Properties

Label 575.4.b.j
Level 575575
Weight 44
Character orbit 575.b
Analytic conductor 33.92633.926
Analytic rank 00
Dimension 1414
Inner twists 22

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [575,4,Mod(24,575)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(575, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 4, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("575.24"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Level: N N == 575=5223 575 = 5^{2} \cdot 23
Weight: k k == 4 4
Character orbit: [χ][\chi] == 575.b (of order 22, degree 11, not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [14,0,0,-54,0,-82] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: 33.926098253333.9260982533
Analytic rank: 00
Dimension: 1414
Coefficient field: Q[x]/(x14+)\mathbb{Q}[x]/(x^{14} + \cdots)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x14+83x12+2715x10+44273x8+372280x6+1482448x4+2136384x2+746496 x^{14} + 83x^{12} + 2715x^{10} + 44273x^{8} + 372280x^{6} + 1482448x^{4} + 2136384x^{2} + 746496 Copy content Toggle raw display
Coefficient ring: Z[a1,,a7]\Z[a_1, \ldots, a_{7}]
Coefficient ring index: 24 2^{4}
Twist minimal: yes
Sato-Tate group: SU(2)[C2]\mathrm{SU}(2)[C_{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,,β131,\beta_1,\ldots,\beta_{13} for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+β1q2+(β8+β6+β1)q3+(β5+β44)q4+(β7+β5+β4+5)q6+(β12β8)q7+(2β10+2β8+3β1)q8++(23β925β7++288)q99+O(q100) q + \beta_1 q^{2} + ( - \beta_{8} + \beta_{6} + \beta_1) q^{3} + (\beta_{5} + \beta_{4} - 4) q^{4} + (\beta_{7} + \beta_{5} + \beta_{4} + \cdots - 5) q^{6} + ( - \beta_{12} - \beta_{8}) q^{7} + (2 \beta_{10} + 2 \beta_{8} + \cdots - 3 \beta_1) q^{8}+ \cdots + (23 \beta_{9} - 25 \beta_{7} + \cdots + 288) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 14q54q482q620q9104q11+84q14170q16+20q19404q21+606q2452q26+910q291380q31+1314q34+408q36+554q39460q41++4286q99+O(q100) 14 q - 54 q^{4} - 82 q^{6} - 20 q^{9} - 104 q^{11} + 84 q^{14} - 170 q^{16} + 20 q^{19} - 404 q^{21} + 606 q^{24} - 52 q^{26} + 910 q^{29} - 1380 q^{31} + 1314 q^{34} + 408 q^{36} + 554 q^{39} - 460 q^{41}+ \cdots + 4286 q^{99}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x14+83x12+2715x10+44273x8+372280x6+1482448x4+2136384x2+746496 x^{14} + 83x^{12} + 2715x^{10} + 44273x^{8} + 372280x^{6} + 1482448x^{4} + 2136384x^{2} + 746496 : Copy content Toggle raw display

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== (2761ν12+193316ν10+4988175ν8+57142094ν6+264366298ν4+132164352)/21505248 ( 2761 \nu^{12} + 193316 \nu^{10} + 4988175 \nu^{8} + 57142094 \nu^{6} + 264366298 \nu^{4} + \cdots - 132164352 ) / 21505248 Copy content Toggle raw display
β3\beta_{3}== (2899ν12+208820ν10+5584101ν8+67767626ν6+365072926ν4++573138432)/21505248 ( 2899 \nu^{12} + 208820 \nu^{10} + 5584101 \nu^{8} + 67767626 \nu^{6} + 365072926 \nu^{4} + \cdots + 573138432 ) / 21505248 Copy content Toggle raw display
