# Properties

 Label 2100.2.f.i Level 2100 Weight 2 Character orbit 2100.f Analytic conductor 16.769 Analytic rank 0 Dimension 16 CM no Inner twists 8

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2100.f (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$16.7685844245$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 8 x^{15} + 22 x^{14} - 4 x^{13} - 80 x^{12} - 84 x^{11} + 1324 x^{10} - 3800 x^{9} + 5642 x^{8} - 4872 x^{7} + 4136 x^{6} - 11608 x^{5} + 30032 x^{4} - 44288 x^{3} + 37232 x^{2} - 16848 x + 3204$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{10}\cdot 5^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{3} + ( \beta_{1} - \beta_{8} ) q^{7} + ( -1 - \beta_{7} - \beta_{9} ) q^{9} +O(q^{10})$$ $$q + \beta_{3} q^{3} + ( \beta_{1} - \beta_{8} ) q^{7} + ( -1 - \beta_{7} - \beta_{9} ) q^{9} + ( \beta_{5} + \beta_{7} + \beta_{9} ) q^{11} + ( -\beta_{1} - 2 \beta_{2} ) q^{13} -\beta_{12} q^{17} + ( \beta_{11} - \beta_{13} ) q^{19} + ( -1 - \beta_{5} + \beta_{6} + \beta_{13} ) q^{21} + ( \beta_{4} + \beta_{8} - \beta_{10} ) q^{23} + ( \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{12} ) q^{27} + ( 2 + 3 \beta_{5} - \beta_{7} + \beta_{9} ) q^{29} -\beta_{13} q^{31} + ( 3 \beta_{2} - \beta_{3} ) q^{33} -\beta_{8} q^{37} + ( -1 + 3 \beta_{5} + 2 \beta_{7} + 2 \beta_{9} ) q^{39} + ( 2 \beta_{6} + \beta_{13} ) q^{41} + ( -2 \beta_{8} - \beta_{15} ) q^{43} + ( -2 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} - 3 \beta_{12} ) q^{47} + ( -3 - 2 \beta_{11} ) q^{49} + ( \beta_{5} - \beta_{7} + 2 \beta_{9} ) q^{51} + ( -\beta_{8} + 2 \beta_{10} - \beta_{15} ) q^{53} + ( -\beta_{4} + \beta_{10} ) q^{57} + ( 2 \beta_{6} + \beta_{13} - \beta_{14} ) q^{59} + ( -2 \beta_{11} - \beta_{13} ) q^{61} + ( -\beta_{1} - \beta_{2} + \beta_{4} + 2 \beta_{8} + \beta_{12} ) q^{63} -3 \beta_{8} q^{67} + ( \beta_{6} + 3 \beta_{11} - 2 \beta_{13} ) q^{69} + ( 2 + \beta_{5} - 3 \beta_{7} - \beta_{9} ) q^{71} + \beta_{1} q^{73} + ( -\beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{8} + \beta_{10} - \beta_{12} ) q^{77} + ( 3 + 2 \beta_{9} ) q^{79} + ( -2 + 2 \beta_{5} + 2 \beta_{7} + 5 \beta_{9} ) q^{81} + ( \beta_{1} - \beta_{2} + 2 \beta_{3} + 2 \beta_{12} ) q^{83} + ( 8 \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{12} ) q^{87} + ( 2 \beta_{6} + \beta_{13} + \beta_{14} ) q^{89} + ( -4 - 2 \beta_{9} + 3 \beta_{11} - 2 \beta_{13} ) q^{91} + ( -\beta_{4} + \beta_{8} ) q^{93} + ( 5 \beta_{1} - 3 \beta_{2} ) q^{97} + ( 4 - 3 \beta_{5} - 2 \beta_{7} - 2 \beta_{9} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q - 24q^{9} + O(q^{10})$$ $$16q - 24q^{9} - 8q^{21} - 24q^{39} - 48q^{49} - 16q^{51} + 48q^{79} - 32q^{81} - 64q^{91} + 72q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 8 x^{15} + 