# Properties

 Label 507.4.b.b.337.1 Level $507$ Weight $4$ Character 507.337 Analytic conductor $29.914$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [507,4,Mod(337,507)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(507, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("507.337");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$507 = 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 507.b (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: $$29.9139683729$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 39) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 337.1 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 507.337 Dual form 507.4.b.b.337.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-3.00000 q^{3} +8.00000 q^{4} -12.0000i q^{5} -2.00000i q^{7} +9.00000 q^{9} +O(q^{10})$$ $$q-3.00000 q^{3} +8.00000 q^{4} -12.0000i q^{5} -2.00000i q^{7} +9.00000 q^{9} +36.0000i q^{11} -24.0000 q^{12} +36.0000i q^{15} +64.0000 q^{16} +78.0000 q^{17} +74.0000i q^{19} -96.0000i q^{20} +6.00000i q^{21} +96.0000 q^{23} -19.0000 q^{25} -27.0000 q^{27} -16.0000i q^{28} +18.0000 q^{29} -214.000i q^{31} -108.000i q^{33} -24.0000 q^{35} +72.0000 q^{36} +286.000i q^{37} -384.000i q^{41} -524.000 q^{43} +288.000i q^{44} -108.000i q^{45} -300.000i q^{47} -192.000 q^{48} +339.000 q^{49} -234.000 q^{51} +558.000 q^{53} +432.000 q^{55} -222.000i q^{57} -576.000i q^{59} +288.000i q^{60} +74.0000 q^{61} -18.0000i q^{63} +512.000 q^{64} +38.0000i q^{67} +624.000 q^{68} -288.000 q^{69} -456.000i q^{71} +682.000i q^{73} +57.0000 q^{75} +592.000i q^{76} +72.0000 q^{77} +704.000 q^{79} -768.000i q^{80} +81.0000 q^{81} -888.000i q^{83} +48.0000i q^{84} -936.000i q^{85} -54.0000 q^{87} +1020.00i q^{89} +768.000 q^{92} +642.000i q^{93} +888.000 q^{95} +110.000i q^{97} +324.000i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 6 q^{3} + 16 q^{4} + 18 q^{9}+O(q^{10})$$ 2 * q - 6 * q^3 + 16 * q^4 + 18 * q^9 $$2 q - 6 q^{3} + 16 q^{4} + 18 q^{9} - 48 q^{12} + 128 q^{16} + 156 q^{17} + 192 q^{23} - 38 q^{25} - 54 q^{27} + 36 q^{29} - 48 q^{35} + 144 q^{36} - 1048 q^{43} - 384 q^{48} + 678 q^{49} - 468 q^{51} + 1116 q^{53} + 864 q^{55} + 148 q^{61} + 1024 q^{64} + 1248 q^{68} - 576 q^{69} + 114 q^{75} + 144 q^{77} + 1408 q^{79} + 162 q^{81} - 108 q^{87} + 1536 q^{92} + 1776 q^{95}+O(q^{100})$$ 2 * q - 6 * q^3 + 16 * q^4 + 18 * q^9 - 48 * q^12 + 128 * q^16 + 156 * q^17 + 192 * q^23 - 38 * q^25 - 54 * q^27 + 36 * q^29 - 48 * q^35 + 144 * q^36 - 1048 * q^43 - 384 * q^48 + 678 * q^49 - 468 * q^51 + 1116 * q^53 + 864 * q^55 + 148 * q^61 + 1024 * q^64 + 1248 * q^68 - 576 * q^69 + 114 * q^75 + 144 * q^77 + 1408 * q^79 + 162 * q^81 - 108 * q^87 + 1536 * q^92 + 1776 * q^95

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/507\mathbb{Z}\right)^\times$$.

 $$n$$ $$170$$ $$340$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 1.00000 $$0$$
−1.00000 $$\pi$$
$$3$$ −3.00000 −0.577350
$$4$$ 8.00000 1.00000
$$5$$ − 12.0000i − 1.07331i −0.843801 0.536656i $$-0.819687\pi$$
0.843801 0.536656i $$-0.180313\pi$$
$$6$$ 0 0
$$7$$ − 2.00000i − 0.107990i −0.998541 0.0539949i $$-0.982805\pi$$
0.998541 0.0539949i $$-0.0171955\pi$$
$$8$$ 0 0
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ 36.0000i 0.986764i 0.869813 + 0.493382i $$0.164240\pi$$
−0.869813 + 0.493382i $$0.835760\pi$$
$$12$$ −24.0000 −0.577350
$$13$$ 0 0
$$14$$ 0 0
$$15$$ 36.0000i 0.619677i
$$16$$ 64.0000 1.00000
$$17$$ 78.0000 1.11281 0.556405 0.830911i $$-0.312180\pi$$
0.556405 + 0.830911i $$0.312180\pi$$
$$18$$ 0 0
$$19$$ 74.0000i 0.893514i 0.894655 + 0.446757i $$0.147421\pi$$
−0.894655 + 0.446757i $$0.852579\pi$$
$$20$$ − 96.0000i − 1.07331i
$$21$$ 6.00000i 0.0623480i
$$22$$ 0 0
$$23$$ 96.0000 0.870321 0.435161 0.900353i $$-0.356692\pi$$
0.435161 + 0.900353i $$0.356692\pi$$
$$24$$ 0 0
$$25$$ −19.0000 −0.152000
$$26$$ 0 0
$$27$$ −27.0000 −0.192450
$$28$$ − 16.0000i − 0.107990i
$$29$$ 18.0000 0.115259 0.0576296 0.998338i $$-0.481646\pi$$
0.0576296 + 0.998338i $$0.481646\pi$$
$$30$$ 0 0
$$31$$ − 214.000i − 1.23986i −0.784659 0.619928i $$-0.787162\pi$$
0.784659 0.619928i $$-0.212838\pi$$
$$32$$ 0 0
$$33$$ − 108.000i − 0.569709i
$$34$$ 0 0
$$35$$ −24.0000 −0.115907
$$36$$ 72.0000 0.333333
$$37$$ 286.000i 1.27076i 0.772200 + 0.635380i $$0.219156\pi$$
−0.772200 + 0.635380i $$0.780844\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ − 384.000i − 1.46270i −0.682002 0.731350i $$-0.738890\pi$$
0.682002 0.731350i $$-0.261110\pi$$
$$42$$ 0 0
$$43$$ −524.000 −1.85835 −0.929177 0.369634i $$-0.879483\pi$$
−0.929177 + 0.369634i $$0.879483\pi$$
$$44$$ 288.000i 0.986764i
$$45$$ − 108.000i − 0.357771i
$$46$$ 0 0
$$47$$ − 300.000i − 0.931053i −0.885034 0.465527i $$-0.845865\pi$$
