LMFDB - Newform orbit 350.4.g.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [350,4,Mod(293,350)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("350.293"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(350, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([3, 2])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 350 = 2 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	350.g (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$20.6506685020$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 284 x^{14} + 31178 x^{12} - 1629024 x^{10} + 39810581 x^{8} - 379411788 x^{6} + \cdots + 3410793604 $ x^16 - 284x^14 + 31178x^12 - 1629024x^10 + 39810581x^8 - 379411788x^6 + 1669993424x^4 - 2718915464*x^2 + 3410793604
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{18} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

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_{7} q^{2} + \beta_{2} q^{3} - 4 \beta_{3} q^{4} - \beta_{9} q^{6} + (\beta_{10} - 3 \beta_{7} - \beta_{4}) q^{7} + 4 \beta_{8} q^{8} + (3 \beta_{5} - 10 \beta_{3}) q^{9} + ( - \beta_1 - 2) q^{11}+ \cdots + ( - 16 \beta_{5} + 372 \beta_{3}) q^{99}+O(q^{100}) $ q + b7 * q^2 + b2 * q^3 - 4b3 q^4 - b9 * q^6 + (b10 - 3b7 - b4) q^7 + 4b8 q^8 + (3b5 - 10b3) * q^9 + (-b1 - 2) * q^11 - 4b4 q^12 + (b13 - b12 + 3b2) q^13 + (b14 + b6 - 2b5 + 12b3) * q^14 - 16 * q^16 + (-b11 + b10 - 13b4) q^17 + (-3b13 - 3b12 + 10b8) q^18 + (2b14 - 2b6) * q^19 + (b15 + 8b9 - 5b1 - 18) * q^21 + (b11 + b10 - 3b7) q^22 + (-7b13 - 7b12 + 24b8) q^23 + 4b6 q^24 + (2b15 - 3b9) * q^26 + (-3b11 + 3b10 - 13b4) q^27 + (4b13 - 12b8 - 4b2) q^28 + (9b5 - 45b3) * q^29 + (8b15 + 10b9) * q^31 - 16b7 q^32 + (b13 - b12 - 13b2) q^33 + (2b14 + 13b6) * q^34 + (-12b1 - 28) q^36 + (-12b11 - 12b10 + 78b7) q^37 + (4b13 - 4b12 + 8b2) q^38 + (5b5 - 139b3) * q^39 + (10b15 + 2b9) * q^41 + (7b11 + 3b10 - 23b7 + 32b4) * q^42 + (13b13 + 13b12 + 36b8) q^43 + (-4b5 + 12b3) * q^44 + (-28b1 - 68) q^46 + (-4b11 + 4b10 + 65b4) q^47 - 16b2 q^48 + (3b14 - 18b6 + 22b5 + 43b3) * q^49 + (-43b1 - 410) q^51 + (4b11 - 4b10 - 12b4) q^52 + (2b13 + 2b12 - 60b8) q^53 + (6b14 + 13b6) * q^54 + (4b15 + 4b9 + 8b1 + 40) q^56 + (-2b11 - 2b10 + 102b7) q^57 + (-9b13 - 9b12 + 45b8) q^58 + (10b14 - 40b6) * q^59 + (16b15 + 14b9) * q^61 + (16b11 - 16b10 + 40b4) q^62 + (4b13 + 21b12 - 201b8 - 46b2) * q^63 + 64b3 q^64 + (2b15 + 13b9) * q^66 + (-11b11 - 11b10 - 174b7) q^67 + (4b13 - 4b12 - 52b2) q^68 + (14b14 + 94b6) * q^69 + (-28b1 + 232) q^71 + (12b11 + 12b10 - 40b7) q^72 + (-16b13 + 16b12 + 84b2) q^73 + (48b5 - 312b3) * q^74 + (8b15 - 8b9) * q^76 + (-7b11 - b10 + 66b7 + 15b4) q^77 + (-5b13 - 5b12 + 139b8) q^78 + (35b5 - 517b3) * q^79 + (30b1 - 157) q^81 + (20b11 - 20b10 + 8b4) q^82 + (-15b13 + 15b12 + 42b2) q^83 + (-4b14 - 32b6 - 20b5 + 92b3) * q^84 + (52b1 - 196) q^86 + (-9b11 + 9b10 - 135b4) q^87 + (4b13 + 4b12 - 12b8) q^88 + 78b6 q^89 + (8b15 + 22b9 - 47b1 + 318) q^91 + (28b11 + 28b10 - 96b7) q^92 + (46b13 + 46b12 - 258b8) q^93 + (8b14 - 65b6) * q^94 + 16b9 q^96 + (-27b11 + 27b10 - 119b4) q^97 + (-16b13 - 28b12 - 43b8 + 72b2) * q^98 + (-16b5 + 372b3) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 40 q^{11} - 256 q^{16} - 328 q^{21} - 544 q^{36} - 1312 q^{46} - 6904 q^{51} + 704 q^{56} + 3488 q^{71} - 2272 q^{81} - 2720 q^{86} + 4712 q^{91}+O(q^{100}) $ 16 * q - 40 * q^11 - 256 * q^16 - 328 * q^21 - 544 * q^36 - 1312 * q^46 - 6904 * q^51 + 704 * q^56 + 3488 * q^71 - 2272 * q^81 - 2720 * q^86 + 4712 * q^91

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 284 x^{14} + 31178 x^{12} - 1629024 x^{10} + 39810581 x^{8} - 379411788 x^{6} + \cdots + 3410793604 $ x^16 - 284*x^14 + 31178*x^12 - 1629024*x^10 + 39810581*x^8 - 379411788*x^6 + 1669993424*x^4 - 2718915464*x^2 + 3410793604 :

