Properties

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

Related objects

Downloads

Learn more

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.

\(f(q)\) \(=\) \( q + 4 i q^{2} + 10 i q^{3} - 16 q^{4} - 40 q^{6} - 49 i q^{7} - 64 i q^{8} + 143 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 4 i q^{2} + 10 i q^{3} - 16 q^{4} - 40 q^{6} - 49 i q^{7} - 64 i q^{8} + 143 q^{9} - 336 q^{11} - 160 i q^{12} + 584 i q^{13} + 196 q^{14} + 256 q^{16} + 1458 i q^{17} + 572 i q^{18} - 470 q^{19} + 490 q^{21} - 1344 i q^{22} - 4200 i q^{23} + 640 q^{24} - 2336 q^{26} + 3860 i q^{27} + 784 i q^{28} - 4866 q^{29} - 7372 q^{31} + 1024 i q^{32} - 3360 i q^{33} - 5832 q^{34} - 2288 q^{36} - 14330 i q^{37} - 1880 i q^{38} - 5840 q^{39} + 6222 q^{41} + 1960 i q^{42} + 3704 i q^{43} + 5376 q^{44} + 16800 q^{46} + 1812 i q^{47} + 2560 i q^{48} - 2401 q^{49} - 14580 q^{51} - 9344 i q^{52} - 37242 i q^{53} - 15440 q^{54} - 3136 q^{56} - 4700 i q^{57} - 19464 i q^{58} - 34302 q^{59} + 24476 q^{61} - 29488 i q^{62} - 7007 i q^{63} - 4096 q^{64} + 13440 q^{66} + 17452 i q^{67} - 23328 i q^{68} + 42000 q^{69} + 28224 q^{71} - 9152 i q^{72} + 3602 i q^{73} + 57320 q^{74} + 7520 q^{76} + 16464 i q^{77} - 23360 i q^{78} - 42872 q^{79} - 3851 q^{81} + 24888 i q^{82} - 35202 i q^{83} - 7840 q^{84} - 14816 q^{86} - 48660 i q^{87} + 21504 i q^{88} - 26730 q^{89} + 28616 q^{91} + 67200 i q^{92} - 73720 i q^{93} - 7248 q^{94} - 10240 q^{96} + 16978 i q^{97} - 9604 i q^{98} - 48048 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 32 q^{4} - 80 q^{6} + 286 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 32 q^{4} - 80 q^{6} + 286 q^{9} - 672 q^{11} + 392 q^{14} + 512 q^{16} - 940 q^{19} + 980 q^{21} + 1280 q^{24} - 4672 q^{26} - 9732 q^{29} - 14744 q^{31} - 11664 q^{34} - 4576 q^{36} - 11680 q^{39} + 12444 q^{41} + 10752 q^{44} + 33600 q^{46} - 4802 q^{49} - 29160 q^{51} - 30880 q^{54} - 6272 q^{56} - 68604 q^{59} + 48952 q^{61} - 8192 q^{64} + 26880 q^{66} + 84000 q^{69} + 56448 q^{71} + 114640 q^{74} + 15040 q^{76} - 85744 q^{79} - 7702 q^{81} - 15680 q^{84} - 29632 q^{86} - 53460 q^{89} + 57232 q^{91} - 14496 q^{94} - 20480 q^{96} - 96096 q^{99}+O(q^{100}) \) 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 10.0000i −16.0000 0 −40.0000 49.0000i 64.0000i 143.000 0
99.2 4.00000i 10.0000i −16.0000 0 −40.0000 49.0000i 64.0000i 143.000 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.d 2
5.b even 2 1 inner 350.6.c.d 2
5.c odd 4 1 14.6.a.a 1
5.c odd 4 1 350.6.a.i 1
15.e even 4 1 126.6.a.f 1
20.e even 4 1 112.6.a.c 1
35.f even 4 1 98.6.a.a 1
35.k even 12 2 98.6.c.d 2
35.l odd 12 2 98.6.c.c 2
40.i odd 4 1 448.6.a.e 1
40.k even 4 1 448.6.a.l 1
60.l odd 4 1 1008.6.a.b 1
105.k odd 4 1 882.6.a.x 1
140.j odd 4 1 784.6.a.i 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.6.a.a 1 5.c odd 4 1
98.6.a.a 1 35.f even 4 1
98.6.c.c 2 35.l odd 12 2
98.6.c.d 2 35.k even 12 2
112.6.a.c 1 20.e even 4 1
126.6.a.f 1 15.e even 4 1
350.6.a.i 1 5.c odd 4 1
350.6.c.d 2 1.a even 1 1 trivial
350.6.c.d 2 5.b even 2 1 inner
448.6.a.e 1 40.i odd 4 1
448.6.a.l 1 40.k even 4 1
784.6.a.i 1 140.j odd 4 1
882.6.a.x 1 105.k odd 4 1
1008.6.a.b 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 \(S_{6}^{\mathrm{new}}(350, [\chi])\):

\( T_{3}^{2} + 100 \) Copy content Toggle raw display
\( T_{11} + 336 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 16 \) Copy content Toggle raw display
$3$ \( T^{2} + 100 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 2401 \) Copy content Toggle raw display
$11$ \( (T + 336)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 341056 \) Copy content Toggle raw display
$17$ \( T^{2} + 2125764 \) Copy content Toggle raw display
$19$ \( (T + 470)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 17640000 \) Copy content Toggle raw display
$29$ \( (T + 4866)^{2} \) Copy content Toggle raw display
$31$ \( (T + 7372)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 205348900 \) Copy content Toggle raw display
$41$ \( (T - 6222)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 13719616 \) Copy content Toggle raw display
$47$ \( T^{2} + 3283344 \) Copy content Toggle raw display
$53$ \( T^{2} + 1386966564 \) Copy content Toggle raw display
$59$ \( (T + 34302)^{2} \) Copy content Toggle raw display
$61$ \( (T - 24476)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 304572304 \) Copy content Toggle raw display
$71$ \( (T - 28224)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 12974404 \) Copy content Toggle raw display
$79$ \( (T + 42872)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 1239180804 \) Copy content Toggle raw display
$89$ \( (T + 26730)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 288252484 \) Copy content Toggle raw display
show more
show less