# Properties

 Label 575.4.b.b.24.1 Level $575$ Weight $4$ Character 575.24 Analytic conductor $33.926$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [575,4,Mod(24,575)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(575, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("575.24");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$575 = 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 575.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: $$33.9260982533$$ 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 23) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 24.1 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 575.24 Dual form 575.4.b.b.24.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.00000i q^{2} +5.00000i q^{3} +4.00000 q^{4} +10.0000 q^{6} -8.00000i q^{7} -24.0000i q^{8} +2.00000 q^{9} +O(q^{10})$$ $$q-2.00000i q^{2} +5.00000i q^{3} +4.00000 q^{4} +10.0000 q^{6} -8.00000i q^{7} -24.0000i q^{8} +2.00000 q^{9} +34.0000 q^{11} +20.0000i q^{12} +57.0000i q^{13} -16.0000 q^{14} -16.0000 q^{16} -80.0000i q^{17} -4.00000i q^{18} +70.0000 q^{19} +40.0000 q^{21} -68.0000i q^{22} -23.0000i q^{23} +120.000 q^{24} +114.000 q^{26} +145.000i q^{27} -32.0000i q^{28} -245.000 q^{29} +103.000 q^{31} -160.000i q^{32} +170.000i q^{33} -160.000 q^{34} +8.00000 q^{36} -298.000i q^{37} -140.000i q^{38} -285.000 q^{39} +95.0000 q^{41} -80.0000i q^{42} -88.0000i q^{43} +136.000 q^{44} -46.0000 q^{46} -357.000i q^{47} -80.0000i q^{48} +279.000 q^{49} +400.000 q^{51} +228.000i q^{52} +414.000i q^{53} +290.000 q^{54} -192.000 q^{56} +350.000i q^{57} +490.000i q^{58} +408.000 q^{59} +822.000 q^{61} -206.000i q^{62} -16.0000i q^{63} -448.000 q^{64} +340.000 q^{66} +926.000i q^{67} -320.000i q^{68} +115.000 q^{69} +335.000 q^{71} -48.0000i q^{72} +899.000i q^{73} -596.000 q^{74} +280.000 q^{76} -272.000i q^{77} +570.000i q^{78} +1322.00 q^{79} -671.000 q^{81} -190.000i q^{82} +36.0000i q^{83} +160.000 q^{84} -176.000 q^{86} -1225.00i q^{87} -816.000i q^{88} +460.000 q^{89} +456.000 q^{91} -92.0000i q^{92} +515.000i q^{93} -714.000 q^{94} +800.000 q^{96} -964.000i q^{97} -558.000i q^{98} +68.0000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 8 q^{4} + 20 q^{6} + 4 q^{9}+O(q^{10})$$ 2 * q + 8 * q^4 + 20 * q^6 + 4 * q^9 $$2 q + 8 q^{4} + 20 q^{6} + 4 q^{9} + 68 q^{11} - 32 q^{14} - 32 q^{16} + 140 q^{19} + 80 q^{21} + 240 q^{24} + 228 q^{26} - 490 q^{29} + 206 q^{31} - 320 q^{34} + 16 q^{36} - 570 q^{39} + 190 q^{41} + 272 q^{44} - 92 q^{46} + 558 q^{49} + 800 q^{51} + 580 q^{54} - 384 q^{56} + 816 q^{59} + 1644 q^{61} - 896 q^{64} + 680 q^{66} + 230 q^{69} + 670 q^{71} - 1192 q^{74} + 560 q^{76} + 2644 q^{79} - 1342 q^{81} + 320 q^{84} - 352 q^{86} + 920 q^{89} + 912 q^{91} - 1428 q^{94} + 1600 q^{96} + 136 q^{99}+O(q^{100})$$ 2 * q + 8 * q^4 + 20 * q^6 + 4 * q^9 + 68 * q^11 - 32 * q^14 - 32 * q^16 + 140 * q^19 + 80 * q^21 + 240 * q^24 + 228 * q^26 - 490 * q^29 + 206 * q^31 - 320 * q^34 + 16 * q^36 - 570 * q^39 + 190 * q^41 + 272 * q^44 - 92 * q^46 + 558 * q^49 + 800 * q^51 + 580 * q^54 - 384 * q^56 + 816 * q^59 + 1644 * q^61 - 896 * q^64 + 680 * q^66 + 230 * q^69 + 670 * q^71 - 1192 * q^74 + 560 * q^76 + 2644 * q^79 - 1342 * q^81 + 320 * q^84 - 352 * q^86 + 920 * q^89 + 912 * q^91 - 1428 * q^94 + 1600 * q^96 + 136 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/575\mathbb{Z}\right)^\times$$.

 $$n$$ $$51$$ $$277$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 2.00000i − 0.707107i −0.935414 0.353553i $$-0.884973\pi$$
0.935414 0.353553i $$-0.115027\pi$$
$$3$$ 5.00000i 0.962250i 0.876652 + 0.481125i $$0.159772\pi$$
−0.876652 + 0.481125i $$0.840228\pi$$
$$4$$ 4.00000 0.500000
$$5$$ 0 0
$$6$$ 10.0000 0.680414
$$7$$ − 8.00000i − 0.431959i −0.976398 0.215980i $$-0.930705\pi$$
0.976398 0.215980i $$-0.0692945\pi$$
$$8$$ − 24.0000i − 1.06066i
$$9$$ 2.00000 0.0740741
$$10$$ 0 0
$$11$$ 34.0000 0.931944 0.465972 0.884799i $$-0.345705\pi$$
0.465972 + 0.884799i $$0.345705\pi$$
$$12$$ 20.0000i 0.481125i
$$13$$ 57.0000i 1.21607i 0.793909 + 0.608037i $$0.208043\pi$$
−0.793909 + 0.608037i $$0.791957\pi$$
$$14$$ −16.0000 −0.305441
$$15$$ 0 0
$$16$$ −16.0000 −0.250000
$$17$$ − 80.0000i − 1.14134i −0.821178 0.570672i $$-0.806683\pi$$
0.821178 0.570672i $$-0.193317\pi$$
$$18$$ − 4.00000i − 0.0523783i
$$19$$ 70.0000 0.845216 0.422608 0.906313i $$-0.361115\pi$$
0.422608 + 0.906313i $$0.361115\pi$$
$$20$$ 0 0
$$21$$ 40.0000 0.415653
$$22$$ − 68.0000i − 0.658984i
$$23$$ − 23.0000i − 0.208514i
$$24$$ 120.000 1.02062
$$25$$ 0 0
$$26$$ 114.000 0.859894
$$27$$ 145.000i 1.03353i
$$28$$ − 32.0000i − 0.215980i
$$29$$ −245.000 −1.56881 −0.784403 0.620252i $$-0.787030\pi$$
−0.784403 + 0.620252i $$0.787030\pi$$
$$30$$ 0 0
$$31$$ 103.000 0.596753 0.298377 0.954448i $$-0.403555\pi$$
0.298377 + 0.954448i $$0.403555\pi$$
$$32$$ − 160.000i − 0.883883i
$$33$$ 170.000i 0.896764i
$$34$$ −160.000 −0.807052
$$35$$ 0 0
$$36$$ 8.00000 0.0370370
$$37$$ − 298.000i − 1.32408i −0.749469 0.662039i $$-0.769691\pi$$
0.749469 0.662039i $$-0.230309\pi$$
$$38$$ − 140.000i − 0.597658i
$$39$$ −285.000 −1.17017
$$40$$ 0 0