$\beta_{1}$	$=$	$ ( - 81940 \nu^{14} + 20350730 \nu^{12} - 1812261472 \nu^{10} + 65231524415 \nu^{8} + \cdots - 88\!\cdots\!47 ) / 865787919726267 $ (-81940v^14 + 20350730v^12 - 1812261472v^10 + 65231524415v^8 - 700176302564v^6 + 3291049356640v^4 - 5400248304152*v^2 - 8814346494926447) / 865787919726267
$\beta_{2}$	$=$	$ ( - 324163433602 \nu^{14} + 88255907784765 \nu^{12} + \cdots + 22\!\cdots\!54 ) / 97\!\cdots\!12 $ (-324163433602v^14 + 88255907784765v^12 - 9179436724016573v^10 + 449112820236289844v^8 - 10436267231352705295v^6 + 115677228609425731129v^4 - 978738100382270649492*v^2 + 2238303424831338230354) / 973003632553489000212
$\beta_{3}$	$=$	$ ( 10874 \nu^{14} - 3153465 \nu^{12} + 357984445 \nu^{10} - 19823933998 \nu^{8} + \cdots - 32502949499236 ) / 28437039855882 $ (10874v^14 - 3153465v^12 + 357984445v^10 - 19823933998v^8 + 541029014183v^6 - 6437287617593v^4 + 28240113514266*v^2 - 32502949499236) / 28437039855882
$\beta_{4}$	$=$	$ ( 2213413924994 \nu^{14} - 614824303733136 \nu^{12} + \cdots - 81\!\cdots\!62 ) / 97\!\cdots\!12 $ (2213413924994v^14 - 614824303733136v^12 + 65592967500374146v^10 - 3284735697162414175v^8 + 74133648628735130414v^6 - 563546149607278384841v^4 + 1730358563282982847710*v^2 - 815449877933893739962) / 973003632553489000212
$\beta_{5}$	$=$	$ ( 1928693300459 \nu^{14} - 568530290194146 \nu^{12} + \cdots - 62\!\cdots\!40 ) / 48\!\cdots\!06 $ (1928693300459v^14 - 568530290194146v^12 + 65702297488606975v^10 - 3704543255136992740v^8 + 102627508066074718034v^6 - 1227899448825946519994v^4 + 5398603007175809716392*v^2 - 6212601898800890722840) / 486501816276744500106
$\beta_{6}$	$=$	$ ( - 19\!\cdots\!89 \nu^{15} + \cdots + 10\!\cdots\!48 \nu ) / 28\!\cdots\!12 $ (-19283249881353589v^15 + 5479240542884544240v^13 - 602360370400340985692v^11 + 31592134941595180065530v^9 - 781062197800320128570857v^7 + 7787924513159113871151898v^5 - 38310665133574980578760354v^3 + 108526464770162091754815248v) / 28412679074194432295190612
$\beta_{7}$	$=$	$ ( 19\!\cdots\!89 \nu^{15} + \cdots - 51\!\cdots\!24 \nu ) / 28\!\cdots\!12 $ (19283249881353589v^15 - 5479240542884544240v^13 + 602360370400340985692v^11 - 31592134941595180065530v^9 + 781062197800320128570857v^7 - 7787924513159113871151898v^5 + 38310665133574980578760354v^3 - 51701106621773227164434024v) / 28412679074194432295190612
$\beta_{8}$	$=$	$ ( 30\!\cdots\!87 \nu^{15} + \cdots + 17\!\cdots\!52 \nu ) / 28\!\cdots\!12 $ (30623155208509687v^15 - 8567708279169251520v^13 + 918861764104292412614v^11 - 46055094302430433480796v^9 + 1027290466718267470949797v^7 - 7289232724646784073551928v^5 + 18228245590978253898814206v^3 + 17794630559963631893859652v) / 28412679074194432295190612
$\beta_{9}$	$=$	$ ( - 62111701515 \nu^{15} + 17641050123700 \nu^{13} + \cdots + 55\!\cdots\!00 \nu ) / 33\!\cdots\!56 $ (-62111701515v^15 + 17641050123700v^13 - 1938856220460802v^11 + 101638789153486116v^9 - 2501458152005202389v^7 + 23909513331435277452v^5 - 88577679130749104734v^3 + 55945981083101808700v) / 33709164058568963556
$\beta_{10}$	$=$	$ ( 21\!\cdots\!78 \nu^{15} + \cdots - 50\!\cdots\!96 ) / 56\!\cdots\!24 $ (214065309374333378v^15 + 1311532169334028692v^14 - 60902781051722802150v^13 - 368179308417845295147v^12 + 6705672975574420388968v^11 + 39682219910670051018360v^10 - 352336969408752981986530v^9 - 2005956761198275211619207v^8 + 8727126965659565029483322v^7 + 45602133971724096081377412v^6 - 87081548871126453110104988v^5 - 347473194399084012044692704v^4 + 428558086317020165930540532v^3 + 1068650153880200327967619152v^2 - 578354655897075995514431560*v - 502026589065754842052976196) / 56825358148388864590381224
$\beta_{11}$	$=$	$ ( 21\!\cdots\!78 \nu^{15} + \cdots + 50\!\cdots\!96 ) / 56\!\cdots\!24 $ (214065309374333378v^15 - 1311532169334028692v^14 - 60902781051722802150v^13 + 368179308417845295147v^12 + 6705672975574420388968v^11 - 39682219910670051018360v^10 - 352336969408752981986530v^9 + 2005956761198275211619207v^8 + 8727126965659565029483322v^7 - 45602133971724096081377412v^6 - 87081548871126453110104988v^5 + 347473194399084012044692704v^4 + 428558086317020165930540532v^3 - 1068650153880200327967619152v^2 - 578354655897075995514431560*v + 502026589065754842052976196) / 56825358148388864590381224
$\beta_{12}$	$=$	$ ( 34\!\cdots\!40 \nu^{15} + \cdots - 13\!\cdots\!76 ) / 56\!\cdots\!24 $ (341024064182904140v^15 + 153181915673465139v^14 - 95525825961841413936v^13 - 43202533689420577500v^12 + 10256495051434103125906v^11 + 4707911460440241423951v^10 - 514598496648613968735544v^9 - 244966856062406265542568v^8 + 11487085373053609155758306v^7 + 6109822573983466728292056v^6 - 81525771031908617453696384v^5 - 69955477458892401264251748v^4 + 203899783728972536854308852v^3 + 602925937551080086725447420v^2 + 199166256573195850173084200*v - 1381501653204897460364030376) / 56825358148388864590381224
$\beta_{13}$	$=$	$ ( 34\!\cdots\!40 \nu^{15} + \cdots + 13\!\cdots\!76 ) / 56\!\cdots\!24 $ (341024064182904140v^15 - 153181915673465139v^14 - 95525825961841413936v^13 + 43202533689420577500v^12 + 10256495051434103125906v^11 - 4707911460440241423951v^10 - 514598496648613968735544v^9 + 244966856062406265542568v^8 + 11487085373053609155758306v^7 - 6109822573983466728292056v^6 - 81525771031908617453696384v^5 + 69955477458892401264251748v^4 + 203899783728972536854308852v^3 - 602925937551080086725447420v^2 + 199166256573195850173084200*v + 1381501653204897460364030376) / 56825358148388864590381224
$\beta_{14}$	$=$	$ ( 24\!\cdots\!74 \nu^{15} + \cdots - 14\!\cdots\!04 \nu ) / 35\!\cdots\!52 $ (2478516889494874v^15 - 705922723281394395v^13 + 77841544572120382592v^11 - 4098622810703965498975v^9 + 101801193722976952797082v^7 - 1016943138150762964291024v^5 + 5008088248747650613279044v^3 - 14191159630267128258187404v) / 350773815730795460434452
$\beta_{15}$	$=$	$ ( 1939479430857 \nu^{15} - 552793809824240 \nu^{13} + \cdots - 17\!\cdots\!20 \nu ) / 10\!\cdots\!68 $ (1939479430857v^15 - 552793809824240v^13 + 60992812746035447v^11 - 3210422061359245368v^9 + 79297342767237017254v^7 - 758966836682890027200v^5 + 2813718538836662905400v^3 - 1776142253119066166120v) / 101127492175706890668

