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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 17.2
Character \(\chi\) \(=\) 230.17
Dual form 230.2.l.a.203.2

$q$-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+(-0.936950 + 0.349464i) q^{2} +(-1.69758 + 0.926949i) q^{3} +(0.755750 - 0.654861i) q^{4} +(-1.50435 + 1.65436i) q^{5} +(1.26661 - 1.46175i) q^{6} +(-0.735871 - 0.983009i) q^{7} +(-0.479249 + 0.877679i) q^{8} +(0.400622 - 0.623380i) q^{9} +(0.831363 - 2.07577i) q^{10} +(3.08362 - 1.40824i) q^{11} +(-0.675923 + 1.81222i) q^{12} +(-5.47285 - 4.09692i) q^{13} +(1.03300 + 0.663869i) q^{14} +(1.02025 - 4.20287i) q^{15} +(0.142315 - 0.989821i) q^{16} +(-0.214766 - 3.00282i) q^{17} +(-0.157514 + 0.724078i) q^{18} +(-1.88225 - 2.17223i) q^{19} +(-0.0535372 + 2.23543i) q^{20} +(2.16040 + 0.986621i) q^{21} +(-2.39707 + 2.39707i) q^{22} +(3.93443 + 2.74231i) q^{23} -1.93417i q^{24} +(-0.473838 - 4.97750i) q^{25} +(6.55951 + 1.92605i) q^{26} +(0.311700 - 4.35813i) q^{27} +(-1.19987 - 0.261015i) q^{28} +(-3.99884 - 3.46501i) q^{29} +(0.512830 + 4.29442i) q^{30} +(-5.92845 + 1.74075i) q^{31} +(0.212565 + 0.977147i) q^{32} +(-3.92933 + 5.24896i) q^{33} +(1.25060 + 2.73844i) q^{34} +(2.73326 + 0.261395i) q^{35} +(-0.105457 - 0.733470i) q^{36} +(-6.54487 + 1.42375i) q^{37} +(2.52269 + 1.37749i) q^{38} +(13.0882 + 1.88180i) q^{39} +(-0.731040 - 2.11319i) q^{40} +(-2.34072 + 1.50429i) q^{41} +(-2.36897 - 0.169432i) q^{42} +(4.04137 + 7.40121i) q^{43} +(1.40824 - 3.08362i) q^{44} +(0.428620 + 1.60056i) q^{45} +(-4.64470 - 1.19446i) q^{46} +(5.79351 + 5.79351i) q^{47} +(0.675923 + 1.81222i) q^{48} +(1.54733 - 5.26972i) q^{49} +(2.18342 + 4.49808i) q^{50} +(3.14805 + 4.89846i) q^{51} +(-6.81902 + 0.487706i) q^{52} +(-8.41893 + 6.30233i) q^{53} +(1.23096 + 4.19228i) q^{54} +(-2.30911 + 7.21993i) q^{55} +(1.21543 - 0.174753i) q^{56} +(5.20882 + 1.94279i) q^{57} +(4.95761 + 1.84909i) q^{58} +(-7.66795 + 1.10249i) q^{59} +(-1.98124 - 3.84444i) q^{60} +(0.704279 + 2.39855i) q^{61} +(4.94633 - 3.70278i) q^{62} +(-0.907593 + 0.0649124i) q^{63} +(-0.540641 - 0.841254i) q^{64} +(15.0109 - 2.89086i) q^{65} +(1.84726 - 6.29117i) q^{66} +(-3.20331 - 8.58840i) q^{67} +(-2.12874 - 2.12874i) q^{68} +(-9.22099 - 1.00827i) q^{69} +(-2.65228 + 0.710264i) q^{70} +(-1.74986 + 3.83166i) q^{71} +(0.355130 + 0.650371i) q^{72} +(0.769439 + 0.0550313i) q^{73} +(5.63467 - 3.62118i) q^{74} +(5.41826 + 8.01048i) q^{75} +(-2.84502 - 0.409052i) q^{76} +(-3.65346 - 