# Properties

 Label 600.4.f.b.49.2 Level $600$ Weight $4$ Character 600.49 Analytic conductor $35.401$ 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] = [600,4,Mod(49,600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(600, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("600.49");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$600 = 2^{3} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 600.f (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: $$35.4011460034$$ 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: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 24) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 49.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 600.49 Dual form 600.4.f.b.49.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+3.00000i q^{3} +24.0000i q^{7} -9.00000 q^{9} +O(q^{10})$$ $$q+3.00000i q^{3} +24.0000i q^{7} -9.00000 q^{9} -28.0000 q^{11} -74.0000i q^{13} -82.0000i q^{17} -92.0000 q^{19} -72.0000 q^{21} +8.00000i q^{23} -27.0000i q^{27} +138.000 q^{29} +80.0000 q^{31} -84.0000i q^{33} -30.0000i q^{37} +222.000 q^{39} +282.000 q^{41} +4.00000i q^{43} -240.000i q^{47} -233.000 q^{49} +246.000 q^{51} -130.000i q^{53} -276.000i q^{57} -596.000 q^{59} -218.000 q^{61} -216.000i q^{63} +436.000i q^{67} -24.0000 q^{69} +856.000 q^{71} -998.000i q^{73} -672.000i q^{77} +32.0000 q^{79} +81.0000 q^{81} -1508.00i q^{83} +414.000i q^{87} +246.000 q^{89} +1776.00 q^{91} +240.000i q^{93} -866.000i q^{97} +252.000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 18 q^{9}+O(q^{10})$$ 2 * q - 18 * q^9 $$2 q - 18 q^{9} - 56 q^{11} - 184 q^{19} - 144 q^{21} + 276 q^{29} + 160 q^{31} + 444 q^{39} + 564 q^{41} - 466 q^{49} + 492 q^{51} - 1192 q^{59} - 436 q^{61} - 48 q^{69} + 1712 q^{71} + 64 q^{79} + 162 q^{81} + 492 q^{89} + 3552 q^{91} + 504 q^{99}+O(q^{100})$$ 2 * q - 18 * q^9 - 56 * q^11 - 184 * q^19 - 144 * q^21 + 276 * q^29 + 160 * q^31 + 444 * q^39 + 564 * q^41 - 466 * q^49 + 492 * q^51 - 1192 * q^59 - 436 * q^61 - 48 * q^69 + 1712 * q^71 + 64 * q^79 + 162 * q^81 + 492 * q^89 + 3552 * q^91 + 504 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/600\mathbb{Z}\right)^\times$$.

 $$n$$ $$151$$ $$301$$ $$401$$ $$577$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-1$$

## 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
$$3$$ 3.00000i 0.577350i
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 24.0000i 1.29588i 0.761692 + 0.647939i $$0.224369\pi$$
−0.761692 + 0.647939i $$0.775631\pi$$
$$8$$ 0 0
$$9$$ −9.00000 −0.333333
$$10$$ 0 0
$$11$$ −28.0000 −0.767483 −0.383742 0.923440i $$-0.625365\pi$$
−0.383742 + 0.923440i $$0.625365\pi$$
$$12$$ 0 0
$$13$$ − 74.0000i − 1.57876i −0.613904 0.789381i $$-0.710402\pi$$
0.613904 0.789381i $$-0.289598\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ − 82.0000i − 1.16988i −0.811077 0.584939i $$-0.801118\pi$$
0.811077 0.584939i $$-0.198882\pi$$
$$18$$ 0 0
$$19$$ −92.0000 −1.11086 −0.555428 0.831565i $$-0.687445\pi$$
−0.555428 + 0.831565i $$0.687445\pi$$
$$20$$ 0 0
$$21$$ −72.0000 −0.748176
$$22$$ 0 0
$$23$$ 8.00000i 0.0725268i 0.999342 + 0.0362634i $$0.0115455\pi$$
−0.999342 + 0.0362634i $$0.988454\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ − 27.0000i − 0.192450i
$$28$$ 0 0
$$29$$ 138.000 0.883654 0.441827 0.897100i $$-0.354331\pi$$
0.441827 + 0.897100i $$0.354331\pi$$
$$30$$ 0 0
$$31$$ 80.0000 0.463498 0.231749 0.972776i $$-0.425555\pi$$
0.231749 + 0.972776i $$0.425555\pi$$
$$32$$ 0 0
$$33$$ − 84.0000i − 0.443107i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ − 30.0000i − 0.133296i −0.997777 0.0666482i $$-0.978769\pi$$
0.997777 0.0666482i $$-0.0212305\pi$$
$$38$$ 0 0
$$39$$ 222.000 0.911499
$$40$$ 0 0
$$41$$ 282.000 1.07417 0.537085 0.843528i $$-0.319525\pi$$
0.537085 + 0.843528i $$0.319525\pi$$
$$42$$ 0 0
$$43$$ 4.00000i 0.0141859i 0.999975 + 0.00709296i $$0.00225778\pi$$
−0.999975 + 0.00709296i $$0.997742\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ − 240.000i − 0.744843i −0.928064 0.372421i $$-0.878528\pi$$
0.928064 0.372421i $$-0.121472\pi$$
$$48$$ 0 0
$$49$$ −233.000 −0.679300
$$50$$ 0 0
$$51$$ 246.000 0.675429
$$52$$ 0 0
$$53$$ − 130.000i − 0.336922i −0.985708 0.168461i $$-0.946120\pi$$
0.985708 0.168461i $$-0.0538797\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ − 276.000i − 0.641353i
$$58$$ 0 0
$$59$$ −596.000 −1.31513 −0.657564 0.753398i $$-0.728413\pi$$
