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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [230,4,Mod(137,230)] 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("230.137"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(230, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([1, 2])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 230 = 2 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 230.e (of order \(4\), degree \(2\), 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: \(13.5704393013\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(36\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 137.5
Character \(\chi\) \(=\) 230.137
Dual form 230.4.e.a.183.5

$q$-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 - 1.41421i) q^{2} +(-3.58924 + 3.58924i) q^{3} +4.00000i q^{4} +(-11.1634 - 0.614616i) q^{5} +10.1519 q^{6} +(16.2976 - 16.2976i) q^{7} +(5.65685 - 5.65685i) q^{8} +1.23476i q^{9} +(14.9183 + 16.6567i) q^{10} -34.5362i q^{11} +(-14.3569 - 14.3569i) q^{12} +(23.5981 - 23.5981i) q^{13} -46.0964 q^{14} +(42.2742 - 37.8622i) q^{15} -16.0000 q^{16} +(-55.6511 + 55.6511i) q^{17} +(1.74622 - 1.74622i) q^{18} +29.0874 q^{19} +(2.45846 - 44.6537i) q^{20} +116.992i q^{21} +(-48.8416 + 48.8416i) q^{22} +(15.5323 + 109.205i) q^{23} +40.6076i q^{24} +(124.244 + 13.7224i) q^{25} -66.7456 q^{26} +(-101.341 - 101.341i) q^{27} +(65.1902 + 65.1902i) q^{28} +293.672i q^{29} +(-113.330 - 6.23951i) q^{30} -254.453 q^{31} +(22.6274 + 22.6274i) q^{32} +(123.959 + 123.959i) q^{33} +157.405 q^{34} +(-191.953 + 171.920i) q^{35} -4.93905 q^{36} +(82.6349 - 82.6349i) q^{37} +(-41.1358 - 41.1358i) q^{38} +169.399i q^{39} +(-66.6267 + 59.6731i) q^{40} -292.497 q^{41} +(165.451 - 165.451i) q^{42} +(295.293 + 295.293i) q^{43} +138.145 q^{44} +(0.758904 - 13.7842i) q^{45} +(132.473 - 176.405i) q^{46} +(277.347 + 277.347i) q^{47} +(57.4278 - 57.4278i) q^{48} -188.220i q^{49} +(-156.302 - 195.115i) q^{50} -399.490i q^{51} +(94.3925 + 94.3925i) q^{52} +(185.792 + 185.792i) q^{53} +286.636i q^{54} +(-21.2265 + 385.543i) q^{55} -184.386i q^{56} +(-104.402 + 104.402i) q^{57} +(415.314 - 415.314i) q^{58} +26.0941i q^{59} +(151.449 + 169.097i) q^{60} +191.833i q^{61} +(359.851 + 359.851i) q^{62} +(20.1236 + 20.1236i) q^{63} -64.0000i q^{64} +(-277.940 + 248.932i) q^{65} -350.608i q^{66} +(94.6227 - 94.6227i) q^{67} +(-222.604 - 222.604i) q^{68} +(-447.712 - 336.214i) q^{69} +(514.595 + 28.3316i) q^{70} +17.9990 q^{71} +(6.98487 + 6.98487i) q^{72} +(-272.645 + 272.645i) q^{73} -233.727 q^{74} +(-495.196 + 396.690i) q^{75} +116.350i q^{76} +(-562.856 - 562.856i) q^{77} +(239.566 - 239.566i) q^{78} -1034.70 q^{79} +(178.615 + 9.83385i) q^{80} +694.137 q^{81} +(413.653 + 413.653i) q^{82} +(819.532 + 819.532i) q^{83} -467.966 q^{84} +(655.461 - 587.053i) q^{85} -835.215i q^{86} +(-1054.06 - 1054.06i) q^{87} +(-195.366 - 195.366i) q^{88} +394.478 q^{89} +(-20.5670 + 18.4205i) q^{90} -769.184i q^{91} +(-436.820 + 62.1294i) q^{92} +(913.292 - 913.292i) q^{93} -784.455i q^{94} +(-324.716 - 17.8776i) q^{95} -162.430 q^{96} +(587.067 - 587.067i) q^{97} +(-266.184 + 266.184i) q^{98} +42.6441 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 72 q - 16 q^{3} - 16 q^{6} - 64 q^{12} + 192 q^{13} - 1152 q^{16} + 32 q^{18} + 276 q^{23} + 880 q^{25} + 304 q^{26} + 728 q^{27} + 608 q^{31} + 688 q^{35} + 2816 q^{36} - 2208 q^{41} - 256 q^{46} + 144 q^{47}+ \cdots - 448 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(47\) \(51\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(-1\)

