# Properties

 Label 7600.2.a.bm.1.3 Level $7600$ Weight $2$ Character 7600.1 Self dual yes Analytic conductor $60.686$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7600,2,Mod(1,7600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7600, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("7600.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$7600 = 2^{4} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7600.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: $$60.6863055362$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.257.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 4x + 3$$ x^3 - x^2 - 4*x + 3 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 950) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$2.19869$$ of defining polynomial Character $$\chi$$ $$=$$ 7600.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+3.03293 q^{3} -2.46980 q^{7} +6.19869 q^{9} +O(q^{10})$$ $$q+3.03293 q^{3} -2.46980 q^{7} +6.19869 q^{9} -0.728896 q^{11} -6.23163 q^{13} +0.563139 q^{17} +1.00000 q^{19} -7.49073 q^{21} -4.63555 q^{23} +9.70142 q^{27} +10.2316 q^{29} -6.06587 q^{31} -2.21069 q^{33} -5.72890 q^{37} -18.9001 q^{39} +4.79476 q^{41} -8.06587 q^{43} -8.12628 q^{47} -0.900112 q^{49} +1.70796 q^{51} -1.53020 q^{53} +3.03293 q^{57} +5.76183 q^{59} +10.9396 q^{61} -15.3095 q^{63} -12.9330 q^{67} -14.0593 q^{69} +4.39738 q^{71} -4.09334 q^{73} +1.80022 q^{77} -15.3370 q^{79} +10.8277 q^{81} -7.85517 q^{83} +31.0318 q^{87} -10.0000 q^{89} +15.3908 q^{91} -18.3974 q^{93} +11.0055 q^{97} -4.51820 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 2 q^{3} - 2 q^{7} + 13 q^{9}+O(q^{10})$$ 3 * q - 2 * q^3 - 2 * q^7 + 13 * q^9 $$3 q - 2 q^{3} - 2 q^{7} + 13 q^{9} - 2 q^{11} - 2 q^{13} - 4 q^{17} + 3 q^{19} - 11 q^{21} - 14 q^{23} + 7 q^{27} + 14 q^{29} + 4 q^{31} + 4 q^{33} - 17 q^{37} - 29 q^{39} - 8 q^{41} - 2 q^{43} - 13 q^{47} + 25 q^{49} + 11 q^{51} - 10 q^{53} - 2 q^{57} + 6 q^{59} + 22 q^{61} - 2 q^{63} + 8 q^{69} + 2 q^{71} - 12 q^{73} - 50 q^{77} - 24 q^{79} - q^{81} - 12 q^{83} + 21 q^{87} - 30 q^{89} + 7 q^{91} - 44 q^{93} - 24 q^{99}+O(q^{100})$$ 3 * q - 2 * q^3 - 2 * q^7 + 13 * q^9 - 2 * q^11 - 2 * q^13 - 4 * q^17 + 3 * q^19 - 11 * q^21 - 14 * q^23 + 7 * q^27 + 14 * q^29 + 4 * q^31 + 4 * q^33 - 17 * q^37 - 29 * q^39 - 8 * q^41 - 2 * q^43 - 13 * q^47 + 25 * q^49 + 11 * q^51 - 10 * q^53 - 2 * q^57 + 6 * q^59 + 22 * q^61 - 2 * q^63 + 8 * q^69 + 2 * q^71 - 12 * q^73 - 50 * q^77 - 24 * q^79 - q^81 - 12 * q^83 + 21 * q^87 - 30 * q^89 + 7 * q^91 - 44 * q^93 - 24 * q^99

## 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
$$3$$ 3.03293 1.75107 0.875533 0.483159i $$-0.160511\pi$$
0.875533 + 0.483159i $$0.160511\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −2.46980 −0.933495 −0.466747 0.884391i $$-0.654574\pi$$
−0.466747 + 0.884391i $$0.654574\pi$$
$$8$$ 0 0
$$9$$ 6.19869 2.06623
$$10$$ 0 0
$$11$$ −0.728896 −0.219770 −0.109885 0.993944i $$-0.535048\pi$$
−0.109885 + 0.993944i $$0.535048\pi$$
$$12$$ 0 0
$$13$$ −6.23163 −1.72834 −0.864171 0.503198i $$-0.832157\pi$$
−0.864171 + 0.503198i $$0.832157\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 0.563139 0.136581 0.0682907 0.997665i $$-0.478245\pi$$
0.0682907 + 0.997665i $$0.478245\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ −7.49073 −1.63461
$$22$$ 0 0
$$23$$ −4.63555 −0.966579 −0.483290 0.875460i $$-0.660558\pi$$
−0.483290 + 0.875460i $$0.660558\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 9.70142 1.86704
$$28$$ 0 0
$$29$$ 10.2316 1.89997 0.949983 0.312303i $$-0.101100\pi$$
0.949983 + 0.312303i $$0.101100\pi$$
$$30$$ 0 0
$$31$$ −6.06587 −1.08946 −0.544731 0.838611i $$-0.683368\pi$$
−0.544731 + 0.838611i $$0.683368\pi$$
$$32$$ 0 0
$$33$$ −2.21069 −0.384832
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −5.72890 −0.941825 −0.470912 0.882180i $$-0.656075\pi$$
−0.470912 + 0.882180i $$0.656075\pi$$
$$38$$ 0 0
$$39$$ −18.9001 −3.02644
$$40$$ 0 0
$$41$$ 4.79476 0.748816 0.374408 0.927264i $$-0.377846\pi$$
0.374408 + 0.927264i $$0.377846\pi$$
$$42$$ 0 0
$$43$$ −8.06587 −1.23003 −0.615017 0.788514i $$-0.710851\pi$$
−0.615017 + 0.788514i $$0.710851\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −8.12628 −1.18534 −0.592670 0.805446i $$-0.701926\pi$$
