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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 43.9
Character \(\chi\) \(=\) 90.43
Dual form 90.11.k.b.67.9

$q$-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+(-5.85641 - 21.8564i) q^{2} +(-181.788 - 161.252i) q^{3} +(-443.405 + 256.000i) q^{4} +(-3058.83 - 639.651i) q^{5} +(-2459.76 + 4917.59i) q^{6} +(141.013 + 526.269i) 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} +(121886. + 24962.2i) q^{12} +(111602. - 416503. i) q^{13} +(10676.5 - 6164.09i) q^{14} +(452914. + 609524. i) q^{15} +(131072. - 227023. i) q^{16} +(1.52316e6 - 1.52316e6i) q^{17} +(1.24012e6 - 497317. i) q^{18} +3.11311e6i q^{19} +(1.52005e6 - 499437. i) q^{20} +(59227.3 - 118408. i) q^{21} +(5.42279e6 + 1.45303e6i) q^{22} +(-851307. + 3.17712e6i) q^{23} +(-168232. - 2.81018e6i) q^{24} +(8.94732e6 + 3.91318e6i) q^{25} -9.75685e6 q^{26} +(8.17312e6 - 1.17937e7i) q^{27} +(-197251. - 197251. i) q^{28} +(-9.18029e6 - 5.30024e6i) q^{29} +(1.06695e7 - 1.34687e7i) q^{30} +(-473144. - 819510. i) q^{31} +(-5.72953e6 - 1.53522e6i) q^{32} +(5.71998e7 - 1.90566e7i) 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} +(6.80413e7 - 1.82316e7i) q^{38} +(-8.74498e7 + 5.77193e7i) q^{39} +(-1.98180e7 - 3.02980e7i) q^{40} +(8.85829e7 + 1.53430e8i) q^{41} +(-2.93483e6 - 601051. i) q^{42} +(-1.64676e8 + 4.41249e7i) q^{43} -1.27032e8i q^{44} +(1.59525e7 - 1.83837e8i) q^{45} +7.44260e7 q^{46} +(8.61047e7 + 3.21347e8i) q^{47} +(-6.04352e7 + 2.01345e7i) q^{48} +(2.44374e8 - 1.41089e8i) q^{49} +(3.31288e7 - 2.18473e8i) q^{50} +(-5.22505e8 + 3.12797e7i) q^{51} +(5.71401e7 + 2.13250e8i) 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} +(5.01994e8 - 5.65925e8i) q^{57} +(-6.20807e7 + 2.31688e8i) q^{58} +(-6.93500e8 + 4.00392e8i) q^{59} +(-3.56863e8 - 1.54320e8i) q^{60} +(-5.70943e8 + 9.88903e8i) q^{61} +(-1.51406e7 + 1.51406e7i) q^{62} +(-2.98603e7 + 1.19746e7i) q^{63} +1.34218e8i q^{64} +(-6.07788e8 + 1.20263e9i) q^{65} +(-7.51493e8 - 1.13858e9i) q^{66} +(-1.15353e9 - 3.09088e8i) q^{67} +(-2.85448e8 + 1.06531e9i) q^{68} +(6.67074e8 - 4.40287e8i) q^{69} +(-3.66005e7 + 1.20257e7i) q^{70} +1.81853e9 q^{71} +(-4.22565e8 + 5.37985e8i) q^{72} +(-1.83162e9 - 1.83162e9i) q^{73} +(-1.91333e9 - 1.10466e9i) q^{74} +(-9.95507e8 - 2.15414e9i) q^{75} +(-7.96955e8 - 1.38037e9i) q^{76} +(-1.30572e8 - 3.49868e7i) q^{77} +(1.77368e9 + 1.57331e9i) 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} +(1.63649e9 - 4.38497e8i) q^{83} +(4.05076e6 + 6.76648e7i) q^{84} +(-5.63339e9 + 3.68481e9i) q^{85} +(1.92882e9 + 3.34082e9i) q^{86} +(8.14192e8 + 2.44386e9i) q^{87} +(-2.77647e9 + 7.43952e8i) q^{88} -4.15330e9i q^{89} +(-4.11145e9 + 7.27962e8i) q^{90} +2.34930e8 q^{91} +(-4.35869e8 - 1.62669e9i) q^{92} +(-4.61356e7 + 2.25272e8i) q^{93} +(6.51923e9 - 3.76388e9i) q^{94} +(1.99130e9 - 9.52248e9i) q^{95} +(7.94001e8 + 1.20298e9i) q^{96} +(-3.39621e9 - 1.26748e10i) 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 \(\left(\mathbb{Z}/90\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −5.85641 − 21.8564i −0.183013 − 0.683013i
