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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [75,8,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, 8); 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, 8, names="a")
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 8 \)
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,-138] 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: \(23.4288769113\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{601})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 301x^{2} + 22500 \) 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 49.2
Root \(-11.7577i\) of defining polynomial
Character \(\chi\) \(=\) 75.49
Dual form 75.8.b.d.49.3

$q$-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)
 
\(f(q)\) \(=\) \(q-8.75765i q^{2} -27.0000i q^{3} +51.3036 q^{4} -236.457 q^{6} +1338.43i q^{7} -1570.28i q^{8} -729.000 q^{9} +7411.55 q^{11} -1385.20i q^{12} +14594.3i q^{13} +11721.5 q^{14} -7185.09 q^{16} +14414.7i q^{17} +6384.33i q^{18} -409.730 q^{19} +36137.6 q^{21} -64907.8i q^{22} +17913.8i q^{23} -42397.5 q^{24} +127812. q^{26} +19683.0i q^{27} +68666.1i q^{28} +10705.4 q^{29} +178691. q^{31} -138071. i q^{32} -200112. i q^{33} +126239. q^{34} -37400.3 q^{36} -427962. i q^{37} +3588.27i q^{38} +394046. q^{39} +53370.3 q^{41} -316480. i q^{42} -89349.4i q^{43} +380239. q^{44} +156883. q^{46} -161273. i q^{47} +193997. i q^{48} -967848. q^{49} +389197. q^{51} +748740. i q^{52} +121833. i q^{53} +172377. q^{54} +2.10170e6 q^{56} +11062.7i q^{57} -93754.2i q^{58} +1.69229e6 q^{59} -1.23924e6 q^{61} -1.56491e6i q^{62} -975714. i q^{63} -2.12887e6 q^{64} -1.75251e6 q^{66} -944402. i q^{67} +739526. i q^{68} +483674. q^{69} -936943. q^{71} +1.14473e6i q^{72} +5.49712e6i q^{73} -3.74794e6 q^{74} -21020.6 q^{76} +9.91983e6i q^{77} -3.45092e6i q^{78} +3.29987e6 q^{79} +531441. q^{81} -467399. i q^{82} +4.16260e6i q^{83} +1.85399e6 q^{84} -782491. q^{86} -289046. i q^{87} -1.16382e7i q^{88} -8.50941e6 q^{89} -1.95334e7 q^{91} +919044. i q^{92} -4.82465e6i q^{93} -1.41237e6 q^{94} -3.72792e6 q^{96} +6.61583e6i q^{97} +8.47607e6i q^{98} -5.40302e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 138 q^{4} + 378 q^{6} - 2916 q^{9} + 6896 q^{11} + 24528 q^{14} - 48990 q^{16} + 99168 q^{19} + 70416 q^{21} - 78246 q^{24} + 432308 q^{26} - 363544 q^{29} + 608464 q^{31} + 879844 q^{34} + 100602 q^{36}+ \cdots - 5027184 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 8.75765i − 0.774074i −0.922064 − 0.387037i \(-0.873499\pi\)
0.922064 − 0.387037i \(-0.126501\pi\)
\(3\) − 27.0000i − 0.577350i
\(4\) 51.3036 0.400809
\(5\) 0 0
\(6\) −236.457 −0.446912
\(7\) 1338.43i 1.47486i 0.675421 + 0.737432i \(0.263962\pi\)
−0.675421 + 0.737432i \(0.736038\pi\)
\(8\) − 1570.28i − 1.08433i
\(9\) −729.000 −0.333333
\(10\) 0 0
\(11\) 7411.55 1.67894 0.839469 − 0.543408i \(-0.182866\pi\)
0.839469 + 0.543408i \(0.182866\pi\)
\(12\) − 1385.20i − 0.231407i
\(13\) 14594.3i 1.84239i 0.389101 + 0.921195i \(0.372786\pi\)
−0.389101 + 0.921195i \(0.627214\pi\)
\(14\) 11721.5 1.14165
\(15\) 0 0
\(16\) −7185.09 −0.438543
\(17\) 14414.7i 0.711598i 0.934563 + 0.355799i \(0.115791\pi\)
−0.934563 + 0.355799i \(0.884209\pi\)
\(18\) 6384.33i 0.258025i
\(19\) −409.730 −0.0137044 −0.00685220 − 0.999977i \(-0.502181\pi\)
−0.00685220 + 0.999977i \(0.502181\pi\)
\(20\) 0 0
\(21\) 36137.6 0.851513
\(22\) − 64907.8i − 1.29962i
\(23\) 17913.8i 0.307002i 0.988148 + 0.153501i \(0.0490549\pi\)
−0.988148 + 0.153501i \(0.950945\pi\)
\(24\) −42397.5 −0.626038
\(25\) 0 0
\(26\) 127812. 1.42615
