Properties

Label 240.10.a.m
Level $240$
Weight $10$
Character orbit 240.a
Self dual yes
Analytic conductor $123.609$
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] = [240,10,Mod(1,240)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(240, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 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("240.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 240.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: \(123.608600679\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{4729}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1182 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3} \)
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 \(\beta = 4\sqrt{4729}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 81 q^{3} - 625 q^{5} + ( - 7 \beta + 5936) q^{7} + 6561 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 81 q^{3} - 625 q^{5} + ( - 7 \beta + 5936) q^{7} + 6561 q^{9} + ( - 244 \beta - 17744) q^{11} + (173 \beta + 71838) q^{13} + 50625 q^{15} + ( - 275 \beta + 192578) q^{17} + (121 \beta + 201648) q^{19} + (567 \beta - 480816) q^{21} + ( - 8121 \beta - 111852) q^{23} + 390625 q^{25} - 531441 q^{27} + ( - 1552 \beta - 37286) q^{29} + (9467 \beta + 2513564) q^{31} + (19764 \beta + 1437264) q^{33} + (4375 \beta - 3710000) q^{35} + (21837 \beta + 2686814) q^{37} + ( - 14013 \beta - 5818878) q^{39} + ( - 58762 \beta + 7105666) q^{41} + (19048 \beta - 13874460) q^{43} - 4100625 q^{45} + (53921 \beta - 47983220) q^{47} + ( - 83104 \beta - 1409975) q^{49} + (22275 \beta - 15598818) q^{51} + ( - 116234 \beta - 32152798) q^{53} + (152500 \beta + 11090000) q^{55} + ( - 9801 \beta - 16333488) q^{57} + ( - 201676 \beta - 93931568) q^{59} + (282086 \beta + 77040030) q^{61} + ( - 45927 \beta + 38946096) q^{63} + ( - 108125 \beta - 44898750) q^{65} + ( - 930520 \beta - 16796188) q^{67} + (657801 \beta + 9060012) q^{69} + (882640 \beta + 114135488) q^{71} + (810026 \beta - 16561158) q^{73} - 31640625 q^{75} + ( - 1324176 \beta + 23905728) q^{77} + ( - 224755 \beta + 466203380) q^{79} + 43046721 q^{81} + (395136 \beta - 103520076) q^{83} + (171875 \beta - 120361250) q^{85} + (125712 \beta + 3020166) q^{87} + (1163274 \beta + 112259082) q^{89} + (524062 \beta + 334801264) q^{91} + ( - 766827 \beta - 203598684) q^{93} + ( - 75625 \beta - 126030000) q^{95} + ( - 5504880 \beta + 193567298) q^{97} + ( - 1600884 \beta - 116418384) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 162 q^{3} - 1250 q^{5} + 11872 q^{7} + 13122 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 162 q^{3} - 1250 q^{5} + 11872 q^{7} + 13122 q^{9} - 35488 q^{11} + 143676 q^{13} + 101250 q^{15} + 385156 q^{17} + 403296 q^{19} - 961632 q^{21} - 223704 q^{23} + 781250 q^{25} - 1062882 q^{27} - 74572 q^{29} + 5027128 q^{31} + 2874528 q^{33} - 7420000 q^{35} + 5373628 q^{37} - 11637756 q^{39} + 14211332 q^{41} - 27748920 q^{43} - 8201250 q^{45} - 95966440 q^{47} - 2819950 q^{49} - 31197636 q^{51} - 64305596 q^{53} + 22180000 q^{55} - 32666976 q^{57} - 187863136 q^{59} + 154080060 q^{61} + 77892192 q^{63} - 89797500 q^{65} - 33592376 q^{67} + 18120024 q^{69} + 228270976 q^{71} - 33122316 q^{73} - 63281250 q^{75} + 47811456 q^{77} + 932406760 q^{79} + 86093442 q^{81} - 207040152 q^{83} - 240722500 q^{85} + 6040332 q^{87} + 224518164 q^{89} + 669602528 q^{91} - 407197368 q^{93} - 252060000 q^{95} + 387134596 q^{97} - 232836768 q^{99}+O(q^{100}) \) 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
34.8839
−33.8839
0 −81.0000 0 −625.000 0 4010.50 0 6561.00 0
1.2 0 −81.0000 0 −625.000 0 7861.50 0 6561.00 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(5\) \(1\)

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 240.10.a.m 2
4.b odd 2 1 15.10.a.c 2
12.b even 2 1 45.10.a.e 2
20.d odd 2 1 75.10.a.g 2
20.e even 4 2 75.10.b.e 4
60.h even 2 1 225.10.a.j 2
60.l odd 4 2 225.10.b.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.10.a.c 2 4.b odd 2 1
45.10.a.e 2 12.b even 2 1
75.10.a.g 2 20.d odd 2 1
75.10.b.e 4 20.e even 4 2
225.10.a.j 2 60.h even 2 1
225.10.b.g 4 60.l odd 4 2
240.10.a.m 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} - 11872T_{7} + 31528560 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(240))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T + 81)^{2} \) Copy content Toggle raw display
$5$ \( (T + 625)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 11872 T + 31528560 \) Copy content Toggle raw display
$11$ \( T^{2} + 35488 T - 4189882368 \) Copy content Toggle raw display
$13$ \( T^{2} - 143676 T + 2896150388 \) Copy content Toggle raw display
$17$ \( T^{2} - 385156 T + 31364196084 \) Copy content Toggle raw display
$19$ \( T^{2} - 403296 T + 39554119280 \) Copy content Toggle raw display
$23$ \( T^{2} + 223704 T - 4977578430720 \) Copy content Toggle raw display
$29$ \( T^{2} + 74572 T - 180861933660 \) Copy content Toggle raw display
$31$ \( T^{2} - 5027128 T - 463313088000 \) Copy content Toggle raw display
$37$ \( T^{2} - 5373628 T - 28861754638220 \) Copy content Toggle raw display
$41$ \( T^{2} + \cdots - 210775232832060 \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots + 165047750825744 \) Copy content Toggle raw display
$47$ \( T^{2} + 95966440 T + 20\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{2} + 64305596 T + 11555844938820 \) Copy content Toggle raw display
$59$ \( T^{2} + 187863136 T + 57\!\cdots\!60 \) Copy content Toggle raw display
$61$ \( T^{2} - 154080060 T - 85608279866044 \) Copy content Toggle raw display
$67$ \( T^{2} + 33592376 T - 65\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( T^{2} - 228270976 T - 45\!\cdots\!56 \) Copy content Toggle raw display
$73$ \( T^{2} + 33122316 T - 49\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{2} - 932406760 T + 21\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{2} + 207040152 T - 10\!\cdots\!68 \) Copy content Toggle raw display
$89$ \( T^{2} - 224518164 T - 89\!\cdots\!40 \) Copy content Toggle raw display
$97$ \( T^{2} - 387134596 T - 22\!\cdots\!96 \) Copy content Toggle raw display
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