−0.592670 + 0.805446i $$0.701926\pi$$
$$48$$ 0 0
$$49$$ −0.900112 −0.128587
$$50$$ 0 0
$$51$$ 1.70796 0.239163
$$52$$ 0 0
$$53$$ −1.53020 −0.210190 −0.105095 0.994462i $$-0.533515\pi$$
−0.105095 + 0.994462i $$0.533515\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 3.03293 0.401722
$$58$$ 0 0
$$59$$ 5.76183 0.750126 0.375063 0.926999i $$-0.377621\pi$$
0.375063 + 0.926999i $$0.377621\pi$$
$$60$$ 0 0
$$61$$ 10.9396 1.40067 0.700336 0.713814i $$-0.253034\pi$$
0.700336 + 0.713814i $$0.253034\pi$$
$$62$$ 0 0
$$63$$ −15.3095 −1.92882
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −12.9330 −1.58002 −0.790012 0.613092i $$-0.789926\pi$$
−0.790012 + 0.613092i $$0.789926\pi$$
$$68$$ 0 0
$$69$$ −14.0593 −1.69254
$$70$$ 0 0
$$71$$ 4.39738 0.521873 0.260937 0.965356i $$-0.415969\pi$$
0.260937 + 0.965356i $$0.415969\pi$$
$$72$$ 0 0
$$73$$ −4.09334 −0.479090 −0.239545 0.970885i $$-0.576998\pi$$
−0.239545 + 0.970885i $$0.576998\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 1.80022 0.205155
$$78$$ 0 0
$$79$$ −15.3370 −1.72554 −0.862772 0.505593i $$-0.831274\pi$$
−0.862772 + 0.505593i $$0.831274\pi$$
$$80$$ 0 0
$$81$$ 10.8277 1.20308
$$82$$ 0 0
$$83$$ −7.85517 −0.862217 −0.431109 0.902300i $$-0.641877\pi$$
−0.431109 + 0.902300i $$0.641877\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 31.0318 3.32696
$$88$$ 0 0
$$89$$ −10.0000 −1.06000 −0.529999 0.847998i $$-0.677808\pi$$
−0.529999 + 0.847998i $$0.677808\pi$$
$$90$$ 0 0
$$91$$ 15.3908 1.61340
$$92$$ 0 0
$$93$$ −18.3974 −1.90772
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 11.0055 1.11744 0.558718 0.829358i $$-0.311294\pi$$
0.558718 + 0.829358i $$0.311294\pi$$
$$98$$ 0 0
$$99$$ −4.51820 −0.454096
$$100$$ 0 0
$$101$$ −9.73436 −0.968605 −0.484302 0.874901i $$-0.660926\pi$$
−0.484302 + 0.874901i $$0.660926\pi$$
$$102$$ 0 0
$$103$$ 8.33151 0.820928 0.410464 0.911877i $$-0.365367\pi$$
0.410464 + 0.911877i $$0.365367\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 15.0264 1.45266 0.726328 0.687348i $$-0.241225\pi$$
0.726328 + 0.687348i $$0.241225\pi$$
$$108$$ 0 0
$$109$$ −13.6619 −1.30858 −0.654288 0.756245i $$-0.727032\pi$$
−0.654288 + 0.756245i $$0.727032\pi$$
$$110$$ 0 0
$$111$$ −17.3754 −1.64920
$$112$$ 0 0
$$113$$ 1.20524 0.113379 0.0566895 0.998392i $$-0.481945\pi$$
0.0566895 + 0.998392i $$0.481945\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ −38.6279 −3.57115
$$118$$ 0 0
$$119$$ −1.39084 −0.127498
$$120$$ 0 0
$$121$$ −10.4687 −0.951701
$$122$$ 0 0
$$123$$ 14.5422 1.31123
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −9.52366 −0.845088 −0.422544 0.906342i $$-0.638863\pi$$
−0.422544 + 0.906342i $$0.638863\pi$$
$$128$$ 0 0
$$129$$ −24.4633 −2.15387
$$130$$ 0 0
$$131$$ 17.4028 1.52049 0.760247 0.649635i $$-0.225078\pi$$
0.760247 + 0.649635i $$0.225078\pi$$
$$132$$ 0 0
$$133$$ −2.46980 −0.214158
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 2.25910 0.193008 0.0965040 0.995333i $$-0.469234\pi$$
0.0965040 + 0.995333i $$0.469234\pi$$
$$138$$ 0 0
$$139$$ −16.8606 −1.43010 −0.715050 0.699073i $$-0.753596\pi$$
−0.715050 + 0.699073i $$0.753596\pi$$
$$140$$ 0 0
$$141$$ −24.6465 −2.07561
$$142$$ 0 0
$$143$$ 4.54221 0.379838
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ −2.72998 −0.225165
$$148$$ 0 0
$$149$$ −13.0055 −1.06545 −0.532724 0.846289i $$-0.678832\pi$$
−0.532724 + 0.846289i $$0.678832\pi$$
$$150$$ 0 0
$$151$$ −17.5895 −1.43142 −0.715708 0.698400i $$-0.753896\pi$$
−0.715708 + 0.698400i $$0.753896\pi$$
$$152$$ 0 0
$$153$$ 3.49073 0.282209
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 9.52366 0.760071 0.380035 0.924972i $$-0.375912\pi$$
0.380035 + 0.924972i $$0.375912\pi$$
$$158$$ 0 0
$$159$$ −4.64101 −0.368056
$$160$$ 0 0
$$161$$ 11.4489 0.902297
$$162$$ 0 0
$$163$$ −13.1921 −1.03329 −0.516644 0.856200i $$-0.672819\pi$$
−0.516644 + 0.856200i $$0.672819\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −1.81331 −0.140318 −0.0701591 0.997536i $$-0.522351\pi$$
−0.0701591 + 0.997536i $$0.522351\pi$$
$$168$$ 0 0
$$169$$ 25.8332 1.98717
$$170$$ 0 0
$$171$$ 6.19869 0.474026
$$172$$ 0 0
$$173$$ −12.5237 −0.952156 −0.476078 0.879403i $$-0.657942\pi$$