−0.657564 + 0.753398i $$0.728413\pi$$
$$60$$ 0 0
$$61$$ −218.000 −0.457574 −0.228787 0.973476i $$-0.573476\pi$$
−0.228787 + 0.973476i $$0.573476\pi$$
$$62$$ 0 0
$$63$$ − 216.000i − 0.431959i
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 436.000i 0.795013i 0.917599 + 0.397507i $$0.130124\pi$$
−0.917599 + 0.397507i $$0.869876\pi$$
$$68$$ 0 0
$$69$$ −24.0000 −0.0418733
$$70$$ 0 0
$$71$$ 856.000 1.43082 0.715412 0.698703i $$-0.246239\pi$$
0.715412 + 0.698703i $$0.246239\pi$$
$$72$$ 0 0
$$73$$ − 998.000i − 1.60010i −0.599935 0.800048i $$-0.704807\pi$$
0.599935 0.800048i $$-0.295193\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ − 672.000i − 0.994565i
$$78$$ 0 0
$$79$$ 32.0000 0.0455732 0.0227866 0.999740i $$-0.492746\pi$$
0.0227866 + 0.999740i $$0.492746\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 0 0
$$83$$ − 1508.00i − 1.99427i −0.0756351 0.997136i $$-0.524098\pi$$
0.0756351 0.997136i $$-0.475902\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 414.000i 0.510178i
$$88$$ 0 0
$$89$$ 246.000 0.292988 0.146494 0.989212i $$-0.453201\pi$$
0.146494 + 0.989212i $$0.453201\pi$$
$$90$$ 0 0
$$91$$ 1776.00 2.04588
$$92$$ 0 0
$$93$$ 240.000i 0.267600i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ − 866.000i − 0.906484i −0.891387 0.453242i $$-0.850267\pi$$
0.891387 0.453242i $$-0.149733\pi$$
$$98$$ 0 0
$$99$$ 252.000 0.255828
$$100$$ 0 0
$$101$$ 270.000 0.266000 0.133000 0.991116i $$-0.457539\pi$$
0.133000 + 0.991116i $$0.457539\pi$$
$$102$$ 0 0
$$103$$ − 1496.00i − 1.43112i −0.698552 0.715560i $$-0.746172\pi$$
0.698552 0.715560i $$-0.253828\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 1692.00i 1.52871i 0.644797 + 0.764354i $$0.276942\pi$$
−0.644797 + 0.764354i $$0.723058\pi$$
$$108$$ 0 0
$$109$$ −406.000 −0.356768 −0.178384 0.983961i $$-0.557087\pi$$
−0.178384 + 0.983961i $$0.557087\pi$$
$$110$$ 0 0
$$111$$ 90.0000 0.0769588
$$112$$ 0 0
$$113$$ 786.000i 0.654342i 0.944965 + 0.327171i $$0.106095\pi$$
−0.944965 + 0.327171i $$0.893905\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 666.000i 0.526254i
$$118$$ 0 0
$$119$$ 1968.00 1.51602
$$120$$ 0 0
$$121$$ −547.000 −0.410969
$$122$$ 0 0
$$123$$ 846.000i 0.620173i
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ − 1744.00i − 1.21854i −0.792962 0.609272i $$-0.791462\pi$$
0.792962 0.609272i $$-0.208538\pi$$
$$128$$ 0 0
$$129$$ −12.0000 −0.00819024
$$130$$ 0 0
$$131$$ 652.000 0.434851 0.217426 0.976077i $$-0.430234\pi$$
0.217426 + 0.976077i $$0.430234\pi$$
$$132$$ 0 0
$$133$$ − 2208.00i − 1.43953i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ − 1530.00i − 0.954137i −0.878866 0.477068i $$-0.841699\pi$$
0.878866 0.477068i $$-0.158301\pi$$
$$138$$ 0 0
$$139$$ −516.000 −0.314867 −0.157434 0.987530i $$-0.550322\pi$$
−0.157434 + 0.987530i $$0.550322\pi$$
$$140$$ 0 0
$$141$$ 720.000 0.430035
$$142$$ 0 0
$$143$$ 2072.00i 1.21167i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ − 699.000i − 0.392194i
$$148$$ 0 0
$$149$$ −1342.00 −0.737859 −0.368929 0.929457i $$-0.620276\pi$$
−0.368929 + 0.929457i $$0.620276\pi$$
$$150$$ 0 0
$$151$$ −424.000 −0.228507 −0.114254 0.993452i $$-0.536448\pi$$
−0.114254 + 0.993452i $$0.536448\pi$$
$$152$$ 0 0
$$153$$ 738.000i 0.389959i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ − 262.000i − 0.133184i −0.997780 0.0665920i $$-0.978787\pi$$
0.997780 0.0665920i $$-0.0212126\pi$$
$$158$$ 0 0
$$159$$ 390.000 0.194522
$$160$$ 0 0
$$161$$ −192.000 −0.0939858
$$162$$ 0 0
$$163$$ − 2292.00i − 1.10137i −0.834713 0.550685i $$-0.814367\pi$$
0.834713 0.550685i $$-0.185633\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 1896.00i 0.878544i 0.898354 + 0.439272i $$0.144764\pi$$
−0.898354 + 0.439272i $$0.855236\pi$$
$$168$$ 0 0
$$169$$ −3279.00 −1.49249
$$170$$ 0 0
$$171$$ 828.000 0.370285
$$172$$ 0 0
$$173$$ − 2874.00i − 1.26304i −0.775359 0.631521i $$-0.782431\pi$$
0.775359 0.631521i $$-0.217569\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ − 1788.00i − 0.759290i
$$178$$ 0 0
$$179$$ 1188.00 0.496063 0.248032 0.968752i $$-0.420216\pi$$
0.248032 + 0.968752i $$0.420216\pi$$
$$180$$ 0 0
$$181$$ −3474.00 −1.42663 −0.713316 0.700843i $$-0.752808\pi$$
−0.713316 + 0.700843i $$0.752808\pi$$
$$182$$ 0 0
$$183$$ − 654.000i − 0.264181i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 2296.00i 0.897862i
$$188$$ 0 0
$$189$$ 648.000 0.249392
$$190$$ 0 0
$$191$$ 192.000 0.0727363 0.0363681 0.999338i $$-0.488421\pi$$
