Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [75,10,Mod(49,75)] 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("75.49"); S:= CuspForms(chi, 10); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(75, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 10, names="a")
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 75.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,-3042] 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: no
Analytic conductor: \(38.6276877123\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{4729})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 2365x^{2} + 1397124 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 15)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.3
Root \(33.8839i\) of defining polynomial
Character \(\chi\) \(=\) 75.49
Dual form 75.10.b.e.49.2

$q$-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)
 
\(f(q)\) \(=\) \(q+24.8839i q^{2} +81.0000i q^{3} -107.207 q^{4} -2015.59 q^{6} +4010.50i q^{7} +10072.8i q^{8} -6561.00 q^{9} +84861.3 q^{11} -8683.74i q^{12} +119425. i q^{13} -99796.8 q^{14} -305541. q^{16} -116934. i q^{17} -163263. i q^{18} +234932. q^{19} -324851. q^{21} +2.11168e6i q^{22} +2.34570e6i q^{23} -815899. q^{24} -2.97176e6 q^{26} -531441. i q^{27} -429953. i q^{28} +464196. q^{29} -5.11766e6 q^{31} -2.44574e6i q^{32} +6.87377e6i q^{33} +2.90976e6 q^{34} +703383. q^{36} -8.69354e6i q^{37} +5.84601e6i q^{38} -9.67345e6 q^{39} -9.05805e6 q^{41} -8.08354e6i q^{42} +8.63491e6i q^{43} -9.09769e6 q^{44} -5.83701e7 q^{46} -3.31511e7i q^{47} -2.47488e7i q^{48} +2.42695e7 q^{49} +9.47161e6 q^{51} -1.28032e7i q^{52} -6.41254e7i q^{53} +1.32243e7 q^{54} -4.03971e7 q^{56} +1.90295e7i q^{57} +1.15510e7i q^{58} -1.49407e8 q^{59} +1.54634e8 q^{61} -1.27347e8i q^{62} -2.63129e7i q^{63} -9.55772e7 q^{64} -1.71046e8 q^{66} -2.72755e8i q^{67} +1.25360e7i q^{68} -1.90002e8 q^{69} -3.56924e8 q^{71} -6.60878e7i q^{72} +2.06253e8i q^{73} +2.16329e8 q^{74} -2.51862e7 q^{76} +3.40337e8i q^{77} -2.40713e8i q^{78} +4.04380e8 q^{79} +4.30467e7 q^{81} -2.25399e8i q^{82} -5.17034e6i q^{83} +3.48262e7 q^{84} -2.14870e8 q^{86} +3.75999e7i q^{87} +8.54793e8i q^{88} -4.32242e8 q^{89} -4.78955e8 q^{91} -2.51475e8i q^{92} -4.14530e8i q^{93} +8.24928e8 q^{94} +1.98105e8 q^{96} +1.32066e9i q^{97} +6.03918e8i q^{98} -5.56775e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 3042 q^{4} + 3078 q^{6} - 26244 q^{9} + 70976 q^{11} + 490392 q^{14} + 1414530 q^{16} + 806592 q^{19} - 1923264 q^{21} - 8042814 q^{24} - 3815092 q^{26} + 149144 q^{29} - 10054256 q^{31} - 17721764 q^{34}+ \cdots - 465673536 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/75\mathbb{Z}\right)^\times\).

