# Properties

 Label 441.6.a.t.1.1 Level $441$ Weight $6$ Character 441.1 Self dual yes Analytic conductor $70.729$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [441,6,Mod(1,441)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(441, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("441.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$441 = 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 441.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

 Self dual: yes Analytic conductor: $$70.7292645375$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{249})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 62$$ x^2 - x - 62 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 21) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-7.38987$$ of defining polynomial Character $$\chi$$ $$=$$ 441.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-6.38987 q^{2} +8.83040 q^{4} -38.7291 q^{5} +148.051 q^{8} +O(q^{10})$$ $$q-6.38987 q^{2} +8.83040 q^{4} -38.7291 q^{5} +148.051 q^{8} +247.474 q^{10} +576.390 q^{11} +391.491 q^{13} -1228.60 q^{16} +1329.70 q^{17} +942.474 q^{19} -341.993 q^{20} -3683.05 q^{22} +1632.08 q^{23} -1625.06 q^{25} -2501.58 q^{26} +1463.54 q^{29} -3912.42 q^{31} +3112.95 q^{32} -8496.61 q^{34} -16300.3 q^{37} -6022.28 q^{38} -5733.86 q^{40} +13103.8 q^{41} +14733.5 q^{43} +5089.75 q^{44} -10428.8 q^{46} +6814.52 q^{47} +10383.9 q^{50} +3457.02 q^{52} +2011.34 q^{53} -22323.0 q^{55} -9351.85 q^{58} -51453.1 q^{59} +41097.8 q^{61} +24999.8 q^{62} +19423.8 q^{64} -15162.1 q^{65} +50578.2 q^{67} +11741.8 q^{68} -39970.6 q^{71} -55686.6 q^{73} +104157. q^{74} +8322.42 q^{76} -63151.4 q^{79} +47582.4 q^{80} -83731.7 q^{82} -45572.4 q^{83} -51498.1 q^{85} -94145.1 q^{86} +85334.9 q^{88} -15686.7 q^{89} +14411.9 q^{92} -43543.9 q^{94} -36501.1 q^{95} +3128.49 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 3 q^{2} + 65 q^{4} + 33 q^{5} + 375 q^{8}+O(q^{10})$$ 2 * q + 3 * q^2 + 65 * q^4 + 33 * q^5 + 375 * q^8 $$2 q + 3 q^{2} + 65 q^{4} + 33 q^{5} + 375 q^{8} + 921 q^{10} + 1137 q^{11} + 925 q^{13} - 895 q^{16} + 324 q^{17} + 2311 q^{19} + 3687 q^{20} + 1581 q^{22} - 1596 q^{23} + 395 q^{25} + 2508 q^{26} + 2217 q^{29} + 4294 q^{31} - 1017 q^{32} - 17940 q^{34} - 19109 q^{37} + 6828 q^{38} + 10545 q^{40} + 12858 q^{41} - 2771 q^{43} + 36579 q^{44} - 40740 q^{46} + 23160 q^{47} + 29352 q^{50} + 33424 q^{52} + 31653 q^{53} + 17889 q^{55} - 2277 q^{58} - 41097 q^{59} + 42052 q^{61} + 102057 q^{62} - 30031 q^{64} + 23106 q^{65} + 30763 q^{67} - 44748 q^{68} - 102096 q^{71} - 28577 q^{73} + 77784 q^{74} + 85192 q^{76} - 18464 q^{79} + 71511 q^{80} - 86040 q^{82} - 61179 q^{83} - 123636 q^{85} - 258510 q^{86} + 212565 q^{88} - 29322 q^{89} - 166908 q^{92} + 109938 q^{94} + 61662 q^{95} - 9791 q^{97}+O(q^{100})$$ 2 * q + 3 * q^2 + 65 * q^4 + 33 * q^5 + 375 * q^8 + 921 * q^10 + 1137 * q^11 + 925 * q^13 - 895 * q^16 + 324 * q^17 + 2311 * q^19 + 3687 * q^20 + 1581 * q^22 - 1596 * q^23 + 395 * q^25 + 2508 * q^26 + 2217 * q^29 + 4294 * q^31 - 1017 * q^32 - 17940 * q^34 - 19109 * q^37 + 6828 * q^38 + 10545 * q^40 + 12858 * q^41 - 2771 * q^43 + 36579 * q^44 - 40740 * q^46 + 23160 * q^47 + 29352 * q^50 + 33424 * q^52 + 31653 * q^53 + 17889 * q^55 - 2277 * q^58 - 41097 * q^59 + 42052 * q^61 + 102057 * q^62 - 30031 * q^64 + 23106 * q^65 + 30763 * q^67 - 44748 * q^68 - 102096 * q^71 - 28577 * q^73 + 77784 * q^74 + 85192 * q^76 - 18464 * q^79 + 71511 * q^80 - 86040 * q^82 - 61179 * q^83 - 123636 * q^85 - 258510 * q^86 + 212565 * q^88 - 29322 * q^89 - 166908 * q^92 + 109938 * q^94 + 61662 * q^95 - 9791 * q^97

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −6.38987 −1.12958 −0.564790 0.825235i $$-0.691043\pi$$
−0.564790 + 0.825235i $$0.691043\pi$$
$$3$$ 0 0
$$4$$ 8.83040 0.275950
$$5$$ −38.7291 −0.692807 −0.346403 0.938086i $$-0.612597\pi$$
−0.346403 + 0.938086i $$0.612597\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 148.051 0.817872
$$9$$ 0 0
$$10$$ 247.474 0.782580
$$11$$ 576.390 1.43627 0.718133 0.695906i $$-0.244997\pi$$
0.718133 + 0.695906i $$0.244997\pi$$
$$12$$ 0 0
$$13$$ 391.491 0.642486 0.321243 0.946997i $$-0.395899\pi$$
0.321243 + 0.946997i $$0.395899\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ −1228.60 −1.19980
$$17$$ 1329.70 1.11592 0.557958 0.829869i $$-0.311585\pi$$
0.557958 + 0.829869i $$0.311585\pi$$
$$18$$ 0 0
$$19$$ 942.474 0.598943 0.299471 0.954105i $$-0.403190\pi$$
0.299471 + 0.954105i $$0.403190\pi$$
$$20$$ −341.993 −0.191180
$$21$$ 0 0
$$22$$ −3683.05 −1.62238
$$23$$ 1632.08 0.643312 0.321656 0.946857i $$-0.395761\pi$$
0.321656 + 0.946857i $$0.395761\pi$$
$$24$$ 0 0
$$25$$ −1625.06 −0.520019
$$26$$ −2501.58 −0.725739
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 1463.54 0.323155 0.161577 0.986860i $$-0.448342\pi$$
0.161577 + 0.986860i $$0.448342\pi$$
$$30$$ 0 0
$$31$$ −3912.42 −0.731208 −0.365604 0.930770i $$-0.619138\pi$$
