Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2975,2,Mod(1,2975)] 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("2975.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2975, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2975 = 5^{2} \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2975.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [17,2,4,22,0,6,17,0,29,0,10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(23.7554946013\)
Analytic rank: \(0\)
Dimension: \(17\)
Coefficient field: \(\mathbb{Q}[x]/(x^{17} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{17} - 2 x^{16} - 26 x^{15} + 52 x^{14} + 265 x^{13} - 532 x^{12} - 1341 x^{11} + 2726 x^{10} + \cdots - 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 595)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.15
Root \(2.49820\) of defining polynomial
Character \(\chi\) \(=\) 2975.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.49820 q^{2} +2.44484 q^{3} +4.24100 q^{4} +6.10770 q^{6} +1.00000 q^{7} +5.59846 q^{8} +2.97726 q^{9} -0.766211 q^{11} +10.3686 q^{12} +4.17292 q^{13} +2.49820 q^{14} +5.50406 q^{16} +1.00000 q^{17} +7.43778 q^{18} -7.60200 q^{19} +2.44484 q^{21} -1.91415 q^{22} +5.65328 q^{23} +13.6873 q^{24} +10.4248 q^{26} -0.0555986 q^{27} +4.24100 q^{28} -8.87682 q^{29} -4.99527 q^{31} +2.55333 q^{32} -1.87327 q^{33} +2.49820 q^{34} +12.6265 q^{36} -5.94051 q^{37} -18.9913 q^{38} +10.2021 q^{39} +0.284773 q^{41} +6.10770 q^{42} -1.97386 q^{43} -3.24950 q^{44} +14.1230 q^{46} +12.7325 q^{47} +13.4566 q^{48} +1.00000 q^{49} +2.44484 q^{51} +17.6973 q^{52} -1.76135 q^{53} -0.138896 q^{54} +5.59846 q^{56} -18.5857 q^{57} -22.1761 q^{58} -4.84190 q^{59} -3.52226 q^{61} -12.4792 q^{62} +2.97726 q^{63} -4.62940 q^{64} -4.67979 q^{66} +3.35353 q^{67} +4.24100 q^{68} +13.8214 q^{69} -5.09643 q^{71} +16.6681 q^{72} -0.685062 q^{73} -14.8406 q^{74} -32.2401 q^{76} -0.766211 q^{77} +25.4869 q^{78} +14.0393 q^{79} -9.06771 q^{81} +0.711420 q^{82} +8.54989 q^{83} +10.3686 q^{84} -4.93111 q^{86} -21.7024 q^{87} -4.28960 q^{88} -0.407105 q^{89} +4.17292 q^{91} +23.9755 q^{92} -12.2127 q^{93} +31.8084 q^{94} +6.24248 q^{96} +12.1590 q^{97} +2.49820 q^{98} -2.28121 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17 q + 2 q^{2} + 4 q^{3} + 22 q^{4} + 6 q^{6} + 17 q^{7} + 29 q^{9} + 10 q^{11} + 10 q^{12} + 11 q^{13} + 2 q^{14} + 40 q^{16} + 17 q^{17} - 8 q^{18} + 20 q^{19} + 4 q^{21} - 12 q^{22} - 11 q^{23} + 24 q^{24}+ \cdots + 44 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.49820 1.76649 0.883247 − 0.468909i \(-0.155353\pi\)
0.883247 + 0.468909i \(0.155353\pi\)
\(3\) 2.44484 1.41153 0.705765 − 0.708446i \(-0.250603\pi\)
0.705765 + 0.708446i \(0.250603\pi\)
\(4\) 4.24100 2.12050
\(5\) 0 0
\(6\) 6.10770 2.49346
\(7\) 1.00000 0.377964
\(8\) 5.59846 1.97935
\(9\) 2.97726 0.992420
\(10\) 0 0
\(11\) −0.766211 −0.231021 −0.115511 − 0.993306i \(-0.536850\pi\)
−0.115511 + 0.993306i \(0.536850\pi\)
\(12\) 10.3686 2.99315
\(13\) 4.17292 1.15736 0.578679 − 0.815555i \(-0.303568\pi\)
0.578679 + 0.815555i \(0.303568\pi\)
\(14\) 2.49820 0.667672
\(15\) 0 0
\(16\) 5.50406 1.37602
\(17\) 1.00000 0.242536
\(18\) 7.43778 1.75310
\(19\) −7.60200 −1.74402 −0.872009 − 0.489489i \(-0.837183\pi\)
−0.872009 + 0.489489i \(0.837183\pi\)
\(20\) 0 0
\(21\) 2.44484 0.533509
\(22\) −1.91415 −0.408098
