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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 17.4
Character \(\chi\) \(=\) 230.17
Dual form 230.2.l.a.203.4

$q$-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+(-0.936950 + 0.349464i) q^{2} +(1.26604 - 0.691312i) q^{3} +(0.755750 - 0.654861i) q^{4} +(-0.584903 + 2.15821i) q^{5} +(-0.944630 + 1.09016i) q^{6} +(1.35459 + 1.80952i) q^{7} +(-0.479249 + 0.877679i) q^{8} +(-0.496969 + 0.773299i) q^{9} +(-0.206194 - 2.22654i) q^{10} +(-0.475362 + 0.217091i) q^{11} +(0.504099 - 1.35154i) q^{12} +(3.11241 + 2.32992i) q^{13} +(-1.90155 - 1.22205i) q^{14} +(0.751486 + 3.13674i) q^{15} +(0.142315 - 0.989821i) q^{16} +(-0.578458 - 8.08789i) q^{17} +(0.195395 - 0.898216i) q^{18} +(4.90659 + 5.66250i) q^{19} +(0.971289 + 2.01410i) q^{20} +(2.96592 + 1.35449i) q^{21} +(0.369525 - 0.369525i) q^{22} +(-0.430826 - 4.77644i) q^{23} +1.44249i q^{24} +(-4.31578 - 2.52469i) q^{25} +(-3.73039 - 1.09534i) q^{26} +(-0.403312 + 5.63904i) q^{27} +(2.20872 + 0.480478i) q^{28} +(1.93734 + 1.67872i) q^{29} +(-1.80028 - 2.67635i) q^{30} +(-0.910979 + 0.267488i) q^{31} +(0.212565 + 0.977147i) q^{32} +(-0.451751 + 0.603469i) q^{33} +(3.36841 + 7.37580i) q^{34} +(-4.69765 + 1.86511i) q^{35} +(0.130819 + 0.909866i) q^{36} +(2.51303 - 0.546675i) q^{37} +(-6.57607 - 3.59081i) q^{38} +(5.55114 + 0.798134i) q^{39} +(-1.61390 - 1.54768i) q^{40} +(-5.19777 + 3.34040i) q^{41} +(-3.25226 - 0.232606i) q^{42} +(-4.84144 - 8.86643i) q^{43} +(-0.217091 + 0.475362i) q^{44} +(-1.37827 - 1.52487i) q^{45} +(2.07286 + 4.32473i) q^{46} +(-0.388392 - 0.388392i) q^{47} +(-0.504099 - 1.35154i) q^{48} +(0.532671 - 1.81411i) q^{49} +(4.92596 + 0.857301i) q^{50} +(-6.32361 - 9.83973i) q^{51} +(3.87797 - 0.277358i) q^{52} +(7.32556 - 5.48385i) q^{53} +(-1.59276 - 5.42444i) q^{54} +(-0.190487 - 1.15291i) q^{55} +(-2.23737 + 0.321685i) q^{56} +(10.1265 + 3.77699i) q^{57} +(-2.40185 - 0.895842i) q^{58} +(-7.55765 + 1.08663i) q^{59} +(2.62206 + 1.87847i) q^{60} +(-3.43203 - 11.6884i) q^{61} +(0.760064 - 0.568977i) q^{62} +(-2.07250 + 0.148228i) q^{63} +(-0.540641 - 0.841254i) q^{64} +(-6.84892 + 5.35447i) q^{65} +(0.212378 - 0.723291i) q^{66} +(-3.30790 - 8.86882i) q^{67} +(-5.73361 - 5.73361i) q^{68} +(-3.84745 - 5.74935i) q^{69} +(3.74967 - 3.38917i) q^{70} +(-2.35355 + 5.15355i) q^{71} +(-0.440537 - 0.806783i) q^{72} +(0.283343 + 0.0202651i) q^{73} +(-2.16354 + 1.39042i) q^{74} +(-7.20931 - 0.212823i) q^{75} +(7.41630 + 1.06630i) q^{76} +(-1.03675 - 