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Properties

Label 230.2.l.a.17.1
Level $230$
Weight $2$
Character 230.17
Analytic conductor $1.837$
Analytic rank $0$
Dimension $240$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [230,2,Mod(7,230)] 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("230.7"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(230, base_ring=CyclotomicField(44)) chi = DirichletCharacter(H, H._module([11, 38])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 230 = 2 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 230.l (of order \(44\), degree \(20\), 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: \(1.83655924649\)
Analytic rank: \(0\)
Dimension: \(240\)
Relative dimension: \(12\) over \(\Q(\zeta_{44})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{44}]$

Embedding invariants

Embedding label 17.1
Character \(\chi\) \(=\) 230.17
Dual form 230.2.l.a.203.1

$q$-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+(-0.936950 + 0.349464i) q^{2} +(-2.43530 + 1.32977i) q^{3} +(0.755750 - 0.654861i) q^{4} +(2.05188 - 0.888703i) q^{5} +(1.81704 - 2.09698i) q^{6} +(-1.58597 - 2.11861i) q^{7} +(-0.479249 + 0.877679i) q^{8} +(2.54045 - 3.95301i) q^{9} +(-1.61194 + 1.54973i) q^{10} +(1.25057 - 0.571117i) q^{11} +(-0.969658 + 2.59975i) q^{12} +(4.76976 + 3.57060i) q^{13} +(2.22635 + 1.43079i) q^{14} +(-3.81516 + 4.89279i) q^{15} +(0.142315 - 0.989821i) q^{16} +(-0.123408 - 1.72547i) q^{17} +(-0.998835 + 4.59157i) q^{18} +(2.57738 + 2.97445i) q^{19} +(0.968730 - 2.01533i) q^{20} +(6.67957 + 3.05046i) q^{21} +(-0.972139 + 0.972139i) q^{22} +(2.48233 + 4.10342i) q^{23} -2.77470i q^{24} +(3.42041 - 3.64702i) q^{25} +(-5.71682 - 1.67861i) q^{26} +(-0.336299 + 4.70208i) q^{27} +(-2.58599 - 0.562548i) q^{28} +(-0.566767 - 0.491106i) q^{29} +(1.86476 - 5.91755i) q^{30} +(8.10734 - 2.38053i) q^{31} +(0.212565 + 0.977147i) q^{32} +(-2.28606 + 3.05382i) q^{33} +(0.718617 + 1.57355i) q^{34} +(-5.13703 - 2.93767i) q^{35} +(-0.668731 - 4.65113i) q^{36} +(5.87592 - 1.27823i) q^{37} +(-3.45434 - 1.88621i) q^{38} +(-16.3639 - 2.35277i) q^{39} +(-0.203365 + 2.22680i) q^{40} +(-7.72122 + 4.96213i) q^{41} +(-7.32445 - 0.523855i) q^{42} +(-5.20194 - 9.52665i) q^{43} +(0.571117 - 1.25057i) q^{44} +(1.69963 - 10.3688i) q^{45} +(-3.75982 - 2.97721i) q^{46} +(-2.78897 - 2.78897i) q^{47} +(0.969658 + 2.59975i) q^{48} +(-0.00107183 + 0.00365032i) q^{49} +(-1.93025 + 4.61239i) q^{50} +(2.59502 + 4.03793i) q^{51} +(5.94299 - 0.425051i) q^{52} +(1.09796 - 0.821926i) q^{53} +(-1.32811 - 4.52314i) q^{54} +(2.05847 - 2.28325i) q^{55} +(2.61953 - 0.376632i) q^{56} +(-10.2320 - 3.81634i) q^{57} +(0.702656 + 0.262077i) q^{58} +(8.33749 - 1.19875i) q^{59} +(0.320789 + 6.19612i) q^{60} +(0.141177 + 0.480803i) q^{61} +(-6.76426 + 