Properties

Label 4005.2.a.o
Level 4005
Weight 2
Character orbit 4005.a
Self dual yes
Analytic conductor 31.980
Analytic rank 0
Dimension 7
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4005 = 3^{2} \cdot 5 \cdot 89 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4005.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(31.9800860095\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 445)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q + ( 1 - \beta_{1} ) q^{2} + ( 2 - \beta_{1} + \beta_{3} ) q^{4} - q^{5} + ( -2 - \beta_{6} ) q^{7} + ( 3 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} - \beta_{6} ) q^{8} +O(q^{10})\) \( q + ( 1 - \beta_{1} ) q^{2} + ( 2 - \beta_{1} + \beta_{3} ) q^{4} - q^{5} + ( -2 - \beta_{6} ) q^{7} + ( 3 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} - \beta_{6} ) q^{8} + ( -1 + \beta_{1} ) q^{10} + ( 1 - \beta_{4} - \beta_{5} + \beta_{6} ) q^{11} + ( -2 - \beta_{1} - \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} ) q^{13} + ( 3 \beta_{1} + 2 \beta_{3} + \beta_{4} - 2 \beta_{6} ) q^{14} + ( 4 - \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} - 3 \beta_{6} ) q^{16} + ( 2 + \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} ) q^{17} + ( -3 \beta_{1} - \beta_{2} + \beta_{6} ) q^{19} + ( -2 + \beta_{1} - \beta_{3} ) q^{20} + ( 1 - 2 \beta_{1} - \beta_{5} + \beta_{6} ) q^{22} + ( 1 - \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{23} + q^{25} + ( -\beta_{1} - 2 \beta_{2} + \beta_{4} + 2 \beta_{5} + \beta_{6} ) q^{26} + ( -2 + \beta_{1} - \beta_{2} + \beta_{4} - 2 \beta_{5} - 3 \beta_{6} ) q^{28} + ( -1 - 2 \beta_{1} - 2 \beta_{3} - 2 \beta_{4} + \beta_{5} + 3 \beta_{6} ) q^{29} + ( -2 + \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{5} + 3 \beta_{6} ) q^{31} + ( 6 + \beta_{2} + 5 \beta_{3} + 2 \beta_{4} - 5 \beta_{6} ) q^{32} + ( 5 + \beta_{2} + 3 \beta_{3} + 2 \beta_{4} - 4 \beta_{5} - 4 \beta_{6} ) q^{34} + ( 2 + \beta_{6} ) q^{35} + ( -1 - \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{4} ) q^{37} + ( 6 - 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + 2 \beta_{6} ) q^{38} + ( -3 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} + \beta_{6} ) q^{40} + ( 2 - 3 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} ) q^{41} + ( -5 + \beta_{1} + \beta_{2} + \beta_{5} ) q^{43} + ( 4 - \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} ) q^{44} + ( 3 - 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + 4 \beta_{5} + \beta_{6} ) q^{46} + ( 2 + 3 \beta_{2} - \beta_{4} - \beta_{6} ) q^{47} + ( 2 + \beta_{2} + \beta_{3} + 4 \beta_{6} ) q^{49} + ( 1 - \beta_{1} ) q^{50} + ( -2 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + \beta_{6} ) q^{52} + ( -\beta_{1} + 2 \beta_{2} - 3 \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} ) q^{53} + ( -1 + \beta_{4} + \beta_{5} - \beta_{6} ) q^{55} + ( 1 + \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - \beta_{5} - \beta_{6} ) q^{56} + ( -3 \beta_{1} - \beta_{2} - 3 \beta_{3} - \beta_{4} + 3 \beta_{5} + 6 \beta_{6} ) q^{58} + ( 2 - 2 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + \beta_{6} ) q^{59} + ( -1 - \beta_{1} - 3 \beta_{2} - 3 \beta_{3} - \beta_{4} + \beta_{6} ) q^{61} + ( -7 + \beta_{2} - 4 \beta_{3} - \beta_{4} - \beta_{5} + 5 \beta_{6} ) q^{62} + ( 7 - \beta_{1} + 4 \beta_{3} + \beta_{4} - 4 \beta_{5} - 7 \beta_{6} ) q^{64} + ( 2 + \beta_{1} + \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{6} ) q^{65} + ( 1 - \beta_{1} + 2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{67} + ( 12 + \beta_{1} + 8 \beta_{3} + 4 \beta_{4} - 6 \beta_{5} - 7 \beta_{6} ) q^{68} + ( -3 \beta_{1} - 2 \beta_{3} - \beta_{4} + 2 \beta_{6} ) q^{70} + ( 2 - 2 \beta_{1} + 2 \beta_{3} - 3 \beta_{4} ) q^{71} + ( 3 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - 4 \beta_{5} - 3 \beta_{6} ) q^{73} + ( 5 + 3 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{74} + ( 7 - 6 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{76} + ( -5 - 2 \beta_{2} + \beta_{4} + \beta_{5} - 3 \beta_{6} ) q^{77} + ( -1 - \beta_{1} + \beta_{2} - 3 \beta_{3} - \beta_{4} + 2 \beta_{5} + 3 \beta_{6} ) q^{79} + ( -4 + \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{5} + 3 \beta_{6} ) q^{80} + ( 6 - 5 \beta_{1} - 3 \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{82} + ( 8 - 4 \beta_{1} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} ) q^{83} + ( -2 - \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} ) q^{85} + ( -8 + 5 \beta_{1} - 2 \beta_{3} ) q^{86} + ( 4 - \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{6} ) q^{88} - q^{89} + ( 5 + 5 \beta_{1} + \beta_{2} + 2 \beta_{3} + 3 \beta_{4} - 5 \beta_{5} - \beta_{6} ) q^{91} + ( -1 - 7 \beta_{1} - 3 \beta_{2} - \beta_{3} + 2 \beta_{5} + 2 \beta_{6} ) q^{92} + ( 8 + \beta_{1} + 2 \beta_{2} + 3 \beta_{3} + 2 \beta_{4} - 3 \beta_{5} - 3 \beta_{6} ) q^{94} + ( 3 \beta_{1} + \beta_{2} - \beta_{6} ) q^{95} + ( 1 - \beta_{1} + \beta_{3} + 2 \beta_{4} - 2 \beta_{6} ) q^{97} + ( -5 - 6 \beta_{1} - 8 \beta_{3} - 4 \beta_{4} - 2 \beta_{5} + 7 \beta_{6} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7q + 4q^{2} + 8q^{4} - 7q^{5} - 16q^{7} + 12q^{8} + O(q^{10}) \) \( 7q + 4q^{2} + 8q^{4} - 7q^{5} - 16q^{7} + 12q^{8} - 4q^{10} + 10q^{11} - 7q^{13} - 3q^{14} + 10q^{16} + 13q^{17} - 7q^{19} - 8q^{20} + 2q^{22} + 13q^{23} + 7q^{25} - q^{26} - 21q^{28} + 4q^{29} + q^{31} + 13q^{32} + 