Properties

Label 13.8.a.b
Level $13$
Weight $8$
Character orbit 13.a
Self dual yes
Analytic conductor $4.061$
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] = [13,8,Mod(1,13)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(13, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("13.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 13 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 13.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: \(4.06100533129\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{337}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 84 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
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 \(\beta = \frac{1}{2}(1 + \sqrt{337})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta - 9) q^{2} + (3 \beta + 21) q^{3} + (19 \beta + 37) q^{4} + ( - 11 \beta - 171) q^{5} + ( - 51 \beta - 441) q^{6} + (9 \beta - 1009) q^{7} + ( - 99 \beta - 777) q^{8} + (135 \beta - 990) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta - 9) q^{2} + (3 \beta + 21) q^{3} + (19 \beta + 37) q^{4} + ( - 11 \beta - 171) q^{5} + ( - 51 \beta - 441) q^{6} + (9 \beta - 1009) q^{7} + ( - 99 \beta - 777) q^{8} + (135 \beta - 990) q^{9} + (281 \beta + 2463) q^{10} + ( - 622 \beta - 594) q^{11} + (567 \beta + 5565) q^{12} + 2197 q^{13} + (919 \beta + 8325) q^{14} + ( - 777 \beta - 6363) q^{15} + ( - 665 \beta + 10573) q^{16} + ( - 2003 \beta - 11679) q^{17} + ( - 360 \beta - 2430) q^{18} + (4582 \beta + 8762) q^{19} + ( - 3865 \beta - 23883) q^{20} + ( - 2811 \beta - 18921) q^{21} + (6814 \beta + 57594) q^{22} + (6792 \beta - 16608) q^{23} + ( - 4707 \beta - 41265) q^{24} + (3883 \beta - 38720) q^{25} + ( - 2197 \beta - 19773) q^{26} + ( - 6291 \beta - 32697) q^{27} + ( - 18667 \beta - 22969) q^{28} + (3544 \beta - 4674) q^{29} + (14133 \beta + 122535) q^{30} + ( - 3496 \beta + 21620) q^{31} + (8749 \beta + 60159) q^{32} + ( - 16710 \beta - 169218) q^{33} + (31709 \beta + 273363) q^{34} + (9461 \beta + 164223) q^{35} + ( - 11250 \beta + 178830) q^{36} + ( - 13135 \beta + 88217) q^{37} + ( - 54582 \beta - 463746) q^{38} + (6591 \beta + 46137) q^{39} + (26565 \beta + 224343) q^{40} + (41178 \beta - 186024) q^{41} + (47031 \beta + 406413) q^{42} + (4357 \beta + 112475) q^{43} + ( - 46118 \beta - 1014690) q^{44} + ( - 13680 \beta + 44550) q^{45} + ( - 51312 \beta - 421056) q^{46} + (4221 \beta - 821373) q^{47} + (15759 \beta + 54453) q^{48} + ( - 18081 \beta + 201342) q^{49} + ( - 110 \beta + 22308) q^{50} + ( - 83109 \beta - 750015) q^{51} + (41743 \beta + 81289) q^{52} + ( - 104610 \beta + 575496) q^{53} + (95607 \beta + 822717) q^{54} + (119738 \beta + 676302) q^{55} + (92007 \beta + 709149) q^{56} + (136254 \beta + 1338666) q^{57} + ( - 30766 \beta - 255630) q^{58} + (45486 \beta - 207822) q^{59} + ( - 164409 \beta - 1475523) q^{60} + ( - 78370 \beta + 2376896) q^{61} + (13340 \beta + 99084) q^{62} + ( - 143910 \beta + 1100970) q^{63} + ( - 62529 \beta - 2629691) q^{64} + ( - 24167 \beta - 375687) q^{65} + (336318 \beta + 2926602) q^{66} + ( - 272038 \beta - 774682) q^{67} + ( - 334069 \beta - 3628911) q^{68} + (113184 \beta + 1362816) q^{69} + ( - 258833 \beta - 2272731) q^{70} + (545931 \beta - 840771) q^{71} + ( - 20250 \beta - 353430) q^{72} + (57000 \beta - 3258142) q^{73} + (43133 \beta + 309387) q^{74} + ( - 22968 \beta + 165396) q^{75} + (423070 \beta + 7637066) q^{76} + (616654 \beta + 129114) q^{77} + ( - 112047 \beta - 968877) q^{78} + ( - 516480 \beta + 221336) q^{79} + (4727 \beta - 1193523) q^{80} + ( - 544320 \beta - 106839) q^{81} + ( - 225756 \beta - 1784736) q^{82} + (163292 \beta - 6132132) q^{83} + ( - 516915 \beta - 5186433) q^{84} + (493015 \beta + 3847881) q^{85} + ( - 156045 \beta - 1378263) q^{86} + (71034 \beta + 794934) q^{87} + (603678 \beta + 5634090) q^{88} + ( - 639712 \beta + 5227386) q^{89} + (92250 \beta + 748170) q^{90} + (19773 \beta - 2216773) q^{91} + (64800 \beta + 10225536) q^{92} + ( - 19044 \beta - 426972) q^{93} + (779163 \beta + 7037793) q^{94} + ( - 930306 \beta - 5732070) q^{95} + (390453 \beta + 3468087) q^{96} + ( - 615988 \beta - 8487850) q^{97} + ( - 20532 \beta - 293274) q^{98} + (451620 \beta - 6465420) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 19 q^{2} + 45 q^{3} + 93 q^{4} - 353 q^{5} - 