Properties

Label 85.4.a.f.1.2
Level $85$
Weight $4$
Character 85.1
Self dual yes
Analytic conductor $5.015$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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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] = [85,4,Mod(1,85)] 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("85.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(85, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 85 = 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 85.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,3,9] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(5.01516235049\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.568.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 6x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.76156\) of defining polynomial
Character \(\chi\) \(=\) 85.1

$q$-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)
 
\(f(q)\) \(=\) \(q+0.135359 q^{2} +9.28467 q^{3} -7.98168 q^{4} -5.00000 q^{5} +1.25676 q^{6} +32.4157 q^{7} -2.16327 q^{8} +59.2051 q^{9} -0.676796 q^{10} +41.8742 q^{11} -74.1073 q^{12} -50.3621 q^{13} +4.38776 q^{14} -46.4234 q^{15} +63.5606 q^{16} -17.0000 q^{17} +8.01395 q^{18} -113.150 q^{19} +39.9084 q^{20} +300.969 q^{21} +5.66806 q^{22} -57.7951 q^{23} -20.0852 q^{24} +25.0000 q^{25} -6.81697 q^{26} +299.014 q^{27} -258.731 q^{28} -110.333 q^{29} -6.28382 q^{30} -25.4263 q^{31} +25.9096 q^{32} +388.788 q^{33} -2.30110 q^{34} -162.078 q^{35} -472.556 q^{36} -124.544 q^{37} -15.3159 q^{38} -467.596 q^{39} +10.8163 q^{40} -16.7597 q^{41} +40.7389 q^{42} -344.435 q^{43} -334.227 q^{44} -296.026 q^{45} -7.82309 q^{46} -188.485 q^{47} +590.139 q^{48} +707.775 q^{49} +3.38398 q^{50} -157.839 q^{51} +401.974 q^{52} +394.600 q^{53} +40.4743 q^{54} -209.371 q^{55} -70.1237 q^{56} -1050.56 q^{57} -14.9346 q^{58} +344.824 q^{59} +370.536 q^{60} +257.635 q^{61} -3.44168 q^{62} +1919.17 q^{63} -504.978 q^{64} +251.811 q^{65} +52.6261 q^{66} +282.255 q^{67} +135.689 q^{68} -536.608 q^{69} -21.9388 q^{70} -696.547 q^{71} -128.076 q^{72} -493.004 q^{73} -16.8582 q^{74} +232.117 q^{75} +903.128 q^{76} +1357.38 q^{77} -63.2934 q^{78} -1173.97 q^{79} -317.803 q^{80} +1177.71 q^{81} -2.26858 q^{82} +902.089 q^{83} -2402.24 q^{84} +85.0000 q^{85} -46.6224 q^{86} -1024.41 q^{87} -90.5851 q^{88} +648.960 q^{89} -40.0698 q^{90} -1632.52 q^{91} +461.302 q^{92} -236.075 q^{93} -25.5132 q^{94} +565.751 q^{95} +240.562 q^{96} -530.420 q^{97} +95.8038 q^{98} +2479.17 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} + 9 q^{3} - q^{4} - 15 q^{5} + 33 q^{6} + 34 q^{7} + 39 q^{8} + 60 q^{9} - 15 q^{10} + 52 q^{11} + 17 q^{12} + 19 q^{13} - 2 q^{14} - 45 q^{15} + 59 q^{16} - 51 q^{17} - 153 q^{19} + 5 q^{20}+ \cdots + 2524 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.135359 0.0478567 0.0239283 − 0.999714i \(-0.492383\pi\)
0.0239283 + 0.999714i \(0.492383\pi\)
\(3\) 9.28467 1.78684 0.893418 − 0.449226i \(-0.148300\pi\)
0.893418 + 0.449226i \(0.148300\pi\)
\(4\) −7.98168 −0.997710
\(5\) −5.00000 −0.447214
\(6\) 1.25676 0.0855120
\(7\) 32.4157 1.75028 0.875141 − 0.483869i \(-0.160769\pi\)
0.875141 + 0.483869i \(0.160769\pi\)
\(8\) −2.16327 −0.0956037
\(9\) 59.2051 2.19278
\(10\) −0.676796 −0.0214022
\(11\) 41.8742 1.14778 0.573889 − 0.818933i \(-0.305434\pi\)
0.573889 + 0.818933i \(0.305434\pi\)
\(12\) −74.1073 −1.78274
\(13\) −50.3621 −1.07446 −0.537229 − 0.843437i \(-0.680529\pi\)
−0.537229 + 0.843437i \(0.680529\pi\)
\(14\) 4.38776 0.0837626
\(15\) −46.4234 −0.799097
\(16\) 63.5606 0.993134
\(17\) −17.0000 −0.242536
\(18\) 8.01395 0.104939
\(19\) −113.150 −1.36623 −0.683116 − 0.730309i \(-0.739376\pi\)
−0.683116 + 0.730309i \(0.739376\pi\)
\(20\) 39.9084 0.446189
\(21\) 300.969 3.12747
