# Properties

 Label 825.6.a.y.1.6 Level $825$ Weight $6$ Character 825.1 Self dual yes Analytic conductor $132.317$ Analytic rank $0$ Dimension $13$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [825,6,Mod(1,825)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(825, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("825.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$825 = 3 \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 825.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: $$132.316651346$$ Analytic rank: $$0$$ Dimension: $$13$$ Coefficient field: $$\mathbb{Q}[x]/(x^{13} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{13} - 306 x^{11} - 206 x^{10} + 34574 x^{9} + 39928 x^{8} - 1788312 x^{7} - 2591628 x^{6} + 42852537 x^{5} + 63733360 x^{4} - 448113518 x^{3} + \cdots + 522579400$$ x^13 - 306*x^11 - 206*x^10 + 34574*x^9 + 39928*x^8 - 1788312*x^7 - 2591628*x^6 + 42852537*x^5 + 63733360*x^4 - 448113518*x^3 - 549984598*x^2 + 1518551280*x + 522579400 Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{9}\cdot 3^{2}\cdot 5^{7}$$ Twist minimal: no (minimal twist has level 165) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.6 Root $$1.88656$$ of defining polynomial Character $$\chi$$ $$=$$ 825.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-0.886559 q^{2} +9.00000 q^{3} -31.2140 q^{4} -7.97903 q^{6} +224.774 q^{7} +56.0430 q^{8} +81.0000 q^{9} +O(q^{10})$$ $$q-0.886559 q^{2} +9.00000 q^{3} -31.2140 q^{4} -7.97903 q^{6} +224.774 q^{7} +56.0430 q^{8} +81.0000 q^{9} +121.000 q^{11} -280.926 q^{12} +1065.64 q^{13} -199.275 q^{14} +949.163 q^{16} -1347.75 q^{17} -71.8113 q^{18} +691.252 q^{19} +2022.96 q^{21} -107.274 q^{22} +3396.13 q^{23} +504.387 q^{24} -944.749 q^{26} +729.000 q^{27} -7016.09 q^{28} +8603.02 q^{29} -320.509 q^{31} -2634.86 q^{32} +1089.00 q^{33} +1194.86 q^{34} -2528.34 q^{36} -1906.23 q^{37} -612.836 q^{38} +9590.72 q^{39} +5821.85 q^{41} -1793.48 q^{42} +1421.42 q^{43} -3776.90 q^{44} -3010.87 q^{46} -6454.51 q^{47} +8542.47 q^{48} +33716.2 q^{49} -12129.7 q^{51} -33262.8 q^{52} +9109.26 q^{53} -646.302 q^{54} +12597.0 q^{56} +6221.27 q^{57} -7627.08 q^{58} +10213.6 q^{59} -34276.3 q^{61} +284.150 q^{62} +18206.7 q^{63} -28037.3 q^{64} -965.463 q^{66} -68436.0 q^{67} +42068.6 q^{68} +30565.2 q^{69} +1883.92 q^{71} +4539.48 q^{72} -87733.7 q^{73} +1689.99 q^{74} -21576.8 q^{76} +27197.6 q^{77} -8502.74 q^{78} -52408.8 q^{79} +6561.00 q^{81} -5161.42 q^{82} +58665.8 q^{83} -63144.8 q^{84} -1260.17 q^{86} +77427.1 q^{87} +6781.20 q^{88} -113843. q^{89} +239527. q^{91} -106007. q^{92} -2884.58 q^{93} +5722.30 q^{94} -23713.8 q^{96} -115329. q^{97} -29891.4 q^{98} +9801.00 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$13 q + 13 q^{2} + 117 q^{3} + 209 q^{4} + 117 q^{6} + 304 q^{7} + 399 q^{8} + 1053 q^{9}+O(q^{10})$$ 13 * q + 13 * q^2 + 117 * q^3 + 209 * q^4 + 117 * q^6 + 304 * q^7 + 399 * q^8 + 1053 * q^9 $$13 q + 13 q^{2} + 117 q^{3} + 209 q^{4} + 117 q^{6} + 304 q^{7} + 399 q^{8} + 1053 q^{9} + 1573 q^{11} + 1881 q^{12} + 986 q^{13} - 610 q^{14} + 3501 q^{16} + 1476 q^{17} + 1053 q^{18} + 270 q^{19} + 2736 q^{21} + 1573 q^{22} + 9084 q^{23} + 3591 q^{24} + 2652 q^{26} + 9477 q^{27} + 10920 q^{28} + 11952 q^{29} + 19096 q^{31} + 11661 q^{32} + 14157 q^{33} - 1302 q^{34} + 16929 q^{36} + 39964 q^{37} + 1574 q^{38} + 8874 q^{39} + 35184 q^{41} - 5490 q^{42} - 96 q^{43} + 25289 q^{44} - 4120 q^{46} + 34984 q^{47} + 31509 q^{48} + 14557 q^{49} + 13284 q^{51} + 39002 q^{52} + 22984 q^{53} + 9477 q^{54} + 59802 q^{56} + 2430 q^{57} + 18896 q^{58} - 9192 q^{59} + 5438 q^{61} + 272 q^{62} + 24624 q^{63} + 106557 q^{64} + 14157 q^{66} + 71508 q^{67} + 127948 q^{68} + 81756 q^{69} + 101700 q^{71} + 32319 q^{72} + 77390 q^{73} + 13676 q^{74} + 139966 q^{76} + 36784 q^{77} + 23868 q^{78} + 93954 q^{79} + 85293 q^{81} + 53284 q^{82} + 185918 q^{83} + 98280 q^{84} + 370930 q^{86} + 107568 q^{87} + 48279 q^{88} - 18418 q^{89} + 174536 q^{91} + 274264 q^{92} + 171864 q^{93} + 64520 q^{94} + 104949 q^{96} + 94312 q^{97} + 145677 q^{98} + 127413 q^{99}+O(q^{100})$$ 13 * q + 13 * q^2 + 117 * q^3 + 209 * q^4 + 117 * q^6 + 304 * q^7 + 399 * q^8 + 1053 * q^9 + 1573 * q^11 + 1881 * q^12 + 986 * q^13 - 610 * q^14 + 3501 * q^16 + 1476 * q^17 + 1053 * q^18 + 270 * q^19 + 2736 * q^21 + 1573 * q^22 + 9084 * q^23 + 3591 * q^24 + 2652 * q^26 + 9477 * q^27 + 10920 * q^28 + 11952 * q^29 + 19096 * q^31 + 11661 * q^32 + 14157 * q^33 - 1302 * q^34 + 16929 * q^36 + 39964 * q^37 + 1574 * q^38 + 8874 * q^39 + 35184 * q^41 - 5490 * q^42 - 96 * q^43 + 25289 * q^44 - 4120 * q^46 + 34984 * q^47 + 31509 * q^48 + 14557 * q^49 + 13284 * q^51 + 39002 * q^52 + 22984 * q^53 + 9477 * q^54 + 59802 * q^56 + 2430 * q^57 + 18896 * q^58 - 9192 * q^59 + 5438 * q^61 + 272 * q^62 + 24624 * q^63 + 106557 * q^64 + 14157 * q^66 + 71508 * q^67 + 127948 * q^68 + 81756 * q^69 + 101700 * q^71 + 32319 * q^72 + 77390 * q^73 + 13676 * q^74 + 139966 * q^76 + 36784 * q^77 + 23868 * q^78 + 93954 * q^79 + 85293 * q^81 + 53284 * q^82 + 185918 * q^83 + 98280 * q^84 + 370930 * q^86 + 107568 * q^87 + 48279 * q^88 - 18418 * q^89 + 174536 * q^91 + 274264 * q^92 + 171864 * q^93 + 64520 * q^94 + 104949 * q^96 + 94312 * q^97 + 145677 * q^98 + 127413 * q^99

