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Properties

Label 230.2.j.a.209.4
Level $230$
Weight $2$
Character 230.209
Analytic conductor $1.837$
Analytic rank $0$
Dimension $120$
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(9,230)] 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("230.9"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(230, base_ring=CyclotomicField(22)) chi = DirichletCharacter(H, H._module([11, 10])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 230 = 2 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 230.j (of order \(22\), degree \(10\), 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: \(120\)
Relative dimension: \(12\) over \(\Q(\zeta_{22})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label 209.4
Character \(\chi\) \(=\) 230.209
Dual form 230.2.j.a.219.4

$q$-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+(-0.540641 + 0.841254i) q^{2} +(0.665993 - 0.0957553i) q^{3} +(-0.415415 - 0.909632i) q^{4} +(-0.171396 + 2.22949i) q^{5} +(-0.279508 + 0.612038i) q^{6} +(1.62329 + 1.40659i) q^{7} +(0.989821 + 0.142315i) q^{8} +(-2.44410 + 0.717653i) q^{9} +(-1.78290 - 1.34954i) q^{10} +(3.00982 - 1.93430i) q^{11} +(-0.363766 - 0.566030i) q^{12} +(-3.40020 + 2.94629i) q^{13} +(-2.06092 + 0.605141i) q^{14} +(0.0993370 + 1.50124i) q^{15} +(-0.654861 + 0.755750i) q^{16} +(5.12683 + 2.34135i) q^{17} +(0.717653 - 2.44410i) q^{18} +(2.28599 + 5.00562i) q^{19} +(2.09922 - 0.770256i) q^{20} +(1.21579 + 0.781342i) q^{21} +3.57778i q^{22} +(-2.94656 - 3.78388i) q^{23} +0.672842 q^{24} +(-4.94125 - 0.764251i) q^{25} +(-0.640289 - 4.45331i) q^{26} +(-3.39515 + 1.55051i) q^{27} +(0.605141 - 2.06092i) q^{28} +(3.52252 - 7.71324i) q^{29} +(-1.31663 - 0.728062i) q^{30} +(-0.681945 + 4.74303i) q^{31} +(-0.281733 - 0.959493i) q^{32} +(1.81930 - 1.57643i) q^{33} +(-4.74144 + 3.04714i) q^{34} +(-3.41421 + 3.37803i) q^{35} +(1.66812 + 1.92511i) q^{36} +(-2.34825 - 7.99739i) q^{37} +(-5.44690 - 0.783146i) q^{38} +(-1.98238 + 2.28779i) q^{39} +(-0.486941 + 2.18240i) q^{40} +(9.99296 + 2.93420i) q^{41} +(-1.31461 + 0.600364i) q^{42} +(3.35224 - 0.481980i) q^{43} +(-3.00982 - 1.93430i) q^{44} +(-1.18109 - 5.57210i) q^{45} +(4.77624 - 0.433085i) q^{46} -10.7053i q^{47} +(-0.363766 + 0.566030i) q^{48} +(-0.339622 - 2.36212i) q^{49} +(3.31437 - 3.74366i) q^{50} +(3.63863 + 1.06840i) q^{51} +(4.09253 + 1.86900i) q^{52} +(-4.00429 - 3.46974i) q^{53} +(0.531182 - 3.69445i) q^{54} +(3.79662 + 7.04190i) q^{55} +(1.40659 + 1.62329i) q^{56} +(2.00177 + 3.11481i) q^{57} +(4.58437 + 7.13343i) q^{58} +(1.25908 + 1.45305i) q^{59} +(1.32431 - 0.713996i) q^{60} +(0.973399 - 6.77014i) q^{61} +(-3.62141 - 3.13797i) q^{62} +(-4.97694 - 2.27289i) q^{63} +(0.959493 + 