Properties

Label 574.2.f.a
Level $574$
Weight $2$
Character orbit 574.f
Analytic conductor $4.583$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [574,2,Mod(155,574)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(574, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("574.155");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 574 = 2 \cdot 7 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 574.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: \(4.58341307602\)
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: \( x^{20} + 40 x^{18} + 666 x^{16} + 6052 x^{14} + 33033 x^{12} + 112020 x^{10} + 235396 x^{8} + 296360 x^{6} + 208336 x^{4} + 71168 x^{2} + 8464 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{5} \)
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.

\(f(q)\) \(=\) \( q + \beta_{14} q^{2} + \beta_{13} q^{3} - q^{4} + \beta_{4} q^{5} - \beta_{15} q^{6} + \beta_{9} q^{7} - \beta_{14} q^{8} + ( - \beta_{18} + \beta_{14} - \beta_{10} - \beta_{9} + \beta_{8} - \beta_{3}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{14} q^{2} + \beta_{13} q^{3} - q^{4} + \beta_{4} q^{5} - \beta_{15} q^{6} + \beta_{9} q^{7} - \beta_{14} q^{8} + ( - \beta_{18} + \beta_{14} - \beta_{10} - \beta_{9} + \beta_{8} - \beta_{3}) q^{9} - \beta_{2} q^{10} + (\beta_{14} + \beta_{13} - \beta_{12} + 1) q^{11} - \beta_{13} q^{12} + (\beta_{18} - \beta_{17} - \beta_{14} - \beta_{13} + \beta_{3} - 1) q^{13} + \beta_{10} q^{14} + ( - \beta_{19} - 2 \beta_{10} + \beta_{8} - \beta_{7} - \beta_{6} - \beta_{4} - \beta_{3} + \beta_{2} - \beta_1) q^{15} + q^{16} + (\beta_{15} + 2 \beta_{10}) q^{17} + ( - \beta_{12} - \beta_{10} + \beta_{9} + \beta_{8} + \beta_{5} - 1) q^{18} + ( - \beta_{18} + \beta_{16} + \beta_{15} - \beta_{14} - \beta_{12} + \beta_{8} + \beta_{5} - \beta_{4} + \beta_{2} + \cdots + 1) q^{19}+ \cdots + ( - \beta_{19} - \beta_{18} - 2 \beta_{16} - 3 \beta_{15} + \beta_{14} - \beta_{12} + \beta_{11} + 2 \beta_{8} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 4 q^{3} - 20 q^{4} + 4 q^{6}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 4 q^{3} - 20 q^{4} + 4 q^{6} + 16 q^{11} + 4 q^{12} - 20 q^{13} + 4 q^{15} + 20 q^{16} - 4 q^{17} - 20 q^{18} + 12 q^{19} - 16 q^{22} + 8 q^{23} - 4 q^{24} - 20 q^{25} + 20 q^{26} - 16 q^{27} + 12 q^{29} + 4 q^{30} - 8 q^{31} - 4 q^{34} - 32 q^{37} + 12 q^{38} - 12 q^{41} - 16 q^{44} - 72 q^{45} + 12 q^{47} - 4 q^{48} + 80 q^{51} + 20 q^{52} - 16 q^{54} - 20 q^{55} + 48 q^{57} - 12 q^{58} - 8 q^{59} - 4 q^{60} + 16 q^{63} - 20 q^{64} + 20 q^{65} - 40 q^{66} - 4 q^{67} + 4 q^{68} - 8 q^{69} - 8 q^{71} + 20 q^{72} + 20 q^{75} - 12 q^{76} + 24 q^{78} + 24 q^{79} + 4 q^{81} + 12 q^{82} - 4 q^{85} + 8 q^{86} + 16 q^{88} - 48 q^{89} - 8 q^{92} - 8 q^{93} + 12 q^{94} + 28 q^{95} + 4 q^{96} + 16 q^{97} - 20 q^{98} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 40 x^{18} + 666 x^{16} + 6052 x^{14} + 33033 x^{12} + 112020 x^{10} + 235396 x^{8} + 296360 