# Properties

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

# Related objects

## Newspace parameters

 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

 Self dual: no Analytic conductor: $$48.6765757547$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{33})$$ Defining polynomial: $$x^{4} + 17 x^{2} + 64$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 33) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + 3 \beta_{2} q^{3} + ( -1 + \beta_{3} ) q^{4} + ( 3 - 3 \beta_{3} ) q^{6} + ( -2 \beta_{1} - 2 \beta_{2} ) q^{7} + ( 7 \beta_{1} + 8 \beta_{2} ) q^{8} -9 q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} + 3 \beta_{2} q^{3} + ( -1 + \beta_{3} ) q^{4} + ( 3 - 3 \beta_{3} ) q^{6} + ( -2 \beta_{1} - 2 \beta_{2} ) q^{7} + ( 7 \beta_{1} + 8 \beta_{2} ) q^{8} -9 q^{9} + 11 q^{11} + 3 \beta_{1} q^{12} + ( -8 \beta_{1} - 42 \beta_{2} ) q^{13} + 16 q^{14} + ( -63 + 7 \beta_{3} ) q^{16} + ( 30 \beta_{1} + 28 \beta_{2} ) q^{17} -9 \beta_{1} q^{18} + ( 36 - 18 \beta_{3} ) q^{19} + 6 \beta_{3} q^{21} + 11 \beta_{1} q^{22} + 112 \beta_{2} q^{23} + ( -3 - 21 \beta_{3} ) q^{24} + ( 30 + 34 \beta_{3} ) q^{26} -27 \beta_{2} q^{27} -16 \beta_{2} q^{28} + ( -134 + 46 \beta_{3} ) q^{29} + ( 32 - 104 \beta_{3} ) q^{31} + ( -7 \beta_{1} + 120 \beta_{2} ) q^{32} + 33 \beta_{2} q^{33} + ( -242 + 2 \beta_{3} ) q^{34} + ( 9 - 9 \beta_{3} ) q^{36} + ( 44 \beta_{1} + 46 \beta_{2} ) q^{37} + ( 36 \beta_{1} - 144 \beta_{2} ) q^{38} + ( 102 + 24 \beta_{3} ) q^{39} + ( -246 - 2 \beta_{3} ) q^{41} + 48 \beta_{2} q^{42} + ( 86 \beta_{1} + 10 \beta_{2} ) q^{43} + ( -11 + 11 \beta_{3} ) q^{44} + ( 112 - 112 \beta_{3} ) q^{46} + ( -48 \beta_{1} + 8 \beta_{2} ) q^{47} + ( 21 \beta_{1} - 168 \beta_{2} ) q^{48} + ( 311 - 4 \beta_{3} ) q^{49} + ( 6 - 90 \beta_{3} ) q^{51} + ( -34 \beta_{1} - 64 \beta_{2} ) q^{52} + ( 128 \beta_{1} + 22 \beta_{2} ) q^{53} + ( -27 + 27 \beta_{3} ) q^{54} + ( 112 + 16 \beta_{3} ) q^{56} + ( -54 \beta_{1} + 54 \beta_{2} ) q^{57} + ( -134 \beta_{1} + 368 \beta_{2} ) q^{58} -196 q^{59} + ( -578 + 52 \beta_{3} ) q^{61} + ( 32 \beta_{1} - 832 \beta_{2} ) q^{62} + ( 18 \beta_{1} + 18 \beta_{2} ) q^{63} + ( -321 - 71 \beta_{3} ) q^{64} + ( 33 - 33 \beta_{3} ) q^{66} + ( 152 \beta_{1} - 388 \beta_{2} ) q^{67} + ( -2 \beta_{1} + 240 \beta_{2} ) q^{68} -336 q^{69} + ( 320 - 184 \beta_{3} ) q^{71} + ( -63 \beta_{1} - 72 \beta_{2} ) q^{72} + ( 252 \beta_{1} - 170 \beta_{2} ) q^{73} + ( -350 - 2 \beta_{3} ) q^{74} + ( -180 + 36 \beta_{3} ) q^{76} + ( -22 \beta_{1} - 22 \beta_{2} ) q^{77} + ( 102 \beta_{1} + 192 \beta_{2} ) q^{78} + ( 152 - 74 \beta_{3} ) q^{79} + 81 q^{81} + ( -246 \beta_{1} - 16 \beta_{2} ) q^{82} + ( 324 \beta_{1} + 336 \beta_{2} ) q^{83} + 48 q^{84} + ( -764 + 76 \beta_{3} ) q^{86} + ( 138 \beta_{1} - 264 \beta_{2} ) q^{87} + ( 77 \beta_{1} + 88 \beta_{2} ) q^{88} + ( -490 + 8 \beta_{3} ) q^{89} + ( -128 - 84 \beta_{3} ) q^{91} + 112 \beta_{1} q^{92} + ( -312 \beta_{1} - 216 \beta_{2} ) q^{93} + ( 440 - 56 \beta_{3} ) q^{94} + ( -381 + 21 \beta_{3} ) q^{96} + ( 420 \beta_{1} + 802 \beta_{2} ) q^{97} + ( 311 \beta_{1} - 32 \beta_{2} ) q^{98} -99 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 2q^{4} + 6q^{6} - 36q^{9} + O(q^{10})$$ $$4q - 2q^{4} + 6q^{6} - 36q^{9} + 44q^{11} + 64q^{14} - 238q^{16} + 108q^{19} + 12q^{21} - 54q^{24} + 188q^{26} - 444q^{29} - 80q^{31} - 964q^{34} + 18q^{36} + 456q^{39} - 988q^{41} - 22q^{44} + 224q^{46} + 1236q^{49} - 156q^{51} - 54q^{54} + 480q^{56} - 784q^{59} - 2208q^{61} - 1426q^{64} + 66q^{66} - 1344q^{69} + 912q^{71} - 1404q^{74} - 648q^{76} + 460q^{79} + 324q^{81} + 192q^{84} - 2904q^{86} - 1944q^{89} - 680q^{91} + 1648q^{94} - 1482q^{96} - 396q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 17 x^{2} + 64$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 9 \nu$$$$)/8$$ $$\beta_{3}$$ $$=$$ $$\nu^{2} + 9$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} - 9$$ $$\nu^{3}$$ $$=$$ $$8 \beta_{2} - 9 \beta_{1}$$