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −0.886559 −0.156723 −0.0783615 0.996925i $$-0.524969\pi$$
−0.0783615 + 0.996925i $$0.524969\pi$$
$$3$$ 9.00000 0.577350
$$4$$ −31.2140 −0.975438
$$5$$ 0 0
$$6$$ −7.97903 −0.0904841
$$7$$ 224.774 1.73380 0.866902 0.498478i $$-0.166108\pi$$
0.866902 + 0.498478i $$0.166108\pi$$
$$8$$ 56.0430 0.309597
$$9$$ 81.0000 0.333333
$$10$$ 0 0
$$11$$ 121.000 0.301511
$$12$$ −280.926 −0.563169
$$13$$ 1065.64 1.74884 0.874420 0.485169i $$-0.161242\pi$$
0.874420 + 0.485169i $$0.161242\pi$$
$$14$$ −199.275 −0.271727
$$15$$ 0 0
$$16$$ 949.163 0.926917
$$17$$ −1347.75 −1.13106 −0.565531 0.824727i $$-0.691329\pi$$
−0.565531 + 0.824727i $$0.691329\pi$$
$$18$$ −71.8113 −0.0522410
$$19$$ 691.252 0.439291 0.219646 0.975580i $$-0.429510\pi$$
0.219646 + 0.975580i $$0.429510\pi$$
$$20$$ 0 0
$$21$$ 2022.96 1.00101
$$22$$ −107.274 −0.0472538
$$23$$ 3396.13 1.33864 0.669321 0.742973i $$-0.266585\pi$$
0.669321 + 0.742973i $$0.266585\pi$$
$$24$$ 504.387 0.178746
$$25$$ 0 0
$$26$$ −944.749 −0.274084
$$27$$ 729.000 0.192450
$$28$$ −7016.09 −1.69122
$$29$$ 8603.02 1.89957 0.949786 0.312900i $$-0.101301\pi$$
0.949786 + 0.312900i $$0.101301\pi$$
$$30$$ 0 0
$$31$$ −320.509 −0.0599013 −0.0299507 0.999551i $$-0.509535\pi$$
−0.0299507 + 0.999551i $$0.509535\pi$$
$$32$$ −2634.86 −0.454866
$$33$$ 1089.00 0.174078
$$34$$ 1194.86 0.177263
$$35$$ 0 0
$$36$$ −2528.34 −0.325146
$$37$$ −1906.23 −0.228913 −0.114457 0.993428i $$-0.536513\pi$$
−0.114457 + 0.993428i $$0.536513\pi$$
$$38$$ −612.836 −0.0688471
$$39$$ 9590.72 1.00969
$$40$$ 0 0
$$41$$ 5821.85 0.540881 0.270440 0.962737i $$-0.412831\pi$$
0.270440 + 0.962737i $$0.412831\pi$$
$$42$$ −1793.48 −0.156882
$$43$$ 1421.42 0.117234 0.0586168 0.998281i $$-0.481331\pi$$
0.0586168 + 0.998281i $$0.481331\pi$$
$$44$$ −3776.90 −0.294106
$$45$$ 0 0
$$46$$ −3010.87 −0.209796
$$47$$ −6454.51 −0.426205 −0.213102 0.977030i $$-0.568357\pi$$
−0.213102 + 0.977030i $$0.568357\pi$$
$$48$$ 8542.47 0.535156
$$49$$ 33716.2 2.00608
$$50$$ 0 0
$$51$$ −12129.7 −0.653019
$$52$$ −33262.8 −1.70589
$$53$$ 9109.26 0.445444 0.222722 0.974882i $$-0.428506\pi$$
0.222722 + 0.974882i $$0.428506\pi$$
$$54$$ −646.302 −0.0301614
$$55$$ 0 0
$$56$$ 12597.0 0.536780
$$57$$ 6221.27 0.253625
$$58$$ −7627.08 −0.297707
$$59$$ 10213.6 0.381989 0.190995 0.981591i $$-0.438829\pi$$
0.190995 + 0.981591i $$0.438829\pi$$
$$60$$ 0 0
$$61$$ −34276.3 −1.17942 −0.589712 0.807614i $$-0.700759\pi$$
−0.589712 + 0.807614i $$0.700759\pi$$
$$62$$ 284.150 0.00938791
$$63$$ 18206.7 0.577935
$$64$$ −28037.3 −0.855629
$$65$$ 0 0
$$66$$ −965.463 −0.0272820
$$67$$ −68436.0 −1.86251 −0.931253 0.364374i $$-0.881283\pi$$
−0.931253 + 0.364374i $$0.881283\pi$$
$$68$$ 42068.6 1.10328
$$69$$ 30565.2 0.772866
$$70$$ 0 0
$$71$$ 1883.92 0.0443524 0.0221762 0.999754i $$-0.492941\pi$$
0.0221762 + 0.999754i $$0.492941\pi$$
$$72$$ 4539.48 0.103199
$$73$$ −87733.7 −1.92690 −0.963451 0.267885i $$-0.913675\pi$$
−0.963451 + 0.267885i $$0.913675\pi$$
$$74$$ 1689.99 0.0358760
$$75$$ 0 0
$$76$$ −21576.8 −0.428501
$$77$$ 27197.6 0.522762
$$78$$ −8502.74 −0.158242
$$79$$ −52408.8 −0.944793 −0.472397 0.881386i $$-0.656611\pi$$
−0.472397 + 0.881386i $$0.656611\pi$$
$$80$$ 0 0
$$81$$ 6561.00 0.111111
$$82$$ −5161.42 −0.0847685
$$83$$ 58665.8 0.934737 0.467369 0.884062i $$-0.345202\pi$$
0.467369 + 0.884062i $$0.345202\pi$$
$$84$$ −63144.8 −0.976426
$$85$$ 0 0
$$86$$ −1260.17 −0.0183732
$$87$$ 77427.1 1.09672
$$88$$ 6781.20 0.0933469
$$89$$ −113843. −1.52346 −0.761728 0.647896i $$-0.775649\pi$$
−0.761728 + 0.647896i $$0.775649\pi$$
$$90$$ 0 0
$$91$$ 239527. 3.03215
$$92$$ −106007. −1.30576
$$93$$ −2884.58 −0.0345840
$$94$$ 5722.30 0.0667961
$$95$$ 0 0
$$96$$ −23713.8 −0.262617
$$97$$ −115329. −1.24454 −0.622271 0.782802i $$-0.713790\pi$$
−0.622271 + 0.782802i $$0.713790\pi$$
$$98$$ −29891.4 −0.314399
$$99$$ 9801.00 0.100504
$$100$$ 0 0
$$101$$ 39906.6 0.389261 0.194631 0.980877i $$-0.437649\pi$$
0.194631 + 0.980877i $$0.437649\pi$$
$$102$$ 10753.7 0.102343
$$103$$ 69320.0 0.643822 0.321911 0.946770i $$-0.395675\pi$$
0.321911 + 0.946770i $$0.395675\pi$$
$$104$$ 59721.4 0.541435
$$105$$ 0 0
$$106$$ −8075.90 −0.0698114
$$107$$ 131149. 1.10740 0.553700 0.832717i $$-0.313216\pi$$
0.553700 + 0.832717i $$0.313216\pi$$
$$108$$ −22755.0 −0.187723
$$109$$ 146929. 1.18451 0.592257 0.805749i $$-0.298237\pi$$
0.592257 + 0.805749i $$0.298237\pi$$
$$110$$ 0 0
$$111$$ −17156.1 −0.132163
$$112$$ 213347. 1.60709
$$113$$ −57346.0 −0.422481 −0.211240 0.977434i $$-0.567750\pi$$
−0.211240 + 0.977434i $$0.567750\pi$$
$$114$$ −5515.52 −0.0397489
$$115$$ 0 0
$$116$$ −268535. −1.85291
$$117$$ 86316.5 0.582947
$$118$$ −9055.00 −0.0598665
$$119$$ −302938. −1.96104
$$120$$ 0 0
$$121$$ 14641.0 0.0909091
$$122$$ 30388.0 0.184843
$$123$$ 52396.7 0.312278
$$124$$ 10004.4 0.0584300
$$125$$ 0 0
$$126$$ −16141.3 −0.0905757
$$127$$ 172649. 0.949852 0.474926 0.880026i $$-0.342475\pi$$
0.474926 + 0.880026i $$0.342475\pi$$
$$128$$ 109172. 0.588963
$$129$$ 12792.8 0.0676848
$$130$$ 0 0
$$131$$ 297723. 1.51577 0.757886 0.652387i $$-0.226233\pi$$
0.757886 + 0.652387i $$0.226233\pi$$
$$132$$ −33992.1 −0.169802
$$133$$ 155375. 0.761645
$$134$$ 60672.6 0.291897
$$135$$ 0 0
$$136$$ −75531.8 −0.350173
