# Properties

 Label 1280.4.d.u.641.1 Level $1280$ Weight $4$ Character 1280.641 Analytic conductor $75.522$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1280,4,Mod(641,1280)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1280, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1280.641");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1280 = 2^{8} \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1280.d (of order $$2$$, degree $$1$$, not minimal)

## 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: no Analytic conductor: $$75.5224448073$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{10})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 25$$ x^4 + 25 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 160) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 641.1 Root $$-1.58114 - 1.58114i$$ of defining polynomial Character $$\chi$$ $$=$$ 1280.641 Dual form 1280.4.d.u.641.4

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-6.32456i q^{3} -5.00000i q^{5} +18.9737 q^{7} -13.0000 q^{9} +O(q^{10})$$ $$q-6.32456i q^{3} -5.00000i q^{5} +18.9737 q^{7} -13.0000 q^{9} -12.6491i q^{11} +38.0000i q^{13} -31.6228 q^{15} +34.0000 q^{17} +101.193i q^{19} -120.000i q^{21} +82.2192 q^{23} -25.0000 q^{25} -88.5438i q^{27} +270.000i q^{29} +341.526 q^{31} -80.0000 q^{33} -94.8683i q^{35} -206.000i q^{37} +240.333 q^{39} +270.000 q^{41} +537.587i q^{43} +65.0000i q^{45} +132.816 q^{47} +17.0000 q^{49} -215.035i q^{51} +258.000i q^{53} -63.2456 q^{55} +640.000 q^{57} -75.8947i q^{59} -250.000i q^{61} -246.658 q^{63} +190.000 q^{65} +815.868i q^{67} -520.000i q^{69} -645.105 q^{71} +1078.00 q^{73} +158.114i q^{75} -240.000i q^{77} +278.280 q^{79} -911.000 q^{81} +1106.80i q^{83} -170.000i q^{85} +1707.63 q^{87} -890.000 q^{89} +720.999i q^{91} -2160.00i q^{93} +505.964 q^{95} -254.000 q^{97} +164.438i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 52 q^{9}+O(q^{10})$$ 4 * q - 52 * q^9 $$4 q - 52 q^{9} + 136 q^{17} - 100 q^{25} - 320 q^{33} + 1080 q^{41} + 68 q^{49} + 2560 q^{57} + 760 q^{65} + 4312 q^{73} - 3644 q^{81} - 3560 q^{89} - 1016 q^{97}+O(q^{100})$$ 4 * q - 52 * q^9 + 136 * q^17 - 100 * q^25 - 320 * q^33 + 1080 * q^41 + 68 * q^49 + 2560 * q^57 + 760 * q^65 + 4312 * q^73 - 3644 * q^81 - 3560 * q^89 - 1016 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1280\mathbb{Z}\right)^\times$$.

 $$n$$ $$257$$ $$261$$ $$511$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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 0
$$3$$ − 6.32456i − 1.21716i −0.793492 0.608581i $$-0.791739\pi$$
0.793492 0.608581i $$-0.208261\pi$$
$$4$$ 0 0
$$5$$ − 5.00000i − 0.447214i
$$6$$ 0 0
$$7$$ 18.9737 1.02448 0.512241 0.858842i $$-0.328816\pi$$
0.512241 + 0.858842i $$0.328816\pi$$
$$8$$ 0 0
$$9$$ −13.0000 −0.481481
$$10$$ 0 0
$$11$$ − 12.6491i − 0.346714i −0.984859 0.173357i $$-0.944539\pi$$
0.984859 0.173357i $$-0.0554614\pi$$
$$12$$ 0 0
$$13$$ 38.0000i 0.810716i 0.914158 + 0.405358i $$0.132853\pi$$
−0.914158 + 0.405358i $$0.867147\pi$$
$$14$$ 0 0
$$15$$ −31.6228 −0.544331
$$16$$ 0 0
$$17$$ 34.0000 0.485071 0.242536 0.970143i $$-0.422021\pi$$
0.242536 + 0.970143i $$0.422021\pi$$
$$18$$ 0 0
$$19$$ 101.193i 1.22185i 0.791687 + 0.610927i $$0.209203\pi$$
−0.791687 + 0.610927i $$0.790797\pi$$
$$20$$ 0 0
$$21$$ − 120.000i − 1.24696i
$$22$$ 0 0
$$23$$ 82.2192 0.745387 0.372693 0.927955i $$-0.378434\pi$$
0.372693 + 0.927955i $$0.378434\pi$$
$$24$$ 0 0
$$25$$ −25.0000 −0.200000
$$26$$ 0 0
$$27$$ − 88.5438i − 0.631121i
$$28$$ 0 0
$$29$$ 270.000i 1.72889i 0.502729 + 0.864444i $$0.332329\pi$$
−0.502729 + 0.864444i $$0.667671\pi$$
$$30$$ 0 0
$$31$$ 341.526 1.97871 0.989353 0.145537i $$-0.0464908\pi$$
0.989353 + 0.145537i $$0.0464908\pi$$
$$32$$ 0 0
$$33$$ −80.0000 −0.422006
$$34$$ 0 0
$$35$$ − 94.8683i − 0.458162i
$$36$$ 0 0
$$37$$ − 206.000i − 0.915302i −0.889132 0.457651i $$-0.848691\pi$$
0.889132 0.457651i $$-0.151309\pi$$
$$38$$ 0 0
$$39$$ 240.333 0.986772
$$40$$ 0 0
$$41$$ 270.000 1.02846 0.514231 0.857652i $$-0.328078\pi$$
0.514231 + 0.857652i $$0.328078\pi$$
$$42$$ 0 0
$$43$$ 537.587i 1.90654i 0.302117 + 0.953271i $$0.402307\pi$$
−0.302117 + 0.953271i $$0.597693\pi$$
$$44$$ 0 0
$$45$$ 65.0000i 0.215325i
$$46$$ 0 0
$$47$$ 132.816 0.412195 0.206097 0.978531i $$-0.433924\pi$$
0.206097 + 0.978531i $$0.433924\pi$$
$$48$$ 0 0
$$49$$ 17.0000 0.0495627
$$50$$ 0 0
$$51$$ − 215.035i − 0.590410i
$$52$$ 0 0
$$53$$ 258.000i 0.668661i 0.942456 + 0.334330i $$0.108510\pi$$