\(22\) 5.66806 0.0549288
\(23\) −57.7951 −0.523961 −0.261981 − 0.965073i \(-0.584376\pi\)
−0.261981 + 0.965073i \(0.584376\pi\)
\(24\) −20.0852 −0.170828
\(25\) 25.0000 0.200000
\(26\) −6.81697 −0.0514199
\(27\) 299.014 2.13131
\(28\) −258.731 −1.74627
\(29\) −110.333 −0.706494 −0.353247 − 0.935530i \(-0.614923\pi\)
−0.353247 + 0.935530i \(0.614923\pi\)
\(30\) −6.28382 −0.0382421
\(31\) −25.4263 −0.147313 −0.0736564 − 0.997284i \(-0.523467\pi\)
−0.0736564 + 0.997284i \(0.523467\pi\)
\(32\) 25.9096 0.143132
\(33\) 388.788 2.05089
\(34\) −2.30110 −0.0116069
\(35\) −162.078 −0.782750
\(36\) −472.556 −2.18776
\(37\) −124.544 −0.553377 −0.276689 − 0.960960i \(-0.589237\pi\)
−0.276689 + 0.960960i \(0.589237\pi\)
\(38\) −15.3159 −0.0653834
\(39\) −467.596 −1.91988
\(40\) 10.8163 0.0427553
\(41\) −16.7597 −0.0638398 −0.0319199 − 0.999490i \(-0.510162\pi\)
−0.0319199 + 0.999490i \(0.510162\pi\)
\(42\) 40.7389 0.149670
\(43\) −344.435 −1.22153 −0.610765 − 0.791812i \(-0.709138\pi\)
−0.610765 + 0.791812i \(0.709138\pi\)
\(44\) −334.227 −1.14515
\(45\) −296.026 −0.980642
\(46\) −7.82309 −0.0250750
\(47\) −188.485 −0.584966 −0.292483 − 0.956271i \(-0.594481\pi\)
−0.292483 + 0.956271i \(0.594481\pi\)
\(48\) 590.139 1.77457
\(49\) 707.775 2.06348
\(50\) 3.38398 0.00957133
\(51\) −157.839 −0.433371
\(52\) 401.974 1.07200
\(53\) 394.600 1.02269 0.511344 − 0.859376i \(-0.329148\pi\)
0.511344 + 0.859376i \(0.329148\pi\)
\(54\) 40.4743 0.101997
\(55\) −209.371 −0.513302
\(56\) −70.1237 −0.167333
\(57\) −1050.56 −2.44123
\(58\) −14.9346 −0.0338105
\(59\) 344.824 0.760886 0.380443 − 0.924804i \(-0.375772\pi\)
0.380443 + 0.924804i \(0.375772\pi\)
\(60\) 370.536 0.797267
\(61\) 257.635 0.540767 0.270383 − 0.962753i \(-0.412850\pi\)
0.270383 + 0.962753i \(0.412850\pi\)
\(62\) −3.44168 −0.00704990
\(63\) 1919.17 3.83799
\(64\) −504.978 −0.986285
\(65\) 251.811 0.480512
\(66\) 52.6261 0.0981488
\(67\) 282.255 0.514671 0.257335 − 0.966322i \(-0.417156\pi\)
0.257335 + 0.966322i \(0.417156\pi\)
\(68\) 135.689 0.241980
\(69\) −536.608 −0.936233
\(70\) −21.9388 −0.0374598
\(71\) −696.547 −1.16430 −0.582148 − 0.813083i \(-0.697787\pi\)
−0.582148 + 0.813083i \(0.697787\pi\)
\(72\) −128.076 −0.209638
\(73\) −493.004 −0.790435 −0.395218 − 0.918588i \(-0.629331\pi\)
−0.395218 + 0.918588i \(0.629331\pi\)
\(74\) −16.8582 −0.0264828
\(75\) 232.117 0.357367
\(76\) 903.128 1.36310
\(77\) 1357.38 2.00893
\(78\) −63.2934 −0.0918790
\(79\) −1173.97 −1.67192 −0.835961 − 0.548789i \(-0.815089\pi\)
−0.835961 + 0.548789i \(0.815089\pi\)
\(80\) −317.803 −0.444143
\(81\) 1177.71 1.61551
\(82\) −2.26858 −0.00305516
\(83\) 902.089 1.19298 0.596489 − 0.802622i \(-0.296562\pi\)
0.596489 + 0.802622i \(0.296562\pi\)
\(84\) −2402.24 −3.12030
\(85\) 85.0000 0.108465
\(86\) −46.6224 −0.0584584
\(87\) −1024.41 −1.26239
\(88\) −90.5851 −0.109732
\(89\) 648.960 0.772918 0.386459 − 0.922307i \(-0.373698\pi\)
0.386459 + 0.922307i \(0.373698\pi\)
\(90\) −40.0698 −0.0469303
\(91\) −1632.52 −1.88060
\(92\) 461.302 0.522761
\(93\) −236.075 −0.263224
\(94\) −25.5132 −0.0279945
\(95\) 565.751 0.610998
\(96\) 240.562 0.255753
\(97\) −530.420 −0.555216 −0.277608 − 0.960694i \(-0.589542\pi\)
−0.277608 + 0.960694i \(0.589542\pi\)
\(98\) 95.8038 0.0987515
\(99\) 2479.17 2.51683
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 85.4.a.f.1.2 ✓ 3
3.2 odd 2 765.4.a.k.1.2 3
4.3 odd 2 1360.4.a.p.1.1 3
5.2 odd 4 425.4.b.h.324.4 6
5.3 odd 4 425.4.b.h.324.3 6
5.4 even 2 425.4.a.f.1.2 3
17.16 even 2 1445.4.a.k.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
85.4.a.f.1.2 ✓ 3 1.1 even 1 trivial
425.4.a.f.1.2 3 5.4 even 2
425.4.b.h.324.3 6 5.3 odd 4
425.4.b.h.324.4 6 5.2 odd 4
765.4.a.k.1.2 3 3.2 odd 2
1360.4.a.p.1.1 3 4.3 odd 2
1445.4.a.k.1.2 3 17.16 even 2