1.99494i) q^{77} +(-12.9206 + 2.81071i) q^{78} +(-0.551491 - 3.83570i) q^{79} +(1.42343 + 1.72448i) q^{80} +(4.43411 + 9.70936i) q^{81} +(1.66744 - 2.22744i) q^{82} +(-2.57122 - 11.8197i) q^{83} +(2.27882 - 0.669122i) q^{84} +(5.29085 + 4.16201i) q^{85} +(-6.37301 - 5.52225i) q^{86} +(10.0002 + 2.17542i) q^{87} +(-0.241837 + 3.38133i) q^{88} +(-10.7709 - 3.16262i) q^{89} +(-0.960932 - 1.34985i) q^{90} +8.39466i q^{91} +(4.76927 - 0.504008i) q^{92} +(8.45044 - 8.45044i) q^{93} +(-7.45285 - 3.40360i) q^{94} +(6.42523 + 0.153881i) q^{95} +(-1.26661 - 1.46175i) q^{96} +(1.36894 - 6.29292i) q^{97} +(0.391808 + 5.47819i) q^{98} +(0.357496 - 2.48644i) 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
\(3\) −1.69758 + 0.926949i −0.980098 + 0.535174i −0.887638 − 0.460542i \(-0.847655\pi\)
−0.0924605 + 0.995716i \(0.529473\pi\)
\(4\) 0.755750 − 0.654861i 0.377875 − 0.327430i
\(5\) −1.50435 + 1.65436i −0.672768 + 0.739854i
\(6\) 1.26661 − 1.46175i 0.517092 − 0.596756i
\(7\) −0.735871 − 0.983009i −0.278133 − 0.371542i 0.639622 − 0.768690i \(-0.279091\pi\)
−0.917755 + 0.397148i \(0.870000\pi\)
\(8\) −0.479249 + 0.877679i −0.169440 + 0.310306i
\(9\) 0.400622 − 0.623380i 0.133541 − 0.207793i
\(10\) 0.831363 − 2.07577i 0.262900 − 0.656417i
\(11\) 3.08362 − 1.40824i 0.929747 − 0.424601i 0.107805 − 0.994172i \(-0.465618\pi\)
0.821942 + 0.569571i \(0.192891\pi\)
\(12\) −0.675923 + 1.81222i −0.195122 + 0.523143i
\(13\) −5.47285 − 4.09692i −1.51790 − 1.13628i −0.949603 − 0.313455i \(-0.898513\pi\)
−0.568292 − 0.822827i \(-0.692396\pi\)
\(14\) 1.03300 + 0.663869i 0.276081 + 0.177426i
\(15\) 1.02025 − 4.20287i 0.263428 − 1.08518i
\(16\) 0.142315 − 0.989821i 0.0355787 − 0.247455i
\(17\) −0.214766 − 3.00282i −0.0520885 − 0.728292i −0.954650 − 0.297731i \(-0.903770\pi\)
0.902561 − 0.430561i \(-0.141684\pi\)
\(18\) −0.157514 + 0.724078i −0.0371263 + 0.170667i
\(19\) −1.88225 − 2.17223i −0.431818 − 0.498344i 0.497583 − 0.867416i \(-0.334221\pi\)
−0.929401 + 0.369072i \(0.879675\pi\)
\(20\) −0.0535372 + 2.23543i −0.0119713 + 0.499857i
\(21\) 2.16040 + 0.986621i 0.471438 + 0.215298i
\(22\) −2.39707 + 2.39707i −0.511057 + 0.511057i
\(23\) 3.93443 + 2.74231i 0.820386 + 0.571810i
\(24\) − 1.93417i − 0.394811i
\(25\) −0.473838 − 4.97750i −0.0947675 − 0.995499i
\(26\) 6.55951 + 1.92605i 1.28643 + 0.377729i
\(27\) 0.311700 − 4.35813i 0.0599867 − 0.838723i
\(28\) −1.19987 − 0.261015i −0.226754 − 0.0493273i
\(29\) −3.99884 − 3.46501i −0.742565 − 0.643436i 0.199103 − 0.979979i \(-0.436197\pi\)
