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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 41.3
Character \(\chi\) \(=\) 230.41
Dual form 230.4.g.b.101.3

$q$-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+(1.91899 + 0.563465i) q^{2} +(0.460964 - 0.531981i) q^{3} +(3.36501 + 2.16256i) q^{4} +(-0.711574 + 4.94911i) q^{5} +(1.18434 - 0.761126i) q^{6} +(-1.15601 - 2.53131i) q^{7} +(5.23889 + 6.04600i) q^{8} +(3.77198 + 26.2347i) q^{9} +(-4.15415 + 9.09632i) q^{10} +(27.7442 - 8.14644i) q^{11} +(2.70159 - 0.793259i) q^{12} +(-9.09159 + 19.9078i) q^{13} +(-0.792063 - 5.50892i) q^{14} +(2.30482 + 2.65990i) q^{15} +(6.64664 + 14.5541i) q^{16} +(-60.0930 + 38.6194i) q^{17} +(-7.54397 + 52.4695i) q^{18} +(82.7987 + 53.2115i) q^{19} +(-13.0972 + 15.1150i) q^{20} +(-1.87949 - 0.551867i) q^{21} +57.8310 q^{22} +(70.5303 + 84.8085i) q^{23} +5.63129 q^{24} +(-23.9873 - 7.04331i) q^{25} +(-28.6640 + 33.0800i) q^{26} +(31.6837 + 20.3618i) q^{27} +(1.58413 - 11.0178i) q^{28} +(9.45320 - 6.07520i) q^{29} +(2.92415 + 6.40300i) q^{30} +(-21.5608 - 24.8825i) q^{31} +(4.55407 + 31.6743i) q^{32} +(8.45533 - 18.5146i) q^{33} +(-137.078 + 40.2498i) q^{34} +(13.3503 - 3.92000i) q^{35} +(-44.0415 + 96.4374i) q^{36} +(1.10983 + 7.71903i) q^{37} +(128.907 + 148.766i) q^{38} +(6.39967 + 14.0133i) q^{39} +(-33.6501 + 21.6256i) q^{40} +(12.6386 - 87.9033i) q^{41} +(-3.29575 - 2.11805i) q^{42} +(-157.323 + 181.560i) q^{43} +(110.977 + 32.5857i) q^{44} -132.523 q^{45} +(87.5601 + 202.488i) q^{46} +397.098 q^{47} +(10.8064 + 3.17304i) q^{48} +(219.546 - 253.370i) q^{49} +(-42.0627 - 27.0320i) q^{50} +(-7.15592 + 49.7705i) q^{51} +(-73.6452 + 47.3289i) q^{52} +(-212.097 - 464.428i) q^{53} +(49.3273 + 56.9267i) q^{54} +(20.5755 + 143.106i) q^{55} +(9.24808 - 20.2505i) q^{56} +(66.4747 - 19.5187i) q^{57} +(21.5637 - 6.33168i) q^{58} +(-43.5998 + 95.4704i) q^{59} +(2.00354 + 13.9349i) q^{60} +(99.5194 + 114.851i) q^{61} +(-27.3544 - 59.8979i) q^{62} +(62.0478 - 39.8757i) q^{63} +(-9.10815 + 63.3486i) q^{64} +(-92.0565 - 59.1611i) q^{65} +(26.6580 - 30.7650i) q^{66} +(-788.734 - 231.593i) q^{67} -285.731 q^{68} +(77.6284 + 1.57288i) q^{69} +27.8278 q^{70} +(-544.680 - 159.933i) q^{71} +(-138.854 + 160.246i) q^{72} +(558.848 + 359.150i) q^{73} +(-2.21966 + 15.4381i) q^{74} +(-14.8042 + 9.51408i) q^{75} +(163.546 + 358.115i) q^{76} +(-52.6937 - 60.8118i) q^{77} +(4.38486 + 30.4974i) q^{78} +(519.770 - 1138.14i) q^{79} +(-76.7594 + 22.5386i) q^{80} +(-661.197 + 194.145i) q^{81} +(73.7837 - 161.564i) q^{82} +(-105.476 - 733.599i) q^{83} +(-5.13105 - 5.92155i) q^{84} +(-148.371 - 324.887i) q^{85} +(-404.204 + 259.766i) q^{86} +(1.12569 - 7.82937i) q^{87} +(194.602 + 125.063i) q^{88} +(317.048 - 365.893i) q^{89} +(-254.309 - 74.6718i) q^{90} +60.9028 q^{91} +(53.9318 + 437.908i) q^{92} -23.1757 q^{93} +(762.025 + 223.751i) q^{94} +(-322.267 + 371.916i) q^{95} +(18.9494 + 12.1780i) q^{96} +(120.229 - 836.214i) q^{97} +(564.071 - 362.506i) q^{98} +(318.370 + 697.134i) 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 \(\left(\mathbb{Z}/230\mathbb{Z}\right)^\times\).

