# Properties

 Label 825.4.c.h Level $825$ Weight $4$ Character orbit 825.c Analytic conductor $48.677$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [825,4,Mod(199,825)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(825, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("825.199");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$825 = 3 \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 825.c (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: $$48.6765757547$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{97})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 49x^{2} + 576$$ x^4 + 49*x^2 + 576 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 33) 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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} - 3 \beta_{2} q^{3} + (\beta_{3} - 17) q^{4} + (3 \beta_{3} - 3) q^{6} + ( - 14 \beta_{2} - 4 \beta_1) q^{7} + (24 \beta_{2} - 9 \beta_1) q^{8} - 9 q^{9}+O(q^{10})$$ q + b1 * q^2 - 3*b2 * q^3 + (b3 - 17) * q^4 + (3*b3 - 3) * q^6 + (-14*b2 - 4*b1) * q^7 + (24*b2 - 9*b1) * q^8 - 9 * q^9 $$q + \beta_1 q^{2} - 3 \beta_{2} q^{3} + (\beta_{3} - 17) q^{4} + (3 \beta_{3} - 3) q^{6} + ( - 14 \beta_{2} - 4 \beta_1) q^{7} + (24 \beta_{2} - 9 \beta_1) q^{8} - 9 q^{9} - 11 q^{11} + (48 \beta_{2} - 3 \beta_1) q^{12} + (14 \beta_{2} - 2 \beta_1) q^{13} + (10 \beta_{3} + 86) q^{14} + ( - 25 \beta_{3} + 113) q^{16} + ( - 60 \beta_{2} - 14 \beta_1) q^{17} - 9 \beta_1 q^{18} + ( - 2 \beta_{3} - 24) q^{19} + ( - 12 \beta_{3} - 30) q^{21} - 11 \beta_1 q^{22} + (72 \beta_{2} + 10 \beta_1) q^{23} + ( - 27 \beta_{3} + 99) q^{24} + ( - 16 \beta_{3} + 64) q^{26} + 27 \beta_{2} q^{27} + (128 \beta_{2} + 54 \beta_1) q^{28} + ( - 6 \beta_{3} + 102) q^{29} + ( - 8 \beta_{3} + 184) q^{31} + ( - 408 \beta_{2} + 41 \beta_1) q^{32} + 33 \beta_{2} q^{33} + (46 \beta_{3} + 290) q^{34} + ( - 9 \beta_{3} + 153) q^{36} + (190 \beta_{2} + 52 \beta_1) q^{37} + ( - 48 \beta_{2} - 24 \beta_1) q^{38} + ( - 6 \beta_{3} + 48) q^{39} + (14 \beta_{3} - 398) q^{41} + ( - 288 \beta_{2} - 30 \beta_1) q^{42} + (206 \beta_{2} + 26 \beta_1) q^{43} + ( - 11 \beta_{3} + 187) q^{44} + ( - 62 \beta_{3} - 178) q^{46} + ( - 96 \beta_{2} + 74 \beta_1) q^{47} + ( - 264 \beta_{2} + 75 \beta_1) q^{48} + ( - 96 \beta_{3} - 141) q^{49} + ( - 42 \beta_{3} - 138) q^{51} + ( - 272 \beta_{2} + 48 \beta_1) q^{52} + ( - 234 \beta_{2} + 54 \beta_1) q^{53} + ( - 27 \beta_{3} + 27) q^{54} + (6 \beta_{3} - 534) q^{56} + (78 \beta_{2} + 6 \beta_1) q^{57} + ( - 144 \beta_{2} + 102 \beta_1) q^{58} + ( - 100 \beta_{3} + 136) q^{59} + ( - 34 \beta_{3} - 372) q^{61} + ( - 192 \beta_{2} + 184 \beta_1) q^{62} + (126 \beta_{2} + 36 \beta_1) q^{63} + (249 \beta_{3} - 529) q^{64} + ( - 33 \beta_{3} + 33) q^{66} + (340 \beta_{2} - 96 \beta_1) q^{67} + (624 \beta_{2} + 178 \beta_1) q^{68} + (30 \beta_{3} + 186) q^{69} + ( - 54 \beta_{3} + 342) q^{71} + ( - 216 \beta_{2} + 81 \beta_1) q^{72} + (662 \beta_{2} + 28 \beta_1) q^{73} + ( - 138 \beta_{3} - 1110) q^{74} + (8 \beta_{3} + 360) q^{76} + (154 \beta_{2} + 44 \beta_1) q^{77} + ( - 144 \beta_{2} + 48 \beta_1) q^{78} + (144 \beta_{3} - 398) q^{79} + 81 q^{81} + (336 \beta_{2} - 398 \beta_1) q^{82} + ( - 240 \beta_{2} - 156 \beta_1) q^{83} + (162 \beta_{3} + 222) q^{84} + ( - 180 \beta_{3} - 444) q^{86} + ( - 288 \beta_{2} + 18 \beta_1) q^{87} + ( - 264 \beta_{2} + 99 \beta_1) q^{88} + (72 \beta_{3} + 342) q^{89} + (36 \beta_{3} - 32) q^{91} + ( - 912 \beta_{2} - 98 \beta_1) q^{92} + ( - 528 \beta_{2} + 24 \beta_1) q^{93} + (170 \beta_{3} - 