Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [75,3,Mod(7,75)] 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("75.7"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(75, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 75.f (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,4,0,0,0,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.04360198270\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 15)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 7.1
Root \(-1.22474 - 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 75.7
Dual form 75.3.f.c.43.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.224745 - 0.224745i) q^{2} +(1.22474 - 1.22474i) q^{3} -3.89898i q^{4} -0.550510 q^{6} +(-3.44949 - 3.44949i) q^{7} +(-1.77526 + 1.77526i) q^{8} -3.00000i q^{9} +11.3485 q^{11} +(-4.77526 - 4.77526i) q^{12} +(5.55051 - 5.55051i) q^{13} +1.55051i q^{14} -14.7980 q^{16} +(17.3485 + 17.3485i) q^{17} +(-0.674235 + 0.674235i) q^{18} +8.69694i q^{19} -8.44949 q^{21} +(-2.55051 - 2.55051i) q^{22} +(-11.5505 + 11.5505i) q^{23} +4.34847i q^{24} -2.49490 q^{26} +(-3.67423 - 3.67423i) q^{27} +(-13.4495 + 13.4495i) q^{28} +35.1464i q^{29} +10.6969 q^{31} +(10.4268 + 10.4268i) q^{32} +(13.8990 - 13.8990i) q^{33} -7.79796i q^{34} -11.6969 q^{36} +(6.04541 + 6.04541i) q^{37} +(1.95459 - 1.95459i) q^{38} -13.5959i q^{39} +0.696938 q^{41} +(1.89898 + 1.89898i) q^{42} +(26.4949 - 26.4949i) q^{43} -44.2474i q^{44} +5.19184 q^{46} +(-44.2474 - 44.2474i) q^{47} +(-18.1237 + 18.1237i) q^{48} -25.2020i q^{49} +42.4949 q^{51} +(-21.6413 - 21.6413i) q^{52} +(0.696938 - 0.696938i) q^{53} +1.65153i q^{54} +12.2474 q^{56} +(10.6515 + 10.6515i) q^{57} +(7.89898 - 7.89898i) q^{58} -39.9342i q^{59} +5.90918 q^{61} +(-2.40408 - 2.40408i) q^{62} +(-10.3485 + 10.3485i) q^{63} +54.5051i q^{64} -6.24745 q^{66} +(45.1010 + 45.1010i) q^{67} +(67.6413 - 67.6413i) q^{68} +28.2929i q^{69} -68.0000 q^{71} +(5.32577 + 5.32577i) q^{72} +(-77.7878 + 77.7878i) q^{73} -2.71735i q^{74} +33.9092 q^{76} +(-39.1464 - 39.1464i) q^{77} +(-3.05561 + 3.05561i) q^{78} -24.4949i q^{79} -9.00000 q^{81} +(-0.156633 - 0.156633i) q^{82} +(-13.1464 + 13.1464i) q^{83} +32.9444i q^{84} -11.9092 q^{86} +(43.0454 + 43.0454i) q^{87} +(-20.1464 + 20.1464i) q^{88} +82.1816i q^{89} -38.2929 q^{91} +(45.0352 + 45.0352i) q^{92} +(13.1010 - 13.1010i) q^{93} +19.8888i q^{94} +25.5403 q^{96} +(24.5959 + 24.5959i) q^{97} +(-5.66403 + 5.66403i) q^{98} -34.0454i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} - 12 q^{6} - 4 q^{7} - 12 q^{8} + 16 q^{11} - 24 q^{12} + 32 q^{13} - 20 q^{16} + 40 q^{17} + 12 q^{18} - 24 q^{21} - 20 q^{22} - 56 q^{23} + 88 q^{26} - 44 q^{28} - 16 q^{31} + 76 q^{32}+ \cdots + 188 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(26\) \(52\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.224745 − 0.224745i −0.112372 − 0.112372i 0.648685 − 0.761057i \(-0.275319\pi\)
−0.761057 + 0.648685i \(0.775319\pi\)
\(3\) 1.22474 − 1.22474i 0.408248 − 0.408248i
\(4\) − 3.89898i − 0.974745i
\(5\) 0 0
\(6\) −0.550510 −0.0917517
\(7\) −3.44949 − 3.44949i −0.492784 − 0.492784i 0.416398 − 0.909182i \(-0.363292\pi\)
