↑

Properties

Label 230.2.l.a.7.9
Level $230$
Weight $2$
Character 230.7
Analytic conductor $1.837$
Analytic rank $0$
Dimension $240$
Inner twists $4$

Related objects

Downloads

Learn more

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 7.9
Character \(\chi\) \(=\) 230.7
Dual form 230.2.l.a.33.9

$q$-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} +(-0.307156 + 1.41198i) q^{3} +(0.989821 - 0.142315i) q^{4} +(1.55520 - 1.60666i) q^{5} +(-0.205645 + 1.43029i) q^{6} +(-1.84581 + 3.38035i) q^{7} +(0.977147 - 0.212565i) q^{8} +(0.829567 + 0.378851i) q^{9} +(1.43661 - 1.71352i) q^{10} +(3.85970 - 3.34445i) q^{11} +(-0.103085 + 1.44132i) q^{12} +(-3.94532 + 2.15431i) q^{13} +(-1.59996 + 3.50342i) q^{14} +(1.79088 + 2.68939i) q^{15} +(0.959493 - 0.281733i) q^{16} +(-2.64144 - 3.52855i) q^{17} +(0.854480 + 0.318705i) q^{18} +(0.596940 + 4.15181i) q^{19} +(1.31071 - 1.81164i) q^{20} +(-4.20602 - 3.64454i) q^{21} +(3.61128 - 3.61128i) q^{22} +(2.37283 - 4.16769i) q^{23} +1.44500i q^{24} +(-0.162734 - 4.99735i) q^{25} +(-3.78158 + 2.43027i) q^{26} +(-3.38760 + 4.52530i) q^{27} +(-1.34595 + 3.60863i) q^{28} +(-5.29900 - 0.761882i) q^{29} +(1.97818 + 2.55478i) q^{30} +(2.10320 + 1.35164i) q^{31} +(0.936950 - 0.349464i) q^{32} +(3.53675 + 6.47707i) q^{33} +(-2.88643 - 3.33112i) q^{34} +(2.56049 + 8.22271i) q^{35} +(0.875039 + 0.256935i) q^{36} +(-3.94954 - 10.5891i) q^{37} +(0.891605 + 4.09864i) q^{38} +(-1.83000 - 6.23240i) q^{39} +(1.17813 - 1.90053i) q^{40} +(-2.05845 - 4.50738i) q^{41} +(-4.45530 - 3.33520i) q^{42} +(-3.43244 - 0.746682i) q^{43} +(3.34445 - 3.85970i) q^{44} +(1.89882 - 0.743648i) q^{45} +(2.06947 - 4.32635i) q^{46} +(-4.84820 - 4.84820i) q^{47} +(0.103085 + 1.44132i) q^{48} +(-4.23527 - 6.59021i) q^{49} +(-0.518826 - 4.97301i) q^{50} +(5.79356 - 2.64583i) q^{51} +(-3.59857 + 2.69386i) q^{52} +(8.03345 + 4.38659i) q^{53} +(-3.05614 + 4.75544i) q^{54} +(0.629185 - 11.4025i) q^{55} +(-1.08508 + 3.69546i) q^{56} +(-6.04560 - 0.432390i) q^{57} +(-5.33986 - 0.381914i) q^{58} +(0.493606 - 1.68107i) q^{59} +(2.15539 + 2.40715i) q^{60} +(-0.855388 + 1.33101i) q^{61} +(2.19426 + 1.19816i) q^{62} +(-2.81187 + 2.10494i) q^{63} +(0.909632 - 0.415415i) q^{64} +(-2.67450 + 9.68917i) q^{65} +(3.98981 + 6.20826i) q^{66} +(0.499531 + 6.98435i) q^{67} +(-3.11672 - 3.11672i) q^{68} +(5.15585 + 4.63052i) q^{69} +(3.14057 + 8.01909i) q^{70} +(-9.09632 + 10.4977i) q^{71} +(0.891139 + 0.193855i) q^{72} +(3.61933 + 2.70939i) q^{73} +(-4.69489 - 10.2804i) q^{74} +(7.10612 + 1.30519i) q^{75} +(1.18173 + 4.02459i) q^{76} +(4.18114 + 19.2204i) q^{77} +(-2.26995 - 6.08597i) q^{78} +(4.81604 + 1.41412i) q^{79} +(1.03955 - 1.97973i) q^{80} +(-3.55743 - 4.10549i) q^{81} +(-2.37476 - 4.34904i) q^{82} +(5.41151 - 2.01839i) q^{83} +(-4.68188 - 3.00886i) q^{84} +(-9.77715 - 1.24368i) q^{85} +(-3.47696 - 0.499912i) q^{86} +(2.70338 - 7.24805i) q^{87} +(3.06058 - 4.08846i) q^{88} +(0.368885 - 0.237068i) q^{89} +(1.84094 - 0.877214i) q^{90} -17.3130i q^{91} +(1.75556 - 4.46296i) q^{92} +(-2.55450 + 2.55450i) q^{93} +(-5.18171 - 4.48998i) q^{94} +(7.59891 + 5.49779i) q^{95} +(0.205645 + 1.43029i) q^{96} +(16.6014 + 6.19201i) q^{97} +(-4.69462 - 6.27128i) q^{98} +(4.46893 - 1.31220i) 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
\(3\) −0.307156 + 1.41198i −0.177337 + 0.815204i 0.799406 + 0.600791i \(0.205148\pi\)
−0.976743 + 0.214413i \(0.931216\pi\)
\(4\) 0.989821 − 0.142315i 0.494911 − 0.0711574i
\(5\) 1.55520 − 1.60666i 0.695505 − 0.718522i
\(6\) −0.205645 + 1.43029i −0.0839541 + 0.583913i
\(7\) −1.84581 + 3.38035i −0.697651 + 1.27765i 0.253023 + 0.967460i \(0.418575\pi\)
−0.950674 + 0.310192i \(0.899607\pi\)
\(8\) 0.977147 − 0.212565i 0.345474 − 0.0751532i
\(9\) 0.829567 + 0.378851i 0.276522 + 0.126284i
\(10\) 1.43661 − 1.71352i 0.454298 − 0.541861i
\(11\) 3.85970 − 3.34445i 1.16374 − 1.00839i 0.163985 − 0.986463i \(-0.447565\pi\)
0.999760 − 0.0219272i \(-0.00698020\pi\)
\(12\) −0.103085 + 1.44132i −0.0297581 + 0.416072i
\(13\) −3.94532 + 2.15431i −1.09423 + 0.597497i −0.921856 − 0.387533i \(-0.873327\pi\)
−0.172379 + 0.985031i \(0.555145\pi\)
\(14\) −1.59996 + 3.50342i −0.427607 + 0.936328i
\(15\) 1.79088 + 2.68939i 0.462403 + 0.694399i
\(16\) 0.959493 − 0.281733i 0.239873 − 0.0704331i
\(17\) −2.64144 − 3.52855i −0.640643 − 0.855799i 0.356264 − 0.934385i \(-0.384050\pi\)
−0.996907 + 0.0785860i \(0.974959\pi\)
\(18\) 0.854480 + 0.318705i 0.201403 + 0.0751194i
\(19\) 0.596940 + 4.15181i 0.136947 + 0.952490i 0.936193 + 0.351486i \(0.114323\pi\)
−0.799246 + 0.601004i \(0.794768\pi\)
\(20\) 1.31071 − 1.81164i 0.293085 − 0.405094i
\(21\) −4.20602 − 3.64454i −0.917829 − 0.795303i
\(22\) 3.61128 − 3.61128i 0.769927 − 0.769927i
\(23\) 2.37283 − 4.16769i 0.494770 − 0.869024i
\(24\) 1.44500i 0.294959i
\(25\) −0.162734 − 4.99735i −0.0325468 − 0.999470i
\(26\) −3.78158 + 2.43027i −0.741629 + 0.476616i
\(27\) −3.38760 + 4.52530i −0.651944 + 0.870895i
\(28\) −1.34595 + 3.60863i −0.254361 + 0.681967i