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 − 1.41421i −0.500000 − 0.500000i
\(3\) −3.58924 + 3.58924i −0.690749 + 0.690749i −0.962397 − 0.271648i \(-0.912431\pi\)
0.271648 + 0.962397i \(0.412431\pi\)
\(4\) 4.00000i 0.500000i
\(5\) −11.1634 − 0.614616i −0.998488 − 0.0549729i
\(6\) 10.1519 0.690749
\(7\) 16.2976 − 16.2976i 0.879985 − 0.879985i −0.113547 − 0.993533i \(-0.536221\pi\)
0.993533 + 0.113547i \(0.0362214\pi\)
\(8\) 5.65685 − 5.65685i 0.250000 − 0.250000i
\(9\) 1.23476i 0.0457320i
\(10\) 14.9183 + 16.6567i 0.471757 + 0.526730i
\(11\) − 34.5362i − 0.946642i −0.880890 − 0.473321i \(-0.843055\pi\)
0.880890 − 0.473321i \(-0.156945\pi\)
\(12\) −14.3569 − 14.3569i −0.345374 − 0.345374i
\(13\) 23.5981 − 23.5981i 0.503457 − 0.503457i −0.409053 − 0.912511i \(-0.634141\pi\)
0.912511 + 0.409053i \(0.134141\pi\)
\(14\) −46.0964 −0.879985
\(15\) 42.2742 − 37.8622i 0.727677 − 0.651732i
\(16\) −16.0000 −0.250000
\(17\) −55.6511 + 55.6511i −0.793963 + 0.793963i −0.982136 − 0.188173i \(-0.939744\pi\)
0.188173 + 0.982136i \(0.439744\pi\)
\(18\) 1.74622 − 1.74622i 0.0228660 − 0.0228660i
\(19\) 29.0874 0.351217 0.175608 − 0.984460i \(-0.443811\pi\)
0.175608 + 0.984460i \(0.443811\pi\)
\(20\) 2.45846 − 44.6537i 0.0274864 − 0.499244i
\(21\) 116.992i 1.21570i
\(22\) −48.8416 + 48.8416i −0.473321 + 0.473321i
\(23\) 15.5323 + 109.205i 0.140814 + 0.990036i
\(24\) 40.6076i 0.345374i
\(25\) 124.244 + 13.7224i 0.993956 + 0.109780i
\(26\) −66.7456 −0.503457
\(27\) −101.341 − 101.341i −0.722338 − 0.722338i
\(28\) 65.1902 + 65.1902i 0.439993 + 0.439993i
\(29\) 293.672i 1.88046i 0.340535 + 0.940232i \(0.389392\pi\)
−0.340535 + 0.940232i \(0.610608\pi\)
\(30\) −113.330 − 6.23951i −0.689704 − 0.0379725i
\(31\) −254.453 −1.47423 −0.737115 − 0.675767i \(-0.763812\pi\)
−0.737115 + 0.675767i \(0.763812\pi\)
\(32\) 22.6274 + 22.6274i 0.125000 + 0.125000i
\(33\) 123.959 + 123.959i 0.653892 + 0.653892i
\(34\) 157.405 0.793963
\(35\) −191.953 + 171.920i −0.927030 + 0.830279i
\(36\) −4.93905 −0.0228660
\(37\) 82.6349 − 82.6349i 0.367165 − 0.367165i −0.499277 − 0.866442i \(-0.666401\pi\)
0.866442 + 0.499277i \(0.166401\pi\)
\(38\) −41.1358 − 41.1358i −0.175608 − 0.175608i
\(39\) 169.399i 0.695525i
\(40\) −66.6267 + 59.6731i −0.263365 + 0.235879i
\(41\) −292.497 −1.11415 −0.557077 − 0.830461i \(-0.688077\pi\)
−0.557077 + 0.830461i \(0.688077\pi\)
\(42\) 165.451 − 165.451i 0.607849 − 0.607849i
\(43\) 295.293 + 295.293i 1.04725 + 1.04725i 0.998827 + 0.0484242i \(0.0154199\pi\)
0.0484242 + 0.998827i \(0.484580\pi\)
\(44\) 138.145 0.473321
\(45\) 0.758904 − 13.7842i 0.00251402 − 0.0456628i
\(46\) 132.473 − 176.405i 0.424611 − 0.565425i
\(47\) 277.347 + 277.347i 0.860748 + 0.860748i 0.991425 − 0.130677i \(-0.0417150\pi\)
−0.130677 + 0.991425i \(0.541715\pi\)
\(48\) 57.4278 − 57.4278i 0.172687 − 0.172687i
