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Properties

Label 70.5.f.b
Level $70$
Weight $5$
Character orbit 70.f
Analytic conductor $7.236$
Analytic rank $0$
Dimension $12$
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] = [70,5,Mod(43,70)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(70, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([3, 0]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("70.43");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 70 = 2 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 70.f (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: \(7.23589741587\)
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: \( x^{12} - 4 x^{11} + 8 x^{10} + 256 x^{9} + 11896 x^{8} - 21136 x^{7} + 22144 x^{6} + 200304 x^{5} + \cdots + 48400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 5^{4}\cdot 7^{2} \)
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.

\(f(q)\) \(=\) \( q + (2 \beta_1 + 2) q^{2} + ( - \beta_{2} + 2 \beta_1 - 2) q^{3} + 8 \beta_1 q^{4} + (\beta_{11} + \beta_{5} - \beta_{2} + \cdots + 3) q^{5}+ \cdots + (7 \beta_{11} + 2 \beta_{10} + \cdots - 48 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (2 \beta_1 + 2) q^{2} + ( - \beta_{2} + 2 \beta_1 - 2) q^{3} + 8 \beta_1 q^{4} + (\beta_{11} + \beta_{5} - \beta_{2} + \cdots + 3) q^{5}+ \cdots + ( - 706 \beta_{11} + 334 \beta_{10} + \cdots - 5900 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 24 q^{2} - 20 q^{3} + 40 q^{5} - 80 q^{6} - 192 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 24 q^{2} - 20 q^{3} + 40 q^{5} - 80 q^{6} - 192 q^{8} - 48 q^{10} - 316 q^{11} - 160 q^{12} + 300 q^{13} + 288 q^{15} - 768 q^{16} - 364 q^{17} + 1104 q^{18} - 512 q^{20} - 196 q^{21} - 632 q^{22} - 2128 q^{23} - 688 q^{25} + 1200 q^{26} + 3508 q^{27} + 1784 q^{30} + 2848 q^{31} - 1536 q^{32} + 1212 q^{33} + 784 q^{35} + 4416 q^{36} - 5480 q^{37} - 2880 q^{38} - 1664 q^{40} - 10064 q^{41} - 392 q^{42} + 5084 q^{43} + 1420 q^{45} - 8512 q^{46} + 6412 q^{47} + 1280 q^{48} - 1336 q^{50} + 19788 q^{51} + 2400 q^{52} + 3508 q^{53} + 8708 q^{55} - 1324 q^{57} - 7784 q^{58} + 4832 q^{60} - 9192 q^{61} + 5696 q^{62} - 3920 q^{63} - 21136 q^{65} + 4848 q^{66} - 3660 q^{67} + 2912 q^{68} + 1960 q^{70} - 4944 q^{71} + 8832 q^{72} + 936 q^{73} - 14212 q^{75} - 11520 q^{76} - 2744 q^{77} + 11608 q^{78} - 2560 q^{80} - 20420 q^{81} - 20128 q^{82} - 16672 q^{83} + 4092 q^{85} + 20336 q^{86} - 18844 q^{87} + 5056 q^{88} + 8304 q^{90} + 11172 q^{91} - 17024 q^{92} - 18228 q^{93} + 18980 q^{95} + 5120 q^{96} + 36980 q^{97} - 8232 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4 x^{11} + 8 x^{10} + 256 x^{9} + 11896 x^{8} - 21136 x^{7} + 22144 x^{6} + 200304 x^{5} + \cdots + 48400 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 28\!\cdots\!03 \nu^{11} + \cdots + 37\!\cdots\!60 ) / 70\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 33\!\cdots\!46 \nu^{11} + \cdots + 39\!\cdots\!00 ) / 14\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 35\!\cdots\!42 \nu^{11} + \cdots + 41\!\cdots\!00 ) / 11\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 48\!\cdots\!04 \nu^{11} + \cdots - 11\!\cdots\!40 ) / 14\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 18\!\cdots\!57 \nu^{11} + \cdots - 87\!\cdots\!00 ) / 49\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 59\!