Newform orbit 225.4.k.c

• Label: 225.4.k.c
• Level: $225$
• Weight: $4$
• Character orbit: 225.k
• Analytic conductor: $13.275$
• Analytic rank: $0$
• Dimension: $12$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	225.k (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(13.2754297513\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(6\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial:	\( x^{12} - 23x^{10} + 198x^{8} - 719x^{6} + 886x^{4} + 585x^{2} + 81 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{6}\cdot 3^{2} \)
Twist minimal:	no (minimal twist has level 45)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + (\beta_{11} - \beta_{7}) q^{2} + (\beta_{11} + \beta_{8} + \beta_{5}) q^{3} + ( - \beta_{6} - 2 \beta_{4} - 4 \beta_{3} - \beta_1) q^{4} + (\beta_{4} + 5 \beta_{3} - 3 \beta_{2} - \beta_1 + 17) q^{6} + (\beta_{11} + 6 \beta_{10} - 5 \beta_{9} - \beta_{7} - 5 \beta_{5}) q^{7} + (3 \beta_{8} - 3 \beta_{7} - 3 \beta_{5}) q^{8} + (\beta_{6} + 4 \beta_{4} + 22 \beta_{3} - 5 \beta_{2} - \beta_1 + 2) q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 22 q^{4} + 168 q^{6} - 114 q^{9}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 23x^{10} + 198x^{8} - 719x^{6} + 886x^{4} + 585x^{2} + 81 \) :

\(\beta_{1}\)	\(=\)	\( ( 17\nu^{10} - 260\nu^{8} + 1311\nu^{6} - 2275\nu^{4} - 1388\nu^{2} + 33786 ) / 7875 \)
\(\beta_{2}\)	\(=\)	\( ( -19\nu^{10} + 445\nu^{8} - 3627\nu^{6} + 9800\nu^{4} + 5566\nu^{2} - 24327 ) / 7875 \)
\(\beta_{3}\)	\(=\)	\( ( -18\nu^{10} + 415\nu^{8} - 3594\nu^{6} + 13350\nu^{4} - 18398\nu^{2} - 5994 ) / 1125 \)
\(\beta_{4}\)	\(=\)	\( ( -368\nu^{10} + 8665\nu^{8} - 77019\nu^{6} + 295225\nu^{4} - 411623\nu^{2} - 133794 ) / 7875 \)
\(\beta_{5}\)	\(=\)	\( ( 181\nu^{11} - 4055\nu^{9} + 33723\nu^{7} - 115325\nu^{5} + 124141\nu^{3} + 132273\nu ) / 23625 \)
\(\beta_{6}\)	\(=\)	\( ( 106\nu^{10} - 2455\nu^{8} + 21423\nu^{6} - 80150\nu^{4} + 111416\nu^{2} + 29448 ) / 1575 \)
\(\beta_{7}\)	\(=\)	\( ( -88\nu^{11} + 2015\nu^{9} - 17154\nu^{7} + 60725\nu^{5} - 67168\nu^{3} - 70929\nu ) / 6750 \)
\(\beta_{8}\)	\(=\)	\( ( -316\nu^{11} + 7355\nu^{9} - 64053\nu^{7} + 233450\nu^{5} - 264751\nu^{3} - 277353\nu ) / 23625 \)
\(\beta_{9}\)	\(=\)	\( ( -1528\nu^{11} + 35465\nu^{9} - 309924\nu^{7} + 1163225\nu^{5} - 1604758\nu^{3} - 514449\nu ) / 23625 \)
\(\beta_{10}\)	\(=\)	\( ( 2818\nu^{11} - 65540\nu^{9} + 575244\nu^{7} - 2181725\nu^{5} + 3102073\nu^{3} + 800019\nu ) / 23625 \)
\(\beta_{11}\)	\(=\)	\( ( -811\nu^{11} + 18830\nu^{9} - 164763\nu^{7} + 620450\nu^{5} - 860821\nu^{3} - 275688\nu ) / 6750 \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/225\mathbb{Z}\right)^\times\).

\(n\)	\(101\)	\(127\)
\(\chi(n)\)	\(\beta_{3}\)	\(-1\)

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

49.1

−2.88506	−	0.500000i
−0.0378788	+	0.500000i
1.98116	−	0.500000i
−1.98116	+	0.500000i
0.0378788	−	0.500000i
2.88506	+	0.500000i
−2.88506	+	0.500000i
−0.0378788	−	0.500000i
1.98116	+	0.500000i
−1.98116	−	0.500000i
0.0378788	+	0.500000i
2.88506	−	0.500000i