\(n\) \(26\) \(52\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 24.8839i 1.09972i 0.835256 + 0.549861i \(0.185319\pi\)
−0.835256 + 0.549861i \(0.814681\pi\)
\(3\) 81.0000i 0.577350i
\(4\) −107.207 −0.209388
\(5\) 0 0
\(6\) −2015.59 −0.634925
\(7\) 4010.50i 0.631332i 0.948870 + 0.315666i \(0.102228\pi\)
−0.948870 + 0.315666i \(0.897772\pi\)
\(8\) 10072.8i 0.869453i
\(9\) −6561.00 −0.333333
\(10\) 0 0
\(11\) 84861.3 1.74760 0.873801 − 0.486283i \(-0.161648\pi\)
0.873801 + 0.486283i \(0.161648\pi\)
\(12\) − 8683.74i − 0.120890i
\(13\) 119425.i 1.15971i 0.814718 + 0.579857i \(0.196892\pi\)
−0.814718 + 0.579857i \(0.803108\pi\)
\(14\) −99796.8 −0.694289
\(15\) 0 0
\(16\) −305541. −1.16554
\(17\) − 116934.i − 0.339562i −0.985482 − 0.169781i \(-0.945694\pi\)
0.985482 − 0.169781i \(-0.0543060\pi\)
\(18\) − 163263.i − 0.366574i
\(19\) 234932. 0.413571 0.206786 − 0.978386i \(-0.433700\pi\)
0.206786 + 0.978386i \(0.433700\pi\)
\(20\) 0 0
\(21\) −324851. −0.364500
\(22\) 2.11168e6i 1.92188i
\(23\) 2.34570e6i 1.74782i 0.486084 + 0.873912i \(0.338425\pi\)
−0.486084 + 0.873912i \(0.661575\pi\)
\(24\) −815899. −0.501979
\(25\) 0 0
\(26\) −2.97176e6 −1.27536
\(27\) − 531441.i − 0.192450i
\(28\) − 429953.i − 0.132193i
\(29\) 464196. 0.121874 0.0609369 − 0.998142i \(-0.480591\pi\)
0.0609369 + 0.998142i \(0.480591\pi\)
\(30\) 0 0
\(31\) −5.11766e6 −0.995277 −0.497638 − 0.867385i \(-0.665799\pi\)
−0.497638 + 0.867385i \(0.665799\pi\)
\(32\) − 2.44574e6i − 0.412321i
\(33\) 6.87377e6i 1.00898i
\(34\) 2.90976e6 0.373423
\(35\) 0 0
\(36\) 703383. 0.0697960
\(37\) − 8.69354e6i − 0.762586i −0.924454 − 0.381293i \(-0.875479\pi\)
0.924454 − 0.381293i \(-0.124521\pi\)
\(38\) 5.84601e6i 0.454813i
\(39\) −9.67345e6 −0.669562
\(40\) 0 0
\(41\) −9.05805e6 −0.500619 −0.250310 − 0.968166i \(-0.580532\pi\)
−0.250310 + 0.968166i \(0.580532\pi\)
\(42\) − 8.08354e6i − 0.400848i
\(43\) 8.63491e6i 0.385168i 0.981281 + 0.192584i \(0.0616867\pi\)
−0.981281 + 0.192584i \(0.938313\pi\)
\(44\) −9.09769e6 −0.365927
\(45\) 0 0
\(46\) −5.83701e7 −1.92212
\(47\) − 3.31511e7i − 0.990964i −0.868618 − 0.495482i \(-0.834991\pi\)
0.868618 − 0.495482i \(-0.165009\pi\)
\(48\) − 2.47488e7i − 0.672927i
\(49\) 2.42695e7 0.601420
\(50\) 0 0
\(51\) 9.47161e6 0.196046
\(52\) − 1.28032e7i − 0.242830i
\(53\) − 6.41254e7i − 1.11632i −0.829733 − 0.558160i \(-0.811508\pi\)
0.829733 − 0.558160i \(-0.188492\pi\)
\(54\) 1.32243e7 0.211642
\(55\) 0 0
\(56\) −4.03971e7 −0.548914
\(57\) 1.90295e7i 0.238775i
\(58\) 1.15510e7i 0.134027i
\(59\) −1.49407e8 −1.60523 −0.802613 − 0.596500i \(-0.796558\pi\)
−0.802613 + 0.596500i \(0.796558\pi\)
\(60\) 0 0
\(61\) 1.54634e8 1.42995 0.714973 − 0.699152i \(-0.246439\pi\)
0.714973 + 0.699152i \(0.246439\pi\)
\(62\) − 1.27347e8i − 1.09453i
\(63\) − 2.63129e7i − 0.210444i
\(64\) −9.55772e7 −0.712106
\(65\) 0 0
\(66\) −1.71046e8 −1.10960
\(67\) − 2.72755e8i − 1.65362i −0.562479 − 0.826811i \(-0.690152\pi\)
0.562479 − 0.826811i \(-0.309848\pi\)
\(68\) 1.25360e7i 0.0711001i
\(69\) −1.90002e8 −1.00911
\(70\) 0 0
\(71\) −3.56924e8 −1.66691 −0.833457 − 0.552584i \(-0.813642\pi\)
−0.833457 + 0.552584i \(0.813642\pi\)
\(72\) − 6.60878e7i − 0.289818i
\(73\) 2.06253e8i 0.850057i 0.905180 + 0.425029i \(0.139736\pi\)
−0.905180 + 0.425029i \(0.860264\pi\)
\(74\) 2.16329e8 0.838632
\(75\) 0 0
\(76\) −2.51862e7 −0.0865968
\(77\) 3.40337e8i 1.10332i
\(78\) − 2.40713e8i − 0.736331i
\(79\) 4.04380e8 1.16807 0.584033 − 0.811730i \(-0.301474\pi\)
0.584033 + 0.811730i \(0.301474\pi\)
\(80\) 0 0
\(81\) 4.30467e7 0.111111
\(82\) − 2.25399e8i − 0.550542i
\(83\) − 5.17034e6i − 0.0119582i −0.999982 − 0.00597912i \(-0.998097\pi\)
0.999982 − 0.00597912i \(-0.00190323\pi\)
\(84\) 3.48262e7 0.0763218
\(85\) 0 0
\(86\) −2.14870e8 −0.423577
\(87\) 3.75999e7i 0.0703639i
\(88\) 8.54793e8i 1.51946i
\(89\) −4.32242e8 −0.730250 −0.365125 − 0.930958i \(-0.618974\pi\)
−0.365125 + 0.930958i \(0.618974\pi\)
\(90\) 0 0
\(91\) −4.78955e8 −0.732165
\(92\) − 2.51475e8i − 0.365973i
\(93\) − 4.14530e8i − 0.574623i
\(94\) 8.24928e8 1.08978
\(95\) 0 0
\(96\) 1.98105e8 0.238054
\(97\) 1.32066e9i 1.51468i 0.653023 + 0.757338i \(0.273501\pi\)
−0.653023 + 0.757338i \(0.726499\pi\)
\(98\) 6.03918e8i 0.661395i
\(99\) −5.56775e8 −0.582534
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 75.10.b.e.49.3 4
3.2 odd 2 225.10.b.g.199.2 4
5.2 odd 4 15.10.a.c.1.1 ✓ 2
5.3 odd 4 75.10.a.g.1.2 2
5.4 even 2 inner 75.10.b.e.49.2 4
15.2 even 4 45.10.a.e.1.2 2
15.8 even 4 225.10.a.j.1.1 2
15.14 odd 2 225.10.b.g.199.3 4
20.7 even 4 240.10.a.m.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.10.a.c.1.1 ✓ 2 5.2 odd 4
45.10.a.e.1.2 2 15.2 even 4
75.10.a.g.1.2 2 5.3 odd 4
75.10.b.e.49.2 4 5.4 even 2 inner
75.10.b.e.49.3 4 1.1 even 1 trivial
225.10.a.j.1.1 2 15.8 even 4
225.10.b.g.199.2 4 3.2 odd 2
225.10.b.g.199.3 4 15.14 odd 2
240.10.a.m.1.1 2 20.7 even 4