LMFDB - Newform orbit 200.3.i.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [200,3,Mod(93,200)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("200.93"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(200, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 2, 3])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 200 = 2^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	200.i (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$5.44960528721$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 12x^{14} + 58x^{12} + 132x^{10} - 51x^{8} - 1128x^{6} + 1372x^{4} - 96x^{2} + 100 $ x^16 + 12x^14 + 58x^12 + 132x^10 - 51x^8 - 1128x^6 + 1372x^4 - 96*x^2 + 100
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 2^{22} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-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 $1,\beta_1,\ldots,\beta_{15}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q - \beta_{7} q^{2} + \beta_{11} q^{3} + (\beta_{15} + \beta_{5}) q^{4} + \beta_{6} q^{6} + (\beta_{13} + 2 \beta_{7} - \beta_{3}) q^{7} + ( - \beta_{14} - \beta_{11} + \beta_{8}) q^{8} + (2 \beta_{15} + \beta_{9} + 3 \beta_{5}) q^{9}+ \cdots + (20 \beta_{15} + 14 \beta_{12} - 3 \beta_{9}) q^{99}+O(q^{100}) $ q - b7 * q^2 + b11 * q^3 + (b15 + b5) * q^4 + b6 * q^6 + (b13 + 2b7 - b3) q^7 + (-b14 - b11 + b8) * q^8 + (2b15 + b9 + 3b5) * q^9 + (2b6 + b1) q^11 + (b13 - 3b10 + b7 - 2b3) * q^12 + (2b11 - 2b8 + 2b2) q^13 + (-2b15 - b12 - 2b9 - 6b5) q^14 + (-2b6 + 2b4 - 2b1 + 2) q^16 + (2b13 - 2b7) * q^17 + (-2b14 - 2b11 + 5b8 + 4b2) * q^18 + (-4b15 - 2b12 + b9) * q^19 + (-4b6 + 4b4) * q^21 + (2b13 - 6b10) * q^22 + (-b14 - 10b8 - 3b2) * q^23 + (2b12 - 2b9 - 12b5) q^24 + (2b6 - 4b4 - 12) * q^26 + (-2b10 - 4b7 - 4b3) q^27 + (b14 + 5b11 - 9b8 - 6b2) q^28 + (-4b15 + 4b12 + 4b9) q^29 + (-8b4 + 4b1 + 8) * q^31 + (8b10 - 4b3) * q^32 + (-2b14 + 22b8 + 8b2) q^33 + (-2b12 - 4b9 + 8b5) q^34 + (-4b6 + 3b4 - 4b1 - 27) q^36 + (-4b10 + 4b7 + 4b3) q^37 + (2b14 + 10b11 + 4b8 + 8b2) * q^38 + (8b15 + 4b9 + 24b5) q^39 + (-2b4 + b1 + 6) q^41 + (16b10 - 4b7 + 8b3) q^42 + (-5b11 + 12b8 - 12b2) q^43 + (-2b15 + 4b12 - 4b9 + 6b5) * q^44 + (-b6 - 6b4 - 2b1 + 34) * q^46 + (-3b13 + 18b7 - 5b3) q^47 + (2b14 - 6b11 - 18b8 - 12b2) * q^48 + (6b15 + 3b9 + 11b5) q^49 + (4b4 + 2b1) * q^51 + (-2b13 - 10b10 + 14b7 - 4b3) * q^52 + (-14b11 - 10b8 + 10b2) q^53 + (8b15 + 2b12 - 24b5) q^54 + (6b6 - 4b4 + 2b1 + 48) q^56 + (-10b13 - 38b7 + 16b3) q^57 + (8b14 - 8b11 + 4b8 + 8b2) * q^58 + (-4b15 + 2b12 + 3b9) q^59 + (8b6 + 4b1) * q^61 + (-8b13 - 8b10 - 16b7 + 16b3) * q^62 + (3b14 - 42b8 - 15b2) q^63 + (4b15 - 8b12 - 28b5) q^64 + (-2b6 + 16b4 - 4b1 - 72) q^66 + (7b10 - 4b7 - 4b3) q^67 + (-2b14 + 6b11 + 2b8 - 12b2) * q^68 + (-4b15 - 12b12 - 4b9) q^69 + (16b4 - 8b1) * q^71 + (-b13 + 15b10 + 31b7 - 8b3) q^72 + (10b14 + 34b8 + 8b2) q^73 + (-8b15 + 4b12 + 24b5) q^74 + (12b6 - 6b4 + 4b1 - 66) q^76 + (22b10 + 14b7 + 14b3) q^77 + (-8b14 - 8b11 + 32b8 + 16b2) * q^78 + (-8b15 - 4b9 - 56b5) q^79 + (-6b4 + 3b1 + 3) * q^81 + (-2b13 - 2b10 - 8b7 + 4b3) * q^82 + (5b11 + 12b8 - 12b2) q^83 + (-4b15 - 16b12 + 60b5) q^84 + (-5b6 + 24b4 + 72) * q^86 + (-4b13 + 28b7 - 8b3) q^87 + (6b14 - 10b11 - 6b8 - 24b2) * q^88 + (-8b15 - 4b9 - 6b5) q^89 + (-16b6 + 4b4 - 6b1) q^91 + (-7b13 - 3b10 - 31b7 - 6b3) * q^92 + (40b11 - 16b8 + 16b2) q^93 + (-10b15 + 3b12 + 6b9 - 62b5) * q^94 + (-4b6 - 8b4 + 4b1 + 96) q^96 + (14b13 + 10b7 - 8b3) q^97 + (-6b14 - 6b11 + 17b8 + 12b2) * q^98 + (20b15 + 14b12 - 3b9) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 32 q^{16} - 192 q^{26} + 128 q^{31} - 432 q^{36} + 96 q^{41} + 544 q^{46} + 768 q^{56} - 1152 q^{66} - 1056 q^{76} + 48 q^{81} + 1152 q^{86} + 1536 q^{96}+O(q^{100}) $ 16 * q + 32 * q^16 - 192 * q^26 + 128 * q^31 - 432 * q^36 + 96 * q^41 + 544 * q^46 + 768 * q^56 - 1152 * q^66 - 1056 * q^76 + 48 * q^81 + 1152 * q^86 + 1536 * q^96

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} + 12x^{14} + 58x^{12} + 132x^{10} - 51x^{8} - 1128x^{6} + 1372x^{4} - 96x^{2} + 100 $ x^16 + 12*x^14 + 58*x^12 + 132*x^10 - 51*x^8 - 1128*x^6 + 1372*x^4 - 96*x^2 + 100 :

