Properties

Label 130.3.i.b
Level $130$
Weight $3$
Character orbit 130.i
Analytic conductor $3.542$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [130,3,Mod(27,130)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(130, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("130.27");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 130 = 2 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 130.i (of order \(4\), degree \(2\), minimal)

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: no
Analytic conductor: \(3.54224343668\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4 x^{11} + 34 x^{10} + 12 x^{9} + 435 x^{8} + 2024 x^{7} + 10862 x^{6} + 44868 x^{5} + 149289 x^{4} + 340080 x^{3} + 619204 x^{2} + 735420 x + 354025 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2\cdot 5 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{3} + 1) q^{2} + \beta_{2} q^{3} - 2 \beta_{3} q^{4} + (\beta_{10} + \beta_{8} + 1) q^{5} + (\beta_{2} + \beta_1) q^{6} + (\beta_{11} + \beta_{7} - \beta_{3} + 1) q^{7} + ( - 2 \beta_{3} - 2) q^{8} + ( - \beta_{10} + \beta_{9} + \beta_{8} - \beta_{7} + \beta_{6} - \beta_{5} + 2 \beta_{4} + 4 \beta_{3} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} + 1) q^{2} + \beta_{2} q^{3} - 2 \beta_{3} q^{4} + (\beta_{10} + \beta_{8} + 1) q^{5} + (\beta_{2} + \beta_1) q^{6} + (\beta_{11} + \beta_{7} - \beta_{3} + 1) q^{7} + ( - 2 \beta_{3} - 2) q^{8} + ( - \beta_{10} + \beta_{9} + \beta_{8} - \beta_{7} + \beta_{6} - \beta_{5} + 2 \beta_{4} + 4 \beta_{3} + \cdots + 1) q^{9}+ \cdots + (9 \beta_{11} + 6 \beta_{10} + 6 \beta_{9} + 6 \beta_{8} + 6 \beta_{7} - 6 \beta_{6} + \cdots + 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 12 q^{2} + 4 q^{5} + 8 q^{7} - 24 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 12 q^{2} + 4 q^{5} + 8 q^{7} - 24 q^{8} - 16 q^{11} - 32 q^{15} - 48 q^{16} - 8 q^{17} + 52 q^{18} - 8 q^{20} + 56 q^{21} - 16 q^{22} - 60 q^{23} + 56 q^{25} - 12 q^{27} - 16 q^{28} - 16 q^{30} + 32 q^{31} - 48 q^{32} + 156 q^{33} + 68 q^{35} + 104 q^{36} + 28 q^{37} - 32 q^{38} - 16 q^{40} - 8 q^{41} + 56 q^{42} - 52 q^{43} + 24 q^{45} - 120 q^{46} - 48 q^{47} - 12 q^{50} - 328 q^{51} + 20 q^{53} - 192 q^{55} - 32 q^{56} - 148 q^{57} - 96 q^{58} + 32 q^{60} - 104 q^{61} + 32 q^{62} - 16 q^{63} + 312 q^{66} + 88 q^{67} + 16 q^{68} + 88 q^{70} + 560 q^{71} + 104 q^{72} + 108 q^{73} - 40 q^{75} - 64 q^{76} + 188 q^{77} - 16 q^{80} - 60 q^{81} - 8 q^{82} + 40 q^{83} - 168 q^{85} - 104 q^{86} + 220 q^{87} + 32 q^{88} + 144 q^{90} + 104 q^{91} - 120 q^{92} - 620 q^{93} + 396 q^{95} - 416 q^{97} + 300 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4 x^{11} + 34 x^{10} + 12 x^{9} + 435 x^{8} + 2024 x^{7} + 10862 x^{6} + 44868 x^{5} + 149289 x^{4} + 340080 x^{3} + 619204 x^{2} + 735420 x + 354025 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 19\!\cdots\!78 \nu^{11} + \cdots - 54\!\cdots\!00 ) / 15\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 19\!\cdots\!48 \nu^{11} + \cdots + 54\!\cdots\!00 ) / 15\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 861103744344 \nu^{11} + 4543005521154 \nu^{10} - 35270655980094 \nu^{9} + 35262839612400 \nu^{8} + \cdots - 25\!\cdots\!00 ) / 62\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 64\!\cdots\!23 \nu^{11} + \cdots - 14\!\cdots\!25 ) / 31\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 59\!\cdots\!63 \nu^{11} + \cdots - 17\!