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Properties

Label 920.2.bb.a.11.5
Level $920$
Weight $2$
Character 920.11
Analytic conductor $7.346$
Analytic rank $0$
Dimension $480$
Inner twists $2$

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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] = [920,2,Mod(11,920)] 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("920.11"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(920, base_ring=CyclotomicField(22)) chi = DirichletCharacter(H, H._module([11, 11, 0, 9])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 920 = 2^{3} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 920.bb (of order \(22\), degree \(10\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [480,2,0,-4,-48] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.34623698596\)
Analytic rank: \(0\)
Dimension: \(480\)
Relative dimension: \(48\) over \(\Q(\zeta_{22})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label 11.5
Character \(\chi\) \(=\) 920.11
Dual form 920.2.bb.a.251.5

$q$-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+(-1.34604 + 0.433784i) q^{2} +(-0.132869 + 0.924126i) q^{3} +(1.62366 - 1.16778i) q^{4} +(-0.959493 + 0.281733i) q^{5} +(-0.222024 - 1.30155i) q^{6} +(2.80604 + 3.23834i) q^{7} +(-1.67895 + 2.27621i) q^{8} +(2.04212 + 0.599622i) q^{9} +(1.16931 - 0.795437i) q^{10} +(-3.09409 + 4.81450i) q^{11} +(0.863445 + 1.65563i) q^{12} +(2.73900 + 2.37336i) q^{13} +(-5.18179 - 3.14173i) q^{14} +(-0.132869 - 0.924126i) q^{15} +(1.27256 - 3.79217i) q^{16} +(4.43451 - 2.02517i) q^{17} +(-3.00889 + 0.0787244i) q^{18} +(-3.44622 - 1.57383i) q^{19} +(-1.22889 + 1.57792i) q^{20} +(-3.36547 + 2.16286i) q^{21} +(2.07633 - 7.82269i) q^{22} +(-1.78810 - 4.45002i) q^{23} +(-1.88042 - 1.85400i) q^{24} +(0.841254 - 0.540641i) q^{25} +(-4.71633 - 2.00650i) q^{26} +(-1.98899 + 4.35529i) q^{27} +(8.33774 + 1.98113i) q^{28} +(2.94715 - 1.34592i) q^{29} +(0.579719 + 1.18628i) q^{30} +(4.96005 - 0.713147i) q^{31} +(-0.0679368 + 5.65645i) q^{32} +(-4.03810 - 3.49903i) q^{33} +(-5.09055 + 4.64958i) q^{34} +(-3.60472 - 2.31661i) q^{35} +(4.01595 - 1.41118i) q^{36} +(6.20384 + 1.82161i) q^{37} +(5.32146 + 0.623535i) q^{38} +(-2.55721 + 2.21583i) q^{39} +(0.969663 - 2.65702i) q^{40} +(-11.1848 + 3.28416i) q^{41} +(3.59186 - 4.37119i) q^{42} +(-4.00647 - 0.576043i) q^{43} +(0.598535 + 11.4304i) q^{44} -2.12834 q^{45} +(4.33721 + 5.21427i) q^{46} -2.40525i q^{47} +(3.33536 + 1.67987i) q^{48} +(-1.61679 + 11.2451i) q^{49} +(-0.897842 + 1.09265i) q^{50} +(1.28230 + 4.36713i) q^{51} +(7.21878 + 0.654971i) q^{52} +(6.44475 + 7.43764i) q^{53} +(0.788015 - 6.72520i) q^{54} +(1.61236 - 5.49118i) q^{55} +(-12.0823 + 0.950098i) q^{56} +(1.91232 - 2.97563i) q^{57} +(-3.38315 + 3.09009i) q^{58} +(-3.78950 + 