22 x^{14} - 4 x^{13} - 80 x^{12} - 84 x^{11} + 1324 x^{10} - 3800 x^{9} + 5642 x^{8} - 4872 x^{7} + 4136 x^{6} - 11608 x^{5} + 30032 x^{4} - 44288 x^{3} + 37232 x^{2} - 16848 x + 3204$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$57403374489 \nu^{15} - 373367002804 \nu^{14} + 701222131854 \nu^{13} + 835306036507 \nu^{12} - 3358106117202 \nu^{11} - 9913427677824 \nu^{10} + 61257088041072 \nu^{9} - 125830353287956 \nu^{8} + 133263830063988 \nu^{7} - 76905276790596 \nu^{6} + 119964804746976 \nu^{5} - 485834790073352 \nu^{4} + 991542180607608 \nu^{3} - 1040225507966284 \nu^{2} + 554801595445380 \nu - 119141715042936$$$$)/ 663475361490$$ $$\beta_{2}$$ $$=$$ $$($$$$-77323022525 \nu^{15} + 626674939546 \nu^{14} - 1652501840140 \nu^{13} - 183401535319 \nu^{12} + 7127194982154 \nu^{11} + 8131441630236 \nu^{10} - 108330014842670 \nu^{9} + 281466093593488 \nu^{8} - 359667781317426 \nu^{7} + 240749551047996 \nu^{6} - 210575615232028 \nu^{5} + 871443150537926 \nu^{4} - 2212235837321132 \nu^{3} + 2826929145871240 \nu^{2} - 1800272872764336 \nu + 457970972564304$$$$)/ 663475361490$$ $$\beta_{3}$$ $$=$$ $$($$$$100088104905 \nu^{15} - 672606326953 \nu^{14} + 1324628699625 \nu^{13} + 1382862362872 \nu^{12} - 6334737350562 \nu^{11} - 16874919289398 \nu^{10} + 111469384632840 \nu^{9} - 234153879206074 \nu^{8} + 251496180131028 \nu^{7} - 145697330406498 \nu^{6} + 213655203154554 \nu^{5} - 883679203029908 \nu^{4} + 1846487578162146 \nu^{3} - 1963073630872930 \nu^{2} + 1054657479570048 \nu - 230259470125842$$$$)/ 663475361490$$ $$\beta_{4}$$ $$=$$ $$($$$$-91435174149446 \nu^{15} + 809758733801354 \nu^{14} - 2380725481593076 \nu^{13} + 471516558604255 \nu^{12} + 9760817625912960 \nu^{11} + 5885969225100684 \nu^{10} - 141700307198985008 \nu^{9} + 402819089391960698 \nu^{8} - 549347824171115100 \nu^{7} + 388411383997627392 \nu^{6} - 287670523503487168 \nu^{5} + 1152719245145689066 \nu^{4} - 3160229026930206008 \nu^{3} + 4324738922526229916 \nu^{2} - 2931975072018951048 \nu + 789943381126797366$$$$)/ 477038784911310$$ $$\beta_{5}$$ $$=$$ $$($$$$-2586850136 \nu^{15} + 16889035175 \nu^{14} - 31860826666 \nu^{13} - 37727839379 \nu^{12} + 153099296514 \nu^{11} + 446964096000 \nu^{10} - 2776835535788 \nu^{9} + 5701391057396 \nu^{8} - 6015190713816 \nu^{7} + 3448029446688 \nu^{6} - 5402044811956 \nu^{5} + 22006940984932 \nu^{4} - 44955950967560 \nu^{3} + 46937768878016 \nu^{2} - 24837347608704 \nu + 5340554258850$$$$)/ 11245345110$$ $$\beta_{6}$$ $$=$$ $$($$$$394067700008 \nu^{15} - 2620048717602 \nu^{14} + 5071372130419 \nu^{13} + 5644186621575 \nu^{12} - 24504385330236 \nu^{11} - 67341552918828 \nu^{10} + 434039817972950 \nu^{9} - 899428368512850 \nu^{8} + 947538852220986 \nu^{7} - 531807903614832 \nu^{6} + 820013525317330 \nu^{5} - 3431662430278170 \nu^{4} + 7093771930177982 \nu^{3} - 7395523643242350 \nu^{2} + 3820930691534844 \nu - 767042048246670$$$$)/ 1617080626818$$ $$\beta_{7}$$ $$=$$ $$($$$$-1508144322 \nu^{15} + 9702247024 \nu^{14} - 17541138297 \nu^{13} - 24143769229 \nu^{12} + 87076046064 \nu^{11} + 271718024049 \nu^{10} - 1594178988846 \nu^{9} + 3145831094029 \nu^{8} - 3140506689666 \nu^{7} + 1657608931605 \nu^{6} - 3002949942144 \nu^{5} + 12564085078403 \nu^{4} - 24827023794558 \nu^{3} + 24461932213372 \nu^{2} - 11723129533632 \nu + 2123885484981$$$$)/ 5622672555$$ $$\beta_{8}$$ $$=$$ $$($$$$28786349208 \nu^{15} - 203158500162 \nu^{14} + 438648529038 \nu^{13} + 314345478065 \nu^{12} - 2022060235560 \nu^{11} - 4393011662922 \nu^{10} + 34058987846944 \nu^{9} - 76610594463144 \nu^{8} + 87791944855140 \nu^{7} - 54064370422086 \nu^{6} + 65812112235744 \nu^{5} - 271315744323048 \nu^{4} + 603346777013884 \nu^{3} - 687096422851968 \nu^{2} + 397058058543384 \nu - 93251629981978$$$$)/ 94146197930$$ $$\beta_{9}$$ $$=$$ $$($$$$-51174056 \nu^{15} + 369355329 \nu^{14} - 833049106 \nu^{13} - 469586340 \nu^{12} + 3760534290 \nu^{11} + 7314904494 \nu^{10} - 62214651188 \nu^{9} + 145032796893 \nu^{8} - 171605433420 \nu^{7} + 108735949992 \nu^{6} - 121388663668 \nu^{5} + 497098132356 \nu^{4} - 1141261635848 \nu^{3} + 1345105698096 \nu^{2} - 802838700408 \nu + 193580214201$$$$)/ 137138355$$ $$\beta_{10}$$ $$=$$ $$($$$$237146115290732 \nu^{15} - 1721592748555913 \nu^{14} + 3865712879810812 \nu^{13} + 2342293690718270 \nu^{12} - 17768161105979250 \nu^{11} - 34730436558706908 \nu^{10} + 291242148865846016 \nu^{9} - 669246871731066416 \nu^{8} + 775475237234401080 \nu^{7} - 477188249024734434 \nu^{6} + 551512082253892276 \nu^{5} - 2318111849938045072 \nu^{4} + 5271595374806164556 \nu^{3} - 6074511014886719072 \nu^{2} + 3508792753291626816 \nu - 814057061299264242$$$$)/ 477038784911310$$ $$\beta_{11}$$ $$=$$ $$($$$$19477599 \nu^{15} - 132952136 \nu^{14} + 273792114 \nu^{13} + 235989785 \nu^{12} - 1271868450 \nu^{11} - 3100677486 \nu^{10} + 22097309352 \nu^{9} - 48356828852 \nu^{8} + 54292687260 \nu^{7} - 33010227468 \nu^{6} + 43220019252 \nu^{5} - 175894423504 \nu^{4} + 380852817012 \nu^{3} - 424709811584 \nu^{2} + 241278803772 \nu - 55853497044$$$$)/36431730$$ $$\beta_{12}$$ $$=$$ $$($$$$9185255669 \nu^{15} - 63157440526 \nu^{14} + 129797340514 \nu^{13} + 116532552115 \nu^{12} - 613884121680 \nu^{11} - 1488031599246 \nu^{10} + 10543806408542 \nu^{9} - 22788180451882 \nu^{8} + 25037732878050 \nu^{7} - 14684014240788 \nu^{6} + 19983226662772 \nu^{5} - 83617995398834 \nu^{4} + 179619253365752 \nu^{3} - 195752741941804 \nu^{2} + 106776542011392 \nu - 23187055339044$$$$)/ 16182325890$$ $$\beta_{13}$$ $$=$$ $$($$$$-63212058907 \nu^{15} + 428243944168 \nu^{14} - 860507807102 \nu^{13} - 