0.566110i) q^{77} +(-5.48006 + 1.19211i) q^{78} +(0.529838 + 3.68510i) q^{79} +(2.05301 + 0.886096i) q^{80} +(2.24214 + 4.90961i) q^{81} +(3.70270 - 4.94623i) q^{82} +(-1.46663 - 6.74199i) q^{83} +(3.12849 - 0.918609i) q^{84} +(17.7937 + 3.48220i) q^{85} +(7.63468 + 6.61549i) q^{86} +(3.61328 + 0.786020i) q^{87} +(0.0372809 - 0.521256i) q^{88} +(9.07615 + 2.66500i) q^{89} +(1.82425 + 0.947073i) q^{90} +8.78807i q^{91} +(-3.45350 - 3.32766i) q^{92} +(-0.968421 + 0.968421i) q^{93} +(0.499632 + 0.228174i) q^{94} +(-15.0908 + 7.27745i) q^{95} +(0.944630 + 1.09016i) q^{96} +(-1.22057 + 5.61087i) q^{97} +(0.134881 + 1.88588i) q^{98} +(0.0683643 - 0.475484i) 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
\(3\) 1.26604 − 0.691312i 0.730950 − 0.399129i −0.0702468 − 0.997530i \(-0.522379\pi\)
0.801197 + 0.598401i \(0.204197\pi\)
\(4\) 0.755750 − 0.654861i 0.377875 − 0.327430i
\(5\) −0.584903 + 2.15821i −0.261577 + 0.965183i
\(6\) −0.944630 + 1.09016i −0.385644 + 0.445056i
\(7\) 1.35459 + 1.80952i 0.511988 + 0.683936i 0.979820 − 0.199881i \(-0.0640557\pi\)
−0.467832 + 0.883817i \(0.654965\pi\)
\(8\) −0.479249 + 0.877679i −0.169440 + 0.310306i
\(9\) −0.496969 + 0.773299i −0.165656 + 0.257766i
\(10\) −0.206194 − 2.22654i −0.0652041 − 0.704094i
\(11\) −0.475362 + 0.217091i −0.143327 + 0.0654553i −0.485787 − 0.874077i \(-0.661467\pi\)
0.342460 + 0.939532i \(0.388740\pi\)
\(12\) 0.504099 − 1.35154i 0.145521 − 0.390156i
\(13\) 3.11241 + 2.32992i 0.863227 + 0.646204i 0.936330 − 0.351121i \(-0.114199\pi\)
−0.0731033 + 0.997324i \(0.523290\pi\)
\(14\) −1.90155 − 1.22205i −0.508211 − 0.326607i
\(15\) 0.751486 + 3.13674i 0.194033 + 0.809903i
\(16\) 0.142315 − 0.989821i 0.0355787 − 0.247455i
\(17\) −0.578458 − 8.08789i −0.140297 − 1.96160i −0.251641 − 0.967821i \(-0.580970\pi\)
0.111345 − 0.993782i \(-0.464484\pi\)
\(18\) 0.195395 − 0.898216i 0.0460550 − 0.211711i
\(19\) 4.90659 + 5.66250i 1.12565 + 1.29907i 0.949170 + 0.314763i \(0.101925\pi\)
0.176478 + 0.984305i \(0.443529\pi\)
\(20\) 0.971289 + 2.01410i 0.217187 + 0.450366i
\(21\) 2.96592 + 1.35449i 0.647217 + 0.295574i
\(22\) 0.369525 − 0.369525i 0.0787830 − 0.0787830i
\(23\) −0.430826 − 4.77644i −0.0898333 − 0.995957i
\(24\) 1.44249i 0.294447i
\(25\) −4.31578 − 2.52469i −0.863155 − 0.504939i
\(26\) −3.73039 − 1.09534i −0.731590 − 0.214814i
\(27\) −0.403312 + 5.63904i −0.0776174 + 1.08523i
\(28\) 2.20872 + 0.480478i 0.417409 + 0.0908017i
\(29\) 1.93734 + 1.67872i 0.359756 + 0.311730i 0.815919 − 0.578166i \(-0.196231\pi\)