0.281733i) q^{64} +(-5.98594 - 8.08568i) q^{65} +(0.342592 + 2.38278i) q^{66} +(-0.464748 + 0.723162i) q^{67} -5.63616i q^{68} +(-2.32472 - 2.23789i) q^{69} +(-0.995922 - 4.69852i) q^{70} +(7.34689 + 4.72156i) q^{71} +(-2.52136 + 0.362516i) q^{72} +(5.78187 - 2.64049i) q^{73} +(7.99739 + 2.34825i) q^{74} +(-3.36402 - 0.0358349i) q^{75} +(3.60364 - 4.15882i) q^{76} +(7.60660 + 1.09366i) q^{77} +(-0.852857 - 2.90456i) q^{78} +(-9.61636 - 11.0979i) q^{79} +(-1.57270 - 1.58954i) q^{80} +(4.31606 - 2.77376i) q^{81} +(-7.87101 + 6.82027i) q^{82} +(1.86485 + 6.35109i) q^{83} +(0.205675 - 1.43050i) q^{84} +(-6.09873 + 11.0289i) q^{85} +(-1.40689 + 3.08066i) q^{86} +(1.60739 - 5.47427i) q^{87} +(3.25447 - 1.48626i) q^{88} +(-0.622136 - 4.32705i) q^{89} +(5.32610 + 2.01891i) q^{90} -9.66375 q^{91} +(-2.21789 + 4.25217i) q^{92} +3.22413i q^{93} +(9.00583 + 5.78770i) q^{94} +(-11.5518 + 4.23865i) q^{95} +(-0.279508 - 0.612038i) q^{96} +(-1.51710 + 5.16676i) q^{97} +(2.17076 + 0.991351i) q^{98} +(-5.96816 + 6.88762i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 120 q + 12 q^{4} - 4 q^{6} + 8 q^{9} + 8 q^{11} - 6 q^{15} - 12 q^{16} - 16 q^{19} - 22 q^{20} + 4 q^{24} - 52 q^{25} - 4 q^{26} - 8 q^{29} - 44 q^{30} + 12 q^{31} + 16 q^{35} - 8 q^{36} - 36 q^{39} - 28 q^{41}+ \cdots + 188 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/230\mathbb{Z}\right)^\times\).

\(n\) \(47\) \(51\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{11}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.540641 + 0.841254i −0.382291 + 0.594856i
\(3\) 0.665993 − 0.0957553i 0.384511 − 0.0552844i 0.0526511 − 0.998613i \(-0.483233\pi\)
0.331860 + 0.943329i \(0.392324\pi\)
\(4\) −0.415415 − 0.909632i −0.207708 − 0.454816i
\(5\) −0.171396 + 2.22949i −0.0766506 + 0.997058i
\(6\) −0.279508 + 0.612038i −0.114109 + 0.249864i
\(7\) 1.62329 + 1.40659i 0.613548 + 0.531642i 0.905257 − 0.424865i \(-0.139678\pi\)
−0.291709 + 0.956507i \(0.594224\pi\)
\(8\) 0.989821 + 0.142315i 0.349955 + 0.0503159i
\(9\) −2.44410 + 0.717653i −0.814700 + 0.239218i
\(10\) −1.78290 − 1.34954i −0.563803 − 0.426762i
\(11\) 3.00982 − 1.93430i 0.907496 − 0.583212i −0.00150867 − 0.999999i \(-0.500480\pi\)
0.909004 + 0.416787i \(0.136844\pi\)
\(12\) −0.363766 − 0.566030i −0.105010 − 0.163399i
\(13\) −3.40020 + 2.94629i −0.943045 + 0.817153i −0.983292 − 0.182037i \(-0.941731\pi\)
0.0402468 + 0.999190i \(0.487186\pi\)
\(14\) −2.06092 + 0.605141i −0.550804 + 0.161731i
\(15\) 0.0993370 + 1.50124i 0.0256487 + 0.387618i
\(16\) −0.654861 + 0.755750i −0.163715 + 0.188937i
\(17\) 5.12683 + 2.34135i 1.24344 + 0.567860i 0.924958 − 0.380069i \(-0.124100\pi\)
0.318482 + 0.947929i \(0.396827\pi\)
\(18\) 0.717653 − 2.44410i 0.169152 − 0.576080i