−0.942456 + 0.334330i $$0.891490\pi$$
$$54$$ 0 0
$$55$$ −63.2456 −0.155055
$$56$$ 0 0
$$57$$ 640.000 1.48719
$$58$$ 0 0
$$59$$ − 75.8947i − 0.167469i −0.996488 0.0837343i $$-0.973315\pi$$
0.996488 0.0837343i $$-0.0266847\pi$$
$$60$$ 0 0
$$61$$ − 250.000i − 0.524741i −0.964967 0.262371i $$-0.915496\pi$$
0.964967 0.262371i $$-0.0845043\pi$$
$$62$$ 0 0
$$63$$ −246.658 −0.493269
$$64$$ 0 0
$$65$$ 190.000 0.362563
$$66$$ 0 0
$$67$$ 815.868i 1.48767i 0.668362 + 0.743837i $$0.266996\pi$$
−0.668362 + 0.743837i $$0.733004\pi$$
$$68$$ 0 0
$$69$$ − 520.000i − 0.907256i
$$70$$ 0 0
$$71$$ −645.105 −1.07831 −0.539154 0.842207i $$-0.681256\pi$$
−0.539154 + 0.842207i $$0.681256\pi$$
$$72$$ 0 0
$$73$$ 1078.00 1.72836 0.864181 0.503182i $$-0.167837\pi$$
0.864181 + 0.503182i $$0.167837\pi$$
$$74$$ 0 0
$$75$$ 158.114i 0.243432i
$$76$$ 0 0
$$77$$ − 240.000i − 0.355202i
$$78$$ 0 0
$$79$$ 278.280 0.396316 0.198158 0.980170i $$-0.436504\pi$$
0.198158 + 0.980170i $$0.436504\pi$$
$$80$$ 0 0
$$81$$ −911.000 −1.24966
$$82$$ 0 0
$$83$$ 1106.80i 1.46370i 0.681468 + 0.731848i $$0.261342\pi$$
−0.681468 + 0.731848i $$0.738658\pi$$
$$84$$ 0 0
$$85$$ − 170.000i − 0.216930i
$$86$$ 0 0
$$87$$ 1707.63 2.10434
$$88$$ 0 0
$$89$$ −890.000 −1.06000 −0.529999 0.847998i $$-0.677808\pi$$
−0.529999 + 0.847998i $$0.677808\pi$$
$$90$$ 0 0
$$91$$ 720.999i 0.830563i
$$92$$ 0 0
$$93$$ − 2160.00i − 2.40840i
$$94$$ 0 0
$$95$$ 505.964 0.546430
$$96$$ 0 0
$$97$$ −254.000 −0.265874 −0.132937 0.991124i $$-0.542441\pi$$
−0.132937 + 0.991124i $$0.542441\pi$$
$$98$$ 0 0
$$99$$ 164.438i 0.166936i
$$100$$ 0 0
$$101$$ − 598.000i − 0.589141i −0.955630 0.294570i $$-0.904823\pi$$
0.955630 0.294570i $$-0.0951766\pi$$
$$102$$ 0 0
$$103$$ −499.640 −0.477971 −0.238985 0.971023i $$-0.576815\pi$$
−0.238985 + 0.971023i $$0.576815\pi$$
$$104$$ 0 0
$$105$$ −600.000 −0.557657
$$106$$ 0 0
$$107$$ 626.131i 0.565704i 0.959164 + 0.282852i $$0.0912806\pi$$
−0.959164 + 0.282852i $$0.908719\pi$$
$$108$$ 0 0
$$109$$ 854.000i 0.750444i 0.926935 + 0.375222i $$0.122433\pi$$
−0.926935 + 0.375222i $$0.877567\pi$$
$$110$$ 0 0
$$111$$ −1302.86 −1.11407
$$112$$ 0 0
$$113$$ 1698.00 1.41358 0.706789 0.707424i $$-0.250143\pi$$
0.706789 + 0.707424i $$0.250143\pi$$
$$114$$ 0 0
$$115$$ − 411.096i − 0.333347i
$$116$$ 0 0
$$117$$ − 494.000i − 0.390345i
$$118$$ 0 0
$$119$$ 645.105 0.496947
$$120$$ 0 0
$$121$$ 1171.00 0.879790
$$122$$ 0 0
$$123$$ − 1707.63i − 1.25180i
$$124$$ 0 0
$$125$$ 125.000i 0.0894427i
$$126$$ 0 0
$$127$$ 234.009 0.163503 0.0817516 0.996653i $$-0.473949\pi$$
0.0817516 + 0.996653i $$0.473949\pi$$
$$128$$ 0 0
$$129$$ 3400.00 2.32057
$$130$$ 0 0
$$131$$ − 1732.93i − 1.15578i −0.816116 0.577888i $$-0.803877\pi$$
0.816116 0.577888i $$-0.196123\pi$$
$$132$$ 0 0
$$133$$ 1920.00i 1.25177i
$$134$$ 0 0
$$135$$ −442.719 −0.282246
$$136$$ 0 0
$$137$$ −1546.00 −0.964115 −0.482057 0.876140i $$-0.660110\pi$$
−0.482057 + 0.876140i $$0.660110\pi$$
$$138$$ 0 0
$$139$$ 328.877i 0.200683i 0.994953 + 0.100342i $$0.0319936\pi$$
−0.994953 + 0.100342i $$0.968006\pi$$
$$140$$ 0 0
$$141$$ − 840.000i − 0.501708i
$$142$$ 0 0
$$143$$ 480.666 0.281086
$$144$$ 0 0
$$145$$ 1350.00 0.773182
$$146$$ 0 0
$$147$$ − 107.517i − 0.0603258i
$$148$$ 0 0
$$149$$ − 3246.00i − 1.78472i −0.451328 0.892358i $$-0.649050\pi$$
0.451328 0.892358i $$-0.350950\pi$$
$$150$$ 0 0
$$151$$ −1505.24 −0.811225 −0.405613 0.914045i $$-0.632942\pi$$
−0.405613 + 0.914045i $$0.632942\pi$$
$$152$$ 0 0
$$153$$ −442.000 −0.233553
$$154$$ 0 0
$$155$$ − 1707.63i − 0.884904i
$$156$$ 0 0
$$157$$ − 1226.00i − 0.623219i −0.950210 0.311610i $$-0.899132\pi$$
0.950210 0.311610i $$-0.100868\pi$$
$$158$$ 0 0
$$159$$ 1631.74 0.813868
$$160$$ 0 0
$$161$$ 1560.00 0.763635
$$162$$ 0 0
$$163$$ − 1448.32i − 0.695960i −0.937502 0.347980i $$-0.886868\pi$$
0.937502 0.347980i $$-0.113132\pi$$
$$164$$ 0 0
$$165$$ 400.000i 0.188727i
$$166$$ 0 0
$$167$$ −2333.76 −1.08139 −0.540694 0.841219i $$-0.681838\pi$$
−0.540694 + 0.841219i $$0.681838\pi$$
$$168$$ 0 0
$$169$$ 753.000 0.342740
$$170$$ 0 0
$$171$$ − 1315.51i − 0.588300i
$$172$$ 0 0
$$173$$ − 3098.00i − 1.36148i −0.732524 0.680742i $$-0.761658\pi$$
0.732524 0.680742i $$-0.238342\pi$$
$$174$$ 0 0
$$175$$ −474.342 −0.204896
$$176$$ 0 0
$$177$$ −480.000 −0.203836
$$178$$ 0 0
$$179$$ − 2352.73i − 0.982411i −0.871044 0.491206i $$-0.836556\pi$$
0.871044 0.491206i $$-0.163444\pi$$
$$180$$ 0 0
$$181$$ − 2182.00i − 0.896060i −0.894019 0.448030i $$-0.852126\pi$$