5.06366i) q^{62} +(-12.4040 + 0.887149i) q^{63} +(-0.540641 - 0.841254i) q^{64} +(12.9602 + 3.08753i) q^{65} +(1.07472 - 3.66017i) q^{66} +(1.15888 + 3.10707i) q^{67} +(-1.22321 - 1.22321i) q^{68} +(-11.5018 - 6.69210i) q^{69} +(5.83975 + 0.957241i) q^{70} +(-5.61249 + 12.2896i) q^{71} +(2.25197 + 4.12417i) q^{72} +(14.5923 + 1.04366i) q^{73} +(-5.05875 + 3.25106i) q^{74} +(-3.48001 + 13.4299i) q^{75} +(3.89570 + 0.560117i) q^{76} +(-3.19335 - 1.74370i) q^{77} +(16.1543 - 3.51416i) q^{78} +(-0.692476 - 4.81627i) q^{79} +(-0.587645 - 2.15747i) q^{80} +(0.422345 + 0.924807i) q^{81} +(5.50031 - 7.34755i) q^{82} +(0.0505310 + 0.232287i) q^{83} +(7.04571 - 2.06881i) q^{84} +(-1.78665 - 3.43078i) q^{85} +(8.20318 + 7.10810i) q^{86} +(2.03330 + 0.442318i) q^{87} +(-0.0980779 + 1.37131i) q^{88} +(-11.7514 - 3.45054i) q^{89} +(2.03105 + 10.3090i) q^{90} -15.7681i q^{91} +(4.56319 + 1.47558i) q^{92} +(-16.5782 + 16.5782i) q^{93} +(3.58776 + 1.63848i) q^{94} +(7.93187 + 3.81269i) q^{95} +(-1.81704 - 2.09698i) q^{96} +(1.32094 - 6.07227i) q^{97} +(-0.000271405 - 0.00379474i) q^{98} +(0.919379 - 6.39442i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 240 q - 8 q^{3} - 8 q^{6} - 8 q^{12} + 16 q^{13} + 24 q^{16} - 72 q^{18} - 80 q^{23} - 8 q^{26} + 16 q^{27} - 44 q^{28} + 24 q^{31} - 44 q^{33} - 8 q^{35} - 32 q^{36} - 88 q^{37} - 24 q^{41} - 8 q^{46}+ \cdots + 24 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/230\mathbb{Z}\right)^\times\).

\(n\) \(47\) \(51\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(e\left(\frac{7}{22}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.936950 + 0.349464i −0.662524 + 0.247108i
\(3\) −2.43530 + 1.32977i −1.40602 + 0.767744i −0.989927 − 0.141579i \(-0.954782\pi\)
−0.416091 + 0.909323i \(0.636600\pi\)
\(4\) 0.755750 − 0.654861i 0.377875 − 0.327430i
\(5\) 2.05188 − 0.888703i 0.917628 − 0.397440i
\(6\) 1.81704 − 2.09698i 0.741804 − 0.856088i
\(7\) −1.58597 − 2.11861i −0.599440 − 0.800759i 0.393506 − 0.919322i \(-0.371262\pi\)
−0.992946 + 0.118563i \(0.962171\pi\)
\(8\) −0.479249 + 0.877679i −0.169440 + 0.310306i
\(9\) 2.54045 − 3.95301i 0.846816 − 1.31767i
\(10\) −1.61194 + 1.54973i −0.509739 + 0.490067i
\(11\) 1.25057 − 0.571117i 0.377062 − 0.172198i −0.217864 − 0.975979i \(-0.569909\pi\)
0.594926 + 0.803781i \(0.297182\pi\)
\(12\) −0.969658 + 2.59975i −0.279916 + 0.750484i
\(13\) 4.76976 + 3.57060i 1.32289 + 0.990306i 0.999022 + 0.0442052i \(0.0140756\pi\)
0.323872 + 0.946101i \(0.395015\pi\)
\(14\) 2.22635 + 1.43079i 0.595018 + 0.382395i
\(15\) −3.81516 + 4.89279i −0.985070 + 1.26331i
\(16\) 0.142315 − 0.989821i 0.0355787 − 0.247455i
\(17\) −0.123408 − 1.72547i −0.0299309 − 0.418488i −0.990223 − 0.139492i \(-0.955453\pi\)
0.960292 − 0.278996i \(-0.0900016\pi\)
\(18\) −0.998835 + 4.59157i −0.235428 + 1.08224i