β4\beta_{4}== (673ν12+46391ν10+1172553ν8+13181345ν6+62463766ν4++29332800)/3584208 ( 673 \nu^{12} + 46391 \nu^{10} + 1172553 \nu^{8} + 13181345 \nu^{6} + 62463766 \nu^{4} + \cdots + 29332800 ) / 3584208 Copy content Toggle raw display
β5\beta_{5}== (673ν1246391ν101172553ν813181345ν662463766ν4++13677696)/3584208 ( - 673 \nu^{12} - 46391 \nu^{10} - 1172553 \nu^{8} - 13181345 \nu^{6} - 62463766 \nu^{4} + \cdots + 13677696 ) / 3584208 Copy content Toggle raw display
β6\beta_{6}== (16975ν131118189ν1126046213ν9244991279ν7++3227461056ν)/1548377856 ( - 16975 \nu^{13} - 1118189 \nu^{11} - 26046213 \nu^{9} - 244991279 \nu^{7} + \cdots + 3227461056 \nu ) / 1548377856 Copy content Toggle raw display
β7\beta_{7}== (7196ν12+487045ν10+12004194ν8+129952549ν6+577259876ν4++189432288)/21505248 ( 7196 \nu^{12} + 487045 \nu^{10} + 12004194 \nu^{8} + 129952549 \nu^{6} + 577259876 \nu^{4} + \cdots + 189432288 ) / 21505248 Copy content Toggle raw display
β8\beta_{8}== (26717ν132037367ν1158609191ν9779396941ν7++1827097920ν)/1548377856 ( - 26717 \nu^{13} - 2037367 \nu^{11} - 58609191 \nu^{9} - 779396941 \nu^{7} + \cdots + 1827097920 \nu ) / 1548377856 Copy content Toggle raw display
β9\beta_{9}== (27685ν12+1912367ν10+48919983ν8+569920133ν6+2981644672ν4++3235561344)/43010496 ( 27685 \nu^{12} + 1912367 \nu^{10} + 48919983 \nu^{8} + 569920133 \nu^{6} + 2981644672 \nu^{4} + \cdots + 3235561344 ) / 43010496 Copy content Toggle raw display
β10\beta_{10}== (1077ν1365807ν111326331ν98251101ν7+27581252ν5++537403648ν)/28673664 ( - 1077 \nu^{13} - 65807 \nu^{11} - 1326331 \nu^{9} - 8251101 \nu^{7} + 27581252 \nu^{5} + \cdots + 537403648 \nu ) / 28673664 Copy content Toggle raw display
β11\beta_{11}== (7253ν13+551887ν11+16165695ν9+229537141ν7+1623583616ν5++4412473344ν)/129031488 ( 7253 \nu^{13} + 551887 \nu^{11} + 16165695 \nu^{9} + 229537141 \nu^{7} + 1623583616 \nu^{5} + \cdots + 4412473344 \nu ) / 129031488 Copy content Toggle raw display
β12\beta_{12}== (20576ν13+1347439ν11+31418445ν9+302528533ν7+890890415ν5+6944411376ν)/193547232 ( 20576 \nu^{13} + 1347439 \nu^{11} + 31418445 \nu^{9} + 302528533 \nu^{7} + 890890415 \nu^{5} + \cdots - 6944411376 \nu ) / 193547232 Copy content Toggle raw display
β13\beta_{13}== (183923ν13+12861853ν11+334482897ν9+3987741031ν7++32486214528ν)/774188928 ( 183923 \nu^{13} + 12861853 \nu^{11} + 334482897 \nu^{9} + 3987741031 \nu^{7} + \cdots + 32486214528 \nu ) / 774188928 Copy content Toggle raw display
ν\nu== β1 \beta_1 Copy content Toggle raw display
ν2\nu^{2}== β5+β412 \beta_{5} + \beta_{4} - 12 Copy content Toggle raw display
ν3\nu^{3}== 2β10+2β810β619β1 2\beta_{10} + 2\beta_{8} - 10\beta_{6} - 19\beta_1 Copy content Toggle raw display
ν4\nu^{4}== 4β723β533β4+4β3+216 4\beta_{7} - 23\beta_{5} - 33\beta_{4} + 4\beta_{3} + 216 Copy content Toggle raw display
ν5\nu^{5}== 8β138β1214β1162β1062β8+334β6+395β1 8\beta_{13} - 8\beta_{12} - 14\beta_{11} - 62\beta_{10} - 62\beta_{8} + 334\beta_{6} + 395\beta_1 Copy content Toggle raw display
ν6\nu^{6}== 160β7+505β5+921β4148β336β24332 -160\beta_{7} + 505\beta_{5} + 921\beta_{4} - 148\beta_{3} - 36\beta_{2} - 4332 Copy content Toggle raw display