\(n\) \(47\) \(51\)
\(\chi(n)\) \(1\) \(e\left(\frac{6}{11}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.91899 + 0.563465i 0.678464 + 0.199215i
\(3\) 0.460964 − 0.531981i 0.0887126 − 0.102380i −0.709658 − 0.704547i \(-0.751150\pi\)
0.798370 + 0.602167i \(0.205696\pi\)
\(4\) 3.36501 + 2.16256i 0.420627 + 0.270320i
\(5\) −0.711574 + 4.94911i −0.0636451 + 0.442662i
\(6\) 1.18434 − 0.761126i 0.0805839 − 0.0517881i
\(7\) −1.15601 − 2.53131i −0.0624187 − 0.136678i 0.875852 − 0.482580i \(-0.160300\pi\)
−0.938271 + 0.345902i \(0.887573\pi\)
\(8\) 5.23889 + 6.04600i 0.231528 + 0.267198i
\(9\) 3.77198 + 26.2347i 0.139703 + 0.971657i
\(10\) −4.15415 + 9.09632i −0.131366 + 0.287651i
\(11\) 27.7442 − 8.14644i 0.760472 − 0.223295i 0.121570 − 0.992583i \(-0.461207\pi\)
0.638902 + 0.769288i \(0.279389\pi\)
\(12\) 2.70159 − 0.793259i 0.0649902 − 0.0190828i
\(13\) −9.09159 + 19.9078i −0.193966 + 0.424725i −0.981478 − 0.191573i \(-0.938641\pi\)
0.787513 + 0.616298i \(0.211368\pi\)
\(14\) −0.792063 − 5.50892i −0.0151206 − 0.105166i
\(15\) 2.30482 + 2.65990i 0.0396735 + 0.0457856i
\(16\) 6.64664 + 14.5541i 0.103854 + 0.227408i
\(17\) −60.0930 + 38.6194i −0.857335 + 0.550976i −0.893855 − 0.448357i \(-0.852009\pi\)
0.0365193 + 0.999333i \(0.488373\pi\)
\(18\) −7.54397 + 52.4695i −0.0987850 + 0.687065i
\(19\) 82.7987 + 53.2115i 0.999754 + 0.642503i 0.934722 − 0.355380i \(-0.115649\pi\)
0.0650320 + 0.997883i \(0.479285\pi\)
\(20\) −13.0972 + 15.1150i −0.146431 + 0.168991i
\(21\) −1.87949 − 0.551867i −0.0195304 − 0.00573463i
\(22\) 57.8310 0.560437
\(23\) 70.5303 + 84.8085i 0.639417 + 0.768860i
\(24\) 5.63129 0.0478951
\(25\) −23.9873 − 7.04331i −0.191899 − 0.0563465i
\(26\) −28.6640 + 33.0800i −0.216210 + 0.249520i
\(27\) 31.6837 + 20.3618i 0.225834 + 0.145135i
\(28\) 1.58413 − 11.0178i 0.0106918 − 0.0743634i
\(29\) 9.45320 − 6.07520i 0.0605315 − 0.0389013i −0.510025 − 0.860160i \(-0.670364\pi\)
0.570557 + 0.821258i \(0.306728\pi\)
\(30\) 2.92415 + 6.40300i 0.0177958 + 0.0389674i
\(31\) −21.5608 − 24.8825i −0.124917 − 0.144162i 0.689846 − 0.723956i \(-0.257678\pi\)
−0.814763 + 0.579794i \(0.803133\pi\)
\(32\) 4.55407 + 31.6743i 0.0251579 + 0.174977i
\(33\) 8.45533 − 18.5146i 0.0446026 − 0.0976660i