0.885034 0.465527i $$-0.154135\pi$$
$$48$$ −192.000 −0.577350
$$49$$ 339.000 0.988338
$$50$$ 0 0
$$51$$ −234.000 −0.642481
$$52$$ 0 0
$$53$$ 558.000 1.44617 0.723087 0.690757i $$-0.242723\pi$$
0.723087 + 0.690757i $$0.242723\pi$$
$$54$$ 0 0
$$55$$ 432.000 1.05911
$$56$$ 0 0
$$57$$ − 222.000i − 0.515870i
$$58$$ 0 0
$$59$$ − 576.000i − 1.27100i −0.772102 0.635498i $$-0.780795\pi$$
0.772102 0.635498i $$-0.219205\pi$$
$$60$$ 288.000i 0.619677i
$$61$$ 74.0000 0.155323 0.0776617 0.996980i $$-0.475255\pi$$
0.0776617 + 0.996980i $$0.475255\pi$$
$$62$$ 0 0
$$63$$ − 18.0000i − 0.0359966i
$$64$$ 512.000 1.00000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 38.0000i 0.0692901i 0.999400 + 0.0346451i $$0.0110301\pi$$
−0.999400 + 0.0346451i $$0.988970\pi$$
$$68$$ 624.000 1.11281
$$69$$ −288.000 −0.502480
$$70$$ 0 0
$$71$$ − 456.000i − 0.762215i −0.924531 0.381107i $$-0.875543\pi$$
0.924531 0.381107i $$-0.124457\pi$$
$$72$$ 0 0
$$73$$ 682.000i 1.09345i 0.837311 + 0.546726i $$0.184126\pi$$
−0.837311 + 0.546726i $$0.815874\pi$$
$$74$$ 0 0
$$75$$ 57.0000 0.0877572
$$76$$ 592.000i 0.893514i
$$77$$ 72.0000 0.106561
$$78$$ 0 0
$$79$$ 704.000 1.00261 0.501305 0.865271i $$-0.332853\pi$$
0.501305 + 0.865271i $$0.332853\pi$$
$$80$$ − 768.000i − 1.07331i
$$81$$ 81.0000 0.111111
$$82$$ 0 0
$$83$$ − 888.000i − 1.17435i −0.809462 0.587173i $$-0.800241\pi$$
0.809462 0.587173i $$-0.199759\pi$$
$$84$$ 48.0000i 0.0623480i
$$85$$ − 936.000i − 1.19439i
$$86$$ 0 0
$$87$$ −54.0000 −0.0665449
$$88$$ 0 0
$$89$$ 1020.00i 1.21483i 0.794385 + 0.607415i $$0.207793\pi$$
−0.794385 + 0.607415i $$0.792207\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 768.000 0.870321
$$93$$ 642.000i 0.715831i
$$94$$ 0 0
$$95$$ 888.000 0.959020
$$96$$ 0 0
$$97$$ 110.000i 0.115142i 0.998341 + 0.0575712i $$0.0183356\pi$$
−0.998341 + 0.0575712i $$0.981664\pi$$
$$98$$ 0 0
$$99$$ 324.000i 0.328921i
$$100$$ −152.000 −0.152000
$$101$$ 990.000 0.975333 0.487667 0.873030i $$-0.337848\pi$$
0.487667 + 0.873030i $$0.337848\pi$$
$$102$$ 0 0
$$103$$ −1208.00 −1.15561 −0.577805 0.816175i $$-0.696090\pi$$
−0.577805 + 0.816175i $$0.696090\pi$$
$$104$$ 0 0
$$105$$ 72.0000 0.0669189
$$106$$ 0 0
$$107$$ 996.000 0.899878 0.449939 0.893059i $$-0.351446\pi$$
0.449939 + 0.893059i $$0.351446\pi$$
$$108$$ −216.000 −0.192450
$$109$$ − 1402.00i − 1.23199i −0.787749 0.615997i $$-0.788754\pi$$
0.787749 0.615997i $$-0.211246\pi$$
$$110$$ 0 0
$$111$$ − 858.000i − 0.733673i
$$112$$ − 128.000i − 0.107990i
$$113$$ 1926.00 1.60339 0.801694 0.597735i $$-0.203932\pi$$
0.801694 + 0.597735i $$0.203932\pi$$
$$114$$ 0 0
$$115$$ − 1152.00i − 0.934127i
$$116$$ 144.000 0.115259
$$117$$ 0 0
$$118$$ 0 0
$$119$$ − 156.000i − 0.120172i
$$120$$ 0 0
$$121$$ 35.0000 0.0262960
$$122$$ 0 0
$$123$$ 1152.00i 0.844491i
$$124$$ − 1712.00i − 1.23986i
$$125$$ − 1272.00i − 0.910169i
$$126$$ 0 0
$$127$$ 988.000 0.690321 0.345161 0.938544i $$-0.387824\pi$$
0.345161 + 0.938544i $$0.387824\pi$$
$$128$$ 0 0
$$129$$ 1572.00 1.07292
$$130$$ 0 0
$$131$$ −2100.00 −1.40059 −0.700297 0.713851i $$-0.746949\pi$$
−0.700297 + 0.713851i $$0.746949\pi$$
$$132$$ − 864.000i − 0.569709i
$$133$$ 148.000 0.0964904
$$134$$ 0 0
$$135$$ 324.000i 0.206559i
$$136$$ 0 0
$$137$$ 2496.00i 1.55655i 0.627922 + 0.778276i $$0.283906\pi$$
−0.627922 + 0.778276i $$0.716094\pi$$
$$138$$ 0 0
$$139$$ −2464.00 −1.50355 −0.751776 0.659418i $$-0.770803\pi$$
−0.751776 + 0.659418i $$0.770803\pi$$
$$140$$ −192.000 −0.115907
$$141$$ 900.000i 0.537544i
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 576.000 0.333333
$$145$$ − 216.000i − 0.123709i
$$146$$ 0 0
$$147$$ −1017.00 −0.570617
$$148$$ 2288.00i 1.27076i
$$149$$ 216.000i 0.118761i 0.998235 + 0.0593806i $$0.0189125\pi$$
−0.998235 + 0.0593806i $$0.981087\pi$$
$$150$$ 0 0
$$151$$ 898.000i 0.483962i 0.970281 + 0.241981i $$0.0777971\pi$$
−0.970281 + 0.241981i $$0.922203\pi$$
$$152$$ 0 0
$$153$$ 702.000 0.370937
$$154$$ 0 0
$$155$$ −2568.00 −1.33075
$$156$$ 0 0
$$157$$ −1510.00 −0.767587 −0.383793 0.923419i $$-0.625383\pi$$
−0.383793 + 0.923419i $$0.625383\pi$$
$$158$$ 0 0
$$159$$ −1674.00 −0.834949
$$160$$ 0 0
$$161$$ − 192.000i − 0.0939858i
$$162$$ 0 0
$$163$$ 394.000i 0.189328i 0.995509 + 0.0946640i $$0.0301777\pi$$
−0.995509 + 0.0946640i $$0.969822\pi$$
$$164$$ − 3072.00i − 1.46270i
$$165$$ −1296.00 −0.611476
$$166$$ 0 0
$$167$$ − 84.0000i − 0.0389228i −0.999811 0.0194614i $$-0.993805\pi$$
0.999811 0.0194614i $$-0.00619515\pi$$
$$168$$ 0 0
$$169$$ 0 0
$$170$$ 0 0
$$171$$ 666.000i 0.297838i
$$172$$ −4192.00 −1.85835
$$173$$ −1194.00 −0.524729 −0.262365 0.964969i $$-0.584502\pi$$
−0.262365 + 0.964969i $$0.584502\pi$$
$$174$$ 0 0
$$175$$ 38.0000i 0.0164145i
$$176$$ 2304.00i 0.986764i
$$177$$ 1728.00i 0.733810i
$$178$$ 0 0