x^{6} + 208336 x^{4} + 71168 x^{2} + 8464 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 55 \nu^{18} + 2741 \nu^{16} + 52253 \nu^{14} + 493339 \nu^{12} + 2480856 \nu^{10} + 6624052 \nu^{8} + 8673880 \nu^{6} + 4454432 \nu^{4} + 373008 \nu^{2} - 75856 ) / 103936 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 233 \nu^{18} + 8231 \nu^{16} + 116659 \nu^{14} + 863681 \nu^{12} + 3653664 \nu^{10} + 9084892 \nu^{8} + 13245544 \nu^{6} + 11092992 \nu^{4} + 4930480 \nu^{2} + \cdots + 845328 ) / 25984 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 325 \nu^{19} - 10033 \nu^{17} - 112559 \nu^{15} - 518567 \nu^{13} - 334964 \nu^{11} + 5233896 \nu^{9} + 17443112 \nu^{7} + 19736768 \nu^{5} + 7720864 \nu^{3} + \cdots + 993904 \nu ) / 298816 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 255 \nu^{19} - 7912 \nu^{18} - 13834 \nu^{17} - 287937 \nu^{16} - 299067 \nu^{15} - 4238854 \nu^{14} - 3371116 \nu^{13} - 32882157 \nu^{12} - 21644252 \nu^{11} + \cdots - 28128080 ) / 597632 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 11 \nu^{18} - 393 \nu^{16} - 5641 \nu^{14} - 42263 \nu^{12} - 179864 \nu^{10} - 442500 \nu^{8} - 613752 \nu^{6} - 451360 \nu^{4} - 154832 \nu^{2} - 18032 ) / 512 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2703 \nu^{18} + 97149 \nu^{16} + 1404741 \nu^{14} + 10614163 \nu^{12} + 45562552 \nu^{10} + 112799700 \nu^{8} + 156174552 \nu^{6} + 112435488 \nu^{4} + \cdots + 4077616 ) / 103936 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 5940 \nu^{19} + 19803 \nu^{18} - 193716 \nu^{17} + 704053 \nu^{16} - 2398572 \nu^{15} + 10062017 \nu^{14} - 13806044 \nu^{13} + 75258323 \nu^{12} + \cdots + 57133104 ) / 2390528 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 6585 \nu^{19} - 9085 \nu^{18} + 241067 \nu^{17} - 325427 \nu^{16} + 3574147 \nu^{15} - 4695703 \nu^{14} + 27947045 \nu^{13} - 35528997 \nu^{12} + 125789496 \nu^{11} + \cdots - 13466960 ) / 2390528 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 6585 \nu^{19} + 9085 \nu^{18} + 241067 \nu^{17} + 325427 \nu^{16} + 3574147 \nu^{15} + 4695703 \nu^{14} + 27947045 \nu^{13} + 35528997 \nu^{12} + 125789496 \nu^{11} + \cdots + 13466960 ) / 2390528 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 240 \nu^{19} - 1035 \nu^{18} + 8588 \nu^{17} - 37881 \nu^{16} + 123960 \nu^{15} - 561545 \nu^{14} + 941052 \nu^{13} - 4390631 \nu^{12} + 4110288 \nu^{11} - 19761784 \nu^{10} + \cdots - 3168112 ) / 82432 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 240 \nu^{19} - 1035 \nu^{18} - 8588 \nu^{17} - 37881 \nu^{16} - 123960 \nu^{15} - 561545 \nu^{14} - 941052 \nu^{13} - 4390631 \nu^{12} - 4110288 \nu^{11} + \cdots - 3168112 ) / 82432 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 395 \nu^{19} + 971 \nu^{18} - 14149 \nu^{17} + 35281 \nu^{16} - 204161 \nu^{15} + 517625 \nu^{14} - 1544739 \nu^{13} + 3988383 \nu^{12} - 6686456 \nu^{11} + \cdots + 2423280 ) / 103936 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 49 \nu^{19} + 