Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [145,2,Mod(59,145)] 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("145.59"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(145, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 145 = 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 145.b (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(1.15783082931\)
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: \( x^{4} - x^{3} - 2x^{2} - 3x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 59.3
Root \(1.68614 + 0.396143i\) of defining polynomial
Character \(\chi\) \(=\) 145.59
Dual form 145.2.b.a.59.2

$q$-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)
 
\(f(q)\) \(=\) \(q+1.73205i q^{2} -0.792287i q^{3} -1.00000 q^{4} +(0.686141 - 2.12819i) q^{5} +1.37228 q^{6} +5.04868i q^{7} +1.73205i q^{8} +2.37228 q^{9} +(3.68614 + 1.18843i) q^{10} +0.627719 q^{11} +0.792287i q^{12} -4.25639i q^{13} -8.74456 q^{14} +(-1.68614 - 0.543620i) q^{15} -5.00000 q^{16} -1.58457i q^{17} +4.10891i q^{18} -4.00000 q^{19} +(-0.686141 + 2.12819i) q^{20} +4.00000 q^{21} +1.08724i q^{22} -3.46410i q^{23} +1.37228 q^{24} +(-4.05842 - 2.92048i) q^{25} +7.37228 q^{26} -4.25639i q^{27} -5.04868i q^{28} +1.00000 q^{29} +(0.941578 - 2.92048i) q^{30} -3.37228 q^{31} -5.19615i q^{32} -0.497333i q^{33} +2.74456 q^{34} +(10.7446 + 3.46410i) q^{35} -2.37228 q^{36} +3.16915i q^{37} -6.92820i q^{38} -3.37228 q^{39} +(3.68614 + 1.18843i) q^{40} -4.74456 q^{41} +6.92820i q^{42} +10.8896i q^{43} -0.627719 q^{44} +(1.62772 - 5.04868i) q^{45} +6.00000 q^{46} -10.8896i q^{47} +3.96143i q^{48} -18.4891 q^{49} +(5.05842 - 7.02939i) q^{50} -1.25544 q^{51} +4.25639i q^{52} -4.25639i q^{53} +7.37228 q^{54} +(0.430703 - 1.33591i) q^{55} -8.74456 q^{56} +3.16915i q^{57} +1.73205i q^{58} +10.7446 q^{59} +(1.68614 + 0.543620i) q^{60} +6.00000 q^{61} -5.84096i q^{62} +11.9769i q^{63} -1.00000 q^{64} +(-9.05842 - 2.92048i) q^{65} +0.861407 q^{66} -1.87953i q^{67} +1.58457i q^{68} -2.74456 q^{69} +(-6.00000 + 18.6101i) q^{70} +6.74456 q^{71} +4.10891i q^{72} +6.92820i q^{73} -5.48913 q^{74} +(-2.31386 + 3.21543i) q^{75} +4.00000 q^{76} +3.16915i q^{77} -5.84096i q^{78} -11.3723 q^{79} +(-3.43070 + 10.6410i) q^{80} +3.74456 q^{81} -8.21782i q^{82} +9.80240i q^{83} -4.00000 q^{84} +(-3.37228 - 1.08724i) q^{85} -18.8614 q^{86} -0.792287i q^{87} +1.08724i q^{88} +0.744563 q^{89} +(8.74456 + 2.81929i) q^{90} +21.4891 q^{91} +3.46410i q^{92} +2.67181i q^{93} +18.8614 q^{94} +(-2.74456 + 8.51278i) q^{95} -4.11684 q^{96} -6.92820i q^{97} -32.0241i q^{98} +1.48913 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} - 3 q^{5} - 6 q^{6} - 2 q^{9} + 9 q^{10} + 14 q^{11} - 12 q^{14} - q^{15} - 20 q^{16} - 16 q^{19} + 3 q^{20} + 16 q^{21} - 6 q^{24} + q^{25} + 18 q^{26} + 4 q^{29} + 21 q^{30} - 2 q^{31}+ \cdots - 40 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/145\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(117\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.73205i 1.22474i 0.790569 + 0.612372i \(0.209785\pi\)