$$41$$ 95.0000 0.361866 0.180933 0.983495i $$-0.442088\pi$$
0.180933 + 0.983495i $$0.442088\pi$$
$$42$$ − 80.0000i − 0.293911i
$$43$$ − 88.0000i − 0.312090i −0.987750 0.156045i $$-0.950125\pi$$
0.987750 0.156045i $$-0.0498745\pi$$
$$44$$ 136.000 0.465972
$$45$$ 0 0
$$46$$ −46.0000 −0.147442
$$47$$ − 357.000i − 1.10795i −0.832532 0.553977i $$-0.813110\pi$$
0.832532 0.553977i $$-0.186890\pi$$
$$48$$ − 80.0000i − 0.240563i
$$49$$ 279.000 0.813411
$$50$$ 0 0
$$51$$ 400.000 1.09826
$$52$$ 228.000i 0.608037i
$$53$$ 414.000i 1.07297i 0.843911 + 0.536484i $$0.180248\pi$$
−0.843911 + 0.536484i $$0.819752\pi$$
$$54$$ 290.000 0.730815
$$55$$ 0 0
$$56$$ −192.000 −0.458162
$$57$$ 350.000i 0.813309i
$$58$$ 490.000i 1.10931i
$$59$$ 408.000 0.900289 0.450145 0.892956i $$-0.351372\pi$$
0.450145 + 0.892956i $$0.351372\pi$$
$$60$$ 0 0
$$61$$ 822.000 1.72535 0.862675 0.505759i $$-0.168788\pi$$
0.862675 + 0.505759i $$0.168788\pi$$
$$62$$ − 206.000i − 0.421968i
$$63$$ − 16.0000i − 0.0319970i
$$64$$ −448.000 −0.875000
$$65$$ 0 0
$$66$$ 340.000 0.634108
$$67$$ 926.000i 1.68849i 0.535957 + 0.844246i $$0.319951\pi$$
−0.535957 + 0.844246i $$0.680049\pi$$
$$68$$ − 320.000i − 0.570672i
$$69$$ 115.000 0.200643
$$70$$ 0 0
$$71$$ 335.000 0.559960 0.279980 0.960006i $$-0.409672\pi$$
0.279980 + 0.960006i $$0.409672\pi$$
$$72$$ − 48.0000i − 0.0785674i
$$73$$ 899.000i 1.44137i 0.693263 + 0.720685i $$0.256173\pi$$
−0.693263 + 0.720685i $$0.743827\pi$$
$$74$$ −596.000 −0.936265
$$75$$ 0 0
$$76$$ 280.000 0.422608
$$77$$ − 272.000i − 0.402562i
$$78$$ 570.000i 0.827433i
$$79$$ 1322.00 1.88274 0.941371 0.337373i $$-0.109538\pi$$
0.941371 + 0.337373i $$0.109538\pi$$
$$80$$ 0 0
$$81$$ −671.000 −0.920439
$$82$$ − 190.000i − 0.255878i
$$83$$ 36.0000i 0.0476086i 0.999717 + 0.0238043i $$0.00757786\pi$$
−0.999717 + 0.0238043i $$0.992422\pi$$
$$84$$ 160.000 0.207827
$$85$$ 0 0
$$86$$ −176.000 −0.220681
$$87$$ − 1225.00i − 1.50958i
$$88$$ − 816.000i − 0.988476i
$$89$$ 460.000 0.547864 0.273932 0.961749i $$-0.411676\pi$$
0.273932 + 0.961749i $$0.411676\pi$$
$$90$$ 0 0
$$91$$ 456.000 0.525294
$$92$$ − 92.0000i − 0.104257i
$$93$$ 515.000i 0.574226i
$$94$$ −714.000 −0.783441
$$95$$ 0 0
$$96$$ 800.000 0.850517
$$97$$ − 964.000i − 1.00907i −0.863393 0.504533i $$-0.831665\pi$$
0.863393 0.504533i $$-0.168335\pi$$
$$98$$ − 558.000i − 0.575168i
$$99$$ 68.0000 0.0690329
$$100$$ 0 0
$$101$$ −310.000 −0.305407 −0.152704 0.988272i $$-0.548798\pi$$
−0.152704 + 0.988272i $$0.548798\pi$$
$$102$$ − 800.000i − 0.776586i
$$103$$ − 1044.00i − 0.998722i −0.866394 0.499361i $$-0.833568\pi$$
0.866394 0.499361i $$-0.166432\pi$$
$$104$$ 1368.00 1.28984
$$105$$ 0 0
$$106$$ 828.000 0.758703
$$107$$ 414.000i 0.374046i 0.982356 + 0.187023i $$0.0598838\pi$$
−0.982356 + 0.187023i $$0.940116\pi$$
$$108$$ 580.000i 0.516764i
$$109$$ −704.000 −0.618633 −0.309316 0.950959i $$-0.600100\pi$$
−0.309316 + 0.950959i $$0.600100\pi$$
$$110$$ 0 0
$$111$$ 1490.00 1.27409
$$112$$ 128.000i 0.107990i
$$113$$ − 952.000i − 0.792537i −0.918135 0.396268i $$-0.870305\pi$$
0.918135 0.396268i $$-0.129695\pi$$
$$114$$ 700.000 0.575097
$$115$$ 0 0
$$116$$ −980.000 −0.784403
$$117$$ 114.000i 0.0900795i
$$118$$ − 816.000i − 0.636601i
$$119$$ −640.000 −0.493014
$$120$$ 0 0
$$121$$ −175.000 −0.131480
$$122$$ − 1644.00i − 1.22001i
$$123$$ 475.000i 0.348206i
$$124$$ 412.000 0.298377
$$125$$ 0 0
$$126$$ −32.0000 −0.0226253
$$127$$ 261.000i 0.182362i 0.995834 + 0.0911811i $$0.0290642\pi$$
−0.995834 + 0.0911811i $$0.970936\pi$$
$$128$$ − 384.000i − 0.265165i
$$129$$ 440.000 0.300309
$$130$$ 0 0
$$131$$ −1441.00 −0.961074 −0.480537 0.876974i $$-0.659558\pi$$
−0.480537 + 0.876974i $$0.659558\pi$$
$$132$$ 680.000i 0.448382i
$$133$$ − 560.000i − 0.365099i
$$134$$ 1852.00 1.19394
$$135$$ 0 0
$$136$$ −1920.00 −1.21058
$$137$$ 1556.00i 0.970351i 0.874417 + 0.485175i $$0.161244\pi$$
−0.874417 + 0.485175i $$0.838756\pi$$
$$138$$ − 230.000i − 0.141876i
$$139$$ −25.0000 −0.0152552 −0.00762760 0.999971i $$-0.502428\pi$$
−0.00762760 + 0.999971i $$0.502428\pi$$
$$140$$ 0 0
$$141$$ 1785.00 1.06613
$$142$$ − 670.000i − 0.395952i
$$143$$ 1938.00i 1.13331i
$$144$$ −32.0000 −0.0185185
$$145$$ 0 0
$$146$$ 1798.00 1.01920
$$147$$ 1395.00i 0.782705i
$$148$$ − 1192.00i − 0.662039i
$$149$$ −822.000 −0.451952 −0.225976 0.974133i $$-0.572557\pi$$
−0.225976 + 0.974133i $$0.572557\pi$$
$$150$$ 0 0
$$151$$ −1489.00 −0.802471 −0.401235 0.915975i $$-0.631419\pi$$
−0.401235 + 0.915975i $$0.631419\pi$$
$$152$$ − 1680.00i − 0.896487i
$$153$$ − 160.000i − 0.0845440i
$$154$$ −544.000 −0.284654
$$155$$ 0 0
$$156$$ −1140.00 −0.585084
$$157$$ − 632.000i − 0.321268i −0.987014 0.160634i $$-0.948646\pi$$
0.987014 0.160634i $$-0.0513539\pi$$
$$158$$ − 2644.00i − 1.33130i
$$159$$ −2070.00 −1.03246
$$160$$ 0 0
$$161$$ −184.000 −0.0900698
$$162$$ 1342.00i 0.650849i
$$163$$ 3043.00i 1.46225i 0.682245 + 0.731123i $$0.261004\pi$$
−0.682245 + 0.731123i $$0.738996\pi$$
$$164$$ 380.000 0.180933
$$165$$ 0 0
$$166$$ 72.0000 0.0336644
$$167$$ − 2224.00i − 1.03053i −0.857031 0.515264i $$-0.827694\pi$$
0.857031 0.515264i $$-0.172306\pi$$
$$168$$ − 960.000i − 0.440867i
