# Properties

 Label 2550.2.a.h.1.1 Level $2550$ Weight $2$ Character 2550.1 Self dual yes Analytic conductor $20.362$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2550,2,Mod(1,2550)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2550, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("2550.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2550 = 2 \cdot 3 \cdot 5^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2550.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$20.3618525154$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 2550.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -3.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -3.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -5.00000 q^{11} +1.00000 q^{12} +2.00000 q^{13} +3.00000 q^{14} +1.00000 q^{16} -1.00000 q^{17} -1.00000 q^{18} +1.00000 q^{19} -3.00000 q^{21} +5.00000 q^{22} -6.00000 q^{23} -1.00000 q^{24} -2.00000 q^{26} +1.00000 q^{27} -3.00000 q^{28} +10.0000 q^{29} +5.00000 q^{31} -1.00000 q^{32} -5.00000 q^{33} +1.00000 q^{34} +1.00000 q^{36} +3.00000 q^{37} -1.00000 q^{38} +2.00000 q^{39} +6.00000 q^{41} +3.00000 q^{42} +1.00000 q^{43} -5.00000 q^{44} +6.00000 q^{46} -3.00000 q^{47} +1.00000 q^{48} +2.00000 q^{49} -1.00000 q^{51} +2.00000 q^{52} +1.00000 q^{53} -1.00000 q^{54} +3.00000 q^{56} +1.00000 q^{57} -10.0000 q^{58} +8.00000 q^{59} -2.00000 q^{61} -5.00000 q^{62} -3.00000 q^{63} +1.00000 q^{64} +5.00000 q^{66} +11.0000 q^{67} -1.00000 q^{68} -6.00000 q^{69} +6.00000 q^{71} -1.00000 q^{72} +12.0000 q^{73} -3.00000 q^{74} +1.00000 q^{76} +15.0000 q^{77} -2.00000 q^{78} +5.00000 q^{79} +1.00000 q^{81} -6.00000 q^{82} -18.0000 q^{83} -3.00000 q^{84} -1.00000 q^{86} +10.0000 q^{87} +5.00000 q^{88} +12.0000 q^{89} -6.00000 q^{91} -6.00000 q^{92} +5.00000 q^{93} +3.00000 q^{94} -1.00000 q^{96} +14.0000 q^{97} -2.00000 q^{98} -5.00000 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 −0.707107
$$3$$ 1.00000 0.577350
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ −1.00000 −0.408248
$$7$$ −3.00000 −1.13389 −0.566947 0.823754i $$-0.691875\pi$$
−0.566947 + 0.823754i $$0.691875\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −5.00000 −1.50756 −0.753778 0.657129i $$-0.771771\pi$$
−0.753778 + 0.657129i $$0.771771\pi$$
$$12$$ 1.00000 0.288675
$$13$$ 2.00000 0.554700 0.277350 0.960769i $$-0.410544\pi$$
0.277350 + 0.960769i $$0.410544\pi$$
$$14$$ 3.00000 0.801784
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ −1.00000 −0.242536
$$18$$ −1.00000 −0.235702
$$19$$ 1.00000 0.229416 0.114708 0.993399i $$-0.463407\pi$$
0.114708 + 0.993399i $$0.463407\pi$$
$$20$$ 0 0
$$21$$ −3.00000 −0.654654
$$22$$ 5.00000 1.06600
$$23$$ −6.00000 −1.25109 −0.625543 0.780189i $$-0.715123\pi$$
−0.625543 + 0.780189i $$0.715123\pi$$
$$24$$ −1.00000 −0.204124
$$25$$ 0 0
$$26$$ −2.00000 −0.392232
$$27$$ 1.00000 0.192450
$$28$$ −3.00000 −0.566947
$$29$$ 10.0000 1.85695 0.928477 0.371391i $$-0.121119\pi$$
0.928477 + 0.371391i $$0.121119\pi$$
$$30$$ 0 0
$$31$$ 5.00000 0.898027 0.449013 0.893525i $$-0.351776\pi$$
0.449013 + 0.893525i $$0.351776\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ −5.00000 −0.870388
$$34$$ 1.00000 0.171499
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ 3.00000 0.493197 0.246598 0.969118i $$-0.420687\pi$$
0.246598 + 0.969118i $$0.420687\pi$$
$$38$$ −1.00000 −0.162221
$$39$$ 2.00000 0.320256
$$40$$ 0 0
$$41$$ 6.00000 0.937043 0.468521 0.883452i $$-0.344787\pi$$
0.468521 + 0.883452i $$0.344787\pi$$
$$42$$ 3.00000 0.462910
$$43$$ 1.00000 0.152499 0.0762493 0.997089i $$-0.475706\pi$$
0.0762493 + 0.997089i $$0.475706\pi$$
$$44$$ −5.00000 −0.753778
$$45$$ 0 0
$$46$$ 6.00000 0.884652
$$47$$ −3.00000 −0.437595 −0.218797 0.975770i $$-0.570213\pi$$
−0.218797 + 0.975770i $$0.570213\pi$$
$$48$$ 1.00000 0.144338
$$49$$ 2.00000 0.285714
$$50$$ 0 0
$$51$$ −1.00000 −0.140028
$$52$$ 2.00000 0.277350
$$53$$ 1.00000 0.137361 0.0686803 0.997639i $$-0.478121\pi$$
0.0686803 + 0.997639i $$0.478121\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ 0 0
$$56$$ 3.00000 0.400892
$$57$$ 1.00000 0.132453
$$58$$ −10.0000 −1.31306
$$59$$ 8.00000 1.04151 0.520756 0.853706i $$-0.325650\pi$$
0.520756 + 0.853706i $$0.325650\pi$$
$$60$$ 0 0
$$61$$ −2.00000 −0.256074 −0.128037 0.991769i $$-0.540868\pi$$
−0.128037 + 0.991769i $$0.540868\pi$$
$$62$$ −5.00000 −0.635001
$$63$$ −3.00000 −0.377964
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 5.00000 0.615457
$$67$$ 11.0000 1.34386 0.671932 0.740613i $$-0.265465\pi$$
0.671932 + 0.740613i $$0.265465\pi$$
$$68$$ −1.00000 −0.121268
$$69$$ −6.00000 −0.722315
$$70$$ 0 0
$$71$$ 6.00000 0.712069 0.356034 0.934473i $$-0.384129\pi$$