$$179$$ −3156.00 −1.31782 −0.658912 0.752220i $$-0.728983\pi$$
−0.658912 + 0.752220i $$0.728983\pi$$
$$180$$ − 864.000i − 0.357771i
$$181$$ 1078.00 0.442691 0.221346 0.975195i $$-0.428955\pi$$
0.221346 + 0.975195i $$0.428955\pi$$
$$182$$ 0 0
$$183$$ −222.000 −0.0896760
$$184$$ 0 0
$$185$$ 3432.00 1.36392
$$186$$ 0 0
$$187$$ 2808.00i 1.09808i
$$188$$ − 2400.00i − 0.931053i
$$189$$ 54.0000i 0.0207827i
$$190$$ 0 0
$$191$$ 3192.00 1.20924 0.604620 0.796514i $$-0.293325\pi$$
0.604620 + 0.796514i $$0.293325\pi$$
$$192$$ −1536.00 −0.577350
$$193$$ − 722.000i − 0.269278i −0.990895 0.134639i $$-0.957012\pi$$
0.990895 0.134639i $$-0.0429875\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 2712.00 0.988338
$$197$$ 2796.00i 1.01120i 0.862767 + 0.505601i $$0.168729\pi$$
−0.862767 + 0.505601i $$0.831271\pi$$
$$198$$ 0 0
$$199$$ 340.000 0.121115 0.0605577 0.998165i $$-0.480712\pi$$
0.0605577 + 0.998165i $$0.480712\pi$$
$$200$$ 0 0
$$201$$ − 114.000i − 0.0400047i
$$202$$ 0 0
$$203$$ − 36.0000i − 0.0124468i
$$204$$ −1872.00 −0.642481
$$205$$ −4608.00 −1.56994
$$206$$ 0 0
$$207$$ 864.000 0.290107
$$208$$ 0 0
$$209$$ −2664.00 −0.881688
$$210$$ 0 0
$$211$$ −1924.00 −0.627742 −0.313871 0.949466i $$-0.601626\pi$$
−0.313871 + 0.949466i $$0.601626\pi$$
$$212$$ 4464.00 1.44617
$$213$$ 1368.00i 0.440065i
$$214$$ 0 0
$$215$$ 6288.00i 1.99460i
$$216$$ 0 0
$$217$$ −428.000 −0.133892
$$218$$ 0 0
$$219$$ − 2046.00i − 0.631305i
$$220$$ 3456.00 1.05911
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 5042.00i 1.51407i 0.653375 + 0.757034i $$0.273352\pi$$
−0.653375 + 0.757034i $$0.726648\pi$$
$$224$$ 0 0
$$225$$ −171.000 −0.0506667
$$226$$ 0 0
$$227$$ − 2676.00i − 0.782433i −0.920299 0.391217i $$-0.872054\pi$$
0.920299 0.391217i $$-0.127946\pi$$
$$228$$ − 1776.00i − 0.515870i
$$229$$ 2410.00i 0.695447i 0.937597 + 0.347723i $$0.113045\pi$$
−0.937597 + 0.347723i $$0.886955\pi$$
$$230$$ 0 0
$$231$$ −216.000 −0.0615228
$$232$$ 0 0
$$233$$ −3726.00 −1.04763 −0.523816 0.851831i $$-0.675492\pi$$
−0.523816 + 0.851831i $$0.675492\pi$$
$$234$$ 0 0
$$235$$ −3600.00 −0.999311
$$236$$ − 4608.00i − 1.27100i
$$237$$ −2112.00 −0.578857
$$238$$ 0 0
$$239$$ 1248.00i 0.337767i 0.985636 + 0.168884i $$0.0540162\pi$$
−0.985636 + 0.168884i $$0.945984\pi$$
$$240$$ 2304.00i 0.619677i
$$241$$ 4210.00i 1.12527i 0.826706 + 0.562635i $$0.190212\pi$$
−0.826706 + 0.562635i $$0.809788\pi$$
$$242$$ 0 0
$$243$$ −243.000 −0.0641500
$$244$$ 592.000 0.155323
$$245$$ − 4068.00i − 1.06080i
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 2664.00i 0.678009i
$$250$$ 0 0
$$251$$ 7692.00 1.93432 0.967161 0.254165i $$-0.0818007\pi$$
0.967161 + 0.254165i $$0.0818007\pi$$
$$252$$ − 144.000i − 0.0359966i
$$253$$ 3456.00i 0.858802i
$$254$$ 0 0
$$255$$ 2808.00i 0.689583i
$$256$$ 4096.00 1.00000
$$257$$ −1326.00 −0.321843 −0.160921 0.986967i $$-0.551447\pi$$
−0.160921 + 0.986967i $$0.551447\pi$$
$$258$$ 0 0
$$259$$ 572.000 0.137229
$$260$$ 0 0
$$261$$ 162.000 0.0384197
$$262$$ 0 0
$$263$$ −6048.00 −1.41801 −0.709003 0.705205i $$-0.750855\pi$$
−0.709003 + 0.705205i $$0.750855\pi$$
$$264$$ 0 0
$$265$$ − 6696.00i − 1.55220i
$$266$$ 0 0
$$267$$ − 3060.00i − 0.701382i
$$268$$ 304.000i 0.0692901i
$$269$$ 6474.00 1.46739 0.733693 0.679481i $$-0.237795\pi$$
0.733693 + 0.679481i $$0.237795\pi$$
$$270$$ 0 0
$$271$$ − 5978.00i − 1.33999i −0.742365 0.669996i $$-0.766296\pi$$
0.742365 0.669996i $$-0.233704\pi$$
$$272$$ 4992.00 1.11281
$$273$$ 0 0
$$274$$ 0 0
$$275$$ − 684.000i − 0.149988i
$$276$$ −2304.00 −0.502480
$$277$$ −8750.00 −1.89797 −0.948983 0.315327i $$-0.897886\pi$$
−0.948983 + 0.315327i $$0.897886\pi$$
$$278$$ 0 0
$$279$$ − 1926.00i − 0.413285i
$$280$$ 0 0
$$281$$ − 8976.00i − 1.90556i −0.303656 0.952782i $$-0.598207\pi$$
0.303656 0.952782i $$-0.401793\pi$$
$$282$$ 0 0
$$283$$ 592.000 0.124349 0.0621745 0.998065i $$-0.480196\pi$$
0.0621745 + 0.998065i $$0.480196\pi$$
$$284$$ − 3648.00i − 0.762215i
$$285$$ −2664.00 −0.553690
$$286$$ 0 0
$$287$$ −768.000 −0.157957
$$288$$ 0 0
$$289$$ 1171.00 0.238347
$$290$$ 0 0
$$291$$ − 330.000i − 0.0664775i
$$292$$ 5456.00i 1.09345i
$$293$$ 4608.00i 0.918779i 0.888235 + 0.459389i $$0.151932\pi$$
−0.888235 + 0.459389i $$0.848068\pi$$
$$294$$ 0 0
$$295$$ −6912.00 −1.36418
$$296$$ 0 0
$$297$$ − 972.000i − 0.189903i
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 456.000 0.0877572
$$301$$ 1048.00i 0.200683i
$$302$$ 0 0
$$303$$ −2970.00 −0.563109
$$304$$ 4736.00i 0.893514i
$$305$$ − 888.000i − 0.166711i
$$306$$ 0 0
$$307$$ 3166.00i 0.588577i 0.955717 + 0.294289i $$0.0950827\pi$$
−0.955717 + 0.294289i $$0.904917\pi$$
$$308$$ 576.000 0.106561
$$309$$ 3624.00 0.667191
$$310$$ 0 0
$$311$$ −2472.00 −0.450721 −0.225361 0.974275i $$-0.572356\pi$$
−0.225361 + 0.974275i $$0.572356\pi$$
