Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [765,4,Mod(1,765)] 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("765.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(765, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 765 = 3^{2} \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 765.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,-3,0,-1,15,0,34,-39,0,-15,-52] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(45.1364611544\)
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: \(\Z[a_1, \ldots, a_{13}]\)
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 \(\chi\) \(=\) 765.1

$q$-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)
 
\(f(q)\) \(=\) \(q-0.135359 q^{2} -7.98168 q^{4} +5.00000 q^{5} +32.4157 q^{7} +2.16327 q^{8} -0.676796 q^{10} -41.8742 q^{11} -50.3621 q^{13} -4.38776 q^{14} +63.5606 q^{16} +17.0000 q^{17} -113.150 q^{19} -39.9084 q^{20} +5.66806 q^{22} +57.7951 q^{23} +25.0000 q^{25} +6.81697 q^{26} -258.731 q^{28} +110.333 q^{29} -25.4263 q^{31} -25.9096 q^{32} -2.30110 q^{34} +162.078 q^{35} -124.544 q^{37} +15.3159 q^{38} +10.8163 q^{40} +16.7597 q^{41} -344.435 q^{43} +334.227 q^{44} -7.82309 q^{46} +188.485 q^{47} +707.775 q^{49} -3.38398 q^{50} +401.974 q^{52} -394.600 q^{53} -209.371 q^{55} +70.1237 q^{56} -14.9346 q^{58} -344.824 q^{59} +257.635 q^{61} +3.44168 q^{62} -504.978 q^{64} -251.811 q^{65} +282.255 q^{67} -135.689 q^{68} -21.9388 q^{70} +696.547 q^{71} -493.004 q^{73} +16.8582 q^{74} +903.128 q^{76} -1357.38 q^{77} -1173.97 q^{79} +317.803 q^{80} -2.26858 q^{82} -902.089 q^{83} +85.0000 q^{85} +46.6224 q^{86} -90.5851 q^{88} -648.960 q^{89} -1632.52 q^{91} -461.302 q^{92} -25.5132 q^{94} -565.751 q^{95} -530.420 q^{97} -95.8038 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} - q^{4} + 15 q^{5} + 34 q^{7} - 39 q^{8} - 15 q^{10} - 52 q^{11} + 19 q^{13} + 2 q^{14} + 59 q^{16} + 51 q^{17} - 153 q^{19} - 5 q^{20} - 64 q^{22} - 162 q^{23} + 75 q^{25} + 81 q^{26} - 278 q^{28}+ \cdots + 893 q^{98}+O(q^{100}) \) 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 \(-0.507617\pi\)
−0.0239283 + 0.999714i \(0.507617\pi\)
\(3\) 0 0
\(4\) −7.98168 −0.997710
\(5\) 5.00000 0.447214
\(6\) 0 0
\(7\) 32.4157 1.75028 0.875141 − 0.483869i \(-0.160769\pi\)
0.875141 + 0.483869i \(0.160769\pi\)
\(8\) 2.16327 0.0956037
\(9\) 0 0
\(10\) −0.676796 −0.0214022
\(11\) −41.8742 −1.14778 −0.573889 − 0.818933i \(-0.694566\pi\)
−0.573889 + 0.818933i \(0.694566\pi\)
\(12\) 0 0
\(13\) −50.3621 −1.07446 −0.537229 − 0.843437i \(-0.680529\pi\)
−0.537229 + 0.843437i \(0.680529\pi\)
\(14\) −4.38776 −0.0837626
\(15\) 0 0
\(16\) 63.5606 0.993134
\(17\) 17.0000 0.242536
\(18\) 0 0
\(19\) −113.150 −1.36623 −0.683116 − 0.730309i \(-0.739376\pi\)
−0.683116 + 0.730309i \(0.739376\pi\)
\(20\) −39.9084 −0.446189
\(21\) 0 0
\(22\) 5.66806 0.0549288
\(23\) 57.7951 0.523961 0.261981 − 0.965073i \(-0.415624\pi\)
0.261981 + 0.965073i \(0.415624\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 6.81697 0.0514199
\(27\) 0 0
\(28\) −258.731 −1.74627
