[N,k,chi] = [688,2,Mod(97,688)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(688, base_ring=CyclotomicField(14))
chi = DirichletCharacter(H, H._module([0, 0, 10]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("688.97");
S:= CuspForms(chi, 2);
N := Newforms(S);
Newform invariants
sage: f = N[0] # Warning: the index may be different
gp: f = lf[1] \\ Warning: the index may be different
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/688\mathbb{Z}\right)^\times\).
\(n\)
\(431\)
\(433\)
\(517\)
\(\chi(n)\)
\(1\)
\(-\beta_{10}\)
\(1\)
For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.
For more information on an embedded modular form you can click on its label.
Refresh table
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} - 5T_{3}^{5} + 11T_{3}^{4} - 13T_{3}^{3} + 9T_{3}^{2} - 3T_{3} + 1 \)
T3^6 - 5*T3^5 + 11*T3^4 - 13*T3^3 + 9*T3^2 - 3*T3 + 1
acting on \(S_{2}^{\mathrm{new}}(688, [\chi])\).
$p$
$F_p(T)$
$2$
\( T^{12} \)
T^12
$3$
\( (T^{6} - 5 T^{5} + 11 T^{4} - 13 T^{3} + \cdots + 1)^{2} \)
(T^6 - 5*T^5 + 11*T^4 - 13*T^3 + 9*T^2 - 3*T + 1)^2
$5$
\( T^{12} - 3 T^{11} + 9 T^{10} + T^{9} + \cdots + 3136 \)
T^12 - 3*T^11 + 9*T^10 + T^9 + 39*T^8 + 79*T^7 - 125*T^6 - 1638*T^5 + 7532*T^4 - 5768*T^3 + 14896*T^2 + 7840*T + 3136
$7$
\( (T^{6} - 27 T^{4} + 7 T^{3} + 152 T^{2} + \cdots - 64)^{2} \)
(T^6 - 27*T^4 + 7*T^3 + 152*T^2 - 112*T - 64)^2
$11$
\( T^{12} + T^{11} + 14 T^{10} + 62 T^{9} + \cdots + 19321 \)
T^12 + T^11 + 14*T^10 + 62*T^9 + 874*T^8 + 1848*T^7 + 2773*T^6 + 1568*T^5 - 2192*T^4 - 5909*T^3 + 16919*T^2 - 17653*T + 19321
$13$
\( T^{12} + 4 T^{11} - 21 T^{10} + \cdots + 19321 \)
T^12 + 4*T^11 - 21*T^10 - 50*T^9 + 703*T^8 - 1624*T^7 + 1737*T^6 - 2128*T^5 + 18541*T^4 + 1798*T^3 + 33299*T^2 - 278*T + 19321
$17$
\( T^{12} - 17 T^{11} + 135 T^{10} + \cdots + 3136 \)
T^12 - 17*T^11 + 135*T^10 - 657*T^9 + 2293*T^8 - 4807*T^7 + 6693*T^6 - 3724*T^5 + 8428*T^4 - 11088*T^3 + 6272*T^2 - 3136*T + 3136
$19$
\( T^{12} + 13 T^{11} + 80 T^{10} + \cdots + 31820881 \)
T^12 + 13*T^11 + 80*T^10 + 226*T^9 - 52*T^8 - 3696*T^7 + 7981*T^6 + 136920*T^5 + 950854*T^4 + 2938423*T^3 + 9489255*T^2 + 17549151*T + 31820881
$23$
\( T^{12} - 14 T^{11} + 135 T^{10} + \cdots + 15816529 \)
T^12 - 14*T^11 + 135*T^10 - 994*T^9 + 6451*T^8 - 26838*T^7 + 86395*T^6 - 104006*T^5 + 920201*T^4 + 1073212*T^3 + 14533845*T^2 + 26725440*T + 15816529
$29$
\( T^{12} + 6 T^{11} + 85 T^{10} + \cdots + 37515625 \)
T^12 + 6*T^11 + 85*T^10 - 176*T^9 - 811*T^8 - 14862*T^7 + 130691*T^6 + 437570*T^5 - 415275*T^4 + 1041250*T^3 + 24224375*T^2 + 15006250*T + 37515625
$31$
\( T^{12} - 23 T^{11} + 271 T^{10} + \cdots + 177241 \)
T^12 - 23*T^11 + 271*T^10 - 2105*T^9 + 11782*T^8 - 47124*T^7 + 138489*T^6 - 293538*T^5 + 504456*T^4 - 693996*T^3 + 757272*T^2 - 532565*T + 177241
$37$
\( (T^{6} + 12 T^{5} + 19 T^{4} - 147 T^{3} + \cdots + 8)^{2} \)
