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 = [4,0,4,8,4,0,1] 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: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.25492.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 8x^{2} - 2x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-2.40538\) 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)
 
\(f(q)\) \(=\) \(q+2.40538 q^{2} +1.00000 q^{3} +3.78585 q^{4} +1.00000 q^{5} +2.40538 q^{6} +3.14782 q^{7} +4.29563 q^{8} +1.00000 q^{9} +2.40538 q^{10} +3.52828 q^{11} +3.78585 q^{12} +2.52828 q^{13} +7.57169 q^{14} +1.00000 q^{15} +2.76093 q^{16} -0.405378 q^{17} +2.40538 q^{18} +3.12291 q^{19} +3.78585 q^{20} +3.14782 q^{21} +8.48686 q^{22} +4.29563 q^{24} +1.00000 q^{25} +6.08148 q^{26} +1.00000 q^{27} +11.9171 q^{28} +2.00000 q^{29} +2.40538 q^{30} -6.23463 q^{31} -1.95018 q^{32} +3.52828 q^{33} -0.975088 q^{34} +3.14782 q^{35} +3.78585 q^{36} -2.01316 q^{37} +7.51177 q^{38} +2.52828 q^{39} +4.29563 q^{40} -12.4205 q^{41} +7.57169 q^{42} -12.3956 q^{43} +13.3575 q^{44} +1.00000 q^{45} -7.86732 q^{47} +2.76093 q^{48} +2.90875 q^{49} +2.40538 q^{50} -0.405378 q^{51} +9.57169 q^{52} +13.2161 q^{53} +2.40538 q^{54} +3.52828 q^{55} +13.5219 q^{56} +3.12291 q^{57} +4.81076 q^{58} -11.9225 q^{59} +3.78585 q^{60} +6.11173 q^{61} -14.9966 q^{62} +3.14782 q^{63} -10.2128 q^{64} +2.52828 q^{65} +8.48686 q^{66} +1.14782 q^{67} -1.53470 q^{68} +7.57169 q^{70} +1.52828 q^{71} +4.29563 q^{72} +2.51512 q^{73} -4.84241 q^{74} +1.00000 q^{75} +11.8228 q^{76} +11.1064 q^{77} +6.08148 q^{78} -14.1020 q^{79} +2.76093 q^{80} +1.00000 q^{81} -29.8760 q^{82} +7.97707 q^{83} +11.9171 q^{84} -0.405378 q^{85} -29.8161 q^{86} +2.00000 q^{87} +15.1562 q^{88} -2.21415 q^{89} +2.40538 q^{90} +7.95857 q^{91} -6.23463 q^{93} -18.9239 q^{94} +3.12291 q^{95} -1.95018 q^{96} +6.32390 q^{97} +6.99665 q^{98} +3.52828 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} + 8 q^{4} + 4 q^{5} + q^{7} - 6 q^{8} + 4 q^{9} + 5 q^{11} + 8 q^{12} + q^{13} + 16 q^{14} + 4 q^{15} + 16 q^{16} + 8 q^{17} + 13 q^{19} + 8 q^{20} + q^{21} - 6 q^{22} - 6 q^{24} + 4 q^{25}+ \cdots + 5 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.40538 1.70086 0.850430 − 0.526089i \(-0.176342\pi\)
0.850430 + 0.526089i \(0.176342\pi\)
\(3\) 1.00000 0.577350
\(4\) 3.78585 1.89292
\(5\) 1.00000 0.447214
\(6\) 2.40538 0.981992
\(7\) 3.14782 1.18976 0.594881 − 0.803813i \(-0.297199\pi\)
0.594881 + 0.803813i \(0.297199\pi\)
\(8\) 4.29563 1.51874
\(9\) 1.00000 0.333333
\(10\) 2.40538 0.760647
\(11\) 3.52828 1.06382 0.531909 − 0.846802i \(-0.321475\pi\)
0.531909 + 0.846802i \(0.321475\pi\)
\(12\) 3.78585 1.09288
\(13\) 2.52828 0.701220 0.350610 − 0.936522i \(-0.385974\pi\)
0.350610 + 0.936522i \(0.385974\pi\)
\(14\) 7.57169 2.02362
\(15\) 1.00000 0.258199
\(16\) 2.76093 0.690233
\(17\) −0.405378 −0.0983187 −0.0491594 − 0.998791i \(-0.515654\pi\)
−0.0491594 + 0.998791i \(0.515654\pi\)
\(18\) 2.40538 0.566953
\(19\) 3.12291 0.716444 0.358222 − 0.933637i \(-0.383383\pi\)
0.358222 + 0.933637i \(0.383383\pi\)
\(20\) 3.78585 0.846541
\(21\) 3.14782 0.686910
\(22\) 8.48686 1.80940
\(23\) 0 0
\(24\) 4.29563 0.876843
\(25\) 1.00000 0.200000