$\beta_{1}$	$=$	$ ( - 1317 \nu^{14} + 4229 \nu^{12} + 180623 \nu^{10} + 1209999 \nu^{8} + 3991926 \nu^{6} + \cdots + 7514640 ) / 1444370 $ (-1317v^14 + 4229v^12 + 180623v^10 + 1209999v^8 + 3991926v^6 + 4674952v^4 - 21370352*v^2 + 7514640) / 1444370
$\beta_{2}$	$=$	$ ( 32443 \nu^{15} + 42979 \nu^{14} + 367409 \nu^{13} + 333577 \nu^{12} + 1480453 \nu^{11} + \cdots - 70770100 ) / 46219840 $ (32443v^15 + 42979v^14 + 367409v^13 + 333577v^12 + 1480453v^11 + 35469v^10 + 1357039v^9 - 8322953v^8 - 12691624v^7 - 44627032v^6 - 55839488v^5 - 93239104v^4 + 62951668v^3 + 187694644v^2 + 151116460*v - 70770100) / 46219840
$\beta_{3}$	$=$	$ ( 32443 \nu^{15} - 42979 \nu^{14} + 367409 \nu^{13} - 333577 \nu^{12} + 1480453 \nu^{11} + \cdots + 70770100 ) / 46219840 $ (32443v^15 - 42979v^14 + 367409v^13 - 333577v^12 + 1480453v^11 - 35469v^10 + 1357039v^9 + 8322953v^8 - 12691624v^7 + 44627032v^6 - 55839488v^5 + 93239104v^4 + 62951668v^3 - 187694644v^2 + 151116460*v + 70770100) / 46219840
$\beta_{4}$	$=$	$ ( - 72993 \nu^{14} - 867809 \nu^{12} - 4100603 \nu^{10} - 8495379 \nu^{8} + 9503944 \nu^{6} + \cdots - 20658220 ) / 11554960 $ (-72993v^14 - 867809v^12 - 4100603v^10 - 8495379v^8 + 9503944v^6 + 98215408v^4 - 86682828*v^2 - 20658220) / 11554960
$\beta_{5}$	$=$	$ ( 46259 \nu^{15} + 556425 \nu^{13} + 2678793 \nu^{11} + 5925565 \nu^{9} - 3569208 \nu^{7} + \cdots + 11152008 \nu ) / 11554960 $ (46259v^15 + 556425v^13 + 2678793v^11 + 5925565v^9 - 3569208v^7 - 56172078v^5 + 58792396v^3 + 11152008v) / 11554960
$\beta_{6}$	$=$	$ ( 243793 \nu^{14} + 2890049 \nu^{12} + 13726363 \nu^{10} + 30213459 \nu^{8} - 15924344 \nu^{6} + \cdots - 49960260 ) / 11554960 $ (243793v^14 + 2890049v^12 + 13726363v^10 + 30213459v^8 - 15924344v^6 - 269012528v^4 + 386081068*v^2 - 49960260) / 11554960
$\beta_{7}$	$=$	$ ( - 337629 \nu^{15} - 32443 \nu^{14} - 4083991 \nu^{13} - 367409 \nu^{12} - 19949891 \nu^{11} + \cdots - 58676780 ) / 46219840 $ (-337629v^15 - 32443v^14 - 4083991v^13 - 367409v^12 - 19949891v^11 - 1480453v^10 - 46047481v^9 - 1357039v^8 + 15862040v^7 + 12691624v^6 + 393537136v^5 + 55839488v^4 - 407387500v^3 - 62951668v^2 - 30539284*v - 58676780) / 46219840
$\beta_{8}$	$=$	$ ( 337629 \nu^{15} - 32443 \nu^{14} + 4083991 \nu^{13} - 367409 \nu^{12} + 19949891 \nu^{11} + \cdots - 58676780 ) / 46219840 $ (337629v^15 - 32443v^14 + 4083991v^13 - 367409v^12 + 19949891v^11 - 1480453v^10 + 46047481v^9 - 1357039v^8 - 15862040v^7 + 12691624v^6 - 393537136v^5 + 55839488v^4 + 407387500v^3 - 62951668v^2 + 30539284*v - 58676780) / 46219840
$\beta_{9}$	$=$	$ ( 43625 \nu^{15} + 564883 \nu^{13} + 3040039 \nu^{11} + 8345563 \nu^{9} + 4414644 \nu^{7} + \cdots + 66623648 \nu ) / 5777480 $ (43625v^15 + 564883v^13 + 3040039v^11 + 8345563v^9 + 4414644v^7 - 46822174v^5 + 27606652v^3 + 66623648v) / 5777480
$\beta_{10}$	$=$	$ ( - 379263 \nu^{15} - 415681 \nu^{14} - 4343801 \nu^{13} - 5012983 \nu^{12} - 19460689 \nu^{11} + \cdots + 36413260 ) / 46219840 $ (-379263v^15 - 415681v^14 - 4343801v^13 - 5012983v^12 - 19460689v^11 - 24429031v^10 - 37392831v^9 - 56910113v^8 + 49529968v^7 + 14429768v^6 + 423317264v^5 + 457235856v^4 - 760169236v^3 - 567044956v^2 + 243929268*v + 36413260) / 46219840
$\beta_{11}$	$=$	$ ( - 379263 \nu^{15} + 415681 \nu^{14} - 4343801 \nu^{13} + 5012983 \nu^{12} - 19460689 \nu^{11} + \cdots - 36413260 ) / 46219840 $ (-379263v^15 + 415681v^14 - 4343801v^13 + 5012983v^12 - 19460689v^11 + 24429031v^10 - 37392831v^9 + 56910113v^8 + 49529968v^7 - 14429768v^6 + 423317264v^5 - 457235856v^4 - 760169236v^3 + 567044956v^2 + 243929268*v - 36413260) / 46219840
$\beta_{12}$	$=$	$ ( 67735 \nu^{15} + 726876 \nu^{13} + 2867163 \nu^{11} + 3589686 \nu^{9} - 16802812 \nu^{7} + \cdots - 96984504 \nu ) / 5777480 $ (67735v^15 + 726876v^13 + 2867163v^11 + 3589686v^9 - 16802812v^7 - 77152368v^5 + 187044084v^3 - 96984504v) / 5777480
$\beta_{13}$	$=$	$ ( - 171321 \nu^{15} - 66697 \nu^{14} - 2132322 \nu^{13} - 847996 \nu^{12} - 10936383 \nu^{11} + \cdots - 23806080 ) / 11554960 $ (-171321v^15 - 66697v^14 - 2132322v^13 - 847996v^12 - 10936383v^11 - 4432177v^10 - 28120072v^9 - 11557266v^8 - 7603554v^7 - 2808604v^6 + 176163888v^5 + 81997132v^4 - 175222032v^3 - 16798972v^2 - 20627744*v - 23806080) / 11554960
$\beta_{14}$	$=$	$ ( - 171321 \nu^{15} + 66697 \nu^{14} - 2132322 \nu^{13} + 847996 \nu^{12} - 10936383 \nu^{11} + \cdots + 23806080 ) / 11554960 $ (-171321v^15 + 66697v^14 - 2132322v^13 + 847996v^12 - 10936383v^11 + 4432177v^10 - 28120072v^9 + 11557266v^8 - 7603554v^7 + 2808604v^6 + 176163888v^5 - 81997132v^4 - 175222032v^3 + 16798972v^2 - 20627744*v + 23806080) / 11554960
$\beta_{15}$	$=$	$ ( 218377 \nu^{15} + 2659757 \nu^{13} + 13170495 \nu^{11} + 31626537 \nu^{9} - 2788888 \nu^{7} + \cdots - 12825904 \nu ) / 11554960 $ (218377v^15 + 2659757v^13 + 13170495v^11 + 31626537v^9 - 2788888v^7 - 239149650v^5 + 261912276v^3 - 12825904v) / 11554960

