Properties

Label 784.2.m.i
Level $784$
Weight $2$
Character orbit 784.m
Analytic conductor $6.260$
Analytic rank $0$
Dimension $20$
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] = [784,2,Mod(197,784)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(784, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("784.197");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 784 = 2^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 784.m (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: \(6.26027151847\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 109x^{16} + 3858x^{12} + 44914x^{8} + 37101x^{4} + 2209 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{11} \)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{4} q^{2} + \beta_{9} q^{3} - \beta_{17} q^{4} + \beta_{2} q^{5} - \beta_{6} q^{6} + (\beta_{8} - 2 \beta_{7} - \beta_{3}) q^{8} + ( - \beta_{18} + \beta_{17} + \beta_{16} - \beta_{14} - \beta_{7} + \beta_{3}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{2} + \beta_{9} q^{3} - \beta_{17} q^{4} + \beta_{2} q^{5} - \beta_{6} q^{6} + (\beta_{8} - 2 \beta_{7} - \beta_{3}) q^{8} + ( - \beta_{18} + \beta_{17} + \beta_{16} - \beta_{14} - \beta_{7} + \beta_{3}) q^{9} + \beta_{10} q^{10} + ( - \beta_{14} + \beta_{13} + \beta_{7} - \beta_{4} + \beta_{3} + 1) q^{11} + (\beta_{11} - \beta_{2} + \beta_1) q^{12} + ( - \beta_{12} + \beta_{10} - \beta_{9} - \beta_{5}) q^{13} + (\beta_{17} - \beta_{14} + 2 \beta_{13} + 2 \beta_{8} + \beta_{4} - \beta_{3}) q^{15} + (\beta_{17} + 2 \beta_{16} - 2 \beta_{14} + \beta_{13}) q^{16} + (\beta_{19} + \beta_{12} - \beta_{6} + \beta_{2} - \beta_1) q^{17} + ( - \beta_{18} + \beta_{16} + \beta_{14} - 2 \beta_{13} - \beta_{8} - 2 \beta_{7} + \beta_{3}) q^{18} + ( - \beta_{12} - \beta_{11} - \beta_{9}) q^{19} + (\beta_{19} + \beta_{15} + \beta_{12} - \beta_{10} + 2 \beta_{9} - \beta_1) q^{20} + ( - \beta_{18} + \beta_{17} - \beta_{16} + \beta_{14} - \beta_{13} + \beta_{8} - 2 \beta_{7} + \beta_{4} + \beta_{3}) q^{22} + (\beta_{16} - \beta_{14} - \beta_{13} + \beta_{8} - 2 \beta_{7} + 2 \beta_{4}) q^{23} + ( - \beta_{15} - \beta_{12} + \beta_{11} + \beta_{6} - \beta_{5} - \beta_{2}) q^{24} + (2 \beta_{16} - 2 \beta_{14} + \beta_{13} - \beta_{8} + 3 \beta_{7} + \beta_{4}) q^{25} + (\beta_{15} + \beta_{12} - \beta_{11} - \beta_{10} - \beta_{2} - 2 \beta_1) q^{26} + ( - \beta_{19} - \beta_{15} + \beta_{10} + \beta_{6} + \beta_{5} + \beta_{2}) q^{27} + (\beta_{18} + \beta_{17} - \beta_{16} - \beta_{14} + \beta_{13} - \beta_{8} + \beta_{7} - \beta_{4} - \beta_{3} - 1) q^{29} + ( - \beta_{18} + \beta_{17} - 3 \beta_{14} + \beta_{13} + \beta_{8} + 3 \beta_{3} + 2) q^{30} + ( - \beta_{19} + \beta_{15} - \beta_{12} - \beta_{11} - \beta_{10} - \beta_{9} - \beta_{6} + \beta_{5} - \beta_{2} + \beta_1) q^{31} + ( - 2 \beta_{13} - 2 \beta_{7} + 2 \beta_{3} - 2) q^{32} + ( - \beta_{19} - \beta_{15} + \beta_{12} + \beta_{6} - \beta_1) q^{33} + ( - \beta_{15} + \beta_{12} + \beta_{11} + 2 \beta_{9} + \beta_{6} + \beta_{5} - \beta_{2} + 2 \beta_1) q^{34} + ( - \beta_{17} + 2 \beta_{16} + 2 \beta_{14} - 2 \beta_{13} - 2 \beta_{8} - 2 \beta_{3}) q^{36} + (\beta_{18} - 2 \beta_{17} + \beta_{14} - 2 \beta_{8} - \beta_{7} - 1) q^{37} + ( - \beta_{19} + 