−0.365604 + 0.930770i $$0.619138\pi$$
$$32$$ 3112.95 0.537399
$$33$$ 0 0
$$34$$ −8496.61 −1.26052
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −16300.3 −1.95746 −0.978729 0.205160i $$-0.934229\pi$$
−0.978729 + 0.205160i $$0.934229\pi$$
$$38$$ −6022.28 −0.676553
$$39$$ 0 0
$$40$$ −5733.86 −0.566627
$$41$$ 13103.8 1.21741 0.608707 0.793395i $$-0.291688\pi$$
0.608707 + 0.793395i $$0.291688\pi$$
$$42$$ 0 0
$$43$$ 14733.5 1.21516 0.607582 0.794257i $$-0.292140\pi$$
0.607582 + 0.794257i $$0.292140\pi$$
$$44$$ 5089.75 0.396337
$$45$$ 0 0
$$46$$ −10428.8 −0.726672
$$47$$ 6814.52 0.449977 0.224989 0.974361i $$-0.427765\pi$$
0.224989 + 0.974361i $$0.427765\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 10383.9 0.587403
$$51$$ 0 0
$$52$$ 3457.02 0.177294
$$53$$ 2011.34 0.0983550 0.0491775 0.998790i $$-0.484340\pi$$
0.0491775 + 0.998790i $$0.484340\pi$$
$$54$$ 0 0
$$55$$ −22323.0 −0.995054
$$56$$ 0 0
$$57$$ 0 0
$$58$$ −9351.85 −0.365029
$$59$$ −51453.1 −1.92434 −0.962170 0.272451i $$-0.912166\pi$$
−0.962170 + 0.272451i $$0.912166\pi$$
$$60$$ 0 0
$$61$$ 41097.8 1.41415 0.707073 0.707141i $$-0.250015\pi$$
0.707073 + 0.707141i $$0.250015\pi$$
$$62$$ 24999.8 0.825958
$$63$$ 0 0
$$64$$ 19423.8 0.592766
$$65$$ −15162.1 −0.445119
$$66$$ 0 0
$$67$$ 50578.2 1.37650 0.688250 0.725473i $$-0.258379\pi$$
0.688250 + 0.725473i $$0.258379\pi$$
$$68$$ 11741.8 0.307937
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −39970.6 −0.941012 −0.470506 0.882397i $$-0.655929\pi$$
−0.470506 + 0.882397i $$0.655929\pi$$
$$72$$ 0 0
$$73$$ −55686.6 −1.22305 −0.611524 0.791226i $$-0.709443\pi$$
−0.611524 + 0.791226i $$0.709443\pi$$
$$74$$ 104157. 2.21110
$$75$$ 0 0
$$76$$ 8322.42 0.165278
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −63151.4 −1.13845 −0.569226 0.822181i $$-0.692757\pi$$
−0.569226 + 0.822181i $$0.692757\pi$$
$$80$$ 47582.4 0.831231
$$81$$ 0 0
$$82$$ −83731.7 −1.37517
$$83$$ −45572.4 −0.726116 −0.363058 0.931766i $$-0.618267\pi$$
−0.363058 + 0.931766i $$0.618267\pi$$
$$84$$ 0 0
$$85$$ −51498.1 −0.773114
$$86$$ −94145.1 −1.37262
$$87$$ 0 0
$$88$$ 85334.9 1.17468
$$89$$ −15686.7 −0.209921 −0.104961 0.994476i $$-0.533472\pi$$
−0.104961 + 0.994476i $$0.533472\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 14411.9 0.177522
$$93$$ 0 0
$$94$$ −43543.9 −0.508285
$$95$$ −36501.1 −0.414951
$$96$$ 0 0
$$97$$ 3128.49 0.0337603 0.0168801 0.999858i $$-0.494627\pi$$
0.0168801 + 0.999858i $$0.494627\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ −14349.9 −0.143499
$$101$$ 169011. 1.64858 0.824292 0.566164i $$-0.191573\pi$$
0.824292 + 0.566164i $$0.191573\pi$$
$$102$$ 0 0
$$103$$ 112820. 1.04784 0.523918 0.851769i $$-0.324470\pi$$
0.523918 + 0.851769i $$0.324470\pi$$
$$104$$ 57960.5 0.525471
$$105$$ 0 0
$$106$$ −12852.2 −0.111100
$$107$$ 22309.5 0.188378 0.0941890 0.995554i $$-0.469974\pi$$
0.0941890 + 0.995554i $$0.469974\pi$$
$$108$$ 0 0
$$109$$ −83819.5 −0.675739 −0.337869 0.941193i $$-0.609706\pi$$
−0.337869 + 0.941193i $$0.609706\pi$$
$$110$$ 142641. 1.12399
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 40928.4 0.301529 0.150764 0.988570i $$-0.451826\pi$$
0.150764 + 0.988570i $$0.451826\pi$$
$$114$$ 0 0
$$115$$ −63208.9 −0.445691
$$116$$ 12923.7 0.0891746
$$117$$ 0 0
$$118$$ 328779. 2.17369
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 171174. 1.06286
$$122$$ −262610. −1.59739
$$123$$ 0 0
$$124$$ −34548.2 −0.201777
$$125$$ 183965. 1.05308
$$126$$ 0 0
$$127$$ 83270.1 0.458120 0.229060 0.973412i $$-0.426435\pi$$
0.229060 + 0.973412i $$0.426435\pi$$
$$128$$ −223730. −1.20698
$$129$$ 0 0
$$130$$ 96883.7 0.502797
$$131$$ 166875. 0.849596 0.424798 0.905288i $$-0.360345\pi$$
0.424798 + 0.905288i $$0.360345\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ −323188. −1.55487
$$135$$ 0 0
$$136$$ 196863. 0.912676
$$137$$ −38223.9 −0.173994 −0.0869969 0.996209i $$-0.527727\pi$$
−0.0869969 + 0.996209i $$0.527727\pi$$
$$138$$ 0 0
$$139$$ 106263. 0.466492 0.233246 0.972418i $$-0.425065\pi$$
0.233246 + 0.972418i $$0.425065\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 255407. 1.06295
$$143$$ 225652. 0.922780
$$144$$ 0 0
$$145$$ −56681.7 −0.223884
$$146$$ 355830. 1.38153
$$147$$ 0 0
$$148$$ −143938. −0.540160
$$149$$ −192556. −0.710543 −0.355271 0.934763i $$-0.615612\pi$$
−0.355271 + 0.934763i $$0.615612\pi$$
$$150$$ 0 0
$$151$$ 141699. 0.505735 0.252868 0.967501i $$-0.418626\pi$$
0.252868 + 0.967501i $$0.418626\pi$$
$$152$$ 139534. 0.489858
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 151524. 0.506586
$$156$$ 0 0
$$157$$ 565771. 1.83186 0.915928 0.401342i $$-0.131456\pi$$
0.915928 + 0.401342i $$0.131456\pi$$
$$158$$ 403529. 1.28597
$$159$$ 0 0
$$160$$ −120562. −0.372314
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −430201. −1.26824 −0.634121 0.773233i $$-0.718638\pi$$
−0.634121 + 0.773233i $$0.718638\pi$$
$$164$$ 115712. 0.335946
