Properties

Label 775.2.f.f
Level $775$
Weight $2$
Character orbit 775.f
Analytic conductor $6.188$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [775,2,Mod(557,775)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(775, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([1, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("775.557");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 775 = 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 775.f (of order \(4\), degree \(2\), 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: no
Analytic conductor: \(6.18840615665\)
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: \( x^{16} - 6 x^{15} + 18 x^{14} - 42 x^{13} + 66 x^{12} - 24 x^{11} - 162 x^{10} + 612 x^{9} - 1349 x^{8} + 1836 x^{7} - 1458 x^{6} - 648 x^{5} + 5346 x^{4} - 10206 x^{3} + \cdots + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: no (minimal twist has level 155)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{4} q^{3} + (\beta_{2} + \beta_1) q^{4} + ( - \beta_{12} - \beta_{10} + \beta_{7}) q^{6} + (\beta_{6} - \beta_1 + 1) q^{7} + (\beta_{11} + \beta_{8} + \beta_{6} - \beta_{3} + \beta_{2} - 1) q^{8} + (3 \beta_{6} + \beta_{3} + \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - \beta_{4} q^{3} + (\beta_{2} + \beta_1) q^{4} + ( - \beta_{12} - \beta_{10} + \beta_{7}) q^{6} + (\beta_{6} - \beta_1 + 1) q^{7} + (\beta_{11} + \beta_{8} + \beta_{6} - \beta_{3} + \beta_{2} - 1) q^{8} + (3 \beta_{6} + \beta_{3} + \beta_{2}) q^{9} + \beta_{10} q^{11} + ( - \beta_{14} - \beta_{13} - \beta_{10} + \beta_{7}) q^{12} - \beta_{12} q^{13} + ( - 2 \beta_{6} + \beta_{5} - \beta_{3} - \beta_{2}) q^{14} + (2 \beta_{8} + 2 \beta_{5} - 2 \beta_{3} - 2 \beta_1) q^{16} + (\beta_{13} + \beta_{10} - \beta_{9} - \beta_{7}) q^{17} + (\beta_{8} + 5 \beta_{5} - 6 \beta_{3}) q^{18} + (2 \beta_{6} - \beta_{3} + \beta_1) q^{19} + (\beta_{12} + \beta_{10} - \beta_{9} - \beta_{7} - \beta_{4}) q^{21} + (\beta_{14} + \beta_{13} + \beta_{10} - \beta_{7}) q^{22} + ( - \beta_{13} + \beta_{12} + \beta_{10}) q^{23} + ( - \beta_{15} - \beta_{14} - 2 \beta_{13}) q^{24} + ( - \beta_{9} - \beta_{4}) q^{26} + ( - \beta_{14} - \beta_{9}) q^{27} + ( - \beta_{8} - \beta_{6} - \beta_{5} + 2 \beta_{3} + 1) q^{28} + ( - \beta_{15} - \beta_{14} - \beta_{13} + \beta_{9} - \beta_{4}) q^{29} + (\beta_{11} - \beta_{10} + \beta_{9} - \beta_{8} + \beta_{5} + \beta_{4} - \beta_{3} - \beta_1) q^{31} + ( - 2 \beta_{8} - 2 \beta_{5} + 2 \beta_1) q^{32} + ( - \beta_{11} - 4 \beta_{8} - 3 \beta_{6} - 2 \beta_{5} + 6 \beta_{3} - \beta_{2} + 3) q^{33} + (\beta_{15} + \beta_{14} + \beta_{13} + \beta_{9} - \beta_{4}) q^{34} + (4 \beta_{11} + 2 \beta_{8} + 5 \beta_{5} - 5 \beta_{3} - 5 \beta_1 - 4) q^{36} + (\beta_{14} + \beta_{13} + \beta_{10} - \beta_{9}) q^{37} + (\beta_{11} + \beta_{6} + 2 \beta_{5} - 2 \beta_{3} + \beta_{2} - 1) q^{38} + ( - 3 \beta_{6} + 5 \beta_{5} - 2 \beta_{3} - \beta_{2} + \beta_1) q^{39} + ( - 2 \beta_{11} + \beta_{5} - \beta_{3} - \beta_1 - 1) q^{41} + (\beta_{14} + 2 \beta_{9} + \beta_{7}) q^{42} + ( - \beta_{15} - 2 \beta_{12} - \beta_{4}) q^{43} + (\beta_{15} + \beta_{14} + 2 \beta_{13} - 2 \beta_{12} - 2 \beta_{7}) q^{44} + ( - \beta_{15} + \beta_{14} + \beta_{12} + 2 \beta_{10} + \beta_{9} - \beta_{7} + \beta_{4}) q^{46} + (2 \beta_{6} - 2 \beta_1 + 2) q^{47} + ( - 2 \beta_{13} + 2 \beta_{12} + 2 \beta_{10} - 2 \beta_{4}) q^{48} + ( - 3 \beta_{6} - 2 \beta_{5} + 2 \beta_{3} + \beta_{2} - \beta_1) q^{49} + ( - 6 \beta_{8} - 8 \beta_{5} + 8 \beta_{3} + 8 \beta_1 - 3) q^{51} + ( - \beta_{13} - \beta_{10}) q^{52} + (\beta_{15} + \beta_{13} + \beta_{12} - \beta_{10} + \beta_{4}) q^{53} + ( - 3 \beta_{13} + 2 \beta_{12} + \beta_{9} + 2 \beta_{7} - \beta_{4}) q^{54} + ( - 2 \beta_{5} + 2 \beta_{3} + 2 \beta_1 - 2) q^{56} + ( - \beta_{13} - \beta_{10} - \beta_{9} + \beta_{7}) q^{57} + ( - \beta_{15} - 2 \beta_{13} + \beta_{12} + 2 \beta_{10} - 2 \beta_{4}) q^{58} + (3 \beta_{6} - 3 \beta_{5} - \beta_{3} + \beta_{2} + 2 \beta_1) q^{59} + (\beta_{15} - \beta_{14} - \beta_{9} - \beta_{4}) q^{61} + ( - \beta_{14} + \beta_{11} + \beta_{8} - \beta_{7} - 2 \beta_{6} + 2 \beta_{5} - \beta_{3} - \beta_{2} + \cdots - 2) q^{62}+ \cdots + (\beta_{15} + \beta_{14} + 4 \beta_{13} - 2 \beta_{12} - \beta_{9} - 2 \beta_{7} + \beta_{4}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4 q^{2} + 12 q^{7} - 16 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 4 q^{2} + 12 q^{7} - 16 q^{8} - 16 q^{18} + 16 q^{28} - 16 q^{31} - 8 q^{32} + 44 q^{33} - 88 q^{36} - 28 q^{38} - 24 q^{41} + 24 q^{47} - 32 q^{51} - 16 q^{56} - 44 q^{62} - 40 q^{63} + 112 q^{66} - 60 q^{67} + 72 q^{71} - 80 q^{72} - 32 q^{76} - 104 q^{78} + 24 q^{81} - 76 q^{82} + 20 q^{87} + 60 q^{93} + 72 q^{97} + 68 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 6 x^{15} + 18 x^{14} - 42 x^{13} + 66 x^{12} - 24 x^{11} - 162 x^{10} + 612 x^{9} - 1349 x^{8} + 1836 x^{7} - 1458 x^{6} - 648 x^{5} + 5346 x^{4} - 10206 x^{3} + \cdots + 6561 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 64 \nu^{15} + 159 \nu^{14} - 363 \nu^{13} + 798 \nu^{12} + 15 \nu^{11} - 1578 \nu^{10} + 5337 \nu^{9} - 14661 \nu^{8} + 13787 \nu^{7} - 10782 \nu^{6} - 6798 \nu^{5} + 69021 \nu^{4} + \cdots - 25515 ) / 729 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 326 \nu^{15} - 1287 \nu^{14} + 3105 \nu^{13} - 6996 \nu^{12} + 6459 \nu^{11} + 6873 \nu^{10} - 38385 \nu^{9} + 116433 \nu^{8} - 188377 \nu^{7} + 181245 \nu^{6} - 74160 \nu^{5} + \cdots - 920727 ) / 2187 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 332 \nu^{15} + 1320 \nu^{14} - 3195 \nu^{13} + 7194 \nu^{12} - 6729 \nu^{11} - 6927 \nu^{10} + 39429 \nu^{9} - 119619 \nu^{8} + 194635 \nu^{7} - 188214 \nu^{6} + \cdots + 960093 ) / 2187 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 139 \nu^{15} + 1293 \nu^{14} - 3312 \nu^{13} + 7539 \nu^{12} - 12900 \nu^{11} - 1362 \nu^{10} + 34830 \nu^{9} - 115686 \nu^{8} + 265757 \nu^{7} - 274428 \nu^{6} + \cdots + 2040471 ) / 2187 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 272 \nu^{15} - 1539 \nu^{14} + 3816 \nu^{13} - 8643 \nu^{12} + 11616 \nu^{11} + 5010 \nu^{10} - 43812 \nu^{9} + 138654 \nu^{8} - 272590 \nu^{7} + 272199 \nu^{6} - 160254 \nu^{5} + \cdots - 1791153 ) / 2187 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 103 \nu^{15} + 572 \nu^{14} - 1425 \nu^{13} + 3228 \nu^{12} - 4299 \nu^{11} - 1806 \nu^{10} + 16224 \nu^{9} - 51480 \nu^{8} + 100589 \nu^{7} - 100972 \nu^{6} + 59310 \nu^{5} + \cdots + 649539 ) / 729 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 63 \nu^{15} + 667 \nu^{14} - 1722 \nu^{13} + 3924 \nu^{12} - 6981 \nu^{11} - 330 \nu^{10} + 17616 \nu^{9} - 59229 \nu^{8} + 139977 \nu^{7} - 145592 \nu^{6} + 106257 \nu^{5} + \cdots + 1091313 ) / 729 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 820 \nu^{15} - 3399 \nu^{14} + 8244 \nu^{13} - 18591 \nu^{12} + 18345 \nu^{11} + 17256 \nu^{10} - 101331 \nu^{9} + 308952 \nu^{8} - 514799 \nu^{7} + 497943 \nu^{6} + \cdots - 2639709 ) / 2187 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 99 \nu^{15} + 491 \nu^{14} - 1210 \nu^{13} + 2733 \nu^{12} - 3306 \nu^{11} - 1923 \nu^{10} + 14232 \nu^{9} - 44364 \nu^{8} + 82071 \nu^{7} - 81175 \nu^{6} + 43370 \nu^{5} + \cdots + 496449 ) / 243 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 92 \nu^{15} + 501 \nu^{14} - 1244 \nu^{13} + 2817 \nu^{12} - 3705 \nu^{11} - 1641 \nu^{10} + 14223 \nu^{9} - 45042 \nu^{8} + 87397 \nu^{7} - 87540 \nu^{6} + 51055 \nu^{5} + \cdots + 560844 ) / 243 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 1682 \nu^{15} + 6996 \nu^{14} - 16974 \nu^{13} + 38271 \nu^{12} - 38004 \nu^{11} - 35286 \nu^{10} + 208332 \nu^{9} - 635346 \nu^{8} + 1062604 \nu^{7} - 1028826 \nu^{6} + \cdots + 5504679 ) / 2187 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 1439 \nu^{15} - 7155 \nu^{14} + 17622 \nu^{13} - 39837 \nu^{12} + 48327 \nu^{11} + 27681 \nu^{10} - 206406 \nu^{9} + 644715 \nu^{8} - 1194337 \nu^{7} + 1182405 \nu^{6} + \cdots - 7217100 ) / 2187 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 1061 \nu^{15} - 4419 \nu^{14} + 10731 \nu^{13} - 24195 \nu^{12} + 24081 \nu^{11} + 22209 \nu^{10} - 131688 \nu^{9} + 401616 \nu^{8} - 672265 \nu^{7} + 651324 \nu^{6} + \cdots - 3481704 ) / 729 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 416 \nu^{15} + 1721 \nu^{14} - 4173 \nu^{13} + 9408 \nu^{12} - 9270 \nu^{11} - 8748 \nu^{10} + 51288 \nu^{9} - 156267 \nu^{8} + 260332 \nu^{7} - 251740 \nu^{6} + \cdots + 1337715 ) / 243 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 3860 \nu^{15} + 16113 \nu^{14} - 39105 \nu^{13} + 88167 \nu^{12} - 87900 \nu^{11} - 81105 \nu^{10} + 480006 \nu^{9} - 1463949 \nu^{8} + 2452492 \nu^{7} + \cdots + 12715218 ) / 2187 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{11} - 2\beta_{9} - 2\beta_{8} + 2\beta_{6} - 3\beta_{5} + \beta_{3} + \beta_{2} + 2\beta _1 + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{15} - \beta_{14} - 2\beta_{13} - \beta_{12} - \beta_{7} - 2\beta_{3} - 2\beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{15} - 3\beta_{11} - \beta_{8} - 2\beta_{6} + \beta_{5} + 2\beta_{4} - 3\beta_{2} + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3\beta_{15} - 3\beta_{14} - \beta_{12} + 2\beta_{11} - 2\beta_{9} + 4\beta_{8} + \beta_{7} - 2\beta_{4} + 22 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -4\beta_{14} + \beta_{11} - 14\beta_{9} - 20\beta_{8} + 6\beta_{7} - 19\beta_{5} - \beta_{3} - \beta_{2} + 16\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 4 \beta_{15} - 4 \beta_{14} - 8 \beta_{13} + 7 \beta_{9} - 2 \beta_{6} - 7 \beta_{4} - 16 \beta_{3} - 19 \beta_{2} - 3 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 26 \beta_{15} + 6 \beta_{13} + 12 \beta_{12} - 53 \beta_{11} - 6 \beta_{10} + 6 \beta_{8} + 28 \beta_{6} - 17 \beta_{5} + 6 \beta_{4} + 11 \beta_{3} - 53 \beta_{2} - 28 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 29 \beta_{15} - 29 \beta_{14} - 17 \beta_{12} + 60 \beta_{11} - 16 \beta_{10} - 20 \beta_{9} + 108 \beta_{8} + 17 \beta_{7} + 12 \beta_{5} - 20 \beta_{4} - 12 \beta_{3} - 12 \beta _1 - 50 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 9 \beta_{14} - 3 \beta_{13} + 17 \beta_{11} - 3 \beta_{10} - 20 \beta_{9} - 38 \beta_{8} + 54 \beta_{7} - 115 \beta_{6} - 21 \beta_{5} - 17 \beta_{3} - 17 \beta_{2} + 23 \beta _1 - 115 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 53 \beta_{15} - 53 \beta_{14} - 94 \beta_{13} + 55 \beta_{12} + 246 \beta_{9} + 55 \beta_{7} - 120 \beta_{6} - 246 \beta_{4} - 46 \beta_{3} - 232 \beta_{2} - 186 \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 178 \beta_{15} + 168 \beta_{13} + 138 \beta_{12} - 213 \beta_{11} - 168 \beta_{10} - 70 \beta_{8} + 616 \beta_{6} - 347 \beta_{5} - 46 \beta_{4} + 417 \beta_{3} - 213 \beta_{2} - 616 ) / 4 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 27 \beta_{15} - 27 \beta_{14} - 91 \beta_{12} + 446 \beta_{11} - 174 \beta_{10} - 26 \beta_{9} + 658 \beta_{8} + 91 \beta_{7} + 240 \beta_{5} - 26 \beta_{4} - 240 \beta_{3} - 240 \beta _1 - 758 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 358 \beta_{14} - 24 \beta_{13} - 97 \beta_{11} - 24 \beta_{10} + 62 \beta_{9} + 422 \beta_{8} + 834 \beta_{7} - 2712 \beta_{6} + 325 \beta_{5} + 97 \beta_{3} + 97 \beta_{2} - 910 \beta _1 - 2712 ) / 4 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 259 \beta_{15} - 259 \beta_{14} - 134 \beta_{13} - 39 \beta_{12} + 2134 \beta_{9} - 39 \beta_{7} - 1460 \beta_{6} + 12 \beta_{5} - 2134 \beta_{4} + 2762 \beta_{3} + 320 \beta_{2} - 2442 \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 377 \beta_{15} + 1185 \beta_{13} + 36 \beta_{12} + 349 \beta_{11} - 1185 \beta_{10} - 1638 \beta_{8} + 1945 \beta_{6} - 506 \beta_{5} + 564 \beta_{4} + 2144 \beta_{3} + 349 \beta_{2} - 1945 ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/775\mathbb{Z}\right)^\times\).