2 \beta_{15} - 2 \beta_{11} - \beta_{10} - 2 \beta_{9} - \beta_{6}) q^{38} + (\beta_{18} + \beta_{17} - 2 \beta_{14} + 2 \beta_{13} - 2 \beta_{8} + 4 \beta_{7} + \beta_{4} + \beta_{3}) q^{39} + ( - 2 \beta_{15} - \beta_{6} + \beta_{5} + 2 \beta_1) q^{40} + (\beta_{19} + \beta_{12} - \beta_{11} - \beta_{10} + \beta_{6} + \beta_{5}) q^{41} + (\beta_{18} + \beta_{17} + \beta_{16} + \beta_{8} + 3 \beta_{7} + 2 \beta_{4} + 3) q^{43} + ( - 2 \beta_{18} + 2 \beta_{16} - 2 \beta_{8} + 2 \beta_{7} + 2) q^{44} + ( - \beta_{19} - \beta_{15} + \beta_{11} - \beta_{9} + \beta_{6} - 2 \beta_{5} - \beta_{2}) q^{45} + ( - \beta_{18} - \beta_{17} + 2 \beta_{16} - \beta_{14} - \beta_{13} - \beta_{8} - 2 \beta_{7} - \beta_{3} + \cdots - 4) q^{46}+ \cdots + (3 \beta_{18} - 3 \beta_{17} - \beta_{16} - 2 \beta_{14} + 4 \beta_{13} + 3 \beta_{8} + 3 \beta_{7} + \cdots - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 8 q^{4} + 4 q^{11} - 16 q^{15} + 16 q^{18} - 4 q^{29} + 8 q^{30} - 40 q^{32} + 40 q^{36} - 20 q^{37} + 60 q^{43} + 56 q^{44} - 64 q^{46} - 56 q^{50} - 16 q^{51} + 28 q^{53} - 72 q^{58} - 24 q^{60} - 32 q^{64} + 16 q^{65} + 12 q^{67} + 16 q^{72} - 16 q^{74} - 88 q^{78} - 72 q^{79} + 12 q^{81} + 32 q^{85} - 40 q^{86} + 80 q^{88} - 24 q^{92} + 48 q^{93} + 64 q^{95} - 108 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 109x^{16} + 3858x^{12} + 44914x^{8} + 37101x^{4} + 2209 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2749\nu^{17} + 304042\nu^{13} + 10922388\nu^{9} + 127519222\nu^{5} + 54101007\nu ) / 17677696 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -3007\nu^{17} - 332778\nu^{13} - 11994704\nu^{9} - 143671702\nu^{5} - 139036209\nu ) / 17677696 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 64559 \nu^{18} + 54755 \nu^{16} - 6987158 \nu^{14} + 4790334 \nu^{12} - 242619188 \nu^{10} + 127845828 \nu^{8} - 2687600778 \nu^{6} + \cdots + 1422237625 ) / 1661703424 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 67855 \nu^{18} - 129203 \nu^{16} - 7619774 \nu^{14} - 14289974 \nu^{12} - 278937804 \nu^{10} - 513352236 \nu^{8} - 3402159906 \nu^{6} + \cdots - 2542747329 ) / 1661703424 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 205444 \nu^{19} + 6063 \nu^{17} + 22353446 \nu^{15} + 675296 \nu^{13} + 787046330 \nu^{11} + 25199426 \nu^{9} + 9008837674 \nu^{7} + \cdots + 2826828959 \nu ) / 830851712 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 257159 \nu^{19} + 86762 \nu^{17} - 28072772 \nu^{15} + 9562902 \nu^{13} - 996846494 \nu^{11} + 336956254 \nu^{9} - 11726435308 \nu^{7} + \cdots + 1449108136 \nu ) / 830851712 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 36737\nu^{18} + 4010396\nu^{14} + 142406642\nu^{10} + 1675205044\nu^{6} + 1742562717\nu^{2} ) / 415425856 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 152567 \nu^{18} - 44321 \nu^{16} - 16442602 \nu^{14} - 4835830 \nu^{12} - 575502048 \nu^{10} - 160560272 \nu^{8} - 6649070358 \nu^{6} + \cdots - 355468943 ) / 1661703424 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -346329\nu^{19} - 37719734\nu^{15} - 1331473472\nu^{11} - 15330074570\nu^{7} - 9477314983\nu^{3} ) / 830851712 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 396396 \nu^{19} + 314007 \nu^{17} + 43122752 \nu^{15} + 