# Properties

 Label 150.11.b.a Level $150$ Weight $11$ Character orbit 150.b Analytic conductor $95.304$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$150 = 2 \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$11$$ Character orbit: $$[\chi]$$ $$=$$ 150.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$95.3035879011$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.3421020160000.10 Defining polynomial: $$x^{8} + 967 x^{4} + 194481$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{26}\cdot 3^{8}$$ Twist minimal: no (minimal twist has level 6) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{2} + ( 10 \beta_{1} + 3 \beta_{3} - \beta_{4} ) q^{3} + 512 q^{4} + ( 1344 + 21 \beta_{2} - 4 \beta_{7} ) q^{6} + ( 5663 \beta_{1} - 18 \beta_{3} + 48 \beta_{4} + 3 \beta_{6} ) q^{7} + 512 \beta_{3} q^{8} + ( -39753 - 1425 \beta_{2} - 21 \beta_{5} - 21 \beta_{7} ) q^{9} +O(q^{10})$$ $$q + \beta_{3} q^{2} + ( 10 \beta_{1} + 3 \beta_{3} - \beta_{4} ) q^{3} + 512 q^{4} + ( 1344 + 21 \beta_{2} - 4 \beta_{7} ) q^{6} + ( 5663 \beta_{1} - 18 \beta_{3} + 48 \beta_{4} + 3 \beta_{6} ) q^{7} + 512 \beta_{3} q^{8} + ( -39753 - 1425 \beta_{2} - 21 \beta_{5} - 21 \beta_{7} ) q^{9} + ( -3801 \beta_{2} - 105 \beta_{5} + 105 \beta_{7} ) q^{11} + ( 5120 \beta_{1} + 1536 \beta_{3} - 512 \beta_{4} ) q^{12} + ( 33925 \beta_{1} + 360 \beta_{3} - 960 \beta_{4} - 60 \beta_{6} ) q^{13} + ( 11230 \beta_{2} - 192 \beta_{5} + 192 \beta_{7} ) q^{14} + 262144 q^{16} + ( 708 \beta_{1} - 74304 \beta_{3} + 1416 \beta_{4} - 708 \beta_{6} ) q^{17} + ( -364800 \beta_{1} - 38745 \beta_{3} - 2688 \beta_{4} + 168 \beta_{6} ) q^{18} + ( 392182 + 582 \beta_{2} + 2328 \beta_{5} + 291 \beta_{7} ) q^{19} + ( 2407002 + 69505 \beta_{2} - 5639 \beta_{5} + 567 \beta_{7} ) q^{21} + ( -959616 \beta_{1} - 5040 \beta_{3} + 13440 \beta_{4} + 840 \beta_{6} ) q^{22} + ( 2478 \beta_{1} + 90336 \beta_{3} + 4956 \beta_{4} - 2478 \beta_{6} ) q^{23} + ( 688128 + 10752 \beta_{2} - 2048 \beta_{7} ) q^{24} + ( 69770 \beta_{2} + 3840 \beta_{5} - 3840 \beta_{7} ) q^{26} + ( -4310649 \beta_{1} - 247779 \beta_{3} + 33579 \beta_{4} - 4797 \beta_{6} ) q^{27} + ( 2899456 \beta_{1} - 9216 \beta_{3} + 24576 \beta_{4} + 1536 \beta_{6} ) q^{28} + ( -765969 \beta_{2} - 8841 \beta_{5} + 8841 \beta_{7} ) q^{29} + ( -5446462 + 18270 \beta_{2} + 73080 \beta_{5} + 9135 \beta_{7} ) q^{31} + 262144 \beta_{3} q^{32} + ( 3266613 \beta_{1} + 1510236 \beta_{3} + 39690 \beta_{4} - 15099 \beta_{6} ) q^{33} + ( -37771776 + 11328 \beta_{2} + 45312 \beta_{5} + 5664 \beta_{7} ) q^{34} + ( -20353536 - 729600 \beta_{2} - 10752 \beta_{5} - 10752 \beta_{7} ) q^{36} + ( 