10q^{34} + 16q^{35} - 5q^{37} + 40q^{38} - 12q^{40} - 5q^{41} - 31q^{43} + 21q^{44} + 16q^{46} + 14q^{47} + 19q^{49} + 4q^{50} + 13q^{53} - 10q^{55} + q^{56} + 17q^{58} + 14q^{59} + 3q^{61} - 26q^{62} + 14q^{64} + 7q^{65} + q^{67} + 35q^{68} + 3q^{70} + 8q^{71} + 9q^{73} + 35q^{74} + 40q^{76} - 42q^{77} + 9q^{79} - 10q^{80} + 29q^{82} + 42q^{83} - 13q^{85} - 35q^{86} + 30q^{88} - 7q^{89} + 31q^{91} - 19q^{92} + 37q^{94} + 7q^{95} - 7q^{97} - 9q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{7} - x^{6} - 8 x^{5} + 6 x^{4} + 19 x^{3} - 10 x^{2} - 12 x + 6\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} - 2 \)
\(\beta_{2}\)\(=\)\( \nu^{3} - \nu^{2} - 4 \nu + 2 \)
\(\beta_{3}\)\(=\)\( \nu^{4} - 5 \nu^{2} + 3 \)
\(\beta_{4}\)\(=\)\( -\nu^{4} + \nu^{3} + 4 \nu^{2} - 2 \nu - 2 \)
\(\beta_{5}\)\(=\)\( \nu^{5} - \nu^{4} - 6 \nu^{3} + 5 \nu^{2} + 8 \nu - 5 \)
\(\beta_{6}\)\(=\)\( \nu^{6} - \nu^{5} - 7 \nu^{4} + 5 \nu^{3} + 13 \nu^{2} - 4 \nu - 4 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{4} + \beta_{3} - \beta_{2} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\(\beta_{1} + 2\)
\(\nu^{3}\)\(=\)\(2 \beta_{4} + 2 \beta_{3} - \beta_{2} + \beta_{1} + 2\)
\(\nu^{4}\)\(=\)\(\beta_{3} + 5 \beta_{1} + 7\)
\(\nu^{5}\)\(=\)\(\beta_{5} + 8 \beta_{4} + 9 \beta_{3} - 2 \beta_{2} + 6 \beta_{1} + 10\)
\(\nu^{6}\)\(=\)\(\beta_{6} + \beta_{5} + 8 \beta_{3} + \beta_{2} + 23 \beta_{1} + 29\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
2.26266
−1.96388
1.89340
−1.49803
−1.07810
0.885013
0.498937
−2.11962 0 2.49279 −1.00000 0 −4.83304 −1.04452 0 2.11962
1.2 −0.856822 0 −1.26586 −1.00000 0 −0.580377 2.79826 0 0.856822
1.3 −0.584976 0 −1.65780 −1.00000 0 −2.74591 2.13973 0 0.584976
1.4 0.755898 0 −1.42862 −1.00000 0 0.0498231 −2.59169 0 −0.755898
1.5 1.83770 0 1.37716 −1.00000 0 −4.72699 −1.14460 0 −1.83770
1.6 2.21675 0 2.91399 −1.00000 0 −3.75132 2.02609 0 −2.21675
1.7 2.75106 0 5.56834 −1.00000 0 0.587818 9.81674 0 −2.75106
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4005.2.a.o 7
3.b odd 2 1 445.2.a.f 7
12.b even 2 1 7120.2.a.bj 7
15.d odd 2 1 2225.2.a.k 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
445.2.a.f 7 3.b odd 2 1
2225.2.a.k 7 15.d odd 2 1
4005.2.a.o 7 1.a even 1 1 trivial
7120.2.a.bj 7 12.b even 2 1

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(1\)
\(89\) \(1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4005))\):

\( T_{2}^{7} - 4 T_{2}^{6} - 3 T_{2}^{5} + 24 T_{2}^{4} - 8 T_{2}^{3} - 29 T_{2}^{2} + 6 T_{2} + 9 \)
\( T_{7}^{7} + 16 T_{7}^{6} + 94 T_{7}^{5} + 236 T_{7}^{4} + 189 T_{7}^{3} - 96 T_{7}^{2} - 76 T_{7} + 4 \)