$$165$$ 0 0
$$166$$ 291201. 0.820206
$$167$$ 240265. 0.666653 0.333327 0.942811i $$-0.391829\pi$$
0.333327 + 0.942811i $$0.391829\pi$$
$$168$$ 0 0
$$169$$ −218028. −0.587212
$$170$$ 329066. 0.873294
$$171$$ 0 0
$$172$$ 130103. 0.335324
$$173$$ 179300. 0.455476 0.227738 0.973722i $$-0.426867\pi$$
0.227738 + 0.973722i $$0.426867\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −708151. −1.72323
$$177$$ 0 0
$$178$$ 100236. 0.237123
$$179$$ 575559. 1.34263 0.671317 0.741170i $$-0.265729\pi$$
0.671317 + 0.741170i $$0.265729\pi$$
$$180$$ 0 0
$$181$$ 581006. 1.31821 0.659105 0.752051i $$-0.270935\pi$$
0.659105 + 0.752051i $$0.270935\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 241630. 0.526147
$$185$$ 631297. 1.35614
$$186$$ 0 0
$$187$$ 766426. 1.60275
$$188$$ 60174.9 0.124171
$$189$$ 0 0
$$190$$ 233237. 0.468721
$$191$$ 660560. 1.31017 0.655087 0.755554i $$-0.272632\pi$$
0.655087 + 0.755554i $$0.272632\pi$$
$$192$$ 0 0
$$193$$ −557310. −1.07697 −0.538485 0.842635i $$-0.681003\pi$$
−0.538485 + 0.842635i $$0.681003\pi$$
$$194$$ −19990.7 −0.0381349
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 761400. 1.39781 0.698904 0.715216i $$-0.253672\pi$$
0.698904 + 0.715216i $$0.253672\pi$$
$$198$$ 0 0
$$199$$ −135860. −0.243197 −0.121598 0.992579i $$-0.538802\pi$$
−0.121598 + 0.992579i $$0.538802\pi$$
$$200$$ −240591. −0.425309
$$201$$ 0 0
$$202$$ −1.07996e6 −1.86221
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −507499. −0.843433
$$206$$ −720906. −1.18361
$$207$$ 0 0
$$208$$ −480985. −0.770856
$$209$$ 543232. 0.860240
$$210$$ 0 0
$$211$$ −991157. −1.53263 −0.766313 0.642467i $$-0.777911\pi$$
−0.766313 + 0.642467i $$0.777911\pi$$
$$212$$ 17761.0 0.0271411
$$213$$ 0 0
$$214$$ −142555. −0.212788
$$215$$ −570615. −0.841873
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 535596. 0.763301
$$219$$ 0 0
$$220$$ −197121. −0.274585
$$221$$ 520566. 0.716960
$$222$$ 0 0
$$223$$ 543344. 0.731666 0.365833 0.930681i $$-0.380784\pi$$
0.365833 + 0.930681i $$0.380784\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ −261527. −0.340601
$$227$$ −16.3341 −2.10393e−5 0 −1.05196e−5 1.00000i $$-0.500003\pi$$
−1.05196e−5 1.00000i $$0.500003\pi$$
$$228$$ 0 0
$$229$$ −77379.0 −0.0975067 −0.0487534 0.998811i $$-0.515525\pi$$
−0.0487534 + 0.998811i $$0.515525\pi$$
$$230$$ 403896. 0.503443
$$231$$ 0 0
$$232$$ 216679. 0.264299
$$233$$ 103657. 0.125086 0.0625432 0.998042i $$-0.480079\pi$$
0.0625432 + 0.998042i $$0.480079\pi$$
$$234$$ 0 0
$$235$$ −263920. −0.311747
$$236$$ −454351. −0.531021
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −689109. −0.780356 −0.390178 0.920739i $$-0.627587\pi$$
−0.390178 + 0.920739i $$0.627587\pi$$
$$240$$ 0 0
$$241$$ 220296. 0.244323 0.122161 0.992510i $$-0.461017\pi$$
0.122161 + 0.992510i $$0.461017\pi$$
$$242$$ −1.09378e6 −1.20058
$$243$$ 0 0
$$244$$ 362910. 0.390234
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 368970. 0.384812
$$248$$ −579236. −0.598035
$$249$$ 0 0
$$250$$ −1.17551e6 −1.18954
$$251$$ −1.43641e6 −1.43912 −0.719558 0.694433i $$-0.755655\pi$$
−0.719558 + 0.694433i $$0.755655\pi$$
$$252$$ 0 0
$$253$$ 940714. 0.923966
$$254$$ −532085. −0.517483
$$255$$ 0 0
$$256$$ 808042. 0.770609
$$257$$ 909197. 0.858668 0.429334 0.903146i $$-0.358748\pi$$
0.429334 + 0.903146i $$0.358748\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ −133887. −0.122830
$$261$$ 0 0
$$262$$ −1.06631e6 −0.959687
$$263$$ −749057. −0.667768 −0.333884 0.942614i $$-0.608359\pi$$
−0.333884 + 0.942614i $$0.608359\pi$$
$$264$$ 0 0
$$265$$ −77897.4 −0.0681410
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 446626. 0.379845
$$269$$ −669945. −0.564493 −0.282246 0.959342i $$-0.591080\pi$$
−0.282246 + 0.959342i $$0.591080\pi$$
$$270$$ 0 0
$$271$$ 540659. 0.447199 0.223599 0.974681i $$-0.428219\pi$$
0.223599 + 0.974681i $$0.428219\pi$$
$$272$$ −1.63367e6 −1.33888
$$273$$ 0 0
$$274$$ 244246. 0.196540
$$275$$ −936668. −0.746885
$$276$$ 0 0
$$277$$ −401910. −0.314723 −0.157362 0.987541i $$-0.550299\pi$$
−0.157362 + 0.987541i $$0.550299\pi$$
$$278$$ −679005. −0.526940
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 429139. 0.324214 0.162107 0.986773i $$-0.448171\pi$$
0.162107 + 0.986773i $$0.448171\pi$$
$$282$$ 0 0
$$283$$ −340927. −0.253044 −0.126522 0.991964i $$-0.540381\pi$$
−0.126522 + 0.991964i $$0.540381\pi$$
$$284$$ −352957. −0.259672
$$285$$ 0 0
$$286$$ −1.44188e6 −1.04235
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 348246. 0.245268
$$290$$ 362188. 0.252895
$$291$$ 0 0
$$292$$ −491735. −0.337500
$$293$$ 388847. 0.264612 0.132306 0.991209i $$-0.457762\pi$$
0.132306 + 0.991209i $$0.457762\pi$$
$$294$$ 0 0
$$295$$ 1.99273e6 1.33319
$$296$$ −2.41328e6 −1.60095
$$297$$ 0 0
$$298$$ 1.23040e6 0.802615
$$299$$ 638945. 0.413319
$$300$$ 0 0
$$301$$ 0 0
$$302$$ −905435. −0.571268
$$303$$ 0 0
$$304$$ −1.15792e6 −0.718612
$$305$$ −1.59168e6 −0.979730
$$306$$ 0 0