8740355 \beta_{1} + 102312 \beta_{3} - 272832 \beta_{4} - 17052 \beta_{6} ) q^{37} + ( 18624 \beta_{1} + 378214 \beta_{3} + 37248 \beta_{4} - 18624 \beta_{6} ) q^{38} + ( -54321810 - 654175 \beta_{2} - 34405 \beta_{5} - 11340 \beta_{7} ) q^{39} + ( -4081086 \beta_{2} + 40242 \beta_{5} - 40242 \beta_{7} ) q^{41} + ( 18190464 \beta_{1} + 2379786 \beta_{3} + 72576 \beta_{4} + 45112 \beta_{6} ) q^{42} + ( -58998875 \beta_{1} + 122094 \beta_{3} - 325584 \beta_{4} - 20349 \beta_{6} ) q^{43} + ( -1946112 \beta_{2} - 53760 \beta_{5} + 53760 \beta_{7} ) q^{44} + ( 47203584 + 39648 \beta_{2} + 158592 \beta_{5} + 19824 \beta_{7} ) q^{46} + ( 43380 \beta_{1} - 4185600 \beta_{3} + 86760 \beta_{4} - 43380 \beta_{6} ) q^{47} + ( 2621440 \beta_{1} + 786432 \beta_{3} - 262144 \beta_{4} ) q^{48} + ( 12514605 + 135336 \beta_{2} + 541344 \beta_{5} + 67668 \beta_{7} ) q^{49} + ( -177144192 + 8233380 \beta_{2} + 133812 \beta_{5} + 295092 \beta_{7} ) q^{51} + ( 17369600 \beta_{1} + 184320 \beta_{3} - 491520 \beta_{4} - 30720 \beta_{6} ) q^{52} + ( 178017 \beta_{1} + 9081864 \beta_{3} + 356034 \beta_{4} - 178017 \beta_{6} ) q^{53} + ( -120415680 - 8578125 \beta_{2} + 307008 \beta_{5} + 134316 \beta_{7} ) q^{54} + ( 5749760 \beta_{2} - 98304 \beta_{5} + 98304 \beta_{7} ) q^{56} + ( 132148060 \beta_{1} - 2830524 \beta_{3} - 392182 \beta_{4} - 54999 \beta_{6} ) q^{57} + ( -194956416 \beta_{1} - 424368 \beta_{3} + 1131648 \beta_{4} + 70728 \beta_{6} ) q^{58} + ( -17268267 \beta_{2} - 69819 \beta_{5} + 69819 \beta_{7} ) q^{59} + ( -296009686 + 117864 \beta_{2} + 471456 \beta_{5} + 58932 \beta_{7} ) q^{61} + ( 584640 \beta_{1} - 5884942 \beta_{3} + 1169280 \beta_{4} - 584640 \beta_{6} ) q^{62} + ( -114115713 \beta_{1} + 28899450 \beta_{3} - 1979652 \beta_{4} + 390171 \beta_{6} ) q^{63} + 134217728 q^{64} + ( 780861312 + 6735120 \beta_{2} + 966336 \beta_{5} + 158760 \beta_{7} ) q^{66} + ( 35972723 \beta_{1} + 898506 \beta_{3} - 2396016 \beta_{4} - 149751 \beta_{6} ) q^{67} + ( 362496 \beta_{1} - 38043648 \beta_{3} + 724992 \beta_{4} - 362496 \beta_{6} ) q^{68} + ( -149067072 + 36175230 \beta_{2} + 468342 \beta_{5} - 368778 \beta_{7} ) q^{69} + ( -14787066 \beta_{2} - 640794 \beta_{5} + 640794 \beta_{7} ) q^{71} + ( -186777600 \beta_{1} - 19837440 \beta_{3} - 1376256 \beta_{4} + 86016 \beta_{6} ) q^{72} + ( 817893001 \beta_{1} - 832032 \beta_{3} + 2218752 \beta_{4} + 138672 \beta_{6} ) q^{73} + ( 18026374 \beta_{2} + 1091328 \beta_{5} - 1091328 \beta_{7} ) q^{74} + ( 200797184 + 297984 \beta_{2} + 1191936 \beta_{5} + 148992 \beta_{7} ) q^{76} + ( 459438 \beta_{1} - 35848848 \beta_{3} + 918876 \beta_{4} - 459438 \beta_{6} ) q^{77} + ( -165992640 \beta_{1} - 53777490 \beta_{3} - 1451520 \beta_{4} + 275240 \beta_{6} ) q^{78} + ( -49820642 + 962178 \beta_{2} + 3848712 \beta_{5} + 481089 \beta_{7} ) q^{79} + ( 364741137 + 76465242 \beta_{2} + 5485914 \beta_{5} + 1192590 \beta_{7} ) q^{81} + ( -1049908992 \beta_{1} + 1931616 \beta_{3} - 5150976 \beta_{4} - 321936 \beta_{6} ) q^{82} + ( -1165077 \beta_{1} + 24958608 \beta_{3} - 2330154 \beta_{4} + 1165077 \beta_{6} ) q^{83} + ( 1232385024 + 35586560 \beta_{2} - 2887168 \beta_{5} + 290304 \beta_{7} ) q^{84} + ( -117346582 \beta_{2} + 1302336 \beta_{5} - 1302336 \beta_{7} ) q^{86} + ( -24612651 \beta_{1} + 136526292 \beta_{3} + 3341898 \beta_{4} - 3055035 \beta_{6} ) q^{87} + ( -491323392 \beta_{1} - 2580480 \beta_{3} + 6881280 \beta_{4} + 430080 \beta_{6} ) q^{88} + ( 120593298 \beta_{2} + 1761186 \beta_{5} - 1761186 \beta_{7} ) q^{89} + ( 2079308020 - 940500 \beta_{2} - 3762000 \beta_{5} - 470250 \beta_{7} ) q^{91} + ( 1268736 \beta_{1} + 46252032 \beta_{3} + 2537472 \beta_{4} - 1268736 \beta_{6} ) q^{92} + ( 3970781780 \beta_{1} - 142128336 \beta_{3} + 5446462 \beta_{4} - 1726515 \beta_{6} ) q^{93} + ( -2126369280 + 694080 \beta_{2} + 2776320 \beta_{5} + 347040 \beta_{7} ) q^{94} + ( 352321536 + 5505024 \beta_{2} - 1048576 \beta_{7} ) q^{96} + ( 4895130239 \beta_{1} + 1435608 \beta_{3} - 3828288 \beta_{4} - 239268 \beta_{6} ) q^{97} + ( 4330752 \beta_{1} + 9266541 \beta_{3} + 8661504 \beta_{4} - 4330752 \beta_{6} ) q^{98} + ( 656728128 + 271435689 \beta_{2} - 293391 \beta_{5} - 6000813 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 4096 q^{4} + 10752 q^{6} - 318024 q^{9} + O(q^{10})$$ $$8 q + 4096 q^{4} + 10752 q^{6} - 318024 q^{9} + 2097152 q^{16} + 3137456 q^{19} + 19256016 q^{21} + 5505024 q^{24} - 43571696 q^{31} - 302174208 q^{34} - 162828288 q^{36} - 434574480 q^{39} + 377628672 q^{46} + 100116840 q^{49} - 1417153536 q^{51} - 963325440 q^{54} - 2368077488 q^{61} + 1073741824 q^{64} + 6246890496 q^{66} - 1192536576 q^{69} + 1606377472 q^{76} - 398565136 q^{79} + 2917929096 q^{81} + 9859080192 q^{84} + 16634464160 q^{91} - 17010954240 q^{94} + 2818572288 q^{96} + 5253825024 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 967 x^{4} + 194481$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$2 \nu^{6} + 2816 \nu^{2}$$$$)/18963$$ $$\beta_{2}$$ $$=$$ $$($$$$-352 \nu^{7} - 7056 \nu^{5} - 192208 \nu^{3} - 3563280 \nu$$$$)/398223$$ $$\beta_{3}$$ $$=$$ $$($$$$352 \nu^{7} - 7056 \nu^{5} + 192208 \nu^{3} - 3563280 \nu$$$$)/398223$$ $$\beta_{4}$$ $$=$$ $$($$$$172 \nu^{7} - 7231 \nu^{6} - 1764 \nu^{5} + 166324 \nu^{3} - 3809680 \nu^{2} - 2483712 \nu$$$$)/132741$$ $$\beta_{5}$$ $$=$$ $$($$$$-680 \nu^{7} - 3528 \nu^{5} + 889056 \nu^{4} - 805736 \nu^{3} - 11338992 \nu + 429858576$$$$)/398223$$ $$\beta_{6}$$ $$=$$ $$($$$$-2048 \nu^{7} - 14448 \nu^{6} + 14112 \nu^{5} - 2276768 \nu^{3} - 7599648 \nu^{2} + 32612832 \nu$$$$)/132741$$ $$\beta_{7}$$ $$=$$ $$($$$$2048 \nu^{7} + 14112 \nu^{5} + 296352 \nu^{4} + 2276768 \nu^{3} + 32612832 \nu + 143286192$$$$)/132741$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$2 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} - 4 \beta_{4} + 15 \beta_{3} + 13 \beta_{2} - 2 \beta_{1}$$$$)/864$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{6} + 16 \beta_{4} - 6 \beta_{3} + 9296 \beta_{1}$$$$)/864$$ $$\nu^{3}$$ $$=$$ $$($$$$44 \beta_{7} - 44 \beta_{6} - 44 \beta_{5} + 88 \beta_{4} - 897 \beta_{3} + 853 \beta_{2} + 44 \beta_{1}$$$$)/864$$ $$\nu^{4}$$ $$=$$ $$($$$$43 \beta_{7} + 344 \beta_{5} + 86 \beta_{2} - 417744$$$$)/864$$ $$\nu^{5}$$ $$=$$ $$($$$$-505 \beta_{7} - 505 \beta_{6} + 505 \beta_{5} + 1010 \beta_{4} - 15978 \beta_{3} - 15473 \beta_{2} + 505 \beta_{1}$$$$)/432$$ $$\nu^{6}$$ $$=$$ $$($$$$-88 \beta_{6} - 1408 \beta_{4} + 528 \beta_{3} - 306047 \beta_{1}$$$$)/54$$ $$\nu^{7}$$ $$=$$ $$($$$$-24026 \beta_{7} + 24026 \beta_{6} + 24026 \beta_{5} - 48052 \beta_{4} + 978531 \beta_{3} - 954505 \beta_{2} - 24026 \beta_{1}$$$$)/864$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/150\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
149.1
 −3.61315 − 3.61315i −3.61315 + 3.61315i 2.90605 − 2.90605i 2.90605 + 2.90605i −2.90605 + 2.90605i −2.90605 − 2.90605i 3.61315 + 3.61315i 3.61315 − 3.61315i
−22.6274 −137.627 200.269i 512.000 0 3114.15 + 4531.57i 23226.5i −11585.2 −21166.4 + 55125.0i 0
149.2 −22.6274 −137.627 + 200.269i 512.000 0 3114.15 4531.57i 23226.5i −11585.2 −21166.4 55125.0i 0
149.3 −22.6274 18.8335 242.269i 512.000 0 −426.153 + 5481.92i 670.530i −11585.2 −58339.6 9125.53i 0
149.4 −22.6274 18.8335 + 242.269i 512.000 0 −426.153 5481.92i 670.530i −11585.2 −58339.6 + 9125.53i 0
149.5 22.6274 −18.8335 242.269i 512.000 0 −426.153 5481.92i 670.530i 11585.2 −58339.6 + 9125.53i 0
149.6 22.6274 −18.8335 + 242.269i 512.000 0 −426.153 + 5481.92i 670.530i 11585.2 −58339.6 9125.53i 0
149.7 22.6274 137.627 200.269i 512.000 0 3114.15 4531.57i 23226.5i 11585.2 −21166.4 55125.0i 0
149.8 22.6274 137.627 + 200.269i 512.000 0 3114.15 + 4531.57i 23226.5i 11585.2 −21166.4 + 55125.0i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 149.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 150.11.b.a 8
3.b odd 2 1 inner 150.11.b.a 8
5.b even 2 1 inner 150.11.b.a 8
5.c odd 4 1 6.11.b.a 4
