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Properties

Label 825.4.c.r
Level $825$
Weight $4$
Character orbit 825.c
Analytic conductor $48.677$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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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: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 83x^{8} + 2275x^{6} + 24517x^{4} + 87636x^{2} + 40000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + \beta_1) q^{2} + 3 \beta_{2} q^{3} + (\beta_{4} - \beta_{3} - 9) q^{4} + (3 \beta_{3} + 3) q^{6} + ( - \beta_{9} + 7 \beta_{2} + \beta_1) q^{7} + (2 \beta_{8} + \beta_{7} + \cdots - 11 \beta_1) q^{8}+ \cdots - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + \beta_1) q^{2} + 3 \beta_{2} q^{3} + (\beta_{4} - \beta_{3} - 9) q^{4} + (3 \beta_{3} + 3) q^{6} + ( - \beta_{9} + 7 \beta_{2} + \beta_1) q^{7} + (2 \beta_{8} + \beta_{7} + \cdots - 11 \beta_1) q^{8}+ \cdots - 99 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 92 q^{4} + 24 q^{6} - 90 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 92 q^{4} + 24 q^{6} - 90 q^{9} + 110 q^{11} - 110 q^{14} + 1268 q^{16} + 674 q^{19} - 228 q^{21} - 432 q^{24} + 718 q^{26} + 306 q^{29} + 526 q^{31} - 1034 q^{34} + 828 q^{36} - 258 q^{39} - 176 q^{41} - 1012 q^{44} + 872 q^{46} - 4762 q^{49} + 900 q^{51} - 216 q^{54} + 422 q^{56} - 820 q^{59} - 2260 q^{61} - 6340 q^{64} + 264 q^{66} + 1206 q^{69} + 2498 q^{71} + 5970 q^{74} - 5112 q^{76} - 4516 q^{79} + 810 q^{81} + 2646 q^{84} - 2870 q^{86} - 694 q^{89} - 1224 q^{91} - 1814 q^{94} + 1680 q^{96} - 990 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 83x^{8} + 2275x^{6} + 24517x^{4} + 87636x^{2} + 40000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{9} + 297\nu^{7} + 28325\nu^{5} + 688783\nu^{3} + 4293124\nu ) / 2869200 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 19\nu^{8} + 1530\nu^{6} + 35665\nu^{4} + 219038\nu^{2} + 2000 ) / 143460 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -19\nu^{8} - 1530\nu^{6} - 35665\nu^{4} - 75578\nu^{2} + 2293360 ) / 143460 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 13\nu^{8} + 921\nu^{6} + 19117\nu^{4} + 131621\nu^{2} + 205736 ) / 9564 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -224\nu^{8} - 12375\nu^{6} - 110900\nu^{4} + 1036247\nu^{2} + 4265120 ) / 143460 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 91\nu^{9} + 8838\nu^{7} + 291625\nu^{5} + 3850322\nu^{3} + 17899796\nu ) / 717300 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -92\nu^{9} - 8541\nu^{7} - 263300\nu^{5} - 2982214\nu^{3} - 8764897\nu ) / 358650 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -773\nu^{9} - 57339\nu^{7} - 1297475\nu^{5} - 10327741\nu^{3} - 21976888\nu ) / 478200 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} + \beta_{3} - 16 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{8} + 4\beta_{7} - 16\beta_{2} - 27\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{6} + 6\beta_{5} - 39\beta_{4} - 77\beta_{3} + 436 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -2\beta_{9} - 88\beta_{8} - 200\beta_{7} + 1244\beta_{2} + 887\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -84\beta_{6} - 328\beta_{5} + 1493\beta_{4} + 3869\beta_{3} - 14368 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 160\beta_{9} + 3330\beta_{8} + 8600\beta_{7} - 62560\beta_{2} - 32267\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 3010\beta_{6} + 15150\beta_{5} - 58547\beta_{4} - 170997\beta_{3} + 522932 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -9130\beta_{9} - 126024\beta_{8} - 355668\beta_{7} + 2766252\beta_{2} + 1236959\beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
− 4.58039i
− 6.35463i
− 3.91216i
− 2.40874i
0.729174i
− 0.729174i
2.40874i
3.91216i
6.35463i
4.58039i
− 5.58039i 3.00000i −23.1407 0 16.7412 34.4181i 84.4912i −9.00000 0
199.2 − 5.35463i − 3.00000i −20.6721 0 −16.0639 − 19.1169i 67.8543i −9.00000 0
199.3 − 4.91216i 3.00000i −16.1293 0 14.7365 − 35.2335i 39.9323i −9.00000 0
199.4 − 1.40874i − 3.00000i 6.01545 0 −4.22622 13.7148i − 19.7441i −9.00000 0
199.5 − 0.270826i 3.00000i 7.92665 0 0.812479 33.4133i − 4.31336i −9.00000 0
199.6 0.270826i − 3.00000i 7.92665 0 0.812479 − 33.4133i 4.31336i −9.00000 0
199.7 1.40874i 3.00000i 6.01545 0 −4.22622 − 13.7148i 19.7441i −9.00000 0
199.8 4.91216i − 3.00000i −16.1293 0 14.7365 35.2335i − 39.9323i −9.00000 0
199.9 5.35463i 3.00000i −20.6721 0 −16.0639 19.1169i − 67.8543i −9.00000 0
199.10 5.58039i − 3.00000i −23.1407 0 16.7412 − 34.4181i − 84.4912i −9.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 199.10
Significant digits:
Format:

