# Properties

 Label 425.4.b.c Level $425$ Weight $4$ Character orbit 425.b Analytic conductor $25.076$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [425,4,Mod(324,425)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(425, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

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

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

chi := DirichletCharacter("425.324");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$425 = 5^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 425.b (of order $$2$$, degree $$1$$, not minimal)

## 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: no Analytic conductor: $$25.0758117524$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 17) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 3 i q^{2} - 8 i q^{3} - q^{4} + 24 q^{6} + 28 i q^{7} + 21 i q^{8} - 37 q^{9} +O(q^{10})$$ q + 3*i * q^2 - 8*i * q^3 - q^4 + 24 * q^6 + 28*i * q^7 + 21*i * q^8 - 37 * q^9 $$q + 3 i q^{2} - 8 i q^{3} - q^{4} + 24 q^{6} + 28 i q^{7} + 21 i q^{8} - 37 q^{9} - 24 q^{11} + 8 i q^{12} - 58 i q^{13} - 84 q^{14} - 71 q^{16} - 17 i q^{17} - 111 i q^{18} - 116 q^{19} + 224 q^{21} - 72 i q^{22} - 60 i q^{23} + 168 q^{24} + 174 q^{26} + 80 i q^{27} - 28 i q^{28} - 30 q^{29} - 172 q^{31} - 45 i q^{32} + 192 i q^{33} + 51 q^{34} + 37 q^{36} + 58 i q^{37} - 348 i q^{38} - 464 q^{39} - 342 q^{41} + 672 i q^{42} - 148 i q^{43} + 24 q^{44} + 180 q^{46} - 288 i q^{47} + 568 i q^{48} - 441 q^{49} - 136 q^{51} + 58 i q^{52} + 318 i q^{53} - 240 q^{54} - 588 q^{56} + 928 i q^{57} - 90 i q^{58} - 252 q^{59} + 110 q^{61} - 516 i q^{62} - 1036 i q^{63} - 433 q^{64} - 576 q^{66} + 484 i q^{67} + 17 i q^{68} - 480 q^{69} - 708 q^{71} - 777 i q^{72} + 362 i q^{73} - 174 q^{74} + 116 q^{76} - 672 i q^{77} - 1392 i q^{78} + 484 q^{79} - 359 q^{81} - 1026 i q^{82} + 756 i q^{83} - 224 q^{84} + 444 q^{86} + 240 i q^{87} - 504 i q^{88} + 774 q^{89} + 1624 q^{91} + 60 i q^{92} + 1376 i q^{93} + 864 q^{94} - 360 q^{96} + 382 i q^{97} - 1323 i q^{98} + 888 q^{99} +O(q^{100})$$ q + 3*i * q^2 - 8*i * q^3 - q^4 + 24 * q^6 + 28*i * q^7 + 21*i * q^8 - 37 * q^9 - 24 * q^11 + 8*i * q^12 - 58*i * q^13 - 84 * q^14 - 71 * q^16 - 17*i * q^17 - 111*i * q^18 - 116 * q^19 + 224 * q^21 - 72*i * q^22 - 60*i * q^23 + 168 * q^24 + 174 * q^26 + 80*i * q^27 - 28*i * q^28 - 30 * q^29 - 172 * q^31 - 45*i * q^32 + 192*i * q^33 + 51 * q^34 + 37 * q^36 + 58*i * q^37 - 348*i * q^38 - 464 * q^39 - 342 * q^41 + 672*i * q^42 - 148*i * q^43 + 24 * q^44 + 180 * q^46 - 288*i * q^47 + 568*i * q^48 - 441 * q^49 - 136 * q^51 + 58*i * q^52 + 318*i * q^53 - 240 * q^54 - 588 * q^56 + 928*i * q^57 - 90*i * q^58 - 252 * q^59 + 110 * q^61 - 516*i * q^62 - 1036*i * q^63 - 433 * q^64 - 576 * q^66 + 484*i * q^67 + 17*i * q^68 - 480 * q^69 - 708 * q^71 - 777*i * q^72 + 362*i * q^73 - 174 * q^74 + 116 * q^76 - 672*i * q^77 - 1392*i * q^78 + 484 * q^79 - 359 * q^81 - 1026*i * q^82 + 756*i * q^83 - 224 * q^84 + 444 * q^86 + 240*i * q^87 - 504*i * q^88 + 774 * q^89 + 1624 * q^91 + 60*i * q^92 + 1376*i * q^93 + 864 * q^94 - 360 * q^96 + 382*i * q^97 - 1323*i * q^98 + 888 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} + 48 q^{6} - 74 q^{9}+O(q^{10})$$ 2 * q - 2 * q^4 + 48 * q^6 - 74 * q^9 $$2 q - 2 q^{4} + 48 q^{6} - 74 q^{9} - 48 q^{11} - 168 q^{14} - 142 q^{16} - 232 q^{19} + 448 q^{21} + 336 q^{24} + 348 q^{26} - 60 q^{29} - 344 q^{31} + 102 q^{34} + 74 q^{36} - 928 q^{39} - 684 q^{41} + 48 q^{44} + 360 q^{46} - 882 q^{49} - 272 q^{51} - 480 q^{54} - 1176 q^{56} - 504 q^{59} + 220 q^{61} - 866 q^{64} - 1152 q^{66} - 960 q^{69} - 1416 q^{71} - 348 q^{74} + 232 q^{76} + 968 q^{79} - 718 q^{81} - 448 q^{84} + 888 q^{86} + 1548 q^{89} + 3248 q^{91} + 1728 q^{94} - 720 q^{96} + 1776 q^{99}+O(q^{100})$$ 2 * q - 2 * q^4 + 48 * q^6 - 74 * q^9 - 48 * q^11 - 168 * q^14 - 142 * q^16 - 232 * q^19 + 448 * q^21 + 336 * q^24 + 348 * q^26 - 60 * q^29 - 344 * q^31 + 102 * q^34 + 74 * q^36 - 928 * q^39 - 684 * q^41 + 48 * q^44 + 360 * q^46 - 882 * q^49 - 272 * q^51 - 480 * q^54 - 1176 * q^56 - 504 * q^59 + 220 * q^61 - 866 * q^64 - 1152 * q^66 - 960 * q^69 - 1416 * q^71 - 348 * q^74 + 232 * q^76 + 968 * q^79 - 718 * q^81 - 448 * q^84 + 888 * q^86 + 1548 * q^89 + 3248 * q^91 + 1728 * q^94 - 720 * q^96 + 1776 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/425\mathbb{Z}\right)^\times$$.