Display $\nu^j$ in terms of $\beta_i$

$\nu$	$=$	$ ( \beta_{7} + \beta_{6} ) / 2 $ (b7 + b6) / 2
$\nu^{2}$	$=$	$ -\beta_{3} - 2\beta_{2} + 3\beta _1 + 34 $ -b3 - 2b2 + 3b1 + 34
$\nu^{3}$	$=$	$ ( 6\beta_{14} - 9\beta_{11} - 9\beta_{10} + 3\beta_{9} + \beta_{8} + 111\beta_{7} + 67\beta_{6} ) / 2 $ (6b14 - 9b11 - 9b10 + 3b9 + b8 + 111b7 + 67b6) / 2
$\nu^{4}$	$=$	$ 12\beta_{13} - 12\beta_{12} + 18\beta_{5} + 4\beta_{4} - 222\beta_{3} - 268\beta_{2} + 213\beta _1 + 2181 $ 12b13 - 12b12 + 18b5 + 4b4 - 222b3 - 268b2 + 213*b1 + 2181
$\nu^{5}$	$=$	$ ( 60 \beta_{15} + 426 \beta_{14} - 30 \beta_{13} - 30 \beta_{12} - 1065 \beta_{11} - 1065 \beta_{10} + \cdots + 4520 \beta_{6} ) / 2 $ (60b15 + 426b14 - 30b13 - 30b12 - 1065b11 - 1065b10 + 670b9 + 370b8 + 11974b7 + 4520b6) / 2
$\nu^{6}$	$=$	$ 1278 \beta_{13} - 1278 \beta_{12} + 60 \beta_{11} - 60 \beta_{10} + 3195 \beta_{5} + 1340 \beta_{4} + \cdots + 146524 $ 1278b13 - 1278b12 + 60b11 - 60b10 + 3195b5 + 1340b4 - 35924b3 - 27144b2 + 14382*b1 + 146524
$\nu^{7}$	$=$	$ ( 8946 \beta_{15} + 28644 \beta_{14} - 7455 \beta_{13} - 7455 \beta_{12} - 100926 \beta_{11} + \cdots + 303386 \beta_{6} ) / 2 $ (8946b15 + 28644b14 - 7455b13 - 7455b12 - 100926b11 - 100926b10 + 95018b9 + 83824b8 + 1129450b7 + 303386b6) / 2
$\nu^{8}$	$=$	$ 115248 \beta_{13} - 115248 \beta_{12} + 11928 \beta_{11} - 11928 \beta_{10} + 403872 \beta_{5} + \cdots + 9780451 $ 115248b13 - 115248b12 + 11928b11 - 11928b10 + 403872b5 + 253392b4 - 4519872b3 - 2442096b2 + 959991*b1 + 9780451
$\nu^{9}$	$=$	$ ( 1038240 \beta_{15} + 1896126 \beta_{14} - 1211760 \beta_{13} - 1211760 \beta_{12} + \cdots + 20086960 \beta_{6} ) / 2 $ (1038240b15 + 1896126b14 - 1211760b13 - 1211760b12 - 8747271b11 - 8747271b10 + 11000688b9 + 13561392b8 + 97871050b7 + 20086960b6) / 2
$\nu^{10}$	$=$	$ 9695334 \beta_{13} - 9695334 \beta_{12} + 1730880 \beta_{11} - 1730880 \beta_{10} + 43825815 \beta_{5} + \cdots + 640268992 $ 9695334b13 - 9695334b12 + 1730880b11 - 1730880b10 + 43825815b5 + 36679680b4 - 490361146b3 - 205430720b2 + 62841372*b1 + 640268992
$\nu^{11}$	$=$	$ ( 107042298 \beta_{15} + 122220984 \beta_{14} - 160788375 \beta_{13} - 160788375 \beta_{12} + \cdots + 1294844404 \beta_{6} ) / 2 $ (107042298b15 + 122220984b14 - 160788375b13 - 160788375b12 - 717914868b11 - 717914868b10 + 1134050016b9 + 1799044666b8 + 8032577652b7 + 1294844404b6) / 2
$\nu^{12}$	$=$	$ 779025360 \beta_{13} - 779025360 \beta_{12} + 214309524 \beta_{11} - 214309524 \beta_{10} + \cdots + 40430917701 $ 779025360b13 - 779025360b12 + 214309524b11 - 214309524b10 + 4334149512b5 + 4540978432b4 - 48493836072b3 - 16506570736b2 + 3968186805*b1 + 40430917701
$\nu^{13}$	$=$	$ ( 10226349744 \beta_{15} + 7507754562 \beta_{14} - 18814322904 \beta_{13} - 18814322904 \beta_{12} + \cdots + 79539994124 \beta_{6} ) / 2 $ (10226349744b15 + 7507754562b14 - 18814322904b13 - 18814322904b12 - 56604000453b11 - 56604000453b10 + 108341901928b9 + 210509371384b8 + 633329511658b7 + 79539994124b6) / 2
$\nu^{14}$	$=$	$ 60357877734 \beta_{13} - 60357877734 \beta_{12} + 23927497776 \beta_{11} - 23927497776 \beta_{10} + \cdots + 2392540678744 $ 60357877734b13 - 60357877734b12 + 23927497776b11 - 23927497776b10 + 402082405725b5 + 506994502352b4 - 4498810986578b3 - 1278909510312b2 + 234821168592*b1 + 2392540678744
$\nu^{15}$	$=$	$ ( 924880566918 \beta_{15} + 421787341632 \beta_{14} - 2018775903885 \beta_{13} + \cdots + 4468579030904 \beta_{6} ) / 2 $ (924880566918b15 + 421787341632b14 - 2018775903885b13 - 2018775903885b12 - 4312981210464b11 - 4312981210464b10 + 9798544554140b9 + 22587637635874b8 + 48256993152328b7 + 4468579030904b6) / 2

Display $\beta_i$ in terms of $\nu^j$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/350\mathbb{Z}\right)^\times$.

