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Properties

Label 90.11.k.b.7.8
Level $90$
Weight $11$
Character 90.7
Analytic conductor $57.182$
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] = [90,11,Mod(7,90)] 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("90.7"); S:= CuspForms(chi, 11); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(90, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([8, 3])) N = Newforms(chi, 11, names="a")
 
Level: \( N \) \(=\) \( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 90.k (of order \(12\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [120,960] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(57.1821527406\)
Analytic rank: \(0\)
Dimension: \(120\)
Relative dimension: \(30\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label 7.8
Character \(\chi\) \(=\) 90.7
Dual form 90.11.k.b.13.8

$q$-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+(21.8564 - 5.85641i) q^{2} +(-161.252 + 181.788i) q^{3} +(443.405 - 256.000i) q^{4} +(975.463 - 2968.85i) q^{5} +(-2459.76 + 4917.59i) q^{6} +(-526.269 + 141.013i) q^{7} +(8192.00 - 8192.00i) q^{8} +(-7044.69 - 58627.3i) q^{9} +(3933.30 - 70601.2i) q^{10} +(-124055. + 214869. i) q^{11} +(-24962.2 + 121886. i) q^{12} +(-416503. - 111602. i) q^{13} +(-10676.5 + 6164.09i) q^{14} +(382407. + 656061. i) q^{15} +(131072. - 227023. i) q^{16} +(1.52316e6 + 1.52316e6i) q^{17} +(-497317. - 1.24012e6i) q^{18} -3.11311e6i q^{19} +(-327501. - 1.56612e6i) q^{20} +(59227.3 - 118408. i) q^{21} +(-1.45303e6 + 5.42279e6i) q^{22} +(3.17712e6 + 851307. i) q^{23} +(168232. + 2.81018e6i) q^{24} +(-7.86257e6 - 5.79202e6i) q^{25} -9.75685e6 q^{26} +(1.17937e7 + 8.17312e6i) q^{27} +(-197251. + 197251. i) q^{28} +(9.18029e6 + 5.30024e6i) q^{29} +(1.22002e7 + 1.20996e7i) q^{30} +(-473144. - 819510. i) q^{31} +(1.53522e6 - 5.72953e6i) q^{32} +(-1.90566e7 - 5.71998e7i) q^{33} +(4.22111e7 + 2.43706e7i) q^{34} +(-94707.9 + 1.69997e6i) q^{35} +(-1.81322e7 - 2.41922e7i) q^{36} +(6.90415e7 + 6.90415e7i) q^{37} +(-1.82316e7 - 6.80413e7i) q^{38} +(8.74498e7 - 5.77193e7i) q^{39} +(-1.63299e7 - 3.23118e7i) q^{40} +(8.85829e7 + 1.53430e8i) q^{41} +(601051. - 2.93483e6i) q^{42} +(4.41249e7 + 1.64676e8i) q^{43} +1.27032e8i q^{44} +(-1.80928e8 - 3.62741e7i) q^{45} +7.44260e7 q^{46} +(-3.21347e8 + 8.61047e7i) q^{47} +(2.01345e7 + 6.04352e7i) q^{48} +(-2.44374e8 + 1.41089e8i) q^{49} +(-2.05768e8 - 8.05463e7i) q^{50} +(-5.22505e8 + 3.12797e7i) q^{51} +(-2.13250e8 + 5.71401e7i) q^{52} +(-4.44165e8 + 4.44165e8i) q^{53} +(3.05633e8 + 1.09566e8i) q^{54} +(5.16905e8 + 5.77898e8i) q^{55} +(-3.15601e6 + 5.46637e6i) q^{56} +(5.65925e8 + 5.01994e8i) q^{57} +(2.31688e8 + 6.20807e7i) q^{58} +(6.93500e8 - 4.00392e8i) q^{59} +(3.37513e8 + 1.93004e8i) q^{60} +(-5.70943e8 + 9.88903e8i) q^{61} +(-1.51406e7 - 1.51406e7i) q^{62} +(1.19746e7 + 2.98603e7i) q^{63} -1.34218e8i q^{64} +(-7.37613e8 + 1.12767e9i) q^{65} +(-7.51493e8 - 1.13858e9i) q^{66} +(3.09088e8 - 1.15353e9i) q^{67} +(1.06531e9 + 2.85448e8i) q^{68} +(-6.67074e8 + 4.40287e8i) q^{69} +(7.88573e6 + 3.77098e7i) q^{70} +1.81853e9 q^{71} +(-5.37985e8 - 4.22565e8i) q^{72} +(-1.83162e9 + 1.83162e9i) q^{73} +(1.91333e9 + 1.10466e9i) q^{74} +(2.32077e9 - 4.95347e8i) q^{75} +(-7.96955e8 - 1.38037e9i) q^{76} +(3.49868e7 - 1.30572e8i) q^{77} +(1.57331e9 - 1.77368e9i) q^{78} +(1.59793e8 + 9.22567e7i) q^{79} +(-5.46143e8 - 6.10587e8i) q^{80} +(-3.38753e9 + 8.26022e8i) q^{81} +(2.83465e9 + 2.83465e9i) q^{82} +(-4.38497e8 - 1.63649e9i) q^{83} +(-4.05076e6 - 6.76648e7i) q^{84} +(6.00783e9 - 3.03626e9i) q^{85} +(1.92882e9 + 3.34082e9i) q^{86} +(-2.44386e9 + 8.14192e8i) q^{87} +(7.43952e8 + 2.77647e9i) q^{88} +4.15330e9i q^{89} +(-4.16686e9 + 2.66765e8i) q^{90} +2.34930e8 q^{91} +(1.62669e9 - 4.35869e8i) q^{92} +(2.25272e8 + 4.61356e7i) q^{93} +(-6.51923e9 + 3.76388e9i) q^{94} +(-9.24236e9 - 3.03672e9i) q^{95} +(7.94001e8 + 1.20298e9i) q^{96} +(1.26748e10 - 3.39621e9i) q^{97} +(-4.51485e9 + 4.51485e9i) q^{98} +(1.34711e10 + 5.75931e9i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 120 q + 960 q^{2} - 128 q^{3} - 8192 q^{6} + 1830 q^{7} + 983040 q^{8} + 105792 q^{10} + 217740 q^{11} - 65536 q^{12} - 3385658 q^{15} + 15728640 q^{16} + 2438244 q^{17} + 2534464 q^{18} + 1692672 q^{20}+ \cdots + 76966514304 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/90\mathbb{Z}\right)^\times\).

\(n\) \(11\) \(37\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 21.8564 − 5.85641i 0.683013 − 0.183013i
\(3\) −161.252 + 181.788i −0.663588 + 0.748098i
\(4\) 443.405 − 256.000i 0.433013 − 0.250000i
\(5\) 975.463 − 2968.85i 0.312148 − 0.950033i
\(6\) −2459.76 + 4917.59i −0.316327 + 0.632406i
\(7\) −526.269 + 141.013i −0.0313125 + 0.00839015i −0.274441 − 0.961604i \(-0.588493\pi\)
0.243129 + 0.969994i \(0.421826\pi\)
\(8\) 8192.00 − 8192.00i 0.250000 − 0.250000i
\(9\) −7044.69 − 58627.3i −0.119302 − 0.992858i
\(10\) 3933.30 − 70601.2i 0.0393330 − 0.706012i
\(11\) −124055. + 214869.i −0.770283 + 1.33417i 0.167124 + 0.985936i \(0.446552\pi\)
−0.937408 + 0.348234i \(0.886781\pi\)
\(12\) −24962.2 + 121886.i −0.100317 + 0.489833i
\(13\) −416503. − 111602.i −1.12176 − 0.300576i −0.350167 − 0.936687i \(-0.613875\pi\)
−0.771597 + 0.636111i \(0.780542\pi\)
\(14\) −10676.5 + 6164.09i −0.0198513 + 0.0114612i
\(15\) 382407. + 656061.i 0.503581 + 0.863948i
\(16\) 131072. − 227023.i 0.125000 − 0.216506i
\(17\) 1.52316e6 + 1.52316e6i 1.07276 + 1.07276i 0.997137 + 0.0756200i \(0.0240936\pi\)
0.0756200 + 0.997137i \(0.475906\pi\)
\(18\) −497317. − 1.24012e6i −0.263191 − 0.656301i
\(19\) − 3.11311e6i − 1.25726i −0.777704 − 0.628631i \(-0.783615\pi\)
0.777704 − 0.628631i \(-0.216385\pi\)
