Properties

Label 350.6.c.i
Level $350$
Weight $6$
Character orbit 350.c
Analytic conductor $56.134$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: \(\Q(i, \sqrt{79})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 39x^{2} + 400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 4 \beta_1 q^{2} + (\beta_{3} - 10 \beta_1) q^{3} - 16 q^{4} + (4 \beta_{2} - 40) q^{6} - 49 \beta_1 q^{7} + 64 \beta_1 q^{8} + (20 \beta_{2} - 173) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 4 \beta_1 q^{2} + (\beta_{3} - 10 \beta_1) q^{3} - 16 q^{4} + (4 \beta_{2} - 40) q^{6} - 49 \beta_1 q^{7} + 64 \beta_1 q^{8} + (20 \beta_{2} - 173) q^{9} + ( - 3 \beta_{2} - 535) q^{11} + ( - 16 \beta_{3} + 160 \beta_1) q^{12} + (29 \beta_{3} - 368 \beta_1) q^{13} - 196 q^{14} + 256 q^{16} + (50 \beta_{3} - 952 \beta_1) q^{17} + ( - 80 \beta_{3} + 692 \beta_1) q^{18} + (114 \beta_{2} - 414) q^{19} + (49 \beta_{2} - 490) q^{21} + (12 \beta_{3} + 2140 \beta_1) q^{22} + ( - 37 \beta_{3} - 827 \beta_1) q^{23} + ( - 64 \beta_{2} + 640) q^{24} + (116 \beta_{2} - 1472) q^{26} + ( - 130 \beta_{3} + 5620 \beta_1) q^{27} + 784 \beta_1 q^{28} + (32 \beta_{2} + 1123) q^{29} + ( - 427 \beta_{2} + 144) q^{31} - 1024 \beta_1 q^{32} + ( - 505 \beta_{3} + 4402 \beta_1) q^{33} + (200 \beta_{2} - 3808) q^{34} + ( - 320 \beta_{2} + 2768) q^{36} + ( - 390 \beta_{3} + 4493 \beta_1) q^{37} + ( - 456 \beta_{3} + 1656 \beta_1) q^{38} + (658 \beta_{2} - 12844) q^{39} + ( - 297 \beta_{2} + 5614) q^{41} + ( - 196 \beta_{3} + 1960 \beta_1) q^{42} + (767 \beta_{3} - 1701 \beta_1) q^{43} + (48 \beta_{2} + 8560) q^{44} + ( - 148 \beta_{2} - 3308) q^{46} + ( - 108 \beta_{3} + 7786 \beta_1) q^{47} + (256 \beta_{3} - 2560 \beta_1) q^{48} - 2401 q^{49} + (1452 \beta_{2} - 25320) q^{51} + ( - 464 \beta_{3} + 5888 \beta_1) q^{52} + (952 \beta_{3} + 8094 \beta_1) q^{53} + ( - 520 \beta_{2} + 22480) q^{54} + 3136 q^{56} + ( - 1554 \beta_{3} + 40164 \beta_1) q^{57} + ( - 128 \beta_{3} - 4492 \beta_1) q^{58} + (331 \beta_{2} + 46464) q^{59} + (521 \beta_{2} + 7586) q^{61} + (1708 \beta_{3} - 576 \beta_1) q^{62} + ( - 980 \beta_{3} + 8477 \beta_1) q^{63} - 4096 q^{64} + ( - 2020 \beta_{2} + 17608) q^{66} + (1665 \beta_{3} + 39483 \beta_1) q^{67} + ( - 800 \beta_{3} + 15232 \beta_1) q^{68} + (457 \beta_{2} + 3422) q^{69} + (2303 \beta_{2} - 10411) q^{71} + (1280 \beta_{3} - 11072 \beta_1) q^{72} + ( - 547 \beta_{3} - 36442 \beta_1) q^{73} + ( - 1560 \beta_{2} + 17972) q^{74} + ( - 1824 \beta_{2} + 6624) q^{76} + (147 \beta_{3} + 26215 \beta_1) q^{77} + ( - 2632 \beta_{3} + 51376 \beta_1) q^{78} + ( - 1923 \beta_{2} + 36053) q^{79} + ( - 2060 \beta_{2} + 55241) q^{81} + (1188 \beta_{3} - 22456 \beta_1) q^{82} + (4631 \beta_{3} + 23700 \beta_1) q^{83} + ( - 784 \beta_{2} + 7840) q^{84} + (3068 \beta_{2} - 6804) q^{86} + (803 \beta_{3} - 1118 \beta_1) q^{87} + ( - 192 \beta_{3} - 34240 \beta_1) q^{88} + (1875 \beta_{2} + 4960) q^{89} + (1421 \beta_{2} - 18032) q^{91} + (592 \beta_{3} + 13232 \beta_1) q^{92} + (4414 \beta_{3} - 136372 \beta_1) q^{93} + ( - 432 \beta_{2} + 31144) q^{94} + (1024 \beta_{2} - 10240) q^{96} + (5859 \beta_{3} - 13396 \beta_1) q^{97} + 9604 \beta_1 q^{98} + ( - 10181 \beta_{2} + 73595) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 64 q^{4} - 160 q^{6} - 692 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 64 q^{4} - 160 q^{6} - 692 q^{9} - 2140 q^{11} - 784 q^{14} + 1024 q^{16} - 1656 q^{19} - 1960 