# Properties

 Label 2800.2.g.s.449.1 Level $2800$ Weight $2$ Character 2800.449 Analytic conductor $22.358$ Analytic rank $0$ Dimension $4$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2800,2,Mod(449,2800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2800, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2800.449");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2800 = 2^{4} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2800.g (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: $$22.3581125660$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 3x^{2} + 1$$ x^4 + 3*x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 175) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 449.1 Root $$-1.61803i$$ of defining polynomial Character $$\chi$$ $$=$$ 2800.449 Dual form 2800.2.g.s.449.4

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-3.23607i q^{3} -1.00000i q^{7} -7.47214 q^{9} +O(q^{10})$$ $$q-3.23607i q^{3} -1.00000i q^{7} -7.47214 q^{9} +0.236068 q^{11} -1.23607i q^{13} +2.47214i q^{17} -4.47214 q^{19} -3.23607 q^{21} +6.23607i q^{23} +14.4721i q^{27} -5.00000 q^{29} -3.70820 q^{31} -0.763932i q^{33} +3.00000i q^{37} -4.00000 q^{39} +4.76393 q^{41} +1.76393i q^{43} +2.00000i q^{47} -1.00000 q^{49} +8.00000 q^{51} -8.47214i q^{53} +14.4721i q^{57} +11.7082 q^{59} -9.70820 q^{61} +7.47214i q^{63} +4.23607i q^{67} +20.1803 q^{69} -8.70820 q^{71} +8.76393i q^{73} -0.236068i q^{77} -11.1803 q^{79} +24.4164 q^{81} -7.70820i q^{83} +16.1803i q^{87} -17.2361 q^{89} -1.23607 q^{91} +12.0000i q^{93} +5.23607i q^{97} -1.76393 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 12 q^{9}+O(q^{10})$$ 4 * q - 12 * q^9 $$4 q - 12 q^{9} - 8 q^{11} - 4 q^{21} - 20 q^{29} + 12 q^{31} - 16 q^{39} + 28 q^{41} - 4 q^{49} + 32 q^{51} + 20 q^{59} - 12 q^{61} + 36 q^{69} - 8 q^{71} + 44 q^{81} - 60 q^{89} + 4 q^{91} - 16 q^{99}+O(q^{100})$$ 4 * q - 12 * q^9 - 8 * q^11 - 4 * q^21 - 20 * q^29 + 12 * q^31 - 16 * q^39 + 28 * q^41 - 4 * q^49 + 32 * q^51 + 20 * q^59 - 12 * q^61 + 36 * q^69 - 8 * q^71 + 44 * q^81 - 60 * q^89 + 4 * q^91 - 16 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2800\mathbb{Z}\right)^\times$$.

 $$n$$ $$351$$ $$801$$ $$2101$$ $$2577$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-1$$

## 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
$$3$$ − 3.23607i − 1.86834i −0.356822 0.934172i $$-0.616140\pi$$
0.356822 0.934172i $$-0.383860\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ − 1.00000i − 0.377964i
$$8$$ 0 0
$$9$$ −7.47214 −2.49071
$$10$$ 0 0
$$11$$ 0.236068 0.0711772 0.0355886 0.999367i $$-0.488669\pi$$
0.0355886 + 0.999367i $$0.488669\pi$$
$$12$$ 0 0
$$13$$ − 1.23607i − 0.342824i −0.985199 0.171412i $$-0.945167\pi$$
0.985199 0.171412i $$-0.0548329\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 2.47214i 0.599581i 0.954005 + 0.299791i $$0.0969168\pi$$
−0.954005 + 0.299791i $$0.903083\pi$$
$$18$$ 0 0
$$19$$ −4.47214 −1.02598 −0.512989 0.858395i $$-0.671462\pi$$
−0.512989 + 0.858395i $$0.671462\pi$$
$$20$$ 0 0
$$21$$ −3.23607 −0.706168
$$22$$ 0 0
$$23$$ 6.23607i 1.30031i 0.759802 + 0.650155i $$0.225296\pi$$
−0.759802 + 0.650155i $$0.774704\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 14.4721i 2.78516i
$$28$$ 0 0
$$29$$ −5.00000 −0.928477 −0.464238 0.885710i $$-0.653672\pi$$
−0.464238 + 0.885710i $$0.653672\pi$$
$$30$$ 0 0
$$31$$ −3.70820 −0.666013 −0.333007 0.942925i $$-0.608063\pi$$
−0.333007 + 0.942925i $$0.608063\pi$$
$$32$$ 0 0
$$33$$ − 0.763932i − 0.132983i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 3.00000i 0.493197i 0.969118 + 0.246598i $$0.0793129\pi$$
−0.969118 + 0.246598i $$0.920687\pi$$
$$38$$ 0 0
$$39$$ −4.00000 −0.640513
$$40$$ 0 0
$$41$$ 4.76393 0.744001 0.372001 0.928232i $$-0.378672\pi$$
0.372001 + 0.928232i $$0.378672\pi$$
$$42$$ 0 0
$$43$$ 1.76393i 0.268997i 0.990914 + 0.134499i $$0.0429424\pi$$
−0.990914 + 0.134499i $$0.957058\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 2.00000i 0.291730i 0.989305 + 0.145865i $$0.0465965\pi$$
−0.989305 + 0.145865i $$0.953403\pi$$
$$48$$ 0 0
$$49$$ −1.00000 −0.142857
$$50$$ 0 0
$$51$$ 8.00000 1.12022
$$52$$ 0 0
$$53$$ − 8.47214i − 1.16374i −0.813283 0.581869i $$-0.802322\pi$$
0.813283 0.581869i $$-0.197678\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 14.4721i 1.91688i
$$58$$ 0 0