0.356034 + 0.934473i $$0.384129\pi$$
$$72$$ −1.00000 −0.117851
$$73$$ 12.0000 1.40449 0.702247 0.711934i $$-0.252180\pi$$
0.702247 + 0.711934i $$0.252180\pi$$
$$74$$ −3.00000 −0.348743
$$75$$ 0 0
$$76$$ 1.00000 0.114708
$$77$$ 15.0000 1.70941
$$78$$ −2.00000 −0.226455
$$79$$ 5.00000 0.562544 0.281272 0.959628i $$-0.409244\pi$$
0.281272 + 0.959628i $$0.409244\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ −6.00000 −0.662589
$$83$$ −18.0000 −1.97576 −0.987878 0.155230i $$-0.950388\pi$$
−0.987878 + 0.155230i $$0.950388\pi$$
$$84$$ −3.00000 −0.327327
$$85$$ 0 0
$$86$$ −1.00000 −0.107833
$$87$$ 10.0000 1.07211
$$88$$ 5.00000 0.533002
$$89$$ 12.0000 1.27200 0.635999 0.771690i $$-0.280588\pi$$
0.635999 + 0.771690i $$0.280588\pi$$
$$90$$ 0 0
$$91$$ −6.00000 −0.628971
$$92$$ −6.00000 −0.625543
$$93$$ 5.00000 0.518476
$$94$$ 3.00000 0.309426
$$95$$ 0 0
$$96$$ −1.00000 −0.102062
$$97$$ 14.0000 1.42148 0.710742 0.703452i $$-0.248359\pi$$
0.710742 + 0.703452i $$0.248359\pi$$
$$98$$ −2.00000 −0.202031
$$99$$ −5.00000 −0.502519
$$100$$ 0 0
$$101$$ 11.0000 1.09454 0.547270 0.836956i $$-0.315667\pi$$
0.547270 + 0.836956i $$0.315667\pi$$
$$102$$ 1.00000 0.0990148
$$103$$ −4.00000 −0.394132 −0.197066 0.980390i $$-0.563141\pi$$
−0.197066 + 0.980390i $$0.563141\pi$$
$$104$$ −2.00000 −0.196116
$$105$$ 0 0
$$106$$ −1.00000 −0.0971286
$$107$$ 5.00000 0.483368 0.241684 0.970355i $$-0.422300\pi$$
0.241684 + 0.970355i $$0.422300\pi$$
$$108$$ 1.00000 0.0962250
$$109$$ 5.00000 0.478913 0.239457 0.970907i $$-0.423031\pi$$
0.239457 + 0.970907i $$0.423031\pi$$
$$110$$ 0 0
$$111$$ 3.00000 0.284747
$$112$$ −3.00000 −0.283473
$$113$$ −1.00000 −0.0940721 −0.0470360 0.998893i $$-0.514978\pi$$
−0.0470360 + 0.998893i $$0.514978\pi$$
$$114$$ −1.00000 −0.0936586
$$115$$ 0 0
$$116$$ 10.0000 0.928477
$$117$$ 2.00000 0.184900
$$118$$ −8.00000 −0.736460
$$119$$ 3.00000 0.275010
$$120$$ 0 0
$$121$$ 14.0000 1.27273
$$122$$ 2.00000 0.181071
$$123$$ 6.00000 0.541002
$$124$$ 5.00000 0.449013
$$125$$ 0 0
$$126$$ 3.00000 0.267261
$$127$$ −12.0000 −1.06483 −0.532414 0.846484i $$-0.678715\pi$$
−0.532414 + 0.846484i $$0.678715\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 1.00000 0.0880451
$$130$$ 0 0
$$131$$ −20.0000 −1.74741 −0.873704 0.486458i $$-0.838289\pi$$
−0.873704 + 0.486458i $$0.838289\pi$$
$$132$$ −5.00000 −0.435194
$$133$$ −3.00000 −0.260133
$$134$$ −11.0000 −0.950255
$$135$$ 0 0
$$136$$ 1.00000 0.0857493
$$137$$ −16.0000 −1.36697 −0.683486 0.729964i $$-0.739537\pi$$
−0.683486 + 0.729964i $$0.739537\pi$$
$$138$$ 6.00000 0.510754
$$139$$ 8.00000 0.678551 0.339276 0.940687i $$-0.389818\pi$$
0.339276 + 0.940687i $$0.389818\pi$$
$$140$$ 0 0
$$141$$ −3.00000 −0.252646
$$142$$ −6.00000 −0.503509
$$143$$ −10.0000 −0.836242
$$144$$ 1.00000 0.0833333
$$145$$ 0 0
$$146$$ −12.0000 −0.993127
$$147$$ 2.00000 0.164957
$$148$$ 3.00000 0.246598
$$149$$ −6.00000 −0.491539 −0.245770 0.969328i $$-0.579041\pi$$
−0.245770 + 0.969328i $$0.579041\pi$$
$$150$$ 0 0
$$151$$ 8.00000 0.651031 0.325515 0.945537i $$-0.394462\pi$$
0.325515 + 0.945537i $$0.394462\pi$$
$$152$$ −1.00000 −0.0811107
$$153$$ −1.00000 −0.0808452
$$154$$ −15.0000 −1.20873
$$155$$ 0 0
$$156$$ 2.00000 0.160128
$$157$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$158$$ −5.00000 −0.397779
$$159$$ 1.00000 0.0793052
$$160$$ 0 0
$$161$$ 18.0000 1.41860
$$162$$ −1.00000 −0.0785674
$$163$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$164$$ 6.00000 0.468521
$$165$$ 0 0
$$166$$ 18.0000 1.39707
$$167$$ 18.0000 1.39288 0.696441 0.717614i $$-0.254766\pi$$
0.696441 + 0.717614i $$0.254766\pi$$
$$168$$ 3.00000 0.231455
$$169$$ −9.00000 −0.692308
$$170$$ 0 0
$$171$$ 1.00000 0.0764719
$$172$$ 1.00000 0.0762493
$$173$$ −20.0000 −1.52057 −0.760286 0.649589i $$-0.774941\pi$$
−0.760286 + 0.649589i $$0.774941\pi$$
$$174$$ −10.0000 −0.758098
$$175$$ 0 0
$$176$$ −5.00000 −0.376889
$$177$$ 8.00000 0.601317
$$178$$ −12.0000 −0.899438
$$179$$ −12.0000 −0.896922 −0.448461 0.893802i $$-0.648028\pi$$
−0.448461 + 0.893802i $$0.648028\pi$$
$$180$$ 0 0
$$181$$ −17.0000 −1.26360 −0.631800 0.775131i $$-0.717684\pi$$
−0.631800 + 0.775131i $$0.717684\pi$$
$$182$$ 6.00000 0.444750
$$183$$ −2.00000 −0.147844
$$184$$ 6.00000 0.442326
$$185$$ 0 0
$$186$$ −5.00000 −0.366618
$$187$$ 5.00000 0.365636
$$188$$ −3.00000 −0.218797
$$189$$ −3.00000 −0.218218
$$190$$ 0 0
$$191$$ 3.00000 0.217072 0.108536 0.994092i $$-0.465384\pi$$
0.108536 + 0.994092i $$0.465384\pi$$
$$192$$ 1.00000 0.0721688
$$193$$ 24.0000 1.72756 0.863779 0.503871i $$-0.168091\pi$$
0.863779 + 0.503871i $$0.168091\pi$$
$$194$$ −14.0000 −1.00514
$$195$$ 0 0
$$196$$ 2.00000 0.142857
$$197$$ −12.0000 −0.854965 −0.427482 0.904024i $$-0.640599\pi$$