$$307$$ −2.35747e6 −1.42758 −0.713789 0.700361i $$-0.753022\pi$$
−0.713789 + 0.700361i $$0.753022\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ −968220. −0.572229
$$311$$ 1.43663e6 0.842254 0.421127 0.907002i $$-0.361635\pi$$
0.421127 + 0.907002i $$0.361635\pi$$
$$312$$ 0 0
$$313$$ −822800. −0.474715 −0.237358 0.971422i $$-0.576281\pi$$
−0.237358 + 0.971422i $$0.576281\pi$$
$$314$$ −3.61520e6 −2.06923
$$315$$ 0 0
$$316$$ −557652. −0.314156
$$317$$ 1.76693e6 0.987580 0.493790 0.869581i $$-0.335611\pi$$
0.493790 + 0.869581i $$0.335611\pi$$
$$318$$ 0 0
$$319$$ 843572. 0.464136
$$320$$ −752264. −0.410672
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 1.25321e6 0.668370
$$324$$ 0 0
$$325$$ −636196. −0.334105
$$326$$ 2.74893e6 1.43258
$$327$$ 0 0
$$328$$ 1.94003e6 0.995689
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −3052.27 −0.00153128 −0.000765638 1.00000i $$-0.500244\pi$$
−0.000765638 1.00000i $$0.500244\pi$$
$$332$$ −402422. −0.200372
$$333$$ 0 0
$$334$$ −1.53526e6 −0.753038
$$335$$ −1.95885e6 −0.953649
$$336$$ 0 0
$$337$$ 2.02939e6 0.973398 0.486699 0.873570i $$-0.338201\pi$$
0.486699 + 0.873570i $$0.338201\pi$$
$$338$$ 1.39317e6 0.663302
$$339$$ 0 0
$$340$$ −454748. −0.213341
$$341$$ −2.25508e6 −1.05021
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 2.18130e6 0.993848
$$345$$ 0 0
$$346$$ −1.14570e6 −0.514496
$$347$$ −3.78218e6 −1.68624 −0.843119 0.537727i $$-0.819283\pi$$
−0.843119 + 0.537727i $$0.819283\pi$$
$$348$$ 0 0
$$349$$ 291147. 0.127953 0.0639763 0.997951i $$-0.479622\pi$$
0.0639763 + 0.997951i $$0.479622\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 1.79427e6 0.771848
$$353$$ −385076. −0.164479 −0.0822394 0.996613i $$-0.526207\pi$$
−0.0822394 + 0.996613i $$0.526207\pi$$
$$354$$ 0 0
$$355$$ 1.54803e6 0.651939
$$356$$ −138520. −0.0579277
$$357$$ 0 0
$$358$$ −3.67775e6 −1.51661
$$359$$ 3.23014e6 1.32277 0.661385 0.750046i $$-0.269969\pi$$
0.661385 + 0.750046i $$0.269969\pi$$
$$360$$ 0 0
$$361$$ −1.58784e6 −0.641268
$$362$$ −3.71255e6 −1.48902
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 2.15669e6 0.847336
$$366$$ 0 0
$$367$$ −479559. −0.185856 −0.0929280 0.995673i $$-0.529623\pi$$
−0.0929280 + 0.995673i $$0.529623\pi$$
$$368$$ −2.00517e6 −0.771847
$$369$$ 0 0
$$370$$ −4.03390e6 −1.53187
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 872666. 0.324770 0.162385 0.986727i $$-0.448081\pi$$
0.162385 + 0.986727i $$0.448081\pi$$
$$374$$ −4.89736e6 −1.81043
$$375$$ 0 0
$$376$$ 1.00889e6 0.368024
$$377$$ 572965. 0.207622
$$378$$ 0 0
$$379$$ −2.43493e6 −0.870742 −0.435371 0.900251i $$-0.643383\pi$$
−0.435371 + 0.900251i $$0.643383\pi$$
$$380$$ −322320. −0.114506
$$381$$ 0 0
$$382$$ −4.22089e6 −1.47995
$$383$$ 3.61169e6 1.25809 0.629047 0.777367i $$-0.283445\pi$$
0.629047 + 0.777367i $$0.283445\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 3.56114e6 1.21652
$$387$$ 0 0
$$388$$ 27625.9 0.00931615
$$389$$ 175232. 0.0587138 0.0293569 0.999569i $$-0.490654\pi$$
0.0293569 + 0.999569i $$0.490654\pi$$
$$390$$ 0 0
$$391$$ 2.17018e6 0.717882
$$392$$ 0 0
$$393$$ 0 0
$$394$$ −4.86525e6 −1.57893
$$395$$ 2.44579e6 0.788727
$$396$$ 0 0
$$397$$ 1.88515e6 0.600303 0.300152 0.953892i $$-0.402963\pi$$
0.300152 + 0.953892i $$0.402963\pi$$
$$398$$ 868126. 0.274710
$$399$$ 0 0
$$400$$ 1.99654e6 0.623920
$$401$$ −1.33983e6 −0.416091 −0.208046 0.978119i $$-0.566710\pi$$
−0.208046 + 0.978119i $$0.566710\pi$$
$$402$$ 0 0
$$403$$ −1.53168e6 −0.469791
$$404$$ 1.49243e6 0.454927
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −9.39535e6 −2.81143
$$408$$ 0 0
$$409$$ 6.58628e6 1.94685 0.973423 0.229013i $$-0.0735499\pi$$
0.973423 + 0.229013i $$0.0735499\pi$$
$$410$$ 3.24285e6 0.952725
$$411$$ 0 0
$$412$$ 996247. 0.289150
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 1.76497e6 0.503058
$$416$$ 1.21869e6 0.345271
$$417$$ 0 0
$$418$$ −3.47118e6 −0.971710
$$419$$ 6.96869e6 1.93917 0.969585 0.244754i $$-0.0787071\pi$$
0.969585 + 0.244754i $$0.0787071\pi$$
$$420$$ 0 0
$$421$$ 3.84041e6 1.05602 0.528010 0.849238i $$-0.322938\pi$$
0.528010 + 0.849238i $$0.322938\pi$$
$$422$$ 6.33336e6 1.73122
$$423$$ 0 0
$$424$$ 297781. 0.0804418
$$425$$ −2.16084e6 −0.580297
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 197002. 0.0519829
$$429$$ 0 0
$$430$$ 3.64615e6 0.950963
$$431$$ −3.03636e6 −0.787337 −0.393668 0.919253i $$-0.628794\pi$$
−0.393668 + 0.919253i $$0.628794\pi$$
$$432$$ 0 0
$$433$$ −941529. −0.241332 −0.120666 0.992693i $$-0.538503\pi$$
−0.120666 + 0.992693i $$0.538503\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −740160. −0.186470
$$437$$ 1.53819e6 0.385307
$$438$$ 0 0
$$439$$ 1.34109e6 0.332122 0.166061 0.986116i $$-0.446895\pi$$
0.166061 + 0.986116i $$0.446895\pi$$
$$440$$ −3.30494e6 −0.813827
$$441$$ 0 0
$$442$$ −3.32635e6 −0.809864
$$443$$ 772341. 0.186982 0.0934910 0.995620i $$-0.470197\pi$$
0.0934910 + 0.995620i $$0.470197\pi$$
