Properties

Label 350.4.c.h
Level $350$
Weight $4$
Character orbit 350.c
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(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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("350.99");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
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: \(20.6506685020\)
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.

\(f(q)\) \(=\) \( q - 2 i q^{2} + i q^{3} - 4 q^{4} + 2 q^{6} + 7 i q^{7} + 8 i q^{8} + 26 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - 2 i q^{2} + i q^{3} - 4 q^{4} + 2 q^{6} + 7 i q^{7} + 8 i q^{8} + 26 q^{9} - 65 q^{11} - 4 i q^{12} - 13 i q^{13} + 14 q^{14} + 16 q^{16} - 73 i q^{17} - 52 i q^{18} + 142 q^{19} - 7 q^{21} + 130 i q^{22} - 130 i q^{23} - 8 q^{24} - 26 q^{26} + 53 i q^{27} - 28 i q^{28} - 111 q^{29} + 256 q^{31} - 32 i q^{32} - 65 i q^{33} - 146 q^{34} - 104 q^{36} - 266 i q^{37} - 284 i q^{38} + 13 q^{39} - 424 q^{41} + 14 i q^{42} - 534 i q^{43} + 260 q^{44} - 260 q^{46} - 269 i q^{47} + 16 i q^{48} - 49 q^{49} + 73 q^{51} + 52 i q^{52} + 132 i q^{53} + 106 q^{54} - 56 q^{56} + 142 i q^{57} + 222 i q^{58} + 224 q^{59} - 572 q^{61} - 512 i q^{62} + 182 i q^{63} - 64 q^{64} - 130 q^{66} - 108 i q^{67} + 292 i q^{68} + 130 q^{69} + 560 q^{71} + 208 i q^{72} - 586 i q^{73} - 532 q^{74} - 568 q^{76} - 455 i q^{77} - 26 i q^{78} - 57 q^{79} + 649 q^{81} + 848 i q^{82} - 252 i q^{83} + 28 q^{84} - 1068 q^{86} - 111 i q^{87} - 520 i q^{88} + 184 q^{89} + 91 q^{91} + 520 i q^{92} + 256 i q^{93} - 538 q^{94} + 32 q^{96} - 605 i q^{97} + 98 i q^{98} - 1690 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{4} + 4 q^{6} + 52 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 8 q^{4} + 4 q^{6} + 52 q^{9} - 130 q^{11} + 28 q^{14} + 32 q^{16} + 284 q^{19} - 14 q^{21} - 16 q^{24} - 52 q^{26} - 222 q^{29} + 512 q^{31} - 292 q^{34} - 208 q^{36} + 26 q^{39} - 848 q^{41} + 520 q^{44} - 520 q^{46} - 98 q^{49} + 146 q^{51} + 212 q^{54} - 112 q^{56} + 448 q^{59} - 1144 q^{61} - 128 q^{64} - 260 q^{66} + 260 q^{69} + 1120 q^{71} - 1064 q^{74} - 1136 q^{76} - 114 q^{79} + 1298 q^{81} + 56 q^{84} - 2136 q^{86} + 368 q^{89} + 182 q^{91} - 1076 q^{94} + 64 q^{96} - 3380 q^{99}+O(q^{100}) \) 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
2.00000i 1.00000i −4.00000 0 2.00000 7.00000i 8.00000i 26.0000 0
99.2 2.00000i 1.00000i −4.00000 0 2.00000 7.00000i 8.00000i 26.0000 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 350.4.c.h 2
5.b even 2 1 inner 350.4.c.h 2
5.c odd 4 1 70.4.a.c 1
5.c odd 4 1 350.4.a.r 1
15.e even 4 1 630.4.a.x 1
20.e even 4 1 560.4.a.i 1
35.f even 4 1 490.4.a.d 1
35.f even 4 1 2450.4.a.bc 1
35.k even 12 2 490.4.e.n 2
35.l odd 12 2 490.4.e.o 2
40.i odd 4 1 2240.4.a.v 1
40.k even 4 1 2240.4.a.r 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.4.a.c 1 5.c odd 4 1
350.4.a.r 1 5.c odd 4 1
350.4.c.h 2 1.a even 1 1 trivial
350.4.c.h 2 5.b even 2 1 inner
490.4.a.d 1 35.f even 4 1
490.4.e.n 2 35.k even 12 2
490.4.e.o 2 35.l odd 12 2
560.4.a.i 1 20.e even 4 1
630.4.a.x 1 15.e even 4 1
2240.4.a.r 1 40.k even 4 1
2240.4.a.v 1 40.i odd 4 1
2450.4.a.bc 1 35.f even 4 1

Hecke kernels

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

\( T_{3}^{2} + 1 \) Copy content Toggle raw display
\( T_{11} + 65 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 4 \) Copy content Toggle raw display
$3$ \( T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 49 \) Copy content Toggle raw display
$11$ \( (T + 65)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 169 \) Copy content Toggle raw display
$17$ \( T^{2} + 5329 \) Copy content Toggle raw display
$19$ \( (T - 142)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 16900 \) Copy content Toggle raw display
$29$ \( (T + 111)^{2} \) Copy content Toggle raw display
$31$ \( (T - 256)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 70756 \) Copy content Toggle raw display
$41$ \( (T + 424)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 285156 \) Copy content Toggle raw display
$47$ \( T^{2} + 72361 \) Copy content Toggle raw display
$53$ \( T^{2} + 17424 \) Copy content Toggle raw display
$59$ \( (T - 224)^{2} \) Copy content Toggle raw display
$61$ \( (T + 572)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 11664 \) Copy content Toggle raw display
$71$ \( (T - 560)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 343396 \) Copy content Toggle raw display
$79$ \( (T + 57)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 63504 \) Copy content Toggle raw display
$89$ \( (T - 184)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 366025 \) Copy content Toggle raw display
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