q^{21} + 2560 q^{24} - 5888 q^{26} + 4492 q^{29} + 576 q^{31} - 15232 q^{34} + 11072 q^{36} - 51376 q^{39} + 22456 q^{41} + 34240 q^{44} - 13232 q^{46} - 9604 q^{49} - 101280 q^{51} + 89920 q^{54} + 12544 q^{56} + 185856 q^{59} + 30344 q^{61} - 16384 q^{64} + 70432 q^{66} + 13688 q^{69} - 41644 q^{71} + 71888 q^{74} + 26496 q^{76} + 144212 q^{79} + 220964 q^{81} + 31360 q^{84} - 27216 q^{86} + 19840 q^{89} - 72128 q^{91} + 124576 q^{94} - 40960 q^{96} + 294380 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 39x^{2} + 400 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} - 19\nu ) / 20 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + 59\nu ) / 10 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 4\nu^{2} - 78 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + 2\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 78 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 19\beta_{2} + 118\beta_1 ) / 4 \) 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
−4.44410 + 0.500000i
4.44410 + 0.500000i
4.44410 0.500000i
−4.44410 0.500000i
4.00000i 27.7764i −16.0000 0 −111.106 49.0000i 64.0000i −528.528 0
99.2 4.00000i 7.77639i −16.0000 0 31.1056 49.0000i 64.0000i 182.528 0
99.3 4.00000i 7.77639i −16.0000 0 31.1056 49.0000i 64.0000i 182.528 0
99.4 4.00000i 27.7764i −16.0000 0 −111.106 49.0000i 64.0000i −528.528 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.i 4
5.b even 2 1 inner 350.6.c.i 4
5.c odd 4 1 350.6.a.r 2
5.c odd 4 1 350.6.a.s yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
350.6.a.r 2 5.c odd 4 1
350.6.a.s yes 2 5.c odd 4 1
350.6.c.i 4 1.a even 1 1 trivial
350.6.c.i 4 5.b even 2 1 inner

Hecke kernels

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

\( T_{3}^{4} + 832T_{3}^{2} + 46656 \) Copy content Toggle raw display
\( T_{11}^{2} + 1070T_{11} + 283381 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} + 832 T^{2} + 46656 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 2401)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 1070 T + 283381)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + \cdots + 16986430224 \) Copy content Toggle raw display
$17$ \( T^{4} + \cdots + 13526620416 \) Copy content Toggle raw display
$19$ \( (T^{2} + 828 T - 3935340)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots + 63164255625 \) Copy content Toggle raw display
$29$ \( (T^{2} - 2246 T + 937545)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 288 T - 57595228)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots + 777102095655601 \) Copy content Toggle raw display
$41$ \( (T^{2} - 11228 T + 3642952)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots + 33\!\cdots\!29 \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 32\!\cdots\!84 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 48\!\cdots\!84 \) Copy content Toggle raw display
$59$ \( (T^{2} - 92928 T + 2124282020)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 15172 T - 28227960)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 46\!\cdots\!21 \) Copy content Toggle raw display
$71$ \( (T^{2} + 20822 T - 1567614723)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{2} - 72106 T + 131273245)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 38\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( (T^{2} - 9920 T - 1086335900)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
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