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Properties

Label 309.8.a.c
Level $309$
Weight $8$
Character orbit 309.a
Self dual yes
Analytic conductor $96.527$
Analytic rank $1$
Dimension $31$
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] = [309,8,Mod(1,309)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(309, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("309.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 309 = 3 \cdot 103 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 309.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: \(96.5269728746\)
Analytic rank: \(1\)
Dimension: \(31\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 31 q - 14 q^{2} - 837 q^{3} + 2062 q^{4} - 695 q^{5} + 378 q^{6} + 146 q^{7} - 1638 q^{8} + 22599 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 31 q - 14 q^{2} - 837 q^{3} + 2062 q^{4} - 695 q^{5} + 378 q^{6} + 146 q^{7} - 1638 q^{8} + 22599 q^{9} - 3427 q^{10} - 13312 q^{11} - 55674 q^{12} + 15443 q^{13} - 14734 q^{14} + 18765 q^{15} + 121910 q^{16} - 65666 q^{17} - 10206 q^{18} - 99832 q^{19} - 69361 q^{20} - 3942 q^{21} - 110607 q^{22} - 40931 q^{23} + 44226 q^{24} + 703136 q^{25} - 413430 q^{26} - 610173 q^{27} - 140101 q^{28} - 562368 q^{29} + 92529 q^{30} - 326869 q^{31} - 564903 q^{32} + 359424 q^{33} + 510646 q^{34} - 464980 q^{35} + 1503198 q^{36} - 654496 q^{37} + 1335684 q^{38} - 416961 q^{39} + 806653 q^{40} - 306934 q^{41} + 397818 q^{42} + 406841 q^{43} + 1700755 q^{44} - 506655 q^{45} - 47779 q^{46} - 588918 q^{47} - 3291570 q^{48} + 5623029 q^{49} + 487643 q^{50} + 1772982 q^{51} + 3980099 q^{52} - 484550 q^{53} + 275562 q^{54} + 1302046 q^{55} + 1642815 q^{56} + 2695464 q^{57} + 3542861 q^{58} - 4238661 q^{59} + 1872747 q^{60} - 1322229 q^{61} - 1437062 q^{62} + 106434 q^{63} + 3969318 q^{64} - 581829 q^{65} + 2986389 q^{66} - 363907 q^{67} - 11960657 q^{68} + 1105137 q^{69} - 2932080 q^{70} - 12347974 q^{71} - 1194102 q^{72} + 1178772 q^{73} - 18172125 q^{74} - 18984672 q^{75} - 18116471 q^{76} - 4914292 q^{77} + 11162610 q^{78} + 1273638 q^{79} + 2722743 q^{80} + 16474671 q^{81} + 43816519 q^{82} - 2713999 q^{83} + 3782727 q^{84} + 2361480 q^{85} + 10856553 q^{86} + 15183936 q^{87} - 14536511 q^{88} - 9100620 q^{89} - 2498283 q^{90} - 17159698 q^{91} - 5855144 q^{92} + 8825463 q^{93} - 22676000 q^{94} - 52728702 q^{95} + 15252381 q^{96} - 4390269 q^{97} - 43456255 q^{98} - 9704448 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −21.3906 −27.0000 329.557 513.732 577.546 −1023.67 −4311.42 729.000 −10989.0
1.2 −21.2059 −27.0000 321.690 −430.405 572.559 −148.000 −4107.38 729.000 9127.13
1.3 −20.9864 −27.0000 312.430 21.4141 566.633 −777.809 −3870.51 729.000 −449.405
1.4 −20.0123 −27.0000 272.491 −150.845 540.331 1798.32 −2891.59 729.000 3018.76
1.5 −14.5726 −27.0000 84.3612 159.587 393.461 −53.3644 635.932 729.000 −2325.60
1.6 −14.5086 −27.0000 82.4997 171.825 391.732 353.546 660.146 729.000 −2492.95
1.7 −14.4547 −27.0000 80.9398 −359.728 390.278 −1439.79 680.244 729.000 5199.78
1.8 −14.3613 −27.0000 78.2462 −256.900 387.754 985.694 714.528 729.000 3689.42
1.9 −14.0544 −27.0000 69.5258 −431.420 379.468 869.385 821.819 729.000 6063.34
1.10 −12.8225 −27.0000 36.4165 204.983 346.208 −1033.85 1174.33 729.000 −2628.39
1.11 −9.14990 −27.0000 −44.2794 232.334 247.047 914.586 1576.34 729.000 −2125.83
1.12 −7.93380 −27.0000 −65.0548 316.055 214.213 1221.21 1531.66 729.000 −2507.51
1.13 −5.96447 −27.0000 −92.4251 165.575 161.041 1575.76 1314.72 729.000 −987.570
1.14 −3.50783 −27.0000 −115.695 −184.629 94.7113 −1321.53 854.840 729.000 647.648
1.15 −2.83716 −27.0000 −119.950 −39.1733 76.6034 −227.468 703.476 729.000 111.141
1.16 0.0195912 −27.0000 −128.000 536.735 −0.528963 44.0616 −5.01534 729.000 10.5153
1.17 1.03028 −27.0000 −126.939 −543.371 −27.8176 −958.253 −262.658 729.000 −559.825
1.18 3.45180 −27.0000 −116.085 340.790 −93.1987 −1791.87 −842.534 729.000 1176.34
1.19 5.47396 −27.0000 −98.0357 −441.272 −147.797 −847.092 −1237.31 729.000 −2415.50
1.20 5.90996 −27.0000 −93.0723 −238.455 −159.569 −274.159 −1306.53 729.000 −1409.26
See all 31 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.31
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(103\) \(-1\)

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 309.8.a.c ✓ 31
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
309.8.a.c ✓ 31 1.a even 1 1 trivial