\(26\) 6.08148 1.19268
\(27\) 1.00000 0.192450
\(28\) 11.9171 2.25213
\(29\) 2.00000 0.371391 0.185695 − 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 2.40538 0.439160
\(31\) −6.23463 −1.11977 −0.559886 − 0.828569i \(-0.689155\pi\)
−0.559886 + 0.828569i \(0.689155\pi\)
\(32\) −1.95018 −0.344746
\(33\) 3.52828 0.614195
\(34\) −0.975088 −0.167226
\(35\) 3.14782 0.532078
\(36\) 3.78585 0.630974
\(37\) −2.01316 −0.330962 −0.165481 − 0.986213i \(-0.552918\pi\)
−0.165481 + 0.986213i \(0.552918\pi\)
\(38\) 7.51177 1.21857
\(39\) 2.52828 0.404849
\(40\) 4.29563 0.679199
\(41\) −12.4205 −1.93976 −0.969880 − 0.243585i \(-0.921677\pi\)
−0.969880 + 0.243585i \(0.921677\pi\)
\(42\) 7.57169 1.16834
\(43\) −12.3956 −1.89031 −0.945156 − 0.326619i \(-0.894091\pi\)
−0.945156 + 0.326619i \(0.894091\pi\)
\(44\) 13.3575 2.01372
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) −7.86732 −1.14757 −0.573784 − 0.819007i \(-0.694525\pi\)
−0.573784 + 0.819007i \(0.694525\pi\)
\(48\) 2.76093 0.398506
\(49\) 2.90875 0.415536
\(50\) 2.40538 0.340172
\(51\) −0.405378 −0.0567643
\(52\) 9.57169 1.32735
\(53\) 13.2161 1.81538 0.907688 − 0.419646i \(-0.137846\pi\)
0.907688 + 0.419646i \(0.137846\pi\)
\(54\) 2.40538 0.327331
\(55\) 3.52828 0.475754
\(56\) 13.5219 1.80694
\(57\) 3.12291 0.413639
\(58\) 4.81076 0.631683
\(59\) −11.9225 −1.55217 −0.776087 − 0.630625i \(-0.782799\pi\)
−0.776087 + 0.630625i \(0.782799\pi\)
\(60\) 3.78585 0.488751
\(61\) 6.11173 0.782526 0.391263 − 0.920279i \(-0.372038\pi\)
0.391263 + 0.920279i \(0.372038\pi\)
\(62\) −14.9966 −1.90458
\(63\) 3.14782 0.396588
\(64\) −10.2128 −1.27660
\(65\) 2.52828 0.313595
\(66\) 8.48686 1.04466
\(67\) 1.14782 0.140228 0.0701141 − 0.997539i \(-0.477664\pi\)
0.0701141 + 0.997539i \(0.477664\pi\)
\(68\) −1.53470 −0.186110
\(69\) 0 0
\(70\) 7.57169 0.904990
\(71\) 1.52828 0.181374 0.0906869 − 0.995879i \(-0.471094\pi\)
0.0906869 + 0.995879i \(0.471094\pi\)
\(72\) 4.29563 0.506245
\(73\) 2.51512 0.294373 0.147186 − 0.989109i \(-0.452978\pi\)
0.147186 + 0.989109i \(0.452978\pi\)
\(74\) −4.84241 −0.562919
\(75\) 1.00000 0.115470
\(76\) 11.8228 1.35617
\(77\) 11.1064 1.26569
\(78\) 6.08148 0.688592
\(79\) −14.1020 −1.58659 −0.793297 − 0.608835i \(-0.791637\pi\)
−0.793297 + 0.608835i \(0.791637\pi\)
\(80\) 2.76093 0.308682
\(81\) 1.00000 0.111111
\(82\) −29.8760 −3.29926
\(83\) 7.97707 0.875597 0.437799 − 0.899073i \(-0.355758\pi\)
0.437799 + 0.899073i \(0.355758\pi\)
\(84\) 11.9171 1.30027
\(85\) −0.405378 −0.0439695
\(86\) −29.8161 −3.21516
\(87\) 2.00000 0.214423
\(88\) 15.1562 1.61566
\(89\) −2.21415 −0.234700 −0.117350 − 0.993091i \(-0.537440\pi\)
−0.117350 + 0.993091i \(0.537440\pi\)
\(90\) 2.40538 0.253549
\(91\) 7.95857 0.834285
\(92\) 0 0
\(93\) −6.23463 −0.646501
\(94\) −18.9239 −1.95185
\(95\) 3.12291 0.320403
\(96\) −1.95018 −0.199039
\(97\) 6.32390 0.642095 0.321047 − 0.947063i \(-0.395965\pi\)
0.321047 + 0.947063i \(0.395965\pi\)
\(98\) 6.99665 0.706768
\(99\) 3.52828 0.354606
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.ba.1.4 yes 4
23.22 odd 2 7935.2.a.z.1.4 ✓ 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7935.2.a.z.1.4 ✓ 4 23.22 odd 2
7935.2.a.ba.1.4 yes 4 1.1 even 1 trivial