$$59$$ 11.7082 1.52428 0.762139 0.647413i $$-0.224149\pi$$
0.762139 + 0.647413i $$0.224149\pi$$
$$60$$ 0 0
$$61$$ −9.70820 −1.24301 −0.621504 0.783411i $$-0.713478\pi$$
−0.621504 + 0.783411i $$0.713478\pi$$
$$62$$ 0 0
$$63$$ 7.47214i 0.941401i
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 4.23607i 0.517518i 0.965942 + 0.258759i $$0.0833136\pi$$
−0.965942 + 0.258759i $$0.916686\pi$$
$$68$$ 0 0
$$69$$ 20.1803 2.42943
$$70$$ 0 0
$$71$$ −8.70820 −1.03347 −0.516737 0.856144i $$-0.672853\pi$$
−0.516737 + 0.856144i $$0.672853\pi$$
$$72$$ 0 0
$$73$$ 8.76393i 1.02574i 0.858466 + 0.512870i $$0.171418\pi$$
−0.858466 + 0.512870i $$0.828582\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ − 0.236068i − 0.0269024i
$$78$$ 0 0
$$79$$ −11.1803 −1.25789 −0.628943 0.777451i $$-0.716512\pi$$
−0.628943 + 0.777451i $$0.716512\pi$$
$$80$$ 0 0
$$81$$ 24.4164 2.71293
$$82$$ 0 0
$$83$$ − 7.70820i − 0.846085i −0.906110 0.423043i $$-0.860962\pi$$
0.906110 0.423043i $$-0.139038\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 16.1803i 1.73471i
$$88$$ 0 0
$$89$$ −17.2361 −1.82702 −0.913510 0.406817i $$-0.866639\pi$$
−0.913510 + 0.406817i $$0.866639\pi$$
$$90$$ 0 0
$$91$$ −1.23607 −0.129575
$$92$$ 0 0
$$93$$ 12.0000i 1.24434i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 5.23607i 0.531642i 0.964022 + 0.265821i $$0.0856430\pi$$
−0.964022 + 0.265821i $$0.914357\pi$$
$$98$$ 0 0
$$99$$ −1.76393 −0.177282
$$100$$ 0 0
$$101$$ 4.76393 0.474029 0.237014 0.971506i $$-0.423831\pi$$
0.237014 + 0.971506i $$0.423831\pi$$
$$102$$ 0 0
$$103$$ 8.47214i 0.834784i 0.908726 + 0.417392i $$0.137056\pi$$
−0.908726 + 0.417392i $$0.862944\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ − 8.00000i − 0.773389i −0.922208 0.386695i $$-0.873617\pi$$
0.922208 0.386695i $$-0.126383\pi$$
$$108$$ 0 0
$$109$$ −8.41641 −0.806146 −0.403073 0.915168i $$-0.632058\pi$$
−0.403073 + 0.915168i $$0.632058\pi$$
$$110$$ 0 0
$$111$$ 9.70820 0.921462
$$112$$ 0 0
$$113$$ 14.4164i 1.35618i 0.734978 + 0.678091i $$0.237192\pi$$
−0.734978 + 0.678091i $$0.762808\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 9.23607i 0.853875i
$$118$$ 0 0
$$119$$ 2.47214 0.226620
$$120$$ 0 0
$$121$$ −10.9443 −0.994934
$$122$$ 0 0
$$123$$ − 15.4164i − 1.39005i
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ − 13.6525i − 1.21146i −0.795670 0.605731i $$-0.792881\pi$$
0.795670 0.605731i $$-0.207119\pi$$
$$128$$ 0 0
$$129$$ 5.70820 0.502579
$$130$$ 0 0
$$131$$ 16.9443 1.48043 0.740214 0.672371i $$-0.234724\pi$$
0.740214 + 0.672371i $$0.234724\pi$$
$$132$$ 0 0
$$133$$ 4.47214i 0.387783i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ − 10.9443i − 0.935032i −0.883985 0.467516i $$-0.845149\pi$$
0.883985 0.467516i $$-0.154851\pi$$
$$138$$ 0 0
$$139$$ 10.6525 0.903531 0.451766 0.892137i $$-0.350794\pi$$
0.451766 + 0.892137i $$0.350794\pi$$
$$140$$ 0 0
$$141$$ 6.47214 0.545052
$$142$$ 0 0
$$143$$ − 0.291796i − 0.0244012i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 3.23607i 0.266906i
$$148$$ 0 0
$$149$$ −3.94427 −0.323127 −0.161564 0.986862i $$-0.551654\pi$$
−0.161564 + 0.986862i $$0.551654\pi$$
$$150$$ 0 0
$$151$$ 20.2361 1.64679 0.823394 0.567470i $$-0.192078\pi$$
0.823394 + 0.567470i $$0.192078\pi$$
$$152$$ 0 0
$$153$$ − 18.4721i − 1.49338i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 0.763932i 0.0609684i 0.999535 + 0.0304842i $$0.00970493\pi$$
−0.999535 + 0.0304842i $$0.990295\pi$$
$$158$$ 0 0
$$159$$ −27.4164 −2.17426
$$160$$ 0 0
$$161$$ 6.23607 0.491471
$$162$$ 0 0
$$163$$ − 1.52786i − 0.119672i −0.998208 0.0598358i $$-0.980942\pi$$
0.998208 0.0598358i $$-0.0190577\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ − 5.23607i − 0.405179i −0.979264 0.202590i $$-0.935064\pi$$
0.979264 0.202590i $$-0.0649357\pi$$
$$168$$ 0 0
$$169$$ 11.4721 0.882472
$$170$$ 0 0
$$171$$ 33.4164 2.55542
$$172$$ 0 0
$$173$$ 11.5279i 0.876447i 0.898866 + 0.438224i $$0.144392\pi$$
−0.898866 + 0.438224i $$0.855608\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ − 37.8885i − 2.84788i
$$178$$ 0 0
$$179$$ −23.4164 −1.75022 −0.875112 0.483920i $$-0.839213\pi$$
−0.875112 + 0.483920i $$0.839213\pi$$
$$180$$ 0 0
$$181$$ 8.18034 0.608040 0.304020 0.952666i $$-0.401671\pi$$
0.304020 + 0.952666i $$0.401671\pi$$
$$182$$ 0 0
$$183$$ 31.4164i 2.32237i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 0.583592i 0.0426765i