$$444$$ 0 0
$$445$$ 607531. 0.145435
$$446$$ −3.47190e6 −0.826475
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −2.25684e6 −0.528304 −0.264152 0.964481i $$-0.585092\pi$$
−0.264152 + 0.964481i $$0.585092\pi$$
$$450$$ 0 0
$$451$$ 7.55291e6 1.74853
$$452$$ 361414. 0.0832069
$$453$$ 0 0
$$454$$ 104.373 2.37655e−5 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 4.28470e6 0.959688 0.479844 0.877354i $$-0.340693\pi$$
0.479844 + 0.877354i $$0.340693\pi$$
$$458$$ 494442. 0.110142
$$459$$ 0 0
$$460$$ −558160. −0.122988
$$461$$ 3.10462e6 0.680387 0.340193 0.940355i $$-0.389507\pi$$
0.340193 + 0.940355i $$0.389507\pi$$
$$462$$ 0 0
$$463$$ −3.53386e6 −0.766121 −0.383060 0.923723i $$-0.625130\pi$$
−0.383060 + 0.923723i $$0.625130\pi$$
$$464$$ −1.79811e6 −0.387722
$$465$$ 0 0
$$466$$ −662356. −0.141295
$$467$$ −2.72459e6 −0.578109 −0.289054 0.957313i $$-0.593341\pi$$
−0.289054 + 0.957313i $$0.593341\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 1.68641e6 0.352143
$$471$$ 0 0
$$472$$ −7.61767e6 −1.57386
$$473$$ 8.49224e6 1.74530
$$474$$ 0 0
$$475$$ −1.53158e6 −0.311461
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 4.40331e6 0.881475
$$479$$ −978685. −0.194896 −0.0974482 0.995241i $$-0.531068\pi$$
−0.0974482 + 0.995241i $$0.531068\pi$$
$$480$$ 0 0
$$481$$ −6.38144e6 −1.25764
$$482$$ −1.40766e6 −0.275982
$$483$$ 0 0
$$484$$ 1.51154e6 0.293296
$$485$$ −121164. −0.0233893
$$486$$ 0 0
$$487$$ 3.92744e6 0.750390 0.375195 0.926946i $$-0.377576\pi$$
0.375195 + 0.926946i $$0.377576\pi$$
$$488$$ 6.08456e6 1.15659
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −2.63241e6 −0.492777 −0.246388 0.969171i $$-0.579244\pi$$
−0.246388 + 0.969171i $$0.579244\pi$$
$$492$$ 0 0
$$493$$ 1.94607e6 0.360614
$$494$$ −2.35767e6 −0.434676
$$495$$ 0 0
$$496$$ 4.80678e6 0.877305
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 2.12544e6 0.382118 0.191059 0.981579i $$-0.438808\pi$$
0.191059 + 0.981579i $$0.438808\pi$$
$$500$$ 1.62449e6 0.290597
$$501$$ 0 0
$$502$$ 9.17850e6 1.62560
$$503$$ −2.60929e6 −0.459835 −0.229917 0.973210i $$-0.573846\pi$$
−0.229917 + 0.973210i $$0.573846\pi$$
$$504$$ 0 0
$$505$$ −6.54564e6 −1.14215
$$506$$ −6.01104e6 −1.04369
$$507$$ 0 0
$$508$$ 735308. 0.126418
$$509$$ 1.00182e7 1.71394 0.856970 0.515366i $$-0.172344\pi$$
0.856970 + 0.515366i $$0.172344\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 1.99607e6 0.336512
$$513$$ 0 0
$$514$$ −5.80965e6 −0.969933
$$515$$ −4.36942e6 −0.725948
$$516$$ 0 0
$$517$$ 3.92782e6 0.646287
$$518$$ 0 0
$$519$$ 0 0
$$520$$ −2.24476e6 −0.364050
$$521$$ 4.17941e6 0.674560 0.337280 0.941404i $$-0.390493\pi$$
0.337280 + 0.941404i $$0.390493\pi$$
$$522$$ 0 0
$$523$$ 3.60525e6 0.576343 0.288172 0.957579i $$-0.406953\pi$$
0.288172 + 0.957579i $$0.406953\pi$$
$$524$$ 1.47357e6 0.234446
$$525$$ 0 0
$$526$$ 4.78637e6 0.754297
$$527$$ −5.20234e6 −0.815967
$$528$$ 0 0
$$529$$ −3.77266e6 −0.586150
$$530$$ 497754. 0.0769707
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 5.13003e6 0.782172
$$534$$ 0 0
$$535$$ −864025. −0.130509
$$536$$ 7.48814e6 1.12580
$$537$$ 0 0
$$538$$ 4.28086e6 0.637640
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −6.68166e6 −0.981501 −0.490751 0.871300i $$-0.663277\pi$$
−0.490751 + 0.871300i $$0.663277\pi$$
$$542$$ −3.45474e6 −0.505146
$$543$$ 0 0
$$544$$ 4.13929e6 0.599692
$$545$$ 3.24625e6 0.468156
$$546$$ 0 0
$$547$$ 8.69076e6 1.24191 0.620954 0.783847i $$-0.286745\pi$$
0.620954 + 0.783847i $$0.286745\pi$$
$$548$$ −337532. −0.0480136
$$549$$ 0 0
$$550$$ 5.98518e6 0.843666
$$551$$ 1.37935e6 0.193551
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 2.56815e6 0.355505
$$555$$ 0 0
$$556$$ 938342. 0.128728
$$557$$ −6.24742e6 −0.853223 −0.426612 0.904435i $$-0.640293\pi$$
−0.426612 + 0.904435i $$0.640293\pi$$
$$558$$ 0 0
$$559$$ 5.76803e6 0.780725
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −2.74214e6 −0.366226
$$563$$ 1.19045e7 1.58285 0.791423 0.611269i $$-0.209341\pi$$
0.791423 + 0.611269i $$0.209341\pi$$
$$564$$ 0 0
$$565$$ −1.58512e6 −0.208901
$$566$$ 2.17848e6 0.285833
$$567$$ 0 0
$$568$$ −5.91768e6 −0.769627
$$569$$ −21414.4 −0.00277284 −0.00138642 0.999999i $$-0.500441\pi$$
−0.00138642 + 0.999999i $$0.500441\pi$$
$$570$$ 0 0
$$571$$ 7.11647e6 0.913428 0.456714 0.889614i $$-0.349026\pi$$
0.456714 + 0.889614i $$0.349026\pi$$
$$572$$ 1.99259e6 0.254641
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −2.65223e6 −0.334534
$$576$$ 0 0
$$577$$ −1.06652e7 −1.33361 −0.666805 0.745232i $$-0.732339\pi$$
−0.666805 + 0.745232i $$0.732339\pi$$
$$578$$ −2.22524e6 −0.277050
$$579$$ 0 0
$$580$$ −500522. −0.0617808
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 1.15932e6 0.141264
$$584$$ −8.24444e6 −1.00030
$$585$$ 0 0
$$586$$ −2.48468e6 −0.298901
$$587$$ 1.30101e7 1.55843 0.779213 0.626759i $$-0.215619\pi$$
0.779213 + 0.626759i $$0.215619\pi$$
$$588$$ 0 0
$$589$$ −3.68735e6 −0.437952