−0.427482 + 0.904024i $$0.640599\pi$$
$$198$$ 5.00000 0.355335
$$199$$ 17.0000 1.20510 0.602549 0.798082i $$-0.294152\pi$$
0.602549 + 0.798082i $$0.294152\pi$$
$$200$$ 0 0
$$201$$ 11.0000 0.775880
$$202$$ −11.0000 −0.773957
$$203$$ −30.0000 −2.10559
$$204$$ −1.00000 −0.0700140
$$205$$ 0 0
$$206$$ 4.00000 0.278693
$$207$$ −6.00000 −0.417029
$$208$$ 2.00000 0.138675
$$209$$ −5.00000 −0.345857
$$210$$ 0 0
$$211$$ 18.0000 1.23917 0.619586 0.784929i $$-0.287301\pi$$
0.619586 + 0.784929i $$0.287301\pi$$
$$212$$ 1.00000 0.0686803
$$213$$ 6.00000 0.411113
$$214$$ −5.00000 −0.341793
$$215$$ 0 0
$$216$$ −1.00000 −0.0680414
$$217$$ −15.0000 −1.01827
$$218$$ −5.00000 −0.338643
$$219$$ 12.0000 0.810885
$$220$$ 0 0
$$221$$ −2.00000 −0.134535
$$222$$ −3.00000 −0.201347
$$223$$ −18.0000 −1.20537 −0.602685 0.797980i $$-0.705902\pi$$
−0.602685 + 0.797980i $$0.705902\pi$$
$$224$$ 3.00000 0.200446
$$225$$ 0 0
$$226$$ 1.00000 0.0665190
$$227$$ 27.0000 1.79205 0.896026 0.444001i $$-0.146441\pi$$
0.896026 + 0.444001i $$0.146441\pi$$
$$228$$ 1.00000 0.0662266
$$229$$ 24.0000 1.58596 0.792982 0.609245i $$-0.208527\pi$$
0.792982 + 0.609245i $$0.208527\pi$$
$$230$$ 0 0
$$231$$ 15.0000 0.986928
$$232$$ −10.0000 −0.656532
$$233$$ 10.0000 0.655122 0.327561 0.944830i $$-0.393773\pi$$
0.327561 + 0.944830i $$0.393773\pi$$
$$234$$ −2.00000 −0.130744
$$235$$ 0 0
$$236$$ 8.00000 0.520756
$$237$$ 5.00000 0.324785
$$238$$ −3.00000 −0.194461
$$239$$ 13.0000 0.840900 0.420450 0.907316i $$-0.361872\pi$$
0.420450 + 0.907316i $$0.361872\pi$$
$$240$$ 0 0
$$241$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$242$$ −14.0000 −0.899954
$$243$$ 1.00000 0.0641500
$$244$$ −2.00000 −0.128037
$$245$$ 0 0
$$246$$ −6.00000 −0.382546
$$247$$ 2.00000 0.127257
$$248$$ −5.00000 −0.317500
$$249$$ −18.0000 −1.14070
$$250$$ 0 0
$$251$$ −18.0000 −1.13615 −0.568075 0.822977i $$-0.692312\pi$$
−0.568075 + 0.822977i $$0.692312\pi$$
$$252$$ −3.00000 −0.188982
$$253$$ 30.0000 1.88608
$$254$$ 12.0000 0.752947
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −2.00000 −0.124757 −0.0623783 0.998053i $$-0.519869\pi$$
−0.0623783 + 0.998053i $$0.519869\pi$$
$$258$$ −1.00000 −0.0622573
$$259$$ −9.00000 −0.559233
$$260$$ 0 0
$$261$$ 10.0000 0.618984
$$262$$ 20.0000 1.23560
$$263$$ −3.00000 −0.184988 −0.0924940 0.995713i $$-0.529484\pi$$
−0.0924940 + 0.995713i $$0.529484\pi$$
$$264$$ 5.00000 0.307729
$$265$$ 0 0
$$266$$ 3.00000 0.183942
$$267$$ 12.0000 0.734388
$$268$$ 11.0000 0.671932
$$269$$ −6.00000 −0.365826 −0.182913 0.983129i $$-0.558553\pi$$
−0.182913 + 0.983129i $$0.558553\pi$$
$$270$$ 0 0
$$271$$ 14.0000 0.850439 0.425220 0.905090i $$-0.360197\pi$$
0.425220 + 0.905090i $$0.360197\pi$$
$$272$$ −1.00000 −0.0606339
$$273$$ −6.00000 −0.363137
$$274$$ 16.0000 0.966595
$$275$$ 0 0
$$276$$ −6.00000 −0.361158
$$277$$ 31.0000 1.86261 0.931305 0.364241i $$-0.118672\pi$$
0.931305 + 0.364241i $$0.118672\pi$$
$$278$$ −8.00000 −0.479808
$$279$$ 5.00000 0.299342
$$280$$ 0 0
$$281$$ −30.0000 −1.78965 −0.894825 0.446417i $$-0.852700\pi$$
−0.894825 + 0.446417i $$0.852700\pi$$
$$282$$ 3.00000 0.178647
$$283$$ 14.0000 0.832214 0.416107 0.909316i $$-0.363394\pi$$
0.416107 + 0.909316i $$0.363394\pi$$
$$284$$ 6.00000 0.356034
$$285$$ 0 0
$$286$$ 10.0000 0.591312
$$287$$ −18.0000 −1.06251
$$288$$ −1.00000 −0.0589256
$$289$$ 1.00000 0.0588235
$$290$$ 0 0
$$291$$ 14.0000 0.820695
$$292$$ 12.0000 0.702247
$$293$$ 18.0000 1.05157 0.525786 0.850617i $$-0.323771\pi$$
0.525786 + 0.850617i $$0.323771\pi$$
$$294$$ −2.00000 −0.116642
$$295$$ 0 0
$$296$$ −3.00000 −0.174371
$$297$$ −5.00000 −0.290129
$$298$$ 6.00000 0.347571
$$299$$ −12.0000 −0.693978
$$300$$ 0 0
$$301$$ −3.00000 −0.172917
$$302$$ −8.00000 −0.460348
$$303$$ 11.0000 0.631933
$$304$$ 1.00000 0.0573539
$$305$$ 0 0
$$306$$ 1.00000 0.0571662
$$307$$ −20.0000 −1.14146 −0.570730 0.821138i $$-0.693340\pi$$
−0.570730 + 0.821138i $$0.693340\pi$$
$$308$$ 15.0000 0.854704
$$309$$ −4.00000 −0.227552
$$310$$ 0 0
$$311$$ −2.00000 −0.113410 −0.0567048 0.998391i $$-0.518059\pi$$
−0.0567048 + 0.998391i $$0.518059\pi$$
$$312$$ −2.00000 −0.113228
$$313$$ 34.0000 1.92179 0.960897 0.276907i $$-0.0893093\pi$$
0.960897 + 0.276907i $$0.0893093\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 5.00000 0.281272
$$317$$ 12.0000 0.673987 0.336994 0.941507i $$-0.390590\pi$$
0.336994 + 0.941507i $$0.390590\pi$$
$$318$$ −1.00000 −0.0560772
$$319$$ −50.0000 −2.79946
$$320$$ 0 0
$$321$$ 5.00000 0.279073
$$322$$ −18.0000 −1.00310
$$323$$ −1.00000 −0.0556415
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 5.00000 0.276501
$$328$$ −6.00000 −0.331295
$$329$$ 9.00000 0.496186
$$330$$ 0 0
$$331$$ −25.0000 −1.37412 −0.687062 0.726599i $$-0.741100\pi$$