\(19\) 2.57738 + 2.97445i 0.591291 + 0.682386i 0.969993 − 0.243133i \(-0.0781750\pi\)
−0.378702 + 0.925519i \(0.623630\pi\)
\(20\) 0.968730 − 2.01533i 0.216615 − 0.450642i
\(21\) 6.67957 + 3.05046i 1.45760 + 0.665665i
\(22\) −0.972139 + 0.972139i −0.207261 + 0.207261i
\(23\) 2.48233 + 4.10342i 0.517601 + 0.855622i
\(24\) − 2.77470i − 0.566383i
\(25\) 3.42041 − 3.64702i 0.684083 − 0.729405i
\(26\) −5.71682 − 1.67861i −1.12116 − 0.329203i
\(27\) −0.336299 + 4.70208i −0.0647209 + 0.904916i
\(28\) −2.58599 − 0.562548i −0.488706 − 0.106312i
\(29\) −0.566767 − 0.491106i −0.105246 − 0.0911961i 0.600648 − 0.799513i \(-0.294909\pi\)
−0.705894 + 0.708317i \(0.749455\pi\)
\(30\) 1.86476 − 5.91755i 0.340457 − 1.08039i
\(31\) 8.10734 − 2.38053i 1.45612 − 0.427556i 0.544561 − 0.838721i \(-0.316696\pi\)
0.911561 + 0.411165i \(0.134878\pi\)
\(32\) 0.212565 + 0.977147i 0.0375766 + 0.172737i
\(33\) −2.28606 + 3.05382i −0.397952 + 0.531601i
\(34\) 0.718617 + 1.57355i 0.123242 + 0.269862i
\(35\) −5.13703 − 2.93767i −0.868317 − 0.496557i
\(36\) −0.668731 − 4.65113i −0.111455 − 0.775188i
\(37\) 5.87592 − 1.27823i 0.965995 − 0.210139i 0.298216 − 0.954499i \(-0.403609\pi\)
0.667779 + 0.744359i \(0.267245\pi\)
\(38\) −3.45434 − 1.88621i −0.560367 − 0.305984i
\(39\) −16.3639 − 2.35277i −2.62032 − 0.376744i
\(40\) −0.203365 + 2.22680i −0.0321548 + 0.352088i
\(41\) −7.72122 + 4.96213i −1.20585 + 0.774954i −0.979959 − 0.199199i \(-0.936166\pi\)
−0.225892 + 0.974152i \(0.572530\pi\)
\(42\) −7.32445 − 0.523855i −1.13019 − 0.0808326i
\(43\) −5.20194 − 9.52665i −0.793289 − 1.45280i −0.889253 − 0.457416i \(-0.848775\pi\)
0.0959639 − 0.995385i \(-0.469407\pi\)
\(44\) 0.571117 − 1.25057i 0.0860992 − 0.188531i
\(45\) 1.69963 − 10.3688i 0.253367 − 1.54569i
\(46\) −3.75982 − 2.97721i −0.554355 − 0.438966i
\(47\) −2.78897 − 2.78897i −0.406813 − 0.406813i 0.473813 − 0.880625i \(-0.342877\pi\)
−0.880625 + 0.473813i \(0.842877\pi\)
\(48\) 0.969658 + 2.59975i 0.139958 + 0.375242i
\(49\) −0.00107183 + 0.00365032i −0.000153119 + 0.000521475i
\(50\) −1.93025 + 4.61239i −0.272979 + 0.652290i
\(51\) 2.59502 + 4.03793i 0.363375 + 0.565423i
\(52\) 5.94299 − 0.425051i 0.824145 − 0.0589440i
\(53\) 1.09796 − 0.821926i 0.150817 − 0.112900i −0.521205 − 0.853432i \(-0.674517\pi\)
0.672022 + 0.740532i \(0.265426\pi\)
\(54\) −1.32811 − 4.52314i −0.180733 − 0.615521i
\(55\) 2.05847 − 2.28325i 0.277564 − 0.307874i
\(56\) 2.61953 − 0.376632i 0.350050 − 0.0503296i
\(57\) −10.2320 − 3.81634i −1.35526 − 0.505487i
\(58\) 0.702656 + 0.262077i 0.0922633 + 0.0344124i
\(59\) 8.33749 − 1.19875i 1.08545 − 0.156064i 0.423710 − 0.905798i \(-0.360728\pi\)
0.661739 + 0.749734i \(0.269819\pi\)
\(60\) 0.320789 + 6.19612i 0.0414137 + 0.799915i
\(61\) 0.141177 + 0.480803i 0.0180758 + 0.0615606i 0.968037 − 0.250807i \(-0.0806960\pi\)
