Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [230,3,Mod(91,230)] 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("230.91"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(230, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 230 = 2 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 230.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.26704608029\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 78x^{14} + 2165x^{12} + 28310x^{10} + 184804x^{8} + 569634x^{6} + 696037x^{4} + 285578x^{2} + 529 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{7} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 91.2
Root \(0.0431371i\) of defining polynomial
Character \(\chi\) \(=\) 230.91
Dual form 230.3.d.a.91.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.41421 q^{2} -5.41949 q^{3} +2.00000 q^{4} +2.23607i q^{5} +7.66432 q^{6} -8.24199i q^{7} -2.82843 q^{8} +20.3709 q^{9} -3.16228i q^{10} -15.8246i q^{11} -10.8390 q^{12} -14.3219 q^{13} +11.6559i q^{14} -12.1184i q^{15} +4.00000 q^{16} +10.1666i q^{17} -28.8088 q^{18} +36.5359i q^{19} +4.47214i q^{20} +44.6674i q^{21} +22.3793i q^{22} +(22.2445 - 5.84663i) q^{23} +15.3286 q^{24} -5.00000 q^{25} +20.2543 q^{26} -61.6245 q^{27} -16.4840i q^{28} +6.46533 q^{29} +17.1379i q^{30} -42.8526 q^{31} -5.65685 q^{32} +85.7611i q^{33} -14.3777i q^{34} +18.4296 q^{35} +40.7418 q^{36} +63.6379i q^{37} -51.6696i q^{38} +77.6175 q^{39} -6.32456i q^{40} -37.0921 q^{41} -63.1692i q^{42} -6.00126i q^{43} -31.6491i q^{44} +45.5507i q^{45} +(-31.4584 + 8.26838i) q^{46} -32.4676 q^{47} -21.6780 q^{48} -18.9303 q^{49} +7.07107 q^{50} -55.0977i q^{51} -28.6438 q^{52} +36.6640i q^{53} +87.1502 q^{54} +35.3848 q^{55} +23.3119i q^{56} -198.006i q^{57} -9.14336 q^{58} +6.65851 q^{59} -24.2367i q^{60} +55.7093i q^{61} +60.6028 q^{62} -167.897i q^{63} +8.00000 q^{64} -32.0248i q^{65} -121.284i q^{66} -4.45984i q^{67} +20.3331i q^{68} +(-120.554 + 31.6858i) q^{69} -26.0634 q^{70} +118.412 q^{71} -57.6176 q^{72} +82.2675 q^{73} -89.9976i q^{74} +27.0975 q^{75} +73.0718i q^{76} -130.426 q^{77} -109.768 q^{78} +133.084i q^{79} +8.94427i q^{80} +150.636 q^{81} +52.4561 q^{82} +67.5614i q^{83} +89.3348i q^{84} -22.7331 q^{85} +8.48707i q^{86} -35.0388 q^{87} +44.7586i q^{88} -104.729i q^{89} -64.4185i q^{90} +118.041i q^{91} +(44.4890 - 11.6933i) q^{92} +232.240 q^{93} +45.9161 q^{94} -81.6968 q^{95} +30.6573 q^{96} +98.6666i q^{97} +26.7715 q^{98} -322.360i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 32 q^{4} - 8 q^{6} + 64 q^{9} + 24 q^{13} + 64 q^{16} - 32 q^{18} + 4 q^{23} - 16 q^{24} - 80 q^{25} + 96 q^{26} - 96 q^{27} - 108 q^{29} - 116 q^{31} + 60 q^{35} + 128 q^{36} + 248 q^{39} - 156 q^{41}+ \cdots + 256 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/230\mathbb{Z}\right)^\times\).