−0.909182 + 0.416398i \(0.863292\pi\)
\(8\) −1.77526 + 1.77526i −0.221907 + 0.221907i
\(9\) − 3.00000i − 0.333333i
\(10\) 0 0
\(11\) 11.3485 1.03168 0.515840 − 0.856685i \(-0.327480\pi\)
0.515840 + 0.856685i \(0.327480\pi\)
\(12\) −4.77526 − 4.77526i −0.397938 − 0.397938i
\(13\) 5.55051 − 5.55051i 0.426962 − 0.426962i −0.460630 − 0.887592i \(-0.652376\pi\)
0.887592 + 0.460630i \(0.152376\pi\)
\(14\) 1.55051i 0.110751i
\(15\) 0 0
\(16\) −14.7980 −0.924872
\(17\) 17.3485 + 17.3485i 1.02050 + 1.02050i 0.999785 + 0.0207127i \(0.00659354\pi\)
0.0207127 + 0.999785i \(0.493406\pi\)
\(18\) −0.674235 + 0.674235i −0.0374575 + 0.0374575i
\(19\) 8.69694i 0.457734i 0.973458 + 0.228867i \(0.0735020\pi\)
−0.973458 + 0.228867i \(0.926498\pi\)
\(20\) 0 0
\(21\) −8.44949 −0.402357
\(22\) −2.55051 − 2.55051i −0.115932 − 0.115932i
\(23\) −11.5505 + 11.5505i −0.502196 + 0.502196i −0.912120 − 0.409924i \(-0.865555\pi\)
0.409924 + 0.912120i \(0.365555\pi\)
\(24\) 4.34847i 0.181186i
\(25\) 0 0
\(26\) −2.49490 −0.0959576
\(27\) −3.67423 − 3.67423i −0.136083 − 0.136083i
\(28\) −13.4495 + 13.4495i −0.480339 + 0.480339i
\(29\) 35.1464i 1.21195i 0.795485 + 0.605973i \(0.207216\pi\)
−0.795485 + 0.605973i \(0.792784\pi\)
\(30\) 0 0
\(31\) 10.6969 0.345063 0.172531 − 0.985004i \(-0.444805\pi\)
0.172531 + 0.985004i \(0.444805\pi\)
\(32\) 10.4268 + 10.4268i 0.325837 + 0.325837i
\(33\) 13.8990 − 13.8990i 0.421181 − 0.421181i
\(34\) − 7.79796i − 0.229352i
\(35\) 0 0
\(36\) −11.6969 −0.324915
\(37\) 6.04541 + 6.04541i 0.163389 + 0.163389i 0.784066 − 0.620677i \(-0.213142\pi\)
−0.620677 + 0.784066i \(0.713142\pi\)
\(38\) 1.95459 − 1.95459i 0.0514366 − 0.0514366i
\(39\) − 13.5959i − 0.348613i
\(40\) 0 0
\(41\) 0.696938 0.0169985 0.00849925 − 0.999964i \(-0.497295\pi\)
0.00849925 + 0.999964i \(0.497295\pi\)
\(42\) 1.89898 + 1.89898i 0.0452138 + 0.0452138i
\(43\) 26.4949 − 26.4949i 0.616160 − 0.616160i −0.328384 − 0.944544i \(-0.606504\pi\)
0.944544 + 0.328384i \(0.106504\pi\)
\(44\) − 44.2474i − 1.00562i
\(45\) 0 0
\(46\) 5.19184 0.112866
\(47\) −44.2474 − 44.2474i −0.941435 − 0.941435i 0.0569424 − 0.998377i \(-0.481865\pi\)
−0.998377 + 0.0569424i \(0.981865\pi\)
\(48\) −18.1237 + 18.1237i −0.377578 + 0.377578i
\(49\) − 25.2020i − 0.514327i
\(50\) 0 0
\(51\) 42.4949 0.833233
\(52\) −21.6413 − 21.6413i −0.416179 − 0.416179i
\(53\) 0.696938 − 0.696938i 0.0131498 − 0.0131498i −0.700501 − 0.713651i \(-0.747040\pi\)
0.713651 + 0.700501i \(0.247040\pi\)
\(54\) 1.65153i 0.0305839i
\(55\) 0 0
\(56\) 12.2474 0.218704
\(57\) 10.6515 + 10.6515i 0.186869 + 0.186869i
\(58\) 7.89898 − 7.89898i 0.136189 − 0.136189i
\(59\) − 39.9342i − 0.676851i −0.940993 − 0.338425i \(-0.890106\pi\)
0.940993 − 0.338425i \(-0.109894\pi\)
\(60\) 0 0
\(61\) 5.90918 0.0968719 0.0484359 − 0.998826i \(-0.484576\pi\)
0.0484359 + 0.998826i \(0.484576\pi\)
\(62\) −2.40408 − 2.40408i −0.0387755 − 0.0387755i
\(63\) −10.3485 + 10.3485i −0.164261 + 0.164261i
\(64\) 54.5051i 0.851642i
\(65\) 0 0
\(66\) −6.24745 −0.0946583
\(67\) 45.1010 + 45.1010i 0.673150 + 0.673150i 0.958441 − 0.285291i \(-0.0920903\pi\)
−0.285291 + 0.958441i \(0.592090\pi\)
\(68\) 67.6413 − 67.6413i 0.994725 − 0.994725i