0.0363681 + 0.999338i $$0.488421\pi$$
$$192$$ 0 0
$$193$$ 4802.00i 1.79096i 0.445100 + 0.895481i $$0.353168\pi$$
−0.445100 + 0.895481i $$0.646832\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ − 1518.00i − 0.549000i −0.961587 0.274500i $$-0.911488\pi$$
0.961587 0.274500i $$-0.0885123\pi$$
$$198$$ 0 0
$$199$$ −5128.00 −1.82670 −0.913352 0.407170i $$-0.866516\pi$$
−0.913352 + 0.407170i $$0.866516\pi$$
$$200$$ 0 0
$$201$$ −1308.00 −0.459001
$$202$$ 0 0
$$203$$ 3312.00i 1.14511i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ − 72.0000i − 0.0241756i
$$208$$ 0 0
$$209$$ 2576.00 0.852563
$$210$$ 0 0
$$211$$ 1084.00 0.353676 0.176838 0.984240i $$-0.443413\pi$$
0.176838 + 0.984240i $$0.443413\pi$$
$$212$$ 0 0
$$213$$ 2568.00i 0.826087i
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 1920.00i 0.600636i
$$218$$ 0 0
$$219$$ 2994.00 0.923816
$$220$$ 0 0
$$221$$ −6068.00 −1.84696
$$222$$ 0 0
$$223$$ 688.000i 0.206600i 0.994650 + 0.103300i $$0.0329402\pi$$
−0.994650 + 0.103300i $$0.967060\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ − 4812.00i − 1.40698i −0.710707 0.703488i $$-0.751625\pi$$
0.710707 0.703488i $$-0.248375\pi$$
$$228$$ 0 0
$$229$$ −2494.00 −0.719686 −0.359843 0.933013i $$-0.617170\pi$$
−0.359843 + 0.933013i $$0.617170\pi$$
$$230$$ 0 0
$$231$$ 2016.00 0.574212
$$232$$ 0 0
$$233$$ 698.000i 0.196255i 0.995174 + 0.0981277i $$0.0312854\pi$$
−0.995174 + 0.0981277i $$0.968715\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 96.0000i 0.0263117i
$$238$$ 0 0
$$239$$ 6320.00 1.71049 0.855244 0.518225i $$-0.173407\pi$$
0.855244 + 0.518225i $$0.173407\pi$$
$$240$$ 0 0
$$241$$ −6510.00 −1.74002 −0.870012 0.493030i $$-0.835889\pi$$
−0.870012 + 0.493030i $$0.835889\pi$$
$$242$$ 0 0
$$243$$ 243.000i 0.0641500i
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 6808.00i 1.75378i
$$248$$ 0 0
$$249$$ 4524.00 1.15139
$$250$$ 0 0
$$251$$ 628.000 0.157924 0.0789622 0.996878i $$-0.474839\pi$$
0.0789622 + 0.996878i $$0.474839\pi$$
$$252$$ 0 0
$$253$$ − 224.000i − 0.0556631i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 4862.00i 1.18009i 0.807370 + 0.590045i $$0.200890\pi$$
−0.807370 + 0.590045i $$0.799110\pi$$
$$258$$ 0 0
$$259$$ 720.000 0.172736
$$260$$ 0 0
$$261$$ −1242.00 −0.294551
$$262$$ 0 0
$$263$$ 5816.00i 1.36361i 0.731533 + 0.681806i $$0.238805\pi$$
−0.731533 + 0.681806i $$0.761195\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 738.000i 0.169157i
$$268$$ 0 0
$$269$$ −3526.00 −0.799197 −0.399599 0.916690i $$-0.630850\pi$$
−0.399599 + 0.916690i $$0.630850\pi$$
$$270$$ 0 0
$$271$$ −256.000 −0.0573834 −0.0286917 0.999588i $$-0.509134\pi$$
−0.0286917 + 0.999588i $$0.509134\pi$$
$$272$$ 0 0
$$273$$ 5328.00i 1.18119i
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ − 142.000i − 0.0308013i −0.999881 0.0154006i $$-0.995098\pi$$
0.999881 0.0154006i $$-0.00490237\pi$$
$$278$$ 0 0
$$279$$ −720.000 −0.154499
$$280$$ 0 0
$$281$$ 8842.00 1.87712 0.938558 0.345122i $$-0.112162\pi$$
0.938558 + 0.345122i $$0.112162\pi$$
$$282$$ 0 0
$$283$$ − 7180.00i − 1.50815i −0.656788 0.754075i $$-0.728085\pi$$
0.656788 0.754075i $$-0.271915\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 6768.00i 1.39199i
$$288$$ 0 0
$$289$$ −1811.00 −0.368614
$$290$$ 0 0
$$291$$ 2598.00 0.523359
$$292$$ 0 0
$$293$$ 7374.00i 1.47029i 0.677912 + 0.735143i $$0.262885\pi$$
−0.677912 + 0.735143i $$0.737115\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 756.000i 0.147702i
$$298$$ 0 0
$$299$$ 592.000 0.114502
$$300$$ 0 0
$$301$$ −96.0000 −0.0183832
$$302$$ 0 0
$$303$$ 810.000i 0.153575i
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ − 1500.00i − 0.278858i −0.990232 0.139429i $$-0.955473\pi$$
0.990232 0.139429i $$-0.0445268\pi$$
$$308$$ 0 0
$$309$$ 4488.00 0.826257
$$310$$ 0 0
$$311$$ −7608.00 −1.38717 −0.693585 0.720374i $$-0.743970\pi$$
−0.693585 + 0.720374i $$0.743970\pi$$
$$312$$ 0 0
$$313$$ − 4758.00i − 0.859227i −0.903013 0.429614i $$-0.858650\pi$$
0.903013 0.429614i $$-0.141350\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ − 4374.00i − 0.774979i −0.921874 0.387489i $$-0.873342\pi$$
0.921874 0.387489i $$-0.126658\pi$$
$$318$$ 0 0
$$319$$ −3864.00 −0.678190
$$320$$ 0 0
$$321$$ −5076.00 −0.882600
$$322$$ 0 0
$$323$$ 7544.00i 1.29956i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ − 1218.00i − 0.205980i
$$328$$ 0 0
$$329$$ 5760.00 0.965225
$$330$$ 0 0
