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Properties

Label 959.6.a.a
Level $959$
Weight $6$
Character orbit 959.a
Self dual yes
Analytic conductor $153.808$
Analytic rank $1$
Dimension $74$
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] = [959,6,Mod(1,959)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(959, 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("959.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 959 = 7 \cdot 137 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 959.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: \(153.808083201\)
Analytic rank: \(1\)
Dimension: \(74\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 74 q - 20 q^{2} - 49 q^{3} + 976 q^{4} - 169 q^{5} - 273 q^{6} + 3626 q^{7} - 747 q^{8} + 4275 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 74 q - 20 q^{2} - 49 q^{3} + 976 q^{4} - 169 q^{5} - 273 q^{6} + 3626 q^{7} - 747 q^{8} + 4275 q^{9} - 1322 q^{10} - 1446 q^{11} - 1466 q^{12} - 1746 q^{13} - 980 q^{14} - 4313 q^{15} + 9208 q^{16} - 3681 q^{17} - 10234 q^{18} - 2860 q^{19} - 7308 q^{20} - 2401 q^{21} - 13879 q^{22} - 13685 q^{23} - 13424 q^{24} + 18155 q^{25} - 9144 q^{26} - 6865 q^{27} + 47824 q^{28} - 19489 q^{29} + 2307 q^{30} - 33560 q^{31} - 27274 q^{32} - 40132 q^{33} - 35811 q^{34} - 8281 q^{35} - 27689 q^{36} - 70663 q^{37} - 37203 q^{38} - 51201 q^{39} - 86817 q^{40} - 67917 q^{41} - 13377 q^{42} - 104475 q^{43} - 45827 q^{44} - 93598 q^{45} - 137776 q^{46} - 43192 q^{47} - 135425 q^{48} + 177674 q^{49} - 73802 q^{50} - 110795 q^{51} - 107131 q^{52} - 99015 q^{53} - 46226 q^{54} - 71678 q^{55} - 36603 q^{56} - 146490 q^{57} - 143069 q^{58} - 12512 q^{59} - 177875 q^{60} - 125581 q^{61} - 75283 q^{62} + 209475 q^{63} - 8449 q^{64} - 95447 q^{65} + 213311 q^{66} - 282713 q^{67} + 191684 q^{68} - 171171 q^{69} - 64778 q^{70} - 189029 q^{71} + 20181 q^{72} - 96401 q^{73} - 96089 q^{74} - 21522 q^{75} - 276776 q^{76} - 70854 q^{77} + 106155 q^{78} - 454125 q^{79} + 253095 q^{80} + 12226 q^{81} + 107086 q^{82} - 168146 q^{83} - 71834 q^{84} - 329524 q^{85} + 191853 q^{86} + 61244 q^{87} - 505209 q^{88} - 325374 q^{89} - 277645 q^{90} - 85554 q^{91} - 189827 q^{92} - 347054 q^{93} - 125581 q^{94} - 343566 q^{95} + 289017 q^{96} - 844266 q^{97} - 48020 q^{98} - 490575 q^{99}+O(q^{100}) \) 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 −10.6693 −5.47184 81.8345 −26.0498 58.3808 49.0000 −531.700 −213.059 277.934
1.2 −10.6119 −15.8970 80.6132 77.6302 168.698 49.0000 −515.880 9.71386 −823.807
1.3 −10.4384 13.3623 76.9593 −73.3008 −139.480 49.0000 −469.301 −64.4491 765.139
1.4 −10.3158 −23.0044 74.4161 88.8475 237.309 49.0000 −437.557 286.201 −916.535
1.5 −10.2765 12.3393 73.6067 42.4969 −126.805 49.0000 −427.572 −90.7428 −436.720
1.6 −9.84789 −9.48815 64.9809 −43.0135 93.4383 49.0000 −324.793 −152.975 423.592
1.7 −9.59106 18.0221 59.9885 20.4003 −172.851 49.0000 −268.440 81.7953 −195.660
1.8 −9.51481 27.1011 58.5316 63.4653 −257.862 49.0000 −252.443 491.470 −603.861
1.9 −9.48708 −16.2130 58.0046 4.30700 153.814 49.0000 −246.708 19.8604 −40.8608
1.10 −9.10689 22.0207 50.9354 −63.5761 −200.540 49.0000 −172.442 241.911 578.980
1.11 −9.00307 5.93592 49.0553 −74.0821 −53.4415 49.0000 −153.550 −207.765 666.966
1.12 −8.78769 −27.2157 45.2235 −72.0581 239.163 49.0000 −116.204 497.697 633.224
1.13 −8.24149 −6.37240 35.9221 53.6806 52.5180 49.0000 −32.3243 −202.393 −442.408
1.14 −7.97218 −0.378329 31.5556 87.0360 3.01611 49.0000 3.54260 −242.857 −693.866
1.15 −7.58056 18.4757 25.4649 4.30721 −140.056 49.0000 49.5394 98.3520 −32.6510
1.16 −6.60885 −2.51328 11.6769 −27.8910 16.6099 49.0000 134.313 −236.683 184.327
1.17 −6.26853 14.6950 7.29447 −96.9509 −92.1159 49.0000 154.867 −27.0577 607.739
1.18 −6.20925 −20.6262 6.55473 −78.7659 128.073 49.0000 157.996 182.440 489.077
1.19 −6.19852 12.7141 6.42163 82.2940 −78.8085 49.0000 158.548 −81.3521 −510.101
1.20 −6.15043 −21.2865 5.82780 −43.2366 130.921 49.0000 160.970 210.115 265.924
See all 74 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.74
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \(-1\)
\(137\) \(-1\)

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 959.6.a.a ✓ 74
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
959.6.a.a ✓ 74 1.a even 1 1 trivial