Properties

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

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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: \(1\)
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.

\(f(q)\) \(=\) \( q - 4 q^{2} + (\beta - 4) q^{3} + 16 q^{4} + ( - 4 \beta + 16) q^{6} + 49 q^{7} - 64 q^{8} + ( - 8 \beta + 89) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 4 q^{2} + (\beta - 4) q^{3} + 16 q^{4} + ( - 4 \beta + 16) q^{6} + 49 q^{7} - 64 q^{8} + ( - 8 \beta + 89) q^{9} + (3 \beta - 3) q^{11} + (16 \beta - 64) q^{12} + (15 \beta - 346) q^{13} - 196 q^{14} + 256 q^{16} + (6 \beta - 252) q^{17} + (32 \beta - 356) q^{18} + ( - 54 \beta - 34) q^{19} + (49 \beta - 196) q^{21} + ( - 12 \beta + 12) q^{22} + (75 \beta - 825) q^{23} + ( - 64 \beta + 256) q^{24} + ( - 60 \beta + 1384) q^{26} + ( - 122 \beta - 1912) q^{27} + 784 q^{28} + ( - 276 \beta + 1425) q^{29} + ( - 21 \beta + 1922) q^{31} - 1024 q^{32} + ( - 15 \beta + 960) q^{33} + ( - 24 \beta + 1008) q^{34} + ( - 128 \beta + 1424) q^{36} + ( - 366 \beta - 5671) q^{37} + (216 \beta + 136) q^{38} + ( - 406 \beta + 6124) q^{39} + ( - 333 \beta + 7644) q^{41} + ( - 196 \beta + 784) q^{42} + (39 \beta - 427) q^{43} + (48 \beta - 48) q^{44} + ( - 300 \beta + 3300) q^{46} + ( - 648 \beta - 7530) q^{47} + (256 \beta - 1024) q^{48} + 2401 q^{49} + ( - 276 \beta + 2904) q^{51} + (240 \beta - 5536) q^{52} + ( - 252 \beta + 9126) q^{53} + (488 \beta + 7648) q^{54} - 3136 q^{56} + (182 \beta - 16928) q^{57} + (1104 \beta - 5700) q^{58} + (51 \beta + 19098) q^{59} + (921 \beta - 7828) q^{61} + (84 \beta - 7688) q^{62} + ( - 392 \beta + 4361) q^{63} + 4096 q^{64} + (60 \beta - 3840) q^{66} + (1023 \beta - 32449) q^{67} + (96 \beta - 4032) q^{68} + ( - 1125 \beta + 27000) q^{69} + (1953 \beta - 20043) q^{71} + (512 \beta - 5696) q^{72} + ( - 2997 \beta + 36680) q^{73} + (1464 \beta + 22684) q^{74} + ( - 864 \beta - 544) q^{76} + (147 \beta - 147) q^{77} + (1624 \beta - 24496) q^{78} + ( - 747 \beta + 17735) q^{79} + (520 \beta - 52531) q^{81} + (1332 \beta - 30576) q^{82} + ( - 3321 \beta - 978) q^{83} + (784 \beta - 3136) q^{84} + ( - 156 \beta + 1708) q^{86} + (2529 \beta - 92916) q^{87} + ( - 192 \beta + 192) q^{88} + (3597 \beta - 9678) q^{89} + (735 \beta - 16954) q^{91} + (1200 \beta - 13200) q^{92} + (2006 \beta - 14324) q^{93} + (2592 \beta + 30120) q^{94} + ( - 1024 \beta + 4096) q^{96} + ( - 21 \beta - 113734) q^{97} - 9604 q^{98} + (291 \beta - 7851) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{2} - 8 q^{3} + 32 q^{4} + 32 q^{6} + 98 q^{7} - 128 q^{8} + 178 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 8 q^{2} - 8 q^{3} + 32 q^{4} + 32 q^{6} + 98 q^{7} - 128 q^{8} + 178 q^{9} - 6 q^{11} - 128 q^{12} - 692 q^{13} - 392 q^{14} + 512 q^{16} - 504 q^{17} - 712 q^{18} - 68 q^{19} - 392 q^{21} + 24 q^{22} - 1650 q^{23} + 512 q^{24} + 2768 q^{26} - 3824 q^{27} + 1568 q^{28} + 2850 q^{29} + 3844 q^{31} - 2048 q^{32} + 1920 q^{33} + 2016 q^{34} + 2848 q^{36} - 11342 q^{37} + 272 q^{38} + 12248 q^{39} + 15288 q^{41} + 1568 q^{42} - 854 q^{43} - 96 q^{44} + 6600 q^{46} - 15060 q^{47} - 2048 q^{48} + 4802 q^{49} + 5808 q^{51} - 11072 q^{52} + 18252 q^{53} + 15296 q^{54} - 6272 q^{56} - 33856 q^{57} - 11400 q^{58} + 38196 q^{59} - 15656 q^{61} - 15376 q^{62} + 8722 q^{63} + 8192 q^{64} - 7680 q^{66} - 64898 q^{67} - 8064 q^{68} + 54000 q^{69} - 40086 q^{71} - 11392 q^{72} + 73360 q^{73} + 45368 q^{74} - 1088 q^{76} - 294 q^{77} - 48992 q^{78} + 35470 q^{79} - 105062 q^{81} - 61152 q^{82} - 1956 q^{83} - 6272 q^{84} + 3416 q^{86} - 185832 q^{87} + 384 q^{88} - 19356 q^{89} - 33908 q^{91} - 26400 q^{92} - 28648 q^{93} + 60240 q^{94} + 8192 q^{96} - 227468 q^{97} - 19208 q^{98} - 15702 q^{99}+O(q^{100}) \) 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 −21.7764 16.0000 0 87.1056 49.0000 −64.0000 231.211 0
1.2 −4.00000 13.7764 16.0000 0 −55.1056 49.0000 −64.0000 −53.2111 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.o 2
5.b even 2 1 350.6.a.t yes 2
5.c odd 4 2 350.6.c.l 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
350.6.a.o 2 1.a even 1 1 trivial
350.6.a.t yes 2 5.b even 2 1
350.6.c.l 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 \(S_{6}^{\mathrm{new}}(\Gamma_0(350))\):