4.37331i) q^{59} +(-1.29492 - 1.34531i) q^{60} +(-0.158817 - 1.10460i) q^{61} +(-6.36708 + 3.11152i) q^{62} +(3.78850 + 8.29565i) q^{63} +(-2.36223 - 7.64329i) q^{64} +(-3.29670 - 1.50555i) q^{65} +(6.95328 + 2.95818i) q^{66} +(-4.57690 - 7.12180i) q^{67} +(4.83518 - 8.46674i) q^{68} +(4.34997 - 1.06116i) q^{69} +(5.85702 + 1.55459i) q^{70} +(-1.73600 - 2.70127i) q^{71} +(-4.79349 + 3.64156i) q^{72} +(5.43154 - 11.8934i) q^{73} +(-9.14083 + 0.239160i) q^{74} +(0.387844 + 0.849259i) q^{75} +(-7.43339 + 1.46906i) q^{76} +(-24.2731 + 3.48995i) q^{77} +(2.48092 - 4.09189i) q^{78} +(-10.6141 + 12.2494i) q^{79} +(-0.152634 + 3.99709i) q^{80} +(1.61085 + 1.03523i) q^{81} +(13.6306 - 9.27242i) q^{82} +(1.16726 - 3.97533i) q^{83} +(-2.93864 + 7.44189i) q^{84} +(-3.68432 + 3.19248i) q^{85} +(5.64276 - 0.962564i) q^{86} +(0.852212 + 2.90237i) q^{87} +(-5.76396 - 15.1261i) q^{88} +(-4.72278 - 0.679033i) q^{89} +(2.86483 - 0.923238i) q^{90} +15.5295i q^{91} +(-8.09994 - 5.13722i) q^{92} +4.67847i q^{93} +(1.04336 + 3.23757i) q^{94} +(3.75002 + 0.539172i) q^{95} +(-5.21824 - 0.814350i) q^{96} +(4.85957 + 16.5502i) q^{97} +(-2.70165 - 15.8377i) q^{98} +(-9.20539 + 7.97652i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 480 q + 2 q^{2} - 4 q^{4} - 48 q^{5} + 5 q^{6} - q^{8} - 48 q^{9} + 2 q^{10} + 3 q^{12} + 32 q^{16} + 7 q^{18} - 4 q^{20} + 8 q^{21} + 4 q^{23} + 2 q^{24} - 48 q^{25} + 7 q^{26} + 12 q^{27} + 5 q^{30}+ \cdots - 98 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/920\mathbb{Z}\right)^\times\).

\(n\) \(231\) \(281\) \(461\) \(737\)
\(\chi(n)\) \(-1\) \(e\left(\frac{9}{22}\right)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.34604 + 0.433784i −0.951796 + 0.306732i
\(3\) −0.132869 + 0.924126i −0.0767121 + 0.533545i 0.914838 + 0.403821i \(0.132318\pi\)
−0.991550 + 0.129724i \(0.958591\pi\)
\(4\) 1.62366 − 1.16778i 0.811831 − 0.583892i
\(5\) −0.959493 + 0.281733i −0.429098 + 0.125995i
\(6\) −0.222024 − 1.30155i −0.0906407 − 0.531356i
\(7\) 2.80604 + 3.23834i 1.06058 + 1.22398i 0.973718 + 0.227757i \(0.0731392\pi\)
0.0868645 + 0.996220i \(0.472315\pi\)
\(8\) −1.67895 + 2.27621i −0.593600 + 0.804760i
\(9\) 2.04212 + 0.599622i 0.680708 + 0.199874i
\(10\) 1.16931 − 0.795437i 0.369768 − 0.251539i
\(11\) −3.09409 + 4.81450i −0.932904 + 1.45163i −0.0411251 + 0.999154i \(0.513094\pi\)
−0.891778 + 0.452472i \(0.850542\pi\)
\(12\) 0.863445 + 1.65563i 0.249255 + 0.477940i
\(13\) 2.73900 + 2.37336i 0.759662 + 0.658250i 0.945975 − 0.324240i \(-0.105109\pi\)
−0.186313 + 0.982490i \(0.559654\pi\)
\(14\) −5.18179 − 3.14173i −1.38489 − 0.839663i
\(15\) −0.132869 − 0.924126i −0.0343067 − 0.238608i
\(16\) 1.27256 − 3.79217i 0.318140 − 0.948044i
\(17\) 4.43451 − 2.02517i 1.07553 − 0.491176i 0.202716 − 0.979238i \(-0.435023\pi\)
0.872809 + 0.488061i \(0.162296\pi\)
