Properties

Label 350.6.a.r
Level $350$
Weight $6$
Character orbit 350.a
Self dual yes
Analytic conductor $56.134$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [350,6,Mod(1,350)]
 
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([0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("350.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 350.a (trivial)

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: yes
Analytic conductor: \(56.1343369345\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{79}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 79 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(\beta = 2\sqrt{79}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 4 q^{2} + (\beta + 10) q^{3} + 16 q^{4} + ( - 4 \beta - 40) q^{6} - 49 q^{7} - 64 q^{8} + (20 \beta + 173) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 4 q^{2} + (\beta + 10) q^{3} + 16 q^{4} + ( - 4 \beta - 40) q^{6} - 49 q^{7} - 64 q^{8} + (20 \beta + 173) q^{9} + (3 \beta - 535) q^{11} + (16 \beta + 160) q^{12} + (29 \beta + 368) q^{13} + 196 q^{14} + 256 q^{16} + ( - 50 \beta - 952) q^{17} + ( - 80 \beta - 692) q^{18} + (114 \beta + 414) q^{19} + ( - 49 \beta - 490) q^{21} + ( - 12 \beta + 2140) q^{22} + ( - 37 \beta + 827) q^{23} + ( - 64 \beta - 640) q^{24} + ( - 116 \beta - 1472) q^{26} + (130 \beta + 5620) q^{27} - 784 q^{28} + (32 \beta - 1123) q^{29} + (427 \beta + 144) q^{31} - 1024 q^{32} + ( - 505 \beta - 4402) q^{33} + (200 \beta + 3808) q^{34} + (320 \beta + 2768) q^{36} + (390 \beta + 4493) q^{37} + ( - 456 \beta - 1656) q^{38} + (658 \beta + 12844) q^{39} + (297 \beta + 5614) q^{41} + (196 \beta + 1960) q^{42} + (767 \beta + 1701) q^{43} + (48 \beta - 8560) q^{44} + (148 \beta - 3308) q^{46} + (108 \beta + 7786) q^{47} + (256 \beta + 2560) q^{48} + 2401 q^{49} + ( - 1452 \beta - 25320) q^{51} + (464 \beta + 5888) q^{52} + (952 \beta - 8094) q^{53} + ( - 520 \beta - 22480) q^{54} + 3136 q^{56} + (1554 \beta + 40164) q^{57} + ( - 128 \beta + 4492) q^{58} + (331 \beta - 46464) q^{59} + ( - 521 \beta + 7586) q^{61} + ( - 1708 \beta - 576) q^{62} + ( - 980 \beta - 8477) q^{63} + 4096 q^{64} + (2020 \beta + 17608) q^{66} + ( - 1665 \beta + 39483) q^{67} + ( - 800 \beta - 15232) q^{68} + (457 \beta - 3422) q^{69} + ( - 2303 \beta - 10411) q^{71} + ( - 1280 \beta - 11072) q^{72} + ( - 547 \beta + 36442) q^{73} + ( - 1560 \beta - 17972) q^{74} + (1824 \beta + 6624) q^{76} + ( - 147 \beta + 26215) q^{77} + ( - 2632 \beta - 51376) q^{78} + ( - 1923 \beta - 36053) q^{79} + (2060 \beta + 55241) q^{81} + ( - 1188 \beta - 22456) q^{82} + (4631 \beta - 23700) q^{83} + ( - 784 \beta - 7840) q^{84} + ( - 3068 \beta - 6804) q^{86} + ( - 803 \beta - 1118) q^{87} + ( - 192 \beta + 34240) q^{88} + (1875 \beta - 4960) q^{89} + ( - 1421 \beta - 18032) q^{91} + ( - 592 \beta + 13232) q^{92} + (4414 \beta + 136372) q^{93} + ( - 432 \beta - 31144) q^{94} + ( - 1024 \beta - 10240) q^{96} + ( - 5859 \beta - 13396) q^{97} - 9604 q^{98} + ( - 10181 \beta - 73595) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{2} + 20 q^{3} + 32 q^{4} - 80 q^{6} - 98 q^{7} - 128 q^{8} + 346 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 8 q^{2} + 20 q^{3} + 32 q^{4} - 80 q^{6} - 98 q^{7} - 128 q^{8} + 346 q^{9} - 1070 q^{11} + 320 q^{12} + 736 q^{13} + 392 q^{14} + 512 q^{16} - 1904 q^{17} - 1384 q^{18} + 828 q^{19} - 980 q^{21} + 4280 q^{22} + 1654 q^{23} - 1280 q^{24} - 2944 q^{26} + 11240 q^{27} - 1568 q^{28} - 2246 q^{29} + 288 q^{31} - 2048 q^{32} - 8804 q^{33} + 7616 q^{34} + 5536 q^{36} + 8986 q^{37} - 3312 q^{38} + 25688 q^{39} + 11228 q^{41} + 3920 q^{42} + 3402 q^{43} - 17120 q^{44} - 6616 q^{46} + 15572 q^{47} + 5120 q^{48} + 4802 q^{49} - 50640 q^{51} + 11776 q^{52} - 16188 q^{53} - 44960 q^{54} + 6272 q^{56} + 80328 q^{57} + 8984 q^{58} - 92928 q^{59} + 15172 q^{61} - 1152 q^{62} - 16954 q^{63} + 8192 q^{64} + 35216 q^{66} + 78966 q^{67} - 30464 q^{68} - 6844 q^{69} - 20822 q^{71} - 22144 q^{72} + 72884 q^{73} - 35944 q^{74} + 13248 q^{76} + 52430 q^{77} - 102752 q^{78} - 72106 q^{79} + 110482 q^{81} - 44912 q^{82} - 47400 q^{83} - 15680 q^{84} - 13608 q^{86} - 2236 q^{87} + 68480 q^{88} - 9920 q^{89} - 36064 q^{91} + 26464 q^{92} + 272744 q^{93} - 62288 q^{94} - 20480 q^{96} - 26792 q^{97} - 19208 q^{98} - 147190 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−8.88819
8.88819
−4.00000 −7.77639 16.0000 0 31.1056 −49.0000 −64.0000 −182.528 0
1.2 −4.00000 27.7764 16.0000 0 −111.106 −49.0000 −64.0000 528.528 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(5\) \( -1 \)
\(7\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 350.6.a.r 2
5.b even 2 1 350.6.a.s yes 2
5.c odd 4 2 350.6.c.i 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
350.6.a.r 2 1.a even 1 1 trivial
350.6.a.s yes 2 5.b even 2 1
350.6.c.i 4 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(350))\):

