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(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 137.13
Character \(\chi\) \(=\) 230.137
Dual form 230.4.e.a.183.13

$q$-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} +(3.25915 - 3.25915i) q^{3} +4.00000i q^{4} +(-7.62223 + 8.17934i) q^{5} -9.21827 q^{6} +(22.5290 - 22.5290i) q^{7} +(5.65685 - 5.65685i) q^{8} +5.75586i q^{9} +(22.3468 - 0.787870i) q^{10} -20.5286i q^{11} +(13.0366 + 13.0366i) q^{12} +(-37.5226 + 37.5226i) q^{13} -63.7217 q^{14} +(1.81570 + 51.4997i) q^{15} -16.0000 q^{16} +(96.6448 - 96.6448i) q^{17} +(8.14001 - 8.14001i) q^{18} +51.6811 q^{19} +(-32.7174 - 30.4889i) q^{20} -146.851i q^{21} +(-29.0318 + 29.0318i) q^{22} +(78.1401 - 77.8532i) q^{23} -36.8731i q^{24} +(-8.80319 - 124.690i) q^{25} +106.130 q^{26} +(106.756 + 106.756i) q^{27} +(90.1161 + 90.1161i) q^{28} -204.476i q^{29} +(70.2638 - 75.3994i) q^{30} -328.130 q^{31} +(22.6274 + 22.6274i) q^{32} +(-66.9057 - 66.9057i) q^{33} -273.353 q^{34} +(12.5511 + 355.994i) q^{35} -23.0234 q^{36} +(86.2739 - 86.2739i) q^{37} +(-73.0881 - 73.0881i) q^{38} +244.584i q^{39} +(3.15148 + 89.3872i) q^{40} +97.1639 q^{41} +(-207.679 + 207.679i) q^{42} +(-268.734 - 268.734i) q^{43} +82.1143 q^{44} +(-47.0791 - 43.8725i) q^{45} +(-220.608 - 0.405832i) q^{46} +(191.459 + 191.459i) q^{47} +(-52.1464 + 52.1464i) q^{48} -672.114i q^{49} +(-163.888 + 188.787i) q^{50} -629.960i q^{51} +(-150.090 - 150.090i) q^{52} +(-256.695 - 256.695i) q^{53} -301.952i q^{54} +(167.910 + 156.473i) q^{55} -254.887i q^{56} +(168.437 - 168.437i) q^{57} +(-289.172 + 289.172i) q^{58} +20.3230i q^{59} +(-205.999 + 7.26280i) q^{60} +350.137i q^{61} +(464.045 + 464.045i) q^{62} +(129.674 + 129.674i) q^{63} -64.0000i q^{64} +(-20.9042 - 592.916i) q^{65} +189.238i q^{66} +(80.2897 - 80.2897i) q^{67} +(386.579 + 386.579i) q^{68} +(0.935268 - 508.406i) q^{69} +(485.701 - 521.201i) q^{70} +711.743 q^{71} +(32.5601 + 32.5601i) q^{72} +(-159.541 + 159.541i) q^{73} -244.020 q^{74} +(-435.073 - 377.692i) q^{75} +206.724i q^{76} +(-462.489 - 462.489i) q^{77} +(345.894 - 345.894i) q^{78} +31.7719 q^{79} +(121.956 - 130.869i) q^{80} +540.462 q^{81} +(-137.410 - 137.410i) q^{82} +(421.261 + 421.261i) q^{83} +587.404 q^{84} +(53.8416 + 1527.14i) q^{85} +760.094i q^{86} +(-666.417 - 666.417i) q^{87} +(-116.127 - 116.127i) q^{88} +1522.01 q^{89} +(4.53487 + 128.625i) q^{90} +1690.70i q^{91} +(311.413 + 312.561i) q^{92} +(-1069.42 + 1069.42i) q^{93} -541.527i q^{94} +(-393.925 + 422.717i) q^{95} +147.492 q^{96} +(-601.487 + 601.487i) q^{97} +(-950.512 + 950.512i) q^{98} +118.160 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
\(3\) 3.25915 − 3.25915i 0.627224 − 0.627224i −0.320145 − 0.947369i \(-0.603732\pi\)
0.947369 + 0.320145i \(0.103732\pi\)
\(4\) 4.00000i 0.500000i
\(5\) −7.62223 + 8.17934i −0.681753 + 0.731582i
