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Properties

Label 578.2.f.b.545.12
Level $578$
Weight $2$
Character 578.545
Analytic conductor $4.615$
Analytic rank $0$
Dimension $224$
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] = [578,2,Mod(35,578)] 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("578.35"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(578, base_ring=CyclotomicField(34)) chi = DirichletCharacter(H, H._module([14])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 578 = 2 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 578.f (of order \(17\), degree \(16\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [224] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.61535323683\)
Analytic rank: \(0\)
Dimension: \(224\)
Relative dimension: \(14\) over \(\Q(\zeta_{17})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{17}]$

Embedding invariants

Embedding label 545.12
Character \(\chi\) \(=\) 578.545
Dual form 578.2.f.b.35.12

$q$-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+(0.739009 - 0.673696i) q^{2} +(0.734416 - 2.58120i) q^{3} +(0.0922684 - 0.995734i) q^{4} +(-1.63281 + 2.16219i) q^{5} +(-1.19621 - 2.40230i) q^{6} +(-3.44007 - 2.13000i) q^{7} +(-0.602635 - 0.798017i) q^{8} +(-3.57259 - 2.21206i) q^{9} +(0.249997 + 2.69790i) q^{10} +(-0.223806 + 2.41525i) q^{11} +(-2.50243 - 0.969446i) q^{12} +(-1.79304 - 2.37437i) q^{13} +(-3.97721 + 0.743469i) q^{14} +(4.38190 + 5.80257i) q^{15} +(-0.982973 - 0.183750i) q^{16} +(-0.830581 - 4.03858i) q^{17} +(-4.13043 + 0.772111i) q^{18} +(4.45541 + 4.06164i) q^{19} +(2.00231 + 1.82535i) q^{20} +(-8.02440 + 7.31520i) q^{21} +(1.46175 + 1.93567i) q^{22} +(-7.41463 - 4.59094i) q^{23} +(-2.50243 + 0.969446i) q^{24} +(-0.640687 - 2.25178i) q^{25} +(-2.92468 - 0.546717i) q^{26} +(-2.38381 + 2.17313i) q^{27} +(-2.43832 + 3.22886i) q^{28} +(0.709028 - 7.65163i) q^{29} +(7.14743 + 1.33609i) q^{30} +(2.01221 + 2.66459i) q^{31} +(-0.850217 + 0.526432i) q^{32} +(6.06989 + 2.35149i) q^{33} +(-3.33458 - 2.42499i) q^{34} +(10.2224 - 3.96020i) q^{35} +(-2.53226 + 3.35325i) q^{36} +(9.49624 - 3.67886i) q^{37} +6.02890 q^{38} +(-7.44557 + 2.88443i) q^{39} +2.70946 q^{40} +(1.79849 - 6.32103i) q^{41} +(-1.00188 + 10.8120i) q^{42} +(-6.58683 + 1.23129i) q^{43} +(2.38430 + 0.445703i) q^{44} +(10.6163 - 4.11276i) q^{45} +(-8.57238 + 1.60246i) q^{46} +(1.09350 - 0.677067i) q^{47} +(-1.19621 + 2.40230i) q^{48} +(4.17699 + 8.38853i) q^{49} +(-1.99049 - 1.23246i) q^{50} +(-11.0344 - 0.822099i) q^{51} +(-2.52968 + 1.56631i) q^{52} +(1.24722 + 0.772247i) q^{53} +(-0.297628 + 3.21192i) q^{54} +(-4.85681 - 4.42757i) q^{55} +(0.373327 + 4.02884i) q^{56} +(13.7561 - 8.51739i) q^{57} +(-4.63089 - 6.13229i) q^{58} +(0.462448 - 