−0.687062 + 0.726599i $$0.741100\pi$$
$$332$$ −18.0000 −0.987878
$$333$$ 3.00000 0.164399
$$334$$ −18.0000 −0.984916
$$335$$ 0 0
$$336$$ −3.00000 −0.163663
$$337$$ −12.0000 −0.653682 −0.326841 0.945079i $$-0.605984\pi$$
−0.326841 + 0.945079i $$0.605984\pi$$
$$338$$ 9.00000 0.489535
$$339$$ −1.00000 −0.0543125
$$340$$ 0 0
$$341$$ −25.0000 −1.35383
$$342$$ −1.00000 −0.0540738
$$343$$ 15.0000 0.809924
$$344$$ −1.00000 −0.0539164
$$345$$ 0 0
$$346$$ 20.0000 1.07521
$$347$$ 23.0000 1.23470 0.617352 0.786687i $$-0.288205\pi$$
0.617352 + 0.786687i $$0.288205\pi$$
$$348$$ 10.0000 0.536056
$$349$$ −14.0000 −0.749403 −0.374701 0.927146i $$-0.622255\pi$$
−0.374701 + 0.927146i $$0.622255\pi$$
$$350$$ 0 0
$$351$$ 2.00000 0.106752
$$352$$ 5.00000 0.266501
$$353$$ 2.00000 0.106449 0.0532246 0.998583i $$-0.483050\pi$$
0.0532246 + 0.998583i $$0.483050\pi$$
$$354$$ −8.00000 −0.425195
$$355$$ 0 0
$$356$$ 12.0000 0.635999
$$357$$ 3.00000 0.158777
$$358$$ 12.0000 0.634220
$$359$$ 15.0000 0.791670 0.395835 0.918322i $$-0.370455\pi$$
0.395835 + 0.918322i $$0.370455\pi$$
$$360$$ 0 0
$$361$$ −18.0000 −0.947368
$$362$$ 17.0000 0.893500
$$363$$ 14.0000 0.734809
$$364$$ −6.00000 −0.314485
$$365$$ 0 0
$$366$$ 2.00000 0.104542
$$367$$ −5.00000 −0.260998 −0.130499 0.991448i $$-0.541658\pi$$
−0.130499 + 0.991448i $$0.541658\pi$$
$$368$$ −6.00000 −0.312772
$$369$$ 6.00000 0.312348
$$370$$ 0 0
$$371$$ −3.00000 −0.155752
$$372$$ 5.00000 0.259238
$$373$$ 18.0000 0.932005 0.466002 0.884783i $$-0.345694\pi$$
0.466002 + 0.884783i $$0.345694\pi$$
$$374$$ −5.00000 −0.258544
$$375$$ 0 0
$$376$$ 3.00000 0.154713
$$377$$ 20.0000 1.03005
$$378$$ 3.00000 0.154303
$$379$$ −8.00000 −0.410932 −0.205466 0.978664i $$-0.565871\pi$$
−0.205466 + 0.978664i $$0.565871\pi$$
$$380$$ 0 0
$$381$$ −12.0000 −0.614779
$$382$$ −3.00000 −0.153493
$$383$$ 8.00000 0.408781 0.204390 0.978889i $$-0.434479\pi$$
0.204390 + 0.978889i $$0.434479\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 0 0
$$386$$ −24.0000 −1.22157
$$387$$ 1.00000 0.0508329
$$388$$ 14.0000 0.710742
$$389$$ −5.00000 −0.253510 −0.126755 0.991934i $$-0.540456\pi$$
−0.126755 + 0.991934i $$0.540456\pi$$
$$390$$ 0 0
$$391$$ 6.00000 0.303433
$$392$$ −2.00000 −0.101015
$$393$$ −20.0000 −1.00887
$$394$$ 12.0000 0.604551
$$395$$ 0 0
$$396$$ −5.00000 −0.251259
$$397$$ −21.0000 −1.05396 −0.526980 0.849878i $$-0.676676\pi$$
−0.526980 + 0.849878i $$0.676676\pi$$
$$398$$ −17.0000 −0.852133
$$399$$ −3.00000 −0.150188
$$400$$ 0 0
$$401$$ −18.0000 −0.898877 −0.449439 0.893311i $$-0.648376\pi$$
−0.449439 + 0.893311i $$0.648376\pi$$
$$402$$ −11.0000 −0.548630
$$403$$ 10.0000 0.498135
$$404$$ 11.0000 0.547270
$$405$$ 0 0
$$406$$ 30.0000 1.48888
$$407$$ −15.0000 −0.743522
$$408$$ 1.00000 0.0495074
$$409$$ −2.00000 −0.0988936 −0.0494468 0.998777i $$-0.515746\pi$$
−0.0494468 + 0.998777i $$0.515746\pi$$
$$410$$ 0 0
$$411$$ −16.0000 −0.789222
$$412$$ −4.00000 −0.197066
$$413$$ −24.0000 −1.18096
$$414$$ 6.00000 0.294884
$$415$$ 0 0
$$416$$ −2.00000 −0.0980581
$$417$$ 8.00000 0.391762
$$418$$ 5.00000 0.244558
$$419$$ 24.0000 1.17248 0.586238 0.810139i $$-0.300608\pi$$
0.586238 + 0.810139i $$0.300608\pi$$
$$420$$ 0 0
$$421$$ 18.0000 0.877266 0.438633 0.898666i $$-0.355463\pi$$
0.438633 + 0.898666i $$0.355463\pi$$
$$422$$ −18.0000 −0.876226
$$423$$ −3.00000 −0.145865
$$424$$ −1.00000 −0.0485643
$$425$$ 0 0
$$426$$ −6.00000 −0.290701
$$427$$ 6.00000 0.290360
$$428$$ 5.00000 0.241684
$$429$$ −10.0000 −0.482805
$$430$$ 0 0
$$431$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$432$$ 1.00000 0.0481125
$$433$$ −21.0000 −1.00920 −0.504598 0.863355i $$-0.668359\pi$$
−0.504598 + 0.863355i $$0.668359\pi$$
$$434$$ 15.0000 0.720023
$$435$$ 0 0
$$436$$ 5.00000 0.239457
$$437$$ −6.00000 −0.287019
$$438$$ −12.0000 −0.573382
$$439$$ 16.0000 0.763638 0.381819 0.924237i $$-0.375298\pi$$
0.381819 + 0.924237i $$0.375298\pi$$
$$440$$ 0 0
$$441$$ 2.00000 0.0952381
$$442$$ 2.00000 0.0951303
$$443$$ 24.0000 1.14027 0.570137 0.821549i $$-0.306890\pi$$
0.570137 + 0.821549i $$0.306890\pi$$
$$444$$ 3.00000 0.142374
$$445$$ 0 0
$$446$$ 18.0000 0.852325
$$447$$ −6.00000 −0.283790
$$448$$ −3.00000 −0.141737
$$449$$ 21.0000 0.991051 0.495526 0.868593i $$-0.334975\pi$$
0.495526 + 0.868593i $$0.334975\pi$$
$$450$$ 0 0
$$451$$ −30.0000 −1.41264
$$452$$ −1.00000 −0.0470360
$$453$$ 8.00000 0.375873
$$454$$ −27.0000 −1.26717
$$455$$ 0 0
$$456$$ −1.00000 −0.0468293
$$457$$ 25.0000 1.16945 0.584725 0.811231i $$-0.301202\pi$$
0.584725 + 0.811231i $$0.301202\pi$$
$$458$$ −24.0000 −1.12145
$$459$$ −1.00000 −0.0466760
$$460$$ 0 0
$$461$$ −15.0000 −0.698620 −0.349310 0.937007i $$-0.613584\pi$$
−0.349310 + 0.937007i $$0.613584\pi$$
