Properties

Label 4.13.b.b
Level 4
Weight 13
Character orbit 4.b
Analytic conductor 3.656
Analytic rank 0
Dimension 4
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 4 = 2^{2} \)
Weight: \( k \) = \( 13 \)
Character orbit: \([\chi]\) = 4.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(3.65597526911\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{12} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( 27 + \beta_{1} ) q^{2} \) \( + ( -\beta_{1} + \beta_{3} ) q^{3} \) \( + ( -380 + 33 \beta_{1} - 7 \beta_{2} - 2 \beta_{3} ) q^{4} \) \( + ( -4590 + 100 \beta_{1} - 20 \beta_{2} ) q^{5} \) \( + ( 3936 - 44 \beta_{1} - 60 \beta_{2} + 56 \beta_{3} ) q^{6} \) \( + ( 1410 \beta_{1} + 128 \beta_{2} - 2 \beta_{3} ) q^{7} \) \( + ( 112752 - 1492 \beta_{1} - 244 \beta_{2} - 216 \beta_{3} ) q^{8} \) \( + ( -341391 - 7320 \beta_{1} + 1464 \beta_{2} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + ( 27 + \beta_{1} ) q^{2} \) \( + ( -\beta_{1} + \beta_{3} ) q^{3} \) \( + ( -380 + 33 \beta_{1} - 7 \beta_{2} - 2 \beta_{3} ) q^{4} \) \( + ( -4590 + 100 \beta_{1} - 20 \beta_{2} ) q^{5} \) \( + ( 3936 - 44 \beta_{1} - 60 \beta_{2} + 56 \beta_{3} ) q^{6} \) \( + ( 1410 \beta_{1} + 128 \beta_{2} - 2 \beta_{3} ) q^{7} \) \( + ( 112752 - 1492 \beta_{1} - 244 \beta_{2} - 216 \beta_{3} ) q^{8} \) \( + ( -341391 - 7320 \beta_{1} + 1464 \beta_{2} ) q^{9} \) \( + ( 237350 - 7950 \beta_{1} - 1120 \beta_{2} - 320 \beta_{3} ) q^{10} \) \( + ( 31273 \beta_{1} + 2816 \beta_{2} - 297 \beta_{3} ) q^{11} \) \( + ( 1729920 - 10336 \beta_{1} - 4704 \beta_{2} + 2752 \beta_{3} ) q^{12} \) \( + ( -1830286 + 10260 \beta_{1} - 2052 \beta_{2} ) q^{13} \) \( + ( -4591296 + 33880 \beta_{1} - 7048 \beta_{2} - 2160 \beta_{3} ) q^{14} \) \( + ( -68050 \beta_{1} - 5760 \beta_{2} + 4690 \beta_{3} ) q^{15} \) \( + ( 9848896 + 63696 \beta_{1} + 19792 \beta_{2} - 10144 \beta_{3} ) q^{16} \) \( + ( 14811714 + 18760 \beta_{1} - 3752 \beta_{2} ) q^{17} \) \( + ( -35663253 - 95439 \beta_{1} + 81984 \beta_{2} + 23424 \beta_{3} ) q^{18} \) \( + ( -547101 \beta_{1} - 52992 \beta_{2} - 35811 \beta_{3} ) q^{19} \) \( + ( 40762440 - 19950 \beta_{1} + 53570 \beta_{2} - 8100 \beta_{3} ) q^{20} \) \( + ( 9806592 + 444720 \beta_{1} - 88944 \beta_{2} ) q^{21} \) \( + ( -102004320 + 756492 \beta_{1} - 139876 \beta_{2} - 61688 \beta_{3} ) q^{22} \) \( + ( 576182 \beta_{1} + 67456 \beta_{2} + 165834 \beta_{3} ) q^{23} \) \( + ( 177007104 + 631936 \beta_{1} - 210816 \beta_{2} + 141056 \beta_{3} ) q^{24} \) \( + ( -107462925 - 918000 \beta_{1} + 183600 \beta_{2} ) q^{25} \) \( + ( -12350394 - 2175022 \beta_{1} - 114912 \beta_{2} - 32832 \beta_{3} ) q^{26} \) \( + ( 5127198 \beta_{1} + 421632 \beta_{2} - 489246 \beta_{3} ) q^{27} \) \( + ( -1248000 - 5701440 \beta_{1} - 240448 \beta_{2} - 226688 \beta_{3} ) q^{28} \) \( + ( -110255310 - 4979420 \beta_{1} + 995884 \beta_{2} ) q^{29} \) \( + ( 224713920 - 1727000 \beta_{1} + 41160 \beta_{2} + 354800 \beta_{3} ) q^{30} \) \( + ( -3837336 \beta_{1} - 269312 \beta_{2} + 874904 \beta_{3} ) q^{31} \) \( + ( -300665088 + 14535360 \beta_{1} + 649408 \beta_{2} - 556416 \beta_{3} ) q^{32} \) \( + ( 436571520 + 11635800 \beta_{1} - 2327160 \beta_{2} ) q^{33} \) \( + ( 467692406 + 14181378 \beta_{1} - 210112 \beta_{2} - 60032 \beta_{3} ) q^{34} \) \( + ( -18737500 \beta_{1} - 1763840 \beta_{2} - 664740 \beta_{3} ) q^{35} \) \( + ( -2726406588 - 20893167 \beta_{1} + 820329 \beta_{2} + 1947678 \beta_{3} ) q^{36} \) \( + ( 549631634 + 16167060 \beta_{1} - 3233412 \beta_{2} ) q^{37} \) \( + ( 1756585440 - 12414204 \beta_{1} + 5116212 \beta_{2} - 1157544 \beta_{3} ) q^{38} \) \( + ( -5622578 \beta_{1} - 590976 \beta_{2} - 878158 \beta_{3} ) q^{39} \) \( + ( -164922400 + 51557400 \beta_{1} + 1807320 \beta_{2} - 76080 \beta_{3} ) q^{40} \) \( + ( 724006242 - 50226800 \beta_{1} + 10045360 \beta_{2} ) q^{41} \) \( + ( 1871462400 - 5136000 \beta_{1} - 4980864 \beta_{2} - 1423104 \beta_{3} ) q^{42} \) \( + ( 36428145 \beta_{1} + 3656192 \beta_{2} + 3789967 \beta_{3} ) q^{43} \) \( + ( -465125760 - 122816672 \beta_{1} - 4099744 \beta_{2} - 5683392 \beta_{3} ) q^{44} \) \( + ( -6895638030 - 540300 \beta_{1} + 108060 \beta_{2} ) q^{45} \) \( + ( -1762741824 + 10511688 \beta_{1} - 13727576 \beta_{2} + 8207408 \beta_{3} ) q^{46} \) \( + ( 116857084 \beta_{1} + 9825536 \beta_{2} - 8776188 \beta_{3} ) q^{47} \) \( + ( 9454295040 + 133697024 \beta_{1} - 18301440 \beta_{2} + 5088256 \beta_{3} ) q^{48} \) \( + ( 4418699809 + 69029280 \beta_{1} - 13805856 \beta_{2} ) q^{49} \) \( + ( -6218049375 - 76618125 \beta_{1} + 10281600 \beta_{2} + 2937600 \beta_{3} ) q^{50} \) \( + ( -28438978 \beta_{1} - 1080576 \beta_{2} + 16552642 \beta_{3} ) q^{51} \) \( + ( 4698780104 - 46905486 \beta_{1} + 15011746 \beta_{2} + 1887644 \beta_{3} ) q^{52} \) \( + ( -3737110446 - 120768140 \beta_{1} + 24153628 \beta_{2} ) q^{53} \) \( + ( -17023470912 + 132837672 \beta_{1} + 5743368 \beta_{2} - 34143888 \beta_{3} ) q^{54} \) \( + ( -395008350 \beta_{1} - 37347200 \beta_{2} - 15810850 \beta_{3} ) q^{55} \) \( + ( 11325090816 - 74451200 \beta_{1} + 50048768 \beta_{2} - 2280960 \beta_{3} ) q^{56} \) \( + ( 27919762560 + 84083400 \beta_{1} - 16816680 \beta_{2} ) q^{57} \) \( + ( -20966541946 + 57053202 \beta_{1} + 55769504 \beta_{2} + 15934144 \beta_{3} ) q^{58} \) \( + ( -85525003 \beta_{1} - 9061376 \beta_{2} - 14150133 \beta_{3} ) q^{59} \) \( + ( 8013792000 + 209019200 \beta_{1} - 10818240 \beta_{2} + 22860160 \beta_{3} ) q^{60} \) \( + ( -20941265998 + 405450900 \beta_{1} - 81090180 \beta_{2} ) q^{61} \) \( + ( 13087146240 - 109594144 \beta_{1} - 37412768 \beta_{2} + 53303616 \beta_{3} ) q^{62} \) \( + ( 416480610 \beta_{1} + 42408576 \beta_{2} + 50013726 \beta_{3} ) q^{63} \) \( + ( -41142533120 - 63726336 \beta_{1} - 50830080 \beta_{2} - 55220736 \beta_{3} ) q^{64} \) \( + ( 20262557700 - 230122000 \beta_{1} + 46024400 \beta_{2} ) q^{65} \) \( + ( 53825249280 + 45608640 \beta_{1} - 130320960 \beta_{2} - 37234560 \beta_{3} ) q^{66} \) \( + ( 48500751 \beta_{1} + 1480960 \beta_{2} - 32210191 \beta_{3} ) q^{67} \) \( + ( 1691370504 + 513459714 \beta_{1} - 99659854 \beta_{2} - 32865156 \beta_{3} ) q^{68} \) \( + ( -140497112832 - 987252720 \beta_{1} + 197450544 \beta_{2} ) q^{69} \) \( + ( 60543166080 - 436405200 \beta_{1} + 138659440 \beta_{2} - 9004000 \beta_{3} ) q^{70} \) \( + ( 1322286546 \beta_{1} + 123349632 \beta_{2} + 34559406 \beta_{3} ) q^{71} \) \( + ( -64303517328 - 2763352212 \beta_{1} + 32984652 \beta_{2} + 151882920 \beta_{3} ) q^{72} \) \( + ( 201025994594 - 271634040 \beta_{1} + 54326808 \beta_{2} ) q^{73} \) \( + ( 73248408486 + 6418418 \beta_{1} - 181071072 \beta_{2} - 51734592 \beta_{3} ) q^{74} \) \( + ( 774298125 \beta_{1} + 52876800 \beta_{2} - 192653325 \beta_{3} ) q^{75} \) \( + ( -62865866880 + 2739096864 \beta_{1} + 271895328 \beta_{2} - 6981696 \beta_{3} ) q^{76} \) \( + ( -209777990400 + 1406130000 \beta_{1} - 281226000 \beta_{2} ) q^{77} \) \( + ( 17705238720 - 117378712 \beta_{1} + 85784136 \beta_{2} - 39721232 \beta_{3} ) q^{78} \) \( + ( -5246886684 \beta_{1} - 457385728 \beta_{2} + 215643676 \beta_{3} ) q^{79} \) \( + ( -104514157440 + 505162400 \beta_{1} - 317850720 \beta_{2} - 96379200 \beta_{3} ) q^{80} \) \( + ( 272153087073 + 1107816120 \beta_{1} - 221563224 \beta_{2} ) q^{81} \) \( + ( -161911214506 + 2411626722 \beta_{1} + 562540160 \beta_{2} + 160725760 \beta_{3} ) q^{82} \) \( + ( -561327629 \beta_{1} - 25686016 \beta_{2} + 278781453 \beta_{3} ) q^{83} \) \( + ( 169795411968 + 908513280 \beta_{1} + 26701824 \beta_{2} - 96460800 \beta_{3} ) q^{84} \) \( + ( -46297406300 + 1395063000 \beta_{1} - 279012600 \beta_{2} ) q^{85} \) \( + ( -116003613024 + 798476140 \beta_{1} - 432144772 \beta_{2} + 153739080 \beta_{3} ) q^{86} \) \( + ( 3727305998 \beta_{1} + 286814592 \beta_{2} - 572345486 \beta_{3} ) q^{87} \) \( + ( 204369200640 - 1797806208 \beta_{1} + 1154409344 \beta_{2} - 85868288 \beta_{3} ) q^{88} \) \( + ( 181751085090 - 1989128120 \beta_{1} + 397825624 \beta_{2} ) q^{89} \) \( + ( -188134222650 - 6877483950 \beta_{1} + 6051360 \beta_{2} + 1728960 \beta_{3} ) q^{90} \) \( + ( -3839153820 \beta_{1} - 354967040 \beta_{2} - 65483620 \beta_{3} ) q^{91} \) \( + ( 288045192960 - 4729613248 \beta_{1} - 911757248 \beta_{2} + 339811200 \beta_{3} ) q^{92} \) \( + ( -780604400640 - 7309185600 \beta_{1} + 1461837120 \beta_{2} ) q^{93} \) \( + ( -386375869056 + 2980093776 \beta_{1} - 23658736 \beta_{2} - 648675104 \beta_{3} ) q^{94} \) \( + ( 10250608950 \beta_{1} + 941270400 \beta_{2} + 103365450 \beta_{3} ) q^{95} \) \( + ( 553459163136 + 6439438336 \beta_{1} - 