\(3\) −181.788 − 161.252i −0.748098 − 0.663588i
\(4\) −443.405 + 256.000i −0.433013 + 0.250000i
\(5\) −3058.83 − 639.651i −0.978827 − 0.204688i
\(6\) −2459.76 + 4917.59i −0.316327 + 0.632406i
\(7\) 141.013 + 526.269i 0.00839015 + 0.0313125i 0.969994 − 0.243129i \(-0.0781738\pi\)
−0.961604 + 0.274441i \(0.911507\pi\)
\(8\) 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 \(0.886781\pi\)
\(12\) 121886. + 24962.2i 0.489833 + 0.100317i
\(13\) 111602. − 416503.i 0.300576 − 1.12176i −0.636111 − 0.771597i \(-0.719458\pi\)
0.936687 − 0.350167i \(-0.113875\pi\)
\(14\) 10676.5 − 6164.09i 0.0198513 − 0.0114612i
\(15\) 452914. + 609524.i 0.596430 + 0.802665i
\(16\) 131072. − 227023.i 0.125000 − 0.216506i
\(17\) 1.52316e6 − 1.52316e6i 1.07276 − 1.07276i 0.0756200 − 0.997137i \(-0.475906\pi\)
0.997137 − 0.0756200i \(-0.0240936\pi\)
\(18\) 1.24012e6 − 497317.i 0.656301 − 0.263191i
\(19\) 3.11311e6i 1.25726i 0.777704 + 0.628631i \(0.216385\pi\)
−0.777704 + 0.628631i \(0.783615\pi\)
\(20\) 1.52005e6 − 499437.i 0.475017 − 0.156074i
\(21\) 59227.3 − 118408.i 0.0145019 − 0.0289924i
\(22\) 5.42279e6 + 1.45303e6i 1.05223 + 0.281943i
\(23\) −851307. + 3.17712e6i −0.132266 + 0.493622i −0.999994 − 0.00341649i \(-0.998912\pi\)
0.867729 + 0.497038i \(0.165579\pi\)
\(24\) −168232. − 2.81018e6i −0.0211276 − 0.352922i
\(25\) 8.94732e6 + 3.91318e6i 0.916205 + 0.400709i
\(26\) −9.75685e6 −0.821189
\(27\) 8.17312e6 − 1.17937e7i 0.569598 − 0.821923i
\(28\) −197251. − 197251.i −0.0114612 − 0.0114612i
\(29\) −9.18029e6 − 5.30024e6i −0.447576 − 0.258408i 0.259230 − 0.965816i \(-0.416531\pi\)
−0.706806 + 0.707408i \(0.749865\pi\)
\(30\) 1.06695e7 − 1.34687e7i 0.439076 − 0.554267i
\(31\) −473144. − 819510.i −0.0165267 − 0.0286250i 0.857644 − 0.514244i \(-0.171928\pi\)
−0.874170 + 0.485619i \(0.838594\pi\)
\(32\) −5.72953e6 − 1.53522e6i −0.170753 − 0.0457532i
\(33\) 5.71998e7 − 1.90566e7i 1.46159 − 0.486940i
\(34\) −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 −0.00435250 − 0.999991i \(-0.501385\pi\)
0.999991 + 0.00435250i \(0.00138545\pi\)
\(38\) 6.80413e7 − 1.82316e7i 0.858726 − 0.230095i
\(39\) −8.74498e7 + 5.77193e7i −0.969250 + 0.639732i
\(40\) −1.98180e7 − 3.02980e7i −0.193535 − 0.295879i
\(41\) 8.85829e7 + 1.53430e8i 0.764594 + 1.32432i 0.940461 + 0.339901i \(0.110394\pi\)
−0.175867 + 0.984414i \(0.556273\pi\)
\(42\) −2.93483e6 − 601051.i −0.0224562 − 0.00459901i
\(43\) −1.64676e8 + 4.41249e7i −1.12018 + 0.300152i −0.770957 − 0.636887i \(-0.780222\pi\)
−0.349225 + 0.937039i \(0.613555\pi\)
\(44\) − 1.27032e8i − 0.770283i
\(45\) 1.59525e7 − 1.83837e8i 0.0864501 − 0.996256i
\(46\) 7.44260e7 0.361356
\(47\) 8.61047e7 + 3.21347e8i 0.375438 + 1.40115i 0.852704 + 0.522394i \(0.174961\pi\)
−0.477267 + 0.878758i \(0.658372\pi\)
\(48\) −6.04352e7 + 2.01345e7i −0.237183 + 0.0790196i
\(49\) 2.44374e8 − 1.41089e8i 0.865115 − 0.499475i
\(50\) 3.31288e7 − 2.18473e8i 0.106012 − 0.699115i
\(51\) −5.22505e8 + 3.12797e7i −1.51440 + 0.0906593i
\(52\) 5.71401e7 + 2.13250e8i 0.150288 + 0.560882i
\(53\) −4.44165e8 − 4.44165e8i −1.06210 − 1.06210i −0.997940 − 0.0641592i \(-0.979563\pi\)
