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Properties

Label 578.2.f.b.35.12
Level $578$
Weight $2$
Character 578.35
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 35.12
Character \(\chi\) \(=\) 578.35
Dual form 578.2.f.b.545.12

$q$-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 \(\left(\mathbb{Z}/578\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(e\left(\frac{7}{17}\right)\)

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 0.820194 + 0.572085i \(0.193865\pi\)
−0.396179 + 0.918173i \(0.629664\pi\)
\(4\) 0.0922684 + 0.995734i 0.0461342 + 0.497867i
\(5\) −1.63281 − 2.16219i −0.730216 − 0.966962i −0.999992 − 0.00403893i \(-0.998714\pi\)
0.269776 − 0.962923i \(-0.413050\pi\)
\(6\) −1.19621 + 2.40230i −0.488349 + 0.980737i
\(7\) −3.44007 + 2.13000i −1.30022 + 0.805064i −0.989374 − 0.145395i \(-0.953555\pi\)
−0.310849 + 0.950459i \(0.600613\pi\)
\(8\) −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 −0.960513 − 0.278235i \(-0.910251\pi\)
0.893033 − 0.449991i \(-0.148573\pi\)
\(12\) −2.50243 + 0.969446i −0.722389 + 0.279855i
\(13\) −1.79304 + 2.37437i −0.497300 + 0.658532i −0.975572 − 0.219680i \(-0.929499\pi\)
0.478272 + 0.878212i \(0.341263\pi\)
\(14\) −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 0.0244492 − 0.999701i \(-0.492217\pi\)
0.997692 + 0.0678959i \(0.0216286\pi\)
\(20\) 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 −0.553859 + 0.832611i \(0.686845\pi\)
−0.992199 + 0.124668i \(0.960214\pi\)
\(24\) −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 0.763519 + 0.645786i \(0.223470\pi\)
−0.631855 + 0.775086i \(0.717706\pi\)
\(30\) 7.14743 − 1.33609i 1.30494 − 0.243935i
\(31\) 2.01221 − 2.66459i 0.361403 − 0.478575i −0.580595 − 0.814192i \(-0.697180\pi\)
0.941998 + 0.335617i \(0.108945\pi\)
\(32\) −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 0.978195 − 0.207688i \(-0.0665938\pi\)
0.582977 + 0.812489i \(0.301888\pi\)
\(38\) 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 0.965955 + 0.258710i \(0.0832974\pi\)
−0.685078 + 0.728469i \(0.740232\pi\)
\(42\) −1.00188 − 10.8120i −0.154593 − 1.66833i
\(43\) −6.58683 − 1.23129i −1.00448 − 0.187770i −0.344287 − 0.938864i \(-0.611879\pi\)
−0.660195 + 0.751094i \(0.729526\pi\)
\(44\) 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 0.603863 − 0.797088i \(-0.293627\pi\)
−0.444359 + 0.895849i \(0.646569\pi\)
\(48\) −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 −0.438094 − 0.898929i \(-0.644346\pi\)
0.609413 + 0.792853i \(0.291405\pi\)
\(54\) −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 0.923222 − 0.384266i \(-0.125545\pi\)
−0.863017 + 0.505175i \(0.831428\pi\)
\(60\) 6.18213 + 3.82781i 0.798109 + 0.494168i
\(61\) 0.0728730 − 0.146349i 0.00933044 − 0.0187380i −0.890449 − 0.455083i \(-0.849610\pi\)
0.899779 + 0.436345i \(0.143727\pi\)
\(62\) 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 −0.491933 − 0.870633i \(-0.663710\pi\)
0.821529 + 0.570166i \(0.193121\pi\)
\(68\) −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 −0.876645 − 0.481138i \(-0.840223\pi\)
0.0399432 + 0.999202i \(0.487282\pi\)
\(72\) 0.387709 − 4.18405i 0.0456920 − 0.493095i
\(73\) 6.22157 − 1.16301i 0.728180 − 0.136120i 0.193410 − 0.981118i \(-0.438045\pi\)
0.534770 + 0.844998i \(0.320398\pi\)
\(74\) 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 −0.728271 − 0.685289i \(-0.759676\pi\)
0.615169 + 0.788395i \(0.289088\pi\)
\(80\) 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 −0.488962 − 0.872305i \(-0.662624\pi\)
0.874934 − 0.484243i \(-0.160905\pi\)
\(84\) 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 0.692349 − 0.721563i \(-0.256576\pi\)
−0.883488 + 0.468454i \(0.844811\pi\)
\(90\) 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 −0.998365 − 0.0571576i \(-0.981796\pi\)
−0.393844 + 0.919177i \(0.628855\pi\)
\(98\) 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.35.12 ✓ 224
289.256 even 17 inner 578.2.f.b.545.12 yes 224
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
578.2.f.b.35.12 ✓ 224 1.1 even 1 trivial
578.2.f.b.545.12 yes 224 289.256 even 17 inner