$$137$$ −45559.8 −0.207387 −0.103693 0.994609i $$-0.533066\pi$$
−0.103693 + 0.994609i $$0.533066\pi$$
$$138$$ −27097.8 −0.121126
$$139$$ −124306. −0.545701 −0.272850 0.962056i $$-0.587966\pi$$
−0.272850 + 0.962056i $$0.587966\pi$$
$$140$$ 0 0
$$141$$ −58090.6 −0.246070
$$142$$ −1670.21 −0.00695104
$$143$$ 128942. 0.527295
$$144$$ 76882.2 0.308972
$$145$$ 0 0
$$146$$ 77781.2 0.301990
$$147$$ 303445. 1.15821
$$148$$ 59501.1 0.223291
$$149$$ −107578. −0.396971 −0.198485 0.980104i $$-0.563602\pi$$
−0.198485 + 0.980104i $$0.563602\pi$$
$$150$$ 0 0
$$151$$ 147513. 0.526487 0.263244 0.964729i $$-0.415208\pi$$
0.263244 + 0.964729i $$0.415208\pi$$
$$152$$ 38739.8 0.136003
$$153$$ −109168. −0.377021
$$154$$ −24112.3 −0.0819288
$$155$$ 0 0
$$156$$ −299365. −0.984893
$$157$$ 231901. 0.750849 0.375425 0.926853i $$-0.377497\pi$$
0.375425 + 0.926853i $$0.377497\pi$$
$$158$$ 46463.5 0.148071
$$159$$ 81983.3 0.257177
$$160$$ 0 0
$$161$$ 763360. 2.32094
$$162$$ −5816.72 −0.0174137
$$163$$ −6862.35 −0.0202304 −0.0101152 0.999949i $$-0.503220\pi$$
−0.0101152 + 0.999949i $$0.503220\pi$$
$$164$$ −181723. −0.527596
$$165$$ 0 0
$$166$$ −52010.7 −0.146495
$$167$$ −10230.0 −0.0283847 −0.0141923 0.999899i $$-0.504518\pi$$
−0.0141923 + 0.999899i $$0.504518\pi$$
$$168$$ 113373. 0.309910
$$169$$ 764285. 2.05844
$$170$$ 0 0
$$171$$ 55991.4 0.146430
$$172$$ −44368.3 −0.114354
$$173$$ 421259. 1.07012 0.535062 0.844813i $$-0.320288\pi$$
0.535062 + 0.844813i $$0.320288\pi$$
$$174$$ −68643.8 −0.171881
$$175$$ 0 0
$$176$$ 114849. 0.279476
$$177$$ 91922.8 0.220541
$$178$$ 100928. 0.238761
$$179$$ 330517. 0.771012 0.385506 0.922705i $$-0.374027\pi$$
0.385506 + 0.922705i $$0.374027\pi$$
$$180$$ 0 0
$$181$$ 668805. 1.51741 0.758705 0.651434i $$-0.225832\pi$$
0.758705 + 0.651434i $$0.225832\pi$$
$$182$$ −212355. −0.475207
$$183$$ −308487. −0.680940
$$184$$ 190329. 0.414439
$$185$$ 0 0
$$186$$ 2557.35 0.00542012
$$187$$ −163078. −0.341028
$$188$$ 201471. 0.415736
$$189$$ 163860. 0.333671
$$190$$ 0 0
$$191$$ −961866. −1.90779 −0.953896 0.300137i $$-0.902967\pi$$
−0.953896 + 0.300137i $$0.902967\pi$$
$$192$$ −252335. −0.493998
$$193$$ −87470.4 −0.169032 −0.0845158 0.996422i $$-0.526934\pi$$
−0.0845158 + 0.996422i $$0.526934\pi$$
$$194$$ 102246. 0.195048
$$195$$ 0 0
$$196$$ −1.05242e6 −1.95680
$$197$$ 799656. 1.46804 0.734019 0.679129i $$-0.237642\pi$$
0.734019 + 0.679129i $$0.237642\pi$$
$$198$$ −8689.17 −0.0157513
$$199$$ −635515. −1.13761 −0.568804 0.822473i $$-0.692594\pi$$
−0.568804 + 0.822473i $$0.692594\pi$$
$$200$$ 0 0
$$201$$ −615924. −1.07532
$$202$$ −35379.6 −0.0610062
$$203$$ 1.93373e6 3.29349
$$204$$ 378618. 0.636980
$$205$$ 0 0
$$206$$ −61456.3 −0.100902
$$207$$ 275086. 0.446214
$$208$$ 1.01146e6 1.62103
$$209$$ 83641.5 0.132451
$$210$$ 0 0
$$211$$ −349594. −0.540578 −0.270289 0.962779i $$-0.587119\pi$$
−0.270289 + 0.962779i $$0.587119\pi$$
$$212$$ −284337. −0.434503
$$213$$ 16955.3 0.0256069
$$214$$ −116271. −0.173555
$$215$$ 0 0
$$216$$ 40855.3 0.0595819
$$217$$ −72042.0 −0.103857
$$218$$ −130261. −0.185641
$$219$$ −789604. −1.11250
$$220$$ 0 0
$$221$$ −1.43621e6 −1.97805
$$222$$ 15209.9 0.0207130
$$223$$ −309538. −0.416824 −0.208412 0.978041i $$-0.566829\pi$$
−0.208412 + 0.978041i $$0.566829\pi$$
$$224$$ −592248. −0.788648
$$225$$ 0 0
$$226$$ 50840.6 0.0662124
$$227$$ −509733. −0.656566 −0.328283 0.944579i $$-0.606470\pi$$
−0.328283 + 0.944579i $$0.606470\pi$$
$$228$$ −194191. −0.247395
$$229$$ −1.28351e6 −1.61737 −0.808684 0.588243i $$-0.799820\pi$$
−0.808684 + 0.588243i $$0.799820\pi$$
$$230$$ 0 0
$$231$$ 244778. 0.301817
$$232$$ 482139. 0.588101
$$233$$ 312877. 0.377558 0.188779 0.982020i $$-0.439547\pi$$
0.188779 + 0.982020i $$0.439547\pi$$
$$234$$ −76524.7 −0.0913612
$$235$$ 0 0
$$236$$ −318809. −0.372607
$$237$$ −471679. −0.545477
$$238$$ 268573. 0.307340
$$239$$ −357254. −0.404559 −0.202280 0.979328i $$-0.564835\pi$$
−0.202280 + 0.979328i $$0.564835\pi$$
$$240$$ 0 0
$$241$$ −680168. −0.754351 −0.377176 0.926142i $$-0.623105\pi$$
−0.377176 + 0.926142i $$0.623105\pi$$
$$242$$ −12980.1 −0.0142475
$$243$$ 59049.0 0.0641500
$$244$$ 1.06990e6 1.15045
$$245$$ 0 0
$$246$$ −46452.8 −0.0489411
$$247$$ 736623. 0.768250
$$248$$ −17962.3 −0.0185452
$$249$$ 527992. 0.539671
$$250$$ 0 0
$$251$$ 157817. 0.158114 0.0790568 0.996870i $$-0.474809\pi$$
0.0790568 + 0.996870i $$0.474809\pi$$
$$252$$ −568303. −0.563740
$$253$$ 410932. 0.403616
$$254$$ −153064. −0.148864
$$255$$ 0 0
$$256$$ 800404. 0.763325
$$257$$ 557296. 0.526324 0.263162 0.964752i $$-0.415235\pi$$
0.263162 + 0.964752i $$0.415235\pi$$
$$258$$ −11341.6 −0.0106078
$$259$$ −428470. −0.396891
$$260$$ 0 0
$$261$$ 696844. 0.633191
$$262$$ −263949. −0.237556
$$263$$ −645647. −0.575580 −0.287790 0.957693i $$-0.592921\pi$$
−0.287790 + 0.957693i $$0.592921\pi$$
$$264$$ 61030.8 0.0538939
$$265$$ 0 0
$$266$$ −137749. −0.119367
$$267$$ −1.02458e6 −0.879568
$$268$$ 2.13616e6 1.81676
$$269$$ −219755. −0.185165 −0.0925825 0.995705i $$-0.529512\pi$$
−0.0925825 + 0.995705i $$0.529512\pi$$
$$270$$ 0 0
$$271$$ 1.54792e6 1.28034 0.640171 0.768233i $$-0.278864\pi$$
0.640171 + 0.768233i $$0.278864\pi$$
$$272$$ −1.27923e6 −1.04840
$$273$$ 2.15574e6 1.75061
$$274$$ 40391.5 0.0325023
$$275$$ 0 0
$$276$$ −954061. −0.753882