\( T_{3}^{2} + 8T_{3} - 300 \) Copy content Toggle raw display
\( T_{13}^{2} + 692T_{13} + 48616 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 4)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 8T - 300 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( (T - 49)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 6T - 2835 \) Copy content Toggle raw display
$13$ \( T^{2} + 692T + 48616 \) Copy content Toggle raw display
$17$ \( T^{2} + 504T + 52128 \) Copy content Toggle raw display
$19$ \( T^{2} + 68T - 920300 \) Copy content Toggle raw display
$23$ \( T^{2} + 1650 T - 1096875 \) Copy content Toggle raw display
$29$ \( T^{2} - 2850 T - 22040991 \) Copy content Toggle raw display
$31$ \( T^{2} - 3844 T + 3554728 \) Copy content Toggle raw display
$37$ \( T^{2} + 11342 T - 10169855 \) Copy content Toggle raw display
$41$ \( T^{2} - 15288 T + 23389812 \) Copy content Toggle raw display
$43$ \( T^{2} + 854T - 298307 \) Copy content Toggle raw display
$47$ \( T^{2} + 15060 T - 75988764 \) Copy content Toggle raw display
$53$ \( T^{2} - 18252 T + 63216612 \) Copy content Toggle raw display
$59$ \( T^{2} - 38196 T + 363911688 \) Copy content Toggle raw display
$61$ \( T^{2} + 15656 T - 206766572 \) Copy content Toggle raw display
$67$ \( T^{2} + 64898 T + 722234437 \) Copy content Toggle raw display
$71$ \( T^{2} + 40086 T - 803568195 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots - 1492892444 \) Copy content Toggle raw display
$79$ \( T^{2} - 35470 T + 138199381 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots - 3484220472 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots - 3994873560 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots + 12935283400 \) Copy content Toggle raw display
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