\(20\) −327501. − 1.56612e6i −0.102344 − 0.489414i
\(21\) 59227.3 − 118408.i 0.0145019 − 0.0289924i
\(22\) −1.45303e6 + 5.42279e6i −0.281943 + 1.05223i
\(23\) 3.17712e6 + 851307.i 0.493622 + 0.132266i 0.497038 − 0.867729i \(-0.334421\pi\)
−0.00341649 + 0.999994i \(0.501088\pi\)
\(24\) 168232. + 2.81018e6i 0.0211276 + 0.352922i
\(25\) −7.86257e6 − 5.79202e6i −0.805127 − 0.593102i
\(26\) −9.75685e6 −0.821189
\(27\) 1.17937e7 + 8.17312e6i 0.821923 + 0.569598i
\(28\) −197251. + 197251.i −0.0114612 + 0.0114612i
\(29\) 9.18029e6 + 5.30024e6i 0.447576 + 0.258408i 0.706806 − 0.707408i \(-0.250135\pi\)
−0.259230 + 0.965816i \(0.583469\pi\)
\(30\) 1.22002e7 + 1.20996e7i 0.502066 + 0.497926i
\(31\) −473144. − 819510.i −0.0165267 − 0.0286250i 0.857644 − 0.514244i \(-0.171928\pi\)
−0.874170 + 0.485619i \(0.838594\pi\)
\(32\) 1.53522e6 − 5.72953e6i 0.0457532 − 0.170753i
\(33\) −1.90566e7 − 5.71998e7i −0.486940 − 1.46159i
\(34\) 4.22111e7 + 2.43706e7i 0.929035 + 0.536378i
\(35\) −94707.9 + 1.69997e6i −0.00180321 + 0.0323669i
\(36\) −1.81322e7 − 2.41922e7i −0.299874 − 0.400095i
\(37\) 6.90415e7 + 6.90415e7i 0.995638 + 0.995638i 0.999991 − 0.00435250i \(-0.00138545\pi\)
−0.00435250 + 0.999991i \(0.501385\pi\)
\(38\) −1.82316e7 − 6.80413e7i −0.230095 − 0.858726i
\(39\) 8.74498e7 − 5.77193e7i 0.969250 − 0.639732i
\(40\) −1.63299e7 − 3.23118e7i −0.159471 − 0.315545i
\(41\) 8.85829e7 + 1.53430e8i 0.764594 + 1.32432i 0.940461 + 0.339901i \(0.110394\pi\)
−0.175867 + 0.984414i \(0.556273\pi\)
\(42\) 601051. − 2.93483e6i 0.00459901 − 0.0224562i
\(43\) 4.41249e7 + 1.64676e8i 0.300152 + 1.12018i 0.937039 + 0.349225i \(0.113555\pi\)
−0.636887 + 0.770957i \(0.719778\pi\)
\(44\) 1.27032e8i 0.770283i
\(45\) −1.80928e8 − 3.62741e7i −0.980488 − 0.196578i
\(46\) 7.44260e7 0.361356
\(47\) −3.21347e8 + 8.61047e7i −1.40115 + 0.375438i −0.878758 − 0.477267i \(-0.841628\pi\)
−0.522394 + 0.852704i \(0.674961\pi\)
\(48\) 2.01345e7 + 6.04352e7i 0.0790196 + 0.237183i
\(49\) −2.44374e8 + 1.41089e8i −0.865115 + 0.499475i
\(50\) −2.05768e8 − 8.05463e7i −0.658457 − 0.257748i
\(51\) −5.22505e8 + 3.12797e7i −1.51440 + 0.0906593i
\(52\) −2.13250e8 + 5.71401e7i −0.560882 + 0.150288i
\(53\) −4.44165e8 + 4.44165e8i −1.06210 + 1.06210i −0.0641592 + 0.997940i \(0.520437\pi\)
−0.997940 + 0.0641592i \(0.979563\pi\)
\(54\) 3.05633e8 + 1.09566e8i 0.665628 + 0.238621i
\(55\) 5.16905e8 + 5.77898e8i 1.02706 + 1.14825i
\(56\) −3.15601e6 + 5.46637e6i −0.00573058 + 0.00992565i
\(57\) 5.65925e8 + 5.01994e8i 0.940556 + 0.834304i
\(58\) 2.31688e8 + 6.20807e7i 0.352992 + 0.0945838i
\(59\) 6.93500e8 − 4.00392e8i 0.970033 − 0.560049i 0.0707866 − 0.997491i \(-0.477449\pi\)
0.899246 + 0.437443i \(0.144116\pi\)
\(60\) 3.37513e8 + 1.93004e8i 0.434044 + 0.248205i
\(61\) −5.70943e8 + 9.88903e8i −0.675995 + 1.17086i 0.300181 + 0.953882i \(0.402953\pi\)
−0.976177 + 0.216976i \(0.930381\pi\)
