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Properties

Label 985.2.a.g
Level 985985
Weight 22
Character orbit 985.a
Self dual yes
Analytic conductor 7.8657.865
Analytic rank 00
Dimension 1717
CM no
Inner twists 11

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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] = [985,2,Mod(1,985)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(985, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("985.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: N N == 985=5⋅197 985 = 5 \cdot 197
Weight: k k == 2 2
Character orbit: [χ][\chi] == 985.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [17,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: 7.865264599107.86526459910
Analytic rank: 00
Dimension: 1717
Coefficient field: Q[x]/(x17−⋯ )\mathbb{Q}[x]/(x^{17} - \cdots)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x17−6x16−8x15+106x14−60x13−698x12+877x11+2076x10−3556x9+⋯+2 x^{17} - 6 x^{16} - 8 x^{15} + 106 x^{14} - 60 x^{13} - 698 x^{12} + 877 x^{11} + 2076 x^{10} - 3556 x^{9} + \cdots + 2 Copy content Toggle raw display
Coefficient ring: Z[a1,…,a7]\Z[a_1, \ldots, a_{7}]
Coefficient ring index: 22 2^{2}
Twist minimal: yes
Fricke sign: −1-1
Sato-Tate group: SU(2)\mathrm{SU}(2)

qq-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)
 

Coefficients of the qq-expansion are expressed in terms of a basis 1,β1,…,β161,\beta_1,\ldots,\beta_{16} for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+β1q2−β6q3+(β2+1)q4−q5+(−β15−β13+β1)q6−β10q7+(β3+β2+β1+1)q8+(−2β15+β14+⋯+2)q9+⋯+(−2β16+2β15+⋯−4)q99+O(q100) q + \beta_1 q^{2} - \beta_{6} q^{3} + (\beta_{2} + 1) q^{4} - q^{5} + ( - \beta_{15} - \beta_{13} + \beta_1) q^{6} - \beta_{10} q^{7} + (\beta_{3} + \beta_{2} + \beta_1 + 1) q^{8} + ( - 2 \beta_{15} + \beta_{14} + \cdots + 2) q^{9}+ \cdots + ( - 2 \beta_{16} + 2 \beta_{15} + \cdots - 4) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr⁡(f)(q)\operatorname{Tr}(f)(q) == 17q+6q2+5q3+18q4−17q5+3q6+7q7+18q8+22q9−6q10+7q11+20q12+3q13+17q14−5q15+28q16+6q17+13q18−23q19+⋯−46q99+O(q100) 17 q + 6 q^{2} + 5 q^{3} + 18 q^{4} - 17 q^{5} + 3 q^{6} + 7 q^{7} + 18 q^{8} + 22 q^{9} - 6 q^{10} + 7 q^{11} + 20 q^{12} + 3 q^{13} + 17 q^{14} - 5 q^{15} + 28 q^{16} + 6 q^{17} + 13 q^{18} - 23 q^{19}+ \cdots - 46 q^{99}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x17−6x16−8x15+106x14−60x13−698x12+877x11+2076x10−3556x9+⋯+2 x^{17} - 6 x^{16} - 8 x^{15} + 106 x^{14} - 60 x^{13} - 698 x^{12} + 877 x^{11} + 2076 x^{10} - 3556 x^{9} + \cdots + 2 : Copy content Toggle raw display

