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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 11.7
Character \(\chi\) \(=\) 920.11
Dual form 920.2.bb.a.251.7

$q$-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+(-1.30481 - 0.545420i) q^{2} +(0.319837 - 2.22451i) q^{3} +(1.40503 + 1.42333i) q^{4} +(-0.959493 + 0.281733i) q^{5} +(-1.63062 + 2.72811i) q^{6} +(0.470129 + 0.542558i) q^{7} +(-1.05698 - 2.62351i) q^{8} +(-1.96769 - 0.577767i) q^{9} +(1.40561 + 0.155720i) q^{10} +(0.0968928 - 0.150768i) q^{11} +(3.61561 - 2.67029i) q^{12} +(4.77506 + 4.13762i) q^{13} +(-0.317505 - 0.964351i) q^{14} +(0.319837 + 2.22451i) q^{15} +(-0.0517542 + 3.99967i) q^{16} +(0.839909 - 0.383573i) q^{17} +(2.25233 + 1.82709i) q^{18} +(-0.153192 - 0.0699606i) q^{19} +(-1.74912 - 0.969834i) q^{20} +(1.35729 - 0.872280i) q^{21} +(-0.208658 + 0.143876i) q^{22} +(4.51547 - 1.61572i) q^{23} +(-6.17409 + 1.51218i) q^{24} +(0.841254 - 0.540641i) q^{25} +(-3.97379 - 8.00320i) q^{26} +(0.886208 - 1.94052i) q^{27} +(-0.111693 + 1.43146i) q^{28} +(4.18618 - 1.91177i) q^{29} +(0.795969 - 3.07700i) q^{30} +(-5.97499 + 0.859074i) q^{31} +(2.24903 - 5.19056i) q^{32} +(-0.304396 - 0.263761i) q^{33} +(-1.30513 + 0.0423860i) q^{34} +(-0.603942 - 0.388130i) q^{35} +(-1.94232 - 3.61246i) q^{36} +(10.1890 + 2.99176i) q^{37} +(0.161728 + 0.174839i) q^{38} +(10.7314 - 9.29884i) q^{39} +(1.75330 + 2.21945i) q^{40} +(-3.27821 + 0.962568i) q^{41} +(-2.24676 + 0.397861i) q^{42} +(-0.788426 - 0.113359i) q^{43} +(0.350731 - 0.0739238i) q^{44} +2.05076 q^{45} +(-6.77305 - 0.354631i) q^{46} -0.573162i q^{47} +(8.88076 + 1.39437i) q^{48} +(0.922856 - 6.41861i) q^{49} +(-1.39255 + 0.246595i) q^{50} +(-0.584631 - 1.99107i) q^{51} +(0.819925 + 12.6100i) q^{52} +(2.71012 + 3.12764i) q^{53} +(-2.21473 + 2.04865i) q^{54} +(-0.0504917 + 0.171959i) q^{55} +(0.926486 - 1.80686i) q^{56} +(-0.204625 + 0.318403i) q^{57} +(-6.50487 + 0.211256i) q^{58} +(5.33984 - 6.16250i) q^{59} +(-2.71684 + 3.58076i) q^{60} +(1.26232 + 8.77960i) q^{61} +(8.26476 + 2.13795i) q^{62} +(-0.611598 - 1.33921i) q^{63} +(-5.76557 + 5.54601i) q^{64} +(-5.74734 - 2.62472i) q^{65} +(0.253317 + 0.510180i) q^{66} +(0.841448 + 1.30932i) q^{67} +(1.72605 + 0.656536i) q^{68} +(-2.14997 - 10.5615i) q^{69} +(0.576333 + 0.835836i) q^{70} +(-5.17146 - 8.04695i) q^{71} +(0.564044 + 5.77294i) q^{72} +(-0.463174 + 1.01421i) q^{73} +(-11.6629 - 9.46094i) q^{74} +(-0.933600 - 2.04430i) q^{75} +(-0.115663 - 0.316341i) q^{76} +(0.127353 - 