$$169$$ −1052.00 −0.478835
$$170$$ 0 0
$$171$$ 140.000 0.0626086
$$172$$ − 352.000i − 0.156045i
$$173$$ − 3230.00i − 1.41949i −0.704457 0.709747i $$-0.748809\pi$$
0.704457 0.709747i $$-0.251191\pi$$
$$174$$ −2450.00 −1.06744
$$175$$ 0 0
$$176$$ −544.000 −0.232986
$$177$$ 2040.00i 0.866304i
$$178$$ − 920.000i − 0.387398i
$$179$$ −369.000 −0.154080 −0.0770401 0.997028i $$-0.524547\pi$$
−0.0770401 + 0.997028i $$0.524547\pi$$
$$180$$ 0 0
$$181$$ −1370.00 −0.562604 −0.281302 0.959619i $$-0.590766\pi$$
−0.281302 + 0.959619i $$0.590766\pi$$
$$182$$ − 912.000i − 0.371439i
$$183$$ 4110.00i 1.66022i
$$184$$ −552.000 −0.221163
$$185$$ 0 0
$$186$$ 1030.00 0.406039
$$187$$ − 2720.00i − 1.06367i
$$188$$ − 1428.00i − 0.553977i
$$189$$ 1160.00 0.446442
$$190$$ 0 0
$$191$$ 4410.00 1.67066 0.835331 0.549747i $$-0.185276\pi$$
0.835331 + 0.549747i $$0.185276\pi$$
$$192$$ − 2240.00i − 0.841969i
$$193$$ 135.000i 0.0503498i 0.999683 + 0.0251749i $$0.00801427\pi$$
−0.999683 + 0.0251749i $$0.991986\pi$$
$$194$$ −1928.00 −0.713517
$$195$$ 0 0
$$196$$ 1116.00 0.406706
$$197$$ 1221.00i 0.441587i 0.975321 + 0.220794i $$0.0708647\pi$$
−0.975321 + 0.220794i $$0.929135\pi$$
$$198$$ − 136.000i − 0.0488136i
$$199$$ 1098.00 0.391131 0.195566 0.980691i $$-0.437346\pi$$
0.195566 + 0.980691i $$0.437346\pi$$
$$200$$ 0 0
$$201$$ −4630.00 −1.62475
$$202$$ 620.000i 0.215956i
$$203$$ 1960.00i 0.677660i
$$204$$ 1600.00 0.549129
$$205$$ 0 0
$$206$$ −2088.00 −0.706203
$$207$$ − 46.0000i − 0.0154455i
$$208$$ − 912.000i − 0.304018i
$$209$$ 2380.00 0.787694
$$210$$ 0 0
$$211$$ −3676.00 −1.19937 −0.599683 0.800238i $$-0.704707\pi$$
−0.599683 + 0.800238i $$0.704707\pi$$
$$212$$ 1656.00i 0.536484i
$$213$$ 1675.00i 0.538822i
$$214$$ 828.000 0.264490
$$215$$ 0 0
$$216$$ 3480.00 1.09622
$$217$$ − 824.000i − 0.257773i
$$218$$ 1408.00i 0.437439i
$$219$$ −4495.00 −1.38696
$$220$$ 0 0
$$221$$ 4560.00 1.38796
$$222$$ − 2980.00i − 0.900921i
$$223$$ − 1656.00i − 0.497282i −0.968596 0.248641i $$-0.920016\pi$$
0.968596 0.248641i $$-0.0799840\pi$$
$$224$$ −1280.00 −0.381802
$$225$$ 0 0
$$226$$ −1904.00 −0.560408
$$227$$ 2940.00i 0.859624i 0.902918 + 0.429812i $$0.141420\pi$$
−0.902918 + 0.429812i $$0.858580\pi$$
$$228$$ 1400.00i 0.406655i
$$229$$ −3612.00 −1.04230 −0.521152 0.853464i $$-0.674498\pi$$
−0.521152 + 0.853464i $$0.674498\pi$$
$$230$$ 0 0
$$231$$ 1360.00 0.387366
$$232$$ 5880.00i 1.66397i
$$233$$ 4325.00i 1.21605i 0.793917 + 0.608026i $$0.208038\pi$$
−0.793917 + 0.608026i $$0.791962\pi$$
$$234$$ 228.000 0.0636958
$$235$$ 0 0
$$236$$ 1632.00 0.450145
$$237$$ 6610.00i 1.81167i
$$238$$ 1280.00i 0.348614i
$$239$$ −2735.00 −0.740219 −0.370110 0.928988i $$-0.620680\pi$$
−0.370110 + 0.928988i $$0.620680\pi$$
$$240$$ 0 0
$$241$$ −6710.00 −1.79348 −0.896741 0.442556i $$-0.854072\pi$$
−0.896741 + 0.442556i $$0.854072\pi$$
$$242$$ 350.000i 0.0929705i
$$243$$ 560.000i 0.147835i
$$244$$ 3288.00 0.862675
$$245$$ 0 0
$$246$$ 950.000 0.246219
$$247$$ 3990.00i 1.02784i
$$248$$ − 2472.00i − 0.632952i
$$249$$ −180.000 −0.0458114
$$250$$ 0 0
$$251$$ −6948.00 −1.74723 −0.873613 0.486621i $$-0.838229\pi$$
−0.873613 + 0.486621i $$0.838229\pi$$
$$252$$ − 64.0000i − 0.0159985i
$$253$$ − 782.000i − 0.194324i
$$254$$ 522.000 0.128950
$$255$$ 0 0
$$256$$ −4352.00 −1.06250
$$257$$ − 4929.00i − 1.19635i −0.801365 0.598176i $$-0.795892\pi$$
0.801365 0.598176i $$-0.204108\pi$$
$$258$$ − 880.000i − 0.212350i
$$259$$ −2384.00 −0.571948
$$260$$ 0 0
$$261$$ −490.000 −0.116208
$$262$$ 2882.00i 0.679582i
$$263$$ − 6138.00i − 1.43911i −0.694437 0.719554i $$-0.744346\pi$$
0.694437 0.719554i $$-0.255654\pi$$
$$264$$ 4080.00 0.951162
$$265$$ 0 0
$$266$$ −1120.00 −0.258164
$$267$$ 2300.00i 0.527182i
$$268$$ 3704.00i 0.844246i
$$269$$ 2063.00 0.467596 0.233798 0.972285i $$-0.424885\pi$$
0.233798 + 0.972285i $$0.424885\pi$$
$$270$$ 0 0
$$271$$ −1064.00 −0.238500 −0.119250 0.992864i $$-0.538049\pi$$
−0.119250 + 0.992864i $$0.538049\pi$$
$$272$$ 1280.00i 0.285336i
$$273$$ 2280.00i 0.505465i
$$274$$ 3112.00 0.686142
$$275$$ 0 0
$$276$$ 460.000 0.100322
$$277$$ 5729.00i 1.24268i 0.783541 + 0.621340i $$0.213411\pi$$
−0.783541 + 0.621340i $$0.786589\pi$$
$$278$$ 50.0000i 0.0107871i
$$279$$ 206.000 0.0442039
$$280$$ 0 0
$$281$$ −960.000 −0.203804 −0.101902 0.994794i $$-0.532493\pi$$
−0.101902 + 0.994794i $$0.532493\pi$$
$$282$$ − 3570.00i − 0.753867i
$$283$$ 114.000i 0.0239456i 0.999928 + 0.0119728i $$0.00381115\pi$$
−0.999928 + 0.0119728i $$0.996189\pi$$
$$284$$ 1340.00 0.279980
$$285$$ 0 0
$$286$$ 3876.00 0.801373
$$287$$ − 760.000i − 0.156311i
$$288$$ − 320.000i − 0.0654729i
$$289$$ −1487.00 −0.302666
$$290$$ 0 0
$$291$$ 4820.00 0.970974
$$292$$ 3596.00i 0.720685i
$$293$$ 7048.00i 1.40529i 0.711543 + 0.702643i $$0.247997\pi$$
−0.711543 + 0.702643i $$0.752003\pi$$
$$294$$ 2790.00 0.553456
$$295$$ 0 0
$$296$$ −7152.00 −1.40440
$$297$$ 4930.00i 0.963191i
$$298$$ 1644.00i 0.319578i
$$299$$ 1311.00 0.253569
$$300$$ 0 0
$$301$$ −704.000 −0.134810
$$302$$ 2978.00i 0.567433i
$$303$$ − 1550.00i − 0.293878i
$$304$$ −1120.00 −0.211304
$$305$$ 0 0
$$306$$ −320.000 −0.0597816