$$312$$ 0 0
$$313$$ −3094.00 −0.558732 −0.279366 0.960185i $$-0.590124\pi$$
−0.279366 + 0.960185i $$0.590124\pi$$
$$314$$ 0 0
$$315$$ −216.000 −0.0386356
$$316$$ 5632.00 1.00261
$$317$$ 2316.00i 0.410345i 0.978726 + 0.205173i $$0.0657756\pi$$
−0.978726 + 0.205173i $$0.934224\pi$$
$$318$$ 0 0
$$319$$ 648.000i 0.113734i
$$320$$ − 6144.00i − 1.07331i
$$321$$ −2988.00 −0.519545
$$322$$ 0 0
$$323$$ 5772.00i 0.994312i
$$324$$ 648.000 0.111111
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 4206.00i 0.711292i
$$328$$ 0 0
$$329$$ −600.000 −0.100544
$$330$$ 0 0
$$331$$ − 4426.00i − 0.734970i −0.930030 0.367485i $$-0.880219\pi$$
0.930030 0.367485i $$-0.119781\pi$$
$$332$$ − 7104.00i − 1.17435i
$$333$$ 2574.00i 0.423587i
$$334$$ 0 0
$$335$$ 456.000 0.0743700
$$336$$ 384.000i 0.0623480i
$$337$$ −866.000 −0.139982 −0.0699911 0.997548i $$-0.522297\pi$$
−0.0699911 + 0.997548i $$0.522297\pi$$
$$338$$ 0 0
$$339$$ −5778.00 −0.925716
$$340$$ − 7488.00i − 1.19439i
$$341$$ 7704.00 1.22345
$$342$$ 0 0
$$343$$ − 1364.00i − 0.214720i
$$344$$ 0 0
$$345$$ 3456.00i 0.539318i
$$346$$ 0 0
$$347$$ −2556.00 −0.395427 −0.197714 0.980260i $$-0.563352\pi$$
−0.197714 + 0.980260i $$0.563352\pi$$
$$348$$ −432.000 −0.0665449
$$349$$ 11014.0i 1.68930i 0.535318 + 0.844650i $$0.320192\pi$$
−0.535318 + 0.844650i $$0.679808\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ − 9720.00i − 1.46556i −0.680465 0.732781i $$-0.738222\pi$$
0.680465 0.732781i $$-0.261778\pi$$
$$354$$ 0 0
$$355$$ −5472.00 −0.818095
$$356$$ 8160.00i 1.21483i
$$357$$ 468.000i 0.0693815i
$$358$$ 0 0
$$359$$ 2988.00i 0.439277i 0.975581 + 0.219639i $$0.0704879\pi$$
−0.975581 + 0.219639i $$0.929512\pi$$
$$360$$ 0 0
$$361$$ 1383.00 0.201633
$$362$$ 0 0
$$363$$ −105.000 −0.0151820
$$364$$ 0 0
$$365$$ 8184.00 1.17362
$$366$$ 0 0
$$367$$ −2068.00 −0.294138 −0.147069 0.989126i $$-0.546984\pi$$
−0.147069 + 0.989126i $$0.546984\pi$$
$$368$$ 6144.00 0.870321
$$369$$ − 3456.00i − 0.487567i
$$370$$ 0 0
$$371$$ − 1116.00i − 0.156172i
$$372$$ 5136.00i 0.715831i
$$373$$ 902.000 0.125211 0.0626056 0.998038i $$-0.480059\pi$$
0.0626056 + 0.998038i $$0.480059\pi$$
$$374$$ 0 0
$$375$$ 3816.00i 0.525486i
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 12818.0i 1.73725i 0.495473 + 0.868623i $$0.334995\pi$$
−0.495473 + 0.868623i $$0.665005\pi$$
$$380$$ 7104.00 0.959020
$$381$$ −2964.00 −0.398557
$$382$$ 0 0
$$383$$ − 1332.00i − 0.177708i −0.996045 0.0888538i $$-0.971680\pi$$
0.996045 0.0888538i $$-0.0283204\pi$$
$$384$$ 0 0
$$385$$ − 864.000i − 0.114373i
$$386$$ 0 0
$$387$$ −4716.00 −0.619452
$$388$$ 880.000i 0.115142i
$$389$$ −3054.00 −0.398056 −0.199028 0.979994i $$-0.563779\pi$$
−0.199028 + 0.979994i $$0.563779\pi$$
$$390$$ 0 0
$$391$$ 7488.00 0.968502
$$392$$ 0 0
$$393$$ 6300.00 0.808633
$$394$$ 0 0
$$395$$ − 8448.00i − 1.07611i
$$396$$ 2592.00i 0.328921i
$$397$$ − 11162.0i − 1.41110i −0.708663 0.705548i $$-0.750701\pi$$
0.708663 0.705548i $$-0.249299\pi$$
$$398$$ 0 0
$$399$$ −444.000 −0.0557088
$$400$$ −1216.00 −0.152000
$$401$$ 14820.0i 1.84557i 0.385310 + 0.922787i $$0.374095\pi$$
−0.385310 + 0.922787i $$0.625905\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 7920.00 0.975333
$$405$$ − 972.000i − 0.119257i
$$406$$ 0 0
$$407$$ −10296.0 −1.25394
$$408$$ 0 0
$$409$$ − 9682.00i − 1.17052i −0.810844 0.585262i $$-0.800992\pi$$
0.810844 0.585262i $$-0.199008\pi$$
$$410$$ 0 0
$$411$$ − 7488.00i − 0.898676i
$$412$$ −9664.00 −1.15561
$$413$$ −1152.00 −0.137255
$$414$$ 0 0
$$415$$ −10656.0 −1.26044
$$416$$ 0 0
$$417$$ 7392.00 0.868076
$$418$$ 0 0
$$419$$ −348.000 −0.0405750 −0.0202875 0.999794i $$-0.506458\pi$$
−0.0202875 + 0.999794i $$0.506458\pi$$
$$420$$ 576.000 0.0669189
$$421$$ 2486.00i 0.287792i 0.989593 + 0.143896i $$0.0459630\pi$$
−0.989593 + 0.143896i $$0.954037\pi$$
$$422$$ 0 0
$$423$$ − 2700.00i − 0.310351i
$$424$$ 0 0
$$425$$ −1482.00 −0.169147
$$426$$ 0 0
$$427$$ − 148.000i − 0.0167734i
$$428$$ 7968.00 0.899878
$$429$$ 0 0
$$430$$ 0 0
$$431$$ − 1812.00i − 0.202508i −0.994861 0.101254i $$-0.967715\pi$$
0.994861 0.101254i $$-0.0322855\pi$$
$$432$$ −1728.00 −0.192450
$$433$$ 6226.00 0.690999 0.345499 0.938419i $$-0.387710\pi$$
0.345499 + 0.938419i $$0.387710\pi$$
$$434$$ 0 0
$$435$$ 648.000i 0.0714235i
$$436$$ − 11216.0i − 1.23199i
$$437$$ 7104.00i 0.777644i
$$438$$ 0 0
$$439$$ 12544.0 1.36376 0.681882 0.731462i $$-0.261162\pi$$
0.681882 + 0.731462i $$0.261162\pi$$
$$440$$ 0 0
$$441$$ 3051.00 0.329446
$$442$$ 0 0
$$443$$ −8556.00 −0.917625 −0.458812 0.888533i $$-0.651725\pi$$
−0.458812 + 0.888533i $$0.651725\pi$$
$$444$$ − 6864.00i − 0.733673i
$$445$$ 12240.0 1.30389
$$446$$ 0 0
$$447$$ − 648.000i − 0.0685668i
$$448$$ − 1024.00i − 0.107990i
$$449$$ − 4116.00i − 0.432619i −0.976325 0.216310i $$-0.930598\pi$$