1707 \nu^{17} + 23595 \nu^{15} + 166805 \nu^{13} + 646568 \nu^{11} + 1352108 \nu^{9} + 1356904 \nu^{7} + 405344 \nu^{5} - 172816 \nu^{3} - 73904 \nu ) / 11776 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 395 \nu^{19} + 971 \nu^{18} + 14149 \nu^{17} + 35281 \nu^{16} + 204161 \nu^{15} + 517625 \nu^{14} + 1544739 \nu^{13} + 3988383 \nu^{12} + 6686456 \nu^{11} + 17588088 \nu^{10} + \cdots + 2423280 ) / 103936 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 2975 \nu^{19} - 4485 \nu^{18} + 109593 \nu^{17} - 167003 \nu^{16} + 1636189 \nu^{15} - 2528735 \nu^{14} + 12883359 \nu^{13} - 20275581 \nu^{12} + 58338344 \nu^{11} + \cdots - 17250000 ) / 341504 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 2975 \nu^{19} - 4485 \nu^{18} - 109593 \nu^{17} - 167003 \nu^{16} - 1636189 \nu^{15} - 2528735 \nu^{14} - 12883359 \nu^{13} - 20275581 \nu^{12} + \cdots - 17250000 ) / 341504 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 30681 \nu^{19} - 1633 \nu^{18} - 1124683 \nu^{17} - 53199 \nu^{16} - 16725555 \nu^{15} - 670611 \nu^{14} - 131580421 \nu^{13} - 4200329 \nu^{12} + \cdots - 20637072 ) / 2390528 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 46525 \nu^{19} - 1694135 \nu^{17} - 24931199 \nu^{15} - 192879897 \nu^{13} - 854612360 \nu^{11} - 2208556284 \nu^{9} - 3240488712 \nu^{7} + \cdots - 111068816 \nu ) / 2390528 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{12} - \beta_{10} + \beta_{9} + \beta_{8} + \beta_{5} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - \beta_{19} - \beta_{18} + \beta_{17} - \beta_{16} - 2 \beta_{15} + 2 \beta_{14} + 2 \beta_{13} - \beta_{10} - \beta_{9} + \beta_{8} - \beta_{3} - 6 \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{15} + \beta_{13} + 10 \beta_{12} + \beta_{11} + 13 \beta_{10} - 13 \beta_{9} - 9 \beta_{8} + 2 \beta_{6} - 9 \beta_{5} + 2 \beta_{3} + 2 \beta_{2} + 28 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 13 \beta_{19} + 11 \beta_{18} - 11 \beta_{17} + 11 \beta_{16} + 26 \beta_{15} - 32 \beta_{14} - 26 \beta_{13} - 2 \beta_{12} - 2 \beta_{11} + 19 \beta_{10} + 19 \beta_{9} - 15 \beta_{8} + 4 \beta_{5} + 2 \beta_{4} + 15 \beta_{3} + 48 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 2 \beta_{17} - 2 \beta_{16} - 23 \beta_{15} - 23 \beta_{13} - 100 \beta_{12} - 15 \beta_{11} - 145 \beta_{10} + 145 \beta_{9} + 85 \beta_{8} - 2 \beta_{7} - 36 \beta_{6} + 85 \beta_{5} - 28 \beta_{3} - 26 \beta_{2} - 244 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 141 \beta_{19} - 119 \beta_{18} + 109 \beta_{17} - 109 \beta_{16} - 282 \beta_{15} + 408 \beta_{14} + 282 \beta_{13} + 42 \beta_{12} + 24 \beta_{11} - 267 \beta_{10} - 267 \beta_{9} + 185 \beta_{8} - 66 \beta_{5} - 26 \beta_{4} + \cdots - 440 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 50 \beta_{17} + 50 \beta_{16} + 351 \beta_{15} + 351 \beta_{13} + 1024 \beta_{12} + 167 \beta_{11} + 1553 \beta_{10} - 1553 \beta_{9} - 857 \beta_{8} + 62 \beta_{7} + 512 \beta_{6} - 