$$462$$ −15.0000 −0.697863
$$463$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$464$$ 10.0000 0.464238
$$465$$ 0 0
$$466$$ −10.0000 −0.463241
$$467$$ −18.0000 −0.832941 −0.416470 0.909149i $$-0.636733\pi$$
−0.416470 + 0.909149i $$0.636733\pi$$
$$468$$ 2.00000 0.0924500
$$469$$ −33.0000 −1.52380
$$470$$ 0 0
$$471$$ 0 0
$$472$$ −8.00000 −0.368230
$$473$$ −5.00000 −0.229900
$$474$$ −5.00000 −0.229658
$$475$$ 0 0
$$476$$ 3.00000 0.137505
$$477$$ 1.00000 0.0457869
$$478$$ −13.0000 −0.594606
$$479$$ 26.0000 1.18797 0.593985 0.804476i $$-0.297554\pi$$
0.593985 + 0.804476i $$0.297554\pi$$
$$480$$ 0 0
$$481$$ 6.00000 0.273576
$$482$$ 0 0
$$483$$ 18.0000 0.819028
$$484$$ 14.0000 0.636364
$$485$$ 0 0
$$486$$ −1.00000 −0.0453609
$$487$$ 40.0000 1.81257 0.906287 0.422664i $$-0.138905\pi$$
0.906287 + 0.422664i $$0.138905\pi$$
$$488$$ 2.00000 0.0905357
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 4.00000 0.180517 0.0902587 0.995918i $$-0.471231\pi$$
0.0902587 + 0.995918i $$0.471231\pi$$
$$492$$ 6.00000 0.270501
$$493$$ −10.0000 −0.450377
$$494$$ −2.00000 −0.0899843
$$495$$ 0 0
$$496$$ 5.00000 0.224507
$$497$$ −18.0000 −0.807410
$$498$$ 18.0000 0.806599
$$499$$ 22.0000 0.984855 0.492428 0.870353i $$-0.336110\pi$$
0.492428 + 0.870353i $$0.336110\pi$$
$$500$$ 0 0
$$501$$ 18.0000 0.804181
$$502$$ 18.0000 0.803379
$$503$$ −6.00000 −0.267527 −0.133763 0.991013i $$-0.542706\pi$$
−0.133763 + 0.991013i $$0.542706\pi$$
$$504$$ 3.00000 0.133631
$$505$$ 0 0
$$506$$ −30.0000 −1.33366
$$507$$ −9.00000 −0.399704
$$508$$ −12.0000 −0.532414
$$509$$ 9.00000 0.398918 0.199459 0.979906i $$-0.436082\pi$$
0.199459 + 0.979906i $$0.436082\pi$$
$$510$$ 0 0
$$511$$ −36.0000 −1.59255
$$512$$ −1.00000 −0.0441942
$$513$$ 1.00000 0.0441511
$$514$$ 2.00000 0.0882162
$$515$$ 0 0
$$516$$ 1.00000 0.0440225
$$517$$ 15.0000 0.659699
$$518$$ 9.00000 0.395437
$$519$$ −20.0000 −0.877903
$$520$$ 0 0
$$521$$ −39.0000 −1.70862 −0.854311 0.519763i $$-0.826020\pi$$
−0.854311 + 0.519763i $$0.826020\pi$$
$$522$$ −10.0000 −0.437688
$$523$$ −20.0000 −0.874539 −0.437269 0.899331i $$-0.644054\pi$$
−0.437269 + 0.899331i $$0.644054\pi$$
$$524$$ −20.0000 −0.873704
$$525$$ 0 0
$$526$$ 3.00000 0.130806
$$527$$ −5.00000 −0.217803
$$528$$ −5.00000 −0.217597
$$529$$ 13.0000 0.565217
$$530$$ 0 0
$$531$$ 8.00000 0.347170
$$532$$ −3.00000 −0.130066
$$533$$ 12.0000 0.519778
$$534$$ −12.0000 −0.519291
$$535$$ 0 0
$$536$$ −11.0000 −0.475128
$$537$$ −12.0000 −0.517838
$$538$$ 6.00000 0.258678
$$539$$ −10.0000 −0.430730
$$540$$ 0 0
$$541$$ −41.0000 −1.76273 −0.881364 0.472438i $$-0.843374\pi$$
−0.881364 + 0.472438i $$0.843374\pi$$
$$542$$ −14.0000 −0.601351
$$543$$ −17.0000 −0.729540
$$544$$ 1.00000 0.0428746
$$545$$ 0 0
$$546$$ 6.00000 0.256776
$$547$$ −22.0000 −0.940652 −0.470326 0.882493i $$-0.655864\pi$$
−0.470326 + 0.882493i $$0.655864\pi$$
$$548$$ −16.0000 −0.683486
$$549$$ −2.00000 −0.0853579
$$550$$ 0 0
$$551$$ 10.0000 0.426014
$$552$$ 6.00000 0.255377
$$553$$ −15.0000 −0.637865
$$554$$ −31.0000 −1.31706
$$555$$ 0 0
$$556$$ 8.00000 0.339276
$$557$$ −17.0000 −0.720313 −0.360157 0.932892i $$-0.617277\pi$$
−0.360157 + 0.932892i $$0.617277\pi$$
$$558$$ −5.00000 −0.211667
$$559$$ 2.00000 0.0845910
$$560$$ 0 0
$$561$$ 5.00000 0.211100
$$562$$ 30.0000 1.26547
$$563$$ 8.00000 0.337160 0.168580 0.985688i $$-0.446082\pi$$
0.168580 + 0.985688i $$0.446082\pi$$
$$564$$ −3.00000 −0.126323
$$565$$ 0 0
$$566$$ −14.0000 −0.588464
$$567$$ −3.00000 −0.125988
$$568$$ −6.00000 −0.251754
$$569$$ −36.0000 −1.50920 −0.754599 0.656186i $$-0.772169\pi$$
−0.754599 + 0.656186i $$0.772169\pi$$
$$570$$ 0 0
$$571$$ −20.0000 −0.836974 −0.418487 0.908223i $$-0.637439\pi$$
−0.418487 + 0.908223i $$0.637439\pi$$
$$572$$ −10.0000 −0.418121
$$573$$ 3.00000 0.125327
$$574$$ 18.0000 0.751305
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ −33.0000 −1.37381 −0.686904 0.726748i $$-0.741031\pi$$
−0.686904 + 0.726748i $$0.741031\pi$$
$$578$$ −1.00000 −0.0415945
$$579$$ 24.0000 0.997406
$$580$$ 0 0
$$581$$ 54.0000 2.24030
$$582$$ −14.0000 −0.580319
$$583$$ −5.00000 −0.207079
$$584$$ −12.0000 −0.496564
$$585$$ 0 0
$$586$$ −18.0000 −0.743573
$$587$$ 18.0000 0.742940 0.371470 0.928445i $$-0.378854\pi$$
0.371470 + 0.928445i $$0.378854\pi$$
$$588$$ 2.00000 0.0824786
$$589$$ 5.00000 0.206021
$$590$$ 0 0
$$591$$ −12.0000 −0.493614
$$592$$ 3.00000 0.123299
$$593$$ −12.0000 −0.492781 −0.246390 0.969171i $$-0.579245\pi$$
−0.246390 + 0.969171i $$0.579245\pi$$
$$594$$ 5.00000 0.205152
$$595$$ 0 0
$$596$$ −6.00000 −0.245770
$$597$$ 17.0000 0.695764
$$598$$ 12.0000 0.490716
$$599$$ 35.0000 1.43006 0.715031 0.699093i $$-0.246413\pi$$
0.715031 + 0.699093i $$0.246413\pi$$
