Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [7935,2,Mod(1,7935)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("7935.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7935, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 7935 = 3 \cdot 5 \cdot 23^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7935.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,6,4,6,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(63.3612940039\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.4507648.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 5x^{4} + 8x^{3} + 7x^{2} - 6x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(0.758419\) of defining polynomial
Character \(\chi\) \(=\) 7935.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.18322 q^{2} +1.00000 q^{3} +2.76645 q^{4} +1.00000 q^{5} +2.18322 q^{6} -1.86907 q^{7} +1.67333 q^{8} +1.00000 q^{9} +2.18322 q^{10} -6.19819 q^{11} +2.76645 q^{12} +3.05229 q^{13} -4.08060 q^{14} +1.00000 q^{15} -1.87966 q^{16} -2.25217 q^{17} +2.18322 q^{18} -1.15766 q^{19} +2.76645 q^{20} -1.86907 q^{21} -13.5320 q^{22} +1.67333 q^{24} +1.00000 q^{25} +6.66382 q^{26} +1.00000 q^{27} -5.17069 q^{28} -4.70966 q^{29} +2.18322 q^{30} -1.11183 q^{31} -7.45036 q^{32} -6.19819 q^{33} -4.91698 q^{34} -1.86907 q^{35} +2.76645 q^{36} -7.23990 q^{37} -2.52742 q^{38} +3.05229 q^{39} +1.67333 q^{40} -9.99175 q^{41} -4.08060 q^{42} -11.0452 q^{43} -17.1470 q^{44} +1.00000 q^{45} +1.25732 q^{47} -1.87966 q^{48} -3.50657 q^{49} +2.18322 q^{50} -2.25217 q^{51} +8.44401 q^{52} +2.47425 q^{53} +2.18322 q^{54} -6.19819 q^{55} -3.12757 q^{56} -1.15766 q^{57} -10.2822 q^{58} +6.79307 q^{59} +2.76645 q^{60} +11.2462 q^{61} -2.42736 q^{62} -1.86907 q^{63} -12.5065 q^{64} +3.05229 q^{65} -13.5320 q^{66} +11.9393 q^{67} -6.23051 q^{68} -4.08060 q^{70} +6.25320 q^{71} +1.67333 q^{72} -7.40669 q^{73} -15.8063 q^{74} +1.00000 q^{75} -3.20261 q^{76} +11.5849 q^{77} +6.66382 q^{78} +7.62932 q^{79} -1.87966 q^{80} +1.00000 q^{81} -21.8142 q^{82} -5.18792 q^{83} -5.17069 q^{84} -2.25217 q^{85} -24.1142 q^{86} -4.70966 q^{87} -10.3716 q^{88} -17.0230 q^{89} +2.18322 q^{90} -5.70496 q^{91} -1.11183 q^{93} +2.74501 q^{94} -1.15766 q^{95} -7.45036 q^{96} +13.1354 q^{97} -7.65561 q^{98} -6.19819 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{3} + 4 q^{4} + 6 q^{5} - 2 q^{7} + 6 q^{9} - 12 q^{11} + 4 q^{12} - 4 q^{13} - 8 q^{14} + 6 q^{15} - 14 q^{17} - 8 q^{19} + 4 q^{20} - 2 q^{21} - 12 q^{22} + 6 q^{25} + 24 q^{26} + 6 q^{27} - 4 q^{28}+ \cdots - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.18322 1.54377 0.771885 − 0.635762i \(-0.219314\pi\)
0.771885 + 0.635762i \(0.219314\pi\)
\(3\) 1.00000 0.577350
\(4\) 2.76645 1.38322
\(5\) 1.00000 0.447214
\(6\) 2.18322 0.891296
\(7\) −1.86907 −0.706443 −0.353221 − 0.935540i \(-0.614914\pi\)
−0.353221 + 0.935540i \(0.614914\pi\)
\(8\) 1.67333 0.591610
\(9\) 1.00000 0.333333
\(10\) 2.18322 0.690395
\(11\) −6.19819 −1.86883 −0.934413 − 0.356192i \(-0.884075\pi\)
−0.934413 + 0.356192i \(0.884075\pi\)
\(12\) 2.76645 0.798605
\(13\) 3.05229 0.846554 0.423277 − 0.906000i \(-0.360880\pi\)
0.423277 + 0.906000i \(0.360880\pi\)
\(14\) −4.08060 −1.09059
\(15\) 1.00000 0.258199
\(16\) −1.87966 −0.469915
\(17\) −2.25217 −0.546231 −0.273116 − 0.961981i \(-0.588054\pi\)
−0.273116 + 0.961981i \(0.588054\pi\)
\(18\) 2.18322 0.514590
\(19\) −1.15766 −0.265585 −0.132793 − 0.991144i \(-0.542394\pi\)
