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Properties

Label 70.5.d.a
Level $70$
Weight $5$
Character orbit 70.d
Analytic conductor $7.236$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [70,5,Mod(69,70)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(70, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("70.69");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 70 = 2 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 70.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: \(7.23589741587\)
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: \( x^{16} - 866 x^{14} + 287813 x^{12} - 45083388 x^{10} + 3271778340 x^{8} - 92428530176 x^{6} + \cdots + 53721628886016 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{20} \)
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.

\(f(q)\) \(=\) \( q - \beta_{5} q^{2} - \beta_{4} q^{3} - 8 q^{4} - \beta_{6} q^{5} - \beta_1 q^{6} + ( - \beta_{12} + \beta_{5}) q^{7} + 8 \beta_{5} q^{8} + ( - \beta_{10} + \beta_{9} + 29) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{2} - \beta_{4} q^{3} - 8 q^{4} - \beta_{6} q^{5} - \beta_1 q^{6} + ( - \beta_{12} + \beta_{5}) q^{7} + 8 \beta_{5} q^{8} + ( - \beta_{10} + \beta_{9} + 29) q^{9} + (\beta_{14} - \beta_{7} - 2 \beta_{4} - \beta_1) q^{10} + ( - \beta_{12} + \beta_{10} + \cdots + 12) q^{11}+ \cdots + (10 \beta_{12} + 2 \beta_{10} + \cdots + 2866) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 128 q^{4} + 468 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 128 q^{4} + 468 q^{9} + 180 q^{11} + 192 q^{14} + 700 q^{15} + 1024 q^{16} - 28 q^{21} - 2520 q^{25} + 564 q^{29} + 2560 q^{30} + 1320 q^{35} - 3744 q^{36} - 12828 q^{39} - 1440 q^{44} + 5248 q^{46} + 7808 q^{49} - 4800 q^{50} - 8684 q^{51} - 1536 q^{56} - 5600 q^{60} - 8192 q^{64} + 24300 q^{65} + 10720 q^{70} + 22128 q^{71} + 16704 q^{74} - 27164 q^{79} + 46088 q^{81} + 224 q^{84} - 14540 q^{85} + 2880 q^{86} - 54748 q^{91} + 24600 q^{95} + 45928 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 866 x^{14} + 287813 x^{12} - 45083388 x^{10} + 3271778340 x^{8} - 92428530176 x^{6} + \cdots + 53721628886016 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 8408733842569 \nu^{14} + \cdots - 37\!\cdots\!28 ) / 23\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 11\!\cdots\!81 \nu^{14} + \cdots + 41\!\cdots\!44 ) / 83\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 12\!\cdots\!11 \nu^{14} + \cdots + 61\!\cdots\!60 ) / 75\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 20\!\cdots\!23 \nu^{15} + \cdots - 20\!\cdots\!28 \nu ) / 11\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 20\!\cdots\!23 \nu^{15} + \cdots - 94\!\cdots\!32 \nu ) / 55\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 48\!\cdots\!61 \nu^{15} + \cdots - 23\!\cdots\!52 ) / 25\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 48\!\cdots\!61 \nu^{15} + \cdots - 23\!\cdots\!52 ) / 25\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 27\!\cdots\!85 \nu^{15} + \cdots - 29\!\cdots\!68 \nu ) / 58\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 65\!\cdots\!81 \nu^{15} + \cdots - 20\!\cdots\!64 ) / 62\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 65\!\cdots\!81 \nu^{15} + \cdots - 15\!\cdots\!60 ) / 62\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 41\!\cdots\!03 \nu^{15} + \cdots - 90\!\cdots\!96 ) / 31\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 50\!\cdots\!91 \nu^{15} + \cdots + 27\!\cdots\!56 \nu ) / 31\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 50\!\cdots\!69 \nu^{15} + \cdots - 79\!\cdots\!84 ) / 25\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 50\!\cdots\!69 \nu^{15} + \cdots - 79\!\cdots\!84 ) / 25\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 32\!\cdots\!69 \nu^{15} + \cdots - 18\!\cdots\!92 ) / 62\!