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(1,75)] 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("75.1"); 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, 0])) N = Newforms(chi, 10, names="a")
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 75.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(38.6276877123\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 469x^{2} + 4449x - 5580 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 15)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(10.9137\) of defining polynomial
Character \(\chi\) \(=\) 75.1

$q$-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)
 
\(f(q)\) \(=\) \(q-14.3372 q^{2} +81.0000 q^{3} -306.446 q^{4} -1161.31 q^{6} -2878.61 q^{7} +11734.2 q^{8} +6561.00 q^{9} -23286.0 q^{11} -24822.1 q^{12} +112501. q^{13} +41271.2 q^{14} -11335.0 q^{16} +115964. q^{17} -94066.2 q^{18} -213578. q^{19} -233168. q^{21} +333855. q^{22} -1.83629e6 q^{23} +950470. q^{24} -1.61295e6 q^{26} +531441. q^{27} +882139. q^{28} +3.67540e6 q^{29} +8.85139e6 q^{31} -5.84540e6 q^{32} -1.88617e6 q^{33} -1.66259e6 q^{34} -2.01059e6 q^{36} -9.17921e6 q^{37} +3.06210e6 q^{38} +9.11258e6 q^{39} -1.17626e7 q^{41} +3.34297e6 q^{42} -3.93230e7 q^{43} +7.13589e6 q^{44} +2.63272e7 q^{46} +3.26882e7 q^{47} -918134. q^{48} -3.20672e7 q^{49} +9.39305e6 q^{51} -3.44754e7 q^{52} -1.04826e8 q^{53} -7.61936e6 q^{54} -3.37782e7 q^{56} -1.72998e7 q^{57} -5.26948e7 q^{58} -1.02836e8 q^{59} -1.65686e8 q^{61} -1.26904e8 q^{62} -1.88866e7 q^{63} +8.96099e7 q^{64} +2.70423e7 q^{66} -1.76387e8 q^{67} -3.55365e7 q^{68} -1.48740e8 q^{69} +1.30237e8 q^{71} +7.69880e7 q^{72} -2.35713e8 q^{73} +1.31604e8 q^{74} +6.54500e7 q^{76} +6.70314e7 q^{77} -1.30649e8 q^{78} +1.56763e8 q^{79} +4.30467e7 q^{81} +1.68642e8 q^{82} -3.38241e7 q^{83} +7.14532e7 q^{84} +5.63780e8 q^{86} +2.97707e8 q^{87} -2.73242e8 q^{88} +4.86766e8 q^{89} -3.23847e8 q^{91} +5.62724e8 q^{92} +7.16963e8 q^{93} -4.68656e8 q^{94} -4.73477e8 q^{96} -1.40255e9 q^{97} +4.59753e8 q^{98} -1.52779e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 3 q^{2} + 324 q^{3} + 597 q^{4} + 243 q^{6} - 9834 q^{7} - 7671 q^{8} + 26244 q^{9} - 35994 q^{11} + 48357 q^{12} - 79998 q^{13} - 208182 q^{14} - 752815 q^{16} - 667878 q^{17} + 19683 q^{18} - 425792 q^{19}+ \cdots - 236156634 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −14.3372 −0.633619 −0.316810 − 0.948489i \(-0.602612\pi\)
−0.316810 + 0.948489i \(0.602612\pi\)
\(3\) 81.0000 0.577350
\(4\) −306.446 −0.598526
\(5\) 0 0
\(6\) −1161.31 −0.365820
\(7\) −2878.61 −0.453150 −0.226575 − 0.973994i \(-0.572753\pi\)
−0.226575 + 0.973994i \(0.572753\pi\)
\(8\) 11734.2 1.01286
\(9\) 6561.00 0.333333
\(10\) 0 0
\(11\) −23286.0 −0.479543 −0.239772 − 0.970829i \(-0.577073\pi\)
−0.239772 + 0.970829i \(0.577073\pi\)
\(12\) −24822.1 −0.345559
\(13\) 112501. 1.09247 0.546237 − 0.837630i \(-0.316060\pi\)
0.546237 + 0.837630i \(0.316060\pi\)
\(14\) 41271.2 0.287125
\(15\) 0 0
\(16\) −11335.0 −0.0432396
\(17\) 115964. 0.336745 0.168373 − 0.985723i \(-0.446149\pi\)
0.168373 + 0.985723i \(0.446149\pi\)
\(18\) −94066.2 −0.211206
\(19\) −213578. −0.375980 −0.187990 − 0.982171i \(-0.560197\pi\)
−0.187990 + 0.982171i \(0.560197\pi\)
\(20\) 0 0
\(21\) −233168. −0.261627
\(22\) 333855. 0.303848
\(23\) −1.83629e6 −1.36825 −0.684127 − 0.729363i \(-0.739816\pi\)
−0.684127 + 0.729363i \(0.739816\pi\)
\(24\) 950470. 0.584773
\(25\) 0 0
\(26\) −1.61295e6 −0.692213
\(27\) 531441. 0.192450
\(28\) 882139. 0.271223
\(29\) 3.67540e6 0.964969 0.482485 − 0.875904i \(-0.339734\pi\)
