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Properties

Label 230.2.l.a.7.12
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 7.12
Character \(\chi\) \(=\) 230.7
Dual form 230.2.l.a.33.12

$q$-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} +(0.544276 - 2.50200i) q^{3} +(0.989821 - 0.142315i) q^{4} +(2.18999 + 0.451614i) q^{5} +(0.364399 - 2.53445i) q^{6} +(-0.470705 + 0.862032i) q^{7} +(0.977147 - 0.212565i) q^{8} +(-3.23486 - 1.47731i) q^{9} +(2.21663 + 0.294232i) q^{10} +(-4.13264 + 3.58095i) q^{11} +(0.182665 - 2.55399i) q^{12} +(-3.04895 + 1.66485i) q^{13} +(-0.408009 + 0.893415i) q^{14} +(2.32190 - 5.23354i) q^{15} +(0.959493 - 0.281733i) q^{16} +(0.102855 + 0.137398i) q^{17} +(-3.33201 - 1.24278i) q^{18} +(-0.241301 - 1.67828i) q^{19} +(2.23197 + 0.135350i) q^{20} +(1.90061 + 1.64689i) q^{21} +(-3.86665 + 3.86665i) q^{22} +(4.29439 - 2.13500i) q^{23} -2.56051i q^{24} +(4.59209 + 1.97806i) q^{25} +(-2.92241 + 1.87812i) q^{26} +(-0.853517 + 1.14017i) q^{27} +(-0.343234 + 0.920246i) q^{28} +(-5.01202 - 0.720619i) q^{29} +(1.94262 - 5.38585i) q^{30} +(-8.53830 - 5.48723i) q^{31} +(0.936950 - 0.349464i) q^{32} +(6.71024 + 12.2889i) q^{33} +(0.112394 + 0.129710i) q^{34} +(-1.42014 + 1.67526i) q^{35} +(-3.41218 - 1.00191i) q^{36} +(1.77232 + 4.75179i) q^{37} +(-0.360413 - 1.65679i) q^{38} +(2.50598 + 8.53460i) q^{39} +(2.23594 - 0.0242217i) q^{40} +(1.70890 + 3.74198i) q^{41} +(2.01325 + 1.50710i) q^{42} +(5.15679 + 1.12179i) q^{43} +(-3.58095 + 4.13264i) q^{44} +(-6.41713 - 4.69621i) q^{45} +(4.13114 - 2.43592i) q^{46} +(-5.74203 - 5.74203i) q^{47} +(-0.182665 - 2.55399i) q^{48} +(3.26295 + 5.07725i) q^{49} +(4.72150 + 1.64542i) q^{50} +(0.399750 - 0.182560i) q^{51} +(-2.78098 + 2.08182i) q^{52} +(9.18578 + 5.01582i) q^{53} +(-0.770004 + 1.19815i) q^{54} +(-10.6676 + 5.97588i) q^{55} +(-0.276710 + 0.942387i) q^{56} +(-4.33039 - 0.309716i) q^{57} +(-5.05066 - 0.361230i) q^{58} +(0.184613 - 0.628733i) q^{59} +(1.55345 - 5.51071i) q^{60} +(2.53267 - 3.94090i) q^{61} +(-8.90800 - 4.86414i) q^{62} +(2.79616 - 2.09318i) q^{63} +(0.909632 - 0.415415i) q^{64} +(-7.42902 + 2.26905i) q^{65} +(7.56982 + 11.7789i) q^{66} +(0.0211304 + 0.295442i) q^{67} +(0.121361 + 0.121361i) q^{68} +(-3.00444 - 11.9066i) q^{69} +(-1.29701 + 1.77231i) q^{70} +(6.31730 - 7.29055i) q^{71} +(-3.47496 - 0.755932i) q^{72} +(-1.54732 - 1.15831i) q^{73} +(2.10680 + 4.61324i) q^{74} +(7.44847 - 10.4128i) q^{75} +(-0.477689 - 1.62686i) q^{76} +(-1.14164 - 