\cdots\!24 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{5} - 2\beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{10} + \beta_{9} + \beta _1 + 108 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 3 \beta_{15} + 2 \beta_{14} - 2 \beta_{13} - 16 \beta_{12} + 3 \beta_{11} - 16 \beta_{8} + \cdots - 402 \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 10 \beta_{14} - 10 \beta_{13} + 35 \beta_{12} - 287 \beta_{10} + 217 \beta_{9} - 35 \beta_{8} + \cdots + 21702 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 1475 \beta_{15} + 1106 \beta_{14} - 1106 \beta_{13} - 5998 \beta_{12} + 1125 \beta_{11} + \cdots - 175 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 53 \beta_{14} - 53 \beta_{13} + 14259 \beta_{12} - 73729 \beta_{10} + 45211 \beta_{9} + \cdots + 4596174 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 557627 \beta_{15} + 355494 \beta_{14} - 355494 \beta_{13} - 1883034 \beta_{12} + \cdots - 104713 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 823868 \beta_{14} + 823868 \beta_{13} + 4118065 \beta_{12} - 17633559 \beta_{10} + 9397429 \beta_{9} + \cdots + 968538282 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 185867679 \beta_{15} + 92920050 \beta_{14} - 92920050 \beta_{13} - 535768578 \beta_{12} + \cdots - 42135849 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 416505501 \beta_{14} + 416505501 \beta_{13} + 1009266279 \beta_{12} - 3927673225 \beta_{10} + \cdots + 198693961830 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 57235646127 \beta_{15} + 21250954886 \beta_{14} - 21250954886 \beta_{13} + \cdots - 14266594881 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 151360840490 \beta_{14} + 151360840490 \beta_{13} + 217336924385 \beta_{12} - 801106330271 \beta_{10} + \cdots + 38399596010922 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 16662325891211 \beta_{15} + 4253023957874 \beta_{14} - 4253023957874 \beta_{13} + \cdots - 4375598326645 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 47616538757383 \beta_{14} + 47616538757383 \beta_{13} + 39971331227247 \beta_{12} + \cdots + 65\!\cdots\!98 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 46\!\cdots\!55 \beta_{15} + 692156319544422 \beta_{14} - 692156319544422 \beta_{13} + \cdots - 12\!\cdots\!93 \beta_{2} ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/70\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(57\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
69.1
−16.0449 + 1.41421i
−12.6864 + 1.41421i
−3.63871 + 1.41421i
−3.06211 + 1.41421i
3.06211 + 1.41421i
3.63871 + 1.41421i
12.6864 + 1.41421i
16.0449 + 1.41421i
−16.0449 − 1.41421i
−12.6864 − 1.41421i
−3.63871 − 1.41421i
−3.06211 − 1.41421i
3.06211 − 1.41421i
3.63871 − 1.41421i
12.6864 − 1.41421i
16.0449 − 1.41421i
− 2.82843i −16.0449 −8.00000 −3.21186 − 24.7928i 45.3818i −43.8181 + 21.9312i 22.6274i 176.438 −70.1247 + 9.08452i
69.2 − 2.82843i −12.6864 −8.00000 −18.0326 + 17.3155i 35.8826i 38.4939 − 30.3187i 22.6274i 79.9449 48.9757 + 51.0038i
69.3 − 2.82843i −3.63871 −8.00000 10.2123 − 22.8190i 10.2918i 44.8401 − 19.7577i 22.6274i −67.7598 −64.5420 − 28.8849i
69.4 − 2.82843i −3.06211 −8.00000 22.2535 + 11.3922i 8.66095i 19.1198 + 45.1158i 22.6274i −71.6235 32.2219 − 62.9424i
69.5 − 2.82843i 3.06211 −8.00000 −22.2535 − 11.3922i − 8.66095i −19.1198 + 45.1158i 22.6274i −71.6235 −32.2219 + 62.9424i
69.6 − 2.82843i 3.63871 −8.00000 −10.2123 + 22.8190i − 10.2918i −44.8401 − 19.7577i 22.6274i −67.7598 64.5420 + 28.8849i
69.7 − 2.82843i 12.6864 −8.00000 18.0326 − 17.3155i − 35.8826i −38.4939 − 30.3187i 22.6274i 79.9449 −48.9757 − 51.0038i
69.8 − 2.82843i 16.0449 −8.00000 3.21186 + 24.7928i − 45.3818i 43.8181 + 21.9312i 22.6274i 176.438 70.1247 − 9.08452i
69.9 2.82843i −16.0449 −8.00000 −3.21186 + 24.7928i − 45.3818i −43.8181 − 21.9312i − 22.6274i 176.438 −70.1247 − 9.08452i