5.c odd 4 1 150.11.d.a 4
15.d odd 2 1 inner 150.11.b.a 8
15.e even 4 1 6.11.b.a 4
15.e even 4 1 150.11.d.a 4
20.e even 4 1 48.11.e.d 4
40.i odd 4 1 192.11.e.g 4
40.k even 4 1 192.11.e.h 4
45.k odd 12 2 162.11.d.d 8
45.l even 12 2 162.11.d.d 8
60.l odd 4 1 48.11.e.d 4
120.q odd 4 1 192.11.e.h 4
120.w even 4 1 192.11.e.g 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.11.b.a 4 5.c odd 4 1
6.11.b.a 4 15.e even 4 1
48.11.e.d 4 20.e even 4 1
48.11.e.d 4 60.l odd 4 1
150.11.b.a 8 1.a even 1 1 trivial
150.11.b.a 8 3.b odd 2 1 inner
150.11.b.a 8 5.b even 2 1 inner
150.11.b.a 8 15.d odd 2 1 inner
150.11.d.a 4 5.c odd 4 1
150.11.d.a 4 15.e even 4 1
162.11.d.d 8 45.k odd 12 2
162.11.d.d 8 45.l even 12 2
192.11.e.g 4 40.i odd 4 1
192.11.e.g 4 120.w even 4 1
192.11.e.h 4 40.k even 4 1
192.11.e.h 4 120.q odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{4} + 539921288 T_{7}^{2} +$$$$24\!\cdots\!76$$ acting on $$S_{11}^{\mathrm{new}}(150, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -512 + T^{2} )^{4}$$
$3$ $$12157665459056928801 + 554440561171812 T^{2} + 11912925798 T^{4} + 159012 T^{6} + T^{8}$$
$5$ $$T^{8}$$
$7$ $$( 242551843253776 + 539921288 T^{2} + T^{4} )^{2}$$
$11$ $$($$$$21\!\cdots\!24$$$$+ 58313211264 T^{2} + T^{4} )^{2}$$
$13$ $$($$$$27\!\cdots\!00$$$$+ 123683520200 T^{2} + T^{4} )^{2}$$
$17$ $$($$$$32\!\cdots\!84$$$$- 7560967182336 T^{2} + T^{4} )^{2}$$
$19$ $$( -1189491369116 - 784364 T + T^{2} )^{4}$$
$23$ $$($$$$61\!\cdots\!24$$$$- 33055507478016 T^{2} + T^{4} )^{2}$$
$29$ $$($$$$20\!\cdots\!00$$$$+ 907304099736960 T^{2} + T^{4} )^{2}$$
$31$ $$( -1294078582786556 + 10892924 T + T^{2} )^{4}$$
$37$ $$($$$$18\!\cdots\!16$$$$+ 9855391123851848 T^{2} + T^{4} )^{2}$$
$41$ $$($$$$28\!\cdots\!04$$$$+ 23561404561257984 T^{2} + T^{4} )^{2}$$
$43$ $$($$$$52\!\cdots\!56$$$$+ 40830669647333192 T^{2} + T^{4} )^{2}$$
$47$ $$($$$$26\!\cdots\!00$$$$- 25124763600230400 T^{2} + T^{4} )^{2}$$
$53$ $$($$$$37\!\cdots\!04$$$$- 212636466457531776 T^{2} + T^{4} )^{2}$$
$59$ $$($$$$20\!\cdots\!04$$$$+ 324064557407447424 T^{2} + T^{4} )^{2}$$
$61$ $$( 32529703648081636 + 592019372 T + T^{2} )^{4}$$
$67$ $$($$$$12\!\cdots\!16$$$$+ 722522134817280968 T^{2} + T^{4} )^{2}$$
$71$ $$($$$$49\!\cdots\!00$$$$+ 1847488216292328960 T^{2} + T^{4} )^{2}$$
$73$ $$($$$$55\!\cdots\!00$$$$+ 5947174014564995720 T^{2} + T^{4} )^{2}$$
$79$ $$( -3668964988480567676 + 99641284 T + T^{2} )^{4}$$
$83$ $$($$$$57\!\cdots\!44$$$$- 5977139602070968704 T^{2} + T^{4} )^{2}$$
$89$ $$($$$$15\!\cdots\!84$$$$+ 27084125311735371264 T^{2} + T^{4} )^{2}$$
$97$ $$($$$$90\!\cdots\!16$$$$+$$$$19\!\cdots\!32$$$$T^{2} + T^{4} )^{2}$$