Properties

Label 98.10.a.l
Level $98$
Weight $10$
Character orbit 98.a
Self dual yes
Analytic conductor $50.474$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 98 = 2 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 98.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(50.4735119441\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Defining polynomial: \( x^{6} - 2x^{5} - 7165x^{4} + 16168x^{3} + 12807408x^{2} - 32180328x - 5372164 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{2}\cdot 7^{6} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 16 q^{2} + (\beta_{3} + \beta_{2}) q^{3} + 256 q^{4} + ( - \beta_{5} - 3 \beta_{3} - 2 \beta_{2}) q^{5} + (16 \beta_{3} + 16 \beta_{2}) q^{6} + 4096 q^{8} + ( - \beta_{4} + 6 \beta_1 + 12094) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 16 q^{2} + (\beta_{3} + \beta_{2}) q^{3} + 256 q^{4} + ( - \beta_{5} - 3 \beta_{3} - 2 \beta_{2}) q^{5} + (16 \beta_{3} + 16 \beta_{2}) q^{6} + 4096 q^{8} + ( - \beta_{4} + 6 \beta_1 + 12094) q^{9} + ( - 16 \beta_{5} - 48 \beta_{3} - 32 \beta_{2}) q^{10} + ( - \beta_{4} + 11 \beta_1 + 18448) q^{11} + (256 \beta_{3} + 256 \beta_{2}) q^{12} + ( - 25 \beta_{5} + 237 \beta_{3} + 140 \beta_{2}) q^{13} + (16 \beta_{4} - 130 \beta_1 - 87170) q^{15} + 65536 q^{16} + ( - 100 \beta_{5} + 984 \beta_{3} + 3271 \beta_{2}) q^{17} + ( - 16 \beta_{4} + 96 \beta_1 + 193504) q^{18} + (3009 \beta_{3} - 995 \beta_{2}) q^{19} + ( - 256 \beta_{5} - 768 \beta_{3} - 512 \beta_{2}) q^{20} + ( - 16 \beta_{4} + 176 \beta_1 + 295168) q^{22} + (112 \beta_{4} + 178 \beta_1 + 40586) q^{23} + (4096 \beta_{3} + 4096 \beta_{2}) q^{24} + ( - 287 \beta_{4} - 50 \beta_1 + 2854320) q^{25} + ( - 400 \beta_{5} + 3792 \beta_{3} + 2240 \beta_{2}) q^{26} + (1580 \beta_{5} + 1976 \beta_{3} + 22876 \beta_{2}) q^{27} + (207 \beta_{4} + 498 \beta_1 + 1744037) q^{29} + (256 \beta_{4} - 2080 \beta_1 - 1394720) q^{30} + (2260 \beta_{5} + 11430 \beta_{3} - 51446 \beta_{2}) q^{31} + 1048576 q^{32} + (2670 \beta_{5} + 36958 \beta_{3} + 53218 \beta_{2}) q^{33} + ( - 1600 \beta_{5} + 15744 \beta_{3} + 52336 \beta_{2}) q^{34} + ( - 256 \beta_{4} + 1536 \beta_1 + 3096064) q^{36} + ( - 653 \beta_{4} + 2938 \beta_1 + 4056625) q^{37} + (48144 \beta_{3} - 15920 \beta_{2}) q^{38} + (210 \beta_{4} - 1500 \beta_1 + 7105410) q^{39} + ( - 4096 \beta_{5} - 12288 \beta_{3} - 8192 \beta_{2}) q^{40} + ( - 900 \beta_{5} - 26244 \beta_{3} + 151337 \beta_{2}) q^{41} + (1153 \beta_{4} - 7883 \beta_1 + 12456752) q^{43} + ( - 256 \beta_{4} + 2816 \beta_1 + 4722688) q^{44} + ( - 13009 \beta_{5} - 241987 \beta_{3} - 564518 \beta_{2}) q^{45} + (1792 \beta_{4} + 2848 \beta_1 + 649376) q^{46} + ( - 6920 \beta_{5} - 258162 \beta_{3} - 16242 \beta_{2}) q^{47} + (65536 \beta_{3} + 65536 \beta_{2}) q^{48} + ( - 4592 \beta_{4} - 800 \beta_1 + 45669120) q^{50} + ( - 1871 \beta_{4} - 3109 \beta_1 + 43373962) q^{51} + ( - 6400 \beta_{5} + 60672 \beta_{3} + 35840 \beta_{2}) q^{52} + (1310 \beta_{4} + 10700 \beta_1 + 22972348) q^{53} + (25280 \beta_{5} + 31616 \beta_{3} + 366016 \beta_{2}) q^{54} + ( - 14268 \beta_{5} - 383064 \beta_{3} - 1048636 \beta_{2}) q^{55} + (995 \beta_{4} + 14050 \beta_1 + 74948345) q^{57} + (3312 \beta_{4} + 7968 \beta_1 + 27904592) q^{58} + (38940 \beta_{5} - 168267 \beta_{3} - 277999 \beta_{2}) q^{59} + (4096 \beta_{4} - 33280 \beta_1 - 22315520) q^{60} + ( - 17575 \beta_{5} - 260397 \beta_{3} + 1618678 \beta_{2}) q^{61} + (36160 \beta_{5} + 182880 \beta_{3} - 823136 \beta_{2}) q^{62} + 16777216 q^{64} + ( - 2305 \beta_{4} - 35710 \beta_1 + 94318275) q^{65} + (42720 \beta_{5} + 591328 \beta_{3} + 851488 \beta_{2}) q^{66} + ( - 6542 \beta_{4} - 21068 \beta_1 - 33828738) q^{67} + ( - 25600 \beta_{5} + 251904 \beta_{3} + 837376 \beta_{2}) q^{68} + (8340 \beta_{5} + 489956 \beta_{3} - 2639644 \beta_{2}) q^{69} + (10270 \beta_{4} + 56350 \beta_1 + 102397488) q^{71} + ( - 4096 \beta_{4} + 24576 \beta_1 + 49537024) q^{72} + ( - 70010 \beta_{5} + 113934 \beta_{3} - 2404929 \beta_{2}) q^{73} + ( - 10448 \beta_{4} + 47008 \beta_1 + 64906000) q^{74} + (67164 \beta_{5} + 2429367 \beta_{3} + 10072083 \beta_{2}) q^{75} + (770304 \beta_{3} - 254720 \beta_{2}) q^{76} + (3360 \beta_{4} - 24000 \beta_1 + 113686560) q^{78} + (14942 \beta_{4} + 27998 \beta_1 - 30390664) q^{79} + ( - 65536 \beta_{5} - 196608 \beta_{3} - 131072 \beta_{2}) q^{80} + ( - 25313 \beta_{4} + 93198 \beta_1 - 72107470) q^{81} + ( - 14400 \beta_{5} - 419904 \beta_{3} + 2421392 \beta_{2}) q^{82} + ( - 43180 \beta_{5} - 119799 \beta_{3} - 6327087 \beta_{2}) q^{83} + ( - 5969 \beta_{4} - 281270 \beta_1 + 344990855) q^{85} + (18448 \beta_{4} - 126128 \beta_1 + 199308032) q^{86} + (52260 \beta_{5} + 2876942 \beta_{3} - 3064078 \beta_{2}) q^{87} + ( - 4096 \beta_{4} + 45056 \beta_1 + 75563008) q^{88} + ( - 113580 \beta_{5} - 1840248 \beta_{3} + 8660755 \beta_{2}) q^{89} + ( - 208144 \beta_{5} - 3871792 \beta_{3} - 9032288 \beta_{2}) q^{90} + (28672 \beta_{4} + 45568 \beta_1 + 10390016) q^{92} + (19806 \beta_{4} + 261084 \beta_1 + 31867458) q^{93} + ( - 110720 \beta_{5} - 4130592 \beta_{3} - 259872 \beta_{2}) q^{94} + (44140 \beta_{4} - 190970 \beta_1 - 218670950) q^{95} + (1048576 \beta_{3} + 1048576 \beta_{2}) q^{96} + (182930 \beta_{5} + 1014690 \beta_{3} + 3117303 \beta_{2}) q^{97} + ( - 70915 \beta_{4} + 323205 \beta_1 + 887229172) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 96 q^{2} + 1536 q^{4} + 24576 q^{8} + 72550 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 96 q^{2} + 1536 q^{4} + 24576 q^{8} + 72550 q^{9} + 110664 q^{11} - 522728 q^{15} + 393216 q^{16} + 1160800 q^{18} + 1770624 q^{22} + 243384 q^{23} + 17125446 q^{25} + 10463640 q^{29} - 8363648 q^{30} + 6291456 q^{32} + 18572800 q^{36} + 24332568 q^{37} + 42635880 q^{39} + 74758584 q^{43} + 28329984 q^{44} + 3894144 q^{46} + 274007136 q^{50} + 260246248 q^{51} + 137815308 q^{53} + 449663960 q^{57} + 167418240 q^{58} - 133818368 q^{60} + 100663296 q^{64} + 565976460 q^{65} - 202943376 q^{67} + 614292768 q^{71} + 297164800 q^{72} + 389321088 q^{74} + 682174080 q^{78} - 182370096 q^{79} - 432881842 q^{81} + 2070495732 q^{85} + 1196137344 q^{86} + 453279744 q^{88} + 62306304 q^{92} + 190722192 q^{93} - 1311555480 q^{95} + 5322586792 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} - 7165x^{4} + 16168x^{3} + 12807408x^{2} - 32180328x - 5372164 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 16557 \nu^{5} + 1210854 \nu^{4} - 100542547 \nu^{3} - 8636476956 \nu^{2} + 22368923366 \nu + 10243942375796 ) / 3101451260 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -10717\nu^{5} + 2982\nu^{4} + 64030099\nu^{3} - 46012148\nu^{2} - 91498140874\nu + 114577250368 ) / 1860870756 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 142991 \nu^{5} - 3910461 \nu^{4} + 1169730299 \nu^{3} + 13263914945 \nu^{2} - 2355511160234 \nu + 2902579423454 ) / 18608707560 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 26311 \nu^{5} - 1222782 \nu^{4} - 155577849 \nu^{3} + 8820525548 \nu^{2} - 21107028046 \nu - 10581294778128 ) / 1240580504 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 374123 \nu^{5} + 15601227 \nu^{4} + 3145018667 \nu^{3} - 58197203515 \nu^{2} - 6442323297722 \nu + 8276645371622 ) / 4652176890 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -2\beta_{4} - 12\beta_{2} - 5\beta _1 + 195 ) / 588 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 28\beta_{5} + 58\beta_{4} - 336\beta_{3} + 16\beta_{2} - 275\beta _1 + 1404621 ) / 588 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 1302\beta_{5} - 7114\beta_{4} + 19656\beta_{3} - 87480\beta_{2} - 18205\beta _1 - 543285 ) / 588 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 202328\beta_{5} + 210234\beta_{4} - 2380896\beta_{3} + 214016\beta_{2} - 983475\beta _1 + 5039754837 ) / 588 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 7715050 \beta_{5} - 25618702 \beta_{4} + 118217400 \beta_{3} - 522316448 \beta_{2} - 65172775 \beta _1 - 3252625455 ) / 588 \) Copy content Toggle raw display