1946) q^{94} + (123 \beta_{3} - 1347) q^{96} + (322 \beta_{2} + 192 \beta_1) q^{97} + ( - 2304 \beta_{2} - 141 \beta_1) q^{98} + 99 q^{99}+O(q^{100})$$ q + b1 * q^2 - 3*b2 * q^3 + (b3 - 17) * q^4 + (3*b3 - 3) * q^6 + (-14*b2 - 4*b1) * q^7 + (24*b2 - 9*b1) * q^8 - 9 * q^9 - 11 * q^11 + (48*b2 - 3*b1) * q^12 + (14*b2 - 2*b1) * q^13 + (10*b3 + 86) * q^14 + (-25*b3 + 113) * q^16 + (-60*b2 - 14*b1) * q^17 - 9*b1 * q^18 + (-2*b3 - 24) * q^19 + (-12*b3 - 30) * q^21 - 11*b1 * q^22 + (72*b2 + 10*b1) * q^23 + (-27*b3 + 99) * q^24 + (-16*b3 + 64) * q^26 + 27*b2 * q^27 + (128*b2 + 54*b1) * q^28 + (-6*b3 + 102) * q^29 + (-8*b3 + 184) * q^31 + (-408*b2 + 41*b1) * q^32 + 33*b2 * q^33 + (46*b3 + 290) * q^34 + (-9*b3 + 153) * q^36 + (190*b2 + 52*b1) * q^37 + (-48*b2 - 24*b1) * q^38 + (-6*b3 + 48) * q^39 + (14*b3 - 398) * q^41 + (-288*b2 - 30*b1) * q^42 + (206*b2 + 26*b1) * q^43 + (-11*b3 + 187) * q^44 + (-62*b3 - 178) * q^46 + (-96*b2 + 74*b1) * q^47 + (-264*b2 + 75*b1) * q^48 + (-96*b3 - 141) * q^49 + (-42*b3 - 138) * q^51 + (-272*b2 + 48*b1) * q^52 + (-234*b2 + 54*b1) * q^53 + (-27*b3 + 27) * q^54 + (6*b3 - 534) * q^56 + (78*b2 + 6*b1) * q^57 + (-144*b2 + 102*b1) * q^58 + (-100*b3 + 136) * q^59 + (-34*b3 - 372) * q^61 + (-192*b2 + 184*b1) * q^62 + (126*b2 + 36*b1) * q^63 + (249*b3 - 529) * q^64 + (-33*b3 + 33) * q^66 + (340*b2 - 96*b1) * q^67 + (624*b2 + 178*b1) * q^68 + (30*b3 + 186) * q^69 + (-54*b3 + 342) * q^71 + (-216*b2 + 81*b1) * q^72 + (662*b2 + 28*b1) * q^73 + (-138*b3 - 1110) * q^74 + (8*b3 + 360) * q^76 + (154*b2 + 44*b1) * q^77 + (-144*b2 + 48*b1) * q^78 + (144*b3 - 398) * q^79 + 81 * q^81 + (336*b2 - 398*b1) * q^82 + (-240*b2 - 156*b1) * q^83 + (162*b3 + 222) * q^84 + (-180*b3 - 444) * q^86 + (-288*b2 + 18*b1) * q^87 + (-264*b2 + 99*b1) * q^88 + (72*b3 + 342) * q^89 + (36*b3 - 32) * q^91 + (-912*b2 - 98*b1) * q^92 + (-528*b2 + 24*b1) * q^93 + (170*b3 - 1946) * q^94 + (123*b3 - 1347) * q^96 + (322*b2 + 192*b1) * q^97 + (-2304*b2 - 141*b1) * q^98 + 99 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 66 q^{4} - 6 q^{6} - 36 q^{9}+O(q^{10})$$ 4 * q - 66 * q^4 - 6 * q^6 - 36 * q^9 $$4 q - 66 q^{4} - 6 q^{6} - 36 q^{9} - 44 q^{11} + 364 q^{14} + 402 q^{16} - 100 q^{19} - 144 q^{21} + 342 q^{24} + 224 q^{26} + 396 q^{29} + 720 q^{31} + 1252 q^{34} + 594 q^{36} + 180 q^{39} - 1564 q^{41} + 726 q^{44} - 836 q^{46} - 756 q^{49} - 636 q^{51} + 54 q^{54} - 2124 q^{56} + 344 q^{59} - 1556 q^{61} - 1618 q^{64} + 66 q^{66} + 804 q^{69} + 1260 q^{71} - 4716 q^{74} + 1456 q^{76} - 1304 q^{79} + 324 q^{81} + 1212 q^{84} - 2136 q^{86} + 1512 q^{89} - 56 q^{91} - 7444 q^{94} - 5142 q^{96} + 396 q^{99}+O(q^{100})$$ 4 * q - 66 * q^4 - 6 * q^6 - 36 * q^9 - 44 * q^11 + 364 * q^14 + 402 * q^16 - 100 * q^19 - 144 * q^21 + 342 * q^24 + 224 * q^26 + 396 * q^29 + 720 * q^31 + 1252 * q^34 + 594 * q^36 + 180 * q^39 - 1564 * q^41 + 726 * q^44 - 836 * q^46 - 756 * q^49 - 636 * q^51 + 54 * q^54 - 2124 * q^56 + 344 * q^59 - 1556 * q^61 - 1618 * q^64 + 66 * q^66 + 804 * q^69 + 1260 * q^71 - 4716 * q^74 + 1456 * q^76 - 1304 * q^79 + 324 * q^81 + 1212 * q^84 - 2136 * q^86 + 1512 * q^89 - 56 * q^91 - 7444 * q^94 - 5142 * q^96 + 396 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 49x^{2} + 576$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 25\nu ) / 24$$ (v^3 + 25*v) / 24 $$\beta_{3}$$ $$=$$ $$\nu^{2} + 25$$ v^2 + 25
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} - 25$$ b3 - 25 $$\nu^{3}$$ $$=$$ $$24\beta_{2} - 25\beta_1$$ 24*b2 - 25*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/825\mathbb{Z}\right)^\times$$.