ν7\nu^{7}== 344β13+296β12+636β11+1578β10+1698β89402β68675β1 -344\beta_{13} + 296\beta_{12} + 636\beta_{11} + 1578\beta_{10} + 1698\beta_{8} - 9402\beta_{6} - 8675\beta_1 Copy content Toggle raw display
ν8\nu^{8}== 96β9+4844β711243β524481β4+4212β3+1832β2+92368 96\beta_{9} + 4844\beta_{7} - 11243\beta_{5} - 24481\beta_{4} + 4212\beta_{3} + 1832\beta_{2} + 92368 Copy content Toggle raw display
ν9\nu^{9}== 11080β138232β1221018β1137942β1043814β8++197115β1 11080 \beta_{13} - 8232 \beta_{12} - 21018 \beta_{11} - 37942 \beta_{10} - 43814 \beta_{8} + \cdots + 197115 \beta_1 Copy content Toggle raw display
ν10\nu^{10}== 5696β9134520β7+254829β5+636777β4109300β3+2054900 - 5696 \beta_{9} - 134520 \beta_{7} + 254829 \beta_{5} + 636777 \beta_{4} - 109300 \beta_{3} + \cdots - 2054900 Copy content Toggle raw display
ν11\nu^{11}== 319896β13+207208β12+614920β11+896002β10+1092906β8+4579603β1 - 319896 \beta_{13} + 207208 \beta_{12} + 614920 \beta_{11} + 896002 \beta_{10} + 1092906 \beta_{8} + \cdots - 4579603 \beta_1 Copy content Toggle raw display
ν12\nu^{12}== 225376β9+3595588β75869375β516347697β4+2723220β3++47101912 225376 \beta_{9} + 3595588 \beta_{7} - 5869375 \beta_{5} - 16347697 \beta_{4} + 2723220 \beta_{3} + \cdots + 47101912 Copy content Toggle raw display
ν13\nu^{13}== 8741576β134995688β1216920198β1121081230β10++108029099β1 8741576 \beta_{13} - 4995688 \beta_{12} - 16920198 \beta_{11} - 21081230 \beta_{10} + \cdots + 108029099 \beta_1 Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/575Z)×\left(\mathbb{Z}/575\mathbb{Z}\right)^\times.

nn 5151 277277
χ(n)\chi(n) 11 1-1

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}
24.1
4.96146i
4.39200i
3.84004i
3.74805i
2.80947i
1.37904i
0.711047i
0.711047i
1.37904i
2.80947i
3.74805i
3.84004i
4.39200i
4.96146i
4.96146i 7.15622i −16.6161 0 −35.5053 3.69237i 42.7482i −24.2114 0
24.2 4.39200i 3.27927i −11.2896 0 −14.4025 13.7978i 14.4481i 16.2464 0
24.3 3.84004i 4.08871i −6.74592 0 15.7008 27.0878i 4.81574i 10.2825 0
24.4 3.74805i 2.78570i −6.04789 0 10.4409 9.96305i 7.31660i 19.2399 0
24.5 2.80947i 9.02517i 0.106856 0 −25.3560 11.4533i 22.7760i −54.4537 0
24.6 1.37904i 5.53432i 6.09824 0 7.63206 8.23397i 19.4421i −3.62871 0
24.7 0.711047i 0.689108i 7.49441 0 0.489988 19.0526i 11.0173i 26.5251 0
24.8 0.711047i 0.689108i 7.49441 0 0.489988 19.0526i 11.0173i 26.5251 0
24.9 1.37904i 5.53432i 6.09824 0 7.63206 8.23397i 19.4421i −3.62871 0
24.10 2.80947i 9.02517i 0.106856 0 −25.3560 11.4533i 22.7760i −54.4537 0
24.11 3.74805i 2.78570i −6.04789 0 10.4409 9.96305i 7.31660i 19.2399 0
24.12 3.84004i 4.08871i −6.74592 0 15.7008 27.0878i 4.81574i 10.2825 0
24.13 4.39200i 3.27927i −11.2896 0 −14.4025 13.7978i 14.4481i 16.2464 0
24.14 4.96146i 7.15622i −16.6161 0 −35.5053 3.69237i 42.7482i −24.2114 0
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 24.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 575.4.b.j 14
5.b even 2 1 inner 575.4.b.j 14
