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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 137.1
Character \(\chi\) \(=\) 230.137
Dual form 230.4.e.a.183.1

$q$-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} +(-7.12210 + 7.12210i) q^{3} +4.00000i q^{4} +(-1.63222 - 11.0606i) q^{5} +20.1443 q^{6} +(-16.1147 + 16.1147i) q^{7} +(5.65685 - 5.65685i) q^{8} -74.4485i q^{9} +(-13.3337 + 17.9503i) q^{10} -27.4497i q^{11} +(-28.4884 - 28.4884i) q^{12} +(38.2364 - 38.2364i) q^{13} +45.5793 q^{14} +(90.3991 + 67.1495i) q^{15} -16.0000 q^{16} +(-24.6785 + 24.6785i) q^{17} +(-105.286 + 105.286i) q^{18} -142.088 q^{19} +(44.2422 - 6.52887i) q^{20} -229.541i q^{21} +(-38.8197 + 38.8197i) q^{22} +(79.3632 + 76.6061i) q^{23} +80.5773i q^{24} +(-119.672 + 36.1065i) q^{25} -108.149 q^{26} +(337.933 + 337.933i) q^{27} +(-64.4589 - 64.4589i) q^{28} +43.7471i q^{29} +(-32.8799 - 222.807i) q^{30} +206.174 q^{31} +(22.6274 + 22.6274i) q^{32} +(195.499 + 195.499i) q^{33} +69.8012 q^{34} +(204.541 + 151.935i) q^{35} +297.794 q^{36} +(-62.6196 + 62.6196i) q^{37} +(200.942 + 200.942i) q^{38} +544.647i q^{39} +(-71.8012 - 53.3347i) q^{40} +104.295 q^{41} +(-324.620 + 324.620i) q^{42} +(6.53188 + 6.53188i) q^{43} +109.799 q^{44} +(-823.441 + 121.516i) q^{45} +(-3.89914 - 220.574i) q^{46} +(-123.349 - 123.349i) q^{47} +(113.954 - 113.954i) q^{48} -176.369i q^{49} +(220.304 + 118.179i) q^{50} -351.525i q^{51} +(152.946 + 152.946i) q^{52} +(414.929 + 414.929i) q^{53} -955.818i q^{54} +(-303.609 + 44.8039i) q^{55} +182.317i q^{56} +(1011.96 - 1011.96i) q^{57} +(61.8678 - 61.8678i) q^{58} +636.533i q^{59} +(-268.598 + 361.597i) q^{60} -411.285i q^{61} +(-291.573 - 291.573i) q^{62} +(1199.72 + 1199.72i) q^{63} -64.0000i q^{64} +(-485.326 - 360.506i) q^{65} -552.956i q^{66} +(-154.446 + 154.446i) q^{67} +(-98.7139 - 98.7139i) q^{68} +(-1110.83 + 19.6364i) q^{69} +(-74.3955 - 504.133i) q^{70} +615.260 q^{71} +(-421.144 - 421.144i) q^{72} +(222.982 - 222.982i) q^{73} +177.115 q^{74} +(595.160 - 1109.47i) q^{75} -568.350i q^{76} +(442.345 + 442.345i) q^{77} +(770.247 - 770.247i) q^{78} -322.803 q^{79} +(26.1155 + 176.969i) q^{80} -2803.47 q^{81} +(-147.495 - 147.495i) q^{82} +(-940.658 - 940.658i) q^{83} +918.165 q^{84} +(313.238 + 232.677i) q^{85} -18.4749i q^{86} +(-311.571 - 311.571i) q^{87} +(-155.279 - 155.279i) q^{88} +834.848 q^{89} +(1336.37 + 992.672i) q^{90} +1232.34i q^{91} +(-306.424 + 317.453i) q^{92} +(-1468.39 + 1468.39i) q^{93} +348.885i q^{94} +(231.918 + 1571.57i) q^{95} -322.309 q^{96} +(-629.315 + 629.315i) q^{97} +(-249.424 + 249.424i) q^{98} -2043.59 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
\(3\) −7.12210 + 7.12210i −1.37065 + 1.37065i −0.511166 + 0.859482i \(0.670786\pi\)
−0.859482 + 0.511166i \(0.829214\pi\)
\(4\) 4.00000i 0.500000i
\(5\) −1.63222 − 11.0606i −0.145990 − 0.989286i