$$331$$ −7804.00 −1.29591 −0.647956 0.761678i $$-0.724376\pi$$
−0.647956 + 0.761678i $$0.724376\pi$$
$$332$$ 0 0
$$333$$ 270.000i 0.0444322i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ − 5106.00i − 0.825346i −0.910879 0.412673i $$-0.864595\pi$$
0.910879 0.412673i $$-0.135405\pi$$
$$338$$ 0 0
$$339$$ −2358.00 −0.377785
$$340$$ 0 0
$$341$$ −2240.00 −0.355727
$$342$$ 0 0
$$343$$ 2640.00i 0.415588i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 4716.00i 0.729591i 0.931088 + 0.364796i $$0.118861\pi$$
−0.931088 + 0.364796i $$0.881139\pi$$
$$348$$ 0 0
$$349$$ −7302.00 −1.11996 −0.559982 0.828505i $$-0.689192\pi$$
−0.559982 + 0.828505i $$0.689192\pi$$
$$350$$ 0 0
$$351$$ −1998.00 −0.303833
$$352$$ 0 0
$$353$$ − 4382.00i − 0.660709i −0.943857 0.330355i $$-0.892832\pi$$
0.943857 0.330355i $$-0.107168\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 5904.00i 0.875274i
$$358$$ 0 0
$$359$$ −7224.00 −1.06203 −0.531014 0.847363i $$-0.678189\pi$$
−0.531014 + 0.847363i $$0.678189\pi$$
$$360$$ 0 0
$$361$$ 1605.00 0.233999
$$362$$ 0 0
$$363$$ − 1641.00i − 0.237273i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ − 1408.00i − 0.200264i −0.994974 0.100132i $$-0.968073\pi$$
0.994974 0.100132i $$-0.0319266\pi$$
$$368$$ 0 0
$$369$$ −2538.00 −0.358057
$$370$$ 0 0
$$371$$ 3120.00 0.436610
$$372$$ 0 0
$$373$$ − 1714.00i − 0.237929i −0.992899 0.118965i $$-0.962043\pi$$
0.992899 0.118965i $$-0.0379575\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ − 10212.0i − 1.39508i
$$378$$ 0 0
$$379$$ −884.000 −0.119810 −0.0599051 0.998204i $$-0.519080\pi$$
−0.0599051 + 0.998204i $$0.519080\pi$$
$$380$$ 0 0
$$381$$ 5232.00 0.703526
$$382$$ 0 0
$$383$$ 10368.0i 1.38324i 0.722263 + 0.691619i $$0.243102\pi$$
−0.722263 + 0.691619i $$0.756898\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ − 36.0000i − 0.00472864i
$$388$$ 0 0
$$389$$ −398.000 −0.0518751 −0.0259375 0.999664i $$-0.508257\pi$$
−0.0259375 + 0.999664i $$0.508257\pi$$
$$390$$ 0 0
$$391$$ 656.000 0.0848474
$$392$$ 0 0
$$393$$ 1956.00i 0.251061i
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 5098.00i 0.644487i 0.946657 + 0.322243i $$0.104437\pi$$
−0.946657 + 0.322243i $$0.895563\pi$$
$$398$$ 0 0
$$399$$ 6624.00 0.831115
$$400$$ 0 0
$$401$$ 10002.0 1.24558 0.622788 0.782391i $$-0.286000\pi$$
0.622788 + 0.782391i $$0.286000\pi$$
$$402$$ 0 0
$$403$$ − 5920.00i − 0.731752i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 840.000i 0.102303i
$$408$$ 0 0
$$409$$ 9270.00 1.12071 0.560357 0.828251i $$-0.310664\pi$$
0.560357 + 0.828251i $$0.310664\pi$$
$$410$$ 0 0
$$411$$ 4590.00 0.550871
$$412$$ 0 0
$$413$$ − 14304.0i − 1.70425i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ − 1548.00i − 0.181789i
$$418$$ 0 0
$$419$$ 6516.00 0.759731 0.379866 0.925042i $$-0.375970\pi$$
0.379866 + 0.925042i $$0.375970\pi$$
$$420$$ 0 0
$$421$$ −2626.00 −0.303999 −0.151999 0.988381i $$-0.548571\pi$$
−0.151999 + 0.988381i $$0.548571\pi$$
$$422$$ 0 0
$$423$$ 2160.00i 0.248281i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ − 5232.00i − 0.592961i
$$428$$ 0 0
$$429$$ −6216.00 −0.699560
$$430$$ 0 0
$$431$$ −4304.00 −0.481012 −0.240506 0.970648i $$-0.577313\pi$$
−0.240506 + 0.970648i $$0.577313\pi$$
$$432$$ 0 0
$$433$$ 11794.0i 1.30897i 0.756076 + 0.654484i $$0.227114\pi$$
−0.756076 + 0.654484i $$0.772886\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ − 736.000i − 0.0805667i
$$438$$ 0 0
$$439$$ 5544.00 0.602735 0.301368 0.953508i $$-0.402557\pi$$
0.301368 + 0.953508i $$0.402557\pi$$
$$440$$ 0 0
$$441$$ 2097.00 0.226433
$$442$$ 0 0
$$443$$ − 3788.00i − 0.406260i −0.979152 0.203130i $$-0.934889\pi$$
0.979152 0.203130i $$-0.0651115\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ − 4026.00i − 0.426003i
$$448$$ 0 0
$$449$$ 13342.0 1.40233 0.701167 0.712997i $$-0.252663\pi$$
0.701167 + 0.712997i $$0.252663\pi$$
$$450$$ 0 0
$$451$$ −7896.00 −0.824408
$$452$$ 0 0
$$453$$ − 1272.00i − 0.131929i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 4390.00i 0.449356i 0.974433 + 0.224678i $$0.0721330\pi$$
−0.974433 + 0.224678i $$0.927867\pi$$
$$458$$ 0 0
$$459$$ −2214.00 −0.225143
$$460$$ 0 0
$$461$$ 5798.00 0.585770 0.292885 0.956148i $$-0.405385\pi$$
0.292885 + 0.956148i $$0.405385\pi$$
$$462$$ 0 0
$$463$$ − 14656.0i − 1.47111i −0.677467 0.735553i $$-0.736922\pi$$
0.677467 0.735553i $$-0.263078\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ − 8412.00i − 0.833535i −0.909013 0.416768i $$-0.863163\pi$$