$$307$$ 3872.00i 0.719826i 0.932986 + 0.359913i $$0.117194\pi$$
−0.932986 + 0.359913i $$0.882806\pi$$
$$308$$ − 1088.00i − 0.201281i
$$309$$ 5220.00 0.961021
$$310$$ 0 0
$$311$$ −4977.00 −0.907459 −0.453730 0.891139i $$-0.649907\pi$$
−0.453730 + 0.891139i $$0.649907\pi$$
$$312$$ 6840.00i 1.24115i
$$313$$ 2536.00i 0.457965i 0.973430 + 0.228983i $$0.0735399\pi$$
−0.973430 + 0.228983i $$0.926460\pi$$
$$314$$ −1264.00 −0.227171
$$315$$ 0 0
$$316$$ 5288.00 0.941371
$$317$$ 1434.00i 0.254074i 0.991898 + 0.127037i $$0.0405467\pi$$
−0.991898 + 0.127037i $$0.959453\pi$$
$$318$$ 4140.00i 0.730062i
$$319$$ −8330.00 −1.46204
$$320$$ 0 0
$$321$$ −2070.00 −0.359926
$$322$$ 368.000i 0.0636889i
$$323$$ − 5600.00i − 0.964682i
$$324$$ −2684.00 −0.460219
$$325$$ 0 0
$$326$$ 6086.00 1.03396
$$327$$ − 3520.00i − 0.595280i
$$328$$ − 2280.00i − 0.383817i
$$329$$ −2856.00 −0.478591
$$330$$ 0 0
$$331$$ 5469.00 0.908167 0.454084 0.890959i $$-0.349967\pi$$
0.454084 + 0.890959i $$0.349967\pi$$
$$332$$ 144.000i 0.0238043i
$$333$$ − 596.000i − 0.0980799i
$$334$$ −4448.00 −0.728694
$$335$$ 0 0
$$336$$ −640.000 −0.103913
$$337$$ − 7796.00i − 1.26016i −0.776529 0.630082i $$-0.783021\pi$$
0.776529 0.630082i $$-0.216979\pi$$
$$338$$ 2104.00i 0.338587i
$$339$$ 4760.00 0.762619
$$340$$ 0 0
$$341$$ 3502.00 0.556141
$$342$$ − 280.000i − 0.0442710i
$$343$$ − 4976.00i − 0.783320i
$$344$$ −2112.00 −0.331022
$$345$$ 0 0
$$346$$ −6460.00 −1.00373
$$347$$ − 10068.0i − 1.55758i −0.627288 0.778788i $$-0.715835\pi$$
0.627288 0.778788i $$-0.284165\pi$$
$$348$$ − 4900.00i − 0.754792i
$$349$$ 7495.00 1.14956 0.574782 0.818306i $$-0.305087\pi$$
0.574782 + 0.818306i $$0.305087\pi$$
$$350$$ 0 0
$$351$$ −8265.00 −1.25685
$$352$$ − 5440.00i − 0.823730i
$$353$$ − 10617.0i − 1.60081i −0.599460 0.800405i $$-0.704618\pi$$
0.599460 0.800405i $$-0.295382\pi$$
$$354$$ 4080.00 0.612569
$$355$$ 0 0
$$356$$ 1840.00 0.273932
$$357$$ − 3200.00i − 0.474403i
$$358$$ 738.000i 0.108951i
$$359$$ −2522.00 −0.370769 −0.185384 0.982666i $$-0.559353\pi$$
−0.185384 + 0.982666i $$0.559353\pi$$
$$360$$ 0 0
$$361$$ −1959.00 −0.285610
$$362$$ 2740.00i 0.397821i
$$363$$ − 875.000i − 0.126517i
$$364$$ 1824.00 0.262647
$$365$$ 0 0
$$366$$ 8220.00 1.17395
$$367$$ 7204.00i 1.02465i 0.858792 + 0.512324i $$0.171215\pi$$
−0.858792 + 0.512324i $$0.828785\pi$$
$$368$$ 368.000i 0.0521286i
$$369$$ 190.000 0.0268049
$$370$$ 0 0
$$371$$ 3312.00 0.463478
$$372$$ 2060.00i 0.287113i
$$373$$ 13310.0i 1.84763i 0.382840 + 0.923815i $$0.374946\pi$$
−0.382840 + 0.923815i $$0.625054\pi$$
$$374$$ −5440.00 −0.752128
$$375$$ 0 0
$$376$$ −8568.00 −1.17516
$$377$$ − 13965.0i − 1.90778i
$$378$$ − 2320.00i − 0.315682i
$$379$$ −12952.0 −1.75541 −0.877704 0.479203i $$-0.840926\pi$$
−0.877704 + 0.479203i $$0.840926\pi$$
$$380$$ 0 0
$$381$$ −1305.00 −0.175478
$$382$$ − 8820.00i − 1.18134i
$$383$$ 2812.00i 0.375161i 0.982249 + 0.187580i $$0.0600645\pi$$
−0.982249 + 0.187580i $$0.939936\pi$$
$$384$$ 1920.00 0.255155
$$385$$ 0 0
$$386$$ 270.000 0.0356027
$$387$$ − 176.000i − 0.0231178i
$$388$$ − 3856.00i − 0.504533i
$$389$$ −1264.00 −0.164749 −0.0823745 0.996601i $$-0.526250\pi$$
−0.0823745 + 0.996601i $$0.526250\pi$$
$$390$$ 0 0
$$391$$ −1840.00 −0.237987
$$392$$ − 6696.00i − 0.862753i
$$393$$ − 7205.00i − 0.924794i
$$394$$ 2442.00 0.312249
$$395$$ 0 0
$$396$$ 272.000 0.0345165
$$397$$ 7119.00i 0.899981i 0.893033 + 0.449990i $$0.148573\pi$$
−0.893033 + 0.449990i $$0.851427\pi$$
$$398$$ − 2196.00i − 0.276572i
$$399$$ 2800.00 0.351317
$$400$$ 0 0
$$401$$ 4262.00 0.530758 0.265379 0.964144i $$-0.414503\pi$$
0.265379 + 0.964144i $$0.414503\pi$$
$$402$$ 9260.00i 1.14887i
$$403$$ 5871.00i 0.725696i
$$404$$ −1240.00 −0.152704
$$405$$ 0 0
$$406$$ 3920.00 0.479178
$$407$$ − 10132.0i − 1.23397i
$$408$$ − 9600.00i − 1.16488i
$$409$$ −229.000 −0.0276854 −0.0138427 0.999904i $$-0.504406\pi$$
−0.0138427 + 0.999904i $$0.504406\pi$$
$$410$$ 0 0
$$411$$ −7780.00 −0.933720
$$412$$ − 4176.00i − 0.499361i
$$413$$ − 3264.00i − 0.388888i
$$414$$ −92.0000 −0.0109216
$$415$$ 0 0
$$416$$ 9120.00 1.07487
$$417$$ − 125.000i − 0.0146793i
$$418$$ − 4760.00i − 0.556984i
$$419$$ −15776.0 −1.83940 −0.919699 0.392623i $$-0.871568\pi$$
−0.919699 + 0.392623i $$0.871568\pi$$
$$420$$ 0 0
$$421$$ −8728.00 −1.01040 −0.505198 0.863003i $$-0.668581\pi$$
−0.505198 + 0.863003i $$0.668581\pi$$
$$422$$ 7352.00i 0.848080i
$$423$$ − 714.000i − 0.0820706i
$$424$$ 9936.00 1.13805
$$425$$ 0 0
$$426$$ 3350.00 0.381005
$$427$$ − 6576.00i − 0.745281i
$$428$$ 1656.00i 0.187023i
$$429$$ −9690.00 −1.09053
$$430$$ 0 0
$$431$$ −2928.00 −0.327232 −0.163616 0.986524i $$-0.552316\pi$$
−0.163616 + 0.986524i $$0.552316\pi$$
$$432$$ − 2320.00i − 0.258382i
$$433$$ 5314.00i 0.589780i 0.955531 + 0.294890i $$0.0952829\pi$$
−0.955531 + 0.294890i $$0.904717\pi$$
$$434$$ −1648.00 −0.182273
$$435$$ 0 0
$$436$$ −2816.00 −0.309316
$$437$$ − 1610.00i − 0.176240i
$$438$$ 8990.00i 0.980728i
$$439$$ −2585.00 −0.281037 −0.140519 0.990078i $$-0.544877\pi$$
−0.140519 + 0.990078i $$0.544877\pi$$
$$440$$ 0 0
$$441$$ 558.000 0.0602527
$$442$$ − 9120.00i − 0.981435i
$$443$$ 2997.00i 0.321426i 0.987001 + 0.160713i $$0.0513794\pi$$