Display $\nu^j$ in terms of $\beta_i$

$\nu$	$=$	$ ( \beta_{8} - \beta_{7} - 4\beta_{5} + \beta_{3} + \beta_{2} ) / 4 $ (b8 - b7 - 4*b5 + b3 + b2) / 4
$\nu^{2}$	$=$	$ ( -2\beta_{8} - 2\beta_{7} + 2\beta_{3} - 2\beta_{2} - \beta _1 - 6 ) / 4 $ (-2b8 - 2b7 + 2b3 - 2b2 - b1 - 6) / 4
$\nu^{3}$	$=$	$ ( \beta_{14} + \beta_{13} - \beta_{11} - \beta_{10} + 3 \beta_{9} - 3 \beta_{8} + 3 \beta_{7} + \cdots - 4 \beta_{2} ) / 4 $ (b14 + b13 - b11 - b10 + 3b9 - 3b8 + 3b7 + 10b5 - 4b3 - 4b2) / 4
$\nu^{4}$	$=$	$ ( - 2 \beta_{14} + 2 \beta_{13} + 2 \beta_{11} - 2 \beta_{10} + 2 \beta_{8} + 2 \beta_{7} - \beta_{6} + \cdots + 7 ) / 2 $ (-2b14 + 2b13 + 2b11 - 2b10 + 2b8 + 2b7 - b6 - b4 - 4b3 + 4b2 + 3*b1 + 7) / 2
$\nu^{5}$	$=$	$ ( - 5 \beta_{15} - 5 \beta_{14} - 5 \beta_{13} + 5 \beta_{12} + 7 \beta_{11} + 7 \beta_{10} + \cdots + 7 \beta_{2} ) / 2 $ (-5b15 - 5b14 - 5b13 + 5b12 + 7b11 + 7b10 - 5b9 + 4b8 - 4b7 - 7b5 + 7b3 + 7b2) / 2
$\nu^{6}$	$=$	$ ( 10 \beta_{14} - 10 \beta_{13} - 22 \beta_{11} + 22 \beta_{10} - 12 \beta_{8} - 12 \beta_{7} + 17 \beta_{6} + \cdots - 9 ) / 2 $ (10b14 - 10b13 - 22b11 + 22b10 - 12b8 - 12b7 + 17b6 + 13b4 + 10b3 - 10b2 - 6*b1 - 9) / 2
$\nu^{7}$	$=$	$ ( 21 \beta_{15} + 16 \beta_{14} + 16 \beta_{13} - 49 \beta_{12} - 62 \beta_{11} - 62 \beta_{10} + \cdots - 12 \beta_{2} ) / 2 $ (21b15 + 16b14 + 16b13 - 49b12 - 62b11 - 62b10 - 34b8 + 34b7 + 37b5 - 12b3 - 12*b2) / 2
$\nu^{8}$	$=$	$ ( - 16 \beta_{14} + 16 \beta_{13} + 160 \beta_{11} - 160 \beta_{10} + 80 \beta_{8} + 80 \beta_{7} + \cdots + 149 ) / 2 $ (-16b14 + 16b13 + 160b11 - 160b10 + 80b8 + 80b7 - 127b6 - 7b4 - 16b3 + 16b2 - 27*b1 + 149) / 2
$\nu^{9}$	$=$	$ ( 105 \beta_{15} + 11 \beta_{14} + 11 \beta_{13} + 303 \beta_{12} + 387 \beta_{11} + 387 \beta_{10} + \cdots + 38 \beta_{2} ) / 2 $ (105b15 + 11b14 + 11b13 + 303b12 + 387b11 + 387b10 + 99b9 + 143b8 - 143b7 - 437b5 + 38b3 + 38b2) / 2
$\nu^{10}$	$=$	$ ( - 110 \beta_{14} + 110 \beta_{13} - 894 \beta_{11} + 894 \beta_{10} - 118 \beta_{8} - 118 \beta_{7} + \cdots - 933 ) / 2 $ (-110b14 + 110b13 - 894b11 + 894b10 - 118b8 - 118b7 + 679b6 - 525b4 + 124b3 - 124b2 + 249*b1 - 933) / 2
$\nu^{11}$	$=$	$ ( - 1749 \beta_{15} - 359 \beta_{14} - 359 \beta_{13} - 1463 \beta_{12} - 2003 \beta_{11} + \cdots - 364 \beta_{2} ) / 2 $ (-1749b15 - 359b14 - 359b13 - 1463b12 - 2003b11 - 2003b10 - 517b9 + 443b8 - 443b7 + 1157b5 - 364b3 - 364b2) / 2
$\nu^{12}$	$=$	$ ( 876 \beta_{14} - 876 \beta_{13} + 4412 \beta_{11} - 4412 \beta_{10} - 2956 \beta_{8} - 2956 \beta_{7} + \cdots - 1343 ) / 2 $ (876b14 - 876b13 + 4412b11 - 4412b10 - 2956b8 - 2956b7 - 3107b6 + 4829b4 - 832b3 + 832b2 - 975*b1 - 1343) / 2
$\nu^{13}$	$=$	$ ( 11869 \beta_{15} + 1851 \beta_{14} + 1851 \beta_{13} + 6643 \beta_{12} + 9651 \beta_{11} + \cdots + 1216 \beta_{2} ) / 2 $ (11869b15 + 1851b14 + 1851b13 + 6643b12 + 9651b11 + 9651b10 + 1911b9 - 11255b8 + 11255b7 + 14831b5 + 1216b3 + 1216b2) / 2
$\nu^{14}$	$=$	$ ( - 3762 \beta_{14} + 3762 \beta_{13} - 21026 \beta_{11} + 21026 \beta_{10} + 34778 \beta_{8} + \cdots + 62283 ) / 2 $ (-3762b14 + 3762b13 - 21026b11 + 21026b10 + 34778b8 + 34778b7 + 14443b6 - 27073b4 - 480b3 + 480b2 + 4595*b1 + 62283) / 2
$\nu^{15}$	$=$	$ ( - 59385 \beta_{15} - 8357 \beta_{14} - 8357 \beta_{13} - 31707 \beta_{12} - 45317 \beta_{11} + \cdots + 11580 \beta_{2} ) / 2 $ (-59385b15 - 8357b14 - 8357b13 - 31707b12 - 45317b11 - 45317b10 - 13707b9 + 95429b8 - 95429b7 - 200435b5 + 11580b3 + 11580b2) / 2

Display $\beta_i$ in terms of $\nu^j$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/200\mathbb{Z}\right)^\times$.