\(n\) \(47\) \(51\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.41421 −0.707107
\(3\) −5.41949 −1.80650 −0.903249 − 0.429117i \(-0.858825\pi\)
−0.903249 + 0.429117i \(0.858825\pi\)
\(4\) 2.00000 0.500000
\(5\) 2.23607i 0.447214i
\(6\) 7.66432 1.27739
\(7\) − 8.24199i − 1.17743i −0.808342 − 0.588713i \(-0.799635\pi\)
0.808342 − 0.588713i \(-0.200365\pi\)
\(8\) −2.82843 −0.353553
\(9\) 20.3709 2.26343
\(10\) − 3.16228i − 0.316228i
\(11\) − 15.8246i − 1.43860i −0.694702 − 0.719298i \(-0.744464\pi\)
0.694702 − 0.719298i \(-0.255536\pi\)
\(12\) −10.8390 −0.903249
\(13\) −14.3219 −1.10169 −0.550843 − 0.834609i \(-0.685694\pi\)
−0.550843 + 0.834609i \(0.685694\pi\)
\(14\) 11.6559i 0.832566i
\(15\) − 12.1184i − 0.807890i
\(16\) 4.00000 0.250000
\(17\) 10.1666i 0.598034i 0.954248 + 0.299017i \(0.0966587\pi\)
−0.954248 + 0.299017i \(0.903341\pi\)
\(18\) −28.8088 −1.60049
\(19\) 36.5359i 1.92294i 0.274904 + 0.961472i \(0.411354\pi\)
−0.274904 + 0.961472i \(0.588646\pi\)
\(20\) 4.47214i 0.223607i
\(21\) 44.6674i 2.12702i
\(22\) 22.3793i 1.01724i
\(23\) 22.2445 − 5.84663i 0.967151 − 0.254201i
\(24\) 15.3286 0.638693
\(25\) −5.00000 −0.200000
\(26\) 20.2543 0.779010
\(27\) −61.6245 −2.28239
\(28\) − 16.4840i − 0.588713i
\(29\) 6.46533 0.222942 0.111471 − 0.993768i \(-0.464444\pi\)
0.111471 + 0.993768i \(0.464444\pi\)
\(30\) 17.1379i 0.571265i
\(31\) −42.8526 −1.38234 −0.691172 − 0.722691i \(-0.742905\pi\)
−0.691172 + 0.722691i \(0.742905\pi\)
\(32\) −5.65685 −0.176777
\(33\) 85.7611i 2.59882i
\(34\) − 14.3777i − 0.422874i
\(35\) 18.4296 0.526561
\(36\) 40.7418 1.13172
\(37\) 63.6379i 1.71994i 0.510341 + 0.859972i \(0.329519\pi\)
−0.510341 + 0.859972i \(0.670481\pi\)
\(38\) − 51.6696i − 1.35973i
\(39\) 77.6175 1.99019
\(40\) − 6.32456i − 0.158114i
\(41\) −37.0921 −0.904685 −0.452342 − 0.891844i \(-0.649412\pi\)
−0.452342 + 0.891844i \(0.649412\pi\)
\(42\) − 63.1692i − 1.50403i
\(43\) − 6.00126i − 0.139564i −0.997562 − 0.0697821i \(-0.977770\pi\)
0.997562 − 0.0697821i \(-0.0222304\pi\)
\(44\) − 31.6491i − 0.719298i
\(45\) 45.5507i 1.01224i
\(46\) −31.4584 + 8.26838i −0.683879 + 0.179747i
\(47\) −32.4676 −0.690800 −0.345400 − 0.938455i \(-0.612257\pi\)
−0.345400 + 0.938455i \(0.612257\pi\)
\(48\) −21.6780 −0.451624
\(49\) −18.9303 −0.386333
\(50\) 7.07107 0.141421
\(51\) − 55.0977i − 1.08035i
\(52\) −28.6438 −0.550843
\(53\) 36.6640i 0.691774i 0.938276 + 0.345887i \(0.112422\pi\)
−0.938276 + 0.345887i \(0.887578\pi\)
\(54\) 87.1502 1.61389
\(55\) 35.3848 0.643360
\(56\) 23.3119i 0.416283i
\(57\) − 198.006i − 3.47379i
\(58\) −9.14336 −0.157644
\(59\) 6.65851 0.112856 0.0564280 − 0.998407i \(-0.482029\pi\)
