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Properties

Label 230.2.l.a.17.7
Level $230$
Weight $2$
Character 230.17
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 17.7
Character \(\chi\) \(=\) 230.17
Dual form 230.2.l.a.203.7

$q$-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} +(-1.94498 + 1.06204i) q^{3} +(0.755750 - 0.654861i) q^{4} +(-1.77597 - 1.35866i) q^{5} +(-1.45120 + 1.67478i) q^{6} +(-3.01546 - 4.02819i) q^{7} +(0.479249 - 0.877679i) q^{8} +(1.03309 - 1.60752i) q^{9} +(-2.13879 - 0.652357i) q^{10} +(1.58353 - 0.723173i) q^{11} +(-0.774429 + 2.07632i) q^{12} +(-0.286592 - 0.214540i) q^{13} +(-4.23304 - 2.72041i) q^{14} +(4.89716 + 0.756414i) q^{15} +(0.142315 - 0.989821i) q^{16} +(-0.152966 - 2.13874i) q^{17} +(0.406183 - 1.86719i) q^{18} +(-0.840634 - 0.970143i) q^{19} +(-2.23192 + 0.136206i) q^{20} +(10.1431 + 4.63220i) q^{21} +(1.23096 - 1.23096i) q^{22} +(-4.06631 + 2.54267i) q^{23} +2.21605i q^{24} +(1.30811 + 4.82585i) q^{25} +(-0.343496 - 0.100860i) q^{26} +(0.172183 - 2.40743i) q^{27} +(-4.91683 - 1.06959i) q^{28} +(1.98168 + 1.71713i) q^{29} +(4.85273 - 1.00266i) q^{30} +(-8.43162 + 2.47575i) q^{31} +(-0.212565 - 0.977147i) q^{32} +(-2.31189 + 3.08832i) q^{33} +(-0.890734 - 1.95044i) q^{34} +(-0.117565 + 11.2509i) q^{35} +(-0.271944 - 1.89141i) q^{36} +(8.59708 - 1.87018i) q^{37} +(-1.12666 - 0.615204i) q^{38} +(0.785264 + 0.112904i) q^{39} +(-2.04359 + 0.907593i) q^{40} +(7.16256 - 4.60310i) q^{41} +(11.1224 + 0.795486i) q^{42} +(-2.22999 - 4.08392i) q^{43} +(0.723173 - 1.58353i) q^{44} +(-4.01880 + 1.45129i) q^{45} +(-2.92135 + 3.80338i) q^{46} +(-1.60762 - 1.60762i) q^{47} +(0.774429 + 2.07632i) q^{48} +(-5.16114 + 17.5772i) q^{49} +(2.91209 + 4.06445i) q^{50} +(2.56894 + 3.99735i) q^{51} +(-0.357085 + 0.0255393i) q^{52} +(6.18386 - 4.62918i) q^{53} +(-0.679983 - 2.31581i) q^{54} +(-3.79484 - 0.867141i) q^{55} +(-4.98061 + 0.716104i) q^{56} +(2.66534 + 0.994122i) q^{57} +(2.45681 + 0.916342i) q^{58} +(12.7331 - 1.83074i) q^{59} +(4.19637 - 2.63530i) q^{60} +(-2.29221 - 7.80653i) q^{61} +(-7.03482 + 5.26620i) q^{62} +(-9.59064 + 0.685936i) q^{63} +(-0.540641 - 0.841254i) q^{64} +(0.217491 + 0.770395i) q^{65} +(-1.08687 + 3.70153i) q^{66} +(1.91069 + 5.12275i) q^{67} +(-1.51618 - 1.51618i) q^{68} +(5.20847 - 9.26400i) q^{69} +(3.82163 + 10.5826i) q^{70} +(5.18399 - 11.3514i) q^{71} +(-0.915780 - 1.67713i) q^{72} +(-9.66299 - 0.691111i) q^{73} +(7.40147 - 4.75664i) q^{74} +(-7.66948 - 7.99692i) q^{75} +(-1.27062 - 