$$277$$ −980724. −0.767976 −0.383988 0.923338i $$-0.625450\pi$$
−0.383988 + 0.923338i $$0.625450\pi$$
$$278$$ 110205. 0.0855239
$$279$$ −25961.2 −0.0199671
$$280$$ 0 0
$$281$$ 2.46283e6 1.86067 0.930334 0.366713i $$-0.119517\pi$$
0.930334 + 0.366713i $$0.119517\pi$$
$$282$$ 51500.7 0.0385648
$$283$$ −838610. −0.622435 −0.311217 0.950339i $$-0.600737\pi$$
−0.311217 + 0.950339i $$0.600737\pi$$
$$284$$ −58804.8 −0.0432630
$$285$$ 0 0
$$286$$ −114315. −0.0826393
$$287$$ 1.30860e6 0.937782
$$288$$ −213424. −0.151622
$$289$$ 396569. 0.279302
$$290$$ 0 0
$$291$$ −1.03796e6 −0.718536
$$292$$ 2.73852e6 1.87957
$$293$$ 1.32689e6 0.902955 0.451478 0.892282i $$-0.350897\pi$$
0.451478 + 0.892282i $$0.350897\pi$$
$$294$$ −269022. −0.181518
$$295$$ 0 0
$$296$$ −106831. −0.0708708
$$297$$ 88209.0 0.0580259
$$298$$ 95374.4 0.0622144
$$299$$ 3.61904e6 2.34107
$$300$$ 0 0
$$301$$ 319498. 0.203260
$$302$$ −130779. −0.0825126
$$303$$ 359159. 0.224740
$$304$$ 656111. 0.407187
$$305$$ 0 0
$$306$$ 96783.6 0.0590878
$$307$$ 1.04842e6 0.634878 0.317439 0.948279i $$-0.397177\pi$$
0.317439 + 0.948279i $$0.397177\pi$$
$$308$$ −848946. −0.509922
$$309$$ 623880. 0.371711
$$310$$ 0 0
$$311$$ −1.53786e6 −0.901601 −0.450801 0.892625i $$-0.648861\pi$$
−0.450801 + 0.892625i $$0.648861\pi$$
$$312$$ 537492. 0.312598
$$313$$ −2.58473e6 −1.49126 −0.745631 0.666359i $$-0.767852\pi$$
−0.745631 + 0.666359i $$0.767852\pi$$
$$314$$ −205594. −0.117675
$$315$$ 0 0
$$316$$ 1.63589e6 0.921587
$$317$$ −1.29146e6 −0.721827 −0.360914 0.932599i $$-0.617535\pi$$
−0.360914 + 0.932599i $$0.617535\pi$$
$$318$$ −72683.1 −0.0403056
$$319$$ 1.04096e6 0.572742
$$320$$ 0 0
$$321$$ 1.18034e6 0.639357
$$322$$ −676764. −0.363745
$$323$$ −931634. −0.496866
$$324$$ −204795. −0.108382
$$325$$ 0 0
$$326$$ 6083.88 0.00317057
$$327$$ 1.32236e6 0.683880
$$328$$ 326274. 0.167455
$$329$$ −1.45080e6 −0.738956
$$330$$ 0 0
$$331$$ 1.77716e6 0.891573 0.445787 0.895139i $$-0.352924\pi$$
0.445787 + 0.895139i $$0.352924\pi$$
$$332$$ −1.83119e6 −0.911778
$$333$$ −154405. −0.0763045
$$334$$ 9069.48 0.00444853
$$335$$ 0 0
$$336$$ 1.92012e6 0.927856
$$337$$ 1.99819e6 0.958435 0.479217 0.877696i $$-0.340921\pi$$
0.479217 + 0.877696i $$0.340921\pi$$
$$338$$ −677584. −0.322605
$$339$$ −516114. −0.243919
$$340$$ 0 0
$$341$$ −38781.6 −0.0180609
$$342$$ −49639.7 −0.0229490
$$343$$ 3.80073e6 1.74434
$$344$$ 79660.7 0.0362951
$$345$$ 0 0
$$346$$ −373471. −0.167713
$$347$$ 2.94954e6 1.31501 0.657507 0.753449i $$-0.271611\pi$$
0.657507 + 0.753449i $$0.271611\pi$$
$$348$$ −2.41681e6 −1.06978
$$349$$ −1.15473e6 −0.507476 −0.253738 0.967273i $$-0.581660\pi$$
−0.253738 + 0.967273i $$0.581660\pi$$
$$350$$ 0 0
$$351$$ 776848. 0.336565
$$352$$ −318819. −0.137147
$$353$$ −829780. −0.354427 −0.177213 0.984172i $$-0.556708\pi$$
−0.177213 + 0.984172i $$0.556708\pi$$
$$354$$ −81495.0 −0.0345639
$$355$$ 0 0
$$356$$ 3.55349e6 1.48604
$$357$$ −2.72644e6 −1.13221
$$358$$ −293023. −0.120835
$$359$$ 3.60644e6 1.47687 0.738435 0.674325i $$-0.235565\pi$$
0.738435 + 0.674325i $$0.235565\pi$$
$$360$$ 0 0
$$361$$ −1.99827e6 −0.807023
$$362$$ −592935. −0.237813
$$363$$ 131769. 0.0524864
$$364$$ −7.47659e6 −2.95767
$$365$$ 0 0
$$366$$ 273492. 0.106719
$$367$$ −403294. −0.156299 −0.0781496 0.996942i $$-0.524901\pi$$
−0.0781496 + 0.996942i $$0.524901\pi$$
$$368$$ 3.22348e6 1.24081
$$369$$ 471570. 0.180294
$$370$$ 0 0
$$371$$ 2.04752e6 0.772313
$$372$$ 90039.4 0.0337346
$$373$$ 3.48177e6 1.29577 0.647884 0.761739i $$-0.275654\pi$$
0.647884 + 0.761739i $$0.275654\pi$$
$$374$$ 144578. 0.0534470
$$375$$ 0 0
$$376$$ −361730. −0.131952
$$377$$ 9.16768e6 3.32205
$$378$$ −145272. −0.0522939
$$379$$ −704863. −0.252062 −0.126031 0.992026i $$-0.540224\pi$$
−0.126031 + 0.992026i $$0.540224\pi$$
$$380$$ 0 0
$$381$$ 1.55384e6 0.548397
$$382$$ 852751. 0.298995
$$383$$ −2.16463e6 −0.754028 −0.377014 0.926208i $$-0.623049\pi$$
−0.377014 + 0.926208i $$0.623049\pi$$
$$384$$ 982551. 0.340038
$$385$$ 0 0
$$386$$ 77547.7 0.0264911
$$387$$ 115135. 0.0390778
$$388$$ 3.59988e6 1.21397
$$389$$ 2.50750e6 0.840168 0.420084 0.907485i $$-0.362001\pi$$
0.420084 + 0.907485i $$0.362001\pi$$
$$390$$ 0 0
$$391$$ −4.57713e6 −1.51409
$$392$$ 1.88955e6 0.621075
$$393$$ 2.67950e6 0.875131
$$394$$ −708942. −0.230075
$$395$$ 0 0
$$396$$ −305929. −0.0980352
$$397$$ −2.75852e6 −0.878417 −0.439208 0.898385i $$-0.644741\pi$$
−0.439208 + 0.898385i $$0.644741\pi$$
$$398$$ 563422. 0.178289
$$399$$ 1.39838e6 0.439736
$$400$$ 0 0
$$401$$ −6.11243e6 −1.89825 −0.949124 0.314901i $$-0.898029\pi$$
−0.949124 + 0.314901i $$0.898029\pi$$
$$402$$ 546053. 0.168527
$$403$$ −341546. −0.104758
$$404$$ −1.24565e6 −0.379700
$$405$$ 0 0
$$406$$ −1.71437e6 −0.516165
$$407$$ −230654. −0.0690200
$$408$$ −679786. −0.202172
$$409$$ 4.96574e6 1.46783 0.733915 0.679241i $$-0.237691\pi$$
0.733915 + 0.679241i $$0.237691\pi$$
$$410$$ 0 0
$$411$$ −410039. −0.119735
$$412$$ −2.16376e6 −0.628008
$$413$$ 2.29576e6 0.662294
$$414$$ −243880. −0.0699320
$$415$$ 0 0
$$416$$ −2.80780e6 −0.795488
$$417$$ −1.11875e6 −0.315061
$$418$$ −74153.2 −0.0207582
$$419$$ −3.30467e6 −0.919589 −0.459794 0.888025i $$-0.652077\pi$$
−0.459794 + 0.888025i $$0.652077\pi$$
$$420$$ 0 0
$$421$$ 5.04577e6 1.38747 0.693733 0.720232i $$-0.255965\pi$$