\(69\) 28.2929i 0.410041i
\(70\) 0 0
\(71\) −68.0000 −0.957746 −0.478873 − 0.877884i \(-0.658955\pi\)
−0.478873 + 0.877884i \(0.658955\pi\)
\(72\) 5.32577 + 5.32577i 0.0739690 + 0.0739690i
\(73\) −77.7878 + 77.7878i −1.06559 + 1.06559i −0.0678931 + 0.997693i \(0.521628\pi\)
−0.997693 + 0.0678931i \(0.978372\pi\)
\(74\) − 2.71735i − 0.0367209i
\(75\) 0 0
\(76\) 33.9092 0.446173
\(77\) −39.1464 − 39.1464i −0.508395 − 0.508395i
\(78\) −3.05561 + 3.05561i −0.0391745 + 0.0391745i
\(79\) − 24.4949i − 0.310062i −0.987910 − 0.155031i \(-0.950452\pi\)
0.987910 − 0.155031i \(-0.0495477\pi\)
\(80\) 0 0
\(81\) −9.00000 −0.111111
\(82\) −0.156633 − 0.156633i −0.00191016 − 0.00191016i
\(83\) −13.1464 + 13.1464i −0.158391 + 0.158391i −0.781853 − 0.623463i \(-0.785725\pi\)
0.623463 + 0.781853i \(0.285725\pi\)
\(84\) 32.9444i 0.392195i
\(85\) 0 0
\(86\) −11.9092 −0.138479
\(87\) 43.0454 + 43.0454i 0.494775 + 0.494775i
\(88\) −20.1464 + 20.1464i −0.228937 + 0.228937i
\(89\) 82.1816i 0.923389i 0.887039 + 0.461695i \(0.152758\pi\)
−0.887039 + 0.461695i \(0.847242\pi\)
\(90\) 0 0
\(91\) −38.2929 −0.420801
\(92\) 45.0352 + 45.0352i 0.489513 + 0.489513i
\(93\) 13.1010 − 13.1010i 0.140871 − 0.140871i
\(94\) 19.8888i 0.211583i
\(95\) 0 0
\(96\) 25.5403 0.266045
\(97\) 24.5959 + 24.5959i 0.253566 + 0.253566i 0.822431 − 0.568865i \(-0.192617\pi\)
−0.568865 + 0.822431i \(0.692617\pi\)
\(98\) −5.66403 + 5.66403i −0.0577962 + 0.0577962i
\(99\) − 34.0454i − 0.343893i
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 75.3.f.c.7.1 4
3.2 odd 2 225.3.g.a.82.2 4
4.3 odd 2 1200.3.bg.k.1057.1 4
5.2 odd 4 15.3.f.a.13.2 yes 4
5.3 odd 4 inner 75.3.f.c.43.1 4
5.4 even 2 15.3.f.a.7.2 ✓ 4
15.2 even 4 45.3.g.b.28.1 4
15.8 even 4 225.3.g.a.118.2 4
15.14 odd 2 45.3.g.b.37.1 4
20.3 even 4 1200.3.bg.k.193.1 4
20.7 even 4 240.3.bg.a.193.2 4
20.19 odd 2 240.3.bg.a.97.2 4
40.19 odd 2 960.3.bg.h.577.1 4
40.27 even 4 960.3.bg.h.193.1 4
40.29 even 2 960.3.bg.i.577.2 4
40.37 odd 4 960.3.bg.i.193.2 4
45.2 even 12 405.3.l.f.28.2 8
45.4 even 6 405.3.l.h.217.1 8
45.7 odd 12 405.3.l.h.28.1 8
45.14 odd 6 405.3.l.f.217.2 8
45.22 odd 12 405.3.l.h.298.2 8
45.29 odd 6 405.3.l.f.352.1 8
45.32 even 12 405.3.l.f.298.1 8
45.34 even 6 405.3.l.h.352.2 8
60.47 odd 4 720.3.bh.k.433.2 4
60.59 even 2 720.3.bh.k.577.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.3.f.a.7.2 ✓ 4 5.4 even 2
15.3.f.a.13.2 yes 4 5.2 odd 4
45.3.g.b.28.1 4 15.2 even 4
45.3.g.b.37.1 4 15.14 odd 2
75.3.f.c.7.1 4 1.1 even 1 trivial
75.3.f.c.43.1 4 5.3 odd 4 inner
225.3.g.a.82.2 4 3.2 odd 2
225.3.g.a.118.2 4 15.8 even 4
240.3.bg.a.97.2 4 20.19 odd 2
240.3.bg.a.193.2 4 20.7 even 4
405.3.l.f.28.2 8 45.2 even 12
405.3.l.f.217.2 8 45.14 odd 6
405.3.l.f.298.1 8 45.32 even 12
405.3.l.f.352.1 8 45.29 odd 6
405.3.l.h.28.1 8 45.7 odd 12
405.3.l.h.217.1 8 45.4 even 6
405.3.l.h.298.2 8 45.22 odd 12
405.3.l.h.352.2 8 45.34 even 6
720.3.bh.k.433.2 4 60.47 odd 4
720.3.bh.k.577.2 4 60.59 even 2
960.3.bg.h.193.1 4 40.27 even 4
960.3.bg.h.577.1 4 40.19 odd 2
960.3.bg.i.193.2 4 40.37 odd 4
960.3.bg.i.577.2 4 40.29 even 2
1200.3.bg.k.193.1 4 20.3 even 4
1200.3.bg.k.1057.1 4 4.3 odd 2