$n$	$101$	$127$
$\chi(n)$	$-1$	$-\beta_{3}$

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.

comment:embeddings in the coefficient field

gp:mfembed(f)

Label

$\iota_m(\nu)$

$ a_{2} $

$ a_{3} $

$ a_{4} $

$ a_{5} $

$ a_{6} $

$ a_{7} $

$ a_{8} $

$ a_{9} $

$ a_{10} $

293.1

−8.92698	+	0.707107i
−2.56012	+	0.707107i
1.14591	+	0.707107i
7.51277	+	0.707107i
8.92698	−	0.707107i
2.56012	−	0.707107i
−1.14591	−	0.707107i
−7.51277	−	0.707107i
−8.92698	−	0.707107i
−2.56012	−	0.707107i
1.14591	−	0.707107i
7.51277	−	0.707107i
8.92698	+	0.707107i
2.56012	+	0.707107i
−1.14591	+	0.707107i
−7.51277	+	0.707107i

−1.41421

1.41421i

−5.81233

5.81233i

−

4.00000i

−

16.4397i

17.8081

−

5.08624i

5.65685

5.65685i

−

40.5663i

293.2

−1.41421

1.41421i

−1.31028

1.31028i

−

4.00000i

−

3.70603i

−16.2403

−

8.90231i

5.65685

5.65685i

23.5663i

293.3

−1.41421

1.41421i

1.31028

−

1.31028i

−

4.00000i

3.70603i

8.90231

16.2403i

5.65685

5.65685i

23.5663i

293.4

−1.41421

1.41421i

5.81233

−

5.81233i

−

4.00000i

16.4397i

5.08624

−

17.8081i

5.65685

5.65685i

−

40.5663i

293.5

1.41421

−

1.41421i

−5.81233

5.81233i

−

4.00000i

16.4397i

−5.08624

17.8081i

−5.65685

−

5.65685i

−

40.5663i

293.6

1.41421

−

1.41421i

−1.31028

1.31028i

−

4.00000i

3.70603i

−8.90231

−

16.2403i

−5.65685

−

5.65685i

23.5663i

293.7

1.41421

−

1.41421i

1.31028

−

1.31028i

−

4.00000i

−

3.70603i

16.2403

8.90231i

−5.65685

−

5.65685i

23.5663i

293.8

1.41421

−

1.41421i

5.81233

−

5.81233i

−

4.00000i

−

16.4397i

−17.8081

5.08624i

−5.65685

−

5.65685i

−

40.5663i

307.1

−1.41421

−

1.41421i

−5.81233

−

5.81233i

4.00000i

16.4397i

17.8081

5.08624i

5.65685

−

5.65685i

40.5663i

307.2

−1.41421

−

1.41421i

−1.31028

−

1.31028i

4.00000i

3.70603i

−16.2403

8.90231i

5.65685

−

5.65685i

−

23.5663i

307.3

−1.41421

−

1.41421i

1.31028

1.31028i

4.00000i

−

3.70603i

8.90231

−

16.2403i

5.65685

−

5.65685i

−

23.5663i

307.4

−1.41421

−

1.41421i

5.81233

5.81233i

4.00000i

−

16.4397i

5.08624

17.8081i

5.65685

−

5.65685i

40.5663i

307.5

1.41421

1.41421i

−5.81233

−

5.81233i

4.00000i

−

16.4397i

−5.08624

−

17.8081i

−5.65685

5.65685i

40.5663i

307.6

1.41421

1.41421i

−1.31028

−

1.31028i

4.00000i

−

3.70603i

−8.90231

16.2403i

−5.65685

5.65685i

−

23.5663i

307.7

1.41421

1.41421i

1.31028

1.31028i

4.00000i

3.70603i

16.2403

−

8.90231i

−5.65685

5.65685i

−

23.5663i

307.8

1.41421

1.41421i

5.81233

5.81233i

4.00000i

16.4397i

−17.8081

−

5.08624i

−5.65685

5.65685i

40.5663i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
5.c	odd	4	2	inner
7.b	odd	2	1	inner
35.c	odd	2	1	inner
35.f	even	4	2	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	350.4.g.a	✓	16
5.b	even	2	1	inner	350.4.g.a	✓	16
5.c	odd	4	2	inner	350.4.g.a	✓	16
7.b	odd	2	1	inner	350.4.g.a	✓	16
35.c	odd	2	1	inner	350.4.g.a	✓	16
35.f	even	4	2	inner	350.4.g.a	✓	16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
350.4.g.a	✓	16	1.a	even	1	1	trivial
350.4.g.a	✓	16	5.b	even	2	1	inner
350.4.g.a	✓	16	5.c	odd	4	2	inner
350.4.g.a	✓	16	7.b	odd	2	1	inner
350.4.g.a	✓	16	35.c	odd	2	1	inner
350.4.g.a	✓	16	35.f	even	4	2	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3}^{8} + 4577T_{3}^{4} + 53824 $ T3^8 + 4577*T3^4 + 53824 acting on $S_{4}^{\mathrm{new}}(350, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{4} + 16)^{4} $ (T^4 + 16)^4
$3$	$ (T^{8} + 4577 T^{4} + 53824)^{2} $ (T^8 + 4577*T^4 + 53824)^2
$5$	$ T^{16} $ T^16
$7$	$ T^{16} + \cdots + 19\!\cdots\!01 $ T^16 - 4892T^12 + 17368809990T^8 - 67711576987292*T^4 + 191581231380566414401
$11$	$ (T^{2} + 5 T - 108)^{8} $ (T^2 + 5*T - 108)^8
$13$	$ (T^{8} + \cdots + 1150831181824)^{2} $ (T^8 + 4321313*T^4 + 1150831181824)^2
$17$	$ (T^{8} + \cdots + 7322695778304)^{2} $ (T^8 + 138180193*T^4 + 7322695778304)^2
$19$	$ (T^{4} - 8272 T^{2} + 6547968)^{4} $ (T^4 - 8272*T^2 + 6547968)^4
$23$	$ (T^{8} + \cdots + 18\!\cdots\!36)^{2} $ (T^8 + 1460256016*T^4 + 184029798122459136)^2
$29$	$ (T^{4} + 21789 T^{2} + 57972996)^{4} $ (T^4 + 21789*T^2 + 57972996)^4