\(19\) 2.28599 + 5.00562i 0.524442 + 1.14837i 0.967730 + 0.251989i \(0.0810848\pi\)
−0.443288 + 0.896379i \(0.646188\pi\)
\(20\) 2.09922 − 0.770256i 0.469399 − 0.172235i
\(21\) 1.21579 + 0.781342i 0.265308 + 0.170503i
\(22\) 3.57778i 0.762786i
\(23\) −2.94656 − 3.78388i −0.614401 − 0.788994i
\(24\) 0.672842 0.137343
\(25\) −4.94125 − 0.764251i −0.988249 − 0.152850i
\(26\) −0.640289 − 4.45331i −0.125571 − 0.873366i
\(27\) −3.39515 + 1.55051i −0.653397 + 0.298396i
\(28\) 0.605141 − 2.06092i 0.114361 − 0.389477i
\(29\) 3.52252 − 7.71324i 0.654115 − 1.43231i −0.233790 − 0.972287i \(-0.575113\pi\)
0.887905 − 0.460026i \(-0.152160\pi\)
\(30\) −1.31663 − 0.728062i −0.240382 − 0.132925i
\(31\) −0.681945 + 4.74303i −0.122481 + 0.851874i 0.832249 + 0.554401i \(0.187053\pi\)
−0.954730 + 0.297472i \(0.903856\pi\)
\(32\) −0.281733 − 0.959493i −0.0498038 − 0.169616i
\(33\) 1.81930 − 1.57643i 0.316700 − 0.274422i
\(34\) −4.74144 + 3.04714i −0.813151 + 0.522580i
\(35\) −3.41421 + 3.37803i −0.577107 + 0.570992i
\(36\) 1.66812 + 1.92511i 0.278019 + 0.320851i
\(37\) −2.34825 − 7.99739i −0.386049 − 1.31476i −0.891936 − 0.452163i \(-0.850653\pi\)
0.505886 − 0.862600i \(-0.331165\pi\)
\(38\) −5.44690 − 0.783146i −0.883604 − 0.127043i
\(39\) −1.98238 + 2.28779i −0.317436 + 0.366340i
\(40\) −0.486941 + 2.18240i −0.0769921 + 0.345068i
\(41\) 9.99296 + 2.93420i 1.56064 + 0.458245i 0.944259 − 0.329204i \(-0.106780\pi\)
0.616380 + 0.787449i \(0.288599\pi\)
\(42\) −1.31461 + 0.600364i −0.202849 + 0.0926381i
\(43\) 3.35224 − 0.481980i 0.511212 − 0.0735012i 0.118118 − 0.993000i \(-0.462314\pi\)
0.393094 + 0.919498i \(0.371405\pi\)
\(44\) −3.00982 − 1.93430i −0.453748 − 0.291606i
\(45\) −1.18109 − 5.57210i −0.176067 − 0.830640i
\(46\) 4.77624 − 0.433085i 0.704218 − 0.0638549i
\(47\) − 10.7053i − 1.56152i −0.624830 − 0.780761i \(-0.714832\pi\)
0.624830 − 0.780761i \(-0.285168\pi\)
\(48\) −0.363766 + 0.566030i −0.0525050 + 0.0816994i
\(49\) −0.339622 − 2.36212i −0.0485174 − 0.337446i
\(50\) 3.31437 − 3.74366i 0.468722 − 0.529433i
\(51\) 3.63863 + 1.06840i 0.509510 + 0.149606i
\(52\) 4.09253 + 1.86900i 0.567532 + 0.259183i
\(53\) −4.00429 − 3.46974i −0.550031 − 0.476605i 0.334947 − 0.942237i \(-0.391282\pi\)
−0.884978 + 0.465632i \(0.845827\pi\)
\(54\) 0.531182 − 3.69445i 0.0722847 − 0.502751i
\(55\) 3.79662 + 7.04190i 0.511936 + 0.949529i
\(56\) 1.40659 + 1.62329i 0.187964 + 0.216922i
\(57\) 2.00177 + 3.11481i 0.265141 + 0.412567i
\(58\) 4.58437 + 7.13343i 0.601958 + 0.936665i
\(59\) 1.25908 + 1.45305i 0.163918 + 0.189171i 0.831766 − 0.555126i \(-0.187330\pi\)
−0.667848 + 0.744297i \(0.732785\pi\)
\(60\) 1.32431 − 0.713996i 0.170967 − 0.0921765i
\(61\) 0.973399 − 6.77014i 0.124631 − 0.866828i −0.827572 − 0.561360i \(-0.810278\pi\)
0.952203 − 0.305467i \(-0.0988127\pi\)