\(6\) 20.1443 1.37065
\(7\) −16.1147 + 16.1147i −0.870114 + 0.870114i −0.992484 − 0.122371i \(-0.960950\pi\)
0.122371 + 0.992484i \(0.460950\pi\)
\(8\) 5.65685 − 5.65685i 0.250000 − 0.250000i
\(9\) − 74.4485i − 2.75735i
\(10\) −13.3337 + 17.9503i −0.421648 + 0.567638i
\(11\) − 27.4497i − 0.752400i −0.926539 − 0.376200i \(-0.877231\pi\)
0.926539 − 0.376200i \(-0.122769\pi\)
\(12\) −28.4884 − 28.4884i −0.685324 − 0.685324i
\(13\) 38.2364 − 38.2364i 0.815759 − 0.815759i −0.169731 − 0.985490i \(-0.554290\pi\)
0.985490 + 0.169731i \(0.0542899\pi\)
\(14\) 45.5793 0.870114
\(15\) 90.3991 + 67.1495i 1.55606 + 1.15586i
\(16\) −16.0000 −0.250000
\(17\) −24.6785 + 24.6785i −0.352083 + 0.352083i −0.860884 − 0.508801i \(-0.830089\pi\)
0.508801 + 0.860884i \(0.330089\pi\)
\(18\) −105.286 + 105.286i −1.37868 + 1.37868i
\(19\) −142.088 −1.71564 −0.857819 − 0.513952i \(-0.828181\pi\)
−0.857819 + 0.513952i \(0.828181\pi\)
\(20\) 44.2422 − 6.52887i 0.494643 − 0.0729950i
\(21\) − 229.541i − 2.38524i
\(22\) −38.8197 + 38.8197i −0.376200 + 0.376200i
\(23\) 79.3632 + 76.6061i 0.719494 + 0.694499i
\(24\) 80.5773i 0.685324i
\(25\) −119.672 + 36.1065i −0.957374 + 0.288852i
\(26\) −108.149 −0.815759
\(27\) 337.933 + 337.933i 2.40871 + 2.40871i
\(28\) −64.4589 − 64.4589i −0.435057 − 0.435057i
\(29\) 43.7471i 0.280125i 0.990143 + 0.140063i \(0.0447304\pi\)
−0.990143 + 0.140063i \(0.955270\pi\)
\(30\) −32.8799 − 222.807i −0.200101 − 1.35596i
\(31\) 206.174 1.19451 0.597256 − 0.802051i \(-0.296258\pi\)
0.597256 + 0.802051i \(0.296258\pi\)
\(32\) 22.6274 + 22.6274i 0.125000 + 0.125000i
\(33\) 195.499 + 195.499i 1.03127 + 1.03127i
\(34\) 69.8012 0.352083
\(35\) 204.541 + 151.935i 0.987819 + 0.733763i
\(36\) 297.794 1.37868
\(37\) −62.6196 + 62.6196i −0.278233 + 0.278233i −0.832403 − 0.554171i \(-0.813036\pi\)
0.554171 + 0.832403i \(0.313036\pi\)
\(38\) 200.942 + 200.942i 0.857819 + 0.857819i
\(39\) 544.647i 2.23624i
\(40\) −71.8012 − 53.3347i −0.283819 − 0.210824i
\(41\) 104.295 0.397271 0.198636 − 0.980073i \(-0.436349\pi\)
0.198636 + 0.980073i \(0.436349\pi\)
\(42\) −324.620 + 324.620i −1.19262 + 1.19262i
\(43\) 6.53188 + 6.53188i 0.0231652 + 0.0231652i 0.718594 − 0.695429i \(-0.244786\pi\)
−0.695429 + 0.718594i \(0.744786\pi\)
\(44\) 109.799 0.376200
\(45\) −823.441 + 121.516i −2.72781 + 0.402546i
\(46\) −3.89914 − 220.574i −0.0124978 − 0.706996i
\(47\) −123.349 − 123.349i −0.382816 − 0.382816i 0.489299 − 0.872116i \(-0.337253\pi\)
−0.872116 + 0.489299i \(0.837253\pi\)
\(48\) 113.954 − 113.954i 0.342662 − 0.342662i
\(49\) − 176.369i − 0.514196i
\(50\) 220.304 + 118.179i 0.623113 + 0.334261i
\(51\) − 351.525i − 0.965163i
\(52\) 152.946 + 152.946i 0.407880 + 0.407880i
\(53\) 414.929 + 414.929i 1.07537 + 1.07537i 0.996918 + 0.0784570i \(0.0249993\pi\)
0.0784570 + 0.996918i \(0.475001\pi\)
\(54\) − 955.818i − 2.40871i
