Properties

Label 45.10.a.d
Level $45$
Weight $10$
Character orbit 45.a
Self dual yes
Analytic conductor $23.177$
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] = [45,10,Mod(1,45)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(45, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("45.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 45 = 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 45.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: \(23.1766126274\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{241}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 60 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 15)
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 \(\beta = \frac{1}{2}(-1 + 3\sqrt{241})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta - 16) q^{2} + (31 \beta + 286) q^{4} - 625 q^{5} + (224 \beta + 7168) q^{7} + ( - 239 \beta - 13186) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta - 16) q^{2} + (31 \beta + 286) q^{4} - 625 q^{5} + (224 \beta + 7168) q^{7} + ( - 239 \beta - 13186) q^{8} + (625 \beta + 10000) q^{10} + (2368 \beta + 11940) q^{11} + ( - 5344 \beta + 9470) q^{13} + ( - 10528 \beta - 236096) q^{14} + (899 \beta + 194082) q^{16} + ( - 7520 \beta + 74718) q^{17} + ( - 5728 \beta - 50812) q^{19} + ( - 19375 \beta - 178750) q^{20} + ( - 47460 \beta - 1474496) q^{22} + ( - 26272 \beta + 354496) q^{23} + 390625 q^{25} + (70690 \beta + 2744928) q^{26} + (279328 \beta + 5813696) q^{28} + (168576 \beta + 1423394) q^{29} + ( - 152736 \beta + 5314848) q^{31} + ( - 85199 \beta + 3158662) q^{32} + (38082 \beta + 2880352) q^{34} + ( - 140000 \beta - 4480000) q^{35} + ( - 198496 \beta + 10884918) q^{37} + (136732 \beta + 3917568) q^{38} + (149375 \beta + 8241250) q^{40} + (492096 \beta - 12784138) q^{41} + ( - 702336 \beta - 3946748) q^{43} + (973980 \beta + 43201976) q^{44} + (39584 \beta + 8567488) q^{46} + ( - 1970528 \beta + 14804856) q^{47} + (3161088 \beta + 38222009) q^{49} + ( - 390625 \beta - 6250000) q^{50} + ( - 1069150 \beta - 87081468) q^{52} + (177728 \beta - 1476694) q^{53} + ( - 1480000 \beta - 7462500) q^{55} + ( - 4613280 \beta - 123533760) q^{56} + ( - 3952034 \beta - 114142496) q^{58} + (4348352 \beta + 19921508) q^{59} + (950208 \beta + 171223774) q^{61} + ( - 3023808 \beta - 2254656) q^{62} + ( - 2340965 \beta - 103730718) q^{64} + (3340000 \beta - 5918750) q^{65} + (1026560 \beta - 143584628) q^{67} + (398658 \beta - 104981692) q^{68} + (6580000 \beta + 147560000) q^{70} + (4545280 \beta - 102870392) q^{71} + (1168192 \beta - 115747446) q^{73} + ( - 7907478 \beta - 66573856) q^{74} + ( - 3035812 \beta - 110774088) q^{76} + (19117952 \beta + 373080064) q^{77} + (19049120 \beta - 2852960) q^{79} + ( - 561875 \beta - 121301250) q^{80} + (5402698 \beta - 62169824) q^{82} + ( - 7260288 \beta + 182410932) q^{83} + (4700000 \beta - 46698750) q^{85} + (14481788 \beta + 443814080) q^{86} + ( - 33512156 \beta - 464186824) q^{88} + ( - 9049152 \beta + 209294982) q^{89} + ( - 34987456 \beta - 580923392) q^{91} + (4290016 \beta - 340036288) q^{92} + (14753064 \beta + 831148480) q^{94} + (3580000 \beta + 31757500) q^{95} + ( - 13450880 \beta + 879104002) q^{97} + ( - 85638329 \beta - 2324861840) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 31 q^{2} + 541 q^{4} - 1250 q^{5} + 14112 q^{7} - 26133 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 31 q^{2} + 541 q^{4} - 1250 q^{5} + 14112 q^{7} - 26133 q^{8} + 19375 q^{10} + 21512 q^{11} + 24284 q^{13} - 461664 q^{14} + 387265 q^{16} + 156956 q^{17} - 95896 q^{19} - 338125 q^{20} - 2901532 q^{22} + 