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Properties

Label 350.4.j.d
Level $350$
Weight $4$
Character orbit 350.j
Analytic conductor $20.651$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [350,4,Mod(149,350)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(350, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 2]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("350.149");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 350.j (of order \(6\), degree \(2\), 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: \(20.6506685020\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 14)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}) q^{2} + \zeta_{12} q^{3} + ( - 4 \zeta_{12}^{2} + 4) q^{4} + 2 q^{6} + (19 \zeta_{12}^{3} - 18 \zeta_{12}) q^{7} - 8 \zeta_{12}^{3} q^{8} - 26 \zeta_{12}^{2} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}) q^{2} + \zeta_{12} q^{3} + ( - 4 \zeta_{12}^{2} + 4) q^{4} + 2 q^{6} + (19 \zeta_{12}^{3} - 18 \zeta_{12}) q^{7} - 8 \zeta_{12}^{3} q^{8} - 26 \zeta_{12}^{2} q^{9} + (35 \zeta_{12}^{2} - 35) q^{11} + ( - 4 \zeta_{12}^{3} + 4 \zeta_{12}) q^{12} + 66 \zeta_{12}^{3} q^{13} + (38 \zeta_{12}^{2} - 36) q^{14} - 16 \zeta_{12}^{2} q^{16} + 59 \zeta_{12} q^{17} - 52 \zeta_{12} q^{18} + 137 \zeta_{12}^{2} q^{19} + (\zeta_{12}^{2} - 19) q^{21} + 70 \zeta_{12}^{3} q^{22} + (7 \zeta_{12}^{3} - 7 \zeta_{12}) q^{23} + ( - 8 \zeta_{12}^{2} + 8) q^{24} + 132 \zeta_{12}^{2} q^{26} - 53 \zeta_{12}^{3} q^{27} + (72 \zeta_{12}^{3} + 4 \zeta_{12}) q^{28} - 106 q^{29} + (75 \zeta_{12}^{2} - 75) q^{31} - 32 \zeta_{12} q^{32} + (35 \zeta_{12}^{3} - 35 \zeta_{12}) q^{33} + 118 q^{34} - 104 q^{36} + (11 \zeta_{12}^{3} - 11 \zeta_{12}) q^{37} + 274 \zeta_{12} q^{38} + (66 \zeta_{12}^{2} - 66) q^{39} - 498 q^{41} + (38 \zeta_{12}^{3} - 36 \zeta_{12}) q^{42} + 260 \zeta_{12}^{3} q^{43} + 140 \zeta_{12}^{2} q^{44} + (14 \zeta_{12}^{2} - 14) q^{46} + ( - 171 \zeta_{12}^{3} + 171 \zeta_{12}) q^{47} - 16 \zeta_{12}^{3} q^{48} + ( - 360 \zeta_{12}^{2} + 323) q^{49} + 59 \zeta_{12}^{2} q^{51} + 264 \zeta_{12} q^{52} + 417 \zeta_{12} q^{53} - 106 \zeta_{12}^{2} q^{54} + (144 \zeta_{12}^{2} + 8) q^{56} + 137 \zeta_{12}^{3} q^{57} + (212 \zeta_{12}^{3} - 212 \zeta_{12}) q^{58} + (17 \zeta_{12}^{2} - 17) q^{59} - 51 \zeta_{12}^{2} q^{61} + 150 \zeta_{12}^{3} q^{62} + ( - 26 \zeta_{12}^{3} + 494 \zeta_{12}) q^{63} - 64 q^{64} + (70 \zeta_{12}^{2} - 70) q^{66} + 439 \zeta_{12} q^{67} + ( - 236 \zeta_{12}^{3} + 236 \zeta_{12}) q^{68} - 7 q^{69} - 784 q^{71} + (208 \zeta_{12}^{3} - 208 \zeta_{12}) q^{72} - 295 \zeta_{12} q^{73} + (22 \zeta_{12}^{2} - 22) q^{74} + 548 q^{76} + ( - 630 \zeta_{12}^{3} - 35 \zeta_{12}) q^{77} + 132 \zeta_{12}^{3} q^{78} - 495 \zeta_{12}^{2} q^{79} + (649 \zeta_{12}^{2} - 649) q^{81} + (996 \zeta_{12}^{3} - 996 \zeta_{12}) q^{82} + 932 \zeta_{12}^{3} q^{83} + (76 \zeta_{12}^{2} - 72) q^{84} + 520 \zeta_{12}^{2} q^{86} - 106 \zeta_{12} q^{87} + 280 \zeta_{12} q^{88} - 873 \zeta_{12}^{2} q^{89} + ( - 1188 \zeta_{12}^{2} - 66) q^{91} + 28 \zeta_{12}^{3} q^{92} + (75 \zeta_{12}^{3} - 75 \zeta_{12}) q^{93} + ( - 342 \zeta_{12}^{2} + 342) q^{94} - 32 \zeta_{12}^{2} q^{96} + 290 \zeta_{12}^{3} q^{97} + ( - 646 \zeta_{12}^{3} - 74 \zeta_{12}) q^{98} + 910 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{4} + 8 q^{6} - 52 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 8 q^{4} + 8 q^{6} - 52 q^{9} - 70 q^{11} - 68 q^{14} - 32 q^{16} + 274 q^{19} - 74 q^{21} + 16 q^{24} + 264 q^{26} - 424 q^{29} - 150 q^{31} + 472 q^{34} - 416 q^{36} - 132 q^{39} - 1992 q^{41} + 280 q^{44} - 28 q^{46} + 572 q^{49} + 118 q^{51} - 212 q^{54} + 320 q^{56} - 34 q^{59} - 102 q^{61} - 256 q^{64} - 140 q^{66} - 28 q^{69} - 3136 q^{71} - 44 q^{74} + 2192 q^{76} - 990 q^{79} - 1298 q^{81} - 136 q^{84} + 1040 q^{86} - 1746 q^{89} - 2640 q^{91} + 684 q^{94} - 64 q^{96} + 3640 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/350\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(-\zeta_{12}^{2}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