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
2.67131
−61.3733
61.1162
58.2878
−58.5449
−0.157117
16.0000 −245.181 256.000 2662.60 −3922.90 0 4096.00 40431.0 42601.6
1.2 16.0000 −143.106 256.000 −474.347 −2289.70 0 4096.00 796.313 −7589.55
1.3 16.0000 −121.370 256.000 −2666.02 −1941.92 0 4096.00 −4952.27 −42656.3
1.4 16.0000 121.370 256.000 2666.02 1941.92 0 4096.00 −4952.27 42656.3
1.5 16.0000 143.106 256.000 474.347 2289.70 0 4096.00 796.313 7589.55
1.6 16.0000 245.181 256.000 −2662.60 3922.90 0 4096.00 40431.0 −42601.6
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 98.10.a.l 6
7.b odd 2 1 inner 98.10.a.l 6
7.c even 3 2 98.10.c.n 12
7.d odd 6 2 98.10.c.n 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
98.10.a.l 6 1.a even 1 1 trivial
98.10.a.l 6 7.b odd 2 1 inner
98.10.c.n 12 7.c even 3 2
98.10.c.n 12 7.d odd 6 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} - 95324T_{3}^{4} + 2418290244T_{3}^{2} - 18134891776800 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(98))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 16)^{6} \) Copy content Toggle raw display
$3$ \( T^{6} - 95324 T^{4} + \cdots - 18134891776800 \) Copy content Toggle raw display
$5$ \( T^{6} - 14422098 T^{4} + \cdots - 11\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{6} \) Copy content Toggle raw display
$11$ \( (T^{3} - 55332 T^{2} + \cdots + 13071858183056)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} - 13450472658 T^{4} + \cdots - 85\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{6} - 375850017702 T^{4} + \cdots - 83\!\cdots\!72 \) Copy content Toggle raw display
$19$ \( T^{6} - 720924860892 T^{4} + \cdots - 81\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( (T^{3} - 121692 T^{2} + \cdots - 95\!\cdots\!60)^{2} \) Copy content Toggle raw display
$29$ \( (T^{3} - 5231820 T^{2} + \cdots + 15\!\cdots\!76)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} - 116478776411856 T^{4} + \cdots - 11\!\cdots\!08 \) Copy content Toggle raw display
$37$ \( (T^{3} - 12166284 T^{2} + \cdots - 24\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} - 386100075738678 T^{4} + \cdots - 53\!\cdots\!68 \) Copy content Toggle raw display
$43$ \( (T^{3} - 37379292 T^{2} + \cdots + 20\!\cdots\!48)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots - 28\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{3} - 68907654 T^{2} + \cdots + 73\!\cdots\!68)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots - 50\!\cdots\!92 \) Copy content Toggle raw display
$61$ \( T^{6} + \cdots - 50\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{3} + 101471688 T^{2} + \cdots - 72\!\cdots\!12)^{2} \) Copy content Toggle raw display
$71$ \( (T^{3} - 307146384 T^{2} + \cdots + 12\!\cdots\!48)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots - 11\!\cdots\!32 \) Copy content Toggle raw display
$79$ \( (T^{3} + 91185048 T^{2} + \cdots - 37\!\cdots\!20)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots - 61\!\cdots\!52 \) Copy content Toggle raw display
$89$ \( T^{6} + \cdots - 13\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{6} + \cdots - 13\!\cdots\!72 \) Copy content Toggle raw display
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