\(34\) −137.078 + 40.2498i −0.691434 + 0.203023i
\(35\) 13.3503 − 3.92000i 0.0644747 − 0.0189315i
\(36\) −44.0415 + 96.4374i −0.203896 + 0.446469i
\(37\) 1.10983 + 7.71903i 0.00493121 + 0.0342973i 0.992138 − 0.125145i \(-0.0399396\pi\)
−0.987207 + 0.159442i \(0.949030\pi\)
\(38\) 128.907 + 148.766i 0.550301 + 0.635081i
\(39\) 6.39967 + 14.0133i 0.0262761 + 0.0575366i
\(40\) −33.6501 + 21.6256i −0.133014 + 0.0854828i
\(41\) 12.6386 − 87.9033i 0.0481418 − 0.334834i −0.951489 − 0.307681i \(-0.900447\pi\)
0.999631 − 0.0271525i \(-0.00864397\pi\)
\(42\) −3.29575 − 2.11805i −0.0121082 − 0.00778148i
\(43\) −157.323 + 181.560i −0.557943 + 0.643900i −0.962715 − 0.270517i \(-0.912805\pi\)
0.404773 + 0.914417i \(0.367351\pi\)
\(44\) 110.977 + 32.5857i 0.380236 + 0.111647i
\(45\) −132.523 −0.439007
\(46\) 87.5601 + 202.488i 0.280653 + 0.649025i
\(47\) 397.098 1.23240 0.616198 − 0.787591i \(-0.288672\pi\)
0.616198 + 0.787591i \(0.288672\pi\)
\(48\) 10.8064 + 3.17304i 0.0324951 + 0.00954142i
\(49\) 219.546 − 253.370i 0.640076 − 0.738687i
\(50\) −42.0627 − 27.0320i −0.118971 − 0.0764582i
\(51\) −7.15592 + 49.7705i −0.0196476 + 0.136652i
\(52\) −73.6452 + 47.3289i −0.196399 + 0.126218i
\(53\) −212.097 − 464.428i −0.549693 − 1.20366i −0.956925 − 0.290336i \(-0.906233\pi\)
0.407231 − 0.913325i \(-0.366494\pi\)
\(54\) 49.3273 + 56.9267i 0.124307 + 0.143458i
\(55\) 20.5755 + 143.106i 0.0504437 + 0.350843i
\(56\) 9.24808 − 20.2505i 0.0220683 − 0.0483229i
\(57\) 66.4747 − 19.5187i 0.154470 − 0.0453565i
\(58\) 21.5637 − 6.33168i 0.0488182 − 0.0143343i
\(59\) −43.5998 + 95.4704i −0.0962071 + 0.210664i −0.951616 − 0.307288i \(-0.900578\pi\)
0.855409 + 0.517953i \(0.173306\pi\)
\(60\) 2.00354 + 13.9349i 0.00431093 + 0.0299832i
\(61\) 99.5194 + 114.851i 0.208888 + 0.241069i 0.850519 − 0.525944i \(-0.176288\pi\)
−0.641632 + 0.767013i \(0.721742\pi\)
\(62\) −27.3544 − 59.8979i −0.0560326 − 0.122694i
\(63\) 62.0478 − 39.8757i 0.124084 − 0.0797439i
\(64\) −9.10815 + 63.3486i −0.0177894 + 0.123728i
\(65\) −92.0565 − 59.1611i −0.175665 − 0.112893i
\(66\) 26.6580 − 30.7650i 0.0497178 − 0.0573774i
\(67\) −788.734 − 231.593i −1.43820 − 0.422293i −0.532576 − 0.846382i \(-0.678776\pi\)
