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Properties

Label 35.5.l.a.2.9
Level $35$
Weight $5$
Character 35.2
Analytic conductor $3.618$
Analytic rank $0$
Dimension $56$
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] = [35,5,Mod(2,35)] 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("35.2"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(35, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([3, 4])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 35 = 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 35.l (of order \(12\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.61794870793\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(14\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label 2.9
Character \(\chi\) \(=\) 35.2
Dual form 35.5.l.a.18.9

$q$-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+(2.34674 - 0.628807i) q^{2} +(-11.5823 - 3.10347i) q^{3} +(-8.74462 + 5.04871i) q^{4} +(11.4928 + 22.2017i) q^{5} -29.1322 q^{6} +(-36.0021 + 33.2392i) q^{7} +(-44.8336 + 44.8336i) q^{8} +(54.3705 + 31.3908i) q^{9} +(40.9311 + 44.8749i) q^{10} +(-90.4787 - 156.714i) q^{11} +(116.951 - 31.3370i) q^{12} +(89.6472 - 89.6472i) q^{13} +(-63.5865 + 100.642i) q^{14} +(-64.2106 - 292.815i) q^{15} +(3.75821 - 6.50941i) q^{16} +(-146.843 + 548.027i) q^{17} +(147.332 + 39.4775i) q^{18} +(102.687 + 59.2862i) q^{19} +(-212.590 - 136.122i) q^{20} +(520.145 - 273.256i) q^{21} +(-310.872 - 310.872i) q^{22} +(-11.0640 - 41.2914i) q^{23} +(658.417 - 380.137i) q^{24} +(-360.832 + 510.319i) q^{25} +(154.008 - 266.750i) q^{26} +(154.471 + 154.471i) q^{27} +(147.010 - 472.429i) q^{28} +15.5584i q^{29} +(-334.809 - 646.784i) q^{30} +(132.134 + 228.862i) q^{31} +(267.290 - 997.540i) q^{32} +(561.596 + 2095.90i) q^{33} +1378.41i q^{34} +(-1151.73 - 417.298i) q^{35} -633.932 q^{36} +(-639.930 + 171.469i) q^{37} +(278.259 + 74.5592i) q^{38} +(-1316.54 + 760.105i) q^{39} +(-1510.65 - 480.120i) q^{40} +880.560 q^{41} +(1048.82 - 968.331i) q^{42} +(-1623.67 + 1623.67i) q^{43} +(1582.40 + 913.601i) q^{44} +(-72.0620 + 1567.88i) q^{45} +(-51.9286 - 89.9430i) q^{46} +(-2616.45 + 701.077i) q^{47} +(-63.7305 + 63.7305i) q^{48} +(191.305 - 2393.37i) q^{49} +(-525.887 + 1424.48i) q^{50} +(3401.57 - 5891.70i) q^{51} +(-331.328 + 1236.53i) q^{52} +(2554.52 + 684.482i) q^{53} +(459.636 + 265.371i) q^{54} +(2439.46 - 3809.86i) q^{55} +(123.870 - 3104.34i) q^{56} +(-1005.36 - 1005.36i) q^{57} +(9.78321 + 36.5114i) q^{58} +(2715.46 - 1567.77i) q^{59} +(2039.83 + 2236.37i) q^{60} +(-2448.25 + 4240.50i) q^{61} +(453.994 + 453.994i) q^{62} +(-3000.86 + 677.098i) q^{63} -2388.78i q^{64} +(3020.62 + 960.027i) q^{65} +(2635.84 + 