\(62\) −1.51406e7 − 1.51406e7i −0.0165267 − 0.0165267i
\(63\) 1.19746e7 + 2.98603e7i 0.0120659 + 0.0300879i
\(64\) − 1.34218e8i − 0.125000i
\(65\) −7.37613e8 + 1.12767e9i −0.635714 + 0.971890i
\(66\) −7.51493e8 − 1.13858e9i −0.600075 − 0.909166i
\(67\) 3.09088e8 − 1.15353e9i 0.228933 − 0.854390i −0.751857 − 0.659326i \(-0.770842\pi\)
0.980790 − 0.195064i \(-0.0624915\pi\)
\(68\) 1.06531e9 + 2.85448e8i 0.732706 + 0.196328i
\(69\) −6.67074e8 + 4.40287e8i −0.426509 + 0.281508i
\(70\) 7.88573e6 + 3.77098e7i 0.00469193 + 0.0224370i
\(71\) 1.81853e9 1.00793 0.503964 − 0.863725i \(-0.331874\pi\)
0.503964 + 0.863725i \(0.331874\pi\)
\(72\) −5.37985e8 − 4.22565e8i −0.278040 − 0.218389i
\(73\) −1.83162e9 + 1.83162e9i −0.883529 + 0.883529i −0.993891 − 0.110363i \(-0.964799\pi\)
0.110363 + 0.993891i \(0.464799\pi\)
\(74\) 1.91333e9 + 1.10466e9i 0.862248 + 0.497819i
\(75\) 2.32077e9 − 4.95347e8i 0.977971 − 0.208739i
\(76\) −7.96955e8 − 1.38037e9i −0.314316 − 0.544411i
\(77\) 3.49868e7 − 1.30572e8i 0.0129256 − 0.0482389i
\(78\) 1.57331e9 − 1.77368e9i 0.544931 − 0.614330i
\(79\) 1.59793e8 + 9.22567e7i 0.0519305 + 0.0299821i 0.525740 − 0.850645i \(-0.323788\pi\)
−0.473810 + 0.880627i \(0.657122\pi\)
\(80\) −5.46143e8 − 6.10587e8i −0.166670 − 0.186336i
\(81\) −3.38753e9 + 8.26022e8i −0.971534 + 0.236901i
\(82\) 2.83465e9 + 2.83465e9i 0.764594 + 0.764594i
\(83\) −4.38497e8 − 1.63649e9i −0.111321 − 0.415455i 0.887665 − 0.460490i \(-0.152326\pi\)
−0.998985 + 0.0450358i \(0.985660\pi\)
\(84\) −4.05076e6 − 6.76648e7i −0.000968589 − 0.0161796i
\(85\) 6.00783e9 − 3.03626e9i 1.35401 − 0.684296i
\(86\) 1.92882e9 + 3.34082e9i 0.410015 + 0.710167i
\(87\) −2.44386e9 + 8.14192e8i −0.490320 + 0.163354i
\(88\) 7.43952e8 + 2.77647e9i 0.140972 + 0.526113i
\(89\) 4.15330e9i 0.743778i 0.928277 + 0.371889i \(0.121290\pi\)
−0.928277 + 0.371889i \(0.878710\pi\)
\(90\) −4.16686e9 + 2.66765e8i −0.705662 + 0.0451768i
\(91\) 2.34930e8 0.0376471
\(92\) 1.62669e9 − 4.35869e8i 0.246811 − 0.0661328i
\(93\) 2.25272e8 + 4.61356e7i 0.0323812 + 0.00663164i
\(94\) −6.51923e9 + 3.76388e9i −0.888295 + 0.512857i
\(95\) −9.24236e9 − 3.03672e9i −1.19444 − 0.392452i
\(96\) 7.94001e8 + 1.20298e9i 0.0973789 + 0.147538i
\(97\) 1.26748e10 − 3.39621e9i 1.47599 − 0.395491i 0.571011 − 0.820943i \(-0.306551\pi\)
0.904981 + 0.425452i \(0.139885\pi\)
\(98\) −4.51485e9 + 4.51485e9i −0.499475 + 0.499475i
\(99\) 1.34711e10 + 5.75931e9i 1.41654 + 0.605612i
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 90.11.k.b.7.8 ✓ 120
5.3 odd 4 inner 90.11.k.b.43.9 yes 120
9.4 even 3 inner 90.11.k.b.67.9 yes 120
45.13 odd 12 inner 90.11.k.b.13.8 yes 120
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
90.11.k.b.7.8 ✓ 120 1.1 even 1 trivial
90.11.k.b.13.8 yes 120 45.13 odd 12 inner
90.11.k.b.43.9 yes 120 5.3 odd 4 inner
90.11.k.b.67.9 yes 120 9.4 even 3 inner