Properties

Label 350.4.e.e
Level $350$
Weight $4$
Character orbit 350.e
Analytic conductor $20.651$
Analytic rank $0$
Dimension $2$
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] = [350,4,Mod(51,350)]
 
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([0, 2]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("350.51");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 350.e (of order \(3\), degree \(2\), 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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) 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: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 \zeta_{6} q^{2} + (5 \zeta_{6} - 5) q^{3} + (4 \zeta_{6} - 4) q^{4} - 10 q^{6} + ( - 14 \zeta_{6} + 21) q^{7} - 8 q^{8} + 2 \zeta_{6} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 \zeta_{6} q^{2} + (5 \zeta_{6} - 5) q^{3} + (4 \zeta_{6} - 4) q^{4} - 10 q^{6} + ( - 14 \zeta_{6} + 21) q^{7} - 8 q^{8} + 2 \zeta_{6} q^{9} + ( - 57 \zeta_{6} + 57) q^{11} - 20 \zeta_{6} q^{12} + 70 q^{13} + (14 \zeta_{6} + 28) q^{14} - 16 \zeta_{6} q^{16} + ( - 51 \zeta_{6} + 51) q^{17} + (4 \zeta_{6} - 4) q^{18} - 5 \zeta_{6} q^{19} + (105 \zeta_{6} - 35) q^{21} + 114 q^{22} + 69 \zeta_{6} q^{23} + ( - 40 \zeta_{6} + 40) q^{24} + 140 \zeta_{6} q^{26} - 145 q^{27} + (84 \zeta_{6} - 28) q^{28} + 114 q^{29} + (23 \zeta_{6} - 23) q^{31} + ( - 32 \zeta_{6} + 32) q^{32} + 285 \zeta_{6} q^{33} + 102 q^{34} - 8 q^{36} - 253 \zeta_{6} q^{37} + ( - 10 \zeta_{6} + 10) q^{38} + (350 \zeta_{6} - 350) q^{39} - 42 q^{41} + (140 \zeta_{6} - 210) q^{42} + 124 q^{43} + 228 \zeta_{6} q^{44} + (138 \zeta_{6} - 138) q^{46} + 201 \zeta_{6} q^{47} + 80 q^{48} + ( - 392 \zeta_{6} + 245) q^{49} + 255 \zeta_{6} q^{51} + (280 \zeta_{6} - 280) q^{52} + (393 \zeta_{6} - 393) q^{53} - 290 \zeta_{6} q^{54} + (112 \zeta_{6} - 168) q^{56} + 25 q^{57} + 228 \zeta_{6} q^{58} + (219 \zeta_{6} - 219) q^{59} + 709 \zeta_{6} q^{61} - 46 q^{62} + (14 \zeta_{6} + 28) q^{63} + 64 q^{64} + (570 \zeta_{6} - 570) q^{66} + ( - 419 \zeta_{6} + 419) q^{67} + 204 \zeta_{6} q^{68} - 345 q^{69} - 96 q^{71} - 16 \zeta_{6} q^{72} + (313 \zeta_{6} - 313) q^{73} + ( - 506 \zeta_{6} + 506) q^{74} + 20 q^{76} + ( - 1197 \zeta_{6} + 399) q^{77} - 700 q^{78} - 461 \zeta_{6} q^{79} + ( - 671 \zeta_{6} + 671) q^{81} - 84 \zeta_{6} q^{82} + 588 q^{83} + ( - 140 \zeta_{6} - 280) q^{84} + 248 \zeta_{6} q^{86} + (570 \zeta_{6} - 570) q^{87} + (456 \zeta_{6} - 456) q^{88} + 1017 \zeta_{6} q^{89} + ( - 980 \zeta_{6} + 1470) q^{91} - 276 q^{92} - 115 \zeta_{6} q^{93} + (402 \zeta_{6} - 402) q^{94} + 160 \zeta_{6} q^{96} + 1834 q^{97} + ( - 294 \zeta_{6} + 784) q^{98} + 114 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 5 q^{3} - 4 q^{4} - 20 q^{6} + 28 q^{7} - 16 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} - 5 q^{3} - 4 q^{4} - 20 q^{6} + 28 q^{7} - 16 q^{8} + 2 q^{9} + 57 q^{11} - 20 q^{12} + 140 q^{13} + 70 q^{14} - 16 q^{16} + 51 q^{17} - 4 q^{18} - 5 q^{19} + 35 q^{21} + 228 q^{22} + 69 q^{23} + 40 q^{24} + 140 q^{26} - 290 q^{27} + 28 q^{28} + 228 q^{29} - 23 q^{31} + 32 q^{32} + 285 q^{33} + 204 q^{34} - 16 q^{36} - 253 q^{37} + 10 q^{38} - 350 q^{39} - 84 q^{41} - 280 q^{42} + 248 q^{43} + 228 q^{44} - 138 q^{46} + 201 q^{47} + 160 q^{48} + 98 q^{49} + 255 q^{51} - 280 q^{52} - 393 q^{53} - 290 q^{54} - 224 q^{56} + 50 q^{57} + 228 q^{58} - 219 q^{59} + 709 q^{61} - 92 q^{62} + 70 q^{63} + 128 q^{64} - 570 q^{66} + 419 q^{67} + 204 q^{68} - 690 q^{69} - 192 q^{71} - 16 q^{72} - 313 q^{73} + 506 q^{74} + 40 q^{76} - 399 q^{77} - 1400 q^{78} - 461 q^{79} + 671 q^{81} - 84 q^{82} + 1176 q^{83} - 700 q^{84} + 248 q^{86} - 570 q^{87} - 456 q^{88} + 1017 q^{89} + 1960 q^{91} - 552 q^{92} - 115 q^{93} - 402 q^{94} + 160 q^{96} + 3668 q^{97} + 1274 q^{98} + 228 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(101\) \(127\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
51.1
0.500000 0.866025i
0.500000 + 0.866025i
1.00000 1.73205i −2.50000 4.33013i −2.00000 3.46410i 0 −10.0000 14.0000 + 12.1244i −8.00000 1.00000 1.73205i 0
151.1 1.00000 + 1.73205i −2.50000 + 4.33013i −2.00000 + 3.46410i 0 −10.0000 14.0000 12.1244i −8.00000 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
7.c even 3 1 inner