$$600$$ 0 0
$$601$$ −22.0000 −0.897399 −0.448699 0.893683i $$-0.648113\pi$$
−0.448699 + 0.893683i $$0.648113\pi$$
$$602$$ 3.00000 0.122271
$$603$$ 11.0000 0.447955
$$604$$ 8.00000 0.325515
$$605$$ 0 0
$$606$$ −11.0000 −0.446844
$$607$$ −20.0000 −0.811775 −0.405887 0.913923i $$-0.633038\pi$$
−0.405887 + 0.913923i $$0.633038\pi$$
$$608$$ −1.00000 −0.0405554
$$609$$ −30.0000 −1.21566
$$610$$ 0 0
$$611$$ −6.00000 −0.242734
$$612$$ −1.00000 −0.0404226
$$613$$ 26.0000 1.05013 0.525065 0.851062i $$-0.324041\pi$$
0.525065 + 0.851062i $$0.324041\pi$$
$$614$$ 20.0000 0.807134
$$615$$ 0 0
$$616$$ −15.0000 −0.604367
$$617$$ 13.0000 0.523360 0.261680 0.965155i $$-0.415723\pi$$
0.261680 + 0.965155i $$0.415723\pi$$
$$618$$ 4.00000 0.160904
$$619$$ −8.00000 −0.321547 −0.160774 0.986991i $$-0.551399\pi$$
−0.160774 + 0.986991i $$0.551399\pi$$
$$620$$ 0 0
$$621$$ −6.00000 −0.240772
$$622$$ 2.00000 0.0801927
$$623$$ −36.0000 −1.44231
$$624$$ 2.00000 0.0800641
$$625$$ 0 0
$$626$$ −34.0000 −1.35891
$$627$$ −5.00000 −0.199681
$$628$$ 0 0
$$629$$ −3.00000 −0.119618
$$630$$ 0 0
$$631$$ −14.0000 −0.557331 −0.278666 0.960388i $$-0.589892\pi$$
−0.278666 + 0.960388i $$0.589892\pi$$
$$632$$ −5.00000 −0.198889
$$633$$ 18.0000 0.715436
$$634$$ −12.0000 −0.476581
$$635$$ 0 0
$$636$$ 1.00000 0.0396526
$$637$$ 4.00000 0.158486
$$638$$ 50.0000 1.97952
$$639$$ 6.00000 0.237356
$$640$$ 0 0
$$641$$ −30.0000 −1.18493 −0.592464 0.805597i $$-0.701845\pi$$
−0.592464 + 0.805597i $$0.701845\pi$$
$$642$$ −5.00000 −0.197334
$$643$$ −34.0000 −1.34083 −0.670415 0.741987i $$-0.733884\pi$$
−0.670415 + 0.741987i $$0.733884\pi$$
$$644$$ 18.0000 0.709299
$$645$$ 0 0
$$646$$ 1.00000 0.0393445
$$647$$ −24.0000 −0.943537 −0.471769 0.881722i $$-0.656384\pi$$
−0.471769 + 0.881722i $$0.656384\pi$$
$$648$$ −1.00000 −0.0392837
$$649$$ −40.0000 −1.57014
$$650$$ 0 0
$$651$$ −15.0000 −0.587896
$$652$$ 0 0
$$653$$ 16.0000 0.626128 0.313064 0.949732i $$-0.398644\pi$$
0.313064 + 0.949732i $$0.398644\pi$$
$$654$$ −5.00000 −0.195515
$$655$$ 0 0
$$656$$ 6.00000 0.234261
$$657$$ 12.0000 0.468165
$$658$$ −9.00000 −0.350857
$$659$$ 16.0000 0.623272 0.311636 0.950202i $$-0.399123\pi$$
0.311636 + 0.950202i $$0.399123\pi$$
$$660$$ 0 0
$$661$$ 8.00000 0.311164 0.155582 0.987823i $$-0.450275\pi$$
0.155582 + 0.987823i $$0.450275\pi$$
$$662$$ 25.0000 0.971653
$$663$$ −2.00000 −0.0776736
$$664$$ 18.0000 0.698535
$$665$$ 0 0
$$666$$ −3.00000 −0.116248
$$667$$ −60.0000 −2.32321
$$668$$ 18.0000 0.696441
$$669$$ −18.0000 −0.695920
$$670$$ 0 0
$$671$$ 10.0000 0.386046
$$672$$ 3.00000 0.115728
$$673$$ −14.0000 −0.539660 −0.269830 0.962908i $$-0.586968\pi$$
−0.269830 + 0.962908i $$0.586968\pi$$
$$674$$ 12.0000 0.462223
$$675$$ 0 0
$$676$$ −9.00000 −0.346154
$$677$$ −12.0000 −0.461197 −0.230599 0.973049i $$-0.574068\pi$$
−0.230599 + 0.973049i $$0.574068\pi$$
$$678$$ 1.00000 0.0384048
$$679$$ −42.0000 −1.61181
$$680$$ 0 0
$$681$$ 27.0000 1.03464
$$682$$ 25.0000 0.957299
$$683$$ 8.00000 0.306111 0.153056 0.988218i $$-0.451089\pi$$
0.153056 + 0.988218i $$0.451089\pi$$
$$684$$ 1.00000 0.0382360
$$685$$ 0 0
$$686$$ −15.0000 −0.572703
$$687$$ 24.0000 0.915657
$$688$$ 1.00000 0.0381246
$$689$$ 2.00000 0.0761939
$$690$$ 0 0
$$691$$ −48.0000 −1.82601 −0.913003 0.407953i $$-0.866243\pi$$
−0.913003 + 0.407953i $$0.866243\pi$$
$$692$$ −20.0000 −0.760286
$$693$$ 15.0000 0.569803
$$694$$ −23.0000 −0.873068
$$695$$ 0 0
$$696$$ −10.0000 −0.379049
$$697$$ −6.00000 −0.227266
$$698$$ 14.0000 0.529908
$$699$$ 10.0000 0.378235
$$700$$ 0 0
$$701$$ 18.0000 0.679851 0.339925 0.940452i $$-0.389598\pi$$
0.339925 + 0.940452i $$0.389598\pi$$
$$702$$ −2.00000 −0.0754851
$$703$$ 3.00000 0.113147
$$704$$ −5.00000 −0.188445
$$705$$ 0 0
$$706$$ −2.00000 −0.0752710
$$707$$ −33.0000 −1.24109
$$708$$ 8.00000 0.300658
$$709$$ 25.0000 0.938895 0.469447 0.882960i $$-0.344453\pi$$
0.469447 + 0.882960i $$0.344453\pi$$
$$710$$ 0 0
$$711$$ 5.00000 0.187515
$$712$$ −12.0000 −0.449719
$$713$$ −30.0000 −1.12351
$$714$$ −3.00000 −0.112272
$$715$$ 0 0
$$716$$ −12.0000 −0.448461
$$717$$ 13.0000 0.485494
$$718$$ −15.0000 −0.559795
$$719$$ 6.00000 0.223762 0.111881 0.993722i $$-0.464312\pi$$
0.111881 + 0.993722i $$0.464312\pi$$
$$720$$ 0 0
$$721$$ 12.0000 0.446903
$$722$$ 18.0000 0.669891
$$723$$ 0 0
$$724$$ −17.0000 −0.631800
$$725$$ 0 0
$$726$$ −14.0000 −0.519589
$$727$$ −2.00000 −0.0741759 −0.0370879 0.999312i $$-0.511808\pi$$
−0.0370879 + 0.999312i $$0.511808\pi$$
$$728$$ 6.00000 0.222375
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −1.00000 −0.0369863
$$732$$ −2.00000 −0.0739221
$$733$$ 16.0000 0.590973 0.295487 0.955347i $$-0.404518\pi$$
0.295487 + 0.955347i $$0.404518\pi$$
$$734$$ 5.00000 0.184553
$$735$$ 0 0
