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Properties

Label 350.4.j.f
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 70)
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.

\(f(q)\) \(=\) \( q + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}) q^{2} + \zeta_{12} q^{3} + ( - 4 \zeta_{12}^{2} + 4) q^{4} + 2 q^{6} + (14 \zeta_{12}^{3} + 7 \zeta_{12}) q^{7} - 8 \zeta_{12}^{3} q^{8} - 26 \zeta_{12}^{2} q^{9} +O(q^{10}) \) 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} + (14 \zeta_{12}^{3} + 7 \zeta_{12}) q^{7} - 8 \zeta_{12}^{3} q^{8} - 26 \zeta_{12}^{2} q^{9} + ( - 30 \zeta_{12}^{2} + 30) q^{11} + ( - 4 \zeta_{12}^{3} + 4 \zeta_{12}) q^{12} - 44 \zeta_{12}^{3} q^{13} + (28 \zeta_{12}^{2} + 14) q^{14} - 16 \zeta_{12}^{2} q^{16} + 24 \zeta_{12} q^{17} - 52 \zeta_{12} q^{18} + 2 \zeta_{12}^{2} q^{19} + (21 \zeta_{12}^{2} - 14) q^{21} - 60 \zeta_{12}^{3} q^{22} + ( - 183 \zeta_{12}^{3} + 183 \zeta_{12}) q^{23} + ( - 8 \zeta_{12}^{2} + 8) q^{24} - 88 \zeta_{12}^{2} q^{26} - 53 \zeta_{12}^{3} q^{27} + ( - 28 \zeta_{12}^{3} + 84 \zeta_{12}) q^{28} + 279 q^{29} + ( - 40 \zeta_{12}^{2} + 40) q^{31} - 32 \zeta_{12} q^{32} + ( - 30 \zeta_{12}^{3} + 30 \zeta_{12}) q^{33} + 48 q^{34} - 104 q^{36} + (76 \zeta_{12}^{3} - 76 \zeta_{12}) q^{37} + 4 \zeta_{12} q^{38} + ( - 44 \zeta_{12}^{2} + 44) q^{39} - 423 q^{41} + (28 \zeta_{12}^{3} + 14 \zeta_{12}) q^{42} - 305 \zeta_{12}^{3} q^{43} - 120 \zeta_{12}^{2} q^{44} + ( - 366 \zeta_{12}^{2} + 366) q^{46} + ( - 456 \zeta_{12}^{3} + 456 \zeta_{12}) q^{47} - 16 \zeta_{12}^{3} q^{48} + (245 \zeta_{12}^{2} - 392) q^{49} + 24 \zeta_{12}^{2} q^{51} - 176 \zeta_{12} q^{52} - 198 \zeta_{12} q^{53} - 106 \zeta_{12}^{2} q^{54} + ( - 56 \zeta_{12}^{2} + 168) q^{56} + 2 \zeta_{12}^{3} q^{57} + ( - 558 \zeta_{12}^{3} + 558 \zeta_{12}) q^{58} + (462 \zeta_{12}^{2} - 462) q^{59} - 281 \zeta_{12}^{2} q^{61} - 80 \zeta_{12}^{3} q^{62} + ( - 546 \zeta_{12}^{3} + 364 \zeta_{12}) q^{63} - 64 q^{64} + ( - 60 \zeta_{12}^{2} + 60) q^{66} + 499 \zeta_{12} q^{67} + ( - 96 \zeta_{12}^{3} + 96 \zeta_{12}) q^{68} + 183 q^{69} - 534 q^{71} + (208 \zeta_{12}^{3} - 208 \zeta_{12}) q^{72} + 800 \zeta_{12} q^{73} + (152 \zeta_{12}^{2} - 152) q^{74} + 8 q^{76} + ( - 210 \zeta_{12}^{3} + 630 \zeta_{12}) q^{77} - 88 \zeta_{12}^{3} q^{78} - 790 \zeta_{12}^{2} q^{79} + (649 \zeta_{12}^{2} - 649) q^{81} + (846 \zeta_{12}^{3} - 846 \zeta_{12}) q^{82} + 597 \zeta_{12}^{3} q^{83} + (56 \zeta_{12}^{2} + 28) q^{84} - 610 \zeta_{12}^{2} q^{86} + 279 \zeta_{12} q^{87} - 240 \zeta_{12} q^{88} + 1017 \zeta_{12}^{2} q^{89} + ( - 308 \zeta_{12}^{2} + 924) q^{91} - 732 \zeta_{12}^{3} q^{92} + ( - 40 \zeta_{12}^{3} + 40 \zeta_{12}) q^{93} + ( - 912 \zeta_{12}^{2} + 912) q^{94} - 32 \zeta_{12}^{2} q^{96} - 1330 \zeta_{12}^{3} q^{97} + (784 \zeta_{12}^{3} - 294 \zeta_{12}) q^{98} - 780 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 \( 4 q + 8 q^{4} + 8 q^{6} - 52 q^{9} + 60 q^{11} + 112 q^{14} - 32 q^{16} + 4 q^{19} - 14 q^{21} + 16 q^{24} - 176 q^{26} + 1116 q^{29} + 80 q^{31} + 192 q^{34} - 416 q^{36} + 88 q^{39} - 1692 q^{41} - 240 q^{44} + 732 q^{46} - 1078 q^{49} + 48 q^{51} - 212 q^{54} + 560 q^{56} - 924 q^{59} - 562 q^{61} - 256 q^{64} + 120 q^{66} + 732 q^{69} - 2136 q^{71} - 304 q^{74} + 32 q^{76} - 1580 q^{79} - 1298 q^{81} + 224 q^{84} - 1220 q^{86} + 2034 q^{89} + 3080 q^{91} + 1824 q^{94} - 64 q^{96} - 3120 q^{99}+O(q^{100}) \) 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 −6.06218 + 17.5000i − 8.00000i −13.0000 + 22.5167i 0