34068326 \nu^{13} + 1523114328 \nu^{11} + 1196106948 \nu^{9} + 17690392640 \nu^{7} + \cdots + 8801176901 \nu ) / 830851712 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 227446 \nu^{19} - 153972 \nu^{17} - 24717307 \nu^{15} - 16696515 \nu^{13} - 870267045 \nu^{11} - 585453761 \nu^{9} - 9999205069 \nu^{7} + \cdots - 2987173971 \nu ) / 415425856 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 227446 \nu^{19} + 153972 \nu^{17} - 24717307 \nu^{15} + 16696515 \nu^{13} - 870267045 \nu^{11} + 585453761 \nu^{9} - 9999205069 \nu^{7} + \cdots + 2987173971 \nu ) / 415425856 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 110211 \nu^{18} - 86762 \nu^{16} + 12031188 \nu^{14} - 9562902 \nu^{12} + 427219926 \nu^{10} - 336956254 \nu^{8} + 5025615132 \nu^{6} + \cdots - 1449108136 ) / 830851712 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 139237 \nu^{18} - 43287 \nu^{16} - 15049980 \nu^{14} - 5425116 \nu^{12} - 526267834 \nu^{10} - 217952630 \nu^{8} - 5963957332 \nu^{6} + \cdots - 1725380387 ) / 830851712 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( -713699\nu^{19} - 77823694\nu^{15} - 2755539892\nu^{11} - 32082125010\nu^{7} - 26072090441\nu^{3} ) / 830851712 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 346329 \nu^{18} + 42629 \nu^{16} + 37719734 \nu^{14} + 3439742 \nu^{12} + 1331473472 \nu^{10} + 77446976 \nu^{8} + 15330074570 \nu^{6} + \cdots - 908013445 ) / 1661703424 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 205444 \nu^{18} - 6063 \nu^{16} - 22353446 \nu^{14} - 675296 \nu^{12} - 787046330 \nu^{10} - 25199426 \nu^{8} - 9008837674 \nu^{6} + \cdots - 1165125535 ) / 830851712 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 103546 \nu^{18} + 42958 \nu^{16} - 11334877 \nu^{14} + 4432429 \nu^{12} - 402602819 \nu^{10} + 147699803 \nu^{8} - 4683058619 \nu^{6} + \cdots + 408683471 ) / 415425856 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 1058380 \nu^{19} - 91979 \nu^{17} + 115227120 \nu^{15} - 9540154 \nu^{13} + 4068854056 \nu^{11} - 320599032 \nu^{9} + 47054920016 \nu^{7} + \cdots - 3644195901 \nu ) / 830851712 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{19} - \beta_{15} - \beta_{12} + \beta_{11} + \beta_{10} + \beta_{6} + \beta_{5} + \beta_{2} + 2\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{13} + \beta_{8} + 3\beta_{7} + \beta_{4} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 5 \beta_{19} + 9 \beta_{15} + 5 \beta_{12} + 5 \beta_{11} + 5 \beta_{10} - 4 \beta_{9} - 5 \beta_{6} + 5 \beta_{5} + 5 \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{18} - 7\beta_{17} - 7\beta_{16} + \beta_{14} - \beta_{13} - \beta_{8} + \beta_{4} + 7\beta_{3} - 17 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 33 \beta_{19} + 29 \beta_{15} + 33 \beta_{12} - 29 \beta_{11} - 29 \beta_{10} + 4 \beta_{9} - 25 \beta_{6} - 29 \beta_{5} - 57 \beta_{2} - 98 \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 11 \beta_{18} - 13 \beta_{17} - 9 \beta_{16} + 11 \beta_{14} + 46 \beta_{13} - 46 \beta_{8} - 107 \beta_{7} - 46 \beta_{4} - 13 \beta_{3} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 173 \beta_{19} - 357 \beta_{15} - 173 \beta_{12} - 225 \beta_{11} - 225 \beta_{10} + 272 \beta_{9} + 173 \beta_{6} - 121 \beta_{5} - 173 \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 96 \beta_{18} + 300 \beta_{17} + 304 \beta_{16} - 92 \beta_{14} + 120 \beta_{13} + 120 \beta_{8} - 68 \beta_{4} - 300 \beta_{3} + 689 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 1545 \beta_{19} - 1049 \beta_{15} - 1529 \beta_{12} + 1033 \beta_{11} + 1049 \beta_{10} - 496 \beta_{9} + 553 \beta_{6} + 1049 \beta_{5} + 2265 \beta_{2} + 4514 \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 692 \beta_{18} + 972 \beta_{17} + 488 \beta_{16} - 768 \beta_{14} - 1953 \beta_{13} + 1953 \beta_{8} + 4483 \beta_{7} + 2029 \beta_{4} + 972 \beta_{3} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 6445 \beta_{19} + 14561 \beta_{15} + 6141 \beta_{12} + 10333 \beta_{11} + 10637 \beta_{10} - 13652 \beta_{9} - 6445 \beta_{6} + 2253 \beta_{5} + 6445 \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 5865 \beta_{18} - 12707 \beta_{17} - 13647 \beta_{16} + 4925 \beta_{14} - 7377 \beta_{13} - 7377 \beta_{8} + 3413 \beta_{4} + 12707 \beta_{3} - 29377 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 73305 \beta_{19} + 40037 \beta_{15} + 69545 \beta_{12} - 36277 \beta_{11} - 40037 \beta_{10} + 33268 \beta_{9} - 6769 \beta_{6} - 40037 \beta_{5} - 94625 \beta_{2} - 199810 \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 33919 \beta_{18} - 53945 \beta_{17} - 23497 \beta_{16} + 43523 \beta_{14} + 82678 \beta_{13} - 82678 \beta_{8} - 193547 \beta_{7} - 92282 \beta_{4} - 53945 \beta_{3} \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 251013 \beta_{19} - 620141 \beta_{15} - 212597 \beta_{12} - 466793 \beta_{11} - 505209 \beta_{10} + 640480 \beta_{9} + 251013 \beta_{6} + 3183 \beta_{5} - 251013 \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 316848 \beta_{18} + 538152 \beta_{17} + 626224 \beta_{16} - 228776 \beta_{14} + 385504 \beta_{13} + 385504 \beta_{8} - 160120 \beta_{4} - 538152 \beta_{3} + 1280481 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( - 3480081 \beta_{19} - 1585777 \beta_{15} - 3127793 \beta_{12} + 1233489 \beta_{11} + 1585777 \beta_{10} - 1894304 \beta_{9} - 308527 \beta_{6} + 1585777 \beta_{5} + \cdots + 8696386 \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 1521912 \beta_{18} + 2713864 \beta_{17} + 1083776 \beta_{16} - 2275728 \beta_{14} - 3505137 \beta_{13} + 3505137 \beta_{8} + 8498947 \beta_{7} + 4258953 \beta_{4} + 2713864 \beta_{3} \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( 10081493 \beta_{19} + 27117305 \beta_{15} + 7066229 \beta_{12} + 20936949 \beta_{11} + 23952213 \beta_{10} - 29211108 \beta_{9} - 10081493 \beta_{6} - 3789227 \beta_{5} + \cdots + 10081493 \beta_{2} ) / 4 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/784\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(687\) \(689\)
\(\chi(n)\) \(\beta_{7}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
197.1
−0.356428 0.356428i
0.356428 + 0.356428i
−1.77012 1.77012i
1.77012 + 1.77012i
−1.83808 1.83808i
1.83808 + 1.83808i
1.55160 + 1.55160i
−1.55160 1.55160i
0.673528 + 0.673528i
−0.673528 0.673528i
−0.356428 + 0.356428i
0.356428 0.356428i
−1.77012 + 1.77012i
1.77012 1.77012i
−1.83808 + 1.83808i
1.83808 1.83808i
1.55160 1.55160i
−1.55160 + 1.55160i
0.673528 0.673528i
−0.673528 + 0.673528i