\(n\) \(251\) \(652\)
\(\chi(n)\) \(-1\) \(\beta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
557.1
1.66787 0.467139i
−0.467139 + 1.66787i
1.66621 + 0.473008i
0.473008 + 1.66621i
−0.00704665 1.73204i
−1.73204 0.00704665i
1.70486 0.305721i
−0.305721 + 1.70486i
1.66787 + 0.467139i
−0.467139 1.66787i
1.66621 0.473008i
0.473008 1.66621i
−0.00704665 + 1.73204i
−1.73204 + 0.00704665i
1.70486 + 0.305721i
−0.305721 1.70486i
−1.15558 1.15558i −2.13501 2.13501i 0.670719i 0 4.93433i 2.15558 + 2.15558i −1.53609 + 1.53609i 6.11651i 0
557.2 −1.15558 1.15558i 2.13501 + 2.13501i 0.670719i 0 4.93433i 2.15558 + 2.15558i −1.53609 + 1.53609i 6.11651i 0
557.3 −0.329998 0.329998i −1.19320 1.19320i 1.78220i 0 0.787510i 1.33000 + 1.33000i −1.24812 + 1.24812i 0.152529i 0
557.4 −0.329998 0.329998i 1.19320 + 1.19320i 1.78220i 0 0.787510i 1.33000 + 1.33000i −1.24812 + 1.24812i 0.152529i 0
557.5 0.759725 + 0.759725i −1.72499 1.72499i 0.845635i 0 2.62104i 0.240275 + 0.240275i 2.16190 2.16190i 2.95118i 0
557.6 0.759725 + 0.759725i 1.72499 + 1.72499i 0.845635i 0 2.62104i 0.240275 + 0.240275i 2.16190 2.16190i 2.95118i 0
557.7 1.72585 + 1.72585i −2.01058 2.01058i 3.95712i 0 6.93991i −0.725850 0.725850i −3.37769 + 3.37769i 5.08484i 0
557.8 1.72585 + 1.72585i 2.01058 + 2.01058i 3.95712i 0 6.93991i −0.725850 0.725850i −3.37769 + 3.37769i 5.08484i 0
743.1 −1.15558 + 1.15558i −2.13501 + 2.13501i 0.670719i 0 4.93433i 2.15558 2.15558i −1.53609 1.53609i 6.11651i 0
743.2 −1.15558 + 1.15558i 2.13501 2.13501i 0.670719i 0 4.93433i 2.15558 2.15558i −1.53609 1.53609i 6.11651i 0
743.3 −0.329998 + 0.329998i −1.19320 + 1.19320i 1.78220i 0 0.787510i 1.33000 1.33000i −1.24812 1.24812i 0.152529i 0
743.4 −0.329998 + 0.329998i 1.19320 1.19320i 1.78220i 0 0.787510i 1.33000 1.33000i −1.24812 1.24812i 0.152529i 0
743.5 0.759725 0.759725i −1.72499 + 1.72499i 0.845635i 0 2.62104i 0.240275 0.240275i 2.16190 + 2.16190i 2.95118i 0
743.6 0.759725 0.759725i 1.72499 1.72499i 0.845635i 0 2.62104i 0.240275 0.240275i 2.16190 + 2.16190i 2.95118i 0
743.7 1.72585 1.72585i −2.01058 + 2.01058i 3.95712i 0 6.93991i −0.725850 + 0.725850i −3.37769 3.37769i 5.08484i 0
743.8 1.72585 1.72585i 2.01058 2.01058i 3.95712i 0 6.93991i −0.725850 + 0.725850i −3.37769 3.37769i 5.08484i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 557.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
31.b odd 2 1 inner
155.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 775.2.f.f 16
5.b even 2 1 155.2.f.b 16
5.c odd 4 1 155.2.f.b 16
5.c odd 4 1 inner 775.2.f.f 16
31.b odd 2 1 inner 775.2.f.f 16
155.c odd 2 1 155.2.f.b 16
155.f even 4 1 155.2.f.b 16
155.f even 4 1 inner 775.2.f.f 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
155.2.f.b 16 5.b even 2 1
155.2.f.b 16 5.c odd 4 1
155.2.f.b 16 155.c odd 2 1