0.693733 + 0.720232i $$0.255965\pi$$
$$422$$ 309936. 0.0847209
$$423$$ −522815. −0.142068
$$424$$ 510510. 0.137908
$$425$$ 0 0
$$426$$ −15031.9 −0.00401318
$$427$$ −7.70441e6 −2.04489
$$428$$ −4.09367e6 −1.08020
$$429$$ 1.16048e6 0.304434
$$430$$ 0 0
$$431$$ 889002. 0.230520 0.115260 0.993335i $$-0.463230\pi$$
0.115260 + 0.993335i $$0.463230\pi$$
$$432$$ 691940. 0.178385
$$433$$ 406627. 0.104226 0.0521131 0.998641i $$-0.483404\pi$$
0.0521131 + 0.998641i $$0.483404\pi$$
$$434$$ 63869.5 0.0162768
$$435$$ 0 0
$$436$$ −4.58623e6 −1.15542
$$437$$ 2.34758e6 0.588054
$$438$$ 700031. 0.174354
$$439$$ −5.04798e6 −1.25013 −0.625067 0.780571i $$-0.714928\pi$$
−0.625067 + 0.780571i $$0.714928\pi$$
$$440$$ 0 0
$$441$$ 2.73101e6 0.668693
$$442$$ 1.27328e6 0.310006
$$443$$ −888258. −0.215045 −0.107523 0.994203i $$-0.534292\pi$$
−0.107523 + 0.994203i $$0.534292\pi$$
$$444$$ 535510. 0.128917
$$445$$ 0 0
$$446$$ 274424. 0.0653259
$$447$$ −968203. −0.229191
$$448$$ −6.30203e6 −1.48349
$$449$$ 3.82230e6 0.894765 0.447383 0.894343i $$-0.352356\pi$$
0.447383 + 0.894343i $$0.352356\pi$$
$$450$$ 0 0
$$451$$ 704444. 0.163082
$$452$$ 1.79000e6 0.412104
$$453$$ 1.32762e6 0.303967
$$454$$ 451909. 0.102899
$$455$$ 0 0
$$456$$ 348658. 0.0785214
$$457$$ 2.10600e6 0.471703 0.235852 0.971789i $$-0.424212\pi$$
0.235852 + 0.971789i $$0.424212\pi$$
$$458$$ 1.13790e6 0.253479
$$459$$ −982509. −0.217673
$$460$$ 0 0
$$461$$ −2.43217e6 −0.533017 −0.266508 0.963833i $$-0.585870\pi$$
−0.266508 + 0.963833i $$0.585870\pi$$
$$462$$ −217011. −0.0473016
$$463$$ 7.53506e6 1.63356 0.816779 0.576951i $$-0.195758\pi$$
0.816779 + 0.576951i $$0.195758\pi$$
$$464$$ 8.16566e6 1.76075
$$465$$ 0 0
$$466$$ −277384. −0.0591720
$$467$$ −3.17287e6 −0.673225 −0.336612 0.941643i $$-0.609281\pi$$
−0.336612 + 0.941643i $$0.609281\pi$$
$$468$$ −2.69428e6 −0.568628
$$469$$ −1.53826e7 −3.22922
$$470$$ 0 0
$$471$$ 2.08711e6 0.433503
$$472$$ 572403. 0.118262
$$473$$ 171992. 0.0353472
$$474$$ 418172. 0.0854887
$$475$$ 0 0
$$476$$ 9.45592e6 1.91287
$$477$$ 737850. 0.148481
$$478$$ 316727. 0.0634038
$$479$$ 3.20281e6 0.637811 0.318905 0.947787i $$-0.396685\pi$$
0.318905 + 0.947787i $$0.396685\pi$$
$$480$$ 0 0
$$481$$ −2.03135e6 −0.400333
$$482$$ 603009. 0.118224
$$483$$ 6.87024e6 1.34000
$$484$$ −457004. −0.0886762
$$485$$ 0 0
$$486$$ −52350.4 −0.0100538
$$487$$ −4.45541e6 −0.851266 −0.425633 0.904896i $$-0.639949\pi$$
−0.425633 + 0.904896i $$0.639949\pi$$
$$488$$ −1.92095e6 −0.365145
$$489$$ −61761.2 −0.0116800
$$490$$ 0 0
$$491$$ 150957. 0.0282586 0.0141293 0.999900i $$-0.495502\pi$$
0.0141293 + 0.999900i $$0.495502\pi$$
$$492$$ −1.63551e6 −0.304608
$$493$$ −1.15947e7 −2.14853
$$494$$ −653060. −0.120403
$$495$$ 0 0
$$496$$ −304216. −0.0555235
$$497$$ 423456. 0.0768984
$$498$$ −468096. −0.0845789
$$499$$ −4.56196e6 −0.820163 −0.410081 0.912049i $$-0.634500\pi$$
−0.410081 + 0.912049i $$0.634500\pi$$
$$500$$ 0 0
$$501$$ −92069.8 −0.0163879
$$502$$ −139914. −0.0247800
$$503$$ −5.18773e6 −0.914234 −0.457117 0.889407i $$-0.651118\pi$$
−0.457117 + 0.889407i $$0.651118\pi$$
$$504$$ 1.02036e6 0.178927
$$505$$ 0 0
$$506$$ −364315. −0.0632559
$$507$$ 6.87857e6 1.18844
$$508$$ −5.38908e6 −0.926521
$$509$$ 6.85927e6 1.17350 0.586750 0.809768i $$-0.300407\pi$$
0.586750 + 0.809768i $$0.300407\pi$$
$$510$$ 0 0
$$511$$ −1.97202e7 −3.34087
$$512$$ −4.20312e6 −0.708593
$$513$$ 503923. 0.0845416
$$514$$ −494076. −0.0824871
$$515$$ 0 0
$$516$$ −399314. −0.0660223
$$517$$ −780995. −0.128506
$$518$$ 379864. 0.0622020
$$519$$ 3.79133e6 0.617836
$$520$$ 0 0
$$521$$ −3.91860e6 −0.632466 −0.316233 0.948682i $$-0.602418\pi$$
−0.316233 + 0.948682i $$0.602418\pi$$
$$522$$ −617794. −0.0992356
$$523$$ −3.64804e6 −0.583183 −0.291592 0.956543i $$-0.594185\pi$$
−0.291592 + 0.956543i $$0.594185\pi$$
$$524$$ −9.29312e6 −1.47854
$$525$$ 0 0
$$526$$ 572405. 0.0902067
$$527$$ 431966. 0.0677521
$$528$$ 1.03364e6 0.161356
$$529$$ 5.09735e6 0.791964
$$530$$ 0 0
$$531$$ 827305. 0.127330
$$532$$ −4.84988e6 −0.742938
$$533$$ 6.20397e6 0.945914
$$534$$ 908355. 0.137849
$$535$$ 0 0
$$536$$ −3.83536e6 −0.576625
$$537$$ 2.97465e6 0.445144
$$538$$ 194826. 0.0290196
$$539$$ 4.07966e6 0.604855
$$540$$ 0 0
$$541$$ −4.23861e6 −0.622630 −0.311315 0.950307i $$-0.600769\pi$$
−0.311315 + 0.950307i $$0.600769\pi$$
$$542$$ −1.37232e6 −0.200659
$$543$$ 6.01924e6 0.876077
$$544$$ 3.55113e6 0.514482
$$545$$ 0 0
$$546$$ −1.91119e6 −0.274361
$$547$$ −513913. −0.0734380 −0.0367190 0.999326i $$-0.511691\pi$$
−0.0367190 + 0.999326i $$0.511691\pi$$
$$548$$ 1.42211e6 0.202293
$$549$$ −2.77638e6 −0.393141
$$550$$ 0 0
$$551$$ 5.94685e6 0.834465
$$552$$ 1.71296e6 0.239277
$$553$$ −1.17801e7 −1.63809
$$554$$ 869470. 0.120359
$$555$$ 0 0
$$556$$ 3.88009e6 0.532297
$$557$$ −1.04431e7 −1.42624 −0.713121 0.701041i $$-0.752719\pi$$
−0.713121 + 0.701041i $$0.752719\pi$$
$$558$$ 23016.2 0.00312930
$$559$$ 1.51472e6 0.205023
$$560$$ 0 0
$$561$$ −1.46770e6 −0.196893
$$562$$ −2.18345e6 −0.291610
$$563$$ 5.42113e6 0.720806 0.360403 0.932797i $$-0.382639\pi$$
0.360403 + 0.932797i $$0.382639\pi$$
$$564$$ 1.81324e6 0.240026
$$565$$ 0 0
$$566$$ 743477. 0.0975499
$$567$$ 1.47474e6 0.192645
$$568$$ 105581. 0.0137313
$$569$$ −4.65152e6 −0.602301 −0.301151 0.953577i $$-0.597371\pi$$
−0.301151 + 0.953577i $$0.597371\pi$$