0.182687i) q^{76} +(-7.68815 - 4.19805i) q^{77} +(0.775209 - 0.168636i) q^{78} +(-1.85277 - 12.8863i) q^{79} +(-1.59757 + 1.56453i) q^{80} +(4.60329 + 10.0798i) q^{81} +(5.10234 - 6.81593i) q^{82} +(-0.422472 - 1.94207i) q^{83} +(10.6991 - 3.14153i) q^{84} +(-2.63415 + 4.00616i) q^{85} +(-3.51657 - 3.04713i) q^{86} +(-5.67798 - 1.23517i) q^{87} +(0.124191 - 1.73641i) q^{88} +(-0.850880 - 0.249841i) q^{89} +(-3.25824 + 2.76421i) q^{90} +1.80138i q^{91} +(-1.40802 + 4.58448i) q^{92} +(13.7700 - 13.7700i) q^{93} +(-2.06807 - 0.944454i) q^{94} +(0.174845 + 2.86507i) q^{95} +(1.45120 + 1.67478i) q^{96} +(-1.33036 + 6.11554i) q^{97} +(1.30688 + 18.2726i) q^{98} +(0.473413 - 3.29266i) 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
\(3\) −1.94498 + 1.06204i −1.12293 + 0.613168i −0.929822 − 0.368009i \(-0.880040\pi\)
−0.193111 + 0.981177i \(0.561858\pi\)
\(4\) 0.755750 − 0.654861i 0.377875 − 0.327430i
\(5\) −1.77597 − 1.35866i −0.794236 − 0.607610i
\(6\) −1.45120 + 1.67478i −0.592451 + 0.683725i
\(7\) −3.01546 − 4.02819i −1.13974 − 1.52251i −0.817571 − 0.575828i \(-0.804680\pi\)
−0.322166 − 0.946683i \(-0.604411\pi\)
\(8\) 0.479249 − 0.877679i 0.169440 − 0.310306i
\(9\) 1.03309 − 1.60752i 0.344364 − 0.535840i
\(10\) −2.13879 − 0.652357i −0.676345 − 0.206293i
\(11\) 1.58353 − 0.723173i 0.477452 − 0.218045i −0.162128 − 0.986770i \(-0.551836\pi\)
0.639580 + 0.768725i \(0.279108\pi\)
\(12\) −0.774429 + 2.07632i −0.223558 + 0.599383i
\(13\) −0.286592 − 0.214540i −0.0794863 − 0.0595027i 0.558796 − 0.829305i \(-0.311264\pi\)
−0.638282 + 0.769803i \(0.720355\pi\)
\(14\) −4.23304 − 2.72041i −1.13133 − 0.727060i
\(15\) 4.89716 + 0.756414i 1.26444 + 0.195305i
\(16\) 0.142315 − 0.989821i 0.0355787 − 0.247455i
\(17\) −0.152966 − 2.13874i −0.0370996 − 0.518721i −0.981692 − 0.190475i \(-0.938997\pi\)
0.944592 − 0.328246i \(-0.106457\pi\)
\(18\) 0.406183 − 1.86719i 0.0957383 − 0.440102i
\(19\) −0.840634 − 0.970143i −0.192855 − 0.222566i 0.651084 − 0.759005i \(-0.274314\pi\)
−0.843939 + 0.536439i \(0.819769\pi\)
\(20\) −2.23192 + 0.136206i −0.499072 + 0.0304565i
\(21\) 10.1431 + 4.63220i 2.21340 + 1.01083i
\(22\) 1.23096 − 1.23096i 0.262442 − 0.262442i
\(23\) −4.06631 + 2.54267i −0.847884 + 0.530182i
\(24\) 2.21605i 0.452349i
\(25\) 1.30811 + 4.82585i 0.261621 + 0.965171i
\(26\) −0.343496 − 0.100860i −0.0673651 − 0.0197802i
\(27\) 0.172183 − 2.40743i 0.0331366 − 0.463310i
\(28\) −4.91683 − 1.06959i −0.929194 − 0.202134i
\(29\) 1.98168 + 1.71713i 0.367988 + 0.318863i 0.819151 − 0.573578i \(-0.194445\pi\)