$31$	$ (T^{4} + 108016 T^{2} + 2399496192)^{4} $ (T^4 + 108016*T^2 + 2399496192)^4
$37$	$ (T^{8} + \cdots + 41\!\cdots\!56)^{2} $ (T^8 + 25896485376*T^4 + 4126932052811513856)^2
$41$	$ (T^{4} + 150736 T^{2} + 347576832)^{4} $ (T^4 + 150736*T^2 + 347576832)^4
$43$	$ (T^{8} + \cdots + 24\!\cdots\!96)^{2} $ (T^8 + 18730374928*T^4 + 24020977881743364096)^2
$47$	$ (T^{8} + \cdots + 46\!\cdots\!44)^{2} $ (T^8 + 83700333217*T^4 + 461165519435760442944)^2
$53$	$ (T^{8} + \cdots + 21\!\cdots\!56)^{2} $ (T^8 + 699875584*T^4 + 21407557176262656)^2
$59$	$ (T^{4} - 704800 T^{2} + 120602880000)^{4} $ (T^4 - 704800*T^2 + 120602880000)^4
$61$	$ (T^{4} + 397168 T^{2} + 21595465728)^{4} $ (T^4 + 397168*T^2 + 21595465728)^4
$67$	$ (T^{8} + \cdots + 29\!\cdots\!36)^{2} $ (T^8 + 124856932624*T^4 + 29317947912460763136)^2
$71$	$ (T^{2} - 436 T - 42048)^{8} $ (T^2 - 436*T - 42048)^8
$73$	$ (T^{8} + \cdots + 13\!\cdots\!04)^{2} $ (T^8 + 311653224704*T^4 + 13054545152467957448704)^2
$79$	$ (T^{4} + 778913 T^{2} + 11999887936)^{4} $ (T^4 + 778913*T^2 + 11999887936)^4
$83$	$ (T^{8} + \cdots + 12\!\cdots\!64)^{2} $ (T^8 + 86690006352*T^4 + 1258615101806738187264)^2
$89$	$ (T^{4} - 1727856 T^{2} + 137399887872)^{4} $ (T^4 - 1727856*T^2 + 137399887872)^4
$97$	$ (T^{8} + \cdots + 53\!\cdots\!44)^{2} $ (T^8 + 1579608405233*T^4 + 538005250960771069394944)^2
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Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	350.g (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\( q + \beta_{7} q^{2} + \beta_{2} q^{3} - 4 \beta_{3} q^{4} - \beta_{9} q^{6} + (\beta_{10} - 3 \beta_{7} - \beta_{4}) q^{7} + 4 \beta_{8} q^{8} + (3 \beta_{5} - 10 \beta_{3}) q^{9} + ( - \beta_1 - 2) q^{11}+ \cdots + ( - 16 \beta_{5} + 372 \beta_{3}) q^{99}+O(q^{100}) \) q + b7 * q^2 + b2 * q^3 - 4b3 q^4 - b9 * q^6 + (b10 - 3b7 - b4) q^7 + 4b8 q^8 + (3b5 - 10b3) * q^9 + (-b1 - 2) * q^11 - 4b4 q^12 + (b13 - b12 + 3b2) q^13 + (b14 + b6 - 2b5 + 12b3) * q^14 - 16 * q^16 + (-b11 + b10 - 13b4) q^17 + (-3b13 - 3b12 + 10b8) q^18 + (2b14 - 2b6) * q^19 + (b15 + 8b9 - 5b1 - 18) * q^21 + (b11 + b10 - 3b7) q^22 + (-7b13 - 7b12 + 24b8) q^23 + 4b6 q^24 + (2b15 - 3b9) * q^26 + (-3b11 + 3b10 - 13b4) q^27 + (4b13 - 12b8 - 4b2) q^28 + (9b5 - 45b3) * q^29 + (8b15 + 10b9) * q^31 - 16b7 q^32 + (b13 - b12 - 13b2) q^33 + (2b14 + 13b6) * q^34 + (-12b1 - 28) q^36 + (-12b11 - 12b10 + 78b7) q^37 + (4b13 - 4b12 + 8b2) q^38 + (5b5 - 139b3) * q^39 + (10b15 + 2b9) * q^41 + (7b11 + 3b10 - 23b7 + 32b4) * q^42 + (13b13 + 13b12 + 36b8) q^43 + (-4b5 + 12b3) * q^44 + (-28b1 - 68) q^46 + (-4b11 + 4b10 + 65b4) q^47 - 16b2 q^48 + (3b14 - 18b6 + 22b5 + 43b3) * q^49 + (-43b1 - 410) q^51 + (4b11 - 4b10 - 12b4) q^52 + (2b13 + 2b12 - 60b8) q^53 + (6b14 + 13b6) * q^54 + (4b15 + 4b9 + 8b1 + 40) q^56 + (-2b11 - 2b10 + 102b7) q^57 + (-9b13 - 9b12 + 45b8) q^58 + (10b14 - 40b6) * q^59 + (16b15 + 14b9) * q^61 + (16b11 - 16b10 + 40b4) q^62 + (4b13 + 21b12 - 201b8 - 46b2) * q^63 + 64b3 q^64 + (2b15 + 13b9) * q^66 + (-11b11 - 11b10 - 174b7) q^67 + (4b13 - 4b12 - 52b2) q^68 + (14b14 + 94b6) * q^69 + (-28b1 + 232) q^71 + (12b11 + 12b10 - 40b7) q^72 + (-16b13 + 16b12 + 84b2) q^73 + (48b5 - 312b3) * q^74 + (8b15 - 8b9) * q^76 + (-7b11 - b10 + 66b7 + 15b4) q^77 + (-5b13 - 5b12 + 139b8) q^78 + (35b5 - 517b3) * q^79 + (30b1 - 157) q^81 + (20b11 - 20b10 + 8b4) q^82 + (-15b13 + 15b12 + 42b2) q^83 + (-4b14 - 32b6 - 20b5 + 92b3) * q^84 + (52b1 - 196) q^86 + (-9b11 + 9b10 - 135b4) q^87 + (4b13 + 4b12 - 12b8) q^88 + 78b6 q^89 + (8b15 + 22b9 - 47b1 + 318) q^91 + (28b11 + 28b10 - 96b7) q^92 + (46b13 + 46b12 - 258b8) q^93 + (8b14 - 65b6) * q^94 + 16b9 q^96 + (-27b11 + 27b10 - 119b4) q^97 + (-16b13 - 28b12 - 43b8 + 72b2) * q^98 + (-16b5 + 372b3) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 40 q^{11} - 256 q^{16} - 328 q^{21} - 544 q^{36} - 1312 q^{46} - 6904 q^{51} + 704 q^{56} + 3488 q^{71} - 2272 q^{81} - 2720 q^{86} + 4712 q^{91}+O(q^{100}) \) 16 * q - 40 * q^11 - 256 * q^16 - 328 * q^21 - 544 * q^36 - 1312 * q^46 - 6904 * q^51 + 704 * q^56 + 3488 * q^71 - 2272 * q^81 - 2720 * q^86 + 4712 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