735264 q^{23} + 781250 q^{25} + 5419166 q^{26} + 11348064 q^{28} + 2678212 q^{29} + 10782432 q^{31} + 6402523 q^{32} + 5722622 q^{34} - 8820000 q^{35} + 21968332 q^{37} + 7698404 q^{38} + 16333125 q^{40} - 26060372 q^{41} - 7191160 q^{43} + 85429972 q^{44} + 17095392 q^{46} + 31580240 q^{47} + 73282930 q^{49} - 12109375 q^{50} - 173093786 q^{52} - 3131116 q^{53} - 13445000 q^{55} - 242454240 q^{56} - 224332958 q^{58} + 35494664 q^{59} + 341497340 q^{61} - 1485504 q^{62} - 205120471 q^{64} - 15177500 q^{65} - 288195816 q^{67} - 210362042 q^{68} + 288540000 q^{70} - 210286064 q^{71} - 232663084 q^{73} - 125240234 q^{74} - 218512364 q^{76} + 727042176 q^{77} - 24755040 q^{79} - 242040625 q^{80} - 129742346 q^{82} + 372082152 q^{83} - 98097500 q^{85} + 873146372 q^{86} - 894861492 q^{88} + 427639116 q^{89} - 1126859328 q^{91} - 684362592 q^{92} + 1647543896 q^{94} + 59935000 q^{95} + 1771658884 q^{97} - 4564085351 q^{98}+O(q^{100}) \) 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
8.26209
−7.26209
−38.7863 0 992.374 −625.000 0 12272.1 −18631.9 0 24241.4
1.2 7.78626 0 −451.374 −625.000 0 1839.88 −7501.08 0 −4866.41
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( -1 \)
\(5\) \( +1 \)

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 45.10.a.d 2
3.b odd 2 1 15.10.a.d 2
5.b even 2 1 225.10.a.k 2
5.c odd 4 2 225.10.b.i 4
12.b even 2 1 240.10.a.r 2
15.d odd 2 1 75.10.a.f 2
15.e even 4 2 75.10.b.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.10.a.d 2 3.b odd 2 1
45.10.a.d 2 1.a even 1 1 trivial
75.10.a.f 2 15.d odd 2 1
75.10.b.f 4 15.e even 4 2
225.10.a.k 2 5.b even 2 1
225.10.b.i 4 5.c odd 4 2
240.10.a.r 2 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 31T_{2} - 302 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(45))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 31T - 302 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T + 625)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 14112 T + 22579200 \) Copy content Toggle raw display
$11$ \( T^{2} + \cdots - 2924934128 \) Copy content Toggle raw display
$13$ \( T^{2} + \cdots - 15338329532 \) Copy content Toggle raw display
$17$ \( T^{2} + \cdots - 24505657916 \) Copy content Toggle raw display
$19$ \( T^{2} + \cdots - 15492203120 \) Copy content Toggle raw display
$23$ \( T^{2} + \cdots - 239117414400 \) Copy content Toggle raw display
$29$ \( T^{2} + \cdots - 13616383922300 \) Copy content Toggle raw display
$31$ \( T^{2} + \cdots + 16415447040000 \) Copy content Toggle raw display
$37$ \( T^{2} + \cdots + 99286893737380 \) Copy content Toggle raw display
$41$ \( T^{2} + \cdots + 38475315093220 \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots - 254550637865456 \) Copy content Toggle raw display
$47$ \( T^{2} + \cdots - 18\!\cdots\!24 \) Copy content Toggle raw display
$53$ \( T^{2} + \cdots - 14677210114460 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots - 99\!\cdots\!20 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots + 28\!\cdots\!96 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots + 20\!\cdots\!64 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots - 147594805309376 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots + 12\!\cdots\!60 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots - 19\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots + 60\!\cdots\!92 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots + 13\!\cdots\!20 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots + 68\!\cdots\!64 \) Copy content Toggle raw display
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