857 \beta_{5} + 324 \beta_{3} + 266 \beta_{2} + \cdots + 2360 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1453 \beta_{19} + 1311 \beta_{18} - 1073 \beta_{17} + 1073 \beta_{16} + 2938 \beta_{15} - 4872 \beta_{14} - 2938 \beta_{13} - 638 \beta_{12} - 204 \beta_{11} + 3331 \beta_{10} + 3331 \beta_{9} - 2153 \beta_{8} + \cdots + 4308 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 846 \beta_{17} - 846 \beta_{16} - 4639 \beta_{15} - 4639 \beta_{13} - 10648 \beta_{12} - 1687 \beta_{11} - 16501 \beta_{10} + 16501 \beta_{9} + 8961 \beta_{8} - 1134 \beta_{7} - 6624 \beta_{6} + 8961 \beta_{5} + \cdots - 24040 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 14777 \beta_{19} - 14563 \beta_{18} + 10669 \beta_{17} - 10669 \beta_{16} - 30458 \beta_{15} + 56536 \beta_{14} + 30458 \beta_{13} + 8482 \beta_{12} + 1420 \beta_{11} - 39399 \beta_{10} - 39399 \beta_{9} + \cdots - 43644 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 12114 \beta_{17} + 12114 \beta_{16} + 57307 \beta_{15} + 57307 \beta_{13} + 111924 \beta_{12} + 16459 \beta_{11} + 175665 \beta_{10} - 175665 \beta_{9} - 95465 \beta_{8} + 16750 \beta_{7} + \cdots + 251504 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 150573 \beta_{19} + 161991 \beta_{18} - 107529 \beta_{17} + 107529 \beta_{16} + 317498 \beta_{15} - 645552 \beta_{14} - 317498 \beta_{13} - 105278 \beta_{12} - 7428 \beta_{11} + 453611 \beta_{10} + \cdots + 451124 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 158478 \beta_{17} - 158478 \beta_{16} - 681751 \beta_{15} - 681751 \beta_{13} - 1186464 \beta_{12} - 159263 \beta_{11} - 1878605 \beta_{10} + 1878605 \beta_{9} + 1027201 \beta_{8} + \cdots - 2670360 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 1546529 \beta_{19} - 1799371 \beta_{18} + 1098637 \beta_{17} - 1098637 \beta_{16} - 3336522 \beta_{15} + 7293912 \beta_{14} + 3336522 \beta_{13} + 1255850 \beta_{12} + \cdots - 4726332 \beta_1 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 1962954 \beta_{17} + 1962954 \beta_{16} + 7922435 \beta_{15} + 7922435 \beta_{13} + 12664604 \beta_{12} + 1548099 \beta_{11} + 20186217 \beta_{10} - 20186217 \beta_{9} + \cdots + 28608496 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 16040533 \beta_{19} + 19944367 \beta_{18} - 11366705 \beta_{17} + 11366705 \beta_{16} + 35343706 \beta_{15} - 81803424 \beta_{14} - 35343706 \beta_{13} - 14614294 \beta_{12} + \cdots + 50011796 \beta_1 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 23470278 \beta_{17} - 23470278 \beta_{16} - 90635343 \beta_{15} - 90635343 \beta_{13} - 135954696 \beta_{12} - 15210823 \beta_{11} - 217826165 \beta_{10} + \cdots - 308327496 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 167991097 \beta_{19} - 220603843 \beta_{18} + 118904229 \beta_{17} - 118904229 \beta_{16} - 377026250 \beta_{15} + 912504200 \beta_{14} + 377026250 \beta_{13} + \cdots - 533296332 \beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/574\mathbb{Z}\right)^\times\).