0.976325 0.216310i $$-0.0694021\pi$$
$$450$$ 0 0
$$451$$ 13824.0 1.44334
$$452$$ 15408.0 1.60339
$$453$$ − 2694.00i − 0.279415i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 6514.00i − 0.666766i −0.942791 0.333383i $$-0.891810\pi$$
0.942791 0.333383i $$-0.108190\pi$$
$$458$$ 0 0
$$459$$ −2106.00 −0.214160
$$460$$ − 9216.00i − 0.934127i
$$461$$ 10500.0i 1.06081i 0.847744 + 0.530405i $$0.177960\pi$$
−0.847744 + 0.530405i $$0.822040\pi$$
$$462$$ 0 0
$$463$$ 5542.00i 0.556282i 0.960540 + 0.278141i $$0.0897183\pi$$
−0.960540 + 0.278141i $$0.910282\pi$$
$$464$$ 1152.00 0.115259
$$465$$ 7704.00 0.768311
$$466$$ 0 0
$$467$$ 5220.00 0.517244 0.258622 0.965979i $$-0.416732\pi$$
0.258622 + 0.965979i $$0.416732\pi$$
$$468$$ 0 0
$$469$$ 76.0000 0.00748263
$$470$$ 0 0
$$471$$ 4530.00 0.443166
$$472$$ 0 0
$$473$$ − 18864.0i − 1.83376i
$$474$$ 0 0
$$475$$ − 1406.00i − 0.135814i
$$476$$ − 1248.00i − 0.120172i
$$477$$ 5022.00 0.482058
$$478$$ 0 0
$$479$$ − 11592.0i − 1.10575i −0.833266 0.552873i $$-0.813532\pi$$
0.833266 0.552873i $$-0.186468\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ 576.000i 0.0542627i
$$484$$ 280.000 0.0262960
$$485$$ 1320.00 0.123584
$$486$$ 0 0
$$487$$ 12170.0i 1.13239i 0.824270 + 0.566196i $$0.191586\pi$$
−0.824270 + 0.566196i $$0.808414\pi$$
$$488$$ 0 0
$$489$$ − 1182.00i − 0.109309i
$$490$$ 0 0
$$491$$ −1812.00 −0.166547 −0.0832733 0.996527i $$-0.526537\pi$$
−0.0832733 + 0.996527i $$0.526537\pi$$
$$492$$ 9216.00i 0.844491i
$$493$$ 1404.00 0.128262
$$494$$ 0 0
$$495$$ 3888.00 0.353036
$$496$$ − 13696.0i − 1.23986i
$$497$$ −912.000 −0.0823115
$$498$$ 0 0
$$499$$ − 1330.00i − 0.119317i −0.998219 0.0596583i $$-0.980999\pi$$
0.998219 0.0596583i $$-0.0190011\pi$$
$$500$$ − 10176.0i − 0.910169i
$$501$$ 252.000i 0.0224721i
$$502$$ 0 0
$$503$$ −2688.00 −0.238274 −0.119137 0.992878i $$-0.538013\pi$$
−0.119137 + 0.992878i $$0.538013\pi$$
$$504$$ 0 0
$$505$$ − 11880.0i − 1.04684i
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 7904.00 0.690321
$$509$$ 5124.00i 0.446203i 0.974795 + 0.223101i $$0.0716181\pi$$
−0.974795 + 0.223101i $$0.928382\pi$$
$$510$$ 0 0
$$511$$ 1364.00 0.118082
$$512$$ 0 0
$$513$$ − 1998.00i − 0.171957i
$$514$$ 0 0
$$515$$ 14496.0i 1.24033i
$$516$$ 12576.0 1.07292
$$517$$ 10800.0 0.918730
$$518$$ 0 0
$$519$$ 3582.00 0.302953
$$520$$ 0 0
$$521$$ −882.000 −0.0741672 −0.0370836 0.999312i $$-0.511807\pi$$
−0.0370836 + 0.999312i $$0.511807\pi$$
$$522$$ 0 0
$$523$$ −2320.00 −0.193970 −0.0969852 0.995286i $$-0.530920\pi$$
−0.0969852 + 0.995286i $$0.530920\pi$$
$$524$$ −16800.0 −1.40059
$$525$$ − 114.000i − 0.00947689i
$$526$$ 0 0
$$527$$ − 16692.0i − 1.37972i
$$528$$ − 6912.00i − 0.569709i
$$529$$ −2951.00 −0.242541
$$530$$ 0 0
$$531$$ − 5184.00i − 0.423666i
$$532$$ 1184.00 0.0964904
$$533$$ 0 0
$$534$$ 0 0
$$535$$ − 11952.0i − 0.965851i
$$536$$ 0 0
$$537$$ 9468.00 0.760846
$$538$$ 0 0
$$539$$ 12204.0i 0.975257i
$$540$$ 2592.00i 0.206559i
$$541$$ − 21422.0i − 1.70241i −0.524833 0.851205i $$-0.675872\pi$$
0.524833 0.851205i $$-0.324128\pi$$
$$542$$ 0 0
$$543$$ −3234.00 −0.255588
$$544$$ 0 0
$$545$$ −16824.0 −1.32231
$$546$$ 0 0
$$547$$ 7040.00 0.550290 0.275145 0.961403i $$-0.411274\pi$$
0.275145 + 0.961403i $$0.411274\pi$$
$$548$$ 19968.0i 1.55655i
$$549$$ 666.000 0.0517745
$$550$$ 0 0
$$551$$ 1332.00i 0.102986i
$$552$$ 0 0
$$553$$ − 1408.00i − 0.108272i
$$554$$ 0 0
$$555$$ −10296.0 −0.787461
$$556$$ −19712.0 −1.50355
$$557$$ 8400.00i 0.638994i 0.947587 + 0.319497i $$0.103514\pi$$
−0.947587 + 0.319497i $$0.896486\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ −1536.00 −0.115907
$$561$$ − 8424.00i − 0.633978i
$$562$$ 0 0
$$563$$ −19044.0 −1.42559 −0.712797 0.701371i $$-0.752572\pi$$
−0.712797 + 0.701371i $$0.752572\pi$$
$$564$$ 7200.00i 0.537544i
$$565$$ − 23112.0i − 1.72094i
$$566$$ 0 0
$$567$$ − 162.000i − 0.0119989i
$$568$$ 0 0
$$569$$ 4698.00 0.346134 0.173067 0.984910i $$-0.444632\pi$$
0.173067 + 0.984910i $$0.444632\pi$$
$$570$$ 0 0
$$571$$ 8728.00 0.639677 0.319838 0.947472i $$-0.396371\pi$$
0.319838 + 0.947472i $$0.396371\pi$$
$$572$$ 0 0
$$573$$ −9576.00 −0.698156
$$574$$ 0 0
$$575$$ −1824.00 −0.132289
$$576$$ 4608.00 0.333333
$$577$$ 2018.00i 0.145599i 0.997347 + 0.0727993i $$0.0231933\pi$$
−0.997347 + 0.0727993i $$0.976807\pi$$
$$578$$ 0 0
$$579$$ 2166.00i 0.155468i
$$580$$ − 1728.00i − 0.123709i
$$581$$ −1776.00 −0.126817
$$582$$ 0 0
$$583$$ 20088.0i 1.42703i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 11376.0i 0.799894i 0.916538 + 0.399947i $$0.130971\pi$$
−0.916538 + 0.399947i $$0.869029\pi$$
$$588$$ −8136.00 −0.570617
$$589$$ 15836.0 1.10783
$$590$$ 0 0
$$591$$ − 8388.00i − 0.583818i
$$592$$ 18304.0i 1.27076i
$$593$$ 25596.0i 1.77252i 0.463192 + 0.886258i $$0.346704\pi$$