\( T_{11}^{7} - 10 T_{11}^{6} + 14 T_{11}^{5} + 149 T_{11}^{4} - 639 T_{11}^{3} + 968 T_{11}^{2} - 608 T_{11} + 128 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - 4 T + 11 T^{2} - 24 T^{3} + 46 T^{4} - 77 T^{5} + 118 T^{6} - 171 T^{7} + 236 T^{8} - 308 T^{9} + 368 T^{10} - 384 T^{11} + 352 T^{12} - 256 T^{13} + 128 T^{14} \)
$3$ \( \)
$5$ \( ( 1 + T )^{7} \)
$7$ \( 1 + 16 T + 143 T^{2} + 908 T^{3} + 4508 T^{4} + 18272 T^{5} + 61958 T^{6} + 177804 T^{7} + 433706 T^{8} + 895328 T^{9} + 1546244 T^{10} + 2180108 T^{11} + 2403401 T^{12} + 1882384 T^{13} + 823543 T^{14} \)
$11$ \( 1 - 10 T + 91 T^{2} - 511 T^{3} + 2672 T^{4} - 10626 T^{5} + 41830 T^{6} - 136602 T^{7} + 460130 T^{8} - 1285746 T^{9} + 3556432 T^{10} - 7481551 T^{11} + 14655641 T^{12} - 17715610 T^{13} + 19487171 T^{14} \)
$13$ \( 1 + 7 T + 47 T^{2} + 168 T^{3} + 622 T^{4} + 2167 T^{5} + 9118 T^{6} + 36132 T^{7} + 118534 T^{8} + 366223 T^{9} + 1366534 T^{10} + 4798248 T^{11} + 17450771 T^{12} + 33787663 T^{13} + 62748517 T^{14} \)
$17$ \( 1 - 13 T + 108 T^{2} - 591 T^{3} + 3017 T^{4} - 14845 T^{5} + 78122 T^{6} - 344702 T^{7} + 1328074 T^{8} - 4290205 T^{9} + 14822521 T^{10} - 49360911 T^{11} + 153344556 T^{12} - 313788397 T^{13} + 410338673 T^{14} \)
$19$ \( 1 + 7 T + 81 T^{2} + 366 T^{3} + 2784 T^{4} + 9257 T^{5} + 61132 T^{6} + 176376 T^{7} + 1161508 T^{8} + 3341777 T^{9} + 19095456 T^{10} + 47697486 T^{11} + 200564019 T^{12} + 329321167 T^{13} + 893871739 T^{14} \)
$23$ \( 1 - 13 T + 144 T^{2} - 1105 T^{3} + 7943 T^{4} - 47855 T^{5} + 274740 T^{6} - 1359354 T^{7} + 6319020 T^{8} - 25315295 T^{9} + 96642481 T^{10} - 309224305 T^{11} + 926833392 T^{12} - 1924466557 T^{13} + 3404825447 T^{14} \)
$29$ \( 1 - 4 T + 113 T^{2} - 625 T^{3} + 6740 T^{4} - 40902 T^{5} + 266814 T^{6} - 1538922 T^{7} + 7737606 T^{8} - 34398582 T^{9} + 164381860 T^{10} - 442050625 T^{11} + 2317759837 T^{12} - 2379293284 T^{13} + 17249876309 T^{14} \)
$31$ \( 1 - T + 77 T^{2} - 34 T^{3} + 3570 T^{4} - 2367 T^{5} + 134894 T^{6} - 138904 T^{7} + 4181714 T^{8} - 2274687 T^{9} + 106353870 T^{10} - 31399714 T^{11} + 2204444627 T^{12} - 887503681 T^{13} + 27512614111 T^{14} \)
$37$ \( 1 + 5 T + 168 T^{2} + 803 T^{3} + 14161 T^{4} + 59989 T^{5} + 771494 T^{6} + 2748646 T^{7} + 28545278 T^{8} + 82124941 T^{9} + 717297133 T^{10} + 1504951283 T^{11} + 11649784776 T^{12} + 12828632045 T^{13} + 94931877133 T^{14} \)
$41$ \( 1 + 5 T + 92 T^{2} + 673 T^{3} + 7237 T^{4} + 46413 T^{5} + 387726 T^{6} + 2290138 T^{7} + 15896766 T^{8} + 78020253 T^{9} + 498781277 T^{10} + 1901737153 T^{11} + 10658770492 T^{12} + 23750521205 T^{13} + 194754273881 T^{14} \)
$43$ \( 1 + 31 T + 687 T^{2} + 10436 T^{3} + 129854 T^{4} + 1291529 T^{5} + 10962460 T^{6} + 77409324 T^{7} + 471385780 T^{8} + 2388037121 T^{9} + 10324301978 T^{10} + 35678607236 T^{11} + 100994800341 T^{12} + 195962254519 T^{13} + 271818611107 T^{14} \)