0.909013 0.416768i $$-0.136837\pi$$
$$468$$ 0 0
$$469$$ −10464.0 −1.03024
$$470$$ 0 0
$$471$$ 786.000 0.0768938
$$472$$ 0 0
$$473$$ − 112.000i − 0.0108875i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 1170.00i 0.112307i
$$478$$ 0 0
$$479$$ −14848.0 −1.41633 −0.708165 0.706047i $$-0.750477\pi$$
−0.708165 + 0.706047i $$0.750477\pi$$
$$480$$ 0 0
$$481$$ −2220.00 −0.210443
$$482$$ 0 0
$$483$$ − 576.000i − 0.0542627i
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ − 18568.0i − 1.72771i −0.503738 0.863857i $$-0.668042\pi$$
0.503738 0.863857i $$-0.331958\pi$$
$$488$$ 0 0
$$489$$ 6876.00 0.635876
$$490$$ 0 0
$$491$$ −14364.0 −1.32024 −0.660120 0.751160i $$-0.729495\pi$$
−0.660120 + 0.751160i $$0.729495\pi$$
$$492$$ 0 0
$$493$$ − 11316.0i − 1.03377i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 20544.0i 1.85417i
$$498$$ 0 0
$$499$$ −21660.0 −1.94316 −0.971578 0.236720i $$-0.923928\pi$$
−0.971578 + 0.236720i $$0.923928\pi$$
$$500$$ 0 0
$$501$$ −5688.00 −0.507228
$$502$$ 0 0
$$503$$ − 17112.0i − 1.51687i −0.651748 0.758436i $$-0.725964\pi$$
0.651748 0.758436i $$-0.274036\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ − 9837.00i − 0.861689i
$$508$$ 0 0
$$509$$ −11478.0 −0.999516 −0.499758 0.866165i $$-0.666578\pi$$
−0.499758 + 0.866165i $$0.666578\pi$$
$$510$$ 0 0
$$511$$ 23952.0 2.07353
$$512$$ 0 0
$$513$$ 2484.00i 0.213784i
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 6720.00i 0.571654i
$$518$$ 0 0
$$519$$ 8622.00 0.729217
$$520$$ 0 0
$$521$$ 13114.0 1.10275 0.551377 0.834256i $$-0.314103\pi$$
0.551377 + 0.834256i $$0.314103\pi$$
$$522$$ 0 0
$$523$$ − 4508.00i − 0.376905i −0.982082 0.188452i $$-0.939653\pi$$
0.982082 0.188452i $$-0.0603471\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ − 6560.00i − 0.542235i
$$528$$ 0 0
$$529$$ 12103.0 0.994740
$$530$$ 0 0
$$531$$ 5364.00 0.438376
$$532$$ 0 0
$$533$$ − 20868.0i − 1.69586i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 3564.00i 0.286402i
$$538$$ 0 0
$$539$$ 6524.00 0.521352
$$540$$ 0 0
$$541$$ 22950.0 1.82384 0.911920 0.410368i $$-0.134600\pi$$
0.911920 + 0.410368i $$0.134600\pi$$
$$542$$ 0 0
$$543$$ − 10422.0i − 0.823666i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 6580.00i 0.514334i 0.966367 + 0.257167i $$0.0827890\pi$$
−0.966367 + 0.257167i $$0.917211\pi$$
$$548$$ 0 0
$$549$$ 1962.00 0.152525
$$550$$ 0 0
$$551$$ −12696.0 −0.981611
$$552$$ 0 0
$$553$$ 768.000i 0.0590573i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ − 7046.00i − 0.535994i −0.963420 0.267997i $$-0.913638\pi$$
0.963420 0.267997i $$-0.0863617\pi$$
$$558$$ 0 0
$$559$$ 296.000 0.0223962
$$560$$ 0 0
$$561$$ −6888.00 −0.518381
$$562$$ 0 0
$$563$$ 8252.00i 0.617727i 0.951106 + 0.308864i $$0.0999486\pi$$
−0.951106 + 0.308864i $$0.900051\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 1944.00i 0.143986i
$$568$$ 0 0
$$569$$ 6838.00 0.503803 0.251901 0.967753i $$-0.418944\pi$$
0.251901 + 0.967753i $$0.418944\pi$$
$$570$$ 0 0
$$571$$ 23316.0 1.70883 0.854417 0.519588i $$-0.173915\pi$$
0.854417 + 0.519588i $$0.173915\pi$$
$$572$$ 0 0
$$573$$ 576.000i 0.0419943i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 10558.0i 0.761760i 0.924625 + 0.380880i $$0.124379\pi$$
−0.924625 + 0.380880i $$0.875621\pi$$
$$578$$ 0 0
$$579$$ −14406.0 −1.03401
$$580$$ 0 0
$$581$$ 36192.0 2.58433
$$582$$ 0 0
$$583$$ 3640.00i 0.258582i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ − 1028.00i − 0.0722830i −0.999347 0.0361415i $$-0.988493\pi$$
0.999347 0.0361415i $$-0.0115067\pi$$
$$588$$ 0 0
$$589$$ −7360.00 −0.514879
$$590$$ 0 0
$$591$$ 4554.00 0.316965
$$592$$ 0 0
$$593$$ 1202.00i 0.0832382i 0.999134 + 0.0416191i $$0.0132516\pi$$
−0.999134 + 0.0416191i $$0.986748\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ − 15384.0i − 1.05465i
$$598$$ 0 0
$$599$$ 3576.00 0.243926 0.121963 0.992535i $$-0.461081\pi$$
0.121963 + 0.992535i $$0.461081\pi$$
$$600$$ 0 0
$$601$$ 8650.00 0.587090 0.293545 0.955945i $$-0.405165\pi$$
0.293545 + 0.955945i $$0.405165\pi$$
$$602$$ 0 0
$$603$$ − 3924.00i − 0.265004i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ − 12656.0i − 0.846279i −0.906065 0.423139i $$-0.860928\pi$$
0.906065 0.423139i $$-0.139072\pi$$
$$608$$ 0 0
$$609$$ −9936.00 −0.661128
$$610$$ 0 0
$$611$$ −17760.0 −1.17593
$$612$$ 0 0
$$613$$ − 3298.00i − 0.217300i −0.994080 0.108650i $$-0.965347\pi$$