−0.463192 + 0.886258i $$0.653296\pi$$
$$594$$ 0 0
$$595$$ −1872.00 −0.128982
$$596$$ 1728.00i 0.118761i
$$597$$ −1020.00 −0.0699260
$$598$$ 0 0
$$599$$ 3480.00 0.237377 0.118689 0.992932i $$-0.462131\pi$$
0.118689 + 0.992932i $$0.462131\pi$$
$$600$$ 0 0
$$601$$ 10010.0 0.679395 0.339698 0.940535i $$-0.389675\pi$$
0.339698 + 0.940535i $$0.389675\pi$$
$$602$$ 0 0
$$603$$ 342.000i 0.0230967i
$$604$$ 7184.00i 0.483962i
$$605$$ − 420.000i − 0.0282238i
$$606$$ 0 0
$$607$$ 3764.00 0.251690 0.125845 0.992050i $$-0.459836\pi$$
0.125845 + 0.992050i $$0.459836\pi$$
$$608$$ 0 0
$$609$$ 108.000i 0.00718618i
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 5616.00 0.370937
$$613$$ 13610.0i 0.896742i 0.893848 + 0.448371i $$0.147996\pi$$
−0.893848 + 0.448371i $$0.852004\pi$$
$$614$$ 0 0
$$615$$ 13824.0 0.906402
$$616$$ 0 0
$$617$$ 6408.00i 0.418114i 0.977903 + 0.209057i $$0.0670394\pi$$
−0.977903 + 0.209057i $$0.932961\pi$$
$$618$$ 0 0
$$619$$ 6694.00i 0.434660i 0.976098 + 0.217330i $$0.0697348\pi$$
−0.976098 + 0.217330i $$0.930265\pi$$
$$620$$ −20544.0 −1.33075
$$621$$ −2592.00 −0.167493
$$622$$ 0 0
$$623$$ 2040.00 0.131189
$$624$$ 0 0
$$625$$ −17639.0 −1.12890
$$626$$ 0 0
$$627$$ 7992.00 0.509043
$$628$$ −12080.0 −0.767587
$$629$$ 22308.0i 1.41411i
$$630$$ 0 0
$$631$$ 27250.0i 1.71918i 0.510981 + 0.859592i $$0.329282\pi$$
−0.510981 + 0.859592i $$0.670718\pi$$
$$632$$ 0 0
$$633$$ 5772.00 0.362427
$$634$$ 0 0
$$635$$ − 11856.0i − 0.740931i
$$636$$ −13392.0 −0.834949
$$637$$ 0 0
$$638$$ 0 0
$$639$$ − 4104.00i − 0.254072i
$$640$$ 0 0
$$641$$ 12630.0 0.778245 0.389122 0.921186i $$-0.372778\pi$$
0.389122 + 0.921186i $$0.372778\pi$$
$$642$$ 0 0
$$643$$ 14798.0i 0.907583i 0.891108 + 0.453792i $$0.149929\pi$$
−0.891108 + 0.453792i $$0.850071\pi$$
$$644$$ − 1536.00i − 0.0939858i
$$645$$ − 18864.0i − 1.15158i
$$646$$ 0 0
$$647$$ −26232.0 −1.59395 −0.796976 0.604012i $$-0.793568\pi$$
−0.796976 + 0.604012i $$0.793568\pi$$
$$648$$ 0 0
$$649$$ 20736.0 1.25417
$$650$$ 0 0
$$651$$ 1284.00 0.0773025
$$652$$ 3152.00i 0.189328i
$$653$$ −30390.0 −1.82121 −0.910607 0.413274i $$-0.864385\pi$$
−0.910607 + 0.413274i $$0.864385\pi$$
$$654$$ 0 0
$$655$$ 25200.0i 1.50328i
$$656$$ − 24576.0i − 1.46270i
$$657$$ 6138.00i 0.364484i
$$658$$ 0 0
$$659$$ −28740.0 −1.69886 −0.849432 0.527698i $$-0.823055\pi$$
−0.849432 + 0.527698i $$0.823055\pi$$
$$660$$ −10368.0 −0.611476
$$661$$ 9214.00i 0.542183i 0.962554 + 0.271092i $$0.0873846\pi$$
−0.962554 + 0.271092i $$0.912615\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ − 1776.00i − 0.103564i
$$666$$ 0 0
$$667$$ 1728.00 0.100312
$$668$$ − 672.000i − 0.0389228i
$$669$$ − 15126.0i − 0.874148i
$$670$$ 0 0
$$671$$ 2664.00i 0.153268i
$$672$$ 0 0
$$673$$ −16598.0 −0.950677 −0.475339 0.879803i $$-0.657674\pi$$
−0.475339 + 0.879803i $$0.657674\pi$$
$$674$$ 0 0
$$675$$ 513.000 0.0292524
$$676$$ 0 0
$$677$$ −8610.00 −0.488788 −0.244394 0.969676i $$-0.578589\pi$$
−0.244394 + 0.969676i $$0.578589\pi$$
$$678$$ 0 0
$$679$$ 220.000 0.0124342
$$680$$ 0 0
$$681$$ 8028.00i 0.451738i
$$682$$ 0 0
$$683$$ − 804.000i − 0.0450428i −0.999746 0.0225214i $$-0.992831\pi$$
0.999746 0.0225214i $$-0.00716938\pi$$
$$684$$ 5328.00i 0.297838i
$$685$$ 29952.0 1.67067
$$686$$ 0 0
$$687$$ − 7230.00i − 0.401516i
$$688$$ −33536.0 −1.85835
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 2270.00i 0.124971i 0.998046 + 0.0624854i $$0.0199027\pi$$
−0.998046 + 0.0624854i $$0.980097\pi$$
$$692$$ −9552.00 −0.524729
$$693$$ 648.000 0.0355202
$$694$$ 0 0
$$695$$ 29568.0i 1.61378i
$$696$$ 0 0
$$697$$ − 29952.0i − 1.62771i
$$698$$ 0 0
$$699$$ 11178.0 0.604851
$$700$$ 304.000i 0.0164145i
$$701$$ −1782.00 −0.0960131 −0.0480066 0.998847i $$-0.515287\pi$$
−0.0480066 + 0.998847i $$0.515287\pi$$
$$702$$ 0 0
$$703$$ −21164.0 −1.13544
$$704$$ 18432.0i 0.986764i
$$705$$ 10800.0 0.576953
$$706$$ 0 0
$$707$$ − 1980.00i − 0.105326i
$$708$$ 13824.0i 0.733810i
$$709$$ 10690.0i 0.566250i 0.959083 + 0.283125i $$0.0913712\pi$$
−0.959083 + 0.283125i $$0.908629\pi$$
$$710$$ 0 0
$$711$$ 6336.00 0.334203
$$712$$ 0 0
$$713$$ − 20544.0i − 1.07907i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −25248.0 −1.31782
$$717$$ − 3744.00i − 0.195010i
$$718$$ 0 0
$$719$$ −11568.0 −0.600019 −0.300009 0.953936i $$-0.596990\pi$$
−0.300009 + 0.953936i $$0.596990\pi$$
$$720$$ − 6912.00i − 0.357771i
$$721$$ 2416.00i 0.124794i
$$722$$ 0 0
$$723$$ − 12630.0i − 0.649675i
$$724$$ 8624.00 0.442691
$$725$$ −342.000 −0.0175194
$$726$$ 0 0
$$727$$ 11644.0 0.594019 0.297010 0.954874i $$-0.404011\pi$$
0.297010 + 0.954874i $$0.404011\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ −40872.0 −2.06800
$$732$$ −1776.00 −0.0896760
$$733$$ − 15010.0i − 0.756353i −0.925733 0.378177i $$-0.876551\pi$$
