Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [725,2,Mod(1,725)] 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("725.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(725, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 725 = 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 725.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,4] 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: yes
Analytic conductor: \(5.78915414654\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{3}, \sqrt{11})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 7x^{2} + 4 \) 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 145)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(0.792287\) of defining polynomial
Character \(\chi\) \(=\) 725.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.73205 q^{2} +0.792287 q^{3} +1.00000 q^{4} +1.37228 q^{6} +5.04868 q^{7} -1.73205 q^{8} -2.37228 q^{9} +0.627719 q^{11} +0.792287 q^{12} +4.25639 q^{13} +8.74456 q^{14} -5.00000 q^{16} -1.58457 q^{17} -4.10891 q^{18} +4.00000 q^{19} +4.00000 q^{21} +1.08724 q^{22} +3.46410 q^{23} -1.37228 q^{24} +7.37228 q^{26} -4.25639 q^{27} +5.04868 q^{28} -1.00000 q^{29} -3.37228 q^{31} -5.19615 q^{32} +0.497333 q^{33} -2.74456 q^{34} -2.37228 q^{36} +3.16915 q^{37} +6.92820 q^{38} +3.37228 q^{39} -4.74456 q^{41} +6.92820 q^{42} -10.8896 q^{43} +0.627719 q^{44} +6.00000 q^{46} -10.8896 q^{47} -3.96143 q^{48} +18.4891 q^{49} -1.25544 q^{51} +4.25639 q^{52} +4.25639 q^{53} -7.37228 q^{54} -8.74456 q^{56} +3.16915 q^{57} -1.73205 q^{58} -10.7446 q^{59} +6.00000 q^{61} -5.84096 q^{62} -11.9769 q^{63} +1.00000 q^{64} +0.861407 q^{66} -1.87953 q^{67} -1.58457 q^{68} +2.74456 q^{69} +6.74456 q^{71} +4.10891 q^{72} -6.92820 q^{73} +5.48913 q^{74} +4.00000 q^{76} +3.16915 q^{77} +5.84096 q^{78} +11.3723 q^{79} +3.74456 q^{81} -8.21782 q^{82} -9.80240 q^{83} +4.00000 q^{84} -18.8614 q^{86} -0.792287 q^{87} -1.08724 q^{88} -0.744563 q^{89} +21.4891 q^{91} +3.46410 q^{92} -2.67181 q^{93} -18.8614 q^{94} -4.11684 q^{96} -6.92820 q^{97} +32.0241 q^{98} -1.48913 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{4} - 6 q^{6} + 2 q^{9} + 14 q^{11} + 12 q^{14} - 20 q^{16} + 16 q^{19} + 16 q^{21} + 6 q^{24} + 18 q^{26} - 4 q^{29} - 2 q^{31} + 12 q^{34} + 2 q^{36} + 2 q^{39} + 4 q^{41} + 14 q^{44} + 24 q^{46}+ \cdots + 40 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.73205 1.22474 0.612372 − 0.790569i \(-0.290215\pi\)
0.612372 + 0.790569i \(0.290215\pi\)
\(3\) 0.792287 0.457427 0.228714 − 0.973494i \(-0.426548\pi\)
0.228714 + 0.973494i \(0.426548\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 1.37228 0.560232
\(7\) 5.04868 1.90822 0.954110 − 0.299456i \(-0.0968053\pi\)
0.954110 + 0.299456i \(0.0968053\pi\)
\(8\) −1.73205 −0.612372
\(9\) −2.37228 −0.790760
\(10\) 0 0
\(11\) 0.627719 0.189264 0.0946322 − 0.995512i \(-0.469833\pi\)
0.0946322 + 0.995512i \(0.469833\pi\)
\(12\) 0.792287 0.228714
\(13\) 4.25639 1.18051 0.590255 − 0.807217i \(-0.299027\pi\)
0.590255 + 0.807217i \(0.299027\pi\)
\(14\) 8.74456 2.33708
\(15\) 0 0
\(16\) −5.00000 −1.25000
\(17\) −1.58457 −0.384316 −0.192158 − 0.981364i \(-0.561549\pi\)
−0.192158 + 0.981364i \(0.561549\pi\)
\(18\) −4.10891 −0.968480
\(19\) 4.00000 0.917663 0.458831 − 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) 4.00000 0.872872
\(22\) 1.08724 0.231800
\(23\) 3.46410 0.722315 0.361158 − 0.932505i \(-0.382382\pi\)
0.361158 + 0.932505i \(0.382382\pi\)
\(24\) −1.37228 −0.280116
\(25\) 0 0