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Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	350.4.g.a

Level	$350$
Weight	$4$
Character orbit	350.g
Analytic conductor	$20.651$
Analytic rank	$0$
Dimension	$16$
Inner twists	$8$

Self dual:	no
Analytic conductor:	\(20.6506685020\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(8\) over \(\Q(i)\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - 284 x^{14} + 31178 x^{12} - 1629024 x^{10} + 39810581 x^{8} - 379411788 x^{6} + \cdots + 3410793604 \) x^16 - 284x^14 + 31178x^12 - 1629024x^10 + 39810581x^8 - 379411788x^6 + 1669993424x^4 - 2718915464*x^2 + 3410793604
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{18} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

\(\beta_{1}\)	\(=\)	\( ( - 81940 \nu^{14} + 20350730 \nu^{12} - 1812261472 \nu^{10} + 65231524415 \nu^{8} + \cdots - 88\!\cdots\!47 ) / 865787919726267 \) (-81940v^14 + 20350730v^12 - 1812261472v^10 + 65231524415v^8 - 700176302564v^6 + 3291049356640v^4 - 5400248304152*v^2 - 8814346494926447) / 865787919726267
\(\beta_{2}\)	\(=\)	\( ( - 324163433602 \nu^{14} + 88255907784765 \nu^{12} + \cdots + 22\!\cdots\!54 ) / 97\!\cdots\!12 \) (-324163433602v^14 + 88255907784765v^12 - 9179436724016573v^10 + 449112820236289844v^8 - 10436267231352705295v^6 + 115677228609425731129v^4 - 978738100382270649492*v^2 + 2238303424831338230354) / 973003632553489000212
\(\beta_{3}\)	\(=\)	\( ( 10874 \nu^{14} - 3153465 \nu^{12} + 357984445 \nu^{10} - 19823933998 \nu^{8} + \cdots - 32502949499236 ) / 28437039855882 \) (10874v^14 - 3153465v^12 + 357984445v^10 - 19823933998v^8 + 541029014183v^6 - 6437287617593v^4 + 28240113514266*v^2 - 32502949499236) / 28437039855882
\(\beta_{4}\)	\(=\)	\( ( 2213413924994 \nu^{14} - 614824303733136 \nu^{12} + \cdots - 81\!\cdots\!62 ) / 97\!\cdots\!12 \) (2213413924994v^14 - 614824303733136v^12 + 65592967500374146v^10 - 3284735697162414175v^8 + 74133648628735130414v^6 - 563546149607278384841v^4 + 1730358563282982847710*v^2 - 815449877933893739962) / 973003632553489000212
\(\beta_{5}\)	\(=\)	\( ( 1928693300459 \nu^{14} - 568530290194146 \nu^{12} + \cdots - 62\!\cdots\!40 ) / 48\!\cdots\!06 \) (1928693300459v^14 - 568530290194146v^12 + 65702297488606975v^10 - 3704543255136992740v^8 + 102627508066074718034v^6 - 1227899448825946519994v^4 + 5398603007175809716392*v^2 - 6212601898800890722840) / 486501816276744500106
\(\beta_{6}\)	\(=\)	\( ( - 19\!\cdots\!89 \nu^{15} + \cdots + 10\!\cdots\!48 \nu ) / 28\!\cdots\!12 \) (-19283249881353589v^15 + 5479240542884544240v^13 - 602360370400340985692v^11 + 31592134941595180065530v^9 - 781062197800320128570857v^7 + 7787924513159113871151898v^5 - 38310665133574980578760354v^3 + 108526464770162091754815248v) / 28412679074194432295190612
\(\beta_{7}\)	\(=\)	\( ( 19\!\cdots\!89 \nu^{15} + \cdots - 51\!\cdots\!24 \nu ) / 28\!\cdots\!12 \) (19283249881353589v^15 - 5479240542884544240v^13 + 602360370400340985692v^11 - 31592134941595180065530v^9 + 781062197800320128570857v^7 - 7787924513159113871151898v^5 + 38310665133574980578760354v^3 - 51701106621773227164434024v) / 28412679074194432295190612
\(\beta_{8}\)	\(=\)	\( ( 30\!\cdots\!87 \nu^{15} + \cdots + 17\!\cdots\!52 \nu ) / 28\!\cdots\!12 \) (30623155208509687v^15 - 8567708279169251520v^13 + 918861764104292412614v^11 - 46055094302430433480796v^9 + 1027290466718267470949797v^7 - 7289232724646784073551928v^5 + 18228245590978253898814206v^3 + 17794630559963631893859652v) / 28412679074194432295190612
\(\beta_{9}\)	\(=\)	\( ( - 62111701515 \nu^{15} + 17641050123700 \nu^{13} + \cdots + 55\!\cdots\!00 \nu ) / 33\!\cdots\!56 \) (-62111701515v^15 + 17641050123700v^13 - 1938856220460802v^11 + 101638789153486116v^9 - 2501458152005202389v^7 + 23909513331435277452v^5 - 88577679130749104734v^3 + 55945981083101808700v) / 33709164058568963556
\(\beta_{10}\)	\(=\)	\( ( 21\!\cdots\!78 \nu^{15} + \cdots - 50\!\cdots\!96 ) / 56\!\cdots\!24 \) (214065309374333378v^15 + 1311532169334028692v^14 - 60902781051722802150v^13 - 368179308417845295147v^12 + 6705672975574420388968v^11 + 39682219910670051018360v^10 - 352336969408752981986530v^9 - 2005956761198275211619207v^8 + 8727126965659565029483322v^7 + 45602133971724096081377412v^6 - 87081548871126453110104988v^5 - 347473194399084012044692704v^4 + 428558086317020165930540532v^3 + 1068650153880200327967619152v^2 - 578354655897075995514431560*v - 502026589065754842052976196) / 56825358148388864590381224
\(\beta_{11}\)	\(=\)	\( ( 21\!\cdots\!78 \nu^{15} + \cdots + 50\!\cdots\!96 ) / 56\!\cdots\!24 \) (214065309374333378v^15 - 1311532169334028692v^14 - 60902781051722802150v^13 + 368179308417845295147v^12 + 6705672975574420388968v^11 - 39682219910670051018360v^10 - 352336969408752981986530v^9 + 2005956761198275211619207v^8 + 8727126965659565029483322v^7 - 45602133971724096081377412v^6 - 87081548871126453110104988v^5 + 347473194399084012044692704v^4 + 428558086317020165930540532v^3 - 1068650153880200327967619152v^2 - 578354655897075995514431560*v + 502026589065754842052976196) / 56825358148388864590381224
\(\beta_{12}\)	\(=\)	\( ( 34\!\cdots\!40 \nu^{15} + \cdots - 13\!\cdots\!76 ) / 56\!\cdots\!24 \) (341024064182904140v^15 + 153181915673465139v^14 - 95525825961841413936v^13 - 43202533689420577500v^12 + 10256495051434103125906v^11 + 4707911460440241423951v^10 - 514598496648613968735544v^9 - 244966856062406265542568v^8 + 11487085373053609155758306v^7 + 6109822573983466728292056v^6 - 81525771031908617453696384v^5 - 69955477458892401264251748v^4 + 203899783728972536854308852v^3 + 602925937551080086725447420v^2 + 199166256573195850173084200*v - 1381501653204897460364030376) / 56825358148388864590381224
\(\beta_{13}\)	\(=\)	\( ( 34\!\cdots\!40 \nu^{15} + \cdots + 13\!\cdots\!76 ) / 56\!\cdots\!24 \) (341024064182904140v^15 - 153181915673465139v^14 - 95525825961841413936v^13 + 43202533689420577500v^12 + 10256495051434103125906v^11 - 4707911460440241423951v^10 - 514598496648613968735544v^9 + 244966856062406265542568v^8 + 11487085373053609155758306v^7 - 6109822573983466728292056v^6 - 81525771031908617453696384v^5 + 69955477458892401264251748v^4 + 203899783728972536854308852v^3 - 602925937551080086725447420v^2 + 199166256573195850173084200*v + 1381501653204897460364030376) / 56825358148388864590381224
\(\beta_{14}\)	\(=\)	\( ( 24\!\cdots\!74 \nu^{15} + \cdots - 14\!\cdots\!04 \nu ) / 35\!\cdots\!52 \) (2478516889494874v^15 - 705922723281394395v^13 + 77841544572120382592v^11 - 4098622810703965498975v^9 + 101801193722976952797082v^7 - 1016943138150762964291024v^5 + 5008088248747650613279044v^3 - 14191159630267128258187404v) / 350773815730795460434452
\(\beta_{15}\)	\(=\)	\( ( 1939479430857 \nu^{15} - 552793809824240 \nu^{13} + \cdots - 17\!\cdots\!20 \nu ) / 10\!\cdots\!68 \) (1939479430857v^15 - 552793809824240v^13 + 60992812746035447v^11 - 3210422061359245368v^9 + 79297342767237017254v^7 - 758966836682890027200v^5 + 2813718538836662905400v^3 - 1776142253119066166120v) / 101127492175706890668