5.24804i) q^{77} +(3.10845 + 8.33408i) q^{78} +(-16.2406 - 4.76868i) q^{79} +(2.22851 - 0.183670i) q^{80} +(-4.59837 - 5.30680i) q^{81} +(1.97150 + 3.61053i) q^{82} +(4.28996 - 1.60007i) q^{83} +(2.11564 + 1.35964i) q^{84} +(0.163199 + 0.347350i) q^{85} +(5.22368 + 0.751051i) q^{86} +(-4.53091 + 12.1478i) q^{87} +(-3.27701 + 4.37757i) q^{88} +(8.23324 - 5.29118i) q^{89} +(-6.73581 - 4.22645i) q^{90} -3.41194i q^{91} +(3.94683 - 2.72443i) q^{92} +(-18.3762 + 18.3762i) q^{93} +(-6.13703 - 5.31777i) q^{94} +(0.229491 - 3.78439i) q^{95} +(-0.364399 - 2.53445i) q^{96} +(-9.77637 - 3.64640i) q^{97} +(3.61684 + 4.83154i) q^{98} +(18.6587 - 5.47869i) 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
\(3\) 0.544276 − 2.50200i 0.314238 − 1.44453i −0.501022 − 0.865435i \(-0.667042\pi\)
0.815260 − 0.579095i \(-0.196594\pi\)
\(4\) 0.989821 − 0.142315i 0.494911 − 0.0711574i
\(5\) 2.18999 + 0.451614i 0.979392 + 0.201968i
\(6\) 0.364399 − 2.53445i 0.148765 − 1.03469i
\(7\) −0.470705 + 0.862032i −0.177910 + 0.325817i −0.951382 − 0.308012i \(-0.900336\pi\)
0.773473 + 0.633830i \(0.218518\pi\)
\(8\) 0.977147 − 0.212565i 0.345474 − 0.0751532i
\(9\) −3.23486 − 1.47731i −1.07829 − 0.492438i
\(10\) 2.21663 + 0.294232i 0.700958 + 0.0930443i
\(11\) −4.13264 + 3.58095i −1.24604 + 1.07970i −0.252336 + 0.967640i \(0.581199\pi\)
−0.993702 + 0.112058i \(0.964256\pi\)
\(12\) 0.182665 − 2.55399i 0.0527308 − 0.737274i
\(13\) −3.04895 + 1.66485i −0.845625 + 0.461746i −0.842786 − 0.538250i \(-0.819086\pi\)
−0.00283996 + 0.999996i \(0.500904\pi\)
\(14\) −0.408009 + 0.893415i −0.109045 + 0.238775i
\(15\) 2.32190 − 5.23354i 0.599511 − 1.35129i
\(16\) 0.959493 − 0.281733i 0.239873 − 0.0704331i
\(17\) 0.102855 + 0.137398i 0.0249459 + 0.0333238i 0.812841 − 0.582486i \(-0.197920\pi\)
−0.787895 + 0.615810i \(0.788829\pi\)
\(18\) −3.33201 − 1.24278i −0.785363 − 0.292925i
\(19\) −0.241301 − 1.67828i −0.0553581 − 0.385024i −0.998599 − 0.0529149i \(-0.983149\pi\)
0.943241 − 0.332109i \(-0.107760\pi\)
\(20\) 2.23197 + 0.135350i 0.499083 + 0.0302652i
\(21\) 1.90061 + 1.64689i 0.414747 + 0.359380i
\(22\) −3.86665 + 3.86665i −0.824372 + 0.824372i
\(23\) 4.29439 − 2.13500i 0.895442 − 0.445179i
\(24\) − 2.56051i − 0.522663i
\(25\) 4.59209 + 1.97806i 0.918418 + 0.395612i
\(26\) −2.92241 + 1.87812i −0.573131 + 0.368329i
\(27\) −0.853517 + 1.14017i −0.164259 + 0.219425i
\(28\) −0.343234 + 0.920246i −0.0648651 + 0.173910i
\(29\) −5.01202 − 0.720619i −0.930708 − 0.133816i −0.339748 − 0.940516i \(-0.610342\pi\)
