Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [315,4,Mod(1,315)] 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("315.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(315, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 315.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-8,0,20] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(18.5856016518\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 315.1

$q$-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)
 
\(f(q)\) \(=\) \(q-5.41421 q^{2} +21.3137 q^{4} +5.00000 q^{5} -7.00000 q^{7} -72.0833 q^{8} -27.0711 q^{10} +52.2548 q^{11} +30.6569 q^{13} +37.8995 q^{14} +219.765 q^{16} -37.2254 q^{17} +80.2254 q^{19} +106.569 q^{20} -282.919 q^{22} -25.8335 q^{23} +25.0000 q^{25} -165.983 q^{26} -149.196 q^{28} -20.9411 q^{29} -314.558 q^{31} -613.186 q^{32} +201.546 q^{34} -35.0000 q^{35} +197.147 q^{37} -434.357 q^{38} -360.416 q^{40} -11.3625 q^{41} -33.8335 q^{43} +1113.74 q^{44} +139.868 q^{46} +361.676 q^{47} +49.0000 q^{49} -135.355 q^{50} +653.411 q^{52} -153.019 q^{53} +261.274 q^{55} +504.583 q^{56} +113.380 q^{58} +616.000 q^{59} +15.2649 q^{61} +1703.09 q^{62} +1561.80 q^{64} +153.284 q^{65} -166.510 q^{67} -793.411 q^{68} +189.497 q^{70} +952.000 q^{71} -148.489 q^{73} -1067.40 q^{74} +1709.90 q^{76} -365.784 q^{77} +857.725 q^{79} +1098.82 q^{80} +61.5189 q^{82} -660.528 q^{83} -186.127 q^{85} +183.182 q^{86} -3766.70 q^{88} +45.7746 q^{89} -214.598 q^{91} -550.607 q^{92} -1958.19 q^{94} +401.127 q^{95} +1682.13 q^{97} -265.296 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{2} + 20 q^{4} + 10 q^{5} - 14 q^{7} - 48 q^{8} - 40 q^{10} + 14 q^{11} + 50 q^{13} + 56 q^{14} + 168 q^{16} + 50 q^{17} + 36 q^{19} + 100 q^{20} - 184 q^{22} - 244 q^{23} + 50 q^{25} - 216 q^{26}+ \cdots - 392 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −5.41421 −1.91421 −0.957107 − 0.289735i \(-0.906433\pi\)
−0.957107 + 0.289735i \(0.906433\pi\)
\(3\) 0 0
\(4\) 21.3137 2.66421
\(5\) 5.00000 0.447214
\(6\) 0 0
\(7\) −7.00000 −0.377964
\(8\) −72.0833 −3.18566
\(9\) 0 0
\(10\) −27.0711 −0.856062
\(11\) 52.2548 1.43231 0.716156 − 0.697941i \(-0.245900\pi\)
0.716156 + 0.697941i \(0.245900\pi\)
\(12\) 0 0
\(13\) 30.6569 0.654052 0.327026 − 0.945015i \(-0.393953\pi\)
0.327026 + 0.945015i \(0.393953\pi\)
\(14\) 37.8995 0.723505
\(15\) 0 0
\(16\) 219.765 3.43382
\(17\) −37.2254 −0.531087 −0.265544 − 0.964099i \(-0.585551\pi\)
−0.265544 + 0.964099i \(0.585551\pi\)
\(18\) 0 0
\(19\) 80.2254 0.968683 0.484341 − 0.874879i \(-0.339059\pi\)
0.484341 + 0.874879i \(0.339059\pi\)
\(20\) 106.569 1.19147
\(21\) 0 0
\(22\) −282.919 −2.74175
\(23\) −25.8335 −0.234202 −0.117101 − 0.993120i \(-0.537360\pi\)
−0.117101 + 0.993120i \(0.537360\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) −165.983 −1.25200
\(27\) 0 0
\(28\) −149.196 −1.00698
\(29\) −20.9411 −0.134092 −0.0670460 − 0.997750i \(-0.521357\pi\)
−0.0670460 + 0.997750i \(0.521357\pi\)
\(30\) 0 0
\(31\) −314.558 −1.82246 −0.911232 − 0.411894i \(-0.864867\pi\)
−0.911232 + 0.411894i \(0.864867\pi\)
\(32\) −613.186 −3.38741
\(33\) 0 0
\(34\) 201.546 1.01661
\(35\) −35.0000 −0.169031
\(36\) 0 0
\(37\) 197.147 0.875968 0.437984 − 0.898983i \(-0.355693\pi\)
0.437984 + 0.898983i \(0.355693\pi\)