69.10 2.82843i −12.6864 −8.00000 −18.0326 − 17.3155i − 35.8826i 38.4939 + 30.3187i − 22.6274i 79.9449 48.9757 − 51.0038i
69.11 2.82843i −3.63871 −8.00000 10.2123 + 22.8190i − 10.2918i 44.8401 + 19.7577i − 22.6274i −67.7598 −64.5420 + 28.8849i
69.12 2.82843i −3.06211 −8.00000 22.2535 − 11.3922i − 8.66095i 19.1198 − 45.1158i − 22.6274i −71.6235 32.2219 + 62.9424i
69.13 2.82843i 3.06211 −8.00000 −22.2535 + 11.3922i 8.66095i −19.1198 − 45.1158i − 22.6274i −71.6235 −32.2219 − 62.9424i
69.14 2.82843i 3.63871 −8.00000 −10.2123 − 22.8190i 10.2918i −44.8401 + 19.7577i − 22.6274i −67.7598 64.5420 − 28.8849i
69.15 2.82843i 12.6864 −8.00000 18.0326 + 17.3155i 35.8826i −38.4939 + 30.3187i − 22.6274i 79.9449 −48.9757 + 51.0038i
69.16 2.82843i 16.0449 −8.00000 3.21186 − 24.7928i 45.3818i 43.8181 − 21.9312i − 22.6274i 176.438 70.1247 + 9.08452i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 69.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.b odd 2 1 inner
35.c odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 70.5.d.a ✓ 16
3.b odd 2 1 630.5.h.a 16
4.b odd 2 1 560.5.p.i 16
5.b even 2 1 inner 70.5.d.a ✓ 16
5.c odd 4 2 350.5.b.e 16
7.b odd 2 1 inner 70.5.d.a ✓ 16
15.d odd 2 1 630.5.h.a 16
20.d odd 2 1 560.5.p.i 16
21.c even 2 1 630.5.h.a 16
28.d even 2 1 560.5.p.i 16
35.c odd 2 1 inner 70.5.d.a ✓ 16
35.f even 4 2 350.5.b.e 16
105.g even 2 1 630.5.h.a 16
140.c even 2 1 560.5.p.i 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.5.d.a ✓ 16 1.a even 1 1 trivial
70.5.d.a ✓ 16 5.b even 2 1 inner
70.5.d.a ✓ 16 7.b odd 2 1 inner
70.5.d.a ✓ 16 35.c odd 2 1 inner
350.5.b.e 16 5.c odd 4 2
350.5.b.e 16 35.f even 4 2
560.5.p.i 16 4.b odd 2 1
560.5.p.i 16 20.d odd 2 1
560.5.p.i 16 28.d even 2 1
560.5.p.i 16 140.c even 2 1
630.5.h.a 16 3.b odd 2 1
630.5.h.a 16 15.d odd 2 1
630.5.h.a 16 21.c even 2 1
630.5.h.a 16 105.g even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(70, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 8)^{8} \) Copy content Toggle raw display
$3$ \( (T^{8} - 441 T^{6} + \cdots + 5143824)^{2} \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 23\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 11\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( (T^{4} - 45 T^{3} + \cdots + 62094424)^{4} \) Copy content Toggle raw display
$13$ \( (T^{8} + \cdots + 59\!\cdots\!24)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + \cdots + 80\!\cdots\!04)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + \cdots + 21\!\cdots\!84)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + \cdots + 11\!\cdots\!76)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 141 T^{3} + \cdots + 110143507576)^{4} \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots + 19\!\cdots\!64)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots + 68\!\cdots\!76)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 37\!\cdots\!04)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots + 56\!\cdots\!76)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots + 16\!\cdots\!64)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 18\!\cdots\!56)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 26\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 11\!\cdots\!84)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 34\!\cdots\!16)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots - 165574624480256)^{4} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 91\!\cdots\!04)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots - 765358135030944)^{4} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 29\!\cdots\!64)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 62\!\cdots\!84)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 12\!\cdots\!04)^{2} \) Copy content Toggle raw display
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