−0.456164 + 0.889896i \(0.650777\pi\)
\(30\) −1.80028 − 2.67635i −0.328685 − 0.488633i
\(31\) −0.910979 + 0.267488i −0.163617 + 0.0480422i −0.362515 − 0.931978i \(-0.618082\pi\)
0.198898 + 0.980020i \(0.436264\pi\)
\(32\) 0.212565 + 0.977147i 0.0375766 + 0.172737i
\(33\) −0.451751 + 0.603469i −0.0786398 + 0.105051i
\(34\) 3.36841 + 7.37580i 0.577678 + 1.26494i
\(35\) −4.69765 + 1.86511i −0.794047 + 0.315260i
\(36\) 0.130819 + 0.909866i 0.0218032 + 0.151644i
\(37\) 2.51303 − 0.546675i 0.413139 − 0.0898729i −0.00119179 − 0.999999i \(-0.500379\pi\)
0.414331 + 0.910126i \(0.364016\pi\)
\(38\) −6.57607 − 3.59081i −1.06678 − 0.582506i
\(39\) 5.55114 + 0.798134i 0.888895 + 0.127804i
\(40\) −1.61390 − 1.54768i −0.255181 − 0.244710i
\(41\) −5.19777 + 3.34040i −0.811755 + 0.521683i −0.879432 − 0.476024i \(-0.842078\pi\)
0.0676770 + 0.997707i \(0.478441\pi\)
\(42\) −3.25226 − 0.232606i −0.501835 − 0.0358919i
\(43\) −4.84144 − 8.86643i −0.738312 − 1.35212i −0.928857 − 0.370439i \(-0.879207\pi\)
0.190544 − 0.981679i \(-0.438975\pi\)
\(44\) −0.217091 + 0.475362i −0.0327276 + 0.0716635i
\(45\) −1.37827 − 1.52487i −0.205460 − 0.227314i
\(46\) 2.07286 + 4.32473i 0.305626 + 0.637646i
\(47\) −0.388392 − 0.388392i −0.0566527 − 0.0566527i 0.678213 − 0.734866i \(-0.262755\pi\)
−0.734866 + 0.678213i \(0.762755\pi\)
\(48\) −0.504099 − 1.35154i −0.0727604 − 0.195078i
\(49\) 0.532671 − 1.81411i 0.0760959 − 0.259159i
\(50\) 4.92596 + 0.857301i 0.696635 + 0.121241i
\(51\) −6.32361 − 9.83973i −0.885482 − 1.37784i
\(52\) 3.87797 − 0.277358i 0.537778 − 0.0384627i
\(53\) 7.32556 − 5.48385i 1.00624 − 0.753264i 0.0369865 − 0.999316i \(-0.488224\pi\)
0.969257 + 0.246051i \(0.0791332\pi\)
\(54\) −1.59276 − 5.42444i −0.216747 − 0.738172i
\(55\) −0.190487 − 1.15291i −0.0256853 − 0.155458i
\(56\) −2.23737 + 0.321685i −0.298981 + 0.0429870i
\(57\) 10.1265 + 3.77699i 1.34129 + 0.500275i
\(58\) −2.40185 − 0.895842i −0.315378 − 0.117630i
\(59\) −7.55765 + 1.08663i −0.983922 + 0.141467i −0.615453 − 0.788174i \(-0.711027\pi\)
−0.368469 + 0.929640i \(0.620118\pi\)
\(60\) 2.62206 + 1.87847i 0.338507 + 0.242510i
\(61\) −3.43203 − 11.6884i −0.439426 − 1.49655i −0.820306 − 0.571924i \(-0.806197\pi\)
0.380880 − 0.924624i \(-0.375621\pi\)
\(62\) 0.760064 − 0.568977i 0.0965282 − 0.0722601i
\(63\) −2.07250 + 0.148228i −0.261110 + 0.0186749i
\(64\) −0.540641 − 0.841254i −0.0675801 − 0.105157i
\(65\) −6.84892 + 5.35447i −0.849504 + 0.664140i
\(66\) 0.212378 − 0.723291i 0.0261419 − 0.0890310i