\cdots\!00 ) / 15\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 60\!\cdots\!41 \nu^{11} + \cdots + 16\!\cdots\!00 ) / 15\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 60\!\cdots\!79 \nu^{11} + \cdots - 17\!\cdots\!00 ) / 15\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 12\!\cdots\!18 \nu^{11} + \cdots + 35\!\cdots\!50 ) / 31\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 88\!\cdots\!69 \nu^{11} + \cdots - 25\!\cdots\!75 ) / 15\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 52\!\cdots\!72 \nu^{11} + \cdots - 14\!\cdots\!50 ) / 63\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 13\!\cdots\!72 \nu^{11} + \cdots + 39\!\cdots\!75 ) / 15\!\cdots\!75 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( - 4 \beta_{11} + 2 \beta_{10} + \beta_{9} + 6 \beta_{8} - 5 \beta_{7} + 5 \beta_{6} - 2 \beta_{5} + 3 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + 6 \beta _1 + 6 ) / 10 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{11} + 2 \beta_{10} + 6 \beta_{9} + \beta_{8} - 15 \beta_{7} + 3 \beta_{5} - 2 \beta_{4} - 8 \beta_{3} - 8 \beta_{2} + 11 \beta _1 - 19 ) / 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 62 \beta_{11} + 34 \beta_{10} + 67 \beta_{9} - 78 \beta_{8} - 25 \beta_{7} + 25 \beta_{6} + 86 \beta_{5} - 59 \beta_{4} - 386 \beta_{3} - 76 \beta_{2} - 18 \beta _1 - 308 ) / 10 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 47 \beta_{11} + 18 \beta_{10} + 17 \beta_{9} - 29 \beta_{8} + 62 \beta_{7} + 32 \beta_{6} + 49 \beta_{5} - 10 \beta_{4} - 145 \beta_{3} - 45 \beta_{2} - 79 \beta _1 - 114 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 1186 \beta_{11} - 418 \beta_{10} - 619 \beta_{9} - 1134 \beta_{8} + 4065 \beta_{7} + 1945 \beta_{6} + 128 \beta_{5} + 1543 \beta_{4} + 2902 \beta_{3} - 1768 \beta_{2} - 4504 \beta _1 - 3544 ) / 10 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 414 \beta_{11} - 5678 \beta_{10} - 2499 \beta_{9} - 3084 \beta_{8} + 6265 \beta_{7} + 210 \beta_{6} - 7002 \beta_{5} + 6808 \beta_{4} + 33392 \beta_{3} + 3937 \beta_{2} - 5554 \beta _1 - 6469 ) / 5 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 29278 \beta_{11} - 75196 \beta_{10} - 17313 \beta_{9} - 20638 \beta_{8} - 2545 \beta_{7} - 89615 \beta_{6} - 94644 \beta_{5} + 40351 \beta_{4} + 360414 \beta_{3} + 143594 \beta_{2} + \cdots - 3058 ) / 10 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 24245 \beta_{11} - 17251 \beta_{10} - 13322 \beta_{9} + 13618 \beta_{8} - 23689 \beta_{7} - 70925 \beta_{6} - 28330 \beta_{5} - 14852 \beta_{4} + 63030 \beta_{3} + 79881 \beta_{2} + \cdots + 83641 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 1550674 \beta_{11} + 992112 \beta_{10} - 1065839 \beta_{9} + 2303826 \beta_{8} - 1863065 \beta_{7} - 2150215 \beta_{6} + 261858 \beta_{5} - 1974517 \beta_{4} - 4290148 \beta_{3} + \cdots + 9280456 ) / 10 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 3202329 \beta_{11} + 5925642 \beta_{10} - 1358279 \beta_{9} + 7095731 \beta_{8} - 6403680 \beta_{7} + 3151345 \beta_{6} + 4094943 \beta_{5} - 4354957 \beta_{4} + \cdots + 23256581 ) / 5 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 3073282 \beta_{11} + 60334874 \beta_{10} + 29181107 \beta_{9} + 30877252 \beta_{8} - 64443015 \beta_{7} + 102687595 \beta_{6} + 59711976 \beta_{5} - 14816339 \beta_{4} + \cdots + 26279292 ) / 10 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/130\mathbb{Z}\right)^\times\).