$$188$$ 0 0
$$189$$ 14.4721 1.05269
$$190$$ 0 0
$$191$$ −6.47214 −0.468307 −0.234154 0.972200i $$-0.575232\pi$$
−0.234154 + 0.972200i $$0.575232\pi$$
$$192$$ 0 0
$$193$$ − 12.4164i − 0.893753i −0.894596 0.446876i $$-0.852536\pi$$
0.894596 0.446876i $$-0.147464\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ − 1.47214i − 0.104885i −0.998624 0.0524427i $$-0.983299\pi$$
0.998624 0.0524427i $$-0.0167007\pi$$
$$198$$ 0 0
$$199$$ 7.23607 0.512951 0.256476 0.966551i $$-0.417439\pi$$
0.256476 + 0.966551i $$0.417439\pi$$
$$200$$ 0 0
$$201$$ 13.7082 0.966902
$$202$$ 0 0
$$203$$ 5.00000i 0.350931i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ − 46.5967i − 3.23870i
$$208$$ 0 0
$$209$$ −1.05573 −0.0730262
$$210$$ 0 0
$$211$$ −12.0000 −0.826114 −0.413057 0.910705i $$-0.635539\pi$$
−0.413057 + 0.910705i $$0.635539\pi$$
$$212$$ 0 0
$$213$$ 28.1803i 1.93089i
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 3.70820i 0.251729i
$$218$$ 0 0
$$219$$ 28.3607 1.91644
$$220$$ 0 0
$$221$$ 3.05573 0.205551
$$222$$ 0 0
$$223$$ 20.1803i 1.35138i 0.737188 + 0.675688i $$0.236153\pi$$
−0.737188 + 0.675688i $$0.763847\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ − 21.4164i − 1.42146i −0.703466 0.710728i $$-0.748365\pi$$
0.703466 0.710728i $$-0.251635\pi$$
$$228$$ 0 0
$$229$$ 4.47214 0.295527 0.147764 0.989023i $$-0.452793\pi$$
0.147764 + 0.989023i $$0.452793\pi$$
$$230$$ 0 0
$$231$$ −0.763932 −0.0502630
$$232$$ 0 0
$$233$$ − 7.94427i − 0.520447i −0.965548 0.260223i $$-0.916204\pi$$
0.965548 0.260223i $$-0.0837962\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 36.1803i 2.35017i
$$238$$ 0 0
$$239$$ −5.52786 −0.357568 −0.178784 0.983888i $$-0.557216\pi$$
−0.178784 + 0.983888i $$0.557216\pi$$
$$240$$ 0 0
$$241$$ −3.52786 −0.227250 −0.113625 0.993524i $$-0.536246\pi$$
−0.113625 + 0.993524i $$0.536246\pi$$
$$242$$ 0 0
$$243$$ − 35.5967i − 2.28353i
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 5.52786i 0.351730i
$$248$$ 0 0
$$249$$ −24.9443 −1.58078
$$250$$ 0 0
$$251$$ −6.47214 −0.408518 −0.204259 0.978917i $$-0.565478\pi$$
−0.204259 + 0.978917i $$0.565478\pi$$
$$252$$ 0 0
$$253$$ 1.47214i 0.0925524i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ − 12.6525i − 0.789240i −0.918844 0.394620i $$-0.870876\pi$$
0.918844 0.394620i $$-0.129124\pi$$
$$258$$ 0 0
$$259$$ 3.00000 0.186411
$$260$$ 0 0
$$261$$ 37.3607 2.31257
$$262$$ 0 0
$$263$$ 16.2361i 1.00116i 0.865691 + 0.500579i $$0.166880\pi$$
−0.865691 + 0.500579i $$0.833120\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 55.7771i 3.41350i
$$268$$ 0 0
$$269$$ 11.7082 0.713862 0.356931 0.934131i $$-0.383823\pi$$
0.356931 + 0.934131i $$0.383823\pi$$
$$270$$ 0 0
$$271$$ −23.7082 −1.44017 −0.720085 0.693885i $$-0.755897\pi$$
−0.720085 + 0.693885i $$0.755897\pi$$
$$272$$ 0 0
$$273$$ 4.00000i 0.242091i
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ − 19.8885i − 1.19499i −0.801874 0.597493i $$-0.796163\pi$$
0.801874 0.597493i $$-0.203837\pi$$
$$278$$ 0 0
$$279$$ 27.7082 1.65885
$$280$$ 0 0
$$281$$ −15.3607 −0.916341 −0.458171 0.888864i $$-0.651495\pi$$
−0.458171 + 0.888864i $$0.651495\pi$$
$$282$$ 0 0
$$283$$ 17.4164i 1.03530i 0.855593 + 0.517649i $$0.173193\pi$$
−0.855593 + 0.517649i $$0.826807\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ − 4.76393i − 0.281206i
$$288$$ 0 0
$$289$$ 10.8885 0.640503
$$290$$ 0 0
$$291$$ 16.9443 0.993291
$$292$$ 0 0
$$293$$ 31.1246i 1.81832i 0.416448 + 0.909160i $$0.363275\pi$$
−0.416448 + 0.909160i $$0.636725\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 3.41641i 0.198240i
$$298$$ 0 0
$$299$$ 7.70820 0.445777
$$300$$ 0 0
$$301$$ 1.76393 0.101671
$$302$$ 0 0
$$303$$ − 15.4164i − 0.885649i
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ − 4.58359i − 0.261599i −0.991409 0.130800i $$-0.958246\pi$$
0.991409 0.130800i $$-0.0417545\pi$$
$$308$$ 0 0
$$309$$ 27.4164 1.55966
$$310$$ 0 0
$$311$$ −24.3607 −1.38137 −0.690684 0.723157i $$-0.742690\pi$$
−0.690684 + 0.723157i $$0.742690\pi$$
$$312$$ 0 0
$$313$$ − 19.5279i − 1.10378i −0.833917 0.551890i $$-0.813907\pi$$
0.833917 0.551890i $$-0.186093\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 25.3607i 1.42440i 0.701978 + 0.712199i $$0.252301\pi$$
−0.701978 + 0.712199i $$0.747699\pi$$
$$318$$ 0 0
$$319$$ −1.18034 −0.0660863
$$320$$ 0 0