0.0183105i) q^{77} +(-19.0742 + 6.28004i) q^{78} +(-3.93088 + 4.53647i) q^{79} +(-1.07718 - 3.85223i) q^{80} +(-9.20892 - 5.91821i) q^{81} +(4.80242 + 0.532034i) q^{82} +(2.87550 - 9.79305i) q^{83} +(3.14859 + 0.706298i) q^{84} +(-0.697821 + 0.604666i) q^{85} +(0.966915 + 0.577934i) q^{86} +(-2.91386 - 9.92368i) q^{87} +(-0.497955 - 0.0948395i) q^{88} +(-7.00700 - 1.00745i) q^{89} +(-2.67585 - 1.11853i) q^{90} +4.53596i q^{91} +(8.64409 + 4.15688i) q^{92} +13.5662i q^{93} +(-0.312614 + 0.747865i) q^{94} +(0.166697 + 0.0239674i) q^{95} +(-10.8272 - 6.66312i) q^{96} +(-5.05401 - 17.2124i) q^{97} +(-4.70498 + 7.87169i) q^{98} +(-0.277764 + 0.240684i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.30481 − 0.545420i −0.922637 − 0.385670i
\(3\) 0.319837 − 2.22451i 0.184658 − 1.28432i −0.660914 − 0.750462i \(-0.729831\pi\)
0.845571 − 0.533862i \(-0.179260\pi\)
\(4\) 1.40503 + 1.42333i 0.702517 + 0.711667i
\(5\) −0.959493 + 0.281733i −0.429098 + 0.125995i
\(6\) −1.63062 + 2.72811i −0.665697 + 1.11375i
\(7\) 0.470129 + 0.542558i 0.177692 + 0.205068i 0.837608 − 0.546272i \(-0.183953\pi\)
−0.659916 + 0.751340i \(0.729408\pi\)
\(8\) −1.05698 − 2.62351i −0.373700 − 0.927550i
\(9\) −1.96769 − 0.577767i −0.655897 − 0.192589i
\(10\) 1.40561 + 0.155720i 0.444494 + 0.0492430i
\(11\) 0.0968928 − 0.150768i 0.0292143 − 0.0454583i −0.826339 − 0.563173i \(-0.809581\pi\)
0.855554 + 0.517714i \(0.173217\pi\)
\(12\) 3.61561 − 2.67029i 1.04374 − 0.770845i
\(13\) 4.77506 + 4.13762i 1.32436 + 1.14757i 0.977800 + 0.209542i \(0.0671972\pi\)
0.346564 + 0.938026i \(0.387348\pi\)
\(14\) −0.317505 − 0.964351i −0.0848569 − 0.257734i
\(15\) 0.319837 + 2.22451i 0.0825815 + 0.574367i
\(16\) −0.0517542 + 3.99967i −0.0129386 + 0.999916i
\(17\) 0.839909 − 0.383573i 0.203708 − 0.0930302i −0.310949 − 0.950427i \(-0.600647\pi\)
0.514656 + 0.857396i \(0.327919\pi\)
\(18\) 2.25233 + 1.82709i 0.530879 + 0.430649i
\(19\) −0.153192 − 0.0699606i −0.0351447 − 0.0160501i 0.397765 − 0.917487i \(-0.369786\pi\)
−0.432910 + 0.901437i \(0.642513\pi\)
\(20\) −1.74912 − 0.969834i −0.391115 − 0.216861i
\(21\) 1.35729 − 0.872280i 0.296186 − 0.190347i
\(22\) −0.208658 + 0.143876i −0.0444861 + 0.0306744i
\(23\) 4.51547 − 1.61572i 0.941540 − 0.336900i
\(24\) −6.17409 + 1.51218i −1.26028 + 0.308673i
\(25\) 0.841254 − 0.540641i 0.168251 − 0.108128i
\(26\) −3.97379 − 8.00320i −0.779325 − 1.56956i
\(27\) 0.886208 − 1.94052i 0.170551 − 0.373454i
\(28\) −0.111693 + 1.43146i −0.0211080 + 0.270521i
