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Properties

Label 155.3.d.b
Level $155$
Weight $3$
Character orbit 155.d
Analytic conductor $4.223$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [155,3,Mod(61,155)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(155, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("155.61");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 155 = 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 155.d (of order \(2\), degree \(1\), 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: \(4.22344409758\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 94 x^{14} + 3636 x^{12} + 74942 x^{10} + 887867 x^{8} + 6061528 x^{6} + 22540376 x^{4} + \cdots + 21377600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{11}\cdot 5 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{9} q^{2} + \beta_1 q^{3} + ( - \beta_{7} + 3) q^{4} + \beta_{2} q^{5} - \beta_{13} q^{6} + ( - \beta_{10} - \beta_{8} + \beta_{7} + \cdots + 1) q^{7}+ \cdots + (\beta_{14} - \beta_{10} + \beta_{9} + \cdots - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{9} q^{2} + \beta_1 q^{3} + ( - \beta_{7} + 3) q^{4} + \beta_{2} q^{5} - \beta_{13} q^{6} + ( - \beta_{10} - \beta_{8} + \beta_{7} + \cdots + 1) q^{7}+ \cdots + (2 \beta_{15} + 3 \beta_{13} + \cdots - 5 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4 q^{2} + 48 q^{4} + 4 q^{7} + 8 q^{8} - 44 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 4 q^{2} + 48 q^{4} + 4 q^{7} + 8 q^{8} - 44 q^{9} + 20 q^{10} - 16 q^{14} - 24 q^{16} - 56 q^{18} + 12 q^{19} - 40 q^{20} + 80 q^{25} - 168 q^{28} + 148 q^{31} + 256 q^{32} - 108 q^{33} + 60 q^{35} - 192 q^{36} + 20 q^{38} + 20 q^{39} - 144 q^{41} + 100 q^{45} + 200 q^{47} - 72 q^{49} + 20 q^{50} + 48 q^{51} + 16 q^{56} - 488 q^{59} + 44 q^{62} + 32 q^{63} - 616 q^{64} + 224 q^{66} - 132 q^{67} - 148 q^{69} + 80 q^{70} + 44 q^{71} + 304 q^{72} + 176 q^{76} + 544 q^{78} - 160 q^{80} - 652 q^{81} + 332 q^{82} + 148 q^{87} - 160 q^{90} + 364 q^{93} + 144 q^{94} + 140 q^{95} + 456 q^{97} - 1292 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 94 x^{14} + 3636 x^{12} + 74942 x^{10} + 887867 x^{8} + 6061528 x^{6} + 22540376 x^{4} + \cdots + 21377600 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 19058982 \nu^{14} + 1652483643 \nu^{12} + 57030683072 \nu^{10} + 997358116714 \nu^{8} + \cdots + 74970645421220 ) / 11178393197380 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 19058982 \nu^{15} - 1652483643 \nu^{13} - 57030683072 \nu^{11} + \cdots - 86149038618600 \nu ) / 11178393197380 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 13190646029 \nu^{15} + 459019515766 \nu^{13} - 9089766740036 \nu^{11} + \cdots + 21\!\cdots\!80 \nu ) / 31\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 16128143991 \nu^{15} - 1303982284154 \nu^{13} - 39381939725516 \nu^{11} + \cdots + 12\!\cdots\!40 \nu ) / 31\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 44164709203 \nu^{15} - 4251308833742 \nu^{13} - 162494816554628 \nu^{11} + \cdots - 29\!\cdots\!60 \nu ) / 31\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 31007756347 \nu^{14} + 2393350743478 \nu^{12} + 72944430721912 \nu^{10} + \cdots + 58\!\cdots\!60 ) / 793665917013980 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 14803859501 \nu^{14} - 1204246736356 \nu^{12} - 38929315517524 \nu^{10} + \cdots - 31\!