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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 183.2
Character \(\chi\) \(=\) 230.183
Dual form 230.4.e.a.137.2

$q$-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} +(-76.6061 + 79.3632i) 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} +(317.453 + 306.424i) 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 \(\left(\mathbb{Z}/230\mathbb{Z}\right)^\times\).

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

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
\(3\) −7.12210 − 7.12210i −1.37065 − 1.37065i −0.859482 − 0.511166i \(-0.829214\pi\)
−0.511166 − 0.859482i \(-0.670786\pi\)
\(4\) − 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 \(-0.0390497\pi\)
−0.122371 + 0.992484i \(0.539050\pi\)
\(8\) 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 0.985490 − 0.169731i \(-0.0542899\pi\)
−0.169731 + 0.985490i \(0.554290\pi\)
\(14\) −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 \(-0.169911\pi\)
−0.508801 + 0.860884i \(0.669911\pi\)
\(18\) −105.286 − 105.286i −1.37868 − 1.37868i
\(19\) 142.088 1.71564 0.857819 − 0.513952i \(-0.171819\pi\)
0.857819 + 0.513952i \(0.171819\pi\)
\(20\) −44.2422 − 6.52887i −0.494643 − 0.0729950i
\(21\) − 229.541i − 2.38524i
\(22\) 38.8197 + 38.8197i 0.376200 + 0.376200i
\(23\) −76.6061 + 79.3632i −0.694499 + 0.719494i
\(24\) − 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 \(-0.955270\pi\)
0.990143 − 0.140063i \(-0.0447304\pi\)
\(30\) 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 \(-0.186964\pi\)
−0.554171 + 0.832403i \(0.686964\pi\)
\(38\) −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 \(-0.755214\pi\)
0.695429 + 0.718594i \(0.255214\pi\)
\(44\) −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 −0.872116 − 0.489299i \(-0.837253\pi\)
0.489299 + 0.872116i \(0.337253\pi\)
\(48\) 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 −0.0784570 + 0.996918i \(0.524999\pi\)
−0.996918 + 0.0784570i \(0.975001\pi\)
\(54\) 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 \(-0.752164\pi\)
0.711897 − 0.702284i \(-0.247836\pi\)
\(60\) 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 \(-0.186186\pi\)
−0.552134 + 0.833755i \(0.686186\pi\)
\(68\) 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 0.862893 − 0.505386i \(-0.168650\pi\)
−0.505386 + 0.862893i \(0.668650\pi\)
\(74\) −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 \(-0.426173\pi\)
0.229862 + 0.973223i \(0.426173\pi\)
\(80\) −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 0.285649 − 0.958334i \(-0.407791\pi\)
0.958334 − 0.285649i \(-0.0922091\pi\)
\(84\) −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 \(-0.665622\pi\)
−0.497156 + 0.867661i \(0.665622\pi\)
\(90\) −1336.37 + 992.672i −1.56518 + 1.16263i
\(91\) 1232.34i 1.41961i
\(92\) 317.453 + 306.424i 0.359747 + 0.347249i
\(93\) −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 \(-0.0957682\pi\)
−0.296346 + 0.955081i \(0.595768\pi\)
\(98\) −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.183.2 yes 72
5.2 odd 4 inner 230.4.e.a.137.1 ✓ 72
23.22 odd 2 inner 230.4.e.a.183.1 yes 72
115.22 even 4 inner 230.4.e.a.137.2 yes 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.4.e.a.137.1 ✓ 72 5.2 odd 4 inner
230.4.e.a.137.2 yes 72 115.22 even 4 inner
230.4.e.a.183.1 yes 72 23.22 odd 2 inner
230.4.e.a.183.2 yes 72 1.1 even 1 trivial