$$590$$ −1.27333e7 −1.50595
$$591$$ 0 0
$$592$$ 2.00265e7 2.34856
$$593$$ −4.26086e6 −0.497578 −0.248789 0.968558i $$-0.580033\pi$$
−0.248789 + 0.968558i $$0.580033\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −1.70034e6 −0.196074
$$597$$ 0 0
$$598$$ −4.08277e6 −0.466877
$$599$$ 1.37958e7 1.57101 0.785507 0.618853i $$-0.212402\pi$$
0.785507 + 0.618853i $$0.212402\pi$$
$$600$$ 0 0
$$601$$ 4.99695e6 0.564311 0.282155 0.959369i $$-0.408951\pi$$
0.282155 + 0.959369i $$0.408951\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 1.25126e6 0.139558
$$605$$ −6.62942e6 −0.736355
$$606$$ 0 0
$$607$$ −3.04946e6 −0.335932 −0.167966 0.985793i $$-0.553720\pi$$
−0.167966 + 0.985793i $$0.553720\pi$$
$$608$$ 2.93387e6 0.321871
$$609$$ 0 0
$$610$$ 1.01706e7 1.10668
$$611$$ 2.66782e6 0.289104
$$612$$ 0 0
$$613$$ −7.03625e6 −0.756293 −0.378147 0.925746i $$-0.623438\pi$$
−0.378147 + 0.925746i $$0.623438\pi$$
$$614$$ 1.50639e7 1.61256
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −1.00066e7 −1.05822 −0.529108 0.848554i $$-0.677473\pi$$
−0.529108 + 0.848554i $$0.677473\pi$$
$$618$$ 0 0
$$619$$ −6.55067e6 −0.687161 −0.343581 0.939123i $$-0.611640\pi$$
−0.343581 + 0.939123i $$0.611640\pi$$
$$620$$ 1.33802e6 0.139792
$$621$$ 0 0
$$622$$ −9.17986e6 −0.951393
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −2.04650e6 −0.209561
$$626$$ 5.25758e6 0.536229
$$627$$ 0 0
$$628$$ 4.99598e6 0.505501
$$629$$ −2.16746e7 −2.18436
$$630$$ 0 0
$$631$$ 2.22672e6 0.222635 0.111317 0.993785i $$-0.464493\pi$$
0.111317 + 0.993785i $$0.464493\pi$$
$$632$$ −9.34960e6 −0.931109
$$633$$ 0 0
$$634$$ −1.12905e7 −1.11555
$$635$$ −3.22497e6 −0.317389
$$636$$ 0 0
$$637$$ 0 0
$$638$$ −5.39031e6 −0.524279
$$639$$ 0 0
$$640$$ 8.66484e6 0.836201
$$641$$ 1.59340e7 1.53172 0.765859 0.643008i $$-0.222314\pi$$
0.765859 + 0.643008i $$0.222314\pi$$
$$642$$ 0 0
$$643$$ −1.49933e7 −1.43011 −0.715056 0.699067i $$-0.753599\pi$$
−0.715056 + 0.699067i $$0.753599\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ −8.00783e6 −0.754977
$$647$$ −1.29805e7 −1.21908 −0.609540 0.792755i $$-0.708646\pi$$
−0.609540 + 0.792755i $$0.708646\pi$$
$$648$$ 0 0
$$649$$ −2.96571e7 −2.76386
$$650$$ 4.06521e6 0.377398
$$651$$ 0 0
$$652$$ −3.79885e6 −0.349972
$$653$$ 1.65492e7 1.51878 0.759389 0.650637i $$-0.225498\pi$$
0.759389 + 0.650637i $$0.225498\pi$$
$$654$$ 0 0
$$655$$ −6.46291e6 −0.588606
$$656$$ −1.60993e7 −1.46066
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −5.86879e6 −0.526423 −0.263212 0.964738i $$-0.584782\pi$$
−0.263212 + 0.964738i $$0.584782\pi$$
$$660$$ 0 0
$$661$$ 7.27687e6 0.647800 0.323900 0.946091i $$-0.395006\pi$$
0.323900 + 0.946091i $$0.395006\pi$$
$$662$$ 19503.6 0.00172970
$$663$$ 0 0
$$664$$ −6.74702e6 −0.593870
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 2.38862e6 0.207889
$$668$$ 2.12164e6 0.183963
$$669$$ 0 0
$$670$$ 1.25168e7 1.07722
$$671$$ 2.36884e7 2.03109
$$672$$ 0 0
$$673$$ −1.82417e7 −1.55248 −0.776241 0.630437i $$-0.782876\pi$$
−0.776241 + 0.630437i $$0.782876\pi$$
$$674$$ −1.29675e7 −1.09953
$$675$$ 0 0
$$676$$ −1.92527e6 −0.162041
$$677$$ 7.76406e6 0.651054 0.325527 0.945533i $$-0.394458\pi$$
0.325527 + 0.945533i $$0.394458\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ −7.62432e6 −0.632308
$$681$$ 0 0
$$682$$ 1.44096e7 1.18629
$$683$$ −8.56324e6 −0.702403 −0.351201 0.936300i $$-0.614227\pi$$
−0.351201 + 0.936300i $$0.614227\pi$$
$$684$$ 0 0
$$685$$ 1.48038e6 0.120544
$$686$$ 0 0
$$687$$ 0 0
$$688$$ −1.81015e7 −1.45796
$$689$$ 787423. 0.0631917
$$690$$ 0 0
$$691$$ 1.64509e7 1.31068 0.655338 0.755336i $$-0.272526\pi$$
0.655338 + 0.755336i $$0.272526\pi$$
$$692$$ 1.58329e6 0.125689
$$693$$ 0 0
$$694$$ 2.41677e7 1.90474
$$695$$ −4.11546e6 −0.323189
$$696$$ 0 0
$$697$$ 1.74242e7 1.35853
$$698$$ −1.86039e6 −0.144533
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 1.66928e7 1.28302 0.641512 0.767113i $$-0.278307\pi$$
0.641512 + 0.767113i $$0.278307\pi$$
$$702$$ 0 0
$$703$$ −1.53626e7 −1.17240
$$704$$ 1.11957e7 0.851370
$$705$$ 0 0
$$706$$ 2.46059e6 0.185792
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 5.61779e6 0.419711 0.209855 0.977732i $$-0.432701\pi$$
0.209855 + 0.977732i $$0.432701\pi$$
$$710$$ −9.89167e6 −0.736417
$$711$$ 0 0
$$712$$ −2.32242e6 −0.171689
$$713$$ −6.38537e6 −0.470395
$$714$$ 0 0
$$715$$ −8.73927e6 −0.639308
$$716$$ 5.08242e6 0.370500
$$717$$ 0 0
$$718$$ −2.06401e7 −1.49417
$$719$$ −1.03718e7 −0.748227 −0.374113 0.927383i $$-0.622053\pi$$
−0.374113 + 0.927383i $$0.622053\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 1.01461e7 0.724363
$$723$$ 0 0
$$724$$ 5.13052e6 0.363760
$$725$$ −2.37835e6 −0.168047
$$726$$ 0 0
$$727$$ 1.15369e7 0.809565 0.404783 0.914413i $$-0.367347\pi$$
0.404783 + 0.914413i $$0.367347\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ −1.37810e7 −0.957134
$$731$$ 1.95911e7 1.35602
$$732$$ 0 0
$$733$$ 1.49470e7 1.02753 0.513764 0.857932i $$-0.328251\pi$$