4565.41i) q^{66} +(-489.598 + 1827.20i) q^{67} +(-1482.74 - 5533.66i) q^{68} +512.587i q^{69} +(-2965.22 - 255.071i) q^{70} -1773.28 q^{71} +(-3844.99 + 1030.26i) q^{72} +(-1939.93 - 519.802i) q^{73} +(-1393.93 + 804.784i) q^{74} +(5763.03 - 4790.84i) q^{75} -1197.28 q^{76} +(8466.47 + 2634.58i) q^{77} +(-2611.62 + 2611.62i) q^{78} +(-2811.90 - 1623.45i) q^{79} +(187.712 + 8.62750i) q^{80} +(-3852.39 - 6672.54i) q^{81} +(2066.45 - 553.702i) q^{82} +(1116.39 - 1116.39i) q^{83} +(-3168.88 + 5015.58i) q^{84} +(-13854.8 + 3038.18i) q^{85} +(-2789.35 + 4831.30i) q^{86} +(48.2850 - 180.202i) q^{87} +(11082.5 + 2969.55i) q^{88} +(5880.67 + 3395.21i) q^{89} +(816.786 + 3724.73i) q^{90} +(-247.684 + 6207.30i) q^{91} +(305.218 + 305.218i) q^{92} +(-820.147 - 3060.83i) q^{93} +(-5699.29 + 3290.49i) q^{94} +(-136.100 + 2961.18i) q^{95} +(-6191.68 + 10724.3i) q^{96} +(-3544.97 - 3544.97i) q^{97} +(-1056.02 - 5736.90i) q^{98} -11360.8i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 56 q - 2 q^{2} - 2 q^{3} + 16 q^{5} - 144 q^{6} + 46 q^{7} + 108 q^{8} - 66 q^{10} + 296 q^{11} - 358 q^{12} - 8 q^{13} - 68 q^{15} + 468 q^{16} + 28 q^{17} - 868 q^{18} - 1032 q^{20} + 1280 q^{21} + 56 q^{22}+ \cdots - 78606 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/35\mathbb{Z}\right)^\times\).

\(n\) \(22\) \(31\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(e\left(\frac{1}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.34674 − 0.628807i 0.586685 − 0.157202i 0.0467472 − 0.998907i \(-0.485114\pi\)
0.539938 + 0.841705i \(0.318448\pi\)
\(3\) −11.5823 − 3.10347i −1.28692 − 0.344830i −0.450433 − 0.892810i \(-0.648730\pi\)
−0.836491 + 0.547980i \(0.815397\pi\)
\(4\) −8.74462 + 5.04871i −0.546539 + 0.315544i
\(5\) 11.4928 + 22.2017i 0.459711 + 0.888069i
\(6\) −29.1322 −0.809227
\(7\) −36.0021 + 33.2392i −0.734737 + 0.678352i
\(8\) −44.8336 + 44.8336i −0.700525 + 0.700525i
\(9\) 54.3705 + 31.3908i 0.671240 + 0.387541i
\(10\) 40.9311 + 44.8749i 0.409311 + 0.448749i
\(11\) −90.4787 − 156.714i −0.747758 − 1.29515i −0.948895 − 0.315591i \(-0.897797\pi\)
0.201138 − 0.979563i \(-0.435536\pi\)
\(12\) 116.951 − 31.3370i 0.812163 − 0.217618i
\(13\) 89.6472 − 89.6472i 0.530457 − 0.530457i −0.390251 − 0.920708i \(-0.627612\pi\)
0.920708 + 0.390251i \(0.127612\pi\)
\(14\) −63.5865 + 100.642i −0.324421 + 0.513481i
\(15\) −64.2106 − 292.815i −0.285380 − 1.30140i
\(16\) 3.75821 − 6.50941i 0.0146805 − 0.0254274i
\(17\) −146.843 + 548.027i −0.508109 + 1.89629i −0.0695669 + 0.997577i \(0.522162\pi\)
−0.438542 + 0.898711i \(0.644505\pi\)
\(18\) 147.332 + 39.4775i 0.454729 + 0.121844i
\(19\) 102.687 + 59.2862i 0.284451 + 0.164228i 0.635437 − 0.772153i \(-0.280820\pi\)