−0.0641592 − 0.997940i \(-0.520437\pi\)
\(54\) −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
\(57\) 5.01994e8 − 5.65925e8i 0.834304 − 0.940556i
\(58\) −6.20807e7 + 2.31688e8i −0.0945838 + 0.352992i
\(59\) −6.93500e8 + 4.00392e8i −0.970033 + 0.560049i −0.899246 − 0.437443i \(-0.855884\pi\)
−0.0707866 + 0.997491i \(0.522551\pi\)
\(60\) −3.56863e8 − 1.54320e8i −0.458928 − 0.198457i
\(61\) −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
\(63\) −2.98603e7 + 1.19746e7i −0.0300879 + 0.0120659i
\(64\) 1.34218e8i 0.125000i
\(65\) −6.07788e8 + 1.20263e9i −0.523824 + 1.03649i
\(66\) −7.51493e8 − 1.13858e9i −0.600075 − 0.909166i
\(67\) −1.15353e9 − 3.09088e8i −0.854390 − 0.228933i −0.195064 − 0.980790i \(-0.562492\pi\)
−0.659326 + 0.751857i \(0.729158\pi\)
\(68\) −2.85448e8 + 1.06531e9i −0.196328 + 0.732706i
\(69\) 6.67074e8 − 4.40287e8i 0.426509 − 0.281508i
\(70\) −3.66005e7 + 1.20257e7i −0.0217770 + 0.00715516i
\(71\) 1.81853e9 1.00793 0.503964 − 0.863725i \(-0.331874\pi\)
0.503964 + 0.863725i \(0.331874\pi\)
\(72\) −4.22565e8 + 5.37985e8i −0.218389 + 0.278040i
\(73\) −1.83162e9 − 1.83162e9i −0.883529 − 0.883529i 0.110363 − 0.993891i \(-0.464799\pi\)
−0.993891 + 0.110363i \(0.964799\pi\)
\(74\) −1.91333e9 − 1.10466e9i −0.862248 − 0.497819i
\(75\) −9.95507e8 − 2.15414e9i −0.419506 − 0.907753i
\(76\) −7.96955e8 − 1.38037e9i −0.314316 − 0.544411i
\(77\) −1.30572e8 − 3.49868e7i −0.0482389 − 0.0129256i
\(78\) 1.77368e9 + 1.57331e9i 0.614330 + 0.544931i
\(79\) −1.59793e8 − 9.22567e7i −0.0519305 − 0.0299821i 0.473810 − 0.880627i \(-0.342878\pi\)
−0.525740 + 0.850645i \(0.676212\pi\)
\(80\) −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
\(83\) 1.63649e9 − 4.38497e8i 0.415455 − 0.111321i −0.0450358 − 0.998985i \(-0.514340\pi\)
0.460490 + 0.887665i \(0.347674\pi\)
\(84\) 4.05076e6 + 6.76648e7i 0.000968589 + 0.0161796i
\(85\) −5.63339e9 + 3.68481e9i −1.26962 + 0.830463i
\(86\) 1.92882e9 + 3.34082e9i 0.410015 + 0.710167i
\(87\) 8.14192e8 + 2.44386e9i 0.163354 + 0.490320i
\(88\) −2.77647e9 + 7.43952e8i −0.526113 + 0.140972i
\(89\) − 4.15330e9i − 0.743778i −0.928277 − 0.371889i \(-0.878710\pi\)
0.928277 − 0.371889i \(-0.121290\pi\)
\(90\) −4.11145e9 + 7.27962e8i −0.696277 + 0.123281i
\(91\) 2.34930e8 0.0376471
\(92\) −4.35869e8 − 1.62669e9i −0.0661328 − 0.246811i
\(93\) −4.61356e7 + 2.25272e8i −0.00663164 + 0.0323812i
\(94\) 6.51923e9 − 3.76388e9i 0.888295 − 0.512857i
\(95\) 1.99130e9 − 9.52248e9i 0.257347 − 1.23064i
\(96\) 7.94001e8 + 1.20298e9i 0.0973789 + 0.147538i
\(97\) −3.39621e9 − 1.26748e10i −0.395491 − 1.47599i −0.820943 − 0.571011i \(-0.806551\pi\)
0.425452 − 0.904981i \(-0.360115\pi\)
\(98\) −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.43.9 yes 120
5.2 odd 4 inner 90.11.k.b.7.8 ✓ 120
9.4 even 3 inner 90.11.k.b.13.8 yes 120
45.22 odd 12 inner 90.11.k.b.67.9 yes 120
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
90.11.k.b.7.8 ✓ 120 5.2 odd 4 inner
90.11.k.b.13.8 yes 120 9.4 even 3 inner
90.11.k.b.43.9 yes 120 1.1 even 1 trivial
90.11.k.b.67.9 yes 120 45.22 odd 12 inner