(T^6 + 12*T^5 + 19*T^4 - 147*T^3 - 494*T^2 - 372*T + 8)^2
$41$
\( T^{12} - 8 T^{11} + 63 T^{10} + \cdots + 171688609 \)
T^12 - 8*T^11 + 63*T^10 - 720*T^9 + 6439*T^8 - 41734*T^7 + 285783*T^6 - 1550206*T^5 + 6900991*T^4 - 22408980*T^3 + 66461451*T^2 - 149583848*T + 171688609
$43$
\( T^{12} - 7 T^{11} + \cdots + 6321363049 \)
T^12 - 7*T^11 - 6*T^10 - 140*T^9 + 3851*T^8 - 13377*T^7 - 13628*T^6 - 575211*T^5 + 7120499*T^4 - 11130980*T^3 - 20512806*T^2 - 1029059101*T + 6321363049
$47$
\( T^{12} + 22 T^{11} + 273 T^{10} + \cdots + 5340721 \)
T^12 + 22*T^11 + 273*T^10 + 1224*T^9 - 225*T^8 - 43596*T^7 - 11171*T^6 + 1110004*T^5 + 8602467*T^4 + 32118260*T^3 + 53999309*T^2 - 4737550*T + 5340721
$53$
\( T^{12} - 27 T^{11} + \cdots + 3037883689 \)
T^12 - 27*T^11 + 385*T^10 - 3564*T^9 + 46150*T^8 - 623231*T^7 + 5694452*T^6 - 32149705*T^5 + 274695385*T^4 - 1054161585*T^3 + 1617959770*T^2 - 904800672*T + 3037883689
$59$
\( T^{12} - 17 T^{11} + \cdots + 2208906001 \)
T^12 - 17*T^11 + 266*T^10 - 2780*T^9 + 25448*T^8 - 269612*T^7 + 2691711*T^6 - 17534258*T^5 + 85559948*T^4 - 309439565*T^3 + 1016953721*T^2 - 2213041913*T + 2208906001
$61$
\( T^{12} - 8 T^{11} + 105 T^{10} + \cdots + 5139289 \)
T^12 - 8*T^11 + 105*T^10 - 2106*T^9 + 10723*T^8 + 89348*T^7 + 1281113*T^6 + 426244*T^5 + 2637613*T^4 - 2007116*T^3 + 6157851*T^2 - 9471526*T + 5139289
$67$
\( T^{12} + 29 T^{11} + \cdots + 5116111729 \)
T^12 + 29*T^11 + 602*T^10 + 8882*T^9 + 115968*T^8 + 1191442*T^7 + 9772267*T^6 + 68309206*T^5 + 392683892*T^4 + 1595696031*T^3 + 4203277001*T^2 + 6594145657*T + 5116111729
$71$
\( T^{12} - 17 T^{11} + 181 T^{10} + \cdots + 11235904 \)
T^12 - 17*T^11 + 181*T^10 - 1633*T^9 + 11769*T^8 - 60669*T^7 + 238001*T^6 - 659960*T^5 + 1230212*T^4 - 2482128*T^3 + 8597536*T^2 - 12362176*T + 11235904
$73$
\( T^{12} + 10 T^{11} + \cdots + 24854468409 \)
T^12 + 10*T^11 + 90*T^10 + 1619*T^9 + 26287*T^8 + 208565*T^7 + 4783234*T^6 - 39406227*T^5 + 74733390*T^4 - 162283284*T^3 + 3847306977*T^2 + 9519245793*T + 24854468409
$79$
\( (T^{6} + 7 T^{5} - 269 T^{4} + \cdots + 318016)^{2} \)
(T^6 + 7*T^5 - 269*T^4 - 2555*T^3 + 9320*T^2 + 141904*T + 318016)^2
$83$
\( T^{12} - 6 T^{11} + 183 T^{10} + \cdots + 292444201 \)
T^12 - 6*T^11 + 183*T^10 + 316*T^9 + 36807*T^8 + 106*T^7 + 2851563*T^6 + 5829558*T^5 + 1968673*T^4 - 13513612*T^3 - 26442213*T^2 - 16758980*T + 292444201
$89$
\( T^{12} + 16 T^{11} + 312 T^{10} + \cdots + 82369 \)
T^12 + 16*T^11 + 312*T^10 + 7393*T^9 + 76785*T^8 + 253565*T^7 + 6603290*T^6 + 85027845*T^5 + 341409936*T^4 + 13926962*T^3 + 18005099*T^2 - 1424381*T + 82369
$97$
\( T^{12} - 10 T^{11} + \cdots + 376889171569 \)
T^12 - 10*T^11 - 14*T^10 - 3067*T^9 + 70493*T^8 - 217203*T^7 + 3577190*T^6 - 83311767*T^5 + 680300906*T^4 - 3532397772*T^3 + 27656291383*T^2 - 47018982757*T + 376889171569
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