$$570$$ 0 0
$$571$$ 836920. 0.107422 0.0537111 0.998557i $$-0.482895\pi$$
0.0537111 + 0.998557i $$0.482895\pi$$
$$572$$ −4.02479e6 −0.514344
$$573$$ −8.65679e6 −1.10146
$$574$$ −1.16015e6 −0.146972
$$575$$ 0 0
$$576$$ −2.27102e6 −0.285210
$$577$$ −1.22020e7 −1.52578 −0.762892 0.646526i $$-0.776221\pi$$
−0.762892 + 0.646526i $$0.776221\pi$$
$$578$$ −351582. −0.0437730
$$579$$ −787233. −0.0975904
$$580$$ 0 0
$$581$$ 1.31865e7 1.62065
$$582$$ 920214. 0.112611
$$583$$ 1.10222e6 0.134307
$$584$$ −4.91686e6 −0.596562
$$585$$ 0 0
$$586$$ −1.17637e6 −0.141514
$$587$$ −4.64629e6 −0.556559 −0.278279 0.960500i $$-0.589764\pi$$
−0.278279 + 0.960500i $$0.589764\pi$$
$$588$$ −9.47175e6 −1.12976
$$589$$ −221553. −0.0263141
$$590$$ 0 0
$$591$$ 7.19690e6 0.847572
$$592$$ −1.80932e6 −0.212184
$$593$$ −3.76104e6 −0.439209 −0.219605 0.975589i $$-0.570477\pi$$
−0.219605 + 0.975589i $$0.570477\pi$$
$$594$$ −78202.5 −0.00909399
$$595$$ 0 0
$$596$$ 3.35794e6 0.387220
$$597$$ −5.71963e6 −0.656799
$$598$$ −3.20849e6 −0.366900
$$599$$ −2.65769e6 −0.302648 −0.151324 0.988484i $$-0.548354\pi$$
−0.151324 + 0.988484i $$0.548354\pi$$
$$600$$ 0 0
$$601$$ 5.84003e6 0.659521 0.329761 0.944065i $$-0.393032\pi$$
0.329761 + 0.944065i $$0.393032\pi$$
$$602$$ −283254. −0.0318555
$$603$$ −5.54331e6 −0.620835
$$604$$ −4.60447e6 −0.513555
$$605$$ 0 0
$$606$$ −318416. −0.0352220
$$607$$ −7.10119e6 −0.782274 −0.391137 0.920332i $$-0.627918\pi$$
−0.391137 + 0.920332i $$0.627918\pi$$
$$608$$ −1.82136e6 −0.199819
$$609$$ 1.74036e7 1.90150
$$610$$ 0 0
$$611$$ −6.87815e6 −0.745364
$$612$$ 3.40756e6 0.367760
$$613$$ −1.10978e7 −1.19285 −0.596423 0.802670i $$-0.703412\pi$$
−0.596423 + 0.802670i $$0.703412\pi$$
$$614$$ −929488. −0.0995000
$$615$$ 0 0
$$616$$ 1.52423e6 0.161845
$$617$$ 1.34866e7 1.42623 0.713114 0.701049i $$-0.247284\pi$$
0.713114 + 0.701049i $$0.247284\pi$$
$$618$$ −553107. −0.0582556
$$619$$ −1.19228e7 −1.25069 −0.625346 0.780347i $$-0.715042\pi$$
−0.625346 + 0.780347i $$0.715042\pi$$
$$620$$ 0 0
$$621$$ 2.47578e6 0.257622
$$622$$ 1.36340e6 0.141302
$$623$$ −2.55888e7 −2.64138
$$624$$ 9.10315e6 0.935902
$$625$$ 0 0
$$626$$ 2.29151e6 0.233715
$$627$$ 752774. 0.0764708
$$628$$ −7.23855e6 −0.732407
$$629$$ 2.56912e6 0.258915
$$630$$ 0 0
$$631$$ −5.56411e6 −0.556317 −0.278159 0.960535i $$-0.589724\pi$$
−0.278159 + 0.960535i $$0.589724\pi$$
$$632$$ −2.93715e6 −0.292505
$$633$$ −3.14635e6 −0.312103
$$634$$ 1.14496e6 0.113127
$$635$$ 0 0
$$636$$ −2.55903e6 −0.250861
$$637$$ 3.59291e7 3.50831
$$638$$ −922877. −0.0897619
$$639$$ 152598. 0.0147841
$$640$$ 0 0
$$641$$ 7.43898e6 0.715103 0.357551 0.933894i $$-0.383612\pi$$
0.357551 + 0.933894i $$0.383612\pi$$
$$642$$ −1.04644e6 −0.100202
$$643$$ 8.56732e6 0.817180 0.408590 0.912718i $$-0.366021\pi$$
0.408590 + 0.912718i $$0.366021\pi$$
$$644$$ −2.38275e7 −2.26394
$$645$$ 0 0
$$646$$ 825949. 0.0778703
$$647$$ −1.65852e7 −1.55762 −0.778808 0.627263i $$-0.784175\pi$$
−0.778808 + 0.627263i $$0.784175\pi$$
$$648$$ 367698. 0.0343996
$$649$$ 1.23585e6 0.115174
$$650$$ 0 0
$$651$$ −648378. −0.0599620
$$652$$ 214202. 0.0197335
$$653$$ −7.12943e6 −0.654292 −0.327146 0.944974i $$-0.606087\pi$$
−0.327146 + 0.944974i $$0.606087\pi$$
$$654$$ −1.17235e6 −0.107180
$$655$$ 0 0
$$656$$ 5.52589e6 0.501352
$$657$$ −7.10643e6 −0.642301
$$658$$ 1.28622e6 0.115811
$$659$$ 1.00795e7 0.904116 0.452058 0.891988i $$-0.350690\pi$$
0.452058 + 0.891988i $$0.350690\pi$$
$$660$$ 0 0
$$661$$ 1.12712e7 1.00338 0.501689 0.865048i $$-0.332712\pi$$
0.501689 + 0.865048i $$0.332712\pi$$
$$662$$ −1.57556e6 −0.139730
$$663$$ −1.29259e7 −1.14203
$$664$$ 3.28780e6 0.289392
$$665$$ 0 0
$$666$$ 136889. 0.0119587
$$667$$ 2.92170e7 2.54285
$$668$$ 319319. 0.0276875
$$669$$ −2.78585e6 −0.240653
$$670$$ 0 0
$$671$$ −4.14743e6 −0.355609
$$672$$ −5.33023e6 −0.455326
$$673$$ −2.71329e6 −0.230918 −0.115459 0.993312i $$-0.536834\pi$$
−0.115459 + 0.993312i $$0.536834\pi$$
$$674$$ −1.77152e6 −0.150209
$$675$$ 0 0
$$676$$ −2.38564e7 −2.00788
$$677$$ −1.70935e6 −0.143337 −0.0716686 0.997429i $$-0.522832\pi$$
−0.0716686 + 0.997429i $$0.522832\pi$$
$$678$$ 457565. 0.0382278
$$679$$ −2.59229e7 −2.15779
$$680$$ 0 0
$$681$$ −4.58760e6 −0.379069
$$682$$ 34382.2 0.00283056
$$683$$ 2.06905e7 1.69715 0.848573 0.529079i $$-0.177462\pi$$
0.848573 + 0.529079i $$0.177462\pi$$
$$684$$ −1.74772e6 −0.142834
$$685$$ 0 0
$$686$$ −3.36957e6 −0.273379
$$687$$ −1.15516e7 −0.933788
$$688$$ 1.34916e6 0.108666
$$689$$ 9.70715e6 0.779011
$$690$$ 0 0
$$691$$ 3.42722e6 0.273053 0.136526 0.990636i $$-0.456406\pi$$
0.136526 + 0.990636i $$0.456406\pi$$
$$692$$ −1.31492e7 −1.04384
$$693$$ 2.20301e6 0.174254
$$694$$ −2.61494e6 −0.206093
$$695$$ 0 0
$$696$$ 4.33925e6 0.339540
$$697$$ −7.84639e6 −0.611770
$$698$$ 1.02373e6 0.0795331
$$699$$ 2.81589e6 0.217983
$$700$$ 0 0
$$701$$ 1.05204e7 0.808608 0.404304 0.914625i $$-0.367514\pi$$
0.404304 + 0.914625i $$0.367514\pi$$
$$702$$ −688722. −0.0527474
$$703$$ −1.31769e6 −0.100560
$$704$$ −3.39251e6 −0.257982
$$705$$ 0 0
$$706$$ 735650. 0.0555468
$$707$$ 8.96995e6 0.674903
$$708$$ −2.86928e6 −0.215124
$$709$$ −5.27905e6 −0.394403 −0.197202 0.980363i $$-0.563185\pi$$
−0.197202 + 0.980363i $$0.563185\pi$$
$$710$$ 0 0
$$711$$ −4.24512e6 −0.314931
$$712$$ −6.38008e6 −0.471657