−0.987001 + 0.160713i $$0.948621\pi$$
$$444$$ 5960.00 0.637047
$$445$$ 0 0
$$446$$ −3312.00 −0.351632
$$447$$ − 4110.00i − 0.434891i
$$448$$ 3584.00i 0.377964i
$$449$$ 16562.0 1.74078 0.870389 0.492365i $$-0.163868\pi$$
0.870389 + 0.492365i $$0.163868\pi$$
$$450$$ 0 0
$$451$$ 3230.00 0.337239
$$452$$ − 3808.00i − 0.396268i
$$453$$ − 7445.00i − 0.772178i
$$454$$ 5880.00 0.607846
$$455$$ 0 0
$$456$$ 8400.00 0.862645
$$457$$ 3924.00i 0.401656i 0.979626 + 0.200828i $$0.0643633\pi$$
−0.979626 + 0.200828i $$0.935637\pi$$
$$458$$ 7224.00i 0.737020i
$$459$$ 11600.0 1.17961
$$460$$ 0 0
$$461$$ −4543.00 −0.458977 −0.229489 0.973311i $$-0.573705\pi$$
−0.229489 + 0.973311i $$0.573705\pi$$
$$462$$ − 2720.00i − 0.273909i
$$463$$ − 9616.00i − 0.965213i −0.875837 0.482606i $$-0.839690\pi$$
0.875837 0.482606i $$-0.160310\pi$$
$$464$$ 3920.00 0.392201
$$465$$ 0 0
$$466$$ 8650.00 0.859879
$$467$$ 7826.00i 0.775469i 0.921771 + 0.387735i $$0.126742\pi$$
−0.921771 + 0.387735i $$0.873258\pi$$
$$468$$ 456.000i 0.0450398i
$$469$$ 7408.00 0.729360
$$470$$ 0 0
$$471$$ 3160.00 0.309140
$$472$$ − 9792.00i − 0.954901i
$$473$$ − 2992.00i − 0.290851i
$$474$$ 13220.0 1.28104
$$475$$ 0 0
$$476$$ −2560.00 −0.246507
$$477$$ 828.000i 0.0794791i
$$478$$ 5470.00i 0.523414i
$$479$$ −11404.0 −1.08781 −0.543906 0.839146i $$-0.683055\pi$$
−0.543906 + 0.839146i $$0.683055\pi$$
$$480$$ 0 0
$$481$$ 16986.0 1.61018
$$482$$ 13420.0i 1.26818i
$$483$$ − 920.000i − 0.0866697i
$$484$$ −700.000 −0.0657400
$$485$$ 0 0
$$486$$ 1120.00 0.104535
$$487$$ − 9267.00i − 0.862275i −0.902286 0.431137i $$-0.858112\pi$$
0.902286 0.431137i $$-0.141888\pi$$
$$488$$ − 19728.0i − 1.83001i
$$489$$ −15215.0 −1.40705
$$490$$ 0 0
$$491$$ −18191.0 −1.67199 −0.835996 0.548735i $$-0.815110\pi$$
−0.835996 + 0.548735i $$0.815110\pi$$
$$492$$ 1900.00i 0.174103i
$$493$$ 19600.0i 1.79055i
$$494$$ 7980.00 0.726796
$$495$$ 0 0
$$496$$ −1648.00 −0.149188
$$497$$ − 2680.00i − 0.241880i
$$498$$ 360.000i 0.0323935i
$$499$$ −19315.0 −1.73278 −0.866391 0.499366i $$-0.833566\pi$$
−0.866391 + 0.499366i $$0.833566\pi$$
$$500$$ 0 0
$$501$$ 11120.0 0.991627
$$502$$ 13896.0i 1.23548i
$$503$$ − 8422.00i − 0.746557i −0.927719 0.373279i $$-0.878234\pi$$
0.927719 0.373279i $$-0.121766\pi$$
$$504$$ −384.000 −0.0339379
$$505$$ 0 0
$$506$$ −1564.00 −0.137408
$$507$$ − 5260.00i − 0.460759i
$$508$$ 1044.00i 0.0911811i
$$509$$ 863.000 0.0751509 0.0375754 0.999294i $$-0.488037\pi$$
0.0375754 + 0.999294i $$0.488037\pi$$
$$510$$ 0 0
$$511$$ 7192.00 0.622613
$$512$$ 5632.00i 0.486136i
$$513$$ 10150.0i 0.873554i
$$514$$ −9858.00 −0.845949
$$515$$ 0 0
$$516$$ 1760.00 0.150154
$$517$$ − 12138.0i − 1.03255i
$$518$$ 4768.00i 0.404428i
$$519$$ 16150.0 1.36591
$$520$$ 0 0
$$521$$ 19260.0 1.61957 0.809785 0.586727i $$-0.199584\pi$$
0.809785 + 0.586727i $$0.199584\pi$$
$$522$$ 980.000i 0.0821713i
$$523$$ 11740.0i 0.981557i 0.871284 + 0.490779i $$0.163288\pi$$
−0.871284 + 0.490779i $$0.836712\pi$$
$$524$$ −5764.00 −0.480537
$$525$$ 0 0
$$526$$ −12276.0 −1.01760
$$527$$ − 8240.00i − 0.681101i
$$528$$ − 2720.00i − 0.224191i
$$529$$ −529.000 −0.0434783
$$530$$ 0 0
$$531$$ 816.000 0.0666881
$$532$$ − 2240.00i − 0.182549i
$$533$$ 5415.00i 0.440056i
$$534$$ 4600.00 0.372774
$$535$$ 0 0
$$536$$ 22224.0 1.79092
$$537$$ − 1845.00i − 0.148264i
$$538$$ − 4126.00i − 0.330640i
$$539$$ 9486.00 0.758054
$$540$$ 0 0
$$541$$ 17741.0 1.40988 0.704940 0.709267i $$-0.250974\pi$$
0.704940 + 0.709267i $$0.250974\pi$$
$$542$$ 2128.00i 0.168645i
$$543$$ − 6850.00i − 0.541366i
$$544$$ −12800.0 −1.00882
$$545$$ 0 0
$$546$$ 4560.00 0.357418
$$547$$ − 6571.00i − 0.513630i −0.966461 0.256815i $$-0.917327\pi$$
0.966461 0.256815i $$-0.0826731\pi$$
$$548$$ 6224.00i 0.485175i
$$549$$ 1644.00 0.127804
$$550$$ 0 0
$$551$$ −17150.0 −1.32598
$$552$$ − 2760.00i − 0.212814i
$$553$$ − 10576.0i − 0.813268i
$$554$$ 11458.0 0.878707
$$555$$ 0 0
$$556$$ −100.000 −0.00762760
$$557$$ − 1372.00i − 0.104369i −0.998637 0.0521845i $$-0.983382\pi$$
0.998637 0.0521845i $$-0.0166184\pi$$
$$558$$ − 412.000i − 0.0312569i
$$559$$ 5016.00 0.379524
$$560$$ 0 0
$$561$$ 13600.0 1.02352
$$562$$ 1920.00i 0.144111i
$$563$$ − 4332.00i − 0.324284i −0.986767 0.162142i $$-0.948160\pi$$
0.986767 0.162142i $$-0.0518403\pi$$
$$564$$ 7140.00 0.533064
$$565$$ 0 0
$$566$$ 228.000 0.0169321
$$567$$ 5368.00i 0.397592i
$$568$$ − 8040.00i − 0.593928i
$$569$$ 3546.00 0.261258 0.130629 0.991431i $$-0.458300\pi$$
0.130629 + 0.991431i $$0.458300\pi$$
$$570$$ 0 0
$$571$$ −6160.00 −0.451468 −0.225734 0.974189i $$-0.572478\pi$$
−0.225734 + 0.974189i $$0.572478\pi$$
$$572$$ 7752.00i 0.566656i
$$573$$ 22050.0i 1.60760i
$$574$$ −1520.00 −0.110529
$$575$$ 0 0
$$576$$ −896.000 −0.0648148
$$577$$ 2953.00i 0.213059i 0.994310 + 0.106529i $$0.0339738\pi$$
−0.994310 + 0.106529i $$0.966026\pi$$
$$578$$ 2974.00i 0.214017i
$$579$$ −675.000 −0.0484491
$$580$$ 0 0
$$581$$ 288.000 0.0205650
$$582$$ − 9640.00i − 0.686582i
$$583$$ 14076.0i 0.999946i
$$584$$ 21576.0 1.52880
$$585$$ 0 0
$$586$$ 14096.0 0.993687
$$587$$ − 2949.00i − 0.207356i −0.994611 0.103678i $$-0.966939\pi$$
0.994611 0.103678i $$-0.0330612\pi$$
