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Properties

Label 350.4.j.b
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.

\(f(q)\) \(=\) \( q + (2 \zeta_{12}^{3} - 2 \zeta_{12}) q^{2} + 5 \zeta_{12} q^{3} + ( - 4 \zeta_{12}^{2} + 4) q^{4} - 10 q^{6} + (7 \zeta_{12}^{3} + 14 \zeta_{12}) q^{7} + 8 \zeta_{12}^{3} q^{8} - 2 \zeta_{12}^{2} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (2 \zeta_{12}^{3} - 2 \zeta_{12}) q^{2} + 5 \zeta_{12} q^{3} + ( - 4 \zeta_{12}^{2} + 4) q^{4} - 10 q^{6} + (7 \zeta_{12}^{3} + 14 \zeta_{12}) q^{7} + 8 \zeta_{12}^{3} q^{8} - 2 \zeta_{12}^{2} q^{9} + ( - 57 \zeta_{12}^{2} + 57) q^{11} + ( - 20 \zeta_{12}^{3} + 20 \zeta_{12}) q^{12} - 70 \zeta_{12}^{3} q^{13} + ( - 14 \zeta_{12}^{2} - 28) q^{14} - 16 \zeta_{12}^{2} q^{16} + 51 \zeta_{12} q^{17} + 4 \zeta_{12} q^{18} + 5 \zeta_{12}^{2} q^{19} + (105 \zeta_{12}^{2} - 35) q^{21} + 114 \zeta_{12}^{3} q^{22} + ( - 69 \zeta_{12}^{3} + 69 \zeta_{12}) q^{23} + (40 \zeta_{12}^{2} - 40) q^{24} + 140 \zeta_{12}^{2} q^{26} - 145 \zeta_{12}^{3} q^{27} + ( - 56 \zeta_{12}^{3} + 84 \zeta_{12}) q^{28} - 114 q^{29} + (23 \zeta_{12}^{2} - 23) q^{31} + 32 \zeta_{12} q^{32} + ( - 285 \zeta_{12}^{3} + 285 \zeta_{12}) q^{33} - 102 q^{34} - 8 q^{36} + ( - 253 \zeta_{12}^{3} + 253 \zeta_{12}) q^{37} - 10 \zeta_{12} q^{38} + ( - 350 \zeta_{12}^{2} + 350) q^{39} - 42 q^{41} + ( - 70 \zeta_{12}^{3} - 140 \zeta_{12}) q^{42} - 124 \zeta_{12}^{3} q^{43} - 228 \zeta_{12}^{2} q^{44} + (138 \zeta_{12}^{2} - 138) q^{46} + (201 \zeta_{12}^{3} - 201 \zeta_{12}) q^{47} - 80 \zeta_{12}^{3} q^{48} + (392 \zeta_{12}^{2} - 245) q^{49} + 255 \zeta_{12}^{2} q^{51} - 280 \zeta_{12} q^{52} + 393 \zeta_{12} q^{53} + 290 \zeta_{12}^{2} q^{54} + (112 \zeta_{12}^{2} - 168) q^{56} + 25 \zeta_{12}^{3} q^{57} + ( - 228 \zeta_{12}^{3} + 228 \zeta_{12}) q^{58} + ( - 219 \zeta_{12}^{2} + 219) q^{59} + 709 \zeta_{12}^{2} q^{61} - 46 \zeta_{12}^{3} q^{62} + ( - 42 \zeta_{12}^{3} + 14 \zeta_{12}) q^{63} - 64 q^{64} + (570 \zeta_{12}^{2} - 570) q^{66} + 419 \zeta_{12} q^{67} + ( - 204 \zeta_{12}^{3} + 204 \zeta_{12}) q^{68} + 345 q^{69} - 96 q^{71} + ( - 16 \zeta_{12}^{3} + 16 \zeta_{12}) q^{72} + 313 \zeta_{12} q^{73} + (506 \zeta_{12}^{2} - 506) q^{74} + 20 q^{76} + ( - 798 \zeta_{12}^{3} + 1197 \zeta_{12}) q^{77} + 700 \zeta_{12}^{3} q^{78} + 461 \zeta_{12}^{2} q^{79} + ( - 671 \zeta_{12}^{2} + 671) q^{81} + ( - 84 \zeta_{12}^{3} + 84 \zeta_{12}) q^{82} - 588 \zeta_{12}^{3} q^{83} + (140 \zeta_{12}^{2} + 280) q^{84} + 248 \zeta_{12}^{2} q^{86} - 570 \zeta_{12} q^{87} + 456 \zeta_{12} q^{88} - 1017 \zeta_{12}^{2} q^{89} + ( - 980 \zeta_{12}^{2} + 1470) q^{91} - 276 \zeta_{12}^{3} q^{92} + (115 \zeta_{12}^{3} - 115 \zeta_{12}) q^{93} + ( - 402 \zeta_{12}^{2} + 402) q^{94} + 160 \zeta_{12}^{2} q^{96} + 1834 \zeta_{12}^{3} q^{97} + ( - 490 \zeta_{12}^{3} - 294 \zeta_{12}) q^{98} - 114 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{4} - 40 q^{6} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 8 q^{4} - 40 q^{6} - 4 q^{9} + 114 q^{11} - 140 q^{14} - 32 q^{16} + 10 q^{19} + 70 q^{21} - 80 q^{24} + 280 q^{26} - 456 q^{29} - 46 q^{31} - 408 q^{34} - 32 q^{36} + 700 q^{39} - 168 q^{41} - 456 q^{44} - 276 q^{46} - 196 q^{49} + 510 q^{51} + 580 q^{54} - 448 q^{56} + 438 q^{59} + 1418 q^{61} - 256 q^{64} - 1140 q^{66} + 1380 q^{69} - 384 q^{71} - 1012 q^{74} + 80 q^{76} + 922 q^{79} + 1342 q^{81} + 1400 q^{84} + 496 q^{86} - 2034 q^{89} + 3920 q^{91} + 804 q^{94} + 320 q^{96} - 456 q^{99}+O(q^{100}) \) 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 4.33013 − 2.50000i 2.00000 + 3.46410i 0 −10.0000 12.1244 − 14.0000i − 8.00000i −1.00000 + 1.73205i 0