\(27\) 19683.0i 0.192450i
\(28\) 68666.1i 0.591139i
\(29\) 10705.4 0.0815099 0.0407549 − 0.999169i \(-0.487024\pi\)
0.0407549 + 0.999169i \(0.487024\pi\)
\(30\) 0 0
\(31\) 178691. 1.07730 0.538649 − 0.842530i \(-0.318935\pi\)
0.538649 + 0.842530i \(0.318935\pi\)
\(32\) − 138071.i − 0.744865i
\(33\) − 200112.i − 0.969335i
\(34\) 126239. 0.550830
\(35\) 0 0
\(36\) −37400.3 −0.133603
\(37\) − 427962.i − 1.38899i −0.719497 − 0.694496i \(-0.755628\pi\)
0.719497 − 0.694496i \(-0.244372\pi\)
\(38\) 3588.27i 0.0106082i
\(39\) 394046. 1.06370
\(40\) 0 0
\(41\) 53370.3 0.120936 0.0604681 − 0.998170i \(-0.480741\pi\)
0.0604681 + 0.998170i \(0.480741\pi\)
\(42\) − 316480.i − 0.659134i
\(43\) − 89349.4i − 0.171377i −0.996322 − 0.0856884i \(-0.972691\pi\)
0.996322 − 0.0856884i \(-0.0273090\pi\)
\(44\) 380239. 0.672933
\(45\) 0 0
\(46\) 156883. 0.237642
\(47\) − 161273.i − 0.226578i −0.993562 − 0.113289i \(-0.963861\pi\)
0.993562 − 0.113289i \(-0.0361387\pi\)
\(48\) 193997.i 0.253193i
\(49\) −967848. −1.17522
\(50\) 0 0
\(51\) 389197. 0.410841
\(52\) 748740.i 0.738447i
\(53\) 121833.i 0.112408i 0.998419 + 0.0562042i \(0.0178998\pi\)
−0.998419 + 0.0562042i \(0.982100\pi\)
\(54\) 172377. 0.148971
\(55\) 0 0
\(56\) 2.10170e6 1.59924
\(57\) 11062.7i 0.00791224i
\(58\) − 93754.2i − 0.0630947i
\(59\) 1.69229e6 1.07274 0.536368 − 0.843984i \(-0.319796\pi\)
0.536368 + 0.843984i \(0.319796\pi\)
\(60\) 0 0
\(61\) −1.23924e6 −0.699040 −0.349520 − 0.936929i \(-0.613655\pi\)
−0.349520 + 0.936929i \(0.613655\pi\)
\(62\) − 1.56491e6i − 0.833908i
\(63\) − 975714.i − 0.491621i
\(64\) −2.12887e6 −1.01512
\(65\) 0 0
\(66\) −1.75251e6 −0.750337
\(67\) − 944402.i − 0.383615i −0.981433 − 0.191807i \(-0.938565\pi\)
0.981433 − 0.191807i \(-0.0614349\pi\)
\(68\) 739526.i 0.285215i
\(69\) 483674. 0.177248
\(70\) 0 0
\(71\) −936943. −0.310677 −0.155338 − 0.987861i \(-0.549647\pi\)
−0.155338 + 0.987861i \(0.549647\pi\)
\(72\) 1.14473e6i 0.361443i
\(73\) 5.49712e6i 1.65389i 0.562286 + 0.826943i \(0.309922\pi\)
−0.562286 + 0.826943i \(0.690078\pi\)
\(74\) −3.74794e6 −1.07518
\(75\) 0 0
\(76\) −21020.6 −0.00549285
\(77\) 9.91983e6i 2.47621i
\(78\) − 3.45092e6i − 0.823386i
\(79\) 3.29987e6 0.753011 0.376506 − 0.926414i \(-0.377126\pi\)
0.376506 + 0.926414i \(0.377126\pi\)
\(80\) 0 0
\(81\) 531441. 0.111111
\(82\) − 467399.i − 0.0936136i
\(83\) 4.16260e6i 0.799083i 0.916715 + 0.399541i \(0.130831\pi\)
−0.916715 + 0.399541i \(0.869169\pi\)
\(84\) 1.85399e6 0.341294
\(85\) 0 0
\(86\) −782491. −0.132658
\(87\) − 289046.i − 0.0470598i
\(88\) − 1.16382e7i − 1.82052i
\(89\) −8.50941e6 −1.27948 −0.639741 − 0.768590i \(-0.720959\pi\)
−0.639741 + 0.768590i \(0.720959\pi\)
\(90\) 0 0
\(91\) −1.95334e7 −2.71728
\(92\) 919044.i 0.123049i
\(93\) − 4.82465e6i − 0.621978i
\(94\) −1.41237e6 −0.175389
\(95\) 0 0
\(96\) −3.72792e6 −0.430048
\(97\) 6.61583e6i 0.736010i 0.929824 + 0.368005i \(0.119959\pi\)
−0.929824 + 0.368005i \(0.880041\pi\)
\(98\) 8.47607e6i 0.909711i
\(99\) −5.40302e6 −0.559646
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 75.8.b.d.49.2 4
3.2 odd 2 225.8.b.n.199.3 4
5.2 odd 4 75.8.a.e.1.2 2
5.3 odd 4 15.8.a.c.1.1 ✓ 2
5.4 even 2 inner 75.8.b.d.49.3 4
15.2 even 4 225.8.a.t.1.1 2
15.8 even 4 45.8.a.i.1.2 2
15.14 odd 2 225.8.b.n.199.2 4
20.3 even 4 240.8.a.p.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.8.a.c.1.1 ✓ 2 5.3 odd 4
45.8.a.i.1.2 2 15.8 even 4
75.8.a.e.1.2 2 5.2 odd 4
75.8.b.d.49.2 4 1.1 even 1 trivial
75.8.b.d.49.3 4 5.4 even 2 inner
225.8.a.t.1.1 2 15.2 even 4
225.8.b.n.199.2 4 15.14 odd 2
225.8.b.n.199.3 4 3.2 odd 2
240.8.a.p.1.1 2 20.3 even 4