−0.476078 + 0.879403i $$0.657942\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 17.4753 1.31352
$$178$$ 0 0
$$179$$ −1.39738 −0.104445 −0.0522226 0.998635i $$-0.516631\pi$$
−0.0522226 + 0.998635i $$0.516631\pi$$
$$180$$ 0 0
$$181$$ 5.72890 0.425825 0.212913 0.977071i $$-0.431705\pi$$
0.212913 + 0.977071i $$0.431705\pi$$
$$182$$ 0 0
$$183$$ 33.1791 2.45267
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −0.410470 −0.0300165
$$188$$ 0 0
$$189$$ −23.9605 −1.74287
$$190$$ 0 0
$$191$$ 27.0198 1.95509 0.977544 0.210733i $$-0.0675849\pi$$
0.977544 + 0.210733i $$0.0675849\pi$$
$$192$$ 0 0
$$193$$ −23.0713 −1.66071 −0.830355 0.557234i $$-0.811862\pi$$
−0.830355 + 0.557234i $$0.811862\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 0.794765 0.0566247 0.0283123 0.999599i $$-0.490987\pi$$
0.0283123 + 0.999599i $$0.490987\pi$$
$$198$$ 0 0
$$199$$ −8.07241 −0.572238 −0.286119 0.958194i $$-0.592365\pi$$
−0.286119 + 0.958194i $$0.592365\pi$$
$$200$$ 0 0
$$201$$ −39.2251 −2.76672
$$202$$ 0 0
$$203$$ −25.2700 −1.77361
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ −28.7344 −1.99718
$$208$$ 0 0
$$209$$ −0.728896 −0.0504188
$$210$$ 0 0
$$211$$ 11.6410 0.801400 0.400700 0.916209i $$-0.368767\pi$$
0.400700 + 0.916209i $$0.368767\pi$$
$$212$$ 0 0
$$213$$ 13.3370 0.913834
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 14.9815 1.01701
$$218$$ 0 0
$$219$$ −12.4148 −0.838917
$$220$$ 0 0
$$221$$ −3.50927 −0.236059
$$222$$ 0 0
$$223$$ 15.6554 1.04836 0.524182 0.851607i $$-0.324371\pi$$
0.524182 + 0.851607i $$0.324371\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −4.80131 −0.318674 −0.159337 0.987224i $$-0.550936\pi$$
−0.159337 + 0.987224i $$0.550936\pi$$
$$228$$ 0 0
$$229$$ 2.79476 0.184683 0.0923416 0.995727i $$-0.470565\pi$$
0.0923416 + 0.995727i $$0.470565\pi$$
$$230$$ 0 0
$$231$$ 5.45996 0.359239
$$232$$ 0 0
$$233$$ 11.5422 0.756155 0.378078 0.925774i $$-0.376585\pi$$
0.378078 + 0.925774i $$0.376585\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ −46.5160 −3.02154
$$238$$ 0 0
$$239$$ 26.2251 1.69636 0.848180 0.529708i $$-0.177699\pi$$
0.848180 + 0.529708i $$0.177699\pi$$
$$240$$ 0 0
$$241$$ −12.0659 −0.777231 −0.388615 0.921400i $$-0.627047\pi$$
−0.388615 + 0.921400i $$0.627047\pi$$
$$242$$ 0 0
$$243$$ 3.73544 0.239629
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −6.23163 −0.396509
$$248$$ 0 0
$$249$$ −23.8242 −1.50980
$$250$$ 0 0
$$251$$ −13.5237 −0.853606 −0.426803 0.904345i $$-0.640360\pi$$
−0.426803 + 0.904345i $$0.640360\pi$$
$$252$$ 0 0
$$253$$ 3.37884 0.212426
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 22.7398 1.41847 0.709235 0.704972i $$-0.249040\pi$$
0.709235 + 0.704972i $$0.249040\pi$$
$$258$$ 0 0
$$259$$ 14.1492 0.879188
$$260$$ 0 0
$$261$$ 63.4227 3.92577
$$262$$ 0 0
$$263$$ 6.67395 0.411533 0.205767 0.978601i $$-0.434031\pi$$
0.205767 + 0.978601i $$0.434031\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ −30.3293 −1.85613
$$268$$ 0 0
$$269$$ −29.7398 −1.81327 −0.906634 0.421917i $$-0.861357\pi$$
−0.906634 + 0.421917i $$0.861357\pi$$
$$270$$ 0 0
$$271$$ −1.11189 −0.0675426 −0.0337713 0.999430i $$-0.510752\pi$$
−0.0337713 + 0.999430i $$0.510752\pi$$
$$272$$ 0 0
$$273$$ 46.6794 2.82517
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 13.1263 0.788682 0.394341 0.918964i $$-0.370973\pi$$
0.394341 + 0.918964i $$0.370973\pi$$
$$278$$ 0 0
$$279$$ −37.6004 −2.25108
$$280$$ 0 0
$$281$$ 22.9265 1.36768 0.683840 0.729632i $$-0.260309\pi$$
0.683840 + 0.729632i $$0.260309\pi$$
$$282$$ 0 0
$$283$$ −0.860634 −0.0511594 −0.0255797 0.999673i $$-0.508143\pi$$
−0.0255797 + 0.999673i $$0.508143\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −11.8421 −0.699016
$$288$$ 0 0
$$289$$ −16.6829 −0.981346
$$290$$ 0 0
$$291$$ 33.3788 1.95670
$$292$$ 0 0
$$293$$ 19.8212 1.15796 0.578982 0.815340i $$-0.303450\pi$$
0.578982 + 0.815340i $$0.303450\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ −7.07133 −0.410320
$$298$$ 0 0
$$299$$ 28.8870 1.67058
$$300$$ 0 0
$$301$$ 19.9210 1.14823
$$302$$ 0 0
$$303$$ −29.5237 −1.69609
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 16.8661 0.962599 0.481299 0.876556i $$-0.340165\pi$$
0.481299 + 0.876556i $$0.340165\pi$$
$$308$$ 0 0
$$309$$ 25.2689 1.43750
$$310$$ 0 0