\(\nu\)	\(=\)	\( ( \beta_{7} + \beta_{6} ) / 2 \) (b7 + b6) / 2
\(\nu^{2}\)	\(=\)	\( -\beta_{3} - 2\beta_{2} + 3\beta _1 + 34 \) -b3 - 2b2 + 3b1 + 34
\(\nu^{3}\)	\(=\)	\( ( 6\beta_{14} - 9\beta_{11} - 9\beta_{10} + 3\beta_{9} + \beta_{8} + 111\beta_{7} + 67\beta_{6} ) / 2 \) (6b14 - 9b11 - 9b10 + 3b9 + b8 + 111b7 + 67b6) / 2
\(\nu^{4}\)	\(=\)	\( 12\beta_{13} - 12\beta_{12} + 18\beta_{5} + 4\beta_{4} - 222\beta_{3} - 268\beta_{2} + 213\beta _1 + 2181 \) 12b13 - 12b12 + 18b5 + 4b4 - 222b3 - 268b2 + 213*b1 + 2181
\(\nu^{5}\)	\(=\)	\( ( 60 \beta_{15} + 426 \beta_{14} - 30 \beta_{13} - 30 \beta_{12} - 1065 \beta_{11} - 1065 \beta_{10} + \cdots + 4520 \beta_{6} ) / 2 \) (60b15 + 426b14 - 30b13 - 30b12 - 1065b11 - 1065b10 + 670b9 + 370b8 + 11974b7 + 4520b6) / 2
\(\nu^{6}\)	\(=\)	\( 1278 \beta_{13} - 1278 \beta_{12} + 60 \beta_{11} - 60 \beta_{10} + 3195 \beta_{5} + 1340 \beta_{4} + \cdots + 146524 \) 1278b13 - 1278b12 + 60b11 - 60b10 + 3195b5 + 1340b4 - 35924b3 - 27144b2 + 14382*b1 + 146524
\(\nu^{7}\)	\(=\)	\( ( 8946 \beta_{15} + 28644 \beta_{14} - 7455 \beta_{13} - 7455 \beta_{12} - 100926 \beta_{11} + \cdots + 303386 \beta_{6} ) / 2 \) (8946b15 + 28644b14 - 7455b13 - 7455b12 - 100926b11 - 100926b10 + 95018b9 + 83824b8 + 1129450b7 + 303386b6) / 2
\(\nu^{8}\)	\(=\)	\( 115248 \beta_{13} - 115248 \beta_{12} + 11928 \beta_{11} - 11928 \beta_{10} + 403872 \beta_{5} + \cdots + 9780451 \) 115248b13 - 115248b12 + 11928b11 - 11928b10 + 403872b5 + 253392b4 - 4519872b3 - 2442096b2 + 959991*b1 + 9780451
\(\nu^{9}\)	\(=\)	\( ( 1038240 \beta_{15} + 1896126 \beta_{14} - 1211760 \beta_{13} - 1211760 \beta_{12} + \cdots + 20086960 \beta_{6} ) / 2 \) (1038240b15 + 1896126b14 - 1211760b13 - 1211760b12 - 8747271b11 - 8747271b10 + 11000688b9 + 13561392b8 + 97871050b7 + 20086960b6) / 2
\(\nu^{10}\)	\(=\)	\( 9695334 \beta_{13} - 9695334 \beta_{12} + 1730880 \beta_{11} - 1730880 \beta_{10} + 43825815 \beta_{5} + \cdots + 640268992 \) 9695334b13 - 9695334b12 + 1730880b11 - 1730880b10 + 43825815b5 + 36679680b4 - 490361146b3 - 205430720b2 + 62841372*b1 + 640268992
\(\nu^{11}\)	\(=\)	\( ( 107042298 \beta_{15} + 122220984 \beta_{14} - 160788375 \beta_{13} - 160788375 \beta_{12} + \cdots + 1294844404 \beta_{6} ) / 2 \) (107042298b15 + 122220984b14 - 160788375b13 - 160788375b12 - 717914868b11 - 717914868b10 + 1134050016b9 + 1799044666b8 + 8032577652b7 + 1294844404b6) / 2
\(\nu^{12}\)	\(=\)	\( 779025360 \beta_{13} - 779025360 \beta_{12} + 214309524 \beta_{11} - 214309524 \beta_{10} + \cdots + 40430917701 \) 779025360b13 - 779025360b12 + 214309524b11 - 214309524b10 + 4334149512b5 + 4540978432b4 - 48493836072b3 - 16506570736b2 + 3968186805*b1 + 40430917701
\(\nu^{13}\)	\(=\)	\( ( 10226349744 \beta_{15} + 7507754562 \beta_{14} - 18814322904 \beta_{13} - 18814322904 \beta_{12} + \cdots + 79539994124 \beta_{6} ) / 2 \) (10226349744b15 + 7507754562b14 - 18814322904b13 - 18814322904b12 - 56604000453b11 - 56604000453b10 + 108341901928b9 + 210509371384b8 + 633329511658b7 + 79539994124b6) / 2
\(\nu^{14}\)	\(=\)	\( 60357877734 \beta_{13} - 60357877734 \beta_{12} + 23927497776 \beta_{11} - 23927497776 \beta_{10} + \cdots + 2392540678744 \) 60357877734b13 - 60357877734b12 + 23927497776b11 - 23927497776b10 + 402082405725b5 + 506994502352b4 - 4498810986578b3 - 1278909510312b2 + 234821168592*b1 + 2392540678744
\(\nu^{15}\)	\(=\)	\( ( 924880566918 \beta_{15} + 421787341632 \beta_{14} - 2018775903885 \beta_{13} + \cdots + 4468579030904 \beta_{6} ) / 2 \) (924880566918b15 + 421787341632b14 - 2018775903885b13 - 2018775903885b12 - 4312981210464b11 - 4312981210464b10 + 9798544554140b9 + 22587637635874b8 + 48256993152328b7 + 4468579030904b6) / 2

