# Properties

 Label 71.11.b.a Level $71$ Weight $11$ Character orbit 71.b Self dual yes Analytic conductor $45.110$ Analytic rank $0$ Dimension $7$ CM discriminant -71 Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [71,11,Mod(70,71)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(71, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1]))

N = Newforms(chi, 11, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("71.70");

S:= CuspForms(chi, 11);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$71$$ Weight: $$k$$ $$=$$ $$11$$ Character orbit: $$[\chi]$$ $$=$$ 71.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$45.1103649398$$ Analytic rank: $$0$$ Dimension: $$7$$ Coefficient field: 7.7.294755098673.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{7} - 14x^{5} + 56x^{3} - 56x - 21$$ x^7 - 14*x^5 + 56*x^3 - 56*x - 21 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{6}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (13 \beta_{4} + 5 \beta_{3}) q^{2} + (65 \beta_{5} - 76 \beta_{2}) q^{3} + (313 \beta_{6} + 221 \beta_1 + 1024) q^{4} + (799 \beta_{6} + 1232 \beta_1) q^{5} + ( - 562 \beta_{6} - 3611 \beta_{5} - 2161 \beta_{2} - 5229 \beta_1) q^{6} + ( - 15540 \beta_{5} + 13312 \beta_{4} + 5120 \beta_{3} - 5099 \beta_{2}) q^{8} + ( - 12554 \beta_{4} + 27087 \beta_{3} + 59049) q^{9}+O(q^{10})$$ q + (13*b4 + 5*b3) * q^2 + (65*b5 - 76*b2) * q^3 + (313*b6 + 221*b1 + 1024) * q^4 + (799*b6 + 1232*b1) * q^5 + (-562*b6 - 3611*b5 - 2161*b2 - 5229*b1) * q^6 + (-15540*b5 + 13312*b4 + 5120*b3 - 5099*b2) * q^8 + (-12554*b4 + 27087*b3 + 59049) * q^9 $$q + (13 \beta_{4} + 5 \beta_{3}) q^{2} + (65 \beta_{5} - 76 \beta_{2}) q^{3} + (313 \beta_{6} + 221 \beta_1 + 1024) q^{4} + (799 \beta_{6} + 1232 \beta_1) q^{5} + ( - 562 \beta_{6} - 3611 \beta_{5} - 2161 \beta_{2} - 5229 \beta_1) q^{6} + ( - 15540 \beta_{5} + 13312 \beta_{4} + 5120 \beta_{3} - 5099 \beta_{2}) q^{8} + ( - 12554 \beta_{4} + 27087 \beta_{3} + 59049) q^{9} + ( - 45012 \beta_{5} + 37321 \beta_{4} + 30434 \beta_{3} - 995 \beta_{2}) q^{10} + (82286 \beta_{6} + 66560 \beta_{5} - 48063 \beta_{4} + \cdots + 9635 \beta_1) q^{12}+ \cdots + (3672178237 \beta_{4} + 1412376245 \beta_{3}) q^{98}+O(q^{100})$$ q + (13*b4 + 5*b3) * q^2 + (65*b5 - 76*b2) * q^3 + (313*b6 + 221*b1 + 1024) * q^4 + (799*b6 + 1232*b1) * q^5 + (-562*b6 - 3611*b5 - 2161*b2 - 5229*b1) * q^6 + (-15540*b5 + 13312*b4 + 5120*b3 - 5099*b2) * q^8 + (-12554*b4 + 27087*b3 + 59049) * q^9 + (-45012*b5 + 37321*b4 + 30434*b3 - 995*b2) * q^10 + (82286*b6 + 66560*b5 - 48063*b4 - 134830*b3 - 77824*b2 + 9635*b1) * q^12 + (253463*b6 - 216858*b4 - 351529*b3 - 120328*b1) * q^15 + (320512*b6 + 374661*b5 - 204689*b2 + 226304*b1 + 1048576) * q^16 + (-461839*b6 + 767637*b4 + 295245*b3 + 1031285*b1 - 1435175) * q^18 + (177454*b4 + 1437263*b3) * q^19 + (818176*b6 + 1374477*b5 - 165881*b2 + 1261568*b1 + 6152849) * q^20 + (-575488*b6 - 3697664*b5 + 3257329*b4 + 85938*b3 - 2212864*b2 - 5354496*b1 - 9549623) * q^24 + (4575873*b5 + 932956*b2 + 9765625) * q^25 + (-3957361*b6 + 3838185*b5 - 4487724*b2 + 6773040*b1) * q^27 + (9247689*b5 + 6677764*b2) * q^29 + (-3880663*b6 - 10189748*b5 + 9283417*b4 - 3635454*b3 - 9516331*b2 - 14143347*b1 - 38721551) * q^30 + (-5106386*b6 - 15912960*b5 + 13631488*b4 + 5242880*b3 - 5221376*b2 - 19871485*b1) * q^32 + (18482337*b6 + 12070636*b5 - 18657275*b4 - 7175875*b3 + 31956461*b2 + 13049829*b1 + 60466176) * q^36 + (7102462*b4 + 39957095*b3) * q^37 + (-2572511*b6 + 56408165*b1 + 51229025) * q^38 + (-23413538*b6 - 46092288*b5 + 79987037*b4 + 30764245*b3 - 1018880*b2 - 47158597*b1) * q^40 + (31963639*b6 + 71205440*b1) * q^43 + (47180151*b6 + 57287137*b5 + 3279246*b4 - 105891241*b3 + 91281620*b2 + 72748368*b1) * q^45 + (84260864*b6 + 68157440*b5 - 124145099*b4 - 47748115*b3 - 79691776*b2 + 9866240*b1 + 493317625) * q^48 + 282475249 * q^49 + (-89829362*b6 - 122259987*b5 + 126953125*b4 + 48828125*b3 + 42649255*b2 - 91650253*b1) * q^50 + (-33185538*b6 - 93286375*b5 - 120471879*b4 + 184013762*b3 + 109073300*b2 - 308767221*b1) * q^54 + (-200702089*b6 - 34232135*b5 - 311514676*b2 + 2549160*b1) * q^57 + (-219880514*b6 - 146445315*b5 + 234753751*b2 + 25638635*b1) * q^58 + (259546112*b6 + 399935185*b5 - 503380163*b4 - 193607755*b3 - 467616524*b2 - 123215872*b1 + 1339993249) * q^60 + (328204288*b6 + 383652864*b5 - 298506479*b4 - 506445838*b3 - 209601536*b2 + 231735296*b1 + 1073741824) * q^64 - 1804229351 * q^71 + (-449209775*b6 - 620725495*b5 + 786060288*b4 + 302330880*b3 + 725771348*b2 - 317173675*b1 - 1469619200) * q^72 + (-544077089*b6 + 1049925560*b1) * q^73 + (-15121463*b6 + 1572531629*b1 + 1751742377) * q^74 + (634765625*b5 - 952892354*b4 + 135007359*b3 - 742187500*b2 + 3268452826) * q^75 + (-338074836*b5 + 665977325*b4 + 256145125*b3 + 1089949789*b2) * q^76 + (374158687*b6 - 2121571096*b1) * q^79 + (1925841737*b6 + 1407464448*b5 - 1148920967*b4 - 1179296446*b3 - 169862144*b2 + 1359779629*b1 + 6300517376) * q^80 + (-418499737*b6 - 741301146*b4 + 1599460263*b3 - 2197904104*b1 + 3486784401) * q^81 + (826932238*b4 - 1817523145*b3) * q^83 + (-1976043636*b5 + 1602609121*b4 + 1787414162*b3 + 354752389*b2) * q^86 + (-1978479338*b4 - 1078580505*b3 + 2757226450) * q^87 + (-698569239*b5 - 2955298964*b2) * q^89 + (-1146704825*b6 - 2516900815*b5 + 2203767729*b4 + 1797097266*b3 + 2942837876*b2 - 1768135600*b1 - 1304984951) * q^90 + (-21645687*b5 + 2196065254*b4 - 2866087009*b3 + 3860720380*b2) * q^95 + (-2989031999*b6 - 3786407936*b5 + 6413129125*b4 + 2466588125*b3 - 2265972736*b2 - 2110466683*b1 - 9778813952) * q^96 + (3672178237*b4 + 1412376245*b3) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$7 q + 7168 q^{4} + 413343 q^{9}+O(q^{10})$$ 7 * q + 7168 * q^4 + 413343 * q^9 $$7 q + 7168 q^{4} + 413343 q^{9} + 7340032 q^{16} - 10046225 q^{18} + 43069943 q^{20} - 66847361 q^{24} + 68359375 q^{25} - 271050857 q^{30} + 423263232 q^{36} + 358603175 q^{38} + 3453223375 q^{48} + 1977326743 q^{49} + 9379952743 q^{60} + 7516192768 q^{64} - 12629605457 q^{71} - 10287334400 q^{72} + 12262196639 q^{74} + 22879169782 q^{75} + 44103621632 q^{80} + 24407490807 q^{81} + 19300585150 q^{87} - 9134894657 q^{90} - 68451697664 q^{96}+O(q^{100})$$ 7 * q + 7168 * q^4 + 413343 * q^9 + 7340032 * q^16 - 10046225 * q^18 + 43069943 * q^20 - 66847361 * q^24 + 68359375 * q^25 - 271050857 * q^30 + 423263232 * q^36 + 358603175 * q^38 + 3453223375 * q^48 + 1977326743 * q^49 + 9379952743 * q^60 + 7516192768 * q^64 - 12629605457 * q^71 - 10287334400 * q^72 + 12262196639 * q^74 + 22879169782 * q^75 + 44103621632 * q^80 + 24407490807 * q^81 + 19300585150 * q^87 - 9134894657 * q^90 - 68451697664 * q^96