5.c odd 4 1 575.4.a.l 7
5.c odd 4 1 575.4.a.m yes 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
575.4.a.l 7 5.c odd 4 1
575.4.a.m yes 7 5.c odd 4 1
575.4.b.j 14 1.a even 1 1 trivial
575.4.b.j 14 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S4new(575,[χ])S_{4}^{\mathrm{new}}(575, [\chi]):

T214+83T212+2715T210+44273T28+372280T26+1482448T24+2136384T22+746496 T_{2}^{14} + 83T_{2}^{12} + 2715T_{2}^{10} + 44273T_{2}^{8} + 372280T_{2}^{6} + 1482448T_{2}^{4} + 2136384T_{2}^{2} + 746496 Copy content Toggle raw display
T314+199T312+14475T310+490274T38+8194505T36+65475300T34++84640000 T_{3}^{14} + 199 T_{3}^{12} + 14475 T_{3}^{10} + 490274 T_{3}^{8} + 8194505 T_{3}^{6} + 65475300 T_{3}^{4} + \cdots + 84640000 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T14+83T12++746496 T^{14} + 83 T^{12} + \cdots + 746496 Copy content Toggle raw display
33 T14+199T12++84640000 T^{14} + 199 T^{12} + \cdots + 84640000 Copy content Toggle raw display
55 T14 T^{14} Copy content Toggle raw display
77 T14++610305007739904 T^{14} + \cdots + 610305007739904 Copy content Toggle raw display
1111 (T7+52T6+935052376)2 (T^{7} + 52 T^{6} + \cdots - 935052376)^{2} Copy content Toggle raw display
1313 T14++22 ⁣ ⁣76 T^{14} + \cdots + 22\!\cdots\!76 Copy content Toggle raw display
1717 T14++74 ⁣ ⁣00 T^{14} + \cdots + 74\!\cdots\!00 Copy content Toggle raw display
1919 (T710T6+451105530840)2 (T^{7} - 10 T^{6} + \cdots - 451105530840)^{2} Copy content Toggle raw display
2323 (T2+529)7 (T^{2} + 529)^{7} Copy content Toggle raw display
2929 (T7+323393919122280)2 (T^{7} + \cdots - 323393919122280)^{2} Copy content Toggle raw display
3131 (T7++38841367049280)2 (T^{7} + \cdots + 38841367049280)^{2} Copy content Toggle raw display
3737 T14++59 ⁣ ⁣04 T^{14} + \cdots + 59\!\cdots\!04 Copy content Toggle raw display
4141 (T7+15 ⁣ ⁣85)2 (T^{7} + \cdots - 15\!\cdots\!85)^{2} Copy content Toggle raw display
4343 T14++31 ⁣ ⁣00 T^{14} + \cdots + 31\!\cdots\!00 Copy content Toggle raw display
4747 T14++19 ⁣ ⁣00 T^{14} + \cdots + 19\!\cdots\!00 Copy content Toggle raw display
5353 T14++29 ⁣ ⁣16 T^{14} + \cdots + 29\!\cdots\!16 Copy content Toggle raw display
5959 (T7+771684969846528)2 (T^{7} + \cdots - 771684969846528)^{2} Copy content Toggle raw display
6161 (T7+67 ⁣ ⁣12)2 (T^{7} + \cdots - 67\!\cdots\!12)^{2} Copy content Toggle raw display
6767 T14++19 ⁣ ⁣00 T^{14} + \cdots + 19\!\cdots\!00 Copy content Toggle raw display
7171 (T7++30 ⁣ ⁣00)2 (T^{7} + \cdots + 30\!\cdots\!00)^{2} Copy content Toggle raw display
7373 T14++13 ⁣ ⁣69 T^{14} + \cdots + 13\!\cdots\!69 Copy content Toggle raw display
7979 (T7+30 ⁣ ⁣36)2 (T^{7} + \cdots - 30\!\cdots\!36)^{2} Copy content Toggle raw display
8383 T14++13 ⁣ ⁣16 T^{14} + \cdots + 13\!\cdots\!16 Copy content Toggle raw display
8989 (T7+28 ⁣ ⁣00)2 (T^{7} + \cdots - 28\!\cdots\!00)^{2} Copy content Toggle raw display
9797 T14++36 ⁣ ⁣24 T^{14} + \cdots + 36\!\cdots\!24 Copy content Toggle raw display
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