\(6\) −9.21827 −0.627224
\(7\) 22.5290 − 22.5290i 1.21645 − 1.21645i 0.247587 − 0.968866i \(-0.420362\pi\)
0.968866 − 0.247587i \(-0.0796377\pi\)
\(8\) 5.65685 − 5.65685i 0.250000 − 0.250000i
\(9\) 5.75586i 0.213180i
\(10\) 22.3468 − 0.787870i 0.706668 − 0.0249146i
\(11\) − 20.5286i − 0.562691i −0.959607 − 0.281345i \(-0.909219\pi\)
0.959607 − 0.281345i \(-0.0907806\pi\)
\(12\) 13.0366 + 13.0366i 0.313612 + 0.313612i
\(13\) −37.5226 + 37.5226i −0.800531 + 0.800531i −0.983178 − 0.182648i \(-0.941533\pi\)
0.182648 + 0.983178i \(0.441533\pi\)
\(14\) −63.7217 −1.21645
\(15\) 1.81570 + 51.4997i 0.0312541 + 0.886478i
\(16\) −16.0000 −0.250000
\(17\) 96.6448 − 96.6448i 1.37881 − 1.37881i 0.532182 − 0.846630i \(-0.321372\pi\)
0.846630 − 0.532182i \(-0.178628\pi\)
\(18\) 8.14001 − 8.14001i 0.106590 − 0.106590i
\(19\) 51.6811 0.624024 0.312012 − 0.950078i \(-0.398997\pi\)
0.312012 + 0.950078i \(0.398997\pi\)
\(20\) −32.7174 − 30.4889i −0.365791 − 0.340877i
\(21\) − 146.851i − 1.52598i
\(22\) −29.0318 + 29.0318i −0.281345 + 0.281345i
\(23\) 78.1401 − 77.8532i 0.708406 − 0.705805i
\(24\) − 36.8731i − 0.313612i
\(25\) −8.80319 − 124.690i −0.0704255 − 0.997517i
\(26\) 106.130 0.800531
\(27\) 106.756 + 106.756i 0.760936 + 0.760936i
\(28\) 90.1161 + 90.1161i 0.608226 + 0.608226i
\(29\) − 204.476i − 1.30932i −0.755925 − 0.654658i \(-0.772813\pi\)
0.755925 − 0.654658i \(-0.227187\pi\)
\(30\) 70.2638 − 75.3994i 0.427612 − 0.458866i
\(31\) −328.130 −1.90109 −0.950546 − 0.310585i \(-0.899475\pi\)
−0.950546 + 0.310585i \(0.899475\pi\)
\(32\) 22.6274 + 22.6274i 0.125000 + 0.125000i
\(33\) −66.9057 − 66.9057i −0.352933 − 0.352933i
\(34\) −273.353 −1.37881
\(35\) 12.5511 + 355.994i 0.0606150 + 1.71926i
\(36\) −23.0234 −0.106590
\(37\) 86.2739 − 86.2739i 0.383334 − 0.383334i −0.488968 − 0.872302i \(-0.662627\pi\)
0.872302 + 0.488968i \(0.162627\pi\)
\(38\) −73.0881 − 73.0881i −0.312012 − 0.312012i
\(39\) 244.584i 1.00422i
\(40\) 3.15148 + 89.3872i 0.0124573 + 0.353334i
\(41\) 97.1639 0.370108 0.185054 − 0.982728i \(-0.440754\pi\)
0.185054 + 0.982728i \(0.440754\pi\)
\(42\) −207.679 + 207.679i −0.762988 + 0.762988i
\(43\) −268.734 − 268.734i −0.953058 − 0.953058i 0.0458884 − 0.998947i \(-0.485388\pi\)
−0.998947 + 0.0458884i \(0.985388\pi\)
\(44\) 82.1143 0.281345
\(45\) −47.0791 − 43.8725i −0.155959 − 0.145336i
\(46\) −220.608 − 0.405832i −0.707106 − 0.00130080i
\(47\) 191.459 + 191.459i 0.594194 + 0.594194i 0.938762 − 0.344567i \(-0.111974\pi\)
−0.344567 + 0.938762i \(0.611974\pi\)
\(48\) −52.1464 + 52.1464i −0.156806 + 0.156806i
\(49\) − 672.114i − 1.95951i
\(50\) −163.888 + 188.787i −0.463546 + 0.533971i
\(51\) − 629.960i − 1.72965i
\(52\) −150.090 − 150.090i −0.400265 − 0.400265i
\(53\) −256.695 − 256.695i −0.665278 − 0.665278i 0.291341 − 0.956619i \(-0.405898\pi\)
−0.956619 + 0.291341i \(0.905898\pi\)
\(54\) − 301.952i − 0.760936i