0.482485 + 0.875904i \(0.339734\pi\)
\(30\) 0 0
\(31\) 8.85139e6 1.72141 0.860704 − 0.509105i \(-0.170024\pi\)
0.860704 + 0.509105i \(0.170024\pi\)
\(32\) −5.84540e6 −0.985460
\(33\) −1.88617e6 −0.276864
\(34\) −1.66259e6 −0.213368
\(35\) 0 0
\(36\) −2.01059e6 −0.199509
\(37\) −9.17921e6 −0.805188 −0.402594 − 0.915379i \(-0.631891\pi\)
−0.402594 + 0.915379i \(0.631891\pi\)
\(38\) 3.06210e6 0.238228
\(39\) 9.11258e6 0.630741
\(40\) 0 0
\(41\) −1.17626e7 −0.650093 −0.325046 − 0.945698i \(-0.605380\pi\)
−0.325046 + 0.945698i \(0.605380\pi\)
\(42\) 3.34297e6 0.165772
\(43\) −3.93230e7 −1.75403 −0.877017 − 0.480459i \(-0.840470\pi\)
−0.877017 + 0.480459i \(0.840470\pi\)
\(44\) 7.13589e6 0.287019
\(45\) 0 0
\(46\) 2.63272e7 0.866952
\(47\) 3.26882e7 0.977126 0.488563 − 0.872529i \(-0.337521\pi\)
0.488563 + 0.872529i \(0.337521\pi\)
\(48\) −918134. −0.0249644
\(49\) −3.20672e7 −0.794655
\(50\) 0 0
\(51\) 9.39305e6 0.194420
\(52\) −3.44754e7 −0.653875
\(53\) −1.04826e8 −1.82486 −0.912428 − 0.409238i \(-0.865794\pi\)
−0.912428 + 0.409238i \(0.865794\pi\)
\(54\) −7.61936e6 −0.121940
\(55\) 0 0
\(56\) −3.37782e7 −0.458977
\(57\) −1.72998e7 −0.217072
\(58\) −5.26948e7 −0.611423
\(59\) −1.02836e8 −1.10486 −0.552432 − 0.833558i \(-0.686300\pi\)
−0.552432 + 0.833558i \(0.686300\pi\)
\(60\) 0 0
\(61\) −1.65686e8 −1.53215 −0.766074 − 0.642752i \(-0.777792\pi\)
−0.766074 + 0.642752i \(0.777792\pi\)
\(62\) −1.26904e8 −1.09072
\(63\) −1.88866e7 −0.151050
\(64\) 8.96099e7 0.667646
\(65\) 0 0
\(66\) 2.70423e7 0.175427
\(67\) −1.76387e8 −1.06937 −0.534686 − 0.845051i \(-0.679570\pi\)
−0.534686 + 0.845051i \(0.679570\pi\)
\(68\) −3.55365e7 −0.201551
\(69\) −1.48740e8 −0.789962
\(70\) 0 0
\(71\) 1.30237e8 0.608234 0.304117 − 0.952635i \(-0.401639\pi\)
0.304117 + 0.952635i \(0.401639\pi\)
\(72\) 7.69880e7 0.337619
\(73\) −2.35713e8 −0.971474 −0.485737 − 0.874105i \(-0.661449\pi\)
−0.485737 + 0.874105i \(0.661449\pi\)
\(74\) 1.31604e8 0.510183
\(75\) 0 0
\(76\) 6.54500e7 0.225034
\(77\) 6.70314e7 0.217305
\(78\) −1.30649e8 −0.399649
\(79\) 1.56763e8 0.452815 0.226407 − 0.974033i \(-0.427302\pi\)
0.226407 + 0.974033i \(0.427302\pi\)
\(80\) 0 0
\(81\) 4.30467e7 0.111111
\(82\) 1.68642e8 0.411911
\(83\) −3.38241e7 −0.0782303 −0.0391151 − 0.999235i \(-0.512454\pi\)
−0.0391151 + 0.999235i \(0.512454\pi\)
\(84\) 7.14532e7 0.156590
\(85\) 0 0
\(86\) 5.63780e8 1.11139
\(87\) 2.97707e8 0.557125
\(88\) −2.73242e8 −0.485709
\(89\) 4.86766e8 0.822366 0.411183 − 0.911553i \(-0.365116\pi\)
0.411183 + 0.911553i \(0.365116\pi\)
\(90\) 0 0
\(91\) −3.23847e8 −0.495055
\(92\) 5.62724e8 0.818936
\(93\) 7.16963e8 0.993856
\(94\) −4.68656e8 −0.619126
\(95\) 0 0
\(96\) −4.73477e8 −0.568956
\(97\) −1.40255e9 −1.60859 −0.804294 − 0.594231i \(-0.797456\pi\)
−0.804294 + 0.594231i \(0.797456\pi\)
\(98\) 4.59753e8 0.503509
\(99\) −1.52779e8 −0.159848
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.a.l.1.2 4
3.2 odd 2 225.10.a.q.1.3 4
5.2 odd 4 15.10.b.a.4.4 ✓ 8
5.3 odd 4 15.10.b.a.4.5 yes 8
5.4 even 2 75.10.a.i.1.3 4
15.2 even 4 45.10.b.c.19.5 8
15.8 even 4 45.10.b.c.19.4 8
15.14 odd 2 225.10.a.u.1.2 4
20.3 even 4 240.10.f.c.49.4 8
20.7 even 4 240.10.f.c.49.8 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.10.b.a.4.4 ✓ 8 5.2 odd 4
15.10.b.a.4.5 yes 8 5.3 odd 4
45.10.b.c.19.4 8 15.8 even 4
45.10.b.c.19.5 8 15.2 even 4
75.10.a.i.1.3 4 5.4 even 2
75.10.a.l.1.2 4 1.1 even 1 trivial
225.10.a.q.1.3 4 3.2 odd 2
225.10.a.u.1.2 4 15.14 odd 2
240.10.f.c.49.4 8 20.3 even 4
240.10.f.c.49.8 8 20.7 even 4