\(23\) 5.65328 1.17879 0.589395 − 0.807845i \(-0.299366\pi\)
0.589395 + 0.807845i \(0.299366\pi\)
\(24\) 13.6873 2.79392
\(25\) 0 0
\(26\) 10.4248 2.04447
\(27\) −0.0555986 −0.0107000
\(28\) 4.24100 0.801473
\(29\) −8.87682 −1.64838 −0.824192 − 0.566310i \(-0.808371\pi\)
−0.824192 + 0.566310i \(0.808371\pi\)
\(30\) 0 0
\(31\) −4.99527 −0.897177 −0.448589 − 0.893738i \(-0.648073\pi\)
−0.448589 + 0.893738i \(0.648073\pi\)
\(32\) 2.55333 0.451369
\(33\) −1.87327 −0.326094
\(34\) 2.49820 0.428438
\(35\) 0 0
\(36\) 12.6265 2.10442
\(37\) −5.94051 −0.976613 −0.488307 − 0.872672i \(-0.662385\pi\)
−0.488307 + 0.872672i \(0.662385\pi\)
\(38\) −18.9913 −3.08080
\(39\) 10.2021 1.63365
\(40\) 0 0
\(41\) 0.284773 0.0444741 0.0222371 − 0.999753i \(-0.492921\pi\)
0.0222371 + 0.999753i \(0.492921\pi\)
\(42\) 6.10770 0.942439
\(43\) −1.97386 −0.301012 −0.150506 − 0.988609i \(-0.548090\pi\)
−0.150506 + 0.988609i \(0.548090\pi\)
\(44\) −3.24950 −0.489880
\(45\) 0 0
\(46\) 14.1230 2.08232
\(47\) 12.7325 1.85723 0.928616 − 0.371041i \(-0.120999\pi\)
0.928616 + 0.371041i \(0.120999\pi\)
\(48\) 13.4566 1.94229
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 2.44484 0.342347
\(52\) 17.6973 2.45418
\(53\) −1.76135 −0.241940 −0.120970 − 0.992656i \(-0.538600\pi\)
−0.120970 + 0.992656i \(0.538600\pi\)
\(54\) −0.138896 −0.0189014
\(55\) 0 0
\(56\) 5.59846 0.748125
\(57\) −18.5857 −2.46174
\(58\) −22.1761 −2.91186
\(59\) −4.84190 −0.630361 −0.315181 − 0.949032i \(-0.602065\pi\)
−0.315181 + 0.949032i \(0.602065\pi\)
\(60\) 0 0
\(61\) −3.52226 −0.450979 −0.225489 − 0.974246i \(-0.572398\pi\)
−0.225489 + 0.974246i \(0.572398\pi\)
\(62\) −12.4792 −1.58486
\(63\) 2.97726 0.375099
\(64\) −4.62940 −0.578676
\(65\) 0 0
\(66\) −4.67979 −0.576042
\(67\) 3.35353 0.409699 0.204849 − 0.978794i \(-0.434330\pi\)
0.204849 + 0.978794i \(0.434330\pi\)
\(68\) 4.24100 0.514296
\(69\) 13.8214 1.66390
\(70\) 0 0
\(71\) −5.09643 −0.604835 −0.302418 − 0.953176i \(-0.597794\pi\)
−0.302418 + 0.953176i \(0.597794\pi\)
\(72\) 16.6681 1.96435
\(73\) −0.685062 −0.0801804 −0.0400902 − 0.999196i \(-0.512765\pi\)
−0.0400902 + 0.999196i \(0.512765\pi\)
\(74\) −14.8406 −1.72518
\(75\) 0 0
\(76\) −32.2401 −3.69819
\(77\) −0.766211 −0.0873178
\(78\) 25.4869 2.88583
\(79\) 14.0393 1.57955 0.789774 − 0.613398i \(-0.210198\pi\)
0.789774 + 0.613398i \(0.210198\pi\)
\(80\) 0 0
\(81\) −9.06771 −1.00752
\(82\) 0.711420 0.0785632
\(83\) 8.54989 0.938472 0.469236 − 0.883073i \(-0.344529\pi\)
0.469236 + 0.883073i \(0.344529\pi\)
\(84\) 10.3686 1.13130
\(85\) 0 0
\(86\) −4.93111 −0.531735
\(87\) −21.7024 −2.32675
\(88\) −4.28960 −0.457273
\(89\) −0.407105 −0.0431530 −0.0215765 − 0.999767i \(-0.506869\pi\)
−0.0215765 + 0.999767i \(0.506869\pi\)
\(90\) 0 0
\(91\) 4.17292 0.437440
\(92\) 23.9755 2.49962
\(93\) −12.2127 −1.26639
\(94\) 31.8084 3.28079
\(95\) 0 0
\(96\) 6.24248 0.637121
\(97\) 12.1590 1.23456 0.617280 − 0.786744i \(-0.288235\pi\)
0.617280 + 0.786744i \(0.288235\pi\)
\(98\) 2.49820 0.252356
\(99\) −2.28121 −0.229270
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 2975.2.a.y.1.15 17
5.2 odd 4 595.2.c.b.239.30 yes 34
5.3 odd 4 595.2.c.b.239.5 ✓ 34
5.4 even 2 2975.2.a.x.1.3 17
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
595.2.c.b.239.5 ✓ 34 5.3 odd 4
595.2.c.b.239.30 yes 34 5.2 odd 4
2975.2.a.x.1.3 17 5.4 even 2
2975.2.a.y.1.15 17 1.1 even 1 trivial