$n$	$101$	$151$	$177$
$\chi(n)$	$-1$	$1$	$\beta_{5}$

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} $

93.1

1.16721	+	1.79852i
0.370982	+	2.36469i
−0.370982	+	2.36469i
1.16721	+	0.201484i
−1.16721	+	1.79852i
0.370982	−	0.364688i
−0.370982	−	0.364688i
−1.16721	+	0.201484i
1.16721	−	1.79852i
0.370982	−	2.36469i
−0.370982	−	2.36469i
1.16721	−	0.201484i
−1.16721	−	1.79852i
0.370982	+	0.364688i
−0.370982	+	0.364688i
−1.16721	−	0.201484i

−1.96572

−

0.368691i

−1.04930

1.04930i

3.72813

1.44949i

2.44949

−

1.67576i

−3.91191

3.91191i

−6.79406

−

4.22383i

6.79796i

93.2

−1.73567

0.993706i

3.30136

−

3.30136i

2.02510

−

3.44949i

−2.44949

9.01065i

6.68558

−

6.68558i

−0.0871225

7.99953i

−

12.7980i

93.3

−0.993706

1.73567i

−3.30136

3.30136i

−2.02510

−

3.44949i

−2.44949

−

9.01065i

6.68558

−

6.68558i

7.99953

−

0.0871225i

−

12.7980i

93.4

−0.368691

−

1.96572i

−1.04930

1.04930i

−3.72813

1.44949i

2.44949

1.67576i

3.91191

−

3.91191i

4.22383

6.79406i

6.79796i

93.5

0.368691

1.96572i

1.04930

−

1.04930i

−3.72813

1.44949i

2.44949

1.67576i

−3.91191

3.91191i

−4.22383

−

6.79406i

6.79796i

93.6

0.993706

−

1.73567i

3.30136

−

3.30136i

−2.02510

−

3.44949i

−2.44949

−

9.01065i

−6.68558

6.68558i

−7.99953

0.0871225i

−

12.7980i

93.7

1.73567

−

0.993706i

−3.30136

3.30136i

2.02510

−

3.44949i

−2.44949

9.01065i

−6.68558

6.68558i

0.0871225

−

7.99953i

−

12.7980i

93.8

1.96572

0.368691i

1.04930

−

1.04930i

3.72813

1.44949i

2.44949

−

1.67576i

3.91191

−

3.91191i

6.79406

4.22383i

6.79796i

157.1

−1.96572

0.368691i

−1.04930

−

1.04930i

3.72813

−

1.44949i

2.44949

1.67576i

−3.91191

−

3.91191i

−6.79406

4.22383i

−

6.79796i

157.2

−1.73567

−

0.993706i

3.30136

3.30136i

2.02510

3.44949i

−2.44949

−

9.01065i

6.68558

6.68558i

−0.0871225

−

7.99953i

12.7980i

157.3

−0.993706

−

1.73567i

−3.30136

−

3.30136i

−2.02510

3.44949i

−2.44949

9.01065i

6.68558

6.68558i

7.99953

0.0871225i

12.7980i

157.4

−0.368691

1.96572i

−1.04930

−

1.04930i

−3.72813

−

1.44949i

2.44949

−

1.67576i

3.91191

3.91191i

4.22383

−

6.79406i

−

6.79796i

157.5

0.368691

−

1.96572i

1.04930

1.04930i

−3.72813

−

1.44949i

2.44949

−

1.67576i

−3.91191

−

3.91191i

−4.22383

6.79406i

−

6.79796i

157.6

0.993706

1.73567i

3.30136

3.30136i

−2.02510

3.44949i

−2.44949

9.01065i

−6.68558

−

6.68558i

−7.99953

−

0.0871225i

12.7980i

157.7

1.73567

0.993706i

−3.30136

−

3.30136i

2.02510

3.44949i

−2.44949

−

9.01065i

−6.68558

−

6.68558i

0.0871225

7.99953i

12.7980i

157.8

1.96572

−

0.368691i

1.04930

1.04930i

3.72813

−

1.44949i

2.44949

1.67576i

3.91191

3.91191i

6.79406

−

4.22383i

−

6.79796i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
5.c	odd	4	2	inner
8.b	even	2	1	inner
40.f	even	2	1	inner
40.i	odd	4	2	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	200.3.i.a	✓	16
4.b	odd	2	1		800.3.m.a		16
5.b	even	2	1	inner	200.3.i.a	✓	16
5.c	odd	4	2	inner	200.3.i.a	✓	16
8.b	even	2	1	inner	200.3.i.a	✓	16
8.d	odd	2	1		800.3.m.a		16
20.d	odd	2	1		800.3.m.a		16
20.e	even	4	2		800.3.m.a		16
40.e	odd	2	1		800.3.m.a		16
40.f	even	2	1	inner	200.3.i.a	✓	16
40.i	odd	4	2	inner	200.3.i.a	✓	16
40.k	even	4	2		800.3.m.a		16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
200.3.i.a	✓	16	1.a	even	1	1	trivial
200.3.i.a	✓	16	5.b	even	2	1	inner
200.3.i.a	✓	16	5.c	odd	4	2	inner
200.3.i.a	✓	16	8.b	even	2	1	inner
200.3.i.a	✓	16	40.f	even	2	1	inner
200.3.i.a	✓	16	40.i	odd	4	2	inner
800.3.m.a		16	4.b	odd	2	1
800.3.m.a		16	8.d	odd	2	1
800.3.m.a		16	20.d	odd	2	1
800.3.m.a		16	20.e	even	4	2
800.3.m.a		16	40.e	odd	2	1
800.3.m.a		16	40.k	even	4	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3}^{8} + 480T_{3}^{4} + 2304 $ T3^8 + 480*T3^4 + 2304 acting on $S_{3}^{\mathrm{new}}(200, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} - 8 T^{12} + \cdots + 65536 $ T^16 - 8T^12 + 144T^8 - 2048*T^4 + 65536
$3$	$ (T^{8} + 480 T^{4} + 2304)^{2} $ (T^8 + 480*T^4 + 2304)^2
$5$	$ T^{16} $ T^16
$7$	$ (T^{8} + 8928 T^{4} + 7485696)^{2} $ (T^8 + 8928*T^4 + 7485696)^2
$11$	$ (T^{4} + 312 T^{2} + 22800)^{4} $ (T^4 + 312*T^2 + 22800)^4
$13$	$ (T^{8} + 44544 T^{4} + 368640000)^{2} $ (T^8 + 44544*T^4 + 368640000)^2
$17$	$ (T^{8} + 163328 T^{4} + 23658496)^{2} $ (T^8 + 163328*T^4 + 23658496)^2
$19$	$ (T^{4} - 1272 T^{2} + 329232)^{4} $ (T^4 - 1272*T^2 + 329232)^4
