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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 183.14
Character \(\chi\) \(=\) 230.183
Dual form 230.4.e.a.137.14

$q$-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} +(77.8532 + 78.1401i) 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} +(312.561 - 311.413i) 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 \(\left(\mathbb{Z}/230\mathbb{Z}\right)^\times\).

\(n\) \(47\) \(51\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(-1\)

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 0.947369 − 0.320145i \(-0.103732\pi\)
−0.320145 + 0.947369i \(0.603732\pi\)
\(4\) − 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 −0.968866 − 0.247587i \(-0.920362\pi\)
−0.247587 − 0.968866i \(-0.579638\pi\)
\(8\) 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 0.182648 − 0.983178i \(-0.441533\pi\)
−0.983178 + 0.182648i \(0.941533\pi\)
\(14\) 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 −0.846630 − 0.532182i \(-0.821372\pi\)
−0.532182 − 0.846630i \(-0.678628\pi\)
\(18\) 8.14001 + 8.14001i 0.106590 + 0.106590i
\(19\) −51.6811 −0.624024 −0.312012 − 0.950078i \(-0.601003\pi\)
−0.312012 + 0.950078i \(0.601003\pi\)
\(20\) 32.7174 − 30.4889i 0.365791 − 0.340877i
\(21\) − 146.851i − 1.52598i
\(22\) 29.0318 + 29.0318i 0.281345 + 0.281345i
\(23\) 77.8532 + 78.1401i 0.705805 + 0.708406i
\(24\) 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 \(0.227187\pi\)
−0.755925 + 0.654658i \(0.772813\pi\)
\(30\) −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 \(-0.337373\pi\)
−0.872302 + 0.488968i \(0.837373\pi\)
\(38\) 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 \(-0.514612\pi\)
0.998947 + 0.0458884i \(0.0146119\pi\)
\(44\) −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 −0.344567 − 0.938762i \(-0.611974\pi\)
0.938762 + 0.344567i \(0.111974\pi\)
\(48\) −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 \(-0.594102\pi\)
0.956619 + 0.291341i \(0.0941016\pi\)
\(54\) 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 \(-0.992862\pi\)
0.999749 − 0.0224223i \(-0.00713785\pi\)
\(60\) 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 \(-0.283011\pi\)
−0.776509 + 0.630107i \(0.783011\pi\)
\(68\) −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 0.567548 − 0.823341i \(-0.307892\pi\)
−0.823341 + 0.567548i \(0.807892\pi\)
\(74\) 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 \(-0.507202\pi\)
−0.0226241 + 0.999744i \(0.507202\pi\)
\(80\) −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 \(-0.878885\pi\)
0.371380 + 0.928481i \(0.378885\pi\)
\(84\) −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 \(-0.861154\pi\)
−0.906364 + 0.422498i \(0.861154\pi\)
\(90\) −4.53487 + 128.625i −0.00531130 + 0.150647i
\(91\) 1690.70i 1.94762i
\(92\) 312.561 − 311.413i 0.354203 − 0.352902i
\(93\) −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 \(-0.103133\pi\)
−0.318363 + 0.947969i \(0.603133\pi\)
\(98\) −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.183.14 yes 72
5.2 odd 4 inner 230.4.e.a.137.13 ✓ 72
23.22 odd 2 inner 230.4.e.a.183.13 yes 72
115.22 even 4 inner 230.4.e.a.137.14 yes 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.4.e.a.137.13 ✓ 72 5.2 odd 4 inner
230.4.e.a.137.14 yes 72 115.22 even 4 inner
230.4.e.a.183.13 yes 72 23.22 odd 2 inner
230.4.e.a.183.14 yes 72 1.1 even 1 trivial