0.994080 0.108650i $$-0.0346528\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ − 5370.00i − 0.350386i −0.984534 0.175193i $$-0.943945\pi$$
0.984534 0.175193i $$-0.0560549\pi$$
$$618$$ 0 0
$$619$$ 16220.0 1.05321 0.526605 0.850110i $$-0.323465\pi$$
0.526605 + 0.850110i $$0.323465\pi$$
$$620$$ 0 0
$$621$$ 216.000 0.0139578
$$622$$ 0 0
$$623$$ 5904.00i 0.379677i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 7728.00i 0.492227i
$$628$$ 0 0
$$629$$ −2460.00 −0.155941
$$630$$ 0 0
$$631$$ −20360.0 −1.28450 −0.642249 0.766496i $$-0.721999\pi$$
−0.642249 + 0.766496i $$0.721999\pi$$
$$632$$ 0 0
$$633$$ 3252.00i 0.204195i
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 17242.0i 1.07245i
$$638$$ 0 0
$$639$$ −7704.00 −0.476941
$$640$$ 0 0
$$641$$ 14498.0 0.893349 0.446674 0.894697i $$-0.352608\pi$$
0.446674 + 0.894697i $$0.352608\pi$$
$$642$$ 0 0
$$643$$ 21612.0i 1.32550i 0.748842 + 0.662748i $$0.230610\pi$$
−0.748842 + 0.662748i $$0.769390\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ − 12184.0i − 0.740344i −0.928963 0.370172i $$-0.879299\pi$$
0.928963 0.370172i $$-0.120701\pi$$
$$648$$ 0 0
$$649$$ 16688.0 1.00934
$$650$$ 0 0
$$651$$ −5760.00 −0.346778
$$652$$ 0 0
$$653$$ − 28122.0i − 1.68530i −0.538464 0.842648i $$-0.680995\pi$$
0.538464 0.842648i $$-0.319005\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 8982.00i 0.533366i
$$658$$ 0 0
$$659$$ 5700.00 0.336935 0.168468 0.985707i $$-0.446118\pi$$
0.168468 + 0.985707i $$0.446118\pi$$
$$660$$ 0 0
$$661$$ −29458.0 −1.73341 −0.866705 0.498822i $$-0.833766\pi$$
−0.866705 + 0.498822i $$0.833766\pi$$
$$662$$ 0 0
$$663$$ − 18204.0i − 1.06634i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 1104.00i 0.0640885i
$$668$$ 0 0
$$669$$ −2064.00 −0.119281
$$670$$ 0 0
$$671$$ 6104.00 0.351181
$$672$$ 0 0
$$673$$ 19810.0i 1.13465i 0.823494 + 0.567325i $$0.192022\pi$$
−0.823494 + 0.567325i $$0.807978\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 10450.0i 0.593244i 0.954995 + 0.296622i $$0.0958601\pi$$
−0.954995 + 0.296622i $$0.904140\pi$$
$$678$$ 0 0
$$679$$ 20784.0 1.17469
$$680$$ 0 0
$$681$$ 14436.0 0.812318
$$682$$ 0 0
$$683$$ 23300.0i 1.30534i 0.757641 + 0.652672i $$0.226352\pi$$
−0.757641 + 0.652672i $$0.773648\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ − 7482.00i − 0.415511i
$$688$$ 0 0
$$689$$ −9620.00 −0.531920
$$690$$ 0 0
$$691$$ −14212.0 −0.782417 −0.391208 0.920302i $$-0.627943\pi$$
−0.391208 + 0.920302i $$0.627943\pi$$
$$692$$ 0 0
$$693$$ 6048.00i 0.331522i
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ − 23124.0i − 1.25665i
$$698$$ 0 0
$$699$$ −2094.00 −0.113308
$$700$$ 0 0
$$701$$ −15978.0 −0.860885 −0.430443 0.902618i $$-0.641643\pi$$
−0.430443 + 0.902618i $$0.641643\pi$$
$$702$$ 0 0
$$703$$ 2760.00i 0.148073i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 6480.00i 0.344704i
$$708$$ 0 0
$$709$$ 8866.00 0.469633 0.234816 0.972040i $$-0.424551\pi$$
0.234816 + 0.972040i $$0.424551\pi$$
$$710$$ 0 0
$$711$$ −288.000 −0.0151911
$$712$$ 0 0
$$713$$ 640.000i 0.0336160i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 18960.0i 0.987551i
$$718$$ 0 0
$$719$$ −7760.00 −0.402502 −0.201251 0.979540i $$-0.564501\pi$$
−0.201251 + 0.979540i $$0.564501\pi$$
$$720$$ 0 0
$$721$$ 35904.0 1.85456
$$722$$ 0 0
$$723$$ − 19530.0i − 1.00460i
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ − 13080.0i − 0.667277i −0.942701 0.333638i $$-0.891724\pi$$
0.942701 0.333638i $$-0.108276\pi$$
$$728$$ 0 0
$$729$$ −729.000 −0.0370370
$$730$$ 0 0
$$731$$ 328.000 0.0165958
$$732$$ 0 0
$$733$$ 16934.0i 0.853304i 0.904416 + 0.426652i $$0.140307\pi$$
−0.904416 + 0.426652i $$0.859693\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ − 12208.0i − 0.610159i
$$738$$ 0 0
$$739$$ 7060.00 0.351429 0.175715 0.984441i $$-0.443776\pi$$
0.175715 + 0.984441i $$0.443776\pi$$
$$740$$ 0 0
$$741$$ −20424.0 −1.01254
$$742$$ 0 0
$$743$$ − 12520.0i − 0.618189i −0.951031 0.309094i $$-0.899974\pi$$
0.951031 0.309094i $$-0.100026\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 13572.0i 0.664757i
$$748$$ 0 0
$$749$$ −40608.0 −1.98102
$$750$$ 0 0
$$751$$ −9792.00 −0.475786 −0.237893 0.971291i $$-0.576457\pi$$
−0.237893 + 0.971291i $$0.576457\pi$$
$$752$$ 0 0
$$753$$ 1884.00i 0.0911777i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ − 13166.0i − 0.632135i −0.948737 0.316068i $$-0.897637\pi$$
0.948737 0.316068i $$-0.102363\pi$$
$$758$$ 0 0
$$759$$ 672.000 0.0321371