−3.96084

+

2.28679i

−3.96084

−

3.36330i

6.45882

−

11.1870i

0

23.3794

+

4.26387i

−17.4197

+

10.0573i

22.4912i

4.37646

+

26.6429i

0

49.2

−3.24635

+

1.87428i

−3.24635

−

4.05724i

3.02587

−

5.24096i

0

18.1432

+

7.08665i

27.1492

−

15.6746i

−

7.30318i

−5.92239

+

26.3425i

0

49.3

−0.151541

+

0.0874923i

−0.151541

−

5.19394i

−3.98469

+

6.90169i

0

0.477395

+

0.773837i

7.32979

−

4.23186i

−

2.79440i

−26.9541

+

1.57419i

0

49.4

0.151541

−

0.0874923i

0.151541

+

5.19394i

−3.98469

+

6.90169i

0

0.477395

+

0.773837i

−7.32979

+

4.23186i

2.79440i

−26.9541

+

1.57419i

0

49.5

3.24635

−

1.87428i

3.24635

+

4.05724i

3.02587

−

5.24096i

0

18.1432

+

7.08665i

−27.1492

+

15.6746i

7.30318i

−5.92239

+

26.3425i

0

49.6

3.96084

−

2.28679i

3.96084

+

3.36330i

6.45882

−

11.1870i

0

23.3794

+

4.26387i

17.4197

−

10.0573i

−

22.4912i

4.37646

+

26.6429i

0

124.1

−3.96084

−

2.28679i

−3.96084

+

3.36330i

6.45882

+

11.1870i

0

23.3794

−

4.26387i

−17.4197

−

10.0573i

−

22.4912i

4.37646

−

26.6429i

0

124.2

−3.24635

−

1.87428i

−3.24635

+

4.05724i

3.02587

+

5.24096i

0

18.1432

−

7.08665i

27.1492

+

15.6746i

7.30318i

−5.92239

−

26.3425i

0

124.3

−0.151541

−

0.0874923i

−0.151541

+

5.19394i

−3.98469

−

6.90169i

0

0.477395

−

0.773837i

7.32979

+

4.23186i

2.79440i

−26.9541

−

1.57419i

0

124.4

0.151541

+

0.0874923i

0.151541

−

5.19394i

−3.98469

−

6.90169i

0

0.477395

−

0.773837i

−7.32979

−

4.23186i

−

2.79440i

−26.9541

−

1.57419i

0

124.5

3.24635

+

1.87428i

3.24635

−

4.05724i

3.02587

+

5.24096i

0

18.1432

−

7.08665i

−27.1492

−

15.6746i

−

7.30318i

−5.92239

−

26.3425i

0

124.6

3.96084

+

2.28679i

3.96084

−

3.36330i

6.45882

+

11.1870i

0

23.3794

−

4.26387i

17.4197

+

10.0573i

22.4912i

4.37646

−

26.6429i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
9.c	even	3	1	inner
45.j	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	225.4.k.c		12
5.b	even	2	1	inner	225.4.k.c		12
5.c	odd	4	1		45.4.e.b	✓	6
5.c	odd	4	1		225.4.e.c		6
9.c	even	3	1	inner	225.4.k.c		12
15.e	even	4	1		135.4.e.b		6
45.j	even	6	1	inner	225.4.k.c		12
45.k	odd	12	1		45.4.e.b	✓	6
45.k	odd	12	1		225.4.e.c		6
45.k	odd	12	1		405.4.a.h		3
45.k	odd	12	1		2025.4.a.s		3
45.l	even	12	1		135.4.e.b		6
45.l	even	12	1		405.4.a.j		3
45.l	even	12	1		2025.4.a.q		3

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
45.4.e.b	✓	6	5.c	odd	4	1
45.4.e.b	✓	6	45.k	odd	12	1
135.4.e.b		6	15.e	even	4	1
135.4.e.b		6	45.l	even	12	1
225.4.e.c		6	5.c	odd	4	1
225.4.e.c		6	45.k	odd	12	1
225.4.k.c		12	1.a	even	1	1	trivial
225.4.k.c		12	5.b	even	2	1	inner
225.4.k.c		12	9.c	even	3	1	inner
225.4.k.c		12	45.j	even	6	1	inner
405.4.a.h		3	45.k	odd	12	1
405.4.a.j		3	45.l	even	12	1
2025.4.a.q		3	45.l	even	12	1
2025.4.a.s		3	45.k	odd	12	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} - 35T_{2}^{10} + 930T_{2}^{8} - 10307T_{2}^{6} + 86710T_{2}^{4} - 2655T_{2}^{2} + 81 \) acting on \(S_{4}^{\mathrm{new}}(225, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	T^12 - 35*T^10 + 930*T^8 - 10307*T^6 + 86710*T^4 - 2655*T^2 + 81
$3$	T^12 + 57*T^10 + 2250*T^8 + 77517*T^6 + 1640250*T^4 + 30292137*T^2 + 387420489
$5$	\( T^{12} \)
$7$	T^12 - 1459*T^10 + 1631674*T^8 - 668166075*T^6 + 205458430878*T^4 - 14156533177983*T^2 + 811313702977761
$11$	(T^6 + 14*T^5 + 3012*T^4 - 126520*T^3 + 7320184*T^2 - 122631168*T + 1896428304)^2