−0.949961 + 0.312368i \(0.898878\pi\)
\(62\) −6.76426 + 5.06366i −0.859062 + 0.643086i
\(63\) −12.4040 + 0.887149i −1.56275 + 0.111770i
\(64\) −0.540641 − 0.841254i −0.0675801 − 0.105157i
\(65\) 12.9602 + 3.08753i 1.60751 + 0.382962i
\(66\) 1.07472 − 3.66017i 0.132289 − 0.450535i
\(67\) 1.15888 + 3.10707i 0.141579 + 0.379589i 0.988315 − 0.152425i \(-0.0487082\pi\)
−0.846736 + 0.532014i \(0.821435\pi\)
\(68\) −1.22321 − 1.22321i −0.148336 − 0.148336i
\(69\) −11.5018 − 6.69210i −1.38466 − 0.805635i
\(70\) 5.83975 + 0.957241i 0.697984 + 0.114412i
\(71\) −5.61249 + 12.2896i −0.666080 + 1.45851i 0.210667 + 0.977558i \(0.432436\pi\)
−0.876746 + 0.480953i \(0.840291\pi\)
\(72\) 2.25197 + 4.12417i 0.265397 + 0.486039i
\(73\) 14.5923 + 1.04366i 1.70789 + 0.122151i 0.890812 − 0.454371i \(-0.150136\pi\)
0.817081 + 0.576522i \(0.195591\pi\)
\(74\) −5.05875 + 3.25106i −0.588067 + 0.377928i
\(75\) −3.48001 + 13.4299i −0.401837 + 1.55076i
\(76\) 3.89570 + 0.560117i 0.446868 + 0.0642499i
\(77\) −3.19335 − 1.74370i −0.363916 − 0.198713i
\(78\) 16.1543 − 3.51416i 1.82912 − 0.397900i
\(79\) −0.692476 − 4.81627i −0.0779096 − 0.541873i −0.990974 − 0.134057i \(-0.957199\pi\)
0.913064 − 0.407816i \(-0.133710\pi\)
\(80\) −0.587645 − 2.15747i −0.0657007 − 0.241212i
\(81\) 0.422345 + 0.924807i 0.0469272 + 0.102756i
\(82\) 5.50031 − 7.34755i 0.607407 − 0.811401i
\(83\) 0.0505310 + 0.232287i 0.00554650 + 0.0254968i 0.979837 − 0.199799i \(-0.0640288\pi\)
−0.974290 + 0.225296i \(0.927665\pi\)
\(84\) 7.04571 − 2.06881i 0.768750 − 0.225725i
\(85\) −1.78665 − 3.43078i −0.193789 − 0.372121i
\(86\) 8.20318 + 7.10810i 0.884572 + 0.766486i
\(87\) 2.03330 + 0.442318i 0.217993 + 0.0474215i
\(88\) −0.0980779 + 1.37131i −0.0104551 + 0.146182i
\(89\) −11.7514 − 3.45054i −1.24565 − 0.365756i −0.408515 − 0.912752i \(-0.633953\pi\)
−0.837136 + 0.546995i \(0.815771\pi\)
\(90\) 2.03105 + 10.3090i 0.214092 + 1.08667i
\(91\) − 15.7681i − 1.65295i
\(92\) 4.56319 + 1.47558i 0.475745 + 0.153839i
\(93\) −16.5782 + 16.5782i −1.71908 + 1.71908i
\(94\) 3.58776 + 1.63848i 0.370050 + 0.168996i
\(95\) 7.93187 + 3.81269i 0.813793 + 0.391174i
\(96\) −1.81704 − 2.09698i −0.185451 − 0.214022i
\(97\) 1.32094 − 6.07227i 0.134121 − 0.616546i −0.860185 − 0.509983i \(-0.829652\pi\)
0.994306 − 0.106563i \(-0.0339845\pi\)
\(98\) −0.000271405 − 0.00379474i −2.74160e−5 − 0.000383326i
\(99\) 0.919379 − 6.39442i 0.0924011 − 0.642664i
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 230.2.l.a.17.1 ✓ 240
5.3 odd 4 inner 230.2.l.a.63.7 yes 240
23.19 odd 22 inner 230.2.l.a.157.7 yes 240
115.88 even 44 inner 230.2.l.a.203.1 yes 240
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.l.a.17.1 ✓ 240 1.1 even 1 trivial
230.2.l.a.63.7 yes 240 5.3 odd 4 inner
230.2.l.a.157.7 yes 240 23.19 odd 22 inner
230.2.l.a.203.1 yes 240 115.88 even 44 inner