$$760$$ 0 0
$$761$$ −23222.0 −1.10617 −0.553086 0.833124i $$-0.686550\pi$$
−0.553086 + 0.833124i $$0.686550\pi$$
$$762$$ 0 0
$$763$$ − 9744.00i − 0.462328i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 44104.0i 2.07628i
$$768$$ 0 0
$$769$$ 39934.0 1.87264 0.936318 0.351154i $$-0.114211\pi$$
0.936318 + 0.351154i $$0.114211\pi$$
$$770$$ 0 0
$$771$$ −14586.0 −0.681325
$$772$$ 0 0
$$773$$ − 17106.0i − 0.795938i −0.917399 0.397969i $$-0.869715\pi$$
0.917399 0.397969i $$-0.130285\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 2160.00i 0.0997292i
$$778$$ 0 0
$$779$$ −25944.0 −1.19325
$$780$$ 0 0
$$781$$ −23968.0 −1.09813
$$782$$ 0 0
$$783$$ − 3726.00i − 0.170059i
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 9956.00i 0.450944i 0.974250 + 0.225472i $$0.0723924\pi$$
−0.974250 + 0.225472i $$0.927608\pi$$
$$788$$ 0 0
$$789$$ −17448.0 −0.787282
$$790$$ 0 0
$$791$$ −18864.0 −0.847948
$$792$$ 0 0
$$793$$ 16132.0i 0.722401i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 9130.00i 0.405773i 0.979202 + 0.202887i $$0.0650323\pi$$
−0.979202 + 0.202887i $$0.934968\pi$$
$$798$$ 0 0
$$799$$ −19680.0 −0.871375
$$800$$ 0 0
$$801$$ −2214.00 −0.0976627
$$802$$ 0 0
$$803$$ 27944.0i 1.22805i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ − 10578.0i − 0.461417i
$$808$$ 0 0
$$809$$ −11482.0 −0.498993 −0.249497 0.968376i $$-0.580265\pi$$
−0.249497 + 0.968376i $$0.580265\pi$$
$$810$$ 0 0
$$811$$ 4612.00 0.199691 0.0998454 0.995003i $$-0.468165\pi$$
0.0998454 + 0.995003i $$0.468165\pi$$
$$812$$ 0 0
$$813$$ − 768.000i − 0.0331303i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ − 368.000i − 0.0157585i
$$818$$ 0 0
$$819$$ −15984.0 −0.681961
$$820$$ 0 0
$$821$$ −35010.0 −1.48826 −0.744128 0.668038i $$-0.767135\pi$$
−0.744128 + 0.668038i $$0.767135\pi$$
$$822$$ 0 0
$$823$$ 13688.0i 0.579749i 0.957065 + 0.289875i $$0.0936136\pi$$
−0.957065 + 0.289875i $$0.906386\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 11668.0i − 0.490612i −0.969446 0.245306i $$-0.921112\pi$$
0.969446 0.245306i $$-0.0788884\pi$$
$$828$$ 0 0
$$829$$ 29306.0 1.22779 0.613896 0.789387i $$-0.289601\pi$$
0.613896 + 0.789387i $$0.289601\pi$$
$$830$$ 0 0
$$831$$ 426.000 0.0177831
$$832$$ 0 0
$$833$$ 19106.0i 0.794698i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ − 2160.00i − 0.0892001i
$$838$$ 0 0
$$839$$ 2664.00 0.109620 0.0548102 0.998497i $$-0.482545\pi$$
0.0548102 + 0.998497i $$0.482545\pi$$
$$840$$ 0 0
$$841$$ −5345.00 −0.219156
$$842$$ 0 0
$$843$$ 26526.0i 1.08375i
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ − 13128.0i − 0.532566i
$$848$$ 0 0
$$849$$ 21540.0 0.870731
$$850$$ 0 0
$$851$$ 240.000 0.00966756
$$852$$ 0 0
$$853$$ 26030.0i 1.04484i 0.852688 + 0.522421i $$0.174971\pi$$
−0.852688 + 0.522421i $$0.825029\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ − 44202.0i − 1.76186i −0.473249 0.880929i $$-0.656919\pi$$
0.473249 0.880929i $$-0.343081\pi$$
$$858$$ 0 0
$$859$$ 32748.0 1.30075 0.650377 0.759612i $$-0.274611\pi$$
0.650377 + 0.759612i $$0.274611\pi$$
$$860$$ 0 0
$$861$$ −20304.0 −0.803668
$$862$$ 0 0
$$863$$ 45344.0i 1.78856i 0.447507 + 0.894280i $$0.352312\pi$$
−0.447507 + 0.894280i $$0.647688\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ − 5433.00i − 0.212819i
$$868$$ 0 0
$$869$$ −896.000 −0.0349767
$$870$$ 0 0
$$871$$ 32264.0 1.25514
$$872$$ 0 0
$$873$$ 7794.00i 0.302161i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 8778.00i 0.337984i 0.985617 + 0.168992i $$0.0540512\pi$$
−0.985617 + 0.168992i $$0.945949\pi$$
$$878$$ 0 0
$$879$$ −22122.0 −0.848870
$$880$$ 0 0
$$881$$ −4142.00 −0.158397 −0.0791984 0.996859i $$-0.525236\pi$$
−0.0791984 + 0.996859i $$0.525236\pi$$
$$882$$ 0 0
$$883$$ 22076.0i 0.841355i 0.907210 + 0.420678i $$0.138208\pi$$
−0.907210 + 0.420678i $$0.861792\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 40376.0i 1.52840i 0.644978 + 0.764201i $$0.276867\pi$$
−0.644978 + 0.764201i $$0.723133\pi$$
$$888$$ 0 0
$$889$$ 41856.0 1.57908
$$890$$ 0 0
$$891$$ −2268.00 −0.0852759
$$892$$ 0 0
$$893$$ 22080.0i 0.827412i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 1776.00i 0.0661080i
$$898$$ 0 0
$$899$$ 11040.0 0.409571
$$900$$ 0 0
$$901$$ −10660.0 −0.394158
$$902$$ 0 0
$$903$$ − 288.000i − 0.0106136i
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 26396.0i 0.966334i 0.875528 + 0.483167i $$0.160514\pi$$
−0.875528 + 0.483167i $$0.839486\pi$$
$$908$$ 0 0