\(18\) −3.00889 + 0.0787244i −0.709203 + 0.0185555i
\(19\) −3.44622 − 1.57383i −0.790616 − 0.361062i −0.0211717 − 0.999776i \(-0.506740\pi\)
−0.769445 + 0.638713i \(0.779467\pi\)
\(20\) −1.22889 + 1.57792i −0.274788 + 0.352833i
\(21\) −3.36547 + 2.16286i −0.734406 + 0.471974i
\(22\) 2.07633 − 7.82269i 0.442674 − 1.66780i
\(23\) −1.78810 − 4.45002i −0.372845 − 0.927894i
\(24\) −1.88042 − 1.85400i −0.383839 − 0.378447i
\(25\) 0.841254 − 0.540641i 0.168251 − 0.108128i
\(26\) −4.71633 − 2.00650i −0.924949 − 0.393508i
\(27\) −1.98899 + 4.35529i −0.382782 + 0.838175i
\(28\) 8.33774 + 1.98113i 1.57568 + 0.374398i
\(29\) 2.94715 − 1.34592i 0.547271 − 0.249930i −0.122538 − 0.992464i \(-0.539103\pi\)
0.669809 + 0.742533i \(0.266376\pi\)
\(30\) 0.579719 + 1.18628i 0.105842 + 0.216584i
\(31\) 4.96005 − 0.713147i 0.890851 − 0.128085i 0.318334 − 0.947978i \(-0.396876\pi\)
0.572516 + 0.819893i \(0.305967\pi\)
\(32\) −0.0679368 + 5.65645i −0.0120096 + 0.999928i
\(33\) −4.03810 − 3.49903i −0.702942 − 0.609103i
\(34\) −5.09055 + 4.64958i −0.873022 + 0.797397i
\(35\) −3.60472 − 2.31661i −0.609309 − 0.391579i
\(36\) 4.01595 − 1.41118i 0.669325 − 0.235196i
\(37\) 6.20384 + 1.82161i 1.01991 + 0.299471i 0.748600 − 0.663022i \(-0.230726\pi\)
0.271306 + 0.962493i \(0.412545\pi\)
\(38\) 5.32146 + 0.623535i 0.863255 + 0.101151i
\(39\) −2.55721 + 2.21583i −0.409481 + 0.354818i
\(40\) 0.969663 − 2.65702i 0.153317 − 0.420112i
\(41\) −11.1848 + 3.28416i −1.74678 + 0.512900i −0.990036 − 0.140817i \(-0.955027\pi\)
−0.756740 + 0.653716i \(0.773209\pi\)
\(42\) 3.59186 − 4.37119i 0.554235 − 0.674489i
\(43\) −4.00647 − 0.576043i −0.610981 − 0.0878458i −0.170123 − 0.985423i \(-0.554417\pi\)
−0.440858 + 0.897577i \(0.645326\pi\)
\(44\) 0.598535 + 11.4304i 0.0902326 + 1.72319i
\(45\) −2.12834 −0.317274
\(46\) 4.33721 + 5.21427i 0.639487 + 0.768802i
\(47\) − 2.40525i − 0.350842i −0.984494 − 0.175421i \(-0.943871\pi\)
0.984494 − 0.175421i \(-0.0561286\pi\)
\(48\) 3.33536 + 1.67987i 0.481418 + 0.242469i
\(49\) −1.61679 + 11.2451i −0.230971 + 1.60644i
\(50\) −0.897842 + 1.09265i −0.126974 + 0.154524i
\(51\) 1.28230 + 4.36713i 0.179558 + 0.611520i
\(52\) 7.21878 + 0.654971i 1.00106 + 0.0908281i
\(53\) 6.44475 + 7.43764i 0.885255 + 1.02164i 0.999602 + 0.0281979i \(0.00897686\pi\)
−0.114347 + 0.993441i \(0.536478\pi\)
\(54\) 0.788015 − 6.72520i 0.107235 − 0.915183i
\(55\) 1.61236 − 5.49118i 0.217410 − 0.740431i
\(56\) −12.0823 + 0.950098i −1.61457 + 0.126962i
\(57\) 1.91232 − 2.97563i 0.253293 − 0.394131i
\(58\) −3.38315 + 3.09009i −0.444229 + 0.405748i
\(59\) −3.78950 + 4.37331i −0.493350 + 0.569357i −0.946758 − 0.321947i \(-0.895663\pi\)
0.453408 + 0.891303i \(0.350208\pi\)
\(60\) −1.29492 − 1.34531i −0.167173 − 0.173678i
\(61\) −0.158817 − 1.10460i −0.0203345 − 0.141429i 0.977125 − 0.212666i \(-0.0682146\pi\)