\(n\):		e.g. 2-40 or 80-90
Embeddings:		e.g. 1-3 or 293.8
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T^{4} + 16)^{4} \) (T^4 + 16)^4
$3$	\( (T^{8} + 4577 T^{4} + 53824)^{2} \) (T^8 + 4577*T^4 + 53824)^2
$5$	\( T^{16} \) T^16
$7$	\( T^{16} + \cdots + 19\!\cdots\!01 \) T^16 - 4892T^12 + 17368809990T^8 - 67711576987292*T^4 + 191581231380566414401
$11$	\( (T^{2} + 5 T - 108)^{8} \) (T^2 + 5*T - 108)^8
$13$	\( (T^{8} + \cdots + 1150831181824)^{2} \) (T^8 + 4321313*T^4 + 1150831181824)^2
$17$	\( (T^{8} + \cdots + 7322695778304)^{2} \) (T^8 + 138180193*T^4 + 7322695778304)^2
$19$	\( (T^{4} - 8272 T^{2} + 6547968)^{4} \) (T^4 - 8272*T^2 + 6547968)^4
$23$	\( (T^{8} + \cdots + 18\!\cdots\!36)^{2} \) (T^8 + 1460256016*T^4 + 184029798122459136)^2
$29$	\( (T^{4} + 21789 T^{2} + 57972996)^{4} \) (T^4 + 21789*T^2 + 57972996)^4
$31$	\( (T^{4} + 108016 T^{2} + 2399496192)^{4} \) (T^4 + 108016*T^2 + 2399496192)^4
$37$	\( (T^{8} + \cdots + 41\!\cdots\!56)^{2} \) (T^8 + 25896485376*T^4 + 4126932052811513856)^2
$41$	\( (T^{4} + 150736 T^{2} + 347576832)^{4} \) (T^4 + 150736*T^2 + 347576832)^4
$43$	\( (T^{8} + \cdots + 24\!\cdots\!96)^{2} \) (T^8 + 18730374928*T^4 + 24020977881743364096)^2
$47$	\( (T^{8} + \cdots + 46\!\cdots\!44)^{2} \) (T^8 + 83700333217*T^4 + 461165519435760442944)^2
$53$	\( (T^{8} + \cdots + 21\!\cdots\!56)^{2} \) (T^8 + 699875584*T^4 + 21407557176262656)^2
$59$	\( (T^{4} - 704800 T^{2} + 120602880000)^{4} \) (T^4 - 704800*T^2 + 120602880000)^4
$61$	\( (T^{4} + 397168 T^{2} + 21595465728)^{4} \) (T^4 + 397168*T^2 + 21595465728)^4
$67$	\( (T^{8} + \cdots + 29\!\cdots\!36)^{2} \) (T^8 + 124856932624*T^4 + 29317947912460763136)^2
$71$	\( (T^{2} - 436 T - 42048)^{8} \) (T^2 - 436*T - 42048)^8
$73$	\( (T^{8} + \cdots + 13\!\cdots\!04)^{2} \) (T^8 + 311653224704*T^4 + 13054545152467957448704)^2
$79$	\( (T^{4} + 778913 T^{2} + 11999887936)^{4} \) (T^4 + 778913*T^2 + 11999887936)^4
$83$	\( (T^{8} + \cdots + 12\!\cdots\!64)^{2} \) (T^8 + 86690006352*T^4 + 1258615101806738187264)^2
$89$	\( (T^{4} - 1727856 T^{2} + 137399887872)^{4} \) (T^4 - 1727856*T^2 + 137399887872)^4
$97$	\( (T^{8} + \cdots + 53\!\cdots\!44)^{2} \) (T^8 + 1579608405233*T^4 + 538005250960771069394944)^2
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\(n\)	\(101\)	\(127\)
\(\chi(n)\)	\(-1\)	\(-\beta_{3}\)