Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [230,4,Mod(31,230)] 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("230.31"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(230, base_ring=CyclotomicField(22)) chi = DirichletCharacter(H, H._module([0, 6])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 230 = 2 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 230.g (of order \(11\), degree \(10\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [60,12,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(13.5704393013\)
Analytic rank: \(0\)
Dimension: \(60\)
Relative dimension: \(6\) over \(\Q(\zeta_{11})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

Embedding invariants

Embedding label 81.2
Character \(\chi\) \(=\) 230.81
Dual form 230.4.g.b.71.2

$q$-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+(-1.68251 + 1.08128i) q^{2} +(-0.719442 + 5.00383i) q^{3} +(1.66166 - 3.63853i) q^{4} +(-4.79746 + 1.40866i) q^{5} +(-4.20008 - 9.19689i) q^{6} +(9.79868 + 11.3083i) q^{7} +(1.13852 + 7.91857i) q^{8} +(1.38563 + 0.406859i) q^{9} +(6.54861 - 7.55750i) q^{10} +(33.3817 + 21.4531i) q^{11} +(17.0111 + 10.9324i) q^{12} +(-17.7157 + 20.4450i) q^{13} +(-28.7138 - 8.43113i) q^{14} +(-3.59721 - 25.0191i) q^{15} +(-10.4778 - 12.0920i) q^{16} +(1.56041 + 3.41682i) q^{17} +(-2.77127 + 0.813717i) q^{18} +(-38.8393 + 85.0461i) q^{19} +(-2.84630 + 19.7964i) q^{20} +(-63.6342 + 40.8952i) q^{21} -79.3618 q^{22} +(71.3386 + 84.1297i) q^{23} -40.4423 q^{24} +(21.0313 - 13.5160i) q^{25} +(7.69996 - 53.5544i) q^{26} +(-59.7338 + 130.799i) q^{27} +(57.4276 - 16.8623i) q^{28} +(-122.698 - 268.672i) q^{29} +(33.1051 + 38.2053i) q^{30} +(-34.3519 - 238.923i) q^{31} +(30.7038 + 9.01544i) q^{32} +(-131.364 + 151.602i) q^{33} +(-6.31995 - 4.06159i) q^{34} +(-62.9384 - 40.4480i) q^{35} +(3.78282 - 4.36560i) q^{36} +(153.661 + 45.1191i) q^{37} +(-26.6115 - 185.087i) q^{38} +(-89.5576 - 103.355i) q^{39} +(-16.6166 - 36.3853i) q^{40} +(-473.706 + 139.093i) q^{41} +(62.8458 - 137.613i) q^{42} +(-40.6616 + 282.808i) q^{43} +(133.527 - 85.8124i) q^{44} -7.22065 q^{45} +(-210.996 - 64.4117i) q^{46} -206.521 q^{47} +(68.0444 - 43.7295i) q^{48} +(16.9509 - 117.896i) q^{49} +(-20.7708 + 45.4816i) q^{50} +(-18.2198 + 5.34982i) q^{51} +(44.9521 + 98.4315i) q^{52} +(-341.231 - 393.802i) q^{53} +(-40.9278 - 284.659i) q^{54} +(-190.368 - 55.8970i) q^{55} +(-78.3894 + 90.4662i) q^{56} +(-397.613 - 255.531i) q^{57} +(496.951 + 319.371i) q^{58} +(-188.191 + 217.184i) q^{59} +(-97.0101 - 28.4847i) q^{60} +(114.116 + 793.697i) q^{61} +(316.140 + 364.845i) q^{62} +(8.97651 + 19.6558i) q^{63} +(-61.4076 + 18.0309i) q^{64} +(56.1902 - 123.039i) q^{65} +(57.0962 - 397.113i) q^{66} +(752.771 - 483.776i) q^{67} +15.0251 q^{68} +(-472.294 + 296.439i) q^{69} +149.630 q^{70} +(120.242 - 77.2748i) q^{71} +(-1.64417 + 11.4355i) q^{72} +(-431.624 + 945.124i) q^{73} +(-307.323 + 90.2382i) q^{74} +(52.5010 + 114.961i) q^{75} +(244.905 + 282.636i) q^{76} +(84.4987 + 587.702i) q^{77} +(262.437 + 77.0585i) q^{78} +(-113.395 + 130.865i) q^{79} +(67.3003 + 43.2513i) q^{80} +(-578.718 - 371.919i) q^{81} +(646.616 - 746.234i) q^{82} +(158.306 + 46.4829i) q^{83} +(43.0600 + 299.489i) q^{84} +(-12.2992 - 14.1940i) q^{85} +(-237.381 - 519.793i) q^{86} +(1432.66 - 420.667i) q^{87} +(-131.872 + 288.760i) q^{88} +(76.4669 - 531.839i) q^{89} +(12.1488 - 7.80756i) q^{90} -404.787 q^{91} +(424.649 - 119.772i) q^{92} +1220.24 q^{93} +(347.473 - 223.308i) q^{94} +(66.5287 - 462.717i) q^{95} +(-67.2013 + 147.150i) q^{96} +(1475.97 - 433.385i) q^{97} +(98.9591 + 216.690i) q^{98} +(37.5264 + 43.3078i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 60 q + 12 q^{2} - 3 q^{3} - 24 q^{4} - 30 q^{5} + 6 q^{6} + 100 q^{7} + 48 q^{8} + 69 q^{9} + 60 q^{10} - 51 q^{11} + 120 q^{12} + 184 q^{13} + 20 q^{14} - 15 q^{15} - 96 q^{16} - 334 q^{17} - 138 q^{18}+ \cdots - 10589 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(47\) \(51\)
\(\chi(n)\) \(1\) \(e\left(\frac{10}{11}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.68251 + 1.08128i −0.594856 + 0.382291i
\(3\) −0.719442 + 5.00383i −0.138457 + 0.962987i 0.795590 + 0.605835i \(0.207161\pi\)
−0.934047 + 0.357151i \(0.883748\pi\)
\(4\) 1.66166 − 3.63853i 0.207708 − 0.454816i
\(5\) −4.79746 + 1.40866i −0.429098 + 0.125995i
\(6\) −4.20008 − 9.19689i −0.285779 − 0.625769i
\(7\) 9.79868 + 11.3083i 0.529079 + 0.610590i 0.955881 − 0.293756i \(-0.0949052\pi\)
−0.426802 + 0.904345i \(0.640360\pi\)
\(8\) 1.13852 + 7.91857i 0.0503159 + 0.349955i
\(9\) 1.38563 + 0.406859i 0.0513198 + 0.0150688i
\(10\) 6.54861 − 7.55750i 0.207085 − 0.238989i
\(11\) 33.3817 + 21.4531i 0.914996 + 0.588032i 0.911202 − 0.411960i \(-0.135156\pi\)
0.00379453 + 0.999993i \(0.498792\pi\)
\(12\) 17.0111 + 10.9324i 0.409223 + 0.262992i
\(13\) −17.7157 + 20.4450i −0.377957 + 0.436185i −0.912576 − 0.408908i \(-0.865910\pi\)
0.534619 + 0.845093i \(0.320455\pi\)
\(14\) −28.7138 − 8.43113i −0.548149 − 0.160951i
\(15\) −3.59721 − 25.0191i −0.0619197 − 0.430661i
\(16\) −10.4778 − 12.0920i −0.163715 − 0.188937i
\(17\) 1.56041 + 3.41682i 0.0222621 + 0.0487472i 0.920436 − 0.390892i \(-0.127834\pi\)
−0.898174 + 0.439639i \(0.855106\pi\)
\(18\) −2.77127 + 0.813717i −0.0362885 + 0.0106553i