$$909$$ −2430.00 −0.0886667
$$910$$ 0 0
$$911$$ 24368.0 0.886222 0.443111 0.896467i $$-0.353875\pi$$
0.443111 + 0.896467i $$0.353875\pi$$
$$912$$ 0 0
$$913$$ 42224.0i 1.53057i
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 15648.0i 0.563514i
$$918$$ 0 0
$$919$$ 5096.00 0.182918 0.0914589 0.995809i $$-0.470847\pi$$
0.0914589 + 0.995809i $$0.470847\pi$$
$$920$$ 0 0
$$921$$ 4500.00 0.160999
$$922$$ 0 0
$$923$$ − 63344.0i − 2.25893i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 13464.0i 0.477040i
$$928$$ 0 0
$$929$$ 18494.0 0.653142 0.326571 0.945173i $$-0.394107\pi$$
0.326571 + 0.945173i $$0.394107\pi$$
$$930$$ 0 0
$$931$$ 21436.0 0.754604
$$932$$ 0 0
$$933$$ − 22824.0i − 0.800883i
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 33222.0i 1.15829i 0.815225 + 0.579144i $$0.196613\pi$$
−0.815225 + 0.579144i $$0.803387\pi$$
$$938$$ 0 0
$$939$$ 14274.0 0.496075
$$940$$ 0 0
$$941$$ 27846.0 0.964669 0.482335 0.875987i $$-0.339789\pi$$
0.482335 + 0.875987i $$0.339789\pi$$
$$942$$ 0 0
$$943$$ 2256.00i 0.0779061i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ − 41052.0i − 1.40867i −0.709868 0.704335i $$-0.751245\pi$$
0.709868 0.704335i $$-0.248755\pi$$
$$948$$ 0 0
$$949$$ −73852.0 −2.52617
$$950$$ 0 0
$$951$$ 13122.0 0.447434
$$952$$ 0 0
$$953$$ 5706.00i 0.193951i 0.995287 + 0.0969756i $$0.0309169\pi$$
−0.995287 + 0.0969756i $$0.969083\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ − 11592.0i − 0.391553i
$$958$$ 0 0
$$959$$ 36720.0 1.23644
$$960$$ 0 0
$$961$$ −23391.0 −0.785170
$$962$$ 0 0
$$963$$ − 15228.0i − 0.509570i
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 39352.0i 1.30866i 0.756209 + 0.654330i $$0.227049\pi$$
−0.756209 + 0.654330i $$0.772951\pi$$
$$968$$ 0 0
$$969$$ −22632.0 −0.750304
$$970$$ 0 0
$$971$$ −33180.0 −1.09660 −0.548299 0.836282i $$-0.684724\pi$$
−0.548299 + 0.836282i $$0.684724\pi$$
$$972$$ 0 0
$$973$$ − 12384.0i − 0.408030i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 4014.00i 0.131442i 0.997838 + 0.0657212i $$0.0209348\pi$$
−0.997838 + 0.0657212i $$0.979065\pi$$
$$978$$ 0 0
$$979$$ −6888.00 −0.224864
$$980$$ 0 0
$$981$$ 3654.00 0.118923
$$982$$ 0 0
$$983$$ 20328.0i 0.659575i 0.944055 + 0.329788i $$0.106977\pi$$
−0.944055 + 0.329788i $$0.893023\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 17280.0i 0.557273i
$$988$$ 0 0
$$989$$ −32.0000 −0.00102886
$$990$$ 0 0
$$991$$ 11728.0 0.375936 0.187968 0.982175i $$-0.439810\pi$$
0.187968 + 0.982175i $$0.439810\pi$$
$$992$$ 0 0
$$993$$ − 23412.0i − 0.748195i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ − 50974.0i − 1.61922i −0.586968 0.809610i $$-0.699679\pi$$
0.586968 0.809610i $$-0.300321\pi$$
$$998$$ 0 0
$$999$$ −810.000 −0.0256529
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 600.4.f.b.49.2 2
3.2 odd 2 1800.4.f.q.649.2 2
4.3 odd 2 1200.4.f.p.49.1 2
5.2 odd 4 24.4.a.a.1.1 1
5.3 odd 4 600.4.a.h.1.1 1
5.4 even 2 inner 600.4.f.b.49.1 2
15.2 even 4 72.4.a.b.1.1 1
15.8 even 4 1800.4.a.bg.1.1 1
15.14 odd 2 1800.4.f.q.649.1 2
20.3 even 4 1200.4.a.u.1.1 1
20.7 even 4 48.4.a.b.1.1 1
20.19 odd 2 1200.4.f.p.49.2 2
35.27 even 4 1176.4.a.a.1.1 1
40.27 even 4 192.4.a.g.1.1 1
40.37 odd 4 192.4.a.a.1.1 1
45.2 even 12 648.4.i.k.433.1 2
45.7 odd 12 648.4.i.b.433.1 2
45.22 odd 12 648.4.i.b.217.1 2
45.32 even 12 648.4.i.k.217.1 2
60.47 odd 4 144.4.a.b.1.1 1
80.27 even 4 768.4.d.b.385.2 2
80.37 odd 4 768.4.d.o.385.1 2
80.67 even 4 768.4.d.b.385.1 2
80.77 odd 4 768.4.d.o.385.2 2
120.77 even 4 576.4.a.u.1.1 1
120.107 odd 4 576.4.a.v.1.1 1
140.27 odd 4 2352.4.a.w.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
24.4.a.a.1.1 1 5.2 odd 4
48.4.a.b.1.1 1 20.7 even 4
72.4.a.b.1.1 1 15.2 even 4
144.4.a.b.1.1 1 60.47 odd 4
192.4.a.a.1.1 1 40.37 odd 4
192.4.a.g.1.1 1 40.27 even 4
576.4.a.u.1.1 1 120.77 even 4
576.4.a.v.1.1 1 120.107 odd 4
600.4.a.h.1.1 1 5.3 odd 4
600.4.f.b.49.1 2 5.4 even 2 inner
600.4.f.b.49.2 2 1.1 even 1 trivial
648.4.i.b.217.1 2 45.22 odd 12
648.4.i.b.433.1 2 45.7 odd 12
648.4.i.k.217.1 2 45.32 even 12
648.4.i.k.433.1 2 45.2 even 12
768.4.d.b.385.1 2 80.67 even 4
768.4.d.b.385.2 2 80.27 even 4
768.4.d.o.385.1 2 80.37 odd 4
768.4.d.o.385.2 2 80.77 odd 4
1176.4.a.a.1.1 1 35.27 even 4
1200.4.a.u.1.1 1 20.3 even 4
1200.4.f.p.49.1 2 4.3 odd 2
1200.4.f.p.49.2 2 20.19 odd 2
1800.4.a.bg.1.1 1 15.8 even 4
1800.4.f.q.649.1 2 15.14 odd 2
1800.4.f.q.649.2 2 3.2 odd 2
2352.4.a.w.1.1 1 140.27 odd 4