−0.350986 + 0.936381i \(0.614153\pi\)
\(20\) −212.590 − 136.122i −0.531475 − 0.340305i
\(21\) 520.145 − 273.256i 1.17947 − 0.619628i
\(22\) −310.872 − 310.872i −0.642299 − 0.642299i
\(23\) −11.0640 − 41.2914i −0.0209149 − 0.0780555i 0.954680 − 0.297635i \(-0.0961980\pi\)
−0.975595 + 0.219580i \(0.929531\pi\)
\(24\) 658.417 − 380.137i 1.14308 − 0.659960i
\(25\) −360.832 + 510.319i −0.577332 + 0.816510i
\(26\) 154.008 − 266.750i 0.227822 − 0.394600i
\(27\) 154.471 + 154.471i 0.211894 + 0.211894i
\(28\) 147.010 − 472.429i 0.187512 − 0.602588i
\(29\) 15.5584i 0.0184998i 0.999957 + 0.00924992i \(0.00294438\pi\)
−0.999957 + 0.00924992i \(0.997056\pi\)
\(30\) −334.809 − 646.784i −0.372011 − 0.718649i
\(31\) 132.134 + 228.862i 0.137496 + 0.238150i 0.926548 − 0.376176i \(-0.122761\pi\)
−0.789052 + 0.614326i \(0.789428\pi\)
\(32\) 267.290 − 997.540i 0.261026 − 0.974161i
\(33\) 561.596 + 2095.90i 0.515699 + 1.92461i
\(34\) 1378.41i 1.19240i
\(35\) −1151.73 − 417.298i −0.940190 − 0.340651i
\(36\) −633.932 −0.489145
\(37\) −639.930 + 171.469i −0.467443 + 0.125251i −0.484850 − 0.874597i \(-0.661126\pi\)
0.0174066 + 0.999848i \(0.494459\pi\)
\(38\) 278.259 + 74.5592i 0.192700 + 0.0516338i
\(39\) −1316.54 + 760.105i −0.865576 + 0.499740i
\(40\) −1510.65 − 480.120i −0.944153 − 0.300075i
\(41\) 880.560 0.523831 0.261916 − 0.965091i \(-0.415646\pi\)
0.261916 + 0.965091i \(0.415646\pi\)
\(42\) 1048.82 − 968.331i 0.594569 − 0.548941i
\(43\) −1623.67 + 1623.67i −0.878133 + 0.878133i −0.993341 − 0.115209i \(-0.963246\pi\)
0.115209 + 0.993341i \(0.463246\pi\)
\(44\) 1582.40 + 913.601i 0.817357 + 0.471901i
\(45\) −72.0620 + 1567.88i −0.0355862 + 0.774264i
\(46\) −51.9286 − 89.9430i −0.0245409 − 0.0425061i
\(47\) −2616.45 + 701.077i −1.18445 + 0.317373i −0.796691 − 0.604387i \(-0.793418\pi\)
−0.387761 + 0.921760i \(0.626751\pi\)
\(48\) −63.7305 + 63.7305i −0.0276608 + 0.0276608i
\(49\) 191.305 − 2393.37i 0.0796773 − 0.996821i
\(50\) −525.887 + 1424.48i −0.210355 + 0.569792i
\(51\) 3401.57 − 5891.70i 1.30779 − 2.26517i
\(52\) −331.328 + 1236.53i −0.122533 + 0.457298i
\(53\) 2554.52 + 684.482i 0.909406 + 0.243675i 0.683052 − 0.730370i \(-0.260652\pi\)
0.226355 + 0.974045i \(0.427319\pi\)
\(54\) 459.636 + 265.371i 0.157625 + 0.0910051i
\(55\) 2439.46 − 3809.86i 0.806433 − 1.25946i
\(56\) 123.870 − 3104.34i 0.0394993 − 0.989904i
\(57\) −1005.36 − 1005.36i −0.309436 − 0.309436i
\(58\) 9.78321 + 36.5114i 0.00290821 + 0.0108536i
\(59\) 2715.46 − 1567.77i 0.780082 − 0.450380i −0.0563774 − 0.998410i \(-0.517955\pi\)
0.836459 + 0.548029i \(0.184622\pi\)
\(60\) 2039.83 + 2236.37i 0.566620 + 0.621215i
\(61\) −2448.25 + 4240.50i −0.657956 + 1.13961i 0.323189 + 0.946335i \(0.395245\pi\)