\(19\) −38.8393 + 85.0461i −0.468965 + 1.02689i 0.516386 + 0.856356i \(0.327277\pi\)
−0.985352 + 0.170535i \(0.945450\pi\)
\(20\) −2.84630 + 19.7964i −0.0318226 + 0.221331i
\(21\) −63.6342 + 40.8952i −0.661244 + 0.424956i
\(22\) −79.3618 −0.769090
\(23\) 71.3386 + 84.1297i 0.646744 + 0.762707i
\(24\) −40.4423 −0.343968
\(25\) 21.0313 − 13.5160i 0.168251 − 0.108128i
\(26\) 7.69996 − 53.5544i 0.0580802 − 0.403957i
\(27\) −59.7338 + 130.799i −0.425769 + 0.932305i
\(28\) 57.4276 − 16.8623i 0.387600 − 0.113810i
\(29\) −122.698 − 268.672i −0.785672 − 1.72038i −0.688642 − 0.725101i \(-0.741793\pi\)
−0.0970304 − 0.995281i \(-0.530934\pi\)
\(30\) 33.1051 + 38.2053i 0.201471 + 0.232510i
\(31\) −34.3519 − 238.923i −0.199025 − 1.38425i −0.807120 − 0.590387i \(-0.798975\pi\)
0.608095 − 0.793864i \(-0.291934\pi\)
\(32\) 30.7038 + 9.01544i 0.169616 + 0.0498038i
\(33\) −131.364 + 151.602i −0.692955 + 0.799712i
\(34\) −6.31995 − 4.06159i −0.0318783 − 0.0204870i
\(35\) −62.9384 − 40.4480i −0.303958 − 0.195342i
\(36\) 3.78282 − 4.36560i 0.0175130 − 0.0202111i
\(37\) 153.661 + 45.1191i 0.682751 + 0.200474i 0.604673 − 0.796473i \(-0.293304\pi\)
0.0780777 + 0.996947i \(0.475122\pi\)
\(38\) −26.6115 − 185.087i −0.113604 − 0.790133i
\(39\) −89.5576 − 103.355i −0.367710 − 0.424360i
\(40\) −16.6166 − 36.3853i −0.0656829 − 0.143825i
\(41\) −473.706 + 139.093i −1.80440 + 0.529820i −0.998096 − 0.0616791i \(-0.980354\pi\)
−0.806306 + 0.591499i \(0.798536\pi\)
\(42\) 62.8458 − 137.613i 0.230888 − 0.505575i
\(43\) −40.6616 + 282.808i −0.144205 + 1.00297i 0.781278 + 0.624183i \(0.214568\pi\)
−0.925483 + 0.378788i \(0.876341\pi\)
\(44\) 133.527 − 85.8124i 0.457498 − 0.294016i
\(45\) −7.22065 −0.0239198
\(46\) −210.996 − 64.4117i −0.676296 − 0.206456i
\(47\) −206.521 −0.640940 −0.320470 − 0.947259i \(-0.603841\pi\)
−0.320470 + 0.947259i \(0.603841\pi\)
\(48\) 68.0444 − 43.7295i 0.204612 − 0.131496i
\(49\) 16.9509 − 117.896i 0.0494196 − 0.343721i
\(50\) −20.7708 + 45.4816i −0.0587486 + 0.128641i
\(51\) −18.2198 + 5.34982i −0.0500252 + 0.0146887i
\(52\) 44.9521 + 98.4315i 0.119880 + 0.262500i
\(53\) −341.231 − 393.802i −0.884372 − 1.02062i −0.999628 − 0.0272833i \(-0.991314\pi\)
0.115256 − 0.993336i \(-0.463231\pi\)
\(54\) −40.9278 − 284.659i −0.103140 − 0.717355i
\(55\) −190.368 − 55.8970i −0.466712 − 0.137039i
\(56\) −78.3894 + 90.4662i −0.187058 + 0.215876i
\(57\) −397.613 − 255.531i −0.923951 − 0.593787i
\(58\) 496.951 + 319.371i 1.12505 + 0.723025i
\(59\) −188.191 + 217.184i −0.415261 + 0.479237i −0.924387 − 0.381455i \(-0.875423\pi\)
0.509126 + 0.860692i \(0.329969\pi\)
\(60\) −97.0101 − 28.4847i −0.208733 − 0.0612894i