−0.905620 + 0.424089i \(0.860594\pi\)
\(68\) −285.731 −0.509558
\(69\) 77.6284 + 1.57288i 0.135440 + 0.00274424i
\(70\) 27.8278 0.0475152
\(71\) −544.680 − 159.933i −0.910446 − 0.267331i −0.207217 − 0.978295i \(-0.566441\pi\)
−0.703229 + 0.710964i \(0.748259\pi\)
\(72\) −138.854 + 160.246i −0.227279 + 0.262294i
\(73\) 558.848 + 359.150i 0.896003 + 0.575826i 0.905602 − 0.424128i \(-0.139419\pi\)
−0.00959942 + 0.999954i \(0.503056\pi\)
\(74\) −2.21966 + 15.4381i −0.00348689 + 0.0242519i
\(75\) −14.8042 + 9.51408i −0.0227926 + 0.0146479i
\(76\) 163.546 + 358.115i 0.246842 + 0.540508i
\(77\) −52.6937 − 60.8118i −0.0779871 − 0.0900019i
\(78\) 4.38486 + 30.4974i 0.00636523 + 0.0442711i
\(79\) 519.770 − 1138.14i 0.740236 − 1.62089i −0.0429293 − 0.999078i \(-0.513669\pi\)
0.783166 − 0.621813i \(-0.213604\pi\)
\(80\) −76.7594 + 22.5386i −0.107275 + 0.0314987i
\(81\) −661.197 + 194.145i −0.906992 + 0.266317i
\(82\) 73.7837 − 161.564i 0.0993664 − 0.217582i
\(83\) −105.476 − 733.599i −0.139487 − 0.970155i −0.932557 − 0.361024i \(-0.882427\pi\)
0.793069 − 0.609131i \(-0.208482\pi\)
\(84\) −5.13105 − 5.92155i −0.00666481 − 0.00769160i
\(85\) −148.371 − 324.887i −0.189331 − 0.414576i
\(86\) −404.204 + 259.766i −0.506818 + 0.325712i
\(87\) 1.12569 − 7.82937i 0.00138721 − 0.00964823i
\(88\) 194.602 + 125.063i 0.235735 + 0.151497i
\(89\) 317.048 − 365.893i 0.377607 − 0.435781i −0.534855 − 0.844944i \(-0.679634\pi\)
0.912461 + 0.409163i \(0.134179\pi\)
\(90\) −254.309 − 74.6718i −0.297850 − 0.0874567i
\(91\) 60.9028 0.0701576
\(92\) 53.9318 + 437.908i 0.0611172 + 0.496251i
\(93\) −23.1757 −0.0258410
\(94\) 762.025 + 223.751i 0.836136 + 0.245512i
\(95\) −322.267 + 371.916i −0.348041 + 0.401660i
\(96\) 18.9494 + 12.1780i 0.0201460 + 0.0129470i
\(97\) 120.229 − 836.214i 0.125850 − 0.875306i −0.824885 − 0.565301i \(-0.808760\pi\)
0.950735 − 0.310005i \(-0.100331\pi\)
\(98\) 564.071 − 362.506i 0.581426 − 0.373660i
\(99\) 318.370 + 697.134i 0.323206 + 0.707723i
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.41.3 ✓ 60
23.9 even 11 inner 230.4.g.b.101.3 yes 60
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.4.g.b.41.3 ✓ 60 1.1 even 1 trivial
230.4.g.b.101.3 yes 60 23.9 even 11 inner