\(55\) 167.910 + 156.473i 0.411654 + 0.383616i
\(56\) − 254.887i − 0.608226i
\(57\) 168.437 − 168.437i 0.391403 − 0.391403i
\(58\) −289.172 + 289.172i −0.654658 + 0.654658i
\(59\) 20.3230i 0.0448446i 0.999749 + 0.0224223i \(0.00713785\pi\)
−0.999749 + 0.0224223i \(0.992862\pi\)
\(60\) −205.999 + 7.26280i −0.443239 + 0.0156271i
\(61\) 350.137i 0.734926i 0.930038 + 0.367463i \(0.119774\pi\)
−0.930038 + 0.367463i \(0.880226\pi\)
\(62\) 464.045 + 464.045i 0.950546 + 0.950546i
\(63\) 129.674 + 129.674i 0.259323 + 0.259323i
\(64\) − 64.0000i − 0.125000i
\(65\) −20.9042 − 592.916i −0.0398899 − 1.13142i
\(66\) 189.238i 0.352933i
\(67\) 80.2897 − 80.2897i 0.146402 − 0.146402i −0.630107 − 0.776509i \(-0.716989\pi\)
0.776509 + 0.630107i \(0.216989\pi\)
\(68\) 386.579 + 386.579i 0.689406 + 0.689406i
\(69\) 0.935268 − 508.406i 0.00163178 − 0.887027i
\(70\) 485.701 − 521.201i 0.829320 − 0.889935i
\(71\) 711.743 1.18970 0.594848 − 0.803838i \(-0.297212\pi\)
0.594848 + 0.803838i \(0.297212\pi\)
\(72\) 32.5601 + 32.5601i 0.0532950 + 0.0532950i
\(73\) −159.541 + 159.541i −0.255793 + 0.255793i −0.823341 − 0.567548i \(-0.807892\pi\)
0.567548 + 0.823341i \(0.307892\pi\)
\(74\) −244.020 −0.383334
\(75\) −435.073 − 377.692i −0.669839 − 0.581494i
\(76\) 206.724i 0.312012i
\(77\) −462.489 − 462.489i −0.684486 − 0.684486i
\(78\) 345.894 − 345.894i 0.502112 − 0.502112i
\(79\) 31.7719 0.0452483 0.0226241 − 0.999744i \(-0.492798\pi\)
0.0226241 + 0.999744i \(0.492798\pi\)
\(80\) 121.956 − 130.869i 0.170438 − 0.182896i
\(81\) 540.462 0.741374
\(82\) −137.410 − 137.410i −0.185054 − 0.185054i
\(83\) 421.261 + 421.261i 0.557101 + 0.557101i 0.928481 − 0.371380i \(-0.121115\pi\)
−0.371380 + 0.928481i \(0.621115\pi\)
\(84\) 587.404 0.762988
\(85\) 53.8416 + 1527.14i 0.0687052 + 1.94872i
\(86\) 760.094i 0.953058i
\(87\) −666.417 − 666.417i −0.821235 − 0.821235i
\(88\) −116.127 − 116.127i −0.140673 − 0.140673i
\(89\) 1522.01 1.81273 0.906364 − 0.422498i \(-0.138846\pi\)
0.906364 + 0.422498i \(0.138846\pi\)
\(90\) 4.53487 + 128.625i 0.00531130 + 0.150647i
\(91\) 1690.70i 1.94762i
\(92\) 311.413 + 312.561i 0.352902 + 0.354203i
\(93\) −1069.42 + 1069.42i −1.19241 + 1.19241i
\(94\) − 541.527i − 0.594194i
\(95\) −393.925 + 422.717i −0.425430 + 0.456525i
\(96\) 147.492 0.156806
\(97\) −601.487 + 601.487i −0.629606 + 0.629606i −0.947969 − 0.318363i \(-0.896867\pi\)
0.318363 + 0.947969i \(0.396867\pi\)
\(98\) −950.512 + 950.512i −0.979757 + 0.979757i
\(99\) 118.160 0.119954
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.137.13 ✓ 72
5.3 odd 4 inner 230.4.e.a.183.14 yes 72
23.22 odd 2 inner 230.4.e.a.137.14 yes 72
115.68 even 4 inner 230.4.e.a.183.13 yes 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.4.e.a.137.13 ✓ 72 1.1 even 1 trivial
230.4.e.a.137.14 yes 72 23.22 odd 2 inner
230.4.e.a.183.13 yes 72 115.68 even 4 inner
230.4.e.a.183.14 yes 72 5.3 odd 4 inner