## 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
199.1
 − 3.37228i − 2.37228i 2.37228i 3.37228i
3.37228i 3.00000i −3.37228 0 10.1168 4.74456i 15.6060i −9.00000 0
199.2 2.37228i 3.00000i 2.37228 0 −7.11684 6.74456i 24.6060i −9.00000 0
199.3 2.37228i 3.00000i 2.37228 0 −7.11684 6.74456i 24.6060i −9.00000 0
199.4 3.37228i 3.00000i −3.37228 0 10.1168 4.74456i 15.6060i −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.i 4
5.b even 2 1 inner 825.4.c.i 4
5.c odd 4 1 33.4.a.d 2
5.c odd 4 1 825.4.a.k 2
15.e even 4 1 99.4.a.e 2
15.e even 4 1 2475.4.a.o 2
20.e even 4 1 528.4.a.o 2
35.f even 4 1 1617.4.a.j 2
40.i odd 4 1 2112.4.a.ba 2
40.k even 4 1 2112.4.a.bh 2
55.e even 4 1 363.4.a.j 2
60.l odd 4 1 1584.4.a.x 2
165.l odd 4 1 1089.4.a.t 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.4.a.d 2 5.c odd 4 1
99.4.a.e 2 15.e even 4 1
363.4.a.j 2 55.e even 4 1
528.4.a.o 2 20.e even 4 1
825.4.a.k 2 5.c odd 4 1
825.4.c.i 4 1.a even 1 1 trivial
825.4.c.i 4 5.b even 2 1 inner
1089.4.a.t 2 165.l odd 4 1
1584.4.a.x 2 60.l odd 4 1
1617.4.a.j 2 35.f even 4 1
2112.4.a.ba 2 40.i odd 4 1
2112.4.a.bh 2 40.k even 4 1
2475.4.a.o 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 $$S_{4}^{\mathrm{new}}(825, [\chi])$$:

 $$T_{2}^{4} + 17 T_{2}^{2} + 64$$ $$T_{7}^{4} + 68 T_{7}^{2} + 1024$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$64 + 17 T^{2} + T^{4}$$
$3$ $$( 9 + T^{2} )^{2}$$
$5$ $$T^{4}$$
$7$ $$1024 + 68 T^{2} + T^{4}$$
$11$ $$( -11 + T )^{4}$$
$13$ $$839056 + 3944 T^{2} + T^{4}$$
$17$ $$52649536 + 15188 T^{2} + T^{4}$$
$19$ $$( -1944 - 54 T + T^{2} )^{2}$$
$23$ $$( 12544 + T^{2} )^{2}$$
$29$ $$( -5136 + 222 T + T^{2} )^{2}$$
$31$ $$( -88832 + 40 T + T^{2} )^{2}$$
$37$ $$237036816 + 33096 T^{2} + T^{4}$$
$41$ $$( 60976 + 494 T + T^{2} )^{2}$$
$43$ $$3591365184 + 124212 T^{2} + T^{4}$$
$47$ $$323424256 + 40064 T^{2} + T^{4}$$
$53$ $$17796627216 + 273864 T^{2} + T^{4}$$
$59$ $$( 196 + T )^{4}$$
$61$ $$( 282396 + 1104 T + T^{2} )^{2}$$
$67$ $$609497344 + 811808 T^{2} + T^{4}$$
$71$ $$( -227328 - 456 T + T^{2} )^{2}$$
$73$ $$190350709264 + 1223048 T^{2} + T^{4}$$
$79$ $$( -31952 - 230 T + T^{2} )^{2}$$
$83$ $$698521522176 + 1792656 T^{2} + T^{4}$$
$89$ $$( 235668 + 972 T + T^{2} )^{2}$$
$97$ $$1220662586896 + 3611528 T^{2} + T^{4}$$