−0.132793 + 0.991144i \(0.542394\pi\)
\(20\) 2.76645 0.618597
\(21\) −1.86907 −0.407865
\(22\) −13.5320 −2.88504
\(23\) 0 0
\(24\) 1.67333 0.341566
\(25\) 1.00000 0.200000
\(26\) 6.66382 1.30688
\(27\) 1.00000 0.192450
\(28\) −5.17069 −0.977169
\(29\) −4.70966 −0.874562 −0.437281 − 0.899325i \(-0.644058\pi\)
−0.437281 + 0.899325i \(0.644058\pi\)
\(30\) 2.18322 0.398600
\(31\) −1.11183 −0.199690 −0.0998448 − 0.995003i \(-0.531835\pi\)
−0.0998448 + 0.995003i \(0.531835\pi\)
\(32\) −7.45036 −1.31705
\(33\) −6.19819 −1.07897
\(34\) −4.91698 −0.843255
\(35\) −1.86907 −0.315931
\(36\) 2.76645 0.461075
\(37\) −7.23990 −1.19023 −0.595116 − 0.803640i \(-0.702894\pi\)
−0.595116 + 0.803640i \(0.702894\pi\)
\(38\) −2.52742 −0.410002
\(39\) 3.05229 0.488758
\(40\) 1.67333 0.264576
\(41\) −9.99175 −1.56045 −0.780224 − 0.625500i \(-0.784895\pi\)
−0.780224 + 0.625500i \(0.784895\pi\)
\(42\) −4.08060 −0.629650
\(43\) −11.0452 −1.68438 −0.842191 − 0.539179i \(-0.818735\pi\)
−0.842191 + 0.539179i \(0.818735\pi\)
\(44\) −17.1470 −2.58500
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) 1.25732 0.183399 0.0916997 − 0.995787i \(-0.470770\pi\)
0.0916997 + 0.995787i \(0.470770\pi\)
\(48\) −1.87966 −0.271306
\(49\) −3.50657 −0.500938
\(50\) 2.18322 0.308754
\(51\) −2.25217 −0.315367
\(52\) 8.44401 1.17097
\(53\) 2.47425 0.339865 0.169932 − 0.985456i \(-0.445645\pi\)
0.169932 + 0.985456i \(0.445645\pi\)
\(54\) 2.18322 0.297099
\(55\) −6.19819 −0.835764
\(56\) −3.12757 −0.417939
\(57\) −1.15766 −0.153336
\(58\) −10.2822 −1.35012
\(59\) 6.79307 0.884382 0.442191 − 0.896921i \(-0.354201\pi\)
0.442191 + 0.896921i \(0.354201\pi\)
\(60\) 2.76645 0.357147
\(61\) 11.2462 1.43992 0.719962 − 0.694013i \(-0.244159\pi\)
0.719962 + 0.694013i \(0.244159\pi\)
\(62\) −2.42736 −0.308275
\(63\) −1.86907 −0.235481
\(64\) −12.5065 −1.56331
\(65\) 3.05229 0.378590
\(66\) −13.5320 −1.66568
\(67\) 11.9393 1.45861 0.729306 − 0.684188i \(-0.239843\pi\)
0.729306 + 0.684188i \(0.239843\pi\)
\(68\) −6.23051 −0.755560
\(69\) 0 0
\(70\) −4.08060 −0.487724
\(71\) 6.25320 0.742118 0.371059 − 0.928609i \(-0.378995\pi\)
0.371059 + 0.928609i \(0.378995\pi\)
\(72\) 1.67333 0.197203
\(73\) −7.40669 −0.866887 −0.433444 − 0.901181i \(-0.642702\pi\)
−0.433444 + 0.901181i \(0.642702\pi\)
\(74\) −15.8063 −1.83744
\(75\) 1.00000 0.115470
\(76\) −3.20261 −0.367364
\(77\) 11.5849 1.32022
\(78\) 6.66382 0.754530
\(79\) 7.62932 0.858365 0.429183 − 0.903218i \(-0.358802\pi\)
0.429183 + 0.903218i \(0.358802\pi\)
\(80\) −1.87966 −0.210152
\(81\) 1.00000 0.111111
\(82\) −21.8142 −2.40897
\(83\) −5.18792 −0.569449 −0.284724 − 0.958609i \(-0.591902\pi\)
−0.284724 + 0.958609i \(0.591902\pi\)
\(84\) −5.17069 −0.564169
\(85\) −2.25217 −0.244282
\(86\) −24.1142 −2.60030
\(87\) −4.70966 −0.504928
\(88\) −10.3716 −1.10562
\(89\) −17.0230 −1.80444 −0.902218 − 0.431279i \(-0.858062\pi\)
−0.902218 + 0.431279i \(0.858062\pi\)
\(90\) 2.18322 0.230132
\(91\) −5.70496 −0.598042
\(92\) 0 0
\(93\) −1.11183 −0.115291
\(94\) 2.74501 0.283126
\(95\) −1.15766 −0.118773
\(96\) −7.45036 −0.760399
\(97\) 13.1354 1.33370 0.666849 − 0.745193i \(-0.267642\pi\)
0.666849 + 0.745193i \(0.267642\pi\)
\(98\) −7.65561 −0.773333
\(99\) −6.19819 −0.622942
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 7935.2.a.bg.1.6 yes 6
23.22 odd 2 7935.2.a.bf.1.6 ✓ 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7935.2.a.bf.1.6 ✓ 6 23.22 odd 2
7935.2.a.bg.1.6 yes 6 1.1 even 1 trivial