933 q^{6} - 2009 q^{7} - 1653 q^{8} - 1845 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 19 q^{2} + 45 q^{3} + 93 q^{4} - 353 q^{5} - 933 q^{6} - 2009 q^{7} - 1653 q^{8} - 1845 q^{9} + 5207 q^{10} - 1810 q^{11} + 11697 q^{12} + 4394 q^{13} + 17569 q^{14} - 13503 q^{15} + 20481 q^{16} - 25361 q^{17} - 5220 q^{18} + 22106 q^{19} - 51631 q^{20} - 40653 q^{21} + 122002 q^{22} - 26424 q^{23} - 87237 q^{24} - 73557 q^{25} - 41743 q^{26} - 71685 q^{27} - 64605 q^{28} - 5804 q^{29} + 259203 q^{30} + 39744 q^{31} + 129067 q^{32} - 355146 q^{33} + 578435 q^{34} + 337907 q^{35} + 346410 q^{36} + 163299 q^{37} - 982074 q^{38} + 98865 q^{39} + 475251 q^{40} - 330870 q^{41} + 859857 q^{42} + 229307 q^{43} - 2075498 q^{44} + 75420 q^{45} - 893424 q^{46} - 1638525 q^{47} + 124665 q^{48} + 384603 q^{49} + 44506 q^{50} - 1583139 q^{51} + 204321 q^{52} + 1046382 q^{53} + 1741041 q^{54} + 1472342 q^{55} + 1510305 q^{56} + 2813586 q^{57} - 542026 q^{58} - 370158 q^{59} - 3115455 q^{60} + 4675422 q^{61} + 211508 q^{62} + 2058030 q^{63} - 5321911 q^{64} - 775541 q^{65} + 6189522 q^{66} - 1821402 q^{67} - 7591891 q^{68} + 2838816 q^{69} - 4804295 q^{70} - 1135611 q^{71} - 727110 q^{72} - 6459284 q^{73} + 661907 q^{74} + 307824 q^{75} + 15697202 q^{76} + 874882 q^{77} - 2049801 q^{78} - 73808 q^{79} - 2382319 q^{80} - 757998 q^{81} - 3795228 q^{82} - 12100972 q^{83} - 10889781 q^{84} + 8188777 q^{85} - 2912571 q^{86} + 1660902 q^{87} + 11871858 q^{88} + 9815060 q^{89} + 1588590 q^{90} - 4413773 q^{91} + 20515872 q^{92} - 872988 q^{93} + 14854749 q^{94} - 12394446 q^{95} + 7326627 q^{96} - 17591688 q^{97} - 607080 q^{98} - 12479220 q^{99}+O(q^{100}) \) 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
9.67878
−8.67878
−18.6788 50.0363 220.897 −277.467 −934.618 −921.891 −1735.20 316.635 5182.74
1.2 −0.321220 −5.03634 −127.897 −75.5334 1.61777 −1087.11 82.1992 −2161.64 24.2629
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(13\) \( -1 \)

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 13.8.a.b 2
3.b odd 2 1 117.8.a.c 2
4.b odd 2 1 208.8.a.g 2
5.b even 2 1 325.8.a.b 2
13.b even 2 1 169.8.a.b 2
13.d odd 4 2 169.8.b.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.8.a.b 2 1.a even 1 1 trivial
117.8.a.c 2 3.b odd 2 1
169.8.a.b 2 13.b even 2 1
169.8.b.b 4 13.d odd 4 2
208.8.a.g 2 4.b odd 2 1
325.8.a.b 2 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 19T_{2} + 6 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(13))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 19T + 6 \) Copy content Toggle raw display
$3$ \( T^{2} - 45T - 252 \) Copy content Toggle raw display
$5$ \( T^{2} + 353T + 20958 \) Copy content Toggle raw display
$7$ \( T^{2} + 2009 T + 1002196 \) Copy content Toggle raw display
$11$ \( T^{2} + 1810 T - 31775952 \) Copy content Toggle raw display
$13$ \( (T - 2197)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 25361 T - 177216678 \) Copy content Toggle raw display
$19$ \( T^{2} + \cdots - 1646636688 \) Copy content Toggle raw display
$23$ \( T^{2} + \cdots - 3712002048 \) Copy content Toggle raw display
$29$ \( T^{2} + \cdots - 1049753004 \) Copy content Toggle raw display
$31$ \( T^{2} - 39744 T - 634808464 \) Copy content Toggle raw display
$37$ \( T^{2} + \cdots - 7868862106 \) Copy content Toggle raw display
$41$ \( T^{2} + \cdots - 115487893152 \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots + 11546069484 \) Copy content Toggle raw display
$47$ \( T^{2} + \cdots + 669689975052 \) Copy content Toggle raw display
$53$ \( T^{2} + \cdots - 648240166944 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots - 140057008272 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots + 4947441275696 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots - 5405517426256 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots - 24787522246284 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots + 10156859198164 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots - 22472459585984 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots + 34361915476704 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots - 10393898367132 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots + 45398949212204 \) Copy content Toggle raw display
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