0.894019 0.448030i $$-0.147874\pi$$
$$182$$ 0 0
$$183$$ −1581.14 −0.638695
$$184$$ 0 0
$$185$$ −1030.00 −0.409336
$$186$$ 0 0
$$187$$ − 430.070i − 0.168181i
$$188$$ 0 0
$$189$$ − 1680.00i − 0.646572i
$$190$$ 0 0
$$191$$ −3023.14 −1.14527 −0.572635 0.819810i $$-0.694079\pi$$
−0.572635 + 0.819810i $$0.694079\pi$$
$$192$$ 0 0
$$193$$ 1298.00 0.484104 0.242052 0.970263i $$-0.422180\pi$$
0.242052 + 0.970263i $$0.422180\pi$$
$$194$$ 0 0
$$195$$ − 1201.67i − 0.441298i
$$196$$ 0 0
$$197$$ − 2846.00i − 1.02928i −0.857405 0.514642i $$-0.827925\pi$$
0.857405 0.514642i $$-0.172075\pi$$
$$198$$ 0 0
$$199$$ 3592.35 1.27967 0.639836 0.768511i $$-0.279002\pi$$
0.639836 + 0.768511i $$0.279002\pi$$
$$200$$ 0 0
$$201$$ 5160.00 1.81074
$$202$$ 0 0
$$203$$ 5122.89i 1.77121i
$$204$$ 0 0
$$205$$ − 1350.00i − 0.459942i
$$206$$ 0 0
$$207$$ −1068.85 −0.358890
$$208$$ 0 0
$$209$$ 1280.00 0.423634
$$210$$ 0 0
$$211$$ 4186.86i 1.36604i 0.730398 + 0.683021i $$0.239334\pi$$
−0.730398 + 0.683021i $$0.760666\pi$$
$$212$$ 0 0
$$213$$ 4080.00i 1.31247i
$$214$$ 0 0
$$215$$ 2687.94 0.852631
$$216$$ 0 0
$$217$$ 6480.00 2.02715
$$218$$ 0 0
$$219$$ − 6817.87i − 2.10369i
$$220$$ 0 0
$$221$$ 1292.00i 0.393255i
$$222$$ 0 0
$$223$$ −4762.39 −1.43010 −0.715052 0.699071i $$-0.753597\pi$$
−0.715052 + 0.699071i $$0.753597\pi$$
$$224$$ 0 0
$$225$$ 325.000 0.0962963
$$226$$ 0 0
$$227$$ − 1663.36i − 0.486348i −0.969983 0.243174i $$-0.921811\pi$$
0.969983 0.243174i $$-0.0781886\pi$$
$$228$$ 0 0
$$229$$ 1050.00i 0.302995i 0.988458 + 0.151498i $$0.0484096\pi$$
−0.988458 + 0.151498i $$0.951590\pi$$
$$230$$ 0 0
$$231$$ −1517.89 −0.432338
$$232$$ 0 0
$$233$$ −2778.00 −0.781085 −0.390543 0.920585i $$-0.627713\pi$$
−0.390543 + 0.920585i $$0.627713\pi$$
$$234$$ 0 0
$$235$$ − 664.078i − 0.184339i
$$236$$ 0 0
$$237$$ − 1760.00i − 0.482381i
$$238$$ 0 0
$$239$$ 2555.12 0.691536 0.345768 0.938320i $$-0.387618\pi$$
0.345768 + 0.938320i $$0.387618\pi$$
$$240$$ 0 0
$$241$$ −5350.00 −1.42997 −0.714987 0.699138i $$-0.753567\pi$$
−0.714987 + 0.699138i $$0.753567\pi$$
$$242$$ 0 0
$$243$$ 3370.99i 0.889913i
$$244$$ 0 0
$$245$$ − 85.0000i − 0.0221651i
$$246$$ 0 0
$$247$$ −3845.33 −0.990577
$$248$$ 0 0
$$249$$ 7000.00 1.78155
$$250$$ 0 0
$$251$$ − 5881.84i − 1.47912i −0.673093 0.739558i $$-0.735034\pi$$
0.673093 0.739558i $$-0.264966\pi$$
$$252$$ 0 0
$$253$$ − 1040.00i − 0.258436i
$$254$$ 0 0
$$255$$ −1075.17 −0.264039
$$256$$ 0 0
$$257$$ 1074.00 0.260678 0.130339 0.991469i $$-0.458393\pi$$
0.130339 + 0.991469i $$0.458393\pi$$
$$258$$ 0 0
$$259$$ − 3908.58i − 0.937711i
$$260$$ 0 0
$$261$$ − 3510.00i − 0.832427i
$$262$$ 0 0
$$263$$ −1486.27 −0.348469 −0.174235 0.984704i $$-0.555745\pi$$
−0.174235 + 0.984704i $$0.555745\pi$$
$$264$$ 0 0
$$265$$ 1290.00 0.299034
$$266$$ 0 0
$$267$$ 5628.85i 1.29019i
$$268$$ 0 0
$$269$$ 406.000i 0.0920233i 0.998941 + 0.0460116i $$0.0146511\pi$$
−0.998941 + 0.0460116i $$0.985349\pi$$
$$270$$ 0 0
$$271$$ −392.122 −0.0878957 −0.0439479 0.999034i $$-0.513994\pi$$
−0.0439479 + 0.999034i $$0.513994\pi$$
$$272$$ 0 0
$$273$$ 4560.00 1.01093
$$274$$ 0 0
$$275$$ 316.228i 0.0693427i
$$276$$ 0 0
$$277$$ − 5934.00i − 1.28715i −0.765385 0.643573i $$-0.777451\pi$$
0.765385 0.643573i $$-0.222549\pi$$
$$278$$ 0 0
$$279$$ −4439.84 −0.952710
$$280$$ 0 0
$$281$$ 1870.00 0.396992 0.198496 0.980102i $$-0.436394\pi$$
0.198496 + 0.980102i $$0.436394\pi$$
$$282$$ 0 0
$$283$$ 4888.88i 1.02690i 0.858118 + 0.513452i $$0.171634\pi$$
−0.858118 + 0.513452i $$0.828366\pi$$
$$284$$ 0 0
$$285$$ − 3200.00i − 0.665093i
$$286$$ 0 0
$$287$$ 5122.89 1.05364
$$288$$ 0 0
$$289$$ −3757.00 −0.764706
$$290$$ 0 0
$$291$$ 1606.44i 0.323612i
$$292$$ 0 0
$$293$$ − 5198.00i − 1.03642i −0.855254 0.518209i $$-0.826599\pi$$
0.855254 0.518209i $$-0.173401\pi$$
$$294$$ 0 0
$$295$$ −379.473 −0.0748942
$$296$$ 0 0
$$297$$ −1120.00 −0.218818
$$298$$ 0 0
$$299$$ 3124.33i 0.604297i
$$300$$ 0 0
$$301$$ 10200.0i 1.95322i
$$302$$ 0 0
$$303$$ −3782.08 −0.717079
$$304$$ 0 0
$$305$$ −1250.00 −0.234671
$$306$$ 0 0
$$307$$ 3750.46i 0.697232i 0.937266 + 0.348616i $$0.113348\pi$$
−0.937266 + 0.348616i $$0.886652\pi$$
$$308$$ 0 0
$$309$$ 3160.00i 0.581767i
$$310$$ 0 0
$$311$$ 6261.31 1.14163 0.570814 0.821079i $$-0.306628\pi$$
0.570814 + 0.821079i $$0.306628\pi$$
$$312$$ 0 0
$$313$$ −2218.00 −0.400539 −0.200270 0.979741i $$-0.564182\pi$$
−0.200270 + 0.979741i $$0.564182\pi$$
$$314$$ 0 0
$$315$$ 1233.29i 0.220597i
$$316$$ 0 0
$$317$$ 4134.00i 0.732456i 0.930525 + 0.366228i $$0.119351\pi$$