\cdots\!16 ) / 317466366805592 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 85972317071 \nu^{14} - 6668081904434 \nu^{12} - 201694307084136 \nu^{10} + \cdots - 84\!\cdots\!40 ) / 15\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 18973662947 \nu^{14} + 1559799944642 \nu^{12} + 50985119316024 \nu^{10} + \cdots + 42\!\cdots\!88 ) / 317466366805592 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 53653183669 \nu^{14} - 3932543510906 \nu^{12} - 111061463738434 \nu^{10} + \cdots - 86\!\cdots\!80 ) / 793665917013980 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 31007756347 \nu^{15} + 2393350743478 \nu^{13} + 72944430721912 \nu^{11} + \cdots + 58\!\cdots\!60 \nu ) / 793665917013980 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 85972317071 \nu^{15} - 6668081904434 \nu^{13} - 201694307084136 \nu^{11} + \cdots - 84\!\cdots\!40 \nu ) / 15\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 249447178423 \nu^{14} + 20019009954812 \nu^{12} + 635069767259428 \nu^{10} + \cdots + 45\!\cdots\!40 ) / 15\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 41545309199 \nu^{15} + 3396395181921 \nu^{13} + 110368454292879 \nu^{11} + \cdots + 95\!\cdots\!40 \nu ) / 396832958506990 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{14} - \beta_{10} + \beta_{9} + \beta_{8} + 2\beta_{2} - 11 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{15} + \beta_{13} - \beta_{12} + \beta_{6} + \beta_{4} + \beta_{3} - 16\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -30\beta_{14} + 2\beta_{11} + 28\beta_{10} - 32\beta_{9} - 28\beta_{8} + 5\beta_{7} - 34\beta_{2} + 174 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -30\beta_{15} - 30\beta_{13} + 29\beta_{12} - 30\beta_{6} + 2\beta_{5} - 28\beta_{4} - 54\beta_{3} + 290\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 785\beta_{14} - 61\beta_{11} - 689\beta_{10} + 848\beta_{9} + 748\beta_{8} - 159\beta_{7} + 419\beta_{2} - 3222 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 785 \beta_{15} + 811 \beta_{13} - 726 \beta_{12} + 750 \beta_{6} - 120 \beta_{5} + 748 \beta_{4} + \cdots - 5741 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 20031 \beta_{14} + 1605 \beta_{11} + 16596 \beta_{10} - 21727 \beta_{9} - 19878 \beta_{8} + \cdots + 66013 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 20031 \beta_{15} - 21574 \beta_{13} + 17624 \beta_{12} - 18201 \beta_{6} + 4887 \beta_{5} + \cdots + 122127 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 509144 \beta_{14} - 41640 \beta_{11} - 401627 \beta_{10} + 554323 \beta_{9} + 524426 \beta_{8} + \cdots - 1455043 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 509144 \beta_{15} + 569605 \beta_{13} - 427089 \beta_{12} + 443267 \beta_{6} - 164439 \beta_{5} + \cdots - 2750306 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 12952588 \beta_{14} + 1082104 \beta_{11} + 9834116 \beta_{10} - 14158470 \beta_{9} - 13728892 \beta_{8} + \cdots + 33789024 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 12952588 \beta_{15} - 14934774 \beta_{13} + 10430415 \beta_{12} - 10916220 \beta_{6} + \cdots + 64621500 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 330091839 \beta_{14} - 28138895 \beta_{11} - 243714889 \beta_{10} + 362165730 \beta_{9} + \cdots - 813078536 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 330091839 \beta_{15} + 389181251 \beta_{13} - 257334588 \beta_{12} + 271853784 \beta_{6} + \cdots - 1564390915 \beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/155\mathbb{Z}\right)^\times\).