\(29\) −5.29900 − 0.761882i −0.984000 − 0.141478i −0.368511 − 0.929623i \(-0.620132\pi\)
−0.615489 + 0.788145i \(0.711042\pi\)
\(30\) 1.97818 + 2.55478i 0.361164 + 0.466437i
\(31\) 2.10320 + 1.35164i 0.377745 + 0.242762i 0.715712 − 0.698396i \(-0.246103\pi\)
−0.337966 + 0.941158i \(0.609739\pi\)
\(32\) 0.936950 − 0.349464i 0.165631 − 0.0617771i
\(33\) 3.53675 + 6.47707i 0.615669 + 1.12751i
\(34\) −2.88643 − 3.33112i −0.495019 − 0.571283i
\(35\) 2.56049 + 8.22271i 0.432802 + 1.38989i
\(36\) 0.875039 + 0.256935i 0.145840 + 0.0428225i
\(37\) −3.94954 − 10.5891i −0.649300 − 1.74084i −0.671412 − 0.741085i \(-0.734312\pi\)
0.0221119 − 0.999756i \(-0.492961\pi\)
\(38\) 0.891605 + 4.09864i 0.144637 + 0.664888i
\(39\) −1.83000 − 6.23240i −0.293034 − 0.997983i
\(40\) 1.17813 − 1.90053i 0.186279 − 0.300500i
\(41\) −2.05845 − 4.50738i −0.321476 − 0.703934i 0.678041 − 0.735024i \(-0.262829\pi\)
−0.999517 + 0.0310905i \(0.990102\pi\)
\(42\) −4.45530 − 3.33520i −0.687468 − 0.514632i
\(43\) −3.43244 − 0.746682i −0.523442 − 0.113868i −0.0569224 − 0.998379i \(-0.518129\pi\)
−0.466520 + 0.884511i \(0.654492\pi\)
\(44\) 3.34445 − 3.85970i 0.504195 − 0.581872i
\(45\) 1.89882 − 0.743648i 0.283060 − 0.110857i
\(46\) 2.06947 − 4.32635i 0.305126 − 0.637885i
\(47\) −4.84820 − 4.84820i −0.707183 − 0.707183i 0.258759 − 0.965942i \(-0.416686\pi\)
−0.965942 + 0.258759i \(0.916686\pi\)
\(48\) 0.103085 + 1.44132i 0.0148790 + 0.208036i
\(49\) −4.23527 − 6.59021i −0.605039 − 0.941458i
\(50\) −0.518826 − 4.97301i −0.0733731 − 0.703290i
\(51\) 5.79356 − 2.64583i 0.811261 − 0.370490i
\(52\) −3.59857 + 2.69386i −0.499032 + 0.373571i
\(53\) 8.03345 + 4.38659i 1.10348 + 0.602545i 0.924452 − 0.381299i \(-0.124523\pi\)
0.179027 + 0.983844i \(0.442705\pi\)
\(54\) −3.05614 + 4.75544i −0.415888 + 0.647134i
\(55\) 0.629185 − 11.4025i 0.0848393 − 1.53752i
\(56\) −1.08508 + 3.69546i −0.145000 + 0.493826i
\(57\) −6.04560 − 0.432390i −0.800759 − 0.0572714i
\(58\) −5.33986 − 0.381914i −0.701157 − 0.0501478i
\(59\) 0.493606 − 1.68107i 0.0642621 − 0.218857i −0.921101 − 0.389323i \(-0.872709\pi\)
0.985364 + 0.170466i \(0.0545273\pi\)
\(60\) 2.15539 + 2.40715i 0.278260 + 0.310762i
\(61\) −0.855388 + 1.33101i −0.109521 + 0.170418i −0.891688 − 0.452651i \(-0.850478\pi\)
0.782166 + 0.623070i \(0.214115\pi\)
\(62\) 2.19426 + 1.19816i 0.278672 + 0.152166i
\(63\) −2.81187 + 2.10494i −0.354263 + 0.265198i
\(64\) 0.909632 − 0.415415i 0.113704 − 0.0519269i
\(65\) −2.67450 + 9.68917i −0.331730 + 1.20179i
\(66\) 3.98981 + 6.20826i 0.491111 + 0.764184i