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.r 10
5.b even 2 1 inner 825.4.c.r 10
5.c odd 4 1 825.4.a.w ✓ 5
5.c odd 4 1 825.4.a.z yes 5
15.e even 4 1 2475.4.a.bf 5
15.e even 4 1 2475.4.a.bm 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
825.4.a.w ✓ 5 5.c odd 4 1
825.4.a.z yes 5 5.c odd 4 1
825.4.c.r 10 1.a even 1 1 trivial
825.4.c.r 10 5.b even 2 1 inner
2475.4.a.bf 5 15.e even 4 1
2475.4.a.bm 5 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 \(S_{4}^{\mathrm{new}}(825, [\chi])\):

\( T_{2}^{10} + 86T_{2}^{8} + 2509T_{2}^{6} + 26364T_{2}^{4} + 44676T_{2}^{2} + 3136 \) Copy content Toggle raw display
\( T_{7}^{10} + 4096T_{7}^{8} + 6208734T_{7}^{6} + 4198643684T_{7}^{4} + 1196094841641T_{7}^{2} + 112858624768036 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + 86 T^{8} + \cdots + 3136 \) Copy content Toggle raw display
$3$ \( (T^{2} + 9)^{5} \) Copy content Toggle raw display
$5$ \( T^{10} \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots + 112858624768036 \) Copy content Toggle raw display
$11$ \( (T - 11)^{10} \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots + 28\!\cdots\!16 \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 59\!\cdots\!64 \) Copy content Toggle raw display
$19$ \( (T^{5} - 337 T^{4} + \cdots + 7593907635)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 35\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( (T^{5} - 153 T^{4} + \cdots - 1390240880)^{2} \) Copy content Toggle raw display
$31$ \( (T^{5} - 263 T^{4} + \cdots - 6663042216)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 37\!\cdots\!36 \) Copy content Toggle raw display
$41$ \( (T^{5} + 88 T^{4} + \cdots - 314650458072)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 10\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 30\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( (T^{5} + 410 T^{4} + \cdots + 950084217280)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} + \cdots + 67775166628112)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 26\!\cdots\!76 \) Copy content Toggle raw display
$71$ \( (T^{5} + \cdots - 246364319624144)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 51\!\cdots\!76 \) Copy content Toggle raw display
$79$ \( (T^{5} + 2258 T^{4} + \cdots - 694623214400)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 36\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( (T^{5} + \cdots - 40747061333440)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 11\!\cdots\!61 \) Copy content Toggle raw display
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