$$311$$ −10.3250 −0.585475 −0.292738 0.956193i $$-0.594566\pi$$
−0.292738 + 0.956193i $$0.594566\pi$$
$$312$$ 0 0
$$313$$ −15.7684 −0.891281 −0.445641 0.895212i $$-0.647024\pi$$
−0.445641 + 0.895212i $$0.647024\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −2.17230 −0.122009 −0.0610043 0.998138i $$-0.519430\pi$$
−0.0610043 + 0.998138i $$0.519430\pi$$
$$318$$ 0 0
$$319$$ −7.45779 −0.417556
$$320$$ 0 0
$$321$$ 45.5741 2.54370
$$322$$ 0 0
$$323$$ 0.563139 0.0313339
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ −41.4358 −2.29140
$$328$$ 0 0
$$329$$ 20.0702 1.10651
$$330$$ 0 0
$$331$$ −11.9791 −0.658429 −0.329215 0.944255i $$-0.606784\pi$$
−0.329215 + 0.944255i $$0.606784\pi$$
$$332$$ 0 0
$$333$$ −35.5117 −1.94603
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −11.1921 −0.609675 −0.304838 0.952404i $$-0.598602\pi$$
−0.304838 + 0.952404i $$0.598602\pi$$
$$338$$ 0 0
$$339$$ 3.65540 0.198534
$$340$$ 0 0
$$341$$ 4.42139 0.239432
$$342$$ 0 0
$$343$$ 19.5117 1.05353
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 20.3843 1.09429 0.547143 0.837039i $$-0.315715\pi$$
0.547143 + 0.837039i $$0.315715\pi$$
$$348$$ 0 0
$$349$$ −0.252557 −0.0135191 −0.00675954 0.999977i $$-0.502152\pi$$
−0.00675954 + 0.999977i $$0.502152\pi$$
$$350$$ 0 0
$$351$$ −60.4556 −3.22688
$$352$$ 0 0
$$353$$ −28.6434 −1.52453 −0.762267 0.647263i $$-0.775914\pi$$
−0.762267 + 0.647263i $$0.775914\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ −4.21832 −0.223257
$$358$$ 0 0
$$359$$ 18.5741 0.980301 0.490151 0.871638i $$-0.336942\pi$$
0.490151 + 0.871638i $$0.336942\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ −31.7509 −1.66649
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 4.79476 0.250285 0.125142 0.992139i $$-0.460061\pi$$
0.125142 + 0.992139i $$0.460061\pi$$
$$368$$ 0 0
$$369$$ 29.7213 1.54723
$$370$$ 0 0
$$371$$ 3.77929 0.196211
$$372$$ 0 0
$$373$$ 15.9485 0.825783 0.412892 0.910780i $$-0.364519\pi$$
0.412892 + 0.910780i $$0.364519\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −63.7597 −3.28379
$$378$$ 0 0
$$379$$ 16.8013 0.863025 0.431513 0.902107i $$-0.357980\pi$$
0.431513 + 0.902107i $$0.357980\pi$$
$$380$$ 0 0
$$381$$ −28.8846 −1.47980
$$382$$ 0 0
$$383$$ −26.7948 −1.36915 −0.684574 0.728943i $$-0.740012\pi$$
−0.684574 + 0.728943i $$0.740012\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −49.9978 −2.54153
$$388$$ 0 0
$$389$$ 18.3424 0.929998 0.464999 0.885311i $$-0.346055\pi$$
0.464999 + 0.885311i $$0.346055\pi$$
$$390$$ 0 0
$$391$$ −2.61046 −0.132017
$$392$$ 0 0
$$393$$ 52.7817 2.66248
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 21.2162 1.06481 0.532404 0.846490i $$-0.321289\pi$$
0.532404 + 0.846490i $$0.321289\pi$$
$$398$$ 0 0
$$399$$ −7.49073 −0.375005
$$400$$ 0 0
$$401$$ −27.8661 −1.39157 −0.695783 0.718252i $$-0.744943\pi$$
−0.695783 + 0.718252i $$0.744943\pi$$
$$402$$ 0 0
$$403$$ 37.8002 1.88296
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 4.17577 0.206985
$$408$$ 0 0
$$409$$ −18.0899 −0.894487 −0.447243 0.894412i $$-0.647594\pi$$
−0.447243 + 0.894412i $$0.647594\pi$$
$$410$$ 0 0
$$411$$ 6.85171 0.337970
$$412$$ 0 0
$$413$$ −14.2305 −0.700239
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ −51.1372 −2.50420
$$418$$ 0 0
$$419$$ 15.3370 0.749260 0.374630 0.927174i $$-0.377770\pi$$
0.374630 + 0.927174i $$0.377770\pi$$
$$420$$ 0 0
$$421$$ −8.42486 −0.410602 −0.205301 0.978699i $$-0.565817\pi$$
−0.205301 + 0.978699i $$0.565817\pi$$
$$422$$ 0 0
$$423$$ −50.3723 −2.44918
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −27.0185 −1.30752
$$428$$ 0 0
$$429$$ 13.7762 0.665122
$$430$$ 0 0
$$431$$ −16.2766 −0.784014 −0.392007 0.919962i $$-0.628219\pi$$
−0.392007 + 0.919962i $$0.628219\pi$$
$$432$$ 0 0
$$433$$ −18.1976 −0.874521 −0.437261 0.899335i $$-0.644051\pi$$
−0.437261 + 0.899335i $$0.644051\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −4.63555 −0.221749
$$438$$ 0 0
$$439$$ −20.4214 −0.974660 −0.487330 0.873218i $$-0.662029\pi$$
−0.487330 + 0.873218i $$0.662029\pi$$
$$440$$ 0 0
$$441$$ −5.57952 −0.265691
$$442$$ 0 0
$$443$$ −21.6685 −1.02950 −0.514750 0.857340i $$-0.672115\pi$$
−0.514750 + 0.857340i $$0.672115\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ −39.4447 −1.86567