\(38\) −434.357 −1.85427
\(39\) 0 0
\(40\) −360.416 −1.42467
\(41\) −11.3625 −0.0432810 −0.0216405 − 0.999766i \(-0.506889\pi\)
−0.0216405 + 0.999766i \(0.506889\pi\)
\(42\) 0 0
\(43\) −33.8335 −0.119990 −0.0599948 − 0.998199i \(-0.519108\pi\)
−0.0599948 + 0.998199i \(0.519108\pi\)
\(44\) 1113.74 3.81598
\(45\) 0 0
\(46\) 139.868 0.448313
\(47\) 361.676 1.12247 0.561233 − 0.827658i \(-0.310327\pi\)
0.561233 + 0.827658i \(0.310327\pi\)
\(48\) 0 0
\(49\) 49.0000 0.142857
\(50\) −135.355 −0.382843
\(51\) 0 0
\(52\) 653.411 1.74254
\(53\) −153.019 −0.396582 −0.198291 − 0.980143i \(-0.563539\pi\)
−0.198291 + 0.980143i \(0.563539\pi\)
\(54\) 0 0
\(55\) 261.274 0.640549
\(56\) 504.583 1.20407
\(57\) 0 0
\(58\) 113.380 0.256681
\(59\) 616.000 1.35926 0.679630 − 0.733555i \(-0.262140\pi\)
0.679630 + 0.733555i \(0.262140\pi\)
\(60\) 0 0
\(61\) 15.2649 0.0320406 0.0160203 − 0.999872i \(-0.494900\pi\)
0.0160203 + 0.999872i \(0.494900\pi\)
\(62\) 1703.09 3.48858
\(63\) 0 0
\(64\) 1561.80 3.05040
\(65\) 153.284 0.292501
\(66\) 0 0
\(67\) −166.510 −0.303618 −0.151809 − 0.988410i \(-0.548510\pi\)
−0.151809 + 0.988410i \(0.548510\pi\)
\(68\) −793.411 −1.41493
\(69\) 0 0
\(70\) 189.497 0.323561
\(71\) 952.000 1.59129 0.795645 − 0.605763i \(-0.207132\pi\)
0.795645 + 0.605763i \(0.207132\pi\)
\(72\) 0 0
\(73\) −148.489 −0.238074 −0.119037 − 0.992890i \(-0.537981\pi\)
−0.119037 + 0.992890i \(0.537981\pi\)
\(74\) −1067.40 −1.67679
\(75\) 0 0
\(76\) 1709.90 2.58078
\(77\) −365.784 −0.541363
\(78\) 0 0
\(79\) 857.725 1.22154 0.610770 − 0.791808i \(-0.290860\pi\)
0.610770 + 0.791808i \(0.290860\pi\)
\(80\) 1098.82 1.53565
\(81\) 0 0
\(82\) 61.5189 0.0828491
\(83\) −660.528 −0.873523 −0.436761 − 0.899577i \(-0.643875\pi\)
−0.436761 + 0.899577i \(0.643875\pi\)
\(84\) 0 0
\(85\) −186.127 −0.237509
\(86\) 183.182 0.229686
\(87\) 0 0
\(88\) −3766.70 −4.56286
\(89\) 45.7746 0.0545180 0.0272590 − 0.999628i \(-0.491322\pi\)
0.0272590 + 0.999628i \(0.491322\pi\)
\(90\) 0 0
\(91\) −214.598 −0.247209
\(92\) −550.607 −0.623965
\(93\) 0 0
\(94\) −1958.19 −2.14864
\(95\) 401.127 0.433208
\(96\) 0 0
\(97\) 1682.13 1.76076 0.880382 − 0.474265i \(-0.157286\pi\)
0.880382 + 0.474265i \(0.157286\pi\)
\(98\) −265.296 −0.273459
\(99\) 0 0
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 315.4.a.f.1.1 2
3.2 odd 2 35.4.a.b.1.2 ✓ 2
5.4 even 2 1575.4.a.z.1.2 2
7.6 odd 2 2205.4.a.u.1.1 2
12.11 even 2 560.4.a.r.1.2 2
15.2 even 4 175.4.b.c.99.4 4
15.8 even 4 175.4.b.c.99.1 4
15.14 odd 2 175.4.a.c.1.1 2
21.2 odd 6 245.4.e.h.116.1 4
21.5 even 6 245.4.e.i.116.1 4
21.11 odd 6 245.4.e.h.226.1 4
21.17 even 6 245.4.e.i.226.1 4
21.20 even 2 245.4.a.k.1.2 2
24.5 odd 2 2240.4.a.bn.1.2 2
24.11 even 2 2240.4.a.bo.1.1 2
105.104 even 2 1225.4.a.m.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.4.a.b.1.2 ✓ 2 3.2 odd 2
175.4.a.c.1.1 2 15.14 odd 2
175.4.b.c.99.1 4 15.8 even 4
175.4.b.c.99.4 4 15.2 even 4
245.4.a.k.1.2 2 21.20 even 2
245.4.e.h.116.1 4 21.2 odd 6
245.4.e.h.226.1 4 21.11 odd 6
245.4.e.i.116.1 4 21.5 even 6
245.4.e.i.226.1 4 21.17 even 6
315.4.a.f.1.1 2 1.1 even 1 trivial
560.4.a.r.1.2 2 12.11 even 2
1225.4.a.m.1.1 2 105.104 even 2
1575.4.a.z.1.2 2 5.4 even 2
2205.4.a.u.1.1 2 7.6 odd 2
2240.4.a.bn.1.2 2 24.5 odd 2
2240.4.a.bo.1.1 2 24.11 even 2