\( T_{3}^{2} - 20T_{3} - 216 \) Copy content Toggle raw display
\( T_{13}^{2} - 736T_{13} - 130332 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 4)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 20T - 216 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( (T + 49)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 1070 T + 283381 \) Copy content Toggle raw display
$13$ \( T^{2} - 736T - 130332 \) Copy content Toggle raw display
$17$ \( T^{2} + 1904 T + 116304 \) Copy content Toggle raw display
$19$ \( T^{2} - 828 T - 3935340 \) Copy content Toggle raw display
$23$ \( T^{2} - 1654 T + 251325 \) Copy content Toggle raw display
$29$ \( T^{2} + 2246 T + 937545 \) Copy content Toggle raw display
$31$ \( T^{2} - 288 T - 57595228 \) Copy content Toggle raw display
$37$ \( T^{2} - 8986 T - 27876551 \) Copy content Toggle raw display
$41$ \( T^{2} - 11228 T + 3642952 \) Copy content Toggle raw display
$43$ \( T^{2} - 3402 T - 183005923 \) Copy content Toggle raw display
$47$ \( T^{2} - 15572 T + 56935972 \) Copy content Toggle raw display
$53$ \( T^{2} + 16188 T - 220879228 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots + 2124282020 \) Copy content Toggle raw display
$61$ \( T^{2} - 15172 T - 28227960 \) Copy content Toggle raw display
$67$ \( T^{2} - 78966 T + 682884189 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots - 1567614723 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots + 1233469320 \) Copy content Toggle raw display
$79$ \( T^{2} + 72106 T + 131273245 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots - 6215296876 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots - 1086335900 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots - 10668157580 \) Copy content Toggle raw display
show more
show less