Properties

Label 70.6.a.g
Level $70$
Weight $6$
Character orbit 70.a
Self dual yes
Analytic conductor $11.227$
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] = [70,6,Mod(1,70)]
 
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([0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("70.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 70 = 2 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 70.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: \(11.2268673869\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3369}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 842 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 \(\beta = \frac{1}{2}(1 + \sqrt{3369})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 4 q^{2} + ( - \beta + 2) q^{3} + 16 q^{4} - 25 q^{5} + ( - 4 \beta + 8) q^{6} - 49 q^{7} + 64 q^{8} + ( - 3 \beta + 603) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 4 q^{2} + ( - \beta + 2) q^{3} + 16 q^{4} - 25 q^{5} + ( - 4 \beta + 8) q^{6} - 49 q^{7} + 64 q^{8} + ( - 3 \beta + 603) q^{9} - 100 q^{10} + (3 \beta + 478) q^{11} + ( - 16 \beta + 32) q^{12} + (15 \beta - 204) q^{13} - 196 q^{14} + (25 \beta - 50) q^{15} + 256 q^{16} + (9 \beta - 1120) q^{17} + ( - 12 \beta + 2412) q^{18} + (6 \beta + 1668) q^{19} - 400 q^{20} + (49 \beta - 98) q^{21} + (12 \beta + 1912) q^{22} + (150 \beta - 300) q^{23} + ( - 64 \beta + 128) q^{24} + 625 q^{25} + (60 \beta - 816) q^{26} + ( - 363 \beta + 3246) q^{27} - 784 q^{28} + ( - 171 \beta - 2172) q^{29} + (100 \beta - 200) q^{30} + (204 \beta - 1120) q^{31} + 1024 q^{32} + ( - 475 \beta - 1570) q^{33} + (36 \beta - 4480) q^{34} + 1225 q^{35} + ( - 48 \beta + 9648) q^{36} + ( - 84 \beta - 922) q^{37} + (24 \beta + 6672) q^{38} + (219 \beta - 13038) q^{39} - 1600 q^{40} + (282 \beta + 9518) q^{41} + (196 \beta - 392) q^{42} + (126 \beta - 7136) q^{43} + (48 \beta + 7648) q^{44} + (75 \beta - 15075) q^{45} + (600 \beta - 1200) q^{46} + ( - 117 \beta - 9814) q^{47} + ( - 256 \beta + 512) q^{48} + 2401 q^{49} + 2500 q^{50} + (1129 \beta - 9818) q^{51} + (240 \beta - 3264) q^{52} + (342 \beta + 13018) q^{53} + ( - 1452 \beta + 12984) q^{54} + ( - 75 \beta - 11950) q^{55} - 3136 q^{56} + ( - 1662 \beta - 1716) q^{57} + ( - 684 \beta - 8688) q^{58} + (816 \beta - 5960) q^{59} + (400 \beta - 800) q^{60} + ( - 1194 \beta + 2374) q^{61} + (816 \beta - 4480) q^{62} + (147 \beta - 29547) q^{63} + 4096 q^{64} + ( - 375 \beta + 5100) q^{65} + ( - 1900 \beta - 6280) q^{66} + (1152 \beta + 32028) q^{67} + (144 \beta - 17920) q^{68} + (450 \beta - 126900) q^{69} + 4900 q^{70} + ( - 672 \beta + 8888) q^{71} + ( - 192 \beta + 38592) q^{72} + (1992 \beta - 19366) q^{73} + ( - 336 \beta - 3688) q^{74} + ( - 625 \beta + 1250) q^{75} + (96 \beta + 26688) q^{76} + ( - 147 \beta - 23422) q^{77} + (876 \beta - 52152) q^{78} + (873 \beta + 35686) q^{79} - 6400 q^{80} + ( - 2880 \beta + 165609) q^{81} + (1128 \beta + 38072) q^{82} + ( - 1524 \beta + 50600) q^{83} + (784 \beta - 1568) q^{84} + ( - 225 \beta + 28000) q^{85} + (504 \beta - 28544) q^{86} + (2001 \beta + 139638) q^{87} + (192 \beta + 30592) q^{88} + (2922 \beta + 39766) q^{89} + (300 \beta - 60300) q^{90} + ( - 735 \beta + 9996) q^{91} + (2400 \beta - 4800) q^{92} + (1324 \beta - 174008) q^{93} + ( - 468 \beta - 39256) q^{94} + ( - 150 \beta - 41700) q^{95} + ( - 1024 \beta + 2048) q^{96} + ( - 2979 \beta - 50824) q^{97} + 9604 q^{98} + (366 \beta + 280656) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 8 q^{2} + 3 q^{3} + 32 q^{4} - 50 q^{5} + 12 q^{6} - 98 q^{7} + 128 q^{8} + 1203 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 8 q^{2} + 3 q^{3} + 32 q^{4} - 50 q^{5} + 12 q^{6} - 98 q^{7} + 128 q^{8} + 1203 q^{9} - 200 q^{10} + 959 q^{11} + 48 q^{12} - 393 q^{13} - 392 q^{14} - 75 q^{15} + 512 q^{16} - 2231 q^{17} + 4812 q^{18} + 3342 q^{19} - 800 q^{20} - 147 q^{21} + 3836 q^{22} - 450 q^{23} + 192 q^{24} + 1250 q^{25} - 1572 q^{26} + 6129 q^{27} - 1568 q^{28} - 4515 q^{29} - 300 q^{30} - 2036 q^{31} + 2048 q^{32} - 3615 q^{33} - 8924 q^{34} + 2450 q^{35} + 19248 q^{36} - 1928 q^{37} + 13368 q^{38} - 25857 q^{39} - 3200 q^{40} + 19318 q^{41} - 588 q^{42} - 14146 q^{43} + 15344 q^{44} - 30075 q^{45} - 1800 q^{46} - 19745 q^{47} + 768 q^{48} + 4802 q^{49} + 5000 q^{50} - 18507 q^{51} - 6288 q^{52} + 26378 q^{53} + 24516 q^{54} - 23975 q^{55} - 6272 q^{56} - 5094 q^{57} - 18060 q^{58} - 11104 q^{59} - 1200 q^{60} + 3554 q^{61} - 8144 q^{62} - 58947 q^{63} + 8192 q^{64} + 9825 q^{65} - 14460 q^{66} + 65208 q^{67} - 35696 q^{68} - 253350 q^{69} + 9800 q^{70} + 17104 q^{71} + 76992 q^{72} - 36740 q^{73} - 7712 q^{74} + 1875 q^{75} + 53472 q^{76} - 46991 q^{77} - 103428 q^{78} + 72245 q^{79} - 12800 q^{80} + 328338 q^{81} + 77272 q^{82} + 99676 q^{83} - 2352 q^{84} + 55775 q^{85} - 56584 q^{86} + 281277 q^{87} + 61376 q^{88} + 82454 q^{89} - 120300 q^{90} + 19257 q^{91} - 7200 q^{92} - 346692 q^{93} - 78980 q^{94} - 83550 q^{95} + 3072 q^{96} - 104627 q^{97} + 19208 q^{98} + 561678 q^{99}+O(q^{100}) \) 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
29.5215
−28.5215
4.00000 −27.5215 16.0000 −25.0000 −110.086 −49.0000 64.0000 514.435 −100.000
1.2 4.00000 30.5215 16.0000 −25.0000 122.086 −49.0000 64.0000 688.565 −100.000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(5\) \( +1 \)
\(7\) \( +1 \)