$$736$$ 6.00000 0.221163
$$737$$ −55.0000 −2.02595
$$738$$ −6.00000 −0.220863
$$739$$ −19.0000 −0.698926 −0.349463 0.936950i $$-0.613636\pi$$
−0.349463 + 0.936950i $$0.613636\pi$$
$$740$$ 0 0
$$741$$ 2.00000 0.0734718
$$742$$ 3.00000 0.110133
$$743$$ −18.0000 −0.660356 −0.330178 0.943919i $$-0.607109\pi$$
−0.330178 + 0.943919i $$0.607109\pi$$
$$744$$ −5.00000 −0.183309
$$745$$ 0 0
$$746$$ −18.0000 −0.659027
$$747$$ −18.0000 −0.658586
$$748$$ 5.00000 0.182818
$$749$$ −15.0000 −0.548088
$$750$$ 0 0
$$751$$ −48.0000 −1.75154 −0.875772 0.482724i $$-0.839647\pi$$
−0.875772 + 0.482724i $$0.839647\pi$$
$$752$$ −3.00000 −0.109399
$$753$$ −18.0000 −0.655956
$$754$$ −20.0000 −0.728357
$$755$$ 0 0
$$756$$ −3.00000 −0.109109
$$757$$ −14.0000 −0.508839 −0.254419 0.967094i $$-0.581884\pi$$
−0.254419 + 0.967094i $$0.581884\pi$$
$$758$$ 8.00000 0.290573
$$759$$ 30.0000 1.08893
$$760$$ 0 0
$$761$$ 36.0000 1.30500 0.652499 0.757789i $$-0.273720\pi$$
0.652499 + 0.757789i $$0.273720\pi$$
$$762$$ 12.0000 0.434714
$$763$$ −15.0000 −0.543036
$$764$$ 3.00000 0.108536
$$765$$ 0 0
$$766$$ −8.00000 −0.289052
$$767$$ 16.0000 0.577727
$$768$$ 1.00000 0.0360844
$$769$$ −37.0000 −1.33425 −0.667127 0.744944i $$-0.732476\pi$$
−0.667127 + 0.744944i $$0.732476\pi$$
$$770$$ 0 0
$$771$$ −2.00000 −0.0720282
$$772$$ 24.0000 0.863779
$$773$$ 34.0000 1.22290 0.611448 0.791285i $$-0.290588\pi$$
0.611448 + 0.791285i $$0.290588\pi$$
$$774$$ −1.00000 −0.0359443
$$775$$ 0 0
$$776$$ −14.0000 −0.502571
$$777$$ −9.00000 −0.322873
$$778$$ 5.00000 0.179259
$$779$$ 6.00000 0.214972
$$780$$ 0 0
$$781$$ −30.0000 −1.07348
$$782$$ −6.00000 −0.214560
$$783$$ 10.0000 0.357371
$$784$$ 2.00000 0.0714286
$$785$$ 0 0
$$786$$ 20.0000 0.713376
$$787$$ −18.0000 −0.641631 −0.320815 0.947142i $$-0.603957\pi$$
−0.320815 + 0.947142i $$0.603957\pi$$
$$788$$ −12.0000 −0.427482
$$789$$ −3.00000 −0.106803
$$790$$ 0 0
$$791$$ 3.00000 0.106668
$$792$$ 5.00000 0.177667
$$793$$ −4.00000 −0.142044
$$794$$ 21.0000 0.745262
$$795$$ 0 0
$$796$$ 17.0000 0.602549
$$797$$ 11.0000 0.389640 0.194820 0.980839i $$-0.437588\pi$$
0.194820 + 0.980839i $$0.437588\pi$$
$$798$$ 3.00000 0.106199
$$799$$ 3.00000 0.106132
$$800$$ 0 0
$$801$$ 12.0000 0.423999
$$802$$ 18.0000 0.635602
$$803$$ −60.0000 −2.11735
$$804$$ 11.0000 0.387940
$$805$$ 0 0
$$806$$ −10.0000 −0.352235
$$807$$ −6.00000 −0.211210
$$808$$ −11.0000 −0.386979
$$809$$ −39.0000 −1.37117 −0.685583 0.727994i $$-0.740453\pi$$
−0.685583 + 0.727994i $$0.740453\pi$$
$$810$$ 0 0
$$811$$ 56.0000 1.96643 0.983213 0.182462i $$-0.0584065\pi$$
0.983213 + 0.182462i $$0.0584065\pi$$
$$812$$ −30.0000 −1.05279
$$813$$ 14.0000 0.491001
$$814$$ 15.0000 0.525750
$$815$$ 0 0
$$816$$ −1.00000 −0.0350070
$$817$$ 1.00000 0.0349856
$$818$$ 2.00000 0.0699284
$$819$$ −6.00000 −0.209657
$$820$$ 0 0
$$821$$ 54.0000 1.88461 0.942306 0.334751i $$-0.108652\pi$$
0.942306 + 0.334751i $$0.108652\pi$$
$$822$$ 16.0000 0.558064
$$823$$ −4.00000 −0.139431 −0.0697156 0.997567i $$-0.522209\pi$$
−0.0697156 + 0.997567i $$0.522209\pi$$
$$824$$ 4.00000 0.139347
$$825$$ 0 0
$$826$$ 24.0000 0.835067
$$827$$ 53.0000 1.84299 0.921495 0.388390i $$-0.126968\pi$$
0.921495 + 0.388390i $$0.126968\pi$$
$$828$$ −6.00000 −0.208514
$$829$$ −30.0000 −1.04194 −0.520972 0.853574i $$-0.674430\pi$$
−0.520972 + 0.853574i $$0.674430\pi$$
$$830$$ 0 0
$$831$$ 31.0000 1.07538
$$832$$ 2.00000 0.0693375
$$833$$ −2.00000 −0.0692959
$$834$$ −8.00000 −0.277017
$$835$$ 0 0
$$836$$ −5.00000 −0.172929
$$837$$ 5.00000 0.172825
$$838$$ −24.0000 −0.829066
$$839$$ −8.00000 −0.276191 −0.138095 0.990419i $$-0.544098\pi$$
−0.138095 + 0.990419i $$0.544098\pi$$
$$840$$ 0 0
$$841$$ 71.0000 2.44828
$$842$$ −18.0000 −0.620321
$$843$$ −30.0000 −1.03325
$$844$$ 18.0000 0.619586
$$845$$ 0 0
$$846$$ 3.00000 0.103142
$$847$$ −42.0000 −1.44314
$$848$$ 1.00000 0.0343401
$$849$$ 14.0000 0.480479
$$850$$ 0 0
$$851$$ −18.0000 −0.617032
$$852$$ 6.00000 0.205557
$$853$$ 37.0000 1.26686 0.633428 0.773802i $$-0.281647\pi$$
0.633428 + 0.773802i $$0.281647\pi$$
$$854$$ −6.00000 −0.205316
$$855$$ 0 0
$$856$$ −5.00000 −0.170896
$$857$$ 21.0000 0.717346 0.358673 0.933463i $$-0.383229\pi$$
0.358673 + 0.933463i $$0.383229\pi$$
$$858$$ 10.0000 0.341394
$$859$$ −23.0000 −0.784750 −0.392375 0.919805i $$-0.628346\pi$$
−0.392375 + 0.919805i $$0.628346\pi$$
$$860$$ 0 0
$$861$$ −18.0000 −0.613438
$$862$$ 0 0
$$863$$ 33.0000 1.12333 0.561667 0.827364i $$-0.310160\pi$$
0.561667 + 0.827364i $$0.310160\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ 0 0
$$866$$ 21.0000 0.713609
$$867$$ 1.00000 0.0339618
$$868$$ −15.0000 −0.509133
$$869$$ −25.0000 −0.848067
$$870$$ 0 0
$$871$$ 22.0000 0.745442
$$872$$ −5.00000 −0.169321