\(62\) −3.62141 − 3.13797i −0.459919 − 0.398522i
\(63\) −4.97694 − 2.27289i −0.627036 − 0.286358i
\(64\) 0.959493 + 0.281733i 0.119937 + 0.0352166i
\(65\) −5.98594 − 8.08568i −0.742464 − 1.00291i
\(66\) 0.342592 + 2.38278i 0.0421701 + 0.293300i
\(67\) −0.464748 + 0.723162i −0.0567780 + 0.0883483i −0.868482 − 0.495721i \(-0.834904\pi\)
0.811704 + 0.584069i \(0.198540\pi\)
\(68\) − 5.63616i − 0.683485i
\(69\) −2.32472 − 2.23789i −0.279863 − 0.269410i
\(70\) −0.995922 − 4.69852i −0.119035 − 0.561581i
\(71\) 7.34689 + 4.72156i 0.871916 + 0.560346i 0.898338 − 0.439305i \(-0.144775\pi\)
−0.0264224 + 0.999651i \(0.508411\pi\)
\(72\) −2.52136 + 0.362516i −0.297145 + 0.0427230i
\(73\) 5.78187 − 2.64049i 0.676717 − 0.309046i −0.0472479 − 0.998883i \(-0.515045\pi\)
0.723965 + 0.689837i \(0.242318\pi\)
\(74\) 7.99739 + 2.34825i 0.929678 + 0.272978i
\(75\) −3.36402 − 0.0358349i −0.388443 − 0.00413786i
\(76\) 3.60364 − 4.15882i 0.413366 − 0.477050i
\(77\) 7.60660 + 1.09366i 0.866852 + 0.124634i
\(78\) −0.852857 − 2.90456i −0.0965670 − 0.328877i
\(79\) −9.61636 − 11.0979i −1.08192 − 1.24861i −0.966876 − 0.255248i \(-0.917843\pi\)
−0.115049 − 0.993360i \(-0.536703\pi\)
\(80\) −1.57270 − 1.58954i −0.175833 − 0.177716i
\(81\) 4.31606 − 2.77376i 0.479562 − 0.308196i
\(82\) −7.87101 + 6.82027i −0.869208 + 0.753173i
\(83\) 1.86485 + 6.35109i 0.204694 + 0.697123i 0.996289 + 0.0860767i \(0.0274330\pi\)
−0.791595 + 0.611047i \(0.790749\pi\)
\(84\) 0.205675 − 1.43050i 0.0224410 − 0.156081i
\(85\) −6.09873 + 11.0289i −0.661500 + 1.19625i
\(86\) −1.40689 + 3.08066i −0.151709 + 0.332196i
\(87\) 1.60739 − 5.47427i 0.172330 − 0.586903i
\(88\) 3.25447 − 1.48626i 0.346927 − 0.158436i
\(89\) −0.622136 − 4.32705i −0.0659463 − 0.458667i −0.995861 − 0.0908909i \(-0.971029\pi\)
0.929915 − 0.367776i \(-0.119881\pi\)
\(90\) 5.32610 + 2.01891i 0.561420 + 0.212812i
\(91\) −9.66375 −1.01304
\(92\) −2.21789 + 4.25217i −0.231231 + 0.443319i
\(93\) 3.22413i 0.334326i
\(94\) 9.00583 + 5.78770i 0.928881 + 0.596955i
\(95\) −11.5518 + 4.23865i −1.18519 + 0.434876i
\(96\) −0.279508 − 0.612038i −0.0285272 − 0.0624659i
\(97\) −1.51710 + 5.16676i −0.154038 + 0.524605i −0.999962 − 0.00869816i \(-0.997231\pi\)
0.845924 + 0.533303i \(0.179049\pi\)
\(98\) 2.17076 + 0.991351i 0.219279 + 0.100142i
\(99\) −5.96816 + 6.88762i −0.599822 + 0.692232i
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.j.a.209.4 ✓ 120
5.4 even 2 inner 230.2.j.a.209.9 yes 120
23.12 even 11 inner 230.2.j.a.219.9 yes 120
115.104 even 22 inner 230.2.j.a.219.4 yes 120
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.j.a.209.4 ✓ 120 1.1 even 1 trivial
230.2.j.a.209.9 yes 120 5.4 even 2 inner
230.2.j.a.219.4 yes 120 115.104 even 22 inner
230.2.j.a.219.9 yes 120 23.12 even 11 inner