−0.930525 + 0.366228i $$0.880649\pi$$
$$318$$ 0 0
$$319$$ 3415.26 0.599429
$$320$$ 0 0
$$321$$ 3960.00 0.688553
$$322$$ 0 0
$$323$$ 3440.56i 0.592687i
$$324$$ 0 0
$$325$$ − 950.000i − 0.162143i
$$326$$ 0 0
$$327$$ 5401.17 0.913411
$$328$$ 0 0
$$329$$ 2520.00 0.422286
$$330$$ 0 0
$$331$$ 11953.4i 1.98495i 0.122443 + 0.992476i $$0.460927\pi$$
−0.122443 + 0.992476i $$0.539073\pi$$
$$332$$ 0 0
$$333$$ 2678.00i 0.440701i
$$334$$ 0 0
$$335$$ 4079.34 0.665308
$$336$$ 0 0
$$337$$ −8014.00 −1.29540 −0.647701 0.761895i $$-0.724269\pi$$
−0.647701 + 0.761895i $$0.724269\pi$$
$$338$$ 0 0
$$339$$ − 10739.1i − 1.72055i
$$340$$ 0 0
$$341$$ − 4320.00i − 0.686044i
$$342$$ 0 0
$$343$$ −6185.42 −0.973706
$$344$$ 0 0
$$345$$ −2600.00 −0.405737
$$346$$ 0 0
$$347$$ − 4484.11i − 0.693716i −0.937918 0.346858i $$-0.887248\pi$$
0.937918 0.346858i $$-0.112752\pi$$
$$348$$ 0 0
$$349$$ 910.000i 0.139574i 0.997562 + 0.0697868i $$0.0222319\pi$$
−0.997562 + 0.0697868i $$0.977768\pi$$
$$350$$ 0 0
$$351$$ 3364.66 0.511659
$$352$$ 0 0
$$353$$ 12962.0 1.95438 0.977192 0.212357i $$-0.0681140\pi$$
0.977192 + 0.212357i $$0.0681140\pi$$
$$354$$ 0 0
$$355$$ 3225.52i 0.482234i
$$356$$ 0 0
$$357$$ − 4080.00i − 0.604864i
$$358$$ 0 0
$$359$$ 12193.7 1.79265 0.896325 0.443398i $$-0.146227\pi$$
0.896325 + 0.443398i $$0.146227\pi$$
$$360$$ 0 0
$$361$$ −3381.00 −0.492929
$$362$$ 0 0
$$363$$ − 7406.05i − 1.07085i
$$364$$ 0 0
$$365$$ − 5390.00i − 0.772947i
$$366$$ 0 0
$$367$$ −3434.23 −0.488462 −0.244231 0.969717i $$-0.578536\pi$$
−0.244231 + 0.969717i $$0.578536\pi$$
$$368$$ 0 0
$$369$$ −3510.00 −0.495185
$$370$$ 0 0
$$371$$ 4895.21i 0.685031i
$$372$$ 0 0
$$373$$ − 4622.00i − 0.641603i −0.947146 0.320802i $$-0.896048\pi$$
0.947146 0.320802i $$-0.103952\pi$$
$$374$$ 0 0
$$375$$ 790.569 0.108866
$$376$$ 0 0
$$377$$ −10260.0 −1.40164
$$378$$ 0 0
$$379$$ − 8449.61i − 1.14519i −0.819838 0.572595i $$-0.805937\pi$$
0.819838 0.572595i $$-0.194063\pi$$
$$380$$ 0 0
$$381$$ − 1480.00i − 0.199010i
$$382$$ 0 0
$$383$$ 1815.15 0.242166 0.121083 0.992642i $$-0.461363\pi$$
0.121083 + 0.992642i $$0.461363\pi$$
$$384$$ 0 0
$$385$$ −1200.00 −0.158851
$$386$$ 0 0
$$387$$ − 6988.63i − 0.917964i
$$388$$ 0 0
$$389$$ 11106.0i 1.44755i 0.690037 + 0.723774i $$0.257594\pi$$
−0.690037 + 0.723774i $$0.742406\pi$$
$$390$$ 0 0
$$391$$ 2795.45 0.361566
$$392$$ 0 0
$$393$$ −10960.0 −1.40677
$$394$$ 0 0
$$395$$ − 1391.40i − 0.177238i
$$396$$ 0 0
$$397$$ − 5754.00i − 0.727418i −0.931513 0.363709i $$-0.881510\pi$$
0.931513 0.363709i $$-0.118490\pi$$
$$398$$ 0 0
$$399$$ 12143.1 1.52360
$$400$$ 0 0
$$401$$ −1118.00 −0.139228 −0.0696138 0.997574i $$-0.522177\pi$$
−0.0696138 + 0.997574i $$0.522177\pi$$
$$402$$ 0 0
$$403$$ 12978.0i 1.60417i
$$404$$ 0 0
$$405$$ 4555.00i 0.558864i
$$406$$ 0 0
$$407$$ −2605.72 −0.317348
$$408$$ 0 0
$$409$$ 11374.0 1.37508 0.687540 0.726146i $$-0.258690\pi$$
0.687540 + 0.726146i $$0.258690\pi$$
$$410$$ 0 0
$$411$$ 9777.76i 1.17348i
$$412$$ 0 0
$$413$$ − 1440.00i − 0.171568i
$$414$$ 0 0
$$415$$ 5533.99 0.654585
$$416$$ 0 0
$$417$$ 2080.00 0.244264
$$418$$ 0 0
$$419$$ 12674.4i 1.47777i 0.673832 + 0.738885i $$0.264647\pi$$
−0.673832 + 0.738885i $$0.735353\pi$$
$$420$$ 0 0
$$421$$ − 1150.00i − 0.133130i −0.997782 0.0665648i $$-0.978796\pi$$
0.997782 0.0665648i $$-0.0212039\pi$$
$$422$$ 0 0
$$423$$ −1726.60 −0.198464
$$424$$ 0 0
$$425$$ −850.000 −0.0970143
$$426$$ 0 0
$$427$$ − 4743.42i − 0.537588i
$$428$$ 0 0
$$429$$ − 3040.00i − 0.342127i
$$430$$ 0 0
$$431$$ −1353.45 −0.151261 −0.0756307 0.997136i $$-0.524097\pi$$
−0.0756307 + 0.997136i $$0.524097\pi$$
$$432$$ 0 0
$$433$$ −7918.00 −0.878787 −0.439394 0.898295i $$-0.644807\pi$$
−0.439394 + 0.898295i $$0.644807\pi$$
$$434$$ 0 0
$$435$$ − 8538.15i − 0.941087i
$$436$$ 0 0
$$437$$ 8320.00i 0.910754i
$$438$$ 0 0
$$439$$ 14217.6 1.54572 0.772858 0.634579i $$-0.218827\pi$$
0.772858 + 0.634579i $$0.218827\pi$$
$$440$$ 0 0
$$441$$ −221.000 −0.0238635
$$442$$ 0 0
$$443$$ 10581.0i 1.13480i 0.823441 + 0.567401i $$0.192051\pi$$
−0.823441 + 0.567401i $$0.807949\pi$$
$$444$$ 0 0
$$445$$ 4450.00i 0.474045i
$$446$$ 0 0
$$447$$ −20529.5 −2.17229
$$448$$ 0 0
$$449$$ 4474.00 0.470247 0.235124 0.971965i $$-0.424450\pi$$
0.235124 + 0.971965i $$0.424450\pi$$
$$450$$ 0 0
$$451$$ − 3415.26i − 0.356582i
$$452$$ 0 0
$$453$$ 9520.00i 0.987392i
$$454$$ 0 0
$$455$$ 3605.00 0.371439
$$456$$ 0 0
$$457$$ −4154.00 −0.425199 −0.212599 0.977139i $$-0.568193\pi$$
−0.212599 + 0.977139i $$0.568193\pi$$
$$458$$ 0 0