155.2.f.b 16 155.f even 4 1
775.2.f.f 16 1.a even 1 1 trivial
775.2.f.f 16 5.c odd 4 1 inner
775.2.f.f 16 31.b odd 2 1 inner
775.2.f.f 16 155.f even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(775, [\chi])\):

\( T_{2}^{8} - 2T_{2}^{7} + 2T_{2}^{6} + 4T_{2}^{5} + 12T_{2}^{4} - 12T_{2}^{3} + 8T_{2}^{2} + 8T_{2} + 4 \) Copy content Toggle raw display
\( T_{3}^{16} + 192T_{3}^{12} + 12182T_{3}^{8} + 279084T_{3}^{4} + 1560001 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} - 2 T^{7} + 2 T^{6} + 4 T^{5} + 12 T^{4} + \cdots + 4)^{2} \) Copy content Toggle raw display
$3$ \( T^{16} + 192 T^{12} + 12182 T^{8} + \cdots + 1560001 \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( (T^{8} - 6 T^{7} + 18 T^{6} - 20 T^{5} + \cdots + 4)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} + 68 T^{6} + 1216 T^{4} + \cdots + 4996)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} + 844 T^{12} + \cdots + 24960016 \) Copy content Toggle raw display
$17$ \( T^{16} + 3820 T^{12} + \cdots + 975000625 \) Copy content Toggle raw display
$19$ \( (T^{8} + 28 T^{6} + 214 T^{4} + 288 T^{2} + \cdots + 49)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + 5068 T^{12} + \cdots + 24960016 \) Copy content Toggle raw display
$29$ \( (T^{8} - 152 T^{6} + 7324 T^{4} + \cdots + 844324)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 8 T^{7} + 60 T^{6} + 376 T^{5} + \cdots + 923521)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + 5636 T^{12} + \cdots + 1560001 \) Copy content Toggle raw display
$41$ \( (T^{4} + 6 T^{3} - 44 T^{2} - 318 T - 469)^{4} \) Copy content Toggle raw display
$43$ \( T^{16} + 22564 T^{12} + \cdots + 44555188561 \) Copy content Toggle raw display
$47$ \( (T^{8} - 12 T^{7} + 72 T^{6} - 160 T^{5} + \cdots + 1024)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} + 16184 T^{12} + \cdots + 2923693034161 \) Copy content Toggle raw display
$59$ \( (T^{8} + 124 T^{6} + 4222 T^{4} + \cdots + 26569)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + 304 T^{6} + 26456 T^{4} + \cdots + 19984)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + 30 T^{7} + 450 T^{6} + \cdots + 89151364)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 18 T^{3} + 58 T^{2} + 260 T - 551)^{4} \) Copy content Toggle raw display
$73$ \( T^{16} + 14488 T^{12} + \cdots + 4408189985761 \) Copy content Toggle raw display
$79$ \( (T^{8} - 324 T^{6} + 29192 T^{4} + \cdots + 4201636)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + 100668 T^{12} + \cdots + 60762165310081 \) Copy content Toggle raw display
$89$ \( (T^{8} - 508 T^{6} + 64512 T^{4} + \cdots + 4996)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} - 36 T^{7} + 648 T^{6} + \cdots + 160000)^{2} \) Copy content Toggle raw display
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