149.2 1.73205 + 1.00000i 0.866025 − 0.500000i 2.00000 + 3.46410i 0 2.00000 6.06218 − 17.5000i 8.00000i −13.0000 + 22.5167i 0
249.1 −1.73205 + 1.00000i −0.866025 − 0.500000i 2.00000 − 3.46410i 0 2.00000 −6.06218 − 17.5000i 8.00000i −13.0000 − 22.5167i 0
249.2 1.73205 − 1.00000i 0.866025 + 0.500000i 2.00000 − 3.46410i 0 2.00000 6.06218 + 17.5000i − 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.f 4
5.b even 2 1 inner 350.4.j.f 4
5.c odd 4 1 70.4.e.a ✓ 2
5.c odd 4 1 350.4.e.g 2
7.c even 3 1 inner 350.4.j.f 4
15.e even 4 1 630.4.k.i 2
20.e even 4 1 560.4.q.e 2
35.f even 4 1 490.4.e.f 2
35.j even 6 1 inner 350.4.j.f 4
35.k even 12 1 490.4.a.k 1
35.k even 12 1 490.4.e.f 2
35.k even 12 1 2450.4.a.m 1
35.l odd 12 1 70.4.e.a ✓ 2
35.l odd 12 1 350.4.e.g 2
35.l odd 12 1 490.4.a.m 1
35.l odd 12 1 2450.4.a.j 1
105.x even 12 1 630.4.k.i 2
140.w even 12 1 560.4.q.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.4.e.a ✓ 2 5.c odd 4 1
70.4.e.a ✓ 2 35.l odd 12 1
350.4.e.g 2 5.c odd 4 1
350.4.e.g 2 35.l odd 12 1
350.4.j.f 4 1.a even 1 1 trivial
350.4.j.f 4 5.b even 2 1 inner
350.4.j.f 4 7.c even 3 1 inner
350.4.j.f 4 35.j even 6 1 inner
490.4.a.k 1 35.k even 12 1
490.4.a.m 1 35.l odd 12 1
490.4.e.f 2 35.f even 4 1
490.4.e.f 2 35.k even 12 1
560.4.q.e 2 20.e even 4 1
560.4.q.e 2 140.w even 12 1
630.4.k.i 2 15.e even 4 1
630.4.k.i 2 105.x even 12 1
2450.4.a.j 1 35.l odd 12 1
2450.4.a.m 1 35.k even 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
\( T_{11}^{2} - 30T_{11} + 900 \) 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
$7$ \( T^{4} + 539 T^{2} + 117649 \) Copy content Toggle raw display
$11$ \( (T^{2} - 30 T + 900)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 1936)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 576 T^{2} + 331776 \) Copy content Toggle raw display
$19$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots + 1121513121 \) Copy content Toggle raw display
$29$ \( (T - 279)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} - 40 T + 1600)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} - 5776 T^{2} + 33362176 \) Copy content Toggle raw display
$41$ \( (T + 423)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 93025)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 43237380096 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 1536953616 \) Copy content Toggle raw display
$59$ \( (T^{2} + 462 T + 213444)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 281 T + 78961)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 62001498001 \) Copy content Toggle raw display
$71$ \( (T + 534)^{4} \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 409600000000 \) Copy content Toggle raw display
$79$ \( (T^{2} + 790 T + 624100)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 356409)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 1017 T + 1034289)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 1768900)^{2} \) Copy content Toggle raw display
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