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 71.6
Character \(\chi\) \(=\) 230.71
Dual form 230.4.g.b.81.6

$q$-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} +(1.10986 + 7.71925i) q^{3} +(1.66166 + 3.63853i) q^{4} +(-4.79746 - 1.40866i) q^{5} +(6.47933 - 14.1878i) q^{6} +(-5.92245 + 6.83487i) q^{7} +(1.13852 - 7.91857i) q^{8} +(-32.4487 + 9.52780i) q^{9} +(6.54861 + 7.55750i) q^{10} +(-5.80260 + 3.72910i) q^{11} +(-26.2425 + 16.8650i) q^{12} +(-30.2489 - 34.9091i) q^{13} +(17.3550 - 5.09588i) q^{14} +(5.54930 - 38.5962i) q^{15} +(-10.4778 + 12.0920i) q^{16} +(-42.3262 + 92.6815i) q^{17} +(64.8974 + 19.0556i) q^{18} +(1.77814 + 3.89358i) q^{19} +(-2.84630 - 19.7964i) q^{20} +(-59.3331 - 38.1311i) q^{21} +13.7951 q^{22} +(-23.3297 - 107.809i) q^{23} +62.3890 q^{24} +(21.0313 + 13.5160i) q^{25} +(13.1474 + 91.4425i) q^{26} +(-22.0899 - 48.3702i) q^{27} +(-34.7100 - 10.1918i) q^{28} +(98.3647 - 215.389i) q^{29} +(-51.0701 + 58.9381i) q^{30} +(29.3484 - 204.123i) q^{31} +(30.7038 - 9.01544i) q^{32} +(-35.2259 - 40.6529i) q^{33} +(171.429 - 110.171i) q^{34} +(38.0408 - 24.4473i) q^{35} +(-88.5858 - 102.234i) q^{36} +(50.5163 - 14.8329i) q^{37} +(1.21833 - 8.47364i) q^{38} +(235.900 - 272.243i) q^{39} +(-16.6166 + 36.3853i) q^{40} +(-238.884 - 70.1427i) q^{41} +(58.5980 + 128.312i) q^{42} +(-31.9283 - 222.066i) q^{43} +(-23.2104 - 14.9164i) q^{44} +169.093 q^{45} +(-77.3193 + 206.615i) q^{46} -398.406 q^{47} +(-104.970 - 67.4601i) q^{48} +(37.1739 + 258.550i) q^{49} +(-20.7708 - 45.4816i) q^{50} +(-762.407 - 223.863i) q^{51} +(76.7544 - 168.069i) q^{52} +(-369.144 + 426.015i) q^{53} +(-15.1353 + 105.269i) q^{54} +(33.0908 - 9.71634i) q^{55} +(47.3796 + 54.6789i) q^{56} +(-28.0820 + 18.0472i) q^{57} +(-398.395 + 256.033i) q^{58} +(-49.7319 - 57.3937i) q^{59} +(149.655 - 43.9425i) q^{60} +(88.8770 - 618.153i) q^{61} +(-270.093 + 311.704i) q^{62} +(127.054 - 278.210i) q^{63} +(-61.4076 - 18.0309i) q^{64} +(95.9430 + 210.086i) q^{65} +(15.3107 + 106.488i) q^{66} +(220.731 + 141.855i) q^{67} -407.556 q^{68} +(806.310 - 299.740i) q^{69} -90.4383 q^{70} +(-442.067 - 284.100i) q^{71} +(38.5031 + 267.795i) q^{72} +(280.473 + 614.151i) q^{73} +(-101.033 - 29.6659i) q^{74} +(-80.9917 + 177.347i) q^{75} +(-11.2122 + 12.9396i) q^{76} +(8.87765 - 61.7454i) q^{77} +(-691.276 + 202.977i) q^{78} +(-233.296 - 269.238i) q^{79} +(67.3003 - 43.2513i) q^{80} +(-419.285 + 269.458i) q^{81} +(326.080 + 376.316i) q^{82} +(-370.476 + 108.782i) q^{83} +(40.1495 - 279.246i) q^{84} +(333.615 - 385.013i) q^{85} +(-186.396 + 408.151i) q^{86} +(1771.81 + 520.250i) q^{87} +(22.9228 + 50.1939i) q^{88} +(-6.22807 - 43.3172i) q^{89} +(-284.500 - 182.837i) q^{90} +417.747 q^{91} +(353.499 - 264.027i) q^{92} +1608.25 q^{93} +(670.320 + 430.789i) q^{94} +(-3.04581 - 21.1841i) q^{95} +(103.669 + 227.004i) q^{96} +(643.790 + 189.034i) q^{97} +(217.020 - 475.209i) q^{98} +(152.757 - 176.291i) 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 \(\left(\mathbb{Z}/230\mathbb{Z}\right)^\times\).