\cdots\!54 \nu^{11} + \cdots - 25\!\cdots\!00 ) / 14\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 64\!\cdots\!11 \nu^{11} + \cdots - 16\!\cdots\!40 ) / 14\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 11\!\cdots\!63 \nu^{11} + \cdots + 49\!\cdots\!60 ) / 26\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 11\!\cdots\!19 \nu^{11} + \cdots + 64\!\cdots\!40 ) / 16\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 98\!\cdots\!87 \nu^{11} + \cdots + 21\!\cdots\!80 ) / 11\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 46\!\cdots\!48 \nu^{11} + \cdots - 61\!\cdots\!60 ) / 49\!\cdots\!92 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{11} - \beta_{8} - \beta_{6} + \beta_{5} + 2\beta_{2} - 2\beta _1 + 2 ) / 5 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 12 \beta_{11} - 7 \beta_{10} - \beta_{9} - \beta_{7} - 5 \beta_{6} - \beta_{5} + 5 \beta_{4} + \cdots - 210 \beta_1 ) / 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 100 \beta_{11} - 22 \beta_{10} + 131 \beta_{9} + 100 \beta_{8} + 122 \beta_{7} + 7 \beta_{6} + \cdots - 328 ) / 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 910 \beta_{10} + 658 \beta_{9} + 1666 \beta_{8} + 16 \beta_{7} + 98 \beta_{6} - 658 \beta_{5} + \cdots - 21532 ) / 5 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 12756 \beta_{11} - 400 \beta_{9} + 12756 \beta_{8} + 15546 \beta_{6} - 15546 \beta_{5} + 400 \beta_{4} + \cdots - 58222 ) / 5 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 42868 \beta_{11} + 21466 \beta_{10} - 5986 \beta_{9} - 2974 \beta_{7} + 17862 \beta_{6} + \cdots + 494688 \beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 1649628 \beta_{11} + 762196 \beta_{10} - 1854286 \beta_{9} - 1649628 \beta_{8} - 1360172 \beta_{7} + \cdots + 9187820 ) / 5 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 12733204 \beta_{10} - 11977476 \beta_{9} - 27421436 \beta_{8} - 3168800 \beta_{7} - 5407844 \beta_{6} + \cdots + 293271824 ) / 5 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 213191520 \beta_{11} - 9503152 \beta_{9} - 213191520 \beta_{8} - 224415364 \beta_{6} + \cdots + 1346460212 ) / 5 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 3510815288 \beta_{11} - 1533958068 \beta_{10} + 841502116 \beta_{9} + 527862636 \beta_{7} + \cdots - 35412247440 \beta_1 ) / 5 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 27516118168 \beta_{11} - 15975988616 \beta_{10} + 27524218636 \beta_{9} + 27516118168 \beta_{8} + \cdots - 188833485160 ) / 5 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(31\) \(57\)
\(\chi(n)\) \(1\) \(\beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1
2.88242 + 2.88242i
−0.137998 − 0.137998i
8.01799 + 8.01799i
0.521375 + 0.521375i
−6.87999 − 6.87999i
−2.40380 − 2.40380i
2.88242 − 2.88242i
−0.137998 + 0.137998i
8.01799 − 8.01799i
0.521375 − 0.521375i
−6.87999 + 6.87999i
−2.40380 + 2.40380i
2.00000 − 2.00000i −12.0748 − 12.0748i − 8.00000i 3.91373 − 24.6918i −48.2990 −13.0958 + 13.0958i −16.0000 − 16.0000i 210.600i −41.5560 − 57.2110i
43.2 2.00000 − 2.00000i −9.68889 − 9.68889i − 8.00000i −14.6123 + 20.2850i −38.7555 13.0958 − 13.0958i −16.0000 − 16.0000i 106.749i 11.3452 + 69.7946i
43.3 2.00000 − 2.00000i −3.84636 − 3.84636i − 8.00000i 24.8737 − 2.51023i −15.3854 13.0958 − 13.0958i −16.0000 − 16.0000i − 51.4111i 44.7268 − 54.7678i