$$588$$ 5580.00i 0.391353i
$$589$$ 7210.00 0.504385
$$590$$ 0 0
$$591$$ −6105.00 −0.424917
$$592$$ 4768.00i 0.331020i
$$593$$ − 16390.0i − 1.13500i −0.823372 0.567501i $$-0.807910\pi$$
0.823372 0.567501i $$-0.192090\pi$$
$$594$$ 9860.00 0.681079
$$595$$ 0 0
$$596$$ −3288.00 −0.225976
$$597$$ 5490.00i 0.376366i
$$598$$ − 2622.00i − 0.179300i
$$599$$ 12920.0 0.881297 0.440648 0.897680i $$-0.354749\pi$$
0.440648 + 0.897680i $$0.354749\pi$$
$$600$$ 0 0
$$601$$ −13835.0 −0.939004 −0.469502 0.882931i $$-0.655567\pi$$
−0.469502 + 0.882931i $$0.655567\pi$$
$$602$$ 1408.00i 0.0953252i
$$603$$ 1852.00i 0.125073i
$$604$$ −5956.00 −0.401235
$$605$$ 0 0
$$606$$ −3100.00 −0.207803
$$607$$ 6004.00i 0.401474i 0.979645 + 0.200737i $$0.0643337\pi$$
−0.979645 + 0.200737i $$0.935666\pi$$
$$608$$ − 11200.0i − 0.747072i
$$609$$ −9800.00 −0.652079
$$610$$ 0 0
$$611$$ 20349.0 1.34735
$$612$$ − 640.000i − 0.0422720i
$$613$$ 16416.0i 1.08162i 0.841143 + 0.540812i $$0.181883\pi$$
−0.841143 + 0.540812i $$0.818117\pi$$
$$614$$ 7744.00 0.508994
$$615$$ 0 0
$$616$$ −6528.00 −0.426982
$$617$$ 3786.00i 0.247032i 0.992343 + 0.123516i $$0.0394170\pi$$
−0.992343 + 0.123516i $$0.960583\pi$$
$$618$$ − 10440.0i − 0.679544i
$$619$$ −15824.0 −1.02750 −0.513748 0.857941i $$-0.671743\pi$$
−0.513748 + 0.857941i $$0.671743\pi$$
$$620$$ 0 0
$$621$$ 3335.00 0.215506
$$622$$ 9954.00i 0.641670i
$$623$$ − 3680.00i − 0.236655i
$$624$$ 4560.00 0.292542
$$625$$ 0 0
$$626$$ 5072.00 0.323830
$$627$$ 11900.0i 0.757959i
$$628$$ − 2528.00i − 0.160634i
$$629$$ −23840.0 −1.51123
$$630$$ 0 0
$$631$$ 17852.0 1.12627 0.563135 0.826365i $$-0.309595\pi$$
0.563135 + 0.826365i $$0.309595\pi$$
$$632$$ − 31728.0i − 1.99695i
$$633$$ − 18380.0i − 1.15409i
$$634$$ 2868.00 0.179657
$$635$$ 0 0
$$636$$ −8280.00 −0.516232
$$637$$ 15903.0i 0.989168i
$$638$$ 16660.0i 1.03382i
$$639$$ 670.000 0.0414785
$$640$$ 0 0
$$641$$ 10324.0 0.636152 0.318076 0.948065i $$-0.396963\pi$$
0.318076 + 0.948065i $$0.396963\pi$$
$$642$$ 4140.00i 0.254506i
$$643$$ 14702.0i 0.901696i 0.892601 + 0.450848i $$0.148878\pi$$
−0.892601 + 0.450848i $$0.851122\pi$$
$$644$$ −736.000 −0.0450349
$$645$$ 0 0
$$646$$ −11200.0 −0.682133
$$647$$ 11939.0i 0.725457i 0.931895 + 0.362728i $$0.118155\pi$$
−0.931895 + 0.362728i $$0.881845\pi$$
$$648$$ 16104.0i 0.976273i
$$649$$ 13872.0 0.839019
$$650$$ 0 0
$$651$$ 4120.00 0.248042
$$652$$ 12172.0i 0.731123i
$$653$$ − 6159.00i − 0.369097i −0.982823 0.184548i $$-0.940918\pi$$
0.982823 0.184548i $$-0.0590823\pi$$
$$654$$ −7040.00 −0.420926
$$655$$ 0 0
$$656$$ −1520.00 −0.0904665
$$657$$ 1798.00i 0.106768i
$$658$$ 5712.00i 0.338415i
$$659$$ 21692.0 1.28225 0.641123 0.767438i $$-0.278469\pi$$
0.641123 + 0.767438i $$0.278469\pi$$
$$660$$ 0 0
$$661$$ 16502.0 0.971034 0.485517 0.874227i $$-0.338631\pi$$
0.485517 + 0.874227i $$0.338631\pi$$
$$662$$ − 10938.0i − 0.642171i
$$663$$ 22800.0i 1.33556i
$$664$$ 864.000 0.0504965
$$665$$ 0 0
$$666$$ −1192.00 −0.0693529
$$667$$ 5635.00i 0.327119i
$$668$$ − 8896.00i − 0.515264i
$$669$$ 8280.00 0.478510
$$670$$ 0 0
$$671$$ 27948.0 1.60793
$$672$$ − 6400.00i − 0.367389i
$$673$$ 27733.0i 1.58845i 0.607622 + 0.794226i $$0.292124\pi$$
−0.607622 + 0.794226i $$0.707876\pi$$
$$674$$ −15592.0 −0.891070
$$675$$ 0 0
$$676$$ −4208.00 −0.239417
$$677$$ − 8814.00i − 0.500369i −0.968198 0.250184i $$-0.919509\pi$$
0.968198 0.250184i $$-0.0804912\pi$$
$$678$$ − 9520.00i − 0.539253i
$$679$$ −7712.00 −0.435875
$$680$$ 0 0
$$681$$ −14700.0 −0.827174
$$682$$ − 7004.00i − 0.393251i
$$683$$ 22999.0i 1.28848i 0.764823 + 0.644240i $$0.222826\pi$$
−0.764823 + 0.644240i $$0.777174\pi$$
$$684$$ 560.000 0.0313043
$$685$$ 0 0
$$686$$ −9952.00 −0.553891
$$687$$ − 18060.0i − 1.00296i
$$688$$ 1408.00i 0.0780225i
$$689$$ −23598.0 −1.30481
$$690$$ 0 0
$$691$$ −12140.0 −0.668346 −0.334173 0.942512i $$-0.608457\pi$$
−0.334173 + 0.942512i $$0.608457\pi$$
$$692$$ − 12920.0i − 0.709747i
$$693$$ − 544.000i − 0.0298194i
$$694$$ −20136.0 −1.10137
$$695$$ 0 0
$$696$$ −29400.0 −1.60116
$$697$$ − 7600.00i − 0.413014i
$$698$$ − 14990.0i − 0.812865i
$$699$$ −21625.0 −1.17015
$$700$$ 0 0
$$701$$ −20024.0 −1.07888 −0.539441 0.842024i $$-0.681364\pi$$
−0.539441 + 0.842024i $$0.681364\pi$$
$$702$$ 16530.0i 0.888725i
$$703$$ − 20860.0i − 1.11913i
$$704$$ −15232.0 −0.815451
$$705$$ 0 0
$$706$$ −21234.0 −1.13194
$$707$$ 2480.00i 0.131924i
$$708$$ 8160.00i 0.433152i
$$709$$ 4956.00 0.262520 0.131260 0.991348i $$-0.458098\pi$$
0.131260 + 0.991348i $$0.458098\pi$$
$$710$$ 0 0
$$711$$ 2644.00 0.139462
$$712$$ − 11040.0i − 0.581098i
$$713$$ − 2369.00i − 0.124432i
$$714$$ −6400.00 −0.335454
$$715$$ 0 0
$$716$$ −1476.00 −0.0770401
$$717$$ − 13675.0i − 0.712276i
$$718$$ 5044.00i 0.262173i
$$719$$ −2760.00 −0.143158 −0.0715790 0.997435i $$-0.522804\pi$$
−0.0715790 + 0.997435i $$0.522804\pi$$
$$720$$ 0 0
$$721$$ −8352.00 −0.431407
$$722$$ 3918.00i 0.201957i
$$723$$ − 33550.0i − 1.72578i
$$724$$ −5480.00 −0.281302
$$725$$ 0 0
$$726$$ −1750.00 −0.0894609
$$727$$ 7746.00i 0.395163i 0.980287 + 0.197581i $$0.0633086\pi$$
−0.980287 + 0.197581i $$0.936691\pi$$
$$728$$ − 10944.0i − 0.557159i