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

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
\(3\) 1.10986 + 7.71925i 0.213593 + 1.48557i 0.761026 + 0.648721i \(0.224696\pi\)
−0.547434 + 0.836849i \(0.684395\pi\)
\(4\) 1.66166 + 3.63853i 0.207708 + 0.454816i
\(5\) −4.79746 − 1.40866i −0.429098 − 0.125995i
\(6\) 6.47933 − 14.1878i 0.440863 − 0.965355i
\(7\) −5.92245 + 6.83487i −0.319782 + 0.369048i −0.892768 − 0.450517i \(-0.851240\pi\)
0.572986 + 0.819565i \(0.305785\pi\)
\(8\) 1.13852 − 7.91857i 0.0503159 − 0.349955i
\(9\) −32.4487 + 9.52780i −1.20180 + 0.352881i
\(10\) 6.54861 + 7.55750i 0.207085 + 0.238989i
\(11\) −5.80260 + 3.72910i −0.159050 + 0.102215i −0.617745 − 0.786379i \(-0.711953\pi\)
0.458695 + 0.888594i \(0.348317\pi\)
\(12\) −26.2425 + 16.8650i −0.631296 + 0.405709i
\(13\) −30.2489 − 34.9091i −0.645350 − 0.744773i 0.334961 − 0.942232i \(-0.391277\pi\)
−0.980311 + 0.197458i \(0.936731\pi\)
\(14\) 17.3550 − 5.09588i 0.331308 − 0.0972808i
\(15\) 5.54930 − 38.5962i 0.0955216 − 0.664367i
\(16\) −10.4778 + 12.0920i −0.163715 + 0.188937i
\(17\) −42.3262 + 92.6815i −0.603860 + 1.32227i 0.322836 + 0.946455i \(0.395364\pi\)
−0.926696 + 0.375813i \(0.877363\pi\)
\(18\) 64.8974 + 19.0556i 0.849803 + 0.249525i
\(19\) 1.77814 + 3.89358i 0.0214701 + 0.0470131i 0.920063 − 0.391770i \(-0.128137\pi\)
−0.898593 + 0.438783i \(0.855410\pi\)
\(20\) −2.84630 − 19.7964i −0.0318226 − 0.221331i
\(21\) −59.3331 − 38.1311i −0.616550 − 0.396233i
\(22\) 13.7951 0.133688
\(23\) −23.3297 − 107.809i −0.211503 − 0.977377i
\(24\) 62.3890 0.530629
\(25\) 21.0313 + 13.5160i 0.168251 + 0.108128i
\(26\) 13.1474 + 91.4425i 0.0991703 + 0.689744i
\(27\) −22.0899 − 48.3702i −0.157452 − 0.344772i
\(28\) −34.7100 − 10.1918i −0.234270 − 0.0687879i
\(29\) 98.3647 − 215.389i 0.629858 − 1.37920i −0.278270 − 0.960503i \(-0.589761\pi\)
0.908128 − 0.418693i \(-0.137512\pi\)
\(30\) −51.0701 + 58.9381i −0.310803 + 0.358686i
\(31\) 29.3484 − 204.123i 0.170036 − 1.18263i −0.708766 − 0.705444i \(-0.750748\pi\)
0.878802 − 0.477186i \(-0.158343\pi\)
\(32\) 30.7038 − 9.01544i 0.169616 − 0.0498038i
\(33\) −35.2259 − 40.6529i −0.185820 − 0.214447i
\(34\) 171.429 − 110.171i 0.864700 − 0.555709i
\(35\) 38.0408 − 24.4473i 0.183716 − 0.118067i
\(36\) −88.5858 − 102.234i −0.410120 − 0.473303i
\(37\) 50.5163 − 14.8329i 0.224455 − 0.0659059i −0.167571 − 0.985860i \(-0.553592\pi\)
0.392026 + 0.919954i \(0.371774\pi\)
\(38\) 1.21833 − 8.47364i 0.00520101 − 0.0361738i
\(39\) 235.900 − 272.243i 0.968571 − 1.11779i
\(40\) −16.6166 + 36.3853i −0.0656829 + 0.143825i
\(41\) −238.884 − 70.1427i −0.909937 − 0.267182i −0.206923 − 0.978357i \(-0.566345\pi\)
−0.703014 + 0.711176i \(0.748163\pi\)
\(42\) 58.5980 + 128.312i 0.215282 + 0.471403i
\(43\) −31.9283 − 222.066i −0.113233 − 0.787552i −0.964739 − 0.263208i \(-0.915220\pi\)
0.851506 − 0.524344i \(-0.175690\pi\)
\(44\) −23.2104 − 14.9164i −0.0795250 − 0.0511076i
\(45\) 169.093 0.560153
\(46\) −77.3193 + 206.615i −0.247828 + 0.662255i
\(47\) −398.406 −1.23646 −0.618228 − 0.785999i \(-0.712149\pi\)
−0.618228 + 0.785999i \(0.712149\pi\)
\(48\) −104.970 − 67.4601i −0.315648 − 0.202855i
\(49\) 37.1739 + 258.550i 0.108379 + 0.753791i
\(50\) −20.7708 − 45.4816i −0.0587486 − 0.128641i