149.2 1.73205 + 1.00000i −4.33013 + 2.50000i 2.00000 + 3.46410i 0 −10.0000 −12.1244 + 14.0000i 8.00000i −1.00000 + 1.73205i 0
249.1 −1.73205 + 1.00000i 4.33013 + 2.50000i 2.00000 − 3.46410i 0 −10.0000 12.1244 + 14.0000i 8.00000i −1.00000 − 1.73205i 0
249.2 1.73205 − 1.00000i −4.33013 − 2.50000i 2.00000 − 3.46410i 0 −10.0000 −12.1244 − 14.0000i − 8.00000i −1.00000 − 1.73205i 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.b 4
5.b even 2 1 inner 350.4.j.b 4
5.c odd 4 1 14.4.c.a ✓ 2
5.c odd 4 1 350.4.e.e 2
7.c even 3 1 inner 350.4.j.b 4
15.e even 4 1 126.4.g.d 2
20.e even 4 1 112.4.i.a 2
35.f even 4 1 98.4.c.a 2
35.j even 6 1 inner 350.4.j.b 4
35.k even 12 1 98.4.a.f 1
35.k even 12 1 98.4.c.a 2
35.k even 12 1 2450.4.a.d 1
35.l odd 12 1 14.4.c.a ✓ 2
35.l odd 12 1 98.4.a.d 1
35.l odd 12 1 350.4.e.e 2
35.l odd 12 1 2450.4.a.q 1
40.i odd 4 1 448.4.i.b 2
40.k even 4 1 448.4.i.e 2
105.k odd 4 1 882.4.g.u 2
105.w odd 12 1 882.4.a.c 1
105.w odd 12 1 882.4.g.u 2
105.x even 12 1 126.4.g.d 2
105.x even 12 1 882.4.a.f 1
140.w even 12 1 112.4.i.a 2
140.w even 12 1 784.4.a.p 1
140.x odd 12 1 784.4.a.c 1
280.br even 12 1 448.4.i.e 2
280.bt odd 12 1 448.4.i.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.4.c.a ✓ 2 5.c odd 4 1
14.4.c.a ✓ 2 35.l odd 12 1
98.4.a.d 1 35.l odd 12 1
98.4.a.f 1 35.k even 12 1
98.4.c.a 2 35.f even 4 1
98.4.c.a 2 35.k even 12 1
112.4.i.a 2 20.e even 4 1
112.4.i.a 2 140.w even 12 1
126.4.g.d 2 15.e even 4 1
126.4.g.d 2 105.x even 12 1
350.4.e.e 2 5.c odd 4 1
350.4.e.e 2 35.l odd 12 1
350.4.j.b 4 1.a even 1 1 trivial
350.4.j.b 4 5.b even 2 1 inner
350.4.j.b 4 7.c even 3 1 inner
350.4.j.b 4 35.j even 6 1 inner
448.4.i.b 2 40.i odd 4 1
448.4.i.b 2 280.bt odd 12 1
448.4.i.e 2 40.k even 4 1
448.4.i.e 2 280.br even 12 1
784.4.a.c 1 140.x odd 12 1
784.4.a.p 1 140.w even 12 1
882.4.a.c 1 105.w odd 12 1
882.4.a.f 1 105.x even 12 1
882.4.g.u 2 105.k odd 4 1
882.4.g.u 2 105.w odd 12 1
2450.4.a.d 1 35.k even 12 1
2450.4.a.q 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 \(S_{4}^{\mathrm{new}}(350, [\chi])\):

\( T_{3}^{4} - 25T_{3}^{2} + 625 \) Copy content Toggle raw display
\( T_{11}^{2} - 57T_{11} + 3249 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$3$ \( T^{4} - 25T^{2} + 625 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 98 T^{2} + 117649 \) Copy content Toggle raw display
$11$ \( (T^{2} - 57 T + 3249)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 4900)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 2601 T^{2} + 6765201 \) Copy content Toggle raw display
$19$ \( (T^{2} - 5 T + 25)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} - 4761 T^{2} + 22667121 \) Copy content Toggle raw display
$29$ \( (T + 114)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 23 T + 529)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots + 4097152081 \) Copy content Toggle raw display
$41$ \( (T + 42)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 15376)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 1632240801 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 23854493601 \) Copy content Toggle raw display
$59$ \( (T^{2} - 219 T + 47961)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 709 T + 502681)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 30821664721 \) Copy content Toggle raw display
$71$ \( (T + 96)^{4} \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 9597924961 \) Copy content Toggle raw display
$79$ \( (T^{2} - 461 T + 212521)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 345744)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 1017 T + 1034289)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 3363556)^{2} \) Copy content Toggle raw display
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