$$448$$ 0 0
$$449$$ 23.8133 1.12382 0.561910 0.827198i $$-0.310067\pi$$
0.561910 + 0.827198i $$0.310067\pi$$
$$450$$ 0 0
$$451$$ −3.49489 −0.164568
$$452$$ 0 0
$$453$$ −53.3479 −2.50650
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 9.15813 0.428399 0.214200 0.976790i $$-0.431286\pi$$
0.214200 + 0.976790i $$0.431286\pi$$
$$458$$ 0 0
$$459$$ 5.46325 0.255003
$$460$$ 0 0
$$461$$ −7.78931 −0.362784 −0.181392 0.983411i $$-0.558060\pi$$
−0.181392 + 0.983411i $$0.558060\pi$$
$$462$$ 0 0
$$463$$ −0.405011 −0.0188225 −0.00941123 0.999956i $$-0.502996\pi$$
−0.00941123 + 0.999956i $$0.502996\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 1.95814 0.0906118 0.0453059 0.998973i $$-0.485574\pi$$
0.0453059 + 0.998973i $$0.485574\pi$$
$$468$$ 0 0
$$469$$ 31.9420 1.47494
$$470$$ 0 0
$$471$$ 28.8846 1.33093
$$472$$ 0 0
$$473$$ 5.87918 0.270325
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ −9.48527 −0.434301
$$478$$ 0 0
$$479$$ 22.9210 1.04729 0.523645 0.851937i $$-0.324572\pi$$
0.523645 + 0.851937i $$0.324572\pi$$
$$480$$ 0 0
$$481$$ 35.7003 1.62780
$$482$$ 0 0
$$483$$ 34.7237 1.57998
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 23.9450 1.08505 0.542527 0.840038i $$-0.317468\pi$$
0.542527 + 0.840038i $$0.317468\pi$$
$$488$$ 0 0
$$489$$ −40.0109 −1.80936
$$490$$ 0 0
$$491$$ −3.86826 −0.174572 −0.0872861 0.996183i $$-0.527819\pi$$
−0.0872861 + 0.996183i $$0.527819\pi$$
$$492$$ 0 0
$$493$$ 5.76183 0.259500
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −10.8606 −0.487166
$$498$$ 0 0
$$499$$ 7.92104 0.354595 0.177297 0.984157i $$-0.443265\pi$$
0.177297 + 0.984157i $$0.443265\pi$$
$$500$$ 0 0
$$501$$ −5.49966 −0.245706
$$502$$ 0 0
$$503$$ 16.6949 0.744388 0.372194 0.928155i $$-0.378606\pi$$
0.372194 + 0.928155i $$0.378606\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 78.3503 3.47966
$$508$$ 0 0
$$509$$ 24.4028 1.08164 0.540818 0.841139i $$-0.318115\pi$$
0.540818 + 0.841139i $$0.318115\pi$$
$$510$$ 0 0
$$511$$ 10.1097 0.447228
$$512$$ 0 0
$$513$$ 9.70142 0.428328
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 5.92321 0.260503
$$518$$ 0 0
$$519$$ −37.9834 −1.66729
$$520$$ 0 0
$$521$$ 26.0659 1.14197 0.570983 0.820962i $$-0.306562\pi$$
0.570983 + 0.820962i $$0.306562\pi$$
$$522$$ 0 0
$$523$$ −17.8726 −0.781516 −0.390758 0.920493i $$-0.627787\pi$$
−0.390758 + 0.920493i $$0.627787\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −3.41593 −0.148800
$$528$$ 0 0
$$529$$ −1.51166 −0.0657243
$$530$$ 0 0
$$531$$ 35.7158 1.54993
$$532$$ 0 0
$$533$$ −29.8792 −1.29421
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ −4.23817 −0.182891
$$538$$ 0 0
$$539$$ 0.656088 0.0282597
$$540$$ 0 0
$$541$$ 23.0604 0.991444 0.495722 0.868481i $$-0.334903\pi$$
0.495722 + 0.868481i $$0.334903\pi$$
$$542$$ 0 0
$$543$$ 17.3754 0.745648
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 27.7584 1.18686 0.593431 0.804885i $$-0.297773\pi$$
0.593431 + 0.804885i $$0.297773\pi$$
$$548$$ 0 0
$$549$$ 67.8111 2.89411
$$550$$ 0 0
$$551$$ 10.2316 0.435882
$$552$$ 0 0
$$553$$ 37.8792 1.61079
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −8.60808 −0.364736 −0.182368 0.983230i $$-0.558376\pi$$
−0.182368 + 0.983230i $$0.558376\pi$$
$$558$$ 0 0
$$559$$ 50.2635 2.12592
$$560$$ 0 0
$$561$$ −1.24493 −0.0525609
$$562$$ 0 0
$$563$$ −7.93959 −0.334614 −0.167307 0.985905i $$-0.553507\pi$$
−0.167307 + 0.985905i $$0.553507\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ −26.7422 −1.12307
$$568$$ 0 0
$$569$$ −1.65757 −0.0694889 −0.0347444 0.999396i $$-0.511062\pi$$
−0.0347444 + 0.999396i $$0.511062\pi$$
$$570$$ 0 0
$$571$$ 14.0528 0.588091 0.294045 0.955791i $$-0.404998\pi$$
0.294045 + 0.955791i $$0.404998\pi$$
$$572$$ 0 0
$$573$$ 81.9494 3.42349
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −1.36445 −0.0568027 −0.0284014 0.999597i $$-0.509042\pi$$
−0.0284014 + 0.999597i $$0.509042\pi$$
$$578$$ 0 0
$$579$$ −69.9738 −2.90801
$$580$$ 0 0
$$581$$ 19.4007 0.804876
$$582$$ 0 0
$$583$$ 1.11536 0.0461935
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 24.6609 1.01786 0.508931 0.860807i $$-0.330041\pi$$
0.508931 + 0.860807i $$0.330041\pi$$
$$588$$ 0 0
$$589$$ −6.06587 −0.249940
$$590$$ 0 0
$$591$$ 2.41047 0.0991535