0.925733 0.378177i $$-0.123449\pi$$
$$734$$ 0 0
$$735$$ 12204.0i 0.612451i
$$736$$ 0 0
$$737$$ −1368.00 −0.0683730
$$738$$ 0 0
$$739$$ − 33410.0i − 1.66307i −0.555474 0.831534i $$-0.687463\pi$$
0.555474 0.831534i $$-0.312537\pi$$
$$740$$ 27456.0 1.36392
$$741$$ 0 0
$$742$$ 0 0
$$743$$ − 6504.00i − 0.321142i −0.987024 0.160571i $$-0.948666\pi$$
0.987024 0.160571i $$-0.0513336\pi$$
$$744$$ 0 0
$$745$$ 2592.00 0.127468
$$746$$ 0 0
$$747$$ − 7992.00i − 0.391448i
$$748$$ 22464.0i 1.09808i
$$749$$ − 1992.00i − 0.0971777i
$$750$$ 0 0
$$751$$ 13912.0 0.675973 0.337987 0.941151i $$-0.390254\pi$$
0.337987 + 0.941151i $$0.390254\pi$$
$$752$$ − 19200.0i − 0.931053i
$$753$$ −23076.0 −1.11678
$$754$$ 0 0
$$755$$ 10776.0 0.519442
$$756$$ 432.000i 0.0207827i
$$757$$ −23974.0 −1.15106 −0.575528 0.817782i $$-0.695204\pi$$
−0.575528 + 0.817782i $$0.695204\pi$$
$$758$$ 0 0
$$759$$ − 10368.0i − 0.495829i
$$760$$ 0 0
$$761$$ 288.000i 0.0137188i 0.999976 + 0.00685939i $$0.00218343\pi$$
−0.999976 + 0.00685939i $$0.997817\pi$$
$$762$$ 0 0
$$763$$ −2804.00 −0.133043
$$764$$ 25536.0 1.20924
$$765$$ − 8424.00i − 0.398131i
$$766$$ 0 0
$$767$$ 0 0
$$768$$ −12288.0 −0.577350
$$769$$ 1514.00i 0.0709964i 0.999370 + 0.0354982i $$0.0113018\pi$$
−0.999370 + 0.0354982i $$0.988698\pi$$
$$770$$ 0 0
$$771$$ 3978.00 0.185816
$$772$$ − 5776.00i − 0.269278i
$$773$$ 15816.0i 0.735915i 0.929843 + 0.367957i $$0.119943\pi$$
−0.929843 + 0.367957i $$0.880057\pi$$
$$774$$ 0 0
$$775$$ 4066.00i 0.188458i
$$776$$ 0 0
$$777$$ −1716.00 −0.0792293
$$778$$ 0 0
$$779$$ 28416.0 1.30694
$$780$$ 0 0
$$781$$ 16416.0 0.752126
$$782$$ 0 0
$$783$$ −486.000 −0.0221816
$$784$$ 21696.0 0.988338
$$785$$ 18120.0i 0.823861i
$$786$$ 0 0
$$787$$ − 10154.0i − 0.459912i −0.973201 0.229956i $$-0.926142\pi$$
0.973201 0.229956i $$-0.0738583\pi$$
$$788$$ 22368.0i 1.01120i
$$789$$ 18144.0 0.818686
$$790$$ 0 0
$$791$$ − 3852.00i − 0.173150i
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ 20088.0i 0.896161i
$$796$$ 2720.00 0.121115
$$797$$ 17442.0 0.775191 0.387596 0.921830i $$-0.373306\pi$$
0.387596 + 0.921830i $$0.373306\pi$$
$$798$$ 0 0
$$799$$ − 23400.0i − 1.03609i
$$800$$ 0 0
$$801$$ 9180.00i 0.404943i
$$802$$ 0 0
$$803$$ −24552.0 −1.07898
$$804$$ − 912.000i − 0.0400047i
$$805$$ −2304.00 −0.100876
$$806$$ 0 0
$$807$$ −19422.0 −0.847196
$$808$$ 0 0
$$809$$ −8778.00 −0.381481 −0.190740 0.981641i $$-0.561089\pi$$
−0.190740 + 0.981641i $$0.561089\pi$$
$$810$$ 0 0
$$811$$ − 430.000i − 0.0186182i −0.999957 0.00930909i $$-0.997037\pi$$
0.999957 0.00930909i $$-0.00296322\pi$$
$$812$$ − 288.000i − 0.0124468i
$$813$$ 17934.0i 0.773644i
$$814$$ 0 0
$$815$$ 4728.00 0.203208
$$816$$ −14976.0 −0.642481
$$817$$ − 38776.0i − 1.66047i
$$818$$ 0 0
$$819$$ 0 0
$$820$$ −36864.0 −1.56994
$$821$$ − 32976.0i − 1.40179i −0.713264 0.700895i $$-0.752784\pi$$
0.713264 0.700895i $$-0.247216\pi$$
$$822$$ 0 0
$$823$$ 1168.00 0.0494701 0.0247351 0.999694i $$-0.492126\pi$$
0.0247351 + 0.999694i $$0.492126\pi$$
$$824$$ 0 0
$$825$$ 2052.00i 0.0865957i
$$826$$ 0 0
$$827$$ 17172.0i 0.722042i 0.932558 + 0.361021i $$0.117572\pi$$
−0.932558 + 0.361021i $$0.882428\pi$$
$$828$$ 6912.00 0.290107
$$829$$ −27146.0 −1.13730 −0.568649 0.822580i $$-0.692534\pi$$
−0.568649 + 0.822580i $$0.692534\pi$$
$$830$$ 0 0
$$831$$ 26250.0 1.09579
$$832$$ 0 0
$$833$$ 26442.0 1.09983
$$834$$ 0 0
$$835$$ −1008.00 −0.0417764
$$836$$ −21312.0 −0.881688
$$837$$ 5778.00i 0.238610i
$$838$$ 0 0
$$839$$ − 30696.0i − 1.26310i −0.775334 0.631552i $$-0.782418\pi$$
0.775334 0.631552i $$-0.217582\pi$$
$$840$$ 0 0
$$841$$ −24065.0 −0.986715
$$842$$ 0 0
$$843$$ 26928.0i 1.10018i
$$844$$ −15392.0 −0.627742
$$845$$ 0 0
$$846$$ 0 0
$$847$$ − 70.0000i − 0.00283970i
$$848$$ 35712.0 1.44617
$$849$$ −1776.00 −0.0717929
$$850$$ 0 0
$$851$$ 27456.0i 1.10597i
$$852$$ 10944.0i 0.440065i
$$853$$ − 24842.0i − 0.997156i −0.866845 0.498578i $$-0.833856\pi$$
0.866845 0.498578i $$-0.166144\pi$$
$$854$$ 0 0
$$855$$ 7992.00 0.319673
$$856$$ 0 0
$$857$$ −11406.0 −0.454634 −0.227317 0.973821i $$-0.572995\pi$$
−0.227317 + 0.973821i $$0.572995\pi$$
$$858$$ 0 0
$$859$$ 20540.0 0.815851 0.407925 0.913015i $$-0.366252\pi$$
0.407925 + 0.913015i $$0.366252\pi$$
$$860$$ 50304.0i 1.99460i
$$861$$ 2304.00 0.0911964
$$862$$ 0 0
$$863$$ 9108.00i 0.359258i 0.983734 + 0.179629i $$0.0574898\pi$$
−0.983734 + 0.179629i $$0.942510\pi$$
$$864$$ 0 0
$$865$$ 14328.0i 0.563198i
$$866$$ 0 0
$$867$$ −3513.00 −0.137610
$$868$$ −3424.00 −0.133892
$$869$$ 25344.0i 0.989340i
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ 990.000i 0.0383808i
$$874$$ 0 0
$$875$$ −2544.00 −0.0982890
$$876$$ − 16368.0i − 0.631305i
$$877$$ − 24046.0i − 0.925856i −0.886396 0.462928i $$-0.846799\pi$$
0.886396 0.462928i $$-0.153201\pi$$