$23$	$ (T^{8} + 520928 T^{4} + 65364080896)^{2} $ (T^8 + 520928*T^4 + 65364080896)^2
$29$	$ (T^{4} - 2784 T^{2} + 364800)^{4} $ (T^4 - 2784*T^2 + 364800)^4
$31$	$ (T^{2} - 16 T - 1472)^{8} $ (T^2 - 16*T - 1472)^8
$37$	$ (T^{8} + 1105920 T^{4} + 12230590464)^{2} $ (T^8 + 1105920*T^4 + 12230590464)^2
$41$	$ (T^{2} - 12 T - 60)^{8} $ (T^2 - 12*T - 60)^8
$43$	$ (T^{8} + \cdots + 147487228602624)^{2} $ (T^8 + 32141280*T^4 + 147487228602624)^2
$47$	$ (T^{8} + \cdots + 9616846018816)^{2} $ (T^8 + 8976608*T^4 + 9616846018816)^2
$53$	$ (T^{8} + \cdots + 220181913206784)^{2} $ (T^8 + 60648960*T^4 + 220181913206784)^2
$59$	$ (T^{4} - 1656 T^{2} + 8208)^{4} $ (T^4 - 1656*T^2 + 8208)^4
$61$	$ (T^{4} + 4992 T^{2} + 5836800)^{4} $ (T^4 + 4992*T^2 + 5836800)^4
$67$	$ (T^{8} + 2181600 T^{4} + 638030317824)^{2} $ (T^8 + 2181600*T^4 + 638030317824)^2
$71$	$ (T^{2} - 6144)^{8} $ (T^2 - 6144)^8
$73$	$ (T^{8} + \cdots + 13\!\cdots\!56)^{2} $ (T^8 + 352378368*T^4 + 1355389581459456)^2
$79$	$ (T^{4} + 9344 T^{2} + 2560000)^{4} $ (T^4 + 9344*T^2 + 2560000)^4
$83$	$ (T^{8} + 56379360 T^{4} + 644753664)^{2} $ (T^8 + 56379360*T^4 + 644753664)^2
$89$	$ (T^{4} + 3144 T^{2} + 2250000)^{4} $ (T^4 + 3144*T^2 + 2250000)^4
$97$	$ (T^{8} + \cdots + 65\!\cdots\!76)^{2} $ (T^8 + 174325248*T^4 + 6551578872446976)^2
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Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 200 = 2^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	200.i (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\( q - \beta_{7} q^{2} + \beta_{11} q^{3} + (\beta_{15} + \beta_{5}) q^{4} + \beta_{6} q^{6} + (\beta_{13} + 2 \beta_{7} - \beta_{3}) q^{7} + ( - \beta_{14} - \beta_{11} + \beta_{8}) q^{8} + (2 \beta_{15} + \beta_{9} + 3 \beta_{5}) q^{9}+ \cdots + (20 \beta_{15} + 14 \beta_{12} - 3 \beta_{9}) q^{99}+O(q^{100}) \) q - b7 * q^2 + b11 * q^3 + (b15 + b5) * q^4 + b6 * q^6 + (b13 + 2b7 - b3) q^7 + (-b14 - b11 + b8) * q^8 + (2b15 + b9 + 3b5) * q^9 + (2b6 + b1) q^11 + (b13 - 3b10 + b7 - 2b3) * q^12 + (2b11 - 2b8 + 2b2) q^13 + (-2b15 - b12 - 2b9 - 6b5) q^14 + (-2b6 + 2b4 - 2b1 + 2) q^16 + (2b13 - 2b7) * q^17 + (-2b14 - 2b11 + 5b8 + 4b2) * q^18 + (-4b15 - 2b12 + b9) * q^19 + (-4b6 + 4b4) * q^21 + (2b13 - 6b10) * q^22 + (-b14 - 10b8 - 3b2) * q^23 + (2b12 - 2b9 - 12b5) q^24 + (2b6 - 4b4 - 12) * q^26 + (-2b10 - 4b7 - 4b3) q^27 + (b14 + 5b11 - 9b8 - 6b2) q^28 + (-4b15 + 4b12 + 4b9) q^29 + (-8b4 + 4b1 + 8) * q^31 + (8b10 - 4b3) * q^32 + (-2b14 + 22b8 + 8b2) q^33 + (-2b12 - 4b9 + 8b5) q^34 + (-4b6 + 3b4 - 4b1 - 27) q^36 + (-4b10 + 4b7 + 4b3) q^37 + (2b14 + 10b11 + 4b8 + 8b2) * q^38 + (8b15 + 4b9 + 24b5) q^39 + (-2b4 + b1 + 6) q^41 + (16b10 - 4b7 + 8b3) q^42 + (-5b11 + 12b8 - 12b2) q^43 + (-2b15 + 4b12 - 4b9 + 6b5) * q^44 + (-b6 - 6b4 - 2b1 + 34) * q^46 + (-3b13 + 18b7 - 5b3) q^47 + (2b14 - 6b11 - 18b8 - 12b2) * q^48 + (6b15 + 3b9 + 11b5) q^49 + (4b4 + 2b1) * q^51 + (-2b13 - 10b10 + 14b7 - 4b3) * q^52 + (-14b11 - 10b8 + 10b2) q^53 + (8b15 + 2b12 - 24b5) q^54 + (6b6 - 4b4 + 2b1 + 48) q^56 + (-10b13 - 38b7 + 16b3) q^57 + (8b14 - 8b11 + 4b8 + 8b2) * q^58 + (-4b15 + 2b12 + 3b9) q^59 + (8b6 + 4b1) * q^61 + (-8b13 - 8b10 - 16b7 + 16b3) * q^62 + (3b14 - 42b8 - 15b2) q^63 + (4b15 - 8b12 - 28b5) q^64 + (-2b6 + 16b4 - 4b1 - 72) q^66 + (7b10 - 4b7 - 4b3) q^67 + (-2b14 + 6b11 + 2b8 - 12b2) * q^68 + (-4b15 - 12b12 - 4b9) q^69 + (16b4 - 8b1) * q^71 + (-b13 + 15b10 + 31b7 - 8b3) q^72 + (10b14 + 34b8 + 8b2) q^73 + (-8b15 + 4b12 + 24b5) q^74 + (12b6 - 6b4 + 4b1 - 66) q^76 + (22b10 + 14b7 + 14b3) q^77 + (-8b14 - 8b11 + 32b8 + 16b2) * q^78 + (-8b15 - 4b9 - 56b5) q^79 + (-6b4 + 3b1 + 3) * q^81 + (-2b13 - 2b10 - 8b7 + 4b3) * q^82 + (5b11 + 12b8 - 12b2) q^83 + (-4b15 - 16b12 + 60b5) q^84 + (-5b6 + 24b4 + 72) * q^86 + (-4b13 + 28b7 - 8b3) q^87 + (6b14 - 10b11 - 6b8 - 24b2) * q^88 + (-8b15 - 4b9 - 6b5) q^89 + (-16b6 + 4b4 - 6b1) q^91 + (-7b13 - 3b10 - 31b7 - 6b3) * q^92 + (40b11 - 16b8 + 16b2) q^93 + (-10b15 + 3b12 + 6b9 - 62b5) * q^94 + (-4b6 - 8b4 + 4b1 + 96) q^96 + (14b13 + 10b7 - 8b3) q^97 + (-6b14 - 6b11 + 17b8 + 12b2) * q^98 + (20b15 + 14b12 - 3b9) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 32 q^{16} - 192 q^{26} + 128 q^{31} - 432 q^{36} + 96 q^{41} + 544 q^{46} + 768 q^{56} - 1152 q^{66} - 1056 q^{76} + 48 q^{81} + 1152 q^{86} + 1536 q^{96}+O(q^{100}) \) 16 * q + 32 * q^16 - 192 * q^26 + 128 * q^31 - 432 * q^36 + 96 * q^41 + 544 * q^46 + 768 * q^56 - 1152 * q^66 - 1056 * q^76 + 48 * q^81 + 1152 * q^86 + 1536 * q^96