−0.590960 + 0.806701i \(0.701251\pi\)
\(30\) 1.94262 − 5.38585i 0.354673 − 0.983317i
\(31\) −8.53830 − 5.48723i −1.53352 − 0.985536i −0.989185 − 0.146670i \(-0.953144\pi\)
−0.544339 − 0.838866i \(-0.683219\pi\)
\(32\) 0.936950 − 0.349464i 0.165631 − 0.0617771i
\(33\) 6.71024 + 12.2889i 1.16810 + 2.13922i
\(34\) 0.112394 + 0.129710i 0.0192755 + 0.0222451i
\(35\) −1.42014 + 1.67526i −0.240048 + 0.283171i
\(36\) −3.41218 − 1.00191i −0.568697 − 0.166984i
\(37\) 1.77232 + 4.75179i 0.291368 + 0.781189i 0.997336 + 0.0729448i \(0.0232397\pi\)
−0.705968 + 0.708244i \(0.749488\pi\)
\(38\) −0.360413 − 1.65679i −0.0584667 − 0.268767i
\(39\) 2.50598 + 8.53460i 0.401279 + 1.36663i
\(40\) 2.23594 − 0.0242217i 0.353533 − 0.00382979i
\(41\) 1.70890 + 3.74198i 0.266886 + 0.584399i 0.994866 − 0.101199i \(-0.0322680\pi\)
−0.727980 + 0.685598i \(0.759541\pi\)
\(42\) 2.01325 + 1.50710i 0.310652 + 0.232551i
\(43\) 5.15679 + 1.12179i 0.786403 + 0.171071i 0.587797 − 0.809008i \(-0.299995\pi\)
0.198605 + 0.980080i \(0.436359\pi\)
\(44\) −3.58095 + 4.13264i −0.539849 + 0.623019i
\(45\) −6.41713 − 4.69621i −0.956610 − 0.700069i
\(46\) 4.13114 − 2.43592i 0.609103 − 0.359157i
\(47\) −5.74203 − 5.74203i −0.837561 − 0.837561i 0.150976 − 0.988537i \(-0.451758\pi\)
−0.988537 + 0.150976i \(0.951758\pi\)
\(48\) −0.182665 − 2.55399i −0.0263654 − 0.368637i
\(49\) 3.26295 + 5.07725i 0.466136 + 0.725321i
\(50\) 4.72150 + 1.64542i 0.667721 + 0.232698i
\(51\) 0.399750 − 0.182560i 0.0559762 − 0.0255635i
\(52\) −2.78098 + 2.08182i −0.385652 + 0.288696i
\(53\) 9.18578 + 5.01582i 1.26176 + 0.688975i 0.964060 − 0.265683i \(-0.0855974\pi\)
0.297704 + 0.954658i \(0.403779\pi\)
\(54\) −0.770004 + 1.19815i −0.104784 + 0.163047i
\(55\) −10.6676 + 5.97588i −1.43842 + 0.805788i
\(56\) −0.276710 + 0.942387i −0.0369769 + 0.125932i
\(57\) −4.33039 − 0.309716i −0.573575 − 0.0410229i
\(58\) −5.05066 − 0.361230i −0.663184 − 0.0474318i
\(59\) 0.184613 − 0.628733i 0.0240345 − 0.0818541i −0.946602 − 0.322403i \(-0.895509\pi\)
0.970637 + 0.240549i \(0.0773275\pi\)
\(60\) 1.55345 − 5.51071i 0.200550 − 0.711430i
\(61\) 2.53267 − 3.94090i 0.324275 − 0.504581i −0.640394 − 0.768046i \(-0.721229\pi\)
0.964669 + 0.263465i \(0.0848654\pi\)
\(62\) −8.90800 − 4.86414i −1.13132 − 0.617746i
\(63\) 2.79616 − 2.09318i 0.352283 − 0.263716i
\(64\) 0.909632 − 0.415415i 0.113704 − 0.0519269i
\(65\) −7.42902 + 2.26905i −0.921457 + 0.281441i
\(66\) 7.56982 + 11.7789i 0.931781 + 1.44988i