\(29\) 110.333 0.706494 0.353247 − 0.935530i \(-0.385077\pi\)
0.353247 + 0.935530i \(0.385077\pi\)
\(30\) 0 0
\(31\) −25.4263 −0.147313 −0.0736564 − 0.997284i \(-0.523467\pi\)
−0.0736564 + 0.997284i \(0.523467\pi\)
\(32\) −25.9096 −0.143132
\(33\) 0 0
\(34\) −2.30110 −0.0116069
\(35\) 162.078 0.782750
\(36\) 0 0
\(37\) −124.544 −0.553377 −0.276689 − 0.960960i \(-0.589237\pi\)
−0.276689 + 0.960960i \(0.589237\pi\)
\(38\) 15.3159 0.0653834
\(39\) 0 0
\(40\) 10.8163 0.0427553
\(41\) 16.7597 0.0638398 0.0319199 − 0.999490i \(-0.489838\pi\)
0.0319199 + 0.999490i \(0.489838\pi\)
\(42\) 0 0
\(43\) −344.435 −1.22153 −0.610765 − 0.791812i \(-0.709138\pi\)
−0.610765 + 0.791812i \(0.709138\pi\)
\(44\) 334.227 1.14515
\(45\) 0 0
\(46\) −7.82309 −0.0250750
\(47\) 188.485 0.584966 0.292483 − 0.956271i \(-0.405519\pi\)
0.292483 + 0.956271i \(0.405519\pi\)
\(48\) 0 0
\(49\) 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 \(-0.670852\pi\)
−0.511344 + 0.859376i \(0.670852\pi\)
\(54\) 0 0
\(55\) −209.371 −0.513302
\(56\) 70.1237 0.167333
\(57\) 0 0
\(58\) −14.9346 −0.0338105
\(59\) −344.824 −0.760886 −0.380443 − 0.924804i \(-0.624228\pi\)
−0.380443 + 0.924804i \(0.624228\pi\)
\(60\) 0 0
\(61\) 257.635 0.540767 0.270383 − 0.962753i \(-0.412850\pi\)
0.270383 + 0.962753i \(0.412850\pi\)
\(62\) 3.44168 0.00704990
\(63\) 0 0
\(64\) −504.978 −0.986285
\(65\) −251.811 −0.480512
\(66\) 0 0
\(67\) 282.255 0.514671 0.257335 − 0.966322i \(-0.417156\pi\)
0.257335 + 0.966322i \(0.417156\pi\)
\(68\) −135.689 −0.241980
\(69\) 0 0
\(70\) −21.9388 −0.0374598
\(71\) 696.547 1.16430 0.582148 − 0.813083i \(-0.302213\pi\)
0.582148 + 0.813083i \(0.302213\pi\)
\(72\) 0 0
\(73\) −493.004 −0.790435 −0.395218 − 0.918588i \(-0.629331\pi\)
−0.395218 + 0.918588i \(0.629331\pi\)
\(74\) 16.8582 0.0264828
\(75\) 0 0
\(76\) 903.128 1.36310
\(77\) −1357.38 −2.00893
\(78\) 0 0
\(79\) −1173.97 −1.67192 −0.835961 − 0.548789i \(-0.815089\pi\)
−0.835961 + 0.548789i \(0.815089\pi\)
\(80\) 317.803 0.444143
\(81\) 0 0
\(82\) −2.26858 −0.00305516
\(83\) −902.089 −1.19298 −0.596489 − 0.802622i \(-0.703438\pi\)
−0.596489 + 0.802622i \(0.703438\pi\)
\(84\) 0 0
\(85\) 85.0000 0.108465
\(86\) 46.6224 0.0584584
\(87\) 0 0
\(88\) −90.5851 −0.109732
\(89\) −648.960 −0.772918 −0.386459 − 0.922307i \(-0.626302\pi\)
−0.386459 + 0.922307i \(0.626302\pi\)
\(90\) 0 0
\(91\) −1632.52 −1.88060
\(92\) −461.302 −0.522761
\(93\) 0 0
\(94\) −25.5132 −0.0279945
\(95\) −565.751 −0.610998
\(96\) 0 0
\(97\) −530.420 −0.555216 −0.277608 − 0.960694i \(-0.589542\pi\)
−0.277608 + 0.960694i \(0.589542\pi\)
\(98\) −95.8038 −0.0987515
\(99\) 0 0
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 765.4.a.k.1.2 3
3.2 odd 2 85.4.a.f.1.2 ✓ 3
12.11 even 2 1360.4.a.p.1.1 3
15.2 even 4 425.4.b.h.324.4 6
15.8 even 4 425.4.b.h.324.3 6
15.14 odd 2 425.4.a.f.1.2 3
51.50 odd 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 3.2 odd 2
425.4.a.f.1.2 3 15.14 odd 2
425.4.b.h.324.3 6 15.8 even 4
425.4.b.h.324.4 6 15.2 even 4
765.4.a.k.1.2 3 1.1 even 1 trivial
1360.4.a.p.1.1 3 12.11 even 2
1445.4.a.k.1.2 3 51.50 odd 2