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	200.3.i.a

Level	$200$
Weight	$3$
Character orbit	200.i
Analytic conductor	$5.450$
Analytic rank	$0$
Dimension	$16$
Inner twists	$8$

Self dual:	no
Analytic conductor:	\(5.44960528721\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(8\) over \(\Q(i)\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} + 12x^{14} + 58x^{12} + 132x^{10} - 51x^{8} - 1128x^{6} + 1372x^{4} - 96x^{2} + 100 \) x^16 + 12x^14 + 58x^12 + 132x^10 - 51x^8 - 1128x^6 + 1372x^4 - 96*x^2 + 100
Coefficient ring:	\(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index:	\( 2^{22} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

\(\beta_{1}\)	\(=\)	\( ( - 1317 \nu^{14} + 4229 \nu^{12} + 180623 \nu^{10} + 1209999 \nu^{8} + 3991926 \nu^{6} + \cdots + 7514640 ) / 1444370 \) (-1317v^14 + 4229v^12 + 180623v^10 + 1209999v^8 + 3991926v^6 + 4674952v^4 - 21370352*v^2 + 7514640) / 1444370
\(\beta_{2}\)	\(=\)	\( ( 32443 \nu^{15} + 42979 \nu^{14} + 367409 \nu^{13} + 333577 \nu^{12} + 1480453 \nu^{11} + \cdots - 70770100 ) / 46219840 \) (32443v^15 + 42979v^14 + 367409v^13 + 333577v^12 + 1480453v^11 + 35469v^10 + 1357039v^9 - 8322953v^8 - 12691624v^7 - 44627032v^6 - 55839488v^5 - 93239104v^4 + 62951668v^3 + 187694644v^2 + 151116460*v - 70770100) / 46219840
\(\beta_{3}\)	\(=\)	\( ( 32443 \nu^{15} - 42979 \nu^{14} + 367409 \nu^{13} - 333577 \nu^{12} + 1480453 \nu^{11} + \cdots + 70770100 ) / 46219840 \) (32443v^15 - 42979v^14 + 367409v^13 - 333577v^12 + 1480453v^11 - 35469v^10 + 1357039v^9 + 8322953v^8 - 12691624v^7 + 44627032v^6 - 55839488v^5 + 93239104v^4 + 62951668v^3 - 187694644v^2 + 151116460*v + 70770100) / 46219840
\(\beta_{4}\)	\(=\)	\( ( - 72993 \nu^{14} - 867809 \nu^{12} - 4100603 \nu^{10} - 8495379 \nu^{8} + 9503944 \nu^{6} + \cdots - 20658220 ) / 11554960 \) (-72993v^14 - 867809v^12 - 4100603v^10 - 8495379v^8 + 9503944v^6 + 98215408v^4 - 86682828*v^2 - 20658220) / 11554960
\(\beta_{5}\)	\(=\)	\( ( 46259 \nu^{15} + 556425 \nu^{13} + 2678793 \nu^{11} + 5925565 \nu^{9} - 3569208 \nu^{7} + \cdots + 11152008 \nu ) / 11554960 \) (46259v^15 + 556425v^13 + 2678793v^11 + 5925565v^9 - 3569208v^7 - 56172078v^5 + 58792396v^3 + 11152008v) / 11554960
\(\beta_{6}\)	\(=\)	\( ( 243793 \nu^{14} + 2890049 \nu^{12} + 13726363 \nu^{10} + 30213459 \nu^{8} - 15924344 \nu^{6} + \cdots - 49960260 ) / 11554960 \) (243793v^14 + 2890049v^12 + 13726363v^10 + 30213459v^8 - 15924344v^6 - 269012528v^4 + 386081068*v^2 - 49960260) / 11554960
\(\beta_{7}\)	\(=\)	\( ( - 337629 \nu^{15} - 32443 \nu^{14} - 4083991 \nu^{13} - 367409 \nu^{12} - 19949891 \nu^{11} + \cdots - 58676780 ) / 46219840 \) (-337629v^15 - 32443v^14 - 4083991v^13 - 367409v^12 - 19949891v^11 - 1480453v^10 - 46047481v^9 - 1357039v^8 + 15862040v^7 + 12691624v^6 + 393537136v^5 + 55839488v^4 - 407387500v^3 - 62951668v^2 - 30539284*v - 58676780) / 46219840
\(\beta_{8}\)	\(=\)	\( ( 337629 \nu^{15} - 32443 \nu^{14} + 4083991 \nu^{13} - 367409 \nu^{12} + 19949891 \nu^{11} + \cdots - 58676780 ) / 46219840 \) (337629v^15 - 32443v^14 + 4083991v^13 - 367409v^12 + 19949891v^11 - 1480453v^10 + 46047481v^9 - 1357039v^8 - 15862040v^7 + 12691624v^6 - 393537136v^5 + 55839488v^4 + 407387500v^3 - 62951668v^2 + 30539284*v - 58676780) / 46219840
\(\beta_{9}\)	\(=\)	\( ( 43625 \nu^{15} + 564883 \nu^{13} + 3040039 \nu^{11} + 8345563 \nu^{9} + 4414644 \nu^{7} + \cdots + 66623648 \nu ) / 5777480 \) (43625v^15 + 564883v^13 + 3040039v^11 + 8345563v^9 + 4414644v^7 - 46822174v^5 + 27606652v^3 + 66623648v) / 5777480
\(\beta_{10}\)	\(=\)	\( ( - 379263 \nu^{15} - 415681 \nu^{14} - 4343801 \nu^{13} - 5012983 \nu^{12} - 19460689 \nu^{11} + \cdots + 36413260 ) / 46219840 \) (-379263v^15 - 415681v^14 - 4343801v^13 - 5012983v^12 - 19460689v^11 - 24429031v^10 - 37392831v^9 - 56910113v^8 + 49529968v^7 + 14429768v^6 + 423317264v^5 + 457235856v^4 - 760169236v^3 - 567044956v^2 + 243929268*v + 36413260) / 46219840
\(\beta_{11}\)	\(=\)	\( ( - 379263 \nu^{15} + 415681 \nu^{14} - 4343801 \nu^{13} + 5012983 \nu^{12} - 19460689 \nu^{11} + \cdots - 36413260 ) / 46219840 \) (-379263v^15 + 415681v^14 - 4343801v^13 + 5012983v^12 - 19460689v^11 + 24429031v^10 - 37392831v^9 + 56910113v^8 + 49529968v^7 - 14429768v^6 + 423317264v^5 - 457235856v^4 - 760169236v^3 + 567044956v^2 + 243929268*v - 36413260) / 46219840
\(\beta_{12}\)	\(=\)	\( ( 67735 \nu^{15} + 726876 \nu^{13} + 2867163 \nu^{11} + 3589686 \nu^{9} - 16802812 \nu^{7} + \cdots - 96984504 \nu ) / 5777480 \) (67735v^15 + 726876v^13 + 2867163v^11 + 3589686v^9 - 16802812v^7 - 77152368v^5 + 187044084v^3 - 96984504v) / 5777480
\(\beta_{13}\)	\(=\)	\( ( - 171321 \nu^{15} - 66697 \nu^{14} - 2132322 \nu^{13} - 847996 \nu^{12} - 10936383 \nu^{11} + \cdots - 23806080 ) / 11554960 \) (-171321v^15 - 66697v^14 - 2132322v^13 - 847996v^12 - 10936383v^11 - 4432177v^10 - 28120072v^9 - 11557266v^8 - 7603554v^7 - 2808604v^6 + 176163888v^5 + 81997132v^4 - 175222032v^3 - 16798972v^2 - 20627744*v - 23806080) / 11554960
\(\beta_{14}\)	\(=\)	\( ( - 171321 \nu^{15} + 66697 \nu^{14} - 2132322 \nu^{13} + 847996 \nu^{12} - 10936383 \nu^{11} + \cdots + 23806080 ) / 11554960 \) (-171321v^15 + 66697v^14 - 2132322v^13 + 847996v^12 - 10936383v^11 + 4432177v^10 - 28120072v^9 + 11557266v^8 - 7603554v^7 + 2808604v^6 + 176163888v^5 - 81997132v^4 - 175222032v^3 + 16798972v^2 - 20627744*v + 23806080) / 11554960
\(\beta_{15}\)	\(=\)	\( ( 218377 \nu^{15} + 2659757 \nu^{13} + 13170495 \nu^{11} + 31626537 \nu^{9} - 2788888 \nu^{7} + \cdots - 12825904 \nu ) / 11554960 \) (218377v^15 + 2659757v^13 + 13170495v^11 + 31626537v^9 - 2788888v^7 - 239149650v^5 + 261912276v^3 - 12825904v) / 11554960

