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Properties

Label 230.2.l.a.7.4
Level $230$
Weight $2$
Character 230.7
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 7.4
Character \(\chi\) \(=\) 230.7
Dual form 230.2.l.a.33.4

$q$-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+(-0.997452 + 0.0713392i) q^{2} +(-0.0823325 + 0.378476i) q^{3} +(0.989821 - 0.142315i) q^{4} +(-0.0392167 + 2.23572i) q^{5} +(0.0551225 - 0.383386i) q^{6} +(1.43832 - 2.63408i) q^{7} +(-0.977147 + 0.212565i) q^{8} +(2.59243 + 1.18392i) q^{9} +(-0.120378 - 2.23283i) q^{10} +(-4.70158 + 4.07395i) q^{11} +(-0.0276317 + 0.386341i) q^{12} +(2.08067 - 1.13613i) q^{13} +(-1.24674 + 2.72998i) q^{14} +(-0.842940 - 0.198915i) q^{15} +(0.959493 - 0.281733i) q^{16} +(2.53420 + 3.38530i) q^{17} +(-2.67029 - 0.995965i) q^{18} +(0.894953 + 6.22454i) q^{19} +(0.279359 + 2.21855i) q^{20} +(0.878518 + 0.761240i) q^{21} +(4.39897 - 4.39897i) q^{22} +(3.39216 - 3.39017i) q^{23} -0.387328i q^{24} +(-4.99692 - 0.175355i) q^{25} +(-1.99432 + 1.28167i) q^{26} +(-1.35788 + 1.81391i) q^{27} +(1.04881 - 2.81197i) q^{28} +(5.28457 + 0.759806i) q^{29} +(0.854983 + 0.138274i) q^{30} +(-1.87848 - 1.20723i) q^{31} +(-0.936950 + 0.349464i) q^{32} +(-1.15480 - 2.11486i) q^{33} +(-2.76925 - 3.19588i) q^{34} +(5.83268 + 3.31898i) q^{35} +(2.73453 + 0.802931i) q^{36} +(0.616132 + 1.65191i) q^{37} +(-1.33673 - 6.14483i) q^{38} +(0.258692 + 0.881025i) q^{39} +(-0.436917 - 2.19297i) q^{40} +(-3.35678 - 7.35032i) q^{41} +(-0.930586 - 0.696628i) q^{42} +(-1.91391 - 0.416347i) q^{43} +(-4.07395 + 4.70158i) q^{44} +(-2.74859 + 5.74953i) q^{45} +(-3.14167 + 3.62353i) q^{46} +(-6.39758 - 6.39758i) q^{47} +(0.0276317 + 0.386341i) q^{48} +(-1.08515 - 1.68852i) q^{49} +(4.99670 - 0.181568i) q^{50} +(-1.48990 + 0.680415i) q^{51} +(1.89780 - 1.42068i) q^{52} +(0.405176 + 0.221243i) q^{53} +(1.22502 - 1.90616i) q^{54} +(-8.92384 - 10.6712i) q^{55} +(-0.845533 + 2.87962i) q^{56} +(-2.42952 - 0.173763i) q^{57} +(-5.32531 - 0.380874i) q^{58} +(3.35242 - 11.4173i) q^{59} +(-0.862669 - 0.0769278i) q^{60} +(3.37239 - 5.24755i) q^{61} +(1.95982 + 1.07014i) q^{62} +(6.84729 - 5.12582i) q^{63} +(0.909632 - 0.415415i) q^{64} +(2.45848 + 4.69636i) q^{65} +(1.30273 + 2.02709i) q^{66} +(-1.02353 - 14.3108i) q^{67} +(2.99018 + 2.99018i) q^{68} +(1.00381 + 1.56297i) q^{69} +(-6.05459 - 2.89443i) q^{70} +(0.153656 - 0.177328i) q^{71} +(-2.78485 - 0.605806i) q^{72} +(-8.90097 - 6.66318i) q^{73} +(-0.732408 - 1.60375i) q^{74} +(0.477777 - 1.87678i) q^{75} +(1.77169 + 6.03382i) q^{76} +(3.96874 + 18.2440i) q^{77} +(-0.320885 - 0.860326i) q^{78} +(9.47630 + 2.78249i) q^{79} +(0.592248 + 2.15621i) q^{80} +(5.02429 + 5.79834i) q^{81} +(3.87259 + 7.09212i) q^{82} +(-6.83823 + 2.55053i) q^{83} +(0.977912 + 0.628466i) q^{84} +(-7.66797 + 5.53301i) q^{85} +(1.93874 + 0.278749i) q^{86} +(-0.722660 + 1.93753i) q^{87} +(3.72816 - 4.98024i) q^{88} +(3.96856 - 2.55044i) q^{89} +(2.33142 - 5.93096i) q^{90} -7.11478i q^{91} +(2.87516 - 3.83842i) q^{92} +(0.611567 - 0.611567i) q^{93} +(6.83768 + 5.92488i) q^{94} +(-13.9514 + 1.75676i) q^{95} +(-0.0551225 - 0.383386i) q^{96} +(14.7600 + 5.50521i) q^{97} +(1.20284 + 1.60681i) q^{98} +(-17.0118 + 4.99511i) 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 \(\left(\mathbb{Z}/230\mathbb{Z}\right)^\times\).