\(67\) 0.0211304 + 0.295442i 0.00258149 + 0.0360940i 0.998580 − 0.0532805i \(-0.0169678\pi\)
−0.995998 + 0.0893745i \(0.971513\pi\)
\(68\) 0.121361 + 0.121361i 0.0147172 + 0.0147172i
\(69\) −3.00444 − 11.9066i −0.361692 − 1.43338i
\(70\) −1.29701 + 1.77231i −0.155023 + 0.211831i
\(71\) 6.31730 − 7.29055i 0.749726 − 0.865230i −0.244816 − 0.969570i \(-0.578728\pi\)
0.994542 + 0.104340i \(0.0332730\pi\)
\(72\) −3.47496 − 0.755932i −0.409528 − 0.0890874i
\(73\) −1.54732 − 1.15831i −0.181100 − 0.135570i 0.504882 − 0.863189i \(-0.331536\pi\)
−0.685982 + 0.727619i \(0.740627\pi\)
\(74\) 2.10680 + 4.61324i 0.244910 + 0.536279i
\(75\) 7.44847 − 10.4128i 0.860075 − 1.20237i
\(76\) −0.477689 − 1.62686i −0.0547947 − 0.186614i
\(77\) −1.14164 − 5.24804i −0.130102 − 0.598070i
\(78\) 3.10845 + 8.33408i 0.351963 + 0.943648i
\(79\) −16.2406 − 4.76868i −1.82722 − 0.536519i −0.827525 − 0.561428i \(-0.810252\pi\)
−0.999690 + 0.0249096i \(0.992070\pi\)
\(80\) 2.22851 − 0.183670i 0.249155 − 0.0205349i
\(81\) −4.59837 − 5.30680i −0.510930 − 0.589645i
\(82\) 1.97150 + 3.61053i 0.217716 + 0.398717i
\(83\) 4.28996 − 1.60007i 0.470885 − 0.175631i −0.102809 − 0.994701i \(-0.532783\pi\)
0.573693 + 0.819070i \(0.305510\pi\)
\(84\) 2.11564 + 1.35964i 0.230835 + 0.148349i
\(85\) 0.163199 + 0.347350i 0.0177015 + 0.0376754i
\(86\) 5.22368 + 0.751051i 0.563283 + 0.0809879i
\(87\) −4.53091 + 12.1478i −0.485765 + 1.30239i
\(88\) −3.27701 + 4.37757i −0.349330 + 0.466651i
\(89\) 8.23324 − 5.29118i 0.872722 − 0.560864i −0.0258624 − 0.999666i \(-0.508233\pi\)
0.898584 + 0.438801i \(0.144597\pi\)
\(90\) −6.73581 − 4.22645i −0.710016 − 0.445507i
\(91\) − 3.41194i − 0.357669i
\(92\) 3.94683 − 2.72443i 0.411486 − 0.284041i
\(93\) −18.3762 + 18.3762i −1.90553 + 1.90553i
\(94\) −6.13703 − 5.31777i −0.632986 − 0.548486i
\(95\) 0.229491 − 3.78439i 0.0235453 − 0.388270i
\(96\) −0.364399 − 2.53445i −0.0371913 − 0.258671i
\(97\) −9.77637 − 3.64640i −0.992640 − 0.370235i −0.199914 − 0.979814i \(-0.564066\pi\)
−0.792726 + 0.609578i \(0.791339\pi\)
\(98\) 3.61684 + 4.83154i 0.365356 + 0.488059i
\(99\) 18.6587 − 5.47869i 1.87527 − 0.550629i
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.7.12 ✓ 240
5.3 odd 4 inner 230.2.l.a.53.6 yes 240
23.10 odd 22 inner 230.2.l.a.217.6 yes 240
115.33 even 44 inner 230.2.l.a.33.12 yes 240
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.l.a.7.12 ✓ 240 1.1 even 1 trivial
230.2.l.a.33.12 yes 240 115.33 even 44 inner
230.2.l.a.53.6 yes 240 5.3 odd 4 inner
230.2.l.a.217.6 yes 240 23.10 odd 22 inner