$$878$$ 0 0
$$879$$ − 13824.0i − 0.530457i
$$880$$ 27648.0 1.05911
$$881$$ −7998.00 −0.305856 −0.152928 0.988237i $$-0.548870\pi$$
−0.152928 + 0.988237i $$0.548870\pi$$
$$882$$ 0 0
$$883$$ −24032.0 −0.915902 −0.457951 0.888978i $$-0.651416\pi$$
−0.457951 + 0.888978i $$0.651416\pi$$
$$884$$ 0 0
$$885$$ 20736.0 0.787608
$$886$$ 0 0
$$887$$ −15648.0 −0.592343 −0.296172 0.955135i $$-0.595710\pi$$
−0.296172 + 0.955135i $$0.595710\pi$$
$$888$$ 0 0
$$889$$ − 1976.00i − 0.0745477i
$$890$$ 0 0
$$891$$ 2916.00i 0.109640i
$$892$$ 40336.0i 1.51407i
$$893$$ 22200.0 0.831909
$$894$$ 0 0
$$895$$ 37872.0i 1.41444i
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ − 3852.00i − 0.142905i
$$900$$ −1368.00 −0.0506667
$$901$$ 43524.0 1.60932
$$902$$ 0 0
$$903$$ − 3144.00i − 0.115865i
$$904$$ 0 0
$$905$$ − 12936.0i − 0.475146i
$$906$$ 0 0
$$907$$ 808.000 0.0295802 0.0147901 0.999891i $$-0.495292\pi$$
0.0147901 + 0.999891i $$0.495292\pi$$
$$908$$ − 21408.0i − 0.782433i
$$909$$ 8910.00 0.325111
$$910$$ 0 0
$$911$$ 39144.0 1.42360 0.711799 0.702383i $$-0.247880\pi$$
0.711799 + 0.702383i $$0.247880\pi$$
$$912$$ − 14208.0i − 0.515870i
$$913$$ 31968.0 1.15880
$$914$$ 0 0
$$915$$ 2664.00i 0.0962504i
$$916$$ 19280.0i 0.695447i
$$917$$ 4200.00i 0.151250i
$$918$$ 0 0
$$919$$ −38248.0 −1.37289 −0.686445 0.727182i $$-0.740830\pi$$
−0.686445 + 0.727182i $$0.740830\pi$$
$$920$$ 0 0
$$921$$ − 9498.00i − 0.339815i
$$922$$ 0 0
$$923$$ 0 0
$$924$$ −1728.00 −0.0615228
$$925$$ − 5434.00i − 0.193155i
$$926$$ 0 0
$$927$$ −10872.0 −0.385203
$$928$$ 0 0
$$929$$ − 54264.0i − 1.91641i −0.286084 0.958205i $$-0.592354\pi$$
0.286084 0.958205i $$-0.407646\pi$$
$$930$$ 0 0
$$931$$ 25086.0i 0.883094i
$$932$$ −29808.0 −1.04763
$$933$$ 7416.00 0.260224
$$934$$ 0 0
$$935$$ 33696.0 1.17859
$$936$$ 0 0
$$937$$ 12206.0 0.425563 0.212782 0.977100i $$-0.431748\pi$$
0.212782 + 0.977100i $$0.431748\pi$$
$$938$$ 0 0
$$939$$ 9282.00 0.322584
$$940$$ −28800.0 −0.999311
$$941$$ 17664.0i 0.611934i 0.952042 + 0.305967i $$0.0989797\pi$$
−0.952042 + 0.305967i $$0.901020\pi$$
$$942$$ 0 0
$$943$$ − 36864.0i − 1.27302i
$$944$$ − 36864.0i − 1.27100i
$$945$$ 648.000 0.0223063
$$946$$ 0 0
$$947$$ 51984.0i 1.78379i 0.452238 + 0.891897i $$0.350626\pi$$
−0.452238 + 0.891897i $$0.649374\pi$$
$$948$$ −16896.0 −0.578857
$$949$$ 0 0
$$950$$ 0 0
$$951$$ − 6948.00i − 0.236913i
$$952$$ 0 0
$$953$$ −13782.0 −0.468460 −0.234230 0.972181i $$-0.575257\pi$$
−0.234230 + 0.972181i $$0.575257\pi$$
$$954$$ 0 0
$$955$$ − 38304.0i − 1.29789i
$$956$$ 9984.00i 0.337767i
$$957$$ − 1944.00i − 0.0656642i
$$958$$ 0 0
$$959$$ 4992.00 0.168092
$$960$$ 18432.0i 0.619677i
$$961$$ −16005.0 −0.537243
$$962$$ 0 0
$$963$$ 8964.00 0.299959
$$964$$ 33680.0i 1.12527i
$$965$$ −8664.00 −0.289020
$$966$$ 0 0
$$967$$ 14618.0i 0.486125i 0.970011 + 0.243063i $$0.0781521\pi$$
−0.970011 + 0.243063i $$0.921848\pi$$
$$968$$ 0 0
$$969$$ − 17316.0i − 0.574066i
$$970$$ 0 0
$$971$$ −18708.0 −0.618299 −0.309149 0.951013i $$-0.600044\pi$$
−0.309149 + 0.951013i $$0.600044\pi$$
$$972$$ −1944.00 −0.0641500
$$973$$ 4928.00i 0.162368i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 4736.00 0.155323
$$977$$ − 48804.0i − 1.59814i −0.601241 0.799068i $$-0.705327\pi$$
0.601241 0.799068i $$-0.294673\pi$$
$$978$$ 0 0
$$979$$ −36720.0 −1.19875
$$980$$ − 32544.0i − 1.06080i
$$981$$ − 12618.0i − 0.410664i
$$982$$ 0 0
$$983$$ 44736.0i 1.45153i 0.687941 + 0.725766i $$0.258515\pi$$
−0.687941 + 0.725766i $$0.741485\pi$$
$$984$$ 0 0
$$985$$ 33552.0 1.08534
$$986$$ 0 0
$$987$$ 1800.00 0.0580493
$$988$$ 0 0
$$989$$ −50304.0 −1.61737
$$990$$ 0 0
$$991$$ −21004.0 −0.673274 −0.336637 0.941635i $$-0.609289\pi$$
−0.336637 + 0.941635i $$0.609289\pi$$
$$992$$ 0 0
$$993$$ 13278.0i 0.424335i
$$994$$ 0 0
$$995$$ − 4080.00i − 0.129995i
$$996$$ 21312.0i 0.678009i
$$997$$ 9038.00 0.287098 0.143549 0.989643i $$-0.454149\pi$$
0.143549 + 0.989643i $$0.454149\pi$$
$$998$$ 0 0
$$999$$ − 7722.00i − 0.244558i
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 507.4.b.b.337.1 2
13.5 odd 4 39.4.a.a.1.1 1
13.8 odd 4 507.4.a.c.1.1 1
13.12 even 2 inner 507.4.b.b.337.2 2
39.5 even 4 117.4.a.a.1.1 1
39.8 even 4 1521.4.a.f.1.1 1
52.31 even 4 624.4.a.g.1.1 1
65.44 odd 4 975.4.a.e.1.1 1
91.83 even 4 1911.4.a.f.1.1 1
104.5 odd 4 2496.4.a.o.1.1 1
104.83 even 4 2496.4.a.f.1.1 1
156.83 odd 4 1872.4.a.m.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.a.a.1.1 1 13.5 odd 4
117.4.a.a.1.1 1 39.5 even 4
507.4.a.c.1.1 1 13.8 odd 4
507.4.b.b.337.1 2 1.1 even 1 trivial
507.4.b.b.337.2 2 13.12 even 2 inner
624.4.a.g.1.1 1 52.31 even 4
975.4.a.e.1.1 1 65.44 odd 4
1521.4.a.f.1.1 1 39.8 even 4
1872.4.a.m.1.1 1 156.83 odd 4
1911.4.a.f.1.1 1 91.83 even 4
2496.4.a.f.1.1 1 104.83 even 4
2496.4.a.o.1.1 1 104.5 odd 4