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 70.6.a.g 2
3.b odd 2 1 630.6.a.u 2
4.b odd 2 1 560.6.a.m 2
5.b even 2 1 350.6.a.q 2
5.c odd 4 2 350.6.c.j 4
7.b odd 2 1 490.6.a.v 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.6.a.g 2 1.a even 1 1 trivial
350.6.a.q 2 5.b even 2 1
350.6.c.j 4 5.c odd 4 2
490.6.a.v 2 7.b odd 2 1
560.6.a.m 2 4.b odd 2 1
630.6.a.u 2 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 3T_{3} - 840 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(70))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 4)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 3T - 840 \) Copy content Toggle raw display
$5$ \( (T + 25)^{2} \) Copy content Toggle raw display
$7$ \( (T + 49)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 959T + 222340 \) Copy content Toggle raw display
$13$ \( T^{2} + 393T - 150894 \) Copy content Toggle raw display
$17$ \( T^{2} + 2231 T + 1176118 \) Copy content Toggle raw display
$19$ \( T^{2} - 3342 T + 2761920 \) Copy content Toggle raw display
$23$ \( T^{2} + 450 T - 18900000 \) Copy content Toggle raw display
$29$ \( T^{2} + 4515 T - 19531926 \) Copy content Toggle raw display
$31$ \( T^{2} + 2036 T - 34014752 \) Copy content Toggle raw display
$37$ \( T^{2} + 1928 T - 5013620 \) Copy content Toggle raw display
$41$ \( T^{2} - 19318 T + 26317192 \) Copy content Toggle raw display
$43$ \( T^{2} + 14146 T + 36655768 \) Copy content Toggle raw display
$47$ \( T^{2} + 19745 T + 85936696 \) Copy content Toggle raw display
$53$ \( T^{2} - 26378 T + 75436792 \) Copy content Toggle raw display
$59$ \( T^{2} + 11104 T - 529992512 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots - 1197584192 \) Copy content Toggle raw display
$67$ \( T^{2} - 65208 T - 54732528 \) Copy content Toggle raw display
$71$ \( T^{2} - 17104 T - 307209920 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots - 3004645004 \) Copy content Toggle raw display
$79$ \( T^{2} - 72245 T + 662931856 \) Copy content Toggle raw display
$83$ \( T^{2} - 99676 T + 527636608 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots - 5491535720 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots - 4737795650 \) Copy content Toggle raw display
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