Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1445,4,Mod(1,1445)] 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("1445.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1445, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1445 = 5 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1445.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [5,2,1,34,-25] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(85.2577599583\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2x^{4} - 20x^{3} + 38x^{2} + 69x - 126 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 85)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.25811\) of defining polynomial
Character \(\chi\) \(=\) 1445.1

$q$-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)
 
\(f(q)\) \(=\) \(q-2.18655 q^{2} -6.08748 q^{3} -3.21900 q^{4} -5.00000 q^{5} +13.3106 q^{6} -10.8418 q^{7} +24.5309 q^{8} +10.0574 q^{9} +10.9327 q^{10} -0.656965 q^{11} +19.5956 q^{12} +48.6749 q^{13} +23.7061 q^{14} +30.4374 q^{15} -27.8860 q^{16} -21.9910 q^{18} +118.448 q^{19} +16.0950 q^{20} +65.9992 q^{21} +1.43649 q^{22} -84.7753 q^{23} -149.331 q^{24} +25.0000 q^{25} -106.430 q^{26} +103.138 q^{27} +34.8998 q^{28} -84.8652 q^{29} -66.5529 q^{30} +264.791 q^{31} -135.273 q^{32} +3.99926 q^{33} +54.2090 q^{35} -32.3747 q^{36} +381.981 q^{37} -258.992 q^{38} -296.307 q^{39} -122.655 q^{40} -102.895 q^{41} -144.311 q^{42} -393.537 q^{43} +2.11477 q^{44} -50.2870 q^{45} +185.365 q^{46} +32.7895 q^{47} +169.756 q^{48} -225.455 q^{49} -54.6637 q^{50} -156.684 q^{52} +90.4169 q^{53} -225.516 q^{54} +3.28482 q^{55} -265.959 q^{56} -721.048 q^{57} +185.562 q^{58} +46.6445 q^{59} -97.9779 q^{60} -385.077 q^{61} -578.980 q^{62} -109.040 q^{63} +518.870 q^{64} -243.374 q^{65} -8.74458 q^{66} -190.810 q^{67} +516.068 q^{69} -118.531 q^{70} -354.934 q^{71} +246.717 q^{72} +105.619 q^{73} -835.221 q^{74} -152.187 q^{75} -381.283 q^{76} +7.12268 q^{77} +647.890 q^{78} +681.326 q^{79} +139.430 q^{80} -899.398 q^{81} +224.985 q^{82} -1106.47 q^{83} -212.452 q^{84} +860.488 q^{86} +516.615 q^{87} -16.1159 q^{88} +330.322 q^{89} +109.955 q^{90} -527.723 q^{91} +272.892 q^{92} -1611.91 q^{93} -71.6958 q^{94} -592.239 q^{95} +823.471 q^{96} +243.284 q^{97} +492.969 q^{98} -6.60735 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 2 q^{2} + q^{3} + 34 q^{4} - 25 q^{5} + 5 q^{6} - 10 q^{7} + 30 q^{8} - 30 q^{9} - 10 q^{10} - 126 q^{11} - 15 q^{12} + 83 q^{13} - 90 q^{14} - 5 q^{15} + 322 q^{16} - 97 q^{18} + 55 q^{19} - 170 q^{20}+ \cdots + 858 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.18655 −0.773062 −0.386531 − 0.922276i \(-0.626327\pi\)
−0.386531 + 0.922276i \(0.626327\pi\)
\(3\) −6.08748 −1.17154 −0.585768 − 0.810479i \(-0.699207\pi\)
−0.585768 + 0.810479i \(0.699207\pi\)
\(4\) −3.21900 −0.402375
\(5\) −5.00000 −0.447214
\(6\) 13.3106 0.905670
\(7\) −10.8418 −0.585402 −0.292701 − 0.956204i \(-0.594554\pi\)
−0.292701 + 0.956204i \(0.594554\pi\)
\(8\) 24.5309 1.08412
\(9\) 10.0574 0.372496
\(10\) 10.9327 0.345724
\(11\) −0.656965 −0.0180075 −0.00900374 − 0.999959i \(-0.502866\pi\)
−0.00900374 + 0.999959i \(0.502866\pi\)
\(12\) 19.5956 0.471397
\(13\) 48.6749 1.03846 0.519230 − 0.854635i \(-0.326219\pi\)
0.519230 + 0.854635i \(0.326219\pi\)
\(14\) 23.7061 0.452552
\(15\) 30.4374 0.523927
\(16\) −27.8860 −0.435719
\(17\) 0 0
\(18\) −21.9910 −0.287963
\(19\) 118.448 1.43020 0.715099 − 0.699023i \(-0.246382\pi\)
0.715099 + 0.699023i \(0.246382\pi\)
\(20\) 16.0950 0.179948
\(21\) 65.9992 0.685820
\(22\) 1.43649 0.0139209
\(23\) −84.7753 −0.768560 −0.384280 − 0.923217i \(-0.625550\pi\)