$$592$$ 0 0
$$593$$ −0.747443 −0.0306938 −0.0153469 0.999882i $$-0.504885\pi$$
−0.0153469 + 0.999882i $$0.504885\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −24.4831 −1.00203
$$598$$ 0 0
$$599$$ −1.40501 −0.0574072 −0.0287036 0.999588i $$-0.509138\pi$$
−0.0287036 + 0.999588i $$0.509138\pi$$
$$600$$ 0 0
$$601$$ 29.7453 1.21334 0.606668 0.794956i $$-0.292506\pi$$
0.606668 + 0.794956i $$0.292506\pi$$
$$602$$ 0 0
$$603$$ −80.1680 −3.26469
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −5.66849 −0.230077 −0.115038 0.993361i $$-0.536699\pi$$
−0.115038 + 0.993361i $$0.536699\pi$$
$$608$$ 0 0
$$609$$ −76.6423 −3.10570
$$610$$ 0 0
$$611$$ 50.6399 2.04867
$$612$$ 0 0
$$613$$ −2.99454 −0.120948 −0.0604742 0.998170i $$-0.519261\pi$$
−0.0604742 + 0.998170i $$0.519261\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −32.3370 −1.30184 −0.650919 0.759147i $$-0.725616\pi$$
−0.650919 + 0.759147i $$0.725616\pi$$
$$618$$ 0 0
$$619$$ 20.9265 0.841107 0.420554 0.907268i $$-0.361836\pi$$
0.420554 + 0.907268i $$0.361836\pi$$
$$620$$ 0 0
$$621$$ −44.9714 −1.80464
$$622$$ 0 0
$$623$$ 24.6980 0.989503
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ −2.21069 −0.0882866
$$628$$ 0 0
$$629$$ −3.22617 −0.128636
$$630$$ 0 0
$$631$$ −22.8002 −0.907663 −0.453831 0.891088i $$-0.649943\pi$$
−0.453831 + 0.891088i $$0.649943\pi$$
$$632$$ 0 0
$$633$$ 35.3064 1.40330
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 5.60916 0.222243
$$638$$ 0 0
$$639$$ 27.2580 1.07831
$$640$$ 0 0
$$641$$ 36.3184 1.43449 0.717246 0.696820i $$-0.245402\pi$$
0.717246 + 0.696820i $$0.245402\pi$$
$$642$$ 0 0
$$643$$ −13.6135 −0.536865 −0.268433 0.963298i $$-0.586506\pi$$
−0.268433 + 0.963298i $$0.586506\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 0.0724126 0.00284683 0.00142342 0.999999i $$-0.499547\pi$$
0.00142342 + 0.999999i $$0.499547\pi$$
$$648$$ 0 0
$$649$$ −4.19978 −0.164856
$$650$$ 0 0
$$651$$ 45.4378 1.78085
$$652$$ 0 0
$$653$$ −43.5346 −1.70364 −0.851820 0.523835i $$-0.824501\pi$$
−0.851820 + 0.523835i $$0.824501\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ −25.3734 −0.989910
$$658$$ 0 0
$$659$$ 33.4512 1.30308 0.651538 0.758616i $$-0.274124\pi$$
0.651538 + 0.758616i $$0.274124\pi$$
$$660$$ 0 0
$$661$$ −2.89465 −0.112589 −0.0562945 0.998414i $$-0.517929\pi$$
−0.0562945 + 0.998414i $$0.517929\pi$$
$$662$$ 0 0
$$663$$ −10.6434 −0.413355
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −47.4292 −1.83647
$$668$$ 0 0
$$669$$ 47.4818 1.83575
$$670$$ 0 0
$$671$$ −7.97382 −0.307826
$$672$$ 0 0
$$673$$ 42.8475 1.65165 0.825826 0.563925i $$-0.190709\pi$$
0.825826 + 0.563925i $$0.190709\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 34.4567 1.32428 0.662139 0.749381i $$-0.269649\pi$$
0.662139 + 0.749381i $$0.269649\pi$$
$$678$$ 0 0
$$679$$ −27.1812 −1.04312
$$680$$ 0 0
$$681$$ −14.5621 −0.558019
$$682$$ 0 0
$$683$$ 2.73436 0.104627 0.0523136 0.998631i $$-0.483340\pi$$
0.0523136 + 0.998631i $$0.483340\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 8.47634 0.323393
$$688$$ 0 0
$$689$$ 9.53566 0.363280
$$690$$ 0 0
$$691$$ 4.74198 0.180394 0.0901968 0.995924i $$-0.471250\pi$$
0.0901968 + 0.995924i $$0.471250\pi$$
$$692$$ 0 0
$$693$$ 11.1590 0.423897
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 2.70012 0.102274
$$698$$ 0 0
$$699$$ 35.0068 1.32408
$$700$$ 0 0
$$701$$ 33.1372 1.25157 0.625787 0.779994i $$-0.284778\pi$$
0.625787 + 0.779994i $$0.284778\pi$$
$$702$$ 0 0
$$703$$ −5.72890 −0.216069
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 24.0419 0.904187
$$708$$ 0 0
$$709$$ 5.37884 0.202006 0.101003 0.994886i $$-0.467795\pi$$
0.101003 + 0.994886i $$0.467795\pi$$
$$710$$ 0 0
$$711$$ −95.0692 −3.56537
$$712$$ 0 0
$$713$$ 28.1187 1.05305
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 79.5390 2.97044
$$718$$ 0 0
$$719$$ −33.0857 −1.23389 −0.616944 0.787007i $$-0.711630\pi$$
−0.616944 + 0.787007i $$0.711630\pi$$
$$720$$ 0 0
$$721$$ −20.5771 −0.766332
$$722$$ 0 0
$$723$$ −36.5950 −1.36098
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −2.62463 −0.0973423 −0.0486711 0.998815i $$-0.515499\pi$$
−0.0486711 + 0.998815i $$0.515499\pi$$
$$728$$ 0 0
$$729$$ −21.1538 −0.783472
$$730$$ 0 0
$$731$$ −4.54221 −0.168000