$$321$$ −25.8885 −1.44496
$$322$$ 0 0
$$323$$ − 11.0557i − 0.615157i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 27.2361i 1.50616i
$$328$$ 0 0
$$329$$ 2.00000 0.110264
$$330$$ 0 0
$$331$$ 24.7082 1.35809 0.679043 0.734099i $$-0.262395\pi$$
0.679043 + 0.734099i $$0.262395\pi$$
$$332$$ 0 0
$$333$$ − 22.4164i − 1.22841i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ − 16.4721i − 0.897294i −0.893709 0.448647i $$-0.851906\pi$$
0.893709 0.448647i $$-0.148094\pi$$
$$338$$ 0 0
$$339$$ 46.6525 2.53381
$$340$$ 0 0
$$341$$ −0.875388 −0.0474049
$$342$$ 0 0
$$343$$ 1.00000i 0.0539949i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ − 20.2361i − 1.08633i −0.839626 0.543165i $$-0.817226\pi$$
0.839626 0.543165i $$-0.182774\pi$$
$$348$$ 0 0
$$349$$ −4.47214 −0.239388 −0.119694 0.992811i $$-0.538191\pi$$
−0.119694 + 0.992811i $$0.538191\pi$$
$$350$$ 0 0
$$351$$ 17.8885 0.954820
$$352$$ 0 0
$$353$$ 2.18034i 0.116048i 0.998315 + 0.0580239i $$0.0184800\pi$$
−0.998315 + 0.0580239i $$0.981520\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ − 8.00000i − 0.423405i
$$358$$ 0 0
$$359$$ −30.1246 −1.58992 −0.794958 0.606664i $$-0.792507\pi$$
−0.794958 + 0.606664i $$0.792507\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ 35.4164i 1.85888i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 37.1246i 1.93789i 0.247278 + 0.968944i $$0.420464\pi$$
−0.247278 + 0.968944i $$0.579536\pi$$
$$368$$ 0 0
$$369$$ −35.5967 −1.85309
$$370$$ 0 0
$$371$$ −8.47214 −0.439851
$$372$$ 0 0
$$373$$ 37.8328i 1.95891i 0.201665 + 0.979454i $$0.435365\pi$$
−0.201665 + 0.979454i $$0.564635\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 6.18034i 0.318304i
$$378$$ 0 0
$$379$$ −11.1803 −0.574295 −0.287148 0.957886i $$-0.592707\pi$$
−0.287148 + 0.957886i $$0.592707\pi$$
$$380$$ 0 0
$$381$$ −44.1803 −2.26343
$$382$$ 0 0
$$383$$ − 33.2361i − 1.69828i −0.528165 0.849142i $$-0.677120\pi$$
0.528165 0.849142i $$-0.322880\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ − 13.1803i − 0.669994i
$$388$$ 0 0
$$389$$ −2.88854 −0.146455 −0.0732275 0.997315i $$-0.523330\pi$$
−0.0732275 + 0.997315i $$0.523330\pi$$
$$390$$ 0 0
$$391$$ −15.4164 −0.779641
$$392$$ 0 0
$$393$$ − 54.8328i − 2.76595i
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 9.05573i 0.454494i 0.973837 + 0.227247i $$0.0729725\pi$$
−0.973837 + 0.227247i $$0.927028\pi$$
$$398$$ 0 0
$$399$$ 14.4721 0.724513
$$400$$ 0 0
$$401$$ 2.52786 0.126236 0.0631178 0.998006i $$-0.479896\pi$$
0.0631178 + 0.998006i $$0.479896\pi$$
$$402$$ 0 0
$$403$$ 4.58359i 0.228325i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 0.708204i 0.0351044i
$$408$$ 0 0
$$409$$ −24.4721 −1.21007 −0.605035 0.796199i $$-0.706841\pi$$
−0.605035 + 0.796199i $$0.706841\pi$$
$$410$$ 0 0
$$411$$ −35.4164 −1.74696
$$412$$ 0 0
$$413$$ − 11.7082i − 0.576123i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ − 34.4721i − 1.68811i
$$418$$ 0 0
$$419$$ −26.1803 −1.27899 −0.639497 0.768794i $$-0.720857\pi$$
−0.639497 + 0.768794i $$0.720857\pi$$
$$420$$ 0 0
$$421$$ −13.0000 −0.633581 −0.316791 0.948495i $$-0.602605\pi$$
−0.316791 + 0.948495i $$0.602605\pi$$
$$422$$ 0 0
$$423$$ − 14.9443i − 0.726615i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 9.70820i 0.469813i
$$428$$ 0 0
$$429$$ −0.944272 −0.0455899
$$430$$ 0 0
$$431$$ −17.5279 −0.844288 −0.422144 0.906529i $$-0.638722\pi$$
−0.422144 + 0.906529i $$0.638722\pi$$
$$432$$ 0 0
$$433$$ 28.3607i 1.36293i 0.731852 + 0.681464i $$0.238656\pi$$
−0.731852 + 0.681464i $$0.761344\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ − 27.8885i − 1.33409i
$$438$$ 0 0
$$439$$ −8.29180 −0.395746 −0.197873 0.980228i $$-0.563403\pi$$
−0.197873 + 0.980228i $$0.563403\pi$$
$$440$$ 0 0
$$441$$ 7.47214 0.355816
$$442$$ 0 0
$$443$$ − 19.4164i − 0.922501i −0.887270 0.461251i $$-0.847401\pi$$
0.887270 0.461251i $$-0.152599\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 12.7639i 0.603713i
$$448$$ 0 0
$$449$$ −20.5279 −0.968770 −0.484385 0.874855i $$-0.660957\pi$$
−0.484385 + 0.874855i $$0.660957\pi$$
$$450$$ 0 0
$$451$$ 1.12461 0.0529559
$$452$$ 0 0
$$453$$ − 65.4853i − 3.07677i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 12.5279i − 0.586029i −0.956108 0.293014i $$-0.905342\pi$$
0.956108 0.293014i $$-0.0946584\pi$$
$$458$$ 0 0
$$459$$ −35.7771 −1.66993
$$460$$ 0 0