1661122560 \beta_{2} - 102436864 \beta_{3} ) q^{96} \) \( + ( 607945078274 + 4358261160 \beta_{1} - 871652232 \beta_{2} ) q^{97} \) \( + ( 368693877627 + 2099316001 \beta_{1} - 773127936 \beta_{2} - 220893696 \beta_{3} ) q^{98} \) \( + ( 7730937753 \beta_{1} + 826315776 \beta_{2} + 1358535783 \beta_{3} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(4q \) \(\mathstrut +\mathstrut 108q^{2} \) \(\mathstrut -\mathstrut 1520q^{4} \) \(\mathstrut -\mathstrut 18360q^{5} \) \(\mathstrut +\mathstrut 15744q^{6} \) \(\mathstrut +\mathstrut 451008q^{8} \) \(\mathstrut -\mathstrut 1365564q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut +\mathstrut 108q^{2} \) \(\mathstrut -\mathstrut 1520q^{4} \) \(\mathstrut -\mathstrut 18360q^{5} \) \(\mathstrut +\mathstrut 15744q^{6} \) \(\mathstrut +\mathstrut 451008q^{8} \) \(\mathstrut -\mathstrut 1365564q^{9} \) \(\mathstrut +\mathstrut 949400q^{10} \) \(\mathstrut +\mathstrut 6919680q^{12} \) \(\mathstrut -\mathstrut 7321144q^{13} \) \(\mathstrut -\mathstrut 18365184q^{14} \) \(\mathstrut +\mathstrut 39395584q^{16} \) \(\mathstrut +\mathstrut 59246856q^{17} \) \(\mathstrut -\mathstrut 142653012q^{18} \) \(\mathstrut +\mathstrut 163049760q^{20} \) \(\mathstrut +\mathstrut 39226368q^{21} \) \(\mathstrut -\mathstrut 408017280q^{22} \) \(\mathstrut +\mathstrut 708028416q^{24} \) \(\mathstrut -\mathstrut 429851700q^{25} \) \(\mathstrut -\mathstrut 49401576q^{26} \) \(\mathstrut -\mathstrut 4992000q^{28} \) \(\mathstrut -\mathstrut 441021240q^{29} \) \(\mathstrut +\mathstrut 898855680q^{30} \) \(\mathstrut -\mathstrut 1202660352q^{32} \) \(\mathstrut +\mathstrut 1746286080q^{33} \) \(\mathstrut +\mathstrut 1870769624q^{34} \) \(\mathstrut -\mathstrut 10905626352q^{36} \) \(\mathstrut +\mathstrut 2198526536q^{37} \) \(\mathstrut +\mathstrut 7026341760q^{38} \) \(\mathstrut -\mathstrut 659689600q^{40} \) \(\mathstrut +\mathstrut 2896024968q^{41} \) \(\mathstrut +\mathstrut 7485849600q^{42} \) \(\mathstrut -\mathstrut 1860503040q^{44} \) \(\mathstrut -\mathstrut 27582552120q^{45} \) \(\mathstrut -\mathstrut 7050967296q^{46} \) \(\mathstrut +\mathstrut 37817180160q^{48} \) \(\mathstrut +\mathstrut 17674799236q^{49} \) \(\mathstrut -\mathstrut 24872197500q^{50} \) \(\mathstrut +\mathstrut 18795120416q^{52} \) \(\mathstrut -\mathstrut 14948441784q^{53} \) \(\mathstrut -\mathstrut 68093883648q^{54} \) \(\mathstrut +\mathstrut 45300363264q^{56} \) \(\mathstrut +\mathstrut 111679050240q^{57} \) \(\mathstrut -\mathstrut 83866167784q^{58} \) \(\mathstrut +\mathstrut 32055168000q^{60} \) \(\mathstrut -\mathstrut 83765063992q^{61} \) \(\mathstrut +\mathstrut 52348584960q^{62} \) \(\mathstrut -\mathstrut 164570132480q^{64} \) \(\mathstrut +\mathstrut 81050230800q^{65} \) \(\mathstrut +\mathstrut 215300997120q^{66} \) \(\mathstrut +\mathstrut 6765482016q^{68} \) \(\mathstrut -\mathstrut 