835204536085 \nu^{12} + 4087346785905 \nu^{11} + 10424199014343 \nu^{10} - 71211400087861 \nu^{9} + 151867933915471 \nu^{8} - 165055373950875 \nu^{7} + 96026202241044 \nu^{6} - 135665741954696 \nu^{5} + 564422972324267 \nu^{4} - 1197020496070456 \nu^{3} + 1289895423572662 \nu^{2} - 695902825241166 \nu + 149299677075222$$$$)/ 98602477245$$ $$\beta_{14}$$ $$=$$ $$($$$$9295948386523 \nu^{15} - 67182473206802 \nu^{14} + 151948666431908 \nu^{13} + 84256855192265 \nu^{12} - 685081077669150 \nu^{11} - 1322694341328882 \nu^{10} + 11320914231054334 \nu^{9} - 26448505828879424 \nu^{8} + 31348735924618530 \nu^{7} - 19889688242778996 \nu^{6} + 22088591186773124 \nu^{5} - 90471698713299898 \nu^{4} + 208123080439317424 \nu^{3} - 245724234499198928 \nu^{2} + 146881976453299224 \nu - 35442084892574268$$$$)/ 8085403134090$$ $$\beta_{15}$$ $$=$$ $$($$$$20502731780 \nu^{15} - 141223574670 \nu^{14} + 291889704550 \nu^{13} + 254773590075 \nu^{12} - 1373424702120 \nu^{11} - 3293595733470 \nu^{10} + 23573948869640 \nu^{9} - 51256745585490 \nu^{8} + 56744878249860 \nu^{7} - 33632622506310 \nu^{6} + 44948659757680 \nu^{5} - 187118268731580 \nu^{4} + 403945266260300 \nu^{3} - 443744604971880 \nu^{2} + 244963342068600 \nu - 54161940862170$$$$)/ 17625670974$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{15} - \beta_{14} + \beta_{13} + \beta_{11} + \beta_{10} + \beta_{8} + \beta_{6} - 2 \beta_{4} - 5 \beta_{3} + 5 \beta_{2} + 5$$$$)/10$$ $$\nu^{2}$$ $$=$$ $$($$$$-6 \beta_{13} + 8 \beta_{11} - 2 \beta_{10} + 10 \beta_{9} + 3 \beta_{8} + 5 \beta_{7} + 10 \beta_{5} - \beta_{4} - 10 \beta_{1} + 15$$$$)/10$$ $$\nu^{3}$$ $$=$$ $$($$$$11 \beta_{15} - 3 \beta_{14} + 6 \beta_{13} - 10 \beta_{12} - 6 \beta_{11} - 5 \beta_{10} - 15 \beta_{9} + 20 \beta_{8} + 8 \beta_{6} - 15 \beta_{5} - 40 \beta_{3} + 15 \beta_{2} - 40 \beta_{1} - 10$$$$)/10$$ $$\nu^{4}$$ $$=$$ $$($$$$-15 \beta_{15} - 16 \beta_{14} - 8 \beta_{13} + 20 \beta_{12} + 28 \beta_{11} - 4 \beta_{10} - 15 \beta_{9} - 29 \beta_{8} + 30 \beta_{7} + 36 \beta_{6} + 30 \beta_{5} - 2 \beta_{4} + 60 \beta_{3} - 40 \beta_{2} + 20 \beta_{1} - 75$$$$)/10$$ $$\nu^{5}$$ $$=$$ $$($$$$5 \beta_{15} + 9 \beta_{14} + 30 \beta_{13} + 10 \beta_{12} - 101 \beta_{11} - 76 \beta_{10} - 75 \beta_{9} + 74 \beta_{8} - 75 \beta_{7} - 44 \beta_{6} - 150 \beta_{5} + 47 \beta_{4} - 20 \beta_{3} - 165 \beta_{2} - 145 \beta_{1} + 130$$$$)/10$$ $$\nu^{6}$$ $$=$$ $$($$$$-117 \beta_{15} - 78 \beta_{14} + 204 \beta_{13} + 120 \beta_{12} - 170 \beta_{11} + 190 \beta_{10} - 455 \beta_{9} - 285 \beta_{8} - 120 \beta_{7} + 228 \beta_{6} - 260 \beta_{5} - 130 \beta_{4} + 240 \beta_{3} + 330 \beta_{2} + 370 \beta_{1} - 635$$$$)/10$$ $$\nu^{7}$$ $$=$$ $$($$$$-290 \beta_{15} + 277 \beta_{14} - 625 \beta_{13} + 435 \beta_{12} + 187 \beta_{11} - 