 $$n$$ $$376$$ $$551$$ $$727$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
199.1
 − 5.42443i − 4.42443i 4.42443i 5.42443i
5.42443i 3.00000i −21.4244 0 −16.2733 7.69772i 72.8199i −9.00000 0
199.2 4.42443i 3.00000i −11.5756 0 13.2733 31.6977i 15.8199i −9.00000 0
199.3 4.42443i 3.00000i −11.5756 0 13.2733 31.6977i 15.8199i −9.00000 0
199.4 5.42443i 3.00000i −21.4244 0 −16.2733 7.69772i 72.8199i −9.00000 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 825.4.c.h 4
5.b even 2 1 inner 825.4.c.h 4
5.c odd 4 1 33.4.a.c 2
5.c odd 4 1 825.4.a.l 2
15.e even 4 1 99.4.a.f 2
15.e even 4 1 2475.4.a.p 2
20.e even 4 1 528.4.a.p 2
35.f even 4 1 1617.4.a.k 2
40.i odd 4 1 2112.4.a.bn 2
40.k even 4 1 2112.4.a.bg 2
55.e even 4 1 363.4.a.i 2
60.l odd 4 1 1584.4.a.bj 2
165.l odd 4 1 1089.4.a.u 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.4.a.c 2 5.c odd 4 1
99.4.a.f 2 15.e even 4 1
363.4.a.i 2 55.e even 4 1
528.4.a.p 2 20.e even 4 1
825.4.a.l 2 5.c odd 4 1
825.4.c.h 4 1.a even 1 1 trivial
825.4.c.h 4 5.b even 2 1 inner
1089.4.a.u 2 165.l odd 4 1
1584.4.a.bj 2 60.l odd 4 1
1617.4.a.k 2 35.f even 4 1
2112.4.a.bg 2 40.k even 4 1
2112.4.a.bn 2 40.i odd 4 1
2475.4.a.p 2 15.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(825, [\chi])$$:

 $$T_{2}^{4} + 49T_{2}^{2} + 576$$ T2^4 + 49*T2^2 + 576 $$T_{7}^{4} + 1064T_{7}^{2} + 59536$$ T7^4 + 1064*T7^2 + 59536

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 49T^{2} + 576$$
$3$ $$(T^{2} + 9)^{2}$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 1064 T^{2} + 59536$$
$11$ $$(T + 11)^{4}$$
$13$ $$T^{4} + 644 T^{2} + 16384$$
$17$ $$T^{4} + 15124 T^{2} + 3779136$$
$19$ $$(T^{2} + 50 T + 528)^{2}$$
$23$ $$T^{4} + 13828 T^{2} + 4260096$$
$29$ $$(T^{2} - 198 T + 8928)^{2}$$
$31$ $$(T^{2} - 360 T + 30848)^{2}$$
$37$ $$T^{4} + \cdots + 1495832976$$
$41$ $$(T^{2} + 782 T + 148128)^{2}$$
$43$ $$T^{4} + 107284 T^{2} + 434972736$$
$47$ $$T^{4} + \cdots + 13248930816$$
$53$ $$T^{4} + 277668 T^{2} + 6718464$$
$59$ $$(T^{2} - 172 T - 235104)^{2}$$
$61$ $$(T^{2} + 778 T + 123288)^{2}$$
$67$ $$T^{4} + \cdots + 5320827136$$
$71$ $$(T^{2} - 630 T + 28512)^{2}$$
$73$ $$T^{4} + \cdots + 160714395664$$
$79$ $$(T^{2} + 652 T - 396572)^{2}$$
$83$ $$T^{4} + \cdots + 317987721216$$
$89$ $$(T^{2} - 756 T + 17172)^{2}$$
$97$ $$T^{4} + \cdots + 710439951376$$