$$461$$ −14.1803 −0.660444 −0.330222 0.943903i $$-0.607124\pi$$
−0.330222 + 0.943903i $$0.607124\pi$$
$$462$$ 0 0
$$463$$ − 13.8885i − 0.645455i −0.946492 0.322728i $$-0.895400\pi$$
0.946492 0.322728i $$-0.104600\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ − 6.94427i − 0.321343i −0.987008 0.160671i $$-0.948634\pi$$
0.987008 0.160671i $$-0.0513659\pi$$
$$468$$ 0 0
$$469$$ 4.23607 0.195603
$$470$$ 0 0
$$471$$ 2.47214 0.113910
$$472$$ 0 0
$$473$$ 0.416408i 0.0191465i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 63.3050i 2.89853i
$$478$$ 0 0
$$479$$ −26.1803 −1.19621 −0.598105 0.801418i $$-0.704079\pi$$
−0.598105 + 0.801418i $$0.704079\pi$$
$$480$$ 0 0
$$481$$ 3.70820 0.169080
$$482$$ 0 0
$$483$$ − 20.1803i − 0.918237i
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ − 5.76393i − 0.261189i −0.991436 0.130594i $$-0.958311\pi$$
0.991436 0.130594i $$-0.0416885\pi$$
$$488$$ 0 0
$$489$$ −4.94427 −0.223588
$$490$$ 0 0
$$491$$ 5.76393 0.260123 0.130061 0.991506i $$-0.458483\pi$$
0.130061 + 0.991506i $$0.458483\pi$$
$$492$$ 0 0
$$493$$ − 12.3607i − 0.556697i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 8.70820i 0.390616i
$$498$$ 0 0
$$499$$ 11.0557 0.494922 0.247461 0.968898i $$-0.420404\pi$$
0.247461 + 0.968898i $$0.420404\pi$$
$$500$$ 0 0
$$501$$ −16.9443 −0.757014
$$502$$ 0 0
$$503$$ − 8.11146i − 0.361672i −0.983513 0.180836i $$-0.942120\pi$$
0.983513 0.180836i $$-0.0578803\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ − 37.1246i − 1.64876i
$$508$$ 0 0
$$509$$ −40.6525 −1.80189 −0.900945 0.433934i $$-0.857125\pi$$
−0.900945 + 0.433934i $$0.857125\pi$$
$$510$$ 0 0
$$511$$ 8.76393 0.387694
$$512$$ 0 0
$$513$$ − 64.7214i − 2.85752i
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 0.472136i 0.0207645i
$$518$$ 0 0
$$519$$ 37.3050 1.63751
$$520$$ 0 0
$$521$$ 22.0000 0.963837 0.481919 0.876216i $$-0.339940\pi$$
0.481919 + 0.876216i $$0.339940\pi$$
$$522$$ 0 0
$$523$$ 16.3607i 0.715403i 0.933836 + 0.357701i $$0.116439\pi$$
−0.933836 + 0.357701i $$0.883561\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ − 9.16718i − 0.399329i
$$528$$ 0 0
$$529$$ −15.8885 −0.690806
$$530$$ 0 0
$$531$$ −87.4853 −3.79654
$$532$$ 0 0
$$533$$ − 5.88854i − 0.255061i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 75.7771i 3.27002i
$$538$$ 0 0
$$539$$ −0.236068 −0.0101682
$$540$$ 0 0
$$541$$ 15.9443 0.685498 0.342749 0.939427i $$-0.388642\pi$$
0.342749 + 0.939427i $$0.388642\pi$$
$$542$$ 0 0
$$543$$ − 26.4721i − 1.13603i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 9.76393i 0.417476i 0.977972 + 0.208738i $$0.0669355\pi$$
−0.977972 + 0.208738i $$0.933064\pi$$
$$548$$ 0 0
$$549$$ 72.5410 3.09598
$$550$$ 0 0
$$551$$ 22.3607 0.952597
$$552$$ 0 0
$$553$$ 11.1803i 0.475436i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ − 9.11146i − 0.386065i −0.981192 0.193032i $$-0.938168\pi$$
0.981192 0.193032i $$-0.0618322\pi$$
$$558$$ 0 0
$$559$$ 2.18034 0.0922186
$$560$$ 0 0
$$561$$ 1.88854 0.0797344
$$562$$ 0 0
$$563$$ 17.4164i 0.734014i 0.930218 + 0.367007i $$0.119618\pi$$
−0.930218 + 0.367007i $$0.880382\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ − 24.4164i − 1.02539i
$$568$$ 0 0
$$569$$ 3.94427 0.165352 0.0826762 0.996576i $$-0.473653\pi$$
0.0826762 + 0.996576i $$0.473653\pi$$
$$570$$ 0 0
$$571$$ −36.5967 −1.53153 −0.765763 0.643123i $$-0.777638\pi$$
−0.765763 + 0.643123i $$0.777638\pi$$
$$572$$ 0 0
$$573$$ 20.9443i 0.874960i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ − 2.00000i − 0.0832611i −0.999133 0.0416305i $$-0.986745\pi$$
0.999133 0.0416305i $$-0.0132552\pi$$
$$578$$ 0 0
$$579$$ −40.1803 −1.66984
$$580$$ 0 0
$$581$$ −7.70820 −0.319790
$$582$$ 0 0
$$583$$ − 2.00000i − 0.0828315i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 24.7639i 1.02212i 0.859546 + 0.511058i $$0.170746\pi$$
−0.859546 + 0.511058i $$0.829254\pi$$
$$588$$ 0 0
$$589$$ 16.5836 0.683315
$$590$$ 0 0
$$591$$ −4.76393 −0.195962
$$592$$ 0 0
$$593$$ 37.3050i 1.53193i 0.642882 + 0.765965i $$0.277739\pi$$
−0.642882 + 0.765965i $$0.722261\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ − 23.4164i − 0.958370i
$$598$$ 0 0
$$599$$ −11.1803 −0.456816 −0.228408 0.973565i $$-0.573352\pi$$
−0.228408 + 0.973565i $$0.573352\pi$$
$$600$$ 0 0
$$601$$ −36.9443 −1.50699 −0.753494 0.657455i $$-0.771633\pi$$
−0.753494 + 0.657455i $$0.771633\pi$$