$$732$$ 0 0
$$733$$ 0.608077 0.0224598 0.0112299 0.999937i $$-0.496425\pi$$
0.0112299 + 0.999937i $$0.496425\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 9.42685 0.347242
$$738$$ 0 0
$$739$$ −36.3974 −1.33890 −0.669450 0.742857i $$-0.733470\pi$$
−0.669450 + 0.742857i $$0.733470\pi$$
$$740$$ 0 0
$$741$$ −18.9001 −0.694313
$$742$$ 0 0
$$743$$ 23.0713 0.846405 0.423202 0.906035i $$-0.360906\pi$$
0.423202 + 0.906035i $$0.360906\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ −48.6918 −1.78154
$$748$$ 0 0
$$749$$ −37.1121 −1.35605
$$750$$ 0 0
$$751$$ −4.75290 −0.173436 −0.0867179 0.996233i $$-0.527638\pi$$
−0.0867179 + 0.996233i $$0.527638\pi$$
$$752$$ 0 0
$$753$$ −41.0164 −1.49472
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 26.8057 0.974269 0.487135 0.873327i $$-0.338042\pi$$
0.487135 + 0.873327i $$0.338042\pi$$
$$758$$ 0 0
$$759$$ 10.2478 0.371971
$$760$$ 0 0
$$761$$ 25.4478 0.922481 0.461241 0.887275i $$-0.347404\pi$$
0.461241 + 0.887275i $$0.347404\pi$$
$$762$$ 0 0
$$763$$ 33.7422 1.22155
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −35.9056 −1.29648
$$768$$ 0 0
$$769$$ −14.6421 −0.528007 −0.264004 0.964522i $$-0.585043\pi$$
−0.264004 + 0.964522i $$0.585043\pi$$
$$770$$ 0 0
$$771$$ 68.9684 2.48384
$$772$$ 0 0
$$773$$ −16.8462 −0.605917 −0.302959 0.953004i $$-0.597974\pi$$
−0.302959 + 0.953004i $$0.597974\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 42.9136 1.53952
$$778$$ 0 0
$$779$$ 4.79476 0.171790
$$780$$ 0 0
$$781$$ −3.20524 −0.114692
$$782$$ 0 0
$$783$$ 99.2613 3.54731
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −18.7367 −0.667893 −0.333946 0.942592i $$-0.608380\pi$$
−0.333946 + 0.942592i $$0.608380\pi$$
$$788$$ 0 0
$$789$$ 20.2416 0.720621
$$790$$ 0 0
$$791$$ −2.97668 −0.105839
$$792$$ 0 0
$$793$$ −68.1714 −2.42084
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −37.9900 −1.34567 −0.672837 0.739791i $$-0.734925\pi$$
−0.672837 + 0.739791i $$0.734925\pi$$
$$798$$ 0 0
$$799$$ −4.57623 −0.161895
$$800$$ 0 0
$$801$$ −61.9869 −2.19020
$$802$$ 0 0
$$803$$ 2.98362 0.105290
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −90.1989 −3.17515
$$808$$ 0 0
$$809$$ 21.8857 0.769461 0.384731 0.923029i $$-0.374294\pi$$
0.384731 + 0.923029i $$0.374294\pi$$
$$810$$ 0 0
$$811$$ −37.8595 −1.32943 −0.664714 0.747098i $$-0.731447\pi$$
−0.664714 + 0.747098i $$0.731447\pi$$
$$812$$ 0 0
$$813$$ −3.37229 −0.118271
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −8.06587 −0.282189
$$818$$ 0 0
$$819$$ 95.4031 3.33365
$$820$$ 0 0
$$821$$ −20.1976 −0.704901 −0.352451 0.935830i $$-0.614652\pi$$
−0.352451 + 0.935830i $$0.614652\pi$$
$$822$$ 0 0
$$823$$ 4.34590 0.151489 0.0757443 0.997127i $$-0.475867\pi$$
0.0757443 + 0.997127i $$0.475867\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 56.8375 1.97643 0.988217 0.153057i $$-0.0489119\pi$$
0.988217 + 0.153057i $$0.0489119\pi$$
$$828$$ 0 0
$$829$$ −17.4543 −0.606214 −0.303107 0.952957i $$-0.598024\pi$$
−0.303107 + 0.952957i $$0.598024\pi$$
$$830$$ 0 0
$$831$$ 39.8111 1.38103
$$832$$ 0 0
$$833$$ −0.506888 −0.0175626
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ −58.8475 −2.03407
$$838$$ 0 0
$$839$$ −1.77622 −0.0613219 −0.0306609 0.999530i $$-0.509761\pi$$
−0.0306609 + 0.999530i $$0.509761\pi$$
$$840$$ 0 0
$$841$$ 75.6862 2.60987
$$842$$ 0 0
$$843$$ 69.5346 2.39490
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 25.8556 0.888408
$$848$$ 0 0
$$849$$ −2.61025 −0.0895834
$$850$$ 0 0
$$851$$ 26.5566 0.910348
$$852$$ 0 0
$$853$$ −16.1187 −0.551892 −0.275946 0.961173i $$-0.588991\pi$$
−0.275946 + 0.961173i $$0.588991\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −47.2031 −1.61243 −0.806213 0.591625i $$-0.798486\pi$$
−0.806213 + 0.591625i $$0.798486\pi$$
$$858$$ 0 0
$$859$$ −37.3239 −1.27347 −0.636737 0.771081i $$-0.719716\pi$$
−0.636737 + 0.771081i $$0.719716\pi$$
$$860$$ 0 0
$$861$$ −35.9163 −1.22402
$$862$$ 0 0
$$863$$ 1.33697 0.0455111 0.0227555 0.999741i $$-0.492756\pi$$
0.0227555 + 0.999741i $$0.492756\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ −50.5981 −1.71840
$$868$$ 0 0
$$869$$ 11.1791 0.379224
$$870$$ 0 0
$$871$$ 80.5939 2.73082
$$872$$ 0 0
$$873$$ 68.2194 2.30888
