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 = [7,-2,4,2,0,-6,-7,0,-1,0,-2] 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: \(1\)
Dimension: \(7\)
Coefficient field: 7.7.173349952.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - 2x^{6} - 6x^{5} + 12x^{4} + 5x^{3} - 12x^{2} - x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
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 1.3
Root \(1.29030\) 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)
 
\(f(q)\) \(=\) \(q-1.29030 q^{2} +2.38157 q^{3} -0.335116 q^{4} -3.07295 q^{6} -1.00000 q^{7} +3.01301 q^{8} +2.67187 q^{9} -0.263809 q^{11} -0.798103 q^{12} -3.79565 q^{13} +1.29030 q^{14} -3.21746 q^{16} +1.00000 q^{17} -3.44753 q^{18} -1.46043 q^{19} -2.38157 q^{21} +0.340394 q^{22} -0.860316 q^{23} +7.17569 q^{24} +4.89755 q^{26} -0.781457 q^{27} +0.335116 q^{28} +1.44225 q^{29} +2.36399 q^{31} -1.87451 q^{32} -0.628279 q^{33} -1.29030 q^{34} -0.895389 q^{36} +8.52654 q^{37} +1.88440 q^{38} -9.03961 q^{39} -6.22489 q^{41} +3.07295 q^{42} -8.69246 q^{43} +0.0884067 q^{44} +1.11007 q^{46} -2.29947 q^{47} -7.66261 q^{48} +1.00000 q^{49} +2.38157 q^{51} +1.27199 q^{52} -10.5696 q^{53} +1.00832 q^{54} -3.01301 q^{56} -3.47811 q^{57} -1.86094 q^{58} -10.8500 q^{59} -7.69267 q^{61} -3.05027 q^{62} -2.67187 q^{63} +8.85362 q^{64} +0.810671 q^{66} +10.3311 q^{67} -0.335116 q^{68} -2.04890 q^{69} -4.06249 q^{71} +8.05038 q^{72} +5.39176 q^{73} -11.0018 q^{74} +0.489414 q^{76} +0.263809 q^{77} +11.6638 q^{78} +0.486482 q^{79} -9.87671 q^{81} +8.03200 q^{82} +8.48224 q^{83} +0.798103 q^{84} +11.2159 q^{86} +3.43482 q^{87} -0.794859 q^{88} -11.7008 q^{89} +3.79565 q^{91} +0.288306 q^{92} +5.63002 q^{93} +2.96701 q^{94} -4.46428 q^{96} -3.94244 q^{97} -1.29030 q^{98} -0.704864 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q - 2 q^{2} + 4 q^{3} + 2 q^{4} - 6 q^{6} - 7 q^{7} - q^{9} - 2 q^{11} + 10 q^{12} + 3 q^{13} + 2 q^{14} + 7 q^{17} - 20 q^{18} - 16 q^{19} - 4 q^{21} + 8 q^{22} - 9 q^{23} - 12 q^{24} - 4 q^{26} + 10 q^{27}+ \cdots - 16 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.29030 −0.912382 −0.456191 − 0.889882i \(-0.650787\pi\)
−0.456191 + 0.889882i \(0.650787\pi\)
\(3\) 2.38157 1.37500 0.687500 − 0.726184i \(-0.258708\pi\)
0.687500 + 0.726184i \(0.258708\pi\)
\(4\) −0.335116 −0.167558
\(5\) 0 0
\(6\) −3.07295 −1.25453
\(7\) −1.00000 −0.377964
\(8\) 3.01301 1.06526
\(9\) 2.67187 0.890624
\(10\) 0 0
\(11\) −0.263809 −0.0795414 −0.0397707 − 0.999209i \(-0.512663\pi\)
−0.0397707 + 0.999209i \(0.512663\pi\)
\(12\) −0.798103 −0.230392
\(13\) −3.79565 −1.05272 −0.526362 − 0.850260i \(-0.676444\pi\)
−0.526362 + 0.850260i \(0.676444\pi\)
\(14\) 1.29030 0.344848
\(15\) 0 0
\(16\) −3.21746 −0.804366
\(17\) 1.00000 0.242536
\(18\) −3.44753 −0.812590
\(19\) −1.46043 −0.335045 −0.167523 − 0.985868i \(-0.553577\pi\)
−0.167523 + 0.985868i \(0.553577\pi\)
\(20\) 0 0
\(21\) −2.38157 −0.519701
\(22\) 0.340394 0.0725722
\(23\) −0.860316 −0.179388 −0.0896942 − 0.995969i \(-0.528589\pi\)
−0.0896942 + 0.995969i \(0.528589\pi\)
\(24\) 7.17569 1.46473
\(25\) 0 0
\(26\) 4.89755 0.960488
\(27\) −0.781457 −0.150391
\(28\) 0.335116 0.0633310