$$729$$ −20917.0 −1.06269
$$730$$ 0 0
$$731$$ −7040.00 −0.356202
$$732$$ 16440.0i 0.830109i
$$733$$ 11976.0i 0.603470i 0.953392 + 0.301735i $$0.0975658\pi$$
−0.953392 + 0.301735i $$0.902434\pi$$
$$734$$ 14408.0 0.724535
$$735$$ 0 0
$$736$$ −3680.00 −0.184302
$$737$$ 31484.0i 1.57358i
$$738$$ − 380.000i − 0.0189539i
$$739$$ −15057.0 −0.749500 −0.374750 0.927126i $$-0.622272\pi$$
−0.374750 + 0.927126i $$0.622272\pi$$
$$740$$ 0 0
$$741$$ −19950.0 −0.989044
$$742$$ − 6624.00i − 0.327729i
$$743$$ − 18532.0i − 0.915038i −0.889200 0.457519i $$-0.848738\pi$$
0.889200 0.457519i $$-0.151262\pi$$
$$744$$ 12360.0 0.609059
$$745$$ 0 0
$$746$$ 26620.0 1.30647
$$747$$ 72.0000i 0.00352656i
$$748$$ − 10880.0i − 0.531834i
$$749$$ 3312.00 0.161573
$$750$$ 0 0
$$751$$ −192.000 −0.00932913 −0.00466457 0.999989i $$-0.501485\pi$$
−0.00466457 + 0.999989i $$0.501485\pi$$
$$752$$ 5712.00i 0.276988i
$$753$$ − 34740.0i − 1.68127i
$$754$$ −27930.0 −1.34901
$$755$$ 0 0
$$756$$ 4640.00 0.223221
$$757$$ − 9830.00i − 0.471965i −0.971757 0.235982i $$-0.924169\pi$$
0.971757 0.235982i $$-0.0758308\pi$$
$$758$$ 25904.0i 1.24126i
$$759$$ 3910.00 0.186988
$$760$$ 0 0
$$761$$ −30219.0 −1.43947 −0.719736 0.694248i $$-0.755737\pi$$
−0.719736 + 0.694248i $$0.755737\pi$$
$$762$$ 2610.00i 0.124082i
$$763$$ 5632.00i 0.267224i
$$764$$ 17640.0 0.835331
$$765$$ 0 0
$$766$$ 5624.00 0.265279
$$767$$ 23256.0i 1.09482i
$$768$$ − 21760.0i − 1.02239i
$$769$$ −1122.00 −0.0526142 −0.0263071 0.999654i $$-0.508375\pi$$
−0.0263071 + 0.999654i $$0.508375\pi$$
$$770$$ 0 0
$$771$$ 24645.0 1.15119
$$772$$ 540.000i 0.0251749i
$$773$$ − 19300.0i − 0.898024i −0.893526 0.449012i $$-0.851776\pi$$
0.893526 0.449012i $$-0.148224\pi$$
$$774$$ −352.000 −0.0163467
$$775$$ 0 0
$$776$$ −23136.0 −1.07028
$$777$$ − 11920.0i − 0.550357i
$$778$$ 2528.00i 0.116495i
$$779$$ 6650.00 0.305855
$$780$$ 0 0
$$781$$ 11390.0 0.521852
$$782$$ 3680.00i 0.168282i
$$783$$ − 35525.0i − 1.62140i
$$784$$ −4464.00 −0.203353
$$785$$ 0 0
$$786$$ −14410.0 −0.653928
$$787$$ − 19396.0i − 0.878517i −0.898361 0.439258i $$-0.855241\pi$$
0.898361 0.439258i $$-0.144759\pi$$
$$788$$ 4884.00i 0.220794i
$$789$$ 30690.0 1.38478
$$790$$ 0 0
$$791$$ −7616.00 −0.342344
$$792$$ − 1632.00i − 0.0732204i
$$793$$ 46854.0i 2.09815i
$$794$$ 14238.0 0.636383
$$795$$ 0 0
$$796$$ 4392.00 0.195566
$$797$$ − 39034.0i − 1.73482i −0.497590 0.867412i $$-0.665782\pi$$
0.497590 0.867412i $$-0.334218\pi$$
$$798$$ − 5600.00i − 0.248418i
$$799$$ −28560.0 −1.26456
$$800$$ 0 0
$$801$$ 920.000 0.0405825
$$802$$ − 8524.00i − 0.375303i
$$803$$ 30566.0i 1.34328i
$$804$$ −18520.0 −0.812376
$$805$$ 0 0
$$806$$ 11742.0 0.513144
$$807$$ 10315.0i 0.449944i
$$808$$ 7440.00i 0.323934i
$$809$$ 10310.0 0.448060 0.224030 0.974582i $$-0.428079\pi$$
0.224030 + 0.974582i $$0.428079\pi$$
$$810$$ 0 0
$$811$$ −40693.0 −1.76193 −0.880965 0.473182i $$-0.843105\pi$$
−0.880965 + 0.473182i $$0.843105\pi$$
$$812$$ 7840.00i 0.338830i
$$813$$ − 5320.00i − 0.229496i
$$814$$ −20264.0 −0.872546
$$815$$ 0 0
$$816$$ −6400.00 −0.274565
$$817$$ − 6160.00i − 0.263784i
$$818$$ 458.000i 0.0195765i
$$819$$ 912.000 0.0389107
$$820$$ 0 0
$$821$$ −13934.0 −0.592326 −0.296163 0.955137i $$-0.595707\pi$$
−0.296163 + 0.955137i $$0.595707\pi$$
$$822$$ 15560.0i 0.660240i
$$823$$ − 6175.00i − 0.261539i −0.991413 0.130770i $$-0.958255\pi$$
0.991413 0.130770i $$-0.0417449\pi$$
$$824$$ −25056.0 −1.05930
$$825$$ 0 0
$$826$$ −6528.00 −0.274986
$$827$$ 28664.0i 1.20525i 0.798023 + 0.602627i $$0.205879\pi$$
−0.798023 + 0.602627i $$0.794121\pi$$
$$828$$ − 184.000i − 0.00772276i
$$829$$ 39590.0 1.65865 0.829323 0.558770i $$-0.188726\pi$$
0.829323 + 0.558770i $$0.188726\pi$$
$$830$$ 0 0
$$831$$ −28645.0 −1.19577
$$832$$ − 25536.0i − 1.06406i
$$833$$ − 22320.0i − 0.928382i
$$834$$ −250.000 −0.0103798
$$835$$ 0 0
$$836$$ 9520.00 0.393847
$$837$$ 14935.0i 0.616761i
$$838$$ 31552.0i 1.30065i
$$839$$ 14316.0 0.589086 0.294543 0.955638i $$-0.404833\pi$$
0.294543 + 0.955638i $$0.404833\pi$$
$$840$$ 0 0
$$841$$ 35636.0 1.46115
$$842$$ 17456.0i 0.714458i
$$843$$ − 4800.00i − 0.196110i
$$844$$ −14704.0 −0.599683
$$845$$ 0 0
$$846$$ −1428.00 −0.0580327
$$847$$ 1400.00i 0.0567941i
$$848$$ − 6624.00i − 0.268242i
$$849$$ −570.000 −0.0230416
$$850$$ 0 0
$$851$$ −6854.00 −0.276089
$$852$$ 6700.00i 0.269411i
$$853$$ − 28366.0i − 1.13861i −0.822127 0.569304i $$-0.807213\pi$$
0.822127 0.569304i $$-0.192787\pi$$
$$854$$ −13152.0 −0.526993
$$855$$ 0 0
$$856$$ 9936.00 0.396735
$$857$$ 19283.0i 0.768605i 0.923207 + 0.384303i $$0.125558\pi$$
−0.923207 + 0.384303i $$0.874442\pi$$
$$858$$ 19380.0i 0.771122i
$$859$$ 26101.0 1.03673 0.518367 0.855158i $$-0.326540\pi$$
0.518367 + 0.855158i $$0.326540\pi$$
$$860$$ 0 0
$$861$$ 3800.00 0.150411
$$862$$ 5856.00i 0.231388i
$$863$$ − 973.000i − 0.0383793i −0.999816 0.0191896i $$-0.993891\pi$$
0.999816 0.0191896i $$-0.00610862\pi$$
$$864$$ 23200.0 0.913519
$$865$$ 0 0
$$866$$ 10628.0 0.417037
$$867$$ − 7435.00i − 0.291241i
$$868$$ − 3296.00i − 0.128887i
$$869$$ 44948.0 1.75461
$$870$$ 0 0
$$871$$ −52782.0 −2.05333
$$872$$ 16896.0i 0.656159i
$$873$$ − 1928.00i − 0.0747456i