149.1
−0.866025 + 0.500000i
0.866025 − 0.500000i
−0.866025 − 0.500000i
0.866025 + 0.500000i
−1.73205 − 1.00000i −0.866025 + 0.500000i 2.00000 + 3.46410i 0 2.00000 15.5885 + 10.0000i − 8.00000i −13.0000 + 22.5167i 0
149.2 1.73205 + 1.00000i 0.866025 − 0.500000i 2.00000 + 3.46410i 0 2.00000 −15.5885 − 10.0000i 8.00000i −13.0000 + 22.5167i 0
249.1 −1.73205 + 1.00000i −0.866025 − 0.500000i 2.00000 − 3.46410i 0 2.00000 15.5885 − 10.0000i 8.00000i −13.0000 − 22.5167i 0
249.2 1.73205 − 1.00000i 0.866025 + 0.500000i 2.00000 − 3.46410i 0 2.00000 −15.5885 + 10.0000i − 8.00000i −13.0000 − 22.5167i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 350.4.j.d 4
5.b even 2 1 inner 350.4.j.d 4
5.c odd 4 1 14.4.c.b ✓ 2
5.c odd 4 1 350.4.e.b 2
7.c even 3 1 inner 350.4.j.d 4
15.e even 4 1 126.4.g.c 2
20.e even 4 1 112.4.i.b 2
35.f even 4 1 98.4.c.e 2
35.j even 6 1 inner 350.4.j.d 4
35.k even 12 1 98.4.a.c 1
35.k even 12 1 98.4.c.e 2
35.k even 12 1 2450.4.a.bf 1
35.l odd 12 1 14.4.c.b ✓ 2
35.l odd 12 1 98.4.a.b 1
35.l odd 12 1 350.4.e.b 2
35.l odd 12 1 2450.4.a.bh 1
40.i odd 4 1 448.4.i.c 2
40.k even 4 1 448.4.i.d 2
105.k odd 4 1 882.4.g.d 2
105.w odd 12 1 882.4.a.p 1
105.w odd 12 1 882.4.g.d 2
105.x even 12 1 126.4.g.c 2
105.x even 12 1 882.4.a.k 1
140.w even 12 1 112.4.i.b 2
140.w even 12 1 784.4.a.l 1
140.x odd 12 1 784.4.a.j 1
280.br even 12 1 448.4.i.d 2
280.bt odd 12 1 448.4.i.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.4.c.b ✓ 2 5.c odd 4 1
14.4.c.b ✓ 2 35.l odd 12 1
98.4.a.b 1 35.l odd 12 1
98.4.a.c 1 35.k even 12 1
98.4.c.e 2 35.f even 4 1
98.4.c.e 2 35.k even 12 1
112.4.i.b 2 20.e even 4 1
112.4.i.b 2 140.w even 12 1
126.4.g.c 2 15.e even 4 1
126.4.g.c 2 105.x even 12 1
350.4.e.b 2 5.c odd 4 1
350.4.e.b 2 35.l odd 12 1
350.4.j.d 4 1.a even 1 1 trivial
350.4.j.d 4 5.b even 2 1 inner
350.4.j.d 4 7.c even 3 1 inner
350.4.j.d 4 35.j even 6 1 inner
448.4.i.c 2 40.i odd 4 1
448.4.i.c 2 280.bt odd 12 1
448.4.i.d 2 40.k even 4 1
448.4.i.d 2 280.br even 12 1
784.4.a.j 1 140.x odd 12 1
784.4.a.l 1 140.w even 12 1
882.4.a.k 1 105.x even 12 1
882.4.a.p 1 105.w odd 12 1
882.4.g.d 2 105.k odd 4 1
882.4.g.d 2 105.w odd 12 1
2450.4.a.bf 1 35.k even 12 1
2450.4.a.bh 1 35.l odd 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(350, [\chi])\):

\( T_{3}^{4} - T_{3}^{2} + 1 \) Copy content Toggle raw display
\( T_{11}^{2} + 35T_{11} + 1225 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$3$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} - 286 T^{2} + 117649 \) Copy content Toggle raw display
$11$ \( (T^{2} + 35 T + 1225)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 4356)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 3481 T^{2} + 12117361 \) Copy content Toggle raw display
$19$ \( (T^{2} - 137 T + 18769)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} - 49T^{2} + 2401 \) Copy content Toggle raw display
$29$ \( (T + 106)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 75 T + 5625)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} - 121 T^{2} + 14641 \) Copy content Toggle raw display
$41$ \( (T + 498)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 67600)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 29241 T^{2} + 855036081 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 30237384321 \) Copy content Toggle raw display
$59$ \( (T^{2} + 17 T + 289)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 51 T + 2601)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 37141383841 \) Copy content Toggle raw display
$71$ \( (T + 784)^{4} \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 7573350625 \) Copy content Toggle raw display
$79$ \( (T^{2} + 495 T + 245025)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 868624)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 873 T + 762129)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 84100)^{2} \) Copy content Toggle raw display
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