\(55\) −303.609 + 44.8039i −0.744338 + 0.109843i
\(56\) 182.317i 0.435057i
\(57\) 1011.96 − 1011.96i 2.35153 − 2.35153i
\(58\) 61.8678 − 61.8678i 0.140063 − 0.140063i
\(59\) 636.533i 1.40457i 0.711897 + 0.702284i \(0.247836\pi\)
−0.711897 + 0.702284i \(0.752164\pi\)
\(60\) −268.598 + 361.597i −0.577931 + 0.778032i
\(61\) − 411.285i − 0.863274i −0.902047 − 0.431637i \(-0.857936\pi\)
0.902047 − 0.431637i \(-0.142064\pi\)
\(62\) −291.573 − 291.573i −0.597256 − 0.597256i
\(63\) 1199.72 + 1199.72i 2.39921 + 2.39921i
\(64\) − 64.0000i − 0.125000i
\(65\) −485.326 − 360.506i −0.926112 − 0.687927i
\(66\) − 552.956i − 1.03127i
\(67\) −154.446 + 154.446i −0.281621 + 0.281621i −0.833755 − 0.552134i \(-0.813814\pi\)
0.552134 + 0.833755i \(0.313814\pi\)
\(68\) −98.7139 − 98.7139i −0.176041 − 0.176041i
\(69\) −1110.83 + 19.6364i −1.93809 + 0.0342601i
\(70\) −74.3955 − 504.133i −0.127028 − 0.860791i
\(71\) 615.260 1.02842 0.514210 − 0.857664i \(-0.328085\pi\)
0.514210 + 0.857664i \(0.328085\pi\)
\(72\) −421.144 − 421.144i −0.689338 − 0.689338i
\(73\) 222.982 − 222.982i 0.357507 − 0.357507i −0.505386 − 0.862893i \(-0.668650\pi\)
0.862893 + 0.505386i \(0.168650\pi\)
\(74\) 177.115 0.278233
\(75\) 595.160 − 1109.47i 0.916308 − 1.70814i
\(76\) − 568.350i − 0.857819i
\(77\) 442.345 + 442.345i 0.654673 + 0.654673i
\(78\) 770.247 − 770.247i 1.11812 − 1.11812i
\(79\) −322.803 −0.459723 −0.229862 − 0.973223i \(-0.573827\pi\)
−0.229862 + 0.973223i \(0.573827\pi\)
\(80\) 26.1155 + 176.969i 0.0364975 + 0.247322i
\(81\) −2803.47 −3.84563
\(82\) −147.495 − 147.495i −0.198636 − 0.198636i
\(83\) −940.658 − 940.658i −1.24398 − 1.24398i −0.958334 − 0.285649i \(-0.907791\pi\)
−0.285649 − 0.958334i \(-0.592209\pi\)
\(84\) 918.165 1.19262
\(85\) 313.238 + 232.677i 0.399711 + 0.296910i
\(86\) − 18.4749i − 0.0231652i
\(87\) −311.571 − 311.571i −0.383953 − 0.383953i
\(88\) −155.279 − 155.279i −0.188100 − 0.188100i
\(89\) 834.848 0.994311 0.497156 − 0.867661i \(-0.334378\pi\)
0.497156 + 0.867661i \(0.334378\pi\)
\(90\) 1336.37 + 992.672i 1.56518 + 1.16263i
\(91\) 1232.34i 1.41961i
\(92\) −306.424 + 317.453i −0.347249 + 0.359747i
\(93\) −1468.39 + 1468.39i −1.63725 + 1.63725i
\(94\) 348.885i 0.382816i
\(95\) 231.918 + 1571.57i 0.250466 + 1.69726i
\(96\) −322.309 −0.342662
\(97\) −629.315 + 629.315i −0.658735 + 0.658735i −0.955081 − 0.296346i \(-0.904232\pi\)
0.296346 + 0.955081i \(0.404232\pi\)
\(98\) −249.424 + 249.424i −0.257098 + 0.257098i
\(99\) −2043.59 −2.07463
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.137.1 ✓ 72
5.3 odd 4 inner 230.4.e.a.183.2 yes 72
23.22 odd 2 inner 230.4.e.a.137.2 yes 72
115.68 even 4 inner 230.4.e.a.183.1 yes 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.4.e.a.137.1 ✓ 72 1.1 even 1 trivial
230.4.e.a.137.2 yes 72 23.22 odd 2 inner
230.4.e.a.183.1 yes 72 115.68 even 4 inner
230.4.e.a.183.2 yes 72 5.3 odd 4 inner