$$602$$ 0 0
$$603$$ − 31.6525i − 1.28899i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 7.12461i 0.289179i 0.989492 + 0.144590i $$0.0461862\pi$$
−0.989492 + 0.144590i $$0.953814\pi$$
$$608$$ 0 0
$$609$$ 16.1803 0.655660
$$610$$ 0 0
$$611$$ 2.47214 0.100012
$$612$$ 0 0
$$613$$ 44.4164i 1.79396i 0.442069 + 0.896981i $$0.354245\pi$$
−0.442069 + 0.896981i $$0.645755\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ − 5.94427i − 0.239307i −0.992816 0.119654i $$-0.961822\pi$$
0.992816 0.119654i $$-0.0381784\pi$$
$$618$$ 0 0
$$619$$ −11.7082 −0.470592 −0.235296 0.971924i $$-0.575606\pi$$
−0.235296 + 0.971924i $$0.575606\pi$$
$$620$$ 0 0
$$621$$ −90.2492 −3.62158
$$622$$ 0 0
$$623$$ 17.2361i 0.690548i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 3.41641i 0.136438i
$$628$$ 0 0
$$629$$ −7.41641 −0.295712
$$630$$ 0 0
$$631$$ −27.6525 −1.10083 −0.550414 0.834892i $$-0.685530\pi$$
−0.550414 + 0.834892i $$0.685530\pi$$
$$632$$ 0 0
$$633$$ 38.8328i 1.54347i
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 1.23607i 0.0489748i
$$638$$ 0 0
$$639$$ 65.0689 2.57409
$$640$$ 0 0
$$641$$ 43.8328 1.73129 0.865646 0.500656i $$-0.166908\pi$$
0.865646 + 0.500656i $$0.166908\pi$$
$$642$$ 0 0
$$643$$ 18.4721i 0.728470i 0.931307 + 0.364235i $$0.118669\pi$$
−0.931307 + 0.364235i $$0.881331\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 19.8885i 0.781899i 0.920412 + 0.390950i $$0.127853\pi$$
−0.920412 + 0.390950i $$0.872147\pi$$
$$648$$ 0 0
$$649$$ 2.76393 0.108494
$$650$$ 0 0
$$651$$ 12.0000 0.470317
$$652$$ 0 0
$$653$$ − 25.0557i − 0.980506i −0.871580 0.490253i $$-0.836904\pi$$
0.871580 0.490253i $$-0.163096\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ − 65.4853i − 2.55482i
$$658$$ 0 0
$$659$$ 17.8885 0.696839 0.348419 0.937339i $$-0.386719\pi$$
0.348419 + 0.937339i $$0.386719\pi$$
$$660$$ 0 0
$$661$$ −42.7214 −1.66167 −0.830834 0.556520i $$-0.812136\pi$$
−0.830834 + 0.556520i $$0.812136\pi$$
$$662$$ 0 0
$$663$$ − 9.88854i − 0.384039i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ − 31.1803i − 1.20731i
$$668$$ 0 0
$$669$$ 65.3050 2.52484
$$670$$ 0 0
$$671$$ −2.29180 −0.0884738
$$672$$ 0 0
$$673$$ − 19.5279i − 0.752744i −0.926469 0.376372i $$-0.877171\pi$$
0.926469 0.376372i $$-0.122829\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ − 14.3607i − 0.551926i −0.961168 0.275963i $$-0.911003\pi$$
0.961168 0.275963i $$-0.0889967\pi$$
$$678$$ 0 0
$$679$$ 5.23607 0.200942
$$680$$ 0 0
$$681$$ −69.3050 −2.65577
$$682$$ 0 0
$$683$$ 14.1246i 0.540463i 0.962795 + 0.270232i $$0.0871003\pi$$
−0.962795 + 0.270232i $$0.912900\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ − 14.4721i − 0.552146i
$$688$$ 0 0
$$689$$ −10.4721 −0.398957
$$690$$ 0 0
$$691$$ 4.18034 0.159028 0.0795138 0.996834i $$-0.474663\pi$$
0.0795138 + 0.996834i $$0.474663\pi$$
$$692$$ 0 0
$$693$$ 1.76393i 0.0670062i
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 11.7771i 0.446089i
$$698$$ 0 0
$$699$$ −25.7082 −0.972374
$$700$$ 0 0
$$701$$ −29.0557 −1.09742 −0.548710 0.836013i $$-0.684881\pi$$
−0.548710 + 0.836013i $$0.684881\pi$$
$$702$$ 0 0
$$703$$ − 13.4164i − 0.506009i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ − 4.76393i − 0.179166i
$$708$$ 0 0
$$709$$ 12.1115 0.454855 0.227428 0.973795i $$-0.426968\pi$$
0.227428 + 0.973795i $$0.426968\pi$$
$$710$$ 0 0
$$711$$ 83.5410 3.13303
$$712$$ 0 0
$$713$$ − 23.1246i − 0.866024i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 17.8885i 0.668060i
$$718$$ 0 0
$$719$$ 16.1803 0.603425 0.301712 0.953399i $$-0.402442\pi$$
0.301712 + 0.953399i $$0.402442\pi$$
$$720$$ 0 0
$$721$$ 8.47214 0.315519
$$722$$ 0 0
$$723$$ 11.4164i 0.424581i
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 3.05573i 0.113331i 0.998393 + 0.0566653i $$0.0180468\pi$$
−0.998393 + 0.0566653i $$0.981953\pi$$
$$728$$ 0 0
$$729$$ −41.9443 −1.55349
$$730$$ 0 0
$$731$$ −4.36068 −0.161286
$$732$$ 0 0
$$733$$ − 4.00000i − 0.147743i −0.997268 0.0738717i $$-0.976464\pi$$
0.997268 0.0738717i $$-0.0235355\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 1.00000i 0.0368355i
$$738$$ 0 0
$$739$$ 25.6525 0.943642 0.471821 0.881694i $$-0.343597\pi$$
0.471821 + 0.881694i $$0.343597\pi$$
$$740$$ 0 0
$$741$$ 17.8885 0.657152
$$742$$ 0 0
$$743$$ − 10.4721i − 0.384185i −0.981377 0.192093i $$-0.938473\pi$$