−0.451163 + 0.892442i \(0.648991\pi\)
\(30\) 4.85273 − 1.00266i 0.885983 − 0.183060i
\(31\) −8.43162 + 2.47575i −1.51436 + 0.444658i −0.930223 − 0.366994i \(-0.880387\pi\)
−0.584141 + 0.811652i \(0.698569\pi\)
\(32\) −0.212565 − 0.977147i −0.0375766 − 0.172737i
\(33\) −2.31189 + 3.08832i −0.402449 + 0.537608i
\(34\) −0.890734 − 1.95044i −0.152760 − 0.334497i
\(35\) −0.117565 + 11.2509i −0.0198721 + 1.90175i
\(36\) −0.271944 − 1.89141i −0.0453241 − 0.315236i
\(37\) 8.59708 − 1.87018i 1.41335 − 0.307456i 0.559871 − 0.828580i \(-0.310851\pi\)
0.853481 + 0.521124i \(0.174487\pi\)
\(38\) −1.12666 − 0.615204i −0.182769 − 0.0997992i
\(39\) 0.785264 + 0.112904i 0.125743 + 0.0180791i
\(40\) −2.04359 + 0.907593i −0.323121 + 0.143503i
\(41\) 7.16256 − 4.60310i 1.11860 − 0.718884i 0.155453 − 0.987843i \(-0.450316\pi\)
0.963151 + 0.268960i \(0.0866799\pi\)
\(42\) 11.1224 + 0.795486i 1.71622 + 0.122746i
\(43\) −2.22999 − 4.08392i −0.340070 − 0.622792i 0.650546 − 0.759467i \(-0.274540\pi\)
−0.990616 + 0.136675i \(0.956358\pi\)
\(44\) 0.723173 − 1.58353i 0.109022 − 0.238726i
\(45\) −4.01880 + 1.45129i −0.599088 + 0.216345i
\(46\) −2.92135 + 3.80338i −0.430730 + 0.560778i
\(47\) −1.60762 − 1.60762i −0.234496 − 0.234496i 0.580071 − 0.814566i \(-0.303025\pi\)
−0.814566 + 0.580071i \(0.803025\pi\)
\(48\) 0.774429 + 2.07632i 0.111779 + 0.299692i
\(49\) −5.16114 + 17.5772i −0.737306 + 2.51103i
\(50\) 2.91209 + 4.06445i 0.411832 + 0.574799i
\(51\) 2.56894 + 3.99735i 0.359723 + 0.559740i
\(52\) −0.357085 + 0.0255393i −0.0495188 + 0.00354166i
\(53\) 6.18386 − 4.62918i 0.849419 − 0.635867i −0.0832788 − 0.996526i \(-0.526539\pi\)
0.932698 + 0.360659i \(0.117448\pi\)
\(54\) −0.679983 − 2.31581i −0.0925340 − 0.315142i
\(55\) −3.79484 − 0.867141i −0.511696 − 0.116925i
\(56\) −4.98061 + 0.716104i −0.665562 + 0.0956934i
\(57\) 2.66534 + 0.994122i 0.353033 + 0.131675i
\(58\) 2.45681 + 0.916342i 0.322595 + 0.120322i
\(59\) 12.7331 − 1.83074i 1.65771 − 0.238342i 0.751060 − 0.660234i \(-0.229543\pi\)
0.906646 + 0.421891i \(0.138634\pi\)
\(60\) 4.19637 − 2.63530i 0.541749 − 0.340215i
\(61\) −2.29221 − 7.80653i −0.293487 − 0.999524i −0.965807 − 0.259263i \(-0.916520\pi\)
0.672320 − 0.740261i \(-0.265298\pi\)
\(62\) −7.03482 + 5.26620i −0.893423 + 0.668808i
\(63\) −9.59064 + 0.685936i −1.20831 + 0.0864198i
\(64\) −0.540641 − 0.841254i −0.0675801 − 0.105157i
\(65\) 0.217491 + 0.770395i 0.0269764 + 0.0955558i
\(66\) −1.08687 + 3.70153i −0.133784 + 0.455627i