\(29\) 1.44225 0.267819 0.133910 − 0.990994i \(-0.457247\pi\)
0.133910 + 0.990994i \(0.457247\pi\)
\(30\) 0 0
\(31\) 2.36399 0.424586 0.212293 − 0.977206i \(-0.431907\pi\)
0.212293 + 0.977206i \(0.431907\pi\)
\(32\) −1.87451 −0.331370
\(33\) −0.628279 −0.109369
\(34\) −1.29030 −0.221285
\(35\) 0 0
\(36\) −0.895389 −0.149231
\(37\) 8.52654 1.40175 0.700877 − 0.713282i \(-0.252792\pi\)
0.700877 + 0.713282i \(0.252792\pi\)
\(38\) 1.88440 0.305689
\(39\) −9.03961 −1.44750
\(40\) 0 0
\(41\) −6.22489 −0.972165 −0.486083 − 0.873913i \(-0.661575\pi\)
−0.486083 + 0.873913i \(0.661575\pi\)
\(42\) 3.07295 0.474166
\(43\) −8.69246 −1.32559 −0.662794 − 0.748802i \(-0.730629\pi\)
−0.662794 + 0.748802i \(0.730629\pi\)
\(44\) 0.0884067 0.0133278
\(45\) 0 0
\(46\) 1.11007 0.163671
\(47\) −2.29947 −0.335412 −0.167706 − 0.985837i \(-0.553636\pi\)
−0.167706 + 0.985837i \(0.553636\pi\)
\(48\) −7.66261 −1.10600
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 2.38157 0.333486
\(52\) 1.27199 0.176393
\(53\) −10.5696 −1.45184 −0.725920 − 0.687779i \(-0.758586\pi\)
−0.725920 + 0.687779i \(0.758586\pi\)
\(54\) 1.00832 0.137215
\(55\) 0 0
\(56\) −3.01301 −0.402630
\(57\) −3.47811 −0.460687
\(58\) −1.86094 −0.244354
\(59\) −10.8500 −1.41255 −0.706276 − 0.707937i \(-0.749626\pi\)
−0.706276 + 0.707937i \(0.749626\pi\)
\(60\) 0 0
\(61\) −7.69267 −0.984946 −0.492473 − 0.870328i \(-0.663907\pi\)
−0.492473 + 0.870328i \(0.663907\pi\)
\(62\) −3.05027 −0.387385
\(63\) −2.67187 −0.336624
\(64\) 8.85362 1.10670
\(65\) 0 0
\(66\) 0.810671 0.0997867
\(67\) 10.3311 1.26215 0.631075 − 0.775722i \(-0.282614\pi\)
0.631075 + 0.775722i \(0.282614\pi\)
\(68\) −0.335116 −0.0406388
\(69\) −2.04890 −0.246659
\(70\) 0 0
\(71\) −4.06249 −0.482129 −0.241065 − 0.970509i \(-0.577497\pi\)
−0.241065 + 0.970509i \(0.577497\pi\)
\(72\) 8.05038 0.948746
\(73\) 5.39176 0.631058 0.315529 − 0.948916i \(-0.397818\pi\)
0.315529 + 0.948916i \(0.397818\pi\)
\(74\) −11.0018 −1.27894
\(75\) 0 0
\(76\) 0.489414 0.0561396
\(77\) 0.263809 0.0300638
\(78\) 11.6638 1.32067
\(79\) 0.486482 0.0547336 0.0273668 − 0.999625i \(-0.491288\pi\)
0.0273668 + 0.999625i \(0.491288\pi\)
\(80\) 0 0
\(81\) −9.87671 −1.09741
\(82\) 8.03200 0.886987
\(83\) 8.48224 0.931046 0.465523 − 0.885036i \(-0.345866\pi\)
0.465523 + 0.885036i \(0.345866\pi\)
\(84\) 0.798103 0.0870802
\(85\) 0 0
\(86\) 11.2159 1.20944
\(87\) 3.43482 0.368251
\(88\) −0.794859 −0.0847322
\(89\) −11.7008 −1.24028 −0.620140 − 0.784491i \(-0.712924\pi\)
−0.620140 + 0.784491i \(0.712924\pi\)
\(90\) 0 0
\(91\) 3.79565 0.397893
\(92\) 0.288306 0.0300580
\(93\) 5.63002 0.583805
\(94\) 2.96701 0.306024
\(95\) 0 0
\(96\) −4.46428 −0.455634
\(97\) −3.94244 −0.400294 −0.200147 − 0.979766i \(-0.564142\pi\)
−0.200147 + 0.979766i \(0.564142\pi\)
\(98\) −1.29030 −0.130340
\(99\) −0.704864 −0.0708415
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.p.1.3 7
5.2 odd 4 595.2.c.a.239.4 ✓ 14
5.3 odd 4 595.2.c.a.239.11 yes 14
5.4 even 2 2975.2.a.q.1.5 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
595.2.c.a.239.4 ✓ 14 5.2 odd 4
595.2.c.a.239.11 yes 14 5.3 odd 4
2975.2.a.p.1.3 7 1.1 even 1 trivial
2975.2.a.q.1.5 7 5.4 even 2