−1.29865 0.559929i −0.925747 + 0.925747i 1.37296 + 1.45430i 2.61733 + 2.61733i 1.72057 0.683864i 0 −0.968683 2.65738i 1.28598i −1.93346 4.86449i
197.2 −1.29865 0.559929i 0.925747 0.925747i 1.37296 + 1.45430i −2.61733 2.61733i −1.72057 + 0.683864i 0 −0.968683 2.65738i 1.28598i 1.93346 + 4.86449i
197.3 −0.428185 1.34783i −1.51587 + 1.51587i −1.63332 + 1.15424i −0.726212 0.726212i 2.69222 + 1.39407i 0 2.25509 + 1.70721i 1.59575i −0.667861 + 1.28977i
197.4 −0.428185 1.34783i 1.51587 1.51587i −1.63332 + 1.15424i 0.726212 + 0.726212i −2.69222 1.39407i 0 2.25509 + 1.70721i 1.59575i 0.667861 1.28977i
197.5 −0.246440 + 1.39258i −0.905952 + 0.905952i −1.87853 0.686372i −2.32368 2.32368i −1.03834 1.48487i 0 1.41877 2.44685i 1.35850i 3.80855 2.66326i
197.6 −0.246440 + 1.39258i 0.905952 0.905952i −1.87853 0.686372i 2.32368 + 2.32368i 1.03834 + 1.48487i 0 1.41877 2.44685i 1.35850i −3.80855 + 2.66326i
197.7 0.739098 + 1.20571i −2.29357 + 2.29357i −0.907469 + 1.78227i 2.00643 + 2.00643i −4.46055 1.07021i 0 −2.81961 + 0.223131i 7.52094i −0.936222 + 3.90212i
197.8 0.739098 + 1.20571i 2.29357 2.29357i −0.907469 + 1.78227i −2.00643 2.00643i 4.46055 + 1.07021i 0 −2.81961 + 0.223131i 7.52094i 0.936222 3.90212i
197.9 1.23417 0.690521i −1.66250 + 1.66250i 1.04636 1.70444i −1.09406 1.09406i −0.903818 + 3.19980i 0 0.114433 2.82611i 2.52780i −2.10574 0.594788i
197.10 1.23417 0.690521i 1.66250 1.66250i 1.04636 1.70444i 1.09406 + 1.09406i 0.903818 3.19980i 0 0.114433 2.82611i 2.52780i 2.10574 + 0.594788i
589.1 −1.29865 + 0.559929i −0.925747 0.925747i 1.37296 1.45430i 2.61733 2.61733i 1.72057 + 0.683864i 0 −0.968683 + 2.65738i 1.28598i −1.93346 + 4.86449i
589.2 −1.29865 + 0.559929i 0.925747 + 0.925747i 1.37296 1.45430i −2.61733 + 2.61733i −1.72057 0.683864i 0 −0.968683 + 2.65738i 1.28598i 1.93346 4.86449i
589.3 −0.428185 + 1.34783i −1.51587 1.51587i −1.63332 1.15424i −0.726212 + 0.726212i 2.69222 1.39407i 0 2.25509 1.70721i 1.59575i −0.667861 1.28977i
589.4 −0.428185 + 1.34783i 1.51587 + 1.51587i −1.63332 1.15424i 0.726212 0.726212i −2.69222 + 1.39407i 0 2.25509 1.70721i 1.59575i 0.667861 + 1.28977i
589.5 −0.246440 1.39258i −0.905952 0.905952i −1.87853 + 0.686372i −2.32368 + 2.32368i −1.03834 + 1.48487i 0 1.41877 + 2.44685i 1.35850i 3.80855 + 2.66326i
589.6 −0.246440 1.39258i 0.905952 + 0.905952i −1.87853 + 0.686372i 2.32368 2.32368i 1.03834 1.48487i 0 1.41877 + 2.44685i 1.35850i −3.80855 2.66326i
589.7 0.739098 1.20571i −2.29357 2.29357i −0.907469 1.78227i 2.00643 2.00643i −4.46055 + 1.07021i 0 −2.81961 0.223131i 7.52094i −0.936222 3.90212i
589.8 0.739098 1.20571i 2.29357 + 2.29357i −0.907469 1.78227i −2.00643 + 2.00643i 4.46055 1.07021i 0 −2.81961 0.223131i 7.52094i 0.936222 + 3.90212i
589.9 1.23417 + 0.690521i −1.66250 1.66250i 1.04636 + 1.70444i −1.09406 + 1.09406i −0.903818 3.19980i 0 0.114433 + 2.82611i 2.52780i −2.10574 + 0.594788i
589.10 1.23417 + 0.690521i 1.66250 + 1.66250i 1.04636 + 1.70444i 1.09406 1.09406i 0.903818 + 3.19980i 0 0.114433 + 2.82611i 2.52780i 2.10574 0.594788i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 197.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
16.e even 4 1 inner
112.l odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 784.2.m.i 20
7.b odd 2 1 inner 784.2.m.i 20