Twists

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

Hecke kernels

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

\( T_{3}^{2} + 5T_{3} + 25 \) Copy content Toggle raw display
\( T_{11}^{2} - 57T_{11} + 3249 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$3$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 28T + 343 \) Copy content Toggle raw display
$11$ \( T^{2} - 57T + 3249 \) Copy content Toggle raw display
$13$ \( (T - 70)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 51T + 2601 \) Copy content Toggle raw display
$19$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$23$ \( T^{2} - 69T + 4761 \) Copy content Toggle raw display
$29$ \( (T - 114)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 23T + 529 \) Copy content Toggle raw display
$37$ \( T^{2} + 253T + 64009 \) Copy content Toggle raw display
$41$ \( (T + 42)^{2} \) Copy content Toggle raw display
$43$ \( (T - 124)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 201T + 40401 \) Copy content Toggle raw display
$53$ \( T^{2} + 393T + 154449 \) Copy content Toggle raw display
$59$ \( T^{2} + 219T + 47961 \) Copy content Toggle raw display
$61$ \( T^{2} - 709T + 502681 \) Copy content Toggle raw display
$67$ \( T^{2} - 419T + 175561 \) Copy content Toggle raw display
$71$ \( (T + 96)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 313T + 97969 \) Copy content Toggle raw display
$79$ \( T^{2} + 461T + 212521 \) Copy content Toggle raw display
$83$ \( (T - 588)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 1017 T + 1034289 \) Copy content Toggle raw display
$97$ \( (T - 1834)^{2} \) Copy content Toggle raw display
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