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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 137.14
Character \(\chi\) \(=\) 230.137
Dual form 230.4.e.a.183.14

$q$-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 \(\left(\mathbb{Z}/230\mathbb{Z}\right)^\times\).

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

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 \(0.579638\pi\)
−0.968866 + 0.247587i \(0.920362\pi\)
\(8\) 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 \(0.0907806\pi\)
−0.959607 + 0.281345i \(0.909219\pi\)
\(12\) 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 \(0.678628\pi\)
−0.846630 + 0.532182i \(0.821372\pi\)
\(18\) 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 \(-0.772813\pi\)
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 −0.872302 − 0.488968i \(-0.837373\pi\)
0.488968 + 0.872302i \(0.337373\pi\)
\(38\) 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 0.998947 − 0.0458884i \(-0.0146119\pi\)
−0.0458884 + 0.998947i \(0.514612\pi\)
\(44\) −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 0.956619 − 0.291341i \(-0.0941016\pi\)
−0.291341 + 0.956619i \(0.594102\pi\)
\(54\) − 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 \(-0.880226\pi\)
0.930038 − 0.367463i \(-0.119774\pi\)
\(62\) 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 −0.776509 − 0.630107i \(-0.783011\pi\)
0.630107 + 0.776509i \(0.283011\pi\)
\(68\) −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 \(-0.507202\pi\)
−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 0.371380 − 0.928481i \(-0.378885\pi\)
−0.928481 + 0.371380i \(0.878885\pi\)
\(84\) −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 −0.318363 − 0.947969i \(-0.603133\pi\)
0.947969 + 0.318363i \(0.103133\pi\)
\(98\) −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.137.14 yes 72
5.3 odd 4 inner 230.4.e.a.183.13 yes 72
23.22 odd 2 inner 230.4.e.a.137.13 ✓ 72
115.68 even 4 inner 230.4.e.a.183.14 yes 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.4.e.a.137.13 ✓ 72 23.22 odd 2 inner
230.4.e.a.137.14 yes 72 1.1 even 1 trivial
230.4.e.a.183.13 yes 72 5.3 odd 4 inner
230.4.e.a.183.14 yes 72 115.68 even 4 inner