\(67\) 1.91069 + 5.12275i 0.233427 + 0.625843i 0.999864 − 0.0164765i \(-0.00524488\pi\)
−0.766437 + 0.642320i \(0.777972\pi\)
\(68\) −1.51618 − 1.51618i −0.183864 − 0.183864i
\(69\) 5.20847 − 9.26400i 0.627026 − 1.11525i
\(70\) 3.82163 + 10.5826i 0.456772 + 1.26486i
\(71\) 5.18399 − 11.3514i 0.615227 − 1.34716i −0.303714 − 0.952763i \(-0.598227\pi\)
0.918941 − 0.394396i \(-0.129046\pi\)
\(72\) −0.915780 − 1.67713i −0.107926 − 0.197651i
\(73\) −9.66299 − 0.691111i −1.13097 − 0.0808884i −0.506717 − 0.862113i \(-0.669141\pi\)
−0.624251 + 0.781224i \(0.714596\pi\)
\(74\) 7.40147 − 4.75664i 0.860404 − 0.552948i
\(75\) −7.66948 − 7.99692i −0.885595 − 0.923405i
\(76\) −1.27062 − 0.182687i −0.145750 − 0.0209557i
\(77\) −7.68815 − 4.19805i −0.876146 − 0.478412i
\(78\) 0.775209 − 0.168636i 0.0877751 − 0.0190943i
\(79\) −1.85277 − 12.8863i −0.208453 − 1.44982i −0.778208 − 0.628007i \(-0.783871\pi\)
0.569755 − 0.821814i \(-0.307038\pi\)
\(80\) −1.59757 + 1.56453i −0.178614 + 0.174920i
\(81\) 4.60329 + 10.0798i 0.511477 + 1.11998i
\(82\) 5.10234 − 6.81593i 0.563460 − 0.752694i
\(83\) −0.422472 − 1.94207i −0.0463723 − 0.213170i 0.948130 − 0.317882i \(-0.102972\pi\)
−0.994503 + 0.104712i \(0.966608\pi\)
\(84\) 10.6991 − 3.14153i 1.16737 − 0.342769i
\(85\) −2.63415 + 4.00616i −0.285714 + 0.434529i
\(86\) −3.51657 − 3.04713i −0.379202 − 0.328580i
\(87\) −5.67798 − 1.23517i −0.608743 − 0.132424i
\(88\) 0.124191 − 1.73641i 0.0132388 − 0.185102i
\(89\) −0.850880 − 0.249841i −0.0901931 − 0.0264831i 0.236325 − 0.971674i \(-0.424057\pi\)
−0.326518 + 0.945191i \(0.605875\pi\)
\(90\) −3.25824 + 2.76421i −0.343449 + 0.291373i
\(91\) 1.80138i 0.188836i
\(92\) −1.40802 + 4.58448i −0.146796 + 0.477965i
\(93\) 13.7700 − 13.7700i 1.42788 − 1.42788i
\(94\) −2.06807 − 0.944454i −0.213305 − 0.0974130i
\(95\) 0.174845 + 2.86507i 0.0179387 + 0.293950i
\(96\) 1.45120 + 1.67478i 0.148113 + 0.170931i
\(97\) −1.33036 + 6.11554i −0.135077 + 0.620939i 0.858973 + 0.512020i \(0.171103\pi\)
−0.994051 + 0.108919i \(0.965261\pi\)
\(98\) 1.30688 + 18.2726i 0.132015 + 1.84581i
\(99\) 0.473413 − 3.29266i 0.0475798 − 0.330925i
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.17.7 ✓ 240
5.3 odd 4 inner 230.2.l.a.63.1 yes 240
23.19 odd 22 inner 230.2.l.a.157.1 yes 240
115.88 even 44 inner 230.2.l.a.203.7 yes 240
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.l.a.17.7 ✓ 240 1.1 even 1 trivial
230.2.l.a.63.1 yes 240 5.3 odd 4 inner
230.2.l.a.157.1 yes 240 23.19 odd 22 inner
230.2.l.a.203.7 yes 240 115.88 even 44 inner