\(n\) \(32\) \(96\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
61.1
− 4.01240i
4.01240i
− 3.40402i
3.40402i
− 2.27500i
2.27500i
− 0.956880i
0.956880i
− 4.08419i
4.08419i
− 5.06303i
5.06303i
− 2.02341i
2.02341i
− 3.71659i
3.71659i
−3.26787 − 4.01240i 6.67899 −2.23607 13.1120i −10.5721 −8.75461 −7.09938 7.30719
61.2 −3.26787 4.01240i 6.67899 −2.23607 − 13.1120i −10.5721 −8.75461 −7.09938 7.30719
61.3 −2.74462 − 3.40402i 3.53293 −2.23607 9.34272i 11.6873 1.28194 −2.58732 6.13715
61.4 −2.74462 3.40402i 3.53293 −2.23607 − 9.34272i 11.6873 1.28194 −2.58732 6.13715
61.5 −2.69500 − 2.27500i 3.26301 2.23607 6.13111i −0.346163 1.98619 3.82439 −6.02620
61.6 −2.69500 2.27500i 3.26301 2.23607 − 6.13111i −0.346163 1.98619 3.82439 −6.02620
61.7 0.304896 − 0.956880i −3.90704 2.23607 − 0.291749i 5.81824 −2.41082 8.08438 0.681768
61.8 0.304896 0.956880i −3.90704 2.23607 0.291749i 5.81824 −2.41082 8.08438 0.681768
61.9 1.19383 − 4.08419i −2.57478 −2.23607 − 4.87581i −3.78040 −7.84915 −7.68059 −2.66948
61.10 1.19383 4.08419i −2.57478 −2.23607 4.87581i −3.78040 −7.84915 −7.68059 −2.66948
61.11 2.39775 − 5.06303i 1.74920 2.23607 − 12.1399i 5.99727 −5.39686 −16.6342 5.36153
61.12 2.39775 5.06303i 1.74920 2.23607 12.1399i 5.99727 −5.39686 −16.6342 5.36153
61.13 3.22842 − 2.02341i 6.42270 2.23607 − 6.53243i −3.76114 7.82149 4.90580 7.21897
61.14 3.22842 2.02341i 6.42270 2.23607 6.53243i −3.76114 7.82149 4.90580 7.21897
61.15 3.58260 − 3.71659i 8.83499 −2.23607 − 13.3150i −3.04306 17.3218 −4.81305 −8.01093
61.16 3.58260 3.71659i 8.83499 −2.23607 13.3150i −3.04306 17.3218 −4.81305 −8.01093
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 61.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 155.3.d.b ✓ 16
5.b even 2 1 775.3.d.g 16
5.c odd 4 2 775.3.c.d 32
31.b odd 2 1 inner 155.3.d.b ✓ 16
155.c odd 2 1 775.3.d.g 16
155.f even 4 2 775.3.c.d 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
155.3.d.b ✓ 16 1.a even 1 1 trivial
155.3.d.b ✓ 16 31.b odd 2 1 inner
775.3.c.d 32 5.c odd 4 2
775.3.c.d 32 155.f even 4 2
775.3.d.g 16 5.b even 2 1
775.3.d.g 16 155.c odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - 2T_{2}^{7} - 26T_{2}^{6} + 48T_{2}^{5} + 217T_{2}^{4} - 378T_{2}^{3} - 548T_{2}^{2} + 996T_{2} - 244 \) acting on \(S_{3}^{\mathrm{new}}(155, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} - 2 T^{7} + \cdots - 244)^{2} \) Copy content Toggle raw display
$3$ \( T^{16} + 94 T^{14} + \cdots + 21377600 \) Copy content Toggle raw display
$5$ \( (T^{2} - 5)^{8} \) Copy content Toggle raw display
$7$ \( (T^{8} - 2 T^{7} + \cdots - 64576)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 9769050137600 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 77\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{8} - 6 T^{7} + \cdots + 1996327024)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 72\!\cdots\!81 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{8} + 72 T^{7} + \cdots - 8699350864)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( (T^{8} - 100 T^{7} + \cdots + 423056036864)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 50\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 8791434874384)^{2} \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 98\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots - 24139589263424)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 1809404986096)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 87\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 49\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 56\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 74\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 101475374621696)^{2} \) Copy content Toggle raw display
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