0.0564280 + 0.998407i \(0.482029\pi\)
\(60\) − 24.2367i − 0.403945i
\(61\) 55.7093i 0.913268i 0.889655 + 0.456634i \(0.150945\pi\)
−0.889655 + 0.456634i \(0.849055\pi\)
\(62\) 60.6028 0.977464
\(63\) − 167.897i − 2.66503i
\(64\) 8.00000 0.125000
\(65\) − 32.0248i − 0.492689i
\(66\) − 121.284i − 1.83764i
\(67\) − 4.45984i − 0.0665648i −0.999446 − 0.0332824i \(-0.989404\pi\)
0.999446 − 0.0332824i \(-0.0105961\pi\)
\(68\) 20.3331i 0.299017i
\(69\) −120.554 + 31.6858i −1.74716 + 0.459214i
\(70\) −26.0634 −0.372335
\(71\) 118.412 1.66777 0.833886 − 0.551937i \(-0.186111\pi\)
0.833886 + 0.551937i \(0.186111\pi\)
\(72\) −57.6176 −0.800245
\(73\) 82.2675 1.12695 0.563476 − 0.826133i \(-0.309464\pi\)
0.563476 + 0.826133i \(0.309464\pi\)
\(74\) − 89.9976i − 1.21618i
\(75\) 27.0975 0.361300
\(76\) 73.0718i 0.961472i
\(77\) −130.426 −1.69384
\(78\) −109.768 −1.40728
\(79\) 133.084i 1.68461i 0.538999 + 0.842307i \(0.318803\pi\)
−0.538999 + 0.842307i \(0.681197\pi\)
\(80\) 8.94427i 0.111803i
\(81\) 150.636 1.85970
\(82\) 52.4561 0.639709
\(83\) 67.5614i 0.813993i 0.913430 + 0.406996i \(0.133424\pi\)
−0.913430 + 0.406996i \(0.866576\pi\)
\(84\) 89.3348i 1.06351i
\(85\) −22.7331 −0.267449
\(86\) 8.48707i 0.0986868i
\(87\) −35.0388 −0.402745
\(88\) 44.7586i 0.508620i
\(89\) − 104.729i − 1.17673i −0.808595 − 0.588365i \(-0.799772\pi\)
0.808595 − 0.588365i \(-0.200228\pi\)
\(90\) − 64.4185i − 0.715761i
\(91\) 118.041i 1.29715i
\(92\) 44.4890 − 11.6933i 0.483576 − 0.127101i
\(93\) 232.240 2.49720
\(94\) 45.9161 0.488470
\(95\) −81.6968 −0.859966
\(96\) 30.6573 0.319347
\(97\) 98.6666i 1.01718i 0.861008 + 0.508591i \(0.169833\pi\)
−0.861008 + 0.508591i \(0.830167\pi\)
\(98\) 26.7715 0.273179
\(99\) − 322.360i − 3.25617i
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 230.3.d.a.91.2 yes 16
3.2 odd 2 2070.3.c.a.91.10 16
4.3 odd 2 1840.3.k.d.321.16 16
5.2 odd 4 1150.3.c.c.1149.3 32
5.3 odd 4 1150.3.c.c.1149.30 32
5.4 even 2 1150.3.d.b.551.16 16
23.22 odd 2 inner 230.3.d.a.91.1 ✓ 16
69.68 even 2 2070.3.c.a.91.15 16
92.91 even 2 1840.3.k.d.321.15 16
115.22 even 4 1150.3.c.c.1149.29 32
115.68 even 4 1150.3.c.c.1149.4 32
115.114 odd 2 1150.3.d.b.551.15 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.3.d.a.91.1 ✓ 16 23.22 odd 2 inner
230.3.d.a.91.2 yes 16 1.1 even 1 trivial
1150.3.c.c.1149.3 32 5.2 odd 4
1150.3.c.c.1149.4 32 115.68 even 4
1150.3.c.c.1149.29 32 115.22 even 4
1150.3.c.c.1149.30 32 5.3 odd 4
1150.3.d.b.551.15 16 115.114 odd 2
1150.3.d.b.551.16 16 5.4 even 2
1840.3.k.d.321.15 16 92.91 even 2
1840.3.k.d.321.16 16 4.3 odd 2
2070.3.c.a.91.10 16 3.2 odd 2
2070.3.c.a.91.15 16 69.68 even 2