\(n\) \(47\) \(51\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(e\left(\frac{19}{22}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.997452 + 0.0713392i −0.705305 + 0.0504444i
\(3\) −0.0823325 + 0.378476i −0.0475347 + 0.218513i −0.994787 − 0.101976i \(-0.967484\pi\)
0.947252 + 0.320489i \(0.103847\pi\)
\(4\) 0.989821 − 0.142315i 0.494911 − 0.0711574i
\(5\) −0.0392167 + 2.23572i −0.0175382 + 0.999846i
\(6\) 0.0551225 − 0.383386i 0.0225037 − 0.156517i
\(7\) 1.43832 − 2.63408i 0.543633 − 0.995590i −0.451039 − 0.892504i \(-0.648947\pi\)
0.994672 − 0.103086i \(-0.0328716\pi\)
\(8\) −0.977147 + 0.212565i −0.345474 + 0.0751532i
\(9\) 2.59243 + 1.18392i 0.864143 + 0.394641i
\(10\) −0.120378 − 2.23283i −0.0380669 − 0.706081i
\(11\) −4.70158 + 4.07395i −1.41758 + 1.22834i −0.481552 + 0.876417i \(0.659927\pi\)
−0.936029 + 0.351924i \(0.885528\pi\)
\(12\) −0.0276317 + 0.386341i −0.00797658 + 0.111527i
\(13\) 2.08067 − 1.13613i 0.577074 − 0.315106i −0.164074 − 0.986448i \(-0.552464\pi\)
0.741148 + 0.671342i \(0.234282\pi\)
\(14\) −1.24674 + 2.72998i −0.333205 + 0.729618i
\(15\) −0.842940 − 0.198915i −0.217646 − 0.0513597i
\(16\) 0.959493 − 0.281733i 0.239873 − 0.0704331i
\(17\) 2.53420 + 3.38530i 0.614634 + 0.821055i 0.994582 − 0.103953i \(-0.0331491\pi\)
−0.379948 + 0.925008i \(0.624058\pi\)
\(18\) −2.67029 − 0.995965i −0.629392 − 0.234751i
\(19\) 0.894953 + 6.22454i 0.205316 + 1.42801i 0.788184 + 0.615440i \(0.211022\pi\)
−0.582868 + 0.812567i \(0.698069\pi\)
\(20\) 0.279359 + 2.21855i 0.0624666 + 0.496083i
\(21\) 0.878518 + 0.761240i 0.191708 + 0.166116i
\(22\) 4.39897 − 4.39897i 0.937864 − 0.937864i
\(23\) 3.39216 − 3.39017i 0.707314 − 0.706899i
\(24\) − 0.387328i − 0.0790630i
\(25\) −4.99692 − 0.175355i −0.999385 − 0.0350711i
\(26\) −1.99432 + 1.28167i −0.391118 + 0.251356i
\(27\) −1.35788 + 1.81391i −0.261324 + 0.349088i
\(28\) 1.04881 − 2.81197i 0.198206 − 0.531411i
\(29\) 5.28457 + 0.759806i 0.981320 + 0.141092i 0.614259 − 0.789105i \(-0.289455\pi\)
0.367061 + 0.930197i \(0.380364\pi\)
\(30\) 0.854983 + 0.138274i 0.156098 + 0.0252452i
\(31\) −1.87848 − 1.20723i −0.337385 − 0.216824i 0.360967 − 0.932579i \(-0.382447\pi\)
−0.698352 + 0.715754i \(0.746083\pi\)
\(32\) −0.936950 + 0.349464i −0.165631 + 0.0617771i
\(33\) −1.15480 − 2.11486i −0.201025 − 0.368149i
