Properties

Label 650.6.a.c
Level $650$
Weight $6$
Character orbit 650.a
Self dual yes
Analytic conductor $104.249$
Analytic rank $0$
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] = [650,6,Mod(1,650)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(650, 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("650.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 650 = 2 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 650.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: \(104.249482878\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{14}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 14 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 130)
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 \(\beta = \sqrt{14}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 4 q^{2} + (3 \beta + 8) q^{3} + 16 q^{4} + ( - 12 \beta - 32) q^{6} + ( - 20 \beta + 126) q^{7} - 64 q^{8} + (48 \beta - 53) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 4 q^{2} + (3 \beta + 8) q^{3} + 16 q^{4} + ( - 12 \beta - 32) q^{6} + ( - 20 \beta + 126) q^{7} - 64 q^{8} + (48 \beta - 53) q^{9} + (137 \beta - 18) q^{11} + (48 \beta + 128) q^{12} + 169 q^{13} + (80 \beta - 504) q^{14} + 256 q^{16} + (314 \beta + 1190) q^{17} + ( - 192 \beta + 212) q^{18} + ( - 41 \beta - 546) q^{19} + (218 \beta + 168) q^{21} + ( - 548 \beta + 72) q^{22} + (861 \beta + 588) q^{23} + ( - 192 \beta - 512) q^{24} - 676 q^{26} + ( - 504 \beta - 352) q^{27} + ( - 320 \beta + 2016) q^{28} + (1256 \beta - 2436) q^{29} + ( - 2127 \beta - 1130) q^{31} - 1024 q^{32} + (1042 \beta + 5610) q^{33} + ( - 1256 \beta - 4760) q^{34} + (768 \beta - 848) q^{36} + ( - 1174 \beta + 8880) q^{37} + (164 \beta + 2184) q^{38} + (507 \beta + 1352) q^{39} + ( - 2090 \beta - 4610) q^{41} + ( - 872 \beta - 672) q^{42} + (199 \beta + 11828) q^{43} + (2192 \beta - 288) q^{44} + ( - 3444 \beta - 2352) q^{46} + (756 \beta - 6430) q^{47} + (768 \beta + 2048) q^{48} + ( - 5040 \beta + 4669) q^{49} + (6082 \beta + 22708) q^{51} + 2704 q^{52} + ( - 538 \beta - 5534) q^{53} + (2016 \beta + 1408) q^{54} + (1280 \beta - 8064) q^{56} + ( - 1966 \beta - 6090) q^{57} + ( - 5024 \beta + 9744) q^{58} + (1301 \beta + 45202) q^{59} + (2604 \beta - 34500) q^{61} + (8508 \beta + 4520) q^{62} + (7108 \beta - 20118) q^{63} + 4096 q^{64} + ( - 4168 \beta - 22440) q^{66} + ( - 4898 \beta - 25398) q^{67} + (5024 \beta + 19040) q^{68} + (8652 \beta + 40866) q^{69} + (5863 \beta - 14334) q^{71} + ( - 3072 \beta + 3392) q^{72} + ( - 2598 \beta + 31344) q^{73} + (4696 \beta - 35520) q^{74} + ( - 656 \beta - 8736) q^{76} + (17622 \beta - 40628) q^{77} + ( - 2028 \beta - 5408) q^{78} + (16438 \beta + 42032) q^{79} + ( - 16752 \beta - 11105) q^{81} + (8360 \beta + 18440) q^{82} + (19772 \beta + 6414) q^{83} + (3488 \beta + 2688) q^{84} + ( - 796 \beta - 47312) q^{86} + (2740 \beta + 33264) q^{87} + ( - 8768 \beta + 1152) q^{88} + (3472 \beta + 46310) q^{89} + ( - 3380 \beta + 21294) q^{91} + (13776 \beta + 9408) q^{92} + ( - 20406 \beta - 98374) q^{93} + ( - 3024 \beta + 25720) q^{94} + ( - 3072 \beta - 8192) q^{96} + (9524 \beta + 37842) q^{97} + (20160 \beta - 18676) q^{98} + ( - 8125 \beta + 93018) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{2} + 16 q^{3} + 32 q^{4} - 64 q^{6} + 252 q^{7} - 128 q^{8} - 106 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 8 q^{2} + 16 q^{3} + 32 q^{4} - 64 q^{6} + 252 q^{7} - 128 q^{8} - 106 q^{9} - 36 q^{11} + 