−0.997459 + 0.0712365i \(0.977306\pi\)
\(62\) −6.36708 + 3.11152i −0.808620 + 0.395163i
\(63\) 3.78850 + 8.29565i 0.477306 + 1.04515i
\(64\) −2.36223 − 7.64329i −0.295279 − 0.955411i
\(65\) −3.29670 − 1.50555i −0.408905 − 0.186741i
\(66\) 6.95328 + 2.95818i 0.855889 + 0.364127i
\(67\) −4.57690 − 7.12180i −0.559158 − 0.870066i 0.440458 − 0.897773i \(-0.354816\pi\)
−0.999616 + 0.0277067i \(0.991180\pi\)
\(68\) 4.83518 − 8.46674i 0.586352 − 1.02674i
\(69\) 4.34997 − 1.06116i 0.523674 − 0.127749i
\(70\) 5.85702 + 1.55459i 0.700047 + 0.185809i
\(71\) −1.73600 − 2.70127i −0.206026 − 0.320582i 0.722829 − 0.691027i \(-0.242842\pi\)
−0.928854 + 0.370445i \(0.879205\pi\)
\(72\) −4.79349 + 3.64156i −0.564919 + 0.429162i
\(73\) 5.43154 − 11.8934i 0.635714 − 1.39202i −0.267805 − 0.963473i \(-0.586298\pi\)
0.903520 − 0.428547i \(-0.140974\pi\)
\(74\) −9.14083 + 0.239160i −1.06260 + 0.0278018i
\(75\) 0.387844 + 0.849259i 0.0447843 + 0.0980640i
\(76\) −7.43339 + 1.46906i −0.852669 + 0.168513i
\(77\) −24.2731 + 3.48995i −2.76618 + 0.397716i
\(78\) 2.48092 − 4.09189i 0.280909 − 0.463315i
\(79\) −10.6141 + 12.2494i −1.19418 + 1.37816i −0.286730 + 0.958011i \(0.592568\pi\)
−0.907454 + 0.420151i \(0.861977\pi\)
\(80\) −0.152634 + 3.99709i −0.0170650 + 0.446888i
\(81\) 1.61085 + 1.03523i 0.178983 + 0.115026i
\(82\) 13.6306 − 9.27242i 1.50525 − 1.02397i
\(83\) 1.16726 − 3.97533i 0.128124 − 0.436349i −0.870297 − 0.492527i \(-0.836073\pi\)
0.998421 + 0.0561782i \(0.0178915\pi\)
\(84\) −2.93864 + 7.44189i −0.320632 + 0.811977i
\(85\) −3.68432 + 3.19248i −0.399621 + 0.346273i
\(86\) 5.64276 − 0.962564i 0.608474 − 0.103796i
\(87\) 0.852212 + 2.90237i 0.0913667 + 0.311166i
\(88\) −5.76396 − 15.1261i −0.614440 − 1.61245i
\(89\) −4.72278 − 0.679033i −0.500613 − 0.0719773i −0.112617 − 0.993638i \(-0.535923\pi\)
−0.387996 + 0.921661i \(0.626833\pi\)
\(90\) 2.86483 − 0.923238i 0.301980 − 0.0973179i
\(91\) 15.5295i 1.62794i
\(92\) −8.09994 − 5.13722i −0.844477 − 0.535592i
\(93\) 4.67847i 0.485134i
\(94\) 1.04336 + 3.23757i 0.107614 + 0.333930i
\(95\) 3.75002 + 0.539172i 0.384744 + 0.0553178i
\(96\) −5.21824 − 0.814350i −0.532585 − 0.0831143i
\(97\) 4.85957 + 16.5502i 0.493414 + 1.68042i 0.710041 + 0.704161i \(0.248677\pi\)
−0.216626 + 0.976255i \(0.569505\pi\)
\(98\) −2.70165 − 15.8377i −0.272908 − 1.59985i
\(99\) −9.20539 + 7.97652i −0.925177 + 0.801670i
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 920.2.bb.a.11.5 ✓ 480
8.3 odd 2 920.2.bb.b.11.14 yes 480
23.21 odd 22 920.2.bb.b.251.14 yes 480
184.67 even 22 inner 920.2.bb.a.251.5 yes 480
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.bb.a.11.5 ✓ 480 1.1 even 1 trivial
920.2.bb.a.251.5 yes 480 184.67 even 22 inner
920.2.bb.b.11.14 yes 480 8.3 odd 2
920.2.bb.b.251.14 yes 480 23.21 odd 22