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== ν2−3 \nu^{2} - 3 Copy content Toggle raw display
β3\beta_{3}== ν3−ν2−5ν+2 \nu^{3} - \nu^{2} - 5\nu + 2 Copy content Toggle raw display
β4\beta_{4}== (−2191ν16+28927ν15−60841ν14−425083ν13+1562505ν12+⋯−270722)/245192 ( - 2191 \nu^{16} + 28927 \nu^{15} - 60841 \nu^{14} - 425083 \nu^{13} + 1562505 \nu^{12} + \cdots - 270722 ) / 245192 Copy content Toggle raw display
β5\beta_{5}== (54273ν16−92866ν15−2278479ν14+5511036ν13+28342853ν12+⋯−24548096)/5639416 ( 54273 \nu^{16} - 92866 \nu^{15} - 2278479 \nu^{14} + 5511036 \nu^{13} + 28342853 \nu^{12} + \cdots - 24548096 ) / 5639416 Copy content Toggle raw display
β6\beta_{6}== (−35046ν16+110817ν15+764848ν14−2404159ν13−6491506ν12+⋯−1062386)/2819708 ( - 35046 \nu^{16} + 110817 \nu^{15} + 764848 \nu^{14} - 2404159 \nu^{13} - 6491506 \nu^{12} + \cdots - 1062386 ) / 2819708 Copy content Toggle raw display
β7\beta_{7}== (−100298ν16+181737ν15+3360696ν14−7555731ν13−37921246ν12+⋯+48232934)/5639416 ( - 100298 \nu^{16} + 181737 \nu^{15} + 3360696 \nu^{14} - 7555731 \nu^{13} - 37921246 \nu^{12} + \cdots + 48232934 ) / 5639416 Copy content Toggle raw display
β8\beta_{8}== (−241552ν16+806883ν15+5112434ν14−17850929ν13−42261732ν12+⋯+2082482)/5639416 ( - 241552 \nu^{16} + 806883 \nu^{15} + 5112434 \nu^{14} - 17850929 \nu^{13} - 42261732 \nu^{12} + \cdots + 2082482 ) / 5639416 Copy content Toggle raw display
β9\beta_{9}== (−246041ν16+1704318ν15+66483ν14−25609808ν13+46293035ν12+⋯−15791344)/5639416 ( - 246041 \nu^{16} + 1704318 \nu^{15} + 66483 \nu^{14} - 25609808 \nu^{13} + 46293035 \nu^{12} + \cdots - 15791344 ) / 5639416 Copy content Toggle raw display
β10\beta_{10}== (350679ν16−1997896ν15−3275133ν14+35199734ν13−11047357ν12+⋯+5393812)/5639416 ( 350679 \nu^{16} - 1997896 \nu^{15} - 3275133 \nu^{14} + 35199734 \nu^{13} - 11047357 \nu^{12} + \cdots + 5393812 ) / 5639416 Copy content Toggle raw display
β11\beta_{11}== (−522455ν16+3194212ν15+3528121ν14−54341730ν13+40686141ν12+⋯−16088228)/5639416 ( - 522455 \nu^{16} + 3194212 \nu^{15} + 3528121 \nu^{14} - 54341730 \nu^{13} + 40686141 \nu^{12} + \cdots - 16088228 ) / 5639416 Copy content Toggle raw display
β12\beta_{12}== (596404ν16−3091147ν15−6671366ν14+54710157ν13+1577112ν12+⋯+11922902)/5639416 ( 596404 \nu^{16} - 3091147 \nu^{15} - 6671366 \nu^{14} + 54710157 \nu^{13} + 1577112 \nu^{12} + \cdots + 11922902 ) / 5639416 Copy content Toggle raw display
β13\beta_{13}== (34331ν16−211845ν15−220207ν14+3614653ν13−2959421ν12+⋯−823298)/245192 ( 34331 \nu^{16} - 211845 \nu^{15} - 220207 \nu^{14} + 3614653 \nu^{13} - 2959421 \nu^{12} + \cdots - 823298 ) / 245192 Copy content Toggle raw display
β14\beta_{14}== (−984739ν16+4852934ν15+12396845ν14−88461580ν13−24382447ν12+⋯+16024912)/5639416 ( - 984739 \nu^{16} + 4852934 \nu^{15} + 12396845 \nu^{14} - 88461580 \nu^{13} - 24382447 \nu^{12} + \cdots + 16024912 ) / 5639416 Copy content Toggle raw display
β15\beta_{15}== (−988531ν16+5841395ν15+7686195ν14−100325551ν13+59472133ν12+⋯+19076038)/5639416 ( - 988531 \nu^{16} + 5841395 \nu^{15} + 7686195 \nu^{14} - 100325551 \nu^{13} + 59472133 \nu^{12} + \cdots + 19076038 ) / 5639416 Copy content Toggle raw display
β16\beta_{16}== (1136939ν16−7098868ν15−7090713ν14+121011790ν13−101282793ν12+⋯−25816036)/5639416 ( 1136939 \nu^{16} - 7098868 \nu^{15} - 7090713 \nu^{14} + 121011790 \nu^{13} - 101282793 \nu^{12} + \cdots - 25816036 ) / 5639416 Copy content Toggle raw display