−0.790569 + 0.612372i \(0.790215\pi\)
\(3\) − 0.792287i − 0.457427i −0.973494 − 0.228714i \(-0.926548\pi\)
0.973494 − 0.228714i \(-0.0734519\pi\)
\(4\) −1.00000 −0.500000
\(5\) 0.686141 − 2.12819i 0.306851 − 0.951757i
\(6\) 1.37228 0.560232
\(7\) 5.04868i 1.90822i 0.299456 + 0.954110i \(0.403195\pi\)
−0.299456 + 0.954110i \(0.596805\pi\)
\(8\) 1.73205i 0.612372i
\(9\) 2.37228 0.790760
\(10\) 3.68614 + 1.18843i 1.16566 + 0.375815i
\(11\) 0.627719 0.189264 0.0946322 − 0.995512i \(-0.469833\pi\)
0.0946322 + 0.995512i \(0.469833\pi\)
\(12\) 0.792287i 0.228714i
\(13\) − 4.25639i − 1.18051i −0.807217 − 0.590255i \(-0.799027\pi\)
0.807217 − 0.590255i \(-0.200973\pi\)
\(14\) −8.74456 −2.33708
\(15\) −1.68614 − 0.543620i −0.435360 − 0.140362i
\(16\) −5.00000 −1.25000
\(17\) − 1.58457i − 0.384316i −0.981364 − 0.192158i \(-0.938451\pi\)
0.981364 − 0.192158i \(-0.0615486\pi\)
\(18\) 4.10891i 0.968480i
\(19\) −4.00000 −0.917663 −0.458831 − 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) −0.686141 + 2.12819i −0.153426 + 0.475879i
\(21\) 4.00000 0.872872
\(22\) 1.08724i 0.231800i
\(23\) − 3.46410i − 0.722315i −0.932505 − 0.361158i \(-0.882382\pi\)
0.932505 − 0.361158i \(-0.117618\pi\)
\(24\) 1.37228 0.280116
\(25\) −4.05842 − 2.92048i −0.811684 − 0.584096i
\(26\) 7.37228 1.44582
\(27\) − 4.25639i − 0.819142i
\(28\) − 5.04868i − 0.954110i
\(29\) 1.00000 0.185695
\(30\) 0.941578 − 2.92048i 0.171908 − 0.533204i
\(31\) −3.37228 −0.605680 −0.302840 − 0.953041i \(-0.597935\pi\)
−0.302840 + 0.953041i \(0.597935\pi\)
\(32\) − 5.19615i − 0.918559i
\(33\) − 0.497333i − 0.0865746i
\(34\) 2.74456 0.470689
\(35\) 10.7446 + 3.46410i 1.81616 + 0.585540i
\(36\) −2.37228 −0.395380
\(37\) 3.16915i 0.521005i 0.965473 + 0.260502i \(0.0838882\pi\)
−0.965473 + 0.260502i \(0.916112\pi\)
\(38\) − 6.92820i − 1.12390i
\(39\) −3.37228 −0.539997
\(40\) 3.68614 + 1.18843i 0.582830 + 0.187907i
\(41\) −4.74456 −0.740976 −0.370488 − 0.928837i \(-0.620810\pi\)
−0.370488 + 0.928837i \(0.620810\pi\)
\(42\) 6.92820i 1.06904i
\(43\) 10.8896i 1.66065i 0.557276 + 0.830327i \(0.311846\pi\)
−0.557276 + 0.830327i \(0.688154\pi\)
\(44\) −0.627719 −0.0946322
\(45\) 1.62772 − 5.04868i 0.242646 − 0.752612i
\(46\) 6.00000 0.884652
\(47\) − 10.8896i − 1.58842i −0.607645 − 0.794208i \(-0.707886\pi\)
0.607645 − 0.794208i \(-0.292114\pi\)
\(48\) 3.96143i 0.571784i
\(49\) −18.4891 −2.64130
\(50\) 5.05842 − 7.02939i 0.715369 − 0.994106i
\(51\) −1.25544 −0.175796
\(52\) 4.25639i 0.590255i
\(53\) − 4.25639i − 0.584660i −0.956318 − 0.292330i \(-0.905569\pi\)