\(n\) \(211\) \(493\)
\(\chi(n)\) \(-\beta_{14}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
155.1
2.90501i
2.17776i
2.11158i
1.98313i
0.789354i
0.488504i
1.07957i
1.30464i
1.93054i
3.31217i
2.90501i
2.17776i
2.11158i
1.98313i
0.789354i
0.488504i
1.07957i
1.30464i
1.93054i
3.31217i
1.00000i −2.05415 + 2.05415i −1.00000 1.47090i 2.05415 + 2.05415i −0.707107 + 0.707107i 1.00000i 5.43908i −1.47090
155.2 1.00000i −1.53991 + 1.53991i −1.00000 4.44564i 1.53991 + 1.53991i 0.707107 0.707107i 1.00000i 1.74263i −4.44564
155.3 1.00000i −1.49311 + 1.49311i −1.00000 2.90089i 1.49311 + 1.49311i −0.707107 + 0.707107i 1.00000i 1.45878i 2.90089
155.4 1.00000i −1.40228 + 1.40228i −1.00000 1.58615i 1.40228 + 1.40228i 0.707107 0.707107i 1.00000i 0.932805i 1.58615
155.5 1.00000i −0.558158 + 0.558158i −1.00000 1.10942i 0.558158 + 0.558158i −0.707107 + 0.707107i 1.00000i 2.37692i −1.10942
155.6 1.00000i −0.345424 + 0.345424i −1.00000 0.0906929i 0.345424 + 0.345424i 0.707107 0.707107i 1.00000i 2.76136i 0.0906929
155.7 1.00000i 0.763370 0.763370i −1.00000 2.03674i −0.763370 0.763370i −0.707107 + 0.707107i 1.00000i 1.83453i 2.03674
155.8 1.00000i 0.922516 0.922516i −1.00000 3.87079i −0.922516 0.922516i 0.707107 0.707107i 1.00000i 1.29793i 3.87079
155.9 1.00000i 1.36510 1.36510i −1.00000 1.10200i −1.36510 1.36510i 0.707107 0.707107i 1.00000i 0.726999i −1.10200
155.10 1.00000i 2.34205 2.34205i −1.00000 2.35731i −2.34205 2.34205i −0.707107 + 0.707107i 1.00000i 7.97044i −2.35731
337.1 1.00000i −2.05415 2.05415i −1.00000 1.47090i 2.05415 2.05415i −0.707107 0.707107i 1.00000i 5.43908i −1.47090
337.2 1.00000i −1.53991 1.53991i −1.00000 4.44564i 1.53991 1.53991i 0.707107 + 0.707107i 1.00000i 1.74263i −4.44564
337.3 1.00000i −1.49311 1.49311i −1.00000 2.90089i 1.49311 1.49311i −0.707107 0.707107i 1.00000i 1.45878i 2.90089
337.4 1.00000i −1.40228 1.40228i −1.00000 1.58615i 1.40228 1.40228i 0.707107 + 0.707107i 1.00000i 0.932805i 1.58615
337.5 1.00000i −0.558158 0.558158i −1.00000 1.10942i 0.558158 0.558158i −0.707107 0.707107i 1.00000i 2.37692i −1.10942
337.6 1.00000i −0.345424 0.345424i −1.00000 0.0906929i 0.345424 0.345424i 0.707107 + 0.707107i 1.00000i 2.76136i 0.0906929
337.7 1.00000i 0.763370 + 0.763370i −1.00000 2.03674i −0.763370 + 0.763370i −0.707107 0.707107i 1.00000i 1.83453i 2.03674
337.8 1.00000i 0.922516 + 0.922516i −1.00000 3.87079i −0.922516 + 0.922516i 0.707107 + 0.707107i 1.00000i 1.29793i 3.87079
337.9 1.00000i 1.36510 + 1.36510i −1.00000 1.10200i −1.36510 + 1.36510i 0.707107 + 0.707107i 1.00000i 0.726999i −1.10200
337.10 1.00000i 2.34205 + 2.34205i −1.00000 2.35731i −2.34205 + 2.34205i −0.707107 0.707107i 1.00000i 7.97044i −2.35731
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 155.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