\(61\) 114.116 + 793.697i 0.239526 + 1.66594i 0.654464 + 0.756093i \(0.272894\pi\)
−0.414938 + 0.909850i \(0.636197\pi\)
\(62\) 316.140 + 364.845i 0.647578 + 0.747345i
\(63\) 8.97651 + 19.6558i 0.0179513 + 0.0393079i
\(64\) −61.4076 + 18.0309i −0.119937 + 0.0352166i
\(65\) 56.1902 − 123.039i 0.107224 − 0.234787i
\(66\) 57.0962 − 397.113i 0.106486 − 0.740624i
\(67\) 752.771 − 483.776i 1.37262 − 0.882130i 0.373654 − 0.927568i \(-0.378105\pi\)
0.998967 + 0.0454384i \(0.0144685\pi\)
\(68\) 15.0251 0.0267950
\(69\) −472.294 + 296.439i −0.824023 + 0.517205i
\(70\) 149.630 0.255489
\(71\) 120.242 − 77.2748i 0.200987 − 0.129167i −0.436276 − 0.899813i \(-0.643703\pi\)
0.637263 + 0.770646i \(0.280066\pi\)
\(72\) −1.64417 + 11.4355i −0.00269121 + 0.0187178i
\(73\) −431.624 + 945.124i −0.692024 + 1.51532i 0.157358 + 0.987542i \(0.449702\pi\)
−0.849382 + 0.527779i \(0.823025\pi\)
\(74\) −307.323 + 90.2382i −0.482778 + 0.141756i
\(75\) 52.5010 + 114.961i 0.0808306 + 0.176994i
\(76\) 244.905 + 282.636i 0.369639 + 0.426586i
\(77\) 84.4987 + 587.702i 0.125059 + 0.869803i
\(78\) 262.437 + 77.0585i 0.380964 + 0.111861i
\(79\) −113.395 + 130.865i −0.161493 + 0.186373i −0.830729 − 0.556677i \(-0.812076\pi\)
0.669236 + 0.743050i \(0.266622\pi\)
\(80\) 67.3003 + 43.2513i 0.0940550 + 0.0604455i
\(81\) −578.718 − 371.919i −0.793851 − 0.510177i
\(82\) 646.616 − 746.234i 0.870814 − 1.00497i
\(83\) 158.306 + 46.4829i 0.209354 + 0.0614718i 0.384728 − 0.923030i \(-0.374295\pi\)
−0.175374 + 0.984502i \(0.556113\pi\)
\(84\) 43.0600 + 299.489i 0.0559313 + 0.389011i
\(85\) −12.2992 − 14.1940i −0.0156945 − 0.0181124i
\(86\) −237.381 − 519.793i −0.297645 − 0.651752i
\(87\) 1432.66 − 420.667i 1.76549 − 0.518394i
\(88\) −131.872 + 288.760i −0.159746 + 0.349795i
\(89\) 76.4669 − 531.839i 0.0910728 − 0.633425i −0.892248 − 0.451546i \(-0.850873\pi\)
0.983321 − 0.181879i \(-0.0582181\pi\)
\(90\) 12.1488 − 7.80756i 0.0142288 − 0.00914432i
\(91\) −404.787 −0.466299
\(92\) 424.649 − 119.772i 0.481225 − 0.135730i
\(93\) 1220.24 1.36057
\(94\) 347.473 − 223.308i 0.381267 − 0.245026i
\(95\) 66.5287 − 462.717i 0.0718495 − 0.499724i
\(96\) −67.2013 + 147.150i −0.0714448 + 0.156442i
\(97\) 1475.97 − 433.385i 1.54497 − 0.453645i 0.605380 − 0.795936i \(-0.293021\pi\)
0.939594 + 0.342291i \(0.111203\pi\)
\(98\) 98.9591 + 216.690i 0.102004 + 0.223357i
\(99\) 37.5264 + 43.3078i 0.0380964 + 0.0439656i
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 230.4.g.b.81.2 yes 60
23.2 even 11 inner 230.4.g.b.71.2 ✓ 60
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.4.g.b.71.2 ✓ 60 23.2 even 11 inner
230.4.g.b.81.2 yes 60 1.1 even 1 trivial