0.928722i) q^{59} +(6.18213 - 3.82781i) q^{60} +(0.0728730 + 0.146349i) q^{61} +(3.28216 + 0.613543i) q^{62} +(7.57828 + 15.2192i) q^{63} +(-0.273663 + 0.961826i) q^{64} +8.06155 q^{65} +(6.06989 - 2.35149i) q^{66} +(2.69786 + 2.45942i) q^{67} +(-4.09799 + 0.454405i) q^{68} +(-17.2956 + 15.7670i) q^{69} +(4.88651 - 9.81344i) q^{70} +(-7.05017 - 4.36528i) q^{71} +(0.387709 + 4.18405i) q^{72} +(6.22157 + 1.16301i) q^{73} +(4.53937 - 9.11628i) q^{74} -6.28284 q^{75} +(4.45541 - 4.06164i) q^{76} +(5.91439 - 7.83192i) q^{77} +(-3.55912 + 7.14767i) q^{78} +(-1.00528 - 0.916431i) q^{79} +(2.00231 - 1.82535i) q^{80} +(-1.76037 - 3.53529i) q^{81} +(-2.92935 - 5.88293i) q^{82} +(3.51636 + 12.3587i) q^{83} +(6.54360 + 8.66513i) q^{84} +(10.0884 + 4.79837i) q^{85} +(-4.03821 + 5.34746i) q^{86} +(-19.2297 - 7.44962i) q^{87} +(2.06229 - 1.27691i) q^{88} +(-1.80320 + 2.38783i) q^{89} +(5.07476 - 10.1915i) q^{90} +(1.11077 + 11.9872i) q^{91} +(-5.25550 + 6.95940i) q^{92} +(8.35565 - 3.23700i) q^{93} +(0.351970 - 1.23704i) q^{94} +(-16.0569 + 3.00156i) q^{95} +(0.734416 + 2.58120i) q^{96} +(-13.7117 - 8.48991i) q^{97} +(8.73815 + 3.38518i) q^{98} +(6.14224 - 8.13364i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 224 q - 14 q^{2} - 2 q^{3} - 14 q^{4} + 28 q^{5} - 2 q^{6} - 4 q^{7} - 14 q^{8} - 24 q^{9} - 6 q^{10} - 18 q^{11} - 2 q^{12} - 12 q^{13} - 4 q^{14} - 22 q^{15} - 14 q^{16} - 16 q^{17} - 24 q^{18} - 12 q^{19}+ \cdots - 162 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/578\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(e\left(\frac{10}{17}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.739009 − 0.673696i 0.522558 − 0.476375i
\(3\) 0.734416 − 2.58120i 0.424015 − 1.49026i −0.396179 − 0.918173i \(-0.629664\pi\)
0.820194 − 0.572085i \(-0.193865\pi\)
\(4\) 0.0922684 − 0.995734i 0.0461342 − 0.497867i
\(5\) −1.63281 + 2.16219i −0.730216 + 0.966962i 0.269776 + 0.962923i \(0.413050\pi\)
−0.999992 + 0.00403893i \(0.998714\pi\)
\(6\) −1.19621 − 2.40230i −0.488349 − 0.980737i
\(7\) −3.44007 − 2.13000i −1.30022 − 0.805064i −0.310849 − 0.950459i \(-0.600613\pi\)
−0.989374 + 0.145395i \(0.953555\pi\)
\(8\) −0.602635 − 0.798017i −0.213064 − 0.282142i
\(9\) −3.57259 − 2.21206i −1.19086 − 0.737352i
\(10\) 0.249997 + 2.69790i 0.0790560 + 0.853150i
\(11\) −0.223806 + 2.41525i −0.0674801 + 0.728226i 0.893033 + 0.449991i \(0.148573\pi\)
−0.960513 + 0.278235i \(0.910251\pi\)
\(12\) −2.50243 − 0.969446i −0.722389 − 0.279855i
\(13\) −1.79304 − 2.37437i −0.497300 − 0.658532i 0.478272 − 0.878212i \(-0.341263\pi\)
−0.975572 + 0.219680i \(0.929499\pi\)
\(14\) −3.97721 + 0.743469i −1.06295 + 0.198701i
\(15\) 4.38190 + 5.80257i 1.13140 + 1.49822i
\(16\) −0.982973 − 0.183750i −0.245743 − 0.0459374i
\(17\) −0.830581 − 4.03858i −0.201445 − 0.979500i