$$713$$ −1.08849e6 −0.0801864
$$714$$ 2.41715e6 0.177443
$$715$$ 0 0
$$716$$ −1.03168e7 −0.752074
$$717$$ −3.21529e6 −0.233572
$$718$$ −3.19732e6 −0.231459
$$719$$ −5.38506e6 −0.388480 −0.194240 0.980954i $$-0.562224\pi$$
−0.194240 + 0.980954i $$0.562224\pi$$
$$720$$ 0 0
$$721$$ 1.55813e7 1.11626
$$722$$ 1.77158e6 0.126479
$$723$$ −6.12151e6 −0.435525
$$724$$ −2.08761e7 −1.48014
$$725$$ 0 0
$$726$$ −116821. −0.00822583
$$727$$ −1.05731e6 −0.0741934 −0.0370967 0.999312i $$-0.511811\pi$$
−0.0370967 + 0.999312i $$0.511811\pi$$
$$728$$ 1.34238e7 0.938743
$$729$$ 531441. 0.0370370
$$730$$ 0 0
$$731$$ −1.91572e6 −0.132598
$$732$$ 9.62911e6 0.664215
$$733$$ 1.02639e6 0.0705589 0.0352794 0.999377i $$-0.488768\pi$$
0.0352794 + 0.999377i $$0.488768\pi$$
$$734$$ 357544. 0.0244957
$$735$$ 0 0
$$736$$ −8.94834e6 −0.608903
$$737$$ −8.28075e6 −0.561566
$$738$$ −418075. −0.0282562
$$739$$ −3.33815e6 −0.224851 −0.112426 0.993660i $$-0.535862\pi$$
−0.112426 + 0.993660i $$0.535862\pi$$
$$740$$ 0 0
$$741$$ 6.62960e6 0.443550
$$742$$ −1.81525e6 −0.121039
$$743$$ −2.15478e7 −1.43196 −0.715981 0.698120i $$-0.754020\pi$$
−0.715981 + 0.698120i $$0.754020\pi$$
$$744$$ −161661. −0.0107071
$$745$$ 0 0
$$746$$ −3.08679e6 −0.203077
$$747$$ 4.75193e6 0.311579
$$748$$ 5.09030e6 0.332652
$$749$$ 2.94787e7 1.92001
$$750$$ 0 0
$$751$$ −2.02621e7 −1.31095 −0.655474 0.755218i $$-0.727531\pi$$
−0.655474 + 0.755218i $$0.727531\pi$$
$$752$$ −6.12638e6 −0.395057
$$753$$ 1.42035e6 0.0912869
$$754$$ −8.12769e6 −0.520641
$$755$$ 0 0
$$756$$ −5.11473e6 −0.325475
$$757$$ 1.23455e7 0.783012 0.391506 0.920176i $$-0.371954\pi$$
0.391506 + 0.920176i $$0.371954\pi$$
$$758$$ 624903. 0.0395039
$$759$$ 3.69838e6 0.233028
$$760$$ 0 0
$$761$$ 9.77671e6 0.611971 0.305986 0.952036i $$-0.401014\pi$$
0.305986 + 0.952036i $$0.401014\pi$$
$$762$$ −1.37758e6 −0.0859465
$$763$$ 3.30257e7 2.05372
$$764$$ 3.00237e7 1.86093
$$765$$ 0 0
$$766$$ 1.91908e6 0.118174
$$767$$ 1.08840e7 0.668038
$$768$$ 7.20364e6 0.440706
$$769$$ −4.31458e6 −0.263101 −0.131551 0.991309i $$-0.541996\pi$$
−0.131551 + 0.991309i $$0.541996\pi$$
$$770$$ 0 0
$$771$$ 5.01567e6 0.303873
$$772$$ 2.73030e6 0.164880
$$773$$ 3.28419e7 1.97688 0.988439 0.151620i $$-0.0484490\pi$$
0.988439 + 0.151620i $$0.0484490\pi$$
$$774$$ −102074. −0.00612440
$$775$$ 0 0
$$776$$ −6.46338e6 −0.385306
$$777$$ −3.85623e6 −0.229145
$$778$$ −2.22304e6 −0.131674
$$779$$ 4.02437e6 0.237604
$$780$$ 0 0
$$781$$ 227955. 0.0133727
$$782$$ 4.05790e6 0.237292
$$783$$ 6.27160e6 0.365573
$$784$$ 3.20021e7 1.85947
$$785$$ 0 0
$$786$$ −2.37554e6 −0.137153
$$787$$ 1.45886e7 0.839608 0.419804 0.907615i $$-0.362099\pi$$
0.419804 + 0.907615i $$0.362099\pi$$
$$788$$ −2.49605e7 −1.43198
$$789$$ −5.81083e6 −0.332312
$$790$$ 0 0
$$791$$ −1.28899e7 −0.732499
$$792$$ 549277. 0.0311156
$$793$$ −3.65261e7 −2.06262
$$794$$ 2.44560e6 0.137668
$$795$$ 0 0
$$796$$ 1.98370e7 1.10967
$$797$$ 4.63451e6 0.258439 0.129220 0.991616i $$-0.458753\pi$$
0.129220 + 0.991616i $$0.458753\pi$$
$$798$$ −1.23974e6 −0.0689168
$$799$$ 8.69905e6 0.482064
$$800$$ 0 0
$$801$$ −9.22126e6 −0.507819
$$802$$ 5.41903e6 0.297499
$$803$$ −1.06158e7 −0.580983
$$804$$ 1.92255e7 1.04891
$$805$$ 0 0
$$806$$ 302801. 0.0164180
$$807$$ −1.97780e6 −0.106905
$$808$$ 2.23648e6 0.120514
$$809$$ 1.28933e6 0.0692618 0.0346309 0.999400i $$-0.488974\pi$$
0.0346309 + 0.999400i $$0.488974\pi$$
$$810$$ 0 0
$$811$$ 6.47343e6 0.345607 0.172803 0.984956i $$-0.444718\pi$$
0.172803 + 0.984956i $$0.444718\pi$$
$$812$$ −6.03595e7 −3.21259
$$813$$ 1.39313e7 0.739205
$$814$$ 204488. 0.0108170
$$815$$ 0 0
$$816$$ −1.15131e7 −0.605294
$$817$$ 982561. 0.0514997
$$818$$ −4.40242e6 −0.230043
$$819$$ 1.94017e7 1.01072
$$820$$ 0 0
$$821$$ 2.48796e7 1.28820 0.644102 0.764939i $$-0.277231\pi$$
0.644102 + 0.764939i $$0.277231\pi$$
$$822$$ 363524. 0.0187652
$$823$$ 2.73317e7 1.40659 0.703295 0.710898i $$-0.251711\pi$$
0.703295 + 0.710898i $$0.251711\pi$$
$$824$$ 3.88490e6 0.199325
$$825$$ 0 0
$$826$$ −2.03533e6 −0.103797
$$827$$ 3.20369e7 1.62887 0.814437 0.580252i $$-0.197046\pi$$
0.814437 + 0.580252i $$0.197046\pi$$
$$828$$ −8.58655e6 −0.435254
$$829$$ −2.54756e7 −1.28747 −0.643736 0.765247i $$-0.722617\pi$$
−0.643736 + 0.765247i $$0.722617\pi$$
$$830$$ 0 0
$$831$$ −8.82652e6 −0.443391
$$832$$ −2.98775e7 −1.49636
$$833$$ −4.54409e7 −2.26900
$$834$$ 991841. 0.0493772
$$835$$ 0 0
$$836$$ −2.61079e6 −0.129198
$$837$$ −233651. −0.0115280
$$838$$ 2.92979e6 0.144121
$$839$$ −2.56179e7 −1.25643 −0.628216 0.778039i $$-0.716215\pi$$
−0.628216 + 0.778039i $$0.716215\pi$$
$$840$$ 0 0
$$841$$ 5.35007e7 2.60837
$$842$$ −4.47338e6 −0.217448
$$843$$ 2.21655e7 1.07426
$$844$$ 1.09122e7 0.527300
$$845$$ 0 0
$$846$$ 463507. 0.0222654
$$847$$ 3.29091e6 0.157619
$$848$$ 8.64617e6 0.412890
$$849$$ −7.54749e6 −0.359363
$$850$$ 0 0
$$851$$ −6.47381e6 −0.306433
$$852$$ −529243. −0.0249779
$$853$$ 8.49389e6 0.399700 0.199850 0.979827i $$-0.435955\pi$$
0.199850 + 0.979827i $$0.435955\pi$$
$$854$$ 6.83042e6 0.320481
$$855$$ 0 0
$$856$$ 7.34996e6 0.342847
$$857$$ 2.29100e7 1.06555 0.532773 0.846258i $$-0.321150\pi$$
0.532773 + 0.846258i $$0.321150\pi$$
$$858$$ −1.02883e6 −0.0477118
$$859$$ 1.05172e7 0.486315 0.243158 0.969987i $$-0.421817\pi$$