Display νj\nu^j in terms of βi\beta_i

ν\nu== β1 \beta_1 Copy content Toggle raw display
ν2\nu^{2}== β2+3 \beta_{2} + 3 Copy content Toggle raw display
ν3\nu^{3}== β3+β2+5β1+1 \beta_{3} + \beta_{2} + 5\beta _1 + 1 Copy content Toggle raw display
ν4\nu^{4}== β15+β13+β10+β8−2β6+β4+2β3+7β2+β1+16 \beta_{15} + \beta_{13} + \beta_{10} + \beta_{8} - 2\beta_{6} + \beta_{4} + 2\beta_{3} + 7\beta_{2} + \beta _1 + 16 Copy content Toggle raw display
ν5\nu^{5}== 2β15−β14+2β13−β12+β10−β9+3β8+β7+⋯+12 2 \beta_{15} - \beta_{14} + 2 \beta_{13} - \beta_{12} + \beta_{10} - \beta_{9} + 3 \beta_{8} + \beta_{7} + \cdots + 12 Copy content Toggle raw display
ν6\nu^{6}== −2β16+9β15−β14+13β13+2β11+9β10+13β8+⋯+99 - 2 \beta_{16} + 9 \beta_{15} - \beta_{14} + 13 \beta_{13} + 2 \beta_{11} + 9 \beta_{10} + 13 \beta_{8} + \cdots + 99 Copy content Toggle raw display
ν7\nu^{7}== −2β16+23β15−12β14+27β13−10β12+β11+⋯+112 - 2 \beta_{16} + 23 \beta_{15} - 12 \beta_{14} + 27 \beta_{13} - 10 \beta_{12} + \beta_{11} + \cdots + 112 Copy content Toggle raw display
ν8\nu^{8}== −29β16+69β15−16β14+128β13−2β12+26β11+⋯+655 - 29 \beta_{16} + 69 \beta_{15} - 16 \beta_{14} + 128 \beta_{13} - 2 \beta_{12} + 26 \beta_{11} + \cdots + 655 Copy content Toggle raw display
ν9\nu^{9}== −38β16+201β15−109β14+279β13−79β12+17β11+⋯+954 - 38 \beta_{16} + 201 \beta_{15} - 109 \beta_{14} + 279 \beta_{13} - 79 \beta_{12} + 17 \beta_{11} + \cdots + 954 Copy content Toggle raw display
ν10\nu^{10}== −299β16+516β15−175β14+1136β13−36β12+243β11+⋯+4501 - 299 \beta_{16} + 516 \beta_{15} - 175 \beta_{14} + 1136 \beta_{13} - 36 \beta_{12} + 243 \beta_{11} + \cdots + 4501 Copy content Toggle raw display
ν11\nu^{11}== −475β16+1605β15−897β14+2602β13−583β12+203β11+⋯+7769 - 475 \beta_{16} + 1605 \beta_{15} - 897 \beta_{14} + 2602 \beta_{13} - 583 \beta_{12} + 203 \beta_{11} + \cdots + 7769 Copy content Toggle raw display
ν12\nu^{12}== −2703β16+3878β15−1639β14+9589β13−428β12+2005β11+⋯+31689 - 2703 \beta_{16} + 3878 \beta_{15} - 1639 \beta_{14} + 9589 \beta_{13} - 428 \beta_{12} + 2005 \beta_{11} + \cdots + 31689 Copy content Toggle raw display
ν13\nu^{13}== −4948β16+12367β15−7054β14+22957β13−4204β12+⋯+61666 - 4948 \beta_{16} + 12367 \beta_{15} - 7054 \beta_{14} + 22957 \beta_{13} - 4204 \beta_{12} + \cdots + 61666 Copy content Toggle raw display
ν14\nu^{14}== −22922β16+29449β15−14175β14+78742β13−4271β12+⋯+226889 - 22922 \beta_{16} + 29449 \beta_{15} - 14175 \beta_{14} + 78742 \beta_{13} - 4271 \beta_{12} + \cdots + 226889 Copy content Toggle raw display
ν15\nu^{15}== −46643β16+93975β15−54224β14+195520β13−30145β12+⋯+481705 - 46643 \beta_{16} + 93975 \beta_{15} - 54224 \beta_{14} + 195520 \beta_{13} - 30145 \beta_{12} + \cdots + 481705 Copy content Toggle raw display
ν16\nu^{16}== −187753β16+225590β15−117146β14+635904β13−38843β12+⋯+1644546 - 187753 \beta_{16} + 225590 \beta_{15} - 117146 \beta_{14} + 635904 \beta_{13} - 38843 \beta_{12} + \cdots + 1644546 Copy content Toggle raw display