$$874$$ −3220.00 −0.124620
$$875$$ 0 0
$$876$$ −17980.0 −0.693479
$$877$$ 5694.00i 0.219239i 0.993974 + 0.109620i $$0.0349633\pi$$
−0.993974 + 0.109620i $$0.965037\pi$$
$$878$$ 5170.00i 0.198723i
$$879$$ −35240.0 −1.35224
$$880$$ 0 0
$$881$$ 45960.0 1.75758 0.878792 0.477205i $$-0.158350\pi$$
0.878792 + 0.477205i $$0.158350\pi$$
$$882$$ − 1116.00i − 0.0426051i
$$883$$ − 17188.0i − 0.655065i −0.944840 0.327532i $$-0.893783\pi$$
0.944840 0.327532i $$-0.106217\pi$$
$$884$$ 18240.0 0.693979
$$885$$ 0 0
$$886$$ 5994.00 0.227283
$$887$$ 8451.00i 0.319906i 0.987125 + 0.159953i $$0.0511343\pi$$
−0.987125 + 0.159953i $$0.948866\pi$$
$$888$$ − 35760.0i − 1.35138i
$$889$$ 2088.00 0.0787731
$$890$$ 0 0
$$891$$ −22814.0 −0.857798
$$892$$ − 6624.00i − 0.248641i
$$893$$ − 24990.0i − 0.936460i
$$894$$ −8220.00 −0.307514
$$895$$ 0 0
$$896$$ −3072.00 −0.114541
$$897$$ 6555.00i 0.243997i
$$898$$ − 33124.0i − 1.23092i
$$899$$ −25235.0 −0.936190
$$900$$ 0 0
$$901$$ 33120.0 1.22463
$$902$$ − 6460.00i − 0.238464i
$$903$$ − 3520.00i − 0.129721i
$$904$$ −22848.0 −0.840612
$$905$$ 0 0
$$906$$ −14890.0 −0.546012
$$907$$ 32774.0i 1.19983i 0.800065 + 0.599913i $$0.204798\pi$$
−0.800065 + 0.599913i $$0.795202\pi$$
$$908$$ 11760.0i 0.429812i
$$909$$ −620.000 −0.0226228
$$910$$ 0 0
$$911$$ −23690.0 −0.861564 −0.430782 0.902456i $$-0.641762\pi$$
−0.430782 + 0.902456i $$0.641762\pi$$
$$912$$ − 5600.00i − 0.203327i
$$913$$ 1224.00i 0.0443686i
$$914$$ 7848.00 0.284014
$$915$$ 0 0
$$916$$ −14448.0 −0.521152
$$917$$ 11528.0i 0.415145i
$$918$$ − 23200.0i − 0.834111i
$$919$$ 30044.0 1.07841 0.539206 0.842174i $$-0.318725\pi$$
0.539206 + 0.842174i $$0.318725\pi$$
$$920$$ 0 0
$$921$$ −19360.0 −0.692653
$$922$$ 9086.00i 0.324546i
$$923$$ 19095.0i 0.680953i
$$924$$ 5440.00 0.193683
$$925$$ 0 0
$$926$$ −19232.0 −0.682508
$$927$$ − 2088.00i − 0.0739794i
$$928$$ 39200.0i 1.38664i
$$929$$ 39705.0 1.40224 0.701119 0.713044i $$-0.252684\pi$$
0.701119 + 0.713044i $$0.252684\pi$$
$$930$$ 0 0
$$931$$ 19530.0 0.687508
$$932$$ 17300.0i 0.608026i
$$933$$ − 24885.0i − 0.873203i
$$934$$ 15652.0 0.548339
$$935$$ 0 0
$$936$$ 2736.00 0.0955438
$$937$$ 17422.0i 0.607419i 0.952765 + 0.303710i $$0.0982253\pi$$
−0.952765 + 0.303710i $$0.901775\pi$$
$$938$$ − 14816.0i − 0.515735i
$$939$$ −12680.0 −0.440677
$$940$$ 0 0
$$941$$ −25292.0 −0.876191 −0.438095 0.898928i $$-0.644347\pi$$
−0.438095 + 0.898928i $$0.644347\pi$$
$$942$$ − 6320.00i − 0.218595i
$$943$$ − 2185.00i − 0.0754543i
$$944$$ −6528.00 −0.225072
$$945$$ 0 0
$$946$$ −5984.00 −0.205662
$$947$$ 33211.0i 1.13961i 0.821779 + 0.569806i $$0.192982\pi$$
−0.821779 + 0.569806i $$0.807018\pi$$
$$948$$ 26440.0i 0.905835i
$$949$$ −51243.0 −1.75281
$$950$$ 0 0
$$951$$ −7170.00 −0.244483
$$952$$ 15360.0i 0.522921i
$$953$$ 14154.0i 0.481105i 0.970636 + 0.240552i $$0.0773286\pi$$
−0.970636 + 0.240552i $$0.922671\pi$$
$$954$$ 1656.00 0.0562002
$$955$$ 0 0
$$956$$ −10940.0 −0.370110
$$957$$ − 41650.0i − 1.40685i
$$958$$ 22808.0i 0.769199i
$$959$$ 12448.0 0.419152
$$960$$ 0 0
$$961$$ −19182.0 −0.643886
$$962$$ − 33972.0i − 1.13857i
$$963$$ 828.000i 0.0277071i
$$964$$ −26840.0 −0.896741
$$965$$ 0 0
$$966$$ −1840.00 −0.0612847
$$967$$ − 46343.0i − 1.54115i −0.637350 0.770574i $$-0.719970\pi$$
0.637350 0.770574i $$-0.280030\pi$$
$$968$$ 4200.00i 0.139456i
$$969$$ 28000.0 0.928266
$$970$$ 0 0
$$971$$ 11710.0 0.387015 0.193508 0.981099i $$-0.438014\pi$$
0.193508 + 0.981099i $$0.438014\pi$$
$$972$$ 2240.00i 0.0739177i
$$973$$ 200.000i 0.00658963i
$$974$$ −18534.0 −0.609720
$$975$$ 0 0
$$976$$ −13152.0 −0.431337
$$977$$ 47854.0i 1.56703i 0.621375 + 0.783513i $$0.286574\pi$$
−0.621375 + 0.783513i $$0.713426\pi$$
$$978$$ 30430.0i 0.994933i
$$979$$ 15640.0 0.510579
$$980$$ 0 0
$$981$$ −1408.00 −0.0458246
$$982$$ 36382.0i 1.18228i
$$983$$ 22078.0i 0.716357i 0.933653 + 0.358178i $$0.116602\pi$$
−0.933653 + 0.358178i $$0.883398\pi$$
$$984$$ 11400.0 0.369328
$$985$$ 0 0
$$986$$ 39200.0 1.26611
$$987$$ − 14280.0i − 0.460524i
$$988$$ 15960.0i 0.513922i
$$989$$ −2024.00 −0.0650753
$$990$$ 0 0
$$991$$ −4288.00 −0.137450 −0.0687249 0.997636i $$-0.521893\pi$$
−0.0687249 + 0.997636i $$0.521893\pi$$
$$992$$ − 16480.0i − 0.527460i
$$993$$ 27345.0i 0.873885i
$$994$$ −5360.00 −0.171035
$$995$$ 0 0
$$996$$ −720.000 −0.0229057
$$997$$ 28966.0i 0.920123i 0.887887 + 0.460061i $$0.152173\pi$$
−0.887887 + 0.460061i $$0.847827\pi$$
$$998$$ 38630.0i 1.22526i
$$999$$ 43210.0 1.36847
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 575.4.b.b.24.1 2
5.2 odd 4 575.4.a.g.1.1 1
5.3 odd 4 23.4.a.a.1.1 1
5.4 even 2 inner 575.4.b.b.24.2 2
15.8 even 4 207.4.a.a.1.1 1
20.3 even 4 368.4.a.d.1.1 1
35.13 even 4 1127.4.a.a.1.1 1
40.3 even 4 1472.4.a.c.1.1 1
40.13 odd 4 1472.4.a.h.1.1 1
115.68 even 4 529.4.a.a.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
23.4.a.a.1.1 1 5.3 odd 4
207.4.a.a.1.1 1 15.8 even 4
368.4.a.d.1.1 1 20.3 even 4
529.4.a.a.1.1 1 115.68 even 4
575.4.a.g.1.1 1 5.2 odd 4
575.4.b.b.24.1 2 1.1 even 1 trivial
575.4.b.b.24.2 2 5.4 even 2 inner
1127.4.a.a.1.1 1 35.13 even 4
1472.4.a.c.1.1 1 40.3 even 4
1472.4.a.h.1.1 1 40.13 odd 4