Properties

Label 350.6.c.b
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 70)
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.

\(f(q)\) \(=\) \( q + 4 i q^{2} + 11 i q^{3} - 16 q^{4} - 44 q^{6} + 49 i q^{7} - 64 i q^{8} + 122 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 4 i q^{2} + 11 i q^{3} - 16 q^{4} - 44 q^{6} + 49 i q^{7} - 64 i q^{8} + 122 q^{9} - 267 q^{11} - 176 i q^{12} + 1087 i q^{13} - 196 q^{14} + 256 q^{16} - 513 i q^{17} + 488 i q^{18} + 802 q^{19} - 539 q^{21} - 1068 i q^{22} + 1290 i q^{23} + 704 q^{24} - 4348 q^{26} + 4015 i q^{27} - 784 i q^{28} - 1779 q^{29} - 2584 q^{31} + 1024 i q^{32} - 2937 i q^{33} + 2052 q^{34} - 1952 q^{36} + 13862 i q^{37} + 3208 i q^{38} - 11957 q^{39} - 11904 q^{41} - 2156 i q^{42} + 598 i q^{43} + 4272 q^{44} - 5160 q^{46} - 17019 i q^{47} + 2816 i q^{48} - 2401 q^{49} + 5643 q^{51} - 17392 i q^{52} - 27852 i q^{53} - 16060 q^{54} + 3136 q^{56} + 8822 i q^{57} - 7116 i q^{58} - 30912 q^{59} - 1780 q^{61} - 10336 i q^{62} + 5978 i q^{63} - 4096 q^{64} + 11748 q^{66} + 25052 i q^{67} + 8208 i q^{68} - 14190 q^{69} - 51984 q^{71} - 7808 i q^{72} - 47690 i q^{73} - 55448 q^{74} - 12832 q^{76} - 13083 i q^{77} - 47828 i q^{78} + 102121 q^{79} - 14519 q^{81} - 47616 i q^{82} + 83676 i q^{83} + 8624 q^{84} - 2392 q^{86} - 19569 i q^{87} + 17088 i q^{88} + 32400 q^{89} - 53263 q^{91} - 20640 i q^{92} - 28424 i q^{93} + 68076 q^{94} - 11264 q^{96} - 148645 i q^{97} - 9604 i q^{98} - 32574 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 32 q^{4} - 88 q^{6} + 244 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 32 q^{4} - 88 q^{6} + 244 q^{9} - 534 q^{11} - 392 q^{14} + 512 q^{16} + 1604 q^{19} - 1078 q^{21} + 1408 q^{24} - 8696 q^{26} - 3558 q^{29} - 5168 q^{31} + 4104 q^{34} - 3904 q^{36} - 23914 q^{39} - 23808 q^{41} + 8544 q^{44} - 10320 q^{46} - 4802 q^{49} + 11286 q^{51} - 32120 q^{54} + 6272 q^{56} - 61824 q^{59} - 3560 q^{61} - 8192 q^{64} + 23496 q^{66} - 28380 q^{69} - 103968 q^{71} - 110896 q^{74} - 25664 q^{76} + 204242 q^{79} - 29038 q^{81} + 17248 q^{84} - 4784 q^{86} + 64800 q^{89} - 106526 q^{91} + 136152 q^{94} - 22528 q^{96} - 65148 q^{99}+O(q^{100}) \) 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 11.0000i −16.0000 0 −44.0000 49.0000i 64.0000i 122.000 0
99.2 4.00000i 11.0000i −16.0000 0 −44.0000 49.0000i 64.0000i 122.000 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.b 2
5.b even 2 1 inner 350.6.c.b 2
5.c odd 4 1 70.6.a.f 1
5.c odd 4 1 350.6.a.d 1
15.e even 4 1 630.6.a.e 1
20.e even 4 1 560.6.a.f 1
35.f even 4 1 490.6.a.l 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.6.a.f 1 5.c odd 4 1
350.6.a.d 1 5.c odd 4 1
350.6.c.b 2 1.a even 1 1 trivial
350.6.c.b 2 5.b even 2 1 inner
490.6.a.l 1 35.f even 4 1
560.6.a.f 1 20.e even 4 1
630.6.a.e 1 15.e even 4 1

Hecke kernels

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

\( T_{3}^{2} + 121 \) Copy content Toggle raw display
\( T_{11} + 267 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 16 \) Copy content Toggle raw display
$3$ \( T^{2} + 121 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 2401 \) Copy content Toggle raw display
$11$ \( (T + 267)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 1181569 \) Copy content Toggle raw display
$17$ \( T^{2} + 263169 \) Copy content Toggle raw display
$19$ \( (T - 802)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 1664100 \) Copy content Toggle raw display
$29$ \( (T + 1779)^{2} \) Copy content Toggle raw display
$31$ \( (T + 2584)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 192155044 \) Copy content Toggle raw display
$41$ \( (T + 11904)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 357604 \) Copy content Toggle raw display
$47$ \( T^{2} + 289646361 \) Copy content Toggle raw display
$53$ \( T^{2} + 775733904 \) Copy content Toggle raw display
$59$ \( (T + 30912)^{2} \) Copy content Toggle raw display
$61$ \( (T + 1780)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 627602704 \) Copy content Toggle raw display
$71$ \( (T + 51984)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 2274336100 \) Copy content Toggle raw display
$79$ \( (T - 102121)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 7001672976 \) Copy content Toggle raw display
$89$ \( (T - 32400)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 22095336025 \) Copy content Toggle raw display
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