41.c even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 574.2.f.a 20
41.c even 4 1 inner 574.2.f.a 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
574.2.f.a 20 1.a even 1 1 trivial
574.2.f.a 20 41.c even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{20} + 4 T_{3}^{19} + 8 T_{3}^{18} + 8 T_{3}^{17} + 134 T_{3}^{16} + 552 T_{3}^{15} + 1168 T_{3}^{14} + 848 T_{3}^{13} + 2217 T_{3}^{12} + 8724 T_{3}^{11} + 19784 T_{3}^{10} + 12344 T_{3}^{9} + 8580 T_{3}^{8} + 27568 T_{3}^{7} + \cdots + 8464 \) acting on \(S_{2}^{\mathrm{new}}(574, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{10} \) Copy content Toggle raw display
$3$ \( T^{20} + 4 T^{19} + 8 T^{18} + 8 T^{17} + \cdots + 8464 \) Copy content Toggle raw display
$5$ \( T^{20} + 60 T^{18} + 1426 T^{16} + \cdots + 3844 \) Copy content Toggle raw display
$7$ \( (T^{4} + 1)^{5} \) Copy content Toggle raw display
$11$ \( T^{20} - 16 T^{19} + 128 T^{18} + \cdots + 1327104 \) Copy content Toggle raw display
$13$ \( T^{20} + 20 T^{19} + \cdots + 325153024 \) Copy content Toggle raw display
$17$ \( T^{20} + 4 T^{19} + 8 T^{18} + 24 T^{17} + \cdots + 16 \) Copy content Toggle raw display
$19$ \( T^{20} - 12 T^{19} + \cdots + 402503962624 \) Copy content Toggle raw display
$23$ \( (T^{10} - 4 T^{9} - 128 T^{8} + \cdots - 300016)^{2} \) Copy content Toggle raw display
$29$ \( T^{20} - 12 T^{19} + \cdots + 657756684484 \) Copy content Toggle raw display
$31$ \( (T^{10} + 4 T^{9} - 152 T^{8} - 96 T^{7} + \cdots - 35896)^{2} \) Copy content Toggle raw display
$37$ \( (T^{10} + 16 T^{9} - 116 T^{8} + \cdots + 25919488)^{2} \) Copy content Toggle raw display
$41$ \( T^{20} + 12 T^{19} + \cdots + 13\!\cdots\!01 \) Copy content Toggle raw display
$43$ \( T^{20} + 580 T^{18} + \cdots + 24908414976 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 390197127614464 \) Copy content Toggle raw display
$53$ \( T^{20} - 356 T^{17} + \cdots + 56\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( (T^{10} + 4 T^{9} - 438 T^{8} + \cdots - 430560736)^{2} \) Copy content Toggle raw display
$61$ \( T^{20} + 768 T^{18} + \cdots + 14\!\cdots\!44 \) Copy content Toggle raw display
$67$ \( T^{20} + 4 T^{19} + \cdots + 40290096112704 \) Copy content Toggle raw display
$71$ \( T^{20} + 8 T^{19} + \cdots + 62\!\cdots\!84 \) Copy content Toggle raw display
$73$ \( T^{20} + 728 T^{18} + \cdots + 976713170944 \) Copy content Toggle raw display
$79$ \( T^{20} - 24 T^{19} + \cdots + 72\!\cdots\!96 \) Copy content Toggle raw display
$83$ \( (T^{10} - 322 T^{8} - 768 T^{7} + \cdots - 9790904)^{2} \) Copy content Toggle raw display
$89$ \( T^{20} + 48 T^{19} + \cdots + 71\!\cdots\!44 \) Copy content Toggle raw display
$97$ \( T^{20} - 16 T^{19} + \cdots + 10806018304 \) Copy content Toggle raw display
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