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 13.1647 0.444539 0.222270 0.974985i $$-0.428653\pi$$
0.222270 + 0.974985i $$0.428653\pi$$
$$878$$ 0 0
$$879$$ 60.1163 2.02767
$$880$$ 0 0
$$881$$ −10.2052 −0.343823 −0.171912 0.985112i $$-0.554994\pi$$
−0.171912 + 0.985112i $$0.554994\pi$$
$$882$$ 0 0
$$883$$ −1.49966 −0.0504674 −0.0252337 0.999682i $$-0.508033\pi$$
−0.0252337 + 0.999682i $$0.508033\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 46.9505 1.57644 0.788222 0.615391i $$-0.211002\pi$$
0.788222 + 0.615391i $$0.211002\pi$$
$$888$$ 0 0
$$889$$ 23.5215 0.788886
$$890$$ 0 0
$$891$$ −7.89227 −0.264401
$$892$$ 0 0
$$893$$ −8.12628 −0.271936
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 87.6125 2.92529
$$898$$ 0 0
$$899$$ −62.0637 −2.06994
$$900$$ 0 0
$$901$$ −0.861719 −0.0287080
$$902$$ 0 0
$$903$$ 60.4192 2.01063
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −16.3668 −0.543452 −0.271726 0.962375i $$-0.587594\pi$$
−0.271726 + 0.962375i $$0.587594\pi$$
$$908$$ 0 0
$$909$$ −60.3403 −2.00136
$$910$$ 0 0
$$911$$ −32.3293 −1.07112 −0.535559 0.844498i $$-0.679899\pi$$
−0.535559 + 0.844498i $$0.679899\pi$$
$$912$$ 0 0
$$913$$ 5.72561 0.189490
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −42.9815 −1.41937
$$918$$ 0 0
$$919$$ 22.8157 0.752620 0.376310 0.926494i $$-0.377193\pi$$
0.376310 + 0.926494i $$0.377193\pi$$
$$920$$ 0 0
$$921$$ 51.1538 1.68557
$$922$$ 0 0
$$923$$ −27.4028 −0.901976
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 51.6445 1.69623
$$928$$ 0 0
$$929$$ −27.2436 −0.893834 −0.446917 0.894575i $$-0.647478\pi$$
−0.446917 + 0.894575i $$0.647478\pi$$
$$930$$ 0 0
$$931$$ −0.900112 −0.0295000
$$932$$ 0 0
$$933$$ −31.3150 −1.02521
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −51.5006 −1.68245 −0.841225 0.540685i $$-0.818165\pi$$
−0.841225 + 0.540685i $$0.818165\pi$$
$$938$$ 0 0
$$939$$ −47.8244 −1.56069
$$940$$ 0 0
$$941$$ −40.8541 −1.33181 −0.665903 0.746039i $$-0.731953\pi$$
−0.665903 + 0.746039i $$0.731953\pi$$
$$942$$ 0 0
$$943$$ −22.2264 −0.723791
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −38.2526 −1.24304 −0.621521 0.783398i $$-0.713485\pi$$
−0.621521 + 0.783398i $$0.713485\pi$$
$$948$$ 0 0
$$949$$ 25.5082 0.828031
$$950$$ 0 0
$$951$$ −6.58845 −0.213645
$$952$$ 0 0
$$953$$ 54.1187 1.75308 0.876538 0.481334i $$-0.159847\pi$$
0.876538 + 0.481334i $$0.159847\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ −22.6190 −0.731168
$$958$$ 0 0
$$959$$ −5.57952 −0.180172
$$960$$ 0 0
$$961$$ 5.79476 0.186928
$$962$$ 0 0
$$963$$ 93.1440 3.00152
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −28.4214 −0.913970 −0.456985 0.889474i $$-0.651071\pi$$
−0.456985 + 0.889474i $$0.651071\pi$$
$$968$$ 0 0
$$969$$ 1.70796 0.0548677
$$970$$ 0 0
$$971$$ −30.8057 −0.988601 −0.494301 0.869291i $$-0.664576\pi$$
−0.494301 + 0.869291i $$0.664576\pi$$
$$972$$ 0 0
$$973$$ 41.6423 1.33499
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 37.6554 1.20470 0.602351 0.798231i $$-0.294231\pi$$
0.602351 + 0.798231i $$0.294231\pi$$
$$978$$ 0 0
$$979$$ 7.28896 0.232956
$$980$$ 0 0
$$981$$ −84.6862 −2.70382
$$982$$ 0 0
$$983$$ 38.7948 1.23736 0.618680 0.785643i $$-0.287668\pi$$
0.618680 + 0.785643i $$0.287668\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 60.8717 1.93757
$$988$$ 0 0
$$989$$ 37.3898 1.18893
$$990$$ 0 0
$$991$$ 15.0295 0.477427 0.238713 0.971090i $$-0.423274\pi$$
0.238713 + 0.971090i $$0.423274\pi$$
$$992$$ 0 0
$$993$$ −36.3317 −1.15295
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 1.18123 0.0374099 0.0187050 0.999825i $$-0.494046\pi$$
0.0187050 + 0.999825i $$0.494046\pi$$
$$998$$ 0 0
$$999$$ −55.5784 −1.75842
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 7600.2.a.bm.1.3 3
4.3 odd 2 950.2.a.m.1.1 yes 3
5.4 even 2 7600.2.a.cb.1.1 3
12.11 even 2 8550.2.a.cj.1.2 3
20.3 even 4 950.2.b.g.799.1 6
20.7 even 4 950.2.b.g.799.6 6
20.19 odd 2 950.2.a.k.1.3 3
60.59 even 2 8550.2.a.co.1.2 3

By twisted newform
Twist Min Dim Char Parity Ord Type
950.2.a.k.1.3 3 20.19 odd 2
950.2.a.m.1.1 yes 3 4.3 odd 2
950.2.b.g.799.1 6 20.3 even 4
950.2.b.g.799.6 6 20.7 even 4
7600.2.a.bm.1.3 3 1.1 even 1 trivial
7600.2.a.cb.1.1 3 5.4 even 2
8550.2.a.cj.1.2 3 12.11 even 2
8550.2.a.co.1.2 3 60.59 even 2