\(67\) 0.499531 + 6.98435i 0.0610274 + 0.853274i 0.932119 + 0.362153i \(0.117958\pi\)
−0.871091 + 0.491121i \(0.836587\pi\)
\(68\) −3.11672 − 3.11672i −0.377958 − 0.377958i
\(69\) 5.15585 + 4.63052i 0.620691 + 0.557449i
\(70\) 3.14057 + 8.01909i 0.375369 + 0.958465i
\(71\) −9.09632 + 10.4977i −1.07953 + 1.24585i −0.111839 + 0.993726i \(0.535674\pi\)
−0.967695 + 0.252123i \(0.918871\pi\)
\(72\) 0.891139 + 0.193855i 0.105022 + 0.0228461i
\(73\) 3.61933 + 2.70939i 0.423610 + 0.317111i 0.789727 − 0.613458i \(-0.210222\pi\)
−0.366117 + 0.930569i \(0.619313\pi\)
\(74\) −4.69489 − 10.2804i −0.545770 − 1.19507i
\(75\) 7.10612 + 1.30519i 0.820544 + 0.150711i
\(76\) 1.18173 + 4.02459i 0.135553 + 0.461653i
\(77\) 4.18114 + 19.2204i 0.476485 + 2.19037i
\(78\) −2.26995 − 6.08597i −0.257021 − 0.689101i
\(79\) 4.81604 + 1.41412i 0.541847 + 0.159101i 0.541192 − 0.840899i \(-0.317973\pi\)
0.000655108 1.00000i \(0.499791\pi\)
\(80\) 1.03955 − 1.97973i 0.116225 − 0.221341i
\(81\) −3.55743 − 4.10549i −0.395270 − 0.456166i
\(82\) −2.37476 − 4.34904i −0.262248 − 0.480271i
\(83\) 5.41151 − 2.01839i 0.593991 − 0.221547i −0.0344476 − 0.999407i \(-0.510967\pi\)
0.628438 + 0.777859i \(0.283694\pi\)
\(84\) −4.68188 − 3.00886i −0.510835 − 0.328294i
\(85\) −9.77715 − 1.24368i −1.06048 − 0.134896i
\(86\) −3.47696 − 0.499912i −0.374931 − 0.0539069i
\(87\) 2.70338 − 7.24805i 0.289833 − 0.777072i
\(88\) 3.06058 − 4.08846i 0.326259 − 0.435831i
\(89\) 0.368885 − 0.237068i 0.0391017 − 0.0251291i −0.520944 − 0.853591i \(-0.674420\pi\)
0.560046 + 0.828462i \(0.310784\pi\)
\(90\) 1.84094 − 0.877214i 0.194052 − 0.0924665i
\(91\) − 17.3130i − 1.81490i
\(92\) 1.75556 − 4.46296i 0.183030 − 0.465296i
\(93\) −2.55450 + 2.55450i −0.264889 + 0.264889i
\(94\) −5.18171 − 4.48998i −0.534453 − 0.463106i
\(95\) 7.59891 + 5.49779i 0.779632 + 0.564061i
\(96\) 0.205645 + 1.43029i 0.0209885 + 0.145978i
\(97\) 16.6014 + 6.19201i 1.68562 + 0.628704i 0.995480 − 0.0949729i \(-0.0302764\pi\)
0.690139 + 0.723676i \(0.257549\pi\)
\(98\) −4.69462 − 6.27128i −0.474228 − 0.633495i
\(99\) 4.46893 − 1.31220i 0.449144 − 0.131881i
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.7.9 ✓ 240
5.3 odd 4 inner 230.2.l.a.53.3 yes 240
23.10 odd 22 inner 230.2.l.a.217.3 yes 240
115.33 even 44 inner 230.2.l.a.33.9 yes 240
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.l.a.7.9 ✓ 240 1.1 even 1 trivial
230.2.l.a.33.9 yes 240 115.33 even 44 inner
230.2.l.a.53.3 yes 240 5.3 odd 4 inner
230.2.l.a.217.3 yes 240 23.10 odd 22 inner