0.956318 − 0.292330i \(-0.0944306\pi\)
\(54\) 7.37228 1.00324
\(55\) 0.430703 − 1.33591i 0.0580760 − 0.180134i
\(56\) −8.74456 −1.16854
\(57\) 3.16915i 0.419764i
\(58\) 1.73205i 0.227429i
\(59\) 10.7446 1.39882 0.699411 − 0.714719i \(-0.253446\pi\)
0.699411 + 0.714719i \(0.253446\pi\)
\(60\) 1.68614 + 0.543620i 0.217680 + 0.0701811i
\(61\) 6.00000 0.768221 0.384111 − 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) − 5.84096i − 0.741803i
\(63\) 11.9769i 1.50894i
\(64\) −1.00000 −0.125000
\(65\) −9.05842 − 2.92048i −1.12356 − 0.362241i
\(66\) 0.861407 0.106032
\(67\) − 1.87953i − 0.229621i −0.993387 − 0.114810i \(-0.963374\pi\)
0.993387 − 0.114810i \(-0.0366261\pi\)
\(68\) 1.58457i 0.192158i
\(69\) −2.74456 −0.330407
\(70\) −6.00000 + 18.6101i −0.717137 + 2.22434i
\(71\) 6.74456 0.800432 0.400216 − 0.916421i \(-0.368935\pi\)
0.400216 + 0.916421i \(0.368935\pi\)
\(72\) 4.10891i 0.484240i
\(73\) 6.92820i 0.810885i 0.914121 + 0.405442i \(0.132883\pi\)
−0.914121 + 0.405442i \(0.867117\pi\)
\(74\) −5.48913 −0.638098
\(75\) −2.31386 + 3.21543i −0.267181 + 0.371286i
\(76\) 4.00000 0.458831
\(77\) 3.16915i 0.361158i
\(78\) − 5.84096i − 0.661359i
\(79\) −11.3723 −1.27948 −0.639741 − 0.768591i \(-0.720958\pi\)
−0.639741 + 0.768591i \(0.720958\pi\)
\(80\) −3.43070 + 10.6410i −0.383564 + 1.18970i
\(81\) 3.74456 0.416063
\(82\) − 8.21782i − 0.907507i
\(83\) 9.80240i 1.07595i 0.842960 + 0.537976i \(0.180811\pi\)
−0.842960 + 0.537976i \(0.819189\pi\)
\(84\) −4.00000 −0.436436
\(85\) −3.37228 − 1.08724i −0.365775 − 0.117928i
\(86\) −18.8614 −2.03388
\(87\) − 0.792287i − 0.0849421i
\(88\) 1.08724i 0.115900i
\(89\) 0.744563 0.0789235 0.0394617 − 0.999221i \(-0.487436\pi\)
0.0394617 + 0.999221i \(0.487436\pi\)
\(90\) 8.74456 + 2.81929i 0.921758 + 0.297179i
\(91\) 21.4891 2.25267
\(92\) 3.46410i 0.361158i
\(93\) 2.67181i 0.277054i
\(94\) 18.8614 1.94541
\(95\) −2.74456 + 8.51278i −0.281586 + 0.873393i
\(96\) −4.11684 −0.420174
\(97\) − 6.92820i − 0.703452i −0.936103 − 0.351726i \(-0.885595\pi\)
0.936103 − 0.351726i \(-0.114405\pi\)
\(98\) − 32.0241i − 3.23492i
\(99\) 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 145.2.b.a.59.3 yes 4
3.2 odd 2 1305.2.c.e.784.1 4
4.3 odd 2 2320.2.d.c.929.3 4
5.2 odd 4 725.2.a.g.1.1 4
5.3 odd 4 725.2.a.g.1.4 4
5.4 even 2 inner 145.2.b.a.59.2 ✓ 4
15.2 even 4 6525.2.a.bk.1.3 4
15.8 even 4 6525.2.a.bk.1.2 4
15.14 odd 2 1305.2.c.e.784.3 4
20.19 odd 2 2320.2.d.c.929.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
145.2.b.a.59.2 ✓ 4 5.4 even 2 inner
145.2.b.a.59.3 yes 4 1.1 even 1 trivial
725.2.a.g.1.1 4 5.2 odd 4
725.2.a.g.1.4 4 5.3 odd 4
1305.2.c.e.784.1 4 3.2 odd 2
1305.2.c.e.784.3 4 15.14 odd 2
2320.2.d.c.929.2 4 20.19 odd 2
2320.2.d.c.929.3 4 4.3 odd 2
6525.2.a.bk.1.2 4 15.8 even 4
6525.2.a.bk.1.3 4 15.2 even 4