7.c even 3 2 784.2.x.n 40
7.d odd 6 2 784.2.x.n 40
16.e even 4 1 inner 784.2.m.i 20
112.l odd 4 1 inner 784.2.m.i 20
112.w even 12 2 784.2.x.n 40
112.x odd 12 2 784.2.x.n 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
784.2.m.i 20 1.a even 1 1 trivial
784.2.m.i 20 7.b odd 2 1 inner
784.2.m.i 20 16.e even 4 1 inner
784.2.m.i 20 112.l odd 4 1 inner
784.2.x.n 40 7.c even 3 2
784.2.x.n 40 7.d odd 6 2
784.2.x.n 40 112.w even 12 2
784.2.x.n 40 112.x odd 12 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(784, [\chi])\):

\( T_{3}^{20} + 168T_{3}^{16} + 7288T_{3}^{12} + 108576T_{3}^{8} + 452752T_{3}^{4} + 565504 \) Copy content Toggle raw display
\( T_{5}^{20} + 376T_{5}^{16} + 44152T_{5}^{12} + 1706272T_{5}^{8} + 9976976T_{5}^{4} + 9048064 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{10} + 2 T^{8} + 2 T^{6} + 4 T^{5} + 4 T^{4} + \cdots + 32)^{2} \) Copy content Toggle raw display
$3$ \( T^{20} + 168 T^{16} + 7288 T^{12} + \cdots + 565504 \) Copy content Toggle raw display
$5$ \( T^{20} + 376 T^{16} + 44152 T^{12} + \cdots + 9048064 \) Copy content Toggle raw display
$7$ \( T^{20} \) Copy content Toggle raw display
$11$ \( (T^{10} - 2 T^{9} + 2 T^{8} + 16 T^{7} + \cdots + 512)^{2} \) Copy content Toggle raw display
$13$ \( T^{20} + 2168 T^{16} + \cdots + 2316304384 \) Copy content Toggle raw display
$17$ \( (T^{10} - 112 T^{8} + 4464 T^{6} + \cdots - 1179136)^{2} \) Copy content Toggle raw display
$19$ \( T^{20} + 4136 T^{16} + \cdots + 6852416114944 \) Copy content Toggle raw display
$23$ \( (T^{10} + 100 T^{8} + 2632 T^{6} + \cdots + 3136)^{2} \) Copy content Toggle raw display
$29$ \( (T^{10} + 2 T^{9} + 2 T^{8} + 128 T^{7} + \cdots + 32)^{2} \) Copy content Toggle raw display
$31$ \( (T^{10} - 256 T^{8} + 23600 T^{6} + \cdots - 57777664)^{2} \) Copy content Toggle raw display
$37$ \( (T^{10} + 10 T^{9} + 50 T^{8} + \cdots + 408608)^{2} \) Copy content Toggle raw display
$41$ \( (T^{10} + 248 T^{8} + 17968 T^{6} + \cdots + 96256)^{2} \) Copy content Toggle raw display
$43$ \( (T^{10} - 30 T^{9} + 450 T^{8} + \cdots + 131072)^{2} \) Copy content Toggle raw display
$47$ \( (T^{10} - 256 T^{8} + 21552 T^{6} + \cdots - 1179136)^{2} \) Copy content Toggle raw display
$53$ \( (T^{10} - 14 T^{9} + 98 T^{8} + \cdots + 20326688)^{2} \) Copy content Toggle raw display
$59$ \( T^{20} + 57256 T^{16} + \cdots + 25\!\cdots\!24 \) Copy content Toggle raw display
$61$ \( T^{20} + 12216 T^{16} + \cdots + 18\!\cdots\!04 \) Copy content Toggle raw display
$67$ \( (T^{10} - 6 T^{9} + 18 T^{8} - 272 T^{7} + \cdots + 8192)^{2} \) Copy content Toggle raw display
$71$ \( (T^{10} + 240 T^{8} + 19744 T^{6} + \cdots + 12845056)^{2} \) Copy content Toggle raw display
$73$ \( (T^{10} + 376 T^{8} + 46592 T^{6} + \cdots + 6160384)^{2} \) Copy content Toggle raw display
$79$ \( (T^{5} + 18 T^{4} + 8 T^{3} - 1264 T^{2} + \cdots + 1888)^{4} \) Copy content Toggle raw display
$83$ \( T^{20} + 81640 T^{16} + \cdots + 14370407252224 \) Copy content Toggle raw display
$89$ \( (T^{10} + 280 T^{8} + 15808 T^{6} + \cdots + 385024)^{2} \) Copy content Toggle raw display
$97$ \( (T^{10} - 592 T^{8} + 120560 T^{6} + \cdots - 3312193024)^{2} \) Copy content Toggle raw display
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