43.4 2.00000 − 2.00000i −0.115508 − 0.115508i − 8.00000i −20.0943 − 14.8734i −0.462032 −13.0958 + 13.0958i −16.0000 − 16.0000i − 80.9733i −69.9355 + 10.4418i
43.5 2.00000 − 2.00000i 6.66442 + 6.66442i − 8.00000i 5.35117 − 24.4206i 26.6577 13.0958 − 13.0958i −16.0000 − 16.0000i 7.82886i −38.1388 − 59.5435i
43.6 2.00000 − 2.00000i 9.06110 + 9.06110i − 8.00000i 20.5681 + 14.2110i 36.2444 −13.0958 + 13.0958i −16.0000 − 16.0000i 83.2069i 69.5582 − 12.7142i
57.1 2.00000 + 2.00000i −12.0748 + 12.0748i 8.00000i 3.91373 + 24.6918i −48.2990 −13.0958 − 13.0958i −16.0000 + 16.0000i − 210.600i −41.5560 + 57.2110i
57.2 2.00000 + 2.00000i −9.68889 + 9.68889i 8.00000i −14.6123 − 20.2850i −38.7555 13.0958 + 13.0958i −16.0000 + 16.0000i − 106.749i 11.3452 − 69.7946i
57.3 2.00000 + 2.00000i −3.84636 + 3.84636i 8.00000i 24.8737 + 2.51023i −15.3854 13.0958 + 13.0958i −16.0000 + 16.0000i 51.4111i 44.7268 + 54.7678i
57.4 2.00000 + 2.00000i −0.115508 + 0.115508i 8.00000i −20.0943 + 14.8734i −0.462032 −13.0958 − 13.0958i −16.0000 + 16.0000i 80.9733i −69.9355 − 10.4418i
57.5 2.00000 + 2.00000i 6.66442 − 6.66442i 8.00000i 5.35117 + 24.4206i 26.6577 13.0958 + 13.0958i −16.0000 + 16.0000i − 7.82886i −38.1388 + 59.5435i
57.6 2.00000 + 2.00000i 9.06110 − 9.06110i 8.00000i 20.5681 − 14.2110i 36.2444 −13.0958 − 13.0958i −16.0000 + 16.0000i − 83.2069i 69.5582 + 12.7142i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 43.6
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 70.5.f.b ✓ 12
5.b even 2 1 350.5.f.c 12
5.c odd 4 1 inner 70.5.f.b ✓ 12
5.c odd 4 1 350.5.f.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.5.f.b ✓ 12 1.a even 1 1 trivial
70.5.f.b ✓ 12 5.c odd 4 1 inner
350.5.f.c 12 5.b even 2 1
350.5.f.c 12 5.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} + 20 T_{3}^{11} + 200 T_{3}^{10} - 916 T_{3}^{9} + 39685 T_{3}^{8} + 707016 T_{3}^{7} + \cdots + 630512100 \) acting on \(S_{5}^{\mathrm{new}}(70, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 4 T + 8)^{6} \) Copy content Toggle raw display
$3$ \( T^{12} + \cdots + 630512100 \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 59\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( (T^{4} + 117649)^{3} \) Copy content Toggle raw display
$11$ \( (T^{6} + \cdots + 1754692842800)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 28\!\cdots\!04 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 89\!\cdots\!24 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 56\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots + 50\!\cdots\!56)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 94\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots + 77\!\cdots\!44)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 31\!\cdots\!84 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 17\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 16\!\cdots\!16 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 32\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots - 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 44\!\cdots\!24 \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots + 26\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 30\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 98\!\cdots\!00 \) Copy content Toggle raw display
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