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{7} - 14x^{5} + 56x^{3} - 56x - 21$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 4$$ v^2 - 4 $$\beta_{3}$$ $$=$$ $$\nu^{4} - \nu^{3} - 8\nu^{2} + 6\nu + 8$$ v^4 - v^3 - 8*v^2 + 6*v + 8 $$\beta_{4}$$ $$=$$ $$-\nu^{4} + 2\nu^{3} + 8\nu^{2} - 12\nu - 8$$ -v^4 + 2*v^3 + 8*v^2 - 12*v - 8 $$\beta_{5}$$ $$=$$ $$\nu^{5} - 10\nu^{3} - 3\nu^{2} + 20\nu + 12$$ v^5 - 10*v^3 - 3*v^2 + 20*v + 12 $$\beta_{6}$$ $$=$$ $$\nu^{6} - 12\nu^{4} + 36\nu^{2} - 5\nu - 16$$ v^6 - 12*v^4 + 36*v^2 - 5*v - 16
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 4$$ b2 + 4 $$\nu^{3}$$ $$=$$ $$\beta_{4} + \beta_{3} + 6\beta_1$$ b4 + b3 + 6*b1 $$\nu^{4}$$ $$=$$ $$\beta_{4} + 2\beta_{3} + 8\beta_{2} + 24$$ b4 + 2*b3 + 8*b2 + 24 $$\nu^{5}$$ $$=$$ $$\beta_{5} + 10\beta_{4} + 10\beta_{3} + 3\beta_{2} + 40\beta_1$$ b5 + 10*b4 + 10*b3 + 3*b2 + 40*b1 $$\nu^{6}$$ $$=$$ $$\beta_{6} + 12\beta_{4} + 24\beta_{3} + 60\beta_{2} + 5\beta _1 + 160$$ b6 + 12*b4 + 24*b3 + 60*b2 + 5*b1 + 160

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/71\mathbb{Z}\right)^\times$$.

 $$n$$ $$7$$ $$\chi(n)$$ $$-1$$

## Embeddings

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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
70.1
 1.64039 −2.61140 1.88136 −0.478208 −2.47768 2.82423 −0.778691
−63.7034 390.171 3034.12 6226.80 −24855.2 0 −128052. 93184.2 −396668.
70.2 −44.5303 −477.255 958.945 −1910.09 21252.3 0 2896.92 168724. 85056.9
70.3 −34.9065 −225.808 194.467 −861.095 7882.18 0 28956.1 −8059.70 30057.9
70.4 8.17505 469.814 −957.169 −5376.73 3840.75 0 −16196.1 161676. −43955.0
70.5 20.1756 16.7214 −616.944 −5843.58 337.364 0 −33107.1 −58769.4 −117898.
70.6 54.7244 −369.320 1970.76 4302.96 −20210.8 0 51810.8 77347.9 235477.
70.7 60.0651 195.677 2583.82 3461.73 11753.4 0 93690.9 −20759.4 207929.
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 70.7 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
71.b odd 2 1 CM by $$\Q(\sqrt{-71})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 71.11.b.a 7
71.b odd 2 1 CM 71.11.b.a 7

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
71.11.b.a 7 1.a even 1 1 trivial
71.11.b.a 7 71.b odd 2 1 CM

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{7} - 7168T_{2}^{5} + 14680064T_{2}^{3} - 7516192768T_{2} + 53684057575$$ acting on $$S_{11}^{\mathrm{new}}(71, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{7} - 7168 T^{5} + \cdots + 53684057575$$
$3$ $$T^{7} - 413343 T^{5} + \cdots + 23\!\cdots\!50$$
$5$ $$T^{7} - 68359375 T^{5} + \cdots - 47\!\cdots\!74$$
$7$ $$T^{7}$$
$11$ $$T^{7}$$
$13$ $$T^{7}$$
$17$ $$T^{7}$$
$19$ $$T^{7} - 42917463804607 T^{5} + \cdots - 18\!\cdots\!98$$
$23$ $$T^{7}$$
$29$ $$T^{7} + \cdots + 45\!\cdots\!02$$
$31$ $$T^{7}$$
$37$ $$T^{7} + \cdots + 56\!\cdots\!50$$
$41$ $$T^{7}$$
$43$ $$T^{7} + \cdots + 47\!\cdots\!50$$
$47$ $$T^{7}$$
$53$ $$T^{7}$$
$59$ $$T^{7}$$
$61$ $$T^{7}$$
$67$ $$T^{7}$$
$71$ $$(T + 1804229351)^{7}$$
$73$ $$T^{7} + \cdots - 30\!\cdots\!50$$
$79$ $$T^{7} + \cdots - 51\!\cdots\!98$$
$83$ $$T^{7} + \cdots - 26\!\cdots\!50$$
$89$ $$T^{7} + \cdots + 15\!\cdots\!98$$
$97$ $$T^{7}$$