$$459$$ − 3010.49i − 0.306138i
$$460$$ 0 0
$$461$$ − 11282.0i − 1.13982i −0.821709 0.569908i $$-0.806979\pi$$
0.821709 0.569908i $$-0.193021\pi$$
$$462$$ 0 0
$$463$$ 5458.09 0.547860 0.273930 0.961750i $$-0.411676\pi$$
0.273930 + 0.961750i $$0.411676\pi$$
$$464$$ 0 0
$$465$$ −10800.0 −1.07707
$$466$$ 0 0
$$467$$ 3775.76i 0.374136i 0.982347 + 0.187068i $$0.0598984\pi$$
−0.982347 + 0.187068i $$0.940102\pi$$
$$468$$ 0 0
$$469$$ 15480.0i 1.52409i
$$470$$ 0 0
$$471$$ −7753.90 −0.758559
$$472$$ 0 0
$$473$$ 6800.00 0.661024
$$474$$ 0 0
$$475$$ − 2529.82i − 0.244371i
$$476$$ 0 0
$$477$$ − 3354.00i − 0.321948i
$$478$$ 0 0
$$479$$ −8930.27 −0.851847 −0.425923 0.904759i $$-0.640051\pi$$
−0.425923 + 0.904759i $$0.640051\pi$$
$$480$$ 0 0
$$481$$ 7828.00 0.742050
$$482$$ 0 0
$$483$$ − 9866.31i − 0.929467i
$$484$$ 0 0
$$485$$ 1270.00i 0.118903i
$$486$$ 0 0
$$487$$ 2422.30 0.225390 0.112695 0.993630i $$-0.464052\pi$$
0.112695 + 0.993630i $$0.464052\pi$$
$$488$$ 0 0
$$489$$ −9160.00 −0.847095
$$490$$ 0 0
$$491$$ 8993.52i 0.826623i 0.910590 + 0.413311i $$0.135628\pi$$
−0.910590 + 0.413311i $$0.864372\pi$$
$$492$$ 0 0
$$493$$ 9180.00i 0.838634i
$$494$$ 0 0
$$495$$ 822.192 0.0746561
$$496$$ 0 0
$$497$$ −12240.0 −1.10471
$$498$$ 0 0
$$499$$ − 3541.75i − 0.317737i −0.987300 0.158868i $$-0.949215\pi$$
0.987300 0.158868i $$-0.0507845\pi$$
$$500$$ 0 0
$$501$$ 14760.0i 1.31622i
$$502$$ 0 0
$$503$$ 2384.36 0.211358 0.105679 0.994400i $$-0.466298\pi$$
0.105679 + 0.994400i $$0.466298\pi$$
$$504$$ 0 0
$$505$$ −2990.00 −0.263472
$$506$$ 0 0
$$507$$ − 4762.39i − 0.417170i
$$508$$ 0 0
$$509$$ 2350.00i 0.204640i 0.994752 + 0.102320i $$0.0326266\pi$$
−0.994752 + 0.102320i $$0.967373\pi$$
$$510$$ 0 0
$$511$$ 20453.6 1.77067
$$512$$ 0 0
$$513$$ 8960.00 0.771138
$$514$$ 0 0
$$515$$ 2498.20i 0.213755i
$$516$$ 0 0
$$517$$ − 1680.00i − 0.142914i
$$518$$ 0 0
$$519$$ −19593.5 −1.65714
$$520$$ 0 0
$$521$$ −858.000 −0.0721491 −0.0360745 0.999349i $$-0.511485\pi$$
−0.0360745 + 0.999349i $$0.511485\pi$$
$$522$$ 0 0
$$523$$ 5799.62i 0.484894i 0.970165 + 0.242447i $$0.0779501\pi$$
−0.970165 + 0.242447i $$0.922050\pi$$
$$524$$ 0 0
$$525$$ 3000.00i 0.249392i
$$526$$ 0 0
$$527$$ 11611.9 0.959813
$$528$$ 0 0
$$529$$ −5407.00 −0.444399
$$530$$ 0 0
$$531$$ 986.631i 0.0806330i
$$532$$ 0 0
$$533$$ 10260.0i 0.833790i
$$534$$ 0 0
$$535$$ 3130.65 0.252991
$$536$$ 0 0
$$537$$ −14880.0 −1.19575
$$538$$ 0 0
$$539$$ − 215.035i − 0.0171841i
$$540$$ 0 0
$$541$$ 20478.0i 1.62739i 0.581292 + 0.813695i $$0.302547\pi$$
−0.581292 + 0.813695i $$0.697453\pi$$
$$542$$ 0 0
$$543$$ −13800.2 −1.09065
$$544$$ 0 0
$$545$$ 4270.00 0.335609
$$546$$ 0 0
$$547$$ 10429.2i 0.815210i 0.913158 + 0.407605i $$0.133636\pi$$
−0.913158 + 0.407605i $$0.866364\pi$$
$$548$$ 0 0
$$549$$ 3250.00i 0.252653i
$$550$$ 0 0
$$551$$ −27322.1 −2.11245
$$552$$ 0 0
$$553$$ 5280.00 0.406019
$$554$$ 0 0
$$555$$ 6514.29i 0.498228i
$$556$$ 0 0
$$557$$ − 13194.0i − 1.00368i −0.864962 0.501838i $$-0.832657\pi$$
0.864962 0.501838i $$-0.167343\pi$$
$$558$$ 0 0
$$559$$ −20428.3 −1.54566
$$560$$ 0 0
$$561$$ −2720.00 −0.204703
$$562$$ 0 0
$$563$$ − 9771.44i − 0.731469i −0.930719 0.365734i $$-0.880818\pi$$
0.930719 0.365734i $$-0.119182\pi$$
$$564$$ 0 0
$$565$$ − 8490.00i − 0.632172i
$$566$$ 0 0
$$567$$ −17285.0 −1.28025
$$568$$ 0 0
$$569$$ −4594.00 −0.338472 −0.169236 0.985576i $$-0.554130\pi$$
−0.169236 + 0.985576i $$0.554130\pi$$
$$570$$ 0 0
$$571$$ − 4389.24i − 0.321688i −0.986980 0.160844i $$-0.948578\pi$$
0.986980 0.160844i $$-0.0514217\pi$$
$$572$$ 0 0
$$573$$ 19120.0i 1.39398i
$$574$$ 0 0
$$575$$ −2055.48 −0.149077
$$576$$ 0 0
$$577$$ −14926.0 −1.07691 −0.538455 0.842654i $$-0.680992\pi$$
−0.538455 + 0.842654i $$0.680992\pi$$
$$578$$ 0 0
$$579$$ − 8209.27i − 0.589233i
$$580$$ 0 0
$$581$$ 21000.0i 1.49953i
$$582$$ 0 0
$$583$$ 3263.47 0.231834
$$584$$ 0 0
$$585$$ −2470.00 −0.174567
$$586$$ 0 0
$$587$$ − 8101.76i − 0.569668i −0.958577 0.284834i $$-0.908061\pi$$
0.958577 0.284834i $$-0.0919385\pi$$
$$588$$ 0 0
$$589$$ 34560.0i 2.41769i
$$590$$ 0 0
$$591$$ −17999.7 −1.25281
$$592$$ 0 0
$$593$$ −26958.0 −1.86683 −0.933417 0.358794i $$-0.883188\pi$$
−0.933417 + 0.358794i $$0.883188\pi$$
$$594$$ 0 0
$$595$$ − 3225.52i − 0.222241i
$$596$$ 0 0
$$597$$ − 22720.0i − 1.55757i
$$598$$ 0 0
$$599$$ −6349.85 −0.433135 −0.216568 0.976268i $$-0.569486\pi$$
−0.216568 + 0.976268i $$0.569486\pi$$
$$600$$ 0 0
$$601$$ −21970.0 −1.49114 −0.745570 0.666427i $$-0.767823\pi$$
−0.745570 + 0.666427i $$0.767823\pi$$