\(49\) − 188.220i − 0.548747i
\(50\) −156.302 − 195.115i −0.442088 − 0.551868i
\(51\) − 399.490i − 1.09686i
\(52\) 94.3925 + 94.3925i 0.251729 + 0.251729i
\(53\) 185.792 + 185.792i 0.481519 + 0.481519i 0.905617 − 0.424097i \(-0.139409\pi\)
−0.424097 + 0.905617i \(0.639409\pi\)
\(54\) 286.636i 0.722338i
\(55\) −21.2265 + 385.543i −0.0520397 + 0.945211i
\(56\) − 184.386i − 0.439993i
\(57\) −104.402 + 104.402i −0.242602 + 0.242602i
\(58\) 415.314 − 415.314i 0.940232 − 0.940232i
\(59\) 26.0941i 0.0575789i 0.999585 + 0.0287895i \(0.00916524\pi\)
−0.999585 + 0.0287895i \(0.990835\pi\)
\(60\) 151.449 + 169.097i 0.325866 + 0.363838i
\(61\) 191.833i 0.402652i 0.979524 + 0.201326i \(0.0645250\pi\)
−0.979524 + 0.201326i \(0.935475\pi\)
\(62\) 359.851 + 359.851i 0.737115 + 0.737115i
\(63\) 20.1236 + 20.1236i 0.0402434 + 0.0402434i
\(64\) − 64.0000i − 0.125000i
\(65\) −277.940 + 248.932i −0.530372 + 0.475019i
\(66\) − 350.608i − 0.653892i
\(67\) 94.6227 − 94.6227i 0.172537 − 0.172537i −0.615556 − 0.788093i \(-0.711068\pi\)
0.788093 + 0.615556i \(0.211068\pi\)
\(68\) −222.604 − 222.604i −0.396982 − 0.396982i
\(69\) −447.712 − 336.214i −0.781133 − 0.586599i
\(70\) 514.595 + 28.3316i 0.878654 + 0.0483753i
\(71\) 17.9990 0.0300857 0.0150429 − 0.999887i \(-0.495212\pi\)
0.0150429 + 0.999887i \(0.495212\pi\)
\(72\) 6.98487 + 6.98487i 0.0114330 + 0.0114330i
\(73\) −272.645 + 272.645i −0.437132 + 0.437132i −0.891046 − 0.453914i \(-0.850027\pi\)
0.453914 + 0.891046i \(0.350027\pi\)
\(74\) −233.727 −0.367165
\(75\) −495.196 + 396.690i −0.762404 + 0.610744i
\(76\) 116.350i 0.175608i
\(77\) −562.856 − 562.856i −0.833031 − 0.833031i
\(78\) 239.566 − 239.566i 0.347763 − 0.347763i
\(79\) −1034.70 −1.47358 −0.736788 − 0.676123i \(-0.763659\pi\)
−0.736788 + 0.676123i \(0.763659\pi\)
\(80\) 178.615 + 9.83385i 0.249622 + 0.0137432i
\(81\) 694.137 0.952177
\(82\) 413.653 + 413.653i 0.557077 + 0.557077i
\(83\) 819.532 + 819.532i 1.08380 + 1.08380i 0.996152 + 0.0876480i \(0.0279351\pi\)
0.0876480 + 0.996152i \(0.472065\pi\)
\(84\) −467.966 −0.607849
\(85\) 655.461 − 587.053i 0.836409 − 0.749116i
\(86\) − 835.215i − 1.04725i
\(87\) −1054.06 − 1054.06i −1.29893 − 1.29893i
\(88\) −195.366 − 195.366i −0.236661 − 0.236661i
\(89\) 394.478 0.469827 0.234914 − 0.972016i \(-0.424519\pi\)
0.234914 + 0.972016i \(0.424519\pi\)
\(90\) −20.5670 + 18.4205i −0.0240884 + 0.0215744i
\(91\) − 769.184i − 0.886070i
\(92\) −436.820 + 62.1294i −0.495018 + 0.0704069i
\(93\) 913.292 − 913.292i 1.01832 − 1.01832i
\(94\) − 784.455i − 0.860748i
\(95\) −324.716 − 17.8776i −0.350685 − 0.0193074i
\(96\) −162.430 −0.172687
\(97\) 587.067 − 587.067i 0.614512 − 0.614512i −0.329606 − 0.944118i \(-0.606916\pi\)
0.944118 + 0.329606i \(0.106916\pi\)
\(98\) −266.184 + 266.184i −0.274374 + 0.274374i
\(99\) 42.6441 0.0432918
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.e.a.137.5 ✓ 72
5.3 odd 4 inner 230.4.e.a.183.6 yes 72
23.22 odd 2 inner 230.4.e.a.137.6 yes 72
115.68 even 4 inner 230.4.e.a.183.5 yes 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.4.e.a.137.5 ✓ 72 1.1 even 1 trivial
230.4.e.a.137.6 yes 72 23.22 odd 2 inner
230.4.e.a.183.5 yes 72 115.68 even 4 inner
230.4.e.a.183.6 yes 72 5.3 odd 4 inner