\(18\) −4.13043 + 0.772111i −0.973551 + 0.181988i
\(19\) 4.45541 + 4.06164i 1.02214 + 0.931805i 0.997692 − 0.0678959i \(-0.0216286\pi\)
0.0244492 + 0.999701i \(0.492217\pi\)
\(20\) 2.00231 + 1.82535i 0.447731 + 0.408160i
\(21\) −8.02440 + 7.31520i −1.75107 + 1.59631i
\(22\) 1.46175 + 1.93567i 0.311646 + 0.412686i
\(23\) −7.41463 − 4.59094i −1.54606 − 0.957278i −0.992199 − 0.124668i \(-0.960214\pi\)
−0.553859 − 0.832611i \(-0.686845\pi\)
\(24\) −2.50243 + 0.969446i −0.510806 + 0.197887i
\(25\) −0.640687 − 2.25178i −0.128137 − 0.450356i
\(26\) −2.92468 − 0.546717i −0.573576 − 0.107220i
\(27\) −2.38381 + 2.17313i −0.458764 + 0.418219i
\(28\) −2.43832 + 3.22886i −0.460800 + 0.610197i
\(29\) 0.709028 − 7.65163i 0.131663 − 1.42087i −0.631855 − 0.775086i \(-0.717706\pi\)
0.763519 − 0.645786i \(-0.223470\pi\)
\(30\) 7.14743 + 1.33609i 1.30494 + 0.243935i
\(31\) 2.01221 + 2.66459i 0.361403 + 0.478575i 0.941998 − 0.335617i \(-0.108945\pi\)
−0.580595 + 0.814192i \(0.697180\pi\)
\(32\) −0.850217 + 0.526432i −0.150299 + 0.0930609i
\(33\) 6.06989 + 2.35149i 1.05663 + 0.409342i
\(34\) −3.33458 − 2.42499i −0.571876 − 0.415882i
\(35\) 10.2224 − 3.96020i 1.72791 − 0.669396i
\(36\) −2.53226 + 3.35325i −0.422043 + 0.558875i
\(37\) 9.49624 − 3.67886i 1.56117 − 0.604801i 0.582977 − 0.812489i \(-0.301888\pi\)
0.978195 + 0.207688i \(0.0665938\pi\)
\(38\) 6.02890 0.978017
\(39\) −7.44557 + 2.88443i −1.19225 + 0.461878i
\(40\) 2.70946 0.428403
\(41\) 1.79849 − 6.32103i 0.280877 − 0.987179i −0.685078 − 0.728469i \(-0.740232\pi\)
0.965955 − 0.258710i \(-0.0832974\pi\)
\(42\) −1.00188 + 10.8120i −0.154593 + 1.66833i
\(43\) −6.58683 + 1.23129i −1.00448 + 0.187770i −0.660195 − 0.751094i \(-0.729526\pi\)
−0.344287 + 0.938864i \(0.611879\pi\)
\(44\) 2.38430 + 0.445703i 0.359447 + 0.0671922i
\(45\) 10.6163 − 4.11276i 1.58258 − 0.613094i
\(46\) −8.57238 + 1.60246i −1.26393 + 0.236269i
\(47\) 1.09350 − 0.677067i 0.159503 − 0.0987603i −0.444359 − 0.895849i \(-0.646569\pi\)
0.603863 + 0.797088i \(0.293627\pi\)
\(48\) −1.19621 + 2.40230i −0.172657 + 0.346743i
\(49\) 4.17699 + 8.38853i 0.596713 + 1.19836i
\(50\) −1.99049 − 1.23246i −0.281498 − 0.174296i
\(51\) −11.0344 − 0.822099i −1.54512 − 0.115117i
\(52\) −2.52968 + 1.56631i −0.350804 + 0.217209i
\(53\) 1.24722 + 0.772247i 0.171319 + 0.106076i 0.609413 − 0.792853i \(-0.291405\pi\)
−0.438094 + 0.898929i \(0.644346\pi\)
\(54\) −0.297628 + 3.21192i −0.0405021 + 0.437087i
\(55\) −4.85681 − 4.42757i −0.654892 − 0.597013i
\(56\) 0.373327 + 4.02884i 0.0498880 + 0.538377i
\(57\) 13.7561 − 8.51739i 1.82203 − 1.12816i
\(58\) −4.63089 − 6.13229i −0.608066 − 0.805209i
\(59\) 0.462448 − 0.928722i 0.0602057 − 0.120909i −0.863017 − 0.505175i \(-0.831428\pi\)
0.923222 + 0.384266i \(0.125545\pi\)
