Properties

Label 350.6.c.h
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.

\(f(q)\) \(=\) \( q - 4 i q^{2} + 23 i q^{3} - 16 q^{4} + 92 q^{6} + 49 i q^{7} + 64 i q^{8} - 286 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - 4 i q^{2} + 23 i q^{3} - 16 q^{4} + 92 q^{6} + 49 i q^{7} + 64 i q^{8} - 286 q^{9} + 555 q^{11} - 368 i q^{12} + 241 i q^{13} + 196 q^{14} + 256 q^{16} - 1491 i q^{17} + 1144 i q^{18} + 2038 q^{19} - 1127 q^{21} - 2220 i q^{22} + 1230 i q^{23} - 1472 q^{24} + 964 q^{26} - 989 i q^{27} - 784 i q^{28} + 5001 q^{29} + 5696 q^{31} - 1024 i q^{32} + 12765 i q^{33} - 5964 q^{34} + 4576 q^{36} - 5602 i q^{37} - 8152 i q^{38} - 5543 q^{39} - 2424 q^{41} + 4508 i q^{42} - 602 i q^{43} - 8880 q^{44} + 4920 q^{46} - 23163 i q^{47} + 5888 i q^{48} - 2401 q^{49} + 34293 q^{51} - 3856 i q^{52} + 25296 i q^{53} - 3956 q^{54} - 3136 q^{56} + 46874 i q^{57} - 20004 i q^{58} - 5724 q^{59} - 36112 q^{61} - 22784 i q^{62} - 14014 i q^{63} - 4096 q^{64} + 51060 q^{66} + 66104 i q^{67} + 23856 i q^{68} - 28290 q^{69} + 16080 q^{71} - 18304 i q^{72} + 80482 i q^{73} - 22408 q^{74} - 32608 q^{76} + 27195 i q^{77} + 22172 i q^{78} + 64147 q^{79} - 46751 q^{81} + 9696 i q^{82} + 106284 i q^{83} + 18032 q^{84} - 2408 q^{86} + 115023 i q^{87} + 35520 i q^{88} + 71676 q^{89} - 11809 q^{91} - 19680 i q^{92} + 131008 i q^{93} - 92652 q^{94} + 23552 q^{96} + 151025 i q^{97} + 9604 i q^{98} - 158730 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 32 q^{4} + 184 q^{6} - 572 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 32 q^{4} + 184 q^{6} - 572 q^{9} + 1110 q^{11} + 392 q^{14} + 512 q^{16} + 4076 q^{19} - 2254 q^{21} - 2944 q^{24} + 1928 q^{26} + 10002 q^{29} + 11392 q^{31} - 11928 q^{34} + 9152 q^{36} - 11086 q^{39} - 4848 q^{41} - 17760 q^{44} + 9840 q^{46} - 4802 q^{49} + 68586 q^{51} - 7912 q^{54} - 6272 q^{56} - 11448 q^{59} - 72224 q^{61} - 8192 q^{64} + 102120 q^{66} - 56580 q^{69} + 32160 q^{71} - 44816 q^{74} - 65216 q^{76} + 128294 q^{79} - 93502 q^{81} + 36064 q^{84} - 4816 q^{86} + 143352 q^{89} - 23618 q^{91} - 185304 q^{94} + 47104 q^{96} - 317460 q^{99}+O(q^{100}) \) 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 23.0000i −16.0000 0 92.0000 49.0000i 64.0000i −286.000 0
99.2 4.00000i 23.0000i −16.0000 0 92.0000 49.0000i 64.0000i −286.000 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.h 2
5.b even 2 1 inner 350.6.c.h 2
5.c odd 4 1 70.6.a.a 1
5.c odd 4 1 350.6.a.n 1
15.e even 4 1 630.6.a.j 1
20.e even 4 1 560.6.a.i 1
35.f even 4 1 490.6.a.i 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.6.a.a 1 5.c odd 4 1
350.6.a.n 1 5.c odd 4 1
350.6.c.h 2 1.a even 1 1 trivial
350.6.c.h 2 5.b even 2 1 inner
490.6.a.i 1 35.f even 4 1
560.6.a.i 1 20.e even 4 1
630.6.a.j 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 \(S_{6}^{\mathrm{new}}(350, [\chi])\):

\( T_{3}^{2} + 529 \) Copy content Toggle raw display
\( T_{11} - 555 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 16 \) Copy content Toggle raw display
$3$ \( T^{2} + 529 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 2401 \) Copy content Toggle raw display
$11$ \( (T - 555)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 58081 \) Copy content Toggle raw display
$17$ \( T^{2} + 2223081 \) Copy content Toggle raw display
$19$ \( (T - 2038)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 1512900 \) Copy content Toggle raw display
$29$ \( (T - 5001)^{2} \) Copy content Toggle raw display
$31$ \( (T - 5696)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 31382404 \) Copy content Toggle raw display
$41$ \( (T + 2424)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 362404 \) Copy content Toggle raw display
$47$ \( T^{2} + 536524569 \) Copy content Toggle raw display
$53$ \( T^{2} + 639887616 \) Copy content Toggle raw display
$59$ \( (T + 5724)^{2} \) Copy content Toggle raw display
$61$ \( (T + 36112)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 4369738816 \) Copy content Toggle raw display
$71$ \( (T - 16080)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 6477352324 \) Copy content Toggle raw display
$79$ \( (T - 64147)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 11296288656 \) Copy content Toggle raw display
$89$ \( (T - 71676)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 22808550625 \) Copy content Toggle raw display
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