256 q^{12} + 338 q^{13} - 1008 q^{14} + 512 q^{16} + 2380 q^{17} + 424 q^{18} - 1092 q^{19} + 336 q^{21} + 144 q^{22} + 1176 q^{23} - 1024 q^{24} - 1352 q^{26} - 704 q^{27} + 4032 q^{28} - 4872 q^{29} - 2260 q^{31} - 2048 q^{32} + 11220 q^{33} - 9520 q^{34} - 1696 q^{36} + 17760 q^{37} + 4368 q^{38} + 2704 q^{39} - 9220 q^{41} - 1344 q^{42} + 23656 q^{43} - 576 q^{44} - 4704 q^{46} - 12860 q^{47} + 4096 q^{48} + 9338 q^{49} + 45416 q^{51} + 5408 q^{52} - 11068 q^{53} + 2816 q^{54} - 16128 q^{56} - 12180 q^{57} + 19488 q^{58} + 90404 q^{59} - 69000 q^{61} + 9040 q^{62} - 40236 q^{63} + 8192 q^{64} - 44880 q^{66} - 50796 q^{67} + 38080 q^{68} + 81732 q^{69} - 28668 q^{71} + 6784 q^{72} + 62688 q^{73} - 71040 q^{74} - 17472 q^{76} - 81256 q^{77} - 10816 q^{78} + 84064 q^{79} - 22210 q^{81} + 36880 q^{82} + 12828 q^{83} + 5376 q^{84} - 94624 q^{86} + 66528 q^{87} + 2304 q^{88} + 92620 q^{89} + 42588 q^{91} + 18816 q^{92} - 196748 q^{93} + 51440 q^{94} - 16384 q^{96} + 75684 q^{97} - 37352 q^{98} + 186036 q^{99}+O(q^{100}) \) 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
−3.74166
3.74166
−4.00000 −3.22497 16.0000 0 12.8999 200.833 −64.0000 −232.600 0
1.2 −4.00000 19.2250 16.0000 0 −76.8999 51.1669 −64.0000 126.600 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(5\) \( +1 \)
\(13\) \( -1 \)

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 650.6.a.c 2
5.b even 2 1 130.6.a.e 2
5.c odd 4 2 650.6.b.c 4
20.d odd 2 1 1040.6.a.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
130.6.a.e 2 5.b even 2 1
650.6.a.c 2 1.a even 1 1 trivial
650.6.b.c 4 5.c odd 4 2
1040.6.a.h 2 20.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 16T_{3} - 62 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(650))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 4)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 16T - 62 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 252T + 10276 \) Copy content Toggle raw display
$11$ \( T^{2} + 36T - 262442 \) Copy content Toggle raw display
$13$ \( (T - 169)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 2380T + 35756 \) Copy content Toggle raw display
$19$ \( T^{2} + 1092 T + 274582 \) Copy content Toggle raw display
$23$ \( T^{2} - 1176 T - 10032750 \) Copy content Toggle raw display
$29$ \( T^{2} + 4872 T - 16151408 \) Copy content Toggle raw display
$31$ \( T^{2} + 2260 T - 62060906 \) Copy content Toggle raw display
$37$ \( T^{2} - 17760 T + 59558536 \) Copy content Toggle raw display
$41$ \( T^{2} + 9220 T - 39901300 \) Copy content Toggle raw display
$43$ \( T^{2} - 23656 T + 139347170 \) Copy content Toggle raw display
$47$ \( T^{2} + 12860 T + 33343396 \) Copy content Toggle raw display
$53$ \( T^{2} + 11068 T + 26572940 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots + 2019524390 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots + 1095318576 \) Copy content Toggle raw display
$67$ \( T^{2} + 50796 T + 309192748 \) Copy content Toggle raw display
$71$ \( T^{2} + 28668 T - 275783210 \) Copy content Toggle raw display
$73$ \( T^{2} - 62688 T + 887951880 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots - 2016220792 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots - 5431908380 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots + 1975849124 \) Copy content Toggle raw display
$97$ \( T^{2} - 75684 T + 162124900 \) Copy content Toggle raw display
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