\(67\) −3.30790 − 8.86882i −0.404124 − 1.08350i −0.966511 − 0.256624i \(-0.917390\pi\)
0.562387 − 0.826874i \(-0.309883\pi\)
\(68\) −5.73361 − 5.73361i −0.695303 − 0.695303i
\(69\) −3.84745 − 5.74935i −0.463179 − 0.692140i
\(70\) 3.74967 − 3.38917i 0.448172 − 0.405083i
\(71\) −2.35355 + 5.15355i −0.279315 + 0.611614i −0.996344 − 0.0854288i \(-0.972774\pi\)
0.717030 + 0.697043i \(0.245501\pi\)
\(72\) −0.440537 − 0.806783i −0.0519177 − 0.0950802i
\(73\) 0.283343 + 0.0202651i 0.0331628 + 0.00237185i 0.0879107 − 0.996128i \(-0.471981\pi\)
−0.0547479 + 0.998500i \(0.517436\pi\)
\(74\) −2.16354 + 1.39042i −0.251506 + 0.161633i
\(75\) −7.20931 − 0.212823i −0.832459 − 0.0245747i
\(76\) 7.41630 + 1.06630i 0.850708 + 0.122313i
\(77\) −1.03675 − 0.566110i −0.118149 − 0.0645142i
\(78\) −5.48006 + 1.19211i −0.620495 + 0.134980i
\(79\) 0.529838 + 3.68510i 0.0596114 + 0.414607i 0.997675 + 0.0681447i \(0.0217080\pi\)
−0.938064 + 0.346462i \(0.887383\pi\)
\(80\) 2.05301 + 0.886096i 0.229533 + 0.0990685i
\(81\) 2.24214 + 4.90961i 0.249127 + 0.545512i
\(82\) 3.70270 − 4.94623i 0.408895 − 0.546219i
\(83\) −1.46663 − 6.74199i −0.160983 − 0.740029i −0.984873 − 0.173278i \(-0.944564\pi\)
0.823889 − 0.566751i \(-0.191800\pi\)
\(84\) 3.12849 − 0.918609i 0.341347 − 0.100228i
\(85\) 17.7937 + 3.48220i 1.93000 + 0.377698i
\(86\) 7.63468 + 6.61549i 0.823269 + 0.713367i
\(87\) 3.61328 + 0.786020i 0.387384 + 0.0842702i
\(88\) 0.0372809 − 0.521256i 0.00397416 − 0.0555660i
\(89\) 9.07615 + 2.66500i 0.962069 + 0.282489i 0.724804 − 0.688956i \(-0.241931\pi\)
0.237266 + 0.971445i \(0.423749\pi\)
\(90\) 1.82425 + 0.947073i 0.192293 + 0.0998303i
\(91\) 8.78807i 0.921240i
\(92\) −3.45350 − 3.32766i −0.360052 − 0.346933i
\(93\) −0.968421 + 0.968421i −0.100421 + 0.100421i
\(94\) 0.499632 + 0.228174i 0.0515331 + 0.0235344i
\(95\) −15.0908 + 7.27745i −1.54828 + 0.746651i
\(96\) 0.944630 + 1.09016i 0.0964109 + 0.111264i
\(97\) −1.22057 + 5.61087i −0.123930 + 0.569698i 0.872742 + 0.488182i \(0.162340\pi\)
−0.996672 + 0.0815157i \(0.974024\pi\)
\(98\) 0.134881 + 1.88588i 0.0136250 + 0.190503i
\(99\) 0.0683643 − 0.475484i 0.00687088 − 0.0477880i
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.17.4 ✓ 240
5.3 odd 4 inner 230.2.l.a.63.10 yes 240
23.19 odd 22 inner 230.2.l.a.157.10 yes 240
115.88 even 44 inner 230.2.l.a.203.4 yes 240
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.l.a.17.4 ✓ 240 1.1 even 1 trivial
230.2.l.a.63.10 yes 240 5.3 odd 4 inner
230.2.l.a.157.10 yes 240 23.19 odd 22 inner
230.2.l.a.203.4 yes 240 115.88 even 44 inner