220 \beta_{10} + 1260 \beta_{9} - 440 \beta_{8} + 210 \beta_{7} - 892 \beta_{6} + 210 \beta_{5} + 120 \beta_{4} + 1220 \beta_{3} - 1085 \beta_{2} + 230 \beta_{1} + 2350$$$$)/10$$ $$\nu^{8}$$ $$=$$ $$($$$$357 \beta_{15} - 4 \beta_{14} + 544 \beta_{13} - 640 \beta_{12} - 956 \beta_{11} + 658 \beta_{10} - 1030 \beta_{9} + 643 \beta_{8} - 700 \beta_{7} + 24 \beta_{6} - 1330 \beta_{5} - 406 \beta_{4} - 2280 \beta_{3} + 2660 \beta_{2} - 220 \beta_{1} - 350$$$$)/5$$ $$\nu^{9}$$ $$=$$ $$($$$$-2255 \beta_{15} + 317 \beta_{14} - 6377 \beta_{13} + 2685 \beta_{12} + 9103 \beta_{11} + 1078 \beta_{10} + 11145 \beta_{9} - 4907 \beta_{8} + 7800 \beta_{7} - 1022 \beta_{6} + 12000 \beta_{5} - 626 \beta_{4} + 7930 \beta_{3} - 1675 \beta_{2} + 5700 \beta_{1} + 4685$$$$)/10$$ $$\nu^{10}$$ $$=$$ $$($$$$5586 \beta_{15} + 1140 \beta_{14} + 888 \beta_{13} - 6600 \beta_{12} - 4604 \beta_{11} - 3589 \beta_{10} - 700 \beta_{9} + 14511 \beta_{8} - 3560 \beta_{7} - 3540 \beta_{6} - 6265 \beta_{5} + 2308 \beta_{4} - 22200 \beta_{3} + 2550 \beta_{2} - 16070 \beta_{1} + 7785$$$$)/5$$ $$\nu^{11}$$ $$=$$ $$($$$$-5570 \beta_{15} - 8349 \beta_{14} + 2994 \beta_{13} + 4915 \beta_{12} + 16869 \beta_{11} + 9605 \beta_{10} - 13035 \beta_{9} - 15020 \beta_{8} + 14355 \beta_{7} + 27454 \beta_{6} + 22275 \beta_{5} - 5775 \beta_{4} + 14320 \beta_{3} + 14550 \beta_{2} + 21445 \beta_{1} - 67565$$$$)/5$$ $$\nu^{12}$$ $$=$$ $$($$$$5799 \beta_{15} + 14338 \beta_{14} - 21214 \beta_{13} + 7090 \beta_{12} - 3310 \beta_{11} - 59480 \beta_{10} + 50145 \beta_{9} + 37775 \beta_{8} - 1920 \beta_{7} - 45228 \beta_{6} - 4710 \beta_{5} + 37190 \beta_{4} + 20820 \beta_{3} - 149450 \beta_{2} - 73520 \beta_{1} + 114765$$$$)/5$$ $$\nu^{13}$$ $$=$$ $$($$$$9313 \beta_{15} - 54120 \beta_{14} + 196806 \beta_{13} - 36820 \beta_{12} - 176263 \beta_{11} + 93127 \beta_{10} - 393315 \beta_{9} - 11168 \beta_{8} - 143910 \beta_{7} + 177650 \beta_{6} - 235755 \beta_{5} - 56444 \beta_{4} - 122560 \beta_{3} + 273750 \beta_{2} + 75055 \beta_{1} - 507940$$$$)/5$$ $$\nu^{14}$$ $$=$$ $$($$$$-285081 \beta_{15} + 87528 \beta_{14} - 390576 \beta_{13} + 411810 \beta_{12} + 404846 \beta_{11} - 129964 \beta_{10} + 764435 \beta_{9} - 581109 \beta_{8} + 335460 \beta_{7} - 278208 \beta_{6} + 539420 \beta_{5} + 82738 \beta_{4} + 1328040 \beta_{3} - 972510 \beta_{2} + 485270 \beta_{1} + 817745$$$$)/5$$ $$\nu^{15}$$ $$=$$ $$($$$$661913 \beta_{15} + 132176 \beta_{14} + 973606 \beta_{13} - 926670 \beta_{12} - 1913782 \beta_{11} + 157210 \beta_{10} - 1573125 \beta_{9} + 1431485 \beta_{8} - 1576980 \beta_{7} - 416366 \beta_{6} - 2554170 \beta_{5} - 92070 \beta_{4} - 3004010 \beta_{3} + 1851230 \beta_{2} - 1263050 \beta_{1} + 580685$$$$)/5$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2100\mathbb{Z}\right)^\times$$.