\(\nu\)	\(=\)	\( ( \beta_{8} - \beta_{7} - 4\beta_{5} + \beta_{3} + \beta_{2} ) / 4 \) (b8 - b7 - 4*b5 + b3 + b2) / 4
\(\nu^{2}\)	\(=\)	\( ( -2\beta_{8} - 2\beta_{7} + 2\beta_{3} - 2\beta_{2} - \beta _1 - 6 ) / 4 \) (-2b8 - 2b7 + 2b3 - 2b2 - b1 - 6) / 4
\(\nu^{3}\)	\(=\)	\( ( \beta_{14} + \beta_{13} - \beta_{11} - \beta_{10} + 3 \beta_{9} - 3 \beta_{8} + 3 \beta_{7} + \cdots - 4 \beta_{2} ) / 4 \) (b14 + b13 - b11 - b10 + 3b9 - 3b8 + 3b7 + 10b5 - 4b3 - 4b2) / 4
\(\nu^{4}\)	\(=\)	\( ( - 2 \beta_{14} + 2 \beta_{13} + 2 \beta_{11} - 2 \beta_{10} + 2 \beta_{8} + 2 \beta_{7} - \beta_{6} + \cdots + 7 ) / 2 \) (-2b14 + 2b13 + 2b11 - 2b10 + 2b8 + 2b7 - b6 - b4 - 4b3 + 4b2 + 3*b1 + 7) / 2
\(\nu^{5}\)	\(=\)	\( ( - 5 \beta_{15} - 5 \beta_{14} - 5 \beta_{13} + 5 \beta_{12} + 7 \beta_{11} + 7 \beta_{10} + \cdots + 7 \beta_{2} ) / 2 \) (-5b15 - 5b14 - 5b13 + 5b12 + 7b11 + 7b10 - 5b9 + 4b8 - 4b7 - 7b5 + 7b3 + 7b2) / 2
\(\nu^{6}\)	\(=\)	\( ( 10 \beta_{14} - 10 \beta_{13} - 22 \beta_{11} + 22 \beta_{10} - 12 \beta_{8} - 12 \beta_{7} + 17 \beta_{6} + \cdots - 9 ) / 2 \) (10b14 - 10b13 - 22b11 + 22b10 - 12b8 - 12b7 + 17b6 + 13b4 + 10b3 - 10b2 - 6*b1 - 9) / 2
\(\nu^{7}\)	\(=\)	\( ( 21 \beta_{15} + 16 \beta_{14} + 16 \beta_{13} - 49 \beta_{12} - 62 \beta_{11} - 62 \beta_{10} + \cdots - 12 \beta_{2} ) / 2 \) (21b15 + 16b14 + 16b13 - 49b12 - 62b11 - 62b10 - 34b8 + 34b7 + 37b5 - 12b3 - 12*b2) / 2
\(\nu^{8}\)	\(=\)	\( ( - 16 \beta_{14} + 16 \beta_{13} + 160 \beta_{11} - 160 \beta_{10} + 80 \beta_{8} + 80 \beta_{7} + \cdots + 149 ) / 2 \) (-16b14 + 16b13 + 160b11 - 160b10 + 80b8 + 80b7 - 127b6 - 7b4 - 16b3 + 16b2 - 27*b1 + 149) / 2
\(\nu^{9}\)	\(=\)	\( ( 105 \beta_{15} + 11 \beta_{14} + 11 \beta_{13} + 303 \beta_{12} + 387 \beta_{11} + 387 \beta_{10} + \cdots + 38 \beta_{2} ) / 2 \) (105b15 + 11b14 + 11b13 + 303b12 + 387b11 + 387b10 + 99b9 + 143b8 - 143b7 - 437b5 + 38b3 + 38b2) / 2
\(\nu^{10}\)	\(=\)	\( ( - 110 \beta_{14} + 110 \beta_{13} - 894 \beta_{11} + 894 \beta_{10} - 118 \beta_{8} - 118 \beta_{7} + \cdots - 933 ) / 2 \) (-110b14 + 110b13 - 894b11 + 894b10 - 118b8 - 118b7 + 679b6 - 525b4 + 124b3 - 124b2 + 249*b1 - 933) / 2
\(\nu^{11}\)	\(=\)	\( ( - 1749 \beta_{15} - 359 \beta_{14} - 359 \beta_{13} - 1463 \beta_{12} - 2003 \beta_{11} + \cdots - 364 \beta_{2} ) / 2 \) (-1749b15 - 359b14 - 359b13 - 1463b12 - 2003b11 - 2003b10 - 517b9 + 443b8 - 443b7 + 1157b5 - 364b3 - 364b2) / 2
\(\nu^{12}\)	\(=\)	\( ( 876 \beta_{14} - 876 \beta_{13} + 4412 \beta_{11} - 4412 \beta_{10} - 2956 \beta_{8} - 2956 \beta_{7} + \cdots - 1343 ) / 2 \) (876b14 - 876b13 + 4412b11 - 4412b10 - 2956b8 - 2956b7 - 3107b6 + 4829b4 - 832b3 + 832b2 - 975*b1 - 1343) / 2
\(\nu^{13}\)	\(=\)	\( ( 11869 \beta_{15} + 1851 \beta_{14} + 1851 \beta_{13} + 6643 \beta_{12} + 9651 \beta_{11} + \cdots + 1216 \beta_{2} ) / 2 \) (11869b15 + 1851b14 + 1851b13 + 6643b12 + 9651b11 + 9651b10 + 1911b9 - 11255b8 + 11255b7 + 14831b5 + 1216b3 + 1216b2) / 2
\(\nu^{14}\)	\(=\)	\( ( - 3762 \beta_{14} + 3762 \beta_{13} - 21026 \beta_{11} + 21026 \beta_{10} + 34778 \beta_{8} + \cdots + 62283 ) / 2 \) (-3762b14 + 3762b13 - 21026b11 + 21026b10 + 34778b8 + 34778b7 + 14443b6 - 27073b4 - 480b3 + 480b2 + 4595*b1 + 62283) / 2
\(\nu^{15}\)	\(=\)	\( ( - 59385 \beta_{15} - 8357 \beta_{14} - 8357 \beta_{13} - 31707 \beta_{12} - 45317 \beta_{11} + \cdots + 11580 \beta_{2} ) / 2 \) (-59385b15 - 8357b14 - 8357b13 - 31707b12 - 45317b11 - 45317b10 - 13707b9 + 95429b8 - 95429b7 - 200435b5 + 11580b3 + 11580b2) / 2