\(34\) −2.76925 − 3.19588i −0.474922 − 0.548089i
\(35\) 5.83268 + 3.31898i 0.985902 + 0.561010i
\(36\) 2.73453 + 0.802931i 0.455755 + 0.133822i
\(37\) 0.616132 + 1.65191i 0.101291 + 0.271573i 0.977704 − 0.209988i \(-0.0673426\pi\)
−0.876412 + 0.481561i \(0.840070\pi\)
\(38\) −1.33673 − 6.14483i −0.216846 − 0.996824i
\(39\) 0.258692 + 0.881025i 0.0414239 + 0.141077i
\(40\) −0.436917 − 2.19297i −0.0690826 − 0.346739i
\(41\) −3.35678 − 7.35032i −0.524241 − 1.14793i −0.967809 − 0.251687i \(-0.919015\pi\)
0.443568 − 0.896241i \(-0.353713\pi\)
\(42\) −0.930586 − 0.696628i −0.143592 − 0.107492i
\(43\) −1.91391 − 0.416347i −0.291869 − 0.0634923i 0.0642467 − 0.997934i \(-0.479536\pi\)
−0.356116 + 0.934442i \(0.615899\pi\)
\(44\) −4.07395 + 4.70158i −0.614171 + 0.708791i
\(45\) −2.74859 + 5.74953i −0.409736 + 0.857089i
\(46\) −3.14167 + 3.62353i −0.463213 + 0.534260i
\(47\) −6.39758 − 6.39758i −0.933183 − 0.933183i 0.0647200 − 0.997903i \(-0.479385\pi\)
−0.997903 + 0.0647200i \(0.979385\pi\)
\(48\) 0.0276317 + 0.386341i 0.00398829 + 0.0557636i
\(49\) −1.08515 − 1.68852i −0.155021 − 0.241217i
\(50\) 4.99670 − 0.181568i 0.706640 − 0.0256776i
\(51\) −1.48990 + 0.680415i −0.208628 + 0.0952772i
\(52\) 1.89780 − 1.42068i 0.263178 − 0.197013i
\(53\) 0.405176 + 0.221243i 0.0556552 + 0.0303900i 0.506836 − 0.862043i \(-0.330815\pi\)
−0.451180 + 0.892433i \(0.648997\pi\)
\(54\) 1.22502 − 1.90616i 0.166704 − 0.259396i
\(55\) −8.92384 − 10.6712i −1.20329 − 1.43891i
\(56\) −0.845533 + 2.87962i −0.112989 + 0.384806i
\(57\) −2.42952 − 0.173763i −0.321798 − 0.0230155i
\(58\) −5.32531 − 0.380874i −0.699247 − 0.0500111i
\(59\) 3.35242 − 11.4173i 0.436448 − 1.48641i −0.388634 − 0.921392i \(-0.627053\pi\)
0.825082 − 0.565013i \(-0.191129\pi\)
\(60\) −0.862669 − 0.0769278i −0.111370 − 0.00993134i
\(61\) 3.37239 − 5.24755i 0.431791 − 0.671879i −0.555371 − 0.831603i \(-0.687424\pi\)
0.987162 + 0.159723i \(0.0510601\pi\)
\(62\) 1.95982 + 1.07014i 0.248897 + 0.135908i
\(63\) 6.84729 − 5.12582i 0.862677 − 0.645792i
\(64\) 0.909632 − 0.415415i 0.113704 − 0.0519269i
\(65\) 2.45848 + 4.69636i 0.304937 + 0.582512i
\(66\) 1.30273 + 2.02709i 0.160355 + 0.249517i
\(67\) −1.02353 − 14.3108i −0.125044 − 1.74834i −0.542874 − 0.839814i \(-0.682664\pi\)
0.417830 − 0.908525i \(-0.362791\pi\)
\(68\) 2.99018 + 2.99018i 0.362613 + 0.362613i
\(69\) 1.00381 + 1.56297i 0.120845 + 0.188160i