$$602$$ 0 0
$$603$$ − 10606.3i − 0.716287i
$$604$$ 0 0
$$605$$ − 5855.00i − 0.393454i
$$606$$ 0 0
$$607$$ 3876.95 0.259243 0.129622 0.991564i $$-0.458624\pi$$
0.129622 + 0.991564i $$0.458624\pi$$
$$608$$ 0 0
$$609$$ 32400.0 2.15585
$$610$$ 0 0
$$611$$ 5047.00i 0.334173i
$$612$$ 0 0
$$613$$ − 2878.00i − 0.189627i −0.995495 0.0948135i $$-0.969775\pi$$
0.995495 0.0948135i $$-0.0302255\pi$$
$$614$$ 0 0
$$615$$ −8538.15 −0.559823
$$616$$ 0 0
$$617$$ −27354.0 −1.78481 −0.892407 0.451231i $$-0.850985\pi$$
−0.892407 + 0.451231i $$0.850985\pi$$
$$618$$ 0 0
$$619$$ − 12547.9i − 0.814771i −0.913256 0.407386i $$-0.866441\pi$$
0.913256 0.407386i $$-0.133559\pi$$
$$620$$ 0 0
$$621$$ − 7280.00i − 0.470429i
$$622$$ 0 0
$$623$$ −16886.6 −1.08595
$$624$$ 0 0
$$625$$ 625.000 0.0400000
$$626$$ 0 0
$$627$$ − 8095.43i − 0.515631i
$$628$$ 0 0
$$629$$ − 7004.00i − 0.443987i
$$630$$ 0 0
$$631$$ 30876.5 1.94798 0.973988 0.226598i $$-0.0727605\pi$$
0.973988 + 0.226598i $$0.0727605\pi$$
$$632$$ 0 0
$$633$$ 26480.0 1.66269
$$634$$ 0 0
$$635$$ − 1170.04i − 0.0731208i
$$636$$ 0 0
$$637$$ 646.000i 0.0401812i
$$638$$ 0 0
$$639$$ 8386.36 0.519185
$$640$$ 0 0
$$641$$ −9430.00 −0.581065 −0.290532 0.956865i $$-0.593832\pi$$
−0.290532 + 0.956865i $$0.593832\pi$$
$$642$$ 0 0
$$643$$ − 9847.33i − 0.603952i −0.953316 0.301976i $$-0.902354\pi$$
0.953316 0.301976i $$-0.0976462\pi$$
$$644$$ 0 0
$$645$$ − 17000.0i − 1.03779i
$$646$$ 0 0
$$647$$ 30048.0 1.82582 0.912911 0.408158i $$-0.133829\pi$$
0.912911 + 0.408158i $$0.133829\pi$$
$$648$$ 0 0
$$649$$ −960.000 −0.0580636
$$650$$ 0 0
$$651$$ − 40983.1i − 2.46737i
$$652$$ 0 0
$$653$$ 18742.0i 1.12317i 0.827418 + 0.561586i $$0.189809\pi$$
−0.827418 + 0.561586i $$0.810191\pi$$
$$654$$ 0 0
$$655$$ −8664.64 −0.516879
$$656$$ 0 0
$$657$$ −14014.0 −0.832174
$$658$$ 0 0
$$659$$ − 8323.11i − 0.491992i −0.969271 0.245996i $$-0.920885\pi$$
0.969271 0.245996i $$-0.0791150\pi$$
$$660$$ 0 0
$$661$$ − 7630.00i − 0.448975i −0.974477 0.224488i $$-0.927929\pi$$
0.974477 0.224488i $$-0.0720708\pi$$
$$662$$ 0 0
$$663$$ 8171.33 0.478655
$$664$$ 0 0
$$665$$ 9600.00 0.559808
$$666$$ 0 0
$$667$$ 22199.2i 1.28869i
$$668$$ 0 0
$$669$$ 30120.0i 1.74067i
$$670$$ 0 0
$$671$$ −3162.28 −0.181935
$$672$$ 0 0
$$673$$ −10878.0 −0.623055 −0.311528 0.950237i $$-0.600841\pi$$
−0.311528 + 0.950237i $$0.600841\pi$$
$$674$$ 0 0
$$675$$ 2213.59i 0.126224i
$$676$$ 0 0
$$677$$ − 126.000i − 0.00715299i −0.999994 0.00357649i $$-0.998862\pi$$
0.999994 0.00357649i $$-0.00113844\pi$$
$$678$$ 0 0
$$679$$ −4819.31 −0.272383
$$680$$ 0 0
$$681$$ −10520.0 −0.591964
$$682$$ 0 0
$$683$$ − 16412.2i − 0.919467i −0.888057 0.459734i $$-0.847945\pi$$
0.888057 0.459734i $$-0.152055\pi$$
$$684$$ 0 0
$$685$$ 7730.00i 0.431165i
$$686$$ 0 0
$$687$$ 6640.78 0.368794
$$688$$ 0 0
$$689$$ −9804.00 −0.542094
$$690$$ 0 0
$$691$$ − 13193.0i − 0.726319i −0.931727 0.363159i $$-0.881698\pi$$
0.931727 0.363159i $$-0.118302\pi$$
$$692$$ 0 0
$$693$$ 3120.00i 0.171023i
$$694$$ 0 0
$$695$$ 1644.38 0.0897483
$$696$$ 0 0
$$697$$ 9180.00 0.498877
$$698$$ 0 0
$$699$$ 17569.6i 0.950707i
$$700$$ 0 0
$$701$$ − 22010.0i − 1.18589i −0.805244 0.592943i $$-0.797966\pi$$
0.805244 0.592943i $$-0.202034\pi$$
$$702$$ 0 0
$$703$$ 20845.7 1.11837
$$704$$ 0 0
$$705$$ −4200.00 −0.224370
$$706$$ 0 0
$$707$$ − 11346.3i − 0.603564i
$$708$$ 0 0
$$709$$ − 550.000i − 0.0291335i −0.999894 0.0145668i $$-0.995363\pi$$
0.999894 0.0145668i $$-0.00463691\pi$$
$$710$$ 0 0
$$711$$ −3617.65 −0.190819
$$712$$ 0 0
$$713$$ 28080.0 1.47490
$$714$$ 0 0
$$715$$ − 2403.33i − 0.125706i
$$716$$ 0 0
$$717$$ − 16160.0i − 0.841710i
$$718$$ 0 0
$$719$$ 17936.4 0.930343 0.465171 0.885221i $$-0.345993\pi$$
0.465171 + 0.885221i $$0.345993\pi$$
$$720$$ 0 0
$$721$$ −9480.00 −0.489672
$$722$$ 0 0
$$723$$ 33836.4i 1.74051i
$$724$$ 0 0
$$725$$ − 6750.00i − 0.345778i
$$726$$ 0 0
$$727$$ −16728.4 −0.853403 −0.426701 0.904393i $$-0.640324\pi$$
−0.426701 + 0.904393i $$0.640324\pi$$
$$728$$ 0 0
$$729$$ −3277.00 −0.166489
$$730$$ 0 0
$$731$$ 18278.0i 0.924808i
$$732$$ 0 0
$$733$$ 2422.00i 0.122044i 0.998136 + 0.0610222i $$0.0194361\pi$$
−0.998136 + 0.0610222i $$0.980564\pi$$
$$734$$ 0 0
$$735$$ −537.587 −0.0269785
$$736$$ 0 0
$$737$$ 10320.0 0.515797
$$738$$ 0 0
$$739$$ 19555.5i 0.973426i 0.873562 + 0.486713i $$0.161804\pi$$
−0.873562 + 0.486713i $$0.838196\pi$$
$$740$$ 0 0
$$741$$ 24320.0i 1.20569i
$$742$$ 0 0
$$743$$ −31059.9 −1.53362 −0.766808 0.641876i $$-0.778156\pi$$