0.243158 + 0.969987i $$0.421817\pi$$
$$860$$ 0 0
$$861$$ 1.17774e7 0.541429
$$862$$ −788153. −0.0361279
$$863$$ 2.93900e7 1.34330 0.671649 0.740870i $$-0.265587\pi$$
0.671649 + 0.740870i $$0.265587\pi$$
$$864$$ −1.92082e6 −0.0875390
$$865$$ 0 0
$$866$$ −360499. −0.0163346
$$867$$ 3.56912e6 0.161255
$$868$$ 2.24872e6 0.101306
$$869$$ −6.34147e6 −0.284866
$$870$$ 0 0
$$871$$ −7.29278e7 −3.25722
$$872$$ 8.23432e6 0.366722
$$873$$ −9.34165e6 −0.414847
$$874$$ −2.08127e6 −0.0921616
$$875$$ 0 0
$$876$$ 2.46467e7 1.08517
$$877$$ −3.81404e7 −1.67451 −0.837253 0.546816i $$-0.815840\pi$$
−0.837253 + 0.546816i $$0.815840\pi$$
$$878$$ 4.47533e6 0.195925
$$879$$ 1.19420e7 0.521321
$$880$$ 0 0
$$881$$ 2.39208e7 1.03833 0.519165 0.854674i $$-0.326243\pi$$
0.519165 + 0.854674i $$0.326243\pi$$
$$882$$ −2.42120e6 −0.104800
$$883$$ −1.95650e6 −0.0844457 −0.0422229 0.999108i $$-0.513444\pi$$
−0.0422229 + 0.999108i $$0.513444\pi$$
$$884$$ 4.48298e7 1.92946
$$885$$ 0 0
$$886$$ 787493. 0.0337025
$$887$$ −2.80192e7 −1.19577 −0.597884 0.801583i $$-0.703992\pi$$
−0.597884 + 0.801583i $$0.703992\pi$$
$$888$$ −961478. −0.0409173
$$889$$ 3.88070e7 1.64686
$$890$$ 0 0
$$891$$ 793881. 0.0335013
$$892$$ 9.66194e6 0.406586
$$893$$ −4.46169e6 −0.187228
$$894$$ 858369. 0.0359195
$$895$$ 0 0
$$896$$ 2.45391e7 1.02115
$$897$$ 3.25713e7 1.35162
$$898$$ −3.38870e6 −0.140230
$$899$$ −2.75735e6 −0.113787
$$900$$ 0 0
$$901$$ −1.22770e7 −0.503825
$$902$$ −624532. −0.0255587
$$903$$ 2.87548e6 0.117352
$$904$$ −3.21384e6 −0.130799
$$905$$ 0 0
$$906$$ −1.17701e6 −0.0476387
$$907$$ −5.99454e6 −0.241957 −0.120978 0.992655i $$-0.538603\pi$$
−0.120978 + 0.992655i $$0.538603\pi$$
$$908$$ 1.59108e7 0.640439
$$909$$ 3.23243e6 0.129754
$$910$$ 0 0
$$911$$ −3.83047e7 −1.52917 −0.764585 0.644523i $$-0.777056\pi$$
−0.764585 + 0.644523i $$0.777056\pi$$
$$912$$ 5.90500e6 0.235089
$$913$$ 7.09856e6 0.281834
$$914$$ −1.86710e6 −0.0739267
$$915$$ 0 0
$$916$$ 4.00634e7 1.57764
$$917$$ 6.69202e7 2.62805
$$918$$ 871052. 0.0341144
$$919$$ −4.41558e7 −1.72464 −0.862321 0.506362i $$-0.830990\pi$$
−0.862321 + 0.506362i $$0.830990\pi$$
$$920$$ 0 0
$$921$$ 9.43580e6 0.366547
$$922$$ 2.15626e6 0.0835360
$$923$$ 2.00757e6 0.0775652
$$924$$ −7.64052e6 −0.294403
$$925$$ 0 0
$$926$$ −6.68028e6 −0.256016
$$927$$ 5.61492e6 0.214607
$$928$$ −2.26678e7 −0.864050
$$929$$ 1.00353e7 0.381499 0.190749 0.981639i $$-0.438908\pi$$
0.190749 + 0.981639i $$0.438908\pi$$
$$930$$ 0 0
$$931$$ 2.33064e7 0.881253
$$932$$ −9.76613e6 −0.368284
$$933$$ −1.38407e7 −0.520540
$$934$$ 2.81294e6 0.105510
$$935$$ 0 0
$$936$$ 4.83743e6 0.180478
$$937$$ 3.04913e7 1.13456 0.567279 0.823526i $$-0.307996\pi$$
0.567279 + 0.823526i $$0.307996\pi$$
$$938$$ 1.36376e7 0.506093
$$939$$ −2.32626e7 −0.860981
$$940$$ 0 0
$$941$$ −4.13865e7 −1.52365 −0.761824 0.647785i $$-0.775696\pi$$
−0.761824 + 0.647785i $$0.775696\pi$$
$$942$$ −1.85034e6 −0.0679399
$$943$$ 1.97718e7 0.724046
$$944$$ 9.69442e6 0.354072
$$945$$ 0 0
$$946$$ −152481. −0.00553973
$$947$$ −9.38834e6 −0.340184 −0.170092 0.985428i $$-0.554406\pi$$
−0.170092 + 0.985428i $$0.554406\pi$$
$$948$$ 1.47230e7 0.532079
$$949$$ −9.34922e7 −3.36984
$$950$$ 0 0
$$951$$ −1.16232e7 −0.416747
$$952$$ −1.69776e7 −0.607132
$$953$$ 3.35369e7 1.19616 0.598081 0.801435i $$-0.295930\pi$$
0.598081 + 0.801435i $$0.295930\pi$$
$$954$$ −654148. −0.0232705
$$955$$ 0 0
$$956$$ 1.11513e7 0.394623
$$957$$ 9.36868e6 0.330673
$$958$$ −2.83948e6 −0.0999597
$$959$$ −1.02406e7 −0.359568
$$960$$ 0 0
$$961$$ −2.85264e7 −0.996412
$$962$$ 1.80091e6 0.0627414
$$963$$ 1.06230e7 0.369133
$$964$$ 2.12308e7 0.735823
$$965$$ 0 0
$$966$$ −6.09088e6 −0.210009
$$967$$ −5.35398e7 −1.84124 −0.920620 0.390459i $$-0.872316\pi$$
−0.920620 + 0.390459i $$0.872316\pi$$
$$968$$ 820525. 0.0281451
$$969$$ −8.38471e6 −0.286866
$$970$$ 0 0
$$971$$ −3.17286e7 −1.07995 −0.539974 0.841682i $$-0.681566\pi$$
−0.539974 + 0.841682i $$0.681566\pi$$
$$972$$ −1.84316e6 −0.0625744
$$973$$ −2.79407e7 −0.946139
$$974$$ 3.94999e6 0.133413
$$975$$ 0 0
$$976$$ −3.25338e7 −1.09323
$$977$$ 5.28990e6 0.177301 0.0886505 0.996063i $$-0.471745\pi$$
0.0886505 + 0.996063i $$0.471745\pi$$
$$978$$ 54754.9 0.00183053
$$979$$ −1.37750e7 −0.459340
$$980$$ 0 0
$$981$$ 1.19012e7 0.394838
$$982$$ −133832. −0.00442877
$$983$$ −2.55796e7 −0.844327 −0.422164 0.906520i $$-0.638729\pi$$
−0.422164 + 0.906520i $$0.638729\pi$$
$$984$$ 2.93647e6 0.0966801
$$985$$ 0 0
$$986$$ 1.02794e7 0.336725
$$987$$ −1.30572e7 −0.426637
$$988$$ −2.29930e7 −0.749380
$$989$$ 4.82733e6 0.156934
$$990$$ 0 0
$$991$$ 4.14167e7 1.33965 0.669825 0.742519i $$-0.266369\pi$$
0.669825 + 0.742519i $$0.266369\pi$$
$$992$$ 844498. 0.0272471
$$993$$ 1.59945e7 0.514750
$$994$$ −375419. −0.0120517
$$995$$ 0 0
$$996$$ −1.64808e7 −0.526415
$$997$$ 3.02210e7 0.962877 0.481438 0.876480i $$-0.340115\pi$$
0.481438 + 0.876480i $$0.340115\pi$$
$$998$$ 4.04445e6 0.128538
$$999$$ −1.38964e6 −0.0440544
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 825.6.a.y.1.6 13
5.2 odd 4 165.6.c.b.34.13 26
5.3 odd 4 165.6.c.b.34.14 yes 26
5.4 even 2 825.6.a.v.1.8 13

By twisted newform
Twist Min Dim Char Parity Ord Type
165.6.c.b.34.13 26 5.2 odd 4
165.6.c.b.34.14 yes 26 5.3 odd 4
825.6.a.v.1.8 13 5.4 even 2
825.6.a.y.1.6 13 1.1 even 1 trivial