0.981377 0.192093i $$-0.0615274\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 57.5967i 2.10735i
$$748$$ 0 0
$$749$$ −8.00000 −0.292314
$$750$$ 0 0
$$751$$ −3.05573 −0.111505 −0.0557526 0.998445i $$-0.517756\pi$$
−0.0557526 + 0.998445i $$0.517756\pi$$
$$752$$ 0 0
$$753$$ 20.9443i 0.763252i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 19.5836i 0.711778i 0.934528 + 0.355889i $$0.115822\pi$$
−0.934528 + 0.355889i $$0.884178\pi$$
$$758$$ 0 0
$$759$$ 4.76393 0.172920
$$760$$ 0 0
$$761$$ 27.7771 1.00692 0.503459 0.864019i $$-0.332060\pi$$
0.503459 + 0.864019i $$0.332060\pi$$
$$762$$ 0 0
$$763$$ 8.41641i 0.304694i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ − 14.4721i − 0.522559i
$$768$$ 0 0
$$769$$ 43.0132 1.55109 0.775547 0.631290i $$-0.217474\pi$$
0.775547 + 0.631290i $$0.217474\pi$$
$$770$$ 0 0
$$771$$ −40.9443 −1.47457
$$772$$ 0 0
$$773$$ − 50.1803i − 1.80486i −0.430835 0.902431i $$-0.641781\pi$$
0.430835 0.902431i $$-0.358219\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ − 9.70820i − 0.348280i
$$778$$ 0 0
$$779$$ −21.3050 −0.763329
$$780$$ 0 0
$$781$$ −2.05573 −0.0735597
$$782$$ 0 0
$$783$$ − 72.3607i − 2.58596i
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ − 40.7639i − 1.45308i −0.687126 0.726539i $$-0.741128\pi$$
0.687126 0.726539i $$-0.258872\pi$$
$$788$$ 0 0
$$789$$ 52.5410 1.87051
$$790$$ 0 0
$$791$$ 14.4164 0.512588
$$792$$ 0 0
$$793$$ 12.0000i 0.426132i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ − 35.4164i − 1.25451i −0.778813 0.627257i $$-0.784178\pi$$
0.778813 0.627257i $$-0.215822\pi$$
$$798$$ 0 0
$$799$$ −4.94427 −0.174916
$$800$$ 0 0
$$801$$ 128.790 4.55058
$$802$$ 0 0
$$803$$ 2.06888i 0.0730093i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ − 37.8885i − 1.33374i
$$808$$ 0 0
$$809$$ −29.4721 −1.03619 −0.518093 0.855325i $$-0.673358\pi$$
−0.518093 + 0.855325i $$0.673358\pi$$
$$810$$ 0 0
$$811$$ 42.7214 1.50015 0.750075 0.661353i $$-0.230017\pi$$
0.750075 + 0.661353i $$0.230017\pi$$
$$812$$ 0 0
$$813$$ 76.7214i 2.69074i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ − 7.88854i − 0.275985i
$$818$$ 0 0
$$819$$ 9.23607 0.322734
$$820$$ 0 0
$$821$$ 28.8328 1.00627 0.503136 0.864207i $$-0.332179\pi$$
0.503136 + 0.864207i $$0.332179\pi$$
$$822$$ 0 0
$$823$$ − 31.6525i − 1.10334i −0.834064 0.551668i $$-0.813992\pi$$
0.834064 0.551668i $$-0.186008\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 41.5410i − 1.44452i −0.691620 0.722261i $$-0.743103\pi$$
0.691620 0.722261i $$-0.256897\pi$$
$$828$$ 0 0
$$829$$ 7.63932 0.265325 0.132662 0.991161i $$-0.457647\pi$$
0.132662 + 0.991161i $$0.457647\pi$$
$$830$$ 0 0
$$831$$ −64.3607 −2.23265
$$832$$ 0 0
$$833$$ − 2.47214i − 0.0856544i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ − 53.6656i − 1.85496i
$$838$$ 0 0
$$839$$ −30.6525 −1.05824 −0.529120 0.848547i $$-0.677478\pi$$
−0.529120 + 0.848547i $$0.677478\pi$$
$$840$$ 0 0
$$841$$ −4.00000 −0.137931
$$842$$ 0 0
$$843$$ 49.7082i 1.71204i
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 10.9443i 0.376050i
$$848$$ 0 0
$$849$$ 56.3607 1.93429
$$850$$ 0 0
$$851$$ −18.7082 −0.641309
$$852$$ 0 0
$$853$$ − 27.4164i − 0.938720i −0.883007 0.469360i $$-0.844485\pi$$
0.883007 0.469360i $$-0.155515\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ − 15.8197i − 0.540389i −0.962806 0.270195i $$-0.912912\pi$$
0.962806 0.270195i $$-0.0870881\pi$$
$$858$$ 0 0
$$859$$ 22.3607 0.762937 0.381468 0.924382i $$-0.375419\pi$$
0.381468 + 0.924382i $$0.375419\pi$$
$$860$$ 0 0
$$861$$ −15.4164 −0.525390
$$862$$ 0 0
$$863$$ 18.3475i 0.624557i 0.949991 + 0.312278i $$0.101092\pi$$
−0.949991 + 0.312278i $$0.898908\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ − 35.2361i − 1.19668i
$$868$$ 0 0
$$869$$ −2.63932 −0.0895328
$$870$$ 0 0
$$871$$ 5.23607 0.177417
$$872$$ 0 0
$$873$$ − 39.1246i − 1.32417i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 30.3607i 1.02521i 0.858625 + 0.512604i $$0.171319\pi$$
−0.858625 + 0.512604i $$0.828681\pi$$
$$878$$ 0 0
$$879$$ 100.721 3.39725
$$880$$ 0 0
$$881$$ 5.81966 0.196069 0.0980347 0.995183i $$-0.468744\pi$$
0.0980347 + 0.995183i $$0.468744\pi$$
$$882$$ 0 0
$$883$$ − 1.40325i − 0.0472232i −0.999721 0.0236116i $$-0.992483\pi$$
0.999721 0.0236116i $$-0.00751650\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 21.3475i 0.716780i 0.933572 + 0.358390i $$0.116674\pi$$
