2} + 435T - 14918$$
$53$ $$T^{2} - 118T - 344$$
$59$ $$T^{2} + 368T - 31492$$
$61$ $$T^{2} + 1058 T + 126416$$
$67$ $$T^{2} + 68T - 227596$$
$71$ $$T^{2} + 131T - 222494$$
$73$ $$T^{2} + 456T - 235316$$
$79$ $$T^{2} + 1008 T + 247216$$
$83$ $$T^{2} + 1958 T + 817664$$
$89$ $$T^{2} + 720T - 510212$$
$97$ $$T^{2} - 928T - 881476$$