−0.384280 + 0.923217i \(0.625550\pi\)
\(24\) −149.331 −1.27009
\(25\) 25.0000 0.200000
\(26\) −106.430 −0.802794
\(27\) 103.138 0.735143
\(28\) 34.8998 0.235551
\(29\) −84.8652 −0.543416 −0.271708 − 0.962380i \(-0.587589\pi\)
−0.271708 + 0.962380i \(0.587589\pi\)
\(30\) −66.5529 −0.405028
\(31\) 264.791 1.53413 0.767063 − 0.641571i \(-0.221717\pi\)
0.767063 + 0.641571i \(0.221717\pi\)
\(32\) −135.273 −0.747285
\(33\) 3.99926 0.0210964
\(34\) 0 0
\(35\) 54.2090 0.261800
\(36\) −32.3747 −0.149883
\(37\) 381.981 1.69723 0.848613 − 0.529014i \(-0.177438\pi\)
0.848613 + 0.529014i \(0.177438\pi\)
\(38\) −258.992 −1.10563
\(39\) −296.307 −1.21659
\(40\) −122.655 −0.484835
\(41\) −102.895 −0.391939 −0.195969 − 0.980610i \(-0.562785\pi\)
−0.195969 + 0.980610i \(0.562785\pi\)
\(42\) −144.311 −0.530181
\(43\) −393.537 −1.39567 −0.697835 − 0.716258i \(-0.745853\pi\)
−0.697835 + 0.716258i \(0.745853\pi\)
\(44\) 2.11477 0.00724576
\(45\) −50.2870 −0.166585
\(46\) 185.365 0.594144
\(47\) 32.7895 0.101762 0.0508812 − 0.998705i \(-0.483797\pi\)
0.0508812 + 0.998705i \(0.483797\pi\)
\(48\) 169.756 0.510461
\(49\) −225.455 −0.657304
\(50\) −54.6637 −0.154612
\(51\) 0 0
\(52\) −156.684 −0.417850
\(53\) 90.4169 0.234334 0.117167 − 0.993112i \(-0.462619\pi\)
0.117167 + 0.993112i \(0.462619\pi\)
\(54\) −225.516 −0.568311
\(55\) 3.28482 0.00805319
\(56\) −265.959 −0.634648
\(57\) −721.048 −1.67553
\(58\) 185.562 0.420095
\(59\) 46.6445 0.102925 0.0514627 − 0.998675i \(-0.483612\pi\)
0.0514627 + 0.998675i \(0.483612\pi\)
\(60\) −97.9779 −0.210815
\(61\) −385.077 −0.808263 −0.404132 − 0.914701i \(-0.632426\pi\)
−0.404132 + 0.914701i \(0.632426\pi\)
\(62\) −578.980 −1.18598
\(63\) −109.040 −0.218060
\(64\) 518.870 1.01342
\(65\) −243.374 −0.464413
\(66\) −8.74458 −0.0163088
\(67\) −190.810 −0.347927 −0.173963 − 0.984752i \(-0.555657\pi\)
−0.173963 + 0.984752i \(0.555657\pi\)
\(68\) 0 0
\(69\) 516.068 0.900395
\(70\) −118.531 −0.202388
\(71\) −354.934 −0.593281 −0.296641 − 0.954989i \(-0.595866\pi\)
−0.296641 + 0.954989i \(0.595866\pi\)
\(72\) 246.717 0.403831
\(73\) 105.619 0.169339 0.0846697 − 0.996409i \(-0.473017\pi\)
0.0846697 + 0.996409i \(0.473017\pi\)
\(74\) −835.221 −1.31206
\(75\) −152.187 −0.234307
\(76\) −381.283 −0.575476
\(77\) 7.12268 0.0105416
\(78\) 647.890 0.940502
\(79\) 681.326 0.970318 0.485159 − 0.874426i \(-0.338762\pi\)
0.485159 + 0.874426i \(0.338762\pi\)
\(80\) 139.430 0.194860
\(81\) −899.398 −1.23374
\(82\) 224.985 0.302993
\(83\) −1106.47 −1.46326 −0.731630 − 0.681702i \(-0.761240\pi\)
−0.731630 + 0.681702i \(0.761240\pi\)
\(84\) −212.452 −0.275957
\(85\) 0 0
\(86\) 860.488 1.07894
\(87\) 516.615 0.636632
\(88\) −16.1159 −0.0195223
\(89\) 330.322 0.393416 0.196708 − 0.980462i \(-0.436975\pi\)
0.196708 + 0.980462i \(0.436975\pi\)
\(90\) 109.955 0.128781
\(91\) −527.723 −0.607917
\(92\) 272.892 0.309249
\(93\) −1611.91 −1.79728
\(94\) −71.6958 −0.0786687
\(95\) −592.239 −0.639604
\(96\) 823.471 0.875471
\(97\) 243.284 0.254657 0.127329 − 0.991861i \(-0.459360\pi\)
0.127329 + 0.991861i \(0.459360\pi\)
\(98\) 492.969 0.508137
\(99\) −6.60735 −0.00670771
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 1445.4.a.l.1.2 5
17.16 even 2 85.4.a.g.1.2 ✓ 5
51.50 odd 2 765.4.a.m.1.4 5
68.67 odd 2 1360.4.a.w.1.1 5
85.33 odd 4 425.4.b.i.324.7 10
85.67 odd 4 425.4.b.i.324.4 10
85.84 even 2 425.4.a.i.1.4 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
85.4.a.g.1.2 ✓ 5 17.16 even 2
425.4.a.i.1.4 5 85.84 even 2
425.4.b.i.324.4 10 85.67 odd 4
425.4.b.i.324.7 10 85.33 odd 4
765.4.a.m.1.4 5 51.50 odd 2
1360.4.a.w.1.1 5 68.67 odd 2
1445.4.a.l.1.2 5 1.1 even 1 trivial