 $$n$$ $$701$$ $$1051$$ $$1177$$ $$1501$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1049.1
 0.365728 + 1.53127i 0.904362 − 0.230889i 0.904362 + 0.230889i 0.365728 − 1.53127i −2.44874 + 1.42852i 1.59751 + 0.247497i 1.59751 − 0.247497i −2.44874 − 1.42852i 1.77273 + 1.86553i 1.07850 − 0.189519i 1.07850 + 0.189519i 1.77273 − 1.86553i 1.93465 + 0.387145i −1.20474 + 0.913233i −1.20474 − 0.913233i 1.93465 − 0.387145i
0 −1.14412 1.30038i 0 0 0 1.41421 2.23607i 0 −0.381966 + 2.97558i 0
1049.2 0 −1.14412 1.30038i 0 0 0 1.41421 + 2.23607i 0 −0.381966 + 2.97558i 0
1049.3 0 −1.14412 + 1.30038i 0 0 0 1.41421 2.23607i 0 −0.381966 2.97558i 0
1049.4 0 −1.14412 + 1.30038i 0 0 0 1.41421 + 2.23607i 0 −0.381966 2.97558i 0
1049.5 0 −0.437016 1.67601i 0 0 0 −1.41421 2.23607i 0 −2.61803 + 1.46489i 0
1049.6 0 −0.437016 1.67601i 0 0 0 −1.41421 + 2.23607i 0 −2.61803 + 1.46489i 0
1049.7 0 −0.437016 + 1.67601i 0 0 0 −1.41421 2.23607i 0 −2.61803 1.46489i 0
1049.8 0 −0.437016 + 1.67601i 0 0 0 −1.41421 + 2.23607i 0 −2.61803 1.46489i 0
1049.9 0 0.437016 1.67601i 0 0 0 1.41421 2.23607i 0 −2.61803 1.46489i 0
1049.10 0 0.437016 1.67601i 0 0 0 1.41421 + 2.23607i 0 −2.61803 1.46489i 0
1049.11 0 0.437016 + 1.67601i 0 0 0 1.41421 2.23607i 0 −2.61803 + 1.46489i 0
1049.12 0 0.437016 + 1.67601i 0 0 0 1.41421 + 2.23607i 0 −2.61803 + 1.46489i 0
1049.13 0 1.14412 1.30038i 0 0 0 −1.41421 2.23607i 0 −0.381966 2.97558i 0
1049.14 0 1.14412 1.30038i 0 0 0 −1.41421 + 2.23607i 0 −0.381966 2.97558i 0
1049.15 0 1.14412 + 1.30038i 0 0 0 −1.41421 2.23607i 0 −0.381966 + 2.97558i 0
1049.16 0 1.14412 + 1.30038i 0 0 0 −1.41421 + 2.23607i 0 −0.381966 + 2.97558i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1049.16 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
7.b odd 2 1 inner
15.d odd 2 1 inner
21.c even 2 1 inner
35.c odd 2 1 inner
105.g even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2100.2.f.i 16
3.b odd 2 1 inner 2100.2.f.i 16
5.b even 2 1 inner 2100.2.f.i 16
5.c odd 4 1 2100.2.d.k 8
5.c odd 4 1 2100.2.d.l yes 8
7.b odd 2 1 inner 2100.2.f.i 16
15.d odd 2 1 inner 2100.2.f.i 16
15.e even 4 1 2100.2.d.k 8
15.e even 4 1 2100.2.d.l yes 8
21.c even 2 1 inner 2100.2.f.i 16
35.c odd 2 1 inner 2100.2.f.i 16
35.f even 4 1 2100.2.d.k 8
35.f even 4 1 2100.2.d.l yes 8
105.g even 2 1 inner 2100.2.f.i 16
105.k odd 4 1 2100.2.d.k 8