\(70\) −6.05459 − 2.89443i −0.723662 − 0.345950i
\(71\) 0.153656 − 0.177328i 0.0182356 − 0.0210450i −0.746559 − 0.665320i \(-0.768295\pi\)
0.764794 + 0.644275i \(0.222841\pi\)
\(72\) −2.78485 − 0.605806i −0.328197 − 0.0713949i
\(73\) −8.90097 − 6.66318i −1.04178 − 0.779866i −0.0658228 − 0.997831i \(-0.520967\pi\)
−0.975957 + 0.217965i \(0.930058\pi\)
\(74\) −0.732408 − 1.60375i −0.0851407 − 0.186432i
\(75\) 0.477777 − 1.87678i 0.0551690 − 0.216712i
\(76\) 1.77169 + 6.03382i 0.203227 + 0.692126i
\(77\) 3.96874 + 18.2440i 0.452280 + 2.07910i
\(78\) −0.320885 − 0.860326i −0.0363331 − 0.0974127i
\(79\) 9.47630 + 2.78249i 1.06617 + 0.313055i 0.767332 − 0.641250i \(-0.221584\pi\)
0.298835 + 0.954305i \(0.403402\pi\)
\(80\) 0.592248 + 2.15621i 0.0662154 + 0.241072i
\(81\) 5.02429 + 5.79834i 0.558254 + 0.644260i
\(82\) 3.87259 + 7.09212i 0.427656 + 0.783194i
\(83\) −6.83823 + 2.55053i −0.750593 + 0.279957i −0.695502 − 0.718525i \(-0.744818\pi\)
−0.0550916 + 0.998481i \(0.517545\pi\)
\(84\) 0.977912 + 0.628466i 0.106699 + 0.0685712i
\(85\) −7.66797 + 5.53301i −0.831708 + 0.600140i
\(86\) 1.93874 + 0.278749i 0.209060 + 0.0300582i
\(87\) −0.722660 + 1.93753i −0.0774773 + 0.207725i
\(88\) 3.72816 − 4.98024i 0.397423 − 0.530895i
\(89\) 3.96856 − 2.55044i 0.420667 − 0.270346i −0.313136 − 0.949708i \(-0.601380\pi\)
0.733803 + 0.679362i \(0.237743\pi\)
\(90\) 2.33142 − 5.93096i 0.245753 − 0.625178i
\(91\) − 7.11478i − 0.745831i
\(92\) 2.87516 − 3.83842i 0.299756 − 0.400183i
\(93\) 0.611567 − 0.611567i 0.0634165 − 0.0634165i
\(94\) 6.83768 + 5.92488i 0.705253 + 0.611105i
\(95\) −13.9514 + 1.75676i −1.43139 + 0.180240i
\(96\) −0.0551225 − 0.383386i −0.00562592 − 0.0391291i
\(97\) 14.7600 + 5.50521i 1.49866 + 0.558969i 0.959122 − 0.282995i \(-0.0913279\pi\)
0.539534 + 0.841964i \(0.318601\pi\)
\(98\) 1.20284 + 1.60681i 0.121505 + 0.162312i
\(99\) −17.0118 + 4.99511i −1.70975 + 0.502027i
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.7.4 ✓ 240
5.3 odd 4 inner 230.2.l.a.53.10 yes 240
23.10 odd 22 inner 230.2.l.a.217.10 yes 240
115.33 even 44 inner 230.2.l.a.33.4 yes 240
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.l.a.7.4 ✓ 240 1.1 even 1 trivial
230.2.l.a.33.4 yes 240 115.33 even 44 inner
230.2.l.a.53.10 yes 240 5.3 odd 4 inner
230.2.l.a.217.10 yes 240 23.10 odd 22 inner