$47$ \( 1 - 14 T + 271 T^{2} - 2621 T^{3} + 31508 T^{4} - 240678 T^{5} + 2207362 T^{6} - 13822254 T^{7} + 103746014 T^{8} - 531657702 T^{9} + 3271255084 T^{10} - 12789643901 T^{11} + 62152496897 T^{12} - 150909014606 T^{13} + 506623120463 T^{14} \)
$53$ \( 1 - 13 T + 269 T^{2} - 2568 T^{3} + 34060 T^{4} - 265121 T^{5} + 2673394 T^{6} - 17174660 T^{7} + 141689882 T^{8} - 744724889 T^{9} + 5070750620 T^{10} - 20262755208 T^{11} + 112494587617 T^{12} - 288136694677 T^{13} + 1174711139837 T^{14} \)
$59$ \( 1 - 14 T + 253 T^{2} - 2446 T^{3} + 31472 T^{4} - 260310 T^{5} + 2563464 T^{6} - 17624256 T^{7} + 151244376 T^{8} - 906139110 T^{9} + 6463687888 T^{10} - 29639065006 T^{11} + 180875847647 T^{12} - 590527470974 T^{13} + 2488651484819 T^{14} \)
$61$ \( 1 - 3 T + 222 T^{2} - 759 T^{3} + 29403 T^{4} - 83651 T^{5} + 2524030 T^{6} - 6527094 T^{7} + 153965830 T^{8} - 311265371 T^{9} + 6673922343 T^{10} - 10508993319 T^{11} + 187500378822 T^{12} - 154561123083 T^{13} + 3142742836021 T^{14} \)
$67$ \( 1 - T + 328 T^{2} - 279 T^{3} + 53091 T^{4} - 39287 T^{5} + 5339384 T^{6} - 3265170 T^{7} + 357738728 T^{8} - 176359343 T^{9} + 15967808433 T^{10} - 5622162759 T^{11} + 442841035096 T^{12} - 90458382169 T^{13} + 6060711605323 T^{14} \)
$71$ \( 1 - 8 T + 187 T^{2} - 1328 T^{3} + 22776 T^{4} - 168048 T^{5} + 2047314 T^{6} - 13028752 T^{7} + 145359294 T^{8} - 847129968 T^{9} + 8151780936 T^{10} - 33746712368 T^{11} + 337390888637 T^{12} - 1024802271368 T^{13} + 9095120158391 T^{14} \)
$73$ \( 1 - 9 T + 286 T^{2} - 1613 T^{3} + 37567 T^{4} - 166989 T^{5} + 3689378 T^{6} - 14466938 T^{7} + 269324594 T^{8} - 889884381 T^{9} + 14614201639 T^{10} - 45806362733 T^{11} + 592898475598 T^{12} - 1362008036601 T^{13} + 11047398519097 T^{14} \)
$79$ \( 1 - 9 T + 430 T^{2} - 2867 T^{3} + 81347 T^{4} - 413191 T^{5} + 9305818 T^{6} - 38366618 T^{7} + 735159622 T^{8} - 2578725031 T^{9} + 40107243533 T^{10} - 111669882227 T^{11} + 1323134251570 T^{12} - 2187787099689 T^{13} + 19203908986159 T^{14} \)
$83$ \( 1 - 42 T + 1008 T^{2} - 16390 T^{3} + 206255 T^{4} - 2129142 T^{5} + 19880600 T^{6} - 179268272 T^{7} + 1650089800 T^{8} - 14667659238 T^{9} + 117933927685 T^{10} - 777841881190 T^{11} + 3970552968144 T^{12} - 13731495681498 T^{13} + 27136050989627 T^{14} \)
$89$ \( ( 1 + T )^{7} \)
$97$ \( 1 + 7 T + 609 T^{2} + 3724 T^{3} + 164856 T^{4} + 856227 T^{5} + 25710150 T^{6} + 108804964 T^{7} + 2493884550 T^{8} + 8056239843 T^{9} + 150459620088 T^{10} + 329683042444 T^{11} + 5229690216513 T^{12} + 5830804034503 T^{13} + 80798284478113 T^{14} \)
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