−0.941668 + 0.336542i \(0.890743\pi\)
\(30\) 0.512830 + 4.29442i 0.0936295 + 0.784051i
\(31\) −5.92845 + 1.74075i −1.06478 + 0.312648i −0.766775 − 0.641916i \(-0.778140\pi\)
−0.298007 + 0.954564i \(0.596322\pi\)
\(32\) 0.212565 + 0.977147i 0.0375766 + 0.172737i
\(33\) −3.92933 + 5.24896i −0.684008 + 0.913727i
\(34\) 1.25060 + 2.73844i 0.214477 + 0.469639i
\(35\) 2.73326 + 0.261395i 0.462006 + 0.0441838i
\(36\) −0.105457 − 0.733470i −0.0175762 − 0.122245i
\(37\) −6.54487 + 1.42375i −1.07597 + 0.234063i −0.715433 − 0.698682i \(-0.753770\pi\)
−0.360538 + 0.932745i \(0.617407\pi\)
\(38\) 2.52269 + 1.37749i 0.409234 + 0.223459i
\(39\) 13.0882 + 1.88180i 2.09580 + 0.301330i
\(40\) −0.731040 − 2.11319i −0.115588 − 0.334125i
\(41\) −2.34072 + 1.50429i −0.365559 + 0.234930i −0.710505 − 0.703692i \(-0.751533\pi\)
0.344946 + 0.938622i \(0.387897\pi\)
\(42\) −2.36897 − 0.169432i −0.365541 − 0.0261440i
\(43\) 4.04137 + 7.40121i 0.616302 + 1.12867i 0.980345 + 0.197291i \(0.0632145\pi\)
−0.364042 + 0.931382i \(0.618604\pi\)
\(44\) 1.40824 − 3.08362i 0.212301 − 0.464873i
\(45\) 0.428620 + 1.60056i 0.0638948 + 0.238597i
\(46\) −4.64470 − 1.19446i −0.684824 − 0.176114i
\(47\) 5.79351 + 5.79351i 0.845070 + 0.845070i 0.989513 − 0.144443i \(-0.0461389\pi\)
−0.144443 + 0.989513i \(0.546139\pi\)
\(48\) 0.675923 + 1.81222i 0.0975611 + 0.261571i
\(49\) 1.54733 − 5.26972i 0.221047 − 0.752817i
\(50\) 2.18342 + 4.49808i 0.308782 + 0.636124i
\(51\) 3.14805 + 4.89846i 0.440815 + 0.685921i
\(52\) −6.81902 + 0.487706i −0.945628 + 0.0676326i
\(53\) −8.41893 + 6.30233i −1.15643 + 0.865692i −0.992676 − 0.120809i \(-0.961451\pi\)
−0.163754 + 0.986501i \(0.552360\pi\)
\(54\) 1.23096 + 4.19228i 0.167513 + 0.570497i
\(55\) −2.30911 + 7.21993i −0.311361 + 0.973535i
\(56\) 1.21543 − 0.174753i 0.162419 − 0.0233523i
\(57\) 5.20882 + 1.94279i 0.689925 + 0.257329i
\(58\) 4.95761 + 1.84909i 0.650966 + 0.242798i
\(59\) −7.66795 + 1.10249i −0.998282 + 0.143531i −0.622036 − 0.782988i \(-0.713694\pi\)
−0.376246 + 0.926520i \(0.622785\pi\)
\(60\) −1.98124 − 3.84444i −0.255777 − 0.496315i
\(61\) 0.704279 + 2.39855i 0.0901737 + 0.307103i 0.992213 − 0.124550i \(-0.0397486\pi\)
−0.902040 + 0.431653i \(0.857930\pi\)
\(62\) 4.94633 − 3.70278i 0.628185 − 0.470253i
\(63\) −0.907593 + 0.0649124i −0.114346 + 0.00817819i
\(64\) −0.540641 − 0.841254i −0.0675801 − 0.105157i
\(65\) 15.0109 − 2.89086i 1.86187 − 0.358567i
\(66\) 1.84726 − 6.29117i 0.227381 − 0.774390i
