Properties

Label 350.6.c.f
Level $350$
Weight $6$
Character orbit 350.c
Analytic conductor $56.134$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [350,6,Mod(99,350)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(350, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("350.99");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 350.c (of order \(2\), degree \(1\), 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: \(56.1343369345\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 14)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 4 i q^{2} + 8 i q^{3} - 16 q^{4} + 32 q^{6} + 49 i q^{7} + 64 i q^{8} + 179 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - 4 i q^{2} + 8 i q^{3} - 16 q^{4} + 32 q^{6} + 49 i q^{7} + 64 i q^{8} + 179 q^{9} - 340 q^{11} - 128 i q^{12} - 294 i q^{13} + 196 q^{14} + 256 q^{16} - 1226 i q^{17} - 716 i q^{18} - 2432 q^{19} - 392 q^{21} + 1360 i q^{22} + 2000 i q^{23} - 512 q^{24} - 1176 q^{26} + 3376 i q^{27} - 784 i q^{28} + 6746 q^{29} + 8856 q^{31} - 1024 i q^{32} - 2720 i q^{33} - 4904 q^{34} - 2864 q^{36} - 9182 i q^{37} + 9728 i q^{38} + 2352 q^{39} - 14574 q^{41} + 1568 i q^{42} + 8108 i q^{43} + 5440 q^{44} + 8000 q^{46} + 312 i q^{47} + 2048 i q^{48} - 2401 q^{49} + 9808 q^{51} + 4704 i q^{52} - 14634 i q^{53} + 13504 q^{54} - 3136 q^{56} - 19456 i q^{57} - 26984 i q^{58} + 27656 q^{59} + 34338 q^{61} - 35424 i q^{62} + 8771 i q^{63} - 4096 q^{64} - 10880 q^{66} - 12316 i q^{67} + 19616 i q^{68} - 16000 q^{69} + 36920 q^{71} + 11456 i q^{72} - 61718 i q^{73} - 36728 q^{74} + 38912 q^{76} - 16660 i q^{77} - 9408 i q^{78} + 64752 q^{79} + 16489 q^{81} + 58296 i q^{82} - 77056 i q^{83} + 6272 q^{84} + 32432 q^{86} + 53968 i q^{87} - 21760 i q^{88} + 8166 q^{89} + 14406 q^{91} - 32000 i q^{92} + 70848 i q^{93} + 1248 q^{94} + 8192 q^{96} - 20650 i q^{97} + 9604 i q^{98} - 60860 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 32 q^{4} + 64 q^{6} + 358 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 32 q^{4} + 64 q^{6} + 358 q^{9} - 680 q^{11} + 392 q^{14} + 512 q^{16} - 4864 q^{19} - 784 q^{21} - 1024 q^{24} - 2352 q^{26} + 13492 q^{29} + 17712 q^{31} - 9808 q^{34} - 5728 q^{36} + 4704 q^{39} - 29148 q^{41} + 10880 q^{44} + 16000 q^{46} - 4802 q^{49} + 19616 q^{51} + 27008 q^{54} - 6272 q^{56} + 55312 q^{59} + 68676 q^{61} - 8192 q^{64} - 21760 q^{66} - 32000 q^{69} + 73840 q^{71} - 73456 q^{74} + 77824 q^{76} + 129504 q^{79} + 32978 q^{81} + 12544 q^{84} + 64864 q^{86} + 16332 q^{89} + 28812 q^{91} + 2496 q^{94} + 16384 q^{96} - 121720 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
99.1
1.00000i
1.00000i
4.00000i 8.00000i −16.0000 0 32.0000 49.0000i 64.0000i 179.000 0
99.2 4.00000i 8.00000i −16.0000 0 32.0000 49.0000i 64.0000i 179.000 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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 350.6.c.f 2
5.b even 2 1 inner 350.6.c.f 2
5.c odd 4 1 14.6.a.b 1
5.c odd 4 1 350.6.a.b 1
15.e even 4 1 126.6.a.c 1
20.e even 4 1 112.6.a.d 1
35.f even 4 1 98.6.a.b 1
35.k even 12 2 98.6.c.b 2
35.l odd 12 2 98.6.c.a 2
40.i odd 4 1 448.6.a.f 1
40.k even 4 1 448.6.a.k 1
60.l odd 4 1 1008.6.a.n 1
105.k odd 4 1 882.6.a.g 1
140.j odd 4 1 784.6.a.h 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.6.a.b 1 5.c odd 4 1
98.6.a.b 1 35.f even 4 1
98.6.c.a 2 35.l odd 12 2
98.6.c.b 2 35.k even 12 2
112.6.a.d 1 20.e even 4 1
126.6.a.c 1 15.e even 4 1
350.6.a.b 1 5.c odd 4 1
350.6.c.f 2 1.a even 1 1 trivial
350.6.c.f 2 5.b even 2 1 inner
448.6.a.f 1 40.i odd 4 1
448.6.a.k 1 40.k even 4 1
784.6.a.h 1 140.j odd 4 1
882.6.a.g 1 105.k odd 4 1
1008.6.a.n 1 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(350, [\chi])\):

\( T_{3}^{2} + 64 \) Copy content Toggle raw display
\( T_{11} + 340 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 16 \) Copy content Toggle raw display
$3$ \( T^{2} + 64 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 2401 \) Copy content Toggle raw display
$11$ \( (T + 340)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 86436 \) Copy content Toggle raw display
$17$ \( T^{2} + 1503076 \) Copy content Toggle raw display
$19$ \( (T + 2432)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 4000000 \) Copy content Toggle raw display
$29$ \( (T - 6746)^{2} \) Copy content Toggle raw display
$31$ \( (T - 8856)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 84309124 \) Copy content Toggle raw display
$41$ \( (T + 14574)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 65739664 \) Copy content Toggle raw display
$47$ \( T^{2} + 97344 \) Copy content Toggle raw display
$53$ \( T^{2} + 214153956 \) Copy content Toggle raw display
$59$ \( (T - 27656)^{2} \) Copy content Toggle raw display
$61$ \( (T - 34338)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 151683856 \) Copy content Toggle raw display
$71$ \( (T - 36920)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 3809111524 \) Copy content Toggle raw display
$79$ \( (T - 64752)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 5937627136 \) Copy content Toggle raw display
$89$ \( (T - 8166)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 426422500 \) Copy content Toggle raw display
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