105.k odd 4 1 2100.2.d.l yes 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2100.2.d.k 8 5.c odd 4 1
2100.2.d.k 8 15.e even 4 1
2100.2.d.k 8 35.f even 4 1
2100.2.d.k 8 105.k odd 4 1
2100.2.d.l yes 8 5.c odd 4 1
2100.2.d.l yes 8 15.e even 4 1
2100.2.d.l yes 8 35.f even 4 1
2100.2.d.l yes 8 105.k odd 4 1
2100.2.f.i 16 1.a even 1 1 trivial
2100.2.f.i 16 3.b odd 2 1 inner
2100.2.f.i 16 5.b even 2 1 inner
2100.2.f.i 16 7.b odd 2 1 inner
2100.2.f.i 16 15.d odd 2 1 inner
2100.2.f.i 16 21.c even 2 1 inner
2100.2.f.i 16 35.c odd 2 1 inner
2100.2.f.i 16 105.g even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(2100, [\chi])$$:

 $$T_{11}^{4} + 16 T_{11}^{2} + 19$$ $$T_{13}^{4} - 36 T_{13}^{2} + 4$$ $$T_{23}^{4} - 80 T_{23}^{2} + 475$$ $$T_{41}^{4} - 90 T_{41}^{2} + 1900$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$( 1 + 6 T^{2} + 22 T^{4} + 54 T^{6} + 81 T^{8} )^{2}$$
$5$ 1
$7$ $$( 1 + 6 T^{2} + 49 T^{4} )^{4}$$
$11$ $$( 1 - 28 T^{2} + 393 T^{4} - 3388 T^{6} + 14641 T^{8} )^{4}$$
$13$ $$( 1 + 16 T^{2} + 82 T^{4} + 2704 T^{6} + 28561 T^{8} )^{4}$$
$17$ $$( 1 - 46 T^{2} + 1062 T^{4} - 13294 T^{6} + 83521 T^{8} )^{4}$$
$19$ $$( 1 - 46 T^{2} + 1126 T^{4} - 16606 T^{6} + 130321 T^{8} )^{4}$$
$23$ $$( 1 + 12 T^{2} - 31 T^{4} + 6348 T^{6} + 279841 T^{8} )^{4}$$
$29$ $$( 1 - 12 T^{2} + 1313 T^{4} - 10092 T^{6} + 707281 T^{8} )^{4}$$
$31$ $$( 1 - 94 T^{2} + 4006 T^{4} - 90334 T^{6} + 923521 T^{8} )^{4}$$
$37$ $$( 1 - 69 T^{2} + 1369 T^{4} )^{8}$$
$41$ $$( 1 + 74 T^{2} + 4606 T^{4} + 124394 T^{6} + 2825761 T^{8} )^{4}$$
$43$ $$( 1 - 82 T^{2} + 3379 T^{4} - 151618 T^{6} + 3418801 T^{8} )^{4}$$
$47$ $$( 1 + 34 T^{2} + 1582 T^{4} + 75106 T^{6} + 4879681 T^{8} )^{4}$$
$53$ $$( 1 + 32 T^{2} + 5374 T^{4} + 89888 T^{6} + 7890481 T^{8} )^{4}$$
$59$ $$( 1 + 76 T^{2} + 3906 T^{4} + 264556 T^{6} + 12117361 T^{8} )^{4}$$
$61$ $$( 1 - 94 T^{2} + 6526 T^{4} - 349774 T^{6} + 13845841 T^{8} )^{4}$$
$67$ $$( 1 - 89 T^{2} + 4489 T^{4} )^{8}$$
$71$ $$( 1 - 164 T^{2} + 13681 T^{4} - 826724 T^{6} + 25411681 T^{8} )^{4}$$
$73$ $$( 1 + 144 T^{2} + 5329 T^{4} )^{8}$$
$79$ $$( 1 - 6 T + 147 T^{2} - 474 T^{3} + 6241 T^{4} )^{8}$$
$83$ $$( 1 - 242 T^{2} + 28294 T^{4} - 1667138 T^{6} + 47458321 T^{8} )^{4}$$
$89$ $$( 1 + 116 T^{2} + 6706 T^{4} + 918836 T^{6} + 62742241 T^{8} )^{4}$$
$97$ $$( 1 + 294 T^{2} + 38222 T^{4} + 2766246 T^{6} + 88529281 T^{8} )^{4}$$