−0.933572 + 0.358390i $$0.883326\pi$$
$$888$$ 0 0
$$889$$ −13.6525 −0.457889
$$890$$ 0 0
$$891$$ 5.76393 0.193099
$$892$$ 0 0
$$893$$ − 8.94427i − 0.299309i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ − 24.9443i − 0.832865i
$$898$$ 0 0
$$899$$ 18.5410 0.618378
$$900$$ 0 0
$$901$$ 20.9443 0.697755
$$902$$ 0 0
$$903$$ − 5.70820i − 0.189957i
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ − 34.8328i − 1.15660i −0.815823 0.578302i $$-0.803715\pi$$
0.815823 0.578302i $$-0.196285\pi$$
$$908$$ 0 0
$$909$$ −35.5967 −1.18067
$$910$$ 0 0
$$911$$ −0.819660 −0.0271566 −0.0135783 0.999908i $$-0.504322\pi$$
−0.0135783 + 0.999908i $$0.504322\pi$$
$$912$$ 0 0
$$913$$ − 1.81966i − 0.0602220i
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ − 16.9443i − 0.559549i
$$918$$ 0 0
$$919$$ −27.7639 −0.915848 −0.457924 0.888991i $$-0.651407\pi$$
−0.457924 + 0.888991i $$0.651407\pi$$
$$920$$ 0 0
$$921$$ −14.8328 −0.488758
$$922$$ 0 0
$$923$$ 10.7639i 0.354299i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ − 63.3050i − 2.07921i
$$928$$ 0 0
$$929$$ −38.2918 −1.25631 −0.628157 0.778087i $$-0.716190\pi$$
−0.628157 + 0.778087i $$0.716190\pi$$
$$930$$ 0 0
$$931$$ 4.47214 0.146568
$$932$$ 0 0
$$933$$ 78.8328i 2.58087i
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 35.2361i 1.15111i 0.817762 + 0.575556i $$0.195214\pi$$
−0.817762 + 0.575556i $$0.804786\pi$$
$$938$$ 0 0
$$939$$ −63.1935 −2.06224
$$940$$ 0 0
$$941$$ −5.23607 −0.170691 −0.0853455 0.996351i $$-0.527199\pi$$
−0.0853455 + 0.996351i $$0.527199\pi$$
$$942$$ 0 0
$$943$$ 29.7082i 0.967432i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ − 34.8328i − 1.13191i −0.824435 0.565957i $$-0.808507\pi$$
0.824435 0.565957i $$-0.191493\pi$$
$$948$$ 0 0
$$949$$ 10.8328 0.351648
$$950$$ 0 0
$$951$$ 82.0689 2.66127
$$952$$ 0 0
$$953$$ − 3.47214i − 0.112474i −0.998417 0.0562368i $$-0.982090\pi$$
0.998417 0.0562368i $$-0.0179102\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 3.81966i 0.123472i
$$958$$ 0 0
$$959$$ −10.9443 −0.353409
$$960$$ 0 0
$$961$$ −17.2492 −0.556427
$$962$$ 0 0
$$963$$ 59.7771i 1.92629i
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 14.1115i 0.453794i 0.973919 + 0.226897i $$0.0728580\pi$$
−0.973919 + 0.226897i $$0.927142\pi$$
$$968$$ 0 0
$$969$$ −35.7771 −1.14933
$$970$$ 0 0
$$971$$ 18.0000 0.577647 0.288824 0.957382i $$-0.406736\pi$$
0.288824 + 0.957382i $$0.406736\pi$$
$$972$$ 0 0
$$973$$ − 10.6525i − 0.341503i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ − 11.4721i − 0.367026i −0.983017 0.183513i $$-0.941253\pi$$
0.983017 0.183513i $$-0.0587470\pi$$
$$978$$ 0 0
$$979$$ −4.06888 −0.130042
$$980$$ 0 0
$$981$$ 62.8885 2.00788
$$982$$ 0 0
$$983$$ − 34.5410i − 1.10169i −0.834608 0.550844i $$-0.814306\pi$$
0.834608 0.550844i $$-0.185694\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ − 6.47214i − 0.206010i
$$988$$ 0 0
$$989$$ −11.0000 −0.349780
$$990$$ 0 0
$$991$$ −13.1803 −0.418687 −0.209344 0.977842i $$-0.567133\pi$$
−0.209344 + 0.977842i $$0.567133\pi$$
$$992$$ 0 0
$$993$$ − 79.9574i − 2.53737i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ − 45.4164i − 1.43835i −0.694828 0.719176i $$-0.744519\pi$$
0.694828 0.719176i $$-0.255481\pi$$
$$998$$ 0 0
$$999$$ −43.4164 −1.37363
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 2800.2.g.s.449.1 4
4.3 odd 2 175.2.b.c.99.2 4
5.2 odd 4 2800.2.a.bh.1.1 2
5.3 odd 4 2800.2.a.bp.1.2 2
5.4 even 2 inner 2800.2.g.s.449.4 4
12.11 even 2 1575.2.d.k.1324.3 4
20.3 even 4 175.2.a.e.1.1 yes 2
20.7 even 4 175.2.a.d.1.2 2
20.19 odd 2 175.2.b.c.99.3 4
28.27 even 2 1225.2.b.k.99.2 4
60.23 odd 4 1575.2.a.n.1.2 2
60.47 odd 4 1575.2.a.s.1.1 2
60.59 even 2 1575.2.d.k.1324.2 4
140.27 odd 4 1225.2.a.n.1.2 2
140.83 odd 4 1225.2.a.u.1.1 2
140.139 even 2 1225.2.b.k.99.3 4

By twisted newform
Twist Min Dim Char Parity Ord Type
175.2.a.d.1.2 2 20.7 even 4
175.2.a.e.1.1 yes 2 20.3 even 4
175.2.b.c.99.2 4 4.3 odd 2
175.2.b.c.99.3 4 20.19 odd 2
1225.2.a.n.1.2 2 140.27 odd 4
1225.2.a.u.1.1 2 140.83 odd 4
1225.2.b.k.99.2 4 28.27 even 2
1225.2.b.k.99.3 4 140.139 even 2
1575.2.a.n.1.2 2 60.23 odd 4
1575.2.a.s.1.1 2 60.47 odd 4
1575.2.d.k.1324.2 4 60.59 even 2
1575.2.d.k.1324.3 4 12.11 even 2
2800.2.a.bh.1.1 2 5.2 odd 4
2800.2.a.bp.1.2 2 5.3 odd 4
2800.2.g.s.449.1 4 1.1 even 1 trivial
2800.2.g.s.449.4 4 5.4 even 2 inner