561988451328q^{69} \) \(\mathstrut +\mathstrut 242172664320q^{70} \) \(\mathstrut -\mathstrut 257214069312q^{72} \) \(\mathstrut +\mathstrut 804103978376q^{73} \) \(\mathstrut +\mathstrut 292993633944q^{74} \) \(\mathstrut -\mathstrut 251463467520q^{76} \) \(\mathstrut -\mathstrut 839111961600q^{77} \) \(\mathstrut +\mathstrut 70820954880q^{78} \) \(\mathstrut -\mathstrut 418056629760q^{80} \) \(\mathstrut +\mathstrut 1088612348292q^{81} \) \(\mathstrut -\mathstrut 647644858024q^{82} \) \(\mathstrut +\mathstrut 679181647872q^{84} \) \(\mathstrut -\mathstrut 185189625200q^{85} \) \(\mathstrut -\mathstrut 464014452096q^{86} \) \(\mathstrut +\mathstrut 817476802560q^{88} \) \(\mathstrut +\mathstrut 727004340360q^{89} \) \(\mathstrut -\mathstrut 752536890600q^{90} \) \(\mathstrut +\mathstrut 1152180771840q^{92} \) \(\mathstrut -\mathstrut 3122417602560q^{93} \) \(\mathstrut -\mathstrut 1545503476224q^{94} \) \(\mathstrut +\mathstrut 2213836652544q^{96} \) \(\mathstrut +\mathstrut 2431780313096q^{97} \) \(\mathstrut +\mathstrut 1474775510508q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4}\mathstrut -\mathstrut \) \(x^{3}\mathstrut +\mathstrut \) \(139\) \(x^{2}\mathstrut +\mathstrut \) \(3741\) \(x\mathstrut +\mathstrut \) \(30480\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} + 6 \nu^{2} + 181 \nu + 3280 \)\()/64\)
\(\beta_{2}\)\(=\)\((\)\( -3 \nu^{3} - 18 \nu^{2} + 1505 \nu - 10352 \)\()/64\)
\(\beta_{3}\)\(=\)\((\)\( -27 \nu^{3} + 350 \nu^{2} - 5399 \nu - 52976 \)\()/64\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2}\mathstrut +\mathstrut \) \(3\) \(\beta_{1}\mathstrut +\mathstrut \) \(8\)\()/32\)
\(\nu^{2}\)\(=\)\((\)\(4\) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(111\) \(\beta_{1}\mathstrut -\mathstrut \) \(2216\)\()/32\)
\(\nu^{3}\)\(=\)\((\)\(-\)\(24\) \(\beta_{3}\mathstrut -\mathstrut \) \(187\) \(\beta_{2}\mathstrut +\mathstrut \) \(839\) \(\beta_{1}\mathstrut -\mathstrut \) \(93112\)\()/32\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
8.65015 15.9147i
8.65015 + 15.9147i
−8.15015 5.14519i
−8.15015 + 5.14519i
−6.60060 63.6587i 292.868i −4008.86 + 840.371i −15342.2 −18643.6 + 1933.10i 129787.i 79957.8 + 249652.i 445669. 101268. + 976664.i
3.2 −6.60060 + 63.6587i 292.868i −4008.86 840.371i −15342.2 −18643.6 1933.10i 129787.i 79957.8 249652.i 445669. 101268. 976664.i
3.3 60.6006 20.5808i 1288.37i 3248.86 2494.41i 6162.19 26515.6 + 78075.9i 44726.1i 145546. 218027.i −1.12845e6 373432. 126823.i
3.4 60.6006 + 20.5808i 1288.37i 3248.86 + 2494.41i 6162.19 26515.6 78075.9i 44726.1i 145546. + 218027.i −1.12845e6 373432. + 126823.i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
4.b Odd 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{3}^{4} \) \(\mathstrut +\mathstrut 1745664 T_{3}^{2} \) \(\mathstrut +\mathstrut 142371717120 \) acting on \(S_{13}^{\mathrm{new}}(4, [\chi])\).