\(60\) 6.18213 − 3.82781i 0.798109 − 0.494168i
\(61\) 0.0728730 + 0.146349i 0.00933044 + 0.0187380i 0.899779 − 0.436345i \(-0.143727\pi\)
−0.890449 + 0.455083i \(0.849610\pi\)
\(62\) 3.28216 + 0.613543i 0.416835 + 0.0779200i
\(63\) 7.57828 + 15.2192i 0.954773 + 1.91744i
\(64\) −0.273663 + 0.961826i −0.0342079 + 0.120228i
\(65\) 8.06155 0.999912
\(66\) 6.06989 − 2.35149i 0.747152 − 0.289448i
\(67\) 2.69786 + 2.45942i 0.329596 + 0.300467i 0.821529 − 0.570166i \(-0.193121\pi\)
−0.491933 + 0.870633i \(0.663710\pi\)
\(68\) −4.09799 + 0.454405i −0.496954 + 0.0551047i
\(69\) −17.2956 + 15.7670i −2.08214 + 1.89812i
\(70\) 4.88651 − 9.81344i 0.584050 − 1.17293i
\(71\) −7.05017 − 4.36528i −0.836701 − 0.518064i 0.0399432 − 0.999202i \(-0.487282\pi\)
−0.876645 + 0.481138i \(0.840223\pi\)
\(72\) 0.387709 + 4.18405i 0.0456920 + 0.493095i
\(73\) 6.22157 + 1.16301i 0.728180 + 0.136120i 0.534770 − 0.844998i \(-0.320398\pi\)
0.193410 + 0.981118i \(0.438045\pi\)
\(74\) 4.53937 − 9.11628i 0.527691 − 1.05975i
\(75\) −6.28284 −0.725479
\(76\) 4.45541 − 4.06164i 0.511071 − 0.465903i
\(77\) 5.91439 − 7.83192i 0.674008 − 0.892530i
\(78\) −3.55912 + 7.14767i −0.402991 + 0.809314i
\(79\) −1.00528 − 0.916431i −0.113102 − 0.103107i 0.615169 − 0.788395i \(-0.289088\pi\)
−0.728271 + 0.685289i \(0.759676\pi\)
\(80\) 2.00231 − 1.82535i 0.223865 − 0.204080i
\(81\) −1.76037 − 3.53529i −0.195596 − 0.392810i
\(82\) −2.92935 − 5.88293i −0.323493 − 0.649661i
\(83\) 3.51636 + 12.3587i 0.385971 + 1.35655i 0.874934 + 0.484243i \(0.160905\pi\)
−0.488962 + 0.872305i \(0.662624\pi\)
\(84\) 6.54360 + 8.66513i 0.713965 + 0.945443i
\(85\) 10.0884 + 4.79837i 1.09424 + 0.520456i
\(86\) −4.03821 + 5.34746i −0.435452 + 0.576631i
\(87\) −19.2297 − 7.44962i −2.06164 − 0.798683i
\(88\) 2.06229 − 1.27691i 0.219840 − 0.136119i
\(89\) −1.80320 + 2.38783i −0.191139 + 0.253109i −0.883488 − 0.468454i \(-0.844811\pi\)
0.692349 + 0.721563i \(0.256576\pi\)
\(90\) 5.07476 − 10.1915i 0.534927 − 1.07428i
\(91\) 1.11077 + 11.9872i 0.116441 + 1.25660i
\(92\) −5.25550 + 6.95940i −0.547923 + 0.725568i
\(93\) 8.35565 − 3.23700i 0.866441 − 0.335661i
\(94\) 0.351970 − 1.23704i 0.0363029 − 0.127591i
\(95\) −16.0569 + 3.00156i −1.64740 + 0.307953i
\(96\) 0.734416 + 2.58120i 0.0749560 + 0.263443i
\(97\) −13.7117 − 8.48991i −1.39221 − 0.862020i −0.393844 − 0.919177i \(-0.628855\pi\)
−0.998365 + 0.0571576i \(0.981796\pi\)
\(98\) 8.73815 + 3.38518i 0.882686 + 0.341955i
\(99\) 6.14224 − 8.13364i 0.617318 − 0.817462i
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 578.2.f.b.545.12 yes 224
289.35 even 17 inner 578.2.f.b.35.12 ✓ 224
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
578.2.f.b.35.12 ✓ 224 289.35 even 17 inner
578.2.f.b.545.12 yes 224 1.1 even 1 trivial