0.513764 + 0.857932i $$0.328251\pi$$
$$734$$ 3.06432e6 0.209939
$$735$$ 0 0
$$736$$ 5.08058e6 0.345715
$$737$$ 2.91528e7 1.97702
$$738$$ 0 0
$$739$$ 9.01364e6 0.607140 0.303570 0.952809i $$-0.401821\pi$$
0.303570 + 0.952809i $$0.401821\pi$$
$$740$$ 5.57460e6 0.374227
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 2.10239e7 1.39714 0.698571 0.715541i $$-0.253820\pi$$
0.698571 + 0.715541i $$0.253820\pi$$
$$744$$ 0 0
$$745$$ 7.45750e6 0.492269
$$746$$ −5.57622e6 −0.366854
$$747$$ 0 0
$$748$$ 6.76785e6 0.442279
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 4.04219e6 0.261527 0.130764 0.991414i $$-0.458257\pi$$
0.130764 + 0.991414i $$0.458257\pi$$
$$752$$ −8.37230e6 −0.539884
$$753$$ 0 0
$$754$$ −3.66117e6 −0.234526
$$755$$ −5.48786e6 −0.350377
$$756$$ 0 0
$$757$$ −1.82059e7 −1.15471 −0.577353 0.816495i $$-0.695914\pi$$
−0.577353 + 0.816495i $$0.695914\pi$$
$$758$$ 1.55589e7 0.983572
$$759$$ 0 0
$$760$$ −5.40402e6 −0.339377
$$761$$ 1.89200e7 1.18429 0.592146 0.805831i $$-0.298281\pi$$
0.592146 + 0.805831i $$0.298281\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 5.83301e6 0.361542
$$765$$ 0 0
$$766$$ −2.30782e7 −1.42112
$$767$$ −2.01434e7 −1.23636
$$768$$ 0 0
$$769$$ −1.12831e7 −0.688037 −0.344019 0.938963i $$-0.611788\pi$$
−0.344019 + 0.938963i $$0.611788\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −4.92127e6 −0.297190
$$773$$ 3.68565e6 0.221853 0.110926 0.993829i $$-0.464618\pi$$
0.110926 + 0.993829i $$0.464618\pi$$
$$774$$ 0 0
$$775$$ 6.35791e6 0.380242
$$776$$ 463176. 0.0276116
$$777$$ 0 0
$$778$$ −1.11971e6 −0.0663219
$$779$$ 1.23500e7 0.729161
$$780$$ 0 0
$$781$$ −2.30387e7 −1.35154
$$782$$ −1.38671e7 −0.810905
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −2.19118e7 −1.26912
$$786$$ 0 0
$$787$$ −1.40748e7 −0.810039 −0.405019 0.914308i $$-0.632735\pi$$
−0.405019 + 0.914308i $$0.632735\pi$$
$$788$$ 6.72347e6 0.385725
$$789$$ 0 0
$$790$$ −1.56283e7 −0.890930
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 1.60894e7 0.908569
$$794$$ −1.20459e7 −0.678090
$$795$$ 0 0
$$796$$ −1.19970e6 −0.0671102
$$797$$ 1.75191e7 0.976937 0.488469 0.872582i $$-0.337556\pi$$
0.488469 + 0.872582i $$0.337556\pi$$
$$798$$ 0 0
$$799$$ 9.06127e6 0.502137
$$800$$ −5.05873e6 −0.279458
$$801$$ 0 0
$$802$$ 8.56134e6 0.470008
$$803$$ −3.20972e7 −1.75662
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 9.78721e6 0.530666
$$807$$ 0 0
$$808$$ 2.50222e7 1.34833
$$809$$ −814521. −0.0437553 −0.0218777 0.999761i $$-0.506964\pi$$
−0.0218777 + 0.999761i $$0.506964\pi$$
$$810$$ 0 0
$$811$$ 1.26533e7 0.675540 0.337770 0.941229i $$-0.390327\pi$$
0.337770 + 0.941229i $$0.390327\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 6.00350e7 3.17573
$$815$$ 1.66613e7 0.878647
$$816$$ 0 0
$$817$$ 1.38859e7 0.727813
$$818$$ −4.20854e7 −2.19912
$$819$$ 0 0
$$820$$ −4.48142e6 −0.232745
$$821$$ −4.27393e6 −0.221294 −0.110647 0.993860i $$-0.535292\pi$$
−0.110647 + 0.993860i $$0.535292\pi$$
$$822$$ 0 0
$$823$$ 1.70463e7 0.877266 0.438633 0.898666i $$-0.355463\pi$$
0.438633 + 0.898666i $$0.355463\pi$$
$$824$$ 1.67031e7 0.856996
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −2.60828e6 −0.132614 −0.0663071 0.997799i $$-0.521122\pi$$
−0.0663071 + 0.997799i $$0.521122\pi$$
$$828$$ 0 0
$$829$$ 2.13865e7 1.08082 0.540409 0.841403i $$-0.318270\pi$$
0.540409 + 0.841403i $$0.318270\pi$$
$$830$$ −1.12780e7 −0.568244
$$831$$ 0 0
$$832$$ 7.60423e6 0.380844
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −9.30525e6 −0.461862
$$836$$ 4.79696e6 0.237383
$$837$$ 0 0
$$838$$ −4.45290e7 −2.19045
$$839$$ −771393. −0.0378330 −0.0189165 0.999821i $$-0.506022\pi$$
−0.0189165 + 0.999821i $$0.506022\pi$$
$$840$$ 0 0
$$841$$ −1.83692e7 −0.895571
$$842$$ −2.45397e7 −1.19286
$$843$$ 0 0
$$844$$ −8.75231e6 −0.422928
$$845$$ 8.44401e6 0.406824
$$846$$ 0 0
$$847$$ 0 0
$$848$$ −2.47113e6 −0.118006
$$849$$ 0 0
$$850$$ 1.38075e7 0.655492
$$851$$ −2.66034e7 −1.25926
$$852$$ 0 0
$$853$$ 2.94032e7 1.38364 0.691818 0.722072i $$-0.256810\pi$$
0.691818 + 0.722072i $$0.256810\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 3.30293e6 0.154069
$$857$$ 2.65010e7 1.23257 0.616283 0.787525i $$-0.288638\pi$$
0.616283 + 0.787525i $$0.288638\pi$$
$$858$$ 0 0
$$859$$ −3.67519e7 −1.69940 −0.849702 0.527264i $$-0.823218\pi$$
−0.849702 + 0.527264i $$0.823218\pi$$
$$860$$ −5.03876e6 −0.232315
$$861$$ 0 0
$$862$$ 1.94020e7 0.889360
$$863$$ −1.18189e7 −0.540196 −0.270098 0.962833i $$-0.587056\pi$$
−0.270098 + 0.962833i $$0.587056\pi$$
$$864$$ 0 0
$$865$$ −6.94413e6 −0.315557
$$866$$ 6.01625e6 0.272603
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −3.63998e7 −1.63512
$$870$$ 0 0
$$871$$ 1.98009e7 0.884382
$$872$$ −1.24095e7 −0.552668
$$873$$ 0 0
$$874$$ −9.82884e6 −0.435235
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 7.93509e6 0.348380 0.174190 0.984712i $$-0.444269\pi$$