\(26\) 7.37228 1.44582
\(27\) −4.25639 −0.819142
\(28\) 5.04868 0.954110
\(29\) −1.00000 −0.185695
\(30\) 0 0
\(31\) −3.37228 −0.605680 −0.302840 − 0.953041i \(-0.597935\pi\)
−0.302840 + 0.953041i \(0.597935\pi\)
\(32\) −5.19615 −0.918559
\(33\) 0.497333 0.0865746
\(34\) −2.74456 −0.470689
\(35\) 0 0
\(36\) −2.37228 −0.395380
\(37\) 3.16915 0.521005 0.260502 − 0.965473i \(-0.416112\pi\)
0.260502 + 0.965473i \(0.416112\pi\)
\(38\) 6.92820 1.12390
\(39\) 3.37228 0.539997
\(40\) 0 0
\(41\) −4.74456 −0.740976 −0.370488 − 0.928837i \(-0.620810\pi\)
−0.370488 + 0.928837i \(0.620810\pi\)
\(42\) 6.92820 1.06904
\(43\) −10.8896 −1.66065 −0.830327 − 0.557276i \(-0.811846\pi\)
−0.830327 + 0.557276i \(0.811846\pi\)
\(44\) 0.627719 0.0946322
\(45\) 0 0
\(46\) 6.00000 0.884652
\(47\) −10.8896 −1.58842 −0.794208 − 0.607645i \(-0.792114\pi\)
−0.794208 + 0.607645i \(0.792114\pi\)
\(48\) −3.96143 −0.571784
\(49\) 18.4891 2.64130
\(50\) 0 0
\(51\) −1.25544 −0.175796
\(52\) 4.25639 0.590255
\(53\) 4.25639 0.584660 0.292330 − 0.956318i \(-0.405569\pi\)
0.292330 + 0.956318i \(0.405569\pi\)
\(54\) −7.37228 −1.00324
\(55\) 0 0
\(56\) −8.74456 −1.16854
\(57\) 3.16915 0.419764
\(58\) −1.73205 −0.227429
\(59\) −10.7446 −1.39882 −0.699411 − 0.714719i \(-0.746554\pi\)
−0.699411 + 0.714719i \(0.746554\pi\)
\(60\) 0 0
\(61\) 6.00000 0.768221 0.384111 − 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) −5.84096 −0.741803
\(63\) −11.9769 −1.50894
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0.861407 0.106032
\(67\) −1.87953 −0.229621 −0.114810 − 0.993387i \(-0.536626\pi\)
−0.114810 + 0.993387i \(0.536626\pi\)
\(68\) −1.58457 −0.192158
\(69\) 2.74456 0.330407
\(70\) 0 0
\(71\) 6.74456 0.800432 0.400216 − 0.916421i \(-0.368935\pi\)
0.400216 + 0.916421i \(0.368935\pi\)
\(72\) 4.10891 0.484240
\(73\) −6.92820 −0.810885 −0.405442 − 0.914121i \(-0.632883\pi\)
−0.405442 + 0.914121i \(0.632883\pi\)
\(74\) 5.48913 0.638098
\(75\) 0 0
\(76\) 4.00000 0.458831
\(77\) 3.16915 0.361158
\(78\) 5.84096 0.661359
\(79\) 11.3723 1.27948 0.639741 − 0.768591i \(-0.279042\pi\)
0.639741 + 0.768591i \(0.279042\pi\)
\(80\) 0 0
\(81\) 3.74456 0.416063
\(82\) −8.21782 −0.907507
\(83\) −9.80240 −1.07595 −0.537976 − 0.842960i \(-0.680811\pi\)
−0.537976 + 0.842960i \(0.680811\pi\)
\(84\) 4.00000 0.436436
\(85\) 0 0
\(86\) −18.8614 −2.03388
\(87\) −0.792287 −0.0849421
\(88\) −1.08724 −0.115900
\(89\) −0.744563 −0.0789235 −0.0394617 − 0.999221i \(-0.512564\pi\)
−0.0394617 + 0.999221i \(0.512564\pi\)
\(90\) 0 0
\(91\) 21.4891 2.25267
\(92\) 3.46410 0.361158
\(93\) −2.67181 −0.277054
\(94\) −18.8614 −1.94541
\(95\) 0 0
\(96\) −4.11684 −0.420174
\(97\) −6.92820 −0.703452 −0.351726 − 0.936103i \(-0.614405\pi\)
−0.351726 + 0.936103i \(0.614405\pi\)
\(98\) 32.0241 3.23492
\(99\) −1.48913 −0.149663
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 725.2.a.g.1.4 4
3.2 odd 2 6525.2.a.bk.1.2 4
5.2 odd 4 145.2.b.a.59.3 yes 4
5.3 odd 4 145.2.b.a.59.2 ✓ 4
5.4 even 2 inner 725.2.a.g.1.1 4
15.2 even 4 1305.2.c.e.784.1 4
15.8 even 4 1305.2.c.e.784.3 4
15.14 odd 2 6525.2.a.bk.1.3 4
20.3 even 4 2320.2.d.c.929.2 4
20.7 even 4 2320.2.d.c.929.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
145.2.b.a.59.2 ✓ 4 5.3 odd 4
145.2.b.a.59.3 yes 4 5.2 odd 4
725.2.a.g.1.1 4 5.4 even 2 inner
725.2.a.g.1.4 4 1.1 even 1 trivial
1305.2.c.e.784.1 4 15.2 even 4
1305.2.c.e.784.3 4 15.8 even 4
2320.2.d.c.929.2 4 20.3 even 4
2320.2.d.c.929.3 4 20.7 even 4
6525.2.a.bk.1.2 4 3.2 odd 2
6525.2.a.bk.1.3 4 15.14 odd 2