\(67\) −3.20331 − 8.58840i −0.391346 − 1.04924i −0.971978 − 0.235073i \(-0.924467\pi\)
0.580631 − 0.814167i \(-0.302806\pi\)
\(68\) −2.12874 − 2.12874i −0.258148 − 0.258148i
\(69\) −9.22099 − 1.00827i −1.11008 − 0.121381i
\(70\) −2.65228 + 0.710264i −0.317008 + 0.0848928i
\(71\) −1.74986 + 3.83166i −0.207670 + 0.454734i −0.984593 − 0.174862i \(-0.944052\pi\)
0.776923 + 0.629596i \(0.216779\pi\)
\(72\) 0.355130 + 0.650371i 0.0418524 + 0.0766470i
\(73\) 0.769439 + 0.0550313i 0.0900560 + 0.00644093i 0.116294 − 0.993215i \(-0.462898\pi\)
−0.0262385 + 0.999656i \(0.508353\pi\)
\(74\) 5.63467 − 3.62118i 0.655017 − 0.420954i
\(75\) 5.41826 + 8.01048i 0.625647 + 0.924970i
\(76\) −2.84502 − 0.409052i −0.326346 − 0.0469215i
\(77\) −3.65346 − 1.99494i −0.416351 − 0.227345i
\(78\) −12.9206 + 2.81071i −1.46297 + 0.318251i
\(79\) −0.551491 − 3.83570i −0.0620476 − 0.431550i −0.997040 − 0.0768830i \(-0.975503\pi\)
0.934993 − 0.354667i \(-0.115406\pi\)
\(80\) 1.42343 + 1.72448i 0.159145 + 0.192803i
\(81\) 4.43411 + 9.70936i 0.492679 + 1.07882i
\(82\) 1.66744 − 2.22744i 0.184138 − 0.245979i
\(83\) −2.57122 − 11.8197i −0.282228 − 1.29738i −0.871879 − 0.489722i \(-0.837098\pi\)
0.589651 − 0.807658i \(-0.299265\pi\)
\(84\) 2.27882 − 0.669122i 0.248640 − 0.0730072i
\(85\) 5.29085 + 4.16201i 0.573873 + 0.451433i
\(86\) −6.37301 − 5.52225i −0.687220 − 0.595479i
\(87\) 10.0002 + 2.17542i 1.07214 + 0.233229i
\(88\) −0.241837 + 3.38133i −0.0257800 + 0.360451i
\(89\) −10.7709 − 3.16262i −1.14171 − 0.335237i −0.344412 − 0.938819i \(-0.611922\pi\)
−0.797302 + 0.603581i \(0.793740\pi\)
\(90\) −0.960932 − 1.34985i −0.101291 − 0.142287i
\(91\) 8.39466i 0.880000i
\(92\) 4.76927 − 0.504008i 0.497231 − 0.0525465i
\(93\) 8.45044 − 8.45044i 0.876270 − 0.876270i
\(94\) −7.45285 − 3.40360i −0.768703 − 0.351055i
\(95\) 6.42523 + 0.153881i 0.659215 + 0.0157878i
\(96\) −1.26661 − 1.46175i −0.129273 − 0.149189i
\(97\) 1.36894 − 6.29292i 0.138995 − 0.638949i −0.853948 − 0.520358i \(-0.825799\pi\)
0.992943 − 0.118591i \(-0.0378378\pi\)
\(98\) 0.391808 + 5.47819i 0.0395786 + 0.553381i
\(99\) 0.357496 − 2.48644i 0.0359297 − 0.249897i
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.17.2 ✓ 240
5.3 odd 4 inner 230.2.l.a.63.8 yes 240
23.19 odd 22 inner 230.2.l.a.157.8 yes 240
115.88 even 44 inner 230.2.l.a.203.2 yes 240
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.l.a.17.2 ✓ 240 1.1 even 1 trivial
230.2.l.a.63.8 yes 240 5.3 odd 4 inner
230.2.l.a.157.8 yes 240 23.19 odd 22 inner
230.2.l.a.203.2 yes 240 115.88 even 44 inner