\(n\)	\(101\)	\(151\)	\(177\)
\(\chi(n)\)	\(-1\)	\(1\)	\(\beta_{5}\)

\(n\):		e.g. 2-40 or 80-90
Embeddings:		e.g. 1-3 or 93.8
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} - 8 T^{12} + \cdots + 65536 \) T^16 - 8T^12 + 144T^8 - 2048*T^4 + 65536
$3$	\( (T^{8} + 480 T^{4} + 2304)^{2} \) (T^8 + 480*T^4 + 2304)^2
$5$	\( T^{16} \) T^16
$7$	\( (T^{8} + 8928 T^{4} + 7485696)^{2} \) (T^8 + 8928*T^4 + 7485696)^2
$11$	\( (T^{4} + 312 T^{2} + 22800)^{4} \) (T^4 + 312*T^2 + 22800)^4
$13$	\( (T^{8} + 44544 T^{4} + 368640000)^{2} \) (T^8 + 44544*T^4 + 368640000)^2
$17$	\( (T^{8} + 163328 T^{4} + 23658496)^{2} \) (T^8 + 163328*T^4 + 23658496)^2
$19$	\( (T^{4} - 1272 T^{2} + 329232)^{4} \) (T^4 - 1272*T^2 + 329232)^4
$23$	\( (T^{8} + 520928 T^{4} + 65364080896)^{2} \) (T^8 + 520928*T^4 + 65364080896)^2
$29$	\( (T^{4} - 2784 T^{2} + 364800)^{4} \) (T^4 - 2784*T^2 + 364800)^4
$31$	\( (T^{2} - 16 T - 1472)^{8} \) (T^2 - 16*T - 1472)^8
$37$	\( (T^{8} + 1105920 T^{4} + 12230590464)^{2} \) (T^8 + 1105920*T^4 + 12230590464)^2
$41$	\( (T^{2} - 12 T - 60)^{8} \) (T^2 - 12*T - 60)^8
$43$	\( (T^{8} + \cdots + 147487228602624)^{2} \) (T^8 + 32141280*T^4 + 147487228602624)^2
$47$	\( (T^{8} + \cdots + 9616846018816)^{2} \) (T^8 + 8976608*T^4 + 9616846018816)^2
$53$	\( (T^{8} + \cdots + 220181913206784)^{2} \) (T^8 + 60648960*T^4 + 220181913206784)^2
$59$	\( (T^{4} - 1656 T^{2} + 8208)^{4} \) (T^4 - 1656*T^2 + 8208)^4
$61$	\( (T^{4} + 4992 T^{2} + 5836800)^{4} \) (T^4 + 4992*T^2 + 5836800)^4
$67$	\( (T^{8} + 2181600 T^{4} + 638030317824)^{2} \) (T^8 + 2181600*T^4 + 638030317824)^2
$71$	\( (T^{2} - 6144)^{8} \) (T^2 - 6144)^8
$73$	\( (T^{8} + \cdots + 13\!\cdots\!56)^{2} \) (T^8 + 352378368*T^4 + 1355389581459456)^2
$79$	\( (T^{4} + 9344 T^{2} + 2560000)^{4} \) (T^4 + 9344*T^2 + 2560000)^4
$83$	\( (T^{8} + 56379360 T^{4} + 644753664)^{2} \) (T^8 + 56379360*T^4 + 644753664)^2
$89$	\( (T^{4} + 3144 T^{2} + 2250000)^{4} \) (T^4 + 3144*T^2 + 2250000)^4
$97$	\( (T^{8} + \cdots + 65\!\cdots\!76)^{2} \) (T^8 + 174325248*T^4 + 6551578872446976)^2
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