−0.766808 + 0.641876i $$0.778156\pi$$
$$744$$ 0 0
$$745$$ −16230.0 −0.798149
$$746$$ 0 0
$$747$$ − 14388.4i − 0.704743i
$$748$$ 0 0
$$749$$ 11880.0i 0.579554i
$$750$$ 0 0
$$751$$ −12155.8 −0.590641 −0.295320 0.955398i $$-0.595426\pi$$
−0.295320 + 0.955398i $$0.595426\pi$$
$$752$$ 0 0
$$753$$ −37200.0 −1.80032
$$754$$ 0 0
$$755$$ 7526.22i 0.362791i
$$756$$ 0 0
$$757$$ 19346.0i 0.928854i 0.885611 + 0.464427i $$0.153740\pi$$
−0.885611 + 0.464427i $$0.846260\pi$$
$$758$$ 0 0
$$759$$ −6577.54 −0.314558
$$760$$ 0 0
$$761$$ 33078.0 1.57566 0.787830 0.615893i $$-0.211205\pi$$
0.787830 + 0.615893i $$0.211205\pi$$
$$762$$ 0 0
$$763$$ 16203.5i 0.768816i
$$764$$ 0 0
$$765$$ 2210.00i 0.104448i
$$766$$ 0 0
$$767$$ 2884.00 0.135769
$$768$$ 0 0
$$769$$ 32530.0 1.52544 0.762719 0.646730i $$-0.223864\pi$$
0.762719 + 0.646730i $$0.223864\pi$$
$$770$$ 0 0
$$771$$ − 6792.57i − 0.317287i
$$772$$ 0 0
$$773$$ 12002.0i 0.558450i 0.960226 + 0.279225i $$0.0900776\pi$$
−0.960226 + 0.279225i $$0.909922\pi$$
$$774$$ 0 0
$$775$$ −8538.15 −0.395741
$$776$$ 0 0
$$777$$ −24720.0 −1.14134
$$778$$ 0 0
$$779$$ 27322.1i 1.25663i
$$780$$ 0 0
$$781$$ 8160.00i 0.373864i
$$782$$ 0 0
$$783$$ 23906.8 1.09114
$$784$$ 0 0
$$785$$ −6130.00 −0.278712
$$786$$ 0 0
$$787$$ − 19954.0i − 0.903789i −0.892071 0.451895i $$-0.850748\pi$$
0.892071 0.451895i $$-0.149252\pi$$
$$788$$ 0 0
$$789$$ 9400.00i 0.424143i
$$790$$ 0 0
$$791$$ 32217.3 1.44819
$$792$$ 0 0
$$793$$ 9500.00 0.425416
$$794$$ 0 0
$$795$$ − 8158.68i − 0.363973i
$$796$$ 0 0
$$797$$ − 32666.0i − 1.45181i −0.687797 0.725903i $$-0.741422\pi$$
0.687797 0.725903i $$-0.258578\pi$$
$$798$$ 0 0
$$799$$ 4515.73 0.199944
$$800$$ 0 0
$$801$$ 11570.0 0.510369
$$802$$ 0 0
$$803$$ − 13635.7i − 0.599246i
$$804$$ 0 0
$$805$$ − 7800.00i − 0.341508i
$$806$$ 0 0
$$807$$ 2567.77 0.112007
$$808$$ 0 0
$$809$$ 23110.0 1.00433 0.502166 0.864771i $$-0.332537\pi$$
0.502166 + 0.864771i $$0.332537\pi$$
$$810$$ 0 0
$$811$$ − 35632.5i − 1.54282i −0.636338 0.771411i $$-0.719552\pi$$
0.636338 0.771411i $$-0.280448\pi$$
$$812$$ 0 0
$$813$$ 2480.00i 0.106983i
$$814$$ 0 0
$$815$$ −7241.62 −0.311243
$$816$$ 0 0
$$817$$ −54400.0 −2.32952
$$818$$ 0 0
$$819$$ − 9372.99i − 0.399901i
$$820$$ 0 0
$$821$$ 8850.00i 0.376208i 0.982149 + 0.188104i $$0.0602343\pi$$
−0.982149 + 0.188104i $$0.939766\pi$$
$$822$$ 0 0
$$823$$ 13895.0 0.588519 0.294259 0.955726i $$-0.404927\pi$$
0.294259 + 0.955726i $$0.404927\pi$$
$$824$$ 0 0
$$825$$ 2000.00 0.0844013
$$826$$ 0 0
$$827$$ 35841.3i 1.50704i 0.657425 + 0.753520i $$0.271646\pi$$
−0.657425 + 0.753520i $$0.728354\pi$$
$$828$$ 0 0
$$829$$ − 23034.0i − 0.965023i −0.875890 0.482511i $$-0.839725\pi$$
0.875890 0.482511i $$-0.160275\pi$$
$$830$$ 0 0
$$831$$ −37529.9 −1.56666
$$832$$ 0 0
$$833$$ 578.000 0.0240414
$$834$$ 0 0
$$835$$ 11668.8i 0.483612i
$$836$$ 0 0
$$837$$ − 30240.0i − 1.24880i
$$838$$ 0 0
$$839$$ 36960.7 1.52089 0.760444 0.649403i $$-0.224981\pi$$
0.760444 + 0.649403i $$0.224981\pi$$
$$840$$ 0 0
$$841$$ −48511.0 −1.98905
$$842$$ 0 0
$$843$$ − 11826.9i − 0.483204i
$$844$$ 0 0
$$845$$ − 3765.00i − 0.153278i
$$846$$ 0 0
$$847$$ 22218.2 0.901328
$$848$$ 0 0
$$849$$ 30920.0 1.24991
$$850$$ 0 0
$$851$$ − 16937.2i − 0.682254i
$$852$$ 0 0
$$853$$ 19122.0i 0.767555i 0.923425 + 0.383778i $$0.125377\pi$$
−0.923425 + 0.383778i $$0.874623\pi$$
$$854$$ 0 0
$$855$$ −6577.54 −0.263096
$$856$$ 0 0
$$857$$ −17786.0 −0.708936 −0.354468 0.935068i $$-0.615338\pi$$
−0.354468 + 0.935068i $$0.615338\pi$$
$$858$$ 0 0
$$859$$ 28713.5i 1.14050i 0.821470 + 0.570251i $$0.193154\pi$$
−0.821470 + 0.570251i $$0.806846\pi$$
$$860$$ 0 0
$$861$$ − 32400.0i − 1.28245i
$$862$$ 0 0
$$863$$ 23748.7 0.936750 0.468375 0.883530i $$-0.344840\pi$$
0.468375 + 0.883530i $$0.344840\pi$$
$$864$$ 0 0
$$865$$ −15490.0 −0.608874
$$866$$ 0 0
$$867$$ 23761.4i 0.930770i
$$868$$ 0 0
$$869$$ − 3520.00i − 0.137408i
$$870$$ 0 0
$$871$$ −31003.0 −1.20608
$$872$$ 0 0
$$873$$ 3302.00 0.128013
$$874$$ 0 0
$$875$$ 2371.71i 0.0916324i
$$876$$ 0 0
$$877$$ − 7706.00i − 0.296708i −0.988934 0.148354i $$-0.952602\pi$$
0.988934 0.148354i $$-0.0473975\pi$$
$$878$$ 0 0
$$879$$ −32875.0 −1.26149
$$880$$ 0 0
$$881$$ 10410.0 0.398095 0.199048 0.979990i $$-0.436215\pi$$
0.199048 + 0.979990i $$0.436215\pi$$
$$882$$ 0 0
$$883$$ − 26822.4i − 1.02225i −0.859506 0.511125i $$-0.829229\pi$$
0.859506 0.511125i $$-0.170771\pi$$
$$884$$ 0 0
$$885$$ 2400.00i 0.0911583i
$$886$$ 0 0
$$887$$ −21130.3 −0.799873 −0.399937 0.916543i $$-0.630968\pi$$