\(29\) 4.18618 − 1.91177i 0.777355 − 0.355006i 0.0130947 − 0.999914i \(-0.495832\pi\)
0.764260 + 0.644908i \(0.223104\pi\)
\(30\) 0.795969 − 3.07700i 0.145323 − 0.561782i
\(31\) −5.97499 + 0.859074i −1.07314 + 0.154294i −0.656160 − 0.754621i \(-0.727821\pi\)
−0.416980 + 0.908916i \(0.636911\pi\)
\(32\) 2.24903 − 5.19056i 0.397575 − 0.917570i
\(33\) −0.304396 − 0.263761i −0.0529886 − 0.0459149i
\(34\) −1.30513 + 0.0423860i −0.223827 + 0.00726915i
\(35\) −0.603942 − 0.388130i −0.102085 − 0.0656059i
\(36\) −1.94232 − 3.61246i −0.323720 − 0.602077i
\(37\) 10.1890 + 2.99176i 1.67506 + 0.491842i 0.974993 − 0.222236i \(-0.0713355\pi\)
0.700066 + 0.714078i \(0.253154\pi\)
\(38\) 0.161728 + 0.174839i 0.0262358 + 0.0283627i
\(39\) 10.7314 − 9.29884i 1.71840 − 1.48901i
\(40\) 1.75330 + 2.21945i 0.277220 + 0.350926i
\(41\) −3.27821 + 0.962568i −0.511970 + 0.150328i −0.527506 − 0.849551i \(-0.676873\pi\)
0.0155360 + 0.999879i \(0.495055\pi\)
\(42\) −2.24676 + 0.397861i −0.346683 + 0.0613912i
\(43\) −0.788426 − 0.113359i −0.120234 − 0.0172870i 0.0819352 − 0.996638i \(-0.473890\pi\)
−0.202169 + 0.979351i \(0.564799\pi\)
\(44\) 0.350731 − 0.0739238i 0.0528747 − 0.0111444i
\(45\) 2.05076 0.305710
\(46\) −6.77305 − 0.354631i −0.998632 − 0.0522875i
\(47\) − 0.573162i − 0.0836043i −0.999126 − 0.0418022i \(-0.986690\pi\)
0.999126 − 0.0418022i \(-0.0133099\pi\)
\(48\) 8.88076 + 1.39437i 1.28183 + 0.201260i
\(49\) 0.922856 − 6.41861i 0.131837 − 0.916944i
\(50\) −1.39255 + 0.246595i −0.196936 + 0.0348738i
\(51\) −0.584631 − 1.99107i −0.0818647 − 0.278806i
\(52\) 0.819925 + 12.6100i 0.113703 + 1.74869i
\(53\) 2.71012 + 3.12764i 0.372264 + 0.429615i 0.910711 − 0.413044i \(-0.135534\pi\)
−0.538448 + 0.842659i \(0.680989\pi\)
\(54\) −2.21473 + 2.04865i −0.301386 + 0.278786i
\(55\) −0.0504917 + 0.171959i −0.00680830 + 0.0231869i
\(56\) 0.926486 − 1.80686i 0.123807 − 0.241452i
\(57\) −0.204625 + 0.318403i −0.0271033 + 0.0421735i
\(58\) −6.50487 + 0.211256i −0.854131 + 0.0277393i
\(59\) 5.33984 − 6.16250i 0.695188 − 0.802290i −0.292906 − 0.956141i \(-0.594622\pi\)
0.988094 + 0.153851i \(0.0491677\pi\)
\(60\) −2.71684 + 3.58076i −0.350743 + 0.462274i
\(61\) 1.26232 + 8.77960i 0.161623 + 1.12411i 0.895575 + 0.444911i \(0.146765\pi\)
−0.733952 + 0.679201i \(0.762326\pi\)
\(62\) 8.26476 + 2.13795i 1.04963 + 0.271520i
\(63\) −0.611598 − 1.33921i −0.0770541 − 0.168725i
\(64\) −5.76557 + 5.54601i −0.720697 + 0.693251i
\(65\) −5.74734 − 2.62472i −0.712870 − 0.325557i
\(66\) 0.253317 + 0.510180i 0.0311812 + 0.0627988i