$$873$$ 14.0000 0.473828
$$874$$ 6.00000 0.202953
$$875$$ 0 0
$$876$$ 12.0000 0.405442
$$877$$ −14.0000 −0.472746 −0.236373 0.971662i $$-0.575959\pi$$
−0.236373 + 0.971662i $$0.575959\pi$$
$$878$$ −16.0000 −0.539974
$$879$$ 18.0000 0.607125
$$880$$ 0 0
$$881$$ −9.00000 −0.303218 −0.151609 0.988441i $$-0.548445\pi$$
−0.151609 + 0.988441i $$0.548445\pi$$
$$882$$ −2.00000 −0.0673435
$$883$$ 20.0000 0.673054 0.336527 0.941674i $$-0.390748\pi$$
0.336527 + 0.941674i $$0.390748\pi$$
$$884$$ −2.00000 −0.0672673
$$885$$ 0 0
$$886$$ −24.0000 −0.806296
$$887$$ −48.0000 −1.61168 −0.805841 0.592132i $$-0.798286\pi$$
−0.805841 + 0.592132i $$0.798286\pi$$
$$888$$ −3.00000 −0.100673
$$889$$ 36.0000 1.20740
$$890$$ 0 0
$$891$$ −5.00000 −0.167506
$$892$$ −18.0000 −0.602685
$$893$$ −3.00000 −0.100391
$$894$$ 6.00000 0.200670
$$895$$ 0 0
$$896$$ 3.00000 0.100223
$$897$$ −12.0000 −0.400668
$$898$$ −21.0000 −0.700779
$$899$$ 50.0000 1.66759
$$900$$ 0 0
$$901$$ −1.00000 −0.0333148
$$902$$ 30.0000 0.998891
$$903$$ −3.00000 −0.0998337
$$904$$ 1.00000 0.0332595
$$905$$ 0 0
$$906$$ −8.00000 −0.265782
$$907$$ 30.0000 0.996134 0.498067 0.867139i $$-0.334043\pi$$
0.498067 + 0.867139i $$0.334043\pi$$
$$908$$ 27.0000 0.896026
$$909$$ 11.0000 0.364847
$$910$$ 0 0
$$911$$ 56.0000 1.85536 0.927681 0.373373i $$-0.121799\pi$$
0.927681 + 0.373373i $$0.121799\pi$$
$$912$$ 1.00000 0.0331133
$$913$$ 90.0000 2.97857
$$914$$ −25.0000 −0.826927
$$915$$ 0 0
$$916$$ 24.0000 0.792982
$$917$$ 60.0000 1.98137
$$918$$ 1.00000 0.0330049
$$919$$ 44.0000 1.45143 0.725713 0.687998i $$-0.241510\pi$$
0.725713 + 0.687998i $$0.241510\pi$$
$$920$$ 0 0
$$921$$ −20.0000 −0.659022
$$922$$ 15.0000 0.493999
$$923$$ 12.0000 0.394985
$$924$$ 15.0000 0.493464
$$925$$ 0 0
$$926$$ 0 0
$$927$$ −4.00000 −0.131377
$$928$$ −10.0000 −0.328266
$$929$$ 27.0000 0.885841 0.442921 0.896561i $$-0.353942\pi$$
0.442921 + 0.896561i $$0.353942\pi$$
$$930$$ 0 0
$$931$$ 2.00000 0.0655474
$$932$$ 10.0000 0.327561
$$933$$ −2.00000 −0.0654771
$$934$$ 18.0000 0.588978
$$935$$ 0 0
$$936$$ −2.00000 −0.0653720
$$937$$ −10.0000 −0.326686 −0.163343 0.986569i $$-0.552228\pi$$
−0.163343 + 0.986569i $$0.552228\pi$$
$$938$$ 33.0000 1.07749
$$939$$ 34.0000 1.10955
$$940$$ 0 0
$$941$$ 22.0000 0.717180 0.358590 0.933495i $$-0.383258\pi$$
0.358590 + 0.933495i $$0.383258\pi$$
$$942$$ 0 0
$$943$$ −36.0000 −1.17232
$$944$$ 8.00000 0.260378
$$945$$ 0 0
$$946$$ 5.00000 0.162564
$$947$$ −61.0000 −1.98223 −0.991117 0.132994i $$-0.957541\pi$$
−0.991117 + 0.132994i $$0.957541\pi$$
$$948$$ 5.00000 0.162392
$$949$$ 24.0000 0.779073
$$950$$ 0 0
$$951$$ 12.0000 0.389127
$$952$$ −3.00000 −0.0972306
$$953$$ 14.0000 0.453504 0.226752 0.973952i $$-0.427189\pi$$
0.226752 + 0.973952i $$0.427189\pi$$
$$954$$ −1.00000 −0.0323762
$$955$$ 0 0
$$956$$ 13.0000 0.420450
$$957$$ −50.0000 −1.61627
$$958$$ −26.0000 −0.840022
$$959$$ 48.0000 1.55000
$$960$$ 0 0
$$961$$ −6.00000 −0.193548
$$962$$ −6.00000 −0.193448
$$963$$ 5.00000 0.161123
$$964$$ 0 0
$$965$$ 0 0
$$966$$ −18.0000 −0.579141
$$967$$ 36.0000 1.15768 0.578841 0.815440i $$-0.303505\pi$$
0.578841 + 0.815440i $$0.303505\pi$$
$$968$$ −14.0000 −0.449977
$$969$$ −1.00000 −0.0321246
$$970$$ 0 0
$$971$$ −38.0000 −1.21948 −0.609739 0.792602i $$-0.708726\pi$$
−0.609739 + 0.792602i $$0.708726\pi$$
$$972$$ 1.00000 0.0320750
$$973$$ −24.0000 −0.769405
$$974$$ −40.0000 −1.28168
$$975$$ 0 0
$$976$$ −2.00000 −0.0640184
$$977$$ 10.0000 0.319928 0.159964 0.987123i $$-0.448862\pi$$
0.159964 + 0.987123i $$0.448862\pi$$
$$978$$ 0 0
$$979$$ −60.0000 −1.91761
$$980$$ 0 0
$$981$$ 5.00000 0.159638
$$982$$ −4.00000 −0.127645
$$983$$ 6.00000 0.191370 0.0956851 0.995412i $$-0.469496\pi$$
0.0956851 + 0.995412i $$0.469496\pi$$
$$984$$ −6.00000 −0.191273
$$985$$ 0 0
$$986$$ 10.0000 0.318465
$$987$$ 9.00000 0.286473
$$988$$ 2.00000 0.0636285
$$989$$ −6.00000 −0.190789
$$990$$ 0 0
$$991$$ 52.0000 1.65183 0.825917 0.563791i $$-0.190658\pi$$
0.825917 + 0.563791i $$0.190658\pi$$
$$992$$ −5.00000 −0.158750
$$993$$ −25.0000 −0.793351
$$994$$ 18.0000 0.570925
$$995$$ 0 0
$$996$$ −18.0000 −0.570352
$$997$$ 43.0000 1.36182 0.680912 0.732365i $$-0.261584\pi$$
0.680912 + 0.732365i $$0.261584\pi$$
$$998$$ −22.0000 −0.696398
$$999$$ 3.00000 0.0949158
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 2550.2.a.h.1.1 1
3.2 odd 2 7650.2.a.bl.1.1 1
5.2 odd 4 2550.2.d.a.2449.1 2
5.3 odd 4 2550.2.d.a.2449.2 2
5.4 even 2 2550.2.a.z.1.1 yes 1
15.14 odd 2 7650.2.a.be.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
2550.2.a.h.1.1 1 1.1 even 1 trivial
2550.2.a.z.1.1 yes 1 5.4 even 2
2550.2.d.a.2449.1 2 5.2 odd 4
2550.2.d.a.2449.2 2 5.3 odd 4
7650.2.a.be.1.1 1 15.14 odd 2
7650.2.a.bl.1.1 1 3.2 odd 2