\(n\) \(27\) \(41\)
\(\chi(n)\) \(-\beta_{3}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
27.1
1.61554 4.65786i
−1.96771 + 1.30974i
−1.24767 + 0.0362500i
−0.272159 + 2.46318i
4.21538 + 3.75303i
−0.343384 2.90434i
1.61554 + 4.65786i
−1.96771 1.30974i
−1.24767 0.0362500i
−0.272159 2.46318i
4.21538 3.75303i
−0.343384 + 2.90434i
1.00000 + 1.00000i −3.71323 + 3.71323i 2.00000i 3.82477 + 3.22042i −7.42646 0.429967 + 0.429967i −2.00000 + 2.00000i 18.5761i 0.604347 + 7.04519i
27.2 1.00000 + 1.00000i −2.22000 + 2.22000i 2.00000i −4.90152 0.987478i −4.44001 −4.73561 4.73561i −2.00000 + 2.00000i 0.856825i −3.91404 5.88900i
27.3 1.00000 + 1.00000i −0.827569 + 0.827569i 2.00000i 4.04912 2.93336i −1.65514 4.16739 + 4.16739i −2.00000 + 2.00000i 7.63026i 6.98248 + 1.11576i
27.4 1.00000 + 1.00000i 0.370523 0.370523i 2.00000i −2.42963 + 4.37000i 0.741047 2.03473 + 2.03473i −2.00000 + 2.00000i 8.72542i −6.79963 + 1.94038i
27.5 1.00000 + 1.00000i 3.04757 3.04757i 2.00000i 4.40191 + 2.37133i 6.09514 −2.53080 2.53080i −2.00000 + 2.00000i 9.57539i 2.03058 + 6.77324i
27.6 1.00000 + 1.00000i 3.34270 3.34270i 2.00000i −2.94465 4.04092i 6.68541 4.63433 + 4.63433i −2.00000 + 2.00000i 13.3473i 1.09626 6.98557i
53.1 1.00000 1.00000i −3.71323 3.71323i 2.00000i 3.82477 3.22042i −7.42646 0.429967 0.429967i −2.00000 2.00000i 18.5761i 0.604347 7.04519i
53.2 1.00000 1.00000i −2.22000 2.22000i 2.00000i −4.90152 + 0.987478i −4.44001 −4.73561 + 4.73561i −2.00000 2.00000i 0.856825i −3.91404 + 5.88900i
53.3 1.00000 1.00000i −0.827569 0.827569i 2.00000i 4.04912 + 2.93336i −1.65514 4.16739 4.16739i −2.00000 2.00000i 7.63026i 6.98248 1.11576i
53.4 1.00000 1.00000i 0.370523 + 0.370523i 2.00000i −2.42963 4.37000i 0.741047 2.03473 2.03473i −2.00000 2.00000i 8.72542i −6.79963 1.94038i
53.5 1.00000 1.00000i 3.04757 + 3.04757i 2.00000i 4.40191 2.37133i 6.09514 −2.53080 + 2.53080i −2.00000 2.00000i 9.57539i 2.03058 6.77324i
53.6 1.00000 1.00000i 3.34270 + 3.34270i 2.00000i −2.94465 + 4.04092i 6.68541 4.63433 4.63433i −2.00000 2.00000i 13.3473i 1.09626 + 6.98557i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 27.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 130.3.i.b 12
5.b even 2 1 650.3.i.c 12
5.c odd 4 1 inner 130.3.i.b 12
5.c odd 4 1 650.3.i.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
130.3.i.b 12 1.a even 1 1 trivial
130.3.i.b 12 5.c odd 4 1 inner
650.3.i.c 12 5.b even 2 1
650.3.i.c 12 5.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} + 4 T_{3}^{9} + 852 T_{3}^{8} - 128 T_{3}^{7} + 8 T_{3}^{6} + 10024 T_{3}^{5} + 123396 T_{3}^{4} + 106888 T_{3}^{3} + 41472 T_{3}^{2} - 59328 T_{3} + 42436 \) acting on \(S_{3}^{\mathrm{new}}(130, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 2 T + 2)^{6} \) Copy content Toggle raw display
$3$ \( T^{12} + 4 T^{9} + 852 T^{8} + \cdots + 42436 \) Copy content Toggle raw display
$5$ \( T^{12} - 4 T^{11} - 20 T^{10} + \cdots + 244140625 \) Copy content Toggle raw display
$7$ \( T^{12} - 8 T^{11} + 32 T^{10} + \cdots + 2624400 \) Copy content Toggle raw display
$11$ \( (T^{6} + 8 T^{5} - 208 T^{4} - 248 T^{3} + \cdots - 74138)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 169)^{3} \) Copy content Toggle raw display
$17$ \( T^{12} + 8 T^{11} + \cdots + 3113728459776 \) Copy content Toggle raw display
$19$ \( T^{12} + 1276 T^{10} + \cdots + 23687202302500 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 630951601562500 \) Copy content Toggle raw display
$29$ \( T^{12} + 5520 T^{10} + \cdots + 60\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{6} - 16 T^{5} - 3172 T^{4} + \cdots + 39973270)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} - 28 T^{11} + \cdots + 56\!\cdots\!44 \) Copy content Toggle raw display
$41$ \( (T^{6} + 4 T^{5} - 7802 T^{4} + \cdots - 1381745000)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 542737815066756 \) Copy content Toggle raw display
$47$ \( T^{12} + 48 T^{11} + \cdots + 53\!\cdots\!44 \) Copy content Toggle raw display
$53$ \( T^{12} - 20 T^{11} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{12} + 12260 T^{10} + \cdots + 54\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{6} + 52 T^{5} - 9512 T^{4} + \cdots - 8632277404)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} - 88 T^{11} + \cdots + 70\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{6} - 280 T^{5} + \cdots + 2676310808950)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} - 108 T^{11} + \cdots + 98\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{12} + 23384 T^{10} + \cdots + 83\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{12} - 40 T^{11} + \cdots + 54\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{12} + 43840 T^{10} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{12} + 416 T^{11} + \cdots + 28\!\cdots\!00 \) Copy content Toggle raw display
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