 $$n$$ $$52$$ $$326$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
324.1
 − 1.00000i 1.00000i
3.00000i 8.00000i −1.00000 0 24.0000 28.0000i 21.0000i −37.0000 0
324.2 3.00000i 8.00000i −1.00000 0 24.0000 28.0000i 21.0000i −37.0000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 425.4.b.c 2
5.b even 2 1 inner 425.4.b.c 2
5.c odd 4 1 17.4.a.a 1
5.c odd 4 1 425.4.a.d 1
15.e even 4 1 153.4.a.d 1
20.e even 4 1 272.4.a.d 1
35.f even 4 1 833.4.a.a 1
40.i odd 4 1 1088.4.a.l 1
40.k even 4 1 1088.4.a.a 1
55.e even 4 1 2057.4.a.d 1
60.l odd 4 1 2448.4.a.f 1
85.f odd 4 1 289.4.b.a 2
85.g odd 4 1 289.4.a.a 1
85.i odd 4 1 289.4.b.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.4.a.a 1 5.c odd 4 1
153.4.a.d 1 15.e even 4 1
272.4.a.d 1 20.e even 4 1
289.4.a.a 1 85.g odd 4 1
289.4.b.a 2 85.f odd 4 1
289.4.b.a 2 85.i odd 4 1
425.4.a.d 1 5.c odd 4 1
425.4.b.c 2 1.a even 1 1 trivial
425.4.b.c 2 5.b even 2 1 inner
833.4.a.a 1 35.f even 4 1
1088.4.a.a 1 40.k even 4 1
1088.4.a.l 1 40.i odd 4 1
2057.4.a.d 1 55.e even 4 1
2448.4.a.f 1 60.l odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(425, [\chi])$$:

 $$T_{2}^{2} + 9$$ T2^2 + 9 $$T_{3}^{2} + 64$$ T3^2 + 64

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 9$$
$3$ $$T^{2} + 64$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 784$$
$11$ $$(T + 24)^{2}$$
$13$ $$T^{2} + 3364$$
$17$ $$T^{2} + 289$$
$19$ $$(T + 116)^{2}$$
$23$ $$T^{2} + 3600$$
$29$ $$(T + 30)^{2}$$
$31$ $$(T + 172)^{2}$$
$37$ $$T^{2} + 3364$$
$41$ $$(T + 342)^{2}$$
$43$ $$T^{2} + 21904$$
$47$ $$T^{2} + 82944$$
$53$ $$T^{2} + 101124$$
$59$ $$(T + 252)^{2}$$
$61$ $$(T - 110)^{2}$$
$67$ $$T^{2} + 234256$$
$71$ $$(T + 708)^{2}$$
$73$ $$T^{2} + 131044$$
$79$ $$(T - 484)^{2}$$
$83$ $$T^{2} + 571536$$
$89$ $$(T - 774)^{2}$$
$97$ $$T^{2} + 145924$$