−0.399937 + 0.916543i $$0.630968\pi$$
$$888$$ 0 0
$$889$$ 4440.00 0.167506
$$890$$ 0 0
$$891$$ 11523.3i 0.433273i
$$892$$ 0 0
$$893$$ 13440.0i 0.503642i
$$894$$ 0 0
$$895$$ −11763.7 −0.439348
$$896$$ 0 0
$$897$$ 19760.0 0.735526
$$898$$ 0 0
$$899$$ 92212.0i 3.42096i
$$900$$ 0 0
$$901$$ 8772.00i 0.324348i
$$902$$ 0 0
$$903$$ 64510.5 2.37738
$$904$$ 0 0
$$905$$ −10910.0 −0.400730
$$906$$ 0 0
$$907$$ 19220.3i 0.703639i 0.936068 + 0.351819i $$0.114437\pi$$
−0.936068 + 0.351819i $$0.885563\pi$$
$$908$$ 0 0
$$909$$ 7774.00i 0.283660i
$$910$$ 0 0
$$911$$ 39402.0 1.43298 0.716491 0.697597i $$-0.245747\pi$$
0.716491 + 0.697597i $$0.245747\pi$$
$$912$$ 0 0
$$913$$ 14000.0 0.507483
$$914$$ 0 0
$$915$$ 7905.69i 0.285633i
$$916$$ 0 0
$$917$$ − 32880.0i − 1.18407i
$$918$$ 0 0
$$919$$ −42931.1 −1.54099 −0.770493 0.637449i $$-0.779990\pi$$
−0.770493 + 0.637449i $$0.779990\pi$$
$$920$$ 0 0
$$921$$ 23720.0 0.848643
$$922$$ 0 0
$$923$$ − 24514.0i − 0.874201i
$$924$$ 0 0
$$925$$ 5150.00i 0.183060i
$$926$$ 0 0
$$927$$ 6495.32 0.230134
$$928$$ 0 0
$$929$$ 34746.0 1.22710 0.613552 0.789654i $$-0.289740\pi$$
0.613552 + 0.789654i $$0.289740\pi$$
$$930$$ 0 0
$$931$$ 1720.28i 0.0605584i
$$932$$ 0 0
$$933$$ − 39600.0i − 1.38955i
$$934$$ 0 0
$$935$$ −2150.35 −0.0752128
$$936$$ 0 0
$$937$$ −21594.0 −0.752876 −0.376438 0.926442i $$-0.622851\pi$$
−0.376438 + 0.926442i $$0.622851\pi$$
$$938$$ 0 0
$$939$$ 14027.9i 0.487521i
$$940$$ 0 0
$$941$$ − 20018.0i − 0.693484i −0.937961 0.346742i $$-0.887288\pi$$
0.937961 0.346742i $$-0.112712\pi$$
$$942$$ 0 0
$$943$$ 22199.2 0.766601
$$944$$ 0 0
$$945$$ −8400.00 −0.289156
$$946$$ 0 0
$$947$$ 46150.3i 1.58361i 0.610771 + 0.791807i $$0.290859\pi$$
−0.610771 + 0.791807i $$0.709141\pi$$
$$948$$ 0 0
$$949$$ 40964.0i 1.40121i
$$950$$ 0 0
$$951$$ 26145.7 0.891517
$$952$$ 0 0
$$953$$ 342.000 0.0116248 0.00581242 0.999983i $$-0.498150\pi$$
0.00581242 + 0.999983i $$0.498150\pi$$
$$954$$ 0 0
$$955$$ 15115.7i 0.512180i
$$956$$ 0 0
$$957$$ − 21600.0i − 0.729602i
$$958$$ 0 0
$$959$$ −29333.3 −0.987718
$$960$$ 0 0
$$961$$ 86849.0 2.91528
$$962$$ 0 0
$$963$$ − 8139.70i − 0.272376i
$$964$$ 0 0
$$965$$ − 6490.00i − 0.216498i
$$966$$ 0 0
$$967$$ 51728.5 1.72025 0.860123 0.510087i $$-0.170387\pi$$
0.860123 + 0.510087i $$0.170387\pi$$
$$968$$ 0 0
$$969$$ 21760.0 0.721395
$$970$$ 0 0
$$971$$ 3099.03i 0.102423i 0.998688 + 0.0512115i $$0.0163083\pi$$
−0.998688 + 0.0512115i $$0.983692\pi$$
$$972$$ 0 0
$$973$$ 6240.00i 0.205596i
$$974$$ 0 0
$$975$$ −6008.33 −0.197354
$$976$$ 0 0
$$977$$ 26226.0 0.858796 0.429398 0.903115i $$-0.358726\pi$$
0.429398 + 0.903115i $$0.358726\pi$$
$$978$$ 0 0
$$979$$ 11257.7i 0.367516i
$$980$$ 0 0
$$981$$ − 11102.0i − 0.361325i
$$982$$ 0 0
$$983$$ −11049.0 −0.358503 −0.179251 0.983803i $$-0.557368\pi$$
−0.179251 + 0.983803i $$0.557368\pi$$
$$984$$ 0 0
$$985$$ −14230.0 −0.460310
$$986$$ 0 0
$$987$$ − 15937.9i − 0.513990i
$$988$$ 0 0
$$989$$ 44200.0i 1.42111i
$$990$$ 0 0
$$991$$ 45928.9 1.47223 0.736115 0.676856i $$-0.236658\pi$$
0.736115 + 0.676856i $$0.236658\pi$$
$$992$$ 0 0
$$993$$ 75600.0 2.41601
$$994$$ 0 0
$$995$$ − 17961.7i − 0.572287i
$$996$$ 0 0
$$997$$ 31026.0i 0.985560i 0.870154 + 0.492780i $$0.164019\pi$$
−0.870154 + 0.492780i $$0.835981\pi$$
$$998$$ 0 0
$$999$$ −18240.0 −0.577666
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 1280.4.d.u.641.1 4
4.3 odd 2 inner 1280.4.d.u.641.3 4
8.3 odd 2 inner 1280.4.d.u.641.2 4
8.5 even 2 inner 1280.4.d.u.641.4 4
16.3 odd 4 320.4.a.p.1.1 2
16.5 even 4 160.4.a.f.1.1 2
16.11 odd 4 160.4.a.f.1.2 yes 2
16.13 even 4 320.4.a.p.1.2 2
48.5 odd 4 1440.4.a.v.1.1 2
48.11 even 4 1440.4.a.v.1.2 2
80.19 odd 4 1600.4.a.ch.1.2 2
80.27 even 4 800.4.c.j.449.1 4
80.29 even 4 1600.4.a.ch.1.1 2
80.37 odd 4 800.4.c.j.449.4 4
80.43 even 4 800.4.c.j.449.3 4
80.53 odd 4 800.4.c.j.449.2 4
80.59 odd 4 800.4.a.p.1.1 2
80.69 even 4 800.4.a.p.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
160.4.a.f.1.1 2 16.5 even 4
160.4.a.f.1.2 yes 2 16.11 odd 4
320.4.a.p.1.1 2 16.3 odd 4
320.4.a.p.1.2 2 16.13 even 4
800.4.a.p.1.1 2 80.59 odd 4
800.4.a.p.1.2 2 80.69 even 4
800.4.c.j.449.1 4 80.27 even 4
800.4.c.j.449.2 4 80.53 odd 4
800.4.c.j.449.3 4 80.43 even 4
800.4.c.j.449.4 4 80.37 odd 4
1280.4.d.u.641.1 4 1.1 even 1 trivial
1280.4.d.u.641.2 4 8.3 odd 2 inner
1280.4.d.u.641.3 4 4.3 odd 2 inner
1280.4.d.u.641.4 4 8.5 even 2 inner
1440.4.a.v.1.1 2 48.5 odd 4
1440.4.a.v.1.2 2 48.11 even 4
1600.4.a.ch.1.1 2 80.29 even 4
1600.4.a.ch.1.2 2 80.19 odd 4