\(51\) −762.407 − 223.863i −2.09330 − 0.614649i
\(52\) 76.7544 − 168.069i 0.204691 − 0.448210i
\(53\) −369.144 + 426.015i −0.956715 + 1.10411i 0.0377762 + 0.999286i \(0.487973\pi\)
−0.994491 + 0.104822i \(0.966573\pi\)
\(54\) −15.1353 + 105.269i −0.0381419 + 0.265282i
\(55\) 33.0908 − 9.71634i 0.0811266 − 0.0238209i
\(56\) 47.3796 + 54.6789i 0.113060 + 0.130478i
\(57\) −28.0820 + 18.0472i −0.0652553 + 0.0419371i
\(58\) −398.395 + 256.033i −0.901929 + 0.579634i
\(59\) −49.7319 − 57.3937i −0.109738 − 0.126644i 0.698223 − 0.715880i \(-0.253974\pi\)
−0.807961 + 0.589236i \(0.799429\pi\)
\(60\) 149.655 − 43.9425i 0.322005 − 0.0945493i
\(61\) 88.8770 − 618.153i 0.186550 − 1.29748i −0.654309 − 0.756227i \(-0.727041\pi\)
0.840859 − 0.541254i \(-0.182050\pi\)
\(62\) −270.093 + 311.704i −0.553256 + 0.638491i
\(63\) 127.054 − 278.210i 0.254085 − 0.556368i
\(64\) −61.4076 − 18.0309i −0.119937 − 0.0352166i
\(65\) 95.9430 + 210.086i 0.183081 + 0.400892i
\(66\) 15.3107 + 106.488i 0.0285547 + 0.198602i
\(67\) 220.731 + 141.855i 0.402487 + 0.258662i 0.726187 − 0.687497i \(-0.241291\pi\)
−0.323700 + 0.946160i \(0.604927\pi\)
\(68\) −407.556 −0.726815
\(69\) 806.310 − 299.740i 1.40679 − 0.522964i
\(70\) −90.4383 −0.154421
\(71\) −442.067 − 284.100i −0.738926 − 0.474879i 0.116248 − 0.993220i \(-0.462913\pi\)
−0.855174 + 0.518341i \(0.826550\pi\)
\(72\) 38.5031 + 267.795i 0.0630227 + 0.438332i
\(73\) 280.473 + 614.151i 0.449684 + 0.984670i 0.989719 + 0.143028i \(0.0456838\pi\)
−0.540035 + 0.841643i \(0.681589\pi\)
\(74\) −101.033 − 29.6659i −0.158714 − 0.0466025i
\(75\) −80.9917 + 177.347i −0.124695 + 0.273044i
\(76\) −11.2122 + 12.9396i −0.0169228 + 0.0195299i
\(77\) 8.87765 − 61.7454i 0.0131390 − 0.0913837i
\(78\) −691.276 + 202.977i −1.00348 + 0.294649i
\(79\) −233.296 − 269.238i −0.332252 − 0.383439i 0.564902 − 0.825158i \(-0.308914\pi\)
−0.897153 + 0.441719i \(0.854369\pi\)
\(80\) 67.3003 − 43.2513i 0.0940550 − 0.0604455i
\(81\) −419.285 + 269.458i −0.575151 + 0.369627i
\(82\) 326.080 + 376.316i 0.439140 + 0.506795i
\(83\) −370.476 + 108.782i −0.489940 + 0.143859i −0.517366 − 0.855765i \(-0.673087\pi\)
0.0274257 + 0.999624i \(0.491269\pi\)
\(84\) 40.1495 − 279.246i 0.0521509 − 0.362717i
\(85\) 333.615 − 385.013i 0.425714 − 0.491300i
\(86\) −186.396 + 408.151i −0.233717 + 0.511768i
\(87\) 1771.81 + 520.250i 2.18343 + 0.641112i
\(88\) 22.9228 + 50.1939i 0.0277679 + 0.0608033i
\(89\) −6.22807 − 43.3172i −0.00741769 − 0.0515912i 0.985776 − 0.168062i \(-0.0537509\pi\)
−0.993194 + 0.116471i \(0.962842\pi\)
\(90\) −284.500 − 182.837i −0.333210 − 0.214141i
\(91\) 417.747 0.481229
\(92\) 353.499 − 264.027i 0.400596 − 0.299204i
\(93\) 1608.25 1.79320
\(94\) 670.320 + 430.789i 0.735513 + 0.472686i
\(95\) −3.04581 − 21.1841i −0.00328941 − 0.0228784i
\(96\) 103.669 + 227.004i 0.110216 + 0.241339i
\(97\) 643.790 + 189.034i 0.673886 + 0.197871i 0.600733 − 0.799450i \(-0.294876\pi\)
0.0731534 + 0.997321i \(0.476694\pi\)
\(98\) 217.020 − 475.209i 0.223698 − 0.489830i
\(99\) 152.757 − 176.291i 0.155077 − 0.178968i
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.71.6 ✓ 60
23.12 even 11 inner 230.4.g.b.81.6 yes 60
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.4.g.b.71.6 ✓ 60 1.1 even 1 trivial
230.4.g.b.81.6 yes 60 23.12 even 11 inner