−0.981144 + 0.193278i \(0.938088\pi\)
\(62\) 453.994 + 453.994i 0.118104 + 0.118104i
\(63\) −3000.86 + 677.098i −0.756074 + 0.170597i
\(64\) − 2388.78i − 0.583198i
\(65\) 3020.62 + 960.027i 0.714939 + 0.227225i
\(66\) 2635.84 + 4565.41i 0.605105 + 1.04807i
\(67\) −489.598 + 1827.20i −0.109066 + 0.407040i −0.998775 − 0.0494888i \(-0.984241\pi\)
0.889709 + 0.456529i \(0.150907\pi\)
\(68\) −1482.74 − 5533.66i −0.320662 − 1.19673i
\(69\) 512.587i 0.107664i
\(70\) −2965.22 − 255.071i −0.605146 − 0.0520554i
\(71\) −1773.28 −0.351771 −0.175885 − 0.984411i \(-0.556279\pi\)
−0.175885 + 0.984411i \(0.556279\pi\)
\(72\) −3844.99 + 1030.26i −0.741702 + 0.198739i
\(73\) −1939.93 − 519.802i −0.364032 − 0.0975421i 0.0721664 − 0.997393i \(-0.477009\pi\)
−0.436198 + 0.899851i \(0.643675\pi\)
\(74\) −1393.93 + 804.784i −0.254552 + 0.146966i
\(75\) 5763.03 − 4790.84i 1.02454 − 0.851705i
\(76\) −1197.28 −0.207284
\(77\) 8466.47 + 2634.58i 1.42798 + 0.444355i
\(78\) −2611.62 + 2611.62i −0.429260 + 0.429260i
\(79\) −2811.90 − 1623.45i −0.450552 − 0.260127i 0.257511 − 0.966275i \(-0.417098\pi\)
−0.708063 + 0.706149i \(0.750431\pi\)
\(80\) 187.712 + 8.62750i 0.0293300 + 0.00134805i
\(81\) −3852.39 − 6672.54i −0.587165 − 1.01700i
\(82\) 2066.45 − 553.702i 0.307324 − 0.0823472i
\(83\) 1116.39 − 1116.39i 0.162054 − 0.162054i −0.621422 − 0.783476i \(-0.713445\pi\)
0.783476 + 0.621422i \(0.213445\pi\)
\(84\) −3168.88 + 5015.58i −0.449104 + 0.710825i
\(85\) −13854.8 + 3038.18i −1.91762 + 0.420509i
\(86\) −2789.35 + 4831.30i −0.377143 + 0.653231i
\(87\) 48.2850 − 180.202i 0.00637930 − 0.0238079i
\(88\) 11082.5 + 2969.55i 1.43111 + 0.383465i
\(89\) 5880.67 + 3395.21i 0.742415 + 0.428634i 0.822947 − 0.568118i \(-0.192328\pi\)
−0.0805315 + 0.996752i \(0.525662\pi\)
\(90\) 816.786 + 3724.73i 0.100838 + 0.459843i
\(91\) −247.684 + 6207.30i −0.0299099 + 0.749583i
\(92\) 305.218 + 305.218i 0.0360608 + 0.0360608i
\(93\) −820.147 − 3060.83i −0.0948256 − 0.353894i
\(94\) −5699.29 + 3290.49i −0.645008 + 0.372396i
\(95\) −136.100 + 2961.18i −0.0150803 + 0.328109i
\(96\) −6191.68 + 10724.3i −0.671840 + 1.16366i
\(97\) −3544.97 − 3544.97i −0.376764 − 0.376764i 0.493170 − 0.869933i \(-0.335838\pi\)
−0.869933 + 0.493170i \(0.835838\pi\)
\(98\) −1056.02 − 5736.90i −0.109956 − 0.597345i
\(99\) − 11360.8i − 1.15915i
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 35.5.l.a.2.9 ✓ 56
5.3 odd 4 inner 35.5.l.a.23.6 yes 56
7.4 even 3 inner 35.5.l.a.32.6 yes 56
35.18 odd 12 inner 35.5.l.a.18.9 yes 56
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.5.l.a.2.9 ✓ 56 1.1 even 1 trivial
35.5.l.a.18.9 yes 56 35.18 odd 12 inner
35.5.l.a.23.6 yes 56 5.3 odd 4 inner
35.5.l.a.32.6 yes 56 7.4 even 3 inner