0.174190 + 0.984712i $$0.444269\pi$$
$$878$$ −8.56939e6 −0.375158
$$879$$ 0 0
$$880$$ 2.74260e7 1.19387
$$881$$ −4.20152e7 −1.82375 −0.911877 0.410464i $$-0.865367\pi$$
−0.911877 + 0.410464i $$0.865367\pi$$
$$882$$ 0 0
$$883$$ 2.12461e7 0.917016 0.458508 0.888690i $$-0.348384\pi$$
0.458508 + 0.888690i $$0.348384\pi$$
$$884$$ 4.59681e6 0.197845
$$885$$ 0 0
$$886$$ −4.93516e6 −0.211211
$$887$$ −1.88490e7 −0.804415 −0.402208 0.915548i $$-0.631757\pi$$
−0.402208 + 0.915548i $$0.631757\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ −3.88204e6 −0.164280
$$891$$ 0 0
$$892$$ 4.79795e6 0.201903
$$893$$ 6.42251e6 0.269511
$$894$$ 0 0
$$895$$ −2.22909e7 −0.930186
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 1.44209e7 0.596762
$$899$$ −5.72600e6 −0.236294
$$900$$ 0 0
$$901$$ 2.67448e6 0.109756
$$902$$ −4.82621e7 −1.97510
$$903$$ 0 0
$$904$$ 6.05948e6 0.246612
$$905$$ −2.25018e7 −0.913264
$$906$$ 0 0
$$907$$ −6.19446e6 −0.250026 −0.125013 0.992155i $$-0.539897\pi$$
−0.125013 + 0.992155i $$0.539897\pi$$
$$908$$ −144.237 −5.80578e−6 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 2.50171e7 0.998712 0.499356 0.866397i $$-0.333570\pi$$
0.499356 + 0.866397i $$0.333570\pi$$
$$912$$ 0 0
$$913$$ −2.62674e7 −1.04290
$$914$$ −2.73787e7 −1.08404
$$915$$ 0 0
$$916$$ −683288. −0.0269070
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 1.54992e7 0.605370 0.302685 0.953091i $$-0.402117\pi$$
0.302685 + 0.953091i $$0.402117\pi$$
$$920$$ −9.35812e6 −0.364518
$$921$$ 0 0
$$922$$ −1.98381e7 −0.768551
$$923$$ −1.56481e7 −0.604587
$$924$$ 0 0
$$925$$ 2.64890e7 1.01791
$$926$$ 2.25809e7 0.865394
$$927$$ 0 0
$$928$$ 4.55594e6 0.173663
$$929$$ −3.75285e7 −1.42667 −0.713333 0.700826i $$-0.752815\pi$$
−0.713333 + 0.700826i $$0.752815\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 915335. 0.0345176
$$933$$ 0 0
$$934$$ 1.74098e7 0.653020
$$935$$ −2.96830e7 −1.11040
$$936$$ 0 0
$$937$$ −1.08298e7 −0.402969 −0.201485 0.979492i $$-0.564577\pi$$
−0.201485 + 0.979492i $$0.564577\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ −2.33052e6 −0.0860267
$$941$$ −3.24591e7 −1.19498 −0.597492 0.801875i $$-0.703836\pi$$
−0.597492 + 0.801875i $$0.703836\pi$$
$$942$$ 0 0
$$943$$ 2.13865e7 0.783177
$$944$$ 6.32151e7 2.30883
$$945$$ 0 0
$$946$$ −5.42643e7 −1.97145
$$947$$ −7.53877e6 −0.273165 −0.136583 0.990629i $$-0.543612\pi$$
−0.136583 + 0.990629i $$0.543612\pi$$
$$948$$ 0 0
$$949$$ −2.18008e7 −0.785792
$$950$$ 9.78656e6 0.351821
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 3.01356e7 1.07485 0.537424 0.843312i $$-0.319398\pi$$
0.537424 + 0.843312i $$0.319398\pi$$
$$954$$ 0 0
$$955$$ −2.55829e7 −0.907697
$$956$$ −6.08510e6 −0.215339
$$957$$ 0 0
$$958$$ 6.25366e6 0.220151
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −1.33221e7 −0.465335
$$962$$ 4.07765e7 1.42060
$$963$$ 0 0
$$964$$ 1.94530e6 0.0674208
$$965$$ 2.15841e7 0.746132
$$966$$ 0 0
$$967$$ 2.88021e6 0.0990509 0.0495255 0.998773i $$-0.484229\pi$$
0.0495255 + 0.998773i $$0.484229\pi$$
$$968$$ 2.53425e7 0.869282
$$969$$ 0 0
$$970$$ 774220. 0.0264201
$$971$$ 2.74490e7 0.934283 0.467142 0.884183i $$-0.345284\pi$$
0.467142 + 0.884183i $$0.345284\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ −2.50958e7 −0.847626
$$975$$ 0 0
$$976$$ −5.04927e7 −1.69669
$$977$$ 5.88524e7 1.97255 0.986274 0.165118i $$-0.0528004\pi$$
0.986274 + 0.165118i $$0.0528004\pi$$
$$978$$ 0 0
$$979$$ −9.04164e6 −0.301502
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 1.68208e7 0.556631
$$983$$ −1.86077e7 −0.614198 −0.307099 0.951678i $$-0.599358\pi$$
−0.307099 + 0.951678i $$0.599358\pi$$
$$984$$ 0 0
$$985$$ −2.94883e7 −0.968410
$$986$$ −1.24352e7 −0.407342
$$987$$ 0 0
$$988$$ 3.25815e6 0.106189
$$989$$ 2.40462e7 0.781729
$$990$$ 0 0
$$991$$ 2.47154e7 0.799435 0.399718 0.916638i $$-0.369108\pi$$
0.399718 + 0.916638i $$0.369108\pi$$
$$992$$ −1.21792e7 −0.392951
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 5.26172e6 0.168488
$$996$$ 0 0
$$997$$ −1.79581e7 −0.572167 −0.286084 0.958205i $$-0.592354\pi$$
−0.286084 + 0.958205i $$0.592354\pi$$
$$998$$ −1.35813e7 −0.431633
$$999$$ 0 0
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 441.6.a.t.1.1 2
3.2 odd 2 147.6.a.i.1.2 2
7.2 even 3 63.6.e.c.46.2 4
7.4 even 3 63.6.e.c.37.2 4
7.6 odd 2 441.6.a.s.1.1 2
21.2 odd 6 21.6.e.b.4.1 4
21.5 even 6 147.6.e.l.67.1 4
21.11 odd 6 21.6.e.b.16.1 yes 4
21.17 even 6 147.6.e.l.79.1 4
21.20 even 2 147.6.a.k.1.2 2
84.11 even 6 336.6.q.e.289.1 4
84.23 even 6 336.6.q.e.193.1 4

By twisted newform
Twist Min Dim Char Parity Ord Type
21.6.e.b.4.1 4 21.2 odd 6
21.6.e.b.16.1 yes 4 21.11 odd 6
63.6.e.c.37.2 4 7.4 even 3
63.6.e.c.46.2 4 7.2 even 3
147.6.a.i.1.2 2 3.2 odd 2
147.6.a.k.1.2 2 21.20 even 2
147.6.e.l.67.1 4 21.5 even 6
147.6.e.l.79.1 4 21.17 even 6
336.6.q.e.193.1 4 84.23 even 6
336.6.q.e.289.1 4 84.11 even 6
441.6.a.s.1.1 2 7.6 odd 2
441.6.a.t.1.1 2 1.1 even 1 trivial