\(67\) 0.841448 + 1.30932i 0.102799 + 0.159959i 0.888843 − 0.458213i \(-0.151510\pi\)
−0.786043 + 0.618171i \(0.787874\pi\)
\(68\) 1.72605 + 0.656536i 0.209315 + 0.0796166i
\(69\) −2.14997 − 10.5615i −0.258826 − 1.27145i
\(70\) 0.576333 + 0.835836i 0.0688850 + 0.0999015i
\(71\) −5.17146 − 8.04695i −0.613739 − 0.954997i −0.999475 − 0.0323877i \(-0.989689\pi\)
0.385736 − 0.922609i \(-0.373948\pi\)
\(72\) 0.564044 + 5.77294i 0.0664732 + 0.680348i
\(73\) −0.463174 + 1.01421i −0.0542104 + 0.118704i −0.934799 − 0.355178i \(-0.884420\pi\)
0.880588 + 0.473882i \(0.157148\pi\)
\(74\) −11.6629 − 9.46094i −1.35578 − 1.09981i
\(75\) −0.933600 − 2.04430i −0.107803 − 0.236055i
\(76\) −0.115663 − 0.316341i −0.0132675 − 0.0362868i
\(77\) 0.127353 − 0.0183105i 0.0145132 − 0.00208668i
\(78\) −19.0742 + 6.28004i −2.15973 + 0.711075i
\(79\) −3.93088 + 4.53647i −0.442258 + 0.510393i −0.932488 − 0.361200i \(-0.882367\pi\)
0.490230 + 0.871593i \(0.336913\pi\)
\(80\) −1.07718 − 3.85223i −0.120432 − 0.430693i
\(81\) −9.20892 − 5.91821i −1.02321 − 0.657579i
\(82\) 4.80242 + 0.532034i 0.530339 + 0.0587533i
\(83\) 2.87550 − 9.79305i 0.315627 − 1.07493i −0.637020 − 0.770847i \(-0.719833\pi\)
0.952647 − 0.304079i \(-0.0983488\pi\)
\(84\) 3.14859 + 0.706298i 0.343539 + 0.0770634i
\(85\) −0.697821 + 0.604666i −0.0756893 + 0.0655852i
\(86\) 0.966915 + 0.577934i 0.104265 + 0.0623202i
\(87\) −2.91386 − 9.92368i −0.312398 − 1.06393i
\(88\) −0.497955 − 0.0948395i −0.0530822 − 0.0101099i
\(89\) −7.00700 − 1.00745i −0.742741 − 0.106790i −0.239452 − 0.970908i \(-0.576968\pi\)
−0.503289 + 0.864118i \(0.667877\pi\)
\(90\) −2.67585 − 1.11853i −0.282059 − 0.117903i
\(91\) 4.53596i 0.475498i
\(92\) 8.64409 + 4.15688i 0.901209 + 0.433385i
\(93\) 13.5662i 1.40675i
\(94\) −0.312614 + 0.747865i −0.0322437 + 0.0771364i
\(95\) 0.166697 + 0.0239674i 0.0171028 + 0.00245901i
\(96\) −10.8272 − 6.66312i −1.10504 − 0.680052i
\(97\) −5.05401 − 17.2124i −0.513157 − 1.74765i −0.652879 − 0.757462i \(-0.726439\pi\)
0.139722 − 0.990191i \(-0.455379\pi\)
\(98\) −4.70498 + 7.87169i −0.475275 + 0.795161i
\(99\) −0.277764 + 0.240684i −0.0279163 + 0.0241896i
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.11.7 ✓ 480
8.3 odd 2 920.2.bb.b.11.25 yes 480
23.21 odd 22 920.2.bb.b.251.25 yes 480
184.67 even 22 inner 920.2.bb.a.251.7 yes 480
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.bb.a.11.7 ✓ 480 1.1 even 1 trivial
920.2.bb.a.251.7 yes 480 184.67 even 22 inner
920.2.bb.b.11.25 yes 480 8.3 odd 2
920.2.bb.b.251.25 yes 480 23.21 odd 22