Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1445,4,Mod(1,1445)] 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("1445.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1445, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1445 = 5 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1445.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,3,-9,-1,15] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(85.2577599583\)
Analytic rank: \(1\)
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: no (minimal twist has level 85)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.76156\) of defining polynomial
Character \(\chi\) \(=\) 1445.1

$q$-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)
 
\(f(q)\) \(=\) \(q+0.135359 q^{2} -9.28467 q^{3} -7.98168 q^{4} +5.00000 q^{5} -1.25676 q^{6} -32.4157 q^{7} -2.16327 q^{8} +59.2051 q^{9} +0.676796 q^{10} -41.8742 q^{11} +74.1073 q^{12} -50.3621 q^{13} -4.38776 q^{14} -46.4234 q^{15} +63.5606 q^{16} +8.01395 q^{18} -113.150 q^{19} -39.9084 q^{20} +300.969 q^{21} -5.66806 q^{22} +57.7951 q^{23} +20.0852 q^{24} +25.0000 q^{25} -6.81697 q^{26} -299.014 q^{27} +258.731 q^{28} +110.333 q^{29} -6.28382 q^{30} +25.4263 q^{31} +25.9096 q^{32} +388.788 q^{33} -162.078 q^{35} -472.556 q^{36} +124.544 q^{37} -15.3159 q^{38} +467.596 q^{39} -10.8163 q^{40} +16.7597 q^{41} +40.7389 q^{42} -344.435 q^{43} +334.227 q^{44} +296.026 q^{45} +7.82309 q^{46} -188.485 q^{47} -590.139 q^{48} +707.775 q^{49} +3.38398 q^{50} +401.974 q^{52} +394.600 q^{53} -40.4743 q^{54} -209.371 q^{55} +70.1237 q^{56} +1050.56 q^{57} +14.9346 q^{58} +344.824 q^{59} +370.536 q^{60} -257.635 q^{61} +3.44168 q^{62} -1919.17 q^{63} -504.978 q^{64} -251.811 q^{65} +52.6261 q^{66} +282.255 q^{67} -536.608 q^{69} -21.9388 q^{70} +696.547 q^{71} -128.076 q^{72} +493.004 q^{73} +16.8582 q^{74} -232.117 q^{75} +903.128 q^{76} +1357.38 q^{77} +63.2934 q^{78} +1173.97 q^{79} +317.803 q^{80} +1177.71 q^{81} +2.26858 q^{82} +902.089 q^{83} -2402.24 q^{84} -46.6224 q^{86} -1024.41 q^{87} +90.5851 q^{88} +648.960 q^{89} +40.0698 q^{90} +1632.52 q^{91} -461.302 q^{92} -236.075 q^{93} -25.5132 q^{94} -565.751 q^{95} -240.562 q^{96} +530.420 q^{97} +95.8038 q^{98} -2479.17 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} - 9 q^{3} - q^{4} + 15 q^{5} - 33 q^{6} - 34 q^{7} + 39 q^{8} + 60 q^{9} + 15 q^{10} - 52 q^{11} - 17 q^{12} + 19 q^{13} + 2 q^{14} - 45 q^{15} + 59 q^{16} - 153 q^{19} - 5 q^{20} + 286 q^{21}+ \cdots - 2524 q^{99}+O(q^{100}) \) 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 \(-0.851700\pi\)
−0.893418 + 0.449226i \(0.851700\pi\)
\(4\) −7.98168 −0.997710
\(5\) 5.00000 0.447214
\(6\) −1.25676 −0.0855120
\(7\) −32.4157 −1.75028 −0.875141 − 0.483869i \(-0.839231\pi\)
−0.875141 + 0.483869i \(0.839231\pi\)
\(8\) −2.16327 −0.0956037
\(9\) 59.2051 2.19278
\(10\) 0.676796 0.0214022
\(11\) −41.8742 −1.14778 −0.573889 − 0.818933i \(-0.694566\pi\)
−0.573889 + 0.818933i \(0.694566\pi\)
\(12\) 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
\(17\) 0 0
\(18\) 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 \(-0.415624\pi\)
0.261981 + 0.965073i \(0.415624\pi\)
\(24\) 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 \(-0.385077\pi\)
0.353247 + 0.935530i \(0.385077\pi\)
\(30\) −6.28382 −0.0382421
\(31\) 25.4263 0.147313 0.0736564 − 0.997284i \(-0.476533\pi\)
0.0736564 + 0.997284i \(0.476533\pi\)
\(32\) 25.9096 0.143132
\(33\) 388.788 2.05089
\(34\) 0 0
\(35\) −162.078 −0.782750
\(36\) −472.556 −2.18776
\(37\) 124.544 0.553377 0.276689 − 0.960960i \(-0.410763\pi\)
0.276689 + 0.960960i \(0.410763\pi\)
\(38\) −15.3159 −0.0653834
\(39\) 467.596 1.91988
\(40\) −10.8163 −0.0427553
\(41\) 16.7597 0.0638398 0.0319199 − 0.999490i \(-0.489838\pi\)
0.0319199 + 0.999490i \(0.489838\pi\)
\(42\) 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
\(51\) 0 0
\(52\) 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 \(-0.587150\pi\)
−0.270383 + 0.962753i \(0.587150\pi\)
\(62\) 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 \(0.417156\pi\)
\(68\) 0 0
\(69\) −536.608 −0.936233
\(70\) −21.9388 −0.0374598
\(71\) 696.547 1.16430 0.582148 − 0.813083i \(-0.302213\pi\)
0.582148 + 0.813083i \(0.302213\pi\)
\(72\) −128.076 −0.209638
\(73\) 493.004 0.790435 0.395218 − 0.918588i \(-0.370669\pi\)
0.395218 + 0.918588i \(0.370669\pi\)
\(74\) 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 \(-0.184911\pi\)
0.835961 + 0.548789i \(0.184911\pi\)
\(80\) 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
\(85\) 0 0
\(86\) −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 \(-0.410458\pi\)
0.277608 + 0.960694i \(0.410458\pi\)
\(98\) 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 1445.4.a.k.1.2 3
17.16 even 2 85.4.a.f.1.2 ✓ 3
51.50 odd 2 765.4.a.k.1.2 3
68.67 odd 2 1360.4.a.p.1.1 3
85.33 odd 4 425.4.b.h.324.3 6
85.67 odd 4 425.4.b.h.324.4 6
85.84 even 2 425.4.a.f.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
85.4.a.f.1.2 ✓ 3 17.16 even 2
425.4.a.f.1.2 3 85.84 even 2
425.4.b.h.324.3 6 85.33 odd 4
425.4.b.h.324.4 6 85.67 odd 4
765.4.a.k.1.2 3 51.50 odd 2
1360.4.a.p.1.1 3 68.67 odd 2
1445.4.a.k.1.2 3 1.1 even 1 trivial