Display βi\beta_i in terms of νj\nu^j

Embeddings

For each embedding ιm\iota_m of the coefficient field, the values ιm(an)\iota_m(a_n) are shown below.

For more information on an embedded modular form you can click on its label.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   ιm(ν)\iota_m(\nu) a2 a_{2} a3 a_{3} a4 a_{4} a5 a_{5} a6 a_{6} a7 a_{7} a8 a_{8} a9 a_{9} a10 a_{10}
1.1
−2.37113
−2.21690
−2.06409
−1.51786
−0.695153
−0.589248
−0.369022
0.0270965
0.290121
0.704734
1.17487
1.68026
1.78623
2.22883
2.55893
2.61726
2.75509
−2.37113 3.22611 3.62225 −1.00000 −7.64953 −3.33938 −3.84656 7.40781 2.37113
1.2 −2.21690 0.781418 2.91464 −1.00000 −1.73232 −1.11516 −2.02767 −2.38939 2.21690
1.3 −2.06409 −1.71472 2.26049 −1.00000 3.53934 3.04480 −0.537666 −0.0597382 2.06409
1.4 −1.51786 −0.903960 0.303910 −1.00000 1.37209 −1.33147 2.57443 −2.18286 1.51786
1.5 −0.695153 −2.78934 −1.51676 −1.00000 1.93902 0.663651 2.44469 4.78042 0.695153
1.6 −0.589248 0.610291 −1.65279 −1.00000 −0.359613 3.82240 2.15240 −2.62754 0.589248
1.7 −0.369022 3.16939 −1.86382 −1.00000 −1.16957 3.62949 1.42584 7.04503 0.369022
1.8 0.0270965 0.151367 −1.99927 −1.00000 0.00410152 −1.42890 −0.108366 −2.97709 −0.0270965
1.9 0.290121 −0.916709 −1.91583 −1.00000 −0.265957 −4.90454 −1.13606 −2.15964 −0.290121
1.10 0.704734 1.91747 −1.50335 −1.00000 1.35131 0.223493 −2.46893 0.676708 −0.704734
1.11 1.17487 −1.75288 −0.619690 −1.00000 −2.05940 −0.641186 −3.07778 0.0725852 −1.17487
1.12 1.68026 −2.94614 0.823263 −1.00000 −4.95027 3.59992 −1.97722 5.67973 −1.68026
1.13 1.78623 2.62867 1.19063 −1.00000 4.69542 2.07657 −1.44572 3.90991 −1.78623
1.14 2.22883 3.02604 2.96767 −1.00000 6.74452 1.28446 2.15678 6.15690 −2.22883
1.15 2.55893 −1.95773 4.54812 −1.00000 −5.00969 −0.330398 6.52046 0.832702 −2.55893
1.16 2.61726 1.86075 4.85003 −1.00000 4.87007 −2.88349 7.45926 0.462402 −2.61726
1.17 2.75509 0.609959 5.59050 −1.00000 1.68049 4.62975 9.89214 −2.62795 −2.75509
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.17
Significant digits:
Format:

Atkin-Lehner signs

p p Sign
55 +1 +1
197197 −1 -1

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 985.2.a.g ✓ 17
3.b odd 2 1 8865.2.a.z 17
5.b even 2 1 4925.2.a.l 17
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
985.2.a.g ✓ 17 1.a even 1 1 trivial
4925.2.a.l 17 5.b even 2 1
8865.2.a.z 17 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T217−6T216−8T215+106T214−60T213−698T212+877T211+⋯+2 T_{2}^{17} - 6 T_{2}^{16} - 8 T_{2}^{15} + 106 T_{2}^{14} - 60 T_{2}^{13} - 698 T_{2}^{12} + 877 T_{2}^{11} + \cdots + 2 acting on S2new(Γ0(985))S_{2}^{\mathrm{new}}(\Gamma_0(985)). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T17−6T16+⋯+2 T^{17} - 6 T^{16} + \cdots + 2 Copy content Toggle raw display
33 T17−5T16+⋯+512 T^{17} - 5 T^{16} + \cdots + 512 Copy content Toggle raw display
55 (T+1)17 (T + 1)^{17} Copy content Toggle raw display
77 T17−7T16+⋯−5912 T^{17} - 7 T^{16} + \cdots - 5912 Copy content Toggle raw display
1111 T17−7T16+⋯+41153536 T^{17} - 7 T^{16} + \cdots + 41153536 Copy content Toggle raw display
1313 T17−3T16+⋯−310096 T^{17} - 3 T^{16} + \cdots - 310096 Copy content Toggle raw display
1717 T17+⋯−349688704 T^{17} + \cdots - 349688704 Copy content Toggle raw display
1919 T17+23T16+⋯+1915904 T^{17} + 23 T^{16} + \cdots + 1915904 Copy content Toggle raw display
2323 T17−49T16+⋯+40562168 T^{17} - 49 T^{16} + \cdots + 40562168 Copy content Toggle raw display
2929 T17+⋯+5029285904 T^{17} + \cdots + 5029285904 Copy content Toggle raw display
3131 T17+⋯+14161566256 T^{17} + \cdots + 14161566256 Copy content Toggle raw display
3737 T17+⋯+376364288 T^{17} + \cdots + 376364288 Copy content Toggle raw display
4141 T17+⋯−4430614816 T^{17} + \cdots - 4430614816 Copy content Toggle raw display
4343 T17+⋯−278500548904 T^{17} + \cdots - 278500548904 Copy content Toggle raw display
4747 T17+⋯−62369461376 T^{17} + \cdots - 62369461376 Copy content Toggle raw display
5353 T17+⋯−17917124608 T^{17} + \cdots - 17917124608 Copy content Toggle raw display
5959 T17+⋯+2605268451328 T^{17} + \cdots + 2605268451328 Copy content Toggle raw display
6161 T17+⋯+3030799356 T^{17} + \cdots + 3030799356 Copy content Toggle raw display
6767 T17+⋯−275592380032 T^{17} + \cdots - 275592380032 Copy content Toggle raw display
7171 T17+⋯−2285257277264 T^{17} + \cdots - 2285257277264 Copy content Toggle raw display
7373 T17+⋯+25494443456 T^{17} + \cdots + 25494443456 Copy content Toggle raw display
7979 T17+⋯−17 ⁣⋯ ⁣16 T^{17} + \cdots - 17\!\cdots\!16 Copy content Toggle raw display
8383 T17+⋯+574612307072 T^{17} + \cdots + 574612307072 Copy content Toggle raw display
8989 T17+⋯−1850800446976 T^{17} + \cdots - 1850800446976 Copy content Toggle raw display
9797 T17+⋯+3084621824 T^{17} + \cdots + 3084621824 Copy content Toggle raw display
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